Women in Commutative Algebra: Proceedings of the 2019 WICA Workshop (Association for Women in Mathematics Series, 29) [1st ed. 2021] 3030919854, 9783030919856

Table of contents :
Preface
Acknowledgments
Contents
On Gerko's Strongly Tor-independent Modules
1 Introduction
2 DG Homological Algebra
3 Perfect DG Modules and Tensor Products
4 Syzygies and Strongly Tor-independent DG Modules
5 Proof of Theorem 1.1
References
Properties of the Toric Rings of a Chordal Bipartite Family of Graphs
1 Introduction
2 The Family of Toric Rings
2.1 The Family F of Graphs
2.2 Toric Rings for F
2.3 Distinction from Join-Meet Ideals of Lattices
3 Properties of the Family of Toric Rings
3.1 Dimension and System of Parameters
3.2 Length, Multiplicity, and Regularity
References
An Illustrated View of Differential Operators of a Reduced Quotient of an Affine Semigroup Ring
1 Introduction
2 Background and Notation
3 Differential Operators on the Rational Normal Curve of Degree 2
4 Differential Operators on the Rational Normal Curve of Degree 3
5 Differential Operators on a Rational Normal Curve
6 Higher Dimensional Examples
7 Non-normal Examples
References
A Hypergraph Characterization of Nearly Complete Intersections
1 Introduction
2 Hypergraphs and Squarefree Monomial Ideals
3 NCI-hypergraphs
4 Resolutions of NCIs
5 Appendix: Vertex-Weighted Graphs
References
The Shape of Hilbert–Kunz Functions
1 Motivation and Outline
2 History in Brief
2.1 Multiplicity
2.2 Functions
3 Techniques in Hilbert–Kunz Functions
3.1 Via Representation Rings and p-Fractals
3.2 Via the Divisor Class Group
3.3 Via Sheaf Theory
3.4 Via Local Chern Characters
3.5 Via Bruns-Gubeladze (BG) Decomposition
3.6 Via Combinatorics
4 Normal Affine Semigroup Rings and Ehrhart Theory
5 Examples in Normal Affine Semigroup Rings
References
Standard Monomial Theory and Toric Degenerations of Richardson Varieties in Flag Varieties
1 Introduction
1.1 Outline of the Paper
2 Preliminaries
2.1 Flag Varieties
2.2 Richardson Varieties
2.3 Ideals of Flag Varieties and Richardson Varieties
2.4 Gröbner Degenerations of In
2.5 Gröbner Degenerations of I(Xwv)
2.6 Permutations, Tableaux and Their Combinatorial Properties
2.7 Standard Monomials
3 Standard Monomials
3.1 Block Structure on Compatible Permutations
3.2 Proof of Main Result
4 Monomial-Free Ideals
5 Toric Degenerations
References
Simplicial Resolutions for the Second Power of Square-Free Monomial Ideals
1 Introduction
2 Background
3 The Quasi-Trees Lq2 and L2(I)
4 A Bound on the Betti Numbers of I2
References
Cohen–Macaulay Fiber Cones and Defining Ideal of Rees Algebras of Modules
1 Introduction
2 Preliminaries
2.1 Rees Algebras and Fiber Cones of Modules
2.2 Generic Bourbaki Ideals
2.3 Iterated Jacobian Duals
3 A Deformation Condition for the Rees Algebra of a Module
4 Cohen–Macaulay Property of Fiber Cones of Modules
4.1 Modules with Cohen–Macaulay Fiber Cone
5 Defining ideal of Rees Algebras
5.1 Almost Linearly Presented Modules of Projective Dimension One
References
Principal Matrices of Numerical Semigroups
1 Introduction
2 Principal Matrix and Look Alikes
3 Embedding Dimension 4 and 5
4 Sufficient Condition for Principal Matrix
5 Examples
6 Appendix
References
A Survey on the Koszul Homology Algebra
1 Introduction
2 Preliminaries
3 Canonical Bases
4 Classical Characterizations
4.1 Complete Intersection and Gorenstein Rings
4.2 Golod Rings
5 Classifications of Koszul Homology Algebras
6 Recent Progess: Koszul Algebras
6.1 Generation by the Linear Strand
6.2 Upper Bounds on Betti Numbers
6.3 Subadditivity of Syzygies
6.4 Towards a Characterization
References
Canonical Resolutions over Koszul Algebras
1 Introduction
2 Koszul Algebras and the Priddy Resolution
3 Resolutions via the Enveloping Algebra
4 The Case of Koszul Algebras
5 Minimal Resolutions for Powers of the Maximal Ideal
6 Examples
References
Well Ordered Covers, Simplicial Bouquets, and Subadditivity of Betti Numbers of Square-Free Monomial Ideals
1 Introduction
2 Background
2.1 Simplicial Complexes and Facet Ideals
2.2 Simplicial Resolutions
3 (Well Ordered) Covers
4 The Subadditivity Property
5 Simplicial Bouquets
6 Reordering Well Ordered Covers
References
A Survey on the Eisenbud-Green-Harris Conjecture
1 Introduction
2 Preliminaries and Macaulay's Theorem on Hilbert Functions
3 The EGH Conjecture
4 Results on the EGH Conjecture
4.1 EGH Depending on the Degrees (a1,…, an)
4.2 EGH via Liaison
4.3 EGH via the Structure of the Regular Sequence
4.4 When a1=…= an=2
5 Open Cases of EGH and More Connections
References
The Variety Defined by the Matrix of Diagonals is F-Pure
1 Introduction and Preliminaries
2 A System of Parameters
3 The Variety Defined by P(X) Is F-Pure
References
Classification of Frobenius Forms in Five Variables
1 Introduction
2 Matrix Factorization of Frobenius Forms
3 Quadratic Forms
4 Frobenius Forms in Five Variables: Classification Statements
5 Classification of Frobenius Forms in Five Variables: Proofs
References
Projective Dimension of Hypergraphs
1 Introduction
2 Preliminaries
2.1 Hypergraph of a Square-Free Monomial Ideal
2.2 Colon Ideals: Key tool
2.3 Splittings: Key Tool
3 Strings with Higher Dimensional Edges
4 Cycles with Higher Dimensional Edges
References
A Truncated Minimal Free Resolution of the Residue Field
1 Introduction
2 Multiplicative Structure on the Homology of the Koszul Algebra
2.1 Products in Degree Two
2.2 Products in Degree Three
2.3 Massey Products in Degree Four
2.4 Products in Degree Four
2.5 Summary
3 Truncated Minimal Free Resolution of the Residue Field
3.1 Construction
4 Applications of the Construction
5 An Example Illustrating the Construction
References

Citation preview

Association for Women in Mathematics Series

Claudia Miller Janet Striuli Emily E. Witt   Editors

Women in Commutative Algebra Proceedings of the 2019 WICA Workshop

Association for Women in Mathematics Series Volume 29

Series Editor Kristin Lauter Facebook Seattle, WA, USA

Focusing on the groundbreaking work of women in mathematics past, present, and future, Springer’s Association for Women in Mathematics Series presents the latest research and proceedings of conferences worldwide organized by the Association for Women in Mathematics (AWM). All works are peer-reviewed to meet the highest standards of scientific literature, while presenting topics at the cutting edge of pure and applied mathematics, as well as in the areas of mathematical education and history. Since its inception in 1971, The Association for Women in Mathematics has been a non-profit organization designed to help encourage women and girls to study and pursue active careers in mathematics and the mathematical sciences and to promote equal opportunity and equal treatment of women and girls in the mathematical sciences. Currently, the organization represents more than 3000 members and 200 institutions constituting a broad spectrum of the mathematical community in the United States and around the world. Titles from this series are indexed by Scopus.

More information about this series at https://link.springer.com/bookseries/13764

Claudia Miller • Janet Striuli • Emily E. Witt Editors

Women in Commutative Algebra Proceedings of the 2019 WICA Workshop

Editors Claudia Miller Department of Mathematics Syracuse University Syracuse, NY, USA

Janet Striuli Department of Mathematics Fairfield University Fairfield, CT, USA

Emily E. Witt Department of Mathematics University of Kansas Lawrence, KS, USA

ISSN 2364-5733 ISSN 2364-5741 (electronic) Association for Women in Mathematics Series ISBN 978-3-030-91985-6 ISBN 978-3-030-91986-3 (eBook) https://doi.org/10.1007/978-3-030-91986-3 Mathematics Subject Classification: 13-XX, 14-XX, 18-XX © The Author(s) and the Association for Women in Mathematics 2021 This work is subject to copyright. All rights are solely and exclusively licensed 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

Preface

This volume is the result of research activities that took place during the first workshop for Women in Commutative Algebra at Banff International Research Station for Mathematical Innovation and Discovery. The workshop brought together researchers from a diverse list of institutions and fostered research collaborations among women at different career stages and from different research backgrounds. This volume has several purposes. First and foremost, it is a celebration of the high-level research activities that took place at the workshop. Further it wants to testify to the successful model behind the high research achievements, a model put in place by several fields in mathematics and, with Emily E. Witt, brought to commutative algebra by the hard work of Karen Smith, Sandra Spiroff, Irena Swanson, to whom we are extremely grateful. We have intended this volume to support and expand the goal of the workshop in re-enforcing the network of collaborations among women in commutative algebra by displaying in one place those connections, and bringing in new ones. The volume indeed contains articles or research advances that were made by the research groups, as well as some contributions related to the area of commutative algebra and, survey articles. Commutative algebra is the study of the properties of rings that historically rose in algebraic and arithmetic geometry. With the development of several techniques and a rich theory, commutative algebra has become a thriving research area that feeds to and from several fields of mathematics such has topology and combinatorics, beyond the classical algebraic and arithmetic geometry. It would not be fair not to mention that significant advances have been made recently in the field with the the breakthrough of new techniques in positive and mixed characteristic methods and homological algebra, and the consequent solution of long-standing conjectures. The volume reflects the ripple effect of such breakthroughs that have inspired a great deal of activities in commutative algebra. Our volume delivers readings that span from case studies to survey articles and cover a wide range of topics in commutative algebra. The study of characteristic p rings is present in this volume through the classification of Frobenius forms in certain dimension, the Frobenius singularities of certain varieties, and through a comprehensive survey of the Hilbert–Kunz function; further, the reader can find results of a homological v

vi

Preface

flavor in the articles that deliver resolutions of powers of the homogeneous maximal ideal of graded Koszul algebras, the construction of the truncated free resolution for the residue field, or the notion of Tor-independent modules; finally, the volume contains several articles that expand on the connection between homological and combinatorial invariants. Syracuse, NY, USA Fairfield, CT, USA Lawrence, KS, USA

Claudia Miller Janet Striuli Emily E. Witt

Acknowledgments

We would like to thank Emily E. Witt’s co-organizers, Karen Smith, Sandra Spiroff, and Irena Swanson, of the Women in Commutative Algebra workshop at the Banff International Research Station for gathering such a great group of 3 researchers

vii

Contents

On Gerko’s Strongly Tor-independent Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hannah Altmann and Keri Sather-Wagstaff

1

Properties of the Toric Rings of a Chordal Bipartite Family of Graphs . . . Laura Ballard

11

An Illustrated View of Differential Operators of a Reduced Quotient of an Affine Semigroup Ring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Christine Berkesch, C-Y. Jean Chan, Patricia Klein, Laura Felicia Matusevich, Janet Page, and Janet Vassilev A Hypergraph Characterization of Nearly Complete Intersections . . . . . . . . Chiara Bondi, Courtney R. Gibbons, Yuye Ke, Spencer Martin, Shrunal Pothagoni, and Ada Stelzer

49

95

The Shape of Hilbert–Kunz Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 C-Y. Jean Chan Standard Monomial Theory and Toric Degenerations of Richardson Varieties in Flag Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Narasimha Chary Bonala, Oliver Clarke, and Fatemeh Mohammadi Simplicial Resolutions for the Second Power of Square-Free Monomial Ideals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Susan M. Cooper, Sabine El Khoury, Sara Faridi, Sarah Mayes-Tang, Susan Morey, Liana M. Sega, ¸ and Sandra Spiroff Cohen–Macaulay Fiber Cones and Defining Ideal of Rees Algebras of Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 Alessandra Costantini Principal Matrices of Numerical Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Papri Dey and Hema Srinivasan

ix

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Contents

A Survey on the Koszul Homology Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 Rachel N. Diethorn Canonical Resolutions over Koszul Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 Eleonore Faber, Martina Juhnke-Kubitzke, Haydee Lindo, Claudia Miller, Rebecca R. G., and Alexandra Seceleanu Well Ordered Covers, Simplicial Bouquets, and Subadditivity of Betti Numbers of Square-Free Monomial Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 Sara Faridi and Mayada Shahada A Survey on the Eisenbud-Green-Harris Conjecture . . . . . . . . . . . . . . . . . . . . . . . . 327 Sema Güntürkün The Variety Defined by the Matrix of Diagonals is F -Pure . . . . . . . . . . . . . . . . . 343 Zhibek Kadyrsizova Classification of Frobenius Forms in Five Variables . . . . . . . . . . . . . . . . . . . . . . . . . 353 Zhibek Kadyrsizova, Janet Page, Jyoti Singh, Karen E. Smith, Adela Vraciu, and Emily E. Witt Projective Dimension of Hypergraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 Kuei-Nuan Lin and Sonja Mapes A Truncated Minimal Free Resolution of the Residue Field . . . . . . . . . . . . . . . . 399 Van C. Nguyen and Oana Veliche

On Gerko’s Strongly Tor-independent Modules Hannah Altmann and Keri Sather-Wagstaff

Keywords Differential graded algebras · Semidualizing modules · Syzygies · Tor-independent modules

1 Introduction We are interested in how existence of certain sequences of modules over a local ring (R, mR ) imposes restrictions on R. Specifically, we investigate what Gerko [6] calls strongly Tor-independent R-modules: A sequence N1 , . . . , Nn of R-modules is strongly Tor-independent provided TorR 1 (Nj1 ⊗R · · · ⊗R Njt , Njt+1 ) = 0 for all distinct j1 , . . . , jt+1 . Gerko is led to this notion in his study of Foxby’s semidualizing modules [5] and Christensen’s semidualizing complexes [3]. In particular, Gerko [6, Theorem 4.5] proves that if R is artinian and possesses a sequence of strongly Tor-independent modules of length n, then mnR = 0. (Note that Gerko’s result only assumes the modules are finitely generated and strongly Torindependent, not necessarily semidualizing.) This generalizes readily from artinian rings to Cohen–Macaulay rings; see Proposition 5.1 below. Our goal in this paper is to prove the following non-Cohen–Macaulay complement to Gerko’s result. Theorem 1.1 Assume (R, mR ) is a local ring. If N1 , . . . , Nn are non-free, strongly Tor-independent R-modules, then n  ecodepth(R).

H. Altmann () College of Arts and Sciences, Dakota State University, Madison, SD, USA e-mail: [email protected] K. Sather-Wagstaff School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Miller et al. (eds.), Women in Commutative Algebra, Association for Women in Mathematics Series 29, https://doi.org/10.1007/978-3-030-91986-3_1

1

2

H. Altmann and K. Sather-Wagstaff

Here ecodepth(R) = β0R (mR ) − depth(R) is the embedding codepth of R, where β0R (mR ) is the minimal number of generators of mR . Note that our result does not recover Gerko’s, but compliments it. Our proof is the subject of Sect. 5 below. Part of the proof of our result is modeled on Gerko’s proof with one crucial difference: where Gerko works over an artinian ring, we work over a finite dimensional DG algebra. See Sects. 2 and 3 for background material and foundational results, including our DG version of Gerko’s notion of strong Tor-independence. Theorem 4.7 is our main result in the DG context, which is the culmination of Sect. 4. Our proof relies on a DG syzygy construction of Avramov et al. [2].

2 DG Homological Algebra Let R be a nonzero commutative Noetherian ring with identity. We work with R-complexes indexed homologically, so for us an R-complex X has differential ∂iX : Xi → Xi−1 . The supremum and infimum of X are respectively sup(X) = sup{i ∈ Z | Xi = 0}

inf(X) = inf{i ∈ Z | Xi = 0}.

The amplitude of X is amp(X) = sup(X) − inf(X). Frequently we consider these invariants applied to the total homology H (X), e.g., as sup(H (X)). As we noted in the introduction, the proof of Theorem 1.1 uses DG techniques which we summarize next. See, e.g., [1, 4] for more details. A differential graded (DG) R-algebra is an R-complex A equipped with an Rlinear chain map A ⊗R A → A denoted a ⊗ a  → aa  that is unital, associative, and graded commutative. We simply write DG algebra when R = Z. The chain map condition here implies that this multiplication is also distributive and satisfies the Leibniz Rule: ∂(aa  ) = ∂(a)a  + (−1)|a| a∂(a  ) where |a| is the homological degree of a. We say that A is positively graded provided Ai = 0 for all i < 0. For example, the trivial R-complex R is a positively graded DG R-algebra, so too is every Koszul complex over R, using the wedge product. The underlying algebra associated to A is the R-algebra A = i∈Z Ai . If R is local, then a positively graded DG R-algebra A is local provided H0 (A) is Noetherian, each H0 (A)-module Hi (A) is finitely generated for all i  0, and the ring H0 (A) is a local R-algebra. Let A be a DG R-algebra. A DG A-module is an R-complex X equipped with an R-linear chain map A ⊗R X → X denoted a ⊗ x → ax that is unital and associative. For instance DG R-modules are precisely R-complexes. We say that X is homologically bounded if amp(H (X)) < ∞, and we say that X is homologically finite if H (X) is finitely generated over H0 (A). We write  n X for the nth shift of X n X . Quasiisomorphisms between obtained by ( n X)i = Xi−n and ∂i X = (−1)n ∂i−n R-complexes, i.e., chain maps that induce isomorphisms on the level of homology, are identified with the symbol . Let A be positively graded and let X be a DG A-module such that inf(X) > −∞. We say that X is semifree if the underlying A -module X is free. In this case

On Gerko’s Strongly Tor-independent Modules

3

a semibasis for X is a set of homogeneous elements of X that is a basis for X over A . A semifree resolution of a DG A-module Y with inf(H (Y )) > −∞ is a quasiisomorphism F − → Y such that F is semifree. The derived tensor product → Y is a semifree of DG A-modules Y and Z is Y ⊗L A Z F ⊗A Z where F −

→ Y such resolution of Y . We say that Y is perfect if it has a semifree resolution F − that F has a finite semibasis. Let A be a local DG R-algebra, and let Y be a homogically finite DG Amodule. By [2, Proposition B.7] Y has a minimal semifree resolution, i.e., a semifree → Y such that the semibasis for F is finite in each homological degree resolution F − and ∂ F (F ) ⊆ mA F .

3 Perfect DG Modules and Tensor Products Throughout this section, let A be a positively graded commutative homologically bounded DG algebra, say amp(H (A)) = s, and assume that A  0. Most of this section focuses on four foundational results on perfect DG modules. Lemma 3.1 Let L be a non-zero semifree DG A-module with a semibasis B concentrated in a single degree n. Then L ∼ =  n A(B) . In particular, inf(H (L)) = n and sup(H (L)) = s + n and amp(H (L)) = s. Proof It suffices to prove that L ∼ =  n A(B) . Apply an appropriate shift to assume without loss of generality that n = 0. The semifree/semibasis assumptions tell us that every element x ∈ L has  the form finite e∈B ae e; the linear independence of the semibasis tells us that this representation is essentially unique. Since A is positively graded, we have L−1 = 0, so ∂ L (e) = 0 for all e ∈ B. Hence, the Leibniz rule for L implies that ∂

L

finite  e∈B

 ae e =

finite  i

∂ A (ae )e +

finite 

finite 

i

i

(−1)|ae | ae ∂ L (e) =

∂ A (ae )e.

From this, it follows that the map A(B) → L given by the identity on B is an isomorphism.

Proposition 3.2 Let L be a non-zero semifree DG A-module with a semibasis B concentrated in degrees n, n + 1, . . . , n + m where n, m ∈ Z and m  0. Then inf(H (L))  n and sup(H (L))  s + n + m, so amp(H (L))  s + m. Proof It suffices to show that inf(H (L))  n and sup(H (L))  s + n + m. We induct on m. The base case m = 0 follows from Lemma 3.1. For the induction step, assume that m  1 and that the result holds for semifree DG A-modules with semibasis concentrated in degrees n, n + 1, . . . , n + m − 1. Set B  = {e ∈ B | |e| < n + m}

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H. Altmann and K. Sather-Wagstaff

and let L denote the semifree submodule of L spanned over A by B  . (See the first paragraph of the proof of [2, Proposition 4.2] for further details.) Note that L has semibasis B  concentrated in degrees n, n + 1, . . . , n + m − 1. In particular, our induction assumption applies to L to give inf(H (L ))  n and sup(H (L ))  s + n + m − 1. If L = L , then we are done by our induction assumption. So assume that L = L . Then the quotient L/L is semifree and non-zero with semibasis concentrated in degree n + m. So, Lemma 3.1 implies that inf(H (L/L )) = n + m and sup(H (L/L )) = s + n + m. Now, consider the short exact sequence 0 → L → L → L/L → 0.

(1)

The desired conclusions for L follow from the associated long exact sequence in homology.

Now, we use the preceding two results to analyze derived tensor products. Lemma 3.3 Let L be a non-zero semifree DG A-module with a semibasis B concentrated in a single degree, say n, and let Y be a homologically bounded DG n (B) . In particular, inf(H (L⊗L Y )) = inf(H (Y ))+n A-module. Then L⊗L AY  Y A L and sup(H (L ⊗A Y )) = sup(H (Y )) + n and amp(H (L ⊗L A Y )) = amp(H (Y )). Proof Immediate from Lemma 3.1.



Proposition 3.4 Let L be a non-zero semifree DG A-module with a semibasis B concentrated in degrees n, n+1,. . . n+m where n, m ∈ Z and m  0, and let Y be a homologically bounded DG A-module. Then inf(H (L ⊗L A Y ))  inf(H (Y )) + n and L Y ))  amp(H (Y )) + m. sup(H (L ⊗L Y ))  sup(H (Y )) + n + m, so amp(H (L ⊗ A A Proof As in the proof of Proposition 3.2, we induct on m. The base case m = 0 follows from Lemma 3.3. For the induction step, assume m  1 and the result holds for semifree DG A-modules with semibasis concentrated in degrees n, n + 1, . . . , n + m − 1 and Y ∈ Db (A). We work with the notation from the proof of Proposition 3.2, and we assume that L = L . The exact sequence (1) of semi-free DG modules gives rise to the following distinguished triangle in D(A). L  L L ⊗ L A Y → L ⊗A Y → (L/L ) ⊗A Y →

Another long exact sequence argument gives the desired conclusion.



We close this section with our DG version of strongly Tor-independent modules. Definition 3.5 The DG A-modules K1 , . . . , Kn are said to be strongly Torindependent if for any subset I ⊂ {1, . . . , n} we have amp(H (⊗L i∈I Ki ))  s. Remark 3.6 It is worth noting that the definition of K1 , . . . , Kn being strongly Torindependent includes amp(H (Ki )))  s for all i = 1, . . . , n. Also, if K1 , . . . , Kn

On Gerko’s Strongly Tor-independent Modules

5

are strongly Tor-independent, then so is any reordering by the commutativity of tensor products.

4 Syzygies and Strongly Tor-independent DG Modules Throughout this section, let (A, mA ) be a local homologically bounded DG algebra, say amp(H (A)) = s, and assume that A  0 and mA = A+ . It follows that A0 is a field. The purpose of this section is to provide a DG version of part of a result of Gerko [6, Theorem 4.5]. Key to this is the following slight modification of the syzygy construction of Avramov et al. mentioned in the introduction. Construction 4.1 Let K be a homologically finite DG A-module. Let F K be a minimal semifree resolution of K, and let E be a semibasis for F . Let F (p) be the semifree DG A-submodule of F spanned by Ep := ∪mp Em .  = τr (F ) for a Set t = sup(H (K)), and consider the soft truncation K  is a surjective fixed integer r  t. Note that the natural morphism F → K  quasiisomorphism of DG A-modules, so we have K F K. Next, set L = F (r) , which is semifree with a finite semibasis Er . Furthermore, the composition π of  is surjective because the morphism the natural morphisms L = F (r) → F → K  F → K is surjective, the morphism L → F is surjective in degrees  r, and i = 0 for all i > r. Set Syzr (K) = ker(π ) ⊆ L and let α : Syzr (K) → L be the K inclusion map. Proposition 4.2 Let K be a homologically finite DG A-module. With the notation of Construction 4.1, there is a short exact sequence of morphisms of DG A-modules α π →0 0 → Syzr (K) − →L− →K

(2)

 K and Im(α) ⊆ A+ L. such that L is semifree with a finite semibasis and where K Proof Argue as in the proof of [2, Proposition 4.2].



Our proof of Theorem 1.1 hinges on the behavior for syzygies documented in the following four results. Lemma 4.3 Let K be a homologically finite DG A-module with amp(H (K))  s and K  = Syzr (K) where r  sup(H (K)). Then sup(H (K  ))  s + r and inf(H (K  ))  r. Therefore, amp(H (K  ))  s. Proof Use the notation from Construction 4.1. Then sup(H (L))  s + r by  = sup(H (K))  r  r +s. Proposition 3.2. Also, by definition we have sup(H (K)) The long exact sequence in homology coming from (2) implies sup(H (K  ))  s +r. Also, inf(H (K  ))  inf(K  )  r because πi is an isomorphism for all i < r by Construction 4.1. So, amp(H (K  )) = sup(H (K  ))−inf(H (K  ))  s +r −r = s.



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H. Altmann and K. Sather-Wagstaff

Proposition 4.4 Let K be a homologically finite DG A-module and set K  = Syzr (K) where r  sup(H (K)). Let Y be a homologically bounded DG A-module and assume that K, Y are strongly Tor-independent. Then sup(H (K  ⊗L A Y ))   ⊗L Y ))  s; sup(H (Y ))+r and inf(H (K  ⊗L Y ))  inf(H (Y ))+r. So, amp(H (K A A in particular, K  , Y are strongly Tor-independent.

 and L be as in Construc→ Y be a semifree resolution of Y . Let K Proof Let G −  ⊗A G))  s. tion 4.1. As K, Y are strongly Tor-independent we have sup(H (K Also, Proposition 3.4 implies sup(H (L ⊗A G))  sup(H (Y )) + r. To conclude the proof, consider the short exact sequence  ⊗A G → 0 0 → K  ⊗A G → L ⊗A G → K and argue as in the proof of Lemma 4.3.

(3)



Proposition 4.5 Let K1 , K2 , . . . , Kn be strongly Tor-independent, homologically finite DG A-modules for n ∈ Z+ and Ki = Syzri (Ki ) where ri  sup(H (Ki )). Then  ,K K1 , . . . , Km m+1 , . . . , Kn are strongly Tor-independent for all m = 1, . . . , n. Proof Induct on m using Proposition 4.4.



Proposition 4.6 Let K1 , K2 , . . . , Kj be strongly Tor-independent DG A-modules, and set Ki = Syzri (Ki ) where ri  sup(H (Ki )) for i = 1, 2, . . . , j . If mnA = 0,  then mH (A) H (⊗L i=1,...,j Ki ) = 0. n−j

Proof Shift Ki if necessary to assume without loss of generality that inf(H (Ki )) = 0 for i = 1, . . . , j . For i = 1, . . . , j let Gi − → Ki be semifree resolutions, and consider the following diagram with notation as in Construction 4.1.

(4) Notice, Im(αi ) ⊆ Ki ⊆ mA Li for i = 1, 2, . . . , j . Set G = ⊗i=1,...,j −1 Gi and consider the following commutative diagram

where θ = ⊗i=1,...,j −1 αi and β is induced by θ ⊗ αj .

On Gerko’s Strongly Tor-independent Modules

7

Claim: H (β) is 1-1. Notice that Hi (G ⊗A Gj ) = 0 for all i < r1 + . . . + rj , so it suffices to show that Hi (β) is 1-1 for all i  r1 + . . . + rj . To this end it suffices to show Hi (G ⊗ αj ) and Hi (θ ⊗ Lj ) are 1-1 for all i  r1 + . . . + rj . First we show this for Hi (G ⊗ αj ). Consider the short exact sequence G⊗αj j → 0. 0 → G ⊗A Gj −−−→ G ⊗A Lj → G ⊗A K

(5)

Proposition 4.4 implies j ))  r1 + . . . + rj . j ))  r1 + . . . + rj −1 + sup(H (K sup(H (G ⊗A K Thus, the long exact sequence in homology associated to (5) implies Hi (G ⊗A αj ) is 1-1 for all i  r1 + . . . + rj as desired. Next, we show Hi (θ ⊗ Lj ) is 1-1 for i  r1 + . . . + rj . Consider the exact sequence θ⊗L

j i ) ⊗A Lj → 0. 0 → G ⊗A Lj −−−→ (⊗i=1,...,j −1 Li ) ⊗A Lj → (⊗i=1,...,j −1 K (6) The first inequality in the next display follows from Proposition 3.4

i ) ⊗A Lj ))  sup(H (⊗i=1,...,j −1 K i )) + rj sup(H ((⊗i=1,...,j −1 K  r1 + . . . + rj −1 + rj . Thus, the long exact sequence in homology associated to (6) implies Hi (θ ⊗ Lj ) is 1-1 for all i  r1 + . . . + rj . This establishes the claim. n−j  L  To complete the proof it remains to show mH (A) H ((⊗L i=1,...,j −1 Ki )⊗A Kj ) = 0.  L  Since H (β) is 1-1, we have H ((⊗L i=1,...,j −1 Ki ) ⊗A Kj ) isomorphic to a submodule j

n−j

of H (mA ((⊗i=1,...,j −1 Li ) ⊗A Lj )). So it suffices to show that mH (A) annihilates j

H (mA ((⊗i=1,...,j −1 Li ) ⊗A Lj )); this annihilation holds because mnA = 0.



Here is the aforementioned version of part of [6, Theorem 4.5]. Theorem 4.7 Let K1 , . . . , Kn be strongly Tor-independent non-perfect DG A-modules. Then mnA = 0, therefore, n  s.  Proof Suppose mnA = 0. Proposition 4.6 implies that 0 = m0H (A) H (⊗L i=1,...,n Ki ) = L  H (⊗i=1,...,n Ki ). Since each Ki has a minimal resolution for i = 1, . . . , n, we must have H (Kl ) = 0 for some l. Hence, Kl has a semifree basis concentrated in a finite number of degrees. This contradicts our assumption that Ki is not perfect for i = 1, . . . , n. Therefore, mnA = 0. Now we show n  s. Soft truncate A to get A A such that sup(A ) = s. Thus, s+1 mA = 0. The sequence of n strongly Tor-independent non-perfect DG A-modules

8

H. Altmann and K. Sather-Wagstaff

gives rise to a sequence of n strongly Tor-independent non-perfect DG A -modules.

Since mnA = 0 and ms+1 A = 0, we have n  s.

5 Proof of Theorem 1.1 Induct on depth(R). Base Case: depth(R) = 0. Let K denote the Koszul complex over R on a minimal generating sequence for mR . The condition depth(R) = 0 implies amp(H (K)) = ecodepth(R) = amp(K).

(7)

L Claim: The sequence K ⊗L R N1 , . . . , K ⊗R Nn is a strongly Tor-independent sequence of DG K-modules. To establish the claim we compute derived tensor products where both L are indexed by i ∈ I :

L

K (K

L ⊗L R Ni ) K ⊗R (

L R

Ni ).

From this we get the first equality in the next display. amp(H (

L

K (K

L ⊗L R Ni ))) = amp(H (K ⊗R (

L R



Ni )))

= amp(H (K ⊗R ( i∈I Ni )))  amp(K ⊗R ( i∈I Ni )) = amp(K) = amp(H (K)) The second equality comes from the strong Tor-independence of the original sequence. The inequality and the third equality are routine, and the final equality is by (7). This establishes the claim. A construction of Avramov provides a local homologically bounded DG algebra (A, mA ) such that A K  0 and mA = A+ ; see [7, 8]. The strongly TorL independent sequence K ⊗L R N1 , . . . , K ⊗R Nn over K gives rise to a strongly Tor-independent sequence M1 , . . . , Mn over A. Now, Theorem 4.7 and (7) imply n  amp(H (A)) = amp(H (K)) = ecodepth(R). This concludes the proof of the Base Case. Inductive Step: Assume depth(R) > 0 and the result holds for local rings S with depth(S) = depth(R)−1. For i = 1, . . . , n let Ni be the first syzygy of Ni . Since the sequence N1 , . . . , Nn is strongly Tor-independent, sequence N1 , . . . , Nn . so is the  Moreover, strong Tor-independence implies that i∈I Ni is a submodule of a free R-module, for each subset i ∈ {1, . . . , n}.

On Gerko’s Strongly Tor-independent Modules

9

Use prime avoidance to find an R-regular element x ∈ mR −m2R . Set R = R/xR. Note that depth(R) = depth(R) − 1 and ecodepth(R) = ecodepth(R). The fact that each i∈I Ni is a submodule of a free R-module implies that x is also i∈I Ni  L  regular. It is straightforward to show that the sequence R ⊗L R N1 , . . . , R ⊗R Nn is strongly Tor-independent over R. By our induction hypothesis we have n  ecodepth(R) = ecodepth(R) as desired.

We conclude with the generalization of Gerko’s result [6, Theorem 4.5] from artinian rings to Cohen–Macaulay rings mentioned in the introduction. In preparation, recall that the Loewy length of a finite length R-module M is

R (M) = min{i  0 | miR M = 0}. The generalized Loewy length of R is then

R (R) = min{

R (R/x) | x is a system of parameters of R}. Notice that when R is artinian, i.e., when R has finite length as an R-module, the generalized Loewy length of R equals the Loewy length of R, so the symbol

R (R) is unambiguous. Proposition 5.1 Assume that R is Cohen–Macaulay and that K1 , . . . , Kn are nonfree, finitely generated, strongly Tor-independent R-modules. Then n 

R (R). Proof We induct on d = dim(R). In the base case d = 0, the ring R is artinian, so Gerko’s result [6, Theorem 4.5] says that mnR = 0. By the definition of Loewy length, this is exactly the desired conclusion. For the induction step, assume that d  1, and that our result holds for Cohen– Macaulay local rings of dimension d − 1. Let x = x1 , . . . , xd be a system of parameters of R such that

R (R) =

R/x (R/x). Since R is Cohen–Macaulay, this is a maximal R-regular sequence. Furthermore, the definition of generalized Loewy length implies that

R/x1  (R/x1 ) 

R/x (R/x) =

R (R). Replace the modules Ki with their first syzygies if necessary to assume without loss of generality that x1 is Ki -regular for i = 1, . . . , n. From this, it is straightforward to use the assumptions on the Ki to conclude that K1 /x1 K1 , . . . , Kn /x1 Kn are non-free, finitely generated, strongly Tor-independent R/x1 -modules. Thus, our induction hypothesis implies that n 

R/x1  (R/x1 ) 

R (R), as desired.

Acknowledgement We are grateful to the anonymous referee for their thoughtful comments.

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H. Altmann and K. Sather-Wagstaff

References 1. L. L. Avramov, H.-B. Foxby, and S. Halperin, Differential graded homological algebra, in preparation. 2. L. L. Avramov, S. B. Iyengar, S. Nasseh, and S. Sather-Wagstaff, Homology over trivial extensions of commutative DG algebras, Comm. Algebra 47 (2019), no. 6, 2341–2356, see also arxiv:1508.00748. MR 3957101 3. L. W. Christensen, Semi-dualizing complexes and their Auslander categories, Trans. Amer. Math. Soc. 353 (2001), no. 5, 1839–1883. MR 2002a:13017 4. Y. Félix, S. Halperin, and J.-C. Thomas, Rational homotopy theory, Graduate Texts in Mathematics, vol. 205, Springer-Verlag, New York, 2001. MR 1802847 5. H.-B. Foxby, Gorenstein modules and related modules, Math. Scand. 31 (1972), 267–284 (1973). MR 48 #6094 6. A. A. Gerko, On the structure of the set of semidualizing complexes, Illinois J. Math. 48 (2004), no. 3, 965–976. MR 2114263 7. A. R. Kustin, Classification of the Tor-algebras of codimension four almost complete intersections, Trans. Amer. Math. Soc. 339 (1993), no. 1, 61–85. MR 1132435 8. S. Nasseh and S. K. Sather-Wagstaff, Applications of differential graded algebra techniques in commutative algebra, preprint (2020), arXiv:2011.02065.

Properties of the Toric Rings of a Chordal Bipartite Family of Graphs Laura Ballard

1 Introduction In recent decades, there has been a growing interest in the investigation of algebraic invariants associated to combinatorial structures. Toric ideals of graphs (and the associated edge rings), a special case of the classical notion of a toric ideal, have been studied by various authors with regard to invariants such as depth, dimension, projective dimension, regularity, graded Betti numbers, Hilbert series, and multiplicity, usually for particular families of graphs (see for example [2, 3, 5, 7– 10, 12, 14, 16–19, 22, 23, 26]). We note in Remarks 2.6 and 2.14 that the family we consider does not overlap at all or for large n with those considered in [5, 8, 9], and [23]; it is more obviously distinct from other families that have been studied. We think it fitting to mention that the recent book by Herzog et al. [15] also investigates toric ideals of graphs as well as binomial ideals coming from other combinatorial structures. In this work, we consider a family of graphs with iterated subfamilies and develop algebraic properties of the toric rings associated to the family which depend only on the number of vertices (equivalently, the number of edges) in the associated graphs. In the development of this project, we were particularly inspired by the work of Jennifer Biermann, Augustine O’Keefe, and Adam Van Tuyl in [3], where they establish a lower bound for the regularity of the toric ideal of any finite simple graph and an upper bound for the regularity of the toric ideal of a chordal bipartite graph. Our goal is to construct as “simple” a family of graphs as possible that still yields

The author was partially supported by the National Science Foundation (DMS-1003384). L. Ballard () Mathematics Department, Syracuse University, Syracuse, NY, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Miller et al. (eds.), Women in Commutative Algebra, Association for Women in Mathematics Series 29, https://doi.org/10.1007/978-3-030-91986-3_2

11

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L. Ballard

interesting toric ideals. It is our hope that our process and results will lead to further generalizations of properties of toric ideals for other (perhaps broader) families of graphs, or for graphs containing or arising from such graphs. Herein, we introduce the infinite family F of chordal bipartite graphs Gtn , where n determines the number of edges and vertices and t determines the structure of the graph, and establish some algebraic properties of the toric rings R(n, t) associated to the graphs Gtn . The use of bipartite graphs makes each R(n, t) normal and Cohen– Macaulay by [25] and [15]; we use the latter in Sect. 3. Our main results prove to be independent of t and depend only on n. In Sect. 2, we construct the family F of graphs Gtn from a family of ladder-like structures Ltn so that the toric ideals of the Gtn are generalized determinantal ideals of the Ltn . The ladder-like structures associated to a subfamily F1 ⊂ F, introduced in Example 2.4, are in fact two-sided ladders (for large n), so that the family of rings R(n, t) is a generalization of the family of ladder determinantal rings coming from F1 . While the rings arising from F1 come from a distributive lattice and have easily derived properties (see for example [15]), we show that the rings associated to F do not naturally arise from any lattice in general, and merit closer study. In Sect. 3, we establish some algebraic properties of the R(n, t), particularly Krull dimension, projective dimension, multiplicity, and regularity. To do so, we prove that the determinantal generators of the defining ideal IGtn are a Gröbner basis (it follows immediately from [15] that R(n, t) is Koszul) and work with the initial ideal in> IGtn . We also develop a system of parameters Xn that allows us to work with Artinian reductions in part of our treatment, and their Hilbert series. Our first result gives an alternate proof for the Krull dimension of the toric ring R(n, t) = S(n)/IGtn , already known due to a result of Villarreal for bipartite graphs [27, Prop 3.2]. Here, the ring S(n) = k[x0 , x2 , x3 , . . . , x2n+3 , x2n+4 ] is the polynomial ring over the edges of Gtn and IGtn is the toric ideal of Gtn . Theorem 1.1 (Theorem 3.4) The dimension of R(n, t) is dim R(n, t) = n + 3. As a corollary, since R(n, t) comes from a bipartite graph and is hence Cohen– Macaulay (Corollary 2.16), we obtain the projective dimension of R(n, t). Corollary 1.2 (Corollary 3.5) The projective dimension of R(n, t) over S(n) is pdS(n) R(n, t) = n + 1. We then develop a linear system of parameters for R(n, t), using differences of elements on antidiagonals of the ladder-like structure Ltn . Proposition 1.3 (Proposition 3.10) Let R(n, t) = S(n)/IGtn . Then the image of Xn = x0 , x2 − x3 , x4 − x5 , . . . , x2n − x2n+1 , x2n+2 − x2n+3 , x2n+4

Properties of the Toric Rings of a Chordal Bipartite Family of Graphs

13

in R(n, t) is a system of parameters for R(n, t). Since R(n, t) is Cohen–Macaulay, the linear system of parameters above is a regular sequence (Corollary 3.12). With the aim of obtaining the multiplicity and regularity of R(n, t), we form an t). We Artinian quotient of R(n, t) by the regular sequence above and call it R(n, t) does not denote the completion, and explain the choice of notation note that R(n, in Definition 3.7. t) established in Lemma 3.13, we Using a convenient vector space basis for R(n, t). show the coefficients of the Hilbert series for R(n, Theorem 1.4 (Theorem 3.16) If R(n, t) R(n, t)/(Xn ), we have

=

t) S(n)/IGtn and R(n,

∼ =

⎧ ⎪ 1 i=0 ⎪ ⎪ ⎪ ⎪ i ⎨1  t))i = (n + j − 2(i − 1)) 1  i  n/2 + 1 dimk (R(n, i! ⎪ ⎪ j =1 ⎪ ⎪ ⎪ ⎩0 i > n/2 + 1. As a corollary, we obtain the regularity of R(n, t), which is equal to the top nonzero t). degree of R(n, Corollary 1.5 (Corollary 3.18) For Gtn ∈ F, reg R(n, t) = n/2 + 1. We include an alternate graph-theoretic proof of the result above at the end of this work. Beginning with an upper bound from [3] (or equivalently for our purposes, one from [14]) and then identifying the initial ideal in> IGtn with the edge ideal of a graph, we use results from [4] (allowing us to use in> IGtn instead of IGtn ) and then [13] for a lower bound which agrees with our upper bound. From a recursion established in Lemma 3.15, we go on to prove a Fibonacci t) in Proposition 3.19, relationship between the lengths of the Artinian rings R(n, and obtain the multiplicity of R(n, t) as a corollary. In the following, we drop t for convenience. Corollary 1.6 (Corollary 3.21) For n  2, there is an equality of multiplicities e(R(n)) = e(R(n − 1)) + e(R(n − 2)). In particular, e(R(n)) = F (n + 3) =

(1 +

√

5)n+3 − (1 − √ 2n+3 5

√ n+3 5)

.

14

L. Ballard

For more background, detail, and motivation, we refer the reader to [1], but note that different notation and indexing conventions have been employed in this work. Throughout, k is a field.

2 The Family of Toric Rings In the following, we define a family of toric rings R(n, t) coming from an iterative chordal bipartite family of graphs, F. We show that although one subfamily of these rings comes from join-meet ideals of a (distributive) lattice and has some easily derived algebraic invariants, this is not true in general. The reader may find the definition of the toric ideal of a graph in Sect. 2.2, when it becomes relevant to the discussion. We recall for the reader that a chordal bipartite graph is a bipartite graph in which every cycle of length greater than or equal to six has a chord.

2.1 The Family F of Graphs Below, we define the family F of chordal bipartite graphs iteratively from a family of ladder-like structures Ltn . We note that the quantities involved in the following definition follow patterns as follows: n n/2 + 2 n/2 + 2 0 2 2 1 2 3 2 3 3 3 3 4 .. .. .. . . . Definition 2.1 For each n  0 and each t ∈ Fn+1 2 , we construct a ladder-like structure Ltn with (n/2 + 2) rows and (n/2 + 2) columns and with nonzero entries in the set {x0 , x2 , x3 , . . . , x2n+4 }. To do so, we use the notation  t ∈ Fn2 for the first n entries of t, that is, all except the last entry. The construction is as follows, where throughout, indices of entries in Ltn are strictly increasing from left to right in each row and from top to bottom in each column. We note that Ltn does not depend on t for n < 2, but does for n  2. • For n = 0, the ladder-like structure L00 = L10 is x0 x2 x3 x4

Properties of the Toric Rings of a Chordal Bipartite Family of Graphs

15

• For n = 1, to create Lt1 (regardless of what t is in F22 ), we add another column with the entries x5 and x6 to the right of Lt0 to obtain x0 x2 x5 x3 x4 x6 • For 2  n ≡ 0 mod 2 (≡ 1 mod 2), to create Ltn , we add another row (column) with the entries x2n+3 , x2n+4 below (to the right of) Ltn−1 in the following way: ◦ The entry x2n+4 is in the ultimate row and column, row n/2 + 2 and column n/2 + 2. ◦ The entry x2n+3 is in the new row (column) in a position directly below (to the right of) another nonzero entry in Ltn . · If the last entry of t is 0, x2n+3 is directly beneath (to the right of) the first nonzero entry in the previous row (column). · If the last entry of t is 1, x2n+3 is directly beneath (to the right of) the second nonzero entry in the previous row (column). In this way, the entries in t determine the choice at each stage for the placement of x2n+3 . Remark 2.2 We note a few things about this construction for n ≡ 0 mod 2 (≡ 1 mod 2), which may be examined in the examples below: • We note that x2n+4 is directly beneath (to the right of) x2n+2 . • We note that the only entries in row n/2 + 1 (column n/2 + 1) of Ltn−1 are x2n−1 , x2n , and x2n+2 , so that the choices listed for placement of x2n+3 are the only cases. In particular, tn+1 = 0 if and only if x2n+3 is directly beneath (to the right of) x2n−1 , and tn+1 = 1 if and only if x2n+3 is directly beneath (to the right of) x2n . • Finally, we note that the only entries in column n/2 + 2 (row n/2 + 2) of Ltn are x2n+1 , x2n+2 , and x2n+4 , and that the only entries in row n/2 + 2 (column n/2 + 2) of Ltn are x2n+3 and x2n+4 . Example 2.3 We have (1,1,1)

L2

x0 x2 x5 = x3 x4 x6 x7 x8

(0,0,0)

L2

x0 x2 x5 = x3 x4 x6 x7 x8

In either of the cases above, we could go on to construct Lt3 and Lt4 in the following way: For n = 3, place x10 to the right of x8 and place x9 to the right of either x5 or x6 , depending whether the last entry of  t is 0 or 1, respectively. Then for n = 4, place x12 below x10 and place x11 below either x7 or x8 , depending whether the last entry of t is 0 or 1, respectively.

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L. Ballard (1,1,...,1)

Example 2.4 In fact, when the entries of t are all ones, we see that Ln has a ladder shape (is a two-sided ladder for n  3), shown below in the case when 2  n ≡ 0 mod 2: x0 x2 x5 x3 x4 x6 x9 x7 x8 x10 x13 x11 x12 x14 x17 x15 x16 x18

..

.

..

. x2n+1 x19 x20 .. .. . . x2n+2 x2n+3 x2n+4 . We denote the subfamily of graphs coming from t = (1, 1, . . . , 1) by F1 ⊂ F. When the entries of t are all zeros, Ln(0,0,...,0) has the following structure, shown below in the case when 2  n ≡ 0 mod 2: x0 x2 x5 x9 x13 x17 x21 · · · x2n+1 x3 x4 x6 x7 x8 x10 x11 x12 x14 x15 x16 x18 x19 x20 x22 . x23 x24 . . .. .. . x2n+2 . x2n+3 x2n+4 . (1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0,0)

For a more varied example, we have L16

x0 x2 x5 x9 x3 x4 x6 x7 x8 x10 x13 x17 x11 x12 x14 x15 x16 x18 x21 x25 x29 x33 x19 x20 x22 x23 x24 x26 x27 x28 x30 x31 x32 x34 x35 x36 .

below:

Properties of the Toric Rings of a Chordal Bipartite Family of Graphs

17

Definition 2.5 If we associate a vertex to each row and each column and an edge to each nonzero entry of Ltn , we have a finite simple connected bipartite graph Gtn . The set Vr of vertices corresponding to rows and the set Vc of vertices corresponding to columns form a bipartition of the vertices of Gtn . We say a graph G is in F if G = Gtn for some n  0 and some t ∈ Fn+1 2 . Remark 2.6 We note that by construction Gtn has no vertices of degree one, since each row and each column of Ltn has more than one nonzero entry. This ensures that for large n our family is distinct from that studied in [5], since a Ferrers graph with bipartitation V1 and V2 with no vertices of degree one must have at least two vertices in V1 of degree |V2 | and at least two vertices in V2 of degree |V1 |, impossible for our graphs when n  3, as the reader may verify. We also use the fact that Gtn has no vertices of degree one for an alternate proof of the regularity of R(n, t) at the end of this work. (1,1,...,1)

Example 2.7 When n = 5, G5

∈ F1 is 3

13

8

10

7

1 5

5

3

0

2

3

4

1

2

4

9

6 14

11

2

12

4 We develop properties of the Ltn which allow us to show in Sect. 2.2 that certain minors of the Ltn are generators for the toric rings of the corresponding graphs Gtn . Definition 2.8 For this work, a distinguished minor of Ltn is a 2-minor involving only (nonzero) entries of the ladder-like structure Ltn , coming from a 2 × 2 subarray of Ltn . Proposition 2.9 For each i  1 and each f ∈ Fi+1 2 , the entry x2i+3 and the entry f x2i+4 each appear in exactly two distinguished minors of Li . For i ≡ 0 mod 2 (≡ 1 mod 2), these minors are of the form

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L. Ballard

s2i := x2i+1 x2i+3 − xj2i x2i+4 coming from the subarray   xj2i x2i+1 x2i+3 x2i+4



xj2i x2i+3 x2i+1 x2i+4



for some j2i ∈ {0, 2, 3, . . . , 2i − 2} and s2i+1 := x2i+2 x2i+3 − xj2i+1 x2i+4 coming from the subarray   xj2i+1 x2i+2 x2i+3 x2i+4



xj2i+1 x2i+3 x2i+2 x2i+4



for some j2i+1 ∈ {2i − 1, 2i}, and the only distinguished minor of Ltn with indices all less than 5 is s1 := x2 x3 − x0 x4 . Proof The last statement is clear by Definition 2.1; we prove the remaining statements by induction on i. For i = 1, we have the distinguished minors s2 = x3 x5 − x0 x6 and s3 = x4 x5 − x2 x6 coming from the subarrays   x0 x5 x3 x6 and   x2 x5 x4 x6 where j2 = 0 ∈ {0} and j3 = 2 ∈ {1, 2}, so we have our base case. Now suppose the statement is true for i with 1  i < n, and let n ≡ 0 mod 2 (≡ 1 mod 2) and t ∈ Fn+1 2 . Case 1: If tn+1 = 0, then by Remark 2.2, x2n+3 is in the same column (row) as x2n−1 . By induction, we have the distinguished minor s2n−2 = x2n−1 x2n+1 − xj2n−2 x2n+2 coming from the subarray 

xj2n−2 x2n+1 x2n−1 x2n+2



  xj2n−2 x2n−1 . x2n+1 x2n+2

Then in fact we have a subarray of the form ⎡

⎤ xj2n−2 x2n+1 ⎣x2n−1 x2n+2 ⎦ x2n+3 x2n+4

  xj2n−2 x2n−1 x2n+3 , x2n+1 x2n+2 x2n+4

Properties of the Toric Rings of a Chordal Bipartite Family of Graphs

19

so that we have the distinguished minors s2n = x2n+1 x2n+3 − xj2n−2 x2n+4 s2n+1 = x2n+2 x2n+3 − x2n−1 x2n+4 with j2n = j2n−2 ∈ {0, 2, 3, . . . , 2n − 4} ⊂ {0, 2, 3, . . . , 2n − 2} by induction and with j2n+1 = 2n − 1 ∈ {2n − 1, 2n}. Since the only entries in row n/2 + 2 (column n/2 + 2) of Ltn are x2n+3 and x2n+4 and since the only entries in column n/2 + 2 (row n/2 + 2) of Ltn are x2n+1 , x2n+2 , and x2n+4 by Remark 2.2, these are the only distinguished minors of Ltn containing either x2n+3 or x2n+4 , as desired. Case 2 for tn+1 = 1 is analogous and yields j2n = j2n−1 ∈ {2n − 3, 2n − 2} ⊂ {0, 2, 3, . . . , 2n − 2} and j2n+1 = 2n ∈ {2n − 1, 2n}.



Definition 2.10 Define the integers j2i , j2i+1 for j2 , . . . , j2n+1 as in the statement of Proposition 2.9. We note in the remark below some properties of the jk . Remark 2.11 From the proof of Proposition 2.9, we note that j2 = 0, j3 = 2, and that for i  2, we have the following: ti+1 = 0 ⇐⇒ j2i = j2i−2 ⇐⇒ j2i+1 = 2i − 1 ti+1 = 1 ⇐⇒ j2i = j2i−1 ⇐⇒ j2i+1 = 2i. For the sake of later proofs, we extend the notion of jk naturally to s1 = x2 x3 −x0 x4 and say that j1 = 0, and note the following properties of the jk for 1  k  2n+1: • We have j2i ∈ {j2i−2 , j2i−1 } and j2i  2i − 2. Indeed, for i = 1, j2 = j1 = 0, and for i  2, this is clear from the statement above. • We have j2i+1 ∈ {2i − 1, 2i}. Indeed, for i = 0, j1 = 0 ∈ {−1, 0}, for i = 1, j3 = 2 ∈ {1, 2}, and for i  2, this follows from the statement above. • The j2i form a non-decreasing sequence. Indeed, for i  2, either j2i = j2i−2 or j2i = j2i−1  2i − 3 > 2i − 4  j2i−2 .

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L. Ballard

Remark 2.12 We also note from the proof above that the following is a subarray of Ltn for all i ≡ 0 mod 2 (≡ 1 mod 2) such that 1  i  n, which we use in the proof of the proposition below: ⎡

⎤ xj2i x2i+1 ⎣xj ⎦ 2i+1 x2i+2 x2i+3 x2i+4



xj2i xj2i+1 x2i+3 x2i+1 x2i+2 x2i+4



Proposition 2.13 For n  0, each graph Gtn ∈ F is chordal bipartite with vertex bipartition Vr ∪ Vc of cardinalities |Vr | =

n

+2 2 n + 2. |Vc | = 2 Proof We already know by Definition 2.5 that every graph Gtn is bipartite for n  0, with the bipartition above coming from the rows and columns of Ltn . The cardinalities of the vertex sets follow from Remark 2.2. We prove the chordal f bipartite property by induction on n. It is clear for i = 0 and i = 1 that Gi is i+1 chordal bipartite for f ∈ F2 , since these graphs have fewer than six vertices. Now f suppose Gi is chordal bipartite for i with 1  i < n ≡ 0 mod 2 (≡ 1 mod 2), and consider Gtn for t ∈ Fn+1 2 . We know that the following array (or its transpose) is a subarray of Ltn by Remark 2.12, and we include for reference the corresponding subgraph of Gtn with vertices labeled by row and column. ⎤ xj2n x2n+1 ⎦ ⎣xj 2n+1 x2n+2 x2n+3 x2n+4 ⎡

1 2

2

2 +3

+1

1

3

2 2 +1

2 +2

2

2 +4

Properties of the Toric Rings of a Chordal Bipartite Family of Graphs

21

We know the only difference between Gtn and Gtn−1 is one vertex r3 corresponding to row n/2 + 2 (column n/2 + 2) and two edges x2n+3 = {r3 , c1 } and x2n+4 = {r3 , c2 }, where c1 corresponds to the column (row) containing x2n+3 and c2 corresponds to column n/2 + 2 (row n/2 + 2). By Remark 2.2, deg r3 = 2, since the only entries in row n/2 + 2 (column n/2 + 2) are x2n+3 and x2n+4 . Then any even cycle containing r3 must also contain x2n+3 and x2n+4 . Similarly, by the same remark, the only other edges with endpoint c2 are x2n+1 and x2n+2 , the entries added to make Ltn−1 , so we know that any even cycle containing x2n+4 and x2n+3 must contain either x2n+1 or x2n+2 . We see that any even cycle containing r3 and x2n+1 is either a 4-cycle or has xj2n as a chord, and any even cycle containing r3 and x2n+2 is either a 4-cycle or has xj2n+1 as a chord. Thus every graph Gtn is chordal bipartite for n  0, with the bipartition above.

Remark 2.14 We note that the previous proposition ensures that our graphs are distinct from those studied in [8, 9], and [23], which are not chordal bipartite except for the first family in [8], in which every four-cycle shares exactly one edge with every other four-cycle (also distinct from our family except for the trivial case with only one four-cycle, corresponding to Gt0 ).

2.2 Toric Rings for F In this section, we develop the toric ring R(n, t) for each of the chordal bipartite graphs Gtn in the family F. We first show that the toric ideal IGtn of the graph Gtn is the same as the ideal I (n, t) generated by the distinguished minors of Ltn . We then demonstrate that for some n and t, these ideals do not arise from the join-meet ideals of lattices in a natural way, so that results in lattice theory do not apply to the general family F in an obvious way. We first define the toric ideal of a graph. For any graph G with vertex set V and edge set E, there is a natural map π : k[E] → k[V ] taking an edge to the product of its endpoints. The polynomial subring in k[V ] generated by the images of the edges under the map π is denoted k[G] and is called the edge ring of G. The kernel of π is denoted IG and is called the toric ideal of G; the ring k[G] is isomorphic to k[E]/IG . In this work, we consider the toric ring k[E]/IG and the toric ideal IG for our particular graphs. It is known in general that IG is generated by binomial expressions coming from even closed walks in G [27, Prop 3.1] and that the toric ideal of a chordal bipartite graph is generated by quadratic binomials coming from the 4-cycles of G (see [20, Th 1.2]). Let S(n) = k[x0 , x2 , x3 , . . . , x2n+4 ] be the polynomial ring in the edges of Gtn . The edge ring for Gtn ∈ F is denoted by k[Gtn ] and is isomorphic to the toric ring R(n, t) :=

S(n) , IGtn

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L. Ballard

where IGtn is the toric ideal of Gtn [15, 5.3]. For the general construction of a toric ideal, we refer the reader to [24, Ch 4]. Our goal is to show that the toric ideal IGtn of Gtn is equal to I (n, t) = ({distinguished minors of Ltn }). Proposition 2.15 Let S(n) = k[x0 , x2 , x3 , . . . , x2n+4 ]. For Gtn ∈ F, we have R(n, t) =

S(n) , I (n, t)

where I (n, t) = ({distinguished minors of Ltn }). Proof To prove this, we need only show that I (n, t) is the toric ideal IGtn of the graph Gtn . By Definition 2.5, it is clear that the distinguished minors of Ltn are in IGtn , corresponding to the 4-cycles of Gtn . Since G is chordal bipartite by Proposition 2.13, these are the only generators of IGtn [15, Cor 5.15].

Corollary 2.16 The rings R(n, t) are normal Cohen–Macaulay rings. Proof By Definition 2.5 and Proposition 2.15, the ring R(n, t) is the toric ring of a finite simple connected bipartite graph, and hence by Corollary 5.26 in [15], R(n, t) is Cohen–Macaulay for each n and t. The fact that each R(n, t) is normal follows from [25, Th 5.9, 7.1].

Because we know the distinguished minors of Ltn , we are now able to characterize the generators for the toric ideal R(n, t) of Gtn . Remark 2.17 By Proposition 2.9, the generators s1 , . . . , s2n+1 for IGtn may be summarized as follows. For integers i such that 1  i  n, set s1 = x2 x3 − xj1 x4 s2i = x2i+1 x2i+3 − xj2i x2i+4 s2i+1 = x2i+2 x2i+3 − xj2i+1 x2i+4 , where the nonnegative integers jk are as in Remark 2.11, that is, j1 = j2 = 0, j3 = 2, and for i  2, we have ti+1 = 0 ⇐⇒ j2i = j2i−2 ⇐⇒ j2i+1 = 2i − 1 ti+1 = 1 ⇐⇒ j2i = j2i−1 ⇐⇒ j2i+1 = 2i. We note that the number of generators depends on n and that the jk depend on t, but we may ignore dependence on t when working with general jk . We sometimes call

Properties of the Toric Rings of a Chordal Bipartite Family of Graphs

23

s1 , . . . , s2n+1 the standard generators of IGtn , and show in Sect. 2.3 that for certain n and t, they are not equal to the usual generators for the join-meet ideal of any lattice D. Example 2.18 We consider the toric ideal of a graph in F1 . For n = 5 and t = (1, 1, . . . , 1), by Remark 2.11 we have j1 = j2 = 0, j3 = 2, j2i = j2i−1 and j2i+1 = 2i for i  2, so that R(5, (1, 1, . . . , 1)) =

k[x0 , x2 , . . . , x14 ] , IG(1,1,...,1) 5

(1,1,...,1)

where IG(1,1,...,1) is generated by the distinguished minors of L5

:

5

s1 = x2 x3 − x0 x4

s2 = x3 x5 − x0 x6

s3 = x4 x5 − x2 x6

s4 = x5 x7 − x2 x8

s5 = x6 x7 − x4 x8

s6 = x7 x9 − x4 x10

s7 = x8 x9 − x6 x10

s8 = x9 x11 − x6 x12

s9 = x10 x11 − x8 x12

s10 = x11 x13 − x8 x14

s11 = x12 x13 − x10 x14 .

2.3 Distinction from Join-Meet Ideals of Lattices We saw in Example 2.4 and Proposition 2.15 that if Gtn ∈ F1 ⊂ F, then IGtn is a ladder determinantal ideal for n  2. It is known that a ladder determinantal ideal is equal to the join-meet ideal of a (distributive) lattice (indeed, with a natural partial ordering which decreases along rows and columns of Ltn we obtain such a lattice). Some algebraic information such as regularity and projective dimension may be easily derived for some join-meet ideals of distributive lattices (see, for example, Chapter 6 of [15]). We spend some time in this section establishing that not all rings R(n, t) ∈ F arise from a lattice in a natural way (see Remark 2.20), and so there does not seem to be any obvious way to obtain our results in Sect. 3 from the literature on join-meet ideals of distributive lattices or on ladder determinantal ideals. The results in Sect. 3 may be viewed as an extension of what may already be derived for the family F1 from the existing literature. The following five lemmas serve to provide machinery to show that there is at least one ring in the family F, namely R(5, (1, 1, 1, 1, 1, 0)), whose toric ideal does not come from a lattice on the set {x0 , x2 , . . . , x14 } in any obvious way. That is, we show that the standard generators of IGt , the sk from Remark 2.17, are not equal to 5 the standard generators (see Definition 2.19) for any lattice D on {x0 , x2 , . . . , x14 }.

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L. Ballard

Before we begin, we introduce some definitions and notation that we use extensively throughout. Definition 2.19 The join-meet ideal of a lattice is defined from the join (least upper bound) x ∨ y and meet (greatest lower bound) x ∧ y of each pair of incomparable elements x, y ∈ L. In this work, a standard generator of the join-meet ideal of a lattice D is a nonzero element of one of the following four forms: xa xb − (xa ∨ xb )(xa ∧ xb ) = xa xb − (xa ∧ xb )(xa ∨ xb ) (xa ∨ xb )(xa ∧ xb ) − xa xb = (xa ∧ xb )(xa ∨ xb ) − xa xb for xa , xb ∈ L. We sometimes refer to such an element as a standard generator of D (the join-meet ideal is defined by analogous generators in the literature, though sometimes a, b ∈ L instead of xa and xb ). We note that for a standard generator, the pair {xa , xb } is an incomparable pair, and the pair {(xa ∨ xb ), (xa ∧ xb )} is a comparable pair. Though we are in a commutative ring, we provide all possible orderings for factors within the terms of a standard generator to emphasize that either factor of the monomial (xa ∨ xb )(xa ∧ xb ) = (xa ∧ xb )(xa ∨ xb ) may be the join or the meet of xa and xb . Remark 2.20 We give an explanation for why it makes sense to focus only on the standard generators of a join-meet ideal. We recall that the standard generators sk for IGtn from Remark 2.17 come from distinct 2 × 2 arrays within the ladder-like structure Ltn and recognize that either monomial of sk determines its 2 × 2 array. Then an element of the form ab − cd in IGt with a, b, c, d ∈ {x0 , x2 , x3 , . . . , x14 } 5 must be equal to ±si for some i, since a nontrivial sum of sk with coefficients in {−1, 1} either has more than two terms or is equal to si for some i, and other coefficients would be extraneous. Then any generating set for IGt where each 5 element has the form ab − cd in IGt with a, b, c, d ∈ {x0 , x2 , x3 , . . . , x14 } must 5 consist of all the sk (up to sign). We conclude that it is natural to check whether the sk are standard generators of a lattice D, instead of non-standard generators. Definition 2.21 Given a standard generator s = uz − wv of a lattice D, where u, v, w, z ∈ D, let Fs ∈ F2 be defined as follows: • If Fs = 0, the elements in the first (positive) monomial of s are not comparable in D (so the elements in the second (negative) monomial of s are comparable in D). • If Fs = 1, the elements in the negative monomial of s are not comparable in D (so the elements in the positive monomial of s are comparable in D).

Properties of the Toric Rings of a Chordal Bipartite Family of Graphs

25

For a given list s1 , s2 , . . . , sm of standard generators of a lattice D, we use F = (F1 , . . . , Fm ) ∈ Fm 2, where Fj = Fsj , to encode the comparability of the variables in these generators. We note that exactly one of Fj = 0 or Fj = 1 happens for each j ; we are merely encoding which monomial in sj corresponds to xa xb , and which to (xa ∨ xb )(xa ∧ xb ) = (xa ∧ xb )(xa ∨ xb ). Notation 2.22 We use the notation u > {w, v} if u > w and u > v in a lattice D, and {w, v} > z if w > z and v > z in D. In the first lemma, we begin by showing what restrictions we must have on a lattice whose join-meet ideal contains the 2-minors of the following array as standard generators: ab e cdf Lemma 2.23 Suppose s1 = bc − ad s2 = ce − af s3 = de − bf are standard generators of a lattice D. Let F ∈ F32 be defined for these three elements as in Definition 2.21. Then up to relabeling of variables, F ∈ {{0, 0, 0}, {0, 0, 1}, {0, 1, 1}}. Proof We first note that some of the cases we consider are equivalent. If we relabel variables according to the permutation (ac)(bd)(ef ), we see that F = {i, j, k} ≡ {1 − i, 1 − j, 1 − k}. This limits the cases we need to consider to F ∈ {{0, 0, 0}, {0, 0, 1}, {0, 1, 0}, {0, 1, 1}}. That is, we only need to show that the case F = {0, 1, 0} is impossible. Let F1 = 0. Then without loss of generality, up to reversing the order in the lattice (which does not affect the join-meet ideal), we have a > {b, c} > d. If F2 = 1, we have e > {a, f } > c, so e > {b, f } > d and hence F3 = 1. We conclude that the case F = {0, 1, 0} is impossible.



26

L. Ballard

In the second lemma, we show what restrictions we must have on a lattice whose join-meet ideal contains the 2-minors of the ladder ab e cdf g h as standard generators, and which meets certain comparability conditions. Lemma 2.24 Suppose s1 = bc − ad s2 = ce − af s3 = de − bf s4 = eg − bh s5 = f g − dh are standard generators of a lattice D, and that {a, g},{a, h},{c, g}, and {c, h} are comparable pairs in D. Let F ∈ F52 be defined for these five elements as in 2.21. Then up to relabeling of variables, F = {0, 0, 0, 0, 0}. Proof We first note that with natural relabeling, both {s1 , s2 , s3 } and {s3 , s4 , s5 } satisfy the hypotheses of Lemma 2.23, so if we let F be defined as in Definition 2.21, this limits the cases we need to consider to 5-tuples whose first three elements and whose last three elements satisfy the conclusion of Lemma 2.23. We note that some of the cases we consider are equivalent, allowing us to reduce to eighteen cases. If we relabel variables according to the permutation (ac)(bd)(ef ), we see that F = {i, j, k, l, m} ≡ {1−i, 1−j, 1−k, m, l}, and if we relabel the variables according to the permutation (be)(df )(gh), we have F = {i, j, k, l, m} ≡ {j, i, 1 − k, 1 − l, 1 − m}. The permutation (ah)(cg)(bf ) yields F = {i, j, k, l, m} ≡ {m, l, k, j, i}. Then we have eighteen cases in four equivalence classes, where the only equivalence class that satisfies the conclusion of the lemma is emboldened. {0, 0, 0, 0, 0} ≡ {1, 1, 1, 0, 0} ≡ {1, 1, 0, 1, 1} ≡ {0, 0, 1, 1, 1} {0, 0, 0, 0, 1} ≡ {1, 1, 1, 1, 0} ≡ {1, 1, 0, 0, 1} ≡ {0, 0, 1, 1, 0} ≡ {0, 1, 1, 0, 0} ≡ {1, 0, 0, 0, 0} ≡ {0, 1, 1, 1, 1} ≡ {1, 0, 0, 1, 1} {0, 0, 0, 1, 1} ≡ {1, 1, 1, 1, 1} ≡ {1, 1, 0, 0, 0} ≡ {0, 0, 1, 0, 0}

Properties of the Toric Rings of a Chordal Bipartite Family of Graphs

27

{0, 1, 1, 1, 0} ≡ {1, 0, 0, 0, 1} Case {0, 0, 0, 0, 1}: Since F1 = 0, without loss of generality (reversing the order on the entire lattice if needed) we have a > {b, c} > d. Then F2 = F3 = F4 = 0 and F5 = 1, with the ordering chosen, yield a > {c, e} > f b > {d, e} > f b > {e, g} > h g > {d, h} > f. If c > g, then a > {b, c} > g > d, but then bc − ad is not a standard generator of D, and this is a contradiction. If c < g, then c < g < b so that both {b, c} and {a, d} from s1 are comparable pairs, but this is a contradiction. We conclude that the case {0, 0, 0, 0, 1} is impossible. In the case F = {0, 0, 0, 1, 1}, the comparability of {c, g} forces either the comparability of {b, c} (a contradiction), or d < {b, c} < a < g, up to reversing the order in the lattice, since s1 is a standard generator. Since {a, h} is a comparable pair, it immediately follows that either {b, h} is comparable (a contradiction) or e < {b, h} < a < g, which is a contradiction since s4 is a standard generator, as the reader may verify. Because of the comparability of the pair {c, g}, the case F = {0, 1, 1, 1, 0} forces comparability of {f, g} or {b, c} and hence yields a contradiction. These cases are compatible with the given relabelings and thus conclude our proof.

In the third lemma, we show what restrictions we must have on a lattice whose join-meet ideal contains the 2-minors of the ladder ab e cdf i g hj as standard generators and which meets certain comparability conditions. Lemma 2.25 Suppose s1 = bc − ad s2 = ce − af s3 = de − bf s4 = eg − bh s5 = f g − dh s6 = gi − dj

28

L. Ballard

s7 = hi − fj are standard generators of a lattice D, and that {a, g}, {a, h}, {c, g}, {c, h}, {b, i}, {b, j }, {e, i}, and {e, j } are comparable pairs in D. Let F ∈ F72 be defined for these seven elements as in Definition 2.21. Then up to relabeling of variables, F = {0, 0, 0, 0, 0, 0, 0}. Proof We first note that with natural relabeling, the subsets {s1 , s2 , s3 , s4 , s5 } and {s3 , s4 , s5 , s6 , s7 } satisfy the hypotheses of Lemma 2.24, so if we let F be defined as in Definition 2.21, this limits the cases we need to consider to 7-tuples whose first five elements and whose last five elements satisfy the conclusion of Lemma 2.24. The only possible cases are {0, 0, 0, 0, 0, 0, 0} and {0, 0, 1, 1, 1, 0, 0}. If we relabel variables according to the permutation (be)(df )(gh), we see that F = {i, j, k, l, m, n, o} ≡ {j, i, 1 − k, 1 − l, 1 − m, o, n}, so that these two cases are equivalent. Then up to relabeling of variables, F = {0, 0, 0, 0, 0, 0, 0}.

In the fourth lemma, we show what restrictions we must have on a lattice whose join-meet ideal contains the 2-minors of the ladder ab e cdf i g hj k l as standard generators, and which meets certain comparability conditions. Lemma 2.26 Suppose s1 = bc − ad s2 = ce − af s3 = de − bf s4 = eg − bh s5 = f g − dh s6 = gi − dj s7 = hi − fj s8 = ik − f l s9 = j k − hl are standard generators of a lattice D, and that {a, g}, {a, h}, {c, g}, {c, h}, {b, i}, {b, j }, {e, i}, {e, j }, {d, k}, {d, l}, {g, k}, and {g, l} are comparable pairs in D. Let F ∈ F92 be defined for these nine elements as in Definition 2.21. Then F = {0, 0, 0, 0, 0, 0, 0, 0, 0}.

Properties of the Toric Rings of a Chordal Bipartite Family of Graphs

29

Proof We first note that with natural relabeling, both {s1 , s2 , s3 , s4 , s5 , s6 , s7 } and {s3 , s4 , s5 , s6 , s7 , s8 , s9 } satisfy the hypotheses of Lemma 2.25, so if we let F be defined as in Definition 2.21, this limits the cases we need to consider to 9-tuples whose first seven entries and whose last seven entries satisfy the conclusion of Lemma 2.25. We see by Lemma 2.25 that F = {0, 0, 0, 0, 0, 0, 0, 0, 0}.

We now have the machinery necessary to show that for t = (1, 1, 1, 1, 1, 0), IGt 5 does not come from a lattice. In our proof, we use the previous four lemmas and the (1,1,1,1,1,0) fact that IGt is generated by the distinguished minors of L5 : 5

x0 x2 x5 x3 x4 x6 x9 x13 x7 x8 x10 x11 x12 x14 Proposition 2.27 Let n = 5 and t = (1, 1, 1, 1, 1, 0). Then the set of standard generators for IGt is not equal to the complete set of standard generators (up to 5 sign) of any (classical) lattice. Proof By Remark 2.17 and choice of n = 5 and t = (1, 1, 1, 1, 1, 0), the generators of IGt are 5

s1 = x2 x3 − x0 x4

s2 = x3 x5 − x0 x6

s3 = x4 x5 − x2 x6

s4 = x5 x7 − x2 x8

s5 = x6 x7 − x4 x8

s6 = x7 x9 − x4 x10

s7 = x8 x9 − x6 x10

s8 = x9 x11 − x6 x12

s9 = x10 x11 − x8 x12

s10 = x11 x13 − x6 x14

s11 = x12 x13 − x9 x14 Suppose a lattice D exists whose complete set of standard generators (up to sign) equals {s1 , . . . , s11 }. We note that if the monomial xi xj does not appear in any of the sk , then {xi , xj } is a comparable pair, since otherwise ±(xi xj − (xi ∨ xj )(xi ∧ xj )) would be in the set of standard generators of D. Thus the pairs {x0 , x7 }, {x0 , x8 }, {x3 , x7 }, {x3 , x8 }, {x2 , x9 }, {x2 , x10 }, {x5 , x9 }, {x5 , x10 }, {x4 , x11 }, {x4 , x12 }, {x7 , x11 }, {x7 , x12 }, {x10 , x13 }, and {x10 , x14 } are comparable pairs in D. Let F ∈ F11 2 be defined as in Definition 2.21. Then with natural relabeling of the first nine relations, this lattice satisfies the hypotheses of Lemma 2.26, so the only cases we need to consider are F = {0, 0, 0, 0, 0, 0, 0, 0, 0, a, b}. Since F1 = 0, without loss of generality, we have x0 > {x2 , x3 } > x4 . Then with the ordering chosen, the fact that F3 = F5 = F7 = F9 = 0 yields x2 > {x4 , x5 } > x6

30

L. Ballard

x4 > {x6 , x7 } > x8 x6 > {x8 , x9 } > x10 x8 > {x10 , x11 } > x12 . The reader may verify that b = 0 and b = 1 both yield contradictions based on inspecting the standard generator s11 in light of the comparability of the pairs {x10 , x13 } and {x10 , x14 }, respectively, using the same technique employed in Case {0, 0, 0, 0, 1} of the proof of Lemma 2.24. We conclude that there is no lattice whose complete set of standard generators (up to sign) equals the set of standard generators of IG(1,1,1,1,1,0) .

5

3 Properties of the Family of Toric Rings In Sect. 2, we defined a family of toric rings, the rings R(n, t) coming from the family F, and we demonstrated some context for these rings in the area of graph theory. Now we investigate some of the algebraic properties of each R(n, t). We develop proofs to establish dimension, regularity, and multiplicity.

3.1 Dimension and System of Parameters We use the degree reverse lexicographic monomial order with x0 > x2 > x3 > · · · throughout this section, and denote it by >. We show that the standard generators sk given in Remark 2.17 are a Gröbner basis for IGtn with respect to >. Lemma 3.1 If s1 , . . . , s2n+1 are as in Remark 2.17, then B = {s1 , . . . , s2n+1 } is a Gröbner basis for IGtn with respect to >. Proof This is a straightforward computation using Buchberger’s Criterion and properties of the jk from Remark 2.11. By Remark 2.17, for 1  i  n the ideal IGtn is generated by s1 = x2 x3 − xj1 x4 s2i = x2i+1 x2i+3 − xj2i x2i+4 s2i+1 = x2i+2 x2i+3 − xj2i+1 x2i+4 , where j1 = j2 = 0, j3 = 2, and for i  2, we have ti+1 = 0 ⇐⇒ j2i = j2i−2 ⇐⇒ j2i+1 = 2i − 1 ti+1 = 1 ⇐⇒ j2i = j2i−1 ⇐⇒ j2i+1 = 2i.

Properties of the Toric Rings of a Chordal Bipartite Family of Graphs

31

If we adopt S-polynomial notation Si,j for the S-polynomial of si and sj , then the cases to consider are S2i−1,2i and S2i,2i+1 for 1  i  n S2i,2i+2 for 1  i  n − 1. To give a flavor of the computation involved, we show the case S2i−1,2i for 1  i  n, and leave the remaining cases to the reader. We show in each subcase that S2i−1,2i is equal to a sum of basis elements with coefficients in S(n), so that the reduced form of S2i−1,2i is zero in each subcase. We have S2i−1,2i = x2i+3 (x2i x2i+1 − xj2i−1 x2i+2 ) − x2i (x2i+1 x2i+3 − xj2i x2i+4 ) = −xj2i−1 x2i+2 x2i+3 + xj2i x2i x2i+4 Case 1: If i  2 and ti+1 = 0, then j2i = j2i−2 and j2i+1 = 2i − 1, so we have S2i−1,2i = −xj2i−1 x2i+2 x2i+3 + xj2i−2 x2i x2i+4 Case 1.1: If in addition i  3 and ti = 0, then j2i−2 = j2i−4 and j2i−1 = 2i − 3, so we have S2i−1,2i = −x2i−3 x2i+2 x2i+3 + xj2i−4 x2i x2i+4 = −x2i−3 (x2i+2 x2i+3 − xj2i+1 x2i+4 ) − x2i+4 (x2i−3 x2i−1 − xj2i−4 x2i ) = −x2i−3 s2i+1 − x2i+4 s2i−4 . Case 1.2: If in addition i = 2 or i  3 and ti = 1, then j2i−2 = j2i−3 and j2i−1 = 2i − 2, so we have S2i−1,2i = −x2i−2 x2i+2 x2i+3 + xj2i−3 x2i x2i+4 = −x2i−2 (x2i+2 x2i+3 − xj2i+1 x2i+4 ) − x2i+4 (x2i−2 x2i−1 − xj2i−3 x2i ) = −x2i−2 s2i+1 − x2i+4 s2i−3 . Case 2: If i = 1 or if i  2 and ti+1 = 1, then j2i = j2i−1 and j2i+1 = 2i, so we have S2i−1,2i = −xj2i x2i+2 x2i+3 + xj2i x2i x2i+4 = −xj2i s2i+1 . This concludes the case S2i−1,2i for 1  i  n − 1. The remaining cases are similar.



32

L. Ballard

Corollary 3.2 The ring R(n, t) is Koszul for all n and all t. Proof Since IGtn has a quadratic Gröbner basis, the ring R(n, t) is Koszul for all n and all t due to [15, Th 2.28].

Corollary 3.3 The initial ideal for IGtn with respect to the degree reverse lexicographic monomial order > is in> IGtn = (x2 x3 , {x2i+1 x2i+3 , x2i+2 x2i+3 | 1  i  n}). We note that in> IGtn does not depend on t, which will be useful for the following sections. Since Gtn is bipartite and has n + 4 vertices, the Krull dimension of R(n, t) is already known to be n + 3 [27, Prop 3.2]. We provide an alternate proof of the Krull dimension using the initial ideal in> IGtn from Corollary 3.3 and direct computation. As a corollary, we obtain the projective dimension of R(n, t). We note that the Krull dimension, like the initial ideal, does not depend on t. We refer the reader to Remark 2.17 for a reminder of how to think of the toric ring R(n, t) =

k[x0 , x2 , x3 , . . . , x2n+4 ] S(n) = IGtn IGtn

in the context of this work. Theorem 3.4 The Krull dimension of R(n, t) is dim R(n, t) = n + 3. Proof Let > be the degree reverse lexicographic monomial order with x0 > x2 > x3 > · · · > x2n+4 . By Corollary 3.3, the initial ideal of IGtn with respect to > is in> IGtn = (x2 x3 , {x2i+1 x2i+3 , x2i+2 x2i+3 | 1  i  n}). Since S(n)/(in> IGtn ) and R(n, t) = S(n)/IGtn are known to have the same Krull dimension (see for example [6, Props 9.3.4 and 9.3.12]), it suffices to prove that dim S(n)/(in> IGtn ) = n + 3. To see that the dimension is at least n + 3, we construct a chain of prime ideals in S(n) containing in> IGtn . Since every monomial generator of in> IGtn contains a variable of odd index, we begin with Pn = ({xk | k odd, 2 < k < 2n + 4}), a prime ideal containing in> IGtn . Then we have the chain of prime ideals Pn  Pn + (x0 )  Pn + (x0 , x2 )  Pn + (x0 , x2 , x4 )  · · ·  Pn + ({x2i | 0  i  n + 2}), so that

Properties of the Toric Rings of a Chordal Bipartite Family of Graphs

33

dim S(n)/(in> IGtn )  n + 3. To see that the dimension is at most n + 3, we find a sequence of n + 3 elements in S(n)/(in> IGtn ) such that the quotient by the ideal they generate has dimension zero. Let Xn = x0 , x2 − x3 , x4 − x5 , . . . , x2n − x2n+1 , x2n+2 − x2n+3 , x2n+4 in S(n), and take the quotient of S(n)/(in> IGtn ) by the image of Xn to obtain the following. In the last step, we rewrite the quotient of S(n) and (in> IGtn ) + (Xn ) by (Xn ) by setting x0 and x2n+4 equal to zero and replacing x2i with x2i+1 for 1  i  n + 1: S(n)/(in> IGtn ) ((in> IGtn ) + (Xn ))/(in> IGtn )

∼ =

S(n) ((in> IGtn ) + (Xn ))

∼ =

S(n)/(Xn ) ((in> IGtn ) + (Xn ))/(Xn )

∼ =

k[x3 , x5 , . . . , x2n+1 , x2n+3 ] 2 2 (x3 , {x2i+1 x2i+3 , x2i+3 |1i

n})

.

Since  2 (x32 , {x2i+1 x2i+3 , x2i+3 | 1  i  n}) = (x3 , x5 , . . . , x2n+3 ), the above ring has dimension zero. Thus, dim S(n)/(in> IGtn )  n + 3. We conclude that dim R(n, t) = dim S(n)/(in> IGtn ) = n + 3.



Corollary 3.5 The projective dimension of R(n, t) over S(n) is pdS(n) R(n, t) = n + 1. Proof We know the Krull dimension of the polynomial ring S(n) is 2n + 4. The result follows from the fact that R(n, t) is Cohen–Macaulay (Corollary 2.16) and from the graded version of the Auslander-Buchsbaum formula.

Remark 3.6 The proof of the previous theorem shows that the image of Xn = x0 , x2 − x3 , x4 − x5 , . . . , x2n − x2n+1 , x2n+2 − x2n+3 , x2n+4 in S(n)/(in> IGtn ) is a system of parameters for S(n)/(in> IGtn ). We prove in the next theorem that the image of Xn in R(n, t) (which we call Xn ) is also a system of

34

L. Ballard

parameters for R(n, t). Before doing so, we introduce some notation and a definition which allows us to better grapple with the quotient ring R(n, t)/(Xn ). Definition 3.7 Consider the isomorphism R(n, t) (Xn )

=

S(n)/(IGtn ) ((I

Gtn

) + (Xn ))/(I

Gtn

)

∼ =

S(n)/(Xn ) ((I ) + (Xn ))/(Xn ) Gtn

We view taking the quotient by Xn as setting x0 and x2n+4 equal to zero and replacing x2i with x2i+1 for 1  i  n + 1 to obtain  := k[x3 , x5 , . . . , x2n+1 , x2n+3 ] ∼ S(n) = S(n)/(Xn ). ∼ By the same process (detailed below), we obtain the ideal I Gtn = (IGtn +(Xn ))/(Xn ). We further define the quotient ∼  I t) := S(n)/ R(n, Gtn = R(n, t)/(Xn ). We find this notation natural since it is often used for the removal of variables, and the quotient by Xn may be viewed as identifying and removing variables. Since this work has no completions in it, there should be no conflict of notation. Definition 3.8 To define I Gtn in particular, we recall the standard generators of ∼ IGtn and introduce further notation to describe the generators of I Gtn = (IGtn + (Xn ))/(Xn ). By Remark 2.17, the standard generators of IGtn are s1 = x2 x3 − xj1 x4 s2i = x2i+1 x2i+3 − xj2i x2i+4 s2i+1 = x2i+2 x2i+3 − xj2i+1 x2i+4 , for 1  i  n, where the nonnegative integers jk are as in Remark 2.11. Let ι be the largest index such that j2ι = 0. By Remark 2.11, we see that the j2i are defined recursively and form a non-decreasing sequence. Then j1 = j2 = j4 = j6 = · · · = j2ι = 0, and since we view taking the quotient by Xn as setting x0 and x2n+4 equal to zero and replacing x2i with x2i+1 for 1  i  n + 1, we define I Gtn by replacing xjk with xJk (defined below) for 1  k < 2n to obtain s1 = x32 − xJ1 x5 s 2i = x2i+1 x2i+3 − xJ2i x2i+5 2 s 2i+1 = x2i+3 − xJ2i+1 x2i+5

Properties of the Toric Rings of a Chordal Bipartite Family of Graphs

35

s 2n = x2n+1 x2n+3 2 s 2n+1 = x2n+3

for 1  i < n, where

xJk

⎧ ⎪ ⎪ ⎨0 = xjk +1 ⎪ ⎪ ⎩x jk

if k is even and k  2 ι, or if k = 1 if 2 ι < k < 2n and jk is even if jk is odd

We note that Jk  k for each 1  k < 2n, since jk  k − 1 by Remark 2.11. By properties of the original jk from Remark 2.11, we know that xJ1 = xJ2 = 0, J3 = 3, and for 2  i < n, ti+1 = 0 ⇐⇒ xJ2i = xJ2i−2 ⇐⇒ J2i+1 = 2i − 1 ti+1 = 1 ⇐⇒ xJ2i = xJ2i−1 ⇐⇒ J2i+1 = 2i + 1. (0,0,0)

Example 3.9 We construct the ring R(2, (0, 0, 0)). For the graph G2 have the toric ring R(2, (0, 0, 0)) =

∈ F, we

k[x0 , x2 , x3 , . . . , x8 ] (x2 x3 − x0 x4 , x3 x5 − x0 x6 , x4 x5 − x2 x6 , x5 x7 − x0 x8 , x6 x7 − x3 x8 )

coming from the ladder-like structure (0,0,0)

L2

x0 x2 x5 = x3 x4 x6 x7 x8

from Example 2.3. We know X2 = x0 , x2 − x3 , x4 − x5 , . . . , x4 − x5 , x6 − x7 , x8 , so that R(2, (0, 0, 0))/(X2 ) is isomorphic to k[x0 , x2 , x3 , x4 , x5 , x6 , x7 , x8 ] (x2 x3 − x0 x4 , x3 x5 − x0 x6 , x4 x5 − x2 x6 , x5 x7 − x0 x8 , x6 x7 − x3 x8 , x0 , x2 − x3 , . . ., x8 )

∼ =

k[x3 , x5 , x7 ] 2 (x3 , x3 x5 , x52 − x3 x7 , x5 x7 , x72 )

= R(2, (0, 0, 0)).

Now we show that Xn is also a system of parameters for R(n, t), and not just for the quotient by the initial ideal.

36

L. Ballard

Proposition 3.10 Let R(n, t) = S(n)/IGtn and let Xn = x0 , x2 − x3 , x4 − x5 , . . . , x2n − x2n+1 , x2n+2 − x2n+3 , x2n+4 so that the image of Xn in S(n)/(in> IGtn ) is the system of parameters from Remark 3.6. Then the image of Xn in R(n, t) is a system of parameters for R(n, t). Proof Let Xn be defined as above. Then by Theorem 3.4 and Definition 3.7 we t) = 0. We have for n = 0 that need only show that dim R(n, t) = k[x3 ] , R(0, (x32 ) for n = 1 t) = R(1,

k[x3 , x5 ] , (x32 , x3 x5 , x52 )

and for n > 1  t) = S(n) = k[x3 , x5 , . . . , x2n+1 , x2n+3 ] , R(n, ({ s1 , s I 2i , s 2i+1 | 1  i  n}) Gtn where s1 = x32 s 2i = x2i+1 x2i+3 − xJ2i x2i+5 2 s 2i+1 = x2i+3 − xJ2i+1 x2i+5

s 2n = x2n+1 x2n+3 2 s 2n+1 = x2n+3

 I t) = dim S(n)/ for 1  i < n from Definition 3.8. We know dim R(n, Gtn =   dim S(n)/ I Gt . We claim that n

 I Gtn = (x3 , x5 , . . . , x2n+1 , x2n+3 ) . This is clear for n ∈ {0, 1}. For n > 1, we prove this by induction. Since s1 = x32  2  t and s = x are in I , we have x , x ∈ I 2n+1 3 2n+3 G Gt . Since 2n+3

n

s3 = x52 − x3 x7 ∈ I Gtn ⊆

n



I Gtn

Properties of the Toric Rings of a Chordal Bipartite Family of Graphs

37

   2 ∈ I  t t and x3 ∈ I , we get x , so that x ∈ I 5 Gn Gn Gtn . Now suppose x2i−1 , x2i+1 ∈ 5  I Gtn for 2  i < n. We have 2  s 2i+1 = x2i+3 − xJ2i+1 x2i+5 ∈ I Gtn ⊆



I Gtn .

 But xJ2i+1 ∈ {x2i−1 , x2i+1 } by Definition 3.8 and {x2i−1 , x2i+1 } ⊆ I Gtn by    induction, so that x 2 ∈ I Gt , and hence x2i+3 ∈ IGt . We conclude that 2i+3

n

(x3 , x5 , . . . , x2n+1 , x2n+3 ) ⊆

n



I Gtn ⊆ (x3 , x5 , . . . , x2n+1 , x2n+3 ) ,

 ∼   ∼ so we have equality. Since S(n)/ I Gtn = k has dimension zero, so does R(n, t) = R(n, t)/(Xn ). Thus, the image of Xn is a system of parameters for R(n, t).



Remark 3.11 We note that as a consequence of the proof of the preceding theorem, t) is Artinian, which will be relevant in Sect. 3.2. the ring R(n, Corollary 3.12 The image of Xn = x0 , x2 − x3 , x4 − x5 , . . . , x2n − x2n+1 , x2n+2 − x2n+3 , x2n+4 in R(n, t) = S(n)/IGtn is a regular sequence for R(n, t). Proof We know by Proposition 3.10 that the image of Xn in R(n, t) is a linear system of parameters. Since the rings R(n, t) are Cohen–Macaulay (Corollary 2.16), we are done.



3.2 Length, Multiplicity, and Regularity In this section, we determine the multiplicity and Castelnuovo-Mumford regularity of the toric rings R(n, t) coming from the associated graphs Gtn ∈ F by computing the length of the Artinian rings t) ∼ R(n, = R(n, t)/(Xn ) from Definition 3.7. We know by Corollary 3.12 that Xn is a linear regular sequence for R(n, t), which allows us to compute the multiplicity of the rings R(n, t). As t), a corollary of Theorem 3.16, which establishes the Hilbert function for R(n, we obtain the multiplicity and regularity of R(n, t). We also develop an alternate

38

L. Ballard

graph-theoretic proof for the regularity of R(n, t), which is included at the end of this section. t), which we We begin with a lemma establishing a vector space basis for R(n, use extensively for our results. Lemma 3.13 The image of all squarefree monomials with only odd indices whose indices are at least four apart, together with the image of 1k , forms a vector space t) over k. basis for R(n, t) and then find the initial ideal Proof We recall for the reader the definition of R(n,  t of IGn and use Macaulay’s Basis Theorem to show that the desired representatives t) as a vector space over k. form a basis for R(n, From Definition 3.7, we have  I t) = S(n)/ R(n, t)/(Xn ) ∼ = R(n, Gtn , where  = k[x3 , x5 , . . . , x2n+1 , x2n+3 ] S(n) and where I 1 , · · · , s 2n+1 as given in Definition 3.8. Gtn has generators s We first show that the image of the monomials with the desired property is a basis  by the initial ideal in> I in the quotient of S(n) Gtn . By Macaulay’s Basis Theorem, t) is also a which is Theorem 1.5.7 in [21], the image of these monomials in R(n, basis. To find the initial ideal of I k are a Gtn , we establish that the given generators s  Gröbner basis for IGtn with respect to the degree reverse lexicographic order >. This is a relatively straightforward computation using Buchberger’s Criterion, with separate cases for when xJk = 0. If we adopt S-polynomial notation Si,j for the S-polynomial of  si and sj , then the cases to consider are S1,2 , S2,3 , S2,4 , S2n−2,2n , S2n−1,2n , S2n,2n+1 S2i−1,2i for 1 < i < n S2i,2i+1 for 1 < i < n S2i,2i+2 for 1 < i < n − 1. To give a flavor of the computation involved, we show the case S2i,2i+2 for 1 < i < n − 1, and leave the remaining cases to the reader. We show in each subcase that  so S2i,2i+2 is equal to zero or to a sum of basis elements with coefficients in S(n), that the reduced form of S2i,2i+2 is zero in each subcase. We have

Properties of the Toric Rings of a Chordal Bipartite Family of Graphs

39

S2i,2i+2 = x2i+5 (x2i+1 x2i+3 − xJ2i x2i+5 ) − x2i+1 (x2i+3 x2i+5 − xJ2i+2 x2i+7 ) 2 + x2i+1 xJ2i+2 x2i+7 = −xJ2i x2i+5

Case 1: If ti+2 = 0, then xJ2i+2 = xJ2i and J2i+3 = 2i + 1, so we have 2 S2i,2i+2 = −xJ2i x2i+5 + xJ2i x2i+1 x2i+7

= −xJ2i s 2i+3 We note that if xJ2i = xJ2i+2 = 0, then S2i,2i+2 = 0. Case 2: If ti+2 = 1, then xJ2i+2 = xJ2i+1 and J2i+3 = 2i + 3, so we have 2 S2i,2i+2 = −xJ2i x2i+5 + x2i+1 xJ2i+1 x2i+7 .

Case 2.1: If in addition ti+1 = 0, then xJ2i = xJ2i−2 and J2i+1 = 2i − 1, so we have 2 + x2i+1 x2i−1 x2i+7 S2i,2i+2 = −xJ2i−2 x2i+5 2 = −xJ2i−2 (x2i+5 − xJ2i+3 x2i+7 ) + x2i+7 (x2i−1 x2i+1 − xJ2i−2 x2i+3 )

= −xJ2i−2 s 2i+3 + x2i+7 s 2i−2 . We note that if xJ2i = xJ2i−2 = 0, then S2i,2i+2 = x2i+7 s 2i−2 . Case 2.2: If in addition ti+1 = 1, then xJ2i = xJ2i−1 and J2i+1 = 2i + 1, so we have 2 2 + x2i+1 x2i+7 S2i,2i+2 = −xJ2i−1 x2i+5 2 2 = −xJ2i−1 (x2i+5 − xJ2i+3 x2i+7 ) + x2i+7 (x2i+1 − xJ2i−1 x2i+3 )

= −xJ2i−1 s 2i+3 + x2i+7 s 2i−1 . This concludes the case S2i,2i+2 for 1 < i < n − 1. The remaining cases are similar. Then the given generators sk are a Gröbner Basis for I Gtn , so that the initial ideal is 2 2 in> (I Gtn ) = (x3 , {x2i+1 x2i+3 , x2i+3 | 1  i  n})

 = k[x3 , x5 , . . . , x2n+1 , x2n+3 ]. Since in> (I in the ring S(n) Gtn ) consists precisely  and all degree two products of variables whose of all squares of variables in S(n) indices differ by exactly two, it follows that the image of the squarefree monomials whose indices are at least four apart, together with the image of 1k , forms a basis for

40

L. Ballard

 S(n) . By Macaulay’s in> I Gtn  S(n) is also a basis. It

t) = Basis Theorem, the image of these monomials in R(n,



Gn

We use the lemma above to establish facts about the vector space dimensions of t), which are applied further below to establish length and degree i pieces of R(n, multiplicity. t))i for the Notation 3.14 Throughout this section, we use dn,i := dimk (R(n,  vector space dimension of the degree i piece of R(n, t), that is, for the ith coefficient t). By Lemma 3.13, these are independent of t. in the Hilbert series of R(n, We establish a recursive relationship between these dimensions by introducing a short exact sequence of vector spaces. Lemma 3.15 For n  2 and i  1, the vector space dimension dn,i = t))i satisfies the recursive relationship dimk (R(n, dn,i = dn−1,i + dn−2,i−1 . Proof We use the vector space basis defined in Lemma 3.13. We note that the basis elements described are actually monomial representatives (which do not depend on t) of equivalence classes (which do depend on t), but we suppress this and speak as if they are monomials, not depending on t. We then take the liberty of suppressing t in what follows, for convenience. We recall for the reader that  = k[x3 , x5 , . . . , x2n−3 , x2n−1 , x2n+1 , x2n+3 ] S(n)  S(n − 1) = k[x3 , x5 , . . . , x2n−3 , x2n−1 , x2n+1 ]  S(n − 2) = k[x3 , x5 , . . . , x2n−3 , x2n−1 ]   i be multiplication by x2n+3 , and let − 2))i−1 → (R(n)) Let x2n+3 : (R(n  i → (R(n  x − 1))i be defined for a basis element b by 2n+3 : (R(n))  x 2n+3 (b) =

b

if x2n+3  b

0

if x2n+3 | b.

We note that these vector space maps are well-defined, since 1k or a squarefree monomial with odd indices at least four apart has an output of 0, 1, or a monomial with the same properties. The following sequence of vector spaces is exact

Properties of the Toric Rings of a Chordal Bipartite Family of Graphs

41

so that dn,i = dn−1,i + dn−2,i−1 .



Applying Lemma 3.15 and induction, we achieve the following closed formula t). for the coefficients of the Hilbert series of R(n, t) ∼ Theorem 3.16 If R(n, t) = S(n)/IGtn and R(n, = R(n, t)/(Xn ), we have ⎧ ⎪ 1 i=0 ⎪ ⎪ ⎪ ⎪ i ⎨1  t))i = (n + j − 2(i − 1)) 1  i  n/2 + 1 dimk (R(n, i! ⎪ ⎪ j =1 ⎪ ⎪ ⎪ ⎩0 i > n/2 + 1. Proof We establish the base cases i, n ∈ {0, 1} and proceed by induction, using Notation 3.14. It is clear that dn,0 = 1, generated by 1k . By Lemma 3.13 and by t) is a graded quotient, every nonzero element of positive degree the fact that R(n, i can be represented uniquely as a sum of degree i squarefree monomials with odd t))1 is generated by the indices whose indices are at least four apart. Then (R(n, images of all the odd variables x3 , x2(1)+3 , . . . , x2n+3 in S(n), so that dn,1 = n + 1 =

1 1  (n + j − 2(1 − 1)) 1! j =1

matches the given formula. Now we establish the base cases n = 0 and n = 1. We recognize that the first monomial of degree two with odd indices at least four apart is x3 x7 , which does not exist until n = 2, so we have ⎧ ⎪ ⎪1 i = 0 ⎨ d0,i = 1 i = 1 ⎪ ⎪ ⎩0 i > 1 and

d1,i =

⎧ ⎪ ⎪ ⎨1 i = 0 2 i=1 ⎪ ⎪ ⎩0 i > 1

42

L. Ballard

which match the given formula. This gives us the following table of base cases for dn,i , which match the given formula: n\i 0 1 2 3 4 .. .

0 1 1 1 1 1 .. .

1 2 3 4 5 6 7 ··· 1 0 0 0 0 0 0 ··· 2 0 0 0 0 0 0 ··· 3 4 5 .. .

We recall by Lemma 3.15 that we have the recursive relationship dn,i = dn−1,i + dn−2,i−1 for n  2 and i  1. We proceed by induction. Suppose that N  2 and 2  I < N/2 + 1 such that the dimension formula holds for all i when n < N. We recognize that we have 1  I  (N − 1)/2 + 1 and 1  I − 1  (N − 2)/2 + 1, so that by our recursion and by induction, we have dN,I = dN −1,I + dN −2,I −1 =

I I −1 1  1 (N − 1 + j − 2(I − 1)) + (N − 2 + j − 2(I − 2)) I! (I − 1)! j =1

=

=

j =1

I −1

I −1

j =1

j =1

  1 1 (N − 2(I − 1)) (N + j − 2(I − 1)) + (I ) (N + j − 2(I − 1)) I! I! I 1  (N + j − 2(I − 1)), I! j =1

as desired. For the special case where I = N/2 + 1, there is only one possible monic monomial of degree N/2 + 1 with odd indices at least four apart, N/2+1 

x4j −1 ,

j =1

so that dN,N/2+1 = 1. This matches the formula, since

Properties of the Toric Rings of a Chordal Bipartite Family of Graphs

1 (N/2 + 1)!

(N/2+1) 

(N + j − 2(N/2)) =

j =1

1 (N/2 + 1)!

43 (N/2+1) 

j = 1.

j =1

The remaining case is I > N/2 + 1. In this case, we have I > (N − 1)/2 + 1 and I −1 > (N −2)/2+1, so that dN,I = 0 by our recursive formula and induction.

Remark 3.17 We note from the proof of the theorem above a few facts for future t)) = 1 + 1 = 2 and (R(1, t)) = reference. By our base cases, we have (R(0, 1 + 2 = 3. Taking the Fibonacci sequence F (n) with F (0) = 0 and F (1) = 1, we have F (2) = 1, F (3) = 2, and F (4) = 3, so that t)) = F (3)

(R(0, t)) = F (4).

(R(1, These facts become useful in Proposition 3.19. We also note that dn,n/2+1 = 1 when n is even, which we use in the following corollary. It follows quickly from Theorem 3.16 that the regularity of R(n, t) is n/2 + 1. For an alternate proof of the regularity of R(n, t) which uses different machinery and more graph-theoretic properties, see the end of this section. Corollary 3.18 For Gtn ∈ F, reg R(n, t) = n/2 + 1. t) is Artinian by Remark 3.11, it is clear that reg R(n, t) is the top Proof Since R(n, t). By Theorem 3.16 and Remark 3.17, we only need show nonzero degree of R(n, that dn,n/2+1 = 0 when n is odd. We have dn,n/2+1 = dn, n+1 = 2

(n+1)/2  1 (j + 1) = 0. ((n + 1)/2)! j =1

t) is an Artinian quotient of R(n, t) by a linear regular sequence, we Since R(n, conclude that t) = n/2 + 1. reg R(n, t) = reg R(n,



t), In the following, we first compute the lengths of the dimension zero rings R(n, and then show a closed form for the multiplicity of our original rings R(n, t) by t) and applying using a Fibonacci relationship between the lengths of the rings R(n, Binet’s formula for F (n), the nth number in the Fibonacci sequence:

44

L. Ballard

F (n) =

(1 +

√ √ n 5) − (1 − 5)n . √ 2n 5

In the theorem and corollaries which follow, we suppress t for convenience, since the statements are independent of t.  satisfy the recursive formula Proposition 3.19 The lengths of the rings R(n)  = (R(n  

(R(n)) − 1)) + (R(n − 2)) for n  2. Consequently, if F (n) is the Fibonacci sequence, with F (0) = 0 and F (1) = 1, then  = F (n + 3) =

(R(n))

(1 +

√

5)n+3 − (1 − √ 2n+3 5

√

5)n+3

.

Proof Again, we use Notation 3.14. By the recursive relationship from Lemma 3.15, since dn,0 = 1 in general, and since dn,i = 0 in general for i > n/2+1 by Theorem 3.16, we have for n  2 that  =

(R(n))

n/2+1 

dn,i = 1 +

n/2+1 

i=0

dn−1,i + dn−2,i−1

!

i=1

=

n/2+1  i=0

dn−1,i +

(n−2)/2+1 

dn−2,i

i=0

  = (R(n − 1)) + (R(n − 2)). Now we show the second statement. For our base cases, we see from Remark 3.17  = F (3) = F (0 + 3) and that (R(1))  = F (4) = F (1 + 3). that (R(0))   Now suppose that (R(n − 1)) = F (n + 2) and (R(n − 2)) = F (n + 1). Then we have  = (R(n  

(R(n)) − 1)) + (R(n − 2)) = F (n + 3) ,  follows directly from Binet’s formula for as desired. The closed form for (R(n)) the Fibonacci sequence.

Corollary 3.20 For n  2, there is an equality of multiplicities  = e(R(n   e(R(n)) − 1)) + e(R(n − 2)).  and hence the Proof We have established the length of the Artinian rings R(n),  multiplicity e(R(n)).

Properties of the Toric Rings of a Chordal Bipartite Family of Graphs

45

Corollary 3.21 For n  2, there is an equality of multiplicities e(R(n)) = e(R(n − 1)) + e(R(n − 2)). In particular, e(R(n)) = F (n + 3) =

(1 +

√

5)n+3 − (1 − √ 2n+3 5

√ n+3 5)

.

 = R(n)/(Xn ), which by Proof To obtain the multiplicity of R(n), we look at R(n) Remark 3.11 and Corollary 3.12 is the Artinian quotient of R(n) by a linear regular sequence. By a standard result, we may calculate length along the obvious short exact sequences coming from multiplication by elements of our regular sequence to obtain the equality HilbR(n) (t)(1 − t)d = Hilb (t), R(n) where d is the Krull dimension of R(n). Defining multiplicity as in and preceding [24, Thm 16.7], it follows immediately that "  e(R(n)) = HilbR(n) (t)(1 − t)d "t=1 = Hilb (1) = (R(n)). R(n)



We are done by Proposition 3.19.

We reintroduce t and spend the remainder of this section providing an alternate graph-theoretic proof for the regularity of R(n, t). Alternate Proof of Corollary 3.18 We show that reg R(n, t) = n/2 + 1 by proving that reg IGtn = n/2 + 2. We first show that reg IGtn  n/2 + 2. We recall by Proposition 2.13 that the graph Gtn is chordal bipartite with vertex bipartition V1 ∪ V2 of cardinalities |V1 | = |V2 | =

n 2 n 2

+2 + 2,

and that Gtn does not have any vertices of degree one by Remark 2.6. Then by Theorem 4.9 of [3], we have reg IGtn  min

# n  2

+ 2,

n

$ n +2 = + 2. 2 2

46

L. Ballard

% & We note that we may equivalently prove reg R(n, t)  n2 + 1 by choosing the %n& 2 + 2 edges whose indices are equivalent to zero modulo 4, one from each row of Ltn , to obtain an edge matching (different from an induced matching, below) and then applying [14, Th 1]. We now show that reg IGtn  n/2 + 2. Since IGtn is homogeneous and in> IGtn consists of squarefree monomials by Corollary 3.3, we have by Corollary 2.7 of [4] that reg in> IGtn = reg IGtn , so it suffices to prove that reg in> IGtn  n/2 + 2. The ideal in> IGtn can be viewed as the edge ideal of a simple graph, a “comb” with n + 1 tines, with consecutive odd variables corresponding to vertices along the spine, as pictured below: 2

4

6

8

3

5

7

9

...

2

2 2

2 1

2 3

We know from Theorem 6.5 of [13] that the regularity of an edge ideal is bounded below by the number of edges in any induced matching plus one, so we choose n/2 + 1 edges (tines) corresponding to certain odd variables that create an induced matching. By beginning with the x3 -tine and choosing every other tine corresponding to the variables x3 , x3+4(1) , . . . , x3+4(n/2) , we obtain n/2 + 1 edges that are an induced matching, so we have reg in> IGtn  n/2 + 2, as desired. We conclude that reg IGtn = n/2 + 2, and hence that reg R(n, t) = n/2 + 1.

Acknowledgments Macaulay2 [11] was used for computation and hypothesis formation. We would like to thank Syracuse University for its support and hospitality and Claudia Miller for her valuable input on the original project in [1] and this condensed version. We would also like to thank the referees for noticing errors in the statement and proof of Theorem 3.16 as well as in the proof of Lemma 3.1, and for highlighting with clarity the connection between Theorem 3.4 and work done by Rafael Villarreal. We acknowledge the partial support of an NSF grant.

References 1. L. Ballard. Properties of the Toric Rings of a Chordal Bipartite Family of Graphs. PhD thesis, Syracuse University, 2020. 2. S. K. Beyarslan, H. T. Há, and A. O’Keefe. Algebraic properties of toric rings of graphs. Communications in Algebra, 47(1):1–16, 2019.

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3. J. Biermann, A. O’Keefe, and A. Van Tuyl. Bounds on the regularity of toric ideals of graphs. Advances in Applied Mathematics, 85:84–102, 2017. 4. A. Conca and M. Varbaro. Square-free Gröbner degenerations. Inventiones Mathematicae, 221(3):713–730, 2020. 5. A. Corso and U. Nagel. Monomial and toric ideals associated to Ferrers graphs. Transactions of the American Mathematical Society, 361(3):1371–1395, 2009. 6. D. Cox, J. Little, and D. O’Shea. Ideals, varieties, and algorithms. Undergraduate Texts in Mathematics. Springer, New York, third edition, 2007. An introduction to computational algebraic geometry and commutative algebra. 7. A. D’Alí. Toric ideals associated with gap-free graphs. Journal of Pure and Applied Algebra, 219(9):3862–3872, 2015. 8. G. Favacchio, G. Keiper, and A. Van Tuyl. Regularity and h-polynomials of toric ideals of graphs. Proceedings of the American Mathematical Society, 148:4665–4677, 2020. 9. F. Galetto, J. Hofscheier, G. Keiper, C. Kohne, M. E. Uribe Paczka, and A. Van Tuyl. Betti numbers of toric ideals of graphs: a case study. Journal of Algebra and its Applications, 18(12):1950226, 2019. 10. I. Gitler and C. E. Valencia. Multiplicities of edge subrings. Discrete Mathematics, 302(1):107–123, 2005. 11. D. R. Grayson and M. E. Stillman. Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/. 12. Z. Greif and J. McCullough. Green-Lazarsfeld condition for toric edge ideals of bipartite graphs. Journal of Algebra, 562:1–27, 2020. 13. H. T. Hà and A. Van Tuyl. Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers. Journal of Algebraic Combinatorics, 27(2):215–245, 2008. 14. J. Herzog and T. Hibi. The regularity of edge rings and matching numbers. Mathematics, 8(1):39, 2020. 15. J. Herzog, T. Hibi, and H. Ohsugi. Binomial ideals, volume 279 of Graduate Texts in Mathematics. Springer, Cham, 2018. 16. T. Hibi, A. Higashitani, K. Kimura, and A. O’Keefe. Depth of edge rings arising from finite graphs. Proceedings of the American Mathematical Society, 139(11):3807–3813, 2011. 17. T. Hibi and L. Katthän. Edge rings satisfying Serre’s condition (R1 ). Proceedings of the American Mathematical Society, 142(7):2537–2541, 2014. 18. T. Hibi, K. Matsuda, and H. Ohsugi. Strongly Koszul edge rings. Acta Mathematica Vietnamica, 41(1):69–76, 2016. 19. T. Hibi, K. Matsuda, and A. Tsuchiya. Edge rings with 3-linear resolutions. Proceedings of the American Mathematical Society, 147(8):3225–3232, 2019. 20. T. Hibi and H. Ohsugi. Toric ideals generated by quadratic binomials. Journal of Algebra, 218(2):509–527, 1999. 21. M. Kreuzer and L. Robbiano. Computational Commutative Algebra 1. Springer, Berlin, Heidelberg, 2000. 22. K. Mori, H. Ohsugi, and A. Tsuchiya. Edge rings with q-linear resolutions. Preprint, arxiv.org/abs/2010.02854. 23. R. Nandi and R. Nanduri. On Betti numbers of toric algebras of certain bipartite graphs. Journal of Algebra and Its Applications, 18(12):1950231, 2019. 24. I. Peeva. Graded Syzygies. Springer London, London, 2011. 25. A. Simis, W. V. Vasconcelos, and R. H. Villarreal. On the ideal theory of graphs. Journal of Algebra, 167(2):389–416, 1994. 26. C. Tatakis and A. Thoma. On the universal Gröbner bases of toric ideals of graphs. Journal of Combinatorial Theory, Series A, 118(5):1540–1548, 2011. 27. R. H. Villarreal. Rees algebras of edge ideals. Communications in Algebra, 23(9):3513–3524, 1995.

An Illustrated View of Differential Operators of a Reduced Quotient of an Affine Semigroup Ring Christine Berkesch, C-Y. Jean Chan, Patricia Klein, Laura Felicia Matusevich, Janet Page, and Janet Vassilev

Keywords Differential operators · Quotient · Reduced · Semigroup ring · Toric · Toric variety

1 Introduction In this paper, we provide illustrative examples and visualizations of some differential operators on the quotient of an affine semigroup ring by a radical monomial ideal. These examples motivate our work in [1]. For a finitely-generated algebra R over a field, let D(R) denote the ring of differential operators of R and use ∗ to denote an action of a differential operator. A foundational result in this area relates

CB was partially supported by NSF DMS 2001101. LFM was partially supported by the Simons Foundation. C. Berkesch () · P. Klein University of Minnesota, Minneapolis, MN, USA e-mail: [email protected]; [email protected] C-Y. J. Chan Department of Mathematics, Central Michigan University, Mt. Pleasant, MI, USA e-mail: [email protected] L. F. Matusevich Texas A & M University, College Station, TX, USA e-mail: [email protected] J. Page University of Michigan, Ann Arbor, MI, USA e-mail: [email protected] J. Vassilev University of New Mexico, Albuquerque, NM, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Miller et al. (eds.), Women in Commutative Algebra, Association for Women in Mathematics Series 29, https://doi.org/10.1007/978-3-030-91986-3_3

49

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C. Berkesch et al.

the ring of differential operators of an arbitrary quotient of a polynomial ring by an ideal J to differential operators on that polynomial ring and the idealizer of J . Proposition 1.1 ([8, Propostion 1.6]) Let S be the coordinate ring of a nonsingular affine variety over an algebraically closed field of characteristic 0, and let J be an S-ideal. Then there is an isomorphism   S ∼ I(J ) D , = J J D(S)

(1.1)

where I(J ) := {δ ∈ D(R) | δ ∗ J ⊂ J }, which is called the idealizer of J . The phrasing of [8, Proposition 1.6] given above differs from the original by making use of the equivalence of the conditions, with notation as above, of δJ D(S) ⊂ J D(S) and δ∗J ⊂ J [9, Lemma 2.3.1]. Traves [11] uses Proposition 1.1 to give concrete descriptions of rings of differential operators of Stanley–Reisner rings, and Saito and Traves [9] use the same to compute rings of differential operators of affine semigroup rings. For a non-regular affine semigroup ring RA over an algebraically closed field of characteristic 0, Proposition 1.1 fails, even for a radical monomial ideal J in RA . However, the differential operators on RA that induce maps on RA /J are precisely those in I(J ). Further, there is an embedding of rings I(J )

→ D D(RA , J )



RA J

 ,

so the description of D(RA ) by Saito and Traves can still be used to compute this subring of D(RA /J ). One primary goal of this article is to visually illustrate the computation of I(J )/D(RA , J ) when RA is an affine semigroup ring and J is a radical monomial ideal in RA ; in [1], we provide an explicit formula for this computation. In particular, the pictures we provide explain how to compute differential operators of the form D(I, J ) := {δ ∈ D(R) | δ ∗ I ⊆ J }, where I and J are subsets of the ring R, and ∗ denotes an action by a differential operator. These sets were originally instrumental in both the work of Smith and Stafford [8] and of Musson [5, Section 1], and they are essential building blocks of our computations, as well. Our second major goal in this paper is to compare J D(RA ) and D(RA , J ) for an affine semigroup ring RA and radical monomial ideal J of RA . Towards this end, the first set of examples we consider consists of quotients of the coordinate rings of rational normal curves. These quotients are all isomorphic to C[x, y]/xy, which is handled in [11]. From the standpoint of comparing J D(RA ) and D(RA , J ), what we will see is that J D(RA ) = D(RA , J ) for the rational normal curves of degrees 1 and 2 but fail to coincide in all degrees larger than 2. For rational normal

An Illustrated View of Differential Operators

51

curves of degree at most 2, J is principally generated and so is J D(RA ). In this case, it is straightforward to see (by definition) that D(RA , J ) is also principal and is isomorphic to J D(RA ). However, for degree n > 2, J has n − 1 generators. We will see that this number of generators greatly impacts I(J )/J D(RA ) but not I(J )/D(RA , J ). In [1], we consider cases where J D(RA ) = D(RA , J ),

(1.2)

for the particular ideal J generated by all monomials corresponding to the interior of the semigroup, i.e., J = ωRA . When RA is Gorenstein and normal, (1.2) holds since ωRA is principal. We show that the converse is also true; that is, (1.2) is equivalent to RA being a Gorenstein ring. To provide intuition beyond the two-dimensional case, we also offer a three-dimensional normal affine semigroup ring RA for which we compute I(J )/D(RA , J ) for two choices of J . Then, we return to the two-dimensional setting to consider a scored but not normal example, as well as a non-scored example, computing differential operators for quotients of both rings. Outline Section 2 fixes notation to be used throughout the article and describes the main result of [9]. Sections 3, 4, and 5 describe I(J )/D(RA , J ) for rational normal cones of degrees 2, 3, and higher, respectively. Section 6 considers a threedimensional normal affine semigroup ring modulo two different choices of radical monomial ideal J , and Sect. 7 computes I(J )/D(RA , J ) for two different nonnormal two-dimensional affine semigroup rings, where J = ωRA is the radical monomial ideal corresponding to the interior of the semigroup NA.

2 Background and Notation In this section, we fix notation and conventions to be assumed throughout the article. Although the results we discuss in this paper hold over any algebraically closed field of characteristic 0, we will use in this illustrated view the field of complex numbers for the sake of simplicity. Having fixed notation, we will then state and discuss [9, Theorem 3.2.2]. Definition 2.1 Let A be a k × matrix with entries in Z. Let NA denote the semigroup in Zk that is generated by the columns of A. The affine semigroup ring determined by A is RA = C[NA] =

' a∈NA

C · ta ,

52

C. Berkesch et al.

where ta = t1a1 t2a2 · · · tkak for a = (a1 , a2 , . . . , ak ) ∈ NA. Throughout this article, we assume that the group generated by the columns of A is the full ambient lattice, so ZA = Zk , and also that the real positive cone over A, R0 A, is pointed, meaning that it is strongly convex. Definition 2.2 A semigroup NA is normal if NA = R0 A ∩ ZA. A semigroup NA is scored if the difference (R0 A∩ZA)\NA consists of a finite union of hyperplane sections of R0 A ∩ ZA that are all parallel to facets of the cone R0 A. An affine semigroup ring C[NA] is said to be normal, or scored, if NA is normal, or scored, respectively. Note that normal semigroups are scored. When we write a facet σ of R0 A (or A or NA), we will always mean the integer points in the corresponding facet of R0 A. When A is normal, this is the same as the semigroup generated by the columns of A that lie in the corresponding facet of R0 A. Throughout, we use  −  to indicate ideals in the commutative rings RA or the polynomial ring C[θ ] = C[θ1 , . . . , θk ]. It will be clear from the context if the ideals are in RA or in C[θ ]. Notice that the Zk -graded prime ideals in RA are in one-to-one correspondence with the faces of A (or R0 A), as a face τ of A corresponds to the multigraded prime RA -ideal ) ( " Pτ := td " d ∈ NA \ τ . In this paper, we compute rings of the form I(J )/D(RA , J ), where J is a radical monomial ideal in RA ; as such, J is always as intersection of primes of the form Pτ , for various faces τ of A. We are mainly following the description of Saito and Traves [9], although Musson and Van den Bergh describe the ring of differential operators of a toric ring C[NA] first in [4, 6] and [7] when viewed as a subring of the ring of differential operators of the Laurent polynomials, i.e., D(C[Zk ]) = C{t1±1 , . . . , tk±1 , ∂1 , . . . , ∂k }/ ∼, where ∂i denotes the differential operator ∂t∂ i and ∼ denotes the usual relations on

the free associative algebra C{t1±1 , . . . , tk±1 , ∂1 , . . . , ∂k } that describe the behavior of differential operators. To explain the differential operators of C[NA] as presented by Saito and Traves, note first that D(C[Zk ]) carries a Zk -grading, where ei denotes the i-th column of the identity matrix I and deg(ti ) = ei = − deg(∂i ). Note also that if ai is a column of A, then deg(tai ) = ai . Set θj = tj ∂j for 1  j  k, and set (d) := {a ∈ NA | a + d ∈ / NA} = NA \ (−d + NA). We note also that for any tm ∈ RA and f (θ ) ∈ C[θ ], f (θ ) ∗ tm = f (m)tm . The idealizer of (d) is defined to be I((d)) := f (θ ) ∈ C[θ ] = C[θ1 , θ2 . . . , θk ] | f (a) = 0 for all a ∈ (d),

An Illustrated View of Differential Operators

53

which is viewed as an ideal in the ring C[θ ], where θi = ti ∂i ∈ D(C[Zd ]) is of degree 0. We will soon see that I((d)) consists of f (θ ) such that td f (θ ) ∈ D(RA ). To compute I((d)) for a normal semigroup ring, consider a facet σ of A, recalling that by this we mean all lattice points on the corresponding facet of the cone R0 A. The primitive integral support function hσ is a uniquely determined linear form on Rk such that: (1) hσ (R0 A)  0, (2) hσ (Rσ ) = 0, and (3) hσ (Zk ) = Z. We write σ1 , σ2 , . . . , σm for the facets of A, so that for the remainder of the paper, we set hj := hj (θ ) = hσj (θ ). For a non-negative integer n, we will use the following notation to denote this descending factorial: (α, n)! :=

n 

(α − i) = α(α − 1) · · · (α − n),

i=0

where α is a function, which could be constant or already evaluated. For example, ! hj , hj (−d) − 1) ! =

hj (−d)−1



(hj − i)

i=0

will be a common expression throughout this article, as it is a factor in the generator of the idealizer I((d)). To streamline our presentation, we make the convention that (α, n)! = 1 for all n < 0. Theorem 2.3 ([9, Theorem 3.2.2]) If RA is a pointed, normal affine semigroup ring with ZA = Zk , then D(RA ) =

'

td · I((d)),

d∈Z

d

=

' d∈Z

d

* t · d



+ ! hj , hj (−d) − 1 ! ,

(2.1)

hj (d) 0, j  n > 0}. Notice in particular that x m ∂xm and y n ∂yn both have degree 0. Setting θx = x∂x and θy = y∂y , from the Weyl algebra relations, it follows that ∂xi = x −i ·

i−1 

(θx − ) = x −i (θx , i − 1)! and ∂y = y −j ·

=0

j

j −1

! (θy − ) = θy , j − 1 !.

=0

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! j j Hence x m ∂xi = x m−i · (θx , i − 1)! and y n ∂y = y n−j · θy , j − 1 !, and x m y n ∂xi ∂y has multidegree (m − i, n − j ) in W . In fact, '

W =

! x m y n ·  (θx , m − 1)! θy , n − 1 !,

(m,n)∈Z

2

which is a presentation of the Weyl algebra using the Saito–Traves approach from Theorem 2.3. From this viewpoint, Traves showed that

I(xy)(m,n)

⎧ m n ⎪ ⎪x y · C[θx , θy ] ⎪ ⎪ ⎨x m y n ·  (θ , −m)! x = ! m n ⎪ x y ·  θy , −n ! ⎪ ⎪ ⎪ ! ⎩ m n x y ·  (θx , −m)! θy , −n !

if m, n  0, if m < 0, n  0, if n < 0, m  0, if m, n < 0.

Further, xyW can be expressed as a multigraded W -ideal that is contained in I(xy) as follows: ⎧ m n ⎪ ⎪ ⎪x y · C[θx , θy ] ⎪ ⎨x m y n ·  (θ , −m)! x = ! ⎪x m y n ·  θy , −n ! ⎪ ⎪ ⎪ ! ⎩ m n x y ·  (θx , −m)! θy , −n !

xyW(m,n)

if m, n > 0, if m  0, n > 0, if n  0, m > 0, if m, n  0.

Now, applying Proposition 1.1 yields a computation for the ring of differential operators for the ordinary double point R = C[x, y]/xy:

 D

C[x, y] xy

 (m,n)

⎧ ⎪ 0 ⎪ ⎪ ⎪ ⎪ C[θx , θy ] ⎪ ⎪ ⎪ ⎪ ⎨ θx θy   (θx , −m)! = xm · ⎪ ⎪ ⎪  (θx , −m)!θ ⎪ ! y ⎪ ⎪ , −n !  θ ⎪ y ⎪y n · ⎪ ! ⎩  θy , −m !θx 

if mn = 0, if m = n = 0, if m = 0, n = 0,

(2.2)

if n = 0, m = 0.

We next fix some notation to be used in the remainder of this paper. Whenever the dimension of the semigroup ring under consideration is k = 2, instead of using t1 , t2 as our variables, we will use s, t. Further, when considering subsets of R2 and R3 that contain the lattice points that describe a set of monomials in our semigroup, such as lines or planes, we will describe them with the variables x, y, and z, for example, the line y = 2x − 1 in R2 or the plane x − z = 2 in R3 . Consider the matrix

An Illustrated View of Differential Operators

57

  1 1 1 ··· 1 An = . 0 1 2 ··· n We call RAn = C[NAn ] = C[s, st, st 2 , . . . st n ] the ring of the rational normal curve of degree n, since it is the coordinate ring of the affine cone of the projective rational normal curve. This ring will be the subject of the next three sections. The two facets of An are   1 = {(x, y) ∈ N2 | x  0, y = 0} and 0   1 σ2 = N = {(x, y) ∈ N2 | x  0, y = nx}. n

σ1 = N

The prime ideal associated to σ1 is P1 = st, st 2 , . . . , st n , and the prime ideal associated to σ2 is P2 = s, st, . . . , st n−1 . We will consider the radical ideal  J = P1 ∩ P2 =

st, st 2 , . . . , st n−1  if n > 1, if n = 1.

s 2 t

∼ C[x, y]/xy for all n > 0. At the end of Sect. 5, we revisit Observe that RAn /J = Example 2.5 and present a C-algebra isomorphism between D(C[x, y]/xy) and D(RAn /J ) and compare it to our calculations for I(J )/D(RAn , J ).

3 Differential Operators on the Rational Normal Curve of Degree 2 In this section, we compute the idealizer, I(I ), along with the subset of differential operators D(RA2 , I ) for the ideals I = P1 = st, st 2  and I = J = st over the ring of the rational normal curve of degree 2, RA2 = C[s, st, st 2 ]. To aid our computations we include illustrations of the lattice representing the multidegrees in the plane broken down into four chambers where the operators will be determined by similar expressions. The facets of A2 are σ1 = {(x, y) ∈ N2 | x  0, y = 0}

and

σ2 = {(x, y) ∈ N2 | x, y  0, y = 2x},

which have primitive integral support functions h1 = θ2

and

h2 = 2θ1 − θ2 .

Figure 2 illustrates the integer lattice, divided into four chambers that are colored as follows:

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Fig. 2 Chambers of D(RA2 )

C1 : The red multidegrees correspond to monomials in J , the gray multidegrees correspond to monomials in P1 \ J and the blue multidegrees correspond to monomials in RA2 \ P1 , C2 : The yellow multidegrees are the d with h1 (d)  0 and h2 (d) < 0, C3 : The violet multidegrees are the d with both h1 (d) < 0 and h2 (d) < 0, and C4 : The green multidegrees are the d with h1 (d) < 0 and h2 (d)  0. Still following the convention (h, n)! = 1 if n < 0, by Theorem 2.3, the graded pieces of D(RA2 ) are D(RA2 )d = s d1 t d2 · (h1 , h1 (−d) − 1)! (h2 , h2 (−d) − 1)! . Broken down by chambers, this amounts to: ⎧ ⎪ s d1 t d2 · C[θ ] ⎪ ⎪ ⎪ ⎨s d1 t d2 ·  (h , −2d + d − 1)! 2 1 2 D(RA2 )d = d d 1 2 ⎪ s t ·  (h1 , −d2 − 1)! (h2 , −2d1 + d2 − 1)! ⎪ ⎪ ⎪ ⎩ d1 d2 s t ·  (h1 , −d2 − 1)!

if d ∈ C1, if d ∈ C2, if d ∈ C3, if d ∈ C4.

Example 3.1 We first compute the graded pieces of the sets of differential operators I(P1 ) and D(RA2 , P1 ). Later, since C[x] ∼ = RA2 /P1 , we will exhibit a C-algebra isomorphism between D(C[x]) and D(RA2 /P1 ). To begin, recall that I(P1 ) = {δ ∈ D(RA2 ) | δ ∗ P1 ⊆ P1 } and D(RA2 , P1 ) = {δ ∈ D(RA2 ) | δ ∗ R ⊆ P1 }. Now if d ∈ C1 and s m1 t m2 ∈ P1 or if d ∈ C1 \ σ1 and s m1 t m2 ∈ RA2 , then for any g(θ ) ∈ C[θ ],

An Illustrated View of Differential Operators

(a)

59

(b)

Fig. 3 Lines parallel to the facets. (a) Lines parallel to σ2 . (b) Lines parallel to σ1

s d1 t d2 · g(θ ) ∗ s m1 t m2 = g(m)s d1 +m1 t d2 +m2 ∈ P1 , so I(P1 )d = D(RA2 )d for all d ∈ C1 and D(RA2 , P1 )d = D(RA2 )d for all d ∈ C1\σ1 . With the aid of Fig. 3, we will explain how to determine I(P1 ) and D(RA2 , P1 ) in the other chambers. The red lattice points in Fig. 3, indicate the monomials in P1 . First, note that if d ∈ C2 and s m1 t m2 ∈ P1 or d ∈ C2 \ (−σ1 ) and m 1 s t m2 ∈ RA2 , and m lies on one of the lines y = 2x − r shown in Fig. 3a, then s d1 t d2 (h2 , −2d1 + d2 − 1)!, the generator of D(RA2 )d , applied to such a monomial will either be a constant times a monomial represented by a red lattice point on one of the lines y = 2x − j for 0  j  r or, when (h2 (m), −2d1 + d2 − 1)! = 0, it will be 0. Hence, I(P1 )d = D(RA2 )d for d ∈ C2

and

D(R, P1 )d = D(RA2 )d for d ∈ C2 \ (−σ1 ).

Now if d ∈ C4 and s m1 t m2 ∈ P1 or d ∈ C4 ∪ σ1 and s m1 t m2 ∈ RA2 and m lies on one of the lines y = r shown in Fig. 3b, then s d1 t d2 (h1 , −d2 − 1)!, the generator of D(RA2 )d , applied to such a monomial will either be a constant times a monomial represented by a red lattice point on one of the lines y = j for 0  j  r or, when (h1 (m), −2d1 + d2 − 1)! = 0, it will be 0. However, the monomials represented by the lattice points on y = 0 do not lie in P1 . The monomials represented by the red lattice points on the line y = −d2 are precisely the monomials whose image is a term represented by a lattice point on σ1 . Hence, we need to further right-multiply any operator in D(RA2 )d by θ2 + d2 , so that I(P1 )d = s d1 t d2 ·  (h1 , −d2 )! for d ∈ C4

and

D(RA2 , P1 ) = s d1 t d2 ·  (h1 , −d2 )! for d ∈ C4 ∪ σ1 .

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Using similar reasoning, we can easily see that I(P1 )d = s d1 t d2 ·  (h1 , −d2 )! (h2 , −2d1 + d2 − 1)! for d ∈ C3 and D(RA2 , P1 )d = s d1 t d2 ·  (h1 , −d2 )! (h2 , −2d1 + d2 − 1)! for d ∈ C3 ∪ (−σ1 ). Putting these all together, the graded pieces of I(P1 ) follows: ⎧ ⎪ s d1 t d2 · C[θ ] ⎪ ⎪ ⎪ ⎨s d1 t d2 ·  (h , −2d + d − 1)! 2 1 2 I(P1 )d = d d 1 2 ⎪ s t ·  (h1 , −d2 )! (h2 , −2d1 + d2 − 1)! ⎪ ⎪ ⎪ ⎩ d1 d2 s t ·  (h1 , −d2 )! ⎧ ⎪ s d1 t d2 ⎪ ⎪ ⎪ ⎨s d1 t d2 D(RA2 , P1 )d = ⎪ s d1 t d2 ⎪ ⎪ ⎪ ⎩ d1 d2 s t

and D(RA2 , P1 ) are as if d ∈ C1 = NA2 , if d ∈ C2, if d ∈ C3, if d ∈ C4,

· C[θ ]

if s d1 t d2 ∈ P1 ,

·  (h2 , −2d1 + d2 − 1)!

if d ∈ C2 \ (−σ1 ),

·  (h1 , −d2 )! (h2 , −2d1 + d2 − 1)! if d ∈ C3 ∪ (−σ1 ), ·  (h1 , −d2 )!

if d ∈ C4 ∪ σ1 .

Now taking the quotient, we obtain: ⎧ ⎪ 0 ⎪ ⎪ ⎪   ⎨ d1 C[θ ] I(P1 ) = s · h1  D(RA2 , P1 ) d ⎪ ⎪  (h2 , −2d1 − 1)! ⎪ ⎪ ⎩s d1 · h1 (h2 , −2d1 − 1)!

if d ∈ / Zσ1 , if d ∈ σ1 , if d ∈ (−σ1 \ 0).

Viewing D(C[x]) as a Z-graded algebra over C[θx ], we note that D(C[x]) =

∞  d=1

∂xd · C[θx ] ⊕

∞ 



x d C[θx ] =

x d (θx , −d − 1)! · C[θx ].

d∈Z

d=0

The map φ:



x d · (θx , −d − 1)! · C[θx ] →

d∈Z

 d∈Z

sd ·

 (h2 , −2d − 1)! , h1 (h2 , −2d − 1)!

which is defined on the generators by φ(x d (θx , −d − 1)!) = s d (h2 , −2d − 1)! + h1 (h2 , −2d − 1)!,

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61

although a C-vector space isomorphism, does not produce a ring isomorphism. The graded pieces in negative degree are generated by polynomials in θ whose degrees are twice as large as large as the degrees in θx given in the Weyl algebra. In fact, ⎧ ⎪ ⎪ ⎪0 ⎪ ⎪ C[θ ] ⎪ ⎪ s d1 · , ⎪ ⎪ h1 ⎪ ⎪ ⎨ 2 , D(RA2 /P1 ) = h2 ⎪ ⎪ ⎪ , −d − 1 ! 1 ⎪ ⎪ 2 ⎪ d 1 ⎪ s ·,  ⎪ ⎪ h1 h2 ⎪ ⎪ ⎩ , −d1 − 1 ! 2 2

if d ∈ / Zσ1 , if d ∈ σ1 ,

if d ∈ (−σ1 \ 0).

Therefore, we can produce an isomorphism of graded rings ψ : D(C[x]) → D(RA2 /P1 ) defined for any m ∈ N by ψ(x ) = s , m

m

and

ψ(∂xm )

=s

−m

 ·

 ,  h 1 h2 h2 ,m − 1 ! + ,m − 1 ! . 2 2 2

Example 3.2 We will now compute the graded pieces of the sets of differential operators I(J ) and D(RA2 , J ), as well as the graded pieces of J D(RA2 ). Recall that I(J ) = {δ ∈ D(RA2 ) | δ ∗ J ⊆ J }

and

D(RA2 , J ) = {δ ∈ D(RA2 ) | δ ∗ R ⊆ J }.

We will soon give a general formula for the graded pieces of I(J ) and D(RA2 , J ); however, for illustrative purposes, first consider the graded piece of D(RA2 ) at (−1, 0): s −1 ·  (h2 , 1)!. Applying s −1 · (h2 , 1)! to a monomial whose exponent lies in the two parallel half-lines σ2 and y = 2x − 1 in NA will yield 0, which certainly lives inside J . However, when we let s −1 · (h2 , 1)! act on a monomial whose exponent is a member of the half-lines y = 2x − 2 or y = 0 lying inside NA2 , we obtain an integer multiple of a monomial whose exponent lies in one of the facets of A2 , σ2 or σ1 respectively, and these are not in J . The remaining monomials in J yield another element of J when they are acted upon by any operator in s −1 ·  (h2 , 1)! . In Fig. 4, the two light blue lines indicate the two half-lines representing the multidegrees of monomials in RA2 that, after application of an element in D(RA2 )(−1,0) , fails to yield an element in J . To correct for this lack of membership in J for the monomials on y = 2x − 2, every element of D(RA2 )(−1,0) should be multiplied by (h2 − 2); applying s −1 · (h2 , 2)! to these monomials yields 0. The application of s −1 · (h2 , 2)! to the remaining monomials in J will output a term inside J . Thus, I(J )(−1,0) = s −1 ·  (h2 , 2)!.

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Fig. 4 Vanishing for d = (−1, 0)

Similarly, for every operator δ ∈ D(RA2 )(−1,0) , δ(h2 − 2)h1 ∗ s m1 t m2 = 0 for every m on y = 2x − 2 or in σ1 . Also, δ(h2 − 2)h1 ∗ s m1 t m2 ∈ J for all other m in NA, so D(RA2 , J )(−1,0) = s −1 · h1 (h2 , 2)!. In fact, using a similar argument applied to any d ∈ C2 in the case of I(J )d and d ∈ C2 \ (−σ1 ) for D(RA2 , J )d , applying an operator in D(RA2 )d to a monomial with exponent on the half-lines y = 2x−j for 0  j < h2 (−d) will give 0; whereas, these operators applied to a monomial with multidegrees on y = 2x + h2 (d) yields a constant multiple of a monomial with exponent in σ2 . Hence, right-multiplying s d1 t d2 · (h2 , h2 (−d) − 1)! by h2 + h2 (d) produces an operator that, when applied to monomials with multidegrees on the lines y = 2x + h2 (d), becomes 0, and I(J )d = s d1 t d2 ·  (h2 , −2d1 + d2 )! for all d ∈ C2,

and

D(RA2 , J )d = s d1 t d2 ·  (h2 , −2d1 + d2 )! for all d ∈ C2 \ (−σ1 ). We will discuss the multidegrees d ∈ C2 ∩ (−σ1 ) momentarily, when we turn to C3. Determining the graded piece of multidegree d for both I(J ) in C4 and D(RA2 , J ) in C4 \ (−σ2 ) is quite similar to the arguments we used to determine I(J )d for d in C2 and D(RA2 , J )d for d in C2 \ (σ1 ), respectively. We will briefly describe I(J )(−1,−2) and D(RA2 , J )(−1,−2) with the aid of Fig. 5 and then immediately describe the general case. Recall that D(RA2 )(−1,−2) = s −1 t −2 ·  (h1 , 1)!. Similar to the argument for degree d = (−1, 0) above, the monomials corresponding to the multidegrees which lie on the two light blue lines (y = 2 and σ2 ) in Fig. 5 are the only exponents of monomials that fail to land inside J after the application of s −1 t −2 · (h1 , 1)!.

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63

Fig. 5 Vanishing for d = (−1, −2)

Fig. 6 Vanishing for d = (−1, −1)

To correct this deficiency, right-multiply by (h1 − 2) for I(J )(−1,−2) and (h1 − 2)h2 for D(RA2 , J )(−1,−2) . Notice that applying the operator s −1 t −2 · (h1 , 2)! to a monomial corresponding to d ∈ NA2 along the half-lines y = 2 or the operator s −1 t −2 ·(h1 , 2)!h2 to a monomial corresponding to d ∈ NA2 along y = 2 or y = 2x now yields 0, and no problems are created for the remaining monomials in J or RA2 , respectively. Thus, I(J )(−1,−2) = s −1 t −2 · (h1 , 2)! and

D(RA2 , J )(−1,−2) = s −1 t −2 · (h1 , 2)!h2 .

In fact, I(J )d = s d1 t d2 ·  (h1 , −d2 )! for all d ∈ C4,

and

D(RA2 , J )d = s d1 t d2 ·  (h1 , −d2 )! for all d ∈ C4 \ (−σ2 ). We will return to D(RA2 , J )d for d ∈ C4 ∩ (−σ2 ) when we turn to C3. Determining I(J )d for d ∈ C3 or D(RA2 , J )d for d ∈ C3 ∪ (−σ1 ) ∪ (−σ2 ) again is akin to the arguments we gave above for D(RA2 , J )(−1,0) and D(RA2 , J )(−1,−2) .

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With the aid of Fig. 6, we will briefly describe I(J )(−1,−1)

and

D(RA2 , J )(−1,−1) .

This explanation can easily be extended from d = (−1, −1) to all multidegrees d in C3 (or in C3 ∪ (−σ1 ) ∪ (−σ2 ) in the case of D(RA2 , J )). If the operator s −1 t −1 · h1 h2 , which is the generator for D(RA2 )(−1,−1) , is applied to any of the monomials corresponding to multidegree d ∈ NA2 along the two light blue halflines in Fig. 6 (the portion of y = 1 or y = 2x − 1 in C1), we obtain an integer multiple of a monomial with exponent in the facets σ1 or σ2 , respectively, which is not in J , and no problems are created for the remaining monomials in J (or RA2 for D(RA2 , J )). Hence, right-multiplying by (h1 − 1)(h2 − 1) yields a new operator s −1 t −1 · (h1 , 1)! (h2 , 1)! that will send to 0 all monomials with multidegrees d ∈ NA2 along the half-lines y = 2x − 1 and y = 1. No problems are created for the remaining monomials in RA2 and we obtain I(J )(−1,−1) = D(RA2 , J )(−1,−1) = s −1 t −1 ·  (h1 , 1)! (h2 , 1)!. In fact, I(J )d = D(RA2 , J )d = s d1 t d2 ·  (h1 , −d2 ))! (h2 , −2d1 + d2 )!

for all d ∈ C3,

and D(RA2 , J )d = s d1 t d2 ·  (h1 , −d2 ))! (h2 , −2d1 + d2 )!

for all d ∈ (−σ1 ) ∪ (−σ2 ).

Hence, the graded pieces of I(J ) and D(RA2 , J ) are as follows: ⎧ ⎪ s d1 t d2 ⎪ ⎪ ⎪ ⎨s d1 t d2 I(J )d = ⎪s d 1 t d 2 ⎪ ⎪ ⎪ ⎩ d1 d2 s t

· C[θ ]

if d ∈ C1 = NA2 ,

·  (h2 , −2d1 + d2 )!

if d ∈ C2,

·  (h1 , −d2 )! (h2 , −2d1 + d2 )! if d ∈ C3, ·  (h1 , −d2 )!

⎧ ⎪ s d1 t d2 · C[θ ] ⎪ ⎪ ⎪ ⎪ d1 d2 ⎪ ⎪ ⎨s t ·  (h2 , −2d1 + d2 )! D(RA2 , J )d = s d1 t d2 · ⎪ ⎪ ⎪ ⎪  (h1 , −d2 )! (h2 , −2d1 + d2 )! ⎪ ⎪ ⎪ ⎩s d1 t d2 ·  (h , −d )! 1

2

Now taking the quotient, we obtain:

if d ∈ C4, if s d1 t d2 ∈ J, if d ∈ (C2 \ (−σ1 )) ∪ (σ2 \ {0}), if d ∈ C3 ∪ (−σ1 ) ∪ (−σ2 ), if d ∈ (C4 \ (−σ2 )) ∪ (σ1 \ {0}).

An Illustrated View of Differential Operators



I(J ) D(RA2 , J )

 = d

⎧ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ s d1 t d2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ d1 d2 ⎪ ⎪ ⎨s t

· ·

⎪ s d1 t d2 · ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ s d1 t d2 · ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩s d 1 t d 2 ·

65

C[θ ] h1 h2  C[θ ] h1  C[θ ] h2   (h2 , −2d1 )! h1 (h2 , −2d1 )!  (h1 , −d2 )! h2 (h1 , −d2 )!

if d ∈ / (Zσ1 ∪ Zσ2 ), if d = 0, if d ∈ σ1 \ {0}, if d ∈ σ2 \ {0}, if d ∈ (−σ1 ), if d ∈ (−σ2 ).

As both J and D(RA2 ) are graded, we can similarly determine J D(RA2 ). Our goal for the remainder of the section is to compute the graded pieces of J D(RA2 ), in order to observe that, in this case, D(RA2 , J ) = J D(RA2 ). To begin, note that for all s d1 t d2 ∈ Z2 , D(RA2 )(d1 −1,d2 −1) = s d1 −1 t d2 −1 ·  (h1 , −d2 )! (h2 , −2d1 + d2 )!. For s m1 t m2 ∈ J the graded piece at the multidegree (d1 − m1 , d2 − m2 ) will be s d1 −m1 t d2 −m2 ·  (h1 , −(d2 − m2 ) − 1)! (h2 , −2(d1 − m1 ) + d2 − m2 − 1)!. Note that since m1 and m2 are both positive, s m1 t m2 s d1 −m1 t d2 −m2 = s d1 t d2 , and  (h1 , −(d2 − c2 ) − 1)! (h2 , −2(d1 − c1 ) + d2 − c2 − 1)! is contained in  (h1 , −d2 )! (h2 , −2d1 + d2 )!. Thus, to determine (J D(RA2 ))d for any d ∈ Z2 , it is enough to (left) multiply D(RA2 )(d1 −1,d2 −1) by st. Hence, from our previous computation, it follows that ⎧ ⎪ s d1 t d2 ⎪ ⎪ ⎪ ⎪ d1 d2 ⎪ ⎪ ⎨s t (J D(RA2 ))d = s d1 t d2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩s d1 t d2

· C[θ ]

if s d1 t d2 ∈ J,

·  (h2 , −2d1 + d2 )!

if d ∈ (C2 \ (−σ1 )) ∪ (σ1 \ {0}),

·  (h1 , −d2 )!· (h2 , −2d1 + d2 )! ·  (h1 , −d2 )!

if d ∈ C3 ∪ (−σ1 ) ∪ (−σ2 ), if d ∈ (C4 \ (−σ2 )) ∪ (σ1 \ {0}).

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Combining all cases together, it follows that for an arbitrary d ∈ Z2 , there is an equality (J D(RA2 ))d = s d1 t d2 · (h1 , h1 (−d))! (h2 , h2 (−d))! = D(RA2 , J )d .

4 Differential Operators on the Rational Normal Curve of Degree 3 In this section, we determine some subsets of the ring of differential operators for the ring of the rational normal curve of degree 3 determined by its interior ideal J . We contrast these computations with the degree 2 case that was determined in Sect. 3. Although the description follows the same reasoning as the degree 2 setting, we ultimately get quite different behavior for the operators that end up in J D(RA3 ). The ring of the rational normal curve of degree 3 is RA3 = C[s, st, st 2 , st 3 ], and we will compute I(J )/D(RA3 , J ), where J = st, st 2  is a radical ideal. The facets of A3 are σ1 = {(x, y) ∈ N2 | x  0, y = 0}

and

σ2 = {(x, y) ∈ N2 | x, y  0, y = 3x},

which have primitive integral support functions h1 = θ2

and

h2 = 3θ1 − θ2 .

Figure 7 illustrates the integer lattice, divided into four chambers that are colored as follows: C1 : The red multidegrees correspond to monomials in J , and the blue multidegrees correspond to monomials in RA2 \ J , C2 : The yellow multidegrees are the d with h1 (d)  0 and h2 (d) < 0, C3 : The violet multidegrees are the d with both h1 (d) < 0 and h2 (d) < 0, and C4 : The green multidegrees are the d with h1 (d) < 0 and h2 (d)  0. Still following the convention (h, n)! = 1 if n < 0, by Theorem 2.3, the graded pieces of D(RA3 ) are D(RA3 )d = s d1 t d2 · (h1 , h1 (−d) − 1)! (h2 , h2 (−d) − 1)! . Broken down by chambers, this amounts to:

An Illustrated View of Differential Operators

⎧ ⎪ s d1 t d2 · C[θ ] ⎪ ⎪ ⎪ ⎨s d1 t d2 ·  (h , −3d + d − 1)! 2 1 2 D(RA3 )d = d d 1 2 ⎪ s t ·  (h1 , −d2 − 1)! (h2 , −3d1 + d2 − 1)! ⎪ ⎪ ⎪ ⎩ d1 d2 s t ·  (h1 , −d2 − 1)!

67

if d ∈ C1, if d ∈ C2, if d ∈ C3, if d ∈ C4.

Determining the graded pieces of I(J ) and D(RA3 , J ) in D(RA3 ) is very similar to our computations in Sect. 3. Here we include some visualizations for d ∈ {(−1, 0), (−1, −3), (−1, −1), (−1, −2)} in Fig. 8 to aid our in our description of how to obtain I(J )d and D(RA3 , J )d . Then, we will list the expressions for I(J )d and D(RA3 , J )d by chamber, as we did in Sect. 3. For each d in the illustrations in Fig. 8, when we apply elements of D(RA3 )d to any of the monomials represented by lattice points along the light blue lines, we obtain an integer multiple of a monomial whose exponent lies on a facet. As in Sect. 3, these lines determine the linear multiples hi − hi (d) we must append to I((d)) to obtain either I(J )d or D(RA3 , J ). Note that the light blue lines hi (x) − hi (d) = 0 that have contain a red dot determine the hi − hi (d) we right-multiply by to obtain I(J )d and that the light blue lines hi (x) − hi (d) = 0 that have non-empty intersection with the entire cone determine the hi − hi (d) we right-multiply by to obtain D(RA3 , J )d . Below we summarize the graded pieces of I(J ) and D(RA3 , J ), as we did in Sect. 3:

Fig. 7 Chambers of D(RA3 )

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Fig. 8 Vanishing at various d. (a) d = (−1, 0). (b) d = (−1, −3). (c) d = (−1, −1). (d) d = (−1, −2)

⎧ ⎪ s d1 t d2 ⎪ ⎪ ⎪ ⎨s d1 t d2 I(J )d = ⎪s d1 t d2 ⎪ ⎪ ⎪ ⎩ d1 d2 s t

D(RA3 , J )d =

· C[θ ]

if d ∈ NA3 ,

·  (h1 , −d2 )!

if d ∈ C4,

·  (h2 , −3d1 + d2 )!

if d ∈ C2,

·  (h1 , −d2 )! (h2 , −3d1 + d2 )! if d ∈ C3,

⎧ ⎪ s d1 t d2 · C[θ ] ⎪ ⎪ ⎪ ⎪ d1 d2 ⎪ ⎪ ⎨s t ·  (h1 , −d2 )!

if s d1 t d2 ∈ J, if d ∈ (C4 \ (−σ2 )) ∪ (σ1 \ {0}),

·  (h2 , −3d1 + d2 )! if d ∈ (C2 \ −σ1 ) ∪ (σ2 \ {0}), ⎪ ⎪ ⎪ d d 1 2 ⎪ s t ·  (h1 , −d2 )!· ⎪ ⎪ ⎪ ⎩ if d ∈ C3 ∪ (−σ1 ) ∪ (−σ2 ). (h2 , −3d1 + d2 )! (4.1) s d1 t d2

An Illustrated View of Differential Operators

69

Taking the quotient, we obtain:



I(J ) D(RA3 , J )

 = d

⎧ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ s d1 t d2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ d1 d2 ⎪ ⎪ ⎨s t

· ·

⎪ s d1 t d2 · ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ s d1 t d2 · ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩s d1 t d2 ·

C[θ ] h1 h2  C[θ ] h1  C[θ ] h2   (h2 , −3d1 )! h1 (h2 , −3d1 )!  (h1 , −d2 )! h2 (h1 , −d2 )!

if d ∈ / (Zσ1 ∪ Zσ2 ), if d = 0, if d ∈ (σ1 \ {0}), if d ∈ (σ2 \ {0}), if d ∈ (−σ1 \ {0}), if d ∈ (−σ2 \ {0}).

In the remainder of the section, we compute the graded pieces of J D(RA3 ) to show that they are not equal to the graded pieces of D(RA3 , J ). This means that Proposition 1.1 does not hold for this ring. To begin the computation, if s m1 t m2 ∈ J , then D(RA3 )(d1 −m1 ,d2 −m2 ) =s d1 −m1 t d2 −m2 ·  (h1 , −(d2 −m2 ) − 1)! (h2 , −3(d1 −m1 )+d2 − m2 − 1)!. Further, since m1 , m2  1, s m1 t m2 s d1 −m1 t d2 −m2 = s d1 t d2 , and there is a containment  (h1 , −(d2 − m2 ) − 1)! (h2 , −3(d1 − m1 ) + d2 − m2 − 1)! ⊆  (h1 , −d2 )! (h2 , −3d1 + d2 + 1)!, (h1 , −d2 + 1)! (h2 , −3d1 + d2 )!. Thus, to determine the graded piece of (J D(RA3 ))d for any d ∈ Z2 , it is enough to consider D(RA3 ) in multidegrees (d1 − 1, d2 − 1) and (d1 − 1, d2 − 2), where D(RA3 )(d1 −1,d2 −1) = s d1 −1 t d2 −1 ·  (h1 , −d2 )! (h2 , −3d1 + d2 + 1)! and D(RA3 )(d1 −1,d2 −2) = s d1 −1 t d2 −2 ·  (h1 , −d2 + 1)! (h2 , −3d1 + d2 )!. We will now break down this computation by chambers from Fig. 7, leaving off some half-lines along the way and addressing them as special cases later on. For C1, Fig. 9a helps to visualize that if s d1 t d2 ∈ J , then at least one of the two multidegrees (d1 − 1, d2 − 1) or (d1 − 1, d2 − 2) lives in NA. Since D(RA3 )m = s m1 t m2 · C[θ ] for all m ∈ NA, (J D(RA3 ))d = s d1 t d2 · C[θ ] for s d1 t d2 ∈ J . We will consider the blue multidegrees in C1 with their neighbors in C2 and C4. Now consider the graded pieces of (J D(RA3 ))d for d in σ2 or C2, excluding the two half-lines in that chamber given by −σ1 and y = 1. Since we excluded the

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Fig. 9 Visualizing (J D(RA3 ))d . (a) (J D(RA3 ))d for td ∈ J . (b) (J D(RA3 ))d for d ∈ C2. (c) (J D(RA3 ))d for d ∈ C4. (d) (J D(RA3 ))d for d ∈ C3. (e) (J D(RA3 ))d for d on exceptional lines in C2 and C4

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x-axis and y = 1, both (d1 − 1, d2 − 1) and (d1 − 1, d2 − 2) lie in C2 for all d under consideration, see Fig. 9b. Since D(RA3 )(d1 −1,d2 −1) = s d1 −1 t d2 −1 ·  (h2 , −3d1 + d2 + 1)! and D(RA3 )(d1 −1,d2 −2) = s d1 −1 t d2 −2 ·  (h2 , −3d1 + d2 )!, and there is a containment (h2 , −3d1 + d2 + 1)! ⊆ (h2 , −3d1 + d2 )!, for such d, (J D(RA3 ))d = s d1 t d2 · (h2 , −3d1 + d2 )! . Now consider (J D(RA3 ))d for d in σ1 or in C4, excluding those multidegrees that lie on −σ2 or the half-line y = 3x − 1. Since we excluded the multidegrees on −σ2 and y = 3x − 1, both (d1 − 1, d2 − 1) and (d1 − 1, d2 − 2) also lie in C4, as seen in Fig. 9c. Since D(RA3 )(d1 −1,d2 −1) = s d1 −1 t d2 −1 ·  (h1 , −d2 )! and D(RA3 )(d1 −1,d2 −2) = s d1 −1 t d2 −2 ·  (h1 , −d2 + 1)!, and there is a containment  (h1 , −d2 + 1)! ⊆  (h1 , −d2 )!, it follows that for such d, (J D(RA3 ))d = s d1 t d2 ·  (h1 , −d2 )!. For (J D(RA3 ))d for d in C3, −σ1 , or −σ2 , see Fig. 9d, D(RA3 )(d1 −1,d2 −1) = s d1 −1 t d2 −1 ·  (h1 , −d2 )! (h2 , −3d1 + d2 + 1)!, D(RA3 )(d1 −1,d2 −2) = s d1 −1 t d2 −2 ·  (h1 , −d2 + 1)! (h2 , −3d1 + d2 )!. Since the ideals  (h1 , −d2 )! (h2 , −3d1 + d2 + 1)! and

 (h1 , −d2 + 1)! (h2 , −3d1 + d2 )!

are incomparable, it follows that for such d, (J D(RA3 ))d =s d1 t d2 ·  (h1 , −d2 )! (h2 , −3d1 +d2 +1)!, (h1 , −d2 +1)! (h2 , −3d1 + d2 )!. The multidegrees d that we have not yet considered for (J D(RA3 ))d are those in C2 along y = 1 and in C4 along y = 3x − 1. For such d, one of (d1 − 1, d2 − 1) or (d1 − 1, d2 − 2) belongs to C3, see Fig. 9e. First, for d in C2 along the line y = 1, (d1 − 1, −1) ∈ C3, and D(RA3 )(d1 −1,−1) = s d1 −1 t −1 · h1 (h2 , −3d1 + 1)!.

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Combining the fact that D(RA3 )(d1 −1,0) = s d1 −1 ·  (h2 , −3d1 + 2)!, we compute that (J D(RA3 ))d = s d1 t ·  (h2 , −3d1 + 2)!, h1 (h2 , −3d1 + 1)!. Second, for d ∈ C4 along the line y = 3x − 1, D(RA3 )(d1 −1,3d1 −2) = s d1 −1 t 3d1 −2 ·  (h1 , −3d1 + 1)!h2 . Connecting with the fact that D(RA3 )(d1 −1,3d1 −3) = s d1 −1 t 3d1 −3 ·  (h1 , −3d1 + 2)!, we determine that (J D(RA3 ))d = s d1 t 3d1 −1 ·  (h1 , −3d1 + 1)!h2 , (h1 , −3d1 + 2)!. Having now computed J D(RA3 ) in all multidegrees, we have that

(J D(RA3 ))d =

⎧ ⎪ s d1 t d2 · C[θ ] ⎪ ⎪ ⎪ ⎪ d1 d2 ⎪ ⎪ ⎨s t ·  (h2 , −3d1 + d2 )!

if s d1 t d2 ∈ J, if d ∈ R2,

·  (h1 , −d2 )! (h2 , −3d1 + d2 + 1)!, ⎪ ⎪ ⎪ ⎪ (h1 , −d2 + 1)! (h2 , −3d1 + d2 )! if d ∈ R3, ⎪ ⎪ ⎪ ⎩s d1 t d2 ·  (h , −d )! if d ∈ R4, 1 2 s d1 t d2

where R2 = (C2 \ (σ1 ∪ {y = 1})) ∪ (σ2 \ {0}), R3 = C3 ∪ (−σ1 ∪ −σ2 ) ∪ (NA3 ),

and

R4 = (C4 \ (σ2 ∪ {y = 3x − 1})) ∪ (σ1 \ {0}). Now compare this with D(RA3 , J ) from (4.1); it becomes clear that (J D(RA3 ))d = D(RA3 , J )d whenever d belongs to any of the following: −σ1 , −σ2 , C3, C2 along the half-line {y = 1}, or C4 along the half-line {y = 3x − 1}. In particular, I(J )/D(RA3 , J ) = I(J )/J D(RA3 ), in contrast to Proposition 1.1.

5 Differential Operators on a Rational Normal Curve The techniques used in Sects. 3 and 4 can also! be applied to compute I(J )d , D(RAn , J )d , (J D(RAn ))d and I(J )/D(RAn , J ) d , where the radical ideal J =

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st, st 2 , . . . , st n−1  is again the intersection of the primes defined by the facets of An . We denote the facets of An by σ1 = {(x, y) ∈ N2 | x  0, y = 0}

and

σ2 = {(x, y) ∈ N2 | x, y  0, y = nx},

which have primitive integral support functions h1 = θ2

h2 = nθ1 − θ2 .

and

Again, we divide Z2 into chambers, analogous to those used in Sects. 3 and 4, so C1 C2 C3 C4

= {d ∈ Z2 = {d ∈ Z2 = {d ∈ Z2 = {d ∈ Z2

| d ∈ NA}, | h1 (d)  0, h2 (d) < 0}, | h1 (d) < 0, h2 (d) < 0}, and | h1 (d) < 0, h2 (d)  0}.

These computations yield the following formulas: ⎧ ⎪ s d1 t d2 ⎪ ⎪ ⎪ ⎨s d1 t d2 I(J )d = ⎪ s d1 t d2 ⎪ ⎪ ⎪ ⎩ d1 d2 s t

· C[θ ]

if d ∈ NAn ,

·  (h1 , −d2 )!

if d ∈ C4,

·  (h2 , −nd1 + d2 )!

if d ∈ C2,

·  (h1 , −d2 )! (h2 , −nd1 + d2 )! if d ∈ C3,

⎧ ⎪ s d1 t d2 ⎪ ⎪ ⎪ ⎨s d1 t d2 D(RAn , J )d = ⎪ s d1 t d2 ⎪ ⎪ ⎪ ⎩ d1 d2 s t

· C[θ ]

if s d1 t d2 ∈ J,

·  (h1 , −d2 )!

if d ∈ C4 ,

·  (h2 , −nd1 + d2 )!

if d ∈ C2 ,

·  (h1 , −d2 )! (h2 , −nd1 + d2 )! if d ∈ C3 ,

where C2 = (C2 \ (−σ1 )) ∪ (σ2 \ {0}), C3 = C3 ∪ (−σ1 ∪ −σ2 ), C4 = (C4 \ (−σ2 )) ∪ (σ1 \ {0});

(J D(RAn ))d =

⎧ ⎪ s d1 t d2 · C[θ ] ⎪ ⎪ ⎪ ⎪ d1 d2 ⎪ ⎪ ⎨s t · (h1 , −d2 )!

if s d1 t d2 ∈ J, if d ∈ C4 ,

s d1 t d2 · (h2 , −nx + y)! ⎪ ⎪ ⎪ ⎪ s d1 t d2 · (h1 , −d2 + j )! · ⎪ ⎪ ⎪ ⎩ (h , −nd + d + n − 2 − j )!n−2 2

1

2

j =0

if d ∈ C2 , if d in C3 ,

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where ⎛ C2 = C2 \ ⎝

⎞

n−2 0

{y = j }⎠ ,

j =0

⎡

⎛

n−2 0

C3 = C3 ∪ ⎣C2 ∩ ⎝ ⎛ C4 = C4 \ ⎝ 

j =0 n−2 0

⎞⎤

⎡

⎛

n−2 0

{y = j }⎠⎦ ∪ ⎣C4 ∩ ⎝ ⎞

⎞⎤ {y = nx − j }⎠⎦ ,

j =0

{y = nx − j }⎠ .

j =0

Recall from Sect. 4 that J D(RA3 ) = D(RA3 , J ); we see that this is true for all rings of rational normal curves RAn = C[s, st, . . . , st n ] with n  3. Specifically, comparing D(RAn , J )d and (J D(RAn ))d for various d ∈ Z2 , the graded pieces differ whenever d belongs to σ1 , σ2 , C3, the half-lines in C2 inside {(x, y) ∈ R2 | x < 0, y = i, 0  i  n − 2}, or the half-lines in C4 inside {(x, y) ∈ R2 | x < 0, y = nx − i, 0  i  n − 2}. Figure 10 has these multidegrees d in blue for the case n = 7. As with the rational normal cones in degrees 2 and 3 from Sects. 3 and 4, for d ∈ Z2 ,

Fig. 10 (J D(RA7 ))d and D(RA7 , J )d differ at blue d

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⎧ ⎪ 0 ⎪ ⎪ ⎪ ⎪ C[θ ] ⎪ ⎪ ⎪   ⎨ h1 h2  I(J ) =  (h2 , −nd1 )! D(RAn , J ) d ⎪ s d1 · ⎪ ⎪ h1 (h2 , −nd1 )! ⎪ ⎪ ⎪  (h1 , −d2 )! ⎪ d d ⎪ 1 2 ⎩s t · h2 (h1 , −d2 )!

if d ∈ Z2 \ (Zσ1 ∪ Zσ2 ), if d = 0, if d ∈ Zσ1 \ {0},

(5.1)

if d ∈ Zσ2 \ {0}.

Example 2.5 considered the ordinary double point C[x, y]/xy, which is isomorphic to RA!n /J for all n  1. Comparing D(C[x, y]/xy))d from (2.2) and I(J )/D(RAn , J ) d from (5.1), we see that there is an isomorphism between the graded components, viewed as C-vector spaces, given by ϕ : D (C[x, y]/xy) → ! D RAn /J with, for m ∈ Z,  ϕ 

C[θx , θy ] θx θy 

 =

C[θ ] , h1 h2 

  (θx , −m)!  (h2 , −nm)! ϕ x = sm · , and  (θx , −m)!θy   (h2 , −nm)!h1   !   θy , −m !  (h1 , −nm)! m ! ϕ y · . = s m t nm ·  (h1 , −nm)!h2   θy , −m !θx  m

However, this isomorphism is not an isomorphism of rings. Again as in Example 3.1, the degree of the polynomial generator in θ1 and θ2 in multidegree d is n times the degree of the polynomial generator of in θx and θy . In fact, noting that along Zσ2 , d2 = nd1 , we have ⎧ ⎪ 0 ⎪ ⎪ ⎪ ⎪ C[θ ] ⎪ ⎪ , ⎪ ⎪ ⎪ h 1 h2 ⎪ ⎪ ⎪ ⎪ n n , ⎪ ⎪ ⎪ ⎪ h2 ⎪ ⎪ , −d1 ! ⎨ ! n D RAn /J d = s d1 · ,  ⎪ h1 h2 ⎪ ⎪ , −d1 ! ⎪ ⎪ n ,n ⎪ ⎪ ⎪ ⎪ h1 ⎪ ⎪ , −d1 ! ⎪ ⎪ n ⎪ d1 t d2 · , ⎪  ⎪ s ⎪ ⎪ h2 h1 ⎪ ⎪ , −d ⎩ 1 ! n n

if d ∈ Z2 \ (Zσ1 ∪ Zσ2 ), if d = 0,

if d ∈ Zσ1 \ {0},

(5.2)

if d ∈ Zσ2 \ {0}.

Now comparing (2.2) and (5.2), we see there is an isomorphism of the ring of differential operators between the two rings given by, for each m ∈ Z,

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ψ((θx , −m)! +  (θx , −m)!θy ) = s !

!

ψ( θy , −m ! +  θy , −m !θx ) = s ψ(x m ) = s m ,

−m

 ·

−m −nm

t

 ,  h2 h2 , −m ! + , −m !h1 , n n 

·

 ,  h1 h1 , −m ! + , −m !h2 , n n

ψ(y m ) = s m t nm .

and

6 Higher Dimensional Examples Thus far, we have considered differential operators determined by radical ideals in two-dimensional semigroup rings. In this section, we turn to looking at some of the differential operators in the three-dimensional semigroup ring given by ⎡

⎤ 1111 A = ⎣0 1 0 1⎦ 0011 and describe some subsets of differential operators of RA given by two different radical ideals J . For both choices of J , we will compute the graded pieces I(J )d , D(RA , J )d and (I(J )/D(RA , J ))d . For the given matrix A, RA = C[NA] = C[t1 , t1 t2 , t1 t3 , t1 t2 t3 ]. The facets of the cone R0 A are σ1 = N{e1 , e1 + e2 },

σ3 = N{e1 + e3 , e1 + e2 + e3 },

σ2 = N{e1 , e1 + e3 },

σ4 = N{e1 + e2 , e1 + e2 + e3 },

and the corresponding primitive integral support functions are h1 = h1 (θ ) = θ3 , h3 = h3 (θ ) = θ1 −θ3 , h2 = h2 (θ ) = θ2 , h4 = h4 (θ ) = θ1 −θ2 . The prime ideals associated to the facets are Pσ1 = t1 t3 , t1 t2 t3 ,

Pσ2 = t1 t2 , t1 t2 t3 ,

Pσ3 = t1 , t1 t2 ,

Pσ4 = t1 , t1 t3 ,

and the prime ideals associated to the rays (or 1-dimensional faces) of the cone are Pσ1 ∩σ2 = t1 t2 , t1 t3 , t1 t2 t3 , Pσ2 ∩σ3 = t1 , t1 t2 , t1 t2 t3 ,

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77

Pσ3 ∩σ4 = t1 , t1 t2 , t1 t3 , Pσ1 ∩σ4 = t1 , t1 t3 , t1 t2 t3 . In contrast with the two dimensional cases considered in Sects. 3, 4, and 5, the increased dimension of the semigroup NA allows for many more choices for radical monomial ideal J in RA . We will compute I(J )/D(RA , J ) for two such choices, J = Pσ1 ∩ Pσ2 ∩ Pσ3 ∩ Pσ4 = t12 t2 t3  and J = Pσ1 ∩ Pσ2 ∩ Pσ3 ∩σ4 = t12 t2 t3 , t12 t22 t3 , t12 t2 t32 . For both choices of J , to compute the graded pieces of I(J ), we will divide up Z3 into 14 chambers, depending on various combinations of signs of hi (d). Table 1 describes the 14 chambers in a list, while Fig. 11 illustrates them, with chamber C1 (given by NA) in the top right with some of the lattice points of NA shown. Note the x-axis is the vertical axis in this picture. We must modify the chambers slightly to compute the graded pieces of D(RA , J ), as we now consider which differential operators, when applied to an element in RA , yield an element in J . Although all of the operators in the graded pieces of I(J ) lying on the facets send J into J , it is not necessarily the case that these operators applied to an element of RA will output an element in J . So we will switch the lattice points on the facets that correspond to monomials that do not lie in J to lie in the adjacent chambers by interchanging  with >. We will describe these new regions within the examples. Example 6.1 First consider the ideal J = Pσ1 ∩ Pσ2 ∩ Pσ3 ∩ Pσ4 = t12 t2 t3  in RA = C[t1 , t1 t2 , t1 t3 , t1 t2 t3 ]. In Fig. 12, the multidegrees in the gray cone whose vertex lies at (2, 1, 1) correspond to the monomials that lie in J , and the monomials in the outer cone lie in RA \ J . Note also that the view of the cone that we see in Fig. 12 is from the side of the xz-plane. Since none of the points on the facets of the cone have monomials that lie in J , the 14 regions that we consider in determining D(RA , J ) are listed in Table 2. If d is in C1 (which are the lattice points corresponding to points in the semigroup) when considering the idealizer or R1 (which are the lattice points corresponding to the monomials in J ) when considering D(RA , J ), then D(RA )d is generated by multiplication by td . Multiplying any element of J by td remains in J making the graded pieces of I(J )d = td · C[θ ] and D(RA , J )d = td · C[θ ]. Since the multidegrees in the chambers and regions corresponding to C2−C5 and R2 − R5 will all need operators adjusted only for monomials in J whose exponents are parallel to a single facet in A, we will only describe the process to determine the

{d ∈ Z3 | d2 > d1  d3  0} {d ∈ Z3 | d3 > d1  d2  0} {d ∈ Z3 | d1  d3  0 > d2 } {d ∈ Z3 | d1  d2  0 > d3 } {d ∈ Z3 | d3  0, d2  0, d2 > d1 , d3 > d1 } {d ∈ Z3 | d2  0 > d3 , d2 > d1  d3 } {d ∈ Z3 | d3  0 > d2 , d3 > d1  d2 } {d ∈ Z3 | 0 > d2 , 0 > d3 , d1  d2 , d1  d3 } {d ∈ Z3 | d2  0 > d3 > d1 } {d ∈ Z3 | d3  0 > d2 > d1 } {d ∈ Z3 | 0 > d3 > d1  d2 } {d ∈ Z3 | 0 > d2 > d1  d3 } −Int(NA)

| h1 (d), h3 (d)  0, h2 (d) > 0, h4 (d) < 0}

{d ∈

{d ∈ Z3 | h2 (d), h4 (d)  0, h1 (d) > 0, h3 (d) < 0}

{d ∈ Z3 | h1 (d), h3 (d)  0, h4 (d) > 0, h2 (d) < 0}

{d ∈ Z3 | h2 (d), h4 (d)  0, h3 (d) > 0, h1 (d) < 0}

{d ∈ Z3 | h1 (d), h2 (d)  0, h3 (d), h4 (d) < 0}

{d ∈ Z3 | h2 (d), h3 (d)  0, h1 (d), h4 (d) < 0}

{d ∈ Z3 | h1 (d), h4 (d)  0, h2 (d), h3 (d) < 0}

{d ∈ Z3 | h3 (d), h4 (d)  0, h1 (d), h2 (d) < 0}

{d ∈ Z3 | h1 (d), h3 (d), h4 (d) < 0, h2 (d)  0}

{d ∈ Z3 | h2 (d), h3 (d), h4 (d) < 0, h1 (d)  0}

{d ∈ Z3 | h1 (d), h2 (d), h3 (d) < 0, h4 (d)  0}

{d ∈ Z3 | h1 (d), h2 (d), h4 (d) < 0, h3 (d)  0}

{d ∈ Z3 | h1 (d), h2 (d), h3 (d), h4 (d) < 0}

C2

C3

C4

C5

C6

C7

C8

C9

C10

C11

C12

C13

C14

NA

| h1 (d), h2 (d), h3 (d), h4 (d)  0}

Z3

{d ∈

Lattice point inequalities

C1

Halfspace inequalities

Z3

Chamber

Table 1 The chambers used to compute I(J ) in Examples 6.1 and 6.2

78 C. Berkesch et al.

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79

Fig. 11 Chambers for C[t1 , t1 t2 , t1 t3 , t1 t2 t3 ]

Fig. 12 Cone of RA with ideal J = t12 t2 t3 

graded pieces of I(J ) and D(RA , J ) for chamber C2 and region R2, respectively, as precisely the same type of argument holds for the other chambers and regions. If d is in C2 in the case of the idealizer or in R2 in the case of D(RA , J ), consider the lattice points corresponding to the monomials in J on the plane x − y + h4 (d) = 0. When an element in I((d)) is applied to such a monomial in J , the result is a constant times a monomial with exponent on the facet σ4 , which is not in J . Hence we need to multiply I((d)) by h4 + h4 (d), so for d in C2, I(J )d = td ·  (h4 , h4 (−d))! and for d in R2, D(RA , J )d = td ·  (h4 , h4 (−d))!. Figure 13a illustrates the plane in C2 that determines the linear form that we multiply by I((d)) to obtain I(J )d . Since the multidegrees in the chambers and regions corresponding to C6−C9 and R6 − R6 will all need operators adjusted only for monomials in J whose exponents

{d ∈ Z3 | d2  d1 > d3 > 0} {d ∈ Z3 | d3  d1 > d2 > 0} {d ∈ Z3 | d1 > d3 > 0  d2 } {d ∈ Z3 | d1 > d2 > 0  d3 } {d ∈ Z3 | d2 > 0, d3 > 0, d2  d1 , d3  d1 } {d ∈ Z3 | d2 > 0  d3 , d2  d1 > d3 } {d ∈ Z3 | d3 > 0  d2 , d3  d1 > d2 } {d ∈ Z3 | d1 > d2 , d1 > d3 , 0  d2 , 0  d3 } {d ∈ Z3 | d2 > 0  d3  d1 } {d ∈ Z3 | d3 > 0  d2  d1 } {d ∈ Z3 | 0  d3  d1 > d2 } {d ∈ Z3 | 0  d2  d1 > d3 } −NA

| h1 (d), h2 (d), h3 (d) > 0, h4 (d)  0}

{d ∈

{d ∈ Z3 | h1 (d), h2 (d), h4 (d) > 0, h3 (d)  0}

{d ∈ Z3 | h1 (d), h3 (d), h4 (d) > 0, h2 (d)  0}

{d ∈ Z3 | h2 (d), h3 (d), h4 (d) > 0, h1 (d)  0}

{d ∈ Z3 | h1 (d), h2 (d) > 0, h3 (d), h4 (d)  0}

{d ∈ Z3 | h2 (d), h3 (d) > 0, h1 (d), h4 (d)  0}

{d ∈ Z3 | h1 (d), h4 (d) > 0, h2 (d), h3 (d)  0}

{d ∈ Z3 | h3 (d), h4 (d) > 0, h1 (d), h2 (d)  0}

{d ∈ Z3 | h1 (d), h3 (d)  0, h4 (d) < 0, h2 (d) > 0}

{d ∈ Z3 | h2 (d), h4 (d)  0, h3 (d) < 0, h1 (d) > 0}

{d ∈ Z3 | h1 (d), h3 (d)  0, h2 (d) < 0, h4 (d) > 0}

{d ∈ Z3 | h2 (d), h4 (d)  0, h1 (d) < 0, h3 (d) > 0}

{d ∈ Z3 | h1 (d), h2 (d), h3 (d), h4 (d)  0}

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12

R13

R14

Int(NA)

| h1 (d), h2 (d), h3 (d), h4 (d) > 0}

Z3

{d ∈

Lattice point inequalities

R1

Halfspace inequalities

Z3

Region

Table 2 Regions to compute D(RA , J ) in Example 6.1

80 C. Berkesch et al.

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Fig. 13 Vanishing planes. (a) d = (1, 2, 1) ∈ C2 and {x − y + h4 (d) = 0}. (b) d = (−1, 0, 0) ∈ C6, {x − y + h4 (d) = 0}, and {x − z + h4 (d) = 0}. (c) d = (−2, 0, −1) ∈ C10, {z + h1 (d) = 0}, {x − z + h3 (d) = 0}, and {x − y + h4 (d) = 0}

are parallel to two facets of A, we will only describe the process to determine the graded pieces of I(J ) and D(RA , J ) for chamber C6 and region R6, respectively, as precisely the same type of argument holds for the other chambers and regions. If d is in C6 in the case of the idealizer or R6 in the case of D(RA , J ), consider the lattice points corresponding to the monomials in J on the planes x − y + h4 (d) = 0 and x − z + h3 (d) = 0. When an element in I((d)) is applied to such monomial in J , the result is a constant times a monomial with exponent on x − y = 0 or x − z = 0, which is not in J . Hence we need to multiply I((d)) by (h3 + h3 (d))(h4 + h4 (d)), so for d in C6, I(J )d = td ·  (h3 , h3 (−d))! (h4 , h4 (−d))!, and for d in R6, D(RA , J )d = td ·  (h3 , h3 (−d))! (h4 , h4 (−d))!. Figure 13b illustrates the two planes in C6 that determine the linear forms that we multiply I((d)) by to obtain I(J )d . Since the multidegrees in the chambers and regions corresponding to C10 − C13 and R10 − R13 will all need operators adjusted only for monomials in J whose exponents are parallel to three facets of A, we will only describe the process to determine the graded piece of I(J ) and D(RA , J ) for chamber C10 and region R10, respectively, as precisely the same type of argument holds for the other chambers and regions. If d is in C10 in the case of the idealizer or R10 in the case of D(RA , J ), consider the lattice points corresponding to the monomials in J on the planes z + h1 (d) = 0, x − z + h3 (d) = 0 and x − y + h4 (d) = 0. When an element of I((d)) is applied to such a monomial in J , the result is a constant times a monomial whose exponent is in σ1 , σ3 , or σ4 , which is not in J . Hence we need to multiply I((d)) by (h1 + h1 (d))(h3 + h3 (d))(h4 + h4 (d)), so for d in C10,

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I(J )d = td ·  (h1 , h1 (−d))! (h3 , h3 (−d))! (h4 , h4 (−d))!, and for d in R10, D(RA , J )d = td ·  (h1 , h1 (−d))! (h3 , h3 (−d))! (h4 , h4 (−d))!. Figure 13c illustrates the three planes in C10 that determine the linear forms that we multiply by I((d)) to obtain I(J )d . If d is in C14 in the case of the idealizer or R14 in the case of D(RA , J ), consider the lattice points corresponding to the monomials in J on the planes z + h1 (d) = 0, y+h2 (d) = 0, x−z+h3 (d) = 0 and x−y+h4 (d) = 0. When an element of I((d)) is applied to such a monomial in J , the result is a constant times a monomial with exponent on one of the facets σi of A, which is not in J . Hence we need to multiply I((d)) by (h1 + h1 (d))(h2 + h2 (d))(h3 + h3 (d))(h4 + h4 (d)), so for d in C14, I(J )d = td ·  (h1 , h1 (−d))! (h2 , h2 (−d))! (h3 , h3 (−d))! (h4 , h4 (−d))!, and for d in R14, D(RA , J )d = td ·  (h1 , h1 (−d))! (h2 , h2 (−d))! (h3 , h3 (−d))! (h4 , h4 (−d))!. Combining the information from all 14 chambers C1-C14 and regions R1-R14, the general formula for the graded piece of the idealizer of J at d ∈ Z3 is *

+



I(J )d = t · d

(hi , hi (−d))! ,

hi (d) d3 } {d ∈ Z3 | d13 < d2  0, d1  d3  0} ∪ {d ∈ Z | d1  d2  0, d1 < d3  0}

{d ∈ Z3 | h1 (d), h2 (d), h4 (d)  0, h3 (d) > 0}

{d ∈ Z3 | h13(d), h2 (d)h3 (d)  0, h4 (d) < 0} ∪ {d ∈ Z | h1 (d), h2 (d)h4 (d)  0, h3 (d) < 0}

R13

R14

{d ∈ Z3 | d2 > 0  d3 > d1 }

{d ∈ Z3 | h1 (d)  0, h3 (d), h4 (d) < 0, h2 (d) > 0}

R10

{d ∈ Z3 | h2 (d), h4 (d) < 0, h1 (d)  0, h3 (d)  0}

{d ∈ Z3 | d13 > d2 , d3 , 0  d2 , d3 } ∪ {d ∈ Z | 0 3> d1 = d2 , 0 > d1 > d3 } ∪ {d ∈ Z | 0 > d1 = d3 , 0 > d1 > d2 }

{d ∈ Z3 |3h3 (d), h4 (d) > 0, h1 (d), h2 (d)  0} ∪ {d ∈ Z | h13(d), h2 (d) < 0, h4 (d) > 0, h3 (d) = 0} ∪ {d ∈ Z | h1 (d), h2 (d < 0, h3 (d) > 0, h4 (d) = 0}

R9

R13

{d ∈ Z3 | d3 > 0  d23, d3 > d1 > d2 } ∪ {d ∈ Z | d3 > 0 > d1 = d2 }

{d ∈ Z3 | h1 (d), h43(d) > 0, h2 (d)  0, h3 (d) < 0} ∪ {d ∈ Z | h1 (d) > 0, h4 (d) = 0, h2 (d) < 0}

R8

{d ∈ Z3 | 0  d2 > d1  d3 }

{d ∈ Z3 | d2 > 0  d33, d2 > d1 > d3 } ∪ {d ∈ Z | d2 > 0 > d1 = d3 }

{d ∈ Z3 | h2 (d), h33(d) > 0, h1 (d)  0, h4 (d) < 0} ∪ {d ∈ Z | h2 (d) > 0, h3 (d) = 0, h1 (d) < 0}

R7

{d ∈ Z3 | h1 (d), h3 (d) < 0, h2 (d)  0, h4 (d)  0}

{d ∈ Z3 | d3 > 0, d2 > 0, d32 > d1 , d3 > d1 } ∪ {d ∈ Z | d1 = d2 = d3 > 0}

{d ∈ Z3 | h13(d), h2 (d) > 0 and h3 (d), h4 (d) < 0} ∪ {d ∈ Z | h1 (d) > 0, h2 (d) > 0, h3 (d) = h4 (d) = 0}

R6

R12

{d ∈ Z3 | d13  d2 > 0  d3 } ∪ {d ∈ C | d1 = d2 > 0 > d3 }

{d ∈ Z3 | h23(d), h3 (d), h4 (d) > 0, h1 (d)  0} ∪ {d ∈ Z | h4 (d) = 0, h2 (d), h3 (d) > 0, h1 (d) < 0}

R5

{d ∈ Z3 | d3 > 0  d2 > d1 }

{d ∈ Z3 | d13 > d3 > 0  d2 } ∪ {d ∈ Z | d1 = d3 > 0 > d2 }

{d ∈ Z3 | h13(d), h3 (d), h4 (d) > 0, h2 (d)  0} ∪ {d ∈ Z | h3 (d) = 0, h1 (d), h4 (d) > 0, h2 (d) < 0}

R4

| h2 (d)  0, h4 (d), h3 (d) < 0, h1 (d) > 0}

{d ∈ Z3 | d3 > d1  d2 > 0}

{d ∈ Z3 | h1 (d), h2 (d) > 0, h4 (d)  0, h3 (d) < 0}

R3

{d ∈

{d ∈ Z3 | d2 > d1  d3 > 0}

{d ∈ Z3 | h1 (d), h2 (d) > 0, h3 (d)  0, h4 (d) < 0}

R2

R11

{d ∈ Z3 | xd ∈ J }

{d ∈ Z3 | h13(d), h2 (d), h3 (d) > 0, h4 (d)  0} ∪ {d ∈ Z | h1 (d), h2 (d), h4 (d) > 0, h3 (d)  0}

R1

Z3

Lattice point inequalities

Halfspace inequalities

Region

Table 3 Regions for D(RA , J ) in Example 6.2

84 C. Berkesch et al.

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If d is in C6 in the case of the idealizer or R6 in the case of D(RA , J ), consider the multidegrees of monomials in J on the planes x−z+h3 (d) = 0 or x−y+h4 (d) = 0. When an element of I((d)) is applied to such a monomial in J , the result is a constant times a monomial on σ3 or σ4 . This is not in J only if this monomial’s exponent is in the intersection of the two planes. Hence, we need to multiply I((d)) by either (h3 + h3 (d)) or (h4 + h4 (d)), so that for d in C6, I(J )d = td · (h4 , h4 (−d) − 1)! (h3 , h3 (−d) − 1)! · (h3 + h3 (d)), (h4 + h4 (d)), and for d in R6, D(RA , J )d = td · (h4 , h4 (−d) − 1)! (h3 , h3 (−d) − 1)! · (h3 + h3 (d)), (h4 + h4 (d)). Since the multidegrees in the chambers and regions corresponding to C10 − C13 and R10 − R13 can will all need operators adjusted only for monomials in J whose exponents are parallel to a three facets of A, we will only describe the process to determine the graded piece of I(J ) and D(RA , J ) for chamber C10 and region R10, respectively, as precisely the same type of argument holds for the other chambers and regions. The reader may refer to Fig. 13c to help visualize the argument below. If d is in C10 in the case of the idealizer or R10 in the case of D(RA , J ), consider the multidegrees of monomials in J on the planes z+h1 (d) = 0, x −z+h3 (d) = 0, and x − y + h4 (d) = 0. When an element in I((d)) is applied to such a monomial in J , the result is a constant times a monomial on σ1 , which is not in J , or a constant times a monomial in σ3 or σ4 , which is not in J only if the monomial is in the intersection of σ3 and σ4 . Hence, we need to multiply I((d)) by either (h1 +h1 (d))(h3 +h3 (d)) or (h1 + h1 (d))(h4 + h4 (d)), so that for d in C10, I(J )d = td · (h1 , h1 (−d))! (h3 , h3 (−d) − 1)! (h4 , h4 (−d) − 1)! · (h3 + h3 (d)), (h4 + h4 (d)), and for d in R10, D(RA , J )d = td · (h1 , h1 (−d))! (h3 , h3 (−d) − 1)! (h4 , h4 (−d) − 1)! · (h3 + h3 (d)), (h4 + h4 (d)). If d is in C14 in the case of the idealizer or R14 in the case of D(RA , J ), consider the multidegrees of the monomials in J on the planes z + h1 (d) = 0, y + h2 (d) = 0, x − z + h3 (d) = 0, and x − y + h4 (d) = 0. When an element of I((d)) is applied to such a monomial in J , the result is a constant times a monomial in σ1 or σ2 , which is not in J , or a constant times a monomial in σ3 or σ4 , which is not in J if this monomial lies in both planes parallel to these faces. Hence we need to multiply I((d)) by (h1 + h1 (d))(h2 + h2 (d))(h3 + h3 (d))(h4 + h4 (d) − 1) or (h1 + h1 (d))(h2 + h2 (d))(h3 + h3 (d) − 1)(h4 + h4 (d)), so that for d in C14,

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I(J )d = td · 4 

(hi , hi (−d) − 1)! ·

* 3 

i=1





2  (hi + hi (d)) , (h4 + h4 (d)) (hi + hi (d))

i=1

+

i=1

and for d in R14, D(RA , J )d = td · 4 

* 3   2 +   (hi + hi (d)) , (h4 + h4 (d)) (hi + hi (d)) . (hi , hi (−d) − 1)! ·

i=1

i=1

i=1

Putting all the information together from all 14 chambers C1–C14 and regions R1–R14, the general formula for the graded piece of the idealizer of J at d ∈ Z3 is I(J )d = td · 4  i=1

⎛ ⎞ ⎛ ⎞ * + ⎜  ⎟ ⎜  ⎟ ⎜ ⎟ ⎜ ⎟ (hi + hi (d))⎠ , ⎝ (hi + hi (d))⎠ , (hi , hi (−d) − 1)! · ⎝ hi (d) edim R − dim R; (2) R is a complete intersection if and only if λ is injective.



This result is applied in the proof of Theorem 6.16, which provides a condition under which the defining ideal of a Koszul algebra is a complete intersection. An analagous characterization of Gorenstein rings in terms of their Koszul homology algebras involves the condition known as Ponicaré duality, which we define below. n Definition 4.3 A graded algebra H = i=0 Hi satisfies Poincaré duality if for each i = 0, . . . , n, the H0 -homomorphism Hi −→ HomH0 (Hn−i , Hn ) h −→ φh

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with φh (x) = hx, is an isomorphism.



Now we state the result of Avramov and Golod which asserts that Gorenstein rings are characterized by their Koszul homology algebras. Theorem 4.4 ([9]) Let R be a local ring, and let n = edim R − depth R. The following conditions are equivalent: 1. R is a Gorenstein ring; 2. H (R) satisfies Poincaré duality; 3. the k-linear map Hn−1 (R) −→ Homk (H1 (R), Hn (R)) induced by the multiplication on H (R) is injective.

We highlight applications of Theorem 4.4 at the end of Sect. 4.2 and in Sect. 5.

4.2 Golod Rings In this section we discuss a classical result which characterizes Golod rings by their Koszul homology algebras, and a more recent result which stems from this classical theorem. At the end of the section, we highlight an application of Theorem 4.4 which provides a connection between the Golod and Gorenstein properties. We begin with a definition of a Golod ring. Definition 4.5 A local ring R is a Golod ring if its Poincaré series satisfies the equality PkR (t) = where n = edim R − depth R.

(1 + t)edim R n , 1 − j =1 rankk Hj (R)t j +1

(4.1)



A classical result of Serre asserts that, for any local ring, the formula in (4.5) provides a coefficient-wise upper bound for its Poincaré series; see for example [28], [43, Theorem 1.3], [5, Proposition 3.3.2], or [30, Corollary 4.2.4]. Thus, roughly speaking, the resolution of the residue field k of a Golod local ring has the fastest growth possible. In [28], Golod introduced some higher order operations, called Massey operations, on Koszul homology, and used them to characterize rings satisfying the equality (4.5), which we now call Golod rings. For a more detailed treatment of Massey operations, see for example [30, Section 4.2]. Definition 4.6 We say that K(R) admits a trivial Massey operation if for some k-basis B = {hλ }λ∈ of H1 (R), there is a function

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μ:

∞ D

Bn → K(R)

n=1

such that μ(hλ ) = zλ is a cycle with [zλ ] = hλ and ∂ K μ(hλ1 , . . . , hλp ) =

p−1 

μ(hλ1 , . . . , hλj )μ(hλj +1 , . . . , hλp )

(4.2)

j =1

where a = (−1)|a|+1 a.



We state Golod’s classical result using the modern terminology of Golod rings, as follows. Theorem 4.7 ([28]) A local ring R is Golod if and only if K(R) admits a trivial Massey operation.

Thus Golod rings are characterized by having highly trivial Koszul homology algebras. Indeed, if K(R) admits a trivial Massey operation, then (4.2) implies that the product on H (R) is trivial; that is, H1 (R) · H1 (R) = 0. Note, however, that a trivial product on H (R) is not enough to imply that R is Golod; the higher Massey products must also be trivial. In [38], Katthän produces an example which illustrates this fact. Theorem 4.8 ([38, Theorem 3.1]) Let Q = k[x1 , x2 , y1 , y2 , z] be a polynomial ring with k a field, let I be the ideal I = (x1 x22 , y1 y22 , z3 , x1 x2 y1 y2 , y22 z2 , x22 z2 , x1 y1 z, x22 y22 z), and let R = Q/I . Then the product on H (R) is trivial, but R is not Golod.



For the remainder of this section, we recognize a few direct applications of the characterizations above. First, we look at an application of Theorem 4.7 and Theorem 3.1 to Golod rings. In [33] Herzog and Huneke exploit the canonical bases of Koszul homologies given in Theorem 3.1 and the characterization of Golod rings in Theorem 4.7 to provide a differential condition for Golodness, and in doing so, they are able to produce large classes of Golod rings, including quotients by powers and symbolic powers of ideals. The differential condition is given in the following result; see also [34, Sections 2 & 3], [31, Theorem 3.5], and [24, Proposition 4.4] for similar applications. Theorem 4.9 Let Q be a standard graded polynomial ring over a field of characteristic zero, let I ⊆ Q be a homogeneous ideal, and let R = Q/I . Let ∂(I ) denote the ideal generated by the partial derivatives of the elements of I . If (∂(I ))2 ⊆ I , then R is a Golod ring.

Now we look at an application of Theorem 4.4 by Avramov and Levin in [44] which establishes a connection between Golod homorphisms and Gorenstein rings.

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The following theorem asserts that factoring a Gorenstein ring by its socle produces a Golod homomorphism; that is, a relative version of the Golod property. We discuss an application of Theorem 4.10 to calculating Poincaré series in the following section. Theorem 4.10 ([44, Theorem 2]) Let R be a local Gorenstein ring with edim R > 1 and dim R = 0. Then R −→ R/(0 : m) is a Golod homomorphism.



Throughout this section, we have seen several classes of local rings which are completely characterized by the algebra structures on their Koszul homology algebras, and we conclude by emphasizing that, even for an arbitrary local ring, the Koszul homology algebra is such a sensitive invariant that it encodes all of the numerical information about the free resolution of its residue field, as we see in the following result of Avramov. Theorem 4.11 ([4, Corollary 5.10]) The Poincaré series PkR (t) of a local ring R with residue field k depends only on the algebra structure on H (R) and its higher order Massey operations.



5 Classifications of Koszul Homology Algebras For a general ring, the algebra structure on its Koszul homology can be quite complicated (see for example [53]); however, there are certain classes of rings whose Koszul homology algebras are completely classified. We outline those cases in this section. We begin with the case of R = Q/I , where Q is a local ring with residue field k, whose defining ideal I is perfect of grade 3; that is, the length of the longest Q-regular sequence contained in I and the projective dimension of R over Q are both equal to 3. In this case, Weyman [57] and Avramov, Kustin, and Miller [10] (see also [6]) provide complete classifications of the algebra TorQ • (R, k). Restricting to the case where Q is regular, this gives the following classification of the possible multiplicative structures on the Koszul homology algebra H (R) via the isomorphism (3.1). Notice that in this case the last nonvanishing Koszul homology module is H3 (R) by the Auslander-Buchsbaum formula and depth sensitivity of the Koszul complex. Theorem 5.1 ([10, Theorem 2.12]) Let R = Q/I with Q a regular local ring. If the projective dimension of R over Q is 3, then there are nonnegative integers p, q, and r and bases {ei }, {fi }, and {gi } of H1 (R), H2 (R), and H3 (R), respectively, such that the multiplication on H (R) is given by one of the following:

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CI: f1 = e2 e3 , f2 = e3 e1 , f3 = e1 e2 , ei fj = δij g1 for 1  i, j  3 TE: f1 = e2 e3 , f2 = e3 e1 , f3 = e1 e2 B: e1 e2 = f3 , e1 f1 = g1 , e2 f2 = g1 G(r): ei fi = g1 for 1  i  r and r  2 H(p, q): ep+1 ei = fi for 1  i  p and ep+1 fp+i = gi for 1  i  q with ej ei = −ei ej , ei2 = 0, and ei fj = fj ei for all i and j , and with the products of basis elements that are not listed above being zero.

The class listed as CI is precisely the class where I is a complete intersection; the class listed as G(r) is the class where I is Gorenstein, but not a complete intersection, and r indicates the minimal number of generators of I ; the class listed as TE consists of rings that are neither Golod, nor complete intersections. The classification in Theorem 5.1 builds on the classification for perfect grade 3 almost complete intersection ideals given by Buchsbaum and Eisenbud in [16, Theorem 5.3]; see also [14] for another special case of this classification. Such classifications for the Tor algebra are also known for grade 4 Gorenstein ideals and grade 4 almost complete intersection ideals; the former is due to Kustin and Miller [42, Theorem 2.2], and the latter is due to Kustin [40, Theorem 1.5]. Again, restricting to the case where the ambient ring is regular, we present the former case here, and we note that the proof applies the characterization of Gorenstein rings in Theorem 4.4. Theorem 5.2 ([42, Theorem 2.2]) Let R = Q/I with Q a regular local ring, and assume that every element in k has a square root in k. If I is a grade 4 Gorenstein ideal that is not a complete intersection, then there are bases e1 , . . . , en for H1 (R);  f1 , . . . , fn−1 , f1 , . . . , fn−1 for H2 (R); g1 , . . . , gn for H3 (R); and h for H4 (R) such that the multiplication Hi (R) · H4−i (R) is given by ei gj = δij h, fi fj = δij h, fi fj = fi fj = 0, with all other products given by one of the following cases: (1) H1 (R) · H1 (R) = 0 and H1 (R) · H2 (R) = 0 (2) All products in H1 (R) · H1 (R) and H1 (R) · H2 (R) are zero except: e1 e2 = f3 , e1 e3 = −f2 , e2 e3 = f1 , e1 f2

=

−e2 f1

= g3 , −e1 f3 =e3 f1 = g2 , e2 f3 = −e3 f2 = g1

(3) There is an integer p such that ep+1 ei = fi , ei fi gp+1 , and ep+1 fi = −gi for 1  i  p, and all other products in H1 (R) · H1 (R) and H1 (R) · H2 (R) are zero.

Not surprisingly, we see that the algebra structure in the grade 4 case is more complicated than that in the grade 3 case. The classifications of Koszul homology algebras in Theorems 5.1 and 5.2 have been instrumental in the study of rationality of Poincaré series, which has been a

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topic of intense interest over the years and remains a central problem in homological commutative ring theory today; see for example [49, Section 6] or [5, Section 4.3] for a complete treatment of this problem. For example, Theorem 5.1 is an important ingredient in the proof of [10, Theorem 6.4], which determines several classes of rings over which the Poincaré series of a finitely generated module is rational; analagously, [37, Theorem A] applies the classification in Theorem 5.2. In fact, much of the recent progress on rationality of Poincaré series relies on the classical characterizations of Koszul homology discussed in Sect. 4. As a direct application of Theorem 4.10, Rossi and Sega ¸ calculate the Poincaré series of a class of Artinian Gorenstein rings in [54, Proposition 6.2] and show that they are indeed rational. For similar applications of the Koszul homology algebra structure in calculating Poincaré series, see [54, Theorem 5.1] and [41, Theorem 7.1]. We finish this section by recognizing one more recent application of the classifications above. In [17], Christensen and Veliche determine minimal cases for which powers of the maximal ideal in a local ring are not Golod. We state their result in embedding dimension 3, which uses the classification in Theorem 5.1; the embedding dimension 4 case [17, Proposition 5.2] uses Theorem 5.2. Theorem 5.3 ([17, Theorem 4.2]) Let R be an Artinian Gorenstein local ring of embedding dimension 3 and socle degree 3 with maximal ideal m. The following conditions are equivalent: (1) (2) (3) (4)

R is a complete intersection; R is compressed and Koszul; R/m3 belongs to the class TE; R/m3 is not Golod.



Such applications demonstrate the usefulness of the Koszul homology algebra H (R) as a tool for learning about R, and discovering its properties.

6 Recent Progess: Koszul Algebras In this section, we focus on recent progress on the Koszul homology algebras of Koszul algebras; that is, k-algebras over which k has a linear resolution. One can check that the defining ideals of Koszul algebras are generated by quadratics; however, not all algebras defined by quadratics are Koszul (see for example [18, Remark 1.10]). Classical examples of Koszul algebras include quotients of polynomial rings by quadratic monomial ideals (e.g. edge ideals), Veronese algebras, and Segre product algebras. Unlike the classes of rings discussed in Sect. 4, Koszul algebras are not known to be characterized solely by the algebra structures on their Koszul homologies; nonetheless, the Koszul property of R is closely connected to the algebra structure on its Koszul homology H (R). To present these connections, we assume throughout this section that R = Q/I is a standard graded k-algebra with Q = k[x1 , . . . , xn ] and k a field (although some

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results can be stated more generally), and we view the Koszul homology algebra H (R) of a Koszul algebra R as a bigraded algebra H (R) =

' Hi (R)j , i,j

where i is the homological degree, and j is the internal degree given by the grading on R. Given the isomorphism (3.1), the rank of Hi (R)j is given by the graded Betti number βij of R. As such, we can view these bigraded pieces of the Koszul homology algebra in the following table, which comes from the Betti table of R: PP i PP 0 j − i PPP P 0 H0,0 1 H0,1 2 H0,2 3 H0,3 .. .. . .

1

2

H1,1 H1,2 H1,3 H1,4 .. .

H2,2 H2,3 H2,4 H2,5 .. .

3 ... H3,3 H3,4 H3,5 H3,6 .. .

... ... ... ...

where Hi,j = Hi (R)j . It is customary to call row one of the Betti table the linear strand and the other rows nonlinear strands. More precisely, we have the following definition. Definition 6.1 We call the subspace '

Hi (R)i+1

i



of H (R) the linear strand of H (R).

Now we recall some useful terminology for describing the shapes of Betti tables, which we will use throughout this section. In [7], Avramov, Conca, and Iyengar introduce the sequence ti (R) = sup{j ∈ Z | Hi (R)j = 0}, which encodes important information about R. For example, the integer ti (R) − i is the height of the ith column of the Betti table. Notice that the regularity of R reg(R) = sup{ti (R) − i} i0

measures the height of the Betti table above. The slope of R,

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?

E ti (R) − t0 (R) , slope(R) = sup i i1 introduced and studied in [7], measures the slope of the Betti table. The last number we will use in this section to describe the shapes of Betti tables is m(R) = min{i ∈ Z | ti (R)  ti+1 (R)}, which is introduced in [8]. The first result we state is a special case of a more general result of Avramov, Conca and Iyengar in [7, Main Theorem]; the inequality in (2) follows from results of Backelin in [11] and of Kempf in [39, Lemma 4]. Theorem 6.2 If R is Koszul, then R satisfies the following conditions: (1) slope(R) = 2; (2) ti (R)  2i for all i  0; (3) reg(R)  pdQ R.



The general result from which Theorem 6.2 follows is one about the slope and regularity of R where Q is Koszul, with no assumptions on R, and the proof is based on the theory of minimal models; see for example [5, Section 7.2]. We will see several other applications of minimal models throughout this section. It follows from Theorem 6.2 that if R is Koszul, then Hi (R)j = 0 for all j > 2i, which we record in Theorem 6.3, and one can easily check that Hi (R)i = 0 for all i  1; in other words, the Betti table above has the form: PP i PP 0 1 2 3 j − i PPP P 0 H0,0 0 0 0 1 0 H1,2 H2,3 H3,4 2 0 0 H2,4 H3,5 3 0 0 0 H3,6 .. .. .. .. .. . . . . .

... ... ... ... ...

Theorem 6.2, along with the other results of [7] and [8], sparked an intensive investigation of the Koszul homology algebra of a Koszul algebra, which we outline throughout the rest of this section.

6.1 Generation by the Linear Strand In this section we discuss a question of Avramov about the generation of the Koszul homology algebra of a Koszul algebra. We begin with the following result of

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Avramov, Conca, and Iyengar which records a consequence of Theorem 6.2 and describes the algebra structure on the main diagonal of the Betti table. This result prompted Question 6.4, which we discuss throughout this section. Theorem 6.3 ([8, Theorem 5.1]) If R is Koszul, then (1) Hi (R)j = 0 for all j > 2i, and (2) Hi (R)2i = (H1 (R)2 )i for all i  0.



In fact, Avramov, Conca, and Iyengar prove a more general version of Theorem 6.3 where the Koszul hypothesis is relaxed to only require linearity in the resolution of k up to degree n. Under this relaxed hypothesis, the equalities in the theorem hold for all i  n − 1. Again, the proof is based on the theory of minimal models. By Theorem 6.3, one can see that the elements of the Koszul homology algebra which lie on the main diagonal are contained in the subalgebra generated by the linear strand. This observation led Avramov to ask the following question. Question 6.4 If R is Koszul, is the Koszul homology algebra of R generated as a k-algebra by the linear strand?

In [12, Theorem 3.1], the authors extend Theorem 6.3 to the next diagonal of the Betti table; their result is stated as follows. Theorem 6.5 ([12, Theorem 3.1]) If R is Koszul, then Hi (R)2i−1 = (H1 (R)2 )i−2 H2 (R)3 for all i  2.



However, the answer to Question 6.4 is negative in general. The first counterexample was found computationally by Eisenbud and Caviglia on Macaulay2 [29], and this example led Conca and Iyengar to consider quotients by edge ideals of ncycles. From this, Boocher, D’Alí, Grifo, Montaño, and Sammartano were able to produce the family of counterexamples in (3) of the following theorem. The proofs of both Theorems 6.5 and 6.6 continue the trend of using the minimal model of R over Q to study H (R). Theorem 6.6 ([12, Theorem 3.15]) Let Q = k[x1 , . . . , xn ], let I be the edge ideal associated to a graph G on the vertices x1 , . . . , xn , and let R = Q/I . (1) If G is an n-path, then H (R) is generated by H1 (R)2 and H2 (R)3 . (2) If G is an n-cycle with n ≡ 1(mod 3), then H (R) is generated by H1 (R)2 and H2 (R)3 . (3) If G is an n-cycle with n ≡ 1(mod 3), then for any 0 = z ∈ H 2n  (R)n , H (R) 3

is generated by H1 (R)2 , H2 (R)3 , and z.

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In [23, Theorem 4.2] the author applies Theorem 3.2, and extends Theorem 6.6(1) to give a broader class of edge ideals for which Question 6.4 has a positive answer. Theorem 6.7 ([23, Theorem 4.2]) Let Q = k[x1 , . . . , xn ] with k a field, let I be the edge ideal associated to a forest on the vertices x1 , . . . , xn , and let R = Q/I . Then H (R) is generated by the linear strand.

The following result of Conca, Katthän, and Reiner gives another class of Koszul algebras whose Koszul homology algebras are generated by the linear strand.  Theorem 6.8 ([19, Corollary 2.4]) Let  Q = k[x1 , . . . , xn ] = i Qi with k a field of characteristic zero, and let R = i Q2i be the second Veronese subalgebra. Then H (R) is generated by the linear strand.

As the authors of [19] point out, Question 6.4 remains open for other classes of Koszul algebras, such as higher Veronese subalgebras and Segre product algebras. Furthermore, the authors of [12] ask the following weaker version of Question 6.4. Question 6.9 ([12, Question 3.16]) Is there a Koszul algebra R which is a domain and whose Koszul homology H (R) is not generated as a k-algebra by the linear strand?

The answer to this question is affirmative; the example below was communicated to the authors of [12] by McCullough. Example 6.10 Let k be a field and let R = k[ae, af, ag, ah, bg, bh, ce, cg, ch, de, df, dg]. Notice that R is the toric edge ring of a bipartite graph, hence a domain, and its defining ideal is generated by quadratic binomials; indeed, R∼ = k[X1 , X2 , X3 , X4 , X5 , X6 , X7 , X8 , X9 , X10 , X11 , X12 ]/I, where I = X3 X11 −X2 X12 , X8 X10 −X7 X12 , X3 X10 −X1 X12 , X2 X10 − X1 X11 ,

! X6 X8 −X5 X9 , X4 X8 −X3 X9 , X4 X7 − X1 X9 , X3 X7 − X1 X8 , X4 X5 − X3 X6 ,

thus R is Koszul by [52]. Using Macaulay2 [29], we see that the Betti table of R is

0 1 2 3

01 1-9 - - -

2 11 10 -

3 26 -

4 15 -

5 1 1

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and it is easy to see that the Koszul homology algebra must have a generator in bidegree (5, 7) for degree reasons. Thus H (R) is not generated by the linear strand.

So far we have seen that Koszulness of R does not imply generation by the linear strand in H (R). It is also the case that generation by the linear strand in H (R) does not imply Koszulness of R, as we see in the example below. Example 6.11 ([21, 7.4]) Let k be a field, and let R = k[X, Y, Z, U ]/(X2 + XY, XZ + Y U, XU, Y 2 , Z 2 , ZU + U 2 ) (see [53, Case 55]). H (R) is generated by the linear strand; indeed, it has only 6 generators in bidegree (1, 2) and 4 generators in bidegree (2, 3). However, R is not Koszul; the resolution of k is not linear.

However, the authors of [21] provide stronger conditions on the algebra structure of H (R) that are enough to imply Koszulness of R. They prove that if H (R) is generated by either a single element of bidegree (1, 2), or by a special set of elements in the linear strand, then R is Koszul. More precisely, their result is as follows. Theorem 6.12 ([21, Theorem 6.1]) Assume one of the following conditions holds: (1) There exists an element [z] of bidegree (1, 2) such that every element in the nonlinear strands of H (R) is a multiple of [z]. (2) R3 = 0 and there is a set of cycles Z representing elements in the linear strand with the property that zz = 0 for all z, z ∈ Z whose homology classes generate the nonlinear strands of H (R).



Then R is Koszul.

In fact, they prove that the condition in (1) implies that R is absolutely Koszul, a condition which implies Koszulness.

6.2 Upper Bounds on Betti Numbers In this section we discuss the following question of Avramov, Conca, and Iyengar about upper bounds of the Betti numbers of R, and equivalently, by (3.1), the ranks of the Koszul homology modules of Koszul algebras. Question 6.13 ([7, Question 6.5]) If R = Q/I is Koszul, and I is minimally generated by g quadrics, does the inequality Q

βi (R) 

  g , i

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! equivalently, dim k Hi (R)  gi , hold for all i? In particular, is the projective dimension of R over Q at most g?

Standard arguments show that the answer to this question is positive for Koszul algebras whose defining ideals are generated by monomials or have Gröbner bases of quadrics. Furthermore, the answer to this question is positive for Koszul algebras whose defining ideals are minimally generated by g  4 quadrics and for Koszul almost complete intersection algebras; that is, algebras with ht I = g − 1, as we outline below. Otherwise, Question 6.13 remains open. The case where g  3 is addressed by Boocher, Hassanzadeh, and Iyengar in the following result. Theorem 6.14 ([13, Theorem 4.5]) If I is generated by 3 quadrics, then the following conditions are equivalent. (1) R is Koszul; (2) H (R) is generated by the linear strand; (3) The Betti table of R over Q is one of the following:

0 1 2 3

0123 1- - -3- - -3- - -1

0123 01- - 1 -312 - -21

012 01- 1 -32

0123 01- - 1 -331



In [13] the authors note that the first Betti table listed in Theorem 6.14 corresponds to a complete intersection of three quadrics; the second is the Betti table of the ring k[x, y, x]/(x 2 , y 2 , xz); the third corresponds to the ideal of minors of a 3 × 2 matrix of linear forms (e.g. k[x, y]/(x, y)2 ); the last is the Betti table of a ring with a linear resolution (e.g. k[x, y, z]/(x 2 , xy, xz)). The implication (1) -⇒ (3) in Theorem 6.14 gives an affirmative answer to Question 6.13 in the case where g  3, and by the isomorphism (3.1), it determines the ranks of the bigraded pieces of H (R) for such algebras; the implication (1) -⇒ (2) provides another class of Koszul algebras for which Question 6.4 has a positive answer. The proofs of the implications (2) -⇒ (1) and (3) -⇒ (1) use D’Alí’s classification of Koszul algebras defined by 3 quadrics in [22], which we state below. Theorem 6.15 ([22, Theorem 3.1]) Let k be an algebraically closed field of characteristic different from 2, let Q be a polynomial ring over k, and let R = Q/I with I a quadratic ideal of Q. If dimk R2 = 3, then R is Koszul if and only if it is not isomorphic as a graded k-algebra (up to trivial fiber extension) to any of these: (1) k[x, y, z]/(y 2 + xy, xy + z2 , xz) (2) k[x, y, z]/(y 2 , xy + z2 , xz) (3) k[x, y, z]/(y 2 , xy + yz + z2 , xz).

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In particular, if R is a non-Koszul algebra with edim R = 3 defined by 3 quadrics, then its Betti table is 0123 01- - 1 -3- 2 - -42



The proof of Theorem 6.14 is also based on some more general results on the algebra structure on H (R). The authors of [13, Theorem 3.3] prove that the diagonal subalgebra (R) = i Hi (R)2i of H (R) is a quotient of the exterior algebra on H1 (R)2 by quadratic relations that depend only on the first syzygies of I . As a consequence of this, they find that the linear strand and the main diagonal of the Betti table for a Koszul algebra satisfy the inequality in Question 6.13, as we see in the following result. We note that the proof of (2) also uses the characterization of complete intersections given by Bruns in Theorem 4.2. Theorem 6.16 ([13, Corollary 3.4, Proposition 4.2]) If R is a Koszul algebra whose defining ideal is minimally generated by g quadrics, then ! (1) dim k Hi (R)i+1  gi for 2  i  g, and if equality holds for i = 2, then I has height 1 and a linear ! resolution of length g. (2) dim k Hi (R)2i  gi for 2  i  g, and if equality holds for some i, then I is a complete intersection.

We note that the inequality in Theorem 6.16 (2) clearly holds for quadratic complete intersections, and the fact that it holds for Koszul algebras which are not complete intersections follows from the contrapositive of the statement in [13, Corollary 3.4]. In fact, this inequality also follows from the proofs of [7, Theorem 3.1] and [7, Corollary 3.2]. Building on the work of Boocher, Hassanzadeh, and Iyengar on the g  3 case, Mantero and Mastroeni describe the Betti tables of Koszul algebras whose defining ideals are generated by 4 quadrics in [45]; their result is as follows. Theorem 6.17 ([45, Main Theorem]) Let R = Q/I be a Koszul algebra with I minimally generated by 4 quadrics and ht I = 2. Then the Betti table of R over Q is one of the following: 0123 01- - 1 -441

0123 01- - 1 -432 - -11

01234 01- - - 1 -4312 - -331

0123 01- - 1 -422 - -44

4 1



Theorem 6.17 gives an affirmative answer to 6.13 in the case that g = 4 and ht I = 2; the case where ht I = 4 (i.e. I is a complete intersection) is clear, the case where ht I = 1 follows from Theorem 6.2, and the case where ht I = 3 is settled by

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Mastroeni in [46, Main Theorem]. Together these results give an affirmative answer to Question 6.13 when g = 4. However, Question 6.13 remains open for g > 4.

6.3 Subadditivity of Syzygies The focus of this section is the following conjecture of Avramov, Conca, and Iyengar from [8] concerning the subadditivity of syzygies, and equivalently the subadditivity of Koszul homologies. Conjecture 6.18 ([8, Conjecture 6.4]) If R is a Koszul algebra, then the following inequality holds ta+b (R)  ta (R) + tb (R)



whenever a + b  pdQ R.

This conjecture is based on the following result, which asserts that a weaker inequality holds. Theorem 6.19 ([8, Theorem 6.2]) Let a and b be non-negative integers such that a+b! is invertible in k and max{a, b}  m(R). If R is Koszul, then the following a inequalities hold ta+1 (R)  ta (R) + 2 ta+b (R)  ta (R) + tb (R) + 1 for b  2.



In fact, both Conjecture 6.18 and Theorem 6.19 are stated more generally in [8]; rather than requiring that R is Koszul, it is only required that the minimal resolution of k is linear up to a certain degree. To our knowledge, Conjecture 6.18 remains open; however, the following results of Eisenbud, Huneke, and Ulrich and of Herzog and Srinivasan provide supporting evidence for its validity. Theorem 6.20 Conjecture 6.18 holds in the following cases: (1) ([25, Corollary 4.1]) dimR  1, depthR = 0, and a + b = rankk R1 ; (2) ([27, Corollary 2.1], [35, Corollary 4]) R has monomial relations and b = 1;

We note that the quadratic monomial case in Theorem 6.20 (2) is treated in [27] and the general monomial case is treated in [35]. Furthermore, in [35] Herzog and Srinivasan prove the following case of Conjecture 6.18 (without the Koszul assumption), which improves a result of McCullough in [48].

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Theorem 6.21 ([35, Corollary 3], [48, Theorem 4.4]) Let pdQ R = p. Then the following inequality holds tp (R)  tp−1 (R) + t1 (R). We finish this section by noting that the question of when the subadditivity property holds is an interesting one, even for non-Koszul algebras. The subadditivity property holds for complete intersection ideals, but not for Gorenstein ideals; these are recent results of McCullough and Seceleanu in [50]. Although the subadditivity property holds for certain classes of monomial ideals (see for example [26]), the question remains open for general monomial ideals (as well as for Koszul algebras, as discussed above).

6.4 Towards a Characterization Although the Koszul property of R has not been shown to be characterized completely by the algebra structure on H (R), recent work of Myers in [51] introduces a property of H (R) that is equivalent to the Koszul property of R under  some extra conditions. Myers defines H (R) to be strand-Koszul if H  (R) = j −i=n Hi (R)j is Koszul in the classical sense; that is, if k has a linear resolution over H  (R). With this terminology, he proves the following result. Theorem 6.22 ([51, Theorem C]) Let R be a standard graded k-algebra with k a field. If H (R) is strand-Koszul, then R is Koszul. Furthermore, the converse holds if R satisfies one of the following conditions: (1) (2) (3) (4) (5)

R has embedding dimension  3; R is Golod; The defining ideal of R is minimally generated by 3 elements; R is a quadratic complete intersection; The defining ideal of R is the edge ideal of a path on  3 vertices.



Despite the connections we have seen between the Koszul property and the algebra structure on Koszul homology throughout this section and the progress towards a characterization highlighted in the theorem above, results of Roos in [53] present potential challenges, or perhaps even obstructions, to finding a characterization of Koszul algebras by the algebra structures on their Koszul homologies alone, as we see in the following theorem. Theorem 6.23 ([53, Section 3]) The Koszul homology algebras of the Koszul algebras R71 = k[x, y, z, u]/(x 2 , y 2 , z2 , u2 , xy, zu, yz − xu)

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and R71v16 = k[x, y, z, u]/(xz + u2 , xy, xu, x 2 , y 2 + z2 , zu, yz), are isomorphic as bigraded k-vector spaces; however, they are not isomorphic as k-algebras.

Roos uses the DGAlgebras package written by Moore to describe and compare the Koszul homology algebra structures of the Koszul algebras A = R71 and B = R71v16 defined above. Roos shows that the Poincaré series PkA (t) and PkB (t), and the Betti tables of A and B over Q = k[x, y, z, u] are both the same, hence H (A) and H (B) are isomorphic as bigraded k-vector spaces; however, the algebra structures on H (A) and H (B) are different. In fact, Roos concludes that the Eilenberg-Moore spectral sequence degenerates over H (A) but not over H (B); thus, by [51, Theorem B], H (A) is strand-Koszul and H (B) is not. We end by noting that the algebra B = R71v16 provides an example of a Koszul algebra whose Koszul homology algebra is not strand-Koszul; thus, the converse of Theorem 6.22 does not hold in general. Acknowledgments The author thanks the anonymous referee for many helpful and insightful comments that greatly improved this survey.

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Canonical Resolutions over Koszul Algebras Eleonore Faber, Martina Juhnke-Kubitzke, Haydee Lindo, Claudia Miller, Rebecca R. G., and Alexandra Seceleanu

Keywords Koszul algebra · Minimal free resolution · Betti numbers

1 Introduction Koszul algebras show up naturally and abundantly in algebra and topology. They were first introduced by Priddy in 1970 as algebras for which the bar resolution, which is normally far from minimal, admits a reduction to a comparatively small subcomplex; see [23]. Priddy’s work explained contemporaneous ideas on restricted Lie algebras in the work of May and, separately, that of Bousfield, Curtis, Kan, Quillen, Rector, and Schlesinger; see [2, 17]. Priddy was an algebraic topologist, but Koszul algebras have since been linked to several fundamental concepts across mathematics where they appear naturally and are studied extensively in fields

E. Faber School of Mathematics, University of Leeds, Leeds, UK e-mail: [email protected] M. Juhnke-Kubitzke Department of Mathematics, University of Osnabrück, Osnabrück, Germany e-mail: [email protected] H. Lindo Department of Mathematics, Harvey Mudd College, Claremont, CA, USA e-mail: [email protected] C. Miller Department of Mathematics, Syracuse University, Syracuse, NY, USA e-mail: [email protected] Rebecca R. G. Department of Mathematical Sciences, George Mason University, Fairfax, VA, USA e-mail: [email protected] A. Seceleanu () Department of Mathematics, University of Nebraska–Lincoln, Lincoln, NE, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Miller et al. (eds.), Women in Commutative Algebra, Association for Women in Mathematics Series 29, https://doi.org/10.1007/978-3-030-91986-3_11

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as diverse as topology [14], representation theory [6], commutative algebra [12], algebraic geometry [4], noncommutative geometry [16], and number theory [21]. For a general overview, see the monograph by Polishchuk and Positselski [22]. An interesting feature of Koszul algebras is that they appear in pairs: every Koszul algebra A has a dual algebra A! which is also a Koszul algebra (see Sect. 2). The prototypical example of such a Koszul pair is a polynomial algebra S over a field, together with the corresponding exterior algebra . The associated theory of Koszul duality is a generalization of the duality underlying the Bernstein– Gelfand–Gelfand correspondence [4] describing coherent sheaves on projective space in terms of modules over the exterior algebra. This exemplifies the philosophy that facts relating the symmetric and exterior algebras often have Koszul duality counterparts. In this paper we extend Priddy’s methods of constructing free resolutions over standard graded Koszul algebras and generalize Buchsbaum and Eisenbud’s resolutions in [3] to resolve powers of the homogeneous maximal ideal over Koszul algebras. Resolutions over Koszul algebras have previously appeared in works of Green and Martínez-Villa [15, Theorem 5.6], Martínez-Villa and Zacharia [20, Proposition 3.2] and in other sources referenced below. Our approach has the advantage of producing explicit minimal resolutions. In particular, in [23] Priddy exploits a natural differential on A ⊗k A! to give an explicit construction for the linear minimal graded free resolution of the residue field of a graded Koszul algebra; see Definition 2.5. In this paper, we extend this construction to a family of acyclic complexes that yields highly structured resolutions of the powers of the homogeneous maximal ideal over standard graded Koszul algebras; see Definition 4.1. Since these complexes are typically not minimal, we also seek to determine their minimal counterparts. To achieve this, we take inspiration from results that describe structured resolutions over a polynomial ring S constructed starting from the Koszul complex (exterior algebra) and its generalizations; see [3]. We provide analogs of these results using any pair of Koszul dual algebras, A and A! , instead of S and . Our main result generalizes the canonical resolutions for the powers of the homogeneous maximal ideal constructed over a polynomial ring S by Buchsbaum and Eisenbud in [3] to obtain minimal free resolutions for powers of the homogeneous maximal ideal of a graded Koszul algebra. In contrast to the situation over S, these are in general infinite resolutions. This allows us to obtain an explicit formula for a the graded Betti numbers defined dimk Tori (ma , k)j and the graded  by βi,j (ma ) = A j Poincaré series Pma (y, z) = i,j 0 βi,j (m )y zi . The following is a combination of Theorem 5.3 and Corollary 5.5. Theorem If A is a graded Koszul algebra with homogeneous maximal ideal m, the complexes ∂n

 ∂n−1

∂1

εa

A LA → LA → LA → ma → 0, a : · · · → Ln,a − n−1,a −−→ . . . − 0,a −

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defined in Eq. (5.3) with the augmentation map εa defined in Eq. (5.4) are minimal free resolutions of the powers ma with a  1. The nonzero graded Betti numbers of the powers of m are given by A βn,n+a (ma ) =

a 

(−1)i+1 dimk ((A! )∗n+i ) dimk (Aa−i ).

i=1

and the graded Poincaré series is A −a Pm HA! ∗ (yz)HA/ma (−yz). a (y, z) = −(−z)

In particular, the minimal graded resolution of ma is a-linear. The Betti numbers presented in the theorem are recovered in a more restricted setting in the recent paper [25] investigating resolutions of monomial ideals over strongly Koszul algebras via different techniques. The paper is structured as follows. In Sect. 2, we provide background on Koszul algebras and the Priddy complex. In Sect. 3, we explain how to obtain a free solution of a module M from a resolution of the ring over its enveloping algebra. In Sect. 4, we rewrite this resolution as the totalization of a double complex, in the case that the module is a power of the homogeneous maximal ideal and the ring is a Koszul algebra. In Sect. 5, we give the minimal resolution and Betti numbers for the powers of the homogeneous maximal ideal over a Koszul algebra. In Sect. 6 we apply our construction to several specific Koszul algebras A to obtain explicit formulas for the Betti numbers of ma .

2 Koszul Algebras and the Priddy Resolution Throughout k is a field and A is a graded k-algebra having finite-dimensional graded components with Ai = 0 for i < 0 and A0 = k. We further assume that A is standard graded, that is, A is generated by A1 as an algebra over A0 = k. Definition 2.1 ([23, Chapter 2]) We say that A is Koszul if k = A/A>0 admits a linear graded free resolution over A, i.e., a graded free resolution P• in which Pi is generated in degree i.

Classes of graded Koszul algebras arise from: quadratic complete intersections [24], quotients of a polynomial ring by quadratic monomial ideals [12], quotients of a polynomial ring by homogeneous ideals which have a quadratic Gröbner basis, and from Koszul filtrations [9]; see, for example, the survey paper by Conca [7]. We now focus on quadratic algebras in order to define Koszul duality. Definition 2.2 ([22, Chapter 1, Section 2]) Let A be a standard graded k-algebra. We say that A is quadratic if A = T (V )/Q, where V is a k-vector space, T (V ) is the tensor algebra of V , and Q is a quadratic ideal of T (V ).

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If A is a quadratic algebra, its quadratic dual algebra is defined by A! =

T (V ∗ ) Q⊥

where V ∗ = Homk (V , k) and Q⊥ is the quadratic ideal generated by the orthogonal complement to Q2 in T (V ∗ )2 = V ∗ ⊗k V ∗ with respect to the natural pairing between V ⊗ V and V ∗ ⊗ V ∗ given by v1 ⊗ v2 , v1∗ ⊗ v2∗  = v1 , v1∗ v2 , v2∗ . Choosing dual bases x1 , . . . , xd and x1∗ , . . . , xd∗ for V and V ∗ respectively yields that T (V ) = kx1 , . . . , xd  and T (V ∗ ) = kx1∗ , . . . , xd∗  are polynomial rings in noncommuting variables of degrees |xi | = 1 and |xi∗ | = −1. This allows one to compute Q⊥ given a quadratic ideal Q ⊆ T (V ) using linear algebra, as described for example in [18, Section 8].

Graded Koszul algebras are quadratic (see, for example, [22, Chapter 2, Definition 1]) and the duality of quadratic algebras restricts well to the class of Koszul algebras since A and A! are Koszul simultaneously [22, Chapter 2, Corollary 3.2 ]. Moreover, (A! )! = A. Example 2.3 The main example of Koszul dual algebras is given by the symmetric algebra on a vector space V S = k[x1 , . . . , xd ] =

kx1 , . . . , xd  (xi xj − xj xi , 1  i < j  d)

and the exterior algebra on V ∗ S! =

kx1∗ , . . . , xd∗  . ((xi∗ )2 , xi∗ xj∗ + xj∗ xi∗ , 1  i  j  d)

=

Example 2.4 For the following commutative Koszul algebra A=

k[x, y, z] kx, y, z = 2 , 2 2 2 (x , xy, y ) (x , xy, y , xz − zx, xy − yx, yz − zy)

the dual algebra is given by A! =

kx ∗ , y ∗ , z∗  . ((z∗ )2 , x ∗ z∗ + z∗ x ∗ , y ∗ z∗ + z∗ y ∗ )

This pair of algebras are further discussed in Example 6.1.



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Definition 2.5 The Priddy complex [23] of a quadratic algebra A is the complex P•A whose i-th term is given by ∗

PiA = A ⊗k (A! )i ,  and the differential is defined by right multiplication by the trace element di=0 xi ⊗ xi∗ , where multiplication by xi∗ ∈ A! on (A! )∗ is defined as the dual of multiplication

by xi∗ on A! . 2.6 The importance of the Priddy complex lies in the fact that P•A is acyclic if and only if A is Koszul; see [22, Chapter 2, Corollary 3.2]. Moreover, when A is Koszul the Priddy complex, also called the generalized Koszul resolution, is a minimal free resolution of the residue field k of A. This will be the base case in the proof that our construction in Sect. 4 is a resolution.

2.7 Duality of Koszul algebras extends to an equivalence of derived categories that goes back to [5] and was developed further in [6]. Let T = A⊗k A! which is an A-A! bimodule. For complexes N• of A! -modules and M• of A-modules define functors L(N• ) = T ⊗A! N• ∼ = Homk (A! , M• ) ∼ = = A ⊗k N• and R(M• ) = HomA (T , M• ) ∼ ! ∗ (A ) ⊗k M• . It is shown in [6, Theorem 2.12.1] that these functors induce an equivalence of categories L : D ↑ (A! )  D ↓ (A) : R where D ↑ (A! ) stands for the derived category of complexes N• of graded A! modules with Ni,j = 0 for i " 0 or i + j 0 0 and D ↓ (A) is the derived category of complexes M• of graded A-modules with Mi,j = 0 for i 0 0 or i + j " 0.

3 Resolutions via the Enveloping Algebra Let A be a (not necessarily commutative) k-algebra where k is a field. In this section, we review how one obtains a free resolution of any A-module M from a resolution of A over its enveloping algebra. In general, one obtains a resolution that is far from minimal. We remedy this in Sects. 4 and 5 over Koszul algebras A for the modules ma (and hence A/ma ). 3.1 Given a k-algebra A, its enveloping algebra is given by Ae = A ⊗k Aop . A left Ae -module structure is equivalent to an A-A-bimodule structure via (a ⊗ b) · m = a · m · b

(3.1)

We consider A as an Ae -module via the action in (3.1), where a, b, m ∈ A and a · m · b represents internal multiplication in A.

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Consider a graded free resolution1 of A over Ae and note that any free left Ae module F can be rewritten as F = Ae ⊗k V = A ⊗k Aop ⊗k V ∼ = A ⊗k V ⊗k A for some vector space V , where the rightmost expression is thought of as an A-Abimodule via the outside two factors. Thus the resolution will be of the form ε

→ A → 0, · · · → A ⊗k V2 ⊗k A → A ⊗k V1 ⊗k A → A ⊗k A −

(3.2)

where the augmentation ε from A ⊗k A to A is given by multiplication across the tensor. We observe that A ⊗k k ⊗k A ∼ = A ⊗k A. Thus setting V0 = k, we may write the resolution as a quasi-isomorphism of Ae -modules

A ⊗k V• ⊗k A − →A Next we show how to construct an A-free resolution for arbitrary A-modules M using (3.2). This is well known; we include it because the construction is the basis of our next step in Sect. 4. In the case of Koszul algebras, it can also be seen using Koszul duality, and, more generally, it follows when the resolution of A comes from an acyclic twisting cochain; see the remarks following the proof.

Proposition 3.2 If M is a graded A-module and A ⊗k V• ⊗k A − → A is a graded Ae -free resolution of A, then the induced map A ⊗k V• ⊗k M − → M is a graded A-free resolution of M where the A-module structure on the latter tensor product is via the first factor.

Proof First note that both A ⊗k V• ⊗k A and A are A-A-bimodules in the obvious ways. Furthermore, the complex A ⊗k V• ⊗k A (considered as an A-module via its righthand factor) and the trivial complex A both consist of free A-modules (although the latter is not a free Ae -module). Therefore the quasi-isomorphism A ⊗k V• ⊗k A − → A is actually a homotopy equivalence of A-modules. Hence it remains a quasi-isomorphism after tensoring over A on the right with arbitrary A-modules. To see this, note that the augmented complex of free A-modules (3.2) is contractible, that is, homotopy equivalent to 0 (equivalently, it is split exact over A). But this complex is the mapping cone of the chain map A ⊗k V• ⊗k A → A. Therefore, upon tensoring (3.2) on the right over A with a left A-module M, one obtains a quasi-isomorphism of left A-modules

(A ⊗k V• ⊗k A) ⊗A M − → A ⊗A M

1 One

can always take the bar resolution of A over its enveloping algebra, but that is usually far from minimal.

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A ⊗k V• ⊗k M − →M As the original resolution of A was a map of A-A-bimodules, this one is still a map of A-modules (via the lefthand factor of the tensor product). Viewing V• ⊗k M as a (rather large) k-vector space, one sees that the complex on the left consists of free A-modules, giving a free A-resolution of M. Remark 3.3 Under the assumption that A is a Koszul algebra, we include here an alternate proof of Proposition 3.2 using Koszul duality. Using the functors from the equivalence described in 2.7, for a graded A-module M, one gets that L(R(M)) M. On the other hand, one computes that R(M) is the complex 0 → (A! )∗ ⊗k M0 → (A! )∗ ⊗k M1 → · · · → (A! )∗ ⊗k Mi → · · · Furthermore, it is clear from the definition that L((A! )∗ ) is simply the Priddy complex P•A . Applying this to the complex above termwise and totalizing gives that L(R(M)) equals the totalization of 0 → P•A ⊗k M0 → P•A ⊗k M1 → · · · → P•A ⊗k Mi → · · · which is exactly the complex described in Proposition 3.2. In the case that M =

A/ma , one gets the double complex XA a described in Corollary 4.5. Remark 3.4 More generally, we now briefly describe this from the perspective of acyclic twisting cochains. Although these go far back, for recent quite general versions of the duality they afford, modeled on that of Dwyer, Greenlees, and Iyengar in [10] and generalizing Koszul duality, and for descriptions of how it specializes to the situation of Koszul algebras, as well as the terms used below, see Avramov’s paper [1], especially Theorem 4.7. Let A be an augmented dg (differential graded) algebra. When there is an augmented dg coalgebra C with a map τ : C → A of degree −1 that is a twisting cochain, that is, a Maurer-Cartan equation ∂A τ + τ ∂C + μ(τ ⊗ τ ) = 0, holds, where  : C → C ⊗ C is the diagonal map and μ is the multiplication map, then one can form tensor products whose differential is “twisted” by τ yielding a natural map A ⊗ τ Cτ ⊗ A → A. If this is a quasi-isomorphism, then τ is called acyclic, in which case the induced map A ⊗ τ Cτ ⊗ M → M is a quasi-isomorphism for all dg A-modules M and a duality generalizing Koszul duality holds. An example is given by the bar construction C = BA with the canonical map τ : BA → A, but in the case of Koszul algebras one can get by with a much smaller complex using Priddy’s construction.

Remark 3.5 Suppose A is local (or standard graded) k-algebra with (homogeneous) maximal ideal m. The resolutions obtained in Proposition 3.2 are in general not

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minimal (respectively, minimal graded) resolutions even when one starts with a minimal (respectively, minimal graded) resolution of A over Ae . For example, for   the explicit resolution XA a given in Corollary 4.5, although ∂ is minimal, ∂ is clearly not.



4 The Case of Koszul Algebras In this section, under the further assumption that A is a Koszul k-algebra, we write the A-free resolution of A/ma obtained in Remark 3.3 as the totalization of a certain double complex. First we recall the minimal graded resolution of A over Ae following the presentation in [26, Section 3]. It is a symmetrization of the resolution of k over A found by Priddy in [23], which is presented in Definition 2.5. Definition 4.1 Let A be a Koszul k-algebra with dual Koszul algebra A! . Let (A! )∗ = Homk (A! , k); thus (A! )∗ is an A! -module where the action of A! on (A! )∗ is the dual of the action of A! on itself. Define free A-modules Fn = A ⊗k (A! )∗n ⊗k A and differentials ∂n = (∂  )n + (−1)n (∂  )n where ∂  = right multiplication by

d 

xi ⊗ xi∗ ⊗ 1

i=0

(which will form our vertical maps) and ∂  = left multiplication by

d 

1 ⊗ xi∗ ⊗ xi

i=0

(which will form our horizontal maps).2 Then the complex ∂n

ε

FA : · · · → Fn − → Fn−1 → · · · F0 − →A→0

2 There

is a misprint in [26] with regards to this map.

(4.1)

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Fig. 1 The minimal resolution of a Koszul algebra A over Ae

∼ A ⊗k A to A is augmented by the multiplication map ε from F0 = A ⊗k k ⊗k A = the minimal graded free resolution of A over Ae [26, Proposition 3.1]. The complex is FA is the totalization of the double complex with differentials ∂  and ∂  depicted in Fig. 1. In particular, Fn is the sum of the modules on the n-th antidiagonal of this double complex.

Remark 4.2 Here is the explicit connection with Priddy’s resolution: tensoring (4.1) on the right over A with k gives Priddy’s minimal resolution of k as a left A-module in Definition 2.5, also called the generalized Koszul resolution. Tensoring (4.1) on the left gives the minimal resolution of k as a right A-module.

4.3 Considering the graded strands of (4.1), one can write this complex as a totalization of an anticommutative double complex, which we also call FA , of free A-modules given by the free A-modules Fij = A ⊗k (A! )∗i ⊗k Aj

(4.2)

where we are using the first tensor factor as “coefficients” and the maps ∂  and ±∂  of Definition 4.1 become the vertical and horizontal maps, respectively in Fig. 1. Note that i is the homological degree in the complex FA .

4.4 One can interpret the complex FA in the language of the functors introduced in 2.7 as FA = L(R(A)), where A is viewed as a complex concentrated in homological degree 0. The discussion in 4.1 shows there is a quasi-isomorphism L(R(A)) A. This was previously shown in [6, Thm 2.12.1] and is a particular case of Remark 3.3.



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Next we apply the discussion from Sect. 3 to the A-module A/ma and arrange its resolution into a double complex similarly to the one shown in Fig. 1 above. Corollary 4.5 Totalization of the truncation of the double complex (4.2) obtained by removing the columns with index j  a  1 gives a graded A-free resolution

! ∗ → A/ma . XA a = A ⊗k (A ) ⊗k Aa−1 −

(4.3)

Proof Applying Proposition 3.2 by tensoring the resolution of A over Ae on the right over A with A/ma gives a graded A-free resolution

! ∗ a XA → A/ma a = A ⊗k (A ) ⊗k A/m −

By means of the k-vector space identification A/ma = Aa−1 the resolution becomes

! ∗ → A/ma XA a = A ⊗k (A ) ⊗k Aa−1 −

Viewing graded strands, one can write this as a totalization of an anticommutative double complex of free A-modules given by the terms Fij = A ⊗k (A! )∗i ⊗k Aj with i  0, 0  j  a − 1 of (4.2) with the differentials inherited from those described in Definition 4.1.

We display the diagram for the double complex that yields the A-free graded a resolution XA a of A/m in Corollary 4.5 in Fig. 2. In the language of 2.7 this a resolution can be described as XA a = L(R(A/m )). A Remark 4.6 The resolution XA a is minimal for a = 1 in which case Xa recovers the Priddy complex without its first term. However for a  2 this resolution is typically non minimal as the rows are split acyclic; see 5.1. The goal of Sect. 5 is to produce a minimal free resolution for A/ma using XA

a.

5 Minimal Resolutions for Powers of the Maximal Ideal We introduce complexes LA a inspired by work of Buchsbaum and Eisenbud [3]. These will turn out to be the minimal resolutions for the powers of the homogeneous maximal ideal of a graded Koszul algebra.

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a Fig. 2 The A-free graded resolution XA a of A/m

5.1 We define free A-modules analogous to the Schur modules used by Buchsbaum and Eisenbud in their resolutions (the case where A is a polynomial ring) in [3]. First note that the rows of the double complex (4.2) except the bottom one are exact; in fact, they can be viewed as the result of applying the exact base change A ⊗k − to the strands of the dual Priddy complex, all of which are exact except the one whose homology is k (in the case where A is a polynomial ring, it is applied to the strands of the tautological Koszul complex; see [19, Section 1.4]). These complexes are contractible, as they consist of free A-modules, and so all kernels, images, and cokernels of the differentials are free as well. Define for a > 0 the following free A-modules LA n,a

  (−1)n+1 ∂  !∗ !∗ = im A ⊗k A n+1 ⊗k Aa−1 −−−−−−→ A ⊗k A n ⊗k Aa

(5.1)

  (−1)n ∂  ∗ ∗ = ker A ⊗k A! n ⊗k Aa −−−−→ A ⊗k A! n−1 ⊗k Aa+1 .

(5.2)

The vertical differentials ∂  in Fig. 2 induce maps on these modules, which we again denote by ∂  , to yield a complex ∂n

 ∂n−1

∂1

A → LA → LA LA a : · · · → Ln,a − n−1,a −−→ . . . − 0,a .

(5.3)

A This complex is minimal in the sense that ∂  (LA n,a ) ⊆ mLn−1,a for all n since the same property holds for the columns of the complex in Fig. 1 viewed as complexes with differential ∂  . The construction of the complex LA a is depicted in Fig. 3.

Lemma 5.2 The complex LA a can be augmented by the evaluation map

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Fig. 3 The construction of the complex LA a ∗

! a εa : LA 0,a = A ⊗k A 0 ⊗k Aa → m

(5.4)

which is the restriction of the multiplication map ∗

ε : A ⊗k A! 0 ⊗k A → A sending r ⊗ v ⊗ s → rvs. Proof As stated in Definition 4.1, ε is an augmentation map FA • → A. We verify explicitly that ε satisfies the required property ε ◦ (∂  − ∂  ) = 0 below: ε ◦ (∂  − ∂  )(r ⊗ v ⊗ s) = ε(rv ⊗ 1 ⊗ s − r ⊗ 1 ⊗ vs) = rvs − rvs = 0. Since ∂  |LA = 0 it follows from the computation above that ε◦∂  (LA 1,a ) = 0, hence 1,a

the complex LA a can be augmented to ∂n

 ∂n−1

∂1

εa

· · · → LA → LA → LA → ma → 0. n,a − n−1,a −−→ . . . − 0,a −



The following is the main result of our paper. The case when A is an exterior algebra has appeared previously in [11, Corollary 5.3]. The proof therein uses the self-injectivity of the exterior algebra in a crucial manner, and therefore does not seem to extend to all Koszul algebras.

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Theorem 5.3 If A is a Koszul algebra, the complexes LA a defined in Eq. (5.3) with the augmentation map εa defined in (5.4) are minimal free resolutions for the powers ma of the maximal ideal with a  1.

Proof The complexes LA a are minimal by the discussion in 5.1. The proof of the remaining claims is by induction on a  1. The definition of Ln,1 in (5.1) shows that there are isomorphisms !∗ !∗ ∼ ∼ LA n,1 = A ⊗k A n+1 ⊗k k = A ⊗k A n+1 ,

since the map ∂  is injective on FA n+1,0 for n  0, the leftmost column of the double complex in Fig. 1 (this column considered by itself is in fact XA 1 ). Therefore there A ) [−1], where (XA ) ∼ is an isomorphism of complexes LA (X = 1 1 1 1 1 denotes the A truncation of the complex X1 by removing the homological degree 0 component. A A A  Since XA 1 = P• is just the Priddy complex (upon noting that ∂ : (X1 )1 → (X1 )0 ∗ A ! agrees with ε under the identification (X1 )0 = A ⊗k A 0 ⊗k A0 ∼ = A), we see ε ε A → m is a resolution of m by 2.6 and the base case that LA → m is a that (X1 )1 − 1 − minimal resolution of m follows. For arbitrary a  2, (5.1) gives a short exact sequence of complexes A A 0 → LA a−1 → P• ⊗k Aa−1 → La [−1] → 0, ∗

where PnA ⊗k Aa−1 = A ⊗k A! n ⊗k Aa−1 is the (a − 1)-st column of the double complex in Fig. 2. The notation signifies that this column can be viewed as the Priddy complex P•A tensored with Aa−1 . From the long exact sequence in homology induced by the short exact sequence of complexes displayed above we deduce  Hi (LA a)=

0 ker

H0 (LA a−1 )

→

H0 (P•A

⊗k Aa−1 )

!

i1 i = 0.

a It remains to show that H0 (LA a ) = m . Indeed, the induced map in homology A A ∼ H0 (LA a−1 ) → H0 (P• ⊗k Aa−1 ) = H0 (P• ) ⊗k Aa−1

can be recovered as the bottom map in the following commutative diagram LA 0,a−1 

A ⊗k (A! )∗0 ⊗k Aa−1 ε0 ⊗idAa−1

εa−1

ma−1

/ k ⊗k Aa−1

 ∼ = ma−1 /ma .

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Commutativity of the diagram yields that for r ⊗ s ⊗ v ∈ LA 0,a the induced map in homology is given by εa (r ⊗ s ⊗ v) = rsv → rsv, where r is the coset of r in k = A/m. Thus we obtain the desired identification 8 9 A ker H0 (LA ) → H (P ⊗ A ) 0 k a−1 • a−1 = εa (Span{r ⊗ s ⊗ v | r ∈ m, s ∈ k, v ∈ Aa−1 }) ∼ = ma .



Remark 5.4 Recall that rows of the double complex (4.2) except the bottom one are exact; in fact, they can be viewed as the result of applying a base change to the strands of the dual Priddy complex; see 5.1. These complexes are contractible, as they consist of free R-modules. Hence for n  1 the (n + a)-th row of Fig. 2, counting from the bottom (as the 0th row), is quasi-isomorphic to LA n,a and the lower rows (numbered 1 through a) are split exact. The acyclic assembly lemma → LA [27, Lemma 2.7.3] yields quasi-isomorphisms (XA a )1 [−1] − a for a  1. As

→ A/ma , hence also shown in Corollary 4.5 there are quasi-isomorphisms XA a − → ma . Transitivity yields a quasi-isomorphism LA → ma . (XA a )1 − a − This approach gives an alternate proof for our main result, but it only determines the augmentation map up to an isomorphism on its target. We prefer the more explicit approach of Theorem 5.3, which specifies the augmentation map εa .

The following corollary of Theorem 5.3 gives an explicit formula for the Betti numbers of powers of the maximal ideal of a Koszul algebra. That ma has an a-linear minimal free resolution also follows from [8, Theorem 3.2]. Once the linearity of this resolution has been established, [22, Chapter 2, Corollary 3.2 (iiiM)] gives an alternate interpretation for the Betti numbers of ma in terms of the graded components of a quadratic dual module for the A-module ma . However this description seems less amenable to explicit computations than our methods. The next result utilizes the Hilbert series H(A! )∗ (t) =



dimk (A! )∗ · t and HA/ma (t) =

a−1 

dimk Aj · t j .

j =0

0

Corollary 5.5 If (A, m) is a Koszul algebra, the nonzero graded Betti numbers of the powers of m are given by A βn,n+a (ma )

=

a  i=1

(−1)i+1 dimk ((A! )∗n+i ) dimk (Aa−i ).

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In particular, the minimal graded resolution of ma is a-linear and its graded Poincaré series is A −a H(A! )∗ (yz)HA/ma (−yz). Pm a (y, z) = −(−z)

Consequently, the nonzero Betti numbers of A/ma are given by A βn,j (A/ma )

⎧ ⎨a (−1)i+1 dim ((A! )∗ k i=1 n+i−1 ) dimk (Aa−i ) = ⎩1

n > 0, j = n + a − 1 n = j = 0.

Proof The fact that minimal free resolution of ma is a-linear follows from the Theorem 5.3 and the description of the differential ∂  of the complex (5.3) in view of the fact that there is a splitting of the map ∂  to each Ln,a identifying a basis of it with part of a basis of the last column of XA a . Consider the rows of the truncated complex XA when augmented to the relevant L n,a as follows. a 0 −→ A ⊗k (A! )∗n+a ⊗k A0 −→ · · · −→ A ⊗k (A! )∗n+1 ⊗k Aa−1 −→ LA n,a −→ 0 The exactness of this complex, as explained in Remark 5.4, yields the identities A (ma ) = rankA (LA βn,n+a n,a ) =

a 8 9  (−1)i+1 rankA A ⊗k (A! )∗n+i ⊗k Aa−i i=1

=

a 

(−1)i+1 dimk (A! )∗n+i dimk Aa−i

i=1

and the vanishing of the remaining Betti numbers is due to the fact that the minimal resolution in Theorem 5.3 is a-linear. Lastly, the coefficients of the series ⎛ −H(A! )∗ (yz)HA/ma (−yz) (−z)a

=

⎜ ⎜ ⎝

n0

⎞ 

⎟ n n+a (−1)j −a dimk (A! )∗ dimk Aj ⎟ , ⎠z y

0j a−1 j + =n+a

A (ma ) by setting j = a − i, = n + i. agree with the preceding expression for βn,n+a



In contrast to Theorem 5.3, for non Koszul algebras the A-free resolution of A/ma afforded by Corollary 4.5 cannot be minimized by the procedure presented in this section. We illustrate the obstructions by means of the following example. Example 5.6 Let A = k[x]/(x 3 ), which is a non Koszul (also non quadratic) algebra. The enveloping algebra is

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Ae = A ⊗k A = k[x]/(x 3 ) ⊗k k[y]/(y 3 ) ∼ = k[x, y]/(x 3 , y 3 ) and the Ae -module structure induced on A by the (surjective) multiplication map ε → A yields the isomorphism A ∼ Ae = A ⊗k A − = Ae /(x − y). Therefore A has the following two-periodic resolution over the complete intersection Ae x 2 +xy+y 2

x−y

x−y

ε

· · · → Ae −−→ Ae −−−−−−→ Ae −−→ Ae − → A → 0. Rewriting this complex in the form of Sect. 3 gives ∂

∂

ε

· · · → A ⊗k V2 ⊗k A − → A ⊗k V1 ⊗k A − → A ⊗k V0 ⊗k A − → A → 0, where each Vi is a one dimensional vector space with basis {ei } and for i > 0  ∂(1 ⊗ ei ⊗ 1) =

x ⊗ ei−1 ⊗ 1 − 1 ⊗ ei−1 ⊗ y x2

for i odd

⊗ ei−1 ⊗ 1 + x ⊗ ei−1 ⊗ y + 1 ⊗ ei−1

⊗ y2

for i even.

The conclusion of Corollary 4.5 still holds and indicates that the truncated a complexes XA a are (non minimal) free resolutions for A/m . But by contrast to the Koszul case, we see that arranging by grading as in (4.2) yields a diagram that is not a bicomplex and whose rows are no longer exact (or even complexes!), and so in the truncated complex (4.3) the rows are no longer acyclic. Correspondingly, the modules Ln,a one could define are no longer free. Thus there is no clear way to minimize the complex XA a in a similar manner to the technique used in this section, except for the case a = 1 where XA

a is already minimal.

6 Examples In this section we provide examples which illustrate our constructions for certain Koszul algebras. For simplicity, all our examples are commutative algebras defined by quadratic monomial ideals, but of course there are plenty of noncommutative examples as well. This class is known to yield Koszul algebras by [12]. Example 6.1 Consider the following pair of dual Koszul algebras from Example 2.4 A=

k[x, y, z] (x 2 , xy, y 2 )

and

A! =

kx ∗ , y ∗ , z∗  . + z∗ x ∗ , y ∗ z∗ + z∗ y ∗ )

((z∗ )2 , x ∗ z∗

The graded pieces (A! )−n are spanned by the words of length n on the alphabet {x ∗ , y ∗ , z∗ } where the first letter is x ∗ , y ∗ or z∗ and the other n − 1 are x ∗ or y ∗ , whence dimk (A! )−n = 3 · 2n−1 for n  1. For n  1, An is spanned by monomials

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of the form (z∗ )n , x ∗ (z∗ )n−1 , and y ∗ (z∗ )n−1 so that dimk A−n = 3. Thus in this case both A and A! are infinite dimensional k-algebras. The Priddy complex P•A (2.5) consists of terms of the form P0 = A ⊗k k Pn = A ⊗k k 3·2

n−1

for n  1

and the resolution of A over Ae viewed as a double complex (4.2) has terms  Fi,j = A ⊗k k

3·2i−1

⊗k Aj =

A3·2

i−1

i−1 A3·2

⊗A A

i  1, j = 0

⊗A

i  1, j  1.

A3

Corollary 5.5 reveals that the Betti numbers of ma are independent of a. Indeed for a  1 and n  0 we have βn,n+a (ma ) =

a−1 

(−1)i+1 · 9 · 2n+i−1 + (−1)a+1 · 3 · 2n+a−1

i=1

= 9 · 2n ·

1 − (−2)a−1 + (−1)a−1 · 3 · 2a+n−1 3

= 3 · 2n . Example 6.2 Consider the following commutative Koszul algebra A=

k[x, y, z] kx, y, z = . (xy, xz) (xy, xz, xz − zx, xy − yx, yz − zy)

The Koszul dual algebra is given by A! =

kx ∗ , y ∗ , z∗  ((x ∗ )2 , (y ∗ )2 , (z∗ )2 , y ∗ z∗ + z∗ y ∗ )

and its Hilbert function satisfies the Fibonacci recurrence dim A!−n−2 = dim A!−n + dim A!−n−1 . Indeed, setting u(n) to be the number of monomials in A! of degree −n ending in x and v(n) to be the number of monomials in A! of degree −n not ending in x ∗ , yields u(n) = v(n − 1) and v(n) = 2u(n − 1) + u(n − 2). The second expression follows because the number of monomials ending in y ∗ or z∗ where the previous letter is x ∗ is 2u(n − 1) and the number of monomials ending in y ∗ z∗ (or equivalently, z∗ y ∗ ) where the previous letter is x is u(n − 2). Thus this leads to

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dim A!−n−2 = u(n + 2) + v(n + 2) = v(n + 1) + 2u(n + 1) + u(n) = v(n + 1) + u(n + 1) + v(n) + u(n) = dim A!−n−1 + dim A!−n . This shows that the Betti numbers of m are the Fibonnacci numbers starting with β0A (m) = 3 and β1A (m) = 5. The identity above in turn implies that the terms of the double complex as well as the free modules in the resolution of ma satisfy similar recurrences rank Fn+2,a = rank Fn+1,a + rank Fn,a ,

rank Ln+2,a = rank Ln+1,a + rank Ln,a .

We conclude that the Fibonacci recurrence holds for Betti numbers A A A (ma ) = βn+1,n+1+a (ma ) + βn,n+a (ma ) for a  1, n  0 βn+2,n+2+a

subject, if a  2, to the initial conditions β0A (ma ) = a + 4 and β1A (ma ) = 2a + 4. Solving the above recurrence yields closed formulas for the Betti numbers of ma with a  2 as follows  A βn,n+a (ma )

=

a + 4 3a + 4 + √ 2 2 5



√ n  √ n  a + 4 3a + 4 1+ 5 1− 5 − √ + . 2 2 2 2 5

We now give an infinite resolution counterpart to a family of square-free monomial ideals that have appeared as ideals of the polynomial ring in work of Galetto [13]. Example 6.3 (See also [25, Example 3.18]) Consider the dual pair of Koszul algebras A=

k[x1 , . . . , xd ] , (x12 , . . . , xd2 )

where dimk (Aj ) = complex (4.2) are

d! j

and

A! =

and dimk (A!−i ) =

kx1∗ , . . . , xd∗  , (xi∗ xj∗ + xj∗ xi∗ , 1  i < j  d) i+d−1! d−1 .

Thus the terms in the double

d i+d−1 Fi,j = A ⊗k k ( d−1 ) ⊗k k (j ) .

Notice that for a  d the ideal ma of A can be described as the ideal generated by all square-free monomials of degree a in A, while for a > d we have ma = 0. We compute the Betti numbers of this family of ideals using Corollary 5.5 as follows βn,n+a (ma ) =

   a  n+i+d −1 d (−1)i+1 d −1 a−i i=1

(6.1)

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  n+i+d −1 is equal to (−1)n−1 times the coefficient of t n+i d −1   d 1 in the Taylor expansion of the rational function (1+t)d around 0. Similarly, a−i 1 is the coefficient of t a−i in the binomial expansion of (1 + t)d . Since (1+t) d · (1 + Note that (−1)i+1

t)d = 1, for n + a > 0, the coefficient of t n+a in their product is 0, i.e. a 

(−1)

n+i

i=−n

   n+i+d −1 d = 0. d −1 a−i

However, when i < a − d, the second binomial coefficient is 0, so this can be restated as a 

(−1)

n+i

i=a−d

   n+i+d −1 d = 0. d −1 a−i

Combined with (6.1), the identity above leads to the more compact formula  0 βn,n+a (m ) =

i=a−d (−1)

a

! d ! i n+i+d−1 d−1 a−i

1ad a  d + 1.

0

This is consistent with ma = 0 for a > d and can be easier to evaluate than (6.1) for some values of a. For example, setting a = d yields βn,n+d (md ) =

  n+d −1 . d −1

Acknowledgments Our work started at the 2019 workshop “Women in Commutative Algebra” hosted by Banff International Research Station. We thank the organizers of this workshop for bringing our team together. We acknowledge the excellent working conditions provided by BIRS and the support of the National Science Foundation for travel through grant DMS-1934391. We thank the Association for Women in Mathematics for funding from grant NSF-HRD 1500481. In addition, we have the following individual acknowledgements for support: Faber was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 789580. Miller was partially supported by the NSF DMS-1003384. R.G.’s travel was partially supported by an AMS-Simons Travel Grant. Seceleanu was partially supported by NSF DMS-1601024. We thank Liana Sega ¸ for helpful comments and for bringing [8] to our attention and Ben Briggs for answering a question and pointing us to [26]. Lastly, we thank the referee for a thorough reading and especially for guiding us to other points of view, cf. Remarks 3.3 and 3.4, as well as for pointing out the papers [15] and [20].

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References 1. Luchezar L. Avramov, (Contravariant) Koszul duality for DG algebras, Algebras, quivers and representations, Abel Symp., vol. 8, Springer, Heidelberg, 2013, pp. 13–58. 2. A. K. Bousfield, E. B. Curtis, D. M. Kan, D. G. Quillen, D. L. Rector, and J. W. Schlesinger, The mod − p lower central series and the Adams spectral sequence, Topology 5 (1966), 331– 342. 3. David A. Buchsbaum and David Eisenbud, Generic free resolutions and a family of generically perfect ideals, Advances in Math. 18 (1975), no. 3, 245–301. 4. I. N. Bernšte˘ın, I. M. Gel’fand, and S. I. Gel’fand, Algebraic vector bundles on Pn and problems of linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 66–67. 5. A. A. Be˘ılinson, V. A. Ginsburg, and V. V. Schechtman, Koszul duality, J. Geom. Phys. 5 (1988), no. 3, 317–350. 6. Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), no. 2, 473–527. 7. Aldo Conca, Koszul algebras and their syzygies, Combinatorial algebraic geometry, Lecture Notes in Math., vol. 2108, Springer, Cham, 2014, pp. 1–31. 8. Liana M. Sega, ¸ Homological properties of powers of the maximal ideal of a local ring, J. Algebra 241 (2001), no. 2, 827–858. 9. Aldo Conca, Ngô Viêt Trung, and Giuseppe Valla, Koszul property for points in projective spaces, Math. Scand. 89 (2001), no. 2, 201–216. 10. W. G. Dwyer, J. P. C. Greenlees, and S. Iyengar, Duality in algebra and topology, Adv. Math. 200 (2006), no. 2, 357–402. 11. David Eisenbud, Gunnar Fløystad, and Frank-Olaf Schreyer, Sheaf cohomology and free resolutions over exterior algebras, Trans. Amer. Math. Soc. 355 (2003), no. 11, 4397–4426. 12. R. Fröberg, Koszul algebras, Advances in commutative ring theory (Fez, 1997), Lecture Notes in Pure and Appl. Math., vol. 205, Dekker, New York, 1999, pp. 337–350. 13. Federico Galetto, On the ideal generated by all squarefree monomials of a given degree, J. Commut. Algebra 12 (2020), no. 2, 199–215. 14. Mark Goresky, Robert Kottwitz, and Robert MacPherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998), no. 1, 25–83. 15. Edward L. Green and Roberto Martínez Villa, Koszul and Yoneda algebras, Representation theory of algebras (Cocoyoc, 1994), CMS Conf. Proc., vol. 18, Amer. Math. Soc., Providence, RI, 1996, pp. 247–297. 16. Yu. I. Manin, Quantum groups and noncommutative geometry, Université de Montréal, Centre de Recherches Mathématiques, Montreal, QC, 1988. 17. J. P. May, The cohomology of restricted Lie algebras and of Hopf algebras, J. Algebra 3 (1966), 123–146. 18. Jason McCullough and Irena Peeva, Infinite graded free resolutions, Commutative algebra and noncommutative algebraic geometry. Vol. I, Math. Sci. Res. Inst. Publ., vol. 67, Cambridge Univ. Press, New York, 2015, pp. 215–257. 19. Claudia Miller and Hamidreza Rahmati, Free resolutions of Artinian compressed algebras, J. Algebra 497 (2018), 270–301. 20. Roberto Martínez-Villa and Dan Zacharia, Approximations with modules having linear resolutions, J. Algebra 266 (2003), no. 2, 671–697. 21. Leonid Positselski, Galois cohomology of a number field is Koszul, J. Number Theory 145 (2014), 126–152. 22. Alexander Polishchuk and Leonid Positselski, Quadratic algebras, University Lecture Series, vol. 37, American Mathematical Society, Providence, RI, 2005. 23. Stewart B. Priddy, Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970), 39–60. 24. John Tate, Homology of Noetherian rings and local rings, Illinois J. Math. 1 (1957), 14–27. 25. K. VandeBogert, Iterated mapping cones for strongly Koszul algebras, arXiv:2104.00037.

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Well Ordered Covers, Simplicial Bouquets, and Subadditivity of Betti Numbers of Square-Free Monomial Ideals Sara Faridi and Mayada Shahada

1 Introduction This paper grew out of investigations into the subadditivity property of syzygies of monomial ideals. For a homogeneous ideal I of a polynomial ring S, suppose the maximum degree j such that βi,j (S/I ) = 0 is denoted by ti . The subadditivity property is said to hold if we have ta+b  ta + tb for all positive values of a and b. While the subadditivity property or related inequalities are known to hold in many special cases—certain cases for ideals of codimension  1 ([5, Corollary 4.1]); some Koszul rings [1]; when I monomial ideal and a = 1 [15]; certain homological degrees for Gorenstein algebras [6]; when a = 1, 2, 3 and I monomial ideal generated in degree 2 [2, 11]; facet ideals of simplicial forests [9]—the problem is open for monomial ideals and is known to fail (see Caviglia’s example in [5, Example 4.5]) for general homogeneous ideals. In the case of monomial ideals, Betti numbers can be calculated as ranks of homology modules of topological objects. In particular, the order complex of the lcm lattice of I (the poset of least common multiples of the minimal monomial generating set of I ordered by division) can be used for this purpose. A nonvanishing Betti number, in this context, corresponds by a result of Baclawski [3] to a “complemented” lcm lattice (see Sect. 4).

S. Faridi () Department of Mathematics & Statistics, Dalhousie University, Halifax, NS, Canada e-mail: [email protected] M. Shahada Department of Mathematics, College of Science, University of Bahrain, Sakheer, Kingdom of Bahrain e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Miller et al. (eds.), Women in Commutative Algebra, Association for Women in Mathematics Series 29, https://doi.org/10.1007/978-3-030-91986-3_12

303

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S. Faridi and M. Shahada

This approach was initiated by the first author in [9]: the existence of complements which have nonvanishing Betti numbers in the “right” homological degrees implies the subadditivity property (Question 4.2). This approach was further pursued by both authors in [10], where Question 4.2 was translated into the context of homological cycles in simplicial complexes breaking into smaller ones. This paper takes yet a different angle. The idea of complements really comes down to the following: take a monomial m in the lcm lattice of I . Among all the complements of m in the lcm lattice, can you pick one, say m , which behaves desirably? If m is a square-free monomial, this question is even simpler: m and m correspond to subsets A and A of the variables {x1 , . . . , xn } such that A ∪ A = {x1 , . . . , xn } and the product of the variables in A ∩ A is not in I . Can we consider the subideals induced on A and A and extract properties from/for them? The main object of study using this approach will be “well ordered covers” of ideals (Definition 3.2). The existence of a well ordered cover of size i is known, via the Lyubeznik resolution, to guarantee a nonvanishing i th Betti number [8]. In this paper we investigate when complements in the lcm lattice produce subideals with well ordered covers of sizes a and b with a + b = i, in order to result in the subadditivity property. Moreover, we introduce strongly disjoint sets of simplicial bouquets, which we show are always well ordered covers, and we demonstrate how they can be broken up to prove subadditivity in certain homological degrees. An advantage of simplicial bouquets is that they are rather easy to spot in simplicial complexes, as they do not rely on an ordering. Rather, one or more orderings are inherent in the definition (see Theorem 5.4). In Sect. 2 we set up the background on simplicial resolutions. Section 3 introduces the reader to well ordered covers of monomials. Section 4 describes the subadditivity property and contains one of the main results of the paper (Theorem 4.7) which considers well ordered covers under complementation. In Sect. 5 we introduce simplicial bouquets and show that certain types of simplicial bouquets are well ordered facet covers. We then apply the results of Sect. 4 to simplicial complexes that contain strongly disjoint sets of bouquets, and show that the subadditivity property holds in degrees that come from the sizes of the bouquets (Theorem 5.7). Section 6 offers ways to optimize the order of monomials in a well ordered cover to get the best possible subadditivity results. With some numerical manipulation, the results of this paper can be adapted to non-square-free monomial ideals via polarization [12], a method that transforms a monomial ideal into a square-free one which retains many of the algebraic properties of the original ideal, including the minimal free resolution.

Well Ordered Covers, Simplicial Bouquets, and Subadditivity of Betti Numbers. . .

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2 Background Throughout let S = k[x1 , . . . , xn ] be a polynomial ring over a field k.

2.1 Simplicial Complexes and Facet Ideals A simplicial complex  on a finite vertex set V () is a set of subsets of V () such that {v} ∈  for every v ∈ V () and if F ∈ , then for every G ⊆ F , we have G ∈ . The elements of  are called faces; the maximal faces with respect to inclusion are called facets, and a simplicial complex contained in  is called a subcomplex of . The set of all facets in  defines  and is denoted by Facets(). If Facets() = {F1 , . . . , Fq }, then we write  = F1 , . . . , Fq . A subcollection of  is a subcomplex of  whose facets are also facets of . If A ⊆ V (), then the induced subcollection [A] is the simplicial complex defined as [A] = F ∈ Facets() : F ⊆ A . We say a facet F contains a free vertex v if F is the only facet of  containing v. Given a simplicial complex  on vertices {x1 , . . . , xn }, we can define the facet ideal of  as  ! xi : F is a facet of  F() = xi ∈F

which is an ideal of S = k[x1 , . . . , xn ]. Conversely, given a square-free monomial ideal I ⊂ S, the facet complex of I is the simplicial complex  6 7 F(I ) = F : xi is a generator of I . xi ∈F

Example 2.1 For I = (xy, yz, zu), the simplicial complex F(I ) is below.

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2.2 Simplicial Resolutions Any monomial ideal I of S = k[x1 , . . . , xn ] admits a minimal graded free resolution, which is an exact sequence of free S-modules 0 → ⊕j ∈N S(−j )βp,j → ⊕j ∈N S(−j )βp−1,j → · · · → ⊕j ∈N S(−j )β1,j → S. For each i and j , the rank βi,j (S/I ) of the free S-modules appearing above are called the graded Betti numbers of the S-module S/I , and the total Betti number in homological degree i is βi (S/I ) =



βi,j (S/I ).

j

If I is generated by monomials, the graded Betti numbers can be further refined into sums of multigraded Betti numbers. For a monomial m in S, the multigraded Betti number of S/I is of the form βi,m (S/I ) and we have βi,j (S/I ) =



βi,m (S/I )

(2.1)

where the sum is taken over all monomials m of degree j that are least common multiples of subsets of the minimal monomial generating set of I . The multigraded Betti numbers βi,m (S/I ) are related to the combinatorics of the ideal I . Given a monomial ideal I minimally generated by m1 , . . . , mq , one can consider a simplicial complex  on q vertices {v1 , . . . , vq }, where each vertex vi is labeled with the monomial generator mi , and each face τ of  is labeled with the monomial ! lcm(τ ) = lcm mi : vi ∈ τ . We say that  supports a free resolution of I if the simplicial chain complex of  can be “homogenized”, using the monomial labels of the faces, to produce a free resolution of I . For details of homogenization of a chain complex see [20, Section 55]. The resulting free resolution is called a simplicial resolution. Taylor’s resolution ([21], see also [20, Construction 26.5]) is an example of a simplicial resolution where the underlying simplicial complex is a full simplex T(I ) over the vertex set labeled with the monomial generators {m1 , . . . , mq } of I , called the Taylor complex of I . It is known that all simplicial complexes supporting a free resolution of I are subcomplexes of the Taylor complex, in other words: all simplicial resolutions are contained in the Taylor resolution ([19]). If  is a simplicial complex supporting a free resolution of I , and m is a monomial in S, the simplicial subcomplex  1, then

(6.5)

Well Ordered Covers, Simplicial Bouquets, and Subadditivity of Betti Numbers. . .

323

(1) for every i ∈ {2, . . . , } the sequence mi , mi+1 , . . . , ms , m1 , . . . , mi−1 is a well ordered cover of I ; (2) ts  ts−i + ti for 1  i  − 1. Proof Statement 1 follows directly from the definition of well ordered covers. For the second statement, by part 1, for 1  i < , we have mi+1 , . . . , ms , m1 , . . . , mi is a well ordered cover for I . By (6.5) mi+1 , . . . , ms is a well ordered cover of I[m] where m = lcm(mi+1 , . . . , ms ). By Proposition 4.5 m1 , . . . , mi is a well ordered cover of I[m ] where m = lcm(m1 , . . . , mi ). By Proposition 4.4, m and m are complements in LCM(I ), and by Theorem 3.4 βs−i,m = 0

and

βi,m = 0,

which together imply that ts  ts−i + ti .



Example 6.3 The ideal I from Example 5.3 has the following well ordered cover M : gy , gx , ge , gf , bcd , gh , gi , abc. In particular β8,11 (S/I ) = 0 and t8 = 11. The reordering in Proposition 6.2 of M yields the following well ordered cover M : gf , bcd , gh , gi , abc , gy , gx , ge, where = 4. By part 2 of Proposition 6.2 and using the new well ordered cover M , we have: (1) t8  t7 +t1 . Here we take m = abcdf ghixy = lcm(gf, bcd, gh, gi, abc, gy, gx) and m = ge.

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(2) t8  t6 + t2 . Here we take m = abcdf ghiy = lcm(gf, bcd, gh, gi, abc, gy) and m = gex = lcm(gx, ge). Note that this inequality was also done in Example 5.9 using simplicial bouquets. (3) t8  t5 + t3 . Here we take m = abcdf ghi = lcm(gf, bcd, gh, gi, abc) and m = gexy = lcm(gy, gx, ge). The only remaining case is a = b = 4. If we take m = bcdfghi = lcm(gf, bcd, gh, gi)

and

m = abcgexy = lcm(abc, gy, gx, ge),

then {abc, gy, gx, ge} is a well ordered cover of I[m ] by Proposition 4.5, and it is easy to check that {gf, bcd, gh, gi} is a well ordered cover of I[m] . Hence β4,m (S/I ) = 0 and β4,m (S/I ) = 0 by Theorem 3.4. Also as m and m are lattice complements (by Proposition 4.4), we get t4 + t4  deg(m) + deg(m ) > 11 = t8 . Acknowledgments The authors are grateful to the referees for their comments, which greatly improved this paper. Sara Faridi’s research is supported by an NSERC Discovery Grant.

References 1. Avramov, L., Conca, A., Iyengar, S.: Subadditivity of syzygies of Koszul algebras. Math. Ann., 361, no.1–2, 511–534 (2015) 2. Abedelfatah, A., Nevo, E.: On vanishing patterns in j -strands of edge ideals. J. Algebraic Combin. 46, no. 2, 287–295 (2017) 3. Baclawski, K.: Galois connections and the Leray spectral sequence. Advances in Math. 25, no. 3, 191–215 (1977) 4. Bayer, D., Sturmfels, B.: Monomial resolutions. Math. Res. Lett. 5, no. 1–2, 31–46 (1998) 5. Eisenbud, D., Huneke, C., Ulrich, B.: The regularity of Tor and graded Betti numbers. Amer. J. Math. 128, no. 3, 573–605 (2006) 6. El Khoury, S., Srinivasan, H.: A note on the subadditivity of Syzygies. Journal of Algebra and its Applications, vol.16, no.9, 1750177 (2017) 7. Erey, N.: Bouquets, Vertex Covers and Edge Ideals. Journal of Algebra and its Applications, vol.16, no.5, 1750084 (2017) 8. Erey, N., Faridi, S.: Betti numbers of monomial ideals via facet covers. J. Pure Appl. Algebra 220, no. 5, 1990–2000 (2016) 9. Faridi, S.: Lattice complements and the subadditivity of syzygies of simplicial forests. Journal of Commutative Algebra, Volume 11, Number 4, 535–546 (2019) 10. Faridi, S., Shahada, M.: Breaking up Simplicial Homology and Subadditivity of Syzygies. J. Algebraic Combin., (2021). https://doi.org/10.1007/s10801-021-01073-3 11. Fernandez-Ramos, O., Gimenez, P.: Regularity 3 in edge ideals associated to bipartite graphs. J. Algebraic Combin., 39 (2014) 12. Fröberg, R.: On Stanley-Reisner rings. Topics in algebra, Banach Center Publications, 26 Part 2, 57–70 (1990) 13. Grayson, D., Stillman, M. : Macaulay2, a software system for research in algebraic geometry. available at http://www.math.uiuc.edu/Macaulay2/. 14. Ha, H. T., Van Tuyl, A.: Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers. J. Algebraic Combin., 27, 215–245 (2008) 15. Herzog, J., Srinivasan, H.: On the subadditivity problem for maximal shifts in free resolutions. Commutative Algebra and Noncommutative Algebraic Geometry, II MSRI Publications

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Volume 68, (2015) 16. Khosh-Ahang, F., Moradi, S.: Codismantlability and projective dimension of the StanleyReisner ring of special hypergraphs. Proc. Indian Acad. Sci. Math. Sci. 128, no. 1, Paper No. 7, 10 pp (2018) 17. Kimura, K.: Non-Vanishingness of Betti Numbers of Edge Ideals. Harmony of Grobner Bases and the Modern Industrial Society, World Scientific, Hackensack 153–168 (2012) 18. Lyubeznik, G.: A new explicit finite free resolution of ideals generate by monomials in an R-sequence. J. Pure Appl. Algebra 51, 193–195 (1998) 19. Mermin, J.: Three simplicial Resolutions. Progress in commutative algebra 1, 127–141, de Gruyter, Berlin (2012) 20. Peeva, I.: Graded syzygies. Algebra and Applications, 14, Springer-Verlag London, Ltd., London (2011) 21. Taylor, D.: Ideals generated by monomials in an R-sequence, Thesis, University of Chicago (1966) 22. Zheng, X.: Resolutions of facet ideals. Comm. Algebra, 32, 2301–2324 (2004)

A Survey on the Eisenbud-Green-Harris Conjecture Sema Güntürkün

Keywords Hilbert functions · Lex-plus-powers ideals · Lexicographic ideals · Regular sequences

1 Introduction Let I be a homogeneous ideal given in a standard graded polynomial ring R in n variables over a field K, and let Id denote the degree d graded component of I. Assuming the K-dimension of Id is known, it sounds a quite simple question to ask what one can say about the dimension of the graded component of I in degree d + 1, and yet it attracts a lot of attention in commutative algebra and algebraic geometry. An answer to this question was given by Macaulay’s breakthrough work [34] by providing a numerical bound for the growth of Hilbert function depending on the value at the preceding degree. He showed that Hilbert functions of special monomial ideals, called lexicographic ideals, describe all possible Hilbert functions of homogeneous ideals in R. Macaulay’s result led to other classical results on Hilbert functions such as Gotzmann’s Persistence Theorem and Green’s Hyperplane Restriction Theorem (see [4] for nice treatments of all these theorems). Generalizations of Macaulay’s result on the extremal behavior of lexicographic ideals for Hilbert functions allows to relate a homogeneous ideal containing the powers of variables with a monomial ideal containing the same powers of variables (see [13, 30, 31]). In their Higher Castelnuovo Theory paper, Eisenbud, Green and Harris conjectured a further generalization of Macaulay’s theorem for homogeneous ideals containing a regular sequence in certain degrees (see Conjecture 3.3). EisenbudGreen-Harris (EGH) conjecture, motivated by Cayley-Bacharach theorems, suggests a refinement of Macaulay’s bound on the growth of the Hilbert function

S. Güntürkün () Amherst College, Department of Mathematics and Statistics, Amherst, MA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Miller et al. (eds.), Women in Commutative Algebra, Association for Women in Mathematics Series 29, https://doi.org/10.1007/978-3-030-91986-3_13

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by involving the information of degrees of the regular sequence contained in the ideal minimally. Although there are notable works done on Eisenbud-Green-Harris (EGH) conjecture, it is still widely open after more than 25 years. A survey on lex-plus-powers ideals by Francisco and Richert [19] provides a very good source to understand these special monomial ideals thoroughly, and it also discusses the EGH conjecture and its equivalent variations in details. Since there have been significant progress on the EGH conjecture since [19], the main intent of our survey paper is to contribute the literature by providing the current state of the EGH conjecture and to assemble the results that have been obtained so far. As plan of this paper, in Sect. 2 we state some preliminaries and review Macaulay’s results on Hilbert function. Section 3 lays out the Eisenbud-GreenHarris (EGH) conjecture and its variations. In Sect. 4, we present the results obtained on the EGH conjecture by grouping them in terms of their approaches. Finally, in Sect. 5 we point out the open cases of the EGH conjecture, and we recall a closely related conjecture known as the Lex-Plus-Powers conjecture. We conclude the final section with some applications of EGH.

2 Preliminaries and Macaulay’s Theorem on Hilbert Functions We let R be the polynomial ring K[x1 , . . . , xn ] over a field K with standard grading R = ⊕i0 Ri where Ri is the i-th graded component. We fix the lexicographic order as x1 >lex x2 >lex . . . >lex xn . Then we define the monomial order between two monomials of the same degree as x1a1 x2a2 · · · xnan >lex x1b1 x2b2 · · · xnbn if ai > bi where i is the smallest index such that ai = bi . For the sake of simplicity, we use > for >lex . Definition 2.1 Let I be a monomial ideal in R minimally generated by monomials m1 , . . . , mk . We call I a lexicographic ideal or simply a lex ideal if it satisfies the following property: for any monomial m in R with deg m = deg mi and m > mi for some i = 1, . . . , k, then m ∈ I as well.

We next define another special type of monomial ideal in our context. Definition 2.2 For given 2  a1  a2  . . .  an , we call a monomial ideal L a lex-plus-powers ideal associated with degree (a1 , . . . , an ) if it can be written as L = (x1a1 , . . . , xnan ) + J where J is a lex ideal in R. For any homogeneous ideal I we define the Hilbert function of R/I as HR/I (t) = dimK (R/I )t = dimK Rt − dimK It



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where Rt is the degree t graded component of R with dimK Rt = the degree t graded component of the ideal I .

n−1+t ! , t

and It is

Example 2.3 In K[x, y, z], consider the monomial ideal I = (x 3 , xy, y 4 , yz, z2 ). Then simple computations give us the graded components of R/I ; (R/I )1 = K − Span{x, y, z}, (R/I )2 = K − Span{x 2 , xz, y 2 }, (R/I )3 = K − Span{x 2 z, y 3 } and dimK (R/I )i = 0 for i  4. Therefore, one expresses the Hilbert function of R/I as (h0 , h1 , h2 , h3 ) = (1, 3, 3, 2) where hi = HR/I (i) and hi = 0 for i  4.

As we see in the above example, it is possible to compute the Hilbert function of monomial ideals even by hand, however for arbitrary homogeneous ideals it becomes challenging to calculate without using a software such as Macaulay 2 (see [33] package LexIdeals, command hilbertFunct). We call a sequence f1 , . . . , fr of forms in R = K[x1 , . . . , xn ] a regular sequence of length r if, for each i = 1, . . . , r, fi is a non-zero-divisor on the ring R/(f1 , . . . , fi−1 ). If r = n, then the regular sequence has full length, in this case, it is referred as a maximal regular sequence. Remark 2.4 Let c be a homogeneous ideal in R generated by a regular sequence f1 , . . . , fn with deg fi = ai for i = 1, . . . , n. We call c a complete intersection ideal of type (a1 , . . . , an ) and the ring R/c is called a complete intersection ring. Then the Hilbert function of R/c is HR/c (i) = HR/(x a1 ,...,x an ) (i) for all i  0. n 1 n  Furthermore, HR/(x a1 ,...,x an ) (i) = HR/(x a1 ,...,x an ) (s − i) where s = (ai − 1). 1

n

1

n

i=1



The following proposition provides a very useful relation between the Hilbert function of an ideal I and the Hilbert function of another ideal generated by a regular sequence contained in I under the liaison (see [15, Theorem 3]). Proposition 2.5 Let I be a homogeneous ideal and c ⊆ I a complete intersection n  ideal, and s = (ai − 1). Then, for all j  0, i=1

HR/I (j ) = HR/c (j ) − HR/(c:I ) (s − j ) For a given two positive integers d and i, the i-th Macaulay representation of d (also known as the i-th binomial expansion of d), denoted d (a) , is given by         mi−1 m2 m1 mi + + ... + + d= i i−1 2 1 where mi > mi−1 > . . . > m1  0 are uniquely determined and are called the i-th Macaulay coefficients of d. In this case, we let d

i

        mi−1 + 1 m2 + 1 m1 + 1 mi + 1 + + ... + + . := i+1 i 3 2

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! ! ! ! To give a simple example, let d = 21, i = 4. Then 21(4) = 64 + 43 + 22 + 11 , ! ! ! ! therefore 214 = 75 + 54 + 33 + 22 = 28. We next state Macaulay’s well-known theorem on Hilbert functions. Theorem 2.6 ([4, 34]) Let I be a homogeneous ideal in the polynomial ring R. (a) There is a lex ideal in R with the same Hilbert function, and this lex ideal is uniquely determined. (b) [Macaulay’s bound] If HR/I (i) = d then HR/I (i + 1)  d i . ! Example 2.7 Notice that HR (i) = n+i−1 = d and so the i-th Macaulay i ! ! . Computing HR (i + 1) = n+i representation of d is d (i) = n−1+i i i+1 shows that R attains exactly Macaulay’s bound d i .

The following important theorem shows when a homogeneous ideal carries a similar behavior of attaining Macaulay’s bound as in Example 2.7. Theorem 2.8 (Gotzmann’s Persistence Theorem [4, 24]) Let I be a homogeneous ideal in R generated by forms of degree  d. If the Hilbert function of R/I achieves Macaulay’s bound in the next degree d + 1, that is HR/I (d + 1) = HR/I (d)d , then HR/I (j ) = HR/I (j − 1)d

for all j  d.

Another classical result on the growth of Hilbert functions worth to mention is given by Green. This result was also used to give an elegant proof of Macaulay’s theorem. Theorem 2.9 (Green’s Hyperplane Restriction Theorem [4, 22]) Let I be a homogeneous ideal in R, and let t  1 be a given degree. Then         mt−1 − 1 m2 − 1 m1 − 1 mt − 1 HR/I +( ) (t)  + + ... + + t t −1 2 1 where  HR/I (t)(t) =

mt t



 +

     mt−1 m2 m1 + ... + + t −1 2 1

is the t-th Macaulay representation of HR/I (t).



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3 The EGH Conjecture A generalization of Macaulay’s results was considered by studying homogeneous ideals in S = R/(x1a1 , . . . , xnan ) instead of homogeneous ideals in the polynomial ring R = K[x1 , . . . , xn ]. The existence of the lex ideal in S with the same Hilbert function was shown by Kruskal [31] and Katona [30] when a1 = a2 = . . . = an = 2 and more generally when 2  a1  . . .  an was done by Clements and Lindström [13] and also by Greene-Kleitman [23]. These results were obtained in set theoretical and combinatorial settings. A question can be raised for a similar behavior for the homogeneous ideals in R/(f1 , . . . , fn ) where f1 , . . . , fn is a regular sequence in R. In [16], Eisenbud, Green and Harris initially stated the following conjecture for the case of regular sequence of quadrics. Conjecture 3.1 Given homogeneous ideal I in R containing a full length regular sequence of quadratic forms. Let HR/I (i) = h and the i-th Macaulay representation of h be         mi−1 m2 m1 mi h(i) = + + ... + + i i−1 2 1 where mi > mi−1 > . . .  m1  0. Then       mi−1 m1 mi + + ... + . HR/I (i + 1)  i+1 i 2 The new bound proposed by Conjecture 3.1 finer Macaulay’s bound. We ! is m ! than m ! can see this using the binomial identity m+1 = + k k k−1 , h

i

      mi−1 + 1 m1 + 1 mi + 1 + + ... + = i+1 i 2             mi mi−1 mi−1 m1 m1 mi + + + + ... + + = i+1 i i i−1 2 1       mi−1 m1 mi + + ... + = h(i) + i+1 i 2       mi−1 m1 mi + + ... + . > i+1 i 2

Example 3.1 Suppose I ⊆ K[x1 , . . . , x7 ] is a homogeneous ideal containing a regular sequence of quadratic forms f1 , . . . , f7 and HR/I (2) = 17. The possible growth for the Hilbert function in degree 3 by Macaulay’s bound is HR/I (3)  38, but Conjecture 3.1 claims that HR/I (3)  21.



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If the homogeneous ideal I is generated by generic quadrics, it is already known that Conjecture 3.1 is true by Herzog and Popescu [27] when the characteristic is zero. When K has arbitrary characteristic, this was shown by Gasharov [20]. The main motivation behind Conjecture 3.1 about homogeneous ideals containing quadratic regular sequence was another conjecture, known as the Generalized Cayley-Bacharach conjecture, stated in [16] in more geometric perspective. Conjecture 3.2 (Generalized Cayley-Bacharach Conjecture for Quadrics) Let  be a complete intersection of quadrics in Pn . Any hypersurface X ⊂ Pn of degree d containing a subscheme  ⊂  of degree strictly greater than 2n − 2n−d must contain .

Conjecture 3.1 implies the Generalized Cayley-Bacharach conjecture for quadrics. In the same article, Eisenbud, Green and Harris dropped the quadratic condition on the regular sequence, and further conjectured the same statement for homogeneous ideals containing regular sequences with any degrees 2  a1  a2 . . .  an . Conjecture 3.3 (Eisenbud-Green-Harris (EGH) Conjecture, [16]) Let I be a homogeneous ideal in R = K[x1 , . . . , xn ] containing a regular sequence f1 , . . . , fn with degrees a1 , . . . , an such that 2  a1  . . .  an . Then there is a lex-pluspowers ideal L = (x1a1 , . . . , xnan ) + J with a lex ideal J in R such that HR/I (i) = HR/L (i)

for all

i  0.

From now on, we will refer to this conjecture as the EGH conjecture. We will also use EGH(a1 ,...,an ),n to emphasize the degrees of the regular sequence and also that it is a full length-n regular sequence in R = K[x1 , . . . , xn ]. Notice that in Remark 2.4, we observed a very trivial version of this statement when I = (f1 , . . . , fn ) and L = (x1a1 , . . . , xnan ). Another statement of the Generalized Cayley-Bacharach conjecture that does not require quadrics was given in [17, Conjecture CB12]. In 2013, Geramita and Kreuzer reformulated this version of Generalized Cayley-Bacharach conjecture for arbitrary degrees by dividing it into intervals. They also strengthened the Conjecture CB12 in [21, Conjecture 3.5]. In P3 , they provided a proof for it. In Pn , they confirmed [21, Conjecture 3.5] for some intervals. The EGH conjecture which is the concern of this paper is stronger than [17, Conjecture CB12] as well. One of the variations of the EGH conjecture in the literature is when one allows to have a regular sequence that is not of full length. Conjecture 3.4 (EGHn,(a1 ,...,ar ),r ) Let I be a homogeneous ideal in R = K[x1 , . . . , xn ] containing a regular sequence of length r < n with degrees a1 , . . . , ar such that 2  a1  . . .  ar . Then there is a lex-plus-powers ideal L = (x1a1 , . . . , xrar ) + J in R with the same Hilbert function as I .

Remark 3.2 The equivalence between Conjectures 3.3 and 3.4 was discussed by Caviglia and Maclagan [9]. For 2  a1  . . .  ar fixed, they showed that if EGH(a1 ,...,an ),n holds for all 2  a1  . . .  an where ai = ai for i = 1, . . . , r

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then EGHn,(a1 ,...,ar ),r holds (see [9, Propositions 9]). They also showed that if EGH(a1 ,...,ar ),r holds then EGHn,(a1 ,...,ar ),r holds for all n  r (see [9, Propositions 10]).



4 Results on the EGH Conjecture Richert [37] proved that the EGH conjecture is true for R = K[x, y]. Thus, for any homogeneous ideal I in two variables containing a regular sequence f1 , f2 with degrees 2  a1  a2 , we have a lex plus powers ideal L = (x a1 , y a2 ) + J such that dimK Ii = dimK Li

for all

i  0.

For K[x1 , . . . , xn ] with n > 2, we put together the known results on EGH depending on the approaches were used.

4.1 EGH Depending on the Degrees (a1 , . . . , an ) Let n  2. For a fixed degree d  1, when the Hilbert function of R/I at degree d is known, the EGH conjecture proposes a maximal growth for degree d +1. One of the adopted approaches in the literature focuses on the growth at certain degree. Hence, the following definition states a partial version of EGH conjecture for consecutive degrees. Definition 4.1 (EGH(a1 ,...,an ),n (d)) For any homogeneous ideal I in R containing a regular sequence f1 , . . . , fn of degrees 2  a1  . . .  an respectively, if there exists a lex-plus-powers ideal L associated with degrees a1 , . . . , an such that HR/I (d) = HR/L (d)

and

then we say that EGH(a1 ,...,an ),n (d) holds.

HR/I (d + 1) = HR/L (d + 1),



The following proposition is given by Francisco [18] for the almost complete intersection ideals. Proposition 4.2 Let I = (f1 , . . . , fn , g) be a homogeneous ideal where f1 , . . . , fn is a regular sequence with degrees a1 , . . . , an and deg g = d  a1 . Then EGH(a1 ,...,an ),n (d) is true for I .

In [6], for an almost complete intersection I = (f1 , f2 , f3 , g) ⊂ K[x1 , . . . , xn ] where f1 , f2 , f3 is a regular sequence of length three with deg fi = ai , i = 1, 2, 3 and deg g = d  a1 +a2 +a3 −3, Caviglia-De Stefani showed HR/I (i)  HR/L (i) for all i  0 where L = (x1a1 , x2a2 , x3a3 , m) with m is the largest monomial of degree

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d with respect to lexicographic order that is not in (x1a1 , x2a2 , x3a3 ). Their work on such almost complete intersections also recovered the result of [21] for P3 . To show that EGH(d,...,d),n (d) holds it suffices to show that the statement in Definition 4.1 holds for the homogeneous ideals I = (f1 , . . . , fn , g1 , . . . , gm ) ⊆ K[x1 , . . . , xn ] generated by degree d forms f1 , . . . , fn , g1 , . . . , gm where f1 , . . . , fn form a regular sequence (see [25, Lemma 2.6].) In other words, it is enough to show the statement for the ideals where not only the regular sequence f1 , . . . , fn in the generators have degree d, but also rest of the generators g1 , . . . , gm have degree d too. Thus, focusing on the case d = 2 we have the following remark. Remark 4.3 In order to show that EGH(2,...,2),n (2) is true, it suffices to study the ideals generated by only quadrics containing a maximal regular sequence.

In [9], Caviglia-Maclagan provided the following lemma about this weaker version of the EGH. Due to its importance as a tool for studying the EGH conjecture, we would like to present its proof given in [9, Lemma 12]. Lemma 4.4 Given 2  a1  . . .  an , set s =

n 

(ai − 1). Let d  1. Then i=1 EGH(a1 ,...,an ),n (d) holds if and only if EGH(a1 ,...,an ),n (s − d − 1) holds. Furthermore, if EGH(a1 ,...,an ),n (d) holds for all 0  d   s−1 2  then EGH(a1 ,...,an ),n



holds.

Proof Suppose that EGH(a1 ,...,an ),n (d) holds. Then given any homogeneous ideal containing a regular sequence with degrees a1 , . . . , an , there is lex-plus-powers ideal associated with degrees a1 , . . . , an such that Hilbert functions of both ideals agree at degrees d and d +1. Let I be a homogeneous ideal in R containing a regular sequence f1 , . . . , fn with deg fi = ai , for i = 1, . . . , n. Then by Proposition 2.5 we get HR/I (j ) = HR/(f1 ,...,fn ) (j ) − HR/((f1 ,...,fn ):I ) (s − j ). Since the colon ideal ((f1 , . . . , fn ) : I ) contains the regular sequence f1 , . . . , fn , then by assumption there is a lex-plus-powers ideal L = (x1a1 , . . . , xnan ) + J  such that HR/((f1 ,...,fn ):I ) (d) = HR/L (d) and HR/((f1 ,...,fn ):I ) (d + 1) = HR/L (d + 1). (1) On the other hand, we also have HR/L (j ) = HR/(x a1 ,...,x an ) (j ) − HR/((x a1 ,...,x an ):L ) (s − j ). 1

n

1

Thus, for j = s − d − 1 and j = s − d, HR/I (j ) = HR/(f1 ,...,fn ) (j ) − HR/((f1 ,...,fn ):I ) (s − j )

n

(2)

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= HR/(f1 ,...,fn ) (j ) − HR/(x a1 ,...,x an ) (s − j ) + HR/((x a1 ,...,x an ):L ) (j ) 1

n

n

1

= HR/((x a1 ,...,x an ):L ) (j ). 1

n

The second equality follows from (1) and (2). Then the last equality is by Remark 2.4 and Proposition 2.5. Finally, since the ideal ((x1a1 , . . . , xnan ) : L ) contains the regular sequence a1 x1 , . . . , xnan , by Clement-Lindström result mentioned previously, there exists a lex-plus-powers ideal L associated with degrees a1 , . . . , an such that HR/((x a1 ,...,x an ):L ) (i) = HR/L (i) for all i  0. n 1 Hence,

HR/I (s − d − 1) = HR/L (s − d − 1) and HR/I (s − d) = HR/L (s − d). Remark 4.5 We know HR/I (0) = 1. If HR/I (1) = n, then any lex-plus-powers ideal L = (x1a1 , . . . , xnan ) + J where the lex ideal J does not contain any linear form has HR/L (0) = 1 and HR/L (1) = n. If HR/I (1) = r < n, that is I has n − r linear generators, then it is enough to pick the lex ideal J containing x1 , . . . , xn−r , then HR/L (1) = r as well. Therefore, we see that EGH(a1 ,a2 ,...,an ),n (0) is always true.

Theorem 4.6 ([7, 9]) For 2  a1  . . .  an such that aj 

j −1

(ai − 1), the

i=1

EGH(a1 ,...,an ),n is true.



The strict inequality was shown by Caviglia-Maclagan in [9]. Their proof used an inductive argument on n, and followed from Lemma 4.4 and Remark 3.2. Very recently, Caviglia-De Stefani [7] extended this degree growth condition by including the equality. Their work actually provided a stronger case. They showed that if a homogeneous ideal I ⊆ R contains a regular sequence f1 , . . . , fn−1 with degrees 2  a1  . . .  an−1 satisfies the EGHn,(a1 ,...,an−1 ),n−1 , then I + (fn ) satisfies the EGH conjecture for any fn where f1 , . . . , fn form a regular sequence and deg fn  n−1  (ai − 1) (see [7, Theorem 3.6]). i=1

The result of Caviglia-Maclagan with the recent improvement by Caviglia-De Stefani provides an affirmative answer for the EGH conjecture for a large case in terms of the degrees a1 , . . . , an . One of the interesting case that is not covered by this result is when a1 = a2 = . . . = an , more specifically as in Conjecture 3.1 when n  4. We will focus on the quadratic case ai = 2 for all i = 1, . . . , n separately (see Sect. 4.4). In [14], Cooper approached the EGH conjecture for n = 3 in a geometric setting by investigating the Hilbert functions of the subsets of complete intersections in P2 and P3 . She showed that the EGH(a1 ,a2 ,a3 ),3 is true for the degrees (2, a, a) for a  2 and (3, a, a) for a  3.

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Another result for n = 3 for the Gorenstein ideals containing a regular sequence with degrees 2  a1  a2  a3 was proven by Chong [12].

4.2 EGH via Liaison Chong’s work covers EGH beyond Gorenstein ideals in K[x, y, z]. It uses the linkage theory and studies a special subclass of licci ideals. First, we would like to review some definitions and concepts related to linkage theory for ideals in R = K[x1 , . . . , xn ] to present Chong’s result. Let I, I  ⊆ R be homogeneous ideals of height r  n. If there exists a regular sequence g1 , . . . , gr such that the complete intersection c = (g1 , . . . , gr ) ⊆ I ∩ I  , and I = c : I  and I  = c : I , then we say that I and I  are linked (algebraically) c via c. We express this linkage as I ∼ I  . If I minimally contains a regular sequence f1 , . . . , fr with degrees 2  a1  . . .  ar and if the link c is a complete intersection of type (a1 , . . . , ar ) then we say c is a minimal link. c c c Suppose that there is a finite sequence of links I ∼1 I1 ∼2 · · · ∼t It where It is a complete intersection, we say that I is in the linkage class (a.k.a. liaison class) of the complete intersection It . An ideal in the linkage class of a complete intersection is called licci. We next state the work of Chong on this. Theorem 4.7 ([12]) Let 2  a1  . . .  an , and I ⊆ R be a homogeneous ideal c c containing a regular sequence of degrees a1 , . . . , an . If I is licci such that I ∼1 I1 ∼2 c · · · ∼t It where each link ci has type a¯ i for i = 1, . . . , n with a¯ 1  a¯ 2  . . .  a¯ n , and c1 is a minimal link, that is a¯ 1 = (a1 , . . . , an ), then there is a lex-plus-powers ideal associated with degrees a1 , . . . , an with the same Hilbert function as I .

In the same paper [12], Chong also proved that EGHn,(a1 ,...,ar ),r holds for the licci ideals where the types of the links satisfy the ascending condition and the first link is minimal. His result on Gorenstein ideals when n = 3 we mentioned in previous subsection is a consequence of this result.

4.3 EGH via the Structure of the Regular Sequence Let I ⊆ R = K[x1 , . . . , xn ] be a homogeneous ideal containing a regular sequence f1 , . . . , fn with degree 2  a1  . . .  an , respectively. By Clements-Lindström’s result, we already know that EGH(a1 ,...,an ),n is true when fi = xiai for all i = 1, . . . , n. Mermin and Murai [35] proved a special case of EGHn,(a1 ,...,ar ),r , r < n, when char K = 0. For the homogeneous ideals containing a regular sequence f1 , . . . , fr formed by monomials with degrees 2  a1  . . .  ar , they showed

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the existence of lex-plus-powers ideal associated with (a1 , . . . , ar ) with the same Hilbert function. Another notable work regarding the structure of the regular sequence contained in the ideal is done by Abedelfatah in [1]. He showed that if a homogeneous ideal I containing a regular sequence f1 , . . . , fn such that deg fi = ai , i = 1, . . . , n and each fi splits into linear factors completely, then I has the same Hilbert function of a lex-plus-powers ideal (x1a1 , . . . , xnan ) + J in R. Shortly after, Abedelfatah extended this result in [2]. Theorem 4.8 Let a be an ideal generated by the product of linear forms and contain a regular sequence f1 , . . . , fr with degrees a1 , . . . , ar . Let I be a homogeneous ideal in R such that (f1 . . . , fr ) ⊂ a ⊂ I then the Hilbert function of I is the same as the lex-plus-powers ideal containing powers x1a1 , . . . , xrar .

The previous result in [1] is simply the case when a = (f1 , . . . , fr ). Let I ⊂ R be a height r monomial ideal containing a regular sequence of degrees a1  . . .  ar , then this theorem of Abedelfatah confirms that I has the same Hilbert functions as a lex-plus-powers containing x1a1 , . . . , xrar . This also improves another related result given by Caviglia-Constantinescu-Varbaro in [5] for height r monomial ideals generated by quadrics.

4.4 When a1 = . . . = an = 2 Finally we focus on the case when the regular sequence is formed by quadrics as originally conjectured by the Eisendbud-Green-Harris as in Conjecture 3.1. For simplicity, we refer it as EGH(2,2,...,2),n = EGH2,n ¯ . We have already mentioned the cases when n = 2 by Richert [37] as his result shows EGH for any degree when n = 2. Moreover, in K[x, y, z], we have seen that the EGH conjecture for the degrees (2, 2, 2) was covered by Cooper [14] and Caviglia-De Stefani [7] separately, and quadratic monomial ideal case by [5]. In terms of the weaker version of the EGH conjecture for consecutive degrees given in Definition 4.1, using Proposition 4.2 given by Francisco, EGH2,n ¯ (2) is true for almost complete intersections I = (f1 , . . . , fn , g) where deg fi = 2 for all i = 1, . . . , n and deg g = 2. More precisely, HR/I (3)  HR/L (3) where L is the lex-plus-powers ideal containing the squares of the variables. An analogous result on EGH2,n ¯ (2) for the ideals generated by a quadratic regular sequence plus two more generators is given in the following theorem. Theorem 4.9 ([25]) For n  5, EGH2,n ¯ (2) holds for homogeneous ideals minimally generated by a regular sequence of quadrics and two more generators whose degrees are at least 2.

Thanks to [25, Lemma 2.6], which is mentioned in Remark 4.3, to prove Theorem 4.9 it was enough to show the statement for an ideal I generated by n + 2 quadrics containing a maximal regular sequence. More precisely, it sufficed to show

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the Hilbert function of the lex-plus-powers ideal (x12 , . . . , xn2 , x1 x2 , x1 x3 ) in degree 3 is greater than or equal to HR/I (3). This was shown by analyzing the linear relations among the generators of the ideal I . For a homogeneous ideal I = (f1 , . . . , fn , g1 , . . . , gm ) containing quadratic regular sequence f1 , . . . , fn , it is easy to see that if each gi has degree > 2 then they don’t contribute the dimension in degree 2 and dimK I2 = n. Therefore, any lex plus power ideal L = (x12 , . . . , xn2 ) + J where J is generated by monomials of degree > 2 gives dimK L2 = n as well. We finish this section by presenting the known cases of the original EGH conjecture for n  4. Theorem 4.10 EGH2,n ¯ is true when (a) n = 4 by Chen [11], (b) n = 5 by the author and Hochster [25].



Proof Notice that when a1 = . . . = an = 2 we get s =

n 

(ai − 1) = n. Then i=1 EGH2,n ¯ (n − 1 − d) holds.

by

Lemma 4.4, we get EGH2,n ¯ (d) holds if and only if Using this symmetry, when n = 4, by Remark 4.5 we trivially have EGH2,4 ¯ (0), therefore we have EGH2,4 ¯ (3). It is enough to show EGH2,4 ¯ (1) and therefore we also get EGH2,4 ¯ (2). Similarly, when n = 5, we know EGH2,5 ¯ (0) holds, so does EGH2,5 ¯ (4). Then we need to show only EGH2,5 ¯ (1) and EGH2,5 ¯ (2) because EGH2,5 ¯ (1) implies EGH2,5 ¯ (3). By [11, Proposition 2.1], we know that EGH2,n ¯ (1) holds for any n  2. Thus this completes the proof of (a). On the other hand, the proof of (b) is done as well since EGH2,5 ¯ (2) is true as a result of Theorem 4.9 and Remark 4.3.



5 Open Cases of EGH and More Connections Although there has been a significant progress on the EGH conjecture, it is fair to say that the conjecture is still broadly open. In this section, we discuss the open cases, and also state another well known conjecture related to the EGH conjecture. Besides Richert’s result on EGH when n = 2 in [37], we still do not know if the EGH conjecture is true when n  3 without assuming any conditions on the degrees or on the homogeneous ideal. Question 5.1 Is EGH(a1 ,a2 ,a3 ),3 true for any given degrees 2  a1  a2  a3 ?



We have seen that the works by Cooper, Caviglia-Maclagan and Caviglia-De Stefani cover many cases of (a1 , a2 , a3 ) already. On the other hand, Chong’s and Abedelfatah’s results require certain conditions on the homogeneous ideals. As a result, we can conclude that it is not known if EGH(a1 ,a2 ,a3 ),3 is true for the ideals in

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the following scenario: Let I be a homogeneous ideal containing a regular sequence f1 , f2 , f3 such that • the degrees deg fi = ai , i = 1, 2, 3 satisfy 4  a1  a2  a3 and a2  a3 < a1 + a2 − 2, • I is not a Gorenstein ideal, and • f1 , f2 , f3 cannot be split into linear factors. For example, EGH(4,4,5),3 and EGH(4,5,5),3 are two open cases with small degrees. Remark 5.2 If we focus on the original EGH conjecture, that is, when a1 = . . . = an = 2, EGH2,n

¯ is still open when n  6. For a given homogeneous ideal I in R = K[x1 , . . . , xn ], the graded Betti number of I is βi,j (R/I ) = dimK (TorR i (R/I, K))j . Another well-known conjecture motivated by the EGH conjecture is given by Evans and Charalambous if these graded Betti numbers are also concerned (see lex-plus-powers ideals survey [19]). This conjecture can be also considered analogous to Bigatti-Hulett-Pardue Theorem [3, 28, 36] which is a generalization of the Macaulay’s theorem for the graded Betti numbers, more precisely, it shows the extremal behavior of lex ideals for the graded Betti numbers among the homogeneous ideals with the same Hilbert function. Conjecture 5.1 (Lex-Plus-Powers (LPP) Conjecture) Let I be a homogeneous ideal containing a regular sequence of degrees 2  a1  . . .  an in R = K[x1 , . . . , xn ]. Suppose that there exists a lex-plus-powers ideal L with the same Hilbert function as I . Then the graded Betti numbers of R/I cannot exceed those of R/L. That is, βi,j (R/I )  βi,j (R/L) for all i and j. Just like the EGH conjecture, the LPP conjecture remains widely open. Nevertheless, there have been remarkable results obtained. In [37], Richert showed the equivalence of the EGH conjecture and the LPP conjecture when n = 2, 3. Therefore, the LPP conjectures holds when n = 2. Similar to Conjecture 3.4 where ideal contains a regular sequence of length  n, one can restate the LPP conjecture by allowing non-maximal regular sequences f1 , . . . , fr with r < n. Caviglia and Kummini [8] showed that this case can be also reduced to Artinian case like the EGH conjecture. In [35], the LPP conjecture is shown to be true when the homogeneous ideal containing monomial regular sequence. Thus, when the regular sequence has full length then the LPP conjecture is true for the homogeneous ideals containing x1a1 , . . . , xnan . In [10], when characteristic of K is 0, it was shown that the LPP conjecture holds for the homogeneous ideals containing a regular sequence with degrees 2  a1  i−1  . . .  an if ai  (aj − 1) + 1 for i  3. j =1

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When i = 1, the numbers β1,j (R/I ) tells us how many generators the homogeneous ideal I has in degree j . For a given homogeneous I containing a regular sequence with degrees 0  a1  . . .  an , the EGH conjecture claims the existence of the lex-plus-powers ideal L associated with degrees 0  a1  . . .  an such that dimK Ij = dimK Lj for all j  0. Then it is well-known that this implies β1,j (R/L)  β1,j (R/I ). Thus, the EGH conjecture covers this particular case of the LPP conjecture when i = 1. Remark 5.3 Let I = (f1 , . . . , fn , g1 , g2 ) ⊆ K[x1 , . . . , xn ] be a homogeneous quadratic ideal where f1 , . . . , fn is a regular sequence. Then, by Theorem 4.9, we see that EGH2,n ¯ (2) holds for such quadratic ideals. Therefore, we get 2 dimK I3  n + 2n − 5 = dimK L3 , where L is the lex-plus-powers ideal (x12 , . . . , xn2 , x1 x2 , x1 x3 ). This shows us that the number of the independent linear relations among the generators f1 , . . . , fn , g1 , g2 is always at most 5. In other words, we obtain β2,3 (R/I )  5 = β2,3 (R/L) as well.

Richert and Sabourin, in [38], showed that the following conjecture, a special case of the LPP conjecture when i = n, is equivalent the EGH conjecture. Conjecture 5.2 Let I be a homogeneous ideal in R containing a regular sequence of degrees 2  a1  . . .  an and let L be a lex-plus-powers ideal associated with degrees ai such that HR/I (i) = HR/L (i) for all i  0. Then the dimension of the socle of I is at most the dimension of socle of L in every degree. In other words, βn,j (R/L)  βn,j (R/I ) for all j  0.

The LPP conjecture seems much harder than the EGH conjecture due to its strong claim on every graded Betti numbers, yet focusing on certain Betti numbers as its special cases opens up many new directions to work on. Finally, there are some interesting applications of the EGH conjecture worth to mention. In [26], Harima-Wachi-Watanabe show that, assuming the EGH conjecture is true, every graded complete intersection has the Sperner property, which simply says for a graded complete intersection A = R/(f1 , . . . , fn ), max{μ(I ) | I is an ideal in A} = max{dimK Ai | i = 0, 1, 2, . . .}, where μ(I ) is the number of minimal generators of I . It is also known that the Sperner property is related to the so-called Weak Lefschetz property. Due to geometric background of the EGH conjecture as a result of its connection to Cayley-Bacharach theory, EGH has applications in more geometric settings as well. For instance, a recent work by Jorgenson [29] points out that an affirmative answer for EGH has an implication on the sequence of secant indices of Veronese varieties of Pn (see Question 3.2 in [29].) Another application of the EGH conjecture coincides with a very famous problem on decomposing real polynomials in n variables as a sum of squares of real polynomials. The cone of real polynomials that can be decomposed as a sum of squares of real polynomials is simply referred as SOS cone. In a recent work by Laplagne and Valdettaro[32], they show that, when EGH holds, for a strictly positive polynomial on the boundary of the SOS cone, they provide bounds for the

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maximum number of polynomials that can appear in a SOS decomposition and the maximum rank of the matrices in the Gram spectrahedron. Acknowledgments The author thanks Mel Hochster for introducing and proposing to work on the EGH conjecture during her postdoctoral research. She is deeply grateful for all of their conversations. The author thanks the referee for their valuable feedback and comments. She also thanks Martin Kreuzer for pointing out their work.

References 1. Abedelfatah, A.: On the Eisenbud-Green-Harris conjecture. Proc. Amer. Math. Soc. 143, no. 1, 105–115 (2015) 2. Abedelfatah, A.:Hilbert functions of monomial ideals containing a regular sequence. Israel J. Math. 214, no. 2, 857–865 (2016) 3. Bigatti, A.: Upper bounds for the betti numbers of a given Hilbert function. Comm. Algebra 21, no. 7, 2317–2334 (1993) 4. Bruns, W., Herzog, J.: Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics. 39, Cambridge University Press, Cambridge (1993) 5. Caviglia, G., Constantinescu, A., Varbaro, M.: On a conjecture by Kalai. Israel J. Math. 204, no. 1, 469–475 (2014) 6. Caviglia, G., De Stefani, A.: A Cayley-Bacharach theorem for points in Pn . Bull. Lond. Math. Soc., 53, no. 4, 1185–1195 (2021) 7. Caviglia, G. De Stefani, A.: The Eisenbud-Green-Harris conjecture for fast-growing degree sequences. Preprint. arXiv:2007.15467v2 (2020) 8. Caviglia, G., Kummini, M.: Poset embeddings of Hilbert functions and Betti numbers. J. Algebra 410, 244–257 (2014) 9. Caviglia, G., Maclagan, D.: Some cases of the Eisenbud-Green-Harris conjecture. Math. Res. Lett. 15, no. 3, 427–433 (2008) 10. Caviglia, G., Sammartano, A.: On the lex-plus-powers conjecture. Adv. Math. 340, 284–299 (2018) 11. Chen, R.-X.: Some special cases of the Eisenbud-Green-Harris conjecture. Illinois J. Math. 56, no. 3, 661–675 (2012) 12. Chong, K.F.E.: An Application of liaison theory to the Eisenbud-Green-Harris conjecture. J. Algebra 445, 221–231 (2016) 13. Clements, G., Lindström, B.: A generalization of a combinatorial theorem of Macaulay. J. Combinatorial Theory 7, 230–238 (1969) 14. Cooper, S. M.: Subsets of complete intersections and the EGH conjecture. Progress in Commutative Algebra 1, de Gruyter, Berlin 167–198 (2012) 15. Davis, E. D., Geramita A. V., Orecchia, F.: Gorenstein algebras and the Cayley– Bacharach theorem. Proc. Amer. Math. Soc. 93, no. 4, 593–597 (1985) 16. Eisenbud, D., Green, M., Harris, J.: Higher Castelnuovo theory. Journées de Géométrie Algébrique d’Orsay (Orsay, 1992), Astérisque 218, 187–202 (1993) 17. Eisenbud, D., Green, M., Harris, J.: Cayley-Bacharach theorems and conjectures. Bull. Amer. Math. Soc. (N.S.) 33, no. 3, 295–324 (1996) 18. Francisco, C.: Almost complete intersections and the lex-plus-powers conjecture. J. Algebra 276, no. 2, 737–760 (2004) 19. Francisco, C. A., Richert, B. P.: Lex-plus-powers ideals. Syzygies and Hilbert functions, Lect. Notes Pure Appl. Math. 254, 113–144 (2007) 20. Gasharov, V.: Hilbert functions and homogeneous generic forms II. Compositio Math. 116, no. 2, 167–172 (1999)

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21. Geramita, A., Kreuzer, M.: On the uniformity of zero-dimensional complete intersections. J. Algebra, 391, 82–92 (2013) 22. Green, M.: Restrictions of linear series to hyperplanes, and some results of Macaulay and Gotzmann. In Algebraic curves and projective geometry (Trento, 1988), Lecture Notes in Math. 1389, 76–86. Springer, Berlin (1989) 23. Greene, C., Kleitman, D.: Proof techniques in the theory of finite sets. Studies in combinatorics, MAA Stud. Math., Math. Assoc. America 17, 22–79 (1978) 24. Gotzmann, G.: Eine Bedingung für die Flachheit und das Hilbertpolynom eines graduierten Ringes. Math. Z. 158, 61–70 (1978) 25. Gunturkun, S., Hochster, M.: The Eisenbud-Green-Harris conjecture for defect two quadratic ideals. Math. Res. Letters 27, no. 5, 1341–1365 (2020) 26. Harima, T., Wachi, A., Watanabe, J.: The EGH conjecture and the Sperner property of complete intersections. Proc. Amer. Math. Soc. 145, no. 4, 1497–1503 (2017) 27. Herzog, J., Popescu, D.: Hilbert functions and generic forms. Compositio Math. 113, no. 1, 1–22 (1998) 28. Hulett, H.A.: Maximum betti numbers of homogeneous ideals with a given Hilbert function. Comm. Algebra 21, no.7, 2335–2350 (1993) 29. Jorgenson, G.: Secant indices of projective varieties. Preprint arXiv:2003.08481 (2020) 30. Katona, G.: A theorem for finite sets. Theory of Graphs (P. Erdös and G. Katona, eds.), Academic Press, New York 187–207 (1968) 31. Kruskal, J.: The number of simplices in a complex. Mathematical Optimization Techniques (R. Bellman, ed.), University of California Press, Berkeley/Los Angeles 251–278 (1963) 32. Laplagne, S., Valdettaro, M.: Strictly positive polynomials in the boundary of the SOS cone. Preprint arXiv:2012.05951 (2020) 33. Grayson, D. R., Stillman, M. E.: Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/ 34. Macaulay, F.: Some properties of enumeration in the theory of modular systems. Proc. London Math. Soc. 26, 531–555 1927) 35. Mermin, J., Murai, S.: The lex-plus-powers conjecture holds for pure powers. Adv. Math. 226, no. 4, 3511–3539 (2011) 36. Pardue, K.: Deformation classes of graded modules and maximal betti numbers. Illinois J. Math. 40, no.4, 564–585 (1996) 37. Richert, B. P.: A study of the lex plus powers conjecture. J. Pure Appl. Algebra 186, no. 2, 169–183 (2004) 38. Richert, B. P., Sabourin, S.: The residuals of lex plus powers ideals and the Eisenbud-GreenHarris conjecture. Illinois J. Math. 52, no. 4, 1355–1384 (2008)

The Variety Defined by the Matrix of Diagonals is F -Pure Zhibek Kadyrsizova

Keywords Frobenius · Singularities · F-purity · System of parameters

1 Introduction and Preliminaries Let X = (xij )1i, j n be a square matrix of size n with indeterminate entries over a field K and R = K[X] be the polynomial ring over K in {xij | 1  i, j  n}. Let D(X) be an n × n matrix whose j th column consists of the diagonal entries of the matrix Xj −1 written from left to right with the convention that X0 is the identity matrix of size n. Define P(X) = det(D(X)). In [8] and [9], H-W.Young studies the varieties of nearly commuting matrices and derives their important properties such as the decomposition into irreducible components through the use of the polynomial P(X). We hope to understand better their Frobenius singularities and as it can be seen in [7] this is also closely related to the singularities of the variety defined by P(X). In this paper we prove that the latter is F -pure and find a homogeneous system of parameters for it. Let us first define the necessary preliminaries. The following notation is fixed for the rest of the paper. Let ⎡

⎤ x11 x12 . . . x1n ⎦ X=⎣ ... ... xn1 xn2 . . . xnn and   X0 0 ˜ X= 0 xnn

Z. Kadyrsizova () Department of Mathematics, Nazarbayev University, Nur-Sultan, Kazakhstan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Miller et al. (eds.), Women in Commutative Algebra, Association for Women in Mathematics Series 29, https://doi.org/10.1007/978-3-030-91986-3_14

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where ⎡

⎤ x11 . . . x1,n−1 ⎦. X0 = ⎣ ... xn−1,1 . . . xn−1,n−1 That is, X˜ is the matrix obtained from X by setting all the entries of its last column and the last row to 0 except for the entry xnn . Let X0 be the matrix obtained from X by deleting its last column and last row. Lemma 1.1 ([9], Lemma 1.3) P(X) is an irreducible polynomial.



˜ = P(X0 )cX0 (xnn ) where cX0 (t) is Lemma 1.2 ([9], Proof of Lemma 1.3) P(X) the characteristic polynomial of X0 .

Remark 1.3 The above lemma is also true when we annihilate either only the last column or only the last row of X with the exception of the entry xnn .



2 A System of Parameters First, we find a homogeneous system of parameters on R/(P(X)). To do so we prove several useful lemmas. Lemma 2.1 Let A be a square matrix of size n with integer entries and with det(A) = ±1. Then the following are equivalent (a) (b) (c) (d)

P(A) = ±1. The diagonals of I, A, A2 , . . . , An−1 span Zn . The diagonals of the elements of Z[A, A−1 ] span Zn . There exist n integer powers of A with the property that their diagonals span Zn .



Proof The equivalence of (a) and (b) is clear as the columns of the matrix D(A) are the diagonals of I, A, A2 , . . . , An−1 .  i By Cayley-Hamilton’s theorem we have that An ∈ n−1 hence I ∈ i=0 ZA and n−1 n i −1 ∈ i=0 ZAi as i=1 ZA . Therefore, since A is invertible, we also have that A −h −1 well as A for all integers h. Thus Z[A] = Z[A, A ]. Finally, M L det diag(I ) diag(A) . . . diag(An−1 ) = ±1 if and only if diag(I ), diag(A), . . . , diag(An−1 ) span Zn if and only if the diagonals of all the integer powers of A span Zn if and only if there exist n integer powers of A with the property that their diagonals span Zn .



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Lemma 2.2 Let ⎡

⎤ 1 1 ... 1 1 ⎢1 1 ... 1 0⎥ ⎥. A=⎢ ⎣ ... ... ⎦ 1 0 ... 0 0 be a square matrix of size n  1 with the property that all the entries strictly below the main anti-diagonal are 0 and the rest are equal to 1. Then P(A) is equal to either 1 or -1.

Proof First observe that det(A) = ±1. Hence the matrix is invertible and it can be shown that ⎡

B = A−1

⎤ 0 0 ... 0 1 ⎢ 0 0 . . . 1 −1 ⎥ ⎥. =⎢ ⎣ ... ... ⎦ 1 −1 . . . 0 0

B has two non-zero anti-diagonals and the rest of the entries are equal to 0. To show that P(A) = ±1 it is necessary and sufficient to show that there exist n powers of B so that their diagonals span Zn , see Lemma 2.1. We claim that for this purpose it is sufficient to take n odd powers of B. Claim 2.3 ⎡

1 ⎢ −1 ⎢ ⎢ B2 = ⎢ ⎢ ⎣ 0 0

−1 2 ... 0 0

0 −1 ... 0 0

⎤ 0 0 ⎥ ⎥ ⎥ ⎥. ⎥ . . . 2 −1 ⎦ . . . −1 2 ... 0 ... 0

We show this by induction with the induction step equal to 2. Cases n = 2 and n = 3 can be easily verified. Write B as ⎡

0 0...0 ⎢0 ⎢ ⎢ ⎢0 ⎢. ⎢. B0 ⎢. ⎢ ⎣0 1 −1 0 . . . 0

then

⎤ 1 −1 ⎥ ⎥ ⎥ 0 ⎥ .. ⎥ ⎥ . ⎥ ⎥ 0 ⎦ 0

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1 ⎢ −1 ⎢ ⎢ ⎢ 0 ⎢ ⎢ 2 B =⎢ ⎢ ⎢ . ⎢ . ⎢ . ⎢ ⎣ 0 0

−1 0 . . . . . . . . . 0 0

⎡

⎤ 1 0 ... 0 B02 + ⎣ 0 0 . . . 0 ⎦ 0 0 ... 0 0 0.........0 − 1

⎤ 0 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥. ⎥ .. ⎥ ⎥ . ⎥ ⎥ −1 ⎦ 2

Claim 2.4 For all 1  j  n − 1, we have that ⎧ if k + l = n − j + 2, ⎨ (−1)j +1 2j −1 )kl = (B if k + l = n + j + 1, (−1)j ⎩ 0 if k + l  n − j + 1 or k + l  n + j + 2. We prove the claim by induction on j . When j = 1, it is true. Suppose that the claim is true for all integers less than or equal to j . Then B 2(j +1)−1 = B 2j +1 = B 2 B 2j −1 . Therefore, (B 2j +1 )kl = (B 2 B 2j −1 )kl =

n  (B 2 )ks (B 2j −1 )sl = s=1

For now assume that k > 1. (B 2 )kk (B 2j −1 )kl + (B 2 )k,k+1 (B 2j −1 )k+1,l + (B 2 )k,k−1 (B 2j −1 )k−1,l . Suppose first that k + l = n − j + 1. Then (B 2j +1 )kl = (B 2 )kk · 0 + (−1)(−1)j +1 + (−1) · 0 = (−1)j +2 . Next, suppose that k + l = n + j + 2 (B 2j +1 )kl = (B 2 )kk · 0 + (−1) · 0 + (−1)(−1)j = (−1)j +1 . Finally, consider the case when k + l  n − j or k + l  n + j + 3. We have that (B 2j +1 )kl = (B 2 )kk · 0 + (−1) · 0 + (−1) · 0 = 0. Thus the formula for B 2j −1 is true.

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Remark 2.5 The case k = 1 goes along the same lines as above with the exception when we have that 1 + l = n + j + 2. Then l = n + j + 1 > n and this is not possible.

Now we are ready to finish the proof of the lemma. Consider the matrix whose columns are the diagonals of the odd powers of B written from left to right. Observe that in each odd power the main diagonal meets only one of the sub-anti-diagonals that we highlighted above. Therefore, we have that L M diag(B) diag(B 3 ) . . . diag(B 2j −1 ) . . . diag(B 2n−1 ) = for matrices of odd sizes ⎡0 0 ⎢0 0 ⎢0 0 ⎢ ⎢ ⎢ ⎢0 0 ⎢ ⎢0 0 ⎢ ⎢1 ⎢ ⎢0 1 ⎢ ⎢0 0 ⎢ ⎢ ⎢ ⎢0 0 ⎣ 0 0 0 0

0 0 0 ... 0 1 ∗ ∗ 0 ... 0 0 0

0 0 0 0 0 0

0 0 ∗

0 ∗ ∗ ∗ 1

∗ ∗ ∗ ∗ ∗

1 ∗ ∗ ... ∗ ∗

0 0 0 0 0 0

∗ 1 0

0 1 ∗ ... ∗ ∗ ∗ ∗ ∗ ... ∗ ∗ ∗

and for matrices of even sizes

0 1⎤ ∗ ∗⎥ ∗ ∗⎥ ⎥ ⎥ ⎥ ∗ ∗⎥ ⎥ ∗ ∗⎥ ⎥ ∗ ∗⎥ ⎥, ∗ ∗⎥ ⎥ ∗ ∗⎥ ⎥ ⎥ ⎥ ∗ ∗⎥ ⎦ ∗ ∗ 1 ∗

⎡ 0 ⎢ 0 ⎢ 0 ⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ −1 ⎢ ⎢ 0 ⎢ ⎢ ... ⎢ ⎢ 0 ⎣ 0 0

0 0 0 0 −1 ∗ 0

0 0 0 ... 0 ∗ ∗ −1

0 0 0 −1 ∗ ∗ ∗

0 0 0

0 0 0

0 0 0

−1 ⎤ ∗ ⎥ ∗ ⎥ ⎥ ⎥ ⎥ ∗ ∗ ⎥ ⎥ ∗ ∗ ⎥ ⎥. ∗ ∗ ⎥ ⎥ ∗ ∗ ⎥ ⎥ ⎥ ... ⎥ ∗ ∗ ∗ ∗ ⎥ ⎦ −1 ∗ ∗ ∗ 0 ∗ −1 ∗ 0 0 ∗

0 0 −1 ∗ ∗ ∗ ... ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

Since every row of the matrices has a pivot position, we conclude that the determinants of these matrices are not zero and thus the columns of each of them span Zn . This finishes the proof of the lemma.

Theorem 2.6 Let X = (xij )1i, j n be a square matrix of size n with indeterminate entries over a field K and R = K[X] be the polynomial ring over K in {xij | 1  i, j  n}. Let = {(i, j ) |1  i  n, n + 1 − i  j  n} and  = {(k, l) |1  k  n − 1, 1  l  n − k, (k, l) = (1, 1)} . Then J S = xij , x11 − xkl | (i, j ) ∈

K , (k, l) ∈ 

is a homogeneous system of parameters and hence a regular sequence on R/(P(X)).

Proof We prove the lemma by induction on n. Consider the first few small cases. Let n = 2. In this case modulo (S) we have that

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x 0 X = 11 0 0



and 

1 x11 P(X) = det 1 0

 = −x11 .

Let n = 3. In this case ⎡

⎡ 2 2 ⎤ ⎤ x11 x11 0 2x11 x11 0 2 x2 0 ⎦ X = ⎣ x11 0 0 ⎦ , X2 = ⎣ x11 11 0 0 0 0 0 0 and ⎡

2 ⎤ 1 x11 2x11 2 ⎦ = x3 . P(X) = det ⎣ 1 0 x11 11 1 0 0

Let n = 4. In this case ⎡

x11 ⎢ x11 X=⎢ ⎣ x11 0 ⎡

x11 x11 0 0

3 5x 3 3x 3 6x11 11 11 ⎢ 5x 3 4x 3 2x 3 3 11 11 11 ⎢ X =⎣ 3 3 x3 3x11 2x11 11 0 0 0

x11 0 0 0

⎤ ⎡ 0 1 ⎢1 0⎥ ⎥ , and P(X) = det ⎢ ⎣1 0⎦ 1 0

n(n−1)/2

Claim 2.7 P(X) = ±x11

⎡ 2 ⎤ 2 3x11 2x11 0 2 2 ⎢ ⎥ 0⎥ 2x11 2x11 , X2 = ⎢ 2 2 ⎣ ⎦ 0 x11 x11 0 0 0

2 x11 2 x11 2 x11 0

⎤ 0 0⎥ ⎥, 0⎦ 0

2 6x 3 ⎤ x11 3x11 11 2 4x 3 ⎥ x11 2x11 6 11 ⎥ 3 ⎦ = −x11 . 2 0 x11 x11 0 0 0

in the quotient ring of R by the ideal generated by

{xij , x11 − xkl | (i, j ) ∈

, (k, l) ∈ }

Remark 2.8 In our proof of the claim we do not establish exactly the sign of the image of P(X). We only show that it is either 1 or -1.

˜ and set the elements strictly below the main antiConsider the matrices X, X0 , X, diagonal of X0 and xnn to 0, that is,

The Variety Defined by the Matrix of Diagonals is F -Pure

⎡

x11 x12 ⎢ x ⎢ 21 x22 ⎢ X˜˜ = ⎢ ... ⎢ ⎣ xn−1,1 0 0 0

349

. . . x1,n−2 x1,n−1 . . . x2,n−2 0 ... 0 ... 0 0 ... 0 0

⎤ 0 0⎥ ⎥ ⎥ 0⎥. ⎥ 0⎦ 0

˜ = P (X0 )cX0 (xnn ) where cX0 is the characteristic polynomial of X0 Since P (X) ˜˜ = P (X˜ )(−1)(n−2)(n−1)/2 3n−1 x (see Lemma 1.2), we have that P (X) 0 i=1 i,n−i . Set (n−1)(n−2)/2 ˜ P (Y0 ), the elements x11 − xkl to zero for all (k, l) ∈ . Then P (X0 ) = x11 where ⎡

⎤ 1 1 ... 1 1 ⎢1 1 ... 1 0⎥ ⎥. Y0 = ⎢ ⎣ ... ... ⎦ 1 0 ... 0 0 Hence ˜˜ = x (n−1)(n−2)/2 P (Y )(−1)(n−2)(n−1)/2 P (X) 0 11

n−1 

n(n−1)/2

x11 = ±x11

P (Y0 ).

i=1

By Lemma 2.2 we have that P (Y0 ) = ±1, which finishes the proof of the claim. n(n−1)/2 ), which has Krull Finally, we have that R/(P(X), S) ∼ = K[x11 ]/(x11 dimension 0. Hence, S is indeed a system of parameters on R/(P(X)) and, since the ring is a complete intersection, it is also a regular sequence.



3 The Variety Defined by P(X) Is F -Pure Definition 3.1 Let S be a ring with positive prime characteristic p. Then S is called F -pure if the Frobenius endomorphism F : S → S with F (s) = s p is pure, that is, for every S-module M we have that F ⊗S idM : S ⊗ M → S ⊗ M is injective. Next is Fedder’s criterion specialized for hypersurfaces and we use it to prove F purity of R/(P(X)). Lemma 3.2 (Fedder’s Criterion, [2], Proposition 2.1) Let S be a polynomial ring over a field K of positive prime characteristic p and let f ∈ S be a homogeneous polynomial. Then S/(f ) is F -pure if the polynomial f p−1 has a non-zero monomial term in which every indeterminate has degree at most p − 1.

We also need the fact that F -purity deforms for Gorenstein rings.

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Lemma 3.3 ([2], Theorem 3.4(2)) Let S be a Gorenstein ring and let x ∈ S be a non-zero divisor. Then if S/xS is F -pure, then so is S.

For more examples of F -pure rings and other related notions the reader may refer to [1, 3, 4, 6]. Computer algebra system Macaulay2, [5], is a great tool in studying rings and their properties. Let S = {xij : 1  i  n, n − i + 1  j  n} ⊆ S. These are the entries of the matrix X on and below the main anti-diagonal. It has been shown in the previous section that S is part of a homogeneous system of parameters and a regular sequence on R/(P(X)). Theorem 3.4 Let K be a field of positive prime characteristic p. Then R/(P(X)) is F -pure for all n  1.

Proof We use the fact that F -purity deforms for Gorenstein rings, [2], and we have that R/(P(X)) is a complete intersection and hence is Gorenstein. Therefore, it is sufficient to show that we have an F -pure ring once we take the quotient of the ring R/(P(X)) by the ideal generated by the regular sequence S . Let us first take a look at what we have for few small values of n. Case n = 1 is trivial. P(X) = 1 and R/(P(X)) = 0.  1 x11 = x22 − x11 . Then R/(P(X)) ∼ Case n = 2: P(X) = det = 1 x22 K[x11 , x12 , x21 ] is regular and hence F -pure. Now we are ready to prove the general statement. We do it by induction on n. First observe the following: if our statement is true for a fixed n, that is, if R/(P(X), S ) is F -pure, then so is R/(P(X), S0 ), where S0 = {xij : 2  i  n, n − i + 2  j  n} ⊆ S , that is, the entries strictly below the main anti-diagonal. Moreover, if a particular monomial term of P(X) has a nonzero coefficient in R/(S ), then it is a monomial term of P(X) with a nonzero coefficient in R/(S0 ). A partial converse is also true. If P(X) in R/(S0 ) has a nonzero monomial term in the entries of X which are strictly above the main anti-diagonal, then so does P(X) in R/(S ). The first non-trivial case is n = 3. Let Xˆ be the matrix X modulo the elements of S0 , that is, ⎤ x11 x12 x13 Xˆ = ⎣ x21 x22 0 ⎦ x31 0 0 ⎡

and ⎡

2 +x x x x ⎤ 1 x11 x11 12 21 13 31 2 +x x ˆ = det ⎣ 1 x22 ⎦= P (X) x22 12 21 1 0 x13 x31

The Variety Defined by the Matrix of Diagonals is F -Pure

351

2 2 = −x11 x13 x31 + x11 x12 x21 − x12 x21 x22 + x11 x22 − x11 x22 .

ˆ has a monomial term x11 x12 x21 with a coefficient 1 modulo p, so Since P (X) does P(X) in R/(S ). Here is our induction hypothesis: for all k < n we have that P(X)p−1 in R/(S ) 3k−i p−1 3 and in R/(S0 ) has a monomial term k−1 with coefficient ±1 modulo i=1 j =1 xij p. In other words, this monomial term is the (p − 1)st power of the product of the entries of X strictly above the main anti-diagonal. The basis of the induction is verified above. ˜˜ As in the Theorem 1, since P (X) ˜ = Now consider the matrices X0 , X˜ and X. P (X0 )cX0 (xnn ) where cX0 is the characteristic polynomial of X0 (see Lemma 1.2) ˜˜ = P (X˜ )(−1)(n−2)(n−1)/2 3n−1 x we have that P (X) 0 i,n−i . By induction hypothesis 3n−1−i i=1 3 p−1 P (X˜0 )p−1 has a monomial term n−2 x with coefficient ±1. i=1 j =1 ij 3 3 ˜˜ p−1 has a monomial term n−1 n−i x p−1 with coefficient ±1. Hence P (X) i=1 j =1 ij Therefore, by Fedder’s criterion we have that R/(P(X), S ) is F -pure and thus so is R/(P(X)).

The next natural question that one can consider is whether the variety defined by the polynomial P(X) is F -regular. It is certainly true when n = 1 and n = 2, but is unknown for larger values of n. Conjecture 3.5 Let K be a field of positive prime characteristic p. Then R/(P(X)) is F -regular for all n  1.

Acknowledgments The author is grateful to Mel Hochster for valuable discussions and comments and thanks the referee for useful suggestions on improving the paper.

References 1. W. Bruns and J. Herzog, Cohen-Macaulay rings, second edition ed., Cambridge Univ. Press, Cambridge, UK, 1998. 2. R. Fedder, F-purity and rational singularity, Trans. Amer. Math. Soc 278 (1983), no. 2, 461– 480. 3. , F-purity and rational singularity in graded complete intersection rings, Transactions of the American Mathematical Society 301 (1987), no. 1, 47–62. 4. R. Fedder and K. Watanabe, A characterization of F-regularity in terms of F-purity, Commutative Algebra, vol. 15, Math. Sci. Res. Inst., Berlin Heidelberg New York Springer, 1989, pp. 227–245. 5. D. Grayson and M. Stillman, Macaulay 2: a computer algebra system for algebraic geometry and commutative algebra, available at http://www.math.uiuc.edu/Macaulay2. 6. M. Hochster and J. L. Roberts, The purity of the Frobenius and local cohomology, Adv. in Math. 21 (1976), 117–172. 7. Z. Kadyrsizova, Nearly commuting matrices, Journal of Algebra 497 (2018), 199–218. 8. Hsu-Wen Vincent Young, Components of algebraic sets of commuting and nearly commuting matrices, Ph.D. thesis, University of Michigan, 2011. 9. , On matrix pairs with diagonal commutators, Journal of Algebra 570 (2021), 437 – 451.

Classification of Frobenius Forms in Five Variables Zhibek Kadyrsizova, Janet Page, Jyoti Singh, Karen E. Smith, Adela Vraciu, and Emily E. Witt

Keywords Frobenius form · Quadratic form · Positive characteristic · Threefold

1 Introduction Fix a field k of prime characteristic p. A Frobenius form over k is a homogeneous polynomial in indeterminates x1 , . . . , xn of the form pe

pe

pe

x1 L1 + x2 L2 + · · · + xn Ln

(1)

where each Li is some linear form and e is a positive integer. Put differently, a Frobenius form is a homogeneous polynomial of degree pe + 1 that is in the ideal pe pe generated by x1 , . . . , xn . Frobenius forms can be compared to quadratic forms: if we allow e = 0 in the expression (1) above, we get a quadratic form. Quadratic forms are well-studied in the classical literature. For example, much is known about the geometry of quadric

Z. Kadyrsizova Department of Mathematics, Nazarbayev University, Nur-Sultan, Kazakhstan e-mail: [email protected] J. Page · K. E. Smith University of Michigan, Ann Arbor, MI, USA e-mail: [email protected]; [email protected] J. Singh Visvesvaraya National Institute of Technology, Nagpur, India e-mail: [email protected] A. Vraciu University of South Carolina, Columbia, SC, USA e-mail: [email protected] E. E. Witt () Department of Mathematics, University of Kansas, Lawrence, KS, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Miller et al. (eds.), Women in Commutative Algebra, Association for Women in Mathematics Series 29, https://doi.org/10.1007/978-3-030-91986-3_15

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hypersurfaces, and, at least over a quadratically closed field, their classification up to linear changes of coordinates is well-known. Both admit a convenient matrix factorization, where the action of changing coordinates behaves similarly. One special but very interesting case of Frobenius forms are those of degree three (which are necessarily defined over a field of characteristic two). A smooth cubic surface is always Frobenius split, it turns out, unless it is defined by a Frobenius form; in particular, non-Frobenius split (smooth) cubic surfaces exist only over fields of characteristic two [8, 5.5]. A detailed examination of nonFrobenius split cubic surfaces of characteristic two, including the non-smooth ones, was undertaken in [10]. In particular, the Frobenius forms of degree three in up to four variables are classified there, up to projective equivalence. In this paper, we extend that classification to arbitrary Frobenius forms in up to five variables. Put differently, we classify the projective equivalence classes of three-dimensional projective hypersurfaces defined by Frobenius forms—a class called extremal threefolds in [11]. Section 4 describes this classification in detail, including a particularly “sparse” equation representing each of the seven types of projective equivalence classes of extremal three-folds. Frobenius forms and the projective schemes they define have various “extreme” properties, both algebraically and geometrically. For example, cubic surfaces defined by Frobenius forms are characterized by the geometric property that they “contain no triangles”—that is, any plane section consisting of three lines must contain a point on all three [10, 5.1]. Analogous extremal configurations of linear subvarieties occur more generally for extremal hypersurfaces of higher degree and dimension; see [11, §8]. Algebraically, reduced Frobenius forms can be characterized as those achieving the minimal possible F -pure threshold among reduced forms of the same degree [11, 1.1].

2 Matrix Factorization of Frobenius Forms Fix a field k of prime characteristic p. Let q denote an integral positive power of p. The beauty of Frobenius forms is that, like quadratic forms, they admit a matrix factorization: a Frobenius form h in n variables can be written as L e e h = x1p x2p

⎡ ⎤ x1 ⎢ ⎥ M ⎢x2 ⎥ pe · · · xn A ⎢ .. ⎥ ⎣.⎦ xn

for some n × n matrix A with entries in k. Because e > 0, the matrix A representing the Frobenius form h is unique. When e = 0, we recover the case of quadratic forms. In the quadratic form case, of course, A is not unique, but we can force uniqueness, for example, by insisting that A be symmetric (when p = 2).

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There is an interesting story of Frobenius forms with “conjugate symmetric” matrices, where conjugacy is defined using a Frobenius automorphism. By definition, a Hermitian Frobenius form over k is a Frobenius form whose matrix A satisfies A1 = A[q] for some q = pe (in particular, such a matrix is defined over Fq 2 ). This special class of Frobenius forms gives a characteristic p analog of Hermitian forms over the complex numbers, and has been studied, for example, in [1, 9, 15] and [12, §35]. Like quadratic forms over quadratically closed fields, there is exactly one non-degenerate Hermitian Frobenius form, up to projective equivalence, in each dimension [1, 4.1] over Fq 2 . As we will see, the classification of more general Frobenius forms is more complicated. To classify Frobenius forms up to projective equivalence, we need to understand how linear changes of coordinates act on them. Let g ∈ GLn (k) be any linear change of coordinates for the polynomial ring k[x1 , . . . , xn ]. We represent the action of g on the variables as ⎡ ⎤ ⎡ ⎤ x1 x1 ⎢x2 ⎥ ⎢x2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ . ⎥ → g ⎢ . ⎥ , ⎣ .. ⎦ ⎣ .. ⎦ xn xn the usual matrix multiplication. In particular, when g acts1 on the Frobenius form h, we have ⎛ ⎜L e e ⎜ g · h = g · ⎜ x1p x2p ⎝

⎡ ⎤⎞ x1 ⎢ ⎥⎟ M x ⎢ 2 ⎥⎟ pe · · · xn A ⎢ . ⎥⎟ ⎣ .. ⎦⎠ xn

L e e = x1p x2p

⎡ ⎤ x1 ⎢ ⎥ M ⎢x2 ⎥ pe [pe ] 1 · · · xn (g ) Ag ⎢ . ⎥ , ⎣ .. ⎦ xn

where g [p ] denotes the matrix whose entries are the pe -th powers of the entries of g and B 1 indicates the transpose of a matrix B. Thus, the action of g on the Frobenius e form h transforms the matrix A representing h into the matrix (g [p ] )1 Ag. A Frobenius form is said to be nondegenerate if it cannot be written, after a linear change of coordinates, in a smaller number of variables, and is otherwise degenerate, similarly as for quadratic forms. We define its embedding dimension to be the smallest number of variables needed to write f up to linear change of coordinates. The rank of a Frobenius form is defined as the rank of the representing matrix, just as the rank of a quadratic form is the rank of the corresponding symmetric matrix when p = 2. The rank of a Frobenius form is invariant under changes of e

write “g · h” to indicate the group action on polynomials, whereas adjacency, “gB,” will indicate the usual matrix multiplication on an n × m matrix B.

1 We

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coordinates, since the rank of a matrix is unchanged by multiplication on the left and right by invertible matrices. The rank of a Frobenius form is equal to the codimension of the singular locus of the corresponding hypersurface [11, 5.3]. This is analogous to the corresponding statement about quadratic forms of non-even characteristic, when we work with the corresponding symmetric matrix. There is precisely one full rank Frobenius form, up to change of coordinates, over an algebraically closed field, in each fixed degree pe + 1 and embedding dimension n: Theorem 2.1 ([11, 6.1][2]) Every full rank Frobenius form over an algebraically closed field k of characteristic p > 0 is represented, in suitable linear coordinates, q+1 q+1 by the diagonal form x1 + · · · + xn , where q is an integral power of p.

Put differently, every smooth projective hypersurface defined by a Frobenius q+1 q+1 q+1 form is projectively equivalent to one defined by x1 + x2 + · · · + xn . This is analogous to the situation for quadratic forms over a quadratically closed field of characteristic not two, though the proof is a bit more involved. In particular, we do not have a complete understanding of the situation over non-closed fields, an interesting open problem. We have no counterexample to the speculation that the full rank Frobenius forms may be “diagonalizable” over a perfect field closed under all degree pe extensions. In the non-full rank case, there are finitely many projective equivalence classes of Frobenius forms in each fixed degree and embedding dimension [11, 7.4]. Indeed, the number of non-degenerate Frobenius forms of embedding dimension n (of fixed degree) is bounded above by the n-th Fibonacci number. However, the paper [11] stopped short of precisely classifying the Frobenius forms in each dimension, a task essentially completed for Frobenius forms in four variables in [10]. The case of five variables is treated here in Sect. 5. For the statement, see Sect. 4.

3 Quadratic Forms To complete our story, we recall the classification of quadratic forms: Proposition 3.1 Let f be a non-degenerate quadratic form in n variables over a quadratically closed2 field. If the characteristic of k is two, then f is projectively equivalent to either 1. x1 x2 + x3 x4 + · · · + xn−2 xn−1 + xn2 if n is odd, or 2. x1 x2 + x3 x4 + · · · + xn−3 xn−2 + xn−1 xn if n is even.

2 By quadratically closed, we mean that every degree two polynomial over k splits. In characteristic two, this is a stronger assumption than requiring that the field contain the square root of every element.

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If the characteristic of k is not two, then f is projectively equivalent to x12 + x22 + · · · + xn2 . In particular, over a quadratically closed field, there is exactly one quadratic form in each embedding dimension.

This classification is well known, but since we could not find a low-tech proof in the modern literature for the case p = 2, we include one here. Alternate discussions using more machinery can be found, for example, in [5, I.16], [7, 12.9] or [6, 7.32], which also contain more refined classifications over non-closed fields. A lower-tech proof can be found in the classic book of [4], but for the convenience of the reader we include a direct and concise proof here.3 Proof The only quadratic form in one variable is x12 . Likewise, the two-variable case is trivial: a degree two form in two variables must factor into two linear forms over a quadratically closed field, so in suitable coordinates, the form is either x1 x2 or x12 (which is degenerate).

Case of Characteristic Not Two It is straightforward to check (even without closure assumptions on k) that a suitable choice of linear change of coordinates puts f in the form λ1 x12 + · · · + λn xn2 , where the λi are nonzero (e.g., see [13]). So, if the ground field is quadratically closed, the change of coordinates taking each xi → √1 xi normalizes the form to x12 +· · ·+xn2 . λi

Case of Characteristic Two Say that n  3. Since f is non-degenerate, it is not the square of a linear form. Thus some square-free term, which we can assume to be x1 x2 , appears with nonzero coefficient. Scaling, we may assume the coefficient of x1 x2 is 1. Now write f in the form L2 + x1 x2 +

n  j =3

a1j x1 xj +

n 

a2j x2 xj + h1 (x3 , . . . , xn )

(2)

j =3

where L is a (possibly zero) linear form in x1 , x2 , and h1 is a quadratic  form in x3 , . . . , xn . Apply the linear change of coordinates sending x2 to x2 + nj=3 a1j xj , fixing the other variables. This transforms (2) into an expression which can be written

3 While the stated classification of quadratic forms in characteristic two is well known, it appears that an elementary proof is not. In his history of quadratic forms [14], Scharlau laments that Dickson’s 1899 work [3], which is elementary but “rather involved,” is not better known, stating “However, one must admit, that this paper—like most of Dickson’s work—is not very pleasant to read. It is entirely algebraic.” We hope the reader will find our straightforward and entirely algebraic proof more pleasant to read.

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L2 + x1 x2 +

n 

 a2j x2 xj + h2 (x3 , . . . , xn ),

(3)

j =3

where again h2 is a quadratic form in x3 , . . . , xn . we may assume the summand nWith another change of coordinates,  is nonzero, then we can assume a  = 1 a x x is zero. Indeed, if some a 2 j j =3 2j 2j 23 after renumbering and if necessary. Now apply the linear transformation nscaling  x , fixing the other variables. This transforms the sending x3 to x3 + j =4 a2j j expression (3) into L2 + (x1 + x3 )x2 + h3 (x3 , . . . , xn ),

(4)

where h3 is quadratic in x3 , . . . , xn . Next, the coordinate change taking x1 to x1 +x3 , fixing the other variables, transforms (4) into the form L2 + x1 x2 + h4 (x3 , . . . , xn ),

(5)

where h4 is quadratic. Finally, by induction, we separately apply linear changes of coordinates to the quadratic L2 + x1 x2 in {x1 , x2 } and the quadratic h4 in {x3 , . . . , xn } to put each into the desired form. It is easy to see, then, that their sum has the desired form as well, depending on the parity of n in the stated way. This completes the proof.

Remark 3.2 Our proof easily adapts to show the well-known basic fact that a quadratic form over an arbitrary field of characteristic two is a sum of binary quadratics in distinct variables (plus a quadratic in one variable if the embedding dimension is odd). Alternatively, our proof adapts to prove Theorem 199 in [4] over any perfect field of characteristic two. Remark 3.3 Quadratic forms behave like Frobenius forms from the point of view of achieving the minimal F -pure threshold. For a reduced form of degree d, it is 1 proved in [11, 1.1] that the F -pure threshold is at least d−1 , with equality if and only if the form is a Frobenius or quadratic form. Another way in which Frobenius forms and quadratic forms are similar is that the corresponding hypersurfaces both contain many high-dimensional linear subvarieties; see [11, §8].

4 Frobenius Forms in Five Variables: Classification Statements Fix an algebraically closed field k of positive characteristic p. Let q be an integral positive power of p.

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Theorem 4.1 There are seven projective equivalence classes of Frobenius forms of a fixed degree q + 1 and embedding dimension five. Specifically, these are represented by the following forms: 1. 2. 3. 4. 5. 6. 7.

q+1

q+1

q+1

q+1

q+1

x1 + x2 + x3 + x4 + x5 (rank 5) q q q+1 q x1 x5 + x2 x4 + x3 + x4 x2 (rank 4) q q q+1 q x1 x5 + x2 x4 + x3 + x4 x1 (rank 4) q q q q x1 x5 + x2 x4 + x3 x2 + x4 x1 (rank 4) q q q q x1 x5 + x2 x3 + x3 x2 + x4 x1 (rank 4) q q q+1 x1 x5 + x2 x4 + x3 (rank 3) q q q x1 x5 + x2 x4 + x3 x2 (rank 3)

For completeness, we also describe degenerate Frobenius forms in five variables in the following two theorems. The proofs are the same as in [10], although that source considered only cubic Frobenius forms. Theorem 4.2 There are five projective equivalence classes of Frobenius forms of a fixed degree q +1 and embedding dimension four. Specifically, these are represented by the following forms: 1. 2. 3. 4. 5.

q+1

q+1

q+1

q+1

x1 + x2 + x3 + x4 (rank 4) q q+1 q x1 x4 + x2 + x3 x1 (rank 3) q q q x1 x4 + x2 x3 + x3 x1 (rank 3) q q q x1 x4 + x2 x3 + x3 x2 (rank 3) q q x1 x4 + x2 x3 (rank 2)

Theorem 4.3 There are three projective equivalence classes of nondegenerate Frobenius forms in three variables in each fixed degree, represented by precisely one of the following forms: q+1

q+1

q+1

1. The diagonal form x1 + x2 + x3 (rank 3) q q+1 2. The cuspidal form x1 x3 + x2 (rank 2) q q 3. The reducible form x1 x3 + x2 x1 (rank 2) Moreover, in two variables, each Frobenius form is projectively equivalent to exactly one of the following: q−1

q−1

1. The form x1 x2 (x1 +x2 ), defining q +1 distinct points 0, ∞ and the (q −1)-st roots of unity in P1 . q 2. The form x1 x2 , defining the union of a q-fold point and a reduced point. q+1 3. The form x1 , defining a (q + 1)-fold point.

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5 Classification of Frobenius Forms in Five Variables: Proofs In this section, we prove Theorem 4.1. For this, we recall the method in [11] for showing that there are only finitely many Frobenius forms of any fixed degree and embedding dimension, up to projective equivalence. A key point is that a Frobenius form is equivalent to one represented by a “sparse matrix:” Theorem 5.1 ([11]) 4 Fix any algebraically closed field of characteristic p > 0. A Frobenius form of embedding dimension n and rank r can be represented by a matrix A with the following properties: 1. All rows beyond the r-th are zero. 2. All columns beyond the n-th are zero. 3. There are exactly r nonzero entries (all of which are 1) occurring in positions (1 j1 ), (2 j2 ), . . . , (r jr ), where j1 > j2 > · · · > jr . In particular, we may assume that all columns of A are zero but for r of them, which are the standard unit basis vectors er , . . . , e1 (in that order, and possibly interspersed with zero columns). For example, any full rank Frobenius form is projectively equivalent to a Frobenius form whose matrix is the anti-diagonal matrix, that is, whose columns are en , . . . , e1 . A Frobenius form whose matrix satisfies the three conditions of Theorem 5.1 will be called a sparse form. Theorem 5.1 implies the following bounds: Corollary 5.2 1. The rank of a Frobenius form of embedding dimension n is at least n2 . 2. The number of non-degenerate n × n matrices of rank r satisfying the three r ! conditions in Theorem (5.1) is n−r . 3. The number of projective equivalence types of Frobenius! forms of rank r and r embedding dimension n (and fixed degree) is at most n−r . Proof Choosing a sparse matrix to represent the Frobenius form, we can assume it looks like pe

pe

x1 L1 + · · · + xr Lr , where L1 , . . . , Lr ∈ {x1 , . . . , xn } are variables that appear in reverse order, each variable appearing at most once. All the variables xr+1 , . . . , xn must appear in the list L1 , . . . , Lr (otherwise the embedding dimension is less than n). In particular, n − r  r. This proves (1).

4 An

older version of [11] contained the statement of Theorem 5.1 explicitly; observe now that it follows easily from Lemma 5.5 in the new version of [11]. The updated version of [11], in fact, classifies Frobenius forms in all dimensions ([11, Thm. 7.1]), so [11] gives a new proof of Theorem 4.1 as well.

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For (2), we continue by observing that conditions (1) and (3) of sparseness (Theorem 5.1) force L1 = xn , L2 = xn−1 , . . . , Ln−r = xr+1 . For the remaining linear forms {Ln−r+1 , . . . , Lr }, we can choose 2r − n variables out of the remaining variables {x1 , . . . , xr }. There are 

   r r = 2r − n n−r



such choices. So (3) follows as well. It is easy to determine the embedding dimension of a sparse Frobenius form: Lemma 5.3 A rank r Frobenius form of the type q

q

q

x1 xj1 + x2 xj2 + · · · + xr xjr

(6)

has embedding dimension equal to the number of distinct variables appearing in the expression (6).

Proof Suppose that a Frobenius form f of the type (6) involves n variables and has rank r. If its embedding dimension is not n, then f could be written as a polynomial in n − 1 independent linear forms, y1 , . . . , yn−1 . Without loss of generality, we can assume that the yi have the following very special property: there is an index j such that every yi is a binomial linear form x i + ai xj for some index i = j and scalar ai . To see this, write ⎡ ⎤ x1 ⎢x2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥=B⎢ . ⎥ ⎣ .. ⎦ ⎣ ⎦ yn−1 xn ⎡

y1 y2 .. .

⎤

where B is an (n − 1) × n matrix of full rank, and then left-multiply by the inverse in GL(n − 1) of a full rank (n − 1) × (n − 1) submatrix of B. This replaces {y1 , . . . , yn−1 } by a set of linear forms spanning the same space and with the desired binomial form. There are two cases to consider, depending on whether or not j ∈ {1, . . . , r}. If j  r, then without loss of generality j = 1, so that yi = ai x1 + xi+1 for each i = 1, . . . , n − 1. Now if f is a Frobenius form in y1 , . . . , yn−1 , then q

q

q

q

q

q

x1 xj1 + x2 xj2 + · · · + xr xjr = y1 L1 + y2 L2 + · · · + yn−1 Ln−1

(7)

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for some linear forms Li in the yi . Because the only terms on the right side of q q q q (7) that involve xr+1 , . . . , xn come from yr , . . . , yn−1 , respectively, it follows that Lr = · · · = Ln−1 = 0. In this case, q

q

q

f = y1 L1 + y2 L2 + · · · + yr−1 Lr−1 , so that f has rank less than r, contrary to our hypothesis. If j > r, then without loss of generality j = n, so that yi = xi + ai xn for each i = 1, . . . , n − 1. Now assuming q

q

q

q

q

q

x1 xj1 + x2 xj2 + · · · + xr xjr = y1 L1 + y2 L2 + · · · + yn−1 Ln−1

(8)

for some linear forms Li in the yj , again it follows that Lr+1 = · · · = Ln−1 = 0 q by looking at the xi terms, with i > r. Moreover, since the only terms on the right q q q q side of (8) that involve x1 , . . . , xr come from y1 , . . . , yr , we see that Li = xji for i  r. But note that xn must appear among the variables xj1 , . . . , xjr , since the original form f involves all n variables. So xn is one of the Li . This says that xn is a linear combination of x1 + a1 xn , . . . , xn−1 + an−1 xn , a contradiction. Combining the two cases, we conclude that f is not a form in y1 , . . . , yn−1 , and so the embedding dimension of f is n.

Remark 5.4 The total number of projective equivalence classes of Frobenius forms of embedding dimension n is bounded above by the n-th Fibonacci number [11, 7.4].5 This follows from Corollary 5.2 (3) simply by adding the bounds for each relevant rank. This upper bound is sharp for n  4, but not in general. There are distinct sparse matrices that define equivalent Frobenius forms starting in five variables:

Example 5.4.1 Consider the Frobenius forms corresponding to the following matrices: ⎡ ⎤ ⎡ ⎤ 00001 00001 ⎢0 0 0 1 0⎥ ⎢0 0 0 1 0⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢0 1 0 0 0⎥ and ⎢1 0 0 0 0⎥ ⎢ ⎥ ⎢ ⎥ ⎣0 0 0 0 0⎦ ⎣0 0 0 0 0⎦ 00000 00000 q

q

q

q

q

q

namely, x1 x5 + x2 x4 + x3 x2 and x1 x5 + x2 x4 + x3 x1 . These are rank three nondegenerate Frobenius forms corresponding to distinct matrices satisfying the three conditions of Theorem 5.1, but they are equivalent by the change of coordinates which swaps x1 ↔ x2 and x4 ↔ x5 , but fixes x3 .



5 In

an updated version of [11], the authors have recently given a precise count of the number of projective equivalence classes with a fixed embedding dimension.

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The proof of Theorem 5.1 in [11] relied on noting that a sparse n × n matrix of rank r will take one of two forms ⎡

⎤ ⎡ ⎤ | 0 | | 0 | ⎣0 B e1 ⎦ or ⎣er B e1 ⎦ | 0 | | 0 | where B is a sparse matrix of rank r − 1 in the first case, and a sparse matrix of rank r − 2 in the second. In light of this, we make the following definition. Definition 5.4.2 For n  3, we say a sparse n × n matrix of rank r < n has type a if its first column is 0, and type b if its first column is er . In particular, a Frobenius form of embedding dimension n and rank r is equivalent, after a linear change of coordinates, to a polynomial of one of two forms: (type a) h = x1 xn + h (x2 , . . . , xn−1 ) where h is a Frobenius form of rank r − 1 and embedding dimension n − 2. In this case, B is non-degenerate of rank r − 1. q q (type b) h = x1 xn + xr x1 + h (x2 , . . . , xn−1 ) where h is a Frobenius form of  rank r − 2. If h is non-degenerate in the n − 2 variables x2 , . . . xn−1 , then B is non-degenerate of rank r − 2. Otherwise, h must be non-degenerate in the n − 3 variables x2 , . . . ,  xr , . . . , xn−1 , so that B is degenerate of rank r − 2. q

The proof of Theorem 4.1 will now follow from the following two propositions. Proposition 5.5.2 There are precisely n − 1 projective equivalence classes of Frobenius forms of embedding dimension n and rank n − 1 (in fixed degree pe + 1). The proof of Proposition 5.5.2 uses the following lemma: Lemma 5.6 Let h1 and h2 be two Frobenius forms of the same degree, both of embedding dimension n and rank r = n − 1, and both represented by matrices (say A1 and A2 ) in sparse form. Then h1 and h2 are projectively equivalent if and only if A1 and A2 are the same type, and their B matrices are projectively equivalent. Proof Without loss of generality, by Theorem 5.1 we may assume q

q

q

q

q

q

q

q

h1 = x1 xn + x2 xj2 + x3 xj3 + · · · + xn−1 xjn−1 , and  h2 = x1 xn + x2 xj2 + x3 xj3 + · · · + xn−1 xjn−1

where xji , xji ∈ {x1 , . . . , xn−1 } for each i. In particular, we note that the only term q containing xn in both h1 and h2 is x1 xn . Now suppose some change of coordinates φ sends h1 to h2 . The singular locus of both h1 and h2 is defined by the vanishing of x1 , x2 , . . . , xn−1 , so the ideal x1 , . . . , xn−1  is stable under φ. In particular, φ must send xn to a linear form involving xn . Supposing that φ maps x1 → λ11 x1 + · · · + λ1,n−1 xn−1 , and xn → λn1 x1 + · · · + λnn xn

with λnn = 0,

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then φ sends h1 to a polynomial q

q

q

q

λ11 λnn x1 xn + · · · + λ1,n−1 λnn xn−1 xn + terms that do not involve xn . In order for this to be equal to h2 , we must have λ1i = 0 for all i > 1. So φ sends x1 → λ11 x1 where λ11 = 0. In particular, if h1 and h2 are equivalent, they must be equivalent after modding out by x1 . We can now see that if h1 and h2 are equivalent, then A1 and A2 have the same type. Letting ∼ denote equivalence up to change of coordinates, if ⎡

⎤ ⎡ ⎤ 0 0 1 0 0 1 ⎣0 B 0⎦ ∼ ⎣er−1 B  0⎦ 0 0 0 0 0 0 then      B0 B 0 ∼ 0 0 0 0 which is the same as saying B ∼ B  . By rank considerations this never happens between type a and type b (see the discussion following Definition 5.4.2). Thus the sparse matrices of projectively equivalent Frobenius forms whose rank is one less than the embedding dimension must have the same type. Furthermore, the argument above also shows that their “B” matrices are projectively equivalent. Because the converse is obvious, the lemma is proved.

Proof (Proof of Proposition 5.5.2) Fix q. Let N (n, r) denote the number of projective equivalence classes of Frobenius forms of degree q + 1, rank r, and embedding dimension n. We want to show that N (n, n − 1) = n − 1. We will induce on n. One readily verifies that N (1, 0) = 0, N (2, 1) = 1, and N (3, 2) = 2. Since type a and type b matrices yield distinct classes for r = n − 1, the number of classes N(n, n − 1) is equal to the number of classes of type a plus the number of type b. The discussion of the types following Definition 5.4.2 informs us, therefore, that for n  4 N(n, n − 1) = N (n − 2, n − 2) + N (n − 2, n − 3) + N (n − 3, n − 3).

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Recalling that there is only one full rank form in each degree and dimension (Theorem 2.1), it follows that N(n, n − 1) = 1 + N (n − 2, n − 3) + 1 = N (n − 2, n − 3) + 2. Finally, by induction on n, N (n, n − 1) = (n − 3) + 2 = n − 1,



as desired.

In light of Proposition 5.5.2, we have now fully classified all Frobenius forms in five variables of ranks four and five. By Corollary 5.2 (1), it only remains ! to analyze the rank three case. Using Corollary 5.2 (2), we see that there are 32 = 3 sparse forms of rank three in five variables, and we have seen in Example 5.4.1 that two of them are projectively equivalent. To complete the classification, it remains to see that the third sparse matrix produces a Frobenius form not equivalent to these. This is accomplished by the following: Proposition 5.7 The following rank three Frobenius forms in five variables are not projectively equivalent: q

q

q+1

q

q

q

f = x1 x5 + x2 x4 + x3 and

g = x1 x5 + x2 x4 + x3 x2 . Proof To see this, note that g ∈ x1 , x2 . Thus, if f and g are projectively equivalent, there must be some linear forms L1 and L2 such that f ∈ L1 , L2 . Since any projective change of coordinates must respect the singular locus, any form in x1 , x2 , x3 must be sent to another form in x1 , x2 , x3 , as the vanishing of these coordinates defines the singular set of both hypersurfaces. This implies that L1 , L2 , being the images of x1 and x2 under our linear change of coordinates, are forms in x1 , x2 , x3 .( ) q+1 Now note that x3 , x4 , x5 = f, x4 , x5  ⊂ L1 , L2 , x4 , x5 . In particular, √ x3 ∈ L1 , L2 , x4 , x5  = L1 , L2 , x4 , x5  , so that x3 ∈ L1 , L2 . Without loss of generality, we may assume L1 , L2  = x3 , L2 , where L2 is a linear form in x1 and x2 , so that f ∈ x3 , L2  . Therefore, q

q

x1 x5 + x2 x4 ∈ x3 , L2 

(9)

as well. Considering the image of the expression (9) under the natural quotient map k[x1 , . . . , x5 ] → k[x1 , . . . , x5 ]/x3 , we see that

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q

x1 x5 + x2 x4 ∈ L2 

(10) q

q

in a polynomial ring in four variables. But the polynomial x1 x5 +x2 x4 is irreducible (for example, by Eisenstein’s criterion), so we arrive at a contradiction. This contradiction ensures that f and g are not projectively equivalent.

This completes our classification of Frobenius forms in up to five variables. Acknowledgments This paper grew out of discussions begun at the AWM-sponsored workshop “Women in Commutative Algebra” at the Banff International Research Station in October 2019, where additional funding was provided by NSF Conference Grant DMS #1934391 and AWM ADVANCE Grant NSF-HRD #1500481. We are grateful to Eloísa Grifo and Jennifer Kenkel, who also participated in those discussions. Zhibek Kadyrsizova was partially supported by FDCRGP Grant 021220FD4151. Jyoti Singh was partially supported by SERB(DST) Grant MTR/2021/000879. Karen E. Smith was partially supported by NSF Grant DMS #1801697 and NSF Grant DMS #2101075. Emily E. Witt was partially supported by NSF CAREER Award DMS #1945611.

References 1. R. C. Bose and I. M. Chakravarti, Hermitian varieties in a finite projective space PG(N, q 2 ), Canad. J. Math. 18 (1966), 1161–1182. 2. A. Beauville, Sur les hypersurfaces dont les sections hyperplanes sont à module constant, The Grothendieck Festschrift, Vol. I, Progr. Math., vol. 86, Birkhäuser Boston, Boston, MA, 1990, With an appendix by David Eisenbud and Craig Huneke, pp. 121–133. MR 1086884 3. L.E. Dickson, Determination of the structure of all linear homogeneous groups in a Galois field which are defined by a quadratic invariant, Amer. J. Math. 21 (1899), no. 3, 193–256. MR 1505798 4. , Linear groups: With an exposition of the Galois field theory, B.G. Teubners Sammlung von Lehrbüchern auf dem Gebiete der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, B. G. Teubner, 1901. 5. J.A. Dieudonné, La géométrie des groupes classiques, Springer-Verlag, Berlin-New York, 1971, Troisième édition, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 5. MR 0310083 6. R. Elman, N. Karpenko, and A. Merkurjev, The algebraic and geometric theory of quadratic forms, American Mathematical Society Colloquium Publications, vol. 56, American Mathematical Society, Providence, RI, 2008. MR 2427530 7. L.C. Grove, Classical groups and geometric algebra, Graduate Studies in Mathematics, vol. 39, American Mathematical Society, Providence, RI, 2002. MR 1859189 8. N. Hara, A characterization of rational singularities in terms of injectivity of Frobenius maps, Amer. J. Math. 120 (1998), no. 5, 981–996. MR 1646049 9. M. Homma and S. J. Kim, The characterization of Hermitian surfaces by the number of points, J. Geom. 107 (2016), no. 3, 509–521. 10. Z. Kadyrsizova, J. Kenkel, J. Page, J. Singh, K. E. Smith, A. Vraciu, and E. E. Witt, Cubic surfaces of characteristic two, Trans. Amer. Math. Soc. 374 (2021), no. 9, 6251–6267. 11. Z. Kadyrsizova, J. Kenkel, J. Page, J. Singh, K. E. Smith, A. Vraciu, and E. E. Witt, Lower bounds on the F-pure threshold and extremal singularities, Trans. Amer. Math. Soc. (2020). 12. J. Kollár, Szemerédi–Trotter-type theorems in dimension 3, Adv. Math. 271 (2015), 30–61.

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13. T. Y. Lam, Introduction to quadratic forms over fields, Graduate Studies in Mathematics, vol. 67, American Mathematical Society, Providence, RI, 2005. MR 2104929 14. W. Scharlau, On the history of the algebraic theory of quadratic forms, Quadratic forms and their applications (Dublin, 1999), Contemp. Math., vol. 272, Amer. Math. Soc., Providence, RI, 2000, pp. 229–259. MR 1803370 15. B. Segre, Forme e geometrie hermitiane, con particolare riguardo al caso finito, Ann. Mat. Pura Appl. 70 (1965), no. 4, 1–201.

Projective Dimension of Hypergraphs Kuei-Nuan Lin and Sonja Mapes

2020 Mathematics Subject Classification 13D02, 05E40

1 Introduction Let R = K[x1 , . . . , xn ] be a polynomial ring over a field K. The minimal free resolution of R/I for an ideal I ⊂ R is an exact sequence of the form 0→

' j

S(−j )βp,j (R/I ) → · · · →

'

S(−j )β1,j (R/I ) → R → R/I → 0

j

The exponents βi,j (R/I ) are invariants of R/I , called the Betti numbers of R/I . In general, finding Betti numbers is still a wide open question. Two other invariants that one can associate to a minimal resolution are the projective dimension of R/I , denoted pd(R/I ), which is defined as follows pd(R/I ) = max{i | βi,j (R/I ) = 0}, and the (Castelnuovo-Mumford) regularity of R/I , denoted reg(R/I ), which is defined as follows reg (R/I ) = max{j − i | βi,j (R/I ) = 0}. Those two invariants play important roles in algebraic geometry, commutative algebra, and combinatorial algebra. In general, one finds the graded minimal free K.-N. Lin () Department of Mathematics, The Penn State University, McKeesport, PA, USA e-mail: [email protected] S. Mapes Department of Mathematics, University of Notre Dame, Notre Dame, IN, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Miller et al. (eds.), Women in Commutative Algebra, Association for Women in Mathematics Series 29, https://doi.org/10.1007/978-3-030-91986-3_16

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resolution of an ideal to obtain those invariants, but the computation can be difficult and computationally expensive. Kimura, Terai and Yoshida define the dual hypergraph of a square-free monomial ideal in order to compute its arithmetical rank [12](see Definition 2.1 for definition of a hypergraph). Since then, there have been a couple of papers using this combinatorial object to study various properties, for example, [8] and [16]. In particular, Lin and Mantero use it to show that ideals with the same dual hypergraph have the same Betti numbers and projective dimension [13] (Theorem 2.4 part 1), which has found use in other papers, such as in [11]. The focus of this work is to use the properties of a hypergraph to compute the projective dimension of the associated square-free monomial ideal without finding the minimal free resolution of the ideal. This is a different focus than various work by others which produce the full resolution, for example, the recent work of Eagon, Miller, and Ordog [4]. More precisely, we find the projective dimension of hypergraphs when their 1-skeleton is a string or cycle. This extends the work of Lin and Mantero in [13]. A key development is the result of Lin and Mapes in [15], which allows us to remove a large class of higher dimensional edges of a hypergraph without impacting its projective dimension (Corollary 4.4 [15]). Given this previous result we can restrict our study in this paper to specific cases which are not previously dealt with. Our main results are summarized by the following theorem. Theorem 1.1 Let H be a hypergraph and F be an edge of H such that dimF > 0. 1. Let HS be an open string of length μ. If H is the union of HS% and & F such that all vertices of F are on HS , then pd(H) = pd(HS ) = μ − μ3 or pd(H) = pd(HS ) + 1 depending on the position of vertices of F . 2. Let HCμ be an open cycle of length μ. If H is the union of HCμ and F such that   .

all vertices of F are on HCμ , then pd(H) = pd(HCμ ) = μ − 1 − μ−2 3 In Sect. 2 of the paper we begin by giving the necessary background to understand the results of this paper. We also establish a new technique for computing the projective dimension using bounds on sub-ideals, which is inspired by methods from [5], namely Betti splittings (Lemma 2.13). We then proceed with our results concerning higher dimensional edges on strings and cycles in Sects. 3 and 4. Through out this paper, ideals are monomial ideals in a polynomial ring R over the field K.

2 Preliminaries 2.1 Hypergraph of a Square-Free Monomial Ideal Kimura, Terai, and Yoshida associate a square-free monomial ideal with a hypergraph in [12], see Definition 2.1. Note that this construction is different from the

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constructions associating ideals to hypergraphs coming from the study of edge ideals. In particular relative to edge ideals, the hypergraph of Kimura, Terai, and Yoshida might be more aptly named the “dual hypergraph”. The construction of the dual hypergraph is first introduced by Berge in [1]. In the edge ideal case, one associates a square-free monomial with a hypergraph by setting variables as vertices and each monomial corresponds to an edge of the hypergraph (see for example [7]). In the following definition, we actually associate variables with edges of the hypergraph and vertices with the monomial generators of the ideal, and in practice this is the dual hypergraph of the hypergraph in the edge ideal construction. Definition 2.1 Let I be a square-free monomial ideal in a polynomial ring with n variables with minimal monomial generating set {m1 , . . . , mμ }. Let V be the set {1, . . . , μ}. We define H(I ) (or H when I is understood) to be the hypergraph associated to I which is defined as {{j ∈ V : xi |mj } : i = 1, 2, . . . , n}. We call the sets {j ∈ V : xi |mj } the edges of the hypergraph.

Note that if you start with a hypergraph you can create a monomial ideal by assigning a variable to each edge, then each vertex (or element in V ) would be assigned the monomial product of the variables corresponding the edges using that vertex. The issue however that doing this will not always produce a minimal generating set. To obtain a minimal generating set the hypergraph needs to be separated, where H is separated if in addition for every 1  j1 < j2  μ, there exist edges F1 and F2 in H so that j1 ∈ F1 ∩ (V − F2 ) and j2 ∈ F2 ∩ (V − F1 ). All hypergraphs in this paper will be separated unless otherwise stated. Example 2.2 Let I =(m1 = abk, m2 = bcl, m3 = cdklm, m4 = dekn, m5 = ef gn, m6 = ghmn, m7 = hikl, m8 = ij k) the Fig. 1 is the hypergraph associated to I via the Definition 2.1 where H(I ) = {a = {1}, b = {1, 2}, c = {2, 3}, d = {3, 4}, e = {4, 5}, f = {5}, g = {5, 6}, h = {6, 7}, i = {7, 8}, j = {8}, k = {1, 3, 4, 7, 8}, l = {2, 3, 7}, m = {3, 6}, n = {4, 5, 6}}.

Some important terminology regarding these hypergraphs is the following. We say a vertex i ∈ V of H is an open vertex if {i} is not in H, and otherwise i is closed. In Fig. 1, we can see that the vertices labeled by a, f and j are all closed, and the rest are open. Moreover, a hypergraph H with V = [μ] is a string if {i, i +1} is in H for all i = 1, . . . , μ−1, and the only edges containing i are {i−1, i}, {i, i+1} and possibly {i}. We say that a string is an open string if all vertices other than 1 and μ are open (Note that for H to be separated 1 and μ must be closed). Also, H

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Fig. 1 Hypergraph of Example 2.2

˜ ∪ {μ, 1} where H ˜ is a string. We say a cycle is an open cycle is a μ-cycle if H = H if all the vertices are open. In Fig. 1 if we were to remove the edges corresponding to the variables k, l, m, and n then we would be left with a string. If we further removed the edge corresponding to the variable f , then it would be an open string. Let Hi = {F ∈ H : |F |  i + 1} denote the i-th dimensional subhypergraph of H where |F | is the cardinality of the F . We call H1 the 1-skeleton of H. Example 2.3 Some simple examples where V = {1, 2, 3} of open strings and open cycles are as follows. Let H1 = {a = {1}, b = {1, 2}, c = {2, 3}, d = {3}}, then H1 is an open string and the corresponding monomial ideal is I1 = (ab, bc, cd). If we let H2 = {a = {1, 2}, b = {2, 3}, c = {1, 3}}, then this is an open cycle and the corresponding monomial ideal is I2 = (ac, ab, bc).

Recently there have been a number of results determining both the projective dimension and the regularity of certain square-free monomial ideals from the associated hypergraph. As this paper focuses more on the projective dimension, we include the statements of some of results that are useful for the rest of the paper here (these appear separately in the literature but we list them all here as part of one statement). We write pd(H) = pd(H(I )) = pd(R/I ) and reg(H) = reg(H(I )) = reg(R/I ) where H(I ) is the hypergraph obtained from a square-free monomial ideal I . We do this for the Betti numbers as well. Notice that for the content of this work, we assume there is a unique variable associated to each face of H. Theorem 2.4 1. [13, Proposition 2.2] If I1 and I2 are square-free monomial ideals associated to the same separated hypergraph H, then the total Betti numbers of the two ideals coincide. 2. [15, Corollary 4.4] Let F be an edge on the hypergraph H. If F is an union of other edges of H, then pd(H) = pd(H\F ). 3. [14, Theorem 2.9 (c)]) If H ⊆ H are hypergraphs with μ(H ) = μ(H), then pd(H )  pd(H) where μ(∗) denotes the number of vertices of ∗. 4. [9, Theorem 7.7.34, Corollary%7.7.35] An openNstring & O hypergraph with μ vertices has projective dimension μ − μ3 , regularity μ3 , and βμ− μ ,μ (H) = 0 when 3 μ(H) = 0.

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5. [10, Corollary 2.2] A hypergraph H with n1 + n2 vertices that is a disjoint union of open string H1 and H2 with n1 and n2 vertices has pd(H) = % & % &hypergraphs,     n1 + n2 − n31 − n32 and β (H) = 0. n1 n2 n1 +n2 −

3

−

3

,n1 +n2

6. [9, Corollary 7.6.30] If H is an open cycle with μ vertices then pd(H) = μ − 1 −  μ−2 3 . 7. [2, Corollary 20.19] If I = mI  where m is a monomial of degree r then pd(R/I ) = pd(R/I  ), reg(R/I ) = r + reg(R/I  ). 8. [2, Corollary 20.19] Let m be a monomial of degree r, then reg(R/(I, m))  max{reg(R/I ), reg(R/(I : m) + r − 1}. Remark 2.5 Note that Theorem 2.4 part 1 allows us to talk about the projective dimension of a hypergraph rather than an ideal. We will use pd(H(I )) in the place of pd(R/I ) throughout the paper. Moreover if an edge is an union of other edges in a hypergraph, and we are considering the projective dimension of the hypergraph, we can just ignore or remove the edge using Theorem 2.4 part 2. For example, in Fig. 1, we can remove the edge k = {1, 3, 4, 7, 8} = {1} ∪ {3, 4} ∪ {7, 8} and n = {4, 5, 6} = {4, 5} ∪ {5, 6}. If a hypergraph H = H(I ) is an union of two disconnected hypergraphs G1 = H(I1 ) and G2 = H(I2 ), we have pd(H) = pd(G1 )+pd(G2 ) and reg(H) = reg(G1 )+reg(G2 ) as one can construct the minimal resolution of R/I using the tensor of minimal resolutions of R/I1 and R/I2 .



2.2 Colon Ideals: Key tool One technique that is used in [13] and [14] which we will need here, is using the short exact sequences obtained by looking at colon ideals. Specifically there are two types of colon ideals that we are interested in, and we explain below what each operation looks like on the associated hypergraphs. Definition 2.6 Let H be a hypergraph, and I = I (H) be the standard square-free monomial ideal associated to it in the polynomial ring R. Let G(I ) = {m1 , . . . , mμ } be the minimal generating set of I . Let F be an edge in H and let xF ∈ R be the variable associated to F . Also let v be a vertex in H and mv ∈ I be the monomial generator associated to it. • The hypergraph Hv : v = Qv is the hypergraph associated to the ideal Iv : mv where Iv = (G(I )\mv ), and Hv = H(Iv ) is the hypergraph associated to the ideal Iv . • The hypergraph H : F , obtained by removing F in H, is the hypergraph associated to the ideal I : xF . • The hypergraph (H, xF ), obtained by adding a vertex corresponding to the variable xF in H, is the hypergraph associated to the ideal (I, xF ).

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Remark 2.7 Given a hypergraph H and a vertex v of H, we consider a short exact sequence 0 → Hv : v = Qv → Hv → H → 0. If pd(Hv ) > pd(Qv ), then with the projective dimension on the short exact sequence that pd(Hv )  max{pd(Qv ), pd(H)} and pd(H)  max{pd(Hv ), pd(Qv ) + 1}, we can conclude that pd(H) = pd(Hv ). We will use this strategy in later sections.

The following result appearing in [14] will be very useful to us in this paper. We put it here for the self-containment of this work and for the reader’s convenience. We need some terminology first. The degree of a vertex is the number of faces containing the vertex. Let v be a vertex in a hypergraph H, and let H1 , ..., Hr be the connected components of Hv ; if one of them, say H1 , is a string hypergraph, we call H1 a branch of H (from v). Theorem 2.8 ([14]) Let H be a one-dimensional hypergraph, w a vertex with degree at least 3 in H, and S be a branch departing from w with v1 , . . . , vn vertices. Suppose vertices of S, v1 , . . . , vn−1 , are open and the end vertex vn , i.e., the leaf of S, is the only closed vertex of S. Let E be the unique edge connecting w to v1 . Then pd(H) = pd(H ), where H is the following hypergraph: (a) if n ≡ 1 mod 3, then H = H : E; (b) if n ≡ 2 mod 3, then H = Hw .

Example 2.9 In Fig. 2, there are 4 hypergraphs, H, H1 = H : E, H2 = Hw2 , H3 = (H, xF ). S1 is a branch of length 1 departing from w1 and S2 is a branch of length 2 departing from w2 . By Theorem 2.8, pd(H) = pd(H1 ) = pd(H2 ). Notice that in H2 , the edge F becomes an one-dimensional edge.



2.3 Splittings: Key Tool In [3] the notion of a splitting of a monomial ideal I was introduced. Definition 2.10 ([3]) A monomial ideal I is splittable if I is the sum of two nonzero monomial ideals J and K, i.e. I = J + K, such that 1. The generating set G(I ) of I , is the disjoint union of G(J ) and G(K). 2. There is a splitting function G(J ∩ K) →

G(J ) × G(K)

w→

(ψ(w), φ(w))

Projective Dimension of Hypergraphs

375

Fig. 2 Hypergraph operations

satisfying (a) (S1) for all w ∈ G(J ∩ K), w = lcm(ψ(w), φ(w)). (b) (S2) for every subset S ⊆ G(J ∩ K), both lcm(ψ(S)) and lcm(φ(S)) strictly divide lcm(S).

If J and K satisfy the above properties they are called a splitting of I .



Now the key reason we are interested in splittings is the following result by both Eliahou-Kervaire and separately Fatabbi. Theorem 2.11 (Eliahou-Kervaire [3] Fatabbi [5]) Suppose I is a splittable monomial ideal with splitting I = J + K. Then for all i, j  0 βi,j (I ) = βi,j (J ) + βi,j (K) + βi−1,j (J ∩ K). It is important to note that not all monomial ideals admit splittings. What is interesting is that there are sometimes monomial ideals that can be decomposed into a sum of ideals J and K which satisfy the conclusions of the previous theorem. This motivates the following definition by Francisco, Ha, and Van Tuyl in [6] Definition 2.12 Let I, J and K be monomial ideals such that G(I ) is the disjoint union of G(J ) and G(K). Then I = J + K is a Betti splitting if βi,j (I ) = βi,j (J ) + βi,j (K) + βi−1,j (J ∩ K) for all i ∈ N and all (multi)degrees j .



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One complication however is that if one wants to use the existence of a Betti splitting to prove something about a resolution, one must first know something about the resolution in question. The key for us will be in dissecting the proof of Fatabbi in order to prove that in some special cases, which may fail condition (S2) in Definition 2.10, that a similar formula for (some) Betti numbers holds. The following lemma is an adaptation of the proof of Fatabbi in a special case where we do not have a splitting. In this case we can show that the necessary conditions hold at the end of the resolution, so that we get a formula like that of Theorem 2.11 for the last Betti numbers. In particular this allows us to prove statements about projective dimension. Lemma 2.13 Let I be a monomial ideal and I = J + K in the ring R = K[x1 , . . . , xn ] over a field K. Suppose we have the following conditions on projective dimension 1. pd(R/J ) < q 2. pd(R/K) = q, and reg(R/K) < r. If βq,q+r (R/J ∩ K) = 0 and pd(R/J ∩ K) = q, then pd(R/I ) = q + 1.

Proof We consider the short exact sequence 0 → J ∩ K → J ⊕ K → I → 0. Let α(w) = (w, w) be the map from J ∩ K to J ⊕ K and π(u, v) = u − v be the map from J ⊕ K to I . There is an induced homology sequence R R · · · → TorR q+1 (J ∩ K, K) → Torq+1 (J, K) ⊕ Torq+1 (K, K) R → TorR q+1 (I, K) → Torq (J ∩ K, K) R → TorR q (J, K) ⊕ Torq (K, K) → · · ·

We have pd(R/J ) < q, pd(R/K) = q, and pd(R/(J ∩ K)) = q, hence the short exact sequence gives pd(R/I )  max{pd(R/(J ∩ K)) + 1, pd(R/J ⊕ K)}  q + 1. The homology sequence becomes R R 0 → 0 → TorR q+1 (I, K) → Torq (J ∩ K, K) → Torq (K, K) → · · ·

Moreover, the q + r graded piece is the following: R R 0 → TorR q+1 (I, K)q+r → Torq (J ∩ K, K)q+r → Torq (K, K)q+r → · · ·

Now using our assumption that reg(R/K) < r and pd(R/K) = q then we have R R ∼ TorR q (K, K)q+r = 0. This shows Torq+1 (I, K)q+r = Torq (J ∩ K, K)q+r = 0 by the fact that βq,q+r (R/J ∩ K) = 0 and hence pd(R/I ) = q + 1.



Projective Dimension of Hypergraphs

377

Remark 2.14 The above lemma can be translated in terms of associated hypergraphs. Let H = H(I ) be a hypergraph with underlying vertex set V , and let V1 and V2 be a partition of the vertices of H such that V1 ∪ V2 = V and V1 ∩ V2 is empty. Now define Ii to be the ideal generated by the generators of I indexed by the elements in Vi for i = 1, 2. Let Gi = H(Ii ) for i = 1, 2 and H(I1 ∩ I2 ) be the hypergraphs corresponding the ideals Ii for i = 1, 2 and the ideal I1 ∩ I2 . Suppose pd(G1 ) < q, pd(G2 ) = q, pd(H(I1 ∩ I2 )) = q, and reg(G2 ) < r and reg(H(I1 ∩ I2 )) = r. Suppose βq,q+r (H(I1 ∩ I2 )) = 0, then pd(H) = q + 1.



3 Strings with Higher Dimensional Edges In this section we are primarily interested in finding the projective dimension of a square-free monomial ideal such that its hypergraph is a string with higher dimensional edges attached to it. Our primary object is described below and we fix the notation now for the easy reference later. Notation 3.1 Let H be a hypergraph and F be an edge of H. 1. Let HSμ be a string hypergraph with vertex set V = {w1 , . . . , wμ }, and an edge F with k > 1 vertices {v1 = wi1 , . . . , vk = wik } ⊆ {w1 , . . . , wμ } with 1  i1 < · · · < ik  μ. When we say H is a string together with an edge consisting of k vertices, we mean H = HSμ ∪ F . 2. Let n1 = i1 − 1, nj = ij − ij −1 − 1 for j = 2, . . . k, and nk+1 = μ − ik be the number of vertices between vertices of F . 3. We write ni = 3li + ri where li are some non-negative integers, and 0  ri  2 for i = 1, . . . , k + 1. Example 3.2 The hypergraph shown in Fig. 3 shows the string hypergraph with a higher dimensional edge with the notation outlined above. In this case, μ = 11, k = 4, n1 = 1, n2 = n3 = n4 = 2, and n5 = 0.



Fig. 3 Hypergraph of Example 3.2

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Because of Theorem 2.4 part 2, we will primarily focus on the higher dimensional edges which are not unions of two or more edges. The vertex wi is assumed to be open for all i unless otherwise  stated. We find the projective dimensions k+1 by considering three different cases: k+1 r < 2k, i i=1 i=1 ri  2k + 1, and k+1 r = 2k in three propositions. We conclude this section with Theorem 3.14 i=1 i which covers all the previous results. Example 3.2 is an example of the cases when k+1 r < 2k. i=1 i  First we will prove two propositions that deal with the case when k+1 i=1 ri < 2k, k+1 or i=1 ri  2k + 1. In these two cases we will show that for a hypergraph H satisfying the hypotheses of the propositions, that the projective dimension will be the same as for HSμ . Before we can get to these propositions though we present couple of technical computations first, and the computational processes will be used in a similar manner  proofs.  We often  use the following  identities   without   in the later 3A 3A+1 3A+2 3A−3 3A−1 = and A − 1 = = = justification A = 3 = 3 3 3 3   3A−2 . 3 Lemma 3.3 Let ni = 3li +ri with 0  ri < 3 for i = 1, . . . , k+1. If then P Q  k+1 k+1 n    k + k+1 i i=1 ni )k+

k+1  i=1

P ni −

k + 2k +

k+1

i=1 (3li )

3

Q

Projective Dimension of Hypergraphs

379

=k+

k+1 

ni − k −

i=1

k+1 

li

i=1

k+1  = (ni − li ) i=1

Lemma 3.4 Let ni = 3li + ri with r1 = 2 and 0  ri < 3 for i = 1, . . . , k + 1. If  k+1 i=1 ri  2k + 1, then R k+1 S % ni &   k+ i=1 ni = k+1 1. k + k+1 n − i=1 i i=1 (ni − 3 ) 3 R k+1 S   % ni &   k+ i=2 ni 2. n1 − n13−2 + k − 2 + k+1 < k+1

n − i=2 i i=1 (ni − 3 ). 3  Proof Notice that 2k + 2  k+1  ri < 3. i=1 ri  2k + 1 by the assumption 0  Where the first inequality assumes that each ri = 2. This means that k+1 i=1 ri is either 2k + 1 or 2k + 2, hence we must have Q P Q P   k + k+1 k + k+1 i=1 ni i=1 (3li + ri ) = 3 3 P k+1 Q  (3l ) + k + k+1 i i=1 i=1 ri = 3 =k+

k+1 

li

i=1

=k+

k+1    ni i=1

3

.

The first equality follows immediately. The inequality (2)requires the extra assumptionr1 = 2, since together with k+1 the assumption that k+1 ri  2k − 1. We write i=1 ri  2k + 1 it implies i=2 k+1 n1 = 3l1 + 2 and ni = 3li + ri , and use the inequality i=2 ri  2k − 1 to obtain the statement’s inequality. P Q  S k+1  k + k+1 n1 − 2 i=2 ni n1 − ni − +k−2+ 3 3 R

i=2

R

3l1 = n1 − 3

S +k−2+

k+1  i=2

P ni −

k+

k+1

i=2 (3li

3

+ ri )

Q

380

K.-N. Lin and S. Mapes

 n1 − l1 + k − 2 +

k+1 

P ni −

i=2

=k−2+

k+1 

ni − k + 1 −

i=1


pd(Qv ), then by Remark 2.7, we conclude that pd(HSμ ) = pd(Hv ) = pd(H). To see the proof of the claim, we use induction on k. When k = 2, Hv is a union of two strings of length n2 + n3 + 1 and n1 . When n2  2 and n3  2, the string of length n2 + n3 + 1 has two open strings with n2 − 1 and n3 − 1 open vertices. When (n2 = 1 and r3 = 2), or (n3 = 1 and r2 = 2), the string of length n2 + n3 + 1 has exactly 3 closed vertices at the ends of string and all other vertices are open. By the work of [13], Theorem 2.4 part 4, we have either pd(Hv ) = n1 − =

n  1

3

R + n2 + n3 + 1 − 2 −

S R S n3 − 2 n2 − 2 − +1 3 3

3 n   i ) = pd(HSμ ) (ni − 3 i=1

when n2  2, n3  2, r2 = 2 and r3 = 2, or pd(Hv ) = n1 − =

n  1

3

S R S n3 − 2 n2 − 2 + n2 + n3 + 1 − 2 − − 3 3 R

3 n   i ) = pd(HSμ ) (ni − 3 i=1

when n2  2, n3  2, (r2 = 1 and r3 = 2) or (r2 = 2 and r3 = 1), or pd(Hv ) = n1 − =

n  1

3

R

n3 − 2 + n2 + n3 + 1 − 1 − 3

S

3 n   i ) = pd(HSμ ) (ni − 3 i=1

when n2 = 1 and r3 = 2, or pd(Hv ) = n1 − =

n  1

3

R + n2 + n3 + 1 − 1 −

3 n   i ) = pd(HSμ ) (ni − 3 i=1

n2 − 2 3

S

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K.-N. Lin and S. Mapes

when n3 = 1 and r2 = 2. On the other hand, Qv is a union of two isolated vertices and two strings of length, n1 − 2, n2 − 2 + n3 + 1, hence we have R pd(Qv ) = 2 + n1 − 2 − 

S S R n1 − 2 n2 + n3 − 1 + n2 + n3 − 1 − 3 3

3 n   i ) − 1 < pd(Hv ). (ni − 3 i=1

For the second inequality above, we use the fact that r2 + r3  3. For the case when k > 2, we use the same exact sequence. Here the hypergraph  Hv is a union of a string of n1 vertices and a string of length μ = k − 1 + k+1 i=2 ni k+1 with a k −2-dimensional edge of k −1 vertices such that i=2 ri % 2(k & −1)+1. By the induction hypothesis and Theorem 2.4 part 4, pd(Hv ) = n1 − n31 +pd(HSμ ) = % ni & k+1 i=1 (ni − 3 ) = pd(HSμ ). On the other hand, Qv is a union of two isolated  closed vertices and two strings of length n1 − 2 and k − 3 + k+1 i=2 ni . Hence we have P Q  S R k+1  k − 3 + k+1 n1 − 2 i=2 ni pd(Qv ) = 2 + n1 − 2 − +k−3+ ni − 3 3 i=2

P Q  S k+1  k + k+1 n1 − 2 i=2 ni +k−2+ = n1 − ni − 3 3 R

i=2


pd(Qv1 ), then by Remark 2.7, we  conclude pd(H) = pd(Hv1 ) = 2k + 2 ki=1 li . Since k > 1, the vertex v1 corresponds to a monomial of degree 3. Notice that Hv1 is the union of a string of length n1 and a hypergraph with exactly the same structure of H (i.e. closed vertex on one end of string and an open vertex coinciding with a vertex of the higher dimensional edge at the other end) such that it has an edge with k − 1 vertices. By Theorem 2.4 part 4 and induction hypothesis, we have pd(Hv1 ) = n1 −

n  1

3

+ 2(k − 1) + 2

k 

li = 2k + 2

i=2

k 

li

i=1

and reg(Hv1 ) 

n  1

3

+k−1+

k 

li = k +

i=2

k 

li .

i=1

Moreover, the hypergraph Qv1 is a union of three isolated vertices and two strings  of length n1 −2 and n2 −2+nk −1+k−2+ k−1 i=3 ni . Therefore by Theorem 2.4 part k 4 with the fact that i=2 ri = 2(k − 1), we have P Q  S k  k − 5 + ki=2 ni n1 − 2 +k−5+ pd(Qv ) = 3 + n1 − 2 − ni − 3 3 R

= 2l1 + k − 2 +

k  i=2

P (3li + 2) −

i=2

k−5+

k

i=2 (3li + 2) 3

Q

384

K.-N. Lin and S. Mapes k  = 2l1 + k − 2 + 2(k − 1) + (3li ) −

P

i=2

= 3k − 4 + 2

k 

li − k + 3 = 2k + 2

i=1

k − 5 + 2(k − 1) + 3

k 

k

i=2 (3li )

Q

li − 1

i=1

and T reg(Qv ) =

W  U V k  k − 5 + ki=2 ni n1 − 2 li − 2. + =k+ 3 3 i=1

We have shown pd(H) = pd(Hv1 ) = 2k + 2

k 

li .

i=1

Using the short exact sequence and Theorem 2.4 part 8, we have and reg(H)  max{k +

k  i=1

li , k +

k 

li − 2 + 3 − 1} = k +

i=1

k 

li .



i=1

In the next lemma we need to consider a specific case which is necessary for the proof of Lemma 3.10. Specifically this will address Hv2 in Lemma 3.10 where H is a string together with an edge consisting of k vertices. Those two special cases are shown in Example 3.11 later. Lemma 3.8 We adapt Notation 3.1. Assume w1 = v1 and wμ = vk , i.e. n1 = nk+1 = 0, and both w1 = v1 and wμ = vk are open vertices. If ni = 2 + 3li for  i = 2, . . . , k. Then the projective dimension of Hv2 is 2(k − 1) + 2 ki=2 li and k reg(Hv2 )  k − 1 + i=2 li . Proof We use induction on k. When k = 2, Hv2 is a string of  n2 + 1. By  length n2 +1 3l2 +3 = 3l2 + 3 − = 2 + 2(l2 ) Theorem 2.4 part 4, pd(Hv2 ) = n2 + 1 − 3 3   and reg(Hv2 ) = n23+1 = 1 + l2 . For the induction step, we consider the short exact sequence 0 → Hv1 ,v2 : v1 → Hv1 ,v2 → Hv2 → 0, where Hv1 ,v2 is the hypergraph obtained from H after removing vertices v1 and v2 . The proof is almost identical to the Lemma 3.7 except that v1 corresponds to a monomial of degree 2. Hv1 ,v2 is a union of a string of length n2 and a hypergraph satisfying the assumptions of Lemma 3.7, and Hv1 ,v2 : v1 is a union of two isolated

Projective Dimension of Hypergraphs

385

vertices,  and two strings of length n2 − 2 and nk − 1 when k = 3, and k nk − 1 + k − 3 + k−1 n when k > 3. We have reg(H : v ) = k − 1 + i v ,v 1 1 2 i=2 li − 1 and i=3  pd(Hv1 ,v2 : v1 ) = 2(k−1)+2 ki=2 li −1. Hence we have pd(Hv2 ) = pd(Hv1 ,v2 ) =    2(k − 1) + 2 ki=2 li and reg(Hv2 )  max{k − 1 + ki=2 li , k − 1 + ki=2 li − 1 + k 2 − 1} = k − 1 + i=2 li by Theorem 2.4 part 8.

Now we will use the splitting type result in Lemma 2.13 to finish our necessary results for the hypergraphs which are a string together with an edge consisting of k vertices, where the spacing between the vertices of the edge are equivalent to 2 modulo 3. The next lemma deals with the intersection ideal in the special case where G1 will correspond to one vertex of a larger hypergraph H. Lemma 3.9 We adapt Notation 3.1. Assume k > 2, w1 = v1 and wμ = vk , i.e. n1 = nk+1 = 0, and both w1 = v1 and wμ = vk are open vertices. Furthermore, ni = 2 + 3li for i = 2, . . . , k. Let G1 = {v2 } and G2 = Hv2 (which is just H with v2 removed) and denote Ii = I (Gi ) as the ideals corresponding to Gi , then I1 ∩ I2 = mv2 I  where mv2 is the monomial corresponding to the vertex v2 and H(I  ) has 4 isolated vertices and 2 strings one of length: • n2 − 3 when n2 > 2, or • 0 if n2 = 2 and the other of length:

 • k − 3 + nk − 1 + n3 − 2 + k−1 i=4 ni when k > 3, or • n3 − 3 when k = 3 and when n3 > 2, or • 0 when k = 3 and n3 = 2



Proof To see this consider the hypergraph H, for notational convenience, let us denote the vertex neighboring v1 as wα , the vertices neighboring v2 as wβ1 and wβ2 , and the vertex neighboring vk as wγ . Now removing v2 from H leaves us with a hypergraph on the same vertex set excluding v2 and all vertices remain open except wβ1 and wβ2 which become closed, together with a k − 1 edge F  that has {v1 , v3 , . . . , vk } as its vertex set (note this also describes the hypergraph Hv2 ). Now we consider the intersection with the ideal generated by G1 . A first step towards finding these new generators is to multiply each generator for G2 by the monomial mv2 corresponding to v2 in the original H. The result on hypergraphs is now we get a hypergraph, which we will denote as mv2 Hv2 , consisting of 2 strings of only open vertices where one string is the part of H consisting of v1 to wβ1 and the other is wβ2 to vk , and all the open vertices are in an edge corresponding to mv2 . Note that the spacing measurements for mv2 Hv2 are the same as for Hv2 . The issue here is that mv2 Hv2 is not separated (i.e. the generators are not minimal). Denote the vertices neighboring each wβi as wβi . It is easy to see that removing vertices wα , wβ1 , wβ2 , and wγ from mv2 Hv2 produces the desired separated hypergraph corresponding to I1 ∩ I2 .

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K.-N. Lin and S. Mapes

Notice that when n2 = 2 then wα = wβ1 and similarly if k = 3 and n3 = 2 then wβ2 = wγ . It is easy to see then that we get 4 isolated vertices corresponding to the original v1 , wβ1 , wβ2 and vk . And the remaining chains of open strings reflect removing v1 , wα , wβ1 and wβ1 from the chain of length n2 + 1 so the result is a chain of length n2 − 3 when n2 > 2, or 0 when n2 = 2. Similarly, after removing vk , wβ2 , wβ2 and wγ from a chain of length k − 2 + n3 + ki=4 ni results a chain of  length k − 3 + n3 − 2 + nk − 1 + k−1 i=4 ni when k > 3, or n3 − 3 when k = 3 and n3 > 2, or 0 when k = 3 and n3 = 2.

Lemma 3.10 We adapt Notation 3.1. Assume w1 = v1 and wμ = vk , i.e. n1 = nk+1 = 0, and both w1 = v1 and wμ = vk are open vertices. If ni = 2 + 3li for 

i = 2, . . . , k. Then the projective dimension of H is 2(k − 1) + 2 ki=2 li + 1. Proof If k = 2, then H is an open  cycleof length 2 + n2 , then by Theorem 2.4 part 6, pd(H) = 2 + n2 − 1 − 2+n32 −2 = 2 + 2l2 + 1. We now assume k >

2. By making vertices v1 , . . . , vk become closed, we obtain a hypergraph H and pd(H)  pd(H ) by Theorem 2.4 part 3. Let H be the hypergraph obtained from H by removing the higher dimensional edge. Then by Theorem 2.4 part 2, we have that pd(H ) = pd(H ) because all the vertices v1 , . . . , vk are closed in H . Note  that H is a string of length k + ki=2 ni with k − 1 open strings of n2 , . . . , nk open vertices. By Theorem 3.4 in [13], the projective dimension is the sum of projective dimension of each open string plus 1, hence we get pd(H ) =

k  i=2

ni −

k    ni i=2

3

+ 1 = 2(k − 1) + 2

k 

li + 1.

i=2

 Thus, we obtain pd(H)  2(k − 1) + 2 ki=2 li + 1. Now we consider H = {v2 } ∪ Hv2 and we will show it satisfies the condition of Remark 2.14 with V1 = {v2 } and V2 as the vertex set of Hv2 . Denote Ii as the ideal generated by the generators of I (H) corresponding to Vi , and G1 = {v2 } = H(I1 ) and G2 = Hv2 = H(I2 ). First notice that pd(G1 ) = 1 and reg(G1 ) = 2, since the degree of the generator corresponding to v2 is 3.Moreover G2 satisfies the condition of Lemma 3.8, hence pd(G2 ) = 2(k − 1) + 2 ki=2 li = q and reg(G2 )  k − 1 + k   isolated i=2 li = r − 1. By Lemma 3.9, I1 ∩ I2 = mv2 I where H(I ) has 4 vertices and two strings of length n2 − 3 and k − 3 + nk − 1 + n3 − 2 + k−1 i=4 ni . Then by ni = 2 + 3li for i = 2, . . . , k, Theorem 2.4 parts 4 and 7 we get pd(H(I1 ∩ I2 ))

P Q  S k  k − 6 + ki=3 ni n2 − 3 +k−6+ ni − = 4 + n2 − 3 − 3 3 R

i=3

Projective Dimension of Hypergraphs

387

P Q  S k  k − 6 + ki=3 (3li + 2) 3l2 − 1 +k−6+ = 3 + 3l2 − (3li + 2) − 3 3 R

i=3

= 2l2 − 2 + k + 2(k − 2) +

k 

P

(3li ) −

i=3

= 2l2 − 2 + k + 2(k − 2) +

k 

(3li ) − k + 4 −

i=3

= 2(k − 1) + 2

k 

k − 6 + 2(k − 2) + 3 k 

k

i=3 (3li )

Q

(li )

i=3

li = q

i=2

and W  U V k − 6 + ki=3 ni n2 − 3 + reg(H(I1 ∩ I2 )) = 3 + 3 3 V W  U T k − 6 + 2(k − 2) + ki=3 (3li ) 3l2 − 1 + =3+ 3 3 T

= 3 + l2 + k − 3 +

k  i=3

li = k +

k 

li = r.

i=2

Moreover, by Theorem 2.4 parts 4 and 5, we have βq,q+r (H(I1 ∩I2 )) = 0. Hence  by Lemma 2.13, pd(H) = 2(k − 1) + 2 ki=2 li + 1. For the smaller cases, one can check similarly.

The following example offers a view of the “splitting” in the proof of Lemma 3.10. Example 3.11 Let H be the whole hypergraph in the left side of Fig. 4. Let V1 be the w4 = v2 (black part) of H and V2 = {w1 , w2 , w3 , w5 , w6 , w7 , w8 , w9 , w10 }

Fig. 4 Splitting hypergraph

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(blue part) of H. Let G1 and G2 be the hypergraphs associated to the vertex sets V1 and V2 . Then the edges {w3 , w4 }, {w4 , w5 }, and {w1 , w4 , w7 , w10 } (the purple part) are the shared edges or edges of G1 and G2 . In the right of Fig. 4, we show the hypergraphs for G1 and G2 separately. Now with Lemma 3.10 we are ready to address the case when the sum of the ni is 2k modulo 3. In this case Lemma 3.12 will be an instance of the special sub-case, and Proposition 3.13 will give the general result. Lemma 3.12 We adapt Notation 3.1. If r1 = 1 = rk+1 , and ri = 2 for all 1 < i < k + 1, then pd(H) = pd(HSμ ) + 1.

R k+1 S   k+ i=1 ni Proof First notice that pd(HSμ ) = k + k+1 and k+1 i=1 ni − i=1 ri = 2k. 3 We simplify pd(HSμ ) first. pd(HSμ ) = k +

k+1 

P ni −

k+

k+1

= k + 2k +

P

k+1 

Q

(3li ) −

k + 2k +

i=1

= 2k +

ni

3

i=1 k+1 

i=1

k+1

i=1 (3li )

Q

3

2li .

i=1

Let xE be the variable corresponding to the edge Ex that connecting v1 = wi1 and the vertex of wi1 −1 and yE be the variable corresponding to the edge Ey that connecting vk = wik and the vertex of wik +1 . We consider the short exact sequences: 0 → (H : Ex ) → H → (H, xE ) → 0, and 0 → ((H : Ex ) : Ey ) → (H : Ex ) → ((H : Ex ), yE ) → 0. Notice that ((H : Ex ) : Ey ) is a union of two isolated vertices, two strings of length n1 − 2 and nk+1 − 2, and a hypergraph that satisfies assumptions of Lemma 3.10. Now with assumptions of ri ’s, we have S S R n1 − 2 nk+1 − 2 + nk+1 − 2 − pd((H : Ex ) : Ey ) = 2 + n1 − 2 − 3 3 R

+ 2(k − 1) + 2

k  i=2

li + 1

Projective Dimension of Hypergraphs

389

= 2 + 2l1 + 2lk+1 + 2(k − 1) + 2

k 

li + 1

i=2

= 2k +

k+1 

2li + 1 = pd(HSμ ) + 1.

i=1

Since (H, xE ) is a  union of an isolated vertex, a string of length n1 − 1, and a string of length k−1+ k+1 such that n2 , . . . , nk+1 i=2 ni with a k−2-dimensional edge  are the numbers of vertices between vertices of F and k+1 i=2 ri = 2k − 1. By Proposition 3.6 and Theorem 2.4 part 4, P Q  S k+1  k − 1 + k+1 n1 − 1 i=2 ni +k−1+ pd(H, xE ) = 1 + n1 − 1 − ni − 3 3 R

k+1  = 2l1 + k + 2k − 1 + (3li ) − i=2

< 2k +

k+1 

P

i=2

k − 1 + 2k − 1 + 3

k+1

i=2 (3li )

Q

2li + 1.

i=1

Moreover, ((H : Ex ), yE ) is a union of two isolated vertices, two strings of length n1 − 2 and nk+1 − 1 and a hypergraph satisfies the assumptions of Lemma 3.7 with a k − 2-dimensional edge. Then by Lemma 3.7 and Theorem 2.4 part 4, S S R n1 − 2 nk+1 − 1 + nk+1 − 1 − 3 3 P Q  k  k − 1 + ki=2 ni +k−1+ ni − 3 R

pd((H : Ex ), yE ) = 2 + n1 − 2 −

i=2

= 1 + 2l1 + 2lk+1 + k + 2(k − 1) + P −

k − 1 + 2k − 2 + 3

< 2k +

k+1 

k

i=2 (3li )

Q

k 

(3li )

i=2

2li + 1 = pd((H : Ex ) : Ey ).

i=1

Since pd((H : Ex ) : Ey ) > pd((H : Ex ), yE ), we have pd(H : Ex )  max{pd((H : Ex ) : Ey ), pd((H : Ex ), yE )} = pd((H : Ex ) : Ey )

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and pd((H : Ex ) : Ey )  max{pd(H : Ex ), pd((H : Ex ), yE ) − 1}. This shows pd(H : Ex ) = pd((H : Ex ) : Ey ). Similarly pd(H, xE ) < pd(H : Ex ) gives pd(H) = pd(H : Ex ) = pd HSμ + 1.

k+1 We are now finally ready for the case, i=1 ri = 2k.  Proposition 3.13 We adapt Notation 3.1 and assume k+1 i=1 ri = 2k with r1 = 2. If ri = 0 for all i, then pd(H) = (pd HSμ ) + 1, otherwise pd(H) = pd(HSμ ). Proof We consider the short exact sequence 0 → Qv1 → Hv1 → H → 0. We will show pd(Hv1 ) > pd(Qv1 ). Moreover, we will show that when ri = 0 for all i, then pd(Hv1 ) = pd(HSμ ) + 1, and otherwise pd(Hv1 ) = pd(HSμ ). These two claims with Remark 2.7 will prove the proposition. We proceed  by induction on k for both claims. k+1 We observe that μ = k + k+1 i=1 ni and pd(HSμ ) = 2k + i=1 2li as in the proof of Lemma 3.12. When k = 2, observe that by Definition 2.6 and Discussion 2.8 in [14], Qv1 is a union of two isolated vertices, and two strings of length n1 − 2 and n2 − 2 + n3 + 1. Hence by Theorem 2.4 part 4, and r1 = 2 implies r2 + r3 = 2, S S R n1 − 2 n2 + n3 − 1 + n2 + n3 − 1 − pd(Qv1 ) = 2 + n1 − 2 − 3 3 R

= 2 + 2l1 + 2l2 + 2l3 + 1 pd(Qv ) and this concludes the case when k = 2. Now suppose F is a k-dimensional edge with k + 1 vertices, v1 , . . . , vk+1 . Suppose ri = 0 for all i then again by Definition 2.6 and discussion 2.8 in [14], Qv1 is either 1.  a union of two isolated vertices, a string of length n1 − 2 and a string of length k+2 i=2 ni + k − 2 when n2  2, or 2. Qv1 is a union of two isolated vertices, a string of length n1 − 2 and a string of  length k+2 i=3 ni + k − 1 when n2 = 1.  For the later case, k+2 i=3 ri = 2k − 1, then by Theorem 2.4 part 4, P Q  S k+2  k − 1 + k+2 n1 − 2 i=3 ni +k−1+ ni − pd(Qv1 ) = 2 + n1 − 2 − 3 3 R

i=3

= 2 + 2l1 + k − 1 + 2k − 1 +

k+2 

P

3li −

i=3

= 2l1 + 2k +

k+2 

k − 1 + 2k − 1 + 3

k+2 i=3

3li

(2li ) + 1

i=3

< 2k + 2 +

k+2  (2li ) + 1. i=1

For the first case, by Theorem 2.4 part 4, we have P Q  S k+2  n k − 2 + k+2 n1 − 2 i i=2 pd(Qv1 ) = 2 + n1 − 2 − +k−2+ ni − 3 3 R

= 2l1 + k + 2k +

k+2 

P 3li −

i=2

= 2k + 1 +

k+2 

i=2

k − 2 + 2k + 3

2li

i=1

< 2k + 2 +

k+2 

2li + 1

i=1

with the fact r1 + . . . + rk+2 = 2k + 2 and r1 = 2.

k+2 i=2

3li

Q

Q

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K.-N. Lin and S. Mapes

Now in both cases consider that Hv1 is a union of a string of length n1 and a string  k+2 of length k+2 i=2 ni + k with a k − 1-dimensional edge. Notice that i=2 ri = 2k and ri = 0 for all i such that 2  i  k + 2. By induction and Theorem 2.4 part 4, we have P Q  k+2 n   k + k+2 1 i=2 ni +k+ pd(Hv1 ) = n1 − ni − +1 3 3 i=2

= 2k + 2 +

k+2 

2li + 1 = pd(HSμ ) + 1

i=1

where we use the fact that r2 + . . . + rk+2 = 2k and r1 = 2. Therefore pd(Qv1 ) < pd(Hv1 ), and pd(H) = pd(Hv1 ) = pd(HSμ ) + 1, satisfying the 2 claims. Now suppose ri = 0 for some i > 1 then rj = 2 for all j = i. Notice again by the discussions in [14] that Qv1 is either 1.  a union of two isolated vertices, a string of length n1 − 2 and a string of length k+2 i=2 ni + k − 2 when n2  2, or 2. Qv1 is a union of two isolated vertices, a string of length n1 − 2 and a string of  length k+2 i=3 ni + k − 2 when n2 = 0.  For the later case, we have k+2 i=3 ri = 2k. By Theorem 2.4 part 4, P Q  S k+2  k − 2 + k+2 n1 − 2 i=3 ni +k−2+ ni − pd(Qv1 ) = 2 + n1 − 2 − 3 3 R

= 2l1 + k + 2k +

k+2 

P 3li −

i=3

= 2k + 1 +

k+2 

i=3

k − 2 + 2k + 3

k+2 i=3

3li

Q

2li

i=1

< 2k + 2 +

k+2 

2li .

i=1

For the first case, by Theorem 2.4 part 4, we have P Q  S k+2  k − 2 + k+2 n1 − 2 i=2 ni +k−2+ ni − pd(Qv1 ) = 2 + n1 − 2 − 3 3 R

= 2l1 + k + 2k +

k+2  i=2

P 3li −

i=2

k − 2 + 2k + 3

k+2 i=2

3li

Q

Projective Dimension of Hypergraphs

= 2k + 1 +

k+2 

393

2li

i=1

< 2k + 2 +

k+2 

2li .

i=1

with the fact that r2 + . . . + rk+2 = 2k and r1 = 2. Similar to the case where ri is never 0, we get that Hv1 is a union of a string of length n1 and a string of length k+2 k+2 i=2 ni + k with a k − 1-dimensional edge. Notice that i=2 ri = 2k and ri = 0 for some 2  i  k + 2. So by induction and Theorem 2.4 part 4, we have pd(Hv1 ) = n1 −

n  1

3

= 2k + 2 +

+k+

k+2  i=2

k+2 

P ni −

k+

k+2 i=2

ni

Q

3

2li

i=1

= pd(HSμ ) where we use the fact that r2 + . . . + rk+2 = 2k and r1 = 2. Hence pd(Qv1 )
2k, we have pd(H) = pd(HSμ ) by Proposition 3.5 and Proposition 3.6. We are left to consider the case, r1 + . . . + rk+1 = 2k. If ri = 0 for some i, then rj = 2 for all j = i, hence Proposition 3.13 applies. If ri = 0 for all i, and r1 = rk+1 = 1, then Lemma 3.12 applies, otherwise, we may assume r1 = 2 and Proposition 3.13 applies again.

Example 3.15 Three hypergraphs in Fig. 5 are strings attached with one extra blue edges (checkerboard region) such that μ = 9, k = 3 and r1 + . . . + rk+1 = 2k = 6. H1 and H2 satisfy ri = 0 for all  but H3 has r4 = 0. Therefore pd(H1 ) =  i, 9 pd(H2 ) = pd(HSμ ) + 1 = 9 − 3 + 1 = 7, and pd(H3 ) = pd(HSμ ) = 6 by Theorem 3.14.

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Fig. 5 Hypergraphs of Example 3.15

Fig. 6 Hypergraph of Example 3.16

Remark 3.16 With Theorem 3.14, one can easily compute the projective dimension of a string with extra edges. We first remove all the edges that are union of other edges using Theorem 2.4 part 2. If we are left with one higher dimensional edge, we can use Theorem 3.14 and Theorem 2.4 part 4 to compute the projective dimension. Example 3.17 illustrates the process.

Example 3.17 Let H be the hypergraph shown in Fig. 6. We can remove all red edges (dashed vertical line and the bricks regions) using Theorem 2.4 part 2 and remove the blue edge (checkerboard regions) using Theorem 3.14. The projective  dimension of the hypergraph is 11 − 11 = 8.

3

4 Cycles with Higher Dimensional Edges Now we examine the case where we have added a higher dimensional edge to an open cycle. Notation 4.1

Let H be a hypergraph and F be an edge of H.

1. Let HCμ be an open cycle with vertex set V = {w1 , . . . , wμ }, and an edge F with k > 1 vertices {v1 = wi1 , . . . , vk = wik } ⊆ {w1 , . . . , wμ } with 1  i1 < · · · < ik  μ. When we say H is an open cycle together with an edge consisting of k vertices, we mean H = HCμ ∪ F . 2. Let nj = ij −ij −1 −1 for j = 1, . . . k be the number of vertices between vertices of F .

Projective Dimension of Hypergraphs

395

3. We write ni = 3li + ri where li are some non-negative integers, and 0  ri  2 for i = 1, . . . , k + 1. We start by showing that the induced hypergraph of HCμ on the complement of VF has smaller projective dimension than that of HCμ when the sum of (the ni modulo 3) is less than 2k − 1. This is necessary in the proof of Theorem 4.4. Lemma 4.2 We adapt Notation 4.1. Let HVF = HCμ ∩ (V \VF ) be the hypergraph obtained by removing all the vertices v1 , . . . , vk . If r1 + . . . + rk < 2k − 1, then pd(HVF ) < pd HCμ .

k Proof Notice that μ = k + i=1 ni and R pd HCμ = μ − 1 − =k−1+

μ−2 3

k 

S

ni −

i=1

>k−1+

k 

P

P ni −

i=1

=

k 

k 

ni −

i=1

Q  k − 2 + ki=1 ni 3 k − 2 + 2k − 1 + 3

k

i=1 3li

Q

li

i=1

by Theorem 2.4 part . . + rk < 2k − 1. On the other & % assumption  r1 + .  6 and the hand, pd HVF = ki=1 (ni − n3i ) = ki=1 ni − ki=1 li because HVF is union of k strings such that each string has ni vertices. Notice that this notation allows that

ni can be 0 for some i. Hence we have pd(HVF ) < pd HCμ . Next, we show that when the sum of the ni modulo 3 is greater than 2k − 1 that the projective dimension of H is the same as for the underlying cycle.  Lemma 4.3 We adapt Notation 4.1. If ki=1 ri  2k − 1, then pd(H) = pd(HCμ ).

k Proof By Theorem 2.4 part 6 and the assumption 2k  i=1 ri  2k − 1, we have pd(HCμ ) = k − 1 +

k 

R ni −

i=1

=k−1+

k 

P ni −

i=1

=

k  i=1

ni −

k  i=1

li .

 S k − 2 + ni=1 ni 3 k−2+

k

i=1 ri

3

+

k

i=1 3li

Q

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K.-N. Lin and S. Mapes

 Since ki=1 ri  2k − 1, ri < 3, and k > 1, we have at most one ri such that ri = 1. We may assume rk = 1 if there is one otherwise ri = 2 for all i. Let Hv1 = Hv be the hypergraph removing the vertex v1 = v from H and let Hv : v1 = Qv . We have a short exact sequence 0 → Qv → Hv → H → 0. We will show that pd(Hv ) > pd(Qv ), then by Remark 2.7, we conclude that pd(H) = pd(Hv ) = pd(HCμ ). When k = 2, Hv is a string of length n1 + n2 + 1 with two open strings of n1 − 1 and n2 − 1 open vertices. Qv is the union of two closed vertices and an open string of n1 − 2 + n2 − 2 + 1 vertices. By Theorem 2.4 part 4 and Theorem 3.4 in [13], we have R S R S n1 − 2 n2 − 2 pd(Hv ) = n1 + n2 + 1 − (2 + + ) + 1 = 2l1 + 2l2 + 4, 3 3 and

R

n1 + n2 − 3 pd(Qv ) = 2 + n1 + n2 − 3 − 3

S = 2l1 + 2l2 + 3

when r1 + r2 = 4, and S R S n2 − 2 n1 − 2 pd(Hv ) = n1 + n2 + 1 − (2 + + ) + 1 = 2l1 + 2l2 + 3, 3 3 R

R

n1 + n2 − 3 pd(Qv ) = 2 + n1 + n2 − 3 − 3 when r1 + r2 = 3. For k > 2, we show pd(Hv ) = k − 1 +

k

R

i=1 ni −

k

S = 2l1 + 2l2 + 2 S  k−1+ ki=1 ni 3

=

k

i=1 ni

k

−

> pd(Qv ). Notice that Hv is a string of length μ − 1 = k − 1 + i=1 ni  attached with a k − 2-dimensional edge of k − 1 vertices. Moreover, ki=1 ri  2k − 1 = 2(k − 1) + 1. By Proposition 3.6, i=1 li

pd(Hv ) = pd(HSμ−1 ) = k − 1 +

k 

P ni −

i=1

Q  k k   k − 1 + ki=1 ni ni − li . = 3 i=1

i=1

On the other hand, Qv isthe union of two closed vertices and an open string of length n1 − 2 + k − 3 + ki=2 ni hence by Theorem 2.4 part 4, pd(Qv ) = 2 + k − 5 +

k  i=1

P ni −

Q  k − 5 + ki=1 ni 3

Projective Dimension of Hypergraphs

k−3+

k 

397

P ni −

i=1

=

k  i=1

when

k

i=1 ri

ni −

k 

k − 5 + 2k − 1 + 3

k

i=1 3li

Q

li − 1 < pd(Hv )

i=1

 2k − 1.



Now we can show that the projective dimension is always preserved H and HCμ differ by a single higher dimensional edge F . Theorem 4.4 Let HCμ be the cycle hypergraph with μ open vertices. Let H = HCμ ∪ F where F is a k − 1-dimensional edge with k vertices. Then pd(H) =   .

pd(HCμ ) = μ − 1 − μ−2 3 Proof The second equality is coming from Theorem 2.4 part 6. Let VF = {v1 , . . . , vk }, which is a subset of the vertex set of HCμ , be the vertex set of F . Let  ni = 3li +ri be the spacing between the vi and defined as before. If ki=1 ri  2k−1 then k the theorem holds by Lemma 4.3. So we need only consider the case when 2k − 1. i=1 ri < k When i=1 ri < 2k − 1, we will use Lemma 4.2 and the short exact sequence 0 → (H : F ) = HCμ → H → (H, xF ) → 0 where xF is the variable corresponding to the edge F . The short exact sequence gives pd(H)  max{pd(HCμ ), pd(H, xF )}. Since (H, xF ) is the union of HVF and an isolated vertex presenting the variable of xF , we have pd(H, xF ) = pd HVF + 1  pd HCμ by Lemma 4.2. By Theorem 2.4 part 3, we have to pd HCμ  pd H. Therefore we have pd H  max{pd(HCμ ), pd(H, xF )} = pd HCμ  pd H.

Remark 4.5 As before, one can compute the projective dimension of an open cycle with extra edges either using Theorem 2.4 part 2 or above theorem.

Remark 4.6 It is natural to conjecture that given an open cycle, no matter how many higher dimensional edges are on the cycle the projective dimension of the hypergraph is the same as the projective dimension of the open cycle. There are more cases that one needs to consider for the proof of the conjecture. In particular, in Theorem 3.14 we have a case for strings where the projective dimension does not stay the same once a higher dimensional edge is removed. The proof of the case when the projective dimension jumped up by one requires a lot of steps. In light of this, new tools must be developed in order to prove the conjecture for cycles.



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Acknowledgments The authors want to express heartfelt thanks to the reviewer for the suggestions that greatly improved this paper.

References 1. Claude Berge, Hypergraphs. Combinatorics of finite sets. Translated from the French. NorthHolland Mathematical Library, 45. North-Holland Publishing Co., Amsterdam, (1989). 2. David Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics 150, Springer-Verlag New York (2004). 3. Shalom Eliahou and Michel Kervaire, Minimal resolutions of some monomial ideals. J. Algebra 129 (1990), 1–25. 4. John Eagon, Ezra Miller and Erika Ordog, Minimal resolutions of monomial ideals. arXiv:1906.08837. 5. Giuliana Fatabbi, On the Resolution of Ideals of Fat Points. J. Algebra 242 (2001), 92–108. 6. Chris A. Francisco, Huy Tài Hà and Adam Van Tuyl, Splittings of monomial ideals. Proc. Amer. Math. Soc. 137 (2009), 3271–3282. 7. Huy Tài Hà, Regularity of squarefree monomial ideals. Connections between algebra, combinatorics, and geometry, Springer Proc. Math. Stat., 76 Springer, New York (2014), 251–267. 8. Huy Tài Hà and Kuei-Nuan Lin, Normal 0-1 polytopes. SIAM J. Discrete Math. 29 (2015), no. 1, 210–223. 9. Sean Jacques, The Betti numbers of graph ideals. Ph.D. Thesis, The University of Sheffield 2004, arXiv.math.AC/0410107. 10. Sean Jacques and Mordechai Katzman, The Betti numbers of forests. arXiv.math.AC/0501226 11. Kyouko Kimura and Paolo Mantero, Arithmetical rank of strings and cycles. J. Commut. Algebra 9 (2017), no. 1, 89–106. 12. Kyouko Kimura, Naoki Terai, and Ken-ichi Yoshida, Arithmetical rank of squarefree monomial ideals of small arithmetic degree. J. Algebraic Combin. 29 (2009), no. 3, 389–404. 13. Kuei-Nuan Lin and Paolo Mantero, Projective dimension of string and cycle hypergraphs. Comm. Algebra 44 (2016), no. 4, 1671–1694. 14. Kuei-Nuan Lin and Paolo Mantero, High projective dimension and 1-dimensional hypergraphs. Int. J. Algebra Comput. 27, No. 6 (2017), 591–617. 15. Kuei-Nuan Lin and Sonja Mapes, Lattices and Hypergraphs associated to square-free monomial ideals arXiv:1804.05919. 16. Kuei-Nuan Lin and Jason McCullough, Hypergraphs and regularity of squarefree monomial ideals. Internat. J. Algebra Comput. 23 (2013), no. 7, 1573–1590.

A Truncated Minimal Free Resolution of the Residue Field Van C. Nguyen and Oana Veliche

Keywords Golod rings · Minimal free resolutions · Koszul complex · Tor algebra · Massey products

1 Introduction Throughout the paper, (R, m, k) is a local Noetherian ring with maximal ideal m and residue field k, of embedding dimension n and codepth c = n − depth R. Let K R be the Koszul complex of R on a minimal set of generators of m. Serre pointed out that there is always a coefficient-wise inequality between the Poincaré series of the R-module k and a rational series involving the ranks of the homologies of K R : PR k (t) :=

∞  i=0

i rankk TorR i (k, k)t 

1−

(1 + t)n . R i+1 i=1 rankk Hi (K )t

c

(1)

In [8], Golod proved that a ring R attains this upper bound if and only if the graded-commutative algebra H(K R ) has trivial multiplications and trivial Massey operations; such a ring is now called a Golod ring. In the same paper, he also constructed the minimal free resolution of the R-module k in terms of K R . In [2, Corollary 5.10], Avramov proved that the Poincaré series PR k (t) is completely determined by the structure of the Koszul homology algebra A := H(K R ) as an algebra with Massey operations. One can now ask the following question:

V. C. Nguyen Department of Mathematics, United States Naval Academy, Annapolis, MD, USA e-mail: [email protected] O. Veliche () Department of Mathematics, Northeastern University, Boston, MA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Miller et al. (eds.), Women in Commutative Algebra, Association for Women in Mathematics Series 29, https://doi.org/10.1007/978-3-030-91986-3_17

399

400

V. C. Nguyen and O. Veliche

Question 1.1 For any local ring, given the knowledge of its Koszul homology algebra, with its products and Massey operations, how does one construct a minimal free resolution of its residue field? In this paper, we answer this question explicitly up to degree five. For any local ring R, using the components of the Koszul complex K R as building blocks and the graded-commutative structure of its homology A, we construct a minimal free resolution of the R-module k, up to degree five, see Construction 3.1 and Theorem 3.1: F :

∂5F

∂4F

∂3F

∂2F

∂1F

F5 −→ F4 −→ F3 −→ F2 −→ F1 −→ F0 → k → 0.

The higher degrees of the resolution can be extended similarly for some special cases of R, but in general it requires an understanding of higher Massey products, which remains elusive and is left for future projects. Indeed, Massey operations can be represented by using the A∞ -algebra structure, which the Koszul homology algebra A possesses, see for example, [12] by Lu, Palmieri, Wu, and Zhang. Moreover, the A∞ -structure was also used by Burke in [6] to construct certain projective resolutions. These connections suggest that perhaps the A∞ -algebra structure of A may play a role in giving a complete answer to Question 1.1 from this perspective. To prepare for Construction 3.1, in Sect. 2 we analyze in detail the multiplicative structure of the algebra A, up to degree four, in terms of bases of Ai and Ai · Aj , and necessary maps. Moreover, in degree four the Massey products appear for the first time, as ternary Massey products A1 , A1 , A1 ; we give a description of the elements of this set in Proposition 2.8. In Sect. 4, we obtain several direct applications of Theorem 3.1. In Corollary 4.1, we explicitly describe the Betti numbers βi := rankk TorR i (k, k), up to degree five, in terms of the multiplicative invariants of the Koszul homology algebra A. Let ai := rankk Ai and consider the difference of series P(t) :=

(1 + t)n  − PR k (t) 1 − ci=1 ai t i+1

that measures how far the ring R is from being Golod, that is, how far the Betti numbers of R are from their maximum possible values. In Proposition 4.2, we compute the first six coefficients of P(t) in terms of multiplicative invariants of A: P0 = P1 = P2 = 0, P5 =

P3 = q11 ,

! n+1! 2 + 2a1 q11

P4 = (n + 1)q11 + q12 ,

+ (n + 1)q12 + a − b,

where qij = rankk (Ai · Aj ), and a, b are described in Summary 2.5. For any local ring R of embedding dimension n with rational Poincaré series of the form PR k (t) =

A Truncated Minimal Free Resolution of the Residue Field (1+t)n d(t) ,

401

we express the denominator, up to degree five, as follows:

d(t) = 1 −

c

i=1 ai t

i+1

!

+ q11 t 3 + (q11 + q12 )t 4 + (q12 − b + a)t 5 + f (t)t 6 ,

for some f (t) ∈ Z[t], see Proposition 4.5. We consider a few classes of rings with such rational Poincaré series, see Corollary 4.10 and Examples 4.11 and 4.12, and using the coefficients of the denominator we obtain algebraic invariants of their Koszul homology algebra A. The Poincaré series PR k (t) can be described by using the deviations εi ’s, see for example [4]. In Corollary 4.13, we give a description of the first five deviations in terms of the algebraic invariants of A. In Sect. 5, we illustrate Construction 3.1 through an example of a ring of codepth 4 examined in [2]. In that example, Avramov provided a nontrivial indecomposable Massey product element in A1 , A1 , A1  that does not come from multiplications of the homology. Using Proposition 2.8, we prove that, up to a scalar, this is the only element with this property, modulo the products in homology.

2 Multiplicative Structure on the Homology of the Koszul Algebra Let (R, m, k) be a local ring of embedding dimension n and codepth c. The differential graded algebra structure on the Koszul complex K = K R on a minimal set of generators of the maximal ideal m induces a graded-commutative algebra structure on the homology of K: A := H(K) = A0 ⊕ A1 ⊕ A2 ⊕ · · · ⊕ Ac . In this section we discuss this multiplicative structure up to degree four. We will use this structure extensively in constructing a truncated minimal free resolution of the residue field over R in the next section. The following notation is used throughout the paper: the differential map on K is denoted by ∂ K , a homogeneous element of degree i in the Koszul complex K is denoted by π i , a representative in Ki of an element in the homology Ai is denoted by pi , and elements of R are denoted by α’s, β’s, and γ ’s. The subscript of a homogeneous element indicates its index in a tuple and the superscript indicates its homological degree. Set ai := rankk (Ai )

and

qij := rankk (Ai · Aj ), for all 0  i, j  c.

(2.1)

It is clear that A0 = k. For each 1  i  a1 we consider zi1 ∈ Ker ∂1K such that {[zi1 ]}i=1,...,a1

is a basis of A1 .

(2.2)

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In particular, every element in Ker ∂1K can be written as a sum of elements in Im ∂2K and a linear combination of {zi1 }i=1,...,a1 .

2.1 Products in Degree Two Let A2 be a k-subspace of A2 such that A2 = (A1 · A1 ) ⊕ A2 . For each 1   a2 − q11 , we consider z 2 ∈ Ker ∂2K such that {[z 2 ]} =1,...,a2 −q11

is a basis of A2 .

(2.3)

In particular, every element in Ker ∂2K can be written as a sum of elements in Im ∂3K , a linear combination of {zi1 ∧ zj1 }i,j =1,...,a1 , and a linear combination of {z 2 } =1,...,a2 −q11 . Consider the multiplication map: φ1 : A1 ⊗ A1 → A2 ,

φ1 ([x] ⊗ [y]) = [x] ∧ [y],

defined by

(2.4)

for all [x], [y] ∈ A1 . As Im φ1 = A1 · A1 , we have rankk (Ker φ1 ) = a12 − q11 . For 1 ∈ Ker ∂ K such that si all 1  i  a1 and 1  s  a12 − q11 , we choose p 1 a1 # $ 1 [zi1 ] ⊗ [ psi ]

s=1,...,a12 −q11

i=1

For each s, we have

a1

1 1] psi i=1 [zi ] ∧ [

is a basis of Ker φ1 .

(2.5)

= 0, so there exists  πs3 ∈ K3 such that

πs3 ) = ∂3K (

a1 

1 zi1 ∧ p si .

(2.6)

i=1

Lemma 2.1 Let p 1 ∈ Ker ∂1K be as in (2.5). ! si 1 1 { [ ps1 ], . . . , [ psa1 ] } s=1,...,a 2 −q11 in Aa11 are linearly independent.

The

1

Proof If αs is in R such that

a12 −q11 s=1

! 1 ], . . . , [ 1 ] ∧ [α ] = 0, then [ ps1 psa s 1

a12 −q11



1 [ psi ] ∧ [αs ] = 0,

for all 1  i  a1 .

s=1

By tensoring with [zi1 ] and taking the sum over i we get

vectors

A Truncated Minimal Free Resolution of the Residue Field a1 

a12 −q11

[zi1 ] ⊗

i=1



1 [ psi ] ∧ [αs ] =

a12 −q11 8 a1  

s=1

s=1

403

9 1 [zi1 ] ⊗ [ psi ] ∧ [αs ] = 0.

i=1



The desired conclusion now follows from (2.5).

Proposition 2.2 Let {[zi1 ]}i=1,...,a1 and {[z 2 ]} =1,...,a2 −q11 be as in (2.2) and (2.3), respectively. If βij and α are in R such that a1 

[zi1 ] ∧ [zj1 ] ∧ [βij ] +

a2 −q11

i,j =1

[z 2 ] ∧ [α ] = 0,

=1

then the following hold: (a) [α ] = 0 for all 1   a2 − q11 ; (b) There exist γs in R such that for all 1  i  a1 a1 −q11 a1   1 1 [zj ] ∧ [βij ] = [ psi ] ∧ [γs ], 2

j =1

s=1

1 is as in (2.5). where p si

Proof (a) The first sum in the hypothesized equality is in A1 · A1 . The elements {[z 2 ]} =1,...,a2 −q11 form a basis in A2 that completes a basis of A1 · A1 , thus [α ] = 0 for all 1   a2 − q11 . (b) By (a) it follows that 0=

a1 

[zi1 ] ∧ [zj1 ] ∧ [βij ] =

i,j =1

a1 a1 8 9  [zi1 ] ∧ [zj1 ] ∧ [βij ] , i=1

j =1

8 9 1 1 a1 1 ] ∧ [β ] is in Ker φ . The desired [zi ] ⊗ [z which implies that ai=1 ij 1 j =1 j conclusion now follows from (2.5).



2.2 Products in Degree Three Let A3 be a k-subspace of A3 such that A3 = (A1 · A2 ) ⊕ A3 = (A1 · A1 · A1 + A1 · A2 ) ⊕ A3 .

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For each 1  t  a3 − q12 , we consider zt3 ∈ Ker ∂3K such that {[zt3 ]}t=1,...,a3 −q12

is basis of A3 .

(2.7)

In particular, every element in Ker ∂3K can be written as a sum of elements in Im ∂4K , a linear combination of {zi1 ∧ zj1 ∧ zk1 }i,j, k=1,...,a1 , a linear combination of {zi1 ∧ z 2 } i=1,...,a1 , and a linear combination of {zt3 }t=1,...,a3 −q12 .

=1,...,a2 −q11

Consider the map: ! ! ψ : A1 ⊗ A1 ⊗ A1 → (A1 · A1 ) ⊗ A1 ⊕ A1 ⊗ (A1 · A1 ) ,

(2.8)

defined by 8 9 ψ([x] ⊗ [y] ⊗ [z]) = [x] ∧ [y] ⊗ [z], [x] ⊗ [y] ∧ [z] for all [x], [y], [z] ∈ A1 and set b = rankk (Coker ψ).

(2.9)

Remark that b = 0 if and only if ψ is surjective. If q11 = 0, then A1 · A1 = 0 and hence b = 0. Lemma 2.3 Let ai , qij be as in (2.1) and b be as in (2.9). The sequence ! ! μ ψ → (A1 · A1 ) ⊗ A1 ⊕ A1 ⊗ A2 − → A3 , A1 ⊗ A1 ⊗ A1 − where 8 9 ψ([x] ⊗ [y] ⊗ [z]) = [x] ∧ [y] ⊗ [z], [x] ⊗ [y] ∧ [z] , and μ(([u] ⊗ [x], [y] ⊗ [v])) = [u] ∧ [x] − [y] ∧ [v], for all [x], [y], [z] ∈ A1 , [u] ∈ (A1 ·A1 ) and [v] ∈ A2 , is a complex whose homology has rank a1 a2 − a1 q11 − q12 + b. Proof We first show that μ ◦ ψ = 0. 88 99 μ(ψ([x] ⊗ [y] ⊗ [z])) = μ [x] ∧ [y] ⊗ [z], [x] ⊗ [y] ∧ [z] = [x] ∧ [y] ∧ [z] − [x] ∧ [y] ∧ [z] = 0.

A Truncated Minimal Free Resolution of the Residue Field

405

Hence, the sequence in the statement of the lemma is a complex. Moreover, we have rankk

8 Ker μ 9 Im ψ

= rankk (Ker μ) − rankk (Im ψ) 8

! !9 (A1 · A1 ) ⊗ A1 ⊕ A1 ⊗ A2 − rankk (Im μ) − rankk (Im ψ) 8 ! !9 = a1 q11 + a1 a2 − q12 − rankk (A1 · A1 ) ⊗ A1 ⊕ A1 ⊗ (A1 · A1 ) + b = rankk

= a1 q11 + a1 a2 − q12 − 2a1 q11 + b = a1 a2 − a1 q11 − q12 + b. The homology of the complex in the statement has the desired rank.



To describe a basis of the homology of the complex in Lemma 2.3, it is enough to observe that Im ψ is generated by elements of the form ([zi1 ] ∧ [zj1 ] ⊗ [zk1 ], [zi1 ] ⊗ [zj1 ] ∧ [zk1 ]) L M for 1  i, j, k  a1 and that in Ker μ/Im ψ we have ([zi1 ] ∧ [zj1 ] ⊗ [zk1 ], 0) = L M − (0, [zi1 ] ⊗ [zj1 ] ∧ [zk1 ]) , as [([zi1 ] ∧ [zj1 ] ⊗ [zk1 ], [zi1 ] ⊗ [zj1 ] ∧ [zk1 ])] = 0. We 2 ∈ Ker ∂ K such that choose p ui 2 a1 #L 8  9M$ 2 0, [zi1 ] ⊗ [ pui ]

u=1,...,a1 a2 −a1 q11 −q12 +b,

i=1

(2.10)

is a basis of the homology defined in Lemma 2.3. By definition of μ,  1 of1 the complex 2 ] = 0 in A . Therefore, for each u there exists [zi ] ∧ [ pui for each u we have ai=1 3  πu4 ∈ K4 such that ∂4K ( πu4 ) =

a1 

2 zi1 ∧ p ui .

i=1 2 from (2.10). The following result gives a method for finding p ui

Proposition 2.4 Consider the map φ2 : A1 ⊗ A2 → A3 , defined by φ2 ([x] ⊗ [v]) = [x] ∧ [v], for all [x] ∈ A1 , [v] ∈ A2 , and set

(2.11)

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V. C. Nguyen and O. Veliche

A=

a1 $ # " 1 and [zi1 ] ⊗ [ psi ] ∧ [zj1 ] " 1  j  a1 and 1  s  a12 − q11 i=1

a1 # $ " 2 " B= [zi1 ] ⊗ [ pui ] 1  u  a1 a2 − a1 q11 − q12 + b , i=1 1 and p 2 are as in (2.5) and (2.10) respectively. Then, the following hold: where p si ui

(a) The set B is linearly independent and Ker φ2 = (Spank A) ⊕ (Spank B). (b) Spank A ⊆ A1 ⊗ (A1 · A1 ) and for b defined in (2.9), b = a1 q11 − rankk (Spank A). (c) If B is a k-subspace of Ker φ2 such that Ker φ2 = (Spank A) ⊕ B, then for every basis B of B the set [(0, B )] is a basis of the homology of the complex in Lemma 2.3. Proof (a) The linear independence of B follows from (2.10). Next, we show that the 2 , it elements of A and B generate the kernel of φ2 . By definitions of p 1 and p ui a1 si1 is clear that the listed elements are in the kernel of φ2 . Let i=1 [zi ] ⊗ [pi2 ] ∈ 1 1 Ker φ2 , for some [pi2 ] ∈ A2 . Then [(0, ai=1 [zi ] ⊗ [pi2 ])] ∈ Ker μ, and thus by 1 ∈ Ker ∂ K such that definition (2.10) for all u, i, j there exist δu ∈ R and pij 1 a1 8  9 0, [zi1 ] ⊗ [pi2 ] = i=1

a1 a2 − a1 q11 −q12 + b



a1 8  9 2 0, [zi1 ] ⊗ [ pui ] ∧ [δu ]

u=1

+

(2.12)

i=1

a1 8 

9 1 1 [zi1 ] ∧ [pij ] ⊗ [zj1 ], [zi1 ] ⊗ [pij ] ∧ [zj1 ] .

i,j =1

Comparing the first components of both sides of (2.12), for each j we have: 0=

a1  1 [zi1 ] ∧ [pij ]. i=1

By (2.5), there exist εj s in R such that for all 1  i, j  a1 we can write:

A Truncated Minimal Free Resolution of the Residue Field

407

a12 −q11 1 [pij ]=



1 [ psi ] ∧ [εj s ].

s=1

Comparing the second components of both sides of (2.12), and using the above 1 ], for each i we have: expression for [pij a1 a2 − a1 q11 −q12 + b

[pi2 ] =



2 [ pui ] ∧ [δu ] +

a1 a2 − a1 q11 −q12 + b

=

u=1

1 [pij ] ∧ [zj1 ]

j =1

u=1



a1 

2 [ pui ] ∧ [δu ] +

a12 −q11 a1  

1 [ psi ] ∧ [zj1 ] ∧ [εj s ].

s=1 j =1

a1

Thus, i=1 [zi1 ] ⊗ [pi2 ] ∈ Ker φ2 is a linear combination of elements of A and 1 we have B. Next, we show that Spank A ∩ Spank B = {0}. By definition of p si a 1 1 1 2 psi ] = 0 for all 1  s  a1 − q11 . By definition of ψ, for all s, j , i=1 [zi ] ∧ [ we get ψ

a1 a1 8 9 8  9 1 1 [zi1 ] ⊗ [ psi ] ⊗ [zj1 ] = 0, [zi1 ] ⊗ [ psi ] ∧ [zj1 ] , i=1

(2.13)

i=1

hence (0, A) ⊆ Im ψ. On the other hand by (2.10), [(0, B)] is a basis of the homology of the complex in Lemma 2.3. Therefore, Spank A and Spank B have no nontrivial elements in common, and part (a) holds. (b) The inclusion follows from the definition of A. Observe that rankk (Ker φ2 ) = a1 a2 −q12 , and by the linear independence of B from part (a), rankk (Spank B) = a1 a2 − q12 − (a1 q11 − b). Therefore, by part (a), rankk (Spank A) = a1 q11 − b. (c) By parts (a) and (b), every basis B of B has a1 a2 − a1 q11 − q12 + b elements. It is clear that (0, B ) ⊆ Ker μ, with μ as in Lemma 2.3 and that the set [(0, B )] has at most a1 a2 − a1 q11 − q12 + b elements. It is enough to show that it is a generating set for the homology of the complex in Lemma 2.3 to conclude that [(0, B )] has exactly a1 a2 − a1 q11 − q12 + b elements, hence it forms a basis for the homology. Since B ⊆ Ker φ2 , every element of B can be written as a linear combination of elements of A and B . In particular, every basis element in [(0, B)] can be written as a linear combination of elements in [(0, A)] and [(0, B )]. However, each element in [(0, A)] is zero in the homology, as remarked in (2.13), thus [(0, B )] is a generating set. Part (c) now holds.

Remark 2.5 In practice, one can apply Proposition 2.4(b) to compute the value b, 2 , instead of using definition (2.9). Similarly, instead of using definition (2.10) for p ui

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1 as defined in (2.5), and then find a basis by Proposition 2.4(c), one can find first p si of a space complementary to

 Spank A = Spank

a1  1 [zi1 ] ⊗ [ psi ] ∧ [zj1 ]

X 1j a1 ,1sa12 −q11

i=1

in the kernel of the multiplication map φ2 , of the form 

a1  2 [zi1 ] ⊗ [ pui ]

X . 1ua1 a2 −a1 q11 −q12 +b

i=1

Proposition 2.6 Let {[zi1 ]}i=1,...,a1 , {[z 2 ]} =1,...,a2 −q11 , and {[zt3 ]}t=1,...,a3 −q12 be as in (2.2), (2.3), and (2.7), respectively. If γij k , βi , and αt are in R such that a1 

[zi1 ]∧[zj1 ]∧[zk1 ]∧[γij k ]+

i,j,k=1

−q11 a1 a2  i=1

[zi1 ]∧[z 2 ]∧[βi ]+

=1

a3 −q12

[zt3 ]∧[αt ] = 0,

t=1

then the following hold: (a) [αt ] = 0 for all 1  t  a3 − q12 ; (b) There exist δu and εj s in R such that a1 

[zj1 ] ∧ [zk1 ] ∧ [γij k ] +

j,k=1

a2 −q11

[z 2 ] ∧ [βi ]

=1 −q11 a1 a1 

a1 a2 − a1 q11 −q12 + b

2

=

1 [ psi ] ∧ [zj1 ] ∧ [εj s ] +

j =1 s=1



2 [ pui ] ∧ [δu ],

u=1

1 and p 2 are as in (2.5) and (2.10) respectively. where p si ui

Proof (a) The first two sums in the hypothesized equality are in A1 · A2 . As {[zt3 ]}t=1,...,a3 −q12 is a basis in A3 that completes a basis of A1 · A2 , this implies [αt ] = 0 for all 1  t  a3 − q12 . (b) By (a) it follows that a2 −q11 a1 a1 8  9  1 1 1 [zi ] ∧ [zj ] ∧ [zk ] ∧ [γij k ] + [z 2 ] ∧ [βi ] = 0, i=1

j,k=1

=1

A Truncated Minimal Free Resolution of the Residue Field

409

! 1 1 a1 a2 −q11 2 1 1 hence ai=1 [zi ] ⊗ j,k=1 [zj ] ∧ [zk ] ∧ [γij k ] +

=1 [z ] ∧ [βi ] is in Ker φ2 . The desired assertion now follows from Proposition 2.4(a).



2.3 Massey Products in Degree Four Massey products occur in degrees four and higher. In degree four, one may obtain only triple Massey products of elements of degree one. The Massey product of a triplet [x], [y], [z] ∈ A1 satisfying [x] ∧ [y] = 0 and

[y] ∧ [z] = 0,

is a subset of A4 , defined as follows: 3 3 3 3 ∧ z + x ∧ πyz ] | ∂3K (πxy ) = x ∧ y and ∂3K (πyz ) = y ∧ z}, [x], [y], [z] = {[πxy 3 , π3 ∈ K . for some πxy 3 yz

Remark 2.7 Let [x], [y], [z] ∈ A1 with [x] ∧ [y] = 0 and [y] ∧ [z] = 0. Choose 3 and π 3 in K such that ∂ K (π 3 ) = x ∧ y and ∂ K (π 3 ) = y ∧ z. Then, every πxy 3 yz xy yz 3 3 element of [x], [y], [z] is of the form 3 3 3 3 3 3 + pxy ) ∧ z + x ∧ (πyz + pyz )] = [πxy ∧ z + x ∧ πyz ] [(πxy 3 3 ] ∧ [z] + [x] ∧ [pyz ], + [pxy 3 and p 3 ∈ Ker ∂ K . Therefore, for some pxy yz 3 3 3 [x], [y], [z] = [πxy ∧ z + x ∧ πyz ] + (A3 · [z]) + ([x] · A3 ). 3 ∧ z + x ∧ π 3 is called a representative of the triple Massey product The element πxy yz [x], [y], [z]. Here, (A3 · [z] + [x] · A3 ) is called the indeterminacy of the Massey operation, see e.g., May [14] and [2, Section 7].

Let A1 , A1 , A1  denote the set of elements in A4 which are Massey products of triplets in A1 . The next result describes the elements of this set. Proposition 2.8 Each element in A1 , A1 , A1  ⊆ A4 has a representative a12 −q11

 s=1

a12 −q11

 πs3

∧ ps1

such that



1 [ psi ∧ ps1 ] = 0, for all i = 1 . . . , a1 ,

s=1

1 and  where ps1 ∈ Ker ∂1K , and p si πs3 are defined in (2.5) and (2.6) respectively.

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Proof Let [x], [y], [z] ∈ A1 such that [x] ∧ [y] = 0 and [y] ∧ [z] = 0. We write a1 a1   [zi1 ] ∧ [αi ] for some αi ∈ R. Since [x] ∧ [y] = [zi1 ] ∧ [x] ∧ [−αi ] = 0, [y] = i=1

i=1

a1  we have [zi1 ] ⊗ [x] ∧ [−αi ] ∈ Ker φ1 . Hence, by (2.5) there exist βs ∈ R such i=1

that [x] ∧ [−αi ] =

a12 −q11 s=1

[x] ∧ [y] =

1 ] ∧ [β ] for all i, so [ psi s

a12 −q11 8 a1   s=1

9 1 [zi1 ∧ p si ] ∧ [βs ]

i=1

a12 −q11

=



[∂3K ( πs3 )] ∧ [βs ]

=

s=1

a12 −q11

 πs3 s=1 a12 −q11

3 = [y] ∧ [z] = 0, we may choose πyz

[z]∧[αi ] = s=1 has a representative

−q11 8 a1 2

∂3K

 πs3 ∧ βs

9M .

s=1

3 = Therefore, we may choose πxy

a12 −q11

L



∧ βs . Similarly, using the equality  πs3 ∧ γs , for some γs ∈ R, where

s=1 1 ]∧[γ ] for [ psi s

all i. It follows that any element in A1 , A1 , A1  a12 −q11

3 πxy

∧z+x

3 ∧ πyz

=



 πs3 ∧ (z ∧ βs − x ∧ γs ).

s=1

If we set ps1 = z ∧ βs − x ∧ γs , then for each 1  i  a1 we have a12 −q11



a12 −q11 1 [ psi

∧ ps1 ]

=

s=1



1 [ psi ] ∧ [z ∧ βs − x ∧ γs ]

s=1 −q11 8 a1 2

=

s=1

1 [ psi

9

∧ βs ] ∧ [z] + [x] ∧

−q11 8 a1 2

1 [ psi ∧ γs ]

9

s=1

= −[x] ∧ [z] ∧ [αi ] + [x] ∧ [z] ∧ [αi ] = 0. Hence the proposition holds.



A Truncated Minimal Free Resolution of the Residue Field

411

2.4 Products in Degree Four Let A4 be a k-subspace of A4 such that A4 = (A1 · A3 + A2 · A2 + Spank A1 , A1 , A1 ) ⊕ A4 and set 8 9 a = rankk A1 · A3 + A2 · A2 + Spank A1 , A1 , A1  .

(2.14)

For some choice of zr4 ∈ Ker ∂4K , let {[zr4 ]}r=1,...,a4 −a

be a basis of A4 .

(2.15)

In particular, every element in Ker ∂4K can be written as a sum of elements in Im ∂5K , a linear combination of {zi1 ∧ pi3 }i=1,...,a1 , a linear combination of {z 2 ∧ p 2 } =1,...,a2 −q11 , a linear combination of { πs3 ∧ ps1 }s=1,...,a 2 −q11 , and a linear 1

combination of {zr4 }r=1,...,a4 −a , where pi3 ∈ Ker ∂3K , p 2 ∈ Ker ∂2K and ps1 ∈ Ker ∂1K , such that ps1 is as in Proposition 2.8 and  πs3 as in (2.6).

2.5 Summary We summarize here all notations, introduced in this section, to be referred to throughout the rest of the paper: A = H(K R ) = A0 ⊕ A1 ⊕ A2 ⊕ · · · A2 = (A1 · A1 ) ⊕ A2 A3 = (A1 · A2 ) ⊕ A3

! A4 = A1 · A3 + A2 · A2 + Spank (A1 , A1 , A1 ) ⊕ A4 ai = rankk Ai , for 1  i  4, (2.1) qij = rankk Ai · Aj , for 1  i  j  4, (2.1)

! a = rankk A1 · A3 + A2 · A2 + Spank A1 , A1 , A1  , (2.14)

{[zi1 ]}i=1,...,a1

is a basis for A1 , (2.2)

{[z 2 ]} =1,...,a2 −q11

is a basis for A2 , (2.3)

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{[zt3 ]}t=1,...,a3 −q12

is a basis for A3 , (2.7)

{[zr4 ]}r=1,...,a4 −a

is a basis for A4 , (2.15)

! ! ψ : A1 ⊗ A1 ⊗ A1 → (A1 · A1 ) ⊗ A1 ⊕ A1 ⊗ (A1 · A1 ) , (2.8) 8 9 ψ([x] ⊗ [y] ⊗ [z]) = [x] ∧ [y] ⊗ [z], [x] ⊗ [y] ∧ [z] for all [x], [y], [z] ∈ A1 b = rankk (Coker ψ), (2.9), see also Proposition 2.4(b) φ1 : A1 ⊗ A1 → A2 , φ1 ([x] ⊗ [y]) = [x] ∧ [y], for all [x], [y] ∈ A1 , (2.4) a1 #

$ 1 [zi1 ] ⊗ [ psi ]

i=1

∂3K ( πs3 ) =

a1 

s=1,...,a12 −q11

is a basis of Ker φ1 , (2.5)

1 zi1 ∧ p si , (2.6)

i=1

φ2 : A1 ⊗ A2 → A3 , φ2 ([x] ⊗ [y]) = [x] ∧ [y], for all [x] ∈ A1 , [y] ∈ A2 , Prop. 2.4 a1 #

$ 2 [zi1 ] ⊗ [ pui ]

i=1

1ua1 a2 −a1 q11 −q12 +b

Ker φ2 = B ⊕ Spank

a1 #

$ 1 [zi1 ] ⊗ [ psi ] ∧ [zj1 ]

i=1

∂4K ( πu4 ) =

a1 

is a basis of B, where

1j a1 ,1sa12 −q11

2 zi1 ∧ p ui , (2.11).

i=1

1 , 2 , and  Lemma 2.9 All the elements zi1 , z 2 , zt3 , zr4 , p si πs3 , p ui πu4 , as in Summary 2.5, are in mK.

Proof From the inclusions Ker ∂jK ⊆ mKj for all j  0, we obtain that 1 , and p 2 are in mK. The assertions for the preimages  si ui πs3 and  πu4 zi1 , z 2 , zt3 , zr4 , p K −1 2 follows from the inclusions (∂j ) (m Kj −1 ) ⊆ mKj for j = 3 and 4 respectively.



3 Truncated Minimal Free Resolution of the Residue Field In this section, we construct the beginning of the minimal free resolution of the residue field k over the local ring (R, m, k), by using the Koszul complex K of R and the graded-commutative structure of the algebra A = H(K) described in Sect. 2.

A Truncated Minimal Free Resolution of the Residue Field

413

3.1 Construction We consider the following sequence of free R-modules: ∂5F

F :

∂3F

∂4F

∂2F

∂1F

F5 −→ F4 −→ F3 −→ F2 −→ F1 −→ F0 ,

where F0 :=K0 F1 :=K1 F2 :=K2 ⊕ K0a1 a −q11

F3 :=K3 ⊕ K1a1 ⊕ K0 2

a −q11

⊕ K0 3

a −q11

⊕ K1 3

F4 :=K4 ⊕ K2a1 ⊕ K1 2 F5 :=K5 ⊕ K3a1 ⊕ K2 2

a a2 −a1 q11 −q12 +b

⊕ K0 1

a 2 −q11

a −q12

⊕ K0 1

a −q12

⊕ K0a4 −a ⊕ K1 1

a 2 −q11

a a2 −a1 q11

⊕ K0 1

.

Using the elements described in Sect. 2, the differential maps are defined by: ∂1F : K1 → K0 ,

∂2F : K2 ⊕ K0a1 → K1 ,

is given by ∂1F := ∂1K .

is given by

(3.1)

! ∂2F := ∂2K z1 ∧ ,

(3.2)

that is  ∂2F

π2

(αi )i=1,...,a1

 := ∂2K (π 2 ) +

a1 

zi1 ∧ αi .

i=1

 ∂3F :

a −q K3 ⊕K1a1 ⊕K0 2 11

that is

→ K2 ⊕K0a1 ,

is given by

∂3F :=

∂3K

z1 ∧ −z2 ∧

0 (∂1K )a1



, 0 (3.3)

414

V. C. Nguyen and O. Veliche

⎛

⎛

⎞

π3

⎞

a2 −q11

a1 

zi1 ∧ πi1 − z 2 ∧ α ⎟ ⎜∂3K (π 3 ) + ⎜ ⎟ ⎟. ∂3F ⎝ (πi1 )i=1,...,a1 ⎠ := ⎜ i=1

=1 ⎠ ⎝ K 1 (α ) =1,...,a2 −q11 (∂ (π ))i=1,...,a 1

a −q11

∂4F : K4 ⊕K2a1 ⊕K1 2

a −q12

⊕K0 3

a 2 −q11

⊕K0 1

i

1

a −q11

→ K3 ⊕K1a1 ⊕K0 2

⎛ K 1 ⎞ ∂4 z ∧ z2 ∧ z3 ∧ − π 3∧ ∂4F := ⎝ 0 (∂2K )a1 0 0 ( p1 ∧)a1 ⎠ , K a −q 2 11 0 0 (∂1 ) 0 0 that is ⎛

π4

, is given by (3.4)

⎞

⎜ ⎟ ⎜ (πi2 )i=1,...,a1 ⎟ ⎜ ⎟ ⎜ ⎟ ∂4F ⎜(π 1 ) =1,...,a2 −q11 ⎟ ⎜ ⎟ ⎜ (αt )t=1,...,a −q ⎟ 3 12 ⎠ ⎝ (βs )s=1,...,a 2 −q11 1 ⎛ ⎞ a12 −q11 a2 −q11 a3 −q12 a1   ⎜∂ K (π 4 ) + zi1 ∧ πi2 + z 2 ∧ π 1 + zt3 ∧ αt −  πs3 ∧ βs ⎟ ⎜ 4 ⎟ ⎜ ⎟ i=1

=1 t=1 s=1 ⎜ ⎟ ⎜ ⎟ 2 a1 −q11 := ⎜ ⎟.  ⎜ ⎟ K 2 1 ⎜ ⎟ (∂2 (πi ) + p si ∧ βs )i=1,...,a1 ⎜ ⎟ ⎝ ⎠ s=1 (∂1K (π 1 )) =1,...,a2 −q11  ∂5F

:

a −q11

K5 ⊕ K3a1 ⊕ K2 2 a 2 −q11

⊕K1 1

a −q12

⊕ K1 3

a a2 −a1 q11 −q12 +b

⊕ K0 1

⊕ K0a4 −a

a a2 −a1 q11

⊕ K0 1

X

 →

a −q11

K4 ⊕ K2a1 ⊕ K1 2 a −q12

⊕K0 3

a 2 −q11

⊕ K0 1

(3.5)

is given by ⎛

⎞

∂5K z1 ∧ −z2 ∧ z3 ∧ z4 ∧ − π 3 ∧ − π 4∧ 0 K a 1 a 2 1 1 0 0 0 ( p ∧) ( p ∧)a1 −(z2 ∧)a1 ⎟ ⎜ 0 (∂3 ) ⎜ ⎟ 0 (∂2K )a2 −q11 0 0 0 0 (z1 ∧)a2 −q11 ⎟ , ∂5F := ⎜ 0 ⎝0 ⎠ 0 0 (∂1K )a3 −q12 0 0 0 0 2 a −q 0 0 0 0 0 0 0 (∂1K ) 1 11

that is

X

A Truncated Minimal Free Resolution of the Residue Field

⎛

415

⎞

π5

⎜ ⎟ (πi3 )i=1,...,a1 ⎜ ⎟ ⎜ ⎟ 2) ⎜ ⎟ (π ⎜ ⎟

=1,...,a2 −q11 ⎜ ⎟ 1 ⎜ ⎟ (π ) t t=1,...,a3 −q12 ⎟ := ∂5F ⎜ ⎜ ⎟ (αr )r=1,...,a4 −a ⎜ ⎟ ⎜ ⎟ 1 ) ⎜ ⎟ (π 2 s s=1,...,a1 −q11 ⎜ ⎟ ⎜ ⎟ ⎝(βu )u=1,...,a1 a2 −a1 q11 −q12 +b ⎠ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

(γ i ) =1,...,a2 −q11 ; i=1,...,a1 a1 a2 − a1 q11 a12 −q11 a3 −q12 a2 −q11 a4 −a −q12 + b      K 5 1 3 2 2 3 1 4 3 ∂5 (π ) + zi ∧ π i − z ∧ π +  πu4 ∧ βu zt ∧ π t + zr ∧ αr −  πs ∧ πs1 − i=1

=1 u=1 t=1 r=1 s=1 a1 

8

∂3K (πi3 ) +

a12 −q11 

1 ∧ π 1 + p si s

a1 a2 − a1 q11 −q12 + b 

s=1

2 ∧β − p ui u

a2 −q11 

u=1 8

∂2K (π 2 ) +

a1 

zi1 ∧ γ i

i=1 8

9

z 2 ∧ γ i

i=1,...,a1

=1 9

=1,...,a2 −q11

9

∂1K (πt1 ) t=1,...,a3 −q12

8

9 ∂1K (πs1 ) s=1,...,a12 −q11

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

Theorem 3.1 Let (R, m, k) be a local ring. The sequence F constructed in 3.1 is a truncated minimal free resolution of k over R, up to homological degree five. Proof The minimality follows from Lemma 2.9. We show exactness at each degree by using the Koszul relations in the complex K, and the basis elements and maps defined in Sect. 2. Exactness at Degree One: Im ∂2F ⊆ Ker ∂1F :  ∂1F

◦ ∂2F



π2

(αi )i=1,...,a1

= ∂1K ∂2K (π 2 ) +

a1

1 i=1 zi

! ∧ αi = 0.

Im ∂2F ⊇ Ker ∂1F : If π 1 ∈ Ker ∂1F , then there exist π 2 ∈ K2 and αi ∈ R such that π = 1

∂2K (π 2 ) +

a1 

 zi1

∧ αi ,

therefore,

i=1

Exactness at Degree Two: Im ∂3F ⊆ Ker ∂2F :

π = 1

∂2F

π2

(αi )i=1,...,a1

 .

416

V. C. Nguyen and O. Veliche

⎛ ∂2F

◦ ∂3F

⎞

π3

⎜ ⎟ F ⎝ (πi1 )i=1,...,a1 ⎠ = ∂2



a1

a2 −q11 2 1 1 z

i=1 zi ∧ πi −

=1 K 1 (∂1 (πi ))i=1,...,a1

∂3K (π 3 ) +

(α ) =1,...,a2 −q11 =

∂2K

8

∂3K (π 3 ) +

a1 

zi1

∧ πi1

−

i=1

=−

a1 

z 2

9

∧ α +

=1

zi1 ∧ ∂1K (πi1 ) +

i=1

a1 

a1 



zi1 ∧ ∂1K (πi1 )

i=1

zi1 ∧ ∂1K (πi1 ) = 0.

i=1





π2

Ker ∂2F :

If

∈ Ker ∂2F , then ∂2K (π 2 ) +

a 1

1 i=1 zi ∧ αi = 0 (αi )i=1,...,a1 1 1 in K1 . In particular, we have ai=1 [zi ] ∧ [αi ] = 0 in A1 . Since {[zi1 ]}i=1,...,a1 is a basis of A1 , we get αi ∈ m. It follows that for each i there exists πi1 ∈ K1 such that

Im ∂3F

⊇

a2 −q11

∧ α

αi = ∂1K (πi1 ).

(3.6)

Hence we have 0 = ∂2K (π 2 ) +

a1 

a1 8 9  zi1 ∧ ∂1K (πi1 ) = ∂2K π 2 − zi1 ∧ πi1 .

i=1

i=1

In particular, there exist π 3 ∈ K3 , pi1 ∈ Ker ∂1K , and β ∈ R such that π − 2

a1 

zi1

∧ πi1

=

∂3K (π 3 ) +

i=1

a1 

zi1

∧ pi1

+

i=1

π 2 = ∂3K (π 3 ) +

a1 

a2 −q11

z 2 ∧ β ,

which implies

=1

zi1 ∧ (πi1 + pi1 ) +

i=1

a2 −q11

z 2 ∧ β .

(3.7)

=1

Combining (3.6) and (3.7) we get  Ker ∂2F 3

π2

(αi )i=1,...,a1



⎛

π3

⎞

⎜ ⎟ = ∂3F ⎝(πi1 + pi1 )i=1,...,a1 ⎠ . −(β ) =1,...,a2 −q11

Exactness at Degree Three: Im ∂4F ⊆ Ker ∂3F : We show that both components of the element

A Truncated Minimal Free Resolution of the Residue Field

⎛

417

⎞

π4

⎜ ⎟ ⎜ (πi2 )i=1,...,a1 ⎟ ⎜ ⎟ ⎜ ⎟ ∂3F ◦ ∂4F ⎜(π 1 ) =1,...,a2 −q11 ⎟ are zero. The first component is: ⎜ ⎟ ⎜ (αt )t=1,...,a −q ⎟ 3 12 ⎠ ⎝ (βs )s=1,...,a 2 −q11 1

∂3K

8

∂4K (π 4 ) +

a1 

zi1

∧ πi2

+

a2 −q11

i=1

+

a1 

=−

zi1

zi1 ∧ ∂2K (πi2 ) +

+

∧ ∂2K (πi2 ) +

a1 

a2 −q11

zi1 ∧



∧ αt −

 πs3 ∧ βs

9

s=1

−q11 9 a2 1 p si ∧ βs − z 2 ∧ ∂1K (π 1 )

s=1

=1

8 

a12 −q11

z 2

∧ ∂1K (π 1 ) −



∂3K ( πs3 ) ∧ βs

s=1 a12 −q11

zi1

a12 −q11

zt3

t=1

=1 a1 

a3 −q12

a12 −q11

i=1

i=1

+

∧ π 1

=1

i=1 a1 

z 2

∧ ∂2K (πi2 ) +

i=1

a1  8 s=1

zi1

1 ∧p si

9

∧ βs −

i=1

a2 −q11

z 2 ∧ ∂1K (π 1 ) = 0.

=1

1 ∈ Ker ∂ K for all The last equality follows from the definition of  πs3 in (2.6). As p si 1 8 9 2 a1 −q11 1 K K 2 si ∧ βs = 0, for each 1  i  a1 , so the i and s, we have ∂1 ∂2 (πi ) + s=1 p second component is zero. ⎞ ⎛ π3 ⎟ ⎜ Im ∂4F ⊇ Ker ∂3F : If ⎝ (πi1 )i=1,...,a1 ⎠ ∈ Ker ∂3F , then

(α ) =1,...,a2 −q11 ∂3K (π 3 ) +

a1 

zi1

i=1

∧ πi1

−

a2 −q11

z 2 ∧ α = 0 in K2 ,

and

(3.8)

=1

∂1K (πi1 ) = 0

for all 1  i  a1 .

(3.9)

It follows that for each i, there exist πi2 ∈ K2 and βij ∈ R such that πi1 = ∂2K (πi2 ) +

a1  j =1

Thus (3.8) becomes:

zj1 ∧ βij .

(3.10)

418

V. C. Nguyen and O. Veliche

∂3K

8

π − 3

a1 

zi1

∧ πi2

9

+

i=1

a1 

zi1

∧

a1 8

zj1

9

∧ βij −

a2 −q11

j =1

i=1

z 2 ∧ α = 0.

(3.11)

=1

a2 −q11 2 ∧ [zj1 ] ∧ [βij ] − =1 [z ] ∧ [α ] = 0. By Propoa 1 a12 −q11 1 sition 2.2, we get [α ] = 0 for all and j =1 [zj1 ] ∧ [βij ] = s=1 [ psi ] ∧ [βs ]

In A2 we obtain

a 1

1 i,j =1 [zi ]

for some βs in R. In particular, there exist π  1 ∈ K1 and πi2 ∈ K2 such that α = ∂1K (π  ), and 1

a1 

zj1 ∧ βij = ∂2K (πi2 ) +

(3.12)

a12 −q11



j =1

1 p si ∧ βs .

s=1

Equation (3.10) becomes: πi1 = ∂2K (πi2 + πi2 ) +

a12 −q11



1 p si ∧ βs ,

(3.13)

s=1

and Eq. (3.11) now becomes:

0=

∂3K

8

π − 3

a1 

zi1

∧ πi2

9

+

i=1 a2 −q11

−

a1 

−q11 a1 a1  2

∧ ∂2K (πi2 ) +

zi1

i=1

i=1

1 zi1 ∧ p si ∧ βs

s=1

z 2 ∧ ∂1K (π  ) 1

=1

8

= ∂3K π 3 −

a1 

zi1 ∧ (πi2 + πi2 ) −

i=1

a2 −q11

a12 −q11

1 z 2 ∧ π 

+

=1



9  πs3 ∧ βs ,

s=1

where the second equality uses the definition of  πs3 from (2.6). Hence, there exist 2 K 1 K 4  π ∈ K4 , pi ∈ Ker ∂2 , p ∈ Ker ∂1 , and αt ∈ R such that π − 3

a1 

zi1

∧ (πi2

+ πi2 ) −

i=1

= ∂4K (π 4 ) +

a2 −q11

z 2

1 ∧ π 

a12 −q11

+

=1 a1  i=1

zi1 ∧ pi2 +

a2 −q11

=1



 πs3 ∧ βs

s=1

z 2 ∧ p 1 +

a3 −q12 t=1

zt3 ∧ αt .

A Truncated Minimal Free Resolution of the Residue Field

419

Thus, π 3 = ∂4K (π 4 ) +

a1 

zi1 ∧ (πi2 + πi2 + pi2 ) +

a2 −q11

i=1

+

a3 −q12

z 2 ∧ (π  + p 1 ) 1

=1

zt3 ∧ αt −

t=1

a12 −q11



 πs3 ∧ βs .

(3.14)

s=1

Equations (3.12), (3.13), and (3.14), now yield ⎛

⎞

π4

⎜ 2 ⎟ ⎜(πi + πi2 + pi2 )i=1,...,a1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ 1 Ker ∂3F 3 ⎝ (πi1 )i=1,...,a1 ⎠ = ∂4F ⎜ (π  + p 1 ) =1,...,a2 −q11 ⎟ . ⎜ ⎟ ⎜ ⎟ (αt )t=1,··· ,a3 −q12 (α ) =1,...,a2 −q11 ⎝ ⎠ (βs )s=1,...,a 2 −q11 ⎛

⎞

π3

1

Exactness at Degree Four: Im ∂5F ⊆ Ker ∂4F : We show that all three components of the element ⎛ ⎞ π5 ⎜ ⎟ (πi3 )i=1,...,a1 ⎜ ⎟ ⎜ ⎟ 2 ⎜ ⎟ (π ) =1,...,a2 −q11 ⎜ ⎟ ⎜ ⎟ 1 ⎜ ⎟ (πt )t=1,...,a3 −q12 ⎟ are zero. The first component is: ∂4F ◦ ∂5F ⎜ ⎜ ⎟ (αr )r=1,...,a4 −a ⎜ ⎟ ⎜ ⎟ 1 ) ⎜ ⎟ (π s s=1,...,a12 −q11 ⎜ ⎟ ⎜ ⎟ ⎝(βu )u=1,...,a1 a2 −a1 q11 −q12 +b ⎠ (γ i ) =1....,a2 −q11 ; i=1,...,a1 a2 −q11 a3 −q12 a1 a 4 −a 8  ∂4K ∂5K (π 5 ) + zi1 ∧ πi3 − z 2 ∧ π 2 + zt3 ∧ πt1 + zr4 ∧ αr i=1 a12 −q11

−



=1

 πs3 ∧ πs1 −

s=1

+

a1  i=1

t=1

a1 a2 − a1 q11 −q12 + b



 πu4 ∧ βu

r=1

9

u=1

a1 −q11 8  1 K 3 1 zi ∧ ∂3 (πi ) + p si ∧ πs1 + 2

s=1

a1 a2 − a1 q11 −q12 + b

 u=1

2 p ui ∧ βu −

a2 −q11

=1

z 2 ∧ γ i

9

420

V. C. Nguyen and O. Veliche

+

a2 −q11

a1 −q11 −q12 a1 9 a3 8   ∧ ∂2K (π 2 ) + zi1 ∧ γi + zt3 ∧ ∂1K (πt1 ) −  πs3 ∧ ∂1K (πs ) 2

z 2

=1

=−

a1 

i=1

zi1 ∧ ∂3K (πi3 ) −

a2 −q11

i=1



∂3K ( πs3 ) ∧ πs1 +

+

zi1 ∧ ∂3K (πi3 ) +



 πs3 ∧ ∂1K (πs1 ) −

a2 −q11  a1

a1 a2 − a1 q11 −q12 + b

zi1 ∧ z 2 ∧ γ i +

9



∂4K ( πu4 ) ∧ βu

a1 a2 − a1 q11 −q12 + b 8



a1 

u=1

i=1

1 ∧ πs1 + zi1 ∧ p si

i=1 a2 −q11

i=1

a3 −q12

zt3 ∧ ∂1K (πt1 )

u=1

a12 −q11 8 a1   s=1

=1

+

a3 −q12

s=1

i=1

−

s=1

t=1

a12 −q11

s=1 a1 

z 2 ∧ ∂2K (π 2 ) −

=1

a12 −q11

−

t=1

z 2 ∧ ∂2K (π 2 ) +

=1

a2 −q11  a1

=1

9 2 ∧ βu zi1 ∧ p ui

z 2 ∧ zi1 ∧ γ i

i=1

a12 −q11

 zt3 ∧ ∂1K (πt1 ) −  πs3 ∧ ∂1K (πs1 ) = 0,

t=1

s=1

1, 2 . For each 1  i  a , as p 1 is in Ker ∂ K and by definitions of  πs3 , p si πu4 , and p ui si 1 1 2 K p ui is in Ker ∂2 , we have a1 a2 − a1 q11 −q12 + b

a1 −q11 8  1 ∂2K ∂3K (πi3 ) + p si ∧ πs1 + 2



s=1 a12 −q11

+



2 p ui ∧ βu −

u=1

a2 −q11

z 2 ∧ γ i

=1

1 p si ∧ ∂1K (πs1 )

s=1 a12 −q11

=−



1 p si

∧ ∂1K (πs1 ) +

s=1

a12 −q11



1 p si ∧ ∂1K (πs1 ) = 0.

s=1

Therefore, the second component is zero. For each 1   a2 − q11 , ∂1K

∂2K (π 2 ) +

a1  i=1

Thus, the third component is zero.

! zi1 ∧ γ i = 0.

9

A Truncated Minimal Free Resolution of the Residue Field

⎛

421

⎞

π4

⎟ ⎜ ⎜ (πi2 )i=1,...,a1 ⎟ ⎟ ⎜ ⎟ ⎜ Im ∂5F ⊇ Ker ∂4F : If ⎜(π 1 ) =1,...,a2 −q11 ⎟ ∈ Ker ∂4F , then ⎟ ⎜ ⎜ (αt )t=1,...,a −q ⎟ 3 12 ⎠ ⎝ (βs )s=1,...,a 2 −q11 1

∂4K (π 4 ) +

a1 

zi1

∧ πi2

+

a2 −q11

i=1

z 2

∧ π 1

+

a3 −q12

=1

a12 −q11

zt3

∧ αt −

t=1



 πs3 ∧ βs = 0,

s=1

(3.15) a12 −q11

∂2K (πi2 ) +



1 p si ∧ βs = 0

for all 1  i  a1 ,

and

(3.16)

s=1

∂1K (π 1 )

=0

for all 1   a2 − q11 .

(3.17)

The equality (3.17) implies that there exist π 2 ∈ K2 and γi ∈ R such that π 1

=

∂2K (π 2 ) +

a1 

zi1 ∧ γi .

(3.18)

i=1

a12 −q11 1 1 ])∧[β ] = 0. Applying ([ ps1 , . . . , [ psa In Aa11 the equality (3.16) becomes s=1 s 1 Lemma 2.1 we obtain [βs ] = 0 for all s, thus βs = ∂1K (πs1 )

for some πs1 ∈ K1 .

(3.19)

The equality (3.16) now becomes a12 −q11

∂2K (πi2 ) +



a1 −q11 8 9  2 1 πi − p si ∧ πs1 = 0. 2

1 p si

∧ ∂1K (πs1 )

=

∂2K

s=1

s=1

Therefore, there exist πi3 in K3 , δij k and γi  in R such that for all i we have: πi2

=

∂3K (πi3 ) +

a1  j,k=1

zj1

∧ zk1

∧ δij k +

a2 −q12

=1

z 2

∧ γi 

a12 −q11

+



1 p si ∧ πs1 .

s=1

Putting together (3.18), (3.19), and (3.20), into the equality (3.15) we get:

(3.20)

422

V. C. Nguyen and O. Veliche 0 = ∂4K (π 4 ) +

a1  i=1

+

−q11 8 a1 2

zi1 ∧

a2 −q11

s=1

zj1 ∧ zk1 ∧ δij k +

a2 −q12

j,k=1

z 2 ∧ γi 

9

=1 a12 −q11

−q12 9 a3 8   2 z 2 ∧ ∂2K (π  ) + zi1 ∧ γi + zt3 ∧ αt −  πs3 ∧ ∂1K (πs1 )

=1

i=1 a12 −q11

t=1

8  = ∂4K π 4 +  πs3 ∧ πs1 −

a1 

s=1

i=1

+

a1 

1 p si ∧ πs1 + ∂3K (πi3 ) +

a1 8 

zi1 ∧ zj1 ∧ zk1 ∧ δij k +

i,j,k=1

zi1 ∧ πi3 +

s=1

a2 −q11

z 2 ∧ π 

2

9

=1 −q11 a1 a2  i=1

a3 −q12

zi1 ∧ z 2 ∧ (γi  + γi ) +

=1

9 zt3 ∧ αt .

t=1

In A3 this reduces to: a1 

[zi1 ]∧[zj1 ]∧[zk1 ]∧[δij k ]+

−q11 a1 a2 

i,j,k=1

i=1

a3 −q12 [zi1 ]∧[z 2 ]∧[γi  +γi ]+ [zt3 ]∧[αt ] = 0.

=1

t=1

By Proposition 2.6, for each t there exists πt1 ∈ K1 , and for each i there exist elements δu , εj s in R, and πi3 ∈ K3 such that αt = ∂1K (πt ) 1

(3.21)

and a1 

zj1 ∧ zk1 ∧ δij k +

j,k=1

a2 −q11

z 2 ∧ (γi + γi  )

=1 a1 a2 − a1 q11 −q12 + b

=

3 ∂3K (πi ) +



−q11 a1 a1  2

2 p ui ∧ δu +

1 p si ∧ zj1 ∧ εj s

j =1 s=1

u=1

Thus, (3.15) further becomes:

0=

∂4K

8

a12 −q11

π + 4



 πs3 ∧ πs1 −

s=1

+

a1  i=1

zi1

8 3 ∧ ∂3K (πi ) +

a1 

zi1 ∧ πi3 +

a2 −q11

i=1

u=1

2

9

=1

a1 a2 − a1 q11 −q12 + b



z 2 ∧ π 

−q11 a1 a1  2

2 p ui

∧ δu

+

j =1 s=1

1 p si ∧ zj1 ∧ εj s

9

A Truncated Minimal Free Resolution of the Residue Field

+

a3 −q12

423

zt3 ∧ ∂1K (πt ) 1

t=1

=

∂4K

8

a12 −q11



π + 4

a1 

 πs3 ∧ πs1 −

s=1 a1 a2 − a1 q11 −q12 + b



+

zi1 ∧ πi3 +

i=1 −q11 a1 a1 

a2 −q11

z 2 ∧ π 2 −

=1

∧ δu

+

a3 −q12

 πs3 ∧ zj1 ∧ εj s −

j =1 s=1

u=1

zi1 ∧ πi

3

i=1

2

 πu4

a1 

zt3 ∧ πt

1

9 .

t=1

The second equality above follows from definitions of  πs3 and  πu4 in (2.6) and 3 K 2 5 (2.11). Therefore, there exist π ∈ K5 , δr ∈ R, pi ∈ Ker ∂3 , p ∈ Ker ∂2K and ps1 ∈ Ker ∂1K with ps1 as in Proposition 2.8, such that a1 

π4 −

a2 −q11

zi1 ∧ (πi3 + πi ) + 3

i=1

+

=

z 2 ∧ π  − 2

=1



a1 9 8   πs3 ∧ πs1 + zj1 ∧ εj s +

s=1

j =1 a1 

zi1

∧ pi3

+

1

a1 a2 − a1 q11 −q12 + b



 πu4 ∧ δu

u=1

a2 −q11

i=1

zt3 ∧ πt

t=1

a12 −q11

∂5K (π 5 ) +

a3 −q12

a12 −q11

z 2

∧ p 2

+

=1



 πs3 ∧ ps1 +

s=1

a 4 −a

zr4 ∧ δr .

r=1

This implies: π 4 = ∂5K (π 5 ) +

a1 

zi1 ∧ (πi3 + πi + pi3 ) − 3

a2 −q11

i=1

+

a3 −q12

zt3

1 ∧ πt

−



2

=1

+

a 4 −a

t=1 a1 a2 − a1 q11 −q12 + b

z 2 ∧ (π  − p 2 )

a12 −q11

zr4

∧ δr −

r=1

 s=1

a1 9 8   πs3 ∧ πs1 + zj1 ∧ εj s − ps1 j =1

 πu4 ∧ δu .

u=1

(3.22) By (3.21), the expression (3.20) becomes:

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V. C. Nguyen and O. Veliche

πi2 = ∂3K (πi3 ) +

a1 

zj1 ∧ zk1 ∧ δij k +

j,k=1

a2 −q12

z 2 ∧ γi  +

a12 −q11

=1



1 p si ∧ πs1

s=1

(3.23) = ∂3K (πi3 + πi ) + 3

a12 −q11



8

1 p si ∧ πs1 +

−

9

a1 a2 − a1 q11 −q12 + b

zj1 ∧ εj s +

j =1

s=1 a2 −q11

a1 



2 p ui ∧ δu

u=1

z 2 ∧ γi .

=1

We conclude that by using (3.18), (3.19), (3.21), (3.22), (3.23), and Proposition 2.8, the chosen kernel element is in the image of ∂5F : ⎛

π5

⎞

⎜ ⎟ (πi3 + πi 3 + pi3 )i=1,...,a1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 2 2 (π − p ) =1,...,a2 −q11 ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎜ (πi2 )i=1,...,a1 ⎟ 1 ⎜ ⎟ ⎟ ⎜ (πt )t=1,...,a3 −q12 ⎟ ⎟ ⎜(π 1 ) F F ⎜ Ker ∂4 3 ⎜ =1,...,a2 −q11 ⎟ = ∂5 ⎜ ⎟. (δr )r=1,...,a4 −a ⎜ ⎟ ⎟ ⎜ ⎜8 ⎟ ⎜ (αt )t=1,...,a −q ⎟ 9 3 12 ⎠ ⎜ 1 a1 1 ⎟ ⎝ 1 ⎜ πs + j =1 zj ∧ εj s − ps ⎟ ⎜ s=1,...,a12 −q11 ⎟ (βs )s=1,...,a 2 −q11 ⎜ ⎟ 1 ⎝ ⎠ (δu )u=1,...,a1 a2 −a1 q11 −q12 +b ⎛

π4

⎞

(γ i ) =1,...,a2 −q11 ; i=1,...,a1 Therefore, the complex F in Construction 3.1 is a minimal free resolution of k up to degree five.

Remark 3.2 If all multiplications on the algebra A are trivial, that is, qij = 0 for all i, j  1, and all Massey operations are zero, Golod’s construction in [8] gives a minimal free resolution of the residue field k of a local ring R in terms of the Koszul complex K of R. Without these assumptions, our complex F in Construction 3.1 generalizes Golod’s resolution up to degree five.

4 Applications of the Construction For a local ring R of embedding dimension n and codepthc, set βi := ∞ i rankk TorR i=0 bi t denote i (k, k) for i  0 for the Betti numbers of k over R. Let the series on the right hand side of the inequality (1). The sequence P := {Pi }i0 defined by:

A Truncated Minimal Free Resolution of the Residue Field

Pi := bi − βi ,

425

for all i  0,

(4.1)

gives the coefficients of the series P(t) defined in the Introduction. The ring R is Golod if and only if Pi = 0 for all i  0. Our goal in this section is to give a description of the sequence {Pi }0i5 in terms of the invariants of the multiplicative structure of the algebra A = H(K R ) and discuss various consequences of Theorem 3.1. First, using Theorem 3.1, we explicitly describe the Betti numbers βi in terms of those invariants, up to degree five. Corollary 4.1 Let R be a local ring of embedding dimension n. Let ai , qij , a and b be as in Summary 2.5. Then the following equalities hold: β0 = 1,

β1 = n,

β2 =

β3 =

n! 3 + na1 + a2 − q11 ,

β5 =

n! n! n! 2 5 + 3 a1 + 2 a2 +na3 +a4 +na1 +2a1 a2 −

β4 =

n! 2 + a1 ,

n! n! 2 4 + 2 a1 + na2 + a3 + a1 − (n + 1)q11 − q12 ,

! n+1! 2 +2a1 q11 −(n+1)q12 +b−a.

Proof This is a direct consequence of Theorem 3.1. The formulas for β0 , . . . , β3 are clear, and we simplify the following expressions for β4 and β5 : β4 = β5 =

n! n! 2 4 + 2 a1 + a1 − q11 + n(a2 − q11 ) + a3 − q12 n! n! 2 5 + 3 a1 + n(a1 − q11 ) + a1 a2 − a1 q11 − q12 + b ! + n2 (a2 − q11 ) + a1 a2 − a1 q11 + n(a3 − q12 ) + a4 − a.



We now obtain the desired expressions from the statement.

Proposition 4.2 Let R be a local ring of embedding dimension n and codepth c. Let ai , qij , a, and b be as in Summary 2.5 and P as in (4.1). Then, the following hold: P0 = P1 = P2 = 0, P5 =

P3 = q11 , ! n+1! 2 + 2a1 q11

P4 = (n + 1)q11 + q12 ,

and

+ (n + 1)q12 + a − b.

∞ i = Proof Recall that the right hand side of the inequality (1) is i=1 bi t n (1+t)  . Comparing the coefficients on both sides of (1) we have the following 1− ci=1 ai t i+1 recursive formulas for bi : b0 = 1,

b1 = n,

bi =

i−1  j =1

In particular, we obtain:

aj bi−j −1 +

n! i ,

for all i  2.

426

b2 = b3 =

V. C. Nguyen and O. Veliche n! 2 + a1 , n! 3 + na1

b4 = + a2 , b5 =

n! 4 + n! 5 +

n! 2 a1 n! 3 a1

+ na2 + a3 + a12 , ! + n2 a2 + na3 + a4 + na12 + 2a1 a2 .

Now, the expression for Pi = bi − βi for 0  i  5 follows from Corollary 4.1. Alternatively, assume qij = 0 for all 1  i +j  4, and Massey products are also trivial, we have a = b = 0. In Construction 3.1 we obtain the maximum possible values of the Betti numbers, so the differences between the actual Betti numbers and these maximum values are exactly the Pi ’s in the proposition.

Corollary 4.3 Let R be a local ring, ai , qij , a, and b be as in Summary 2.5 and P as in (4.1). (a) If q11 = q12 = 0, then Pi = 0 for all 0  i  4 and P5 = a. ! (b) If q11 = q12 = q13 = q22 = 0, then P5 = rankk Spank A1 , A1 , A1  . (c) If codepth R = 4, then R is Golod if and only if Pi = 0 for all 1  i  5. Proof (a) and (b) are straightforward from Proposition 4.2. For (c), result in [8] showed that R is Golod if and only if all products on A and all ternary Massey products are trivial. If codepth R = 4, then the r-ary Massey products are trivial for all r  4. Thus, part (c) follows from Proposition 4.2.

Remark 4.4 A result of Burke [6, Corollary 6.10] implies that for a local ring R of codepth c the following implication holds: If Pi = 0 for all 0  i  c + 1, then Pi = 0 for all i  0. Thus, Corollary 4.3(c) is a consequence of this result as well. We now examine some local rings with rational Poincaré series of certain forms and describe PR k (t) in terms of the algebraic invariants of A. Proposition 4.5 Let (R, m, k) be a local ring of embedding dimension n and codepth c. Let ai , qij , a, and b be as in Summary 2.5. If the Poincaré series of (1+t)n R is rational of the form PR k (t) = d(t) , then c 9 8  d(t) = 1 − ai t i+1 + q11 t 3 + (q11 + q12 )t 4 + (q12 − b + a)t 5 + f (t)t 6 , i=1

for some f (t) ∈ Z[t]. Proof Let Pi be as in (4.1). Set P(t) =

∞  i=0

Then

Pi t i ,

α(t) = 1 −

c  i=1

ai t i+1

and

γ (t) = d(t) − α(t).

A Truncated Minimal Free Resolution of the Residue Field

P(t) =

427

(1 + t)n (1 + t)n · γ (t) (1 + t)n − = ⇐⇒ α(t) d(t) α(t) · (α(t) + γ (t))

γ (t) P(t) · α(t) ⇐⇒ = α(t) + γ (t) (1 + t)n P(t) · α(t) γ (t) = ⇐⇒ α(t) (1 + t)n − P(t) · α(t) γ (t) =

P(t) · (α(t))2 . (1 + t)n − P(t) · α(t)

We compare the coefficients of t i for all 0  i  5 on both sides of the equality: 9 8 γ (t) · (1 + t)n − P(t) · α(t) = P(t) · (α(t))2 . By Proposition 4.2, P0 = P1 = P2 = 0, and thus the left and right hand sides of the above equation become: LHS = (γ0 + γ1 t + γ2 t 2 + γ3 t 3 + γ4 t 4 + γ5 t 5 + · · · ) ! ! ! ! ! · 1+ nt+ n2 t 2 + n3 − P3 t 3 + n4 − P4 t 4 +

! 5 ! n! 5 + P3 a1 − P5 t + · · · ,

RHS = (P3 t 3 + P4 t 4 + P5 t 5 + . . . )

! · 1 − 2a1 t 2 − 2a2 t 3 + (a12 − 2a4 )t 4 + (a1 a2 − 2a5 )t 5 + · · · .

It is clear that γ0 = γ1 = γ2 = 0 and comparing the coefficients of t 3 we get: γ3 = P3 = q11 . Comparing the coefficients of t 4 and by Proposition 4.2 we get: γ4 + nγ3 = P4 ⇐⇒ γ4 = P4 − nγ3 = (n + 1)q11 + q12 − nq11 = q11 + q12 . Finally, comparing the coefficients of t 5 and by Proposition 4.2 we get: γ5 + nγ4 +

n! 2 γ3

= P5 − 2a1 P3 ⇐⇒ ! γ5 = P5 − 2a1 P3 − nγ4 − n2 γ3 ! ! = n+1 2 + 2a1 q11 + (n + 1)q12 − b + a − 2a1 q11 − n(q11 + q12 ) −

n! 2 q11

= q12 − b + a. Therefore, the expression for d(t) = α(t) + γ (t) in the statement holds.



There are many classes of local rings for which the Poincaré series is rational of n the form PR k (t) = (1+t) /d(t). In light of Proposition 4.5, we write the coefficients of the polynomial d(t) in terms of the invariants ai , qij , b and a for some special cases and provide a uniform expression of the Poincaré series in these cases.

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Corollary 4.6 If (R, m, k) is a local ring of embedding dimension n and codepth at most 3, and ai , qij , b are as in Summary 2.5, then PR k (t) =

(1 + t)n−1 . 1 − t − (a1 − 1)t 2 − (a3 − q11 )t 3 + q12 t 4 − bt 5

Proof By [3, Theorem 3.5] the Poincaré series of the ring R is rational given by n PR k (t) = (1 + t) /d(t) with deg d(t)  6 and d(−1) = 0. Since codepth R  3 we have a4 = 0 and a = 0. Thus, Proposition 4.5 gives: PkR (t) =

1 − a1

t2

− (a2 − q11

)t 3

(1 + t)n . − (a3 − q11 − q12 )t 4 − (b − q12 )t 5 − bt 6

Simplifying the fraction by the common factor (1+t), we get the desired conclusion.

Remark 4.7 In [5, Lemma 3.6] Avramov defined the invariant τ for non-Gorenstein ring R of codepth 3 as follows: τ = 1 if R is of class T, and τ = 0 otherwise. By comparing [5, (3.6.2)] and the Poincaré expression in Corollary 4.6 we see that τ is our b = rankk (Coker ψ) in (2.9), which is a multiplicative invariant of the homology algebra A. It follows from [3, Theorem 3.5] that b = 1 for a ring of class C(3) or T, and b = 0 for rings of class C(1), C(2), S, B, G(r), and H(q11 , q12 ). Corollary 4.8 If (R, m, k) is a Gorenstein local ring, not complete intersection, of embedding dimension n and codepth 4, and ai , qij , a, b are as in Summary 2.5, then PR k (t) =

1 − 2t − (a1

− 3)t 2

+ (q11

− 2)t 3

(1 + t)n−2 . + (−q11 + q12 − 1)t 4 − (q11 − q12 − b + a − 1)t 5

Proof Since R is Gorenstein we have a2 = 2a1 − 2, a3 = a1 , and a4 = 1. The statement follows from [3, Theorem 3.5] and Proposition 4.5.

Remark 4.9 Using the Avramov’s notation from [3] for the classification of Gorenstein local rings R of codepth 4 given by Kustin and Miller [11] and comparing the Poincaré expressions from [3, Theorem 3.5] and Corollary 4.8, we obtain the following table of algebra invariants of a Gorenstein local ring R: Class C(4) GT GS GH(p)

q11 q12 6 4 3 3 0 0 p p+1

q22 1 1 1 1

q13 1 1 1 1

a 1 1 1 1

b 4 1 0 0

We provide more examples of rings for which we can calculate the multiplicative invariants qij , a, b from the Poincaré series of the ring.

A Truncated Minimal Free Resolution of the Residue Field

429

Corollary 4.10 Let k be a field, I be an ideal of k[x, y, z, w] such that (x, y, z, w)3 ⊆ I ⊆ (x, y, z, w)2 , and set R = k[x, y, z, w]/I . Let ai , qij , a, b be as in Summary 2.5. If R has a4 = 3 and its Poincaré series is of the form 1 PR , then k (t) = (1 − t)(1 − 3t) q11 = a2 − 8,

q12 = 8,

a = 3,

and

b = 0.

Proof By hypothesis, PR k (t) =

(1 + t)4 (1 + t)4 = . (1 − t)(1 − 3t)(1 + t)4 1 − 7t 2 − 8t 3 + 3t 4 + 8t 5 + 3t 6

Proposition 4.5 thus gives: a1 = 7,

a2 − q11 = 8,

a3 − q11 − q12 = −3,

a4 − q12 + b − a = −8. (4.2) From the second equality we obtain q11 = a2 − 8 and from the third we get q12 = a3 − a2 + 11. Since a4 = 3 and a1 = 7, we get a2 = a3 + 3 and thus q12 = 8. The last equality in (4.2) becomes a − b = 3. By the definitions of a and b we have 0  a  a4 = 3 and b  0, thus a = 3 and b = 0.

and

Example 4.11 Rings satisfying the hypotheses of Corollary 4.10 are discussed by Christensen and Veliche in [7] and Yoshino in [17]. For examples, consider the following ideals in Q = Q[x, y, z, w]: I1 = (yw, xw + zw + w 2 , z2 + w 2 , xz + zw + w 2 , y 2 + yz, xy + zw, x 2 + zw), I2 = (zw + w 2 , yw, z2 + w 2 , yz + xw + w 2 , xz + w 2 , xy + y 2 + xw + w 2 , x 2 + xw + w 2 ), I3 = (zw, yw, xw − w 2 , yz, xz, xy − z2 , x 2 − y 2 ), I4 = (w 2 , yw + zw, xw, yz + z2 , y 2 + zw, xy + xz, x 2 + zw).

The rings Q/Ii satisfy the hypotheses of Corollary 4.10 and have a2 = 10 + i, for 1  i  4. Example 4.12 In an unpublished note, Roos [16], inspired by a paper of Katthän [10], constructed several examples of non-Golod rings R of codepth 4 with trivial algebra multiplications on A. We provide here two of them. For each one, there exists a Golod homomorphism from a complete intersection ring, hence it has rational Poincaré series. The algebra multiplication on A was checked using the DGAlgebras package [15] of Macaulay2 [13]. Proposition 4.5 confirms that indeed q11 = q12 = 0, and moreover it gives us the exact size of the space generated by the triple Massey products A1 , A1 , A1 , known to be nonzero. Consider the following ideals in Q = Q[x, y, z, w]: J1 = (w 3 , xy 2 , xz2 + yz2 , x 2 w, x 2 y + y 2 w, y 2 z + z2 w),

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V. C. Nguyen and O. Veliche

J2 = (w 3 , xy 2 , xz2 + yz2 , x 2 w + zw 2 , y 2 w + xzw, y 2 z + yz2 ). The rings Q/Ji with i = 1, 2 are non-Artinian of codepth 4 with 2 3 4 PQ Q/Ji (t) = 1 + 6t + (10 + i)t + (7 + i)t + 2t , Q/Ji

PQ

(t) =

(1 + t)4 . 1 − 6t 2 − (10 + i)t 3 − (7 + i)t 4 − t 5 + t 6

For both rings Q/Ji , q11 = 0 = q12 by Proposition 4.5, hence b = 0 and a = 1. Since q22 = q13 = 0, the space spanned by the Massey products A1 , A1 , A1  has rank one. By [1, Example 7.1], not all local rings have rational Poincaré series. However, for every local ring R with residue field k, there exists a unique sequence of integers {εi }i0 such that the Poincaré series of R can be expressed as PkR (t)

3∞ 2i−1 )ε2i−1 i=1 (1 + t = 3 , ∞ 2i ε2i i=1 (1 − t )

and εi is called the i-th deviation of R; see e.g., [4, Remark 7.1.1]. Theorem 3.1 allows us to describe the first five deviations in terms of the algebraic invariants of A. Corollary 4.13 Let R be a local ring of embedding dimension n and ai , qij be as in Summary 2.5. Then the first five deviations of R are: ε1 = n,

ε2 = a1 , ε3 = a2 − q11 , ! ε4 = a3 − q12 + a21 − q11 , ε5 = a4 + a1 a2 − a1 q11 − q12 + b − a. Proof By comparing the coefficients on the left and right sides of the equality ∞ 

(1 + t 2i−1 )ε2i−1 =

i=1

∞  (1 − t 2i )ε2i · (β0 + β1 t + β2 t 2 + β3 t 3 + β4 t 4 + β5 t 5 + · · · ) i=1

one obtains the following relations between the Betti numbers {βi }1i5 and the deviations {εi }1i5 : ! β2 = ε2 + ε21 , ! 2 + ε2 β4 = ε4 + ε3 ε1 + 1+ε 2

β1 = ε1 ,

β3 = ε3 + ε2 ε1 + ! ε1 ε1 ! 2 + 4 , ! ! β5 = ε5 + ε4 ε1 + ε3 ε2 + ε3 ε21 + ε22 ε1 − ε1 ε22 + ε2

ε1 ! 3 , ε1 ! ε1 ! 3 + 5 .

A Truncated Minimal Free Resolution of the Residue Field

431

The first four relations were also given by Avramov [4, page 62] and Gulliksen and Levin [9, Proposition 3.3.4, Theorem 4.4.3]. Note that we have corrected the expression for β4 , compared to that given in [4, page 62]. The expressions for εi described in the statement follow from these relations and Corollary 4.1.

Remark 4.14 The formulas for ε2 , ε3 , ε4 in Corollary 4.13 were previously given in [2, Corollary 6.2] and [9, Proposition 3.3.4]. The expression for ε5 obtained by Avramov in [2, Corollary 6.2] is, in our notations: ε5 = a4 + a1 a2 + a1 q11 − q12 − a13 + b − a, where b := rankk (Ker ) with → A2 ⊗ A1 ⊕ A1 ⊗ A2 ,  :A1 ⊗ A1 ⊗ A1 − ([x], [y], [z]) = ([x] ∧ [y], [y] ∧ [z]),

defined by for all [x], [y], [z] ∈ A1 .

The map ψ in (2.8), that defines our invariant b = rankk (Coker ψ), differs from  just by its codomain. Thus, one can relate b in [2] and our b by: b = rankk (Ker ψ) = a13 − 2a1 q11 + b. Note that in both references [2] and [9] there is a shift in the indexing of the deviations, their εi is our εi+1 .

5 An Example Illustrating the Construction In this section, we consider the Artinian local ring of codepth 4 from [2, Section 7]: R = Q[x, y, z, w]/(x 3 , y 3 , z3 − xy 2 , x 2 z2 , xyz2 , y 2 w, w 2 ). This ring is of a particular interest to us, since its Koszul homology algebra A = H(K R ) has nontrivial multiplication and a nontrivial ternary Massey product that does not come from this multiplication. The free modules {Fi }i=0,...,5 are given in terms of the Koszul algebra components {Ki }i=0,...,5 , the ranks ai of Ai , and the multiplicative invariants qij , a and b. The differential maps of the complex F 1 , 2 , and in Construction 3.1 are given in terms of elements zi1 , z 2 , zt3 , zr4 , p si πs3 , p ui 4  πu , see Summary 2.5. We explicitly describe them all for this ring. The bases of Ai = Hi (K) are computed with Macaulay2 [13]. We use our results from Sect. 2 to obtain the other elements needed in Construction 3.1.

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5.1 The ring R is graded Artinian with Ri = 0 for all i  6. The other graded components have the following basis elements: R0 : 1 R1 : x, y, z, w R2 : x 2 , xy, xz, xw, y 2 , yz, yw, z2 , zw R3 : x 2 y, x 2 z, x 2 w, xyz, xyw, xz2 , xzw, y 2 z, yz2 , yzw, z3 , z2 w R4 : x 2 yz, x 2 yw, x 2 zw, xyzw, xz3 , xz2 w, y 2 z2 , yz2 w, z4 R5 : x 2 yzw, xz4 . < 5.2 The Koszul algebra of R has the form K = (R 4 ) ∼ = R⊕R 4 ⊕R 6 ⊕R 4 ⊕R. Let {T1 , T2 , T3 , T4 } be the standard ordered vector basis of the free module K1 = R 4 . Set Tij = Ti ∧ Tj and consider the ordered basis {T12 , T13 , T23 , T14 , T24 , T34 } for < the free module K2 = 2 (R 4 ) ∼ = R 6 . Set Tij k = Ti ∧ Tj ∧ Tk and consider the < ordered basis {T123 , T124 , T134 , T234 } for the free module K3 = 3 (R 4 ) ∼ = R4. Set T1234 < = T1 ∧ T2 ∧ T3 ∧ T4 and consider the basis {T1234 } for the free module K4 = 4 (R 4 ) ∼ = R. By Macaulay2, we obtain bases of Ai = Hi (K) and ranks a1 = 7,

a2 = 15,

a3 = 14,

a4 = 5.

5.3 A basis of A1 is {[zi1 ]}i=1,...,7 where: z11 = wT4 ,

z21 = x 2 T1 ,

z31 = ywT2 ,

z51 = y 2 T1 − z2 T3 ,

z61 = yz2 T1 ,

z71 = xz2 T1 .

z41 = y 2 T2 ,

5.4 Using Koszul and ring relations, we obtain that the space A1 · A1 has rank q11 = 7 and its basis is given by the classes of z11 ∧ z21 = −x 2 wT14 ,

z11 ∧ z51 = z2 wT34 ,

z11 ∧ z61 = −yz2 wT14 ,

z11 ∧ z71 = −xz2 wT14 ,

z21 ∧ z31 = x 2 ywT12 ,

z31 ∧ z51 = −yz2 wT23 ,

z41 ∧ z51 = −y 2 z2 T23 . Therefore, a basis of A2 is {[z 2 ]} =1,...,8 , where z12 = ywT24 ,

z22 = y 2 T24 ,

z32 = xz2 T12 ,

z42 = yz2 T13 ,

z52 = x 2 yT12 − xz2 T13 ,

z62 = x 2 zT13 ,

z72 = yz2 wT12 ,

z82 = z4 T23 .

5.5 All other products among the basis elements of A1 are zero in A2 , except for the Koszul relations of the nonzero products A1 · A1 above. It follows that the kernel

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of the multiplication map φ1 : A1 ⊗ A1 → A2 is generated by a12 − q11 = 42 elements. We record the indices of basis elements of A1 · A1 by the set S = {(1, 2), (1, 5), (1, 6), (1, 7), (2, 3), (3, 5), (4, 5)}. A basis of Ker φ1 , as defined in (2.5), is given by elements of two types: (1) [zi1 ] ⊗ [zj1 ], for 1  i, j  7, (i, j ) ∈ S and (2) [zi1 ] ⊗ [zj1 ] + [zj1 ] ⊗ [zi1 ], for (i, j ) ∈ S.

(j, i) ∈ S;

Therefore, according to the types above, one defines p s1(i,j ) = ( ps1(i,j ) k )k=1,...,7 as:  z1 if k = i 1 (1) p s(i,j ) k = j for all 1  i, j  7, (i, j ) ∈ S and (j, i) ∈ S; 0 if k = i, ⎧ 1 ⎪ ⎪ ⎨zj if k = i (2) p s1(i,j ) k = zi1 if k = j for all (i, j ) ∈ S. ⎪ ⎪ ⎩0 if k = i, j, 5.6 Next, we find the nonzero elements  πs3 ∈ K3 defined in (2.6) as follows. For the elements coming from Koszul relations in K, we choose  πs3 = 0. The only nonzero products in K2 come from elements of type (1) and they are z21 ∧ z41 = x 2 y 2 T12 = −z41 ∧ z21

and

z51 ∧ z71 = xz4 T13 = −z71 ∧ z51 .

Therefore, we choose the following nontrivial liftings in K3 :  πs3(2,4) = xz2 T123 = − πs3(4,2)

and

 πs3(5,7) = −x 2 yzT123 = − πs3(7,5) .

5.7 We describe next the elements of A3 . Claim 1 All the products in A1 · A1 · A1 are zero. Proof First, all the products in A1 · A1 · A1 not involving [z11 ] have the coefficients of [Tij k ] of degree six, and R6 = 0. Thus, all such products are zero. Second, the only pairs (i, j ) with 1 < i < j  7 such that [zi1 ] ∧ [zj1 ] is nonzero in A1 · A1 are in the set {(2, 3), (3, 5), (4, 5)}, but [z11 ∧ z31 ] = [z11 ∧ z41 ] = 0. Hence, all products involving [z11 ] at least once are zero. The claim now follows.

Therefore, A1 · A2 = A1 · A2 , it has rank q12 = 10, and its basis is given by the classes of z11 ∧ z32 = xz2 wT124 ,

z11 ∧ z42 = yz2 wT134 ,

z11 ∧ z52 = x 2 ywT124 − xz2 wT134 ,

z11 ∧ z62 = x 2 zwT134 ,

z21 ∧ z12 = x 2 ywT124 ,

z21 ∧ z22 = x 2 y 2 T124 ,

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z31 ∧ z62 = −x 2 yzwT123 ,

z51 ∧ z12 = yz2 wT234 ,

z51 ∧ z22 = y 2 z2 T234 ,

z51 ∧ z32 = −xz4 T123 = z41 ∧ z62 .

A basis of A3 is {[zt3 ]}t=1,...,4 , where z13 = yz2 wT123 ,

z23 = y 2 z2 T123 ,

z33 = yz2 wT124 ,

z43 = z4 T234 .

2 , for 1  u  a a − 5.8 Using Proposition 2.4, we describe the elements p ui 1 2 a1 q11 − q12 + b = 46 + b and 1  i  7, as defined in (2.10).

Claim 2 Let A, B be as in Proposition 2.4 and let b be as in Summary 2.5. Then Spank A = A1 ⊗ (A1 · A1 )

and

b = 0.

Proof It is clear that Spank A ⊆ A1 ⊗ (A1 · A1 ). For 1  i, j, h  7, any nonzero element [zi1 ] ⊗ [zj1 ] ∧ [zh1 ] in A1 ⊗ (A1 · A1 ) has (j, h) ∈ S or (h, j ) ∈ S. We show that each such element is in Spank A. In the case {(i, j ), (i, h)} ⊆ S we have [zi1 ] ⊗ [zj1 ] ∧ [zh1 ] =

7  [zk1 ] ⊗ [ ps1(i,j ) k ] ∧ [zh1 ], k=1

as in case (1) of 5.5. By Koszul relation, the case (i, h) ∈ S reduces to the case (i, j ) ∈ S in which we have [zi1 ]⊗[zj1 ]∧[zh1 ] = [zi1 ]⊗[zj1 ]∧[zh1 ]+[zj1 ]⊗[zi1 ]∧[zh1 ] =

7  [zk1 ]⊗[ ps1(i,j ) k ]∧[zh1 ], k=1

as in case (2) of 5.5. The first equality above follows from the proof of Claim 1. This implies SpanQ A = A1 ⊗ (A1 · A1 ), so rankQ (Spank A) = a1 q11 . By Proposition 2.4(b) we get b = 0.

Remark that Claim 2 implies Claim 1, since SpanQ A ⊆ Ker φ2 . By definition of A2 as in Summary 2.5 and Claim 2, the following holds: ! A1 ⊗ A2 = A1 ⊗ (A1 · A1 ) ⊕ (A1 ⊗ A2 ) = (SpanQ A) ⊕ (A1 ⊗ A2 ). By Proposition 2.4, Ker φ2 = (SpanQ A) ⊕ B, where B = (Ker φ2 ) ∩ (A1 ⊗ A2 ). In order to find a basis for B, we record the indices of basis elements of A1 · A2 by the set U = {(1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (3, 6), (5, 1), (5, 2), (5, 3), (4, 6)}.

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A basis of B is given by elements of two types: (1 ) [zi1 ] ⊗ [z 2 ] for all 1  i  7 and 1   8 such that (i, ) ∈ U ; (2 ) [z41 ] ⊗ [z62 ] − [z51 ] ⊗ [z32 ]. pu2(i, ) k )k=1,...,7 as: Therefore, according to the types above, one defines p u2(i, ) = (  z2 if k = i 2  (1 ) p u(i, ) k =

for all 1  i  7 and 1   8 such that (i, ) ∈ U ; 0 if k = i, ⎧ 2 ⎪ if k = 4 ⎪ ⎨z6 2  u(4,6) k = −z32 if k = 5 (2 ) p ⎪ ⎪ ⎩0 if k = 4, 5. 5.9 It is easy to check that in K3 we have the equalities: (1 ) zi1 ∧ z 2 = 0 for all 1  i  7 and 1   8 such that (i, ) ∈ U ; (2 ) z41 ∧ z62 − z51 ∧ z32 = 0. Therefore, we may choose  πu4 = 0 for all 1  u  46. 5.10 A similar argument as in the proof of Claim 1 gives A1 · A3 = 0, and hence q13 = 0. Moreover, A2 · A2 has rank q22 = 2 and its basis is given by the classes of z12 ∧ z62 = −x 2 yzwT1234

and

z22 ∧ z62 = −xz4 T1234 .

As A1 · A3 = 0, any element in A1 , A1 , A1  ⊆ A4 has a representative given by 42 

 πs3 ∧ ps1

s=1

such that

42  1 [ psk ∧ ps1 ] = 0, for all 1  k  7,

(5.1)

s=1

1 and  where ps1 ∈ Ker ∂1K is as in Proposition 2.8, and p si πs3 are defined in (2.5) and (2.6) respectively. As described above, the only nontrivial lifting elements  πs3 ∈ K3 3 3 3 3 are  πs(2,4) = − πs(4,2) and  πs(5,7) = − πs(7,5) . Since all of them contain T123 , only the component of ps1 that contains z11 = wT4 contributes to a nonzero Massey product. Thus, there exist α, β ∈ R such that 42  [ πs3 ∧ ps1 ] = [ πs3(2,4) ∧ ps1(2,4) ] + [ πs3(5,7) ∧ ps1(5,7) ] s=1

πs3(5,7) ∧ z11 ] ∧ [β] = [ πs3(2,4) ∧ z11 ] ∧ [α] + [ = [xz2 wT1234 ] ∧ [α] + [x 2 yzwT1234 ] ∧ [β] = [z21 ], [z41 ], [z11 ] ∧ [α] − [z12 ∧ z62 ] ∧ [β].

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The last equality follows from the following computation: [z21 ], [z41 ], [z11 ] = {[(∂3K )−1 (z21 ∧ z41 ) ∧ z11 + z21 ∧ (∂3K )−1 (z41 ∧ z11 )]} = {[ πs3(2,4) ∧ z11 + z21 ∧  πs3(4,1) ]} = {[xz2 wT1234 ]}. By abuse of notation, we write [z21 ], [z41 ], [z11 ] = [xz2 wT1234 ], which is not in (A1 · A3 + A2 · A2 ), as showed in [2, Section 7]. It is clear now that the rank of the Q-vector space A1 · A3 + A2 · A2 + SpanQ A1 · A1 · A1  is a = 3 and its basis is given by [z12 ] ∧ [z62 ],

[z22 ] ∧ [z62 ],

and

[z21 ], [z41 ], [z11 ].

Thus, a basis of A4 is {[zr4 ]}r=1,2 where z14 = yz2 wT1234

and

z24 = y 2 z2 T1234 .

5.11 By Proposition 4.2, we conclude that the ring R has the following invariants: a1 = 7,

a2 = 15,

q11 = 7,

q12 = 10,

a = 3,

b = 0,

P3 = 7,

P4 = 45,

a3 = 14, q13 = 0,

a4 = 5, q22 = 2,

P5 = 221.

Acknowledgments The authors thank Hailong Dao for inspiring them to work on this project and for very fruitful discussions. They also thank Frank Moore for helping them with the DGAlgebras package [15] to check the computations, and thank the referee for helpful suggestions. The first author was supported by the Naval Academy Research Council in Summer 2020.

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