Determinants, Gröbner Bases and Cohomology (Springer Monographs in Mathematics) 3031054792, 9783031054792

Table of contents :
Preface
Contents
1 Gröbner Bases, Initial Ideals and Initial Algebras
1.1 Monomial Orders
1.2 Gröbner Bases
1.3 Sagbi Bases
1.4 Initial Subspaces and Gradings
1.5 Initial Subspaces for Weights. Homogenization
1.6 Deformation to the Initial Object
2 More on Gröbner Deformations
2.1 Minimal and Associated Primes
2.2 Connectedness
2.3 Optimal Gröbner Deformations
2.4 Square-Free Gröbner Deformations
3 Determinantal Ideals and the Straightening Law
3.1 Minors and Determinantal Ideals
3.2 Standard Bitableaux and the Straightening Law
3.3 Refinement of the Straightening Law
3.4 Determinantal Rings
3.5 Powers and Products of Determinantal Ideals
3.6 Representation Theory
4 Gröbner Bases of Determinantal Ideals
4.1 An Instructive Case: the Ideal of 2-Minors
4.2 The Robinson–Schensted–Knuth Correspondence
4.3 RSK and Gröbner Bases of Ideals
4.4 Cohen–Macaulayness and Hilbert Series of Determinantal Rings
4.5 Gröbner Bases via Secants
4.6 Multigraded Structures and the Multidegree
5 Universal Gröbner Bases
5.1 Universal Gröbner Bases
5.2 Universal Gröbner Bases for Maximal Minors
5.3 Universal Gröbner Bases for 2-Minors
6 Algebras Defined by Minors
6.1 A Recap of Toric Algebra
6.2 Algebras Defined by Maximal Minors
6.3 Rees Algebras of Determinantal Ideals
6.4 The Algebra of Minors
6.5 Algebraic Relations between Minors
6.6 Exterior Powers of Linear Maps
6.7 Sagbi Deformations of Determinantal Rings
7 F-singularities of Determinantal Rings
7.1 F-purity
7.2 F-regularity
7.3 F-rationality and Gröbner Deformations
7.4 Equivariant Deformations
7.5 F-pure Thresholds
8 Castelnuovo–Mumford Regularity
8.1 Castelnuovo–Mumford Regularity over General Base Rings
8.2 Castelnuovo–Mumford Regularity and Multigraded Structures
8.3 Vanishing over Nonstandard Multigraded Rings
8.4 Regularity of Powers and Products of Ideals
8.5 Linear Powers and Linear Products
8.6 A Family of Determinantal Ideals with Linear Products
8.7 Maximal Minors of Matrices of Linear Forms and Linear Powers
9 Grassmannians, Flag Varieties, Schur Functors and Cohomology
9.1 Fundamental Multilinear Algebra Constructions
9.2 Affine and Projective Bundles, Grassmannians, Flag Varieties
9.3 Basic Calculus with Direct Images
9.4 The Kempf Vanishing Theorem
9.5 Characteristic-Free Vanishing for some Nondominant Weights
9.6 A Sharp Characteristic-Free Vanishing on Projective Bundles
9.7 Algebraic Groups and Representations
9.8 Schur Functors
9.9 Grothendieck Duality
9.10 Cohomology Vanishing on Grassmannians
10 Asymptotic Regularity for Symbolic Powers of Determinantal Ideals
10.1 Quotients of Ideals Defined by Shape
10.2 Filtrations Associated with Ideals Defined by Shape
10.3 An algorithm to Compute Z(Σ)
10.4 The Filtration for Symbolic Powers of Determinantal Ideals
10.5 Sheaves Defined by Shape and Their Quotients
10.6 A Geometric Realization of the Modules J(σ)l
10.7 Regularity of Symbolic Powers of Determinantal Ideals
10.8 Some Geometric Properties of Determinantal Varieties
11 Cohomology and Regularity in Characteristic Zero
11.1 The Borel–Weil–Bott Theorem
11.2 The Optimality of Theorem 9.5.1
11.3 Borel–Weil–Bott on Grassmannians
11.4 Effective Vanishing of Cohomology in Characteristic Zero
11.5 Cauchy's Formula Revisited
11.6 Explicit Description of Ext Modules for J(σ)l
11.7 Ext Modules for Symbolic Powers
11.8 The Basics on GL-Invariant Ideals
11.9 Ext Modules for GL-Invariant Ideals
11.10 Local Cohomology with Determinantal Support
11.11 Syzygies of GL-Invariant Ideals
Appendix List of Symbols
References
Index

Citation preview

Springer Monographs in Mathematics

Winfried Bruns Aldo Conca Claudiu Raicu Matteo Varbaro

Determinants, Gröbner Bases and Cohomology

Springer Monographs in Mathematics Editors-in-Chief Minhyong Kim, School of Mathematics, Korea Institute for Advanced Study, Seoul, South Korea, International Centre for Mathematical Sciences, Edinburgh, UK Katrin Wendland, School of Mathematics, Trinity College Dublin, Dublin, Ireland Series Editors Sheldon Axler, Department of Mathematics, San Francisco State University, San Francisco, CA, USA Mark Braverman, Department of Mathematics, Princeton University, Princeton, NY, USA Maria Chudnovsky, Department of Mathematics, Princeton University, Princeton, NY, USA Tadahisa Funaki, Department of Mathematics, University of Tokyo, Tokyo, Japan Isabelle Gallagher, Département de Mathématiques et Applications, Ecole Normale Supérieure, Paris, France Sinan Güntürk, Courant Institute of Mathematical Sciences, New York University, New York, NY, USA Claude Le Bris, CERMICS, Ecole des Ponts ParisTech, Marne la Vallée, France Pascal Massart, Département de Mathématiques, Université de Paris-Sud, Orsay, France Alberto A. Pinto, Department of Mathematics, University of Porto, Porto, Portugal Gabriella Pinzari, Department of Mathematics, University of Padova, Padova, Italy Ken Ribet, Department of Mathematics, University of California, Berkeley, CA, USA René Schilling, Institute for Mathematical Stochastics, Technical University Dresden, Dresden, Germany Panagiotis Souganidis, Department of Mathematics, University of Chicago, Chicago, IL, USA Endre Süli, Mathematical Institute, University of Oxford, Oxford, UK Shmuel Weinberger, Department of Mathematics, University of Chicago, Chicago, IL, USA Boris Zilber, Mathematical Institute, University of Oxford, Oxford, UK

This series publishes advanced monographs giving well-written presentations of the “state-of-the-art” in fields of mathematical research that have acquired the maturity needed for such a treatment. They are sufficiently self-contained to be accessible to more than just the intimate specialists of the subject, and sufficiently comprehensive to remain valuable references for many years. Besides the current state of knowledge in its field, an SMM volume should ideally describe its relevance to and interaction with neighbouring fields of mathematics, and give pointers to future directions of research.

Winfried Bruns · Aldo Conca · Claudiu Raicu · Matteo Varbaro

Determinants, Gröbner Bases and Cohomology

Winfried Bruns Institut für Mathematik Universität Osnabrück Osnabrück, Germany

Aldo Conca Dipartimento di Matematica Università di Genova Genova, Italy

Claudiu Raicu Department of Mathematics University of Notre Dame Notre Dame, IN, USA

Matteo Varbaro Dipartimento di Matematica Università di Genova Genova, Italy

ISSN 1439-7382 ISSN 2196-9922 (electronic) Springer Monographs in Mathematics ISBN 978-3-031-05479-2 ISBN 978-3-031-05480-8 (eBook) https://doi.org/10.1007/978-3-031-05480-8 Mathematics Subject Classification: 13A35, 13A50, 13C40, 13C70, 13D02, 13D07, 13D10, 13D40, 13D45, 13F50, 13F65, 13H10, 13H15, 13P10, 14B15, 14L30, 14M12, 14M15, 14M25, 20G05, 20G15 © Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Winfried For Philipp, Helene and Jakob Aldo For Maria Grazia, Fabrizio and Carolina Claudiu For my family Matteo For Giulia, Annaluisa, Rocco and Roberto

Preface

Determinantal rings and varieties have been a central topic of commutative algebra and algebraic geometry for several decades. Numerous researchers have contributed to the development of their theory, and they have served as a testing ground for new notions and tools. Among the areas of mathematics that have intertwined with determinantal rings and varieties, one finds algebraic geometry, commutative algebra, combinatorics, invariant and representation theory of the general linear group. They all play major roles in this book, and their interaction is a ubiquitous motive. When the senior author was a Ph.D. student exactly half a century ago, the Eagon– Northcott complex had been constructed 10 years earlier, and the paper of Hochster and Eagon, which established the Cohen–Macaulay property of determinantal rings in general, had just appeared. Meanwhile, the literature on determinantal rings and ideals has grown to a collection of several hundreds of articles of which the references cited in this book represent only a selection. The monograph Determinantal Rings by Bruns and Vetter appeared in 1988. It gave a comprehensive overview of the algebraic theory available at the time of its writing. Now, more than 30 years later, this area has seen a number of new developments, and geometric techniques have contributed many, very precise results. One of the major themes that came at the end of the 80s is Gröbner bases of ideals in polynomial rings and Sagbi bases of subalgebras. Chapter 1 gives a compact, but a quite complete introduction to Gröbner bases and Sagbi bases in general. The focus is on the structural aspects, namely the use of Gröbner and Sagbi degenerations in the transfer of homological and enumerative information from Stanley–Reisner and/or toric rings to those objects that degenerate to them. This has been a major theme in the last 30 years, not only for determinantal ideals. Chapter 2 pushes this development forward in several directions: finer control of minimal and associated primes, connectedness, tight connections between an ideal and its squarefree initial ideals, should the latter exist. Some of these results were very recently established by Conca and Varbaro. Chapter 3 gives a short introduction to standard bitableaux and the straightening law, following De Concini, Eisenbud and Procesi. This powerful technique has been

vii

viii

Preface

dealt with extensively in the monograph of Bruns and Vetter, but is of central importance in this book as well: for the computation of Gröbner and Sagbi bases on the one hand and for the representation theoretic approach on the other. The straightening law is also the basis for the primary decomposition of powers of determinantal ideals. Nevertheless, we keep the chapter to a minimum, and for some proofs we only indicate the basic ideas. Chapter 4 presents the computation of Gröbner bases, based on standard bitableaux and the Robinson–Schensted–Knuth correspondence to which we give a short introduction. This development was initiated by Sturmfels. Then Herzog and Trung derived several structural and enumerative theorems on determinantal ideals from the shellability of the simplicial complex defined by the initial ideal. Among them is Giambelli’s formula for the degree of a determinantal variety, published in 1903. The connection between the Robinson–Schensted–Knuth correspondence and determinantal ideals was carried on by Bruns and Conca whose article Gröbner bases and determinantal ideals can be considered the nucleus of this book. We also include the simple approach via secants by Sturmfels and Sullivant. It is a theorem of Bernstein, Sturmfels and Zelevinsky that the maximal minors are a universal Gröbner basis. As an addendum to Chapter 4, we present the surprisingly simple approach to this result by Conca, De Negri and Gorla in Chapter 5. A second, simpler case, in which the universal Gröbner basis can be described, is that of 2-minors as a special instance of binomial ideals. That maximal minors and 2-minors have special properties had already been foreshadowed by the last section of Chapter 4: exactly for them the determinantal ring has squarefree multidegrees in the “grading by columns”, a theme going back to van der Waerden’s work in 1929. The homogeneous coordinate ring of the Grassmannian is an algebra generated by minors, and its investigation is a paradigm for the exploitation of toric deformations. Since the results on Gröbner bases of symbolic and ordinary powers of determinantal ideals allow us to find the initial algebras of the corresponding Rees algebras, the basic arithmetical and homological properties become accessible as well in Chapter 6. Algebras generated by lower order minors are coordinate rings of varieties parametrized by exterior powers of linear maps. They are much more complicated than Grassmannians and not yet fully explored. Their initial algebras turn out to be normal monoid domains, and this allows one to show that they are normal Cohen– Macaulay domains, and even the Gorenstein ones among them can be determined. We also describe their orbit structure. However, their defining ideals are only partially understood, and the best general information has been converted to a conjecture by Bruns, Conca and Varbaro. Only for the case of 2-minors one now has a complete system of relations by the recent work of Huang, Perlman, Polini, Raicu and Sammartano. For reasons of space, we must content ourselves with a survey on this difficult topic. At the end of the chapter, we return to firm ground and analyze the determinantal rings via their initial algebras in a suitable embedding. This helps us to understand their Cohen–Macaulay rank 1 modules. Chapter 7 treats ring theoretic properties derived from the Frobenius functor in positive characteristics that were pioneered by Hochster and Huneke in connection with tight closure. We develop them far enough to prove that determinantal rings are

Preface

ix

strongly F-regular. In order to apply deformation arguments, one needs F-rationality. It is closely related to the rationality of singularities in characteristic 0 by a theorem of Hara and Smith, so that we can at least briefly discuss this property for our rings. The F-rationality of normal monoid domains makes it a very handy tool for the exploitation of toric deformations to such algebras, and helps to apply equivariant deformation based on so-called U -invariants. We conclude the chapter by computing F-pure thresholds for determinantal rings and ideals. For standard graded algebras over fields, Castelnuovo–Mumford regularity has become an indispensable invariant. Chapter 8 develops this notion from scratch, but in a more general version for standard graded algebras over Noetherian base rings. In this generality, the Janus-faced nature of regularity still survives: it is defined in terms of graded local cohomology, but can also be computed from minimal graded free resolutions if these are suitably defined. A third facet in the general case is Koszul homology. The main advantage of the general version is the significantly larger flexibility, for example in applications to Rees algebras. The theorem of Cutkosky– Herzog–Trung and Kodiyalam on the regularity of powers of ideals remains true. In the context of determinantal rings, we are mainly interested in linear free resolutions of powers of ideals of maximal minors in the non-generic case, exemplified by ideals of rational normal scrolls. With Chapter 9, the scenery changes significantly from mainly algebraic to predominantly geometric methods. This chapter develops the basic tools to be applied in the chapters that follow: multilinear algebra, Schur functors, tautological vector bundles on flag varieties, their cohomology and the celebrated Kempf vanishing theorem. The theory of algebraic group schemes is outlined to the necessary extent. Most of the material in Chapter 9 is classical, but the vanishing theorems from Sections 9.5 and 9.6 are new, and they play a crucial role in obtaining bounds for Castelnuovo–Mumford regularity in Chapter 10. Chapter 10 applies the methods of Chapter 9 to study the ideals “defined by shape” that were introduced already in Chapter 3. Many of the ideas were present in the work of Raicu in characteristic zero, but here we develop the theory in a characteristic-free fashion. For ideals defined by shape, we exhibit natural filtrations where the composition factors arise as global sections of vector bundles on desingularizations of determinantal varieties, which leads via Grothendieck duality to a description of the corresponding Ext modules as sheaf cohomology groups. The filtrations take a particularly nice form for symbolic powers of determinantal ideals, where the vanishing theorems from Chapter 9, combined with the characterization from Chapter 8 of Castelnuovo–Mumford regularity via Ext modules, allow us to determine an explicit formula for the asymptotic regularity. We end Chapter 10 with a brief survey of several other homological and arithmetic properties of determinantal ideals that can be derived in a compact way via geometric arguments. The goal of Chapter 11 is to extend and prove sharper versions of the results of the preceding chapter, when working over a field of characteristic zero. Two crucial advantages in this setting are the linear reductivity of the general linear group, and the Borel–Weil–Bott theorem describing the cohomology of line bundles on flag varieties. The class of ideals defined by shape can be enlarged to that of GL-invariant

x

Preface

ideals, to which the theory of filtrations from Chapter 10 can be extended, and for which all the Ext modules can be calculated exactly as GL-representations. The consequences of the Borel–Weil–Bott theorem are vastly superior to the vanishing theorems used in the characteristic-free setting, and in particular they lead to effective versions of the earlier asymptotic results on Castelnuovo–Mumford regularity. As an application of the calculation of Ext modules, we explain how to describe the GL-structure for the cohomology with support in determinantal ideals, and briefly discuss how this fits in with the D-module structure of the said cohomology groups. Finally, we conclude the book with a quick survey of the important topic of free resolutions of determinantal ideals, which was pioneered by Kempf and Lascoux, and is treated at length in Weyman’s Cohomology of vector bundles and syzygies. We are grateful to all our friends and colleagues who have helped us by providing valuable information and suggestions: Aldo Brigaglia, Marc Chardin, Dale Cutkosky, Michele D’Adderio, Christian Krattenthaler, András Lörincz, Mike Perlman, Peter Schenzel, Lisa Seccia, Anurag Singh, Keller VandeBogert and Jerzy Weyman. We cordially thank Dinh Van Le who read large parts of the book. He not only corrected uncountably many typos but also helped to improve the text and the mathematics. Also, Manolis Tsakiris, Alessandro De Stefani and Alessio Caminata supported us by reading selected chapters and suggesting valuable comments. Our thanks go to Ulrich von der Ohe for his TEXnical advice. We thank Alessio D’Alì who has assisted us in checking the galley proofs. The authors thank Remi Lodh of Springer London for the excellent cooperation. Osnabrück, Germany Genova, Italy Notre Dame, USA Genova, Italy January 2022

Winfried Bruns Aldo Conca Claudiu Raicu Matteo Varbaro

Contents

1

Gröbner Bases, Initial Ideals and Initial Algebras . . . . . . . . . . . . . . . . 1.1 Monomial Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Gröbner Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Sagbi Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Initial Subspaces and Gradings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Initial Subspaces for Weights. Homogenization . . . . . . . . . . . . . . 1.6 Deformation to the Initial Object . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 4 13 22 26 33 41

2

More on Gröbner Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Minimal and Associated Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Optimal Gröbner Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Square-Free Gröbner Deformations . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 44 50 56 64 67

3

Determinantal Ideals and the Straightening Law . . . . . . . . . . . . . . . . . 69 3.1 Minors and Determinantal Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.2 Standard Bitableaux and the Straightening Law . . . . . . . . . . . . . . 71 3.3 Refinement of the Straightening Law . . . . . . . . . . . . . . . . . . . . . . . 78 3.4 Determinantal Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.5 Powers and Products of Determinantal Ideals . . . . . . . . . . . . . . . . 88 3.6 Representation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4

Gröbner Bases of Determinantal Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 An Instructive Case: the Ideal of 2-Minors . . . . . . . . . . . . . . . . . . 4.2 The Robinson–Schensted–Knuth Correspondence . . . . . . . . . . . 4.3 RSK and Gröbner Bases of Ideals . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Cohen–Macaulayness and Hilbert Series of Determinantal Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Gröbner Bases via Secants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105 106 113 122 128 138

xi

xii

Contents

4.6 Multigraded Structures and the Multidegree . . . . . . . . . . . . . . . . . 146 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 5

Universal Gröbner Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Universal Gröbner Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Universal Gröbner Bases for Maximal Minors . . . . . . . . . . . . . . . 5.3 Universal Gröbner Bases for 2-Minors . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

159 159 161 163 169

6

Algebras Defined by Minors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 A Recap of Toric Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Algebras Defined by Maximal Minors . . . . . . . . . . . . . . . . . . . . . . 6.3 Rees Algebras of Determinantal Ideals . . . . . . . . . . . . . . . . . . . . . 6.4 The Algebra of Minors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Algebraic Relations between Minors . . . . . . . . . . . . . . . . . . . . . . . 6.6 Exterior Powers of Linear Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Sagbi Deformations of Determinantal Rings . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

171 172 177 187 195 198 207 217 227

7

F-singularities of Determinantal Rings . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 F-purity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 F-regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 F-rationality and Gröbner Deformations . . . . . . . . . . . . . . . . . . . . 7.4 Equivariant Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 F-pure Thresholds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

229 230 243 250 259 264 273

8

Castelnuovo–Mumford Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Castelnuovo–Mumford Regularity over General Base Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Castelnuovo–Mumford Regularity and Multigraded Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Vanishing over Nonstandard Multigraded Rings . . . . . . . . . . . . . 8.4 Regularity of Powers and Products of Ideals . . . . . . . . . . . . . . . . . 8.5 Linear Powers and Linear Products . . . . . . . . . . . . . . . . . . . . . . . . 8.6 A Family of Determinantal Ideals with Linear Products . . . . . . . 8.7 Maximal Minors of Matrices of Linear Forms and Linear Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

275

9

Grassmannians, Flag Varieties, Schur Functors and Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Fundamental Multilinear Algebra Constructions . . . . . . . . . . . . . 9.2 Affine and Projective Bundles, Grassmannians, Flag Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Basic Calculus with Direct Images . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 The Kempf Vanishing Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . .

276 287 289 295 305 308 313 322 325 325 329 333 340

Contents

Characteristic-Free Vanishing for some Nondominant Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 A Sharp Characteristic-Free Vanishing on Projective Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Algebraic Groups and Representations . . . . . . . . . . . . . . . . . . . . . 9.8 Schur Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9 Grothendieck Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.10 Cohomology Vanishing on Grassmannians . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

9.5

10 Asymptotic Regularity for Symbolic Powers of Determinantal Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Quotients of Ideals Defined by Shape . . . . . . . . . . . . . . . . . . . . . . 10.2 Filtrations Associated with Ideals Defined by Shape . . . . . . . . . . 10.3 An algorithm to Compute Z() . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 The Filtration for Symbolic Powers of Determinantal Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Sheaves Defined by Shape and Their Quotients . . . . . . . . . . . . . . 10.6 A Geometric Realization of the Modules Jl(σ ) . . . . . . . . . . . . . . . 10.7 Regularity of Symbolic Powers of Determinantal Ideals . . . . . . . 10.8 Some Geometric Properties of Determinantal Varieties . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Cohomology and Regularity in Characteristic Zero . . . . . . . . . . . . . . . 11.1 The Borel–Weil–Bott Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Optimality of Theorem 9.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Borel–Weil–Bott on Grassmannians . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Effective Vanishing of Cohomology in Characteristic Zero . . . . 11.5 Cauchy’s Formula Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Explicit Description of Ext Modules for Jl(σ ) . . . . . . . . . . . . . . . . 11.7 Ext Modules for Symbolic Powers . . . . . . . . . . . . . . . . . . . . . . . . 11.8 The Basics on GL-Invariant Ideals . . . . . . . . . . . . . . . . . . . . . . . . . 11.9 Ext Modules for GL-Invariant Ideals . . . . . . . . . . . . . . . . . . . . . . 11.10 Local Cohomology with Determinantal Support . . . . . . . . . . . . . 11.11 Syzygies of GL-Invariant Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

343 345 350 361 369 374 377 379 380 381 388 391 392 401 408 409 414 415 416 421 424 428 433 435 445 449 455 465 470 477

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499

Chapter 1

Gröbner Bases, Initial Ideals and Initial Algebras

The monomials in a polynomial ring K [X ] in a single indeterminate over a field K are naturally ordered by degree, and this order coincides with the divisibility order. It makes K [X ] a Euclidean domain in which one can compute as efficiently as in K itself. But as soon as there is more than one indeterminate, the natural order of monomials is lost: for efficient computation one must impose an additional structure, an order that is as compatible with divisibility as far as possible, but cannot be derived from it, and is therefore not uniquely determined. A Gröbner basis of an ideal I in a polynomial ring R = K [X 1 , . . . , X n ] is a system of generators that allows one to decide algorithmically whether an element f belongs to I and to find a representation as a linear combination of the Gröbner basis if f ∈ I . The reduction principle on which the algorithm relies has revolutionized efficient computation in commutative algebra and algebraic geometry. But there is a second equally important use of monomial orders: the assignment of an ideal of monomials to I , its initial ideal in(I ), generated by the leading monomials of the elements of I . Monomial ideals come with a natural K -basis, namely the monomials they contain, and therefore their theory is dominated by the analysis of sets of monomials, which have manifold combinatorial interpretations. Many homological and enumerative invariants can be transferred from R/ in(I ) to R/I . This process of Gröbner deformation gives insight into the algebraic structure of R/I that may be difficult to access by other methods. From this perspective the determination of a Gröbner basis can be viewed as the computation of the initial monomial ideal. In the same way as one assigns an initial monomial ideal to an ideal I of R, one can define the initial algebra in(A) of a subalgebra A of R. Initial algebras are derived from Sagbi bases: the acronym stands for “subalgebra analog to Gröbner bases for ideals”. These may be infinite and are difficult to compute, even when they are finite. Therefore they play a lesser role for efficient methods. However, if the initial algebra

© Springer Nature Switzerland AG 2022 W. Bruns et al., Determinants, Gröbner Bases and Cohomology, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-05480-8_1

1

2

1 Gröbner Bases, Initial Ideals and Initial Algebras

is finitely generated, then the principle of Gröbner deformation can be used for the analysis of A: via in(A) the structure of A becomes accessible to toric algebra and geometry.

1.1 Monomial Orders Throughout this section let K be a field. The arithmetic of the polynomial ring K [X ] in a single variable X is governed by division of polynomials with remainder: for f, g ∈ K [X ], g = 0, there exist uniquely determined q, r ∈ K [X ] such that f = qg + r,

deg r < deg g.

The division algorithm that yields q and r depends on two basic facts: (1) g has a unique initial (or leading) term a X n with a ∈ K , a = 0; (2) X m is divisible by a X n ⇐⇒ m ≥ n. Now let us go to the polynomial ring R = K [X 1 , . . . , X n ] in severalvariables X i . n X iu i with A monomial (or power product) of R is an element of the form X u = i=1 n u = (u 1 , . . . , u n ) ∈ N . A term is an element of the form aμ where a is a nonzero element of K and μ is a monomial. Let Mon(R) be the K -basis of R consisting of all the monomials of R. Every polynomial f ∈ R can be written as a sum of terms: there exists a unique (finite) subset supp( f ) of Mon(R) such that f =



aμ μ,

aμ ∈ K , aμ = 0 for μ ∈ supp( f ).

μ∈supp( f )

The only lack of uniqueness in this representation is the order of the terms. If we impose a total order on the set Mon(R), then the representation is uniquely determined, once we require that the monomials are listed in order, from the largest to the smallest. So we can at least rescue property (1) of K [X ] by introducing a total order of the monomials. That there is no hope of rescuing the full validity of property (2) is immediately clear: as soon as n ≥ 2, there exist monomials μ and ν such that neither μ | ν nor ν | μ. This section discusses to what extent the division of monomials can nevertheless be transferred to K [X 1 , . . . , X n ]. The set Mon(R) is a monoid (or semigroup, naturally isomorphic to Nn ) and the weakest condition on a total order ≤ on Mon(R) that at least preserves the implication =⇒ in (2) above is μ | ν =⇒ μ ≤ ν. We go a step further. Definition 1.1.1 A monomial order is a total order < on the setMon(R) which satisfies the following conditions:

1.1 Monomial Orders

3

(a) 1 ≤ μ for all the monomials μ ∈ Mon(R).1 (b) If μ1 , μ2 , μ3 ∈ Mon(R) and μ1 ≤ μ2 , then μ1 μ3 ≤ μ2 μ3 . If it is necessary, we distinguish different monomial orders by adding a subscript to X n . More generally, for every total order of the indeterminates one can consider the lex, deglex and revlex orders extending the order of the indeterminates; just change the above definition correspondingly. There are of course many more monomial orders (for example, see Kreuzer and Robbiano [224, 1.4]). As a rule of thumb one can say that in the lex order the large factors X i make a monomial μ large, whereas in Revlex the small factors X i make it small. Example 1.1.3 Let R = K [X, Y, Z ]. In the lexicographic monomial order determined by X > Y > Z one has X 2 > X Z > Y 2 , whereas in the reverse lexicographic order with X > Y > Z the order of the three monomials changes to X 2 > Y 2 > X Z . From the theoretical as well as from the computational point of view it is important that descending chains in Mon(R) terminate: Proposition 1.1.4 A monomial order on the set Mon(R) is a well-order, i.e., every nonempty subset of Mon(R) has a (unique) minimal element. Equivalently, there are no infinite descending chains in Mon(R). Proof Let N ⊂ Mon(R) be nonempty and let I be the ideal in R generated by N . By Hilbert’s basis theorem,2 I is finitely generated. An ideal generated by monomials is generated by those monomials in it that are minimal with respect to divisibility. For I these belong to N , and among them is the minimal element of N since ≤ refines divisibility. 

1

In certain contexts one must replace condition (a) by the reverse relation or omit it at all; for example, see Greuel and Pfister [157]. 2 It is customary to use Dickson’s lemma at this point, which is a special version of the older basis theorem for monomial ideals.

4

1 Gröbner Bases, Initial Ideals and Initial Algebras

From now on we fix a monomial order < on (the monomials of) R. Every polynomial f = 0 has a unique representation f = a1 μ1 + a2 μ2 + · · · + ak μk where ai ∈ K \ {0} and μ1 , . . . , μk are monomials such that μ1 > · · · > μk . Definition 1.1.5 The initial monomial of f with respect to < is denoted by in( f ) and is, by definition, μ1 . The initial term is init( f ) = a1 μ1 , and the initial coefficient is inic( f ) = a1 . For example, the initial monomial of the polynomial f = X 1 + X 2 X 4 + X 32 with respect to the lex order is X 1 , with respect to deglex it is X 2 X 4 , and with respect to revlex it is X 32 . In order to avoid cumbersome case distinctions, we enlarge Mon(R) to Mon(R) ∪ {−∞} and set in(0) = −∞, (−∞)μ = −∞ for all μ ∈ Mon(R) ∪ {−∞}. After this extension one has in( f g) = in( f ) in(g), (1.1) in( f + g) ≤ max{in( f ), in(g)}, for all f, g ∈ R. Remark 1.1.6 For theoretical reasons and especially for efficient computations it is necessary to extend a monomial order from R to the free R-module R r with its canonical basis e1 , . . . , er of unit vectors. The monomials in R r are the products μei , μ ∈ Mon(R), i = 1, . . . , r . A direct way to extend the monomial order is μei < νe j

⇐⇒

i < j or i = j and μ < ν.

Alternatively, one can give precedence to the monomial order on R. (See [224, pp. 54, 55] for a more general discussion.) A less direct approach uses the symmetric algebra R[Y1 , . . . , Yr ] in which R r appears as the R-module generated by Y1 , . . . , Yr . It is easy to see that a monomial order on R can be extended to R[Y1 , . . . , Yr ] (in many ways). The restriction to R r then is a monomial order on R r .

1.2 Gröbner Bases Throughout this section we assume that the underlying ring is R = K [X 1 , . . . , X n ] endowed with a fixed monomial order. Let I be an ideal of R. Then the K -vector space in(I ) that is spanned by the monomials in( f ), f ∈ I , f = 0, is an ideal as well; it is called the initial ideal of I . In fact, a vector subspace is an ideal if it is closed under multiplication by monomials, and μ in( f ) = in(μf ) by (1.1).

1.2 Gröbner Bases

5

Definition 1.2.1 The polynomials f 1 , . . . , f m = 0 of I form a Gröbner basis of I if the monomials in( f 1 ), . . . , in( f m ) generate in(I ) as an ideal. Since in(I ) is a finitely generated ideal, every ideal I has a Gröbner basis. Example 1.2.2 Let R = K [X, Y, Z ], f 1 = X 2 − Y 2 and f 2 = X Z − Y 2 . First we consider the lexicographic monomial order determined by X > Y > Z . Then f 1 , f 2 is not a Gröbner basis of I = ( f 1 , f 2 ) since g = Z f 1 − X f 2 = X Y 2 − Z Y 2 ∈ I and / (in( f 1 ), in( f 2 )) = (X 2 , X Z ). The reatherefore in(g) = X Y 2 ∈ in(I ), but X Y 2 ∈ son for the existence of g with in(g) ∈ / (in( f 1 ), in( f 2 )) is that the leading terms of Z f 1 and X f 2 cancel in the linear combination that yields g. Let us now change the order to the reverse lexicographic one with the same order of the indeterminates. Then in( f 1 ) = X 2 , in( f 2 ) = Y 2 , and f 1 , f 2 is indeed a Gröbner basis of I ! This will follow immediately from Proposition 1.2.12 below. This simple example shows that the property of being a Gröbner basis in general depends on the monomial order. Usually an ideal is given as the kernel of a K -algebra homomorphism or by a system of generators. In the latter case the famous Buchberger algorithm finds a Gröbner basis of I and therefore computes in(I ) (and indirectly it also computes the kernel of a K -algebra homomorphism). In its basic version the algorithm will be developed in this section. Definition 1.2.3 Let f 1 , . . . , f m ∈ R. A polynomial r is a reduction of g ∈ R modulo f 1 , . . . , f m if there exist q1 , . . . , qm ∈ R satisfying the following conditions: (a) g = q1 f 1 + · · · + qm f m + r , (b) in(qi f i ) ≤ in(g) for all i = 1, . . . , m, (c) no monomial μ ∈ supp(r ) is divisible by any in( f i ), i = 1, . . . , m. The letter q has been chosen since the qi can be thought of as “quotients” of g modulo the f i and r is the “remainder”. Proposition 1.2.4 Every polynomial g has a reduction modulo f 1 , . . . , f m . Proof Let J = (in( f 1 ), . . . , in( f m )). We start with r = g and apply the reduction algorithm: (1) If supp(r ) ∩ J = ∅, we are done: r is the desired reduction. (2) Otherwise choose μ ∈ supp(r ) ∩ J maximal and let b ∈ K be the coefficient of μ in the monomial representation of r . Choose i such that in( f i ) | μ and set r = r − aν f i where ν = μ/ in( f i ) and a = b/ inic( f i ). Then replace r by r and go to (1). This algorithm terminates after finitely many steps since it replaces the monomial μ by a linear combination of monomials that are smaller in the monomial order. Such replacement can be applied only finitely many times since all descending chains of monomials in R terminate. We obtain the “quotient” qi by accumulating the terms aν f i that arise in step (2). It is easy to check that they satisfy condition (b) of the definition. 

6

1 Gröbner Bases, Initial Ideals and Initial Algebras

Example 1.2.5 As in Example 1.2.2 we take R = K [X, Y, Z ], f 1 = X 2 − Y 2 and f 2 = X Z − Y 2 , and again we consider the lexicographic monomial order with X > Y > Z . Set g = X 2 Z . Then g = Z f 1 + Y 2 Z , but g = X f 2 + X Y 2 as well. Both these equations yield reductions of g, namely X Y 2 and Y 2 Z . Thus a polynomial can have several reductions modulo f 1 , f 2 . The ambiguity shown by Example 1.2.5 can be avoided if one requires that the polynomial f i in the proof of Proposition 1.2.4 is chosen in such a way that i is minimal. Therefore we may speak of the reduction algorithm. The proof of Proposition 1.2.4 is the paradigm for proofs by descent in the monomial order that we will encounter many times in the following. A consequence of Proposition 1.2.4: the (residue classes of the) monomials in Mon(R) \ J form a generating set of R/I as a K -vector space. In a distinguished case, they are also linearly independent: Proposition 1.2.6 Let f 1 , . . . , f m ∈ R be polynomials, I = ( f 1 , . . . , f m ) and J = (in( f 1 ), . . . , in( f m )). Then the following are equivalent: (a) f 1 , . . . , f m form a Gröbner basis of I ; (b) every g ∈ I reduces to 0 modulo f 1 , . . . , f m ; (c) the monomials μ, μ ∈ / J , are linearly independent modulo I . If the equivalent conditions (a), (b), (c) hold, then: (d) Every element of R has a unique reduction modulo f 1 , . . . , f m . (e) Moreover, the reduction depends only on I and the monomial order. Proof (a) =⇒ (c) Assume that (c) is false. Then there exists a polynomial r ∈ I which is a nonzero linear combination of monomials μ ∈ / J . On the other hand, in(r ) is divisible by in( f i ) for some i since f 1 , . . . , f m are a Gröbner basis. This is a contradiction. (c) =⇒ (b) Let r be a reduction of g ∈ I modulo f 1 , . . . , f m . Then r ∈ I as well, and r vanishes modulo I . But it is a linear combination of monomials μ ∈ / J . Since these are linearly independent modulo I by assumption, we conclude r = 0. (b) =⇒ (a) Let g ∈ I , g = 0. If g reduces to 0, then we have an equation g = q1 f 1 + · · · + qm f m such that in(qi f i ) ≤ in(g) for all i. But the monomial in(g) must appear on the right hand side as well, and this is only possible if in(g) = in(qi f i ) = in( f i ) in(gi ) for at least one i. In other words, in(g) must be divisible by in( f i ) for some i. Hence in(I ) = J , and f 1 , . . . , f m are a Gröbner basis by definition. (c) =⇒ (d) Let r1 , r2 be reductions of g ∈ R. Then, on the one hand, r1 − r2 ∈ I , and, on the other hand, r1 − r2 is a linear combination of monomials μ ∈ / in(I ). By (c) we must have r1 = r2 . (e) Only the monomial ideal in(I ) has been used in the proof of uniqueness. 

1.2 Gröbner Bases

7

In view of (d) one may speak of the reduction of a polynomial modulo I , once the monomial order is fixed. An immediate consequence of Proposition 1.2.6(b): Corollary 1.2.7 A Gröbner basis of I generates I . The list of equivalent conditions in Proposition 1.2.6 can be extended. For example, (d) is also sufficient for f 1 , . . . , f m to be a Gröbner basis; see [224, 2.2.5].

Lifting of Syzygies Suppose that f 1 , . . . , f m are a Gröbner basis of I , and let g ∈ I . As just proved, the reduction algorithm yields a representation g = a1 f 1 + · · · + am f m

(1.2)

such that in(g) = in(ai f i ) for some i, and, moreover, all other monomials appearing in any of the a j f j are smaller in the monomial order. What distinguishes a nonreduction representation g = b1 f 1 + · · · + bm f m from a reduction? Well, we must have in(g) < in(bi f i ) for at least one i. Set si = ai − bi . Then (1.3) s1 f 1 + · · · + sm f m = 0. In well-established terminology, (s1 , . . . , sm ) ∈ R m is a syzygy of f 1 , . . . , f m . The terms of the si f i in each monomial degree sum to zero. In particular, the “top” terms sum to zero, and we obtain a syzygy of the initial terms init( f i ). In the terminology to be introduced below, this “top” syzygy is “lifted” by (s1 , . . . , sm ) to a syzygy of f1 , . . . , fm . Now suppose that f 1 , . . . , f m are an arbitrary system of generators of I and set J = (in( f 1 ), . . . , in( f m )). Then, given an element g ∈ I , we can write it as a linear combination g = c1 f 1 + · · · + cm f m . The discussion above shows: in order that f 1 , . . . , f m be a Gröbner basis, we must be able to transform this representation into one satisfying the same condition as (1.2) above, and we must be able to reach it by successively “peeling off” terms that are larger than in(g). Our very informal discussion must be made precise. Let F = R m . We have Rlinear maps

8

1 Gröbner Bases, Initial Ideals and Initial Algebras

ϕ : F → R,

ϕ(ei ) = f i ,

ψ : F → R,

ψ(ei ) = init( f i ).

Clearly, Im(ϕ) = I , Im(ψ) = J . For x = (x1 , . . . , xm ) ∈ F, we set inϕ (x) = max in(xi f i ). i

Then we can assign x an initial term init ϕ (x) ∈ F by  init(xi ) if in(xi f i ) = inϕ (x), initϕ (x)i = 0 else,

i = 1, . . . , m.

Without any assumption on f 1 , . . . , f m one gets s ∈ Ker ϕ

=⇒

initϕ (s) ∈ Ker ψ

as we see by reading the equation s1 f 1 + · · · + sm f m = 0 monomial by monomial. In the opposite direction, let us say that s ∈ Ker ϕ lifts t ∈ Ker ψ if initϕ (t) = initϕ (s). Example 1.2.8 Let us discuss a significant example. The matrix  X=

X 11 X 12 X 13 X 21 X 22 X 23



is defined over the polynomial ring R = K [X i j : i = 1, 2, j = 1, 2, 3]. Then the 2-minors, the three determinants   X 1u X 1v , u < v, [u v] = det X 2u X 2v generate an ideal that is paradigmatic for the class of ideals discussed in this book. Let us choose the lexicographic monomial order in which X i j < X pq if i < p or i = p and j < q. Then the “diagonal” X 1u X 2v is the initial monomial (and term) of [u v]. We take F = R 3 and map the basis e1 , e2 , e3 by ϕ and ψ to [1 2], [1 3], [2 3] and in[1 2], in[1 3], in[2 3], respectively. The kernel of ψ contains (0, X 12 , −X 11 )

and

(X 23 , −X 22 , 0)

and

(X 23 , −X 22 , X 13 ),

These are lifted by the elements (−X 13 , X 12 , −X 11 )

respectively, of Ker ϕ. We check this for (0, X 12 , −X 11 ):

1.2 Gröbner Bases

9

X 13 in[1 2] = X 11 X 22 X 13 < X 11 X 12 X 23 = X 12 in[1 3] = X 11 in[2 3]. So the “top” monomials appear in the second and third coordinate as claimed. We return to our general discussion. As above, let I = ( f 1 , . . . , f m ) and J = (in( f 1 ), . . . , in( f m )). Proposition 1.2.9 Let f 1 , . . . , f m ∈ R. Then the following are equivalent: (a) f 1 , . . . , f m is a Gröbner basis of I = ( f 1 , . . . , f m ); (b) for every g ∈ I there exists x ∈ F such that g = ϕ(x) and inϕ (x) = in(g); (c) every t ∈ Ker ψ can be lifted to a syzygy s ∈ Ker ϕ. Proof The equivalence of (a) and (b) is the equivalence of Proposition 1.2.6(a) and (b) in terms of the new notation. (c) =⇒ (b) follows from the discussion above. More formally, suppose we have a representation g = a1 f 1 + · · · + am f m of g ∈ I . If inϕ (a1 , . . . , am ) > in(g), then t = initϕ (a1 , . . . , am ) ∈ Ker ψ, as we see by evaluating the equation g = a1 f 1 + · · · + am f m in the monomial degree inϕ (a1 , . . . , am ). But t can be lifted to a syzygy s ∈ Ker ϕ by assumption. With bi = ai − si we have g = b1 f 1 + · · · + bm f m , and inϕ (b1 , . . . , bm ) < inϕ (a1 , . . . , am ). After finitely many operations of this type we reach a representation of g that satisfies (b). (b) =⇒ (c) Let t ∈ Ker ψ, and set g = ϕ(t). Since g ∈ I and f 1 , . . . , f m is a Gröbner basis by hypothesis, there is a reduced representation g = a1 f 1 + · · · + am f m of g. This is true for an arbitrary choice of t ∈ F, but t ∈ Ker ψ implies that  in(g) < inϕ (t), and therefore (t1 − a1 , . . . , tm − am ) lifts (t1 , . . . , tm ). In order to convert Proposition 1.2.9 into an effective test, we must understand Ker ψ. This is not difficult. Let μ, ν ∈ Mon(R). Two (or more) monomials have a well-defined least common multiple. Clearly lcm(μ, ν) lcm(μ, ν) μ− ν = 0, μ ν   so that lcm(μ,ν) is a syzygy of μ, ν, the divided Koszul syzygy. For , − lcm(μ,ν) μ ν f 1 , . . . , f m we define κi j ∈ F by

(κi j )k =

⎧ ⎪ ⎨

lcm(in( f i ),in( f j )) , init( f i ) lcm(in( f i ),in( f j )) − , init( f j ) ⎪

⎩

0

k = i, k = j,

for k = 1, . . . , m.

else,

This definition takes care of the initial coefficients. The elements κi j are the divided Koszul syzygies of init( f 1 ), . . . , init( f m ).

10

1 Gröbner Bases, Initial Ideals and Initial Algebras

Proposition 1.2.10 (a) The divided Koszul syzygies κi j generate Ker ψ as an R-module. (b) The following are equivalent: (i) each κi j can be lifted to a syzygy σi j of f 1 , . . . , f m ; (ii) every element of Ker ψ can be lifted to a syzygy of f 1 , . . . , f m . (c) Under the equivalent conditions in (b), the syzygies σi j generate Ker ϕ. Proof Let t = (t1 , . . . , tm ) be a nonzero element of Ker ψ. We must show that it is an R-linear combination of the divided Koszul syzygies κi j . For an inductive proof it is enough that (t1 , . . . , tm ) = initϕ (t) is such a linear combination. Let v be the smallest among the indices i such that ti = 0. There must be a second index w > v such that tw = 0. Both tv and tw are terms, and evidently tw in( f w ) and tv in( f v ) have the same monomial degree. Since this monomial is a common multiple of in( f w ) and in( f v ), it is divided by their least common multiple. Hence, for a suitable term τ one of the following cases applies to t = t − τ κvw : (1) inϕ (t ) < inϕ (t), or (2) inϕ (t ) = inϕ (t), but the first nonzero component of initϕ (t ) has index > v. We are done by induction. In (b) the implication (ii) =⇒ (i) is trivial, and (i) =⇒ (ii) follows by the same induction that was used to show (a): in case (1) τ σvw lifts t, in case (2) this is done by τ σvw + s where s lifts t . For (c) the induction applies as well.  Example 1.2.11 At this point we can prove that the three 2-minors [u v] introduced in Example 1.2.8 form a Gröbner basis of the ideal I generated by them. There are 3 divided Koszul relations, and 2 of them have already been lifted in Example 1.2.8. The third is the Koszul relation (in[2 3], 0, − in[1 2]). It has the trivial lifting ([2 3], 0, −[1 2]). As an exercise the reader should extend our findings to all 2 × n matrices of indeterminates. Even the case of 2-minors of m × n matrices is elementary, as we will see in Sect. 4.1. As a consequence of Proposition 1.2.10 we note a fact that often helps to identify regular sequences and initial ideals. Proposition 1.2.12 Let R = K [X 1 , . . . , X n ], I ⊂ R an ideal and g1 , . . . , gr ∈ R. If sequence on in(g1 ), . . . , in(gr ) are a regular  R/ in(I ), then  g1 , . . . , gr are a regular  sequence on R/I , and in I + (g1 , . . . , gr ) = in(I ) + in(g1 ), . . . , in(gr ) . Proof It is enough to prove the assertion for r = 1. Then the general case follows by induction. We set g = g1 . Assume that hg ∈ I , but h ∈ / I . Applying reduction modulo I to h, we can assume that in(h) ∈ / in(I ). But in(hg) = in(h) in(g) ∈ in(I ), in contradiction to our hypothesis. This proves the first statement. For the second we choose a minimal Gröbner basis f 1 , . . . , f m of I . Then in(g) and in( f i ) must be coprime monomials for all i; otherwise in(g) would be a zero-divisor modulo in(I ). But then the Koszul syzygy

1.2 Gröbner Bases

11

of g and f i lifts the (divided) Koszul syzygy of in(g) and in( f i ). Together with the assumption on f 1 , . . . , f m we obtain from Propositions 1.2.10 and 1.2.9 that  f 1 , . . . , f m , g is a Gröbner basis of I + (g). An important special case is I = 0: if in(g1 ), . . . , in(gr ) are a regular sequence in R, then g1 , . . . , gr are a regular sequence as well and a Gröbner basis of the ideal generated by them.

The Buchberger Criterion and Algorithm Theorem 1.2.13 Let f 1 , . . . , f m ∈ R generate the ideal I . Then the following are equivalent: (a) f 1 , . . . , f m form a Gröbner basis of I ; (b) all κi j , 1 ≤ i < j ≤ m, can be lifted to syzygies of f 1 , . . . , f m ; (c) the S-polynomials Si j =

lcm(in( f i ), in( f j )) lcm(in( f i ), in( f j )) fi − fj, init( f i ) init( f j )

1 ≤ i < j ≤ m,

reduce to 0 modulo f 1 , . . . , f m . Proof The equivalence of (a) and (b) follows from Propositions 1.2.9 and 1.2.10, and (a) =⇒ (c)is covered by Proposition 1.2.6. For (c) =⇒ (b) we note that the reduction algorithm, applied to Si j modulo f 1 , . . . , f m , yields a lifting of κi j when all terms are collected on one side and  bundled to a linear combination of f 1 , . . . , f m . The polynomial Si j in the theorem is called the S-polynomial of f i and f j . It will also be denoted by S( f i , f j ). The theorem suggests the Buchberger algorithm for the computation of a Gröbner basis. Suppose that f 1 , . . . , f m generate the ideal I . Then proceed as follows: (1) Set G = { f 1 , . . . , f m }. (2) Set G = ∅. (3) For all i and j, i < j, apply the reduction algorithm to Si j modulo G. If Si j reduces to h = 0, replace G by G ∪ {h}. (4) If G = ∅, G is the desired Gröbner basis. (5) If G = ∅, replace G by G ∪ G and go to (2). This algorithm terminates after finitely many iterations since the ideal (in(g) : g ∈ G) strictly increases if G = ∅. In the Noetherian ring R, any strictly increasing chain of ideals is finite. Example 1.2.14 We take up Example 1.2.2 again: R = K [X, Y, Z ] with the lexicographic term order with X > Y > Z , f 1 = X 2 − Z 2 , f 2 = X Y − Z 2 . We have

12

1 Gröbner Bases, Initial Ideals and Initial Algebras

already computed f 3 = S12 = Z f 1 − X f 2 = X Y 2 − Y 2 Z . Since f 3 is irreducible modulo f 1 , f 2 , we have found a new Gröbner basis element. In the next step we must form S13 and S23 . Since S13 = Y 2 f 2 , it reduces to 0. But S23 = Y 2 (Z 2 − Y 2 ) does not, and we obtain another Gröbner basis element f 4 = S23 . Fortunately the 3 polynomials Si4 , i = 1, 2, 3, reduce to 0 and we are done: f 1 , f 2 , f 3 , f 4 are a Gröbner basis of I . Already this toy example shows that the Buchberger algorithm is very complex and requires careful bookkeeping. Since it is frustrating to do Gröbner basis computations by hand, we refrain from giving more examples. But it is not difficult to extend Example 1.2.8 to Theorem 4.1.1: the 2-minors of an m × n-matrix of indeterminates are a Gröbner basis of the ideal they generate. Its proof uses only the Buchberger criterion. The algorithm given above is the simplest version of the Buchberger algorithm. For effective use it must be refined in various ways. We refer the reader to Eder and Faugère [114] for further information. The computer algebra systems CoCoA [1], Macaulay2 [154] and Singular [109] are built around (refined versions of) the Buchberger algorithm. Remark 1.2.15 As an important side effect we note that the algorithm yields a system of generators of the syzygy module Ker ϕ for a Gröbner basis f 1 , . . . , f m . In order to extend the algorithm to a computation of a free resolution, one must extend the whole theory to initial submodules and Gröbner bases of submodules of free modules. See Ene and Herzog [124], Greuel and Pfister [157] and Kreuzer and Robbiano [224] for a full treatment. We have already indicated in Remark 1.1.6 how to extend a monomial order to the free module R m via the symmetric algebra R[Y1 , . . . , Ym ]. Under the identification Rm ∼ = RY1 + · · · + RYm , a submodule U of R m is embedded into R[Y1 , . . . , Ym ] where it generates an ideal I . The Gröbner basis theory as developed above can be applied to I and then restricted to U . The elements g1 , . . . , gs of the Gröbner basis of I that belong to U then form a Gröbner basis of U in the sense that the initial submodule generated by the initial monomials of the elements of U is generated by in(g1 ), . . . , in(gs ). For the practical computation it would be foolish to really compute in the symmetric algebra and find a Gröbner basis of I . Instead, the Buchberger criterion and algorithm are restricted to R m . It is not hard to work this out in detail.

Reduced Gröbner Bases Evidently Gröbner bases of ideals are not uniquely determined (apart from trivial exceptions). However, every monomial ideal has a unique minimal system of monomial generators, namely the set of elements in it that are minimal with respect to divisibility. One says that f 1 , . . . , f m form a reduced Gröbner basis of I if the following conditions hold: (i) each f i has initial coefficient 1, (ii) in( f 1 ), . . . , in( f m ) is

1.3 Sagbi Bases

13

the unique minimal monomial system of generators of in(I ), (iii) each f i − in( f i ) is a K -linear combination of monomials μ ∈ / in(I ). Proposition 1.2.16 Every ideal I has a unique reduced Gröbner basis. Proof We first show uniqueness. Let G be a reduced Gröbner basis and let μ belong to the minimal system of monomial generators of in(I ). There is a unique element f ∈ G such that in( f ) = μ, and g = μ − f is a reduction of μ modulo G since supp(g) ⊂ Mon(R) \ in(I ). By Proposition 1.2.6(d) g is uniquely determined, and so is f . For the existence of the reduced Gröbner basis we choose it exactly in the way that the proof of uniqueness has told us: let μ1 , . . . , μm be the minimal monomial system of generators of in(I ), let gi be the reduction of μi modulo I for i = 1, . . . , m, and  set f i = μi − gi . Then f 1 , . . . , f m are the reduced Gröbner basis of I . This is not the place to discuss the manifold computational applications of Gröbner bases in detail. But two of them must be mentioned. Remark 1.2.17 (a) The monomial order defines a unique reduction of a polynomial f modulo a given ideal I . We can effectively compute it by the reduction algorithm based on a Gröbner basis f 1 , . . . , f m . In particular we can decide whether f ∈ I , and thereby solve the ideal membership problem. Moreover, we can compute effectively in the residue class ring R/I since we have a normal form and a rewriting algorithm that computes the normal form as a linear combination of the monomials μ ∈ Mon(R) \ in(I ). (b) The second application is the elimination of variables. Let R = K [X 1 , . . . , X m , Y1 , . . . , Yn ] and let I be an ideal in R. The task is to eliminate the variables Y1 , . . . , Yn from I , in other words: to compute I ∩ S for S = K [X 1 , . . . , X m ]. To this end one computes a Gröbner basis G of I with respect to the lex order in which X 1 < · · · < X m < Y1 < · · · < Yn . An element of I belongs to S if and only if its monomial support consists of monomials only involving X 1 , . . . , X m , and this is the case if in( f ) does so—clearly the choice of the monomial order is critical. This shows: G ∩ S is a Gröbner basis of I ∩ S under the restriction of the monomial order to S. By elimination of variables one computes the kernel of a K -algebra homomorphism ϕ : S → T , S = K [X 1 , . . . , X m ], T = K [Y1 , . . . , Yn ]. Choosing R as above,  the task amounts to eliminate Y1 , . . . , Yn from X 1 − ϕ(X 1 ), . . . , X m − ϕ(X m ) .

1.3 Sagbi Bases Many K -algebras we are interested in are defined as K -subalgebras of polynomial rings or can be realized as such K -subalgebras. In this section we transfer the notion of Gröbner basis from ideals to subalgebras (and ideals in them). Since in this section we deal only with K -algebras and their K -subalgebras and K -algebra homomorphisms,

14

1 Gröbner Bases, Initial Ideals and Initial Algebras

we will often skip the prefix and simply refer to them as algebras, subalgebras and algebra homomorphisms. Let A be a subalgebra of R = K [X 1 , . . . , X n ]. Since in( f ) in(g) = in( f g) and A is closed under multiplication, it makes sense to consider the algebra in(A) whose K -basis is given by the initial monomials of the nonzero polynomials f ∈ A. It is called the initial (sub)algebra. (Again we assume that R is endowed with a monomial order.) The main difference between initial ideals and initial algebras is that the latter need not be finitely generated, not even if A is finitely generated as a subalgebra. Nevertheless our discussion of Sagbi bases closely follows that of Gröbner bases, and at several places we can leave the transfer of arguments to the reader. We have learned in the previous section that the lifting of syzygies is the crucial aspect. This is true here as well. While subalgebras of R are not necessarily finitely generated, they are always countably generated since R has “only” countably many monomials. Therefore it is enough to consider families F = ( f i : i ∈ N ) of polynomials f i ∈ R where N is either the set {1, . . . , n} for some n ∈ N or the set of all positive natural numbers. Since we have to take initial monomials, we will always assume f i = 0 for all i. Definition 1.3.1 With the notation introduced, a family F of elements of A is a Sagbi basis of A if the monomials in( f i ), i ∈ N , generate in(A) as a subalgebra of R. The acronym “Sagbi”, introduced by Robbiano and Sweedler [269], stands for “Subalgebra analog to Gröbner bases of ideals”. The choice of this name emphasizes the analogy to Gröbner bases that will become evident in the following. Remark 1.3.2 Kaveh and Manon [203] have introduced Khovanskii bases as a generalization of Sagbi bases. Several authors have adopted the new terminology, but we keep the traditional name. Example 1.3.3 By a theorem of Newton and Lagrange the elementary symmetric functions  X i1 · · · X ik , k = 1, . . . , n, εk = 1≤i 1 X n . Let a1 , . . . , ak be a nonincreasak a1 · · · X π(k) for all ing sequence of natural numbers. Then one has X 1a1 · · · X kak ≥ X π(1) permutations π ∈ Sn (not only in the lexicographic order)! Now choose an arbitrary symmetric function f with initial monomial μ. Then ak a1 · · · X π(k) with a nonincreasthere exists a permutation π such that μ = X π(1) ing sequence a1 , . . . , ak . Applying the substitution defined by π −1 we see that ν = X 1a1 · · · X kak ∈ supp( f ) as well. But then ν is the initial monomial of f by our

1.3 Sagbi Bases

15

preliminary discussion. Evidently ν = μ can be written as a product of powers of the monomials in(εi ), i = 1, . . . , k, and our claim is proved. Since a Sagbi basis of A generates A, as will be shown below, the theorem of Newton and Lagrange follows. Example 1.3.4 Let A ⊂ K [X, Y ] be the subalgebra generated by X + Y, X Y, X Y 2 . We want to show that A has no finitely generated initial algebra, regardless of the monomial order [269]. Let us first assume that X > Y . Taking the initial monomials of the generators, we get X, X Y, X Y 2 ∈ in(A). But X Y 3 = X Y 2 (X + Y ) − (X Y )2 ∈ A as well, and certainly X Y 3 ∈ in(A). Continuing we see that also X Y k ∈ in(A) for all k. The algebra B generated by all these monomials is contained in in(A). It is easy to see that B is not finitely generated, and, as we will show next, in(A) = B. In fact, any strict monomial overalgebra of B must contain a power Y k for some k. But since Y k is the smallest monomial among all monomials of degree k, it would have to be contained in A itself in order to appear as an initial monomial ( A is a graded subalgebra). That this is not the case is left to the reader. Since X 2 Y = X Y (X + Y ) − X Y 2 ∈ A, the algebra A is invariant under the exchange of X and Y , and by symmetry we are done in the case Y > X as well. For the transfer from Gröbner bases to Sagbi bases we must replace R-linear combinations by polynomial expressions of the polynomials f i ∈ F . We write these expressions as K -linear combinations of monomials. Let e = (ei : i ∈ N ) be a family of natural numbers such that ei = 0 for all but finitely many i ∈ N . For such e we set

e fi i . Fe = i∈N

The element F e is called a monomial in F . (In general it does not belong to Mon(R).) Note that in(F e ) = (in(F ))e where in(F ) is the family of initial monomials of the elements of F . The terminology in the next definition is borrowed from geology: Definition 1.3.5 Let g ∈ R. Then r ∈ R is a subduction of g modulo F if there exist monomials F e1 , . . . , F em and coefficients ai ∈ K such that the following hold: (a) g = a1 F e1 + · · · + am F em + r ; (b) in(F ei ) ≤ in(g) for i = 1, . . . .m; (c) no monomial μ ∈ supp(r ) is of type in(F e ). The following is the Sagbi basis analog of Gröbner basis reduction Definition 1.2.3: Proposition 1.3.6 Let F be a family of polynomials in R and g ∈ R. Then g has a subduction modulo F . Proof The proof follows the pattern of the proof of Proposition 1.2.4: if there exists a monomial μ ∈ supp(g) such that μ = in(F e ) for some exponent vector e, then we pass from g to g − aF e where a ∈ K is chosen such that the initial terms cancel

16

1 Gröbner Bases, Initial Ideals and Initial Algebras

each other. This operation is compatible with condition (b) in the definition and only introduces monomials ν < μ. We are done by descending induction over the monomial order.  Now we can mimic Proposition 1.2.6. Proposition 1.3.7 Let B = K [in(F )] be the subalgebra of R generated by the initial monomials in( f i ), i ∈ N . Then the following are equivalent: (a) F is a Sagbi basis of A; (b) every f ∈ A subduces to 0 modulo F ; (c) the monomials μ ∈ / B are linearly independent modulo A. If the equivalent conditions (a), (b), (c) hold, then: (d) Every element of A has a unique subduction modulo F . (e) Moreover, the subduction depends only on A and the monomial order. The proof is a straightforward variation of the proof of Proposition 1.2.6 that we leave to the reader. In Proposition 1.2.6 the algebra structure of R/I was not used, only its vector space structure, and one needs only the vector space structure of R/A in the proof of Proposition 1.3.7. (It is another matter that we are often primarily interested in R/I , but presumably never in R/A.) Corollary 1.3.8 A Sagbi basis of the subalgebra A ⊂ R generates A (as an algebra).

The Lifting of Binomial Relations Next we want to discuss the Sagbi analogs of the lifting of syzygies and of the Buchberger criterion. This requires some preparation, and a somewhat more formal approach is advisable. For simplicity we will make the harmless assumption that the polynomials f i with i ∈ N are nonzero and monic, i.e. inic( f i ) = 1. Let P = K [Yi : i ∈ N ]. We consider the algebra homomorphisms ϕ : P → R, ψ : P → R,

ϕ(Yi ) = f i , ψ(Yi ) = in( f i ).

Clearly ψ(ζ ) = in(ϕ(ζ )) for all monomials ζ ∈ P. For the discussion of Sagbi bases one must pull back the monomial structure and order of R to P in the best possible way. For monomials ζ, η ∈ P we could try the definition ζ < η ⇐⇒ ψ(ζ ) < ψ(η). In general, this relation is not defined everywhere since we may have a tie: ψ(ζ ) = ψ(η). Nevertheless we can define a

1.3 Sagbi Bases

17

replacement of the initial monomial and term with respect to ϕ. Let F ∈ P, F = 0,  F = i ai Y ei with ai ∈ K \ {0} for all i. We set inϕ (F) = max in(ϕ(Y ei )), i  ai Y ei . initϕ (F) = in(ϕ(Y ei ))=inϕ (F)

It is important not to confuse inϕ (F) and in(ϕ(F)): in general, we can have in(ϕ(F)) < inϕ (F) since there may be cancellation among the top monomials of the polynomials ϕ(ai Y ei ). The polynomial ring R = K [X 1 , . . . , X n ] is naturally Zn -multigraded: the multidegree of a monomial is its exponent vector. Via ψ we can pull the multigrading back to the ψ-multigrading on P: deg ζ = deg ψ(ζ ). (As observed, different monomials in P may have the same multidegree in the ψ-multigrading.) Evidently F ∈ P is ψ-multihomogeneous if and only if F = initϕ (F). Let F ∈ Ker ϕ. Then initϕ (F) ∈ Ker ψ, as we can see when the equation ϕ(F) = 0 is split into its monomial components. As in the case of Gröbner bases, let us say that F ∈ Ker ϕ lifts G ∈ Ker ψ if initϕ (F) = initϕ (G). Example 1.3.9 Consider the matrix  X=

X 11 X 12 X 13 X 14 X 21 X 22 X 23 X 24



with entries in the polynomial ring R = K [X i j : i = 1, 2, j = 1, . . . , 4]. We transfer the notation from Example 1.2.8 to the bigger matrix and consider the subalgebra A generated by the 6 determinants [i j], 1 ≤ i < j ≤ 4. For them one has the famous Plücker relation [1 2][3 4] − [1 3][2 4] + [1 4][2 3] = 0. General Plücker relations will be studied in Sect. 3.2, but the very first one above can of course be verified by hand. We represent the left hand side by the polynomial F = Y12 Y34 − Y13 Y24 + Y14 Y23 in P = K [Yi j : 1 ≤ i < j ≤ 4] with respect to the substitution ϕ : Yi j → [i j]. Then initϕ (F) = −Y13 Y24 + Y14 Y23 , and so this element of Ker ψ, a relation of the initial monomials ψ(Yi j ) = init[i j], is lifted by F. Proposition 1.3.10 The following are equivalent: (a) F is a Sagbi basis of A; (b) for every f ∈ A, f = 0, there exists F ∈ P such that f = ϕ(F) and inϕ (F) = in( f ); (c) every G ∈ Ker ψ can be lifted to a polynomial F ∈ Ker ϕ.

18

1 Gröbner Bases, Initial Ideals and Initial Algebras

Again it is justified to say that the proof is a variation of the proof of Proposition 1.2.9. Now we transfer Proposition 1.2.10. At this point another difference of Sagbi bases to Gröbner bases becomes visible: one cannot predict a system of generators of Ker ψ. But at least there is a system of generators of easily understood elements, namely polynomials of type ζ − η with ζ, η ∈ Mon(P). These are called binomials. In general one would have to allow differences of terms, but in this book the more general version is not used. An ideal is binomial if it is generated by binomials. It is evident that the Buchberger algorithm produces a Gröbner basis of binomials if its input is a binomial system of generators of an ideal, independently of the monomial order. The reader may check that the reduced Gröbner basis of a binomial ideal consists of binomials. Let us single out two arguments that will be useful in the proof of the next proposition. (1) Let F ∈ Ker ψ be ψ-multihomogeneous and ζ ∈ Mon(P). If F is lifted by G, then ζ F is lifted by ζ G. (2) Let F1 , F2 be ψ-multihomogeneous elements in Ker ψ, lifted by G 1 and G 2 , respectively. If F1 + F2 = 0, then G 1 + G 2 lifts F1 + F2 . Proposition 1.3.11 (a) The kernel of ψ is generated by binomials. (b) Let B be a binomial system of generators of Ker ψ. The following are equivalent: (i) every β ∈ B can be lifted to a polynomial Fβ ∈ Ker ϕ; (ii) every G ∈ Ker ψ can be lifted to a polynomial F ∈ Ker ϕ. (c) Under the equivalent conditions in (b), Ker ϕ is generated by the polynomials Fβ , β ∈ B. Proof (a) Let M = {ψ(ζ ) : ζ ∈ Mon(P)}. The algebra B = Im ψ has M as a K vector space basis. For μ ∈ M set Cμ = {ζ : ψ(ζ ) = μ}. The sets Cμ partition Mon(P). For each of them we choose a representative ζμ . Set B=



{η − ζμ : η ∈ Cμ }.

μ∈M

We claim that B is a system  of generators of Ker ψ. To this end  let G = aη η ∈ Ker ψ. Replacing each η by ζμ , μ = ψ(η), amounts to subtracting aη (η − ζψ(η) ) from G. The resulting polynomial H is a linear combination of the representatives ζμ . But these are mapped bijectively to a K -basis of B, and therefore are linearly independent themselves. We conclude that H = 0, and G has turned out to be a linear combination of the binomials in B. (b) For the nontrivial implication let G ∈ Ker ψ, G = 0. Clearly initϕ (G) ∈ Ker ψ as well. For the existence of a lifting, we can therefore replace G by initϕ (G), and so assume that G is ψ-multihomogeneous. The binomials in Ker ψ are also ψmultihomogeneous.

1.3 Sagbi Bases

19

 We have an equation G = i ai ζi βi where ai ∈ K , ζi is a monomial in P and βi ∈ B. By multihomogeneity we can assume that all summands have the same ψmultidegree as G. Since all βi are liftable, the rules (1) and (2) above show that G is liftable as well.  For (c) one uses the standard induction over inϕ . Example 1.3.12 We look again at Example 1.3.9 and consider the algebra A generated by the determinants [i j] of the 2 × 4 matrix X . Two of the six monomials in[i j], namely in[1 2] and in[3 4], contain an indeterminate as a factor that does not appear in any other monomial. Therefore they cannot occur in any binomial relation that belongs to a minimal system B of generators of Ker ψ (in the notation introduced above). Omit one of the remaining four monomials. Then the argument just given shows that the other three are algebraically independent. Altogether we conclude that dim in(A) ≥ 5 and dim in(A) < 6 because in(A) is generated by 6 monomials and −Y13 Y24 + Y14 Y23 . is in the kernel of ψ. Therefore the kernel of ψ must be generated by a single irreducible binomial, i.e. the binomial above. It can be lifted to an element of Ker ϕ, as we have seen in Example 1.3.9. Conclusion: the determinants [i j] are a Sagbi basis of A. As we will see in Sect. 6.2, this result can be substantially generalized. The obstruction for an immediate generalization to 2 × n matrices is finding a minimal system B of binomial relations for n > 4. For n = 3 the initial monomials of the three determinants [1 2], [1 3], [2 3] are algebraically independent, and there is no binomial relation to be lifted. Ideals such as Ker ψ in Proposition 1.3.11 are called toric ideals: they are the defining ideals of monomial subalgebras of a polynomial ring (or, more generally, of a Laurent polynomial ring). Also Proposition 1.2.12 has an analog for Sagbi bases: Proposition 1.3.13 Let A be a subalgebra of R = K [X 1 , . . . , X n ] and g1 , . . . , gr ∈ R such that in(g1 ), . . . , in(gr ) are algebraically independent over in(A). Then   , . . . , g are algebraically independent over A, and in A[g , . . . , g ] = in(A) g 1 r 1 r   in(g1 ), . . . , in(gr ) . This follows immediately from the lifting criterion. An important special case is A = K : if in(g1 ), . . . , in(gr ) are algebraically independent, then g1 , . . . , gr are a Sagbi basis of K [g1 , . . . , gr ].

The Sagbi Criterion and Algorithm We are now ready to state the analog of the Buchberger criterion:

20

1 Gröbner Bases, Initial Ideals and Initial Algebras

Theorem 1.3.14 With the notation introduced, suppose that F generates A as an algebra, and let B be a binomial system of generators of Ker ψ. Then the following are equivalent: (a) F is a Sagbi basis of A; (b) every element of B can be lifted to an element of Ker ϕ. (c) the polynomials ϕ(β), β ∈ B, subduce to 0 modulo F . The equivalence of (a) and (b) has already been stated above, and the subduction of ϕ(β) can be reformulated as the lifting of β. The ϕ(β) take the place of the Spolynomials in the Buchberger criterion. (In the romantic language of [269] the pair (ϕ(ζ ), ϕ(η)), β = ζ − η ∈ B is called a tête-a-tête.) In the same way as the Buchberger criterion suggests an algorithm for the computation of Gröbner bases, Theorem 1.3.14 suggests an algorithm for the computation of Sagbi bases: Let the finite family F0 generate the subalgebra A ⊂ R. Then one proceeds as follows: (1) Set i = 0. (2) Set F = ∅ and compute a binomial system of generators Bi of the kernel of ψi : Pi → K [Fi ], Pi = K [Y F : F ∈ Fi ], ψi (Y F ) = in(F). (3) For all β ∈ Bi compute the subduction r of ϕi (β) modulo Fi , ϕi given by the substitution Y F → F, F ∈ Fi . If r = 0, make r monic and add it to F . (4) If F = ∅, set F j = Fi , P j = Pi , B j = Bi for all j ≥ i and stop. (5) Otherwise set Fi+1 = Fi ∪ F , i = i + 1 and go to (2). Proposition 1.3.15 ∞ (a) F = i=0 Fi is a Sagbi basis of A. (b) If A has a finite Sagbi basis, then the algorithm terminates.   Proof (a) We can set P = Pi and B = Bi since Pi+1 extends Pi and Bi+1 extends Bi . Moreover, the ϕi and ψi define homomorphisms ϕ, ψ : P → R such that B is a binomial system of generators of Ker ψ. The augmentation of Fi to Fi+1 is chosen in such a way that all polynomials ϕ(β), β ∈ Bi , subduce to 0 modulo Fi+1 . In total, all ϕ(β), β ∈ B, subduce to 0 modulo F , and we are done by Theorem 1.3.14. (b) A monomial subalgebra C of R has a unique minimal system of generators, namely the set I of irreducible monomials: μ ∈ C is irreducible if it cannot be written as the product of monomials ρ, σ ∈ C, ρ, σ = 1. What we have just stated, is easily proved by induction on the total degree of μ, in the same way as one shows that every positive integer is the product of prime numbers. So, if A has a finite Sagbi basis, the elements f for which in( f ) is irreducible in in(A) form a minimal Sagbi basis. Since the set F constructed by the algorithm is a Sagbi basis, the union of the sets in(Fi ) contains the irreducible monomials of in(A). But the latter are finite in number, so there is an i such that all of them belong to Fi , and then Fi is a Sagbi basis. Hence one has F = ∅, and the algorithm terminates. 

1.3 Sagbi Bases

21

The algorithm certainly has very limited practical value since all its steps have high complexity. At least three of them are difficult: (i) the computation of Bi , which requires the use of Gröbner bases, (ii) the check whether the initial monomial of a polynomial belongs to the subalgebra generated by in(Fi ), which is needed for a subduction, and (iii) subduction itself. After having assigned an initial algebra to a subalgebra A ⊂ R, we can assign an initial ideal to ideals I of A: the initial ideal in(I ) is the K -vector space spanned by the initial monomials in( f ), f ∈ I (which is indeed an ideal of in(A)). The notion of Gröbner basis can be transferred to this relative setting as well. It is not difficult to generalize the results of Sect. 1.2 accordingly.

Lifting Gröbner Bases Let A be a subalgebra of R with finite Sagbi basis f 1 , . . . , f n . Again we consider the algebra homomorphisms ϕ : P → R, ψ : P → R,

ϕ(Yi ) = f i , ψ(Yi ) = in( f i ),

so that A = ϕ(P) and in(A) = ψ(P). In Proposition 1.3.11(c) we have stated that Ker ϕ is generated by the polynomials that lift the binomials in a system of generators of Ker ψ. This statement can be extended to Gröbner bases, provided one chooses a monomial order on P that is compatible with the monomial order < on R that defines in(A): let us say that a monomial order ≺ on P lifts < if in≺ (F) = in≺ (initϕ (F)) for every F ∈ P. Proposition 1.3.16 Suppose that the monomial order ≺ on P lifts the monomial order < on R, and let G be a binomial Gröbner basis of Ker ψ. Then the polynomials in Ker ϕ which lift the elements of G are a Gröbner basis of Ker ϕ. Moreover, in≺ (Ker ϕ) = in≺ (Ker ψ). The easy proof is left to the reader. As a consequence, the monomials in P \ in≺ (Ker ψ) are mapped bijectively to K -bases of A and of in(A) by ϕ and ψ, respectively.

Initial Algebras of Tensor Products Let A ⊂ R and B ⊂ S K -subalgebras of the polynomial rings R and S which are endowed with monomial orders < and ≺, respectively. Since tensor product over K is flat, A ⊗ K B can be considered a subalgebra of the polynomial ring R ⊗ K S in a

22

1 Gröbner Bases, Initial Ideals and Initial Algebras

natural way. Tensor product becomes the “ordinary” product: after the identification of f ⊗ 1 with f ∈ R and 1 ⊗ g with g ∈ S, the tensor f ⊗ g is identified with f g ∈ R ⊗ K S. Under this identification each monomial in R ⊗ K S is a product μν of monomials μ ∈ R and ν ∈ S in a unique way. Therefore we can define a monomial order  on R ⊗ K S by choosing μν  μ ν if μ < μ or μ = μ and ν ≺ ν . Proposition 1.3.17 With the notation introduced, one has in (A ⊗ K B) = in< (A) ⊗ K in≺ (B). Proof For every monomial μ ∈ in< (A) we choose an element f μ ∈ A such that μ = in( f μ ), and make the analogous choice of gν ∈ B for the monomials ν ∈ in≺ (B). Then μν = in ( f μ gν ), showing in< (A) ⊗ K in≺ (B) ⊂ in (A ⊗ K B). For the converse inclusion one observes   that any element h ∈ A ⊗ K B has a unique representation of type h = μ ν aμ,ν f μ gν with coefficients aμ,ν ∈ K of which only finitely many are nonzero. Since the initial monomials in ( f μ gν ) = μν are pairwise different, one of them is the initial monomial of h, and this proves the other inclusion.  The reader may check that the proposition can be generalized in two ways: (i) It is enough that the monomial order on R ⊗ K S restricts to the given monomial orders on R and S. (ii) In the situation on which A and B are subalgebras of a common polynomial ring R one only needs that in the Laurent extension of R the subgroups of monomials generated by Mon(in(A)) and Mon(in(B)) intersect in {1}.

1.4 Initial Subspaces and Gradings So far we have discussed initial ideals and initial subalgebras in the polynomial ring R = K [X 1 , . . . , X n ]. Before we continue, it is useful to take a more general viewpoint by considering arbitrary vector subspaces of the polynomial ring. We assume that a monomial order has been chosen for R, and we suppress explicit reference to it unless it is really necessary. Given a K -subspace V = 0 of R, we define In(V ) = {in( f ) : f ∈ V, f = 0} and set in(V ) = the K -subspace of R generated by In(V ). Of course In(V ) is a K -basis of the initial subspace in(V ) since every set of monomials is linearly independent, and therefore is a basis of the subspace it generates. (For an ideal or subalgebra V there is no ambiguity since the vector space in(V ) is an ideal or subalgebra, respectively.) Proposition 1.4.1 Let V be a K -subspace of R. (a) Mon(R) \ In(V ) is a K -basis of R/V (and not only of R/ in(V )).

1.4 Initial Subspaces and Gradings

23

(b) For every μ ∈ In(V ) there exists a unique f μ ∈ V satisfying the following conditions: (i) in( f μ ) = μ,

(ii) inic( f μ ) = 1,

(iii) supp( f μ ) ∩ In(V ) = {μ}.

(c) The set { f μ : μ ∈ In(V )} is a K -basis of V . (d) If V has finite dimension, then dim(V ) = dim(in(V )). (e) Let < and  monomial orders. If in< (V ) ⊂ in (V ), then in< (V ) = in (V ). Proof (a) That Mon(R) \ In(V ) generates R/V , follows by the same induction that has already been applied in the reduction and subduction algorithms: we do not change the residue class of f ∈ R modulo V if we subtract an element g ∈ V from f . By such an operation we can replace the largest monomial in supp( f ) ∩ In(V ) by smaller monomials. Since descending chains of monomials terminate, the process stops with a representative of f modulo V that is supported in Mon(R) \ In(V ). The linear independence of Mon(R) \ In(V ) is immediate from the definition of In(V ). (b) Using (a) one argues in the same way as for the uniqueness of a reduced Gröbner basis. (c) If we start with an element of V , then finitely many reductions as in (a) must terminate at 0. Moreover, in the reduction it is sufficient to use the f μ and their K -multiples. Thus the f μ generate V . Since they have pairwise different initial monomials, they are automatically linearly independent. (d) and (e) are obvious consequences of (c).  If V has finite dimension, then the computation of in(V ) amounts to writing a basis of V as a matrix with respect to the monomial basis of R and reducing this matrix to row echelon form. We complement Proposition 1.4.1 by a relative version. Proposition 1.4.2 Let V1 ⊂ V2 be K -subspaces of R. (a) Then in(V1 ) ⊂ in(V2 ). If (gμ ) is a family of elements of V2 with in(gμ ) = μ for all μ ∈ In(V2 ) \ In(V1 ), then the (residue classes of the) gμ form a K -basis of the quotient space V2 /V1 . (b) If in(V1 ) = in(V2 ), then V1 = V2 . Proof For (a) we complement the family (gμ ) by additionally choosing polynomials h ν ∈ V1 such that in(h ν ) = ν for all ν ∈ In(V1 ). Let f ∈ V2 . If μ = in( f ) ∈ In(V2 ) \ In(V1 ), we subtract a suitable scalar multiple of gμ from f . If μ ∈ In(V1 ) we do the same with h μ . The resulting polynomial f satisfies in( f ) < μ. Iterating this procedure we must end at 0 since decreasing chains in Mon(R) terminate. In total we have written f as a K -linear combination of the gμ with μ ∈ In(V2 ) \ In(V1 ) and the h ν ∈ V1 . For (a) it remains to show that the gμ are linearly independent modulo V1 . But this is a standard argument that we have seen already several times. (b) follows easily from (a). 

24

1 Gröbner Bases, Initial Ideals and Initial Algebras

One should note a special case of statement (a): If V2 = in(V2 ), then the monomials μ ∈ V2 \ In(V1 ) are a K -basis of V2 /V1 . Next we relate monomial orders and gradings. Any vector g = (g1 , . . . , gn ) ∈ Nn>0 induces a graded structure on R, called the g-grading. With respect to the ggrading the indeterminate X i has degree g(X i ) = gi . The monomial X e has g-degree  ei gi , and the g-degree g( f ) of a nonzero ∞ polynomial f ∈ R is the largest g-degree Ri where Ri is the g-graded component of a monomial in supp( f ). Then R = i=0 of R of degree i, i.e., the K -span of the monomials of g-degree i. With respect to this decomposition, R has the structure of a positively graded K -algebra; see Bruns and Herzog [52, Sect. 1.5]. The elements of Ri are g-homogeneous of g-degree i. We say that a vector subspace V of R is g-graded (or g-homogeneous) if it is generated, as a vector ∞ space, by homogeneous elements. This amounts to the decomposition Vi where Vi = V ∩ Ri . The decomposition of V into its g-homogeneous V = i=0 components induces a corresponding decomposition of any polynomial f ∈ V into  its g-homogeneous components: f = i f i where f i ∈ Vi . The Hilbert function of V (with respect to the given grading) is H (V, i) = dim K Vi .

H (V, ) : Z → N,

Often the Hilbert function is best dealt with it via its generating function, the Hilbert series ∞  H (V, i)t i . HV (t) = i∈Z

Polynomial rings with g-gradings are generalized by finitely generated, ∞positively graded K algebras R. These K -algebras have a decomposition R = i=0 Ri such = K and R R ⊂ R . A graded R-module M has a decomposition M= that R 0 i j i+ j ∞ j∈Z M j satisfying the condition Ri M j ⊂ Mi+ j for all i and j. The terminology of graded components etc. generalizes in a straightforward way. The Hilbert function and series of M, and especially that of R, is defined as above. If M is finitely generated, the Hilbert series is the Laurent expansion of a rational function in t at the origin. See [52, Chap. 4] for an extensive treatment of Hilbert functions and series. Proposition 1.4.3 Let g  ∈ Nn>0 . Suppose V is g-graded with decomposition V = ∞ ∞ i=0 Vi . Then in(V ) = i=0 in(Vi ). In particular, V and in(V ) have the same Hilbert function. The quotient spaces R/V and R/ in(V ) have the same Hilbert function as well. Proof It is really enough to observe that V contains the g-homogeneous components of each of its elements.  Example 1.4.4 Hilbert functions are a very useful tool for the computation of Gröbner and Sagbi bases. Let us take up Example 1.3.12 once more. We have found the Plücker relation [1 2][3 4] − [1 3][2 4] + [1 4][2 3] = 0 that is represented by p = Y12 Y34 − Y13 Y24 + Y14 Y23 in the kernel of ϕ : P → R, ϕ(Yi j ) = [i j],

1.4 Initial Subspaces and Gradings

25

with image A = K [[i j] : 1 ≤ i < j ≤ 4]. In every degree k we therefore have H (A, k) ≤ H (P/( p), k) because the kernel of ϕ is potentially larger than ( p). On the other hand we know that the kernel of ψ : P → R with image B = K [in[i j] : 1 ≤ i < j ≤ 4] is generated by the single degree 2 binomial β = −Y13 Y24 + Y14 Y23 . Hence H (B, k) = H (P/(β), k) = H (P/( p), k). Since B ⊂ in(A), one has H (B, k) ≤ H (in(A), k). Combining the inequalities with Proposition 1.4.3 yields the chain H (B, k) ≤ H (in(A), k) = H (A, k) ≤ H (P/( p), k) = H (P/(β), k) = H (B, k). Thus equality holds everywhere, and this implies B = in(A). In a similar style one can prove the conclusion of Example 1.2.11 that the determinants form a Gröbner basis of the ideal they generate. The simple principle on which the previous example is based can be formulated as follows: Proposition 1.4.5 With the notation of Proposition 1.4.3 suppose that W ⊂ in(V ) is a g-graded K -vector space generated by monomials. If H (V, k) ≤ H (W, k) for all degrees k,then in(V ) = W . The proof uses the chain of inequalities that we have already seen in the example. In the next section vectors such as g will play a second important role by defining initial terms and spaces. Remark 1.4.6 The results of this section hold more generally for (graded) subspaces U of (graded) free modules R m over R. As indicated in Remarks 1.1.6 and 1.2.15, one way to extend them is to go through the symmetric algebra of R m and check that all data can be restricted to R m and to U . With some additional information, one can replace the Hilbert function in Proposition 1.4.5 by a single number, the multiplicity. For its definition we refer the reader to [52, Sect. 4.1]. The standard grading of R is defined by g = (1, ..., 1). The multiplicity of a finitely generated graded module M over a standard graded ring is denoted by e(M). The crucial condition in the next proposition is unmixedness of the ideal J : One says that an ideal is unmixed if all its associated prime ideals have the same height. Proposition 1.4.7 Let the polynomial ring R be standard graded. Given a graded ideal I and homogeneous elements f 1 , . . . , fr ∈ I , let J be the ideal generated by in( f 1 ), . . . , in( fr ). Suppose that (a) dim R/I = dim R/J , (b) e(R/I ) = e(R/J ), (c) J is unmixed. Then J = in(I ) (and I is unmixed as well).

26

1 Gröbner Bases, Initial Ideals and Initial Algebras

Proof First we note that dim R/ in(I ) = dim R/I and e(R/I ) = e(R/ in(I )) since they have the same Hilbert functions. In other words, we can replace R/I by R/ in(I ) in (a) and (b). The multiplicity of a residue class ring R/H with respect to an arbitrary graded ideal H coincides with its “local” multiplicity e(m, R/H ) where m is the graded maximal ideal of R; see [52, Sect. 4.7]. Therefore we can apply the associativity formula [52, 4.7.8] for multiplicity: e(R/H ) =



((R/H )p )e(R/p)

p⊃H,dim R/p=dim R/H

where we use  to denote the length of a module and p denotes a prime ideal. Apply the associativity formula to R/J and R/ in(I ). Since J ⊂ in(I ), (a) implies that all prime ideals p that appear in the summation for in(I ) also appear for J . Moreover, ((R/J )p ) ≥ ((R/ in(I )p ). The equality in (b) can only hold if all the lengths are equal and the same summands appear for both ideals. This means that in(I ) and J have the same minimal prime ideals of maximal dimension and the same primary components with respect to them. But (c) guarantees that J is their intersection. It is impossible that other primary components appear in the decomposition of in(I ) since then in(I ) would be properly contained in J , a contradiction. That I is unmixed along with in(I ) will be stated explicitly in Corollary 2.1.4. 

1.5 Initial Subspaces for Weights. Homogenization The use of initial ideals and initial subalgebras in the structural theory of ideals I and algebras A depends on deformation arguments that allow the transfer of homological, arithmetical and enumerative invariants from the discrete initial objects in(I ) and in(A) to I and A themselves. The deformations are based on weights and initial objects defined by them. As before we consider the polynomial ring R = K [X 1 , . . . , X n ]. A weight vector on R is a vector w ∈ Nn with positive entries. We have already encountered such vectors in the preceding section where they are used to define a grading. They do it here as well, but play another role. Nevertheless we keep the terminology developed  for gradings by weight vectors. The w-initial term or form of f = aμ μ is initw ( f ) =



aμ μ.

w(μ)=w( f )

We will often use that initw ( f g) = initw ( f ) initw (g) for f, g ∈ R. Note that w defines a K -algebra homomorphism ϕ : R → K [Y ], ϕ(X i ) = Y wi , and in conformity with our earlier notation we have init w ( f ) = initϕ ( f ). Clearly f is w-homogeneous if and only if f = initw ( f ).

1.5 Initial Subspaces for Weights. Homogenization

27

As a simple example let us consider f = X 2 − X Y + Z 2 ∈ K [X, Y, Z ]. For the weight vector w = (2, 1, 2) one has initw ( f ) = X 2 + Z 2 . If we change w to (2, 2, 1), then initw ( f ) = X 2 − X Y , and finally initw ( f ) = X 2 for w = (4, 2, 1). For a K -vector subspace V we let inw (V ) denote the w-initial subspace generated by all initial forms init w ( f ), f ∈ V . If A is a subalgebra and I ⊂ A an ideal in A, then inw (A) is a subalgebra and inw (I ) is an ideal in inw (A). This follows from the multiplicativity of initw .

Lifting of Syzygies As in the case of monomial orders it is useful to have a criterion by which one can decide whether inw (I ) = (initw ( f 1 ), . . . , init w ( f m )) for the ideal I = ( f 1 , . . . , f m ). It is not surprising that this property again relies on the lifting of syzygies. In this subsection we simply write init for initw and in for inw . Let F = R m . We have R-linear maps ϕ : F → R, ψ : F → R,

ϕ(ei ) = f i , ψ(ei ) = init( f i ).

Clearly, Im(ϕ) = I , Im(ψ) = in(I ). For x = (x1 , . . . , xm ) ∈ F, we set w(x) = max w(xi f i ). i

Then we can assign x an initial term init ϕ (x) ∈ F by  initϕ (x)i =

init(xi ) if w(xi f i ) = w(x), 0 else,

i = 1, . . . , m.

Clearly s ∈ Ker ϕ

=⇒

init(s) ∈ Ker ψ.

As in the case of monomial orders, we say that s ∈ Ker ϕ lifts t ∈ Ker ψ if initϕ (t) = initϕ (s). Proposition 1.5.1 Let f 1 , . . . , f m ∈ R and I = ( f 1 , . . . , f m ). Then the following are equivalent:   (a) in(I ) = init( f 1 ), . . . , init( f m ) ; (b) for every g ∈ I there exists x ∈ F such that g = ϕ(x) and w(x) = w(g); (c) every t ∈ Ker ψ can be lifted to a syzygy s ∈ Ker ϕ. The proof is essentially a repetition of the proof of Proposition 1.2.9 that we leave to the reader. As an analog of Proposition 1.2.12 we obtain:

28

1 Gröbner Bases, Initial Ideals and Initial Algebras

Proposition 1.5.2 Let R = K [X 1 , . . . , X n ], I ⊂ R an ideal and g1 , . . . , gr ∈ R. If sequence onR/ in(I ), then init(g1 ), . . . , init(gr ) are a regular  g1 , . . . , gr are a reg  ular sequence on R/I , and in I + (g1 , . . . , gr ) = in(I ) + init(g1 ), . . . , init(gr ) . Like in the proof of Proposition 1.2.12 one uses induction. It is of course no longer possible to work with divided Koszul syzygies. After the choice of a minimal system of elements f 1 , . . . , f m such that in(I ) = (init( f 1 ), . . . , init( f m )) one obtains the syzygies of init( f 1 ), . . . , init( f m ), init(g) (with g = g1 ) from the syzygies of init( f 1 ), . . . , init( f m ) and the Koszul syzygies of init( f i ) and g, i = 1, . . . .m. In the same manner one can transfer the theorems on initial algebras and Sagbi bases to weights.

From Subquotients to Quotients Many structural results about algebras of type A/J , where A is a subalgebra of a polynomial ring R and J an ideal in A, are based on their relation to inw (A)/ inw (J ). The following lemma helps us to replace the subquotient A/J by a true quotient of a polynomial ring: Lemma 1.5.3 Let A be a K -subalgebra of R and J an ideal of A. Assume that initw (A) is generated by initw ( f 1 ), . . . , initw ( f k ) with f 1 , . . . , f k ∈ A. Let P = K [Y1 , . . . , Yk ] and take presentations ϕ : P → A/J

and ψ : P → inw (A)/ inw (J )

defined by the substitutions ϕ(Yi ) = f i mod(J ) and ψ(Yi ) = initw ( f i ) mod(inw (J )). Set v = (w( f 1 ), . . . , w( f k )) ∈ Nk . Then inv (Ker ϕ) = Ker ψ. elements init v ( p) with p ∈ Proof As a vector space, inv (Ker ϕ) is generated   by the ei gj Y + b Y where v(Y ei ) = Ker ϕ. Set u = v( p). Then we may write p = a i j   gj ei gj u and v(Y ) < u. The image F = ai f + b j f belongs to J , and, hence, initw (F) ∈ inw (J ). The weight vector v is chosen in such a way that for every exponent vector h  one has v(Y1h 1 · · · Ykh k ) = w( f 1h 1 · · · f kh k ) . It follows that initw (F) = ai initw ( f ei ). Thus initv ( p) ∈ Ker ψ, and this proves the inclusion ⊂. For the converse inclusion we must find a system of generators of Ker ψ that is contained in inv (Ker ϕ). To this end, we lift ϕ and ψ to presentations ρ:P→A

and

σ : P → inw (A),

mapping Yi to f i and to initw ( f i ), respectively. Take a system S1 ⊂ P of vhomogeneous generators of the ideal Ker σ of P and a system S2 ⊂ inw (A) of w-homogeneous generators of the ideal inw (J ) of inw (A).

1.5 Initial Subspaces for Weights. Homogenization

29

Every s ∈ S2 , being w-homogeneous of  degree u = w(s), is of the form s = J . Then s = ai f ei + b j f g j with w( f ei ) = u and w( f g j ) < inw (s ), with s ∈  ei ). We choose the σ -preimage of s suggested by the u. Therefore s = ai initw ( f  given representation, i.e., ts = ai Y ei . Then the set S1 ∪ {ts : s ∈ S2 } generates the ideal Ker ψ.  s ∈ S2 and s as above, the canonical preimage of s , i.e. t = ai Y ei +  For all b j Y g j is in Ker ϕ, and one has ts = initv (t) ∈ inv (Ker ϕ). (Ker ϕ) for s ∈ S1 . Every s ∈ S1 is homogeneous, It remains to show that s ∈ inv  ei ei s ∈ Ker σ means say of degree u, and hence s =  ai Yei with  v(Yg j ) = u. That  ei gj a f = b f with w( f ) < u by Lemma ai initw ( f ) = 0. Therefore i j   1.5.4 below. That is, s = ai Y ei − b j Y g j is in Ker ρ. In particular, s ∈ Ker ϕ  and s = initv (s ). For monomial orders the next lemma, already used in the proof above, is part of Proposition 1.3.7, and the argument in the proof is essentially the same. Nevertheless we repeat it. (The finite generation of inw (A) is inessential.) Lemma 1.5.4 Let A be a K -subalgebra of R. Assume that inw (A) is generated by initw ( f 1 ), . . . , initw ( f k ) with f 1 , . . . , f k ∈ A. Then every F ∈ A has a representation F=



ai f ei

where ai ∈ K \ {0} and w(F) ≥ w( f ei ) for all i. Proof By decreasing induction on w(F). The case w(F) = 0 being trivial, we assume w(F) > 0. Since F ∈ A, we have initw (F) ∈ inw (A) = K [initw ( f 1 ), . . . , initw ( f k )]. Since initw (F) is a w-homogeneous element of the w-graded algebra inw (A), we may write  initw (F) = ai initw ( f ei ) where w(init w ( f ei )) = w(initw (F)) for all i. We set F1 = F − by induction since w(F1 ) < w(F) if F1 = 0.



ai f ei and conclude 

The Approximation of Monomial Orders by Weight Vectors In order to apply the previous results to initial objects defined by monomial orders we have to approximate such orders by weight vectors. This is indeed possible, provided only finitely many monomials must be considered. As a preparatory step we characterize those monomials μ in the support of a polynomial f that can appear as initial monomials with respect to a monomial order. The Newton polytope P f of f is the polytope in Rn spanned by the exponent vectors of the monomials μ ∈ supp( f ). For the notion of “polytope” and the elementary facts of polyhedral geometry that are used in the following we refer the reader to Bruns and Gubeladze [50].

30

1 Gröbner Bases, Initial Ideals and Initial Algebras

Fig. 1.1 The separation of P and Q

Lemma 1.5.5 Let f ∈ R be a polynomial. Then the following are equivalent for a monomial μ ∈ supp( f ): (a) there exists a monomial order < on R such that μ = in< ( f ); (b) the exponent vector y of μ is a vertex of P f and P f ∩ (y + Rn+ ) = {y}; (c) there exists a weight w such that μ = inw ( f ). Proof (a) =⇒ (b): Let supp( f ) \ {μ} = {ν1 , . . . , νk }and z i ,i = 1, . . . , k, be the exponent vector of νi . We set P = P f and Q = y + Rn+ . Figure 1.1 illustrates these choices. We must prove that P ∩ Q = {y}. If not, then P ∩ Q contains a rational point x = y. This follows since P ∩ Q is the intersection of rational polyhedra. After multiplication with a positive common denominator of the coordinates of x we can assume that x ∈ Nn : both P and Q are replaced by homothetic images; in our original monomial setting this means raising all monomials to the same power. Let ξ be the monomial with exponent vector x. Since x ∈ y + Rn+ and x = y, we have μ < ξ . On the one hand, ξ > μ > νi , i = 1, . . . , k in the monomial order. On the other z 1 , . . . , z k , with hand, by the definition of  P, x is a convex combination of z 0 = y, k k (ai /bi )z i with ai , bi ∈ Z+ and i=0 ai /bi = 1. rational coefficients: x = i=0 Passing to monomials, we obtain a binomial relation ξ b = μa0 ν1a1 · · · νkak , contradicting the fact that ξ is larger than every factor on the right hand side, and the numbers of factors coincide. To sum up: we have proved that P ∩ Q = {y}. The same argument that has just implied P ∩ Q = {y}, shows that y is not a convex combination of z 1 , . . . , z k . Hence y is a vertex of P. (b) =⇒ (c): The hypothesis (b) says that P and Q intersect in the common face {y}. By a classical separation theorem of convex geometry there exists a rational hyperplane H such that y ∈ H , P \ {y} is in the interior of one of the two halfspaces defined by H and Q \ {y} lies in the interior of the other halfspace. For example, see [50, Theorem 1.32]. The hyperplane H is defined by a vector w ∈ Zn and w0 ∈ Z via the equation n H = {u : i=1 wi u i = w0 }. n wi u i ≥ w0 for all u ∈ Q. Then Replacing w by −w if necessary, we have i=1 the definition of Q implies that wi > 0 for all i = 1, . . . , n. So, since {y} = P ∩ H , w is our desired weight vector.

1.5 Initial Subspaces for Weights. Homogenization

31

(c) =⇒ (a): This follows immediately from the refinement of the weight w by a monomial order that will be discussed in the next subsection.  Lemma 1.5.6 Let < be a monomial order on R. (a) Let {(μ1 , ν1 ), . . . , (μk , νk )} be a finite set of pairs of monomials such that μi > νi for all i. Then there exists a weight vector w ∈ Nn>0 such that w(μi ) > w(νi ) for all i. (b) Let A be a K -subalgebra of R and I1 , . . . , Ih ideals of A. Assume that in(A) is finitely generated as a K -algebra. Then there exists a weight vector w ∈ Nn>0 such that in(A) = inw (A) and in(Ii ) = inw (Ii ) for all i = 1, . . . , h. Proof (a) One can assume μ1 = · · · = μk = μ. To this end we multiply both μ1 and ν1 by μ2 · · · μk , then μ2 and ν2 by μ1 μ3 · · · μk etc. This process affects neither the hypothesis nor the conclusion, as follows easily from the monotonicity of the monomial order and the linearity of weights. Now we set f = μ + ν1 + · · · + νk and use the implication (a) =⇒ (c) of Lemma 1.5.5. (b) LetF0 be a finite Sagbi basis of A, let Fi be a finite Gröbner basis of Ii and set F = i Fi . Consider the set U of pairs of monomials (in( f ), μ) where f ∈ F and μ is any noninitial monomial of f . Since U is finite, by (a) there exists w ∈ Nn>0 such that initw ( f ) = in( f ) for every f ∈ F. We show w has the desired property. Set V0 = A and Vi = Ii . By construction the (algebra for i = 0 and ideal for i > 0) generators of the in(Vi ) belong to inw (Vi ) so that in(Vi ) ⊂ inw (Vi ). But then, by  Proposition 1.5.7(b), we may conclude that in(Vi ) = inw (Vi ). There is not much point in finding explicit weight vectors in the situation of the lemma, and if one does, one may be forced to choose astronomical weights. As an example let us find a weight vector that picks the largest monomial in every subset of monomials of degree ≤ d in K [X, Y, Z ] for the lexicographic order determined by X > Y > Z . We give weight 1 to Z . Since Y > Z d , we give weight d + 1 to Y . Since X > Y d and w(Y d ) = d(d + 1), we must choose w(X ) = d(d + 1) + 1. It is not hard to check that w = (d(d + 1) + 1, d + 1, 1) indeed solves our problem.

The Refinement of Weights by Monomial Orders As we have just seen, one can approximate a monomial order by a weight vector. For the definition of a monomial order that is compatible with a weight, the degrees w(μ) allow us to compare monomials, but, unless n = 1, there are always monomials μ = ν with w(μ) = w(ν). In other words, the weight vector alone does not define a monomial order. We need a tiebreaker given by a monomial order < on R. It refines the partial order defined by w as follows:  μ 0 a weight vector. If inw (I ) is unmixed, so is I . Proof By Propositions 1.6.2 and 2.1.3, height I = height inw (I ) = Bigheight inw (I ) ≥ Bigheight I ≥ height I, therefore the inequalities are equalities, from which we conclude.



2.1 Minimal and Associated Primes

45

Minimal Prime Ideals Now we want to study the behavior of the minimal primes under Gröbner deformations. Let R = K [X 1 , . . . , X n ], w = (w1 , . . . , wn ) ∈ N>0 N n , and P = R[t] where t is a homogenizing variable. We can supply P with the grading given by deg(X i ) = wi and deg(t) = 1. We say that an element/ideal of P is (w, 1)-homogeneous if it is homogeneous with respect to this grading. Clearly homw ( f ) ∈ P is (w, 1)homogeneous for any f ∈ R and homw (I ) ⊂ P is (w, 1)-homogeneous for any ideal I ⊂ R. It is useful to define the dehomogenization homomorphism: π : P −→ R, F(X 1 , . . . , X n , t) → F(X 1 , . . . , X n , 1). We list some easy properties that we will use freely: (a) For all f ∈ R we have π(homw ( f )) = f . In particular, π(homw (I )) = I . (b) Let F ∈ P \ t P be a (w, 1)-homogeneous polynomial. Then homw (π(F)) = F; moreover, for all r ∈ N, if G = t r F we have homw (π(G))t r = G. F ∈ homw (I ) there exist f 1 , . . . , f m ∈ (c) If F ∈ homw (I ), then π(F) ∈ I ; in fact, if m ri homw ( f i ), and hence I and r1 , . . . , rm ∈ P such that F = i=1 π(F) =

m  i=1

π(ri )π(homw ( f i )) =

m 

π(ri ) f i ∈ I.

i=1

Lemma 2.1.5 Let I and J be ideals of R = K [X 1 , . . . , X n ] and w ∈ Nn>0 . Then homw (I ∩ J ) = homw (I ) ∩ homw (J ). I is a √ prime  ideal √ if and only if homw (I ) is prime. homw I = homw (I ). I = J if and only if homw (I ) = homw (J ). p1 , . . . , ps are the minimal primes of I if and only if homw (p1 ), . . . , homw (ps ) are the minimal primes of homw (I ). (f) height I = height homw (I ). (g) bigheight I = bigheight homw (I ). (a) (b) (c) (d) (e)

Proof (a) Clearly homw (I ∩ J ) ⊂ homw (I ) ∩ homw (J ). To check the other inclusion, since homw (I ) ∩ homw (J ) is a (w, 1)-homogeneous ideal of P = R[t], it is enough to prove the membership for (w, 1)-homogeneous elements: let F ∈ P be a (w, 1)-homogeneous polynomial in homw (I ) ∩ homw (J ). If r is the largest natural number such that t r |F, then F = t r homw (π(F)) ∈ homw (I ∩ J ). (b) Let f, g ∈ R such that f g ∈ I . Then homw ( f ) homw (g) ∈ homw (I ). If homw (I ) is prime, then either homw ( f ) or homw (g) belongs to homw (I ). Say homw ( f ) ∈ homw (I ). Then f = π(homw ( f )) ∈ I . On the other hand, let F, G ∈ P be (w, 1)-homogeneous polynomials such that F G ∈ homw (I ). Then π(F)π(G) = π(F G) ∈ I , and if I is prime, either π(F) or

46

2 More on Gröbner Deformations

π(G) belongs to I . Say π(F) ∈ I . If r is the maximum natural number such that t r |F, so F = t r homw (π(F)) ∈ homw (I ). (c) It is enough to show that, if I is radical, so is homw (I ), and this follows from (a) and (b). (d) If homw (I ) = homw (J ), then I = π(homw (I )) = π(homw (J )) = J . The other direction is clear by definition. √ s (e) If ps 1 , . . . , ps are the√minimal primes of I , then i=1 pi = I . So (a) and (c) imply i=1 homw (pi ) =  homw (I ). Then by (b)all minimal primes of homw (I ) are contained in the set homw (p1 ), . . . , homw (ps ) . Moreover, by (d) all the primes in this set are minimal for homw (I ). (p1 ), . . . , homw (ps ) are the minimal primes of Conversely, if homw√ w (I hom  ), s s homw (pi ) = homw (I ). So by (a) and (c) follows that homw i=1 pi = then i=1 √ √  s pi = I , so using (b) we have that all the minimal primes homw I ; by (d) i=1 of I are contained in the set {p1 , . . . , ps }. Again using (d), the primes in this set are all minimal for I . (f) If I is prime, this follows from (b) and (d); (c) allows to extend it in general. (The fact height I = height homw (I ) follows also from the proof of Proposition 1.6.2 (a). (g) follows from (e) and (f). Proposition 2.1.6 Let I be an ideal of R = K [X 1 , . . . , X n ] and w ∈ Nn>0 . Then bigheight inw (I ) ≤ bigheight I. Proof Let P = R[t] and S = P/ homw (I ). Then R/ inw (I ) ∼ = S/t S by Proposition 1.6.1, so if p ∈ Spec R is minimal over inw (I ), then there must be q ∈ Spec P minimal over homw (I ) + t P such that p ∼ = q/t P. By Krull’s principal ideal theorem height q ≤ bigheight homw (I ) + 1. So, by Lemma 2.1.5(g), height q ≤ bigheight I + 1. Since t is a nonzerodivisor on S by Proposition 1.6.1, height p < height q, so we are done.   The inequality  in Proposition 2.1.6 is strict in general. Consider the ideal X (X + Y ), X (Y + Z ) in K [X, Y, Z ] endowed with a monomial order in which X > Y > Z . Then in(I ) = (X 2 , X Y ), and 1 = bigheight in(I ) < bigheight I = 2. As a corollary of Proposition 2.1.6, we infer a good property of inw (I ) from the same property of I . √ n Corollary √ 2.1.7 Let I be an ideal of K [X 1 , . . . , X n ] and w ∈ N>0 . If I is unmixed, so is inw (I ).

Radical Initial Ideals The following is an immediate consequence of Propositions 2.1.3 and 2.1.6:

2.1 Minimal and Associated Primes

47

Corollary 2.1.8 Let I be an ideal of K [X 1 , . . . , X n ], w ∈ Nn>0 , and assume inw (I ) is radical. Then Bigheight I = Bigheight inw (I ). In particular, I is unmixed if and only if inw (I ) is unmixed. Corollary 2.1.8 shows that the relationship between I and inw (I ) is tighter when inw (I ) is radical. We will see in Sect. 2.4 that, if inw (I ) is a radical monomial ideal, then such a relationship is even tighter. (A monomial is squarefree if it is the product of pairwise different indeterminates, and since a monomial ideal is radical if and only if it is generated by squarefree monomials, we will also call it squarefree.) The latter class of ideals will often come up in this book: we think this is a good place to introduce them and their relation with simplicial complexes. A simplicial complex  on a finite set V is a collection of subsets of the set V such that F ∈ , G ⊂ F =⇒ G ∈ . We also say that  is a simplicial complex on |V | vertices, and often by doing so we identify V and {1, 2, . . . , |V |}. The elements of  are called faces, and the faces of  maximal with respect to inclusion are called facets. The dimension of a face F ∈  is equal to the cardinality of F minus 1, and the dimension of  is the maximum dimension of its faces. A simplicial complex is pure if its facets have the same dimension. Example 2.1.9 Simplicial complexes can be depicted as shown in Fig. 2.1. Here 1 and 2 are the following simplicial complexes on 4 and 5 vertices: 1 = {∅, {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}, {3, 4}}, 2 = {∅, {1}, {2}, {3}, {4}, {5}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}, {3, 4}, {3, 5}, {4, 5}, {3, 4, 5}}. Note that the facets of 1 are {1, 2, 3} and {3, 4}, so 1 is a simplicial complex of dimension 2 which is not pure. The facets of 2 instead are {1, 2, 3} and {3, 4, 5}, so 2 is pure of dimension 2. There is a 1-1 correspondence between squarefree monomial ideals I ⊂ R = K [X 1 , . . . , X n ] and simplicial complexes on n vertices , given by the rule i∈F

Fig. 2.1 Two simplicial complexes

Xi ∈ I

⇐⇒

F∈ / ,

F ⊂ {1, . . . , n}.

48

2 More on Gröbner Deformations

This correspondence is known as the Stanley–Reisner correspondence: Usually, if  is known, I is denoted by I , while if I is known,  is denoted with (I ). One calls I the face ideal or Stanley–Reisner ideal of . Furthermore the Stanley–Reisner ring R/I will be denoted by K []. It is easy to see that, if  is a simplicial complex on n vertices with set of facets {F1 , . . . , Fs }, then Min(I ) =



  X i : i ∈ {1, . . . , n} \ F j : j = 1, . . . , s .

In particular, dim K [] = dim  + 1, and  is a pure simplicial complex if and only if I is an unmixed ideal. Proposition 2.1.10√Let < be a monomial order on R = K [X 1 , . . . , X n ] and I ⊂ R an ideal. Then ( in(I )) is a pure simplicial complex provided I is an unmixed ideal. Moreover, if in(I ) is a squarefree monomial ideal, (in(I )) is a pure simplicial complex if and only if I is an unmixed ideal. Proof By Lemma 1.5.6(b), exists w ∈ Nn>0 such that in(I ) = inw (I ). So Corol√ there √ lary 2.1.7 implies that in(I ) = inw (I ) is an unmixed ideal, implying the first part of the statement. For the second part, use Corollary 2.1.8.  We wish to conclude this section by showing a further special feature of ideals having a radical initial ideal. Lemma 2.1.11 Let I1 , . . . , Ic be ideals of K [X 1 , . . . , X n ] and let w either an element of Nn>0 or a monomial order. Then one has:   (a) inw (i Ii ) ⊂ i inw (Ii ), (b) inw ( i Ii ) ⊃ i inw (Ii ), (c) inw ( i Ii ) ⊃ i inw (Ii ), (d) inw (I1 ∩ I2 ) = inw (I1 ) ∩ inw (I2 ) if and only if inw (I1 + I2 ) = inw (I1 ) + inw (I2 ), Proof We prove the statements for a w ∈ Nn>0 . The proofs for a monomial order are analogous. We deal firstly with (a): inw ( i Ii ) is generated by initw ( f ) with f ∈ ∩i Ii . As init w ( f ) ∈ inw (Ii ) for all i, the desired inclusion follows. Statements (b) and (c) are proved by similar arguments. Now we prove (d). If I1 and I2 are homogeneous ideals then the statement follows easily comparing Hilbert functions (Proposition 1.4.3). For nonhomogeneous ideals, a slightly more complicated argument is needed. For ⇒ we observe that the inclusion inw (I1 + I2 ) ⊃ inw (I1 ) + inw (I2 ) holds by (b). For the other inclusion, let m = initw ( f + g) for some f ∈ I1 , g ∈ I2 . If degw ( f ) = degw (g), then m = initw ( f ) or m = initw (g). Otherwise, if init w ( f ) = − init w (g), m = initw ( f ) + initw (g). If initw ( f ) = − init w (g), init w ( f ) ∈ in(I1 ) ∩ inw (I2 ) = inw (I1 ∩ I2 ). Hence, there exists a polynomial h ∈ I1 ∩ I2 such that initw (h) = initw ( f ). It follows that f + g = f − h + h + g = ( f − h) + (g + h) = f  + g  , for some f  ∈ I1 and g  ∈ I2 , with degw ( f ) > degw ( f  ) and degw (g) > degw (g  ). Since m = initw ( f  + g  ), we can repeat the procedure on f  and g  until we find ˜ and initw ( f˜) = − init w (g). ˜ f˜ ∈ I1 and g˜ ∈ I2 such that m = initw ( f˜ + g)

2.1 Minimal and Associated Primes

49

Finally, for ⇐, we observe that the inclusion inw (I1 ∩ I2 ) ⊂ inw (I1 ) ∩ inw (I2 ) holds by (a). For the other inclusion, let m = initw ( f ) = initw (g), for some f ∈ I1 and g ∈ I2 . Let { f 1 , . . . , f a } and {g1 , . . . , gb } be Gröbner bases of I1 and I2 , respectively. By hypothesis, { f 1 , . . . , f a , g1 , . . . , gb } is a Gröbner basis of I1 + I2 . Therefore, the reduction of f − g by f 1 , . . . , f a , g1 , . . . , gb is zero. Hence, f − g =  a b u f + i=1 i i j=1 v j g j , where, for every i = 1, . . . , a, and j = 1, . . . , b, degw (u i f i ) ≤ degw ( f − g) < degw f and degw (v j g j ) ≤ degw ( f − g) < degw f. Then, setting h= f −

a 

u i fi = g +

i=1

b 

vjgj,

j=1

we conclude that h ∈ I1 ∩2 and initw (h) = m ∈ inw (I1 ∩ I2 ).



Proposition 2.1.12 Let I, J ⊂ K [X 1 , . . . , X n ] be ideals and w ∈ Nn>0 a weight. If inw (I ∩ J ) is radical, then inw (I ∩ J ) = inw (I ) ∩ inw (J ) and inw (I + J ) = inw (I ) + inw (J ). Proof We prove only the first equality, as this is equivalent to the second one by Lemma 2.1.11. The inclusion in(I ∩ J ) ⊂ in(I ) ∩ in(J ) is always true. For the other inclusion, if m = init( f ) = init(g) for f ∈ I , g ∈ J , then m 2 = init( f g) ∈ in(I ∩ J ). Since in(I ∩ J ) is radical, we have that m ∈ in(I ∩ J ).  Example 2.1.13 Let R = K [X i j : i, j = 1, 2, 3] and ⎛

⎞ X 11 X 12 X 13 X = ⎝ X 21 X 22 X 23 ⎠ . X 31 X 32 X 33 For 1 ≤ a < b ≤ 3 and 1 ≤ c < d ≤ 3 set  [a b | c d] = det

 X ac X ad . X bc X bd

Let I ⊂ R be the ideal generated by [1 2 | c d] such that 1 ≤ c < d ≤ 3, and J ⊂ R be the ideal generated by [a b | 2 3] such that 1 ≤ a < b ≤ 3. Then I ∩ J = ([1 2 | 2 3], det(X )). The inclusion ⊃ is obvious, the other will be clear later: as a consequence of the theory of determinantal ideals, both I and J are prime ideals of height 2 and multiplicity 3 (see Sect. 4.1). On the other hand ([1 2 | 2 3], det(X )) is a complete intersection ideal of height 2 and multiplicity 2 · 3 = 6, so the equality follows by the additivity of the multiplicity (see Proposition 1.4.7 for this type of argument).

50

2 More on Gröbner Deformations

Now, by choosing the lexicographical monomial order with X i j < X pq if i < p or i = p and j < q, we have that in(I ∩ J ) = (X 12 X 23 , X 11 X 22 X 33 ) is radical, so by Proposition 2.1.12 in(I + J ) = in(I ) + in(J ).

2.2 Connectedness We recall that, given a ring R, the closed subsets of the set of prime ideals Spec R endowed with the Zariski topology are of the form V (I ) = {p ∈ Spec R : p ⊃ I } for some ideal I ⊂ R. If I ⊂ R is an ideal, we also set U (I ) = Spec R \ V (I ) = {p ∈ Spec R : p ⊃ I }. Note that V (I ) = ∅ √ ⇐⇒ U√(I ) = Spec R ⇐⇒ I = R and that U (I ) = ∅ ⇐⇒ V (I ) = Spec R ⇐⇒ I = (0). For the next definition, we adopt the standard definitions that the Krull dimension of the trivial ring {0} equals −∞, and that the empty set is not connected. Before introducing the main definition of this section let us see concretely what it means that Spec R is not connected. Lemma 2.2.1 Let R be a Noetherian ring and X = Spec R. Then X is not connected if and only if R has nontrivial idempotents. In particular, a domain has a connected spectrum. Proof Assume X is not connected. Then there are two closed nonempty subsets A, B ⊂ X such that A ∩ B = ∅ and A ∪ B = X . Let I and J be ideals of R such that A = V (I ) and B = V (J ). We have: (a) I, J  R; (b) √ I + J = R:√ (c) I ∩ J = (0). By replacing I and J with high enough powers we can assume that I ∩ J = (0). Therefore there exist x ∈ I , y ∈ J such that x + y = 1 (i.e. x = 1 − y). Note that, since both x and y are different from 1, they are also different from 0. Furthermore x y = 0. So x 2 = x and x is neither 0 nor 1. Hence R contains a nontrivial idempotent. On the other hand, if R contains a nontrivial idempotent x, then with I = (x) and J = (1 − x), we have that A = V (I ) and B = V (J ) are nonempty subsets of X such that A ∩ B = ∅ and A ∪ B = X . Hence Spec R is not connected. 

Connectedness in Codimension k Definition 2.2.2 Let R be a d-dimensional Noetherian ring and k ∈ N. We say that R is connected in codimension k if U (I ) is connected (with respect to the topology inherited from Spec R) whenever I ⊂ R is an ideal such that dim R/I < d − k.

2.2 Connectedness

51

Remark 2.2.3 Let us see some basic properties of this notion: (a) If R is connected in codimension k, then R is connected in codimension h for all h ≥ k. (b) Spec R is connected if and only if R is connected in√codimension d. (c) R is connected in codimension k if and only if R/ (0) is connected in codimension k. (d) If R is a domain, then R is connected in codimension 0. Indeed, if I ⊂ R is a nonzero ideal, then U (I ) = ∅. Any nonempty open subset U ⊂ U (I ) is of the form U = U (J ) ∩ U (I ) where J is a nonzero ideal of R. Since R is a domain, U (J ) is dense in Spec R, and so U is dense in U (I ). Therefore a nonempty open subset U ⊂ U (I ) is closed if and only if U = U (I ), namely U (I ) is connected. Proposition 2.2.4 Let R be a d-dimensional Noetherian ring and k ∈ N. If Min R = {p1 , . . . , ps }, the following are equivalent: (a) R is connected in codimension  k;   (b) dim R/ i∈A pi + j∈B p j ≥ d − k for all nontrivial partitions (A, B) of {1, . . . , s}. (c) for all p, q ∈ {1, . . . , s}, there exist m ∈ N and p = i 0 , i 1 , . . . , i m = q all belonging to {1, . . . , s} such that dim R/(pir + pir −1 ) ≥ d − k for r = 1, . . . , m. Proof (a) =⇒  (b): Let (A, B) be a nontrivial partition of {1, . . . , s}, and let I A =  p , I = B i∈A i i∈B pi and I = I A + I B . Then V (I A ) ∩ U (I ) and V (I B ) ∩ U (I ) provide a decomposition of U (I ). So dim R/I ≥ d − k. (b) =⇒ (a): Let I ⊂ R be an ideal such that U (I ) is not connected. Then there are ideals I1 , I2 ⊂ R such that C1 = V (I1 ) ∩ U (I ) and C2 = V (I2 ) ∩ U (I ) provide a decomposition of U (I ). In particular: √ (i) I ⊂ I1 + I2 (⇐⇒ C1 ∩ C2 = ∅). (ii) Any p ∈ Min R ∩ U (I ) contains I1 ∩ I2 (⇐⇒ C1 ∪ C2 = U (I )). (iii) 0 < | Min R ∩ C1 | < | Min R ∩ U (I )| (⇐⇒ ∅  C1  U (I )). . , s} \ A, we get So letting A = {i ∈ {1, . . . , s} : pi ∈ C1 } and B = {1, . .  √ a nontrivial = p and I = p , then I1 ⊂ I A and partition (A, B) of {1, . . . , s}. If I A i B i i∈A i∈B √ √ √ I2 ∩ I ⊂ I B . Note that I ⊂ I1 + I2 implies I ⊂ I1 + I2 ∩ I : indeed if x ∈ I is such that x n = x1 + x2 where xi ∈ Ii , x n+1 = x x1 + x x2 ∈ I1 + I2 ∩ I . So we have   I ⊂ I1 + I2 ∩ I ⊂ I A + I B , which yields dim R/I ≥ dim R/(I A + I B ) ≥ d − k. (b) =⇒ (c):  for all  If p, q ∈ {1, . . . , s} contradict (c), let us define A0 = { p} and, i ≥ 1, A i = j ∈ {1, . . . , s} : dim R/(p j + pt ) ≥ d − k for some t ∈ Ai−1 . Then set A = i∈N Ai and B = {1, . . . , s} \ A. Notice that B is nonempty, since q belongs to it: so (A, B) is a nontrivial partition of {1, . . . , s}, and by construction the dimen  sion of R/ i∈A pi + i∈B pi is less than d − k, that contradicts (b). (c) =⇒ (b): Bycontradiction,  let (A, B) be a nontrivial partition of {1, . . . , s} such that dim R/( i∈A pi + i∈B pi ) < d − k. Then for all i ∈ A and j ∈ B one

52

2 More on Gröbner Deformations

has dim R/(pi + p j ) < d − k. In particular, any pair ( p, q) ∈ A × B contradicts (c).  Example 2.2.5 Let R = K [X 1 , . . . , X 6 ]/J where J = J1 ∩ J2 ∩ J3 with J1 = (X 5 , X 6 ), J2 = (X 1 , X 2 , X 6 ) and J3 = (X 1 , X 2 , X 3 , X 4 ). Then dim R = 4, and denoting the residue class of Ji by pi ⊂ R for i = 1, 2, 3, we have        min dim R/ p1 + (p2 ∩ p3 ) , dim R/(p2 + p1 ∩ p3 ) , dim R/ p3 + (p1 ∩ p2 ) = 1, so, by Proposition 2.2.4, R is connected in codimension 3, but not in codimension 2. Notice that        max height p1 + (p2 ∩ p3 ) , height p2 + (p1 ∩ p3 )) , height p3 + (p1 ∩ p2 ) = 2 = 4 − 1, so in the definition of R being connected in codimension k we cannot replace “dim R/I < dim R − k” by “height I > k”. For the next result, let us recall that a Noetherian ring R is catenary if, for any prime ideals p  p , all maximal chains of prime ideals p = p0  p1  . . .  pm = p have the same length m. Every Cohen–Macaulay ring is catenary (see Bruns and Herzog [52, 2.1.12]), and clearly any quotient or localization of a catenary ring is catenary. Proposition 2.2.6 Let R be a d-dimensional catenary Noetherian ring satisfying the Serre condition (S2 ) and such that Spec R is connected. Then dim R/p = d for any p ∈ Ass R, and R is connected in codimension 1. Proof Set Min R = {p1 , . . . , ps }. Since R satisfies (S1 ), Min R = Ass R, so assume by contradiction that dim R/pi < d for some i ∈ {1, . . . , s}. Setting A = {i ∈ {1, . . . , s} : dim R/pi = d} and B = {1, . . . , s} \ A, we get a nontrivial partition (A, B) of {1, . . . , s}. Since R is connected in codimension d, byProposition 2.2.4there must be a maximal ideal containing I A + I B where I A = i∈A pi and I B = i∈B pi . So we can assume R is local and deduce by Hartshorne’s Connectedness√(See Eisenbud Theorem √ [116, 18.12]) that height(I A + I B ) ≤ 1 (since I A ∩ I B = (0) and I A , I B  (0)). Choose a prime ideal P ⊃ I A + I B of height 1. Then P must strictly contain p j for some j ∈ B, so dim R/P < d − 1. On the other hand P must also strictly contain pi for some i ∈ A, and since height P = 1, there is no prime ideal between pi and P. So, by the catenarity of R, dim R/P = d − 1, that is a contradiction. To see that R is connected incodimension 1, arguing similarly as above, one  can show that height( i∈A pi + i∈B pi ) = 1 for any nontrivial partition (A, B) of {1, . . . , s}. Since by the previous part i = d for all i ∈ {1, . . . , s}, we infer  dim R/p by the catenarity of R that dim R/( i∈A pi + i∈B pi ) = d − 1, and thus conclude that R is connected in codimension 1 by Proposition 2.2.4. 

2.2 Connectedness

53

Remark 2.2.7 A simplicial complex  on n vertices with facets F1 , . . . , Fs is strongly connected if, for all p, q ∈ {1, . . . , s}, there exist m ∈ N and i 0 = p, i 1 , . . . , i m = q ∈ {1, . . . , s} such that Fi j ∩ Fi j+1 is a codimension one face of  (i.e. dim Fi j ∩ Fi j+1 = dim  − 1) for j = 0, . . . , m − 1. (Note that a strongly connected simplicial complex must be pure; for a pure simplicial complex not strongly connected see Example 2.1.9). So by Proposition 2.2.4,  is strongly connected if and only if K [] is connected in codimension 1. Lemma 2.2.8 Let I be an ideal of R = K [X 1 , . . . , X n ], w ∈ Nn>0 and k ∈ N. If R/I is connected in codimension k, then R[t]/ homw (I ) is connected in codimension k. Proof Let p1 , . . . , ps be the minimal primes of I . Then homw (p1 ), . . . , homw (ps ) are the minimal primes of homw (I ) by Lemma 2.1.5(e). We conclude by Proposition 2.2.4 because for all i, j the dimension of R[t]/(homw (pi ) + homw (p j )) is at least  dim R[t]/ homw (pi + p j ) = dim R/(pi + p j ) + 1.

From a Graded Ring to Its Completion Before going ahead, we need to recall some basic notions and facts related to the completion of a ring. In a ring R a descending chain of ideals F : R = I0 ⊃ I1 ⊃ I2 ⊃ . . . is a filtration ∞if Ii I j ⊂ Ii+ j for all i, j ∈ N. The associated graded ring gr F (R) is defined as m=0 Im /Im+1 . The filtration F defines a linear topology on R: that is, a topology on R such that both the addition and the multiplication from R × R (endowed with the product topology) to R are continuous maps, and such that 0 has a fundamental system of neighborhoods consisting of ideals. In this case, F is the desired fundamental system of neighborhoods of 0, and the topology is obtained by considering as fundamental system of neighborhoods of an element r ∈ R the set r + F = {r + Im }m∈N . Given a filtration F on a ring R, we can thus consider the completion  R F . When the filtration is {I m }m∈N for some ideal I , the completion is called the I -adic completion of R. We will use the following facts:

(a) A filtration of ideals F = {Im }m∈N on R yields a filtration of ideals G = {Im∗ }m∈N on  R F , where Im∗ is the kernel of the projection from  R F = lim R/Im ⊂ ← ∗ are isomorphic RR/Im onto R/Im . As it turns out, Im /Im+1 and Im∗ /Im+1 modules for all m ∈ N, and gr (R) and gr ( R F ) are isomorphic rings. F

G

(b) Two filtrations F = {Im }m∈N and G = {Jm }m∈N on R are cofinal if, for all n ∈ N, there exist r, s ∈ N such that Ir ⊂ Jn and Js ⊂ In . In this case, the topologies G are isomorphic rings. defined by F and G on R are the same, so  R F and R (c) If R is Noetherian, its I -adic completion is Noetherian for any ideal I ⊂ R (see Matsumura [243, 8.12]).  is a faithfully (d) If (R, m) is a Noetherian local ring, then the m-adic completion R flat R-algebra [243, 8.14].

54

2 More on Gröbner Deformations

If the m-adic completion of a Noetherian local ring (R, m) is a domain, then R is a domain as well. In general, the converse fails, however it is true in the following case: Lemma 2.2.9 Let R be a positively graded finitely generated algebra over a field K  with homogeneous maximal ideal m, and denote by R m the mRm -adic completion of   Rm . If p is a graded prime ideal of R, then p Rm is a prime ideal of R m . In particular,  if R is a domain, then Rm is a domain as well. Proof If p is a graded prime ideal of R, then R/p is still a positively graded finitely m ∼ generated algebra over K with homogeneous maximal ideal m. Since (R/p) =    /p R [243, 8.11], it is enough to show that if R is a domain then R is a domain R m m m as well. Consider the filtration F = {Im Rm }m∈N on Rm , where the ideals Im ⊂ R are defined as Im = i≥m Ri . Obviously there is an isomorphism of K -algebras F denote the completion of R with respect to between R and gr F (Rm ). Now, let  Rm m F . One has that gr (R ) ∼ Rm the filtration F, and let G be the filtration {Im∗ }m∈N on  m = F   F F gr G Rm . By these considerations we can assert that gr G ( Rm ) is a domain; then  ∗  F is a domain as well (since Rm m Im = (0)). Hence, to conclude it is enough to  F ∼  check that Rm = Rm , which is indeed the case because the filtrations {I j Rm } j∈N and  {m j Rm } j∈N are cofinal. Corollary 2.2.10 Let R be a positively graded finitely generated algebra over a field  K with homogeneous maximal ideal m, and let R m denote the mRm -adic completion  of Rm . For all k ∈ N, R is connected in codimension k if and only if R m is connected in codimension k. Proof Let p ∈ Min R. Then p is graded [52, 1.5.6], and pRm ∈ Min Rm . Moreover,   by Lemma 2.2.9, p R m is a prime ideal of Rm and we claim that it is minimal. Assume   by contradiction there is a prime q ⊂ R m strictly contained in p Rm . Then q ∩ Rm is   which is pR by the faithfully flatness of R contained in the contraction of p R m m m  over Rm . Hence q ∩ Rm = pRm and extending to R we get a contradiction. m  On the other hand, if q ∈ Min R m , then q ∩ R is a prime ideal of R. Let p be a minimal prime ideal (in particular graded) of R contained in q ∩ R; by Lemma    2.2.9 p R m is a prime ideal of Rm contained in q, so q = p Rm . This shows that   = {p R : p ∈ Min R}. Min R m m     We have dim R/(p + p ) = dim R m /(p Rm + p Rm ) for all p, p ∈ MinR; since    Min R m = {p Rm : p ∈ Min R} and dim Rm = dim R, we conclude by Proposition 2.2.4. 

The Grothendieck Connectedness Theorem and Its Consequences We are now ready to prove the following version of the Grothendieck Connectedness theorem that will give the main result of this section. For the reader’s convenience,

2.2 Connectedness

55

we state the connectedness theorem in its original form given in [160, Exposé XIII, 2.1] (adapted to our notation): Theorem 2.2.11 Let (R, m) be a complete local ring of dimension d, k a positive integer, and f 1 , . . . , f m ∈ m. Suppose that R is connected in codimension k. Then R/( f 1 , . . . , f m ) is connected in codimension k + d − m − dim R/( f 1 , . . . , f m ). Grothendieck’s theorem is also proved in Brodmann and Sharp [33, 19.2.12]. We need the graded version. Corollary 2.2.12 Let R be a positively graded finitely generated algebra over a field K , k a positive integer and t ∈ R a homogeneous element of positive degree not contained in any minimal prime of R. If R is connected in codimension k, then R/t R is connected in codimension k.  Proof As usual let m be the homogeneous maximal ideal of R and R m the mRm  adic completion of Rm . By Corollary 2.2.10, Rm is a Noetherian complete local ring connected in codimension k. Moreover, as deduced in the proof of Corol  lary 2.2.10, Min R m = {p Rm : p ∈ Min R}. Hence t is not in any minimal prime     of Rm . So dim Rm /t Rm = dim R m − 1. Therefore, by the Connectedness Theo   ∼  rem of Grothendieck, Rm /t Rm is connected in codimension k. Since R m /t Rm =  (R/t R)m , using again Corollary 2.2.10 we conclude that R/t R is connected in codimension k.  Theorem 2.2.13 Let I be an ideal of R = K [X 1 , . . . , X n ], k a positive integer and w ∈ Nn>0 a weight vector. If R/I is connected in codimension k, then R/ inw (I ) is connected in codimension k. Proof By Lemma 2.2.8, since R/I is connected in codimension k, the positively graded finitely generated K -algebra S = R[t]/ homw (I ) is connected in codimension k as well. Notice that the residue class of t does not belong to any minimal prime of S by Proposition 1.6.1(a). So by Corollary 2.2.12 S/t S ∼ = R/ inw (I ) is connected in codimension k.  Corollary 2.2.14 Let < be a monomial order on R =√K [X 1 , . . . , X n ] and I ⊂ R an ideal. If R/I is connected in codimension 1, then ( in(I )) is a strongly connected simplicial complex. In particular, if I is prime or if it is homogeneous and R/I is √ Cohen–Macaulay, then ( in(I )) is strongly connected. Proof By Lemma 1.5.6 (b), there exists w ∈ Nn>0 such that in(I ) = inw (I ). So √ Theoinw (I ) rem 2.2.13 yields that R/ in(I ) is connected in codimension 1. Therefore R/ √ is connected in codimension 1 as well, so that ( in(I )) is strongly connected by Remark 2.2.7. For the second part, use Remark 2.2.3(d) and Proposition 2.2.6 to infer that in both cases R/I is connected in codimension 1.  √ Corollary 2.2.14 implies that ( in(I )) is strongly connected for any monomial order on R = K [X 1 , . . . , X n ] provided I is a homogeneous ideal such that R/I is Cohen–Macaulay. On the other hand, by Proposition 2.2.6, we know that

56

2 More on Gröbner Deformations

  K [( in(I ))] Cohen–Macaulay =⇒ ( in(I )) strongly connected. So we can ask whether the conclusion of Corollary 2.2.14 can be strengthened, under √the assumption that I is homogeneous and R/I is Cohen–Macaulay, to K [( in(I ))] being Cohen–Macaulay. The next example shows that this is a too optimistic expectation. The rest of this chapter is devoted√to proving the surprising fact that such an expectation is correct provided in(I ) = in(I ). Example 2.2.15 Consider the homogeneous (with respect to the standard grading) ideal of R = K [X 1 , . . . , X 6 ]: I = (X 1 X 5 + X 2 X 6 + X 42 , X 1 X 4 + X 32 − X 4 X 5 , X 12 + X 1 X 2 + X 2 X 5 ). It turns out that I is a complete intersection, so in particular S/I is Cohen–Macaulay. If < is the lexicographical monomial order in which X 1 > X 2 > · · · > X 6 , we have: 

in(I ) = (X 1 , X 2 , X 3 ) ∩ (X 1 , X 3 , X 6 ) ∩ (X 1 , X 2 , X 5 ) ∩ (X 1 , X 4 , X 5 ).

√ As expected, ( in(I √ )) is strongly connected, however one can check that √ K [( in(I ))] = R/ in(I ) is not Cohen–Macaulay.

2.3 Optimal Gröbner Deformations The goal of this and the next section is to show that the relationship between an ideal I of a polynomial ring and its initial ideal in(I ) is tighter if in(I ) satisfies additional conditions. As we will see later, such a condition is satisfied if in(I ) is a squarefree monomial ideal. To this purpose, let us fix some notation that will be used in the next results: R denotes the polynomial ring K [X 1 , . . . , X n ] over the field K , I ⊂ R is an ideal, w ∈ Nn>0 , P = R[t] and S = P/ homw (I ). Our goal now is to prove that, under some assumption on R/ inw (I ), the Pmodules ExtiP (S, P) are flat K [t]-modules for all i ∈ N. Retracing the proof of Proposition 1.6.7, we will then see the significance of this fact in Theorem 2.3.7. Lemma 2.3.1 The following are equivalent: (a) ExtiP (S, P) is a flat K [t]-module for all i ∈ N. (b) ExtiP/t m P (S/t m S, P/t m P) is a flat K [t]/(t m )-module for all i, m ∈ N. Proof (a) =⇒ (b): Consider the short exact sequence ·t m

0 → P −→ P → P/t m P → 0, that induces the long exact sequence:

2.3 Optimal Gröbner Deformations

57 ·t m

· · · → ExtiP (S, P) −→ ExtiP (S, P) → ExtiP (S, P/t m P) ·t m

→ Exti+1 → Exti+1 P (S, P) − P (S, P) → . . . ·t m

By (a), Ext kP (S, P) does not have t-torsion for all k ∈ N, so the maps Ext kP (S, P) −→ Ext kP (S, P) are injective for all k ∈ N. Therefore, for all i ∈ N, we have a short exact sequence ·t m

0 → ExtiP (S, P) −→ ExtiP (S, P) → ExtiP (S, P/t m P) → 0. ExtiP (S, P) . On the other hand, by Propositions 1.6.1 and t m ExtiP (S, P) 1.6.5, one has ExtiP (S, P/t m P) ∼ = ExtiP/t m P (S/t m S, P/t m P). So So ExtiP (S, P/t m P) ∼ =

ExtiP/t m P (S/t m S, P/t m P) ∼ =

ExtiP (S, P) . t m ExtiP (S, P)

Now just notice that any short exact sequence C. of K [t]/(t m )-modules is also a short exact sequence of K [t]-modules, so C. ⊗ K [t] ExtiP (S, P) is exact by (a); on the other hand Exti (S, P) . C. ⊗ K [t] ExtiP (S, P) ∼ = C. ⊗ K [t]/(t m ) m P i t Ext P (S, P) Exti (S, P) is flat over K [t]/(t m ). So ExtiP/t m P (S/t m S, P/t m P) ∼ = m Pi t Ext P (S, P) (b) =⇒ (a): By contradiction, suppose ExtiP (S, P) is not flat over K [t]. Because K [t] is a principal ideal domain, ExtiP (S, P) has nontrivial torsion. Since Ext iP (S, P) is a (w, 1)-graded P-module, it is a standard graded K [t]-module: hence there exists a nontrivial class [ϕ] ∈ ExtiP (S, P) and k ∈ N such that t k [ϕ] = 0. Let us take a P-free resolution F. of S. Then ExtiP (S, P) ∼ = H i (Hom P (F. , P)). Let (G . , ∂ . ) denote the complex Hom P (F. , P). Then ϕ ∈ Ker(∂ i ) \ Im(∂ i−1 ) and t k ϕ is in Im(∂ i−1 ). Now, by Proposition 1.6.1(a), t m is S-regular for all positive integers m; so F. /t m F. is a P/t m P-free resolution of S/t m S. Therefore ExtiP/t m P (S/t m S, P/t m P) ∼ = H i (Hom P/t m P (F. /t m F. , P/t m P)). Let (G . , ∂ . ) denote the complex Hom P/t m P (F. /t m F. , P/t m P), and π . the resulting map of complexes from G . to G . . Of course π i (ϕ) ∈ Ker(∂ i ) and t k π i (ϕ) ∈ Im(∂ i−1 ). Now, it is enough to show that there is a positive integer m such that π i (ϕ) neither belongs to Im(∂ i−1 ) nor to t m−k G i . Indeed, in this case x = [π i (ϕ)] would be

58

2 More on Gröbner Deformations

an element of ExtiP/t m P (S/t m S, P/t m P) but not in t m−k ExtiP/t m P (S/t m S, P/t m P) such that t k x = 0, and this would contradict the flatness of Ext iP/t m P (S/t m S, P/t m P) over K [t]/(t m ). / Im(∂ i−1 ), so Krull’s If π i (ϕ) ∈ Im(∂ i−1 ), then ϕ ∈ Im(∂ i−1 ) + t m G i . But ϕ ∈ i i−1 / Im(∂ ) for all m  0. If intersection theorem [243, 8.9] tells us that π (ϕ) ∈ π i (ϕ) ∈ t m−k G i , then ϕ ∈ t m−k G i + t m G i = t m−k G i . But ϕ = 0, so, again using / t m−k G i for all m  0.  Krull’s intersection theorem, π i (ϕ) ∈

Cohomologically Full Rings Definition 2.3.2 Let A be a positively graded finitely generated algebra over K . We say that A is cohomologically full if, whenever B is a positively graded finitely generated algebra over K with homogeneous maximal ideal m such that B/J ∼ =A for some nilpotent homogeneous ideal J , the natural maps Hmi (B) → Hmi (A) are surjective for all i ∈ Z. Remark 2.3.3 Using the notation above, there exists a polynomial ring √ T over √K and homogeneous ideals H ⊂ L such that T /L ∼ = A, T /H ∼ = B and H = L. Since the Matlis duality functor ∨ = ∗ Hom K ( , K ) is exact, if A is cohomologically full, by graded local duality we have that the natural maps ExtiT (T /L , T ) → ExtiT (T /H, T ) are injective for all i ∈ Z. For this see [52, 3.6.17–19]. This argument shows that, for a positively graded finitely generated algebra over K , the following are equivalent: (a) A is cohomologically full. (b) Whenever T is a polynomial√ ring over √ K , and H ⊂ L are homogeneous ideals such that T /L ∼ = A and H = L, the natural maps ExtiT (T /L , T ) → ExtiT (T /H, T ) are injective for all i ∈ Z. Example 2.3.4 Using the notation of Remark 2.3.3, if c = height L, then ExtiT (T /L , T ) and ExtiT (T /H, T ) vanish for all i < c, and further Ext cT (T /L , T ) → Ext cT (T /H, T ) is automatically injective, because the kernel is Ext c−1 T (L/H, T ) which vanishes since Supp L/H ⊂ V (L). Hence A is cohomologically full whenever it is Cohen– Macaulay. Another large class of cohomologically full rings is the class of Stanley–Reisner rings, as we will deduce in Corollary 2.4.2. Before proving the next results, it is convenient to introduce some further notation: For all positive integers m, we set (a) Pm = P/t m P; (b) Sm = S/t m S;

2.3 Optimal Gröbner Deformations

59

α j : t j+1 Sm → t j Sm the natural inclusion for each j = 0, . . . , m − 1; α = α0 ◦ · · · ◦ αm−2 : t m−1 Sm → Sm ; β j : t j Sm → t m−1 Sm the multiplication by t m−1− j for each j = 0, . . . , m − 1; E mi (−) the contravariant functor ExtiPm (−, Pm ) for each i ∈ N. Notice that t m−1 Sm ∼ = S1 ∼ = R/ inw (I ) and that α ◦ β0 : Sm → Sm is the multiplication by t m−1 on Sm . Also, E mi (Sk ) ∼ = Exti+1 P (Sk , P) for all k ≤ m by a lemma of Rees [52, 3.1.16]. In particular, (c) (d) (e) (f)

E mi (Sk ) ∼ = E ki (Sk )

for all k ≤ m.

Furthermore, for each j = 0, . . . , m − 1 the following is an exact sequence of Pm modules: αj βj 0 → t j+1 Sm − → t j Sm − → t m−1 Sm → 0. As a last thing notice that R/ inw (I ) is a positively graded finitely generated algebra over K (setting deg X i = wi ), so it makes sense to assume that it is cohomologically full. Lemma 2.3.5 If R/ inw (I ) is cohomologically full, then the following complex of Pm -modules is exact for all j = 0, . . . , m − 1 and i ∈ N: E mi (β j )

E mi (α j )

0 → E mi (t m−1 Sm ) −−−→ E mi (t j Sm ) −−−→ E mi (t j+1 Sm ) → 0. Proof For any k = 1, . . . , m, set Jk = homw (I ) + t k P ⊂ P, and denote J1 by J ; notice that P/Jk ∼ = t m−k Sm . The projection P/Jm− j → P/J can be identiβj

fied with t j Sm − → t m−1 Sm , therefore the homomorphism ϕ : Exti+1 P (P/J, P) → i+1 Ext P (P/Jm− j , P) is nothing but E mi (β j ) (under the identifications given by ∼ i m−1 Sm ) and Exti+1 (P/Jm− j , P) ∼ Exti+1 = E mi (t j Sm )). Now, since P (P/J, P) = E m (t P (I ) is cohomologically full, ϕ is injective by Remark 2.3.3, and so P/J ∼ R/ in = w must be E mi (β j ). We conclude by looking at the long exact sequence E mi (β j )

E mi (α j )

· · · → E mi−1 (t j+1 Sm ) → E mi (t m−1 Sm ) −−−→ E mi (t j Sm ) −−−→ E mi (t j+1 Sm ) E mi+1 (β j )

→ E mi+1 (t m−1 Sm ) −−−−→ E mi+1 (t j Sm ) → . . . , that splits in short exact sequences because E mk (β j ) is injective for all k ∈ N.



Theorem 2.3.6 If R/ inw (I ) is cohomologically full, then ExtiP (S, P) is a flat K [t]module for all i ∈ N. Proof By Lemma 2.3.1, it is enough to show that E mi (Sm ) is a flat K [t]/(t m )-module for all positive integers m. Of course this is true for m = 1 (since K [t]/(t) is a field), so let us fix an integer m ≥ 2 and prove the statement by induction on m, using i (Sm−1 ) is a flat the local flatness criterion ([243, 22.3]): hence, assuming that E m−1 m−1 K [t]/(t )-module, we need to show that:

60

2 More on Gröbner Deformations

(a) E mi (Sm )/t m−1 E mi (Sm ) is a flat K [t]/(t m−1 )-module, and (b) the natural map  : (t m−1 )/(t m ) ⊗ K [t]/(t m ) E mi (Sm ) → t m−1 E mi (Sm ), sending t m−1 ⊗ ϕ to t m−1 ϕ, is an isomorphism. In order to prove (a) and (b), we must collect some facts. Since R/ inw (I ) is cohomologically full, by Lemma 2.3.5 the following is an exact sequence of Pm modules for any j = 0, . . . , m − 1 and i ∈ N: E mi (β j )

E mi (α j )

0 → E mi (t m−1 Sm ) −−−→ E mi (t j Sm ) −−−→ E mi (t j+1 Sm ) → 0.

(2.1)

In particular, E mi (α) = E mi (αm−2 ) ◦ . . . ◦ E mi (α0 ) : E mi (Sm ) → E mi (t m−1 Sm ) is surjective. So Im(E mi (β0 )) = t m−1 E mi (Sm ), because E mi (α)

E mi (β0 )

E mi (α ◦ β0 ) : E mi (Sm ) −−−−− E mi (t m−1 Sm ) −−−→ E mi (Sm ) is the multiplication by t m−1 on E mi (Sm ). Therefore Ker(E mi (α0 )) = t m−1 E mi (Sm ). The same argument shows that, for all j = 0, . . . , m − 1, Ker(E mi (α j )) = t m−1− j E mi (t j Sm ). So, by (2.1), for all j = 0, . . . , m − 1: E mi (t j+1 Sm ) ∼ =

E mi (t j Sm ) . m−1− j E i (t j S ) t m m

E mi (Sm ) ∼ i i (Sm−1 ) is a flat = E mi (Sm−1 ) ∼ = E m−1 = E m (t Sm ) ∼ t m−1 E mi (Sm ) K [t]/(t m−1 )-module by the inductive hypothesis, which shows (a). Furthermore, for any j = 0, . . . , m − 1, we thus get: In particular,

t m−1− j E i (Sm ) . Ker(E mi (α j )) = t m−1− j E mi (t j Sm ) ∼ = m− j i m t E m (Sm ) Since E mi (α) = E mi (αm−2 ) ◦ . . . ◦ E mi (α0 ), the equation above yields that the kerE mi (α)

E mi (β0 )

nel of E mi (α) is equal to t E mi (Sm ). As seen, E mi (Sm ) −−−−− E mi (t m−1 Sm ) −−−→ E mi (Sm ) is the multiplication by t m−1 on E mi (Sm ). So we have 0 : Emi (Sm ) t m−1 = Ker(E mi (α)) = t E mi (Sm ). Since Ker() = {t m−1 ⊗ ϕ : ϕ ∈ 0 : Emi (Sm ) t m−1 }, we infer that  is injective. Being obviously surjective, it is an isomorphism, which finishes the proof of (b) and hence of the theorem. 

2.3 Optimal Gröbner Deformations

61

Consequences Next we draw the main consequences of Theorem 2.3.6. To this purpose let us provide R = K [X 1 , . . . , X n ] with the grading given by a positive integral vector g = (g1 , . . . , gn ) ∈ Nn . Recall that, if I ⊂ R is a g-homogeneous ideal, then inw (I ) ⊂ R is a g-homogeneous ideal as well. Finally, notice that m = (X 1 , . . . , X n ) is the only maximal g-homogeneous ideal of R. Theorem 2.3.7 If I is a g-homogeneous ideal of R = K [X 1 , . . . , X n ] and R/ inw (I ) is cohomologically full, we have dim K ExtiR (R/I, R) j = dim K ExtiR (R/ inw (I ), R) j , dim K Hmi (R/I ) j = dim K Hmi (R/ inw (I )) j for all i, j ∈ Z. Proof Set P = R[t], and let us introduce on P a graded structure given by deg X i = gi and deg t = 0. If S = P/ homw (I ), by Theorem 2.3.6, we have that E i = ExtiP (S, P) is a flat K [t]-module for any i ∈ Z. For all j ∈ Z, let E i j denote the degree j part of E i . Then E i j is a finitely generated flat K [t]-module. So E i j is a finitely generated free K [t]-module for any i, j ∈ Z, say E i j ∼ = K [t]ri j . Letting x be t or t − 1 we have the short exact sequence ·x

→ P → P/x P → 0. 0→P− The long exact sequence of Ext P (S, −) associated to it, gives us the following short exact sequence for all i ∈ Z: 0 → Coker μi,x → ExtiP (S, P/x P) → Ker μi+1,x → 0, where μk,x is the multiplication by x on E k . Being E i j = K [t]ri j , independently of x we have Ker μi,x = 0 and (Coker μi−1,x ) j = K ri j . So, using Propositions 1.6.5 and 1.6.1, dim K Ext P/x P (S/x S, P/x P) j = dim K ExtiP (S, P/x P) j = ri j (independently of x). Using again Proposition 1.6.1, there are graded isomorphisms ExtiR (R/I, R) ∼ = ExtiP/(t−1)P (S/(t − 1)S, P/(t − 1)P) and ExtiR (R/ inw (I ), R) ∼ = i Ext P/t P (S/t S, P/t P), so the first equation is proved. The first equation implies the second by graded local duality. Indeed for every ideal J of R one has a graded isomorphism Hmi (R/J )∨ ∼ = Ext R (R/J, R(−|g|)), where |g| = g1 + · · · + gn , induced by the Matlis duality functor ∨ = ∗ Hom K ( , K ). Here R(−|g|) is the graded canonical module of R and ∗ Hom K is the graded Hom functor. Furthermore one observes that Matlis duality preserves the K -dimensions of the graded components since they are finite. For all this see [52, 3.6.17–19]. 

62

2 More on Gröbner Deformations

Let I be a g-homogeneous ideal of R = K [X 1 , . . . , X n ]. Recall that depth R/I = min{i ∈ N : Hmi (R/I ) = 0} and reg R/I = max{i + j : Hmi (R/I ) j = 0}. Also, R/I is called generalized Cohen–Macaulay if the R-modules Hmi (R/I ) have finite length for all i < dim R/I . In the following corollary we gather some immediate consequences of Theorem 2.3.7: Corollary 2.3.8 If I is a g-homogeneous ideal of R = K [X 1 , . . . , X n ] and R/ inw (I ) is cohomologically full, we have: (a) (b) (c) (d)

R/I is Cohen–Macaulay if and only if R/ inw (I ) is Cohen–Macaulay; depth R/I = depth R/ inw (I ); reg R/I = reg R/ inw (I ); R/I is generalized Cohen–Macaulay if and only if R/ inw (I ) is generalized Cohen–Macaulay.

For the following, note that the Krull dimension of the zero module is −∞. Also, a ring R is equidimensional if dim R/p = dim R < +∞ for all p ∈ Min(R). Lemma 2.3.9 For k ∈ N, if I is an unmixed ideal of an equidimensional Gorenstein ring R with all the maximal ideals of height n, the following are equivalent: (a) R/I satisfies Serre condition (Sk ); (b) dim Ext n−i R (R/I, R) ≤ i − k for all i, 0 ≤ i < dim R/I . Proof (a) =⇒ (b): Let p be a prime ideal of R of height h < n − i + k containing I . We claim that (R/I )p has depth greater than h − n + i: indeed h − n + i < n − i + k − n + i = k and, exploiting the fact that R and R/I are catenary, equidimensional, and with all the maximal ideals of the same height, dim(R/I )p = h − n + dim R/I > h − n + i, so: depth(R/I )p ≥ min{k, dim(R/I )p } > h − n + i. Since the formation of Ext commutes with localizations (see Rotman [271, 7.39]), the local duality of Grothendieck [52, 3.5.8] implies n−i ∼ Ext n−i R (R/I, R)p = Ext Rp ((R/I )p , Rp ) = 0.

So p is not in the support of Ext n−i R (R/I, R) whenever height p < n − (i − k), that is, the Krull dimension of Ext n−i R (R/I, R) is less than or equal to i − k. (b) =⇒ (a): Let c = height I , and for all h = c, . . . , n let us denote by Vh the set of prime ideals of R of height h containing I . For each h = c, . . . , n, using local duality again we have the following chain of equivalences: depth(R/I )p ≥ min{k, dim(R/I )p } = b for all p ∈ Vh Ext h−i Rp ((R/I )p ,

⇐⇒

Rp ) = 0 for all p ∈ Vh and i < b

⇐⇒

Ext h−i R (R/I, R)p = 0 for all p ∈ Vh , and i < b

⇐⇒

dim Ext h−i R (R/I,

R) < n − h for all i < b.

2.3 Optimal Gröbner Deformations

63

Since R and R/I are catenary, equidimensional, and with all the maximal ideals of the same height, dim(R/I )p = h − c for all p ∈ Vh . So i < b ≤ h − c, which implies n − h + i < n − c = dim R/I . If dim Ext n−i R (R/I, R) ≤ i − k for all i, 0 ≤ i < dim R/I , n−(n−h+i) (R/I, R) ≤ n − h + i − k < n − h dim Ext h−i R (R/I, R) = dim Ext R

for all h = c, . . . , n (because i < b ≤ k), and this lets us conclude.



Lemma 2.3.10 For any ideal I of an equidimensional Gorenstein ring R with all the maximal ideals of height n, we have dim ExtiR (R/I, R) ≤ n − i for all i ∈ N. Furthermore, I has an associated prime of height h

⇐⇒ dim Ext hR (R/I, R) = n − h.

In particular, I is unmixed if and only if dim ExtiR (R/I, R) < n − i for all i = dim R/I . Proof Let p be a prime ideal of R containing I such that height(p) = k < i. Then ExtiR (R/I, R)p ∼ = ExtiRp ((R/I )p , Rp ) is zero since Rp has injective dimension equal to dim Rp = k [52, 3.1.17]. This means that dim ExtiR (R/I, R) ≤ n − i. Furthermore, dim Ext hR (R/I, R) = n − h if and only if there exists a prime ideal p ⊂ R containing I such that height(p) = h and Ext hR (R/I, R)p = 0. But, by the local duality of Grothendieck [52, 3.5.8], Ext hR (R/I, R)p ∼ = Ext hRp ((R/I )p , Rp ) is not zero if and only if depth(R/I )p = 0, which is the case if and only if p is an associated prime of I .  Corollary 2.3.11 For a g-homogeneous ideal I ⊂ R = K [X 1 , . . . , X n ], the following are equivalent for all k ≥ 2: (a) R/I satisfies Serre condition (Sk ); (b) dim Ext n−i R (R/I, R) ≤ i − k for all i, 0 ≤ i < dim R/I . Proof Notice that Spec R/I is connected for any g-homogeneous ideal I . So it turns out that I ⊂ R is an unmixed ideal both under condition (a) (Corollary 2.2.6), and under condition (b) (Lemma 2.3.10). So Lemma 2.3.9 applies.  Finally we have another surprising consequence of Theorem 2.3.7: Corollary 2.3.12 For a g-homogeneous ideal I ⊂ R = K [X 1 , . . . , X n ] such that R/ inw (I ) is cohomologically full, the following are equivalent for any integer k ≥ 2: (a) R/I satisfies Serre’s condition (Sk ); (b) R/ inw (I ) satisfies Serre’s condition (Sk ). n−i Proof The Krull dimensions of Ext n−i R (R/I, R) and of Ext R (R/ inw (I ), R) are determined by their Hilbert functions, so they are the same by Theorem 2.3.7. Hence the claim follows by Corollary 2.3.11. 

64

2 More on Gröbner Deformations

Notice that by Lemma 2.3.10 and Theorem 2.3.7 it also follows that Bigheight(I ) = Bigheight(inw (I )) if R/ inw (I ) is cohomologically full. In particular, if R/ inw (I ) is cohomologically full, then I is unmixed if and only if inw (I ) is unmixed. Remark 2.3.13 In Corollary 2.3.12 the assumption k ≥ 2 could be removed: in fact, one can show that is cohomologically full rings always satisfy (S1 ) (Dao et al. [100, 3.7]). We decided to not prove this fact because the proof is involved, while in the case of main interest for this book (that is when inw (I ) is a squarefree monomial ideal) it is trivial.

2.4 Square-Free Gröbner Deformations Let R = K [X 1 , . . . , X n ], and I ⊂ R a monomial ideal. Let u 1 , . . . , u r be the (unique) minimal system of monomial generators of I . For all subsets σ ⊂ {1, . . . , r } we define the monomial m(I, σ ) ∈ R as follows: m(I, σ ) := lcm(u i : i ∈ σ ). (We use the convention that m(I, ∅) = 1). For v ∈ σ we set sign(v, σ ) := (−1)q−1 ∈ K if v is the qth element of σ . Let us consider the graded complex of free R-modules F. (I ) = (Fi , ∂i )i=0,...,r with Fi :=



R(− deg m(I, σ )),

σ

where the sum runs over the subsets σ ⊂ {1, . . . , r } of cardinality i and the differentials are defined by 1σ →

 v∈σ

sign(v, σ )

m(I, σ ) · 1σ \{v} . m(I, σ \ {v})

As it turns out, F. (I ) is a graded free R-resolution of R/I [116, Exercise 7.11]. It is called the Taylor resolution. In general, the Taylor resolution is far from being minimal, nevertheless it will be enough to achieve the goal of this section, that is to show that rings defined by squarefree monomial ideals is cohomologically full. Let I be a monomial ideal of R = K [X 1 , . . . , X n ], generated by monomials u 1 , . . . , u r . For any positive integer k we introduce the monomial ideal I [k] = (u k1 , . . . , u rk ). Notice that u 1 , . . . , u r are the minimal system of monomial generators of I if and only if u k1 , . . . , u rk are the minimal system of monomial generators of I [k] . In particular, m(I [k] , σ ) = m(I, σ )k for any σ ⊂ {1, . . . , r }.

2.4 Square-Free Gröbner Deformations

65

Theorem 2.4.1 Let R = K [X 1 , . . . , X n ] and I ⊂ R be a squarefree monomial ideal. For all k ∈ N and i ∈ N the maps ExtiR (R/I [k] , R) → ExtiR (R/I [k+1] , R), induced by the projections R/I [k+1] → R/I [k] , are injective. Proof Let u 1 , . . . , u r be the minimal system of monomial generators of I , and for all k ∈ Z+ set m σ [k] := m(I [k] , σ ) and m σ := m σ [1] for all σ ⊂ {1, . . . , r }. Of course m σ is a squarefree monomial and, from what we said above, m σ [k] = m kσ . The module ExtiR (R/I [k] , R) is the ith cohomology of the complex of R-modules G . [k] = Hom R (F. [k], R) where F. [k] = F. (I [k] ) = (Fi , ∂i [k])i=0,...,r is the Taylor resolution of R/I [k] . For all i ∈ {0, . . . , r }, consider the map f i : Fi → Fi sending 1σ to m σ · 1σ . Since m σ [k] = m kσ , the collection f . = ( f i )i=0,...,r : F. [k + 1] → F. [k] is a morphism of complexes lifting the projection R/I [k+1] → R/I [k] . The maps ExtiR (R/I [k] , R) → ExtiR (R/I [k+1] , R) are the homomorphisms → H i (G . [k + 1]) H i (G . [k]) − gi

induced by g . = Hom( f . , R) : G . [k] → G . [k + 1]. In order to show that g i is injective, we need to understand how it acts: if G . [k] = (G i , ∂ i [k]), then G i = Hom R (Fi , R) can be identified with Fi (ignoring the grading) and ∂ i [k] : G i −→ G i+1  1σ →

 sign(v, σ ∪ {v})

v∈{1,...,r }\σ

m σ ∪{v} mσ

k · 1σ ∪{v} ,

where σ is a subset of {1, . . . , r } of cardinality i. The map g i : G i → G i , under the identification G i ∼ = Fi , is just the map sending 1σ to m σ · 1σ . Now, let x ∈ Ker(∂ i [k]) such that g i (x) ∈ Im(∂ i−1 [k + 1]). To finish we need to show that x∈ Im(∂ i−1 [k]). Let y ∈ G i−1 such that ∂ i−1 [k + 1](y) = g i (x). Let us write y = σ yσ · 1σ where σ varies among the subsets of {1, . . . , r } of cardinality   i − 1 and yσ ∈ R. We can write yσ uniquely as y in σ + m σ yσ where no monomial     the support of yσ is divided by m σ . Putting y = σ yσ · 1σ and y  = σ yσ · 1σ , we have y = y  + g i−1 (y  ). So g i (x) = ∂ i−1 [k + 1](y) = ∂ i−1 [k + 1](y  ) + ∂ i−1 [k + 1](g i−1 (y  )) = ∂

i−1



[k + 1](y ) + g (∂ i

i−1

(2.2)



[k](y )).

 Now, if z = ∂ i−1 [k + 1](y  ) ∈ G i , let us write z = σ z σ · 1σ where σ varies among the subsets of {1, . . . , r } of cardinality i and z σ ∈ R. For all such σ , we have zσ =

 v∈σ



mσ sign(v, σ ) m σ \{v}

k+1

yσ \{v} .

66

2 More on Gröbner Deformations

Since I is squarefree and m σ \{v} does not divide yσ \{v} for any v ∈ σ , m σ cannot divide z σ unless it is zero. On the other hand, m σ must divide z σ by (2.2). Therefore z σ = 0, and since σ was arbitrary z = 0, that is: g i (x) = g i (∂ i−1 [k](y  )). Since g i : G i → G i is obviously injective, we have found x = ∂ i−1 [k](y  ).



Corollary 2.4.2 Let R = K [X 1 , . . . , X n ] and I ⊂ R be a squarefree monomial ideal. Then R/I is cohomologically full. Proof Using Remark 2.3.3 we have to show that the maps ExtiT (T /L , T ) → ExtiT (T /H, T ) are injective for all i ∈ Z, where√T is a polynomial ring over K and H ⊂ L are ideals such that T /L ∼ = R/I and H = L. Since, up to change of coordinates, L is a squarefree monomial ideal of T , it is harmless to take T = R and L = I. √ So, let H ⊂ R be an ideal such that H = I . Then, there exists a positive integer k such that I [k] ⊂ H . By Theorem 2.4.1 the natural map ExtiR (R/I, R) → ExtiR (R/I [k] , R) is injective for all i ∈ N. Since it factors as ExtiR (R/I, R) → ExtiR (R/H, R) → ExtiR (R/I [k] , R), the natural map ExtiR (R/I, R) → ExtiR (R/H, R) is injective as well for any i ∈ N.  By the results in the previous section and Corollary 2.4.2, we immediately get: Theorem 2.4.3 Let R = K [X 1 , . . . , X n ], I ⊂ R a g-homogeneous ideal and < a monomial order such that in(I ) = in< (I ) is a squarefree monomial ideal. Then, for all i, j ∈ Z, dim K Hmi (R/I ) j = dim K Hmi (R/ in(I )) j . In particular: R/I is Cohen–Macaulay if and only if R/ in(I ) is Cohen–Macaulay. depth R/I = depth R/ in(I ). reg R/I = reg R/ in(I ). R/I is generalized Cohen–Macaulay if and only if R/ in(I ) is generalized Cohen–Macaulay. (e) For all positive integers k, R/I satisfies Serre’s condition (Sk ) if and only if R/ in(I ) satisfies Serre’s condition (Sk ).

(a) (b) (c) (d)

Proof By Lemma 1.5.6(b), there exists a weight vector w ∈ Nn>0 such that in(I ) = inw (I ). So the main part of the thesis and the points (a), (b), (c) and (d) follow by Theorem 2.3.7 and Corollaries 2.3.8 and 2.4.2. Point (e) follows by Corollaries 2.3.12 and 2.4.2 for k ≥ 2. Furthermore, being reduced rings, both R/ in(I ) and R/I  satisfy Serre’s condition (S1 ), and so (e) trivially holds also for k = 1.

Notes

67

Notes The fact that Bigheight does not go down by passing to the initial ideal is implicit in the work of Sturmfels et al. [299]. That bigheight does not go up by passing to the initial ideal was essentially proved by Kalkbrener and Sturmfels [200] to solve a conjecture of Kredel and Weispfenning [223] concerning the initial ideal of a prime ideal. Corollary 2.2.14 was proved in [200] for prime ideals over an algebraically closed field. The general formulation that the connectivity properties do not get worse under Gröbner deformations is due to Varbaro [308]. That the depth does not change by passing to a radical monomial initial ideal has been suspected for a long time, and officially conjectured by Jürgen Herzog. Theorem 2.4.3, solving Herzog’s conjecture, has been proved by Conca and Varbaro [95]. In this context Theorem 2.3.6 is crucial; its proof given here is an adaptation to Gröbner deformations of a general result of Kollár and Kovács [217] on deformations of log canonical singularities. The notion of cohomologically full local rings has been introduced in [100], and the one used here is the variation for positively graded algebras finitely generated over K . A similar notion to cohomologically full local rings, namely local rings with liftable local cohomology, has been studied afterwards in [218], that is an expanded version of [217]. The fact that Stanley–Reisner rings are cohomologically full had essentially already been proved by Lyubeznik [236].

Chapter 3

Determinantal Ideals and the Straightening Law

Determinants are highly complicated polynomials. It would be impossible to investigate the ideal generated by the t-minors of a matrix X = (X i j ) of indeterminates if one had to expand its generators into linear combinations of monomials. Instead one introduces the powerful machinery of standard bitableaux and the straightening law. In this approach all the minors of X (and not just the 1-minors X i j ) are considered as algebra generators of the polynomial ring K [X ] over the field K . Then products of minors appear as “monomials” in these generators. The price to be paid, of course, is that one has to choose a proper subset of all such “monomials” as a linearly independent K -basis of K [X ]. Because of their pictorial representation by pairs of standard Young tableaux, the elements in the distinguished basis are called standard bitableaux. They are products over chains of a natural partial order of the minors. The straightening law allows one to rewrite products of minors as linear combinations of standard bitableaux in a combinatorially controlled way. Affine space can be seen as an open set in a suitable Grassmannian. The polynomial ring then becomes the dehomogenization of the coordinate ring of the Grassmannian which itself is generated by maximal minors of a larger matrix. This allows one to first derive the straightening law for maximal minors and then “dehomogenize” it to the general case. So the crucial step is to understand the Plücker relations for the Grassmannian. The class of algebras with straightening law (ASL) axiomatizes this approach for determinantal ideals, extending its applicability to more general rings, and providing access to important structural results. The theory of ASL has been analyzed extensively in Bruns and Vetter [61]. In this book, we will derive the fundamental results of the theory in Chap. 4, via Gröbner bases and deformations to the initial ideal. Symbolic powers of the determinantal ideals have K -bases consisting of standard bitableaux. If the characteristic of the underlying field is large enough, the ordinary powers of determinantal ideals are intersections of symbolic powers, and therefore

© Springer Nature Switzerland AG 2022 W. Bruns et al., Determinants, Gröbner Bases and Cohomology, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-05480-8_3

69

70

3 Determinantal Ideals and the Straightening Law

have bases of standard bitableaux as well. This fact is the starting point for the investigation of algebras defined by minors in later chapters, as well as for the computation of Gröbner bases and initial ideals in Chap. 4.

3.1 Minors and Determinantal Ideals The main actors in this book are the ideals generated by the t-minors of a matrix of indeterminates ⎞ ⎛ X 11 · · · X 1n ⎜ .. ⎟ . X = ⎝ ... . ⎠ X m1 · · · X mn

Usually we assume that m ≤ n. The indeterminates X i j live in the polynomial ring K [X ] = K [X i j : i = 1, . . . , m, j = 1, . . . , n] over a commutative ring K . The letter K indicates that the ring denoted by it is often a field. The t-minors are the polynomials [a1 . . . at | b1 . . . bt ] = det(X ai b j : i = 1, . . . , t, j = 1, . . . , t). The degree t of a t-minor is called its size. The ideal It (X ) is generated by the t-minors. Often we will simply write It for It (X ). By convention, the value of the empty minor [ | ] is 1 ∈ K . We denote the set of all nonempty minors of X by M(X ) and the subset of minors of size t by Mt (X ). If no confusion can arise, we simplify these notations to M and Mt . The definition of [a1 . . . at | b1 . . . bt ] does not require any condition on the indices ai , bi of [a1 . . . at | b1 . . . bt ], and we cannot always insist that a1 < · · · < at and b1 < · · · < bt . If it is necessary to specify the matrix from which a minor is taken, we add a subscript as in [a1 . . . at | b1 . . . bt ] X . If m ≤ n, the maximal minors of X are the m-minors. Their row indices are automatically fixed to be 1, . . . , m and therefore we can use the shorthand notation [b1 . . . bm ] = [1 . . . m | b1 . . . bm ]. The first ideal It (X ) with t > 1 and more than one generator, namely I2 (X ) for m = 2 and n = 3, has come up in Example 1.2.8. It is generated by [1 2], [1 3] and [2 3].

3.2 Standard Bitableaux and the Straightening Law

71

3.2 Standard Bitableaux and the Straightening Law In what follows we will often consider sequences of integers with a monotonicity property. We say that a sequence (ri ) is increasing if ri < ri+1 for all i. It is nonincreasing if ri ≥ ri+1 for all i. Our standard symbol for a product δ1 · · · δw of minors is , and we assume that the sizes |δi | (i.e., the number of rows of the submatrix of X whose determinant is δi ) are nonincreasing, |δ1 | ≥ · · · ≥ |δw |. The shape || of  is the sequence (|δ1 |, . . . , |δw |). If necessary we may add factors [ | ] at the right hand side of the product, and accordingly extend the shape by a sequence of entries 0. A product of minors is also called a bitableau. The choice of the term “bitableau” is motivated by the graphical description of a product  as a pair of Young tableaux as in Fig. 3.1. Every product of minors is represented by a bitableau and, conversely, every bitableau stands for a product of minors:  = δ1 · · · δw ,

δi = [ai1 . . . aiti | bi1 . . . biti ], i = 1, . . . , w.

For aesthetic reasons we use symmetric figures so that the indices in the left tableau must be read from right to left. Sometimes it is necessary to separate the two tableaux from which  is formed; we then write  = (R | C). Figure 3.2 shows an example of a bitableau. It represents the product [1 3 4 | 2 3 5][1 2 3 | 1 4 6][2 | 3] of minors. To be completely accurate, one should consider the bitableaux as purely combinatorial objects (as we will do in Sect. 4.2) and distinguish them from the ring-theoretic

Fig. 3.1 A general bitableau Fig. 3.2 A bitableau of shape (3, 3, 2)

72

3 Determinantal Ideals and the Straightening Law

Fig. 3.3 A standard bitableau

objects represented by them. However, since there is no real danger of confusion, we use the same terminology for both classes of objects. Each side of the bitableau is called a tableau. While we cannot associate an algebraic object with a tableau in our present context, tableaux are of course meaningful combinatorial objects. In order to define the notion of a standard bitableau, we consider the partial order on M(X ) defined by [a1 . . . at | b1 . . . bt ]  [c1 . . . cu | d1 . . . du ] ⇐⇒

t ≥ u and ai ≤ ci , bi ≤ di ,

i = 1, . . . , u.

(Thus [ | ] is maximal among all minors.) A product  = δ1 · · · δw is called a standard bitableau if δ1  · · ·  δw , in other words, if in each column of the bitableau the indices are nondecreasing from top to bottom. (The empty product is also standard.) Figure 3.3 shows a standard bitableau. The letter  is reserved for standard bitableaux. In the literature, standard bitableaux are often called standard monomials; however, since we will use the ordinary monomials in K [X ] so often, we reserve the term “monomial” for products of the indeterminates X i j . Note that the poset of minors is a distributive lattice with the partial order just introduced. Therefore we may form the meet δ1 ∧ δ2 and the join δ1 ∨ δ2 of minors δ1 and δ2 . These operations are associative, commutative, and distributive over each other. To be standard makes of course sense for (single) tableaux: the sides of standard bitableaux are standard tableaux: they have increasing rows and nondecreasing columns. The fundamental straightening law of Doubilet–Rota–Stein [112] says that every element of K [X ] has a unique presentation as a K -linear combination of standard bitableaux: Theorem 3.2.1 (a) The standard bitableaux form a K -vector space basis of K [X ]. (b) (Straightening relations) If the product γδ of two minors is not a standard bitableau, then it has a representation γδ =



xi εi ηi ,

xi ∈ K , xi = 0,

3.2 Standard Bitableaux and the Straightening Law

73

where εi ηi is a standard bitableau, εi ≺ γ, δ ≺ ηi (here we must allow that ηi = [ | ] = 1). (c) The standard representation of an arbitrary bitableau , i.e. its representation as a linear combination of standard bitableaux , can be found by successive application of the straightening relations in (b). (d) Let  be a bitableau and  = σ1 · · · σw be a standard bitableau appearing in the standard representation of . Then σ1  δ for all factors δ of . The field K can be replaced by an arbitrary commutative ring in the straightening law. We outline the argument in Remark 3.2.9. Example 3.2.2 An example of a nontrivial standard representation for m = 3 and n = 6 is [1 4 6][2 3 5] = [1 3 5][2 4 6] − [1 2 5][3 4 6] − [1 2 3][4 5 6].

(3.1)

It is hard to check correctness by hand. Below we will explain how to find the representation.

Plücker Relations For the proof of the theorem we can assume that m ≤ n, passing to the transpose of X if necessary. We derive the theorem from its “restriction” to the subalgebra K [Mm ] generated by the m-minors. As introduced in Sect. 3.1, each m-minor [b1 . . . bm ] is determined by its column indices b1 , . . . , bm . The algebra K [Mm ] is the homogeneous coordinate ring of the Grassmann variety parameterizing m-dimensional vector subspaces of K n . The m-minors satisfy the famous Plücker relations. In their description we use sign(i 1 . . . i s ) to denote the sign of the permutation of {1, . . . , s} represented by the sequence i 1 , . . . , i s . Lemma 3.2.3 For all indices a1 , . . . , a p , bq , . . . , bm , c1 , . . . , cs ∈ {1, . . . , n} such that s = m − p + q − 1 > m and t = m − p > 0 one has 

sign(i 1 . . . i s )[a1 . . . a p ci1 . . . cit ][cit+1 . . . cis bq . . . bm ] = 0.

i 1 σ. Part (a) has been justified above. Part (b) follows from the fact that there is no other tableau of shape ≥ σ that has the same content as In(σ). It is much more difficult to show (c), and we forgo a proof; for example, see [61, (11.4)]. Remark 3.3.2 In Remark 3.2.9 we have pointed out that Theorem 3.2.1 holds for arbitrary coefficient rings B. In the same way, Theorem 3.3.1 can be generalized: its validity for Z as the base ring follows immediately from its validity over Q, and the general case is then obtained by tensoring with B over Z.

Coordinate Rings of Flag Varieties It follows from Theorem 3.3.1 that the straightening law can be restricted to certain subalgebras of K [X ] that generalize K [Mm ]. For a sequence τ = (t1 , . . . , t p ) of natural numbers, with 1 ≤ t1 < · · · < t p ≤ m, we set Mτ =

p

{[1 . . . ti | c1 . . . cti ] : 1 ≤ c1 < · · · < cti ≤ n}. i=1

For τ = (m) one has Mτ = Mm . Proposition 3.3.3 Every polynomial f ∈ K [Mτ ] is a linear combination of standard bitableaux  ∈ K [Mτ ]. Proof We can assume that f is a product of k minors of type [1 . . . t | c1 . . . ct ]. Then f = (In(σ) | C) where we obtain σ by stacking the sequences (1, . . . , t) in order of nonincreasing length. Now the claim follows from Theorem 3.3.1(b). 

80

3 Determinantal Ideals and the Straightening Law

In algebraic geometry K [Mτ ] is the coordinate ring of the variety of flags of K -vector subspaces U1 ⊂ · · · ⊂ U p of K n such that dim Ui = ti , i = 1, . . . , p. So K [Mτ ] is the coordinate ring of a partial flag variety. In the case m = n and τ = (1, . . . , n) it is the coordinate ring of the full (or complete) flag variety of K n .

Initial Monomials and the Straightening Law Already in Example 1.2.8 we have introduced diagonal monomial orders, i.e., monomial orders under which the initial monomial of a minor is the product of the indeterminates along its main diagonal. For example, one can choose the lexicographic order that extends the following order of the indeterminates: X i j < X pq if i < p or i = p and j < q. The diagonal X a1 b1 · · · X at bt is then the initial monomial (and term) of [a1 . . . at | b1 . . . bt ]. Such monomials will simply be called t-diagonals. In view of the major role that diagonals play for diagonal monomial orders, we introduce a notation for them: if δ = [a1 . . . at | b1 . . . bt ] then we let δ = a1 . . . at | b1 . . . bt  =

t 

X ai bi ,

i=1

and for a bitableau  = δ1 . . . δw we set  = δ1 · · · δw  =

w 

δ j .

j=1

With respect to a diagonal monomial order one has in(δ) = δ and in() = . An idea that comes to mind when thinking about Gröbner bases for the determinantal ideals It is to exploit the straightening law and the initial monomials of standard bitableaux. Unfortunately, since the assignment  →  is not injective, the initial monomials of standard bitableaux cannot form the monomial basis of in(It ). There are two ways to overcome this obstacle: (a) In Chap. 4 we will introduce the Robinson–Schensted–Knuth correspondence RSK. It is injective, and the image monomial RSK() is combinatorially close enough to  to allow structural conclusions. (b) In Sect. 6.7 K [X ] will be embedded into a larger polynomial ring with a suitable monomial order for which taking initial monomials of standard bitableaux is injective. Although injectivity fails on the set of all standard bitableaux, it holds on some distinguished subsets. Proposition 3.3.4 (a) Let C, C be standard tableaux of shape σ. If C = C , then In(σ) | C = In(σ) | C .

3.3 Refinement of the Straightening Law

81

(b) In particular, if the standard bitableaux ,  are products of maximal minors and  =  , then  =  . (c) If f ∈ K [Mm ], f = 0, then there exists a standard bitableau  ∈ K [Mm ] for which in( f ) = . Proof (a) It is enough to reconstruct C from μ = In(σ) | C. Factor μ = μ1 · · · μr where μi collects all indeterminates from row i and r is the size of the first factor δ1 of (In(σ) | C). Let X ici be the factor of μi with the smallest column index. Then δ1 = [1 . . . r | c1 . . . cr ], and we are done by induction on the number of rows in (In(σ) | C). (b) is a special case of (a). (c) By the straightening law f is a K -linear combination of standard bitableaux i in K [Mm ]. Because of (b), the largest monomial among the in(i ) appears only once in the linear combination, and therefore it is the initial monomial of f .  It is an immediate corollary of Proposition 3.3.4 that the standard bitableaux in K [Mm ] are linearly independent and, as indicated in Remark 3.2.8, this implies the linear independence of the standard bitableaux in K [X ]. It is another immediate consequence that Mm is a Sagbi basis of K [Mm ], and more generally, that Mτ is a Sagbi basis of K [Mτ ]. We postpone the precise statement until Sect. 6.2. Remark 3.3.5 Proposition 3.3.4 suggests a straightening algorithm for elements in K [Mm (X )] (and therefore also in K [X ]): given f ∈ K [Mm (X )], find the standard bitableau  with in( f ) = in() and pass to f − α where α is the initial coefficient of f . After finitely many of these reduction steps one must reach 0. The crucial argument that justifies our last statement is that in( f − α) < in( f ), and this follows since the initial monomials of different standard bitableaux of maximal minors are different: they cannot cancel in a standard representation. In our situation, the standard bitableaux  =  that appear in the standard representation of f satisfy in( ) < in(). The advantage of this algorithm, as compared to the application of Plücker relations, is that only standard bitableaux are introduced that appear in the standard representation of f . On the other hand, it needs expansion of minors and the multiplication of these polynomials, whereas Plücker relations only use linear algebra of degree 2 polynomials in a large number of variables.

Predicting a Standard Bitableau It is very difficult to predict what standard bitableaux will appear in the standard representation of a bitableau without running the straightening algorithm. However, there is a distinguished standard bitableau that must appear with coefficient 1. For any tableau T , we let T0 denote the tableau that we obtain by sorting the columns of T in nondecreasing order (from top to bottom), as illustrated by Fig. 3.4. It is easy to check that T0 has increasing rows, and therefore it is a standard tableau.

82

3 Determinantal Ideals and the Straightening Law

Fig. 3.4 T and T0

Given a tableau (R | C), we want to show that (R0 | C0 ) appears with coefficient 1 in the standard representation of (R | C). (Notice that in the Examples (3.1) and (3.2), the first summand is (R0 | C0 ).) Theorem 3.3.6 Let R and C be tableaux of shape σ. (a) In(σ) | C0  = In(σ) | C, and (In(σ) | C0 ) appears with coefficient 1 in the standard representation of (In(σ) | C). (b) The standard bitableau (R0 | C0 ) appears with coefficient 1 in the standard representation of (R | C). Proof The equality In(σ) | C0  = In(σ) | C follows from the definition of C0 . By Theorem 3.3.1(b), (In(σ) | C) is a linear combination of (In(σ) | C j ), j = 0, 1, . . . . Using Proposition 3.3.4 we obtain that (In(σ) | C0 ) appears with coefficient 1. Part (b) now follows from Theorem 3.3.1(c) if we observe that the analogue of (a) holds for (R | In(σ)).  Note that R | C = R0 | C0  in general (find an example!). Nevertheless, we can describe the factors of (R0 | C0 ) in terms of the poset of minors. We leave the proof of the next result to the reader. Proposition 3.3.7 Let (R | C) = δ1 · · · δw . Then (R0 | C0 ) = η1 · · · ηw with ηk =



δi .

I ⊂{1,...,w} i∈I |I |=w−k+1

In particular, if w = 2 then η1 = δ1 ∧ δ2 , η2 = δ1 ∨ δ2 .

3.4 Determinantal Rings An ideal in a partially ordered set (M, ≤) is a subset N such that N contains all elements x ≤ y if y ∈ N . Let N be an ideal in the partially ordered set M(X ) and consider the ideal I = N K [X ] generated by N . Every element of I is a K -linear combination of elements δx with x ∈ K [X ] and δ ∈ N . By Theorem 3.2.1(d), every standard bitableau  = γ1 · · · γv in the standard representation of δx has γ1 ≤ δ, and therefore γ1 ∈ N . This shows

3.4 Determinantal Rings

83

Proposition 3.4.1 Let N be an ideal in the partially ordered set M(X ). The standard bitableaux  = γ1 · · · γu with γ1 ∈ N form a K -basis of the ideal I = N K [X ] in the ring K [X ], and the (residue classes of the) standard bitableaux  = δ1 · · · δv with δ j ∈ / N for all j form a K -basis of K [X ]/I . Corollary 3.4.2 The standard bitableaux  = γ1 · · · γu such that |γ1 | ≥ t form a K -basis of It , and (the residue classes of) the standard bitableaux  = δ1 · · · δv with |δ j | ≤ t − 1 for all j form a K -basis of K [X ]/It . In fact, It is generated by all minors of size ≥ t, and these form the ideal   δ ∈ M(X ) : δ  [1 . . . t − 1 | 1 . . . t − 1] . The ideals It are special instances of the so-called 1-cogenerated ideals Iγ , γ ∈ M(X ), that are generated by all minors δ  γ. We come back to these ideals in Theorem 6.7.9. Remark 3.4.3 The straightening law and its refined version in Theorem 3.3.1 can be used in various approaches to the theory of determinantal ideals and rings. (a) The straightening law implies that K [X ] and K [Mm ] are algebras with straightening law (ASL) on the partially ordered sets M(X ) and Mm (X ), respectively. This notion is defined as follows. Let B be a commutative ring, A = k≥0 Ak a positively graded B-algebra, and  ⊂ A a subset of homogeneous elements of positive degrees, endowed with a partial order ≤. Then the following hold: (a) A is a free B-module for which the standard monomials π1 · · · πm , with m ≥ 0, π1 , . . . , πm ∈ , π1 ≤ · · · ≤ πm , are a basis; (b) For all incomparable ρ and σ in  the representation ρσ =



bμ μ,

bμ ∈ B, bμ = 0,

of the product ρσ has as a linear combination of standard monomials μ has the following property: every μ appearing in it contains a factor π ∈  such that π ≤ ρ and π ≤ σ. The ASL property is passed on the residue class rings modulo the class of ideals considered in Proposition 3.4.1. Similarly one sees that the coordinate rings K [Mτ ] of partial or full flag varieties are algebras with straightening law on the posets of minors that generate them. All these posets are distributive lattices, and algebras with straightening law on such posets are Cohen–Macaulay. See De Concini et al. [104] and [61, Sect. 5] for the theory of algebras with straightening law. (b) The “filtration by shapes” as indicated in Theorem 3.3.1 can be used for a deformation process. This allows one to deduce properties of the determinantal ring K [X ]/It from the “semigroup of shapes” occurring in it. See Sect. 7.4.

84

3 Determinantal Ideals and the Straightening Law

Localization and Induction on Size The straightening law allows us to prove basic properties about the ideals It without much effort. Additionally we need an induction on the size of the minors by the inversion of matrix entries. Let us first state a version of the “standard inversion argument” that works for matrices with arbitrary entries. Lemma 3.4.4 Let S be a ring (needless to say commutative and unitary) and let X = (xi j ) be a m × n matrix with entries in S. Assume xmn is invertible in S. Then for every t ∈ N one has It (X ) = It−1 (Y ) where Y = (yi j ) is the (m − 1) × (n − 1) −1 . matrix with entries yi j = xi j − xin xm j xmn Proof It is easy to see that the ideals of minors of a matrix do not change under elementary row and column operations on the matrix. Clearly X can be transformed with row and column operations into the matrix X with block decomposition 

Y 0 X = 0 1



so that It (X ) = It (X ). Finally the equality It (X ) = It−1 (Y ) follows by Laplace expansion.  If the entries of the matrix are indeterminates, we can be more precise. Lemma 3.4.5 Let X = (X i j ) and Y = (Yi j ) be matrices of indeterminates over K of sizes m × n and (m − 1) × (n − 1), respectively. Then the substitution −1 , X i j → Yi j + X m j X in X mn X m j → X m j , X in → X in ,

1 ≤ i ≤ m − 1, 1 ≤ i ≤ m,

1 ≤ j ≤ n − 1, 1 ≤ j ≤ n,

induces an isomorphism −1 ∼ −1 ] = K [Y ][X m1 , . . . , X mn , X 1n , . . . , X m−1,n ][X mn ] K [X ][X mn

(3.5)

under which the extension of It (X ) is mapped to the extension of It−1 (Y ). Therefore one has an isomorphism −1 ∼ −1 ] = (K [Y ]/It−1 (Y ))[X m1 , . . . , X mn , X 1n , . . . , X m−1,n ][X mn ] (K [X ]/It (X ))[xmn

(here xmn denotes the residue class of X mn in K [X ]/It (X )). −1 , Xmj → Proof The substitution has an inverse, namely Yi j → X i j − X m j X in X mn X m j , X in → X in , and so induces the isomorphism (3.5). −1 ], one can apply elementary row and column Since X mn is invertible in K [X ][X mn operations to the matrix X with pivot element X mn . Now one just “writes” Yi j for the

3.4 Determinantal Rings

85

entries in the rows 1, . . . , m − 1 and the columns 1, . . . , n − 1 of the transformed matrix. With this identification one has It (X ) = X mn It−1 (Y ) = It−1 (Y ) in the ring −1 ].  K [X ][X mn Lemma 3.4.5 reveals its usefulness in the proof of Theorem 3.4.6 (a) The ring K [X ]/It (X ) is a normal domain of dimension (m + n − t + 1)(t − 1). (b) Its singular locus is the variety of the ideal It−1 /It . Proof The case t = 1 is trivial. Suppose that t > 1. Set R = K [X ]/It . We claim that the residue class xmn of X mn is a nonzerodivisor on R. In fact, the product of X mn = [m | n] and a standard bitableau is a standard bitableau, and if x is a linear combination of standard bitableaux without a factor of size ≥ t, then so is X mn x. Corollary 3.4.2 now implies our claim. The argument shows even more: X mn is a nonzerodivisor modulo every ideal of K [X ] generated by an ideal N in the partially / N. ordered set M(X ) such that X mn ∈ −1 ], In order to verify that R is a domain, it suffices to prove this property for R[xmn and to the latter ring we can apply induction via Proposition 3.4.5. The dimension formula follows by the same induction, since an affine domain does not change dimension upon the inversion of a nonzero element. Let m be the maximal ideal of R generated by the residue classes xi j of the indeterminates. Clearly Rm is not regular if t ≥ 2. Every other prime ideal p does / p. Then Rp is of not contain one of the xi j , and by symmetry we can assume xmn ∈ −1 ]q with S = K [Y ]/It−1 (Y ). It follows that Rp is regular if the form S[X in , X m j , X mn and only if Sq∩S is regular. Moreover, p contains It−1 (X )/It (X ) if and only if q ∩ S contains It−2 (Y )/It−1 (Y ). Again we can apply induction to prove the claim about the singular locus. For normality we use Serre’s normality criterion, which has already been used in the proof of Proposition 1.6.2. We know the singular locus, and by the dimension formula it has codimension ≥ 2. Now it is enough to show that depth Rp ≥ 2 if dim Rp ≥ 2. If p = m, we obtain this by induction as above, and it remains to show that depth Rm ≥ 2. The set of minors δ of size < t has a smallest element with respect to , namely ε = [1 . . . t − 1 | 1 . . . t − 1]. The same argument that we have applied to X mn shows that ε is a nonzerodivisor modulo It . Moreover, the ideal J = It + (ε) is generated by an ideal in M(X ), and so X mn is a nonzerodivisor modulo J . It follows that m contains a regular R-sequence of length 2.  Remark 3.4.7 Theorem 3.4.6 can be generalized: the domain property stated in (a) holds for domains in place of the coefficient rings, and normality only requires a normal domain; see [61, Sect. 6.A]. For a more general version of (b) see [61, (6.10)](b). Remark 3.4.8 Theorem 3.4.6 can be used for the computation of the graded automorphism group of K [X ]/It . An arbitrary K -automorphism of K [X ]/It must leave

86

3 Determinantal Ideals and the Straightening Law

the prime ideal It−1 /It invariant since it defines the singular locus of K [X ]/It . Thus it induces a (graded) automorphism of K [X ]/It−1 . This argument reduces the problem to the case t = 2 for which a toric approach is available. See Bruns and Gubeladze [49]. The connected component of unity G 0 is isomorphic to GL(m, K ) × GL(n, K )/K ∗ where GL(m, K ) and GL(n, K ) act by linear substitutions (see Sect. 3.6) and K ∗ is embedded diagonally. If m = n the group of graded automorphism is connected and for m = n it is given by G 0 ∪ τ G 0 where τ is induced by the transposition of X . This result goes back to Frobenius [130, Vol. III, p. 99]. See Waterhouse [316] for a group scheme theoretical approach in a somewhat more general context. It has also been applied to minors of symmetric matrices and pfaffians of alternating ones. (The symmetric case can be reduced to the toric situation as above.)

The Cohen–Macaulay Property The key property of the rings K [X ]/It is their Cohen–Macaulayness. It has been proved in several, partly overlapping ways: (a) The first proof in the general case is an induction over a “principal radical system” (Hochster and Eagon [179]; also see [61, Chap. 12]). (b) The ASL approach by De Concini et al. [104, 115]; it is extensively discussed in [61]. (c) K [X ]/It can be realized as a ring of invariants of a reductive linear group acting linearly on a polynomial ring; see [61, Sect. 7]. In characteristic 0 it is therefore a direct summand of such polynomial ring. Therefore the Hochster–Roberts theorem [52, Theorem 6.5.1] applies to it. Variants of this argument use that determinantal rings define rational singularities or are F-regular in characteristic p > 0. See Chap. 7. (d) Cohomology computation based on combinatorial and geometric methods; see Sect. 10.8. (e) Explicit minimal free resolutions constructed by Eagon and Northcott ([113], [61, Chap. 2]) for maximal minors in arbitrary characteristic and Lascoux [231] in characteristic 0 via representation theory; these will be reviewed in Sect. 11.11. (f) Proofs by deformation, namely (i) Gröbner deformation (Chap. 4), (ii) toric deformation (Sect. 6.7), (iii) equivariant deformation (Sect. 7.4). The Cohen–Macaulay property of K [X ]/It can be generalized: under suitable hypotheses, K [X ] can be replaced by a more general Noetherian ring R, and X can be replaced by an m × n matrix A with entries in R. The crucial hypothesis is that grade It (A) attains the same value as It (X ), namely (m − t + 1)(n − t + 1). This is the maximal height of It (A) in general: if It (A) = R, then

3.4 Determinantal Rings

87

height It (A) ≤ (m − t + 1)(n − t + 1); see [113, Theorem 3] or [61, (2.1)]. If one has equality in this bound for grade instead of height, then It (A) is a perfect ideal, namely an ideal of finite projective dimension equal to its grade: Theorem 3.4.9 (a) Let B be a Noetherian ring, X an m × n matrix of indeterminates over B, and 1 ≤ t ≤ min(m, n) an integer. Then It (X ) is a perfect ideal of grade (m − t + 1)(n − t + 1) in B[X ]. (b) Let R be a Noetherian ring, A = (ai j ) an m × n matrix with entries in R, and 1 ≤ t ≤ min(m, n) an integer. If grade It (A) = (m − t + 1)(n − t + 1), then It (A) is perfect. Proof The proof of (b) is based on classical arguments of “generic perfection” developed in [61, Sect. 3]. Let X be a matrix of indeterminates over Z. Then Z[X ]/It (X ) is a free Z-module. This follows from the straightening law: the Plücker relations are defined over Z. Therefore the standard bitableaux generate Z[X ] as a Z-module. Their linear independence over Z follows from their linear independence over Q that we have proved already. In the same way as for a field of coefficients, it follows that Z[X ]/It (X ) is a free Z-module with Z-basis given by the standard bitableaux whose row lengths are bounded by t − 1. Let p be a prime number. Then Z[X ]/It (X ) ⊗ (Z/ pZ) ∼ = (Z/ pZ)[X ]/It (X ) is a Cohen–Macaulay ring. But this means that It (X ) ⊗ (Z/ pZ) is a perfect ideal; see [52, Corollary 2.2.15]. Now it follows from [61, (3.3)] that Z[X ]/It (X ) is a perfect Z[X ]-module. We make R a Z[X ]-algebra by the substitution X i j → ai j , 1 ≤ i ≤ m, 1 ≤ j ≤ n. This yields the isomorphism Z[X ]/It (X ) ⊗ R ∼ = R/It (A). Finally one uses the hypothesis on grade It (A) and [61, (3.5)] to conclude the proof. For (a) we can directly quote [61, (3.3)], using that It (X ) is perfect in (Z/ pZ)[X ].  We can now apply [52, Theorem 2.1.5]: Corollary 3.4.10 With the notation and hypotheses of Theorem 3.4.9 assume that R is a Cohen–Macaulay ring. Then R/It (A) is Cohen–Macaulay. Note that the hypothesis grade It (A) = (m − t + 1)(n − t + 1) implies It (A) = R, since grade R = ∞ by definition. As we will see in Remark 4.4.14, K [X ]/It is Gorenstein if (and only if) m = n. This property can be generalized in the same fashion to the general case: Corollary 3.4.11 With the notation and hypotheses of Theorem 3.4.9 assume that R is a Gorenstein ring. If m = n, then R/It (A) is Gorenstein. The reader is invited to deduce the corollary from [61, (3.6)(c)]. Both corollaries cover the situation in which R = B[X ], A = X , and B is a Cohen–Macaulay or Gorenstein ring, respectively. This follows from Theorem 3.4.9(a).

88

3 Determinantal Ideals and the Straightening Law

3.5 Powers and Products of Determinantal Ideals In this section we want to determine the primary decomposition of powers and, more generally, of products J = It1 · · · Itu , of determinantal ideals. It is easy to see that only the ideals It containing J can be associated to J . In fact, suppose p is a prime ideal in R = K [X ]/J different from the irrelevant maximal ideal m. Then p does not contain one of the xi j , and so we may invert xi j without losing the extension of p as an associated prime ideal of (R/J )[xi−1 j ]. But now the Induction Lemma 3.4.5 applies: by symmetry we can assume (i, j) = (m, n). Thus we have to find primary components of J with respect to the ideals It . Immediate, and as we will see, optimal candidates are the symbolic powers of the ideals It . We determine them first.

Symbolic Powers of Determinantal Ideals The ideal p = It is a prime ideal in the regular ring A = K [X ]. With each such prime ideal one associates a valuation on the quotient field of A as follows. We pass to the localization P = Ap and let q = pP. Now we set vp (x) = max{i : x ∈ qi } ∞ i i+1 q /q is a for all x ∈ P, x = 0, and vp (0) = ∞. The associated graded ring i=0 polynomial ring over the field P/q: P is a regular local ring, and its maximal ideal q is therefore generated by a P-sequence. (See [52, 1.1.18]. We only need that the associated graded ring is an integral domain). This implies vp (x y) = vp (x) + vp (y) for all x, y ∈ P, and vp (x + y) ≥ min(vp (x), vp (y)). To sum up: vp is a discrete valuation on P and can be extended to the quotient field QF(P) = QF(A). By definition, the ith symbolic power is p(i) = pi P ∩ A. With the help of vp we can also describe it as p(i) = {x ∈ A : vp (x) ≥ i}. If p = m is the maximal irrelevant ideal of K [X ] and f is a homogeneous polynomial, then vp ( f ) is the ordinary total degree of f . The description of I = p(i) by vp implies immediately that this ideal is integrally closed: an equation x n + a1 x n−1 + · · · + an−1 x + an = 0 with a j ∈ I j for j = 1, . . . , n cannot hold for x ∈ A, x ∈ / I . (See Swanson and Huneke [304] for a comprehensive introduction to integral dependence.) For the choice A = K [X ], p = It , we denote vp by γt . For a minor δ we claim  γt (δ) =

0, |δ| < t, |δ| − t + 1, |δ| ≥ t.

In this definition we assume that t ≥ 1. For t = 1, this follows immediately, since γ1 (δ) = |δ| is the total degree of δ. Assume now that t > 1. If |δ| < t, then δ ∈ / It , and

3.5 Powers and Products of Determinantal Ideals

89

therefore γt (δ) = 0. Suppose that |δ| ≥ t. It is useful to note that γt (δ) only depends on |δ|: minors δ and δ of the same size are conjugate under an isomorphism of K [X ] that leaves p invariant. We can therefore assume that δ = [m − t + 1 . . . m | n − t + 1 . . . n]. The substitution in the induction Lemma 3.4.5 maps δ to a minor of size |δ| − 1, and reduces t by 1, so we conclude by induction. We extend the function γt to sequences of integers: γt (s1 , . . . , su ) =

u 

max(si − t + 1, 0).

i=1

For instance, if λ = (4, 3, 3, 1), then γ4 (λ) = 1, γ3 (λ) = 4, γ2 (λ) = 7, γ1 (λ) = 11. Figure 3.5 illustrates the computation of γ3 : we must count all boxes to the right of the vertical line. It is clear that γt (s1 , . . . , su ) = γt (δ1 · · · δu ) for minors δi with |δi | = si , i = 1 . . . , u. In Sect. 3.3 we have introduced the partial order of shapes based on the functions αk . We can also use the functions γt for such a comparison, and the next result shows that we get the same partial order: Lemma 3.5.1 Let ρ = (r1 , . . . , ru ) and σ = (s1 , . . . , sv ) be nonincreasing sequences of integers. We have ρ ≤ σ if and only if γt (ρ) ≤ γt (σ) for all t. Proof We use induction on u. The case u = 1 is trivial. Furthermore, there is nothing to show if ri ≤ si for all i. It remains the case in which r j > s j for some j. Next, if ri = si > 0 for some i, then we can remove ri and si and compare the shortened sequences by induction. Thus we may assume that ri = si for all i with ri > 0. Let k be the smallest index with sk < rk . It is easy to see that k ≥ 2 if ρ ≤ σ or γs1 (ρ) ≤ γs1 (σ). Extending σ by 0 if necessary we may assume that k ≤ v. We have sk−1 > rk−1 ≥ rk > sk ; in particular, sk−1 ≥ sk + 2. Define si = si for

= sk−1 − 1, sk = sk + 1, and σ = (s1 , . . . , sv ). Pictorially: we i = k − 1, k, sk−1 move a box from row k − 1 to row k, as illustrated by Fig. 3.6. Suppose first that ρ ≤ σ. It follows easily that ρ ≤ σ , and moreover, that γt (σ ) ≤ γt (σ) for all t. A second inductive argument allows us to assume that γt (ρ) ≤ γt (σ ) for all t, which implies that γt (ρ) ≤ γt (σ) for all t, as desired.

Fig. 3.5 Counting boxes for γ3

90

3 Determinantal Ideals and the Straightening Law

Fig. 3.6 Moving a box from row k −1 to row k

Conversely, suppose that γt (ρ) ≤ γt (σ) for all t. Since σ ≤ σ, it is enough to show that γt (ρ) ≤ γt (σ ) for all t. One has γt (σ ) = γt (σ) for t ≤ sk + 1, and γt (σ ) = γt (σ) − 1 for t = sk + 2, . . . , sk−1 . It is easy to check that γt (ρ) < γt (σ) for t > rk , so it suffices to verify that γt (ρ) < γt (σ) when sk + 2 ≤ t ≤ rk . Suppose that γt (ρ) = γt (σ) for some t in this range. Since sk < t − 1, the only parts that contribute to γt−1 (σ) are s1 , . . . , sk−1 . Since rk ≥ t, at least r1 , . . . , rk contribute to γt−1 (ρ). It follows that k − 1 = γt−1 (σ) − γt (σ) ≥ γt−1 (ρ) − γt (ρ) ≥ k, 

a contradiction that concludes our proof. Setting

I σ = Is1 · · · Isu

for σ = (s1 , . . . , su ) we now describe the symbolic powers of the ideals It . Theorem 3.5.2 One has

It(k) =



I σ.

σ=(s1 ,...,su ) γt (σ)≥k

It(k) has a K -basis consisting of all the standard bitableaux  with γt () ≥ k. Proof If t = 1 then the right hand side is just the ideal of elements of total degree ≥ k, and there is nothing to prove. If t > 1, then X mn is a nonzerodivisor modulo It(k) (by definition of the symbolic power). As we will see, X mn is also a nonzerodivisor modulo the right hand side and therefore we can invert it. This transforms all sizes in the right way, and equality follows by induction on t. We have noticed in Theorem 3.3.1 that straightening does not decrease shape. In conjunction with Lemma 3.5.1 this implies γt () ≥ γt () for all standard bitableaux  in the standard representation of a bitableau . Thus the right hand side has a K basis consisting of all  with γt () ≥ k. Since multiplication by X mn maps standard bitableaux to standard bitableaux, without affecting the value of γt (since t > 1), the result follows.  Only finitely many summands are needed to describe It(k) in Theorem 3.5.2. The simplest nontrivial case is t = k = 2, when we have

3.5 Powers and Products of Determinantal Ideals

91

I2(2) = I22 + I3 , since every other summand is contained in I22 or I3 . If m ≤ 2 or n ≤ 2, then I22 = I2(2) . This observation is easily generalized to the following result of Trung [252]. Corollary 3.5.3 The symbolic powers of the ideal of maximal minors coincide with the ordinary ones. Proof Suppose that m = min(m, n). Then a bitableau  has γm () ≥ k if and only if the first k factors have size exactly m.  Remark 3.5.4 In the proof of Theorem 10.7.1 we will need that there is at least one element of total degree kt in a minimal system of generators of the symbolic power It(k) for an arbitrary matrix X of indeterminates. Choose the doubly initial bitableau  of shape (t k ) and assume it lies in I1 It(k) . Clearly  ∈ Itk ⊂ It(k) . Let X be the t × t submatrix in the left upper corner of X , and consider the natural K -algebra retract K [X ] → K [X ]. It respects standardness of a bitableau (or maps it to 0) as well as the function γt . If  ∈ I1 (X )It (X )(k) , then  ∈ I1 (X )It (X )(k) = I1 (X ). But this clearly impossible. (The reader is invited to find an argument for an arbitrary standard bitableau of shape (t k ).)

Primary Decomposition of Products and Powers The primary decomposition of products of the ideals It depends on the characteristic of K . This indicates that the straightening law alone is not sufficient to prove it. In fact, the straightening law enters the proof of the next result only via Theorem 3.5.2. Theorem 3.5.5 Let ρ = (r1 , . . . , ru ) be a nonincreasing sequence of integers and suppose that char K = 0 or char K > min(ri , m − ri , n − ri ) for i = 1, . . . , u. We have r1  (γ (ρ)) Iρ = It t . t=1

In particular, I ρ is integrally closed. The theorem was proved by De Concini et al. [103] in characteristic 0 and generalized in [61]. The inclusion ⊂ is independent of the characteristic, and it follows from the fact that γt (x) ≥ γt (ρ) for all x ∈ I ρ . Before we indicate the proof of the reverse inclusion, let us have a look at the first nontrivial case, namely I22 = I14 ∩ (I3 + I22 ). For equality we must prove that the degree ≥ 4 elements in I3 , namely those of I1 I3 , are contained in I22 . This type of containment, stated in Lemma 3.5.6, is the crucial

92

3 Determinantal Ideals and the Straightening Law

point in the proof of the theorem. It is the ideal-theoretic analogue of the passage from σ to σ in the proof of Lemma 3.5.1. In view of Theorem 3.5.2 it is enough to show that a product  = δ1 · · · δ p is in I ρ if γt () ≥ γt (ρ) for all t. For the analogy to the proof of Lemma 3.5.1 set σ = ||. Then the same induction works, since Lemma 3.5.6 implies in the critical case that  is a linear combination of bitableaux  , | | = σ . As pointed out above, the symbolic powers are integrally closed, and so is their intersection. Lemma 3.5.6 Let u, v be integers, 0 ≤ u ≤ v − 2. Suppose that char K = 0 or char K > min(u + 1, m − (u + 1), n − (u + 1)). Then Iu Iv ⊂ Iu+1 Iv−1 . Lemma 3.5.6 is, in a sense, the opposite of straightening: while straightening does not increase shape, Lemma 3.5.6 does exactly this. The proof of the lemma is based on a symmetrization argument, which is responsible for the condition on the characteristic. We refer the reader to [61, (10.10)] for a complete proof of the lemma. We content ourselves here with illustrating the basic step, and then explaining the idea of the proof by means of the first significant case. Lemma 3.5.7 Suppose that 0 ≤ u ≤ v − 2. Then: (a) u+1  (−1)i [a1 . . . ai . . . au+1 |b1 . . . bu ][ai , c2 . . . cv |d1 . . . dv ] i=1

is a Z-linear combination of products κλ of minors κ and λ, |κ| = u + 1, |λ| = v − 1. (b) Suppose that {a1 , . . . , au } ⊂ {c1 , . . . , cv }. Then [a1 . . . au |b1 . . . bu ][c1 . . . cv |d1 . . . dv ] is a Z-linear combination of products κλ of minors κ and λ, |κ| = u + 1, |λ| = v − 1. Proof (a) By Laplace expansion and contraction one derives the identity u+1 

(−1)i [a1 . . . ai . . . au+1 | b1 . . . bu ][ai , c2 . . . cv | d1 . . . dv ]

i=1

= (−1)u

v  (−1) j+1 [a1 . . . au+1 | b1 . . . bu d j ][c2 . . . cv | d1 . . . dj . . . dv ]. j=1

(b) We may assume that c1 = ai for i = 1, . . . , u, in which case (b) becomes a special case of (a) with c1 in the role of a1 : only the first summand in (a) is = 0 since the second factor in all other summands is the determinant of a matrix with two equal rows. 

3.5 Powers and Products of Determinantal Ideals

93

It is clear from the discussion above that the first critical case of Lemma 3.5.6 is to prove the containment I1 I3 ⊂ I22 , under the hypothesis that 2 is a unit. To this end, we prove that (3.6) 2[1 | 1][2 3 4 | [2 3 4] ∈ I22 . We check this containment over the base ring Z. By Lemma 3.5.7(a) one has [1 | 1][2 3 4 | 2 3 4] − [2 | 1][1 3 4 | 2 3 4] ∈ I22 and we claim that [1 | 1][2 3 4 | [2 3 4] + [2 | 1][1 3 4 | 2 3 4] ∈ I22

(3.7)

as well. By addition we then obtain (3.6). In order to prove (3.7) we set ⎛

⎞ X 11 + X 21 . . . X 14 + X 24  ... X 34 ⎠ . X = ⎝ X 31 X 41 ... X 44 (This is the symmetrization step.) Then, by Lemma 3.5.7(b), [1 | 1]  X [1 2 3 | 2 3 4]  X ∈ X )2 . Note that I2 (  X )2 ⊂ I22 . Moreover, I2 (  [1 | 1]  X [1 2 3 | 2 3 4]  X =

[1 | 1][1 3 4 | 2 3 4] + [2 | 1][1 3 4 | 2 3 4] + [1 | 1][2 3 4 | 2 3 4] + [2 | 1][2 3 4 | 2 3 4].

Now (3.6) is proved since the first and the last summand lie in I22 , which follows again by Lemma 3.5.7(b). Because of Lemma 3.5.1 we can replace the γ-functions by α-functions (introduced in (3.3)) in the description of the standard bases of products and powers. For powers we obtain a very simple statement (of course, αk () = αk (||)): Proposition 3.5.8 Suppose that char K = 0 or char K > min(t, m − t, n − t). Then Itk has a basis consisting of all standard bitableaux  with αk () ≥ kt. It is possible to derive this result from Theorem 3.5.5, but it is easier to prove it directly. Let V be the K -vector space generated by all standard bitableaux  = δ1 · · · δu with αk () ≥ kt The inclusion Itk ⊂ V follows as before from the fact that straightening does not decrease shape. The reverse inclusion follows from Lemma 3.5.6: if |δi | < t for some i, 1 ≤ i ≤ k, then there is also an index j in this range such that |δ j | > t. Lemma 3.5.6 allows us to increase |δi | at the expense of |δ j |. Remark 3.5.9 (a) Since we are mainly interested in asymptotic properties, we do not discuss when the decomposition in Theorem 3.5.5 is irredundant; see [61, (10.12)] and [61, (10.13)] for a precise result. Roughly speaking, the It -primary component is irredundant if It appears in the product or if the number of factors Iu with t < u
(σ)

and

whose bases are the standard bitableaux of shape ≥ σ and > σ, respectively. The theorem implies that I (σ) and I>(σ) are in fact ideals generated by all bitableaux of shape ≥ σ and > σ respectively. By Lemma 3.5.1 and Theorem 3.5.2 one has I (σ) =



(γt (σ))

It

.

t

Moreover, I (σ) = I σ in characteristic 0, as stated in Theorem 3.5.5. We want to interpret Theorem 3.3.1 as a description of the G-module I (σ) /I>(σ) . To this end we introduce two vector subspaces L σ and σL of K [X ]: L σ is generated by all bitableaux (R | In(σ)) and σL is generated by all bitableaux (In(σ) | C). Evidently, L σ is a GL(m, K )-submodule of K [X ] and σL is a GL(n, K )-submodule. The group G acts on L σ ⊗ σL by

96

3 Determinantal Ideals and the Straightening Law

(g, h)(x ⊗ y) = gx ⊗ hy. Let [m] = (1, . . . , m). We have already introduced the symbol M[m] to denote the set of all row initial minors, that is M[m] = {[1 . . . k | c1 . . . ck ] : k = 1, . . . , m, 1 ≤ c1 < · · · < ck ≤ n}. Similarly the set of all column initial standard minors will be denoted by N[n] : explicitly, N[n] = {[r1 < · · · < rk | 1 . . . k] : k = 1, . . . , n, 1 ≤ r1 < · · · < rk ≤ m}. Then, by the definition of L σ and σL we have: Proposition 3.6.1  σ

L σ = K [N[n] ]

and



σL

= K [M[m] ].

σ

The algebras K [M[m] ] and K [N[n] ] will play an important role in Chap. 6. All results about K [M[m] ] can be transferred to K [N[m] ] via the transposition of the matrix X . Theorem 3.3.1 can now be reformulated as follows: Theorem 3.6.2 (a) The standard bitableaux (R | In(σ)) are a K -basis of L σ . An analogous statement holds for σL.   (b) The assignment (R | In(σ), (In(σ) | C) → (R | C) induces an isomorphism L σ ⊗ σL ∼ = I (σ) /I>(σ) of G-modules. In addition to Theorem 3.3.1 one must only check that the vector space isomorphismin (b) is compatible with the G-actions on both partners. Note that the assignment (R | In(σ)), (In(σ) | C) → (R | C) also defines an embedding of L σ ⊗ σL into K [X ], but this embedding is not a homomorphism of G-modules: the vector subspace spanned by all standard bitableaux of shape σ is not closed under the action of G in general. In representation theory, Theorem 3.6.2 is the characteristic free version of Cauchy’s formula. It will be generalized to sheaves in Theorem 9.8.7. Remark 3.6.3 Theorem 3.3.1 holds for general rings B of coefficients, as pointed out in Remark 3.3.2. Consequently Theorem 3.6.2 generalizes in the same way, including the action of G.

3.6 Representation Theory

97

∼ I (σ) /I (σ) , especially in characterisIn order to analyze the structure of L σ ⊗ σL = > − tic 0, we introduce the subgroup U (n, K ) consisting of all lower triangular matrices in GL(n, K ) that have entries equal to 1 in all diagonal positions. Analogously one defines U + (n, K ). The key argument is − Lemma 3.6.4 Let K be an infinite field. Let M ⊂ K [X ] be a U (m, K )-submodule of K [X ] and f ∈ M \ {0}. Let f = ∈S a  be the unique representation of f as sum of standard bitableaux  = (R | C ) with a ∈ K \ {0} for all  ∈ S. Then there exists a nonempty subset S0 ⊂ S for which the bitableaux (In(||) | C ),  ∈ S0 , are pairwise different, and b ∈ K , b = 0, such that

g=



b (In(||) | C ) ∈ M ∩

∈S0



τL .

τ

The proof of the U − (n, K )-variant is given in full detail in the Lemmas (4.8) and (11.6) of [61], even if only a consequence of Lemma 3.6.4, namely Lemma 3.6.5(a), is stated in [61, (11.6)]. In order to prove Lemma 3.6.4 one applies elementary row transformations in U − (m, K ) to f . We explain the procedure by an example. Let f = [1 3 | 1 3]. For w ∈ K , w = 0, we apply the elementary transformation X 3i → X 3i + w X 2i , i = 1, 2, 3 and X i j → X i j else. Then [1 3 | 1 3] → [1 3 | 1 3] + w[1 2 | 1 3], and it follows immediately that [1 2 | 1 3] ∈ M. (In the general case one needs the infinity of K .) Lemma 3.6.5 Let K be an infinite field. (a) Every nonzero U − (m, K )-submodule of K [X ] intersects

 σ

σL

nontrivially.

(b) Every nonzero U − (m, K )-submodule V of L σ contains (In(σ) | In(σ)). (c) L σ and L τ are isomorphic GL(m, K )-modules if and only if σ = τ . Proof (a) is a less precise version of Lemma 3.6.4.   (b) By (a) one has 0 = V ∩ τL ⊂ L σ ∩ τL, and both L σ and τL have bases of standard bitableaux. The only standard bitableau in the intersection is (In(σ) | In(σ)). (c) Let σ = (s1 , . . . , su ). Any nonzero element of L σ that is U − (m, K )-invariant spans a one dimensional U − (m, K )-submodule and, hence by (b), must contain (In(σ) | In(σ)). Since (In(σ) | In(σ)) is itself invariant under the action of U − (m, K ) on L σ , then it spans the subspace of U − (m, K )-invariants in L σ . This subspace is preserved by the action of the diagonal matrices, and in order to prove the desired conclusion, it suffices to check that this action determines the shape σ. We let σ ∗ = (s1∗ , . . . , sv∗ ) denote the shape dual to σ, where si∗ = |{ j : s j ≥ i}| and v = s1 . A s∗ s∗ diagonal matrix (d1 , . . . , dm ) acts on (In(σ) | In(σ)) with eigenvalue d11 · · · dv v and  thus it determines σ ∗ . Since σ = σ ∗∗ , this in turns determines the shape σ.

98

3 Determinantal Ideals and the Straightening Law

The lemma shows that the groups U − (m, K ) and U + (n, K ) can be used to detect shapes. It allows us to precisely describe the elements of K [X ] that are invariants for U = U − (m, K ) × U + (n, K ). Theorem 3.6.6 Let K be an infinite field. (a) The subalgebra of polynomials in K [X ] that are invariant under the action of U is generated by the bi-initial minors [1 . . . t | 1 . . . t], t = 1, . . . , m = min(m, n). It ring in m indeterminates.  is therefore isomorphic to a polynomial − (b) σ σL = K [M[m] ] is the ring of U (m, K )-invariants, and σ L σ = K [N[m] ] is the ring of U + (n, K )-invariants. Proof (a) follows from (b) since the intersection of the algebras in (b) that both have standard bitableaux bases is the subalgebra whose K -basis is given by the doubly initial bitableaux. For (b) it is enough to consider U − (m, K ). It is obvious that K [N[m] ] is invariant generates under the action of U − (m, K ). Conversely, an invariant polynomial f  a 1-dimensional U − (m, K )-submodule, and therefore it is contained in σ σL by Lemma 3.6.5.  The groups GL(m, K ) and their products are linearly reductive in characteristic 0: a linear algebraic group H is linearly reductive if every finite-dimensional H -module splits into a direct sum of irreducible (or simple) H -modules, i.e., H -modules U whose only H -submodules are U and 0. For every irreducible H -submodule U , the sum of all H -submodules W of V that are isomorphic to U as H -modules evidently depends only on the isomorphism type of U , and can therefore be called an isotypic component of V . Linear reductivity has the following consequences: (a) Every H -submodule U of an H -module V has a complement: V = U ⊕ W for an H -submodule W of V . (b) An H -module V decomposes into the direct sum of its isotypic components. Since the group actions we are interested in restrict to the graded components of the polynomial ring, the assumption of finite dimension is automatic for us. Remark 3.6.7 It follows from Lemma 3.6.5 that in characteristic 0, the GL(m, K )modules L σ are irreducible. If not, then there is a decomposition L σ = U ⊕ W , where U, W are nonzero GL(m, K )-submodules. It follows that U, W are also U − (m, K )submodules, so they both contain (In(σ) | In(σ)), which is impossible. Similar reasoning shows that σL is irreducible as a GL(n, K )-module. Theorem 3.6.8 Let K be a field of characteristic 0. (a) There exists a G-submodule Mσ of K [X ] such that I (σ) = Mσ ⊕ I>(σ) and (consequently) Mσ ∼ = L σ ⊗ σL. (b) Mσ is an irreducible G-module, containing (and therefore generated as a Gmodule by) the bitableau (In(σ) | In(σ)). In particular, Mσ is unique.

3.6 Representation Theory

99

(c) Mσ and Mτ are isomorphic as G-modules if and only if σ = τ . (d)  Mσ . K [X ] = σ

is the decomposition of K [X ] into its isotypic G-components. In representation theory, Theorem 3.6.8 is Cauchy’s formula in characteristic zero. It will be generalized to a sheaf version in Sect. 11.5, where also other descriptions of the G-modules Mσ will be given. Proof of Theorem 3.6.8 (a) The existence of a G-submodule Mσ such that I (σ) = Mσ ⊕ I>(σ) follows from the linear reductivity of G. Using Theorem 3.6.2, Mσ ∼ = I (σ) /I>(σ) ∼ = L σ ⊗ σL. (b) It suffices to check that a G-submodule  V of Mσ contains (In(σ) | In(σ)). Consider any such V , and note that V ∩ τ (τL) = 0 by Lemma 3.6.5. Since V is a GL(n, K )-module, and τL are irreducible pairwise nonisomorphic GL(n, K )modules, we have V ⊃ τL for some τ . Since I (σ) ⊃ τL (resp. I>(σ) ⊃ τL) if and only if τ ≥ σ (resp. τ > σ), it follows that V ⊃ σL, and thus V contains (In(σ) | In(σ)). (c) We have an isomorphism of G-modules Mσ ∼ = Mτ if and only if L σ ∼ = L τ as ∼ GL(m, K )-modules, and σL  = τL as GL(n, K )-modules. This is equivalent to σ = τ . (d) By (b) and (c), Mσ ∩ τ =σ Mτ = 0, so one only needs to show that the direct sum fills K [X ]. This is a dimension argument (degree by degree): K [X ] is the direct sum of the subspaces Bσ spanned by the standard bitableaux of size σ. As a vector  space, Bσ is isomorphic to L σ ⊗ σL, hence it has the same dimension as Mσ . At this point one can proceed to the study of G-stable ideals of K [X ] in characteristic 0 in Sect. 11.8 (also see [61, 11.D]).

Subspaces Defined by Shape The subspaces Mσ of K [X ] do in general not have bases of standard bitableaux. The application of any g ∈ G to a standard bitableau of shape σ produces a K linear combination of bitableaux of shape σ, but their straightening usually requires standard bitableaux of shape > σ. In characteristic 0 we want to characterize those subspaces of K [X ] that are simultaneously G-stable and have a standard bitableaux basis. In order to cover many cases that we will encounter in later chapters and for a characteristic free approach we say that a K -subspace V of K [X ] is defined by shape if the standard bitableaux contained in V span V as a vector space and, for any bitableau , the membership  ∈ V only depends on the shape ||. Lemma 3.6.9 Let V ⊂ K [X ] be a subspace defined by shape and  ∈ V be a bitableau. Then V contains all bitableaux  such that γ1 () = γ1 () and || ≥ ||.

100

3 Determinantal Ideals and the Straightening Law

Proof Let σ = || and τ = ||. By induction in the poset of all shapes ρ with γ1 (ρ) = γ1 (σ) it is enough to prove that  ∈ V if τ is a cover of σ in this poset. (By definition, τ is a cover of σ if σ < τ and there is no η strictly between σ and τ . ) By the definition of V we are free to choose  and it is enough to find a single  ∈ V as desired. There are two possibilities for a cover of σ. Both are shown in Fig. 3.7. Pictorially speaking, a box of σ travels (i) one row up or (ii) one column to the right. (Without the restriction on γ1 there is further cover: we add a row with a single box at the bottom of σ.) Case (i) is slightly more complicated and we discuss it in detail. Let σ = (s1 , . . . , su ) and suppose that τ = (s1 , . . . , sv−1 + 1, sv − 1, sv+1 , . . . , su ). For simpler notation we set p = sv and q = sv−1 . We choose the minors δ1 , . . . , δu of  as follows: all of them except δv−1 and δv are row and column initial and δv−1 = [1 . . . p − 1 p + 1 . . . q + 1 | 1 . . . q],

δv = [1 . . . p | 1 . . . p − 1 q + 1].

If one straightens the product δv−1 δv and substitutes the resulting linear combination of standard bitableaux for δv−1 δv in the product , one obtains the standard representation of . In order to show that a standard bitableau of shape τ appears in this standard representation, it is therefore enough to discuss the case in which u = 2, v = 2. A further simplification is possible: we substitute 1 for the diagonal indeterminates X 11 , . . . , X p−1 p−1 and 0 for all other indeterminates X i j with i ≤ p − 1 or j ≤ q − 1. this reduces our claim to the product [2 . . . q + 1 | 1 . . . q] · [1 | q + 1], for which we must show that [1 . . . q + 1 | 1 . . . q + 1] appears with a coefficient ±1 in its standard representation. The only nonstandard product in the Laplace expansion of this minor along column q + 1 is indeed [2 . . . q + 1 | 1 . . . q] · [1 | q + 1], whence its coefficient of [1 . . . q + 1 | 1 . . . q + 1] is (−1)q . By a similar simplification case (ii) is reduced to the consideration of [1|2][2|1][1|1]k in whose standard representation [1 2 | 1 2][1|1]k appears with coefficient −1.  We need a variant of Lemma 3.6.9 in a situation where we cannot freely choose a tableau like  above, but must force its appearance by using a linear transformation.

Fig. 3.7 Covers with respect to ≤

3.6 Representation Theory

101

Lemma 3.6.10 Let char K = 0, σ be a shape, and τ a cover of σ in the poset of all shapes ρ with γ1 (ρ) = γ1 (σ). Then there exists a linear transformation g ∈ G such that a standard bitableau  of shape τ appears with a nonzero coefficient in the standard representation of g In(σ) | In(σ) . Proof The potential covers of σ are specified in the proof of Lemma 3.6.9. Again we discuss the more difficult case in which the cover is obtained by moving a box one row up. With the same notation as inthe proof of Lemma 3.6.9 we apply the  following linear transformation g ∈ G to In(σ) | In(σ) : row q + 1 of X is added to row p and column q + 1 is added to column p. This transformation affects only δv−1 and δv since either row/column p does not appear in δi or both indices p and q appear in it. Repeating the arguments in the proof of Lemma 3.6.9, one first reduces the claim to the case in which u = v = 2 and then to the case σ = (q, 1),   In(σ) | In(σ) = [1 . . . q | 1 . . . q] · [1 | 1]. Our linear transformation now adds row and column q + 1 to row and column 1, respectively. As in the proof of Lemma 3.6.9, it is to be shown that the determinant [1 . . . q + 1 | 1 . . . q + 1] appears with a nonzero coefficient in the standard representation of g In(σ) | In(σ) . Each of the factors is transformed into a sum of 4 minors, and their product is a sum of 16 bitableaux of shape σ. Only bitableaux that have all row and column indices 1, . . . , q + 1 are critical, and there are four of them; the sign takes care of the transpositions that are needed to rearrange the sequence q + 1, 2 . . . , q in ascending order: π1 = [1 . . . q | 1 . . . q][q + 1 | q + 1], π2 = (−1)q−1 [2 . . . q + 1 | 1 . . . q][1 | q + 1], π3 = [2 . . . q + 1 | 2 . . . q + 1][1|1], π4 = (−1)q−1 [1 . . . q | 2 . . . q + 1][q + 1|1].

The first bitableau π1 is standard. The summands π2 and π4 are transposes of each other, and, as we know from the proof of Lemma 3.6.9, [1 . . . q + 1 | 1 . . . q + 1] has the coefficient (−1)q−1 (−1)q = −1 in their standard representations. The summand [2 . . . q + 1 | 2 . . . q + 1] · [1|1] yields the coefficient 1 − q. This can be seen as follows. For shorter formulas let us write Q for the sequence 1, . . . , q + 1 and Q i for the sequence 1, . . . , i − 1, i + 1, . . . , q + 1. The Laplace expansion of [Q | Q] along all rows (or columns) yields q+1 

(−1)i+ j [Q i | Q j ] · [i | j] = (q + 1)[Q | Q].

i, j=1

Then we subtract the expansions along row 1 and along column 1 from both sides and multiply by −1. Accumulating all summands except [Q 1 | Q 1 ] · [1|1] on the right hand side then gives

102

3 Determinantal Ideals and the Straightening Law

[Q 1 | Q 1 ] · [1|1] =

q+1 

(−1)i+ j [Q i | Q j ] · [i | j] + (1 − q)[Q | Q].

i, j=2

The bitableaux on the right hand side are pairwise different and standard. This finishes case (i). The case (ii) in the proof of Lemma 3.6.9 is left to the reader.  In analogy with the definition of the ideal I (σ) we let V (σ) be the vector subspace whose basis are the standard bitableaux  with γ1 () = γ1 (σ) and || ≥ σ. It is the degree γ1 (σ) component of I (σ) . Theorem 3.6.11 Consider the following properties of a K -subspace V of K [X ]: (a) V is defined by shape; (b) V is a sum of subspaces V (σ) ; (c) V is G-stable and has a basis of standard bitableaux. Then (a) ⇐⇒ (b) =⇒ (c). In characteristic 0, (c) =⇒ (b) as well. Proof The implication (a) =⇒ (b) is Lemma 3.6.9. The converse holds since each V (σ) is defined by shape and this property is preserved by taking sums. (b) =⇒ (c) results from the fact that the operation of G transforms a bitableau into a linear combination of bitableaux of the same shape. Let char K = 0 now. Since the G-submodules Mσ are the isotypic components  of K [X ] (Theorem 3.6.8), V is the direct sum of some of them: V = σ∈S Mσ . We claim that V = σ∈S V (σ) . Let σ ∈ S. Let us first show the following auxiliary statement: if  ∈ V is a standard bitableau, then (i) || ∈ S and (ii)  ∈ V for every standard bitableau  with || = ||. In fact, by Theorem 3.6.8 there is a unique representation  = x + y,

x ∈ Mσ , y ∈ I>(σ) , σ = ||.

That  ∈ / I>(σ) implies that x = 0, and since I>(σ) is itself a direct sum of certain Mτ with τ = σ, this is only possible if σ ∈ S. Similarly we write  = x + y with x ∈ Mσ ⊂ V and y ∈ I>(σ) . So x =  − y , and  must appear in the standard representation of x . The hypothesis on V implies  ∈ V . Now we must show that  ∈ V for every standard bitableau  with γ1 () = γ1 (σ) and || > σ. The auxiliary statement above allows us to use induction in the partially ordered set of the shapes under consideration. As in the proof of Lemma 3.6.9, it is enough to consider the case in which || is a cover of σ, and this case is covered by Lemma 3.6.10.  The implication (c) =⇒ (b) does not hold in arbitrary characteristic p > 0. Simply p p take m = 1, n > 1 and the subspace with basis X 11 , . . . , X 1n . Remark 3.6.12 If one changes “vector subspace” into “ideal” and V (σ) to I (σ) , one obtains a variant of Theorem 3.6.11 for ideals. This variant appears in Theorem 5.2

3.6 Representation Theory

103

of Bruns and Conca [41] with a slightly different definition of “defined by shape”. The proof of Lemma 3.6.10 corrects a mistake in [41].

G-Stable Prime Ideals In Sect. 6.6 we want to determine the G-stable prime ideals in certain subalgebras of K [X ]. In characteristic 0, Theorem 3.6.8 is completely sufficient, but in positive characteristic we must refine Lemma 3.6.5 to overcome the lack of linear reductivity of the group G. In characteristic 0 the following lemma is only a pale shadow of Theorem 3.6.8, but it will turn out strong enough for our purpose. Lemma 3.6.13 Suppose K is an infinite field. Let M ⊂ K [X ] be a G-submodule, and f ∈ M, f = 0. Let ∈S a  be the representation of f as a linear combination of standard bitableaux, a = 0 for  ∈ S. Then there exists 0 ∈ S such that (In(σ) | In(σ)) ∈ M for σ = |0 |. Proof First  we apply Lemma 3.6.4 to produce the element g = ∈S0 b (In(σ)  | C ) ∈ M ∩ τ τL . By hypothesis M is a U + (n, K ) submodule as well, and τ τL + is a U + (n, K ) submodule by construction. So we can apply the U (n, K )-variant of Lemma 3.6.4, and find a nonzero linear combination h = ρ cρ (In(ρ) | In(ρ)) ∈ M. It remains to separate h into its multigraded components, and this can finally be done by applying diagonal matrices. (At this point the infinity of K is used again.) Since M is stable under their action, we do not leave it.  As a consequence we obtain that G-stable prime ideals in subalgebras defined by shape are also defined by shape and have a basis of standard bitableaux. Proposition 3.6.14 Let A be a subalgebra of K [X ] defined by shape. Let p be a G-stable prime ideal of A. Then the following hold: (a) A bitableau  belongs to p if and only if (In(σ)| In(σ)) ∈ p for σ = ||; (b) p has a K -basis of standard bitableaux. Proof (a) If  ∈ p, then Lemma 3.6.13 immediately implies that (In(σ)| In(σ)) ∈ p for σ = ||. For the converse we observe that the product of all bitableaux that we can produce from  by applying row and column permutations splits off (In(σ)| In(σ)) as a factor. The complementary factor can be grouped into a product of bitableaux that all have the same shape as . So they are all in A by hypothesis. Since p is G-stable, either none or all of the permutation conjugates of  belong to p. Since p is prime and their product is in p, we conclude that all conjugates are in p. (b) Let f ∈ p, f = ∈S a  with standard bitableaux  ∈ S and a ∈ K , a = 0. We must show that  ∈ p for all  ∈ S. We use induction on |S|. From Lemma 3.6.13 we conclude that (In(σ) | In(σ)) ∈ p for σ = |0 | with 0 ∈ S. Then (a) implies that 0 ∈ p as well. So we can pass to f − a0 0 and apply induction. 

104

3 Determinantal Ideals and the Straightening Law

In Chaps. 9–11 it will be useful to extend the results of this section to the case, in which K is replaced by a more general ring, or by the structure sheaf of an algebraic variety (a so-called relative situation). In that context it is more convenient to use the theory of algebraic groups (or algebraic group schemes) and their representations, together with the theory of Schur functors. For instance, the general linear group GL(n, K ) and the unipotent subgroups U ± (n, K ) will be replaced by the corresponding algebraic group schemes in Sect. 9.7, ideals and subspaces defined by shapes will be replaced by corresponding relative (sheaf theoretic) constructions— for instance, compare Theorem 3.6.2 with Lemma 10.5.1.

Notes The early history of determinantal ideals can be found in the sections 2.E, 3.E and 4.E of [61]. Here we content ourselves with naming some breakthroughs and in particular those sources on which this chapter has been based. Eagon and Northcott [113] proved the classical height bound for ideals of maximal minors, constructed their free resolutions and proved that they are perfect ideals. Perfection was then generalized by Hochster and Eagon [179] to all determinantal ideals by an induction over a larger class of ideals called “principal radical system” (see also [61, Chap. 12]). The standard monomial theory for the Grassmannian was established by Hodge [186]. The straightening law is due to Doubilet et al. [112]. It was used and refined by De Concini and Procesi [105] for the proof of the main theorems of invariant theory in all characteristics. Our treatment follows De Concini et al. [103]. In Bruns and Vetter [61] it has been discussed in great detail. In characteristic 0 the primary decomposition of determinantal ideals was found in [103]. It was generalized to sufficiently large positive characteristics in [61]. Theorem 3.5.10 on integral closures was proved by Bruns [37]. The basic results of Sect. 3.6 are taken from [103]. A more extensive treatment can be found in [61, Sect. 11]. Ideals defined by shape were introduced by Bruns and Conca [41]. Miró-Roig’s book [248] covers aspects of determinantal ideals complementing those treated in this book: liaison and deformation theory.

Chapter 4

Gröbner Bases of Determinantal Ideals

The starting point of this chapter is ideals of 2-minors. The computation of their Gröbner bases with respect to a diagonal monomial order by the Buchberger criterion does not require much patience: the 2-minors themselves are a Gröbner basis. The first section contains this completely elementary access. The squarefree initial ideal of the length-2 diagonals defines a shellable simplicial complex whose faces can be interpreted as paths in a rectangular grid. Since the combinatorial details are rather simple, this section is a good place to introduce the structural and enumerative properties of shellable simplicial complexes and to transfer them via Gröbner deformation to determinantal rings: on the homological side one concludes the Cohen–Macaulay property, and on the enumerative side this approach gives simple formulas for the multiplicity and even the Hilbert series. However, there seems to be no transparent elementary access to the Gröbner basis of the ideal of t-minors for t > 2. One obvious reason, already mentioned in Chap. 3, is the failure of injectivity in the assignment of initial monomials to standard bitableaux. As Sturmfels discovered, this failure can be remedied: instead of the initial monomial one assigns the image under the Robinson–Schensted–Knuth correspondence (RSK) to a standard bitableau. Then it is crucial that the RSK image is contained in the initial ideal, and this is guaranteed by Schensted’s theorem: the “witchcraft” Knuth [213, p. 60] of the RSK saves one from tracing the Buchberger algorithm through tedious inductions. The RSK method can be extended to symbolic powers, ordinary powers and arbitrary products of determinantal ideals. This is made possible by Greene’s theorem that, roughly speaking, shows that the valuations defined by determinantal ideals are preserved under the passage to RSK images so that the latter indeed lie in the respective initial ideals. With some more combinatorial effort we then extend the results on 2-minors to the general case, where now simplicial complexes of nonintersecting paths come into play. It turns out that also these are shellable. While the homological properties of determinantal rings had been derived before the advent of Gröbner bases, the © Springer Nature Switzerland AG 2022 W. Bruns et al., Determinants, Gröbner Bases and Cohomology, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-05480-8_4

105

106

4 Gröbner Bases of Determinantal Ideals

access to enumerative results is difficult via standard bitableaux since their factors have different degrees. For multiplicities and Hilbert series Gröbner bases are clearly superior, and already this fact justifies their discussion. While the computation of Gröbner bases via standard bitableaux and the RSK is undoubtedly a very elegant approach, it requires considerable sophistication. As Sturmfels and Sullivant found out, one can avoid both of these advanced tools by considering determinantal ideals (or their varieties) as secant ideals of ideals of 2minors (or their varieties). The crucial point is that this relationship holds true for the simplicial complexes defined by the diagonals of 2-minors and t-minors in general. To get started, one must deal with the 2-minors first, and this is another reason why we have singled them out in the first section. All in all, the secant method can be regarded as the most elementary approach to Gröbner bases of determinantal ideals. The last section gives another application of the Gröbner basis approach to an enumerative problem, namely the computation of the multidegrees of a determinantal ring with respect to the multigrading by columns. In this context, “multidegrees” means a multigraded refinement of the ordinary multiplicity mentioned above. It turns out that 2-minors and maximal minors are distinguished in this regard. We will observe a related phenomenon in Chap. 5, namely the existence of a squarefree universal Gröbner basis .

4.1 An Instructive Case: the Ideal of 2-Minors In Example 1.2.8 we have seen that the 2-minors of a 2 × 3-matrix of indeterminates form a Gröbner basis of the ideal they generate. The argument given there can immediately be extended to 2 × n matrices or m × 2-matrices. But it is not much more difficult to show Theorem 4.1.1 Let X be an m × n-matrix of indeterminates. Then the 2-minors [i j | p q] are a Gröbner basis of I2 (X ) with respect to a diagonal monomial order. In particular, in(I2 (X )) is generated by the products X i p X jq , i < j, p < q. Proof According to the Buchberger criterion, Theorem 1.2.13, we must show that the divided Koszul relations of the initial monomials can be lifted, or, equivalently, that the S-polynomials of the pairs of 2-minors [i j | p q], [k l | u v] can be reduced modulo the set of 2-minors. If the two diagonals X i p X jq and X ku X lv have no factor in common, then the Koszul relation of the two minors lifts the (divided) Koszul relation of the initial monomials, as already observed in Example 1.2.11. If both minors live together in a 2 × 3-matrix or a 3 × 2-matrix, we have seen in Example 1.2.8 that the divided Koszul relation can be lifted. In the remaining case the two diagonals share a factor, but “span” a 3 × 3 submatrix of X , and we can right away assume that X is a 3 × 3-matrix. In order to reduce the cases to be considered, one observes that the simultaneous exchange of rows 1 and 3

4.1 An Instructive Case: the Ideal of 2-Minors

107

and columns 1 and 3 preserves the diagonals of the minors. Taking this symmetry into account and discarding all cases in which the diagonals have no factor in common, we get three critical pairs: [1 2 | 1 2], [1 3 | 1 3];

[1 2 | 1 3], [1 3 | 1 2];

[1 2 | 1 2], [2 3 | 2 3].

For the first pair we must reduce X 33 [1 2 | 1 2] − X 22 [1 3 | 1 3] = −X 12 X 21 X 33 + X 13 X 22 X 31 modulo the 2-minors. The further course of the computation depends on which of the two monomials is larger in the monomial order. We assume that X 12 X 21 X 33 is larger: −X 12 X 21 X 33 + X 13 X 22 X 31 + X 21 [1 3 | 2 3] = −X 13 [2 3 | 1 2], and we are done. We leave the other case and the other two pairs to the reader.



It is a remarkable fact that in(I2 ) is generated by squarefree monomials of degree 2 (with respect to some monomial order). Thus in(I2 ) is a radical ideal, and by Proposition 1.6.2 one immediately concludes that I2 is a radical ideal. Note that X m1 and X 1n do not appear in a diagonal of length 2. It follows that X m1 , X 1n is a regular sequence on K [X ]/I2 (X ) so that the localization argument used for Theorem 3.4.6 can be applied and yields (again) that K [X ]/I2 (X ) is a normal domain. A new consequence is Theorem 4.1.2 K [X ]/I2 (X ) is a Koszul algebra. It follows immediately from Corollary 1.6.10. The Koszul property of K [X ]/I2 (X ) can be derived from the straightening law as well. This was observed by Kempf [208]; also see Bruns et al. [53].

The Segre Product of Polynomial Rings There is another elementary approach to compute in(I2 (X )) that does not even need the Buchberger criterion. In addition, it implies that I2 (X ) is prime. The  Segre product S # T of positively graded K -algebras S and T is the subalgebra ∞ k=0 Sk ⊗ K Tk of S ⊗ K T . The Segre product is itself graded with (S # T )k = Sk ⊗ K Tk . If S = K [Y1 , . . . , Ym ] and T = K [Z 1 , . . . , Z n ] with their standard gradings, then S # T = K [Yi Z j : i = 1, . . . , m, j = 1, . . . , n] ⊂ K [Y1 , . . . , Ym , Z 1 . . . , Z n ].

108

4 Gröbner Bases of Determinantal Ideals

Proposition 4.1.3 Let X be an m × n matrix of indeterminates, R = K [X ] and S and T the polynomial rings just introduced. I2 (X ) is the kernel of the K-algebra homomorphism σ : R → S # T , σ(X i j ) = Yi Z j (and therefore a prime ideal). Proof Evidently I2 (X ) ⊂ Ker σ. Let J be the ideal generated by the monomials X i p X jq , i < j, p < q. The set Mon(R) \ J consists of all monomials X r1 s1 · · · X r p s p , p ∈ N, such that r1 ≤ · · · ≤ r p and s1 ≥ · · · ≥ s p . We leave the verification of this statement to the reader, as well as the fact that σ is injective on Mon(R) \ J . The injectivity of σ on Mon(R) \ J implies that Mon(R) \ J is linearly independent not only modulo J , but even modulo Ker σ. So Proposition 1.2.6 implies that the 2-minors are a Gröbner basis of Ker σ with respect to a diagonal monomial order  and that I2 (X ) = Ker σ.

The Simplicial Complex Defined by the Initial Ideal A squarefree monomial ideal defines a simplicial complex, and vice versa. The analysis of the simplicial complex defined by in(I2 ) yields significant information on I2 , as we will see in the following. The terminology related to simplicial complexes was introduced in Sect. 2.1. Let us first determine the simplicial complex 2 defined by I2 . It consists of all the subsets of V = {1, . . . , m} × {1, . . . , n} that do not contain a diagonal of length 2, i.e., a set {(i, p), ( j, q)} with i < j and p < q. (In the figures (Fig. 4.1) below we will use Cartesian coordinates. Note that the usual notation of matrix positions differs from Cartesian coordinates by a 90◦ rotation!) The facets of 2 can pictorially be described in terms of paths. To this end we give V a poset structure (certainly not the most natural one). We set (i, j) ≤ (h, k)

⇐⇒

i ≤ h and j ≥ k.

A subset W of V is said to be a chain if each two elements of W are comparable in the poset V , and it is an antichain if it does not contain a pair of comparable elements. It is easy to see that an antichain with 2 elements corresponds to the main diagonal of a 2-minor. So 2 coincides with the order (or chain) complex of V : its faces are the chains and its facets are the maximal chains of the poset V . Note that the poset V is a planar distributive lattice. A maximal chain of V can be described as a path in V . A path P in V from point p to point q, with p ≤ q, is, by definition, an unrefinable chain with minimum p and maximum q. It can be written as a sequence P:

p = (a1 , b1 ), (a2 , b2 ), . . . , (ad , bd ) = q

4.1 An Instructive Case: the Ideal of 2-Minors

109

Fig. 4.1 A path with 2 right turns (m = 4, n = 5)

where (ai+1 , bi+1 ) − (ai , bi ) = (1, 0) or (0, −1) for all i. A point (ak , bk ) is said to be a right turn of the path P if 1 < k < d and (ak+1 , bk+1 ) − (ak , bk ) = (0, −1), (ak , bk ) − (ak−1 , bk−1 ) = (1, 0). If one describes the lattice V using either the Cartesian or the matrix notation, then a right turn of P is exactly a point where the path turns to the right if one walks from the minimal to the maximal element. The length of a path from p = (x1 , x2 ) to a point q = (y1 , y2 ) depends only on p and q and is equal to y1 − x1 + x2 − y2 + 1. We want to extract combinatorial information from the set F(2 ) of facets of 2 . Evidently all maximal paths in 2 have the same length m + n − 1; so 2 is pure. (The purity of 2 could also be derived a priori from Proposition 2.1.10.) By induction one easily shows that   m+n−2 . |F(2 )| = m−1 Every maximal path is determined by its right turns, and therefore we can count them also by the number of subsets of V that appear as right turns. After fixing the number r of right turns we can freely choose their coordinates (ai , bi ) subject to the conditions 1 < a1 < · · · < ar ≤ m and second coordinates n ≥ b1 > · · · > br > 1. Therefore |F(2 )| =

    p   m−1 n−1 m+n−2 = , k k m−1 k=0

p = min(m, n) − 1.

The size of the facets and their number |F(2 )| have algebraic interpretations: Lemma 4.1.4 Let  be a simplicial complex. The Krull dimension of K [] is dim  + 1, and the multiplicity of K [] equals the number of facets of maximal dimension of .

110

4 Gröbner Bases of Determinantal Ideals

Proof This follows from the fact that the defining ideal of K [] is radical and its / F) where F is a facet of . The formula minimal primes are of the form (X i : i ∈ for the dimension can be found in Bruns and Herzog [52, 5.1.4]. For the multiplicity one uses [52, 4.7.8].  We will state the consequences for K [X ]/I2 (X ) in Proposition 4.1.9 below.

Shellable Simplicial Complexes In order to exploit 2 further, we must analyze its structure. As already observed, 2 is pure. Indeed, since the ideal of 2-minors is prime, 2 is even strongly connected by Corollary 2.2.14. So it makes sense to check for shellability. In order to describe this property, we write F1 , . . . , F p for the smallest subcomplex containing the faces F1 , . . . , F p . Definition 4.1.5 A pure simplicial complex  is called shellable if one of the following equivalent conditions is satisfied: the facets of  can be given a linear order F1 , . . . , Fm in such a way that (a) the maximal faces of Fi ∩ F1 , . . . , Fi−1 are codimension one faces of the simplex Fi for all i, 2 ≤ i ≤ m; (b) the set F1 , . . . , Fi \ F1 , . . . , Fi−1 , has a unique minimal element G i for all i, 2 ≤ i ≤ m; (c) for all i, j, 1 ≤ j < i ≤ m, there exists some v ∈ Fi \ F j and some k ∈ {1, 2, . . . , i − 1} with Fi \ Fk = {v}. A linear order of the facets satisfying the equivalent conditions (a), (b), and (c) is called a shelling of . Shellability means that the simplicial complex  can be built inductively by attaching facets Fi in such a way that the intersection of F1 , . . . , Fi−1 and Fi is topologically simple: it is the union of codimension one faces of the simplex Fi , as expressed by condition (a). Such a union is topologically either homeomorphic to a ball or to a sphere. Condition (b) says that the complement of the intersection has a unique minimal face G i , and this is obviously equivalent to (a): the minimal face is spanned by the vertices that are the complement in Fi of the facets of Fi ∩ F1 , . . . , Fi−1 . Condition (c) is a more technical version of (b), as the reader may check: G i is spanned by the vertices v in (c). (See [52, 5.1.11] for a more formal discussion.) Example 4.1.6 For a simplicial complex , being strongly connected (see Remark 2.2.7) or shellable might look like similar notions. Obviously a shellable simplicial complex must be strongly connected. Furthermore, if dim  ≤ 1, strong connectedness and shellability are actually equivalent. In dimension 2, however, there

4.1 An Instructive Case: the Ideal of 2-Minors

111

Fig. 4.2 A strongly connected simplicial complex which is not shellable

are strongly connected simplicial complexes which are not shellable. The simplicial complex  = {1, 2, 4}, {2, 4, 5}, {2, 3, 5}, {1, 3, 5} on 5 vertices of Fig. 4.2 is such an example. Shellability is a strong property, suited for inductive arguments. Suppose that F1 , . . . , Fm is a shelling of the simplicial complex , let i = F1 , . . . , Fi , and set i∗ = Fi ∩ i . Then i is obviously shellable and one has a short exact sequence 0 → K [i ] → K [i−1 ] ⊕ K [Fi ] → K [i∗ ] → 0 where K [Fi ] is the Stanley–Reisner ring of the simplex defined by Fi , i.e. K [Fi ] is the polynomial ring on the set of vertices of Fi . The sequence is constructed in [52, 5.1.13]. Condition (b) in Definition 4.1.5 translates immediately into an algebraic property: K [i∗ ] is defined by a single monomial. Its degree is the cardinality of G i . This implies Theorem 4.1.7 Let  be a shellable simplicial complex of dimension d − 1 with shelling F1 , . . . , Fm . Then: (a) The Stanley–Reisner ring K [] is Cohen–Macaulay.  d j (b) The Hilbert series of K []  has the form h(z)/(1 − z) with h(z) = h j z ∈   Z[x], h 0 = 1 and h j = {i ∈ {1, . . . , m} : |G i | = j} . Proof We sketch the proof, referring the reader to [52, 5.1.13] for a complete version. By induction one can assume that K [i−1 ] ⊕ K [Fi ] is a Cohen–Macaulay module of dimension d (over the polynomial ring representing all vertices of ). Moreover, K [i∗ ] is of type K [X 1 , . . . , X d ]/(μ) where the monomial μ is supported in the variables representing G i . So K [i∗ ] is Cohen–Macaulay of dimension d − 1. Now the behavior of depth along exact sequences yields that K [i ] is Cohen–Macaulay as well. The exact sequence implies HK [i ] (t) = HK [i−1 ] (t) + HK [Fi ] (t) − HK [i∗ ] (t) and the claim follows by induction. Note that this equation can be considered as an inclusion-exclusion formula counting monomials in the involved K -algebras. 

112

4 Gröbner Bases of Determinantal Ideals

Shellability and Consequences for 2-Minors The set V = {1, . . . , m} × {1, . . . , n} with its partial order is a planar distributive lattice, and 2 is its order complex. Therefore it is shellable by a theorem of Björner [19] that holds for partially ordered sets with much weaker properties (also see [52, 5.1.12]). So we can immediately conclude that K [2 ] and, therefore, K [X ]/I2 are Cohen–Macaulay. But for the computation of the Hilbert series we must find the faces G i in Definition 4.1.5(b). So a direct proof of shellability is necessary (and much simpler in this case). First we define a partial order for paths in V : partial order of pathsP ≤ Q if P is on the right of Q as one walks from (1, m) to (n, 1): the rectangle in the plane spanned by the four corners of V is subdivided by Q into two connected subsets (intersecting in Q). That P ≤ Q means that P is contained in the connected component to which (1, 1) belongs. We extend the partial order to a total order that is denoted by the same symbol.  , be the paths in V , P1 ≤ · · · ≤ Pe . Then Lemma 4.1.8 Let P1 , . . . , Pe , e = m+n−2 m−1 P1 , . . . , Pe is a shelling, for which G i is the set of right turns of Pi , i = 1, . . . , e. Proof If P < Q in the total order, P need not be on the right of Q, but certainly Q cannot be on the right of P. So there must be a point of P that is strictly on the right of Q. But then there exists a right turn of Q that does not belong to P, as demonstrated by Figure 4.3. (Formalize this argument!) On the other hand, given a right turn (a, b) of Q we can easily find a path P that is on the right of Q and differs from Q exactly in (a, b): simply replace (a, b) by (a − 1, b − 1).  We have already counted the sets of right turns by cardinality, and so we can summarize the consequences for K [X ]/I2 (X ) that can be derived from the analysis of 2 as follows: Proposition 4.1.9 Let X be an m × n matrix of indeterminates, m, n ≥ 2 and set R = K [X ]/I2 (X ). Then:  (a) dim R = m + n − 1 and e(R) = m+n−2 ; m−1 (b) R is Cohen–Macaulay;

Fig. 4.3 Paths P, Q, P < Q (or Q < P)

4.2 The Robinson–Schensted–Knuth Correspondence

(c)

m−1 n−1 k t k k , m+n−1 (1 − t)

113

p H R (t) =

k=0

p = min(m, n) − 1;

(d) R is Gorenstein if and only if m = n. Only (d) needs a justification. Instead of going via 2 one uses that R is an integral domain (as already stated in Theorem 3.4.6) and the characterization of the Gorenstein property of Cohen–Macaulay domains by their Hilbert series, namely by the symmetry of the numerator polynomial of H R (t); see [52, 4.4.6] for this theorem of Stanley. We will generalize Proposition 4.1.9 to K [X ]/It (X ) for all t in Sect. 4.4. While the structural arguments will not change, the combinatorial details are considerably more complicated for t > 2, as we will see in the next sections. If one has proved the Cohen–Macaulayness of K [X ]/I2 (X ) in another way, one can conclude the Cohen–Macaulayness of K [X ]/ in(I2 ) from Theorem 2.4.3. However, for combinatorial implications the shellability of 2 is an important stronger property.

4.2 The Robinson–Schensted–Knuth Correspondence The Robinson–Schensted–Knuth correspondence (in our context) sets up a bijection between standard bitableaux and monomials in the ring K [X ].

Deletion and Insertion The passage from bitableaux to monomials is based on the deletion algorithm. Definition 4.2.1 Deletion takes a standard tableau A = (ai j ) of shape (s1 , s2 , . . . ), and an index p such that s p > s p+1 , and constructs from them a standard tableau B and a number x, determined as follows: (1) Define the sequence k p , k p−1 , . . . , k1 by setting k p = s p and choosing ki for i < p to be the largest integer ≤ si such that aiki ≤ ai+1,ki+1 . (2) Define B to be the standard tableau obtained from A by (i) removing a ps p from the pth row, and (ii) replacing the entry aiki of the ith row by ai+1,ki+1 , i = 1, . . . , p − 1. (3) Set x = a1k1 . Informally one could say that the entry a ps p is pushed to the row p − 1 (unless p = 1) and itself pushes the largest entry of that row that is ≤ a ps p to row p − 2 etc. One obtains a trail of “bumps”, as illustrated by Fig. 4.4.

114

4 Gröbner Bases of Determinantal Ideals

Fig. 4.4 Deletion

The reader should check that B is again a standard tableau. It has the same shape as A, except that its row p is shorter by one entry. Deletion has an inverse: Definition 4.2.2 Insertion takes a standard tableau A = (ai j ) of shape (s1 , s2 , . . . ), and an integer x, and constructs from them a standard tableau B and an index p determined as follows: (1) Set i = 1 and B = A. (2) If si = 0 or x > aisi , then add x at the end of the ith row of B, set p = i and terminate. (3) Otherwise let ki be the smallest j such that x ≤ a jsi , replace bki si with x, set x = aki si and i = i + 1. Then go to (2). Again it is easily checked that B is a standard tableau whose shape coincides with that of A, except that the row p of B is longer by one entry. Deletion and Insertion are clearly inverse to each other: if Deletion applied to input (A, p) gives output (B, x), then Insertion applied to (B, x) gives output (A, p) and vice versa.

Robinson-Schensted–Knuth Correspondence This correspondence, RSK for short, is initially defined as a bijective correspondence between the set of the standard bitableaux (as combinatorial objects) and the set of two-line arrays of a certain type. The two-line array RSK() is constructed from the standard bitableau  by an iteration of the following RSK-step: Definition 4.2.3 Let  = (A | B) = (ai j | bi j ) be a nonempty standard bitableau. Then RSK-step constructs a pair of integers (, r ) and a standard bitableau   as follows. (1) Choose the largest entry  in the left tableau of ; suppose that {(i 1 , j1 ), . . . , (i u , ju )}, i 1 < · · · < i u , is the set of indices (i, j) such that  = ai j . Set p = i u and q = ju . (We call ( p, q) the pivot position.) (2) Let A be the standard tableau obtained by removing a pq from A.

4.2 The Robinson–Schensted–Knuth Correspondence

115

(3) Apply Deletion to the pair (B, p). The output is a standard tableau B  and an element r . (4) set   = (A , B  ). Now RSK() is constructed from the outputs of a sequence of RSK-steps: Definition 4.2.4 Let  be a nonempty standard bitableau of shape s1 , s2 , . . . s p . Set k = s1 + · · · + s p and define the two-line array RSK() =

  1 2 . . . k−1 k r1 r2 . . . rk−1 rk

as follows. Starting from k = , the RSK-step Definition 4.2.3, applied to i for i = k, k − 1, . . . , 1, produces the bitableau i−1 and the pair (i , ri ). We give an example in Fig. 4.5. The shaded boxes in the left tableau mark the pivot position, those in the right mark the chains of “bumps” given in Definition 4.2.1(2): The two-line array RSK() has the following properties: (a) i ≤ i+1 for all i, (b) if i = i+1 then ri ≥ ri+1 . Property (a) is clear since the algorithm chooses i+1 ≥ i . If i = i+1 then the pivot position of the (i + 1)th deletion step lies left of (or above) the pivot position of the ith deletion step. Now it is easy to see that the element pushed out by the (i + 1)th step is not larger than that pushed out by the ith step. RSK gives a correspondence between standard bitableaux and two-line arrays with properties (a) and (b). It is bijective since it has an inverse. For the inversion

Fig. 4.5 The RSK algorithm

116

4 Gröbner Bases of Determinantal Ideals

of RSK one just applies the Insertion algorithm to the bottom line of the array to build the right tableau: at step i it inserts ri in the tableau obtained after the (i − 1)th step. Simultaneously one accumulates the left tableau by placing the element i in the position which is added to the right tableau by the ith insertion. It remains to explain how we can interpret any two-line array satisfying (a) and (b) as a monomial: we associate the monomial X 1 r1 X 2 r2 ··· X k rk to it, clearly establishing the desired bijection. To sum up, we have constructed a bijective correspondence between standard bitableaux and monomials. If we restrict our attention to standard bitableaux and monomials where the entries and the indeterminates come from an m × n matrix, we get Theorem 4.2.5 The map RSK is a bijection between the set of standard bitableaux on {1, . . . , m} × {1, . . . , n} and the monomials of K [X ]. This theorem proves the second half of the straightening law: the RSK correspondence says that in every degree d there are as many standard bitableaux as monomials. Since we know already that the standard bitableaux generate the space of homogeneous polynomials of degree d, we may conclude that they must be linearly independent. Other, more classical proofs of the linear independence of standard bitableaux have been indicated in Remark 3.2.8. Remark 4.2.6 In the fundamental paper [212] Knuth extensively treats the RSK correspondence for column standard bitableaux with increasing columns and nondecreasing rows. (Deletion still bumps the entries row-wise, but the condition aiki ≤ ai+1,ki+1 in Definition 4.2.1(1) must be replaced by aiki < ai+1,ki+1 .) The same point of view is taken in Fulton [133], Knuth [213], Sagan [273] and Stanley [292]. The version we are using is the “dual” one (see [212, Sect. 5 and p. 724]). The notes to Chap. 7 of [292] contain a detailed historical discussion of the correspondence. Below we will consider decompositions of sequences into increasing and nonincreasing sequences. For column standard bitableaux these attributes must be exchanged. Remark 4.2.7 We have just seen that RSK is a bijection between two bases of the same vector space. Therefore we can extend RSK to a K -linear automorphism of K [X ]. We still denote it by RSK, i.e. RSK( f ) makes sense now for every polynomial f ∈ K [X ]. The automorphism RSK preserves the degree and the content: no column or row index gets lost. Furthermore, since RSK is a K -linear automorphism, for every V, W K -subspaces of K [X ] we have RSK(V + W ) = RSK(V ) + RSK(W ) and RSK(V ∩ W ) = RSK(V ) ∩ RSK(W ) Note however that RSK is not a K -algebra isomorphism: it acts as the identity on polynomials of degree 1 but it is not the identity map. Question 4.2.8 Despite to the fact that combinatorial properties of RSK has been deeply studied, not so much is know about RSK as linear map. For example, what are the eigenvectors and eigenvalues of RSK?

4.2 The Robinson–Schensted–Knuth Correspondence

117

Remark 4.2.9 We note some important properties of RSK: (a) RSK commutes with transposition of the matrix X : Let X  be a n × m matrix of indeterminates, and let τ : K [X ] → K [X  ] denote the K -algebra isomorphism induced by the substitution X i j → X ji ; then RSK(τ ( f )) = τ (RSK( f )) for all f ∈ K [X ]. Note that it suffices to prove the equality when f is a standard bitableau. Herzog and Trung [172, Lemma 1.1] point out how to translate Knuth’s argument from column to row standard tableaux. (b) All the powers  k of a standard bitableau are again standard, and one has RSK( k ) = RSK()k . This is not hard to check: k successive deletion steps on  k act like a single deletion step on k copies of . (c) If  is a single minor [a1 a2 . . . at | b1 b2 . . . bt ] then RSK() is just (the product of the elements on) the main diagonal of . Moreover, one has RSK(In(σ) | C) = In(σ) | C and RSK(R | In(σ)) = R | In(σ) for all standard tableaux C, R of shape σ (the initial tableau In(σ) was defined in Sect. 3.3, and  is the initial monomial of a standard bitableau  with respect to a diagonal monomial order as introduced above Proposition 3.3.4). This is easy to see in both cases. However, without further conditions it is not even enough that one of the two tableaux of  = (R | C) is “nested”, i.e., the set of entries in each row contains the entries in the next row. In general, RSK() need not be one of the monomials which appear in the expansion of . In other words, RSK does not simply select a monomial of the polynomial ; there is no algebraic relation between RSK() and the monomials appearing in .

RSK Invariants In the application of RSK to Gröbner bases of determinantal ideals it will be important to relate the shape of a standard bitableau  to “shape” invariants of RSK(). In the RSK correspondence the right tableau, and hence its shape, is determined solely by the bottom line of the corresponding two-line array. Therefore we are led to the following problem: Let r = r1 , r2 , . . . , rk be a sequence of integers and let Ins(r ) be the standard tableau determined by the iterated insertions of the ri . What is the relationship between the shape of Ins(r ) and the sequence r ? A subsequence v of r is determined by a subset U of {1, . . . , k}: if U = {i 1 , . . . , i t } with i 1 < · · · < i t then v = v(U, r ) = ri1 , . . . , rit . The length of v is simply the cardinality of U . A first answer to the question above was given by Schensted and had indeed been a motivation of his studies. Theorem 4.2.10 The length of the first row of Ins(r ) is the length of the longest increasing subsequence of r .

118

4 Gröbner Bases of Determinantal Ideals

Does the length of the ith row for i > 1 have a similar meaning? Actually, these lengths cannot be interpreted individually, but some of their combinations reflect properties of the decompositions of the sequence r into increasing subsequences. This is the content of Greene’s extension of Schensted theorem, which we will now explain. A decomposition of r into subsequences corresponds to a partition U = (U1 , U2 , . . . , Us ) of the set {1, . . . , k}. The shape of the decomposition is (|U1 |, |U2 |, . . . , |Us |): we always assume that |U1 | ≥ |U2 | ≥ · · · ≥ |Us |. An inc-decomposition of r is given by a partition (U1 , U2 , . . . , Us ) of the set {1, . . . , k} such that the associated subsequences are increasing. Example 4.2.11 The sequence 4, 1, 2, 5, 6, 3 has the inc-decompositions 1 2 5 6 | 4 | 3 and 4 5 6 | 1 2 3 and many more. The first of them confirms Schensted’s theorem; see Fig. 4.5. k In Sect. 3.2 we have defined the functions αk , namely αk (λ) = i=1 λi . Now we introduce a variant

αk for sequences of integers r , setting

αk (r ) = max{αk (λ) : r has a inc-decomposition of shape λ}. Similarly we set αk (P) = αk (λ)

and

αk (U ) = αk (λ)

for every tableau P and every inc-decomposition U of shape λ. Example 4.2.12 For the sequence r = 4, 1, 2, 5, 6, 3 of Example 4.2.11 one obtains α2 (r ) = 6,

αi (r ) = 6, i ≥ 3.

α1 (r ) = 4,

It attracts attention that

αi (r ) = αi (Ins(r )) for all i, although there is no incdecomposition of shape (4, 2)! This is not an accident, as we will see below. Inc-decompositions are crucial for us since they describe realizations of a monomial as an initial monomial of a product of minors with respect to diagonal monomial orders. This will be made more precise in the next section. However, in Sect. 6.3 it will turn out useful to also have a measure for decompositions into nonincreasing subsequences, which we call noninc-decompositions. As we will see, this has to do

Fig. 4.6 A shape and its dual

4.2 The Robinson–Schensted–Knuth Correspondence

119

with dualization. For a shape λ = (s1 , . . . , st ) we define λ∗ to be the dual shape (Fig. 4.6): the ith component of λ∗ = (s1∗ , . . . , ss∗1 ) counts the number of boxes in the ith column of a tableau of shape σ; formally si∗ = |{k : sk ≥ i}|. The functions αk are dualized to αk∗ (λ) = αk (λ∗ )

and

αk∗ (P) = αk∗ (λ)

if P is a tableau of shape λ. Analogously one defines αk∗ (U ) for a noninc-decomposition. However, the passage from inc-decompositions to noninc-decompositions contains already a dualization, and we set

αk∗ (r ) = max{αk (λ) : r has a noninc-decomposition of shape λ}. Example 4.2.13 Our standard example r = 4, 1, 2, 5, 6, 3 has no nonincsubsequence of length 3, but contains two disjoint such sequences of length 2, for example 4, 1 and 5, 3. It cannot be decomposed into 3 such sequences. Therefore α2∗ (r ) = 4,

α3∗ (r ) = 5,

α4∗ (r ) = 6.

α1∗ (r ) = 2,

Despite of the small size of this example, it is at least suspicious that

αi∗ (r ) = ∗ αi (Ins(r ) ) for all i. We can rephrase the definition of

αk and

αk∗ for sequences r as follows:

αk (r ) ∗ (respectively,

αk (r )) is the length of the longest subsequence of r that can be decomposed into k increasing (nonincreasing) subsequences. Greene’s theorem relates the invariants that have been introduced. Theorem 4.2.14 For every sequence of integers r and every k ≥ 0 we have (a)

αk (r ) = αk (Ins(r )),

(b)

αk∗ (r ) = αk∗ (Ins(r )).

For k = 1 one obtains Schensted’s theorem from (a). (Schensted also proved (b) for k = 1.)

Knuth Relations The proof of Greene’s theorem is based on Knuth’s basic relations: two sequences of integers r and s differ by a Knuth relation if one of the following conditions holds: (1) r = x1 , . . . , xi , y, z, w, xi+4 , . . . , xk and s = x1 , . . . , xi , y, w, z, xi+4 , . . . , xk , with z ≤ y < w;

120

4 Gröbner Bases of Determinantal Ideals

(2) r = x1 , . . . , xi , z, w, y, xi+4 , . . . , xk and s = x1 , . . . , xi , w, z, y, xi+4 , . . . , xk , with z < y ≤ w. For every standard tableau P there is a canonical sequence r P associated with P such that Ins(r P ) = P, defined as follows: r P is obtained by listing the rows of P from bottom to top, i.e. r P = pt1 pt2 . . . ptst pt−11 pt−12 . . . pt−1st−1 . . . p11 p12 . . . p1s1 ; here (s1 , . . . , st ) is the shape of P. Knuth’s theorem explains the relationship between sequences that differ by a Knuth relation. Theorem 4.2.15 Let r and s be sequences. Then Ins(r ) = Ins(s) if and only if r is obtained from s by a sequence of Knuth relations. In particular, the canonical sequence associated with Ins(r ) is obtained from r by a succession of Knuth relations. A detailed proof of the theorem for column standard tableaux is contained in [212]. On [212, p. 724] one finds an explanation how to modify the Knuth relations and statements for row standard tableaux. Now Greene’s theorem is proved as follows: for (a) one shows that (i) αk (P) =

αk (r P ) for the canonical sequence r P associated to a standard tableau P, and (ii)

αk has the same value on sequences that differ by a Knuth relation. The same scheme works for (b). We confine ourselves to (a), leaving (b) to the reader. For the proof of (i) set P = ( pi j ) and let σ = (s1 , . . . , st ) be the shape of P. First of all, r P has the inc-decomposition into the sequences pi1 , . . . , pisi , i = 1, . . . , t. The αk (r P ). On the other hand, decomposition has the same shape as P. Hence αk (P) ≤

note that the columns of P partition r P into nonincreasing subsequences. Therefore an increasing subsequence can contain at most one element from each column, and the total s1number of ∗elements in a disjoint union of k increasing subsequences is at min(k, si ) = αk (P). most i=1 For the proof of (ii) we consider sequences a and b of integers that differ by αk (b) it suffices to show that for a Knuth relation. In order to prove that

αk (a) =

every inc-decomposition G of a there exists an inc-decomposition H of b such that αk (G) ≤ αk (H ). So let G be an inc-decomposition of a and let z, w and y as in the definition of the Knuth relations. If z and w belong to distinct subsequences in the decomposition G, then the increasing subsequences of G are not affected by the Knuth relation, and we may take H equal to G. (Strictly speaking, we must change the partition of the index set underlying the sequences by exchanging the positions of z and w.) It remains the case in which z and w belong to the same increasing subsequence. Then a must play the role of r (in the Knuth relation), and y must belong to another subsequence in its decomposition G. Let u = p1 , z, w, p2 and v = p3 , y, p4 denote

4.2 The Robinson–Schensted–Knuth Correspondence

121

those subsequences in G that contain z, w and y. Here the pi are increasing subsequences of a. Assume first the Knuth relation is of type (1). We can rearrange the elements of the sequences u and v into increasing subsequences of b in three ways:  (I)



u = p1 , z, p4 v = p3 , y, w, p2

⎧  ⎨ u = p1 , w, p2 (II) v = p3 , y, p4 ⎩  w =z

⎧  ⎨ u = p1 , y, w, p2 (III) v = p3 , p4 ⎩  w = z.

Suppose that both u and v contribute to αk (G). Then we replace u and v by the sequences u  and v defined in (I), obtaining an inc-decomposition H of b with αk (H ) ≥ αk (G): in fact, H contains k subsequences whose lengths sum up to αk (G). (Find an example for which αk (H ) > αk (G).) If u does not contribute to αk (G), then we replace u and v by the sequences u  , v , w in (II). The inc-decomposition H of b consisting of the remaining subsequences of a and u  , v , w has αk (H ) = αk (G). Now suppose that u contributes to αk (G), but v does not. Then we replace u and v by the three sequences defined in (III), with the same result as in the previous case. The dual argument works in the case of a Knuth relation of type (2). It completes the proof of Greene’s theorem.

Further RSK Invariants Another  series of very useful functions is given by the γt introduced in Sect. 3.5, γt (λ) = i max(λi − t + 1, 0), where t is a nonnegative integer and λ is a shape. We extend the γt to sequences in the same way as the αk :  

γt (r ) = max γt (λ) : r has an inc-decomposition of shape λ . Like the αk , the γt are invariant under RSK. Theorem 4.2.16 For every sequence of integers r we have

γt (r ) = γt (Ins(r )). One can prove Theorem 4.2.16 by arguments completely analogous with those leading to Greene’s theorem. This approach has been chosen by Bruns and Conca [39]. Alternatively one can use the following lemma. Lemma 4.2.17 For each shape λ the following holds: γt (λ) ≥ u

⇐⇒

αk (λ) ≥ (t − 1)k + u for some k, 1 ≤ k ≤ t.

We leave the easy proof to the reader, as well as the dual version of Theorem 4.2.16.

122

4 Gröbner Bases of Determinantal Ideals

Remark 4.2.18 In spite of Theorems 4.2.14 and 4.2.16, in general there does not exist an inc-decomposition of a sequence r with the same shape as Ins(r ). We have already observed this for the sequence 4, 1, 2, 5, 6, 3 in Example 4.2.12.

4.3 RSK and Gröbner Bases of Ideals In this section we fix a diagonal monomial order, as introduced above t Proposition X ai bi is the 3.3.4: with respect to such a monomial order the diagonal δ = i=1 initial monomial of the minor δ = [a1 . . . at | b1 . . . bt ]. The power of RSK in the study of Gröbner bases of determinantal ideals was detected by Sturmfels [294]. His simple, but crucial observation is the following. Assume that I is an ideal of K [X ] which has a K -basis of standard bitableaux B. Then RSK(I ) is a vector subspace of K [X ] that has two of the properties of an initial ideal: it has a basis of monomials and it has the same Hilbert function as I . Can we conclude that RSK(I ) is the initial ideal of I ? Not in general, but if RSK(I ) ⊂ in(I ) or the other way round, then equality is forced by the Hilbert function. In connection with Proposition 1.4.5 this argument yields Lemma 4.3.1 (a) Let I be an ideal of K [X ] which has a K -basis, say B, of standard bitableaux, and let S be a subset of I . Assume that for all  ∈ B there exists s ∈ S such that in(s) | RSK(). Then S is a Gröbner basis of I and in(I ) = RSK(I ). (b) Let I and J be homogeneous ideals such that in(I ) = RSK(I ) and in(J ) = RSK(J ). Then in(I ) + in(J ) = in(I + J ) = RSK(I + J ) and in(I ) ∩ in(J ) = in(I ∩ J ) = RSK(I ∩ J ). Proof (a) Let J be the ideal generated by the monomials in(s) with s ∈ S. The hypothesis implies that RSK(I ) ⊂ J ⊂ in(I ). Since the first and third terms have the same Hilbert function it follows that RSK(I ) = J = in(I ). For (b) one uses Lemma 2.1.11 and Remark 4.2.7 to obtain RSK(I + J ) = RSK(I ) + RSK(J ) = in(I ) + in(J ) ⊂ in(I + J ), RSK(I ∩ J ) = RSK(I ) ∩ RSK(J ) = in(I ) ∩ in(J ) ⊃ in(I ∩ J ), and concludes equality from the Hilbert function argument. Sturmfels applied Schensted’s Theorem 4.2.10 to prove Theorem 4.3.2 (a) The t-minors of X form a Gröbner basis of It ; (b) RSK(It ) = in(It ).



4.3 RSK and Gröbner Bases of Ideals

123

Proof The set B of standard bitableaux whose first row has length ≥ t is a K -basis of It (Corollary 3.4.2). Let S be the set of the minors of size t. By Lemma 4.3.1, it is enough to show that for every  ∈ B there exists δ ∈ S such that in(δ) | RSK(). Let  and r be the  top and bottom vector, respectively, of the RSK image of  so that RSK() = X i ri . By Schensted’s Theorem 4.2.10 we can find an increasing subsequence r of length t, say ri1 < ri2 < · · · < rit with i 1 < i 2 < · · · < i t . But then it follows from property (b) of the RSK-image. In other words, i1 < i2 < · · · k, then we cancel γt () − k boxes in the bitableau with the  corresponding entries. In this way we get 1 . An important consequence of Theorem 4.3.6, by virtue of Lemma 4.3.4 and Theorem 3.5.5, is: Theorem 4.3.7 Let t1 , . . . , tr be positive integers and set I = It1 · · · Itr and gi = γi (t1 , . . . , tr ). If char K = 0 or char K > min(ti , m − ti , n − ti ) for all i, then I is in-RSK and in(I ) is generated, as a K-vector space, by the monomials μ with

γi (μ) ≥ gi for all i. Theorem 4.3.7 is satisfactory if one only wants to determine the initial ideal of the product It1 · · · Itr , but it does not tell us how to find a Gröbner basis. A natural guess is that any such ideal is G-RSK, i.e. a Gröbner basis of It1 · · · Itr is given by the products of minors (standard or not) which are in It1 · · · Itr . Unfortunately this is wrong in general. Example 4.3.8 Suppose that m ≥ 4 and char K = 0 or > 3, and consider the ideal I4 I2 . The monomial M = X 11 X 13 X 22 X 34 X 43 X 45 has

γ4 (M) = 1,

γ3 (M) = 2, γ1 (M) = 6. We have seen a similar example already in Remark 4.2.18.

γ2 (M) = 4,

Hence, by virtue of Theorem 4.3.7, we know that M ∈ in(I4 I2 ), but this can also seen by computation:  M = in [1 2 3 4 | 1 2 3 5][1 4 | 3 4] − [1 2 3 4 | 1 2 3 4][1 4 | 3 5] .

4.3 RSK and Gröbner Bases of Ideals

125

The products of minors of degree 6 in I4 I2 have the shapes (6), (5, 1) or (4, 2). Clearly M is not the initial monomial of a product of minors of shape (6) or of shape (5, 1). The only initial monomial of a 4-minor that divides M is X 11 X 22 X 34 X 45 but the remaining factor X 13 X 43 is not the initial monomial of a 2-minor. Hence M is not the initial monomial of a product of minors that belongs to I4 I2 . Nevertheless, if we confine our attention to powers of determinantal ideals, the result is optimal: Theorem 4.3.9 Suppose that char K = 0 or char K > min(t, m − t, n − t). Then Itk is G-RSK and in(Itk ) is generated, as a K-vector space, by the monomials M with

αk (M) ≥ kt. In particular, a Gröbner basis of Itk is given by the products of minors  such that  has at most k factors and αk () = deg  = kt. Therefore Itk has a minimal system of generators which is a Gröbner basis. Proof Let S1 be the set of the products of minors  with αk () ≥ kt. By Proposition 3.5.8 we know that S1 ⊂ Itk . Greene’s Theorem 4.2.14 and Remark 4.3.5 imply that for every standard bitableau  with αk () ≥ kt there exists  in S1 with in () | RSK(). Thus it follows from Theorem 4.3.1 that S1 is a Gröbner basis of Itk and Itk is G-RSK. It remains to show that for every product of minors  with αk () ≥ kt there exists a product of minors 1 with at most k factors and αk (1 ) = kt such that in(1 )| in(). Then deg 1 = αk (1 ) = kt since 1 has at most k factors. This step is as easy as the corresponding one in the proof of Theorem 4.3.6: 1 is obtained from  by skipping the rows of index > k (if any) and deleting αk () − kt boxes from the first k rows (in any way).  Example 4.3.10 As we have seen in Example 4.3.8, Theorem 4.3.9 cannot be generalized to arbitrary products: I4 I2 is not G-RSK (under the standard hypothesis on characteristic). This does not exclude the possibility that there is a minimal system of generators that is simultaneously a Gröbner basis. Indeed, by a somewhat tedious analysis of inc-decompositions one can show that in(I4 I2 ) is generated in degree 6, and therefore the ideal has a reduced Gröbner basis in degree 6, which then is automatically a minimal system of generators. However, this observation cannot be generalized to arbitrary products: the initial ideal of I5 I3 for a matrix X of size at least 5 × 7 is not generated in degree 8. Consider the monomial μ = X 11 X 22 X 43 X 54 X 34 X 14 X 25 X 46 X 57 . It arises from the two line array 

 123444567 . 124531245

This bottom row has the inc-subsequence (1 2 3 4 5) of length 5 and two nonoverγ4 (μ) = 2,

γ3 (μ) = 4, lapping inc-subsequences (1 2 4 5). Moreover,

γ5 (μ) = 1,

γ2 (μ) = 6 and

γ1 (μ) = 9. It follows from Theorem 4.3.7 that μ ∈ in(I5 I3 ).

126

4 Gröbner Bases of Determinantal Ideals

No proper divisor ν of μ reaches

γ5 (ν) ≥ 1 and

γ3 (ν) ≥ 4 simultaneously: ν must contain X 34 since otherwise the corresponding bottom sequence of the two line array does not have an increasing subsequence of length 5. So one of the other 8 factors must be missing in ν, with the consequence that one of the two increasing subsequences of length 4 is lost, and

γ3 (ν) ≥ 4 is not reachable since no subsequence has an inc-decomposition of shape (5, 3). Example 4.3.11 In (3.6) and Remark 3.5.9(b) we have described the different behavior of I2 (X )2 for a 4 × 4-matrix X in characteristic 0 or > 2 and characteristic 2. With respect to a diagonal monomial order one has / in(I22 ) in characteristic 2, μ = X 41 X 21 X 32 X 43 ∈ but in all other characteristics the monomial belongs to in(I22 ). For the ideal Im (X ) of maximal minors one can give a much more precise result: Corollary 4.3.12 The standard bitableaux  with k factors of size m form a Gröbner basis of Imk . In particular one has in(Imk ) = in(Im )k for all k ≥ 1. Proof By Theorem 4.3.9 Imk has a Gröbner basis of bitableaux  forming a minimal system of generators of Imk . Such  must have k factors of size m: they have total degree km and satisfy αk () ≥ km. By Proposition 3.3.4 there exists a standard bitableau  of the same shape as  with the same initial monomial so that we can replace  by  in the Gröbner basis.  As we will see in Sect. 6.2 one can prove the corollary without using the RSK.

Variants and Generalizations In the following we list some generalizations of the results above, especially of Corollary 4.3.12. Remark 4.3.13 (a) We can obviously generalize Theorems 4.3.7 and 4.3.9 as follows: Let c1 , . . . , ct ∈ N, t = min(m, n), and V be the vector space spanned by all standard bitableaux  with γi () ≥ ci (or αi () ≥ ci ) for all i; then V is an ideal and in-RSK. In fact, each of the inequalities defines a G-RSK ideal in K [X ]. We can even intersect V with a homogeneous component of K [X ] (with respect to the total degree or the Zm ⊕ Zn -grading) to obtain an in-RSK vector space. (b) In [41] Bruns and Conca have further analyzed the properties of being GRSK or in-RSK. If char K = 0, then all ideals I defined by shape are in-RSK. They are characterized in Remark 3.6.12 as the ideals are stable under the action of GL(m, K ) × GL(n, K ) on K [X ]. They are sums of intersections of symbolic powers of the ideals It , and so they are in-RSK (compare Theorem 3.6.11.)

4.3 RSK and Gröbner Bases of Ideals

127

Furthermore, an ideal defined by shape is G-RSK exactly if it is the sum of ideals J (k, d) ∩ I1u and, if m = n, (J (k, d) ∩ I1u )Imv where the J (k, d) play the same role for the α-functions as the symbolic powers do for the γ-functions: J (k, d) is generated by all bitableaux  with αk () ≥ d. Remark 4.3.14 In the article [45] by Bruns and Conca the northeast ideals It (a) of maximal minors were defined as follows: It (a) is generated by the t-minors of the t × (n − a + 1) northeast submatrix X t (a) = (xi j : 1 ≤ i ≤ t, a ≤ j ≤ n). Let It1 (a1 ), . . . , Itw (aw ) be northeast ideals of maximal minors, and let I be their product. Then (a) I has a linear resolution. (b) in(I ) = in(It1 (a1 )) · · · in(Itw (aw )), and the natural generators of I form a Gröbner basis with respect to a diagonal monomial order. (c) I is integrally closed, and it has a primary decomposition whose components are powers of ideals It (a). The preceding article [16] by Berget, Bruns and Conca investigates the subclass of products u  Isi (1) Jσ = j=1

where σ = (s1 , . . . , su ) is a shape with s1 ≥ · · · ≥ su . The ideal J σ has a standard basis given by the bitableaux  that “contain” a standard bitableau (In(σ) | C). The (In(σ) | C) form a Gröbner basis of J σ . This class of ideals is investigated in Sect. 8.6. While the investigation of the J σ can be based more or less directly on the straightening law and the RSK, one needs a substantial extension of the straightening law for the general case of northeast ideals of maximal minors. Remark 4.3.15 Let T be a new indeterminate, and consider the polynomial ring R  = K [X, T ] where X , as usually, is an m × n matrix of indeterminates with m ≤ n. The RSK-invariance of the functions αk has found another application to the ideal J = Im + Im−1 T + · · · + I1 T m−1 + (T m ). and its powers; see Bruns and Kwieci´nski [54]. With the results accumulated so far, the reader can show that J k = R

 km

 J (k, d)T km−d ,

d=0

with J (k, d) defined as in Remark 4.3.13. Let us extend the diagonal monomial order from K [X ] to R  by first comparing total degrees and, in the case of equal total

128

4 Gröbner Bases of Determinantal Ideals

degree, the X -factors of the monomials. It follows that in(J k ) = in(J )k and J k has a Gröbner basis of products T km−d where  is a bitableaux of total degree d such that αk () = d. The technique by which we explore the Rees algebra of the ideal It in Sect. 6.3 can also be applied to the Rees algebra of J ; see [54]. Remark 4.3.16 (a) Another approach to the proof of Theorem 4.3.2(a) is via Proposition 1.4.7. The degree of the determinantal rings 4.4.2 can be computed without using Gröbner bases, see Harris and Tu [163], Arbarello et al. [10], and Giambelli [141]. One can then use this information to prove Theorem 4.3.2(a) by showing that the ideal Jt generated by the diagonal monomials of the t-minors is unmixed of the correct codimension and multiplicity. We will prove unmixedness in Theorem 4.4.5. This point of view is taken by Knutson and Miller [214] and by Miller and Sturmfels [245] within a more general frame where the ideala of minors It are seen as special members of a class of ideals introduced by Fulton [132], the Schubert determinantal ideals. In this context the authors do not a priori know the degree of the associated Schubert determinantal variety, but they identify a recursive formula that it satisfies. A deep combinatorial analysis is needed to prove that the ideals of initial terms satisfy the same type of recursion. (b) A similar approach is taken by Gorla et al. [149] and applied to ideals of minors of ladders. The key recursion here is organized around the notion of linkage and used to prove that the ideal of initial terms has the correct Hilbert function. (c) A further class of ideals of minors that has recently been introduced by Fløystad in [128, Sect. 6]. In his setting the underlying matrix has entries that are either distinct variables or 0 located in specific places. The ideals under consideration are generated by the (relative maximal) minors of certain submatrices. It is shown that their initial ideals with respect to a diagonal monomial belong to the well-behaved class of letterplace ideals, and hence they are radical and define Cohen–Macaulay rings.

4.4 Cohen–Macaulayness and Hilbert Series of Determinantal Rings The goal of this section is to exploit the information on in(It (X )) for general t in the same way as we have done it for t = 2 in Sect. 4.1. Two tasks must be accomplished: (i) the identification of the simplicial complex t defined by in(It (X )); (ii) the verification that t is shellable, including the determination of the minimal faces G i that appear in Definition 4.4.4(b). In the following we use the terminology and notation of Sect. 4.1; in particular the partial order on V = {1, . . . , m} × {1, . . . , n} is again important. Recall that (i, j) ≤ (h, k)

⇐⇒

i ≤ h and j ≥ k.

4.4 Cohen–Macaulayness and Hilbert Series of Determinantal Rings

129

The Simplicial Complex In view of Theorem 4.3.2 we must describe the simplicial complex simplicial complex defined by initial ideal of It t that consists of all subsets of V that do not contain a t-diagonal. Given two sets P = p1 , . . . , ps and Q = q1 , . . . , qs of s points in V , a set F ⊂ V is said to be a family of nonintersecting paths from P to Q if it can be decomposed as F = P1 ∪ P2 · · · ∪ Ps where Pi is a path from pi to qi and Pi ∩ P j = ∅ if i = j. We identify a family of paths with the set of points on its paths. In the present setting this is allowed because the decomposition above is unique. We will then say that a point is a right turn of F if it is a right turn of the path to which it belongs (Fig. 4.7). Proposition 4.4.1 The facets of t are exactly the families of nonintersecting paths from (1, n), (2, n), . . . , (t − 1, n) to (m, 1), (m, 2), . . . , (m, t − 1). Proof A family of nonintersecting paths is in t since an antichain intersects a chain in at most one point. That it is a facet can be easily proved directly, but follows also from the fact that any such family has dimension (m + n − t + 1)(t − 1) − 1, which is, by Theorem 3.4.6, the dimension of t since dim t = dim K [t ] − 1 = dim K [X ]/It − 1. It remains to show that every facet G ∈ t is a family of nonintersecting paths from (1, n), (2, n), . . . , (t − 1, n) to (m, 1), (m, 2), . . . , (m, t − 1). The points of W = {(a, b) : b − a ≥ n − t + 1 or a − b ≥ m − t + 1} do not belong to any t-antichain: there is not enough room. So G must contain W . We put (a, b) ≺ (c, d) ⇐⇒ a < c and b < d. By construction, distinct points P, Q are either comparable with respect to < or comparable with respect to ≺, but not both. So a chain with respect to ≺ is an antichain with respect to 1.

130

4 Gröbner Bases of Determinantal Ideals

This is called the light and shadow decomposition (the light here comes from the point (1, 1)). The family of the G i satisfies the following conditions: (a) Every G i is a chain since otherwise G i would contain two t − 1 since otherwise we would get a t-antichain in G by (b). It follows that G is the disjoint union of the chains G 1 , . . . , G t−1 and that each chain G i contains (i, n) and (m, i). We prove that each G i is indeed a path from (i, n) and (m, i). Clearly G i cannot contain points which are smaller than (i, n) or larger than (m, i) since those points belong already to the G j with j < i. So it remains to show that G i is saturated. Recall that q is said to be an upper neighbor of p if p < q and there is no r with p < r < q. We have to show the following Claim. If p = (a, b) and q = (c, d) belong to G i and q > p, but q is not an upper neighbor, then there exists r0 ∈ G i such that p < r0 < q. We set

⎧ ⎪ ⎨(c, b) r = (c, d + 1) ⎪ ⎩ (a + 1, b)

if c = a and b = d, if c = a and b = d, if c = a and b = d.

Since p < r < q, if r ∈ G i , then we are done; just set r0 = r . If r ∈ / G i , then there are three possible cases: (1) If r ∈ G j for some j < i, then, by (b), we may find t1 , t2 ∈ G j such that t1 ≺ p and t2 ≺ q. But t1 and t2 must be i, then by (b) there is a t ∈ G i such that t ≺ r . If p < t < q, then we set r0 = t. Otherwise either t ≤ p or t ≥ q. This is impossible since p < r < q. (3) If r ∈ / G then G ∪ {r } does not contain a t-antichain since it has a decomposition into t − 1 chains; just add r to G i . This contradicts the maximality of G and concludes the proof. 

The Multiplicity It follows from the above description that every facet of t has exactly (m + n + 1 − t)(t − 1) elements, so that t is pure. (Since It is a prime ideal by Theorem

4.4 Cohen–Macaulayness and Hilbert Series of Determinantal Rings

131

3.4.6, purity is also implied by Corollary 2.1.8.) Therefore the multiplicity of K [t ] and, hence, that of K [X ]/It is given by the number of families of nonintersecting paths from (1, n), (2, n), . . . , (t − 1, n) to (m, 1), (m, 2), . . . , (m, t − 1). This number can be computed by the Gessel–Viennot determinantal formula [136]: given two sets of points P = p1 , . . . , ps and Q = q1 , . . . , qs , the number Paths(P, Q) of families of nonintersecting paths from P to Q is Paths(P, Q) = det(Paths( pi , q j ))i, j=1,...,s

(4.1)

provided there is no family of nonintersecting paths from P to any nontrivial permutation of Q. Here Paths( pi , q j ) denotes the number of paths from pi to q j .  j For pi = (i, n) and q j = (m, j) we know that Paths( pi , q j ) = m−i+n− and m−i hence   m −i +n− j (4.2) e(K [X ]/It ) = det m −i i, j=1,...,t−1. In proving Eq. 4.2 we have followed Herzog and Trung [172]. They have generalized this approach (including Schensted’s theorem 4.2.10) to the 1-cogenerated ideals introduced in Sect. 3.4. We want to transform the expression (4.2) into a determinant free form. To this end it is useful to set r = t − 1. Since   m −i +n− j (m − i + n − j)! = m −i (m − i)!(n − j)! one can extract the factor r  i=1

 1 1 . (m − i)! j=1 (n − j)! r

If we simultaneously invert the order of the rows and columns, then we must find ⎛

⎞ (N − 2r )! . . . (N − (r + 1))! ⎜ ⎟ .. .. det ⎝ ⎠, . . (N − (r + 1))! . . . (N − 2)!

N = m + n.

In each column k the top entry (N − 2r + k − 1)! divides all the others in an obvious way, and so the last determinant equals ⎛

1 r ⎜ N − 2r +1  ⎜ (N − 2r + k − 1)! · det ⎜(N − 2r + 1)(N − 2r + 2) ⎝ k=1 .. .

⎞ ... 1 ⎟ ... N −r ⎟ . . . . (N − r )(N − r + 1)⎟ ⎠ .. ... .

132

4 Gröbner Bases of Determinantal Ideals

The first two rows remind us of the Vandermonde matrix, and if we subtract the second from the third row, we obtain the third row of the Vandermonde. Continuing (exercise: how?) one sees that the last matrix can be converted into the Vandermonde matrix for N − 2r + 1, . . . , N − r , by a unimodular row transformation. Therefore its determinant equals r −1  i! i=0

Collecting all our factors, we get Theorem 4.4.2 e(K [X ]/It ) =

r  (m + n − 2r + i − 1)!(i − 1)! , (m − i)!(n − i)! i=1

r = t − 1.

Remark 4.4.3 (a) Theorem 4.4.2 is Giambelli’s formula. We present its history in the notes of this chapter. Harris and Tu [163] derive the formula e(K [X ]/It ) =

n−t 

m+i t−1 t+i−1

i=0

(4.3)

t−1

from a different determinant. See also [10]. Harris and Tu also give multiplicity formulas for the cases of symmetric and alternating matrices of indeterminates. It is not hard to see  that the expressions for e(K [X ]/It ) agree if one uses the k−1 i!. In fact, hyperfactorial H (k) = i=0 t−1  (m + n − 2t + i + 1)! (i − 1)! (m − i)! (n − i)! i=1

= whereas

n−t  i=0

H (m + n − t + 1) H (m − t + 1) H (n − t + 1) H (t − 1) , H (m + n − 2t + 2) H (m) H (n)

m+i t−1 t+i−1 = t−1

(m+i)! (t−1)! (m+i−t+1)! (t+i−1)! (t−1)! i! i=0

n−t 

=

H (m+n−t+1) H (m−t+1) H (m) H (m+n−2t+2) H (n) H (t−1) H (n−t+1)

,

and so the two expressions are equal. (b) It is much easier to compute the multiplicity in the case t = m ≤ n of maximal minors. In this case Im (X ) reduces modulo a suitable regular sequence of degree 1 elements to the mth power of the maximal ideal in a polynomial ring with n − m + 1 n ; see [61, (2.16)]. indeterminates, and therefore e(K [X ]/Im ) = m−1

4.4 Cohen–Macaulayness and Hilbert Series of Determinantal Rings

133

Shellability and the Cohen–Macaulay Property Next we show that t is shellable. Theorem 4.4.4 The simplicial complex t is shellable. More precisely, the facets of t can be given a total order such that c(F) is the set of right turns of F for each facet F of t . Proof First we remind the reader of the partial order on the set of paths connecting points p, q with p ≤ q: for two paths P1 and Q 1 from p to q we write P1 < Q 1 if P1 is “on the right” of Q 1 as one goes from p to q (in Cartesian as well as in matrix notation). This partial order was already used in the proof of Lemma 4.1.8. Let pi = (i, n) and qi = (m, i) for i = 1, . . . , t − 1. Given two families of nonintersecting paths P = P1 , . . . , Pt−1 and Q = Q 1 , . . . , Q t−1 from p1 , . . . , pt−1 to q1 , . . . , qt−1 we set P < Q if Pi < Q i for the largest i such that Pi = Q i . We extend this partial order on the set of families arbitrarily to a total order. To prove that the resulting total order is indeed a shelling, one takes two families Q and P with P < Q and lets i denote the largest index j with P j = Q j . Then Q i is not on the right of Pi . We already know that there exists a right turn, say r , of Q i which is (strictly) on the left of Pi . By the choice of i, the point r does not belong to P j for all j. So it suffices to show that for every right turn r = (x, y) of Q i there is a family R which is < Q in the total order such that Q \ R = {r }. This is easy if either (x − 1, y − 1) does not belong to Q i−1 or i = 1: just replace (x, y) with (x − 1, y − 1) in Q i to get a path Q i , and then set R = R1 , . . . , Rt−1 with R j = Q j if j = i and Ri = Q i . By construction R < Q in the total order. It is a little more complicated to define R when (by bad luck) the element (x − 1, y − 1) belongs to Pi−1 . But if this is the case, then (x − 1, y − 1) must be a right turn of Pi−1 (draw a picture). If (x − 2, y − 2) does not belong to Pi−2 , we may repeat the construction above: define R as the family obtained from Q by replacing (x, y) with (x − 1, y − 1) in Q i and (x − 1, y − 1) with (x − 2, y − 2) in Q i−1 . The general case follows by the same construction.  Theorem 4.4.4 is a special case of a more general theorem due to Björner [19, Theorem 7.1]. It has two important consequences. The first is Theorem 4.4.5 The algebras K [t ] and K [X ]/It are Cohen–Macaulay. Proof By Theorem 4.4.4 t is shellable, and hence K [t ] is Cohen–Macaulay by Theorem 4.1.7. Since K [t ] = K [X ]/ in(It ), it follows from Theorem 1.6.11 that  K [X ]/It is Cohen–Macaulay as well.

The Hilbert Series The second consequence of shellability is a combinatorial interpretation of the Hilbert series of determinantal rings. We need more notation for it. Given two sets of s points

134

4 Gröbner Bases of Determinantal Ideals

P and Q, let Paths(P, Q)k denote the number of families of nonintersecting paths  from P to Q with exactly k right turns, and set Paths(P, Q, z) = k Paths(P, Q)k z k . In the case of just one start point p and one end point q, we denote this polynomial simply by Paths( p, q, z). As an immediate consequence of Theorems 4.1.7 and 4.4.4 one gets Proposition 4.4.6 The Hilbert series Ht (z) of K [t ] and K [X ]/It is given by Ht (z) =

Paths(P, Q, z) (1 − z)d

where d = (m + n + 1 − t)(t − 1) is the Krull dimension, P = (1, n), (2, n), . . . , (t − 1, n) and Q = (m, 1), (m, 2), . . . , (m, t − 1). Without a formula for Paths(P, Q, z) the proposition remains a purely formal statement. But such a formula exists: Theorem 4.4.7 Let P = (a1 , n), . . . , (ar , n), 1 ≤ a1 < · · · < ar ≤ m, and Q = (m, b1 ), . . . , (m, br ), 1 ≤ bi < · · · < br ≤ n. Then Paths(P, Q)k =



det

   m − a j − i + j n − bi + i − j , ki ki + i − j i, j=1,...,r r 

where the sum is taken over all sequences (k1 , . . . , kr ) such that

ku = k.

u=1

Remark 4.4.8 Theorem 4.4.7 was proved by Krattenthaler [220] (in more general form) and Kulkarni [226]. However, Kulkarni derives it from Abhyankar’s formula [3] for the Hilbert polynomial of determinantal rings that was proved by counting standard bitableaux—opposite to the direction we want to take. With the help of Theorem 4.4.7 we obtain a first satisfactory expression for the Hilbert series: Theorem 4.4.9 With the notation of Proposition 4.4.6 one has    m − i  n − j  1 k z Ht (z) = det k k +i − j (1 − z)d k

,

i, j=1,...,r

where r = t − 1 and d = (m + n − r )r .

 Proof The product of binomial coefficients in Theorem 4.4.7 simplifies to m−i ki  n− j since now a = j and b = i for all i and j. In order to check that the j i ki +i− j coefficient of z u is indeed given by Theorem 4.4.7 for all u, one writes every row of our determinant as a sum in the powers of z and uses multilinearity. 

4.4 Cohen–Macaulayness and Hilbert Series of Determinantal Rings

135

By analogy with the Gessel–Viennot formula (4.1) one may wonder whether the polynomial Paths(P, Q, z) has a determinantal expression as  det Paths( pi , q j , z) i, j=1,...,r . (4.4) This is obviously true if there is just one starting and ending point, but cannot be true in general since Paths(P, Q, 0) = 1 and Paths( pi , q j , 0) = 1 for all i, j. But, very surprisingly, equality holds after a shift of degree: Theorem 4.4.10 The Hilbert series Ht (z) of K [t ] and K [X ]/It is    m − i n − j  1 k z Ht (z) = r det , k k z (2) (1 − z)(m+n−r )r k

r = t − 1.

i, j=1,...,r

Proof We must show that det

   m − i  n − j  tk k k +i − j k

i, j=1,...,r

=

   m − i n − j  1 k det t r k k t (2) k

.

i, j=1,...,r



This identity is dealt with by the following lemma. Lemma 4.4.11 The r × r -matrices of polynomials         m − i  n − j  1  m −i n− j k k  t , and Hi j = i−1 t , Hi j = t k k +i − j k k k

k

have the same determinant. Proof Let        1 i− j i − 1 j−i j − 1 A = (−1) and B = (−1) . j − 1 t i− j i −1 We claim that H  = AH B. In particular, since A and B are triangular matrices whose diagonal elements are all 1, it follows that det H  = det H . For the critical matrix equation one uses the identity       p−1 a−q a−p = (−1) p−1 (−1) p−q q −1 b−q b−1 q≥1 which in its turn is proved by induction on p.



Remark 4.4.12 Krattenthaler’s determinantal formula for the enumeration of families of nonintersecting paths with a given number of right turns (Theorem 4.4.7) holds no matter how the starting and end points are located. But in general it is not

136

4 Gröbner Bases of Determinantal Ideals

equal to the polynomial (4.4) even if one allows a shift in degree. So the formula of Theorem 4.4.10 should be regarded as an “accident” while the combinatorial description of Proposition 4.4.6 holds more generally, for instance, in the case of algebras defined by 1-cogenerated ideals. On the other hand Krattenthaler and Prohaska [221] were able to show that the same “accident” takes place if the paths are restricted to certain subregions called one-sided ladders. This proves a conjecture of Conca and Herzog on the Hilbert series of one-sided ladder determinantal rings; see [91].

Related Numerical Invariants Several numerical invariants of a graded algebra are defined by the Hilbert series or at least related to it. We deal with them in a series of remarks. Remark 4.4.13 The a-invariant of a positively graded Cohen–Macaulay algebra R with graded maximal ideal m is defined as the top degree of the local cohomology Hmd (R), d = dim R. By graded local duality it can equivalently be described by a(R) = − min{i : ω(R)i = 0} where ω(R) is the graded canonical module of R; see [52, p. 141]. A third description is a(R) = deg H R (z) by the degree of the Hilbert series as a rational function. Consequently there are several ways for its computation. For K [X ]/It two of them were done by Bruns and Herzog [51], namely an ASL approach to the lowest degree of the canonical module and the description as the maximal number of right turns in a family of nonintersecting paths: if 2 ≤ t ≤ m ≤ n, then a(K [X ]/It ) = −r n,

r = t − 1.

Both approaches can be generalized to the case in which the indeterminates deg X i j = ei + f j where ei + f j ≥ 0 for all i and j; see [51]. For the standard graded case, a(K [X ]/It ) was also determined by Gräbe [152], and Conca [80] generalized it to the 1-cogenerated case. For standard graded Cohen–Macaulay algebras the Castelnuovo–Mumford regularity is simply the degree of the numerator polynomial of the Hilbert series. So, in our case, reg K [X ]/It = (m + n − r )r − nr = (m − r )r again under the assumption that 2 ≤ t ≤ m ≤ n and r = t − 1. Another approach to the reg K [X ]/It will be in Example 11.6.2.

4.4 Cohen–Macaulayness and Hilbert Series of Determinantal Rings

137

Remark 4.4.14 Since the rings K [X ]/It are Cohen–Macaulay domains, the Gorenstein ones among them are exactly those with a symmetric numerator polynomial in the Hilbert series [52, 4.4.6]. By a tedious analysis of the formula for the Hilbert series (see [91]) one can prove that K [X ]/It , t ≥ 2, is Gorenstein if and only if m = n, a result due to Svanes [302]. It is however more informative to determine the canonical module ω(K [X ]/It ) for all shapes of matrices; as we will see in Theorem 6.7.13, ω(K [X ]/It ) = qn−m where q is the prime ideal in K [X ]/It generated by the r -minors of the first (or any) r = t − 1 columns of X in K [X ]/It . For the precise description of the graded canonical module we must shift qn−m so that the generators get degree −a(K [X ]/It ) = r n: ω∼ = qn−m (−r m)

(4.5)

is the graded canonical module of K [X ]/It . By a theorem of Stanley one can derive the Hilbert series of the graded canonical module ω(R) from that of R by Hω(R) (z) = (−1)d H R (z −1 ) (as rational functions); see [52, 4.4.5]. This duality can be used for an explicit description of Hω(R) (z) in the case R = K [X ]/It , as done by Conca and Herzog [91]. Once the canonical module of R is known, one can (in principle) compute its minimal number of generators, the type of R. There are several ways to find it for K [X ]/It : (i) it is the value at n − m of the Hilbert function of K [Mr (X  )] for an r × n matrix of indeterminates that was computed by Hodge and Pedoe [188]; (ii) by the hook formula for irreducible representations of GL(n, K ) (for example, see Fulton [133, p. 53]), or (iii) by computing the leading coefficient of the numerator polynomial of HK [X ]/It (z). (The latter method works since K [X ]/It is level: all generators of qn−m have the same degree.) J. Brennan communicated the value n−m+r

 · · · n−1 r r m−1 r , · · · r r

computed by method (ii) (see [61, (9.20)]).

Variations and Generalizations Remark 4.4.15 Ladder determinantal rings are an important generalization of the classical determinantal rings. They are defined by the minors coming from certain subregions, called ladders, of a generic matrix. These objects have been introduced by Abhyankar in his study of the singularities of Schubert varieties of flag manifolds and have been investigated by many authors, including Conca [81, 82], Conca and Herzog [92], Ghorpade [138], Glassbrenner and Smith [144], Gonciulea and Lakshmibai [146, 147], Herzog and Trung [172], Knutson and Miller [214], Krattenthaler and Prohaska [221], Krattenthaler and Rubey [222], Kulkarni [225], and Mulay [249].

138

4 Gröbner Bases of Determinantal Ideals

Ladder determinantal rings share many properties with classical determinantal rings, for instance they are Cohen–Macaulay normal domains, the Gorenstein ones are completely characterized in terms of the shape of the ladder, and there are determinantal formulas for the Hilbert series and functions. Many of these results are derived from the combinatorial structure of the initial ideals of the ideals under consideration. Remark 4.4.16 The ideal of t-minors of a symmetric n × n-matrix of indeterminates and the ideal Pf t of 2t-Pfaffians of an alternating n × n matrix of indeterminates can also be treated by Gröbner basis methods based on suitable variants of RSK. Again we set r = t − 1. For Pfaffians the method was introduced by Herzog and Trung [172]. They used it to compute the multiplicity of K [X ]/ Pf t . A determinantal formula for the Hilbert series can be found in De Negri [107]; see also Ghorpade and Krattenthaler [139]. Pfaffian rings are known to be factorial (Avramov [11]) Cohen–Macaulay domains of dimension r (r + 1)/2 + (n − r )r Kleppe and Laksov [209], and are therefore Gorenstein by a theorem of Murthy (see [52, 3.3.19]). Their a-invariants have been computed by Bruns and Herzog [51], by the approaches via the ASL structure and the exploitation of shellability. Conca [78] transferred the method of Herzog and Trung to the symmetric case, introducing a suitable version of RSK. He derived formulas for the Hilbert series [79] and the multiplicity [78]. The a-invariant was computed by Barile using a variant of the ASL method [14]. In the symmetric case the algebra is a Cohen–Macaulay normal domain of dimension r (r + 1)/2 + (n − r )r (Kutz [229]). It is Gorenstein if and only if t ≡ n mod 2. In the non-Gorenstein case the ideal p defined as in Remark 4.4.14 is the canonical module Goto [150]. Conca [83] attacked another class of determinantal ideals by Gröbner basis methods, the ideals of minors of a Hankel matrix. This case, like that of maximal minors, is “easy” since different standard products of minors have different initial terms so that the essential point is to define the standard products.

4.5 Gröbner Bases via Secants We first collect some results about joins of ideals in polynomial rings.

Joins, Secants and Initial Ideals Let I, J be ideals in a polynomial ring S = K [X 1 , . . . , X n ]. The join I ∗ J of I and J is by definition the kernel of the K -algebra map S → S/I ⊗ K S/J

4.5 Gröbner Bases via Secants

139

that sends X i to xi ⊗ 1 + 1 ⊗ xi for i = 1, . . . , n. Here xi denotes the coset of X i in the corresponding quotient ring. Note that the map sends any linear form L ∈ S to  ⊗ 1 + 1 ⊗  where  denotes the coset of L in the corresponding quotient ring. Hence the join ideal I ∗ J is defined independently of the chosen system of variables. Introducing auxiliary variables Y = Y1 , . . . , Yn and  Z = Z 1 , . . . , Z n the tensor product S/I ⊗ K S/J can be represented as K [Y, Z ]/ I (Y ) + J (Z ) where I (Y ) is the image of I under the K -algebra map S → K [Y, Z ] sending X i to Yi and J (Z ) is defined similarly. Therefore I ∗ J can be identified with the kernel of the K -algebra map  S → K [Y, Z ]/ I (Y ) + J (Z ) sending X i to the coset of Yi + Z i and hence it can be computed as an elimination ideal:  I ∗ J = I (Y ) + J (Z ) + (X i − Yi − Z i : i = 1, . . . , n) ∩ S or equivalently as   I ∗ J = f ∈ S : f (Y + Z ) ∈ I (Y ) + J (Z ) .

(4.6)

We start with a simple observation about the Krull dimension of S/I ∗ J . Lemma 4.5.1 We have dim S/(I ∗ J ) ≤ dim S/I + dim S/J and equality holds if dim S/(I + J ) = 0 Proof Set R = S/I ⊗ K S/J . The first assertion is an immediate consequence of the definition because S/(I ∗ J ) is a subalgebra of R and dim R = dim S/I + dim S/J . For the second assertion it is enough to prove that S → R is a finite extension, i.e. R is a finitely generated S-module. Equivalently, it is enough to prove that R/m S R is a finite dimensional K -vector space. But R/m S R  S/(I + J ) and hence the conclusion follows.  Remark 4.5.2 If K is algebraically closed and I and J are homogeneous ideals defining projective varieties X and Y in Pn−1 then the join (ideal) I ∗ J defines the join variety X ∗ Y of X and Y , i.e. the Zariski closure of the union of the lines through the points p and q with p ∈ X and q ∈ Y . Translating the statements of Lemma 4.5.1 in terms of algebraic dimensions we have dim X ∗ Y ≤ dim X + dim Y + 1 and equality holds if X ∩ Y = ∅. Remark 4.5.3 If the ideals I and J are given as kernels of K -algebra maps ϕ1 : S → A1 and ϕ2 : S → A2 then I ∗ J is the kernel of the K -algebra map ϕ : S → A1 ⊗ K A2 defined by ϕ(X i ) = ϕ1 (X i ) ⊗ 1 + 1 ⊗ ϕ2 (X i ). In particular, if I and J are the kernels of K -algebra maps ϕ1 : S → K [T1 , . . . , Tm ] with ϕ1 (X i ) = Fi (T )

140

4 Gröbner Bases of Determinantal Ideals

and ϕ2 : S → K [W1 , . . . , Wu ] with ϕ2 (X i ) = G i (W ), then I ∗ J is the kernel of the K -algebra map ϕ : S → K [T, W ] with ϕ(X i ) = Fi (T ) + G i (W ). Here we write Fi (T ) and G i (W ) instead of simply Fi and G i to clearly indicate the indeterminates appearing in these polynomials. More generally, given ideals I1 , . . . , Ir of S one defines the join ideal I1 ∗ I2 ∗ · · · ∗ Ir as the kernel of the map S → S/I1 ⊗ K S/I2 × · · · ⊗ K S/Ir sending X i to xi ⊗ 1 ⊗ · · · ⊗ 1 + 1 ⊗ xi ⊗ · · · ⊗ 1 + · · · + 1 ⊗ 1 ⊗ · · · ⊗ xi or equivalently as the elimination ideal I1 ∗ I2 ∗ · · · ∗ Ir =

 r

I j (Y j ) + (X i −

j=1

r 

 Y ji : i = 1, . . . , n) ∩ S.

j=1

where Y j = Y j1 , . . . , Y jn are new variables for j = 1, . . . , r . Equation (4.6) generalizes to   I1 ∗ I2 ∗ · · · ∗ Ir = f ∈ S : f (Y1 + · · · + Yr ) ∈ I1 (Y1 ) + · · · + Ir (Yr ) .

(4.7)

The r th secant ideal of an ideal I is the join I ∗ · · · ∗ I (with r factors) and it will be denoted by I {r } with the convention that I {1} = I . Geometrically it defines the so-called r -secant variety. The following proposition is the first result connecting ideals of minors and secants. Proposition 4.5.4 Let X be an m × n-matrix of indeterminates. Then {r }

Ir +1 ⊂ I2 for 1 ≤ r ≤ min(m, n) − 1.

Proof We introduce r further m × n matrices Y1 , . . . .Yr of indeterminates. In view of Eq. (4.7) we must show Ir +1 (Y1 + · · · + Yr ) ⊂ I2 (Y1 ) + · · · + I2 (Yr ). This follows from multilinearity: every (r + 1)-minor of Y1 + · · · + Yr is the sum of determinants of matrices with rows Yi ji where Yik denotes the k-th row of Yi . Since we have r + 1 rows, but only r choices for i, one i must appear at least twice.  In the following lemmata we collect some important features of join ideals. First we prove the distributivity over intersections and a natural inclusion.

4.5 Gröbner Bases via Secants

141

Lemma 4.5.5 Let I, J, H be ideals of S. Then (a) (I ∩ J ) ∗ H = (I ∗ H ) ∩ (J ∗ H ). (b) If J ⊂ H then I ∗ J ⊂ I ∗ H . Proof (a) Let α : S/(I ∩ J ) → S/I ⊕ S/J be the injective K -algebra map defined by α( f¯) = ( f¯, f¯). We tensor (over K ) α with S/H and obtain an injective map β : S/(I ∩ J ) ⊗ S/H → (S/I ⊗ S/H ) ⊕ (S/J ⊗ S/H ). Now the map γ : S → S/(I ∩ J ) ⊗ S/H defined by γ(X i ) = xi ⊗ 1 + 1 ⊗ xi composed with β gives the K -algebra map: β ◦ γ : S → (S/I ⊗ S/H ) ⊕ (S/J ⊗ S/H ) with β ◦ γ(X i ) = (xi ⊗ 1 + 1 ⊗ xi , xi ⊗ 1 + 1 ⊗ xi ). Since β is injective we have Ker(γ) = Ker(β ◦ γ), i.e. (I ∩ J ) ∗ H = (I ∗ H ) ∩ (J ∗ H ). (b) It follows immediately from the definition.  The next lemma states the associativity of the join operation. Lemma 4.5.6 Let I1 , . . . , Ir be ideals of S and set I = I1 ∗ I2 ∗ · · · ∗ Ir . For a given j with 1 ≤ j < r we set J = I1 ∗ I2 ∗ · · · ∗ I j and H = I j+1 ∗ · · · ∗ Ir . Then I = J ∗ H. j

Proof The ideal J is the kernel of the K -algebra map S → ⊗i=1 S/Ii sending X i to xi ⊗ 1 ⊗ · · · ⊗ 1 + · · · + 1 ⊗ 1 ⊗ · · · ⊗ xi . Tensoring (over K ) the induced injective j map S/J → ⊗i=1 S/Ii with S/H we get an injective K -algebra map: j

α : S/J ⊗ S/H → ⊗i=1 S/Ii ⊗ S/H. j

Similarly tensoring the injective map S/H → ⊗ri= j+1 S/Ii with ⊗i=1 S/Ii we obtain an injective map j β : ⊗i=1 S/Ii ⊗ S/H → ⊗ri=1 S/Ii . Composing β ◦ α with the map S → S/J ⊗ S/H whose kernel is J ∗ H one obtains exactly the map S → ⊗ri=1 S/Ii whose kernel is I . Since α and β are injective one deduces that I = J ∗ H .  Lemma 4.5.7 Let I1 , . . . , Ir be ideals of S and set I = I1 ∗ I2 ∗ · · · ∗ Ir . Then: (a) If I1 , . . . , Ir are homogeneous, then I is homogeneous.

142

4 Gröbner Bases of Determinantal Ideals

(b) If K is algebraically closed and I1 , . . . , Ir are prime, then I is prime. (c) If K is perfect and I1 , . . . , Ir are radical, then I is radical. Proof (a) The assertion follows immediately from the definition. (b) Over an algebraically closed field K the tensor product of K -algebra domains is a domain. (For extension fields of K this is stated in Zariski and Samuel [318, Cor. 1, p. 198], and the general case follows by embedding both algebras into their fields of fractions.) Hence the ideal I , being the kernel of a K -algebra map from S to a K -algebra domain, is prime. (c) Over a perfect field K the tensor product of reduced Noetherian K -algebras is reduced. (Again the case of extension fields is sufficient since a reduced Noetherian K -algebra can be embedded into a direct product of finitely many extension fields of K , and the tensor product of extension fields is reduced [318, Corollary, p. 196]). Hence the ideal I , the kernel of a K -algebra map from S to a reduced K -algebra, is radical.  In the next lemma (V ) is the ideal generated by a vector subset V of S. Geometrically it says that the join of linear subspaces is the smallest linear subspaces containing the “factors”. Lemma 4.5.8 Let V1 , V2 , . . . , Vr be vector subspaces of the K -vector space S1 of the homogeneous polynomials of degree 1. Then (V1 ) ∗ (V2 ) ∗ · · · ∗ (Vr ) = (V1 ∩ V2 ∩ · · · ∩ Vr ). Proof By virtue of Lemma 4.5.6 it is enough to prove the statement for r = 2. After a change of variables we may assume that V1 is generated by the X i with i ∈ A ⊂ {1, . . . , n} and V2 is generated by the X i with i ∈ B ⊂ {1, . . . , n}. Then the map S → S/(V1 ) ⊗ S/(V2 ) sends X i to 0 if i ∈ A ∩ B and the images of the X i with i ∈ {1, . . . , n} \ (A ∩ B) are algebraically independent in S/(V1 ) ⊗ S/(V 2). Hence the kernel of S → S/(V1 ) ⊗ S/(V2 ) is generated by {X i : i ∈ A ∩ B}, i.e.,  (V1 ) ∗ (V2 ) = (V1 ∩ V2 ).

The Join of Simplicial Complexes Given simplicial complexes , 1 , . . . , r over a common set of vertices we define the join simplicial complex as  r   1 ∗ · · · ∗ r = Fi : Fi ∈ i , i=1

and the r th secant of  as

 {r } =

r  i=1

 Fi : Fi ∈  .

4.5 Gröbner Bases via Secants

143

The combinatorial join operation of simplicial complexes commutes with taking Stanley–Reisner ideals and their joins: Lemma 4.5.9 With the notation above one has I1 ∗ I2 · · · ∗ Ir = I with  = 1 ∗ · · · ∗ r . In particular, {r }

I = I{r } . Proof By induction on r and using Lemma 4.5.6 it is enough to prove the assertion for r = 2. We assume that the complexes have vertices contained in {v1 , . . . , vn } so that their Stanley–Reisner ideals live in S = K [X 1 , . . . , X n ]. For a subset F of / F). By Lemma 4.5.8 one has PF ∗ PG = PF∪G . {1, . . . , n} we set PF = (X i : i ∈ Then      PF ∗ PG = (PF ∗ PG ) I 1 ∗ I 2 = F∈1

F∈1 ,G∈2

G∈2



=

PF∪G = I1 ∗2

F∈1 ,G∈2



where the second equality follows from Lemma 4.5.5.

Let us apply the preceding lemmata to the simplicial complexes defined by diagonals of minors. It is important to observe that Proposition 4.4.1 was proved without using any information on initial ideals: t is the simplicial complex whose minimal nonfaces are the diagonals of length t, and its faces are the unions of families of t − 1 nonintersecting paths. Proposition 4.5.10 Let X be an m × n matrix of indeterminates and consider the ideals Jt generated by the diagonals of t-minors, t = 2, . . . , min(m, n). Moreover, let t be the simplicial complex with Stanley–Reisner ideal Jt . Then {r }

r +1 = 2

and

{r }

Jr +1 = J2

for 1 ≤ r ≤ min(m, n) − 1. Proof In view of Lemma 4.5.9 it is enough to show the first equation. {r } One inclusion r +1 ⊂ 2 is evident since every face of r +1 is the union of r nonintersecting paths and therefore a union of r faces of 2 . For the converse inclusion one observes that the union of r paths of which each meets a diagonal in at most 1 point cannot contain an (r + 1)-diagonal. 

144

4 Gröbner Bases of Determinantal Ideals

Joins and Initial Ideals For monomial orders the next lemma is an obvious consequence of the Buchberger criterion, and for weight orders it is reduced to the case of monomial orders. Lemma 4.5.11 Let I, J be ideals in K [X 1 , . . . , X n ]. Assume that the set of the variables in the supports of the generators of I is disjoint from the set of the variables in the supports of the generators of J . Then for every weight w ∈ Nn>0 one has inw (I + J ) = inw (I ) + inw (J ). Proof We may assume the generators of I involve only X 1 , . . . , X m and the generators of J involve only X m+1 , . . . , X n . To compute inw (I + J ) one refines w to a monomial order < on K [X ], computes a Gröbner basis G of I + J with respect to < and then one has inw (I + J ) = (inw (F) : F ∈ G). Let G 1 be the reduced Gröbner basis of I and G 2 the reduced Gröbner basis of J . Hence G 1 involves only X 1 , . . . , X m and G 2 involves only X m+1 , . . . , X n . Therefore for every F1 ∈ G 1 and F2 ∈ G 2 one has gcd(in< (F1 ), in< (F2 )) = 1. Hence G 1 ∪ G 2 is a Gröbner basis of I + J and the desired equality follows.  Taking initial ideals does in general not commute with joins and secants, but one has at least the expected inclusion: Lemma 4.5.12 Let < be a monomial order on S and I, I1 , . . . , Ir ideals of S. Then in< (I1 ∗ · · · ∗ Ir ) ⊂ in< (I1 ) ∗ · · · ∗ in< (Ir ) and, in particular,

 in< I {r } ⊂ in< (I ){r }

for all r . Proof We argue by induction on r , starting with the case r = 2. For I = I1 and J = I2 we have to prove that in< (I ∗ J ) ⊂ in< (I ) ∗ in< (J ). Let F ∈ I ∗ J and set μ = in< (F). Let w ∈ Nn>0 be such that inw (F) = in< (F), inw (I ) = in< (I ) and inw (J ) = in< (J ). Such a weight w is guaranteed by Lemma 1.5.6. By assumption F(Y + Z ) ∈ I (Y ) + J (Z ). In the polynomial ring K [Y, Z ] we introduce the weight v = (w, w) ∈ N2n >0 and observe that for every monomial ν in K [X ] that ν(Y + Z ) is homogeneous with respect to v of weight equal v(ν). Therefore inv (F(Y + Z )) = μ(Y + Z ). Furthermore, by Lemma 4.5.11 one has in  v (I (Y ) + J (Z )) = inv (I (Y )) + inv (J (Z )). Finally one observes that inv (I (Y )) = in< (I ) (Y ) and inv (J (Z )) = in< (J ) (Z ). Summing up, we have μ(Y + Z ) ∈ in< (I )(Y ) + in< (J )(Z ), i.e., μ ∈ in< (I ) ∗ in< (J ). Hence in< (I ∗ J ) ⊂ in< (I ) ∗ in< (J ). For r > 2 we now set I = I1 ∗ · · · ∗ Ir −1 . By virtue of Lemma 4.5.6 one has I1 ∗ · · · ∗ Ir = I ∗ Ir . Then, by the statement for r = 2,

4.5 Gröbner Bases via Secants

145

in< (I1 ∗ · · · ∗ Ir ) = in< (I ∗ Ir ) ⊂ in< (I ) ∗ in< (Ir ). Furthermore, by induction in< (I ) ⊂ in< (I1 ) ∗ · · · ∗ in< (Ir −1 ), and applying the monotonicity rule Lemma 4.5.5(b) one obtains in< (I ) ∗ in< (Ir ) ⊂ (in< (I1 ) ∗ · · · ∗ in< (Ir −1 )) ∗ in< (Ir ). The associativity of joins as stated in Lemma 4.5.6 then implies  in< (I1 ) ∗ · · · ∗ in< (Ir −1 ) ∗ in< (Ir ) = in< (I1 ) ∗ · · · ∗ in< (Ir −1 ) ∗ in< (Ir ). Summing up, in< (I1 ∗ · · · ∗ Ir ) ⊂ in< (I1 ) ∗ · · · ∗ in< (Ir ). 

Initial Ideals of Ideal of Minors We have now collected all tools that are necessary to give a new proof that the t-minors form a Gröbner basis of It . Theorem 4.5.13 Let X be an m × n-matrix of indeterminates. Then the t-minors form a Gröbner basis of It (X ) for t = 1, . . . , min(m, n) with respect to any diagonal monomial order. Proof For t = 1 there is nothing to prove, and the case t = 2 is covered by Theorem 4.1.1 (with its completely elementary proof). Let t > 2 and Jt be the monomial ideal generated by the diagonals of length t. Then we have the following chain of inclusions and equalities:  {r } {r } {r } = J2 = Jr +1 . Jr +1 ⊂ in(Ir +1 ) ⊂ in(I2 ) ⊂ in(I2 ) The first inclusion in obvious and the second follows from Proposition 4.5.4. The third is Lemma 4.5.12, whereas the fourth is the case t = 2. Finally, the fifth is Proposition 4.5.10. It follows that all inclusions are in fact equalities. In particular, Jr +1 is the initial ideal of Ir +1 , and this statement is obviously equivalent to the assertion on Gröbner bases.  The chain of inclusions in the preceding proof also implies {r }

Corollary 4.5.14 Ir +1 = I2 .

146

4 Gröbner Bases of Determinantal Ideals {r }

In fact we have an equality of initial ideals, and since Ir +1 ⊂ I2 , the equality of the ideals themselves follows from Proposition 1.4.2. All arguments leading to the corollary are combinatorial, except Proposition 4.5.4. But one should of course note that there is also a rather easy geometric fact: a matrix has rank ≤ r if and only if is the sum of r matrices of rank ≤ 1. In connection with Lemma 4.5.7 the corollary implies that It is a prime ideal since we can first extend the ground field to its algebraic closure, and then conclude from Proposition 4.1.3 that It is a prime ideal. Remark 4.5.15 Sturmfels and Sullivant who developed the method of secants in [298] also treated minors of symmetric and Pfaffians of alternating matrices. Later on, Sullivant extended and varied the combinatorial approach to secant ideals. In [301] he applies it to symbolic powers, also for minors of symmetric matrices of indeterminates and to Pfaffians of alternating such matrices, in addition to many other classes of ideals.

4.6 Multigraded Structures and the Multidegree As observed already by Lasker [232] and Van der Waerden [306] at the beginning of the 20th century, many notions and algebraic invariants associated to ideals over standard graded polynomial rings can be extended to ideals and modules over standard multigraded polynomial rings. With the goal of describing the multidegree (i.e., the multigraded analogous of the multiplicity) of determinantal rings, we first revise part of the terminology related to multigraded Hilbert polynomials and related invariants. For a modern point of view on multidegrees the reader can consult [245], Castillo et al. [67], and Cid-Ruiz [77]. A polynomial ring S = K [X 1 , . . . , X r ] can be given a graded structure over an Abelian group G, i.e., a G-graded structure, by assigning to each variable X i a degree deg X i ∈ G in an arbitrary way. If G = Zn and the set {deg X i : i = 1, . . . , r } equals the canonical basis {e1 , . . . , en } of Zn we then say that S is equipped with a standard Zn -graded structure. In this case, if necessary, we may rename the variables and denote by Yi,0 , . . . , Yi,di those variables of degree ei . Here di ≥ 0 and r = d1 + d2 + · · · + dn + n. Such Zn -graded polynomial ring can be seen as the coordinate ring of the product of projective spaces P(d1 ,...,dn ) = Pd1 × · · · × Pdn . A multigraded prime ideal of p of S is said to be irrelevant if p contains S(1,1,...,1) and relevant otherwise. When K is algebraically closed relevant primes ideals correspond to irreducible subvarieties of P(d1 ,...,dn ) .

Multigraded Hilbert Functions  A finitely generated Zn -graded module M = a∈Zn Ma over a standard Zn -graded polynomial ring S has a multigraded Hilbert function, i.e., the function H(M, −) that

4.6 Multigraded Structures and the Multidegree

147

associates the number H(M, a) = dim K Ma to a = (a1 , . . . , an ) ∈ Zn . As in the Zgraded case, for a  0 the multigraded Hilbert function agrees with a polynomial in n variables PM (Z ) = PM (Z 1 , . . . , Z n ), the multigraded Hilbert polynomial of M. Let d(M) be the total degree of PM (Z ). Under mild assumptions, for example when all the minimal primes of M are relevant, one has d(M) = dim(M) − n. The homogeneous component of degree d(M) of PM (Z ) can be written as 

eb (M) Z b1 · · · Z nbn b1 !b2 ! · · · bn ! 1

where the sum is over all b ∈ Nn with bi ≤ di such that |b| = d(M). The numbers eb (M) are called mixed multiplicities (or multidegrees) of M. It turns out that they are nonnegative integers. The multidegree of M is the polynomial Deg M (Z 1 , . . . , Z n ) =



eb (M)Z 1b1 · · · Z nbn

where the sum is over all b ∈ Nn such that |b| = d(M).  One can consider M as Z-graded module by Mv = |a|=v Ma . With respect to this Z-grading M has an ordinary multiplicity e(M): Proposition 4.6.1 If all the minimal primes of M are relevant, one has e(M) =



eb (M)

b

where the sum runs over all the b ∈ Nn such that |b| = dim(M) − n. This proved in [77, Theorem 2.8] while a special case appears already in [307]. When M is the coordinate ring of an irreducible multiprojective variety X ⊂ P(d1 ,...,dn ) over an algebraically closed base field, the mixed multiplicities eb (M) have a geometric interpretation. Indeed, in this case eb (M) is the number of points of P(d1 ,...,dn ) that one gets by intersecting X with L 1 × L 2 × · · · × L n where L i is a general linear subspace of Pdi of codimension bi .

Monomial Ideals If J is a radical monomial ideal of S with associated simplicial complex  ⊂ 2[r ] one can describe the multigraded Hilbert polynomial of S/J in terms of  as follow. For each F ∈  and i ∈ [n] we set ci (F) = |{ j ∈ F : deg X j = ei }|

148

4 Gröbner Bases of Determinantal Ideals

and c(F) = (c1 (F), . . . , cn (F)) ∈ Nn . A face F is relevant if the corresponding / F) is relevant, i.e., ci (F) > 0 for every i = 1, . . . , n. Let prime ideal (X j : j ∈ R() denote the set of the relevant faces of , i.e.,   R() = F ∈  : ci (F) > 0 for all i ∈ [n] . Lemma 4.6.2 For every a = (a1 , . . . , an ) ∈ Nn+ one has H(S/J, a) =

 n    ai − 1 ci (F) − 1 F∈R() i=1

and, in particular,  n    Zi − 1 PS/J (Z 1 , . . . , Z n ) = . ci (F) − 1 F∈R() i=1 Proof First we observe that H(S/J, a) is the number of degree a monomials of S not contained in J . To any monomial X v = X 1v1 · · · X rvr we may associate its support F(X v ) = { j : v j > 0}. Since ai > 0 for all i by hypothesis we have a monomial X v of degree a does not belong to J if and only if F(X v ) ∈ R(). We partition the monomials of degree a and not in J by theirsupport. The monomials of degree a and supported on F ∈ R() have the form ( j∈F X j )X v where X v is a monomial with in F and degree a − c(F). The number of these monomials  ai −1contained nsupport .  is i=1 ci (F)−1 As before F() denotes the set of facets of . Recall that  is a pure simplicial complex if all the facets of  have the same dimension. As an immediate corollary we have. Lemma 4.6.3 Assume  is a pure simplicial complex and that F() ∩ R() = ∅. Then  Z 1c1 (F)−1 · · · Z ncn (F)−1 . Deg S/J (Z 1 , . . . , Z n ) = F∈F()∩R()

We also observe Lemma 4.6.4 Let I be a Zn -graded ideal of S and J the initial ideal of I with respect to a monomial order. Then S/I and S/J have the same multigraded Hilbert function, the same multigraded Hilbert polynomial and the same multidegree.

Determinantal Ideals Let us now turn our attention to determinantal ideals. The polynomial ring R = K [X i j : i = 1, . . . , m, j = 1, . . . , n] can be given a Zn -graded by “columns”, i.e., by

4.6 Multigraded Structures and the Multidegree

149

setting deg(X i j ) = e j ∈ Zn . With respect to this Zn -grading the determinantal ideals It (X ) are homogeneous. We have seen that the ordinary multiplicity of R/It (X ) equals the numbers of facets of the complex t described in Sect. 4.4 and that the facets can be represented as nonintersecting paths as described in Proposition 4.4.1. Combining Lemmas 4.6.3 and 4.6.4 we can derive a formula for the multidegree of R/It (X ). To this end let us introduce the generating function associated to the statistics c(F). Given a collection U of subsets of [m] × [n] we set W (U, Z 1 , . . . , Z n ) =



Z 1c1 (F) · · · Z ncn (F)

F∈U

where ci (F) = |{(a, b) ∈ F : b = i}|. Since t is pure and F(t ) ⊂ R(t ), combining Lemma 4.6.3 and 4.6.4 we get Proposition 4.6.5 Deg R/It (X ) (Z 1 , . . . , Z n ) = (Z 1 · · · Z n )−1 W (F(t ), Z 1 , . . . , Z n ). Next we want to describe a determinantal formula for W (F(t ), Z 1 , . . . , Z n ). Recall that by Proposition 4.4.1 the facets of t are exactly the families of nonintersecting paths from p1 = (1, n), p2 = (2, n), . . . , pt−1 = (t − 1, n) to q1 = (m, 1), q2 = (m, 2), . . . , qt−1 = (m, t − 1). One observes that the Gessel–Viennot formula (4.1) is compatible with any weight if, as in our case, it is given to the lattice points. Hence one gets immediately: Lemma 4.6.6 One has:  W (F(t ), Z 1 , . . . , Z n ) = det W (Paths( pi , q j ), Z 1 , . . . , Z n ) i, j=1,...,t−1 where Paths( pi , q j ) is the set of paths from pi to q j . In the following, given a finite subset L of elements of a ring, the symbol h v (L) will denote the complete symmetric function of degree v in L, i.e., the sum of all the power products of the elements of L for which v is the sum of exponents. The function h v (L) is evidently symmetric under all permutations of L. Lemma 4.6.7 Given p = (a, b) and q = (c, d) with a ≤ c and b ≥ d we have W (Paths( p, q), Z 1 , . . . , Z n ) =

 b 

 Z i h c−a (Z d , Z d+1 , . . . , Z b ).

i=d

Proof Any path P form p to q is uniquely determined by the number of points of intersection with the columns. Any such path must have at least one point on column j with d ≤ j ≤ b and no points on the other columns. The only other constrain is that the path has (b − d) + (c − a) + 1 points in total. In terms of c(P) = (c1 (P), . . . , cn (P)) the constrains are c j (P) > 0 if and only if d ≤ j ≤ b  and bj=d c j (P) = (b − d) + (c − a) + 1. Expressing this in terms of generating functions one gets the desired results. 

150

4 Gröbner Bases of Determinantal Ideals

We deduce the following: Theorem 4.6.8 Let K be a field, X be the m × n matrix of variables and 2 ≤ t ≤ min(m, n). The multidegree of the determinantal ring R/It (X ) with respect to the Zn -graded structure given by columns is  Deg R/It (X ) (Z 1 , . . . , Z n ) = (Z 1 · · · Z n )t−2 det h −i+ j (Z 1 , . . . , Z n ) i, j=1,2,...,t−1 where  = m + 1 − t. Proof Combining Proposition 4.6.5, Lemmas 4.6.6, 4.6.7 and observing that F(t ) ⊂ R(t ) we have Deg R/It (X ) (Z 1 , . . . , Z n )

⎛⎛

= (Z 1 · · · Z n )−1 det ⎝⎝

n 

⎞

⎞

Z k ⎠ h m−i (Z j , . . . , Z n )⎠

k= j

.

i, j=1,2,...,t−1

 For j = 1, . . . , t − 1 the factor nk= j Z k can be extracted from the determinant and hence the monomial in front of the determinant becomes t−3 t−2 t−2 Z t−1 Z t · · · Z nt−2 Z 2 Z 32 · · · Z t−2

while the determinant becomes  det h m−i (Z j , . . . , Z n ) i, j=1,2,...,t−1 . It is now enough to prove that the latter equals to  Z 1t−2 Z 2t−3 · · · Z t−2 det h −i+ j (Z 1 , . . . , Z n ) . Let us explain this last transition in full details for the case t = 4 and hence  = m − 3 which is general enough to show all the relevant features and small enough to avoid to be drowned in indices. We have to prove the equality: ⎛

⎞ h m−1 (Z 1 , . . . , Z n ) h m−1 (Z 2 , . . . , Z n ) h m−1 (Z 3 , . . . , Z n ) det ⎝h m−2 (Z 1 , . . . , Z n ) h m−2 (Z 2 , . . . , Z n ) h m−2 (Z 3 , . . . , Z n )⎠ h m−3 (Z 1 , . . . , Z n ) h m−3 (Z 2 , . . . , Z n ) h m−3 (Z 3 , . . . , Z n ) ⎛ ⎞ h m−3 (Z 1 , . . . , Z n ) h m−2 (Z 1 , . . . , Z n ) h m−1 (Z 1 , . . . , Z n ) = Z 12 Z 2 det ⎝h m−4 (Z 1 , . . . , Z n ) h m−3 (Z 1 , . . . , Z n ) h m−2 (Z 1 , . . . , Z n )⎠ . h m−5 (Z 1 , . . . , Z n ) h m−4 (Z 1 , . . . , Z n ) h m−3 (Z 1 , . . . , Z n ) Let us elaborate the first term of the equality. We subtract the second column from the third column and the first column from the second column. Since h v (Z j+1 , . . . , Z n ) −

4.6 Multigraded Structures and the Multidegree

151

h v (Z j , . . . , Z n ) = −Z j h v−1 (Z j , . . . , Z n ), in this way we can collect −Z 2 from the third column and −Z 1 from the second column and the first term becomes ⎛ ⎞ h m−1 (Z 1 , . . . , Z n ) h m−2 (Z 1 , . . . , Z n ) h m−2 (Z 2 , . . . , Z n ) Z 1 Z 2 det ⎝h m−2 (Z 1 , . . . , Z n ) h m−3 (Z 1 , . . . , Z n ) h m−3 (Z 2 , . . . , Z n )⎠ . h m−3 (Z 1 , . . . , Z n ) h m−4 (Z 1 , . . . , Z n ) h m−4 (Z 2 , . . . , Z n ) Then we subtract the second column from the third column and collect −Z 1 to get ⎛

⎞ h m−1 (Z 1 , . . . , Z n ) h m−2 (Z 1 , . . . , Z n ) h m−3 (Z 1 , . . . , Z n ) −Z 12 Z 2 det ⎝h m−2 (Z 1 , . . . , Z n ) h m−3 (Z 1 , . . . , Z n ) h m−4 (Z 1 , . . . , Z n )⎠ . h m−3 (Z 1 , . . . , Z n ) h m−4 (Z 1 , . . . , Z n ) h m−5 (Z 1 , . . . , Z n ) Finally we exchange columns one and three and obtain ⎛

⎞ h m−3 (Z 1 , . . . , Z n ) h m−2 (Z 1 , . . . , Z n ) h m−1 (Z 1 , . . . , Z n ) Z 12 Z 2 det ⎝h m−4 (Z 1 , . . . , Z n ) h m−3 (Z 1 , . . . , Z n ) h m−2 (Z 1 , . . . , Z n )⎠ h m−5 (Z 1 , . . . , Z n ) h m−4 (Z 1 , . . . , Z n ) h m−3 (Z 1 , . . . , Z n ) 

which is the desired result.

The determinant that appears in Theorem 4.6.8 is actually a Schur polynomial. The reader is refereed to Macdonald [241] and Stanley [292] for a complete treatment of the theory of symmetric functions and Schur polynomials. We collect only the definitions and the properties that we will use in the following. A partition λ is a weakly decreasing sequence of nonnegative integers λ1 , λ2 , . . . , λr . The conjugate partition λ∗ = λ∗1 , λ∗2 , . . . , is defined by λi∗ = |{ j : λ j ≥ i}|. Given a partition λ = λ1 , λ2 , . . . , λr the Schur polynomial sλ (Z ) associated to λ and with respect to variables Z = Z1, . . . , Zn is

 sλ (Z ) = det h λi −i+ j (Z ) i, j=1,...,r .

By r construction, sλ (Z ) is a symmetric homogeneous polynomial of degree |λ| = i=1 λi with integral coefficients. Let m μ (Z ) denote the monomial symmetric polynomial associated with the partition μ = μ1 ≥ μ2 ≥ · · · ≥ μn ≥ 0: it is the sum of μ μ all conjugates of Z 1 1 · · · Z n n under the action of the full permutation group Sn . Since the m μ (Z ) form a K -basis of the space of symmetric polynomials, one can express sλ (Z ) as  sλ (Z ) = K λ,μ m μ (Z ). μ : |μ|=|λ|

The coefficients K λ,μ are known as the Kostka numbers of the pair of partitions λ, μ. We recall here their main properties:

152

4 Gröbner Bases of Determinantal Ideals

Proposition 4.6.9 For all partitions λ = λ1 , λ2 , . . . , λr and μ = μ1 , μ2 , . . . , μn with |λ| = |μ| one has: (a) K λ,μ ∈ N. s s λi ≥ i=1 μi (b) K λ,μ > 0 if and only if λ ≥ μ in the dominance order, i.e., i=1 for every s = 1, . . . , r . (c) K λ,μ is the number of standard tableaux of shape λ∗ with entries 1, . . . , n and multiplicities given by μ (i.e. μ1 entries are equal to 1, μ2 entries are equal to 2 and so on). Note that in the combinatorial literature, for example in [241, 292], the columns and rows are exchanged for tableaux and that our attribute “standard” is only “semistandard’ there. Proposition 4.6.9 allows us to reformulate Theorem 4.6.8 in terms of Schur polynomials. Theorem 4.6.10 The multidegree of the determinantal ring R/It (X ) associated to the m × n matrix of variables X with 2 ≤ t ≤ min(m, n) and with respect to the Zn -graded structure given by columns is Deg R/It (X ) (Z 1 , . . . , Z n ) = (Z 1 · · · Z n )t−2 sλ (Z ) where

λ = (t−1) = , , . . . ,  with  = m + 1 − t. !" # (t−1)−times

Summing up, we have the following combinatorial descriptions of the mixed multiplicities of the determinantal ring. Theorem 4.6.11 Let R/It (X ) be the determinantal ring defined by the m × n matrix of variables X and 2 ≤ t ≤ min(m, n), equipped with the Zn -graded structure given by columns. Set λ = (t−1) with  = m + 1 − t. For b ∈ Nn with |b| = dim R/It (X ) − n the mixed multiplicity eb (R/It (X )) satisfies the following properties. (a) eb (R/It (X )) is a symmetric function of b. (b) eb (R/It (X )) > 0 if and only if t − 2 ≤ bi ≤ m − 1 for every i = 1, . . . , n. (c) Set ci = bi + 1 for i = 1, . . . , n. Then eb (R/It (X )) is the number of families of nonintersecting paths from p1 = (1, n), p2 = (2, n), . . . , pt−1 = (t − 1, n) to q1 = (m, 1), q2 = (m, 2), . . . , qt−1 = (m, t − 1) with exactly ci points on the i-th column for i = 1, . . . , n (i.e. ci counts the lattice points on the i-th column of all paths in the family). (d) Set μi = bi − (t − 2) for i = 1, . . . , n. Then eb (R/It (X )) equals the Kostka number K λ,μ , i.e., the number of standard tableaux of shape λ∗ with entries 1, . . . , n and multiplicities given by μ.

4.6 Multigraded Structures and the Multidegree

153

Proof (a) The fact that eb (R/It (X )) is a symmetric function of b follows from the fact that It (X ) is invariant under the permutation of the columns or from Theorem 4.6.8. Furthermore (c) and (d) are reformulations of Proposition 4.6.5, Lemmas 4.6.6, 4.6.7 and Theorem 4.6.10 respectively. Finally, (b) follows by applying Proposition 4.6.9(b) to the present situation.  Example 4.6.12 For m = n = 4 and t = 3 we have 4that  = m + 1 − t = 2, t − 1 = 2 and λ = 2, 2. With Z = Z 1 , . . . , Z 4 and z = i=1 Z i we have:  Deg R/I3 (X ) (Z ) = z s2,2 (Z ) = z m (2,2,0,0) + m (2,1,1,0) + 2m (1,1,1,1) = m (3,3,1,1) + m (3,2,2,1) + 2m (2,2,2,2) . By Theorem 4.6.11, the coefficients appearing in the expression have two combinatorial interpretations. For example, the coefficient 2 of m (2,2,2,2) is the Kostka number K 22,1111 , i.e. the number of standard tableaux of shape 2, 2 with entries 1, . . . , 4 and multiplicities given by (1, 1, 1, 1),

1

2

1

3

3

4

2

4

and it is also the number of families of nonintersecting paths from p1 = (1, 4), p2 = (2, 4) to q1 = (4, 1), q2 = (4, 2) with 3 points on each column as shown in Fig. 4.8. Finally we have also a geometric interpretation of the coefficient of m (2,2,2,2) . That coefficient is the number of points that one gets by intersecting the variety of 4 × 4 matrices of rank at most 2 with a product L 1 × · · · × L 4 where L i is a generic linear space of dimension codimension 2 of P3 . Interpreting the four columns of the matrix as points in P3 , the rank 2 condition means that the four points belong to a line in P3 that must intersect the four generic lines L 1 , . . . , L 4 . How many lines intersect four general lines in P3 ? The answer is 2 and this a classical instance of Schubert calculus, see Eisenbud and Harris [119, 3.4.1] for a modern exposition. The computation Example 4.6.12 can be easily generalized.

Fig. 4.8 Two families of nonintersecting paths with 3 points on each column

154

4 Gröbner Bases of Determinantal Ideals

Example 4.6.13 For m = n and t = n− 1 we have  = m + 1 − t = 2 and λ = n Z i we have: 2(n−2) . With Z = Z 1 , . . . , Z n and z = i=1  Deg R/In−1 (X ) (Z ) = z n−3 s2(n−2) (Z ) = z n−3 m μ1 + m μ2 + 2m μ3 with

⎧ (n−4) ⎪ , 2, 2, 0, 0) ⎨(2 μi = (2(n−4) , 2, 1, 1, 0) ⎪ ⎩ (n−4) , 1, 1, 1, 1) (2

if i = 1 if i = 2 if i = 3.

Remark 4.6.14 One can consider R/It (X ) as multigraded by rows instead of by columns. The corresponding multidegree and mixed multiplicities are obtained by exchanging the role of m and n is the expressions in Theorems 4.6.8 and 4.6.11. Remark 4.6.15 In [245, Chap. 15] the authors compute the multidegree for a large family of determinantal ideals, the Schubert determinantal ideals, with respect to the (finest possible) multigrading deg X i j = ei − f j ∈ Zm ⊕ Zn with {e1 , . . . , em } and { f 1 , . . . , f n } being the canonical bases of Zm and Zn , as introduced in Section 3.3. The ideal of minors It is a special Schubert determinantal ideal. For Schubert determinantal ideals the authors observe that the multidegree is given by a Schur polynomial [245, 15.39] which is (at least apparently) different from the Schur polynomial we have identified. Definition 4.6.16 A Zn -graded module M is said to have multiplicity free multidegree if eb (M) ∈ {0, 1} for all b with |b| = d(M). There are only few determinantal rings with multiplicity free multidegree: Proposition 4.6.17 Let R/It (X ) be a determinantal ring defined by the m × n matrix of variables X , equipped with the Zn -graded structure given by columns. Assume that 2 ≤ t ≤ min(m, n). Then R/It (X ) has a multiplicity free multidegree if and only if t = 2 or t = min(m, n). Proof First we show: if t = 2 or t = min(m, n) then eb (R/It (X )) ∈ {0, 1} for all b ∈ Zn with |b| = dim R/It (X ) − n. For t = 2 the claim follows immediately from Theorem 4.6.8. For t = m ≤ n it follows from the description of eb (R/It (X )) in Theorem 4.6.11(d) in terms of standard tableaux since the corresponding shape is a single row. The case t = n ≤ m can be treated as follows. By 4.6.11(b) eb (R/In (X )) > 0 if and only if n − 2 ≤ bi ≤ m − 1 and |b| = dim R/In (X ) − n. Setting bi = (m − 1) − ci and rewriting the conditions with respect to (c 1 , . . . , cn ), we conclude that n n only if (c , . . . , c ) ∈ N and eb (R/In (X )) > 0 if and 1 n i=1 ci = m − n + 1. Hence m n there are exactly n−1 elements b ∈ N such that eb (R/In (X )) > 0. By Theorem m  . It 4.6.1 b eb (R/In (X )) gives the ordinary multiplicity of R/In (X ), which is n−1 follows that eb (R/In (X )) = 1 whenever eb (R/In (X )) > 0.

Notes

155

It remains to show: if 2 < t < min(m, n) then there exists b ∈ Zn with |b| = dim R/It (X ) − n such that eb (R/It (X )) > 1. Set λ = (t−1) with  = m + 1 − t. Taking into consideration Proposition 4.6.11(d), we may as well identify an element μ ∈ Nn such that: (a) |μ| = (t − 1), (b) there are at least two standard tableaux of shape λ∗ and entries 1, . . . , n with multiplicity given by μ. We define such a μ as follows: ⎧  ⎪ ⎪ ⎪ ⎨ − 1 μi = ⎪ 1 ⎪ ⎪ ⎩ 0

if i if i if i if i

= 1, 2, . . . , t − 3 = t − 2, t − 1 = t, t + 1 = t + 2, . . . , n.

For i = 1, . . . , t − 3, the i-th column of any standard tableau of shape λ∗ and content given by μ consists of exactly  entries equal to i. So we may simply assume that t = 3. Similarly we may assume that n = t + 1 = 4 so that μ = ( − 1,  − 1, 1, 1). Now for j = 1, . . . , m − 4 the j-th row must have entries 1 and 2. Again we may then assume that m = 4 and hence  = 2. Now it is clear that there are exactly two tableaux of shape 2, 2 and content (1, 1, 1, 1), that is, those described in Example 4.6.12. 

Notes That the t-minors of X form a Gröbner basis of It has been proved by several authors. To the best of our knowledge, the result was first published by Narasimhan [251]. Independently, a proof was given by Caniglia et al. [65]. The result was re-proved by Ma [239]. In the fundamental paper [212] Knuth extensively treats the RSK correspondence for column standard bitableaux with increasing columns and nondecreasing rows. The same point of view is taken in Fulton [133], Knuth [213], Sagan [273] and Stanley [292]. The notes to Chap. 7 of [292] contain a detailed historical discussion of the correspondence. Sturmfels [294] found the elegant access to the Gröbner bases of the ideals It (X ) via the Robinson–Schensted–Knuth correspondence RSK. Later on the method was extended by Herzog and Trung [172] to the 1-cogenerated ideals, ladder determinantal ideals and ideals of Pfaffians. Bruns and Conca applied the RSK method to (symbolic) powers and products of the ideals It . Herzog and Trung developed the approach to the homological properties of determinantal ideals via Gröbner deformation and proved a formula for multiplicities. The treatment of the Hilbert series given in Sect. 4.4 is due to Conca and Her-

156

4 Gröbner Bases of Determinantal Ideals

zog [91]. Explicit formulas for the Hilbert series had previously been obtained by Abhyankar [3]. The multiplicity formulas (4.2) and Theorem 4.4.2 are most likely the oldest results on determinantal rings and varieties in this book. Therefore we discuss their interesting history. On 1900 Corrado Segre published the paper [279] that starts with a determinantal formula for the degree of a certain variety introduced by Schubert. Segre says that Schubert (who was a highschool teacher and not a university professor) had “guessed and then published the formula “as correct”, however, without proof. Indeed Schubert had developed powerful methods to solve enumerative problems, but fell short to put his techniques on a solid mathematical ground. This led Hilbert to formulate the problem of establishing “rigorously and with an exact determination of the limits of their validity those geometrical numbers which Schubert especially has determined on the basis of the so-called principle of special position”. This is listed as the 15th problem in his famous lecture at the International Congress of Mathematicians 1900 in Paris [175]. Returning to Segre’s paper, in the footnote he says that he had been “informed by Mister Schubert that he (Schubert) does not intend to publish his proof now”, and hence he gave his student Giovanni Zeno Giambelli the task to “find Schubert’s proof and make further investigations in the field”. Segre goes on and explains that a special case of Schubert’s formula gives the determinantal expression for the degree of the determinantal variety, and then deduces a compact expression for it . After all the transformations necessary to make it compatible with our notation, the determinantal formula is exactly (4.2) and the compact version is Theorem 4.4.2. In 1903 Giambelli’s note [141] was presented at the Acccademia dei Lincei by Segre himself. It gives a formula in terms of a sum of determinants for the degree of the deteminantal variety in the case the (i, j)-entry of the matrix is a general form of degree pi + q j . Giambelli explains that the formula in Segre’s note [279] is a special case of his formula. For lack of space the proof of Giambelli’s results appeared in [142]. Giambelli was the first to publish a complete proof of (4.2) and Thoerem 4.4.2, and hence these formulas are rightly attributed to him. The proof that we presented above follows [172]. The Remarks 4.4.15 and 4.4.16 list several other articles on Gröbner bases of determinantal and related ideals and their applications to enumerative invariants. The idea of using secant ideals for the computation of Gröbner bases was developed by Sturmfels and Sullivant [298]. Instead of using the simplicial complexes t , they apply Dilworth’s theorem on chains and antichains in posets. Initial ideals of joins and joins of monomial ideals had previously been studied by Simis and Ulrich. Their article [284] and [298] are our main sources for the general results on joins developed in Sect. 4.5. The computation in Example 4.6.12 appears also in [307], a paper that Van der Waerden wrote 50 years after his first paper on the subject [306]. Van der Waerden mentions that V. Strassen “during a walk in 1976” raised the questions of computing the degree of a determinantal variety and suggested the approach via mixed multiplicities.

Notes

157

That R/I2 (X ) has a multiplicity free multidegree has been observed by Cartwright and Sturmfels [66], while for R/It (X ) with t = min(m, n) it has been proved by different arguments in Conca et al. [86], and the same authors computed the multidegree explicitly in [87].

Chapter 5

Universal Gröbner Bases

A universal Gröbner basis of an ideal I in a polynomial ring is a subset of I that is a Gröbner basis with respect to every monomial order. Despite to the fact that there are infinitely many monomial orders, one can show that every ideal I has only a finite number of distinct initial ideals and this implies that I has a finite universal Gröbner basis. The goal of the chapter is to discuss universal Gröbner bases for ideals of minors. For maximal minors the result is optimal: the maximal minors themselves are a universal Gröbner basis. For 2-minors a universal Gröbner basis can be described as the set of cycles of the complete bipartite graph. In both cases, maximal minors and 2-minors, every initial ideal is radical and defines a Cohen– Macaulay ring. For minors of size t with 2 < t < m we do not have a description of a universal Gröbner basis and we provide examples of nonradical initial ideals that define nonCohen–Macaulay rings.

5.1 Universal Gröbner Bases Let I be an ideal in the polynomial ring R = K [X 1 , . . . , X n ]. Definition 5.1.1 A universal Gröbner basis of I is a subset of I that is a Gröbner basis with respect to every monomial order. Lemma 5.1.2 Let V ⊂ R be a vector subspace. Then the set of the monomial ideals J such that Mon(R) \ J is a K -basis of R/V is finite. Proof For a proof by contradiction assume A0 is an infinite set of monomial ideals such that Mon(R) \ J is a K -basis of R/V for every J ∈ A0 . Let f 1 ∈ V be nonzero. Since f 1 ≡ 0 mod V , for every J ∈ A0 there exists a monomial μ ∈ supp( f 1 ) such that μ ∈ J . Since A0 is infinite, we can find a monomial μ1 in the support of f 1 such that A1 = {J ∈ A0 : μ1 ∈ J } is infinite. Now (μ1 )  J for every J ∈ A1 , because © Springer Nature Switzerland AG 2022 W. Bruns et al., Determinants, Gröbner Bases and Cohomology, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-05480-8_5

159

160

5 Universal Gröbner Bases

otherwise we would have a strict inclusion among K -bases for R/V . Therefore the monomials in Mon(R) \ (μ1 ) are linearly dependent mod V . Hence there exists a polynomial f 2 ∈ V with supp( f 2 ) ⊂ Mon(R) \ (μ1 ). Arguing as above, one gets at least one monomial μ2 in the support of f 2 such that A2 = {J ∈ A1 : μ2 ∈ J } is infinite. By construction, (μ1 )  (μ1 , μ2 ), and an iteration of the argument creates a strictly ascending chain of monomial ideals, which is impossible in the Noetherian ring R.  Despite to the fact that there are infinitely many monomial orders we have: Lemma 5.1.3 Let I ⊂ R be an ideal. The set of initial ideals of I is finite. Proof If J is an initial ideal of I with respect to some monomial order then Mon(R) \ J is a K -basis of R/I by Proposition 1.2.6. Hence Lemma 5.1.2 implies that there are only finitely many initial ideals of I .  In the next lemma we say that μ is a minimal monomial generator of a monomial ideal J it is a monomial that belongs to the unique minimal monomial set of generators of J . In part (c) of the lemma we assign to a polynomial f ∈ R a monomial μ ∈ supp( f ) that we call its mark, and the say that f is marked by μ. This allows us to distinguish different monomial orders: if f belongs to the reduced Gröbner basis for more than a single monomial order, then its initial monomial can depend on the monomial order. Lemma 5.1.4 Let I ⊂ R be an ideal and J be a monomial ideal such that Mon(R) \ J is a K -basis of R/I . (a) For every monomial μ of J there exists a unique polynomial f μ ∈ I of the form f μ = μ + gμ with supp(gμ ) ⊂ Mon(R) \ J . (b) The set of polynomials f μ with μ a minimal monomial generator of J forms a reduced Gröbner basis of I with respect to every monomial order < such that in< (I ) = J . (c) There is a one-to-one correspondence between initial ideals of I and reduced Gröbner bases of I marked by initial monomials. Proof Part (a) follows from the fact that Mon(R) \ J is a K -basis of R/I . Part (b) follows immediately from definition of reduced Gröbner basis. Finally (c) is a restatement of (b).  As a consequence we have: Proposition 5.1.5 Every ideal I has a finite universal Gröbner basis. Proof By Lemma 5.1.3 the ideal I has finitely many initial ideals, say J1 , . . . , Jv . polynomials Fi in I as described in Lemma 5.1.4(a). By For each Ji we pick a set of v Fi is a finite universal Gröbner basis of I .  Lemma 5.1.4(b) the union i=1 There are few classes of ideals for which universal Gröbner bases are known. Of course, for principal ideals and monomial ideals universal Gröbner bases are easy to detect. For an ideal I generated by linear forms the set of “circuits”, i.e., linear forms in the ideal with minimal support , is a universal Gröbner basis [296, 1.6]. The goal of the next sections is to discuss universal Gröbner bases for ideals of minors.

5.2 Universal Gröbner Bases for Maximal Minors

161

5.2 Universal Gröbner Bases for Maximal Minors We use our standard setup: K is a field, X = (X i j ) is a matrix of indeterminates of size m × n with m ≤ n, K [X ] = K [X i j : i = 1, . . . , m, j = 1, . . . , n] and Im (X ) is the ideal of K [X ] generated by the maximal minors of X . The goal of this section is to prove the following result: Theorem 5.2.1 The set of maximal minors of X is a universal Gröbner basis of Im (X ). Let [c1 , . . . , cm ]W denote the minor with column indices c1 , . . . , cm of an m × n matrix W . In the following lemma we compare two Hilbert series: the inequality HN (t) ≥ HM (t) is to be understood degree by degree, i.e., H (N , k) ≥ H (M, k) for all degrees k. Lemma 5.2.2 Let R be a positively graded K -algebra, let M and N be finitely generated graded R-modules, and x1 , . . . , xs ∈ R homogeneous elements of positive degree. Set J = (x1 , . . . , xs ). Suppose that (a) HN (t) ≥ HM (t), (b) HN /J N (t) = HM/J M (t), (c) x1 , . . . , xs is an M-regular sequence. Then HM (t) = HN (t) and x1 , . . . , xs is an N -regular sequence. Proof For i = 1, . . . , s set Ji = (x1 , . . . , xi ), Ni = N /Ji N , di = deg xi and gi (t) = s (1 − t d j ) ∈ Q[t]. Furthermore set N0 = N , and for i = 0, . . . , s − 1 let K i+1 j=i denote the submodule {y ∈ Ni : xi+1 y = 0} of Ni shifted by −di+1 . The multiplication by xi+1 on Ni induces the exact sequence 0 → K i+1 → Ni (−di+1 ) → Ni → Ni+1 → 0. It yields HNi+1 (t) = (1 − t di+1 )HNi (t) + HK i+1 (t) and hence HN /J N (t) = g1 (t)HN (t) +

s 

g j+1 (t)HK j (t).

j=1

Since HN /J N (t) = HM/J M (t) and HM/J M (t) = g1 (t)HM (t) by hypothesis, we have g1 (t) (HN (t) − HM (t)) +

s 

g j+1 (t)HK j (t) = 0.

j=1

The power series HN (t) − HM (t) and HK j (t) have nonnegative coefficients and the least degree terms of the polynomials gi (t) have coefficient 1. Therefore HN (t) =

162

5 Universal Gröbner Bases

HM (t) and K j = 0 for j = 1, . . . , s. In particular, this proves that x1 , . . . , xs is an N -regular sequence.  Now we are ready to prove Theorem 5.2.1. Proof of Theorem 5.2.1 Gröbner bases and, hence, universal Gröbner bases are insensitive to base field extensions. Therefore we may assume without loss of generality that the base field K is infinite. Let A = (ai j ) be an m × n matrix with entries in K ∗ , such that all its m-minors are nonzero. It exists because K is infinite. Set R = K [X ] and consider the K -algebra map  : R → K [Y1 , . . . , Yn ] induced by the assignment (X i j ) = ai j Y j for all i and j. By construction, Ker  is generated by n(m − 1) linear forms. Let Y be the matrix (X ), that is Y = (ai j Y j ). By construction ([c1 , . . . , cm ] X ) = [c1 , . . . , cm ]Y = [c1 , . . . , cm ] A Yc1 · · · Ycm . By assumption [c1 , . . . , cm ] A ∈ K ∗ . Hence (Im (X )) = Im (Y ) = (Yc1 · · · Ycm : 1 ≤ c1 < · · · < cm ≤ n) i.e., Im (Y ) is generated by all the square-free monomials in Y1 , . . . , Yn of total degree m. In particular Im (Y ) has height n − m + 1. To sum up: Ker  is generated by n(m − 1) linear forms and has height n(m − 1) in R/Im (X ). Since R/Im (X ) is Cohen– Macaulay, this implies that Ker  is generated by an R/Im (X )-regular sequence. Now let ≺ be any monomial order on R and let D be the ideal generated by the initial terms of the maximal minors of X with respect to ≺. We have D ⊂ in≺ (Im (X )) and    in≺ ([c1 , . . . , cm ] X ) = (X σ1 c1 · · · X σm cm ) = aσ1 c1 · · · aσm cm Yc1 · · · Ycm for some σ in the symmetric group Sm . Hence (D) = Im (Y ). We apply Lemma 5.2.2 to the following data: M = R/Im (X ),

N = R/D, and J = Ker .

We have already checked that J is generated by a regular sequence on M. That HM (t) ≤ HN (t) follows from the fact that by construction D ⊂ in≺ (Im (X )). Finally we have M/J M = K [Y1 , . . . , Yn ]/Im (Y ) = N /J N . Hence we deduce from

5.3 Universal Gröbner Bases for 2-Minors

163

Lemma 5.2.2 that M and N have the same Hilbert series, that is, D and Im (X ) have the same Hilbert series. Since D ⊂ in≺ (Im (X )) and Im (X ) and in≺ (Im (X ))  have the same Hilbert series, it follows that D = in≺ (Im (X )). A corollary (of the proof) of Theorem 5.2.1 is the following: Corollary 5.2.3 Let D be any initial ideal of Im (X ). Then: (a) D is radical, (b) R/D is Cohen–Macaulay, (c) βi j (R/D) = βi j (R/Im (X )) for all i, j. Part (a) of Corollary 5.2.3 is an immediate consequence of Theorem 5.2.1 since every term in every minor of X is a square free monomial. Part (c) implies part (b) because R/Im (X ) is Cohen–Macaulay, and (c) itself follows directly from the proof of Theorem 5.2.1 since (using the notation of the proof) R/D and R/Im (X ) are isomorphic mod J and J is generated by a regular sequence on both R/D and R/Im (X ). Finally let us observe that part (b) of Corollary 5.2.3 is also a special instance of Theorem 2.4.3 since R/Im (X ) is Cohen–Macaulay. It should be noted that the graded minimal free resolution of R/Im (X ) is the wellknown Eagon–Northcott complex (see Bruns and Vetter [61, Sect. 2.C]): Im (X ) has a linear free resolution.

5.3 Universal Gröbner Bases for 2-Minors As before, let K be a field and X = (X i j ) be an m × n matrix of indeterminates. In Proposition 4.1.3 we have seen that I2 (X ) is the kernel of the K -algebra homomorphism σ(X i j ) = Yi Z j . σ : K [X ] → K [Y1 , . . . , Ym , Z 1 . . . , Z n ] For every pair of sequences I = i1 , . . . , is ,

and

J = j1 , . . . , js

with pairwise different indices i u , 1 ≤ i 1 , . . . , i s ≤ m and pairwise different jv , 1 ≤ j1 , . . . , js ≤ n, the polynomial FI,J = X i1 j1 X i2 j2 · · · X is js − X i2 j1 X i3 j2 · · · X is js−1 X i1 js is clearly in I2 (X ). Theorem 5.3.1 The set of the binomials FI,J is a universal Gröbner basis of I2 (X ).

164

5 Universal Gröbner Bases

Proof In order to prepare the proof of the claim, we use a graph-theoretical encoding of our data. Let G be the graph with m + n vertices, labeled by Y1 , . . . , Ym and Z 1 , . . . , Z n and edges labeled by X i j connecting Yi and Z j for all i and j: we have constructed the complete bipartite graph K m,n . Consider the binomial FI,J . The monomial X i1 j1 X i2 j2 · · · X is js corresponds to a set of edges to that we assign the color blue, and the second monomial X i2 j1 X i3 j2 · · · X is js−1 X i1 js defines a second set of edges, colored red. We start at the vertex Yi1 and walk along the blue edge X i1 j1 to the vertex Z j1 . From there the red edge X i2 j1 gets us to Yi2 . Continuing we construct a path of alternating blue and red edges that traverses each vertex at most once and is finally closed by the last red edge that brings us back to Yi1 where we stop. Now we consider an arbitrary binomial ζ − ξ ∈ Ker σ with gcd(ζ, ξ) = 1, say β = X t1 u 1 · · · X tr ur − X v1 w1 · · · X vr wr . Again we construct a path whose edges we color alternatively blue and red (the color is defined by β). Namely, we start from Yt1 and reach Z u 1 via X t1 u 1 . Since Z u 1 divides σ(ζ), it must divide σ(ξ) = σ(ζ) as well: u 1 = w p for some p, and reordering the factors of ξ we can assume that p = 1, i.e. u 1 = w1 . Now Yv1 divides σ(ξ) and therefore it divides σ(ζ): u q = v1 for some q. It is impossible that q = 1 because otherwise ζ and ξ would have the common factor X t1 u 1 . We can assume q = 2, and from Yt2 we reach Z u 2 . Continuing our construction we obtain a zigzag path of alternating colors between the vertices Yi and Z j . We stop it as soon as a vertex is reached a second time. This must happen because there are only finitely many edges and the vertex that is visited twice can be on one side or on the other: it can be a vertex of type Yc or Z d . The two sets of blue and red edges, respectively, of the cycle starting and ending at Yc or at Z d define two monomials and their difference clearly is one of the FI,J . Now assume the theorem is wrong and let < be a monomial order for which the FI,J do not form a Gröbner basis. It has been observed on p. 18, that the Buchberger algorithm generates a Gröbner basis B of binomials for the binomial ideal I2 (X ), and that its reduced Gröbner basis is binomial. Therefore there is a binomial β = ζ − ξ in the reduced Gröbner basis such that ζ = in< (β) is not divisible by any of the / in< (I2 (X )). monomials in< (FI,J ) and ξ ∈ However, the preceding argument shows that there exists FI,J such that its first monomial divides ζ and its second monomial divides ξ. Which of the two is in< (FI,J ) does not matter: both alternatives lead to a contradiction.  Remark 5.3.2 Clearly, for different pairs of sequences I, J and H, K one may have FI,J = ±FH,K . A canonical choice can be made as follows: let A be the set of polynomials FI,J where I = i 1 , . . . , i s ∈ and J = j1 , . . . , js are sequences of distinct elements such that: i 1 < i k for k = 2, . . . , s and j1 < j2 .

5.3 Universal Gröbner Bases for 2-Minors

165

The set A is indeed a minimal universal Gröbner basis of I2 (X ), that is, no proper subset is a universal Gröbner basis of I2 (X ). Here it is important to note that the polynomials FI,J are irreducible and are “circuits", i.e., the set of the variables involved in FI,J is minimal among all the nonzero polynomials in I2 (X ). Assume that A \ {FI,J } is a universal Gröbner basis of I2 (X ) for some FI,J ∈ A. Let < be a lexicographic monomial order for which the variables not in the support of FI,J are larger than those in the support. Then there exists FH,K ∈ A, FH,K = FI,J , such that the initial monomial of FH,K divides that of FI,J . Because of the choice of X 21 X 12 ,

X 12 X 23 > X 22 X 13 ,

X 13 X 21 > X 23 X 11

yield a contradiction (the product of the left sides equals the product of the right sides). Then we may assume d = 4 and proceed. We leave the formal continuation of the argument to the reader.  Remark 5.3.6 It follows from Proposition 5.3.5 and Theorem 6.2.1 that the set M2 of 2-minors of the 2 × n matrix X is a universal Sagbi basis of K [M2 ]. This is a truly exceptional case: already for the 3 × 6 matrix X the set M3 of the 3-minors is not a universal Sagbi basis of K [M3 ], see Speyer and Sturmfels [290, Corollary 5.6.]. An almost equivalent formulation is that for X of size 2 × n and for all monomial orders > one has in> (I2 (X )k ) = in> (I2 (X ))k while the same assertion does not hold for minors of size 3 in a 3 × 6 matrix. Remark 5.3.7 The Betti numbers of the initial ideals of I2 (X ) are in general strictly larger than those of I2 (X ). For example, as the argument in Remark 5.3.2 indicates for m = n = 3, with respect to a suitable lexicographic monomial order the initial ideal D of I2 (X ) has a generator in degree 3 so that β13 (R/D) > β13 (R/I2 (X )) = 0. For m=n = 4 there is no initial ideal D of I2 (X ) with β2, j (R/D)=β2, j (R/I2 (X )) for all j. As reported by Conca et al. [94, Ex. 1.4], this can be checked by computations with specialized software Gfan [197] and CoCoA [1]. This behavior, at least for large m and n, is explained by the main result of Dao et al. [101]. Indeed, if there exists an initial ideal D of I2 (X ) with β2, j (R/D) = β2, j (R/I2 (X )) for all j then D is generated in degree 2 with linear first syzygies because I2 (X ) has linear first syzygies (see Remark 11.11.4). For such an ideal it is proved in [101, Theorem 4.1] that reg D is bounded above by logα ( f (mn)) + β where the numbers α > 1 and β and the linear function f are explicitly given. It would then follow that reg I2 (X ) ≤ logα ( f (mn)) + β which is a contradiction since reg I2 (X ) = reg K [X ]/I2 (X ) + 1 = m; see Remark 4.4.13.

Universal Revlex Gröbner Bases for 2-Minors While the 2-minors do not form a universal Gröbner basis of I2 (X ) in general, they are universal within a restricted class of monomial orders: Proposition 5.3.8 The 2-minors of X are a universal revlex Gröbner basis, i.e., they are a Gröbner basis for all (mn)! reverse lexicographic orders.

5.3 Universal Gröbner Bases for 2-Minors

167

Proof Suppose < is any reverse lexicographic order. In view of the Buchberger criterion it is enough to prove that for every pair of 2-minors F, G with gcd(in(F), in(G)) = 1 the corresponding S-polynomial reduces to 0. Clearly F, G involve in total at most 3 row indices and 3 column indices, i.e., they both live in a 3 × 3 submatrix of X . Hence we can assume right away that the matrix is 3 × 3 and test with the help of a computer algebra system that the 2-minors are a Gröbner basis for all the 9! reverse lexicographic monomial orders (this number can be substantially reduced if one identifies monomial orders that differ only by a row and column permutation). One can also perform the S-polynomial reduction “by hand” so that one can understand the role of the reverse lexicographic order. Let gcd(in(F), in(G)) = X i j . Up to sign, we may assume that F = X i j X ab − X ib X a j and G = X i j X cd − X id X cj with min{X ib , X a j } < min{X i j , X ab } and min{X id , X cj } < min{X i j , X cd }. The S-polynomial of F and G is S(F, G) = X ib X a j X cd − X id X cj X ab . Now if X a j X cd ≥ X cj X ad then we may rewrite S(F, G) via X a j X cd − X cj X ad as X ib X cj X ad − X id X cj X ab = X cj (X ib X ad − X id X ab ) and finally via X ib X ad − X id X ab we get remainder 0. We get similar reductions if either X ib X cd ≥ X cb X id

or

X id X ab ≥ X ib X ad

or

X cj X ab ≥ X cb X a j .

Hence it remains to discuss the case in which the following inequalities hold simultaneously: min{X a j , X cd } < min{X cj , X ad }, min{X id , X ab } < min{X ib , X ad }, min{X ib , X a j } < min{X i j , X ab },

min{X ib , X cd } < min{X cb , X id }, min{X cj , X ab } < min{X cb , X a j }, min{X id , X cj } < min{X i j , X cd }.

168

5 Universal Gröbner Bases

Together these inequalities imply min{X a j , X cd , X ib , X id , X ab ,X cj } < min{X cj , X ad , X cb , X id , X ib , X a j , X i j , X ab , X cd } 

which is clearly a contradiction.

Further Examples of Gröbner Bases of Determinantal Ideals A universal Gröbner basis of It (X ) is unknown for 2 < t < m. The examples below show that It (X ) has nonradical and nonCohen–Macaulay initial ideals in general. Example 5.3.9 (a) For t = 3 and m = n = 4 the ideal It (X ) has a nonradical lexicographic initial ideal. (b) For t = 3 and m = 4, n = 5 the ideal √It (X ) has a nonradical lexicographic initial ideal J such that both R/J and R/ J are not Cohen–Macaulay. (c) For t = 4 and m = n = 5 the ideal It (X ) has a nonradical reverse lexicographic initial ideal. (d) For t = 4 and m = 5, n = 6 the ideal It (X ) has√a nonradical reverse lexicographic initial ideal J such that both R/J and R/ J are not Cohen–Macaulay. Examples as described in (a)–(d) are given by means of total orders on the entries of X . This is made explicit by a matrix with entry 1 in the place where the largest variable is placed, then entry 2 for the second and so on. The computations that verified the assertions were done with CoCoA [1]. Here are matrices that realize the initial ideals of (a)–(d): ⎛

1 ⎜8 (a) ⎜ ⎝13 16 ⎛

13 ⎜25 ⎜ (c) ⎜ ⎜24 ⎝23 18

5 2 9 14

11 6 3 10

⎞ 15 12⎟ ⎟ 7⎠ 4

22 16 3 19 5

17 4 7 2 14

15 21 20 9 1

⎛

3 6 14 5

16 19 2 18

13 17 15 8

⎞ 12 9⎟ ⎟ 20⎠ 10

⎛ ⎞ 23 24 8 ⎜25 14 12⎟ ⎜ ⎟ ⎜ 11⎟ ⎟ (d) ⎜10 4 ⎝8 3 ⎠ 10 13 17 6

27 29 12 9 1

28 6 11 26 19

18 5 7 20 2

11 ⎜7 (b) ⎜ ⎝4 1

⎞ 30 21⎟ ⎟ 16⎟ ⎟ 15⎠ 22

Remark 5.3.10 For t = 3 no example as in Example 5.3.9(a) or (b) has been found for reverse lexicographic orders. It is possible that the ideal I3 (X ) has a square-free universal revlex Gröbner basis. This would indeed explain, from a different perspective, a result proved by Conca and Welker [96] asserting that the ideal of 3-minors of a m × n matrix of variables remains radical after setting an arbitrary subset of the variables to 0. For m = n = 4

Notes

169

we have found 69 distinct revlex initial ideals up to the natural S4 × S4 and Z/2Z symmetry and they are all square-free. Remark 5.3.11 Let R be a polynomial ring with a standard Zn -graded structure. A multigraded ideal I of R is a Cartwright–Sturmfels ideal if, after performing a multigraded generic change of coordinates, it has a radical initial ideal. This property has very strong consequences: if I is a Cartwright–Sturmfels ideal then all initial ideals are square-free, all reduced Gröbner bases and universal Gröbner bases consist of elements of degree bounded by (1, 1, . . . , 1). Furthermore every multigraded specialization or elimination of an Cartwright–Sturmfels ideal gives a Cartwright– Sturmfels ideal. It turns out that if I is a Cartwright–Sturmfels ideal, then R/I has multiplicity free multidegree and the converse implication holds if I is primes. See Conca et al. [86–89] for generalities and further results on Cartwright–Sturmfels ideals. As we have see in Sect. 4.6, the polynomial ring R = K [X i j : i = 1, . . . , m j = 1, . . . , n] can be given a Zn -graded by setting deg(X i j ) = e j ∈ Zn and the determinantal ideals It (X ) are homogeneous. In Proposition 4.6.17 we have characterized when R/It (X ) has multiplicity free multidegree: only for t = 2 and t = min(m, n). Hence for these values It (X ) is a Cartwright–Sturmfels ideals. This observation shads new light on the results of this section: the existence of easily described universal Gröbner bases is related to the Cartwright–Sturmfels property. Remark 5.3.12 If R/I is Gorenstein, it is reasonable to ask whether there exists a monomial order < such that R/ in(I ) is Gorenstein as well. In [94, Sect. 4] it has been proved that there exists a degree reverse lexicographic monomial order < such that D = in< (I2 (X )) is generated in degree 2 and defines a Gorenstein ring when I2 (X ) defines a Gorenstein ring (i.e., m = n). More generally, as proved by Bruns and Römer [56], a toric ideal P that defines a Gorenstein ring and has a squarefree initial ideal, also has a square-free initial ideal defining a Gorenstein ring. In a different direction, Soll and Welker [289] proved that for m = n the ideal of t-minors It (X ) has a square-free initial ideal generated in degree t that defines a Gorenstein ring. However, their proof depends on a combinatorial identity that is only available in the preprint [272] of Rubey and Stump.

Notes The original proof of Theorem 5.2.1, obtained combining work of Sturmfels and Zelevinsky [300] and of Bernstein and Zelevinsky [17] is based on a detailed analysis the Newton polytope associated to the product of the maximal minors and on the introduction of the notion of matching fields. Indeed, in [300] the proof of Theorem 5.2.1 is reduced to a conjectural assertion on matching fields that is then proved in [17]. The proof of 5.2.1 we present is taken from the paper [86] of Conca, De Negri

170

5 Universal Gröbner Bases

and Gorla and it is significantly simpler than the original. Corollary 5.2.3(c) was first proved by Boocher [21]. Theorem 5.3.1 is a special case of a theorem of Villarreal [313, 10.1.11] describing the universal Gröbner basis of a toric ideal associated to the edge algebra of a graph. It is also a special case of a result of Sturmfels describing the universal Gröbner basis of a toric ideal associated to a unimodular configuration that one obtains from [296, 4.11 and 8.11]. The statement in Proposition 5.3.8 goes back to Restuccia and Rinaldo [268] and Proposition 5.3.5 is from [300].

Chapter 6

Algebras Defined by Minors

In Chap. 4 we have studied the Gröbner deformations of determinantal ideals defined by their initial ideals. We now turn to the study of algebras generated by minors through their initial algebras. Since the initial algebras are normal monoid domains, toric algebra can be applied to them. Since normal monoid domains are very well understood, we can draw strong consequences for the algebras defined by minors. The chapter starts with a recap of toric algebra, and its results allow us to conclude that our algebras, grosso modo, are normal Cohen–Macaulay domains. The classical paradigm of an algebra generated by minors is the coordinate ring of a Grassmannian studied in Sect. 6.2. With respect to a diagonal monomial order on the polynomial ring, its initial algebra is generated by the diagonal monomials. There are at least two approaches to this basic result, namely by a direct determination of the Sagbi basis and by lifting binomial relations. We discuss both of them in detail, and draw the structural consequences for the coordinate ring of the Grassmannian. The generalization to the coordinate ring of a partial or total flag algebra is then very easy. It is characteristic for this class of rings that the straightening law is completely sufficient for finding the initial algebra. We have already indicted the connection in Sect. 3.3. The Rees algebra of the ideal of maximal minors is more complicated, but the two approaches, namely the direct determination of a Sagbi basis by means of the RSK correspondence and the lifting of binomial relations is still possible. The next level of complication is reached with the Rees algebras of ideals of minors of a fixed size in Sect. 6.3, or even Rees algebras of their products. But we can exploit the knowledge of the initial ideals of such ideals that has already been developed in Chap. 4. Now assumptions on the characteristic of the underlying field can no longer be avoided since they have come up in the primary decomposition of powers and products of ideals of minors. The determination of a Sagbi basis is no longer possible, but the RSK invariants are strong enough for structural consequences. This applies as well to the subalgebras of the polynomial ring generated by the minors of a fixed size (Sect. 6.4). Though we can even determine the Gorenstein rings among them, their defining ideals are not known in general. For nonmaximal minors © Springer Nature Switzerland AG 2022 W. Bruns et al., Determinants, Gröbner Bases and Cohomology, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-05480-8_6

171

172

6 Algebras Defined by Minors

they are not generated by the Plücker relations, and cubic minimal generators appear. As it is discussed in Sect. 6.5, it is conjectured that these ideals are generated in degree less than or equal to 3, but this is a theorem only for 2-minors in characteristic 0. In general these algebras are coordinate rings of varieties of exterior powers of linear maps, and as an illustration of a more geometric approach we determine the orbits of the natural group action on such a variety in Sect. 6.6. In its turn this allows us to find the prime ideals in the algebras that are stable under the group action. The whole chapter, but especially Sect. 6.6 depends on the study of the interplay between the combinatorics of Young diagrams and the straightening law. The last section of this chapter applies the program of toric deformation to the residue class rings of the polynomial ring modulo the ideals of minors of a fixed size. Of course, to this end the residue class ring must first be embedded into a polynomial ring. But such an embedding is classically known: it realizes the determinantal ring as an invariant ring. We exploit the toric degeneration in the computation of the divisor class group and the canonical module of a determinantal ring. A by-product is the determination of the rank 1 Cohen–Macaulay and Ulrich modules of a determinantal ring.

6.1 A Recap of Toric Algebra This section is a very brief collection of notions and results for toric rings. The reader will find a detailed treatment in several books, for example Bruns and Gubeladze [50], Bruns and Herzog [52], or Miller and Sturmfels [245]. Let K be a field and A a K -subalgebra of the polynomial ring K [X 1 , . . . , X n ]. The initial algebra in(A) of A with respect to a monomial order is generated by monomials, and its monomials are a K -basis. The algebra structure of in(A) is completely determined by the multiplicative structure on its subset of monomials: it is the K -bilinear extension of the map (X a , X b ) → X a X b = X a+b . The set of monomials of in(A) is a monoid under multiplication: it is closed under multiplication and contains 1 as its neutral element. In the commutative algebra literature monoids are often called semigroups, tacitly understanding that they have a neutral element. If we pass from monomials to their exponent vectors, then multiplication is turned into addition, and the exponent vectors form a submonoid M of Nn . In a similar way as the properties of the Stanley–Reisner ring of a simplicial complex  depend on the combinatorial structure of , the monomial algebra is governed by the monoid structure of M. In the following we will denote both monoids, the monomials and their exponent vectors, by M. The algebra K [M] is a monoid algebra. Evidently M is a grading monoid for K [M], and each grading component has K -dimension 1: K [M] =

 μ∈M

K μ.

6.1 A Recap of Toric Algebra

173

For more flexibility, in particular to cover rings of quotients, one considers finitely generated submonoids M of Zn . They are called affine monoids and correspond to finitely generated monomial subalgebras K [M] of the Laurent polynomial ring K [X 1±1 , . . . , X n±1 ]. An affine monoid M has a well-defined group gp(M) of differences. It is just the subgroup of Zn generated by M. The algebra K [gp(M)] arises from K [M] if one inverts all monomials. The (algebraic) torus (K ∗ )n operates on K [M] as a group of K -algebra automorphisms via the assignment (α, X a ) → (αX a ),

α ∈ (K ∗ )n ,

X a ∈ M,

and its K -linear extension. For this reason K [M] is also called a toric K -algebra. If K is infinite, then a subalgebra of the Laurent polynomial ring is generated by monomials if and only if it is stable under the torus action.

Rational Cones and Normal Monoids We call a subset of Rn conical if it contains all linear combinations of its elements with nonnegative coefficients. A conical set C is called a cone if it is finitely generated, i.e., there exist x1 , . . . , xm ∈ Rn such that   C = C(x1 , . . . , xm ) = α1 x1 + · · · + αm xm : m ∈ N, α1 , . . . , αm ∈ R+ . A submonoid M of Zn is a subset of Rn , and as such it generates a conical subset of Rn , namely   C(M) = α1 x1 + · · · + αm xm : m ∈ N, α1 , . . . , αm ∈ R+ , x1 , . . . , xm ∈ M . If M is finitely generated, then C(M) cone. One of the main goals of toric algebra is to relate the algebraic structure of K [M] to the geometry and combinatorial structure of the cone C(M). The starting point of this relation is the theorem of Minkowski–Weyl (for example, see [50, 1.15]). Theorem 6.1.1 For a subset C of Rn the following are equivalent: (a) C is a cone; (b) there exist linear forms σ1 , . . . , σs on Rn such that C=

s 

Hi+ ,

Hi+ = {x ∈ Rn : σi (x) ≥ 0}.

(6.1)

i=1

The sets Hi+ are closed linear halfspaces. A cone is rational if it is generated by rational or, equivalently, by integral vectors. Therefore the cones C(M) are rational.

174

6 Algebras Defined by Minors

Rationality can also be controlled by the halfspace representation in (6.1): C(M) is rational if and only if the linear forms σi can be chosen to have rational coefficients. In this case one can chose σ1 , . . . , σs in such a way that they have integral values on gp(M) and map gp(M) surjectively onto Z. The crucial finiteness result for rational cones is Gordan’s lemma [50, 2.9]: Theorem 6.1.2 Let C be a rational cone and L ⊂ Zn a sublattice. Then the monoid C ∩ L is finitely generated. The monoids appearing in Gordan’s lemma are normal: an affine monoid is normal if x ∈ M for all x ∈ gp(M) such that x k ∈ M for some k > 0. The equation x k = y ∈ M is the monomial version of integral dependence. The monoid M = C(M) ∩ gp(M) is the normalization of M. This name is justified [50, 4.40]: Proposition 6.1.3 Let M be an affine monoid, and K a field. Then K [M] is the normalization of K [M]. In particular, M is a normal monoid if and only if K [M] is normal. An important characterization of normal monoid algebras is their direct summand property (see [50, Theorem 4.43]): Theorem 6.1.4 Let M be a positive affine monoid. Then K [M] is normal if there is an embedding K [M] ⊂ P into a polynomial ring P over K such that K [M] is a direct summand of the K [M]-module P. In general, the intersection of finitely generated subalgebras of a Laurent polynomial ring need not be finitely generated. However, as a consequence of Gordan’s lemma, this holds for affine monoids and their algebras ([50, 2.11]). Proposition 6.1.5 Let M and N be affine submonoids of Zn . Then M ∩ N is an affine monoid, and K [M ∩ N ] = K [M] ∩ K [N ] is an affine monoid algebra. The prime ideals of K [M] generated by monomials can be described combinatorially (see [50, 4.D]). Proposition 6.1.6 Let K [M] be an affine monoid algebra. Then an ideal p generated by monomials is prime if and only if M \ p = M ∩ F for a face F of the cone R+ M.

Cohen–Macaulayness and Canonical Module Our main tool in proving Cohen–Macaulayness of an initial algebra in(A), and therefore of A itself, is Hochster’s theorem: Theorem 6.1.7 Let K be a field and M a normal affine monoid. Then K [M] is Cohen–Macaulay.

6.1 A Recap of Toric Algebra

175

There are several ways to prove Theorem 6.1.7. See [52, Sect. 6.2] for an approach via local cohomology that also yields Theorem 6.1.9 below. The most important consequence for us is Corollary 6.1.8 Let A be a K -subalgebra of R = K [X 1 , . . . , X n ] and consider a weight or a monomial order on R. Suppose that in(A) is finitely generated. If it is generated by monomials (e.g., if it is taken with respect to a monomial order) and normal, then A is normal and Cohen–Macaulay. Proof By Theorem of 6.1.7 the normal monoid algebra in(A) is Cohen–Macaulay. Theorem 1.6.11 transfers normality and Cohen–Macaulayness to A itself.  Not only the Cohen–Macaulayness of a normal affine monoid algebra is known, but also its canonical module by the theorem of Danilov–Stanley, in which relint(C) is the interior of C relative to its affine hull: Theorem 6.1.9 Let K be a field, M a normal affine monoid and C = C(M). Then the ideal generated by the monomials in the intersection relint(C) =

s 

Hi> ,

Hi> = {x ∈ Rn : σi (x) > 0},

i=1

is the canonical module of K [M]. Corollary 6.1.10 Suppose that the representation of C = C(M) in (6.1) is irredundant. Then the following are equivalent: (a) K [M] is Gorenstein; (b) the exists a monomial x ∈ M such that σi (x) = 1 for i = 1, . . . , s. We are entitled to call the monomial x in Corollary 6.1.10(b) the generator of the interior ideal.

Divisorial Ideas and Divisor Class Group For the theory of divisor class groups and divisorial ideals we refer the reader to Fossum [129] or Bourbaki [26]. An extensive treatment for monoid algebras can be found in [50, 4.F]. Let K [M] be a normal affine monoid algebra. For the description of the monomial divisorial ideals we choose linear forms σ1 , . . . , σs as in Theorem 6.1.9. Proposition 6.1.11 The monomials bases of the divisorial fractionary ideals of K [M] are the subsets of gp(M) given by inequalities σi (μ) ≥ ki for k1 , . . . , ks .

176

6 Algebras Defined by Minors

It is not difficult to see that K [M][μ−1 ] is isomorphic to a Laurent polynomial ring over K if μ is a monomial in the relative interior of C(M). The minimal prime ideals of μ are defined by the facets of C(M) as described in Proposition 6.1.6. This is the central argument for Chouinard’s theorem: Theorem 6.1.12 Let K [M] be a normal affine monoid and K a field. (a) Every divisorial ideal of K [M] is isomorphic to a monomial divisorial ideal. (b) The divisor class group of K [M] is isomorphic to Zs /σ(gp(M)) where σ(x) = (σ1 (x), . . . σs (x)).

Gradings and Hilbert Series The monoid of lattice points in a rational cone is a classical object of enumerative combinatorics. In the situation of Theorem 6.1.9 the Hilbert series of K [M] with respect to a nonnegative Z-grading is called the Ehrhart series of C, and the reciprocity between the Hilbert series of a Cohen–Macaulay ring and its canonical module becomes Ehrhart reciprocity. Suppose that K [M] is standard graded. Then the degree 1 monomials, or rather the corresponding lattice points, form a subset of the set of lattice points of a polytope P with integer vertices. The polytope P is obtained by intersecting the cone C(M) with the hyperplane of degree 1 elements. If M contains all lattice points of P, then we call K [M] polytopal. If M is normal, then it is polytopal, and P is a normal polytope. A subpolytope S of P is a simplex if its vertices are affinely independent points. A simplex is unimodular if its vertices (as elements of M) are a Z basis of L = gp(M). If P is the union of unimodular simplices, then it is normal.

Triangulations and Gröbner Bases Suppose that R is a polynomial ring whose indeterminates are mapped to the monomials generating K [M] so that K [M] becomes a residue class ring of R. We assume that K [M] is standard graded and let P be the polytope spanned by the generators of M. Then the defining ideal J of K [M] is generated by binomials (see Proposition 1.3.11), and these binomials correspond to linear relations of lattice points. A weight w on R gives weights to the lattice points of P that can be extended to a piecewise affine convex function on P. Such functions induce polyhedral subdivisions of P, and if the weight is “generic”, the polyhedral subdivision is a triangulation w with Stanley–Reisner ideal I (w ). In general, inw (J ) = I (w ) since the initial ideal need not be radical. However, by the Sturmfels correspondence [295] the relationship of the two ideals is as close as it can be.

6.2 Algebras Defined by Maximal Minors

177

Theorem 6.1.13 With the notation introduced, suppose that the ideal inw (J ) is generated by monomials. Then: (a) w induces a triangulation w of P. (b) I (w ) is the radical of inw (J ). (c) I (w ) = inw (J ) if and only if w is a unimodular triangulation. In particular, if inw (J ) is squarefree, then K [M] is normal and, hence, Cohen– Macaulay. With the necessary modifications, Theorem 6.1.13 holds without the assumption that K [M] is standard graded; see [50, 7.A]. But the version above suffices for our applications.

6.2 Algebras Defined by Maximal Minors The first mother of all K -algebras defined by minors is the algebra K [Mm ] generated by the m-minors of an m × n matrix X of indeterminates, m ≤ n. It is the coordinate ring of the Grassmannian of m-dimensional vector subspaces of K n , and we have already studied it in Sect. 3.2 from the viewpoint of the straightening law. We observed the following theorem in Sect. 3.3 without stating it: Theorem 6.2.1 Let K be a field and X an m × n matrix of indeterminates, m ≤ n. Consider a diagonal monomial order on the polynomial ring K [X ]. Then the initial algebra in(K [Mm ]) of the subalgebra K [Mm ] of K [X ] is generated by the diagonals of the m-minors. In other words, the m-minors form a Sagbi basis of the subalgebra they generate. Proof Let f ∈ K [Mm ]. By Proposition 3.3.4(c) there exists a standard bitableau  in K [Mm ] such that in( f ) =  , where, as the reader may recall, δ denotes the diagonal (monomial) of a minor δ and  is the product of the diagonals of the factors of a bitableau .  The proof only repeats, more or less verbatim, what was stated in Sect. 3.3 already. As we will see in Example 6.4.9, Theorem 6.2.1 cannot be generalized to minors of size t in general. As mentioned in Remark 5.3.6, the m-minors do in general not form a universal Sagbi basis of K [Mm ]. Independently of the matrix size, this is only true for m = 2.

Lifting Binomial Relations One can prove Theorem 6.2.1 also by lifting the binomial relations of the diagonals μ , μ ∈ Mm . This approach will give us further insight into the structure of K [Mm ]

178

6 Algebras Defined by Minors

and its initial algebra. The situation under discussion is a paradigm for the lifting technique. Therefore we treat it in great detail. Set R = K [X ]. As in Sect. 1.3 we consider the algebra homomorphisms ϕ : P → R, ψ : P → R,

ϕ(Tμ ) = μ, ψ(Tμ ) = μ .

Here P is the polynomial ring over K in the indeterminates Tμ , μ ∈ Mm . Our first goal is to find Ker ψ. To this end we come back to the combinatorics of minors. The set M of nonempty minors of X is partially ordered, and as in Sect. 3.2 we denote the partial order by . With respect to this partial order, M and its subset Mm are distributive lattices. On Mm the lattice operations are given by [a1 . . . am ] ∨ [b1 . . . bm ] = [max(a1 , b1 ) . . . max(am , bm )], [a1 . . . am ] ∧ [b1 . . . bm ] = [min(a1 , b1 ) . . . min(am , bm )]. We have to discuss ideals in the polynomial ring P on which we introduce an auxiliary monomial order. First we transfer the partial order of minors by setting Tμ Tν if μ ν, then we refine this partial order to a linear order of the indeterminates Tμ , and finally we choose the induced degree reverse lexicographical monomial order on P. Proposition 6.2.2 (a) Ker ψ is generated by the binomials Sμ,ν = Tμ Tν − Tμ∧ν Tμ∨ν for which μ and ν are incomparable elements in Mm . (b) With respect to the monomial order on P introduced above, the binomials Sμ,ν form a Gröbner basis of Ker ψ. (c) Each of them can be lifted to Ker ϕ. Proof Since μ ν = μ ∧ ν μ ∨ ν , our binomials belong to Ker ψ. After this observation (a) follows from (b). For (b) one observes that the initial monomial of Sμ,ν is Tμ Tν since μ ∧ ν < μ, ν. Let J be the monomial ideal generated by all the Tμ Tν . The monomials in J represent nonstandard bitableaux in K [Mm ]: Tμ1 · · · Tμs ∈ J if and only if at least two of the μi are incomparable with respect to . So the monomials in P \ J are exactly those representing standard bitableaux in K [Mm ]. The initial monomials of the standard bitableaux of m-minors are pairwise different, and therefore linearly independent. Thus the monomials in P \ J are linearly independent modulo Ker ψ over K since their images in K [Mm ] are so. By Proposition 1.2.6 we are done with (b). For (c) we choose μ, ν ∈ Mm incomparable. Their product has the standard representation

6.2 Algebras Defined by Maximal Minors

μν = (μ ∧ ν)(μ ∨ ν) +

u 

ai γi δi ,

179

γi ≤ δi , ai ∈ K , ai = 0, i = 1, . . . , u.

i=1

In fact, the standard bitableau (μ ∧ ν)(μ ∨ ν) must appear with coefficient 1; see Theorem 3.3.6. In the polynomial ring P this means sμ,ν = Tμ Tν − Tμ∧ν Tμ∨ν −

u 

ai Tγi Tδi ∈ Ker ϕ.

i=1

  Moreover, in the diagonal monomial order on R, we have in (μ ∧ ν)(μ ∨ ν) > in(γi δi ) for all i. This was observed in Remark 3.3.5. Therefore initϕ (sμ,ν ) = Tμ Tν − Tμ∧ν Tμ∨ν as desired.



Proposition 6.2.2 proves Theorem 6.2.1 once more, and Theorem 6.2.15 can be derived in the same manner. The arguments in the proof of Proposition 6.2.2 imply a strengthening of Corollary 3.2.7: Proposition 6.2.3 With respect to the monomial order on the polynomial ring P above the following hold: (a) The polynomials sμ,ν introduced in the proof of Proposition 6.2.2 form the reduced Gröbner basis of Ker ϕ, with initial ideal generated by the monomials Tμ Tν , μ, ν incomparable in the partial order . (b) The Plücker relations (read as polynomials in P) form a Gröbner basis of K er ϕ. Proof (a) That the polynomials sμ,ν form a Gröbner basis is a consequence of Proposition 6.2.2 by Proposition 1.3.16. Since their leading monomials are pairwise different, the initial coefficients are 1, and the noninitial monomials of every sμ,ν are not in Ker ϕ, they are indeed the reduced Gröbner basis. (b) To distinguish the partial order of the Tμ defined by the partial order of minors, we write ≤ for the extension to the monomial order on P that we are using. Assume that μ and ν are incomparable in the partial order . We can assume that μ < ν. Now we must look into the proof of the straightening law (p. 75). There we find a Plücker relation that involves the product μν and whose other products αβ have α ≺ μ. But this implies Tα Tβ < Tμ Tν for all these products αβ since we have extended the partial order to a reverse lexicographical order. This shows that Tμ Tν is the initial monomial of a polynomial representing a Plücker relation.  Remark 6.2.4 Let L be a lattice, i.e., a partially ordered set with operations ∧ and ∨ such that a ∧ b is the unique largest element simultaneously below a and b, and a ∨ b satisfies the dual condition. Given a lattice L, it is natural to define a K -algebra K [L] whose generators X a represent the elements of L and whose relations are defined by X a X b = X a∧b X a∨b .

180

6 Algebras Defined by Minors

Hibi [174] has proved that K [L] is an ASL if L is a distributive lattice, and then it is isomorphic to a normal monoid domain. Algebras of this type are therefore called Hibi rings. Proposition 6.2.2 shows that in(K [Mm ]) is the Hibi ring for the poset of maximal minors. Remark 6.2.5 Conca [85] related ASLs to Gröbner bases. The fact that the straightening relations are a Gröbner basis with respect to a reverse lexicographic monomial order refining the partial order of the set of generators of the ASL immediately generalizes Proposition 6.2.3. Remark 6.2.6 The ASL K [Mm ] is symmetric: its generators have degree 1 and all products appearing on the right hand side of a straightening relation have two factors. Such ASLs are always Koszul. For example, see Bruns et al. [53]. Via Gröbner bases this follows from Remark 6.2.5. Remark 6.2.7 Toric deformations of Grassmannians and flag varieties are the subject of a large number of articles. We mention only Bossinger, Lamboglia, Mincheva and Mohammadi [23]. This paper contains several references.

Cohen–Macaulayness, Normality and Factoriality Now we draw consequences of Proposition 6.2.2 beyond Theorem 6.2.1. The crucial argument is Lemma 6.2.8 in(K [Mm ]) is a normal monoid domain, hence Cohen–Macaulay, and Koszul. Proof Let us use a maximum of combinatorial arguments. As proved above, the defining ideal of in(K [Mm ]) has a squarefree monomial initial ideal. By Theorem 6.1.13 it follows immediately that in(K [Mm ]) is normal and Cohen–Macaulay. By Corollary 1.6.10 Koszulness follows from the existence of a degree 2 initial ideal.  There are at least two more arguments that can be used for proving Lemma 6.2.8. The first is the direct verification of the condition in the definition of normality. The second is to find the linear inequalities that define the cone C in (6.1). Both arguments are left to the reader, but the second will be applied in a closely related case; see Sect. 6.7. Theorem 6.2.9 The algebras K [Mm ] and in(K [Mm ]) (defined as in Theorem 6.2.1) are normal, Gorenstein and Koszul, and K [Mm ] is even factorial. Proof Corollary 6.1.8 transfers normality, the Cohen–Macaulay property and Koszulness from in(K [Mm ]) to K [Mm ] itself so that these properties of K [Mm ] result from Lemma 6.2.8. (There are other proofs discussed in Remark 6.2.11.) Factoriality of K [Mm ] must be proved by classical ring-theoretic arguments, but the “initial” structure helps with the first step. Let μ = [1 . . . m] be the minimal

6.2 Algebras Defined by Maximal Minors

181

element of the poset Mm . We claim that μ is a prime element in K [Mm ]. By Theorem 1.6.11 it is enough that in(μ) = μ is a prime element in in(K [Mm ]). In other words: we claim that Tμ is a prime element modulo Ker ψ. But this is clear since it does not appear in any of the polynomials Tμ Tν − Tμ∧ν Tμ∨ν : the poset Mm \ {μ} has a single minimal element, namely [1 . . . m − 1 m + 1]. By the theorem of Gauß–Nagata [129, Sect. 7], it is enough for the factoriality of K [Mm ] that K [Mm ][μ−1 ] is a factorial domain. It is in fact a Laurent polynomial ring over K . In addition to μ, the indeterminates in this Laurent polynomial ring are the minors [d1 . . . dm ] such that |{d1 , . . . , dm } ∩ {1, . . . , m}| = m − 1. It is a good exercise in Plücker relations to prove this claim, but one can find it in Bruns and Vetter [61, (6.1)]. Since the Cohen–Macaulay domain K [Mm ] is factorial, it is a Gorenstein ring by a theorem of Murthy (for example, see [52, 3.3.19]). The Gorenstein property of  K [Mm ] implies that of in(K [Mm ]) by Corollary 1.6.12. Remark 6.2.10 Since the minors μ ∈ Mm differ from each other only by permutations of the columns of the matrix X , the minor μ in the proof of Theorem 6.2.9 can be replaced by any other minor ν ∈ Mm : K [Mm ][ν −1 ] is isomorphic to a Laurent polynomial ring over K . Algebraically this implies that the localizations of K [Mm ] with respect to prime ideals different from the graded maximal ideal are regular local rings since they are localizations of Laurent polynomial rings. Geometrically it means that the Grassmannian of m-dimensional subspaces of K n is covered by affine spaces. Remark 6.2.11 Both the Cohen–Macaulay property and the normality of K [Mm ] and its initial algebra can be proved in the framework of algebras with straightening law. For K [Mm ] this is carried out in [61], and the initial algebra is a Hibi ring, as pointed out in Remark 6.2.4. Remark 6.2.12 The a-invariant of K [Mm ] has been computed by Bruns and Herzog [51] by the ASL approach: a(K [Mm ]) = −n. (An “initial” proof will be sketched in Remark 6.4.8.) It follows that reg K [Mm ] = a(K [Mm ]) + dim K [Mm ] = −n + n(n − m) + 1 = n(n − (m + 1)) + 1.

Remark 6.2.13 A further important connection in which K [Mm ] appears as a major player is invariant theory. The group SL(m, K ) of invertible m × m matrices of determinant 1 acts on K [X ] via the linear substitution X i j → (AX )i j , A ∈ SL(m, K ), i = 1, . . . , m, j = 1, . . . , n. Evidently the m-minors are invariant under this action, and K [Mm ] is the subalgebra of invariants if K has infinitely many elements. We refer the reader to [61, Sect. 7] for an extensive discussion of this classical theorem of invariant theory. Since the group SL(m, K ) is connected and has no nontrivial characters, it follows (again) that K [Mm ] is factorial. See [52, p. 297] for the argument.

182

6 Algebras Defined by Minors

Remark 6.2.14 As noticed in Sect. 6.1 and used in the proof of Theorem 6.1.13, one can associate a lattice polytope with a standard graded affine monoid algebra. This connection is generalized by Newton–Okounkov bodies. They were introduced by Okounkov [253]. For a systematic study see Lazarsfeld and Musta¸ta˘ [233] or Kaveh and Khovanskii [202].

Coordinate Rings of Flag Varieties In Sect. 3.3 we also observed the following generalization that is covered by Proposition 3.3.4(a): Theorem 6.2.15 With the notation of Theorem 6.2.1, let τ = (t1 , . . . , t p ) be a sequence of natural numbers, 1 ≤ t1 < · · · < t p ≤ m. Then the minors in Mτ are a Sagbi basis of the coordinate ring K [Mτ ] of the partial flag variety determined by τ . As for the special case K [Mm ], the central argument is Lemma 6.2.16 in(K [Mτ ]) is a normal monoid domain, hence Cohen–Macaulay, and Koszul. The proof of Lemma 6.2.8 can be transferred almost verbatim. Theorem 6.2.17 The algebras K [Mτ ] and in(K [Mτ ]) are normal and Gorenstein, and K [Mτ ] is even factorial. In addition to the grading they inherit from K [X ], the algebras in(K [Mτ ]) and K [Mτ ] carry a standard grading since they are defined by quadratic polynomials that even form a Gröbner basis. Hence they are Koszul in the standard grading. The proof of Theorem 6.2.17 directly generalizes that of Theorem 6.2.9: the straightening law degenerates to Hibi relations for the poset of minors generating K [Mτ ]. Only factoriality needs some additional arguments. Set γ = (1, . . . , m). We claim that K [Mτ ] is a retract of K [Mγ ]. It is evidently a subalgebra, and the vector subspace J spanned by all row initial (standard) bitableaux that contain a minor of size ∈ / {t1 , . . . , t p } is an ideal! Therefore the vector space epimorphism K [Mγ ] → K [Mτ ] that is the identity on K [Mτ ] and has kernel J is a ring homomorphism. Since retracts of factorial rings are factorial (prove it or look up Costa [97, 1.8]), it is enough to prove factoriality for K [Mγ ]. One can now choose between an invariant theoretic argument similar to Remark 6.2.13 or a localization argument as in the proof of Theorem 6.2.9. For the latter one first proves that X 1n is a prime element, using the straightening law. By symmetry we −1 ] is a Laurent polynomial can exchange it with X 11 . Then one shows that K [Mγ ][X 11 extension of K [M(1,....,m−1) ] and concludes by the Gauß–Nagata theorem. The details are left to the reader. The prototype for the localization argument is Lemma 3.4.5. For the argument by invariant theory one has two choices. The first is a direct generalization of Remark 6.2.13 (directly for K [Mτ ]); see Fulton [133, p. 139,

6.2 Algebras Defined by Maximal Minors

183

Exerc. 4]. The other is to observe that K [Mγ ] is the ring of invariants of K [X ] under the action of U − (m, K ), as stated in Theorem 3.6.6. The group U − (m, K ) is connected without nontrivial characters. Remark 6.2.18 In the same way as “ordinary” ideals of maximal minors have been generalized to northeast ideals by Bruns and Conca [45] (see Remark 4.3.14), the algebras K [Mτ ] have been generalized to “multi-fiber” rings of northeast ideals. In general these rings are no longer factorial or at least Gorenstein, but [45] gives conditions under which these properties hold.

The Rees Algebra The main tool for the study of asymptotic properties of the powers of an ideal I of the ring R is the Rees algebra ∞  R(I ) = I k. k=0

As an R-algebra it is generated by I ⊂ R(I ). Therefore, if R is Noetherian, the Rees algebra is Noetherian as well and a finitely generated R-algebra. The Rees algebra is naturally embedded into the polynomial ring R[T ] if we identify I k with I k T k : R(I ) ∼ =

∞ 

I kT k.

k=0

In the following we will always use this identification. The techniques that we have used for the analysis of K [Mm ] can be applied to the Rees algebra R(Im (X )) in quite the same manner. Here X is an m × n matrix and m ≤ n so that we again deal with maximal minors. In order to speak about Sagbi bases we must extend our monomial order to the polynomial ring K [X, T ]: for monomials α, β ∈ Mon(K [X ]) we set αT i < βT k

⇐⇒

i X 24 X 33 , and this holds in a diagonal monomial order. By similar arguments it follows that X 23 1 2 4 > X 21 2 3 4 . In the following we do not want to use any previous knowledge about the Gröbner bases of Im and its powers. Therefore we must start from the ideal generated by the diagonals of the maximal minors.

6.2 Algebras Defined by Maximal Minors

185

Lemma 6.2.20 Let R = K [X ]. Consider the ideal J generated by the diagonals μ of the maximal minors and the epimorphism ψ : R[Yμ : μ ∈ Mm ] → R(J ),

ψ(Yμ ) = μ .

Choose a reverse lexicographic order on P = R[Yμ : μ ∈ Mm ] that refines the partial order of the maximal minors and has X i j > Yμ for all i, j and μ. Then the binomials Sμν , μ, ν incomparable in Mm (as in Proposition 6.2.2), together with the binomials L k,a1 ,...,am+1 = X kak+1 Y[a1 ...ak ak+2 ...am+1 ] − X kak Y[a1 ...ak−1 ak+1 ...am+1 ] form a Gröbner basis of Ker ψ. Proof The proof follows the pattern that we have already seen in the proofs of Propositions 4.1.3 and 6.2.2. Given a product π = νμ1 · · · μk T k ∈ R(J ) we find a “normal form” in P and show that our Gröbner basis candidates let us “rewrite” an arbitrary monomial preimage of π in normal form if we successively use the binomials Sμ,ν and L k,a1 ,...,am+1 . In this rewriting we must of course replace the initial monomial by the trailing one. The normal form is defined as follows. First we use the straightening law to see that there exists a monomial ν  in the indeterminates X i j and a monomial Yμ1 · · · Yμk such that and μ1 · · · μ . (6.4) ψ(ν  Yμ1 · · · Yμk ) = π Among all candidates in (6.4) we choose the one for which Yμ1 · · · Yμk is smallest. The reader can check that our choice is the smallest monomial in ψ −1 (π). Of course, it remains to show that the passage from νYμ1 · · · Yμk to the smallest monomial in the fiber of π can be done by successive rewriting with our Gröbner basis candidates. A possible rewriting path is given by the description of the normal form. Roughly speaking, the standard bitableau is reached by the straightening algorithm, and this amounts to successive “applications” of the binomials Sμ,ν . If the normal form has not yet been reached, one can exchange one of the X i j appearing in ν  with a diagonal entry of one of the minors. Then the straightening law is applied again, and after finitely many steps the normal form is reached.  As already observed, the binomials Sμ,ν and L k,a1 ,...,am+1 can be lifted to the kernel of ϕ : R[Yμ : μ ∈ Mm ] → R(Im ),

ϕ(Yμ ) = μ,

and we obtain Proposition 6.2.21 The polynomials in R[Yμ : μ ∈ Mm ] representing the straightening relations and the row relations of the maximal minors form a Gröbner basis of Ker ϕ.

186

6 Algebras Defined by Minors

Now we have arrived at the same point as before Theorem 6.2.9, and with the same arguments we can derive Theorem 6.2.22 With the notation of Theorem 6.2.19, both R(Im ) and in(R(Im )) are normal, Cohen–Macaulay, and Koszul. In the proofs of Lemma 6.2.20 and Proposition 6.2.22 we have not even used that the maximal minors form a Gröbner basis of Im . Therefore we get a new proof of this fact, and even of Corollary 4.3.12, that is independent of the RSK correspondence. Remark 6.2.23 The normality of the Rees algebra R(Im ) follows already from the integral closedness of the powers Imk ; see Swanson and Huneke [304, Sect. 5.2]. Remark 6.2.24 Eisenbud and Huneke [120] showed that the Rees algebra R(Im ) is an ASL, so that ASL methods can be used for its analysis. They yield Cohen– Macaulayness and normality. This approach was generalized to “relative” maximal minors in [61, Sect. 9]. Bruns, Simis and Trung studied “straightening closed ideals” systematically in [59]. Remark 6.2.25 We have also proved that the row relations generate the syzygy module of Im , and we can even conclude that the syzygy module of Imk is generated by elements that are linear in the indeterminates X i j . Remark 6.2.26 For the ideal Im and its powers, linear free resolutions have been constructed. For Im the resolution is the classical Eagon–Northcott complex of [113]; also see [61, Sect. 2.C]. For the higher powers the resolution was constructed by Akin, Buchsbaum and Weyman [6]. These constructions are independent of the characteristic of K . We will show in Sect. 8.6 that the linearity of the resolutions of the powers of Im can be seen as a corollary of Proposition 6.2.21.

The Extended Rees Algebra and The Associated Graded Ring The extended Rees algebra R(I ) = R[I T, T −1 ] ⊂ R[T, T −1 ] and the associated graded ring gr I (R) =

∞ 

I k /I k+1

k=0

for an ideal I in a ring R are closely related to the Rees algebra R(I ). One has isomorphisms

6.3 Rees Algebras of Determinantal Ideals

R∼ = R(I )/I T R(I ),

187

gr I (R) ∼ = R(I )/I R(I ),

R(I ). gr(R) ∼ R(I )/T −1 = (6.5) They allow one to transfer properties between these algebras when we use that I R(I ) and I T R(I ) are isomorphic ideals of R(I ). However, for ideals of maximal minors one can apply the ASL theory to the extended Rees algebra and the associated graded ring directly. See [61, Sect. 9.C].

6.3 Rees Algebras of Determinantal Ideals Let R = K [X ] be as usual, namely with a field K and an m × n matrix X of indeterminates, m ≤ n. Moreover, we write It for It (X ) and fix a diagonal monomial order on R. There are two related types of Rees algebras: (a) the Rees algebra R(I ) where I is one of the ideals It or a product It1 · · · Itu , and

(k) (b) the symbolic Rees algebra Rsymb (It ) = ∞ k=0 It . In view of our goals and the approach via initial algebras, the symbolic Rees algebra is the crucial case. We embed all these algebras into a common polynomial ring R[T ], obtained by adjoining an auxiliary variable T to R = K [X ]. For an arbitrary ideal

I of K [X ] the Rees algebra R(I ) of I is embedded by the identification R(I ) = k I k T k ⊂ R[T ] as in the previous section. The symbolic Rees algebra of It is identified with a

subalgebra of R[T ] via Rsymb (It ) = k It(k) T k ⊂ R[T ]. The classical case is t = m. Since the powers of Im coincide with the symbolic powers, we need not distinguish the ordinary and the symbolic Rees algebras. Therefore this case is completely covered by Sect. 6.2. Nevertheless we do not exclude the case t = m in this section. However, representations of the Rees algebra or its initial algebra that are irredundant for t < m, usually become redundant for t = m, in the same way as the primary decomposition of the powers of It in Theorem 3.5.5 becomes redundant for the powers of Im . The other extreme case is t = 1. Also in this case the Rees algebra and its symbolic version are identical since powers of a maximal ideals are always symbolic. More generally, let R momentarily be an arbitrary polynomial ring and m be the ideal generated by its indeterminates X 1 , . . . , X n . Then we extend R by n new indeterminates Y1 , . . . , Yn and consider the substitution X i → X i , Yi → X i T , i = 1, . . . , n. It yields the isomorphism R(m) ∼ = K [X, Y ]/I2 (U ) where  U=

X1 . . . Xn . Y1 . . . Yn

For the isomorphism it is enough to note that the 2-minors of U are mapped to 0 by the substitution and that I2 (U ) is a prime ideal of height n − 1 so that dim K [X, Y ]/I2 (U ) = n + 1 = dim R(m). It follows that the Rees algebra is a normal Cohen–Macaulay domain. It is Gorenstein only in the cases n = 1, 2.

188

6 Algebras Defined by Minors

Initial Ideals of Symbolic Powers We need a combinatorial result in order to describe the initial monomials of the symbolic powers by linear inequalities. Before we state it let us remind the reader of the Stanley–Reisner complex t = (It ) of It defined in Sect. 4.4. The minimal prime ideals of the Stanley–Reisner ring correspond bijectively to the facets of the simplicial complex: identifying the indeterminates with the vertices of t , we can say that the prime ideal PF corresponding to a facet F of t is generated by the indeterminates that do not belong to F. The initial ideal in(It ) is the intersection of the prime ideals PF . This representation carries over to the symbolic powers, as we will state in Lemma 6.3.1. (For ideals generated by indeterminates the symbolic powers coincide with the ordinary ones.). To simplify notation we identify monomials of R with their exponent vectors in Rm×n . For every subset F of {1, . . . , m} × {1, . . . , n} we define a linear form λ F on / F and 0 otherwise. Rm×n by setting λ F (X i j ) = 1 if (i, j) ∈ Lemma 6.3.1 Let Ft = F(t ) denote the set of facets of t . (a) Then

   PFk . in It(k) = F∈Ft

  (b) Equivalently, γt (μ) = inf λ F (μ) : F ∈ Ft for all μ.   Proof We have seen in Theorem 4.3.6 that in It(k) is generated by the monomials μ with γt (μ) ≥ k. On the other hand, a monomial belongs to PFk if and only if λ F (μ) ≥ k. This shows the equivalence of (a) and (b). s X ai bi is in PFk if and We continue in the language of (a). A monomial μ = i=1 / F} is ≥ k. Equivalently, μ is in PFk if and only only if the cardinality of {i : (ai , bi ) ∈ if the cardinality of {i : (ai , bi ) ∈ F} is ≤ deg(μ) − k. As a measure we therefore introduce   wt (μ) = max |A| : A ⊂ {1, . . . , s} and {(ai , bi ) : i ∈ A} ∈ t .  Then a monomial μ is in F∈Ft PFk if and only if wt (μ) ≤ deg(μ) − k, or, equiva lently, deg(μ) − wt (μ) ≥ k. Now Lemma 6.3.1 follows from Lemma 6.3.2. Lemma 6.3.2 Let μ be a monomial. Then γt (μ) + wt (μ) = deg(μ). We reduce this lemma to a combinatorial statement on sequences of integers. For such a sequence b we set γt (c) = 0}, wt (b) = max{(c) : c is a subsequence of b and

s X ai bi be a monomial. and (b) denotes the length of the sequence b. Let μ = i=1 We order the indices as in the RSK correspondence, namely ai ≤ ai+1 for every i

6.3 Rees Algebras of Determinantal Ideals

189

and bi+1 ≥ bi whenever ai = ai+1 . By Remark 4.3.5 we have wt (μ) = wt (b). Hence it suffices to prove the following lemma. Lemma 6.3.3 One has γt (b) + wt (b) = (b) for every sequence b of integers. Proof We use part (b) of Greene’s Theorem 4.2.14: the sum αk∗ (Ins(b)) of the lengths of the first k columns of the insertion tableau Ins(b) of b is the length of the longest subsequence of b that can be decomposed into k nonincreasing subsequences. It follows that a sequence a has no increasing subsequence of length t if and only if it can be decomposed into t − 1 nonincreasing subsequences. In fact, the sufficiency of the condition is obvious, whereas its necessity follows from Schensted’s Theorem 4.2.10 and the just quoted result of Greene: if a has no increasing subsequence of length t, then all the rows in the insertion tableau Ins(a) have length at most t − 1. So ∗ (Ins(a)) is the length of a, and a can be decomposed into t − 1 nonincreasing αt−1 subsequences by Greene’s theorem. Consequently wt (b) is the maximal length of a subsequence of b that can be decomposed into t − 1 decreasing subsequences. Applying Greene’s theorem once ∗ (Ins(b)). On the other hand, γt (b) = γt (Ins(b)) by more, we see that wt (b) = αt−1 Theorem 4.2.16. Since γt (Ins(b)) is the sum of the lengths of the columns of Ins(b) γt (b) = (b).  of index ≥ t, one has wt (b) + Remark 6.3.4 The assertion of Lemma 6.3.1 can be restated as    (k) in It(k) = in It and appears in this form in [41, Theorem 7.1]. It has been reproved by De Stefani, Montaño and Núñez-Betancourt in the recent preprint [108] using mainly characteristic p methods partially inspired by Seccia’s use in [278] of Knutson ideals and ideas.

The Symbolic Rees Algebra We turn to the general case and start with the symbolic Rees algebra of It . The description of the symbolic powers in Theorem 3.5.2 yields the finite generation of the symbolic Rees algebra:   Rsymb (It ) = R It T, It+1 T 2 , . . . , Im T m−t+1 . Replacing the ideals Iu in this equation by their natural generating sets yields a finite set of generators for Rsymb (It ).

The initial algebra in(Rsymb (It )) of Rsymb (It ) then is k in(It(k) )T k . The descrip (k)  in Theorem 4.3.6 yields tion of in It Proposition 6.3.5 The initial algebra in(Rsymb (It )) of the symbolic Rees algebra Rsymb (It ) is equal to

190

6 Algebras Defined by Minors

  R in(It )T, in(It+1 )T 2 , . . . , in(Im )T m−t+1 . In particular, it is finitely generated. γt (μ) ≥ k. Moreover, a monomial μT k is in in(Rsymb (It )) if and only if The next step is to show that in(Rsymb (It )) is normal. The fastest argument γ (μ) for all k so that μ belongs to the underlying uses Remark 4.2.9(b): γ (μk ) = k monoid of monomials if μk does. All conclusions below on normality and Cohen– Macaulayness of the algebras under consideration could now be drawn. γ (μ) follows also directly from Lemma 6.3.1. We use The equation γ (μk ) = k this stronger statement to describe the initial algebras by linear inequalities for its monomials. These monomials in R[T ] are identified with lattice points in Rm×n ⊕ R. Theorem 6.3.6 We extend λ F for F ∈ Ft to a linear form  F on Rm×n ⊕ R by setting  F (T ) = −1. Then: (a) A monomial μT k is in the initial algebra in(Rsymb (It )) if and only if it has nonnegative exponents and  F (μT k ) ≥ 0 for all F ∈ Ft . (b) The initial algebra in(Rsymb (It )) is normal and Cohen–Macaulay. (c) The symbolic Rees algebra Rsymb (It ) is normal and Cohen–Macaulay. Proof (a) is a restatement of Lemma 6.3.1. Part (b) follows from (a), Proposition 6.1.3 and Theorem 6.1.7. Finally (c) is a consequence of (b) and Corollary 6.1.8.  Remark 6.3.7 The extended “symbolic” Rees algebra and the “symbolic” associated graded ring, in whose definitions the powers of an ideal are replaced by the symbolic powers, can be attacked by ASL methods. See [61, Sect. 10.B].

The Rees Algebra of a Product of Determinantal Ideals Now we want to go from the symbolic Rees algebras of the determinantal ideals It to the ordinary Rees algebra of a product It1 · · · Itr . Its powers are intersections of symbolic powers of the It by Theorems 3.5.5, and 4.3.7 says the same for the initial ideals. In order to exploit

∞ these results for the Rees algebras, we must use Veronese subalgebras: let S = i=0 Si be a positively graded ring; then the kth Veronese ring

S . is ∞ j=0 k j Theorem 6.3.8 Let t1 , . . . , tr a sequence of integers, 1 ≤ ti ≤ m for i = 1, . . . , r . Suppose that char K = 0 or char K > min(ti , m − ti , n − ti ) for all i. Then (a) in(R(It1 · · · Itr )) is finitely generated and normal, and (b) R(It1 · · · Itr ) is Cohen–Macaulay and normal.

Proof Set J = It1 · · · Itr . One has in(R(J )) = k≥0 in(J k )T k , and, by Theorem  (kg )   4.3.7, in(J k ) = 1≤ j≤m in I j j , g j = γ j (t1 , . . . , tr ). Hence

6.3 Rees Algebras of Determinantal Ideals

in(R(J )) =

191

   (kg j )  T k. in I j

(6.6)

1≤ j≤m k≥0

 (kg ) 

The monomial algebra k≥0 in I j j T k is isomorphic to the g j th Veronese sub (kg ) 

algebra k≥0 in I j j T kg j of the monomial algebra in(Rsymb (I j )) (in the relevant case g j > 0 and equal to R[T ] otherwise). By Theorem 6.3.6 the latter is normal  (kg ) 

and finitely generated, and therefore k≥0 in I j j T k is a normal affine monomial algebra as well. Thus in(R(J )) is finitely generated and normal. In fact, the intersection of a finite number of finitely generated normal monomial algebras is finitely generated by Proposition 6.1.5 and evidently normal. For (b) one applies Corollary 6.1.8 again.  We single out the most important case. Corollary 6.3.9 Suppose that char K = 0 or char K > min(t, m − t, n − t). Then R(It ) is Cohen–Macaulay and normal. The set of the elements T k where  is a standard bitableau with γi () ≥ k(t + 1 − i) for all i = 1, . . . , t is a K -basis of R(It ). Remark 6.3.10 (a) In Corollary 6.3.9 the hypothesis on the characteristic is essential. If m = n = 4 and char K = 2 then R(I2 ) has dimension 17 and depth 1; see Bruns [37]. (b) In order to obtain a version of Corollary 6.3.9 that is valid in arbitrary characteristic one must replace the Rees algebra by its integral closure. The integral closure is always equal to the intersection of symbolic Rees algebras that in nonexceptional characteristic gives the Rees algebra itself. This follows from Theorem 3.5.10. Remark 6.3.11 One can describe the hyperplanes defining the initial algebra of the Rees algebra of a product of It1 · · · Itr in terms of proper extensions of the linear forms λ F . For every j set g j = γ j (t1 , . . . , tr ) and for every F ∈ F j extend λ F to  F by setting  F (T ) = −g j . Then the initial algebra of the Rees algebra of It1 · · · Itr is given by the inequalities  F (μT k ) ≥ 0 for all F ∈ F j and for all j (and the nonnegativity of the exponents of μT k ). Remark 6.3.12 We are not aware of a direct structural approach to the extended Rees algebra or the associated graded ring of a determinantal ideal It , but one can use the isomorphisms in (6.5) to conclude that they are Cohen–Macaulay. We will discuss the relationship between the associated graded ring and the extended Rees algebra in Theorem 7.4.3 where it will be an essential tool.

The Canonical Module of the Rees Algebra In the computation of the canonical module we restrict ourselves to a single ideal It , although it could be carried out for a product of such ideals. Again we start with

192

6 Algebras Defined by Minors

the initial algebra and use linear inequalities in the first step. The functions  F are defined in Remark 6.3.11. Lemma 6.3.13 (a) The monomials in in(R(It )) are exactly those monomials in R[T ] that satisfy the inequalities  F (ν) ≥ 0 for every F ∈ Fi and for i = 1, . . . , t. (b) The canonical module of in(R(It )) is the ideal of in(R(It )) whose K -basis is the set of the monomials ν of R[T ] with all exponents ≥ 1 and  F (ν) ≥ 1 for every F ∈ Fi and for i = 1, . . . , t. Proof Part (a) is just a restatement of Remark 6.3.11 for a single ideal It . For the canonical module we must not forget the inequalities that cut out the positive orthant from Rm×n ⊕ R. The combination of the strict inequalities defines the relative interior of the cone generated by the monomials in in(R(It )), so that Theorem 6.1.9 can be applied.  Every monomial in R[T ] that has positive exponents with respect to every indeterminate is a multiple of XT where X is the product of all the variables X i j with (i, j) ∈ {1, . . . , m} × {1, . . . , n}. Proposition 6.3.14 The canonical module ω(in(R(It ))) of in(R(It )) is the ideal of in(R(It )) whose K -basis is the set of the monomials μT k of R[T ] where μ is a γi (μ) ≥ (t + 1 − i)k + 1 for all monomial of R for which XT | μT k in R[T ] and i = 1, . . . , t. Proof It suffices to show that the conditions given define the monomials listed in Lemma 6.3.13. Let ν = μT k be a monomial, μ ∈ K [X ]. Then, for a given i, one has  F (ν) ≥ 1 for every F ∈ Fi if and only if λ F (μ) ≥ k(t + 1 − i) + 1 for every γi (μ) ≥ k(t + 1 − i) + 1. The role F ∈ Fi . By Lemma 6.3.1 this is equivalent to to of XT has already been explained.  For “de-initialization” we must understand X better. Lemma 6.3.15 (a) γi (X) = (m − i + 1)(n − i + 1). γi (X) + γi (μ) for every i = (b) Let μ be a monomial in K [X ]. Then γi (Xμ) = 1, . . . , m. Proof Let μ be a monomial in the indeterminates X i j . Lemma 6.3.1(b) says that γi (μ) = inf λ F (μ) : F ∈ Fi . Note that i is a pure simplicial complex of dimension dim K [X ]/Ii − 1. Thus λ F (X) = (m − i + 1)(n − i + 1) for every facet F of γi (X) = (m − i + 1)(n − i + 1). i . In particular, We leave (b) as an exercise to the reader. It only matters that X has constant value  under all λ F .

6.3 Rees Algebras of Determinantal Ideals

193

Fig. 6.1 The diagonals of the factors of D for a 5 × 7 matrix

Now we apply the above results to the canonical module of R(It ). Let us try to γi (X) for all i. Since find a product of minors D such that in(D) = X and γi (D) = we have just computed γi (X), we can determine the shape of D, which turns out to be 12 , 22 , . . . , (m − 1)2 , m (n−m+1) . In other words, D must be the product of 2 minors of size 1, 2 minors of size 2, . . . , 2 minors of size m − 1 and n − m + 1 minors of size m. Now it is not difficult to show that D is uniquely determined, the 1-minors are [m|1] and [1|n], the 2-minors are [m − 1, m|1, 2] and [1, 2|n − 1, n] and so on. Explicitly, D=

m−1  i=1

[m − i + 1 . . . m|1 . . . i] · [1 . . . i|n − i + 1 . . . n] ·

n−m+1 

[1 . . . m|i . . . i + m − 1].

i=1

Figure 6.1 shows the diagonals of the factors for a 5 × 7 matrix. The element D will play an important role in Chap. 7. Theorem 6.3.16 Let H be the K -subspace of R[T ] whose K -basis is the set of the elements T k where  is a standard bitableau with γi (D) ≥ (k + 1)(t + 1 − i) for all i = 1 . . . , t. Set J = DT H . Then J is an ideal of R(It ) that is the canonical module of R(It ). Proof That J is indeed an ideal in the Rees algebra follows by the evaluation of shapes and Corollary 6.3.9. Next we show that in(J ) is the canonical module of in(R(It )). It is enough to check that in(J ) is exactly the ideal described in Lemma 6.3.13(b). Note that in(J ) = in(DT ) in(H ) = XT in(H ). Furthermore, by virtue of Lemma 6.3.15, the canonical module of in(R(It )) can be written as XT G where γi (μ) + γi (X) ≥ G is the space with basis the set of the monomials μT k such that (k + 1)(t + 1 − i) for all i = 1, . . . , t. The space H is defined by the same inequalities involving the γ functions for bitableaux. As pointed out in Remark 4.3.13(a), such a vector space is in-RSK. This implies in(G) = H . As just proved, in(J ) is the canonical module of in(R(It )). Now the claim follows from Theorem 1.6.13. 

194

6 Algebras Defined by Minors

Rees Valuations By Theorem 3.5.5 the primary decomposition of a power of It is completely described in terms of the valuations γk , k = 1, . . . , t. These valuations can be extended to the polynomial ring R[T ] and then yield a representation of the Rees algebra R(It ) as an intersection of R[T ] with finitely many valuation rings. Corollary 6.3.9 is exactly this representation, except that the terminology of valuations is not used explicitly. For the theory of valuations we refer the reader to Bourbaki [26, Chap. VI], Swanson and Huneke [304], or Zariski and Samuel [319]. The valuations that we will construct are special cases of Rees valuations; see [304, Chap. 10]. Let L be a field, v a valuation on L with value group G and g ∈ G. Then there ring exists a unique extension v of v to the quotient field L(T ) of the polynomial    j a T (for L[T ] such that v ( j a j T j ) = inf j v(a j ) + jg for all polynomials j j example, see [26, Chap. VI, Sect. 10, no. 1, Lemma 1]). We extend γk , k = 1, . . . , t, to a valuation on R[T ] and its quotient field by setting γk (T ) = −(t + 1 − k).

(6.7)

Note that the extension depends on t, even if the notation hides this dependence. t {x ∈ R[T ] : γk (x) ≥ 0}. Corollary 6.3.9 can now be reformulated as R(It ) = i=1 The valuation theoretic viewpoint and its consequences are discussed by Bruns and Conca [40] and by Bruns and Restuccia [55]. Remark 6.3.17 If we pass to initial monomials and replace the valuation γi in Corollary 6.3.9 by γi , we get the initial algebra of R(It ). However, since the equation γi (μ) + γi (ν) does not always hold, γi cannot be interpreted as a valuation. γi (μν) = But it is the infimum of the valuations induced by the linear functions λ F : the valuation γt is broken into a “piecewise valuation” under the passage from bitableaux to initial monomials. Remark 6.3.18 (a) In [40] Bruns and Conca have translated the combinatorial description of the canonical module into a divisorial one. If t < m ≤ n, the divisor class group of R(It ) is free of rank t, generated by the classes of the prime ideals P1 , . . . , Pt where Pk = {x ∈ R(It ) : γk (x) ≥ 1}. Then the canonical module of R(It ) has divisor class t   i=1

t   (1 − height Ii ) cl(Pi ). 2 − (m − i + 1)(n − i + 1) + t − i cl(Pi ) = cl(It R) + i=1

(b) If t = m = n, then It is a principal ideal, and R(It ) is isomorphic to a polynomial ring over K . (c) Let t = m < n. In this case a theorem of Herzog and Vasconcelos [173] yields that the divisor class group of R(Im ) is free of rank 1, generated by the extension P of Im to R(Im ). Moreover it implies that the canonical module has class (2 − (n − m + 1))P.

6.4 The Algebra of Minors

195

(d) The expression for the class of the canonical module given in (a) can be generalized to a larger class of Rees algebras; see [55]. In particular one obtains results for the Rees algebras of (i) minors of symmetric matrices of indeterminates, (ii) ideals generated by Pfaffians of alternating such matrices, and (iii) minors of Hankel matrices. The latter case has been treated by “initial methods” in [40]. Remark 6.3.19 Bae¸tica [62] has extended the results of this section to the pfaffian case.

6.4 The Algebra of Minors We recall our standard assumptions: K is a field, X is an m × n matrix of indeterminates with m ≤ n, and R = K [X ]. The polynomial ring R is endowed with a diagonal monomial order. In this section the algebra of minors At = K [Mt ] is the prime object. The characteristic of the field K will be either 0 or > min(t, m − t, n − t) throughout. There is nothing to say about A1 = R, and the case t = m < n has been thoroughly discussed in Sect. 6.2. We have seen in Corollary 3.2.6 that dim Am = m(n − m) + 1. However, for t < m, the dimension rises to the maximal possible value for a subalgebra of K [X ] (independently of the characteristic of K ): Proposition 6.4.1 If t < m, then dim At = dim K [X ] = mn. Proof The Krull dimension of an affine domain over K is the transcendence degree of its fraction field over K . Therefore it is enough to show that the indeterminates X i j are algebraic over the fraction field of At . The general case follows from that of a (t + 1) × (t + 1) matrix X . The adjoint matrix X˜ is the (t + 1) × (t + 1) matrix whose entry at position (i, j) is (−1)i+ j [1 . . . t + 1 \ i | 1 . . . t + 1 \ j] where 1 . . . t + 1 \ i is a shorthand for 1, . . . , i − 1, i + 1, . . . t + 1. Since X X˜ is the diagonal matrix with the entry det X in all diagonal places, it follows that det X˜ = (det X )t . Moreover, X −1 = (det X )−1 X˜ , and so the entries of X −1 are algebraic over the fraction field ofAt . Playing the same game again, we conclude the algebricity for  the entries of X = (X −1 )−1 over At . Incidentally, this discussion has revealed another simple case: If t = m − 1 = n − 1, then At is generated by mn = dim At elements, and so is isomorphic to a polynomial ring over K .

196

6 Algebras Defined by Minors

For the other cases we can use the methods and results of the previous section. Let Vt be the subalgebra of R[T ] generated by all monomials of the form μT where μ is a monomial in K [X ] of degree t (thus Vt is isomorphic to the tth Veronese subalgebra of K [X ]). We can identify At with K [Mt T ], and will freely use this identification in the following. It helps us to relate At and R(It ). Compared to R(It ), there is a somewhat surprising simplification in the description of At and its initial algebra. Lemma 6.4.2 (a) A K -basis of At is given by the set of the elements  where  is a standard bitableau with deg() = kt for some k ≥ 0 and γ2 () ≥ k(t − 1). (b) A K -basis of in(At ) is given by the monomials μ of K [X ] such that deg(μ) = kt for some k ≥ 0 and γ2 (μ) ≥ k(t − 1). Proof We use the identification At = K [Mt T ]. Since At = Vt ∩ R(It ), a K basis of At is given by the elements T k in the basis of R(It ) with deg() = kt. Similarly, since in(At ) = Vt ∩ in(R(It )), a K basis of in(At ) is given by the elements μT k in the basis of in(R(It )) with deg(μ) = kt. Now (a) and (b) result from the following statement: if σ is a shape such that i σi = kt and γ2 (σ) ≥ k(t − 1), then γi (σ) ≥ k(t + 1 − i) for all i = 1, . . . , t. The proof is left as an exercise to the reader, but it can be found in [40]. We will come back to the proposition after Lemma 6.6.4.  We know that in(R(It )) is normal. Since Vt is also a normal monoid algebra, the normality of in(At ) follows immediately. By the arguments that have now become familiar, we obtain Theorem 6.4.3 If the characteristic of K is either 0 or > min(t, m − t, n − t), then the algebra At of minors is normal and Cohen–Macaulay. The description of the canonical module of R(It ) by the relative interior of its underlying cone works for in(At ) as well. Using the product X of the indeterminates in R again, we obtain Lemma 6.4.4 The canonical module ω(in(At )) of in(At ) is the ideal of in(At ) whose K -basis is the set of the monomials μ of K [X ], deg μ = kt for some k ≥ 0, X | μ, and γ2 (μ) ≥ (t − 1)k + 1. Also the bitableau D with in(D) = X is used again (D was defined before Theorem 6.3.16): Theorem 6.4.5 Under the hypotheses of Theorem 6.4.3, let H be the K -subspace of R whose K -basis is the set of the elements of the form  where  is a standard bitableau with γ2 (D) ≥ (k + 1)(t − 1) and deg(D) = t (k + 1) for some k ≥ 0. Set J = DH . Then J is an ideal of At that is the canonical module of At . The proof repeats that of Theorem 6.3.16, and can be left as an exercise to the reader.

6.4 The Algebra of Minors

197

Remark 6.4.6 (a) By Theorem 6.2.9 Am is factorial, A1 is a polynomial ring over K , and for the same reason At is factorial if t = m − 1 = n − 1. These rings are Gorenstein. (b) In all the cases different from those in (a), the ring At is not factorial. Its divisor class group is free of rank 1, generated by the class of a single prime ideal q that can be chosen as q = ( f )R[T ] ∩ At where f is a (t + 1)-minor of X . Then (mn − mt − nt) cl(q). is the class of the canonical module; see [40]. As a corollary we have Theorem 6.4.7 Under the hypotheses of Theorem 6.4.3 the ring At is Gorenstein if and only if one of the following conditions is satisfied: (a) t = 1; in this case At = K [X ]. (b) t = min(m, n); in this case At is the coordinate ring of a Grassmannian. (c) t = m − 1 and m = n; in this case At is isomorphic to a polynomial ring. (d) 1 1 1 = + . t m n Proof The cases (a), (b) and (c) are those discussed in Remark 6.4.6(a). For (d) we cab assume that 1 < t < min(m, n) and t = m − 1 if m = n. Now Remark 6.3.18(b) completes the proof. However, since we have not treated divisorial methods in detail, let us indicate how to prove (d) by combinatorial methods. In view of Corollary 1.6.12 it makes no difference whether one works in At or in(At ). We choose in(At ). Suppose that 1t = m1 + n1 , or, equivalently, mn = t (m + n). Then XT m+n not only γ2 (X) − (t − belongs to in(At ), but even to its canonical module ω. Moreover, 1)(m + n) = 1. Using the “linearity” of X proved in Lemma 6.3.15, it is now easy to see that the ideal ω is generated by X. So it is isomorphic to in(At ), and in(At ) is Gorenstein. In all the cases not covered by (a)–(d) one has to show that in(J ) is not a principal ideal. We leave this to the reader as an exercise.  Remark 6.4.8 As the reader may check, X also generates the canonical module of in(Am ). Thus a(K [Mm ]) = a(in(K [Mm ])) = n, as mentioned already in Remark 6.2.12. Example 6.4.9 We have seen as a consequence of Proposition 6.4.1 that A2 is isomorphic to a polynomial ring over K if m = n = 3. This would hold as well for in(At ) if in(At ) were standard graded (after giving degree 1 to the 2-minors), or,

198

6 Algebras Defined by Minors

equivalently, if the diagonals of the 2-minors would generate in(A2 ). This is impossible since these diagonals involve only 7 of the 9 indeterminates and dim A2 = 9 for m = n = 3. The products of the diagonal 1 2 3 | 1 2 3 with 1 | 3 = X 13 and 3 | 1 = X 31 belong to in(A2 ), but are not products of diagonals of length 2. Adding them to the 9 diagonals gives a Sagbi basis of in(A2 ) for m = n = 3. We leave this claim as an exercise for the reader. In the example just discussed the maximum degree of an element in a minimal Sagbi basis is 2 = m − 1. This is not an accident. Let us standardize the degrees in At by giving degree 1 to the t-minors. Varbaro [309, 3.3.2] proved1 Theorem 6.4.10 Suppose that m ≤ n. Then the maximum degree of an element in a homogeneous Sagbi basis of At for m ≥ 2 is bounded by m − 1. If gcd(t − 1, m − 1) > 1 it is bounded by m − 2. While the divisorial approach has made it easy to determine the Gorenstein rings among the At , it only gives the canonical module up to isomorphism. A welcome byproduct of its “absolute” computation above, is the Castelnuovo–Mumford regularity. We refer the reader to Bruns et al. [47, 3.1] for the proof of Theorem 6.4.11 Except in the cases (a), (b) and (c) of Theorem 6.4.7, we have: (a) If m + n − 1 < mn/t, then reg(At ) = mn − mn/t. (b) if m + n − 1 ≥ mn/t, then reg(At ) = mn − m(n + k0 )/t. where k0 = (tm + tn − mn)/(m − t). Remark 6.4.12 Sometimes the following duality is useful: At (m, m) ∼ = Am−t (m, m). For a proof see [47, 1.3]. Remark 6.4.13 Algebras of minors of generic Hankel matrices have been analyzed in [40]. De Negri [106] has investigated algebras generated by Pfaffians.

6.5 Algebraic Relations between Minors In this section we want to survey the state of the art concerning the defining ideals of the algebras At (m, n). More precisely, let Pt (m, n) denote the polynomial ring K [Tμ : μ ∈ Mt (X )], where X is an m × n matrix, and consider 1

Theorem 6.4.10 corrects a typo in [309, 3.3.2].

6.5 Algebraic Relations between Minors

199

ϕ : Pt (m, n) → At (m, n),

ϕ(Tμ ) = μ.

We will discuss the minimal generators of the ideal Jt (m, n) = Ker ϕ, admitting from the beginning that in general they are not yet understood. More precisely, the aim is to describe a minimal generating system of Jt (m, n) by identifying elements that are part of it and whom we call minimal generators. Throughout this section we will assume that K has characteristic 0. Furthermore, we always assume m ≤ n, and we will just write Pt , At and Jt when m and n are not relevant. We are already familiar with the following situations: (a) t = 1. In this case ϕ is obviously an isomorphism, so J1 (m, n) = 0. (b) t = m − 1 = n − 1. In this case ϕ is an isomorphism by Theorem 6.4.7(c), so Jn−1 (n, n) = 0. (c) t = m. In this case the Kernel of ϕ is generated by the Plücker relations (see Corollary 3.2.7). Indeed, the Plücker relations even form a Gröbner basis Ker ϕ (see Proposition 6.2.3). In particular, Jm (m, n) is generated by quadrics. For example,   J2 (2, 4) = T[12] T[34] − T[13] T[24] + T[14] T[23] . In the first case not appearing in the list above, that is t = 2, m = 3, n = 4, notice that we have the following equality: ⎛ ⎞ [12 | 12] [12 | 13] [12 | 14] det ⎝[13 | 12] [13 | 13] [13 | 14]⎠ = 0. (6.8) [23 | 12] [23 | 13] [23 | 14] Indeed, the above determinant, by Laplace expansion, is equal to 1/6 multiplied by the determinant of the 6 × 6-matrix ⎛ ⎞ X 11 X 12 X 11 X 13 X 11 X 14 ⎜ X 21 X 22 X 21 X 23 X 21 X 24 ⎟ ⎜ ⎟ ⎜ X 11 X 12 X 11 X 13 X 11 X 14 ⎟ ⎜ ⎟ ⎜ X 31 X 32 X 31 X 33 X 31 X 34 ⎟ , ⎜ ⎟ ⎝ X 21 X 22 X 21 X 23 X 21 X 24 ⎠ X 31 X 32 X 31 X 33 X 31 X 34 which is obviously 0 (for example because the first and third columns of the matrix above are the same). Therefore (6.8) corresponds to a degree 3 polynomial in J2 (3, 4), and it turns out that such a cubic polynomial is a minimal generator of J2 (3, 4) (this was first noticed in [37]). By Theorem 3.6.8 the decomposition of the polynomial ring R = R(m, n) = K [X ] in its isotypic G-components is R(m, n) =

 σ

Mσ ,

200

6 Algebras Defined by Minors

where the direct sum runs over the shapes σ = (s1 , . . . , sk ) with s1 ≤ m. This decomposition behaves well with the grading, indeed, R(m, n)e =



Mσ ,

σ

where the direct sum runs over the shapes σ = (s1 , . . . , sk ) with s1 ≤ m and |σ| = s1 + . . . + sk = e. We want to understand At (m, n) in terms of such decomposition: Definition 6.5.1 For any positive integer d, a shape σ = (s1 , . . . , sk ) is (t, d)admissible if |σ| = td and k ≤ d. Rescaling the degrees in order to have a standard grading on At (m, n) and exploiting Proposition 3.5.8 we have At (m, n)d =



Mσ

σ

where the direct sum runs over the (t, d)-admissible shapes σ = (s1 , . . . , sk ) with s1 ≤ m. At this point it is useful to “separate rows and columns”: it was observed in Remark 3.6.7 that, for a shape σ = (s1 , . . . , sk ) with s1 ≤ m, the K -vector subspace L σ ⊂ R(m, n) generated by all bitableaux (R| In(σ)) is an irreducible GL(m, K )module. We also know: (a) The set of standard bitableaux (R| In(σ)) is a K -basis of L σ (Theorem 3.6.2(a)). (b) L σ and L τ are isomorphic GL(m, K )-modules if and only if σ = τ (Lemma 3.6.5(c)). (c) Every irreducible GL(m, K )-module is isomorphic to some L σ as a GL(m, K )module. Analog statements hold true for the irreducible GL(n, K )-submodules σL ⊂ R(m, n) generated by all bitableaux (In(σ)|R). One should notice that in order to get all the GL(n, K )-modules as σL must take m = n (so that any shape σ = (s1 , . . . , sk ) with s1 ≤ n is good). Remark 6.5.2 To be precise, we should say that in this section by a GL(m, K )module E we mean a polynomial GL(m, K )-representation: that is, the group homomorphism GL(m, K ) → GL(E) is polynomial in the original m 2 variables. Indeed, even if K has characteristic 0, there exist nonpolynomial GL(m, K )-representations which are indecomposable but not irreducible: For example, one can check that in the representation GL(1, R) → GL(2, R) mapping  a →

1 ln |a| , 0 1

{(x, 0) : x ∈ R} is the only proper GL(1, R)-submodule of R2 : Hence R2 is a reducible indecomposable GL(1, R)-representation.

6.5 Algebraic Relations between Minors

201

Analogously, by a GL(n, K )-module we mean the dual of a polynomial GL(n, K )representation. Since the present section wants only to be a survey, we will confine the discussion to this remark: the interested reader can find more in Procesi [260]. Since Mσ ∼ = L σ ⊗ σL as G-modules (see Theorem 3.6.7(a)), we have the decomposition in isotypic G-components At (m, n)d ∼ =



L σ ⊗ σL

σ

where the direct sum runs over the (t, d)-admissible shapes σ = (s1 , . . . , sk ) with s1 ≤ m. Since a t-minor of X is mapped by any g ∈ G to a K -linear combination of tminors of X , Pt (m, n) is also a G-module, and the decomposition of its degree d component into isotypic G-components is necessarily Pt (m, n)d ∼ =



(L σ ⊗ τL)mult(σ|τ )

σ,τ

where the direct sum runs over the shapes σ = (s1 , . . . , sk ) and τ = (t1 , . . . , th ) with |σ| = |τ | = td, s1 ≤ m and t1 ≤ n. The multiplicities mult (σ|τ ) are not known: to know them would solve some completely open plethysm problems. At least, exploiting Pieri’s rule (see [260, p. 289]), one can show that if mult(σ|τ ) > 0 than both σ and τ must be (t, d)-admissible. We refer to Bruns et al. [47] for more details and for the proofs of the unexplained facts that follow. The map ϕ : Pt (m, n) → At (m, n) is a map of G-modules, so any copy of L σ ⊗ τL is mapped to 0 or isomorphically to itself (in the latter case, we necessarily have σ = τ ). Therefore Jt (m, n) is a G-module as well. At this point it is convenient to extend our notation as follows: given a G-module E, by mult(σ|τ ) (E) we denote the number of copies isomorphic (as G-modules) to L σ ⊗ τL occurring in E. Summarizing what has been said above, for shapes σ = (s1 , . . . , sk ) and τ = (t1 , . . . , th ) (or more concisely for a bi-shape (σ|τ )) we have: For any G-module E, if mult(σ|τ ) (E) > 0 then s1 ≤ m and t1 ≤ n. mult (σ|τ ) = mult (σ|τ ) (Pt ) (by definition). If mult (σ|τ ) (Pt ) > 0, then |σ| = |τ | and both σ and τ are (t, |σ|)-admissible. mult (σ|τ ) (R) = 1 if σ = τ and s1 ≤ m, while mult(σ|τ ) (R) = 0 otherwise mult (σ|τ ) (At ) = 1 if σ = τ is (t, |σ|)-admissible and s1 ≤ m, while mult(σ|τ ) (At ) = 0 otherwise. (f) If σ and τ are (t, d)-admissible, s1 ≤ m and t1 ≤ n, then we have mult (σ|τ ) (Jt ) = mult(σ|τ ) (Pt ) if σ = τ and, otherwise, mult (σ|σ) (Jt ) = mult (σ|σ) (Pt ) − 1.

(a) (b) (c) (d) (e)

The last point might seem promising, because it provides the isotypic decomposition of Jt in terms of that of Pt . However this is not such good news, since the isotypic decomposition of Pt , as already mentioned, is very hard to understand. Moreover, we are interested in the minimal generators of Jt ; in other words, we are interested

202

6 Algebras Defined by Minors

in the G-module Jtmin = Jt ⊗ Pt K (we will write Jtmin (m, n) when we want to keep track of m and n). For this goal it is useful to introduce the following notion. Definition 6.5.3 We say that a shape α = (a1 , . . . , ah ) is a predecessor of a (t, d)admissible shape σ = (s1 , . . . , sk ) if α is (t, d − 1)-admissible, a1 ≤ s1 ≤ a1 + t and ai ≤ si ≤ ai−1 for all i ≥ 2. In particular, k ≤ h + 1. We say that (α|β) is a bi-predecessor of (σ|τ ) if α is a predecessor of σ and β is a predecessor of τ . Let V ⊂ Pt a G-stable vector subspace. As it turns out, again exploiting Pieri’s rule, if mult (σ|τ ) (V · (Pt )1 ) > 0 then there exists a bi-predecessor (α|β) of (σ|τ ) such that mult (α|β) (V ) > 0. Similarly, if all predecessors (α|β) of (σ|τ ) satisfy mult(α|β) (V ) = mult(α|β) (Pt ), then mult(σ|τ ) (V · (Pt )1 ) = mult (σ|τ ) (Pt ). The last observation allows one to prove: Theorem 6.5.4 Suppose that all bi-predecessors (α|β) of a bi-shape (σ|τ ) are such that α = β. Then mult(σ|τ ) (Jtmin ) = 0. In particular, for every bi-shape (σ|τ ) we have mult (σ|τ ) (Jtmin (m, m + t)) = mult(σ|τ ) (Jtmin (m, n)) for any n ≥ m + t. As a consequence, the Castelnuovo-Mumford regularity of At (m, n) (see Theorem 6.4.11) yields an upper bound that is quadratic in m and independent of n for the degree of a minimal generator of Jt (m, n).

Shapes of Single Exterior Type We introduce a notion that is useful for finding some minimal generators of Jt . Definition 6.5.5 A shape σ is called of single exterior type if mult(σ|σ) (Pt ) = 1. A bi-shape (σ|τ ) is of single exterior type if both σ and τ are shapes of single exterior type. Remark 6.5.6 In the above definition m and n are not specified, but it is understood that one should take m at least as the length of the first row of σ, otherwise mult(σ|σ) (Pt ) = 0. From now on we will not specify m and n anymore: if they are large enough with respect to t, all considered bi-shapes will occur, otherwise the reader should remove those bi-shapes (σ|τ ) whose first rows are too long. The shapes σ of single exterior type have been classified by Bruns and Varbaro [60, Theorem 3.5], where a more precise version of the following is proved: Theorem 6.5.7 Let d ∈ N and σ = (s1 , . . . , sk ) be a (t, d)-admissible shape with s1 ≤ m. Then σ is of single exterior type if and only if (at least) one of the following holds: (a) sd ≥ t − 1. (b) s1 ≤ t + 1.

6.5 Algebraic Relations between Minors

203

(c) sd ≥ s2 − 1. (d) sd−1 ≥ s1 − 1. One can prove that mult(σ|τ ) (Pt ) = 1 if (σ|τ ) is a bi-shape of single exterior type. However there are bi-shapes (σ|τ ) with mult(σ|τ ) (Pt ) = 1 that are not of single exterior type. The notion of “single exterior type” allows to find minimal generators of Jt for strong combinatorial reasons: Theorem 6.5.8 Let (σ|τ ) be a bi-shape of single exterior type such that σ = τ . If σ and τ share the same predecessors (necessarily of single exterior type), then mult(σ|τ ) (Jtmin ) = 1; otherwise mult(σ|τ ) (Jtmin ) = 0. Remark 6.5.9 In [47, Theorem 1.23] more precise versions of Theorems 6.5.8 and 6.5.4 have been proved. However, to state them (indeed, also   to prove Theorem  6.5.8), one must introduce and study the natural action of GL( mt , K ) × GL( nt , K ) on Pt (m, n). Let us introduce the following shapes in order to exploit Theorem 6.5.8: (a) For u ∈ {0, . . . , t} let

τu = (t + u, t − u).

(b) For u ∈ {1, . . . , t/2} let γu = (t + u, t + u, t − 2u) and λu = (t + 2u, t − u, t − u). (c) For each u ∈ {2, . . . , t/2} let ρu = (t + u, t + u − 1, t − 2u + 1) and σu = (t + 2u − 1, t − u + 1, t − u).

One can check, using Theorem 6.5.7, that all the shapes above are of single exterior type. Let us define the following sets:   A = (τu |τv ) : 0 ≤ u, v ≤ t, u + v even, u = v ,   B = (γu |λu ), (λu |γu ) : 1 ≤ u ≤ t/2   ∪ ((ρv |σv ), (σv |ρv ) : 2 ≤ v ≤ t/2 . One can check that every bi-shape in A ∪ B occurs in Theorem 6.5.8, so it provides minimal generators of Jt . Example 6.5.10 For any possible u, v, the only bi-predecessor of (τu |τv ) is ((t)|(t)). For t ≥ 2, m ≥ t + 2 and n ≥ t + 3 the shapes ρ2 = (t + 2, t + 1, t − 3) and σ2 = (t + 3, t − 1, t − 2) have the same set of predecessors, namely {(t + 1, t − 1), (t + 2, t − 2)}. To sum up, Theorem 6.5.8 implies that

204

6 Algebras Defined by Minors



L σ ⊗ τL ⊂ Jtmin (m, n).

(6.9)

(σ|τ )∈A∪B

In [47] it was conjectured that the inclusion above is an equality. In particular, Jt is generated by quadrics and cubics. Since the time in which [47] was written, there has been some progress towards the solution of the conjecture, though it still remains open. In particular, the conjecture has been solved positively in the case t = 2 by Huang et al. [192]. We summarize the state of the art in the following. Theorem 6.5.11 In characteristic 0 we have:

(a) (Jtmin )2 = (Jt )2 = (σ|τ )∈A L σ ⊗ τL [47].

(b) (Jtmin )3 = (σ|τ )∈B L σ ⊗ τL for t = 2 and t = 3 [47, Sect. 3.4]. (c) (J2min )d = 0 for d ≥ 4 (see [47, Sect. 3.4] for d = 4 and [192] for d > 4). (d) For any t, if Jtmin (m, m + t)d = 0 then Jtmin (m, n)d = 0 for any n by Theorem 6.5.4. In particular, (Jtmin (m, n))d = 0 for d ≥ 2(m 2 − m) by Theorem 6.4.11. (e) For any t, if (σ|τ ) gives a minimal relation for the strong combinatorial reasons explained in Theorem 6.5.8, namely σ = τ are both of single exterior type and share the same predecessors, then (σ|τ ) ∈ A ∪ B [60]. Remark 6.5.12 The proof of Theorem 6.5.11(c) from [192] combines geometric and representation-theoretic tools, as well as an inductive procedure depending on the parameters m and n. Using transpose duality, which is a representation-theoretic involution that replaces L σ with L σ∗ , one can show that the problem of determining relations between minors is equivalent to that of determining relations between permanents. One advantage that the algebra At generated by permanents has over At is that the ambient polynomial ring K [X ] is finite as a module over At , but infinite over At . In the case when t = 2, [192] explains how to control the presentation of K [X ] as an A2 -module, and shows that the minimal relations must lie in low degree. Moreover, one can study a natural finite filtration of K [X ] by A2 -submodules, and bound their initial syzygies using the Kempf-Weyman geometric technique. By moving down from K [X ] to A2 along the filtration (peeling off one layer at a time), one can then conclude that the defining relations for A2 have low degree. Applying induction on m, n and using the theory of subspace varieties, which is a common tool in the study of subvarieties in spaces of tensors, it is shown in [185] that all the (potentially) new relations between permanents that arise when we increase the parameters m, n lie in high degrees. Combined with the prior bounds on the degrees of relations, one concludes that in fact no new relations appear when we increase m, n, besides the ones that propagate via the group action from low values of the parameters, and are listed in Theorem 6.5.11(a), (b). Although many of the steps above can in principle be extended to higher values of t, the details of the arguments become more involved, so it is likely that new ideas would be necessary to extend the proofs to arbitrary t. We note that the duality in Remark 6.4.12 and the main result of [192] imply that the inclusion (6.9) is an equality also if t = m − 2 = n − 2.

6.5 Algebraic Relations between Minors

205

The actual minimal relations corresponding to the bi-shapes in A ∪ B are explicitly described in [47], but we think it is pointless to reproduce them here in general: they would not add any understanding to the problem. We limit ourselves to t = 2, 3.

The Case t = 2 As said, the case t = 2 is fully understood thanks to [192]. We have J2min = (J2min )2 ⊕ (J2min )3 where (J2min )2 = L (2,2) ⊗ (4)L ⊕ L (4) ⊗ (2,2)L and (J2min )3 = L (3,3) ⊗ (4,1,1)L ⊕ L (4,1,1) ⊗ (3,3)L . In other words, there are 4 minimal generators of J2 , up to the action of G on P2 : two quadrics q, q  and two cubics g, g  (we are tacitly assuming m ≥ 4). More precisely, one can now show: (a) L (2,2) ⊗ (4)L is generated as a G-module by the quadric q = T[12 | 12] T[12 | 34] − T[12 | 13] T[12 | 24] + T[12 | 14] T[12 | 23] . (b) L (4) ⊗ (2,2)L is generated as a G-module by the quadric q  = T[12 | 12] T[34 | 12] − T[13 | 12] T[24 | 12] + T[14 | 12] T[23 | 12] . (c) L (3,3) ⊗ (4,1,1)L is generated as a G-module by the cubic ⎛

⎞ T[12 | 12] T[12 | 13] T[12 | 14] g = det ⎝T[13 | 12] T[13 | 13] T[13 | 14] ⎠ . T[23 | 12] T[23 | 13] T[23 | 14] (d) L (4,1,1) ⊗ (3,3)L is generated as a G-module by the cubic ⎛

⎞ T[12 | 12] T[13 | 12] T[14 | 12] g  = det ⎝T[12 | 13] T[13 | 13] T[14 | 13] ⎠ . T[12 | 23] T[13 | 23] T[14 | 23] Remark 6.5.13 Notice that q and q  are Plücker relations, simply by fixing the first two rows or columns. If t ≥ 3, the G-module (Jt )2 is not generated by Plücker relations in general: more precisely, L τu ⊗ τvL is generated, as G-module, by a Plücker relation if and only if u = 0 or v = 0.

206

6 Algebras Defined by Minors

The cubic relations g and g  are determinants of 3 × 3-matrices of variables. Also in this case, if t ≥ 3, the G-module (Jt )3 is not generated by determinantal relations in general.

The Case t = 3 The t = 3 case is the first open one (if m ≥ 4 and n ≥ 6). We have (J3min )2 = L (3,3) ⊗ (5,1)L ⊕ L (5,1) ⊗ (3,3)L ⊕ L (4,2) ⊗ (6)L ⊕ L (6) ⊗ (4,2)L and (J3min )3 = L (4,4,1) ⊗ (5,2,2)L ⊕ L (5,2,2) ⊗ (4,4,1)L ⊕ L (5,4) ⊗ (6,2,1)L ⊕ L (6,2,1) ⊗ (5,4)L .

However this time we only have an inclusion (J3min )2 ⊕ (J3min )3 ⊂ J3min , and the equality is just conjectured. Using Theorem 6.5.11(b) one can see that there are 8 degree ≤ 3 minimal generators of J3 , up to the action of G on P3 : 4 quadrics q0 , q0 , q1 , q1 and 4 cubics g, g  , h, h  (we are tacitly assuming m ≥ 6). More precisely one can show: (a) L (3,3) ⊗ (5,1)L is generated as a G-module by the quadric q0 = T[123 | 123] T[123 | 145] − T[123 | 124] T[123 | 135] + T[123 | 125] T[123 | 134] . (b) L (4,2) ⊗ (6)L is generated as a G-module by the quadric q1 = T[123 | 123] T[124 | 456] − T[123 | 124] T[124 | 356] + T[123 | 125] T[124 | 346] − T[123 | 126] T[124 | 345] + T[123 | 134] T[124 | 256] − T[123 | 135] T[124 | 246] + T[123 | 136] T[124 | 245] + T[123 | 145] T[124 | 236] − T[123 | 146] T[124 | 235] + T[123 | 156] T[124 | 234] − T[124 | 123] T[123 | 456] + T[124 | 124] T[123 | 356] − T[124 | 125] T[123 | 346] + T[124 | 126] T[123 | 345] − T[124 | 134] T[123 | 256] + T[123 | 135] T[124 | 246] − T[123 | 136] T[124 | 245] − T[123 | 145] T[124 | 236] + T[124 | 146] T[123 | 235] − T[124 | 156] T[123 | 234] . (c) L (4,4,1) ⊗ (5,2,2)L is generated as a G-module by the cubic ⎛

⎞ T[123 | 123] T[123 | 124] T[123 | 125] g = det ⎝T[124 | 123] T[124 | 124] T[124 | 125] ⎠ . T[234 | 123] T[234 | 124] T[234 | 125] (d) L (5,4) ⊗ (6,2,1)L is generated as a G-module by the cubic

6.6 Exterior Powers of Linear Maps

207

h=



h abcd ,

1≤a 1 is empty, and so only the cases sr(x) = 0 and sr(x) = 1 occur. Proof of Proposition 6.6.2 Since m ≤ n, we can identify V with a subspace of W , and Lt (V, V ) with a subspace of Lt (V, W ). Rank and small rank do not change if we extend elements from Lt (V, V ) to Lt (V, W ) in a trivial way. Therefore we can assume that m = n, identify V and W , and consider the elements of L as endomorphisms. Let e1 , . . . , em be a basis of V . For sr(x) = rank x = 0 we choose x = d 0,0 = 0. For sr(x) = rank x = 1 we choose x = d 1,1 = t (ϕ) where ϕ(ei ) = ei , i = 1, . . . , t and ϕ(ei ) = 0 for i > t. Let 2 ≤ u ≤ t + 1, 1 ≤ k ≤ m − t, and set v = t + 1 − u. We will now identify an element  u+k−1 . d u,u+k−1 ∈ X t (m, n), sr(d u,u+k−1 ) = u, rank(d u,u+k−1 ) = u−1 We consider the morphism α : K ∗ → L, where α(κ) is the diagonal matrix with the entries ⎧ −(t−v) ⎪ , 1 ≤ i ≤ v, ⎨κ α(κ)ii = κv , v + 1 ≤ i ≤ v + u + k − 1, ⎪ ⎩ 0, else. Then t ◦ α extends to a morphism α¯ : K → Lt for which α(0) ¯ is a diagonal matrix d u,u+k−1 with entries 1 or 0 on the diagonal. Clearly d u,u+k−1 lies in X t . Furthermore d u,u+k−1 has exactly  u+k−1 u−1 entries equal to 1 on the diagonal and they sit in the positions with indices [1 . . . v, I | 1 . . . v, I ] where I varies over the (t − v)-subsets of {v + 1, . . . , v + u + k − 1} = {v + 1, . . . , t + k}. It remains to show that  sr(d u,u+k−1 ) = u. Consider the subspace V  = K e1 + · · · +K et+1 . We identify t V  with the subspace generated by the basis elements e J = j∈J e j where J is a t-subset of {1, . . . , t + 1}. The linear map d u,u+k−1 sends t + 1 − v = u elements of this basis to themselves, namely those for which J contains {1, . . . , v}. Therefore sr(d u,u+k−1 ) ≥ u. For the opposite inequality we choose elements f 1 , . . . , f t+1 in V , and represent them in the basis e1 , . . . , em :

6.6 Exterior Powers of Linear Maps

211

fi =

m 

ai j e j .

j=1

 Then the restriction of d u,u+k−1 to t V  , V  = K f 1 + · · · + K f t+1 , is given by a matrix A whose entries are t-minors of A = (ai j ). In a row of A we find the t-minors of A whose row indices leave out a given index i = 1, . . . , t + 1 and whose column indices correspond to those t-subsets of {1, . . . , t + k} that contain 1, . . . , v. Such a matrix has rank ≤ t + 1 − v. In fact, the rank is maximal when the entries of A are indeterminates, and then there exist v linearly independent relations of the t + 1 rows of A , given by the columns of A with indices 1, . . . , v (with appropriate signs), resulting from Laplace expansion of a t-minor with two equal columns (namely the jth, j = 1, . . . , v).  We can already observe that Yt is a proper subset of X t if 1 < t < m. In fact, let ϕ ∈ L. If rank ϕ < t, then t (ϕ) = 0, and if rank ϕ = t, then rank t (ϕ) = 1. If rank ϕ ≥ t + 1, then sr(t (ϕ)) = t + 1. We need some functions which help us to determine small rank. Lemma 6.6.3 Let δ = [1 . . . v | 1 . . . v], 0 ≤ v ≤ t + 1, and η = [1 . . . t + 1 | 1 . . . t + 1] (with δ = 1 if v = 0) and set f v = δη t−v ∈ At . Then f v (d u,u+k−1 ) = 0

⇐⇒ u < t + 1 − v.

Proof First we have to express f v in the coordinates of Lt . We claim that   f v = det (−1)i+ j [1 . . . t + 1 \ i | 1 . . . t + 1 \ j]i, j=v+1,...,t+1 .

(6.10)

Note that this equation generalizes the formula for the determinant of the adjoint matrix in the proof of Proposition 6.4.1 (which it contains for v = 0). There are several ways to prove Eq. (6.10). We choose a representation theoretic argument. It is enough to prove the equation, which is an identity defined over Z, for the field of complex numbers. Both sides of the equation are invariant under the action of the direct product U = U − (m, K ) × U + (n, K ). Furthermore they have the same content. The space of such forms is 1-dimensional by Theorem 3.6.6, and so both sides must differ by a scalar. That it is 1, follows if we evaluate them on the unit matrix.  Finally we evaluate f v on the elements d u,u+k−1 using Eq. (6.10).

G-stable Prime Ideals in At By definition At is a subalgebra of the polynomial ring R = K [X ]. But we can also see it as a subalgebra of R(It ). Both embeddings are natural sources of G-stable prime ideals.

212

6 Algebras Defined by Minors

Let us first consider At ⊂ R. Then the ideals qk = Ik (X ) ∩ At ,

k = t, . . . , m,

are certainly prime and G-stable. Second, we can restrict the Rees valuations to At . The Rees valuations have been introduced in Eq. (6.7). In order to avoid notational chaos we must give their restrictions to At a new name and rewrite them avoiding the variable T : deg x (t + 1 − i). πi (x) = γi (x) − t In fact, for k = (deg x)/t, x ∈ At is identified with x T k ∈ R(It ) if we identify At and K [Mt T ] ⊂ R[T ]. The Rees valuations define the G-stable prime ideals   pi = x ∈ At : πi+2 (x) ≥ 1 ,

i = 0, . . . , t − 2.

Let σ be a shape. Its weight is ε(σ) = (ε1 (σ), . . . , εm (σ))

where

εi (σ) = |{ j : σ j = i}|.

(6.11)

In the following it is very useful that we can express the valuation πi by π2 and the weight. Moreover it translates Lemma 6.4.2 into the new notation. The proof is left to the reader. Lemma 6.6.4

 2 (a) For u = 3, . . . , t one has πu = (u − 1)π2 + u−2 k=1 (u − 1 − k)εk . In particular, π j (x) ≥ 0 for j ≥ 3 and all x ∈ At . (b) A homogeneous polynomial x ∈ R belongs to At if and only if its degree is divisible by t and π2 (x) ≥ 0. In addition to the pi and q j , the sums pi + q j are also G-stable, and they are prime: Theorem 6.6.5 Suppose that m ≤ n and char K = 0 or char K > min(t, m − t, n − t). There exist exactly t (m − t) + 2 G-stable prime ideals in At , namely (1) pi + q j , i = −1, . . . , t − 2, j = t + 2, . . . , m + 1, (2) qt+1 and qt , where p−1 = qm+1 = 0. As in Proposition 6.6.2 we do not exclude the case t = m in the theorem. For t = m, the list in (1) is empty, and only the G-stable prime ideals qm and qm+1 = 0 remain. The prime ideals p0 , . . . , pm−2 are all equal to qm . For t = 1 the theorem says that exactly the ideals I1 (X ), . . . , Im (X ) and 0 are G-stable and prime. 2

In [43, 3.2] the factor u − 1 − k was forgotten.

6.6 Exterior Powers of Linear Maps

213

Example 6.6.6 (a) The residue class ring At /qt+1 is the subalgebra generated by the t-minors in R/It+1 . In [61, Sect. 9.A] these rings are discussed in detail. (b) Among the G-stable prime ideals that are properly contained in qt+1 , pt−2 + qt+2 is maximal with respect to inclusion. The standard bitableaux of shape ((t + 1)u , (t)v , (t − 1)u ), u, v ≥ 0, are a K -basis of the residue class ring. (The exponent indicates the number of repetitions.) That the ideals listed in Theorem 6.6.5 are indeed prime will be proved in the next subsection. Our first goal is to show that there are no other G-stable prime ideals. Again a toric approach helps us to structure the arguments. For a G-submodule H of K [X ] we set   suppG (H ) = ε(λ) : Mλ ⊂ H . The G-submodules Mλ of R have been defined in Section 3.6 where they appear as the summands in the unique decomposition of R into a direct sum of irreducible G-submodules. Let St = suppG (At ). One has ε = (ε1 , . . . , εm ) ∈ St if and only if the following conditions hold: (i)

m 

iεi ≡ 0

mod t,

(ii) π2 (ε) ≥ 0,

(iii) εi ≥ 0, i = 1, . . . , m. (6.12)

i=1

For (ii) we have transferred π2 from shapes to weights: m  (i − 1)εi − π2 (ε) = i=1



iεi (t + 1 − i). t

It follows that St is a normal submonoid of Zm . In fact, it is the intersection of the sublattice defined by the congruence (i) and the cone cut out from Rm by the inequalities (ii) and (iii) in (6.12). Set U = K [[1 . . . k | 1 . . . k] : k = 1, . . . , m]. It is the algebra of U-invariants in R where U = U − (m, K ) × U + (n.K ). It has been introduced in Sect. 3.6. The monomials in U can be identified with the shapes λ of the standard bitableaux in R by the correspondence λ ←→ ε(λ). Moreover H ∩ U = K suppG H ⊂ U. for every G-submodule H of R. In particular, H ∩ U is a monomial subalgebra of U if H is a subalgebra of R, and p ∩ U is a monomial (prime) ideal if p is a (prime) ideal in H . By Proposition 6.1.6, the monomials in the complement of a monomial prime ideal are those contained in a face of the cone generated by the monoid.

214

6 Algebras Defined by Minors

Proposition 6.6.7 Let p ⊂ At be a G-stable prime ideal. Then there exists a face F (p) of the cone R+ St such that (a) suppG p = St \ F (p); (b) the standard bitableaux  for which (In(σ) | In(σ))) ∈ St \ F (p), σ = ||, are a K -basis of p. The proposition follows from Proposition 3.6.14, together with the description of monomial prime ideals. The facets of R+ St are determined by the inequalities in (6.12) They are F0 = {ε ∈ R+ St : π2 (ε) = 0}, i = 1, . . . , m. Fi = {ε ∈ R+ St : εi = 0}, The subalgebra U captures very little of the algebra structure of R. Therefore we cannot expect that all faces of R+ St correspond to prime ideals in R. The next lemma describes the interactions between the facets containing F (p) for the G-stable prime ideals p. Lemma 6.6.8 Let p be a G-stable prime ideal. (a) If F (p) ⊂ Fi for some i = 0, . . . , m, then  ∈ p for all bitableaux  of shape λ with ε(λ) ∈ / Fi . (b) If F (p) ⊂ Fi for some i, 1 ≤ i ≤ t, then F (pi ) ⊂ Fi−1 as well. (c) If F (p) ⊂ Fi for some i, t ≤ i ≤ m − 1, then F (pi ) ⊂ Fi+1 as well. (d) If F (p) ⊂ Ft , then p = qt , and F (qt ) = F0 ∩ · · · ∩ Fm . (e) If F (p) ⊂ Ft−1 or F (p) ⊂ Ft+1 for p = qt , then p = qt+1 and F (qt+1 ) = F0 ∩ · · · ∩ Ft−1 ∩ Ft+1 ∩ · · · ∩ Fm . Proof (a) This follows from Proposition 3.6.14. (b) Let  be a bitableau of shape λ with ε(λ) ∈ / Fi−1 . We must show that  ∈ p. Let i = 1 first. Then π2 () > 0. It follows that  has a factor η of size k > t. We apply Laplace expansion to η and substitute η in  by this expression. All bitableaux  resulting from this substitution satisfy π2 () = π2 () − 1 ≥ 0 (and the total degree in R has remained unchanged). It follows that all summands  are in R and ε1 () > 0. So  ∈ p, and it follows from (a) that  ∈ p. Now let i > 1. Since εi−1 (λ) > 0,  has a size i − 1 factor η. But i − 1 < t, and so  must also have a factor ζ of size k > t. By Lemma 3.5.6 we can write ηζ as a linear combination of bitableaux of shape (i, k − 1). Again we substitute this expression into the product . This time the value under π2 has not even changed, and we can again apply (a). (c) follows by similar arguments and is left to the reader. (d) is obvious: if F (p) ⊂ Ft , then p contains all t-minors and therefore the full generating set of the K -algebra At . (e) Suppose that  is a bitableau whose weight is not contained in F0 ∩ · · · ∩ Ft−1 ∩ Ft+1 ∩ · · · ∩ Fm . If εi () > 0 for some i > t, then clearly  ∈ qt+1 . But if

6.6 Exterior Powers of Linear Maps

215

ε j () > 0 for some j < t, then π2 () ≥ 0 implies that εi () > 0 for some i > t as well. Moreover, π2 () > 0 also implies that  has a factor of size > t. Since F (qt+1 ) is properly contained in F (qt ), it follows that F (qt+1 ) = F0 ∩ · · · ∩ Ft−1 ∩ Ft+1 ∩ · · · ∩ Fm . Now we consider a G-stable prime ideal p = qt . If F (p) ⊂ Ft+1 , then F (p) ⊂ Fi for i = t + 2, . . . , m by (c), and it follows that qt+1 ⊂ p. Assume that F (p) ⊂ Ft−1 . We must show that a bitableau  ∈ qt+1 is in p. If  contains a factor of size < t or π2 () > 0, then  ∈ p since F (p) ⊂ Fi for i = 0, . . . , t − 1 by (b). The remaining case is that π2 () = 0 and all factors of  have size ≥ t. But those of size t do not belong to p, since otherwise p = qt . We can omit them without leaving p or At since π2 (δ) = 0 for a t-minor δ. This leaves us with a product  of minors of size > t such that π2 ( ) = 0. This is only possible  if  is the empty product. Proposition 6.6.9 Set p−1 = qm+1 = 0. Then F (pi + q j ) = F0 ∩ · · · ∩ Fi ∩ F j ∩ · · · ∩ Fm for all i = −1, . . . , t − 2 and j = t + 2, . . . , m + 1 (where the empty intersection is the full cone). Furthermore F (qt+1 ) = F0 ∩ · · · ∩ Ft−1 ∩ Ft+1 ∩ · · · ∩ Fm , and F (qt ) = F0 ∩ · · · ∩ Fm . Before saying anything about the proof of Proposition 6.6.9, let us observe that Lemma 6.6.8 leaves only the t (m − t) + 2 values for F (p) that are listed in the proposition. There are two routes we can now follow: (1) We accept that all the ideals pi + q j of Theorem 6.6.5 are prime. Their number is t (m − t) + 2. There cannot be any further G-stable prime ideals, and Theorem 6.6.5 is proved. Proposition 6.6.9 follows by observing the inclusions between the G-stable prime ideals. (2) We do not use that the ideals pi + q j of Theorem 6.6.5 are prime (except for the pi and q j ). By Lemma 6.6.8 we know that there are at most t (m − t) + 2 G-stable prime ideals, and the proposition, once proved, implies that only the pi + q j are candidates for them. Then we need another argument that there are at least t (m − t) + 2 G-stable prime ideals, and we will give it below. Proof of Proposition 6.6.9 Note that it is enough to discuss the ideals pi and q j since F (pi + q j ) = F (pi ) ∩ F (q j ). Our arguments so far have shown that these ideals are prime so that Lemma 6.6.8 can be applied. This reduces the combinatorial complexity somewhat. Moreover, the claims about qt and qt+1 are contained in Lemma 6.6.8. We consider p0 . By definition, F (p0 ) ⊂ F0 . In order to show that F j ⊃ F (p0 ) for j > 0 it is enough to consider the cases j = 1 and j = m in view of Lemma 6.6.8. For j = 1 we want to find a bitableau  ∈ / p, equivalently π2 () = 0, with ε1 () > 0. We take a product δη t−1 where δ has size 1 and η has size t + 1. Similar arguments (together with Proposition 6.6.4) work for j = m and in the all the other cases. We leave them to the reader as an exercise. 

216

6 Algebras Defined by Minors

Prime Ideals and Orbits We now connect our knowledge of orbits and G-stable prime ideals. The facts used in the proof of Theorem 6.6.10 can be found in Humphreys [193, Sect. 8.3], Kraft [219, Sect. II.2] or Steinberg [293, Sect. 1.13]. Theorem 6.6.10 Let K be an algebraically closed field of characteristic 0 or > min(t, m − t) and m ≤ n. (a) The closure X t of Yt decomposes into t (m − t) + 2 orbits. (b) Each orbit is characterized by a pair (sr(x), rank x) listed in Proposition 6.6.2. (c) With respect to suitable bases in V and W , respectively, every x ∈ X t is given by a matrix d u,u+k−1 . (d) The coordinate ring At of X t contains exactly t (m − t) + 2 G-stable prime ideals, namely the ideals listed in Theorem 6.6.5. Proof Let x ∈ X t . Then the orbit Gx is open and dense in its Zariski closure Gx, which itself is the union of Gx and orbits of smaller dimensions. The ideal of functions vanishing on Gx is G-stable since Gx is the union of orbits. It is a prime ideal since our group G is irreducible as a variety: if Gx would decompose into more than one irreducible components, then also Gx must decompose, and therefore G must decompose in a nontrivial way. We have proved in Proposition 6.6.2 that there are at least t (m − t) + 2 orbits, and we have seen in Proposition 6.6.9 that there are at most t (m − t) + 2 prime ideals. Altogether this establishes a bijection between the sets (i) of the orbits, (ii) of their closures and (iii) of the G-stable prime ideals.  Theorem 6.6.10 finishes the proof of Theorem 6.6.5 where we have not assumed that K is algebraically closed: Let K be the algebraic closure of K ; if K ⊗ K R is an integral domain for a K -algebra R, then R is an integral domain. The residue class rings of At modulo the prime ideals pi + q j are actually Cohen– Macaulay integral domains, as will be documented in Remark 7.4.6. The correspondence between orbits and prime ideals is made explicit in Corollary 6.6.11 The G-stable prime ideal corresponding to the orbit {0} is qt , and qt+1 corresponds to the orbit of rank 1 elements in X t (m, n). The prime ideal corresponding to the orbit of elements of small rank u, 2 ≤ u ≤ t + 1, and rank u+k−1 , k = 1, . . . , m − t, is pt−u + qt+1+k (where again p−1 = qm+1 = 0). u−1 Proof The G-stable prime ideal p corresponds to the orbit Gx if and only if I (Gx) = p. It is obvious that the orbit {0} and qt correspond to each other. In order to show that qt+1 and the rank 1 matrices correspond to each other, we can use the element f t−1 constructed in Lemma 6.6.3. As an bitableau of shape (t + 1, t − 1), it belongs to qt+1 and vanishes only on the rank 1 element d 1,t . Thus only the orbit of rank 1 matrices can be contained in V (qt+1 ) (in addition to {0}). Using similarly the function f v which belongs to pv for v = 0, . . . , t − 2 (and counting the G-stable prime ideals contained in pv ), we see that exactly the orbits

6.7 Sagbi Deformations of Determinantal Rings

217

given by small rank at most t − v are contained in V (pv ). It follows that only one of the prime ideals pt−u + q j , j = t + 2, . . . , m + 1, can correspond to an orbit with small rank u, and now it is enough to order these orbits by the inclusion of their  closures and compare them to the sequence of prime ideals pt−u + q j . In [43] Bruns and Conca have analyzed the structure of the orbits. Roughly speaking, they prove that the orbits in X t \ Yt can be produced from orbits in Yu for u < t by simple algebraic operations. Additionally the singular locus has been determined by Bruns and Conca [43]: Theorem 6.6.12 Suppose that 1 < t < min(m, n), and that t = m − 1 if m = n. Then the singular locus of X t (m, n) consists of the points x such that rank x ≤ 1.

6.7 Sagbi Deformations of Determinantal Rings As usual, let K be a field and X an m × n matrix of indeterminates. A priori, it makes no sense to speak of a Sagbi deformation of the determinantal ring Rr +1 = K [X ]/Ir +1 (X ),

r = 0, . . . , min(m, n),

since this K -algebra is not embedded into a polynomial ring over K . But such an embedding is classically known. Take an m × r matrix Y of indeterminates and an r × n matrix of indeterminates. Then the product matrix Y Z has rank r over K [Y, Z ], and therefore the algebra homomorphism ϕ : K [X ] → K [Y, Z ],

X i j → (Y Z )i j , i = 1, . . . , m; j = 1, . . . , n,

factors through Rr +1 in a natural way since the product matrix Y Z has rank r . We will see below that ϕ is injective. Therefore it is an isomorphism between Rr +1 and K [Y Z ], the K -algebra generated by the entries of Y Z . There is a geometric observation behind this construction. Let V and W be K vector spaces of dimension m and n, respectively. A linear map f : V → W of rank ≤ r can be factored through K r , and conversely every pair of maps g : V → K r and h : K r → W composes to a linear map of rank ≤ r from V to W .

The Sagbi Basis On K [Y, Z ] we introduce a monomial order by first listing the variables of Y column by column from bottom to top, starting with the first column, and then the entries of Z row by row from right to left: Ym1 > Ym−11 > · · · > Y11 > Ym2 > · · · > Y1r > Z 1n > · · · > Z 11 > Z 2n > · · · > Z r 1 .

218

6 Algebras Defined by Minors

This total order is then extended to the induced degree reverse lexicographic order on K [Y, Z ]. Note that the restrictions of the monomial orders to K [Y ] and K [Z ] are diagonal. For convenience we transfer our notation for minors and diagonals from X to the new environment: [a1 . . . at | b1 . . . bt ]Y Z is the determinant of the t × t submatrix of Y Z in which the rows a1 , . . . , at and the columns b1 , . . . , bt of Y Z intersect. Moreover we set a1 . . . at Y = a1 , . . . , at | 1 . . . t Y and b1 . . . bt Z = 1 . . . t | b1 . . . bt Z . So a1 . . . at Y b1 . . . bt Z =

t 

Yai i Z ibi .

i=1

Lemma 6.7.1 in[a1 . . . at | b1 . . . bt ]Y Z = a1 . . . at Y b1 . . . bt Z . Proof First assume that t = r . Then [a1 . . . ar | b1 . . . br ]Y Z = [a1 , . . . , ar | 1 . . . r ]Y [1 . . . r | b1 . . . br ] Z by the multiplicativity of determinants. Taking initial monomials on both sides and using that the orders on K [Y ] and K [Z ] are diagonal we obtain the claim in the case t = r. For the general case we use that our monomial order is revlex. It implies that all terms in the expansion of [a1 . . . at | b1 . . . bt ]Y Z that involve an indeterminate Z i j with i > t can be ignored, provided at least one term survives. Note that the corresponding truncation of our minor is the product of maximal minors of the  and  truncated matrices Y Z that consist of the first t columns of Y and the first t rows of Z , respectively. This reduces the lemma to the case r = t that we have discussed already.  In the following we do not distinguish notationally between (standard) bitableaux of minors of size ≤ r and their residue classes modulo Ir +1 (X ). The set Sr of standard bitableaux of minors of size ≤ r is a K -basis of Rr +1 by Corollary 3.4.2. In analogy with the notation introduced above, Y Z is the “evaluation” of the standard bitableau on the product matrix Y Z or, equivalently, Y Z = ϕ(). Proposition 6.7.2 (a) The initial monomial of the standard bitableau =(a

u i j | bi j ), with ai1 . . . aiti Y i=1, . . . , u, j=1, . . . , ti , r ≥ t1 ≥ . . . ≥ tu , is the monomial i=1 bi1 . . . biti Z . (b) If ,   ∈ Sr ,  =   , then in(Y Z ) = in(Y Z ). In particular, the polynomials Y Z are K -linearly independent. Proof Part (a) follows immediately from Lemma 6.7.1. For part (b) we observe that both the row tableau R and the column tableau C of  = (R | C) can be rebuilt from

6.7 Sagbi Deformations of Determinantal Rings

219

in(Y Z ). This was already observed in Proposition 3.3.4(a) for C and follows by symmetry for R.  Corollary 6.7.3 The homomorphism ϕ : Rr +1 → K [Y Z ] is an isomorphism. Proof Only the injectivity is critical. It follows since the K -basis Sr of Rr +1 is mapped to a linearly independent set.  For r = min(m, n) we have obtained an embedding of K [X ] into the polynomial ring K [Y, Z ] whose monomial order is perfectly adapted to the straightening law. Whereas inside K [X ] different standard bitableaux in general have identical initial monomials with respect to a diagonal order, the initial monomials in K [Y, Z ] are different. For r = 1 we obtain the realization of R1 as the Segre product of polynomials rings that has already been discussed in Sect. 4.1. Remark 6.7.4 Corollary 6.7.3 only uses that the standard bitableaux generate Rr +1 as a K -vector space. The linear independence of the standard bitableaux  can be derived from the linear independence of their images ϕ() = Y Z . This observation is a new proof of the linear independence of the standard bitableaux. Remark 6.7.5 The straightening algorithm for maximal minors that is discussed in Remark 3.3.5 can now be generalized to all minors. Given an element f ∈ Rr +1 (so f ∈ K [X ] if r = min(m, n)), we map it to K [Y Z ]. Then the initial monomial of ϕ( f ) is determined. It is the initial monomial of a unique standard bitableau . Next  is expanded in Rr +1 (of course, not in K [Y Z ]!), and we replace f by f − α where α is the initial coefficient of ϕ( f ). Since f − α = 0 or in(ϕ( f − α)) < in(ϕ( f )), an iteration of the procedure must terminate after finitely many steps. For a bitableau  the standard bitableau  for which in(ϕ()) = in(ϕ()) is exactly the bitableau (R0 | C0 ) that has appeared in Theorem 3.3.6, as follows immediately if one compares their constructions. Corollary 6.7.3 allows us to identify Rr +1 and K [Y Z ] in the following. We let F(Y ) denote the algebra generated by the column initial bitableaux of Y and F  (Z ) the corresponding algebra for the row initial bitableaux of Z . Theorem 6.7.6 Let K be an arbitrary field. (a) The determinants [a1 . . . at | b1 . . . bt ]Y Z , 1 ≤ t ≤ r , are a Sagbi basis of Rr +1 ⊂ K [Y, Z ]. (b) The monomials in(Y Z ),  ∈ Sr , are a K -basis of the initial algebra Dr +1 = in(Rr +1 ). (c) Dr +1 ∼ = in(F(Y )) # in(F  (Z )) is a normal monoid ring. (The Segre product is taken with respect to the grading by weight.) Proof (a) follows from (b), and (b) results from Proposition 6.7.2 and the standard Hilbert function argument. (c) follows if we compare the monomial bases of Dr +1 and in(F(Y )) # in(F  (Z )). The Segre product has already been studied in Sect. 7.4. 

220

6 Algebras Defined by Minors

We suppress the obvious consequences for the structure of Rr +1 that have been stated already. Remark 6.7.7 (a) The embedding Rr +1 ⊂ K [Y, Z ] is a classical construction of invariant theory. It allows us to see Rr +1 as the ring of invariants of a linear action of G = GL(r, K ). The group G acts on K [Y, Z ] by the substitution Yi j → (Y g −1 )i j , i = 1, . . . , m, j = 1, . . . , r , and Z i j → (g Z )i j , i = 1, . . . , t, j = 1, . . . , n. Since (Y g −1 )(g Z ) = Y Z , the subalgebra K [Y Z ] is contained in the ring K [Y, Z ]G of invariants. For the converse inclusion the normality of Rr +1 is a crucial argument. See [61, Chap. 7] for a detailed discussion. (b) If one shrinks GL(r, K ) to SL(r, K ), then the ring of invariants increases from K [Y Z ] to K [Y Z , Mr (Y ), Mr (Z )]. This is discussed in [61, Chap. 7] as well. The initial algebra is generated by the monomials of in(K [Y Z ]) together with the diagonals of the r -minors of Y and the r -minors of Z . See Bruns et al. [57]. Remark 6.7.8 We will use the deformation of Rr +1 to Dr +1 below to compute the canonical module of Rr +1 . In characteristic 0 it is also possible to determine it by invariant theory. For a linear algebraic group G acting linearly on a K -algebra R and a character χ of G let Rχ denote the semi-invariants associated with the character χ: Rχ = {x ∈ R : g(x) = χ(x)x for all g ∈ G} (χ : G → K ∗ is a group homomorphism). Evidently Rχ is a module over the ring R G of invariants. Under a mild nondegeneracy hypothesis on the action of G, the canonical module of R G is given by R χ , χ = det −1 . This was proved by Knop [211]. Moreover, Knop gave an estimate for the a-invariant of R G : a(R G ) ≤ − dim R G . An application to the action of GL(r, K ) on K [Y, Z ] yields the canonical module of Rr +1 . We leave the details as an exercise to the reader.

1-Cogenerated Ideals Let δ be a minor of X and let I (X ; δ) be the ideal generated by all minors γ ≥ δ. These ideals are called 1-cogenerated since the underlying generating system of minors is an ideal in the poset of all minors, and we are taking the largest ideal not containing the single element δ. These ideals have been briefly mentioned in Sect. 3.4. They generalize the ideals Ir +1 (X ) = I (X ; δ) for δ = [1 . . . r | 1 . . . r ]. The 1cogenerated ideals are extensively studied in [61]. Using the approach via initial algebras that has been introduced in this section one obtains their basic properties almost effortless. Theorem 6.7.9 For any δ the K -algebra R(X ; δ) = K [X ]/I (X ; δ) is a normal Cohen–Macaulay domain. Proof It is enough to show that the initial ideal of I (X ; δ) is a monomial prime ideal in the normal monoid algebra in(K [X ]) = in(Rr +1 ) for r = min(m, n). The generators of in(I (X ; δ)) are exactly those monomials that contain one of certain

6.7 Sagbi Deformations of Determinantal Rings

221

indeterminates of K [Y, Z ] as a factor. These monomials evidently generate a prime ideal. One possibility for γ ≥ δ with δ = [a1 . . . at | b1 . . . bt ], γ = [c1 . . . cu | d1 . . . du ] is ci < ai for some i. This holds if and only if in(γ) contains the factor Yci i for  ci < ai . Similarly one discusses the other cases in which γ ≥ δ. The algebras R(X ; δ) can be realized as rings of invariants. See [61, Chap. 7].

Inequalities for the Initial Algebra Let μ ∈ K [Y, Z ] be a monomial, μ=

 i, j

α

Yi j i j



βuv Z uv .

uv

Then the exponents fill a pair of matrices [(αi j ), (βuv )] ∈ (Rm×r ) ⊕ (Rr ×n ). For an application below we describe the set E of vectors [(αi j ), (βuv )] ∈ (Rm×r ) ⊕ (Rr ×nn ) that appear as exponent vectors of elements in Dr +1 = in(Rr +1 ). It is not hard to check that E is the set of lattice points in the cone defined by the following linear equations and inequalities: αi j , βuv ≥ 0, αi j = βuv = 0, k−1 

αi j−1 −

k 

i= j−1

i= j

w−1 

w 

βu−1t t=u−1 n 

−

αi j −

i=1

t=u n 

i > j, v > u and i = j = u = v = r, j > i, u > v,

(6.13) (6.14)

αi j ≥ 0,

j = 2, . . . , r, k = j, . . . , m,

(6.15)

βut ≥ 0,

t = 2, . . . , r, w = u, . . . , n,

(6.16)

β jv = 0

j = 1, . . . , r.

(6.17)

v=1

The inequalities in (6.13) say that all our exponent vectors must be nonnegative. The equations (6.14) express that the exponents are 0 for variables that do not belong to the diagonals starting in the first column of Y or the first row of Z . The equations in (6.17) force the equality of the shapes on the Y side and the Z side. The remaining inequalities in (6.15) and (6.16) ensure that our monomials are products of diagonals starting in the first column of Y and the first column of Z .

The Divisor Class Group and The Canonical Module Later in this section and in Chap. 7 we will need precise knowledge of the divisor class group and the canonical module of Rr +1 . This topic is discussed extensively in [61],

222

6 Algebras Defined by Minors

but for the reader’s convenience we include a treatment based on our methodological framework. For the theory of divisor class groups the reader is referred to [129]. Theorem 6.7.10 Let K be a field, and X an m × n matrix of indeterminates and r an integer, 1 ≤ r < min(m, n). (a) The divisor class group Cl(Rr +1 ) of Rr +1 = K [X ]/Ir +1 (X ) is isomorphic to Z. (b) It is generated by the class cl(p) of the prime ideal p generated by the r -minors of the first r rows of the matrix X modulo Ir +1 (X ), as well as by the class of the prime ideal q generated by the minors of the first r columns. (c) One has cl(p) = − cl(q). Proof The ideal in K [X ] generated by the (r + 1)-minors of X and the r -minors of the first r rows is a 1-cogenerated ideal, as the reader is asked to check. Therefore it is prime by Theorem 6.7.9. This implies that p is a prime ideal in Rr +1 . The same argument applies to q. Consider the residue class δ of the r -minor [1 . . . r | 1 . . . r ] in Rr +1 . The bitableaux basis of Rr +1 is described in Proposition 3.4.1. It follows that the standard bitableaux basis of p consists of those standard bitableaux that do only contain factors of size ≤ r and at least one factor with rows 1, . . . , r . Since these minors form an ideal in the poset of all nonzero minors of Rr +1 , the minimal minor in a standard bitableaux appearing in the standard representation of an element of of p is an r -minor of the first r rows. The same applies to q if we replace rows by columns. The only minor in the intersection of the two bases is [1 . . . r | 1 . . . r ]. So one has p ∩ q = (δ). This implies cl(p) + cl(q) = cl(δ) = 0. Let S = Rr +1 [δ −1 ]. By the theorem of Gauß–Nagata there is an exact sequence 0 → U → Cl(Rr +1 ) → Cl(S) → 0 where U is the subgroup of Cl(Rr +1 ) generated by the classes of the minimal prime ideals of δ. Set  = {xi j : i ≤ r or j ≤ r } where xi j is the residue class of X i j in Rr +1 . Then δ = [1 . . . r | 1 . . . r ] ∈ K [] and S = K [, δ −1 ], as one sees by elementary transformations or expansion of r + 1-minors of the matrix of residue classes. So K [] has the same Krull dimension as Rr +1 , and therefore  is algebraically independent over K . It follows that K [] and S are factorial, and Cl(Rr +1 ) = U = Z cl(p) = Z cl(q). Now it is clear that Cl(Rr +1 ) is a cyclic group and that its elements are given by the multiples of cl(p) and cl(q). In other words, every divisorial ideal of Rr +1 is isomorphic to a symbolic power of p or q. If none of them is a principal ideal, then Cl(Rr +1 ) ∼ = Z as claimed. There are various ways to prove this. We refer to Proposition 6.7.11 below where we show that the symbolic powers coincide with the ordinary ones, and these are certainly not principal ideals.  One can of course replace first the r rows or columns in the theorem by any choice of r rows or columns.

6.7 Sagbi Deformations of Determinantal Rings

223

Proposition 6.7.11 Let the notation be as in Theorem 6.7.10 and k ≥ 0 an integer. Then the following hold: (a) pk and qk have K -vector space bases consisting of standard bitableaux all of whose factors have size ≤ r and contain k factors with rows 1, . . . , r and columns 1, . . . , r , respectively. (b) p(k) = pk and q(k) = qk . Proof (a) We discuss pk . The claim for qk follows by symmetry. All the mentioned bitableaux belong to pk since they contain k factors from p. Therefore it is enough to show that the standard bitableaux showing up in the standard representation of an element of pk contain k of these bitableaux. Using the Plücker relations one sees that the standard bitableaux with exactly k factors from p generate pk as an Rr +1 -module. Pick x ∈ pk . Then x is a linear combination of the standard bitableaux just mentioned with coefficients from Rr +1 . It is enough to show that every standard bitableau that is produced by the straightening procedure from the summands of the linear combination again contains k factors from p. In fact, if we straighten a product of two minors of which one, say δ, is in p, then the right hand side of the straightening relation must contain such a factor as well since all bitableaux γ ∈ Rr +1 , γ = 0, γ ≤ δ, belong to p. If both factors lie in p, then the product is a linear combination of factors from p as the straightening law in the Grassmannian shows. (b) Part (a) enables us to compute the initial ideal of pk . We represent a monomial in K [Y, Z ] by a table [(αi j ), (βuv )] as above. Again we let E denote the monoid of the exponent vectors of the monomials in Dr +1 . Let E k be the set of initial monomials of ϕ(pk ). It follows from (a) that E k = { [(αi j ), (βuv )] ∈ E | αii ≥ k, i = 1, . . . , r } = { [(αi j ), (βuv )] ∈ E | αrr ≥ k }. Therefore the initial ideal in(ϕ(pk )) is a divisorial ideal in Dr +1 . This follows from Proposition 6.1.11 and the fact that in(ϕ(pk )) is a height 1 ideal in Dr +1 by Proposition 1.6.2. Since in(ϕ(pk )) is a divisorial ideal in the normal domain Dr +1 , it has no embedded prime divisors. In other words, Dr +1 / in(ϕ(pk )) satisfies the Serre condition (S1 ). But according to Proposition 1.6.3 this holds for Rr +1 /pk as well. (We have used Lemma 1.5.3 to get into a position in which we can apply Propositions 1.6.2 and 1.6.3.) So pk has no embedded prime divisors, and so it coincides with its  p-primary component, which is p(k) . Remark 6.7.12 The statements about the powers of p and q can be generalized to straightening closed ideals and ideals of virtual maximal minors. See [61, Sect. 9], Bruns et al. [59] and Huneke [194]. Huneke relates our topic to the theory of d-sequences. Theorem 6.7.10 and Proposition 6.7.11 allow us to compute the canonical module of Rr +1 . There are three basic facts that must be observed. First, the canonical module of a Cohen–Macaulay ring is unique up to multiplication by an invertible ideal

224

6 Algebras Defined by Minors

of R; see [52, 3.3.17]. Next, the only invertible ideals of a graded ring R are the principal ideals. Finally, the canonical module of a normal Cohen–Macaulay domain is isomorphic to a divisorial ideal [52, 3.3.19]. This implies: the canonical module of Rr +1 is unique up to isomorphism and therefore given by a unique class in the divisor class group. Theorem 6.7.13 Let K be a field, X an m × n matrix of indeterminates and r an integer, 1 ≤ r < min(m, n). Suppose that m ≤ n. (a) Then qn−m is the canonical module of Rr +1 (up to isomorphism). (b) With respect to the standard grading, qn−m (−r m) is the graded canonical module (up to graded isomorphism). (c) Rr +1 is a Gorenstein ring if and only if m = n. Proof Let us first consider the case r = 1. Since R2 = D2 is a normal monoid domain, we can apply Theorem 6.1.9: the canonical module is (up to isomorphism) the monomial ideal whose monomial basis is given by all lattice points in the interior of the cone defined by the monoid. For r = 1 only the sign inequalities for the from exponents of Yi1 , i = 1, . . . , m, and Z 1, j , j = 1, . . . , n remain the inequalities  (6.13)–(6.16), and (6.17) reduces to the single equation i αi1 = j β1 j . Therefore the prime ideals defined by the facets of the cone in our case are the monomial ideals Pi generated by the entries of the ith row of X in R2 , i = 1, . . . , m, and the corresponding ideals Q 1 , . . . , Q n of the columns. The ideal of the interior is their intersection. The divisor class of ω is therefore given by cl(ω) =

m  i=1

cl(Pi ) +

n 

cl(Q j ).

j=1

Since cl(Pi ) = cl(p) for i = 1, . . . , m and cl(Q j ) = cl(q) for j = 1, . . . , n, one obtains cl(ω) = m cl(p) + n cl(q) = (n − m) cl(q). In conjunction with Proposition 6.7.11 this proves (a) and (c) for r = 1. The reader is invited to prove (a) for r > 1 from the toric deformation to Dr +1 similarly to our proof for the equality of the ordinary and symbolic powers of p and q, but is simpler to use the induction argument Lemma 3.4.5: −1 , X i j → Yi j + X m j X in X mn X m j → X m j , X in → X in ,

1 ≤ i ≤ m − 1, 1 ≤ i ≤ m,

1 ≤ j ≤ n − 1, 1 ≤ j ≤ n,

with an (m − 1) × (n − 1) matrix of indeterminates. The formation of the canonical module commutes with localization as well as with polynomial extensions [52, 3.3.21]. Moreover, under the localization above, the ideal of the r -minors of the first r rows or columns is equal to the Laurent polynomial extension of the corresponding (r − 1)-minors of Y . We are done by induction for (a) and (c). (b) has already been justified in Remark 4.4.14, using (a). 

6.7 Sagbi Deformations of Determinantal Rings

225

Of course, if m ≥ n in Theorem 6.7.13, one must replace q by p. Let us indicate a very fast argument showing that Rr +1 is Gorenstein if m = n. In this case the transposition of the matrix X induces an automorphism of Rr +1 by the linear substitution X i j → X ji for all i, j ≤ n. This automorphism exchanges p(k) and q(k) . But the canonical module is mapped to the canonical module which is given by a (symbolic) power of the ideal of the first r -columns or first r rows, in the same way for X as well as for its transpose. The only symbolic power that is identical for both selections is the zeroth.

Cohen–Macaulay and Ulrich Ideals Using the description of Dr +1 by inequalities for its monomials, we can completely describe the rank 1 Cohen–Macaulay modules of Rr +1 . A Cohen–Macaulay module M of positive rank over a standard graded algebra R is called Ulrich if its number of generators μ(M) is maximal relative to its multiplicity. More precisely, μ(M) ≤ e(M) for all Cohen–Macaulay modules M, and the Ulrich modules are those for which one has equality (see Brennan et al. [28] or [52, 4.7.11] for the local version). In the following we assume that we are in the nontrivial case in which r < min(m, n). Over a normal Noetherian domain R a Cohen–Macaulay module of rank 1 is isomorphic to a divisorial ideal so that one needs to search only among a system of ideals representing the divisor classes. For Rr +1 this system is given by Proposition 6.7.11. Since the minimal numbers of generators of pk and qk are strictly growing with k, it is a priori clear that there are only finitely many Cohen–Macaulay classes. Let us first determine the maximal exponents beyond which the powers are definitely not Cohen–Macaulay. Proposition 6.7.14 For any integer k ≥ 1 one has  μ(pk ) = det

k+n− j n−i



 and μ(qk ) = det

1≤i, j≤r

k+m− j m −i

. 1≤i, j≤r

(6.18) Proof By symmetry it is enough to consider pk . This ideal is generated by its lowest degree component, and the vector space (pk )kr has the standard bitableaux of shape (r k ) in an r × n matrix as a K -basis. These standard bitableaux are counted by the Hilbert function of the coordinate ring of the corresponding Grassmannian. A version that gives our formulas is contained in Ghorpade [137, Theorem 6].  Proposition 6.7.15 One has μ(pm−r ) = μ(qn−r ) = e(Rr +1 ). Therefore all powers pk with k > m − r are not Cohen–Macaulay. Nor are the powers qk with k > n − r . Proof The second statement follows from the first since the minimal number of generators is strictly growing with the exponent.

226

6 Algebras Defined by Minors

For the equations we quote (4.2):  e(Rr +1 ) = det

m −i +n− j m −i



 = det

i, j=1,...,r

m −i +n− j n− j

. i, j=1,...,r

(6.19) m−r , but by the binomial This is not exactly the expression appearing in (6.18) for p    a  a+1 = b+1 one can transform the matrix of (6.18) into the transpose identity ab + b+1 of the matrix in (6.19). The equation for qn−r follows by symmetry.  It remains to show that those powers of p and q that are not excluded from Cohen– Macaulayness by Proposition 6.7.15 are indeed Cohen–Macaulay. To this end we show that their initial ideals are Cohen–Macaulay. Let S = K [M] be a normal monoid domain and suppose that M is embedded in the vector space V = R gp(M) so that M is the intersection of the cone C = R+ M and gp(M). A fractional ideal I of S is called conic if I has a basis B of monomials μ ∈ gp(M) such that B = (C + w) ∩ gp(M) for some w ∈ V . Less formally: the monomial basis of I is cut out from gp(M) by a translate of the cone C. Conic ideals are Cohen–Macaulay; see [50, 6.68]. Proposition 6.7.16 The initial ideals in(ϕ(pk )) are conic for 0 ≤ k ≤ m − r , as well as the initial ideals in(ϕ(qk )) for 0 ≤ k ≤ n − r . Proof Set Jk = in(ϕ(pk )). We want to show that Jk is a conic ideal in Dr +1 . To this end we have to find wk ∈ RE such that E k = ZE ∩ (wk + R+ E) where E k is defined as in the proof of Proposition 6.7.11. We choose a positive real number ε < 1 and define wk = [(αi j ), (βuv )] by setting ⎧ ⎪ if i = j, ⎨k − ε, αi j = −(k − ε)/(m − r ), if j < i ≤ m − r + j, ⎪ ⎩ 0, otherwise. and βuv = 0 for all u, v. It is clear that wk ∈ RE. Since −(k − ε)/(m − r ) > −1 (this is the point where we need k ≤ m − r !) we have ZE ∩ (wk + R+ E) = E k . So  Jk is indeed a conic ideal. Corollary 6.7.17 The ideals pk are Cohen–Macaulay for 0 ≤ k ≤ m − r , as well as the ideals qk for 0 ≤ k ≤ n − r . Moreover, pm−r and qn−r are Ulrich ideals. Proof The Cohen–Macaulayness of the ideals follows from the Cohen–Macaulayness of their initial ideals, which in its turn holds since the ideals are conic. That the two “boundary” cases are Ulrich is implied by Proposition 6.7.15, which also excludes the higher powers from being Cohen–Macaulay.  Remark 6.7.18 Kleppe and Miró-Roig [210] have investigated the categories of indecomposable Cohen–Macaulay and Ulrich modules over the rings Rr +1 , r = min(m, n): the representation type for Cohen–Macaulay modules is wild. The paper also discusses Ulrich modules.

Notes

227

Notes In [297, Theorem 3.2.9] Sturmfels stated that the maximal minors form a Sagbi basis of the coordinate ring of the Grassmannian, and most likely this is the first connection between determinantal rings and Sagbi bases. From a more general viewpoint, toric deformations of flag and Schubert varieties were studied by Gonciulea and Lakshmibai [145]. The ring theoretic properties of the coordinate ring of Grassmannians and flag varieties have all been proved by other techniques before, for example by the ASL approach developed by De Concini et al. [104]. To mention at least one classical source: the normality of the Grassmannian is a theorem of Igusa [195]. Sections 6.3, 6.4 and 6.6 are based on the work of Bruns and Conca [39–41, 43]. Section 6.7 follows Bruns et al. [57]. The divisor class group was first computed by Bruns [34], as well as the canonical module [35]. Section 6.5 surveys Bruns et al. [47] and Huang et al. [192].

Chapter 7

F-singularities of Determinantal Rings

If the characteristic of the field K is positive, the linear group GL(m, K ) × GL(n, K ) is not linearly reductive, so many tools coming from representation theory for the study of the determinantal rings become less effective. However, in this case there is the Frobenius map: for a ring R of characteristic p > 0, the Frobenius map is F : R → R, r → r p . Of course F is a homomorphism of rings, which is injective if and only if R is reduced. A celebrated theorem of Kunz asserts that F is flat if and only if R is a regular ring. This theorem demonstrates that properties of the map F reflect properties of the ring R: such properties are the so-called F-singularities. In the first section we are going to study the F-split (often equivalent to F-pure) notion, namely when F splits as a map of R-modules. We will prove a criterion due to Fedder to establish which homogeneous polynomial ideals define an F-split quotient, and on the way we will show that many of these ideals admit a squarefree monomial initial ideal. We conclude this section by proving that determinantal rings are F-split, describing an explicit F-splitting. In the second section we will study F-regular rings. We will mainly focus on strongly F-regular rings, but we will also explain the relationship with (weakly) Fregular rings and with the theory of tight closure. We will show that determinantal rings are strongly F-regular, exploiting an extension of Fedder’s criterion. Strong F-regularity does not behave well under Gröbner or Sagbi deformation. This is however the case for F-rationality, introduced in the third section. Together with the fact that Gorenstein F-rational rings are strongly F-regular, this opens a second way to prove strong F-regularity of determinantal rings. The connections to the representations and invariants of the general linear group will come back in Sect. 7.4 which deviates slightly from the general theme of Gröbner © Springer Nature Switzerland AG 2022 W. Bruns et al., Determinants, Gröbner Bases and Cohomology, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-05480-8_7

229

230

7 F-singularities of Determinantal Rings

and Sagbi deformations by introducing equivariant deformations. In our case their structure depends on the monoids of shapes of the standard bitableaux appearing in the algebra under consideration, and once the connection has been established, it is perhaps the combinatorially simplest approach to all our algebras. The algebraic structure of the equivariant deformations is determined from their toric deformations, and therefore they are in line with the main theme of the book. In the last section we discuss the F-singularities of pairs: for a pair we mean a couple (I, R) where I is an ideal of the ring R. We focus on the F-pure threshold fpt R (I ), and we will compute fpt K [X ] (It ) and fpt K [X ]/It (n), where n is the unique maximal homogeneous ideal of the determinantal ring K [X ]/It . While an essential ingredient for the first computation is again a Fedder’s type criterion (and special products of minors already exploited in the previous sections), the second computation relies on a connection between F-pure threshold at the homogeneous maximal ideal and the divisor class group of a normal Cohen–Macaulay standard graded K -algebra.

7.1

F-purity

Let p be a prime number, and R a ring of characteristic p. The Frobenius map on R is the ring homomorphism F : R → R, r → r p . Let F∗ R denote the R-module obtained by restricting the scalars: (a) F∗ R = R as additive group (in particular as a set); (b) r · x = r p x for all r ∈ R and x ∈ F∗ R. In this way we can think of F as the map of R-modules F : R → F∗ R sending r ∈ R to r p ∈ F∗ R = R. We introduce a fundamental concept. Definition 7.1.1 The ring R is called F-finite if it is Noetherian and F∗ R is a finitely generated R-module. In this section we do not assume a priori that our rings are F-finite. However, such an assumption would make the study of F-singularities easier (as it will be clear to the reader), and we will adopt it starting from the next section. Among the reasons why such an assumption will be useful is that Hom R (F∗ R, R) may not commute with flat base change if R is not F-finite. The following example shows that several natural rings of prime characteristic are indeed F-finite. Example 7.1.2 Let K be a field of characteristic p. Obviously K is F-finite if it is perfect. Furthermore, if K is F-finite and R is a finitely generated K -algebra, then R is F-finite. Moreover, if a ring R is F-finite, then all its localizations are F-finite.

7.1 F-purity

231

Remark 7.1.3 The Frobenius map carries important information about the ring R: (a) It is straightforward to check that R is reduced if and only if F is injective; (b) If R is Noetherian, Kunz proved in [227] that R is a regular ring if and only if F∗ R is a (necessarily faithfully) flat R-module. In this section, we want to study the rings R such that the Frobenius map is pure, which is a notion between being injective and being faithfully flat. We will also inquire on the related notion that the Frobenius map splits as a map of R-modules. Before it is convenient to quickly review some basic facts related to pure maps.

Pure Maps For this subsection we leave the “characteristic p world”. So, let R be a ring of any characteristic. We say that a map of R-modules φ : N → M is pure if φ ⊗ id L : N ⊗ R L → M ⊗ R L is injective for any R-module L. Remark 7.1.4 Using the above notation, we have that: (a) If ϕ is a pure map, then φ is injective (simply choose L = R). (b) If φ is a split-inclusion of R-modules (i.e., if there exists ψ ∈ Hom R (M, N ) such that ψ ◦ φ = id N ), then φ is a pure map, because − ⊗ R L is a functor. (c) If φ is injective and M/φ(N ) is a flat R-module, then φ is a pure map: to see this, consider the exact sequence Tor 1R (M/φ(N ), L) = 0 → N ⊗ R L → M ⊗ R L φ

→ M → M/φ(N ) → 0. coming from the short exact sequence 0 → N − (d) The map φ is pure if and only if φ is injective and, for every m × n-matrix (ri j ) ∈ R m×n , the following holds: whenever nj=1 ri j x j ∈ φ(N ) for all i =   1, . . . , m and x j ∈ M, there exist y j ∈ N such that nj=1 ri j x j = nj=1 ri j φ(y j ) (see Matsumura [243, Theorem 7.13]). Remark 7.1.4(b) shows that a split-inclusion is pure. Although the two notions are different in general, they coincide in a number of natural situations: Lemma 7.1.5 Suppose that R is a Noetherian ring and φ : N → M is a map of R-modules such that M/φ(N ) is a finitely generated R-module. Then the following are equivalent: (a) φ is a pure map. (b) φ is a split-inclusion. Proof (b) =⇒ (a) is always true, as noticed in Remark 7.1.4(b). For (a) =⇒ (b), since φ is injective, we can consider the short exact sequence φ

π

0→N − →M− → M/φ(N ) → 0.

232

7 F-singularities of Determinantal Rings

Proving that φ is a split-inclusion of R-modules is equivalent to showing that π is a split-surjection of R-modules, namely that there exists ι ∈ Hom R (M/φ(N ), M) such that π ◦ ι = id M/φ(N ) . We are going to construct such a ι. Take x1 , . . . , xn ∈ M such that x1 , . . . , xn generate M/φ(N ) as an R-module. free R-module with basis e1 , . . . , en and U So M/φ(N ) ∼ = R n /U , where R n is the  is the submodule of R n generated by nj=1 ri j e j for some m × n-matrix (ri j ) ∈ n r x ∈ φ(N ), by Remark 7.1.4(d) there exist y j ∈ N such R m×n . Since j=1 n inj j n that j=1 ri j x j = j=1 ri j φ(y j ). Hence j=1 ri j (x j − φ(y j )) = 0, and the map ι : M/φ(N ) → M sending x j to x j − φ(y j ) is a well-defined map of R-modules  satisfying π ◦ ι = id M/φ(N ) . For the next results we assume that N = R, so that φ : R → M is a map of R-modules. Lemma 7.1.6 Let (R, m) be a Noetherian complete local ring. For a map of Rmodules φ : R → M, the following are equivalent: (a) φ is a split-inclusion. (b) φ is a pure map. (c) φ ⊗ id E : E → M ⊗ R E is injective where E is the injective hull of R/m. Proof (a) =⇒ (b) =⇒ (c) are obvious implications. To see (c) =⇒ (a), if φ ⊗ id E : E → M ⊗ R E is injective, since E is an injective R-module, then the following is surjective: Hom R (M, Hom R (E, E)) ∼ = Hom R (M ⊗ R E, E) → Hom R (E, E) Hence, since by Matlis duality Hom R (E, E) ∼ = R ([52, 3.2.13]), the corresponding map α : Hom R (M, R) → R is surjective. So there exists θ ∈ Hom R (M, R) such that α(θ) = 1. On the other hand, by construction α(θ) = θ(φ(1)), hence we have  θ ◦ φ = id R . Remark 7.1.7 If in Lemma 7.1.6 we do not assume that (R, m) is complete, the equivalence (b) ⇐⇒ (c) is still true: (b) =⇒ (c) is obvious, and for (c) =⇒ (b),  ⊗R E ∼ noting that R = E and using Lemma 7.1.6, we obtain that the map → R  ⊗R M R  is a faithfully flat R-module, the map R → M is pure. is pure. Since R We get the following useful corollary. Corollary 7.1.8 Let R be a Noetherian ring and φ : R → M a map of R-modules. Suppose that we are in one of the following situations: (a) M is a finitely generated R-module. (b) R is a complete local ring.

7.1 F-purity

233

Then φ is a pure map if and only if it is a split-inclusion. For ring homomorphisms purity follows from faithful flatness. Before proving it we observe the following: Remark 7.1.9 Let φ : R → S be a ring homomorphism which is a pure map of R-modules (where the R-module structure of S is given by φ). Then I S ∩ R = I for any ideal I ⊂ R. To see this, just take L = R/I : the fact that R/I ∼ = R ⊗R L → S ⊗R L ∼ = S/I S is injective means exactly that I S ∩ R = I . Although the purity of φ is stronger than the condition that I S ∩ R = I for any ideal I ⊂ R, in many situations (e.g. if R is an excellent ring), they are equivalent notions (see Hochster [178] for a precise discussion). Proposition 7.1.10 Let φ : R → S be a faithfully flat ring homomorphism. Then φ is a pure map (where the R-module structure of S is given by φ). Proof Let M be an R-module and consider the map of R-modules φ ⊗ id M : M → S ⊗ R M sending x to 1 ⊗ x. Since S is a flat R-module, S ⊗ R Rx is isomorphic to the submodule S(1 ⊗ x) of S ⊗ R M for all x ∈ M. Since S is faithfully flat, x = 0 ⇐⇒ Rx = 0 ⇐⇒ S ⊗ R Rx = 0 ⇐⇒ 1 ⊗ x = 0. Hence φ ⊗ id M is injective.



F-pure Rings and F-split Rings Let p be a prime number, and R a Noetherian ring of characteristic p. In this subsection we will compare the following two concepts: Definition 7.1.11 The ring R is: (a) F-pure if F : R → F∗ R is a pure map of R-modules. (b) F-split if F : R → F∗ R is a split-inclusion of R-modules. In this case, a map θ ∈ Hom R (F∗ R, R) such that θ(1) = 1 is called an F-splitting of R. Of course, F-pure and F-split rings must be reduced. For certain ideals I of R it is easy to see that R/I is F-split together with R: Definition 7.1.12 If θ : F∗ R → R is an F-splitting, we say that an ideal I ⊂ R is compatibly split with respect to θ if θ(I ) ⊂ I . The condition θ(I ) ⊂ I could be replaced by the formally stronger θ(I ) = I . In fact, θ ◦ F = id R if θ splits F, and since F(I ) ⊂ I for every ideal, θ(I ) ⊃ I for all ideals I of R.

234

7 F-singularities of Determinantal Rings

Remark 7.1.13 In the following proposition, and throughout this chapter, given an ideal I ⊂ R we consider it as a subset of F∗ R. Of course I is an R-submodule of F∗ R, and has not to be confused with the extension I F∗ R (F∗ R is a ring, indeed F∗ R = R as rings). Some authors denote the R-submodule I ⊂ F∗ R with F∗ I , but we will not do it: it should always be clear when we consider I as an ideal of R or as an R-submodule of F∗ R. Notice that the quotient F∗ R/I is an R/I -module and can be identified with F∗ (R/I ). Proposition 7.1.14 Let θ be an F-splitting of R and I a compatibly split ideal with respect to θ. Then the naturally induced map θ : (F∗ R)/I = F∗ (R/I ) → R/I defines an F-splitting of R/I ; in particular R/I is F-split. Proof It is enough to observe that θ really induces a map F∗ (R/I ) → R/I . This is clear since θ(I ) ⊂ I .  The next proposition gives a simple way to produce reduced rings that are not F-split: it is enough that R/I is not reduced for an ideal I ∈ Z (R), a set of ideals defined in the proposition. An explicit example is given in Example 7.1.16. Proposition 7.1.15 Let R be F-split, and consider the smallest class Z (R) of ideals of R such that (a) (0) ∈ Z (R); (b) if I ∈ Z (R) then I : J ∈ Z (R) for all ideals J ⊂ R; (c) if I, J ∈ Z (R) then I + J ∈ Z (R) and I ∩ J ∈ Z (R). Then all ideals I ∈ Z (R) are compatibly split for any F-splitting of R. In particular, R/p is F-split for every minimal prime ideal p of R. Proof Every ideal I ∈ Z (R) is produced from (0) by a chain of the operations under which Z (R) is closed. So the essential point is to observe that these operations preserve compatible splittings. We leave this to the reader. The last statement follows since the minimal prime ideals of R belong to Z (R) if R is Noetherian.  Example 7.1.16 Let K be a field of positive characteristic. The reduced ring R = K [X, Y ]/(X (X + Y )Y ) is not F-split: in fact, using the notation of Proposition 7.1.15 and putting x = X and y = Y , the ideal I = (x(x + y), y) = (x 2 , y) is in Z (R), but it is not even radical, so R/I cannot be F-split. Clearly an F-split ring R is F-pure. The converse, in general, is false. We discuss this in the next remark. Beforehand it is convenient to introduce a further fundamental concept: Definition 7.1.17 If I ⊂ R is an ideal, the Frobenius power of I is I [ p] = F(I )R ⊂ p p R. Equivalently, for a system of generators r1 , . . . , rk of I , I [ p] = (r1 , . . . , rk ) ⊂ R.

7.1 F-purity

235

Remark 7.1.18 If R is a regular ring, then it is F-pure. To see this, if R is regular then F is flat by the theorem of Kunz ([52, Corollary 8.2.8]). It is even faithfully flat because mF∗ R = m[ p]  F∗ R for any maximal ideal of R ([243, Theorem 7.2]), so by Proposition 7.1.10 F is a pure map. In contrast, there are regular rings that are not F-split; a nonexcellent discrete valuation ring with F-finite fraction field is an example (see Datta and Smith [102, Corollary 4.2.2]; for an explicit example see [102, Ex. 4.5.1]). In spite of the discussion above, for many rings the two concepts “F-split” and “F-pure” are equivalent: Proposition 7.1.19 Let R be a Noetherian ring of characteristic p satisfying one of the following conditions: (a) R is F-finite. (b) R is a complete local ring. Then R is F-pure if and only if it is F-split. In the situation (b), these two properties are furthermore equivalent to the fact that F ⊗ id E : E → F∗ R ⊗ R E is injective where E is the injective hull of R/m. 

Proof This follows immediately from Lemmas 7.1.5 and 7.1.6.

In general, if a K -algebra R is F-pure, there might be field extensions K ⊂ L such that R ⊗ K L is not F-pure, as shown by the following example. Example 7.1.20 Let K = (Z/ pZ)(X ) and R = K [Y ]/(Y p − X ). Since R is a field, it is certainly F-pure (even F-split). However, if L = R, R  = R ⊗ K L is not even reduced, hence not F-pure. Even if F-purity is not preserved under field extensions, we have the following descent property: Lemma 7.1.21 Let φ : R → S be a pure map of Noetherian rings of characteristic p. If S is F-pure, then R is so. Proof Notice that the function φ : F∗ R → F∗ S is a map of R-modules. Let M be an R-module and consider the commutative diagram S ⊗R M

F⊗id S⊗ R M

F∗ S ⊗ S (S ⊗ R M) ∼ = F∗ S ⊗ R M

φ⊗id M

M

φ⊗id M F⊗id M

.

F∗ R ⊗ R M

Since both the maps on the top and on the left hand side are injective, also the map on the bottom must be injective. That is, R is F-pure.  We have the following useful consequence:

236

7 F-singularities of Determinantal Rings

Corollary 7.1.22 Let R be a finitely generated algebra over a field K of positive characteristic. If R ⊗ K L is F-pure for some extension of fields K ⊂ L, then R is F-pure. Remark 7.1.23 In Corollary 7.1.22, assuming that R is positively graded the reverse implication is also true. Namely, for a positively graded finitely generated algebra R over a field K of positive characteristic the following are equivalent: (a) R is F-pure; (b) R ⊗ K L is F-pure for any extension of fields K ⊂ L. For the use in this book the implication (b) =⇒ (a), already proved, is sufficient. The reverse implication is quite technical. So we only sketch the proof within this remark. Notice that R is defined over a field K  ⊂ K that is isomorphic to (Z/ pZ)(α1 , . . . , αr ), where the αi belong to K (some of the αi may be algebraic over Z/ pZ, some not). That is, R ∼ = R  ⊗ K  K for some positively graded finitely   generated algebra R over K . By Corollary 7.1.22, R  is F-pure. Since K  is an F-finite field by construction, R  is F-finite as well. Hence, if m denotes the unique  maximal homogeneous ideal of R  , the completion  Rm  is F-pure by Fedder [125,  ∼      Prop. 1.3(4)]. Notice that Rm = S /I where S = K [[X 1 , . . . , X n ]] for some n ∈ N and ideal I  ⊂ S  . If L  is the perfect closure of L, setting S = L  [[X 1 , . . . .X n ]] and I = I  S, we infer that S/I is F-pure exploiting [125, Theorem 1.12]. Notice that S/I ∼ L) ⊗ L L  )m where m is the unique maximal homogeneous ideal of = ((R ⊗ K (R ⊗ K L) ⊗ L L  . So we deduce that (R ⊗ K L) ⊗ L L  is F-pure by [125, Proposition 1.3] (since L  is perfect, (R ⊗ K L) ⊗ L L  is F-finite). Finally, R ⊗ K L is F-pure by Corollary 7.1.22.

F-splittings of the Polynomial Ring For this subsection, K will be a perfect field of positive characteristic p and R = K [X 1 , . . . , X n ] the polynomial ring in n variables over K . It is easy to see that F∗ R is the free R-module generated by the monomials X 1i1 · · · X nin with i j < p for all j. In particular, R is F-split. We want to describe all the F-splittings θ : F∗ R → R, and more generally the elements of Hom R (F∗ R, R). Of course the latter is a free R-module generated by the dual basis of X 1i1 · · · X nin with i j < p for all j, say φi1 ,...,in . But our purpose is to understand the structure of Hom R (F∗ R, R) as an F∗ R-module. To this end, let us introduce the fundamental element Tr ∈ Hom R (F∗ R, R), which is defined simply as Tr = φ p−1, p−1,..., p−1 . We claim that Hom R (F∗ R, R), as an F∗ Rmodule, is generated by Tr. More precisely, the following is an isomorphism of F∗ R-modules:  : F∗ R → Hom R (F∗ R, R), f → f  Tr : g → Tr( f g).

7.1 F-purity

237

The fact that  is an injective map of F∗ R-modules is clear. For the surjectivity, just p−i −1 p−i −1 notice that φi1 ,...,in = X 1 1 · · · X n n  Tr if i 1 , . . . , i n are natural numbers such that i j < p for all j: since Hom R (F∗ R, R) is generated, as an R-module, by φi1 ,...,in with i j < p for all j, the surjectivity follows. Because two basis elements of the rank 1 free F∗ R-module Hom R (F∗ R, R) differ by unit in F∗ R, that is, a nonzero element in K , we did actually not have much choice for the generator. Remark 7.1.24 (a) The notation Tr suggests that it is a “trace”. It would lead us too far astray to discuss this interpretation. Instead we refer the reader to Brion and Kumar [31, Sect. 1.3]. Note that it is not the R-valued trace defined on Hom R (F∗ R, R). (b) Nevertheless we can sketch a more general framework that Tr fits in. Let us write S for F∗ R. As a K -algebra, S = K [X 1 , . . . , X n ]. Then we must identify R p p with F(R) = K [X 1 , . . . , X n ]. Clearly R is a graded Noether normalization of S. Since S is Cohen–Macaulay, it is a free R-module, as observed already. Since R is Gorenstein, it is isomorphic to its own canonical module ω R (for simplicity we disregard the graded structure), and Hom R (S, R) is the canonical module of S. p p Set S = S/(X 1 , . . . , X n ). It is a graded Artinian K -algebra. Its socle is generated p−1 p−1 by the residue class of X 1 · · · X n . The homomorphism S → K induced by Tr ∨ generates the graded K -dual S which is the canonical module of S. Such a generator can be lifted to a generator of ω S . For the background of this discussion we refer the reader to [52, Chap. 3]. At least we see that Tr appears naturally in the theory of canonical modules. Remark 7.1.25 Observe that f  Tr is an F-splitting of R for f ∈ R if and only if the following two conditions hold: p−1

p−1

(a) X 1 · · · X n ∈ supp( f ) and its coefficient in f is 1. (b) If X 1u 1 · · · X nu n ∈ supp( f ) and u 1 ≡ · · · ≡ u n ≡ −1 (mod p), then u i = p − 1 for all i. p−1

p−1

Proposition 7.1.26 The map θ = X 1 · · · X n  Tr ∈ Hom R (F∗ R, R) is an Fsplitting of R, and the compatibly split ideals with respect to θ are exactly the squarefree monomial ideals of R. Proof That θ is an F-splitting is clear, and it is easy to check that a squarefree monomial ideal is compatibly s split with respect to θ. Vice versa, let g = i=1 ai μi ∈ I , where μi ∈ Mon(R) and ai ∈ K \ {0}. Set μi = X 1u i1 · · · X nu in and pick i ∈ {1, . . . , s}. Our purpose is to show that μi ∈ I if I is a compatibly split ideal with respect to θ. Clearly we can find N ∈ N with the following property: for all k ∈ {1, . . . , s}, k = i, there exists j ∈ {1, . . . , n} such that u k j ≡ u i j (mod p N ). For each j = 1, . . . , n, pick v j such that 0 ≤ v j < p N and u i j ≡ −v j modulo p N , and put h = X v1 · · · X vn g ∈ I . Since I is a compatibly split ideal with respect to θ, it follows that θ N (h) ∈ I . Notice that the monomials in the support of θ N (h) correspond to those k ∈ {1, . . . , s} such that u k j ≡ u i j modulo p N for all j = 1, . . . , n. Hence θ N (h) is a monomial, precisely

238

7 F-singularities of Determinantal Rings

θ N (h) =

 ai X u i1 +v1 · · · X u in +vn .

pN

Since (u i j + v j )/ p N ≤ u i j for any j = 1, . . . , n (check it!), μi is a multiple of θ N (h) ∈ I , so that μi ∈ I . This shows that I is a monomial ideal. That I is radical follows from the fact that R/I is F-split.  Next we want to give an useful criterion to check whether an ideal I ⊂ R is compatibly split for a given F-splitting. More generally, we are going to find a criterion for an ideal I ⊂ R to be fixed by a given element of Hom R (F∗ R, R). Proposition 7.1.27 For any f ∈ R and any ideal I ⊂ R, we have: ( f  Tr)(I ) ⊂ I

f ∈ I [ p] : I.

⇐⇒

Proof Certainly Tr( f I ) ⊂ I if and only if for all g ∈ I one has Tr( f g) ∈ I , which is the case if and only if Im( f g  Tr) ⊂ I for all g ∈ I . Choose g ∈ I . Then Im( f g  Tr) ⊂ I if and only if f g  Tr(X 1i1 · · · X nin ) ∈ I whenever i 1 , . . . , i n are natural numbers with i j < p for all j. Namely, for all g ∈ I , if we call f g  Tr(X 1i1 · · · X nin ) = gi1 ,...,in , one has f g  Tr =



  p−i −1 gi1 ,...,in · X 1 1 · · · X np−in −1  Tr   p−i −1 = gi1 ,...,in · X 1 1 · · · X np−in −1  Tr   p p−i −1 gi1 ,...,in X 1 1 · · · X np−in −1  Tr . =

gi1 ,...,in · φi1 ,...,in =



For a single g this is equivalent to f g =



p

p−i 1 −1

gi1 ,...,in X 1

p−i n −1

· · · Xn

 . Letting g

range over I , we get the equivalence to f ∈ I [ p] : I . In the transition we have replaced the action of R on F∗ R, indicated by ·, by its  effect in F∗ R. Proposition 7.1.28 Let w = (w1 , . . . , wn ) ∈ (N>0 )n be a weight vector. Then, for any g ∈ R, either Tr(initw (g)) = 0 or Tr(initw (g)) = initw (Tr(g)). Proof Given two vectors (u 1 , . . . , u n ), (v1 , . . . , vn ) ∈ Rn clearly we have: n  i=1

u i wi ≥

n  i=1

vi wi

⇐⇒

n  n    ui + 1 vi + 1 − 1 wi ≥ − 1 wi . p p i=1 i=1 (7.1)

 Recall that w(μ) = nj=1 w j u j for a monomial μ = X 1u 1 · · · X nu n , and that w( f ) = max{w(ν) :  ν ∈ supp( f )} for any f ∈ R, f = 0. s ai μi ∈ I , where μi = X u i1 · · · X u in ∈ Mon(R) and ai ∈ K \ {0}. Let g = i=1 w(g) and If Tr(initw (g)) = 0, then there exists i ∈ {1, . . . , s} such that w(μ  i ) = 1/ p u i j ≡ −1 (mod p) for all j = 1, . . . , n. In this case Tr(initw (g)) = k∈A ak Tr(μk )

7.1 F-purity

239

where A = {k ∈ {1, . . . , s} : w(μk ) = w(g) and u k j ≡ −1 for all j = 1, . . . , n}. By our assumption A is nonempty, i ∈ A.  indeed 1/ p On the other hand, Tr(g) = k∈B ak Tr(μk ) where B = {k ∈ {1, . . . , s} : u k j ≡ −1 for all j = 1, . . . , n}. Of course A ⊂ B ⊂ {1, . . . , s}. Furthermore, by (7.1)



 k ∈ B : w(Tr(μk )) is maximal = k ∈ B : w(μk ) is maximal = A.

So initw (Tr(g)) =



1/ p

k∈A

ak

Tr(μk ) = Tr(initw (g)).



Corollary 7.1.29 Let f ∈ R be such that there is a monomial order < with p−1 p−1 init< ( f ) = X 1 · · · X n . Then f  Tr is an F-splitting of R, and in< (I ) ⊂ R is a squarefree monomial ideal for any ideal I ⊂ R that is compatibly split with respect to f  Tr. Proof The fact that f  Tr is an F-splitting follows from Remark 7.1.25. So by Proposition 7.1.26, it is enough to show that in< (I ) is a compatibly split ideal p−1 p−1 with respect to X 1 · · · X n  Tr. Notice that in< (I ), as an R-submodule of F∗ R, is generated by finitely many monomials, say μ1 , . . . , μk . So to check that p−1 p−1 in< (I ) is compatibly split with respect to X 1 · · · X n  Tr it is enough to verp−1 p−1 ify that Tr(X 1 · · · X n μi ) ∈ in< (I ) for all i = 1, . . . , k. By definition of initial ideal, for any i = 1, . . . , k there is gi ∈ I such that init< (gi ) = in< (gi ) = μi . By Lemma 1.5.6, there exists a weight vector w ∈ (Nn>0 ) such that initw (gi ) = init< (gi ) for any i = 1, . . . , k and initw ( f ) = init< ( f ) (so initw (I ) = in< (I )). Then either p−1 p−1 Tr(X 1 · · · X n μi ) = 0 or, using Proposition 7.1.28, p−1

Tr(X 1

· · · X np−1 μi ) = Tr(initw ( f ) initw (gi )) = Tr(initw ( f gi ))

= initw (Tr( f gi ))) ∈ initw (Tr( f I )) ⊂ initw (I ) = in< (I ).



We conclude by noticing that to any F-splitting of R/I where I is an ideal of R contained in m = (X 1 , . . . , X n ), one can associate an element of Hom R (F∗ R, R) that is close to an F-splitting. Indeed, suppose that there is a homomorphism of R/I modules φ : F∗ (R/I ) → (R/I ) such that φ(1) = 1. Moreover, if  : R → R/I is a surjective map of R-modules, the function  : F∗ R → F∗ (R/I ) is a surjective map of R-modules as well. Therefore, since F∗ R is a free R-module, the homomorphism of R-modules φ : F∗ (R/I ) → R/I can be lifted to a homomorphism of R-modules θ : F∗ R → R. By what has been said, θ = f  Tr for some f ∈ R. Since θ(I ) ⊂ I , by Proposition 7.1.27 it follows that f ∈ I [ p] : I . Furthermore, φ(1) = 1 implies p−1 p−1 1 ∈ supp(θ(1)), that is X 1 · · · X n ∈ supp( f ) and its coefficient in f is 1. Hence we have proved: Proposition 7.1.30 Let R = K [X 1 , . . . , X n ] be a polynomial ring in n variables over a perfect field K of characteristic p. Given an ideal I ⊂ m, if R/I is F-split, then I [ p] : I ⊂ m[ p] . Similarly we have:

240

7 F-singularities of Determinantal Rings

Proposition 7.1.31 Let R = K [X 1 , . . . , X n ] be the polynomial ring in n variables over a perfect field K of prime characteristic p, I ⊂ R an ideal and A = R/I . [ p] Then the homomorphism of F∗ A-modules φ : I I [ p]:I → Hom A (F∗ A, A) sending f to f  Tr is an isomorphism. Proof We already know from Proposition 7.1.27 that φ is a well-defined map of F∗ A-modules. The same argument used for the proof of Proposition 7.1.27 shows that, given any f ∈ R, Tr( f R) ⊂ I if and only if f R ⊂ I [ p] , i.e. φ is injective. To show that φ is surjective, as explained above, any element of Hom A (F∗ A, A) can be lifted to an element of Hom R (F∗ R, R), that clearly maps I to I , so we conclude by using again Proposition 7.1.27. 

Fedder’s Criterion and the F-split Property of Determinantal Rings We remind the reader that when we speak of the grading of R = K [X 1 , . . . , X n ] we mean the g-grading for some vector g = (g1 , . . . , gn ) ∈ (N>0 )n (thus X i has degree gi ), unless we specify it otherwise. In this situation m denotes the unique maximal homogeneous ideal m = (X 1 , . . . , X n ). The following result is the graded version of Fedder’s criterion [125, Theorem 1.12]. Theorem 7.1.32 Let R = K [X 1 , . . . , X n ] be the polynomial ring in n variables over a perfect field K of prime characteristic p, and I ⊂ R be a homogeneous ideal. Then R/I is F-split if and only if I [ p] : I ⊂ m[ p] . Proof The “only if” part follows from Proposition 7.1.30. Now suppose there exists f ∈ (I [ p] : I ) \ m[ p] . We can take f homogeneous. Since f ∈ / m[ p] , there must exist a n p−1 monomial μ in the support of f dividing i=1 X i . If c ∈ K \ {0} is the coefficient n n p−1 p−1 of μ in f , substituting f with f · i=1 X i /cμ we can assume that i=1 Xi occurs in f with coefficient 1, that is: f =

n

i=1

p−1

Xi

+

s 

ci μi

i=1

n p−1 where ci ∈ K \ {0} and μi = i=1 X i . We claim that f  Tr is an F-splitting of R. s s n p−1 1/ p 1/ p + i=1 ci Tr(μi ) = 1 + i=1 ci Tr(μi ), Indeed, f  Tr(1) = Tr i=1 X i 0 for all i = 1, . . . , s. If Tr(μi ) = 0, so it is enough to show that Tr(μ n i ) =p−1 a multiple of X , which would imply that deg(μi ) > thenμi would be i=1 i  n p−1 , contradicting the fact that f is homogeneous. So f  Tr is an deg i=1 X i F-splitting of R.

7.1 F-purity

241

Moreover f  Tr(I ) = Tr( f I ) ⊂ I by Proposition 7.1.27, so f  Tr is a welldefined map of R/I -modules from F∗ (R/I ) to R/I , and f  Tr(1) = 1. In particular, R/I is F-split.  We can combine Theorem 7.1.32 and Corollary 7.1.29 as follows: Theorem 7.1.33 Let R = K [X 1 , . . . , X n ] be the polynomial ring in n variables over a perfect field K of positive characteristic p, and I ⊂ R be a homogeneous ideal. Then the following are equivalent: (a) R/I is F-pure; (b) R/I is F-split; p−1 p−1 (c) there exists f ∈ S such that f I ⊂ I [ p] and X 1 · · · X n ∈ supp( f ). p−1

p−1

If, furthermore, there exists a monomial order < on R such that X 1 · · · X n the initial monomial of f , then in< (I ) is a squarefree monomial ideal.

is

Proof The equivalence between (b) and (c) follows at once from Theorem 7.1.32, while the equivalence between (a) and (b) follows from Proposition 7.1.19 because R/I is F-finite. Finally, the last statement follows from Corollary 7.1.29.  Example 7.1.34 Consider the polynomial ring R = K [X 1 , X 2 , X 3 , X 4 ], equipped with the standard grading, over a perfect field K of positive characteristic p, f = (X 1 X 3 − X 22 ) p−1 (X 2 X 4 − X 32 ) p−1 , and choose lex as a monomial order.Then in( f ) = (X 1 X 2 X 3 X 4 ) p−1 . The ideal I = (X 1 X 3 − X 22 , X 1 X 4 − X 2 X 3 , X 2 X 4 − X 32 ) ⊂ R is such that f I ⊂ I [ p] (check it!), so R/I is F-split and in(I ) is a squarefree monomial ideal by Theorem 7.1.33. Indeed one can check that in(I ) = (X 1 X 3 , X 1 X 4 , X 2 X 4 ). Remark 7.1.35 The statements in Theorems 7.1.32 and 7.1.33 are true even without assuming that the base field K is perfect. This follows by Remark 7.1.23. Next we want to show that determinantal rings over a field of positive characteristic are F-pure. To this end, let X be an m × n matrix of indeterminates over a field K , assume m ≤ n and take a positive integer t ≤ m. We now introduce a variant of the product D that was introduced for Theorem 6.3.16 (see Fig. 6.1) by omitting all factors of size < t: Dt =

m−1

i=t

[m − i + 1 . . . m|1 . . . i] · [1 . . . i|n − i + 1 . . . n] ·

n−m+1

[1 . . . m|i . . . i + m − 1].

i=1

For a diagonal monomial order in(Dt ) divides the product X of all indeterminates in the matrix X . Therefore it is a squarefree monomial. Recall that by It we mean the ideal of R = K [X ] generated by the t-minors of X . In particular I1 = m = (X i j : i = 1, . . . , m, j = 1, . . . , n). Before proving that determinantal rings are F-split, we want to highlight the following crucial lemma: Lemma 7.1.36 For h = height It we have Dk ∈ It(h) for all k = 1, . . . , t.

242

7 F-singularities of Determinantal Rings

Proof This follows at once by Theorem 3.5.2, because  γt (Dk ) = (m − t + 1)(n − m + 1) + 2 m−t j=1 j = (m − t + 1)(n − m + 1) + (m − t + 1)(m − t) = (m − t + 1)(n − t + 1) and height It = (m − t + 1)(n − t + 1) (e.g., see Theorem 3.4.6). p−1

Theorem 7.1.37 If K is a field of characteristic p > 0. Then Dk m[ p] for all k, 1 ≤ k ≤ t. In particular, R/It is F-pure.

 [ p]

∈ (It

p−1

Proof As just observed, in(Dk ) is a squarefree monomial. Therefore Dk We use the localization R It . By Lemma 7.1.36, we have p−1

Dk

∈ (It R It )h( p−1) ,

: It ) \

∈ / m[ p] .

h = height It = dim R It .

Therefore, since It R It is generated by h elements, p−1

Dk

f ∈ (It R It )[ p]

for all f ∈ It .

 p−1  [ p] So, for any f ∈ It , there exists g ∈ R \ It such that Dk f g ∈ It . In particular p−1 [ p] (Dk f )g p ∈ It . Since F∗ R is a free R-module, we have [ p]

It

    [ p] : g p = It · F∗ R : g · F∗ R = (It : g) · F∗ R = (It : g)[ p] = It ,

where · stands for the multiplication on F∗ R by an element of R. The last equality p−1 [ p] p−1 / It . So Dk f ∈ It for any f ∈ It , that is, Dk ∈ holds because It is prime and g ∈ [ p] It : It . That R is F-pure follows immediately by Theorem 7.1.32 in the perfect case. If K is not perfect, take the perfect closure K  of K . We have shown that R/It ⊗ K K  is F-pure. Hence Corollary 7.1.22 implies that R/It is F-pure.   p−1  m Remark 7.1.38 Fix a diagonal monomial order. Since in D1 = i=1 n p−1 j=1 X i j , combining Theorems 7.1.37 and 7.1.33 we get an alternative proof, in positive characteristic, of the fact that in(It ) is a squarefree monomial ideal. Remark 7.1.39 An interesting new technique to prove that determinantal rings are F-pure has been developed by Seccia [278] and is based on the notion of Knutson ideals. Such a new technique allows to prove that any sum of ideals of minors insisting in an interval of rows and/or of columns defines an F-pure ring, and that its natural set of generators form a Gröbner basis.

7.2 F-regularity

7.2

243

F-regularity

From now on in this chapter we will work with Noetherian rings of prime characteristic p which are F-finite, unless we specify otherwise. This assumption could be avoided in most cases, but it would significantly increase the difficulty of the exposition. Our intention is to give an overview on what is known about the F-singularities of determinantal rings, and not to give an exhaustive account on F-singularities in general (even if we are trying to be as accurate as possible for the F-singularities we treat). For this reason we decided to use the assumption of F-finiteness, loosing some generality, but greatly gaining in the clarity of the exposition.

Strong F-regularity Let R be an F-finite ring of characteristic p. By R ◦ we mean the set of elements of R not belonging to any minimal prime of R. Before giving the definition of “strong F-regularity”, we need to iterate some basic definitions given in Sect. 7.1. For any positive integer e, F∗e R is the R-module defined as follows: (a) F∗e R = R as an additive group (in particular as a set); e (b) r · x = r p x for all r ∈ R and x ∈ F∗e R. In this way we can also consider the ring homomorphism F e : R → R as the following map of R-modules: F e : R → F∗e R, e

r → r p . Notice that F∗e R is a finitely generated R-module for all e ∈ N, since we are assuming that R is F-finite. Given an ideal I ⊂ R, the eth iterated Frobenius power e of I is I [ p ] = F e (I )R ⊂ R. Equivalently, for an arbitrary system of generators e pe pe r1 , . . . , rk of I , I [ p ] = (r1 , . . . , rk ) ⊂ R. Definition 7.2.1 The ring R is strongly F-regular if for every c ∈ R ◦ there exists e > 0 such that cF e : R → F∗e R splits as a map of R-modules. Equivalently, R is strongly F-regular if for every c ∈ R ◦ there exist e > 0 and θ ∈ Hom R (F∗e R, R) such that θ(c) = 1. Remark 7.2.2 It would be tempting to extend the definition of “strongly F-regular” to non-F-finite rings R of prime characteristic by asking that for every c ∈ R ◦ , there exists e > 0 such that cF e : R → F∗e R is a pure map of R-modules. This concept actually has been studied by Hashimoto [168], and a ring R satisfying it is called very strongly F-regular. The definition of strongly F-regular rings in the non-Ffinite case uses the notion of tight closure of a submodule in a given module, and

244

7 F-singularities of Determinantal Rings

we will briefly discuss it in the next subsection. Once the definitions are given it is not difficult to prove that “very strongly F-regular” implies “strongly F-regular”. Beyond the F-finite case, they are known to be equivalent notions for local rings by [168, Lemma 3.6]. We list some useful basic properties: Lemma 7.2.3 Let R be a ring of characteristic p. (a) If there exist c ∈ R ◦ and e > 0 such that the map cF e : R → F∗e R splits as a map of R-modules, then R is F-split. (b) Let s, t, e be positive integers and φ : F∗s R → F∗t R a map of R-modules. Then the function φ : F∗s+e R → F∗t+e R is a map of R-modules. As a consequence, the latter is a split-inclusion of R-modules if the former is so. (c) If there exist c ∈ R ◦ and e > 0 such that the map cF e : R → F∗e R splits as a   map of R-modules, then cF e : R → F∗e R splits as a map of R-modules for all e ≥ e. Proof Clearly, if f i : Mi → Mi+1 is a map of R-modules for i = 0, . . . , e and f e ◦ f e−1 ◦ · · · ◦ f 0 : M0 → Me+1 is a split-inclusion, then f 0 : M0 → M1 is a splitinclusion. Hence, for (a) we are done by putting Mi = F∗i R for all i = 0, . . . , e, Me+1 = F∗e R, f i = F for all i = 0, . . . , e − 1 and let f e be the multiplication by c on F∗e R.  Part (b) is straightforward to check. Concerning (c), notice that cF e : F∗e −e R →    F∗e R is a split-inclusion of R-modules by (b). Moreover, F e −e : R → F∗e −e R is   a split-inclusion of R-modules by (a). So their composition cF e : R → F∗e R is a split-inclusion of R-modules as well.  As an immediate consequence we get: Corollary 7.2.4 A strongly F-regular ring is F-split. The above corollary shows that strongly F-regular rings are F-split. On the other hand, if R is a local F-finte regular ring, Kunz’s theorem [52, Corollary 8.2.8] says that F∗e R is a free R-module for any e ∈ N. So, if c ∈ R ◦ , we can choose e ∈ N big enough such that c ∈ F∗e R is part of an R-basis of F∗e R. In this way cF e : R → F∗e R splits as a map of R-modules. Proposition 7.2.5 An F-finite regular ring is strongly F-regular. Proof In the local case we already explained it. In general the result follows by the fact that “strong F-regularity” is a local property, as we are going to show in the next result.  The following result says that “strong F-regularity” is a local property. This is perhaps the main motivation that convinced us to deal with the “strongly F-regular” concept instead of the “(weakly) F-regular” one, that is not known to localize (see the next subsection for a discussion).

7.2 F-regularity

245

Proposition 7.2.6 Let R be an F-finite ring of characteristic p. The following are equivalent: (a) (b) (c) (d)

R is strongly F-regular. T −1 R is strongly F-regular for any multiplicative system T ⊂ R. Rp is strongly F-regular for all prime ideals p ⊂ R. Rm is strongly F-regular for all maximal ideals m ⊂ R.

Proof (a) =⇒ (b): Let c ∈ (T −1 R)◦ be given. Then c = d/t with d ∈ R ◦ and t ∈ T . Choose e > 0 such that there exists α ∈ Hom R (F∗e R, R) such that α(d) = 1. Define β : T −1 (F∗e R) → T −1 R sending r/u to α(tr )/u for any r ∈ F∗e R, u ∈ T . It is straightforward to check that β is a well-defined map of T −1 R-modules sending e t p −1 d/t to 1. It is simple to check that φ : T −1 (F∗e R) → F∗e (T −1 R), sending r/u e to r/u p for any r ∈ F∗e R = R, u ∈ T , is an isomorphism of T −1 R-modules. Then θ = β ◦ φ−1 ∈ Hom T −1 R (F∗e (T −1 R), T −1 R) is such that θ(c) = 1. That is, T −1 R is strongly F-regular. (b) =⇒ (c) and (c) =⇒ (d) are obvious. ◦ for any maximal ideal m of R. Given (d) =⇒ (a): Let c ∈ R ◦ . Then c/1 ∈ Rm a maximal ideal m, choose e(m) ∈ N such that cF e(m) : Rm → F∗e(m) (Rm ) splits. e(m) e(m) Since F∗e(m) (Rm ) ∼ = (F∗ R)m as R-modules, there exists φ ∈ Hom Rm ((F∗ R)m , Rm ) such that φ(c/1) = 1. If F∗e(m) R is generated by x1 , . . . , xk as R-module, then φ is a well-defined map of Rn -modules from (F∗e(m) R)n to Rn for all maximal ideals n ∈ U := U (( f 1 f 2 · · · f k )) ⊂ Spec(R) where φ(xi /1) = yi / f i where yi ∈ R and f i ∈ R \ m. Since R is Noetherian, we can cover Spec(R) with finitely many such open neighborhoods U . Taking the maximum among the e(m)’s over the finitely maximal ideals m used to construct such a covering, call it e, we have that cF e : Rm → F∗e (Rm ) ∼ = (F∗e R)m splits for all maximal ideals m of R. So the map cF e : e R → F∗ R is a pure map of R-modules when localizing at any maximal ideal of R, hence it must be pure itself. Therefore, under our assumptions, the map cF e : R → F∗e R splits by Lemma 7.1.5. Because we started from an arbitrary c ∈ R ◦ , R is strongly F-regular. 

Tight Closure It is not our intent to discuss in this book the theory of tight closure, initiated by Hochster and Huneke in the 1990s with the paper [180], but we give some information for the interested reader. Throughout this subsection the ring R is Noetherian and has prime characteristic p, but is not assumed to be F-finite. Given an ideal I ⊂ R, the tight closure of I is the ideal I ∗ ⊂ R consisting of the e e elements r ∈ R such that there exists c ∈ R ◦ with cr p ∈ I [ p ] for all e  0. The ideal I is tightly closed if I = I ∗ . The ring R is weakly F-regular if every ideal is tightly closed. More generally, given an R-module M and an R-submodule N ⊂ M, the tight ∗ of N inside M is the R-submodule of M consisting of those elements closure N M m ∈ M such that there exists c ∈ R ◦ with

246

7 F-singularities of Determinantal Rings

 id F∗e R ⊗ι c ⊗ m ∈ Im F∗e R ⊗ R N −−−−→ F∗e R ⊗ R M for all e  0, where ι is the inclusion N ⊂ M. This definition is consistent with the one given above for ideals, in the sense that I R∗ = I ∗ for any ideal I ⊂ R. ∗ = N . It turns out that the The R-submodule N is called tightly closed in M if N M following conditions are equivalent (Hochster and Huneke [182, Proposition 8.7]): (a) R is weakly F-regular. (b) N is tightly closed in M for any finitely generated R-module M and any N ⊂ M. On the other hand, when R is F-finite the following are equivalent; see Hochster [177, p. 166]: (a) R is strongly F-regular. (b) N is tightly closed in M for any R-module M and any N ⊂ M. Let us prove the implication (a) =⇒ (b) in the latter equivalence. Fix an inclusion of R-modules ι : N ⊂ M, c ∈ R ◦ and m ∈ M such that  id F∗e R ⊗ι e e c ⊗ m ∈ Im F∗ R ⊗ R N −−−−→ F∗ R ⊗ R M for all e  0. Since R is strongly F-regular there exists θ ∈ Hom R (F∗e R, R) such that θ(c) = 1 for e big enough. Writing c ⊗ m = a1 ⊗ n 1 + · · · + ak ⊗ n k for some ai ∈ F∗e R and n i ∈ N , we have m = θ(c)m = θ(a1 )n 1 + · · · + θ(ak )n k . So N is tightly closed in M. The latter equivalence leads one to define the notion of “strongly F-regular” for all Noetherian rings of prime characteristic (not necessarily F-finite). A Noetherian ring R of prime characteristic is said to be strongly F-regular if N is tightly closed in M for any R-module M and any N ⊂ M. It is not known whether being “weakly F-regular” localizes, and this fact led to the following definition: R is F-regular if T −1 R is weakly F-regular for any multiplicative system T ⊂ R. This is reminiscent of an analogue question about regular rings finally settled independently by Serre and by Auslander and Buchsbaum in the 1950s: is the localization of a regular ring always regular? The answer is positive; see [52, 2.2.9]. Contrarily, the property of being “strongly F-regular” is known to localize in general (indeed it is even a local property, as we showed in the F-finite case in Proposition 7.2.6, see [168, Lemma 3.6]), so we have: R is strongly F-regular =⇒ R is F-regular =⇒ R is weakly F-regular. Conjecturally, the three notions are equivalent, but this is not known in general. Remarkably, the upstream more general question whether tight closure (of ideals) commutes with localization, which was hoped to be true for several years, turned out to have a negative answer by the work of Brenner and Monsky [29]. On the

7.2 F-regularity

247

other hand, when R is a positively graded algebra over a field, the three notions are known to agree, as proved by Lyubeznik and Smith [238]. Also, using the recent developments on the Minimal Model Program in prime characteristic, Aberbach and Polstra proved in [2] that “strongly F-regular” is equivalent to “F-regular” for rings of dimension at most 4 essentially of finite type over a field of characteristic p > 5. In the same paper can be found other references to results concerning this problem. As noticed in Remark 7.2.2, there is also a fourth notion, that of “very strongly F-regular”. It is easy to check that it implies “strongly F-regular”, but, even if it localizes, it is not known to be a local property, so it is not a fully desirable new member of the family.

Back to Strong F-regularity We go back to our assumption that R is F-finite. Proposition 7.2.7 If R is a strongly F-regular ring, then it is normal and Cohen– Macaulay. Proof At this point we allow ourselves a forward reference to the next section where we will discuss F-rationality. A strongly F-regular ring is F-rational, so it is normal by Proposition 7.3.2(a). Since R is F-finite, it is a quotient of a regular ring by Gabber [134, Remark 13.6]. So, R is Cohen–Macaulay by Proposition 7.3.2(d).  The following result is extremely useful because it says that, in order to show that a ring R is strongly F-regular, it is enough to check the splitting property only for some c: Theorem 7.2.8 Suppose that Rc is strongly F-regular for some c ∈ R ◦ . If there exists e > 0 such that the map cF e : R → F∗e R splits, then R is strongly F-regular. Proof First of all we notice that R is reduced by Lemma 7.2.3(a) since the map cF e : R → F∗e R splits. Thus c is a nonzerodivisor of R, and we can think of R as a subring of Rc . Let d ∈ R ◦ be given. Then d belongs to (Rc )◦ as well, so by the assumption there exists t > 0 and α ∈ Hom Rc (F∗t Rc , Rc ) such that α(d) = 1. Since F∗t R is a finitely s generated R-module, there exists s ∈ N such that c p α ∈ Hom R (F∗t R, R). So we s t ps get a map β ∈ Hom R (F∗ R, R) such that β(d) = c (simply choose β = c p α). By Lemma 7.2.3(b), β is also a map of R-modules from F∗e+s+t R to F∗e+s R, and we will think of β in this manner. On the other hand, combining statements (a) and (c) of Lemma 7.2.3, we know that F s : R → F∗s R is a split-inclusion of R-modules. Hence there exists s γ ∈ Hom R (F∗s R, R) such that γ(1) = 1, hence γ(c p ) = c. By Lemma 7.2.3(b), γ ∈ Hom R (F∗e+s R, F∗e R), and by assumption there is θ ∈ Hom R (F∗e R, R) sending c to 1. Then

248

7 F-singularities of Determinantal Rings

θ ◦ γ ◦ β ∈ Hom R (F∗e+s+t R, R) sends d to 1. That is, d F e+s+t : R → F∗e+s+t R splits. Since d is an arbitrary element of R ◦ , R is strongly F-regular.  As well as F-purity, strong F-regularity descends along pure maps: Lemma 7.2.9 Let φ : R → S be a map of F-finite Noetherian rings of characteristic p which is pure as a map of R-modules. If S is strongly F-regular, then R is so. Proof First we reduce to the case that S is a domain. Notice that it is harmless to assume that R is local, with maximal ideal m. Denoting by E the injective hull of R/m, since φ is a pure map of R-modules, the map φ E = φ ⊗ id E : E → S ⊗ R E is injective. In particular, if x is the generator of the R-module 0 : E m ∼ = R/m, we have φ E (x) = 0. By the hypothesis that S is strongly F-regular, S must be normal by Proposition 7.2.7, and so a product of domains. Say S = S1 × · · · × Sr . r Since φ E (x) = 0 in S ⊗ R E ∼ = i=1 (Si ⊗ R E), there exists i ∈ {1, . . . , r } such that π E,i (φ E (x)) = 0, where πi : S → Si is the natural projection and π E,i = πi ⊗ id E . Because E is an essential extension of R/m, the map π E,i ◦ φ E = (πi ◦ φ) ◦ id E : E → E ⊗ Si is injective. Therefore, by Remark 7.1.7 the map of R-modules πi ◦ φ : R → Si is pure. It remains to show that Si is strongly F-regular: for 0 = c ∈ Si , consider d = (1, . . . , c, . . . , 1) ∈ S (all entries 1 but to the ith which is c). Note that d ∈ S ◦ , so there exists e > 0 and θ ∈ Hom S (F∗e S, S) such that θ(d) = 1. Because F∗e S = F∗e S1 × · · · × F∗e Sr , the restriction of θ to F∗e Si is a map in Hom Si (F∗e Si , Si ) sending c to 1. So Si is strongly F-regular. Therefore we can assume that S (and so R) is a domain. Hence φ(c) ∈ S ◦ for any c ∈ R ◦ . Therefore if c ∈ R ◦ there exists e > 0 and θ ∈ Hom S (F∗e S, S) such that θ(φ(c)) = 1; in particular φ(c)F e : S → F∗e S is a pure map of S-modules. We notice that the function φ : F∗e R → F∗e S is a map of R-modules and, for any R-module M, consider the commutative diagram S ⊗R M

φ(c)F e ⊗id S⊗ R M

F∗e S ⊗ S (S ⊗ R M) ∼ = F∗e S ⊗ R M

φ⊗id M

φ⊗id M cF ⊗id M e

M

.

F∗e R ⊗ R M

Since both of the maps on the top and on the left hand side are injective, also the map on the bottom must be injective. So the map cF e : R → F∗e R is a pure map of R-modules, and since F∗e R is a finitely generated R-module, it must be a split-inclusion. That is, R is strongly F-regular.  Proposition 7.2.10 If R is an F-finite ring, then R is strongly F-regular if and only if R[X ] is strongly F-regular. Proof The “if part” follows by Lemma 7.2.9. For the “only if part”, using Proposition 7.2.6 we can assume that R is local. In particular, by Proposition 7.2.7 R, and so

7.2 F-regularity

249

R[X ], is a domain. Hence we must show that for any nonzero polynomial f ∈ R[X ] there exists e > 0 such that f F e : R[X ] → F∗e (R[X ]) splits as a map of R[X ]modules. If d is the degree of f and c ∈ R \ {0} its leading coefficient, choose e > 0 such that d < p e . Then there exists θ ∈ Hom R (F∗e R, R) with θ(c) = 1 (since R is strongly F-regular such an e exists by Lemma 7.2.3). For any i ∈ N set α(X i ) = 0 e + d with d 0. Fix a positive integer e. Then  : F∗e R → Hom R (F∗e R, R), f → f  Tr e : g → Tr e ( f g), is an isomorphism of F∗e R-modules. Moreover, given an ideal I ⊂ R, the following are equivalent: (a) ( f  Tr e )(I ) ⊂ I . e (b) f ∈ I [ p ] : I . Furthermore, given c ∈ R different from 0, f  Tr e is a splitting of cF e : R → (i.e., ( f  Tr e )(c) = 1) if and only if

F∗e R

pe −1

pe −1

(c) X 1 · · · Xn ∈ supp(c f ) and its coefficient in c f is 1. (d) If X 1u 1 · · · X nu n ∈ supp(c f ) and u 1 ≡ · · · ≡ u n ≡ −1 (mod p e ), then u i = p e − 1 for all i.

250

7 F-singularities of Determinantal Rings

Finally, if I ⊂ R is a homogeneous ideal and c ∈ R that is not in any minimal prime ideal of I , the following are equivalent: (e) The map cF e : R/I → F∗e (R/I ) is a split-inclusion. e e (f) c(I [ p ] : I ) ⊂ m[ p ] , where m is the unique maximal homogeneous ideal of R. Theorem 7.2.13 Let K be an F-finite field of characteristic p > 0, X an m × n matrix of indeterminates over K and 1 ≤ t ≤ min{m, n}. Then R = K [X ]/It is strongly F-regular. Proof By Lemma 7.2.9 it is harmless to assume that K is perfect. Let c = [1 . . . t − 1|n − t + 2 . . . n]. Then Rc is an F-finite regular ring by Theorem 3.4.6; so by Proposition 7.2.5, Rc is strongly F-regular. Therefore, by Theorem 7.2.8 it is enough to show that the map cF e : R → F∗e R splits for some positive integer e. By Propo[ pe ] [ pe ] sition 7.2.12, the latter condition is equivalent to c(It : It ) ⊂ I1 . It turns out p−1 [ p] that this is true even for e = 1: indeed, by Theorem 7.1.37, Dt ∈ (It : It ), so p−1 [ p] p−1 cDt ∈ (It : It ) as well. Moreover, for a diagonal monomial order, in(cDt ) = [ p] p−1 [ p] / I1 .  in(c) in(Dt ) p−1 is not in I1 , so cDt ∈

7.3

F-rationality and Gröbner Deformations

In this section we provide a different proof that determinantal rings are strongly F-regular. This proof requires to control the behaviour of the F-singularities via Gröbner and Sagbi deformations.

F-rationality Before going ahead, we must discuss another very important F-singularity. Again, throughout this section the ring R is Noetherian and has prime characteristic p, but is not assumed to be F-finite. Definition 7.3.1 The ring R is F-rational if every ideal I ⊂ R generated by height I elements of R is tightly closed. Of course, weakly F-regular rings, and thus strongly F-regular rings, are Frational. Under mild assumptions, F-rationality localizes, and it is even a local property. We summarize some important properties of F-rational rings: Proposition 7.3.2 Let R be a Noetherian ring of prime characteristic p. (a) If R is F-rational, then it is normal [52, Proposition 10.3.2]. (b) If R is a homomorphic image of a Cohen–Macaulay ring, then R is F-rational if and only if Rp is F-rational for every prime ideal p ⊂ R [52, Proposition 10.3.6, 10.3.10].

7.3 F-rationality and Gröbner Deformations

251

 (c) If R = i∈Z Ri is a Z-graded ring having a unique maximal homogeneous ideal, say m ⊂ R, and (R0 , m0 ) is a complete local ring, then R is F-rational if and only if Rm is F-rational. (d) If R is a homomorphic image of a Cohen–Macaulay ring and it is F-rational, then R is Cohen–Macaulay [52, Corollary 10.3.4]. (e) If R is a homomorphic image of a Cohen–Macaulay ring, it is local with maximal ideal m and x ∈ m is an R-regular element such that R/x R is F-rational, then R is F-rational [52, Proposition 10.3.11]. (f) If R is Gorenstein, then it is F-rational if and only if it is strongly F-regular ([52, Cororollary 10.3.7] and Hochster and Huneke [181, Theorem 3.1.(f)]). Proof We just prove (c), since the other items are classical. Notice that under the assumptions of (c) R is a homomorphic image of a Cohen–Macaulay ring, so the “only if” direction follows from (b). For the other direction we start by noting that R is a domain since Rm is a domain by (a). First let us assume that R0 is infinite. Then also the multiplicative group R0 \ m0 is infinite (otherwise m0 would be infinite, and the injective function m0 → R \ m0 sending x to 1 + x would contradict the finiteness of R \ m0 ). Since R is reduced and finitely generated over the excellent local ring R0 , Velez [312, Theorem 3.5] says that U = {p ∈ Spec R : Rp isF-rational} is an open subset of Spec R. Let I ⊂ R be the radical ideal such that V (I ) = {p ∈ Spec R : p ⊃ I } is the complement Spec R \ U . Since m ∈ U , we are done once we show that I is homogeneous. Consider the action (R0 \ m0 ) × R → R defined by λ · f = λd f whenever f ∈ Rd , extended by additivity. Because R0 \ m0 is infinite, it can be easily checked that I is homogeneous if and only if it is stable under this action. Let us see that this is indeed true: of course x ∈ I if and only if Rx is Frational. Because φλ : R → R sending f to λ · f is an automorphism of R for all λ ∈ R0 \ m0 , Rx is F-rational if and only if Rλ·x is F-rational for all λ ∈ R0 \ m0 . Hence, if x ∈ I , then λ · x ∈ I for any λ ∈ R0 \ m0 , and this concludes the proof in the case in which R0 \ m0 is infinite. If R0 is finite, then being a domain, it must be a perfect field. Hence R0 → L is a separable extension where L is an algebraic closure of R0 . Consider R  = R ⊗ R0 L. Then Rm → Rn , where n is the only maximal homogeneous ideal of R  , is a faithfully flat smooth extension, so by [312, Theorem 3.1], Rn is F-rational. Now, since L = R0 is infinite, by what has previously been said, R  is F-rational, and therefore R is Frational by Lemma 7.3.3 below.  We remark that Proposition 7.3.2(e), which says that deformation holds for Frationality, is not true if one replaces “F-rational” by “strongly F-regular” (or “Fsplit”). The first example is given in Singh [286, Theorem 1.1]. The deformation for F-rationality, together with Proposition 7.3.2(c), will be needed to prove that F-rationality “Gröbner deforms”. To this task we also need the following lemma, saying that F-rationality descends along faithfully flat maps.

252

7 F-singularities of Determinantal Rings

Lemma 7.3.3 Let R → S be a faithfully flat map of Noetherian rings of characteristic p. If S is F-rational, then so is R. Proof Let I ⊂ R be an ideal generated by h = height I elements of R. So I S is a proper ideal of S (by faithfully flatness) generated by h elements. Hence height I S ≤ h by Krull’s Hauptidealsatz. On the other hand, since for each minimal prime ideal p of I S we have p ∩ R ⊃ I , and the going-down theorem holds between R and S [243, Theorem 9.5], height I S ≥ h. Hence I S is tightly closed, that is, (I S)∗ = I S. So, using the trivial inclusion I ∗ S ⊂ (I S)∗ , Proposition 7.1.10 and Remark 7.1.9, we get I ∗ = I ∗ S ∩ R ⊂ (I S)∗ ∩ R = I S ∩ R = I. Hence I is tightly closed, and so R is F-rational.



Remark 7.3.4 Unlike F-purity and strong F-regularity (see Lemmas 7.1.21 and 7.2.9), F-rationality does not descend to pure maps. An example is given by Watanabe [315, Ex. 3.3(3)]. This fact, in contrast to the already mentioned one that F-rationality, unlike strong F-regularity, does deform, makes the interplay between these two properties fascinating. Hopefully, this will be more clear later, when we give a second proof that determinantal rings are strongly F-regular.

F-rational Type and Rational Singularities A last fact we want to mention about F-rationality is a fundamental result of Smith and Hara relating it with the concept of having rational singularities in characteristic 0, which is a very important kind of singularity in birational geometry. For its definition we refer the reader to Kempf et al. [205]. Before stating the result we need the following definition: Definition 7.3.5 Let K be a field of characteristic 0 and R a finitely generated K -algebra. We say that R is of F-rational type if there exist a finitely generated Z-algebra A and a finitely generated A-algebra R A such that A → R A is flat and (a) (A → R A ) ⊗ A K is isomorphic to K → R, (b) R A ⊗ A A/m is F-rational for every maximal ideal m ⊂ A in a dense open subset of Spec A. Remark 7.3.6 Let us add some comments to Definition 7.3.5. (a) Since A is a finitely generated Z-algebra, A/m is a finite field for any maximal ideal m ⊂ A. In particular, A/m is a perfect field of positive characteristic. (b) It is always possible to choose A and R A satisfying condition (a). In fact, if R = K [X 1 , . . . , X n ]/I , fix a monomial order < on K [X 1 , . . . , X n ] and a Gröbner basis f 1 , . . . , f m of I such that inic( f i ) = 1 for all i = 1, . . . , m (e.g. the reduced Gröbner basis of I ) and set

7.3 F-rationality and Gröbner Deformations

253

A = Z[α : α occurs among the coefficients of the f i ] ⊂ K . Then R A = A[X 1 , . . . , X n ]/( f 1 , . . . , f m ) is a free A-module with basis the monomials not divisible by any in( f i ): to see this, one can argue as in Proposition 1.2.6, since every polynomial of A[X 1 , . . . , X n ] has a reduction modulo f 1 , . . . , f m ∈ A[X 1 , . . . , X n ] (the algorithm proposed in Proposition 1.2.4 works because inic( f i ) = 1 for all i). Finally, by construction R A ⊗ A K ∼ = R. (c) It can be shown that the notion of being of F-rational type does not depend on the choice of A. See Remark 7.3.8. (d) In particular, with the notation of (b), if I has a reduced Gröbner basis f 1 , . . . , f m ∈ Z[X 1 , . . . , X n ], then R is of F-rational type if and only if (Z/ pZ)[X 1 , . . . , X n ]/( f 1 , . . . , f m ) is F-rational for all but finitely many prime numbers p. (The necessity follows from (c).) Theorem 7.3.7 Let R be a finitely generated algebra over a field of characteristic 0. The following are equivalent: (a) R has only rational singularities. (b) R is of F-rational type. Remark 7.3.8 In Theorem 7.3.7 Smith proved (b) =⇒ (a) in [288], and Hara proved (a) =⇒ (b) in [162]. Actually, Hara proved something more: if R has only rational singularities, then for any finitely generated Z-algebra A and for any finitely generated A-algebra R A such that A → R A is flat and (A → R A ) ⊗ A K is isomorphic to K → R, we have that R A ⊗ A A/m is F-rational for every maximal ideal m ⊂ A in a dense open subset of Spec A. This shows, as we anticipated in Remark 7.3.6(c), that the definition of being of F-rational type does not depend on the choice of A and of R A . We remark that, contrarily, for a fixed prime number p, the property that R A ⊗ A A/m is F-rational may depend on the chosen maximal ideal m ⊂ A containing p. For example, let R = C[X ]/(X 2 + 2X + 2i). Since R ∼ = C × C, it has only rational singularities, and so it is of F-rational type by Theorem 7.3.7. On the other hand, we can choose A = Z[i] and R A = A[X ]/(X 2 + 2X + 2i). If p = 5, there are two maximal ideals of A containing p, namely m1 = (i + 2) and m2 = (i − 2). We have R A ⊗ A A/m1 ∼ = R A ⊗ A A/m2 ∼ =

(Y +

2, Y 2

Z[Y, X ] (Z/5Z)[X ] ∼ , = 2 + 1, X + 2X + 2Y ) X 2 + 2X + 1

(Y −

2, Y 2

Z[Y, X ] (Z/5Z)[X ] ∼ . = 2 + 1, X + 2X + 2Y ) X 2 + 2X − 1

Hence R A ⊗ A A/m1 , being not reduced, is not F-rational, while R A ⊗ A A/m2 is F-rational because it is a field.

254

7 F-singularities of Determinantal Rings

F-singularities and Gröbner Deformations Now we want to study the problem of which kind of F-singularities passes from R/ inw (I ) to R/I . Bad news is that both “F-pure” and “strongly F-regular” do not belong to this class: Example 7.3.9 Set R = K [X 1 , . . . , X 5 ] where K has characteristic p > 2, and let I be the ideal generated by the 2-minors of the matrix: 

X 42 + X 53 X 3 X2 . X1 X 42 X 34 − X 2

Note that, if deg(X 4 ) = 3, deg(X 1 ) = deg(X 3 ) = 6, deg(X 2 ) = 24 and deg(X 5 ) = 2, the ideal I is homogeneous. By [286, Proposition 4.5] R/I is not strongly Fregular (if p > 3, R/I is not even F-pure). However, considering the weight vector w = (4, 16, 4, 2, 1), we leave as an exercise to the reader to prove that inw (I ) is the ideal of 2-minors of the matrix:  2 X2 X4 X3 . X 1 X 42 X 34 − X 2 By [286, Proposition 4.3] R/ inw (I ) is strongly F-regular (in particular R/ inw (I ) is F-pure). Fortunately there is also good news: F-rationality does belong to this privileged class. To prove it we show the analog statement we already proved for normality in Proposition 1.6.3. Proposition 7.3.10 Let R be a positively graded finitely generated algebra over a field of positive characteristic. Let t be a homogeneous degree 1 nonzerodivisor of R. Then F-rationality ascends from R/(t) to R and descends from R to R/(t − 1). Proof Assume that R/(t) is F-rational. If we denote the maximal homogeneous ideal of R by m, then Rm /t Rm is F-rational by Proposition 7.3.2(b). Hence Rm is F-rational by Proposition 7.3.2(e). Since R is a positively graded algebra over the field R0 , then R is F-rational by Proposition 7.3.2(c). This proves that F-rationality ascends from R/(t) to R. To see that F-rationality descends from R to R/(t − 1), note that by Proposition 7.3.2(b), Rt is F-rational. Since Rt ∼ = R  [X, X −1 ], where R  = R/(t − 1) (we already noticed this isomorphism, for example, in the proof of Proposition 1.6.2), and the inclusion R  → R  [X, X −1 ] is faithfully flat, we conclude that R/(t − 1) = R  is F-rational from Lemma 7.3.3.  Theorem 7.3.11 Let R = K [X 1 , . . . , X n ] be a polynomial ring over a field K of positive characteristic and w ∈ Nn>0 . If I ⊂ R is an ideal such that R/ inw (I ) is F-rational, then R/I is F-rational.

7.3 F-rationality and Gröbner Deformations

255

Proof Set S = R[t]/ homw (I ). Then S is a positively graded finitely generated algebra over a field of positive characteristic, t is a homogeneous degree 1 nonzerodivisor of S, S/t S ∼ = R/I (for all this see Proposition 1.6.1). = R/ inw (I ) and S/(t − 1) ∼ Hence we can immediately conclude by Proposition 7.3.10.  Remark 7.3.12 In this book we did not discuss F-injective rings, and at least we want to mention some properties of this class of rings within a remark: If I is an ideal of a Noetherian ring R of prime characteristic, the local cohomology functor with support in I induces a map of R-modules F : HIi (R) → HIi (F∗ R) for all i ∈ N. As Abelian groups, it is not difficult to check that HIi [ p] (R) ∼ = HIi (F∗ R), hence, since [ p] I and I share the same radical, we have the following Frobenius action on local cohomology: F : HIi (R) → HIi (R). If R is F-split, then F : HIi (R) → HIi (F∗ R) splits as a map of R-modules. In particular, F : HIi (R) → HIi (R) is injective. The latter fact turned out to be very powerful since the works of Hochster and Roberts [185] and of Mehta and Ramanathan [244], so it has been natural to introduce the following definition: a Noetherian ring R of prime characteristic is F-injective if the map F : Hmi (R) → Hmi (R) is injective for any maximal ideal m ⊂ R and i ∈ N. While we have already noticed after Proposition 7.3.2 that the “F-split” property does not deform, it is still an open problem whether the “F-injective” property deforms. Quite surprisingly, it has been recently shown by Horiuchi et al. [190] that the “F-split” property deforms to “F-injectivity”; in particular, a similar argument to the one used to prove Theorem 7.3.11, shows that if I is an ideal of the polynomial ring R = K [X 1 , . . . , X n ] over a field K of positive characteristic such that in(I ) is a squarefree monomial ideal for some monomial order, then R/I is F-injective (see [216] for the details). In particular we could have inferred from Theorem 4.3.2, that a determinantal ring of prime characteristic is F-injective (of course by now we know much more, i.e. that determinantal rings are strongly F-regular). More generally, any algebra with straightening law (see Remark 3.4.3 for the definition) defined over a field of positive characteristic is F-injective, since by the definition it is easy to prove that the defining ideal of an ASL admits a squarefree initial ideal. We end the remark noticing that, unfortunately, one cannot deduce that R/I is F-split if in(I ) is a squarefree monomial ideal for some monomial order: consider the same ideal I of Example 7.3.9 and lex (or revlex) as monomial order. Somehow even more surprisingly, one can find examples of ASL defined over a field of positive characteristic which are not F-pure: In [285] is proved that the ring A = K [X 1 .X 2 , X 3 , X 4 ]/ X 1 X 2 , X 1 X 3 , X 2 (X 3 − X 42 ) is not F-pure for every field K of positive characteristic. However, letting deg X 1 = deg X 2 = deg X 4 = 1, deg X 3 = 2 and  the partially order set {X 1 .X 2 , X 3 , X 4 } with X 4 < X 1 , X 2 , X 3 and X 1 , X 2 , X 3 incomparable between each other, one can see A is an ASL on . Corollary 7.3.13 Let R be a K -subalgebra of the polynomial ring K [X 1 , . . . , X n ] where K is a field. Fix a monomial order < and suppose that in< (R) is a finitely generated normal domain.

256

7 F-singularities of Determinantal Rings

(a) If K has positive characteristic, R is F-rational. (b) If K has characteristic 0, R has only rational singularities. Proof Let r1 , . . . , rk ∈ R be a Sagbi basis of R. By Proposition 6.1.3 in< (R) = K [in< (r1 ), . . . , in< (rk )] = K [M] where M is a normal submonoid of Nn . By Lemma 1.5.6 there exists a weight vector w ∈ Nn>0 such that in< (R) = inw (R). Let P = K [Y1 , . . . , Yk ] be the polynomial ring in k variables over K . By Lemma 1.5.3, if φ : P → R and ψ : P → initw (R) are the maps of K -algebras defined by φ(Yi ) = ri and ψ(Yi ) = initw (ri ), then there exists a weight vector v ∈ Nk>0 such that inv (Ker φ) = Ker ψ. Let us call I = Ker φ: in particular we have that P/I ∼ = = R and P/ inv (I ) ∼ K [M], hence inv (I ) = (μ−ν : μ, ν ∈ Mon(P), ψ(μ) = ψ(ν)); see Bruns and Gubeladze [50, Proposition 4.26]. To show (a), first of all by Lemma 7.3.3 it is harmless to assume that K is perfect. Since P/ inv (I ) is a normal monoid ring over a perfect field, it is strongly F-regular, and hence F-rational, by Corollary 7.2.11. Now Theorem 7.3.11 lets us conclude. For (b), we can refine the partial order on the monomials of P given by v ∈ Nk>0 by considering a monomial order 0. Then E # E  and E ⊗ K E  are strongly F-regular. Proof The Segre product is a direct summand of the tensor product: E ⊗ K E  = E # E  ⊕ N where N is the (direct) sum of all K -vector spaces L ρ ⊗ σL with ρ = σ. Since strong F-regularity passes to direct summands by Lemma 7.2.9, it is enough to discuss the tensor product. Both E and E  are subalgebras of polynomial rings P and P  , respectively, in which they have normal affine monoid rings as initial algebras; see Lemma 6.2.16. In this situation Proposition 1.3.17 says that the formation of the tensor product commutes with taking initial algebras, and the tensor product of the initial algebras is again normal: it is the algebra defined by the direct sum of the normal monoids which is normal in its turn. Now we can apply Theorem 7.3.13 which implies that E ⊗ K E  is F-rational. This ring is factorial, and therefore Gorenstein. We have not stated this fact so far, but the reader can prove it by the technique applied in the proof of Theorem 6.2.17, using that E is factorial by that theorem. Together with F-rationality, Gorensteinness implies strong F-regularity. See Proposition 7.3.2(f).  Let us return to R = K [X ] with an m × n matrix X of indeterminates, m ≤ n. In order to cover the case of Rees algebras and similar constructions, we extend the action of G to a polynomial extension R[T1 , . . . , T p ] by letting G act trivially on by shape monomials in T1 , . . . , T p . Also the notion of K -vector subspace defined is extended “trivially”: V ⊂ R[T1 , . . . , TP ] is defined by shape if V = μ Vμ · μ where the sum runs over the monomials in T1 , . . . , T p , and every Vμ is a subspace of R defined by shape. The group U in the following theorem is defined as in Sect. 3.6: U = U − (m, K ) × U + (n, K ). Theorem 7.4.5 Let K be a field of positive characteristic and R = K [X ]. Consider a subalgebra A ⊂ R[T1 , . . . , T p ] and an ideal I ⊂ A, both defined by shape. Let U be the algebra of U-invariants of A/I . If U is a normal monoid domain, then A/I is F-rational (and therefore Cohen–Macaulay and normal).

7.4 Equivariant Deformations

263

Proof Let (Jk ) be the family of ideals defining the filtration F extended to R[T1 , . . . , T p ]. Then we set Jk = Jk ∩ A and Jk = (Jk + I )/I . The filtration G = (Jk ) on A/I is again separated and Noetherian, so that it is enough that gr G (A/I ) is F-rational. We claim hat   gr G (A/I ) ∼ = U # (E # E  )[T1 , . . . , T p ]

(7.4)

where the outer Segre product on the right hand side is taken with respect to the Zm × Z p -grading in which Zm presents the weights of bitableaux and Z p the multidegrees with respect to the monomials in T1 , . . . , T p . For the proof one proceeds in principle as for that of Proposition 7.4.2. We use Theorem 3.3.1 in its reformulation, Theorem 3.6.2, and the description of the U-invariants in Theorem 3.6.6. Both sides of (7.4) are Zm × Z p -graded vector spaces so that we can prove their vector space isomorphism comparing them degree by degree. Let us fix a graded component. On the left it coincides with the same graded component of A/I , and that has a basis of products μ where the monomial μ in T1 , . . . , T p is fixed by the Z p -degree and  varies over all standard bitableaux whose weight is given by the Zm -degree. This holds because both A and I are defined by shape, and split into their Z p -graded components within which they are vector spaces defined by shape. Also U is Zm × Z p -graded, and its nonzero graded components live in the same degrees are as those of A/I or, equivalently, in gr G (A/I ). These have dimension 1 as K -vector spaces. Thus in the Segre product each nonzero component of U simply selects the subspace of the same degree in (E # E  )[T1 , . . . , T p ]. Since A and I are defined by shape, all products μ of the given Zm × Z p -degree live on both sides. It remains to compare the multiplication tables for proving that both sides are isomorphic K -algebras. Let us look at a product 1 μ1 · 2 μ2 , first on the left hand side. In A we must apply the straightening law to 1 2 and multiply the result by μ1 μ2 . Modulo I , all standard bitableaux whose products with μ1 μ2 belong to I drop out, and if we take the associated graded vector space, then all standard bitableaux whose weight is not (1 ) + (2 ) are suppressed. On the right hand side, standard bitableaux of “wrong” shape are not produced by the straightening in E # E  , and those whose products with μ1 μ2 are in I , are cut away by theSegre product with U. In view of Theorem 7.4.3 it remains to show that U # (E # E  )[T1 , . . . , T p ] is F-rational. We claim that it is even strongly F-regular. There are several direct summands visible: (1) The Segre products are direct summands of the corresponding tensor products. (2) U is a direct summand of a polynomial extension of K since it is a normal monoid domain (see Theorem 6.1.4).   In total this implies that U # (E # E  )[T1 , . . . , T p ] is a direct summand of a polynomial extension of E # E  (or of E ⊗ K E  ). Since strong F-regularity descends to direct summands (Lemma 7.2.9) and ascends to polynomial extensions (Proposition 7.2.10), we conclude by applying Proposition 7.4.4.  Remark 7.4.6 The theorems on Cohen–Macaulayness and normality of the preceding chapters can be derived from Theorem 7.4.5:

264

7 F-singularities of Determinantal Rings

(a) The minors [1 . . . k | 1 . . . k], k ≤ r , generate the algebra of U-invariants of Rr +1 (X ). So it is the polynomial ring over K in r indeterminates. This gives us a new, combinatorially simple proof of the Cohen–Macaulayness of the determinantal rings. (b) The description of the various types of Rees algebras in Sect. 6.3 in terms of the valuations γk shows that these algebras are defined by shape, and so the statements on Cohen–Macaulayness and normality follow from Theorem 7.4.5. (c) The same applies to the algebra At of t-minors and its residue class rings At /(pi + q j ). Remark 7.4.7 The crucial point in the proof of Theorem 7.4.5 is the use of a property that descends to direct summands. The Hochster–Roberts theorem [184] states that a graded affine algebra R over a field K that is a direct summand of a polynomial ring S over K is Cohen–Macaulay. More generally, it is enough to assume that R is a pure subalgebra of S; see, [52, 10.5]. Even if we would restrict ourselves to discussing Cohen–Macaulayness, the Hochster–Roberts theorem would not solve our problem since the assumption on the “ambient” ring is too strong: it was noticed by Chow [76] that Segre products of Cohen–Macaulay rings are not Cohen–Macaulay in general. (See Goto and Watanabe [151] for a discussion of the Cohen–Macaulayness of Segre products.) The Hochster–Roberts theorem was strengthened in characteristic 0 by Boutot [27]: an affine algebra over a field of characteristic 0 that is direct summand (or just a pure subring) of an affine algebra with rational singularities has rational singularities. The optimal replacement of rational singularities in positive characteristics is the notion of (strong) F-regularity, as we have seen. Remark 7.4.8 The group actions that are relevant for us are multiplicity free: every irreducible representation occurs with multiplicity ≤ 1. This makes the ring of U invariants a monoid ring. In the light of the preceding remark it is clear that one can generalize Theorem 7.4.5 to more general group actions if the ring of U -invariants has rational singularities (char K = 0) or is strongly F-regular (char K = p > 0). A general version of Theorem 7.4.5 was proved by Brion in his thesis [30] and made public by Kraft [219]: we refer the reader to Kraft’s book for an extensive discussion of the theory of U -invariants. The transfer to positive characteristics was developed by Grosshans [159].

7.5

F-pure Thresholds

In this section we study the F-pure thresholds (corresponding to the log-canonical thresholds in characteristic 0), of some determinantal objects the continue to consider Noetherian rings of prime characteristic p that are F-finite. In particular in what follows R is such a ring.

7.5 F-pure Thresholds

265

Definition 7.5.1 Let a be an ideal of R such that a ∩ R ◦ = ∅. Given a real number t, in the following the symbol at is a shorthand for the pair (a, t). (a) For a real number t ≥ 0, the pair (R, at ) is F-pure if for all e  0 there exists e c ∈ at ( p −1) such that the map cF e : R → F∗e R splits as a map of R-modules. (b) Suppose that R is F-pure. Then the F-pure threshold fpt R (a) of a is defined as fpt R (a) = sup{t ∈ R≥0 : (R, at ) isF-pure}. Remark 7.5.2 Notice that the following are equivalent: (a) (b) (c) (d) (e)

R is F-pure. The pair (R, R t ) is F-pure for every t ≥ 0. The pair (R, R t ) is F-pure for some t ≥ 0. There is an ideal a ⊂ R such that the pair (R, at ) is F-pure for some t ≥ 0. There exists an ideal a ⊂ R such that fpt R (a) ≥ 0.

(a) =⇒ (b). For Noetherian F-finite rings being F-pure is equivalent to being F-split by Proposition 7.1.19. So for any e > 0, it is enough to choose c = 1 ∈ e R t ( p −1) = R by Lemma 7.2.3(c). (b) =⇒ (c) =⇒ (d) ⇐⇒ (e) are obvious. (d) =⇒ (a). If there exist t ≥ 0, q = p e and c ∈ at (q−1) such that cF e : R → F∗e R splits as a map of R-modules, then R is F-split by Lemma 7.2.3(a). The above remark can be interpreted as the fact that F-pure thresholds measure “how much a ring of positive characteristic is F-pure”. Let R = K [X 1 , . . . , X n ] be the polynomial ring in n variables over a perfect field K of characteristic p > 0 and I ⊂ R be a homogeneous ideal. Given e ∈ N and a proper homogeneous ideal a ⊂ R containing I and not contained in any minimal prime of I , we define νeI (a) ∈ N ∪ {−∞} by

e e  νeI (a) := sup r ∈ N : ar (I [ p ] : I ) ⊂ m[ p ] . The following result is an analogue of Fedder’s criterion Theorem 7.1.32. Proposition 7.5.3 With the notation just introduced, if A = R/I we have: (a) For every real number t ≥ 0, the pair (A, (aA)t ) is F-pure if and only if νeI (a) ≥ ( p e − 1)t for every e  0. 

(b) If A is F-pure, the sequence of real numbers νeI (a)/ p e e∈N is nondecreasing. In particular fpt A (aA) = lime→∞ νeI (a)/ p e . Proof Concerning (a), the pair (A, (aA)t ) is F-pure if and only if for every e  0 e there exists c ∈ at ( p −1) such that the map cF e : A → F∗e A splits as a map of Amodules, where c ∈ A = R/I denotes the residue class of c. By Proposition 7.2.12, e e this is the case if and only if for all e  0 there exists c ∈ at ( p −1) such that c(I p : e I ) ⊂ m[ p ] . Finally, this is the case if and only if νeI (a) ≥ ( p e − 1)t for all e  0.

266

7 F-singularities of Determinantal Rings

As for (b), first we claim the following: for g ∈ R and an ideal J ⊂ R, if g p (I [ p] : I ) ⊂ J [ p] then g ∈ J . Indeed, since A is F-pure, by Theorem 7.1.33 there exists p−1 p−1 f ∈ I [ p] : I such that X 1 · · · X n ∈ supp( f ). Of course we can choose f homop−1 p−1 geneous and such that X 1 · · · X n occurs with coefficient 1 in it. Write g p f = p p h 1 a1 + · · · + h m am where J = (h 1 , . . . , h m ). Then we have p

g = g Tr( f ) = Tr(g · f ) = Tr(g p f ) = Tr(h 1 a1 + · · · + h mp am ) = Tr(h 1 · a1 + · · · + h m · am ) = h 1 Tr(a1 ) + · · · + h m Tr(am ) ∈ J. Next let νe = νeI (a). By what has just been proved, aνe (I [ p ] : I ) ⊂ m[ p ] =⇒ (aνe )[ p] (I [ p ] : I )[ p] (I [ p] : I ) ⊂ m[ p e

e

e

e+1

]

.

Because aνe p (I [ p

e+1

]

: I ) ⊃ (aνe )[ p] (I [ p

e+1

]

: I [ p] )(I [ p] : I )

= (aνe )[ p] (I [ p ] : I )[ p] (I [ p] : I ), e

where the equality follows because F∗ R is a free R-module, we get νe+1 ≥ pνe . So the sequence of real numbers {νeI (a)/ p e }e∈N is nondecreasing. In particular by part  (a) we infer that fpt A (aA) = lime→∞ νeI (a)/ p e . Remark 7.5.4 With the notation of Proposition 7.5.3, if R is furthermore standard graded and a = m = (X 1 , . . . , X n ), then

 e e e νeI (m) = sup r ∈ N : (I [ p ] : I ) ⊂ m[ p ] + mn( p −1)+1−r . / Indeed, if mr (I [ p ] : I ) ⊂ m[ p ] , then there is f ∈ I [ p ] : I and g ∈ mr such that g f ∈ e e e e e / mn( p −1)+1−r . m[ p ] . Since mn( p −1)+1 ⊂ m[ p ] , g f is not in mn( p −1)+1 , and hence f ∈ e On the other hand, if f ∈ I [ p ] : I is a homogeneous polynomial which is not in e e e m[ p ] + mn( p −1)+1−r , then there is a monomial α ∈ supp( f ) such that α is not in m[ p ] e e p −1 p −1 and has degree less than or equal to n( p e − 1) − r . Setting μ = X 1 · · · X n /α r [ pe ] [ pe ] : I) \ m . we have deg(μ) ≥ r , and so μ f ∈ m (I e

e

e

The F-pure Threshold fpt R (It ) Let X be an m × n matrix of indeterminates over a perfect field K of characteristic p > 0, assume m ≤ n and take a positive integer t ≤ m. Set R = K [X ] and It the ideal of R generated by the t-minors of X . Next, we aim at computing fpt R (It ). Theorem 7.5.5 The F-pure threshold of It ⊂ R is

7.5 F-pure Thresholds

267

 fpt R (It ) = min

 (m − k)(n − k) : k = 0, . . . , t − 1 . t −k

Before proving the theorem we need the following lemma. Lemma 7.5.6 Let k be the least integer in the set {0, 1, . . . , t − 1} such that (m − k)(n − k) (m − k − 1)(n − k − 1) ≤ ; t −k t −k−1 interpreting the quotient of a positive integer and zero as infinity, such a k indeed exists. Set u = t (m + n − 2k) − mn + k 2 . Then t − k − u ≥ 0. Moreover, if k is nonzero, then t − k + u > 0. If k = 0, then t (m + n − 1) ≤ mn. Proof Rearranging the inequality above, we have t (m + n − 2k − 1) ≤ mn − k 2 − k, which gives t − k − u ≥ 0. If k is nonzero, then the minimality of k implies that t (m + n − 2k + 1) > mn − k 2 + k, equivalently, that t − k + u > 0. If k = 0, the inequality is readily verified.



Proof of Theorem 7.5.5 We first show that for each k with 0 ≤ k ≤ t − 1, one has fpt(It ) ≤

(m − k)(n − k) . t −k

Let δk and δt be minors of size k and t respectively. Notice that δkt−k−1 δt is a product of t − k minors and has degree (k + 1)(t − k), so Remark 3.5.11 implies that t−k , δkt−k−1 δt ∈ Ik+1 t−k . By the Briançon– and hence that δkt−k−1 It is contained in the integral closure of Ik+1 Skoda theorem, see, for example, [52, Theorem 10.2.6], there exists an integer N such that  t−k−1  N +l (t−k)(l+1) It ∈ Ik+1 δk

for each natural number l (it is enough to take N to be the minimal number of t−k ). Localizing at the prime ideal Ik+1 of R, one has generators of Ik+1 (t−k)(l+1) R Ik+1 ItN +l R Ik+1 ⊂ Ik+1

for each l ≥ 1,

268

7 F-singularities of Determinantal Rings

as the element δk is a unit in R Ik+1 . Since R Ik+1 is a regular local ring of dimension (m − k)(n − k), with maximal ideal Ik+1 R Ik+1 , it follows that [q]

(t−k)(l+1) ItN +l ⊂ Ik+1 R Ik+1 ⊂ Ik+1 R Ik+1

for natural numbers l and q = p e satisfying (t − k)(l + 1) > (q − 1)(m − k)(n − k). Returning to the polynomial ring R, since F∗ R is a free R-module we have, by an [q] [q] argument already used in the proof of Theorem 7.1.37, that Ik+1 R Ik+1 ∩ R = Ik+1 , hence [q] ItN +l ⊂ Ik+1 for all natural numbers q = p e , l satisfying the above inequality. This implies that νe (It ) := νe{0} (It ) ≤ N +

(q − 1)(m − k)(n − k) . t −k

Thanks to Proposition 7.5.3, dividing by q and passing to the limit as e goes to infinity, one obtains (m − k)(n − k) . fpt R (It ) ≤ t −k Next, fix k and u be as in Lemma 7.5.6. Using the notation of Lemma 7.1.36, let ⎧ t ⎪ ⎨D1 t−k−u  = Dt−k k+1 · [m − k + 1, . . . , m : 1, . . . , k] ⎪ ⎩ (Dk+1 · [m − k + 1, . . . , m : 1, . . . , k])t−k+u · D−u k

if if if

k = 0, k ≥ 1 and u ≥ 0, k ≥ 1 and u < 0.

We claim that  belongs to the integral closure of the ideal It(m−k)(n−k) . By straightforward computations, in each case, deg  = t (m − k)(n − k). Moreover, if k ≥ 1, again by straightforward computations we have that  is a product of (m − k)(n − k) minors, whereas if k = 0 then  is a product of t (m + n − 1) minors and by Lemma 7.5.6 t (m + n − 1) ≤ mn. Therefore Theorem 3.5.10 tells us that , having degree t (m − k)(n − k) and being the product of no more than (m − k)(n − k) minors, belongs to the integral closure of the ideal It(m−k)(n−k) .

7.5 F-pure Thresholds

269

Let m be the homogeneous maximal ideal of R. For a positive integer s, that is not necessarily a power of p, we set m[s] = (X isj : i = 1, . . . , m, j = 1, . . . , n). Given a diagonal monomial order, each variable X i j occurs in the monomial init() with exponent at most t − k. It follows that ∈ / m[t−k+1] . As  belongs to the integral closure of It(m−k)(n−k) , it must satisfy an equation of the form  N + f 1  N −1 + f 2  N −2 + · · · + f N = 0,

f i ∈ Iti(m−k)(n−k) .

If f = f N −1 , it is simple to see that f l ∈ It(m−k)(n−k)l for all l ≥ N . But then

f l ∈ It(m−k)(n−k)l \ m[q]

for all integers l ≥ N and q = p e with deg f + l(t − k) ≤ q − 1. Hence, νe (It ) ≥ (m − k)(n − k)l

for all integers l ≥ N with l ≤

q − 1 − deg f . t −k

Thus, if e is big enough,  νe (It ) ≥ (m − k)(n − k)

q − 1 − deg f −1 , t −k

and passing to the limit as e goes to infinite, one obtains fpt R (It ) ≥ which completes the proof.

(m − k)(n − k) , t −k 

Remark 7.5.7 Given a shape σ = (s1 , . . . , su ), recall that I (σ) ⊂ R = K [X ] is the ideal generated by all the standard bitableaux of shape ≥ σ. For any set of shapes ,  we define the ideal I () = σ∈ I (σ) ⊂ R. These are exactly the ideals I defined by shape: the standard bitableaux belonging to I span it as a K -vector space and for any bitableau , the membership  ∈ I depends only on the shape || (see Theorem 3.6.11 and Remark 3.6.12). They will be investigated more deeply in Chap. 10. In [169] the F-pure threshold of any ideal defined by shape has been computed. Given a

270

7 F-singularities of Determinantal Rings

shape σ, we denote the vector (γ1 (σ), γ2 (σ), . . . , γm (σ)) ∈ Rm by γ(σ), and for a set of shapes , the polyhedron P is the convex hull of γ() = {γ(σ) : σ ∈ } ⊂ Rm . With this notation,      (m − i + 1)(n − i + 1) min max : ai > 0 . (7.5) fpt R I () = (a1 ,...,am )∈P ai In the case that  = {(t)}, P consists of the single point (t, t − 1, . . . , 1, 0m−t ) ∈ Rm ; hence, since in this case I () = It and Eq. (7.5) recovers Theorem 7.5.5. In order to prove Eq. (7.5), in [169] the generalized test ideals of any ideal defined by shape have been computed. A crucial ingredient follows from Lemma 3.5.1 and Theorem 3.5.2: m  (γ (σ)) It t , I (σ) = t=1

where on the right hand side we just have symbolic powers. Hence, if σ = (s1 , . . . , su ), the ideal I (σ) is the integral closure of Is1 · · · Isu in arbitrary characteristic (see the discussion around Theorem 3.5.10) and the two ideals coincide in characteristic 0 or high enough by Theorem 3.5.5. The polyhedron P comes into play as follows: given two vectors v = (v1 , . . . , vm ) and w = (w1 , . . . , wm ) of Rm , we write v ≤ w if vi ≤ wi for any i = 1, . . . , m (notice that given two shapes σ and τ , by Lemma 3.5.1 we have σ ≤ τ if and only if γ(σ) ≤ γ(τ )). Then an argument similar to [103, Theorem 8.2] yields that for any set of shapes  the integral closure of I () is the sum of I (τ ) such that γ(τ ) ≥ x for some x ∈ P . A second important ingredient is the use of the polynomial Dt with its properties of having a squarefree (height It ) initial monomial and of belonging to It (Lemma 7.1.36). All this is explained in [310] in greater generality. Interestingly, in characteristic 0, Eq. (7.5) implies the analog formula for the log-canonical threshold of any GL-invariant ideal of R = K [X ]. These are also determined by a set of shapes , and will be denoted by I in Chap. 11. We have:    (m − i + 1)(n − i + 1) min : ai > 0 . lct R (I ) = max (a1 ,...,am )∈P ai

(7.6)

The point is that, even if I and I () do not agree in general, their integral closures agree, as shown in [103, Theorem 8.2].

The F-pure Threshold fpt R/It (n) In this subsection we want to compute the F-pure threshold of a determinantal ring at its homogeneous maximal ideal. To this purpose, the usual argument exploiting the product of minors Dt allows to prove the following:

7.5 F-pure Thresholds

271

Proposition 7.5.8 Let X be an m × n matrix of indeterminates over a perfect field K of characteristic p > 0, 1 ≤ t ≤ m ≤ n, R = K [X ] and It the ideal of R generated by the t-minors of X . Furthermore, let n = I1 /It be the unique maximal homogeneous ideal of R/It . Then fpt R/It (n) ≥ t (t − 1). pe −1

Proof The same argument used to prove Theorem 7.1.37 shows Dt On the other hand, considering the monomial of R μ=

t−1

[ pe ]

∈ It

: It .

X m−k+1,1 · · · X m,k · X 1,n−k+1 · · · X k,n

k=1

of degree t (t − 1), we have for a diagonal monomial order that in(μDt ) = m n e [ pe ] pe −1 ∈ (It : It ) \ m[ p ] , where m = I1 is the μ in(Dt ) = i=1 j=1 X i j . So, (μDt ) homogeneous maximal ideal of R. Therefore, Proposition 7.5.3 implies νeIt (m) ≥  t (t − 1)( p e − 1), and so fpt R/It (n) ≥ t (t − 1). At this point, one could be tempted to prove that the lower bound of Proposition 7.5.8 is actually an equality. As it will be clear in the following, however, this is true only in the case that t = 1 or t = m. To compute fpt R/It (n) in general, we need to [ pe ] [ pe ] achieve a deep understanding of the R/It -module (It : It )/It . For the next discussion let us fix R = K [X 1 , . . . , X n ] a standard graded polynomial ring over a perfect field K , and m = (X 1 , . . . , X n ) the unique homogeneous maximal ideal of R. Remark 7.5.9 Let I be a proper nonzero homogeneous ideal of R such that R/I e e is F-pure. We know that I [ p ] : I is not contained in m[ p ] by Proposition 7.1.30; in e [ pe ] [ pe ] particular, (It : It )/It is not zero in some degrees ≤ n( p e − 1), as mn( p −1)+1 ⊂ e e e [p ] [p ] is minimally generated in m[ p ] . Suppose that the R/I -module (It : It )/It degrees n( p e − 1) − d1 < n( p e − 1) − d2 < · · · < n( p e − 1) − ds . Then the minimal degree of an element which is in I [ p ] : I but not in m[ p ] must e belong to the set {n( p e − 1) − d1 , n( p e − 1) − d2 , . . . , n( p e − 1) − ds } and I [ p ] : e I ⊂ mn( p −1)−d1 . By Remark 7.5.4 e

e



e e e νeI (m) = max r ∈ N : (I [ p ] : I ) ⊂ m[ p ] + mn( p −1)+1−r , so it follows that νeI (m) ∈ {d1 , . . . , ds }. Let I ⊂ R be a homogeneous ideal. Assuming that A := R/I is a Cohen–Macaulay normal domain, the canonical module ω A can be seen as a divisorial ideal (c.f. [52, Proposition 3.3.18]). So it makes sense to compute the kth symbolic e e power of ω A for any integer k. We are going to relate the A-module (I [ p ] : I )/I [ p ] with a suitable symbolic power of the canonical module of A. Let us start with the following remark:

272

7 F-singularities of Determinantal Rings

Remark 7.5.10 Let A be a Noetherian normal domain, and M be a reflexive Amodule (i.e., the natural map of A-modules M → Hom A (Hom A (M, A), A) is an isomorphism). Let N , N  be finitely generated A-modules such that there is a homomorphism of A-modules N → N  which becomes an isomorphism when localizing at prime ideals of A of height 1. Then there is a homomorphism of A-modules Hom A (N  , M) → Hom A (N , M) which is an isomorphism when localizing at prime ideals of A of height 1. Since both Hom A (N , M) and Hom A (N  , M) are reflexive, such homomorphism must be an isomorphism. Theorem 7.5.11 Let R = K [X 1 , . . . , X n ] be a standard graded polynomial ring over a perfect field K of characteristic p > 0. Let I ⊂ R be a homogeneous ideal such that A = R/I is an F-pure Cohen–Macaulay normal domain. Let ω A denote the graded canonical module of A. Then, for each e ∈ N, one has a graded isomorphism of A-modules e I [p ] : I ∼ e = (ω A (n))(1− p ) . e] [ p I In particular, if m is the homogeneous maximal ideal of R, then −νeI (m) equals the (1− pe ) degree of a minimal generator of ω A . Proof By the obvious extension of Proposition 7.1.31, and since the homomorphism Tr e ∈ Hom A (F∗e A, A) has degree n − np e , we have a graded isomorphism of F∗ Amodules e I [p ] : I Hom A (F∗e A, A)(n − np e ) ∼ . = I [ pe ] Next, note that there are graded isomorphisms Hom A (F∗e A, A) ∼ = Hom A (F∗e A, Hom A (ω A , ω A )) ∼ = Hom A (F e A ⊗ A ω A , ω A ) ∗

(p ) ∼ = Hom A (ω F∗e A , ω A ) e

(p ) ∼ = Hom A (ω F∗e A ⊗ F∗e A F∗e A, ω A ) e

(p ) ∼ = Hom F∗e A (ω F∗e A , Hom A (F∗e A, ω A )) e

(p ) ∼ = Hom F∗e A (ω F∗e A , ω F∗e A ) e

(1− p ) ∼ = ω F∗e A , e

where the first isomorphism follows by [52, Theorem 3.3.4, Proposition 3.6.9], the third follows by Remark 7.5.10 and the sixth by [52, Proposition 3.6.12]. The last isomorphism results from the properties of the divisor class group of A. The last sentence is justified in Remark 7.5.9.  As a corollary of Theorem 7.5.11 we can achieve our goal, namely to compute the F-pure threshold of a determinantal ring at its homogeneous maximal ideal.

7.5 F-pure Thresholds

273

Corollary 7.5.12 Let X be an m × n matrix of indeterminates over a perfect field K of characteristic p > 0, 1 ≤ t ≤ m ≤ n, R = K [X ] and It the ideal of R generated by the t-minors of X . Furthermore, let n = I1 /It be the maximal homogeneous ideal of R/It . Then fpt R/It (n) = m(t − 1). Proof Set A = R/It . By Theorem 6.7.13, where r = t − 1, the graded canonical module of A is given by ωA ∼ = qn−m (m − mt) where q ⊂ A is the prime ideal generated by the size t − 1 minors of the first t − 1 columns of X . Let p be the ideal of A generated by the (t − 1)-minors of the first t − 1 rows of X . By Theorem 6.7.10 the inverse of cl(p) is cl(q). If we take the grading into account, then q−1 ∼ = p(−(t − 1)). Therefore = (p(t − 1))n−m (mt − m). ω (−1) A (1− pe )

In total, ω A = ((p(t − 1))n−m (mt − m)) p −1 , which has all of its minimal generators in degree −m(t − 1)( p e − 1). Theorem 7.5.11 gives e

νeIt (m) = m( p e − 1)(t − 1), and Proposition 7.5.3 implies fpt A (n) = lime→∞ νeIt (a)/ p e = m(t − 1).



Remark 7.5.13 In Corollary 7.5.12 we actually computed the values νeIt (m) for all natural numbers e. We also see that the lower bound given in Proposition 7.5.8 is tight only in the cases t = 1 or t = m.

Notes The fact that determinantal rings are F-pure, and more generally strongly F-regular, in positive characteristic has been proved, for the first time, by Hochster and Huneke in [183, Theorem 7.14]. The proofs of Theorems 7.1.37 and 7.2.13 we provided are deeply different from the original proof. The second proof we gave of the fact that determinantal rings are strongly F-regular, after Corollary 7.3.13, is more similar, but still different, to the original one by Hochster and Huneke. The latter proof is essentially the one given by Bruns and Conca in [38, Theorem 3.1]. This paper is also the basis of Sect. 7.4. In [183, Teorem 7.14] Hochster and Huneke proved also that the coordinate ring of the Grassmannian is strongly F-regular, corresponding to the case of maximal minors in Theorem 7.3.18.

274

7 F-singularities of Determinantal Rings

The computation of the F-pure threshold fpt K [X ] (It ) (Theorem 7.5.5) is due to Miller, Singh and Varbaro in [246, Theorem 1.2]. The F-pure threshold fpt K [X ]/It (I1 /It ) (Corollary 7.5.12) is due to Singh, Takagi and Varbaro in [287, Proposition 4.3]. In both cases, the approaches are the same as in the original papers.

Chapter 8

Castelnuovo–Mumford Regularity

In this chapter we give first a short and self contained  introduction to the Castelnuovo– Mumford regularity for standard graded rings R = i∈N Ri over general base rings R0 . Regularity is originally defined in terms of vanishing of local cohomology modules. But it receives its power from equivalent descriptions based on the vanishing of Koszul homology and the shifts in a minimal free resolution. One of the important features of the regularity of a graded module M is its characterization by the linearity of the resolutions of the “truncations” of M. We then discuss how to access regularity in multigraded structures and their various specializations to standard gradings. This is the basis of investigating regularity of powers of ideals via Rees algebras. A key argument is the linear nature of the nonvanishing degree in bigraded modules. After these preparations we present a concise proof of a classical result on regularity for powers due originally to Cutkosky, Herzog and Trung and, independently, to Kodiyalam and later generalized by Trung and Wang. The result asserts that the regularity of powers I v of a homogeneous ideal I of R is eventually a linear function in v. Our discussion includes the relationship of this function with the reductions of the ideal. For a family of ideals one can form products of powers of the members, and the regularity of these ideals again depends essentially linearly on the exponent vector of the power product. This variant requires the introduction of multi-Rees rings. Next we give a characterization of ideals, whose powers and products have linear free resolutions, in terms of the regularity of the associated Rees rings. These characterizations are then applied to two families of determinantal ideals. The first family is derived from matrices with indeterminate entries that we usually consider in this book. However, the last section deals with ideals of maximal minors for matrices of linear forms that are not generic in general, nevertheless have powers with linear resolutions.

© Springer Nature Switzerland AG 2022 W. Bruns et al., Determinants, Gröbner Bases and Cohomology, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-05480-8_8

275

276

8 Castelnuovo-Mumford Regularity

8.1 Castelnuovo–Mumford Regularity over General Base Rings Castelnuovo–Mumford regularity was introduced in the early eighties of the twentieth century by Eisenbud and Goto in [117] and by Ooishi [254] as an algebraic counterpart of the notion of regularity for coherent sheaves on projective spaces discussed by Mumford in [250]. One of the most important features of Castelnuovo–Mumford regularity is that it can be equivalently defined in terms of (and hence it bounds) the vanishing of local cohomology modules, the vanishing of Koszul homology modules and the vanishing of syzygies. This triple nature of Castelnuovo–Mumford regularity is usually stated for standard graded rings over a base field but indeed it holds for general base rings as we will show in this section. For general properties of local cohomology modules we refer the reader to Brodmann and Sharp [33], Bruns and Herzog [52] and to the Barcelona notes of Schenzel [277].  Let R = i∈N Ri be an N-graded ring with R0 commutative and Noetherian. We assume that R is standard graded, i.e., it is generated as an R0 -algebra by finitely many elements x1 , . . . , xn of degree 1. Let S = R0 [X 1 , . . . , X n ] be the polynomial ring with N-graded structure induced by the assignment deg X i = 1. The R0 -algebra map S → R sending X i to xi induces an S-module structure on R and hence on every R-module.  Let M = i∈Z Mi be a finitely generated graded R-module. Given s ∈ Z we let M(s) denote the module that is obtained from M by shifting the degrees by s, i.e., M(s)i = Mi+s . The Castelnuovo–Mumford regularity of M is defined in terms of local cohomology modules HQi R (M) with support on Q R = R+ = (x1 , . . . , xn ). In our setting the module HQi R (M) is Z-graded and its homogeneous component HQi R (M) j of degree j ∈ Z vanishes for large j. Furthermore one has HQi R (M) = 0 for i > n and   grade(Q R , M) = min i : HQi R (M) = 0 . For i = 0, . . . , n we set   ai (M) = sup j : HQi R (M) j = 0 and, in particular, ai (M) = −∞ if HQi R (M) = 0. We use these invariants to give the (first) definition of Castelnuovo–Mumford regularity. Definition 8.1.1 The Castelnuovo–Mumford regularity reg M of M or, simply, the regularity of M is defined as

8.1 Castelnuovo–Mumford Regularity over General Base Rings

277

    reg M = max ai (M) + i : 0 ≤ i ≤ n = max ai (M) + i : g ≤ i ≤ n where g = grade(Q R , M). We may as well consider M as an S-module by means of the map S → R and local cohomology supported on Q = (X 1 , . . . , X n ). Since HQi (M) = HQi R (M), the resulting regularity is the same. The following are simple properties of Castelnuovo–Mumford regularity that we will freely use. They are either straightforward consequences of the definition or can be found, in the current setting, in [33, Chap.16]. (a) reg M(s) = reg M − s. (b) HQi (S) = 0 for i = n and HQn (S) = (X 1 · · · X n )−1 R0 [X 1−1 , . . . , X n−1 ]. In particular reg S = 0. (c) If 0 → N → M → L → 0 is a short exact sequence of finitely generated graded R-modules with maps of degree 0 then reg N ≤ max{reg M, reg L + 1}, reg M ≤ max{reg L , reg N }, reg L ≤ max{reg M, reg N − 1}. (d) If M j = 0 for all j  0 then HQi R (M) = 0 for i > 0 and HQ0 R (M) = M. In particular, reg M = a0 (M) = max{ j : M j = 0}. A minimal set of generators of M is, by definition, a set of generators that is minimal with respect to inclusion. Unless R0 is a local ring, the number of elements in a minimal set of generators is not uniquely determined, but the set of the degrees of the elements in a minimal set of homogeneous generators of M is uniquely determined because it coincides with the set of i ∈ Z such that [M/Q R M]i = 0. So we have a well defined notion of largest degree of a minimal generator of M that we denote by t0 (M), i.e.,    t0 (M) = min i ∈ Z : M is generated by Mj   = max i ∈ Z : [M/Q R M]i = 0

j≤i

if M = 0, and we set t0 (M) = −∞ if M = 0. We use t0 because M/Q R M ∼ = Tor 0R (M, R0 ) = Tor 0S (M, R0 ), and t reminds us of Tor.

278

8 Castelnuovo-Mumford Regularity

The following result establishes the crucial link between the Castelnuovo– Mumford regularity and the largest degree of a minimal generator of a graded module M. Lemma 8.1.2 For every finitely generated graded R-module M one has t0 (M) ≤ reg M. Proof We may assume that R = S. Furthermore we may assume M = 0 and let v = t0 (M). Then the R0 -module [M/Q M]v is nonzero. Therefore there is a prime ideal P of R0 such that [M/Q M]v localized at P is nonzero. In other words, the localization M of M at the multiplicatively closed set R0 \ P is a finitely generated Z-graded module over (R0 ) P [X 1 , . . . , X n ] with t0 (M ) = t0 (M). Since the local cohomology modules behave well with respect to localization we have reg M ≤ reg M. Hence we may assume right away that R0 is local with maximal ideal m. Similarly, using [33, 16.2.4], we may also assume that the residue  field of R0 is infinite. If M j = 0 for j  0 then M = HQ0 (M) and reg M = max j : M j = 0 ≥ t0 (M). If M = HQ0 (M), set M = M/HQ0 (M). By construction, HQi (M) = HQi (M ) for i > 0 and HQ0 (M ) = 0. Hence t0 (HQ0 (M)) ≤ reg M and reg M ≤ reg M. Since t0 (M) ≤ max{t0 (M ), t0 (HQ0 (M))} it is enough to prove the statement for M . That is to say we may assume that grade(Q, M) > 0. Because the residue field of R0 is infinite, there exists  ∈ S1 \ mS1 such that  is a nonzero divisor on M. By a change of coordinates we may assume that  is indeed one of the variables, say  = X n . The short exact sequence 0 → M(−1) → M → M = M/(X n )M → 0 implies that reg M ≤ reg M (it is actually equal but we do not need equality). As M is a finitely generated graded module over R0 [X 1 , . . . , X n−1 ], we may assume, by induction on the number of variables, that it is generated in degrees ≤ reg M. But clearly t0 (M) = t0 (M), and then it follows that M is generated in degrees ≤ reg M. 

Regularity by Koszul Homology and Free Resolutions We now define the regularity in terms of the vanishing of Koszul homology and in terms of free resolutions. In Theorem 8.1.3 we will prove that all these definitions are indeed equivalent. We consider the (graded) Koszul homology H (Q R , M) associated to the ideal Q R of R with coefficients in the module M. Clearly H (Q R , M) = H (Q, M) and hence in the following we will mainly use the notation H (Q, M). For generalities on Koszul homology we refer the reader to [52]. Here we recall n Hi (Q, M) where Hi (Q, M) is a finitely generated graded that H (Q, M) = i=0 R-module annihilated by Q. Furthermore

8.1 Castelnuovo–Mumford Regularity over General Base Rings

Hi (Q, M) = 0

⇐⇒

279

i = 0, . . . , n − grade(Q, M).

Since the Koszul complex K (Q, S) is a graded free resolution of R0 as an S-module, we have the isomorphism Hi (Q, M) ∼ = ToriS (M, R0 ) of graded modules. We extend the definition of t0 (M) and define ti (M) for i ≥ 0 as follows:   ti (M) = max j : Hi (Q, M) j = 0 if i = 0, . . . , n − grade(Q, M) and ti (M) = −∞ otherwise. Now we set   reg1 (M) = max ti (M) − i : i ∈ N

  = max ti (M) − i : i = 0, . . . , n − grade(Q, M) .

Here index 1 in reg1 is used, just temporarily, to distinguish the regularity defined in terms of local cohomology from the one defined in terms of Koszul homology. We will see in Theorem 8.1.3 that reg1 (M) = reg M and then there will be no need to use the index anymore. Since H0 (Q, M) ∼ = M/Q M, the assertion t0 (M) ≤ reg1 (M) is obvious. Now, let F : · · · → Fc → Fc−1 → · · · → F0 → 0 be a graded S-free resolution of M, i.e., each Fi is graded and S-free of finite rank, the maps have degree 0 and Hi (F) = 0 for all i with the exception of H0 (F) ∼ = M. We say that F is minimal if F0 → M maps a basis of F0 to a minimal set of homogeneous generators of M, F1 → F0 maps a basis of F1 to a minimal set of homogeneous generators of the kernel of F0 → M and for i ≥ 2 the image of a basis of Fi is a minimal set of homogeneous generators of the kernel of Fi−1 → Fi−2 . If R0 is a field then a (finite) minimal S-free resolution not only exists, but is unique up to an isomorphism of complexes. For general R0 , it is still true that every module has a minimal free graded resolution but it is, in general, not finite, and furthermore it is not unique up to an isomorphism of complexes. Given a minimal graded S-free resolution F of M we set:   reg2 (F) = max t0 (Fi ) − i : i = 0, . . . , n − grade(Q, M) and

  reg3 (F) = max t0 (Fi ) − i : i ∈ N .

280

8 Castelnuovo-Mumford Regularity

Obviously we have t0 (M) = t0 (F0 ) ≤ reg2 (F) ≤ reg3 (F). We are ready to establish the following fundamental result: Theorem 8.1.3 Let M be a finitely generated graded R-module and let F be a minimal S-free resolution of M. With the notation above, reg M = reg1 (M) = reg2 (F) = reg3 (F). Before we embark on the proof of the theorem some remarks and an example are advisable. In the “classical” situation in which R0 = K is a field, the equalities reg1 (M) = reg2 (F) = reg3 (F) are quite obvious. In this case Hilbert’s syzygy theorem applies to the polynomial ring S = K [X 1 , . . . , X n ], and n − grade(Q R , M) is the length of the uniquely determined minimal free resolution so that reg2 (F) = reg3 (F), and the number evidently depends only on M. Next we can compare reg2 (F) and reg1 (M) immediately if we observe that they are uniquely determined by the graded Betti numbers of M, given by dim K ToriS (M, K ), and these can be computed either from a minimal free resolution of M or a minimal free resolution of S/Q S ∼ = K , namely the Koszul complex. After a field as the base ring, the next is that of a local ring R0 with ∞ best situation Ri is ∗ local in the sense of [52, Sect. 1.5], maximal ideal m0 . In this case R = i=0 and (m0 , Q R ) is the ∗ maximal ideal. It is even a maximal ideal in the plain sense, and therefore the graded Betti numbers of M are well-defined; see [52, Proposition 1.5.16]. There is a unique choice for F, up to graded isomorphism. However, even in this case the comparison to the Koszul homology H (Q R , M) is not straightforward, as the following example shows. Example 8.1.4 Let K be a field, R0 = K [T ]/(T 2 ) and R = S = R0 [X ] the polynomial ring over R0 . Let t = T¯ ∈ R0 . The R-module R/(t) has the periodic infinite minimal free resolution ··· → R → ··· → R → 0 in which all maps are multiplication by the degree 0 element t. We set N = (R/(t))(−2). This gives N the minimal free resolution · · · → R(−2) → · · · → R(−2) → 0. Then let M = N ⊕ R/(X ) with minimal free resolution F : · · · → R(−2) → · · · → R(−2) → R(−2) ⊕ R(−1) → R(−2) ⊕ R → 0. Hence t0 (F1 ) = 2, but t1 (M) = 1 since H1 ((X ), M) = (R/(X ))(−1).

8.1 Castelnuovo–Mumford Regularity over General Base Rings

281

  Proof of Theorem 8.1.3 Set g = grade(Q, M). Recall that g = min i : HQi (M) = 0   and n − g = max i : Hi (Q, M) = 0 . We first prove that reg M ≤ reg1 (M) by decreasing induction on g. We consider the minimal presentation 0 → N → F0 → M → 0

(8.1)

of M. Suppose g = n. Then HQi (M) = 0 for i = n, and the short exact sequence above induces an exact sequence HQn (F0 ) → HQn (M) → HQn+1 (N ) = 0 i.e., the map on the left is surjective. It follows that reg M = n + an (M) ≤ n + an (F0 ). On the other hand, an (F0 ) = t0 (F0 ) + an (S) = t0 (F0 ) − n, and t0 (F0 ) = t0 (M) because the presentation is minimal. Summing up, reg M ≤ t0 (F0 ) = t0 (M) = reg1 (M). Now assume that g < n. We have grade(Q, N ) = g + 1 and   reg M ≤ max reg F0 , reg N − 1 . By induction reg N ≤ reg1 (N ) and, since reg F0 = t0 (F) = t0 (M), we have   reg M ≤ max t0 (M), reg1 (N ) − 1 . The long exact sequence of Koszul homology arising from the short exact sequence (8.1) and the fact that F0 is S-free imply that Hi (Q, N ) ∼ = Hi+1 (Q, M) for i > 0 and furthermore we have an exact complex 0 → H1 (Q, M) → H0 (Q, N ) → H0 (Q, F0 ) → H0 (Q, M) → 0. This implies ti (N ) = ti+1 (M)

for i > 0,

282

8 Castelnuovo-Mumford Regularity

and furthermore that t0 (N ) ≤ max{t1 (M), t0 (F0 )} = max{t1 (M), t0 (M)}. Therefore   reg1 (N ) = max ti (N ) − i : i ∈ N   = max t0 (M), max{ti+1 (M) − i : i ∈ N} ≤ reg1 (M) + 1 Summing up, we have reg M ≤ max{t0 (M), reg1 (N ) − 1} ≤ max{t0 (M), reg1 (M)} = reg1 (M). Second, we prove reg1 (M) ≤ reg2 (F). As we have already observed, Hi (Q, M) ∼ = ToriS (M, R0 ). On the other hand, ToriS (M, R0 ) can be computed from an S-free resolution of M, i.e., ToriS (M, R0 ) = Hi (F ⊗ R0 ). Hence Hi (Q, M) = Hi (F ⊗ R0 ) so that Hi (Q, M) is a subquotient of Fi ⊗ R0 = Fi /Q Fi and hence ti (M) ≤ t0 (Fi ). Furthermore, Hi (Q, M) = 0 if i > n − g. Therefore reg1 (M) ≤ reg2 (F). That reg2 (F) ≤ reg3 (F) is obvious by definition, so it remains to prove reg3 (F) ≤ reg M. Set M0 = M, M1 = N and Mi = Ker(Fi → Fi−1 ) for i > 1. We consider the short exact sequences 0 → Mi+1 → Fi → Mi → 0 with i ≥ 0. By the minimality of F we have t0 (Fi ) = t0 (Mi ) and t0 (Mi ) ≤ reg Mi because of Lemma 8.1.2. Furthermore reg Fi = t0 (Fi ) because Fi is S-free. Hence reg Mi+1 ≤ max{reg Fi , reg Mi + 1} = max{t0 (Mi ), reg Mi + 1} = reg Mi + 1 for all i ≥ 0. It follows that for all i ≥ 0. Then

reg Mi ≤ reg M + i

8.1 Castelnuovo–Mumford Regularity over General Base Rings

283

t0 (Fi ) = t0 (Mi ) ≤ reg Mi ≤ reg M + i for every i, that is, t0 (Fi ) − i ≤ reg M. It follows that reg3 (F) ≤ reg M.  Remark 8.1.5 Let ϕ : T → R0 be a finite ring map, i.e., R0 is finitely generated as T -module via ϕ. Then ϕ extends uniquely to a ring map ϕ : S = T [X 1 , . . . , X n ] → S = R0 [X 1 , . . . , X n ] with ϕ (X i ) = X i . Therefore a finitely generated graded Rmodule M can be regarded as a finitely generated graded S -module. Hence the regularity of M can be computed also using a graded minimal free resolution as an S -module.

Linear Resolutions Definition 8.1.6 Let M be a nonzero finitely generated graded R-module generated by elements of the same degree d. In general we have reg M ≥ d. We say that M has a linear resolution (or that it has minimal regularity) if reg M = d. Equivalently, M has a linear resolution if and only if for a (and hence for all) minimal S-free resolution F of M one has t0 (Fi ) ≤ d + i for all i. If R0 is a field, then M has a linear resolution if and only if all the matrices in the minimal S-free resolution of M have nonzero entries of degree 1. But for general R0 this need not be the case because, for example, the matrices might have nonzero entries of degree 0. Nevertheless the following holds: Proposition 8.1.7 Let M be a finitely generated graded R-module generated by elements of the same degree d. Then M has a linear resolution if and only if Hi (Q, M) j = 0 for every j = d + i. Proof If Hi (Q, M) j = 0 for j = d + i then ti (M) = d + i or ti (M) = −∞ and hence reg M = d. Vice versa, if reg M = d then ti (M) ≤ d + i, i.e., Hi (Q, M) j = 0  if j > d + i. Furthermore Hi (Q, M) is a subquotient of i S(−1)n ⊗ M which is  generated in degree d + i. Hence Hi (Q, M) j = 0 for j < d + i.  For a Z-graded module M and j ∈ Z one sets M≥ j = i≥ j Mi , which is clearly a submodule of M. If j ≥ t0 (M) then either M≥ j = 0 or it is generated by elements of degree j. More generally one has: Proposition 8.1.8 Let M be a nonzero finitely generated graded R-module. Then   reg M = min j ≥ t0 (M) : M≥ j is either 0 or has a linear resolution .

284

8 Castelnuovo-Mumford Regularity

Proof We first prove that if j ≥ reg M then M≥ j is either 0 or has a linear resolution. Suppose M≥ j = 0, then M≥ j is generated by elements of degree j because j ≥ t0 (M). We have a short exact sequence: 0 → M≥ j → M → M/M≥ j → 0 and hence reg M≥ j ≤ max{reg M, reg(M/M≥ j ) + 1}. Since M/M≥ j is 0 in degrees ≥ j, one has reg M/M≥ j < j. As reg M ≤ j by assumption, we conclude that reg M≥ j ≤ j, i.e., it has a linear resolution. If reg M = t0 (M) then we can conclude already the desired equality. So we assume that reg M > t0 (M) and choose j such that t0 (M) ≤ j < reg M. We must show that M≥ j cannot be 0 and does not have a linear resolution. Let us first rule out the first option: if M≥ j was 0, then reg M = max{i : Mi = 0} < j, a contradiction. For the second option, assume, by contradiction that M≥ j has a linear resolution, i.e., reg M≥ j = j. The short exact sequence above now implies that   reg M ≤ max reg M≥ j , reg M/M≥ j = max{ j, j − 1} = j, again a contradiction.



Remark 8.1.9 One can establish more precise relations between the ai (M), the ti (M) and the t0 (Fi ). For example, in the proof of Theorem 8.1.3 we have observed that ti (M) ≤ t0 (Fi ) and equality holds if R0 is a field. Furthermore it is easy to see that ai (M) + i ≤ t0 (Fn−i ) for all i = 0, . . . , n. When R0 is a field, by [277, Theorem 3.11.] the following very interesting fact holds: for every c = 0, . . . , n one has max{ai (M) + i : i = 0, . . . , c} = max{ti (M) − i : i = n − c, . . . , n}. As far as we know, this relation might hold as well over general base rings. A special case of Schenzel’s result discussed in Remark 8.1.9 is part (b) in the following: Lemma 8.1.10 Suppose R0 is a filed, and let M be a finitely generated graded R-module. Set h = n − dim M. Then (a) th (M) > th−1 (M) > · · · > t0 (M), (b) reg M = max{ti (M) − i : i = h, . . . , n}. Proof Assertion (b) follows immediately from (a). To prove (a) we assume that i ∈ N satisfies ti+1 (M) ≤ ti (M) and we prove that i ≥ h. Let F be a minimal graded free resolution of M. Since R0 is a field, t0 (Fi ) = ti (M). By assumption, there is a basis element, say e, of Fi whose degree is larger than or equal to the degree of every basis element of Fi+1 . In the free complex G = Hom(F, S) the matrices are simply the transposes of the matrices in F, and hence have entries in Q = (X 1 , . . . , X n ).

8.1 Castelnuovo–Mumford Regularity over General Base Rings

285

Hence e corresponds to a basis element e∗ of G i = Hom(Fi , S) that is a cycle by degree reasons and that cannot be a boundary because G has matrices with entries in Q. It follows that ExtiS (M, S) = 0. From [52, Lemma 1.2.4] one easily derives that j Ext S (M, S) = 0 for j < h. Hence i ≥ h. 

Basic Rules for Regularity If the “left” module lives only in finitely many degrees, one can precisely compute the regularity of the ‘middle’ module in a short exact sequence. Proposition 8.1.11 Let 0 → N → M → P → 0 be a short exact sequence of finitely generated graded R-modules and assume that N j = 0 for j  0. Then reg M = max{reg N , reg P}. Proof Since HQi (N ) = 0 for i > 0 the long exact sequence of local cohomology modules gives a short exact sequence 0 → HQ0 (N ) → HQ0 (M) → HQ0 (P) → 0 and isomorphisms HQi (M) ∼ = HQi (P) for i > 0. It follows that a0 (M) = max{a0 (N ), a0 (P)} = max{reg N , a0 (P)} and ai (M) = ai (P) for i > 0. Hence   reg M = sup ai (M) + i : i ≥ 0   = sup reg N , ai (P) + i : i ≥ 0 = max{reg N , reg P}.  If a submodule of the zeroth local cohomology contains the “socle”, one knows its regularity: Proposition 8.1.12 Let M be a finitely generated R-module and let W ⊂ HQ0 (M) be a graded submodule of M containing 0 : M Q and set a = a0 (M). Then reg W = a, and if HQ0 (M) = 0, then (0 : M Q)a = Wa = HQ0 (M)a . Proof If HQ0 (M) = 0 the statement is obvious. If HQ0 (M) = 0, then a ∈ Z and Q HQ0 (M)a = 0. Hence (0 : M Q)a = HQ0 (M)a and the assertion follows.  A homogeneous element z ∈ R of positive degree is said to be almost regular with respect to a graded R-module M, almost M-regular for short, if 0 : M z vanishes in high degrees. Similarly, a sequence z 1 , . . . , z c of elements of R of positive

286

8 Castelnuovo-Mumford Regularity

degrees is an almost M-regular sequence if z i is almost regular with respect to M/(z 1 , . . . , z i−1 )M for i = 1, . . . , c. Proposition 8.1.13 Let M be a finitely generated graded R-module and let z ∈ R be an almost M-regular element of degree d > 0. Set T = 0 : M z. Then a0 (M) = reg T and reg M = max{reg M/z M − d + 1, a0 (M)}. Proof Since d > 0 we have T ⊃ 0 : M Q. Furthermore, since T vanishes in high degrees, it is annihilated by some power of Q and hence T ⊂ HQ0 (M). Then we have reg T = a0 (M) by Proposition 8.1.12. Next we observe that the exact complex z

0 → T (−d) → M(−d) → M → M/z M → 0 gives two short exact sequences 0 → T (−d) → M(−d) → z M → 0, 0 → z M → M → M/z M → 0.

(8.2) (8.3)

Set α = reg M, β = reg M/z M, γ = reg z M. Since T ⊂ HQ0 (M) we can apply Proposition 8.1.11 to (8.2) and get α + d = max{a0 (M) + d, γ}, that is α = max{a0 (M), γ − d},

(8.4)

γ ≤ max{α, β + 1}.

(8.5)

while (8.3) gives

Combining (8.4) and (8.5) we obtain α ≤ max{a0 (M), α − d, β + 1 − d}.

(8.6)

Taking into account that d > 0, (8.6) implies α ≤ max{a0 (M), β + 1 − d}. Now (8.3) also yields β ≤ max{α, γ − 1}, and combining it with (8.4) we have β ≤ max{γ − d, a0 (M), γ − 1} = max{a0 (M), γ − 1},

(8.7)

8.2 Castelnuovo–Mumford Regularity and Multigraded Structures

287

and hence max{a0 (M), β + 1 − d} ≤ max{a0 (M), a0 (M) + 1 − d, γ − d} = max{a0 (M), γ − d} = α. (8.8) Combining (8.7) and (8.8) we obtain the desired equality.



A particular case of Proposition 8.1.13 gives the following. Proposition 8.1.14 Let M be a finitely generated graded R-module and let z ∈ R be an M-regular element of degree d > 0. Then reg M = reg M/z M − d + 1. We conclude with a remark concerning the characterization of the regularity in terms of vanishing of Ext modules obtained via the graded local duality and that holds if the base ring is a field. Remark 8.1.15 Assume that R0 = K is a field so that S = K [X 1 , . . . , X n ] and Q = (X 1 , . . . , X n ). Let M be a finitely generated graded R-module M. As we have already seen in the proof of Theorem 2.3.7, from the graded version of the local duality [52, 3.6.19] one obtains dim K HQi (M) j = dim K Ext n−i S (M, S)− j−n . In particular, HQi (M) j = 0

⇐⇒

Ext n−i S (M, S)− j−n  = 0.

Therefore one can express the regularity of M in terms of vanishing properties of the Ext modules of M. Working out the simple details one obtains j

reg M = max{−k − j : Ext S (M, S)k = 0}.

8.2 Castelnuovo–Mumford Regularity and Multigraded Structures  Assume now R = (i, j)∈N2 R(i, j) is N2 -graded with R0 = R(0,0) commutative and Noetherian and that R is generated as an R0 -algebra by elements x1 , . . . , xn , y1 , . . . , ym , where the xi are homogeneous of degree (1, 0) and the y j are homogeneous of degree (0, 1).  We let R (∗,0) denote the R0 -subalgebra i R(i,0) of R and by Q (1,0) the ideal of R (∗,0) generated by R(1,0) , i.e., by x1 , . . . , xn . Similarly R (0,∗) is the R0 -subalgebra  (0,∗) generated by R(0,1) , i.e., by y1 , . . . , ym . j R(0, j) of R and Q (0,1) the ideal of R

288

8 Castelnuovo-Mumford Regularity

We have (at least) three ways of getting an N-graded structure on R out of the N2 graded structure: (a) (1, 0)-graded structure: the homogeneous component of degree i ∈ N of R is  (0,∗) R and the ideal of the given by R (i,∗) = j (i, j) . The degree 0 part of R is R homogeneous elements of positive degree is Q (1,0) R = (x1 , . . . , xn ). (b) (0, 1)-graded  structure: the homogeneous component of degree j ∈ N is given by R (∗, j) = i R(i, j) . The degree 0 part is R (∗,0) and the ideal of the homogeneous elements of positive degree is Q (0,1) R = (y1 , . . . , ym ).  (c) total degree: the homogeneous component of degree u ∈ N of R is i+ j=u R(i, j) . The degree 0 part is R0 and the ideal of the homogeneous elements of positive degree is (x1 , . . . , xn , y1 , . . . , ym ). Remark 8.2.1 A notion of Castelnuovo–Mumford regularity for multigraded modules can be defined without coarsening the grading as above, see Maclagan and Smith [242], Hoffman and Hao [189], and Botbol and Chardin [24]. We will not use this notion of multigraded regularity in this chapter.  Let M = (i, j)∈Z2 M(i, j) be a Z2 -graded R-module. Given i, j ∈ Z we set    (∗, j) M (i,∗) = v M(i,v) and M = v M(v, j) . Clearly M = i M (i,∗) as a Z-graded  (∗, j) as a Z-graded R (∗,0) -module. R (0,∗) -module and M = j M Lemma 8.2.2 If M is a finitely generated Z2 -graded R-module, then M (i,∗) is a finitely generated graded R (0,∗) -module for all i ∈ Z and M (∗, j) is a finitely generated graded R (∗,0) -module for all j ∈ Z. Proof Suppose M is generated by homogeneous elements m 1 , . . . , m c of degree deg m v = (αv , βv ). For a given i the module M (i,∗) is generated, as an R (0,∗) -module, by f m v with αv ≤ i and where f is a monomial of degree i − αv in x1 , . . . , xn .  These are finitely many elements. One argues similarly for M (∗, j) . As we have done for R also M = ⊕M(i, j) can be turned into a Z-graded module by regrading it with respect to the (1, 0)-grading, the (0, 1)-grading or the total degree. Hence we may consider the Castelnuovo–Mumford regularity of M with respect to any of these different graded structures. To distinguish them we let reg(1,0) M denote the regularity of M with respect to the (1, 0)-graded structure. Similarly, reg(0,1) M is the regularity of M with respect to the (0, 1)-graded structure. Let S = R0 [X 1 , . . . , X n , Y1 , . . . , Ym ] with the N2 -graded structure induced by the assignment deg X i = (1, 0) and deg Y j = (0, 1). We have: Proposition 8.2.3 Let M be a finitely generated Z2 -graded R-module. Let F be a bigraded S-free minimal resolution of M. (a) Let αi ∈ Z be the largest integer α for which Fi has a minimal generator in degree (α, ∗). Then   max reg M (∗, j) : j ∈ Z = reg(1,0) M = max{αi − i : i = 0, . . . , n} where reg M (∗, j) is the regularity as Z-graded R (∗,0) -module.

8.3 Vanishing over Nonstandard Multigraded Rings

289

(b) Let βi ∈ Z be the largest integer β for which Fi has a minimal generator in degree (∗, β). Then  max{reg M ( j,∗) : j ∈ Z = reg(0,1) M = max{βi − i : i = 0, . . . , m} where reg M ( j,∗) is the regularity as Z-graded R (0,∗) -module. Hence a minimal bigraded resolution of M as an S-module can be used to compute both the (1, 0)- and the (0, 1)-regularity. Proof By symmetry, it is enough to prove (a). Set Q = Q (1,0) , i.e., Q is the ideal of R (∗,0) generated by R(1,0) . The (1, 0)-regularity of M is defined by means of the local cohomology HQ∗ R (M). We may regard M as an R (∗,0) -module, so that  c (∗, j) HQc R (M) = HQc (M) = ) for all c. This explains the first equality. For j H Q (M the second equality, by Theorem 8.1.3 the regularity reg(1,0) M can be computed from any graded minimal free resolution of M as an R (0,1) [X 1 , . . . , X n ]-module, but we have observed in Remark 8.1.5 that it can be as well computed from a minimal free resolution of M as an S-module. Note that n ≥ grade(Q, M)!  The notation and results of this section can be generalized, without complications, to Zr -graded rings and modules. We just state the main result in the form it will be used. Proposition 8.2.4 Let M be a finitely generated Zr -graded module over a standard Zr -graded ring R with R0 an arbitrary Noetherian ring. Consider a presentation of R as a quotient of a standard Zr -graded polynomial ring S over R0 . Let F be a Zr -graded S-free minimal resolution of M. Let αi be the largest integer α such that Fi has a minimal generator in degree (α, ∗) ∈ Zr . Let n be the number of generators of R(1,0) , where (1, 0) = (1, 0, . . . , 0) ∈ Zr . Then max{reg M (∗,v) : v ∈ Zr −1 } = reg(1,0) M = max{αi − i : i = 0, . . . , n}.

8.3 Vanishing over Nonstandard Multigraded Rings For later applications we consider in this section a polynomial ring A = A0 [Y1 , . . . , Yg ] over a Noetherian ring A0 . A (nonstandard) Z2 -graded structure on A given by deg Y j = (d j , 1), for j = 1, . . . , g and deg x = (0, 0) for x ∈ A0 .

d j ∈ N,

290

8 Castelnuovo-Mumford Regularity

For every Z2 -graded A-module N =

 (i,v)∈Z2

N(i,v) and for every v ∈ Z we set

  ρ N (v) = sup i ∈ Z : N(i,v) = 0 ∈ Z ∪ {±∞}. The function ρ N (v) is called the top degree function of N . We will study the behavior of ρ N (v) as a function of v. In the following, if we say that an element x of a module M is a minimal generator, then we assume that a minimal set of generators of M has been chosen implicitly and that x belongs to it. We first observe the following Lemma 8.3.1 Let N be a Z2 -graded finitely generated A-module. Then ρ N (v) is bounded above by a linear function of v. a

a1 g 2 Proof If w is a homogeneous element of N of degree (α, β) ∈ Z , then Y1 · · · Yg w has j a j d j + α, j a j + β). Hence N(i,v) is nonzero only if (i, v) =  degree (  ( j a j d j + α, j a j + β) for some (a1 , . . . , ag ) ∈ Ng and the degree (α, β) of some minimal generator of N . If we set dmax = max{d1 , . . . , dg }, then N(i,v) = 0 implies i ≤ (v − β)dmax + α = vdmax + α − dmax β

for the degree (α, β) of a minimal generator of N . By hypothesis N is finitely generated. Hence we may consider the degrees (α1 , β1 ), . . . , (αc , βc ) of its minimal generators. Summing up, for C = max{αi − βi dmax : i = 1, . . . , c} we have ρ N (v) ≤ vdmax + C

for all v.



The next goal is to prove that ρ N (v) is not only linearly bounded, but is, in the long run, actually a linear function of v. To this end we will need two general facts. The first is Lemma 8.3.2 Given a chain of submodules 0 = N0 ⊂ N1 ⊂ N2 ⊂ · · · ⊂ N p = N of Z2 -graded A-modules one has ρ N (v) = max{ρ Ni /Ni−1 (v) : i = 1, . . . , p} for all v. Proof We first observe that the function ρ N (v) behaves well on short exact sequences. That is, if 0→U → M →T →0 is an exact sequence of Z2 -graded A-modules with maps of degree 0, then for every (i, v) we get a short exact sequence 0 → U(i,v) → M(i,v) → T(i,v) → 0 of A0 -modules and hence M(i,v) = 0 if and only if U(i,v) = 0 and T(i,v) = 0. Therefore ρ M (v) = max{ρU (v), ρT (v)}.

8.3 Vanishing over Nonstandard Multigraded Rings

291

Now the statement follows by induction on p using the short exact sequences 0 → Ni−1 → Ni → Ni /Ni−1 → 0 associated to the chain of submodules.



The second fact is the behavior of the function ρ N (v) under Gröbner deformation. Most of the notions we have introduced in Chap. 1 can be extended to polynomial rings over general base rings. The major difference is that, in the generalized setting, the coefficients of the polynomials play a prominent role. The ideals generated by terms are more complicated than ordinary monomial ideals, see the paper of Vipera [314] for an overview. In particular, to check whether a term belongs to an ideal generated by terms, it is not enough to test if it is divisible by a minimal generator, for example in Z[X, Y ] we have X Y ∈ (2X, 3Y ) and X Y is not divisible by 2X or 3Y . For generalities on Gröbner bases in this context we refer the reader to Adams and Loustaunau [4, Chap. 4]. In the following, we introduce the minimal amount of notions that allows us to reach our goal. Let F be a finitely generated Z2 -graded free A-module with basis e1 , . . . , e p . As a an A0 -module F is free with basis given by the monomials Y a ei = Y1a1 · · · Yg g ei with a = (a1 , . . . , ag ) ∈ Ng and 1 ≤ i ≤ p. Let < be a monomial order on F (defined as in Definition 1.1.1). For every Z2 -graded A-submodule U of F we let in(U ) denote the A0 -submodule of F generated by the leading terms (the largest monomial in the support but with its coefficient!) of the nonzero elements in U . Since U is an Asubmodule of F, it turns out that in(U ) is an A-submodule of F as well. Furthermore for every term αY a ei in in(U ) (with α ∈ A0 \ {0}) there exists an element u ∈ U such that init(u) = αY a ei . Lemma 8.3.3 With the notation above, for every (i, v) ∈ Z2 we have U(i,v) = F(i,v) if and only if in(U(i,v) ) = F(i,v) . In particular, ρ F/U (v) = ρ F/ in(U ) (v) for all v. Proof The “only if” implication is obvious. For the “if” implication, we argue by contradiction. Suppose in(U(i,v) ) = F(i,v) and U(i,v) = F(i,v) . As F(i,v) is generated by the elements Y a ei that have degree (i, v), there must be at least one such element not in U(i,v) . Hence let Y a ei be the smallest (with respect to the monomial order) monomial of degree (i, v) which is not in U(i,v) . Since Y a ei ∈ in(U(i,v) ) there exists u ∈ U such that init(u) = Y a ei . We may assume that u is homogeneous of degree (i, v). If not, we simply replace u with the degree (i, v) homogeneous component of u, which is in U since U is graded. So u = Y a ei + u 1 where u 1 is an A0 -linear combination of monomials of degree (i, v) that are < Y a ei . Hence, by assumption, u 1 ∈ U(i,v) . It follows that Y a ei = u − u 1 ∈ U(i,v) , a contradiction. The equality ρ F/U (v) = ρ F/ in(U ) (v) is an immediate consequence.  We are ready to prove the main result of the section: Theorem 8.3.4 Let N be a Z2 -graded and finitely generated A-module. (a) Either ρ N (v) = −∞ for v  0 or ρ N (v) is eventually a linear function of v with leading coefficient in {d1 , . . . , dg }.

292

8 Castelnuovo-Mumford Regularity

(b) More precisely, in the second case there exist b ∈ Z and v0 ∈ N such that ρ N (v) = δv + b

for v ≥ v0

/ δ = max{d j : Y j ∈

√ Ann N }.

with

Proof To prove that ρ N (v) is either eventually linear in v or −∞, we present N as F/U where F is a finitely generated A-free bigraded module and U is a bigraded A-submodule of F. Let < be a monomial order on F. By Lemma 8.3.3 we then have ρ F/U (v) = ρ F/ in(U ) (v). Hence we may assume right away that U is generated by terms (monomials with coefficients). We can consider a chain of bigraded submodules 0 = N0 ⊂ N1 ⊂ N2 ⊂ · · · ⊂ Nq = N with cyclic quotients Ci = Ni /Ni−1 annihilated by a monomial prime ideal, i.e., an ideal of the form P A + J where P is a prime ideal of A0 and J is an ideal generated by a subset of the variables Y1 , . . . , Yg . Hence ρ N (v) = sup{ρCi (v) : i = 1, . . . , q}. Since the supremum of finitely many eventually linear functions in one variable is eventually linear itself, it is enough to prove the statement for each Ci . Up to a shift (− j, −w) ∈ Z2 , the module Ci has the form A/P where P = P0 A + J such that P0 is a prime ideal of A0 and J is generated by a subset of the variables. With / P} and d = sup{dk : k ∈ G}, we have G = {k : Yk ∈  ρCi (v) =

dv + (−dw + j) −∞

if G = ∅ and v ≥ w, if G = ∅ and v > w.

So we have proved that ρ N (v) = −∞ or eventually a linear function with leading coefficient in {d1 , . . . , dg }, say ρ N (v) = di v + b for v  0 and some i and this shows (a). √ For (b) we assume√that ρ N (v) = −∞, i.e., (Y1 , . . . , Yg ) ⊂ Ann N . We choose dk = max{d j : Y√j ∈ / Ann N }. To conclude we must prove that di = dk . By its very definition Yk ∈ / Ann N , and hence there exists a generator w ∈ N such that Ykv w = 0 for all v ∈ N. Let (a, v0 ) be the degree of w. Then deg Ykv w = (vdk + a, v + v0 ) and so N(dk v+a,v+v0 ) = 0, i.e., ρ N (v + v0 ) ≥ dk v + a for all v ∈ N or, equivalently, ρ N (v) ≥ dk v − dk v0 + a for all v ≥ v0 . As ρ N (v) = di v + b for v  0 this implies that di ≥ dk . [Y j : Y j ∈ / √ Vice versa, we observe that N is a finitely generated module over A0√ Ann N ]. Hence, by the assertion (a) applied to N as an A [Y : Y ∈ / Ann N ]0 j j √ module, we have di ∈ {d j : Y j ∈ / Ann N } and so di ≤ dk . 

8.3 Vanishing over Nonstandard Multigraded Rings

293

Remark 8.3.5 In principle the linear form δv + b and the value v0 discussed in Theorem 8.3.4 can be computed following the strategy outlined in the proof: (a) Compute a presentation of the module N = F/U . (b) Compute an initial module in(U ) of U . (c) Compute a chain of submodules of F/ in(U ) with factors that are, up to shift, of the form A/P with P = P0 + J where J is an ideal generated by variables and P0 is a prime ideal of A0 . (d) Read out from the shift and ideal J a linear form and the associated v0 according to the description given in the proof of Theorem 8.3.4. (e) Among the linear forms collected in (d) pick those with largest leading coefficient (this gives δ), and, among them, those with largest constant term (this gives b), and among them the one associated with the shift (− j, −w) with w minimal (this gives v0 = w). When the base ring A0 is a field K , we may replace (c) and (d) with (c’) and (d’):  (c’) Consider the bigraded Hilbert series HN (z, t) = (i,v) dim K N(i,v) z i t v of N and compute a rational expression for it a(z, t) di i=1 (1 − z t)

HN (z, t) = g

with a(z, t) ∈ Z[z, t]. (d’) Decompose HN (z, t) as sum of rational series of type αz j t w di i∈G (1 − z t)



with α ∈ N>0 . Then to the data G and (− j, −w) we associate a linear form as in the proof of Theorem 8.3.4. Then proceed with (e). Remark 8.3.6 Let N be a Z2 -graded and finitely generated A-module. If T, U are submodules of N such that N = T + U , then clearly ρ N (v) = max{ρT (v), ρU (v)}. Therefore if N is generated by n 1 , . . . , n p then ρ N (v) = max{ρ Ni (v) : i = 1, . . . , p} where Ni the submodule of N generated by n i . This reduces the problem of computing ρ N (v) to principal modules, i.e., modules of type A/I up to shift. One can employ this idea for practical computations and to make slight variation of the argument in the proof Theorem 8.3.4, i.e., first reduce to a principal module A/I , replace I with in(I ) and then proceed as in the proof of Theorem 8.3.4.

A Multigraded Variant For later applications we need a generalization of the results presented so far. Let A = A0 [Yi j : 1 ≤ i ≤ m, 1 ≤ j ≤ gi ] with a Z × Zm -graded structure induced by

294

8 Castelnuovo-Mumford Regularity

deg(Yi j ) = (di j , ei ) ∈ Z × Zm where di j ∈ N and ei is the ith element of the canonical basis of Zm . Given a Z × Zm -graded A-module N=



N(i,v) ,

(i,v)∈Z×Zm

for every v = (v1 , . . . , vm ) ∈ Zm we set ρ N (v) = sup{i ∈ Z : N(i,v) = 0}, which is a function from Zm to Z ∪ {±∞}. Our goal is to prove that ρ N (v) is eventually a piecewise linear function if N is a finitely generated Z × Zm -graded A-module. Also in the multigraded case ρ N (v) is called the top degree function. . , vm ), w = (w1 , . . . , wm ) ∈ Rm we let v · w In the following, for v = (v1 , . .  m vi wi , and we set v ≥ w if vi ≥ wi for denote the ordinary scalar product i=1 i = 1, . . . , m. Furthermore we say that a property depending on v ∈ Rm holds for v  0 if there exists v0 ∈ Rm such that the property holds for all v ≥ v0 . With these conventions we get Theorem 8.3.7 Let N be a finitely generated Z × Zm -graded A-module. Then either ρ N (v) = −∞ for v  0 or there exist w1 , . . . , wr ∈ Nm and c1 , . . . , cr ∈ Z such that   ρ N (v) = sup wk · v + ck : 1 ≤ k ≤ r for every v ∈ Zm with v  0. More precisely, wk = (wk1 , . . . , wkm ) and wki ∈ {di1 , . . . , digi } for i = 1, . . . , m. Proof The proof follows exactly the same path of the proof of Theorem 8.3.4. We just sketch it. First of all we present N as F/U where F is a finitely generated Z × Zm -graded A-free module and U is a Z × Zm -graded submodule of it. Then we replace U with its initial submodule in(U ) with respect to some monomial order. Next we take a chain of submodules of F/U so that the successive quotients are shifted copies of quotients of A by monomial prime ideals. Hence ρ N (v) is the supremum of finitely many functions of type ρC (v) where C = (A/P)(−u, −v0 ) with P a monomial prime of A and (−u, −v0 ) ∈ Z × Zm . The ideal P has the form / G) where P0 is a prime ideal of A0 and G is a subset of the set P0 A + (Yi j : Yi j ∈ of variables {Yi j : i = 1, . . . , m, j = 1, . . . , gi }. If there exists i such that G ∩ {Yi1 , . . . , Yigi } = ∅, then clearly ρC (v) = −∞ for v  0. Otherwise if G ∩ {Yi1 , . . . , Yigi } is not empty for all i we set   w = (max di j : Yi j ∈ G : i = 1, . . . , m) and obtain ρ A/P (v) = w · v and hence

if v ≥ 0,

8.4 Regularity of Powers and Products of Ideals

295

ρC (v) = w · v + u − w · v0

if v ≥ v0 .

Setting c = u − w · v0 yields ρC (v) = w · v + c

if v ≥ v0 .

Summing up, ρ N (v) is the supremum of finitely many functions that are eventually linear. Finally one observes that the ith coefficient of the vector w is by construction  in {di1 , . . . , digi }, and this completes the proof. Example 8.4.14 will show that one cannot expect linearity in Theorem 8.3.7.

8.4 Regularity of Powers and Products of Ideals We return to the notation of Sect. 8.1. That is, R0 is a commutative Noetherian ring and R = i∈N Ri is an N-graded ring, generated, as an R0 -algebra, by elements x1 , . . . , xn of degree 1. Furthermore S denotes the polynomial ring R0 [X 1 , . . . , X n ] with N-graded structure induced by the assignment deg X i = 1, projecting onto R by the map that sends X i to xi .

Regularity of Powers For a finitely generated graded R-module M and a homogeneous ideal I of R we will study the behavior of reg I v M as a function of v ∈ N. For simplicity we will tacitly assume throughout that I v M = 0 for every v. We consider the Rees algebra R(I ) of I :  R(I ) = Iv v∈N

with its natural bigraded structure given by R(I )(i,v) = (I v )i . A convenient way to describe the elements of R(I ) makes use of a new variable T and regards R(I ) as a subalgebra of the polynomial ring R[T ]: R(I ) =

 v

ai T : v ∈ N, ai ∈ I for all i ⊂ R[T ]. i

i

i=0

Here the variable T is a place holder, it is used to keep track on the grading associated to the powers of I . By its very definition, R(I ) is an R-algebra generated by

296

8 Castelnuovo-Mumford Regularity

f 1 T, . . . , f g T where f 1 , . . . , f g is a minimal set of homogeneous generators of I of degrees d1 , . . . , dg ∈ N. Hence R(I ) = R[ f 1 T, . . . , f g T ]. The Rees module of the pair I, M is by definition R(I, M) =



IvM

v∈N

which is clearly an R(I )-module naturally bigraded by R(I, M)(i,v) = (I v M)i . Let m 1 , . . . , m h be a minimal set of homogeneous generators of M of degrees, say, w1 , . . . , wh ∈ Z. The elements m 1 , . . . , m h are also the generators of R(I, M) as an R(I )-module and, as elements of R(I, M), they have bidegree (w1 , 0), . . . , (wh , 0). We may present R(I ) as a quotient of B = R[Y1 , . . . , Yg ] via the map ψ : B → R(I ),

Yi → f i T ∈ R(I )(di ,1) .

Actually B can be naturally bigraded by assigning the bidegree (i, 0) to x ∈ Ri and setting deg Y j = (d j , 1) so that ψ preserves the degrees. Consider Q R = (x1 , . . . , xn ) and its extension Q R B to B. We consider the Koszul homology H (Q R B, R(I, M)) = H (Q R , R(I, M)) =



H (Q R , I v M).

v∈N

Since Q R H (Q R , R(I, M)) = 0, the Koszul homology H (Q R , R(I, M)) naturally acquires the structure of a finitely generated Z2 -graded B/Q R B-module. Here B/Q R B = R0 [Y1 , . . . , Yg ] has a bigraded structure defined in Sect. 8.3. Theorem 8.4.1 Let I be a homogeneous ideal of R minimally generated by homogeneous elements of degree d1 , . . . , dg and M be a finitely generated graded R-module. Then: (a) For i = 0, . . . , n we have that either Hi (Q R , I v M) = 0 for v  0 or Hi (Q R , I v M) = 0 for v  0. In other words, grade(Q R , I v M) is constant, say equal to s, for v  0.

8.4 Regularity of Powers and Products of Ideals

297

(b) For i = 0, . . . , n − s there exist δi ∈ {d1 , . . . , dg } and ci ∈ Z such that ti (I v M) = δi v + ci

for v  0.

(c) δ0 ≥ δ1 ≥ · · · ≥ δn−s . (d) there exists c ∈ Z such that reg I v M = δ0 v + c

for v  0.

Proof For i = 0, . . . , n we consider the ith Koszul homology module Hi = Hi (Q R , R(I, M)) =



Hi (Q R , I v M).

v∈N

As already observed, Hi is a finitely generated Z2 -graded B/Q R B-module and B/Q R B is a polynomial ring with the grading considered in Sect. 8.3. Hence, the top degree function ρ discussed in Sect. 8.3 can be applied to the B/Q R B-module Hi and, by the definition of ti , we get ρ Hi (v) = ti (I v M). Therefore we may apply Theorem 8.3.4 and conclude that either Hi (Q R , I v M) = 0 for v  0 or Hi (Q R , I v M) = 0 for v  0. In the second case, ti (I v M) is a linear function for large v with leading coefficient in {d1 , . . . , dg }. This proves √ (a) and (b). / Ann Hi }. The More precisely, ti (I v M) = δi v + ci where δi = max{d j : Y j ∈ rigidity property √ √ of Koszul homology modules, see [52, 1.6.31], implies that Ann Hi ⊂ Ann Hi+1 and hence δi ≥ δi+1 , proving (c). Finally, as reg I v M = max{ti (I v M) − i : i = 0, . . . , n}, we conclude that  reg I v M is eventually a linear function in v with leading coefficient δ0 . A particular case that deserves special attention is the following: Theorem 8.4.2 Let I be a homogeneous ideal of R minimally generated by homogeneous elements of degree d and M be a finitely generated graded R-module. Then there exists c ∈ Z such that reg I v M = dv + c

for v  0.

Regularity and Reductions Our next goal is to give another description of the number δ0 that appears in the statement of Theorem 8.4.1, a description in terms of reductions of the ideal I . Let us recall that an ideal J of R is said to be a reduction of an ideal I if J ⊂ I and

298

8 Castelnuovo-Mumford Regularity

J I v = I v+1 for some number v. We have an inclusion of Rees algebras R(J ) ⊂ R(I ) and the fact that J is reduction of I implies (it is indeed equivalent) that R(I ) is a finitely generated R(J )-module. Furthermore, if M is a finitely generated R-module then R(I, M) is finitely generated as an R(J )-module. Indeed, for the finite generation of R(I, M) over R(J ), we just need to require that J is a reduction of I with respect to the module M, that is, J I u M = I u+1 M for some number u. Returning to our setting, since we deal with graded ideals, we will consider homogeneous reductions. Remark 8.4.3 Given a homogeneous reduction J of I with respect to M, the Koszul homology H (Q R , R(I, M)) is finitely generated over R(J )/Q R R(J ). Hence the proof and the conclusion of Theorem 8.4.1 holds with {d1 , . . . , dg } replaced by the set of the degrees of the generators of J . In particular, δ0 ≤ t0 (J ) for every homogeneous reduction J of I with respect to M. More precisely: Theorem 8.4.4 The number δ0 in Theorem 8.4.1 has the following description:   δ0 = min j : I≤ j is a reduction of I with respect to M where I≤ j denotes the subideal of I generated by the elements of I of degree ≤ j. Proof Set   γ = min j : I≤ j is a reduction of I with respect to M . If I≤ j is a reduction of I with respect to M then we have already argued in Remark 8.4.3 that δ0 ≤ t0 (I≤ j ), and obviously t0 (I≤ j ) ≤ j. Hence δ0 ≤ γ. For a contradiction assume that δ0 < γ. Set j = γ − 1, J = I≤ j and let H be the ideal generated by the minimal generators of I of degrees > j, i.e., in degree ≥ γ. For every v a minimal generator of I v either contains at least one factor in J or is the product of v elements of H , that is, I v = J I v−1 + H v . Multiplication by M yields I v M = J I v−1 M + H v M.

8.4 Regularity of Powers and Products of Ideals

299

By Theorem 8.4.1(b), t0 (I v M) = δ0 v + c0 for large v. By construction H v M is generated in degree ≥ γv + e where e is the smallest integer such that Me = 0. Since δ0 < γ, there exists v  0 such that δ0 v + c0 < γv + e. Hence the elements of H v M cannot be minimal generators of I v M, and therefore I v M = J I v−1 M. But then J is a reduction of I with respect to M, a contradiction because j < γ. 

Examples and Open Questions Remark 8.4.5 We have proved that ti (I v M) is eventually a linear function in v. One might ask whether the same behavior occurs for ai (I v M). It turns out that the individual ai (I v M) can have a more complicated nature. Indeed, already in Cutkosky et al. [99] one finds examples of ideals I in polynomial rings over a field for which reg sat(I v ) is not a linear function for large v. Here sat(J ) denotes the saturation of an ideal J with respect to the maximal homogeneous ideal. As reg sat(I v ) = sup{ai (I v ) + i : i ≥ 2} it turns out that not all the ai (I v ) can be eventually linear. An even more complicated example is presented in Cutkosky [98]: there exists a homogeneous prime ideal I in a 4-dimensional polynomial ring such that √ reg sat(I v ) = 9 + 2. v→+∞ v lim

On the positive side, in the setting of Theorem 8.4.1, Chardin proved in [71] that for every i the limit lim ai (I v M) + i − δ0 v v→+∞

exists and it is either −∞ or an integer, and the second case occurs for at least one i. This implies that at least for one i one has that ai (I v M) is eventually linear. Actually, for the smallest i such that HQi (I v M) = 0 for v  0, it follows from Theorem 8.4.1 and Remark 8.1.9 that ai (I v M) is eventually linear. Furthermore, combining Theorem 8.4.1 with Remark 8.1.9 one has, at least if the base ring is a field, that max{ai (I v M) + i : i = 0, . . . , c} is eventually linear for every c. Remark 8.4.6 When S = K [X 1 , . . . , X n ] and I is a homogeneous ideal of S of height h, then Lemma 8.1.10 implies t0 (S/I ) < t1 (I ) < · · · < th (S/I ) and hence that t0 (I ) < t1 (I ) < · · · < th−1 (I ). As height I v = height I , we have t0 (I v ) < t1 (I v ) < · · · < th−1 (I v ) for every v. In particular, with the notation of Theorem 8.4.1, δi = δ0 for i = 0, . . . , h − 1. Hence ti (I v ) = δ0 v + ci for i = 0, . . . , h − 1 and v  0 and c0 < c1 < · · · < ch−1 . Examples 8.4.8 and 8.4.9 show that δi can be smaller that δ0 for i > h − 1.

300

8 Castelnuovo-Mumford Regularity

Remark 8.4.7 The proof of Theorem 8.4.1 can be turned into a procedure to compute the asymptotic values of ti (I v M) as follows: (a) Compute a presentation of R = R(I, M) as a quotient of a free B-module where B is a polynomial ring over R. (b) Compute the Koszul homology Hi = Hi (Q R , R). (c) Compute the asymptotic value of ti (I v M) = ρ Hi (v) following the strategies outlined in Remark 8.3.5. Especially when the base ring R0 is a field, this approach can be implemented and, for “small” input data, the computation can be successfully completed. For instance, this is done in Examples 8.4.8 and 8.4.9. Example 8.4.8 Let I = (X 13 , X 1 X 22 , X 23 X 3 ) be the ideal of R = Q[X 1 , X 2 , X 3 ] and Q = (X 1 , X 2 , X 3 ). One has t0 (I v ) = 4v

for v ≥ 1,

t1 (I ) = 4v + 1 t2 (I v ) = 3v + 3

for v ≥ 1, for v ≥ 2.

v

These data are computed following the strategy outlined in Remark 8.4.7. In this case R = R(I ) is a quotient of R[Y1 , Y2 , Y3 ] with deg(Y1 ) = deg(Y2 ) = (3, 1) and deg(Y3 ) = (4, 1) and a presentation can be computed. A presentation of the Koszul homology H1 = H1 (Q, R) can be computed as well and it has bigraded Hilbert series H H1 (z, t) =

−2z 12 t 3 − z 9 t 2 + z 8 t 2 + 2z 5 t (1 − z 3 t)2 (1 − z 4 t)

with the decomposition H H1 (z, t) =

2z 5 t z8t 2 z9t 2 . + + (1 − z 3 t)2 (1 − z 3 t)2 (1 − z 3 t)(1 − z 4 t)

The three summands contribute to t1 (I v ) for different ranges of v, in which we must take the maximum of the corresponding linear functions—two of them coincide: ⎧ ⎪ ⎨3v + 2 v t1 (I ) = max 3v + 2 ⎪ ⎩ 4v + 1

for v ≥ 1, for v ≥ 2, for v ≥ 2.

Here is another interesting example with δ0 = δh−1 > δh > δh+1 when h = height I . Again the values are obtained following the strategy in Remark 8.4.7. Example 8.4.9 Let I = (X Y 2 , X 4 , X 2 Y T 2 , Y 3 Z 3 ) ⊂ Q[X, Y, Z , T ]. One has height I = 2 and

8.4 Regularity of Powers and Products of Ideals

t0 (I v ) = 6v,

t1 (I v ) = 6v + 1,

301

t2 (I v ) = 5v + 2,

t3 (I v ) = 4v + 6,

for all v ≥ 3. Remark 8.4.10 An important general question is the following. One observes that, for obvious reasons, t0 (I v ) ≤ vt0 (I ) holds for all v and hence one might ask if reg I v ≤ v reg I holds for all v. This turns out to be false in general and many counterexamples have been described in Conca [84]. The smallest one, in terms of number of generators, is the ideal I = (X 2 Y, X 2 Z , X Z 2 , Y Z 2 , X Z T ) of the polynomial ring S = Q[X, Y, Z , T ] for which reg I = 3 and reg I 2 = 7; see [84, Ex. 2.5]. As to determinantal ideals, at least in characteristic 0, the ideal I generated by the 3-minors of a generic symmetric 4 × 4 matrix has reg I = 3 and reg I 2 = 7, [84, Ex. 2.8]. This example is particularly interesting because reg I v = 3v for v = 3, 4. The question whether reg I v = 3v for v = 2 is mentioned by Bruns, Conca and Varbaro at the end of [48] and it is still unanswered. The analysis of this type of behavior is the subject of the paper of Borna [22]. On the other hand, the inequality reg I v ≤ v reg I for all v holds if I is an ideal of a polynomial ring S over a field with dim S/I ≤ 1. This result and its generalizations are discussed by Caviglia [68], Chandler [69], Chardin [70], Eisenbud et al. [121], Geramita et al. [135], Sidman [283].

Regularity of Products and Powers Now we consider a family of homogeneous ideals I1 , . . . , Im of R and a finitely generated graded R-module M. We will study the behavior of reg I1v1 · · · Imvm M as a function of v = (v1 , . . . , vm ) ∈ Nm . To this end we introduce the associated multi-Rees algebra: R(I1 , . . . , Im ) =



I1v1 · · · Imvm

v∈Nm

that can  be realized as the subalgebra of R[T1 , . . . , Tm ] whose elements have the form v∈Nm Fv T v with Fv ∈ I v where we have set T v = T1v1 · · · Tmvm

and

I v = I1v1 · · · Imvm

for v ∈ Nm to simplify the exposition. The polynomial ring R[T1 , . . . , Tm ] is naturally Z × Zm -graded if we extend the Z-grading on R by using the first coordinate to record the grading of R and then set deg T j = (0, e j ) where e j denotes the jth unit vector in Zm . In other words, the degree (i, v) ∈ Z × Zm homogeneous component of R[T1 , . . . , Tm ] is Ri T v . Evidently R(I1 , . . . , Im ) is a Z × Zm -graded subalgebra of R[T1 , . . . , Tm ] with: R(I1 , . . . , Im )(i,v) = (I v )i T v .

302

8 Castelnuovo-Mumford Regularity

Let f i1 , . . . , f igi be a homogeneous generating system of Ii . Set di j = deg f i j . So R(I1 , . . . , Im ) = R[ f i j Ti : i = 1, . . . , m and j = 1, . . . , gi ]. Then we consider B = R[Yi j : 1 ≤ i ≤ m, 1 ≤ j ≤ gi ] so that R(I1 , . . . , Im ) has a structure of B-module via the surjective R-algebra map B → R(I1 , . . . , Im ) sending Yi j to f i j ti . This map is indeed Z × Zm -graded if we equip B with the graded structure induced by the assignment deg(x) = ( j, 0) if x ∈ R j and deg(Yi j ) = (di j , ei ). Furthermore we let A = R0 [Yi j : 1 ≤ i ≤ m, 1 ≤ j ≤ gi ] ⊂ B with the induced Z × Zm -graded structure; A is canonically identified with B/Q R B. We may as well define the multi-Rees module R(I1 , . . . , Im , M) =



IvM

v∈Nm

which is finitely generated as R(I1 , . . . , Im )-module and hence as B-module. Now we consider the Koszul homology H (Q R , R(I1 , . . . , Im , M)). By construction  Hi (Q R , I v M) Hi = Hi (Q R , R(I1 , . . . , Im , M)) = v∈Zm

is a finitely generated Z × Zm graded B-module annihilated by Q R . Hence Hi is a finitely generated Z × Zm -graded A-module. Therefore the function ρ defined in Sect. 8.3, applied to the module Hi , gives ρ Hi (v) = sup{ j : Hi (Q R , I v M) j = 0} = ti (I v M) for every v ∈ Nv . This implies Theorem 8.4.11 (a) For i = 0, . . . , n, either Hi (Q R , I v M) = 0 for v  0 or Hi (Q R , I v M) = 0 for v  0. (b) grade(Q R , I v M) is constant, say equal to s, for v  0. (c) For i = 0, . . . , n − s and v  0, ti (I v M) is the supremum of finitely many linear functions of v. (d) For v  0 the regularity reg I v M is the supremum of finitely many linear functions of v.

8.4 Regularity of Powers and Products of Ideals

303

(e) Each linear function appearing in (c) and in (d) has the form w · v + c with c ∈ Z, w = (w1 , . . . , wm ) ∈ Nm and wi ∈ {di1 , . . . , digi }. Proof The theorem follows by applying Theorem 8.3.7 to the Koszul homology modules Hi (Q, R(I, M)), arguing exactly as in the proof of Theorem 8.4.1.  If each ideal I j is generated in a single degree, say d j , then there is only one choice for the vector w appearing in Theorem 8.4.11, i.e., (d1 , . . . , dm ). But then the supremum in Theorem 8.4.11 is attained by a single linear function, the one with the largest constant term. This implies Corollary 8.4.12 If each I j is generated in a single degree, say d j , then for every i = 0, . . . , n − s there exists ci ∈ Z such that ti (I v M) = d · v + ci

for v  0

where d = (d1 , . . . , dm ). In particular, there exists c ∈ Z such that reg I v M = d · v + c

for v  0.

Remark 8.4.13 For i = 1, . . . , m let Ji be a graded subideal of Ii . Then R(J1 , . . . , Jm ) is an R-subalgebra of R(I1 , . . . , Im ). If R(I1 , . . . , Im , M) is finitely generated as an R(J1 , . . . , Jm )-module (e.g., each Ji is a reduction of Ii with respect to M) then the Koszul homology H (Q R , R(I1 , . . . , Im , M)) is finitely generated over R(J1 , . . . , Jm )/Q R R(J1 , . . . , Jm ). As an R0 -algebra the latter is generated by elements of degrees (a, ei ) ∈ Z × Zm where a is the degree of a minimal generator of Ji . Hence the coefficients of the linear forms in Theorem 8.4.11 can be taken in the set of the degrees of the generators of Ji . Therefore Corollary 8.4.12 holds more generally if each Ii has a reduction generated in a single degree.

Two Examples for Nonlinearity We conclude the section by presenting two examples showing that, in general, t j (I v ) and reg I v need not be linear functions of v ∈ Zm (for v  0). Example 8.4.14 Consider the ideals I1 = (X, Y 2 ), I2 = (X 2 , Y ) in the polynomial ring R = K [X, Y ]. We claim that

ti (I1v1 I2v2 ) =

 max{2v1 + v2 , v1 + 2v2 } + i 0

if i = 0, 1 and (v1 , v2 ) ∈ N2 \ {(0, 0)}, if i = 0 and (v1 , v2 ) = (0, 0).

304

8 Castelnuovo-Mumford Regularity

So that reg I1v1 I2v2 = max{2v1 + v2 , v1 + 2v2 }

for all (v1 , v2 ) ∈ N2 .

One could compute these invariants by the strategy outlined in Remark 8.4.7 but for this simple case we can as well describe directly a minimal resolution of I1v1 I2v2 . First we observe that I1v1 I2v2 is minimally generated by the 1 + v1 + v2 elements X v1 Y v2 , X v1 −1 Y v2 +2 , X v1 −2 Y v2 +4 , . . . , Y v2 +2v1 , || X v1 Y v2 , X v1 +2 Y v2 −1 , X v1 +4 Y v2 −2 , . . . , X v1 +2v2 . This confirms our formula t0 (I1v1 I2v2 ) = max{2v1 + v2 , v1 + 2v2 }

for all v = (v1 , v2 ) ∈ N2 .

By Hilbert’s syzygy theorem I1v1 I2v2 has a free resolution of length 1 (if v1 + v2 > 0). We claim the syzygy module of I1v1 I2v2 is generated by the rows of the following (v1 + v2 ) × (v1 + v2 + 1) matrix ⎛

⎞ Y 2 −X ⎜ ⎟ Y 2 −X ⎜ ⎟ ⎜ ⎟ . . .. .. ⎜ ⎟ ⎜ ⎟ 2 ⎜ ⎟ Y −X ⎜ 2 ⎟ ⎟. −Y X ϕ=⎜ ⎜ ⎟ 2 ⎜ ⎟ −Y X ⎜ ⎟ 2 ⎜ ⎟ −Y X ⎜ ⎟ ⎜ ⎟ . . .. .. ⎝ ⎠ 2 X −Y In fact, the rows of ϕ are syzygies. Therefore we have a complex ϕ

0 → R v1 +v2 − → R 1+v1 +v2 → I1v1 I2v2 → 0. Both X v1 +2v2 and Y 2v1 +v2 appear as maximal minors of ϕ. Thus the ideal of maximal minors of ϕ has grade 2 in R. Now the Buchsbaum-Eisenbud exactness criterion (for example, see [52, 1.4.3]) implies that our complex is exact, and so we know the first syzygy module of I1v1 I2v2 . Keeping track of the degrees one obtains that   t1 (I1v1 I2v2 ) = max 2v1 + v2 + 1, v1 + 2v2 + 1 for all (v1 , v2 ) ∈ N2 \ {(0, 0)}.

8.5 Linear Powers and Linear Products

305

In the example above the regularity and the invariants ti (I v ) are given by the supremum of two linear functions for v  0. But even for two ideals, i.e., m = 2, the number of linear functions needed can be larger as the next example shows. Example 8.4.15 Consider the ideals I1 = (X, Y 2 , Z 3 ) and I2 = (X 4 , Y 3 , Z ) in R = K [X, Y, Z ]. For every v ≥ (1, 1) one has: t0 (I1v1 I2v2 ) = t1 (I1v1 I2v2 ) = t2 (I1v1 I2v2 ) = reg I1v1 I2v2 =

max{v1 + 4v2 + 1, 2v1 + 3v2 , 3v1 + v2 max{v1 + 4v2 + 2, 2v1 + 3v2 + 1, 3v1 + v2 + 2 max{v1 + 4v2 + 3, 2v1 + 3v2 + 2, 3v1 + v2 + 3 max{v1 + 4v2 + 1, 2v1 + 3v2 , 3v1 + v2 + 1

}, }, }, }.

The computation has been done implementing the strategy outlined in Remark 8.4.7.

8.5 Linear Powers and Linear Products We keep the notation of Sect. 8.4. Assume now that the minimal generators of the ideal I have all degree d and that the minimal generators of M have all degree d0 . Under the current assumptions, R(I ) and R(I, M) can be given a compatible and “normalized” Z2 -graded structure: R(I )(i,v) = (I v )vd+i , R(I, M)(i,v) = (I v M)vd+d0 +i . From the presentation point of view, this amounts to assigning deg Yi = (0, 1) so that B = R[Y1 , . . . , Yg ] is a Z2 -graded R0 -algebra with generators in degree (1, 0), namely the elements of R1 , and in degree (0, 1), the Yi . Remark 8.5.1 Let J be the kernel of the presentation map B → R(I ). The following types of elements of J play a prominent role: (1) the elements of degree (a, 1) for some a ∈ N. They correspond to the syzygies among the generators of I and all together generate the defining ideal of the symmetric algebra of I ; (2) the elements of degree (0, a) for some a ∈ N. They correspond to the algebraic relations among the generators of I , i.e., the defining equations of the “fiber ring” R(I )/Q R(I ) ∼ = R0 [Id ] as quotient of R0 [Y1 , . . . , Yg ]. The ideal I is said to be of fiber type if J is generated by the types of relations (1) and (2). This notion was introduced by Herzog et al. [171]. An example of an ideal of fiber type is the ideal of maximal minors, see Proposition 6.2.21. Returning to the general setting we observe that I v M is generated by elements of degree vd + d0 and therefore reg I v M ≥ vd + d0 for every v.

306

8 Castelnuovo-Mumford Regularity

Definition 8.5.2 We say that I has linear powers with respect to M if I v M has a linear resolution for all v ∈ N, that is, reg I v M = vd + d0 for every v ∈ N. We will give a characterization of linear powers in terms of the homological properties of the Rees module R(I, M). With the notation introduced in Sect. 8.2, R(I, M)(∗,v) = (I v M)(vd + d0 ). Applying Proposition 8.2.3 one gets   reg(1,0) R(I, M) = max reg R(I, M)(∗,v) : v ∈ N   = max reg I v M − vd − d0 : v ∈ N . Summing up, we have proved Theorem 8.5.3 (a) reg I v M ≤ vd + d0 + reg(1,0) R(I, M) for all v and the equality holds for at least one v. (b) I has linear powers with respect to M if and only if reg(1,0) R(I, M) = 0. Both from the computational and theoretical point of view, it can be difficult to access the invariant reg(1,0) R(I, M). On the other hand, we have the following criterion which turns out to be quite useful in concrete applications. Proposition 8.5.4 Let R = K [X 1 , . . . , X n ] and J ⊂ B = K [X 1 , . . . , X n , Y1 , . . . , Yg ] be the kernel of the presentation map B → R(I ). If there is a monomial order < on B such that in(J ) is generated by elements of degree at most 1 in the X i , then reg(1,0) R(I ) = 0. Proof By virtue of Proposition 8.2.3, the (1, 0)-regularity of R(I ) can be computed via the B-free minimal Z2 -graded resolution F of R(I ). From this point of view, the claim reg(1,0) R(I ) = 0 is equivalent to the assertion that Fi is generated by elements whose degree in Z2 has first coordinate bounded by i. By Corollary 1.6.9 we may replace F by the B-free minimal Z2 -graded resolution G of B/ in(J ). Furthermore we may also replace G by a nonminimal resolution T of B/ in(J ) as long as we prove that Ti is generated by elements whose degree satisfies the required condition. We choose T to be the Taylor resolution of B/ in(J ), see Miller and Sturmfels [245, Sect. 4.3.2]. By definition, every generator of Ti has a degree that is bounded above by the degree of the product of i generators of in(J ) and, by assumption, the first coordinate of such a degree is ≤ i.  We have also Proposition 8.5.5 Let R = K [X 1 , . . . , X n ] and suppose there exists a monomial order < on R such that in(I ) is generated by monomials of degree d and in(I v ) = in(I )v for all v ≥ 1. Then reg(1,0) R(I ) ≤ reg(1,0) R(in(I )). The combination of Propositions 8.5.4 and 8.5.5 yields

8.5 Linear Powers and Linear Products

307

Proposition 8.5.6 Let R = K [X 1 , . . . , X n ] and suppose there exists a monomial order on R such that in(I ) = (μ1 , . . . , μg ) with deg μi = d and in(I v ) = in(I )v for all v. Assume furthermore that the kernel J1 of the R-algebra presentation B → R(in(I )) sending Yi to μi T has an initial ideal in(J1 ), with respect to a monomial order on B, that is generated by elements of degree at most 1 in the X i . Then reg(1,0) R(I ) = 0. The condition in(I v ) = in(I )v for all v, or the equivalent formulation in(R(I )) = R(in(I )), can be tested directly or by the lifting criterion for Sagbi bases, Theorem 1.3.14, as we have seen in the proof of Theorem 6.2.19 for maximal minors. Another instance that allows us to conclude that an ideal has linear powers is Blum’s theorem [20]: Theorem 8.5.7 Let I be an ideal in the polynomial ring K [X 1 , . . . , X n ] that is generated by elements of constant degree. If the Rees algebra R(I ) is Koszul, then all powers of I have linear resolutions. The definitions and the results just proved can be extended to a collection of ideals. Indeed, suppose we are given homogeneous ideals I1 , . . . , Im of R such that Ii is generated by gi elements of degree di and a graded m R-module M generated by di vi + d0 for every v ∈ Nm . elements of degree d0 . We have reg I1v1 · · · Imvm M ≥ i=1 Definition 8.5.8 We say that I1 , . . . , Im have linear products with respect to M if I1v1 · · · Imvm M has alinear resolution for all v = (v1 , . . . , vm ) ∈ Nm , that is, m di vi + d0 for all v. reg I1v1 · · · Imvm M = i=1 To simplify notation we set d = (d1 , . . . , dm ) ∈ Nm , I v = I1v1 · · · Imvm for v ∈ Nm , R(I ) = R(I1 , . . . , Im ) and R(I, M) = R(I1 , . . . , Im , M). In this setting R(I ) and R(I, M) can be given a compatible and “normalized” Z × Zm -graded structure as follows. For (i, v) ∈ Z × Zm we put R(I )(i,v) = (I v )v·d+i , R(I, M)(i,v) = (I v M)v·d+d0 +i . From the presentation point of view, this amounts to setting deg Yi j = (0, ei ) so that B = R[Yi j : 1 ≤ i ≤ m and 1 ≤ j ≤ gi ] is a Z × Zm -graded R0 -algebra with generators in degree (1, 0) ∈ Z × Zm , namely the elements of R1 , and in degree (0, ei ), the Yi j . This makes the presentation map B → R(I ) compatible to the Z × Zm structure. With the notation introduced in Sect. 8.2, we have R(I, M)(∗,v) = (I v M)(v · d + d0 ) for all v ∈ Nm . So, applying Proposition 8.2.4 yields   reg(1,0) R(I, M) = max reg R(I, M)(∗,v) : v ∈ Nm   = max reg I v M − v · d − d0 : v ∈ Nm . This implies

308

8 Castelnuovo-Mumford Regularity

Theorem 8.5.9 (a) reg I v M ≤ v · d + d0 + reg(1,0) R(I, M) for all v ∈ Nm and the equality holds for at least one v ∈ Nm . (b) The ideals I1 , . . . , Im have linear products with respect to the module M if and only if reg(1,0) R(I, M) = 0. Also Propositions 8.5.4 and 8.5.5 can be extended to the general setting giving a concrete way of checking that a family of ideals has linear products. In Sect. 8.6 we will see these ideas in action for the families of ideals of maximal minors.

8.6 A Family of Determinantal Ideals with Linear Products The goal of this section is to present families of determinantal ideals with linear products. Let K be a field and X an n × n matrix of indeterminates X i j over K . We write K [X ] for the polynomial ring in the variables X i j . For each t, 1 ≤ t ≤ n, let Jt ⊂ K [X ] denote the ideal generated by the t-minors of the first t rows of X . By construction, each Jt is the ideal of maximal minors of a submatrix of X and hence we know its properties very well. In particular, we have seen in Remark 6.2.26, Corollary 3.5.3 and Theorem 6.2.22 that (a) (b) (c) (d)

Jt has linear powers, Jt has primary powers, R(Jt ) is Cohen–Macaulay and normal, R(Jt ) is defined by a Gröbner basis of quadrics. In particular R(Jt ) is Koszul.

Now let R(J ) = R(J1 , . . . , Jn ) be the multi-Rees algebra associated to the family of ideals J = (J1 , J2 , . . . , Jn ). Our goal is to prove that: Theorem 8.6.1 With the notation above, we have: (a) The family of ideals J has linear products; (b) for every a1 , . . . , an ∈ N one has the following primary decomposition: n  t=1

Jtat =

n  t=1

Jtbt ,

bt =

n 

ai ;

i=t

(c) the multi-Rees algebra R(J ) is Cohen–Macaulay and normal; (d) R(J ) is defined by a Gröbner basis of quadrics. In particular R(J ) is Koszul. The main tool we will use is the Sagbi deformation, i.e., we will study the initial algebra of R(J ) via lifting of relations as we have done in Proposition 6.2.21 for a single ideal of maximal minors.

8.6 A Family of Determinantal Ideals with Linear Products

309

In K [X ] we consider a diagonal monomial order, for example the lexicographic order induced by X 11 > X 12 > · · · > X 1n > X 21 > · · · > X nn , and any extension of it to the polynomial ring K [X, Y ] = K [X ][Y1 , . . . , Yn ]. The multi-Rees algebra R(J ) = K [X ][J1 Y1 , . . . , Jn Yn ] is a K -subalgebra of K [X, Y ] and we will consider its initial algebra with respect to the given term order. Let Hi be the ideal generated by the main diagonals of the i minors of the first i rows of X and H = (H1 , . . . , Hn ). The first aim is to find the defining equations of R(H ) = K [X ][H1 Y1 , . . . , Hn Yn ]. We consider the sets At = {(a1 , . . . , at ) : 1 ≤ a1 < a2 < · · · < at ≤ n}, A=

n 

At ,

t=1

and observe that A is partially ordered by (a1 , . . . , at ) ≤ (b1 , . . . , bs )

⇐⇒

t ≥ s and ai ≤ bi for all i = 1, . . . , s.

Actually A is a distributive lattice with respect to this order with (a1 , . . . , at ) ∧ (b1 , . . . , bs ) = (min(a1 , b1 ), min(a2 , b2 ), . . . , min(as , bs ), as+1 , . . . , at ), (a1 . . . at ) ∨ (b1 . . . bs ) = (max(a1 , b1 ), max(a2 , b2 ), . . . , max(as , bs )). if t ≥ s and similarly for t ≤ s. For each a = (a1 , . . . , at ) ∈ A we set [a1 . . . at ] = [1 . . . t|a1 . . . at ], as usual and observe the partial order on A actually corresponds to the partial order on the minors. Furthermore, for each a = (a1 , . . . , at ) ∈ A we use our standard notation a1 , . . . , at  = in([a1 . . . at ]) = X 1a1 · · · X tat , and introduce a variable Pa . This allows us to define K [X ]-algebra homomorphisms  : K [X ][Pa : a ∈ A] → R(H ) = K [X ][H1 Y1 , . . . , Hn Yn ] and

310

8 Castelnuovo-Mumford Regularity

 : K [X ][Pa : a ∈ A] → R(J ) = K [X ][J1 Y1 , . . . , Jn Yn ] defined by (Pa ) = a1 , . . . , at Yt and  (Pa ) = [a1 , . . . , at ]Yt for every a = (a1 , . . . , at ) ∈ A. First we determine the kernel of . Proposition 8.6.2 The ideal Ker  is generated by (1) the Hibi relations Pa Pb − Pa∧b Pa∨b with a, b ∈ A incomparable, and (2) the linear relations, i.e., relations of the form X i j Pa − X ik Pb with a = (a1 . . . , ai , . . . , as ), ai−1 < j ≤ ai , b = (a1 . . . , ai−1 , j, ai+1 , . . . , as ), k = ai , and by convention a0 = 0. These polynomials form a Gröbner basis of Ker  with respect to monomial orders for which the underlined terms are initial. Proof The elements in (1) and (2) are clearly in Ker , and it is enough to prove that they are a Gröbner basis of Ker . First note that a monomial order selecting the underlined monomials is given by taking the reverse lexicographic order associated to a total order of the Pa that refines the partial order on A. To prove the assertion it suffices to show that for an arbitrary monomial in the image of , say W Ys1 · · · Yse ,

s1 ≥ · · · ≥ se , W a monomial in the X i j ,

the preimage −1 (W Ys1 · · · Yse ) contains exactly one monomial that is not divisible by the initial terms we have identified. A monomial U Pa1 · · · Pae with ai = (ai,1 , ai,2 , . . . , ai,si ) ∈ A for i = 1, . . . , e and where U is a monomial in the X i j is not divisible by the initial terms we have identified if and only if it satisfies (i) a1 ≤ a2 ≤ · · · ≤ ae in the poset A; (ii) for every X i j dividing U and for every k, 1 ≤ k ≤ e, one has either j ≥ ak,i or j ≤ ak,i−1 where, by convention, ak,0 = 0. Conditions (i) and (ii) uniquely determine a1 as the minimum of the b ∈ As1 such that b|W . Similarly a2 is uniquely determined as the minimum of the b ∈ As2 such  that a1 b|W and so on. Example 8.6.3 For n = 4 the generators of Ker  are

8.6 A Family of Determinantal Ideals with Linear Products

X 1,3 P4 − X 1,4 P3 X 1,2 P3 − X 1,3 P2 X 2,3 P24 − X 2,4 P23 X 1,2 P34 − X 1,3 P24 X 2,2 P13 − X 2,3 P12 X 2,2 P134 − X 2,3 P124 P34 P1 − P14 P3 P14 P23 − P13 P24 P234 P13 − P134 P23

X 1,2 P4 − X 1,4 P2 X 1,1 P3 − X 1,3 P1 X 2,3 P14 − X 2,4 P13 X 1,1 P34 − X 1,3 P14 X 1,1 P23 − X 1,2 P13 X 1,1 P234 − X 1,2 P134 P24 P1 − P14 P2 P234 P1 − P134 P2 P234 P12 − P124 P23

311

X 1,1 P4 − X 1,4 P1 X 1,1 P2 − X 1,2 P1 X 2,2 P14 − X 2,4 P12 X 1,1 P24 − X 1,2 P14 X 3,3 P124 − X 3,4 P123 P34 P2 − P24 P3 P23 P1 − P13 P2 P234 P14 − P134 P24 P134 P12 − P124 P13 .

For use in the next proposition we set (1) X = {X i j : 1 ≤ i, j ≤ n}, (2) F = {[a1 , . . . , at ]Yt : 1 ≤ t ≤ n and a ∈ A} and proceed to prove Proposition 8.6.4 The set X ∪ F is a Sagbi basis of R(J ). In particular one has in(R(J )) = R(H ). Proof The statement follows from Theorem 1.3.14 and Proposition 8.6.2, provided we can “lift” the Hibi relations and the linear relations. That the linear relations lift can be seen directly as we have done in the discussion before Lemma 6.2.20 or by using the fact that the maximal minors form a Gröbner basis with respect to a diagonal order by Theorem 4.3.2. So we are left with the Hibi relations. That they can be lifted, can be seen as in the case of the Grassmannian (Proposition 6.2.2). The only difference is that we must now look at products of two arbitrary row initial bitableaux instead of two initial bitableaux of the same size. The liftability of the Hibi relations follows also from Theorem 6.2.15. The details are left to the reader.  Proof of Theorem 8.6.1 By Proposition 8.6.2 the defining ideal Ker  of the toric algebra R(H ) has a quadratic square-free initial ideal. Hence by 6.1.13 R(H ) is normal and hence Cohen–Macaulay. Then statement (c) follows because by Theorem 8.6.4 R(H ) is the initial algebra of R(J ) and hence by Theorem 1.6.11 the Cohen–Macaulayness and normality deform. Furthermore by Proposition 1.3.16 since Ker  has a quadratic initial ideal the same property holds also for the defining ideal Ker  of R(J ), i.e., (d) holds. Furthermore (the multigraded variant of) Proposition 8.5.4 applies and we have that reg(1,0) R(J ) = 0 and hence J has linear products by Theorem 8.5.9, that is (a) holds. It remains to prove the assertion (b). It is clear that the right hand side of the equality is indeed a primary decomposition since each Jt is a prime ideal with primary powers. Here we observe that, since Jt ⊂ Js for s ≤ t we have that nt=1 Jtat ⊂ Jsbs for every s, i.e., the inclusion n n   at Jt ⊂ Jtbt (8.9) t=1

t=1

holds. Now we have the following chain of inclusions:

312

8 Castelnuovo-Mumford Regularity n 

(i) Htat ⊂

in

 n 

t=1

 Jtat

(ii)



⊂ in

t=1

n 

 Jtbt

(iii)

⊂

t=1

n 

Htbt ,

where (i) and (iii) hold by Lemma 2.1.11 and (ii) holds by Eq. (8.9). If we prove that n n   Htbt ⊂ Htat t=1

(8.10)

t=1

(8.11)

t=1

then it follows that all the inclusions are actually equalities in Eq. (8.10). In particular in

 n  t=1

 Jtat

= in

 n 

 Jtbt

t=1

that Proposition 1.4.2 implies the desired equality. So it remains to prove the inclusion (8.11). We argue by induction on |a| = a1 + · · · + an . Let k = max{i : ai > 0}, so . . , 0) and that bi = 0 for i > k and bk = ak > 0. Let a = (a1 , . . . , ak−1 , ak − 1, 0, . b = (b1 − 1, . . . , bk−1 − 1, bk − 1, 0, . . . , 0). Let M be a monomial in nt=1 Htbt . In particular, M ∈ Hk . Hence there exists c = (c1 , . . . , ck ) ∈ A such that c|M and we may assume that c is the smallest (with respect to the partial order of At ) monomial of that form with this property. It is well defined since if c  and c  divide M, so does c ∧ c .  We have M = cN . If we prove that N ∈ kt=1 Htbt −1 then, by induction, it k−1 at ak −1 follows that N ∈ ( t=1 Ht )Hk , and hence M ∈ kt=1 Htat as desired. To see that N ∈ Htbt −1 for all t ≤ k, let us observe that M ∈ Htbt and hence M is divisible by  f 1  · · ·  f bt  with f 1 , . . . , f bt ∈ At . As  f i  f j  =  f i ∧ f j  f i ∨ f j  we may assume that f 1 ≤ f 2 ≤ · · · ≤ f bt in the partial order of A. We may also assume that f 1 is the smallest element of At with  f 1 |M. But then, by the choice of f 1 and c, f 1 must coincide with the first t coordinates of c, so that  f 2  · · ·  f bt  divides N . That is, N ∈ Htbt −1 and we are done.  Remark 8.6.5 In Remark 6.2.4 we have observed that the Hibi rings and Hibi relations show up in the toric deformation of the Grassmannian. In the present context, the Hibi ring associated to A coincides with the toric deformation of the coordinate ring K [[a1 , . . . , at ]Yt : a ∈ A] ∼ = K [[a1 , . . . , at ] : a ∈ A] of the complete flag variety that is a direct summand of the multi-Rees algebra R(J ). The coordinate ring of the complete flag variety and its toric deformation have been studied in Theorem 6.2.17. Remark 8.6.6 The results of Theorem 8.6.1 can be generalized to the larger class of northeast ideals of maximal minors introduced in Remark 4.3.14. See Bruns and Conca [45] for details. In particular, the family of northeast ideals has linear products. Remark 8.6.7 The multigraded Rees rings R(J ) and R(H ) are actually Gorenstein rings. Since in(R(J )) = R(H ) by Corollary 1.6.12, it is enough to prove the statement for R(H ) or for R(J ). For R(H ) one can either exploit the toric nature and use

8.7 Maximal Minors of Matrices of Linear Forms and Linear Powers

313

Danilov–Stanley theorem and its Corollary 6.1.10. Alternatively one can prove by decreasing induction on t that R(Jt , Jt+1 , . . . , Jn ) is Gorenstein and make use a theorem of Herzog and Vasconcelos [173, Theorem(c), p. 183], see [45, Theorem 6.1] where this strategy is applied to the larger class of northeast ideals introduced in Remark 4.3.14.

8.7 Maximal Minors of Matrices of Linear Forms and Linear Powers As we have recalled at the beginning of Sect. 8.6, the ideal of maximal minors Im (X ) of the m × n matrix of variables X = (X i j ) has linear powers, i.e., reg Im (X )k = km for all k ∈ N. Under which conditions does the same statement hold if we replace X with a matrix of linear forms? More precisely: Question 8.7.1 Let L = (L i j ) be an m × n matrix of linear forms of a polynomial ring S = K [X 1 , . . . , X c ]. Under which conditions on L does Im (L) have linear powers? It is harmless to assume that I1 (L) = (X 1 , . . . , X c ). Then clearly S/Im (L) is obtained from K [X ]/Im (X ) by modding out a sequence of linear forms that, if L is general enough, can be S/Im (L)k -regular for all k. If this is the case the linear resolution of Im (X )k specializes to a linear resolution of Im (L)k and hence Im (L) has linear powers. But when is L general enough to allow this specialization? A result due to Eisenbud and Huneke [120, Theorem 3.5] answers this question in terms of specialization of the Rees algebras. As a corollary of Eisenbud and Huneke’s result we have: Theorem 8.7.2 Let S be a Cohen-Macaulay domain and let L = (L i j ) be an m × n matrix with entries in S. Suppose that height I j (L) ≥ (m + 1 − j)(n − m) + 1 for every j = 1, . . . , m, then proj dim Im (L)k ≤ m(n − m) for all k and R(Im (L)) is Cohen-Macaulay. If furthermore S = K [X 1 , . . . , X c ] and the entries of L are linear forms then (a) Im (L) is of fiber type. (b) Im (L) has linear powers. (c) βiS (Im (L)k ) = βiK [X ] (Im (X )k ) for all i, k. Remark 8.7.3 The statement in Theorem 8.7.2 about the projective dimension appears in Bruns and Vetter [61, 9.27] for the matrix of variables X . Remark 8.7.4 Under the assumption of Theorem 8.7.2, when S = K [X 1 , . . . , X c ] and the entries of L are linear forms one can say, more precisely, that R(Im (L)) is defined by equations of degree (1, 1), that is the linear relations among the maximal minors and the Plücker relations in degree (0, 2). Furthermore proj dim Im (L)k = min{k, m}(n − m), as can be derived from Akin et al. [5, Theorem 5.4]; see [48, 3.2.] for details.

314

8 Castelnuovo-Mumford Regularity

If Im (L) has the expected height, height Im (L) = n − m + 1, then reg Im (L) = m because then Im (L) is resolved by the Eagon–Northcott complex. But the assumption height Im (L) = n − m + 1 alone is not enough to guarantee that Im (L) has linear powers, as the examples in Examples 8.7.5 and 8.7.6 show. Example 8.7.5 The ideal I3 (L) of 3-minors of the 3 × 5 matrix ⎛

⎞ X1 0 0 X2 X4 L = ⎝ 0 0 X 3 X 2 X 5⎠ 0 X2 X1 X3 X3 with linear entries in Q[X 1 , X 2 , X 3 , X 4 , X 5 ] has the expected height 3, and hence it has a linear resolution. Its square I3 (L)2 does not have a linear resolution. Actually one has reg I3 (L)2 = 7. Example 8.7.6 The ideal I4 (L) of 4-minors of the following 4 × 5 matrix ⎛

X1 ⎜ 0 L=⎜ ⎝ 0 0

0 X2 0 0

0 0 X2 X1

0 0 X3 X4

⎞ X3 X4 ⎟ ⎟ 0 ⎠ X3

with linear entries in Q[X 1 , X 2 , X 3 , X 4 ] has the expected height 2 and hence it has a linear resolution. Its square I4 (L)2 does not have a linear resolution. Actually one has reg I4 (L)2 = 9. The following general results will give us a way to address Question 8.7.1. Lemma 8.7.7 Let S = K [X 1 , . . . , X c ] with c > 1 and let I be a homogeneous ideal generated by elements of degree m and with a linear resolution. Let z ∈ S1 be an almost S/I -regular element. Set R = S/(z) and J = I + (z)/(z). Then: (a) J has a linear resolution. (b) The Betti numbers βiR (J ) of J as an R-module depend only on the Betti numbers βiS (I ) of I as an S-module. Indeed, βiR (J ) = βiS (I ) −

  c−1 S βc−1 (I ). i

Proof (a) Denote the homogeneous maximal ideal of S by m S . With M = S/I we may apply Proposition 8.1.13 and obtain reg M = max{reg M/z M, a0 (M)}. By hypothesis reg M = reg I − 1 = m − 1. Since M/z M = R/J , one has m − 1 = max{reg R/J, a0 (S/I )}.

(8.12)

8.7 Maximal Minors of Matrices of Linear Forms and Linear Powers

315

That R is isomorphic to a polynomial ring in c − 1 variables implies reg R/J = reg J − 1. Hence (8.12) entails reg J ≤ m. But reg J ≥ m holds since J is generated in degree m (and J = 0 because c > 1). Therefore reg J = m, i.e., J has a linear resolution. (b) Note that a0 (S/I ) ≤ m − 1. By (8.12), Proposition 8.1.12 implies that (I : z)/I and (I : m S )/I both vanish in degree ≥ m, coincide in degree m − 1, and are obviously 0 in degrees < m − 1. Hence I : z = I : m S . Observe that I ∩ (z) = z(I : z) = z(I : m S ). Therefore the short exact sequence 0 → I ∩ (z) → I → J → 0 can be rewritten as

z

0 → I : m S (−1) → I → J → 0 and hence we have the following relation among the Hilbert series: H J (t) = HI (t) − t HI :mS (t).

(8.13)

We have already observed that I : m S and I coincide from degree m on and since the cth Koszul homology Hc (m S , S/I ) equals (I : m S /I )(−c) we have dim K (I : m S )m−1 = βc (S/I ) = βc−1 (I ). Hence HI :mS (t) = βc−1 (I )t m−1 + HI (t) and we may rewrite (8.13) as H J (t) = (1 − t)HI (t) − βc−1 (I )t m .

(8.14)

Since both I and J have linear resolutions, we have c−1  (−1)i βi (I )t m+i

HI (t) = and

i=0

c−2 

H J (t) =

(8.15)

(1 − t)c

(−1)i βi (J )t m+i

i=0

(1 − t)c−1

.

(8.16)

Substituting (8.15) and (8.16) in (8.14), clearing the denominators and dividing both sides with t m yields

316

8 Castelnuovo-Mumford Regularity c−2 c−1   (−1)i βi (J )t i = (−1)i βi (I )t i − (1 − t)c−1 βc−1 (I ) i=0

i=0

and, substituting t with −t, one obtains c−2 

βi (J )t i =

i=0

c−1 

βi (I )t i − (1 + t)c−1 βc−1 (I ).

i=0

Now comparing the coefficients one gets the desired formula.



We now strengthen the hypothesis of Lemma 8.7.7. Lemma 8.7.8 Let S = K [X 1 , . . . , X c ] with c > 1 and let I be a homogeneous ideal generated by elements of degree m with linear powers. Let z ∈ S1 be an almost S/I k -regular element for all k. Set R = S/(z) and J = I + (z)/(z). Then: (a) J has linear powers. (b) For all k one has βiR (J k ) = βiS (I k ) −

  c−1 S βc−1 (I k ). i

(c) If I is of fiber type, then J is of fiber type. Proof Since J k = I k + (z)/(z), one may apply Lemma 8.7.7 to each power I k of I and hence (a) and (b) follow. For (c), consider the Rees rings R(I ) and R(J ) with the standard Z2 -graded structure described in Sect. 8.5 and represent them as quotients of R = S[Y1 , . . . , Yv ] where v is the number of generators of I . We have a short exact sequence: 0→V =

 I k ∩ (z) → R(I )/z R(I ) → R(J ) → 0. zIk k∈N

As we have seen already in the proof of Lemma 8.7.7, I k : z = I k : m S and I k : z/I z vanishes in degree km and higher. Hence I k ∩ (z) = z(I k : z) = z(I k : m S ) and V =

 z(I k : m S ) zIk k∈N

so that V is annihilated by m S . Keeping track of the degrees, one sees that V(i,k) = 0 for every i > 0, i.e., V gets identified with the kernel of the map K [Im ] → K [Jm ]. There is a short exact sequence Tor 1R (R(I )/z R(I ), K ) → Tor 1R (R(J ), K ) → V /(Y )V.

8.7 Maximal Minors of Matrices of Linear Forms and Linear Powers

317

By assumption, Tor 1R (R(I ), K )(i,k) = 0 if k > 1 and i > 0. Since z is a nonzero divisor of degree (1, 0) on the domain R(I ) we get that Tor 1R (R(I )/z R(I ), K )(i,k) = 0 if k > 1 and i > 0. Furthermore V /(Y )V vanishes in degree (i, k) with i > 0. Sum ming up, Tor 1R (R(J ), K )(i,k) = 0 if k > 1 and i > 0, i.e., J is of fiber type. Remark 8.7.9 In the setting of Lemma 8.7.8, z ∈ S1 is almost S/I k -regular for all k if and only if z ∈ / P for all P ∈ ∪k Ass S (S/I k ) with P = m S = (X 1 , . . . , X c ). A classical theorem of Brodmann [32] asserts that the set ∪k Ass S (S/I k ) is finite. Therefore, by the usual prime avoidance, if K is infinite then there actually exists a z ∈ S1 that is almost S/I k -regular for all k. We will employ Lemma 8.7.8 to approach Question 8.7.1 and, to this end, we have to be able to identify an element z that is almost S/I k -regular for all k where I = Im (L). As observed in Remark 8.7.9, this can be done by keeping ∪k Ass S (S/I k ) under control. For this goal, the following result will play an important role. We state it for general Noetherian Cohen–Macaulay rings because this makes it easier to do an induction. Lemma 8.7.10 Let S be a Cohen–Macaulay domain and L an m × n matrix with entries in S. Suppose that height Im (L) = n − m + 1 and height I j (L) ≥ (m + 1 − j)(n − m) + 2 for every j = 1, . . . , m − 1. Then Ass S/Im (L)k = Min Im (L) for every k > 0. Proof We argue by induction on m. Let m = 1. Then I1 (L) is generated by a regular sequence. Then for every k the quotient S/Im (L)k has projective dimension ≤ n by Theorem 8.7.2 or by [61, Proposition 2.14]. Therefore S/Im (L)k is Cohen-Macaulay and the associated primes of S/I1 (L)k are the minimal primes of I1 (L)k , i.e., the minimal primes of I1 (L). Now let m ≥ 2, and let P be any prime ideal of S associated to S/Im (L)k . Observe that by Remark 8.7.3 the projective dimension of S/Im (L)k is at most m(n − m) + 1. If P contains I1 (L) then height P ≥ height I1 (L) ≥ m(n − m) + 2. But this is a contradiction since the projective dimension of S P /Im (L)k S P is ≤ m(n − m) + 1 and P S P is associated to S P /Im (L)k S P . Now suppose that P does not contain I1 (L). Then one entry of L, say u = L mn , does not belong to P. We can apply the standard inversion argument Lemma 3.4.4 by inverting u. So I j (L)Su = I j−1 (L ) where L is an (m − 1) × (n − 1) matrix with entries in Su , and height I j−1 (L ) ≥ height I j (L). The assumption about height I j (L) depends only on differences of n − m and m − j hence it holds for L as well. Since

318

8 Castelnuovo-Mumford Regularity

the inversion of an element outside P does not affect the property of P being an associated prime, we can apply the induction hypothesis. We obtain that P Su ∈  Min Im (L)Su and hence P ∈ Min Im (L). Corollary 8.7.11 In the setting of Lemma 8.7.10, suppose that for some number v with 0 ≤ v ≤ m − 1 we have (i) height Im (L) = n − m + 1, (ii) height I j (L) ≥ (m + 1 − j)(n − m) + 2 for all j = v + 1, . . . , m − 1. Then Ass S/Im (L)k ⊂ Ass S/Im (L) ∪ {P : P ⊃ Iv (L)} for every k > 0 (where I0 (L) = S by convention). Proof We argue by induction on m. The case m = 1 is a special case of Lemma 8.7.10 and the same is true for the case m > 1 and v = 0. Hence we may assume that m > 1, v > 0. Let P ∈ Ass S/Im (L)k . The assertion we have to prove is obvious if P contains Iv (L). Otherwise if P does not contain Iv (L), then, in particular, it does not contain I1 (L). Now we can proceed as in the proof of Lemma 8.7.10 by inverting one entry of L and conclude by induction on m.  The next ingredient that we need is a sort of deformation lemma. Lemma 8.7.12 Let L be an m × n matrix whose entries are linear forms in a polynomial ring S = K [X 1 , . . . , X c ]. Let Y be a new variable. Then: (a) For every A ∈ Mmn (K ) and for every j one has height I j (L) ≤ height I j (L + Y A) ≤ 1 + height I j (L). (b) If K is infinite then there exists A ∈ Mmn (K ) such that height I j (L + Y A) = min{(m + 1 − j)(n + 1 − j), height I j (L) + 1} for every j. Proof (a) Set R = S[Y ]. One has I j (L + Y A) ⊂ (Y ) + I j (L)R and hence height I j (L + Y A) ≤ height((Y ) + I j (L)R) = 1 + height I j (L). For the other inequality we consider the weight vector w on R that gives weight 2 to the variables of S and weight 1 to Y . One has inw (I j (L + Y A)) ⊃ I j (L)R. Since by Proposition 1.6.2 the height does not change by taking ideals of the initial forms, we conclude that height I j (L) ≤ height I j (L + Y A). (b) Set h j = height I j (L), g j = (m + 1 − j)(n + 1 − j) and T = { j : h j < g j }. Recall that height I j (H ) ≤ g j holds for every m × n matrix H in a Noetherian ring such that I j (H ) is a proper ideal. The inequality height I j (L + Y A) ≤ min{(m + 1 − j)(n + 1 − j), height I j (L) + 1}

8.7 Maximal Minors of Matrices of Linear Forms and Linear Powers

319

follows from (a) and holds for all A ∈ Mmn (K ). /T By virtue of (a) we have height I j (L + Y A) ≥ height I j (L) and hence if j ∈ then height I j (L + Y A) = (m + 1 − j)(n + 1 − j) holds for all A ∈ Mmn (K ). To conclude it is enough to show that there exists A ∈ Mmn (K ) such that height I j (L + Y A) ≥ h j + 1 for every j ∈ T . Because the base field K is infinite, after a change of coordinates, we may assume that X 1 , . . . , X c satisfy the following property: for every j the set E j = {X k : k > h j } (L). In other words, I j (L) has height is a system of parameters for the ring S/I j c X i Ai with Ai ∈ Mmn (K ). Since h j also modulo (E j ). We may write L = i=1 h j h j X i Ai ) = h j , that is, the L = i=1 X i Ai mod (E j ), we deduce that height I j ( i=1 h j radical of I j ( i=1 X i Ai ) is (X 1 , . . . , X h j ), for every j. If we prove that there exists A ∈ Mmn (K ) such that ⎛ height I j ⎝

hj 

⎞ X i Ai + Y A⎠ = h j + 1

(8.17)

i=1

c for every j ∈ T then (a) implies height I j ( i=1 X i Ai + Y A) ≥ h j + 1, and so we are done. Let K¯ be the algebraic closure of K . We now consider the subvariety V j of the projective space P(Mmn ( K¯ )) of the matrices of rank < j. Furthermore we consider the linear space W j = Ai : i = 1, . . . , h j  ⊂ P(Mmn ( K¯ )). We claim that W j ∩ V j = ∅. By contradiction, assume there exists an element of the form h j h j ¯ i=1 λi Ai ∈ P(Mmn ( K )) that belongs to V j , i.e., the rank of i=1 λi Ai is < j. Then h j X i Ai ) the point (λ1 , . . . , λh j ) is in the subvariety of Ph j −1 ( K¯ ) defined by I j ( i=1 which is empty by construction. Now consider the join W j ∗ V j of W j and V j , that is W j ∗ V j is the Zariski closure of ∪BC where BC is the line joining B and C and (B, C) varies in W j × V j . As we have seen in Remark 4.5.2, dim W j ∗ V j ≤ dim W j + dim V j + 1 = mn − 1 − (g j − h j ) < mn − 1. (8.18) (Actually, by Remark 4.5.2, equality holds in (8.18), but we will not need this.) ¯  Therefore W j ∗ V j is a proper subvariety of P(Mmn ( K )) if j ∈ T , and so is (W ∗ V ). Because K is infinite, we can therefore take A ∈ Mmn (K ) and A ∈ / j j j∈T W j ∗ V j for every j ∈ T . We claim that with this choice of A the formula in (8.17) h j X i Ai + holds. Suppose, by contradiction, that for a j ∈ T one has height I j ( i=1 Y A) < h j + 1. Then there is a point (a1 , . . . , ah j , b) in the projective subvariety of h j X i Ai + Y A) where the last coordinate corresponds Ph j defined by the ideal I j ( i=1 h j ai Ai + b A has rank < j. If b = 0 we obtain to the variable Y . In other words i=1 A ∈ W j ∗ V j , a contradiction. If b = 0 we have (a1 , . . . , ah j ) ∈ W j ∩ V j which is also a contradiction.  We are ready to state and prove the main theorem of this section.

320

8 Castelnuovo-Mumford Regularity

Theorem 8.7.13 Let L be an m × n matrix with m ≤ n whose entries are linear forms in a polynomial ring S over a field K . Suppose that height Im (L) = n − m + 1 and   height I j (L) ≥ min (m + 1 − j)(n − m) + 1, c for every j = 2, . . . , m − 1 where c = height I1 (L).Then (a) Im (L) has linear powers; (b) Im (L) is of fiber type; (c) the Betti numbers of Im (L)k depend only on m, n, k and c. Proof It is harmless to assume that the base field is infinite. Set c = height I1 (L). Replacing S with the K -algebra generated by the entries of L, we may assume that c = dim S. We have c ≤ mn and we prove the statement by decreasing induction on c. If c ≥ m(n − m) + 1 then L satisfies the assumption of Theorem 8.7.2 and hence the statements hold by Theorem 8.7.2. We now assume that c < m(n − m) + 1, consider a new variable Y , and set R = S[Y ]. In view of Lemma 8.7.12 we may take A ∈ Mmn (K ) such that the ideals of minors of the matrix L = L + Y A satisfy height I j (L ) = min{(m + 1 − j)(n + 1 − j), height I j (L) + 1} for every j. In particular height I1 (L ) = min{mn, c + 1} = c + 1. Since, by assumption, height I j (L) ≥ min{(m + 1 − j)(n − m) + 1, c} we can deduce that height I j (L ) ≥ min{(m + 1 − j)(n + 1 − j), (m + 1 − j)(n − m) + 2, c + 1}. (8.19) In particular, height I j (L ) ≥ min{(m + 1 − j)(n − m) + 1, c + 1} holds for every j. Hence, L satisfies the assumption and by induction we may assume that the conclusions hold for L . Since R/Im (L ) ⊗ R S = S/Im (L), one can apply Theorem 8.7.8 and deduce that the conclusions hold for Im (L) as well, provided we show that Y is almost R/Im (L )k -regular for all k ≥ 1. Now (8.19) implies height I j (L ) ≥ min{(m + 1 − j)(n − m) + 2, c + 1} for every j ≤ m − 1. The inequality c + 1 < (m + 1 − v)(n − m) + 2 holds for v = 1 by assumption and does not hold for v = m because height Im (L) = n − m + 1 implies that n − m + 1 ≤ c. Hence let v be the largest number, necessarily ≤ m − 1, such that (m + 1 − v)(n − m) + 2 > c + 1. Hence height I j (L ) ≥ (m + 1 − j)(n − m) + 2 for j = v + 1, . . . , m − 1. From Corollary 8.7.11 we deduce that Ass R/Im (L )k = Ass R/Im (L ) ∪ {P : P ⊃ Iv (L )}.

Notes

321

By construction, height Iv (L ) = c + 1 and hence the radical of Iv (L ) is the maximal homogeneous ideal m R of R. Therefore Ass R/Im (L )k ⊂ Ass R/Im (L ) ∪ {m R }.

(8.20)

Note that Ass R/Im (L ) = Min Im (L ) because R/Im (L ) is Cohen–Macaulay. Together with (8.20) this implies: to prove that Y is almost R/Im (L )k -regular for all k ≥ 1, we have to show that Y does not belong to any P ∈ Min Im (L ). Suppose by contradiction that Y ∈ P for some P ∈ Min Im (L ). Then P ⊃ (Y ) + Im (L ) = (Y ) + Im (L) therefore

height P ≥ 1 + height Im (L) = 1 + height Im (L )

which is a contradiction because R/Im (L ) is Cohen–Macaulay and hence height P =  height Im (L ). A special and important case of Theorem 8.7.13 is Theorem 8.7.14 Let L be a 2 × n matrix with m ≤ n whose entries are linear forms in a polynomial ring S over a field K . Assume that height I2 (L) = n − 1. Then I2 (L) has linear powers, it is of fiber type and the Betti numbers of I2 (L) depend only on n, k and c = height I1 (L). We conclude with an interesting example. Example 8.7.15 Let S = Q[X 1 , X 2 , X 3 , X 4 ] and R = S[Y1 , Y2 , Y3 , Y4 ]. Let ⎛

⎞ X1 X2 0 0 L = ⎝ 0 X1 X3 0 ⎠ . 0 0 X4 X2 Then I3 (L) = (X 22 X 3 , X 1 X 2 X 3 , X 12 X 2 , X 12 X 4 ) has the expected height 2 and height I2 (L) is 2 (and not at least 3, as required in Theorem 8.7.13). Indeed I3 (L) and I2 (L) have the minimal prime (X 1 , X 2 ) in common. The defining ideal P of R(I3 (L)) as a quotient of R is minimally generated by X 1 Y1 − X 2 Y2 , X 1 Y2 − X 3 Y3 , X 4 Y3 − X 2 Y4 , X 4 Y22 − X 3 Y1 Y4 and in(P) = (X 4 Y3 , X 1 Y1 , X 1 Y2 , X 2 Y22 , X 4 Y22 ). Therefore by Proposition 8.5.4, R(I3 (L)) has (1, 0)-regularity equal to 0 and hence I3 (L) has linear powers, and it is not of fiber type.

322

8 Castelnuovo-Mumford Regularity

Notes Lemma 8.1.2 appears in Ooishi [254, Theorem 2], where it is attributed to Mumford, and it appears also in Brodmann and Sharp [33, Theorem 16.3.1]. The validity of the equality reg M = reg1 (M) is due to Eisenbud and Goto [117] when the base ring is a field. According to Chardin [72], Jouanolou observed that the equality it is valid even for non Noetherian base rings, as actually proved by Chardin, Jouanolou and Rahimi [75]. Lemma 8.3.1 is a special case of results proved by Bagheri et al. [13] and by Chardin, Ghosh and Nemati [74]. That the regularity of powers is asymptotically a linear function of the exponent has been proved in Cutkosky et al. [99] and Kodiyalam [215] when R is a polynomial ring over a field and by Trung and Wang [305] for general base rings. This development has been inspired by earlier results of Bertram et al. [18] asserting that if I is the defining ideal of a smooth complex variety then reg I v is linearly bounded and by Swanson [303] who extended the result to any homogeneous ideal in a polynomial ring as a consequence of an important statement on the existence of uniform primary decompositions for powers. The proof of Theorem 8.4.1, parts (a), (b), (d) we present is a modification of the one given in [99] and it appears in [46] by Bruns and Conca in its variant for many ideals and when the base ring is a field. The assertion 8.4.1(c) seems to be new. The characterization of δ0 in terms of reductions appears in [215] when the base ring is a field and in [305] in general. The proof we present follows the arguments given in [305]. The nature of the other invariants arising from Theorem 8.4.1, i.e., the constant terms of the linear forms and the starting value of the asymptotic linearity, have been deeply investigated by Bagheri et al. [13], Chardin [71, 73], Eisenbud and Harris [118], Eisenbud and Ulrich [122] and Römer [270]. They are relatively well understood in small dimension, but remain largely unknown in general. Other important examples and developments are discussed by Chardin [70] and by Borna [22]. In particular for every d > 2 in [84] Conca constructs an ideal J generated by monomials of degree d in 4 variables such that reg J v = dv for every v < d − 1 and reg J d−1 > d(d − 1). Theorem 8.4.12 is proved, in a more general form, by Bagheri et al. [13, Theorem 4.6] and by Bruns and Conca [46, Theorem 2.2.] over a base field while a weaker version of it is proved by Ghosh [140]. Examples 8.4.14 and 8.4.15 are taken from [46]. The characterization of linear powers by means of the (1, 0)-regularity Theorem 8.5.3 appears in Römer [270] (over a field and only one direction), in Bruns et al. [48] (over a field and both directions) and in Chardin [72, Theorem 5] in general. The relation between linear products and uniform primary decompositions is explored by Bruns and Conca [44]. The results of Sect. 8.6 are taken from [16] by Berget, Bruns, and Conca. In that paper both the “lifting of relations” and the RSK approach are used to control the

Notes

323

Sagbi deformation and to establish the main result Theorem 8.6.1 which is then used to describe generators of a natural Grothendieck group. The results of Sect. 8.7 are taken from the paper [48]. The main motivation of [48] was to prove that ideals defining a rational normal scrolls have linear powers, which is now a special case of Theorem 8.7.14. Sammartano [276] continued the investigation by describing the defining equations of the Rees algebra of such ideals, completing a program started in 1996 by Conca et al. [93].

Chapter 9

Grassmannians, Flag Varieties, Schur Functors and Cohomology

In this chapter we introduce the basic theory of flag varieties, and describe general results regarding cohomology groups of line bundles, with an emphasis on vanishing statements. We develop the theory in the relative setting, which offers significant flexibility for the inductive arguments that we employ. We also introduce Schur functors and explain their relationship to direct images of line bundles associated to dominant weights. We end with a brief discussion of Grothendieck duality and some applications, that will be instrumental in our calculation of Ext modules in subsequent chapters.

9.1 Fundamental Multilinear Algebra Constructions Let V be a finitely generated free module of rank n over a commutative ring R. For a positive integer d we consider the (dth) tensor power T d V = V ⊗d = V ⊗ R V ⊗ R · · · ⊗ R V, which is free of rank n d . The elements of T d V are called (d-)tensors. The symmetric group Sd acts on T d V by permuting the factors. The invariants of this action are the (dth) divided powers D d V = (T d V )Sd = {w ∈ T d V : σ (w) = w for all σ ∈ Sd }. The (dth) symmetric powers are defined by Symd V = (T d V )Sd =

TdV d (V )

© Springer Nature Switzerland AG 2022 W. Bruns et al., Determinants, Gröbner Bases and Cohomology, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-05480-8_9

325

326

9 Grassmannians, Flag Varieties, Schur Functors and Cohomology

where d (V ) is the submodule generated by w − σ (w) with σ ∈ Sd , w ∈ T d V . For any σ ∈ Sd , the tensors v1 ⊗ v2 ⊗ · · · ⊗ vd

and

vσ (1) ⊗ vσ (2) ⊗ · · · ⊗ vσ (d)

yield the same equivalence class in Symd V . It is then customary to represent this class as a monomial v1 v2 · · · vd .  We write Sym. (V ) = d≥0 Symd V for the symmetric algebra of V , which upon a choice of basis for V may be identified with a polynomial ring in n variables over d. R, with the standard grading where Symd V is placed   in degree . There is a natural map Both D d V and Symd V are free modules of rank d+n−1 d from divided to symmetric powers, obtained as the composition ϕd : D d V → T d V  Symd V.

(9.1)

If R contains a field of characteristic zero or larger than d, then the map ϕd is an isomorphism. In general however, this may fail to be the case, as the following example shows. Example 9.1.1 Suppose that R = Z/2Z is the field with two elements, V is an Rvector space with dim V ≥ 2, and consider linearly independent elements v1 , v2 ∈ V . We have 0 = w = v1 ⊗ v2 + v2 ⊗ v1 ∈ D 2 V, and moreover ϕ2 (w) = v1 v2 + v2 v1 = 2v1 v2 = 0. Consider now the submodule d (V ) ⊂ T d V spanned by tensors v1 ⊗ · · · ⊗ vd , where vi = v j for some indices i, j with 1 ≤ i < j ≤ d. The (dth) exterior power of V is defined by d  TdV V = . d (V )    It is a free R-module of rank dn , and particular d V = 0 for d > n. We define in the determinant of V to be det V = n V .  The equivalence class of v1 ⊗ · · · ⊗ vd in d V is denoted v1 ∧ · · · ∧ vd , and we have for each permutation σ ∈ Sd that

9.1 Fundamental Multilinear Algebra Constructions

327

vσ (1) ∧ · · · ∧ vσ (d) = sign(σ ) · v1 ∧ · · · ∧ vd , where sign(σ ) denotes the sign of the permutation σ . It is often convenient to realize  d V as a subspace of T d V , via the map d 

V → T d V, v1 ∧ · · · ∧ vd −→



sign(σ ) · vσ (1) ⊗ · · · ⊗ vσ (d) .

σ ∈Sd

The composition d 

 TdV V = d (V ) d

V → T d V 

is multiplication by d!, so it is an isomorphism only when d! is a unit in R. We define the dual of V to be V ∨ = Hom R (V, R), which is a free module of rank n, and we have a perfect pairing V × V ∨ −→ R, (v, f ) → f (v). This induces for each d ≥ 0 a perfect pairing T d V × T d (V ∨ ), (v1 ⊗ · · · ⊗ vd , f 1 ⊗ · · · ⊗ f d ) → f 1 (v1 ) . . . f d (vd ). get an induced Via this pairing, we get that D d V is orthogonal to d (V ∨ ), and we  d d d ∨ V × Sym (V ) −→ R. Likewise, if we think of V ⊂ TdV, perfect pairing D   d d V is orthogonal to d (V ∨ ), giving a perfect pairing V ⊗ d (V ∨ ) −→ then R. We get natural isomorphisms Sym (V ) ∼ = (D d V )∨ , d

∨

 d ∨ d   ∨ ∼ (V ) = V .

(9.2)

 Because of (9.2), the notation d V ∨ is unambiguous, allowing us to omit daddiV× tional parentheses. We will also consider for 0 ≤ d ≤ n the perfect pairing  n−d V −→ n V , which gives rise to an isomorphism d 

V ∼ = det V ⊗

n−d 

V ∨.

(9.3)

The formation of tensor, divided, symmetric, exterior powers, and duals is functorial, hence it can be applied to coherent locally free sheaves E on an algebraic variety B (see Hartshorne [165, Sect. II.5] for the basic theory). We write T d E, D d E,

328

9 Grassmannians, Flag Varieties, Schur Functors and Cohomology

 Symd E, d E,  E∨ for the resulting sheaves, which are also locally free. If E has rank n then det E = n E is an invertible sheaf, called the determinant of E. A number of elementary properties of these constructions are discussed in [165, Exercise II.5.16], and will be used in later sections. To simplify the notation, we make the following conventions regarding the tensor product notation. If B is a variety over a field K , V is a K -vector space, and E, F are sheaves of OB -modules, then we can also think of E, F as sheaves of K -vector spaces. We will write E ⊗ V for E ⊗ K V,

and

E ⊗ F for E ⊗OB F ,

and leave it to the reader to interpret the symbol ⊗ based on the context. We will never consider E ⊗ K F (unless B = Spec K ), so our notation should cause no confusion. n Notice that if V = K n , and F = O⊕n B = OB ⊗ K , then we have E⊕n = E ⊗ V = E ⊗ F , where the first ⊗ is ⊗ K , and the second one is ⊗OB . If B = B1 × B2 , with projection maps πi : B −→ Bi , and if Ei denotes a sheaf of OBi -modules, then we will write E1  E2 = (π1∗ E1 ) ⊗ (π2∗ E2 ). We end the section with a brief discussion of some Koszul complexes. Lemma 9.1.2 Suppose that we have a short exact sequence 0 −→ E −→ F −→ G −→ 0 of locally free sheaves of finite rank on an algebraic variety B. For each d ≥ 0, the sheaf Symd G has a Koszul resolution K. −→ Sym G −→ 0, with K j = d

j 

E ⊗ Symd− j F for 0 ≤ j ≤ d.

Proof If we write ι : E −→ F then the differential in K. is defined by (e1 ∧ · · · ∧ e j ) ⊗ ( f 1 · · · f d− j ) −→

j 

(−1)i (e1 ∧ . . . e i · · · ∧ e j ) ⊗ (ι(ei ) · f 1 · · · f d− j ).

i=1

To check exactness, it suffices to work on an affine chart U = Spec R, and assume that F = (U, F ) has R-basis {x1 , . . . , xn }, E = (U, E) has basis {x1 , . . . , xk }, and G = (U, G) has basis {xk+1 , . . . , xn }. We let S = R[x1 , . . . , xn ] = Sym. (F), consider the ideal I = E = x1 , . . . , xk , and note that

. S/I ∼ = R[xk+1 , . . . , xn ] = Sym (G).

9.2 Affine and Projective Bundles, Grassmannians, Flag Varieties

329

Since x1 , . . . , xk form a regular sequence, S/I has a free resolution given by the Koszul complex, see [52, Sect. 1.6] or [116, Sect. 17], . . . −→

j 

E ⊗ S −→ . . . −→ S −→ S/I −→ 0,

and restricting to the degree d component we obtain an exact complex · · · −→

j 

E ⊗ Symd− j F −→ . . . −→ Symd F −→ Symd G −→ 0.

This shows that (U, K. ) resolves (U, Symd G), concluding the proof.



9.2 Affine and Projective Bundles, Grassmannians, Flag Varieties Let B be an algebraic variety, and let E be a locally free sheaf of finite rank on B. The geometric affine bundle associated to E [165, Exercise II.5.18] is defined using the relative Spec construction in [165, Exerc II.5.17] as AB (E) = SpecO Sym. (E), where Sym. (E) = OB ⊕ E ⊕ Sym2 E ⊕ · · · B

Over a point b ∈ B, AB (E) parametrizes elements in E∨b , the dual of the fiber Eb (recall that Eb is defined as E ⊗ κ(b), where κ(b) = OB,b /mb is the residue field of B at b, OB,b is the local ring of B at b, and mb is the maximal ideal of OB,b ). Any surjection E  F induces a closed immersion AB (F ) → AB (E). In the special case when B = Spec K , and E = V is a finite dimensional K -vector space, the affine bundle AB (E) is an affine space naturally identified with V ∨ . We define in a similar way, using the relative Proj construction in [165, Sect. II.7], the projective bundle PB (E) = ProjO Sym. (E), B

parametrizing rank one quotients of E (over a point b ∈ B, it parametrizes nonzero elements of E∨b up to scaling). More generally, suppose that rank(E) = n and consider an integer 0 ≤ l ≤ n. We let GB (l; E) denote the (relative) Grassmannian of rank l quotients of E. It is also common to refer to GB (l; E) as a Grassmann bundle. If we write π : GB (l; E) −→ B for the structure morphism, then we have a tautological exact sequence (see also [317, Sect. 3.3]) E −→ π ∗ E −→ QlE −→ 0, (9.4) 0 −→ Rn−l

330

9 Grassmannians, Flag Varieties, Schur Functors and Cohomology

E where QlE is the tautological rank l quotient sheaf of π ∗ E and Rn−l is the tautological ∗ rank (n − l) subsheaf of π E. Parametrizing rank l quotients of E is equivalent to parametrizing rank (n − l) quotients of E∨ (a quotient E  Q corresponds to coker(Q∨ → E∨ )), and this gives a natural identification

GB (l; E) ∼ = GB (n − l; E∨ ).

(9.5)

Via this identification, the tautological sequence on GB (n − l; E∨ ) arises as the dual of (9.4), and we have  E ∨  ∨ ∨ ∨ , RlE ∼ QEn−l ∼ = Rn−l = QlE .

(9.6)

When no confusion is to be expected, we drop E from the notation and write R, Q instead of Rn−l , Ql . When l = 1, we have that GB (l; E) = PB (E) is a projective bundle, Q = OPB (E) (1) is the tautological line bundle on PB (E), and R = PB (E)/B (1) is a twist of the sheaf of relative differentials for the morphism π [165, Theorem II.8.13]. For any l we have more generally that GB (l;E)/B ∼ = Q∨ ⊗ R is locally free of rank l · (n − l), so the relative canonical sheaf is given by ωGB (l;E)/B ∼ = det(Q∨ ⊗ R) =

l·(n−l) 

(Q∨ ⊗ R)

∼ = det(Q)−n+l ⊗ det(R)l ∼ = det(Q)−n ⊗ π ∗ (det(E)l ),

(9.7)

where the last isomorphism follows by taking determinants in (9.4) and using the fact that π ∗ respects tensor operations [165, Exercise II.5.16]. Formula (9.7) is perhaps most familiar when l = 1, B = Spec K , E = K n , in which case GB (l; E) ∼ = Pn−1 ∼ is the usual projective space, with canonical sheaf ωPn−1 = O(−n). To identify this with (9.7) note that Q = det(Q) = O(1), and π ∗ (det(E)l ) = det(K n ) ⊗ OPn−1 which is (noncanonically) isomorphic to OPn−1 . Each  Grassmann bundle GB (l; E) embeds into a corresponding projective bundle PB ( l E) via the Plücker  embedding, which associates to a rank l quotient E  Q, the rank one quotient l E  det(Q). We write  l  E ι : GB (l; E) → PB for the embedding, and  P = det(Q) = ι∗ OPB (l E) (1) . The invertible sheaf P is called the Plücker line bundle on GB (l; E).

9.2 Affine and Projective Bundles, Grassmannians, Flag Varieties

331

Iterating the construction of relative Grassmannians leads to relative flag varieties as follows. For any subset L = {l1 > l2 > · · · > lr } ⊂ {1, . . . , n − 1}

(9.8)

we let FB (L; E) denote the (relative, partial) flag variety, or flag bundle, parametrizing flags of quotients of E of ranks l1 , l2 , . . . , lr . It comes with tautological rank li locally free sheaves QlEi and quotient maps π ∗ E  QlE1  · · ·  QlEr , where π : FB (L; E) −→ B is the structure morphism of FB (L; E). The above sequence of maps is equivalent (by taking kernels of the quotient maps) to a filtration E E E ⊂ Rn−l ⊂ · · · ⊂ Rn−l ⊂ π ∗ E, Rn−l 1 2 r which we call the tautological filtration on π ∗ E. It is sometimes convenient to let l0 = n, lr +1 = 0, and write Ql0 = π ∗ E = Rn−lr +1 , Qlr +1 = 0 = Rn−l0 . The constructions above lead to further tautological (subquotient) sheaves  E E RlEi ,l j = ker QlEi  QlEj = Rn−l /Rn−l , for 0 ≤ i < j ≤ r, j i noting that rank(RlEi ,l j ) = li − l j , and that RlEi ,0 = QlEi . For L = {l} we have a natural E identification FB (L; E) = GB (l; E), under which Rn,l corresponds to the tautological E subsheaf Rn−l from (9.4). To see how flag varieties arise as iterated Grassmannians, consider L  = L \ {li } for some 1 ≤ i ≤ r , and let f : FB (L; E) −→ FB (L  ; E) be the natural map given by forgetting the ith element in the flag. We note that given a surjection p : Q 1  Q 2 , with rank(Q i ) = qi , the choice of a quotient Q of rank q sitting in a flag Q 1  Q  Q 2 extending p is equivalent to that of a rank (q − q2 ) quotient of ker( p). This implies that the morphism f identifies   FB (L; E) ∼ = GFB (L  ;E) li − li+1 , RlEi−1 ,li+1 .

(9.9)

There is an inevitable abuse of notation when working with several flag varieties of the same sheaf E: for j = i, there exists a sheaf that we denoted by QlEj on both

FB (L; E) and FB (L  ; E), so perhaps a more appropriate notation would be QlE,L in j 

the first case, and QlE,L in the latter. Nevertheless, we have j

332

9 Grassmannians, Flag Varieties, Schur Functors and Cohomology

  = QlE,L f ∗ QlE,L j j

and

  = QlE,L f ∗ QlE,L , j j

and for most purposes making a notational distinction will be unnecessary. Our main interest will be in complete flag varieties, when L = {1, 2, . . . , n − 1}, E , which is a line in which case we write FB (E) for FB (L , E). We write LiE for Ri,i−1 bundle, and define OEFB (E) (y) = (LE1 ) y1 ⊗ · · · ⊗ (LEn ) yn for y = (y1 , . . . , yn ) ∈ Zn . Since LE1 , . . . , LEn are the composition factors for the tautological filtration on π ∗ (E), it follows that π ∗ (det E) ∼ = det(π ∗ E) ∼ = LE1 ⊗ · · · ⊗ LEn = OEFB (E) (1, . . . , 1).

(9.10)

In fact, by similar reasoning we have that for all i = 1, . . . , n det(QiE ) ∼ = LE1 ⊗ · · · ⊗ LiE = OEFB (E) (1i , 0n−i ), det(RE ) ∼ = LE ⊗ · · · ⊗ LE = OE (0 j , 1i− j , 0n−i ). i, j

j+1

i

FB (E)

E For more general flag varieties FB (L , E) we also write LiE = Ri,i−1 whenever we have i, i − 1 ∈ L. In analogy with (9.5) and (9.6), we have an identification

∨  ∨ FB (E∨ ) ∼ = FB (E), where LiE ↔ LEn−i .

(9.11)

Remark 9.2.1 The construction of partial flag bundles that we discuss here is the natural generalization of that of classical partial flag varieties considered in Sect. 3.3. More precisely, for τ = (t1 , . . . , t p ), the partial flag variety with coordinate ring K [Mτ ] in Sect. 3.3 is recovered as follows: we let B = Spec K , E = K n , and define L as in (9.8), where r = p and li = n − ti for i = 1, . . . , p. The choice of a subspace U of K n of dimension t is equivalent to that of a quotient K n  K n /U of dimension n − t, and therefore a partial flag of subspaces U1 ⊂ · · · ⊂ U p with dim Ui = ti corresponds uniquely to a flag of quotients K n  Q1  · · ·  Q p, where dim Q i = li = n − ti . As the notation can get quite involved, it is often preferable to simplify it by dropping E when it is understood from the context, or simply writing O instead of OEFB (E) . We will do so when there is no danger of confusion. To get a feeling about

9.3 Basic Calculus with Direct Images

333

the various interactions between the spaces and sheaves introduced so far, we prove the following. Lemma 9.2.2 The relative canonical sheaf of the map π : FB (E) −→ B is given by ωFB (E)/B ∼ = OEFB (E) (−n + 1, −n + 3, . . . , n − 3, n − 1). Proof We let Fi = FB ({i, i − 1, . . . , 1}, E) for i = 0, . . . , n − 1, with F0 = B. We drop E from the notation, and make the notational abuses alluded to earlier. For each i = 1, . . . , n − 1 we have that Fi = PFi−1 (Rn,i−1 ) is a projective bundle, with tautological exact sequence (9.4) given by 0 −→ Rn,i −→ Rn,i−1 −→ Li −→ 0. It follows from (9.7) that ωFi /Fi−1 = Li−(n−i) ⊗ det(Rn,i ). Using the fact that det(Rn,i ) ∼ = Li+1 ⊗ · · · ⊗ Ln , and the isomorphism ωFn /B =

n−1

ωFi /Fi−1 ,

i=1

where again we abused notation by writing ωFi /Fi−1 for its pullback to Fn , we get ωFB (E)/B ∼ =

n−1 

n Li−(n−i) ⊗ Li+1 ⊗ · · · ⊗ Ln ∼ Li−(n−i)+(i−1) , =

i=1

i=1



as desired.

9.3 Basic Calculus with Direct Images Suppose that f : X −→ Y is a morphism of algebraic varieties, F is a quasi-coherent sheaf on X , and E is a locally free sheaf of finite rank on Y . There is a natural isomorphism, called the projection formula [165, Exercise III.8.3]: R i f ∗ (F ⊗ f ∗ E) ∼ = R i f ∗ (F ) ⊗ E

for all i,

334

9 Grassmannians, Flag Varieties, Schur Functors and Cohomology

where R i f ∗ denotes the higher (derived) direct image functor. It follows that if f ∗ O X = OY and R i f ∗ O X = 0 for i = 0 (which will be true for most of the maps that we are concerned with) then taking F = O X yields f ∗ ( f ∗ E) = E,

and

R i f ∗ ( f ∗ E) = 0 for i = 0.

It will be useful to adopt the standard derived category notation, and write R f∗F = G

whenever

f∗ F = G

and

R i f ∗ F = 0 for i = 0.

Even though we are developing the theory in the relative setting (where the target of f is an arbitrary variety), as dictated by the inductive arguments that we want to employ, we are ultimately interested in concrete calculations of cohomology groups. These occur when Y = Spec K (or more generally an affine variety), when we have H 0 (Y, R i f ∗ F ) = H i (X, F )

for all i.

For a composition of morphisms f : X −→ Y , g : Y −→ Z , we have the relative Leray spectral sequence, see for instance the Stacks Project [291, tag 01F6], p,q

E2

= R p g∗ (R q f ∗ F ) =⇒ R p+q (g ◦ f )∗ F .

If Z = Spec K , this specializes to the absolute Leray spectral sequence p,q

= H p (Y, R q f ∗ F ) =⇒ H p+q (X, F ).

E2

We will often be in the situation where R f ∗ F = G, when the spectral sequence degenerates on the second page. We get Rg∗ G = R(g ◦ f )∗ F ,

that is, R p g∗ G = R p (g ◦ f )∗ F

for all p.

(9.12)

Direct Images of Tautological Sheaves on Flag Bundles If E is locally free of rank n on B as in the previous section, π : PB (E) −→ B and Q = O(1) is the tautological line bundle, then [165, Exercise III.8.4] yields   d Symd E if d ≥ 0, π∗ Q = 0 otherwise,   R i π∗ Qd = 0

for all i ∈ / {0, n − 1}, and all d ∈ Z,

(9.13)

(9.14)

9.3 Basic Calculus with Direct Images

R

n−1



π∗ Q

d



 =

335

(Sym−d−n E)∨ ⊗ det(E)∨ if d ≤ −n, 0 otherwise,

(9.15)

In particular, we have Rπ∗ Qd = Symd E for d ≥ 0, and Rπ∗ Qd = 0 for

(9.16)

− n < d < 0.

Lemma 9.3.1 Let E be a locally free sheaf of rank n on an algebraic variety B. (a) If π : FB (L , E) −→ B, L ⊂ {1, . . . , n − 1} denotes the structure morphism of a flag variety, then Rπ∗ OFB (L ,E) = OB . (b) For every L  ⊂ L ⊂ {1, . . . , n − 1}, if we write f : FB (L , E) −→ FB (L  , E) for the natural map forgetting elements of a flag, then R f ∗ OFB (L ,E) = OFB (L  ,E) . (c) If FB (L , E) = FB (E) then Rπ∗ OFB (E) (d, 0, . . . , 0) = Symd E for all d ≥ 0. (d) If π, L are as in part (a) and f : FB (E) −→ FB (L , E) is the natural map, then for all d ≥ 0 we have R f ∗ OFB (E) (d, 0, . . . , 0) = Symd Ql , where l = min(L). (e) If π is as in part (a) and l ∈ L, then Rπ∗ (Symd Ql ) = Symd E for all d ≥ 0. Proof To prove (a), consider first the case when L = {1}. We have FB (L , E) = PB (E), and the conclusion follows by taking d = 0 in (9.16). Consider next the case when L = {1, . . . , n − 1}, when FB (L , E) = FB (E) is the full flag variety. We can factor π as a composition π n−1

π n−2

π2

π1

FB (E) = Fn−1 −→ Fn−2 −→ · · · −→ F1 −→ F0 = B,

(9.17)

where F j = FB ({1, . . . , j}, E). Note that as a special case of (9.9), we have that each F j = PF j−1 (Rn, j−1 ) is a projective bundle over F j−1 with structure morphism π j , j and therefore Rπ∗ OF j = OF j−1 . Iterating (9.12), we get the desired conclusion for the composition π = π 1 ◦ π 2 ◦ · · · ◦ π n−1 . To prove (a) in general, we apply descending induction on |L|. Suppose that L  {1, . . . , n − 1} as in (9.8), and choose i with li−1 − li ≥ 2 (recall the convention l0 = n, lr +1 = 0). Define L  = L ∪ {li + 1}  L, and consider the commutative diagram f

F = FB (L  , E)

g

F = FB (L , E)

π

B

Since F = PF (Rli−1 ,li ) is a projective bundle, we have Rg∗ OF = OF . Since |L  | > |L|, we know by induction that R f ∗ OF = OB . Using (9.12) for the composition f = π ◦ g yields Rπ∗ OF = OB , as desired.

336

9 Grassmannians, Flag Varieties, Schur Functors and Cohomology

To prove (b), we may reduce via (9.12) to the case L  = L \ {li } for some li ∈ L. The conclusion follows by applying part (a) to the Grassmann bundle in (9.9). To prove (c), we use the factorization (9.17). We let F = FB (E) and consider h = π 2 ◦ · · · ◦ π n−1 : F −→ F1 . We observe that F1 = PB (E) with tautological quotient (line) bundle Q = Q1 , and that h ∗ (Qd ) = OF (d, 0, . . . , 0). Since Rh ∗ OF = OF1 , the projection formula yields Rh ∗ OF (d, 0, . . . , 0) = Rh ∗ (h ∗ (Qd )) = Qd . Using (9.16) for the map π 1 , together with (9.12) for the composition π = π 1 ◦ h, we obtain the desired conclusion. To prove (d), we factor f as f

F = FB (E)

g

F = FB ({l, l + 1, . . . , n − 1}, E)

h

F = FB (L , E)

Since F = FF (Ql ), it follows from part (c) that Rg∗ OF (d, 0, . . . , 0) = Symd Ql . We know from part (b) that Rh ∗ OF = OF . Since h ∗ (Symd Ql ) = Symd Ql (with the usual abuse of notation), the projection formula then yields Rh ∗ (Symd Ql ) = Symd Ql , and we conclude, using (9.12) for f = h ◦ g. To prove (e), suppose first that l = min(L), let f : FB (E) −→ FB (L , E) as in (d), and let g = π ◦ f . We know from (d) that R f ∗ OFB (E) (d, 0, . . . , 0) = Symd Ql , and from (c) that Rg∗ OFB (E) (d, 0, . . . , 0) = Symd E. The conclusion follows by applying (9.12) to g = π ◦ f . If l = min(L), let L  = L ∩ {l, l + 1, . . . , n − 1}, and factor π as π

F = FB (L , E)

g

F = FB (L  , E)

h

B

Since l = min(L  ), we know that Rh ∗ (Symd Ql ) = Symd E. It follows from (b) that Rg∗ OF = OF . Since g ∗ (Symd Ql ) = Symd Ql , the projection formula implies that Rg∗ (Symd Ql ) = Symd Ql . We conclude as usual by applying (9.12) to π = h ◦ g.  For every d ∈ Z, it follows from (9.10) that

9.3 Basic Calculus with Direct Images

337

OFB (E) (d n ) = π ∗ (det(E)d ), so using Lemma 9.3.1(a), it follows from the projection formula that   Rπ∗ OFB (E) (d n ) = det(E)d . If we now let F = OFB (E) (y) in the projection formula, then we get   R i π∗ OFB (E) (y + (d n )) = R i π∗ OFB (E) (y) ⊗ det(E)d

for all i.

(9.18)

 It is an important problem to understand the higher direct images R i π∗ OFB (E) (y) for an arbitrary weight y ∈ Zn , which in characteristic zero is completely resolved by the Borel–Weil–Bott theorem (Sect. 11.1). Much less is known in positive characteristic, and we will discuss some partial results in the rest of the chapter.

Vanishing Results for (Higher) Direct Images If B = Spec K and E = V  is an n-dimensional vector space, then the flag variety F = FB (E) has dimension n2 , and it follows from the Grothendieck vanishing theorem [165, Theorem III.2.7] that H (F, OF (y)) = 0 j

for all

  n j> 2

and all

y ∈ Zn .

To prove the relative version of this vanishing statement, we use the following. Lemma 9.3.2 If P = PB (E) with structure morphism π : P −→ B, and F is any quasi-coherent sheaf on P, then R j π∗ F = 0 for j > n − 1. Proof Since the vanishing result is local on B, we may assume that B = Spec R is affine, (B, E) = R n is a free R-module, and P = Proj(R[x1 , . . . , xn ]) ∼ = Pn−1 R . It follows that P has an open affine cover with n charts (defined by xi = 0 for each ˇ i = 1, . . . , n), and since we can compute the sheaf cohomology of F using the Cech complex as in [165, Proposition III.8.7], it follows that H j (P, F ) = 0 for j > n − 1, concluding the proof.



Corollary 9.3.3 Let F = FB (E), with structure morphism π : F −→ B. For every quasi-coherent sheaf F on F we have Grothendieck vanishing for flag bundles:

338

9 Grassmannians, Flag Varieties, Schur Functors and Cohomology

R j π∗ (F ) = 0 for j >

  n . 2

Proof We consider the factorization (9.17) of π , and identify Fl = PFl−1 (Rn,l−1 ) via π l . It follows from Lemma 9.3.2 that for every quasi-coherent sheaf F on Fl we have R j π∗l F = 0 for j > rank(Rn,l−1 ) − 1 = n − l. Iterating the Leray spectral sequence, it follows that R j π∗ (F ) = 0 for j > 1 + 2 + · · · + (n − 1) =

  n , 2 

as desired.

We next identify a set of weights y ∈ Zn for which all the (higher) direct images of the line bundles OFB (y) vanish identically. We call such weights singular. Lemma 9.3.4 Suppose that y ∈ Zn satisfies y j − y j+1 = −1 for some 1 ≤ j ≤ n − 1, and write π : FB (E) −→ B for the structure morphism of the full flag variety of E, where rank(E) = n. We have that Rπ∗ (OFB (E) (y)) = 0. Proof We consider first the case when n = 2, in which case we have y = (y1 , y2 ) with y1 − y2 = −1. We obtain using (9.18) that     R i π∗ OFB (E) (y1 , y2 ) = R i π∗ OFB (E) (y1 − y2 , 0) ⊗ det(E) y2   = R i π∗ OFB (E) (−1, 0) ⊗ det(E) y2 . Since rank(E) = 2, we have that FB (E) = PB (E), with tautological sheaves R, Q of rank one, and OFB (E) (−1, 0) = Q−1 . Applying (9.16) with d = −1 we get the desired vanishing. Suppose now that n ≥ 3, and fix 1 ≤ j ≤ n − 1 with y j − y j+1 = −1. Consider the set L  = {1, . . . , j − 1, j + 1, . . . , n − 1}, let F = FB (L  , E), and factor π as π

F = FB (E)

f

F

g

B

(9.19)

Using (9.9), we can identify F = FF (R j+1, j−1 ), with structure morphism f . Moreover, we have for all z 1 , z 2 ∈ Z that R

OF j+1, j−1 (z 1 , z 2 ) = OEF (0 j−1 , z 1 , z 2 , 0n− j−1 ), and f ∗ (Rt,t−1 ) = Lt for 1 ≤ t ≤ n − 1, t = j, j + 1. It follows that if we consider the sheaf

9.3 Basic Calculus with Direct Images

F =

339

t Rt,t−1 on F ,

y

t= j, j+1

then we obtain ⎛ OF (y) = OEF (0 j−1 , y j , y j+1 , 0n− j−1 ) ⊗ ⎝

t = j, j+1

⎞ y Lt t ⎠

R

= OF j+1, j−1 (y j , y j+1 ) ⊗ f ∗ (F ).

Since rank(R j+1, j−1 ) = 2 and y j − y j+1 = −1, we have seen earlier in the proof that  R R f ∗ OF j+1, j−1 (y j , y j+1 ) = 0. The projection formula then implies that R f ∗ (OF (y)) = 0, and (9.12) applied to the  composition π = g ◦ f yields Rπ∗ (OF (y)) = 0, as desired. We record one more vanishing result and a consequence that will be used later on (we leave it to the interested reader to reformulate Lemma 9.3.5 below in terms of Rπ∗ OF (y), for y = (0n−1 , j), 1 ≤ j ≤ n). Lemma 9.3.5 Consider the Grassmannian G = GB (n − 1, E) with structure morphism π : G −→ B and tautological subsheaf R of rank one. We have Rπ∗ (R j ) = 0 for 1 ≤ j ≤ n − 1, and  det(E) if i = n − 1; i n R π∗ (R ) = 0 otherwise.   ∼ QE∨ − j , so the conProof Using (9.5), (9.6), we get that G ∼ = PB (E∨ ), and R j = 1 clusion follows from (9.13)–(9.16) and the isomorphism (det E∨ )∨ ∼  = det(E). Corollary 9.3.6 With the notation in Lemma 9.3.5, suppose that F is a locally free sheaf of rank r ≤ (n − 1) on B, and let F˜ = π ∗ F . We have Rπ∗ (Symd (Q ⊗ F˜ )) = Symd (E ⊗ F ) for all d ≥ 0. If rank(F ) = n then the same conclusion holds when 0 ≤ d ≤ n − 1, while for d ≥ n we have a short exact sequence 0 → det(E) ⊗ det(F )⊗ Symd−n (E ⊗ F ) → Symd (E ⊗ F ) → π∗ (Symd (Q ⊗ F˜ )) → 0. Proof Tensoring the tautological exact sequence on G with F˜ we get 0 −→ R ⊗ F˜ −→ E˜ ⊗ F˜ −→ Q ⊗ F˜ −→ 0,

340

9 Grassmannians, Flag Varieties, Schur Functors and Cohomology

where E˜ = π ∗ E. For d ≥ 0, we get from Lemma 9.1.2 a resolution K. of Symd (Q ⊗ F˜ ) with Kj =

j  

 R ⊗ F˜ ⊗ Symd− j E˜ ⊗ F˜

= Rj ⊗

j 

 F˜ ⊗ Symd− j E˜ ⊗ F˜

for 0 ≤ j ≤ d.

 Notice that j F˜ = 0 for j > r = rank(F ), so K j = 0 for j ≥ min(r, d) + 1. Using the projection formula, together with the isomorphism j 

 j   F˜ ⊗ Symd− j E˜ ⊗ F˜ ∼ F ⊗ Symd− j (E ⊗ F ) , = π∗

it follows from Lemma 9.3.5 that Rπ∗ K j = (Rπ∗ R j ) ⊗

j 

F ⊗ Symd− j (E ⊗ F ) = 0

if 1 ≤ j ≤ n − 1 or j > min(r, d). If r ≤ n − 1 or d ≤ n − 1, this shows that Rπ∗ K j = 0 for j > 0, and we obtain Rπ∗ (Symd (Q ⊗ F˜ )) = Rπ∗ K0 = Symd (E ⊗ F ), where the second isomorphism follows from Lemma 9.3.1(a) and the projection formula. If r = n and d ≥ n then Rπ∗ K j = 0 precisely when j = 0 and j = n, and we have Rπ∗ K0 = Symd (E ⊗ F ) as before. Moreover, Lemma 9.3.5 and the projection formula imply that R n−1 π∗ Kn = det(E) ⊗ det(F ) ⊗ Symd−n (E ⊗ F ) , and R i π∗ Kn = 0 for i = n − 1. The spectral sequence associated with the derived  direct image of K. implies then the desired short exact sequence.

9.4 The Kempf Vanishing Theorem The goal of this section is to discuss a celebrated vanishing theorem due to Kempf [206], for cohomology of line bundles on flag varieties. A quick and beautiful proof in characteristic p is explained in [161] by Haboush. Here we take a different approach, that permits some further generalizations to be explained in subsequent sections. As usual, we will work in the relative setting, with the flag variety of a locally free sheaf. A weight y ∈ Zn is said to be dominant if

9.4 The Kempf Vanishing Theorem

341

y1 ≥ · · · ≥ yn . We write Zndom for the collection of all dominant weights in Zn . We will refer to the next result as the Kempf vanishing theorem. Theorem 9.4.1 Suppose that E is a locally free sheaf of rank n on an algebraic variety B, let F = FB (E) be the corresponding flag variety, and let π : F −→ B denote the structure morphism. If y ∈ Zndom then R j π∗ (OF (y)) = 0 for j = 0. Note that we have already established a special case of Theorem 9.4.1 in Lemma 9.3.1(c). If n = 2, this is sufficient to prove the result in general: indeed, the projection formula (9.18) yields R j π∗ (OF (y1 , y2 )) = R j π∗ (OF (y1 − y2 , 0)) ⊗ det(E) y2 = 0 for j = 0. For an arbitrary n, the same formula (9.18) shows that Kempf vanishing only depends on y up to a translation by multiples of (1n ). Therefore, we may always assume that yn ≥ 0 (or even yn = 0). If y ∈ Zndom , yn ≥ 0, we will refer to y as a partition. We define the height of a partition y to be h(y) = max{h : yh = 0}. We prove that Kempf vanishing holds for a partition y by induction, using the lexicographic ordering on the pairs 

|y|, −h(y) .

The base case of the induction is y = (0n ) and follows from Lemma 9.3.1(a). Suppose now that y has height h > 0, and define y  = (y1 − 1, . . . , yh − 1, 0n−h ), L = OF (y), and L = OF (y  ) = L ⊗ (L1 ⊗ · · · ⊗ Lh )−1 , where Li are as defined in Sect. 9.2. Lemma 9.4.2 For h and L as above, we let F =

h  (π ∗ E) ⊗ L .

There exists a short exact sequence 0 −→ F  −→ F −→ L −→ 0,

342

9 Grassmannians, Flag Varieties, Schur Functors and Cohomology

where the sheaf F  admits a finite filtration with composition factors OF (z), with |z| = |y| and z i = 0 for some i > h. Moreover, for each such z we either have z 1 ≥ z 2 ≥ · · · ≥ z n ≥ 0, or z i − z i+1 = −1 for some 1 ≤ i ≤ n − 1. Proof Consider the short exact sequence 0 −→ Rn,h −→ π ∗ E −→ Qh −→ 0. By [165, Exercise II.5.16(d)], we get a surjection h h   (π ∗ E)  Qh = L1 ⊗ · · · ⊗ Lh ,

  whose kernel has a filtration with composition factors i Rn,h ⊗ h−i Qh , i > 0. Since Qh has a filtration with composition factors L1 , . . . , Lh , and Rn,h has one with composition factors Lh+1 , . . . , Ln , we get a short exact sequence 0 −→ G −→

h  (π ∗ E) −→ L1 ⊗ · · · ⊗ Lh −→ 0,

(9.20)

 where G has a filtration with composition factors of the form t∈T Lt , where T varies among the subsets of {1, . . . , n} with |T | = h, T = {1, . . . , h}. In particular, each such T contains some t > h. Tensoring (9.20) with L and letting F  = G ⊗ L , we get an exact sequence of the desired form, where F  has composition factors OF (z), with z = y, z i − yi ∈ {0, 1} for all i, and z i = z i − yi = 1 for some i > h. It is clear that z i ≥ 0 for all i. If z fails  , we obtain to be a partition, we must have z i < z i+1 for some i. Since yi ≥ yi+1  ) ≥ 0 − 1 = −1, z i − z i+1 ≥ (z i − yi ) − (z i+1 − yi+1

so we must have z i − z i+1 = −1, as desired.



(Conclusion of Proof of Theorem 9.4.1) By induction, we know that Kempf vanishing holds for L , so R j π∗ F = 0 for j > 0. The short exact sequence in Lemma 9.4.2 leads to a long exact sequence · · · −→ R j π∗ F −→ R j π∗ L −→ R j+1 π∗ F  −→ R j+1 π∗ F −→ · · · ∼ R j+1 π∗ F  for j > 0. It is then enough to verify that if j > 0 then hence R j π∗ L = j R π∗ (OF (z)) = 0 for each of the composition factors of F  in Lemma 9.4.2. If we have z i − z i+1 = −1 for some i, the vanishing follows from Lemma 9.3.4. Otherwise, z is a partition with |z| = |y| and h(z) > h(y). It follows by induction that Kempf vanishing holds for z, concluding our proof. 

9.5 Characteristic-Free Vanishing for some Nondominant Weights

343

9.5 Characteristic-Free Vanishing for some Nondominant Weights In this section we establish a vanishing result for higher direct images of special line bundles on flag varieties, that holds uniformly in all characteristics, just as the Kempf vanishing Theorem 9.4.1. The line bundles that we consider depend on a number of parameters, as follows. We consider integers 0 ≤ l ≤ n, and d ≥ c ≥ 0, and weights b ∈ Zl , y ∈ Zn−l , with b1 > · · · >bl > 0, c = y1 ≥ y2 ≥ · · · ≥ yn−l ≥ 0, d ≥ |y| = y1 + · · · + yn−l .

(9.21)

We let F = FB (E) and π : F −→ B as in Theorem 9.4.1. Theorem 9.5.1 Suppose that d  0 and consider any l, c, b, y satisfying (9.21). If we let b = |b| = b1 + · · · + bl , and L = OF (c − b1 , . . . , c − bl , y1 , . . . , yn−l ), then we have (d − c) · l + b . (9.22) R j π∗ L = 0 for all j > 2 In Sect. 11.1 we will see various examples showing that this vanishing result is often sharp. The downside of Theorem 9.5.1 is that it does not provide an effective lower bound for the parameter d. In characteristic zero, we will find such an effective bound as a consequence of the Borel–Weil–Bott theorem. The rest of this section will focus on the proof of Theorem 9.5.1. Proof (Proof of Theorem 9.5.1) Notice that when l = 0, we have that b is the empty sequence, b = 0, and the result follows from Theorem 9.4.1. We may therefore assume that l > 0, and we start by making a reduction on the sequences b that may occur in Theorem 9.5.1. More precisely, we show that we may assume bi − bi+1 ≥ 2 for i = 1, . . . , l − 1.

(9.23)

Indeed, suppose that bi − bi+1 = 1 for some i. It follows that (c − bi ) − (c − bi+1 ) = −1, so by Lemma 9.3.4, Rπ∗ L = 0. We therefore assume from now on that (9.23) holds. To restrict our parameters even further, we notice that if either d − c or b is large, say d − c ≥ n 2 , or b ≥ n 2 , then   n2 n (d − c) · l + b ≥ > , 2 2 2 and the desired vanishing follows from Corollary 9.3.3. We may therefore assume that d − c, b < n 2 are bounded in terms of n. Because of our hypothesis d  0, we obtain that c  0. Since

344

9 Grassmannians, Flag Varieties, Schur Functors and Cohomology

d − c = d − y1 ≥ y2 + · · · + yn−l , there are only finitely many possibilities allowed for the values of y2 , . . . , yn−l . Since b < n 2 , there are also finitely many possibilities for b. We may therefore fix y2 , . . . , yn−l and b, and since d − c ≥ 0, it is sufficient to prove that for c  0 we have b R j π∗ L = 0 when j > . 2 We observe next that by our earlier reductions we have already eliminated the case l = n: indeed, if l = n then by combining bn > 0 with (9.23), we get bn ≥ 1, bn−1 ≥ 3, . . . , b1 ≥ 2n − 1, and therefore b ≥ n 2 , contradicting the inequality b < n 2 . For the rest of the proof we may then assume that 1 ≤ l ≤ n − 1. Note that if bl = 1 then (c − bl ) − y1 = −1, and we conclude by Lemma 9.3.4. Using (9.23), we may therefore assume that bl ≥ 2, bl−1

  l +1 ≥ 4, . . . , b1 ≥ 2l, and thus b ≥ l · (l + 1) = 2 · . 2

  n  = 2 , and (9.22) Notice that if l = n − 1 then the above inequality implies b2 ≥ l+1 2 follows again from Corollary 9.3.3. For arbitrary l, to prove (9.22) it suffices to show that for fixed y2 , . . . , yn−l , b, and for c  0, we have R j π∗ L = 0 when j >

  l +1 . 2

We let G = GB (l + 1, E), and factor π as π

F

f

G

g

B

We consider the tautological sequence on G 0 −→ R −→ g ∗ E −→ Q −→ 0, and let P = det(Q) denote the Plücker line bundle on G. We have   f ∗ P = OF 1l+1 , 0n−l−1 , and since c = y1 , we get L∼ = f ∗ (Pc ) ⊗ L , where L = OF (−b1 , . . . , −bl , 0, y2 , . . . , yn−l ),

(9.24)

9.6 A Sharp Characteristic-Free Vanishing on Projective Bundles

345

and the projection formula yields R j f ∗ L = Pc ⊗ R j f ∗ L . Since P is g-relatively ample, and since c  0, it then follows from Serre vanishing [165, Theorem III.5.2] that R i g∗ (R j f ∗ L) = R i g∗ (Pc ⊗ R j f ∗ L ) = 0 for i > 0. It follows that the Leray spectral sequence for the composition π = g ◦ f degenerates on the second page, and therefore R j π∗ L = g∗ (R j f ∗ L) for all j. To prove (9.24), it is then enough to check that R j f ∗ L = 0 for j > we let F = FB ({l + 1, l, . . . , 1}; E) and factor f as

l+1 2

. To do so,

f

F

α

F

β

G

Via the map α we can identify F = FF (Rn,l ), and we note that the line bundles L j on F are pulled back from F for j ≤ l + 1, so (with the usual abuse of notation) R

c 1 ⊗ · · · ⊗ Ll+1 ). L = OF n,l (y2 , . . . , yn−l ) ⊗ α ∗ (Lc−b 1

Using the projection formula and the Kempf vanishing Theorem 9.4.1, it follows from the inequalities y2 ≥ · · · ≥ yn−l that R i α∗ L = 0 for i > 0, and the Leray spectral sequence for f = β ◦ α yields R j f ∗ (L) = R j β∗ (α∗ L) for all j. Via the map β, we can identify F = FG (Q), so Corollary 9.3.3 implies   l +1 , R β∗ (α∗ L) = 0 for j > 2 j

and therefore R j f ∗ L = 0 for j >

l+1 , as desired. 2



9.6 A Sharp Characteristic-Free Vanishing on Projective Bundles In this section we consider the special case l = 1 of Theorem 9.5.1, and prove that the vanishing (9.22) holds provided that d ≥ n − 2. The examples in Sect. 11.1 show that this bound is the best possible.

346

9 Grassmannians, Flag Varieties, Schur Functors and Cohomology

Theorem 9.6.1 Suppose that d, B ≥ 0, y ∈ Zn−1 with y1 ≥ · · · ≥ yn−1 ≥ 0, and that d ≥ |y| and d > B − n. If we let L = OF (d − B, y1 , . . . , yn−1 ) then R j π∗ L = 0 for j >

B . 2

The proof of Theorem 9.6.1 is based on a careful induction on the weights y, similar to the one employed to verify Kempf vanishing. Before going into more details, we make some preliminary observations, and establish the connection with Theorem 9.5.1. We let P = PB (E), with tautological exact sequence 0 −→ R −→ π ∗ E −→ Q −→ 0, where Q = OP (1). We have the usual factorization of π π

F

f

P

g

B

and via f we identify F = FP (R). We set L = OEF (0, y1 , . . . , yn−1 ) = ORF (y1 , . . . , yn−1 ), and note that f ∗ Q = OEF (1, 0, . . . , 0), so L = f ∗ (Qd−B ) ⊗ L . Using the projection formula and Theorem 9.4.1, we obtain R f ∗ (L) = Qd−B ⊗ f ∗ (L ), and the Leray spectral sequence yields R j π∗ L = R j g∗ (Qd−B ⊗ f ∗ (L )) for all j. Corollary 9.6.2 If l = 1 then Theorem 9.5.1 holds for all d ≥ n − 2. Proof We let B = b + d − c = b1 + d − c ≥ 0, so that (d − c)l + b B = , 2 2

(9.25)

9.6 A Sharp Characteristic-Free Vanishing on Projective Bundles

347

and the line bundles L in Theorems 9.5.1 and 9.6.1 coincide. If d − B > −n then (9.22) follows from Theorem 9.6.1. If d − B ≤ −n then B ≥ d + n ≥ 2n − 2, hence

B ≥ n − 1, 2

and (9.22) follows from (9.25) using Lemma 9.3.2.



Moving towards the proof of Theorem 9.6.1, we observe that if d − B ≥ y1 then the desired conclusion follows from Kempf vanishing. We also record the special case when y = (0n−1 ), which will be the base case of our induction. Lemma 9.6.3 Theorem 9.6.1 holds for y = (0n−1 ). Proof In light of (9.25), it suffices to check that R j g∗ (Qd−B ) = 0 for j > 0, which follows from the hypothesis d − B > −n and (9.14), (9.15).  In analogy with the proof of Kempf vanishing in Sect. 9.4, we prove Theorem 9.6.1 by lexicographic induction on the pairs 

|y|, −h(y) .

The base case y = (0n−1 ) corresponds to the smallest such tuple, and was verified in Lemma 9.6.3. Suppose now that y has height h > 0, and define L = OF (d − B, y1 − 1, . . . , yh − 1, 0n−h−1 ) = L ⊗ (L2 ⊗ · · · ⊗ Lh+1 )−1 . Lemma 9.6.4 For h and L as above, we let F =

h  (π ∗ E) ⊗ L .

There exists a filtration 0 ⊂ F1 ⊂ F2 ⊂ F satisfying the following properties: (a) F2 /F1 ∼ = L. (b) If h = n − 1 then F1 = 0. Otherwise, F1 has a filtration with composition factors of the form OF (d − B, z), where |z| = |y| and z i = 0 for some i > h. Moreover, for each such z we either have z 1 ≥ z 2 ≥ · · · ≥ z n−1 ≥ 0, or z i − z i+1 = −1 for some i with 1 ≤ i ≤ n − 2. (c) F /F2 has a filtration with composition factors of the form OF (d + 1 − B, w), with |w| = |y| − 1. Moreover, for each such w we either have w1 ≥ w2 ≥ · · · ≥ wn−1 ≥ 0, or wi − wi+1 = −1

348

9 Grassmannians, Flag Varieties, Schur Functors and Cohomology

for some 1 ≤ i ≤ n − 2.  Proof We define F2 = h Rn,1 ⊗ L , and use the inclusion Rn,1 ⊂ π ∗ E to get that F2 ⊂ F . Since Rn,1 has composition factors L2 , . . . , Ln , it follows that F2 has composition factors Li1 ⊗ · · · ⊗ Lih ⊗ L , with 2 ≤ i 1 < · · · < i h ≤ n.

(9.26)

This shows that F2 has composition factors of the form OF (d − B, z), where |z| = |y| and z i − yi ∈ {0, 1} for all i, where y  = (y1 − 1, . . . , yh − 1, 0n−h−1 ). It follows that  z i − z i+1 ≥ (z i − yi ) − (z i+1 − yi+1 ) ≥ −1,

so either z is a partition or z i − z i+1 = −1 for some i. To define F1 , we consider the surjection ϕ : Rn,1  Rh+1,1 , induced by the tautological quotient map π ∗ E  Qh+1 . Since rank(Rh+1,1 ) = h and Rh+1,1 has a filtration with composition factors L2 , . . . , Lh+1 , it follows that h 

By taking

h

Rh+1,1 = det(Rh+1,1 ) = L2 ⊗ · · · ⊗ Lh+1 .

ϕ and twisting by L we obtain a surjection F2 =

h 

Rn,1 ⊗ L  L2 ⊗ · · · ⊗ Lh+1 ⊗ L = L,

whose kernel we define to be F1 . By construction, we have F2 /F1 = L, so part (a) holds. Moreover, by construction we have that the composition factors of F1 are those elements in (9.26) for which (i 1 , . . . , i h ) = (2, 3, . . . , h + 1), so (b) holds. To verify (c), we note that Q1 = L1 , so we have a short exact sequence 0 −→ Rn,1 −→ π ∗ E −→ L1 −→ 0, which, since L1 has rank one, induces a short exact sequence 0 −→

h 

Rn,1 −→

h h−1   (π ∗ E) −→ Rn,1 ⊗ L1 −→ 0.

Twisting by L we get an identification F /F2 = that F /F2 has composition factors

h−1

Rn,1 ⊗ L1 ⊗ L . It follows

L1 ⊗ Li2 ⊗ · · · ⊗ Lih ⊗ L , with 2 ≤ i 2 < · · · < i h ≤ n,

9.6 A Sharp Characteristic-Free Vanishing on Projective Bundles

349

and conclusion (c) follows by a similar argument to (b).



We next use our induction hypothesis to prove cohomology vanishing in the appropriate range for the sheaves F1 , F2 constructed in Lemma 9.6.4. Lemma 9.6.5 With the notation in Lemma 9.6.4, if B ≥ 2 then we have R j π∗ (F1 ) = R j π∗ (F2 ) = 0 for j >

B . 2

Proof To prove the vanishing for F1 , we may assume that F1 = 0, and we consider a composition factor OF (d − B, z) of F1 as in Lemma 9.6.4(b). If z is not dominant, then z i − z i+1 = −1 for some i, and we conclude by Lemma 9.3.4. If z is dominant, then using the fact that    |y|, −h(y) >lex |z|, −h(z) , we get by induction that R j π∗ (OF (d − B, z)) = 0 for j > B/2. This proves that R j π∗ (F1 ) = 0 for j > B/2, as desired. To prove the vanishing for F2 , consider the short exact sequence 0 −→ F2 −→ F −→ F /F2 −→ 0. Applying the long exact sequence in cohomology, it suffices to check that R j π∗ (F ) = 0 for j > B/2, and R j π∗ (F /F2 ) = 0 for j > B/2 − 1. (9.27) Since y  = (y1 − 1, . . . , yh − 1, 0n−h−1 ) satisfies |y  | = |y| − h < |y|, we know by induction that R j π∗ (L ) = 0 for j > B/2, so the same vanishing holds for F . To prove the vanishing (9.27) for F /F2 , we may again ignore the composition factors OF (d + 1 − B, w) in Lemma 9.6.4(c) where w is not dominant. We then assume that w is dominant, and note that |w| = |y| − 1, so in particular    |y|, −h(y) >lex |w|, −h(w) . We have that d + 1 − B = (d − 1) − (B − 2), d − 1 ≥ |w|, and (d − 1) > (B − 2) − n, so in order to apply induction we have to check that d − 1, B − 2 ≥ 0. Since d ≥ |y| and y is nontrivial, we get d ≥ 1, while the inequality B ≥ 2 is part of the hypothesis. It follows by induction that R j π∗ (OF (d + 1 − B, w)) = 0 for j > (B − 2)/2 = B/2 − 1, so the same vanishing holds for F /F2 . This completes the verification of (9.27) and concludes our proof. 

350

9 Grassmannians, Flag Varieties, Schur Functors and Cohomology

(Conclusion of Proof of Theorem 9.6.1) To finish the inductive step we note that Lemma 9.6.4(a) yields a short exact sequence 0 −→ F1 −→ F2 −→ L −→ 0. If B ≥ 2, then applying the long exact sequence for higher direct images, and using Lemma 9.6.5, we conclude that R j π∗ (L) = 0 for j > B/2, as desired. If B = 0, or B = 1 and d − 1 ≥ y1 , then the weight (d − B, y1 , . . . , yn−1 ) is dominant, so R j π∗ (L) = 0 for j ≥ 1 > B/2 by Kempf vanishing. If B = 1 and d − B < y1 , then the inequality d ≥ |y| forces y1 = d, yi = 0 for i > 0. We get (d − B, y1 , . . . , yn−1 ) = (d − 1, d, 0, . . . , 0) and therefore R j π∗ (L) = 0 for all j by Lemma 9.3.4.



9.7 Algebraic Groups and Representations Following Jantzen [196] and Milne [247], we define an algebraic K -group to be a group object in the category of affine algebraic varieties over K . More precisely, G is an affine algebraic variety G over K , together with morphisms • (multiplication) m G : G × G −→ G, • (inverse) i G : G −→ G, • (unit element) 1G : Spec K −→ G, such that if we write id G : G −→ G for the identity map on G, then the following diagrams commute: G×G×G

id G ×m G

G×G

m G ×id G

mG mG

G×G (Spec K ) × G

1G ×id G

G×G

G id G ×1G

mG

∼ =

G × (Spec K )

∼ =

G G

(i G ,id G )

G×G

(id G ,i G )

G

mG

(Spec K )

1G

G

1G

(Spec K )

9.7 Algebraic Groups and Representations

351

The first diagram gives the usual associativity condition g1 · (g2 · g3 ) = (g1 · g2 ) · g3 , the second diagram is the condition that g · 1 = 1 · g = g, while the third diagram says that g −1 · g = g · g −1 = 1. We write K [G] for the coordinate ring of G. Using the contravariant correspondence between affine varieties and their coordinate rings, we get K -algebra homomorphisms: • (comultiplication) G = m G : K [G] −→ K [G] ⊗ K [G], • (coinverse) σG = i G : K [G] −→ K [G],  • (counit) εG = 1G : K [G] −→ K , satisfying (id K [G] ⊗G ) ◦ G = (G ⊗ id K [G] ) ◦ G , (ε G ⊗ id K [G] ) ◦ G = id K [G] = (id K [G] ⊗ε G ) ◦ G , (σG ⊗ id K [G] ) ◦ G = ε G = (id K [G] ⊗σG ) ◦ G , where f ⊗g denotes the map sending a ⊗ b −→ f (a)g(b), and ε G : K [G] −→ K [G] is the composition of the inclusion map K → K [G] with the counit εG . These axioms make K [G] into a commutative Hopf algebra, and are equivalent to the original axioms for G.

First Examples of Algebraic Groups Example 9.7.1 The polynomial ring K [T ] can be endowed with a Hopf algebra structure by letting , σ, ε be the K -algebra homomorphisms induced by: (T ) = T ⊗ 1 + 1 ⊗ T, σ (T ) = −T, ε(T ) = 0. The resulting algebraic group is called the additive group Ga . Example 9.7.2 The Laurent polynomial ring K [T, T −1 ] can be endowed with a Hopf algebra structure by letting , σ, ε be the K -algebra homomorphisms induced by: (T ) = T ⊗ T, σ (T ) = T −1 , ε(T ) = 1. The resulting algebraic group is called the multiplicative group Gm . Example 9.7.3 Generalizing Example 9.7.2, consider independent variables Ti, j , 1 ≤ i, j ≤ n, and let A = K [Ti, j ]det denote the localization of the polynomial ring K [Ti, j ] at det = det(Ti, j ). We can endow A with a Hopf algebra structure via

352

9 Grassmannians, Flag Varieties, Schur Functors and Cohomology

(Ti, j ) =

n 

Ti,k ⊗ Tk, j , ε(Ti, j ) = δi, j , and

k=1

σ (Ti, j ) = the (i, j)-entry of the inverse of (Ti, j ). The resulting algebraic group is the general linear group GLn . We have Gm ∼ = GL1 .

Morphisms of Algebraic Groups and Characters A morphism of algebraic K -groups is a morphism ϕ : G −→ G  of algebraic varieties, making the diagram G×G

mG

G ϕ

ϕ×ϕ 

× G

m G

G

commute. Equivalently, if ϕ  : K [G  ] −→ K [G] is the induced map on coordinate rings then (9.28) G ◦ ϕ  = (ϕ  ⊗ ϕ  ) ◦ G  . Example 9.7.4 There is a morphism of algebraic groups ϕ : GLn −→ Gm , defined by ϕ  : K [T, T −1 ] −→ K [Ti, j ]det , ϕ  (T ) = det, which amounts to checking that (det) = det ⊗ det in Example 9.7.3. We have ⎛



(det) =  ⎝

sign(σ ) ·

σ ∈Sn

=



i=1

sign(σ ) ·

=

n  k1 ,...,kn =1

σ ∈Sn n 

n 



sign(σ ) ·

k1 ,...,kn =1 σ ∈Sn

⎞



Ti,σ (i) ⎠ =  n 

σ ∈Sn



⊗

Ti,ki

i=1

 n 

sign(σ ) ·

Ti,ki

i=1



⊗

 n 

Ti,k ⊗ Tk,σ (i)

k=1

Tki ,σ (i)

i=1



i=1

 n 

 n n  

Tki ,σ (i) .

i=1

If ka = kb for some a = b, then if we consider the transposition τ = (a, b) we have for each permutation σ that sign(σ ) = − sign(σ τ ), and n  i=1

Tki ,σ (i) =

n  i=1

Tki ,σ τ (i) ,

9.7 Algebraic Groups and Representations

353

so by grouping the permutations of Sn into pairs (σ, σ τ ) with σ even we get 

sign(σ ) ·

σ ∈Sn

 n 



 ⊗

Ti,ki

i=1

n 

Tki ,σ (i)

= 0.

i=1

We may therefore restrict our attention to the case when ki are distinct, so ki = κ(i) for some κ ∈ Sn . We get (det) =

 

sign(σ ) ·

κ∈Sn σ ∈Sn

 n 

Ti,κ(i) ⊗

 n 

i=1

Tκ(i),σ (i) .

i=1

n n Using the fact that i=1 Tκ(i),σ (i) = i=1 Ti,σ κ −1 (i) , and sign(σ ) = sign(κ) · sign(σ κ −1 ), we can make the substitution τ = σ κ −1 and obtain ⎛ (det) = ⎝



κ∈Sn

sign(κ) ·

n 

⎞

⎛

Ti,κ(i) ⎠ ⊗ ⎝



sign(τ ) ·

τ ∈Sn

i=1

n 

⎞ Ti,τ (i) ⎠ = det ⊗ det,

i=1

as desired. To put Example 9.7.4 in context, we define a character of an algebraic group G to be a homomorphism χ : G −→ Gm , and prove the following. Lemma 9.7.5 There is a bijective correspondence between characters χ of G and invertible elements a = a(χ ) ∈ K [G] with G (a) = a ⊗ a. The set X (G) of characters of G has a natural structure of a commutative group, isomorphic to a subgroup of the group K [G]× of invertible elements of K [G]. Proof A character corresponds to a K -algebra homomorphism χ  : K [T, T −1 ] −→ K [G] satisfying (9.28). If we let a = χ  (T ) then a is invertible with inverse χ  (T −1 ). Moreover, (9.28) implies that G (a) = G ◦ χ  (T ) = (χ  ⊗ χ  ) ◦ Gm (T ) = (χ  ⊗ χ  )(T ⊗ T ) = a ⊗ a. Conversely, any a ∈ K [G]× determines a ring homomorphism K [T, T −1 ] −→ K [G], and the condition G (a) = a ⊗ a implies (9.28). Since G is a ring homomorphism, it follows that the elements a ∈ K [G]× with G (a) = a ⊗ a form a subgroup of K [G]× . This operation corresponds to the multiplication of characters given by χ1 ×χ2

ϕ

χ1 ⊗ χ2 : G × G −→ Gm × Gm −→ Gm ,

354

9 Grassmannians, Flag Varieties, Schur Functors and Cohomology

where ϕ corresponds to the K -algebra homomorphism ϕ  : K [T, T −1 ] −→ K [T1 , T2 , T1−1 , T2−1 ] given by ϕ  (T ) = T1 · T2 . Since a(χ1 ⊗ χ2 ) = a(χ1 ) · a(χ2 ), we get the desired iden tification of X (G) with a subgroup of K [G]× .

Algebraic Subgroups An algebraic subgroup is a closed subvariety H ⊂ G, such that the inclusion map is a morphism of algebraic K -groups. The ideal I ⊂ K [G] of H in G satisfies G (I ) ⊂ I ⊗ K [G] + K [G] ⊗ I, σG (I ) ⊂ I, εG (I ) = 0, and the corresponding maps  H , σ H , ε H are then induced by those on G. Example 9.7.6 The surjective ring homomorphism ϕ  : K [Ti, j ]det −→ K [T, T −1 ], defined by ϕ  (Ti, j ) = 0 if i = j, ϕ  (Ti,i ) = T , corresponds to a closed immersion ϕ : Gm −→ GLn . Since ϕ  satisfies (9.28), ϕ identifies Gm with a subgroup of GLn . In the next example we record three more subgroups of GLn that will be of interest in our subsequent discussion of Schur functors. Example 9.7.7 The subgroup Tn ⊂ GLn of upper triangular matrices is defined by the ideal ITn = Ti, j : 1 ≤ j < i ≤ n ⊂ K [GLn ]. The subgroup Un ⊂ GLn of unipotent matrices is defined by the ideal IUn = Ti, j : 1 ≤ j < i ≤ n + Ti,i − 1 : 1 ≤ i ≤ n ⊂ K [GLn ]. The subgroup Dn ⊂ GLn of diagonal matrices is defined by the ideal IDn = Ti, j : 1 ≤ i = j ≤ n ⊂ K [GLn ]. Since ITn ⊂ IUn and ITn ⊂ IDn , we have that Un and Dn are subgroups of Tn . If we write Ti = Ti,i then K [Dn ] ∼ = K [T1 , . . . , Tn ]T1 ···Tn ∼ = = K [T1 , . . . , Tn , T1−1 , . . . , Tn−1 ] ∼

n

K [Ti , Ti−1 ]

i=1

so Dn ∼ = Gm×n is a direct product of n copies of the multiplicative group Gm .

9.7 Algebraic Groups and Representations

355

It is an elementary fact in group theory that every morphism ϕ : G −→ A to an Abelian group factors through the Abelianization G/[G, G]. Since (at least settheoretically) [Tn , Tn ] = Un , and Tn /Un ∼ = Dn , and Gm is Abelian, it follows that characters of Tn and Dn are naturally identified. Moreover, since [Un , Un ] = Un , the group Un has no nontrivial characters. We next verify this using our formalism. Lemma 9.7.8 The inclusion of Dn → Tn induces an isomorphism X (Tn ) −→ X (Dn ), and the two groups are isomorphic to Zn , while X (Un ) is trivial. Proof As seen in Example 9.7.7, K [Dn ] = K [T1 , . . . , Tn , T1−1 , . . . , Tn−1 ], so   K [Dn ]× = c · T1e1 · · · Tnen : ei ∈ Z for all i, 0 = c ∈ K . Moreover, for each a = c · T1e1 · · · Tnen in K [Dn ]× we have Dn (a) = c−1 · (a ⊗ a), so a corresponds to a character of Dn if and only if c = 1. The assignment Zn −→ X (Dn ),

(e1 , . . . , en ) → a = T1e1 · · · Tnen

gives a group isomorphism X (Dn ) ∼ = Zn . Using again Example 9.7.7, we have K [Tn ] = K [Ti, j : 1 ≤ i ≤ j ≤ n]T1,1 ···Tn,n = R[Ti, j : 1 ≤ i < j ≤ n], −1 −1 where R = K [T1,1 , . . . , Tn,n , T1,1 , . . . , Tn,n ]. Since R is a domain and K [Tn ] is a polynomial ring over R, we get

  e1 en · · · Tn,n : ei ∈ Z for all i, 0 = c ∈ K . K [Tn ]× = R × = c · T1,1 The inclusion ϕ : Dn → Tn corresponds to a surjection ϕ  : K [Tn ] −→ K [Dn ], ϕ  (Ti,i ) = Ti , and ϕ  (Ti, j ) = 0 if i = j, which then restricts to an isomorphism K [Tn ]× ∼ = K [Dn ]× . Moreover, for each elee1 en × ment a = c · T1,1 · · · Tn,n in K [Tn ] , we have Tn (a) = a ⊗ a proving that X (Tn ) ∼ = X (Dn ).

⇐⇒

Dn (ϕ  (a)) = ϕ  (a) ⊗ ϕ  (a),

356

9 Grassmannians, Flag Varieties, Schur Functors and Cohomology

To prove that X (Un ) is trivial, note that K [Un ] is a polynomial ring over K by Example 9.7.7. We get K [Un ]× = K × , and for every c ∈ K × we have Un (c) = c · (1 ⊗ 1), which equals c ⊗ c precisely when c = 1. 

Modules, Invariants, and Eigenspaces Using the Hopf algebra structure on K [G], we define a G-module (or K [G]comodule) to be a K -vector space M endowed with a K -linear map  M : M −→ M ⊗ K [G], satisfying

(id M ⊗G ) ◦  M = ( M ⊗ id K [G] ) ◦  M , (id M ⊗ε G ) ◦  M = id M .

(9.29)

The first condition encodes the familiar property for a group action (g1 · g2 ) · m = g1 · (g2 · m), while the second one corresponds to the property 1 · m = m. A G-submodule of a G-module M is a subspace N ⊂ M with the property that  M (N ) ⊂ N ⊗ K [G]. Notice that the restriction  N = ( M )| N makes N into a G-module. An important example of a G-submodule is given by taking invariants, as follows. For a G-module M, the set of G-fixed points (or G-invariants) is defined by M G = {m ∈ M :  M (m) = m ⊗ 1}.

(9.30)

More generally, for every character χ : G −→ Gm we let a(χ ) ∈ K [G] denote the corresponding invertible element from Lemma 9.7.5, and define the χ -eigenspace of M to be (9.31) MχG = {m ∈ M :  M (m) = m ⊗ a(χ )}. It is a general fact that the elements a(χ ), χ ∈ X (G), are K -linearly independent, and the group G is called diagonalizable if they form a basis of K [G]. For G diagonalizable and M a G-module, one has a decomposition M=



MχG ,

χ∈X (G)

and the notion of a G-module is equivalent to that of a X (G)-graded vector space. We verify this next for G = Dn , and note that the special case n = 1 is the assertion that Gm -modules are precisely Z-graded vector spaces.

9.7 Algebraic Groups and Representations

357

Lemma 9.7.9 The notion of a Dn -module M is equivalent to that of a Zn -graded vector space M = e∈Zn Me . Moreover, under this identification we have M Dn = M0 . Proof Suppose  M : M −→ M ⊗ K [Dn ] satisfies (9.29), and make the identification X (Dn ) = Zn as in Lemma 9.7.8. For a character e ∈ Zn let T e = T1e1 · · · Tnen and consider the corresponding eigenspace Me = {m ∈ M :  M (m) = m ⊗ T e }. Since the monomials T1e1 · · · Tnen form a K -basis of K [Dn ], we can write for m ∈ M  M (m) =



me ⊗ T e,

e∈Z

and applying the first identity in (9.29) yields 

me ⊗ T e ⊗ T e =

e∈Zn



 M (m e ) ⊗ T e ,

e∈Zn

which forces  M (m e ) = m e ⊗ T e , so m e ∈ Me . This shows that M = To prove that the sum is direct, suppose by contradiction that 

 e∈Zn

Me .

m e = 0.

e∈Zn

Applying  M yields



m e ⊗ T e = 0,

e∈Zn

 forcing m e = 0 for all e ∈ Zn , and proving that M = e∈Zn Me .  Conversely, suppose that M = e∈Zn Me , and define  M : M −→ M ⊗ K [Dn ] by ⎛ ⎞   M ⎝ me⎠ = me ⊗ T e. e

e

The first identity in (9.29) follows from the fact that  M (m e ) = m e ⊗ T e and Dn (T e ) = T e ⊗ T e . Since εDn (T e ) = 1, it follows that (id M ⊗εDn )(m e ⊗ T e ) = m e , so the second condition follows as well, proving that M is a Dn -module.

358

9 Grassmannians, Flag Varieties, Schur Functors and Cohomology

By definition, we have m ∈ M Dn ⇐⇒ (m) = m ⊗ 1 ⇐⇒ (m) = m ⊗ T 0 ⇐⇒ m ∈ M0 , 

concluding the proof.

To contrast the conclusions of Lemma 9.7.9 with the “set-theoretic” notion of invariants, suppose that K = Z/2Z is a field with two elements, so there is only one K -valued point of Gm = D1 , namely the identity. Since Gm (K ) is trivial, the notion of a Gm (K )-module is no different from that of a K -vector space, and the operation of taking Gm (K )-invariants is trivial.

Eigenspaces for Matrix Group Actions To study further the interplay between Tn , Un , Dn , we show next that the formation of Tn -eigenspaces can be broken down in two steps. It is easy to see set-theoretically that since Un is a normal subgroup of Tn , for every Tn -module the invariants M Un form a Tn -submodule, and since Tn = Dn Un , we have M Tn = (M Un )Dn . We formalize this, as well as the corresponding statement for eigenspaces, as follows. Proposition 9.7.10 Suppose that M is a Tn -module, and identify X (Tn ) = X (Dn ) as in Lemma 9.7.8. We have that M Un is a Tn -module (and thus a Dn -module) and n for all χ ∈ X (Tn ) = X (Dn ). MχTn = (M Un )D χ

Proof To distinguish between various actions on M, we let GM : M −→ M ⊗ K [G] denote the K [G]-comodule structure map of M, where G = Tn , Dn , or Un . We write −1 : 1 ≤ i ≤ j ≤ n], K [Tn ] = K [Ti, j , T(i,i)

K [Un ] = K [Ti, j : 1 ≤ i < j ≤ n],

K [Dn ] = K [Ti , Ti−1 : 1 ≤ i ≤ n]. The inclusions of Un and Dn into Tn induce surjective homomorphisms πU : K [Tn ] −→ K [Un ], π D : K [Tn ] −→ K [Dn ], with πU (Ti,i ) = 1 and π D (Ti,i ) = Ti for 1 ≤ i ≤ n, πU (Ti, j ) = Ti, j and π D (Ti, j ) = 0 for i = j. Moreover, we have UMn = (id M ⊗πU ) ◦ TMn , DMn = (id M ⊗π D ) ◦ TMn .

(9.32)

9.7 Algebraic Groups and Representations

359

The defining ideals of Un and Dn in K [Tn ] are given by IU = ker(πU ) = Ti,i − 1 : 1 ≤ i ≤ n,

I D = ker(π D ) = Ti, j : 1 ≤ i < j ≤ n.

We define α to be the composition α

K [Tn ]

Tn

K [Tn ] ⊗ K [Tn ]

π D ⊗πU

K [Dn ] ⊗ K [Un ]

and note that α(Ti,i ) = Ti ⊗ 1 for 1 ≤ i ≤ n, α(Ti, j ) = Ti ⊗ Ti, j for 1 ≤ i < j ≤ n. In particular α is an isomorphism (which is equivalent to the fact that the multiplication map Dn × Un −→ Tn is an isomorphism of algebraic varieties). For e ∈ Zn , we abuse notation and write T e for both e1 en · · · Tn,n in K [Tn ], and for T1e1 · · · Tnen in K [Dn ]. T1,1

We prove now that m ∈ M Un ⇐⇒ TMn (m) =



m e ⊗ T e for m e ∈ M.

(9.33)

e∈Zn

For the implication “⇐=”, note that since εTn (T e ) = 1, it follows from (9.29) that m=



me.

(9.34)

e∈Zn

Since πU (T e ) = 1, we then conclude using (9.32) that UMn (m) =



m e ⊗ 1 = m ⊗ 1,

e∈Zn

so m ∈ M Un . For the implication “=⇒”, using that K [Tn ] = K [Ti,i±1 ] ⊕ I D as K vector spaces, we can write TMn (m) =



me ⊗ T e +

e∈Zn

r 

m i ⊗ fi ,

(9.35)

i=1

where m i are linearly independent, and 0 = f i ∈ I D . Since I D ⊂ ker(εTn ), it follows from (9.29) that (9.34) still holds. Applying id M ⊗π D to (9.35) and using (9.32) we get  DMn (m) = me ⊗ T e, e∈Zn

360

9 Grassmannians, Flag Varieties, Schur Functors and Cohomology

which in turn forces DMn (m e ) = m e ⊗ T e as in Lemma 9.7.9. Applying id M ⊗πU to (9.35), and using that m ∈ M Un together with (9.32) and πU (T e ) = 1, it follows that m ⊗ 1 = UMn (m) =



me ⊗ 1 +

e∈Zn

r 

m i ⊗ πU ( f i ) = m ⊗ 1 +

i=1

r 

m i ⊗ πU ( f i ).

i=1

Since m i are linearly independent, this forces f i ∈ ker(πU ) for all i. Using (9.29), it follows from (9.35) that 

me ⊗ T e ⊗ T e +

e∈Zn

r 

m i ⊗ Tn ( f i ) =



T

 Mn (m e ) ⊗ T e +

e∈Zn

i=1

r  T  Mn (m i ) ⊗ f i . i=1

Applying id M ⊗π D ⊗ πU to this identity, we get an equality in M ⊗ K [Un ] ⊗ K [Dn ]: 

me ⊗ T e ⊗ 1 +

e∈Zn

r 

m i ⊗ α( f i ) =

i=1





DMn (m e ) ⊗ 1 =

e∈Zn

m e ⊗ T e ⊗ 1.

e∈Zn

Since α is injective, f i = 0, and m i are linearly independent, we must have r = 0, so (9.33) holds. Combining (9.33), (9.29) and Tn (T e ) = T e ⊗ T e , we have that for m ∈ M Un 

TMn (m e ) ⊗ T e =

e∈Zn



me ⊗ T e ⊗ T e,

e∈Zn

which forces TMn (m e ) = m e ⊗ T e , so m e ∈ M Un . We get TMn (M Un ) ⊂ M Un ⊗ K [Tn ], so M Un is a Tn -submodule of M. Suppose now that χ ∈ X (Tn ) = X (Dn ) corresponds to e ∈ Zn as in Lemma 9.7.8. We have m ∈ MχTn ⇐⇒ TMn (m) = m ⊗ T e  (9.33), Lemma 9.7.9  ⇐⇒ m ∈ M Un and m ∈ MχDn n ⇐⇒ m ∈ (M Un )D χ ,

concluding our proof.



Sheaves of G-Modules We end this section with an elementary result on sheaves of G-modules, which will be used in establishing the relationship between Schur functors and cohomology on flag varieties in the following sections.

9.8 Schur Functors

361

Lemma 9.7.11 If F is a sheaf of G-modules on a space B, and χ : G −→ Gm is a character of G, then the presheaf FχG defined by FχG (U ) = F (U )χG is a sheaf. Moreover, for every continuous map f : B −→ B we have   f ∗ FχG = ( f ∗ F )χG . Proof Since − ⊗ K K [G] is exact, it follows that the sheaf F ⊗ K [G] satisfies (F ⊗ K [G])(U ) = F (U ) ⊗ K [G] for every open set U . The hypothesis then implies that there exists a morphism of sheaves F : F −→ F ⊗ K [G], which on an open set U is given by F (U ) = F (U ) : F (U ) −→ F (U ) ⊗ K [G], the structure map for the K [G]-comodule F (U ). The inclusion K → K [G] given by 1 → a(χ ), induces a map of sheaves idF ⊗a(χ ) : F −→ F ⊗ K [G] and we have using (9.31) that F (U )χG = ker ((F − idF ⊗a(χ ))(U )). Our conclusion now follows since the presheaf kernel of a morphism of sheaves is a sheaf. We can reformulate the discussion above as the existence of an exact sequence 0 −→ FχG −→ F −→ F ⊗ K [G]. For a continuous map f : B −→ B , the direct image functor f ∗ is left exact. Using the identification f ∗ (F ⊗ K [G]) = ( f ∗ F ) ⊗ K [G], we obtain an exact sequence   0 −→ f ∗ FχG −→ f ∗ F −→ ( f ∗ F ) ⊗ K [G],   which implies f ∗ FχG = ( f ∗ F )χG , as desired.



9.8 Schur Functors For every m ≥ 0, we consider the vector space K m with standard basis e1 , . . . , em , and endow K m with a Tm -module structure given by  K m (ei ) =

i  j=1

e j ⊗ T j,i .

362

9 Grassmannians, Flag Varieties, Schur Functors and Cohomology 

For m ≤ m  we think of K m ⊂ K m as the span of the first m vectors in the standard  basis of K m . We also think of K [Tm ] ⊂ K [Tm  ] as a subring via the natural inclusion (corresponding to the group homomorphism Tm  −→ Tm obtained by truncating an m  × m  matrix to the upper left m × m block). Note that we have a commutative diagram where the vertical maps are induced by the aforementioned inclusions. Km

Km



K m

K m

K m ⊗ K [Tm ]



K m ⊗ K [Tm  ]

Schur Functors as Eigenspaces Suppose that F is a locally free sheaf on an algebraic variety B, and that v = (v1 ≥ v2 ≥ · · · ≥ 0) is a partition of size |v| = v1 + v2 + · · · = m. With the identification X (Tm ) ∼ = Zm in Lemma 9.7.8, we let Sv F = (Symm (F ⊗ K m ))Tv m , where the action of Tm on Symm (F ⊗ K m ) is induced from the natural Tm -action on K m . We call Sv the Schur functor associated to v. Proposition 9.8.1 Suppose that F is locally free of rank r , v∗ = λ = ( p1 , p2 , . . . ) is the shape dual to v, and l ≥ p1 . We have (Symm (F ⊗ K l ))Tv l is locally free, of rank equal to the number of standard tableaux of shape λ and entries in {1, . . . , r }. In particular, if vi = 0 for i > r then (Symm (F ⊗ K l ))Tv l = 0. Proof Suppose that U = Spec A is an open affine subset of B where F|U is free, and choose a basis f 1 , . . . , fr for the free A-module F = (U, F ). If we write xi, j = f i ⊗ e j , for 1 ≤ i ≤ r, 1 ≤ j ≤ l, then M = (U, Symm (F ⊗ K l )) = Symm (F ⊗ K l ) is a free A-module with basis indexed by the monomials of degree m in the variables xi, j . It follows from Theorem 3.2.1 (see also Remark 3.2.9) that M has an A-basis consisting of standard bitableaux of shape σ , where |σ | = m, and M Ul is the A-submodule spanned by the standard bitableaux (R| In(σ )) where |σ | = m and R has entries in {1, . . . , r }. Since

9.8 Schur Functors

363

(R| In(σ )) is a σ ∗ -eigenvector for the Dl -action on M Ul , it follows from Proposition 9.7.10 that l MvTl = (M Ul )D v has an A-basis consisting of standard bitableaux (R| In(λ)), from which the desired conclusions follow.  Corollary 9.8.2 Suppose that l ≥ 0 is such that vi = 0 for i > l. For every d ≥ l, the inclusion Symm (F ⊗ K l ) ⊂ Symm (F ⊗ K d ) induces an isomorphism (Symm (F ⊗ K l ))Tv l ∼ = (Symm (F ⊗ K d ))Tv d .

(9.36)

In particular, if we take d = m in the equation above, then we get (Symm (F ⊗ K l ))Tv l ∼ = Sv F . We conclude that (Symm (F ⊗ K d ))Tv d ∼ = Sv F for all d ≥ l. Proof With the notation in Proposition 9.8.1, the assumption vi = 0 for i > l is equivalent to p1 ≤ l. Since d ≥ l ≥ p1 , the proof of Proposition 9.8.1 applies to both sides of (9.36), providing the desired identification.  Based on the proof of Proposition 9.8.1, we can show that for every morphism f : B −→ B we have (9.37) f ∗ (Sv F ) = Sv ( f ∗ F ). Indeed, the proof of Proposition 9.8.1 shows that the inclusion Sv F ∼ = (Symm (F ⊗ K l ))Tv l → Symm (F ⊗ K l ) is locally split (MvTl is generated by a subset of a basis of M). It follows that f ∗ (Sv F ) → f ∗ Symm (F ⊗ K l ) = Symm ( f ∗ F ⊗ K l ) is also locally split. The functoriality of f ∗ shows that f ∗ (Sv F ) → Sv ( f ∗ F ) =   ker Symm ( f ∗ F ⊗ K l ) −→ Symm ( f ∗ F ⊗ K l ) ⊗ K [G] , and since Sv ( f ∗ F ) is locally split in Symm ( f ∗ F ⊗ K l ), it follows that the inclusion f ∗ (Sv F ) → Sv ( f ∗ F ) is locally split as well. Since the two sheaves are locally free of the same rank, we get the desired isomorphism (9.37). m Lemma 9.8.3 If v = (1m ) then Sv F ∼ F . If v = (m) then Sv F ∼ = = Symm (F ). m Proof We have an inclusion ι : F −→ Symm (F ⊗ K m ) defined locally by ι( f 1 ∧ · · · ∧ f m ) =

 σ ∈Sm

sign(σ ) ·

m  i=1

f σ (i) ⊗ ei ,

364

9 Grassmannians, Flag Varieties, Schur Functors and Cohomology

and one can check that the image of ι is contained in (Sym m (F ⊗ K m ))T(1mm ) = S(1m ) F . If r = rank(F ) then Proposition 9.8.1 implies that rank

m 

F

=

  r = rank(S(1m ) F ), m

 so ι establishes an isomorphism m F ∼ = S(1m ) F , as desired. The conclusion S(m) F ∼ = m Sym (F ) follows by applying Corollary 9.8.2 with l = 1, d = m, and using the iso morphism F ∼ = F ⊗ K 1. Based on Lemma 9.8.3, one can view (9.37) as a generalization of the fact that symmetric and exterior powers are compatible with pullbacks. The next result will allow us to extend the definition of Schur functors from partitions to arbitrary dominant weights. Lemma 9.8.4 If rank(F ) = r and v satisfies vi = 0 for i > r , then we have an isomorphism det(F ) ⊗ Sv F ∼ = Sv+(1r ) F . Proof Consider the inclusion ι : det(F ) −→ Symr (F ⊗ K r ) from the proof of Lemma 9.8.3, and recall that Im(ι) = (Symr (F ⊗ K r ))T(1rr ) . If m = |v| then the map ψ defined by the commutative diagram det(F ) ⊗ Symm (F ⊗ K r )

ι⊗id

Symr (F ⊗ K r ) ⊗ Symm (F ⊗ K r )

ψ

Symr +m (F ⊗ K r ) restricts to an isomorphism Tr det(F ) ⊗ (Symm (F ⊗ K r ))Tv r ∼ = (Symr +m (F ⊗ K r ))v+(1 r ).

Indeed, if we work in a local chart U = Spec A as in the proof of Proposition 9.8.1, we have that (U, det F) = A · δ is free of rank one, where δ = [1, . . . , r |1, . . . , r ] is the unique standard bitableau of shape (r ), and ψ|U is given by ψ(δ ⊗ g) = δ · g. If we write v ∗ = σ = (s1 , s2 , . . . ) and (v + (1r ))∗ = σ˜ = (r, s1 , s2 , . . . ), then multiplication by δ establishes a bijection between the standard bitableaux of shape σ with entries in {1, . . . , r } and those of shape σ˜ . Since such bitableaux give the bases r of (U, (Symm (F ⊗ K r ))Tv r ) and (U, (Symr +m (F ⊗ K r ))Tv+(1 r ) ) respectively, the desired isomorphism follows. 

9.8 Schur Functors

365

Using Lemma 9.8.4, for F of rank r we can extend the definition of the Schur functors Sv F to an arbitrary dominant weight v ∈ Zrdom : we write v = u + (d r ) for some d ∈ Z, u = (u 1 ≥ · · · ≥ u r ≥ 0), and let Sv F = Su F ⊗ (det F )d .

Schur Functors as Direct Images The next result establishes the relationship between Schur functors and direct images for line bundles on flag varieties. Theorem 9.8.5 Suppose that E is a locally free sheaf of rank n on an algebraic variety B, let F = FB (E) be the corresponding flag variety, and let π : F −→ B denote the structure morphism. If y ∈ Zndom then π∗ (OF (y)) = S y E, and if y ∈ Zn \ Zndom then π∗ (OF (y)) = 0. Proof Suppose first that y ∈ Zn \ Zndom , and choose j such that y j < y j+1 . We factor π as in (9.19). Using the notation in the proof of Lemma 9.3.4, and π∗ = g∗ ◦ f ∗ , it is enough to check that  R f ∗ OF j+1, j−1 (y j , y j+1 ) = 0. Using the isomorphism R R j+1, j−1 (y j − y j+1 , 0) ⊗ f ∗ (det(R j+1, j−1 ) y j+1 ) OF j+1, j−1 (y j , y j+1 ) ∼ = OF

and the projection formula, this further reduces to checking that  R f ∗ OF j+1, j−1 (y j − y j+1 , 0) = 0, which follows from (9.13) and the fact that F = PF (R j+1, j−1 ). Suppose now that y ∈ Zndom . Using the projection formula, we conclude that π∗ (OF (y)) = π∗ (OF (y  )) ⊗ (det E) yn , where yi = yi − yn for 1 ≤ i ≤ n. It follows from Lemma 9.8.4 that we may assume without loss of generality that yn = 0. We factor π as π

F = FB (E)

f

B

g

B

366

9 Grassmannians, Flag Varieties, Schur Functors and Cohomology

where B = GB (n − 1, E). The morphism f then identifies F with FB (Q), where Q is the tautological rank (n − 1) quotient sheaf on B . Moreover, since yn = 0 we have OF (y) = OEF (y) = OQF (y1 , . . . , yn−1 ), so using induction on n, we deduce that f ∗ (OF (y)) = S y Q, , so it suffices to verify that g∗ (S y Q) = S y E. Using Corollary 9.3.6 with F = O⊕(n−1) B we have that g∗ (S y Q) = g∗ (Symm (Q ⊗ K n−1 )Ty n−1 ) = Symm (E ⊗ K n−1 )Ty n−1 = S y E, 

as desired. ∨

Using the identification (9.11), we get that OE (y1 , . . . , yn ) corresponds to OE (−yn , . . . , −y1 ), and therefore by Theorem 9.8.5 we have an isomorphism   S y E∨ ∼ = S(−yn ,...,−y1 ) E.

(9.38)

Combining Theorem 9.8.5 with Kempf vanishing, we obtain the following. Corollary 9.8.6 For 1 ≤ l ≤ n − 1, write y ∈ Zndom as the concatenation of x = (y1 , . . . , yl ) ∈ Zldom and z = (yl+1 , . . . , yn ) ∈ Zn−l dom . If we let G = G(l, E) with structure morphism f : G −→ B, and tautological sheaves Q of rank l, and R of rank (n − l), then R f ∗ (Sx Q ⊗ Sz R) = S y E. Proof With the notation in Theorem 9.8.5, we factor π as π

F = FB (E)

h

F

g

G

f

B

where F = F({l, l − 1, . . . , 1}, E). We have F = FF (R), where R = g ∗ R is the tautological rank (n − l) subsheaf on F , using the usual abuse of notation, and F = FG (Q). Moreover,   OEF (y) = ORF (z) ⊗ h ∗ OQF (x) .

9.8 Schur Functors

367

Using the projection formula, and Theorems 9.8.5 and 9.4.1, we obtain  (9.37) Rh ∗ OEF (y) = Sz R ⊗ OQF (x) = g ∗ (Sz R) ⊗ OQF (x). Using the projection formula, (9.12), and Theorems 9.8.5 and 9.4.1, we get  R(g ◦ h)∗ OEF (y) = Sz R ⊗ Sx Q. Finally, using (9.12), π = f ◦ g ◦ h, and Theorems 9.8.5 and 9.4.1, we conclude  R f ∗ (Sz R ⊗ Sx Q) = Rπ∗ OEF (y) = S y E, 

as desired.

Cauchy’s Formula The theory of Schur functors gives rise to a natural filtration on symmetric powers of a tensor product of locally free sheaves, as in the next result, which is a characteristic-free version of Cauchy’s formula (see also Corollary 11.5.3, and [317, Theorem 2.3.2, Corollary 2.3.3]). Theorem 9.8.7 Suppose that F1 , F2 are locally free sheaves of finite rank on an algebraic variety B. For each d ≥ 0 there exists a (Cauchy) filtration on Symd (F1 ⊗ F2 ) with composition factors Sv F1 ⊗ Sv F2 , for v = (v1 ≥ v2 ≥ · · · ≥ 0), |v| = d.

(9.39)

Proof We let ri = rank(Fi ), and prove the statement by induction on the triple (d, r1 , r2 ). If d = 0 then there is nothing to prove. Without loss of generality, we may assume that r1 ≥ r2 . If r2 = 1 then F2 is an invertible sheaf, and we have by Lemma 9.8.3 that Symd (F1 ⊗ F2 ) = Symd (F1 ) ⊗ F2d = S(d) F1 ⊗ S(d) F2 , as desired. For the induction step, we assume r1 ≥ r2 ≥ 2, and consider G = G(r1 − 1, F1 ), with π : G −→ B the structure morphism, and tautological sheaves R of rank one, Q of rank r1 − 1. If we let F˜ i = π ∗ Fi , then we have by induction that Sym d (Q ⊗ F˜ 2 ) admits a filtration with composition factors Sv Q ⊗ Sv F˜ 2 , for v = (v1 ≥ v2 ≥ · · · ≥ 0), |v| = d.

(9.40)

368

9 Grassmannians, Flag Varieties, Schur Functors and Cohomology

ByCorollary 9.8.6, every factor in (9.40) has vanishing higher direct images, so π∗ Symd (Q ⊗ F˜ 2 ) admits a filtration with composition factors    (9.37) π∗ Sv Q ⊗ Sv F˜ 2 = π∗ Sv Q ⊗ Sv F2  Sv F1 ⊗ Sv F2 if vi = 0 for i ≥ r1 ; = . 0 otherwise.

(9.41)

Suppose first that r1 > r2 , so that rank(F2 ) ≤ r1 − 1. By Corollary 9.3.6 we have  π∗ Symd (Q ⊗ F˜ 2 ) = Symd (F1 ⊗ F2 ), and the conclusion follows from (9.41) and the fact that Sv Q ⊗ Sv F˜ 2 = 0 ⇐⇒ Sv F1 ⊗ Sv F2 = 0 ⇐⇒ vi = 0 for i > r2 . If r1 = r2 = r , then Corollary 9.3.6 yields an exact sequence 0 −→ det(F1 ) ⊗ det(F2 ) ⊗ Symd−r (F1 ⊗ F2 ) −→ Symd (F1 ⊗ F2 ) −→  −→ π∗ Symd (Q ⊗ F˜ 2 ) −→ 0, and the conclusion follows by induction, combining (9.41) and Lemma 9.8.4: the subsheaf det(F1 ) ⊗ det(F2 ) ⊗ Symd−r (F1 ⊗ F2 ) has composition factors Sv F1 ⊗ Sv F2 , d ˜ with |v| = d and vr = 0, while the quotient π∗ Sym (Q ⊗ F2 ) has composition factors Sv F1 ⊗ Sv F2 , with |v| = d and vr = 0.



We write P(d) for the set of partitions of d, write λ for an element of P(d) when we think of the shape of a tableau, and v when we think of the weight of a Schur functor Sv . We write Nd = |P(d)| for the number of partitions of d, and order the elements of P(d) lexicographically: λ1 >lex λ2 >lex · · · >lex λ Nd .

(9.42)

We write vi = (λi )∗ for the dual partition as usual. Keeping track of the inductive proof of Theorem 9.8.7, we can write the filtration of Symd (F1 ⊗ F2 ) as 0 = G0 ⊂ G1 ⊂ · · · ⊂ G Nd = Symd (F1 ⊗ F2 ), with

Gi /Gi−1 ∼ = Svi F1 ⊗ Svi F2 for i = 1, . . . , Nd .

(9.43)

9.9 Grothendieck Duality

369

Notice that Gi = Gi−1 if vri = 0 for some r > min(rank F1 , rank F2 ). Since vri = 0 is equivalent to λi1 ≥ r , this condition will hold at the beginning of the filtration, and if we write i 0 for the first index when Gi0 = Gi0 −1 then we get strict inclusions 0  Gi0  Gi0 +1  · · ·  G Nd = Symd (F1 ⊗ F2 ). Example 9.8.8 Suppose that d = 4, rank F1 = 3, and rank F2 = 2. We have Nd = 5 and the corresponding partitions λi , vi are given as follows: 1 2 3 4 5 i i λ (4) (3, 1) (2, 2) (2, 1, 1) (1, 1, 1, 1) i (4) v (1, 1, 1, 1) (2, 1, 1) (2, 2) (3, 1) Since rank F2 = 2, it follows from Proposition 9.8.1 that S(1,1,1,1) F2 = S(2,1,1) F2 = 0, and therefore G0 = G1 = G2 = 0. We have G3 = S(2,2) F1 ⊗ S(2,2) F2 , G4 /G3 = S(3,1) F1 ⊗ S(3,1) F2 , G5 /G4 = S(4) F1 ⊗ S(4) F2 = Sym4 (F1 ) ⊗ Sym4 (F2 ). We note that G5 = Sym4 (F1 ⊗ F2 ), and the quotient map G5  G5 /G4 generalizes for any d to a quotient map Symd (F1 ⊗ F2 )  Symd (F1 ) ⊗ Symd (F2 ). In Sect. 10.5, we will give a different perspective on the filtration (9.43), based on the straightening law from Chap. 3. In fact, we will produce one such filtration for any ordering of the partitions in P(d) which is compatible with ≥ (in the same way as (9.42) is).

9.9 Grothendieck Duality In this section we formulate a general duality result, usually referred to by the name Grothendieck duality, that will be used in several cohomology calculations in subsequent chapters. We follow Hartshorne [164], making simplifying assumptions that fit more closely into our context. We write D + (resp. D − ) for the derived category of bounded below (resp. above) complexes with quasi-coherent cohomology. The next result (the Grothendieck duality theorem) follows from [164, Theorem III.8.7, III.11.1].

370

9 Grassmannians, Flag Varieties, Schur Functors and Cohomology

Theorem 9.9.1 Let f : X −→ Y denote a projective morphism of algebraic varieties over the field K . There exists a functor f ! : D + (Y ) −→ D + (X ), called the exceptional inverse image functor, with the property that for every F . ∈ D − (X ) and G. ∈ D + (Y ) we have

om Y (R f∗F ., G.) ∼= R f∗(R Hom X (F ., f !G.)).

RH

(9.44)

Moreover, we have: (a) If f is smooth of relative dimension n, then

. f ! G. ∼ = f ∗ G ⊗ ω X/Y [n], where ω X/Y [n] is the complex consisting of the relative canonical sheaf, concentrated in cohomological degree −n. (b) If f is a closed immersion, then

om Y ( f∗O X , G.))|

f ! G. ∼ = (R H

X

,

We explain next several important consequences of Theorem 9.9.1. If B is a projective variety over K , and if we write f : B −→ Spec K , then f ! K is called the dualizing complex of B. If in addition B is smooth, then part (a) implies f ! K = ωB [n]. Taking F . = F for some coherent sheaf F , and G. = K , we get from (9.44)

om X (F , ωB[n])) = R Hom X (F , ωB)[n].

R Hom(R f ∗ F , K ) ∼ = R f ∗ (R H

Taking cohomology in degree (− j) yields an isomorphism H j (B, F )∨ ∼ = Ext n− j (F , ωB ),

(9.45)

which is the Serre duality Theorem [165, Sect. III.7]. Going back to a general projective morphism f : X −→ Y , if F . = F is locally free of finite rank, then (9.44) becomes

om Y (R f∗F , G.) ∼= R f∗(F ∨ ⊗ f !G.),

RH

(9.46)

so we can rewrite for instance (9.45) as H j (B, F )∨ ∼ = H n− j (B, F ∨ ⊗ ωB ). We get the following relative version of Serre duality.

9.9 Grothendieck Duality

371

Lemma 9.9.2 Suppose that f : X −→ Y is a smooth projective morphism of relative dimension n, and that F is locally free of finite rank, such that for some r we have R r f ∗ F is locally free, and R j f ∗ F = 0 for j = r . There exists an isomorphism (R r f ∗ F )∨ ∼ = R n−r f ∗ (F ∨ ⊗ ω X/Y ), and R j f ∗ (F ∨ ⊗ ω X/Y ) = 0 for j = n − r . Proof If we take G. = OY then it follows from Theorem 9.9.1(a) that f ! G. = ω X/Y [n], so (9.46) yields (R f ∗ F )∨ ∼ = R f ∗ (F ∨ ⊗ ω X/Y )[n], and the desired conclusion follows by computing cohomology on both sides.



Relative Serre Duality for Flag Bundles The reader may verify using Lemma 9.9.2 that (9.15) is a consequence of (9.13), (9.14), and the special case ωPB (E)/B ∼ = Q−n ⊗ π ∗ (det(E)) of (9.7). We record for later use another consequence of Lemma 9.9.2. Corollary 9.9.3 With π : F −→ B as in Theorem 9.4.1, suppose that x ∈ Zn satisfies x1 < x2 < · · · < xn . If x j − x j+1 = −1 for some j with 1 ≤ j ≤ n − 1, then R i π∗ (OF (x)) = 0 for all i. Otherwise, let z = (−x1 − (n − 1), −x2 − (n − 3), . . . , −xn−1 + (n − 3), −xn + (n − 1)). We have that z ∈ Zndom , and   n n ∨ i ( ) R 2 π∗ (OF (x)) = (Sz E) , R π∗ (OF (x)) = 0 for i = . 2 Proof If x j − x j+1 = −1 for some j, the conclusion follows from Lemma 9.3.4. Otherwise, the hypothesis x j < x j+1 implies that x j − x j+1 ≤ −2 for 1 ≤ j ≤ n − 1. In particular z j − z j+1 = −x j + x j+1 − 2 ≥ 0, so z is dominant. It follows from Theorem 9.4.1 that π∗ OF (z) = Sz E, and R i π∗ (OF (z)) = 0 for i = 0.

372

9 Grassmannians, Flag Varieties, Schur Functors and Cohomology

Notice that by Lemma 9.2.2 we have OF (z) ∼ = (OF (x))∨ ⊗ ωF/B . We can then apply Lemma 9.9.2, with f = π , r = 

R (2) π∗ (OF (x)) n

and R i π∗ (OF (x)) = 0 for i =

∨

n  , F = OF (x), to get 2

∼ = π∗ OF (z) = Sz E,

n  , as desired. 2



Describing Ext Groups Via Sheaf Cohomology For the remainder of the section, we focus on a geometric setting leading to many explicit calculations of Ext groups, that we will detail in subsequent chapters. Suppose that B is a smooth projective variety over the field K , V is a finite dimensional K -vector space, and ξ, η are locally free sheaves on B sitting in a short exact sequence 0 −→ ξ −→ V ⊗ K OB −→ η −→ 0.

(9.47)

We consider the following diagram of spaces and maps: π

Y = AB (η)

ι

X = AB (V ⊗ K OB ) ϕ

q

B

(9.48)

p

A = A K (V ) We write S = Sym. (V ) so that A = Spec(S), and note that X ∼ = A × B, with p, q denoting the projections to the two factors. We have that X is a trivial vector bundle over B, and Y is a subbundle, locally cut out by the sections in ξ . It follows that OY is resolved as an OX -module by a Koszul complex on ξ . More precisely, if we let r = rank(ξ ) then we have an exact sequence r 

(q ∗ ξ ) −→

r −1

(q ∗ ξ ) −→ . . . −→ (q ∗ ξ ) −→ OX −→ ι∗ OY −→ 0. (9.49) Since A is affine, we will make the usual identification between quasi-coherent sheaves on A and S-modules. It follows that for every quasi-coherent sheaf M on Y and integer j we have 0 −→

9.9 Grothendieck Duality

373

R j ϕ∗ (M) = H j (Y , M)

is an S-module.

The following special case of the Grothendieck duality theorem will be our main tool for computing Ext modules in later chapters. Theorem 9.9.4 Suppose that M is a locally free sheaf of finite rank on Y , let M = H 0 (Y , M), and suppose H i (Y , M) = 0 for i > 0. If we let dB denote the dimension of B, then we have for all j ∈ Z that   j Ext S (M, S) ∼ = H j+dB −rank(ξ ) Y , M∨ ⊗ π ∗ (ωB ⊗ det(ξ )∨ ) . Proof The hypothesis of the theorem implies an isomorphism Rϕ∗ M ∼ = M,

(9.50)

where M and M are interpreted as complexes concentrated in cohomological degree 0. By Theorem 9.9.1, we have  R Hom S (Rϕ∗ M, S) ∼ = Rϕ∗ R H

om Y (M, ϕ! S) ∼= Rϕ∗ M∨ ⊗ ϕ! S ,

(9.51)

where the second isomorphism follows from the fact that M is locally free. In light of (9.50), the jth cohomology of the complex on the left of (9.51) is precisely j Ext S (M, S), so in order to prove our result, we need to identify the right hand side. We use the factorization ϕ = p ◦ ι to get that ϕ ! = ι! ◦ p ! . Since p : A × B −→ A is the projection map, it follows that p ! (S) ∼ = q ∗ (ωB [dB ]), where ωB [dB ] is the dualizing complex on B. For a locally free sheaf E on X , we have ι! (E) ∼ = E OrX (ι∗ OY , E)[−r ],

xt

where r = rank(ξ ) is the codimension of Y in X , and the right hand side is interpreted as a sheaf on Y in the natural way. We can therefore use (9.49) to compute ι! (E), and we obtain r  ι! (E) ∼ (q ∗ ξ )∨ ⊗ E[−r ]|Y . = Since q ∗ commutes with tensor operations, and let E = q ∗ ωB then we obtain

r

ξ = det(ξ ), it follows that if we

ϕ! S ∼ = ι! (E)[dB ] ∼ = π ∗ (det(ξ )∨ ⊗ ωB )[dB − r ].

374

9 Grassmannians, Flag Varieties, Schur Functors and Cohomology

Plugging this into (9.51) and taking cohomology yields   Ext j (M, S) ∼ = R j+dB −r ϕ∗ M∨ ⊗ π ∗ (det(ξ )∨ ⊗ ωB ) , 

as desired.

In the cases of interest to us, the sheaf M in Theorem 9.9.4 will be pulled back from B, in which case we can rephrase the formula for Ext groups as a cohomology calculation on B. Corollary 9.9.5 With the notation and assumptions in Theorem 9.9.4, suppose that M = π ∗ V for some locally free sheaf V on B. Thinking of V as a Z-graded sheaf concentrated in degree v, we have that M = H 0 (B, V ⊗ Sym. (η))   is a graded S-module with Mk = H 0 B, V ⊗ Symk−v (η) . Moreover, with the induced grading on Ext groups, we get isomorphisms   Ext j (M, S)k ∼ = H j+dB −rank(ξ ) B, V∨ ⊗ det(ξ )∨ ⊗ ωB ⊗ Symk+rank(ξ )+v η   ∨ ∨ ∼ . = H rank(ξ )− j B, V ⊗ det(ξ ) ⊗ Symk+rank(ξ )+v η Proof Since π∗ OY = Sym. (η), it follows from the projection formula that M = H 0 (Y , π ∗ V) = H 0 (B, V ⊗ π∗ OY ) = H 0 (B, V ⊗ Sym. (η)) , as desired. The grading follows by interpreting (9.47) as a sequence of Z-graded sheaves concentrated in degree 1. The first description of Ext j (M, S)k follows from Theorem 9.9.4 and the projection formula, noting that V∨ is concentrated in degree −v, det(ξ )∨ in degree −r , and ωB in degree 0. The second description of Ext j (M, S)k follows from the first one by applying Serre duality. 

9.10 Cohomology Vanishing on Grassmannians In this section we reformulate Theorem 9.5.1 as a vanishing result on a Grassmannian, which is the form that will be used later on in Sect. 10.7. Theorem 9.10.1 Suppose that V is an n-dimensional K -vector space, 0 ≤ l ≤ n, and let G = G(l, V ), with tautological sheaves Q, R as in Sect. 9.2. Suppose that d  0, and v ∈ Zldom , y ∈ Zn−l dom , and c ≥ 0 are such that c = y1 ≥ y2 ≥ · · · ≥ yn−l ≥ 0, d ≥ |y|.

9.10 Cohomology Vanishing on Grassmannians

375

If we let v = |v| = v1 + · · · + vl then we have  (d − c + 1)l + v H j G, (det Q)c−l ⊗ (Sv Q)∨ ⊗ S y R = 0 for j > . 2 If l = 1 then the condition d  0 may be replaced with d ≥ n − 2. Proof Consider the full and partial flag varieties F = F(V ) and F = F({l, l − 1, . . . , 1}, V ), and consider the natural maps f

F

g

F

π

G

Notice that via the map π , we can identify F = FG (Q) as the full flag variety of Q. Similarly, via g we identify F = FF (Rn,l ), and note that Rn,l = π ∗ R. We define b1 = v1 + (2l − 1), b2 = v2 + (2l − 3), . . . , bl = vl + 1, and note that b1 > b2 > · · · > bl > 0, so that the line bundle L = OFV (c − b1 , c − b2 , . . . , c − bl , y1 , . . . , yn−l ) satisfies the hypotheses of Theorem 9.5.1. If we let L = OFV (c − b1 , c − b2 , . . . , c − bl ), then we get R

L = g ∗ L ⊗ OF n,l (y1 , . . . , yn−l ). It follows from the projection formula that g∗ L = L ⊗ S y Rn,l = L ⊗ π ∗ (S y R), and R i g∗ L = 0 for i = 0. Moreover, it follows from Corollary 9.9.3 that l R (2) π∗ L = (det Q)c−l ⊗ (Sv Q)∨ , and R i π∗ L = 0 for i =

  l . 2

Using the projection formula, together with the Leray spectral sequence for the composition f = π ◦ g, we obtain l R (2) f ∗ L = (det Q)c−l ⊗ (Sv Q)∨ ⊗ S y R, and R i f ∗ L = 0 for i =

  l . 2

Using once again the Leray spectral sequence, now to compute the cohomology of L, we get

376

9 Grassmannians, Flag Varieties, Schur Functors and Cohomology

  l l H j+(2) (F, L) = H j G, R (2) f ∗ L = H j G, (det Q)c−l ⊗ (Sv Q)∨ ⊗ S y R . By Theorem 9.5.1, these cohomology groups vanish for   (d − c)l + v + l 2 (d − c + 1)l + v l (d − c)l + b = ⇐⇒ j > , j+ > 2 2 2 2 as desired. If l = 1 then we can apply Corollary 9.6.2 to conclude that d ≥ n − 2 suffices to get the desired vanishing.  Using the Cauchy formula and Künneth’s formula [291, Lemma 0BED], Theorem 9.10.1 yields the following vanishing on a product of Grassmannians. Corollary 9.10.2 Let V1 , V2 denote K -vector spaces with dim V1 = m, dim V2 = n, and suppose that 0 ≤ l ≤ m ≤ n, and consider the Grassmannians Gi = G(l, Vi ), with tautological sheaves i Q, i R. If y ∈ Zm−l dom satisfies y1 ≥ · · · ≥ ym−l ≥ 0, we can also think of y ∈ Zn−l dom by adding trailing zeroes. We let c = y1 and define     c Vi = det i Q ⊗ S y i R , and let V = V1  V2 , and η = 1 Q  2 Q. If d  0 and y ∈ Zm−l dom satisfies |y| ≤ d − 1 then   H j G1 × G2 , V ⊗ det(η∨ ) ⊗ (Symv η)∨ = 0 for j > (d − c) · l + v. If l = 1 then the condition d  0 may be replaced with d ≥ n − 1.

  Proof By Theorem 9.8.7, Symv η has a filtration with composition factors Sv 1 Q  2  Sv Q , where v ∈ Zldom satisfies |v| = v. Moreover, since rank(i Q) = l, we get   −l  2 −l  det Q . det(η∨ ) = det 1 Q  ∨ If we let Mi = (det(i Q))−l ⊗ Vi ⊗ Sv (i Q) , it is then sufficient to prove that H j (G1 × G2 , M1  M2 ) = 0 for j > (d − c) · l + v. By the Künneth formula, this further reduces to proving that H j (Gi , Mi ) = 0 for j >

(d − c) · l + v , 2

which follows from Theorem 9.10.1, where d is replaced by d − 1, and G = Gi . 

Notes

377

Notes For an introduction to the basic theory of Grassmannians and flag varieties, the reader may also consult Fulton [133, Chap. 9], Weyman [317, Chap. 3], or Lakshmibai and Brown [230, Chap. 12]. The original reference for the Kempf vanishing theorem is [206], and alternative proofs can be found in Jantzen [196, Sect. II.4] and Haboush [161]. Detailed accounts of the theory of algebraic group schemes and their representations can be found in [196] and Milne [247]. For a more extensive discussion of Schur functors, we refer the reader to Akin, Buchsbaum, and Weyman [6, 133], [317, Chap. 3], or Procesi [260, Chap. 9]. A thorough development of duality theory can be found in Hartshorne [164] or [291, Tag 0DWE].

Chapter 10

Asymptotic Regularity for Symbolic Powers of Determinantal Ideals

Throughout this chapter K is a field of arbitrary characteristic, S = K [X ] is the polynomial ring in the entries of the generic m × n matrix, and I p denotes the ideal of p × p minors of the generic matrix. Our goal is to compute explicitly the regularity of high symbolic powers of the ideals I p , which is achieved in Sect. 10.7. To that end, we introduce in Sect. 10.2 a natural filtration on S/I for every ideal I ⊂ S defined by shape, in Sect. 10.3 we describe an algorithm for computing this filtration, and in Sect. 10.4 we specialize the construction to the symbolic powers I = I p(d) . The composition factors of the filtration are constructed combinatorially in Sect. 10.1, and they are given a geometric realization in Sect. 10.6. This realization depends on establishing a theory of sheaves and ideals defined by shape, discussed in Sect. 10.5, which generalizes the notions of subspaces and ideals defined by shape from Chap. 3. The geometric description of the composition factors leads via Grothendieck duality to a description of associated Ext modules in terms of sheaf cohomology groups on products of Grassmannians. The vanishing results from Chap. 9 can then be used to control the regularity of I p(d) , and prove in Theorem 10.7.1 that reg I p(d) = p · d for d  0. When p = 2, we are able to show that the above equality holds for d ≥ n − 1, and in the next chapter we will prove that the same lower bound holds for all p if the field K has characteristic zero. For general p, the problem of finding an effective bound for d in positive characteristic remains open. We conclude the chapter with a discussion of some natural desingularizations of determinantal varieties in Sect. 10.8, and how they allow us to recover basic properties such as normality, Cohen–Macaulayness, or rational singularities.

© Springer Nature Switzerland AG 2022 W. Bruns et al., Determinants, Gröbner Bases and Cohomology, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-05480-8_10

379

380

10 Asymptotic Regularity for Symbolic Powers of Determinantal Ideals

10.1 Quotients of Ideals Defined by Shape Recall from Sect. 3.6 the definition of the ideal I (σ ) , which has a K -vector space basis consisting of all the standard bitableaux of shape ≥ σ . In this section we introduce certain S-module quotients Jl(σ ) of the ideals I (σ ) that have nice combinatorial, geometric, and cohomological properties: they have bases indexed by standard bitableaux, are scheme theoretically supported on determinantal varieties, and admit an accessible description of the modules Ext S (Jl(σ ) , S). Definition 10.1.1 Suppose that σ = (s1 , . . . , su ) is a shape, and that l ≥ 0. We define the set of l-successors of σ as ⎧ ⎫ there exists i ≤ u with ti > si ⎬ ⎨ or succ(σ, l) = τ = (t1 , . . . , tv ) : τ ≥ σ, and . ⎩ ⎭ tu+1 ≥ l + 1 We note that an ideal has a basis of standard bitableaux if and only if it is spanned as a vector space by such bitableaux. This property is then preserved by taking sums, and therefore it is satisfied by the ideals that we introduce next. For any set  of shapes, we define  I (σ ) . (10.1) I () = σ ∈

We will say that τ is a shape in I () if τ ≥ σ for some σ ∈ . Definition 10.1.2 For any shape σ and integer l ≥ 0, we let Jl(σ ) =

I (σ ) I (succ(σ,l))

.

If we take σ to be the empty shape (u = 0), then we get from Definition 10.1.1 that (10.2) succ(∅, l) = {τ = (t1 , . . . , tv ) : t1 ≥ l + 1}, and therefore I (succ(∅,l)) = Il+1 is the determinantal ideal of (l + 1)-minors. Since I (∅) = S is the whole ring, it follows that Jl(∅) = S/Il+1

(10.3)

is the coordinate ring of the variety of rank ≤ l matrices. By Corollary 3.4.2, Jl(∅) has a basis of standard tableaux of shape τ = (t1 , t2 , . . . ), with t1 ≤ l. Our next goal is to identify analogous properties for a general Jl(σ ) . We define the concatenation of two shapes σ = (s1 , . . . , su ), τ = (t1 , . . . , tv ), to be σ  τ = (s1 , . . . , su , t1 , . . . , tv ).

10.2 Filtrations Associated with Ideals Defined by Shape

381

In what follows it will be useful to set s0 = ∞ for a general partition σ = (s1 , . . . , su ), so that the inequality su ≥ l makes sense (and is trivially satisfied) when σ is the empty partition. Lemma 10.1.3 If σ = (s1 , . . . , su ) satisfies su ≥ l, then the S-module Jl(σ ) has a basis indexed by standard bitableaux of shape σ  τ , where τ = (t1 , t2 , . . . ) satisfies l ≥ t 1 ≥ t2 ≥ · · · . Proof When u = 0, the conclusion follows from (10.3), so we assume that u > 0. If we write  = succ(σ, l), it follows from Definition 10.1.2 that Jl(σ ) has a K basis consisting of the standard bitableaux that are contained in I (σ ) but not in I () . Moreover, this condition only depends on the shape μ of the bitableau. Suppose that μ = (m 1 , m 2 , . . . ) is a shape in I (σ ) , but not in I () , and let i denote the minimal index for which m i = si . Since μ ≥ σ , we have m i > si . If i ≤ u, then it follows from Definition 10.1.1 that μ is a shape in I () , a contradiction. We may therefore assume that i > u, and in particular μ = σ  τ for some partition τ . Since μ is not a shape in I () , it follows from Definition 10.1.1 that t1 = m u+1 ≤ l, as desired. Conversely, if μ = σ  τ with t1 ≤ l, then the condition su ≥ l guarantees that μ is a partition. Since μ ≥ σ , we have that μ is a shape in I (σ ) . Moreover, since  l ≥ t1 = m u+1 , we get that μ is not a shape in I () by Definition 10.1.1. Lemma 10.1.4 With notation as in Lemma 10.1.3, the annihilator of the module Jl(σ ) is Ann(Jl(σ ) ) = Il+1 . Proof We first show that Il+1 ⊂ Ann(Jl(σ ) ). For this, it is enough to check that δ ·  ∈ I () for  = succ(σ, l), provided δ ∈ Ml+1 (X ) is an (l + 1)-minor, and  is a bitableau of shape μ, where μ = σ  τ as in Lemma 10.1.3. Since the shape μ˜ of δ ·  is obtained by arranging the sequence μ  {l + 1} in nonincreasing order, it follows that μ˜ ∈  and therefore δ ·  ∈ I () , as desired. / Il+1 . Let 0 = Suppose now that there exists an element f ∈ Ann(Jl(σ ) ) ∈ (In(σ )| In(σ )) denote the initial bitableau of shape σ , and note that 0 ∈ I (σ ) . We / I () . After possibly subtracting will reach a contradiction by showing that f · 0 ∈ an element of Il+1 from f , we may assume that f is a nontrivial linear combination of standard bitableaux  of shape τ = (t1 , t2 , . . . ), where t1 ≤ l ≤ su . For each such bitableau , we have that 0 ·  is standard of shape σ  τ . It follows that f · 0 is a nontrivial linear combination of standard bitableaux outside I () , so f · 0 ∈ / I () , as desired. 

10.2 Filtrations Associated with Ideals Defined by Shape The goal of this section is to construct a filtration S = F 0 ⊃ F 1 ⊃ · · · ⊃ F r = I ()

(10.4)

382

10 Asymptotic Regularity for Symbolic Powers of Determinantal Ideals

for every ideal I () , with the property that the successive quotients F i /F i+1 are isomorphic to modules of the form Jl(σ ) . Given shapes σ = (s1 , . . . , su ), τ = (t1 , . . . , tv ), we write σ ⊃ τ if si ≥ ti for all i (with the convention that si = 0 for i > u, ti = 0 for i > v). For c ≥ 0, we define the (c-) truncation of σ to be σ (c) = (s1 , . . . , sc ). Note that σ (c) = σ if c ≥ u. We write Pm for the set of shapes σ with s1 ≤ m. The next definition describes the combinatorial data governing the filtration (10.4). Definition 10.2.1 For a subset  ⊂ Pm we define Z(I () ) (or Z()) to be the set consisting of pairs (σ, l), where σ = (s1 , s2 , . . . ) ∈ Pm and l ≥ 0 are such that if we write c = max{i : si = 0} for the number of parts of σ , then the following hold: (a) There exists a shape λ = ( p1 , p2 , . . . ) in I () such that λ(c) ⊂ σ and pc+1 ≤ l + 1. (b) For every shape λ in I () satisfying (a) we have pc+1 = l + 1. We note that conditions (a) and (b) can only be true if sc ≥ l + 1. Indeed, consider any λ that satisfies (a) and (b). We have sc ≥ pc ≥ pc+1 = l + 1.

(10.5)

We are now ready to state the main result regarding the filtrations (10.4). Theorem 10.2.2 For every subset  ⊂ Pm , we have that Z() is finite. If r = |Z()| then there exists a filtration as in (10.4) with successive quotients given (in some order) by the modules Jl(σ ) for (σ, l) ∈ Z(). We begin by illustrating Theorem 10.2.2 with an example. Example 10.2.3 Suppose that m = 3 and  = {(3), (2, 2)}, so that I () = I2(2) is the symbolic square of the ideal of 2 × 2 minors of a generic 3 × 3 matrix. We have Z() = {(∅, 1), ((2), 1)} and the corresponding filtration is given by S ⊃ I2 ⊃ I2(2) . The equality S/I2 = J1(∅) is a special case of (10.3), while the isomorphism I2 /I2(2) ∼ = J1((2)) follows from Definition 10.1.2 and the fact that τ is a 1-successor of σ = (2) if and only if τ ≥ (3) or τ ≥ (2, 2).

10.2 Filtrations Associated with Ideals Defined by Shape

383

Combinatorics of Z() We record next a number of basic properties of the sets Z() from Definition 10.2.1. To simplify the discussion, we will say that σ is a shape in Z() if (σ, l) ∈ Z() for some value of l. Lemma 10.2.4 Suppose that σ = (s1 , . . . , sc ) is a shape in Z() with sc > 0. There exists a unique value of l for which (σ, l) ∈ Z(), determined by

l = min pc+1 − 1 : λ = ( p1 , p2 , . . . ) is a shape in I () with λ(c) ⊂ σ . (10.6) Proof If (σ, l) ∈ Z(), it follows from condition (a) in Definition 10.2.1 that there exists a shape λ in I () with λ(c) ⊂ σ and pc+1 ≤ l + 1, which implies that

l ≥ min pc+1 − 1 : λ = ( p1 , p2 , . . . ) is a shape in I () with λ(c) ⊂ σ . If the inequality were strict, then condition (b) in Definition 10.2.1 would fail, contradicting (σ, l) ∈ Z().  The following slight modification of Lemma 10.2.4 proves the finiteness statement in Theorem 10.2.2. Lemma 10.2.5 Let  0 denote the set of minimal shapes in I () with respect to the ⊂ order. The set  0 is finite, and with the notation in Lemma 10.2.4, we have

l = min pc+1 − 1 : λ = ( p1 , p2 , . . . ) ∈  0 with λ(c) ⊂ σ . In particular, the set Z() is a finite set. Proof The fact that  0 is finite is a consequence of Dickson’s lemma, see e.g. Herzog and Hibi [170, Theorem 2.1.1]. The formula for l follows from (10.6), by noting that if λ = ( p1 , p2 , . . . ), μ = (m 1 , m 2 , . . . ) satisfy λ ⊂ μ, then λ(c) ⊂ μ(c) and pc+1 − 1 ≤ m c+1 − 1, so we can restrict (10.6) to shapes that are minimal with respect to ⊂. To prove the finiteness of Z(), let C denote the maximum number of parts of a shape λ ∈  0 . If σ = (s1 , . . . , sc ) has c parts and (σ, l) ∈ Z(), we claim that c < C. Indeed, if c ≥ C then pc+1 = 0 for every λ = ( p1 , p2 , . . . ) ∈  0 , which yields l = −1, a contradiction. Since s1 ≤ m and c < C, it follows that there are only finitely many possible shapes σ in Z(), so the set Z() is finite by Lemma 10.2.4.  Lemma 10.2.6 If σ is a shape in I () , then σ is not in Z(). Proof Suppose by contradiction that σ is a shape of I () and (σ, l) ∈ Z() for some l ≥ 0. With the notation in Lemma 10.2.4, if we take λ = σ then λ(c) ⊂ σ and  pc+1 − 1 = −1. Using (10.6), we get that l ≤ −1, a contradiction.

384

10 Asymptotic Regularity for Symbolic Powers of Determinantal Ideals

Lemma 10.2.7 Suppose that σ = (s1 , . . . , sc ) is a shape not in I () , with sc > 0. If there exists a shape λ in I () with σ ⊃ λ(c), then σ is a shape in Z(). Proof The hypothesis implies that the set in (10.6) is nonempty, so we can define l using formula (10.6), and in particular conditions (a) and (b) in Definition 10.2.1 are satisfied. If σ is not a shape in Z(), then l is negative, hence there exists a shape λ = ( p1 , p2 , . . . ) in I () with pc+1 = 0 and σ ⊃ λ(c) = λ. This implies that σ ≥ λ, and therefore σ is a shape in I () , contradicting the hypothesis. It follows that l ≥ 0 and therefore (σ, l) ∈ Z().  Lemma 10.2.8 Suppose that σ = (s1 , . . . , sc ), τ = (t1 , . . . , tc ), with sc , tc > 0 and σ ⊂ τ . If σ is a shape in Z() and τ is not in I () , then τ is a shape in Z(). Proof Since σ is in Z(), there exists a shape λ in I () with λ(c) ⊂ σ . It follows that λ(c) ⊂ τ , and since τ is not in I () , we get from Lemma 10.2.7 that τ is a shape in Z(), as desired.  Lemma 10.2.9 If σ = (s1 , . . . , sc ) is a shape in Z(), then all the truncations σ (i), i ≥ 0, are also in Z(). Proof Since σ (i) = σ for i ≥ c, we may assume that i < c. We may also assume that sc > 0. If σ is in Z(), then it follows from Lemma 10.2.6 that σ is not in I () , and therefore none of the truncations σ (i) is in I () . To prove that σ (i) is in Z() we apply Lemma 10.2.7. Consider a shape λ in I () with σ ⊃ λ(c). We have σ (i) ⊃ (λ(c))(i) = λ(i), where the last equality follows from i < c. Since σ (i) has exactly i parts, it follows from Lemma 10.2.7 that σ (i) is in Z().  Lemma 10.2.10 If I () = S and if we let

p = min p1 : λ = ( p1 , p2 , . . . ) is a shape in I () , then (∅, p − 1) ∈ Z(). Moreover, for every (σ, l) ∈ Z() we have l ≤ p − 1. Proof Since I () = S, σ = ∅ is not a shape in I () . Moreover, we have σ has c = 0 parts, and λ(0) ⊂ σ for all shapes λ in I () . It follows from Lemma 10.2.7 and Lemma 10.2.4 that (∅, p − 1) ∈ Z(). The fact that l ≤ p − 1 for (σ, l) ∈ Z() follows from (10.6). For the last assertion of the lemma, suppose by contradiction that (σ, l) ∈ Z() with l ≥ p. By (10.5), it follows that sc ≥ p + 1. Consider any shape λ = ( p1 , p2 , . . . ) in I () , with p1 = p. For a sufficiently large d we have ( p d ) ≥ λ, so μ = ( p d ) is also a shape in I () . Since μ(c) = ( p c ) ⊂ σ , it follows that μ satisfies condition (a) in Definition 10.2.1. However, μ fails condition (b) since p < l + 1, contradicting the assumption (σ, l) ∈ Z(). 

10.2 Filtrations Associated with Ideals Defined by Shape

385

Construction of the Filtration To prove Theorem 10.2.2, we will construct a filtration as in (10.4) inductively. We will employ the lexicographic ordering on shapes, given by σ = (s1 , s2 , . . . ) >lex λ = ( p1 , p2 , . . . ) if σ = λ, and the minimal index i for which si = pi satisfies si > pi . We will use freely the implications σ ⊃λ

=⇒

σ ≥λ

=⇒

σ ≥lex λ.

The inductive step of our construction is based on the following proposition. Proposition 10.2.11 Suppose that I () = S and let σ be lexicographically maximal among the shapes in Z(). If we let l ≥ 0 be such that (σ, l) ∈ Z() and let   =  ∪ {σ } then we have an isomorphism 

I ( ) ∼ (σ ) = Jl . I () 

Proof Notice that I ( ) = I () + I (σ ) , so that the image of the natural map f : I (σ ) −→

S I ()



is equal to I ( ) /I () . To prove the desired conclusion, it suffices to show that Ker( f ) = I (succ(σ,l)) . Since Ker( f ) =



I (τ ) ,

τ ≥σ τ a shape in I ()

it is enough to check that {τ : τ ≥ σ and τ is a shape in I () } = succ(σ, l). We write σ = (s1 , . . . , sc ), with sc > 0. For the forward inclusion “⊂”, suppose by contradiction that τ = (t1 , t2 , . . . ) ≥ σ is a shape in I () which is not in succ(σ, l). It follows that ti ≤ si for i ≤ c, and tc+1 ≤ l. Since τ (c) ⊂ σ and tc+1 ≤ l + 1, τ satisfies condition (a) in Definition 10.2.1, but it fails (b), contradicting the fact that (σ, l) ∈ Z().

386

10 Asymptotic Regularity for Symbolic Powers of Determinantal Ideals

For the reverse inclusion “⊃”, let τ ∈ succ(σ, l) and suppose by contradiction that τ is not a shape in I () . Suppose first that ti > si for some i ≤ c, and choose i to be minimal, noting that τ (i) >lex σ . Consider any shape λ in I () with λ(c) ⊂ σ , which exists by Definition 10.2.1. Since i ≤ c, we have τ (i) ⊃ σ (i) ⊃ λ(i), and τ (i) ≤ τ is not a shape in I () . It follows from Lemma 10.2.7 that τ (i) is a shape in Z(), contradicting the maximality of σ . We are left with the case in which ti = si for i ≤ c and tc+1 ≥ l + 1. We note as before that τ (c + 1) ≤ τ is not in I () , and τ (c + 1) >lex σ . By Definition 10.2.1, there exists a shape λ = ( p1 , p2 , . . . ) in I () with λ(c) ⊂ σ and pc+1 ≤ l + 1. Since τ (c) = σ ⊃ λ(c) and tc+1 ≥ l + 1 ≥ pc+1 , it follows that τ (c + 1) ⊃ λ(c + 1). By Lemma 10.2.7, τ (c + 1) is a shape in Z(), contradicting the maximality of σ , and concluding our proof.  Proof (Proof of Theorem 10.2.2) The finiteness of the set Z() was established in Lemma 10.2.5. We prove the existence of the desired filtration by induction on r = |Z()|. If r = 0 then it follows from Lemma 10.2.10 that I () = S, and the conclusion is trivial. We assume that r > 0, let σ be the lexicographically maximal shape in Z(), let l such that (σ, l) ∈ Z(), and define   =  ∪ {σ }. By Proposition 10.2.11, in order to prove the inductive step it suffices to show that Z(  ) = Z() \ {(σ, l)}.

(10.7) 

For the forward inclusion “⊂” we first note that σ is a shape in I ( ) , so by Lemma 10.2.6, σ is not in Z(  ). Consider any (τ, u) ∈ Z(  ), τ = (t1 , . . . , td ), with td > 0. To show that (τ, u) ∈ Z(), we will verify that (i) τ is a shape in Z();  (ii) for every shape μ = (m 1 , m 2 , . . . ) in I ( ) with μ(d) ⊂ τ , there exists a shape λ = ( p1 , p2 , . . . ) in I () with λ(d) ⊂ τ and pd+1 ≤ m d+1 . Once (i) and (ii) are established, we can apply (10.6) to obtain

 u = min m d+1 − 1 : μ = (m 1 , m 2 , . . . ) is a shape in I ( ) with μ(d) ⊂ τ

(ii) = min pd+1 − 1 : λ = ( p1 , p2 , . . . ) is a shape in I () with λ(d) ⊂ τ , which by Lemma 10.2.4 proves that (τ, u) ∈ Z().  To prove (i), we first note that since τ is a shape in Z(  ), it is not in I ( ) by Lemma 10.2.6, and therefore also not in I () . By Lemma 10.2.7, it is then enough

10.2 Filtrations Associated with Ideals Defined by Shape

387

to find a shape λ in I () with τ ⊃ λ(d). Suppose by contradiction that no such  shape exists (and in particular that τ is not in Z()). Since I ( ) = I () + I (σ )  and τ is a shape of Z( ), it follows that there exists a shape μ ≥ σ in I (σ ) with τ ⊃ μ(d) ≥ σ (d). If τ = σ (d) then τ is a shape in Z() by Lemma 10.2.9, a contradiction. Otherwise, let i ≤ d denote the minimal index for which ti > si . We have τ (i) ⊃ σ (i), τ (i) is not in I () , and σ (i) is in Z() by Lemma 10.2.9, so τ (i) is also in Z() by Lemma 10.2.8. Since τ (i) >lex σ , this contradicts the maximality of σ , and concludes the verification of (i).  To prove (ii), consider any μ in I ( ) with μ(d) ⊂ τ . If μ is in I () , then we can take λ = μ. Otherwise, μ ≥ σ and therefore τ ≥ σ (d) as in the previous paragraph, which also showed that we must in fact have τ = σ (d), and thus μ(d) = σ (d). Since τ = σ , we also have d < c. Consider any shape λ in I () with λ(c) ⊂ σ . Since c > d, we have pd+1 ≤ sd+1 . Since μ ≥ σ and μ(d) = σ (d), it follows that m d+1 ≥ sd+1 . This shows pd+1 ≤ m d+1 and λ(d) ⊂ σ (d) = μ(d) ⊂ τ , as desired. It remains to prove the reverse inclusion “⊃” in (10.7). Let (τ, v) ∈ Z() with τ = (t1 , . . . , td ), td > 0. We know by Lemma 10.2.6 that τ is not in I () . By the  maximality of σ , we have τ  σ , so τ is not in I (σ ) . Since I ( ) = I () + I (σ ) , we  conclude that τ is not in I ( ) . Moreover, since τ is in Z(), there exists a shape λ  in I () ⊂ I ( ) such that τ ⊃ λ(d). Applying Lemma 10.2.7, we conclude that τ is a shape in Z(  ). Let u ≥ 0 be such that (τ, u) ∈ Z(  ). By the already established inclusion “⊂” in (10.7), we have that (τ, u) ∈ Z(). Lemma 10.2.4 shows u = v,  so (τ, v) ∈ Z(  ), concluding our proof.

The Radical of an Ideal Defined By Shape We record one final consequence of the results in this section, which the reader may also verify using the theory developed in Chap. 3. Lemma 10.2.12 If p is as in Lemma 10.2.10, then I p is the radical of I () . √ Proof We first show that I p ⊃ I () . Since I () annihilates S/I () , it follows that it also annihilates its composition factors Jl(σ ) , for (σ, l) ∈ Z(). By Lemma 10.2.10, (∅) one of these factors is J p−1 = S/I p . We conclude that I p ⊃ I () , and since I p is √ prime, that I p ⊃ I () . √ We prove the reverse inclusion I p ⊂ I () using Lemmas 10.1.4, 10.2.10, and Theorem 10.2.2. If we let r = |Z()|, then we obtain 

I () ⊃

Il+1 ⊃ I pr ,

(σ,l)∈Z()

showing that I p ⊂

√

I () , as desired.



388

10 Asymptotic Regularity for Symbolic Powers of Determinantal Ideals

10.3 An algorithm to Compute Z() In this section we present an algorithm for determining the set Z(), and illustrate it with an example. The general strategy can be summarized by the following steps: (a) Construct the finite set  0 in Lemma 10.2.5. (b) Generate recursively all the shapes σ in Z(), based on Lemmas 10.2.7 and 10.2.9. (c) For each shape σ in Z(), determine the value of l such that (σ, l) ∈ Z() based on Lemma 10.2.5. To construct  0 , one can proceed as follows: for each λ ∈  we can list all shapes μ ∈ Pm with μ ≥ λ and |μ| = |λ|, and then pick out among all of them those that are minimal with respect to inclusion. This is implemented in Algorithm 10.1. Algorithm 10.1: Function to generate  0 1 2 3 4 5

Input: A finite subset  of Pm Output: The set  0 of shapes in I () that are minimal with respect to inclusion function generateSigma0() D←∅ for λ ∈  do Dλ ← shapes of size |λ| dominating λ D ← D ∪ Dλ

 0 ← minimal shapes in D with respect to ⊂ 7 return  0 6

Example 10.3.1 Suppose that m = 3 and  = {(3, 1), (2, 2, 1, 1)}. The only shape μ ∈ P3 with |μ| = 4 that dominates λ = (3, 1) is μ = λ. The shapes μ ∈ P3 with |μ| = 6 that dominate λ = (2, 2, 1, 1) are listed in Fig. 10.1. Since (3, 1) is contained in each of (3, 1, 1, 1), (3, 2, 1), and (3, 3), it follows that  0 = {(3, 1), (2, 2, 1, 1), (2, 2, 2)}. We will next need a function to compute the value of l in Lemma 10.2.5. For a given shape σ = (s1 , . . . , sc ), our algorithm will generate the subset  of  0

Fig. 10.1 Shapes μ ∈ P3 with |μ| = 6 that dominate λ = (2, 2, 1, 1)

10.3 An algorithm to Compute Z()

389

consisting of those λ for which λ(c) ⊂ σ , and Algorithm 10.2 will be called with σ and  as inputs, in order to determine the value of l. Algorithm 10.2: Function to compute l Input: A shape σ = (s1 , . . . , sc ) in Pm , with sc > 0 A non-empty set  of shapes in Pm Output: The minimum value of pc+1 − 1 for λ = ( p1 , p2 , . . . ) in  1 function computel(σ, ) 2 l ← min{ pc+1 − 1 : λ = ( p1 , p2 , . . . ) ∈ } 3 return l

The main part of the algorithm consists of the recursive generation of all the shapes in Z(). It follows from Lemma 10.2.9 that if σ = (s1 , . . . , sc ) is a shape in Z(), then all of its truncations (s1 ), (s1 , s2 ), . . . , are also in Z(). This suggests that in order to generate all shapes of Z(), we need to understand when (and how) a given shape σ in Z() with c parts can be extended to a shape in Z() with (c + 1) parts. This can be done based on Lemmas 10.2.6 and 10.2.7. In Algorithm 10.3,  will be the set of all shapes λ ∈  0 with λ(c) ⊂ σ . The set Z is a partial approximation of Z(), that will be updated with every iteration of the function. Algorithm 10.3: Function to generate Z() recursively 1 2 3 4 5 6 7 8 9 10

Input: A shape σ = (s1 , . . . , sc ) in Pm , with sc > 0 A non-empty set  of shapes in Pm function shape(σ, ) l ← computel(σ, ) if l ≥ 0 then Z ← Z ∪ {(σ, l)} for s = 1, . . . , sc do σ˜ ← (s1 , . . . , sc , s) ˜ =  for λ ∈  do if λ(c + 1) ⊂ σ˜ then ˜ ← ˜ \ {λ}  ˜ = ∅ then if  ˜ shape (σ˜ , )

11 12 13

return

To understand the function shape in Algorithm 10.3 better, we note that σ will always be a candidate for a shape in Z(), and  will always be a nonempty subset of  0 , so that σ ⊃ λ(c) for some λ ∈  0 . Using Lemmas 10.2.6 and 10.2.7, it follows that σ is a shape in Z() if and only if there is no λ ∈  with λ ⊂ σ , or equivalently, with λc+1 = 0. This is further equivalent to the fact that the quantity l computed in

390

10 Asymptotic Regularity for Symbolic Powers of Determinantal Ideals

line 2 is nonnegative. If this is the case, then the condition in line 3 is satisfied, (σ, l) is added to Z() in line 4, and the function proceeds to construct further candidate ˜ shapes σ˜ in Z() by adding a new part to σ . For each such σ˜ one computes the set  of shapes λ with λ(c + 1) ⊂ σ˜ , and if nonempty, the recursive procedure continues ˜ The algorithm to generate the full set Z() consists with (σ, ) replaced by (σ˜ , ). then of initializing σ to ∅ (the minimal shape in Z() by Lemma 10.2.10),  to  0 (since for every λ ∈  0 we have λ(0) = ∅), and then calling the function shape. This is summarized in Algorithm 10.4. Algorithm 10.4: Algorithm to compute Z() 1 2 3 4

Input: A finite subset  of Pm Output: The set Z() in Definition 10.2.1  0 ← generateSigma0() Z←∅ shape(∅,  0 ) Z() ← Z

Example 10.3.2 With the notation in Example 10.3.1, we keep track in Table 10.1 of how each call of the function shape modifies the approximation Z of Z(). In conclusion, we have that if m = 3 and  = {(3, 1), (2, 2, 1, 1)} then Z() = {(∅, 1), ((2), 1), ((2, 2), 0), ((2, 2, 1), 0), ((3), 0)}. The reader may have noticed that since (3, 1) was a shape in I () , the same is true for (3, 2) and (3, 3), so the last calls of the function shape in the table above are not necessary (see also Lemma 10.2.7). To keep things simple, we do not implement this additional check into our function. Table 10.1 Results of iterations of the function shape for Example 10.3.2 σ



c

l

Z

∅

(3, 1), (2, 2, 1, 1), (2, 2, 2)

0

1

(∅, 1)

(2)

(2, 2, 1, 1), (2, 2, 2)

1

1

(∅, 1), ((2), 1)

(2, 2)

(2, 2, 1, 1), (2, 2, 2)

2

0

(∅, 1), ((2), 1), ((2, 2), 0)

(2, 2, 1)

(2, 2, 1, 1)

3

0

(∅, 1), ((2), 1), ((2, 2), 0), ((2, 2, 1), 0)

(2, 2, 1, 1)

(2, 2, 1, 1)

4

−1

(∅, 1), ((2), 1), ((2, 2), 0), ((2, 2, 1), 0)

(2, 2, 2)

(2, 2, 1, 1), (2, 2, 2)

3

−1

(∅, 1), ((2), 1), ((2, 2), 0), ((2, 2, 1), 0)

(3)

(3, 1), (2, 2, 1, 1), (2, 2, 2)

1

0

(∅, 1), ((2), 1), ((2, 2), 0), ((2, 2, 1), 0), ((3), 0)

(3, 1)

(3, 1)

2

−1

(∅, 1), ((2), 1), ((2, 2), 0), ((2, 2, 1), 0), ((3), 0)

(3, 2)

(3, 1), (2, 2, 1, 1), (2, 2, 2)

2

−1

(∅, 1), ((2), 1), ((2, 2), 0), ((2, 2, 1), 0), ((3), 0)

(3, 3)

(3, 1), (2, 2, 1, 1), (2, 2, 2)

2

−1

(∅, 1), ((2), 1), ((2, 2), 0), ((2, 2, 1), 0), ((3), 0)

10.4 The Filtration for Symbolic Powers of Determinantal Ideals

391

We end this section by indicating the filtration constructed in the proof of Theorem 10.2.2 for m = 3 and  = {(3, 1), (2, 2, 1, 1)}. The shapes of Z() are ordered lexicographically by (3) >lex (2, 2, 1) >lex (2, 2) >lex (2) >lex ∅ and the corresponding filtration is I () ⊂ I (1 ) ⊂ I (2 ) ⊂ I (3 ) ⊂ I (4 ) ⊂ S,

(10.8)

where 1 =  ∪ {(3)}, 2 = 1 ∪ {(2, 2, 1)}, 3 = 2 ∪ {(2, 2)}, 4 = 3 ∪ {(2)}. The reader may wish to apply the algorithm of this section to compute the sets Z(i ), and confirm that we have a chain of inclusions Z() ⊃ Z(1 ) ⊃ Z(2 ) ⊃ Z(3 ) ⊃ Z(4 ) ⊃ ∅, where each inclusion corresponds to dropping the largest lexicographic shape in Z(i ). Finally, we note that the shape (2) is contained in every other shape in 4 , so in fact I (4 ) = I2 is the ideal of 2 × 2 minors, which is also the radical of I () (see Lemma 10.2.12).

10.4 The Filtration for Symbolic Powers of Determinantal Ideals The goal of this section is to determine the composition factors of the filtration from Theorem 10.2.2 in the case when I () = I p(d) is a symbolic power of the determinantal ideal of p × p minors. The following theorem generalizes Example 10.2.3 (recall the definition of γ p (σ ) from Sect. 3.5). Theorem 10.4.1 For 1 ≤ p ≤ m ≤ n and d ≥ 1 one has Z(I p(d) ) = Z(d) p , where Z(d) p = {(σ, p − 1) : σ = (s1 , . . . , sc ) with sc ≥ p and γ p (σ ) ≤ d − 1}. Proof Recall from Theorem 3.5.2 that λ is a shape in I p(d) if and only if γ p (λ) ≥ d. We first check the inclusion Z(I p(d) ) ⊂ Z(d) p . We need to show that for every (σ, l) ∈ (d) Z(I p ) we have γ p (σ ) ≤ d − 1 and l = p − 1, which by (10.5) implies sc ≥ p. If γ p (σ ) ≥ d, then σ is a shape in I p(d) , contradicting Lemma 10.2.6. To prove l = p − 1, note that Lemma 10.2.10 implies l ≤ p − 1. Suppose by contradiction that l ≤ p − 2, and consider any λ satisfying condition (a) in Definition 10.2.1. Since

392

10 Asymptotic Regularity for Symbolic Powers of Determinantal Ideals

λc+1 ≤ l + 1 ≤ p − 1, it follows that γ p (λ) = γ p (λ(c)), and therefore λ(c) satisfies condition (a) in Definition 10.2.1. Using the inequality l ≥ 0, we get a contradiction by noting that λ(c) fails condition (b) of Definition 10.2.1. (d) It remains to check that every (σ, p − 1) ∈ Z(d) p belongs to Z(I p ). Since sc ≥ p, it follows that λ = σ  ( p d ) is a partition. Moreover, γ p (λ) ≥ γ p (( p d )) = d, so λ is a shape in I p(d) . Writing λ = ( p1 , p2 , . . . ), we have in addition pc+1 = p = ( p − 1) + 1, so λ satisfies condition (a) in Definition 10.2.1. To check condition (b) in Definition 10.2.1, consider any λ = ( p1 , p2 , . . . ) satisfying condition (a). If pc+1 = p then pc+1 < p and therefore d ≤ γ p (λ) = γ p (λ(c)) ≤ γ p (σ ) ≤ d − 1, a contradiction that concludes our proof.



10.5 Sheaves Defined by Shape and Their Quotients In order to give a more geometric description of the modules Jl(σ ) , we need to work with a relative version of the polynomial ring K [X ], the ideals I () and the quotients Jl(σ ) . To that end, we consider a base algebraic variety B, and locally free sheaves F1 , F2 on B, with rank Fk = rk for k = 1, 2. We begin by considering a generalization to sheaves of the notion of subspaces defined by shape in Chap. 3, and relate it to the discussion of Schur functors in Sect. 9.8. We then define sheaves of ideals I(σ ) that generalize the ideals I (σ ) , and discuss the corresponding generalizations of the quotients Jl(σ ) . To begin, we consider the sheaf of OB -algebras S = Sym. (F1 ⊗ F2 ). The sheaf S is Z-graded, with degree d component equal to Symd (F1 ⊗ F2 ) for d ≥ 0 (and 0 otherwise). For any open affine subset U = Spec A of B where (Fk )|U is free, we choose bases { f ik } for the free A-modules Fk = (U, Fk ), k = 1, 2, as in Sect. 9.8. If we write xi, j = f i1 ⊗ f j2 , for 1 ≤ i ≤ r1 , 1 ≤ j ≤ r2 , then (U, S) = A[xi, j ] is a (standard graded) polynomial ring. In particular, when B = Spec K , we can identify S with its ring of global sections, which is K [X ]. In general, we can then think of S as a relative version of K [X ], and our goal is to extend the constructions considered for K [X ] to this setting.

10.5 Sheaves Defined by Shape and Their Quotients

393

Sheaves Defined By Shape and Their Quotients We fix d ≥ 0 and a shape σ of size |σ | = d, and define V(σ ) (U ) ⊂ (U, Symd (F1 ⊗ F2 ))

(10.9)

to be the A-submodule generated by all the bitableaux of shape τ ≥ σ , with |τ | = d. By Theorem 3.6.11, the definition of V(σ ) (U ) is independent on the choice of bases on F1 , F2 . Moreover, it is compatible with the restriction to affine subsets of U , hence (10.9) defines a subsheaf V(σ ) ⊂ Symd (F1 ⊗ F2 ). Since V(σ ) (U ) is a free A-module (generated by standard bitableaux), it follows that V(σ ) is locally free. Moreover, we have V(σ ) ⊃ V(τ ) for all σ ≤ τ with |σ | = |τ |. We define V() =



V (σ ) for  ⊂ P(d),

(10.10)

σ ∈

and set

(σ ) V> = V(P(d)>σ ) , where P(d)>σ = {τ ∈ P(d) : τ > σ }.

We refer to V() as a sheaf defined by shape, and we say that a shape τ belongs to V() if V (τ ) ⊂ V() , or equivalently, if |τ | = d and τ ≥ σ for some σ ∈ . It will sometimes be convenient to work with the saturation of  ⊂ P(d), which is defined as (10.11)  sat = {τ ∈ P(d) : τ ≥ σ for some σ ∈ }, and has the property that V() = V( ) . We say that  is saturated if  =  sat . We note that if ,   ⊂ P(d) are saturated, then sat





V() ∩ V( ) = V(∩ ) .

(10.12)

If r = min(rank F1 , rank F2 ) then there are no standard bitableaux of shape τ = (t1 , t2 , . . . ) when t1 > r . It follows that V() = V() , where  = {σ ∈  : s1 ≤ r },

(10.13)

so we may freely ignore shapes τ = (t1 , t2 , . . . ) with t1 > r when convenient. Since S is a sheaf of algebras, we can talk about sheaves of ideals in S (or equivalently, ideal sheaves on AB (F1 ⊗ F2 )). We let I(σ ) denote the sheaf of ideals in S generated by V(σ ) (the relative version of I (σ ) ), and let I() =

 σ ∈

so in particular we have

) (P(d)>σ ) I(σ ) , I(σ , > =I

394

10 Asymptotic Regularity for Symbolic Powers of Determinantal Ideals

I(∅) = S.

(10.14)

We note that if we restrict to the degree e component, then we get = V(e ) , where e = {τ ∈ P(e) : τ ≥ σ for some σ ∈ }. I() e

(10.15)

We also note that if r = min(rank F1 , rank F2 ), then I() = 0

if

s1 > r for all σ = (s1 , s2 , . . . ) ∈ .

(10.16)

Lemma 10.5.1 If v = σ ∗ then we have an isomorphism (σ ) ∼ (σ ) ) ∼ V(σ ) /V> = I /I(σ > = Sv F1 ⊗ Sv F2 . ) Proof The first isomorphism follows by definition, since I(σ ) and I(σ >σ agree in all degrees > |σ |. The second isomorphism is the natural generalization of Theorem 3.6.2(b). Suppose that U = Spec A is an open affine subset as before, and recall from the proof of Proposition 9.8.1 that (Sv F1 )(U ) is the free A-module spanned by the standard bitableaux (R| In(σ )) where R has entries in {1, . . . , r1 }. By the symmetry of the tensor product, we can identify in a similar way (Sv F2 )(U ) with the free A-module spanned by the standard bitableaux (In(σ )|C) where C has entries (σ ) )(U ) is the free A-module spanned by in {1, . . . , r2 }. By construction, (V(σ ) /V> standard bitableaux (R|C). It follows from Theorem 3.6.2(b) that the assignment

(R | In(σ ), (In(σ ) | C) → (R | C) induces an isomorphism (σ ) )(U ), (Sv F1 ⊗ Sv F2 )(U ) ∼ = (V(σ ) /V>

which is independent on the choice of bases for F1 (U ), F2 (U ). This is enough to conclude that we have the desired isomorphism at the level of sheaves.  Example 10.5.2 We will often consider the case when σ = (d), which is the unique ((d)) maximal element of P(d) with respect to ≥. It follows that V> = 0, and combining Lemma 10.5.1 with Lemma 9.8.3, we get V((d)) =

d 

F1 ⊗

d 

F2 .

Notice that V((d)) = 0 if rank F1 < d or rank F2 < d, and that V((d)) = det(F1 ) ⊗ det(F2 ) if rank F1 = rank F2 = d, which is an invertible sheaf, locally generated by the standard initial bitableau (In((d))| In((d))). As noted in (10.13), if rank F1 = rank F2 = d, then, when considering V() , we may assume that s1 ≤ d for all σ = (s1 , s2 , . . . ) ∈ . We get

10.5 Sheaves Defined by Shape and Their Quotients

395

V((d)) ⊗ V() = V( ) , for  d| = {(d, s1 , s2 , . . . ) : σ = (s1 , s2 , . . . ) ∈ }. (10.17) d|

It will be useful to have the following slight generalization of Lemma 10.5.1. Lemma 10.5.3 Let  ⊂ P(d) and consider a shape σ ∈  which is minimal with respect to ≤. We define  =σ = {τ ∈ P(d) : τ ≥ ρ for some ρ ∈ , and τ = σ } =  sat \ {σ }. If v = σ ∗ then we have an isomorphism V() /V( =σ ) ∼ = Sv F1 ⊗ Sv F2 . Proof By construction,

V(σ ) + V( =σ ) = V() ,

so we get an isomorphism V(σ ) V() ∼ . = V(σ ) ∩ V( =σ ) V( =σ ) Since a shape τ ∈ P(d) with τ ≥ σ belongs to V( =σ ) if and only if τ > σ , we get (σ ) , V(σ ) ∩ V( =σ ) = V>

and the desired isomorphism now follows from Lemma 10.5.1.



Filtrations and Cauchy’s Formula for Sheaves Defined By Shape Lemma 10.5.3 leads to the following extension of Theorem 9.8.7. Corollary 10.5.4 For every subset  ⊂ P(d), there exists a filtration of V() with composition factors

Sv F1 ⊗ Sv F2 : v ∗ ∈  sat . Proof We prove the statement by induction on | sat |, with base case | sat | = 0 being trivial. If we let σ denote a shape of  which is minimal with respect to ≤, it follows from Lemma 10.5.1 that V() /V( =σ ) ∼ = Sv F1 ⊗ Sv F2 , where v = σ ∗ . Moreover, sat sat \ {σ },  = σ =  =σ = 

396

10 Asymptotic Regularity for Symbolic Powers of Determinantal Ideals

sat sat ( =σ ) hence | = has a filtration with composition factors σ | < | |. By induction, V



Sv F1 ⊗ Sv F2 : v ∗ ∈  sat \ {σ } , which combined with the inclusion V( =σ ) ⊂ V() provides the desired filtration on  V() . As the proof of Corollary 10.5.4 shows, there are choices involved in the filtration of V() , since in general  may contain several minimal shapes. This leads to the following generalizations of the filtration of Symd (F1 ⊗ F2 ) = V(P(d)) constructed in Sect. 9.8. If μ1 , . . . , μ Nd is any enumeration of the partitions in P(d), with the property that (10.18) μ j ≥ μi =⇒ i ≥ j, then if we let  i = {μ1 , . . . , μi } for i = 1, . . . , Nd , it follows from (10.18) that each  i is saturated, and that μi is minimal in  i . Moreover, we have ( i ) =μi =  i−1 , and therefore the filtration 0 ⊂ V( ) ⊂ · · · ⊂ V( 1

Nd

)

= Symd (F1 ⊗ F2 )

(10.19)

has composition factors given by V( ) /V( i

i−1

)

∼ = Svi F1 ⊗ Svi F2 , where v i = (μi )∗

for

i = 1, . . . , Nd .

The filtration (9.43) is obtained as the special case of (10.19), if we let μi = λi as i in (9.42), and let Gi = V( ) for all i. In analogy with Definition 10.1.2, we let Jl(σ ) =

I(σ ) I(succ(σ,l))

,

which is a sheaf of (finitely generated, graded) S-modules (thus it may be identified with a coherent sheaf on AB (F1 ⊗ F2 )). It is the relative version of Jl(σ ) . By (10.14), (10.2) and (10.16), we have that if l ≥ min(rank F1 , rank F2 ) then Jl(∅) = S.

(10.20)

In the special case when rank F1 = rank F2 = r , we get from (10.17) that if σ = (s1 , s2 , . . . ) and r ≥ s1 (so that I(σ ) and Jl(σ ) do not trivially vanish) we have V((r )) ⊗ I(σ ) = I(r,s1 ,s2 ,... ) and V((r )) ⊗ Jl(σ ) = Jl(r,s1 ,s2 ,... ) .

(10.21)

10.5 Sheaves Defined by Shape and Their Quotients

397

Functorial Properties for Sheaves Defined By Shape When it is necessary to clarify to which sheaves F1 , F2 we apply the constructions above, we will write S(F1 , F2 ), V(σ ) (F1 , F2 ), I(σ ) (F1 , F2 ), Jl(σ ) (F1 , F2 )

etc.

We note that all the constructions above are functorial, so if F1 , F2 are locally free sheaves, and f k : Fk −→ Fk are morphisms of OB -modules, then there exists an induced map f : S(F1 , F2 ) −→ S(F1 , F2 ), with

f V(σ ) (F1 , F2 ) ⊂ V(σ ) (F1 , F2 ),



f I(σ ) (F1 , F2 ) ⊂ I(σ ) (F1 , F2 ),

etc.

The constructions are also compatible with base change, as follows. Lemma 10.5.5 Suppose that π : B −→ B is a morphism of algebraic varieties. The following hold for V(σ ) (as well as V() , I(σ ) , I() , Jl(σ ) ): (a) If F1 , F2 are locally free sheaves on B then

π ∗ V(σ ) (F1 , F2 ) = V(σ ) (π ∗ F1 , π ∗ F2 ). (b) If F1 , F2 are locally free sheaves on B then there exists a natural map

V(σ ) (π∗ F1 , π∗ F2 ) −→ π∗ V(σ ) (F1 , F2 ) . Proof We prove the results for V(σ ) , and leave it to the reader to repeat the arguments in the other cases. We may assume that B = Spec A, B = Spec A are affine, in which case we have to show that A ⊗ A V(σ ) (F1 , F2 ) = V(σ ) (A ⊗ A F1 , A ⊗ A F2 ), when F1 , F2 are free A-modules. This follows from the (local) definition of V(σ ) , and the fact that any A-basis of Fk is also an A -basis of A ⊗ A Fk . For (b), we let Fk = π∗ (Fk ) for k = 1, 2. By adjunction, we have canonical maps ∗ π Fk −→ Fk , and using (a) and the functoriality of V(σ ) , we get a natural map

π ∗ V(σ ) (F1 , F2 ) = V(σ ) (π ∗ F1 , π ∗ F2 ) −→ V(σ ) (F1 , F2 ). Using adjunction again, this yields a natural map

V(σ ) (F1 , F2 ) −→ π∗ V(σ ) (F1 , F2 ) , as desired.



398

10 Asymptotic Regularity for Symbolic Powers of Determinantal Ideals

Direct Images of Sheaves Defined By Shape We next extend Corollary 9.3.6 to any subsheaf of Symd (E ⊗ F ) defined by shape. Proposition 10.5.6 Let E be a locally free sheaf of rank n on B, and consider the Grassmannian G = GB (n − 1, E) with structure morphism π : G −→ B and tautological quotient sheaf Q (of rank n − 1). Suppose that F is a locally free sheaf of rank r ≤ n on B, and let F˜ = π ∗ F . For  ⊂ P(d), we have:   (a) R i π∗ V() (Q, F˜ ) = 0 for all i > 0. (b) If r ≤ n − 1 then   π∗ V() (Q, F˜ ) = V() (E, F ). (c) If r = n then we have a short exact sequence   0 −→ V(+ ) (E, F ) −→ V() (E, F ) −→ π∗ V() (Q, F˜ ) −→ 0, where + = {τ = (t1 , t2 , . . . ) ∈ P(d) : τ ≥ σ for some σ ∈  and t1 ≥ n}. Proof To prove (a), we consider the filtration of V() (Q, F˜ ) constructed in Corollary 10.5.4, and note that it suffices to prove R i π∗ vanishes on each of the composition factors, when i > 0. For that, we use the projection formula and (9.37) to get R i π∗ (Sv Q ⊗ Sv F˜ ) = (R i π∗ Sv Q) ⊗ Sv F = 0, where the vanishing of R i π∗ Sv Q follows from Corollary 9.8.6. For (b) and (c), we note that π∗ Q = E by Lemma 9.3.1, and π∗ F˜ = F by the projection formula, hence Lemma 10.5.5(b) provides a natural map   V() (E, F ) −→ π∗ V() (Q, F˜ ) .

(10.22)

We may assume that  =  sat , and we prove (b) and (c) by induction on ||, the case || = 0 being trivial. We let σ = (s1 , s2 , . . . ) ∈  minimal with respect to ≤, let v = σ ∗ , and consider the commutative diagram

0

V(\{σ }) (E, F ) α

0

  π∗ V(\{σ }) (Q, F˜ )

V() (E, F ) β

  π∗ V() (Q, F˜ )

Sv E ⊗ Sv F

0

γ

  π∗ Sv Q ⊗ Sv F˜

0

10.5 Sheaves Defined by Shape and Their Quotients

399

The first row is exact which also shows that the second row is  by Lemma 10.5.3,  1 (\{σ }) ˜ (Q, F ) = 0 by part (a). exact, since R π∗ V To prove (b), we note that α is an isomorphism by induction. If s1 ≥ n > r then vn = 0, hence Sv F = Sv F˜ = 0 by the last part of Proposition 9.8.1. If s1 ≤ n − 1 then vi = 0 for i ≥ n, and π∗ (Sv Q) = Sv E by Corollary 9.8.6. It follows from the projection formula that γ is an isomorphism, hence β is also an isomorphism, concluding the inductive step. To prove (c), we first establish the surjectivity of (10.22). The argument in the previous paragraph shows that γ is always surjective (it is an isomorphism if vi = 0 for i ≥ n, while if vn = 0 then Sv Q = 0). Moreover, α is surjective by induction, hence β is surjective as well. To conclude the proof of (c), we have to check that ker(β) = V(+ ) (E, F ). We note that if d < n then + = ∅, and every v ∈ P(d) satisfies vi = 0 for i ≥ n. It follows that γ is an isomorphism for all such v, hence we get by induction that β is also an isomorphism. Therefore, ker(β) = 0 = V(+ ) (E, F ), as desired. We are left with considering the case d ≥ n. By the functoriality of V() , the surjectivity of β, Corollary 9.3.6 and (10.17), we have a commutative diagram

V() (E, F )

0

V(P(d)+ ) (E, F )

Symd (E ⊗ F )

β

π∗ (V() (Q, F˜ ))

0

π∗ (Symd (Q ⊗ F˜ ))

0

where the rows are exact. It follows that ker(β) = V() (E, F ) ∩ V(P(d)+ ) (E, F ) = V(+ ) (E, F ), where the last equality follows from (10.12), since  and P(d)+ are saturated, and  since by definition we have  ∩ P(d)+ = + . Corollary 10.5.7 With the notation in Proposition 10.5.6, suppose that   ⊂  ⊂ P(d) satisfy + = + . We have  Rπ∗

V() (Q, F˜ ) V(  ) (Q, F˜ )

 =

V() (E, F ) . V(  ) (E, F )

Proof Using the long exact sequence for higher direct images, it follows that V() (Q,F˜ ) R i π∗ V (  ) (Q,F ˜ ) = 0 for i > 0, and that there is a short exact sequence      0 → π∗ V( ) (Q, F˜ ) → π∗ V() (Q, F˜ ) → π∗



V() (Q, F˜ ) V(  ) (Q, F˜ )

 → 0. (10.23)

400

10 Asymptotic Regularity for Symbolic Powers of Determinantal Ideals 

Noting that V(+ ) (E, F ) = V(+ ) (E, F ) = 0 when r ≤ n − 1, we can interpret Proposition 10.5.6(b) as a special case of (c). Applying the snake lemma to the commutative diagram

0

V(+ ) (E, F )

V( ) (E, F )

   π∗ V( ) (Q, F˜ )

0

0

V(+ ) (E, F )

V() (E, F )

  π∗ V() (Q, F˜ )

0





whose rows are exact (by Proposition 10.5.6(c)), and using (10.23), we get the desired conclusion.  Proposition 10.5.8 With the notation in Proposition 10.5.6, suppose that l ≥ 0 and σ = (s1 , . . . , sc ) is a shape with min(n − 1, r ) ≥ s1 ≥ · · · ≥ sc ≥ l. We have that

  Rπ∗ Jl(σ ) (Q, F˜ ) = Jl(σ ) (E, F ).

Proof By Lemma 10.5.5, there is a natural map   Jl(σ ) (E, F ) −→ π∗ Jl(σ ) (Q, F˜ ) .

(10.24)

Since Jl(σ ) (E, F ) and Jl(σ ) (Q, F˜ )are Z-graded, to prove that (10.24) is an isomorphism, and that R i π∗ J (σ ) (Q, F˜ ) = 0 for i > 0, it suffices to do so in each degree. l

Fix a degree e, and let  = {τ ∈ P(e) : τ ≥ σ },   = {τ ∈ P(e) : τ ≥ σ  for some σ  ∈ succ(σ, l)}. By the definition of Jl(σ ) and (10.15), we have    V() (E, F )  (σ ) V() (Q, F˜ ) , Jl (Q, F˜ ) = Jl(σ ) (E, F ) = (  ) . e e V (E, F ) V(  ) (Q, F˜ ) The desired conclusions now follow from Corollary 10.5.7, provided that we verify + = + . Since  ⊃   , it follows that + ⊃ + , so we only have to verify the reverse inclusion. Let τ = (t1 , t2 , . . . ) ∈ + , so that t1 ≥ n and τ ≥ σ . If c = 0, then since t1 ≥ n > s1 , it follows from Definition 10.1.1 that τ ∈ succ(σ, l). If c = 0, then we can use t1 ≥ n ≥ l + 1 and Definition 10.1.1 to conclude again that τ ∈ succ(σ, l). Since in addition τ ∈ P(e) and t1 ≥ n, we conclude that τ ∈ + , as desired. 

(σ )

10.6 A Geometric Realization of the Modules Jl

401

Corollary 10.5.9 Suppose that F1 , F2 are locally free sheaves on B of rank r . We define Gk = G(r − 1, Fk ), with tautological (rank r − 1) quotient bundle Qk , for k = 1, 2. Suppose l ≥ 0 and σ = (s1 , . . . , sc ) is a shape satisfying r > s1 and sc ≥ l. If we write π : G1 ×B G2 −→ B for the morphism induced by the structure maps Gk −→ B, then   Rπ∗ Jl(σ ) (Q1 , Q2 ) = Jl(σ ) (F1 , F2 ). Proof We factor π as f

g

G1 ×B G2 −→ G1 −→ B. Since G1 ×B G2 = GG1 (r − 1, g ∗ F2 ), it follows from Proposition 10.5.8, with E = g ∗ F2 and F = Q1 , that   R f ∗ Jl(σ ) (Q1 , Q2 ) = Jl(σ ) (Q1 , g ∗ F2 ). Similarly, applying Proposition 10.5.8 to E = F1 and F = F2 , we have Rg∗ Jl(σ ) (Q1 , g ∗ F2 ) = Jl(σ ) (F1 , F2 ), and the desired conclusion follows from the Leray spectral sequence.



10.6 A Geometric Realization of the Modules Jl(σ ) In this section we assume that m ≤ n, we let F1 = K m , F2 = K n , and identify the polynomial ring K [X ] = K [X i j ] from Chap. 3 with the symmetric algebra Sym(F1 ⊗ F2 ). Our goal is to give a more geometric realization of the modules Jl(σ ) from Definition 10.1.2, one that will allow us to apply the Grothendieck duality theory as in Theorem 9.9.4. We begin by observing that with the notation in Sect. 10.5, we can identify Jl(σ ) = Jl(σ ) (F1 , F2 ) as a sheaf of Sym(F1 ⊗ F2 ) = S(F1 , F2 )-modules on Spec K . By considering the sheaves Jl(σ ) over more general bases than Spec K (that is, in the relative setting), we will identify Jl(σ ) with the global sections of a locally free sheaf on a smooth variety. One can think of this informally as a way of desingularizing the module Jl(σ ) . More precisely, we will prove the following. Theorem 10.6.1 Suppose that σ = (s1 , . . . , sc ) is a partition, and 0 ≤ l ≤ sc . Consider the product of Grassmannians G = G(l, F1 ) × G(l, F2 ) and the tautological sheaves i Q, i R on G(l, Fi ). Consider the affine bundle

402

10 Asymptotic Regularity for Symbolic Powers of Determinantal Ideals

Y = AG

1

Q  2Q

and write g : Y −→ G for its structure morphism. Define the sequence ∗ , . . . , sm∗ ), y = (sl+1

(10.25)

where σ ∗ = (s1∗ , s2∗ , . . . ) is the partition dual to σ , and consider the locally free sheaves c Mi = det i Q ⊗ S y i R on G(l, Fi ). If we let M = g ∗ (M1  M2 ) then we have  H (Y , M) = j

Jl(σ ) j = 0; 0 j > 0.

For an example, consider the case when σ = ∅. We have that Jl(σ ) = S/Il+1 , and M = OY is the structure sheaf of Y . The variety Y is a resolution of singularities of the determinantal variety Z l of matrices of rank ≤ l. The assertion that H 0 (Y , OY ) = S/Il+1 is equivalent to the fact that Z l is normal. The vanishing of higher cohomology for OY is the statement that Z l has rational singularities. We will pursue these remarks in more detail in Sect. 10.8.

The Significance of Theorem 10.6.1 The context of Theorem 10.6.1 fits in with the discussion in Sect. 9.9. We can specialize the diagram (9.48) by taking V = F1 ⊗ F2 , B = G, V = M1  M2 , η = 1 Q  2 Q. The conclusion of Theorem 10.6.1 allows us to apply Theorem 9.9.4 and Corollary 9.9.5 to the modules Jl(σ ) . To match the natural grading on Jl(σ ) , we place the sheaf V in degree |σ | = s1 + · · · + sc . Since rank(η) = l 2 , it follows that rank(ξ ) = mn − l 2 , and Corollary 9.9.5 yields  ∨    ∨ 2 2 j (σ ) Ext S Jl , S ∼ = H mn−l − j G, V ⊗ det η ⊗ Symk+mn−l +|σ | η ⊗ det(V ) , k

(10.26) where we have used the isomorphism det(ξ ) ∼ = det(η)∨ ⊗ det(V ). In characteristic zero, the above formula combined with the Borel–Weil–Bott theorem yields explicit j formulas for all the modules Ext S (Jl(σ ) , S). Similar formulas are not known in positive characteristic, but vanishing theorems such as those developed in Chap. 9 may be used to deduce corresponding vanishing results for the Ext groups. We will pursue this idea further in Sect. 10.7 below, in order to give an asymptotic formula for the regularity of symbolic powers of determinantal ideals.

(σ )

10.6 A Geometric Realization of the Modules Jl

403

The Outline of the Proof of Theorem 10.6.1 To prove Theorem 10.6.1, we will set up an inductive procedure, using the generalizations of the modules Jl(σ ) to the relative setting. For i ≤ j, we consider the interval [i, j] = {i, i + 1, . . . , j} and define (q)

(q)

(q)

(q)

B1 = F([q, m], F1 ), B2 = F([q, n], F2 ), and B(q) = B1 × B2 , for q ≤ m. We consider the natural maps (q)

πk

(q)

: Bk

(q+1)

−→ Bk

(q)

, and π (q) = π1

(q)

× π2

: B(q) −→ B(q+1) , for q ≤ m − 1.

(m) (m) We note that B(m) = F([m, n], F2 ) is a 1 = Spec K , B2 = F([m, n], F2 ), so B partial flag variety. We denote the structure map of this flag variety by

π (m) : B(m) −→ Spec K . If n = m then B(m) = Spec K , and π (m) is the identity map. When q ≤ p ≤ m, we (q) have tautological rank p quotient sheaves k Q p on Bk . We point out again the abuse (q) k of notation discussed in Sect. 9.2: Q p exists as a sheaf on Bk whenever p ≥ q, and we have

(q) (πk )∗ k Q p = k Q p for p > q, k = 1, 2. (q)

Moreover, the map πk

gives an identification

(q) Bk ∼ = G q, k Qq+1 for q ≤ m − 1, k = 1, 2, (q)

so Bk can be considered as a Grassmann bundle of the type considered in Proposition 10.5.6 and the subsequent results from Sect. 10.5. We now summarize the proof strategy for Theorem 10.6.1, and then proceed with verifying the details in the rest of the section. (a) We use the usual identification between OY -modules and sheaves of Sym(1 Q  2 Q)-modules on G, given by F ↔ g∗ F = Rg∗ F . Notice that



g∗ M = (M1  M2 ) ⊗ Sym 1 Q  2 Q = (M1  M2 ) ⊗ S 1 Q, 2 Q , (10.27) and that H j (Y , M) = H j (G, g∗ M) for all j. (b) To compute the cohomology of g∗ M, we consider the commutative diagram

404

10 Asymptotic Regularity for Symbolic Powers of Determinantal Ideals

B(l)

π (l)

B(l+1)

π (l+1)

...

π (m−1)

B(m) π (m)

f

Y

g

G

Spec K

where f is induced by the natural map B(l) k −→ G(l, Fk ) forgetting the intermediate flags. (c) We construct a sheaf N on B(l) with R f ∗ N = g∗ M, and compute step by step the iterated direct image of N along π (l) , π (l+1) , . . . , π (m−1) , using Corollary 10.5.9, and along π (m) using Proposition 10.5.8. We show that the end result is precisely Jl(σ ) , and all the higher direct images vanish, which is a restatement of the conclusion of our Theorem 10.6.1.

The Construction of the Sheaf N There is nothing to check regarding parts (a) and (b) of the above outline, so we begin with the definition and properties of the sheaf N in part (c). We consider the line bundles k

ω p = det

k

(q) Q p on Bk for q ≤ p ≤ m, and k = 1, 2.

For a shape τ = (t1 , . . . , tv ) with m ≥ t1 ≥ · · · ≥ tv ≥ q, we consider the line bundles k

ω(τ ) =

v 

k

(q)

ωti on Bk , for k = 1, 2, and ω(τ ) = 1 ω(τ )  2 ω(τ ) on B(q) .

i=1

For σ as in Theorem 10.6.1, we consider the sheaf

N = ω(σ ) ⊗ S 1 Ql , 2 Ql on B(l) .

(10.28)

Lemma 10.6.2 We have a natural identification f ∗ N = g∗ M, and R i f ∗ N = 0 for all i > 0. Proof If we write f = f 1 × f 2 , with f k : B(l) k −→ G(l, Fk ), then we have f k∗



k k

Q = Ql and therefore f ∗ S 1 Q, 2 Q = S 1 Ql , 2 Ql .

Using the projection formula and (10.27), it suffices to check that

R f ∗ ω(σ ) = M1  M2 ,

(σ )

10.6 A Geometric Realization of the Modules Jl

or equivalently, that R f k∗

k

405

ω(σ ) = Mk for k = 1, 2.

Since the situation is symmetric for k = 1, 2, we let Q = k Q, let

R = ker Fk ⊗ OG(l,Fk )  Q the tautological subsheaf on G(l, Fk ), which has rank r = m − l when k = 1, and r = n − l when k = 2. We note that B(l) k = FG(l,Fk ) (R) is the full flag variety of R. For simplicity, we denote this flag variety by F, let γ = f k : F −→ G(l, Fk ) denote its structure morphism, and let Li = LiR the tautological line bundles on F. We have for each p ≥ l a short exact sequence 0 −→ R p,l −→ k Q p −→ γ ∗ Q −→ 0, and by taking determinants we get k

ω p = L1 ⊗ · · · ⊗ L p−l ⊗ γ ∗ (det(Q)).

It follows that  k

ω

(σ )

=

c 

 L1 ⊗ · · · ⊗ Lsi −l



⊗ γ ∗ det(Q)c

i=1 ∗ sl+1 L1 ⊗



sm∗ · · · ⊗ Lm−l ⊗ γ ∗ det(Q)c

= OF (y) ⊗ γ ∗ det(Q)c . =

Using the projection formula, Kempf vanishing, and Theorem 9.8.5, it follows that Rγ∗

k

 

ω(σ ) = γ∗ OF (y) ⊗ det(Q)c = S y R ⊗ det(Q)c = Mk , 

concluding our proof.

Pushing Forward Along the Diagram in Step (b) of the Proof Strategy To proceed with step (c) of our strategy, we first use (10.20) to rewrite (10.28) as N = ω(σ ) ⊗ Jl(∅)

1

Ql , 2 Ql .

For each j = l, l + 1, . . . , m, we express σ as the concatenation of two shapes

406

10 Asymptotic Regularity for Symbolic Powers of Determinantal Ideals

λ j = (s1 , . . . , si ), μ j = (si+1 , . . . , sc ), where si > j ≥ si+1 , with the convention s0 = ∞ (so that i = 0 and λ j is empty if j ≥ s1 ). We then define (μ j ) 1

N j = ω(λ ) ⊗ Jl j

Q j , 2 Q j on B( j) , for j = l, l + 1, . . . , m.

Lemma 10.6.3 We have that Nl = N, and moreover Rπ∗( j) N j = N j+1 for all j = l, l + 1, . . . , m − 1. Proof We first check that Nl = N. We consider the index i for which si > l ≥ si+1 , and we note that the hypothesis sc ≥ l implies that si+1 = si+2 = · · · = sc = l, so μl = (l c−i ). It follows that

N = ω(σ ) ⊗ Jl(∅) 1 Ql , 2 Ql



c−i l = ω(λ ) ⊗ det(1 Ql )  det(2 Ql ) ⊗ Jl(∅) 1 Ql , 2 Ql

(10.21) (λl ) (μl ) 1 = ω ⊗ Jl Ql , 2 Ql = Nl , as desired. Suppose now that l ≤ j ≤ m − 1, and note that since all parts of λ j j j are larger than j, it follows that ω(λ ) = (π ( j) )∗ ω(λ ) (with the standard abuse of notation). It follows from the projection formula and Corollary 10.5.9 that (μ j ) 1

Rπ∗( j) N j = ω(λ ) ⊗ Jl j

Q j+1 , 2 Q j+1 .

(10.29)

If λ j = λ j+1 , μ j = μ j+1 , then the right side of (10.29) is N j+1 , and the desired conclusion follows. Otherwise, we list the parts of σ of size j + 1 as si+1 = · · · = si  = j + 1, i < i  , so that si > j + 1, si  +1 ≤ j, and the pair (λ j+1 , μ j+1 ) is obtained from (λ j , μ j ) by removing the last (i  − i) parts from λ j and appending them to the beginning of μ j . We get using (10.29) that

i  −i

(μ j ) 1 ⊗ det(1 Q j+1 )  det(2 Q j+1 ) ⊗ Jl Q j+1 , 2 Q j+1

(10.21) (λ j+1 ) (μ j+1 ) 1 = ω ⊗ Jl Q j+1 , 2 Q j+1 = N j+1 ,

Rπ∗( j) N j = ω(λ

j+1

)

concluding our proof.



(σ )

10.6 A Geometric Realization of the Modules Jl

407

The Conclusion of the Proof of Theorem 10.6.1 We now have all the tools in place to prove Theorem 10.6.1. Proof (Proof of Theorem 10.6.1) We have that for all j H j (Y , M) = H j (G, g∗ M) = H j (G, f ∗ N) = H j (B(l) , N), where the last equality follows from Lemma 10.6.2 and the Leray spectral sequence. Applying Lemma 10.6.3 and the Leray spectral sequence, we get H j (B(l) , N) = H j (B(l) , Nl ) = H j (B(l+1) , Nl+1 ) = · · · = H j (B(m) , Nm ). Notice that our assumption that m ≥ s1 implies λm = ∅ and μm = σ, and therefore

Nm = Jl(σ )

1

Qm , 2 Qm .

If m = n, then B(m) = Spec K and k Qm = Fk for k = 1, 2, so  (m)

H (B j

, Nm ) =

Jl(σ ) j = 0, 0 j > 0,



as desired. If m < n, then B(m) = F([m, n], F2 ), 1 Q1 = F1 , so Nm = Jl(σ ) F1 , 2 Qm . We can factor π (m) as a composition γ (m)

γ (m+1)

γ (n−1)

F([m, n], F2 ) −→ F([m + 1, n], F2 ) −→ . . . −→ F([n, n], F2 ) = Spec K . It follows that if we define

Nt = Jl(σ ) F1 , 2 Qt on F([t, n], F2 ), for t = m, m + 1, . . . , n, then Nn = Jl(σ ) (F1 , F2 ) on Spec K , and applying Proposition 10.5.8 we get Rγ∗(t) Nt = Nt+1 for t = m, . . . , n − 1. We obtain as before that

408

10 Asymptotic Regularity for Symbolic Powers of Determinantal Ideals

H j (B(m) , Nm ) = H j (F([m, n], F2 ), Nm ) = · · ·  J (σ ) j = 0, j = H (F([n, n], F2 ), Nn ) = l 0 j > 0, 

which concludes our proof.

10.7 Regularity of Symbolic Powers of Determinantal Ideals In this section we put together the combinatorial and cohomological results developed in earlier sections to prove the following. Theorem 10.7.1 Suppose that 1 ≤ p ≤ m ≤ n. The Castelnuovo–Mumford regularity for the symbolic powers of the ideal I p satisfies reg I p(d) = p · d for d  0. If p = 1 or p = m then we can take d ≥ 0. If p = 2 then we can take d ≥ m − 1. Proof It follows from Theorem 3.5.2 that I p(d) has minimal generators of degree p · d (see Remark 3.5.4), so reg I p(d) ≥ p · d. In order to prove Theorem 10.7.1, we then need to establish the reverse inequality. Recall from Remark 8.1.15 that for a graded S-module M j

reg M = max{−k − j : Ext S (M, S)k = 0},

(10.30)

so we need to show that for d  0 Ext S (I p(d) , S)k = 0 when j

− k − j > p · d.

Based on Theorems 10.2.2 and 10.4.1, it suffices to prove the above vanishing after replacing I p(d) by Jl(σ ) with (σ, l) ∈ Z(d) p . Recall that for each such (σ, l) we have l = p − 1, which we assume for the rest of the proof. In light of (10.26), this further reduces to proving that for d  0 and (σ, l) ∈ Z(d) p , we have H mn−l

2

−j

  ∨  2 G, V ⊗ det(η∨ ) ⊗ Symk+mn−l +|σ | η = 0 if

− k − j > p · d.

(10.31) Recall that for each shape σ in Z(d) p we have sc ≥ p = l + 1, which implies ∗ = c. y1 = sl+1

10.8 Some Geometric Properties of Determinantal Varieties

409

Moreover, d − 1 ≥ γ p (σ ) = |σ | − ( p − 1) · c = |y|, which implies pd > (d − c) · l + |σ |.

(10.32)

Using Corollary 9.10.2 with v = k + mn − l 2 + |σ |, in order to prove (10.31) it is enough to verify the inequality mn − l 2 − j > (d − c)l + k + mn − l 2 + |σ |

⇐⇒

−k − j > (d − c)l + |σ |.

This follows by combining the inequality −k − j > p · d in (10.31) with (10.32), and concludes the proof of the general case. If p = 1 or p = m then I p(d) = I pd has a linear free resolution for all d (when p = 1, I p is the maximal ideal, while for p = m this has been discussed several times already; see Remark 6.2.26 and Sect. 8.6). If p = 2, then l = p − 1 = 1, and the last part of Corollary 9.10.2 shows that we may replace the condition d  0 with d ≥ m − 1. 

10.8 Some Geometric Properties of Determinantal Varieties We end this chapter by explaining a more geometric approach to verifying that the determinantal rings K [X ]/It are normal, Cohen–Macaulay, with rational singularities, as well as computing their divisor class group. These properties have been proved by other methods in preceding chapters: • • • •

dimension, codimension and normality: Theorem 3.4.6; rational singularities and strong F-regularity: Theorems 7.3.15, 7.2.13; Cohen–Macaulayness: discussion in Sect. 3.4 with numerous references; divisor class group: Theorem 6.7.10.

We employ the notation from Sect. 10.6: we let F1 = K m , F2 = K n , and identify K [X ] = S = Sym(F1 ⊗ F2 ). We write A(F1 ⊗ F2 ) = Spec S for the affine variety K m×n , and recall from Sect. 9.2 that the K -points of A(F1 ⊗ F2 ) parametrize elements of (F1 ⊗ F2 )∨ = Hom K (F1 , F2∨ ). For the remainder of the section we will informally identify our varieties with the set of K -points to make the discussion more transparent (the reader may wish to restate the arguments in terms of the functor of points of our varieties and scheme theory, or assume that K is algebraically closed). We consider the diagram (9.48) specialized to the setting of Theorem 10.6.1:

410

10 Asymptotic Regularity for Symbolic Powers of Determinantal Ideals g

Y = AG

1

Q  2Q

ι

X = AG (F1  F2 )

q

G

p

φ

A = A(F1 ⊗ F2 ) Recall that G(l, Fk ) parametrizes l-dimensional quotients f k : Fk  Q k , so we can think of X = A × G set-theoretically as a collection of triples

X = ( f, Q 1 , Q 2 ) : Q k ∈ G(l, Fk ) and f ∈ Hom K (F1 , F2∨ ) , and the projection maps p, q are given by p( f, Q 1 , Q 2 ) = f and q( f, Q 1 , Q 2 ) = (Q 1 , Q 2 ). Via g, we can think of the geometric affine bundle Y as a set of triples

Y = (γ , Q 1 , Q 2 ) : γ ∈ Hom K (Q 1 , Q ∨2 ) , with g(γ , Q 1 , Q 2 ) = (Q 1 , Q 2 ). With these conventions, the inclusion ι : Y → X is given by ι(γ , Q 1 , Q 2 ) = ( f, Q 1 , Q 2 ), where f : F1 −→ F2∨ is obtained as the composition γ

f1

f 2∨

f : F1  Q 1 −→ Q ∨2 → F2∨ . In particular, we have that φ(γ , Q 1 , Q 2 ) = f 2∨ ◦ γ ◦ f 1 . Notice that f 1 and f 2∨ have maximal rank, hence rank( f ) = rank(γ ) ≤ l. It follows that if Z l ⊂ A denotes the variety of matrices of rank ≤ l, then φ(Y ) = Z l . Moreover, we have a divisor D = φ −1 (Z l−1 ) ⊂ Y , defined by the condition det(γ ) = 0. The map φ establishes an isomorphism Y \D∼ = Z l \ Z l−1 , where for f ∈ Z l \ Z l−1 , the inverse φ −1 ( f ) = (Q 1 , Q 2 , γ ) is determined by

10.8 Some Geometric Properties of Determinantal Varieties

411

Q 1 = F1 /(ker f ), Q 2 = (Im f )∨ , γ : F1 /(ker f ) −→ Im f, the map induced by f. We conclude that φ : Y −→ Z l is birational. We also have that φ is a projective morphism, since it is the composition of the closed immersion ι with the projection A × G −→ A , and since G is a projective variety. Since Y is a geometric affine bundle over the nonsingular variety G, it follows that Y is also nonsingular. We conclude that φ : Y −→ Z l is a desingularization of Z l . We next record some immediate consequences of the discussion above and of Theorem 10.6.1. Corollary 10.8.1 We have that dim Z l = l(m + n − l) and codimA (Z l ) = (m − l)(n − l). Proof Since φ : Y −→ Z l is birational, we have dim Z l = dim Y = dim G + rank

1

Q  2 Q = l(m − l) + l(n − l) + l 2 = l(m + n − l).

Since dim A = mn, we get codimA (Z l ) = mn − l(m + n − l) = (m − l)(n − l), as desired.  Corollary 10.8.2 The variety Z l is a normal variety, and Il+1 is the ideal of polynomials in S vanishing along Z l . In particular, K [X ]/Il+1 is the coordinate ring of Z l and it is normal. Proof We make the usual identification between sheaves on an affine space and their global sections. Since φ is a resolution of singularities for Z l , it follows that φ∗ OY is the normalization of O Z l . Moreover, we have natural maps S = OA  O Z l → φ∗ OY = H 0 (Y , OY ) = S/Il+1 , where the last equality follows from the case σ = ∅ of Theorem 10.6.1, and (10.3). Since the composition of the above maps is the surjection S  S/Il+1 , we must have O Z l = φ∗ OY = S/Il+1 . This proves that Il+1 is the defining ideal of Z l , and that Z l (and hence O Z l = K [X ]/Il+1 ) is normal.  Corollary 10.8.3 In characteristic zero, the variety Z l has rational singularities. Proof Since φ : Y −→ Z l is a resolution of singularities, the conclusion is equivalent to the fact that R i φ∗ OY = 0 for i > 0. This follows from the special case σ = ∅  of Theorem 10.6.1, and the identification R i φ∗ OY = H i (Y , OY ). Our main interest in considering the variety Y was the study of the modules Jl(σ ) for all σ , but if one is only concerned with Jl(∅) = S/Il+1 , smaller desingularizations may also be used. For instance, let G = G(l, F1 ), Y  = AG

1

Q ⊗ F2 , with structure map g  : Y  −→ G .

412

10 Asymptotic Regularity for Symbolic Powers of Determinantal Ideals

Set-theoretically, we have

Y  = (γ  , Q 1 ) : Q 1 ∈ G and γ  ∈ Hom K (Q 1 , F2∨ ) . We have another commutative diagram g

Y  = AG

1

Q ⊗ F2

ι

φ

X  = AG (F1  F2 )

q

G

p

A = A(F1 ⊗ F2 ) with X  = G × A , p (Q 1 , f ) = f, q  (Q 1 , f ) = Q 1 , ι (Q 1 , γ  ) = (Q 1 , γ  ◦ f 1 ), φ  (Q 1 , γ  ) = γ  ◦ f 1 .

As before, if f = γ  ◦ f 1 ∈ A then we have that rank( f ) = rank(γ  ) ≤ l, with equality for a general point (Q 1 , γ  ) ∈ Y  . If we let D  = (φ  )−1 (Z l−1 ) (no longer a divisor in Y  ) we get as before an isomorphism Y  \ D ∼ = Z l \ Z l−1 .

(10.33)

It follows using our earlier arguments that Y  is a desingularization of Z l . We note that the map φ factors through φ  , via the morphism Y −→ Y  , (Q 1 , Q 2 , γ ) −→ (Q 1 , f 2∨ ◦ γ ), and this is the sense in which Y  is smaller.

By identifying g∗ OY  = Jl(∅) 1 Q, F2 , the reader may verify using our earlier arguments that Rφ∗ OY  = S/Il+1 . We give another proof as follows. Lemma 10.8.4 We have that H 0 (Y  , OY  ) = S/Il+1 and H j (Y  , OY  ) = 0 for j > 0. Proof Since Y  is a resolution of singularities of Z l , it follows that H 0 (Y  , OY  ) is the normalization of O Z l , which by Corollary 10.8.2 is equal to S/Il+1 . Since

Rg∗ OY  = Sym 1 Q ⊗ F2 , it follows from the Leray spectral sequence that H j (Y  , OY  ) = 0 for j > 0 is equivalent to

10.8 Some Geometric Properties of Determinantal Varieties

413



H j G , Symr 1 Q ⊗ F2 = 0 for j > 0, r ≥ 0. Using Theorem 9.8.7, this vanishing reduces to showing that for every partition z = (z 1 ≥ z 2 ≥ · · · ≥ 0) we have



H j G , Sz 1 Q ⊗ Sz F2 = H j G , Sz 1 Q ⊗ Sz F2 = 0 for j > 0, 

which follows from Corollary 9.8.6.

We next use the desingularization Y  to show that Z l is Cohen–Macaulay (which is also a consequence of the fact that Z l has rational singularities). Corollary 10.8.5 The variety Z l (and hence the ring S/Il+1 ) is Cohen–Macaulay. Proof Since codimA (Z l ) = (m − l)(n − l), the Cohen–Macaulay property is equivalent to j Ext S (S/Il+1 , S) = 0 for j = (m − l)(n − l). Arguing as in the proof of Theorem 10.7.1 and using Lemma 10.8.4, we can apply Corollary 9.9.5 with B = G , M = OG , and η = 1 Q ⊗ F2 . Using det(ξ )∨ ∼ = det(η) ∼ = det

1 n −m Q , ωG ∼ , = det 1 Q

dG = l(m − l), and rank(ξ ) = (m − l)n, our desired vanishing is equivalent to   n−m ⊗ Symr (η) = 0 for j = 0, r ≥ 0. H j G , det 1 Q We can now conclude as in the proof of Lemma 10.8.4: by Theorem 9.8.7, it suffices to show that   n−m ⊗ Sz 1 Q = 0 when j > 0 and z is a partition. H j G , det 1 Q n−m ⊗ Sz 1 Q = Sz+(n−m)l 1 Q , and z + (n − m)l is still a partition Since det 1 Q (because n ≥ m), the desired vanishing follows from Corollary 9.8.6.  We conclude with a quick computation of the divisor class group of Z l , which we denote by Cl(Z ), see Hartshorne [165, Sect. II.6]. Since Z 0 = {0} and Z m = K m×n , we have Cl(Z 0 ) = Cl(Z m ) = 0, so we may assume that 0 < l < m. Lemma 10.8.6 If 0 < l < m then Cl(Z l ) ∼ = Z. Proof Recall that if X is a normal variety, and U ⊂ X is an open subset with codim X (X \ U ) ≥ 2, then Cl(X ) ∼ = Cl(U ). Applying this to X = Z l and U = Z l \ Z l−1 , and using the fact that (see Corollary 10.8.1)

414

10 Asymptotic Regularity for Symbolic Powers of Determinantal Ideals

dim Z l − dim Z l−1 = l(m + n − l) − (l − 1)(m + n − l + 1) = m + n − 2l + 1 ≥ 3,

∼ Cl(Z l \ Z l−1 ), which in turn is isomorphic to Cl(Y  \ D  ) by we get Cl(Z l ) = (10.33). Since Y  is a geometric vector bundle over G , and since Cl(G ) ∼ = Z (generated by the Plücker divisor), it follows that Cl(Y  ) ∼ = Cl(G ) ∼ = Z. To conclude, it suffices to check that Cl(Y  \ D  ) ∼ = Cl(Y  ), for which, as remarked  earlier, it is enough to show that codimY  (D ) ≥ 2. Working relative to the base G , we can think of Y  as a space of l × n matrices, while D  ⊂ Y  is the subvariety of matrices of rank ≤ l − 1. It follows from Corollary 10.8.1 that codimY  (D  ) = (l − (l − 1))(n − (l − 1)) = n − l + 1 ≥ 2, as desired.



Notes The theory of filtrations for ideals defined by shape is adapted from Raicu [262], where it is developed only in characteristic zero, for ideals that are equivariant with respect to the natural action by row and column operations on matrices (we will explain this in more detail in Chap. 11). In particular, the determination of the asymptotic regularity of symbolic powers of determinantal ideals is new in positive characteristic. The inspiration for considering the modules Jl(σ ) comes from the Kempf–Weyman geometric technique, which is discussed extensively in Weyman [317], with the principal purpose of studying syzygy modules (see also Sect. 11.11). As it is apparent from the results here, the method is equally fruitful in the study of Ext modules (see also Raicu et al. [267], Raicu and Weyman [263], Perlman [256], L˝orincz and Raicu [234]). Sources for the results in Sect. 10.8 have been specified in the notes to Chaps. 3, 6 and 7.

Chapter 11

Cohomology and Regularity in Characteristic Zero

Throughout this chapter, K is a field of characteristic zero. Our goal is to refine some of the earlier characteristic-free results, and develop new aspects of the theory that are only valid in characteristic zero. In Sect.11.1 we present the Borel–Weil–Bott theorem regarding the cohomology of line bundles on flag varieties, and use it in Sect. 11.2 to compute some examples indicating that the vanishing in Theorem 9.5.1 is optimal even when restricted to characteristic zero. Furthermore, in Sect. 11.4 we explain how to find a sharp effective bound for the parameter d  0 in Theorem 9.5.1. Part of the proof uses a version of the Borel–Weil–Bott theorem for Grassmannians, which we explain in Sect. 11.3. In Sect. 11.5 we refine Theorem 9.8.7 by showing that Symd (F1 ⊗ F2 ) is isomorphic to the direct sum of the composition factors in (9.39). The Borel–Weil–Bott theorem and Cauchy’s formula are then used in Sect. 11.6 j to give an explicit description of all the modules Ext S (Jl(σ ) , S) as representations of the group GL = GL(F1 ) × GL(F2 ). In Sect. 11.7, this leads to a calculation of all the j modules Ext S (S/I p(d) , S) for symbolic powers of determinantal ideals. In Sect. 11.8 we establish basic results regarding the classification of GL-invariant ideals, and explain how the theory of filtrations for ideals defined by shape (from Sect. 10.2) extends to this setting. Based on this extension, in Sect. 11.9 we give a recipe to j compute Ext S (S/I, S) for an arbitrary GL-invariant ideal I , and to understand the natural maps on Ext for any inclusion of such ideals. Using the description of local cohomology as a direct limit of Ext modules, we explain in Sect. 11.10 how to j compute all the local cohomology modules HI p (S) with support in determinantal ideals. We conclude the chapter with a discussion of syzygies of GL-invariant ideals, starting with Lascoux’ theorem, the Kempf–Weyman geometric technique, and ending with the recent progress obtained based on the connections with the representation theory of the general linear Lie superalgebra.

© Springer Nature Switzerland AG 2022 W. Bruns et al., Determinants, Gröbner Bases and Cohomology, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-05480-8_11

415

416

11 Cohomology and Regularity in Characteristic Zero

11.1 The Borel–Weil–Bott Theorem The goal of this section is to prove the Borel–Weil–Bott theorem, describing (in characteristic zero) the cohomology of line bundles on a flag variety. As usual, we will formulate our results in the relative setting, where B is a base algebraic variety, E is a locally free sheaf of rank n on B, and F = FB (E) is the corresponding flag variety (see Sect. 9.2), with structure morphism π : F −→ B. It y ∈ Zn , then we write OF (y) for the corresponding line bundle. We recall that the description of R j π∗ (OF (y)) when y is dominant, and the vanishing π∗ (OF (y)) = 0 when y is not dominant, are covered by Theorems 9.4.1 and 9.8.5. To compute the higher direct images of OF (y) for a general y, we consider the symmetric group Sn of permutations of {1, . . . , n}, with the (left) action on Zn given by σ (y1 , . . . , yn ) = (yσ −1 (1) , . . . , yσ −1 (n) ). We let ρ = (n − 1, n − 2, . . . , 1, 0) ∈ Zn

(11.1)

and consider the . -action of Sn on Zn , defined by σ . y = σ (y + ρ) − ρ for y ∈ Zn .

(11.2)

We also define the length of a permutation σ ∈ Sn by (σ ) = |{(i, j) : i < j and σ (i) > σ ( j)}|. If we call a pair (i, j) with i < j and σ (i) > σ ( j) an inversion of σ , then (σ ) computes the number of inversions in σ . Example 11.1.1 Suppose that n = 3, so that ρ = (2, 1, 0), let y = (−1, 3, 2) and take σ to be the transposition (1, 2). We have σ . y = σ (1, 4, 2) − (2, 1, 0) = (4, 1, 2) − (2, 1, 0) = (2, 0, 2). If we now take τ = (2, 3) then τ . (2, 0, 2) = τ (4, 1, 2) − (2, 1, 0) = (4, 2, 1) − (2, 1, 0) = (2, 1, 1), which is a dominant weight. It follows that the 3-cycle τ σ = (1, 3, 2) has the property that (τ σ ) . y is dominant. Notice that (σ ) = (τ ) = 1, and (τ σ ) = 2. We can now state the Borel–Weil–Bott theorem.

11.1 The Borel–Weil–Bott Theorem

417

Theorem 11.1.2 Let F = FB (E), let y ∈ Zn , and let ρ be as in (11.1). (a) If y + ρ has repeated entries, then R j π∗ (OF (y)) = 0 for all j. (b) If y + ρ has distinct entries, let σ ∈ Sn denote the unique permutation such that σ (y + ρ) is decreasing. We have  R j π∗ (OF (y)) =

Sσ . y (E) if j = (σ ), 0 otherwise.

We note that if y ∈ Zndom then y + ρ is decreasing, so the permutation σ in Theorem 11.1.2(b) is the identity permutation, whose length is (σ ) = 0, and for which σ . y = y. The conclusion of Theorem 11.1.2(b) agrees then with our earlier characteristic-free results. The next example deals with a nondominant weight. Example 11.1.3 Let n = 3 and let y = (−1, 3, 2) as in Example 11.1.1. We have y + ρ = (1, 4, 2), and the unique σ ∈ S3 making σ (1, 4, 2) decreasing is the 3-cycle σ = (1, 3, 2), of length (σ ) = 2. It follows from Theorem 11.1.2(b) that  R π∗ (OF (−1, 3, 2)) = j

S(2,1,1) (E) = E ⊗ det(E) if j = 2, 0 otherwise.

Consequences of a Self-Duality Statement for Symmetric Powers of Rank 2 Bundles A key ingredient in our proof of Theorem 11.1.2 is the isomorphism (9.1) that holds under our assumption that char K = 0. Combining this with (9.2), and making the usual extension to sheaves, it follows that for any locally free sheaf F on a variety over Spec K , we have isomorphisms    ∨ Symd F ∨  Symd F

for all d ≥ 0.

(11.3)

In the special case when rank(F ) = 2, we get from (9.3) a natural isomorphism F ∨  F ⊗ det(F )−1 . It then follows from (11.3) that for every locally free sheaf F of rank two,    ∨ Symd F  Symd F ⊗ det(F )−1  Symd (F ) ⊗ det(F )−d .

(11.4)

418

11 Cohomology and Regularity in Characteristic Zero

As a consequence, we get the following. Lemma 11.1.4 Suppose that rank(E) = n = 2, and let y = (y1 , y2 ) ∈ Z2 . If we let σ be the transposition (1, 2) then the following hold. (a) If y1 ≥ y2 then R j π∗ (OF (y)) = R j+1 π∗ (OF (σ . y)) for j ∈ Z, and is 0 for j = 0. (b) If y1 ≤ y2 − 2 then R j π∗ (OF (y)) = R j−1 π∗ (OF (σ . y)) for j ∈ Z, and is 0 for j = 1. (c) If y1 = y2 − 1 then y = σ . y, and R j π∗ (OF (y)) = 0 for all j ∈ Z. Proof One has that σ . y = σ (y1 + 1, y2 ) − (1, 0) = (y2 , y1 + 1) − (1, 0) = (y2 − 1, y1 + 1). It follows that if y1 = y2 − 1, then σ . y = y, and the vanishing Rπ∗ (OF (y)) = 0 follows from Lemma 9.3.4, proving (c). If we let z = σ . y, then we have z 1 − z 2 = −(y1 − y2 ) − 2, so the condition y1 ≥ y2 is equivalent to z 1 ≤ z 2 − 2, while y1 ≤ y2 − 2 is equivalent to z 1 ≥ z 2 . This shows that (a) and (b) are equivalent, so it suffices to verify (a). Note that since y is dominant, we have π∗ (OF (y)) = S y E, and R j π∗ (OF (y)) = 0 for j = 0. Moreover, since z 1 ≤ z 2 − 2, we can apply Corollary 9.9.3 to z to get (using

n  2

= 1)

 ∨ R 1 π∗ (OF (z)) = S(−z1 −1,−z2 +1) E , and R j π∗ (OF (z)) = 0 for j = 1.  ∨ To conclude, we have to check that S(−z1 −1,−z2 +1) E is isomorphic to S y E. To this end, we note that  ∨ ∨  S(−z1 −1,−z2 +1) E = Sym−z1 +z2 −2 E ⊗ det(E)−z2 +1  ∨ = Sym−z1 +z2 −2 E ⊗ det(E)z2 −1 (11.4)

= Sym−z1 +z2 −2 E ⊗ det(E)z1 −z2 +2 ⊗ det(E)z2 −1 = Sym y1 −y2 E ⊗ det(E) y2 = S y E, as desired.



Corollary 11.1.5 Suppose that n ≥ 2 and let σi denote the transposition (i, i + 1). (a) If yi ≥ yi+1 then R j π∗ (OF (y)) = R j+1 π∗ (OF (σi . y)) for j ∈ Z. (b) If yi ≤ yi+1 − 2 then R j π∗ (OF (y)) = R j−1 π∗ (OF (σi . y)) for j ∈ Z. (c) If yi = yi+1 − 1 then R j π∗ (OF (y)) = R j π∗ (OF (σi . y)) = 0 for j ∈ Z.

11.1 The Borel–Weil–Bott Theorem

419

Proof If we consider the factorization π = g ◦ f as in (9.19), where F = FB ({1, 2, . . . , i − 1, i + 1, . . . , n}, E), then F = FF (R), where R = Ri+1,i−1 has rank 2. Moreover, as in the proof of Lemma 9.3.4 we have OF (y) = ORF (yi , yi+1 ) ⊗ f ∗ (F ), OF (σi . y) = ORF (yi+1 − 1, yi + 1) ⊗ f ∗ (F ) for some sheaf F on F . Using the projection formula and Lemma 11.1.4, it follows that in case (a) R j π∗ (OF (y)) = R j g∗ ( f ∗ OF (y)) = R j g∗ (R 1 f ∗ OF (σi . y)) = R j+1 π∗ (OF (σi . y)), as desired. Applying part (a) with y replaced by σi . y yields (b), while part (c) is a consequence of Lemma 9.3.4. 

Proof of Borel–Weil–Bott We are now ready to verify Theorem 11.1.2. Proof (Proof of Theorem 11.1.2) For part (a), suppose that y + ρ has repeated entries, say yi + (n − i) = yi + (n − i ) for some i < i . If i = i + 1 then yi − yi+1 = −1 and the conclusion follows Corollary 11.1.5(c). Otherwise, we consider the cyclic permutation

from

σ = (i + 1, i + 2, . . . , i ) = σi+1 σi+2 · · · σi −1 . It follows that if we let z = σ . y then z i = yi , z i+1 = yi + (i − i ) + 1, hence z i − z i+1 = (yi − yi ) − (i − i ) − 1 = −1.

It follows that R j π∗ (OF (z)) = 0 for all j, and by iterating Corollary 11.1.5 we get for an appropriate choice of signs that R j π∗ (OF (y)) = R j±1 π∗ (OF (σi −1 . y)) = R j±1±1 π∗ (OF (σi −2 σi −1 . y)) = R j±1±1±···±1 π∗ (OF (z)) = 0, as desired. We prove part (b) by induction on (σ ). If (σ ) = 0 then σ is the identity permutation, hence yi + (n − i) > yi+1 + (n − i − 1) for all i. Equivalently, yi ≥ yi+1 and

420

11 Cohomology and Regularity in Characteristic Zero

therefore y ∈ Zndom . As noted earlier, the desired conclusion is then a consequence of Theorems 9.4.1 and 9.8.5. Suppose now that (σ ) > 0, so there exists an index i with yi + (n − i) < yi+1 + (n − i − 1), or equivalently, yi ≤ yi+1 − 2. If we let z = σi . y then z i ≥ z i+1 . It follows that the corresponding partition τ with τ (z + ρ) decreasing is given by τ = σ σi , and it has the property that (τ ) = (σ ) − 1. Applying Corollary 11.1.5(b) and our induction hypothesis, we get  R π∗ (OF (y)) = R j

j−1

π∗ (OF (z)) =

Sτ .z (E) if j − 1 = (τ ), 0 otherwise.

Since τ . z = σ σi σi . y = σ . y, and the equality j − 1 = (τ ) = (σ ) − 1 is equivalent to j = (σ ), the desired conclusion follows. 

Schur Functors and Duals As a consequence of Theorem 11.1.2, we can derive the following compatibility between Schur functors and duals (which may fail over a field of positive characteristic, for instance (Sym2 E)∨  Sym2 (E∨ ) in characteristic 2). Corollary 11.1.6 For every y ∈ Zndom there exists an isomorphism  ∨   S y E  S(−yn ,−yn−1 ,...,−y1 ) E  S y E∨ . Proof Using Corollary 9.9.3, it follows that 

Sy E

∨

= R (2) π∗ OF (x), n

where x = (−y1 − (n − 1), −y2 − (n − 3), . . . , −yn−1 + (n − 3), −yn + (n − 1)). The permutation that makes   x + ρ decreasing is given by σ (i) = n + 1 − i for all i, and has length (σ ) = n2 . Moreover, we have σ . x = (xn − (n − 1), xn−1 − (n − 3), . . . , x1 + (n − 1)) = (−yn , −yn−1 , . . . , −y1 ),

hence by Theorem 11.1.2 we obtain R (2) π∗ OF (x)  S(−yn ,−yn−1 ,...,−y1 ) E. n

The desired conclusion now follows from (9.38).



11.2 The Optimality of Theorem 9.5.1

421

11.2 The Optimality of Theorem 9.5.1 In this section we apply the Borel–Weil–Bott theorem to compute the higher direct images for certain families of line bundles, indicating the extent to which the characteristic-free vanishing Theorem 9.5.1 is optimal. Example 11.2.1 With the notation in Theorem 9.5.1, suppose that c = d ≥ l, b = (2l, 2l − 2, . . . , 2), and y = (d, 0, . . . , 0). It follows that L = OF (d − 2l, d − 2l + 2, . . . , d − 2, d, 0, . . . , 0). We consider the permutation σ ∈ Sn defined by  l +2−i σ (i) = i

for i = 1, . . . , l + 1, for i > l + 1,

(11.5)

note that (i, j) is an inversion of σ if and only if 1 ≤ i < j ≤ l + 1, so (σ ) = and l+1 . Applying the . -action we obtain 2   σ . (d − 2l, d − 2l + 2, . . . , d − 2, d, 0, . . . , 0) = (d − l)l+1 , 0n−1−l , and using Theorem 11.1.2 we get 

  S(d−l)l+1 E if j = l+1 , 2 R π∗ L = 0 otherwise. j

Notice that in our example we have b = l · (l + 1), so vanishing in Theorem 9.5.1 is sharp.

(d−c)·l+b 2

=

l+1 , and the 2

In the next example we show that a lower bound for d is necessary in Theorem 9.5.1; otherwise the vanishing of cohomology may fail for small values of d. Example 11.2.2 With the notation in Theorem 9.5.1, suppose that c = 0, 1 ≤ l ≤ n − 2, y = (0n−l ), 0 ≤ d ≤ n − l − 2, and that b = (n + l − 1, n + l − 3, . . . , n − l + 3, n − l + 1). It follows that b = nl and L = OF (−n − l + 1, −n − l + 3, . . . , −n + l − 3, −n + l − 1, 0, . . . , 0). If we consider the permutation σ ∈ Sn defined by

422

11 Cohomology and Regularity in Characteristic Zero

 σ (i) =

n+1−i i −l

for i = 1, . . . , l, for i ≥ l + 1.

then the set of inversions in σ is given by {(i, j) : 1 ≤ i < j ≤ l} ∪ {(i, j) : i ≤ l, j ≥ l + 1},   l (σ ) = + l(n − l). 2

and therefore we have

Using the . -action we get σ . (−n − l + 1, −n − l + 3, . . . , −n + l − 3, −n + l − 1, 0, . . . , 0) = (−l, −l, . . . , −l),

and Theorem 11.1.2 shows that 

S(−l n ) E if j = l(n − l) + R π∗ L = 0 otherwise. j

l  , 2

Moreover,     dl + nl (n − l − 2)l + nl l l l (d − c) · l + b = ≤ = l(n − l) + − < l(n − l) + 2 2 2 2 2 2

and therefore the vanishing in Theorem 9.5.1 is violated. We end the section with two more examples showing the failure of (9.22) when d ≤ n − 3. The proof of Theorem 9.5.1 shows that the requirement d  0 is not necessary for l = 0, l = n − 1 and l = n, so the relevant values of l are 1 ≤ l ≤ n − 2. For each l in this range and for l − 1 ≤ d ≤ n − 3, we have the following counterexample to (9.22). Example 11.2.3 With the notation in Theorem 9.5.1, suppose that l − 1 ≤ d ≤ n − 3, let c = d and y = (d, 0, . . . , 0). Let b1 = d + n and bi = 2(l − i) + 2 for i = 2, . . . , l. We have b=2

  l +1 + (d + n − 2l) and L = OF (−n, d − 2l + 2, . . . , d − 2, d, 0, . . . , 0). 2

If we consider the permutation σ ∈ Sn defined by ⎧ ⎪ ⎨n σ (i) = l + 2 − i ⎪ ⎩ i −1

for i = 1, for i = 2, . . . , l + 1, for i > l + 1,

11.2 The Optimality of Theorem 9.5.1

423

then the set of inversions in σ is given by {(1, i) : i = 2, . . . , n} ∪ {(i, j) : 2 ≤ i < j ≤ l + 1}, and therefore (σ ) = n − 1 +

    l l +1 = + (n − 1 − l). 2 2

Using the . -action we get   σ . (−n, d − 2l + 2, . . . , d − 2, d, 0, . . . , 0) = (d − l)l , (−1)n−l , which is dominant since d − l ≥ −1. It follows from Theorem 11.1.2 that    S((d−l)l ,(−1)n−l ) E if j = l+1 + (n − 1 − l), j 2 R π∗ L = 0 otherwise. Moreover, we have that     b l +1 d + n − 2l l +1 (d − c) · l + b = = + < + (n − 1 − l), 2 2 2 2 2 hence the vanishing (9.22) does not hold in this case. When d ≤ l − 1, we have the following counterexample to (9.22), which in the case d = l − 1 overlaps with the construction from Example 11.2.3. Example 11.2.4 With the notation in Theorem 9.5.1, suppose that d ≤ l − 1 and l ≤ n − 2, let c = d and y = (d, 0, . . . , 0). Let k = l − d ≥ 1, and define  bi = We have b = 2

l+1 2

n + l + 1 − 2i for i = 1, . . . , k, 2(l − i) + 2 for i = k + 1, . . . , l.

+ k(n − 1 − l) and L = OF (y), where

y = (d + 1 − n − l, d + 3 − n − l, . . . , d + 2k − 1 − n − l, d + 2k − 2l, . . . , d − 2, d, 0, . . . , 0).

If we consider the permutation σ ∈ Sn defined by ⎧ ⎪ ⎨n + 1 − i σ (i) = l + 2 − i ⎪ ⎩ i −k

for i ≤ k, for k + 1 ≤ i ≤ l + 1, for i > l + 1,

then the set of inversions in σ is given by

424

11 Cohomology and Regularity in Characteristic Zero

{(i, j) : 1 ≤ i < j ≤ l + 1} ∪ {(i, j) : i ≤ k, j ≥ l + 2}, and therefore we have (σ ) =

  l +1 + k(n − 1 − l). 2

Since k = l − d, we get σ . y = ((d − l)n ), and it follows from Theorem 11.1.2 that 

  S((d−l)n ) E if j = l+1 + k(n − 1 − l), 2 R π∗ L = 0 otherwise. j

Moreover, we have that     b l +1 k(n − 1 − l) l +1 (d − c) · l + b = = + < + k(n − 1 − l), 2 2 2 2 2 hence the vanishing (9.22) does not hold in this case.

11.3 Borel–Weil–Bott on Grassmannians In this section we explain how Theorem 11.1.2 applies to compute the higher direct images sheaves of the form (11.6) Sx Q ⊗ S y R on a Grassmannian, generalizing Corollary 9.8.6. As usual, we assume that E is locally free of rank n on B, and we let G = G(l, E) denote the corresponding Grassmann bundle for 0 < l < n, with structure morphism γ : G −→ B, tautological quotient sheaf Q, and tautological subsheaf R. In (11.6), we assume that x ∈ Zldom

and

y ∈ Zn−l dom ,

(11.7)

with no further relation between x and y. Theorem 11.3.1 With the notation above, we let z = (x1 , . . . , xl , y1 , . . . , yn−l ) ∈ Zn ,

u i = { j ∈ [n − l] : xi − i < y j − (l + j)} , for i ∈ [l],

t j = {i ∈ [l] : xi − i < y j − (l + j)} , for j ∈ [n − l]. xi − i = y j −  (l + j) Rγ∗ Sx Q ⊗ S y R = 0.

(a) If

for

some

i ∈ [l]

and

j ∈ [n − l]

(11.8) (11.9) then

11.3 Borel–Weil–Bott on Grassmannians

425

(b) If xi − i = y j − (l + j) for every i ∈ [l] and j ∈ [n − l], then the assignment σ (i) = i + u i for i ∈ [l], σ (l + j) = (l + j) − t j for j ∈ [n − l], defines a permutation σ ∈ Sn . Moreover, σ has the property that σ . z ∈ Zndom , (σ ) = u 1 + · · · + u l = t1 + · · · + tn−l , and σ . z can be computed as (σ . z)i+u i = xi + u i for i ∈ [l], and (σ . z)l+ j−t j = y j − t j for j ∈ [n − l].

Finally, we have    Sσ .z (E) if j = (σ ), R j γ∗ S x Q ⊗ S y R = 0 otherwise. Proof We let F = FB (E) with structure morphism π : F −→ B, and we factor π = γ ◦ α where α : F −→ G is the map forgetting all but the l-dimensional quotient in a complete flag. The argument in the proof of Corollary 9.8.6 implies that Rα∗ (OF (z)) = Sx Q ⊗ S y R, hence it follows from the Leray spectral sequence that   Rγ∗ Sx Q ⊗ S y R = Rπ∗ (OF (z)),

(11.10)

which we can compute using Theorem 11.1.2. Notice that (11.7) implies z 1 + (n − 1) > · · · > zl + (n − l) and zl+1 + (n − l − 1) > · · · > z n , (11.11) which means that z + ρ has repeated entries if and only if z i + (n − i) = zl+ j + (n − l − j) for some i ∈ [l] and j ∈ [n − l]. This is further equivalent to xi − i = y j − (l + j), proving part (a). For part (b), we let σ ∈ Sn denote the unique permutation such that σ (z + ρ) is decreasing, and note that in light of (11.10) and Theorem 11.1.2, it suffices to show that σ = σ and to determine the formula for its length. By definition, σ (i) = 1 + |{k ∈ [n] : z i + (n − i) < z k + (n − k)}| = 1 + |{k ∈ [n] : z i − i < z k − k}| . Suppose first that i ∈ [l]. If k ≤ l then the condition xi − i = z i − i < z k − k = xk − k is equivalent to k < i, hence it holds for i − 1 values of k. If k = l + j > l then xi − i = z i − i < z k − k = y j − (l + j) and it holds for exactly u i values of k.

426

11 Cohomology and Regularity in Characteristic Zero

We get

σ (i) = 1 + (i − 1) + u i = σ (i) for i ∈ [l].

If i = l + j for j ∈ [n − l], then a similar argument shows that σ (l + j) = 1 + ( j − 1) + (l − t j ) = σ (l + j), so σ = σ . To conclude, we note that because of (11.11), the only inversions of σ are of the form (i, l + j) for i ∈ [l] and j ∈ [n − l]. Moreover, (i, l + j) is an inversion if and only if xi − i < y j − (l + j), hence (σ ) = u 1 + · · · + u l = t1 + · · · + tn−l , as desired. To conclude, we have to compute σ . z explicitly. We have for i ∈ [l] that (σ . z)i+u i = z σ −1 (i+u i ) + (n − σ −1 (i + u i )) − (n − i − u i ) = z i + u i = xi + u i , and for j ∈ [n − l] that (σ

. z)l+ j−t

j

= z σ −1 (l+ j−t j ) + (n − σ −1 (l + j − t j ) − (n − l − j + t j ) = zl+ j − t j = y j − t j ,



concluding our proof. We next illustrate Theorem 11.3.1 with an example.

Example 11.3.2 Suppose that n = 6, l = 2, x = (5, −5) and y = (11, 4, 3, −5). One has x1 − 1 = 4, x2 − 2 = −7, y1 − 3 = 8, y2 − 4 = 0, y3 − 5 = −2, y4 − 6 = −11. It follows that u 1 = 1, u 2 = 3, t1 = 2, t2 = 1, t3 = 1 and t4 = 0. The permutation σ is given by i 123456 σ (i) 2 5 1 3 4 6 and it has length (σ ) = 4 = 1 + 3 = 2 + 1 + 1. Moreover, we have that z = (5, −5, 11, 4, 3, −5) and therefore σ . z = (9, 6, 3, 2, −2, −5). By inspection, this agrees with the formula for σ . z given in Theorem 11.3.1:

11.3 Borel–Weil–Bott on Grassmannians

i 1 2 i + ui 2 5 xi + u i 6 −2

427

123 4 j l + j − tj 1 3 4 6 y j − t j 9 3 2 −5

Finally,    S(9,6,3,2,−2,−5) (E) if j = 4, R γ∗ S(5,−5) Q ⊗ S(11,4,3,−5) R = 0 otherwise. j

We end the section by making some general observations that we can extrapolate from the example above. Firstly, we have that u 1 ≤ u 2 ≤ · · · ≤ u l ≤ n − l and l ≥ t1 ≥ t2 ≥ · · · ≥ tn−l , or equivalently, σ (1) < σ (2) < · · · < σ (l) and σ (l + 1) < · · · < σ (n). A permutation σ with the above property is usually called a Grassmannian permutation, and is uniquely determined by either of the sequences u or t. Moreover, if we identify u and t with partitions, then they are conjugate to each other: (u l , . . . , u 1 ) = (t1 , . . . , tn−l )∗ .

(11.12)

To see this, note that since the sequence xi − i is decreasing, it follows that t j ≥ i ⇐⇒ xl−i+1 − (l − i + 1) < y j − (l + j), hence by the definition of the conjugate partition, we see that ti∗ = |{ j : t j ≥ i}| = |{ j : xl−i+1 − (l − i + 1) < y j − (l + j)}| = u l+1−i . The following observations will be useful in Sect. 11.6. Remark 11.3.3 Since the sequences xi − i and y j − (l + j) are decreasing, we can use (11.8) to reformulate the hypothesis of Theorem 11.3.1(b) as the existence of a chain of strict inequalities y1 − (l + 1) > · · · > yu 1 − (l + u 1 ) > x1 − 1 > > yu 1 +1 − (l + u 1 + 1) > · · · > yu 2 − (l + u 2 ) > x2 − 2 > .. . > yul−1 +1 − (l + u l−1 + 1) > · · · > yul − (l + u l ) > xl − l > > yul +1 − (l + u l + 1) > · · · > yn−l − n.

428

11 Cohomology and Regularity in Characteristic Zero

Remark 11.3.4 If we are in the situation of Theorem 11.3.1(b) and y j = y j+1 for some j ≤ n − l − 1 then t j = t j+1 . Indeed, the assumption implies that y j − (l + j) and y j+1 − (l + 1 + j) are consecutive integers, hence for each i we either have xi − i > y j − (l + j) > y j+1 − (l + 1 + j) or y j − (l + j) > y j+1 − (l + 1 + j) > xi − i, which forces t j = t j+1 , as desired.

11.4 Effective Vanishing of Cohomology in Characteristic Zero Using the Borel–Weil–Bott theorem, we can now revisit the proof of Theorem 9.5.1 to obtain an effective lower bound for d (in terms of n) that is sufficient for the conclusion of the theorem to hold. In light of Examples 11.2.3 and 11.2.4, a necessary lower bound is d ≥ n − 2. A careful analysis in the case l = 1 shows that this bound is also sufficient and is explained in Sect. 9.6. The goal of this section is to show that d ≥ n − 2 is sufficient for arbitrary l, provided we work in characteristic zero. Theorem 11.4.1 In characteristic zero, Theorem 9.5.1 holds when d ≥ n − 2. In particular, if 1 ≤ p ≤ m ≤ n, then we have reg I p(d) = p · d for all d ≥ m − 1.

Quick Effective Vanishing We begin by considering a weaker version of Theorem 11.4.1, which has a simple proof and gives an effective, but not optimal bound for d. Lemma 11.4.2 In characteristic zero, Theorem 9.5.1 holds when d ≥ 3n 2 . Proof As in the proof of Theorem 9.5.1, we may assume d − c, b < n 2 , and that (9.23) holds. Under these circumstances, we need to verify the vanishing (9.24). Since d ≥ 3n 2 , it follows that c ≥ 2n 2 . Since d − c ≥ y2 + y3 + · · · ,

11.4 Effective Vanishing of Cohomology in Characteristic Zero

429

and d − c < n 2 , we obtain n 2 > y2 . Using n 2 > b ≥ b1 , we conclude that c + l ≥ 2n 2 > b1 + y2 . If we consider the permutation σ in (11.5), then σ . (c − b1 , . . . , c − bl , c, y2 , . . . , yn−l ) = (c − l, c − l + 2 − bl , . . . , c + l − b1 , y2 , . . . , yn−l ),   which is a dominant weight since c + l − b1 > y2 . Since (σ ) = l+1 , it follows 2 l+1 j  from Theorem 11.1.2 that R π∗ L = 0 for j = 2 , proving (9.24).

An Optimization Problem To prove the stronger statement in Theorem 11.4.1, we will need to solve an optimization problem, which we describe next. For d ≥ 0 and 0 ≤ l ≤ n, we consider   |y| ≤ d, l = t1 ≥ · · · ≥ tn−l n−l YT (l, n, d) = (y, t) ∈ Zn−l × Z : ≥0 ≥0 y1 − t1 ≥ · · · ≥ yn−l − tn−l

(11.13)

and note that for any (y, t) ∈ YT (l, n, d), y is dominant: yi − yi+1 ≥ ti − ti+1 ≥ 0. We consider the problem of maximizing the function gl,n (y, t) = ly1 + |t| −

n−l 

ti · ((yi−1 − ti−1 ) − (yi − ti ))

(11.14)

i=2

over the set YT (l, n, d). Proposition 11.4.3 If we let   Rl,n,d = max gl,n (y, t) : (y, t) ∈ YT (l, n, d) then for all d ≥ n − 2 one has Rl,n,d = l(d + 1). Proof Observe first that if we take y1 = d, t1 = l, yi = ti = 0 for 2 ≤ i ≤ n − l, then (y, t) ∈ YT (l, n, d) and gl,n (y, t) = l(d + 1),

430

11 Cohomology and Regularity in Characteristic Zero

so Rl,n,d ≥ l(d + 1). We then need to verify that Rl,n,d ≤ l(d + 1), which we prove by induction on (l, n). We fix a pair (y, t) ∈ YT (l, n, d) for which gl,n (y, t) = Rl,n,d , and break up our analysis into two cases. Case 1: yn−l ≥ 1. Using the fact that y2 ≥ · · · ≥ yn−l ≥ 1, we have d ≥ |y| = y1 + (y2 + · · · + yn−l ) ≥ y1 + (n − l − 1).

(11.15)

Since (yi−1 − ti−1 ) − (yi − ti ) ≥ 0 for (y, t) ∈ YT (l, n, d), we get gl,n (y, t)

(11.14)

≤

ly1 + |t| ≤ ly1 + l(n − l)

(11.15)

≤

l(d − n + l + 1) + l(n − l) = l(d + 1).

Case 2: yn−l = 0. If tn−l = 0 then if we let y = (y1 , . . . , yn−l−1 ), t = (t1 , . . . , tn−l−1 ), we have (y , t ) ∈ YT (l, n − 1, d), and it follows from (11.14) that gl,n (y, t) = gl,n−1 (y , t ) ≤ Rl,n−1 = l(d + 1), where the last equality follows by induction. We may therefore assume from now on that tn−l = k > 0. If we define n = n − k, l = l − k, t = (t1 − k, . . . , tn−l − k) then n − l = n − l and it follows from (11.13) that (y, t ) ∈ YT (l , n , d). Moreover, gl,n (y, t) − gl ,n (y, t ) = ky1 + k(n − l) − k

n−l 

((yi−1 − ti−1 ) − (yi − ti ))

i=2

= k(y1 + n − l − (y1 − t1 ) + (yn−l − tn−l )) = k(n − k), where the last equality uses the fact that t1 = l, tn−l = k, and yn−l = 0. It follows from the above equality and induction that Rl,n,d = gl,n (y, t) = gl ,n (y, t ) + k(n − k) ≤ l (d + 1) + k(n − k). To conclude, we have to verify that k(n − k) ≤ (l − l )(d + 1) = k(d + 1), which (since k > 0) is equivalent to d ≥ n − k − 1. This follows from our hypothesis d ≥ n − 2, and the fact that k > 0, concluding our proof. 

11.4 Effective Vanishing of Cohomology in Characteristic Zero

431

Sharp Effective Vanishing We are now ready to verify the statement of Theorem 11.4.1. Proof (Proof of Theorem 11.4.1) As in the proof of Theorem 9.5.1, if bl = 1 or bi − bi+1 = 1 for some i, then Rπ∗ L = 0 by Lemma 9.3.4. We will therefore assume that bi − bi+1 ≥ 2 for i = 1, . . . , l − 1, and bl ≥ 2. We let G = GB (l, E) and F = FB ({l, l − 1, . . . , 1}; E), and factor π : F −→ B as π

F

α

F

β

G

γ

B

We write Q (resp. R) for the tautological rank l (resp. n − l) quotient sheaf (resp. subsheaf) on G, and note that F = FG (Q) and F = FF (β ∗ R). Using the fact that y ∈ Zn−l dom , we get     Rα∗ L = OF (c − b1 , . . . , c − bl ) ⊗ S y β ∗ R = OF (c − b1 , . . . , c − bl ) ⊗ β ∗ S y R .

Theorem 11.1.2 for the map β and the projection formula imply R (2) β∗ (Rα∗ L) = Sx Q ⊗ S y R, and R j β∗ (Rα∗ L) = 0 for j = l

  l , 2

where x = (c − bl − (l − 1), c − bl−1 − (l − 3), . . . , c − b2 + (l − 3), c − b1 + (l − 1)). Using the Leray spectral sequence for β ◦ α we get R j (β ◦ α)∗ L = 0 for j = and using it again for π = γ ◦ (β ◦ α), we conclude that

l  , 2

  l R j π∗ L = R j−(2) γ∗ Sx Q ⊗ S y R for all j. To compute the right hand side, we apply Theorem 11.3.1: in case (a) all the higher direct images vanish, so we may assume that we are in case (b), so R

j−(2l )

    l γ∗ Sx Q ⊗ S y R = 0 for j −

= (σ ) = |t|, 2

432

11 Cohomology and Regularity in Characteristic Zero

where t and σ are as in Theorem 11.3.1. To conclude, we have to verify that   l (d − c)l + b . |t| + ≤ 2 2 Using the fact that |x| = lc − b, we can rewrite the inequality above as |x| ≤ l(d + 1) − l 2 − 2|t|.

(11.16)

Since x1 − 1 = c − bl − l < c − l − 1 = y1 − l − 1, it follows from (11.9) that t1 = l. Moreover, since σ . z is dominant, it follows that y1 − t1 ≥ y2 − t2 ≥ · · · , and using our hypothesis d ≥ |y|, (11.13) implies that (y, t) ∈ YT (l, n, d). In analogy with Remark 11.3.3, we have a chain of strict inequalities y1 − (l + 1) > x1 − 1 > · · · > xl−t2 − (l − t2 ) > y2 − (l + 2) > xl−t2 +1 − (l − t2 + 1) > · · · > xl−t3 − (l − t3 ) > .. . yn−l − n > xl−tn−l +1 − (l − tn−l + 1) > · · · > xl − l. It follows that |x| is maximized when (with the convention tn−l+1 = 0) xl−t j +1 = xl−t j +2 = · · · = xl−t j+1 = y j − (t j + j) for j = 1, . . . , n − l, in which case we have |x| =

n−l 

⎛ ⎞ n−l  (t j − t j+1 ) · (y j − (t j + j)) = ⎝ (t j − t j+1 ) · (y j − t j )⎠ − |t|,

j=1

j=1

 where the last equality uses the identity n−l j=1 (t j − t j+1 ) · j = t1 + · · · + tn−l . To prove (11.16), it is then enough to show that l 2 + |t| +

n−l 

(t j − t j+1 ) · (y j − t j ) ≤ l(d + 1),

j=1

which by Proposition 11.4.3 reduces further to verifying the identity n−l  l + |t| + (t j − t j+1 ) · (y j − t j ) = gl,n (y, t). 2

j=1

11.5 Cauchy’s Formula Revisited

433

Note that since t1 = l, we have l 2 + t1 (y1 − t1 ) = ly1 , so based on (11.14) it remains to check that −t2 (y1 − t1 ) +

n−l n−l   (t j − t j+1 ) · (y j − t j ) = − ti · ((yi−1 − ti−1 ) − (yi − ti )), j=2

i=2

which follows by inspection. Since Theorem 9.5.1 holds in characteristic zero for d ≥ n − 2, it follows that Corollary 9.10.2 holds for d ≥ n − 1. The proof of Theorem 10.7.1 then yields the desired formula for regularity in the effective range d ≥ m − 1. 

11.5 Cauchy’s Formula Revisited In this section we will explain how to refine Theorem 9.8.7 in characteristic zero, showing that Symd (F1 ⊗ F2 ) is isomorphic to the direct sum of the composition factors in (9.39). We begin with a more concrete take on the isomorphism (11.3). Let F be a locally free sheaf of rank r on a variety B, let U = Spec A ⊂ B be an open affine subset where F is free, and let x1 , . . . , xr be an A-basis of F = F (U ). If we write ∂1 , . . . , ∂r for the dual basis of F ∨ = F ∨ (U ), then Sym(F) = A[x1 , . . . , xr ] and Sym(F ∨ ) = A[∂1 , . . . , ∂r ]. Each ∂i gives rise to an A-derivation on A[x1 , . . . , xr ] (the partial derivative with respect to xi ), and we get a natural action of Sym(F ∨ ) on Sym(F) by differentiation. We can rephrase this in terms of a pairing −, − : Sym(F) × Sym(F ∨ ) −→ A, ( f, P) → P( f ). If we write

x α = x1α1 · · · xrαr , ∂ α = ∂1α1 · · · ∂rαr , 

then we have α

β

x , ∂  =

α1 ! · · · αr ! if α = β, 0 otherwise.

(11.17)

This restricts for each d ≥ 0 to a perfect pairing between Symd (F) and Symd (F ∨ ), which is independent on the choice of basis for F. It follows that it extends to a perfect pairing (11.18) −, − : Symd (F ) × Symd (F ∨ ) −→ OB , which gives the isomorphism (11.3). Remark 11.5.1 The construction of the pairing (11.18) commutes with base-change (since B ⊗ A A[x1 , . . . , xr ] = B[x1 , . . . , xr ]), which will often allow us to derive properties of the pairing by reducing to the case when A = Q is the field of rational numbers. In that case, if

434

11 Cohomology and Regularity in Characteristic Zero

f (x) =



cα x α ∈ Q[x1 , . . . , xr ],

α

then we let f (∂) =



cα ∂ α ∈ Q[∂1 , . . . , ∂r ],

α

and note that because of (11.17), we have that if f = 0 then some cα = 0, hence  f (x), f (∂) =

 α

cα2 · α1 ! · · · αr ! > 0,

and in particular  f (x), f (∂) = 0. Lemma 11.5.2 For each  ⊂ P(d), the pairing (11.18) restricts to a perfect pairing between V() (F1 , F2 ) and V() (F1∨ , F2∨ ). Proof The question is local, so we may assume that B = Spec A, and Fk = Fk (B) is a free A-module for k = 1, 2. We choose bases on F1 , F2 and write xi, j for the induced basis of F1 ⊗ F2 , and ∂i, j for the dual basis of F1∨ ⊗ F2∨ . Since both the construction of the pairing (11.18) and that of V() are functorial and commute with base change, we may further reduce to the case when A = Q. To conclude, it is now sufficient to show that for A = Q, the induced pairing between V() (F1 , F2 ) and V() (F1∨ , F2∨ ) is nondegenerate. Suppose that f (xi, j ) ∈ V() (F1 , F2 ) satisfies  f (xi, j ), − = 0. If f (xi, j ) = 0 then if we take f (∂i, j ) ∈ V() (F1∨ , F2∨ ), it follows from Remark 11.5.1 that  f (xi, j ), f (∂i, j ) = 0, a contradiction. By symmetry, we have that −, f (∂i, j ) = 0 if f (∂i, j ) = 0, so the form is nondegenerate, as desired. 

Direct Sum Decomposition of Sheaves Defined By Shape If we consider the orthogonal complement V() (F1∨ , F2∨ )⊥ ⊂ Symd (F1 ⊗ F2 ), then it follows from Lemma 11.5.2 that we have a direct sum decomposition Symd (F1 ⊗ F2 ) = V() (F1 , F2 ) ⊕ V() (F1∨ , F2∨ )⊥ .

(σ )

11.6 Explicit Description of Ext Modules for Jl

435

We define for each σ ∈ P(d) the sheaf (σ ) (F1∨ , F2∨ )⊥ , Mσ (F1 , F2 ) = V(σ ) (F1 , F2 ) ∩ V>

(11.19)

and notice that by Lemma 11.5.2, we can interpret Mσ (F1 , F2 ) ⊂ V(σ ) (F1 , F2 ) (σ ) (F1∨ , F2∨ ) ⊂ V(σ ) (F1∨ , F2∨ ) for the restricted as the orthogonal complement of V> ∨ ∨ (σ ) (σ ) pairing V (F1 , F2 ) × V (F1 , F2 ) −→ OB . It follows that we have a direct sum decomposition (compare with Theorem 3.6.8) (σ ) (F1 , F2 ). V(σ ) (F1 , F2 ) = Mσ (F1 , F2 ) ⊕ V>

(11.20)

Using the notation in Lemma 10.5.3, the inclusions Mσ (F1 , F2 ) ⊂ V(σ ) (F1 , F2 ) ⊂ V() (F1 , F2 ) induce isomorphisms Mσ (F1 , F2 ) 

V(σ ) (F1 , F2 ) (σ ) V> (F1 , F2 )



V() (F1 , F2 )  Sv F1 ⊗ Sv F2 , V( =σ ) (F1 , F2 )

and in particular V() (F1 , F2 ) = Mσ (F1 , F2 ) ⊕ V( =σ ) (F1 , F2 ). An easy induction argument leads then to the following (see also Theorem 3.6.8(d)). Corollary 11.5.3 We have a direct sum decomposition 

V() (F1 , F2 ) =

Mσ (F1 , F2 ).

σ ∈ sat

In particular, there is an isomorphism (Cauchy’s formula) Symd (F1 ⊗ F2 ) =



Sv F1 ⊗ Sv F2 .

v∈P(d)

11.6 Explicit Description of Ext Modules for Jl(σ ) The goal of this section is to give a concrete description of the groups Ext S (Jl(σ ) , S). We will use the notation in Theorem 10.6.1, where σ = (s1 , . . . , sc ), 0 ≤ l ≤ sc , and j

∗ , . . . , sn∗ ) ∈ Zn−l y = (sl+1 dom .

436

11 Cohomology and Regularity in Characteristic Zero

We assume that m ≤ n, and that m ≥ s1 (so that Jl(σ ) = 0), which implies that ym−l+1 = · · · = yn−l = 0. We will show that each of the groups Ext S (Jl(σ ) , S) decomposes into a direct sum of j

n Sa F1 ⊗ Sb F2 , with a ∈ Zm dom , b ∈ Zdom , |a| = |b|,

(11.21)

j

each term appearing with an appropriate multiplicity m a,b ≥ 0. The description of the summands and their multiplicities is somewhat involved, so a careful bookkeeping is necessary to simplify the formulas. First, a term in (11.21) will only contribute j to the degree |a| = |b| component of Ext S (Jl(σ ) , S), and this will be implicit in all the formulas from this section. Second, we introduce a formal variable q that keeps track of the cohomological grading, and interpret formal expressions 

Ext S (Jl(σ ) , S) · q j = j

j≥0

 j ⊕m a,b Sa F1 ⊗ Sb F2 · q j, j,a,b

to mean that for each j we have a decomposition Ext S (Jl(σ ) , S) = j

 j ⊕m a,b Sa F1 ⊗ Sb F2 . a,b

We define for z ∈ Zm dom and 0 ≤ r ≤ m, with z r ≥ r − m and z r +1 ≤ r − n, an associated dominant weight z(r ) ∈ Zndom via ⎛

⎞

z(r ) = ⎝z 1 , . . . , zr , r − m, . . . , r − m , zr +1 + (n − m), . . . , z m + (n − m)⎠    n−m

(11.22) For each sequence l ≥ t1 ≥ · · · ≥ tm−l ≥ s ≥ 0,

(11.23)

we write t = (t1 , . . . , tm−l ) and consider the set ⎧ ⎪ ⎨ Al(σ ) (t, s) = a ∈ Zm dom ⎪ ⎩

⎧ ⎫ ⎪ ⎪ ⎨am ≥ l − c − n, ⎬ : am+1− j−l+t j = t j − y j − n for j ∈ [m − l], ⎪ ⎪ ⎩ ⎭ as ≥ s − m, as+1 ≤ s − n. (11.24) Throughout this section (and when there is no danger of confusion) we write A(t, s) instead of Al(σ ) (t, s). We are now ready to formulate the explicit description of j Ext S (Jl(σ ) , S), which will be explained in detail in the rest of this section.

(σ )

11.6 Explicit Description of Ext Modules for Jl

437

Theorem 11.6.1 With the notation above, we have     2 j Sa F1 ⊗ Sa(s) F2 · q mn−l −s(n−m)−2|t| . Ext S (Jl(σ ) , S) · q j = j≥0

l≥t1 ≥···≥tm−l ≥s≥0 a∈A(t,s)

Ext Groups for the Determinantal Rings Before proving Theorem 11.6.1, we examine an important special case. Example 11.6.2 Suppose that σ = ∅, so that Jl(σ ) = S/Il+1 . We note that c = 0 and y j = 0 for all j, and we analyze the conditions (11.24). For j ∈ [m − l], we have that m − j − l ≥ 0, hence m + 1 − j − l + t j ≥ 1 + t j ≥ s + 1, which implies that if a ∈ A(t, s) then s − n ≥ as+1 ≥ am+1− j−l+t j = t j − n ≥ s − n. It follows that if A(t, s) is nonempty, then we must have t1 = · · · = tm−l = s. In addition, since t j ≤ l, we get m + 1 − j − l + t j ≤ m + 1 − j ≤ m for j ∈ [m − l], which implies that if a ∈ A(t, s) then l − n ≥ t j − n = am+1− j−l+t j ≥ am ≥ l − n, forcing t j = l for all j ∈ [m − l]. We conclude that A(t, s) is nonempty if and only if t = (l m−l ) and s = l, in which case A((l m−l ), l) = {a ∈ Zm dom : al ≥ l − m, al+1 = al+2 = · · · = am = l − n}. j

It follows from Theorem 11.6.1 that Ext S (S/Il+1 , S) = 0 unless j = mn − l 2 − l(n − m) − 2l(m − l) = (m − l)(n − l). Notice that this vanishing is equivalent to the fact that S/Il+1 is Cohen–Macaulay of codimension (m − l)(n − l), which was already established several times in this book (see Theorems 3.4.9, 4.4.5, and Sect. 10.8).

438

11 Cohomology and Regularity in Characteristic Zero

The minimal value of |a| for a ∈ A((l m−n ), l) is attained by a1 = · · · = al = l − m, al+1 = · · · = am = l − n, when we have |a| = l(l − m) + (m − l)(l − n) = −(m − l)n. It follows from Theorem 11.6.1 that   (S/I , S)

= 0 = −(m − l)n, min r : Ext (m−l)(n−l) l+1 r S which combined with (10.30) shows that reg S/Il+1 = −(m − l)(n − l) + (m − l)n = l(m − l). For an alternative derivation of the regularity formula, see Remark 4.4.13. Remark 11.6.3 In the situations when we apply Theorem 11.6.1, we will have by (10.5) that sc ≥ l + 1, which is equivalent to y1 = c. In this case, the set A(t, s) in (11.24) can only be nonempty when t1 = l, in which case every a ∈ A(t, s) satisfies am = l − c − n. Indeed, we have l − c − n ≥ t1 − c − n = t1 − y1 − n = am−l+t1 ≥ am ≥ l − c − n, so equality must hold throughout.

Proof of Theorem 11.6.1 Our strategy to compute Ext S (Jl(σ ) , S) is based on (10.26), Cauchy’s formula (Corollary 11.5.3), and the Borel–Weil–Bott theorem for Grassmannians (Theorem 11.3.1). We break up our calculations into several steps. j

Step 1. With the notation used in (10.26), we have     l  l  det(η) = det 1 Q  det 2 Q = S(l l ) 1 Q  S(l l ) 2 Q , and using the last part of Corollary 11.5.3, we get Sym(η) =

 w1 ≥···≥wl ≥0

Sw

1    Q  Sw 2 Q .

(σ )

11.6 Explicit Description of Ext Modules for Jl

439

It follows from Lemma 9.8.4 that 

det(η) ⊗ Sym(η) =

Sw

1    Q  Sw 2 Q .

(11.25)

w1 ≥···≥wl ≥l

Step 2. We have using Lemma 9.8.4 and Corollary 11.1.6 that  k c  ∨   det Q ⊗ Sw k Q = S(c−wl ,...,c−w1 ) k Q for k = 1, 2. Moreover, if we let x = (c − wl , . . . , c − w1 ) then we have the equivalence w1 ≥ · · · ≥ wl ≥ l If we define Skx,y = Sx

⇐⇒

c − l ≥ x 1 ≥ · · · ≥ xl .

k    Q ⊗ S y k R for k = 1, 2,

then it follows from (10.26) and (11.25) that Ext S (Jl(σ ) , S) = j



H mn−l

2

−j

 ∨ G, S1x,y  S2x,y ⊗ det(V )∨ .

(11.26)

c−l≥x1 ≥···≥xl

Step 3. We know by Theorem 11.3.1 that there exists at most one index lk for which   H lk G(l, Fk ), Skx,y = 0, k = 1, 2.

(11.27)

Moreover, we have that (11.27) holds for k = 1, 2 if and only if xi − i = y j − (l + j) for all i ∈ [l], j ∈ [n − l], and we recall that y j = 0 for j > m − l (note that the nonvanishing (11.27) for k = 1 implies that for k = 2). If (11.27) holds, then Künneth’s formula yields       H l1 +l2 G, S1x,y  S2x,y = H l1 G(l, F1 ), S1x,y ⊗ H l2 G(l, F2 ), S2x,y . (11.28) If we define t j as in Theorem 11.3.1, then since y j = 0 for j > m − l, it follows from Remark 11.3.4 that tm−l+1 = · · · = tn−l . We write  t = (t1 , . . . , tm−l ) and s =

tm−l+1 = · · · = tn−l if m < n, 0 if m = n.

and note that l1 , l2 are determined by the formulas l1 = |t| and l2 = |t| + s(n − m).

440

11 Cohomology and Regularity in Characteristic Zero

It follows from (11.26) that (11.28) contributes to Ext S (Jl(σ ) , S) only when j

mn − l 2 − j = 2|t| + s(n − m). We also consider 0 ≤ u 1 ≤ · · · ≤ u l ≤ n − l as in Theorem 11.3.1, and recall that {u 1 + 1, . . . , u l + l, (l + 1) − t1 , . . . , m − tm−l , (m + 1) − s, . . . , n − s} = [n]. (11.29) It follows that we must have (see also (11.12)) u l−s+1 = u l−s+2 = · · · = u l = n − l, and u l−s ≤ m − l,

(11.30)

which is trivially satisfied when m = n and s = 0. If we consider instead the sequence 0 ≤ u 1 ≤ · · · ≤ u l ≤ m − l with (u l , . . . , u 1 ) = (t1 , . . . , tm−l )∗ , then  u i

=

ui if i ≤ l − s, m − l otherwise.

We can then apply Theorem 11.3.1 to obtain   H lk G(l, Fk ), Skx,y = Sz k Fk for k = 1, 2, n 2 where z 1 ∈ Zm dom and z ∈ Zdom are given as follows:

1 z i+u

= x i + u i for i ∈ [l]

2 z i+u = xi + u i for i ∈ [l] i

1 z l+ j−t j = y j − t j for j ∈ [m − l]

2 z l+ j−t j = y j − t j for j ∈ [n − l]

i

(11.31)

In fact, z 1 and z 2 are closely related, as we show next: if we append (n − m) zeros to z 1 then we get a weight (z 1 , 0n−m ) ∈ Zn which is in the same Sn -orbit as z 2 under the . -action. Since z 2 is dominant, this shows that z 1 determines z 2 . More concretely, we have the following. Lemma 11.6.4 With the notation (11.22), we have z 2 = z 1 (m − s), that is, ⎛

⎞

1 1 , −s, −s, . . . , −s , z m−s+1 + (n − m), . . . , z m1 + (n − m)⎠ . z 2 = ⎝z 11 , . . . , z m−s    n−m

Proof The reader may verify the assertion using the . -action of Sn . We give here a direct argument based on the explicit formulas (11.31). Using the fact that y j = 0 and t j = s for m − l + 1 ≤ j ≤ n − l, it follows from (11.31) that

(σ )

11.6 Explicit Description of Ext Modules for Jl

441

2 z l+ j−s = −s for j = m − l + 1, . . . , n − l,

which gives the block of size n − m where each term is equal to −s in z 2 . 1 2 = z i+u by (11.31). Similarly, if j ∈ If i ∈ [l − s] then u i = u i and hence z i+u i i 1 2 [m − l] then z l+ j−t j = z l+ j−t j . Since i + u i ≤ (l − s) + (m − l) for i ∈ [l − s], l + j − t j ≤ m − s for j ∈ [m − l],

and (l − s) + (m − l) = m − s, it follows that {i + u i : i ∈ [l − s]} ∪ {l + j − t j : j ∈ [m − l]} = [m − s], hence zr1 = zr2 for r ∈ [m − s], which gives the first (m − s) entries of z 2 . Finally, if l − s + 1 ≤ i ≤ l then u i = m − l and u i = n − l, hence by (11.31) we have 1 2 = xi + (m − l), and z i+n−l = xi + (n − l), z i+m−l which shows that 2 1 = z m−s+1 + (n − m), . . . , z n2 = z m1 + (n − m), z n−s+1



and this gives the last s entries of z 2 , concluding our proof.

Step 4. Since the sequences x in (11.26) are allowed to vary, it is more convenient to regroup them according to the corresponding pair (t, s). For this reason, for (t, s) satisfying (11.23) we let u = (u 1 ≤ u 2 ≤ · · · ≤ u l ) such that (11.29) holds, and with the conventions x0 = y0 = ∞ and xl+1 = yn−l+1 = −∞, we define the set

X(t, s) =

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

x ∈ Zldom

⎧ ⎪ ⎪x1 ≤ c − l, ⎪ ⎨x ≤ y − (l + u ) + (i − 1) for i ∈ [l], i ui i : ⎪xi ≥ yu i +1 − (l + u i ) + i for i ∈ [l], ⎪ ⎪ ⎩ xl−s ≥ l − s − m, xl−s+1 ≤ l − s − n.

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

(11.32)

The first three conditions in (11.32) are transparent in light of Remark 11.3.3, but the last condition is perhaps more mysterious. As we will see shortly, it is trivially satisfied when m < n, and it is introduced in order to treat both the cases m < n and m = n uniformly. Since s ≤ tm−l , it follows from (11.12) that (11.30) also holds when s = 0 and m = n. Lemma 11.6.5 (a) If m < n then the last condition in (11.32) is superfluous. (b) If m = n then

442

11 Cohomology and Regularity in Characteristic Zero

⎧ ⎪ ⎨ ⎪ ⎩

x ∈ Zldom

⎧ ⎫ ⎪ ⎪ ⎨x1 ≤ c − l, ⎬ tm−l : xi ≤ yu i − (l + u i ) + (i − 1) for i ∈ [l], = X(t, s). ⎪ ⎪ ⎩ ⎭ s=0 for i ∈ [l]. xi ≥ yu i +1 − (l + u i ) + i

Proof (a) If we let i = l − s then we have by (11.30) that u i ≤ m − l, hence xl−s = xi ≥ yu i +1 − (l + u i ) + i ≥ 0 − (l + m − l) + (l − s) = l − s − m. If we take i = l − s + 1 then by (11.30) we have u i = n − l, and since yn−l = 0, we get xl−s+1 = xi ≤ yu i − (l + u i ) + (i − 1) = 0 − (l + n − l) + (l − s) = l − s − n. (b) The containment ⊃ follows from the definition of the sets X(t, s). To see that these sets are disjoint, we note that if s > s then the last conditions in (11.32) are incompatible. Indeed, if they hold for the same x ∈ Zldom then we have l − s − m ≥ xl−s+1 ≥ xl−s ≥ l − s − m, which is equivalent to s ≥ s, a contradiction. To prove ⊂, we have to verify that each weight x ∈ Zldom that satisfies the first three conditions in (11.32), also satisfies the last condition for some 0 ≤ s ≤ tm−l . Suppose by contradiction that this isn’t the case for some weight x. We prove by descending induction on s that xl−s ≥ l − s − m for tm−l ≥ s ≥ 0. Taking i = l − tm−l , and using the fact that yu i +1 ≥ 0 and u i ≤ m − l, we get that xl−tm−l = xi ≥ yu i +1 − (l + u i ) + i ≥ 0 − (l + m − l) + (l − tm−l ) = l − tm−l − m,

hence xl−s ≥ l − s − m holds for s = tm−l , verifying the base case of the induction. / X(t, s), we get xl−s+1 > l − Suppose now that s > 0, xl−s ≥ l − s − m. Since x ∈ s − m, or equivalently xl−(s−1) ≥ l − (s − 1) − m, proving the induction step. Taking now s = 0, we know that xl−s ≥ l − s − m, while xl−s+1 = −∞ ≤ l − s − m is trivially satisfied, which means that x ∈ X(t, s), a contradiction that concludes the proof of the inclusion ⊂.  It follows from Lemma 11.6.5 and Step 3 that if we write z 1 (x, y) ∈ Zm dom and z 2 (x, y) ∈ Zndom for the weights computed in (11.31) then we can rewrite (11.26) as Ext S (Jl(σ ) , S) = j





Sz 1 (x,y) F1 ⊗ Sz 2 (x,y) F2

∨

⊗ det(V )∨ .

l≥t1 ≥···≥tm−l ≥s≥0 2|t|+s(n−m)=mn−l 2 − j x∈X(t,s)

(11.33)

(σ )

11.6 Explicit Description of Ext Modules for Jl

443

Step 5. Since z 2 (x, y) is determined by z 1 (x, y) using Lemma 11.6.4, in order to better understand (11.33), we need to determine which weights z 1 (x, y) occur as we vary x inside X(t, s). We define

Z(t, s) =

⎧ ⎪ ⎨ ⎪ ⎩

z ∈ Zm dom

⎧ ⎪ ⎨z 1 ≤ c − l, : zl+ j−t j = y j − t j for j ∈ [m − l], ⎪ ⎩ z m−s ≥ −s, z m−s+1 ≤ −s + m − n.

⎫ ⎪ ⎬ ⎪ ⎭

(11.34)

and prove the following. Lemma 11.6.6 With the notation as before, we have a bijection X(t, s)  Z(t, s), given by x → z 1 (x, y). Proof We consider first x ∈ X(t, s), define z = z 1 (x, y) using (11.31), and show that it satisfies the conditions in (11.34). By definition, we have that zl+ j−t j = y j − t j for j ∈ [m − l]. If t1 = l then using the fact that c ≥ y1 , we obtain z 1 = y1 − t1 ≤ c − l. If t1 < l then u 1 = 0 by (11.12), hence u 1 = 0 and by (11.31) and (11.32) we have z 1 = x1 + u 1 = x1 ≤ c − l, proving the first condition in (11.34). For the last condition, note that since m − s + 1 > m − tm−l , we have m − s + 1 = i + u i for i = l − s + 1. As noticed earlier, this implies that u i = m − l, and therefore

z m−s+1 = xl−s+1 + u l−s+1

(11.32)

≤ (l − s − n) + (m − l) = −s + m − n, (11.35)

as desired. To show that z m−s ≥ −s we consider two cases. If s = tm−l then by taking j = m − l we get m − s = l + j − t j and therefore z m−s = zl+ j−t j = y j − t j = ym−l − tm−l ≥ 0 − s = −s.

= m − l and If s < tm−l then we have m − s = i + u i for i = l − s, hence u l−s

z m−s = xl−s + u l−s

(11.32)

≥ (l − s − m) + (m − l) = −s.

(11.36)

It follows that the assignment x → z 1 (x, y) gives a well-defined map X(t, s) → Z(t, s). By (11.31), this map is injective since we can recover x via xi = z i+u i − u i for i ∈ [l].

(11.37)

444

11 Cohomology and Regularity in Characteristic Zero

To conclude, we need to prove the surjectivity of X(t, s) → Z(t, s), or equivalently, that for every z ∈ Z(t, s), if we define x via (11.37) then x ∈ X(t, s). We have x1 = z 1+u 1 − u 1 ≤ z 1 ≤ c − l, proving the first condition in (11.32). For the last conditions, we rewrite (11.35) as

≤ −s + m − n − (m − l) = l − s − n. xl−s+1 = z m−s+1 − u l−s+1

≤ m − l we get (l − s) + u l−s ≤ m − s, hence Moreover, since u l−s

− u l−s ≥ z m−s − (m − l) ≥ −s − (m − l) = l − s − m. xl−s = z (l−s)+ul−s

To prove the middle conditions in (11.32) we consider two cases. If i ≤ l − s then we have u i = u i ≤ m − l and by (11.31) we get yu i = zl+u i −tui + tu i . Moreover, using (11.12) we have tu i > l − i, hence i + u i > l + u i − tu i , and since z is dominant, this yields z i+u i − (i + u i ) < zl+u i −tui − (l + u i − tu i ), which in turn implies that xi = z i+u − u i = z i+u i − u i ≤ zl+u i −tu + tu i − (l + u i ) + (i − 1) = yu i − (l + u i ) + (i − 1), i i

proving the second inequality in (11.32). Similarly, (11.12) shows that tu i +1 ≤ l − i, hence i + u i < l + u i + 1 − tu i +1 . When u i + 1 ≤ m − l, this implies as before that xi − i = z i+u i − (i + u i ) > zl+u i +1−tui +1 − (l + u i + 1 − tu i +1 ) = yu i +1 − (l + u i ) − 1,

which is equivalent to third inequality xi ≥ yu i +1 − (l + u i ) + i in (11.32). If instead u i = m − l, then i + u i ≤ (l − s) + (m − l) = m − s, and therefore xi − i = z i+u i − (i + u i ) ≥ z m−s − (m − s) ≥ −s − (m − s) = −m = yu i +1 − (l + u i ),

where the last equality follows since ym−l+1 = 0. We get again xi ≥ yu i +1 − (l + u i ) + i, concluding the analysis of the case i ≤ l − s. Consider now the case when i ≥ l − s + 1, so u i = m − l and u i = n − l. Since yu i +1 = yn−l+1 = −∞, the third inequality in (11.32) is vacuous, so we only need to verify the second one. Since m − l + i ≥ m − l + (l − s + 1) = m − s + 1, it follows that

11.7 Ext Modules for Symbolic Powers

445

xi − i = z m−l+i − (m − l + i) ≤ z m−s+1 − (m − s + 1) ≤ −s + m − n − (m − s + 1) = −n − 1.

Since yu i = yn−l ≥ 0 and l + u i = n, it follows from the above inequality that xi ≤ yu i − (l + u i ) + (i − 1), 

concluding our proof. Step 6. Using Lemma 9.8.4 and Corollary 11.1.6, we have that ∨  Sz F1 ⊗ det(F1 )n = Sa F1 , where ai = −z m+1−i − n for i ∈ [m].

(11.38)

Similarly (assuming that z m−s ≥ −s and z m−s+1 ≤ −s + m − n), we have ∨  Sz(m−s) F2 ⊗ det(F2 )m = Sb F1 , where ⎧ ⎪ ⎨−(z m+1−i + (n − m)) − m = ai for 1 ≤ i ≤ s, bi = s − m for s + 1 ≤ i ≤ s + n − m, ⎪ ⎩ for s + n − m + 1 ≤ i ≤ n. −z m+1−i − m = ai + (n − m) Comparing with (11.22), this is equivalent to b = a(s). Using the isomorphism det(V )  det(F1 )n ⊗ det(F2 )m , it follows that ∨  Sz F1 ⊗ Sz(m−s) F2 ⊗ det(V )∨ = Sa F1 ⊗ Sa(s) F2 ,

(11.39)

where a is as in (11.38). Notice that the formula (11.38) transforms the set Z(t, s) into the set A(t, s) defined in (11.24). It follows from Lemma 11.6.6 that if we let x vary within X(t, s) then z 1 (x, y) = z varies within Z(t, s), and z 2 (x, y) = z(m − s) by Lemma 11.6.4. Using (11.39), we can therefore rewrite (11.33) as Ext S (Jl(σ ) , S) = j



Sa F1 ⊗ Sa(s) F2 ,

l≥t1 ≥···≥tm−l ≥s≥0 2|t|+s(n−m)=mn−l 2 − j a∈A(t,s)

which concludes the proof of our theorem.

11.7 Ext Modules for Symbolic Powers The goal of this section is to explain how Theorem 11.6.1 can be used to give a j (d) , S) for all d ≥ 0. The main ingredients will be concrete description of Ext S (S/Il+1 Theorem 10.2.2, and basic facts about the representation theory of the group GL = (d) , and GL(F1 ) × GL(F2 )  GLm (K ) × GLn (K ). Since all the symbolic powers Il+1

446

11 Cohomology and Regularity in Characteristic Zero

more generally ideals of the form I () are invariant under the GL-action, it follows j j (d) , S) and Ext S (Jl(σ ) , S) is naturally a GL-module. that each of the groups Ext S (S/Il+1 We will prove the following. Theorem 11.7.1 With the notation in Theorem 10.4.1, for every j ≥ 0, d ≥ 1, we have an isomorphism of GL-modules (preserving the natural grading on the Ext groups)  j j (d) , S)  Ext S (Jl(σ ) , S). Ext S (S/Il+1 (d) (σ,l)∈Zl+1

We can make this result more explicit using Theorem 11.6.1. For (t, s) satisfying (11.23), we define ⎫ ⎧ ⎧ m−l ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎬ ⎨ ⎨ (t − a j m+1− j−l+t j − n) < d, (d) m (11.40) A(t, s) = a ∈ Zdom : j=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ ⎩a ≥ s − m, a ≤ s − n. s

s+1

Corollary 11.7.2 With the notation above, we have (note that l = t1 below) 

(d) Ext S (S/Il+1 , S) · q j = j



  2 Sa F1 ⊗ Sa(s) F2 · q mn−l −s(n−m)−2|t| .

l=t1 ≥t2 ≥···≥tm−l ≥s≥0 a∈(d) A(t,s)

j≥0

(d) , we have sc ≥ l + 1, so by Remark 11.6.3, the set Proof For every (σ, l) ∈ Zl+1 Al(σ ) (t, s) is empty unless l = t1 . It follows from Theorems 11.7.1 and 11.6.1 that in order to prove our formula, it suffices to check that for l = t1 ≥ · · · ≥ tm−l ≥ s, we have (d) A(t, s) = Al(σ ) (t, s). (d) (σ,l)∈Zl+1

(d) Suppose first that a ∈ Al(σ ) (t, s) for (σ, l) ∈ Zl+1 , so as ≥ s − m, as+1 ≤ s − n, and (11.41) t j − am+1− j−l+t j − n = y j for j ∈ [m − l]. (d) It follows from the definition of the set Zl+1 that m−l c  (10.25)  (t j − am+1− j−l+t j − n) = |y| = (si − l) = γl+1 (σ ) < d, j=1

i=1

(d) are disjoint, since so a ∈ (d) A(t, s). Moreover, the sets Al(σ ) (t, s) with (σ, l) ∈ Zl+1 a determines y via (11.41), and

11.7 Ext Modules for Symbolic Powers

σ = (cl , y1 , y2 , . . . , ym−l )∗ , where c = y1 .

447

(11.42)

Suppose now that a ∈ (d) A(t, s), define y j via (11.41), and c = y1 = l − am − n. We first check that y is dominant: since the sequence (t j ) is weakly decreasing, (m + 1 − j − l + t j ) is decreasing, hence (am+1− j−l+t j ) is weakly increasing, and hence y is weakly decreasing, as desired. Moreover, tm−l ≥ s implies m + 1 − (m − l) − l + tm−l ≥ 1 + s, hence ym−l ≥ tm−l − as+1 − n ≥ s − (s − n) − n = 0, (d) so y is a partition. If we define σ via (11.42), then (σ, l) ∈ Zl+1 and a ∈ Al(σ ) (t, s), concluding our proof. 

From now on, we will use freely the fact that GL(Fk ) is reductive, and that Sv Fk are irreducible, pairwise nonisomorphic, GL(Fk )-modules (see Remark 3.6.7, where Sv F1 was denoted L σ for σ = v ∗ ). In particular, if we write HomGL for the functor of GL-module homomorphisms, then   HomGL Sa F1 ⊗ Sb F2 , Sa F1 ⊗ Sb F2 = 0 if a = a or b = b .

(11.43)

Combining (11.43) with Theorem 11.6.1, we obtain the following. (d) Lemma 11.7.3 For distinct pairs (σ, l) and (σ , l) in Zl+1 , we have

 

j j+1 HomGL Ext S (Jl(σ ) , S), Ext S (Jl(σ ) , S) = 0. Proof Suppose by contradiction that this isn’t the case. By (11.43) and Theorem 11.6.1, there exist s ≥ 0 and a ∈ Zm dom , such that Sa F1 ⊗ Sa(s) F2 is a summand j j+1 (σ ) (σ ) in both Ext S (Jl , S) and Ext S (Jl , S). We note that s is uniquely determined by a: if a(s) = a(s ) ∈ Zndom for s < s , then we have s − n ≥ as+1 ≥ as ≥ s − m, which implies s ≥ s + (n − m) ≥ s , a contradiction. Using Theorem 11.6.1, we can find t, t ∈ Zm−l such that ⎧ (σ ) (σ ) ⎪ ⎨a ∈ Al (t, s) ∩ Al (t , s), j = mn − l 2 − s(n − m) − 2|t|, ⎪ ⎩ j + 1 = mn − l 2 − s(n − m) − 2|t |. This forces 1 = ( j + 1) − j = 2(|t| − |t |), which is impossible.



(d) | and consider the filtration conProof (Proof of Theorem 11.7.1) We let r = |Zl+1 (d) (d) lexicostructed in Theorem 10.2.2 for I () = Il+1 . If we order the shapes in Zl+1 graphically,

448

11 Cohomology and Regularity in Characteristic Zero

σ 0

is 1-dimensional, spanned by the determinant in the dual variables. A degree 2 polynomial f (xi, j ) is in (∂1,1 ∂2,2 − ∂1,2 ∂2,1 )⊥ if and only if the coefficients of x1,1 x2,2 and x1,2 x2,1 agree, hence 2 2 2 2 , x1,2 , x2,1 , x2,2 , x1,1 x1,2 , M(1,1) = K · {x1,1

x1,1 x2,1 ,x1,2 x2,2 , x2,1 x2,2 , x1,1 x2,2 + x1,2 x2,1 }. Note that all but the last generator of M(1,1) is represented by a standard bitableau, whereas x1,1 x2,2 + x1,2 x2,1 = 2[1|1][2|2] − [1, 2|1, 2], and in fact M(1,1) does not admit a basis of standard bitableaux (see Theorem 3.6.11). If we define a generalized 2 × 2 subpermanent of (xi, j ) to be the permanent of the 2 × 2 matrix obtained by selecting 2 rows and 2 columns (not necessarily distinct) from the matrix (xi, j ), then it is easy to see that the generators of M(1,1) are, up to scaling, the 2 × 2 subpermanents of (xi, j ). For instance, selecting rows {1, 1} and columns {1, 2}, we get " x1,1 x1,2 = 2 · x1,1 x1,2 . perm x1,1 x1,2 !

In the same way, one can interpret I(1 p ) as the ideal of p × p generalized subpermanents of (xi, j ), which is a close analogue of the ideal I p of p × p minors (but is nonzero even when p  0). We note that if we instead consider the ideal of p × p

11.8 The Basics on GL-Invariant Ideals

453

permanents (without repeating rows and columns), then we do not in general get a GL-invariant ideal. Using Corollary 11.8.3, it is easy to find the intersection of GL-invariant ideals I ∩ J : it is given as the sum of Mτ where τ is a shape in both I and J . In particular, Iσ ∩ Iτ = Isup(σ,τ ) , where sup(σ, τ ) = (max(s1 , t1 ), max(s2 , t2 ), . . . ).

(11.50)

Note that the log-canonical threshold of GL-invariant ideals has been determined in Remark 7.5.7.

Quotients of GL-Invariant Ideals We next consider a generalization of the modules Jl(σ ) to quotients of GL-invariant ideals that are not necessarily defined by shape. To introduce them, we begin with the following analogue of Definition 10.1.1. Definition 11.8.6 Suppose that σ = (s1 , . . . , su ) is a partition, and that l ≥ 0. We define the set of l-successors of σ with respect to ⊂ as ⎧ ⎨

⎫ there is 1 ≤ i ≤ u with ti > si ⎬ or succ⊂ (σ, l) = τ = (t1 , . . . , tv ) : τ ⊃ σ, and . ⎩ ⎭ tu+1 ≥ l + 1 For this reason, to emphasize the distinction from Definition 10.1.1, we may write succ≤ (σ, l) for the set succ(σ, l). Since ≤ refines the order ⊂, it follows that succ⊂ (σ, l) ⊂ succ≤ (σ, l). We can now consider the following analogue of Definition 10.1.2. Definition 11.8.7 For any partition σ and integer l ≥ 0, we let Jσ,l =

Iσ . Isucc⊂ (σ,l)

Lemma 11.8.8 Suppose that σ = (s1 , . . . , sc ), and 0 ≤ l ≤ sc . The inclusion of Iσ into I (σ ) induces an isomorphism of S-modules Jσ,l  Jl(σ ) . Moreover, Jσ,l and Jl(σ ) are isomorphic as GL-modules to  τ ∈T (σ,l)

Mτ , where T (σ, l) = {τ = (t1 , t2 , . . . ) : ti = si for i ≤ c and tc+1 ≤ l}.

454

11 Cohomology and Regularity in Characteristic Zero

Proof The description of Jσ,l and Jl(σ ) as GL-modules is equivalent to T (σ, l) = {τ : τ ⊃ σ } \ succ⊂ (σ, l) = {τ : τ ≥ σ } \ succ≤ (σ, l) which follows from the definitions of the sets involved. Let f : Iσ −→ Jl(σ ) denote the map induced by the inclusion Iσ ⊂ I (σ ) followed by the projection I (σ )  Jl(σ ) . ≤ To prove that f is surjective, we have to show that Iσ + I (succ (σ,l)) = I (σ ) , or equivalently, that {τ : τ ⊃ σ } ∪ succ≤ (σ, l) = {τ : τ ≥ σ }. The inclusion ⊂ follows by definition, whereas ⊃ follows since {τ : τ ⊃ σ } contains the complement T (σ, l) of succ≤ (σ, l) in {τ : τ ≥ σ }. Using the isomorphism Im( f )  Iσ / Ker( f ), we are left with verifying that Ker( f ) = Isucc⊂ (σ,l) . Equivalently, {τ : τ ⊃ σ } ∩ succ≤ (σ, l) = succ⊂ (σ, l), which is immediate from the definitions.



Generalization of Z(I) to Arbitrary GL-Invariant Ideals We define Z(I ) for every GL-invariant ideal I ⊂ S, by replacing I () with I in Definition 10.2.1. Notice that the notation Z() may now be ambiguous: for this reason, we set Z≤ () = Z(I () ) and Z⊂ () = Z(I ). The algorithm to compute Z≤ () in Sect. 10.3 applies to compute Z⊂ () if we replace I () with I everywhere, with the first step of computing  0 being in fact easier: the set  0 of minimal shapes in I with respect to ⊂ is simply  ◦ . We summarize the steps in Algorithm 11.1. Algorithm 11.1: Algorithm to compute Z⊂ () Input: A finite subset  of Pm Output: The set Z⊂ () = Z(I ) 1 Z←∅ 2 shape(∅,  ◦ ) 3 Z⊂ () ← Z

To illustrate this, we consider the following analogue of Example 10.3.2. Example 11.8.9 We let m = 3 and  = {(3, 1), (2, 2, 1, 1)} as in Example 10.3.2, and note that  ◦ = . Table 11.1 keeps track of how the recursive calls of the function shape modify the approximation Z of Z⊂ ().

11.9 Ext Modules for GL-Invariant Ideals

455

Table 11.1 Results of iterations of the function shape for Example 11.8.9 σ



c

∅

(3,1),(2,2,1,1)

0

1

Z (∅, 1)

(2)

(2,2,1,1)

1

1

(∅, 1), ((2), 1)

(2,2)

(2,2,1,1)

2

0

(∅, 1), ((2), 1), ((2, 2), 0)

(2,2,1)

(2,2,1,1)

3

0

(∅, 1), ((2), 1), ((2, 2), 0), ((2, 2, 1), 0)

(2,2,1,1)

(2,2,1,1)

4

−1

(∅, 1), ((2), 1), ((2, 2), 0), ((2, 2, 1), 0)

(2,2,2)

(2,2,1,1)

3

0

(∅, 1), ((2), 1), ((2, 2), 0), ((2, 2, 1), 0), ((2, 2, 2), 0)

(2,2,2,1)

(2,2,1,1)

4

−1

(∅, 1), ((2), 1), ((2, 2), 0), ((2, 2, 1), 0), ((2, 2, 2), 0)

(2,2,2,2)

(2,2,1,1)

4

−1

(∅, 1), ((2), 1), ((2, 2), 0), ((2, 2, 1), 0), ((2, 2, 2), 0)

(3)

(3,1),(2,2,1,1)

1

0

(∅, 1), ((2), 1), ((2, 2), 0), ((2, 2, 1), 0), ((2, 2, 2), 0), ((3), 0)

(3,1)

(3,1)

2

−1

(∅, 1), ((2), 1), ((2, 2), 0), ((2, 2, 1), 0), ((2, 2, 2), 0), ((3), 0)

(3,2)

(3,1),(2,2,1,1)

2

−1

(∅, 1), ((2), 1), ((2, 2), 0), ((2, 2, 1), 0), ((2, 2, 2), 0), ((3), 0)

(3,3)

(3,1),(2,2,1,1)

2

−1

(∅, 1), ((2), 1), ((2, 2), 0), ((2, 2, 1), 0), ((2, 2, 2), 0), ((3), 0)

l

In conclusion, we have that if m = 3 and  = {(3, 1), (2, 2, 1, 1)} then Z⊂ () = Z(I ) = {(∅, 1), ((2), 1), ((2, 2), 0), ((2, 2, 1), 0), ((2, 2, 2), 0), ((3), 0)}.

Notice that Z⊂ () \ Z≤ () = {((2, 2, 2), 0)}, I ⊂ I () , and I () /I  J(2,2,2),0 . We can therefore extend the filtration (10.8) to get a filtration I ⊂ · · · ⊂ S with composition factors Jσ,l , for (σ, l) ∈ Z⊂ (). This suggests that Theorem 10.2.2 has an analogue for ideals of the form I , which we will show to be the case shortly. The combinatorial results for Z() = Z≤ () from Sect. 10.2 apply essentially without change to Z⊂ (), and we will use them freely in the next section (this is in fact the context in which the theory was first developed in [262]).

11.9 Ext Modules for GL-Invariant Ideals The goal of this section is to provide generalizations of Theorems 10.2.2 and 11.7.1 to arbitrary GL-invariant ideals. More precisely, we show that S/I has a filtration with j composition factors Jσ,l for (σ, l) ∈ Z(I ), and describe all the groups Ext S (S/I, S) and the natural maps between them. All the ideals and the Ext modules considered have a natural Z-grading, and the isomorphisms that we construct preserve the grading. To simplify the discussion, we will avoid mentioning this grading in what follows–it suffices to remember that a Schur functor Sv will always appear in degree |v|; in fact, this grading can be described more conceptually as in Lemma 9.7.9, in terms of the eigenspace decomposition relative to the action of the center of either GL(F1 ) or GL(F2 ), which is isomorphic to the multiplicative group Gm .

456

11 Cohomology and Regularity in Characteristic Zero

Theorem 11.9.1 For I ⊂ S a GL-invariant ideal and (σ, l) ∈ Z(I ), there exist GLsubmodules j j Eσ,l (I ) ⊂ Ext S (S/I, S), with the following properties. j

j

(a) Eσ,l (I )  Ext S (Jσ,l , S) as GL-modules, for all (σ, l) ∈ Z(I ). (b) We have a direct sum decomposition 

j

Ext S (S/I, S) =

j

Eσ,l (I ).

(σ,l)∈Z(I )

(c) For every inclusion I ⊃ J of GL-invariant ideals, if we consider the induced map j j f : Ext S (S/I, S) −→ Ext S (S/J, S), j

j

then f maps isomorphically Eσ,l (I ) onto Eσ,l (J ) if (σ, l) ∈ Z(I ) ∩ Z(J ), and j otherwise it sends Eσ,l (I ) to 0. (d) For every inclusion I ⊃ J as in (c), if Z(I ) ⊂ Z(J ), then the induced map j j Ext S (S/I, S) −→ Ext S (S/J, S) is injective for all j. (e) For every inclusion I ⊃ J as in (c), we have an isomorphism j

Ext S (I /J, S) ⎛  ⎝

⎞ j Ext S (Jσ,l ,

S)⎠



⎛



⎝

(σ,l)∈Z(J )\Z(I )

⎞ j+1 Ext S (Jσ,l ,

S)⎠ .

(σ,l)∈Z(I )\Z(J )

Using the isomorphism Jσ,l  Jl(σ ) in Lemma 11.8.8, along with Theorem 11.6.1, we can derive based on Theorem 11.9.1 an explicit description of all the groups j Ext S (I /J, S) as well as the natural maps between them. Part (e) of Theorem 11.9.1 follows from the long exact sequence for Ext associated with the short exact sequence 0 −→ I /J −→ S/J −→ S/I −→ 0, and from part (c), which implies that Ker( f ) =



j

Ext S (Jσ,l , S),

(σ,l)∈Z(I )\Z(J )

Coker( f ) =



j

Ext S (Jσ,l , S).

(σ,l)∈Z(J )\Z(I )

The above description of Ker( f ) shows that f is injective when Z(I ) ⊂ Z(J ), j proving part (d). Therefore, our focus in this section will be on constructing Eσ,l (I ), and establishing properties (a), (b), and (c). We also note here that in light of (10.5), we

11.9 Ext Modules for GL-Invariant Ideals

457

will only be concerned with modules Jσ,l where σ = (s1 , . . . , sc ) satisfies sc ≥ l + 1. We break up our analysis into several steps. Step 1. Suppose that σ = (s1 , . . . , sc ) and sc ≥ l + 1. Our first goal is to understand the GL-invariant ideals I for which the composition of natural maps Iσ → S  S/I induces an injection Jσ,l → S/I . This condition is equivalent to the fact that I ∩ Iσ = Isucc⊂ (σ,l) .

(11.51)

It is clear that I = Isucc⊂ (σ,l) is the unique minimal GL-invariant ideal satisfying (11.51), and we claim that there is also a unique maximal one. We consider the set of rectangular shapes σ,l = {((l + 1)c+1 )} ∪ {((si + 1)i ) : 1 ≤ i ≤ c}.

(11.52)

For instance, when σ = (5, 4, 4, 3) and l = 2, we have c = 4 and σ,l = {(35 ), (6), (52 ), (53 ), (44 )}.

Lemma 11.9.2 A GL-invariant ideal I satisfies (11.51) if and only if Isucc⊂ (σ,l) ⊂ I ⊂ Iσ,l . Proof We firsts check that Iσ,l ⊃ Isucc⊂ (σ,l) , by showing that every τ = (t1 , t2 , . . . ) in succ⊂ (σ, l) is a shape in Iσ,l . If ti > si then τ ⊃ ((si + 1)i ), while if tc+1 ≥ l + 1 then τ ⊃ ((l + 1)c+1 ), so τ is a shape in Iσ,l , as desired. Using the inclusions Isucc⊂ (σ,l) = Isucc⊂ (σ,l) ∩ Iσ ⊂ I ∩ Iσ ⊂ Iσ,l ∩ Iσ , in order to check the if implication, it suffices that  Iσ,l ∩ Iσ ⊂ Isucc⊂ (σ,l) ,  to verify i + 1) ), σ then τ ⊃ σ and ti > si , which we do based on (11.50). If τ = sup ((s i   so τ ∈ succ⊂ (σ, l). If τ = sup ((l + 1)c+1 ), σ then τ ⊃ σ and tc+1 ≥ l + 1, so τ ∈ succ⊂ (σ, l), proving the desired inclusion. For the only if implication, we have already seen that I ⊃ Isucc⊂ (σ,l) , so it remains to check that I ⊂ Iσ,l . If τ is a shape in I then (11.51) and (11.50) imply that sup(τ, σ ) ∈ succ⊂ (σ, l). If max(ti , si ) > si then ti > si and therefore τ ⊃ ((si + 1)i ), while if max(tc+1 , sc+1 ) ≥ l + 1 then tc+1 ≥ l + 1 since sc+1 = 0, hence τ ⊃ ((l + 1)c+1 ). We conclude that τ is a shape in Iσ,l , so I ⊂ Iσ,l , as desired.  We will also need the following property of Iσ,l .

458

11 Cohomology and Regularity in Characteristic Zero

Lemma 11.9.3 If (σ, l) ∈ Z(I ), then I ⊂ Iσ,l . Proof We have to show that every shape λ = ( p1 , p2 , . . . ) in I is also in Iσ,l . If pi > si for some i ≤ c, then λ ⊃ ((si + 1)i ), hence λ is in Iσ,l , as desired. Otherwise, we have pi ≤ si for i ≤ c, hence σ ⊃ λ(c). Since (σ, l) ∈ Z(I ), it follows from Definition 10.2.1 that pc+1 ≥ l + 1, hence λ ⊃ ((l + 1)c+1 ), showing again that λ is  in Iσ,l and concluding our proof. Step 2. Our next goal is to generalize Theorem 10.2.2 to arbitrary GL-invariant ideals: Theorem 11.9.4 For every GL-invariant ideal I ⊂ S, there exists a filtration of S/I with composition factors given by the modules Jσ,l , for (σ, l) ∈ Z(I ). The proof strategy is to find an appropriate inclusion Jσ,l → S/I as in Step 1, to consider the quotient S S/I = , Jσ,l I + Iσ and to proceed by induction. The key part of the argument is the following. Proposition 11.9.5 If I satisfies (11.51) and (σ, l) ∈ Z(I ) then Z(I + Iσ ) = Z(I ) \ {(σ, l)}. / Z(I + Proof Since σ is a shape in I + Iσ , we know by Lemma 10.2.6 that (σ, l) ∈ Iσ ). Suppose by contradiction that (τ, u) ∈ Z(I + Iσ ) \ Z(I ), with τ = (t1 , . . . , td ). Since I + Iσ has more shapes than I , condition (b) in Definition 10.2.1 for Z(I + Iσ ) implies the corresponding condition for Z(I ). Since (τ, u) ∈ Z(I + Iσ ) \ Z(I ), it follows that (τ, u) fails condition (a) for Z(I ): there is no shape λ = ( p1 , p2 , . . . ) in I with τ ⊃ λ(d) and pd+1 ≤ u + 1. (11.53) Let λ be a shape in I + Iσ satisfying τ ⊃ λ(d) and pd+1 = u + 1. Since λ is not in I , we have λ ⊃ σ , so τ ⊃ σ (d) and sd+1 = u + 1. Since si = 0 for i > c, we get d < c, and since sd+1 ≥ sc ≥ l + 1, we get u ≥ l. Since (σ, l) ∈ Z(I ), we can find a shape μ = (m 1 , m 2 , . . . ) in I with σ ⊃ μ(c). This implies d c. If we take λ = ( p1 , p2 , . . . ) a shape in I with τ ⊃ λ(d), then pi ≤ si for i ≤ c and pc+1 ≤ tc+1 ≤ l, so λ is not a shape in Iσ,l , contradicting the  inclusion I ⊂ Iσ,l from Lemma 11.9.2. Example 11.9.6 We let I = I , where  = {(3, 1), (2, 2, 1, 1)} Example 11.8.9. If we take σ = (2, 2, 2) and l = 0, then we have

as

in

σ,l = {(14 ), (3), (32 ), (33 )}, so Iσ,l = I(14 ) + I(3) . Moreover, Isucc⊂ (σ,l) = I(3,2,2) + I(2,2,2,1) , hence Isucc⊂ (σ,l) ⊂ I ⊂ Iσ,l . By Lemma 11.9.2, we have an inclusion Jσ,l → S/I , and by Proposition 11.9.5 we have Z(I + Iσ ) = Z(I ) \ {(σ, l)}. Since I + Iσ = I () and I () /I = Jσ,l , these facts were already established in Examples 11.8.9 and 10.3.2. We could also choose σ = (3) and l = 0, so m=3

Iσ,l = I(4) + I(1,1) = I(1,1) and Isucc⊂ (σ,l) = I(3,1) . Since Isucc⊂ (σ,l) ⊂ I ⊂ Iσ,l , we get again Jσ,l → S/I with Z(I + Iσ ) = Z(I ) \ {(σ, l)}. The reader can check that no other choice (σ, l) ∈ Z(I ) gives an inclusion Jσ,l → S/I . For instance, if σ = (2) and l = 1, then we have Iσ,l = I(3) + I(2,2) = Isucc⊂ (σ,l) , hence the inclusion Isucc⊂ (σ,l) ⊂ I in Lemma 11.9.2 fails. In fact, the only GLinvariant ideal I with J(2),1 → S/I is I = I(3) + I(2,2) . Proof (Proof of Theorem 11.9.4) We do induction on |Z(I )|. As in the proof of Theorem 10.2.2, if |Z(I )| = 0 then I = S, so S/I = 0 and we have nothing to prove. Suppose now that |Z(I )| > 0, and choose (σ, l) ∈ Z(I ) with σ lexicographically maximal. We prove that Jσ,l → S/I using Lemma 11.9.2 (see also the proof of Proposition 10.2.11). If τ ∈ succ⊂ (σ, l) then τ >lex σ , hence τ is not a shape in Z(I ).

460

11 Cohomology and Regularity in Characteristic Zero

It follows from Lemma 10.2.8 that τ is a shape in I , hence Isucc⊂ (σ,l) ⊂ I . Suppose by contradiction that λ = ( p1 , p2 , . . . ) is a shape in I that is not in Iσ,l . It follows that pi ≤ si for i ≤ c, or equivalently σ ⊃ λ(c), and pc+1 ≤ l. This contradicts condition (b) of Definition 10.2.1, hence no such λ exists and I ⊂ Iσ,l . From Lemma 11.9.2 we have an exact sequence 0 −→ Jσ,l −→

S S −→ −→ 0, I I + Iσ

(11.56)

and by Proposition 11.9.5 we have Z(I + Iσ ) = Z(I ) \ {(σ, l)}. By induction, there is a filtration of S/(I + Iσ ) with composition factors Jτ,u with (τ, u) ∈ Z(I ) \ {(σ, l)}, which together with the inclusion Jσ,l → S/I gives the desired filtration on S/I .  j

j

Step 3. Our next step is to construct the GL-modules Eσ,l (I ) ⊂ Ext S (S/I, S). We begin with some basic observations on GL-modules. Since GL is reductive, if 0 −→ A −→ B −→ C −→ 0 is a short exact sequence of GL-modules, then B  A ⊕ C. If we only know that β

α

A −→ B −→ C is exact, then 0 → Im(α) → B → Im(β) → 0 is a short exact sequence, hence we get as above that B  Im(α) ⊕ Im(β). Splitting the surjection A  Im(α), we get that B is (noncanonically) isomorphic to a submodule of A ⊕ C. If we apply this reasoning to the exact sequence j

j

j

Ext S (S/(I + Iσ ), S) −→ Ext S (S/I, S) −→ Ext S (Jσ,l , S) obtained from (11.56), then we see that there exists a GL-equivariant inclusion j

j

j

Ext S (S/I, S) → Ext S (S/(I + Iσ ), S) ⊕ Ext S (Jσ,l , S). It follows then by induction that there exists a GL-equivariant inclusion j

Ext S (S/I, S) →



j

Ext S (Jσ,l , S).

(11.57)

(σ,l)∈Z(I )

The content of Theorem 11.9.1 is that (11.57) is in fact an isomorphism, that can be chosen compatibly for all GL-invariant ideals I . As we will see shortly, it is essential the existence of the unique GL-invariant ideal Iσ,l , which is maximal among those I with the property (σ, l) ∈ Z(I ) (see Lemma 11.9.3 and the next result). Lemma 11.9.7 One has Z(Iσ,l ) \ {(σ, l)}, then

that

(σ, l) ∈ Z(Iσ,l ).

Moreover,

if

(τ, u) ∈

11.9 Ext Modules for GL-Invariant Ideals

461

  j j+1 HomGL Ext S (Jσ,l , S), Ext S (Jτ,u , S) = 0. Proof If we let λ = ( p1 , p2 , . . . ) = ((l + 1)c+1 ) ∈ σ,l , then since sc ≥ l + 1, we have σ ⊃ λ(c). Moreover, λ is unique with this property among the minimal weights of Iσ,l . It follows that (σ, pc+1 − 1) = (σ, l) is in Z(Iσ,l ), as desired. Consider now any (τ, u) = (σ, l) in Z(Iσ,l ), and write τ = (t1 , . . . , td ). We note that since every shape in σ,l has only parts of size ≥ l + 1, it follows from Lemma 10.2.5 that u ≥ l. We consider λ = ( p1 , p2 , . . . ) a minimal shape in Iσ,l with τ ⊃ λ(d), so that λ ∈ σ,l and pd+1 = u + 1 > 0. Since every shape in σ,l has at most c + 1 parts, this shows that d ≤ c. We conclude that l − c ≤ u − d,

with equality if and only if l = u and c = d.

(11.58)

To prove that last assertion, we suppose by contradiction that there exists a sumj j+1 mand Sa F1 ⊗ Sa(s) F2 in both Ext S (Jσ,l , S) and Ext S (Jτ,u , S). It follows from Remark 11.6.3 that l − c − n = am = u − d − n, ) hence by (11.58) we have l = u. If t and t are such that a ∈ Al(σ ) (t, s) ∩ A(τ u (t , s), then it follows from Theorem 11.6.1 that

j = mn − l 2 − s(n − m) − 2|t| and j + 1 = mn − u 2 − s(n − m) − 2|t |, and since l = u, we get that the difference j − ( j + 1) is even, a contradiction.  j

j

Corollary 11.9.8 There exists a GL-submodule Eσ,l (Iσ,l ) ⊂ Ext S (S/Iσ,l , S) with the property that the natural map j

j

Ext S (S/Iσ,l , S) −→ Ext S (Jσ,l , S), j

j

induced by Jσ,l → S/Iσ,l sends Eσ,l (Iσ,l ) isomorphically onto Ext S (Jσ,l , S). Proof We write I = Iσ,l , consider (11.56), and the associated long exact sequence j

f

j

δ

j+1

· · · −→ Ext S (S/I, S) −→ Ext S (Jσ,l , S) −→ Ext S (S/(I + Iσ ), S) −→ · · · It follows from Proposition 11.9.5 that Z(I + Iσ ) = Z(I ) \ {(σ, l)}, which comj+1 bined with (11.57) shows that Ext S (S/(I + Iσ ), S) is isomorphic to a GLsubmodule of  j+1 Ext S (Jτ,u , S). (τ,u)∈Z(I )\{(σ,l)}

Using Lemma 11.9.7, we conclude that δ = 0 and therefore f is surjective. Choosing j  a splitting of f gives the desired GL-module Eσ,l (Iσ,l ).

462

11 Cohomology and Regularity in Characteristic Zero j

j

We fix once and for all a GL-module Eσ,l (Iσ,l ) ⊂ Ext S (S/Iσ,l , S) for each pair (σ, l) with sc ≥ l + 1. If I is any GL-invariant ideal with (σ, l) ∈ Z(I ), then we consider the natural surjection S/I  S/Iσ,l given by Lemma 11.9.3, and the induced j

j

f : Ext S (S/Iσ,l , S) −→ Ext S (S/I, S).   j j Eσ,l (I ) = f Eσ,l (Iσ,l ) .

We let

(11.59)

Suppose now that I ⊂ J are GL-invariant ideals with (σ, l) ∈ Z(I ) ∩ Z(J ). By functoriality, we have natural maps f

j

g

j

j

Ext S (S/Iσ,l , S) −→ Ext S (S/I, S) −→ Ext S (S/J, S),   j j and by definition we have Eσ,l (J ) = (g ◦ f ) Eσ,l (Iσ,l ) , hence   j j g Eσ,l (I ) = Eσ,l (J ).

(11.60)

j

j

Part of Theorem 11.9.1 states that in fact g maps Eσ,l (I ) isomorphically onto Eσ,l (J ), which will be explained next. One can think of (11.60) as a functoriality property of j the assignment I −→ Eσ,l (I ). Step 4. Our next goal is to prove parts (a) and (b) of Theorem 11.9.1. We start with the following extension of Corollary 11.9.8. Lemma 11.9.9 Suppose that (σ, l) ∈ Z(I ) and Jσ,l → S/I . The natural map h

j

Ext S (S/I, S) −→ Ext j (Jσ,l , S) j

is surjective, and h sends Eσ,l (I ) isomorphically onto Ext j (Jσ,l , S). Proof Lemma 11.9.3 implies that I ⊂ Iσ,l , and the natural maps Jσ,l → S/I  S/Iσ,l induce natural maps on Ext groups f j

Ext S (S/Iσ,l , S)

g

j

Ext S (S/I, S)

h

Ext j (Jσ,l , S) j

By construction, f induces an isomorphism between Eσ,l (Iσ,l ) and Ext j (Jσ,l , S). Since f = h ◦ g is surjective, this shows  that h is also  surjective. Moreover, h induces j j an isomorphism between Eσ,l (I ) = g Eσ,l (Iσ,l ) and Ext j (Jσ,l , S), as desired. 

11.9 Ext Modules for GL-Invariant Ideals

463

Proof (Proof of Theorem 11.9.1(a),(b)) We do induction on |Z(I )|, the case |Z(I )| = 0 being trivial. Suppose that |Z(I )| > 0 and choose (σ, l) ∈ Z(I ) such that Jσ,l → S/I (as in the proof of Theorem 11.9.4, it suffices to take σ to be lexicographically maximal among the shapes in Z(I )). Using the long exact sequence for Ext associated with (11.56), it follows from Lemma 11.9.9 that for each j we have a short exact sequence ι

j

h

j

j

0 −→ Ext S (S/(I + Iσ ), S) −→ Ext S (S/I, S) −→ Ext S (Jσ,l , S) −→ 0. j

Since h maps Eσ,l (I ) isomorphically onto Ext j (Jσ,l , S), this implies that we have a direct sum decomposition   j j j Eσ,l (I ). Ext S (S/I, S) = ι Ext S (S/(I + Iσ ), S)

(11.61)

Since Z(I + Iσ ) = Z(I ) \ {(σ, l)} by Proposition 11.9.5, we have by induction that 

⎛



ι Ext S (S/(I + Iσ ), S) = ι ⎝ j



⎞ j Eτ,u (I + Iσ )⎠

(τ,u)∈Z(I +Iσ ) (11.60)

=



j Eτ,u (I ),

(τ,u)∈Z(I )\{(σ,l)}

which combined with (11.61) shows the inductive step of the proof of j Theorem 11.9.1(b). We already know that Eσ,l (I )  Ext j (Jσ,l , S) by Lemma 11.9.9. Moreover, since ι is injective, it follows from (11.60) that for each (τ, u) ∈ Z(I + j j Iσ ), ι maps Eτ,u (I + Iσ ) isomorphically onto Eτ,u (I ). By induction, we know j j j that Eτ,u (I + Iσ )  Ext (Jτ,u , S), hence Eτ,u (I )  Ext j (Jτ,u , S) for all (τ, u) ∈ Z(I ) \ {(σ, l)}, concluding the inductive step of the proof of Theorem 11.9.1(a).  Step 5. Our final goal is to verify Theorem 11.9.1(c). Similarly to (11.52), we consider

= {((si + 1)i ) : 1 ≤ i ≤ c}, σ,l

(11.62)

⊂ σ,l , we have Iσ,l ⊂ Iσ,l . We have the following. and note that since σ,l

. Moreover, we Lemma 11.9.10 If I ⊃ J and (σ, l) ∈ Z(I ) \ Z(J ), then J ⊂ Iσ,l have   j j

, S) = 0. HomGL Ext S (Jσ,l , S), Ext S (S/Iσ,l

. Since (σ, l) ∈ Z(I ) and I has more shapes than J , Proof We first prove J ⊂ Iσ,l it follows that (σ, l) must satisfy condition (b) in the Definition 10.2.1 for Z(J ). Since (σ, l) ∈ / Z(J ), it must therefore fail condition (a). It follows that each shape λ = ( p1 , p2 , . . . ) of J either satisfies pi > si for some i ≤ c, in which case λ is a

464

11 Cohomology and Regularity in Characteristic Zero

, or it satisfies σ ⊃ λ(c) and p shape in Iσ,l c+1 ≤ l. If J contains any such shape, then J ⊂ Iσ,l , contradicting J ⊂ I ⊂ Iσ,l , where the last inclusion comes from Lemma 11.9.3. To prove the last assertion, note that by (11.57) it suffices to check that

  j j HomGL Ext S (Jσ,l , S), Ext S (Jτ,u , S) = 0

). If we write τ = (t , t , . . . , t ) then as in the proof of for every (τ, u) ∈ Z(Iσ,l 1 2 d

have at most c parts) and u > l Lemma 11.9.7 we have d ≤ c (since all shapes in σ,l

have parts of size si + 1 ≥ sc + 1 > l + by Lemma 10.2.5 (since the shapes in σ,l j 1). If we assume by contradiction that Sa F1 ⊗ Sa(s) F2 appears in both Ext S (Jσ,l , S) j and Ext S (Jτ,u , S), then we know by Remark 11.6.3 that

l − c − n = am = u − d − n, so 0 > l − u = c − d ≥ 0, a contradiction.



Proof (Proof of Theorem 11.9.1(c)) Suppose first that (σ, l) ∈ Z(I ) ∩ Z(J ), so that j j Eσ,l (I ) and Eσ,l (J ) are both isomorphic to Ext j (Jσ,l , S) by Theorem 11.9.1(a). In j j particular, degree by degree, Eσ,l (I ) and Eσ,l (J ) are finite dimensional vector spaces j of the same dimension. We know by (11.60) that the map f : Ext S (S/I, S) −→ j j j Ext S (S/J, S) sends Eσ,l (I ) onto Eσ,l (J ) (and preserves degrees), hence it must be an isomorphism. Finally, suppose that (σ, l) ∈ Z(I ) \ Z(J ). It follows from Lemmas 11.9.3 and 11.9.10 that we have a commutative diagram ⊃

I

⊂

⊂ ⊃

Iσ,l

S/I

S/J

S/Iσ,l

S/Iσ,l

J or equivalently

Iσ,l

Passing to Ext groups, this yields a commutative diagram j

Eσ,l (I )

⊂

j

Ext S (S/I, S)

f

Ext S (S/J, S)

h

, S) Ext S (S/Iσ,l

j

g j

Eσ,l (Iσ,l )

⊂

j

Ext S (S/Iσ,l , S)

j

  j By Lemma 11.9.10, we have that h Eσ,l (Iσ,l ) = 0, while by (11.60) we have that     j j j g Eσ,l (Iσ,l ) = Eσ,l (I ). The commutativity of the diagram forces f Eσ,l (I ) = 0, concluding our proof. 

11.10 Local Cohomology with Determinantal Support

465

11.10 Local Cohomology with Determinantal Support The goal of this section is to give an explicit description of the local cohomology j groups HIl+1 (S) of S with support in the ideal Il+1 ; we refer to Brodmann and Sharp [33] and Bruns and Herzog [52] for generalities on local cohomology. We assume that m ≤ n and for 0 ≤ s ≤ m we let A(s) = {a ∈ Zm dom : as ≥ s − m and as+1 ≤ s − n},

(11.63)

where we make the convention a0 = +∞ and am+1 = −∞. We also consider, for u ≥ v ≥ 0, the q-binomial coefficients (or Gauß polynomials)   u (1 − q u )(1 − q u−1 ) · · · (1 − q u−v+1 ) = (1 − q v )(1 − q v−1 ) · · · (1 − q) v q  q | p| . =

(11.64)

0≤ p1 ≤···≤ pu−v ≤v

If we specialize q to a prime power in (11.64), then we get the number of subspaces of dimension v in a vector space of dimension u over the field with q elements. Another interpretation of (11.64) is as the Poincaré polynomial of the Grassmannian of v-planes in u-space, since the coefficient of q r counts the number of Schubert cells of (co)dimension r in the Grassmannian. Our interest in the q-binomial coefficients comes from the following result. Theorem 11.10.1 With the notation above,  j≥0

j H Il+1 (S) · q j

=

l  s=0

⎛ ⎝



⎞ Sa F1 ⊗ Sa(s) F2 ⎠ · q (m−l)

2 +(m−s)(n−m)

a∈A(s)

·

  m−1−s . l −s q2

Proof It is a standard fact that (d) , S), HIl+1 (S) = lim Ext S (S/Il+1 − → j

j

(11.65)

d

(d) (d+1) ⊂ Zl+1 for and the directed limit is a union by Theorem 11.9.1(d), since Zl+1 (d) all d. Moreover, with the notation (11.40), we have that the sets A(t, s) form an increasing chain, and # (d) A(t, s) = A(s). d,t

Using Corollary 11.7.2 and the fact that (11.65) is a union, we have

466

11 Cohomology and Regularity in Characteristic Zero





j

HIl+1 (S) · q j =



 2 Sa F1 ⊗ Sa(s) F2 · q mn−l −s(n−m)−2|t| .

l=t1 ≥t2 ≥···≥tm−l ≥s≥0 a∈A(s)

j≥0

To conclude, we must check that if we fix s and allow t j to vary, then 

q mn−l

2

−s(n−m)−2|t|

= q (m−l)

2

+(m−s)(n−m)

l=t1 ≥t2 ≥···≥tm−l ≥s

  m−1−s . (11.66) l −s q2

Making the substitution p j = l − t j+1 for j ∈ [m − l − 1], we get 0 ≤ p1 ≤ p2 ≤ · · · ≤ pm−l−1 ≤ l − s, and mn − l 2 − s(n − m) − 2|t| = (m − l)2 + (m − s)(n − m) + 2| p|. The formula (11.66) now follows from the fact that 

q 2| p| =

0≤ p1 ≤···≤ pm−l−1 ≤l−s

    m −l −1+l −s m−1−s = , l −s l −s q2 q2

which follows from (11.64) by taking u = m − 1 − s, v = l − s, and q  q 2 .



Cohomological Dimension and Arithmetic Rank An immediate consequence of Theorem 11.10.1 is the following characterization of the (non)vanishing of local cohomology groups with determinantal support. j

Corollary 11.10.2 We have that HIl+1 (S) = 0 if and only if j = (m − l)2 + (m − s)(n − m) + 2r for some 0 ≤ s ≤ l and 0 ≤ r ≤ (l − s)(m − 1 − l). In particular, we have   j max j : HIl+1 (S) = 0 = mn − (l + 1)2 + 1. Proof The first assertion follows from Theorem 11.10.1 and (11.64), since the coef ficient of q i in uv q is nonzero if and only if 0 ≤ i ≤ v(u − v). For the last assertion, we note that

11.10 Local Cohomology with Determinantal Support

467

(m − l)2 + (m − s)(n − m) + 2(l − s)(m − 1 − l) is a decreasing function of s, hence it is maximized when s = 0, when we get (m − l)2 + m(n − m) + 2l(m − 1 − l) = mn − (l + 1)2 + 1.



For an ideal I ⊂ S, the largest degree c for which HIc (S) = 0 is called the cohomological dimension of I , and it is denoted cd S (I ). If we let t = l + 1 then Corollary 11.10.2 shows that (11.67) cd(It ) = mn − t 2 + 1, which was first proved by Bruns and Schwänzl [58]. One important property of cohomological dimension is its relationship with the arithmetic rank of the ideal I . This invariant is defined as the minimal number of generators of I up to radical, that is,  √ √  ara(I ) = min r : there exists an ideal J =  f 1 , . . . , fr  with I = J . √ √ The condition I = J is equivalent to the fact that the ideals I, J define the same closed subset of Spec(S) in the Zariski topology, denoted V (I ) = V (J ). We can then say that ara(I ) is the smallest number of equations required to cut out V (I ) set-theoretically (that is, as a topological space, but not necessarily as a scheme). To relate back to local cohomology, we note that if J =  f 1 , . . . , fr  then H Ji (R) can be computed as the ith cohomology of the complex Cˇ . ( f 1 , . . . , fr ) :

S −→



S fi −→



S fi f j −→ . . . −→ S f1 ··· fr −→ 0,

i< j

and since Cˇ . ( f 1 , . . . , fr ) has length r , we have H Ji (S) = 0 for i > r . Moreover, ˇ complex associated with the covering of Cˇ . ( f 1 , . . . , fr ) is the (extended) Cech Spec(S) \ V (J ) by the basic open affine sets D( f i ), and hence its cohomology only depends on Spec(S) \ V (J ) and not on the covering. Since V (I ) = V (J ), we conclude that Cˇ . ( f 1 , . . . , fr ) also computes HIi (S), and in particular HIi (S) = 0 for i > ara(I ) if we take r to be minimal. This discussion shows that cd S (I ) ≤ ara(I ), hence the cohomological dimension of I gives a lower bound for the number of equations required to cut out V (I ) set-theoretically. Remarkably, for generic determinantal ideals it is shown in [58, Theorem 2] that the two invariants agree, namely ara(It ) = mn − t 2 + 1.

(11.68)

468

11 Cohomology and Regularity in Characteristic Zero

Since S/It is a Cohen–Macaulay algebra, it follows from Peskine and Szpiro [258, Proposition III.4.1] and Corollary 10.8.1 that in characteristic p > 0 we have cd(It ) = (m − t + 1) · (n − t + 1), and in particular (11.67) fails when 1 < t ≤ m = min(m, n). Nevertheless, formula (11.68) remains true even in this case: this is explained in [58] by replacing the study of local cohomology with étale cohomology, and using the characteristic-free results on algebras with straightening law from Bruns [36]. The fact that formula (11.67) is quite special for generic determinantal varieties in characteristic zero is explained in a very compelling manner in Lyubeznik, Singh and Walther [237]. It is shown in [237, Theorem 5.1] that if A is any commutative Noetherian ring, M = (m i j ) is an m × n matrix with entries in A, and It is the ideal of t × t minors of M, then HImn−t t

2

+1

(A) = 0 if 1 ≤ t ≤ m = min(m, n) and dim A ⊗Z Q < mn.

D-Module Structure of Local Cohomology A more conceptual interpretation of the formula in Theorem 11.10.1, whose details go beyond the scope of this book, is based on the fact that As =



Sa F1 ⊗ Sa(s) F2

(11.69)

a∈A(s)

is the GL-representation underlying a simple GL-equivariant W-module, supported on the determinantal variety of rank ≤ s matrices (here W = S∂i, j  is the Weyl algebra of differential operators with polynomial coefficients). When s = m, we get a W-module with full support, namely the polynomial ring S. To say that Am is the GL-module underlying S is just a restatement of Cauchy’s formula. Theorem 11.10.1 j is then a reflection of the fact that the local cohomology groups HIl+1 (S) are GLequivariant W-modules, and as such they admit finite filtrations, with composition factors corresponding to (11.69), each appearing with certain multiplicity. Example 11.10.3 Suppose that m = n, and let det = det(xi, j ). The localization Sdet is a W-module, and it has a natural filtration 0  S  det −1 W  · · ·  det −m W = Sdet ,

(11.70)

where det −i W denotes the W-submodule of Sdet generated by det −i . The composition factors for the above filtration are Ds =

det s−m W , for s = 0, . . . , m − 1, and Dm = S, det s−m+1 W

11.10 Local Cohomology with Determinantal Support

469

and we have Ds  As as a GL-module. To see how this is reflected in Theorem 11.10.1, we take l = m − 1, and since Il+1 = det S is a hypersurface, it follows that Sdet j HI1m (S) = and HIm (S) = 0 for j = 1. S   Since m − 1 = l, we have m−1−s = 1, hence Theorem 11.10.1 implies l−s q 2 

j

HIm (S) · q j =

m−1 

As · q 1 .

s=0

j≥0 j

This shows the vanishing of HIm (S) for j = 1, and the existence of an isomorphism HI1m (S)  A0 ⊕ A1 ⊕ · · · ⊕ Am−1 . of GL-modules. This is consistent with the discussion of W-modules, since quotienting out by S = Dm in (11.70) leaves us with the W-module composition factors D0 , . . . , Dm−1 . It is interesting to consider instead the ideal Im of maximal minors for m < n. Example 11.10.4 Suppose that m < n and l = m − 1. Using (11.69), we get from Theorem 11.10.1 that 

j

HIm (S) · q j =

l 

As · q 1+(m−s)(n−m) ,

s=0

j≥0

or equivalently, the nonzero local cohomology groups are given by (S) = Am−1 , HI1+2(n−m) (S) = Am−2 , . . . , HI1+m(n−m) (S) = A0 . HI1+n−m m m m (S) is a simIn terms of the W-module structure, we have that each HI1+(m−s)(n−m) m ple W-module. Notice that overall the same simple modules appear as in Example 11.10.3, but they are now separated into distinct cohomological degrees. We conclude with exhibiting the minimal example when multiplicities > 1 occur in the formula from Theorem 11.10.1. Since     u u = = 1 + q + · · · + q u−1 , 1 q u−1 q we need a Gauß polynomial

u  v q

with v ≥ 2 and u − v ≥ 2, and the minimal one is

  4 = 1 + q + 2q 2 + q 3 + q 4 . 2 q

470

11 Cohomology and Regularity in Characteristic Zero

Example 11.10.5 Suppose that m = n = 5 and l = 2, so that we consider local cohomology with support in the ideal of 3 × 3 minors of a 5 × 5 generic matrix. We have      j 4 3 HIm (S) · q j = A0 · q 9 · + A1 · q 9 · + A2 · q 9 , 2 1 2 2 q q j≥0 hence the nonzero local cohomology groups are (compare with Corollary 11.10.2) j j HI3 (S)

9 11 13 15 17 ⊕2 A0 ⊕ A1 ⊕ A2 A0 ⊕ A1 A0 ⊕ A1 A0 A0

where the table above records the GL-module (but not the W-module) structure.

11.11 Syzygies of GL-Invariant Ideals The goal of this section is to give a brief description of the syzygies of the determinantal ideals obtained in the work of Lascoux, and to summarize important recent progress on the corresponding problem for general GL-invariant ideals. We continue to assume that char K = 0 and that m ≤ n. For a GL-invariant ideal I ⊂ S we consider the basic problem of describing the groups Bi, j (I ) = ToriS (I, K ) j

(11.71)

as GL-representations. It is common to refer to the groups (11.71) as (minimal generators of the) syzygies of I . The simplest example is when I = I1 is the maximal homogeneous ideal, which is resolved by the Koszul complex. In that case, we have Bi,i+1 (I1 ) =

i+1 $

(F1 ⊗ F2 ) ,

Bi, j (I1 ) = 0 for j = i + 1.

(11.72)

% Already in this simple case a question arises of how to decompose i+1 (F1 ⊗ F2 ) into a sum of irreducible representations. The answer is classical, and comes from a version of Cauchy’s Formula for exterior powers (compare with Corollary 11.5.3 and see [317, Theorem 2.3.2]): if F1 , F2 are locally free sheaves then we have a decomposition d  $ Sv F1 ⊗ Sv∗ F2 . (11.73) (F1 ⊗ F2 ) = v∈P(d)

Example 11.11.1 If we specialize (11.73) to the case when F j = F j are vector spaces and d = 3, then we get that

11.11 Syzygies of GL-Invariant Ideals

B2,3 (I1 ) =

3 $

&

(F1 ⊗ F2 )

= Sym F1 ⊗ 3

3 $

' F2

471

& 3 ' $   3 ⊕ S(2,1) F1 ⊗ S(2,1) F2 ⊕ F1 ⊗ Sym F2 .

In order to give equally explicit formulas for other GL-invariant ideals, we introduce more notation. We first note that, just as in the case of Ext modules, the groups Bi, j (I ) decompose into irreducible representations (11.21), where |a| = |b| = j. For this reason, the internal degree j is implicit in the GL-module decomposition and we will ignore it to simplify notation. We thus write Bi (I ) = ToriS (I, K ) and encode these groups into the equivariant Betti polynomial B I (q) =



Bi (I ) · q i ,

(11.74)

i∈Z

where q is an indeterminate keeping track of the homological index, and the coefficients of B I (q) are GL-representations (which are finite dimensional, unlike in the case of local cohomology modules). For instance, we have that B I1 (q) =

i+1 $

(F1 ⊗ F2 ) · q i .

i≥0

Notice that in general, the degree of the polynomial B I (q) measures the projective dimension of the ideal I . If r, s are positive integers, α = (a1 , a2 , . . . , ar ) is a partition with at most r parts, and β = (b1 , b2 , . . . ) is a partition with parts of size at most s (that is, b1 ≤ s), we construct the weight v(r, s; α, β) = (s + a1 , . . . , s + ar , b1 , b2 , . . . ).

(11.75)

This is easiest to visualize in terms of Young diagrams: one starts with an r × s rectangle, and attach α to the right and β to the bottom of the rectangle. If r = 4, s = 5, α = (4, 2, 1), β = (3, 2), then αααα αα α

v(r, s; α, β) = (9, 7, 6, 5, 3, 2) ←→ β β β β β We consider the set of partitions

(11.76)

472

11 Cohomology and Regularity in Characteristic Zero

Ar,s = {α = (a1 , a2 , . . . ) : a1 ≤ n − r, a1∗ ≤ min(r, s)}, Br,s = {β = (b1 , b2 , . . . ) : b1 ≤ min(r, s), b1∗ ≤ m − r }, and define the polynomials h r ×s (q) (with coefficients GL-representations) via h r ×s (q) =



(Sv(r,s;α,β) F1 ⊗ Sv(r,s;β ∗ ,α∗ ) F2 ) · q |α|+|β| .

(11.77)

α∈Ar,s ,β∈Br,s

Notice that the restrictions on α and β guarantee that v(r, s; α, β) has at most m parts and v(r, s; β ∗ , α ∗ ) has at most n parts. In addition, |v(r, s; α, β)| = |v(r, s; β ∗ , α ∗ )| = r · s + |α| + |β|. The description of the syzygies of determinantal ideals was first obtained in the seminal work of Lascoux [231] (see also [317, Chap. 6]). With our notation, it can be summarized as follows. Theorem 11.11.2 For each t = 1, . . . , m, we have that B It (q) =

m−t 

h (t+i)×(1+i) (q) · q i

2

+2i

.

i=0

Notice that the largest partition in Ar,s has size |α| = (n − r ) · min(r, s), and similarly, the largest partition in Br,s has size |β| = (m − r ) · min(r, s). It follows from (11.77) that   deg h (t+i)×(1+i) (q) = (n − t − i) · (1 + i) + (m − t − i) · (1 + i), and therefore   deg B It (q) = max {(n − t − i) · (1 + i) + (m − t − i) · (1 + i) + i 2 + 2i} 0≤i≤m−t

= max {(i + 1) · (m + n − 2t − i + 1)} − 1 0≤i≤m−t

= (m − t + 1) · (n − t + 1) − 1. Since the projective dimension of It is one less than that of S/It , this implies that S/It has projective dimension (m − t + 1) · (n − t + 1), which is true more generally in arbitrary characteristic by Corollary 10.8.5. Furthermore, using the characterization of regularity in terms of syzygies in Theorem 8.1.3, we get reg(It ) = max {(t + i) · (1 + i) − (i 2 + 2i)} 0≤i≤m−t

= max {(i + 1) · (t − 1)} − 1 0≤i≤m−t

= (m − t + 1) · (t − 1) − 1.

11.11 Syzygies of GL-Invariant Ideals

473

Since reg(It ) = reg(S/It ) − 1, this agrees with the calculation in Example 11.6.2.

The Geometric Technique to Compute Syzygies The idea of using (higher) direct images of the Koszul complex in order to compute syzygies originates the work of Kempf in the 70s [204, 207], and was popularized with the comprehensive treatment in Weyman’s book [317]. It is closely related to the Koszul cohomology approach to syzygies in Green’s work in the 80s [155, 156]. To explain it in a simple example, which is powerful enough to prove Lascoux’ theorem, consider the short exact sequence (9.47) where B is a (not necessarily smooth) %i+ j projective variety. For i, j ≥ 0, we can obtain a (right) Koszul resolution of ξ given by ( 0→

i+ j $

ξ→

) i+ j $

V ⊗ OB → · · · →

i $

V ⊗ Sym j η → · · · → Symi+ j η → 0

(11.78) If we assume that the symmetric powers of η have vanishing higher cohomology, that is, (11.79) H k (B, Symd η) = 0 for all d ≥ 0, k > 0, % then it follows that the cohomology of i+ j ξ , or equivalently, the hypercohomology of its resolution (11.78), coincides with the cohomology of the complex obtained by global sections i+ $j

V ⊗ H 0 (B, OB ) → · · · →

i $

V ⊗ H 0 (B, Sym j η) → · · · → H 0 (B, Symi+ j η) → 0

If we consider the graded rings R=



H 0 (B, Sym j η), S =

j≥0



Sym j V,

j≥0

then the complex above takes the form i+ j $

V ⊗ R0 → · · · →

i $

V ⊗ R j → · · · → Ri+ j → 0

and hence it agrees with the degree (i + j) strand in the Koszul complex computing the syzygies of R as an S-module. Our discussion then yields the following sheaf cohomological description of the said syzygies. Theorem 11.11.3 Under the assumption (11.79), we have that

474

11 Cohomology and Regularity in Characteristic Zero

& ToriS (R,

K )i+ j = H

j

B,

i+ j $

' ξ

for all i, j ≥ 0.

Notice that in terms of the diagram (9.48), and with the usual identification of quasi-coherent sheaves on affine space with their global sections, we have that S = OA , and R = ϕ∗ OY = H 0 (Y , OY ), giving a natural geometric interpretation for the rings R and S. Proof (Proof of Theorem 11.11.2) We specialize the discussion above to the setup in the proofs of Lemma 10.8.4 and Corollary 10.8.5, with t = l + 1, V = F1 ⊗ F2 , B = G , η = 1 Q  F2 , ξ = 1 R  F2 . The proof of Lemma 10.8.4 shows that R = S/Il+1 = S/It , and that (11.79) holds, so the conclusion of Theorem 11.11.3 applies to give & Tor Sp (It ,

K ) p+ j =

Tor Sp+1 (R,

(11.73)

=

K ) p+ j = H



j−1

G,

p+ j $

' ξ

  H j−1 G(l, F1 ), S y 1 R ⊗ S y ∗ F2 .

y∈P( p+ j)

The desired conclusion now follows from a direct application of Theorem 11.3.1. More precisely, if S y 1 R has non-trivial cohomology, then it is described as follows. Let k denote the unique index such that yk − (l + k) ≥ 0 and yk+1 − (l + k + 1) ≤ −(l + 1), let z = (0l , y1 , . . . , ym−l ), and define u i , t j as in (11.8), (11.9). We have u 1 = · · · = u l = k, t1 = · · · = tk = l, tk+1 = · · · = tm−l = 0. The associated permutation σ satisfies (σ ) = kl and σ . z = (y1 − l, . . . , yk − l, k l , yk+1 , . . . , ym−l ). We can now define partitions α, β by α = (y1 − k − l, . . . , yk − k − l) and β = (yk+1 , . . . , ym−l ), and we get that σ . z = v(k + l, k; α, β) and y ∗ = v(k + l, k; β ∗ , α ∗ ).

(11.80)

11.11 Syzygies of GL-Invariant Ideals

475

It follows that   H kl G(l, F1 ), S y 1 R ⊗ S y ∗ F2 = Sv(k+l,k;α,β) F1 ⊗ Sv(k+l,k;β ∗ ,α∗ ) F2 .

(11.81)

Writing j = kl + 1 and p = |y| − j then it follows that every summand in Tor Sp (It , K ) p+ j is a representation as in (11.81) for some partition y. Moreover, recalling that t = l + 1 and setting i = k − 1, it follows that the representation 2 (11.81) also appears in h (t+i)×(1+i) (q) · q i +2i as a contribution to the coefficient of 2 2 2 2 q |α|+|β|+i +2i = q |y|−k(k+l)+(i+1) −1 = q p+kl+1−k −kl+k −1 = q p . It is now easy to revert this construction: for every representation Sv(t+i,1+i;α,β) F1 ⊗ Sv(t+i,1+i;β ∗ ,α∗ ) F2

(11.82)

appearing as a contribution to the coefficient to q p in h (t+i)×(1+i) (q) · q i +2i , we define k = i + 1, y = v(t + i, 1 + i; β ∗ , α ∗ )∗ , so that (11.80) holds. We set j = kl − 1, and the previous calculation shows that (11.82) contributes to Tor Sp (It , K ) p+ j , concluding our proof.  2

Remark 11.11.4 Theorem 11.11.3 holds over a field of arbitrary characteristic, and hence the same is true about the initial step of the proof of Theorem 11.11.2, including the equalities & Tor Sp (It ,

K ) p+ j =

Tor Sp+1 (R,

K ) p+ j = H

j−1

G,

p+ j $

' ξ .

The first part of our argument that fails in positive characteristic is (11.73), which remains true only up to a filtration (compare with Corollary 10.5.4). The more serious issue is the failure of the Borel-Weil-Bott theorem in positive characteristic, and in particular that of Theorem 11.3.1. It is known that the syzygies of It are independent on the characteristic when t = 1 (the Koszul complex), when t = m (the Eagon– Northcott complex), and when t = m − 1 (by [5]). In general however, they do depend on the characteristic, as shown in several examples by Hashimoto [166, 167] (see also [317, Theorem 6.2.6]). The fact that the first syzygy module of It (the case p = 1) is generated by linear syzygies, for all t and all characteristics, was proved by Kurano [228]: more precisely, the linear syzygies are obtained as row or column relations and by the difference of row and column Laplace expansions of minors of size t + 1. This fact was initially conjectured by Sharpe, who proved it in the case when t = 2 [281, 282].

476

11 Cohomology and Regularity in Characteristic Zero

The General Linear Super Lie Algebra and Syzygies For arbitrary GL-invariant ideals I ⊂ S the geometric technique to compute syzygies no longer applies (or at least not in an obvious way), so new input is necessary. The key idea, first appearing in work of Pragacz–Weyman [259] and Akin–Weyman [7], is to look for additional structure on the syzygy modules, coming from the action of the general linear Lie superalgebra gl(m|n). From this perspective, the polynomials (11.77) have a very natural interpretation: they encode the GL-representations underlying certain irreducible gl(m|n)-modules. It is shown in [266] that the syzygies of an arbitrary GL-invariant ideal I ⊂ S can be naturally assembled into (not necessarily irreducible) representations of gl(m|n). With this perspective in mind, and using the explicit calculations of Ext modules described in earlier sections, the following generalization of Theorem 11.11.2 is proved in [265, Theorem 3.1]. Theorem 11.11.5 For positive integers a, b, consider the shape σ = (s1 , s2 , . . . ) defined by s1 = · · · = sb = a, si = 0 for i > b. The equivariant Betti polynomial of the ideal Iσ (defined in (11.44)) is B Iσ (q) =

m−a  i=0

h (a+i)×(b+i) (q) · q i

2

+2i

·

  i + min(a, b) − 1 i q2

One notes that Theorem 11.11.2 is obtained as the special case when a = t and b = 1, in which case Iσ = It . Another special case of Theorem 11.11.5 occurs by setting a = m and letting b be any positive integer. In that case Iσ = Imb is a power of the ideal of maximal minors, whose minimal resolution was first computed in [5]. To further generalize Lascoux’ theorem, the next step is to consider arbitrary ideals Iσ . A conjectural description of the structure of the syzygies of Iσ in terms of simple gl(m|n)-module composition factors is given in [266], and the conjecture is proved in [191]. As noted in [191, Remark 3.12], the methods can be extended further to give a description for the syzygies of arbitrary GL-invariant ideals. The formulas nevertheless get increasingly less explicit as one moves away towards more general ideals I . One reason is that the simple gl(m|n)-modules involved have a more complicated GL-module structure than the ones encoded by the polynomials (11.77). In particular, the decomposition of a simple gl(m|n)-module into irreducible GL-representations is in general not multiplicity free. In retrospect, it is then quite remarkable that the structure of Ext modules was so well-behaved!

11.11 Syzygies of GL-Invariant Ideals

477

Notes The Borel–Weil–Bott theorem is valid more generally on a homogeneous space G/B, where G is a reductive linear algebraic group, and B is a Borel subgroup. The proof in this generality can be found in Weyman [317, Sect. 4.3] or Jantzen [196, Sect. II.5]. Some of the classical references regarding cohomology of line bundles on homogeneous spaces include Serre [280], Bott [25], Demazure [110, 111]. The structure of GL-invariant ideals was first studied by De Concini, Eisenbud and Procesi in [103], and the description of the corresponding Ext modules is given by Raicu in [262], which also computes the regularity of powers and symbolic powers of determinantal ideals. The local cohomology groups with determinantal support were first computed (using a different directed limit than the one coming from symbolic powers) by Raicu and Weyman [263], which extended the case of maximal minors due to Raicu et al. [267]. A fairly complete analysis of the W-module structure of local cohomology with determinantal support is further explained in Raicu [261], L˝orincz and Raicu [234], Perlman and Raicu [257], and L˝orincz and Walther [235]. The analogous results for Ext modules and Castelnuovo–Mumford regularity in the case of generic skew-symmetric matrices are discussed by Perlman [256], while some further results on the structure of local cohomology with support in ideals of symmetric minors and Pfaffians are considered by Raicu and Weyman [264], and by Perlman [255]. The syzygies of generic determinantal varieties were first computed by Lascoux [231]. The study of Lie superalgebras was initiated by Kac [198, 199], who classified the finite dimensional simple Lie superalgebras over a field of characteristic zero. The Lie superalgebra action on syzygies of determinantal varieties goes back to Pragacz–Weyman [259], and was explored further in several works, including [7–9, 191, 265, 266, 274, 275].

List of Symbols

∨ ∧

0 νeI (a) OB OB,b OFEB (E) (y) ω R , ω(R) ωFB (E)/B ωGB (l;E)/B

Stanley–Reisner ring of simplicial complex , 48 kernel of a homomorphism Kostka number of partitions λ, μ, 151 monoid algebra, 172 coordinate ring of flag variety, 79 dual of shape λ, 119 map taking a linear map to its tth exterior power, 208 exterior power of locally free sheaf, 328 exterior power of a free module (or vector space) exterior power of V ∨ , also dual of the exterior power of V , 327 least common multiple tautological line bundle, 332 subspaces generated by all right resp. left initial bitableaux, 95 length of a permutation σ , 416 maximal ideal of OB,b , 329 G-invariants of M, 356 multiplication for an algebraic group G, 350 ith homogeneous component of M w.r.t. (1, 0)-grading, 288 jth homogeneous component of M w.r.t. (0, 1)-grading, 288  truncated module i≥ j Mi , 283 set of monomials in R, 2 irreducible G-representation, 98 a direct summand of Symd (F1 ⊗ F2 ) (in characteristic zero), 435 set of bitableaux generating coordinate ring of flag variety, 79 set of t-minors of X , 70 minimal number of generators of module M, 225 monomial μ1 precedes μ2 in monomial order, 3 multiplicity of bi-shape (σ |τ ) in Pt (m, n), 201 multiplicity of bi-shape (σ |τ ) occurring in E, 201 set of nonempty minors, 70 set of nonnegative integers set of  positive integers e e  sup r ∈ N : ar (I [ p ] : I ) ⊂ m[ p ] , 265 structure sheaf of a variety B local ring at the point b, 329 line bundle associated to the weight y ∈ Zn , 332 (graded) canonical module of R, 136 relative canonical sheaf for a complete flag bundle, 333 relative canonical sheaf for a Grassmann bundle, 330

484

List of Symbols

GB (l;E)/B PB (E)/B ω X/Y Paths(P, Q) Paths(P, Q, z) PB (E) P(d) pi

2

t πi Pm P≤Q proj dim M Proj Pt (m, n), Pt Q Q (0,1) Q (1,0)  u v q qk QlE QR (R|C) Rχ R◦ R() R(F) R(I ) R(I, M) R(I1 , . . . , Im ) R(I1 , . . . , Im , M)  ) R(I Rsymb (It ) reg M

sheaf of relative differentials for a Grassmann bundle, 330 sheaf of relative differentials for a projective bundle, 330 relative canonical sheaf number of families of nonintersecting paths from P to Q, 131 generating function of families of paths from P to Q, 134 projective bundle, 329 set of partitions of d G-stable prime ideal in At , 212 simplicial complex defined by initial ideal of I2 , 108 simplicial complex defined by initial ideal of It , 128 valuation on At , 212 set of shapes with parts of size ≤ m, 382 partial order of paths, 112 projective dimension of module M relative Proj, 329 polynomial ring in indeterminates representing the tminors, 198 often ideal (X 1 , . . . , X n ), 277 ideal of R (0,∗) generated by R(0,1) , 287 ideal of R (∗,0) generated by R(1,0) , 287 q-binomial coefficient, 465 G-stable prime ideal in At , 212 tautological rank l quotient sheaf, 330 ideal generated by R1 , 276 bitableau with row tableau R and column tableau C, 71 semi-invariants of R for character χ , 220 set of elements of R not in any minimal prime ideal, 243 set of relevant faces of simplicial complex , 148 Rees algebra of filtration F, 260 Rees algebra of ideal I , 183 Rees module of ideal I and module M, 296 multi-Rees algebra of ideals I1 , . . . , Im , 301 multi-Rees module of ideals I1 , . . . , Im and module M, 302 extended Rees algebra of ideal I , 186 symbolic Rees algebra of ideal It , 187 regularity of module M (generalized on p. 276), 39

List of Symbols

485

reg(0,1) M reg(1,0) M relint(C)

RF  R m ρ N (v) Ri f∗ (Rk ) RlEi ,l j R(m, n) REn−l R#S RSK() RSK( f ) RSK(I ) R (i,∗) R (∗, j) R (∗,0) R (0,∗) sat(I ) Sd sign shape σ ≤τ σ (c) d (V ) σG  sat σ,l (σ |τ ) (Sk ) SL(r, K ) Spec √ I sr(ψ) succ≤ (σ, l) succ(σ, l) succ⊂ (σ, l) supp( f ) suppG (H )

regularity of M with respect to the (0, 1)-grading, 288 regularity of M with respect to the (1, 0)-grading, 288 relative interior of cone C, 175 completion of ring R w.r.t. filtration F, 53 completion of Rm w.r.t. ideal mRm , 54 supremum of degrees i of nonvanishing components N(i,v) , 290 higher direct image functor, 334 Serre condition, 35 tautological subquotient sheaf, 331 K [X ] for m × n matrix X , 199 tautological rank (n − l) subsheaf, 330 Segre product of graded algebras R and S, 107 monomial (or 2-line array) obtained from  by RSK, 115 value of f under linear map RSK, 116 image of ideal I under linear map RSK, 122 ith homogeneous component of R w.r.t. (1, 0)-grading, 288 jth homogeneous component of R w.r.t. (0, 1)-grading, 288  subring i R(i,0) of bigraded ring, 287 subring j R(0, j) of bigraded ring, 287 saturation of ideal I , 299 symmetric group sign of permutation recursive function to generate shapes in Z(), 390 σ precedes τ in dominance order, 78 truncation of the shape σ , 382 submodule of symmetric power, 326 coinverse on K [G], 351 saturation of a set of shapes in P(d), 393 set of rectangular shapes associated to (σ, l), 457 bi-shape, 201 Serre condition, 35 group of r × r matrices of determinant 1, 220 relative Spec, 329 radical of I small rank of ψ, 209 another notation for succ(σ, l), 453 l-successors of the shape σ with respect to ≤, 380 l-successors of σ with respect to ⊂, 453 set of monomials with nonzero coefficient in f , 2 set of shapes in G-module H , 213

486

List of Symbols

Sv (−) Sym. (E) Sym. (V ) Symd E Symd V t0 (M) T dE d (T V )Sd TdV d (V ) ti (M) Tn Tr U (I ) Un U + (n, K ) U − (n, K ) U V∨ V (I ) vp v(r, s; α, β) V (σ ) , V () W X (G) X t , X t (m, n) Yt , Yt (m, n) Z(I () ), Z() Z(I ), Z ⊂ () Zndom Z (d) p z(r ) Z ≤ ()

Schur functor associated to the weight v, 362 symmetric algebra of locally free sheaf, 329 symmetric algebra of free module, 326 symmetric power of locally free sheaf, 328 symmetric power of free module, 325 minimum degree needed to generate module M, 277 tensor power of locally free sheaf, 327 coinvariants for the symmetric group action on tensors, 325 tensor power of free module, 325 submodule of tensor power, 326 largest degree in ith Koszul homology of M, 279 algebraic group of upper triangular matrices (group scheme), 354 distinguished map F∗ R → R, 236 open subset of spectrum defined by ideal I , 50 unipotent group (group scheme), 354 upper unipotent n × n matrices, 97 lower unipotent n × n matrices, 97 U − (m, K ) × U + (n, K ), 98 dual of a free module, 327 closed subset of spectrum defined by ideal I , 50 valuation defined by prime ideal p, 88 weight used to describe syzygies of determinantal ideals, 471 sheaf defined by shape, 393 Weyl algebra, 468 character group of G, 353 Zariski closure of Yt , 208 set of exterior powers of linear maps, 208 combinatorial set indexing a natural filtration on S/I () , 382 combinatorial set indexing a natural filtration on S/I , 454 set of dominant weights, 341 special notation for the set Z(I p(d) ), 391 modification of the weight z, 436 another notation for Z(I () ), 454

References

1. J. Abbott and A. M. Bigatti. CoCoA-5: a system for doing Computations in Commutative Algebra. Available at http://cocoa.dima.unige.it. 2. I. Aberbach and T. Polstra. Local cohomology bounds and test ideals. arXiv e-prints (Sept. 2019), arXiv:1909.11231. 3. S. S. Abhyankar. Enumerative combinatorics of Young tableaux, vol. 115 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, Inc., New York, 1988. 4. W. W. Adams and P. Loustaunau. An introduction to Gröbner bases, vol. 3 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1994. 5. K. Akin, D. A. Buchsbaum, and J. Weyman. Resolutions of determinantal ideals: the submaximal minors. Adv. Math. 39, 1 (1981), 1–30. 6. K. Akin, D. A. Buchsbaum, and J. Weyman. Schur functors and Schur complexes. Adv. Math. 44, 3 (1982), 207–278. 7. K. Akin and J. Weyman. Minimal free resolutions of determinantal ideals and irreducible representations of the Lie superalgebra gl(m|n). J. Algebra 197, 2 (1997), 559–583. 8. K. Akin and J. Weyman. The irreducible tensor representations of gl(m|1) and their generic homology. J. Algebra 230, 1 (2000), 5–23. 9. K. Akin and J. Weyman. Primary ideals associated to the linear strands of Lascoux’s resolution and syzygies of the corresponding irreducible representations of the Lie superalgebra gl(m|n). J. Algebra 310, 2 (2007), 461–490. 10. E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris. Geometry of algebraic curves. Vol. I, vol. 267 of Grundlehren der Mathematichen Wissenschaften. Springer-Verlag, New York, 1985. 11. L. L. Avramov. A class of factorial domains. Serdica 5, 4 (1979), 378–379. 12. J. Backelin and R. Fröberg. Koszul algebras, Veronese subrings and rings with linear resolutions. Rev. Roumaine Math. Pures Appl. 30, 2 (1985), 85–97. 13. A. Bagheri, M. Chardin, and H. T. Hà. The eventual shape of Betti tables of powers of ideals. Math. Res. Lett. 20, 6 (2013), 1033–1046. 14. M. Barile. The Cohen-Macaulayness and the a-invariant of an algebra with straightening laws on a doset. Comm. Algebra 22, 2 (1994), 413–430. 15. D. A. Bayer. The division algorithm and the Hilbert scheme. ProQuest LLC, Ann Arbor, MI, 1982. Thesis (Ph.D.)–Harvard University. 16. A. Berget, W. Bruns, and A. Conca. Ideals generated by superstandard tableaux. In Commutative algebra and noncommutative algebraic geometry. Vol. II, vol. 68 of Math. Sci. Res. Inst. Publ. Cambridge Univ. Press, New York, 2015, pp. 43–62. 17. D. Bernstein and A. Zelevinsky. Combinatorics of maximal minors. J. Algebraic Combin. 2, 2 (1993), 111–121. 18. A. Bertram, L. Ein, and R. Lazarsfeld. Vanishing theorems, a theorem of Severi, and the equations defining projective varieties. J. Amer. Math. Soc. 4, 3 (1991), 587–602. © Springer Nature Switzerland AG 2022 W. Bruns et al., Determinants, Gröbner Bases and Cohomology, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-05480-8

487

488

References

19. A. Björner. Shellable and Cohen-Macaulay partially ordered sets. Trans. Amer. Math. Soc. 260, 1 (1980), 159–183. 20. S. Blum. Subalgebras of bigraded Koszul algebras. J. Algebra 242, 2 (2001), 795–809. 21. A. Boocher. Free resolutions and sparse determinantal ideals. Math. Res. Lett. 19, 4 (2012), 805–821. 22. K. Borna. On linear resolution of powers of an ideal. Osaka J. Math. 46, 4 (2009), 1047–1058. 23. L. Bossinger, S. Lamboglia, K. Mincheva, and F. Mohammadi. Computing toric degenerations of flag varieties. In Combinatorial algebraic geometry, vol. 80 of Fields Inst. Commun. Fields Inst. Res. Math. Sci., Toronto, ON, 2017, pp. 247–281. 24. N. Botbol and M. Chardin. Castelnuovo Mumford regularity with respect to multigraded ideals. J. Algebra 474 (2017), 361–392. 25. R. Bott. Homogeneous vector bundles. Ann. of Math. (2) 66 (1957), 203–248. 26. N. Bourbaki. Commutative algebra. Chapters 1–7. Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1998. 27. J.-F. Boutot. Singularités rationnelles et quotients par les groupes réductifs. Invent. Math. 88, 1 (1987), 65–68. 28. J. P. Brennan, J. Herzog, and B. Ulrich. Maximally generated Cohen-Macaulay modules. Math. Scand. 61, 2 (1987), 181–203. 29. H. Brenner and P. Monsky. Tight closure does not commute with localization. Ann. of Math. (2) 171, 1 (2010), 571–588. 30. M. Brion. Sur la théorie des invariants, vol. 45 of Publ. Math. Univ. Pierre et Marie Curie, 1981. 31. M. Brion and S. Kumar. Frobenius splitting methods in geometry and representation theory, vol. 231 of Progress in Mathematics. Birkhäuser Boston, Inc., Boston, MA, 2005. 32. M. Brodmann. Asymptotic stability of Ass(M/I n M). Proc. Amer. Math. Soc. 74, 1 (1979), 16–18. 33. M. P. Brodmann and R. Y. Sharp. Local cohomology: an algebraic introduction with geometric applications, vol. 60 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2013. 34. W. Bruns. Die Divisorenklassengruppe der Restklassenringe von Polynomringen nach Determinantenidealen. Rev. Roumaine Math. Pures Appl. 20, 10 (1975), 1109–1111. 35. W. Bruns. The canonical module of a determinantal ring. In Commutative algebra: Durham 1981 (Durham, 1981), vol. 72 of London Math. Soc. Lecture Note Ser. Cambridge Univ. Press, Cambridge-New York, 1982, pp. 109–120. 36. W. Bruns. Additions to the theory of algebras with straightening law. In Commutative algebra (Berkeley, CA, 1987), vol. 15 of Math. Sci. Res. Inst. Publ. Springer, New York, 1989, pp. 111– 138. 37. W. Bruns. Algebras defined by powers of determinantal ideals. J. Algebra 142, 1 (1991), 150–163. 38. W. Bruns and A. Conca. F-rationality of determinantal rings and their Rees rings. Michigan Math. J. 45, 2 (1998), 291–299. 39. W. Bruns and A. Conca. KRS and powers of determinantal ideals. Compositio Math. 111, 1 (1998), 111–122. 40. W. Bruns and A. Conca. Algebras of minors. J. Algebra 246, 1 (2001), 311–330. 41. W. Bruns and A. Conca. KRS and determinantal ideals. In Geometric and combinatorial aspects of commutative algebra (Messina, 1999), vol. 217 of Lecture Notes in Pure and Appl. Math. Dekker, New York, 2001, pp. 67–87. 42. W. Bruns and A. Conca. Gröbner bases and determinantal ideals. In Commutative algebra, singularities and computer algebra (Sinaia, 2002), vol. 115 of NATO Sci. Ser. II Math. Phys. Chem. Kluwer Acad. Publ., Dordrecht, 2003, pp. 9–66. 43. W. Bruns and A. Conca. The variety of exterior powers of linear maps. J. Algebra 322, 9 (2009), 2927–2949. 44. W. Bruns and A. Conca. Linear resolutions of powers and products. In Singularities and computer algebra. Springer, Cham, 2017, pp. 47–69.

References

489

45. W. Bruns and A. Conca. Products of Borel fixed ideals of maximal minors. Adv. in Appl. Math. 91 (2017), 1–23. 46. W. Bruns and A. Conca. A remark on regularity of powers and products of ideals. J. Pure Appl. Algebra 221, 11 (2017), 2861–2868. 47. W. Bruns, A. Conca, and M. Varbaro. Relations between the minors of a generic matrix. Adv. Math. 244 (2013), 171–206. 48. W. Bruns, A. Conca, and M. Varbaro. Maximal minors and linear powers. J. Reine Angew. Math. 702 (2015), 41–53. 49. W. Bruns and J. Gubeladze. Polytopal linear groups. J. Algebra 218, 2 (1999), 715–737. 50. W. Bruns and J. Gubeladze. Polytopes, rings, and K-theory. Springer Monographs in Mathematics, Springer, Dordrecht, 2009. 51. W. Bruns and J. Herzog. On the computation of a-invariants. Manuscripta Math. 77, 2-3 (1992), 201–213. 52. W. Bruns and J. Herzog. Cohen-Macaulay rings. Rev. ed. Cambridge: Cambridge University Press, 1998. 53. W. Bruns, J. Herzog, and U. Vetter. Syzygies and walks. In Commutative algebra (Trieste, 1992). World Sci. Publ., River Edge, NJ, 1994, pp. 36–57. 54. W. Bruns and M. Kwieci´nski. Generic graph construction ideals and Greene’s theorem. Math. Z. 233, 1 (2000), 115–126. 55. W. Bruns and G. Restuccia. Canonical modules of Rees algebras. J. Pure Appl. Algebra 201, 1-3 (2005), 189–203. 56. W. Bruns and T. Römer. h-vectors of Gorenstein polytopes. J. Combin. Theory Ser. A 114, 1 (2007), 65–76. 57. W. Bruns, T. Römer, and A. Wiebe. Initial algebras of determinantal rings, Cohen-Macaulay and Ulrich ideals. Michigan Math. J. 53, 1 (2005), 71–81. 58. W. Bruns and R. Schwänzl. The number of equations defining a determinantal variety. Bull. London Math. Soc. 22, 5 (1990), 439–445. 59. W. Bruns, A. Simis, and N. V. Trung. Blow-up of straightening-closed ideals in ordinal Hodge algebras. Trans. Amer. Math. Soc. 326, 2 (1991), 507–528. 60. W. Bruns and M. Varbaro. Partitions of single exterior type. Transform. Groups 19, 4 (2014), 969–978. 61. W. Bruns and U. Vetter. Determinantal rings, vol. 1327 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1988. 62. C. B˘ae¸tic˘a. Rees algebra of ideals generated by Pfaffians. Comm. Algebra 26, 6 (1998), 1769– 1778. 63. B. Buchberger. Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. Innsbruck: Univ. Innsbruck, Mathematisches Institut (Diss.), 1965. 64. B. Buchberger. Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems. Aequationes Math. 4 (1970), 374–383. 65. L. Caniglia, J. A. Guccione, and J. J. Guccione. Ideals of generic minors. Comm. Algebra 18, 8 (1990), 2633–2640. 66. D. Cartwright and B. Sturmfels. The Hilbert scheme of the diagonal in a product of projective spaces. Int. Math. Res. Not. IMRN, 9 (2010), 1741–1771. 67. F. Castillo, Y. Cid-Ruiz, B. Li, J. Montaño, and N. Zhang. When are multidegrees positive? Adv. Math. 374 (2020), 107382, 34. 68. G. Caviglia. Bounds on the Castelnuovo-Mumford regularity of tensor products. Proc. Amer. Math. Soc. 135, 7 (2007), 1949–1957. 69. K. A. Chandler. Regularity of the powers of an ideal. Comm. Algebra 25, 12 (1997), 3773– 3776. 70. M. Chardin. Some results and questions on Castelnuovo-Mumford regularity. In Syzygies and Hilbert functions, vol. 254 of Lect. Notes Pure Appl. Math. Chapman & Hall/CRC, Boca Raton, FL, 2007, pp. 1–40.

490

References

71. M. Chardin. Powers of ideals and the cohomology of stalks and fibers of morphisms. Algebra Number Theory 7, 1 (2013), 1–18. 72. M. Chardin. Powers of ideals: Betti numbers, cohomology and regularity. In Commutative algebra. Springer, New York, 2013, pp. 317–333. 73. M. Chardin. Regularity stabilization for the powers of graded M-primary ideals. Proc. Amer. Math. Soc. 143, 8 (2015), 3343–3349. 74. M. Chardin, D. Ghosh, and N. Nemati. The (ir)regularity of Tor and Ext. Trans. Amer. Math. Soc. 375, 1 (2022), 47–70. 75. M. Chardin, J.-P. Jouanolou, and A. Rahimi. The eventual stability of depth, associated primes and cohomology of a graded module. J. Commut. Algebra 5, 1 (2013), 63–92. 76. W. L. Chow. On unmixedness theorem. Amer. J. Math. 86 (1964), 799–822. 77. Y. Cid-Ruiz. Mixed multiplicities and projective degrees of rational maps. J. Algebra 566 (2021), 136–162. 78. A. Conca. Gröbner bases of ideals of minors of a symmetric matrix. J. Algebra 166, 2 (1994), 406–421. 79. A. Conca. Symmetric ladders. Nagoya Math. J. 136 (1994), 35–56. 80. A. Conca. The a-invariant of determinantal rings. Math. J. Toyama Univ. 18 (1995), 47–63. 81. A. Conca. Ladder determinantal rings. J. Pure Appl. Algebra 98, 2 (1995), 119–134. 82. A. Conca. Gorenstein ladder determinantal rings. J. London Math. Soc. (2) 54, 3 (1996), 453–474. 83. A. Conca. Straightening law and powers of determinantal ideals of Hankel matrices. Adv. Math. 138, 2 (1998), 263–292. 84. A. Conca. Regularity jumps for powers of ideals. In Commutative algebra, vol. 244 of Lect. Notes Pure Appl. Math. Chapman & Hall/CRC, Boca Raton, FL, 2006, pp. 21–32. 85. A. Conca. Linear spaces, transversal polymatroids and ASL domains. J. Algebraic Combin. 25, 1 (2007), 25–41. 86. A. Conca, E. De Negri, and E. Gorla. Universal Gröbner bases for maximal minors. Int. Math. Res. Not. IMRN, 11 (2015), 3245–3262. 87. A. Conca, E. De Negri, and E. Gorla. Multigraded generic initial ideals of determinantal ideals. In Homological and computational methods in commutative algebra, vol. 20 of Springer INdAM Ser. Springer, Cham, 2017, pp. 81–96. 88. A. Conca, E. De Negri, and E. Gorla. Cartwright-Sturmfels ideals associated to graphs and linear spaces. J. Comb. Algebra 2, 3 (2018), 231–257. 89. A. Conca, E. De Negri, and E. Gorla. Universal Gröbner bases and Cartwright-Sturmfels ideals. Int. Math. Res. Not. IMRN, 7 (2020), 1979–1991. 90. A. Conca, E. De Negri, and M. E. Rossi. Koszul algebras and regularity. In Commutative algebra. Springer, New York, 2013, pp. 285–315. 91. A. Conca and J. Herzog. On the Hilbert function of determinantal rings and their canonical module. Proc. Amer. Math. Soc. 122, 3 (1994), 677–681. 92. A. Conca and J. Herzog. Ladder determinantal rings have rational singularities. Adv. Math. 132, 1 (1997), 120–147. 93. A. Conca, J. Herzog, and G. Valla. Sagbi bases with applications to blow-up algebras. J. Reine Angew. Math. 474 (1996), 113–138. 94. A. Conca, S. Ho¸sten, and R. R. Thomas. Nice initial complexes of some classical ideals. In Algebraic and geometric combinatorics, vol. 423 of Contemp. Math. Amer. Math. Soc., Providence, RI, 2006, pp. 11–42. 95. A. Conca and M. Varbaro. Square-free Gröbner degenerations. Invent. Math. 221, 3 (2020), 713–730. 96. A. Conca and V. Welker. Lovász-Saks-Schrijver ideals and coordinate sections of determinantal varieties. Algebra Number Theory 13, 2 (2019), 455–484. 97. D. L. Costa. Retracts of polynomial rings. J. Algebra 44, 2 (1977), 492–502. 98. S. D. Cutkosky. Irrational asymptotic behaviour of Castelnuovo-Mumford regularity. J. Reine Angew. Math. 522 (2000), 93–103.

References

491

99. S. D. Cutkosky, J. Herzog, and N. V. Trung. Asymptotic behaviour of the CastelnuovoMumford regularity. Compositio Math. 118, 3 (1999), 243–261. 100. H. Dao, A. De Stefani, and L. Ma. Cohomologically full rings. Int. Math. Res. Not. IMRN, 17 (2021), 13508–13545. 101. H. Dao, C. Huneke, and J. Schweig. Bounds on the regularity and projective dimension of ideals associated to graphs. J. Algebraic Combin. 38, 1 (2013), 37–55. 102. R. Datta and K. E. Smith. Frobenius and valuation rings. Algebra Number Theory 10, 5 (2016), 1057–1090. 103. C. De Concini, D. Eisenbud, and C. Procesi. Young diagrams and determinantal varieties. Invent. Math. 56, 2 (1980), 129–165. 104. C. De Concini, D. Eisenbud, and C. Procesi. Hodge algebras, vol. 91 of Astérisque. Société Mathématique de France, Paris, 1982. 105. C. De Concini and C. Procesi. A characteristic free approach to invariant theory. Adv. Math. 21, 3 (1976), 330–354. 106. E. De Negri. K -algebras generated by Pfaffians. Math. J. Toyama Univ. 19 (1996), 105–114. 107. E. De Negri. Some results on Hilbert series and a-invariant of Pfaffian ideals. Math. J. Toyama Univ. 24 (2001), 93–106. 108. A. De Stefani, J. Montaño, and L. Núñez-Betancourt. Blowup algebras of determinantal ideals in prime characteristic. arXiv e-prints (Sept. 2021), arXiv:2109.00592. 109. W. Decker, G.-M. Greuel, G. Pfister, and H. Schönemann. Singular 4-1-0 — A computer algebra system for polynomial computations. http://www.singular.uni-kl.de, 2016. 110. M. Demazure. Une démonstration algébrique d’un théorème de Bott. Invent. Math. 5 (1968), 349–356. 111. M. Demazure. A very simple proof of Bott’s theorem. Invent. Math. 33, 3 (1976), 271–272. 112. P. Doubilet, G.-C. Rota, and J. Stein. On the foundations of combinatorial theory. IX. Combinatorial methods in invariant theory. Studies in Appl. Math. 53 (1974), 185–216. 113. J. A. Eagon and D. G. Northcott. Ideals defined by matrices and a certain complex associated with them. Proc. Roy. Soc. Ser. A 269 (1962), 188–204. 114. C. Eder and J.-C. Faugère. A survey on signature-based algorithms for computing Gröbner bases. J. Symbolic Comput. 80, part 3 (2017), 719–784. 115. D. Eisenbud. Introduction to algebras with straightening laws. In Ring theory and algebra, III (Proc. Third Conf., Univ. Oklahoma, Norman, Okla., 1979), vol. 55 of Lecture Notes in Pure and Appl. Math., Dekker, New York, 1980, pp. 243–268. 116. D. Eisenbud. Commutative algebra, vol. 150 of Graduate Texts in Mathematics. SpringerVerlag, New York, 1995. 117. D. Eisenbud and S. Goto. Linear free resolutions and minimal multiplicity. J. Algebra 88, 1 (1984), 89–133. 118. D. Eisenbud and J. Harris. Powers of ideals and fibers of morphisms. Math. Res. Lett. 17, 2 (2010), 267–273. 119. D. Eisenbud and J. Harris. 3264 and all that—a second course in algebraic geometry. Cambridge University Press, Cambridge, 2016. 120. D. Eisenbud and C. Huneke. Cohen-Macaulay Rees algebras and their specialization. J. Algebra 81, 1 (1983), 202–224. 121. D. Eisenbud, C. Huneke, and B. Ulrich. The regularity of Tor and graded Betti numbers. Amer. J. Math. 128, 3 (2006), 573–605. 122. D. Eisenbud and B. Ulrich. Notes on regularity stabilization. Proc. Amer. Math. Soc. 140, 4 (2012), 1221–1232. 123. R. Elkik. Singularités rationnelles et déformations. Invent. Math. 47, 2 (1978), 139–147. 124. V. Ene and J. Herzog. Gröbner bases in commutative algebra, vol. 130 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2012. 125. R. Fedder. F-purity and rational singularity. Trans. Amer. Math. Soc. 278, 2 (1983), 461–480. 126. R. Fedder and K. Watanabe. A characterization of F-regularity in terms of F-purity. In Commutative algebra (Berkeley, CA, 1987), vol. 15 of Math. Sci. Res. Inst. Publ. Springer, New York, 1989, pp. 227–245.

492

References

127. H. Flenner. Rationale quasihomogene Singularitäten. Arch. Math. (Basel) 36, 1 (1981), 35–44. 128. G. Fløystad. Poset ideals of P-partitions and generalized letterplace and determinantal ideals. Acta Math. Vietnam. 44, 1 (2019), 213–241. 129. R. M. Fossum. The divisor class group of a Krull domain, vol. 74 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, New York-Heidelberg, 1973. 130. F. G. Frobenius. Gesammelte Abhandlungen. Bände I, II, III. Springer-Verlag, Berlin-New York, 1968. 131. R. Fröberg. Determination of a class of Poincaré series. Math. Scand. 37, 1 (1975), 29–39. 132. W. Fulton. Flags, Schubert polynomials, degeneracy loci, and determinantal formulas. Duke Math. J. 65, 3 (1992), 381–420. 133. W. Fulton. Young tableaux, vol. 35 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1997. 134. O. Gabber. Notes on some t-structures. In Geometric aspects of Dwork theory. Vol. I, II. Walter de Gruyter, Berlin, 2004, pp. 711–734. 135. A. V. Geramita, A. Gimigliano, and Y. Pitteloud. Graded Betti numbers of some embedded rational n-folds. Math. Ann. 301, 2 (1995), 363–380. 136. I. Gessel and G. Viennot. Binomial determinants, paths, and hook length formulae. Adv. Math. 58, 3 (1985), 300–321. 137. S. R. Ghorpade. A note on Hodge’s postulation formula for Schubert varieties. In Geometric and combinatorial aspects of commutative algebra (Messina, 1999), vol. 217 of Lecture Notes in Pure and Appl. Math. Dekker, New York, 2001, pp. 211–219. 138. S. R. Ghorpade. Hilbert functions of ladder determinantal varieties. Discrete Math. 246, 1-3 (2002), 131–175. 139. S. R. Ghorpade and C. Krattenthaler. The Hilbert series of Pfaffian rings. In Algebra, arithmetic and geometry with applications (West Lafayette, IN, 2000). Springer, Berlin, 2004, pp. 337– 356. 140. D. Ghosh. Asymptotic linear bounds of Castelnuovo-Mumford regularity in multigraded modules. J. Algebra 445 (2016), 103–114. 141. G. Z. Giambelli. Ordini della varietá rappresentata coll’annullare tutti i minori di dato ordine estratti da una data matrice di forme. Atti Reale Accad. Lincei (5) XII (1903), 294–297. 142. G. Z. Giambelli. Ordine di una varietà piú ampia di quella rappresentata coll’annullare tutti i minori di dato ordine estratti da una data matrice generica di forme. Mem. del R. Ist. Lombardo Classe di Scienze 20 (1905), 101–135. 143. D. Glassbrenner. Strong F-regularity in images of regular rings. Proc. Amer. Math. Soc. 124, 2 (1996), 345–353. 144. D. Glassbrenner and K. E. Smith. Singularities of certain ladder determinantal varieties. J. Pure Appl. Algebra 101, 1 (1995), 59–75. 145. N. Gonciulea and V. Lakshmibai. Degenerations of flag and Schubert varieties to toric varieties. Transform. Groups 1, 3 (1996), 215–248. 146. N. Gonciulea and V. Lakshmibai. Schubert varieties, toric varieties, and ladder determinantal varieties. Ann. Inst. Fourier (Grenoble) 47, 4 (1997), 1013–1064. 147. N. Gonciulea and V. Lakshmibai. Singular loci of ladder determinantal varieties and Schubert varieties. J. Algebra 229, 2 (2000), 463–497. 148. P. Gordan. Les invariants des formes binaires. J. de Math. Pure et Appl. 6 (1900). 149. E. Gorla, J. C. Migliore, and U. Nagel. Gröbner bases via linkage. J. Algebra 384 (2013), 110–134. 150. S. Goto. On the Gorensteinness of determinantal loci. J. Math. Kyoto Univ. 19, 2 (1979), 371–374. 151. S. Goto and K. Watanabe. On graded rings. I. J. Math. Soc. Japan 30, 2 (1978), 179–213. 152. H.-G. Gräbe. Streckungsringe. Dissertation B, Erfurt/Mühlhausen, 1988. 153. H. Grauert. Über die Deformation isolierter Singularitäten analytischer Mengen. Invent. Math. 15 (1972), 171–198. 154. D. R. Grayson and M. E. Stillman. Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/.

References

493

155. M. L. Green. Koszul cohomology and the geometry of projective varieties. J. Differential Geom. 19, 1 (1984), 125–171. 156. M. L. Green. Koszul cohomology and the geometry of projective varieties. II. J. Differential Geom. 20, 1 (1984), 279–289. 157. G.-M. Greuel and G. Pfister. A Singular introduction to commutative algebra, extended ed. Springer, Berlin, 2008. 158. W. Gröbner. Über die algebraischen Eigenschaften der Integrale von linearen Differentialgleichungen mit konstanten Koeffizienten. Monatsh. Math. 47 (1939), 247–284. 159. F. D. Grosshans. Contractions of the actions of reductive algebraic groups in arbitrary characteristic. Invent. Math. 107, 1 (1992), 127–133. 160. A. Grothendieck. Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), vol. 4 of Documents Mathématiques (Paris) [Mathematical Documents (Paris)]. Société Mathématique de France, Paris, 2005. 161. W. J. Haboush. A short proof of the Kempf vanishing theorem. Invent. Math. 56, 2 (1980), 109–112. 162. N. Hara. A characterization of rational singularities in terms of injectivity of Frobenius maps. Amer. J. Math. 120, 5 (1998), 981–996. 163. J. Harris and L. W. Tu. On symmetric and skew-symmetric determinantal varieties. Topology 23, 1 (1984), 71–84. 164. R. Hartshorne. Residues and duality, vol. 20 of Lecture Notes in Mathematics. SpringerVerlag, Berlin-New York, 1966. 165. R. Hartshorne. Algebraic geometry, vol. 52 of Graduate Texts in Mathematics. SpringerVerlag, New York-Heidelberg, 1977. 166. M. Hashimoto. Determinantal ideals without minimal free resolutions. Nagoya Math. J. 118 (1990), 203–216. 167. M. Hashimoto. Determinantal ideals and their Betti numbers—a survey. In Modern aspects of combinatorial structure on convex polytopes. Proceedings of a conference held at the Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan, June 14–18, 1993. Kyoto: Kyoto Univ., Research Institute for Mathematical Sciences, 1994, pp. 40–50. 168. M. Hashimoto. F-pure homomorphisms, strong F-regularity, and F-injectivity. Comm. Algebra 38, 12 (2010), 4569–4596. 169. I. B. D. A. Henriques and M. Varbaro. Test, multiplier and invariant ideals. Adv. Math. 287 (2016), 704–732. 170. J. Herzog and T. Hibi. Monomial ideals, vol. 260 of Graduate Texts in Mathematics. SpringerVerlag London, Ltd., London, 2011. 171. J. Herzog, T. Hibi, and M. Vl˘adoiu. Ideals of fiber type and polymatroids. Osaka J. Math. 42, 4 (2005), 807–829. 172. J. Herzog and N. V. Trung. Gröbner bases and multiplicity of determinantal and Pfaffian ideals. Adv. Math. 96, 1 (1992), 1–37. 173. J. Herzog and W. V. Vasconcelos. On the divisor class group of Rees-algebras. J. Algebra 93, 1 (1985), 182–188. 174. T. Hibi. Distributive lattices, affine semigroup rings and algebras with straightening laws. In Commutative algebra and combinatorics (Kyoto, 1985), vol. 11 of Adv. Stud. Pure Math. North-Holland, Amsterdam, 1987, pp. 93–109. 175. D. Hilbert. Mathematical problems. Bull. Amer. Math. Soc. 8 (1902), 437–479. Translation of the German original by Mary Winston Newson. 176. H. Hironaka. Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II. Ann. of Math. (2) 79 (1964), 109–203; ibid. (2) 79 (1964), 205–326. 177. M. Hochster. Foundations of tight closure theory. Lecture notes from a course taught at the University of Michigan, Fall 2007. 178. M. Hochster. Cyclic purity versus purity in excellent Noetherian rings. Trans. Amer. Math. Soc. 231, 2 (1977), 463–488. 179. M. Hochster and J. A. Eagon. Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci. Amer. J. Math. 93 (1971), 1020–1058.

494

References

180. M. Hochster and C. Huneke. Tightly closed ideals. Bull. Amer. Math. Soc. (N.S.) 18, 1 (1988), 45–48. 181. M. Hochster and C. Huneke. Tight closure and strong F-regularity. Mém. Soc. Math. France (N.S.), 38 (1989), 119–133. 182. M. Hochster and C. Huneke. Tight closure, invariant theory, and the Briançon-Skoda theorem. J. Amer. Math. Soc. 3, 1 (1990), 31–116. 183. M. Hochster and C. Huneke. Tight closure of parameter ideals and splitting in module-finite extensions. J. Algebraic Geom. 3, 4 (1994), 599–670. 184. M. Hochster and J. L. Roberts. Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay. Adv. Math. 13 (1974), 115–175. 185. M. Hochster and J. L. Roberts. The purity of the Frobenius and local cohomology. Adv. Math. 21, 2 (1976), 117–172. 186. W. V. D. Hodge. Some enumerative results in the theory of forms. Proc. Cambridge Philos. Soc. 39 (1943), 22–30. 187. W. V. D. Hodge and D. Pedoe. Methods of algebraic geometry. Vol. II. Book III: General theory of algebraic varieties in projective space. Book IV: Quadrics and Grassmann varieties. Cambridge, at the University Press, 1952. 188. W. V. D. Hodge and D. Pedoe. Methods of algebraic geometry. Vol. I. Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1994. 189. J. W. Hoffman and H. H. Wang. Castelnuovo-Mumford regularity in biprojective spaces. Adv. Geom. 4, 4 (2004), 513–536. 190. J. Horiuchi, L. E. Miller, and K. Shimomoto. Deformation of F-injectivity and local cohomology. Indiana Univ. Math. J. 63, 4 (2014), 1139–1157. With an appendix by Karl Schwede and Anurag K. Singh. 191. H. Huang. Syzygies of Determinantal Thickenings. arXiv e-prints (Aug. 2020), arXiv:2008.02690. 192. H. Huang, M. Perlman, C. Polini, C. Raicu, and A. Sammartano. Relations between the 2 × 2 minors of a generic matrix. Adv. Math. 386 (2021), Paper No. 107807, 29. 193. J. E. Humphreys. Linear algebraic groups, vol. 21 of Graduate Texts in Mathematics. SpringerVerlag, New York-Heidelberg, 1975. 194. C. Huneke. Powers of ideals generated by weak d-sequences. J. Algebra 68, 2 (1981), 471– 509. 195. J.-i. Igusa. On the arithmetic normality of the Grassmann variety. Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 309–313. 196. J. C. Jantzen. Representations of algebraic groups, second ed., vol. 107 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2003. 197. A. Jensen. Gfan: a software system for Gröbner fans. available at http://home.imf.au.dk/ jensen/software/gfan/gfan.html.. 198. V. G. Kac. Lie superalgebras. Adv. Math 26, 1 (1977), 8–96. 199. V. G. Kac. A sketch of Lie superalgebra theory. Comm. Math. Phys. 53, 1 (1977), 31–64. 200. M. Kalkbrener and B. Sturmfels. Initial complexes of prime ideals. Adv. Math. 116, 2 (1995), 365–376. 201. D. Kapur and K. Madlener. A completion procedure for computing a canonical basis for a k-subalgebra. In Computers and mathematics (Cambridge, MA, 1989). Springer, New York, 1989, pp. 1–11. 202. K. Kaveh and A. G. Khovanskii. Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory. Ann. of Math. (2) 176, 2 (2012), 925–978. 203. K. Kaveh and C. Manon. Khovanskii bases, higher rank valuations, and tropical geometry. SIAM J. Appl. Algebra Geom. 3, 2 (2019), 292–336. 204. G. Kempf. On the geometry of a theorem of Riemann. Ann. of Math. (2) 98 (1973), 178–185. 205. G. Kempf, F. F. Knudsen, D. Mumford, and B. Saint-Donat. Toroidal embeddings. I, vol. 339 of Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 1973. 206. G. R. Kempf. Linear systems on homogeneous spaces. Ann. of Math. (2) 103, 3 (1976), 557–591.

References

495

207. G. R. Kempf. On the collapsing of homogeneous bundles. Invent. Math. 37, 3 (1976), 229–239. 208. G. R. Kempf. Some wonderful rings in algebraic geometry. J. Algebra 134, 1 (1990), 222–224. 209. H. Kleppe and D. Laksov. The algebraic structure and deformation of Pfaffian schemes. J. Algebra 64, 1 (1980), 167–189. 210. J. O. Kleppe and R. M. Miró-Roig. The representation type of determinantal varieties. Algebr. Represent. Theory 20, 4 (2017), 1029–1059. 211. F. Knop. Der kanonische Modul eines Invariantenrings. J. Algebra 127, 1 (1989), 40–54. 212. D. E. Knuth. Permutations, matrices, and generalized Young tableaux. Pacific J. Math. 34 (1970), 709–727. 213. D. E. Knuth. The art of computer programming. Vol. 3. Addison-Wesley, Reading, MA, 1998. 214. A. Knutson and E. Miller. Gröbner geometry of Schubert polynomials. Ann. of Math. (2) 161, 3 (2005), 1245–1318. 215. V. Kodiyalam. Asymptotic behaviour of Castelnuovo-Mumford regularity. Proc. Amer. Math. Soc. 128, 2 (2000), 407–411. 216. M. Koley and M. Varbaro. Gröbner deformation and F-singularities. arXiv e-prints (July 2021), arXiv:2107.12116. 217. J. Kollár and S. J. Kovács. Deformations of log canonical singularities. arXiv e-prints (Mar. 2018), arXiv:1803.03325. 218. J. Kollár and S. J. Kovács. Deformations of log canonical and F-pure singularities. Algebr. Geom. 7, 6 (2020), 758–780. 219. H. Kraft. Geometrische Methoden in der Invariantentheorie. Aspects of Mathematics, D1, Friedr. Vieweg & Sohn, Braunschweig, 1984. 220. C. Krattenthaler. Counting nonintersecting lattice paths with turns. Sém. Lothar. Combin. 34 (1995), Art. B34i, approx. 17 pp. 221. C. Krattenthaler and M. Prohaska. A remarkable formula for counting nonintersecting lattice paths in a ladder with respect to turns. Trans. Amer. Math. Soc. 351, 3 (1999), 1015–1042. 222. C. Krattenthaler and M. Rubey. A determinantal formula for the Hilbert series of one-sided ladder determinantal rings. In Algebra, arithmetic and geometry with applications (West Lafayette, IN, 2000). Springer, Berlin, 2004, pp. 525–551. 223. H. Kredel and V. Weispfenning. Computing dimension and independent sets for polynomial ideals. J. Symbolic Comput. 6, 2-3 (1988), 231–247. 224. M. Kreuzer and L. Robbiano. Computational commutative algebra. 1. Springer-Verlag, Berlin, 2000. 225. D. M. Kulkarni. Hilbert polynomial of a certain ladder-determinantal ideal. J. Algebraic Combin. 2, 1 (1993), 57–71. 226. D. M. Kulkarni. Counting of paths and coefficients of the Hilbert polynomial of a determinantal ideal. Discrete Math. 154, 1-3 (1996), 141–151. 227. E. Kunz. Characterizations of regular local rings of characteristic p. Amer. J. Math. 91 (1969), 772–784. 228. K. Kurano. The first syzygies of determinantal ideals. J. Algebra 124, 2 (1989), 414–436. 229. R. E. Kutz. Cohen-Macaulay rings and ideal theory in rings of invariants of algebraic groups. Trans. Amer. Math. Soc. 194 (1974), 115–129. 230. V. Lakshmibai and J. Brown. Flag varieties, vol. 53 of Texts and Readings in Mathematics. Hindustan Book Agency, Delhi, 2018. 231. A. Lascoux. Syzygies des variétés déterminantales. Adv. Math. 30, 3 (1978), 202–237. 232. E. Lasker. Zur Theorie der Moduln und Ideale. Math. Ann. 60, 1 (1905), 20–116. 233. R. Lazarsfeld and M. Musta¸ta˘ . Convex bodies associated to linear series. Ann. Sci. Éc. Norm. Supér. (4) 42, 5 (2009), 783–835. 234. A. C. L˝orincz and C. Raicu. Iterated local cohomology groups and Lyubeznik numbers for determinantal rings. Algebra Number Theory 14, 9 (2020), 2533–2569. 235. A. C. L˝orincz and U. Walther. On categories of equivariant D-modules. Adv. Math. 351 (2019), 429–478. 236. G. Lyubeznik. On the local cohomology modules Hai (R) for ideals a generated by monomials in an R-sequence. In Complete intersections (Acireale, 1983), vol. 1092 of Lecture Notes in Math. Springer, Berlin, 1984, pp. 214–220.

496

References

237. G. Lyubeznik, A. K. Singh, and U. Walther. Local cohomology modules supported at determinantal ideals. J. Eur. Math. Soc. (JEMS) 18, 11 (2016), 2545–2578. 238. G. Lyubeznik and K. E. Smith. Strong and weak F-regularity are equivalent for graded rings. Amer. J. Math. 121, 6 (1999), 1279–1290. 239. Y. Ma. On the minors defined by a generic matrix. J. Symbolic Comput. 18, 6 (1994), 503–518. 240. F. S. Macaulay. Some properties of enumeration in the theory of modular systems. Proc. London Math. Soc. 26 (1927), 531–555. 241. I. G. Macdonald. Symmetric functions and Hall polynomials, second ed. Oxford Classic Texts in the Physical Sciences, The Clarendon Press, Oxford University Press, New York, 2015. 242. D. Maclagan and G. G. Smith. Multigraded Castelnuovo-Mumford regularity. J. Reine Angew. Math. 571 (2004), 179–212. 243. H. Matsumura. Commutative ring theory, second ed., vol. 8 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1989. 244. V. B. Mehta and A. Ramanathan. Frobenius splitting and cohomology vanishing for Schubert varieties. Ann. of Math. (2) 122, 1 (1985), 27–40. 245. E. Miller and B. Sturmfels. Combinatorial commutative algebra, vol. 227 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2005. 246. L. E. Miller, A. K. Singh, and M. Varbaro. The F-pure threshold of a determinantal ideal. Bull. Braz. Math. Soc. (N.S.) 45, 4 (2014), 767–775. 247. J. S. Milne. Algebraic groups, vol. 170 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2017. 248. M. Miró-Roig. Determinantal ideals, vol. 264 of Progress in Mathematics. Birkhäuser Verlag, Basel, 2008. 249. S. B. Mulay. Determinantal loci and the flag variety. Adv. Math. 74, 1 (1989), 1–30. 250. D. Mumford. Lectures on curves on an algebraic surface. With a section by G. M. Bergman. Annals of Mathematics Studies, No. 59, Princeton University Press, Princeton, N.J., 1966. 251. H. Narasimhan. The irreducibility of ladder determinantal varieties. J. Algebra 102, 1 (1986), 162–185. 252. Ngô Viêt Trung. On the symbolic powers of determinantal ideals. J. Algebra 58 (1979), 361–369. 253. A. Okounkov. Brunn-Minkowski inequality for multiplicities. Invent. Math. 125, 3 (1996), 405–411. 254. A. Ooishi. Castelnuovo’s regularity of graded rings and modules. Hiroshima Math. J. 12, 3 (1982), 627–644. 255. M. Perlman. Lyubeznik numbers for Pfaffian rings. J. Pure Appl. Algebra 224, 5 (2020), 106247, 24. 256. M. Perlman. Regularity and cohomology of pfaffian thickenings. J. Commut. Algebra (2020). 257. M. Perlman and C. Raicu. Hodge ideals for the determinant hypersurface. Selecta Math. (N.S.) 27, 1 (2021), Paper No. 1. 258. C. Peskine and L. Szpiro. Dimension projective finie et cohomologie locale. Applications à la démonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck. Inst. Hautes Études Sci. Publ. Math., 42 (1973), 47–119. 259. P. Pragacz and J. Weyman. Complexes associated with trace and evaluation. Another approach to Lascoux’s resolution. Adv. Math. 57, 2 (1985), 163–207. 260. C. Procesi. Lie groups. Universitext, Springer, New York, 2007. 261. C. Raicu. Characters of equivariant D-modules on spaces of matrices. Compos. Math. 152, 9 (2016), 1935–1965. 262. C. Raicu. Regularity and cohomology of determinantal thickenings. Proc. Lond. Math. Soc. (3) 116, 2 (2018), 248–280. 263. C. Raicu and J. Weyman. Local cohomology with support in generic determinantal ideals. Algebra Number Theory 8, 5 (2014), 1231–1257. 264. C. Raicu and J. Weyman. Local cohomology with support in ideals of symmetric minors and Pfaffians. J. Lond. Math. Soc. (2) 94, 3 (2016), 709–725.

References

497

265. C. Raicu and J. Weyman. The syzygies of some thickenings of determinantal varieties. Proc. Amer. Math. Soc. 145, 1 (2017), 49–59. 266. C. Raicu and J. Weyman. Syzygies of determinantal thickenings and representations of the general linear Lie superalgebra. Acta Math. Vietnam. 44, 1 (2019), 269–284. 267. C. Raicu, J. Weyman, and E. E. Witt. Local cohomology with support in ideals of maximal minors and sub-maximal Pfaffians. Adv. Math. 250 (2014), 596–610. 268. G. Restuccia and G. Rinaldo. On certain classes of degree reverse lexicographic Gröbner bases. Int. Math. Forum 2, 21-24 (2007), 1053–1068. 269. L. Robbiano and M. Sweedler. Subalgebra bases. In Commutative algebra (Salvador, 1988), vol. 1430 of Lecture Notes in Math. Springer, Berlin, 1990, pp. 61–87. 270. T. Römer. Homological properties of bigraded algebras. Illinois J. Math. 45, 4 (2001), 1361– 1376. 271. J. J. Rotman. An introduction to homological algebra, second ed. Universitext, Springer, New York, 2009. 272. M. Rubey and C. Stump. Crossings and nestings in set partitions of classical types. arXiv e-prints (Apr. 2009), arXiv:0904.1097. 273. B. E. Sagan. The symmetric group, second ed., vol. 203 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2001. 274. S. V. Sam. Derived supersymmetries of determinantal varieties. J. Commut. Algebra 6, 2 (2014), 261–286. 275. S. V. Sam and A. Snowden. Cohomology of flag supervarieties and resolutions of determinantal ideals. arXiv e-prints (Aug. 2021), arXiv:2108.00504. 276. A. Sammartano. Blowup algebras of rational normal scrolls. Trans. Amer. Math. Soc. 373, 2 (2020), 797–818. 277. P. Schenzel. On the use of local cohomology in algebra and geometry. In Six lectures on commutative algebra (Bellaterra, 1996), vol. 166 of Progr. Math. Birkhäuser, Basel, 1998, pp. 241–292. 278. L. Seccia. Knutson ideals of generic matrices. Proc. Amer. Math. Soc. 150, 5 (2022), 1967– 1973. 279. C. Segre. Gli ordini delle varietà che annullano i determinanti dei diversi gradi estratti da una data matrice. Atti Reale Accad. Lincei (5) IX (1900), 253–260. 280. J.-P. Serre. Représentations linéaires et espaces homogènes kählériens des groupes de Lie compacts (d’après Armand Borel et André Weil). In Séminaire Bourbaki, Vol. 2, Exp. No. 100. Soc. Math. France, Paris, 1995, pp. 447–454. 281. D. W. Sharpe. On certain polynomial ideals defined by matrices. Quart. J. Math. Oxford Ser. (2) 15 (1964), 155–175. 282. D. W. Sharpe. The syzygies and semi-regularity of certain ideals defined by matrices. Proc. London Math. Soc. (3) 15 (1965), 645–679. 283. J. Sidman. On the Castelnuovo-Mumford regularity of products of ideal sheaves. Adv. Geom. 2, 3 (2002), 219–229. 284. A. Simis and B. Ulrich. On the ideal of an embedded join. J. Algebra 226, 1 (2000), 1–14. 285. A. K. Singh. Deformation of F-purity and F-regularity. J. Pure Appl. Algebra 140, 2 (1999), 137–148. 286. A. K. Singh. F-regularity does not deform. Amer. J. Math. 121, 4 (1999), 919–929. 287. A. K. Singh, S. Takagi, and M. Varbaro. A Gorenstein criterion for strongly F-regular and log terminal singularities. Int. Math. Res. Not. IMRN, 21 (2017), 6484–6522. 288. K. E. Smith. F-rational rings have rational singularities. Amer. J. Math. 119, 1 (1997), 159– 180. 289. D. Soll and V. Welker. Type-B generalized triangulations and determinantal ideals. Discrete Math. 309, 9 (2009), 2782–2797. 290. D. Speyer and B. Sturmfels. The tropical Grassmannian. Adv. Geom. 4, 3 (2004), 389–411. 291. Stacks Project. https://stacks.math.columbia.edu, 2021. 292. R. P. Stanley. Enumerative combinatorics. Vol. 2, vol. 62 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1999.

498

References

293. R. Steinberg. Conjugacy classes in algebraic groups, vol. 386 of Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 1974. 294. B. Sturmfels. Gröbner bases and Stanley decompositions of determinantal rings. Math. Z. 205, 1 (1990), 137–144. 295. B. Sturmfels. Gröbner bases of toric varieties. Tohoku Math. J. (2) 43, 2 (1991), 249–261. 296. B. Sturmfels. Gröbner bases and convex polytopes, vol. 8 of University Lecture Series. American Mathematical Society, Providence, RI, 1996. 297. B. Sturmfels. Algorithms in invariant theory, second ed. Texts and Monographs in Symbolic Computation, Springer, Vienna, 2008. 298. B. Sturmfels and S. Sullivant. Combinatorial secant varieties. Pure Appl. Math. Q. 2, 3, Special Issue: In honor of Robert D. MacPherson. Part 1 (2006), 867–891. 299. B. Sturmfels, N. V. Trung, and W. Vogel. Bounds on degrees of projective schemes. Math. Ann. 302, 3 (1995), 417–432. 300. B. Sturmfels and A. Zelevinsky. Maximal minors and their leading terms. Adv. Math. 98, 1 (1993), 65–112. 301. S. Sullivant. Combinatorial symbolic powers. J. Algebra 319, 1 (2008), 115–142. 302. T. Svanes. Coherent cohomology on Schubert subschemes of flag schemes and applications. Adv. Math. 14 (1974), 369–453. 303. I. Swanson. Powers of ideals. Primary decompositions, Artin-Rees lemma and regularity. Math. Ann. 307, 2 (1997), 299–313. 304. I. Swanson and C. Huneke. Integral closure of ideals, rings, and modules, vol. 336 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2006. 305. N. V. Trung and H.-J. Wang. On the asymptotic linearity of Castelnuovo-Mumford regularity. J. Pure Appl. Algebra 201, 1-3 (2005), 42–48. 306. B. L. van der Waerden. On Hilbert’s function, series of composition of ideals and a generalization of the theorem of Bézout. Proc. R. Acad. Amst. 31 (1929), 749–770. 307. B. L. van der Waerden. On varieties in multiple-projective spaces. Nederl. Akad. Wetensch. Indag. Math. 40, 2 (1978), 303–312. 308. M. Varbaro. Grobner deformations, connectedness and cohomological dimension. J. Algebra 322, 7 (2009), 2492–2507. 309. M. Varbaro. Cohomological and Combinatorial Methods in the Study of Symbolic Powers and Equations defining Varieties. PhD thesis, Università degli studi di Genova, 2011. Available at https://arxiv.org/pdf/1105.5507.pdf. 310. M. Varbaro. F-thresholds, integral closure and convexity. In Homological and computational methods in commutative algebra, vol. 20 of Springer INdAM Ser. Springer, Cham, 2017, pp. 249–256. 311. W. V. Vasconcelos. Computational methods in commutative algebra and algebraic geometry, vol. 2 of Algorithms and Computation in Mathematics. Springer-Verlag, Berlin, 1998. 312. J. D. Vélez. Openness of the F-rational locus and smooth base change. J. Algebra 172, 2 (1995), 425–453. 313. R. H. Villarreal. Monomial algebras, second ed. Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2015. 314. M. C. Vipera. Monomial ideals of polynomials over Noetherian rings. Rend. Sem. Mat. Univ. Politec. Torino 47, 2 (1989), 125–137 (1992). 315. K.-i. Watanabe. F-rationality of certain Rees algebras and counterexamples to “Boutot’s theorem” for F-rational rings. J. Pure Appl. Algebra 122, 3 (1997), 323–328. 316. W. C. Waterhouse. Automorphisms of det(X i j ): the group scheme approach. Adv. Math. 65, 2 (1987), 171–203. 317. J. Weyman. Cohomology of vector bundles and syzygies, vol. 149 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2003. 318. O. Zariski and P. Samuel. Commutative algebra. Vol. 1. Springer-Verlag, New YorkHeidelberg-Berlin, 1975. 319. O. Zariski and P. Samuel. Commutative algebra. Vol. II. Springer-Verlag, New YorkHeidelberg, 1975.

Index

Symbols (0, 1)-graded structure, 288 (1, 0)-graded structure, 288 1-cogenerated ideal, 83, 220 determinantal ring defined by, 220

A Abelianization, 355 Affine monoid, 173 Affine monoid algebra normal and squarefree initial ideal, 177 canonical module, 175 Cohen–Macaulay, 174 divisor class group, 176 divisorial ideals, 175 Gorenstein, 175 strongly F-regular, 249 polytopal, 176 prime ideal of, 174 ai (M), 276, 284, 299 Algebra of maximal minors. see Grassmannian Algebra of minors, 195 bound for Sagbi basis, 198 canonical module, 196 Castelnuovo–Mumford regularity, 198 class group, 197 class of canonical module, 197 Cohen–Macaulayness, 196 dimension, 195 Gorensteinness, 197 G-stable prime ideal, 212 normality, 196 rational singularities, 258 relations, 198–207

cubic, 199 t = 2, 205 t = 3, 206 Sagbi basis, 19 singular locus, 217 strongly F-rational, 258 Algebra of permanents, 204 Algebra with straightening law, 83 and Gröbner basis, 180 symmetric, 180 Algebraic group, 350 additive Ga , 351 character, 353 diagonalizable, 356 eigenspace, 356 general linear, 352 invariants, 356 inverse, 350 module, 356 morphism, 352 multiplication, 350 multiplicative Gm , 351 subgroup, 354 unit element, 350 Almost regular element, 285, 314, 316 sequence, 286 Antichain, 108 Arithmetic rank, 467 ASL. see algebra with straightening law Associated graded ring, 186 defined by filtration, 261 Associated primes of initial ideal, 44–46

B Bi-shape, 201

© Springer Nature Switzerland AG 2022 W. Bruns et al., Determinants, Gröbner Bases and Cohomology, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-05480-8

499

500 single exterior type, 202 Bigheight, 44, 46 Bigraded module, 288 ring, 287 Binomial, 18 ideal, 18 Gröbner basis, 18 Bitableau, 71 content, 78 partial order, 72 standard, 72 Blum’s theorem, 307 Borel–Weil–Bott theorem, 417 for Grassmannians, 424 Boutot’s theorem, 264 Brion’s theorem, 264 Buchberger algorithm, 11 and syzygies, 12 criterion, 11 Bundle affine, 329 flag, 331 Grassmann, 329 projective, 329

C Canonical module. see specific type of ring Canonical sequence, 120 Canonical sheaf flag bundle (relative), 333 Grassmann bundle (relative), 330 projective space, 330 Cartwright–Sturmfels ideal, 169 Castelnuovo–Mumford regularity. see regularity Catenary, 52 Cauchy’s formula characteristic free, 96, 367 characteristic zero, 99, 435 exterior powers, 470 ˇ Cech complex, 337 local cohomology, 467 Chain, 108 Chain complex, 108 Closed immersion, 329 Cofinal, 53 Cohen–Macaulay ring. see specific type of ring Cohomological dimension, 467 Cohomologically full rings, 58–66

Index and flatness of Ext, 59 Cohen–Macaulay rings, 58 consequences of, 61 Stanley–Reisner rings, 66 Coinverse, 351 Column standard, 116 Compatibly split, 233 Composition factors Cauchy filtration, 367 tautological filtration, 332 Comultiplication, 351 Concatenation of shapes, 380 Cone, 173 Conical, 173 Connected in codimension 1 and (S2 ), 52 in codimension k, 50–56 Connectedness, 50–56 theorem of Grothendieck, 55 Content, 78 Counit, 351 Cover (in a poset), 100

D Danilov–Stanley theorem, 175 Decomposition of polynomial ring into irreducibles, 98 Defined by shape, 99–103 Deglex, 3 Degree lexicographic order, 3 Deletion, 113 Derivation, 433 Desingularization determinantal variety, 411 module, 401 Determinant free module, 326 locally free sheaf, 328 Determinantal ideal and secants of 2-minors, 140, 145 arithmetic rank, 467 cohomological dimension, 467 cohomology with support in, 466 first syzygy module, 475 F-pure threshold of, 266 free resolution, 470 Gröbner basis, 8, 145 2-minors, 106–113 of power for maximal minors, 126 for t-minors, 122 of power, 125 of product, 124

Index of symbolic power, 124 height bound, 86 initial ideal. see Gröbner basis integral closure of power, 94 of Hankel matrix, 138 of maximal minors and linear powers, 313, 320 of matrix of linear forms, 313–321 symbolic and ordinary powers, 91 of symmetric matrix, 138 perfection, 87 primary decomposition of power, 91 Rees algebra canonical module, 193 Cohen–Macaulayness, 191 normality, 191 Rees algebra of product Cohen–Macaulayness, 190 initial algebra, 190 normality, 190 standard bitableaux basis of power, 93 symbolic power, 88–91 initial ideal, 188 regularity, 408, 428 standard bitableaux basis, 90 symbolic Rees algebra, 189 Cohen–Macaulayness, 190 initial algebra, 190 normality, 190 valuation defined by, 88 Determinantal ring, 82–87 a-invariant, 136 as ring of invariants, 220 canonical module, 136, 224 Castelnuovo–Mumford regularity, 136, 438 Cohen–Macaulay ideals in, 226 Cohen–Macaulayness, 86, 87, 133, 413 divisor class group, 222, 413 divisorial ideals, 223 F-pure threshold of maximal ideal, 273 free resolution, 470 F-split, 242 Gorensteinness, 87, 136 graded automorphism group, 85 Hilbert series, 134, 135 Krull dimension, 85, 411 localization, 84 mixed multiplicities, 152 multidegree, 150, 152 multiplicity, 132 multiplicity free multidegree, 154 normal domain, 85, 411

501 properties for 2-minors, 112 rational singularities, 257, 411 representation type, 226 singular locus, 85 standard bitableaux basis, 83 strongly F-regular, 250 toric deformation, 219 initial algebra, 219 Sagbi basis, 219 type, 137 Ulrich ideals in, 226 w.r.t. 2-minors is Koszul, 107 Diagonal, 80 monomial order, 80 Dickson’s lemma, 383 Dimension of simplicial complex, 47 Divided Koszul syzygy, 10 Divided powers free module, 325 locally free sheaf, 327 Divisor class group, 176, 222, 413 Dominance order, 79 Dominant weights, 340 Dual free module, 327 locally free sheaf, 328 Schur functor, 420 Dual shape, 119 Dualizing complex, 370

E Eagon–Northcott complex, 186 Ehrhart reciprocity, 176 series, 176 Elementary symmetric function, 14 Elimination of variables, 13 Equivariant Betti polynomial, 471 Equivariant deformation, 261 Exceptional inverse image, 370 Extended Rees algebra. see Rees algebra Exterior powers free module, 326 linear maps, 208 locally free sheaf, 327

F Face, 47 relevant, 148 Face ideal, 48 Facet, 47

502 Fedder’s criterion, 240 F-finite, 230 Fiber locally free sheaf, 329 Fiber type, 305 Filtration, 53 Cauchy, 367 GL-invariant ideals, 455 ideals defined by shape, 381 of polynomial ring by shape, 261 sheaves defined by shape, 395 Flag variety, 79 Cohen–Macaulayness, 182 complete (full), 80, 332 factoriality, 182 Gorensteinness, 182 initial algebra, 182 Koszulness, 182 normality, 182 partial, 80, 331 rational singularities, 257 relative, 331 Sagbi basis, 182 strongly F-rational, 257 F-pure, 233, 235, 241 does not deform, 254 for pair, 265 threshold, 265 F-rational, 250 and Gröbner deformation, 254 and Sagbi deformation, 255 F-rational type, 252 and rational singularities, 253 F-regular, 246 Frobenius map, 230 Frobenius power, 234 iterated, 243 F-split, 233, 235, 241 counterexample, 234 regular counterexample, 235 F-splitting, 233 of polynomial ring, 236 and initial ideal, 239 and squarefree monomial ideals, 237

G Gauß polynomials, 465 General linear Lie superalgebra, 476 Generalized Cohen–Macaulay, 62 Generalized permanents, 452 Geometric technique, 473 Gessel–Viennot formula, 131

Index g-graded. see graded GL-invariant ideals classification, 451 definition, 449 intersection, 453 log-canonical threshold, 270 quotients, 453 G-module, 95 Gordan’s lemma, 174 Gorenstein ring. see specific type of ring Graded algebra, 24 component, 24 module, 24 subspace, 24 Grading, 24 monoid, 172 Grassmannian, 73 a-invariant, 181 as ring of invariants, 181 Castelnuovo–Mumford regularity, 181 Cohen–Macaulayness, 180 defining ideal, 77 Gröbner basis, 179 dimension, 77 factoriality, 180 Gorensteinness, 180 initial algebra, 177 iterated, 331 Koszulness, 180 normality, 180 Plücker relations, 77 relative, 329 Sagbi basis, 177 Greene’s theorem, 119 Gröbner basis, 5–13 and linear independence of monomials, 6 and regular sequence, 10 and unique reduction, 6 generates ideal, 7 lifting criterion, 9 lifting Koszul syzygies, 10 of determinantal ideal. see determinantal ideal reduced, 12 universal. see universal Gröbner basis Gröbner deformation. see also Gröbner or Sagbi deformation Bigheight, 44 bigheight, 46 connectedness, 55 unmixedness, 44

Index with squarefree initial ideal, 66 Gröbner or Sagbi deformation, 33–41 canonical module, 41 homological invariants, 37, 39 Koszul property, 39 ring-theoretic properties, 40 Serre properties, 35 Grothendieck connectedness theorem, 55 Grothendieck duality, 369 Grothendieck vanishing theorem absolute, 337 for flag bundles, 337 Group of characters, 353 G-RSK, 123 G-stable prime ideal and face of cone of weights, 213

H Height of partition, 341 Higher direct image functor, 334 Hilbert function, 24 multigraded, 146 of initial subspace, 24 polynomial multigraded, 147 series, 24 Hochster’s theorem, 174 Hochster–Roberts theorem, 264 Homogeneous, 24 Homogenization, 32, 45 Homogenizing variable, 32 Hopf algebra commutative, 351 comodule, 356

I Ideal defined by shape, 102 and RSK, 126 filtration of, 381 F-pure threshold, 270 quotients of, 380 Ideal membership problem, 13 Ideal of minors. see determinantal ideal Inc-decomposition, 118 Increasing, 71 Initial coefficient, 4 monomial, 4, 8 term (or form), 4, 8, 27

503 w.r.t. weight, 26 Initial algebra. see also Gröbner or Sagbi deformation, 14 of tensor product, 22 Initial ideal, 4 2-minors, 165 and reg(1,0) , 306 and regular sequence, 28 Betti numbers 2-minors, 166 Bigheight, 47 bigheight, 47 connected in codim k, 55 Gorenstein, 169 in subalgebra, 21 maximal minors, 163 multiplicity criterion, 25 of determinantal ideal. see determinantal ideal radical, 46–50 set is finite, 160 squarefree properties preserved by, 66 sum and intersection, 48 unmixed, 48 Initial subalgebra. see initial algebra Initial subspace, 22 and Hilbert function, 25 Initial tableau, 79 in-RSK, 123 Insertion, 114 Inversion in permutation, 416 Irreducible, 98 Irrelevant prime ideal, 146 Isotypic component, 98, 449 J Join, 138 and initial ideal, 144 of simplicial complexes, 142 properties, 141 K Kempf vanishing theorem, 341 Khovanskii basis, 14 Knuth relation, 119 Kostka number, 151 Koszul algebra, 39 Koszul homology, 278 of Rees module, 296 Koszul resolution locally free sheaves, 328

504 Künneth formula, 376 L Ladder determinantal ring, 136, 137 Length path, 109 permutation, 416 Leray spectral sequence absolute, 334 degenerate, 334 relative, 334 Lex (order), 3 Lexicographic order, 3 Lifting binomial relations, 16–19 syzygies, 7–11, 27 Light and shadow decomposition, 130 Linear powers, 306 and reg(1,0) , 306 and almost regular element, 316 products, 307 example of family with, 308–313 Linear halfspace, 173 Linear resolution, 283 and almost regular element, 314 of truncated module, 283 Linear topology, 53 Linearly reductive, 98 Local cohomology, 465 with support, 276 Localization argument, 84 Locally split inclusion, 363 M Minimal free resolution, 279 generator, 290 generators (of ideal of relations), 199 set of generators, 277 Minimal primes of initial ideal, 44–46 Minkowski–Weyl theorem, 173 Minor, 70 Mixed multidegreess, 147 multiplicities, 147 Monoid, 172 affine, 173 algebra. see also affine monoid algebra, 172 Monomial, 2 in a family of ring elements, 15

Index Monomial ideal minimal generator, 160 Monomial order, 2 approximation by weight vector, 31 deglex, 3 lex, 3 on free module, 4 revlex, 3 Monomial symmetric polynomial, 151 Multi-Rees algebra, 301 module, 302 Multidegree, 147 multiplicity free, 154 of Stanley–Reisner ring, 148 Multiplicity, 25 Multiplicity free, 264

N Newton–Okounkov body, 182 Noninc-decomposition, 118 Nonincreasing, 71 Nonintersecting paths, 129 Normalization, 174 Northeast ideal of maximal minors, 127, 183, 313

O Orbits in variety of exterior powers of linear maps, 216 Order complex, 108

P Partition, 341 Path, 108 Perfect ideal, 87 Perfect pairing complementary exterior powers, 327 divided and symmetric powers, 327 exterior powers and duals, 327 sheaves defined by shape, 434 symmetric powers characteristic 0, 433 tensors and duals, 327 Permutation Grassmannian, 427 Pfaffian ideal, 138 Pieri’s rule, 449 Pivot position, 114 Plücker embedding, 330 line bundle, 330

Index relations, 73–74 Polynomial mark, 160 Predecessor (of shape), 202 Presheaf, 361 Projection formula, 333 Pure map, 231 Q q-binomial coefficients, 465 R Rank divided power, 326 exterior power, 326 Schur functor, 362 sheaf of differentials on Grassmannian, 330 symmetric power, 326 tensor power, 325 Rational cone, 173 Rational singularities, 252 and Sagbi deformation, 255 Reduced Gröbner basis, 12 Reduction algorithm, 5 of polynomial, 5 unique, 6 Reduction (of ideal), 297 with respect to module, 298 Rees algebra, 183, 295 extended, 186 Koszulness and linear resolution of powers, 307 of maximal ideal, 187 of maximal minors Cohen–Macaulayness, 186 defining ideal of, 185 initial ideal, 183 Koszulness, 186 normality, 186 Sagbi basis, 183 Rees valuations, 194 Reflexive module, 272 Regular sequence, 329 Hilbert function criterion, 161 Regularity, 276 along chain of submodules, 290 and almost regular element, 286 and linear resolution, 283 and multigraded structures, 287–289 and reduction, 298 and regular element, 287

505 and short exact sequence, 277, 285 comparison to free resolution, 280 comparison to Koszul homology, 280 minimal, 283 of powers, 296, 297 of product examples of nonlinearity, 303 for constant generator degree, 303 of products, 302 of saturation, 299 symbolic powers, 408, 428 Relative ideal defined by shape, 393 polynomial ring, 392 quotient defined by shape, 396 setting, 334 Relevant face, 148 Representation, 95 Residue field at a point on a variety, 329 Reverse lexicographic order, 3 Revlex (order), 3 Right turn, 109 Ring of invariants, 220 canonical module, 220 Robinson–Schensted–Knuth correspondence. see also RSK, 113–122 RSK algorithm, 115 and initial ideal, 122 invariance of α, 119 invariance of γ , 121 invariants, 117–122 is bijection, 116 properties, 117 step, 114 S Sagbi basis, 13–26 algebraically independent initials, 19 algorithm, 20 and Gröbner basis, 21 and linear independence of monomials, 16 and unique subduction, 16 criterion, 20 example of infinite, 15 generates subalgebra, 16 lifting criterion, 17, 178, 184, 311 universal 2-minors, 166 higher order minors, 166 Sagbi deformation. see Gröbner or Sagbi deformation

506 Saturation, 299 shapes, 393 Schensted’s theorem, 117 Schur functor, 362 as direct image, 365 as eigenspace, 362 Secant ideal, 140 Segre product, 107 Semigroup. see monoid Serre duality absolute, 370 relative, 370 Shape, 71 admissible, 200 predecessor, 202 partial order, 78, 89 single exterior type, 202 Sheaf coherent, 327 defined by shape, 393 locally free, 327 of G-modules, 361 of ideals, 392 Sheaf of relative differentials Grassmann bundle, 330 projective bundle, 330 Shellable, 110 Shelling, 110 Sign of permutation, 327 Simplex, 176 Simplicial complex, 47, 108 dimension, 47 face, 47 facet, 47 join, 142 pure, 47, 48 strongly connected, 53, 55 Single exterior type, 202 Singular weights, 338 Size, 70 Small rank, 209 Split-inclusion, 231 S-polynomial, 11 Stable prime ideal, 103 Standard Zn -grading, 146 Standard bitableau, 72 left initial, 96 right initial, 96 Standard grading, 25, 326 Standard inversion argument, 84 Stanley–Reisner correspondence, 48 ideal, 48

Index Stanley–Reisner ring, 48 Cohen–Macaulayness, 111 Hilbert series for shellable complex, 111 Krull dimension, 109 multiplicity, 109 Straightening algorithm, 81 Straightening law, 72, 74–78 and initial monomials, 80–81 refinement, 78–82 Straightening relations, 72 Strongly connected, 110 strongly F-regular, 243 does not deform, 254 F-finite regular ring is, 244 implies F-split, 244 implies normal and Cohen–Macaulay, 247 monoid algebra, 249 Noetherian ring, 246 very, 243 Structure morphism flag bundle, 331 Grassmann bundle, 329 Sturmfels correspondence, 176 Subduction, 15 unique, 16 Subspace defined by shape, 99 characterization, 102 Successors, 380 with respect to ≤, 453 with respect to ⊂, 453 Symbolic Rees algebra, 187 Symmetric algebra free module, 326 locally free sheaf, 329 Symmetric group action on skew-symmetric tensors, 326 on tensors, 325 Symmetric powers locally free sheaf, 327 Syzygies, 470

T t0 (I v ), 300 t0 (M), 277 Tableau, 72 initial, 79 standard, 72 Tautological exact sequence, 329 Tautological filtration, 331 Tautological sheaf line bundle, 330, 332

Index quotient, 330 sub, 330 subquotient, 331 Taylor resolution, 64 Tensor powers free module, 325 locally free sheaf, 327 Term, 2 tête-a-tête, 20 Tight closure, 245 Tightly closed, 245 submodule, 246 ti (I v ), 300 ti (I v M), 300, 302, 303 ti (M), 279, 284, 299 Top degree function, 290, 294 of bigraded module, 291 of multigraded module, 294 Toric algebra, 172–177 Toric ideal, 19 Torus, 173 Total degree, 288 Truncation of shape, 382 Two-line array, 115 Type, 137 U U-invariants, 98, 262

507 Unimodular, 176 Unipotent group, 97, 354 Universal Gröbner basis, 159–170 2-minors, 163 revlex, 166 definition, 159 examples, 168 finiteness, 160 maximal 2-minors, 165 maximal minors, 161 Unmixed, 25, 44 Unmixedness, 46 Upper triangular group, 354

W Weakly F-regular, 245 Weight, 212 refinement by monomial order, 31 vector, 26 Weyl algebra, 468 W -module, 468 GL-equivariant, 468

Z Zariski topology, 50