Skew PBW Extensions: Ring and Module-theoretic Properties, Matrix and Gröbner Methods, and Applications 3030533778, 9783030533779

Table of contents :
Preface
Contents
Part I Ring and Module-Theoretic Properties of Skew PBW Extensions
Chapter 1 Skew PBW Extensions
1.1 Definition and Some Examples
1.2 Universal Property and Characterization
1.3 Existence
Chapter 2 Examples
2.1 PBW Extensions
2.2 Ore Extensions of Bijective Type
2.3 Operator Algebras
2.4 Algebras of Diffusion Type
2.5 Quantum Algebras
2.6 Quadratic Algebras in 3 Variables
Chapter 3 Basic Properties
3.1 Hilbert's Basis Theorem
3.2 Radicals
3.3 The Center
Chapter 4 Rings of Fractions
4.1 Preliminary Key Results
4.2 Ore's Theorem
4.3 Goldie's Theorem
4.4 Skew Quantum Polynomials
4.5 The Gelfand–Kirillov Conjecture
4.6 The Center of the Total Division Ring of Fractions
Chapter 5 Prime Ideals
5.1 Invariant Ideals
5.2 Extensions of Derivation Type
5.3 Extensions of Automorphism Type
5.4 Extensions of Mixed Type
Chapter 6 Minimal Prime Ideals
6.1 Skew Armendariz Rings
6.2 Wedderburn, Lower Nil, Levitzky and Upper Nil Radicals
6.3 Köthe's Conjecture
6.4 Description of Minimal Prime Ideals
Chapter 7 Dimensions
7.1 Global Dimension
7.2 Krull Dimension
7.3 Goldie Dimension
7.4 Gelfand–Kirillov Dimension
Chapter 8 Transfer of Homological Properties
8.1 Regularity
8.2 Serre's Theorem
8.3 Auslander Regularity
8.4 Cohen–Macaulayness
8.5 Strongly Noetherian Algebras
8.6 K-theory
8.7 Summary and Remarks
Part II Projective Modules Over Skew PBW Extensions
Chapter 9 Stably Free Modules
9.1 RC and IBN rings
9.2 Characterizations of Stably Free Modules
9.3 Stafford's Theorem: A Constructive Proof
9.4 The Projective Dimension of a Module
Chapter 10 Hermite Rings
10.1 Matrix Descriptions of Hermite Rings
10.2 Matrix Characterization of PF Rings
10.3 Some Important Subclasses of Hermite Rings
10.4 Products and Quotients
10.5 Localizations
10.6 Examples
Chapter 11 d-Hermite Rings
11.1 d-Hermite Rings
11.2 Stable Rank
11.3 Kronecker's Theorem
Chapter 12 Extended Rings
12.1 Extended Modules and Rings
12.2 Extended Rings and Ore Extensions
12.3 Vaserstein's Theorem
12.4 Quillen's Patching Theorem
12.5 The Quillen–Suslin Theorem
12.6 An Elementary Matrix Proof of the Quillen–Suslin Theorem
Part III Matrix and Gröbner Methods for Skew PBW Extensions
Chapter 13 Gröbner Bases for Skew PBW Extensions
13.1 Monomial Orders in Skew PBW Extensions
13.2 Reduction in Skew PBW Extensions
13.3 Gröbner Bases of Left Ideals
13.4 Buchberger's Algorithm for Left Ideals
Chapter 14 Gröbner Bases of Modules
14.1 Monomial Orders on Mon(Am)
14.2 Reduction in Am
14.3 Gröbner Bases for Submodules of Am
14.4 Buchberger's Algorithm for Modules
14.5 Right Skew PBW Extensions and Right Gröbner Bases
Chapter 15 Elementary Applications of Gröbner Theory
15.1 The Membership Problem
15.2 Computing Syzygies
15.3 Intersections
15.4 Quotients
15.5 Presentation of a Module
15.6 Computing Free Resolutions
15.7 The Kernel and Image of a Homomorphism
Chapter 16 Computing Tor and Ext
16.1 Centralizing Bimodules
16.2 Computation of MN
16.3 Computation of Tor
16.4 Computation of Hom
16.5 Computation of Ext
16.6 Some Applications
Chapter 17 Matrix Computations Using Gröbner Bases
17.1 Computing the Inverse of a Matrix
17.2 Computing the Projective Dimension
17.3 Test for Stably-freeness
17.4 Computing Minimal Presentations
17.5 Computing Free Bases
Part IV Applications: The Noncommutative Algebraic Geometry of Skew PBW Extensions
Chapter 18 Semi-graded Rings
18.1 Semi-graded Rings and Modules
18.2 Generalized Hilbert Series and Hilbert Polynomials
18.3 Gelfand–Kirillov Dimension for FSG 18.3 Rings
18.4 Noncommutative Schemes Associated to SG Rings
18.5 The Serre–Artin–Zhang–Verevkin theorem for semi-graded rings
18.6 Point Modules and the Point Functor
Chapter 19 Semi-graded Algebras
19.1 Definition
19.2 Examples of FSG Algebras
19.3 Koszulity
19.4 Artin–Schelter Regularity
19.5 Classification of Skew PBW Algebras
Chapter 20 The Zariski Cancellation Problem for Skew PBW Extensions
20.1 The Zariski Cancellation Problem
20.2 The Center and the Zariski Cancellation Problem
20.3 Gelfand–Kirillov Dimension for Rings
20.4 Makar-Limanov Invariants
20.5 The Discriminant and the Divisor Algebra as Tools for the Cancellation Problem
20.6 Noncommutative Cancellative Algebras: Nondomain Examples
20.7 The Zariski Cancellation Problem for Rings
20.8 Skew PBW Cancellation
20.9 Examples
Appendices
Appendix A Noncommutative Algebraic Geometry of Graded Algebras
A.1 Finitely Graded Algebras
A.2 Graded Hom and Ext
A.3 Geometry Via Point Modules
A.4 Functorial Characterization of Point Modules
A.5 Geometry Via Noncommutative Schemes
Appendix B Koszul and Artin–Schelter Regular N-Graded Algebras
B.1 Koszul Algebras
B.2 Artin–Schelter Regular Algebras
Appendix C Implementation of Skew PBW Extensions With Maple
C.1 Gröbner Theory of Skew PBW Extensions With Maple
C.1.1 Defining Skew PBW Extensions
C.1.2 Division Algorithm for Left Ideals
C.1.3 Buchberger's Algorithm for Left Ideals
C.1.4 The Division Algorithm for Modules
C.1.5 Buchberger's Algorithm for Modules
C.2 Some Homological Computations
C.2.1 Computation of Syzygies
C.2.2 Computation of Free Resolutions
C.2.3 Computing the Left Inverse of a Matrix
C.3 Algorithm for the Quillen–Suslin Theorem
Appendix D Maple Library Documentation
D.1 The Package SPBWETools
D.1.1 Skew PBW Extensions
Calling Sequence
Parameters
Remark
D.1.2 Some Useful Functions With Skew
D.1.2.1 SkewProd
Calling Sequence
Parameters
D.1.2.2 SkewSum
Calling Sequence
Parameters
D.1.2.3 SkewSubs
Calling Sequence
Parameters
D.1.2.4 SkewRelation
Calling Sequence
Parameters
D.1.2.5 deg
Calling Sequence
Parameters
D.1.2.6 CanonicalVector
Calling Sequence
Parameters
D.1.2.7 SkewScalarProd
Calling Sequence
Parameters
D.1.2.8 SkewPointedProd
Calling Sequence
Parameters
D.1.2.9 SkewSumVector
Calling Sequence
Parameters
D.1.2.10 SkewMinusVector
Calling Sequence
Parameters
D.1.2.11 GeneratePolyMatrix
Calling Sequence
Parameters
D.1.2.12 SkewProdMatrix
Calling Sequence
Parameters
D.1.2.12 SkewProdMatrix
Calling Sequence
Parameters
D.1.2.14 SkewSubsMatrix
Calling Sequence
Parameters
D.2 The Package SPBWEGrobner
D.2.0.1 lcVector
Calling Sequence
Parameters
D.2.0.2 ltVector
Calling Sequence
Parameters
D.2.0.3 lmVector
Calling Sequence
Parameters
D.2.0.4 PrintSkewPolyVector
Calling Sequence
Parameters
D.2.0.5 DivisionAlgorithm
Calling Sequence
Parameters
D.2.0.6 BuchbergerAlgSkewPoly
Calling Sequence
Parameters
D.2.0.7 SyzModule
Calling Sequence
Parameters
D.2.0.8 FreeResolution
Calling Sequence
Parameters
D.2.0.9 HQMatrices
Calling Sequence
Parameters
D.2.0.10 LeftInverseMatrix
Calling Sequence
Parameters
Appendix E Examples of Skew PBW Extensions in SPBWE.lib
E.1 PBW Extensions
Calling Sequence
Parameters
E.2 The Dispin Algebra
Calling Sequence
Parameters
E.3 The Manin Algebra of 22 Quantum Matrices
Calling Sequence
Parameters
E.4 The Woronowicz Algebra
Calling Sequence
Parameters
E.5 The Heisenberg Algebra
Calling Sequence
Parameters
Remark
E.6 The Univariate Skew Polynomial Ring R[x;,]
Calling Sequence
Parameters
E.7 The Additive Analogue of the Weyl Algebra
Calling Sequence
Parameters
Remark
E.8 The Multiplicative Analogue of the Weyl Algebra
Calling Sequence
Parameters
E.9 The Witten Algebra
Calling Sequence
Parameters
E.10 The σ-Multivariate Ore Extension
Calling Sequence
Parameters
Remark
References
Index

Citation preview

Algebra and Applications

William Fajardo · Claudia Gallego  Oswaldo Lezama · Armando Reyes  Héctor Suárez · Helbert Venegas

Skew PBW Extensions

Ring and Module-theoretic Properties, Matrix and Gröbner Methods, and Applications

Algebra and Applications Volume 28

Series Editors Michel Broué, Université Paris Diderot, Paris, France Alice Fialowski, Eötvös Loránd University, Budapest, Hungary Eric Friedlander, University of Southern California, Los Angeles, CA, USA Iain Gordon, University of Edinburgh, Edinburgh, UK John Greenlees, Sheffield University, Sheffield, UK Gerhard Hiß, Aachen University, Aachen, Germany Ieke Moerdijk, Utrecht University, Nijmegen, Utrecht, The Netherlands Christoph Schweigert, Hamburg University, Hamburg, Germany Mina Teicher, Bar-Ilan University, Ramat-Gan, Israel Alain Verschoren, University of Antwerp, Antwerp, Belgium

Algebra and Applications aims to publish well-written and carefully refereed monographs with up-to-date expositions of research in all fields of algebra, including its classical impact on commutative and noncommutative algebraic and differential geometry, K-theory and algebraic topology, and further applications in related domains, such as number theory, homotopy and (co)homology theory through to discrete mathematics and mathematical physics. Particular emphasis will be put on state-of-the-art topics such as rings of differential operators, Lie algebras and super-algebras, group rings and algebras, Kac-Moody theory, arithmetic algebraic geometry, Hopf algebras and quantum groups, as well as their applications within mathematics and beyond. Books dedicated to computational aspects of these topics will also be welcome. More information about this series at http://www.springer.com/series/6253

William Fajardo • Claudia Gallego Oswaldo Lezama • Armando Reyes Héctor Suárez • Helbert Venegas

Skew PBW Extensions Ring and Module-theoretic Properties, Matrix and Gröbner Methods, and Applications

William Fajardo Instituto Politécnico Grancolombiano Bogotá, Colombia

Claudia Gallego Universidad Sergio Arboleda Bogotá, Colombia

Oswaldo Lezama National University of Colombia Bogota, Colombia

Armando Reyes National University of Colombia Bogota, Colombia

Héctor Suárez Universidad Pedagógica y Tecnológica de Colombia Tunja, Colombia

Helbert Venegas Universidad de la Sabana Chía, Colombia

ISSN 1572-5553 ISSN 2192-2950 (electronic) Algebra and Applications ISBN 978-3-030-53377-9 ISBN 978-3-030-53378-6 (eBook) https://doi.org/10.1007/978-3-030-53378-6 Mathematics Subject Classification (2020): 16S36, 16U20, 16D40, 16E05, 16E65, 16S38, 16S80, 16W70, 16Z05 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Dedicated to Universidad Nacional de Colombia

Preface

The usual commutative polynomial ring in several variables over a commutative ring R, R[x1 , . . . , xn ], can be generalized to a noncommutative context by changing R to any noncommutative ring, but preserving the basic rules of multiplication, i.e., xi xj = xj xi and rxi = xi r for 1 ≤ i, j ≤ n and any r ∈ R. However, in some areas of mathematics and its applications, such as algebras of differential operators in differential equations and linear multidimensional control systems in algebraic analysis (see [91], [92], [93], [95], [316], [318], [319], [320], [317] and [323]), it is necessary to consider wider classes of rings of polynomial type in which the variables do not commute or the coefficients do not commute with the variables. For example, take a homogeneous linear ordinary differential equation with coefficients in Q[t], pn (t)y (n) + · · · + p1 (t)y 0 + p0 (t)y = 0,

pi (t) ∈ Q[t],

0 ≤ i ≤ n;

d considering the differential operator ∂ := dt this equation can be interpreted as [pn (t)∂ n + · · · + p1 (t)∂ + p0 (t)](y) = 0,

where y is in the set S of solutions of this equation, and in turn, S is contained in a D-module F , where D is a Q-algebra generated by two variables t, ∂. Observe that for p ∈ Q[t] and y ∈ F , we must have (∂p)(y) = ∂(py) = p∂(y)+∂(p)y = py 0 +p0 y = (p∂ +p0 )(y), i.e., ∂p = p∂ +p0 , in particular, if p = t, then in D we have the rule of multiplication ∂t = t∂ + 1. Hence, in the new “polynomial ring” D = Q[t, ∂], the variables do not commute. These important types of rings are particular cases of the skew polynomial rings of injective type, and these last ones are a subclass of the skew P BW extensions that we will introduce and study in the present monograph. Skew P BW extensions represent a generalization of P BW (Poincar´e– Birkhoff–Witt) extensions defined by Bell and Goodearl ([48]), and include the usual polynomials rings and a lot of other important classes of rings vii

viii

Preface

such as Weyl algebras, enveloping algebras of Lie algebras, and many examples of quantum algebras such as the Manin algebra of quantum matrices, q-Heisenberg algebra, Hayashi algebra, Witten’s deformation of U (sl(2, K), etc. Most of the examples of algebras and rings described in the present work as skew P BW extensions have been investigated by many other authors, but they interpreted them in a different way, for example, Levandovsky in [231] defined the G-algebras, which also include as important particular cases the quantum algebras mentioned before. In a similar way, Bueso, G´omezTorrecillas and Verschoren in [74] introduced the P BW rings, which give an alternative way of interpreting most of the quantum algebras and rings arising in mathematical physics. It is also important to note that the Hopf algebras are another very useful way of interpreting all of these noncommutative algebras (see [201]). Thus, the skew P BW extensions can be considered as an alternative technique for studying a very wide class of modern noncommutative algebras of polynomial type which have recently arisen in quantum mechanics and mathematical physics. Our point of view is very general since for all of these rings of polynomial type the ring of coefficients is not necessarily a field, but an arbitrary ring. This monograph is divided into four parts; the first part is concerned with the ring-module-theoretic and homological properties of skew P BW extensions. We start with the definition and the universal property that characterizes this new class of noncommutative rings. In the second chapter we have included many important examples of rings and algebras that can be interpreted as skew P BW extensions. This chapter contains a new short proof of the Poincar´e–Birkhoff–Witt theorem on the bases of the universal enveloping algebra of a finite-dimensional Lie algebra. The proof is supported by the universal characterization and the theorem of existence of skew P BW extensions studied in the first chapter. Some of the most classical topics of ring-module theory will be considered for the skew P BW extensions in this first part. We will study the Hilbert basis theorem, rings of fractions and the Ore and Goldie theorems. A particular collection of prime ideals will be computed, as well as the Jacobson and the prime radicals in some special cases. Some dimensions will be estimated, in particular, the global dimension, the Krull dimension, the Goldie dimension and the Gelfand–Kirillov dimension. An introduction to the Gelfand–Kirillov conjecture will be presented, and for this, we will compute the center of some skew P BW extensions. The theory of regularity will be considered, in particular, Serre’s theorem, Auslander regularity and the Cohen–Macaulay condition. We will compute the Quillen K-groups for bijective skew PBW extensions, in particular, the Grothendieck, Bass and Milnor groups. Most of the computations and results of the first part are supported by the following basic facts: we will prove that skew P BW extensions are positively filtered rings and the corresponding graded rings are iterated skew polynomial rings, hence, the well-known results in the literature (see [159] and [278]) about ring-theoretic and homological properties of skew polynomials, and the transfer theorems from graded to filtered, can be applied. The first important theorem that we will prove using this technique

Preface

ix

is the Hilbert basis theorem, which states if the ring of coefficients of a bijective skew P BW extensions is left noetherian, then the extension is left noetherian. The second part is dedicated to the investigation of finitely generated projective modules over skew P BW extensions from a matrix point of view. As preparatory material, we consider first stably freeness, Stafford’s theorem about the stable rank, and the Hermite condition for arbitrary noncommutative rings, and then, we apply these preliminaries to study extended modules over skew P BW extensions. An elementary matrix-constructive proof of the Quillen–Suslin theorem for single Ore extensions of bijective type over fields is included. In order to make constructive the theory of projective modules studied in the second part, in the third part we will construct the theory of Gr¨obner bases of left (right) ideals and modules for bijective skew P BW extensions. We will extend some results of [421] (compare also with [73], [231] and [257]). We will present some applications in noncommutative homological algebra, as was done in [239] for commutative polynomial rings (see also [236], [237] and [238]). For example, we will compute syzygies and the Ext and Tor modules over bijective skew P BW extensions. Matrix computations using Gr¨obner bases are included in this part. We will calculate inverses of matrices as well as algorithms for testing stably-freeness, and we will compute minimal presentations of stably-free modules over skew P BW extensions. The Gr¨obner theory and some of the mentioned computations have been implemented in Maple in [117] and [118]. This implementation is based on a library specialized for working with bijective skew P BW extensions. The library has utilities to calculate Gr¨obner bases, and it includes some functions that compute the module of syzygies, free resolutions and left inverses of matrices, among other things. In addition, another independent library was created that makes it possible to execute the Quillen–Suslin theorem for K[x; σ, δ], with K a field, σ a K-automorphism and δ a σ-derivation. The reader interested in these computational aspects of the skew P BW extensions can consult Appendices C, D and E at the end of the monograph (see also Chapter 4 and Appendix A in [117]). The last part of the monograph is dedicated to the application of skew P BW extensions in the investigation of some key topics of the noncommutative algebraic geometry of quantum algebras. For this purpose, we will introduce the semi-graded rings, and for them we will study the noncommutative version of the Serre–Artin–Zhang–Verevkin theorem on the equivalence of a certain quotient abelian category of finitely generated semi-graded modules with the noncommutative version of the category of coherent sheaves. The semi-graded Koszul algebras and the semi-graded Artin–Schelter regular algebras will be defined and investigated. In order to understand better the semi-graded rings, we include in Appendix A a quick review of the noncommutative algebraic geometry of finitely graded algebras and in Appendix B a review of Koszul and Artin–Schelter regular algebras. Another classical topic arising in commutative algebraic geometry is the Zariski cancellation

x

Preface

problem. We will investigate it for noncommutative rings and algebras in the context of skew P BW extensions. We will identify families of skew P BW extensions that are cancellative. As was mentioned above, the present monograph is based on known results in the theory of skew polynomial rings, and also on algebraic and homological properties of filtered and graded rings. Moreover, for the last part concerning the noncommutative algebraic geometry of skew P BW extensions, we use basic properties of finitely graded algebras over fields. Thus, in general, the monograph is not self-contained, and the wide list of references included at the end contains many theoretical facts that we have used in the proofs of properties and results on the skew P BW extensions. However, the used facts have been properly inserted and cited in the text. A very reduced list of the most used references, and that have mainly influenced the present work, is [20], [22], [23], [30], [51], [52], [74], [130], [138], [139], [140], [151], [159], [215], [216], [231], [257], [264], [278], [281], [291], [292], [300], [311], [322], [323], [344], [373], [379], [421]. The four parts of the monograph are almost independent, but we recommend the readers to cover the first three chapters in order to understand better any of the topics studied. Some problems (probably open) which arose during the writing of this monograph have been included. The readers are invited to work on them. In this monograph we will use the following notation: in general, all rings are noncommutative, but always with unity; if nothing contrary is said, all modules are left modules; K is a field, S is an arbitrary ring; S ∗ represents the group of invertible elements of S; Mr×s (S) is the set of rectangular matrices over S of r rows and s columns; GLr (S) := Mr×r (S)∗ is the general linear group of order r over S. If F is a matrix in Mr (S) := Mr×r (S), then its transpose is denoted by F T . Since we will mainly deal with skew P BW extensions, we reserve the capital letter A to denote these types of rings. Oswaldo Lezama Bogot´a, Colombia June, 2020

Contents

Part I Ring and Module-Theoretic Properties of Skew P BW Extensions 1

Skew P BW Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 Definition and Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Universal Property and Characterization . . . . . . . . . . . . . . . . . . 17 1.3 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 P BW Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Ore Extensions of Bijective Type . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Operator Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Algebras of Diffusion Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Quantum Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Quadratic Algebras in 3 Variables . . . . . . . . . . . . . . . . . . . . . . . .

25 25 27 28 30 31 39

3

Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Hilbert’s Basis Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 41 49 51

4

Rings of Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Preliminary Key Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Ore’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Goldie’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Skew Quantum Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 The Gelfand–Kirillov Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 The Center of the Total Division Ring of Fractions . . . . . . . . .

65 65 76 79 83 89 91

5

Prime Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Invariant Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Extensions of Derivation Type . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Extensions of Automorphism Type . . . . . . . . . . . . . . . . . . . . . . .

99 99 102 104 xi

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5.4 Extensions of Mixed Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6

Minimal Prime Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Skew Armendariz Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Wedderburn, Lower Nil, Levitzky and Upper Nil Radicals . . . 6.3 K¨othe’s Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Description of Minimal Prime Ideals . . . . . . . . . . . . . . . . . . . . . .

111 111 118 122 123

7

Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Global Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Krull Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Goldie Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Gelfand–Kirillov Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

127 127 130 132 133

8

Transfer of Homological Properties . . . . . . . . . . . . . . . . . . . . . . . 8.1 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Serre’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Auslander Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Cohen–Macaulayness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Strongly Noetherian Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Summary and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

137 137 138 139 141 142 144 148

Part II Projective Modules Over Skew P BW Extensions 9

Stably Free Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 RC and IBN rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Characterizations of Stably Free Modules . . . . . . . . . . . . . . . . . . 9.3 Stafford’s Theorem: A Constructive Proof . . . . . . . . . . . . . . . . . 9.4 The Projective Dimension of a Module . . . . . . . . . . . . . . . . . . . .

161 161 166 173 178

10 Hermite Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Matrix Descriptions of Hermite Rings . . . . . . . . . . . . . . . . . . . . . 10.2 Matrix Characterization of P F Rings . . . . . . . . . . . . . . . . . . . . . 10.3 Some Important Subclasses of Hermite Rings . . . . . . . . . . . . . . 10.4 Products and Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Localizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

183 183 186 193 200 203 205

11 d-Hermite Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 d-Hermite Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Stable Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Kronecker’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

209 210 211 213

Contents

12 Extended Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Extended Modules and Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Extended Rings and Ore Extensions . . . . . . . . . . . . . . . . . . . . . . 12.3 Vaserstein’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Quillen’s Patching Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 The Quillen–Suslin Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 An Elementary Matrix Proof of the Quillen–Suslin Theorem .

xiii

217 218 219 222 226 230 231

Part III Matrix and Gr¨ obner Methods for Skew P BW Extensions 13 Gr¨ obner Bases for Skew P BW Extensions . . . . . . . . . . . . . . . . 13.1 Monomial Orders in Skew P BW Extensions . . . . . . . . . . . . . . . 13.2 Reduction in Skew P BW Extensions . . . . . . . . . . . . . . . . . . . . . . 13.3 Gr¨obner Bases of Left Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Buchberger’s Algorithm for Left Ideals . . . . . . . . . . . . . . . . . . . .

237 237 239 248 250

14 Gr¨ obner Bases of Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Monomial Orders on Mon(Am ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Reduction in Am . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Gr¨obner Bases for Submodules of Am . . . . . . . . . . . . . . . . . . . . . 14.4 Buchberger’s Algorithm for Modules . . . . . . . . . . . . . . . . . . . . . . 14.5 Right Skew P BW Extensions and Right Gr¨obner Bases . . . . .

261 261 263 273 275 283

15 Elementary Applications of Gr¨ obner Theory . . . . . . . . . . . . . . 15.1 The Membership Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Computing Syzygies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 Presentation of a Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6 Computing Free Resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7 The Kernel and Image of a Homomorphism . . . . . . . . . . . . . . . .

287 287 291 305 308 309 310 311

16 Computing Tor and Ext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 Centralizing Bimodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Computation of M ⊗ N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Computation of Tor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4 Computation of Hom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5 Computation of Ext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.6 Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

317 317 318 323 324 327 330

17 Matrix Computations Using Gr¨ obner Bases . . . . . . . . . . . . . . 17.1 Computing the Inverse of a Matrix . . . . . . . . . . . . . . . . . . . . . . . 17.2 Computing the Projective Dimension . . . . . . . . . . . . . . . . . . . . . 17.3 Test for Stably-freeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4 Computing Minimal Presentations . . . . . . . . . . . . . . . . . . . . . . . . 17.5 Computing Free Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

335 335 342 344 346 347

xiv

Contents

Part IV Applications: The Noncommutative Algebraic Geometry of Skew P BW Extensions 18 Semi-graded Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1 Semi-graded Rings and Modules . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Generalized Hilbert Series and Hilbert Polynomials . . . . . . . . . 18.3 Gelfand–Kirillov Dimension for F SG Rings . . . . . . . . . . . . . . . . 18.4 Noncommutative Schemes Associated to SG Rings . . . . . . . . . . 18.5 The Serre–Artin–Zhang–Verevkin theorem for semi-graded rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.6 Point Modules and the Point Functor . . . . . . . . . . . . . . . . . . . . .

370 380

19 Semi-graded Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 Examples of F SG Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3 Koszulity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4 Artin–Schelter Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.5 Classification of Skew P BW Algebras . . . . . . . . . . . . . . . . . . . . .

387 387 389 393 399 411

20 The Zariski Cancellation Problem for Skew P BW Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1 The Zariski Cancellation Problem . . . . . . . . . . . . . . . . . . . . . . . . 20.2 The Center and the Zariski Cancellation Problem . . . . . . . . . . 20.3 Gelfand–Kirillov Dimension for Rings . . . . . . . . . . . . . . . . . . . . . 20.4 Makar-Limanov Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.5 The Discriminant and the Divisor Algebra as Tools for the Cancellation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.6 Noncommutative Cancellative Algebras: Nondomain Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.7 The Zariski Cancellation Problem for Rings . . . . . . . . . . . . . . . . 20.8 Skew P BW Cancellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.9 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

357 357 361 363 367

413 413 414 417 431 440 444 455 458 462

Appendices A

B

Noncommutative Algebraic Geometry of Graded Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Finitely Graded Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Graded Hom and Ext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Geometry Via Point Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Functorial Characterization of Point Modules . . . . . . . . . . . . . . A.5 Geometry Via Noncommutative Schemes . . . . . . . . . . . . . . . . . .

465 465 477 481 492 499

Koszul and Artin–Schelter Regular N-Graded Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 B.1 Koszul Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 B.2 Artin–Schelter Regular Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 506

Contents

xv

Implementation of Skew P BW Extensions With Maple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.1 Gr¨obner Theory of Skew P BW Extensions With Maple . . . . . C.1.1 Defining Skew P BW Extensions . . . . . . . . . . . . . . . . . . . C.1.2 Division Algorithm for Left Ideals . . . . . . . . . . . . . . . . . . C.1.3 Buchberger’s Algorithm for Left Ideals . . . . . . . . . . . . . . C.1.4 The Division Algorithm for Modules . . . . . . . . . . . . . . . . C.1.5 Buchberger’s Algorithm for Modules . . . . . . . . . . . . . . . . C.2 Some Homological Computations . . . . . . . . . . . . . . . . . . . . . . . . . C.2.1 Computation of Syzygies . . . . . . . . . . . . . . . . . . . . . . . . . . C.2.2 Computation of Free Resolutions . . . . . . . . . . . . . . . . . . . C.2.3 Computing the Left Inverse of a Matrix . . . . . . . . . . . . . C.3 Algorithm for the Quillen–Suslin Theorem . . . . . . . . . . . . . . . . .

509 509 510 514 515 517 520 522 522 526 529 533

D

Maple Library Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.1 The Package SPBWETools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.1.1 Skew P BW Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . D.1.2 Some Useful Functions With Skew P BW Extensions . . D.2 The Package SPBWEGrobner . . . . . . . . . . . . . . . . . . . . . . . . . . . .

543 543 544 545 550

E

Examples of Skew P BW Extensions in SPBWE.lib . . . . . . . E.1 P BW Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.2 The Dispin Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.3 The Manin Algebra of 2 × 2 Quantum Matrices . . . . . . . . . . . . E.4 The Woronowicz Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.5 The Heisenberg Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.6 The Univariate Skew Polynomial Ring R[x; σ, δ] . . . . . . . . . . . . E.7 The Additive Analogue of the Weyl Algebra . . . . . . . . . . . . . . . E.8 The Multiplicative Analogue of the Weyl Algebra . . . . . . . . . . . E.9 The Witten Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.10 The σ-Multivariate Ore Extension . . . . . . . . . . . . . . . . . . . . . . . .

555 555 556 556 557 557 558 559 559 560 561

C

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579

Part I Ring and Module-Theoretic Properties of Skew P BW Extensions

Chapter 1

Skew P BW Extensions

In this first chapter we introduce the skew P BW extensions and we present some examples of this new class of rings that generalizes the P BW (Poincar´e– Birkhoff–Witt) extensions (see [48]). Skew P BW extensions include wellknown classes of Ore algebras, operator algebras and also a lot of quantum rings and algebras. The universal property that characterizes these objects and its existence will be studied.

1.1 Definition and Some Examples P BW extensions were defined by Bell and Goodearl in 1988 in [48]; let R and A be rings; we say that A is a Poincar´e–Birkhoff–Witt extension of R, abbreviated ‘P BW extension’, if the following conditions hold: (i) R ⊆ A. (ii) There exist finitely many elements x1 , . . . , xn ∈ A such that A is a left R-free module with basis Mon(A) := Mon{x1 , . . . , xn } αn n 1 := {xα = xα 1 · · · xn | α = (α1 , . . . , αn ) ∈ N }. In this case we say that A is a ring of left polynomial type over R with respect to {x1 , . . . , xn } and Mon(A) is the set of standard monomials of A. Moreover, x01 · · · x0n := 1 ∈ Mon(A). We say that Mon(A) is a P BW basis of A. (iii) xi r − rxi ∈ R, for each r ∈ R and 1 ≤ i ≤ n. (iv) xi xj − xj xi ∈ R + Rx1 + · · · + Rxn , for any 1 ≤ i, j ≤ n. In this situation we write A = Rhx1 , . . . , xn i. Many important classes of rings and algebras are P BW extensions, for example: © Springer Nature Switzerland AG 2020 W. Fajardo et al., Skew PBW Extensions, Algebra and Applications 28, https://doi.org/10.1007/978-3-030-53378-6_1

3

4

1 Skew P BW Extensions

(a) If A = R[t1 , . . . , tn ] is the usual polynomial ring, so ti r − rti = 0 and ti tj − tj ti = 0, for any r ∈ R and 1 ≤ i, j ≤ n. The R-free basis is Mon(A). (b) Recall that the skew polynomial ring A = R[x; σ, δ] is the noncommutative polynomial ring with product defined by xr = σ(r)x + δ(r), where σ : R → R is an endomorphism of R and δ is a σ-derivation of R, i.e., δ(r + r0 ) = δ(r) + δ(r0 ) and δ(rr0 ) = σ(r)δ(r0 ) + δ(r)r0 , for any r, r0 ∈ R. Any skew polynomial ring of derivation type, i.e., when σ = iR , is a P BW extension: for this ring, xr − rx = δ(r) and xx − xx = 0; the R-free basis is {xl |l ≥ 0} (see [278] and [300], or also [98] and [242]). (c) Consider the iterated skew polynomial ring A := R[x1 ; σ1 , δ1 ] · · · [xn ; σn , δn ], where σi , δi are defined on R[x1 ; σ1 , δ1 ] · · · [xi−1 ; σi−1 , δi−1 ], i.e., σi , δi : R[x1 ; σ1 , δ1 ] · · · [xi−1 ;σi−1 , δi−1 ] → R[x1 ; σ1 , δ1 ] · · · [xi−1 ; σi−1 , δi−1 ]; a particular case of iterated skew polynomial rings are the Ore extensions, i.e., when the following conditions hold: σi (xj ) = xj , j < i, δi (xj ) = 0, j < i, σi σj = σj σi , 1 ≤ i, j ≤ n, δi δj = δj δi , 1 ≤ i, j ≤ n, σi δj = δj σi , 1 ≤ i 6= j ≤ n. In [242] it is proved that these conditions are equivalent to the following: σi (xj ) = xj , j < i, δi (xj ) = 0, j < i, σi σ1 = σ1 σi , 1 ≤ i ≤ n, δi δ1 = δ1 δi , 1 ≤ i ≤ n, and in turn, these last ones are equivalent to xi xj = xj xi , 1 ≤ i, j ≤ n, σi (R), δi (R) ⊆ R, 1 ≤ i ≤ n. Thus, in order to define σi , δi in an Ore extension we only need to know the values in R of σi (r) and δi (r), for any r ∈ R. In other words, in any Ore extension σi and δi can be understood as functions from R to R, i.e., σi , δi : R → R. Important examples of Ore extensions are the Ore algebras, i.e., when R := K[t1 , . . . , tm ], m ≥ 0, K a field (or more generally, a commutative ring), and σi , δi are K-linear; observe that this

1.1 Definition and Some Examples

5

is equivalent to σi (k) = k, δi (k) = 0 for every k ∈ K; moreover, note that in fact A is a K-algebra. Any Ore extension of derivation type, i.e., such that σi = iR , for any 1 ≤ i ≤ n, is a P BW extension: xi r − rxi = δi (r), xi xj − xj xi = 0; from the definition of iterated skew polynomial rings it is clear that the R-free basis is Mon(A). Note that the usual polynomial ring R[x1 , . . . , xn ] is an Ore extension of derivation type, and also any Ore algebra of derivation type is a P BW extension. (d) The Weyl algebra An (K) = K[t1 , . . . , tn ][x1 ; σ1 , δ1 ] · · · [xn ; σn , δn ], K a field, is another important example of a P BW extension: in fact, the Weyl algebra An (K) is an Ore algebra of derivation type with σi := iK[t1 ,...,tn ] and δi := ∂/∂ti , for 1 ≤ i ≤ n. In this situation, xi p = pxi + ∂p/∂ti , xi xj − xj xi = 0, for any p ∈ K[t1 , . . . , tn ] and 1 ≤ i, j ≤ n (see [278], or also [242]). An (K) can be generalized to the extended Weyl algebra Bn (K) := K(t1 , . . . , tn )[x1 ; σ1 , δ1 ] · · · [xn ; σn , δn ], with σi := iK(t1 ,...,tn ) and δi := ∂/∂ti , for 1 ≤ i ≤ n (K(t1 , . . . , tn ) is the field of fractions of K[t1 , . . . , tn ]). Bn (K) is also an Ore algebra of derivation type, and hence, a P BW extension. Another generalization of the Weyl algebra An (K) can be obtained by assuming that R is an arbitrary ring, i.e., we have the Weyl ring An (R) := R[t1 , . . . , tn ][x1 ; ∂/∂t1 ] · · · [xn ; ∂/∂tn ]. Observe that An (R) is an Ore extension of derivation type. Moreover, note that An+m (R) ∼ = An (Am (R)). (e) Let K be a field (or more generally, a commutative ring) and G a finitedimensional Lie algebra over K with basis {x1 , . . . , xn }; the universal enveloping algebra of G, U (G), is a P BW extension of K (see [159], [201], [257]), [278] and [421]). In this case, for any k ∈ K and 1 ≤ i, j ≤ n, xi k − kxi = 0 and xi xj − xj xi = [xi , xj ] ∈ G = Kx1 + · · · + Kxn ⊆ K + Kx1 + · · · + Kxn . In Chapter 2 we will prove that the standard monomials Mon{x1 , . . . , xn } in fact conform a K-basis of U (G), i.e., we will prove the classical Poincar´e–Birkhoff–Witt theorem. Some concrete examples of universal enveloping algebras are the following: ˆ The universal enveloping algebra of the Lie algebra sl(2, K) ([201] V. 3); U (sl(2, K)) is the K-algebra generated by the variables x, y, z subject to the relations

[x, y] = z,

[x, z] = −2x,

[y, z] = 2y.

(1.1.1)

Recall that sl(2, K) is the Lie algebra of matrices over K of size 2 × 2 with null trace. ˆ U (so(3, K)) is the K-algebra generated by the variables x, y, z subject to the relations [x, y] = z,

[x, z] = −y,

[y, z] = x.

6

1 Skew P BW Extensions

so(3, K) is the Lie algebra of matrices over K of size 3 × 3 that satisfy F + F T = 0. (f) Let K, G, {x1 , . . . , xn } and U (G) be as in (e); let R be a K-algebra containing K. The tensor product A := R⊗K U (G) of algebras R and U (G) αn 1 is a P BW extension of R. In this case {1 ⊗ xα 1 · · · xn |αi ≥ 0, 1 ≤ i ≤ n} is an R-free basis of A; moreover, (r ⊗ 1)(1 ⊗ xi ) − (1 ⊗ xi )(r ⊗ 1) = 0 and observe that (1 ⊗ xi )(1 ⊗ xj ) − (1 ⊗ xj )(1 ⊗ xi ) = 1 ⊗ [xi , xj ] ∈ 1 ⊗ G ⊆ R ⊗ 1 + R ⊗ x1 + · · · + R ⊗ xn . The tensor product is a particular case of a more general construction, the crossed product R ∗ U(G) of R by U (G), which is also a P BW extension of R, see [278], or also [242]. We observe that if in the example (b) above σ 6= iR , then the skew polynomial ring R[x; σ, δ] is not a P BW extension since xr − rx = (σ(r) − r)x + δ(r) ∈ R + Rx and σ(r) − r 6= 0 for some r ∈ R. A similar situation can occur for iterated skew polynomial rings. In [136] the following generalization of P BW extensions was introduced. Definition 1.1.1. Let R and A be rings, we say that A is a skew PBW extension of R (also called a σ-PBW extension) if the following conditions hold: (i) R ⊆ A. (ii) There exist finitely many elements x1 , . . . , xn ∈ A such that A is a left R-free module with basis Mon(A) := Mon{x1 , . . . , xn } = {xα αn n 1 := xα 1 · · · xn |α = (α1 , . . . , αn ) ∈ N }. (iii) For every 1 ≤ i ≤ n and r ∈ R − {0} there exists a ci,r ∈ R − {0} such that xi r − ci,r xi ∈ R. (1.1.2) (iv) For every 1 ≤ i, j ≤ n there exists ci,j ∈ R − {0} such that xj xi − ci,j xi xj ∈ R + Rx1 + · · · + Rxn .

(1.1.3)

Under these conditions we will write A = σ(R)hx1 , . . . , xn i, and R will be called the ring of coefficients of the extension. Remark 1.1.2. (i) In general, for i 6= j the elements xi and xj do not commute. Since Mon(A) is an R-basis for A, in the above definition the elements ci,r and ci,j are unique. Note that for i = j, ci,i = 1: in fact, x2i − ci,i x2i = r0 + r1 x1 + · · · + rn xn , with rk ∈ R for 0 ≤ k ≤ n, hence rk = 0 and ci,i = 1. (ii) If r = 0, then we define ci,0 = 0: indeed, 0 = xi 0 = ci,0 xi + r0 , with 0 r ∈ R, but since Mon(A) is an R-basis, then r0 = 0 and ci,0 = 0. (iii) Condition (iv) in Definition 1.1.1 is equivalent to the following: for every 1 ≤ i < j ≤ n there exists left invertible ci,j ∈ R such that xj xi − ci,j xi xj ∈ R + Rx1 + · · · + Rxn .

(1.1.4)

1.1 Definition and Some Examples

7

In fact, from (1.1.3) there exist cj,i , ci,j ∈ R such that xi xj − cj,i xj xi ∈ R + Rx1 + · · · + Rxn and xj xi − ci,j xi xj ∈ R + Rx1 + · · · + Rxn , but since Mon(A) is an R-basis then 1 = cj,i ci,j , whence, for every 1 ≤ i < j ≤ n, ci,j is left invertible. Conversely, assuming that ci,j is left invertible for 1 ≤ i < j ≤ n, let c0i,j ∈ R such that c0i,j ci,j = 1, so from (1.1.4), xi xj − c0i,j xi xj ∈ R + Rx1 + · · · + Rxn , hence cj,i := c0i,j 6= 0 and condition (iv) in Definition 1.1.1 holds. (iv) The elements of Mon(A) will also be denoted by capital letters, thus, xα ∈ Mon(A) will be represented also as X if it is not important to highlight the exponents α1 , . . . , αn in xα . (v) Each element f ∈ A − {0} has a unique representation in the form f = c1 X1 + · · · + ct Xt , with ci ∈ R − {0} and Xi ∈ Mon(A), 1 ≤ i ≤ t. The following proposition justifies the notation that we have introduced for the skew P BW extensions. Proposition 1.1.3. Let A be a skew P BW extension of R. Then, for 1 ≤ i ≤ n, there exist an injective ring endomorphism σi : R → R and a σi -derivation δi : R → R such that xi r = σi (r)xi + δi (r), for every r ∈ R. Proof. For 1 ≤ i ≤ n and every r ∈ R we have elements ci,r , ri ∈ R such that xi r = ci,r xi + ri ; since Mon(A) is an R-basis of A then ci,r and ri are unique for r, so we define σi , δi : R → R by σi (r) := ci,r , δi (r) := ri . From xi (r + s) = xi r + xi s and xi (rs) = (xi r)s, with r, s ∈ R, we get σi (r + s)xi + δi (r + s) = σi (r)xi + δi (r) + σi (s)xi + δi (s), xi (rs) = σi (rs)xi + δi (rs), (xi r)s = σi (r)σi (s)xi + σi (r)δi (s) + δi (r)s, so σi is a ring endomorphism and δi is a σi -derivation of R, i.e., δi (r + s) = δi (r) + δi (s) and δi (rs) = σi (r)δi (s) + δi (r)s, for any r, s ∈ R. Moreover, by Definition 1.1.1 (iii), ci,r 6= 0 for r 6= 0. This means that σi is injective. t u The case when all derivations δi are zero defines a particular class of skew P BW extensions. Another interesting case is when all σi are bijective and the constants ci,j are invertible. We have the following definition. Definition 1.1.4. Let A be a skew P BW extension. (a) A is quasi-commutative if conditions (iii) and (iv) in Definition 1.1.1 are replaced by (iii0 ) For every 1 ≤ i ≤ n and r ∈ R − {0} there exists a ci,r ∈ R − {0} such that xi r = ci,r xi . (1.1.5)

8

1 Skew P BW Extensions

(iv 0 ) For every 1 ≤ i, j ≤ n there exists ci,j ∈ R − {0} such that xj xi = ci,j xi xj .

(1.1.6)

(b) A is bijective if σi is bijective for every 1 ≤ i ≤ n and ci,j is invertible for any 1 ≤ i, j ≤ n. Some examples of skew P BW extensions are the following. Example 1.1.5. (i) Any P BW extension is a bijective skew P BW extension since in this case σi = iR for every 1 ≤ i ≤ n, and ci,j = 1 for every 1 ≤ i, j ≤ n. (ii) Any skew polynomial ring R[x; σ, δ] of injective type, i.e., with σ injective, is a skew P BW extension; in this case we have R[x; σ, δ] = σ(R)hxi. If additionally δ = 0, then R[x; σ] is quasi-commutative. (iii) Let R[x1 ; σ1 , δ1 ] · · · [xn ; σn , δn ] be an iterated skew polynomial ring of injective type, i.e., if the following conditions hold: For For For For

1 ≤ i ≤ n, σi is injective. every r ∈ R and 1 ≤ i ≤ n, σi (r), δi (r) ∈ R. i < j, σj (xi ) = cxi + d, with c, d ∈ R and c is left invertible. i < j, δj (xi ) ∈ R + Rx1 + · · · + Rxi .

Then, R[x1 ; σ1 , δ1 ] · · · [xn ; σn , δn ] is a skew P BW extension (as was observed before, from the definition of iterated skew polynomial rings it is clear that the R-free basis is Mon(A)). Under these conditions we have R[x1 ; σ1 , δ1 ] · · · [xn ; σn , δn ] = σ(R)hx1 , . . . , xn i. In particular, any Ore extension R[x1 ; σ1 , δ1 ] · · · [xn ; σn , δn ] of injective type, i.e., for 1 ≤ i ≤ n, σi is injective, is a skew P BW extension. In fact, in Ore extensions for every r ∈ R and 1 ≤ i ≤ n, σi (r), δi (r) ∈ R, and for i < j, σj (xi ) = xi and δj (xi ) = 0. An important subclass of Ore extensions of injective type are the Ore algebras of injective type, i.e., when R = K[t1 , . . . , tm ], m ≥ 0, and σi , δi are K-linear. Thus, we have K[t1 , . . . , tm ][x1 ; σ1 , δ1 ] · · · [xn ; σn , δn ] = σ(K[t1 , . . . , tm ])hx1 , . . . , xn i. Some concrete examples of Ore algebras of injective type are the following. The algebra of shift operators: Let K be a field and h ∈ K. The algebra of shift operators is defined by Sh := K[t][xh ; σh , δh ], where σh (p(t)) := p(t−h), and δh := 0 (observe that Sh also be considered as a skew polynomial ring of injective type). Thus, Sh is a quasi-commutative bijective skew P BW extension (note that σh is surjective, and since K[t] is noetherian, then σh is injective). The mixed algebra Dh : Again let K be a field and h ∈ K. The mixed d ][xh ; σh , δh ], where σh (x) := x. algebra Dh is defined by Dh := K[t][x; iK[t] , dt Then, Dh is a bijective skew P BW extension. The algebra for multidimensional discrete linear systems is defined by D := K[t1 , . . . , tn ][x1 ; σ1 , 0] · · · [xn ; σn , 0], where K is a field and

1.1 Definition and Some Examples

9

σi (p(t1 , . . . , tn )) := p(t1 , . . . , ti−1 , ti + 1, ti+1 , . . . , tn ), σi (xi ) = xi , 1 ≤ i ≤ n. Thus, D is a quasi-commutative bijective skew P BW extension. Observe that all of these examples are not P BW extensions. (iv) The additive analogue of the Weyl algebra: Let K be a field. The Kalgebra An (q1 , . . . , qn ) is generated by x1 , . . . , xn , y1 , . . . , yn and subject to the relations: xj xi = xi xj , yj yi = yi yj , 1 ≤ i, j ≤ n, yi xj = xj yi , i 6= j, yi xi = qi xi yi + 1, 1 ≤ i ≤ n, where qi ∈ K − {0}. We observe that An (q1 , . . . , qn ) is isomorphic to the iterated skew polynomial ring A := K[x1 , . . . , xn ][y1 ; σ1 , δ1 ] · · · [yn ; σn , δn ] over the commutative polynomial ring K[x1 , . . . , xn ], where σj (yi ) := yi , δj (yi ) := 0, 1 ≤ i < j ≤ n, σi (xj ) := xj , δi (xj ) := 0, i = 6 j, σi (xi ) := qi xi , δi (xi ) := 1, 1 ≤ i ≤ n. In fact, σi , δi can be defined using recurrently the universal property of skew polynomial rings with σi (k) = k and δi (k) = 0 for all k ∈ K (see [278], or also [242]); the function {x1 , . . . , xn , y1 , . . . , yn } → A,

xi 7→ xi , yi 7→ yi , 1 ≤ i ≤ n,

induces a K-algebra homomorphism α : K{x1 , . . . , xn , y1 , . . . , yn } → A, with xi 7→ xi , yi 7→ yi , 1 ≤ i ≤ n; since An (q1 , . . . , qn ) = K{x1 , . . . , xn , y1 , . . . , yn }/I, where I is the ideal of the above relations, and α(I) = 0, then we get an algebra homomorphism, also denoted by α, α : An (q1 , . . . , qn ) → A, with xi 7→ xi , yi 7→ yi , 1 ≤ i ≤ n. On the other hand, using again recurrently the universal property of skew polynomial rings, we have the algebra homomorphism β : A → An (q1 , . . . , qn ),

xi 7→ xi ,

yj 7→ yj ,

1 ≤ i ≤ n,

1 ≤ j ≤ n − 1,

moreover, the element yn ∈ An (q1 , . . . , qn ) satisfies yn β(p) = β(σn (p))yn + β(δn (p)), for every p ∈ A, then β induces an homomorphism, also denoted by β, from A to An (q1 , . . . , qn ) such that β(yn ) = yn . From this we get the claimed isomorphism An (q1 , . . . , qn ) ∼ = A. An (q1 , . . . , qn ) satisfies the conditions in (iii) and it is a bijective skew P BW extension: An (q1 , . . . , qn ) = σ(K[x1 , . . . , xn ])hy1 , . . . , yn i.

10

1 Skew P BW Extensions

Note that An (q1 , . . . , qn ) can also be interpreted as a skew P BW extension of K, An (q1 , . . . , qn ) = σ(K)hx1 , . . . , xn , y1 , . . . , yn i. (v) The multiplicative analogue of the Weyl algebra: Let K be a field. The K-algebra On (λji ) is generated by x1 , . . . , xn and subject to the relations: xj xi = λji xi xj , 1 ≤ i < j ≤ n, where λji ∈ K − {0}. As in (iv), it can be proved that On (λji ) is isomorphic to the iterated skew polynomial ring K[x1 ][x2 ; σ2 ] · · · [xn ; σn ], with σj (xi ) := λji xi , 1 ≤ i < j ≤ n. Thus, On (λji ) satisfies the conditions of (iii), and hence, On (λji ) is an iterated skew polynomial ring of injective type (but it is not Ore of injective type). Therefore, On (λji ) = σ(K[x1 ])hx2 , . . . , xn i. Finally, observe that On (λji ) can also be viewed as a skew P BW extension of K, On (λji ) = σ(K)hx1 , . . . , xn i. Note that On (λji ) is quasi-commutative and bijective. (vi) q-Heisenberg algebra: Let K be a field. The K-algebra Hn (q) is generated by x1 , . . . , xn , y1 , . . . , yn , z1 , . . . , zn and subject to the relations: xj xi = xi xj , zj zi = zi zj , yj yi = yi yj , 1 ≤ i, j ≤ n, zj yi = yi zj , zj xi = xi zj , yj xi = xi yj , i 6= j, zi yi = qyi zi , zi xi = q −1 xi zi + yi , yi xi = qxi yi , 1 ≤ i ≤ n, with q ∈ K − {0}. Note that Hn (q) is isomorphic to the iterated skew polynomial ring K[x1 , . . . , xn ][y1 ; σ1 ] · · · [yn ; σn ][z1 ; θ1 , δ1 ] · · · [zn ; θn , δn ] on the commutative polynomial ring K[x1 , . . . , xn ], where θj (zi ) := zi , δj (zi ) := 0, σj (yi ) := yi , 1 ≤ i < j ≤ n, θj (yi ) := yi , δj (yi ) := 0, θj (xi ) := xi , δj (xi ) := 0, σj (xi ) := xi , i 6= j, θi (yi ) := qyi , δi (yi ) := 0, θi (xi ) := q −1 xi , δi (xi ) := yi , σi (xi ) := qxi , 1 ≤ i ≤ n. Since δi (xi ) = yi ∈ / K[x1 , . . . , xn ], not all of the conditions in (iii) are satisfied, however, with respect to K, Hn (q) satisfies the conditions in (iii), and hence, Hn (q) is a bijective skew P BW extension of K: Hn (q) = σ(K)hx1 , . . . , xn ; y1 , . . . , yn ; z1 , . . . , zn i. The above examples of skew P BW extensions can be summarized in the following way:

1.1 Definition and Some Examples

11

1. P BW extensions: Rhx1 , . . . , xn i (all are bijective) a. Ore extensions of derivation type: R[x1 ; iR , δ1 ] · · · [xn ; iR , δn ] i. Polynomial rings: R[x1 , . . . , xn ] ii. Ore algebras of derivation type: K[t1 , . . . , tm ][x1 ; δ1 ] · · · [xn ; δn ] Weyl algebras An (K) = K[t1 , . . . , tn ][x1 ; ∂1 /∂t1 ] · · · [xn ; ∂n /∂tn ] Bn (K) = K(t1 , . . . , tn )[x1 ; ∂1 /∂t1 ] · · · [xn ; ∂n /∂tn ] iii. Weyl rings: An (R) := R[t1 , . . . , tn ][x1 ; ∂/∂t1 ] · · · [xn ; ∂/∂tn ] b. Crossed product: R ∗ U (G) Tensor product: R ⊗K U (G) Enveloping algebra of a Lie algebra: U (G) 2. Iterated skew polynomial rings of injective type a. Ore extensions of injective type Ore algebras of injective type i. The algebra of shift operators: Sh (quasi-commutative and bijective) ii. The mixed algebra: Dh (bijective) iii. The algebra for multidimensional discrete linear systems: D (quasi-commutative and bijective) iv. Additive analogue of the Weyl algebra: An (q1 , . . . , qn ) (bijective) b. Multiplicative analogue of the Weyl algebra: On (λji ) (quasi-commutative and bijective) c. q-Heisenberg algebra: Hn (q) (bijective). Remark 1.1.6. (i) Many other prominent examples of bijective skew P BW extensions and some other classes of noncommutative rings of polynomial type, closely related to such extensions, will be presented in Chapter 2 and Section 4.4. Therein we will exhibit examples that are neither P BW extensions nor iterated skew polynomial rings of injective type. (ii) We want to remark that the skew P BW extensions are not a subclass of the collection of iterated skew polynomial rings, for example the algebra U 0 (so(3, K)) of Chapter 2 (see also the 3-dimensional skew polynomial algebras and diffusion algebras in Chapter 2). On other hand, the skew polynomial rings are not included in the class of skew P BW extensions, take R[x; σ, δ], with σ not injective, for example, let K be a field, R := K[t], σ(p(t)) := p(0) and δ = 0. (iii) As An (q1 , . . . , qn ) and On (λji ) show, a given ring A can be interpreted as a skew P BW extension in different ways:

12

1 Skew P BW Extensions

An (q1 , . . . , qn ) = σ(K[x1 , . . . , xn ])hy1 , . . . , yn i = σ(K)hx1 , . . . , xn , y1 , . . . , yn i. On (λji ) = σ(K[x1 ])hx2 , . . . , xn i = σ(K)hx1 , . . . , xn i. (iv) Note that any ring A has the trivial interpretation as a skew P BW extension by taking R = A and x := 1, but of course, this interpretation is not useful for studying A. So, from now on in this monograph, we will assume that R is a proper subring of A and, from condition (ii) in Definition 1.1.1, x1 , . . . , xn ∈ A − R. (v) It is possible that a given skew P BW extension can be classified in different subclasses, for example, the Weyl algebra An (K) is both a P BW extension and an iterated skew polynomial ring of injective type. (vi) As was observed above, for rings that can be interpreted as iterated skew polynomial rings of injective type, the condition (ii) in Definition 1.1.1 holds automatically. However, for other classes of skew P BW extensions the verification of this condition is not trivial; in Section 1.2 we will study a method for checking this condition. Definition 1.1.7. Let A be a skew P BW extension of R with endomorphisms σi as in Proposition 1.1.3, 1 ≤ i ≤ n. (i) For α = (α1 , . . . , αn ) ∈ Nn , σ α := σ1α1 · · · σnαn , |α| := α1 + · · · + αn . If β = (β1 , . . . , βn ) ∈ Nn , then α + β := (α1 + β1 , . . . , αn + βn ). (ii) For X = xα ∈ Mon(A), exp(X) := α and deg(X) := |α|. (iii) Let 0 6= f ∈ A. If t(f ) is the finite set of terms that conform f , i.e., if f = c1 X1 + · · · + ct Xt , with Xi ∈ Mon(A) and ci ∈ R − {0}, then t(f ) := {c1 X1 , . . . , ct Xt }. (iv) Let f be as in (iii), then deg(f ) := max{deg(Xi )}ti=1 . The skew P BW extensions can be characterized in a similar way as was done in [73] and [74] for P BW rings. Theorem 1.1.8. Let A be a ring of a left polynomial type over R w.r.t. {x1 , . . . , xn }. A is a skew P BW extension of R if and only if the following conditions hold: (a) For every xα ∈ Mon(A) and every 0 6= r ∈ R there exist unique elements rα := σ α (r) ∈ R − {0} and pα,r ∈ A such that xα r = rα xα + pα,r ,

(1.1.7)

where pα,r = 0 or deg(pα,r ) < |α| if pα,r 6= 0. Moreover, if r is left invertible, then rα is left invertible. (b) For every xα , xβ ∈ Mon(A) there exist unique elements cα,β ∈ R and pα,β ∈ A such that xα xβ = cα,β xα+β + pα,β , (1.1.8) where cα,β is left invertible, pα,β = 0 or deg(pα,β ) < |α + β| if pα,β 6= 0. Proof. ⇒) We divide the proof of (a) into three steps. Step 1. For each 1 ≤ i ≤ n, 0 6= r ∈ R and for every k ∈ N,

1.1 Definition and Some Examples

13

xki r = rk xki + pk,r , where rk := σik (r) ∈ R − {0}, pk,r ∈ A and pk,r = 0 or deg(pk,r ) < k. Moreover, if r is left invertible, then rk is left invertible. In fact, we will prove this by induction on k: for k = 0 we have r0 = σi0 (r) = r and p0,r = 0; for k = 1 we have xi r = σi (r)xi + δi (r), so r1 := σi (r) 6= 0 and p1,r = δi (r), with δi (r) = 0 or deg(δi (r)) = 0 < 1 (if r is left invertible, then σi (r) is left invertible). By induction we have xk+1 r = xi xki r = xi (rk xki +pk,r ), i k where rk = σi (r) ∈ R − {0}, pk,r ∈ A, pk,r = 0 or deg(pk,r ) < k (if r is left invertible, then rk is left invertible). So, xk+1 r = (xi rk )xki + xi pk,r = i k+1 k (σi (rk )xi + δi (rk ))xi + xi pk,r = σi (rk )xi + δi (rk )xki + xi pk,r . Note that k+1 k rk+1 := σi (rk ) = σi (σi (r)) = σi (r) 6= 0 since rk 6= 0 and σi is injective; moreover, pk+1,r := δi (rk )xki +xi pk,r = 0 or deg(pk+1,r ) < k+1 since pk,r = 0 or deg(xi pk,r ) ≤ k < k+1 (if rk is left invertible, then σi (rk ) is left invertible). Step 2. We complete the proof by induction on the number of variables involved in xα . For one variable only, the proof is the content of step 1. Then, αn−1 αn−1 α1 α1 αn αn αn 1 xα r = xα 1 · · · xn r = x1 · · · xn−1 (xn r) = x1 · · · xn−1 (rαn xn + pαn ,r ), αn with rαn := σn (r) 6= 0 and pαn ,r ∈ A, pαn ,r = 0 or deg(pαn ,r ) < αn (if r is left invertible, then rαn is left invertible). So by induction α

α

α1 n−1 n−1 αn 1 xα r = (xα 1 · · · xn−1 rαn )xn + x1 · · · xn−1 pαn ,r αn−1 αn−1 α1 αn 1 = (rα xα 1 · · · xn−1 + qα,rαn )xn + x1 · · · xn−1 pαn ,r αn−1 αn 1 = rα x α 1 · · · xn−1 xn + pα,r = rα xα + pα,r , α

α

n−1 n−1 αn where rα := σ1α1 · · · σn−1 (rαn ) = (σ1α1 · · · σn−1 σn )(r) = σ α (r) 6= 0 n (by induction and since rαn 6= 0), qα,rαn ∈ A and pα,r := qα,rαn xα n + αn−1 1 xα · · · x p ∈ A (if r is left invertible, then r is left invertible); αn α 1 n−1 αn ,r note that pα,r = 0 or deg(pα,r ) < |α| = α1 + · · · + αn since pαn ,r = 0 or deg(pαn ,r ) < αn , and, qα,rαn = 0 or deg(qα,rαn ) < α1 + · · · + αn−1 . Step 3. Since Mon(A) is an R-basis for A, rα and pα,r are unique. Now we will consider the proof of (b). We also divide the proof into three steps. Step 3.1. We will prove first that for i < j and k, m ≥ 0

m k xkj xm i = ck,m xi xj + pk,m ,

with ck,m ∈ R left invertible, pk,m ∈ A, pk,m = 0 or deg(pk,m ) < k + m. For this we will use double induction, on k and on m. For k = 0 we have c0,m := 1, p0,m := 0. k = 1: for i < j and m ≥ 0 we will prove by induction on m that m xj xm i = c1,m xi xj + p1,m ,

with c1,m ∈ R left invertible, p1,m ∈ A, p1,m = 0 or deg(p1,m ) < 1 + m. For m = 0, c1,0 = 1, p1,0 = 0. Let m = 1, then xj xi = ci,j xi xj + p1,1 , with ci,j ∈ R left invertible (see Definition 1.1.2 (iv)), p1,1 ∈ A, p1,1 = 0 or

14

1 Skew P BW Extensions

deg(p1,1 ) ≤ 1 < 1 + 1. Now we use the induction hypothesis, so xj xm+1 = i m xj xm x = (c x x + p )x , with c ∈ R left invertible, p ∈ A, 1,m i j 1,m i 1,m 1,m i i p1,m = 0 or deg(p1,m ) < 1 + m. Then, xj xm+1 = c1,m xm i xj xi + p1,m xi = i m m c1,m xm i (ci,j xi xj + p1,1 ) + p1,m xi = c1,m xi ci,j xi xj + c1,m xi p1,1 + p1,m xi = m c1,m (rm xm +p )x x +c x p +p x , where r ∈ R is left invertible m,ci,j i j 1,m i 1,1 1,m i m i since ci,j is left invertible (part (a)); moreover pm,ci,j ∈ A, pm,ci,j = 0 or deg(pm,ci,j ) < m. Hence, xj xm+1 = c1,m+1 xm+1 xj + p1,m+1 , with c1,m+1 := i i c1,m rm ∈ R left invertible, p1,m+1 := c1,m pm,ci,j xi xj + c1,m xm i p1,1 + p1,m xi ∈ A, p1,m+1 = 0 or deg(p1,m+1 ) ≤ m + 1 < m + 2. This completes the proof for k = 1. k m m k k + 1: xk+1 xm i = xj xj xi = xj (ck,m xi xj + pk,m ), with ck,m ∈ R left j invertible, pk,m ∈ A, pk,m = 0 or deg(pk,m ) < k + m. Thus, xk+1 xm i = j m k m k (xj ck,m )xi xj + xj pk,m = (r1 xj + p1,ck,m )xi xj + xj pk,m , with r1 ∈ R left m k invertible, p1,ck,m = 0 or deg(p1,ck,m ) < 1; then, xk+1 xm i = r1 x j x i x j + j m k m k m k p1,ck,m xi xj + xj pk,m = r1 (c1,m xi xj + p1,m )xj + p1,ck,m xi xj + xi pk,m , by induction c1,m ∈ R is left invertible, p1,m ∈ A, p1,m = 0 or deg(p1,m ) < 1+m, m k+1 hence xk+1 xm + pk+1,m , with ck+1,m := r1 c1,m ∈ R left i = ck+1,m xi xj j k invertible, pk+1,m := r1 p1,m xkj + p1,ck,m xm i xj + xj pk,m ∈ A, pk+1,m = 0 or deg(pk+1,m ) ≤ k + m < k + 1 + m. This completes Step 3.1. Step 3.2. The proof is by induction on the number of variables involved in xα or xβ . If xα and xβ include only one variable, then we apply Step 3.1. So by induction we assume that (1.1.8) is true when the number of variables of xα or xβ is ≤ n − 1. Then, β2 α1 αn β1 βn αn β1 βn 1 xα xβ = xα 1 · · · xn x1 · · · xn = (x1 · · · xn x1 )x2 · · · xn β2 βn 1 +β1 α2 n = (c1 xα x2 · · · xα n + p1 )(x2 · · · xn ), 1 c1 ∈ R left invertible, p1 ∈ A, p1 = 0 or deg(p1 ) < α1 + · · · + αn + β1 ≤ |α + β| β2 β2 βn βn 1 +β1 α2 n = c1 (xα x2 · · · xα n )(x2 · · · xn ) + p1 x2 · · · xn 1 1 +β1 α2 +β2 n +βn = c1 (c2 xα x2 · · · xα + p2 ) + p1 xβ2 2 · · · xβnn , n 1 c2 ∈ R left invertible, p2 ∈ A, p2 = 0 or deg(p2 ) < α1 + β1 + α2 + · · · + αn + β2 + · · · + βn = |α + β|

= cα,β xα+β + pα,β , with cα,β := c1 c2 ∈ R, left invertible, pα,β := c1 p2 + p1 xβ2 2 · · · xβnn ∈ A, pα,β = 0 or deg(pα,β ) < |α + β|. Step 3.3. Since Mon(A) is an R-basis for A, cα,β and pα,β are unique. ⇐) The condition (iii) of Definition 1.1.1 is a particular case of (1.1.7), and the condition (iv) is a particular case of (1.1.8), thus (a) and (b) implies that A is a skew P BW extension. t u

1.1 Definition and Some Examples

15

Remark 1.1.9. (i) A left inverse of cα,β will be denoted by c0α,β . We observe that if in (b) of the previous theorem α = 0 or β = 0, then cα,β = 1 and hence c0α,β = 1. (ii) Let θ, γ, β ∈ Nn and c ∈ R, then we have the following identities: σ θ (cγ,β )cθ,γ+β = cθ,γ cθ+γ,β , σ θ (σ γ (c))cθ,γ = cθ,γ σ θ+γ (c). In fact, since xθ (xγ xβ ) = (xθ xγ )xβ , xθ (cγ,β xγ+β + pγ,β ) = (cθ,γ xθ+γ + pθ,γ )xβ , σ θ (cγ,β )cθ,γ+β xθ+γ+β + p = cθ,γ cθ+γ,β xθ+γ+β + q, with p = 0 or deg(p) < |θ + γ + β|, and, q = 0 or deg(q) < |θ + γ + β|. From this we get the first identity. For the second, xθ (xγ c) = (xθ xγ )c, and hence xθ (σ γ (c)xγ + pγ,c ) = (cθ,γ xθ+γ + pθ,γ )c, σ θ (σ γ (c))cθ,γ xθ+γ + p = cθ,γ σ θ+γ (c)xθ+γ + q, with p = 0 or deg(p) < |θ + γ|, and, q = 0 or deg(q) < |θ + γ|. This proves the second identity. (iii) We observe if A is quasi-commutative, then from the proof of Theorem 1.1.8 we conclude that pα,r = 0 and pα,β = 0 for every 0 6= r ∈ R and every α, β ∈ Nn . Moreover, the evaluation function at 0, i.e., A → R, f ∈ A 7→ f (0) ∈ R, is a surjective ring homomorphism with kernel hx1 , . . . , xn i the two-sided ideal generated by x1 , . . . , xn . Thus, A/hx1 , . . . , xn i ∼ = R as rings. (iv) From the proof of Theorem 1.1.8 we get also that if A is bijective, then cα,β is invertible for any α, β ∈ Nn . (v) In Mon(A) we define xα  xβ ⇐⇒  α β   x = x    or xα 6= xβ but |α| > |β|   or    xα 6= xβ , |α| = |β| but ∃ i with

α1 = β1 , . . . , αi−1 = βi−1 , αi > βi .

It is clear that this is a total order on Mon(A), called deglex order. If xα  xβ but xα 6= xβ , we write xα  xβ . Each element f ∈ A−{0} can be represented in a unique way as f = c1 xα1 + · · · + ct xαt , with ci ∈ R − {0}, 1 ≤ i ≤ t, and xα1  · · ·  xαt . We say that xα1 is the leading monomial of f and we write lm(f ) := xα1 ; c1 is the leading coefficient of f , lc(f ) := c1 , and c1 xα1 is the leading term of f denoted by lt(f ) := c1 xα1 . If f = 0, we define lm(0) := 0, lc(0) := 0, lt(0) := 0, and we set X  0 for any X ∈ Mon(A) (see also Section 13.1 in Part III). We observe that

16

1 Skew P BW Extensions

xα  xβ ⇒ lm(xγ xα xλ )  lm(xγ xβ xλ ), for every xγ , xλ ∈ Mon(A). Example 1.1.10. We will illustrate Theorem 1.1.8 in some particular examples of skew P BW extensions. (i) We consider first the skew polynomial ring of injective type Z12 [t][x; σ, δ], with σ(p(t)) := p(t − 1) and δ(t) := 1 (it is easy to prove that if σ is an endomorphism of a ring R, then for every a ∈ R, δa (r) := ar − σ(r)a is a σ-derivation of R; whence, taking a = 1 in Z12 [t] we get that δ(t) = 1; finally, note that σ is injective, actually bijective). We have x2 (6t2 + 7) = xx(6t2 + 7) = x[σ(6t2 + 7)x + δ(6t2 + 7)] = x[(6(t − 1)2 + 7)x + 12t − 6] = x(6(t − 1)2 + 7)x − 6x = [σ(6(t − 1)2 + 7)x + δ(6(t − 1)2 + 7)]x − 6x = (6(t − 2)2 + 7)x2 + 6x − 6x = (6t2 − 24t + 24 + 7)x2 = (6t2 + 7)x2 . We observe that 6t2 + 7 is invertible in Z12 [t]. For t + 4 we have x2 (t + 4) = (t + 2)x2 + 2x. Note that t + 4 is not invertible in Z12 [t]. (ii) Now we consider A2 (q1 , q2 ), with K := Q and q1 = 2 = q2 ; then R := Q[x1 , x2 ] and in A2 (2, 2) = σ(R)hy1 , y2 i we have y1 y22 (3x21 x2 ) = 3y1 y22 (x21 x2 ) = 3y1 (y22 x21 )x2 = 3(y1 x21 )(y22 x2 ) = 3(y1 x1 x1 )(y2 y2 x2 ) = 3[(2x1 y1 + 1)x1 ][y2 (2x2 y2 + 1)] = 3[2x1 y1 x1 + x1 ][2y2 x2 y2 + y2 ] = 3[2x1 (2x1 y1 + 1) + x1 ][2(2x2 y2 + 1)y2 + y2 ] = 3(4x21 y1 + 3x1 )(4x2 y22 + 3y2 ) = 48x21 x2 y1 y22 + 36x21 y1 y2 + 36x1 x2 y22 + 27x1 y2 ; with the notation of Theorem 1.1.8 we have r = 3x21 x2 , y α = y1 y22 , rα = 48x21 x2 , pα,r = 36x21 y1 y2 + 36x1 x2 y22 + 27x1 y2 . On the other hand, since in A2 (2, 2), y1 y2 = y2 y1 , then for every α, β ∈ N2 , y α y β = y α+β ,

cα,β = 1 and pα,β = 0.

(iii) For On (λji ) we consider the following illustration of Theorem 1.1.8: n := 3, K := Q, R := Q[x1 ], x3 x1 = 2x1 x3 , x3 x2 = 2x2 x3 , x2 x1 = −x1 x2 , this algebra will be denoted by O3 (2, 2, −1); then for r := 5x21 and xα := x2 x23 we get

1.2 Universal Property and Characterization

17

x2 x23 (5x21 ) = 5x2 x3 (x3 x1 )x1 = 5x2 x3 (2x1 x3 )x1 = 10x2 (x3 x1 )(x3 x1 ) = 10x2 (2x1 x3 )(2x1 x3 ) = 40x2 x1 x3 x1 x3 = 80x2 x21 x23 = 80x21 x2 x23 , thus rα = 80x21 and pα,r = 0. Let now xβ := x22 x3 , then xα xβ = (x2 x23 )(x22 x3 ) = 16x32 x33 , thus cα,β = 16 and pα,β = 0. (iv) We end this example by illustrating Theorem 1.1.8 for Hn (q): we take n := 2, q := 2, K := Q, then h2 (2) = σ(Q)hx1 , x2 , y1 , y2 , z1 , z2 i; in this case rα = r and pα,r = 0 for any xα ; let xα := x1 x2 y1 y2 z1 z2 and xβ := x22 z22 , then using the relations defining H2 (2) we get xα xβ = (x1 x2 y1 y2 z1 z2 )(x22 z22 ) = x1 x32 y1 y2 z1 z23 + 5x1 x22 y1 y22 z1 z22 , thus cα,β = 1, pα,β = 5x1 x22 y1 y22 z1 z22 , deg(pα,β ) = 9.

deg(xα+β ) = deg(x1 x32 y1 y2 z1 z23 ) = 10,

A natural and useful result that we will use later is the following property. Proposition 1.1.11. Let A be a bijective skew P BW extension of a ring R. Then, AR is free with basis Mon(A). Proof. First note that AR is a module where the product f · r is defined by the multiplication in A: f ·r := f r, f ∈ A, r ∈ R. We prove next that Mon(A) is a system of generators of A. Let f ∈ A, then f is a finite sum of terms like rxα , with r ∈ R and xα ∈ Mon(A), so it is enough to prove that each of these terms is a right linear R-combination of elements of Mon(A). From Theorem 1.1.8, rxα = xα σ −α (r) − pα,σ−α (r) , with deg(pα,σ−α (r) ) < |α| if pα,σ−α (r) 6= 0, so by induction on |α| we get the result. Now we will show that Mon(A) is linearly independent: let xα1 r1 + · · · xαt rt = 0, with xα1  · · ·  xαt for the total order  on Mon(A) defined in the previous remark, then σ α1 (r1 )xα1 +pα1 ,r1 +· · ·+σ αt (rt )xαt +pαt ,rt = 0, with deg(pαi ,ri ) < |αi | if pαi ,ri 6= 0, 1 ≤ i ≤ t; hence, σ α1 (r1 ) = 0 and from this r1 = 0. By induction on t we obtain the result. t u

1.2 Universal Property and Characterization Skew P BW extensions can be characterized by functions, a universal property and a finite set of constants, in a similar way as is done for skew polynomial rings. This problem, formulated in the seminar Constructive Algebra

18

1 Skew P BW Extensions

SAC 2 (sites.google.com/a/ unal.edu.co/sac2), was solved in [4] and [2], where skew P BW extensions were generalized to infinite sets of generators. In the present section we will adapt the results in [4] to the finite case. If A = σ(R)hx1 , . . . , xn i is a skew P BW extension of the ring R, then, as was observed in Proposition 1.1.3, A induces injective endomorphisms σk : R → R and σk -derivations δk : R → R, 1 ≤ k ≤ n; moreover, by Definition 1.1.1 and Remark 1.1.2, there exists a unique finite set of constants (k) ci,j , dij , aij ∈ R such that (1)

(n)

xj xi = ci,j xi xj + aij x1 + · · · + aij xn + dij , for every 1 ≤ i < j ≤ n. (1.2.1) Definition 1.2.1. Let A = σ(R)hx1 , . . . , xn i be a skew P BW extension. R, (k) n, σk , δk , ci,j , dij , aij , with 1 ≤ i < j ≤ n, 1 ≤ k ≤ n, defined as before, are called the parameters of A. Theorem 1.2.2 (Universal property). Let A = σ(R)hx1 , . . . , xn i be a (k) skew P BW extension with parameters R, n, σk , δk , ci,j , dij , aij , 1 ≤ i < j ≤ n, 1 ≤ k ≤ n. Let B be a ring with a homomorphism ϕ : R → B and elements y1 , . . . , yn ∈ B such that (i) yk ϕ(r) = ϕ(σk (r))yk + ϕ(δk (r)), for every r ∈ R, 1 ≤ k ≤ n. (1) (n) (ii) yj yi = ϕ(ci,j )yi yj + ϕ(aij )y1 + · · · + ϕ(aij )yn + ϕ(dij ), 1 ≤ i < j ≤ n. Then, there exists a unique ring homomorphism ϕ e : A → B such that ϕι e =ϕ and ϕ(x e i ) = yi , where ι is the inclusion of R in A, 1 ≤ i ≤ n. Proof. Since A is a free left R-module with basis Mon(A), we define the R-homomorphism ϕ e : A → B,

r1 xα1 + · · · + rt xαt 7→ ϕ(r1 )y α1 + · · · + ϕ(rt )y αt ,

where y θ := y1θ1 · · · ynθn , with θ := (θ1 , . . . , θn ) ∈ Nn . Note that ϕ(1) e = 1. ϕ e is multiplicative: in fact, applying induction on the degree |α + β| we have α ϕ(ax e α bxβ ) = ϕ(a[σ e (b)xα xβ + pα,b xβ ])

= ϕ[aσ e α (b)[cα,β xα+β + pα,β ] + apα,b xβ ] = ϕ(a)ϕ(σ α (b))ϕ(cα,β )y α+β + ϕ(a)ϕ(σ α (b))ϕ(pα,β )(y) + ϕ(a)ϕ(pα,b )(y)y β , where ϕ(pα,β )(y) is the element in B obtained by replacing each monomial xθ in pα,β by y θ and every coefficient c by ϕ(c). In a similar way we have for ϕ(pα,b )(y) (observe that the degree of each monomial of pα,b xβ is < |α + β|). On the other hand, applying (i) and (ii) we get ϕ(ax e α )ϕ(bx e β ) = ϕ(a)y α ϕ(b)y β = ϕ(a)[ϕ(σ α (b))y α + ϕ(pα,b )(y)]y β = ϕ(a)ϕ(σ α (b))y α y β + ϕ(a)ϕ(pα,b )(y)y β

1.3 Existence

19

= ϕ(a)ϕ(σ α (b))[ϕ(cα,β )y α+β + ϕ(pα,β )(y)] + ϕ(a)ϕ(pα,b )(y)y β = ϕ(a)ϕ(σ α (b))ϕ(cα,β )y α+β + ϕ(a)ϕ(σ α (b))ϕ(pα,β )(y) + ϕ(a)ϕ(pα,b )(y)y β . It is clear that ϕι e = ϕ and ϕ(x e i ) = yi . Moreover, note that ϕ e is the only ring homomorphism that satisfies these two conditions. t u Corollary 1.2.3. Let B be a ring with a homomorphism ϕ : R → B and elements y1 , . . . , yn ∈ B such that the conditions (i)–(ii) in Theorem 1.2.2 (k) are satisfied with respect to the parameters R, n, σk , δk , ci,j , dij , aij , 1 ≤ i < j ≤ n, 1 ≤ k ≤ n. If B satisfies the universal property, then B ∼ = A = σ(R)hx1 , . . . , xn i. Moreover, the monomials y1α1 · · · ynαn , αi ≥ 0, 1 ≤ i ≤ n form an R-basis of B as a left R-module. Proof. By the universal property of A there exists a ϕ e such that ϕι e = ϕ; by the universal property of B there exists an e ι such that e ιϕ = ι. Note that e ιϕι e = ι and ϕe eιϕ = ϕ. The uniqueness gives that e ιϕ e = iA and ϕe eι = iB . Moreover, in the proof of Theorem 1.2.2 we observed that ϕ e is not only a ring homomorphism but also an R-homomorphism, whence ϕ(Mon(A)) e = {y1α1 · · · ynαn |αi ≥ 0, 1 ≤ i ≤ n} is an R-basis of B as a left R-module.

t u

Corollary 1.2.4. Let B be a ring that satisfies the following conditions with (k) respect to the system of parameters R, n, σk , δk , ci,j , dij , aij , 1 ≤ i < j ≤ n, 1 ≤ k ≤ n. (i) There exists a ring homomorphism ϕ : R → B. (ii) There exist elements y1 , . . . , yn ∈ B such that B is a left free R-module with basis Mon(y1 , . . . , yn ), and the product is given by r · b := ϕ(r)b, r ∈ R, b ∈ B. (iii) The conditions (i) and (ii) in Theorem 1.2.2 hold. Then B ∼ = A = σ(R)hx1 , . . . , xn i. Proof. According to the universal property of A, there exists a ring homomorphism ϕ e : A → B given by r1 xα1 +· · ·+rt xαt 7→ ϕ(r1 )y α1 +· · ·+ϕ(rt )y αt ; from (ii) we get that ϕ e is bijective. t u

1.3 Existence Given a system of parameters, we will consider now the existence of a skew P BW extension corresponding to this system. Thus, assume given: (P1) A ring R. (P2) For 1 ≤ i ≤ n, functions σi , δi : R → R.

20

1 Skew P BW Extensions (k)

(P3) A system of constants ci,j ∈ R, left invertible, aij , dij ∈ R, with 1 ≤ i < j ≤ n, 1 ≤ k ≤ n. With this data, we will define some functions. 1. Let W be the free monoid in the alphabet X∪R, with X := {x1 , . . . , xn }, X ∩R = ∅, and let Z{X ∪R} be the free algebra over Z in the alphabet X ∪R; then the function p : W → Z{X ∪ R}, is defined by induction on the complexity of each word w ∈ W , in the following way: (a) The complexity of w, denoted c(w), is a triple of nonnegative integers (a, b, c), where a is the number of x’s in w, b is the number of inversions involving only x’s, and c is the number of inversions of type xi r. In other words, the complexity of a word indicates the changes (inversions) needed to put w in the standard form r1 · · · rl xi1 · · · xim , with i1 ≤ · · · ≤ im , the r’s in R and the x’s in X. For example, if n = 4 and w = x3 x1 r1 x2 x3 r2 x2 x1 , then c(w) = (6, 8, 6). (b) The triples are ordered with the lexicographic order, i.e., (a, b, c) ≤ (d, e, f ) if and only if a < d, or, a = d and b < e, or, a = d, b = e and c ≤ f . Note that ≤ is a well order. (c) Let T := {w ∈ W |c(w) = (a, 0, 0)} and ZT be the Z-submodule of Z{X ∪ R} generated by T . (d) The function p is defined as follows: If w ∈ T , then p(w) := w. If w = v1 x i r v2 , where r ∈ R, v1 ∈ W and rv2 ∈ T , then p(w) := p(v1 σi (r)xi v2 ) + p(v1 δi (r)v2 ).

(1.3.1)

If w = v1 x j x i v2 , where v1 ∈ W , xi v2 ∈ T , with i < j, then p(w) := p(v1 ci,j xi xj v2 ) +

n X

(k)

p(v1 aij xk v2 ) + p(v1 dij v2 ).

(1.3.2)

k=1

2. We also denote by p the Z-linear extension of p to Z{X ∪ R}, thus we have the homomorphism of Z-modules p : Z{X ∪ R} → Z{X ∪ R}, and note that Im(p) ⊆ ZT . 3. Consider Mon{x1 , . . . , xn } and let FR (Mon) be the left free R-module with basis Mon{x1 , . . . , xn }. We define q : ZT → FR (Mon) to be the Z-linear extension of the function q(r1 · · · rl xi1 · · · xim ) := r1 · · · rl xi1 · · · xim ,

1.3 Existence

21

with rk ∈ R, 1 ≤ k ≤ l, i1 ≤ · · · ≤ im . 4. Finally, we define h : Z{X ∪ R} → FR (Mon)

, h := qp.

Theorem 1.3.1 ([4]). With the above notation and parameters (P 1)–(P 3), there exists a skew P BW extension A satisfying the following conditions: (i) (ii) (iii) (iv)

R ⊂ A, A is a left R-free module with basis Mon{x1 , . . . , xn }, xi r = σi (r)xi + δi (r), 1 ≤ i ≤ n, r ∈ R, (1) (n) xj xi = ci,j xi xj + aij x1 + · · · + aij xn + dij , 1 ≤ i < j ≤ n,

if and only if (1) for 1 ≤ i ≤ n, σi is an injective ring homomorphism and δi is a σi derivation, (2) h(xj xi r) = h(p(xj xi )r), for i < j and r ∈ R, (3) h(xk xj xi ) = h(p(xk xj )xi ), for i < j < k. Remark 1.3.2. (i) The previous theorem, and its proof in [4], can be used to check if a given ring can be interpreted as a skew P BW extension. The main idea in the proof of part (⇐) consists in showing that the free R-module A := FR (Mon) is endowed with a product that converts it into a ring, and satisfies conditions (i)–(iv) of the theorem, in particular, it is proved that this product is associative. Hence, from the definition, Mon{x1 , . . . , xn } is an R-basis of A. Next below we will illustrate the theorem with one example (see also Chapter 2). (ii) There are other methods for testing if a given finitely generated algebra has a P BW basis, or more exactly, if the standard monomials in the finite set of generators conform a P BW basis. In [231], Chapter 1, one of these methods is given which involves the computation of Gr¨obner bases of twosided ideals of free algebras. Bergman’s Diamond Lemma is another very used method (see [61]); in [328] this method is applied in order to give criteria and some algorithms which decide whether a given algebra with some variables and relations can be described as a skew P BW extension; the procedures in [328] were illustrated with some examples. Example 1.3.3. Let K be a field and consider the K-algebra U 0 (so(3, K)) generated by I1 , I2 , I3 subject to the relations I2 I1 − qI1 I2 = −q 1/2 I3 ,

I3 I1 − q −1 I1 I3 = q −1/2 I2 ,

I3 I2 − qI2 I3 = −q 1/2 I1 ,

where q ∈ K −{0}. We will prove that U 0 (so(3, K)) is a skew P BW extension of K. For this, we apply first Theorem 1.3.1; we take R := K, X := {x1 := I1 , x2 := I2 , x3 := I3 }, σi := iR , δi := 0 for 1 ≤ i ≤ 3, (1)

(2)

(3)

c1,2 := q, a12 := 0, a12 := 0, a12 := −q 1/2 , d12 := 0, (1)

(2)

(3)

c1,3 := q −1 , a13 := 0, a13 := q −1/2 , a13 := 0, d13 := 0,

22

1 Skew P BW Extensions (1)

(2)

(3)

c2,3 := q, a23 := −q 1/2 , a23 := 0, a23 := 0, d23 := 0. Clearly σi , δi satisfy condition (1) of Theorem 1.3.1. Let r ∈ R, then h(x2 x1 r) = q(p(x2 x1 r)) = q(p(x2 σ1 (r)x1 ) + p(x2 δ1 (r))) = q(p(x2 rx1 )) = q(p(rx2 x1 )) (1)

(2)

(3)

= q(p(rc1,2 x1 x2 ) + p(ra12 x1 + ra12 x2 + ra12 x3 ) + p(rd12 )) 1

= q(p(rqx1 x2 ) + p(−rq 2 x3 )) 1

= q(rqx1 x2 − rq 2 x3 ) 1

= q(rqx1 x2 ) − q(rq 2 x3 ) 1

= rqx1 x2 − rq 2 x3 ; (2)

(1)

(3)

h(p(x2 x1 )r) = h((p(c1,2 x1 x2 ) + p(a12 x1 + a12 x2 + a12 x3 ))r) 1

= h((qx1 x2 − q 2 x3 )r) 1

= h(qx1 x2 r) − h(q 2 x3 r) 1

= qp(qx1 x2 r) − qp(q 2 x3 r) 1

= qp(rqx1 x2 ) − qp(rq 2 x3 ) 1

= q(rqx1 x2 ) − q(rq 2 x3 ) 1

= rqx1 x2 − rq 2 x3 .

Similarly we can prove that h(x3 x1 r) = h(p(x3 x1 )r), h(x3 x2 r) = h(p(x3 x2 )r). Thus, condition (2) of Theorem 1.3.1 holds. Now,

h(x3 x2 x1 ) = q(p(x3 x2 x1 )) (3)

= q(p(x3 c1,2 x1 x2 ) + p(x3 a12 x3 )) 1

= q(p(x3 qx1 x2 ) − p(x3 q 2 x3 )) 1

= q(p(qx3 x1 x2 ) − q 2 x23 ) 1

1

= q(p(q(q −1 x1 x3 )x2 ) + p(q(q − 2 x2 )x2 )) − q 2 x23 1

1

= q(p(x1 x3 x2 ) + p(q 2 x22 )) − q 2 x23 1

1

1

= q(p(x1 qx2 x3 ) − p(x1 q 2 x1 )) + q 2 x22 − q 2 x23 1

1

1

= qx1 x2 x3 − q 2 x21 + q 2 x22 − q 2 x23 ;

1.3 Existence

23 1

h(p(x3 x2 )x1 ) = h((p(qx2 x3 ) + p(−q 2 x1 ))x1 ) 1

= h(qx2 x3 x1 ) − q 2 x21 1

= qp(qx2 x3 x1 ) − q 2 x21 1

1

= q(p(qx2 q −1 x1 x3 ) + p(qx2 q − 2 x2 )) − q 2 x21 1

1

= q(p(x2 x1 x3 )) + q 2 x22 − q 2 x21 1

1

1

= q(p(qx1 x2 x3 ) − p(x3 q 2 x3 )) + q 2 x22 − q 2 x21 1

1

1

= qx1 x2 x3 − q 2 x23 + q 2 x22 − q 2 x21 . This proves (3) of Theorem 1.3.1, and hence, there exists a skew P BW extension A of K with the above parameters (recall that A = FK (Mon)). The universal property of A (Theorem 1.2.2) and the inclusion ϕ : K → U 0 (so(3, K)) induces a K-algebra homomorphism ϕ e : A → U 0 (so(3, K)) with xi 7→ Ii , k 7→ k, 1 ≤ i ≤ 3, k ∈ K. On the other hand, the universal property of the free K-algebra K{I1 , I2 , I3 } induces a K-algebra homomorphism φ : K{I1 , I2 , I3 } → A, where Ii 7→ xi , k 7→ k, 1 ≤ i ≤ 3, k ∈ K; let J be the two-sided ideal of relations that defines U 0 (so(3, K)), since φ(J) = 0, then φ induces a K-algebra homomorphism φe : U 0 (so(3, K)) → A, where Ii 7→ xi , k 7→ k, 1 ≤ i ≤ 3, k ∈ K. Therefore, U 0 (so(3, K)) ∼ = A, and ϕ(Mon(X)) e = Mon{I1 , I2 , I3 } is a K-basis of U 0 (so(3, K)).

Chapter 2

Examples

In this chapter we classify some remarkable examples of bijective skew P BW extensions. Some examples were considered in Chapter 1, however, for completeness we will repeat them here. In every example below we will highlight in bold the ring of coefficients. However, it is important to note that there are different ways to choose this ring (and hence, the finite set of generators) as well as the subclass to which each example belongs. Moreover, as was observed before (see Remark 1.1.6), for rings that can be interpreted as iterated skew polynomial rings, the condition (ii) in Definition 1.1.1 holds automatically. However, for other classes of skew P BW extensions the verification of this condition can be realized using Theorem 1.3.1; in some examples below we will apply this technique for checking the condition (see also Example 1.3.3). In particular, we will present a short proof of the classical Poincar´e– Birkhoff–Witt theorem in Theorem 2.1.1. If nothing contrary is said, K will represent a field and R an arbitrary ring.

2.1 P BW Extensions Any PBW extension is a bijective skew P BW extension since in this case σi = iR for each 1 ≤ i ≤ n, and ci,j = 1 for every 1 ≤ i, j ≤ n. Thus, for P BW extensions we have A = i(R)hx1 , . . . , xn i. Examples of PBW extensions are the following (see Section 1.1): (a) The usual polynomial ring A = R[t1 , . . . , tn ]. (b) Any skew polynomial ring of derivation type A = R[x; σ, δ], i.e., with σ = iR . In general, any Ore extension of derivation type R[x1 ; σ1 , δ1 ] · · · [xn ; σn , δn ], i.e., such that σi = iR , for any 1 ≤ i ≤ n. In particular, any Ore algebra of derivation type, i.e., when R := K[t1 , . . . , tm ], m ≥ 0. (c) The Weyl algebra An (K) := K[t1 , . . . , tn ][x1 ; ∂/∂t1 ] · · · [xn ; ∂/∂tn ]. The extended Weyl algebra Bn (K) := K(t1 , . . . , tn )[x1 ; ∂/∂t1 ] · · · [xn ; ∂/∂tn ], where K(t1 , . . . , tn ) is the field of fractions of K[t1 , . . . , tn ], is also a P BW © Springer Nature Switzerland AG 2020 W. Fajardo et al., Skew PBW Extensions, Algebra and Applications 28, https://doi.org/10.1007/978-3-030-53378-6_2

25

26

2 Examples

extension. These algebras are also known as algebras of linear partial differential operators, see Section 2.3 below. The Weyl algebra An (K) can be generalized assuming that R is an arbitrary ring, i.e., we have the Weyl ring An (R) := R[t1 , . . . , tn ][x1 ; ∂/∂t1 ] · · · [xn ; ∂/∂tn ]. (d) Let K be a commutative ring and G be a finite-dimensional Lie algebra over K with basis {x1 , . . . , xn }; the universal enveloping algebra of G, U (G), is a P BW extension of K (see [257]), [278] and [421]). In this case, xi r − rxi = 0 and xi xj − xj xi = [xi , xj ] ∈ G = Kx1 + · · · + Kxn ⊆ K + Kx1 + · · · + Kxn , for any r ∈ K and 1 ≤ i, j ≤ n. Using Theorem 1.3.1, we will now give a new short proof of the Poincar´e– Birkhoff–Witt theorem on the bases of enveloping algebras of finite-dimensional Lie algebras. Theorem 2.1.1 (Poincar´ e–Birkhoff–Witt theorem). The standard αn 1 monomials xα · · · x , α ≥ 0, 1 ≤ i ≤ n, conform a K-basis of U (G). i n 1 Proof. For U (G) we take: X := {x1 , . . . , xn }, ci,j := 1,

dij := 0,

σi := iK ,

δi := 0, (1)

(n)

[xi , xj ] = aij x1 + · · · + aij xn ,

1 ≤ i < j ≤ n.

We want to prove that conditions (1)–(3) in Theorem 1.3.1 hold. Condition (1) trivially holds. Recalling that h = qp and applying (1.3.1) and (1.3.2), we have: For (2), let i < j and r ∈ K, then h(xj xi r) = h(xj rxi ) = h(rxj xi ) = h(rxi xj ) + h(r[xj , xi ]) = rxi xj + r[xj , xi ]; h(p(xj xi )r) = h(xi xj r) + h([xj , xi ]r) = h(xi rxj ) + h(r[xj , xi ]) = rxi xj + r[xj , xi ]. Condition (3) of Theorem 1.3.1 also holds: in fact, for i < j < k, h(p(xk xj )xi ) = h(xj xk xi ) + h([xk , xj ]xi ) = h(xj xi xk ) + h(xj [xk , xi ]) + h([xk , xj ]xi ) = xi xj xk + h([xj , xi ]xk ) + h(xj [xk , xi ]) + h([xk , xj ]xi ), but observe that expanding the brackets, a direct computation shows that h([xj , xi ]xk ) = h(xk [xj , xi ]) + h([[xj , xi ], xk ]), h(xj [xk , xi ]) = h([xk , xi ]xj ) + h([xj , [xk , xi ]]), h([xk , xj ]xi ) = h(xi [xk , xj ]) + h([[xk , xj ], xi ]); therefore,

2.2 Ore Extensions of Bijective Type

27

h(p(xk xj )xi ) = xi xj xk + h(xk [xj , xi ]) + h([[xj , xi ], xk ]) + h([xk , xi ]xj ) + h([xj , [xk , xi ]]) + h(xi [xk , xj ]) + h([[xk , xj ], xi ]) =xi xj xk + h([[xj , xi ], xk ] + [xj , [xk , xi ]] + [[xk , xj ], xi ]) + h(xk [xj , xi ]) + h([xk , xi ]xj ) + h(xi [xk , xj ]) =xi xj xk + h(xk [xj , xi ]) + h([xk , xi ]xj ) + h(xi [xk , xj ]), where the last equality holds by the Jacobi identity. On the other hand, h(xk xj xi ) = h(xk xi xj ) + h(xk [xj , xi ]) = h(xi xk xj ) + h([xk , xi ]xj ) + h(xk [xj , xi ]) = h(xi xj xk ) + h(xi [xk , xj ]) + h([xk , xi ]xj ) + h(xk [xj , xi ]) = xi xj xk + h(xi [xk , xj ]) + h([xk , xi ]xj ) + h(xk [xj , xi ]). From Theorem 1.3.1 we get that there exists a skew P BW extension αn 1 A = σ(K)hx1 , . . . , xn i, in particular, the monomials xα 1 · · · xn , αi ≥ 0, 1 ≤ i ≤ n, conform a K-basis of A. Applying the universal property of A and the universal property of the free algebra K{x1 , . . . , xn } we can αn 1 prove (see Example 1.3.3), that U (G) ∼ = A, whence xα 1 · · · xn , αi ≥ 0, 1 ≤ i ≤ n, is a K-basis of U (G). t u (e) Let K, G, {x1 , . . . , xn } and U (G) be as in the previous example; let R be a K-algebra containing K. The tensor product A := R ⊗K U (G) is a P BW extension of R, and it is a particular case of a more general construction, the crossed product R ∗ U (G) of R by U (G), which is also a P BW extension of R (cf. [278]). In the following sections we consider a lot of skew P BW extensions that are not PBW extensions.

2.2 Ore Extensions of Bijective Type Any skew polynomial ring R[x; σ, δ] of bijective type, i.e., with σ bijective, is a bijective skew P BW extension (see Section 1.1). In this case we have R[x; σ, δ] ∼ = σ(R)hxi. If additionally δ = 0, then R[x; σ] is quasi-commutative. More generally, let R[x1 ; σ1 , δ1 ] · · · [xn ; σn , δn ] be an iterated skew polynomial ring of bijective type, i.e., the following conditions hold: ˆ ˆ ˆ ˆ

For For For For

1 ≤ i ≤ n, σi is bijective. every r ∈ R and 1 ≤ i ≤ n, σi (r), δi (r) ∈ R. i < j, σj (xi ) = cxi + d, with c, d ∈ R and c is invertible. i < j, δj (xi ) ∈ R + Rx1 + · · · + Rxn .

Then, R[x1 ; σ1 , δ1 ] · · · [xn ; σn , δn ] is a bijective skew P BW extension. Under these conditions we have R[x1 ; σ1 , δ1 ] · · · [xn ; σn , δn ] ∼ = σ(R)hx1 , . . . , xn i.

28

2 Examples

In particular, any Ore extension R[x1 ; σ1 , δ1 ] · · · [xn ; σn , δn ] of bijective type, i.e., for 1 ≤ i ≤ n, σi is bijective, is a skew bijective P BW extension. In fact, in Ore extensions for every r ∈ R and 1 ≤ i ≤ n, σi (r), δi (r) ∈ R, and for i < j, σj (xi ) = xi and δj (xi ) = 0. An important subclass of Ore extensions of bijective type are the Ore algebras of bijective type, i.e., when R = K[t1 , . . . , tm ], m ≥ 0. Thus, we have K[t1 , . . . , tm ][x1 ; σ1 , δ1 ] · · · [xn ; σn , δn ] ∼ = σ(K[t1 , . . . , tm ])hx1 , . . . , xn i. Some concrete examples of Ore algebras of bijective type are the following. (a) The algebra of q-differential operators Dq,h [x, y]: Let q, h ∈ K, q 6= 0; consider K[y][x; σ, δ], σ(y) := qy and δ(y) := h. By definition of skew polynomial ring we have xy = σ(y)x + δ(y) = qyx + h, and hence xy − qyx = h. Therefore, Dq,h ∼ = σ(K[y])hxi. (b) The algebra of shift operators Sh : Let h ∈ K. The algebra of shift operators is defined by Sh := K[t][xh ; σh , δh ], where σh (p(t)) := p(t − h), and δh := 0. Thus, Sh ∼ = σ(K[t])hxh i. (c) The mixed algebra Dh : Let h ∈ K. The algebra Dh is defined by d Dh := K[t][x; iK[t] , dt ][xh ; σh , δh ], where σh (x) := x. Then Dh ∼ = σ(K[t])hx, xh i. (d) The algebra for multidimensional discrete linear systems is defined by D := K[t1 , . . . , tn ][x1 ; σ1 ] · · · [xn ; σn ], where σi (p(t1 , . . . , tn )) : = p(t1 , . . . , ti−1 , ti + 1, ti+1 , . . . , tn ), σi (xi ) = xi , 1 ≤ i ≤ n. So, D is a quasi-commutative bijective skew P BW extension of K[t1 , . . . , tn ] and D ∼ = σ(K[t1 , . . . , tn ])hx1 , . . . , xn i.

2.3 Operator Algebras In this section we recall some important and well-known operator algebras (cf. [90], [402]); as in Example 1.3.3, it is easy to check that they are skew P BW extensions. (a) Algebra of linear partial differential operators. The nth Weyl algebra An (K) over K coincides with the K-algebra of linear partial differential operators with polynomial coefficients K[t1 , . . . , tn ]. As we have seen, the generators of An (K) satisfy the following relations t i t j = t j ti , ∂ i ∂ j = ∂ j ∂ i , ∂j ti = ti ∂j + δij ,

1 ≤ i < j ≤ n, 1 ≤ i, j ≤ n,

(2.3.1) (2.3.2)

2.3 Operator Algebras

29

where δij is the Kronecker symbol. Let K(t1 , . . . , tn ) be the field of rational functions in n variables. Then the K-algebra of linear partial differential operators with rational function coefficients is the algebra Bn (K) = K(t1 , . . . , tn )[∂1 , . . . , ∂n ], where the generators satisfy the above relations. (b) Algebra of linear partial shift operators. The K-algebra of linear partial shift (recurrence) operators with polynomial, respectively with rational coefficients, is K[t1 , . . . , tn ][E1 , . . . , Em ], respectively K(t1 , . . . , tn )[E1 , . . . , Em ],

n ≥ m,

subject to the relations: 1 ≤ i < j ≤ n,

t j t i = ti t j ,

1 ≤ i ≤ m,

Ei ti = (ti + 1)Ei = ti Ei + Ei ,

i 6= j,

E j t i = ti E j ,

1 ≤ i < j ≤ m.

Ej E i = Ei Ej ,

(c) Algebra of linear partial difference operators. The K-algebra of linear partial difference operators with polynomial, respectively with rational coefficients, is K[t1 , . . . , tn ][∆1 , . . . , ∆m ], respectively K(t1 , . . . , tn )[∆1 , . . . , ∆m ],

n ≥ m,

subject to the relations: 1 ≤ i < j ≤ n,

t j t i = t i tj ,

1 ≤ i ≤ m,

∆i ti = (ti + 1)∆i + 1 = ti ∆i + ∆i + 1,

i 6= j,

∆ j ti = t i ∆ j ,

1 ≤ i < j ≤ m.

∆j ∆i = ∆ i ∆ j ,

(d) Algebra of linear partial q-dilation operators. For a fixed q ∈ K − {0}, the K-algebra of linear partial q-dilation operators with polynomial coefficients, respectively, with rational coefficients, is (q)

(q) K[t1 , . . . , tn ][H1 , . . . , Hm ],

respectively (q)

(q) ], K(t1 , . . . , tn )[H1 , . . . , Hm

subject to the relations:

n ≥ m,

30

2 Examples

1 ≤ i < j ≤ n,

t j ti = ti t j , (q)

(q)

1 ≤ i ≤ m,

(q) ti H j ,

i 6= j,

Hi ti = qti Hi , (q) Hj ti (q)

(q)

Hj H i

=

(q)

(q)

1 ≤ i < j ≤ m.

= Hi H j ,

(e) Algebra of linear partial q-differential operators. For a fixed q ∈ K − {0}, the K-algebra of linear partial q-differential operators with polynomial coefficients, respectively with rational coefficients, is (q)

(q) K[t1 , . . . , tn ][D1 , . . . , Dm ],

respectively the ring (q)

(q) K(t1 , . . . , tn )[D1 , . . . , Dm ],

n ≥ m,

subject to the relations: 1 ≤ i < j ≤ n,

t j t i = ti t j , (q) D i ti (q) D j ti (q) (q) Dj Di

=

(q) qti Di (q) ti D j ,

=

(q) (q) Di Dj ,

=

1 ≤ i ≤ m,

+ 1,

i 6= j, 1 ≤ i < j ≤ m.

Note that if n = m, then this operator algebra coincides with the additive analogue An (q1 , . . . , qn ) of the Weyl algebra An (q) (Section 2.5 (a)).

2.4 Algebras of Diffusion Type In [185] the diffusion algebras were introduced; following this notion we define, for n ≥ 2, the algebra A generated by 2n variables {Di , xi | 1 ≤ i ≤ n} over K with relations xi xj = xj xi , xi Dj = Dj xi , 1 ≤ i, j ≤ n, λij Di Dj − λji Dj Di = xj Di − xi Dj , i < j, λij , λji ∈ K ∗ .

(2.4.1) (2.4.2)

We rewrite (2.4.2) as −1 −1 Dj Di = λ−1 ji λij Di Dj − λji xj Di + λji xi Dj .

In this case we have R := K[x1 , . . . , xn ], and for i < j

X := {xi := Di },

σi := iR ,

δi := 0 for 1 ≤ i ≤ n,

2.5 Quantum Algebras

31

ci,j : = λ−1 ji λij ; (1)

(i−1)

aij = · · · = aij (i)

aij : = −λ−1 ji xj ,

(i+1)

= aij

(j−1)

= · · · = aij

(j+1)

= aij

(n)

= · · · = aij = 0 = dij ;

(j)

aij := λ−1 ji xi .

Clearly σi , δi satisfy the condition (1) of Theorem 1.3.1; from (2.4.1) the condition (2) holds trivially. Now, by a direct computation we can check that the condition (3) of Theorem 1.3.1 is satisfied if and only if for all i < j < k the following identities hold: λik + λkj λij + λjk λji + λik λjk + λki λij λki + λij

= λjk + λki ; = λkj + λji ; = λki + λij ; = λkj + λji ; = λjk ; = λkj + λji .

Under these conditions A ∼ = σ(K[x1 , . . . , xn ])hD1 , . . . , Dn i. Observe that the previous identities are equivalent to the following: λij = λjk = λki ,

λji = λkj = λik ,

2λjk = 2λkj .

If n ≥ 3 and char(K) 6= 2, then A is a P BW extension since for all i < j, λij = λji , and hence, ci,j = 1. A concrete easy example that we will use later in the present monograph is when K = Q, n = 2, λ12 = −2 and λ21 = −1, i.e., D2 D1 = 2D1 D2 +x2 D1 −x1 D2 . Note that A is neither a P BW extension nor an iterated skew polynomial ring of injective type.

2.5 Quantum Algebras (a) The additive analogue of the Weyl algebra. The K-algebra An (q1 , . . . , qn ) is generated by x1 , . . . , xn , y1 , . . . , yn subject to the relations: xj xi y j yi yi x j yi x i

= xi xj , = y i yj , = x j yi , = qi xi yi + 1,

1 ≤ i, j ≤ n, 1 ≤ i, j ≤ n, i 6= j, 1 ≤ i ≤ n,

where qi ∈ K − {0}. From the relations above we have An (q1 , . . . , qn ) ∼ = σ(K)hx1 , . . . , xn ; y1 , . . . , yn i ∼ σ(K[x1 , . . . , xn ])hy1 , . . . , yn i. =

(2.5.1) (2.5.2) (2.5.3) (2.5.4)

32

2 Examples

(b) The multiplicative analogue of the Weyl algebra. The K-algebra On (λji ) is generated by x1 , . . . , xn subject to the relations: xj xi = λji xi xj , 1 ≤ i < j ≤ n, with λji ∈ K − {0}. Note that On (λji ) ∼ = σ(K)hx1 , . . . , xn i ∼ = σ(K[x1 ])hx2 , . . . , xn i. (c) The quantum algebra U 0 (so(3, K)) (cf. [168] and [184]). This is the Kalgebra generated by I1 , I2 , I3 subject to the relations I2 I1 −qI1 I2 = −q 1/2 I3 , I3 I1 −q −1 I1 I3 = q −1/2 I2 , I3 I2 −qI2 I3 = −q 1/2 I1 , where q ∈ K − {0}. Thus, U 0 (so(3, K)) ∼ = σ(K )hI1 , I2 , I3 i (see Example 1.3.3). (d) The 3-dimensional skew polynomial algebra A. This is given by the relations yz − αzy = λ,

zx − βxz = µ,

xy − γyx = ν,

(2.5.5)

such that λ, µ, ν ∈ K + Kx + Ky + Kz, and α, β, γ ∈ K − {0}. Let −γ −1 ν := a0 + a1 x + a2 y + a3 z, µ := c0 + c1 x + c2 y + c3 z;

−α−1 λ := b0 + b1 x + b2 y + b3 z,

we take R := K, X := {x1 := x, x2 := y, x3 := z}, σi := iR , δi := 0 for 1 ≤ i ≤ n, c1,2 := γ −1 ,

(1)

c1,3 := β, c2,3 := α

−1

(1)

a12 := a1 ,

(2)

a13 := c1 , ,

(1) a23

(2)

a13 := c2 ,

:= b1 ,

(3)

a12 := a2 , (2) a23

a12 := a3 , (3)

a13 := c3 ,

:= b2 ,

(3) a23

d12 := a0 ,

d13 := c0 ,

:= b3 ,

d23 := b0 .

Clearly σi , δi satisfy condition (1) of Theorem 1.3.1; since A is a Kalgebra, then condition (2) holds trivially. Now, by a direct computation we can check that condition (3) of Theorem 1.3.1 is satisfied if and only if the following identities hold: b1 = γ −1 βb1 , α−1 c2 = γ −1 c2 , α−1 βa3 = a3 , α−1 c1 γ −1 + b2 γ −1 = α−1 βb2 + γ −1 c1 , α−1 βa1 + b3 β = γ −1 βb3 + a1 β, α−1 βa2 + α−1 c3 = γ −1 c3 α−1 + a2 α−1 ,

2.5 Quantum Algebras

33

α−1 c1 a1 + b0 + b2 a1 + b3 c1 = γ −1 βb0 + γ −1 c3 b1 + a1 c1 + a2 b1 , α−1 c0 + α−1 c1 a2 + b2 a2 + b3 c2 = γ −1 c0 + γ −1 c3 b2 + a1 c2 + a2 b2 , α−1 βa0 + α−1 c1 a3 + b2 a3 + b3 c3 = γ −1 c3 b3 + a0 + a1 c3 + a2 b3 , α−1 c1 a0 + b2 a0 + b3 c0 = γ −1 c3 b0 + a1 c0 + a2 b0 .

Thus, under these conditions, A ∼ = σ(K )hx, y, zi. In [347] (see also [49]) it is shown that up to isomorphism there are fifteen 3-dimensional skew polynomial algebras. For example, if α = β = γ 6= 1, then one of these algebras is yz − αzy = b1 x + b0 ,

zx − αxz = c2 y + c0 ,

xy − αyx = a3 z + a0 .

Taking b1 , c2 , a3 6= 0, A is neither a P BW extension nor an iterated skew polynomial ring of injective type. In Example 19.2.1 we have included the complete classification. (e) The dispin algebra U (osp(1, 2)). This is generated by x, y, z over the commutative ring K satisfying the relations yz − zy = z,

xy − yx = x.

zx + xz = y,

We will prove that U (osp(1, 2)) is a skew P BW extension of K. For this, first we apply Theorem 1.3.1; we take R := K, X := {x1 := x, x2 := y, x3 := z}, σi := iR , δi := 0 for 1 ≤ i ≤ n, (1)

(2)

(3)

c1,2 := 1,

a12 := −1,

a12 := 0,

a12 := 0,

d12 := 0,

c1,3 := −1,

(1) a13 := 0, (1) a23 := 0,

(2) a13 := 1, (2) a23 := 0,

(3) a13 := 0, (3) a23 := −1,

d13 := 0,

c2,3 := 1,

d23 := 0.

Clearly σi , δi satisfy condition (1) of Theorem 1.3.1; since U (osp(1, 2)) is an R-algebra, condition (2) holds trivially. Now, h(x3 x2 x1 ) = q(p(x3 x2 x1 )) = q(p(x3 x1 x2 ) − p(x3 x1 )) = q(p(−x1 x3 x2 ) + p(x2 x2 ) − p(−x1 x3 ) − p(x2 )) = −q(p(x1 x3 x2 )) + x22 + x1 x3 − x2 = −x1 x2 x3 + 2x1 x3 + x22 − x2 ; h(p(x3 x2 )x1 ) = h((x2 x3 − x3 )x1 ) = h(x2 x3 x1 − x3 x1 ) = q(p(x2 x3 x1 ) − p(x3 x1 ))

34

2 Examples

= qp(x2 x3 x1 ) + x1 x3 − x2 = q(p(x2 (x3 x1 ))) + x1 x3 − x2 = −x1 x2 x3 + x1 x3 + x22 + x1 x3 − x2 = −x1 x2 x3 + 2x1 x3 + x22 − x2 . As in Example 1.3.3, U (osp(1, 2)) ∼ = σ(K )hx, y, zi. (f) The Woronowicz algebra Wν (sl(2, K)). This algebra was introduced by Woronowicz in [403] and redefined in [347] over an arbitrary field K; it is generated by x, y, z subject to the relations xz − ν 4 zx = (1 + ν 2 )x,

xy − ν 2 yx = νz,

zy − ν 4 yz = (1 + ν 2 )y,

where ν ∈ K−{0} is not a root of unity. We will prove that Wν (sl(2, K)) ∼ = σ(K )hx, y, zi. For this, first we apply Theorem 1.3.1; we take R := K, X := {x1 := x, x2 := y, x3 := z}, σi := iR , δi := 0 for 1 ≤ i ≤ n, (1)

(2)

(3)

c1,2 := v −2 , a12 := 0, a12 := 0, a12 := −v −1 , d12 := 0, (1)

(2)

(3)

c1,3 := v −4 , a13 := −v −4 (1 + v 2 ), a13 := 0, a13 := 0, d13 := 0, (1)

(2)

(3)

c2,3 := v 4 , a23 := 0, a23 := 1 + v 2 , a23 := 0, d23 := 0. Obviously σi , δi satisfy condition (1) of Theorem 1.3.1; moreover, since Wν (sl(2, K)) is an R-algebra, condition (2) holds. Now, in a similar way as in the previous example, we can check that h(x3 x2 x1 ) = v −2 xyz − v −1 z 2 = h(p(x3 x2 )x1 ). ∼ σ(K )hx, y, zi. As in Example 1.3.3, Wν (sl(2, K)) = (g) The complex algebra Vq (sl3 (C)). Let q be a complex number such that q 8 6= 1. Consider the complex algebra generated by e12 , e13 , e23 , f12 , f13 , f23 , k1 , k2 , l1 , l2 with the following relations (cf. [408]): e13 e12 = q −2 e12 e13 ,

f13 f12 = q −2 f12 f13 ,

e23 e12 = q 2 e12 e23 − qe13 ,

f23 f12 = q 2 f12 f23 − qf13 ,

e23 e13 = q

−2

e13 e23 ,

e12 f12 = f12 e12 +

k12 − l12 , q 2 − q −2

e12 f13 = f13 e12 + qf23 k12 ,

f23 f13 = q −2 f13 f23 , e12 k1 = q −2 k1 e12 ,

k1 f12 = q −2 f12 k1 ,

e12 k2 = qk2 e12 ,

k2 f12 = qf12 k2 ,

k1 e13 ,

k1 f13 = q −1 f13 k1 ,

e13 k2 = q −1 k2 e13 ,

k2 f13 = q −1 f13 k2 ,

e23 k1 = qk1 e23 ,

k1 f23 = qf23 k1 ,

e13 f23 = f23 e13 + qk22 e12 ,

e23 k2 = q −2 k2 e23 ,

k2 f23 = q −2 f23 k2 ,

e23 f12 = f12 e23 ,

e12 l1 = q 2 l1 e12 ,

l1 f12 = q 2 f12 l1 ,

e12 f23 = f23 e12 ,

e13 k1 = q

e13 f12 = f12 e13 − q −1 l12 e23 , e13 f13 = f13 e13 −

k12 k22 − l12 l22 , q 2 − q −2

−1

2.5 Quantum Algebras

35

e23 f13 = f13 e23 − q −1 f12 l22 ,

e12 l2 = q −1 l2 e12 ,

l2 f12 = q −1 f12 l2 ,

e13 l1 = ql1 e13 ,

l1 f13 = qf13 l1 ,

e13 l2 = ql2 e13 ,

l2 f13 = qf13 l2 ,

e23 l1 = q −1 l1 e23 ,

l1 f23 = q −1 f23 l1 ,

e23 l2 = q 2 l2 e23 ,

l2 f23 = q 2 f23 l2 ,

k22 − l22 , q 2 − q −2

e23 f23 = f23 e23 +

l1 k 1 = k 1 l 1 ,

l 2 k1 = k1 l2 ,

k 2 k1 = k 1 k2 ,

l1 k 2 = k 2 l 1 ,

l 2 k2 = k2 l2 ,

l 2 l1 = l 1 l 2 .

We can see from these relations that this algebra is a bijective skew P BW extension of the polynomial ring C[l1 , l2 , k1 , k2 ], that is, Vq (sl3 (C)) ∼ = σ(C[l1 , l2 , k1 , k2 ])he12 , e13 , e23 , f12 , f13 , f23 i. (h) The algebra U. Let U be the algebra generated over the field K = C by the set of variables xi , yi , zi , 1 ≤ i ≤ n subject to the relations: xj xi = xi xj , yj yi = yi yj , zj zi = zi zj , 1 ≤ i, j ≤ n, yj xi = q δij xi yj , zj xi = q −δij xi zj , 1 ≤ i, j ≤ n, zj y i = y i zj , i = 6 j, zi yi = q 2 yi zi − q 2 x2i , 1 ≤ i ≤ n, where q ∈ C − {0}. From the relations above we see that the algebra U is a bijective skew P BW extension of C[x1 , . . . , xn ], that is, U∼ = σ(C[x1 , . . . , xn ])hy1 , . . . , yn ; z1 , . . . , zn i. (i) The coordinate algebra of the quantum matrix space Mq (2). This algebra is also known as the Manin algebra of 2 × 2 quantum matrices (cf. [257] and [271]). By definition, Oq (M2 (K)), also denoted O(Mq (2)), is the coordinate algebra of the quantum matrix space Mq (2), it is the K-algebra generated by the variables x, y, u, v satisfying the relations yu = q −1 uy,

xu = qux,

vu = uv,

(2.5.6)

and xv = qvx,

yx − xy = −(q − q −1 )uv,

vy = qyv,

(2.5.7)

where q ∈ K − {0}. Using Theorem 1.3.1 we will prove that Oq (M2 (K)) is a skew P BW extension of K[u]. We have X := {x1 := x, x2 := y, x3 := v},

R := K[u], σ1 (u) := qu, c1,2 := 1, c1,3 := q −1 ,

σ2 (u) := q −1 u,

(1) a12

:= 0,

(1) a13

:= 0,

(2) a12

σ3 := iR ,

:= 0,

(2) a13

:= 0,

(3) a12

δi := 0 for 1 ≤ i ≤ 3,

:= −(q − q −1 )u,

(3) a13

:= 0,

d13 := 0,

d12 := 0,

36

2 Examples (1)

c2,3 := q,

a23 := 0,

(2)

a23 := 0,

(3)

a23 := 0,

d23 := 0.

Clearly σi , δi satisfy condition (1) of Theorem 1.3.1; since Oq (M2 (K)) is a K-algebra, then condition (2) holds for any r ∈ K; now, it is easy to check that h(xj xi uk ) = h(p(xj xi )uk ) for i < j and k ≥ 1, so h(xj xi r) = h(p(xj xi )r) for every r ∈ R. On the other hand, h(x3 x2 x1 ) = q(p(x3 x2 x1 )) = q(p(x3 x1 x2 ) − p(x3 (q − q −1 )ux3 )) = qp(x3 x1 x2 ) − (q − q −1 )ux23 = q(p(q −1 x1 x3 x2 )) − (q − q −1 )ux23 = x1 x2 x3 − (q − q −1 )ux23 ; h(p(x3 x2 )x1 ) = h(qx2 x3 x1 ) = qp(qx2 q −1 x1 x3 ) = qp(x2 x1 x3 ) = x1 x2 x3 − (q − q −1 )ux23 . As in Example 1.3.3, O(Mq (2)) ∼ = σ(K[u])hx, y, vi. Due to the last relation in (2.5.7), we remark that it is not possible to consider O(Mq (2)) as a skew P BW extension of K. This algebra can be generalized to n variables, Oq (Mn (K)), and coincides with the coordinate algebra of the quantum group SLq (2), see [74] for more details. (j) The q-Heisenberg algebra. The K-algebra Hn (q) is generated by the set of variables x1 , . . . , xn , y1 , . . . , yn , z1 , . . . , zn subject to the relations: xj xi = xi xj , zj y i = y i zj ,

z j zi = zi z j , z j x i = x i zj ,

yj yi = yi yj , 1 ≤ i, j ≤ n, (2.5.8) y j x i = x i yj , i = 6 j, (2.5.9)

zi yi = qyi zi ,

zi xi = q −1 xi zi + yi ,

yi xi = qxi yi , 1 ≤ i ≤ n, (2.5.10)

with q ∈ K − {0}. Note that Hn (q) ∼ = σ(K)hx1 , . . . , xn ; y1 , . . . , yn ; z1 , . . . , zn i ∼ = σ(K[y1 , . . . , yn ])hx1 , . . . , xn ; z1 , . . . , zn i. (k) The quantum enveloping algebra of sl(2, K). Uq (sl(2, K)) is defined as the algebra generated by x, y, z, z −1 with relations zz −1 = z −1 z = 1, xz = q

−2

zx,

2

yz = q zy,

(2.5.11) (2.5.12)

−1

xy − yx =

z−z , q − q −1

(2.5.13)

2.5 Quantum Algebras

37

where q 6= 1, −1. From the previous relations, and as in Example 1.3.3, it is easy to check that Uq (sl(2, K)) is a skew P BW extension: Uq (sl(2, K)) = σ(K[z, z −1 ])hx, yi. (l) The Hayashi algebra Wq (J). T. Hayashi in [169] defined the quantized Kalgebra Wq (J) generated by xi , yi , zi , 1 ≤ i ≤ n, subject to the relations (2.5.8)–(2.5.10) replacing zi xi = q −1 xi zi + yi by (zi xi − qxi zi )yi = 1 = yi (zi xi − qxi zi ), i = 1, . . . , n,

(2.5.14)

with q ∈ K − {0}. Note that Wq (J) is a σ-P BW extension of K[y1±1 , . . . , yn±1 ]. In fact, xi yj−1 = yj−1 xi , zi yj−1 = yj−1 zi ,

yj yj−1 = yj−1 yj = 1,

zi xi = qxi zi + yi−1 ,

1 ≤ i, j ≤ n.

±1 ±1 Hence, Wq (J) ∼ ])hx1 , . . . , xn ; z1 , . . . , zn i. = σ(K[y1 , . . . , yn (m) The quantum Weyl algebra of Maltsiniotis Aq,λ n . Let K be a commutative ring and q = [qij ] a matrix over K such that qij qji = 1 and qii = 1 for all 1 ≤ i, j ≤ n. Fix an element λ := (λ1 , . . . , λn ) of (K ∗ )n . By definition, this algebra is generated by xi , yj , 1 ≤ i, j ≤ n, subject to the relations

(a) For any 1 ≤ i < j ≤ n, xi xj = λi qij xj xi , xi yj = qji yj xi ,

yi yj = qij yj yi , y i xj =

λ−1 i qji xj yi .

(2.5.15) (2.5.16)

(b) For any 1 ≤ i ≤ n, x i yi − λi y i x i = 1 +

X

(λj − 1)yj xj .

(2.5.17)

1≤j 0 and h ∈ K ∗ . 2 Then, Z(Dh ) = K[xp , xph , tp − tp ]. t u

Proof. See [395].

Next we present another interesting example of a skew P BW extension with trivial center. Let K be a field. The Jordan plane (see Example 1.10, [344]) is the K-algebra defined by J := K{x, y}/hyx − xy − y 2 i;

3.3 The Center

57

note that J is a skew P BW extension of K[y]: in fact, J = σ(K[y])hxi, with rule of multiplication given by xy = yx − y 2 . Observe that another presentation of the Jordan plane is J = K{x, y}/hyx − xy − x2 i (changing x to y and y to −x, see [344]), in this case J = σ(K[x])hyi. Proposition 3.3.11. Let K be a field with char(K) = 0. Then Z(J ) = K. Proof. This proof can also be found in [188]. In J we have y m xn =

n   X n (m + n − l − 1)!

(m − 1)!

l

l=0

xl y m+n−l ;

in particular, for n = 1, y m x =P xy m + my m+1 ; n n l n+1−l for m = 1, yx = l=0 n! . l! x y Ps

Let w =

i=0

Pt

j=0

s X t X

αij x

αij xi y j ∈ Z(J ), then xw = wx, and hence, i+1 j

y = xw = wx =

i=0 j=0

s X t X

αij xi (jy j+1 + xy j ),

i=0 j=0

therefore s X

xi

i=0

t X

jαij y j+1 = 0,

j=0

but J is a K[y]-free right module with basis {xk }k≥0 , so for 0 ≤ i ≤ s we Pt have j=1 jαij y j+1 = 0, also, K[y] is a K-vector space with basis {y k }k≥0 , so jαij = 0, and since char(K) = 0, we have αij = 0 for 0 ≤ i ≤ s and 1 ≤ j ≤ t. P s Thus, w = i=0 αi0 xi ; considering that wy = yw, we obtain s X

αi0 xi y = wy = yw

i=0

=

s X

αi0

i=0

=

s X

i X i! l=0 i

αi0

s X i=1

αi0

xl y i+1−l

xy+

i=0

whence

l!

i−1 X i! l=0

i−1 X i! l=0

l!

l!

! l i+1−l

xy

,

xl y i+1−l = 0.

Using this relation we will show by induction on s that αi0 = 0 for 1 ≤ i ≤ s. Note that the monomials {xn y m | n, m ∈ N} conform a K-basis of J , hence,

58

3 Basic Properties

for s = 2 we have α10 y 2 +P 2α20 y 3 +P2α20 xy 2 = 0, so αi0 = 0 for i = 1, 2. s i−1 Assume by induction that i=1 αi0 l=0 i!l! xl y i+1−l = 0 implies αi0 = 0 for 1 ≤ i ≤ s; then 0=

=

s+1 X

αi0

i−1 X i!

i=1

l=0

s X

i−1 X

αi0

i=1

l=0

l!

xl y i+1−l s

X (s + 1)! i! l i+1−l xy + αs+1,0 xl y s+2−l , l! l! l=0

whence the coefficient of y s+2 is (s + 1)!αs+1,0 = 0, but char(K) = 0 so αs+1,0 = 0, and hence s X

αi0

i=1

i−1 X i!

l!

l=0

xl y i+1−l = 0,

by induction αi0 = 0 for 1 ≤ i ≤ s + 1. From this, w = α00 ∈ K, and hence Z(J) = K. t u Proposition 3.3.12. Let K be a field with char(K) = p > 0. Then Z(J ) = K[xp , y p ]. Proof. The proof has been taken from [362], Theorem 2.2 (see also [188]). Observe first that K[xp , y p ] ⊆ Z(J ): in fact, since char(K) = p, y p x = p xy + py p+1 = xy p , so y p ∈ Z(J ). On the other hand, yxp = xp y + Pp−1 p! l p+1−l p p , but for l = 0, . . . , p − 1 we have p | p! l=0 l! x y l! , so yx = x y p p and hence K[x , y ] ⊆ Z(J ). Ps Pt Let w = i=0 j=0 αij xi y j ∈ Z(J ), from xw = wx we have s X

xi

i=0

t X

jαij y j+1 = 0,

j=1

but J is a free right K[y]-module with basis {xk }k≥0 , so for i = 0, 1, . . . , s Pt we get that j=1 jαij y j+1 = 0 and K[y] is a free left K-module with basis {y k }k≥0 , therefore jαij = 0. Thus, if p - j, then αij = 0, for i = 0, . . . , s; therefore, taking βik := αi,pk and t0 := max{j ∈ {1, . . . , t} | p | j}, we get Pt0 Ps w = k=0 i=0 βik xi y pk . On the other hand, from wy = yw we have 0

t X s X

0

βik xi y pk+1 =

k=0 i=0

=

t0 X s X k=0 i=0

so

t X s X k=0 i=0

βik

i−1 X l=0

βik

i X i! l=0

l!

xl y i+1−l+pk

i! l i+1−l+pk xy + xi y pk+1 l!

! ,

3.3 The Center

59 0

t X s X

βik

i−1 X i!

k=0 i=1

l=0

l!

xl y i+1−l+pk = 0.

(3.3.2)

From this we will show that βik = 0 for k = 0, 1, . . . , t0 , when p - i. The proof is by induction on s. Recall that {xn y m |n, m ∈ N} is a K-basis of J , therefore for s = 2 we have ! t0 1 X X 2! l 3−l+pk 2+pk 0= β1k y + β2k xy l! k=0

=

t0 X

l=0



 β1k y 2+pk + 2β2k (y 3+pk + xy 2+pk ) ,

k=0

so β1k = 0 and 2β2k = 0 for k = 0, . . . , t0 . It is clear that p - 1, and if p - 2, then β2k = 0 for k = 0, . . . , t0 . Assume by induction that for s, the equation (3.3.2) implies βik = 0 for k = 0, . . . , t0 when p - i. Then, 0

0=

t X s+1 X

βik

k=0 i=1

i−1 X i! l=0

l!

0

l i+1−l+pk

xy

=

t X s X

βik

k=0 i=1

+

i−1 X i! l=0

t0 X

βs+1,k

k=0

0

0=

t X s X k=0 i=1

βik

i−1 X i! l=0

l!

+

xl y i+1−l+pk

s X (s + 1)! l=0

l!

xl y s+2−l+pk

xl y i+1−l+pk

0

t X

l!

βs+1,k

k=0

s−1 X (s + 1)! l=0

l!

! l s+2−l+pk

xy

s 2+pk

+ (s + 1)x y

,

thus, the coefficient of xs y 2+pk is (s + 1)βs+1,k = 0, and two possibilities arise: If p - s + 1, then βs+1,k = 0 for k = 0, . . . , t0 . If p | s + 1, then

(s+1)! l!

= 0 for l = 0, . . . , s.

In any of these two cases we have 0

0=

t X s X k=0 i=1

βik

i−1 X i! l=0

l!

xl y i+1−l+pk ,

so by induction βik = 0 for k = 0, . . . , t0 , when p - i with i = 1, . . . , s + 1.

60

3 Basic Properties

We can conclude the proof; let γhk := βph,k and s0 := max{i ∈ {1, . . . , s} | P t 0 P s0 ph pk p | i}, then w = ∈ K[xp , y p ], and hence, Z(J ) = h=0 k=0 γhk x y p p K[x , y ]. t u Proposition 3.3.13. Let K be a field and q ∈ K − {0}. Let A := Kq [x1 , . . . , xn ] be the skew P BW extension defined by xj xi = qxi xj for all 1 ≤ i < j ≤ n, n ≥ 2. If q is not a root of unity, then Z(A) = K. Proof. This proof can also be found in [188] or in [251]. Let p=

t X

αin i1 ri x α ∈ Z(A), 1 · · · xn

i=1

with αij ≥ 1 and ri ∈ K − {0} for i = 1, 2, . . . , t and j = 1, 2, . . . , n. Since x1 p = px1 , we have t X i=1

i1 +1 in ri x α · · · xα = n 1

t X

i1 +1 in ri q αi2 +···+αin xα · · · xα n . 1

i=1

From this we get that ri = ri q αi2 +···+αin , so 1 = q αi2 +···+αin for i = 1, 2, . . . , t. Since q is not a root of unity, then αi2 + · · · + αin = 0, so αij = 0 Pt i1 for i = 1, 2, . . . , t and j = 2, 3, . . . , n. Thus, p = i=1 ri xα 1 , but xn p = pxn , Pt P t αi1 αi1 αi1 αi1 so i=1 ri q x1 xn = i=1 ri x1 xn , whence 1 = q for i = 1, 2, . . . , t, Pt i.e., αi1 = 0 and p = i=1 ri ∈ K. Therefore, Z(A) ⊆ K ⊆ Z(A). t u Taking n = 2 in the previous proposition we obtain that the quantum plane (see Section 4.4) has trivial center if q is not a root of unity. Next we will consider the case when q 6= 1 is a root of unity of degree m ≥ 2 (if q = 1, then the quantum plane is K[x, y] and its center is K[x, y]). Proposition 3.3.14 ([362]). If q 6= 1 is an arbitrary root of unity of degree m ≥ 2, then the center of the quantum plane A := Kq [x, y] is the subalgebra generated by xm and y m , i.e., Z(Kq [x, y]) = K[xm , y m ]. Proof. Note first that in A the following rule of multiplication holds: y r xs = q rs xs y r , for r, s ≥ 0. K[xm , y m ] ⊆ Z(A): This inclusion is trivial since xxm = xm x, yxm = q m xm y = xm y, y m y = yy m , y m x = q m xy m = xy m . Pn Z(A) ⊆ K[xm , y m ]: Let p = i=1 ki xri y si ∈ Z(A), with ki 6= 0, then Pn Pn xp = px ⇒ i=1 ki xri +1 y si = i=1 ki q si xri +1 y si , so ki = ki q si , whence si is multiple of m; similarly, ri is multiple of m. Thus, p ∈ K[xm , y m ]. t u

3.3 The Center

61

Proposition 3.3.15. Let K be a field and q ∈ K − {0}. Let A := Kq [x1 , . . . , xn ] be the skew P BW extension defined by xj xi = qxi xj for all 1 ≤ i < j ≤ n. If n ≥ 2 and q 6= 1 is a root of unity of degree m ≥ 2, then (i) If q = −1, then Z(A) = K[x21 , . . . , x2n ] when n is even, Z(A) = K[x21 , . . . , x2n , x1 . . . xn ] when n is odd. (ii) q 6= −1, then m Z(A) = K[xm 1 , . . . , xn ] when n is even, m m K[x1 , . . . , xn ] ( Z(A) when n is odd.

Proof. See [82], Lemma 4.1, and [395].

t u

For the next two propositions we need the following quasi-commutative skew P BW extension (see Example 4.4.2): Let R be a commutative do−1 main, q := [qij ] ∈ Mn (R), with qij ∈ R∗ and qji := qij , qii := 1, then Rq [x1 , . . . , xn ] is defined by σi = iR , δi = 0, 1 ≤ i ≤ n, xj xi = qij xi xj , 1 ≤ i < j ≤ n. Proposition 3.3.16. Let R be a commutative domain. Assume that for every 1 ≤ i, j ≤ n, qij is not a root of unity. Then, (i) Z(Rq [x1 , . . . , xn ]) = R. (ii) Z(Q(Rq [x1 , . . . , xn ])) ∼ = Q(R). (iii) Q(Z(Rq [x1 , . . . , xn ])) ∼ = Z(Q(Rq [x1 , . . . , xn ])). Proof. See [395].

t u

Proposition 3.3.17. Let R be a commutative domain. Assume that for every 1 ≤ i, j ≤ n, qij is a nontrivial root of unity of degree dij < ∞. Let kij ∈ Z √ kij such that |kij | < dij , lcd(kij , dij ) = 1 and qij = exp(2π −1 dij ) (choosing kji := −kij ). Let Li := lcm{dij |j = 1, . . . , n}. Then Z(Rq [x1 , . . . , xn ]) is a Ln 1 polynomial ring if and only if it is of the form R[xL 1 , . . . , xn ]. Proof. See [84], Theorem 0.3.

t u

Proposition 3.3.18. Let R be a commutative domain and let Aq be the qquantum Weyl R-algebra generated by x, y with rule of multiplication yx = qxy + a, where q ∈ R∗ and a ∈ R. Then (i) Aq is a bijective skew P BW extension of R. (ii) If q is an m-th primitive root of unity for some m ≥ 2, then Z(Aq ) = R[xm , y m ] and Aq is free over Z(Aq ) with basis {xi y j |0 ≤ i, j ≤ m − 1}.

62

3 Basic Properties

Proof. (i) This a trivial consequence of Theorem 1.3.1. (ii) See [84], Section 2.

t u

Proposition 3.3.19. Let R be a commutative domain and let V be the generalized quantum Weyl R-algebra generated by n ≥ 3 variables x1 , . . . , xn with relations xj xi = qij xi xj + aij , qij ∈ R∗ , aij ∈ R, 1 ≤ i < j ≤ n. (i) V is a bijective skew P BW extension of R if and only if the following identities hold for all i < j < k: ajk = qij qik ajk , qjk aik = qij qik aik , qjk qik aij = aij . (ii) Assuming that V satisfies the identities in (i), then Gr(V ) ∼ = Rq [x1 , . . . , xn ]. (iii) Assume that V satisfies the following conditions: (a) Gr(V ) ∼ = Rq [x1 , . . . , xn ]. (b) Every qij is a nontrivial root of unity. (c) For any pair (i, j), aij = 0 whenever qij = 1. Then, (1) If Z(Gr(V )) is a polynomial ring, then so is V , and Z(V ) ∼ = Z(Gr(V )). (2) If Z(V ) is a polynomial ring, then V is a finite module over its center. Ln 1 (3) If Z(Rq [x1 , . . . , xn ]) = R[xL 1 , . . . , xn ], then Ln 1 Z(V ) = R[xL 1 , . . . , xn ]

and V is finitely generated and free over Z(V ). Proof. (i) Since V is an R-algebra, this follows from condition (3) of Theorem 1.3.1 by direct computation. (ii) This a direct consequence of (3.1.2) in the proof of Theorem 3.1.2. (iii) See [84], Theorem 0.3 and Proposition 6.1. t u From the above examples and partial results we observe that the description of the center of an arbitrary skew P BW extension is a difficult task. In [251] the center and central subalgebras of some other important examples of skew P BW extensions were computed. We conclude this section with two general theorems about the center of some skew P BW extensions that are K-algebras, K a field. Theorem 3.3.20. Let A = σ(K)hx1 , . . . , xn i be a skew P BW extension of K defined by (1)

(n)

xj xi = qij xi xj + aij x1 + · · · + aij xn + dij ,

3.3 The Center

63 l

with 1 ≤ i < j ≤ n, and qij = kijij , (lij , lrs ) = (kij , krs ) = (lij , krs ) = 1 for i, j, r, s ∈ {1, 2, . . . , n}. If char(K) = 0 and qij is not a root of unity, then Z(A) = K. Proof. See [251], Theorem 2.5.

t u

Theorem 3.3.21. Let A = σ(K[x1 , . . . , xn ])hy1 , . . . , yn i be the skew P BW extension defined by yi yj = yj yi 1 ≤ i < j ≤ n, yj x i = x i yj i = 6 j, yi xi = qi xi yi + di yi + ai , 1 ≤ i ≤ n. If char(K) = 0, then Z(A) = K in any of the following cases: (a) For 1 ≤ i ≤ n, qi is not a root of unity. (b) If for some i, qi = 1, then di = 6 0 or ai 6= 0. Proof. See Theorem 2.6 in [251].

t u

Chapter 4

Rings of Fractions

Recall that if R is a ring and S is a multiplicative subset of R (i.e., 1 ∈ S, 0 ∈ / S, ss0 ∈ S, for s, s0 ∈ S) then the left ring of fractions of R exists if and only if two conditions hold: (i) given a ∈ R and s ∈ S such that as = 0, then there exists an s0 ∈ S such that s0 a = 0; (ii) (the left Ore condition) given a ∈ R and s ∈ S there exist s0 ∈ S and a0 ∈ R such that s0 a = a0 s. Note that any domain R satisfies (i) with respect to any S, and R is said to be a left Ore domain if it satisfies (ii) with respect to S := R − {0}. The elements of the ring R that are nonzero divisors are called regular and the set of regular elements of R will be denoted by S0 (R). In this chapter we will investigate the localization of skew P BW extensions with respect to different kinds of multiplicative subsets. We will consider multiplicative subsets of the ring of coefficients R as well as sets of regular elements of the extension A = σ(R)hx1 , . . . , xn i. In particular, Ore’s and Goldie’s theorems will be studied. Some additional questions about rings of fractions that arise in the investigation of the Gelfand–Kirillov conjecture will be also analyzed (see [144]). For example, the center of A and isomorphisms between rings of fractions will be also considered. In addition, the skew quantum polynomials are introduced, generalizing in this way the quantum polynomials (see [20]).

4.1 Preliminary Key Results In this first section we localize iterated skew polynomial rings and skew P BW extensions by multiplicative subsets of the ring of coefficients. The basic results presented here will used in other sections of the present chapter. We start by recalling a couple of well-known facts. Proposition 4.1.1. Let σ be an automorphism of R and R[x; σ, δ] the left skew polynomial ring. Then, the right skew polynomial ring R[x; σ −1 , −δσ −1 ]r is isomorphic to R[x; σ, δ]. Proof. See 1.2.6 in [278], or also [241].

t u

© Springer Nature Switzerland AG 2020 W. Fajardo et al., Skew PBW Extensions, Algebra and Applications 28, https://doi.org/10.1007/978-3-030-53378-6_4

65

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Proposition 4.1.2. Let R be a ring and S ⊂ R a multiplicative subset. If Q := S −1 R exists, then any finite set {q1 , . . . , qn } of elements of Q possess a common denominator, i.e., there exist r1 , . . . , rn ∈ R and s ∈ S such that qi = rsi , 1 ≤ i ≤ n. t u

Proof. See [278], Lemma 2.1.8. Theorem 4.1.3. Let R be a ring and S ⊂ R a multiplicative subset. (a) If S −1 R exists and σ(S) ⊆ S, then S −1 (R[x; σ, δ]) ∼ = (S −1 R)[x; σ, δ],

(4.1.1)

with σ

S −1 R − → S −1 R a σ(a) 7→ s σ(s)

δ

S −1 R − → S −1 R a δ(s) a δ(a) 7→ − + . s σ(s) s σ(s)

(b) If RS −1 exists and σ is bijective with σ(S) = S, then e (R[x; σ, δ])S −1 ∼ e, δ], = (RS −1 )[x; σ

(4.1.2)

with σ e

RS −1 − → RS −1 a σ(a) 7→ s σ(s)

e δ

RS −1 − → RS −1 a σ(a) δ(s) δ(a) 7→ − + . s σ(s) s s

(c) If S −1 R and RS −1 exist and σ is bijective with σ(S) = S, then e ∼ S −1 (R[x; σ, δ]) ∼ e, δ] = (S −1 R)[x; σ, δ] ∼ = (RS −1 )[x; σ = (R[x; σ, δ])S −1 . (d) If R is commutative and σ = iR , then δ = δe and S −1 (R[x; δ]) ∼ = (S −1 R)[x; δ] = (RS −1 )[x; δ] ∼ = (R[x; δ])S −1 . Proof. (a) In the proof as , bt will represent a couple of elements of S −1 R. (i) σ is well-defined : If as = bt , then there exist c, d ∈ R such that ca = db and cs = dt ∈ S,

(4.1.3)

σ(c)σ(a) = σ(d)σ(b)

(4.1.4)

so and σ(cs) = σ(c)σ(s) = σ(d)σ(t) = σ(dt) ∈ S. 0

(4.1.5) 0

The above equalities imply that there exist c := σ(c) and d := σ(d) such σ(b) that c0 σ(a) = d0 σ(b) and c0 σ(s) = d0 σ(t), that is, σ(a) σ(s) = σ(t) .

4.1 Preliminary Key Results

67

(ii) σ is an endomorphism of S −1 R: The left Ore condition applied to the elements s and t determines the existence of c ∈ S and d ∈ R such that cs = dt =: u ∈ S. By (4.1.4) and (4.1.5), σ(c)σ(a) σ(a) = , σ(u) σ(s) σ(d)σ(b) σ(b) = σ(u) σ(t) and hence       a a b ca + db σ(c)σ(a) + σ(d)σ(b) b σ + =σ = =σ +σ . s t u σ(u) s t Now, by the Ore condition applied to the elements a and t we guarantee the existence of elements c ∈ R and u ∈ S such that ua = ct. Then, σ(u)σ(a) = σ(c)σ(t), with σ(u) ∈ S, and from this we get σ(c)σ(b) σ(a) σ(b) = ; σ(u)σ(s) σ(s) σ(t) therefore,  σ

ab st



 =σ

cb us



a σ(c)σ(b) = =σ σ σ(u)σ(s) s

  b . t

Note that σ(1) = 1. (iii) δ is well-defined : From relations (4.1.3), (4.1.4) and (4.1.5) we have the following identities: δ(ca) = σ(c)δ(a) + δ(c)a = δ(db) = σ(d)δ(b) + δ(d)b, δ(cs) = σ(c)δ(s) + δ(c)s = δ(dt) = σ(d)δ(t) + δ(d)t, and also δ(s) σ(c)δ(s) δ(t) σ(d)δ(t) δ(a) σ(c)δ(a) δ(b) σ(d)δ(b) = , = , = , = . σ(s) σ(cs) σ(t) σ(dt) σ(s) σ(cs) σ(t) σ(dt) (4.1.6)

From these relations we have −

and

δ(s) a δ(cs) − δ(c)s a σ(d)δ(t) + δ(d)t − δ(c)s a =− =− σ(s) s σ(dt) s σ(dt) s σ(d)δ(t) a δ(d)t − δ(c)s a δ(t) a δ(d)t − δ(c)s a =− − =− − , σ(dt) s σ(dt) s σ(t) s σ(dt) s

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δ(a) δ(ca) − δ(c)a δ(db) − δ(c)a σ(d)δ(b) + δ(d)b − δ(c)a = = = σ(s) σ(dt) σ(dt) σ(dt) σ(d)δ(b) δ(d)b − δ(c)a δ(b) δ(d)b − δ(c)a = + = + . σ(dt) σ(dt) σ(t) σ(dt) Then a

δ

s

δ(t) a δ(d)t − δ(c)s a δ(b) δ(d)b − δ(c)a − + + σ(t) s σ(dt) s σ(t) σ(dt) δ(t) b δ(b) δ(d)t a δ(c)s a δ(d)b δ(c)a =− + − + + − ; σ(t) t σ(t) σ(dt) s σ(dt) s σ(dt) σ(dt)

=−

and since 1δ(c)s = δ(c)s, we have So, δ

a s

=−

δ(c)s a σ(dt) s

=

δ(c)a σ(dt) .

Similarly,

δ(d)t b σ(dt) t

δ(t) b δ(b) δ(d)t b δ(d)b δ(t) b δ(b) + − + =− + =δ σ(t) t σ(t) σ(dt) t σ(dt) σ(t) t σ(t)

=

δ(d)b σ(dt) .

  b . t

(iv) δ is additive: In the notation of the first part of (ii) we have     a b ca + db δ + =δ = A + B, s t u

with A := −

δ(u) ca + db δ(ca + db) , B := . σ(u) u σ(u)

Computing A, B, using the relations (4.1.6), and by the fact that ca a db b = , = , δ(u) = σ(c)δ(s) + δ(c)s = σ(d)δ(t) + δ(d)t, cs s dt t we obtain     a b δ(s) a δ(a) δ(t) b δ(b) a b δ + =− + − + =δ ) + δ( . s t σ(s) s σ(s) σ(t) t σ(t) s t (v) δ is a σ-derivation: In the notation of the second part of (ii) we have     ab cb δ(us) cb δ(cb) δ =δ = C + D, with C := − , D := . st us σ(us) us σ(us) In order to compute C and D we have δ(us) = σ(u)δ(s) + δ(u)s, δ(ua) = σ(u)δ(a) + δ(u)a = δ(ct) = σ(c)δ(t) + δ(c)t, δ(u)s a δ(u)a δ(c)t b δ(c)b = , = , σ(us) s σ(us) σ(us) t σ(us)

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69

σ(c)δ(t) σ(a) δ(t) σ(c)δ(b) σ(a) δ(b) σ(u)δ(a) δ(a) = , = , = . σ(us) σ(s) σ(t) σ(us) σ(s) σ(t) σ(us) σ(s) Hence, δ(u)s cb σ(u)δ(s) cb − σ(us) us σ(us) us     δ(s) a b δ(u)s a b =− − σ(s) s t σ(us) s t     δ(s) a b δ(u)s a b =− − σ(s) s t σ(us) s t   δ(s) a b δ(u)a b − =− σ(s) s t σ(us) t   δ(s) a b σ(c)δ(t) + δ(c)t − σ(u)δ(a) b =− − σ(s) s t σ(us) t   δ(s) a b σ(a) δ(t) b δ(c)b δ(a) b =− − − + ; σ(s) s t σ(s) σ(t) t σ(us) σ(s) t

C=−

σ(c)δ(b) + δ(c)b σ(us) σ(c)δ(b) δ(c)b + = σ(us) σ(us) σ(a) δ(b) δ(c)b = + . σ(s) σ(t) σ(us)

D=

From this we get that       ab σ(a) δ(t) b δ(b) δ(s) a δ(a) b δ = − + + − + st σ(s) σ(t) t σ(t) σ(s) s σ(s) t a b a b =σ δ +δ . s t s t (vi) The results above guarantee the existence of (S −1 R)[x; σ, δ]. Now we will prove the isomorphism (4.1.1). With this purpose in mind we will verify the four conditions that define a left ring of fractions (see [376] or [241]). Consider the ring homomorphism f : R → B, with B := (S −1 R)[x; σ, δ] and f (r) := 1r , r ∈ R. Note that f satisfies xf (r) = f (σ(r))x + f (δ(r)); the universal property of R[x; σ, δ] (cf. [159], see also [241]) guarantees the Pn i f : R[x; σ, δ] → B defined by r x 7→ existence of a ring homomorphism i i=0 Pn P n ri i ∗ f (ri )xi = i=0 P i=0 1 x . It is clear that f (S) ⊆ B ; if f (a) = 0, with n rn 0 i a := i=0 ri x , in particular, 1 = 1 , and hence, there exist cn , dn ∈ R such that cn rn = dn 0 = 0 and cn 1 = dn 1 = cn ∈ S. Therefore, cn a = cn r0 + cn r1 x + · · · + cn rn−1 xn−1 . Let a0 := cn a. Then f (a0 ) = 0 and we find cn−1 ∈ S such that cn−1 cn a = cn−1 cn r0 + cn−1 cn r1 x + · · · + cn−1 cn rn−2 xn−2 .

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Following this reasoning, we find c := c0 c1 · · · cn−1 cn ∈ S such that ca = 0. This proves the third condition. Now, let z := as00 + as11 x + · · · + asnn xn ∈ B. By Proposition 4.1.2 (ii), there exist elements a00 , . . . , a0n ∈ R and s ∈ S such a0 a0 a0 a0 a0 a0 that z = s0 + s1 x + · · · + sn xn = 1s ( 10 + 11 x + · · · + 1n xn ) = f (s)−1 f (a00 ), with a00 := a00 + a01 x + · · · + a0n xn ∈ R[x; σ, δ]. This shows that S −1 (R[x; σ, δ]) exists and establishes the isomorphism (4.1.1). (b) From Proposition 4.1.1, we have R[x; σ −1 , −δσ −1 ]d ∼ = R[x; σ, δ]. Let θ := σ −1 and γ := −δσ −1 , then R[x; θ, γ]r ∼ = R[x; σ, δ], so (R[x; θ, γ]r )S −1 ∼ = (R[x; σ, δ])S −1 . Repeating the proof of part (a), but for the right side (the inclusion σ −1 (S) ⊂ S is guaranteed by the condition σ(S) = S), we obtain eγ e a ) := (R[x; θ, γ]r )S −1 ∼ e]r , with θ( = (RS −1 )[x; θ, s γ e( as ) := − as γ(s) θ(s) +

θ(a) θ(s) ,

γ(a) θ(s) .

e −1 , −e e −1 ]. But note that (θ) e −1 = σ Hence, (R[x; σ, δ])S −1 ∼ γ (θ) e = (RS −1 )[x; (θ) −1 e e e and −e γ (θ) = δ, where σ e, δ are defined as in the statement of the theorem. e a ) = − σ(θ(a)) δ(θ(s)) + δ(θ(a)) = a γ(s) − γ(a) = In fact, σ eθe = iRS −1 , and δeθ( s σ(θ(s)) θ(s) θ(s) s θ(s) θ(s) −e γ ( as ). Finally, observe that (c) and (d) are consequences of (a) and (b). t u The previous theorem can be extended to iterated skew polynomial rings. Corollary 4.1.4. Let R be a ring and A := R[x1 ; σ1 , δ1 ] · · · [xn ; σn , δn ] be the iterated skew polynomial ring. Let S be a multiplicative system of R. (a) If S −1 R exists and σi (S) ⊆ S for every 1 ≤ i ≤ n, then S −1 A ∼ = (S −1 R)[x1 ; σ1 , δ1 ] · · · [xn ; σn , δn ], with (S −1 R)[x1 ; σ1 , δ1 ] · · ·[xi−1 ; σi−1 , δi−1 ] σ

i −→ (S −1 R)[x1 ; σ1 , δ1 ] · · · [xi−1 ; σi−1 , δi−1 ]

a σi (a) 7→ s σi (s) (S −1 R)[x1 ; σ1 , δ1 ] · · ·[xi−1 ; σi−1 , δi−1 ] δ

i −→ (S −1 R)[x1 ; σ1 , δ1 ] · · · [xi−1 ; σi−1 , δi−1 ]

a δi (s) a δi (a) 7→ − + . s σi (s) s σi (s) (b) If RS −1 exists and σi is bijective with σi (S) = S for every 1 ≤ i ≤ n, then AS −1 ∼ f1 , δe1 ] · · · [xn ; σ fn , δen ], = (RS −1 )[x1 ; σ with

4.1 Preliminary Key Results

71

g (RS −1 )[x1 ; σ f1 , δe1 ] · · ·[xg i−1 ; σg i−1 , δi−1 ] σe

i g −→ (RS −1 )[x1 ; σ f1 , δe1 ] · · · [xg i−1 ; σg i−1 , δi−1 ]

a σi (a) 7→ s σi (s) g (RS −1 )[x1 ; σ f1 , δe1 ] · · ·[xg i−1 ; σg i−1 , δi−1 ] δe

i g −→ (RS −1 )[x1 ; σ f1 , δe1 ] · · · [xg i−1 ; σg i−1 , δi−1 ]

a σi (a) δi (s) δi (a) 7→ − + . s σi (s) s s (c) If S −1 R and RS −1 exist and σi is bijective with σi (S) = S for every 1 ≤ i ≤ n, then ∼ f f1 , δe1 ] · · · [xn ; σf S −1 A ∼ = (RS −1 )[x1 ; σ = (S −1 R)[x1 ; σ1 , δ1 ] · · · [xn ; σn , δn ] ∼ n, δ n] = AS −1 .

(d) If R is commutative and σi = iR for every 1 ≤ i ≤ n, then δi = δei and S −1 A ∼ = (S −1 R)[x1 ; δ1 ] · · · [xn ; δn ] = (RS −1 )[x1 ; δ1 ] · · · [xn ; δ1 ] ∼ = AS −1 .

Proof. The corollary follows from Theorem 4.1.3 by iteration and observing that (S −1 R)[x1 ; σ1 , δ1 ] · · · [xi−1 ; σi−1 , δi−1 ] ∼ = S −1 (R[x1 ; σ1 , δ1 ] · · · [xi−1 ; σi−1 , δi−1 ]), thus any element of (S −1 R)[x1 ; σ1 , δ1 ] · · · [xi−1 ; σi−1 , δi−1 ] can be represented as a fraction as , with a ∈ R[x1 ; σ1 , δ1 ] · · · [xi−1 ; σi−1 , δi−1 ] and s ∈ S. The same remark applies for the other functions involved. t u Corollary 4.1.5. Let A := R[z1 ; σ1 ] · · · [zn ; σn ] be a quasi-commutative skew P BW extension of a ring R and let S be a multiplicative system of R. (a) If S −1 R exists and σi (S) ⊆ S for every 1 ≤ i ≤ n, then S −1 A is a quasi-commutative skew P BW extension of RS −1 and S −1 A ∼ = (S −1 R)[z1 ; σ1 ] · · · [zn ; σn ]. In addition, if A is bijective with σi (S) = S for every i, then S −1 A is a quasi-commutative bijective skew P BW extension of S −1 R. (b) If RS −1 exists and A is bijective with σi (S) = S for every 1 ≤ i ≤ n, then AS −1 is a quasi-commutative bijective skew P BW extension of RS −1 and AS −1 ∼ f1 ] · · · [xn ; σ fn ]. = (RS −1 )[x1 ; σ (c) If S −1 R and RS −1 exist and A is bijective with σi (S) = S for every 1 ≤ i ≤ n, then S −1 A ∼ = AS −1 is a quasi-commutative bijective skew −1 P BW extension of S R ∼ = RS −1 .

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Proof. This is a consequence of the previous corollary, taking into account σi (a) 0 that each σi is injective. Indeed, if σσii(a) = 01 , therefore (s) = 1 , then 1 there exists c, d ∈ R[x1 ; σ1 ] · · · [xi−1 ; σi−1 ] such that cσi (a) = d0 = 0 and c1 = d1 ∈ S, thus actually c ∈ S and there exists a c0 ∈ S such that σi (c0 ) = c, so σi (c0 a) = 0, whence c0 a = 0 and hence a1 = 10 , so 1s a1 = 0, i.e., a 0 s = 1. Assuming that each σi is bijective and σi (S) = S, then as before, σi is injective, and it is clear that σi is surjective. Finally, if the constants ci,j c −1 that define A are invertible (see Definition 1.1.4), then i,j A are also 1 ∈ S invertible. For part (b) the proof is analogous. (c) is consequence of (a) and (b). t u Now we consider arbitrary skew P BW extensions and S a multiplicative subset of R. The next powerful theorem generalizes Lemma 14.2.7 of [278]. Theorem 4.1.6. Let R be a ring and A := σ(R)hx1 , . . . , xn i be a skew P BW extension of R. Let S be a multiplicative subset of R such that σi (S) = S, for every 1 ≤ i ≤ n, where σi is defined by Proposition 1.1.3. (a) If S −1 R exists, then S −1 A exists and is a skew P BW extension of S −1 R with S −1 A = σ(S −1 R)hx01 , . . . , x0n i, where x0i := ci,j 1 ,

xi 1 and σi (r) σi (s) , 1

the system of constants of S −1 R is given by c0i,j :=

c0i, r := ≤ i, j ≤ n. Moreover, if A is bijective, then S −1 A is s bijective. (b) If RS −1 exists and A is bijective, then AS −1 exists and is a bijective skew P BW extension of RS −1 , with AS −1 = σ(RS −1 )hx001 , . . . , x00n i, where x00i := ci,j 00 1 , ci, rs −1

xi 1 and the system σi (r) σi (s) , 1 ≤ i, j ≤ n. −1

of constants of RS −1 is given by c00i,j :=

:= (c) If S R, RS exist and A is bijective, then S −1 A ∼ = AS −1 is a bijective −1 −1 skew P BW extension of S R ∼ = RS . Proof. We will use the notation given in Definition 1.1.7 and Remark 1.1.9. In particular, we will consider the deglex order  on Mon(A). (a) Let f ∈ A and s ∈ S be such that f s = 0. We have to prove that there exists a u ∈ S such that uf = 0. If f = 0 we take u = 1. So, assume that f = 6 0; the proof is by induction on lm(f ). Let f = cxα + f 0 , with 0 lm(f ) ≺ lm(f ) = xα ; from f s = 0 we get that cσ α (s) = 0, but since σ α (s) ∈ S, there exists a u0 ∈ S such that u0 c = 0. Whence, u0 f = u0 f 0 , and hence, u0 f 0 s = 0, where lm(u0 f 0 ) ≺ lm(f ). By induction there exists a u00 ∈ S such that u00 u0 f 0 = 0, so u00 u0 f = 0, i.e., uf = 0 with u := u00 u0 . Now, let again f ∈ A and s ∈ S. We have to find u ∈ S and g ∈ A such that uf = gs. If f = 0 we take u = 1 and g = 0. Let f 6= 0 and again lt(f ) := cxα , then there exists u1 ∈ S and r ∈ R such that u1 c = rσ α (s).

4.1 Preliminary Key Results

73

Consider u1 f − rxα s; if u1 f − rxα s = 0, then the Ore condition is satisfied. Let u1 f −rxα s 6= 0, then lm(u1 f −rxα s) ≺ lm(f ). By induction on lm, there exist u2 ∈ S and g 0 ∈ A such that u2 (u1 f − rxα s) = g 0 s. Thus, uf = gs, with u := u2 u1 and g := u2 rxα + g 0 . This proves that S −1 A exists. Let R0 := S −1 R and A0 := S −1 A; from R ⊆ A we get that R0 ,→ A0 . In fact, the correspondence rs 7→ rs is an injective ring homomorphism since if r 0 sr r 0 0 0 s = 1 in A , then 1 s = 1 = 1 in A . There exists a c ∈ S such that cr = 0, r 0 r 0 so 1 = 1 in R . This implies that s = 01 in R0 . We define x0i := x1i ∈ A0 for every 1 ≤ i ≤ n. It is clear that Mon{x01 , . . . , x0n } generates A0 as a left R0 -module. Let x0α1 , . . . , x0αt ∈ Mon{x01 , . . . , x0n } and asii ∈ R0 , 1 ≤ i ≤ t, such that as11 x0α1 + · · · + astt x0αt = 0. From Proposition 4.1.2 we αcan assume, without lost of generality, that all αt 1 a1 xα1 +···at xαt tx si are equal, and hence a1 x +···a = 0, i.e., = 0. Whence, s 1 there exists a u ∈ S such that uai = 0 for every i, i.e., asii = 0 for every i. This proves that Mon{x01 , . . . , x0n } is a left R0 -basis of A0 . c 0 Let c0i,j := i,j 1 . Since ci,j is left invertible, then ci,j is also left invertible 0 0 0 0 0 0 0 0 0 0 and xj xi − ci,j xi xj ∈ R + R xi + · · · + R xn , for every 1 ≤ i < j ≤ n. In addition, if ci,j is invertible, then c0i,j is invertible. The endomorphisms σi of R and the σi -derivations δi that define A (Proposition 1.1.3) induce endomorphisms σi of R0 and σi -derivations δi of R0 (see Theorem 4.1.3). Since σi is injective and σi (S) = S then each σi is injective, moreover, if σi is bijective, then σi is bijective
(the proof
is similar to that of Corollary 4.1.5). We claim that x0i rs = σi rs x0i + δi rs . Indeed, r r σi (r)xi δi (r) δi (s) r σi ( )x0i + δi ( ) = + − s s σi (s) σi (s) σi (s) s and x0i

r xi r c(x)r = = , with u ∈ S, c(x) ∈ A and uxi = c(x)s. s 1 s u

Therefore, deg(c(x)) = 1 and c(x) involves only xi . In fact, let lt(c(x)) = bxα , with xα 6= xi , then bσ α (s) = 0. Since σ α (s) ∈ S, there exists a u0 ∈ S such that u0 b = 0. Whence, u0 uxi = c0 (x)s, with lm(c0 (x)) ≺ lm(c(x)). Thus, we can assume that lm(c(x)) = xi . If some other term of c(x) involves xj , with j 6= i, then, in the same way, we find u0 ∈ S such that u0 uxi = c0 (x)s, where in c0 (x) this term disappears. Thus, c(x) = c1 xi + c0 , where c0 , c1 ∈ R. From this we get the relations u = c1 σi (s),

c1 δi (s) + c0 s = 0.

Therefore, x0i but

r c 1 xi r + c 0 r c1 σi (r)xi c1 δi (r) c0 r = = + + , s u c1 σi (s) c1 σi (s) c1 σi (s)

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c1 σi (r)xi σi (r)xi = , c1 σi (s) σi (s) c1 δi (r) δi (r) = and c1 σi (s) σi (s) δi (s) r c1 δi (s) r −c0 s r c0 r − =− =− = . σi (s) s c1 σi (s) s c1 σi (s) s c1 σi (s) This proves the claim. Thus, given rs ∈ R0 − {0} there exists a c0i, r := σi ( rs ) ∈ s R0 − {0} such that x0i rs − c0i, r x0i ∈ R0 . This completes the proof that S −1 A is s a skew P BW extension of S −1 R. (b) The proof is similar to the proof of (a), but we include it anyway. Let f ∈ A and s ∈ S be such that sf = 0, we have to prove that there exists a u ∈ S such that f u = 0. If f = 0 we take u = 1, so assume that f 6= 0; the proof is by induction on lm(f ). Let f = cxα + f 0 , with lm(f 0 ) ≺ lm(f ) = xα . From sf = 0 we get that sc = 0, there exists a u0 ∈ S such that cu0 = 0. Whence, f σ −α (u0 ) = cxα σ −α (u0 ) + f 0 σ −α (u0 ) = cu0 xα + cpα,σ−α (u0 ) + f 0 σ −α (u0 ) = cpα,σ−α (u0 ) + f 0 σ −α (u0 ), where lm(cpα,σ−α (u0 ) + f 0 σ −α (u0 )) ≺ lm(f ). Note that 0 = sf σ −α (u0 ) = s[cpα,σ−α (u0 ) + f 0 σ −α (u0 )]; by induction, there exists a u00 ∈ S such that [cpα,σ−α (u0 ) + f 0 σ −α (u0 )]u00 = 0, so f σ −α (u0 )u00 = 0, i.e., f u = 0 with u := σ −α (u0 )u00 ∈ S. This proves the first condition for the existence of AS −1 . Now, we have to find u ∈ S and g ∈ A such that f u = sg. If f = 0 we take u := 1 and g := 0. Let f 6= 0, lt(f ) := cxα ; there exist u1 ∈ S and r ∈ R such that cu1 = sr. Consider f σ −α (u1 ) − srxα ; if f σ −α (u1 ) = srxα , then the Ore condition is satisfied. Let f σ −α (u1 ) − srxα 6= 0, then lm(f σ −α (u1 ) − srxα ) ≺ lm(f ). By induction on lm(f ), there exist u2 ∈ S and g 0 ∈ A such that (f σ −α (u1 ) − srxα )u2 = sg 0 . Then f u = sg, with u := σ −α (u1 )u2 and g := rxα u2 + g 0 . This proves that AS −1 exists. Let R00 := RS −1 and A00 := AS −1 ; from R ⊆ A we get that R00 ,→ A00 . In fact, the correspondence rs 7→ rs is an injective ring homomorphism since if 0 rs r 0 r 00 00 s = 1 in A , then s 1 = 1 = 1 in A . There exists a c ∈ S such that rc = 0, so 1r = 01 in R00 . This implies that rs = 01 in R00 . c

We define x00i := x1i ∈ A00 for every 1 ≤ i ≤ n. Let c00i,j := i,j 1 , then is invertible and x00j x00i − c00i,j x00i x00j ∈ R00 + R00 x00i + · · · + R00 x00n for every 1 ≤ i < j ≤ n. c00i,j

The endomorphisms σi of R and the σi -derivations δi that define A (see Proposition 1.1.3) induce endomorphisms σei of R00 and σei -derivations δei of R00 (see Theorem 4.1.3). Note that each σei is bijective. We claim that x00i rs = σei ( rs )x00i + δei ( rs ). Indeed, x00i

r xi r xi r σi (r)xi + δi (r) σi (r)xi δi (r) = = = = + . s 1 s s s s s

On the other hand,

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σei

r

 r  σ (r) x σi (r) δi (s) δi (r) i i x00i + δei = − + . s s σi (s) 1 σi (s) s s

Thus, we must prove that σi (r)xi σi (r) xi σi (r) δi (s) = − . s σi (s) 1 σi (s) s Applying the right Ore condition to σi (s) and xi we get that xi u = σi (s)c(x), with u ∈ S and c(x) ∈ A. As in part (a), c(x) = cxi + d, with c, d ∈ R, so σi (u) = σi (s)c and δi (u) = σi (s)d. Therefore, σi (r)(cxi + d) σi (r)cxi σi (r)d σi (r) xi = = + σi (s) 1 u u u and hence σi (r) xi σi (r) δi (s) σi (r)cxi σi (r)d σi (r) δi (s) − = + − , σi (s) 1 σi (s) s u u σi (s) s but u = sσi−1 (c), so σi (r)cxi σi (r) cxi σi (r) xi σi−1 (c) − δi (σi−1 (c)) = = u 1 u 1 sσi−1 (c) =

σi (r) xi σi−1 (c) σi (r) δi (σi−1 (c)) − 1 sσi−1 (c) 1 sσi−1 (c)

=

σi (r) xi σi (r)δi (σi−1 (c)) σi (r)xi σi (r)δi (σi−1 (c)) − = − . −1 1 s s sσi (c) sσi−1 (c)

Hence, the problem is reduced to proving the equality σi (r)d σi (r) δi (s) σi (r)δi (σi−1 (c)) − = u σi (s) s sσi−1 (c) or equivalently, to proving σi (r)d σi (r)δi (σi−1 (c)) σi (r) δi (s) − = . u u σi (s) s Note that δi (u) = σi (s)δi (σi−1 (c)) + δi (s)σi−1 (c) = σi (s)d, i.e., δi (s)σi−1 (c) = σi (s)(d − δi (σi−1 (c))). But this relation indicates that σi (r) δi (s) σi (r)(d − δi (σi−1 (c))) σi (r)d σi (r)δi (σi−1 (c)) = = − . −1 σi (s) s u u sσi (c) This proves the claim. Thus, given rs ∈ R00 −{0} there exists a c00i, r := σei ( rs ) ∈ s R00 − {0} such that x00i rs − c00i, r x00i ∈ R00 . s

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Now we will show that A00 is a free left R00 -module with basis Mon{x001 , . . . , x00n }. First note that A00 is generated by Mon{x001 , . . . , x00n }. In α1 αt ) tx fact, let z ∈ A00 , then z has the form z = (c1 x +···+c , with ci ∈ R, s αi x ∈ Mon{x1 , . . . , xn }, 1 ≤ i ≤ t, and s ∈ S. It is enough to show α that each summand cxs is generated by Mon{x001 , . . . , x00n }. But observe that α α cx cx 1 c 00α 1 s = 1 1 s = 1x s and, as in the proof of part (a) of Theorem 1.1.8, 00α 1 α 1 x s = (e σ ) ( s )x00α + pα, 1s , where pα, 1s is a left linear combination of elements of Mon{x001 , . . . , x00n } with coefficients in R00 (observe that in that proof we did not use the condition we are proving, there we only used the inversion with the σ’s). Thus, A00 is left generated over R00 by Mon{x001 , . . . , x00n }. Now let sr11 , . . . , srtt ∈ R00 and x00α1 , . . . , x00αt ∈ Mon{x001 , . . . , x00n } such that r1 00α1 + · · · + srtt x00αt = 0. Taking a common denominator, and without loss s1 x α1 αt of generality, we can write rs1 x1 + · · · + rst x1 = 0, with s ∈ S; moreover, we can assume that xα1  xα2  · · ·  xαt . There exist ui ∈ S and gi ∈ A such that xαi ui = sgi , 1 ≤ i ≤ t, so ru1 g11 + · · · + rut gtt = 0. Note that every gi 6= 0. Applying repeatedly the Ore condition we find elements ai ∈ R such that ui ai = u ∈ S and r1 gu1 a1 + · · · + rt gut at = 0. From this we find w ∈ S such that r1 g1 a1 w + · · · + rt gt at w = 0. Let gi = ci xβi + gi0 , with lt(gi ) = ci xβi 6= 0 and gi0 ∈ A. From xαi ui = sgi we get that σ αi (ui ) = sci and αi = βi . In particular, σ α1 (u1 ) = sc1 ; moreover, r1 c1 σ β1 (a1 w) = 0, then r1 c1 σ β1 (a1 )σ β1 (w) = 0. Thus, we have r1 (c1 σ β1 (a1 )σ β1 (w)) = 0 and s(c1 σ β1 (a1 )σ β1 (w)) = σ α1 (u1 )σ β1 (a1 )σ β1 (w) = σ α1 (u1 a1 )σ β1 (w) = σ α1 (u)σ β1 (w) ∈ S. This means that rs1 = 0. By induction on t we get that every rsi = 0. This completes the proof that AS −1 is a skew P BW extension of RS −1 . (c) follows from (a) and (b). t u

4.2 Ore’s Theorem This section deals with establishing sufficient conditions for a skew P BW extension A of a ring R to be a left (right) Ore domain, and hence, have a left (right) total division ring of fractions. A first elementary result in this direction is the following proposition. Proposition 4.2.1. If R is a left (right) noetherian domain and A is a bijective skew P BW extension of R, then A is a left (right) Ore domain, and hence, the left (right) division ring of fractions of A exists. Proof. It is well known that left (right) noetherian domains are left (right) Ore domains (see [278], Theorem 2.1.15). The result is consequence of Proposition 3.2.1 and Theorem 3.1.5. t u The purpose of the present section is to replace the noetherianity in proposition 4.2.1 by the Ore condition. A preliminary result is needed.

4.2 Ore’s Theorem

77

Proposition 4.2.2. Let B be a domain and S a multiplicative subset of B such that S −1 B exists. Then, B is a left Ore domain if and only if S −1 B is a left Ore domain. In such case Ql (B) ∼ = Ql (S −1 B). The right side version of the proposition holds too. Proof. (i) ⇒) Note first that S −1 B is a domain: Let as , bt ∈ S −1 B such that ab 0 cb 0 s t = 1 . There exist u ∈ S and c ∈ B such that ua = ct and us = 1 . Hence, 0 0 0 0 0 there exist c , d ∈ B such that c cb = 0 and c us = d 1 ∈ S. Since B is a domain cb = 0, so b = 0 or c = 0, and in this last case we get that a = 0. Thus, as = 01 or bt = 01 . Let again as , bt ∈ S −1 B with bt 6= 10 , then b 6= 0 and there exist p = 6 0 and ps a pa qb qt b ps 0 q in B such that pa = qb. Then, 1 s = 1 = 1 = 1 t , with 1 6= 1 . ⇐) Let a, u ∈ B, u 6= 0, then a1 , u1 ∈ S −1 B, with u1 6= 01 . There exist c d du −1 A, with ct 6= 10 such that ct a1 = ds u1 , i.e., ca t, s ∈ S t = s . There exist 0 0 0 0 0 0 c , d ∈ B such that c ca = d du and c t = d s ∈ S. Note that c0 c 6= 0 since c0 6= 0 and c 6= 0. (ii) The function ϕ : S −1 B → Ql (B) b b 7→ s s satisfies the four conditions that define a left total ring of fractions, i.e., ϕ is an injective ring homomorphism, the nonzero elements of S −1 B are invertible in Ql (B) and each element ub of Ql (B) can be written as ub = ϕ( u1 )−1 ϕ( 1b ). t u Proposition 4.2.3. If R is a left Ore domain and σ is injective, then R[x; σ, δ] is a left Ore domain and Ql (R[x; σ, δ]) ∼ = Ql (Ql (R)[x; σ, δ]).

(4.2.1)

If R is a right Ore domain and σ is bijective, then R[x; σ, δ] is a right Ore domain and e Qr (R[x; σ, δ]) ∼ e, δ]). (4.2.2) = Qr (Qr (R)[x; σ If R is an Ore domain (left and right) and σ is bijective, then R[x; σ, δ] is an Ore domain and e Q(R[x; σ, δ]) ∼ e, δ]). = Q(Q(R)[x; σ, δ]) ∼ = Q(Q(R)[x; σ

(4.2.3)

If R is an integral domain and σ = iR , then δ = δe and Q(R[x; δ]) ∼ = Q(Q(R)[x; δ]).

(4.2.4)

Proof. The conditions in (a) of Theorem 4.1.3 are trivially satisfied for S := R − {0}. Thus, Ql (R)[x; σ, δ] is a well-defined skew polynomial ring

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over the division ring Ql (R) and we have the isomorphism S −1 (R[x; σ, δ]) ∼ = Ql (R)[x; σ, δ]. Note that σ is injective, and hence Ql (R)[x; σ, δ] is a left noetherian domain and therefore a left Ore domain. From this we get that S −1 (R[x; σ, δ]) is a left Ore domain. From Proposition 4.2.2, R[x; σ, δ] is a left Ore domain and Ql (R[x; σ, δ]) ∼ = Ql (S −1 (R[x; σ, δ])) ∼ = Ql (Ql (R)[x; σ, δ]). This proves (4.2.1). For the second statement note that if R is a right Ore domain, then the right skew polynomial ring is a right Ore domain. Therefore, Proposition 4.1.1 guarantees that if R is a right Ore domain, then R[x; σ, δ] is a right Ore domain; from Proposition 4.2.2 and (4.1.2) of Theorem 4.1.3 we get e The last two statements of the proposiQr (R[x; σ, δ]) ∼ e, δ]). = Qr (Qr (R)[x; σ tion are also consequences of Theorem 4.1.3 and Proposition 4.2.2. t u Corollary 4.2.4. Let R be a left Ore domain and A := R[x1 ; σ1 , δ1 ] · · · [xn ; σn , δn ], with σi injective for every 1 ≤ i ≤ n. Then, A is a left Ore domain and Ql (A) ∼ = Ql (Ql (R)[x1 ; σ1 , δ1 ] · · · [xn ; σn , δn ]). If R is a right Ore domain and σi is bijective for every 1 ≤ i ≤ n, then A is a right Ore domain and Qr (A) ∼ e, δe1 ] · · · [xn ; σ e, δen ]). = Qr (Qr (R)[x1 ; σ If R is an Ore domain (left and right) and σi is bijective for every 1 ≤ i ≤ n, then A is an Ore domain and Q(A) ∼ e, δe1 ] · · · [xn ; σ e, δen ]). = Q(Q(R)[x1 ; σ1 , δ1 ] · · · [xn ; σn , δn ]) ∼ = Q(Q(R)[x1 ; σ If R is an integral domain and σ = iR , then for every 1 ≤ i ≤ n, δi = δei and Q(A) ∼ = Q(Q(R)[x1 ; δ1 ] · · · [xn ; δn ]). Proof. The result follows from Proposition 4.2.3 by iteration.

t u

Theorem 4.2.5 (Ore’s theorem: quasi-commutative case). Let R be a left Ore domain and A := R[x1 ; σ1 ] · · · [xn ; σn ] be a quasi-commutative skew P BW extension of R. Then A is a left Ore domain, and hence, A has a left total division ring of fractions such that Ql (A) ∼ = Ql (Ql (R)[x1 ; σ1 ] · · · [xn ; σn ]). If R is a right Ore domain and σi is bijective for every 1 ≤ i ≤ n, then A is a right Ore domain and Qr (A) ∼ f1 ] · · · [xn ; σ fn ]). = Qr (Qr (R)[x1 ; σ If R is an Ore domain (left and right) and σi is bijective for every 1 ≤ i ≤ n, then A is an Ore domain and

4.3 Goldie’s Theorem

79

Q(A) ∼ f1 ] · · · [xn ; σ fn ]). = Q(Q(R)[x1 ; σ1 ] · · · [xn ; σn ]) ∼ = Q(Q(R)[x1 ; σ Proof. This follows from Corollary 4.2.4 since for any skew P BW extension, the endomorphisms σ’s are always injective, see Proposition 1.1.3. t u Now we consider the previous theorem for bijective extensions, extending in this way Theorem 4.2.1 to left (right) Ore domains. Theorem 4.2.6 (Ore’s theorem: bijective case). Let A = σ(R)hx1 , . . . , xn i be a bijective skew P BW extension of a left Ore domain R. Then A is also a left Ore domain, and hence, A has a left total division ring of fractions such that Ql (A) ∼ = Ql (σ(Ql (R))hx01 , . . . , x0n i). If R is a right Ore domain, then A is also a right Ore domain, and hence, A has a right total division ring of fractions such that Qr (A) ∼ = Qr (σ(Qr (R))hx001 , . . . , x00n i). If R is an Ore domain (left and right), then A is an Ore domain and Q(A) ∼ = Q(σ(Q(R))hx01 , . . . , x0n i) ∼ = Q(σ(Q(R))hx001 , . . . , x00n i). Proof. With S := R − {0} in Theorem 4.1.6, S −1 A = σ(Ql (R))hx01 , . . . , x0n i is a left Ore domain. In fact, we have that Ql (R) is a division ring, so from Proposition 3.2.1 and Theorem 3.1.5 we obtain that σ(Ql (R))hx01 , . . . , x0n i is a left noetherian domain, and hence, a left Ore domain. From Proposition 4.2.2 we get that A is a left Ore domain and Ql (A) ∼ = Ql (S −1 A) ∼ = 0 0 Ql (σ(Ql (R))hx1 , . . . , xn i). The proof for the right side is analogous. t u

4.3 Goldie’s Theorem Goldie’s theorem says that a ring B has semisimple left (right) total rings of fractions if and only if B is semiprime and left (right) Goldie. In particular, B has simple left (right) artinian left (right) total ring of fractions if and only if B is prime and left (right) Goldie (see [159]). In this section we study this classical result for skew P BW extensions. The first step is the following proposition. Proposition 4.3.1. Let R be a prime left (right) noetherian ring and let A be a bijective skew PBW extension of R. Then A has a left (right) total ring of fractions which is simple and left (right) artinian.

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Proof. By Theorem 3.1.5 we know that A is left (right) noetherian and hence left (right) Goldie. Now, Theorem 3.2.6 implies that A is a prime ring. The assertion follows from Goldie’s theorem. t u Next we want to extend the previous proposition to the case when the ring R of coefficients is semiprime and left (right) Goldie. Proposition 4.3.2. Let R be a semiprime left Goldie ring and let σ be injective. Then, R[x; σ, δ] is semiprime left Goldie, and hence, Ql (R[x; σ, δ]) exists and is semisimple. If R is right Goldie and σ is bijective, then R[x; σ, δ] is semiprime right Goldie, and hence, Qr (R[x; σ, δ]) exists and is semisimple. Proof. See [229], Theorem 3.8. For the second part we use also Proposition 4.1.1. t u Corollary 4.3.3. Let R be a semiprime left Goldie ring and σi injective for every 1 ≤ i ≤ n. Then, A := R[x1 ; σ1 , δ1 ] · · · [xn ; σn , δn ] is semiprime left Goldie, and hence, Ql (A) exists and is semisimple. If R is right Goldie and every σi is bijective, then A is semiprime right Goldie, and hence, Qr (A) exists and is semisimple. Proof. Direct consequence of the previous proposition by iteration.

t u

Theorem 4.3.4 (Goldie’s theorem: quasi-commutative case). Let R be a semiprime left Goldie ring and A a quasi-commutative skew P BW extension of R. Then, A is semiprime left Goldie, and hence, Ql (A) exists and is semisimple. If R is right Goldie and every σi is bijective, then A is semiprime right Goldie, and hence, Qr (A) exists and is semisimple. Proof. This follows from Theorem 3.1.4 and the previous corollary.

t u

Next we consider Goldie’s theorem for bijective extensions. Some preliminaries are needed. Recall that an element x of a ring B is left regular if bx = 0 implies that b = 0 for b ∈ B. We start by considering rings for which the set of left regular elements coincides with the set of regular elements. The set of regular elements of B will be denoted by S0 (B). One remarkable subclass of this class of rings is the semiprime left Goldie rings (see [278], Proposition 2.3.4). Proposition 4.3.5. Let B be a semiprime left Goldie ring. Then the set of left regular elements coincides with S0 (B). A similar statement is true for the right side. Proof. Let x be a left regular element of B, then Bx ≤ B and B ∼ = Bx as left B-modules, so udim(Bx) = udim(B B), and hence, Bx ≤e B, whence x is regular. t u Proposition 4.3.6. Let B be a ring and S ⊆ S0 (B) a multiplicative system of B such that S −1 B exists. Suppose that any left regular element of S −1 B is regular, then the same holds for B. The right side version of the proposition is also true.

4.3 Goldie’s Theorem

81

Proof. Let a ∈ B be a left regular element, and let b ∈ B such that ab = 0. 0 Then a1 is a left regular element of S −1 B. In fact, if uc a1 = 01 , then ca u = 1 , i.e., ca 1 = 0, but since S has no zero divisors, we have ca = 0. This implies that c = 0, i.e., uc = 0. Now, from ab = 0 we get a1 1b = 0, and by the hypothesis, b t u 1 = 0, i.e., b = 0. The following property is a generalization of Proposition 4.2.2. Proposition 4.3.7. Let B be a ring such that the set of left regular elements coincides with S0 (B). Let S ⊆ S0 (B) be a multiplicative system of B such that S −1 B exists. Then, (i) Ql (B) exists if and only if Ql (S −1 B) exists. In this case, Ql (B) ∼ = Ql (S −1 B). (ii) B is semiprime left Goldie if and only if S −1 B is semiprime left Goldie. The right side version of the proposition holds. Proof. (i) ⇒) Let as ∈ S −1 B and bt ∈ S0 (S −1 B). Note that b ∈ S0 (B). In fact, if bc = 0 for some c ∈ B, then bt 1c = 01 and hence 1c = 01 . Since S has no zero divisors, c = 0. On the other hand, if db = 0 for some d ∈ B, then dt b db 0 dt 0 1 t = 1 = 1 . This implies that 1 = 1 , and hence, d = 0. By the hypothesis, there exist z ∈ S0 (B) and z 0 ∈ B such that za = z 0 b. a z0 t b From this we obtain zs 1 s = 1 t , but observe that zs ∈ S0 (B) and hence, zs −1 B). In fact, we will show that if u ∈ S0 (B), then u1 ∈ S0 (S −1 B). 1 ∈ S0 (S p pu 0 0 −1 Let v ∈ S B such that vp u1 = 01 , then pu v = 1 , so 1 = 1 and hence p = 0, p q 0 u q 0 −1 i.e., v = 1 . Now, let w ∈ S B be such that 1 w = 1 . There exist v ∈ S and 0 x ∈ B such that vu = xw and xq u = 1 , i.e., xq = 0. Note that x is left regular since vu is regular, then by the hypothesis x is regular, and hence, q = 0, i.e., q 0 w = 1. This proves that Ql (S −1 B) exists. ⇐) Let a ∈ B and u ∈ S0 (B), then a1 , u1 ∈ S −1 B and, as above, u1 ∈ 0 0 S0 (S −1 B). By the hypothesis, there exist zs , zs0 ∈ S −1 B with zs0 ∈ S0 (S −1 B) 0 0 0 such that zs0 a1 = zs u1 , i.e., zs0a = zu s , so there exist c, d ∈ B such that cz a = dzu and cs0 = ds ∈ S. In order to complete the proof of the left Ore condition z0 0 we have to show that cz 0 ∈ S0 (B). If xcz 0 = 0 for some x ∈ B, then xc 1 1 = 1, 0 0 0 0 0 s 1 z 0 xcs z 0 xcs 0 0 i.e., xc 1 1 s0 1 = 1 , so 1 s0 = 1 , and hence 1 = 1 . This means xcs = 0, 0 0 p 0 0 p so x = 0. Now, if cz 0 p = 0 for some p ∈ B, then 1c s1 s10 z1 1 = 10 = cs1 zs0 1 , but 0 0 since cs0 ∈ S we get that cs1 ∈ S0 (S −1 B), and hence zs0 p1 = 01 , and from this we obtain p1 = 01 , i.e., p = 0. This proves that Ql (B) exists. The function ϕ : S −1 B → Ql (B) b b 7→ s s

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satisfies the four conditions that define a left total ring of fractions, i.e., (a) ϕ is a ring homomorphism. (b) ϕ(S0 (S −1 B)) ⊆ Ql (B)∗ : In fact, let bt ∈ S0 (S −1 B), then as we observed at the beginning of the proof, b ∈ S0 (B), and hence, ϕ( bt ) = bt is invertible in Ql (B) with inverse bt . (c) sb ∈ ker(ϕ) if and only if d1 sb = 01 with d1 ∈ S0 (S −1 B): in fact, if sb ∈ ker(ϕ), then there exist c, d ∈ B such that cb = 0 and cs = d, with d ∈ S0 (B), so d1 sb = 01 and, as above, d1 ∈ S0 (S −1 B). (d) each element ub of Ql (B) can be written as ub = ϕ( u1 )−1 ϕ( 1b ). (ii) This is a direct consequence of (i) and Goldie’s theorem. t u Proposition 4.3.8. Let B be a positively filtered ring. If Gr(B) is semiprime, then B is semiprime. Proof. Let I be a two-sided ideal of B such that I 2 = 0. Then, Gr(I)Gr(I) ⊆ Gr(I 2 ) = 0 and hence Gr(I) = 0. This implies that I = 0. t u Theorem 4.3.9 (Goldie’s theorem: bijective case). Let R be a semiprime left Goldie ring and A = σ(R)hx1 , . . . , xn i a bijective skew PBW extension of R. Then, A is semiprime left Goldie, and hence, Ql (A) exists and is semisimple. The right side version of the theorem also holds. Proof. By Goldie’s theorem, Ql (R) = S0 (R)−1 R exists and is semisimple. Note that for every 1 ≤ i ≤ n, σi (S0 (R)) = S0 (R). By Theorem 4.1.6, S0 (R)−1 A exists and is a bijective extension of Ql (R), i.e., S0 (R)−1 A = σ(Ql (R))hx01 , . . . , x0n i. Since Ql (R) is left noetherian, then by Theorem 3.1.5, S0 (R)−1 A is left noetherian, i.e, left Goldie. By Theorem 3.1.2, Gr(S0 (R)−1 A) = Gr(σ(Ql (R))hx01 , . . . , x0n i) is a quasi-commutative (and bijective) extension of the semiprime left Goldie ring Ql (R), so by Theorem 4.3.4, Gr(S0 (R)−1 A) is semiprime (left Goldie). Proposition 4.3.8 says that S0 (R)−1 A is semiprime. In order to apply Proposition 4.3.7 and conclude the proof it only remains to observe that S0 (R) ⊆ S0 (A) and the left regular elements of A coincide with S0 (A). The last statement can be justified in the following way: since S0 (R)−1 A is semiprime left Goldie, the left regular elements of S0 (R)−1 A coincide with its regular elements, so by Proposition 4.3.6 the same is true for A. t u For a complementary result we need a preliminary property. Proposition 4.3.10. Let B be a positively filtered ring. If Gr(B) is left (right) noetherian and has a left (right) total ring of fractions which is left (right) artinian, then so does B. Proof. See [190], Theorem 2.2.

t u

Theorem 4.3.11. Let R be a left (right) noetherian ring. If A is a bijective skew P BW extension of R such that Gr(A) has a left (right) total ring of fractions which is left (right) artinian, then so does A.

4.4 Skew Quantum Polynomials

83

Proof. We know that A is positively filtered; Theorem 3.1.5 guarantees that Gr(A) is left (right) noetherian. The result follows from Proposition 4.3.10. t u Remark 4.3.12. (i) Theorem 4.3.11 generalizes [190], Corollary 2.3, where the result is established for skew polynomial rings of automorphism type. (ii) The noetherian condition in Proposition 4.3.10 is essential and cannot be replaced by either “commutative” or “semiprime.” For some examples that illustrate this situation see [190], p. 234.

4.4 Skew Quantum Polynomials In this section we present the algebra of quantum polynomials (see [20]) and its generalization that we have called the ring of skew quantum polynomials. This new class of rings can be defined as a quasi-commutative bijective skew P BW extension of the r-multiparameter quantum torus, or also, as a localization of a quasi-commutative bijective skew P BW extension. For this purpose we present first the skew Laurent polynomial ring (see [159], [278]). Example 4.4.1. Let R be a ring and σ be an automorphism of R, the skew Laurent polynomial ring over R, also called the skew Laurent extension of R, is a ring T that satisfies the following conditions: (i) R ⊆ T . (ii) There exists an x ∈ T ∗ such that (a) T is a free left R-module with basis {xk |k ∈ Z}. (b) xr = σ(r)x, for all r ∈ R. We prove first that T exists. In fact, let B := R[x; σ] be the skew polynomial ring over R, where σ is a given automorphism of R, then S := {xi |i ∈ N} is a multiplicative subset of B; note that if f ∈ B and xm ∈ S are such that f xm = 0, then f = 0 and hence xm f = 0; moreover, S satisfies the left Ore condition: in fact, given f := f0 + f1 x + · · · + fn xn ∈ B and xm ∈ S we have xm f = gxm , with g := σ m (f0 ) + σ m (f1 )x + · · · + σ m (fn )xn . Thus, S −1 B exists. In a similar way, and since σ is bijective, we can prove that BS −1 also exists, and hence, S −1 B ∼ = BS −1 . Note that BS −1 satisfies the above conditions (i)–(ii). Moreover, any ring T with these conditions is isomorphic to BS −1 : by (i) and (ii)(a), and using the universal property of R[x; σ], we have the ring homomorphism ι : R[x; σ] → T , ι(r) := r, ι(x) := x; by (ii), ι(S) ⊆ T ∗ , then by the universal property of BS −1 there exists a ring homomorphism γ : BS −1 → T defined by γ( p(x) ) := p(x)x−k . Condition (ii)(a) says that γ xk is bijective. We have proved that there exists only one ring, up to isomorphism, that satisfies (i)–(ii). This ring is denoted by R[x±1 ; σ], thus,

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4 Rings of Fractions

R[x±1 ; σ] := S −1 (R[x; σ]) ∼ = (R[x; σ])S −1 . If σ = iR , then R[x±1 ; σ] = R[x±1 ] is the usual Laurent polynomial ring over R, and if R = K is a field, then K[x±1 ] is the algebra of Laurent polynomials. Example 4.4.2. Let R be a ring with a fixed matrix of parameters q := [qij ] ∈ Mn (R), n ≥ 2, such that qii = 1 = qij qji = qji qij for every 1 ≤ i, j ≤ n, and suppose also given a system σ := [σ1 , . . . , σn ] of automorphisms of R. The quasi-commutative bijective skew P BW extension Rq,σ [x1 , . . . , xn ] defined by xj xi = qij xi xj , xi r = σi (r)xi , r ∈ R, 1 ≤ i < j ≤ n is called the n-multiparametric skew quantum space over R. Recall from Theorem 1.3.1 that the following identities hold for every r ∈ R and i < j < k: σj σi (r)qij = qij σi σj (r); σk (qij )qik σi (qjk ) = qjk σj (qik )qij . When all automorphisms of the extension are trivial, i.e., σi = iR , 1 ≤ i ≤ n, we write Rq [x1 , . . . , xn ] and call it the n-multiparametric quantum space over R. If R = K is a field, then Kq,σ [x1 , . . . , xn ] is simply called the n-multiparametric skew quantum space, and the particular case n = 2 is called the skew quantum plane; for trivial automorphisms we have the nmultiparametric quantum space and the quantum plane (see [159]). Note that the multiplicative analogue of the Weyl algebra in Example 1.1.5 coincides with the n-multiparametric quantum space. From Theorem 3.1.4 we conclude that Rq,σ [x1 , . . . , xn ] is isomorphic to R[z1 ; θ1 ] · · · [zn ; θn ], where   θ1 := σ1 ; θj : R[z1 ; θ1 ] · · · R[zj−1 ; θj−1 ] → R[z1 ; θ1 ] · · · R[zj−1 ; θj−1 ],   θj (zi ) := qij zi , 1 ≤ i < j ≤ n, θj (r) = σj (r), for r ∈ R. Thus,

Rq,σ [x1 , . . . , xn ] ∼ = R[z1 ; θ1 ][z2 ; θ2 ] · · · [zn ; θn ].

(4.4.1)

∼ R[z1 ][z2 ; θ2 ] · · · [zn ; θn ], and for the quantum In particular, Rq [x1 , . . . , xn ] = plane we have Q2q (K) := K[y][z; θ], where θ is an automorphism of K[y] defined by θ(k) := k and θ(y) := qy for some q 6= 0 in K, i.e., zy = qyz. Finally, and before introducing the skew quantum polynomials, we present the n-multiparametric skew quantum torus. Example 4.4.3. Let R be a ring with a fixed matrix of parameters q := [qij ] ∈ Mn (R), n ≥ 2, such that qii = 1 = qij qji = qji qij for every 1 ≤ i, j ≤ n, and suppose also given a system σ := [σ1 , . . . , σn ] of automorphisms of R. The n-multiparametric skew quantum torus over R is a ring T satisfying the following conditions:

4.4 Skew Quantum Polynomials

85

(i) R ⊆ T . (ii) There exist elements x1 , . . . , xn ∈ T ∗ that satisfy the following relations: xi r = σi (r)xi , xj xi = qij xi xj ,

r ∈ R, 1 ≤ i ≤ n, 1 ≤ i < j ≤ n.

(iii) T is a free left R-module with basis {xα |α ∈ Zn }. We will show that such a ring T exists, proving that it is the ring of fractions of B := Rq,σ [x1 , . . . , xn ] with respect to the multiplicative subset S := {rxα |r ∈ R∗ , xα ∈ Mon{x1 , . . . , xn }}. In fact, if f ∈ B and rxα ∈ S are such that f rxα = 0, then 0 = f rxα = f xα [(σ α )−1 (r)], so 0 = f xα since (σ α )−1 (r) ∈ R∗ , and hence, f = 0. From this we get that rxα f = 0. S satisfies the left (right) Ore condition: if f = c1 xβ1 + · · · + ct xβt , then grxα = xα f , where g := d1 xβ1 + · · · + dt xβt with βi −1 di := σ α (ci )cα,βi c−1 ), and cα,βi , cβi ,α are the elements of R that we βi ,α σ (r obtain when we apply Theorem 1.1.8 to B (for the right Ore condition g is defined in a similar way). This means that S −1 B exists (BS −1 also exists), and hence, S −1 B ∼ = BS −1 . It is easy to prove that BS −1 satisfies conditions (i)–(iii): in fact, R ,→ BS −1 , r 7→ 1r ; for 1 ≤ i ≤ n, the elements x1i ∈ (BS −1 )∗ x x satisfy (ii); note that (iii) holds since 1j ( x1i )−1 = σ−11(q ) ( x1i )−1 1j , with i < j, i

ij

and this implies that if c ∈ R, r ∈ R∗ and xβ , xα ∈ Mon{x1 , . . . , xn }, then cxβ d x β−α , for some d ∈ R (see Example 4.4.4 for more details). rxα = 1 ( 1 ) Finally, any ring T with these conditions is isomorphic to BS −1 : indeed, by the universal property of B (see Theorem 1.2.2), there exists a ring homomorphism h : B → T defined by h(xj ) := xj , h(r) := r, 1 ≤ j ≤ n, r ∈ R; note that h(S) ⊆ T ∗ , so by the universal property of the ring of fractions there exists a ring homomorphism e h : BS −1 → T defined by p(x1 ,...,xn ) α −1 e h( rxα ) := h(p(x1 , . . . , xn ))h(rx ) = p(x1 , . . . , xn )(xα )−1 r−1 . The condition (ii) implies that e h is bijective. We have proved that there exists only one ring, up to isomorphism, that ±1 satisfies (i)–(iii). This ring is denoted by Rq,σ [x±1 1 , . . . , xn ], thus, ±1 −1 Rq,σ [x±1 (Rq,σ [x1 , . . . , xn ]) ∼ = (Rq,σ [x1 , . . . , xn ])S −1 . 1 , . . . , xn ] := S

In particular, if K is a field, ±1 −1 Kq,σ [x±1 (Kq,σ [x1 , . . . , xn ]) ∼ = (Kq,σ [x1 , . . . , xn ])S −1 . 1 , . . . , xn ] = S ±1 When σi = iR for 1 ≤ i ≤ n, we write Rq [x±1 1 , . . . , xn ] and call it the n-multiparametric quantum torus over R. If R = K is a field, then ±1 Kq,σ [x±1 1 , . . . , xn ] is simply called the n-multiparametric skew quantum torus, and the particular case n = 2 is called the skew quantum torus; for trivial automorphisms we have the n-multiparametric quantum torus and the quantum torus (see [159]).

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4 Rings of Fractions

Example 4.4.4. Let R be a ring with a fixed matrix of parameters q := [qij ] ∈ Mn (R), n ≥ 2, such that qii = 1 = qij qji = qji qij for every 1 ≤ i, j ≤ n, and suppose also given a system σ := [σ1 , . . . , σn ] of automorphisms of R. The skew quantum polynomials over R is a ring T that satisfies the following conditions: (i) R ⊆ T . (ii) There exist elements x1 , . . . , xn ∈ T such that x1 , . . . , xk ∈ T ∗ for some 0 ≤ k ≤ n, xi r = σi (r)xi , r ∈ R, 1 ≤ i ≤ n, xj xi = qij xi xj , 1 ≤ i < j ≤ n. (iii) T is a free left R-module with basis αn 1 {xα 1 · · · xn |αi ∈ Z for 1 ≤ i ≤ k and αi ∈ N for k + 1 ≤ i ≤ n}. (4.4.2)

We will show that there exists only one ring, up to isomorphism, that satisfies ±1 (i)–(iii). This ring is denoted by Rq,σ [x±1 1 , . . . , xk , xk+1 , . . . , xn ]. The idea ±1 ±1 is to prove that Rq,σ [x1 , . . . , xk , xk+1 , . . . , xn ] is a localization of a skew P BW extension. In fact, we have the quasi-commutative bijective skew P BW extension A := σ(R)hx1 , . . . , xn i, with xi r = σi (r)xi , r ∈ R, 1 ≤ i ≤ n, and xj xi = qij xi xj , 1 ≤ i < j ≤ n; (4.4.3) recall that the identities in Example 4.4.2 must be satisfied. If we set S := {rxα |r ∈ R∗ , xα ∈ Mon{x1 , . . . , xk }}, then S is a multiplicative subset of A and we will show that ±1 −1 Rq,σ [x±1 A∼ = AS −1 . 1 , . . . , xk , xk+1 , . . . , xn ] := S

(4.4.4)

This can be proved by repeating the reasoning used in Example 4.4.3. If f ∈ A and rxα ∈ S are such that f rxα = 0, then 0 = f rxα = f xα [(σ α )−1 (r)], so 0 = f xα since (σ α )−1 (r) ∈ R∗ , and hence, f = 0. From this we get that rxα f = 0. S satisfies the left (right) Ore condition: if f = c1 xβ1 +· · ·+ct xβt , then grxα = βi −1 xα f , where g := d1 xβ1 + · · · + dt xβt with di := σ α (ci )cα,βi c−1 ), and βi ,α σ (r cα,βi , cβi ,α are the elements of R that we obtain when we apply Theorem 1.1.8 to A (for the right Ore condition g is defined in a similar way). This means that S −1 A exists (AS −1 also exists, and hence, S −1 A ∼ = AS −1 ). −1 As in Example 4.4.3, AS satisfies conditions (i)–(iii): indeed, R ,→ AS −1 , r 7→ 1r ; for 1 ≤ i ≤ n, the elements x1i satisfy (ii). For (iii) we 1 +···+ct Xt have: any element of AS −1 has the form c1 X , where cl ∈ R, α α rx 1 ···x k 1

k

Xl ∈ Mon{x1 , . . . , xn }, 1 ≤ l ≤ t, r ∈ R∗ and αi ∈ N, 1 ≤ i ≤ k, so consider an element of the form rxα1cX := α , with c ∈ R and X ···x k 1

β

k+1 xβ1 1 · · · xβkk xk+1 · · · xβnn ∈ Mon{x1 , . . . , xn }:

k

4.4 Skew Quantum Polynomials

87

 x β n cX cxβ1 1 · · · xβnn 1 c  x 1 β 1 1 n = = · · · αk αk α1 α1 α1 k rx1 · · · xk 1 rx1 · · · xk 1 1 1 rx1 · · · xα k     c x1 β1 xn βn 1 1 ··· , = α1 k −α 1 1 1 σ (r) x1 · · · xα k with α := (α1 , . . . , αk ), but exists a c0 ∈ R such that

1 σ −α (r)

=

r0 1,

with r0 := [σ −α (r)]−1 ∈ R∗ , so there

 x β n cX c 0  x 1 β 1 1 n ··· , αk = 1 k · · · xk 1 1 1 xα · · · xα 1 k

1 rxα 1

moreover, there exists a c00 ∈ R∗ such that  α 1  α k  x −αk c00 1 c00  x1 −α1 1 1 k = · · · = ··· , αk α1 x1 · · · xk 1 x1 xk 1 1 1 but we have the identity  x −1 x xj  xi −1 1 i j = −1 , with i < j, 1 ≤ i ≤ k, 1 1 1 σi (qij ) 1 therefore there exists d ∈ R and γ ∈ Zn such that d  x γ cX , with γi ∈ N, for k + 1 ≤ i ≤ n. αk = · · · xk 1 1

1 rxα 1

From this we obtain (iii). Finally, any ring T satisfying conditions (i)–(iii) is isomorphic to AS −1 : indeed, by the universal property of A (see Theorem 1.2.2), there exists a ring homomorphism h : A → T defined by h(xj ) := xj , h(r) := r, 1 ≤ j ≤ n, r ∈ R; note that h(S) ⊆ T ∗ , so by the universal property of the ring of fractions there exists a ring homomorphism e h : AS −1 → T defined by   p(x1 , . . . , xn ) αk −1 1 e h : = h(p(x1 , . . . , xn ))h(rxα αk 1 · · · xk ) 1 rxα 1 · · · xk αk −1 −1 1 = p(x1 , . . . , xn )(xα r . 1 · · · xk )

Condition (ii) implies that e h is bijective. When all automorphisms are trivial, we write ±1 Rq [x±1 1 , . . . , xk , xk+1 , . . . , xn ],

and this ring is called the ring of quantum polynomials over R, moreover, if R = K is a field, then we get the algebra of quantum polynomials, denoted by Oq (see [20]).

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Remark 4.4.5. When k = 0, ±1 Rq,σ [x±1 1 , . . . , xk , xk+1 , . . . , xn ] = Rq,σ [x1 , . . . , xn ]

is the n-multiparametric skew quantum space over R, and when k = n, ±1 it coincides with Rq,σ [x±1 1 , . . . , xn ], i.e., with the n-multiparametric skew ±1 ±1 quantum torus over R. In this case, if n = 1, Rq,σ [x±1 ; σ], 1 , . . . , xn ] = R[x i.e., this ring coincides with the skew Laurent polynomial ring over R (see Example 4.4.1). From this we conclude that the ring of skew quantum polynomials over R generalizes the rings considered in Examples 4.4.1, 4.4.2 and 4.4.3. In the next example we will summarize some algebraic basic properties of the skew quantum polynomial ring. Example 4.4.6. (i) If R is a left (right) noetherian ring, then ±1 ±1 Qk,n q,σ (R) := Rq,σ [x1 , . . . , xk , xk+1 , . . . , xn ]

is left (right) noetherian: this is a consequence of Theorem 3.1.5 and the behavior of the noetherianity under the ring of fractions ([47]; see also [241]). (ii) If R is a domain, then Qk,n q,σ (R) is also a domain (see Proposition 3.2.1, (4.4.4) and the fact that any localization of a domain is a domain). (iii) If R is prime, then Qk,n q,σ (R) is also prime. This follows from Theorem 3.2.6 and the fact that any localization of a prime ring is a prime ring. Thus, rad(Qk,n q,σ (R)) = 0. (iv) If R is a domain, then Rad(Qk,n q,σ (R)) = 0 when 1 ≤ k ≤ n − 1. In k,n fact, Qq,σ (R) can be interpreted also as a quasi-commutative bijective skew P BW extension of the skew quantum torus, namely, ±1 ±1 Qk,n q,σ (R) = σ(Rq0 ,σ 0 [x1 , . . . , xk ])hxk+1 , . . . , xn i,

(4.4.5)

with q0 ∈ Mk (R), σi0 := σi , 1 ≤ i ≤ k. For this, observe that A in (4.4.3) can be viewed as a quasi-commutative bijective extension of the kmultiparametric skew quantum space over R, in other words, A = σ(Rq0 ,σ0 [x1 , . . . , xk ])hxk+1 , . . . , xn i, moreover, since σj (S) = S for k + 1 ≤ j ≤ n, then we can apply Theorem 4.1.6, and we get −1 Qk,n A = σ(S −1 (Rq0 ,σ0 [x1 , . . . , xk ]))hxk+1 , . . . , xn i q,σ (R) = S ±1 = σ(Rq0 ,σ0 [x±1 1 , . . . , xk ])hxk+1 , . . . , xn i.

For this interpretation, using the identities of Example 4.4.2, we have extended σj by σj (xi ) := qij xi for k + 1 ≤ j ≤ n. Therefore, since ±1 Rq0 ,σ0 [x±1 1 , . . . , xk ] is a domain, then from Theorem 3.2.3 we obtain that the Jacobson radical of Qk,n q,σ (R) is trivial.

4.5 The Gelfand–Kirillov Conjecture

89

For k = 0, Qk,n q,σ (R) = Rq,σ [x1 , . . . , xn ], so by Theorem 3.2.3 the radical is trivial. For k = n, the n-multiparameter skew quantum torus ±1 Rq,σ [x±1 1 , . . . , xn ]

also has trivial radical: this follows from [287] by induction on n since every domain is skew-Armendariz and as before, using the identities of Example 4.4.2, we can extend every σj by σj (xi ) := qij xi for 1 ≤ i < j ≤ n. (v) If R is a left (right) Ore domain, then Qk,n q,σ (R) is a left (right) Ore domain: this follows from Proposition 4.2.2. (vi) Let R be a semiprime left (right) Goldie ring, from Theorem 4.3.4 (we can also use Theorem 4.3.9) we get that A in (4.4.3) is a semiprime left (right) Goldie ring. In addition, note that the set S in (4.4.4) satisfies the hypothesis of Proposition 4.3.7: in fact, since A is semiprime left (right) Goldie, any left (right) regular element is regular; S ⊆ S0 (A) since if rxα ∈ S and p = c1 xβ1 + · · · + ct xβt ∈ A are such that rxα p = 0 or prxα = 0, then p = 0 since r and the constants cα,β are invertible. Proposition 4.3.7 says that Qk,n q,σ (R) is semiprime left (right) Goldie.

4.5 The Gelfand–Kirillov Conjecture The description of rings of fractions of skew P BW extensions considered in the previous sections was also motivated by the famous Gelfand–Kirillov conjecture ([144]). In this section we will study this problem for bijective skew P BW extensions. We start by recalling the conjecture and some well-known cases, classical and quantum, where the conjecture has positive answer. In what follows, Z(R) will denote the center of the ring R. Conjecture 4.5.1 (Gelfand–Kirillov, [144]). Let G be an algebraic Lie algebra of finite dimension over a field L, with char(L) = 0. Then, there exist integers n, k ≥ 1 such that Q(U (G)) ∼ = Q(An (L[s1 , . . . , sk ])).

(4.5.1)

Recall that G is algebraic if G is the Lie algebra of a linear affine algebraic group. A group G is linear affine algebraic if G is an affine algebraic variety such that the multiplication and the inversion in G are morphisms of affine algebraic varieties. Example 4.5.2 ([144], Lemma 7). Let G be the algebra of all n × n matrices over a field L, i.e., G = Mn (L), with char(L) = 0. Then, G is algebraic and (4.5.1) holds. The same is true if G is the algebra of matrices of null trace. Example 4.5.3 ([144], Lemma 8; see also [298]). Let G be a finite-dimensional nilpotent Lie algebra over a field L, with char(L) = 0. Then, G is algebraic and (4.5.1) holds. Moreover, Q(Z(U (G))) ∼ = Z(Q(U(G))).

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4 Rings of Fractions

Example 4.5.4 ([193], Theorem 3.2). Let G be a finite-dimensional solvable algebraic Lie algebra over the field C of complex numbers. Then, G satisfies the conjecture (4.5.1). Remark 4.5.5. (i) Other cases of algebraic Lie algebras for which the conjecture holds are discussed in several papers. For example, in [13] it has been proved that if G is an algebraic Lie algebra over an algebraically closed field L of characteristic zero with dim(G) ≤ 8, then G satisfies the Gelfand–Kirillov conjecture. Other works that should be mentioned are [65], [128], [194] and [299]. (ii) There are examples of non-algebraic Lie algebras for which the conjecture is false. However, other examples show that the conjecture holds for some non-algebraic Lie algebras (see [144], Section 8). We are interested in the so-called quantum version of the Gelfand–Kirillov conjecture, in this case the Weyl algebra An (L[s1 , . . . , sk ]) in (4.5.1) is replaced by a suitable n-multiparametric quantum space as is shown the following examples. Example 4.5.6 ([80]). Let K be a field and A := K[x1 ][x2 ; σ2 , δ2 ] · · · [xn ; σn , δn ] be an iterated skew polynomial ring with some extra adequate conditions on σ’s and δ’s. Then there exists a q := [qij ] ∈ Mn (K) with qii = 1 = qij qji , for every 1 ≤ i, j ≤ n, such that Q(A) ∼ = Q(Kq [x1 , . . . , xn ]). Example 4.5.7 ([11], Theorem 3.5). Let AQ,Γ n (K) be the multiparameter quantized Weyl algebra (see Section 2.5); in particular, consider the case when there exists a parameter q ∈ K ∗ that is not a root of unity, such that every parameter in Q = [q1 , . . . , qn ] and Γ = [γij ] is a power of q, and in addition, qi 6= 1, 1 ≤ i ≤ n. Under these conditions, there exists a q := [qij ] ∈ M2n (K) with qii = 1 = qij qji , and qij is a power of q, 1 ≤ i, j ≤ n, such that ∼ Q(AQ,Γ n (K)) = Q(Kq [x1 , . . . , x2n ]), Z(Q(Kq [x1 , . . . , x2n ])) = K. Example 4.5.8 ([11], Theorem 2.15). Let Uq+ (slm ) be the quantum enveloping algebra of the Lie algebra of strictly superior triangular matrices of size m×m over a field K, where m ≥ 3 and q ∈ K − {0} is not a root of unity. (i) If m = 2n + 1, then Q(Uq+ (slm )) ∼ = Q(Kq [x1 , . . . , x2n2 ]), where K := Q(Z(Uq+ (slm ))) and q := [qij ] ∈ M2n2 (K), with qii = 1 = qij qji , and qij is a power of q for every 1 ≤ i, j ≤ 2n2 . (ii) If m = 2n, then Q(Uq+ (slm )) ∼ = Q(Kq [x1 , . . . , x2n(n−1) ]),

4.6 The Center of the Total Division Ring of Fractions

91

where K := Q(Z(Uq+ (slm ))) and q := [qij ] ∈ M2n(n−1) (K), with qii = 1 = qij qji , and qij is a power of q for every 1 ≤ i, j ≤ 2n(n − 1). Moreover, in both cases Q(Z(Uq+ (slm ))) ∼ = Z(Q(Uq+ (slm ))). Our first result in this section is the next theorem on the Gelfand–Kirillov conjecture for the skew quantum polynomials (see Example 4.4.6). Theorem 4.5.9. If R is an Ore domain (left and right), then ∼ Q(Qk,n q,σ (R)) = Q(Qq,σ [x1 , . . . , xn ]), with Q := Q(R).

(4.5.2)

Proof. From Proposition 4.2.2 and Theorem 4.2.5 (or also using Theorem 4.2.6) if R is an Ore domain, then Qk,n q,σ (R) is an Ore domain, and hence ∼ Qk,n (R) has total division ring of fractions Q(Qk,n q,σ q,σ (R)) = Q(A), with A as in (4.4.3). Then, with the notation of the previous sections, we have 0 0 ∼ ∼ ∼ Q(Qk,n q,σ (R)) = Q(A) = Q(σ(Q(R))hx1 , . . . , xn i) = Q(Qq,σ [x1 , . . . , xn ]),

where Q := Q(R) and we define xi := x0i = 1 ≤ i ≤ n.

xi 1

and σi := σi : Q → Q, t u

We have proved that the total ring of fractions of Qk,n q,σ (R) is the total ring of fractions of the n-multiparametric skew quantum space over Q(R), thus, in this case, the n-multiparametric quantum space was replaced by the n-multiparametric skew quantum space. Corollary 4.5.10. Let A := σ(R)hx1 , . . . , xn i be a skew P BW extension. Then, (i) Gr(A) = Rq,σ [x1 , . . . , xn ], where the system of constants q := [cij ] and endomorphisms σ := [σ1 , . . . , σn ] are as in Definition 1.2.1. (ii) If R is an Ore domain, then Q(Gr(A)) ∼ = Q(Qq,σ [x1 , . . . , xn ]), with Q := Q(R). Proof. (i) This follows from (3.1.2) and Example 4.4.2. (ii) This follows from (i) and (4.5.2).

t u

Problem 4.5.11. Study the Gelfand–Kirillov conjecture for bijective skew P BW extensions over Ore domains.

4.6 The Center of the Total Division Ring of Fractions Given an Ore domain A, it is interesting to know when Z(Q(A)) ∼ = Q(Z(A)), where Z(A) is the center of A and Q(A) is the total ring of fractions of A.

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This question became important after the formulation of the Gelfand–Kirillov conjecture in [144]. In this section we study the isomorphism Z(Qr (A)) ∼ = Q(Z(A)), where A is a right Ore domain, Z(A) is the center of A and Qr (A) is the right total ring of fractions of A. The main tool that we will use is the classical Gelfand–Kirillov dimension over fields, so we will assume that A is a K-algebra, where K is an arbitrary field. The principal result is Theorem 4.6.3. This result can also be interpreted as a way of computing the center of Qr (A). Many examples that illustrate the theorem are included. We start with an easy proposition. We include the proof for completeness. Proposition 4.6.1. Let A be a right Ore domain. (i) If

p q

∈ Z(Qr (A)) then

(a) pq = qp. (b) For every s ∈ A, psq = qsp. (c) p ∈ Z(A) if and only if q ∈ Z(A). (ii) Let p ∈ A. Then, p1 ∈ Z(Qr (A)) if and only if p ∈ Z(A). Thus, Z(A) ,→ Z(Qr (A)). (iii) If K is a field and A is a K-algebra such that Z(Qr (A)) = K, then Q(Z(A)) = K = Z(Qr (A)). pq qp q pq qp Proof. (i) (a) We have pq 1q = 1q pq , so p1 = qp q , whence 1 1 = q 1 , i.e., 1 = 1 , thus pq = qp. (b) For s = 0 is clear. Let s ∈ A−{0}, then pq = ps qs , so by (a), psqs = qsps, whence psq = qsp. (c) If pq = 0, then pq = 01 and the claimed trivially holds. We can assume that p 6= 0; by (b), for every s ∈ A − {0}, psq = qsp, hence if p ∈ Z(A), then sqp = qsp, whence sq = qs, i.e., q ∈ Z(A). On the other hand, since q p ∈ Z(Qr (A)), then if q ∈ Z(A) we get p ∈ Z(A). (ii) If p1 ∈ Z(Qr (A)), then by (i), ps = sp for every s 6= 0, whence p ∈ Z(A). Conversely, let p ∈ Z(A) − {0} (for p = 0, p1 ∈ Z(Qr (A))), then for every pa pa acp a ap ac s ∈ Qr (A) we have 1 s = s and s 1 = r = rp , where sc = pr = rp, with acp apc ap pa c, r = 6 0. From this we get rp = sc = s = s , i.e., p1 ∈ Z(Qr (A)). (iii) From (ii), K ⊆ Z(A) ⊆ Z(Qr (A)) = K, so Z(A) = K, and hence Q(Z(A)) = K = Z(Qr (A)). t u

The next example illustrates part (iii) of Proposition 4.6.1. Example 4.6.2. We consider the quantum plane A := Kq [x, y], where q is not a root of unity. We will show that Z(Qr (A)) = K. Let ps ∈ Z(Qr (A)) − {0}, Pt Pl θj γ j i βi where p := i=1 ri xα 1 x2 and s := j=1 uj x1 x2 , with ri , uj ∈ K − {0}. From px1 s = sx1 p and since q is not a root of unity, we get βi + βi θj = γj + γj αi for every 1 ≤ i ≤ t and 1 ≤ j ≤ l. Similarly, from px2 s = sx2 p we obtain θj βi + θj = αi γj + αi for all i, j, whence βi + αi = γj + θj , so fixing i and then fixing j we conclude that p and s are homogeneous of the same / Z(Qr (A))). Now, degree (this condition is not enough since xx12 ∈

93

4.6 The Center of the Total Division Ring of Fractions

Pt Pt i +1 βi ri x α i x β i ri x α x2 p x1 x1 p 1 = , i.e., Pl i=1 θ1j −12 γj = Pi=1 , θ γ l j s 1 1 s uj x1 x2 uj x1 x2 j j=1

j=1

k hence there exist c := xm 1 pm (x2 ) + · · · + p0 (x2 ), d := x1 qk (x2 ) + · · · + q0 (x2 ) ∈ A − {0} such that ! ! t t X X αi βi αi +1 βi ri x 1 x 2 c = ri x 1 x2 d, i=1



l X

 j=1

i=1



  l X θ −1 γ θ γ uj x1j x2j  c =  uj x1j x2j  d. j=1

Since p and q are homogeneous, we can assume α1 > · · · > αt and θ1 > · · · > θl , whence β1 < · · · < βt and γ1 < · · · < γl . Then, m mβ1 α1 +m β1 1 β1 (r1 xα x1 x2 pm (x2 ), 1 x2 )(x1 pm (x2 )) = r1 q 1 +1 β1 k 1 +1+k β1 r1 x α x2 x1 qk (x2 ) = r1 q kβ1 xα x2 qk (x2 ), 1 1

whence α1 + m = α1 + 1 + k, i.e., m = k + 1. Moreover, let pm be the leading coefficient of pm (x2 ) and qk be the leading coefficient of qk (x2 ), then q β1 pm = qk . Similarly, we can prove that q γ1 pm = qk , but since q is not a root of unity, β1 = γ1 . From α1 + β1 = θ1 + γ1 we get that α1 = θ1 (considering instead the identity ps x12 = x12 ps we obtain the same result). Thus, we have α1 = θ1 and β1 = γ1 . Notice that p p − r1 u−1 1 s = r1 u−1 , with r1 u−1 1 + 1 ∈ K ⊆ Z(Qr (A)), s s p−r u−1 s

p−r u−1 s

1 1 1 1 hence ∈ Z(Qr (A)). But observe that = 0. In fact, if not, s s we could repeat the previous procedure and find that there exists either i ≥ 2 such that αi = θ1 = α1 , βi = γ1 = β1 , or j ≥ 2 such that θj = θ1 , γj = γ1 , a contradiction. Thus, ps = r1 u−1 1 ∈ K, and hence, Z(Qr (A)) = K.

∼ The previous example shows that the proof of the isomorphism Z(Qr (A)) = Q(Z(A)) by direct computation of the center of the total ring of fractions is tedious. An alternative more practical method is given by the following theorem. Theorem 4.6.3. Let K be a field and A be a right Ore domain. If A is a finitely generated K-algebra such that GKdim(A) < GKdim(Z(A)) + 1, then   p Z(Qr (A)) = | p, q ∈ Z(A), q 6= 0 ∼ = Q(Z(A)). q Proof. We divide the proof into three steps.

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Step 1. As in the proof of Theorem 4.12 in [210], we will show that   p −1 Qr (A) ∼ A(Z(A) ) = | p ∈ A, q ∈ Z(A) = 0 0 , q with Z(A)0 := Z(A) − {0}. First observe that Z(A)0 is a right Ore set of A, so A(Z(A)0 )−1 exists. From the canonical injection Q(Z(A)) ,→ A(Z(A)0 )−1 , pq 7→ pq , we get that A(Z(A)0 )−1 is a vector space over Q(Z(A)), moreover, A(Z(A)0 )−1 = AQ(Z(A)). We will show that the dimension of this vector space is finite. Let n V be S a frame that generates A. Since S {V n }n≥0 is a filtration of A, we have n A = n≥0 V and AQ(Z(A)) = n≥0 V Q(Z(A)). Two possibilities arise: either there exists an n ≥ 0 such that V n Q(Z(A)) = V n+1 Q(Z(A)), or else Q(Z(A)) ( V Q(Z(A)) ( V 2 Q(Z(A)) ( · · · In the first case AQ(Z(A)) = V n Q(Z(A)) and the claim is proved. In the second case, dimQ(Z(A)) Q(Z(A)) dimQ(Z(A)) V Q(Z(A)) dimQ(Z(A)) V 2 Q(Z(A)) · · · and we will show that this produces a contradiction. In fact, for every n ≥ 0, dimQ(Z(A)) V n Q(Z(A)) ≥ n + 1; let u1 , . . . , ud(n) be a Q(Z(A))-basis of V n Q(Z(A)), thus, d(n) ≥ n + 1; we can assume that ui ∈ V n for every 1 ≤ i ≤ d(n); let W be an arbitrary K-subspace of Z(A) of finite dimension, then (V + W )2n ⊇ V n W n ⊇ u1 W n ⊕ · · · ⊕ ud(n) W n (the sum is direct since the elements ui are linearly independent over Z(A)); from this we get dimK (V + W )2n ≥ d(n) dimk (W n ) ≥ (n + 1) dimK (W n ), but since V + W is a frame of A, then GKdim(A) ≥ 1 + GKdim(Z(A)), false. Now we can prove the claimed isomorphism. For this consider the canonical injective homomorphism g : A → A(Z(A)0 )−1 , a 7→ a1 . If a ∈ A − {0}, then a1 is invertible in A(Z(A)0 )−1 . In fact, the map h : A(Z(A)0 )−1 → A(Z(A)0 )−1 , pq 7→ a1 pq , is an injective Q(Z(A))-homomorphism since A is a domain, but as was observed above, A(Z(A)0 )−1 is finite-dimensional over Q(Z(A)), therefore h is surjective, whence, there exists pq ∈ A(Z(A)0 )−1 such that

ap 1q

pa q 1 0 pp 1 = 0 1q 1,

= 11 . Observe that

A(Z(A)0 )

−1

such that

= 11 : in fact, since p 6= 0, there exists and since

p q

=

1p q 1,

p0 q0

we have

a 1 p p0 1 p0 a1 p0 a1q p0 q a p0 q = , i.e., = , so = , whence = , so 1 q 1 q0 1 q0 1q q0 1q1 q0 1 1 q0 1

in

4.6 The Center of the Total Division Ring of Fractions

95

pa p p0 q 1 p p0 q 1q 1 = = = = . q1 q q0 1 q 1 q0 1 q1 1 In order to conclude the proof of the isomorphism A(Z(A)0 )−1 ∼ = Qr (A), observe that any element pq ∈ A(Z(A)0 )−1 can be written as pq = g(p)g(q)−1 . Step 2. Let C := { pq | p, q ∈ Z(A), q 6= 0}. If pq ∈ Z(Qr (A)), then by the first step we can assume that q ∈ Z(A)0 , and from part (i)(c) of Proposition 4.6.1, we get p ∈ Z(A). Therefore, Z(Qr (A)) ⊆ C. Conversely, let pq ∈ C, then p, q ∈ Z(A), with q 6= 0, whence, by part (ii) of Proposition 4.6.1, p q p p1 1 1 , 1 ∈ Z(Qr (A)), so q ∈ Z(Qr (A)), and hence, q = 1 q ∈ Z(Qr (A)). Thus, C ⊆ Z(Qr (A)). Step 3. According to part (ii) of Proposition 4.6.1, we have the canonical ι injective homomorphism Z(A) → − Z(Qr (A)), p 7→ p1 , that sends invertible elements of Z(A) to invertible elements of Z(Qr (A)), moreover, by step 2, every element pq ∈ Z(Qr (A)) can be written pq = ι(p)ι(q)−1 . This proves the isomorphism Z(Qr (A)) ∼ t u = Q(Z(A)). Next we present many examples of K-algebras that satisfy the hypotheses of Theorem 4.6.3, most of them within the skew P BW extensions. Example 4.6.4. (i) Any domain A such that dimK A < ∞. For example, the real algebra H of quaternions since dimR (H) = 4. (ii) Any right Ore domain A finitely generated as a Z(A)-module. Example 4.6.5. Applying Theorems 4.2.6 and 7.4.1 below, we will check next that the following skew P BW extensions are K-algebras that satisfy the hypotheses of Theorem 4.6.3. (i) Consider a skew polynomial ring A := R[x; σ], with R a commutative domain R that is a K-algebra generated by a subspace V of finite dimension such that σ(V ) ⊆ V , σ is K-linear of finite order m, Rσ = K and GKdim(R) = 0. Then, GKdim(A) = 1, and from Proposition 3.3.6, Z(A) = K[xm ], and hence, GKdim(Z(A)) = 1. Thus, Z(Qr (A)) ∼ = Q(Z(A)) = K(xm ). A particular case of this general example is A := C[x; σ] as an R-algebra, with σ(z) := z, z ∈ C (here C and R are the fields of complex and real numbers, respectively). In this case the order of σ is two and GKdim(C) = 0 since dimR (C) = 2. (ii) Let char(K) = p > 0 and A := Sh := K[t][xh ; σh ] be the algebra of shift operators. Then, GKdim(A) = 2. Moreover, for every k ≥ 0, σhk (t) = t − kh, then σhp (t) = t, i.e., the order of σh is p, therefore, Z(A) = K[t]σh [xph ]. Since K[tp ] ⊆ K[t]σh ⊆ K[t] and K[t] is finitely generated over K[tp ], GKdim(K[tp ]) = GKdim(K[t]) = 1, whence GKdim(K[t]σh ) = 1. Therefore, p GKdim(Z(A)) = 2. Thus, Z(Qr (A)) ∼ = Q(Z(A)) = Q(K[t]σh )(xh ). d (iii) Let char(K) = p > 0 and A := K[t][x; dt ][xh ; σh ] be the algebra of 2 shift differential operators. Then, GKdim(A) = 3 and Z(A) = K[xp , xph , tp − p t ] (see [395]). Since GKdim(Z(A)) = 3, we have Z(Qr (A)) ∼ = Q(Z(A)) = p p2 p p K(x , xh , t − t ). (iv) Let char(K) = p > 0 and A := An (K) be the Weyl algebra. Since GKdim(A) = 2n and Z(A) = K[tp1 , . . . , tpn , xp1 , . . . , xpn ] (see [231],

96

4 Rings of Fractions

Example 1.3.), then GKdim(Z(A)) = 2n. So, Z(Qr (A)) ∼ = Q(Z(A)) = K(tp1 , . . . , tpn , xp1 , . . . , xpn ). (v) Let char(K) = p > 0 and A := J := K{x, y}/hyx−xy−x2 i be the Jordan algebra. Since GKdim(J ) = 2 and Z(A) = K[xp , y p ] (see Theorem 2.2 in [362]), then GKdim(Z(A)) = 2, whence Z(Qr (A)) ∼ = Q(Z(A)) = K(xp , y p ). (vi) Consider the quantum plane A := Kq [x, y], with q 6= 1 a root of unity of degree m ≥ 2. Then GKdim(A) = 2 and Z(A) = K[xm , y m ] (see [362]). Therefore, Z(Qr (A)) ∼ = Q(Z(A)) = K(xm , y m ). (vii) The previous example can be extended to the quantum polynomials A := Kq [x1 , . . . , xn ], where n ≥ 2 and q ∈ K − {0, 1}, defined by xj xi = qxi xj , with 1 ≤ i < j ≤ n. If q is a root of unity of degree m ≥ 2, then it is known (see a comm plete proof in [395]) that if n even, then Z(A) = K[xm 1 , . . . , xn ]. Therefore, GKdim(Z(A)) = n = GKdim(A) and hence m Z(Qr (A)) ∼ = Q(Z(A)) = K(xm 1 , . . . , xn ).

(viii) Let Aq be the quantum Weyl algebra generated by x, y with rule of multiplication yx = qxy + a, where q, a ∈ K − {0}. If q is a primitive root of unity of degree m ≥ 2, then Z(Aq ) = K[xm , y m ] (see [84]). Since GKdim(Aq ) = 2, then Z(Qr (Aq )) ∼ = Q(Z(A)) = K(xm , y m ). (ix) In [251] the center of the following algebras are computed (the precise definition of these algebras can be found in [248]). In every example we assume that the q-parameters are roots of unity of degree l ≥ 2, or li ≥ 2, appropriately: (a) The algebra of q-differential operators: Z(A) = K[xl , y l ] and GKdim(A) = 2, so Z(Qr (A)) ∼ = Q(Z(A)) = K(xl , y l ). (b) The additive analogue of the Weyl algebra: Z(A) = K[xl11 , . . . , xlnn , y1l1 , . . . , ynln ] and GKdim(A) = 2n, so Z(Qr (A)) ∼ = Q(Z(A)) = K(xl11 , . . . , xlnn , y1l1 , . . . , ynln ). (c) The algebra of linear partial q-dilation operators: In this case we have GKdim(A) = 2n and Z(A) = K[tl1 , . . . , tln , H1l , . . . , Hnl ]. Then, Z(Qr (A)) ∼ = Q(Z(A)) = K(tl1 , . . . , tln , H1l , . . . , Hnl ). (d) The algebra of linear partial q-differential operators: In this case GKdim(A) = 2n and Z(A) = K[tl1 , . . . , tln , D1l , . . . , Dnl ]. Hence, Z(Qr (A)) ∼ = Q(Z(A)) = K(tl1 , . . . , tln , D1l , . . . , Dnl ). (x) Let sl(n, K) be the Lie algebra of 2 × 2 matrices with null trace with K-basis e, f, h. If char(K) = 2, then Z(U (sl(2, K))) = K[e2 , f 2 , h] (see [231], p. 147). Moreover, GKdim(U(sl(2, K))) = 3. Thus, Z(Qr (A)) ∼ = Q(Z(A)) = K(e2 , f 2 , h).

4.6 The Center of the Total Division Ring of Fractions

97

Remark 4.6.6. As occurs for the Gelfand–Kirillov conjecture (see [144]), if the hypotheses of Theorem 4.6.3 fail, then the isomorphism Z(Qr (A)) ∼ = Q(Z(A)) could hold or fail. Thus, the hypotheses are not necessary conditions. For example, (a) H is not finitely generated as a Q-algebra, however Qr (H) = H and Z(Qr (H)) ∼ =R∼ = Q(Z(H)). (b) Let K be a field with char(K) = 0, and let G be a three-dimensional completely solvable Lie algebra with basis x, y, z such that [y, x] = y, [z, x] = λz and [y, z] = 0, λ ∈ K − {0} (see Example 14.4.2 in [278]). If λ ∈ K − Q, then Z(U (G)) = K and GKdim(U (G)) = 3, thus, in this case GKdim(U(G)) ≮ GKdim(Z(U (G))) + 1, and 14.4.7 in [278] shows that Z(Qr (U (G)))  Q(Z(U (G))). (c) Let A := Uq+ (slm ) be the quantum enveloping algebra of the Lie algebra of strictly upper triangular matrices of size m × m over a field K, where q ∈ K − {0} is not a root of unity. In [11] it was proved that Z(A) is the classical commutative polynomial algebra over K in n variables, with m = 2n or m = 2n + 1, whence, GKdim(Z(A)) = n. On the other hand, according to [11], p. 236, A is an iterated skew polynomial ring of K of m(m−1) variables, 2 therefore GKdim(A) = m(m−1) ([179], Lemma 2.2; see also Theorem 20.3.4 in 2 Section 20.3). Thus, GKdim(A) ≮ GKdim(Z(A)) + 1, however, Z(Qr (A)) ∼ = Q(Z(A)).

Chapter 5

Prime Ideals

In this chapter we describe particular families of prime ideals of some important classes of skew P BW extensions. For this purpose we will consider the techniques that we found in [48], [156] and [159]. However, in several of our results, some substantial modifications to these techniques should be introduced. We will use the notation of Definition 1.1.7 and Remark 1.1.9, in particular, the total order  defined on Mon(A) is needed. Some extra terminology we will be introduced.

5.1 Invariant Ideals Let A = σ(R)hx1 , . . . , xn i be a skew P BW extension of a ring R. By Proposition 1.1.3, we know that xi r − σi (r)xi = δi (r) for all r ∈ R, where σi is an injective endomorphism of R and δi is a σi -derivation of R, 1 ≤ i ≤ n, i.e., {σi , δi }ni=1 are some of the parameters of A (see Definition 1.2.1). This motivates the following notions. Definition 5.1.1. Let R be a ring and (Σ, ∆) a system of endomorphisms and Σ-derivations of R, with Σ := {σ1 , . . . , σn } and ∆ := {δ1 , . . . , δn }. (i) If I is an ideal of R, I is called Σ-invariant if σi (I) ⊆ I, for every 1 ≤ i ≤ n. The ∆-invariant ideals are defined similarly. If I is both Σ and ∆-invariant, we say that I is (Σ, ∆)-invariant. (ii) A proper Σ-invariant ideal I of R is Σ-prime if whenever a product of two Σ-invariant ideals is contained in I, one of the ideals is contained in I. R is a Σ-prime ring if the ideal 0 is Σ-prime. The ∆-prime and (Σ, ∆)-prime ideals and rings are defined similarly. (iii) The system Σ is commutative if σi σj = σj σi for every 1 ≤ i ≤ n. Commutativity for ∆ is defined similarly. The system (Σ, ∆) is commutative if both Σ and ∆ are commutative. The following proposition describes the behavior of these properties when we pass to a quotient ring. © Springer Nature Switzerland AG 2020 W. Fajardo et al., Skew PBW Extensions, Algebra and Applications 28, https://doi.org/10.1007/978-3-030-53378-6_5

99

100

5 Prime Ideals

Proposition 5.1.2. Let R be a ring, (Σ, ∆) a system of endomorphisms and Σ-derivations of R, I a proper ideal of R and R := R/I. (i) If I is (Σ, ∆)-invariant, then over R := R/I is induced a system (Σ, ∆) of endomorphisms and Σ-derivations defined by σi (r) := σi (r) and δi (r) := δi (r), 1 ≤ i ≤ n. If σi is bijective and σi (I) = I, then σi is bijective. (ii) Let I be Σ-invariant. If Σ is commutative, then Σ is commutative. Similar properties are valid for ∆ and (Σ, ∆). (iii) Let I be Σ-invariant. I is Σ-prime if and only if R is Σ-prime. Similar properties are valid for ∆ and (Σ, ∆). Proof. All statements follow directly from the definitions.

t u

According to the properties of Σ and ∆, we need to introduce some special classes of skew P BW extensions. Definition 5.1.3. Let A be a skew P BW extension of a ring R with system of endomorphisms Σ := {σ1 , . . . , σn } and Σ-derivations ∆ := {δ1 , . . . , δn }. (i) If σi = iR for every 1 ≤ i ≤ n, we say that A is a skew PBW extension of derivation type. (ii) If δi = 0 for every 1 ≤ i ≤ n, we say that A is a skew PBW extension of endomorphism type. In addition, if every σi is bijective, A is a skew PBW extension of automorphism type. (iii) A is Σ-commutative if the system Σ is commutative. The ∆ and (Σ, ∆)-commutativity of A are defined similarly. Related to the previous definition, we have the following two interesting results. The second one extends Lemma 1.5.(c) in [156]. Proposition 5.1.4. Let A be a skew P BW extension of derivation type of a ring R. Then, for any θ, γ, β ∈ Nn and c ∈ R, the following identities hold: cγ,β cθ,γ+β = cθ,γ cθ+γ,β , ccθ,γ = cθ,γ c. In particular, the system of constants ci,j are central, and hence, A is bijective. Proof. This is a direct consequence of Remark 1.1.9, part (ii).

t u

Proposition 5.1.5. Let A be a skew P BW extension of a ring R. If for every 1 ≤ i ≤ n, δi is inner, then A is a skew P BW extension of R of endomorphism type. Proof. Let ai ∈ R such that δi = δai is inner, 1 ≤ i ≤ n. We will prove that A = σ(R)hz1 , . . . , zn i, where zi := xi − ai , the system of endomorphisms coincides with the original system Σ and every σi -derivation is equal zero. We will check the conditions in Definition 1.1.1. It is clear that R ⊆ A. Let r ∈ R, then zi r = (xi − ai )r = xi r − ai r = σi (r)xi + δai (r) − ai r = σi (r)xi + ai r − σi (r)ai − ai r = σi (r)(xi − ai ) = σi (r)zi . Thus, the systems of constants

5.1 Invariant Ideals

101

ci,r of σ(R)hz1 , . . . , zn i coincides with the original one, and the same is true for the system of endomorphisms. Note that the system of Σ-derivations is trivial, i.e., each one is equal zero. This means that σ(R)hz1 , . . . , zn i is of endomorphism type. zj zi = (xj − aj )(xi − ai ) = xj xi − xj ai − aj xi + aj ai = ci,j xi xj + r0 + r1 x1 + · · · + rn xn − xj ai − aj xi + aj ai , for some r0 , r1 , . . . , rn ∈ R. Replacing xi by zi + ai for every 1 ≤ i ≤ n, we conclude that zj zi − ci,j zi zj ∈ R + Rz1 + · · · + Rzn . Finally, note that Mon{z1 , . . . , zn } := {z α = z1α1 · · · znαn |α = (α1 , . . . , αn ) ∈ Nn } is a left R-basis of A. In fact, it is clear that Mon{z1 , . . . , zn } generates A as a left R-module. Let c1 , . . . , ct ∈ R such that c1 z α1 + · · · + ct z αt = 0 with z αi ∈ Mon{z1 , . . . , zn }, 1 ≤ i ≤ n. Then, using the deglex order in Remark 1.1.9, we conclude that c1 xα1 + · · · + ct xαt should be zero, whence c1 = · · · = ct = 0. t u In the next proposition we study quotients of skew P BW extensions by (Σ, ∆)-invariant ideals. Proposition 5.1.6. Let A be a skew P BW extension of a ring R and I be a (Σ, ∆)-invariant ideal of R. Then, (i) IA is an ideal of A and IA ∩ R = I. IA is proper if and only if I is proper. Moreover, if for every 1 ≤ i ≤ n, σi is bijective and σi (I) = I, then IA = AI. (ii) If I is proper and σi (I) = I for every 1 ≤ i ≤ n, then A/IA is a skew P BW extension of R/I. Moreover, if A is of automorphism type, then A/IA is of automorphism type. If A is bijective, then A/IA is bijective. In addition, if A is Σ-commutative, then A/IA is Σ-commutative. Similar properties are true for the ∆ and (Σ, ∆) commutativity. (iii) Let A be of derivation type and I proper. Then, IA = AI and A/IA is a skew P BW extension of derivation type of R/I. (iv) Let R be left (right) noetherian and σi bijective for every 1 ≤ i ≤ n. Then, σi (I) = I for every i and IA = AI. If I is proper and A is bijective, then A/IA is a bijective skew P BW extension of R/I. Proof. (i) It is clear that IA is a right ideal, but since I is (Σ, ∆)-invariant, then IA is also a left ideal of A. It is obvious that IA ∩ R = I. From this last equality we get also that IA is proper if and only if I is proper. Using again that I is (Σ, ∆)-invariant, we get that AI ⊆ IA. Assuming that σi is bijective and σi (I) = I for every i, then IA ⊆ AI. (ii) According to (i), we only have to show that A := A/IA is a skew P BW extension of R := R/I. For this we will verify the four conditions of Definition 1.1.1 (another way of proving this is to use Theorem 1.3.1 for σi , δi : R → R defined by σi (r) := σi (r), δi (r) := δi (r) with r ∈ R). It is clear that R ⊆ A. Moreover, A is a left R-module with generating set Mon{x1 , . . . , xn }. Next we

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show that Mon{x1 , . . . , xn } is independent. Consider the expression r1 X1 + · · · + rn Xn = 0, where Xi ∈ Mon(A) for each i. We have r1 X1 + · · · + rn Xn ∈ IA and hence r1 X1 + · · · + rn Xn = r10 X1 + · · · + rn0 Xn , for some ri0 ∈ I, i = 1, . . . , n. Thus, (r1 −r10 )X1 +· · ·+(rn −rn0 )Xn = 0, so ri ∈ I, i.e., ri = 0 for i = 1, . . . , n. Let r 6= 0 with r ∈ R. Then r ∈ / IA, and hence, r ∈ / I, in particular, r 6= 0 and there exists ci,r := σi (r) 6= 0 such that xi r = ci,r xi + δi (r). Thus, xi r = ci,r xi + δi (r). Observe that ci,r = 6 0, but ci,r = σi (r) ∈ IA ∩ R = I = σi (I), i.e, r ∈ I, a contradiction. This completes the proof of condition (iii) in Definition 1.1.1. Pn In A we have xj xi − ci,j xi xj ∈ R + t=1 Rxt , with ci,j ∈ R − {0}, so in A we get xj xi − ci,j xi xj ∈ R +

Pn

t=1

R xt .

Since I is proper and ci,j is left invertible for i < j, then ci,j 6= 0. This completes the proof of condition (iv) in Definition 1.1.1. By Proposition 5.1.2, if σi is bijective, then σi is bijective. It is obvious that if every constant ci,j is invertible, then ci,j is invertible. The statements about the commutativity follow from Proposition 5.1.2. (iii) This is direct consequence of (i) and (ii). (iv) From the Noether condition and the chain I ⊆ σi−1 (I) ⊆ σi−2 (I) ⊆ · · · we get that σi (I) = I for every i (note that for this equality we have not used that I is ∆-invariant). The other statements follow from (i) and (ii). t u

5.2 Extensions of Derivation Type Now we pass to describe the prime ideals of skew P BW extensions of derivation type. Two technical propositions are needed. The total order introduced in Remark 1.1.9 will be used in what follows. The right annihilator of a subset X of a ring R will be denoted by rannR (X). Proposition 5.2.1. Let A be a bijective skew P BW extension of a ring R. Let J be a nonzero ideal of A. If f is a nonzero element of J of minimal leading monomial xαt and σ αt (r) = r for any r ∈ rannR (lc(f )), then rannA (f ) = (rannR (lc(f )))A. Proof. Consider 0 6= f = m1 X1 + · · · + mt Xt , an element of J of minimal leading monomial Xt = xαt , with Xt  Xt−1  · · ·  X1 . By definition of the right annihilator, rannR (lc(f )) := {r ∈ R | mt r = 0}. From Theorem 1.1.8 we have f r = m1 X1 r + · · · + mt (σ αt (r)xαt + pαt ,r ),

5.2 Extensions of Derivation Type

103

where pαt ,r = 0 or deg(pαt ,r ) < |αt | if pαt ,r 6= 0. Note that if r ∈ rannR (lc(f )), then f r = 0. In fact, if the contrary is assumed, since σ αt (r) = r, we get lm(f r) ≺ Xt with f r ∈ J, but this is a contradiction since Xt is minimal. Thus, f rannR (lc(f )) = 0 and f rannR (lc(f ))A = 0. Therefore rannR (lc(f ))A ⊆ rannA (f ). Next we will show that rannA (f ) ⊆ rannR (lc(f ))A. Let u = r1 Y1 + · · · + rk Yk be an element of rannA (f ), with Yk  Yk−1  · · ·  Y1 , then f u = (m1 X1 + · · · + mt Xt )(r1 Y1 + · · · + rk Yk ) = 0, which implies that mt Xt rk Yk = 0, whence mt σ αt (rk )Xt Yk = 0, but A is bijective, so mt σ αt (rk ) = 0, which means σ αt (rk ) ∈ rannR (mt ). Let σ αt (rk ) := s. Then s ∈ rannR (lc(f )). Note that rk = σ −αt (s) = s; moreover σ αt (s) = s implies s = σ −αt (s) and hence rk ∈ rannR (lc(f )). This shows that rk Yk ∈ rannR (lc(f ))A ⊆ rannA (f ), but since u ∈ rannA (f ), we have u − rk Yk ∈ rannA (f ). Continuing in this way we obtain that rk−1 Yk−1 , rk−2 Yk−2 , . . . , r1 Y1 ∈ rannR (lc(f ))A, which guarantees that u ∈ rannR (lc(f ))A. Thus, we have proved that rannA (f ) ⊆ rannR (lc(f ))A. u t Proposition 5.2.2. Let A be a skew P BW extension of derivation type of a ring R and let K 6= 0 be an ideal of A. Let K 0 be the ideal of R generated by the coefficients of terms of all polynomials of K. Then K 0 is a ∆-invariant ideal of R. Proof. Let k ∈ K 0 , then k is a finite sum of elements of the form rcr0 , with r, r0 ∈ R and c is the coefficient of one term of some polynomial of K. It is enough to prove that for every 1 ≤ i ≤ n, δi (rcr0 ) ∈ K 0 . We have δi (rcr0 ) = rcδi (r0 ) + rδi (c)r0 + δi (r)cr0 . Note that rcδi (r0 ), δi (r)cr0 ∈ K 0 , so it only remains to prove that δi (c) ∈ K 0 . There exists a p ∈ K such that p = cxα + p0 , with p0 ∈ A and xα does not appear in p0 . Note that the coefficients of all terms of p0 are also in K 0 . Observe that xi p ∈ K and we have xi p = xi cxα + xi p0 = cxi xα + δi (c)xα + xi p0 ; from the previous expression we conclude that the coefficient of xα in xi p is δi (c) + cr + r0 , where r is the coefficient of xα in xi xα and r0 is the coefficient of xα in xi p0 . Since c ∈ K 0 we only have to prove that r0 ∈ K 0 . Let p0 = c1 xβ1 + · · · + ct xβt , then c1 , . . . , ct ∈ K 0 and xα ∈ / {xβ1 , . . . , xβt }. We have xi p0 = (c1 xi + δi (c1 ))xβ1 + · · · + (ct xi + δi (ct ))xβt = c1 xi xβ1 + δi (c1 )xβ1 + · · · + ct xi xβt + δi (ct )xβt ; from the previous expression we get that r0 has the form r0 = c1 r1 +· · ·+ct rt , where rj is the coefficient of xα in xi xβj , 1 ≤ j ≤ t. This proves that r0 ∈ K 0. t u The following theorem gives a description of prime ideals of skew P BW extensions of derivation type without assuming any conditions on the ring of

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coefficients. This result generalizes the description of prime ideals of classical P BW extensions given in Proposition 6.2 of [48]. Compare also with [278], Proposition 14.2.5 and Corollary 14.2.6. Theorem 5.2.3. Let A be a skew P BW extension of derivation type of a ring R. Let I be a proper ∆-invariant ideal of R. I is a ∆-prime ideal of R if and only if IA is a prime ideal of A. In such case, IA = AI and IA ∩ R = I. Proof. (i) By Proposition 5.1.6 we know that A/IA is a skew P BW extension of R/I of derivation type, IA = AI and IA ∩ R = I. Then we may assume that I = 0. Note that if R is not ∆-prime, then A is not prime. Indeed, there exist I, J 6= 0 ∆-invariant ideals of R such that IJ = 0, so IA, JA 6= 0 and IAJA = IJA = 0, i.e., A is not prime. Suppose that R is ∆-prime. We need to show that if J, K are nonzero ideals of A, then JK 6= 0. Let K 0 be as in Proposition 5.2.2, then K 0 6= 0 and it is ∆-invariant. Now let j be a nonzero element of J of minimal leading monomial. If jK = 0, then taking f = j in Proposition 5.2.1 we get K ⊆ rannA (j) = rannR (lc(j))A. Therefore lc(j)K 0 = 0 and hence lannR (K 0 ) 6= 0. We have lannR (K 0 )K 0 = 0, and note that lannR (K 0 ) is also ∆-invariant. In fact, let a ∈ lannR (K 0 ) and k 0 ∈ K 0 , then δi (a)k 0 = δi (ak 0 ) − aδi (k 0 ) = δi (0) − aδi (k 0 ) = 0 since δi (k 0 ) ∈ K 0 . Thus, R is not ∆-prime, a contradiction. In this way jK 6= 0 and so JK 6= 0, which concludes the proof. t u

5.3 Extensions of Automorphism Type In this section we consider the characterization of prime ideals for extensions of automorphism type over left (right) noetherian rings such that the system {ci,j } of constants of Definition 1.2.1 are central. Proposition 5.3.1. Let A be a bijective skew P BW extension of a ring R. Suppose that given a, b ∈ R − {0} there exists θ ∈ Nn such that either aRσ θ (b) 6= 0 or aRδ θ (b) 6= 0. Then, A is a prime ring. Proof. Suppose that A is not a prime ring, then there exist nonzero ideals I, J of A such that IJ = 0. We can assume that I := lannA (J) and J := rannA (I). Let u be a nonzero element of I with minimal leading monomial xα and leading coefficient cu . We will prove first that σ −α (cu ) ∈ I, i.e., σ −α (cu )J = 0. Since rannA (I) ⊆ rannA (u), it is enough to show that σ −α (cu )rannA (u) = 0. Suppose that σ −α (cu )rannA (u) 6= 0, let v ∈ rannA (u) of minimal leading monomial xβ and leading coefficient cv such that σ −α (cu )v 6= 0. Since uv = 0 and cα,β is invertible (see Theorem 1.1.8 and Remark 1.1.9), then cu σ α (cv ) = 0, whence lm(ucv ) ≺ xα . The minimality of xα implies that ucv = 0, and hence u(v−cv xβ ) = 0. Moreover, v−cv xβ ∈ rannA (u) and lm((v−cv xβ ) ≺ xβ ,

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105

so σ −α (cu )(v − cv xβ ) = 0. However, cu σ α (cv ) = 0, so we have σ −α (cu )cv = 0 and hence σ −α (cu )v = 0, a contradiction. Thus, I ∩ R 6= 0, and by symmetry, J ∩ R 6= 0. Let 0 6= a ∈ I ∩ R and 0 6= b ∈ J ∩ R. By the hypothesis there exists θ ∈ Nn and r ∈ R such that arσ θ (b) 6= 0 or arδ θ (b) 6= 0. If θ = (0, . . . , 0), then arb 6= 0, and hence IJ 6= 0, a contradiction. If θ 6= (0, . . . , 0), then arxθ b = ar(σ θ (b)xθ + pθ,b ), but note that the independent term of pθ,b is δ θ (b) (see Theorem 1.1.8, part (b)). Thus, arxθ b 6= 0, i.e., IJ 6= 0, a contradiction. t u Remark 5.3.2. Note if R is a prime ring and A is a bijective skew P BW extension of R, given a, b ∈ R − {0}, then aRσ θ (b) 6= 0 for every θ ∈ Nn . Thus, from the previous proposition, A is prime. This coincides with Theorem 3.2.6. Lemma 5.3.3. Let A be a bijective Σ-commutative skew P BW extension of automorphism type of a left (right) noetherian ring R. Let I be a Σ-invariant proper ideal of R. I is a Σ-prime ideal of R if and only if IA is a prime ideal of A. In such case, IA = AI and IA ∩ R = I. Proof. By Proposition 5.1.6, IA = AI is a proper ideal of A, I = IA ∩ R and A := A/IA is a bijective skew P BW extension of R := R/I of automorphism type. In addition, observe that I is Σ-prime if and only if R is Σ-prime (Proposition 5.1.2). Thus, we can assume that I = 0, and hence, we have to prove that R is Σ-prime if and only if A is prime. ⇒) Suppose that R is Σ-prime, i.e., 0 is Σ-prime. According to Proposition 5.3.1, we have to show that given a, b ∈ R − {0} there exists a θ ∈ Nn such that aRσ θ (b) 6= 0. Let L be the ideal generated by the elements σ θ (b), θ ∈ Nn ; observe that L 6= 0, and since A is Σ-commutative, L is Σ-invariant. But R is left (right) noetherian and A is bijective, then σi (L) = L for every 1 ≤ i ≤ n (see Proposition 5.1.6). This implies that AnnR (L) is Σ-invariant, but 0 is Σ-prime, therefore AnnR (L) = 0. Thus, aL 6= 0, so there exists a θ ∈ Nn such that aσ θ (b) 6= 0. ⇐) Note that if R is not Σ-prime, then A is not prime. Indeed, there exist K, J 6= 0 Σ-invariant ideals of R such that KJ = 0, so KA, JA 6= 0 and since J is Σ-invariant, then AJ = JA and hence KAJA = KJA = 0, i.e., A is not prime. t u Proposition 5.3.4. Let A be a bijective skew P BW extension of a ring R such that the system {ci,j } of constants of Definition 1.2.1 are central. Then A is Σ-commutative. Proof. For i = j it is clear that σj σi = σi σj . Let i 6= j, say i < j, then for any r ∈ R we have lc(xj xi r) = σj σi (r)ci,j = ci,j σi σj (r), but since ci,j ∈ Z(R)∗ , then σj σi (r) = σi σj (r). t u Theorem 5.3.5. Let A be a bijective skew P BW extension of automorphism type of a left (right) noetherian ring R. Assume that the system {ci,j } of constants of Definition 1.2.1 are central. Let I be a proper Σ-invariant ideal of R. I is a Σ-prime ideal of R if and only if IA is a prime ideal of A. In such case, IA = AI and IA ∩ R = I.

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Proof. This follows from Lemma 5.3.3 and Propositions 5.1.6 and 5.3.4.

t u

Example 5.3.6. Let R and {ci,j } be as in the previous theorem, then we obtain a description of prime ideals for the n-multiparametric skew quantum space A = Rq,σ [x1 , . . . , xn ]. Thus, we get a description of prime ideals for Qr,n q,σ (R) (see Example 4.4.4) since it is well known that there exists a bijective correspondence between the prime ideals of S −1 A and the prime ideals of A with empty intersection with S.

5.4 Extensions of Mixed Type Our next task is to give a description of prime ideals of bijective skew P BW extensions of mixed type, i.e., when both systems Σ and ∆ could be nontrivial. We will assume that the ring R is commutative, noetherian and semiprime. The proof of the main theorem (Theorem 5.4.7) is as in Lemma 5.3.3 but first we need some preliminary technical propositions. Definition 5.4.1. Let R be a commutative ring, I a proper ideal of R and R := R/I. We define S(I) := {a ∈ R | a := a + I is regular}. By a regular element we mean a nonzero divisor. Note that S(0) is the set of regular elements of R, this multiplicative system is also denoted by S0 (R) (see Chapter 4). Next we will describe the behavior of the properties introduced in Definition 5.1.1 when we pass to the total ring of fractions. Proposition 5.4.2. Let R be a commutative ring with total ring of fractions Q(R) and let (Σ, ∆) be a system of automorphisms and Σ-derivations of R. Then, e ∆) e of automorphisms and Σ-derive (i) Over Q(R) is induced a system (Σ, σi (a) δi (s) a δi (a) a a e and δi ( ) := − + . ations defined by σei ( ) := s

σi (s)

s

σi (s) s

σi (s)

e (ii) Q(R) is Σ-prime if and only if R is Σ-prime. The same is valid for ∆ and (Σ, ∆). Proof. (i) This part can be proved not only for commutative rings but also in the noncommutative case (see Chapter 4, Theorem 4.1.3). (ii) ⇒) Let I, J be Σ-invariant ideals of R such that IJ = 0, then e IJS(0)−1 = IS(0)−1 JS(0)−1 = 0, but note that IS(0)−1 , JS(0)−1 are Σ−1 −1 invariant, so IS(0) = 0 or JS(0) = 0, i.e., I = 0 or J = 0. e ⇐) K, L be two Σ-invariant ideals of Q(R) such that KL = 0, then K = IS(0)−1 and L = JS(0)−1 , with I := {a ∈ R| a1 ∈ K} and J := {b ∈ R| 1b ∈ L}. Note that IJ = 0, moreover, I, J are Σ-invariant. In fact, if a ∈ I, then a ei ( a1 ) = σi1(a) ∈ K, i.e., σi (a) ∈ I, for every 1 ≤ i ≤ n. 1 ∈ K and hence σ Analogously for L. Since R is Σ-prime, then I = 0 or J = 0, i.e., K = 0 or L = 0. The proofs for ∆ and (Σ, ∆) are analogous. t u

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107

We will use the following special notation: let m ≥ 0 be an integer, then σ(m) will denote the product of m endomorphisms taken from Σ in any order, and probably with repetitions, i.e., σ(m) = σi1 · · · σim , with i1 , . . . , im ∈ {1, . . . , n}. For m = 0 we will understand that this product is the identical isomorphism of R. Proposition 5.4.3. Let R be a commutative ring and (Σ, ∆) a system of endomorphisms and Σ-derivations of R. Let I be a Σ-invariant ideal of R. Set I0 := R, I1 := I, and for j ≥ 2, Ij := {r ∈ I|δi1 σ(m(1))δi2 σ(m(2)) · · · δil σ(m(l))(r) ∈ I, for all l = 1, . . . , j − 1; i1 , . . . , il ∈ {1, . . . , n} and m(1), . . . , m(l) ≥ 0}. Then, (i) (ii) (iii) (iv)

I0 ⊇ I1 ⊇ I2 ⊇ · · · . δi (Ij ) ⊆ Ij−1 , for every 1 ≤ i ≤ n and any j ≥ 1. Ij is a Σ-invariant ideal of R, for any j ≥ 0. IIj ⊆ Ij+1 , for any j ≥ 0.

Proof. (i) This is evident. (ii) It is clear that for every i, δi (I1 ) ⊆ I0 . Let j ≥ 2 and let r ∈ Ij , then δi1 σ(m(1))δi2 σ(m(2)) · · · δil σ(m(l))(r) ∈ I, for all l = 1, . . . , j − 1. From this obtain that δi1 σ(m(1))δi2 σ(m(2)) · · · δil σ(m(l))δi σ(0)(r) ∈ I for all l = 1, . . . , j − 2. This means that δi (r) ∈ Ij−1 . (iii) It is clear that I0 , I1 are Σ-invariant ideals. Let j ≥ 2 and let r ∈ Ij , then for every σi we have δi1 σ(m(1))δi2 σ(m(2)) · · · δil σ(m(l))(σi (r)) = δi1 σ(m(1))δi2 σ(m(2)) · · · δil σ(m(l) + 1)(r) ∈ I. This means that σi (r) ∈ Ij , i.e., Ij is Σ-invariant. Let us prove that Ij is an ideal of R. By induction we assume that Ij is an ideal. It is obvious that if a, a0 ∈ Ij+1 , then a + a0 ∈ Ij+1 ; let r ∈ R, then a ∈ Ij and hence ra ∈ Ij , therefore δi1 σ(m(1))δi2 σ(m(2)) · · · δik σ(m(k))(ra) ∈ I for all k < j. Consider any σ(m(j)), we have σ(m(j))(a) ∈ Ij+1 , so σ(m(j))(a), δi (σ(m(j))(a)) ∈ Ij for every i. Therefore, δi (σ(m(j))(ra)) = δi (σ(m(j))(r)σ(m(j))(a)) = σi σ(m(j))(r)δi (σ(m(j))(a)) + δi (σ(m(j))(r))σ(m(j))(a) ∈ Ij . This implies that δi1 σ(m(1))δi2 σ(m(2)) · · · δij σ(m(j))(ra) ∈ I. This means that ra ∈ Ij+1 . This proves that Ij+1 is an ideal. (iv) Of course II0 ⊆ I1 . Let j ≥ 1, suppose that for IIj−1 ⊆ Ij , we have to prove that IIj ⊆ Ij+1 . Let a ∈ I and b ∈ Ij , then b ∈ Ij−1 and ab ∈ Ij . Therefore, δi1 σ(m(1))δi2 σ(m(2)) · · · δik σ(m(k))(ab) ∈ I for i < j. For any σ(m(j)) and every δi we have, as above,

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δi (σ(m(j))(ab)) = δi (σ(m(j))(a)σ(m(j))(b)) = σi σ(m(j))(a)δi (σ(m(j))(b)) + δi (σ(m(j))(a))σ(m(j))(b) ∈ IIj−1 + RIj ⊆ Ij . From this we conclude that δi1 σ(m(1))δi2 σ(m(2)) · · · δij σ(m(j))(ab) ∈ I, and this means that ab ∈ Ij+1 . This completes the proof. t u Proposition 5.4.4. Let R be a commutative noetherian ring and Σ a system of automorphisms of R. Then, any Σ-prime ideal of R is semiprime. √ Proof. Let I be a Σ-prime ideal of R. Since R is √ noetherian, I is finitely generated,√so there exists an m ≥ 1√such that ( √I)m ⊆ I. Since I is Σinvariant, I is Σ-invariant, whence I ⊆ I, i.e., I = I. This means that I is an intersection of prime ideals, i.e., I is semiprime. t u Proposition 5.4.5. Let R be a commutative noetherian ring and (Σ, ∆) a system of automorphisms and Σ-derivations of R. Let rad(R) be the prime radical of R. If R is (Σ, ∆)-prime, then (i) rad(R) is Σ-prime. (ii) S(0) = S(rad(R)). (iii) Q(R) is artinian. Proof. (i) The set of Σ-invariant ideals I of R such that AnnR (I) 6= 0 is not empty since 0 satisfies these conditions (AnnR (X) denotes the annihilator of subset X of the commutative ring R). Since R is assumed to be noetherian, let I be maximal with these conditions. Let K, L be Σ-invariant ideals of R such that I K and I L, then AnnR (K) = 0 = AnnR (L), and hence AnnR (KL) = 0. This implies that KL * I. This proves that I is Σ-prime. We will prove that I = rad(R). By Proposition 5.4.3, we have the descending chain of Σ-invariant ideals I0 ⊇ I1 ⊇ · · · , and the ascending chain AnnR (I0 ) ⊆ AnnR (I1 ) ⊆ · · · . There exists an m ≥ 1 such that AnnR (Im ) = AnnR (Im+1 ) (since I0 = R and I1 = I, see Proposition 5.4.3, m 6= 0). Note that AnnR (Im ) is Σ-invariant since σi (Im ) = Im for every i (here we have used again that R is noetherian, Proposition 5.1.6). Let b ∈ AnnR (Im ). For a ∈ Im+1 we have a ∈ Im , moreover, by Proposition 5.4.3, for every i, δi (a) ∈ Im , so ab = 0 = δi (a)b, therefore 0 = δi (ab) = σi (a)δi (b). From this we obtain that σi (Im+1 )δi (b) = 0, i.e., Im+1 δi (b) = 0. Thus, δi (b) ∈ AnnR (Im+1 ) = AnnR (Im ). We have proved that AnnR (Im ) is (Σ, ∆)invariant. Let H := AnnR (AnnR (Im )), we shall see that H is also (Σ, ∆)-invariant. In fact, let x ∈ H, then xAnnR (Im ) = 0 and for every i we have σi (x)σi (AnnR (Im )) = 0 = σi (x)AnnR (Im ), thus σi (x) ∈ H. Now let y ∈ AnnR (Im ), then xy = 0 and for every i we have δi (xy) = 0 = σi (x)δi (y) + δi (x)y, but δi (y) ∈ AnnR (Im ), so σi (x)δi (y) = 0. Thus, δi (x)y = 0, i.e., δi (x) ∈ H.

5.4 Extensions of Mixed Type

109

Since R is (Σ, ∆)-prime and HAnnR (Im ) = 0, we have H = 0 or AnnR (Im ) = 0. From Proposition 5.4.3, AnnR (Im ) ⊇ AnnR (I) 6= 0, whence H = 0. Since Im ⊆ H, we have Im = 0. Again, from Proposition 5.4.3 we obtain that I m ⊆ Im , so I m = 0, and hence I ⊆ rad(R). On the other hand, since I is Σ-prime, I is semiprime (Proposition 5.4.4), but rad(R) is the smallest semiprime ideal of R, whence I = rad(R). Thus, rad(R) is Σ-prime. (ii) The inclusion S(0) ⊆ S(rad(R)) is well known (see [278], Proposition 4.1.3). The other inclusion is equivalent to proving that R is S(rad(R))torsion free. Since Im = 0, it is enough to prove that every factor Ij /Ij+1 is S(rad(R))-torsion free. In fact, in general, if M is an R-module and N is an submodule of M such that M/N and N are S-torsion free (with S an arbitrary system of R), then M is S-torsion free. Thus, the assertion follows from R = I0 /Im , I0 /I2 /I1 /I2 I0 /Im /Im−1 /Im

∼ = I0 /I1 , I0 /I3 /I2 /I3 ∼ = I0 /I2 , . . . , ∼ = I0 /Im−1 .

I0 /I1 = R/rad(R). It is clearly S(rad(R))-torsion free. By induction, we assume that Ij−1 /Ij is S(rad(R))-torsion free. Let a ∈ Ij and r ∈ S(rad(R)) such that rb a = b 0 in Ij /Ij+1 , i.e., ra ∈ Ij+1 . From Proposition 5.4.3 we get that for every i and any σ(m), σ(m)(a) ∈ Ij , σ(m)(ra) ∈ Ij+1 and δi (σ(m)(ra)) ∈ Ij , then δi (σ(m)(ra)) = δi (σ(m)(r)σ(m)(a)) = σi (σ(m)(r))δi (σ(m)(a)) + δi (σ(m)(r))σ(m)(a) ∈ Ij , whence, σi (σ(m)(r))δi (σ(m)(a)) ∈ Ij . For every k, σk (rad(R)) = rad(R), then we can prove that σk (S(rad(R))) = S(rad(R)), so σi (σ(m)(r)) ∈ S(rad(R)); moreover, δi (σ(m)(a)) ∈ Ij−1 , then by induction δi (σ(m)(a)) ∈ Ij . But this is valid for any i and any σ(m), then a ∈ Ij+1 . This proves that Ij−1 /Ij is S(rad(R))-torsion free. (iii) This follows from (ii) and Small’s Theorem (see [278], Corollary 4.1.4). t u Corollary 5.4.6. Let R be a commutative noetherian semiprime ring and (Σ, ∆) a system of automorphisms and Σ-derivations of R. If R is (Σ, ∆)prime, then R is Σ-prime. Proof. Since R is semiprime, then by Proposition 5.4.5, 0 = rad(R) is Σprime, i.e., R is Σ-prime. t u Theorem 5.4.7. Let R be a commutative noetherian semiprime ring and A a bijective skew P BW extension of R. Let I be a semiprime (Σ, ∆)-invariant ideal of R. I is a (Σ, ∆)-prime ideal of R if and only if IA is a prime ideal of A. In such case, IA = AI and I = IA ∩ R. Proof. The proof is exactly as in Lemma 5.3.3, anyway we will repeat it. By Proposition 5.1.6, IA = AI is a proper ideal of A, I = IA∩R and A := A/IA

110

5 Prime Ideals

is a bijective skew P BW extension of the commutative noetherian semiprime ring R := R/I. In addition, observe that I is (Σ, ∆)-prime if and only if R is (Σ, ∆)-prime (see Propositions 5.1.2 and 5.1.6). Thus, we can assume that I = 0, and hence, we have to prove that R is (Σ, ∆)-prime if and only if A is prime. ⇒) From Corollary 5.4.6, we know that R is Σ-prime, i.e., 0 is Σ-prime. Let a, b ∈ R−{0} and L be the ideal generated by the elements σ θ (b), θ ∈ Nn ; observe that L = 6 0, and since A is Σ-commutative (Proposition 5.3.4), L is Σ-invariant. But R is noetherian and A is bijective, then σi (L) = L for every 1 ≤ i ≤ n (see the proof of part (iv) in Proposition 5.1.6). This implies that AnnR (L) is Σ-invariant, therefore AnnR (L) = 0. Thus, aL 6= 0, so there exists a θ ∈ Nn such that aσ θ (b) 6= 0. From Proposition 5.3.1 we get that A is prime. ⇐) Note that if R is not (Σ, ∆)-prime, then A is not prime. Indeed, there exist I, J 6= 0 (Σ, ∆)-invariant ideals of R such that IJ = 0, so IA, JA 6= 0 and IAJA = IJA = 0, i.e., A is not prime. t u

Chapter 6

Minimal Prime Ideals

In this chapter we investigate the minimal prime ideals of skew P BW extensions assuming certain restrictions on the ring of coefficients related to the class of skew Armendariz rings. We also study some radicals of skew PBW extensions in this context: the Wedderburn radical, lower nil radical, Levitzky radical and the upper nil radical. In addition, we will prove K¨othe’s conjecture for skew P BW extensions. We will use in the present chapter some notation of the previous ones, in particular Definitions 1.1.7 and 5.1.1.

6.1 Skew Armendariz Rings A commutative ring B is called Armendariz (the term was introduced by Rege and Chhawchharia in [326]) if for polynomials f (x) = a0 + a1 x + · · · + an xn , g(x) = b0 + b1 x + · · · + bm xm of B[x] which satisfy f (x)g(x) = 0, then ai bj = 0, for every i, j. As we can appreciate, the interest of this notion lies in its natural and its useful role in understanding the relation between the annihilators of the ring B and the annihilators of the polynomial ring B[x]. For instance, in Lemma 1 of [17], Armendariz showed that a reduced ring (i.e., a ring without nonzero nilpotent elements) always satisfies this condition (reduced rings are abelian, that is, every idempotent is central, and also semiprime, i.e., its prime radical is trivial). For noncommutative rings, more exactly, for skew polynomial rings, the notion of Armendariz ring has been also studied. Commutative and noncommutative treatments have been investigated in several papers (see for example [16], [17], [176], [180], [202], [227], [275], [326]). For a ring B with a ring endomorphism σ : B → B and a σ-derivation δ : B → B, Krempa in [211] considered the skew polynomial ring B[x; σ, δ], and defined σ as a rigid endomorphism if bσ(b) = 0 implies b = 0, for b ∈ B. Krempa called B σ-rigid if there exists a rigid endomorphism σ of B. Some other related properties such as Baer, quasi-Baer, p.p., and p.q.-Baer over σ-rigid rings have been also investigated in the literature (see [178], [211], [337], and others). © Springer Nature Switzerland AG 2020 W. Fajardo et al., Skew PBW Extensions, Algebra and Applications 28, https://doi.org/10.1007/978-3-030-53378-6_6

111

112

6 Minimal Prime Ideals

For skew P BW extensions we have an adequate notion of rigidness. Definition 6.1.1. If B is a ring and Σ is a family of endomorphisms of B, then Σ is called a rigid endomorphisms family if rσ α (r) = 0 implies r = 0, for every r ∈ B and α ∈ Nn . A ring B is said to be Σ-rigid if there exists a rigid endomorphisms family Σ of B. Note that if Σ is a rigid endomorphisms family, then every element σi ∈ Σ is a monomorphism. In this way, we consider the family of injective endomorphisms Σ and the family ∆ of Σ-derivations of a skew P BW extension A of a ring R (see Proposition 1.1.3). Σ-rigid rings are reduced rings: if B is a Σ-rigid ring and r2 = 0, for r ∈ B, then 0 = rσ α (r2 )σ α (σ α (r)) = rσ α (r)σ α (r)σ α (σ α (r)) = rσ α (r)σ α (rσ α (r)), i.e., rσ α (r) = 0, and so r = 0, that is, B is reduced. By [332], Corollary 3.4, if A is a skew P BW extension of a Σ-rigid ring R, the equality ab = 0, for a, b ∈ R, implies axα bxβ = 0 in A, for any α, β ∈ Nn . Recall that if A is a skew P BW extension of a ring R, then R is Σ-rigid if and only if A is a reduced ring ([332], Proposition 3.5). With the purpose of generalizing the notion of σ-rigid ring and studying the properties of being Baer, quasiBaer, p.p. and p.q.-Baer over this more general structure, several notions of skew-Armendariz rings have been established in the literature (see [176], [285], [289]). For the case of skew PBW extensions we have the following definitions. Definition 6.1.2 ([294]). Let A be a skew PBW extension of a ring R. We say that R is a (Σ, ∆)-Armendariz ring if for polynomials f = a0 + a1 X1 + · · · + am Xm and g = b0 + b1 Y1 + · · · + bt Yt in A, the equality f g = 0 implies ai Xi bj Yj = 0, for every i, j. We say that R is a (Σ, ∆)-weak Armendariz ring if, for linear polynomials f = a0 + a1 x1 + · · · + an xn and g = b0 + b1 x1 + · · · + bn xn in A, the equality f g = 0 implies ai xi bj xj = 0, for every i, j. Every Σ-rigid ring is a (Σ, ∆)-skew Armendariz ring ([294], Proposition 3.6). Definition 6.1.3. Let A be a skew PBW extension of aPring R. R is m called a Σ-skew Armendariz ring if, for elements f = i=0 ai Xi and Pt αi g = b Y in A, the equality f g = 0 implies a σ (b ) i j = 0, for all j=0 j j 0 ≤ i ≤ m and 0 ≤ j ≤ t, where αi = exp(X ). R is called a weak Σ-skew Pn i P n Armendariz ring if, for elements f = i=0 ai xi and g = j=0 bj xj in A (x0 := 1), the equality f g = 0 implies ai σi (bj ) = 0, for all 0 ≤ i, j ≤ n (σ0 := idR ). Note that every Σ-skew Armendariz ring is a weak Σ-skew Armendariz ring. If A is a skew P BW extension of a ring R, and if R is Σ-rigid, then R is Σ-skew Armendariz ([339], Proposition 3.4). The converse of this proposition is false as the following remark shows.

6.1 Skew Armendariz Rings

113

Remark 6.1.4. We have the following facts: 

  at | a ∈ Z, t ∈ Q . Then B is a com0a mutative ring,  and if we consider the automorphism σ of R given by   at a t/2 σ = . In [176], Example 1, it was shown that R is 0a 0 a σ-skew Armendariz and is not σ-rigid. Since Σ-rigid and Σ-skew Armendariz are generalizations of σ-rigid and σ-skew Armendariz, respectively, this example shows that there exists an example of a Σ-skew Armendariz ring which is not Σ-rigid. (ii) Let B = Z2 [x] be the commutative polynomial ring over Z2 , and σ the endomorphism of B = Z2 [x] defined by σ(f (x)) = f (0). Then B = Z2 [x] is σ-skew Armendariz and is not σ-rigid ([176], Example 5). (i) Consider the ring B =

From the definitions above we can establish the following relations Σ-rigid $ (Σ, ∆)-Armendariz $ (Σ, ∆)-weak Armendariz, Σ-rigid $ Σ-skew Armendariz $ weak Σ-skew Armendariz, (Σ, ∆)-Armendariz $ Σ-skew Armendariz, (Σ, ∆)-weak Armendariz $ weak Σ-skew Armendariz. As we can appreciate, the more general class of rings consists of the weak Σ-skew Armendariz. Two new notions of Armendariz for skew P BW extensions are skew-Armendariz (Definition 6.1.5) and a more general notion, the weak skew-Armendariz (Definition 6.1.6). These definitions generalize the treatments developed for both classical polynomial rings and Ore extensions of injective type (see [16], [17], [174], [176], [178], [180], [202], [227], [275], [282], [285], [289], [290] and [326]. In particular, our Definitions 6.1.5 and 6.1.6 are motivated and generalize [290], Definitions 2.1 and 2.4, respectively. We show also that the families of Armendariz rings defined in [294] and [339] are contained in the family of skew-Armendariz and weak skew-Armendariz, but the converse is false. Definition 6.1.5. Let R be a ring and A a skew PBW extension of R. We say that R is a skew-Armendariz ring if, for polynomials f = a0 + a1 X1 + · · · + am Xm and g = b0 + b1 Y1 + · · · + bt Yt in A, f g = 0 implies a0 bk = 0, for each 0 ≤ k ≤ t. Note that every Armendariz ring is skew-Armendariz, where σi = idR and δi = 0 (1 ≤ i ≤ n), and every Σ-skew Armendariz ring is also a skewArmendariz ring. If R is Σ-rigid, the elements ci,j are invertible (Definition 1.1.1 (iv)), and they are in the center of R, then from [332], Proposition 3.6 we know that R is skew-Armendariz.

114

6 Minimal Prime Ideals

Definition 6.1.6. Let R be a ring and A a skew PBW extension of R. We say that R is a weak skew-Armendariz ring, if for linear polynomials f = a0 + a1 x1 + · · · + an xn , and g = b0 + b1 x1 + · · · + bn xn in A, f g = 0 implies a0 bk = 0, for every 0 ≤ k ≤ n. We can see that every skew-Armendariz ring is weak skew-Armendariz. However, a weak Armendariz ring is not necessarily Armendariz. As an illustration of this fact in the case of Ore extensions, see [227], Example 3.2. Of course, every weak Σ-skew Armendariz ring is a weak skew-Armendariz ring. So, we have the relations Σ-rigid $ (Σ, ∆)-Armendariz $ Σ-skew Armendariz $ skew-Armendariz, Σ-rigid $ (Σ, ∆)-weak Armendariz $ weak Σ-skew Armendariz,

and of course, weak Σ-skew Armendariz $ weak skew-Armendariz. Lemma 6.1.7. If R is a weak skew-Armendariz ring, the equality ab = 0 implies σ α (a)δ α (b) = δ α (a)b = 0, for each α ∈ Nn . Proof. We only show the case σi (a)δi (b) = δi (a)b = 0, for i = 1, . . . , n. Since ab = 0, we have 0 = δi (ab) = σi (a)δi (b) + δi (a)b, or equivalently, δi (a)b = −σi (a)δi (b). Let f, g ∈ A be given by f = δi (a) + 0x1 + · · · + 0xi−1 + σi (a)xi + 0xi+1 + · · · + 0xn , and g = b + bx1 + · · · + bxn , respectively. Note that f g = 0: f g = δi (a)b + δi (a)bx1 + · · · + δi (a)bxn + σi (a)xi b + σi (a)xi bx1 + · · · + σi (a)xi bxn = δi (a)b + δi (a)bx1 + · · · + δi (a)bxn + σi (a)[σi (b)xi + δi (b)] + σi (a)[σi (b)xi + δi (b)]x1 + · · · + σi (a)[σi (b)xi + δi (b)]xn = δi (a)b + δi (a)bx1 + · · · + δi (a)bxn + σi (a)σi (b)xi + σi (a)δi (b) + σi (a)σi (b)xi x1 + σi (a)δi (b)x1 + · · · + σi (a)σi (b)xi xn + σi (a)δi (b)xn = 0.

From Definition 6.1.6 we obtain δi (a)b = 0, so σi (a)δi (b) = 0.

t u

In [89] and [275] the authors give a positive answer to the following question formulated in [176], p. 115: let σ be a monomorphism (or automorphism) of a (commutative) reduced ring B and B be σ-skew Armendariz. Is B σrigid? The content of Theorem 6.1.9 is the generalization of this answer to skew P BW extensions. We suppose that the elements ci,j in Definition 1.1.1 (iv) are invertible and central. These assumptions are satisfied for a lot of algebras, for example: any Ore extension R[x; σ, δ], the additive analogue of the Weyl algebra, the multiplicative analogue of the Weyl algebra, the quantum algebra U (so(3, K)), the 3-dimensional skew polynomial algebras, the dispin algebra U (osp(1, 2)), the Woronowicz algebra W (sl(2, K)), the complex algebra Vq (sl3 (C)), the q-Heisenberg algebra, the quantum enveloping algebra of sl(2, K), Uq (sl(2, K)), P BW extensions, diffusion algebras, and others. It is clear that any skew PBW extension over a field K satisfies these assumptions. The following remark will be often used from now on.

6.1 Skew Armendariz Rings

115

Remark 6.1.8. Let A = σ(R)hx1 , . . . , xn i be a skew P BW extension. With respect to Definition 1.1.1 and Proposition 1.1.3, we have the following facts: (i) If i < j and d0i , b0j ∈ R, then d0i xi b0j xj = d0i [σi (b0j )xi + δi (b0j )]xj = d0i σi (b0j )xi xj + d0i δi (b0j )xj . On the other hand, since xj xi = ci,j xi xj + r(i,j) +

(i,j) xk , k=1 rk

Pn

we have d0j xj b0i xi = d0j [σj (b0i )xj + δj (b0i )]xi = d0j σj (b0i )xj xi + d0j δj (b0i )xi = d0j σj (b0i )(ci,j xi xj + r(i,j) +

n X

(i,j)

rk

xk ) + d0j δj (b0i )xi .

k=1

Thus, d0i xi b0j xj + d0j xj b0i xi = [d0i σi (b0j ) + d0j σj (b0i )ci,j ]xi xj (i,j)

+ [d0j δj (b0i ) + d0j σj (b0i )ri

(i,j)

+ [d0i δi (b0j ) + d0j σj (b0i )rj + d0j σj (b0i )

n X

(i,j)

rk

]xi

]xj

xk + d0j σj (b0i )r(i,j) .

k=1; k6=i,j

(ii) If β

αin i1 Xi := xα and Yj := x1 j1 · · · xβnjn , 1 · · · xn

then we can see that when we compute every summand of ai Xi bj Yj we obtain products of the coefficient ai with several evaluations of bj in σ’s and δ’s depending on the coordinates of αi . This assertion follows from the equation ai Xi bj Yj = ai σ αi (bj )xαi xβj αin βj i2 + ai pαi1 ,σαi2 (···(σαin (b))) xα 2 · · · xn x i2

in

αi3 αin βj i1 α α + ai xα 1 pαi2 ,σ3 i3 (···(σ in (b))) x3 · · · xn x in

+ + +

αi4 αin βj i1 αi2 α α ai xα 1 x2 pαi3 ,σi4i4 (···(σinin (b))) x4 · · · xn x αi(n−2) αin βj i1 αi2 α · · · + ai xα 1 x2 · · · xi(n−2) pαi(n−1) ,σinin (b) xn x αi(n−1) βj i1 ai xα 1 · · · xi(n−1) pαin ,b x .

(iii) If α = (α1 , . . . , αn ) ∈ Nn and r is an element of a ring R, then

116

6 Minimal Prime Ideals α

α

α1 n−1 αn n−1 1 α2 xα r = xα 1 x2 · · · xn−1 xn r = x1 · · · xn−1

X αn

n −j δ (σ j−1 (r))xj−1 xα n n n n



j=1 α

αX n−1

α

αX n−2

n−2 1 + xα 1 · · · xn−2



α

−j

j−1 αn αn δn−1 (σn−1 (σn (r)))xj−1 n−1 xn

α

−j

j−1 n−1 n−1 αn αn δn−2 (σn−2 (σn−1 (σn (r))))xj−1 n−2 xn−1 xn + · · ·

n−1 xn−1

j=1 n−3 1 + xα 1 · · · xn−3

n−2 xn−2



α

α

j=1 1 + xα 1

X α2



α

n−1 αn αn 2 −j 3 α4 xα δ2 (σ2j−1 (σ3α3 (σ4α4 (· · · (σn (r))))))xj−1 xα 2 2 3 x4 · · · xn−1 xn

j=1 αn αn 1 + σ1α1 (σ2α2 (· · · (σn (r))))xα 1 · · · xn ,

σj0 := idR for 1 ≤ j ≤ n.

Theorem 6.1.9. If A is a skew PBW extension of a ring R, then the following statements are equivalent: (i) R is reduced and skew-Armendariz; (ii) R is Σ-rigid; (iii) A is reduced. Proof. (i) ⇒ (ii) Suppose that R is reduced, skew-Armendariz and is not Σ-rigid. Then there exists a β ∈ Nn with aσ β (a) = 0 and a 6= 0. Note that σ β (a)σ β (σ β (a)) = σ β (aσ β (a)) = 0. Using that R is reduced, the equality (σ β (a)a)2 = σ β (a)aσ β (a)a = 0 implies σ β (a)a = 0. Equivalently, since a 6= 0, σ β is injective, and R is reduced, σ β (a) 6= 0 and (σ β (a))2 6= 0. With this in mind, consider the elements f = σ β (a) + σ β (a)xβ , g = a − σ β (a)xβ . Then f g = (σ β (a) + σ β (a)xβ )(a − σ β (a)xβ ) = σ β (a)a − (σ β (a))2 xβ + σ β (a)xβ a − σ β (a)xβ σ β (a)xβ = −(σ β (a))2 xβ + σ β (a)[σ β (a)xβ + pβ,a ] − σ β (a)[σ β (σ β (a))xβ + qβ,σβ (a) ]xβ = σ β (a)pβ,a − σ β (aσ β (a))xβ xβ − σ β (a)qβ,σβ (a) xβ = σ β (a)pβ,a −σ β (a)qβ,σβ (a) xβ , where pβ,a = 0 or deg(pβ,a ) < |β|, if pβ,r 6= 0, and qβ,σβ (a) = 0 or deg(qβ,σβ (a) ) < |β|, if qβ,σβ (a) 6= 0. Since aσ β (a) = σ β (a)a = 0, Remark 6.1.8(ii) and Lemma 6.1.7 guarantee that σ β (a)pβ,a = σ β (a)qβ,σβ (a) xβ = 0, so f g = 0. By assumption, R is skew-Armendariz, that is, −(σ β (a))2 = 0, but −(σ β (a))2 6= 0, i.e., we have obtained a contradiction. Hence, R is Σ-rigid. (ii) ⇒ (i) From [332] we know that a Σ-rigid ring is reduced, and as we saw above, every Σ-rigid ring is also skew-Armendariz. (ii) ⇒ (iii) Let R be Σ-rigid and suppose that A is not reduced. Then there exists a non-zero element f ∈ A such that f 2 = 0. Since R is reduced, f∈ / R. Following Definition 1.1.7, we consider f = a0 + a1 X1 + · · · + am Xm , αin i1 where ai ∈ R, 0 ≤ i ≤ m, am 6= 0, with Xi = xαi = xα 1 · · · xn , 2 and Xm  Xm−1  · · ·  X1 . By Theorem 1.1.8 (b) we have f = (am Xm + · · · + a1 X1 + a0 )(am Xm + · · · + a1 X1 + a0 ) = am Xm am Xm + other terms of order less than Xm Xm = am [σ αm (am )Xm + pαm ,am ]Xm + · · · = am σ αm (am )Xm Xm + am pαm ,am Xm + · · · = am σ αm (am )[cαm ,αm x2αm + pαm ,αm ] + am pαm ,am Xm + · · · , where pαm ,am = 0 or deg(pαm ,am ) < |αm | if pαm ,am 6= 0, and pαm ,αm = 0 or deg(pαm ,αm = 0) < |αm + αm | if pαm ,αm 6= 0. From the equality lc(f 2 ) = am σ αm (am )cαm ,αm = 0 we obtain am σ αm (am ) = 0 (A is bijective). From [332], Lemma 3.3 (iv), we obtain

6.1 Skew Armendariz Rings

117

a2m = 0, and so am = 0 (R is reduced), which is a contradiction. Hence, A is reduced. (iii) ⇒ (ii) If A is reduced, R is also reduced as a subring. Let us see that R is Σ-rigid. If a ∈ R and aσ α (a) = 0, then 0 = σ α (a)xα aσ α (a)xα a = (σ α (a)xα a)2 , and so σ α (a)xα a = 0. Thus, 0 = σ α (a)xα a = σ α (a)[σ α (a)xα + pα,a ] = (σ α (a))2 xα + σ α (a)pα,a , with pα,a = 0, or deg(pα,a ) < |α| if pα,a 6= 0 (Theorem 1.1.8). Hence (σ α (a))2 = 0, that is, σ α (a) = 0. Now, since σ α is injective, we obtain a = 0, which shows that R is Σ-rigid. t u Corollary 6.1.10. If A is a skew PBW extension of a ring R, then the following statements are equivalent: (i) R is reduced and (Σ, ∆)-Armendariz (Σ-skew Armendariz). (ii) R is Σ-rigid. (iii) A is reduced. Remark 6.1.11. We know that Σ-rigid rings ( (Σ, ∆)-Armendariz rings and Σ-rigid rings ( Σ-skew Armendariz rings, so Σ-rigid rings ( skewArmendariz rings. Hence, [294], Theorem 3.9, [339], Theorem 3.6, and Theorem 6.1.9, show that if we assume that the ring R is reduced, then for Σ-rigid rings the notions of (Σ, ∆)-Armendariz, Σ-skew Armendariz, and skew-Armendariz coincide. This fact shows the importance of considering skew PBW extensions over nonreduced rings with the aim of obtaining ring theoretical properties more general than those established in these papers. Our goal now is to prove that if the ring R of coefficients of a skew P BW extension A is skew-Armendariz, then A is abelian. Proposition 6.1.12. If R is a weak skew-Armendariz ring, and e ∈ R is an idempotent element, we have σi (e) = e and δi (e) = 0, for every i = 1, . . . , n. Proof. See Proposition 4.7 in [337].

t u

Proposition 6.1.13. Let R be a skew-Armendariz ring. If e2 = e ∈ A, with Pm e = i=0 ei Xi , then e ∈ R. Proof. Let e = e0 + e1 X1 + · · · + em Xm be an element of A with e2 = e. Since (e0 + e1 X1 + · · · en Xn )((1 − e0 ) − e1 X1 − · · · − en Xn ) = ((1 − e0 ) − e1 X1 − · · · − en Xn )(e0 + e1 X1 + · · · en Xn ), the assumption on R implies e0 (1 − e0 ) = (1 − e0 )ei = e0 ei (1 ≤ i ≤ n) = 0. Hence ei = 0, for every i, which shows that e = e0 = e20 . t u Proposition 6.1.14. Every weak skew-Armendariz ring is abelian. Proof. See Proposition 4.9 in [337].

t u

Corollary 6.1.15. If R is a skew-Armendariz ring R, then A is an abelian ring. Proof. Propositions 6.1.12, 6.1.13 and 6.1.14 imply the result.

t u

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6 Minimal Prime Ideals

6.2 Wedderburn, Lower Nil, Levitzky and Upper Nil Radicals Let B be a ring, following the notation presented in [288], the Wedderburn radical (the largest nilpotent ideal in B), the lower nil radical (the intersection of all prime ideals), the Levitzky radical (the sum of all locally nilpotent ideals), the upper nil radical (the sum of all nil ideals), will be denoted by N0 (B), Nil∗ (B), L-rad(B), Nil∗ (B), respectively. In this section we compute these radicals for the skew P BW extensions, under the assumptions of Σskew Armendariz and (Σ, ∆)-compatibility. Definition 6.2.1. Consider a ring R with a family of endomorphisms Σ and a family of Σ-derivations ∆. Then, (i) R is said to be Σ-compatible if, for each a, b ∈ R, aσ α (b) = 0 if and only if ab = 0, for every α ∈ Nn . (ii) R is said to be ∆-compatible if, for each a, b ∈ R, ab = 0 implies aδ β (b) = 0, for every β ∈ Nn . (iii) If R is both Σ-compatible and ∆-compatible, R is called (Σ, ∆)compatible. Proposition 6.2.2. Let Σ be a family of endomorphisms of a ring R, and let ∆ be a family of Σ-derivations of R. If R is Σ-rigid, then R is (Σ, ∆)compatible. Proof. If R is a Σ-rigid ring, then R is reduced, and ab = 0 if and only if ba = 0. In this way, aσ α (b)σ α (aσ α (b)) = aσ α (ba)σ α (σ α (b)) = 0, whence aσ α (b) = 0. Using a similar reasoning, we can see that the equality ba = 0 implies σ α (a)b = 0: 0 = σ α (σ α (a))σ α (ba)b = σ α (σ α (a))σ α (b)σ α (a)b = σ α (σ α (a)b)σ α (a)b ⇒ σ α (a)b = 0. Now, if aσ α (b) = 0, then baσ α (b)σ α (a) = baσ α (ba) = 0, whence ba = 0, and then ab = 0. Finally, for every i, 0 = δi (ba) = δi (b)a + σi (b)δi (a), that is, (σi (b)δi (a))2 = −δi (b)aσi (b)δi (a) = 0, and since R is reduced, σi (b)δi (a) = 0, i.e., δi (b)a = 0, which shows that aδi (b) = 0. t u Next we present several properties that we need for proving the main results of this section. We include only a few proofs, the other proofs are cited. Proposition 6.2.3. Let R be a (Σ, ∆)-compatible ring. For every a, b ∈ R, we have: (i) If ab = 0, then aσ θ (b) = σ θ (a)b = 0, for each θ ∈ Nn . (ii) If σ β (a)b = 0 for some β ∈ Nn , then ab = 0. (iii) If ab = 0, then σ θ (a)δ β (b) = δ β (a)σ θ (b) = 0, for every θ, β ∈ Nn . Proof. The proofs are similar to those in [166], Lemma 2.4.

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The converse of Proposition 6.2.2 is false, as [338], Example 3.6 shows. However, if the ring R is assumed to be reduced, these two notions coincide.

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Proposition 6.2.4. If A is a skew PBW extension over a ring R, then the following statements are equivalent: (i) R is reduced and (Σ, ∆)-compatible. (ii) R is Σ-rigid. (iii) A is reduced. Proof. (i) ⇒ (ii) Suppose that R is reduced and (Σ, ∆)-compatible. Consider an element r ∈ R such that rσ α (r) = 0. From Proposition 6.2.3 (i) we obtain σ α (r)σ α (r) for every α ∈ N, and using the injectivity of σ α and the assumption on R, we have r = 0. (ii) ⇒ (i) This follows from Proposition 6.2.2. (ii) ⇔ (iii) This was proved in Theorem 6.1.9. t u Example 6.2.5. From Corollary 6.1.10 and Proposition 6.2.4 we can observe that the concepts of Σ-rigid, Σ-skew Armendariz and (Σ, ∆)-compatible coincide when R is a reduced ring. Since skew PBW extensions over domains are also domains (see Proposition 3.2.1), and every domain is a reduced ring, all these extensions are reduced rings, and hence, their coefficient rings are reduced, Σ-skew Armendariz, (Σ, ∆)-compatible and Σ-rigid. All important examples of skew P BW extensions correspond to this situation. Definition 6.2.6. Let A be a skew PBW extension of R. We say that R satisfies the condition (SA1) if, whenever f g = 0 for f = a0 + a1 X1 + · · · + am Xm and g = b0 + b1 Y1 + · · · + bt Yt elements of A, then ai bj = 0, for every i, j. Every Σ-rigid ring satisfies the condition (SA1) ([332], Proposition 3.6). Proposition 6.2.7. If A is a skew PBW extension of a ring R, then R is (Σ, ∆)-compatible and Σ-skew Armendariz if and only if R satisfies (SA1). Proof. See [341], Proposition 3.2.

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Proposition 6.2.8. If A is a skew PBW extension of a ring R, then R is (Σ, ∆)-compatible and weak Σ-skew Armendariz if and only if R satisfies (SA1). Proof. See [341], Proposition 3.3.

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For f ∈ A, let Cf be the set of coefficients of f . Lemma 6.2.9. Let A be a skew PBW extension over a (Σ, ∆)-compatible and Σ-skew Armendariz ring R. If p1 , . . . , pl are elements of A such that p1 · · · pl = 0, then a1 · · · al = 0, where ai ∈ Cfi , for each i. Pm Proof. We proceed by induction. If l = 2, let p1 = i=0 ai Xi , p2 = Pt αi j=0 bj Yj . By assumption we have ai σ (bj ) = 0, for every i, j, and using the Σ-compatibility of R, ai bj = 0, for every value of i and j, so the assertion follows.

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Let l > 2. If h := p2 p3 · · · pl , then p1 h = 0, and by the reasoning above, a1 ah = 0, where a1 ∈ Cp1 , ah ∈ Ch . Having in mind the form of the elements of h, that is, ah = a2 · · · al , where a2 ∈ Cf2 , . . . , al ∈ Cfl (which is due to the fact that R is Σ-skew Armendariz and Σ-compatible), then we obtain a1 · · · al = 0. t u Lemma 6.2.10. Let A be a skew PBW extension over a (Σ, ∆)-compatible and weak Σ-skew Armendariz ring R. If ab = cm = 0, for some m ∈ N and elements a, b, c of R, then acm−1 b = 0. Proof. See [341], Lemma 3.5.

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Proposition 6.2.11. If R is a (Σ, ∆)-compatible weak Σ-skew Armendariz ring, and ab = cm = 0, for some positive integer m and elements a, b, c ∈ R, then acb = 0. Proof. See [341], Proposition 3.6.

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Proposition 6.2.12. Let A be a skew PBW extension over a (Σ, ∆)-compatible and Σ-skew Armendariz ring R. If for any elements f, g, h ∈ A, f s = 0, for some s ∈ N, and gh = 0, then gf h = 0. Pm Pp Pt Proof. We consider f = i=0 ai Xi , g = j=0 bj Yj and h = l=0 cl Zl . Since s s f = 0, from Proposition 6.2.9 we know that ai = 0, for every 0 ≤ i ≤ m. Now, by assumption gh = 0, and using that R is (Σ, ∆)-compatible and Σskew Armendariz, we have bj cl = 0, for every j, l. In this way, by Proposition 6.2.11 we can assert bj ai cl = 0, for each value of i, j, l. Finally, Proposition 6.2.3 with Remark 6.1.8 (ii) and (iii) allow us to conclude that gf h = 0. t u Now we are ready for the computations of radicals of skew PBW extensions, and thus, to extend the results of [288]. Theorem 6.2.13. If A is a skew PBW extension over a (Σ, ∆)-compatible and Σ-skew Armendariz ring R, then N0 (A) = Nil∗ (A) = L-rad(A) = Nil∗ (A). Proof. It is sufficient to show that Nil∗ (A) ⊆ N0 (A). Let f be an element of a summand of Nil∗ (A). By definition, there exists an m ∈ N such that f m = 0. Let us see that (Af A)2m−1 = 0. With this objective, note that Af A ⊆ Nil∗ (A) (since f m−1 f = 0), and so for every g ∈ Af A, we obtain f m−1 gf = 0 (Proposition 6.2.12). This fact means that f m−1 Af Af = 0, whence f m−2 f Af Af = 0. As we saw before, f m−2 Af Af Af Af = 0. Repeating this argument we obtain f Af Af Af A · · · f Af , and hence (Af A)2m−1 . Thus, f ∈ N0 (B), which concludes the proof. t u Theorem 6.2.14. If R is a (Σ, ∆)-compatible and weak Σ-skew Armendariz ring, then N0 (R) = Nil∗ (R) = L-rad(R) = Nil∗ (R). Proof. The assertions follow from Proposition 6.2.11 using a similar reasoning to that in Theorem 6.2.13. t u

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With the aim of establishing other results about radicals of skew PBW extensions, we consider Definition 5.1.1 and we write P(Σ,∆) := Spec(R; Σ, ∆) for the set of all (Σ, ∆)-prime ideals of R, and \ Nil∗ (R; Σ, ∆) :=

P

P ∈P(Σ,∆)

for the (Σ, ∆)-prime radical of R. R is a Σ-semiprime ring if Nil∗ (R; Σ) = 0. In a similar way we define ∆-semiprime and (Σ, ∆)-semiprime rings. About these sets we have the following theorem. Theorem 6.2.15. If R is a (Σ, ∆)-compatible and weak Σ-skew Armendariz ring, then Nil∗ (R; Σ, ∆) = Nil∗ (R) = Nil∗ (R; Σ) = Nil∗ (R; ∆). Proof. From the definitions above, the inclusions Nil∗ (R; Σ, ∆) ⊆ Nil∗ (R; ∆) and Nil∗ (R; ∆) ⊆ Nil∗ (R) follow, so Nil∗ (R; Σ, ∆) ⊆ Nil∗ (R). Consider an element a of Nil∗ (R). Theorem 6.2.14 tells us that a ∈ N0 (R), which means that haim = 0, for some m ∈ N, where hai is the two-sided ideal of R generated by a. Let I be the (Σ, ∆)-ideal generated by a. Then haim = 0 implies that r1 ar2 a · · · arm arm+1 = 0, for any elements r1 , . . . , rm+1 ∈ R, whence I m = 0 (Proposition 6.2.3). If P ∈ Spec(R; Σ, ∆), it is clear that I ⊆ P and so a ∈ Nil∗ (R; Σ, ∆), which shows that Nil∗ (R; Σ, ∆) = Nil∗ (R). Using a similar reasoning we can prove the equalities Nil∗ (R; Σ) = Nil∗ (R) and Nil∗ (R; ∆) = Nil∗ (R), which concludes the proof. t u If D ⊆ R and A is a skew PBW extension over R, then DA will denote the set of elements f of A with coefficients in D, that is, DA := {f ∈ A | f = d0 + d1 X1 + · · · + dm Xm , di ∈ D, for every i}. Theorem 6.2.16. If A is a skew PBW extension over a (Σ, ∆)-compatible and Σ-skew Armendariz ring R, then Nil∗ (A) = Nil∗ (R)A = Nil∗ (R; Σ, ∆)A = Nil∗ (R; Σ)A = Nil∗ (R; ∆)A. Proof. One can see that Nil∗ (R; Σ, ∆)A ⊆ Nil∗ (A), and together with Theorem 6.2.15, we can assert that Nil∗ (R)A = Nil∗P (R; Σ, ∆)A ⊆ Nil∗ (A). On m the other hand, let f ∈ Nil∗ (A) be given by f = i=0 ai Xi . From Theorem t 6.2.13 we obtain that f ∈ Nil0 (A), that is hf i = 0, for some t ∈ N, where hf i is the two-sided ideal of A generated by f . In this way, Lemma 6.2.9 guarantees that (Rai R)t = 0, for every value of i, which means that ai ∈ Nil∗ (R). Hence f ∈ Nil∗ (R)A, and so Nil∗ (A) = Nil∗ (R)A. The other equalities follow from Theorem 6.2.15. t u The above properties guarantee the following fact.

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Corollary 6.2.17. If A is a skew PBW extension over a (Σ, ∆)-compatible and Σ-skew Armendariz ring R, then we have the following equalities: N0 (R)A = N0 (A) = Nil∗ (A) = Nil∗ (R)A = L-rad(A) = L-rad(R)A = Nil∗ (A) = Nil∗ (R)A = Nil∗ (R; Σ, ∆)A = Nil∗ (R; Σ)A = Nil∗ (R; ∆)A. Remark 6.2.18. If R is (Σ, ∆)-compatible and Σ-skew Armendariz, then the following statements are equivalent: R is semiprime; R is (Σ; ∆)-semiprime; R is Σ-semiprime; R is ∆-semiprime; A is semiprime. Amitsur [14] asked the following question: if B is a nil ring, is the polynomial ring B[x]? A negative answer to this question was given by Smoktunowicz [369]. Nevertheless, this question formulated for skew PBW extensions over (Σ, ∆)-compatible and Σ-skew Armendariz rings has a positive answer, as the following result shows. Proposition 6.2.19. If A is a skew PBW extension over a (Σ, ∆)-compatible and Σ-skew Armendariz ring R, then A is a nil ring. Proof. The assertion follows from the results above.

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Proposition 6.2.20. If R is a semiprime (Σ, ∆)-compatible and weak Σskew Armendariz ring, then R has no nonzero nil one-sided ideal. Proof. We follow the ideas presented in [288], Proposition 2.16. Suppose that R has a nonzero nil one-sided ideal. Let I be a nonzero nil right ideal of R (the left case is similar). In the case that there exists a nonzero element a of I with am = 0 and am−1 6= 0, for some m > 2, then every element in aR is nilpotent, since aR is a nil right ideal. Hence, aaRam−1 = 0 (Proposition 6.2.11) which is obviously a contradiction. Then, for every r ∈ I, r2 = 0, and so there exists 0 6= a ∈ I with a2 = 0. Let us see that (aR)3 = 0. If r1 , r2 , r3 are elements of R, then ar2 r1 ar2 r1 = 0 (ar2 r1 ∈ I). So, (r1 ar2 )3 = 0 and a2 = 0. Proposition 6.2.11 guarantees that ar1 ar2 a = 0, which implies that ar1 ar2 ar3 = 0, that is, (aR)3 = 0, and using that R is semiprime, aR = 0 whence a = 0, a contradiction. Therefore our initial assumption is false and R has no nonzero nil right ideal. t u

6.3 K¨ othe’s Conjecture K¨othe’s conjecture, posed in 1930, conjectured that a ring B with no nonzero nil (two-sided) ideals would also have no nonzero nil one-sided ideals (see [207] for the original formulation and [218] for equivalent statements of the conjecture). It is known that this conjecture holds for several classes of rings such as noetherian rings (both left and right noetherian), Goldie rings, rings with right (left) Krull dimension, monomial algebras, P I rings and algebras over uncountable fields. Propositions 6.3.1 and 6.3.3 establish that the conjecture is also true for coefficient rings which are (Σ, ∆)-compatible and weak Σ-skew Armendariz, and skew PBW extensions over these rings.

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Proposition 6.3.1. (Σ, ∆)-compatible and weak Σ-skew Armendariz rings satisfy K¨ othe’s conjecture. Proof. Note that if Nil∗ (R) = 0, then R is semiprime and Proposition 6.2.20 implies that R has no nonzero nil one-sided ideal. t u Proposition 6.3.2. If A is a skew PBW extension of a ring R which is semiprime, (Σ, ∆)-compatible and Σ-skew Armendariz, then A has no nonzero nil one-sided ideal. Proof. We proceed by contradiction. Let I be a P nil right ideal of A and m consider a nonzero element f of A given by f = i=0 ai Xi . Note that for every element r of R, f r ∈ I and so (f r)l = 0, for some l ∈ N. From Lemma 6.2.9, (am r)l = 0, which shows that am R is a nil right ideal of R, i.e., am R = 0 (Proposition 6.2.20). If we repeat this argument we obtain that f = 0, so the assertion follows. t u Theorem 6.3.3. If A is a skew PBW extension of a (Σ, ∆)-compatible and Σ-skew Armendariz ring, then A satisfies K¨ othe’s conjecture. Proof. Consider Nil∗ (A) = 0. Then A is semiprime, and by Remark 6.2.18, R is also semiprime. Since R is Σ-skew Armendariz, Proposition 6.3.2 asserts that the skew PBW extension A has no nonzero nil one-sided ideal, in other words, K¨othe’s conjecture is true for A. t u Example 6.3.4. From Theorem 3.1.5 we know that the Hilbert basis theorem is valid for bijective skew PBW extensions. In this way, for bijective skew PBW extensions over noetherian rings K¨ othe’s conjecture is true. With this in mind, the importance of Theorem 6.3.3 is precisely that we establish that the conjecture is also valid for skew PBW extensions which are not noetherian (if R is noetherian but some of the injective endomorphisms σi ∈ Σ is not bijective then A is not necessarily noetherian). Let us see some situations where Theorem 6.3.3 can be applied: (i) As we observed in Example 6.2.5, since skew PBW extensions over domains are also domains, and every domain is a reduced ring, all these extensions are reduced rings and hence their coefficient rings are reduced, Σ-skew Armendariz, (Σ, ∆)-compatible and Σrigid. (ii) If A is a constant skew PBW extension of a ring R, then it is clear that R is (Σ, ∆)-compatible. So, if R is reduced, then R will be (Σ, ∆)compatible and Σ-skew Armendariz, whence a skew PBW extension over A will satisfy the Koth¨e’s conjecture.

6.4 Description of Minimal Prime Ideals Assuming that the ring R of coefficients of a skew P BW extension A satisfies some of the conditions studied in the present chapter, in this final section we will describe the minimal prime ideals of A. We start with a result that establishes a property for right and left annihilators on (Σ, ∆)-compatible and Σ-skew Armendariz rings.

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Theorem 6.4.1. If A is a skew PBW extension of a (Σ, ∆)-compatible and Σ-skew Armendariz ring R, then (i) ϕ : {rannR (U ) | U ⊆ R} → {rannA (U ) | U ⊆ A}; C → CA is bijective. (ii) ψ : {lannR (U ) | U ⊆ R} → {lannA (U ) | U ⊆ A}; D 7→ AD is bijective. Proof. (i) Let ϕ : {rannR (U ) | U ⊆ R} → {rannA (U ) | U ⊆ A} be defined by C → CA, for every C ∈ {rannR (U ) | U ⊆ R}, and ϕ0 : {rannA (U ) | U ⊆ A} → {rannR (U ) | U ⊆ R} be defined by B 7→ B ∩ R, for all B ∈ {rannA (U ) | U ⊆ A}. Since rannR (U )A = rannA (U ), for every U ⊆ R, ϕ is well defined. By the assumption of (Σ, ∆)-compatibility of R, it follows that rannA (V )∩R = rannR (V0 ), for every V ⊆ A, considering V0 as the set of coefficients of all elements of V . This fact guarantees that ϕ0 is well defined. In this way, ϕ0 ϕ = idR , and so ϕ is injective. Now, if B ∈ {rannA (U ) | U ⊆ A}, then B = rannA (J), for some J ⊆ A. If B1 and J1 are the sets of coefficients of elements of B and J, respectively, let us see that rannR (J1 ) = B1 R. Let Pm Pt f = i=0 ai Xi ∈ J and g = j=0 bj Yj ∈ B. Then f g = 0, and using that R is Σ-skew Armendariz and (Σ, ∆)-compatible, ai bj = 0, for every ai and bj (Proposition 6.2.7). Hence J1 B1 = 0, and so B1 ⊆ rannR (J1 ). The (Σ, ∆)compatibility of R implies that rannR (J1 ) ⊆ B1 R, whence rannR (J1 ) = B1 R, i.e., rannA (J) = B1 A, which shows that ϕ is surjective. Therefore ϕ is bijective. Similarly, we can prove (ii). t u From Theorem 6.4.1 we have immediately the following consequence. Corollary 6.4.2. If A is a skew PBW extension over a (Σ, ∆)-compatible and Σ-skew Armendariz ring R, then R satisfies the ascending chain condition on right (left) annihilators if and only if so does A. Lemma 6.4.3. If R is a semiprime (Σ, ∆)-compatible ring, then every annihilator (Σ, ∆)-prime ideal P of R is a minimal (Σ, ∆)-prime ideal of R. Proof. Consider P 0 , a (Σ, ∆)-prime ideal of R with P 0 ⊆ P . If lannR (P ) ⊆ P 0 holds, then lannR (P ) ⊆ P , and so lannR (P )lR (P ) = 0. Using that R is semiprime, lannR (P ) = 0, a contradiction, so lannR (P ) * P 0 . Nevertheless, lannR (P )P = 0 and P is a (Σ, ∆)-ideal of R. By the (Σ, ∆)-compatibility of R, lannR (P ) is an (Σ, ∆)-ideal of R, and hence P ⊆ P 0 , that is, P = P 0 . t u Lemma 6.4.4. Let A be a skew PBW extension over a ring R which is semiprime, (Σ, ∆)-compatible, Σ-skew Armendariz and satisfies the ascending chain condition on right annihilators. If P is a minimal prime ideal of A, then P ∩ R is a minimal (Σ, ∆)-prime ideal of R. Proof. From Theorem 6.4.1 and Remark 6.2.18 we know that A is semiprime and has the ascending chain condition on right annihilators. By [278], Section 2.2.14, P = rannA (U ) for some subset U of A. Consider the set V consisting of all coefficients of elements of U . Using the (Σ, ∆)-compatibility of R we obtain P ∩ R = rannR (V ). Now, for every a ∈ P ∩ R and v ∈ V , av = 0 and

6.4 Description of Minimal Prime Ideals

125

hence σ θ (a)v = δ θ (a)v = 0, for every θ ∈ N (Proposition 6.2.3), which shows that P ∩ R is a (Σ, ∆)-ideal of R. If IJ ⊆ P ∩ R, for a pair of (Σ, ∆)-ideals I, J of R, then IAJA ⊆ P and hence I ⊆ P ∩ R, whence I ⊆ P ∩ R or J ⊆ P ∩ R. This fact means that P ∩ R is an annihilator (Σ, ∆)-prime ideal of R, and Lemma 6.4.3 guarantees the result. t u Proposition 6.4.5. If R is a semiprime (Σ, ∆)-compatible ring with the ascending chain condition on right annihilators, then every minimal prime ideal of R is a minimal (Σ, ∆)-prime ideal of R and conversely. Proof. From [278], Section 2.2.14, we know that a prime ideal of R is minimal if and only if it is an annihilator ideal. Let P be a minimal prime ideal of R and P = rannR (U ), for some subset U of R. Then U P = 0, which means that for every element r ∈ P and each u ∈ U , ur = 0. By assumption R is (Σ, ∆)-compatible, so uσ θ (r) = uδ θ (r) = 0, for any θ ∈ Nn (Proposition 6.2.3), so U δ θ (r) = U σ θ (r) = 0, that is, P is (Σ, ∆)-prime and so P is a minimal (Σ, ∆)-prime ideal of R (Lemma 6.4.3). Conversely, T let P be a minimal (Σ, ∆)-prime ideal of R. We know that rad(P ) = Qi , where Qi is a minimal prime ideal of R with P ⊆ Qi , for every i. It is clear that P ⊆ rad(P ) ⊆ Qi , for each i, and since Qi is a minimal prime ideal of R, it is also a minimal (Σ, ∆)-prime, as we saw above. Therefore P = Qi and so rad(P ) = P , which proves that P is a minimal prime ideal of R. t u Proposition 6.4.6. Let A be a skew PBW extension of R which is semiprime, (Σ, ∆)-compatible, Σ-skew Armendariz and satisfies the ascending chain condition on right annihilators. If P is a minimal (Σ, ∆)-prime ideal of R, then P A is a minimal prime ideal of A. Proof. Let P be a minimal (Σ, ∆)-prime ideal of R. Using Proposition 6.4.5 we have that P is a minimal prime ideal of R. One can prove that P A is a prime ideal of A. If Q is a minimal prime ideal of A with Q ⊆ P A, by Proposition 6.4.4, Q∩R is a minimal (Σ, ∆)-prime ideal of R and Q∩R ⊆ P , whence Q ∩ R = P . Therefore, Q = (Q ∩ R)A = P A, which concludes the proof. t u Theorem 6.4.7. If A is a skew PBW extension over a ring R which is semiprime, (Σ, ∆)-compatible, Σ-skew Armendariz and satisfies the ascending chain condition on right annihilators, then A has finitely many minimal prime ideals Q1 , . . . , Qm with Q1 · · · Qm = 0, and Qi = pi A = qi A, for each i, where {p1 , . . . , pm } is the set of all minimal prime ideals of R and {q1 , . . . , qm } is the set of all minimal (Σ, ∆)-prime ideals of R. Proof. Let R be a semiprime ring which satisfies the ascending chain condition on right annihilators. Then R has finitely many minimal prime ideals p1 , . . . , pm ([278], Section 2.2.15). From Remark 6.2.18 and Theorem 6.4.1 we know that A is semiprime and has the ascending chain condition on right annihilators. Again, by [278], Section 2.2.15, A has finitely many minimal prime

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ideals Q1 , . . . , Qt , say, with Q1 Q2 · · · Qt = 0. Note that if P is a minimal prime ideal of R, then P is a minimal (Σ, ∆)-prime ideal of R (Proposition 6.4.5), and P A is a minimal prime ideal of A (Proposition 6.4.6). In this way, if Q is a minimal prime ideal of A, then Q∩R is a minimal prime ideal of R (Lemma 6.4.4). Therefore m = t and Proposition 6.4.5 implies the theorem. t u Remark 6.4.8. Recently, in [295], the minimal prime ideals of skew PBW extensions over other important classes of rings have been characterized.

Chapter 7

Dimensions

Now we will compute the global, Krull, Goldie and Gelfand–Kirillov dimensions for bijective skew P BW extensions. All of these computations will be done also for the skew quantum polynomials.

7.1 Global Dimension We start by estimating the global dimension of bijective skew P BW extensions. Note that this includes the particular case of P BW extensions. For this we need the following well known preliminary results. Proposition 7.1.1. Let R ⊆ S be rings such that SR (R S) is faithfully flat and R S (SR ) is projective. Then, lgld(R) ≤ lgld(S), if lgld(R) < ∞, rgld(R) ≤ rgld(S), if rgld(R) < ∞. t u

Proof. See [278], Theorem 7.2.6; see also [241]. Proposition 7.1.2. Let S be a filtered ring, then lgld(S) ≤ lgld(Gr(S)), rgld(S) ≤ rgld(Gr(S)).

t u

Proof. See [278], Corollary 7.6.18; see also [242]). Proposition 7.1.3. Let R be a ring and σ an automorphism of R. Then, lgld(R[x; σ]) = lgld(R) + 1, rgld(R[x; σ]) = rgld(R) + 1. Proof. See [278], Theorem 7.5.3; see also [242].

© Springer Nature Switzerland AG 2020 W. Fajardo et al., Skew PBW Extensions, Algebra and Applications 28, https://doi.org/10.1007/978-3-030-53378-6_7

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Theorem 7.1.4. Let A be a bijective skew P BW extension of a ring R. Then, lgld(R) ≤ lgld(A) ≤ lgld(R) + n, if lgld(R) < ∞, rgld(R) ≤ rgld(A) ≤ rgld(R) + n, if rgld(R) < ∞. If A is quasi-commutative, then lgld(A) = lgld(R) + n, rgld(A) = rgld(R) + n. In particular, if R is semisimple, then lgld(A) = n = rgld(A). Proof. Since A is a filtered ring, by Proposition 7.1.2 lgld(A) ≤ lgld(Gr(A)), rgld(A) ≤ rgld(Gr(A)); according to Theorems 3.1.2 and 3.1.4, Gr(A) is isomorphic to a iterated skew polynomial ring of automorphism type. From Proposition 7.1.3 we get the right inequalities. By Proposition 1.1.11, AR is free, and hence projective and faithfully flat (by definition R A is free, and hence, projective and faithfully flat). From Proposition 7.1.1 we get the left inequalities. If A is quasi-commutative, Theorem 3.1.4 and Proposition 7.1.3 give the equalities. Finally, if R is semisimple, then lgld(R) = 0 = lgld(R). t u Example 7.1.5. The previous results can be applied to estimate the global dimension of skew P BW extensions presented in Chapter 2. In particular, we have the following computations: (i) Let R be a ring, then lgld(R) + n ≤ lgld(An (R)) ≤ lgld(R) + 2n, if lgld(R) < ∞, rgld(R) + n ≤ rgld(An (R)) ≤ rgld(R) + 2n, if rgld(R) < ∞. In addition, if K is a field, in [278] it is proved (see also [242]) that if char(K) = p > 0, then lgld(An (K)) = 2n = rgld(An (K)). If char(K) = 0, then lgld(An (K)) = n = rgld(An (K)). (ii) rgld(Sh ) = 2 = lgld(Sh ). (iii) Let K be a field, if char(K) = p > 0, then rgld(Dh ) = 3 = lgld(Dh ); if char(K) = 0, then rgld(Dh ) = 2 = lgld(Dh ). (iv) rgld(D) = 2n = lgld(D). (v) n ≤ rgld(An (q1 , . . . , qn )) ≤ 2n; n ≤ lgld(An (q1 , . . . , qn )) ≤ 2n. (vi) rgld(On (λji )) = n = lgld(On (λji )). (vii) rgld(Hn (q)) ≤ 3n; lgld(Hn (q)) ≤ 3n. (viii) Let K be a commutative ring, G a finite-dimensional Lie algebra over K and R a K-algebra with K ⊆ R. Then,

7.1 Global Dimension

129

lgld(R) ≤ lgld(R ∗ U (G)) ≤ lgld(R) + dimK (G), if lgld(R) < ∞, rgld(R) ≤ rgld(R ∗ U (G)) ≤ rgld(R) + dimK (G), if rgld(R) < ∞, lgld(R) ≤ lgld(R ⊗ U (G)) ≤ lgld(R) + dimK (G), if lgld(R) < ∞, rgld(R) ≤ rgld(R ⊗ U (G)) ≤ rgld(R) + dimK (G), if rgld(R) < ∞, gld(K) ≤ lgld(U(G)) ≤ gld(K) + dimK (G), if gld(K) < ∞, gld(K) ≤ rgld(U(G)) ≤ gld(K) + dimK (G), if gld(K) < ∞. Moreover, if G is completely solvable, then in [278], Theorem 7.5.7 it is proved that lgld(R ⊗ U (G)) = lgld(R) + dimK (G), rgld(R ⊗ U (G)) = rgld(R) + dimK (G), lgld(U(G)) = gld(K) + dimK (G) = rgld(U (G)). Next we will compute the global dimension of skew quantum polynomials over a ring R. For this we need the following result. Proposition 7.1.6. Let B be a ring and S a multiplicative system of B such that S −1 B (BS −1 ) exists. Then lgld(S −1 B) ≤ lgld(B), rgld(BS −1 ) ≤ rgld(B). Proof. See [278], Corollary 7.4.3; see also [241].

t u

Example 7.1.7. Let R be a ring. Then, (i) lgld(R) ≤ lgld(Qr,n q,σ (R)) ≤ lgld(R) + n, if lgld(R) < ∞. rgld(R) ≤ rgld(Qr,n q,σ (R)) ≤ rgld(R) + n, if rgld(R) < ∞. In fact, condition (ii) of Example 4.4.4 says that Qr,n q,σ (R) is left free over R, but Qr,n (R) is right free over R, also with basis (4.4.2): rxα = xα (σ α )−1 (r) q,σ −1 −1 −1 and xα r = σ α (r)xα ; note that in Qr,n q,σ (R) the identity xi r = σi (r)xi holds, for 1 ≤ i ≤ r. Then the result follows from Proposition 7.1.1, Proposition 7.1.6 and Theorem 7.1.4. (ii) From (i), if R is semisimple, 0 ≤ lgld(R[x±1 ; σ]), rgld(R[x±1 ; σ]) ≤ 1, but R[x±1 ; σ] is not left (right) artinian, then lgld(R[x±1 ; σ]) = 1 = rgld(R[x±1 ; σ]), for R semisimple. Compare this result with [278], Theorem 7.5.3. (iii) From Theorem 7.1.4 we also get that lgld(Rq,σ [x1 , . . . , xn ]) = lgld(R) + n, rgld(Rq,σ [x1 , . . . , xn ]) = rgld(R) + n. If R is semisimple, then lgld(Rq,σ [x1 , . . . , xn ]) = n = rgld(Rq,σ [x1 , . . . , xn ]). (iv) From (i), if R is semisimple, then ±1 ±1 ±1 lgld(Rq,σ [x±1 1 , . . . , xn ]), rgld(Rq,σ [x1 , . . . , xn ] ≤ n.

130

7 Dimensions

7.2 Krull Dimension Now we want to compute the Krull dimension of bijective skew P BW extensions of left (right) noetherian rings. For this we need the following well-known preliminary results. Proposition 7.2.1. Let R ⊆ S be left (right) noetherian rings such that SR (R S) is faithfully flat. Then, lKdim(R) ≤ lKdim(S), rKdim(R) ≤ rKdim(S). Proof. See [278], Corollary 6.5.3, [159], Exercise 15U; see also [241].

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Proposition 7.2.2. Let S be a left (right) noetherian filtered ring. Then, lKdim(S) ≤ lKdim(Gr(S)), rKdim(S) ≤ rKdim(Gr(S)). Proof. See [278], Lemma 6.5.6, see also [242].

t u

Proposition 7.2.3. Let R be a left (right) noetherian ring and σ an automorphism of R. Then, lKdim(R[x; σ]) = lKdim(R) + 1, rKdim(R[x; σ]) = rKdim(R) + 1. Proof. See [278], Proposition 6.5.4; see also [242].

t u

From these propositions and from Theorems 3.1.2, 3.1.4 and 3.1.5 we get the following result. Theorem 7.2.4. Let A be a bijective skew P BW extension of a left (right) noetherian ring R. Then, lKdim(R) ≤ lKdim(A) ≤ lKdim(R) + n, rKdim(R) ≤ rKdim(A) ≤ rKdim(R) + n. If A is quasi-commutative, then lKdim(A) = lKdim(R) + n, rKdim(A) = rKdim(R) + n. In particular, if R is left (right) artinian, then lKdim(A) = n = rKdim(A). Proof. From Theorem 3.1.5 we know that A is a left (right) noetherian ring; by Proposition 1.1.11, AR is free, and hence faithfully flat (by definition R A is free, and hence, faithfully flat). By Propositions 7.2.1 and 7.2.2 we get lKdim(R) ≤ lKdim(A) ≤ lKdim(Gr(A)), rKdim(R) ≤ rKdim(A) ≤ rKdim(Gr(A)). By Theorems 3.1.2 and 3.1.4, Gr(A) is isomorphic to an iterated skew polynomial ring of automorphism type, and hence, we complete the proof of the first part with Proposition 7.2.3. If A is quasi-commutative we apply directly Theorem 3.1.4 and Proposition 7.2.3. t u

7.2 Krull Dimension

131

Example 7.2.5. The results of this section can be applied for computing the Krull dimension of skew P BW extensions presented in Chapter 2. In particular, we have: (i) If R is left (right) noetherian ring, then lKdim(R) + n ≤ lKdim(An (R)) ≤ lKdim(R) + 2n, rKdim(R) + n ≤ rKdim(An (R)) ≤ rKdim(R) + 2n. In addition, if K is a field, in [278] it is proved (see also [242]) that if char(K) = p > 0, then lKdim(An (K)) = 2n = rKdim(An (K)). If char(K) = 0, then lKdim(An (K)) = n = rKdim(An (K)). (ii) rKdim(Sh ) = 2 = lKdim(Sh ). (iii) Let K be a field. Then, rKdim(Dh ) = 3 = lKdim(Dh ) if char(K) = p > 0, rKdim(Dh ) = 2 = lKdim(Dh ) if char(K) = 0. (iv) rKdim(D) = 2n = lKdim(D). (v) n ≤ rKdim(An (q1 , . . . , qn )) ≤ 2n; n ≤ lKdim(An (q1 , . . . , qn )) ≤ 2n. (vi) rKdim(On (λji )) = n = lKdim(On (λji )). (vii) rKdim(Hn (q)) ≤ 3n; lKdim(Hn (q)) ≤ 3n. (viii) Let K be a commutative ring, G a finite-dimensional Lie algebra over K and R a K-algebra with K ⊆ R. If R is left (right) noetherian, then lKdim(R) ≤ lKdim(R ∗ U (G)) ≤ lKdim(R) + dimK (G), rKdim(R) ≤ rKdim(R ∗ U (G)) ≤ rKdim(R) + dimK (G), lKdim(R) ≤ lKdim(R ⊗ U (G)) ≤ lKdim(R) + dimK (G), rKdim(R) ≤ rKdim(R ⊗ U (G)) ≤ rKdim(R) + dimK (G), and for K noetherian Kdim(K) ≤ lKdim(U(G)) ≤ Kdim(K) + dimK (G), Kdim(K) ≤ rKdim(U(G)) ≤ Kdim(K) + dimK (G). Moreover, if G is solvable, then in [278] is proved that lKdim(R ⊗ U (G)) = lKdim(R) + dimK (G), rKdim(R ⊗ U (G)) = rKdim(R) + dimK (G), lKdim(U(G)) = Kdim(K) + dimK (G) = rKdim(U (G)). Next we will compute the Krull dimension of skew quantum polynomials over a ring R. The following result is needed. Proposition 7.2.6. Let B be a left (right) noetherian ring and S a multiplicative system of B such that S −1 B (BS −1 ) exists. Then lKdim(S −1 B) ≤ lKdim(B), rKdim(BS −1 ) ≤ rKdim(B). Proof. See [278], Lemma 6.5.3; see also [241].

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132

7 Dimensions

Example 7.2.7. Let R be a left (right) noetherian ring. Then, (i) lKdim(R) ≤ lKdim(Qr,n q,σ (R)) ≤ lKdim(R) + n, rKdim(R) ≤ rKdim(Qr,n q,σ (R)) ≤ rKdim(R) + n. In fact, as we saw in Example 7.1.7, Qr,n q,σ (R) is left and right free over R, so the result follows from Theorem 3.1.5, Proposition 7.2.1, Proposition 7.2.6 and Theorem 7.2.4. (ii) From (i), if R is left (right) artinian, 0 ≤ lKdim(R[x±1 ; σ]), rKdim(R[x±1 ; σ]) ≤ 1, but R[x±1 ; σ] is not left (right) artinian, so lKdim(R[x±1 ; σ]) = 1 = rKdim(R[x±1 ; σ]), for R left (right) artinian. Compare this result with [278], Proposition 6.5.4. (iii) From Theorem 7.2.4 we get that lKdim(Rq,σ [x1 , . . . , xn ]) = lKdim(R) + n, rKdim(Rq,σ [x1 , . . . , xn ]) = rKdim(R) + n. In particular, lKdim(Rq,σ [x1 , . . . , xn ]) = n = rKdim(Rq,σ [x1 , . . . , xn ]), for R left (right) artinian. Compare this result with [159], Corollary 15.20. (iv) From (i), if R is left (right) artinian, then ±1 ±1 ±1 lKdim(Rq,σ [x±1 1 , . . . , xn ]), rKdim(Rq,σ [x1 , . . . , xn ] ≤ n.

Compare this result with [22], Theorem 3.1.4.

7.3 Goldie Dimension Now we want to compute the uniform dimension, also called the Goldie dimension, of skew P BW extensions. Since all interesting examples of skew P BW extensions have coefficients in a domain (see Chapter 2), we will suppose that R is a domain. For this we need the following preliminary result. Proposition 7.3.1. Let R be a domain. Then, ludim(R) = 1 if and only if R is a left Ore domain. Also, rudim(R) = 1 if and only if R is a right Ore domain. Proof. See [278], Example 2.2.11; see also [242].

t u

Theorem 7.3.2. Let A be a bijective skew PBW extension of a ring R. If R is a left (right) Ore domain, then the Goldie dimension of A is 1, that is, ludim(A) = 1 = rudim(A).

7.4 Gelfand–Kirillov Dimension

133

Proof. This follows from Theorem 4.2.6.

t u

Next we will compute the Goldie dimension of skew quantum polynomials over a domain R. The following result is needed. Example 7.3.3. If R is a left (right) Ore domain, then ludim(Qr,n q,σ (R)) = 1 = r,n rudim(Qr,n q,σ (R)). This follows from Proposition 7.3.1 since Qq,σ (R) is a left (right) Ore domain (see Example 4.4.6).

7.4 Gelfand–Kirillov Dimension In this section we compute the classical Gelfand–Kirillov dimension (GKdim for short) of a bijective skew PBW extension A of R, where R is a finitely generated K-algebra, K a field; all automorphisms and derivations here are K-linear, so A is a K-algebra. Matczuk ([276], Theorem A) established that if R is finitely generated over K (also called an affine K-algebra) and A is a Poincar´e-Birkhoff-Witt extension of R, GKdim(A) = GKdim(R) + n. Matczuk’s result generalizes Proposition 8.2.10 in [278]. We generalize Matczuk’s result for skew P BW extensions. We recall the classical definition of GKdim (see also Section A.1). If A is an arbitrary K-algebra, the classical Gelfand–Kirillov dimension of A is defined by GKdim(A) := sup lim logn (dimK V n ), (7.4.1) V

n→∞

where V vanishes over all frames of A and V n := K hv1 · · · vn |vi ∈ V i (a frame V of A is a finite-dimensional K-subspace of A such that 1 ∈ V ). Next we formulate one of the main results of this section. Consider the automorphisms σi of R in Proposition 1.1.3, 1 ≤ i ≤ n. Theorem 7.4.1. Let R be a K-algebra with a finite-dimensional generating subspace V and let A = σ(R)hx1 , . . . , xn i be a bijective skew P BW extension of R. If σi , δi are K-linear and σi (V ) ⊆ V , for 1 ≤ i ≤ n, then GKdim(A) = GKdim(R) + n. Proof. This theorem was considered first in [329] and [328], however the result has been extended in Section 20.3 to algebras over commutative domains, i.e., assuming that K is a commutative domain. See Theorem 20.3.4 (or also [253]). t u We will also compute the GKdim for skew quantum polynomials (Example 4.4.6). Recall that for a K-algebra B, an automorphism σ of B is said to be locally algebraic if for any b ∈ B the set {σ m (b) | m ∈ N} is contained in a finite-dimensional subspace of B (cf. [230] or [418]). We present next a useful result about rings with a locally algebraic automorphism.

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7 Dimensions

Lemma 7.4.2. If σ is a locally algebraic automorphism of a K-algebra B, then GKdim(B[x; σ]) = GKdim(B) + 1 = GKdim(B[x±1 ; σ]). Proof. See [230], Proposition 1. This result was extended in Section 20.3 to algebras over commutative domains, see Theorem 20.3.4. t u Lemma 7.4.3. Let R be a K-algebra with a finite-dimensional generating subspace V and suppose that σj (V ) ⊆ V for k + 1 ≤ j ≤ n, then ±1 GKdim(Rq,σ [x±1 1 , . . . , xk , xk+1 , . . . , xn ] = ±1 GKdim(Rq0 ,σ0 [x1 , . . . , x±1 k ]) + n − k. ±1 Proof. Recall that Rq,σ [x±1 1 , . . . , xk , xk+1 , . . . , xn ] is as a quasi-commutative bijective skew P BW extension of the k-multiparametric skew quantum torus over R, i.e., ±1 ∼ Rq,σ [x±1 1 , . . . , xk , xk+1 , . . . , xn ] = σ(T )hxk+1 , . . . , xn i, with ±1 T := Rq0 ,σ0 [x1 , . . . , x±1 k ].

Observe that for T , as in Example 4.4.6, σ 0 is defined by σi0 = σi , 1 ≤ i ≤ k. Moreover, T is a K-algebra with a finite-dimensional generating subspace V X ±1 , such that σj (V X ±1 ) ⊆ V X ±1 , for k + 1 ≤ j ≤ n, where X ±1 := ±1 {1, x±1 t u 1 , . . . , xk }. The result follows from Theorem 7.4.1. ±1 Lemma 7.4.4. Let Rq,σ [x±1 1 , . . . , xk ] be the k-multiparametric skew quantum torus. If R is a K-algebra and for every 1 ≤ i ≤ k, the automorphisms σi are locally algebraic, then ±1 GKdim(Rq,σ [x±1 1 , . . . , xk ]) = GKdim(R) + k.

Proof. The statement follows from Lemma 7.4.2 since for every 1 ≤ j ≤ k, the ±1 0 0 automorphism σj0 of Rq,σ [x±1 1 , . . . , xj−1 ] given by σj (r) := σj (r), σj (xi ) := qij xi , for r ∈ R and 1 ≤ i < j, is locally algebraic. t u Theorem 7.4.5. Let R be a K-algebra with a finite-dimensional generating subspace V . If σi is locally algebraic for 1 ≤ i ≤ k and σj (V ) ⊆ V for k + 1 ≤ j ≤ n, then ±1 GKdim(Rq,σ [x±1 1 , . . . , xk xk+1 , . . . , xn ]) = GKdim(R) + n.

Proof. The assertion follows from Lemmas 7.4.3 and 7.4.4.

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Remark 7.4.6. (i) Computations of Gelfand–Kirillov dimension in the literature agree with our Theorems 7.4.1 and 7.4.5 (see [74], [125], [231], [257] and [278]). For instance, GKdim(R[x]) = GKdim(R) + 1, GKdim(Kq [x, y]) = 2, GKdim(U q (sl(2, K))) = 3,

7.4 Gelfand–Kirillov Dimension

135

GKdim(U q (sl(3, K))) = 8, GKdim(A2 (Ja,b )) = 4. Also, GKdim(Am (K)) = 2m, GKdim(An (R)) = GKdim(R) + n, GKdim(U(G)) = dim(G), GKdim(R ⊗ U (G)) = GKdim(R) + dim(G) for a finite-dimensional Lie algebra G. Finally, if R is finitely generated, GKdim(R ∗ U(G)) = GKdim(R) + dim(G). (ii) The restriction in Theorems 7.4.1 and 7.4.5 to bijective skew P BW extensions is necessary as the next examples show. Let R = K[y ±1 , z ±1 ] and consider the skew P BW extensions of R given by A = K[y ±1 , z ±1 ][x; σ] and T = K[y ±1 , z ±1 ][x±1 ; σ], where σ(y) := yz and σ(z) := z. Then, GKdim(R) = 2 and GKdim(A) = GKdim(T ) = 4. See [278], Example 8.2.16 for more details. Anther example is given by Ore extensions B[x; σ, δ] with B a K-algebra and σ a K-endomorphism. Huh and Kim ([179]) showed that GKdim(B[x; σ, δ]) ≥ GKdim(B) + 1, where equality holds whenever each finite dimensional subspace of B is contained in a finitely generated subalgebra of B that is stable under both σ and δ. In general, the difference GKdim(B[x; σ, δ]) − GKdim(B) may be an arbitrary natural number, and it may be infinite.

Chapter 8

Transfer of Homological Properties

In this chapter we study the transfer of some important homological properties from the ring of coefficients R to the extension A = σ(R)hx1 , . . . , xn i. In particular, we will investigate the regularity of these rings and Serre’s theorem on the stably freeness of every finitely generated projective module. Moreover, we will compute the Quillen’s K-groups for these extensions.

8.1 Regularity In order to prove the main result of this section about the regularity of any bijective skew P BW extension we need some preliminary results. Recall that a ring S is left regular if every finitely generated left S-module has finite projective dimension (see [278]). Right regularity is defined in a similar way. Proposition 8.1.1. Let S be a filtered ring. If Gr(S) is left (right) regular, then S is left (right) regular. Proof. See [278], Proposition 7.7.4; see also [242].

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Proposition 8.1.2. If R is a left (right) regular and left (right) noetherian ring and σ is an automorphism, then R[x; σ] is left (right) regular. Proof. See [278], Theorem 7.7.5; see also [242].

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Theorem 8.1.3. Let A be a bijective skew P BW extension of a ring R. If R is a left (right) regular and left (right) noetherian ring, then A is left (right) regular. Proof. Theorems 3.1.2 and 3.1.4 say that Gr(A) is isomorphic to a iterated skew polynomial ring of automorphism type with coefficients in R, then the result follows from Propositions 8.1.2 and 8.1.1. t u Example 8.1.4. The previous theorem can be applied to skew P BW extensions presented in Chapter 2. In particular, we have: © Springer Nature Switzerland AG 2020 W. Fajardo et al., Skew PBW Extensions, Algebra and Applications 28, https://doi.org/10.1007/978-3-030-53378-6_8

137

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8 Transfer of Homological Properties

(a) If R is a left (right) regular and left (right) noetherian ring, then: (i) An (R) is left (right) regular. Moreover, if K is a field, then Bn (K) is left (right) regular. (ii) Let K be a commutative ring and G a finite-dimensional Lie algebra over K. If R is a K-algebra, with K ⊆ R, then R ∗ U (G) is left (right) regular. In particular, R ⊗ U (G) is left (right) regular. If K is noetherian and regular, then U (G) is left (right) regular. (b) The following rings are left (right) regular: Sh , Dh , D, An (q1 , . . . , qn ), On (λji ), Hn (q). Now we will consider the regularity of skew quantum polynomials over a ring R. For this we need the following preliminary result. Proposition 8.1.5. Let B be a left (right) regular ring and S a multiplicative system of B such that S −1 B (BS −1 ) exists, then S −1 B (BS −1 ) is left (right) regular. Proof. See [278], Proposition 7.7.3; see also [242].

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Example 8.1.6. Let R be a left (right) regular and left (right) noetherian ring, then the ring Qr,n q,σ (R) of skew quantum polynomials over R is left (right) regular: this follows from Theorem 8.1.3 and Proposition 8.1.5. Compare this result with [22] and [278].

8.2 Serre’s Theorem A very important consequence of Theorems 3.1.5 and 8.1.3 for bijective skew P BW extensions is the famous Serre’s theorem. Definition 8.2.1. A ring B is PSF if each finitely generated projective Bmodule is stably free. A module M is stably free if there exist integers r, s ≥ 0 such that B r ∼ = Bs ⊕ M . We will consider again stably free modules and PSF rings (see Corollary 9.2.5) in Part II of the present monograph from a matrix point of view. Proposition 8.2.2. If B is a filtered ring with filtration {Bp }p≥0 such that Gr(B) is left noetherian, left regular, and flat as a right B0 -module, then B is PSF when B0 is PSF . Proof. See [278], Theorem 12.3.2.

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Theorem 8.2.3 (Serre’s theorem). Let A be a bijective skew P BW extension of a ring R such that R is left (right) noetherian, left (right) regular and PSF . Then A is PSF .

8.3 Auslander Regularity

139

Proof. By Theorem 3.1.2, A is filtered, A0 = R, and Gr(A) is a quasicommutative bijective skew P BW extension of R; Theorem 3.1.5 says that Gr(A) is left noetherian, and Theorem 8.1.3 implies that Gr(A) is left regular. Moreover, Gr(A) is flat as a right R-module (see Proposition 1.1.11), then assuming that R is PSF we get from Proposition 8.2.2 that A is PSF . t u Proposition 8.2.4. Let B be a left (right) noetherian, left (right) regular and PSF ring, and let S be a multiplicative system of B such that S −1 B (BS −1 ) exists, then S −1 B (BS −1 ) is PSF . Proof. See [278], Proposition 12.1.12.

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Example 8.2.5. If R is left noetherian, left regular and P SF , then the ring Qr,n q,σ (R) of skew quantum polynomials over R is PSF . This follows from Theorem 8.2.3 and Proposition 8.2.4.

8.3 Auslander Regularity In this section we will study the Auslander regularity for skew P BW extensions, for this we will follow the presentation given in [250] and [393]. We start by recalling the notion of the grade of a module (see also Section 16.6). Definition 8.3.1. Let B be a ring. (i) The grade j(M ) of a left (or right) B-module M is defined by j(M ) := min{i | ExtiB (M, B) 6= 0} or ∞ if no such i exists. (ii) B satisfies the Auslander condition if for every noetherian left (or right) B-module M and for all i ≥ 0, j(N ) ≥ i for all submodules N ⊆ ExtiB (M, B). (iii) B is Auslander–Gorenstein (AG) if B is two-sided noetherian, satisfies the Auslander condition, id(B B) < ∞ and id(BB ) < ∞. (iv) B is Auslander regular (AR) if it is AG and gld(B) < ∞. Remark 8.3.2. (i) We recall that if a ring B is two-sided noetherian, then gld(B) := rgld(B) = lgld(B). (ii) If B is two-sided noetherian and if id(B B) < ∞ and id(BB ) < ∞, then id(B B) = id(BB ) (see [414]). Some preliminaries are needed for the main result of the present section. Proposition 8.3.3. If B is AG, respectively AR, then the skew polynomial ring B[x; σ, δ], with σ bijective, is also AG, respectively AR. Proof. See [112], Theorem 4.2.

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8 Transfer of Homological Properties

Definition 8.3.4. Let B be a filtered ring with filtration {Fn (B)}n∈Z . (i) The Rees ring associated to B is a graded ring defined by e := L B n∈Z Fn (B). (ii) The filtration {Fn (B)}n∈Z is left (right) Zariskian, and B is called a left (right) Zariski ring if F−1 (B) ⊆ Rad(F0 (B)) and the associated e is left (right) noetherian. Rees ring B Proposition 8.3.5. Let B be a left and right Zariski ring. If its associated graded ring Gr(B) is AG, respectively AR, then so too is B. Proof. See [63], Theorem 3.9.

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Proposition 8.3.6. Let B be an N-filtered ring such that Gr(B) is left (right) noetherian. Then, B is left (right) Zariskian. t u

Proof. See [256].

Proposition 8.3.7. Let B be an AG ring and S a multiplicative Ore set of regular elements of B. Then, S −1 B (and also BS −1 ) is AG. Proof. See [8], Proposition 2.1.

t u

Lemma 8.3.8. If A is a bijective skew P BW extension of a noetherian ring R, then A is a left and right Zariski ring. Proof. Since A is N-filtered, 0 = F−1 (A) ⊆ Rad(F0 (A)) = Rad(R). By Theorem 3.1.4, Gr(A) is isomorphic to an iterated skew polynomial ring R[z1 ; θ1 ] · · · [zn ; θn ], with θi is bijective, 1 ≤ i ≤ n. Whence Gr(A) is noetherian. Proposition 8.3.6 says that A is a left and right Zariski ring. t u Now we can prove the main result of the present section. Theorem 8.3.9. Let A be a bijective skew P BW extension of a ring R such that R is AG, respectively AR, then A is AG, respectively AR. Proof. According to Theorem 3.1.2, Gr(A) is a quasi-commutative skew P BW extension, and by the hypothesis, Gr(A) is also bijective. By Theorem 3.1.4, Gr(A) is isomorphic to an iterated skew polynomial ring R[z1 ; θ1 ] · · · [zn ; θn ] such that each θi is bijective, 1 ≤ i ≤ n. Proposition 8.3.3 says that Gr(A) is AG, respectively AR. From Lemma 8.3.8, A is a left and right Zariski ring, so by Proposition 8.3.5, A is AG, respectively AR. t u Corollary 8.3.10. If R is AG, respectively AR, then the ring of skew quantum polynomials ±1 ±1 Qr,n q,σ (R) = Rq,σ [x1 , . . . , xr , xr+1 , . . . , xn ]

is AG, respectively AR. Proof. Let R be AG; according to Example 4.4.4, Qr,n q,σ (R) is a localization of a bijective skew P BW extension A of the ring R by a multiplicative Ore set of regular elements of A. From Theorem 8.3.9 and Proposition 8.3.7 we get that Qr,n q,σ (R) is AG. If R is AR, then R is AG and gld(R) < ∞, whence r,n Qr,n q,σ (R) is AG, but from Example 7.1.7 we get that gld(Qq,σ (R)) < ∞, so r,n Qq,σ (R) is AR. t u

8.4 Cohen–Macaulayness

141

8.4 Cohen–Macaulayness In this section we study the Cohen–Macaulay property for skew P BW extensions following the presentation given in [250] and [393]. For the notion of the Gelfand–Kirillov dimension of a module see [210], or also [241]. In this section K is a field, B is a K-algebra and the grade of a B-module M is denoted by jB (M ). Definition 8.4.1. Let B be an algebra over a field K. We say that B is Cohen–Macaulay (CM) with respect to the classical Gelfand–Kirillov dimension if GKdim(B) = jB (M ) + GKdim(M ) for every nonzero noetherian B-module M . Proposition 8.4.2. Suppose that B is a left and right Zariskian ring, and Gr(B) is AG. Then jB (M ) = jGr(B) (Gr(M )) for every non-zero noetherian B-module M . Proof. If M is a finitely generated B-module and if {Fn (M )} is a good filtration on M (see [63], [112], or also [393]), then in general jB (M ) ≤ jGr(B) (Gr(M )), but if B is a left and right Zariski ring and Gr(B) is AG then jB (M ) ≥ jGr(B) (Gr(M )), so jB (M ) = jGr(B) (Gr(M )) (see [63], Proof of Theorem 3.9). t u Now it is possible to prove the following proposition. Proposition 8.4.3. Let B be a left and right Zariski ring with finite filtration and such that Gr(B) is AG. If Gr(B) is CM , then B is CM . Proof. Let M 6= 0 be a noetherian B-module, then GKdim(B) = GKdim(Gr(B)) = GKdim(Gr(B) Gr(M )) + jGr(B) (M ) = GKdim(B M ) + jB (M ) ( [210], P roposition 6.6 ). Therefore B is CM .

t u

Proposition 8.4.4. Suppose that R is an AR (AG) and CM ring. Let R[x; σ, δ] be an Ore extension with σ bijective. If R = ⊕i≥0 Ri is a connected graded K-algebra (i.e., R0 = K) such that σ(Ri ) ⊆ Ri for each i ≥ 0. Then R[x; σ, δ] is CM . Proof. See [233], Lemma, Part (ii).

t u

Definition 8.4.5. Let B be a K-algebra. We say that x ∈ B is a local normal element if for every frame V ⊂ B, there is a frame V 0 ⊃ V such that xV 0 = V 0 x. It is clear that every central element is local. The next proposition says that the CM property is preserved under certain localizations.

142

8 Transfer of Homological Properties

Proposition 8.4.6. Let B be an AG ring, and S a multiplicatively closed set of local normal elements in B. If B is CM , so is S −1 B. Proof. cf. [8], Theorem 2.4.

t u

Theorem 8.4.7. Let A be a P bijective skew P BW extension of a ring R such that R is AG, CM , and R = i≥0 ⊕Ri is a connected graded K-algebra such that σj (Ri ) ⊆ Ri for each i ≥ 0 and 1 ≤ j ≤ n, then A is CM . Proof. It is clear that A is a K-algebra with a finite filtration. By Theorem 3.1.2, Gr(A) is a quasi-commutative skew P BW extension, and by the hypothesis, Gr(A) is also bijective. By Theorem 3.1.4, Gr(A) is isomorphic to an iterated skew polynomial ring R[z1 ; θ1 ] · · · [zn ; θn ] such that each θi is bijective, 1 ≤ i ≤ n, moreover, θj (r) = σj (r) for every r ∈ R. Proposition 8.3.3 says that Gr(A) is AG. From Lemma 8.3.8, A is a left and right Zariski ring, and by Proposition 8.4.4 Gr(A) is CM , so from Proposition 8.4.3 we get that A is CM . t u P Corollary 8.4.8. Let R = i≥0 ⊕Ri be a connected graded K-algebra such that σj (Ri ) ⊆ Ri for every i ≥ 0 and all σj in the definition of skew quantum r,n polynomials Qr,n q,σ . If R is AG and CM , then Qq,σ is CM . Proof. By Example 4.4.4, Qr,n q,σ is a localization of a bijective skew P BW extension A of R by a multiplicative Ore set of regular elements of A, and the multiplicative set generated by x1 , . . . , xr consists of monomials, which are local normal elements. From Theorem 8.4.7 and Proposition 8.4.6 we get that Qr,n t u q,σ is CM .

8.5 Strongly Noetherian Algebras In this section K will denote a commutative ring. We will study the strongly noetherianity for skew P BW extensions (see also [250] and [393]). Definition 8.5.1. Let B be a left noetherian K-algebra. We say that B is left strongly noetherian if for any commutative noetherian K-algebra C, C ⊗K B is left noetherian. Proposition 8.5.2. Let R be a left strongly noetherian K-algebra. Then, R[x; σ, δ] is left strongly noetherian when σ is bijective. Proof. It is clear that A := R[x; σ, δ] is a left noetherian K-algebra; let C be a commutative noetherian K-algebra, we have to show that C ⊗K A is left noetherian. For this it is enough to prove the isomorphism C ⊗K A ∼ = (C ⊗K R)[x; σ, δ], where σ, δ : C ⊗K R → C ⊗K R are defined in the following natural way, σ := iC ⊗ σ, δ := iC ⊗ δ; it is easy to check that σ is a bijective ring homomorphism and δ is a σ-derivation.

8.5 Strongly Noetherian Algebras

143

For proving the claimed isomorphism firstly we apply the universal property of the skew polynomial ring (C ⊗K R)[x; σ, δ], thus, we consider the ring homomorphism f : C ⊗K R → C ⊗K A induced by the bilinear Kbalanced function f 0 : C × R → C ⊗K A defined by f 0 (c, r) := c ⊗ r; let y := 1⊗x ∈ C ⊗K A, note that y satisfies yf (c⊗r) = f (σ(c⊗r))y+f (δ(c⊗r)), so there exists a ring homomorphism f : (C ⊗K R)[x; σ, δ] → C ⊗K A defined by f (c ⊗ r) := f (c ⊗ r) = c ⊗ r and f (x) := 1 ⊗ x. To conclude the proof we have to define a ring homomorphism g : C ⊗K A → (C ⊗K R)[x; σ, δ] such that f g = iC⊗K A and g f = i(C⊗K R)[x;σ,δ] . The homomorphism g is induced by the bilinear K-balanced function g 0 : C × A → (C ⊗K R)[x; σ, δ] defined by g 0 (c, r0 + r1 x + · · · + rn xn ) := (c ⊗ r0 ) + (c ⊗ r1 )x + · · · + (c ⊗ rn )xn , so g(c ⊗ (r0 + r1 x + · · · + rn xn )) := (c ⊗ r0 ) + (c ⊗ r1 )x + · · · + (c ⊗ rn )xn and it is easy to check that g satisfies the mentioned equalities.

t u

Proposition 8.5.3. Let B be an N-filtered K-algebra. If Gr(B) is left strongly noetherian, then B is left strongly noetherian. Proof. Let {Fn (B)}n∈N be a filtration of B and let C be a commutative noetherian K-algebra. We have to show that C ⊗K B is left noetherian. {C ⊗ Fn (B)}n∈N is a filtration of C ⊗K B and we have a surjective ring homomorphism C ⊗K Gr(B) → Gr(C ⊗K B), therefore Gr(C ⊗K B) is left noetherian, and hence, C ⊗K B is left noetherian. t u Proposition 8.5.4. Let B be a left strongly noetherian K-algebra, and let S be a multiplicative left Ore set of B. Then S −1 B is left strongly noetherian. Proof. Let C be a commutative noetherian K-algebra and consider the canonical ring homomorphism φ : C ⊗K B → C ⊗K S −1 B; let B 0 := Im(φ) and S 0 := φ(1 ⊗ S), note that S 0 is a multiplicative subset of B 0 and C ⊗K S −1 B ∼ = S 0−1 B 0 . This isomorphism can be proved using the inclu0 sion ι : B → C ⊗K S −1 B: In fact, ι is an injective ring homomorphism, ι(S 0 ) ⊆ (C ⊗K S −1 B)∗ and every element z ∈ C ⊗K S −1 B can be represented as z = ι(s0 )−1 ι(z 0 ), with z 0 ∈ B 0 . Since C ⊗K B is left noetherian, B 0 is left noetherian, whence S 0−1 B 0 is left noetherian. t u We are ready to prove the main results of the section. Theorem 8.5.5. Let A = σ(R)hx1 , . . . , xn i be a bijective skew P BW extension of a left strongly noetherian K-algebra R. Then A is left strongly noetherian. Proof. As in the proof of Theorem 8.3.9, Gr(A) is isomorphic to an iterated skew polynomial ring R[z1 ; θ1 ] · · · [zn ; θn ] such that each θi is bijective, 1 ≤ i ≤ n. The result follows from Propositions 8.5.2 and 8.5.3 t u

144

8 Transfer of Homological Properties

Corollary 8.5.6. If R is a left strongly noetherian K-algebra, then the ring of skew quantum polynomials ±1 ±1 Qr,n q,σ (R) := Rq,σ [x1 , . . . , xr , xr+1 , . . . , xn ]

is left strongly noetherian. Proof. The result follows from Theorem 8.5.5 and Proposition 8.5.4.

t u

8.6 K-theory In this section we compute Quillen’s K-groups for bijective skew PBW extensions, in particular, we compute Grothendieck, Bass and Milnor groups for these extensions (see also [248]). For this we use the following deep result due to Quillen. Theorem 8.6.1. Let B be a filtered ring with filtration {Bp }p≥0 such that B0 and Gr(B) are left noetherian left regular rings. Then the functor B ⊗B0 − induces isomorphisms Ki (B) ∼ = Ki (B0 ),

for all i ≥ 0. t u

Proof. [325], see also [175].

Theorem 8.6.2. Let R be a left noetherian left regular ring. If A is a bijective skew PBW extension of R, then Ki (A) ∼ = Ki (R) for all i ≥ 0. Proof. Theorem 3.1.2 describes explicitly the filtration of A with F0 = R. The proof of Theorem 3.1.5 shows that Gr(A) is left noetherian and the proof of Theorem 8.1.3 guarantees that Gr(A) is left regular (see [248]). The assertion follows from Quillen’s result. t u According to Theorem 8.6.2, to effectively calculate Quillen’s K-groups for bijective skew PBW extensions we have to compute Ki (R), but for many remarkable examples of bijective skew PBW extensions the ring R of coefficients is an iterated Laurent polynomial ring. Thus, Theorem 8.6.2 can be complemented with the next proposition, which can be used to effective compute Quillen’s K-groups for the examples considered in [248]. Proposition 8.6.3. Let B be a left noetherian left regular ring. Then ±1 ∼ Km (B[x±1 1 , . . . , xn ]) =

m M

[Kj (B)]n Cm−j ,

j=0

 where

n Cm−j :=

 n . m−j

(8.6.1) ±1 In particular, Grothendieck, Bass and Milnor groups of B[x±1 , . . . , x n ] are 1 given by ±1 ∼ (i) K0 (B[x±1 1 , . . . , xn ]) = K0 (B).

8.6 K-theory

145

±1 ∼ n (ii) K1 (B[x±1 1 , . . . , xn ]) = [K0 (B)] ⊕ K1 (B). n(n−1) ±1 ±1 ∼ (iii) K2 (B[x1 , . . . , xn ]) = [K0 (B)] 2 ⊕ [K1 (B)]n ⊕ K2 (B).

Proof. With the conditions on B, the proof is by induction on m and n and using the following well-known identity ([325] Theorem 8 or [370] Corollary 5.5): Ki (B[x±1 ]) ∼ = Ki (B) ⊕ Ki−1 (B), where K−1 (B) := 0.

(8.6.2)

For m = 0 and n = 0 (8.6.1) it is trivial. Let m = 0 and n > 0. By induction on n: ±1 ±1 ±1 ∼ ±1 K0 (B[x±1 1 , . . . , xn ]) = K0 (B[x1 , . . . , xn−1 ][xn ]) ±1 ±1 ±1 ±1 ±1 ∼ = K0 (B[x±1 1 , . . . , xn−1 ][xn ]) ⊕ K−1 (B[x1 , . . . , xn−1 ][xn ]) ∼ = K0 (B).

Let m > 0. By induction on m: ±1 ±1 ±1 ∼ ±1 Km (B[x±1 1 , . . . , xn ]) = Km (B[x1 , . . . , xn−1 ][xn ]) ∼ = Km (B[x±1 , . . . , x±1 ]) ⊕ Km−1 (B[x±1 , . . . , x±1 ]) 1

n−1

1

±1 ±1 ∼ = Km (B[x±1 1 , . . . , xn−2 ][xn−1 ]) ⊕

m−1 M

n−1

[Kj (B)]n−1 Cm−1−j

j=0 ±1 ±1 ±1 ∼ = Km (B[x±1 1 , . . . , xn−2 ]) ⊕ Km−1 (B[x1 , . . . , xn−2 ])

⊕

m−1 M

[Kj (B)]n−1 Cm−1−j

j=0 ±1 ∼ = Km (B[x±1 1 , . . . , xn−2 ]) ⊕

m−1 M

[Kj (B)]n−2 Cm−1−j ⊕

m−1 M

j=0

[Kj (B)]n−1 Cm−1−j

j=0

±1 ±1 ∼ = Km (B[x±1 1 , . . . , xn−3 ][xn−2 ]) ⊕

m−1 M

[Kj (B)]n−2 Cm−1−j

j=0

⊕

m−1 M

[Kj (B)]n−1 Cm−1−j

j=0 ±1 ±1 ±1 ∼ = Km (B[x±1 1 , . . . , xn−3 ]) ⊕ Km−1 (B[x1 , . . . , xn−3 ])

⊕

m−1 M

[Kj (B)]n−2 Cm−1−j ⊕

j=0

m−1 M j=0

±1 ∼ = Km (B[x±1 1 , . . . , xn−3 ]) ⊕

m−1 M j=0

⊕

m−1 M j=0

[Kj (B)]n−1 Cm−1−j

[Kj (B)]n−1 Cm−1−j .

[Kj (B)]n−3 Cm−1−j ⊕

m−1 M j=0

[Kj (B)]n−2 Cm−1−j

146

8 Transfer of Homological Properties

Following this procedure we obtain ±1 Km (B[x±1 1 , . . . , xn ])

∼ = Km (B[x±1 1 ]) ⊕

m−1 M

[Kj (B)]1 Cm−1−j ⊕

j=0

⊕ ··· ⊕

m−1 M

m−1 M

[Kj (B)]2 Cm−1−j

j=0

[Kj (B)]n−2 Cm−1−j ⊕

j=0

m−1 M

[Kj (B)]n−1 Cm−1−j

j=0

∼ = Km (B) ⊕ Km−1 (B) ⊕

m−1 M

[Kj (B)]1 Cm−1−j +2 Cm−1−j +···+n−2 Cm−1−j +n−1 Cm−1−j

j=0

∼ = Km (B) ⊕ [Km−1 (B)]n ⊕

m−2 M

[Kj (B)]1 Cm−1−j +2 Cm−1−j +···+n−2 Cm−1−j +n−1 Cm−1−j .

j=0

Note that

n−1 X t=1

t m−1−j

 =

n−1 X t=m−1−j



t m−1−j



and using the equality           m−1−j m−j n−2 n−1 n + +· · ·+ + = , m−1−j m−1−j m−1−j m−1−j m−j we have ±1 ∼ n Km (B[x±1 1 , . . . , xn ]) = Km (B) ⊕ [Km−1 (B)] ⊕

m−2 M

[Kj (B)]n Cm−j

j=0

or equivalently ±1 ∼ Km (B[x±1 1 , . . . , xn ]) =

m M [Kj (B)]n Cm−j . j=0

t u Next we compute Quillen’s K-groups for skew quantum polynomials. We will see that these computations generalize the results presented in [22]. Lemma 8.6.4. If R is a left noetherian left regular ring, then ±1 ±1 ±1 ∼ Ki (Rq,σ [x±1 1 , . . . , xr , xr+1 , . . . , xn ]) = Ki (Rq,σ [x1 , . . . , xr ]), i ≥ 0. (8.6.3)

8.6 K-theory

147

±1 Proof. Note that Rq,σ [x±1 1 , . . . , xr , xr+1 , . . . , xn ] is a quasi-commutative bijective skew P BW extension of the r-multiparametric skew quantum torus over R, i.e., ±1 ∼ Rq,σ [x±1 1 , . . . , xr , xr+1 , . . . , xn ] = σ(T )hxr+1 , . . . , xn i, with ±1 T := Rq,σ [x1 , . . . , x±1 r ].

Note that for T , σ is defined by the σj in Example 4.4.4, with 1 ≤ j ≤ r. Then the result follows from Theorem 8.6.2 since T is left noetherian left regular (see Theorems 3.1.5 and 8.1.3). t u This lemma says that computing Quillen’s K-groups is reduced to com±1 puting Ki (Rq,σ [x±1 1 , . . . , xr ]) for i ≥ 0, but note that ±1 ±1 ∼ ±1 Rq,σ [x±1 1 , . . . , xr ] = R[x1 ; σ1 ] · · · [xr ; σr ], xj r = σj (r)xj , 1 ≤ j ≤ r, σj (xi ) := qij xi , 1 ≤ i < j ≤ r, so ±1 ±1 ∼ ±1 Ki (Rq,σ [x±1 1 , . . . , xr ]) = Ki (R[x1 ; σ1 ] · · · [xr ; σr ]),

i ≥ 0.

(8.6.4)

±1 In other words, we need to compute Ki (R[x±1 1 ; σ1 ] · · · [xr ; σr ]), i ≥ 0. With this purpose in mind, we present the following proposition, which is a generalization of (8.6.2) for skew Laurent polynomial rings.

Proposition 8.6.5. Let B be a left noetherian left regular ring. If σ is an automorphism of B that acts trivially on the K-theory of B, then Ki (B[x±1 ; σ]) ∼ = Ki (B) ⊕ Ki−1 (B), where K−1 (B) := 0. Proof. The idea is to apply the following long exact sequence of K-groups (see [409] Corollary 2.2.): 1−σ

∗ · · · → Ki (B) −−−→ Ki (B) → Ki (B[x±1 ; σ]) → Ki−1 (B) → · · · .

(8.6.5)

The assumption that σ acts trivially on the K-theory of B implies that from the long exact sequence (8.6.5) we can extract a short exact sequence 0 → Ki (B) → Ki (B[x±1 ; σ]) → Ki−1 (B) → 0 (cf. [325], [370] or [409]), and hence Ki (B[x±1 ; σ]) ∼ = Ki (B) ⊕ Ki−1 (B).

t u

Proposition 8.6.6. Suppose that R is left noetherian left regular and that σj , 1 ≤ j ≤ r − 1, acts trivially on the K-theory of the j − 1-multiparametric ±1 skew quantum torus R[x±1 1 ; σ1 ] · · · [xj−1 ; σj−1 ]. Then ±1 Km (R[x±1 1 ; σ1 ] · · · [xr ; σr ])

∼ =

m M

[Kj (R)]r Cm−j .

(8.6.6)

j=0

Proof. Similar to the proof of Proposition 8.6.3 and using Proposition 8.6.5. t u

148

8 Transfer of Homological Properties

Proposition 8.6.7. If R is left noetherian left regular, then for the r-multi±1 parametric skew quantum torus Rq,σ [x±1 1 , . . . , xr ] we have ±1 ∼ r K1 (Rq,σ [x±1 1 , . . . , xr ]) = [K0 (R)] ⊕ K1 (R).

Proof. Similar to the proof of Proposition 8.6.3 and using the fact that if B is a left noetherian left regular ring with an automorphism σ and i : B → B[x±1 ; σ] is the natural embedding, then the sequence i

∂

∗ 1 → K1 (B) −→ K1 (B[x±1 ; σ]) − → K0 (B) → 0

is exact (see [325], p. 122 and [36], Ch. 9. Theorem 6.3).

t u

Corollary 8.6.8. If R is left noetherian left regular then for the skew quantum polynomial we have ±1 r ∼ K1 (Rq,σ [x±1 1 , . . . , xr , xr+1 , . . . , xn ]) = [K0 (R)] ⊕ K1 (R).

Proof. Follows from Lemma 8.6.4 and Proposition 8.6.7.

t u

Now we can establish the main result for skew quantum polynomials over rings. Theorem 8.6.9. Under the same conditions of Proposition 8.6.6, we have ±1 ∼ Km (Rq,σ [x±1 1 , . . . , xr , xr+1 , . . . , xn ]) =

m M

[Kj (R)]r Cm−j .

j=0

In particular, ±1 ∼ K0 (Rq,σ [x±1 1 , . . . , xr , xr+1 , . . . , xn ]) = K0 (R); ±1 ±1 ∼ [K0 (R)]r ⊕ K1 (R); K1 (Rq,σ [x , . . . , x , xr+1 , . . . , xn ]) = 1

r

±1 K2 (Rq,σ [x±1 1 , . . . , xr , xr+1 , . . . , xn ])

r(r−1) ∼ = [K0 (R)] 2 ⊕ [K1 (R)]r ⊕ K2 (R).

Proof. Follows from Lemma 8.6.4 and Proposition 8.6.6.

t u

8.7 Summary and Remarks In this section we summarize the main results related to the basic theoretic properties of skew P BW extensions studied in the first part of this monograph. The conclusions given next generalize or agree with many results of the literature, in particular, compare with [20], [22], [36], [40], [41], [73], [74], [159], [231], [257], [276] [278] and [370]. Some additional remarks are included. 1. From Proposition 3.2.1 we get that if R and K are domains (in particular, if K is a field), then all rings and algebras presented in Chapter 2 are

8.7 Summary and Remarks

2.

3.

4.

5.

6.

7.

149

domains. Moreover, from Example 4.4.6 we get that the ring of skew quantum polynomials Qr,n q,σ (R) is also a domain. From Theorem 3.2.6 we can conclude that if R and K are prime rings, then all rings and algebras of Chapter 2 are prime. From this we get also that all of these rings have trivial prime radical. In addition, from Example 4.4.6 we conclude that Qr,n q,σ (R) is also a prime ring and hence rad(Qr,n q,σ (R)) = 0. From Theorem 3.2.3 we get that if R and K are domains, then all rings and algebras of Chapter 2 have trivial Jacobson radical. In addition, from Example 4.4.6 we conclude that Rad(Qr,n q,σ (R)) = 0. From Theorems 4.2.5 and 4.2.6, and assuming that R and K are left (right) Ore domains, all rings and algebras of Chapter 2 are left (right) Ore domains, and hence, they have left (right) division ring of fractions. Moreover, Qr,n q,σ (R) is also a left (right) Ore domain, and hence, it has left (right) division ring of fractions (see Example 4.4.6). From Theorem 4.3.9 we conclude that if R and K are semiprime left (right) Goldie rings, then all rings and algebras of Chapter 2 are semiprime left (right) Goldie, and hence, Ql (A) (or Qr (A)) exists and is semisimple. r,n Moreover, Qr,n q,σ (R) is also semiprime left (right) Goldie and Ql (Qq,σ (R)) r,n (or Qr (Qq,σ (R))) is semisimple (see Example 4.4.6) . From Proposition 4.3.1 we get also that R and K are prime left (right) noetherian rings, then all rings and algebras of Chapter 2 have simple left (right) artinian total ring of fractions. Moreover, Ql (Qr,n q,σ (R)) (or Qr (Qr,n q,σ (R))) is simple left (right) artinian. From Theorems 3.1.5, 8.1.3, 8.2.3 and assuming that R and K in the previous section are left (right) noetherian, left (right) regular and P SF rings, we can conclude that all rings and algebras presented in Chapter 2 are left (right) noetherian, left (right) regular and P SF rings. Note also that Qr,n q,σ (R) is left (right) noetherian, left (right) regular and P SF (see Examples 4.4.6, 8.1.6 and 8.2.5). Compare this result with [20] and [22]. Li ([257]) shows that linear solvable polynomial algebras are left noetherian, left regular and PSF rings (all these algebras considered over a field). We remark that the results obtained here generalize the results of Li because we deal with extensions of rings more general than fields. In a similar way as we saw in Examples 7.1.5 and 7.2.5, we can estimate the global and Krull dimensions of all rings and algebras of Chapter 2 using Theorems 7.1.4 and 7.2.4 (see also [248]). The global and Krull dimensions of skew quantum polynomials were computed in Examples 7.1.7 and 7.2.7. In the tables below, dim represents the global or Krull dimension, R is a left noetherian ring and K is a field (or a commutative noetherian ring in some particular examples). Our estimations agree with some exact computations that we found in the literature. For example: (i) lK.dim(U q (sl(2, K))) = 2 if q is not a root of unity, and the value is 3 otherwise (cf. [40]); (ii) lK.dim(W ν (sl(2, K))) = 3 (cf. [40] and [41]); (iii) lK.dim(Oq (M (2, K))) = 4 (cf. [40] and [41]);

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8 Transfer of Homological Properties

(iv) (v) (vi) (vii)

lK.dim(Oq (SL2 (K))) = 3 (cf. [40] and [41]); lK.dim(H1 (q)) = 3 (cf. [40] and [41]); lK.dim(U (sl(2, C))) = 2 (cf. [367]); lgld(An (±1, pi,j )) = lKdim(An (±1, pi,j )) = n with char(K) = 0 (cf. [125]); (viii) lgld(An (q, pi,j )) = lKdim(An (q, pi,j )) = 2n if q 2 6= 1 or char(K) 6= 0 (cf. [145]); (ix) lKdim(R) ≤ lKdim(R[x±1 ; σ]) ≤ Kdim(R) + 1 with σ an automorphism of R (cf. [40]); (x) For Witten’s deformation of U (sl(2, K)), its Krull dimension is 3 (cf. [40] and [41]). 8. In Theorem 7.3.2 we proved that if A is a bijective skew P BW extension of a left (right) noetherian domain R, then the Goldie dimension of A is 1, that is, ludim(A) = 1 = rudim(A). In Example 7.3.3 we observed that if R is a left (right) noetherian domain, then ludim(Qr,n q,σ (R)) = 1 = (R)). rudim(Qr,n q,σ 9. Under the conditions of Theorems 7.4.1 and 7.4.5 we have computed in the tables below the Gelfand–Kirillov dimension of all rings and algebras of Chapter 2, as well as the Gelfand–Kirillov dimension of skew quantum polynomials. Recall that in this context, R is an algebra over a field, and K is a field or a commutative ring, in this last case, it is understood that K is an algebra over some base field K 0 . 10. All rings and algebras presented in Chapter 2 are AR (AG) rings assuming that the ring of coefficients (R or K) is AR (AG). The same is true for the skew quantum polynomials (see Theorem 8.3.9 and Corollary 8.3.10). 11. With respect to the CM property, all rings and algebras of Chapter 2 satisfying the hypothesis of Theorem 8.4.7 are CM . Similarly for the skew quantum polynomials (see Corollary 8.4.8). 12. All rings and algebras presented in Chapter 2 are left strongly noetherian algebras assuming that the ring of coefficients is a left strongly noetherian algebra. The same is true for the skew quantum polynomials (see Theorem 8.5.5 and Corollary 8.5.6). 13. Related to the description of prime ideals given by Theorems 5.2.3, 5.3.5 and 5.4.7, in table 8.5 below we have included the list of examples of rings and algebras of Chapter 2. For each example we have indicated by X that the corresponding theorem that can be applied, or by x if it cannot. If R is a commutative noetherian ring, then Theorem 5.3.5 gives a description of prime ideals for the n-multiparametric skew quantum space A = Rq,σ [x1 , . . . , xn ]. With this, we get a description of prime ideals for Qr,n q,σ (R) since it is well known that there exists a bijective correspondence between the prime ideals of S −1 A and the prime ideals of A with empty intersection with S (see Example 4.4.4). Under certain conditions related to skew Armendariz rings, in Chapter 6 we describe the minimal prime ideals of skew P BW extensions as well as the Wedderburn, lower nil, Levitzky and upper nil radicals.

8.7 Summary and Remarks Ring

151 L. B.

Usual polynomial ring R[x1 , . . . , xn ] dim(R) + n Ore extension of bijective type R[x1 ; σ1 , δ1 ] · · · [xn ; σn , δn ] dim(R) Weyl algebra An (K) n Extended Weyl algebra Bn (K) 0 Universal enveloping algebra of a Lie algebra G, U (G), K commutative ring dim(K) Tensor product R ⊗K U (G) dim(R) Crossed product R ∗ U (G) dim(R) Algebra of q-differential operators Dq,h [x, y] 1 Algebra of shift operators Sh 2 Mixed algebra Dh 1 Discrete linear systems K[t1 , . . . , tn ][x1 , σ1 ] · · · [xn ; σn ] 2n Linear partial shift operators K[t1 , . . . , tn ][E1 , . . . , Em ] n Linear partial shift operators K(t1 , . . . , tn )[E1 , . . . , Em ] 0 L. P. Differential operators K[t1 , . . . , tn ][∂1 , . . . , ∂n ] n L. P. Differential operators K(t1 , . . . , tn )[∂1 , . . . , ∂n ] 0 L. P. Difference operators K[t1 , . . . , tn ][∆1 , . . . , ∆m ] n L. P. Difference operators K(t1 , . . . , tn )[∆1 , . . . , ∆m ] 0 (q) (q) L. P. q-dilation operators K[t1 , . . . , tn ][H1 , . . . , Hm ] n+m (q) (q) L. P. q-dilation operators K(t1 , . . . , tn )[H1 , . . . , Hm ] m (q) (q) L. P. q-differential operators K[t1 , . . . , tn ][D1 , . . . , Dm ] n (q) (q) L. P. q-differential operators K(t1 , . . . , tn )[D1 , . . . , Dm ] 0 Algebras of diffusion type n Additive analogue of the Weyl algebra An (q1 , . . . , qn ) n Multiplicative analogue of the Weyl algebra O n (λji ) n Quantum algebra U 0 (so(3, K)) 3 3-dimensional skew polynomial algebras 3 Dispin algebra U (osp(1, 2)) 3 Woronowicz algebra W ν (sl(2, K)) 3 Complex algebra Vq (sl3 (C)) 4 Algebra U n Manin algebra O q (M2 (K)) 2 Coordinate algebra of the quantum group SLq (2) 2 q-Heisenberg algebra Hn (q) n Quantum enveloping algebra of sl(2, K), U q (sl(2, K)) 1 Hayashi algebra Wq (J) n Differential operators on a quantum space Sq , Dq (Sq ) n Witten’s Deformation of U (sl(2, K) 1 q,λ Quantum Weyl algebra of Maltsiniotis An , K commutative ring dim(K) Quantum Weyl algebra An (q, pij ) 1 Multiparameter Weyl algebra AQ,Γ (K) 1 n Quantum symplectic space O q (sp(K 2n )) 1 Quadratic algebras in 3 variables 1

U. B. dim(R) + n dim(R) + n 2n n dim(K) + n dim(R) + n dim(R) + n 2 2 3 2n n+m m 2n n n+m m n+m m n+m m 2n 2n n 3 3 3 3 10 3n 4 4 3n 3 3n 2n 3 dim(K) + 2n 2n 2n 2n 3

Table 8.1 Global and Krull dimension for some examples of bijective skew P BW extensions.

The investigation of prime ideals for some examples listed in the above tables has been considered in the classical and specialized literature by other authors. The most remarkable results are briefly summarized next. (i) If K is a field, K[x] is a principal ideal domain, and hence, the prime ideals are {0, hf (x)i}, with f (x) irreducible. When K is algebraically closed, the methods of affine varieties of commutative algebraic geometry tells us that the prime ideals of K[x1 , . . . , xn ] are

152

8 Transfer of Homological Properties

Ring

L.B.

U.B.

Skew Laurent extension R[x±1 ; σ1 ] Skew Laurent polynomials K[x±1 ; σ1 ] Usual Laurent polynomial ring R[x±1 ] Algebra of Laurent polynomials K[x±1 ] n-Multiparametric skew quantum space Rq,σ [x1 , . . . , xn ] n-Multiparametric quantum space Rq [x1 , . . . , xn ] n-Multiparametric skew quantum space Kq,σ [x1 , . . . , xn ] n-Multiparametric quantum space Kq [x1 , . . . , xn ] ±1 k-Multiparametric skew quantum torus Rq,σ [x±1 1 , . . . , xk ] ±1 ±1 k-Multiparametric quantum torus Rq [x1 , . . . , xk ] ±1 k-Multiparametric skew quantum torus Kq,σ [x±1 1 , . . . , xk ] ±1 ±1 Ring of skew quantum polynomials Rq,σ [x1 , . . . , xk , xk+1 , . . . , xn ] ±1 Ring of quantum polynomials Rq [x±1 1 , . . . , xk , xk+1 , . . . , xn ] ±1 ±1 Skew quantum polynomials Kq,σ [x1 , . . . , xk , xk+1 , . . . , xn ] ±1 Algebra of quantum polynomials O q = Kq [x±1 1 , . . . , xk , xk+1 , . . . , xn ]

dim(R) 0 dim(R) 0 dim(R) dim(R) 0 0 dim(R) dim(R) 0 dim(R) dim(R) 0 0

dim(R) + 1 1 dim(R) + 1 1 dim(R) + n dim(R) + n n n dim(R) + k dim(R) + k k dim(R) + n dim(R) + n n n

Table 8.2 Global and Krull dimension of skew quantum polynomials.

{0, hx1 − a1 , . . . , xn − an i}, ai ∈ K, 1 ≤ i ≤ n. (ii) In [74] all primes ideals of B := C[x; σ] are described, where σ is the conjugation automorphism, i.e., σ(r) := r, for any r ∈ C. The primes are given by Spec(B) = {0} ∪ {Bf |f = x or f = g(x2 ), where g ∈ R[t] is irreducible, g 6= t}. (iii) If R is a left (right) noetherian ring and σ is an automorphism of R, then the prime ideals of B := R[x; σ] are given in the following way (see [149], [278] and [321]): (a) Let P be a prime ideal of B. If x ∈ P , then P = (P ∩ R) + Bx, and P ∩ R is a prime ideal of R. If x∈ / P , then x is a regular element of B/P and σ(P ) = P , where σ is extended to B in the canonical way, i.e., σ(r0 + r1 x + · · · + rn xn ) := σ(r0 ) + σ(r1 )x + · · · + σ(rn )xn . Moreover, P ∩ R is a σ-prime ideal of R and (P ∩ R)B is a prime ideal of B. (b) If Q is a prime ideal of R, then Q + Bx is a prime ideal of B. If Q is σ-prime, then QB is prime when x ∈ QB, and QB is σ-prime if x ∈ / QB. (iv) Let R be a noetherian commutative Q-algebra and B := R[x; δ]. Then, (a) If P is a prime ideal of B, then P ∩ R is a δ-prime ideal of R. (b) If Q is δ-prime ideal of R, then QB is a prime ideal of B and QB ∩ R = Q (see [159]). (iii) The contraction-extension method presented in the two previous items was used in [156] for describing the prime ideals of the general skew polynomial ring R[x; σ, δ]. (iv) In [74] all primes ideals of the quantum plane Kq [x, y] are described, where K is a field and q 6= 0 is not a root of unity. In this case the prime ideals of Kq [x, y] are: {0, hxi, hyi, hx − a, yi, hx, y − b, i for any a, b ∈ K}.

8.7 Summary and Remarks Ring

153 GKdim

Usual polynomial ring R[x1 , . . . , xn ] GKdim(R) + n Ore extension of bijective type R[x1 ; σ1 , δ1 ] · · · [xn ; σn , δn ] GKdim(R) + n Weyl algebra An (K) 2n Extended Weyl algebra Bn (K) n Universal enveloping algebra of a Lie algebra G, U (G), K commutative ring GKdim(K) + n Tensor product R ⊗K U (G) GKdim(R) + n Crossed product R ∗ U (G) GKdim(R) + n Algebra of q-differential operators Dq,h [x, y] 2 Algebra of shift operators Sh 2 Mixed algebra Dh 3 Algebra of discrete linear systems K[t1 , . . . , tn ][x1 , σ1 ] · · · [xn ; σn ] 2n Linear partial shift operators K[t1 , . . . , tn ][E1 , . . . , Em ] n+m Linear partial shift operators K(t1 , . . . , tn )[E1 , . . . , Em ] m Linear partial differential operators K[t1 , . . . , tn ][∂1 , . . . , ∂n ] 2n Linear partial differential operators K(t1 , . . . , tn )[∂1 , . . . , ∂n ] n Linear partial difference operators K[t1 , . . . , tn ][∆1 , . . . , ∆m ] n+m Linear partial difference operators K(t1 , . . . , tn )[∆1 , . . . , ∆m ] m (q) (q) L. Partial q-dilation operators K[t1 , . . . , tn ][H1 , . . . , Hm ] n+m (q) (q) L. Partial q-dilation operators K(t1 , . . . , tn )[H1 , . . . , Hm ] m (q) (q) L. Partial q-differential operators K[t1 , . . . , tn ][D1 , . . . , Dm ] n+m (q) (q) L. Partial q-differential operators K(t1 , . . . , tn )[D1 , . . . , Dm ] m Algebras of diffusion type 2n Additive analogue of the Weyl algebra An (q1 , . . . , qn ) 2n Multiplicative analogue of the Weyl algebra O n (λji ) n Quantum algebra U 0 (so(3, K)) 3 3-dimensional skew polynomial algebras 3 Dispin algebra U (osp(1, 2)) 3 Woronowicz algebra W ν (sl(2, K)) 3 Complex algebra Vq (sl(3, C)) 10 Algebra U 3n Manin algebra O q (M2 (K)) 4 Coordinate algebra of the quantum group SLq (2) 4 q-Heisenberg algebra Hn (q) 3n Quantum enveloping algebra of sl(2, K), U q (sl(2, K)) 3 Hayashi algebra Wq (J) 3n Differential operators on a quantum space Sq , Dq (Sq ) 2n Witten’s deformation of U (sl(2, K) 3 Quantum Weyl algebra of Maltsiniotis Aq,λ GKdim(K) + 2n n , K commutative ring Quantum Weyl algebra An (q, pij ) 2n Multiparameter Weyl algebra AQ,Γ (K) 2n n Quantum symplectic space O q (sp(K 2n )) 2n Quadratic algebras in 3 variables 3

Table 8.3 Gelfand–Kirillov dimension for some examples of skew P BW extensions.

We can also find this result in [130], Section 4.2, but assuming that K is algebraically closed (see p. 5 in [130] for this hypothesis on K; see also Remark A.4.4). Observe that the quantum plane can be interpreted as a skew polynomial ring, Kq [x, y] = K[x][y; σ], with σ(x) := qx. This description can be extended to the n-parametric quantum space Kq [x1 , . . . , xn ] (see [397]), thus the prime ideals are {0, hxj1 , . . . , xjr i, hx1 , . . . , xi − a, . . . , xn i, 1 ≤ r ≤ n − 1, 1 ≤ i ≤ n, a ∈ K}.

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8 Transfer of Homological Properties

Ring

GKdim

Skew Laurent extension R[x±1 ; σ1 ] Skew Laurent polynomials K[x±1 ; σ1 ] Usual Laurent polynomial ring R[x±1 ] Algebra of Laurent polynomials K[x±1 ] n-Multiparametric skew quantum space Rq,σ [x1 , . . . , xn ] n-Multiparametric quantum space Rq [x1 , . . . , xn ] n-Multiparametric skew quantum space Kq,σ [x1 , . . . , xn ] n-Multiparametric quantum space Kq [x1 , . . . , xn ] ±1 k-Multiparametric skew quantum torus Rq,σ [x±1 1 , . . . , xk ] ±1 ±1 k-Multiparametric quantum torus Rq [x1 , . . . , xk ] ±1 k-Multiparametric skew quantum torus Kq,σ [x±1 1 , . . . , xk ] ±1 Ring of skew quantum polynomials Rq,σ [x±1 , . . . , x , x k+1 , . . . , xn ] 1 k ±1 Ring of quantum polynomials Rq [x±1 1 , . . . , xk , xk+1 , . . . , xn ] ±1 Skew quantum polynomials Kq,σ [x±1 1 , . . . , xk , xk+1 , . . . , xn ] ±1 Algebra of quantum polynomials O q = Kq [x±1 1 , . . . , xk , xk+1 , . . . , xn ]

GKdim(R) + 1 1 GKdim(R) + 1 1 GKdim(R) + n GKdim(R) + n n n GKdim(R) + k GKdim(R) + k k GKdim(R) + n GKdim(R) + n n n

Table 8.4 Gelfand–Kirillov dimension of skew quantum polynomials

(v) In [156] all prime ideals of the quantized Weyl algebra A1 (K, q) are listed, where K is a field and q 6= 0 is not a root of unity. Recall that T := A1 (K, q) = K[x][y; σ, δ], where σ(x) := qx, δ(x) := 1 and σ(k) := k, δ(k) := 0, for all k ∈ K. In table 8.5, this algebra corresponds to the additive analogue of the Weyl algebra, A1 (q), or also to the algebra of q-differential operators Dq,1 [x, y]. The primes are given by Spec(T ) = {0} ∪ {uT + QT |Q ∈ Spec(K[x]) and x ∈ / Q}, where u := yx − xy. (vi) In [9] the prime ideals of the quantum Weyl algebra of Maltsiniotis q,λ Aq,λ n are studied. It is proved that An has infinitely many maximal ideals, however the number of non-maximal prime ideals is finite. (vii) An interesting description of completely prime ideals is given in [74] for the general skew polynomial ring R[x; σ, δ], where R is a noetherian (left and right) domain. Let I be a proper ideal of R[x; σ, δ] such that I ∩ R is σ-invariant. Then the following conditions are equivalent: (a) I is completely prime (b) π(I) is a completely prime ideal of (R/I ∩ R)[x; σ, δ], where π : R[x; σ, δ] → (R/I ∩ R)[x; σ, δ] is the canonical homomorphism given by π(rxn ) := rxn . In such case, I ∩ R is also completely prime. In general, this description is not true for prime ideals: indeed, for Cq [x, y] = C[x][y; σ] the ideal P := hx2 − 1, y 2 + 1i is prime but P ∩ C[x] is not. Another description involving the total ring of fractions Q(R) of R is as follows. Let I be a proper ideal of R[x; σ, δ] such that I ∩ R is σ-invariant. Then the following conditions are equivalent: (a0 ) e is completely prime in I is completely prime (b0 ) Ql (R)[x; σ e, δ]I e e Ql (R)[x; σ e, δ] and Ql (R)[x; σ e, δ]I ∩ R[x; σ, δ] = I.

8.7 Summary and Remarks Ring Usual polynomial ring R[x1 , . . . , xn ] Ore extension of bijective type R[x1 ; σ1 , δ1 ] · · · [xn ; σn , δn ], R commutative noetherian semiprime, δi δj = δj δi Weyl algebra An (K) Extended Weyl algebra Bn (K) Universal enveloping algebra of a Lie algebra G, U (G), K commutative ring Tensor product R ⊗K U (G) Crossed product R ∗ U (G) Algebra of q-differential operators Dq,h [x, y] Algebra of shift operators Sh Mixed algebra Dh Discrete linear systems K[t1 , . . . , tn ][x1 , σ1 ] · · · [xn ; σn ] Linear partial shift operators K[t1 , . . . , tn ][E1 , . . . , Em ] Linear partial shift operators K(t1 , . . . , tn )[E1 , . . . , Em ] L. P. Differential operators K[t1 , . . . , tn ][∂1 , . . . , ∂n ] L. P. Differential operators K(t1 , . . . , tn )[∂1 , . . . , ∂n ] L. P. Difference operators K[t1 , . . . , tn ][∆1 , . . . , ∆m ] L. P. Difference operators K(t1 , . . . , tn )[∆1 , . . . , ∆m ] (q) (q) L. P. q-dilation operators K[t1 , . . . , tn ][H1 , . . . , Hm ] (q) (q) L. P. q-dilation operators K(t1 , . . . , tn )[H1 , . . . , Hm ] (q) (q) L. P. q-differential operators K[t1 , . . . , tn ][D1 , . . . , Dm ] (q) (q) L. P. q-differential operators K(t1 , . . . , tn )[D1 , . . . , Dm ] Algebras of diffusion type Additive analogue of the Weyl algebra An (q1 , . . . , qn ) Multiplicative analogue of the Weyl algebra O n (λji ) Quantum algebra U 0 (so(3, K)) 3-dimensional skew polynomial algebras Dispin algebra U (osp(1, 2)), Woronowicz algebra W ν (sl(2, K)) Complex algebra Vq (sl3 (C)) Algebra U Manin algebra O q (M2 (K)) Coordinate algebra of the quantum group SLq (2) q-Heisenberg algebra Hn (q) Quantum enveloping algebra of sl(2, K), U q (sl(2, K)) Hayashi algebra Wq (J) Differential operators on a quantum space Sq , Dq (Sq ) Witten’s Deformation of U (sl(2, K) Quantum Weyl algebra of Maltsiniotis Aq,λ n , K commutative ring Quantum Weyl algebra An (q, pij ) Multiparameter Weyl algebra AQ,Γ (K) n Quantum symplectic space O q (sp(K 2n )) Quadratic algebras in 3 variables

155 5.2.3 5.3.5 5.4.7

X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X x x x x x x x

x x x x x x x

x x x x x x x

Table 8.5 Prime ideals of skew P BW extensions.

(viii) For the following algebras, prime and completely prime ideals coincide: (a) U (g), where g is a finite-dimensional completely solvable Lie algebra ([278]) (b) Oq (M2 (K)) and Kq [x1 , . . . , xn ] ([157]) (c) R[x1 ; δ1 ] · · · [xn ; δn ], where R is a commutative noetherian Q-algebra (see [159]). 14. It is well known that for a division ring D, K1 (D) ∼ = D∗ /[D∗ , D∗ ]. In this way Corollary 8.6.8 generalizes the result presented in [22] Corol-

156

8 Transfer of Homological Properties −1 ∼ lary 4.48 which establishes that K1 (DQ,α [x−1 i1 , . . . , xir , x1 , . . . , xn , ]) = ∗ r D∗ /[D∗ , D∗ ] × Zr , and K1 (Oq ) ∼ K ⊕ Z . Theorem 8.6.9 generalizes the = result presented in [22] Theorem 3.17, K0 (Oq ) ∼ = Z. Using Theorem 8.6.2 and Proposition 8.6.3, we have computed in table 8.6 K0 , K1 and K2 for all examples considered in the previous sections, and with Theorem 8.6.9 we have also computed these groups for skew quantum polynomials. R will denote a left noetherian left regular ring and K a field (for U (g), K is a noetherian regular commutative ring). Note that the computations of K0 groups in these tables agree with the conclusion in item 6 (see [97], Proposition 0.3.4).

Ring

K0

Usual polynomial ring R[t1 , . . . , tn ] Ore extension of bijective type R[x1 ; σ1 , δ1 ] · · · [xn ; σn , δn ] Weyl algebra An (K) Extended Weyl algebra Bn (K) Universal enveloping algebra of a Lie algebra G U (G), K commutative ring Tensor product R ⊗K U (G) Crossed product R ∗ U (G) Algebra of q-differential operators Dq,h [x, y] Algebra of shift operators Sh Mixed algebra Dh Algebra of discrete linear systems K[t1 , . . . , tn ][x1 ; σ1 ] · · · [xn ; σn ] Linear partial shift operators K[t1 , . . . , tn ][E1 , . . . , Em ] Linear partial shift operators K(t1 , . . . , tn )[E1 , . . . , Em ] Linear partial differential operators K[t1 , . . . , tn ][∂1 , . . . , ∂n ] Linear partial differential operators K(t1 , . . . , tn )[∂1 , . . . , ∂n ] Linear partial difference operators K[t1 , . . . , tn ][∆1 , . . . , ∆m ] Linear partial difference operators K(t1 , . . . , tn )[∆1 , . . . , ∆m ] (q) (q) L. Partial q-dilation operators K[t1 , . . . , tn ][H1 , . . . , Hm ] (q) (q) L. Partial q-dilation operators K(t1 , . . . , tn )[H1 , . . . , Hm ] (q) (q) L. Partial q-differential operators K[t1 , . . . , tn ][D1 , . . . , Dm ] (q) (q) L. Partial q-differential operators K(t1 , . . . , tn )[D1 , . . . , Dm ] Algebras of diffusion type Additive analogue of the Weyl algebra An (q1 , . . . , qn ) Multiplicative analogue of the Weyl algebra O n (λji ) Quantum algebra U 0 (so(3, K)) 3-dimensional skew polynomial algebras Dispin algebra U (osp(1, 2)) Woronowicz algebra W ν (sl(2, K)) Complex algebra Vq (sl(3, C)) Algebra U Manin algebra O q (M2 (K)) Coordinate algebra of the quantum group SLq (2) q-Heisenberg algebra Hn (q) Quantum enveloping algebra of sl(2, K), U q (sl(2, K))

K0 (R) K1 (R) K0 (R) K1 (R) Z K∗ Z K(t1 , . . . , tn )∗

K2 (R) K2 (R) K2 (K) K2 (K(t1 , . . . , tn ))

K0 (K) K1 (K) K0 (R) K1 (R) K0 (R) K1 (R) Z K∗ Z K∗ Z K∗ Z K∗ Z K∗ Z K(t1 , . . . , tn )∗ Z K∗ Z K(t1 , . . . , tn )∗ Z K∗ Z K(t1 , . . . , tn )∗ Z K∗ Z K(t1 , . . . , tn )∗ Z K∗ Z K(t1 , . . . , tn )∗ Z K∗ Z K∗ Z K∗ Z K∗ Z K∗ Z K∗ Z K∗ Z K∗ Z K∗ Z K∗ Z K∗ Z K∗ Z K∗ ⊕ Z

K2 (K) K2 (R) K2 (R) K2 (K) K2 (K) K2 (K) K2 (K) K2 (K) K2 (K(t1 , . . . , tn )) K2 (K) K2 (K(t1 , . . . , tn )) K2 (K) K2 (K(t1 , . . . , tn )) K2 (K) K2 (K(t1 , . . . , tn )) K2 (K) K2 (K(t1 , . . . , tn )) K2 (K) K2 (K) K2 (K) K2 (K) K2 (K) K2 (K) K2 (K) K2 (K) K2 (K) K2 (K) K2 (K) K2 (K) K2 (K)

Hayashi algebra Wq (J) Z Differential operators Dq (Sq ) on a quantum space Sq Z Witten’s deformation of U (sl(2, K) Z q,λ Quantum Weyl algebra of Maltsiniotis An , K commutative ring K0 (K) Quantum Weyl algebra An (q, pij ) Z Multiparameter Weyl algebra AQ,Γ (K) Z n 2n Quantum symplectic space O q (sp(K )) Z Quadratic algebras in 3 variables Z

K1

K ∗ ⊕ Zn K∗ K∗ K1 (K) K∗ K∗ K∗ K∗

K2

K2 (K) ⊕ [K ∗ ]n ⊕ Z K2 (K) K2 (K) K2 (K) K2 (K) K2 (K) K2 (K) K2 (K)

n(n−1) 2

Table 8.6 Grothendieck, Bass and Milnor groups for some bijective skew PBW extensions of left noetherian left regular rings

8.7 Summary and Remarks

157

Ring

K0

R[x±1 ; σ], σ acts trivially on K0,2,... (R) K[x±1 ; σ] σ acts trivially on K0,2,... (R) R[x±1 ] K[x±1 ] Rq,σ [x1 , . . . , xn ] Rq [x1 , . . . , xn ] Kq,σ [x1 , . . . , xn ] Kq [x1 , . . . , xn ]

K0 (R) K0 (R) ⊕ K1 (R) Z Z ⊕ K∗ K0 (R) K0 (R) ⊕ K1 (R) Z Z ⊕ K∗ K0 (R) K1 (R) K0 (R) K1 (R) Z K∗ Z K∗

±1 Rq,σ [x±1 1 , . . . , xr ]

K0 (R) [K0 (R)]r ⊕ K1 (R) [K0 (R)]

±1 Rq,σ [x±1 1 , . . . , xr , xr+1 , . . . , xn ] ±1 Kq,σ [x1 , . . . , x±1 r , xr+1 , . . . , xn ]

K1

K2

K0 (R) [K0

(R)]r

K∗

Zr

Z

⊕

K1 (R) ⊕ K2 (R) K ∗ ⊕ K2 (K) K1 (R) ⊕ K2 (R) K ∗ ⊕ K2 (K) K2 (R) K2 (R) K2 (K) K2 (K)

⊕ K1 (R) [K0 (R)]

Z

r(r−1) 2

r(r−1) 2

⊕ [K1 (R)]r ⊕ K2 (R)

r(r−1) 2

⊕ [K1 (R)]r ⊕ K2 (R)

⊕

[K ∗ ]n

⊕ K2 (K)

Table 8.7 Grothendieck, Bass and Milnor groups for skew quantum polynomials

Part II Projective Modules Over Skew P BW Extensions

Chapter 9

Stably Free Modules

Serre’s Theorem for bijective skew P BW extensions (Theorem 8.2.3) says that if M is an f.g. projective module over a bijective skew P BW extension A of a left noetherian, left regular P SF ring R, then M is stably free. In the same way, Example 8.2.5 establishes that if M is an f.g. projective module over the ring Qr,n q,σ (R) of skew quantum polynomials over R, where R satisfies the same above conditions, then M is stably free. The following natural question arises: when are stably free modules over A (or over Qr,n q,σ (R)) free? First we have to observe that not any stably free module over a bijective skew P BW extension is free. The next trivial example shows this ([216], p. 36): if K is a division ring (i.e., a noncommutative field), then S := K[x, y] has a module M such that M ⊕S ∼ = S 2 , but M is not free. In a more general framework, and as preparatory material for posterior studies in the coming chapters, we are interested in studying when stably free modules over noncommutative rings are free. A well-known result in this direction is Stafford’s Theorem, which we will prove in this chapter. Many characterizations of stably free modules will also be presented. There are different techniques to investigate stably free modules. We will combine homological and matrix constructive methods.

9.1 RC and IBN rings In this section we recall some notations and well-known elementary properties of linear algebra for left modules. All rings are noncommutative and modules will be considered on the left; S will represent an arbitrary noncommutative f ring; S r is the left S-module of columns of size r × 1; if S s − → S r is an Shomomorphism then there is a matrix associated to f in the canonical bases of S r and S s , denoted F := m(f ), and disposed by columns, i.e., F ∈ Mr×s (S). In fact, if f is given by f

Ss − → S r , e j 7→ f j © Springer Nature Switzerland AG 2020 W. Fajardo et al., Skew PBW Extensions, Algebra and Applications 28, https://doi.org/10.1007/978-3-030-53378-6_9

161

9 Stably Free Modules

162

where {e 1 , . . . , e s } is the canonical basis of S s , f can be represented by a  T matrix, i.e., if f j := f1j . . . frj , then the matrix of f in the canonical bases of S s and S r is   f11 · · · f1s    . ...  F := f 1 · · · f s =  ..  ∈ Mr×s (S). fr1 · · · frs Note that Im(f ) is the column module of F , i.e., the left S-module generated by the columns of F , denoted by hF i: Im(f ) = hf (e 1 ), . . . , f (e s )i = hf 1 , . . . , f s i = hF i. Moreover, observe that if a := (a1 , . . . , as )T ∈ S s , then f (a) = (a T F T )T .

(9.1.1)

In fact, f (a) = a1 f (e 1 ) + · · · + as f (e s ) = a1 f 1 + · · · + as f s     f11 f1s  ..   ..  = a1  .  + · · · + as  .  fr1   =

frs 

a1 f11 + · · · + as f1s  ..  .

a1 fr1 + · · · + as frs  f11 · · ·  .  = ( a1 · · · as  ..

 fr1 .. )T . 

f1s · · · frs T

T T

= (a F ) . Note that the function m : HomS (S s , S r ) → Mr×s (S) is bijective; moreover, g → S p is a homomorphism, then the matrix of gf in the canonical bases if S r − is m(gf ) = (F T GT )T . Thus, f : S r → S r is an isomorphism if and only if F T ∈ GLr (S). Finally, let C ∈ Mr (S); the columns of C conform a basis of S r if and only if C T ∈ GLr (S). We recall also that Syz({f 1 , . . . , f s }) := {a := (a1 , . . . , as )T ∈ S s |a1 f 1 + · · · + as f s = 0}. Note that Syz({f 1 , . . . , f s }) = ker(f ),

(9.1.2)

but Syz({f 1 , . . . , f s }) 6= ker(F ) since we have a ∈ Syz({f 1 , . . . , f s }) ⇔ a T F T = 0.

(9.1.3)

9.1 RC and IBN rings

163

A matrix characterization of f.g. projective modules can be formulated in the following way. Proposition 9.1.1. Let S be an arbitrary ring and M be an S-module. Then, M is an f.g. projective S-module if and only if there exists a square matrix F over S such that F T is idempotent and M = hF i. Proof. ⇒) If M = 0, then F = 0; let M 6= 0, there exists s ≥ 1 and an M 0 such that S s = M ⊕ M 0 ; let f : S s → S s be the projection on M and F the matrix of f in the canonical basis of S s . Then, f 2 = f and (F T F T )T = F , so F T F T = F T ; note that M = Im(f ) = hF i. ⇐) Let f : S s → S s be the homomorphism defined by F (see (9.1.1)); from F T F T = F T we get that f 2 = f , moreover, since M = hF i, we have Im(f ) = M and hence M is a direct summand of S s , i.e., M is f.g. projective (observe that the complement M 0 of M is ker(f ) and f is the projection on M ). t u Remark 9.1.2. (i) When S is commutative, or when we consider right modules instead of left modules, (9.1.1) says that f (a) = F a. Moreover, in such cases Syz({f 1 , . . . , f s }) = ker(F ) and the matrix of a composite homomorphism gf is given by m(gf ) = m(g)m(f ). Note that f : S r → S r is an isomorphism if and only if F ∈ GLr (S); moreover, C ∈ GLr (S) if and only if its columns conform a basis of S r . In addition, Proposition 9.1.1 says that M is an f.g. projective S-module if and only if there exists a square matrix F over S such that F is idempotent and M = hF i. (ii) When the matrices of homomorphisms of left modules are disposed by rows instead of by columns, i.e., if S 1×s is the left free module of row vectors f of length s and the matrix of the homomorphism S 1×s − → S 1×r is defined by  0    0 f11 · · · fr1 f11 · · · f1r  ..  :=  .. ..  ∈ M (S), F 0 =  ...  . s×r .  .  0 0 fs1 · · · fsr

f1s · · · frs

then f (a1 , . . . , as ) = (a1 , . . . , as )F 0 ,

(9.1.4)

i.e., f (a T ) = a T F T . Thus, the values given by (9.1.4) and (9.1.1) agree since F 0 = F T . Moreover, the composed homomorphism gf means that g acts first and then f , and hence, the matrix of gf is given by m(gf ) = m(g)m(f ). Note that f : S 1×r → S 1×r is an isomorphism if and only if m(f ) ∈ GLr (S); moreover, C ∈ GLr (S) if and only if its rows conform a basis of S 1×r . This left-row notation is used by some authors (cf. [97]). Observe that with this notation, the proof of Proposition 9.1.1 says that M is an f.g. projective Smodule if and only if there exists a square matrix F over S such that F is idempotent and M = hF i, but in this case hF i represents the module generated by the rows of F . Note that Proposition 9.1.1 could have been formulated this way: in fact, the set of idempotent matrices of Ms (S) coincides with the set {F T |F ∈ Ms (S), F T idempotent}.

164

9 Stably Free Modules

Definition 9.1.3 ([216]). Let S be a ring. (i) S satisfies the rank condition (RC) if for any integers r, s ≥ 1, given an f

epimorphism S r − → S s , then r ≥ s. (ii) S is an IBN (Invariant Basis Number) ring if for any integers r, s ≥ 1, Sr ∼ = S s if and only if r = s. Proposition 9.1.4. Let S be a ring. (i) S is RC if and only if given any matrix F ∈ Ms×r (S) the following condition holds: if F has a right inverse then r ≥ s. (ii) S is RC if and only if given any matrix F ∈ Ms×r (S) the following condition holds: if F has a left inverse then s ≥ r. Proof. (i) ⇒) Let G be a right inverse of F , F G = Is ; let f : S r → S s and g : S s → S r such that m(f ) = F and m(g) = G. Then, ((F T )T (GT )T )T = Is ; let f T : S s → S r and g T : S r → S s such that m(f T ) = F T and m(g T ) = GT , then m(g T f T ) = m(iS s ) and hence g T f T = iS s , i.e., g T is surjective. Since S is RC, we have r ≥ s. f

g

⇐) Let S r − → S s be an epimorphism, there exists S s − → S r such that f g = iS s ; let F := m(f ) ∈ Ms×r (S) and G := m(g) ∈ Mr×s (S), then m(f g) = (GT F T )T = Is , so GT F T = Is , i.e., GT has right inverse, and by hypothesis r ≥ s. This means that S is RC. (ii) ⇒) Let G ∈ Mr×s (S) be a left inverse of F , then G has a right inverse, and by (i), s ≥ r. f

⇐) Let S r − → S s be an epimorphism; as in (i), GT F T = Is , so F T ∈ Mr×s (S) has a left inverse and by the hypothesis r ≥ s. Thus, S is RC. t u Proposition 9.1.5. RC ⇒ IBN . f

Proof. Let S r − → S s be an isomorphism, then f is an epimorphism, and hence r ≥ s; considering f −1 we get that s ≥ r. t u Example 9.1.6. Most rings are RC, and hence, IBN . f

(i) Any field K is RC: let K r − → K s be an epimorphism, then dim(K r ) = r = dim(ker(f )) + s, so r ≥ s. f

(ii) Let S and T be rings and let S − → T be a ring homomorphism, if T is an RC ring then S is also an RC ring. In fact, T is a right S-module, t·s := tf (s); f

iT ⊗f

suppose that S r − → S s is an epimorphism, then T ⊗S S r −−−→ T ⊗S S s is also an epimorphism of left T -modules, i.e., we have an epimorphism T r → T s , so r ≥ s (a similar result and proof is valid for the IBN property). (iii) We can apply the property proved in (ii) in many situations. For example, any commutative ring S is RC: let J be a maximal ideal of S, then

9.1 RC and IBN rings

165

the canonical homomorphism S → S/J shows that S is RC since S/J is a field. (iv) Any ring S with finite uniform dimension (Goldie dimension, see [278] f

and [159]) is RC: in fact, suppose that S r − → S s is an epimorphism, then s Sr ∼ S ⊕ M and hence r udim(S) = s udim(S) + udim(M ), so r ≥ s. = (v) Since any left noetherian ring S has finite uniform dimension, then S is RC. In particular, any left artinian ring is RC. Since the objects studied in the present monograph are the skew P BW extensions, it is natural to investigate the IBN and RC properties for these rings. Proposition 9.1.7. Let B be a filtered ring. If Gr(B) is RC (IBN ), then B is RC (IBN ). Proof. Let {Bp }p≥0 be the filtration of B and f : B r → B s an epimorphism. For M := B r we consider the standard positive filtration given by F0 (M ) := B0 · e1 + · · · + B0 · er , Fp (M ) := Bp F0 (M ), p ≥ 1, where {ei }ri=1 is the canonical basis of B r . Let e0i := f (ei ), then B s is generated by {e0i }ri=1 and N := B s has a standard positive filtration given by F0 (N ) := B0 · e01 + · · · + B0 · e0r , Fp (N ) := Bp F0 (N ), p ≥ 1. Note that f is filtered and strict: in fact, f (Fp (M )) = Bp f (F0 (M )) = Bp (B0 · f (e1 ) + · · · + B0 · f (er )) = Bp (B0 · e01 + · · · + B0 · e0r ) = Bp F0 (N ) = Fp (N ). Gr(f )

This implies that Gr(M ) −−−→ Gr(N ) is surjective. If we prove that Gr(M ) and Gr(N ) are free over Gr(B) with bases of r and s elements, respectively, then from the hypothesis we conclude that r ≥ Psr and hence B is RC. Since every ei ∈ F0 (M ) and Fp (M ) = i=1 ⊕Bp · ei , M is filteredfree with filtered-basis {ei }ri=1 , so Gr(M ) is graded-free with graded-basis {ei }ri=1 , ei := ei + F−1 (M ) = ei (recall that by definition of positive filtration, F−1 (M ) := 0). For Gr(N ) note that N is also filtered-free with respect to the filtration {Fp (N )}p≥0 given above: indeed, we will show next that the canonical basis {fj }sj=1 of N is a filtered basis. If fj = xj1 · e01 + · · · + xjr · e0r , with xji ∈ Bpij , let p := max{pij }, 1 ≤ i ≤ r, 1 ≤ j ≤ s, then fj ∈ Fp (N ), moreover, for every q, Bq−p ·f1 ⊕· · ·⊕Bq−p ·fs ⊆ Bq−p Fp (N ) ⊆ Fq (N ) (recall that for k < 0, Bk = 0); in turn, let x ∈ Fq (N ), then x = b1 · f1 + · · · + bs · fs and in Gr(N ) we have x ∈ Gr(N )q , x = b1 · f1 + · · · + bs · fs . If bj ∈ Buj , let u := max{uj }, so bj · fj ∈ Gr(N )u+p , so q = u + p, i.e., u = q − p and hence x ∈ Bq−p · f1 ⊕ · · · ⊕ Bq−p · fs . Thus, we have proved that Bq−p · f1 ⊕ · · · ⊕ Bq−p · fs = Fq (N ), for every q, and consequently, {fj }sj=1 is a filtered basis of N . From this we conclude that Gr(N ) is graded-free with graded-basis {fj }sj=1 , fj := fj + Fp−1 (N ). We can repeat the previous proof for the IBN property but assuming that f is an isomorphism. t u

166

9 Stably Free Modules

Corollary 9.1.8. Let A be a skew P BW extension of a ring R. Then, A is RC (IBN ) if and only if R is RC (IBN ). Proof. We consider only the proof for RC, the case IBN is completely analogous. ⇒) Since R ,→ A, Example 9.1.6 shows that if A is RC, then R is RC. ⇐) We consider first the skew polynomial ring R[x; σ] of endomorphism type, then R[x; σ] → R given by p(x) → p(0) is a ring homomorphism, so R[x; σ] is RC since R is RC. By Theorem 3.1.4, Gr(A) is isomorphic to an iterated skew polynomial ring R[z1 ; θ1 ] · · · [zn ; θn ], so Gr(A) is RC. Proposition 9.1.7 completes the proof. t u Remark 9.1.9. (i) The condition IBN for rings is independent of the side we are considering for the modules. In fact, if we define left IBN rings and right IBN rings, depending on left or right free S-modules, then S is left IBN if and only if S is right IBN (see [240]). The same is true for the RC property. (ii) From now on we will assume that all rings considered in the present monograph are RC.

9.2 Characterizations of Stably Free Modules Definition 9.2.1. Let M be an S-module and t ≥ 0 be an integer. M is stably free of rank t ≥ 0 if there exist an integer s ≥ 0 such that S s+t ∼ = Ss ⊕ M . The rank of M is denoted by rank(M). Note that any stably free module M is finitely generated and projective. Moreover, as we will show in the next proposition, rank(M) is well defined, i.e., rank(M) is unique for M . Proposition 9.2.2. Let t, t0 , s, s0 ≥ 0 be integers such that S s+t ∼ = Ss ⊕ M s0 +t0 ∼ s0 0 and S = S ⊕ M . Then, t = t. 0 0 0 0 0 Proof. We have S s ⊕ S s+t ∼ = S s ⊕ S s ⊕ M and S s ⊕ S s +t ∼ = Ss ⊕ Ss ⊕ M , then since S is an IBN ring, s0 + s + t = s + s0 + t0 , and hence t0 = t. t u

Corollary 9.2.3. M is stably free of rank t ≥ 0 if and only if there exist integers r, s ≥ 0 such that S r ∼ = S s ⊕ M , with r ≥ s and t = r − s. Proof. If M is stably free of rank t, then S s+t ∼ = S s ⊕ M for some integers s, t ≥ 0, then taking r := s + t we get the result. Conversely, if there exist integers r, s ≥ 0 such that S r ∼ = S s ⊕ M , with r ≥ s, then S s+r−s ∼ = Ss ⊕ M , i.e., M is stably free of rank r − s. t u Proposition 9.2.4. Let M be an S-module and let r, s ≥ 0 be integers such that S r ∼ = S s ⊕ M . Then r ≥ s. Proof. The canonical projection S r → S s is an epimorphism, but since we are assuming that S is RC, then r ≥ s. t u

9.2 Characterizations of Stably Free Modules

167

Corollary 9.2.5. M is stably free if and only if there exist integers r, s ≥ 0 such that S r ∼ = Ss ⊕ M . Proof. This is a direct consequence of Corollary 9.2.3 and Proposition 9.2.4. t u Proposition 9.2.6. Let M1 , M2 be stably free modules of ranks p, q, respectively. Then, M1 ⊕ M2 is stably free of rank p + q. Proof. We have S s+p ∼ = S s ⊕ M1 and S r+q ∼ = S r ⊕ M2 , then S s+p ⊕M2 ∼ = S s ⊕M1 ⊕M2 and also S s+p ⊕S r ⊕M2 ∼ = S s ⊕S r ⊕M1 ⊕M2 . Hence, S s+p ⊕ S r+q ∼ = S s+r ⊕ M1 ⊕ M2 , i.e., S s+r+p+q ∼ = S s+r ⊕ M1 ⊕ M2 . t u Remark 9.2.7. Let S be a ring with finite uniform dimension and let M be stably free, then udim(M) rank(M) = . (9.2.1) udim(S) In fact, from S s+t ∼ = S s ⊕M we have (s+t) udim(S) = s udim(S)+udim(M ), and this proves the equality. Next we will prove many characterizations of stably free modules over noncommutative rings (compare with [221], Chapter 21, [262], and [278], Chapter 11). Theorem 9.2.8. Let M be an S-module. Then, the following conditions are equivalent (i) M is stably free. (ii) M is projective and has a finite free resolution: fk

fk−1

f2

f1

f0

0 → S tk −→ S tk−1 −−−→ · · · −→ S t1 −→ S t0 −→ M → 0. In this case rank(M ) =

k X

(−1)i ti .

(9.2.2)

i=0

(iii) M is isomorphic to the kernel of an epimorphism of free modules: M ∼ = ker(π), π : S r → S s . f1

f0

(iv) M is projective and has a finite presentation S s −→ S r −→ M → 0, where ker(f0 ) is stably free. f1

f0

(v) M has a finite presentation S s −→ S r −→ M → 0, where f1 has a left inverse.

168

9 Stably Free Modules

Proof. (i) ⇒ (ii) If S s+t ∼ = S s ⊕ M for some integers s, t ≥ 0, then M is projective and we have the finite free resolution ι

π

0 → Ss → − S s+t − → M → 0, where ι is the canonical inclusion and π is the canonical projection on M . (ii) ⇒ (i) Let fk−1

fk

f2

f1

f0

0 → S tk −→ S tk−1 −−−→ · · · −→ S t1 −→ S t0 −→ M → 0 be a finite free resolution of M . By induction on k we will prove that M is stably free and (9.2.2) holds. If k = 0 then M ∼ = S t0 is free of finite dimension t0 , and hence, stably free of rank t0 . Let k ≥ 1 and let M1 = ker(f0 ). We get the exact sequence ι

f0

0 → M1 → − S t0 −→ M → 0, and hence S t0 ∼ = M ⊕ M1 since M is a projective module. This implies that M1 is also projective and we have the finite free resolution of M1 fk−1

fk

f2

f1

0 → S tk −→ S tk−1 −−−→ · · · −→ S t1 −→ M1 → 0. Pk By induction, M1 is stably free of rank(M1 ) = i=1 (−1)i−1 ti := p. There exists a q ≥ 0 such that S q+p ∼ = S q ⊕M1 , and hence, S t0 ⊕S q ∼ = M ⊕M1 ⊕S q ∼ = q+p t0 +q ∼ q+p M ⊕S , i.e., S . By Proposition 9.2.4, t0 + q ≥ q + p, i.e., = M ⊕S q+p+(t0 −p) ∼ tP = M ⊕ S q+p , i.e., M is stably free of rank t0 − p = 0 ≥ p, so S k i i=0 (−1) ti . (i) ⇒ (iii) By Proposition 9.2.5, there exist integers r, s ≥ 0 such that Sr ∼ = S s ⊕ M , with r ≥ s. Hence M ∼ = ker(π), where π is the canonical projection of S r on S s . π (iii) ⇒ (i) Let S r − → S s be an epimorphism such that M ∼ = ker(π). Then we have the exact sequence ι

π

0→M → − Sr − → S s → 0, but S s is projective and hence S r ∼ = Ss ⊕ M . r ∼ s (i) ⇒ (iv) Let S = S ⊕M for some integers r, s ≥ 0, then M is projective f1

f0

and we have the exact sequence 0 → S s −→ S r −→ M → 0, and also the f1 f0 finite presentation S s −→ S r −→ M → 0, where f0 is the canonical projection and f1 is the canonical injection of S s in S r . But ker(f0 ) = Im(f1 ) ∼ = Ss, thus ker(f0 ) is free, and hence, stably free. f1

f0

(iv) ⇒ (i) Let M be projective and S s −→ S r −→ M → 0 be a finite presentation of M with ker(f0 ) stably free. Then S r ∼ = M ⊕ ker(F0 ). There exist some integers p, q ≥ 0 with p ≥ q such that S p ∼ = S q ⊕ ker(F0 ), and p q+r ∼ hence, S ⊕ M = S . By Corollary 9.2.5, M is stably free. (i) ⇒ (v) Let S r ∼ = S s ⊕ M for some integers r, s ≥ 0, then we have the f1

f0

f1

exact sequence 0 → S s −→ S r −→ M → 0 and the finite presentation S s −→ f0 S r −→ M → 0, where f0 is the canonical projection and f1 is the canonical injection of S s in S r . Since M is projective there exists an h0 : M → S r such

9.2 Characterizations of Stably Free Modules

169

that f0 h0 = iM , and hence, S r = ker(f0 ) ⊕ Im(h0 ) = Im(f1 ) ⊕ Im(h0 ). For x ∈ S r we have x = f1 (y) + h0 (z) with y ∈ S s and z ∈ M , we note that y and z are unique for x since f1 and h0 are injective, so we define g1 : S r → S s by g1 (x) = y. It is clear that g1 is an S-homomorphism and g1 f1 = iS s . (v) ⇒ (i) Let g1 : S r → S s such that g1 f1 = iS s , then f1 is injective and f1

f0

M has the finite free resolution 0 → S s −→ S r −→ M → 0. M is projective since this sequence splits; by (ii) and (i) M is stably free. t u Definition 9.2.9. A finite presentation f1

f0

S s −→ S r −→ M → 0

(9.2.3)

of an S-module M is minimal if f1 has a left inverse. Corollary 9.2.10. Let M be an S-module. Then, M is stably free if and only if M has a minimal presentation. Proof. See the proof of Theorem 9.2.8, part (i)⇔(v).

t u

Unimodular matrices are closely related to the stably free modules. Definition 9.2.11. Let F be a matrix over S of size r × s. Then (i) Let r ≥ s. F is unimodular if and only if F has a left inverse. (ii) Let s ≥ r. F is unimodular if and only if F has a right inverse. The set of unimodular column matrices of size r × 1 is denoted by Umc (r, S). Umr (s, S) is the set of unimodular row matrices of size 1 × s. Remark 9.2.12. Note that a column matrix is unimodular if and only if the left ideal generated by its entries coincides with S, and a row matrix is unimodular if and only if the right ideal generated by its entries is S. We can add some others characterizations of stably free modules (compare with [323], Lemma 16). Corollary 9.2.13. Let M be an S-module. Then the following conditions are equivalent: (i) M is stably free. (ii) M is projective and has a finite system of generators f1 , . . . , fr such that Syz{f1 , . . . , fr } is the module generated by the columns of a matrix F1 of size r × s such that F1T has a right inverse. (iii) M is projective and has a finite system of generators f1 , . . . , fr such that Syz{f1 , . . . , fr } is the module generated by the columns of a matrix F1 of size r × s such that F1T is unimodular. Proof. (i) ⇒ (ii) By (v) of Theorem 9.2.8, M is projective and has a finite f1

f0

presentation S s −→ S r −→ M → 0, where f1 has a left inverse. Let f i = f0 (e i ), where {e i }1≤i≤r is the canonical basis of S r . Then M = hf 1 , . . . , f r i and Im(f1 ) = ker(f0 ) = Syz{f 1 , . . . , f r }, but Im(f1 ) is the module generated

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9 Stably Free Modules

by the columns of the matrix F1 defined by f1 in the canonical bases. Thus, let g1 : S r → S s be a left inverse of f1 , then g1 f1 = iS s and the matrix of g1 f1 in the canonical bases is Is = (F1T GT1 )T , so Is = F1T GT1 . (ii) ⇒ (i) Let f 1 , . . . , f r be a set of generators of M such that Syz{f 1 , . . . , f r } is the module generated by the columns of a matrix F1 of size r × s such that F1T has a right inverse. We have the exact sequence ι

f0

0 → ker(f0 ) → − S r −→ M → 0, where ι is the canonical injection and f0 is defined as above. We have ker(f0 ) = Syz{f 1 , . . . , f r } = hF1 i, and we get the finite presentation f1

f0

S s −→ S r −→ M → 0, where f1 (e j ) is the j th column of F1 , 1 ≤ j ≤ s. By hypothesis, F1T has a right inverse, F1T GT1 = Is , so Is = (F1T GT1 )T . Let g1 : S r → S s be the homomorphism defined by G1 ∈ Ms×r (S) in the canonical bases, then g1 f1 = iS s and f1 is injective. This implies that the sequence f1 f0 0 → S s −→ S r −→ M → 0 is exact. By Theorem 9.2.8, M is stably free. (ii) ⇔ (iii) This is a direct consequence of Definition 9.2.11. t u Corollary 9.2.14. Let M be an S-module. (i) If M is stably free, then for any free resolution of M , fk+1

fk

fk−1

f2

f1

f0

· · · −−−→ S sk −→ S sk−1 −−−→ · · · −→ S s1 −→ S s0 −→ M −→ 0, Im(fk ) is stably free for each k ≥ 0. (ii) If there exists a free resolution of M as in (i) such that Im(fk ) is stably free for some k ≥ 0 and Im(fk−1 ), . . . , Im(f0 ) are projective, then M is stably free. Proof. (i) We will prove this by induction on k. For k = 0 we have Im(f0 ) = f0

M . For k = 1 we have the exact sequence 0 → ker(f0 ) → S s0 −→ M → 0, then S s0 ∼ = M ⊕ ker(f0 ) since M is projective. But S q ⊕ M = S p since M is stably free, then S s0 +q ∼ = S p ⊕ ker(f0 ), thus ker(f0 ) = Im(f1 ) is stably free. We assume that Im(fk−1 ) is stably free and we consider the exact sequence fk−1

0 → ker(fk−1 ) → S sk−1 −−−→ Im(fk−1 ) → 0, then S sk−1 ∼ = Im(fk−1 ) ⊕ ker(fk−1 ), and hence there exist l, t ≥ 0 such that S l ⊕ Im(fk−1 ) ∼ = S t , therefore S sk−1 +l ∼ = S t ⊕ ker(fk−1 ). Thus, ker(fk−1 ) = Im(fk ) is stably free. (ii) If k = 0 there is nothing to prove. Let k ≥ 1. We consider the presentafk

fk−1

tion S sk −→ S sk−1 −−−→ Im(fk−1 ) → 0. By (iv) of Theorem 9.2.8, Im(fk−1 ) is stably free. Similarly, we can prove that Im(fk−2 ), . . . , Im(f1 ), Im(f0 ) = M are stably free. t u Another interesting result about stably free modules over arbitrary RC rings is presented next (see [92], Proposition 12). For this, we recall that if M f1

f0

is a finitely presented left S-module with presentation S s −→ S r −→ M → 0

171

9.2 Characterizations of Stably Free Modules

and F1 is the matrix of f1 in the canonical bases, then the right S-module M T defined by M T := S s /Im(f1T ), where f1T : S r → S s is the homomorphism of right free S-modules induced by the matrix F1T , is called the transposed fT

module of M . Thus, M T is given by the presentation S r −−1→ S s → M T → 0. Theorem 9.2.15. Let M be an S-module with exact sequence f1

f0

0 → S s −→ S r −→ M → 0. Then, M T ∼ = Ext1S (M, S) and the following conditions are equivalent: (i) (ii) (iii) (iv) (v)

M is stably free. M is projective. M T = 0. F1T has a right inverse. f1 has a left inverse.

∼ Ext1 (M, S): from the left complex Proof. We first prove that M T = S f1

0 → S s −→ S r → 0 we get the right complex f∗

0

1 0 → HomS (S r , S) −→ HomS (S s , S) − → HomS (0, S) → · · · ,

i.e., f∗

0

1 → 0 → ···, 0 → S r −→ Ss −

so Ext1S (M, S) = ker(0)/Im(f1∗ ) = S s /Im(f1∗ ). But Im(f1∗ ) ∼ = Im(f1T ) under s r r the isomorphisms HomS (S , S) ∼ = S s . In fact, we = S and HomS (S , S) ∼ have the following diagram f∗

HomS (S r , S) −−−1−→ HomS (S s , S)    β αy y Sr

fT

−−−1−→

(9.2.4)

Ss

where the vertical rows are isomorphisms of right S-modules defined by α(h) := (h(e 1 ), . . . , h(e r ))T , β(g) := (g(e 1 ), . . . , g(e s ))T , and moreover f1∗ (h) := hf1 and f1T ((x1 , . . . , xr )T ) := F1T (x1 , . . . , xr )T . Note that the diagram is commutative: βf1∗ (h) = β(hf1 ) = (hf1 (e 1 ), . . . , hf1 (e s ))T = (h((e T1 F1T )T ), . . . , h((e sT F1T )T ))T

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9 Stably Free Modules









f11 f1s  ..   ..  T = (h( . ), . . . , h( . )) ; fr1 frs f1T α(h) = f1T ((h(e 1 ), . . . , h(e r ))T )     h(e 1 ) f11 h(e 1 ) + · · · + fr1 h(e r )     .. = F1T  ...  =   . f1s h(e 1 ) + · · · + frs h(e r )      h(f11 e 1 + · · · + fr1 e r ) f11 f1s      . . .  T .. =  = (h( .. ), . . . , h( .. )) . h(e r )



h(f1s e 1 + · · · + frs e r )

fr1

frs

From this we conclude that Ext1S (M, S) ∼ = S s /Im(f1T ) = M T . (i)⇒(ii) This is obvious. (ii)⇒(i) This is a direct consequence of Theorem 9.2.8. (ii)⇒(iii) Since M is projective, Ext1S (M, S) = 0 and hence M T = 0. (iii)⇒(i) If M T = 0, then Ext1S (M, S) = 0. From the given exact sequence of left modules we get the exact sequence of right modules f∗

f∗

0 1 0 → HomS (M, S) −→ HomS (S r , S) −→ HomS (S s , S) → Ext1S (M, S) → . . . ,

fT

i.e., we have the exact sequence 0 → M ∗ → S r −−1→ S s → 0; but since S s gT

1 is projective, this sequence splits, i.e., f1T has right inverse, say S s −−→ Sr, T T T i.e., f1 g1 = iS s . Let G1 of size s × r be such that G1 is the matrix of the right homomorphism g1T , then m(f1T g1T ) = m(f1T )m(g1T ) = m(iS s ), i.e., g1 F1T GT1 = Is . Let S r −→ S s be the left homomorphism corresponding to G1 , T T T then m(g1 f1 ) = (F1 G1 ) = Is = m(iS s ), so g1 f1 = iS s , i.e., f1 has a left inverse. This means that the exact sequence

f1

f0

0 → S s −→ S r −→ M → 0 splits, so M is stably free. f1

f0

(ii)⇔(iv) If M is projective, then the exact sequence 0 → S s −→ S r −→ M → 0 splits, so there exists a g1 such that g1 f1 = iS s , and hence, as before, F1T has a right inverse. Conversely, if F1T GT1 = Is , then g1 f1 = iS s , where g1 S r −→ S s is the left homomorphism corresponding to G1 , so the previous sequence splits, and hence, M is projective. (iv)⇔(v) From the above discussion we get that f1 has a left inverse if and t u only if F1T has a right inverse. Remark 9.2.16. Theorem 9.2.15 gives procedures for testing stably freeness if we have algorithms for computing the right inverse of a matrix and the Ext modules. These algorithms will be considered in Part IV of the present work.

9.3 Stafford’s Theorem: A Constructive Proof

173

9.3 Stafford’s Theorem: A Constructive Proof A well-known result due to Stafford says that any left ideal of the Weyl algebras D := An (K) or Bn (K), with char(K) = 0, is generated by two elements, (see [372] and [323]). From Stafford’s Theorem it follows that any stably free left module M over D with rank(M ) ≥ 2 is free. In [323] a constructive proof of this result is presented that we want to study for arbitrary RC rings. Actually, we will consider the generalization given in [323] stating that any stably free left S-module M with rank(M ) ≥ sr(S) is free, where sr(S) denotes the stable rank of the ring S. Our proof has been adapted from [323], however we do not need the involution of the ring S used in [323] because of our left notation for modules and column representation for homomorphisms. This could justify our special left-column notation. In order to apply the main result of this section to bijective skew P BW extensions we will estimate the stable rank of such extensions. In Section 17.5 we will complement these results, presenting algorithms for computing the corresponding free bases.  T Definition 9.3.1. Let S be a ring and v := v1 . . . vr ∈ Umc (r, S) a unimodular column vector. v is called stable (reducible) if there exists  T a1 , . . . , ar−1 ∈ S such that v0 := v1 + a1 vr . . . vr−1 + ar−1 vr is unimodular. We say that the left stable rank of S is d ≥ 1, denoted sr(S) = d, if d is the least positive integer such that every unimodular column vector of length d + 1 is stable. We say that sr(S) = ∞ if for every d ≥ 1 there exists a non-stable unimodular column vector of length d + 1. Remark 9.3.2. The right stable rank of S is defined in a similar way, however, both ranks coincide; we list next some well-known properties of the stable rank (see [22], [38], [88], [278], [323], [372], [373]), [391], or also [141]). (i) sr(S) = sr(S op ). (ii) If T is a division ring, then sr(T ) = 1. (iii) If I is a two-sided ideal of S, then sr(S/I) ≤ sr(S). Moreover, if 1 + I ⊆ S ∗ , then sr(S/I) = sr(S). In particular, sr(S/Rad(S)) = sr(S). (iv) If S is a local ring, then sr(S) = 1. (v) If {Si }i∈C is a nonempty family of rings, then Y sr( Si ) = sup{sr(Si )}i∈C . i∈C

(vi) If sr(S) = 1, then sr(Mn (S)) = 1, for any n ≥ 1. (vii) If S is simple artinian, semisimple or semilocal, then sr(S) = 1. (viii) If S is a Dedekind domain, then sr(S) = 2. In particular, if K is a field, then sr(K[x]) = 2; thus, sr(Q[x]) = sr(R[x]) = sr(C[x]) = 2. (ix) If K is a field with char(K) = 0 then sr(An (K)) = 2 = sr(Bn (K)). (x) If S is a left noetherian ring, then sr(S) ≤ Kdim(S) + 1. In particular, if S is a left artinian ring, then sr(S) = 1. (xi) Let n ≥ 3. If n > sr(S), then En (S) E GLn (S).

9 Stably Free Modules

174

T  a unimodular Proposition 9.3.3. Let S be a ring and v := v1 . . . vr stable column vector over S, then there exists a U ∈ Er (S) such that U v = e1 . Proof. There exist elements a1 , . . . , ar−1 ∈ S such that 0 )T ∈ Umc (r − 1, S), with vi0 := vi + ai vr , 1 ≤ i ≤ r − 1. v 0 := (v10 , . . . , vr−1 (9.3.1) Consider the matrix   1 0 0 · · · 0 a1   0 1 0 · · · 0 a2   .. .. .. .. .. ..  E1 :=  . . . . . .  ∈ Er (S); (9.3.2)   0 0 0 · · · 1 ar−1  0 0 0 ··· 0 1 0 0 ) ∈ Umc (r − 1, S), then E1 v = (v10 , . . . , vr−1 , vr )T . Since v 0 := (v10 , . . . , vr−1 Pr−1 Pr−1 0 there exists b1 , . . . , br−1 ∈ S such that i=1 bi vi = 1, and hence, i=1 (v10 − 1 − vr )bi vi0 = v10 − 1 − vr . Let vi00 := (v10 − 1 − vr )bi , 1 ≤ i ≤ r − 1 and   1 0 0 ··· 0 0  0 1 0 · · · 0 0    .. .. .. .. .. ..  ∈ E (S); (9.3.3) E2 :=  . . . .  r . .    0 0 0 · · · 1 0 00 1 v100 v200 v300 · · · vr−1 0 , v10 − 1)T . Moreover, let then E2 E1 v = (v10 , . . . , vr−1



1 0 0 ··· 0 1 0 · · ·   E3 :=  ... ... ... ...  0 0 0 · · · 0 0 0 ···

 0 −1 0 0  .. ..  ∈ E (S), r . .   1 0 0 1

(9.3.4)

0 , v10 − 1)T . Finally, let then E3 E2 E1 v = (1, v20 , . . . , vr−1

 00  0 0    .. ..  ∈ E (S), E4 :=   r . .   0  −vr−1 0 0 · · · 1 0 −v10 + 1 0 0 · · · 0 1 

1 −v20 .. .

0 0 ··· 1 0 ··· ... ... ...

then E4 E3 E2 E1 v = e 1 and U := E1 E2 E3 E4 ∈ Er (S).

(9.3.5)

t u

Next we present two lemmas that give some elementary matrix characterizations of free modules. The second one is needed for the proof of the main theorem of the present section.

9.3 Stafford’s Theorem: A Constructive Proof

175

Lemma 9.3.4. Let S be a ring and let M = hf1 , . . . , fs i be a finitely generated S-module. Then, (i) M is free with basis {f1 , . . . , fs } if and only if Syz({f1 , . . . , fs }) = 0. (ii) M is free if and only if there exist matrices P of size r × s and Q of size s × r such that M ∼ = hP i and QT P T = Ir , with s ≥ r, i.e., M is isomorphic to the column module of a matrix such that its transpose is unimodular. Thus, M is isomorphic to the image of an S-module epimorphism of free modules of finite dimension. Proof. (i) Evident. g (ii) ⇒) There exists an isomorphism M − → S r ; from this we get the epigh

h

morphism S s −→ S r , where S s − → M is defined by h(e i ) := f i , 1 ≤ i ≤ s, and {e 1 , . . . , e s } is the canonical basis of S s . Thus, we get the epimorphism p := gh : S s → S r ; let P be the matrix of p in the canonical bases of S s and S r , then P is of size r × s and hP i ∼ = M . In fact, let {x1 , . . . , xr } be a basis of M . We choose zj ∈ S s such that h(zj ) = xj , 1 ≤ j ≤ r. We define the homomorphism t : M → Im(p) = hP i by t(xj ) := p(zj ). t is injective since if t(a1 ·x1 +· · · +ar ·xr ) = 0 with aj ∈ A, then a1 ·p(z1 )+· · · +ar ·p(zr ) = 0 and hence a1 · gh(z1 ) + · · · + ar · gh(zr ) = 0, so g(a1 · h(z1 ) + · · · + ar · h(zr )) = 0, but g is injective, so a1 ·h(z1 )+· · ·+ar ·h(zr ) = 0, i.e., a1 ·x1 +· · ·+ar ·xr = 0 and from this a1 = · · · = ar = 0. Now, if p(z) ∈ Im(p), with z ∈ S s , then p(z) = gh(z) = g(b1 · x1 + · · · + br · xr ) for some bj ∈ A, so p(z) = g(b1 · h(z1 ) + · · · br · h(zr )) = b1 · gh(z1 ) + · · · + br · gh(zr ) = b1 · p(z1 ) + · · · + br · p(zr ) = t(b1 · x1 + · · · + br · xr ), and this proves that t is surjective. q → S s such that Since S r is projective there exists a homomorphism S r − pq = iS r and hence QT P T = Ir , with s ≥ r. ⇐) Now we assume that hP i ∼ = M and QT P T = Ir , where P is of size r × s and Q is of size s × r, with s ≥ r. If p, q are the homomorphisms defined by P and Q, we have pq = iS r and S r = Im(iS r ) ⊆ Im(p) ⊆ S r , i.e., u t M∼ = Im(p) = S r . Lemma 9.3.5. Let S be a ring and M a stably free S-module given by a f1 f0 minimal presentation S s −→ S r −→ M → 0. Let g1 : S r → S s such that g1 f1 = iS s . Then the following conditions are equivalent: (i) M is free of dimension r − s.   Is , where G1 0 is the matrix of g1 in the canonical bases. In such case, the last r − s columns of U T conform a basis for M . Moreover, the first s columns of U T conform the matrix F1 of f1 in the canonical bases. (iii) There exists a matrix V ∈ GLr (S) such that G1T coincides with the first s columns of V , i.e., G1T can be completed to an invertible matrix V of GLr (S). (ii) There exists a matrix U ∈ GLr (S) such that U GT1 =

9 Stably Free Modules

176

f1

f0

Proof. By the hypothesis, the exact sequence 0 → S s −→ S r −→ M → 0 splits, so F1T admits a right inverse G1T , where F1 is the matrix of f1 in the canonical bases and G1 is the matrix of g1 : S r → S s , with g1 f1 = iS s , i.e., F1T G1T = Is . Moreover, there exists a g0 : M → S r such that f0 g0 = iM . g0 g1 From this we get also the split sequence 0 → M −→ S r −→ S s → 0. Note ∼ ker(g1 ). that M = (i) ⇒ (ii) We have S r = ker(g1 ) ⊕ Im(f1 ); by the hypothesis ker(g1 ) is free. If s = r then ker(g1 ) = 0 and hence f1 is an isomorphism, so f1 g1 = iS r , i.e., G1T F1T = Ir . Thus, we can take U := F1T . Let r > s; if {e 1 , . . . , e s } is the canonical basis of S s , then {u 1 , . . . , u s } is a basis of Im(f1 ) with u i := f1 (e i ), 1 ≤ i ≤ s; let {v 1 , . . . , v p } be a basis of ker(g1 ) with p = r − s. Then, {v 1 , . . . , v p , u 1 , . . . , u s } is a basis of S r . h

→ S r by h(e i ) := u i for 1 ≤ i ≤ s, and h(e s+j ) = v j for We define S r − 1 ≤ j ≤ p. Clearly h is bijective; moreover, g1 h(e  i ) = g1 (u i ) = g1 f1 (e i ) = e i Is T T . Let U := H T , so we observe and g1 h(e s+j ) = g1 (v j ) = 0, i.e., H G1 = 0 ∼ M and the first that the last p columns of U T conform a basis of ker(g1 ) = T s columns of U conform F1 . (ii) ⇒ (i) Let U(k) be the k-th row of U , then U G1T = [U(1) · · · U(s) · · · U(r) ]T G1T =

  Is , 0

so U(i) G1T = e Ti , 1 ≤ i ≤ s, U(s+j) G1T = 0, 1 ≤ j ≤ p with p := r − s. This means that (U(s+j) )T ∈ ker(g1 ) and hence h(U(s+j) )T |1 ≤ j ≤ pi ⊆ ker(g1 ). T −1 T T T r On the other hand,   let c ∈ ker(g1 ) ⊆ S , then c G1 = 0 and c U U G1 = I 0, thus c T U −1 s = 0 and hence (c T U −1 )T ∈ ker(l), where l : S r → S s is 0   the homomorphism with matrix Is 0 . Let d = [d1 , . . . , dr ]T ∈ ker(l), then   I [d1 , . . . , dr ] s = 0 and from this we conclude that d1 = · · · = ds = 0, 0 i.e., ker(l) = he s+1 , e s+2 , . . . , e s+p i. From (c T U −1 )T ∈ ker(l) we get that (c T U −1 )T = a1 ·e s+1 +· · ·+ap ·e s+p , so c T U −1 = (a1 ·e s+1 +· · ·+ap ·e s+p )T , i.e., c T = (a1 · e s+1 + · · · + ap · e s+p )T U and from this we get that c ∈ h(U(s+j) )T |1 ≤ j ≤ pi. This proves that ker(g1 ) = h(U(s+j) )T |1 ≤ j ≤ pi; but since U is invertible, ker(g1 ) is free of dimension p. We have also proved that a basis for ker(g1 ) ∼ the last p columns of U T conform = M .  Is −1 Is T T (ii) ⇔ (iii) U G1 = if and only if G1 = U , but the first s 0 0   I columns of U −1 s coincide with the first s columns of U −1 ; taking V := 0 U −1 we get the result. t u Theorem 9.3.6. Let S be a ring. Any stably free S-module M with rank(M ) ≥ sr(S) is free with dimension equal to rank(M ).

9.3 Stafford’s Theorem: A Constructive Proof

177

Proof. Since M is stably free it has a minimal presentation, and hence, it is given by an exact sequence f1

f0

0 → S s −→ S r −→ M → 0; moreover, note that rank(M ) = r − s. Since this sequence splits, F1T admits a right inverse GT1 , where F1 is the matrix of f1 in the canonical bases and G1 is the matrix of g1 : S r → S s , with g1 f1 = iS s .The idea of the proof is I to find a matrix U ∈ GLr (S) such that U GT1 = s and then apply Lemma 0 9.3.5. We have F1T GT1 = Is and from this we get that the first column g 1 of GT1 is unimodular, but since r > r − s ≥ sr(S), g 1 is stable, and by Proposition 9.3.3, there exists a U1 ∈ Er (S) such that U1 g 1 = e 1 . If s = 1 we are finished since GT1 = g 1 . Let s ≥ 2; we have   1 ∗ T U 1 G1 = , F2 ∈ M(r−1)×(s−1) (S). 0 F2 Note that U1 GT1 has a left inverse (for instance F1T U1−1 ), and the form of this left inverse is   1 ∗ L= , L2 ∈ M(s−1)×(r−1) (S), 0 L2 and hence L2 F2 = Is−1 . The first column of F2 is unimodular and since r − 1 > r − s ≥ sr(S) we apply again Proposition 9.3.3 and obtain a matrix U20 ∈ Er−1 (S) such that   1 ∗ U20 F2 = , F3 ∈ M(r−2)×(s−2) (S). 0 F3 Let 

 1 0 U2 := ∈ Er (S), 0 U20 then we have 

 1∗ ∗ U2 U1 GT1 = 0 1 ∗ . 0 0 F3 By induction on s and multiplying on the left by elementary matrices we get a matrix U ∈ Er (S) such that   I T U G1 = s . 0 t u Corollary 9.3.7 (Stafford). Let D := An (K) or Bn (K), with char(K) = 0. Then, any stably free left D-module M satisfying rank(M ) ≥ 2 is free.

178

9 Stably Free Modules

Proof. The results follows from Theorem 9.3.6 since sr(D) = 2.

t u

9.4 The Projective Dimension of a Module Closely related to the study of stably free modules is the computation of the projective dimension of a given module M . Next we will present some theoretical results that will be used in Chapter 17 for computing the projective dimension of finitely presented left modules over some key examples of skew P BW extensions. The first one only requires the computation of arbitrary free resolutions of M ; the second one allows moreover the computations of a minimal presentation of a finitely presented module M when a finite free resolution of M is given, and also, it makes it possible to check if M is stably free (see [323]). Recall that S denotes an arbitrary noncommutative RC ring. We start with the following theorem, which can be used for testing if a finitely presented module is projective (compare with [238], Theorem 4). Theorem 9.4.1. Let M be an S-module given by a presentation f0

0 → K → S n −→ M → 0, where K is f.g. Then, the following conditions are equivalent: (i) M is projective. (ii) Ext1S (M, K) = 0. Proof. (i) ⇒ (ii) This implication is well known, see [350]. (ii) ⇒ (i) From the given sequence we get the exact sequence (f0 )∗

0 → HomS (M, K) → HomS (M, S n ) −−−→ HomS (M, M ) → Ext1S (M, K) = 0, see [350], Theorem 7.3. Then, (f0 )∗ is surjective and there exists an f ∈ HomS (M, S n ) such that (f0 )∗ (f ) = iM , i.e., f0 f = iM . This means that Sn ∼ t u = K ⊕ M , i.e., M is projective. Let fr+1

fr

fr−1

f2

f1

f0

· · · −−−→ Pr −→ Pr−1 −−−→ · · · −→ P1 −→ P0 −→ M −→ 0 be a projective resolution of M ; recall that ker(fi ) is called the i-th syzygy of M . When Pi := S si is free of finite dimension, we get a free resolution of M. Theorem 9.4.2. Let M be an S-module and fr+1

fr

fr−1

f2

f1

f0

· · · −−−→ Pr −→ Pr−1 −−−→ · · · −→ P1 −→ P0 −→ M −→ 0

(9.4.1)

be a projective resolution of M . Let r be the smallest integer such that Im(fr ) is projective. Then r does not depend on the resolution and pd(M ) = r.

9.4 The Projective Dimension of a Module

179

Proof. It is well known that pd(M ) ≤ r if and only if there exists a projective resolution of M where the (r − 1)-th syzygy is projective if and only if for every projective resolution of M the (r − 1)-th syzygy is projective (see [350], Theorem 9.5). Let r be the smallest integer such Im(fr ) is projective, since Im(fr ) = ker(fr−1 ) = (r − 1)-th syzygy, then pd(M ) ≤ r. Suppose that pd(M ) = t < r, then the (t − 1)-th syzygy of (9.4.1) is projective, but this means that r is not minimal. Thus, pd(M ) = r. Let 0 fs+1

0 fs−1

f0

f0

f0

f0

s 2 1 0 0 · · · −−−→ Ps0 −→ Ps−1 −−−→ · · · −→ P10 −→ P00 −→ M −→ 0

be another projective resolution of M , where s is the smallest integer such that Im(fs0 ) is projective. Then pd(M ) ≤ s and hence r ≤ s. Suppose that r < s. The (r − 1)-th syzygy of M in the previous resolution is projective since pd(M ) = r, but this is impossible since s is minimal, hence r = s. t u Next we present a second result that also allows us to compute the projective dimension of a module given by a finite free resolution. For this we follow [323]. Theorem 9.4.3. Let M be an S-module and fm

fm−1

fm−2

f2

f1

f0

0 → Pm −−→ Pm−1 −−−→ Pm−2 −−−→ · · · −→ P1 −→ P0 −→ M −→ 0 (9.4.2) be a projective resolution of M . If m ≥ 2 and there exists a homomorphism gm : Pm−1 → Pm such that gm fm = iPm , then we have the following projective resolution of M : hm−2

hm−1

fm−3

0 → Pm−1 −−−→ Pm−2 ⊕ Pm −−−→ Pm−3 −−−→ f2

f1

f0

· · · −→ P1 −→ P0 −→ M −→ 0

(9.4.3)

with 

hm−1

 fm−1 := , gm

  hm−2 := fm−2 0 .

Proof. Im(hm−1 ) ⊆ ker(hm−2 ): we have     fm−1 hm−2 hm−1 = fm−2 0 = 0. gm ker(hm−2 ) ⊆ Im(hm−1 ): let (a, b)T ∈ ker(hm−2 ), then a ∈ Pm−2 , b ∈ Pm and hm−2 [(a, b)T ] = 0 = fm−2 (a). Then there exists a c ∈ Pm−1 such that a = fm−1 (c); we define   d := iPm−1 − fm gm fm (c, b)T = c − (fm gm )(c) + fm (b) ∈ Pm−1 . Then, the image of d under hm−1 is

9 Stably Free Modules

180







 fm−1 (c) − fm−1 (fm (gm (c))) + fm−1 (fm (b)) fm−1 (c) = = gm (c) − ((gm fm )gm )(c) + gm fm (b) gm (c) − gm (c) + b   a . b

hm−1 is injective: if d ∈ ker(hm−1 ), then hm−1 (d) = 0, so fm−1 (d) = 0 and gm (d) = 0; we consider the exact sequence fm−1

fm

0 → Pm −−→ Pm−1 −−−→ Im(fm−1 ) → 0, since gm fm = iPm this sequence splits, i.e., there exists a homomorphism km−1 : Im(fm−1 ) → Pm−1 such that iPm−1 = fm gm + km−1 fm−1 . Hence, d = fm gm (d) + km−1 fm−1 (d) = 0. Finally, Im(hm−2 ) = hm−2 (Pm−2 ⊕ Pm ) = fm−2 (Pm−2 ) = Im(fm−2 ) = u t ker(fm−3 ). Corollary 9.4.4. Let M be an S-module and fm−1

fm

fm−2

f1

f2

f0

0 → S sm −−→ S sm−1 −−−→ S sm−2 −−−→ · · · −→ S s1 −→ S s0 −→ M −→ 0 (9.4.4) a finite free resolution of M . Let Fi be the matrix of fi in the canonical bases, 1 ≤ i ≤ m. Then, (i) If m ≥ 3 and there exists a homomorphism gm : S sm−1 → S sm such that gm fm = iS sm , then we have the following finite free resolution of M : hm−2

hm−1

fm−3

f

f

0 1 M −→ 0 0 → S sm−1 −−−−→ S sm−2 +sm −−−−→ S sm−3 −−−−→ · · · − S s0 − − → − → (9.4.5)

with  hm−1 :=

 fm−1 , gm

  hm−2 := fm−2 0 .

In a matrix notation, if Gm is the matrix of gm and Hj is the matrix of hj in the canonical bases, j = m − 1, m − 2, then  T    T F T T GTm , Hm−2 := Fm−1 := m−2 . Hm−1 0 (ii) If m = 2 and there exists a homomorphism g2 : S s1 → S s2 such that g2 f2 = iS s2 , then we have the following finite presentation of M : f0

h

1 0 → S s1 −→ S s0 +s2 −→ M → 0,

with   f h1 := 1 , g2

  h0 := f0 0 .

In a matrix notation,   H1T := F1T GT2 ,

H0T =



 F0T . 0

(9.4.6)

9.4 The Projective Dimension of a Module

181

Proof. This is an obvious consequence of the previous theorem.

t u

Theorem 9.4.5. Let M be an S-module and n ≥ 1. pd(M ) = n if and only if there exists a finite projective resolution of M as (9.4.2) where fn is non-split, i.e., there exists no homomorphism gn : Pn−1 → Pn such that gn fn = iPn . Proof. ⇒) There exists a finite projective resolution of M as in (9.4.2) with fn

fn−1

m = n; we have the exact sequence 0 → Pn −→ Pn−1 −−−→ Im(fn−1 ) → 0. If fn splits, then Im(fn−1 ) is projective, and by Theorem 9.4.2, pd(M ) ≤ n − 1, false. Thus, fn is non-split. ⇐) If M has a finite projective resolution as in (9.4.2), with m = n, which is non-split, then pd(M ) ≤ n and Im(fn−1 ) is not projective. Suppose that there exists a k ≤ n − 2 such that Im(fk ) is projective; we have the exact sequence ι

fk

0 → Im(fk+1 ) → − Pk −→ Im(fk ) → 0, where ι is the canonical inclusion, and hence, Im(fk+1 ) is also projective. We can repeat this reasoning and we get that Im(fn−1 ) is projective, false. Thus, the smallest r such that Im(fr ) is projective is r = n, and by Theorem 9.4.2, pd(M ) = n. t u Remark 9.4.6. The results above will be used in Chapter 17 to construct algorithms for computing the projective dimension of modules over some bijective skew P BW extensions, and also, for computing minimal presentations and testing stably-freeness.

Chapter 10

Hermite Rings

Rings for which all stably free modules are free have attracted special attention in homological algebra. In this chapter we will consider a matrixconstructive interpretation of such rings and some other closely related classes. We will also study some classical algebraic constructions as quotients, products and rings of fractions of these rings. The material presented here can be considered as preparatory for the next chapter when we will study the Hermite condition for skew P BW extensions. Recall that all rings considered are RC (see Remark 9.1.9).

10.1 Matrix Descriptions of Hermite Rings Definition 10.1.1. Let S be a ring. (i) S is a PF ring if every f.g. projective S-module is free. (ii) S is a PSF ring if every f.g. projective S-module is stably free. (iii) S is a Hermite ring, a property denoted by H, if any stably free Smodule is free. The right versions of the above rings (i.e., for right modules) are defined in a similar way and denoted by P Fr , P SFr and Hr , respectively. We say that S is a PF ring if S is P F and P Fr simultaneously; similarly, we define the properties PSF and H. However, we will prove below later that these properties are left-right symmetric, i.e., they can be denoted simply by PF , PSF and H. For domains we will write PFD, PSFD and HD. From Definition 10.1.1 we get that H ∩ P SF = P F.

(10.1.1)

The following theorem gives a matrix description of H rings (see [97] and compare with [239] for the particular case of commutative rings. In [88] a different and independent proof of this theorem is presented for right modules).

© Springer Nature Switzerland AG 2020 W. Fajardo et al., Skew PBW Extensions, Algebra and Applications 28, https://doi.org/10.1007/978-3-030-53378-6_10

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184

Theorem 10.1.2. Let S be a ring. Then, the following conditions are equivalent. (i) S is H. (ii) For every r ≥ 1, any unimodular row matrix u over S of size 1 × r can be completed to an invertible matrix of GLr (S) by adding r −1 new rows. (iii) For every r ≥ 1, if u is a unimodular row matrix of size 1 × r, then there exists a matrix U ∈ GLr (S) such that uU = (1, 0, . . . , 0). (iv) For every r ≥ 1, given a unimodular matrix F of size s × r, r ≥ s, there exists a U ∈ GLr (S) such that   F U = Is | 0 . Proof. (i) ⇒ (ii) Let u := [u1 · · · ur ] and v := [v1 · · · vr ]T such that uv = 1, i.e., u1 v1 + · · · + ur vr = 1; we define α

→S Sr − e i 7→ vi where {e 1 , . . . , e r } is the canonical basis of the left free module S r of column vectors. Observe that α(u T ) = 1; we define the homomorphism β : S → S r by β(1) := u T , then αβ = iS . From this we get that S r = Im(β) ⊕ ker(α), ∼ S and Im(β) is free with basis {u T }. This β is injective, hu T i = Im(β) = r ∼ implies that S = S ⊕ ker(α), i.e., ker(α) is stably free of rank r − 1, so by the hypothesis, ker(α) is free of dimension r − 1; let {x 1 , . . . , x r−1 } be a basis of ker(α), then {u T , x 1 , . . . , x r−1 } is a basis of S r . This means that T  T u x 1 · · · x r−1 ∈ GLr (S), i.e., u can be completed to an invertible matrix of GLr (S) by adding r − 1 rows. (ii) ⇒) (i) Let M be a stably free S-module, then there exist integers ∼ S s ⊕ M . It is enough to prove that M is free when r, s ≥ 0 such that S r = r ∼ s s = 1. In fact, S = S ⊕ M = S ⊕ (S s−1 ⊕ M ) is free and hence S s−1 ⊕ M is free; repeating this reasoning we conclude that S ⊕ M is free, so M is free. Let r ≥ 1 such that S r ∼ = S ⊕ M , let π : S r −→ S be the canonical projection with kernel isomorphic to M and let {e 1 , . . . , e r } be the canonical basis of S r ; there exists a µ : S −→ S r such that πµ = iS and S r = ker(π) ⊕ Im(µ). Let µ(1) := u T := [u1 · · · ur ]T ∈ S r , then π(u T ) = 1 = u1 π(e 1 ) + · · · + ur π(e r ), i.e., v := [π(e 1 ) · · · π(e r )]T is such that uv = 1, moreover, S r = ker(π) ⊕ hu T i. By the hypothesis, there exists a U ∈ GLr (S) such that e 1T U = u. Let f T : S r −→ S r be the homomorphism defined by U T , then f T (e 1 ) = T u and f T (e i ) = u i for i ≥ 2, where u 2 , . . . , u r are the other columns of U T (i.e., the transpose of the other rows of U ). Since U = (U T )T , we see that f T is an isomorphism. If we prove that f T (e i ) ∈ ker(π) for each i ≥ 2, then ker(π) is free, and consequently, M is free. In fact, let f 0 be the restriction of f T to he 2 , . . . , e r i, i.e., f 0 : he 2 , . . . , e r i −→ ker(π). Then f 0 is bijective: of course f 0 is injective; let w be any vector of S r , then there exists an x ∈ S r such that f T (x ) = w . We write x := [x1 · · · xr ]T = x1 e 1 + z , with z = x2 e 2 + · · · + xr e r . We have f T (x ) = f T (x1 e 1 + z ) =

10.1 Matrix Descriptions of Hermite Rings

185

x1 f T (e 1 ) + f T (z ) = x1 u T + f T (z ) = w . In particular, if w ∈ ker(π), then w − f T (z ) ∈ ker(π) ∩ hu T i = 0, so w = f T (z ) and hence w = f 0 (z ), i.e., f 0 is surjective. In order to conclude the proof we will show that f T (e i ) ∈ ker(π) for each i ≥ 2. Since f T was defined by U T , the idea is to change U T in such a way that its first column was u T and for the others columns, u i ∈ ker(π), 2 ≤ i ≤ r. Let π(u i ) := ri ∈ S, i ≥ 2, and u 0i := u i − ri u T ; then adding to the i-th column of U T the first column multiplied by −ri , we get a new matrix U T such that its first column is again u T and for the others we have π(u i0 ) = π(u i ) − ri π(u T ) = ri − ri = 0, i.e., u 0i ∈ ker(π). (ii) ⇔ (iii) u can be completed to an invertible matrix of GLr (S) if and only if there exists a V ∈ GLr (S) such that (1, 0, . . . , 0)V = u if and only if (1, 0, . . . , 0) = uV −1 ; thus U := V −1 . (iii) ⇒) (iv) The proof will be done by induction on s. For s = 1 the result is trivial. We assume that (iv) is true for unimodular matrices with l ≤ s − 1 rows. Let F be a unimodular matrix of size s × r, r ≥ s, then there exists a matrix B such that F B = Is . This implies that the first row u of F is unimodular; by (iii) there exists a U 0 ∈ GLr (S) such that uU 0 = (1, 0, . . . , 0) = e 1T , and hence F U 0 = F 00 ,  T e 00 F = 10 , F with F 0 a matrix of size (s − 1) × r. Since F B = Is , we have Is = F 00 (U 0−1 B), i.e., F 00 is a unimodular matrix; let F 000 be the matrix eliminating the first with r−1 ≥ s−1, column of F 0 , then F 000 is unimodular of size   (s−1)×(r−1), 0 ∗ since the right inverse of F 00 has the form . By induction, there exists ∗ G000   000 a matrix C ∈ GLr−1 (S) such that F C = Is−1 | 0 . From this we get,    F U 0 = F 00 =  

1 a011 ... a0s−11

 0 ··· 0    a012 · · · a01r  1 0 . ..  = ∗ F 000 , .  .. 0 0 as−12 · · · as−1r

and hence FU

0



      1 0 1 0 1 0 0 1 0 = = . 0C ∗ F 000 0 C ∗ Is−1 0

Multiplying the last matrix on the right by elementary matrices we get (iv). (iv) ⇒) (iii) Taking s = 1 and F = u in (iv) we get (iii). t u From the proof of the previous theorem we get the following result. Corollary 10.1.3. Let S be a ring. Then, S is H if and only if any stably free S-module M of type S r ∼ = S ⊕ M is free.

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Remark 10.1.4. (a) If we consider right modules and the right S-module structure on the module S r of column vectors, the conditions of the previous theorem can be formulated in the following way: (i)r S is Hr . (ii)r For every r ≥ 1, any unimodular column matrix v over S of size r × 1 can be completed to an invertible matrix of GLr (S) by adding r − 1 new columns. (iii)r For every r ≥ 1, given an unimodular column matrix v over S of size r × 1 there exists a matrix U ∈ GLr (S) such that U v = e 1 . (iv)r For every r ≥ 1, given an unimodular matrix F of size r × s, r ≥ s, there exists a U ∈ GLr (S) such that   I UF = s . 0 The proof is as in the commutative case, see [239]. Corollary 10.1.3 can be formulated in this case as follows: S is Hr if and only if any stably free right S-module M of type S r ∼ = S ⊕ M is free. (b) Considering again left modules and disposing the matrices of homomorphisms by rows and composing homomorphisms from the left to the right (see Remark 9.1.2), we can repeat the proof of Theorem 10.1.2 and obtain the equivalence of conditions (i)–(iv). Thus, we do not need to take transposes in the proof of Theorem 10.1.2. (c) If S is a commutative ring, of course, left and right conditions are equivalent, see [239]. This follows from the fact that (F G)T = GT F T for any matrices F ∈ Mr×s (S), G ∈ Ms×r (S). However, as we remarked before, the Hermite condition is left-right symmetric for general rings (Proposition 10.2.7). Another independent proof of this fact can be found in [88], Theorem 11.4.4.

10.2 Matrix Characterization of P F Rings In [97] are given some matrix characterizations of projective-free rings, in this section we present another matrix interpretation of this important class of rings. The main result presented here (Corollary 10.2.4) extends Theorem 6.2.2 in [239]; this theorem has also been proved independently in [88], Proposition 11.4.9. A matrix proof of a theorem of Kaplansky about finitely generated projective modules over local rings is also included. Theorem 10.2.1. Let S be a Hermite ring and M an f.g. projective module given by the column module of a matrix F ∈ Ms (S), with F T idempotent. Then, M is free with dim(M ) = r if and only if there exists a matrix U ∈ Ms (S) such that U T ∈ GLs (S) and (U T )−1 F T U T =



0 0 0 Ir

T .

(10.2.1)

10.2 Matrix Characterization of P F Rings

187

In such case, a basis of M is given by the last r rows of (U T )−1 . Proof. ⇒) As in the proof of Proposition 9.1.1, let f : S s → S s be the homomorphism defined by F and S s = M ⊕ M 0 with Im(f ) = M and M 0 = ker(f ); by the hypothesis, M is free with dimension r, so r ≤ s (recall that S is RC). Let h : M → S r be an isomorphism and {z 1 , . . . , z r } ⊂ M such that h(z i ) = e i , 1 ≤ i ≤ r, then {z 1 , . . . , z r } is a basis of M . Since S is a Hermite ring, M 0 is free, let {w 1 , . . . , w s−r } be a basis of M 0 (recall that S is IBN ). Then {w 1 , . . . , w s−r ; z 1 , . . . , z r } is a basis for S s . With this we define u in the following way: u(w j ) := e j , for 1 ≤ j ≤ s − r, u(z i ) := e s−r+i , for 1 ≤ i ≤ r. Note that u is an isomorphism and we get the following commutative diagram f

S s −−−−→   uy

Ss  u y

t

S s −−−0−→ S s where t is given by t0 (e j ) := 0 if 1 ≤ j ≤ s − r, and t0 (e s−r+i ) = e s−r+i if 1 ≤ i ≤ r; so the matrix of t0 in the canonical basis is   0 0 T0 = . 0 Ir Thus, uf = t0 u and hence F T U T = U T T0T . Note that (U T )−1 exists since u is an isomorphism, hence (U T )−1 F T U T = T0T . From u(z i ) := e s−r+i we get that (z Ti U T )T = e s−r+i , so z Ti U T = e Ts−r+i and hence z i = e Ts−r+i (U T )−1 , i.e., the basis of M coincides with the last r rows of (U T )−1 . ⇐) Let f, u be the homomorphisms defined by F and U , then m(uf ) = m(t0 u), where t0 is the homomorphism defined by T0 . This means that uf = t0 u, but by the hypothesis U T is invertible, so u is an isomorphism; from this we conclude that Im(f ) ∼ = Im(t0 ), i.e., M = Im(f ) ∼ = Im(t0 ) = hT0 i ∼ = Sr. Note that this part of the proof does not use that S is a Hermite ring. t u From the previous theorem we get the following matrix description of P F rings. Corollary 10.2.2. Let S be a ring. S is P F if and only if for each s ≥ 1, given a matrix F ∈ Ms (S), with F T idempotent, there exists a matrix U ∈ Ms (S) such that U T ∈ GLs (S) and T −1

(U )

where r = dim(hF i), 0 ≤ r ≤ s.

T

F U

T



0 0 = 0 Ir

T ,

(10.2.2)

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Proof. ⇒) Let F ∈ Ms (S), with F T idempotent, and let M be the S-module generated by the columns of F . By Proposition 9.1.1, M is an f.g. projective module, and by the hypothesis, M is free. Since S is H, we can apply Theorem 10.2.1. If r = dim(M ), then r = dim(hF i). ⇐) Let M be a finitely generated projective S-module, so there exists an f

s ≥ 1 such that S s = M ⊕ M 0 ; let S s − → S s be the canonical projection on T M , so F is idempotent and, by the hypothesis, there exists a U ∈ Ms (S) such that U T ∈ GLs (S) and (10.2.2) holds. From the second part of the proof of Theorem 10.2.1 we get that M is free. t u Remark 10.2.3. (i) If we consider right modules instead of left modules, then the previous corollary can be reformulated in the following way: S is P Fr if and only if for each s ≥ 1, given an idempotent matrix F ∈ Ms (S), there exists a matrix U ∈ GLs (S) such that   0 0 U F U −1 = , (10.2.3) 0 Ir where r = dim(hF i), 0 ≤ r ≤ s, and hF i represents the right S-module generated by the columns of F . The proof is as in the commutative case, see [239]. (ii) Considering again left modules and disposing the matrices of homomorphisms by rows and composing homomorphisms from the left to the right (Remark 9.1.2), we can repeat the proofs of Theorem 10.2.1 and Corollary 10.2.2 and get the characterization (10.2.3) for the P F property; with this row notation we do not need to take transposes in the proofs. However, observe that in this case hF i represents the left S-module generated by the rows of F . Note that Corollary 10.2.2 can be formulated this way: in fact,  T   0 0 0 0 = 0 Ir 0 Ir and we can rewrite (10.2.2) as (10.2.3), changing F T by F (see Remark 9.1.2) and (U T )−1 by U . In this case, a basis of M is given by the last r rows of U . (iii) If S is a commutative ring, of course P F = P Fr = PF . However, we will prove in Corollary 10.2.5 that the projective-free property is left-right symmetric for general rings. Corollary 10.2.4. S is P F if and only if for each s ≥ 1, given an idempotent matrix F ∈ Ms (S), there exists a matrix U ∈ GLs (S) such that   0 0 −1 UFU = , (10.2.4) 0 Ir where r = dim(hF i), 0 ≤ r ≤ s, and hF i represents the free left S-module M generated by the rows of F . Moreover, the final r rows of U form a basis of M. Proof. This is the content of the part (ii) in the previous remark.

t u

10.2 Matrix Characterization of P F Rings

189

Corollary 10.2.5. Let S be a ring. S is P F if and only if S is P Fr , i.e., P F = P Fr = PF . Proof. Let F ∈ Ms (S) be an idempotent matrix, if S is P F , then there exists a P ∈ GLs (S) such that   0 0 U F U −1 = , 0 Ir where r is the dimension of the left S-module generated by the rows of F . Observe that U F U −1 is also idempotent, moreover, the matrices X := U F and Y := U −1 satisfy U F U −1 = XY and F = Y X. Then from Proposition 0.3.1 in [97] we conclude that the left S-module generated by the rows of U F U −1 coincides with the left S-module generated by the rows of F , and also, the right S-module generated by the columns of U F U −1 coincides with the right S-module generated by the columns of F . This implies that the S-module generated by the rows of F coincides with the right S-module generated by the columns of F . This means that S is P Fr . The symmetry of the problem completes the proof. t u Another interesting matrix characterization of PF rings is given in [97], Proposition 0.4.7: a ring S is PF if and only if given an idempotent matrix F ∈ Ms (S) there exist matrices X ∈ Ms×r (S), Y ∈ Mr×s (S) such that F = XY and Y X = Ir . A similar matrix interpretation can be given for P SF rings using Proposition 0.3.1 in [97] and Corollary 9.2.5. Proposition 10.2.6. Let S be a ring. Then, (i) S is P SF if and only if given an idempotent matrix F ∈ Mr (S) there exist s ≥ 0 and matrices X ∈ M(r+s)×r (S), Y ∈ Mr×(r+s) (S) such that   F 0 = XY and Y X = Ir . 0 Is (ii) P SF = P SFr = PSF . Proof. Direct consequence of Proposition 0.3.1 in [97] and Corollary 9.2.5. t u For the H property we have a similar characterization that proves the symmetry of this condition. Proposition 10.2.7. Let S be a ring. Then, (i) S is H if and only if given an idempotent matrix F ∈ Mr (S) with factorization   F 0 = XY and Y X = Ir , 0 1 for some matrices X ∈ M(r+1)×r (S), Y ∈ Mr×(r+1) (S), there exist matrices X 0 ∈ Mr×(r−1) (S), Y 0 ∈ M(r−1)×r (S) such that F = X 0 Y 0 and Y 0 X 0 = Ir−1 . (ii) H = Hr = H.

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Proof. Direct consequence of Propositions 0.3.1 and 0.4.7 in [97] and Corollary 10.1.3. t u We conclude this section by giving a matrix constructive proof of a wellknown theorem of Kaplansky. Proposition 10.2.8. Any local ring S is PF . Proof. Let M be an f.g. projective left S-module. By Remark 9.1.2, part (ii), there exists an idempotent matrix F = [fij ] ∈ Ms (S) such that the module generated by the rows of F coincides with M . According to Corollary 10.2.4, we need to show that there exists a U ∈ GLs (S) such that the relation (10.2.4) holds. The proof is by induction on s. s = 1: in this case F = [fij ] = [f ]; since S is local, its idempotents are trivial, then f = 1 or f = 0 and hence M is free. s = 2: since S is local, two possibilities may arise: f11 is invertible. Then, one can find G ∈ GL2 (S) such that   10 GF G−1 = , 0f for some f ∈ S. For this it is enough to take   −1 1 f11 f12 G= . −1 −f21 f11 1 To show that this matrix is invertible with inverse   f −f12 G−1 = 11 , −1 f21 −f21 f11 f12 + 1 we can use the relations that exist between the entries of F . To see, for example, that GG−1 = I2 : −1 2 f11 + f11 f12 f21 = 1 because f11 + f12 f21 = f11 and f11 is invertible; −1 −1 −1 −1 −1 −f12 − f11 f12 f21 f11 f12 + f11 f12 = −f12 + (1 − f11 f12 f21 )f11 f12 −1 = −f12 + f11 f11 f12 = 0; −1 −f21 f11 f11 + f21 = 0; −1 −1 f21 f11 f12 − f21 f11 f12 + 1 = 1.

Similar calculations show that G−1 G = I2 . Since F is idempotent, so is f ; applying the case s = 1 we get the result. 1 − f11 is invertible. In the same way, we can find H ∈ GL2 (S) such that   00 HF H −1 = ; 0g for this it is enough to take   1 −(1 − f11 )−1 f12 H= . f21 −f21 (1 − f11 )−1 f12 + 1 Note that

10.2 Matrix Characterization of P F Rings

H −1 =



191

1 − f11 (1 − f11 )−1 f12 . −f21 1 

Indeed HH −1 = I2 : 1−f11 +(1−f11 )−1 f12 f21 = 1−f11 +f11 = 1 because f12 f21 = (1−f11 )f11 ; (1 − f11 )−1 f12 − (1 − f11 )−1 f12 = 0; f21 (1 − f11 ) + f21 (1 − f11 )−1 f12 f21 − f21 = f21 (1 − f11 ) + f21 f11 − f21 = 0; f21 (1 − f11 )−1 f12 − f21 (1 − f11 )−1 f12 + 1 = 1. An analogous calculation shows that H −1 H = I2 . Note that g is an idempotent of S, then g = 0 or g = 1, and the statement follows. Suppose that the result holds for s − 1 and consider the two possibilities for f11 : if f11 is invertible, taking   −1 −1 −1 f12 f11 1 f11 f1s f13 · · · f11 −1   0  ··· 1 0 −f21 f11 −f31 f −1  0 1 ··· 0  11 G=   ..   ··· . −1 −fs1 f11

0

0

···

we have that G ∈ GLs (S) and its inverse is:  −f13 f11 −f12 −1 −1 f21 −f21 f11 + 1 −f f 21 f11 f13 12  −1 −1  G−1 = f31 −f31 f11 f12 −f31 f11 f13 + 1  ..  . fs1

−1 f12 −fs1 f11

−1 −fs1 f11 f13

··· ··· ···

1

−f1s −1 f1s −f21 f11 −1 f1s −f31 f11

··· −1 · · · −fs1 f11 f1s + 1

    .  

In fact, to see that GG−1 = Is : −1 −1 f11 + f11 f12 f21 + · · · + f11 f1s fs1 = 1 because 2 + f12 f21 + · · · + f1s fs1 = f11 ; f11 −1 −1 −1 −1 −1 f12 − f11 f12 + f11 f12 − · · · − f12 − f11 f12 f21 f11 f13 f31 f11 −1 −1 f1s fs1 f11 f12 − f11 s X −1 −1 −1 f12 = 0; f12 = −f12 + f11 f11 f1i fi1 )f11 = −f12 + (1 − f11 i=2

. .. −1 −1 −1 −1 f13 f31 f11 − f1s − f11 f12 f21 f11 f1s − f11 f1s − · · · −1 −1 −1 f1s fs1 f11 f1s − f11 f1s + f11 s X −1 −1 −1 = −f1s + (1 − f11 f1s = 0; f1s = −f1s + f11 f11 f1i fi1 )f11 i=2

10 Hermite Rings

192

−1 −1 −1 − f21 f11 f12 + 1 = 1; f12 − f21 f11 f11 + f21 = 0; f21 f11 −1 −1 f21 f11 f1i = 0 for every 3 ≤ i ≤ s; f1i − f21 f11 .. . −1 f11 + fs1 = 0; − fs1 f11 −1 −1 f1i − fs1 f11 fs1 f11 f1i = 0 for every 2 ≤ i ≤ s − 1 −1 −1 and, finally, fs1 f11 f1s − fs1 f11 f1s + 1 = 1.

Similarly, G−1 G = Is . Moreover, GF G

−1

 =

1 0s−1,1

 01,s−1 , F1

where F1 ∈ Ms−1 (S) is an idempotent matrix. It only remains to apply the induction hypothesis. If 1 − f11 is invertible, taking −(1 − f11 )−1 f13 1 −(1 − f11 )−1 f12 f21 −f21 (1 − f11 )−1 f12 + 1 −f21 (1 − f11 )−1 f13  −1 −1  H = f31 −f31 (1 − f11 ) f12 −f31 (1 − f11 ) f13 + 1  .  .. −fs1 (1 − f11 )−1 f13 fs1 −fs1 (1 − f11 )−1 f12



we have that H ∈ GLs (S) with inverse given by:  1 − f11 (1 − f11 )−1 f12 (1 − f11 )−1 f13  −f21 1 0   −f31 −1 0 1 H =  ..  . −fs1

0

0

··· ··· ···

−(1 − f11 )−1 f1s −f21 (1 − f11 )−1 f1s   −f31 (1 − f11 )−1 f1s 

··· · · · −fs1 (1 − f11 )−1 f1s + 1



  

 · · · (1 − f11 )−1 f1s  0 ···   0 ··· .   ··· 1 ···

In fact, note that HH −1 = Is : Ps Ps 1−f11 +(1−f11 )−1 i=2 f1i fi1 = 1−f11 +f11 = 1 because i=2 f1i fi1 = (1 − f11 )f11 and (1 − f11 ) is invertible; also (1 − f11 )−1 f1i − (1 − f11 )−1 f1i for 2 ≤ i ≤ s; Ps f21 (1 − f11 ) + f21 i=1 (1 − f11 )−1 f1i fi1 − f21 = −f21 f11 + f21 f11 = 0; f21 (1 − f11 )−1 f12 − f21 (1 − f11 )−1 f12 + 1 = 1; and f21 (1 − f11 )−1 f1i − f21 (1 − f11 )−1 f1i = 0 for 3 ≤ i ≤ s. .. . Ps fs1 (1 − f11 ) + fs1 i=1 (1 − f11 )−1 f1i fi1 − fs1 = −fs1 f11 + f21 f11 = 0; fs1 (1 − f11 )−1 f1i − fs1 (1 − f11 )−1 f1i = 0 for 3 ≤ i ≤ s − 1 and, finally, fs1 (1 − f11 )−1 f1s − fs1 (1 − f11 )−1 f1s + 1 = 1. Similarly, we can to show that H −1 H = Is . Furthermore, we also have

10.3 Some Important Subclasses of Hermite Rings

HF H −1 =



193



0 01,s−1 , 0s−1,1 F2

with F2 ∈ Ms−1 (S) an idempotent matrix. One more time we apply the induction hypothesis. t u

10.3 Some Important Subclasses of Hermite Rings There are some other classes of rings closely related to Hermite rings that we will consider next (see [97], [197], [216] and [412]). Definition 10.3.1. Let S be a ring. (i) S is an elementary divisor ring (ED) if for any r, s ≥ 1, given a rectangular matrix F ∈ Mr×s (S) there exist invertible matrices P ∈ GLr (S) and Q ∈ GLs (S) such that P F Q is a Smith normal diagonal matrix, i.e., there exist d1 , d2 , . . . , dl ∈ S, with l = min{r, s}, such that P F Q = diag(d1 , d2 , . . . , dl ), with Sdi+1 S ⊆ Sdi ∩ di S for 1 ≤ i ≤ l, where SdS denotes the two-sided ideal generated by d. (ii) S is an ID ring if for any s ≥ 1, given an idempotent matrix F ∈ Ms (S) there exists an invertible matrix P ∈ GLs (S) such that P F P −1 is a Smith normal diagonal matrix. (iii) S is a left K-Hermite ring (KH) if given a, b ∈ S there exist U ∈  T  T GL2 (S) and d ∈ S such that   U a b = d 0 . S is a right K-Hermite ring (KHr ) if a b U = d 0 . The ring S is KH if S is KH and KHr . (iv) S is a left B´ ezout ring (B) if every f.g. left ideal of S is principal. S is a right B´ ezout ring (Br ) if every f.g. right ideal of S is principal. S is a B ring if S is B and Br . (v) S is a left cancellable ring (C) if for any f.g. projective left S-modules P, P 0 holds: P ⊕S ∼ = P 0 ⊕S ⇔ P ∼ = P 0 . S is right cancellable (Cr ) if for any f.g. projective right S-modules P, P 0 holds: P ⊕S ∼ = P 0 ⊕S ⇔ P ∼ = P 0. S is cancellable (C) if S is (C) and (Cr ). From Proposition 0.3.1 of [97] it is easy to give a matrix interpretation of C rings, and also, we can deduce that C = Cr = C. Proposition 10.3.2. Let S be a ring. Then, (i) S is C if and only if given idempotent matrices F ∈ Ms (S), G ∈ Mr (S) the following statement is true: the matrices     F 0 G0 and 0 1 0 1 can be factorized as 

   F 0 G0 = X 0Y 0, = Y 0X 0, 0 1 0 1

194

10 Hermite Rings

for some matrices X 0 ∈ M(s+1)×(r+1) (S), Y 0 ∈ M(r+1)×(s+1) (S), if and only if F = XY , G = Y X, for some matrices X ∈ Ms (S), Y ∈ Mr (S). (ii) C = Cr = C. Proof. Direct consequence of Proposition 0.3.1 in [97].

t u

For domains, the above classes of rings are denoted by EDD, IDD, KHD, KHDr , KHD, BD, BDr , BD and CD, respectively. Theorem 10.3.3. (i) ED ⊆ KH ⊆ B. (ii) KHD = BD ⊆ PFD. (iii) PF ⊆ ID (iv) ID = PF for rings without nontrivial idempotents. Thus, IDD = PF D. (v) PF ⊆ C ⊆ H. Similar relations hold for KHr , KH, Br and B. Proof. (i) It is clear that ED ⊆ KH. Let a, b ∈ S. We want to prove that any left ideal Sa + Sb is principal. There exist U ∈ GL2 (S) and d ∈ S  T  T such that U a b = d 0 . This implies that Sd ⊆ Sa + Sb, but since  T  T a b = U −1 d 0 , we have Sa + S ⊆ Sd. This proves that KH ⊆ B. (ii) KHD = BD was proved by Amitsur in [15]. In order to prove the inclusion BD ⊆ PFD we show first that if S is BD then each finitely generated left ideal of S is free: let I be a left ideal of S, if I = 0, then I is free; let I 6= 0, then I = Sa, for some a 6= 0, but since S has no zero divisors, I is free with basis {a}. Next we will prove that each finitely generated submodule of a free Smodule is free: let M be a free S-module with basis X and let N = Sz1 + · · · + Szt be a finitely generated submodule of M (if M = 0 or N = 0 there is nothing to prove). Each zi defines a finite subset Xi of X, 1 ≤ i ≤ t, so N ⊆ h∪ti=1 Xi }, and hence, there exists a finite sequence x1 , . . . , xn of elements of X such that N ⊆ Sx1 ⊕ · · · ⊕ Sxn , i.e., N is a submodule of a free module with a basis of n elements, so we can complete the proof of freeness of N by induction. For n = 1 we have N ⊆ Sx1 ∼ = S, so N is isomorphic to a finitely generated left ideal of S, hence N is free. Consider again that N ⊆ Sx1 ⊕ · · · ⊕ Sxn and define the function f : N → S by x = s1 x1 + · · · + sn xn 7→ sn . Note that f is a homomorphism and f (N ) is a finitely generated left ideal of S, i.e., f (N ) is free. We have the exact sequence 0 → N ∩ (Sx1 ⊕ · · · ⊕ Sxn−1 ) → N → f (N ) → 0, but since f (N ) is projective, this sequence splits, so N ∼ = f (N ) ⊕ (N ∩ (Sx1 ⊕ · · · ⊕ Sxn−1 )). Note that N ∩ (Sx1 ⊕ · · · ⊕ Sxn−1 ) is a finitely generated submodule of a free module with a basis of n − 1 elements. By induction N ∩ (Sx1 ⊕ · · · ⊕ Sxn−1 ) is free, so N is free. Now we are able to prove that S is PF . Let M be a finitely generated projective S-module, then M is a finitely generated submodule (as a free summand) of a free module, hence M is free. (iii) Using permutation matrices, it is clear that PF ⊆ ID (see Corollary 10.2.4).

10.3 Some Important Subclasses of Hermite Rings

195

(iv) Let S be an ID ring and let F = [fij ] ∈ Ms (S) be an idempotent matrix over S; by the hypothesis, there exists a P ∈ GLs (S) such that P F P −1 is diagonal. Let D := P F P −1 = diag(d1 , d2 , . . . , ds ); since P F P −1 is idempotent, then each di is idempotent, so di = 0 or di = 1 for each 1 ≤ i ≤ s. By permutation matrices we can assume that   0 0 P F P −1 = . 0 Ir In addition, note that r is the dimension of the left S-module generated by the rows of F . Then, S is PF . (v) Let P, P 0 be f.g. S-modules such that P ⊕ S ∼ = P 0 ⊕ S; since S is0 PF 0 n 0 ∼ n0 ∼ there exists n, n such that P = S , P = S , and hence, S n ⊕ S ∼ = S n ⊕ S, 0 0 ∼ so n + 1 = n + 1, i.e., P = P . Let now M be a stably free module. M ⊕ S s ∼ = S r , since r ≥ s and S is r−s left cancellable, so M ∼ . t u =S From Theorem 10.3.3 we conclude that for domains the following inclusions hold: EDD ⊆ KHD = BD ⊆ PFD = IDD ⊆ CD ⊆ HD.

(10.3.1)

Similar relations hold for the right side. The next proposition gives an alternative characterization of KH rings and will be used to prove that KH ⊆ H for commutative rings. Proposition 10.3.4. Let S be a ring. S is KH if and only if for every r ≥ 2, given elements b1 , . . . , br ∈ S, there exists a U ∈ GLr (S) and d ∈ S such that  T  T U b1 · · · br = d · · · 0 . A similar characterization holds for KHr rings. Proof. ⇒) By induction over r. The case r = 2 is direct consequence of the definition. Suppose that the result holds for any row of size < r and let  T  T U0 ∈ GL2 (S) such that U0 br−1 br = d0 0 , for some d0 ∈ S. We have  T  T U1 b1 · · · br−2 br−1 br = b1 · · · br−2 d0 0 , with 

I 0 U1 := r−2 0 U0

 ∈ GLr (S).

Applying the induction hypothesis to b1 , . . . , br−2 , d0 we find U2 ∈ GLr−1 (S) such that  T  T U2 b1 · · · br−2 d0 = d · · · 0 , for some d ∈ S. Let U 0 :=



U2 0 0 1

 ∈ GLr (S),

196

10 Hermite Rings

 T  T then U := U 0 U1 ∈ GLr (S) satisfies U b1 · · · br = d · · · 0 . ⇐) Trivial.

t u

Corollary 10.3.5. For commutative rings, KH ⊆ H.  T Proof. Let S be a commutative KH ring and let u = u1 · · · ur be a unimodular column vector. By Proposition 10.3.4 there exists a U ∈ GLr (S)  T such that U u = d · · · 0 , for some d ∈ S. This implies that Sd = Su1 + · · · + Sur = S, i.e., d is left invertible, and hence, invertible. From this we get that d−1 U u = e 1 . t u The following characterization of ID rings for which all idempotents are central will be used below (see [268] and [239] for the particular case of commutative rings). Proposition 10.3.6. Let S be a ring such that all idempotents are central. Then the following conditions are equivalent: (i) S is ID. (ii) Any idempotent matrix over S is similar to a diagonal matrix. (iii) Given an idempotent matrix F ∈ Mr (S) there exists a unimodular vector v = [v1 , . . . , vr ]T over S and an invertible matrix U ∈ GLr (S) such that U v = e1 and F v = av, for some a ∈ S. Proof. (i) ⇒ (ii) is trivial. (ii) ⇒ (iii) Let F ∈ Mr (S) be idempotent. There exists a P ∈ GLr (S) such that P F P −1 = diag(d1 , . . . , dr ). Note that each di is idempotent (see the proof of part (iv) in Theorem 10.3.3); the canonical vector e 1 is unimodular, moreover P F P −1 e 1 = d1 e 1 . Let v := P −1 e 1 , then v is unimodular, F v = d1 v and P v = e 1 . Thus, the result is valid with U = P and a = d1 . (iii) ⇒ (ii) Let F ∈ Mr (S) be idempotent. We will prove that there exists a Q ∈ GLr (S) such that QF Q−1 is diagonal. The proof is by induction on r. For r = 1, if f ∈ S with f 2 = f , then there exist v, u ∈ S ∗ such that uv = 1 and f v = av, for some a ∈ S, hence f = a, i.e., 1f 1−1 = a. Suppose that any idempotent matrix of size < r is similar to a diagonal matrix. Let F ∈ Mr (S) be idempotent; if F = 0 there is nothing to prove. Let F 6= 0. By the hypothesis, there exist a unimodular vector v = [v1 , . . . , vr ]T over S and an invertible matrix U ∈ GLr (S) such that U v = e 1 and F v = d1 v , for some d1 ∈ S. Then, F is similar to the matrix Fe := U F U −1 , and Fe has the form   d1 a12 · · · a1r  0 a22 · · · a2r    Fe =  . . . . .  .. .. .. ..  0 ar2 . . . arr In fact, Fe e 1 = U F U −1 e 1 = U F v = U d1 v = d1 U v = d1 e 1 . But Fe is idempotent since F is idempotent, so d21 = d1 and the submatrix H := [aij ], with 2 ≤ i, j ≤ r, is idempotent of size (r − 1) × (r − 1). By induction,

10.3 Some Important Subclasses of Hermite Rings

197

there exists Q0 ∈ GLr−1 (S) and d2 , d3 , . . . , dr ∈ S such that Q0 HQ0−1 = diag(d2 , d3 , . . . , dr ). From this we get that F is similar to the matrix Fb , where   d 1 b2 b 3 · · · b r        0 d2 0 . . . 0  1 0 e 1 0  0 0 d3 . . . 0  b F := F = , 0 Q0 0 Q0−1  .. .. . . ..  . . . . 0 0 0 . . . dr for some b2 , . . . , br ∈ S. Since F is idempotent, Fb is idempotent, and hence, d2i = di , for each 1 ≤ i ≤ r. Moreover, for each 2 ≤ j ≤ r, bj (d1 + dj − 1) = 0.

(10.3.2)

Now we consider for a moment S r as a right S-module of column vectors (see Remark 9.1.2 (i)); the idea is to make a change of basis of S r and to prove that F is similar to the matrix diag(d1 . . . , dr ). For this we have to construct a basis {u 1 , u 2 , . . . , u r } of S r such that Fb u i = di u i , 1 ≤ i ≤ r. We consider the vectors u 1 = e 1 , u 2 = (a2 , 1, 0, . . . , 0)T , u 3 = (a3 , 0, 1, . . . , 0)T , . . . , u r = (ar , 0, 0, . . . , 1)T , where a2 , . . . , ar ∈ S must be defined. For 2 ≤ j ≤ r, from condition Fb u j = dj u j , the aj ’s must satisfy bj = (dj − d1 )aj .

(10.3.3)

(10.3.2) implies that bj (d1 − dj + 2dj − 1) = 0, and hence bj (d1 − dj ) = bj (1 − 2dj ), but (1 − 2dj )2 = 1, so bj (d1 − dj )(1 − 2dj ) = bj , thus aj := bj (2dj − 1) satisfies (10.3.3). With this change of basis we get H Fb H −1 = diag(d1 . . . , dr ), where   1 −a2 −a3 · · · −ar 0 1 0 ··· 0    0 0 1 ··· 0  H :=  , with aj := bj (2dj − 1), 2 ≤ j ≤ r.  .. .. .. ..  . . . .  0 0 0 ··· 1 Thus, we have proved that F is similar to the matrix diag(d1 , d2 , . . . , dr ), i.e., there exists a P ∈ GLr (S) such that P F P −1 = diag(d1 , d2 , . . . , dr ). (ii) ⇒ (i) Let F ∈ GLr (S) be an idempotent matrix. Then there exists a Q ∈ GLr (S) such that QF Q−1 = D := diag(d1 , d2 , . . . , dr ); as we saw before, each di is idempotent. We will prove that there exists a P ∈ GLr (S) such that P DP −1 is a diagonal Smith normal matrix. We divide this proof into three steps. Step 1. We observe first that there exist idempotents f1 , . . . , fr ∈ S and  T a ∈ S such that f = f1 · · · fr is unimodular and afi = di , for 1 ≤ i ≤ r. In fact, we define

10 Hermite Rings

198

a :=d1 + · · · + dr +

r X

Y

(−1)j+1 (

di1 · · · dij ),

i1 · · · , t u

but this is impossible since deg(Xi1 ) is finite.

From now on we will assume that Mon(A) is endowed with some monomial order. Definition 13.1.3. Let xα , xβ ∈ Mon(A). We say that xα divides xβ , denoted by xα | xβ , if there exists xγ , xλ ∈ Mon(A) such that xβ = lm(xγ xα xλ ). We will also say that any monomial xα ∈ Mon(A) divides the zero polynomial. Proposition 13.1.4. Let xα , xβ ∈ Mon(A) and f, g ∈ A − {0}. Then, (a) lm(xα g) = lm(xα lm(g)) = xα+exp(lm(g)) , i.e., exp(lm(xα g)) = α + exp(lm(g)). In particular, lm(lm(f )lm(g)) = xexp(lm(f ))+exp(lm(g)) , i.e., exp(lm(lm(f )lm(g))) = exp(lm(f )) + exp(lm(g)) and lm(xα xβ ) = xα+β , i.e., exp(lm(xα xβ )) = α + β.

(13.1.1)

(b) The following conditions are equivalent: (i) xα | xβ . (ii) There exists a unique xθ ∈ Mon(A) such that xβ = lm(xθ xα ) = xθ+α and hence β = θ + α. (iii) There exists a unique xθ ∈ Mon(A) such that xβ = lm(xα xθ ) = xα+θ and hence β = α + θ. (iv) βi ≥ αi for 1 ≤ i ≤ n, with β := (β1 , . . . , βn ) and α := (α1 , . . . , αn ).

13.2 Reduction in Skew P BW Extensions

239

Proof. (a) Let g = cxβ + p, with 0 6= c ∈ R, p = 0 or deg(p) < |β|. Then, xα g = xα (cxβ + p)= (xα c)xβ + xα p = (cα xα + q)xβ + xα p, with 0 6= cα ∈ R, q ∈ A, q = 0 or deg(q) < |α|. So, xα g = cα xα xβ + qxβ + xα p= cα (cα,β xα+β + t) + qxβ + xα p, with cα,β ∈ R left invertible, t ∈ A, t = 0 or deg(t) < |α + β|. Thus, xα g = c0 xα+β + p0 , with c0 := cα cα,β ∈ R − {0}, p0 := cα t + qxβ + xα p ∈ A, p0 = 0 or deg(p0 ) < |α + β|. This implies that lm(xα g) = xα+β , but xα lm(g) = xα xβ = cα,β xα+β + t, so lm(xα lm(g)) = xα+β . (b) This part is a direct consequence of part (a). t u Remark 13.1.5. (i) Let  be a monomial order on Mon(A); if there exists an f = c1 xγ1 + · · · + ct xγt ∈ A − {0} such that xβ = xα f , then by Proposition 13.1.4, xβ = xα+γ1 , i.e., xα | xβ . Of course, the converse is not true. On the other hand, if xβ = f xα , then xβ = c1 cγ1 ,α xγ1 +α + c1 pγ1 ,α + · · · + ct cγt ,α xγt +α + ct pγt ,α ; if ci cγi ,α = 0 for every 1 ≤ i ≤ t, then we cannot assert that xα |xβ , however, if A is bijective, then xβ = xα+γ1 , i.e., xα x |β . In this case again the converse is not true. (ii) We note that there exists a least common multiple of two elements of Mon(A): in fact, let xα , xβ ∈ Mon(A), then lcm(xα , xβ ) = xγ ∈ Mon(A), where γ = (γ1 , . . . , γn ) with γi := max{αi , βi } for each 1 ≤ i ≤ n.

13.2 Reduction in Skew P BW Extensions Some natural computational conditions on R will be assumed in the rest of this chapter (compare with [236]). Definition 13.2.1. A ring R is left Gr¨ obner soluble (LGS) if the following conditions hold: (i) R is left noetherian. (ii) Given a, r1 , . . . , rm ∈ R there exists an algorithm which decides whether a is in the left ideal Rr1 + · · · + Rrm , and if so, finds b1 , . . . , bm ∈ R such that a = b1 r1 + · · · + bm rm . (iii) Given r1 , . . . , rm ∈ R there exists an algorithm which finds a finite set of generators of the left R-module SyzR [r1 · · · rm ] := {(b1 , . . . , bm ) ∈ Rm | b1 r1 + · · · + bm rm = 0}. Remark 13.2.2. The above three conditions imposed on R are needed in order to guarantee a Gr¨ obner theory in the rings of coefficients, in particular, to have an effective solution of the membership problem in R (see (ii) in Definition 13.2.3 below). From now on in this chapter we will assume that A = σ(R)hx1 , . . . , xn i is a skew P BW extension of R, where R is a LGS ring and Mon(A) is endowed with some monomial order. Definition 13.2.3. Let F be a finite set of nonzero elements of A, and let F f, h ∈ A. We say that f reduces to h by F in one step, denoted f −−→ h, if there exist elements f1 , . . . , ft ∈ F and r1 , . . . , rt ∈ R such that

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13 Gr¨ obner Bases for Skew P BW Extensions

(i) lm(fi ) | lm(f ), 1 ≤ i ≤ t, i.e., there exists an xαi ∈ Mon(A) such that lm(f ) = lm(xαi lm(fi )), i.e., αi + exp(lm(fi )) = exp(lm(f )). (ii) lc(f ) = r1 σ α1 (lc(f1 ))cα1 ,f1 + · · · + rt σ αt (lc(ft ))cαt ,ft , where cαi ,fi are defined as in Theorem 1.1.8, i.e., cαi ,fi := cαi ,exp(lm(fi )) . Pt (iii) h = f − i=1 ri xαi fi . F

We say that f reduces to h by F , denoted f −−→+ h, if there exist h1 , . . . , ht−1 ∈ A such that F

F

F

F

F

f −−−−→ h1 −−−−→ h2 −−−−→ · · · −−−−→ ht−1 −−−−→ h. f is reduced (also called minimal) w.r.t. F if f = 0 or there is no one step reduction of f by F , i.e., one of the conditions (i) or (ii) fails. Otherwise, we F will say that f is reducible w.r.t. F . If f −−→+ h and h is reduced w.r.t. F , then we say that h is a remainder for f w.r.t. F . Remark 13.2.4. (i) By Theorem 1.1.8, the coefficients cαi ,fi in the previous definition are unique and satisfy xαi lm(fi ) = cαi ,fi xαi +exp(lm(fi )) + pαi ,fi , where pαi ,fi = 0 or deg(pαi ,fi ) < |αi + exp(lm(fi ))|, 1 ≤ i ≤ t. (ii) lm(f )  lm(h) and f − h ∈ hF i, where hF i is the left ideal of A generated by F . (iii) The remainder of f is not unique. F (iv) By definition we will assume that 0 − → 0. From the reduction relation we get the following interesting properties. Proposition 13.2.5. Let A be a skew P BW extension such that cα,β is invertible for each α, β ∈ Nn . Let f, h ∈ A, θ ∈ Nn and F = {f1 , . . . , ft } be a finite set of nonzero polynomials of A. Then, F

(i) If f −−→ h, then there exists a p ∈ A with p = 0 or lm(xθ f )  lm(p) F such that xθ f + p −−→ xθ h. In particular, if A is quasi-commutative, then p = 0. F (ii) If f −−→+ h and p ∈ A is such that p = 0 or lm(h)  lm(p), then F f + p −−→+ h + p. F (iii) If f −−→+ h, then there exists a p ∈ A with p = 0 or lm(xθ f )  lm(p) F such that xθ f + p −−→+ xθ h. If A is quasi-commutative, then p = 0. F (iv) If f −−→+ 0, then there exists a p ∈ A with p = 0 or lm(xθ f )  lm(p) F such that xθ f + p −−→+ 0. If A is quasi-commutative, then p = 0. Proof. (i) If f = 0, then h = 0 = p. Let f 6= 0 and let lm(f ) := xλ ; there exist f1 , . . . , ft ∈ F and r1 . . . , rt ∈ R such that lm(fi ) | lm(f ), i.e., λ = αi + βi , with lm(fi ) := xβi and αi ∈ Nn , 1 ≤ i ≤ t, lc(f ) = r1 σ α1 (lc(f1 ))cα1 ,β1 + · · · + rt σ αt (lc(ft ))cαt ,βt , Pt h = f − i=1 ri xαi fi .

13.2 Reduction in Skew P BW Extensions

241

We note that lt(xθ f ) = σ θ (lc(f ))cθ,λ xθ+λ , thus lm(xθ f ) = xθ+λ , so lm(fi )|lm(xθ f ), with θ + λ = (θ + αi ) + βi ; we observe that lc(xθ f ) = σ θ (lc(f ))cθ,λ =

t X

σ θ (ri )σ θ (σ αi (lc(fi )))σ θ (cαi ,βi )cθ,λ ,

i=1

and by Remark 1.1.9 lc(xθ f ) =

t X

θ σ θ (ri )cθ,αi σ θ+αi (lc(fi ))c−1 θ,αi σ (cαi ,βi )cθ,λ

i=1

=

t X

θ σ θ (ri )cθ,αi σ θ+αi (lc(fi ))c−1 θ,αi σ (cαi ,βi )cθ,αi +βi

i=1

=

=

t X i=1 t X

σ θ (ri )cθ,αi σ θ+αi (lc(fi ))cθ+αi ,βi ri0 σ θ+αi (lc(fi ))cθ+αi ,βi , with ri0 := σ θ (ri )cθ,αi .

i=1

Moreover, xθ h = xθ f −

t X

x θ ri x α i f i

i=1

= xθ f −

t X

(σ θ (ri )xθ + pθ,ri )xαi fi

i=1

= xθ f − = xθ f −

t X i=1 t X

σ θ (ri )xθ xαi fi + pθ,ri xαi fi σ θ (ri )cθ,αi xθ+αi fi + p

i=1

= xθ f −

t X

ri0 xθ+αi fi + p

i=1

= xθ f + p −

t X

ri0 xθ+αi fi ,

i=1

with p :=

t X i=1

pθ,ri xαi fi +

t X i=1

σ θ (ri )pθ,αi fi ;

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13 Gr¨ obner Bases for Skew P BW Extensions

note that p = 0 or deg(p) < |θ + αi + βi | = |θ + λ| = deg(xθ f ), so lm(xθ f )  lm(p). Moreover, lm(xθ f + p) = lm(xθ f ) and lc(xθ f + p) = lc(xθ f ), so by F the previous discussion xθ f + p −−→ xθ h. If A is quasi-commutative, then from Remark 1.1.9 (iii), p = 0. (ii) Let F

F

F

F

F

f −−−−→ h1 −−−−→ h2 −−−−→ · · · −−−−→ ht−1 −−−−→ ht := h; (13.2.1) F we start with f − → h1 , if f = 0, then h1 = 0 = p and there is nothing to prove. Let f 6= 0. If h1 = 0 then p = 0 and hence lm(f )  lm(p); if h1 6= 0, then lm(f )  lm(h1 )  lm(p), and hence lm(f + p) = lm(f ), lc(f P + p) = lc(f ), t and as in the proof of the first part of (i), h1 + p = f + p − i=1 ri xαi fi ; F

but lm(f + p) = lm(f ) and lc(f + p) = lc(f ), then f + p −−→ h1 + p. Since F lm(hi )  lm(p) we can repeat this reasoning for hi −−→ hi+1 for 1 ≤ i ≤ t−1. This completes the proof of (ii). (iii) By (i) and using (13.2.1), there exists a p1 ∈ A with p1 = 0 or F lm(xθ f )  lm(p1 ) such that xθ f + p1 −−→ xθ h1 ; there exists a p2 ∈ A with F p2 = 0 or lm(xθ h1 )  lm(p2 ) such that xθ h1 + p2 −−→ xθ h2 ; by (ii) we get F F that xθ f + p1 + p2 −−→ xθ h1 + p2 −−→ xθ h2 , i.e., p00 := p1 + p2 ∈ A is such that F xθ f + p00 −−→+ xθ h2 , with p00 = 0 or lm(xθ f )  lm(p00 ) since lm(xθ f )  lm(p1 ) and lm(xθ f )  lm(p2 ) . By induction on t we find p0 ∈ A such that F

xθ f + p0 −−→+ xθ ht−1 , with p0 = 0 or lm(xθ f )  lm(p0 ). By (i) there exists a pt ∈ A such that F xθ ht−1 + pt −−→ xθ h, with pt = 0 or lm(xθ ht−1 )  lm(pt ). By (ii), xθ f + p0 + F F pt −−→+ xθ ht−1 + pt −−→ xθ h. Thus, F

xθ f + p −−→+ xθ h, with p := p0 + pt = 0 or lm(xθ f )  lm(p) since lm(xθ f )  lm(p0 ) and lm(xθ f )  lm(pt ). From the just proved we observe that if A is quasicommutative then p = 0. (iv) This is a direct consequence of (iii) taking h = 0. t u The next theorem is the theoretical support of the Division Algorithm for skew P BW extensions. Theorem 13.2.6. Let F = {f1 , . . . , ft } be a finite set of nonzero polynomials of A and f ∈ A, then the Division Algorithm below produces polynomials F q1 , . . . , qt , h ∈ A, with h reduced w.r.t. F , such that f −−→+ h and f = q1 f1 + · · · + qt ft + h,

13.2 Reduction in Skew P BW Extensions

243

with lm(f ) = max{lm(lm(q1 )lm(f1 )), . . . , lm(lm(qt )lm(ft )), lm(h)}. Proof. We first note that the Division Algorithm is the iteration of the reduction process. If f is reduced with respect to F := {f1 , . . . , ft }, then h = f, q1 = · · · = qt = 0 and lm(f ) = lm(h). If f is not reduced, then P F αj we make the first reduction, f −−→ h1 , where f = j∈J1 rj1 x fj + h1 , with J1 := {j | lm(fj ) divides lm(f )} and rj1 ∈ R. If h1 is reduced with respect to F , then the cycle WHILE ends and we have that qj = rj1 xαj for j ∈ J1 and qj = 0 for j ∈ / J1 . Moreover, lm(f )  lm(h1 ) and lm(f ) = lm(lm(qj )lm(fj )) for j ∈ J1 such that rj1 6= 0, hence, lm(f ) = max1≤j≤t {lm(lm(qj )lm(fj )), lm(h1 )}. If h1 is not reduced, we make the secP F ond reduction with respect to F , h1 −−→ h2 , with h1 = j∈J2 rj2 xαj fj + h2 , J2 := {j | lm(fj ) divides lm(h1 )} and rj2 ∈ R. We have X X f= rj1 xαj fj + rj2 xαj fj + h2 . j∈J1

j∈J2

If h2 is reduced with respect to F the procedure ends and we get that qj = qj for j ∈ / J2 and qj = qj + rj2 xαj for j ∈ J2 . We know that lm(f )  lm(h1 )  lm(h2 ). This implies that the algorithm produces polynomials qj with monomials ordered according to the monomial order fixed, and again we have lm(f ) = max1≤j≤t {lm(lm(qj )lm(fj )), lm(h2 )}. We can continue this way and the algorithm ends since Mon(A) is well ordered. t u

Example 13.2.7. In A2 (2, 2), with K := Q, we consider the order deglex with y1  y2 ; let f1 := x21 x2 y1 y2 , f2 := x2 y1 , f3 := x1 y2 and f := x1 x22 y12 y2 + x21 x2 y1 . For f we will find q1 , q2 , q3 and h, following the previous algorithm. We will use the relations of Example 1.1.5 (iv); in particular, since yi yj = yj yi , then cα,β = 1 for every xα , xβ ∈ Mon(A2 (2, 2)). For j = 1, 2, 3, we define αj := (αj1 , αj2 ) ∈ N2 . Step 1. We start with h := f , q1 := 0, q2 := 0, q3 := 0; since lm(fj ) | lm(h) for j = 1, 2, 3, we compute αj such that αj + exp(lm(fj )) = exp(lm(h)), and also, σ αj (lc(fj )):

(α11 , α12 ) + (1, 1) = (2, 1) ⇒ α11 = 1, α12 = 0, σ α1 (lc(f1 )) = σ1 σ20 (lc(f1 )) = σ1 (x21 x2 ) = 4x21 x2 , (α21 , α22 ) + (1, 0) = (2, 1) ⇒ α21 = 1, α22 = 1, σ α2 (lc(f2 )) = σ1 σ2 (x2 ) = 2x2 , (α31 , α32 ) + (0, 1) = (2, 1) ⇒ α31 = 2, α32 = 0, σ α3 (lc(f3 )) = σ12 σ20 (x1 ) = 4x1 .

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13 Gr¨ obner Bases for Skew P BW Extensions

Division algorithm in A INPUT: f, f1 , . . . , ft ∈ A with fj 6= 0 (1 ≤ j ≤ t) OUTPUT: q1 , . . . , qt , h ∈ A with f = q1 f1 + · · · + qt ft + h, h reduced w.r.t. {f1 , . . . , ft } and lm(f ) = max{lm(lm(q1 )lm(f1 )), . . . , lm(lm(qt )lm(ft )), lm(h)} INITIALIZATION: q1 := 0, q2 := 0, . . . , qt := 0, h := f WHILE h = 6 0 and there exists j such that lm(fj ) divides lm(h) DO Calculate J := {j | lm(fj ) divides lm(h)} FOR j ∈ J DO Calculate αj ∈ Nn such that αj + exp(lm(fj )) = exp(lm(h)) IF the equation lc(h) = j∈J rj σ αj (lc(fj ))cαj ,fj is soluble, where cαj ,fj are defined as in Theorem 1.1.8 THEN

P

Calculate one solution (rj )j∈J h := h −

P

j∈J

rj xαj fj

FOR j ∈ J DO qj := qj + rj xαj ELSE Stop

Now we solve the equation lc(h) = x1 x22 = r1 4x21 x2 + r2 2x2 + r3 4x1 ⇒ r1 = 1, r2 = 12 x1 x2 , r3 = −x1 x2 , and with the relations defining A2 (2, 2) we compute h = h − (r1 y α1 f1 + r2 y α2 f2 + r3 y α3 f3 ) 1 = h − y1 f1 − x1 x2 y1 y2 f2 + x1 x2 y12 f3 2 1 2 = h − y1 x1 x2 y1 y2 − x1 x2 y1 y2 x2 y1 + x1 x2 y12 x1 y2 2 1 2 2 = − x 1 x 2 y 1 + x 1 x 2 y1 . 2 We compute also q1 = y1 , q2 = 12 x1 x2 y1 y2 , q3 = −x1 x2 y12 . Step 2. lm(h) = y12 , lc(h) = − 12 x1 x2 ; since lm(f2 )|lm(h), we compute α2 such that α2 + exp(lm(f2 )) = exp(lm(h)), and also, σ α2 (lc(f2 )): (α21 , α22 ) + (1, 0) = (2, 0) ⇒ α21 = 1, α22 = 0, σ α2 (lc(f2 )) = σ1 σ20 (x2 ) = x2 ,

13.2 Reduction in Skew P BW Extensions

245

and from this we get 1 1 − x 1 x 2 = r2 x 2 ⇒ r2 = − x 1 , 2 2 1 1 α2 h = h − r2 y f2 = h + x1 y1 f2 = h + x1 y1 x2 y1 = x21 x2 y1 , 2 2 1 1 q1 = y1 , q2 = x1 x2 y1 y2 − x1 y1 , q3 = −x1 x2 y12 . 2 2 Step 3. lm(h) = y1 , lc(h) = x21 x2 ; in this case we have (α21 , α22 ) + (1, 0) = (1, 0) ⇒ α21 = 0, α22 = 0, σ α2 (lc(f2 )) = σ10 σ20 (x2 ) = x2 , x21 x2 = r2 x2 ⇒ r2 = x21 , h = h − r2 y α2 f2 = h − x21 f2 = h − x21 x2 y1 = 0, 1 1 q1 = y1 , q2 = x1 x2 y1 y2 − x1 y1 + x21 , q3 = −x1 x2 y12 . 2 2 We can check that f = q1 f1 + q2 f2 + q3 f3 , i.e., f = y1 f1 + ( 12 x1 x2 y1 y2 − 12 x1 y1 + x21 )f2 + (−x1 x2 y12 )f3 ; we observe that max{lm(lm(q1 )lm(f1 )), lm(lm(q2 )lm(f2 )), lm(lm(q3 )lm(f3 ))} = max{y12 y2 , y12 y2 , y12 y2 } = y12 y2 = lm(f ). Example 13.2.8. We consider the diffusion algebra (see Section 2.4) with n = 2, K = Q, λ12 = −2 and λ21 = −1. In this bijective skew P BW extension, D2 D1 = 2D1 D2 + x2 D1 − x1 D2 and the automorphisms σ1 and σ2 are the identity. We consider the deglex order with D1  D2 and the polynomials f1 := x1 x2 D1 D2 , f2 := x2 D1 , f3 = x1 D2 , f = x1 x22 D12 D2 + x21 x2 D2 . We want to divide f by the polynomials f1 , f2 and f3 . Step 1. We start with h := f , q1 := 0, q2 := 0, q3 := 0. Since lm(fj ) | lm(f ) for j = 1, 2, 3, we compute αj = (αj1 , αj2 ) ∈ N2 such that αj +exp(lm(fj )) = exp(lm(h)) and the corresponding value of σ αj (lc(fj ))cαj ,βj , where βj = exp(lm(fj )): (α11 , α12 ) + (1, 1) = (2, 1) ⇒ α11 = 1, α12 = 0, σ α1 (lc(f1 ))cα1 ,β1 = x1 x2 , (α21 , α22 ) + (1, 0) = (2, 1) ⇒ α21 = 1, α22 = 1, σ α2 (lc(f2 ))cα2 ,β2 = 2x2 , (α31 , α32 ) + (0, 1) = (2, 1) ⇒ α31 = 2, α32 = 0, σ α3 (lc(f3 ))cα3 ,β3 = x1 . Now, we solve the equation

13 Gr¨ obner Bases for Skew P BW Extensions

246

lc(h) = x1 x22 = r1 (x1 x2 ) + r2 (2x2 ) + r3 (x1 ) ⇒ r1 = 3x2 , r2 = − 21 x1 x2 , r3 = −x22 , and with the relations defining A, we replace h by h =h − (r1 Dα1 f1 + r2 Dα2 f2 + r3 Dα3 f3 ) 1 =h − 3x1 x22 D12 D2 + x1 x22 (2D12 D2 + x2 D12 − x1 D1 D2 ) + x1 x22 D12 D2 2 1 1 2 2 3 2 = x1 x2 D1 − x1 x2 D1 D2 + x21 x2 D2 . 2 2 We compute also q1 := 3x2 D1 , q2 := − 12 x1 x2 D1 D2 , q3 := −x22 D12 . Step 2. lm(h) = D12 , lc(h) = 12 x1 x32 . In this case, lm(fj ) | lm(f ) only for j = 2 and we have that α2 = (α21 , α22 ) ∈ N2 such that αj + exp(lm(fj )) = exp(lm(h)) is α2 = (1, 0); moreover, σ α2 (lc(f2 ))cα2 ,β2 = x2 and r = 12 x1 x22 is such that lc(h) = rx2 . Thus we have: h =h − rDα2 f2 1 = − x21 x22 D1 D2 + x21 x2 D2 . 2 and q1 := 3x2 D1 , q2 := − 21 x1 x2 D1 D2 + 12 x1 x22 D1 , q3 := −x22 D12 . Step 3. Note that lm(h) = D1 D2 and lm(fj ) | lm(h) for j = 1, 2, 3. In this case we have: (α11 , α12 ) + (1, 1) = (1, 1) ⇒ α11 = 0, α12 = 0, σ α1 (lc(f1 ))cα1 ,β1 = x1 x2 , (α21 , α22 ) + (1, 0) = (1, 1) ⇒ α21 = 0, α22 = 1, σ α2 (lc(f2 ))cα2 ,β2 = 2x2 , (α31 , α32 ) + (0, 1) = (1, 1) ⇒ α31 = 1, α32 = 0, σ α3 (lc(f3 ))cα3 ,β3 = x1 . We solve − 12 x21 x22 = r1 x1 x2 +r2 (2x2 )+r3 x1 ⇒ r1 = 3x1 x2 , r2 = −x21 x2 , r3 = − 32 x1 x22 ; thus we replace h and we get h =h − (r1 Dα1 f1 + r2 Dα2 f2 + r3 Dα3 f3 ) 3 =h − (3x21 x22 D1 D2 − x21 x22 (2D1 D2 + x2 D1 − x1 D2 ) − x21 x22 D1 D2 ) 2 =x21 x32 D1 + (x21 x2 − x31 x22 )D2 and also

13.2 Reduction in Skew P BW Extensions

247

q1 := 3x2 D1 + 3x1 x2 , q2 := − 21 x1 x2 D1 D2 + 12 x1 x22 D1 − x21 x2 D2 , q3 := −x22 D12 − 32 x1 x22 D1 . Step 4. Observe that lm(h) = D1 and only lm(f2 ) | lm(h). In this case we have: (α21 , α22 ) + (1, 0) = (1, 0) ⇒ α21 = 0, α22 = 0, σ α2 (lc(f2 ))cα2 ,β2 = x2 . We solve x21 x32 = r2 x2 ⇒ r2 = x21 x22 ; thus we replace h and we obtain h =h − r2 Dα2 f2 =h − x21 x22 f2 =x21 x32 D1 + (x21 x2 − x31 x22 )D2 − x21 x22 (x2 D1 ) =(x21 x2 − x31 x22 )D2 and also q1 := 3x2 D1 + 3x1 x2 , q2 := − 12 x1 x2 D1 D2 + 21 x1 x22 D1 − x21 x2 D2 + x21 x22 , q3 := −x22 D12 − 32 x1 x22 D1 . Step 5. Now lm(h) = D2 and only lm(f3 ) | lm(h). Thus, we have: (α31 , α32 ) + (0, 1) = (0, 1) ⇒ α31 = 0, α32 = 0, σ α3 (lc(f3 ))cα3 ,β3 = x1 . We solve x21 x2 − x31 x22 = r3 x1 ⇒ r3 = x1 x2 − x21 x22 . Hence, we change h and we get h =h − r3 Dα3 f3 = h − (x1 x2 − x21 x22 )x1 D2 = h − (x21 x2 − x31 x22 )D2 = 0. So, q1 := 3x2 D1 + 3x1 x2 , q2 := − 12 x1 x2 D1 D2 + 21 x1 x22 D1 − x21 x2 D2 + x21 x22 , q3 := −x22 D12 − 32 x1 x22 D1 and f = q1 f 1 + q2 f 2 + q3 f 3 , with max{lm(lm(q1 )lm(f1 )), lm(lm(q2 )lm(f2 )), lm(lm(q3 )lm(f3 ))} = max{D12 D2 , D12 D2 , D12 D2 } = D12 D2 = lm(f ).

248

13 Gr¨ obner Bases for Skew P BW Extensions

13.3 Gr¨ obner Bases of Left Ideals Our next purpose is to define Gr¨ obner bases for the left ideals of a skew P BW extension A = σ(R)hx1 , . . . , xn i. Definition 13.3.1. Let I 6= 0 be a left ideal of A and let G be a nonempty finite subset of nonzero polynomials of I. We say that G is a Gr¨obner basis for I if each element 0 6= f ∈ I is reducible w.r.t. G. We will say that {0} is a Gr¨ obner basis for I = 0. Theorem 13.3.2. Let I 6= 0 be a left ideal of A and let G be a finite subset of nonzero polynomials of I. Then the following conditions are equivalent: (i) G is a Gr¨ obner basis for I. (ii) For any polynomial f ∈ A, G

f ∈ I if and only if f −−→+ 0. (iii) For any 0 6= f ∈ I there exist g1 , . . . , gt ∈ G such that lm(gj ) | lm(f ), 1 ≤ j ≤ t, (i.e., there exist αj ∈ Nn such that αj + exp(lm(gj )) = exp(lm(f ))) and lc(f ) ∈ hσ α1 (lc(g1 ))cα1 ,g1 , . . . , σ αt (lc(gt ))cαt ,gt }. (iv) For α ∈ Nn , let hα, I} be the left ideal of R defined by hα, I} := hlc(f ) | f ∈ I, exp(lm(f )) = α}. Then, hα, I} = J, with J := hσ β (lc(g))cβ,g | g ∈ G, with β + exp(lm(g)) = α}. G

Proof. (i)⇒ (ii) Let f ∈ I. If f = 0, then by definition f −−→+ 0. If f 6= 0, G then there exists an h1 ∈ A such that f −−→ h1 , with lm(f )  lm(h1 ) and f − h1 ∈ hG} ⊆ I, hence h1 ∈ I; if h1 = 0, so we end. If h1 6= 0, then we can repeat this reasoning for h1 , and since Mon(A) is well ordered, we get that G f −−→+ 0. G

Conversely, if f −−→+ 0, then by Theorem 13.2.6, there exist g1 , . . . , gt ∈ G and q1 , . . . , qt ∈ A such that f = q1 g1 + · · · + qt gt , i.e., f ∈ I. (ii) ⇒ (i) Evident. (i) ⇔ (iii) This is a direct consequence of Definition 13.2.3. (iii) ⇒ (iv) Since R is a left noetherian ring, there exist r1 , . . . , rs ∈ R, f1 , . . . , fl ∈ I such that hα, I} = hr1 , . . . , rs }, lm(fi ) = xα , 1 ≤ i ≤ l, with hr1 , . . . , rs } ⊆ hlc(f1 ), . . . , lc(fl )}, then hlc(f1 ), . . . , lc(fl )} = hα, I}. Let r ∈ hα, I}. There exist a1 , . . . , al ∈ R such that r = a1 lc(f1 ) + · · · + al lc(fl ); by (iii), for each i there exist g1i , . . . , gti i ∈ G and bji ∈ R such that lc(fi ) = b1i σ α1i (lc(g1i ))cα1i ,g1i + · · · + bti i σ αti i (lc(gti i ))cαti i ,gti i ,

13.3 Gr¨ obner Bases of Left Ideals

249

with exp(lm(fi )) = αji + exp(lm(gji )), thus hα, I} ⊆ J. Conversely, if r ∈ J, then r = b1 σ β1 (lc(g1 ))cβ1 ,g1 + · · · + bt σ βt (lc(gt ))cβt ,gt , with bi ∈ R, βi ∈ Nn , gi ∈ G such that βi + exp(lm(gi )) = α for any 1 ≤ i ≤ t; note that xβi gi ∈ I, exp(lm(xβi gi )) = α, lc(xβi gi ) = σ βi (lc(gi ))cβi ,gi , for 1 ≤ i ≤ t, and r = b1 lc(xβ1 g1 ) + · · · + bt lc(xβt gt ), i.e., r ∈ hα, I}. (iv)⇒ (iii) Let 0 6= f ∈ I and let α = exp(lm(f )), then lc(f ) ∈ hα, I}; by (iv) lc(f ) = b1 σ β1 (lc(g1 ))cβ1 ,g1 + · · · + bt σ βt (lc(gt ))cβt ,gt , with bi ∈ R, βi ∈ Nn , gi ∈ G such that βi + exp(lm(gi )) = α for any 1 ≤ i ≤ t. From this we conclude that lc(f ) ∈ hσ β1 (lc(g1 ))cβ1 ,g1 , . . . , σ βt (lc(gt ))cβt ,gt }. t u From this theorem we get the following consequences. Corollary 13.3.3. Let I 6= 0 be a left ideal of A. Then, (i) If G is a Gr¨ obner basis for I, then I = hG}. G (ii) Let G be a Gr¨ obner basis for I. If f ∈ I and f −−→+ h, with h reduced, then h = 0. (iii) Let G = {g1 , . . . , gt } be a set of nonzero polynomials of I with lc(gi ) ∈ R∗ for each 1 ≤ i ≤ t. Then, G is a Gr¨ obner basis of I if and only if given 0 6= r ∈ I there exists an i such that lm(gi ) divides lm(r). Proof. (i) This is a direct consequence of Theorem 13.3.2. G

(ii) Let f ∈ I and f −−→+ h, with h reduced; since f − h ∈ hG} = I, we have h ∈ I; if h 6= 0 then h can be reduced by G, but this is not possible since h is reduced. (iii) If G is a Gr¨ obner basis of I, then given 0 6= r ∈ I, r is reducible w.r.t. G, hence there exists an i such that lm(gi ) divides lm(r). Conversely, if this condition holds for some i, then r is reducible w.r.t. G since the equation lc(r) = r1 σ αi (lc(gi ))cαi ,gi , with αi + exp(lm(gi )) = exp(lm(r)), is soluble with solution r1 = lc(r)c0αi ,gi (σ αi (lc(gi )))−1 , where c0αi ,gi is a left inverse of cαi ,gi . t u Corollary 13.3.4. Let G be a Gr¨ obner basis for a left ideal I. Given g ∈ G, if g is reducible w.r.t. G0 = G − {g}, then G0 is a Gr¨ obner basis for I. Proof. We will show that every f ∈ I is reducible w.r.t. G0 . Let f be a nonzero polynomial in I; since G is a Gr¨ obner basis for I, f is reducible w.r.t. G and there exist elements g1 , . . . , gt ∈ G satisfying the conditions (i), (ii) and (iii) in Definition 13.2.3. If gi 6= g for each 1 ≤ i ≤ t, then we are finished. Suppose that for some j ∈ {1, . . . , t}, gj = g and let βi := exp(gi ) for i 6= j, β := exp(g), and αi , α ∈ Nn such that αi + βi = exp(f ) = α + β. Thus, lc(f ) = r1 σ α1 (lc(g1 ))cα1 ,β1 + · · · + rj σ α (lc(g))cα,β + · · · + rt σ αt (lc(gt ))cαt ,βt . On the other hand, since g is reducibleP w.r.t. G0 , there exist g10 , . . . , gs0 ∈ G0 0 s 0 such that lm(gl ) | lm(g) and lc(g) = l=1 rl0 σ αl (lc(gl0 ))cα0l ,βl0 , where βl0 := exp(gl0 ), αl0 ∈ Nn and αl0 + βl0 = exp(g) = β. So, α + αl0 + βl0 = α + β = exp(f )

250

13 Gr¨ obner Bases for Skew P BW Extensions

and hence lm(gl0 ) | lm(f ) for 1 ≤ i ≤ s. Moreover, using the identities of Remark 1.1.9, we have that σ α (lc(g))cα,β = σ α (

s X

0

rl0 σ αl (lc(gl0 ))cα0l ,βl0 )cα,β

l=1 0

= σ α (r10 )σ α σ α1 (lc(g10 ))σ α (cα01 ,β10 )cα,β + · · · 0

+ σ α (rs0 )σ α σ αs (lc(gs0 ))σ α (cα0s ,βs0 )cα,β 0

α 0 0 = σ α (r10 )cα,α01 σ α+α1 (lc(g10 ))c−1 α,α0 σ (cα1 ,β1 )cα,β + · · · 1

0

α + σ α (rs0 )cα,α0s σ α+αs (lc(gs0 ))c−1 α,α0s σ (cα0s ,βs0 )cα,β 0

= σ α (r10 )cα,α01 σ α+α1 (lc(g10 ))cα+α01 ,β10 + · · · 0

+ σ α (rs0 )cα,α0s σ α+αs (lc(gs0 ))cα+α0s ,βs0 . Further, if there exists a gk ∈ {g1 , . . . , gt } such that gk = gl0 for some l ∈ {1, . . . , s}, then βl0 = βk and α + αl0 = αk , hence, in the representation of lc(f ) the following term appears (rk + rj σ α (rl0 )cα,α0l )σ αk (lc(gk ))cαk ,βk . From this we get that f is reducible w.r.t. G0 , so G0 is a Gr¨obner basis for I. t u

13.4 Buchberger’s Algorithm for Left Ideals In [136] Buchberger’s algorithm for computing Gr¨obner bases of left ideals for the particular case of quasi-commutative bijective skew P BW extensions was constructed. In this section we extend Buchberger’s procedure to the general case of bijective skew P BW extensions without assuming that they are quasi-commutative. Complementing Remark 13.2.2, from now on in this chapter we will assume that A = σ(R)hx1 , . . . , xn i is bijective. We start by fixing some notation and proving a preliminary general result for arbitrary skew P BW extensions. Definition 13.4.1. Let F := {g1 , . . . , gs } ⊆ A, XF the least common multiple of {lm(g1 ), . . . , lm(gs )}, θ ∈ Nn , βi := exp(lm(gi )) and γi ∈ Nn such that γi + βi = exp(XF ), 1 ≤ i ≤ s. BF,θ will denote a finite set of generators in Rs of SF,θ := SyzR [σ γ1 +θ (lc(g1 ))cγ1 +θ,β1 · · · σ γs +θ (lc(gs ))cγs +θ,βs )]. For θ = 0 := (0, . . . , 0), SF,θ will be denoted by SF and BF,θ by BF . Remark 13.4.2. Let (b1 , . . . , bs ) ∈ SF,θ . Since A is bijective, there exists a unique (b01 , . . . , b0s ) ∈ SF such that bi = σ θ (b0i )cθ,γi for 1 ≤ i ≤ s: in fact, the existence and uniqueness of (b01 , . . . , b0s ) follows from the bijectivity of

13.4 Buchberger’s Algorithm for Left Ideals

251

Ps A. Now, since (b1 , . . . , bs ) ∈ SF,θ , we have i=1 bi σ θ+γi (lc(gi ))cθ+γi ,βi = 0. Replacing bi by σ θ (b0i )cθ,γi in the last equation, we obtain s X

σ θ (b0i )cθ,γi σ θ+γi (lc(gi ))c−1 θ,γi cθ,γi cθ+γi ,βi = 0;

i=1

multiplying by c−1 θ,γi +βi we get s X

−1 σ θ (b0i )cθ,γi σ θ+γi (lc(gi ))c−1 θ,γi cθ,γi cθ+γi ,βi cθ,γi +βi = 0;

i=1

now we can use the identities of Remark 1.1.9, so s X

σ θ (b0i )σ θ (σ γi (lc(gi )))σ θ (cγi ,βi ) = 0,

i=1

and since σ θ is injective, s X

b0i σ γi (lc(gi ))cγi ,βi = 0, i.e., (b01 , . . . , b0s ) ∈ SF .

i=1

Lemma 13.4.3. Let g1 , . . . , gs ∈ A , c1 , . . . , cs ∈ R − {0} and Ps α 1 , . . . , α s ∈ Nn such that δ := α1 + exp(g1 ) = · · · = αs + exp(gs ). If lm( i=1 ci xαi gi ) ≺ xδ , then there exist r1 , . . . , rk ∈ R and l1 , . . . , ls ∈ A such that s X i=1

ci xαi gi =

k X

rj xδ−exp(XF )

j=1

X s

bji xγi gi



i=1

+

s X

li gi ,

i=1

where XF is the least common multiple of lm(g1 ), . . . , lm(gs ), γi ∈ Nn is such that γi + exp(gi ) = exp(XF ), 1 ≤ i ≤ s, and BF = {b1 , . . . , bk } := {(b11 , . . . , b1s ), . . . , (bk1 , . . . , bks )}. Ps Moreover, lm(xδ−exp(XF ) i=1 bji xγi gi ) ≺ xδ for every 1 ≤ j ≤ k, and lm(li gi ) ≺ xδ for every 1 ≤ i ≤ s. Proof. Let xβi := lm(gi ) for 1 ≤ i ≤ s; since xδ = lm(xα i lm(gi )), we have lm(gi ) | xδ and hence XF | xδ , so there exists a θ ∈ Nn such that exp(XF ) + θ = δ. On the other hand, γi + β i = exp(XF ) and αi + βi = δ, P s so α = γ + θ for every 1 ≤ i ≤ s. Now, lm( xαi gi ) ≺ xδ implies that i i Ps Ps i=1 ciθ+γ αi i (lc(gi ))cθ+γi ,βi = 0. i=1 ci σ (lc(gi ))cαi ,βi = 0. So we have i=1 ci σ This implies that (c1 , . . . , cs ) ∈ SF,θ ; from Remark 13.4.2 we know that there exists a unique (c01 , . . . , c0s ) ∈ SF such that ci = σ θ (c0i )cθ,γi . Then, s X i=1

ci xαi gi =

s X i=1

σ θ (c0i )cθ,γi xαi gi .

13 Gr¨ obner Bases for Skew P BW Extensions

252

Now, xθ ci0 xγi = (σ θ (c0i )xθ + pθ,ci0 )xγi = σ θ (ci0 )xθ xγi + pθ,c0i xγi = σ θ (c0i )cθ,γi xθ+γi + σ θ (ci0 )pθ,γi + pc0i ,θ xγi = σ θ (ci0 )cθ,γi xθ+γi + p0i where p0i := σ θ (ci0 )pθ,γi + pc0i ,θ xγi ; note that p0i = 0 or lm(p0i ) ≺ xθ+γi for each i. Thus, σ θ (ci0 )cθ,γi xθ+γi = xθ ci0 xγi + pi , with pi = 0 or lm(pi ) ≺ xθ+γi . Hence, s X

ci xαi gi =

s X

σ θ (c0i )cθ,γi xαi gi

i=1

i=1

=

s X

(xθ c0i xγi + pi )gi

i=1

=

s X

xθ ci0 xγi gi +

i=1

s X

pi gi ,

i=1

with pi gi = 0 or lm(pi gi ) ≺ xθ+γi +βi = xδ . On the other hand, since (c01 , . . . , c0s ) ∈ SF , there exist r10 , . . . , rk0 ∈ R such that (c10 , . . . , c0s ) = r10 b 1 + Pk · · · + rk0 b k = r10 (b11 , . . . , b1s ) + · · · + rk0 (bk1 , . . . , bks ), thus c0i = j=1 rj0 bji . Using this, we have s X

xθ ci0 xγi gi =

s X

xθ

i=1

i=1

=

s X

k X


rj0 bji xγi gi

j=1 k X


xθ rj0 bji xγi gi

i=1 j=1

=

s X k X


(σ θ (rj0 )xθ + pθ,rj0 )bji xγi gi

i=1 j=1

=

k s X X

σ θ (rj0 )xθ bji xγi gi +

=

σ θ (rj0 )xθ bji xγi gi +

=

j=1

where qi :=

Pk

j=1

σ θ (rj0 )xθ

s X k X




pθ,rj0 bji xγi gi

i=1 j=1

j=1 i=1 k X

pθ,rj0 bji xγi gi

j=1

i=1 j=1 k X s X

k X

s X i=1

bji xγi gi +

s X

qi gi ,

i=1

prj0 ,θ bji xγi = 0 or lm(qi ) ≺ xθ+γi . Therefore,

13.4 Buchberger’s Algorithm for Left Ideals s X

αi

ci x gi =

i=1

k X

rj x

j=1

θ

253 s X

γi

bji x gi +

i=1

s X

li gi ,

i=1

with li := pi +qi for 1 ≤ i ≤ s and rj := σ θ (rj0 ) for 1 ≤ j ≤ k. Finally, it is easy Ps Ps γi δ γi γi +βi to see that lm(xθ , i=1 bji x gi )) ≺ x since lm( i=1 bji x gi ) ≺ x δ and lm(li gi ) = lm(pi gi + qi gi ) ≺ x . t u With the notation of Definition 13.4.1 and Lemma 13.4.3, we can prove the main result of the present section. Theorem 13.4.4. Let I 6= 0 be a left ideal of A and let G be a finite subset of nonzero generators of I. Then the following conditions are equivalent: (i) G is a Gr¨ obner basis of I. (ii) For all F := {g1 , . . . , gs } ⊆ G, and for any (b1 , . . . , bs ) ∈ BF , s X

G

bi xγi gi −−→+ 0.

i=1

Proof.. (i) ⇒ (ii) We observe that f :=

Ps

i=1 bi x

γi

gi ∈ I, so by Theorem

G

13.3.2 f −−→+ 0. (ii) ⇒ (i) Let 0 6= f ∈ I. We will prove that the condition (iii) of Theorem 13.3.2 holds. Let G := {g1 , . . . , gt }, then there exist h1 , . . . , ht ∈ A such that f = h1 g1 + · · · + ht gt and we can choose {hi }ti=1 such that xδ := max{lm(lm(hi )lm(gi ))}ti=1 is minimal. Let lm(hi ) := xαi , ci := lc(hi ), lm(gi ) = xβi for 1 ≤ i ≤ t and F := {gi ∈ G | lm(lm(hi )lm(gi )) = xδ }; renumbering the elements of G we can assume that F = {g1 , . . . , gs }. We will consider two possible cases. Case 1: lm(f ) = xδ . Then lm(gi ) | lm(f ) for 1 ≤ i ≤ s and lc(f ) = c1 σ α1 (lc(g1 ))cα1 ,β1 + · · · + cs σ αs (lc(gs ))cαs ,βs , i.e., condition (iii) of Theorem 13.3.2 holds. Case 2: lm(f ) ≺ xδ . We will prove that this produces a contradiction. To begin, note that f can be written as f=

s X

ci xαi gi +

i=1

s X

(hi − ci xαi )gi +

i=1

t X

hi gi ;

(13.4.1)

i=s+1

we have lm((hi − ci xαi )gi ) ≺ xδ for every 1 ≤ i ≤ s and lm(hi gi ) ≺ xδ for every s + 1 ≤ i ≤ t, so lm(

Ps

and hence lm( have

i=1 (hi

Ps

− ci xαi )gi ) ≺ xδ and lm(

i=1 ci x

αi

Pt

i=s+1

hi gi ) ≺ xδ ,

gi ) ≺ xδ . By Lemma 13.4.3 (and its notation), we

254

13 Gr¨ obner Bases for Skew P BW Extensions s X

αi

ci x gi =

i=1

k X

rj x

δ−exp(XF )

j=1

where lm(xδ−exp(XF )

s X

γi




bji x gi +

i=1

Ps

s X

li gi ,

(13.4.2)

i=1

gi ) ≺ xδ for every 1 ≤ j ≤ k and lm(li gi ) ≺ Ps G xδ for 1 ≤ i ≤ s. By the hypothesis, i=1 bji xγi gi −−→+ 0, whence, by ThePs Pt orem 13.2.6, q1 , . . . , qt ∈ A such that i=1 bji xγi gi = i=1 qi gi , Ps there γexist t i with lm( P i=1 bji x gi ) = max{lm(lm(qi )lm(gi ))}i=1 , but (bj1 , . . . , bjs ) ∈ s γi BF , so lm( i=1 bji x gi ) ≺ XF and hence lm(lm(qi )lm(gi )) ≺ XF for every 1 ≤ i ≤ t. Thus, k X

i=1 bji x

rj xδ−exp(XF )

j=1

s X

γi

k t X
X
bji xγi gi = rj xδ−exp(XF ) qi gi

i=1

j=1

=

i=1

t X k X

rj xδ−exp(XF ) qi gi

i=1 j=1

=

t X

qei gi ,

i=1

Pk with qei := rj xδ−exp(XF ) qi and lm(e qi gi ) ≺ xδ for every 1 ≤ i ≤ t. Pj=1 P P s t s Substituting i=1 ci xαi gi = i=1 qei gi + i=1 li gi into equation (13.4.1), we obtain t s s t X X X X f= qei gi + (hi − ci xαi )gi + li gi + hi gi , i=1

i=1

i=1

i=s+1

and hence we have expressed f as a combination of polynomials g1 , . . . , gt , where every term has leading monomial ≺ xδ . This contradicts the minimality of xδ and we have finished the proof. t u Remark 13.4.5. For P BW extensions, the proof of (i)⇔(ii) in Theorem 13.4.4 involves the proof of Theorem 3.3.4 in [421], since in that case σi = idR for all i and cα,β = 1 for all α, β ∈ Nn . Corollary 13.4.6. Let F = {f1 , . . . , fs } be a set of nonzero polynomials of A. The algorithm below produces a Gr¨ obner basis for the left ideal hF } of A (P (X) denotes the set of subsets of the set X). From Theorem 3.1.5 and the previous corollary we get the following direct conclusion. Corollary 13.4.7. Each left ideal of A has a Gr¨ obner basis. Next we will illustrate Buchberger’s algorithm with some examples. We start with an elementary verification of the algorithm.

13.4 Buchberger’s Algorithm for Left Ideals

255

Buchberger’s algorithm for bijective skew P BW extensions INPUT: F := {f1 , . . . , fs } ⊆ A, fi 6= 0, 1 ≤ i ≤ s OUTPUT: G = {g1 , . . . , gt } a Gr¨ obner basis for hF } INITIALIZATION: G := ∅, G0 := F WHILE G0 6= G DO D := P (G0 ) − P (G) G := G0 FOR each S := {gi1 , . . . , gik } ∈ D DO Compute BS FOR each b = (b1 , . . . , bk ) ∈ BS DO G0

Reduce kj=1 bj xγj gij −−−→+ r, with r reduced with respect to G0 and γj defined as in Definition 13.4.1

P

IF r 6= 0 THEN G0 := G0 ∪ {r}

Example 13.4.8. In O3 (2, 2, 2) = σ(R)hx2 , x3 i, with R := Q[x1 ] and x2 x1 = 2x1 x2 , x3 x1 = 2x1 x3 , x3 x2 = 2x2 x3 , we consider the order deglex with x2  x3 ; let f1 := x2 + x1 and f2 := x3 + 1. Using the previous algorithm we will construct a Gr¨obner basis for the left ideal I := hf1 , f2 }. We observe that in this example the endomorphisms σ1 , σ2 : R → R coincide, and are defined by k 7→ k, x1 7→ 2x1 , for any k ∈ Q. We start with G := ∅ and G0 := {f1 , f2 }. Since G0 6= G we have D = {S1 , S2 , S1,2 }, with S1 = {f1 }, S2 = {f2 }, S1,2 = {f1 , f2 }. We make G = G0 . For S1 we compute BS1 , a system of generators of SyzR [σ γ1 (lc(f1 ))cγ1 ,β1 ]. Simplifying the notation, we will write BS1 = SyzR [σ γ1 (lc(f1 ))cγ1 ,β1 ]. We have XS1 = lcm{lm(f1 )} = lm(f1 ), β1 = exp(lm(f1 )) = (1, 0), so γ1 = (0, 0) and cγ1 ,β1 = 1. Thus, BS1 = SyzR [1] = 0, and hence, we do not add a new polynomial to G0 . For S2 we have the same situation. In general, since R has no zero divisors, if S has only one element, then BS = 0 and no new element is added. For S1,2 we compute BS1,2 = SyzR [σ γ1 (lc(f1 ))cγ1 ,β1 σ γ2 (lc(f2 ))cγ2 ,β2 ]: XS1,2 = lcm{x2 , x3 } = x2 x3 , so γ1 = (0, 1) and x3 x2 = 2x2 x3 , thus cγ1 ,β1 = 2; in a similar way γ2 = (1, 0) and cγ2 ,β2 = 1. Then, BS1,2 = SyzR [σ10 σ21 (1)2 σ11 σ20 (1)1] = SyzR [2 1] = {(1, −2)} and hence 1xγ1 f1 − 2xγ2 f2 = x3 (x2 + x1 ) − 2x2 (x3 + 1) = −2x2 + 2x1 x3 ;

256

13 Gr¨ obner Bases for Skew P BW Extensions

using the division algorithm of Theorem 13.2.6 we get that −2x2 + 2x1 x3 can be reduced to 0 by f1 and f2 , so G = {f1 , f2 } is a Gr¨obner basis of I. Example 13.4.9. In A2 (2, 2) = σ(Q[x1 , x2 ])hy1 , y2 i we consider the deglex order with y1  y2 . Let f1 := x21 x2 y1 y2 , f2 := x2 y1 ; we will apply the algorithm for computing a Gr¨ obner basis for the left ideal I := hf1 , f2 }. We start with G := ∅ and G0 := {f1 , f2 }. Step 1. Since G0 6= ∅ we have D = {F1 , F2 , F1,2 }, with F1 = {f1 },

F2 = {f2 },

F1,2 = {f1 , f2 }.

We make G = G0 . For F1 and F2 we found BF1 = {0} = BF2 . For F1,2 we have BF1,2 = {(1, − 21 x21 )} and γ1 = (0, 0), γ2 = (0, 1), and hence 1 1 1xγ1 f1 − x21 xγ2 f2 = x21 x2 y1 y2 − x21 y2 x2 y1 2 2 1 = x21 x2 y1 y2 − x21 (2x2 y2 + 1)y1 2 1 = x21 x2 y1 y2 − (2x21 x2 y2 y1 + x21 y1 ) 2 1 = x21 x2 y1 y2 − x21 x2 y1 y2 − x21 y1 2 1 = − x21 y1 . 2 We observe that f3 := − 12 x21 y1 is not reducible w.r.t. G0 : in fact, although lm(f2 ) | lm(f3 ) there is no r ∈ Q[x1 , x2 ] such that − 12 x21 = lc(f3 ) = rlc(f2 ) = rx2 . Then we make G0 := {f1 , f2 , f3 }. Step 2. Since G0 6= G we compute D = P (G0 )−P (G) = {F3 , F1,3 , F2,3 , F1,2,3 } and we make G = G0 . Making all computations for the elements of D we conclude that no new elements arise in G0 . This means that the algorithm stops and G = {f1 , f2 , f3 } is a Gr¨ obner basis of I. Example 13.4.10. In H2 (2) = σ(Q)hx1 , x2 ; y1 , y2 ; z1 , z2 i we consider the order deglex with x1  x2  y1  y2  z1  z2 ; let f1 := x1 x2 y1 y2 , f2 := x2 y1 , f3 := x1 z2 . Using Buchberger’s algorithm we will construct a Gr¨obner basis for the left ideal I := hf1 , f2 , f3 }. We recall first that this algebra is bijective and all endomorphisms σi are trivial, i.e., σi = idQ for every 1 ≤ i ≤ 6 (see Proposition 1.1.3 and Example 1.1.5 (vi)). We start with G := ∅ and G0 := {f1 , f2 , f3 }. Step 1. Since G0 = 6 G we have D = {S1 , S2 , S3 , S1,2 , S1,3 , S2,3 , S1,2,3 }, with S1 = {f1 }, S2 = {f2 }, S3 = {f3 }, S1,2 = {f1 , f2 }, S1,3 = {f1 , f3 }, S2,3 = {f2 , f3 }, S1,2,3 = {f1 , f2 , f3 }.

13.4 Buchberger’s Algorithm for Left Ideals

257

We make G = G0 . Since the ring of coefficients R = Q has nonzero divisors, S1 , S2 and S3 does not add new elements to G0 . For S1,2 we compute BS1,2 = SyzQ [σ γ1 (lc(f1 ))cγ1 ,β1 σ γ2 (lc(f2 ))cγ2 ,β2 ]: XS1,2 = lcm{x1 x2 y1 y2 , x2 y1 } = x1 x2 y1 y2 , so γ1 = (0, 0, 0, 0, 0, 0), cγ1 ,β1 = 1, γ2 + (0, 1, 1, 0, 0, 0) = (1, 1, 1, 1, 0, 0), i.e., γ2 = (1, 0, 0, 1, 0, 0) and hence x1 y2 x2 y1 = 2x1 x2 y2 y1 = 2x1 x2 y1 y2 , so cγ2 ,β2 = 2. Thus, BS1,2 = SyzQ [1 2] = {(1, − 21 )} and hence 1xγ1 f1 − 12 xγ2 f2 = x1 x2 y1 y2 − 12 x1 y2 x2 y1 = x1 x2 y1 y2 − 12 x1 (2x2 y2 )y1 = 0, i.e., we do not add a new polynomial to G0 . For S1,3 we compute BS1,3 = SyzQ [σ γ1 (lc(f1 ))cγ1 ,β1 σ γ3 (lc(f3 ))cγ3 ,β3 ]: XS1,3 = lcm{x1 x2 y1 y2 , x1 z2 } = x1 x2 y1 y2 z2 , so γ1 = (0, 0, 0, 0, 0, 1) and from this we have z2 x 1 x 2 y 1 y 2 = x 1 z2 x 2 y 1 y 2 1 = x1 ( x2 z2 + y2 )y1 y2 2 1 = x1 x2 y1 z2 y2 + x1 y1 y22 2 1 = x1 x2 y1 (2y2 z2 ) + x1 y1 y22 2 = x1 x2 y1 y2 z2 + x1 y1 y22 , so cγ1 ,β1 = 1. Note that γ3 = (0, 1, 1, 1, 0, 0), so x2 y1 y2 x1 z2 = 2x1 x2 y1 y2 z2 , i.e., cγ3 ,β3 = 2. Thus, BS1,3 = SyzQ [1 2] = {(1, − 12 )} and hence 1 1 1xγ1 f1 − xγ3 f3 = z2 x1 x2 y1 y2 − x2 y1 y2 x1 z2 2 2 = x1 x2 y1 y2 z2 + x1 y1 y22 − x1 x2 y1 y2 z2 = x1 y1 y22 . Note that f4 := x1 y1 y22 cannot be reduced with G0 , so we make G0 := {f1 , f2 , f3 , f4 }. For S2,3 we get in a similar way that γ2 = (1, 0, 0, 0, 0, 1), γ3 = (0, 1, 1, 0, 0, 0) and BS2,3 = {(1, − 14 )}, so 1xγ2 f2 − 14 xγ3 f3 = x1 y1 y2 . Note that f5 := x1 y1 y2 cannot be reduced with G0 , so we define G0 := {f1 , f2 , f3 , f4 , f5 }. For S1,2,3 we have γ1 = (0, 0, 0, 0, 0, 1), γ2 = (1, 0, 0, 1, 0, 1), γ3 = (0, 1, 1, 1, 0, 0) BS1,2,3 = {(1, 0, − 12 ), (0, 1, − 12 )},

258

13 Gr¨ obner Bases for Skew P BW Extensions

so 1 1xγ1 f1 − xγ3 f3 = x1 y1 y22 2 1 γ2 1x f2 − xγ3 f3 = x1 y1 y22 , 2 but x1 y1 y22 = f4 , so we do not add a new polynomial to G0 . Step 2. Since G0 6= G we compute D = P (G0 ) − P (G) and we make G = G0 . D has 24 different subsets of G0 , and for each of these subsets the reduction procedure produces zero, f4 or x1 y1 y23 . But note that x1 y1 y23 can be reduced to zero with f4 : in fact, lm(f4 ) = x1 y1 y22 divides lm(x1 y1 y23 ), moreover lc(x1 y1 y23 ) = 1 = rσ a (lc(f4 ))ca,β4 , with a = (0, 0, 0, 1, 0, 0); then y2 x1 y1 y22 = x1 y2 y1 y22 = x1 y1 y23 and hence ca,β4 = 1, thus r = 1 and h = x1 y1 y23 −1y2 x1 y1 y22 = 0. Thus, the algorithm stops and G = {f1 , f2 , f3 , f4 , f5 } is a Gr¨obner basis for I. Example 13.4.11. We consider the diffusion algebra (see Section 2.4) with n = 2, K = Q, λ12 = −2 and λ21 = −1. In this bijective skew P BW extension, D2 D1 = 2D1 D2 + x2 D1 − x1 D2 and the automorphisms σ1 and σ2 are the identity. We consider the deglex order with D1  D2 and the polynomials f1 = x21 x2 D12 D2 , f2 = x22 D1 D22 . We will calculate a Gr¨obner basis for the left ideal generated by f1 and f2 . We start by taking G := ∅ and G0 := {f1 , f2 }. Step 1. Since G0 6= G, we have D = {S1 , S2 , S1,2 }. We make G = G0 . Since R = Q[x1 , x2 ] has no zero divisors, S1 and S2 do not add any polynomial to G0 . For S1,2 , we compute BS1,2 , a generator set of SyzR [σ γ1 (lc(f1 ))cγ1 ,β1

σ γ2 (lc(f2 ))cγ2 ,β2 ] :

X1,2 = lcm{D12 D2 , D1 D22 } = D12 D22 , so γ1 = (0, 1) and D2 (D12 D2 ) = 4D12 D22 + 3x2 D12 D2 − 3x1 D1 D22 − x1 x2 D1 D2 + x1 D22 , thus cγ1 ,β1 = 4; in a similar way, γ2 = (1, 0) and cγ2 ,β2 = 1. Whence, BS1,2 = {( 14 x2 , −x21 )} and we have 1 4 x2 D 2 f1

− x21 D1 f2 = 34 x21 x32 D12 D2 − x31 x22 D1 D22 − 14 x31 x32 D1 D2 + 14 x41 x22 D22 .

Since 3 2 3 2 4 x1 x2 D 1 D 2

G

− x31 x22 D1 D22 − 14 x31 x32 D1 D2 + 14 x41 x22 D22 −−→+ − 41 x31 x32 D1 D2 + 14 x41 x22 D22 =: f3

and f3 is reduced with respect to G, we add the polynomial f3 and we make G0 := {f1 , f2 , f3 }. Step 2. Since G0 6= G, we compute D = P (G0 ) − P (G) and we make G = G0 . In D we only need to consider three subsets: S1,3 = {f1 , f3 }, S2,3 = {f2 , f3 }, S1,2,3 = {f1 , f2 , f3 }. For S1,3 , XS1,3 = D12 D2 so γ1 = (0, 0), cγ1 ,β1 = 1; in the same way, γ3 = (1, 0) and cγ3 ,β3 = 1. Thus, we must calculate a generator set for SyzR [x21 x2 − 1 3 3 2 4 x1 x2 ]. We have BS1,3 = {(x1 x2 , 4)} and

13.4 Buchberger’s Algorithm for Left Ideals

259

x1 x22 f1 + 4D1 f3 = x41 x22 D1 D22 can be reduced to 0 by f2 . For S2,3 , XS2,3 = D1 D22 , so γ2 = (0, 0) and cγ2 ,β2 = 1; in the same way, γ3 = (0, 1) and, since D2 D1 D2 = 2D1 D22 + x2 D1 D2 − x1 D22 , we have cγ3 ,β3 = 2. Thus, a set of generators for SyzR [x22 − 21 x31 x32 ] is BS2,3 = {(x31 x2 , 2)}, and x31 x2 f2 + 2D2 f3 = 12 x41 x22 D23 − 12 x31 x42 D1 D2 + 12 x41 x32 D22 =: f4 . Since f4 is reduced with respect to G, we add f4 and we make G0 := {f1 , f2 , f3 , f4 }. For S1,2,3 , XS1,2,3 = D12 D22 and hence γ1 = (0, 1), γ2 = (1, 0) and γ3 = (1, 1). So, cγ1 ,β1 = 4, cγ2 ,β2 = 1 and, since D1 D2 D1 D2 = 2D12 D22 + x2 D12 D2 − x1 D1 D22 , we have cγ3 ,β3 = 2. Therefore, a system of generators for SyzR [4x21 x2 x22 − 12 x31 x32 ] is BS1,2,3 = {( 14 x2 , −x21 , 0), ( 14 x1 x22 , 0, 2)}; for the first generator we obtain a polynomial that can be reduced to 0 by f1 , f2 and f3 (in this case, we have the same calculations as in step one). For the second generator, we obtain the following polynomial: 1 2 4 x1 x2 D2 f1 +2D1 D2 f3

= 14 x31 x42 D12 D2 − 12 x41 x32 D1 D22 − 14 x41 x42 D1 D2 + 14 x51 x32 D22

which can be reduced to 0 by f1 , f2 and f3 . In consequence, we do not add any polynomial. Step 3. Again, G 6= G0 . Thus, we compute D = P (G0 ) − P (G) and we make G = G0 . In this case, we only need to consider the following subsets: S1,4 , S2,4 , S3,4 , S1,2,4 , S1,3,4 , S2,3,4 , S1,2,3,4 . For S1,4 , XS1,4 = D12 D23 , and γ1 = (0, 2), γ4 = (2, 0). Now, since D22 D12 D2 = 16D12 D23 + 24x2 D12 D22 − 24x1 D1 D23 + 9x22 D12 D2 − 26x1 x2 D1 D22 + 9x21 D23 − 4x1 x22 D1 D2 + 4x21 x2 D22 , we have cγ1 ,β1 = 16. As cγ4 β4 = 1, a generator set for SyzR [16x21 x2 1 2 is BS1,4 = {( 16 x1 x2 , −2)}. With this single generator, we obtain

1 4 2 2 x1 x2 ]

1 2 1 3 x x2 D22 f1 − 2D12 f4 = x31 x42 D13 D2 − x41 x32 D12 D22 − x51 x22 D1 D23 16 1 2 2 9 4 4 2 13 5 3 9 6 2 3 1 5 4 1 2 + x1 x2 D1 D2 − x1 x2 D1 D2 + x1 x2 D2 − x1 x2 D1 D2 + x61 x32 D2 , 16 8 16 4 4 a polynomial reducible to 0 by f1 , f2 , f3 and f4 . For S2,4 , XS2,4 = D1 D23 , so γ2 = (0, 1) and γ4 = (1, 0). As D2 D1 D22 = 2D1 D23 + x2 D1 D22 − x1 D23 , we have cγ2 ,β2 = 2. Thus, BS2,4 = {( 12 x41 , −2)} is a system of generators of SyzR [2x22 , 12 x41 x22 ], and we have 1 4 2 x1 D 2 f2

− 2D1 f4 = x31 x42 D12 D2 + 12 x41 x32 D1 D22 − 12 x51 x22 D23 ,

which is also reducible to 0 w.r.t. f1 , f2 , f3 and f4 . For S3,4 , XS3,4 = D1 D23 , whence γ3 = (0, 2) and γ4 = (1, 0). Since D22 D1 D2 = 4D1 D23 + 4x2 D1 D22 − 3x1 D23 + x22 D1 D2 − x1 x2 D22 , we have cγ3 ,β3 = 4. Thus, a generator set for SyzR [−x31 x32 12 x41 x22 ] is BS3,4 = {(−x1 , −2x2 )}; therefore,

260

13 Gr¨ obner Bases for Skew P BW Extensions

−x1 D22 f3 − 2x2 D1 f4 = − 14 x51 x22 D24 + x31 x52 D12 D2 − 34 x51 x32 D23 + 14 x41 x52 D1 D2 − 14 x51 x42 D22 . Since this last polynomial is reducible to 0 through f2 , f3 and f4 , no polynomial is added. For S1,2,4 we have XS1,2,4 = D12 D23 , hence γ1 = (0, 2), γ2 = (1, 2) and γ4 = (2, 0). Thus, cγ1 ,β1 = 16, cγ2 ,β2 = 2, cγ4 ,β4 = 1 and, hence, BS1,2,4 = {(

1 1 1 x2 , − x21 , 0), ( x21 x2 , 0, −2)}. 16 2 16

For these generators, we obtain polynomials that are reducible to 0 by f1 , f2 , f3 , and f4 . For S1,3,4 , XS1,3,4 = D12 D23 ; thus γ1 = (0, 2), γ3 = (1, 2) and γ4 = (2, 0). In consequence, cγ1 ,β1 = 16, cγ3 ,β3 = 4, cγ4 ,β4 = 1 and a set of generators for SyzR [16x21 x2 − x31 x32 12 x41 x22 ] is BS1,3,4 = 1 1 2 x1 x22 , 1, 0), ( 16 x1 x2 , 0, −2)}. It is not difficult to show that these gen{( 16 erators produce polynomials which can be reducible to 0 w.r.t. f1 , f2 , f3 , and f4 . For S2,3,4 , we obtain a similar situation. Finally, for S1,2,3,4 we have that XS1,2,3,4 = D12 D23 , γ1 = (0, 2), γ2 = (1, 1), γ3 = (1, 2) and γ4 = (2, 0). Thus, cγ1 ,β1 = 16, cγ2 ,β2 = 2, cγ3 ,β3 = 4, cγ4 ,β4 = 1 1 2 1 x2 , − 12 x21 , 0, 0), ( 16 x1 x22 , 0, 1, 0), ( 16 x1 x2 , 0, 0, −2)}. 1, and BS1,2,3,4 = {( 16 Once again, the polynomials obtained through these generators are reducible to 0 by f1 , f2 , f3 and f4 . Therefore, G = {f1 , f2 , f3 , f4 } is a Gr¨obner basis for I := hf1 , f2 }. Remark 13.4.12. If I is a left ideal of a bijective skew P BW extension A and G = {g1 , . . . , gt } is a subset of nonzero polynomials in I, then Corollary 13.3.3 gives us a quick tool to verify if G is a Gr¨ obner basis for I when lc(gi ) ∈ R∗ for each 1 ≤ i ≤ t. For example, consider the extension O3 (2, 2, 2) = σ(Q[x1 ])hx2 , x3 i of Example 13.4.8, where I = hf1 , f2 }, with f1 := x2 + x1 and f2 := x3 + 1. Observe that G := {f1 , f2 } is a Gr¨obner basis of I since β for every f ∈ I we have lm(f ) = xα 2 x3 , with α ≥ 1 or β ≥ 1; in either case, lm(f ) is divisible by lm(f1 ) or lm(f2 ).

Chapter 14

Gr¨ obner Bases of Modules

In this chapter we construct the general theory of Gr¨obner bases for submodules of Am , m ≥ 1, where A = σ(R)hx1 , . . . , xn i is a bijective skew P BW extension of R, with R a LGS ring (see Definition 13.2.1) and Mon(A) endowed with some monomial order (see Definition 13.1.1). This theory was studied in [191] and [192], but now we will extend Buchberger’s algorithm to the general bijective case without assuming that A is quasi-commutative. Am is the left free A-module of column vectors of length m ≥ 1; since A is a left noetherian ring (Theorem 3.1.5), A is an IBN ring (Invariant Basis Number, see [240]), and hence, all bases of the free module Am have m elements. Note moreover that Am is left noetherian, and hence, any submodule of Am is finitely generated. The plan is to define and calculate Gr¨obner bases for submodules of Am , thus, we will define the monomials in Am , orders on the monomials, and the concept of reduction, we will construct a division algorithm, we will give equivalent conditions in order to define Gr¨obner bases, and finally, we will compute Gr¨ obner bases using a procedure similar to Buchberger’s algorithm for the general case of bijective skew P BW extensions. The results presented in this chapter are an easy generalization of those of the previous chapter, i.e., taking m = 1 we get the theory of Gr¨obner bases for the left ideals of A. We will include only some proofs since most of them can be found in [191] and [192] or they are an easy adaptation of those of the previous chapter. The theory that we will present has been also studied by G´omez-Torrecillas et al. (see [73] , [74]) for left P BW algebras over division rings and assuming some special commutativity conditions.

14.1 Monomial Orders on Mon(Am ) We will often represent the elements of Am also as row vectors, if this does not cause confusion. We recall that the canonical basis of Am is e 1 = (1, 0, . . . , 0), e 2 = (0, 1, 0, . . . , 0), . . . , e m = (0, 0, . . . , 1). © Springer Nature Switzerland AG 2020 W. Fajardo et al., Skew PBW Extensions, Algebra and Applications 28, https://doi.org/10.1007/978-3-030-53378-6_14

261

262

14 Gr¨ obner Bases of Modules

Definition 14.1.1. A monomial in Am is a vector X = Xei , where X = xα ∈ Mon(A) and 1 ≤ i ≤ m, i.e., X = Xei = (0, . . . , X, . . . , 0), where X is in the i-th position, named the index of X, ind(X) := i. A term is a vector cX, where c ∈ R. The set of monomials of Am will be denoted by Mon(Am ). Let Y = Y ej ∈ Mon(Am ). We say that X divides Y if i = j and X divides Y . We will say that any monomial X ∈ Mon(Am ) divides the null vector 0 . The least common multiple of X and Y, denoted by lcm(X, Y), is 0 if i 6= j, and U ei , where U = lcm(X, Y ), if i = j. Finally, we define exp(X) := exp(X) = α and deg(X) := deg(X) = |α|. We now define monomial orders on Mon(Am ). Definition 14.1.2. A monomial order on Mon(Am ) is a total order  satisfying the following three conditions: (i) lm(xβ xα )ei  xα ei , for every monomial X = xα ei ∈ Mon(Am ) and any monomial xβ in Mon(A). (ii) If Y = xβ ej  X = xα ei , then lm(xγ xβ )ej  lm(xγ xα )ei for every monomial xγ ∈ Mon(A). (iii)  is degree compatible, i.e., deg(X) ≥ deg(Y) ⇒ X  Y. If X  Y but X 6= Y we will write X  Y. Y  X means that X  Y. Proposition 14.1.3. Every monomial order on Mon(Am ) is a well order. Proof. We can repeat the proof of Proposition 13.1.2: suppose that we have a monomial order  on Mon(Am ) that is not a well order. This means that we have an infinite sequence of monomials X1  X2  X3  ··· and since  is degree compatible, we have an infinite subsequence deg(X i1 ) > deg(X i2 ) > deg(X i3 ) > · · · , but this is impossible since deg(X i1 ) is finite.

t u

Given a monomial order  on Mon(A), we can define two natural orders on Mon(Am ). Definition 14.1.4. Let X = Xei and Y = Y ej ∈ Mon(Am ). (i) The TOP (term over position) order is defined by   X  Y X  Y ⇐⇒ or   X = Y and i > j. (ii) The TOPREV order is defined by

14.2 Reduction in Am

263

  X  Y X  Y ⇐⇒ or   X = Y and

i < j.

Remark 14.1.5. (i) Note that with TOP we have e m  e m−1  · · ·  e 1 and e1  e2  · · ·  em for TOPREV. (ii) The POT (position over term) and POTREV orders defined in [6] and [236] for modules over classical polynomial commutative rings are not degree compatible. (iii) Other examples of monomial orders in Mon(Am ) are considered in [74], e.g., orders with weights. We fix a monomial order on Mon(A), let f 6= 0 be a vector of Am , then we may write f as a sum of terms in the following way f = c1 X 1 + · · · + ct X t , where c1 , . . . , ct ∈ R − 0 and X 1  X 2  · · ·  X t are monomials of Mon(Am ). Definition 14.1.6. With the above notation, we say that (i) lt(f) := c1 X1 is the leading term of f. (ii) lc(f) := c1 is the leading coefficient of f. (iii) lm(f) := X1 is the leading monomial of f. For f = 0 we define lm(0) = 0, lc(0) = 0, lt(0) = 0, and if  is a monomial order on Mon(Am ),Sthen we define X  0 for any X ∈ Mon(Am ). So, we extend  to Mon(Am ) {0}.

14.2 Reduction in Am The reduction process in Am is defined as follows. Definition 14.2.1. Let F be a finite set of nonzero vectors of Am , and let F f, h ∈ Am . We say that f reduces to h by F in one step, denoted f −−→ h, if there exist elements f1 , . . . , ft ∈ F and r1 , . . . , rt ∈ R such that (i) lm(fi ) | lm(f), 1 ≤ i ≤ t, i.e., ind(lm(fi )) = ind(lm(f)) and there exists xαi ∈ Mon(A) such that αi + exp(lm(fi )) = exp(lm(f)). (ii) lc(f) = r1 σ α1 (lc(f1 ))cα1 ,f1 + · · · + rt σ αt (lc(ft ))cαt ,ft , where cαi ,fi := cαi ,exp(lm(fi )) . Pt (iii) h = f − i=1 ri xαi fi .

264

14 Gr¨ obner Bases of Modules F

We say that f reduces to h by F , denoted f −−→+ h, if and only if there exist vectors h1 , . . . , ht−1 ∈ Am such that F

F

F

F

F

f −−−−→ h1 −−−−→ h2 −−−−→ · · · −−−−→ ht−1 −−−−→ h . f is reduced (also called minimal) w.r.t. F if f = 0 or there is no one step reduction of f by F , i.e., one of the conditions (i) or (ii) fails. Otherwise, we F will say that f is reducible w.r.t. F . If f −−→+ h and h is reduced w.r.t. F , then we say that h is a remainder for f w.r.t. F . Remark 14.2.2. Related to the previous definition we have the following remarks: (i) By Theorem 1.1.8, the coefficients cαi ,f i in the previous definition are unique and satisfy xαi xexp(lm(f i )) = cαi ,f i xαi +exp(lm(f i )) + pαi ,f i , where pαi ,f i = 0 or deg(lm(pαi ,f i )) < |αi + exp(lm(f i ))|, 1 ≤ i ≤ t. (ii) lm(f )  lm(h) and f − h ∈ hF i, where hF i is the submodule of Am generated by F . (iii) The remainder of f is not unique. F

(iv) By definition we will assume that 0 − → 0. (v) t X lt(f ) = ri lt(xαi lt(f i )), i=1

Proposition 14.2.3. Let A be a skew P BW extension such that cα,β is invertible for each α, β ∈ Nn . Let f, h ∈ Am , θ ∈ Nn and F = {f1 , . . . , ft } be a finite set of nonzero vectors of Am . Then, F

(i) If f −−→ h, then there exists a p ∈ Am with p = 0 or lm(xθ f)  lm(p) F such that xθ f+p −−→ xθ h. In particular, if A is quasi-commutative, then p = 0. F (ii) If f −−→+ h and p ∈ Am is such that p = 0 or lm(h)  lm(p), then F f + p −−→+ h + p. F (iii) If f −−→+ h, then there exists a p ∈ Am with p = 0 or lm(xθ f)  lm(p) F such that xθ f + p −−→+ xθ h. If A is quasi-commutative, then p = 0. F (iv) If f −−→+ 0, then there exists a p ∈ Am with p = 0 or lm(xθ f)  lm(p) F such that xθ f + p −−→+ 0. If A is quasi-commutative, then p = 0. Proof. (i) If f = 0, then h = 0 = p. Let f 6= 0, then there exist f 1 , . . . , f t ∈ F and r1 . . . , rt ∈ R such that lm(f i ) | lm(f ), 1 ≤ i ≤ t, i.e., ind(lm(f i )) = ind(lm(f )) and there exists an xαi ∈ Mon(A) such that αi + exp(lm(f i )) = exp(lm(f )). Moreover, lc(f ) = r1 σ α1 (lc(f 1 ))cα1 ,f 1 + · · · + rt σ αt (lc(f t ))cαt ,f t , Pt and h = f − i=1 ri xαi f i . Let λ := exp(lm(f )) and βi := exp(lm(f i )). Note that ind(lm(f )) = ind(lm(xθ f )) and exp(xθ f ) = θ + λ, so

14.2 Reduction in Am

265

lm(f i )|lm(xθ f ), with θ + λ = (θ + αi ) + βi . We observe that lc(xθ f ) = σ θ (lc(f ))cθ,λ =

t X

σ θ (ri )σ θ (σ αi (lc(f i )))σ θ (cαi ,βi )cθ,λ ,

i=1

and by Remark 1.1.9 lc(xθ f ) =

=

t X i=1 t X

θ σ θ (ri )cθ,αi σ θ+αi (lc(f i ))c−1 θ,αi σ (cαi ,βi )cθ,λ

θ σ θ (ri )cθ,αi σ θ+αi (lc(f i ))c−1 θ,αi σ (cαi ,βi )cθ,αi +βi

i=1

=

t X

σ θ (ri )cθ,αi σ θ+αi (lc(f i ))cθ+αi ,βi

i=1

=

t X

ri0 σ θ+αi (lc(f i ))cθ+αi ,βi , with ri0 := σ θ (ri )cθ,αi .

i=1

Moreover, θ

θ

x h =x f −

t X

x θ ri x α i f i

i=1

= xθ f −

t X

(σ θ (ri )xθ + pθ,ri )xαi f i

i=1

= xθ f −

t X

σ θ (ri )xθ xαi f i + pθ,ri xαi f i

i=1

= xθ f − = xθ f −

t X i=1 t X

σ θ (ri )cθ,αi xθ+αi f i + p ri0 xθ+αi f i + p

i=1

= xθ f + p −

t X

ri0 xθ+αi f i ,

i=1

with p :=

Pt

i=1

pθ,ri xαi f i +

Pt

i=1

σ θ (ri )pθ,αi f i ;

note that p = 0 or deg(lm(p)) < |θ + αi + βi | = |θ + λ| = deg(lm(xθ f )), so lm(xθ f )  lm(p). Moreover, lm(xθ f + p) = lm(xθ f ) and lc(xθ f + p) =

266

14 Gr¨ obner Bases of Modules F

lc(xθ f ), so by the previous discussion xθ f + p −−→ xθ h. If A is quasicommutative, then from Remark 1.1.9 (iii), p = 0. (ii) Let F

F

F

F

F

f −−−−→ h 1 −−−−→ h 2 −−−−→ · · · −−−−→ h t−1 −−−−→ h t := h; (14.2.1) F we start with f − → h 1 , if f = 0, then h 1 = 0 = p and there is nothing to prove. Let f 6= 0, if h 1 = 0 then p = 0 and hence lm(f )  lm(p); if h 1 6= 0, then lm(f )  lm(h 1 )  lm(p), and hence lm(f + p) = lm(f ), lc(f + p)P= lc(f ), and as in the proof of the first part of (i), h 1 + p = t f + p − i=1 ri xαi f i ; but lm(f + p) = lm(f ) and lc(f + p) = lc(f ), then F

f + p −−→ h 1 + p. Since lm(h i )  lm(p) we can repeat this reasoning for F h i −−→ h i+1 for 1 ≤ i ≤ t − 1. This completes the proof of (ii). (iii) By (i) and using (14.2.1), there exists a p 1 ∈ Am with p 1 = 0 or F lm(xθ f )  lm(p 1 ) such that xθ f + p 1 −−→ xθ h 1 ; there exists a p 2 ∈ Am F with p 2 = 0 or lm(xθ h 1 )  lm(p 2 ) such that xθ h 1 + p 2 −−→ xθ h 2 ; by (ii) we F F get that xθ f + p 1 + p 2 −−→ xθ h 1 + p 2 −−→ xθ h 2 , i.e., p 00 := p 1 + p 2 ∈ Am is such that F

xθ f + p 00 −−→+ xθ h 2 , with p 00 = 0, or lm(xθ f )  lm(p 00 ) since, as we noted above, lm(xθ f )  lm(p 1 ) and lm(xθ f )  lm(p 2 ) . By induction on t we find p 0 ∈ Am such that F

xθ f + p 0 −−→+ xθ h t−1 , with p 0 = 0 or lm(xθ f )  lm(p 0 ). By (i) there exists a p t ∈ Am such F that xθ h t−1 + p t −−→ xθ h, with p t = 0 or lm(xθ h t−1 )  lm(p t ). By (ii), F F xθ f + p 0 + p t −−→+ xθ h t−1 + p t −−→ xθ h. Thus, F

xθ f + p −−→+ xθ h, with p := p 0 + p t = 0, or lm(xθ f )  lm(p) since lm(xθ f )  lm(p 0 ) and lm(xθ f )  lm(p t ). From the just proved we observe that if A is quasicommutative then p = 0. (iv) This is a direct consequence of (iii) taking h = 0. t u Theorem 14.2.4. Let F = {f1 , . . . , ft } be a set of nonzero vectors of Am and f ∈ Am , then the Division Algorithm below produces polynomials q1 , . . . , qt ∈ F A and a reduced vector h ∈ Am w.r.t. F such that f −−→+ h and f = q1 f1 + · · · + qt ft + h with lm(f) = max{lm(lm(q1 )lm(f1 )), . . . , lm(lm(qt )lm(ft )), lm(h)}.

14.2 Reduction in Am

267

Division algorithm in Am INPUT: f, f1 , . . . , ft ∈ Am with fj 6= 0 (1 ≤ j ≤ t) OUTPUT: q1 , . . . , qt ∈ A , h ∈ Am with f = q1 f1 + · · · + qt ft + h, h reduced w.r.t. {f1 , . . . , ft } and lm(f) = max{lm(lm(q1 )lm(f1 )), . . . , lm(lm(qt )lm(ft )), lm(h)} INITIALIZATION: q1 := 0, q2 := 0, . . . , qt := 0, h := f WHILE h 6= 0 and there exists a j such that lm(fj ) divides lm(h) DO Calculate J := {j | lm(fj ) divides lm(h)} FOR j ∈ J DO Calculate αj ∈ Nn such that αj + exp(lm(fj )) = exp(lm(h)) IF the equation lc(h) = j∈J rj σ αj (lc(fj ))cαj ,fj is soluble, where cαj ,fj are defined as in Definition 14.2.1

P

THEN Calculate one solution (rj )j∈J h := h −

P

j∈J

r j xα j f j

FOR j ∈ J DO qj := qj + rj xαj ELSE Stop

Proof. We first note that the Division Algorithm is the iteration of the reduction process. If f is reduced with respect to F := {f 1 , . . . , f t }, then h = f , q1 = · · · = qt = 0 and lm(f ) = lm(h). If f is not reduced, then P F αj we make the first reduction, f −−→ h 1 , with f = j∈J1 rj1 x f j + h 1 , with J1 := {j | lm(f j ) divides lm(f )} and rj1 ∈ R. If h 1 is reduced with respect to F , then the cycle WHILE ends and we have that qj = rj1 xαj for j ∈ J1 and qj = 0 for j ∈ / J1 . Moreover, lm(f )  lm(h 1 ) and lm(f ) = lm(lm(qj )lm(f j )) for j ∈ J1 such that rj1 6= 0, hence, lm(f ) = max1≤j≤t {lm(lm(qj )lm(f j )), lm(h 1 )}. If h 1 is not reduced, we make the secP F ond reduction with respect to F , h 1 −−→ h 2 , with h 1 = j∈J2 rj2 xαj f j +h 2 , J2 := {j | lm(f j ) divides lm(h 1 )} and rj2 ∈ R. We have P P f = j∈J1 rj1 xαj f j + j∈J2 rj2 xαj f j + h 2 . If h 2 is reduced with respect to F the procedure ends and we get that qj = qj for j ∈ / J2 and qj = qj + rj2 xαj for j ∈ J2 . We know that lm(f )  lm(h 1 )  lm(h 2 ). This implies that the algorithm produces polynomials qj with monomials ordered according to the fixed monomial order, and again we have lm(f ) = max1≤j≤t {lm(lm(qj )lm(f j )), lm(h 2 )}. We can continue this way and the algorithm ends since Mon(Am ) is well ordered. t u

268

14 Gr¨ obner Bases of Modules

Example 14.2.5. We consider the Heisenberg algebra A := H1 (2) = σ(Q)hx, y, zi with deglex order and x > y > z in Mon(A) and the TOPREV order in Mon(A3 ) with e1  e2  e3 . Let f := x2 yze1 + y 2 ze2 + xze1 + z 2 e3 , f 1 := xze1 +xe3 +ye2 and f 2 := xye1 +ze2 +ze3 . Following the Division Algorithm we will compute q1 , q2 ∈ A and h ∈ A3 such that f = q1 f 1 + q2 f 2 + h, with lm(f ) = max{lm(lm(q1 )lm(f 1 )), lm(lm(q2 )lm(f 2 )), lm(h)}. We will represent the elements of Mon(A) by tα instead of xα . For j = 1, 2, we define αj := (αj1 , αj2 , αj3 ) ∈ N3 . Step 1 : we start with h := f , q1 := 0 and q2 := 0; since lm(f 1 ) | lm(h) and lm(f 2 ) | lm(h), we compute αj such that αj + exp(lm(fj )) = exp(lm(h)). ˆ lm(tα1 lm(f 1 )) = lm(h), i.e.,

lm(tα1 (xze1 )) = x2 yze1 , so lm(xα11 y α12 z α13 xz) = x2 yz, and hence α11 = 1; α12 = 1; α13 = 0. Thus, tα1 = xy. ˆ lm(tα2 lm(f 2 )) = lm(h), i.e., lm(tα2 (xye1 )) = x2 yze1 , so lm(xα21 y α22 z α23 xy) = x2 yz, and hence α21 = 1; α22 = 0; α23 = 1. Thus, tα2 = xz. Next, for j = 1, 2 we compute cαj ,fj : ˆ tα1 texp(lm(f 1 )) = (xy)(xz) = x(2xy)z = 2x2 yz. Thus, cα1 ,f 1 = 2. ˆ tα2 texp(lm(f 2 )) = (xz)(xy) = x( 21 xz + y)y = 12 x2 zy + xy 2 = x2 yz + xy 2 . So, cα2 ,f 2 = 1. We must solve the equation 1 = lc(h) = r1 σ α1 (lc(f 1 ))cα1 ,f 1 + r2 σ α2 (lc(f 2 ))cα2 ,f 2 = r1 σ α1 (1)2 + r2 σ α2 (1)1 = 2r1 + r2 , then r1 = 0 and r2 = 1. We make h := h − (r1 tα1 f 1 + r2 tα2 f 2 ), i.e., h : = h − (xz(xye1 + ze2 + ze3 )) = h − (xzxye1 + xz 2 e2 + xz 2 e3 ) = h − ((x2 yz + xy 2 )e1 + xz 2 e2 + xz 2 e3 ) = x2 yze1 + xze1 + y 2 ze2 + z 2 e3 − x2 yze1 − xy 2 e1 − xz 2 e2 − xz 2 e3 = −xy 2 e1 − xz 2 e2 − xz 2 e3 + y 2 ze2 + xze1 + z 2 e3 . In addition, we have q1 := q1 + r1 tα1 = 0 and q2 := q2 + r2 tα2 = xz.

14.2 Reduction in Am

269

Step 2 : h := −xy 2 e1 − xz 2 e2 − xz 2 e3 + y 2 ze2 + xze1 + z 2 e3 , so lm(h) = xy 2 e1 and lc(h) = −1; moreover, q1 = 0 and q2 = xz. Since lm(f 2 ) | lm(h), we compute α2 such that α2 + exp(lm(f 2 )) = exp(lm(h)): ˆ lm(tα2 lm(f 2 )) = lm(h), i.e., lm(tα2 (xye1 )) = xy 2 e1 , then we have lm(xα21 y α22 z α23 xy) = xy 2 , so α21 = 0; α22 = 1; α23 = 0. Thus, tα2 = y.

We compute cα2 ,f 2 : tα2 texp(lm(f 2 )) = y(xy) = 2xy 2 . Then, cα2 ,f 2 = 2. We solve the equation −1 = lc(h) = r2 σ α2 (lc(f 2 ))cα2 ,f 2 = r2 σ α2 (1)2 = 2r2 , thus, r2 = − 21 . We define h := h − r2 tα2 f 2 , i.e., 1 h : = h + y(xye1 + ze2 + ze3 ) 2 1 1 1 = h + yxye1 + yze2 + yze3 2 2 2 = −xy 2 e1 − xz 2 e2 − xz 2 e3 + y 2 ze2 + xze1 + z 2 e3 + xy 2 e1 1 1 + yze2 + yze3 2 2 1 1 2 = −xz e2 − xz 2 e3 + y 2 ze2 + xze1 + yze2 + yze3 + z 2 e3 . 2 2 We have also that q1 := 0 and q2 := q2 + r2 tα2 = xz − 12 y. Step 3 : h = −xz 2 e2 − xz 2 e3 + y 2 ze2 + xze1 + 12 yze2 + 12 yze3 + z 2 e3 , so lm(h) = xz 2 e2 and lc(h) = −1; moreover, q1 = 0 and q2 = xz − 12 y. Since lm(f 1 ) - lm(h) and lm(f 2 ) - lm(h), h is reduced with respect to {f 1 , f 2 }, so the algorithm stops. Thus, we get q1 , q2 ∈ A and h ∈ A3 is reduced such that f = q1 f 1 + q2 f 2 + h. In fact, 

 1 q1 f 1 + q2 f 2 + h = 0f 1 + xz − y f 2 + h 2 1 = (xz − y)(xye1 + ze2 + ze3 ) − xz 2 e2 − xz 2 e3 + y 2 ze2 + xze1 2 1 1 + yze2 + yze3 + z 2 e3 2 2 1 1 = x2 yze1 + xy 2 e1 − xy 2 e1 + xz 2 e2 − yze2 + xz 2 e3 − yze3 2 2 1 1 2 2 2 − xz e2 − xz e3 + y ze2 + xze1 + yze2 + yze3 + z 2 e3 2 2 = x2 yze1 + y 2 ze2 + xze1 + z 2 e3 = f ,

270

14 Gr¨ obner Bases of Modules

and max{lm(lm(q1 )lm(f 1 )), lm(lm(q2 )lm(f 2 )), lm(h)} = max{0, x2 yze1 , xz 2 e2 } = x2 yze1 = lm(f ). Example 14.2.6. In this additional example we illustrate the algorithm for the diffusion algebra. In this case, we will take K = Q, n = 2, c12 = −2, c21 = −1, deglex order on Mon(A) with D1  D2 , and TOPREV on Mon(A2 ), with e 1 > e 2 . Note that in this ring the endomorphisms σi are the identity. Let f 1 = (D1 D22 , D22 + x1 D1 D2 ), f 2 = (x1 D1 D2 + x1 D1 , D22 ), f 3 = (x1 D1 , D22 + x2 ), f 4 = (D2 , D12 ) and f = ((x1 x2 + 1)D12 D22 + x1 D12 , D1 D2 + x2 D22 ). We will divide f by f 1 , f 2 , f 3 and f 4 . Step 1. We start with h := f , q1 := 0, q2 := 0, q3 := 0, q4 := 0. Since lm(f j ) | lm(f ) for j = 1, 2, we compute α = (αj1 , αj2 ) ∈ N2 such that αj + exp(lm(f j )) = exp(lm(h)) and the corresponding value of σ αj (lc(f j ))cαj ,βj , where βj := exp(lm(f j )): (α11 , α12 ) + (1, 2) = (2, 2) ⇒ α11 = 1, α12 = 0, D1 D1 D22 = D12 D22 ⇒ cα1 ,β1 = 1, (α21 , α22 ) + (1, 1) = (2, 2) ⇒ α21 = 1, α22 = 1, D1 D2 D1 D2 = 2D12 D22 + x2 D12 D2 − x1 D1 D22 ⇒ cα2 ,β2 = 2. Now, we solve the equation lc(h) = x1 x2 + 1 = r1 + 2r2 x1 ⇒ r1 = 1, r2 = 21 x2 , and with the relations defining the algebra, we compute h = h − (r1 xα1 f 1 + r2 xα2 f 2 ) 1 x2 D1 D2 (x1 D1 D2 e 1 + D22 e 2 + x1 D1 e 1 ) 2 1 1 1 = − x2 D1 D23 e 2 − ( x1 x22 + x1 x2 )D12 D2 e 1 − x1 D12 D2 e 2 + x21 x2 D1 D22 e 1 − D23 e 2 2 2 2 1 1 − x1 x22 D12 e 1 + x21 x2 D1 D2 e 1 . 2 2 = h − D1 (D1 D22 e 1 + x1 D1 D2 e 2 + D22 e 2 ) −

We also compute q1 := D1 , q2 := 12 x2 D1 D2 , q3 := 0, q4 := 0. Step 2. lm(h) = D1 D23 e 2 , lc(h) = − 12 x2 . In this case, lm(fj ) | lm(f ) just for j = 3, and we must compute α = (α31 , α32 , α33 ) ∈ N3 such that α3 + exp(lm(f3 )) = exp(lm(h)): (α31 , α32 ) + (0, 2) = (1, 3) ⇒ α31 = 1, α32 = 1, D1 D2 D22 = D1 D23 ⇒ cα3 ,β3 = 1, and we have lc(h) = − 12 x2 = r3 . Thus,

14.2 Reduction in Am

271

1 h = h + x2 D 1 D 2 f 3 2 1 1 1 = − x1 x22 D12 D2 e 1 − x1 D12 D2 e 2 + x21 x2 D1 D22 e 1 − D23 e 2 + x22 D1 D2 e 2 , 2 2 2 and q1 := D1 , q2 := 12 x2 D1 D2 , q3 := − 12 x2 D1 D2 , q4 := 0. Step 3. Note that lm(h) = D12 D2 e 1 and lm(f j ) | lm(h) for j = 2. In this case, we have: (α21 , α22 ) + (1, 1) = (2, 1) ⇒ α21 = 1, α22 = 0, D1 D1 D2 = D12 D2 ⇒ cα2 ,β2 = 1. and r2 = − 12 x22 . Therefore, h =h+

1 2 x D1 f 2 2 2

= −x1 D12 D2 e 2 +

1 2 1 1 1 x x2 D1 D22 e 1 + x22 D1 D22 e 2 − D23 e 2 + x2 D12 e 1 + x22 D1 D2 e 2 , 2 1 2 2 2

and q1 := D1 , q2 := 12 x2 D1 D2 − 12 x22 D1 , q3 := − 12 x2 D1 D2 , q4 := 0. Step 4. lm(h) = D12 D2 e 2 and lm(f j ) | lm(h) just for j = 4. So, (α41 , α42 ) + (2, 0) = (2, 1) ⇒ α21 = 0, α22 = 1, D2 D12 = 4D12 D2 + 3x2 D12 − 4x1 D1 D2 − x1 x2 D1 + x21 D2 ⇒ cα4 ,β4 = 4. and r4 = 14 x1 . Therefore, 1 h = h + x1 D 2 f 4 4 1 2 1 1 3 = x1 x2 D1 D22 e 1 + x22 D1 D22 e 2 − D23 e 2 + x2 D12 e 1 + x1 x2 D12 e 2 2 2 2 4
1 2 1 2 1 3 2 + x2 − x1 D 1 D 2 e 2 − x1 x2 D 1 e 2 + x1 D 2 e 2 , 2 4 4 and q1 := D1 , q2 := 12 x2 D1 D2 − 12 x22 D1 , q3 := − 12 x2 D1 D2 , q4 := − 14 x1 D2 . Step 5. lm(h) = D1 D22 e 1 and lm(fj ) | lm(h) for j = 1, 2. So, (α11 , α12 ) + (1, 2) = (1, 2) ⇒ α11 = 0, α12 = 0, D1 D1 D22 = D12 D22 ⇒ cα1 ,β1 = 1, (α21 , α22 ) + (1, 1) = (1, 2) ⇒ α21 = 0, α22 = 1, D2 D1 D2 = 2D1 D22 + x2 D1 D2 − x1 D22 ⇒ cα2 ,β2 = 2. Now, we solve the equation

272

14 Gr¨ obner Bases of Modules

lc(h) = 21 x21 x2 = r1 + 2r2 x1 ⇒ r1 = 12 x21 x2 , r2 = 0, and with the relations defining A, we compute 1 h = h − x21 x2 f 1 2 1 2 1 3 = x2 D1 D22 e 2 − D23 e 2 + x2 D12 e 1 + x1 x2 D12 e 2 2 2 4
1 3 1 2 + − x1 x2 + x2 − x21 D1 D2 e 2 2 2 1 2 1 1 − x1 x2 D22 e 2 − x21 x2 D1 e 2 + x31 D2 e 2 . 2 4 4 Further, q1 := D1 , q2 := 12 x2 D1 D2 − 12 x22 D1 − 12 x21 x2 , q3 := − 12 x2 D1 D2 , q4 := − 14 x1 D2 . Step 6. lm(h) = D1 D22 e 2 and lm(f j ) | lm(h) for j = 3. We have, (α31 , α32 ) + (0, 2) = (1, 2) ⇒ α31 = 1, α32 = 0, D1 D22 = D1 D22 ⇒ cα3 ,β3 = 1, and r3 = 12 x22 . Hence, h =h −

1 2 x D1 f 3 2 2


1 1 3 1 1 = − D23 e 2 + (− x1 x22 + x2 )D12 e 1 + x1 x2 D12 e 2 + − x31 x2 + x22 − x21 D1 D2 e 2 2 2 4 2 2 1 1 1 1 − x1 x2 D22 e 2 − (x32 + x21 x2 )D1 e 2 + x31 D2 e 2 . 2 2 2 4

Moreover, q1 := D1 , q2 := 12 x2 D1 D2 − 12 x22 D1 − 12 x21 x2 , q3 := − 12 x2 D1 D2 + 12 x22 D1 , q4 := − 14 x1 D2 . Step 7. Finally, lm(h) = D23 e 2 and lm(f j ) | lm(h) for j = 3. We have, (α31 , α32 ) + (0, 2) = (0, 3) ⇒ α31 = 0, α32 = 1, D2 D22 = D23 ⇒ cα3 ,β3 = 1, and r3 = −1. Hence, h = h + D2 f 3 1 1 3 = (− x1 x22 + x2 )D12 e 1 + x1 x2 D12 e 2 + 2x1 D1 D2 e 1 2 2 4
1 3 1 2 1 2 + − x1 x2 + x2 − x1 D1 D2 e 2 − x1 x2 D22 e 2 + x2 D1 e 1 2 2 2 1 3 1 2 1 − (x2 + x1 x2 )D1 e 2 − x1 D2 e 1 + ( x31 + x2 )D2 e 2 . 2 2 4

14.3 Gr¨ obner Bases for Submodules of Am

273

Observe that h is reduced with respect to {f 1 , f 2 , f 3 , f 4 } and f = q1 f 1 + q2 f 2 + q3 f 3 + q4 f 4 + h, with q1 := D1 , q2 := 21 x2 D1 D2 − 12 x22 D1 − 12 x21 x2 , q3 := − 12 x2 D1 D2 + 12 x22 D1 − D2 , q4 := − 14 x1 D2 . We also see that max{lm(lm(q1 )lm(f 1 )), lm(lm(q2 )lm(f 2 )), lm(lm(q3 )lm(f 3 )), lm(lm(q4 )lm(f 4 ), lm(h)} = max{D12 D22 e1 , D12 D22 e1 , D1 D23 e2 , D12 D2 e2 , D12 e1 } = D12 D22 e1 = lm(f ).

14.3 Gr¨ obner Bases for Submodules of Am Our next purpose is to define Gr¨obner bases for submodules of Am . Definition 14.3.1. Let M 6= 0 be a submodule of Am and let G be a nonempty finite subset of nonzero vectors of M . We say that G is a Gr¨ obner basis for M if each element 0 6= f ∈ M is reducible w.r.t. G. We will say that {0} is a Gr¨obner basis for M = 0. Theorem 14.3.2. Let M 6= 0 be a submodule of Am and let G be a finite subset of nonzero vectors of M . Then the following conditions are equivalent: (i) G is a Gr¨ obner basis for M . (ii) For any vector f ∈ Am , G

f ∈ M if and only if f −−→+ 0. (iii) For any 0 6= f ∈ M there exist g1 , . . . , gt ∈ G such that lm(gj ) | lm(f), 1 ≤ j ≤ t, (i.e., ind(lm(gj )) = ind(lm(f)) and there exist αj ∈ Nn such that αj + exp(lm(gj )) = exp(lm(f))) and lc(f) ∈ hσ α1 (lc(g1 ))cα1 ,g1 , . . . , σ αt (lc(gt ))cαt ,gt }. (iv) For α ∈ Nn and 1 ≤ u ≤ m, let hα, M }u be the left ideal of R defined by hα, M }u := hlc(f) | f ∈ M, ind(lm(f)) = u, exp(lm(f)) = α}. Then, hα, M }u = Ju , with Ju := hσ β (lc(g))cβ,g | g ∈ G, ind(lm(g)) = u and β + exp(lm(g)) = α}. G

Proof. (i)⇒ (ii) Let f ∈ M . If f = 0, then by definition f −−→+ 0. If f 6= 0, G then there exists an h 1 ∈ Am such that f −−→ h 1 , with lm(f )  lm(h 1 ) and f − h 1 ∈ hGi ⊆ M , hence h 1 ∈ M ; if h 1 = 0, so we end. If h 1 6= 0, then we can repeat this reasoning for h 1 , and since Mon(Am ) is well ordered, we get G that f −−→+ 0.

274

14 Gr¨ obner Bases of Modules G

Conversely, if f −−→+ 0, then by Theorem 14.2.4, there exist g 1 , . . . , g t ∈ G and q1 , . . . , qt ∈ A such that f = q1 g 1 + · · · + qt g t , i.e., f ∈ M . (ii)⇒ (i) Evident. (i)⇔ (iii) This is a direct consequence of Definition 14.2.1. (iii)⇒ (iv) Since R is left noetherian, there exist r1 , . . . , rs ∈ R, f 1 , . . . , f l ∈ M such that hα, M }u = hr1 , . . . , rs }, ind(lm(f i )) = u and exp(lm(f i )) = α for each 1 ≤ i ≤ l, with hr1 , . . . , rs } ⊆ hlc(f 1 ), . . . , lc(f l )}. Then, hlc(f 1 ), . . . , lc(f l )} = hα, M }u . Let r ∈ hα, M }u . There exist a1 , . . . , al ∈ R such that r = a1 lc(f 1 ) + · · · + al lc(f l ); by (iii), for each i, 1 ≤ i ≤ l, there exist g 1i , . . . , g ti i ∈ G and bji ∈ R such that lc(f i ) = b1i σ α1i (lc(g 1i ))cα1i ,g 1i + · · · + bti i σ αti i (lc(g ti i ))cαti i ,g ti i , with u = ind(lm(f i )) = ind(lm(g ji )) and exp(lm(f i )) = αji + exp(lm(g ji )), thus hα, M }u ⊆ Ju . Conversely, if r ∈ Ju , then r = b1 σ β1 (lc(g 1 ))cβ1 ,g 1 + · · · + bt σ βt (lc(g t ))cβt ,g t , with bi ∈ R, βi ∈ Nn , g i ∈ G such that ind(lm(g i )) = u and βi +exp(lm(g i )) = α for any 1 ≤ i ≤ t. Note that xβi g i ∈ M , ind(lm(xβi g i )) = u, exp(lm(xβi g i )) = α, lc(xβi g i ) = σ βi (lc(g i ))cβi ,g i , for 1 ≤ i ≤ t, and r = b1 lc(xβ1 g 1 ) + · · · + bt lc(xβt g t ), i.e., r ∈ hα, M }u . (iv)⇒ (iii) Let 0 6= f ∈ M and let u = ind(lm(f )), α = exp(lm(f )), then lc(f ) ∈ hα, M }u ; by (iv) lc(f ) = b1 σ β1 (lc(g 1 ))cβ1 ,g 1 +· · ·+bt σ βt (lc(g t ))cβt ,g t , with bi ∈ R, βi ∈ Nn , g i ∈ G such that u = ind(lm(g i )) and βi + exp(lm(g i )) = α for any 1 ≤ i ≤ t. From this we conclude that lm(g j )|lm(f ), 1 ≤ j ≤ t. t u From this theorem we get the following consequences. Corollary 14.3.3. Let M 6= 0 be a submodule of Am . Then, (i) If G is a Gr¨ obner basis for M , then M = hGi. G (ii) Let G be a Gr¨ obner basis for M , if f ∈ M and f −−→+ h, with h reduced, then h = 0. (iii) Let G = {g1 , . . . , gt } be a set of nonzero polynomials of M with lc(gi ) ∈ R∗ for each 1 ≤ i ≤ t. Then, G is a Gr¨ obner basis of M if and only if given 0 6= r ∈ M there exists an i such that lm(gi ) divides lm(r). Proof. (i) This is a direct consequence of Theorem 14.3.2. G (ii) Let f ∈ M and f −−→+ h, with h reduced; since f − h ∈ hGi = M , we have h ∈ M ; if h 6= 0 then h can be reduced by G, but this is not possible since h is reduced. (iii) If G is a Gr¨ obner basis of M , then given 0 6= r ∈ M , r is reducible w.r.t. G, hence there exists an i such that lm(g i ) divides lm(r ). Conversely, if this condition holds for some i, then r is reducible w.r.t. G since the equation lc(r ) = r1 σ αi (lc(g i ))cαi ,g i , with αi + exp(lm(g i )) = exp(lm(r )), is soluble αi −1 with solution r1 = lc(r )c−1 . t u αi ,g i (σ (lc(g i ))) Note that the remainder of f ∈ Am with respect to a Gr¨obner basis is not unique. Moreover, changing the term order, a Gr¨obner basis could not be again a Gr¨obner basis. In fact, a counterexample was given in [236] for the trivial case when A = R[x1 , . . . , xn ] is the commutative polynomial ring.

14.4 Buchberger’s Algorithm for Modules

275

Of course, there exists a version of Corollary 13.3.4 for the module case. Corollary 14.3.4. Let G be a Gr¨ obner basis for a left A-module M . Given g ∈ G, if g is reducible w.r.t. G0 = G − {g}, then G0 is a Gr¨ obner basis for M. Proof. We will show that every f ∈ M is reducible w.r.t. G0 . Let f be a nonzero vector in M ; since G is a Gr¨ obner basis for M , f is reducible w.r.t. G and there exist elements g 1 , . . . , g t ∈ G satisfying the conditions (i), (ii) and (iii) in the Definition 14.2.1. If g 6= g i for each 1 ≤ i ≤ t, then we are finished. Suppose that g = g j for some j ∈ {1, . . . , t} and let βi = exp(g i ) for i 6= j, β = exp(g ), and αi , α ∈ Nn such that αi + βi = exp(f ) = α + β. Thus, lc(f ) = r1 σ α1 (lc(g 1 ))cα1 ,β1 + · · · + rj σ α (lc(g ))cα,β + · · · + rt σ αt (lc(g t ))cαt ,βt . On the other hand, since g is reducible P w.r.t. G0 , there exist g 01 , . . . , g 0s ∈ G0 0 s 0 such that lm(g l ) | lm(g ) and lc(g ) = l=1 rl0 σ αl (lc(g 0l ))cα0l ,βl0 , where βl0 = 0 0 n 0 0 exp(g l ), αl ∈ N and αl + βl = exp(g ) = β. So, lm(g0l ) | lm(f) for 1 ≤ i ≤ s; moreover, using the identities of Remark 1.1.9, we have that σ α (lc(g))cα,β = σ α (

s X

0

rl0 σ αl (lc(g0l ))cα0l ,βl0 )cα,β

l=1 0

= σ α (r10 )σ α σ α1 (lc(g01 ))σ α (cα01 ,β10 )cα,β + · · · 0

+ σ α (rs0 )σ α σ αs (lc(g0s ))σ α (cα0s ,βs0 )cα,β 0

α 0 0 = σ α (r10 )cα,α01 σ α+α1 (lc(g01 ))c−1 α,α0 σ (cα1 ,β1 )cα,β + · · · 1

0

α + σ α (rs0 )cα,α0s σ α+αs (lc(g0s ))c−1 α,α0s σ (cα0s ,βs0 )cα,β 0

= σ α (r10 )cα,α01 σ α+α1 (lc(g01 ))cα+α01 ,β10 + · · · 0

+ σ α (rs0 )cα,α0s σ α+αs (lc(g0s ))cα+α0s ,βs0 . Since α + β = exp(f), we have α + αl0 + βl0 = exp(f). Further, if there exists a gk ∈ {g1 , . . . , gt } such that gk = g0l for some l ∈ {1, . . . , s}, then βl0 = βk and α + αl0 = αk ; therefore, in the representation of lc(f) the term (rk + rj σ α (rl0 )cα,α0l )σ αk (lc(gk ))cαk ,βk appears. Therefore, f is reducible w.r.t. G0 , and hence G0 is a Gr¨ obner basis for M . t u

14.4 Buchberger’s Algorithm for Modules Recall that we are assuming that A is a bijective skew P BW extension. We will prove in the present section that every submodule M of Am has a Gr¨obner basis, and also, we will construct Buchberger’s algorithm for computing such bases. The results obtained here improve those of [192] and [191] and generalize the results obtained in Section 13.4 for left ideals.

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14 Gr¨ obner Bases of Modules

We start by fixing some notation and proving a preliminary general result. Definition 14.4.1. Let F := {g1 , . . . , gs } ⊆ Am be such that the least common multiple of {lm(g1 ), . . . , lm(gs )}, denoted by XF , is nonzero. Let θ ∈ Nn , βi := exp(lm(gi )) and γi ∈ Nn such that γi + βi = exp(XF ), 1 ≤ i ≤ s. BF,θ will denote a finite set of generators of SF,θ := SyzR [σ γ1 +θ (lc(g1 ))cγ1 +θ,β1 · · · σ γs +θ (lc(gs ))cγs +θ,βs )]. For θ = 0 := (0, . . . , 0), SF,θ will be denoted by SF and BF,θ by BF . Lemma 14.4.2. Let g1 , . . . , gs ∈ Am , c1 , . . . , cs ∈ R − {0} and α1 , . . . , αs ∈ Nn be such that lm(xα1 lm(g1 )) = · · · = lm(xαs lm(gs )) =: Xδ . Ps If lm( i=1 ci xαi gi ) ≺ Xδ , then there exist r1 , . . . , rk ∈ R and l1 , . . . , ls ∈ A such that s X

αi

c i x gi =

i=1

k X

rj x

δ−exp(XF )

j=1

X s

γi



bji x gi

i=1

+

s X

l i gi ,

i=1

where XF is the least common multiple of lm(g1 ), . . . , lm(gs ), γi ∈ Nn is such that γi + exp(gi ) = exp(XF ), 1 ≤ i ≤ s, and BF := {b1 , . . . , bk } := {(b11 , . . . , b1s ), . . . , (bk1 , . . . , bks )}. Ps Moreover, lm(xδ−exp(XF ) i=1 bji xγi gi ) ≺ Xδ for every 1 ≤ j ≤ k, and lm(li gi ) ≺ Xδ for every 1 ≤ i ≤ s. Proof.. Let βi := exp(lm(g i )) for 1 ≤ i ≤ s; since X δ = lm(xαi lm(g i )), we have lm(g i ) | X δ and hence X F | X δ , so there exists a θ ∈ Nn such that exp(X F ) + θ = δ, with δ = exp(Xδ ). On the other hand, γi +P βi = exp(X F ) and αi + βi = δ, so P αi = γi + θ for every 1 ≤ i ≤ s. Now, s s αi αi lm( P c x g ) ≺ X implies that δ i i=1 i i=1 ci σ (lc(g i ))cαi ,βi = 0. So we s θ+γi (lc(g i ))cθ+γi ,βi = 0. This implies that (c1 , . . . , cs ) ∈ SF,θ ; have i=1 ci σ from Remark 13.4.2 we know that there exists (c01 , . . . , c0s ) ∈ SF such that ci = σ θ (c0i )cθ,γi . Then, s X i=1

c i xαi g i =

s X

σ θ (c0i )cθ,γi xαi g i .

i=1

Now, xθ c0i xγi = (σ θ (c0i )xθ + pc0i ,θ )xγi = σ θ (c0i )xθ xγi + pc0i ,θ xγi = σ θ (c0i )cθ,γi xθ+γi + σ θ (c0i )pθ,γi + pc0i ,θ xγi = σ θ (c0i )cθ,γi xθ+γi + p0i

14.4 Buchberger’s Algorithm for Modules

277

where p0i := σ θ (ci0 )pθ,γi + pc0i ,θ xγi ; note that p0i = 0 or lm(p0i ) ≺ xθ+γi for each i. Thus, σ θ (c0i )cθ,γi xθ+γi = xθ c0i xγi + pi , with pi = 0 or lm(pi ) ≺ xθ+γi . Hence, s X

c i xαi g i =

s X

i=1

σ θ (ci0 )cθ,γi xαi g i

i=1

=

s X

(xθ ci0 xγi + pi )g i

i=1

=

s X

xθ c0i xγi g i +

i=1

s X

pi g i ,

i=1

with pi g i = 0 or lm(pi g i ) ≺ xθ+γi +βi = xδ . On the other hand, since (c01 , . . . , cs0 ) ∈ SF , then there exist r10 , . . . , rk0 ∈ R such that (c10 , . . . , c0s ) = r10 b 1 + · · · + rk0 b k = r10 (b11 , . . . , b1s ) + · · · + rk0 (bk1 , . . . , bks ), thus ci0 = Pk 0 j=1 rj bji . Using this, we have s X

xθ ci0 xγi g i

=

i=1

s X

x

k X

θ

j=1

i=1

=


rj0 bji xγi g i

s X k X


xθ rj0 bji xγi g i

i=1 j=1

=

k s X X


(σ θ (rj0 )xθ + prj0 ,θ )bji xγi g i

i=1 j=1

=

s X k X

σ θ (rj0 )xθ bji xγi g i +

=

σ θ (rj0 )xθ bji xγi g i +

=

σ θ (rj0 )xθ

where qi :=

Pk

j=1

s X

bji xγi g i +

i=1

j=1




k s X X

prj0 ,θ bji xγi g i

i=1 j=1

j=1 i=1 k X

prj0 ,θ bji xγi g i

j=1

i=1 j=1 s k X X

k X

s X

qi g i ,

i=1

prj0 ,θ bji xγi = 0 or lm(qi ) ≺ xθ+γi . Therefore,

s X i=1

c i xαi g i =

k X j=1

rj x θ

s X i=1

bji xγi g i +

s X

li g i ,

i=1

with li := pi + qi for 1 ≤ i ≤ s and rj := σ θ (rj0 ) for 1 ≤ j ≤ k. Finally, it Ps Ps γi γi is easy to see that lm(xθ i=1 bji x g i )) ≺ Xδ since lm( i=1 bji x g i ) ≺ γi lm(x lm(gi )), and lm(li g i ) = lm(pi g i + qi g i ) ≺ Xδ . t u

278

14 Gr¨ obner Bases of Modules

Theorem 14.4.3. Let M 6= 0 be a submodule of Am and let G be a finite subset of nonzero generators of M . Then the following conditions are equivalent: (i) G is a Gr¨ obner basis of M . (ii) For all F := {g1 , . . . , gs } ⊆ G, with XF 6= 0, and for any (b1 , . . . , bs ) ∈ BF , Ps G γi −→+ 0. i=1 bi x gi − Proof. (i) ⇒ (ii) We observe that f :=

Ps

i=1 bi x

γi

g i ∈ M , so by Theorem

G

14.3.2 f −−→+ 0. (ii) ⇒ (i) Let 0 6= f ∈ M . We will prove that condition (iii) of Theorem 14.3.2 holds. Let G := {g1 , . . . , g t }. Then there exist h1 , . . . , ht ∈ A such that f = h1 g 1 + · · · + ht gt , and we can choose {hi }ti=1 such that Xδ := max{lm(lm(hi )lm(gi ))}ti=1 is minimal. Let lm(hi ) := xαi , ci := lc(hi ), exp(lm(gi )) = βi for 1 ≤ i ≤ t and F := {gi ∈ G | lm(lm(hi )lm(gi )) = Xδ }; renumbering the elements of G we can assume that F = {g1 , . . . , g s }. We will consider two possible cases. Case 1 : lm(f) = Xδ . Then lm(gi ) | lm(f) for 1 ≤ i ≤ s and lc(f) = c1 σ α1 (lc(g1 ))cα1 ,β1 + · · · + cs σ αs (lc(gs ))cαs ,βs , i.e., the condition (iii) of Theorem 14.3.2 holds. Case 2 : lm(f) ≺ Xδ . We will prove that this produces a contradiction. To begin, note that f can be written as f=

s X

c i x α i gi +

i=1

s X

(hi − ci xαi )gi +

i=1

t X

hi gi ;

(14.4.1)

i=s+1

Ps Pt αi we see i=1 (hi − ci x )gi ) ≺ Xδ and lm( i=s+1 hi gi ) ≺ Xδ , thus Ps that lm( lm( i=1 ci xαi gi ) ≺ Xδ ; by Lemma 14.4.2, we have s X

c i x α i gi =

i=1

k X

rj xδ−exp(XF )

j=1

s X

s
X bji xγi gi + li gi ,

i=1

(14.4.2)

i=1

Ps G where lm(li gi ) ≺ Xδ for 1 ≤ i ≤ s. By the hypothesis, i=1 bji xγi gi −−→+ 0, whence, by Theorem 14.2.4, there exist q1 , . . . , qt ∈ A such that s X

γi

bji x gi =

i=1

lm(

s X

t X

qi gi , with

i=1

bji xγi gi ) = max{lm(lm(qi )lm(gi ))}ti=1 ,

i=1

but (bj1 , . . . , bjs ) ∈ SF , so lm(

Ps

i=1 bji x

γi

gi ) ≺ XF and hence

14.4 Buchberger’s Algorithm for Modules

279

lm(lm(qi )lm(gi )) ≺ XF for every 1 ≤ i ≤ t. Thus, Pk

j=1 rj x

with qei :=


Pk γi δ−exp(XF ) i=1 bji x gi = j=1 rj x Pt Pk P t δ−exp(XF ) qi gi = i=1 qei gi , i=1 j=1 rj x

δ−exp(XF )

Pk

j=1 rj x

Ps

δ−exp(XF )

Pt

i=1 qi gi




=

qi and lm(e qi gi ) ≺ Xδ . Substituting

s X

c i x α i gi =

i=1

t X

qei gi

i=1

into equation (14.4.1), we obtain f=

t X i=1

qei gi +

s X

(hi − ci xαi )gi +

i=1

s X

l i gi +

i=1

t X

hi gi ,

i=s+1

and so we have expressed f as a combination of the vectors g1 , . . . , gt , where every term has leading monomial ≺ Xδ . This contradicts the minimality of Xδ and we have finished the proof. t u Corollary 14.4.4. Let F = {f1 , . . . , fs } be a set of nonzero vectors of Am . The algorithm on the following page produces a Gr¨ obner basis for the submodule hf1 , . . . , fs i (P (X) denotes the set of subsets of the set X). From Theorem 3.1.5 and the previous corollary we get the following direct conclusion. Corollary 14.4.5. Every submodule of Am has a Gr¨ obner basis. Now we will illustrate Buchberger’s algorithm. Example 14.4.6. We will consider the multiplicative analogue of the Weyl algebra   1 A := O3 (λ21 , λ31 , λ32 ) = O3 2, , 3 = σ(Q[x1 ])hx2 , x3 i, 2 hence we have the relations x2 x1 = λ21 x1 x2 = 2x1 x2 , so σ2 (x1 ) = 2x1 , 1 1 x1 x3 , so σ3 (x1 ) = x1 , 2 2 x3 x2 = λ32 x2 x3 = 3x2 x3 , so c2,3 = 3,

x3 x1 = λ31 x1 x3 =

and for r ∈ Q, σ2 (r) = r = σ3 (r). We choose in Mon(A) the deglex order with x2 > x3 and in Mon(A2 ) the TOPREV order with e1 > e2 .

280

14 Gr¨ obner Bases of Modules

Buchberger’s algorithm for modules over bijective skew P BW extensions INPUT: F := {f1 , . . . , fs } ⊆ Am , fi 6= 0 , 1 ≤ i ≤ s OUTPUT: G = {g1 , . . . , gt } a Gr¨ obner basis for hF i INITIALIZATION: G := ∅, G0 := F WHILE G0 6= G DO D := P (G0 ) − P (G) G := G0 FOR each S := {gi1 , . . . , gik } ∈ D, with XS 6= 0 , DO Compute BS FOR each b = (b1 , . . . , bk ) ∈ BS DO G0

Reduce kj=1 bj xγj gij −−−→+ r, with r reduced with respect to G0 and γj defined as in Definition 14.4.1

P

IF r 6= 0 THEN G0 := G0 ∪ {r}

Let f 1 = x21 x22 e1 + x2 x3 e2 , lm(f 1 ) = x22 e1 and f 2 = 2x1 x2 x3 e1 + x2 e2 , lm(f 2 ) = x2 x3 e1 . We will construct a Gr¨ obner basis for the module M := hf 1 , f 2 i. Step 1. We start with G := ∅, G0 := {f 1 , f 2 }. Since G0 6= G, we make D := P(G0 ) − P(G), i.e., D := {S1 , S2 , S1,2 }, where S1 := {f 1 }, S2 := {f 2 }, S1,2 := {f 1 , f 2 }. We also make G := G0 , and for every S ∈ D such that X S = 6 0 we compute BS :  For S1 we have SyzQ[x1 ] [σ γ1 (lc(f 1 ))cγ1 ,β1 ], where β1 = exp(lm(f 1 )) = (2, 0); X S1 = l.c.m.{lm(f 1 )} = lm(f 1 ) = x22 e1 ; exp(X S1 ) = (2, 0); γ1 = exp(X S1 ) − β1 = (0, 0); xγ1 xβ1 = x22 , so cγ1 ,β1 = 1. Then, σ γ1 (lc(f 1 ))cγ1 ,β1 = σ γ1 (x21 )1 = σ20 σ30 (x21 ) = x21 . Thus, SyzQ[x1 ] [x21 ] = {0} and BS1 = {0}, i.e., we do not add any vector to G0 .  For S2 we have an identical situation.  For S1,2 we compute SyzQ[x1 ] [σ γ1 (lc(f 1 ))cγ1 ,β1 σ γ2 (lc(f 2 ))cγ2 ,β2 ], where β1 = exp(lm(f 1 )) = (2, 0) and β2 = exp(lm(f 2 )) = (1, 1); X S1,2 = l.c.m.{lm(f 1 ), lm(f 2 )} = l.c.m.(x22 e1 , x2 x3 e1 ) = x22 x3 e1 ;

14.4 Buchberger’s Algorithm for Modules

281

exp(X S1,2 ) = (2, 1); γ1 = exp(X S1,2 ) − β1 = (0, 1); γ2 = exp(X S1,2 ) − β2 = (1, 0); xγ1 xβ1 = x3 x22 = 3x2 x3 x2 = 9x22 x3 , so cγ1 ,β1 = 9. In a similar way xγ2 xβ2 = x22 x3 , i.e., cγ2 ,β2 = 1. Then, σ γ1 (lc(f 1 ))cγ1 ,β1 = σ γ1 (x21 )9 = σ20 σ3 (x21 )9 = (σ3 (x1 )σ3 (x1 ))9 =

9 2 x 4 1

and σ γ2 (lc(f 2 ))cγ2 ,β2 = σ γ2 (2x1 )1 = σ2 σ30 (2x1 ) = σ2 (2x1 ) = 4x1 . Hence SyzQ[x1 ] [ 94 x21 4x1 ] = {(b1 , b2 ) ∈ Q[x1 ]2 | b1 ( 94 x21 ) + b2 (4x1 ) = 0} and BS1,2 = {(4, − 94 x1 )}. From this we get 9 9 4xγ1 f 1 − x1 xγ2 f 2 = 4x3 (x21 x22 e1 + x2 x3 e2 ) − x1 x2 (2x1 x2 x3 e1 + x2 e2 ) 4 4 9 9 2 2 = 4x3 x1 x2 e1 + 4x3 x2 x3 e2 − x1 x2 2x1 x2 x3 e1 − x1 x22 e2 4 4 9 2 2 2 2 2 2 = 9x1 x2 x3 e1 + 12x2 x3 e2 − 9x1 x2 x3 e1 − x1 x2 e2 4 9 2 2 = 12x2 x3 e2 − x1 x2 e2 := f 3 , 4 so lm(f 3 ) = x2 x23 e2 . We observe that f 3 is reduced with respect to G0 . We make G0 := G0 ∪ {f 3 }, i.e., G0 = {f 1 , f 2 , f 3 }. Step 2. Since G = {f 1 , f 2 } 6= G0 = {f 1 , f 2 , f 3 }, we make D := P(G0 ) − P(G), i.e., D := {S3 , S1,3 , S2,3 , S1,2,3 }, where S3 := {f 3 }, S1,3 := {f 1 , f 3 }, S2,3 := {f 2 , f 3 }, S1,2,3 := {f 1 , f 2 , f 3 }. We make G := G0 , and for every S ∈ D such that X S = 6 0 we must compute BS . Since X S1,3 = X S2,3 = X S1,2,3 = 0, we only need to consider S3 . We have to compute SyzQ[x1 ] [σ γ3 (lc(f 3 ))cγ3 ,β3 ], where β3 = exp(lm(f 3 )) = (1, 2); X S3 = l.c.m.{lm(f 3 )} = lm(f 3 ) = x2 x23 e2 ; exp(X S3 ) = (1, 2); γ3 = exp(X S3 ) − β3 = (0, 0); xγ3 xβ3 = x2 x23 , so cγ3 ,β3 = 1. Hence σ γ3 (lc(f 3 ))cγ3 ,β3 = σ γ3 (12)1 = σ20 σ30 (12) = 12, and SyzQ[x1 ] [12] = {0}, i.e., BS3 = {0}. This means that we not add any vector to G0 and hence G = {f 1 , f 2 , f 3 } is a Gr¨obner basis for M . Example 14.4.7. In this example we consider the additive analogue of the Weyl algebra An (q1 , . . . , qn ) (see Example 1.1.5, (iv)). We will take n = 2, K = Q, q1 = 21 , q2 = 13 so A := A2 ( 12 , 13 ). On Mon(A) we take the order deglex with y1  y2 and in Mon(A2 ) the TOPREV order with e 1 > e 2 .

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14 Gr¨ obner Bases of Modules

Let f 1 = x1 y12 e 1 + x2 y2 e 2 and f 2 = x2 y22 e 1 + x1 y1 e 2 . We will construct a Gr¨obner basis for the module M := hf 1 , f 2 i. Step 1. We start with G := ∅, G0 := {f 1 , f 2 }. Since G0 6= G, we make D := P (G0 ) − P (G), i.e., D := {S1 , S2 , S1,2 }, where S1 := {f 1 }, S2 := {f 2 }, S1,2 := {f 1 , f 2 }. We also make G := G0 , and for every S ∈ D such that X S 6= 0 we compute BS :  For S1 we have SyzQ[x1 ,x2 ] [σ γ1 (lc(f 1 ))cγ1 ,β1 ], where β1 = exp(lm(f 1 )) = (2, 0), γ1 = (0, 0) and cγ1 ,β1 = 1; thus BS1 = {0} and we do not add any vector to G0 .  For S2 we have an identical situation.  For S1,2 we compute SyzQ[x1 ,x2 ] [σ γ1 (lc(f 1 ))cγ1 ,β1 σ γ1 (lc(f 2 ))cγ2 ,β2 ], where β1 = exp(lm(f 1 )) = (2, 0), β2 = exp(lm(f 2 )) = (0, 2); X S1,2 = lcm{lm(f 1 ), lm(f 2 )} = y12 y22 e 1 ; γ1 = (0, 2); y γ1 y β1 = cγ1 ,β1 = 1 and σ γ1 (lc(f 1 )) = x1 ; analogously, γ2 = (2, 0), cγ2 ,β2 σ γ2 (lc(f 2 )) = x2 . Hence, SyzQ[x1 ,x2 ] [x1 x2 ] = h(x2 , −x1 )i and {(x2 , −x1 )}. From this we get

we have y12 y22 , so = 1 and BS1,2 =

x2 y γ1 f 1 − x1 y12 f 2 = x2 y22 (x1 y12 e 1 + x2 y2 e 2 ) − x1 y12 (x2 y22 e 1 + x1 y1 e 2 ) = x1 x2 y22 y12 y22 e 1 + x2 y22 x2 y2 e 2 − x1 x2 y12 y12 y2 e 1 − x1 y12 x1 y1 e 2 1 1 3 4 = − x21 y13 e 2 + x22 y23 e 2 − x1 y12 e 2 + x2 y22 e 2 := f 3 , 4 9 2 3 We observe that f3 is reduced with respect to G0 . We make G0 := {f 1 , f 2 , f 3 }. Step 2. Since G = {f 1 , f 2 } = 6 G0 = {f 1 , f 2 , f 3 }, we make D := P(G0 ) − P(G), i.e., D := {S3 , S1,3 , S2,3 , S1,2,3 }, where S1 := {f 1 }, S1,3 := {f 1 , f 3 }, S2,3 := {f 2 , f 3 }, S1,2,3 := {f 1 , f 2 , f 3 }. We make G := G0 , and for every S ∈ D such that X S 6= 0 we must compute BS . Since X S1,3 = X S2,3 = X S1,2,3 = 0, we only need to consider S3 .  We have to compute SyzQ[x1 ,x2 ] [σ γ3 (lc(f 3 ))cγ3 ,β3 ], where β3 = exp(lm(f 3 )) = (0, 3); X S3 = lcm{lm(f 3 )} = lm(f 3 ) = y13 e2 ; exp(X S3 ) = (0, 3); γ3 = exp(X S3 ) − β3 = (0, 0); xγ3 xβ3 = y13 , so cγ3 ,β3 = 1. Hence σ γ3 (lc(f 3 ))cγ3 ,β3 = σ γ3 (−x21 )1 = σ20 σ30 (−x21 ) = −x21 , and SyzQ[x1 ,x2 ] [−x21 ] = {0}, i.e., BS3 = {0}. This means that we not add any vector to G0 and hence G = {f 1 , f 2 , f 3 } is a Gr¨obner basis for M . Finally, we get the following direct consequence of Theorem 14.4.3. Corollary 14.4.8. Let G = {g1 , . . . , gt } be a generator set of a module M . obner basis for M . If ind(gi ) = 6 ind(gj ) for every i 6= j, then G is a Gr¨ Proof. If we have that ind(g i ) 6= ind(g j ) for every i 6= j, then X F = 0 for each subset F of G. In this way, the condition (ii) in Theorem 14.4.3 trivially holds; thus G = {g 1 , . . . , g t } is a Gr¨ obner basis for M . t u

14.5 Right Skew P BW Extensions and Right Gr¨ obner Bases

283

14.5 Right Skew P BW Extensions and Right Gr¨ obner Bases Our definition of a skew P BW extension A of a ring R depends on the assumption that A is a free left R-module over the standard monomials Mon(A) (Definition 1.1.1). However, if A is bijective, then A is a right free R-module with basis Mon(A) (Proposition 1.1.11). Definition 14.5.1. Let A and R be rings with R ⊆ A. Let x1 , . . . , xn be finitely many elements of A. We say that A is a ring of right polynomial type over R w.r.t. {x1 , . . . , xn } if A is a right R-free module with basis αn n 1 Mon(A) := Mon{x1 , . . . , xn } := {xα = xα 1 · · · xn |α = (α1 , . . . , αn ) ∈ N }.

Moreover, we say that A is a ring of polynomial type over R w.r.t. {x1 , . . . , xn } if Mon(A) is a basis for A as a left and as a right R-module. Thus, if A is a ring of polynomial type w.r.t. {x1 , . . . , xn }, every element f ∈ A has a standard representation, both left and right, in the following way: Ps Pt f = i=1 ci xαi = j=1 xβj dj , for some ci , dj ∈ R and xαi , xβj ∈ Mon(A), 1 ≤ i ≤ s, 1 ≤ j ≤ t. Given a monomial order on Mon(A) (e.g., deglex order), we can rewrite f with the property that xα1  · · ·  xαs and xβ1  · · ·  xβt . Thus, the left and right leading monomials of f are, respectively, lml (f ) := xα1 and lmr (f ) := xβ1 . Since the usual definition of skew P BW extensions considers left representations (see Definition 1.1.1), we could call them “left skew P BW extensions”. Thus, using the right polynomial ring notion, we can establish the definition of “right skew P BW extension”, as follows. Definition 14.5.2. Let R and A be rings, we say that A is a right skew P BW extension of R if the following conditions hold: (i) R ⊆ A. (ii) There exists a finite set of elements x1 , . . . , xn ∈ A such that A is a right R-free module with basis αn n 1 Mon(A) := {xα = xα 1 · · · xn |α = (α1 , . . . , αn ) ∈ N }.

(iii) For every 1 ≤ i ≤ n and r ∈ R − {0} there exists a di,r ∈ R − {0} such that rxi − xi di,r ∈ R. (14.5.1) (iv) For every 1 ≤ i, j ≤ n there exists a di,j ∈ R − {0} such that xj xi − xi xj di,j ∈ R + x1 R + · · · + xn R. Under these conditions we will write A = σ r (R)hx1 , . . . , xn i. The right version of Theorem 1.1.8 is as follows.

(14.5.2)

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14 Gr¨ obner Bases of Modules

Theorem 14.5.3. Let A be a ring of right polynomial type over R w.r.t. {x1 , . . . , xn }. A is a right skew P BW extension of R if and only if the following conditions hold: (a) For every xα ∈ Mon(A) and every 0 6= r ∈ R there exist unique elements rα ∈ R − {0} and qα,r ∈ A such that rxα = xα rα + qα,r ,

(14.5.3)

where qα,r = 0 or deg(qα,r ) < |α| if qα,r 6= 0. Moreover, if r is right invertible, then rα is right invertible. (b) For every xα , xβ ∈ Mon(A) there exist unique elements dα,β ∈ R and qα,β ∈ A such that xα xβ = xα+β dα,β + qα,β , (14.5.4) where dα,β is right invertible, qα,β = 0 or deg(qα,β ) < |α + β| if qα,β 6= 0. Remark 14.5.4. (i) All properties studied in Section 1.1 can be established for right skew P BW extensions. For example, the elements di,j in (14.5.2) are right invertible for i < j: indeed, let i < j, by (14.5.2) there exist dj,i , di,j ∈ R such that xi xj − xj xi dj,i ∈ R + x1 R + · · · + xn R and xj xi − xi xj di,j ∈ R + x1 R + · · · + xn R. So, xi xj − xi xj di,j dj,i ∈ R + x1 R + · · · + xn R and since Mon(A) is an R-basis for AR , 1 = di,j dj,i , i.e., for every 1 ≤ i < j ≤ n, di,j has a right inverse and dj,i has a left inverse. (ii) In a similar way as quasi-commutative and bijective left skew P BW extensions were defined, it is also possible to define the same notions in the right case. Hence, if A is a right skew P BW extension of a ring R, then A is bijective if the endomorphisms induced by the elements di,r in (14.5.1) are automorphism of R, and the coefficients di,j in (14.5.2) are invertible (compare with Definition 1.1.4). (iii) If A is a bijective right skew P BW extension of a right noetherian ring R, then adapting the proof of Theorem 3.1.5 we get that A is right noetherian. Lemma 14.5.5. Let A be a ring of polynomial type over R w.r.t. {x1 , . . . , xn }. If A is a left or right skew P BW extension of R, then lml (f ) = lmr (f ) for every f ∈ A. Proof. Suppose that A is a left skew P BW extension of R. If f = 0 there is nothing to prove. If f 6= 0 with lmr (f ) = xβ1 , then f has a right representation in the form f = xβ1 d1 + · · · + xβt dt , with xβ1  · · ·  xβt and 0 6= di ∈ R, for 1 ≤ i ≤ t. From Theorem 1.1.8 we obtain that f = σ β1 (d1 )xβ1 + pβ1 ,d1 + · · · + σ βt (dt )xβt + pβt ,dt , where pβi ,di = 0 or deg(pβi ,di ) < |βi | if pβ1 ,d1 6= 0. From this we get that lml (f ) = xβ1 . A similar proof holds if we suppose that A is a right skew P BW extension of R. t u The following theorem allows us to establish the theory of Gr¨obner bases for right ideals and right modules of bijective left skew P BW extensions.

14.5 Right Skew P BW Extensions and Right Gr¨ obner Bases

285

Theorem 14.5.6. Let A and R be rings such that R ⊆ A and let x1 , . . . , xn be nonzero elements in A. Suppose that Mon(A) is ordered by some monomial order. Consider the following statements: (i) A is a ring of right polynomial type over R w.r.t. {x1 , . . . , xn } and a left skew P BW extension of R. (ii) A is a ring of left polynomial type over R w.r.t. {x1 , . . . , xn } and a right skew P BW extension of R. (iii) A is a bijective left skew P BW extension of R. (iv) A is a bijective right skew P BW extension of R. Then, (i) ⇔ (ii), (iii) ⇔ (iv) and (iii) ⇒ (i). Further, if in (i) we add to the first condition that A is also a right skew P BW extension of R, then (i) ⇒ (iii). Proof. (i) ⇔ (ii) Since A is a left skew P BW extension of R, Mon(A) is a basis for R A, i.e., A is a ring of left polynomial type over R w.r.t. x1 , . . . , xn . Now, since A is a ring of right polynomial type over R w.r.t. x1 , . . . , xn , A satisfies (ii) in Definition 14.5.2. On the other hand, given 0 6= r ∈ R and 1 ≤ i ≤ n, we have that rxi = xi di,r + pi,r for some 0 6= di,r ∈ R and pi,r ∈ R (see Lemma 14.5.5). Similarly, for 1 ≤ i, j ≤ n, we have that xj xi = ci,j xi xj + pi,j = xi xj di,j + qi,j for some 0 6= di,j ∈ R and qi,j ∈ R + x1 R + · · · + xn R. The proof of (ii) ⇒ (i) is analogous. (iii) ⇔ (iv) From Proposition 1.1.11 we have that A is a right free R-module with basis Mon(A). It only remains to show that there exist elements di,r and di,j in R satisfying (iii) and (iv) in Definition 14.5.2, and that with these elements A turns out to be bijective. Since A is bijective, each endomorphism σi in Proposition 1.1.3 is an automorphism, thus, given r ∈ R and 1 ≤ i ≤ n, rxi − xi σi−1 (r) ∈ R, so it is enough to take di,r := σi−1 (r). We define σi0 : R → R as σi0 := σi−1 . Thus, (iii) in Definition 14.5.2 holds and, of course, each σi0 is bijective. For 1 ≤ i, j ≤ n, we have that xj xi = ci,j xi xj + pi,j , where ci,j is invertible and pi,j ∈ R + Rx1 + · · · + Rxn . Using again Lemma 14.5.5, as in the first part of the proof, xj xi = xi xj di,j + qi,j for some di,j 6= 0 and qi,j ∈ R + x1 R + · · · + xn R. So, (iv) in Definition 14.5.2 holds. Moreover, observe that xi xj di,j = xi [σj (di,j )xj + r] = xi σj (di,j )xj + xi r = [σi (σj (di,j ))xi + s]xj + xi r = σi (σj (di,j ))xi xj + sxj + σi (r)xi + u, with r, s, u ∈ R, whence, ci,j = σi (σj (di,j )), i.e., di,j = σj−1 (σi−1 (ci,j )) is invertible. We have proved that A is a bijective right skew P BW extension of R. The reverse implication can be proved similarly. The implication (iii) ⇒ (i) is immediate. Finally, if A is a left and right skew P BW extension of R, then the endomorphism σi is bijective for each 1 ≤ i ≤ n: in fact, since for r ∈ R we have

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14 Gr¨ obner Bases of Modules

0 0 rxi = xi σi0 (r) + qi,r = σi (σi0 (r))xi + qi,r for certain qi,r ∈ R. Uniqueness in the standard representation implies that r = σi (σi0 (r)); i.e., σi σi0 = iR and hence σi is surjective, but according to Proposition 1.1.3, σi is injective. So, σi is bijective and σi0 = σi−1 . Now, as above, di,j = σj−1 (σi−1 (ci,j )) and di,j is right invertible (see Remark 14.5.4), so ci,j is right invertible, i.e., ci,j is invertible for 1 ≤ i, j ≤ n. t u

Remark 14.5.7. (i) Adapting the conditions (i), (ii) and (iii) in Definition 13.2.1, we can define the right Gr¨ obner soluble rings (RGS). The Gr¨obner theory for bijective right skew P BW extensions of RGS rings can of course be easy adapted. (ii) The equivalence (iii)⇔(iv) in the previous theorem leads us to the following key conclusion: Let A be a bijective skew P BW extension of a ring R (we always mean left in the present work), then A is also a bijective right skew P BW extension of R. So, assuming that R is also RGS, we have a right division algorithm. Obviously, if the elements of A are given by their left standard representation, we may have to rewrite them in their right standard representation, in order to be able to perform right divisions. A right version of Buchberger’s algorithm is also available. Thus, the theory of Gr¨obner bases for left ideals and submodules of left free modules developed in this and in the previous chapter has its counterpart on the right for A.

Chapter 15

Elementary Applications of Gr¨ obner Theory

There are some classical and elementary applications of Gr¨obner theory that we will study in this chapter. We will consider the membership problem, and we will compute the syzygy module, free resolutions of modules, the intersection and quotient of ideals and submodules, the matrix presentation of a finitely presented module, and the kernel and the image of homomorphism between modules. Recall that A = σ(R)hx1 , . . . , xn i represents a bijective skew P BW extension of an LGS ring R and Mon(A) is endowed with some monomial order (see Definition 13.1.1).

15.1 The Membership Problem Let F = {f1 , . . . , fs } ⊂ A and I := hF } be the left ideal generated by F . The membership problem asks whether one may effectively decide if an element f ∈ A belongs to I. Gr¨ obner theory provides an easy answer to this problem. Indeed, let G be a Gr¨ obner basis of I; making use of the division algorithm (Theorem 13.2.6), it is possible to obtain polynomials h1 , . . . , ht , h ∈ A, with G h reduced w.r.t. G, such that f −−→+ h and f = q1 f1 + · · · + qt ft + h, but according to Corollary 13.3.3 if h 6= 0, then f ∈ / I; and if h = 0, then f ∈ I. The next theorem complements the answer allowing us to write f as an A-linear combination of f1 , . . . , fs when f ∈ I.

Theorem 15.1.1. Let F = {f1 , . . . , fs } be a subset of A and G = {g1 , . . . , gt } be a Gr¨ obner basis of I := hF }. Then, there exist matrices H = [hij ] ∈ Ms×t (A) and Q = [qij ] ∈ Mt×s (A) such that GT = H T F T and F T = QT GT ,     where G := g1 · · · gt , F := f1 · · · fs , © Springer Nature Switzerland AG 2020 W. Fajardo et al., Skew PBW Extensions, Algebra and Applications 28, https://doi.org/10.1007/978-3-030-53378-6_15

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15 Elementary Applications of Gr¨ obner Theory

288



h11 · · ·  .. . . H :=  . .



 q11 · · · h1t ..  and Q :=  .. . .  . . . 

 q1s ...  .

qt1 · · · qts

hs1 · · · hst

Proof. We start by showing how Buchberger’s algorithm allows us to compute the matrix H. For this, we take G−1 := ∅ G0 := F 

Gi+1 := Gi ∪ r 6= 0 |

k X

G →+ r, for (b1 , . . . , bk ) ∈ BS , bj xγj gij −

j=1

where S = {gi1 , . . . , gik } ∈ P (Gi )−P (Gi−1 ) and Gi := {g1 , . . . , gti }. Suppose that      g1 f1 h11 · · · hs1    ..   .. . . . .  . = . . ..   ..  gti

h1ti · · · hsti

fs

Pk Gi and let gti +1 be an element in A − {0} such that j1 bj xγj gij −−→ + gti +1 ; Pk γj then, j1 bj x gij = a1 g1 + · · · + ati gti + gti +1 , and thus gti +1 =

k X

bj xγj gij + (−a1 )g1 + · · · + (−ati )gti

j=1

= (−a1 )g1 + · · · + (b1 xγ1 − ai1 )gi1 + · · · + (bk xγk − aik )gik + · · · + (−ati )gti = (−a1 )(h11 f1 + · · · + hs1 fs ) + · · · + (b1 xγ1 − ai1 )(h1i1 f1 + · · · + hsi1 fs ) + · · · + (bk xγk − aik )(h1ik f1 + · · · + hsik fs ) + · · · + (−ati )(h1ti f1 + · · · + hsti fs ) = (−a1 h11 + · · · + (b1 xγ1 − ai1 )h1i1 + · · · + (bk xγk − aik )h1ik + · · · − ati h1ti )f1 + · · · + (−a1 hs1 + · · · + (b1 xγ1 − ai1 )hsi1 + · · · + (bk xγk − aik )hsik + · · · − ati hsti )fs = h1ti +1 f1 + · · · + hsti +1 fs , with hrti +1 := −a1 hr1 + · · · + (b1 xγ1 − ai1 )hri1 + · · · + (bk xγk − aik )hrik + · · · − ati hrti , for 1 ≤ r ≤ s. With this we have   h11 · · · h1ti +1   ...  Htk +1 =  ... . . . . hs1 · · · hsti +1

15.1 The Membership Problem

289

Iterating this construction we will obtain a matrix H with the required properties. In order to obtain the matrix Q, it is enough to remember that if G = G {g1 , . . . , gt } is a Gr¨ obner basis for hF }, then fi − →+ 0 for any 1 ≤ i ≤ s; the division algorithm implies that fi = q1i g1 + · · · + qti gt for all 1 ≤ i ≤ s, and thus the matrix   q11 · · · q1s   Q =  ... . . . ...  qt1 · · · qts t u

satisfies the assertion.

The membership problem can be extended for modules: Let F = {f 1 , . . . , f s } be a set of nonzero vectors in Am and M := hf 1 , . . . , f s i the A-submodule of Am generated by f 1 , . . . , f s ; let G = {g1 , . . . , gt } be a Gr¨ obner basis for M and f ∈ Am , applying the division algorithm we find l1 , . . . , lt , ∈ A and a reduced vector h ∈ Am w.r.t. F such that f = l1 g 1 + · · · + lt g t + h; then, f ∈ M if and only if h = 0. In addition, Theorem 15.1.1 can be formulated and proved for modules. Theorem 15.1.2. Let F = {f1 , . . . , fs } be a subset of nonzero vectors of Am , and G = {g1 , . . . , gt } be a Gr¨ obner basis of M := hF i. Then, there exist matrices H = [hij ] ∈ Ms×t (A) and Q = [qij ] ∈ Mt×s (A) such that GT = H T F T and F T = QT GT ,     where G := g1 · · · gt , F := f1 · · · fs and 

h11 · · ·  .. . . H :=  . . hs1 · · ·

  h1t q11 · · · ..  , Q :=  .. . .  . . .  hst qt1 · · ·

(15.1.1)

 q1s ..  . .  qts

Therefore, (15.1.1) allow us to write f as an A-linear combination of f1 , . . . , fs when f ∈ M . As an application of the membership problem, given two ideals I and J of A generated by {f1 , . . . , fm } and {g1 , . . . , gn } respectively, we can decide effectively whether I = J: it is enough to check if fi ∈ J for all i ≤ i ≤ m, and if gj ∈ I for all 1 ≤ j ≤ n. We can proceed similarly for modules. Example 15.1.3. As in Example 13.4.11, we consider the diffusion algebra. We want to know if the polynomial f = x21 x2 D1 D22 + 23 x21 x22 D1 D2 − x21 x32 D1 + 1 2 2 x1 x2 D2 is in the left ideal I := hf1 , f2 }, where f1 = x1 D1 D2 + x2 , f2 = 2 x2 D2 . For this task, we calculate a Gr¨ obner basis for I and we check if f can be reduced to 0 with respect to {f1 , f2 }. We consider the order deglex on Mon(A), with D1  D2 . We start by taking G := ∅ and G0 := {f1 , f2 }. Step 1. Since G0 6= G, we have D = {S1 , S2 , S1,2 }.

15 Elementary Applications of Gr¨ obner Theory

290

We make G = G0 . Since R has no zero divisors, S1 and S2 do not add any polynomial to G0 . For S1,2 , we compute BS1,2 , a generator set of SyzR [σ γ1 (lc(f1 ))cγ1 ,β1 , σ γ2 (lc(f2 ))cγ2 ,β2 ] : X1,2 = lcm{D1 D2 , D22 } = D1 D22 , so γ1 = (0, 1), D2 (D1 D2 ) = 2D1 D22 + x2 D1 D2 − x1 D22 , and whence, cγ1 ,β1 = 2; in a similar way, γ2 = (1, 0) and cγ2 ,β2 = 1. Therefore, BS1,2 = {( 21 x2 , −x1 )} and we have 1 2 x2 D 2 f1

− x1 D1 f2 = 12 x1 x22 D1 D2 − 12 x21 x2 D22 + 12 x22 D2 .

Since 1 2 2 x1 x2 D 1 D 2

G

− 12 x21 x2 D22 + 12 x22 D2 −−→+ 12 x22 D2 − 12 x32 =: f3

and f3 is reduced with respect to G, we add the polynomial f3 and we make G0 := {f1 , f2 , f3 }. Step 2. Since G0 6= G, we compute D = P (G0 ) − P (G) and we make G = G0 . In D we only need to consider three subsets: S1,3 = {f1 , f3 }, S2,3 = {f2 , f3 }, S1,2,3 = {f1 , f2 , f3 }. For S1,3 we have X1,3 = D1 D2 , and hence, γ1 = (0, 0) and γ3 = (1, 0). From this it follows that BS1,3 = {(x22 , −2x1 )}, and we obtain x22 f1 − 2x1 D1 f3 = x1 x32 D1 + x32 =: f4 and f4 is reduced with respect to G, we add the polynomial f4 and we make G0 := {f1 , f2 , f3 , f4 }. For S2,3 , XS2,3 = D22 , so γ2 = (0, 0) and cγ2 ,β2 = 1; in the same way, γ3 = (0, 1) and cγ3 ,β3 = 1. Thus BS2,3 = {(x2 , −2)}, and G

x2 f2 − 2D2 f3 = x32 D2 −−→ x42 =: f5 . Since f5 is reduced with respect to G, we add f5 and we make G0 := {f1 , f2 , f3 , f4 , f5 }. For S1,2,3 we have that γ1 = (0, 1), γ2 = (1, 0), γ3 = (1, 1), and hence, BS1,2,3 = {(0, x2 , −2), ( 21 x2 , −x1 , 0)}; for the first generator we obtain a polynomial that can be reduced to 0 by f1 , f2 and f3 . The same applies for the second generator. Therefore, we do not add any polynomial to G0 . Step 3. Again, G 6= G0 . Thus, we compute D = P (G0 ) − P (G) and we make G = G0 . In this case, we need to consider 14 sets in D. For these subsets we obtain polynomials that are reducible to 0 by G = {f1 , f2 , f3 , f4 , f5 }. Thus, G is a Gr¨obner basis for I := hf1 , f2 }. Finally, by the division algorithm, f reduces to 0 with respect to {f1 , f2 , f3 , f4 , f5 }. Moreover, we have that 1 1 f = ( x1 x2 D2 + x1 x22 )f1 + x31 f2 − x1 f3 . 2 2

15.2 Computing Syzygies

291

Remark 15.1.4. Of course, Theorems 15.1.1 and 15.1.2 have their right version (see Remark 9.1.2): Let F = {f 1 , . . . , f s } be a subset of Am and G = {g 1 , . . . , g t } be a Gr¨ obner basis of M := hF i. Then, there exist matrices H = [hij ] ∈ Ms×t (A) and Q = [qij ] ∈ Mt×s (A) such that



where G := g 1 · · · g t



G = F H and F = GQ,   and F := f 1 · · · f s .

15.2 Computing Syzygies Now we will compute the syzygy module of a finite set of polynomials of A, and more generally, of a finite set of elements of Am . Given a left ideal I of A, with I = hf1 , . . . .fs }, we may define the following A-homomorphism: φ : As → I,

(h1 , . . . , hs )T 7→

s X

hi f i .

i=1

Note that φ is surjective, therefore I ∼ = As / ker(φ). Definition 15.2.1. The kernel of φ is called the syzygy module of the ma  T trix f1 · · · fs . It is denoted by Syz(f  1 , . . . , fs ).  An element (h1 , . . . , hs ) ∈ Syz(f1 , . . . , fs ) is called a syzygy of f1 · · · fs and satisfies h1 f1 + · · · + hs fs = 0. Note that φ can be viewed as a matrix multiplication:   f1   .  φ(h1 , . . . , hs ) = h1 · · · hs  ..  , fs and Syz(f1 , . . . fs ) as the set of all solutions (h1 , . . . , hs )T ∈ As of the linear equation   f1   .  h1 · · · hs  ..  = 0. fs Since A is a left noetherian ring, Syz(f1 , . . . , fs ) is a finitely generated left A-module. We will compute a system of generators for Syz(f1 , . . . , fs ) for any f1 , . . . , fs ∈ A. For this, we first compute a Gr¨obner basis G = {g1 , . . . , gt } for I = hf1 , . . . , fs }. Next, we obtain a set of generators for Syz(g1 , . . . , gt ) and, finally, we will obtain a system of generators for Syz(f1 , . . . , fs ) from one of Syz(g1 , . . . , gt ).

15 Elementary Applications of Gr¨ obner Theory

292

obner basis for I, F = {gi1 , . . . , gik } ⊆ G Thus, let G = {g1 , . . . , gt } be a Gr¨ and b = (b1 , . . . , bk ) ∈ BF ; recall that BF is a set of generators of SyzR [σ γj (lc(gij ))cγj ,exp(gij ) | 1 ≤ j ≤ k]. Pk G →+ 0 and hence there exist h1 , . . . , hs ∈ A We know that j=1 bj xγj gij − Pt Pk γj such that j=1 bj x gij = i=1 hi gi . For each b ∈ BF , we define sbF :=

k X

bj xγj eij − (h1 , . . . , ht ) ∈ At .

j=1

Then, sbF ∈ Syz(g1 , . . . , gt ): in fact,     g1 g1 k X     sbF  ...  = [ bj xγj eij − (h1 , . . . , ht )]  ...  j=1

gt =

k X j=1

bj xγj gij −

gt t X

hi gi = 0.

i=1

Definition 15.2.2. Let X1 , . . . , Xt ∈ Mon(A) and J ⊆ {1, . . . , t}. Let XJ := lcm{Xj | j ∈ J}. We say that J is saturated with respect to {X1 , . . . , Xt }, if Xj | XJ ⇒ j ∈ J, for any j ∈ {1, . . . , t}. The saturation J 0 of J consists of all j ∈ {1, . . . , t} such that Xj | XJ . Theorem 15.2.3. With the above notations, a generating set for Syz(g1 , . . . , gt ) is S := {svJ | J ⊆ {1, . . . , t} is saturated w.r.t. {lm(g1 ), . . . , lm(gt )}, 1 ≤ v ≤ lJ }, where sJv :=

X

J γj bvj x ej − (h1v , . . . , htv ),

j∈J n

with γj ∈ N such that γj + βj = exp(XJ ), βj = exp(gj ) for j ∈ J, BJ := {b1J , . . . , blJJ } a system of generators for SJ := SyzR [σ γj (lc(gj ))cγj ,βj | j ∈ J], and bvJ := (bJvj )j∈J for 1 ≤ v ≤ lJ .

15.2 Computing Syzygies

293

Proof. We have already seen that hSi ⊆ Syz(g1 , . . . , gt ). Suppose that there exists a u = (u1 , . . . , ut ) ∈ Syz(g1 , . . . , gt ) − hSi. We can choose u such that xδ := max {lm(lm(ui )lm(gi ))} is minimal with respect to . Let 1≤i≤t

J := {j ∈ {1, . . . , t} | lm(lm(uj )lm(gj )) = xδ }. Pt P Since i=1 ui gi = 0, we have j∈J lc(uj )σ αj (lc(gj ))cαj ,βj = 0, where αi := exp(ui ) for 1 ≤ i ≤ t. If XJ := lcm{lm(gj ) | j ∈ J}, then XJ | xδ and there is a θ ∈ Nn with θ + exp(XJ ) = δ. But αj + βj = δ and γj + βj = exp(XJ ) for all j ∈ J, then θ + γj + βj = αj + βj , i.e., θ + γj = αj . Thus, (lc(uj ))j∈J ∈ SJ,θ := SyzR [σ θ+γj (lc(gj ))cθ+γj ,βj | j ∈ J]. If J 0 is the saturation of J, then XJ = XJ 0 and w = (wj )j∈J 0 given by ( lc(uj ), if j ∈ J, wj = 0, if j ∈ J 0 − J is an element of SJ 0 ,θ . According to Remark 13.4.2, there exists (bj )j∈J 0 ∈ SJ 0 := SyzR [σ γj (lc(gj ))cγj ,βj | j ∈ J 0 ] such that wj = σ θ (bj )cθ,γj for j ∈ J 0 . P lJ 0 0 J 0 This implies that bj = 0 for j ∈ J 0 − J. Now, (bj )j∈J 0 = v=1 rv bv , with J0 0 0 0 0 BJ := {bv | 1 ≤ v ≤ lJ } a system of generators for SJ and rv ∈ R for 1 ≤ P lJ 0 0 J 0 P lJ 0 θ 0 θ J 0 v ≤ lJ 0 . Hence, bj = v=1 rv bvj and thus wj = v=1 σ (rv )σ (bvj )cθ,γj for P lJ 0 0 0 θ J0 all j ∈ J . Define u := u − v=1 rv x sv , with rv := σ θ (rv0 ) for 1 ≤ v ≤ lJ 0 ; P lJ 0 0 then u0 ∈ Syz(g1 , . . . , gt ) since v=1 rv xθ sJv ∈ hSi. Note that lJ 0 X

0

0

0

rv xθ sJv = r1 xθ sJ1 + · · · + rlJ 0 xθ sJlJ 0

v=1

= r1 x θ [

0

X

bJ1j xγj ej − (h11 , . . . , h1t )] + · · ·

j∈J 0

+ rl J 0 x θ [

0

X

l

l

bJlJ 0 j xγj ej − (h1J 0 , . . . , htJ 0 )]

j∈J 0

= r1 [

X

θ

0

(σ (bJ1j )cθ,γj xθ+γj + p1j )ej − (h11 , . . . , h1t )] + · · ·

j∈J 0

+ rl J 0 [

X

0

l

l

l

(σ θ (bJlJ 0 j )cθ,γj xθ+γj + pjJ 0 )ej − (h1J 0 , . . . , htJ 0 )].

j∈J 0

Thus, for j ∈ J we have that   lJ 0 lJ 0 lJ 0 X X X 0 u0j = uj −  rv σ θ (bJvj )cθ,γj xθ+γj + pvj − hvj  v=1

 = uj − 

lJ 0 X v=1

v=1 0

σ θ (rv0 )σ θ (bJvj )cθ,γj xαj +

lJ 0 X v=1

v=1

pvj −

lJ 0 X v=1

 hvj 

294

15 Elementary Applications of Gr¨ obner Theory

= uj − lc(uj )x

αj

−

lJ 0 X v=1

pvj

+

lJ 0 X

hvj

v=1

P lJ 0 θ 0 θ J 0 since j ∈ J, γj + θ = αj and wj = lc(uj ) = v=1 σ (rv )σ (bvj )cθ,γj . Here pvj = 0 or deg(pvj ) < |θ + γj | for every 1 ≤ v ≤ lJ 0 . Then, lm(lm(uj − lc(uj )xαj )lm(gj )) ≺ lm(lm(uj )lm(gj )) = xδ , lm(pvj gj ) ≺ xθ+γj +βj = xδ , and P 0 lm(lm(hvj )lm(gj ))  lm( j∈J 0 bJvj xγj gj ) ≺ XJ 0 = XJ  xδ , so lm(lm(u0j )lm(gj )) ≺ xδ . Now, if j ∈ J 0 − J, then wj =

lJ 0 X

0

σ θ (rv0 )σ θ (bJvj )cθ,γj = 0 and

v=1

lm(lm(uj )lm(gj )) ≺ xδ , P lJ 0 v thus lm(lm(u0j )lm(gj )) ≺ xδ . Finally, if j ∈ / J 0 , then u0j = uj + v=1 hj and 0 δ 0 δ lm(lm(uj )lm(gj )) ≺ x . So, lm(lm(ui )lm(gi )) ≺ x for every 1 ≤ i ≤ t and, by minimality of u, we have that u0 ∈ hSi and hence, u ∈ hSi, a contradiction. Therefore, hSi = Syz(g1 , . . . , gt ). t u Now, we return to the initial problem of calculating a system of generators for Syz(f1 , . . . , fs ), where {f1 , . . . , fs } is a collection of nonzero polynomials, which do not necessarily form a Gr¨ obner basis for I = hf1 , . . . , fs }. As we saw in Theorem 15.1.1, there exist H ∈ Ms×t(A) and Q  ∈ Mt×s  (A) such that GT = H T F T and F T = QT GT , where G := g1 · · · gt , F := f1 · · · fs and G is a Gr¨obner basis for I. By Theorem 15.2.3, we may compute a set of generators {s1 , . . . , sl } for Syz(g1 , . . . , gt ). Thus, for each 1 ≤ i ≤ l we have that si H T F T = si GT = 0, and therefore, hsi H T | 1 ≤ i ≤ li ⊆ Syz(f1 , . . . , fs ). Further,         f1 f1 f1 0    ..   ..  T T  ..  T T  ..  Is − Q H  .  =  .  − Q H  .  =  .  , fs fs fs 0 and thereby the rows r1 , . . . , rs of Is − QT H T also belong to Syz(f1 , . . . , fs ). Theorem 15.2.4. With the above notation, we have Syz(f1 , . . . , fs ) = hs1 H T , . . . , sl H T , r1 , . . . , rs i ≤ As . Proof.. Let s = (a1 , . . . , as )T be an element in Syz(f1 , . . . , fs ), then 0 = s T F T = s T Q T GT ,

15.2 Computing Syzygies

295

and therefore sT QT ∈ Syz(g1 , . . . , gt ). Thus, sT QT = Pl pi ∈ A. Thereby, sT QT H T = i=1 pi (si H T ) and

Pl

i=1

pi si for some

sT = sT − sT QT H T + sQT H T = sT (Is − QT H T ) +

l X

pi (si H T )

i=1

=

s X i=1

ai ri +

l X

pi (si H T );

i=1

thus, s T ∈ hs1 H T , . . . , sl H T , r 1 , . . . , r s i and we obtain the required equality. t u Example 15.2.5. We consider again the diffusion algebra with n = 2, K = Q, c12 = −2 and c21 = −1. In this ring, we have D2 D1 = 2D1 D2 + x2 D1 − x1 D2 and the automorphisms σ1 and σ2 are the identity. We consider the order deglex with D1  D2 and the polynomials f1 = x21 x2 D12 D2 , f2 = x22 D1 D22 . As we saw in Example 13.4.11, G = {f1 , f2 , f3 , f4 } is a Gr¨obner basis for I := hf1 , f2 }, where f3 = − 41 x31 x32 D1 D2 + 14 x41 x22 D22 , f4 = x31 x2 f2 + 2D2 f3 = 1 4 2 3 1 3 4 1 4 3 2 2 x1 x2 D2 − 2 x1 x2 D1 D2 + 2 x1 x2 D2 . We will use this for computing a system of generators for SyzA {f1 , f2 }. By Theorem 15.2.3, we must consider the saturated subsets of {1, 2, 3, 4} w.r.t. {lm(fi )}4i=1 ; these sets are: J3 = {3}, J4 = {4}, J1,3 = {1, 3}, J2,3 = {2, 3}, J1,2,3 = {1, 2, 3}, J2,3,4 = {2, 3, 4} and J1,2,3,4 = {1, 2, 3, 4}. We have: • For J3 = {1} we compute a system BJ3 of generators of SyzR [σ γ1 (lc(f3 ))]cγ3 ,β3 , where γ1 = XJ3 − β3 = (0, 0). Then BJ3 = {0}, and hence we have only one generator bJ1 3 = (bJ113 ) = 0 and sJ1 3 = bJ113 xγ3 e e3 − (0, 0, 0, 0) = (0, 0, 0, 0), with e e1 = (0, 0, 0, 0)T . • For J4 = {4} the situation is similar. • For J1,3 : XJ1,3 = D12 D2 and γ1 = (0, 0), γ3 = (1, 0); thus, cγ1 ,β1 = 1 and cγ3 ,β3 = 1. A system of generators of SyzR [σ γ1 (lc(f1 ))cγ1 ,β1 σ γ3 (lc(f3 ))cγ3 ,β3 ] = SyzR [x21 x2 is BJ1,3 = {(x1 x22 , 4)}. J Thus, we only have one generator b1 1,3 = (x1 x22 , 4). Since x1 x22 f1 + 4D1 f3 = x41 f2 , we have

− 41 x31 x2 ]

15 Elementary Applications of Gr¨ obner Theory

296 J

s1 1,3 = x1 x22 e e1 + 4D1 e e3 − (0, x14 , 0, 0)   x1 x22  −x41   =  4D1  . 0 • For J2,3 : XJ1,3 = D1 D22 and γ2 = (0, 0), γ3 = (0, 1); thus, cγ2 ,β2 = 1. Since D2 (D1 D2 ) = 2D1 D22 + x2 D1 D2 − x1 D22 , then cγ3 ,β3 = 2. A system of generators of SyzR [σ γ2 (lc(f2 ))cγ2 ,β2 σ γ3 (lc(f3 ))cγ3 ,β3 ] = SyzR [x2

− 21 x31 x32 ]

is BJ1,3 = {(x31 x2 , 2)}. Therefore, x31 x2 f2 + 2D2 f3 = f4 , and J

s1 2,3 = x31 x2 e e2 + 2D2 e e3 − (0, 0, 0, 1)   0 x31 x2   =  2D2  . −1 • For J1,2,3 : XJ1,2,3 = D12 D22 and γ1 = (0, 1), γ2 = (1, 0) and γ3 = (1, 1). Now, since D2 D12 D2 = 4D12 D22 + 3x2 D12 D2 − 4x1 D1 D22 − x1 x2 D1 D2 + x21 D22 , D1 D2 D1 D2 = 2D12 D22 + x2 D12 D2 − x1 D1 D22 , we have cγ1 ,β1 = 4, cγ2 ,β2 = 1 and cγ3 ,β3 = 2. We have SyzR [4x21 x2 x22

− 12 x31 x32 ] = h( 14 x2 , −x21 , 0), ( 14 x1 x22 , 0, 2)i.

J

For b1 1,2,3 = ( 14 x2 , −x21 , 0) we found 1 4 x2 D 2 f1

− x21 D1 f2 = 34 x22 f1 − x31 f2 + f3

and J

1 3 x2 D 2 e e1 − x21 D1 e e2 − ( x22 , −x31 , 1, 0) 4 4   1 3 2 4 x2 D 2 − 4 x2  −x21 D1 + x31  . =   −1 0

s1 1,2,3 =

J

For b2 1,2,3 = ( 14 x1 x22 , 0, 2) we have 1 2 4 x1 x2 D 2 f1

+ 2D1 D2 f3 = 34 x1 x32 f1 − x41 x2 f2 + x1 x2 f3 + D1 f4

15.2 Computing Syzygies

297

and J

1 3 x1 x22 D2 e e1 + 2D1 D2 e e2 − ( x1 x32 , −x41 x2 , x1 x2 , D1 ) 4 4 1  3 2 3 x x D − x x 4 1 2 2 4 1 2   x41 x2  . = 2D1 D2 − x1 x2  −D1

s1 1,2,3 =

• For J2,3,4 : XJ2,3,4 = D1 D23 , so γ2 = (0, 1), γ3 = (0, 2) and γ4 = (1, 0). Now, since D22 D1 D2

D2 D1 D22 = 2D1 D23 + x2 D1 D22 − x1 D23 , = 4D1 D23 + 4x2 D1 D22 − 3x1 D23 + x22 D1 D2 − x1 x2 D22 ,

we have cγ2 ,β2 = 2, cγ3 ,β3 = 4 and cγ4 ,β4 = 1. We have SyzR [2x22 , −x31 x32 , 12 x41 x22 ] = h( 12 x31 x2 , 1, 0), ( 12 x41 , 0, −2)i. J

For b1 2,3,4 = ( 12 x31 x2 , 1, 0) the following equality holds 1 3 2 x1 x2 D 2 f2

+ D22 f3 = 12 D2 f4

and J

1 3 1 x1 x2 D 2 e e2 + D22 e e3 − (0, 0, 0, D2 ) 2 2   0  1 x21 x2  2  =  D22  . 1 2 D2

s1 2,3,4 =

J

For b2 2,3,4 = ( 12 x41 , 0, −2), 1 4 2 x1 D 2 f2

− 2D1 f4 = x1 x32 f1 − 12 x41 x2 f2 − 2x1 x2 f3 − x1 f4

and hence J

1 4 1 x1 D 2 e e2 − 2D1 e e4 − (x1 x32 , − x41 x2 , −2x1 x2 , −x1 ) 2 2   −x1 x32  1 x41 + 1 x41 x2  2 2 . =   2x1 x2 −2D1 + x1

s2 2,3,4 =

• For J1,2,3,4 : XJ1,2,3,4 = D12 D23 , so γ1 = (0, 2), γ2 = (1, 1), γ3 = (1, 2) and γ4 = (2, 0). In this case, cγ1 ,β1 = 16, cγ2 ,β2 = 2, cγ3 ,β3 = 4 and cγ4 ,β4 = 1. We have SyzR [16x21 x2 , 2x22 , −x31 x32 , 12 x41 x22 ] = 1 1 2 1 1 2 h( 16 x2 , − 2 x1 , 0, 0), ( 16 x1 x22 , 0, 1, 0), ( 16 x1 x2 , 0, 0, −2)i.

15 Elementary Applications of Gr¨ obner Theory

298 J

1 For b1 1,2,3,4 = ( 16 x2 , − 12 x21 , 0, 0) we obtain 9 3 16 x2 f1

+

1 1 2 2 16 x2 D2 f1 − 2 x1 D1 D2 f2 = 1 3 21 2 3 (x1 x2 D1 − 2 x1 D2 − 17 8 x1 x2 )f2 + 4 x2 f3

+

17 8 f4 ,

thereby 1 1 x2 D22 e e1 − x21 D1 D2 e e2 16 2 9 1 17 21 17 − ( x32 , x21 x2 D1 − x31 D2 − x31 x2 , x2 , ) 2 8 4 8 16 1 9 3 2 x D − x 2 2 2 16 16 − 1 x21 D1 D2 − x21 x2 D1 + 1 x31 D2 + 17 x31 x2  2 2 8 . =   − 21 4 x2 17 −8

J

s1 1,2,3,4 =

J

1 For b2 1,2,3,4 = ( 16 x1 x22 , 0, 1, 0), 9 4 16 x1 x2 f1

−

1 2 2 2 16 x1 x2 D2 f1 + D1 D2 f3 = 13 4 2 13 1 2 8 x1 x2 f2 + 4 x1 x2 f3 + ( 2 D 1 D 2 −

x2 D1 + 98 x1 x2 )f4

and 1 x1 x22 D22 e e1 + D1 D22 e e3 16 9 13 13 1 9 − ( x1 x42 , − x41 x22 , x1 x22 , D1 D2 − x2 D1 + x1 x2 ) 8 4 8  161  2 9 2 2 4 x x D − x x 1 1 2 2 2 16 16 13 4 2   8 x1 x2 . = 13 2 2   D 1 D 2 − 4 x1 x2 1 9 − 2 D 1 D 2 + x2 D 1 − 8 x1 x2

J

s2 1,2,3,4 =

J

1 2 For b3 1,2,3,4 = ( 16 x1 x2 , 0, 0, −2),

1 2 x x2 D22 f1 − 2D12 f4 16 1 33 1 17 = (x1 x32 D1 + x21 x32 )f1 + ( x41 x2 D1 − x51 x2 )f2 16 2 8 11 2 9 2 + x1 x2 f3 + (−3x1 D1 + x1 )f4 2 8 and J

1 2 x x2 D22 e e1 − 2D12 e e4 16 1 33 1 17 11 9 − (x1 x32 D1 + x21 x32 , x41 x2 D1 − x51 x2 , x21 x2 , −3x1 D1 + x21 ) 16 2 2 8 1 2 8 33 2 3 2 3 x x D − x x D − x x 2 1 1 2 2 16 1 16 1 2 5   − 12 x41 x2 D1 + 17 8 x1 x2 . = 11 2   − 2 x1 x2 9 2 2 −2D1 + 3x1 D1 − 8 x1

s3 1,2,3,4 =

15.2 Computing Syzygies

299

In consequence, J

J

J

J

J

J

J

J

J

S = {s1 1,3 , s1 2,3 , s1 1,2,3 , s2 1,2,3 , s1 2,3,4 , s2 2,3,4 , s1 1,2,3,4 , s2 1,2,3,4 , s3 1,2,3,4 } is a set of generators for Syz(G). To compute a generator set for Syz(M ) we use Theorem 15.2.4. In this case the matrices H and Q in Theorem 15.1.2 are:   10  0 1  Q=  0 0 ; 00   1 1 0 14 x2 D2 − 34 x22 x2 D22 − 32 x22 D2 2 H= . 0 1 −x21 D1 + x31 −4x21 D1 D2 − 2x21 x2 D1 + 4x31 D2 + x31 x2   00 T T Since I2 − Q H = , the generators for Syz(f1 , f2 ) are given by sH T 00   x2 D1 D2 − 3x22 D1 + x1 x22 J1,3 T for each s ∈ S. Therefore: • s 1 := s 1 H = −4x21 D12 + 4x31 D1 − x41 J

J

J

• s 1 2,3 H T = s 1 1,2,3 H T = s 2 1,2,3 H T = 0 • s2 := s J1 2,3,4 H T

1 x D 3 − 32 x22 D22 2 2 2 −8x21 D1 D22 − 8x21 x2 D1 D2 + 7x31 D22 − 2x21 x22 D1 + 52 x31 x2 D2 + 12 x21 x2 J s3 := s 2 2,3,4 H T  −x2 D1 D22 + 3x22 D1 D2 + 12 x1 x2 D12 − x1 x22 D2 − 52 x1 x32 = 8x21 D12 D2 − 12x31 D1 D2 + 4x21 x2 D12 − 6x31 x2 D1 + 4x41 D2 + 52 x41 x2 + 12 x41   x2 D2 + 27 x3 −x2 D22 + 15 J 8 8 2 s 4 := s 1 1,2,3,4 H T = 17 2 21 3 2 3 8x D D + x x D − 8x1 D2 − 4 x1 x2  11 12 2 2 2 31 32 1  x D D − 2 x2 D1 D2 − 12 x1 x22 D22 + 78 x1 x32 D2 + 15 x x4 J1,2,3,4 T 2 2 1 2 8 1 2 s 5 := s 2 H = 9 4 11 4 2 2 2 3 2 2 2 −4x1 x2 D1 D2 + 8x1 x2 D1 D2 − 2x1 x2 D1 − 2 x1 x2 D2 − 4 x1 x2 J s 6 := s 3 1,2,3,4 H T =   −x2 D12 D22 + 3x22 D12 D2 + 32 x1 x2 D1 D22 − 92 x1 x22 D1 D2 − 12 x21 x2 D22 − x1 x32 D1 5 2 2   + 16 x1 x2 D2 + 33 x2 x3  16 1 2   8x21 D13 D2 − 20x31 D12 D2 + 4x21 x2 D13 − 8x31 x2 D12 + 33 x41 D1 D2 + 41 x41 x2 D1 . 2 4 − 92 x51 D2 − 92 x51 x2



=

•

• •

•

Hence, {s 1 , s 2 , s 3 , s 4 , s 5 , s 6 } is a generator set for Syz(f1 , f2 ). Corollary 15.2.6. Let R be an LGS ring. If A = σ(R)hx1 , . . . , xn i is a bijective skew P BW extension of R, then A is LGS. Proof. This follows from the Hilbert Basis Theorem (Theorem 3.1.5), the discussion at the beginning of previous section, Theorem 15.1.1, and from Theorem 15.2.4. t u Now we can generalize the method described above for computing the   syzygy module of a submodule M = hf 1 , . . . , f s i of Am . Let F := f 1 · · · f s .
T We recall that Syz(M ) := Syz(F ) consists of column vectors h = h1 · · · hs in As such that

300

15 Elementary Applications of Gr¨ obner Theory

h1 f 1 + · · · + hs f s = 0, i.e., h T F T = 0, and it is also the kernel of the homomorphism f : As → M , e i 7→ f i , where {e i }si=1 is the canonical basis of As (see Section 9.1). We note that Syz(F ) is a submodule of As and we can set a matrix with its generators, so sometimes we will refer to Syz(F ) as a matrix. We will also write Syz(M ) = Syz(F ) = Syz({f 1 , . . . , f s }). (15.2.1) The computation of Syz(F ) is done in two steps. First, we consider a Gr¨obner basis G = {g 1 , . . . , g t } for M and we compute Syz(G) := Syz({g 1 , . . . , g t }) ≤ At , and then, we obtain a system of generators for Syz(F ) from one of Syz(G). For F = {g i1 , . . . , g ik } ⊆ G and (b1 , . . . , bk ) ∈ BF , with BF a set of generators of SyzR (σ γj (lc(g ij ))cγj ,exp(g i ) | 1 ≤ j ≤ k), we have that j Pk G γj b x g − − → 0, and hence, there exist h1 , . . . , hs ∈ A such that j + ij Pj=1 Pt k γj j=1 bj x g ij = i=1 hi g i . For each b ∈ BF , we define sbF :=

k X

bj xγj eij − (h1 , . . . , ht ) ∈ At ;

j=1

then sbF ∈ Syz(g 1 , . . . , g t ): in fact,     g1 g1 k X  ..   ..  γj sbF  .  = [ bj x eij − (h1 , . . . , ht )]  .  j=1

gt =

k X j=1

bj xγj g ij −

gt t X

hi g i = 0.

i=1

Definition 15.2.7. Let X1 , . . . , Xt ∈ Mon(Am ) and J ⊆ {1, . . . , t}. Let XJ := lcm{Xj | j ∈ J}. We say that J is saturated with respect to {X1 , . . . , Xt }, if Xj | XJ ⇒ j ∈ J, for any j ∈ {1, . . . , t}. The saturation J 0 of J consists of all j ∈ {1, . . . , t} such that Xj | XJ . Theorem 15.2.8. With the above notations, a generating set for Syz(g1 , . . . , gt ) is

15.2 Computing Syzygies

301

S := {sJv | J ⊆ {1, . . . , t} is saturated w.r.t. {lm(g1 ), . . . , lm(gt )}, 1 ≤ v ≤ lJ }, where sJv :=

X

bJvj xγj ej − (hv1 , . . . , hvt ),

j∈J

with γj ∈ Nn such that γj + βj = exp(XJ ), βj = exp(gj ), j ∈ J, B J := {bJ1 , . . . , bJlJ } is a system of generators for S J := SyzR [σ γj (lc(gj ))cγj ,βj | j ∈ J], and bJv := (bJvj )j∈J . Proof. We have already seen that hSi ⊆ Syz(g 1 , . . . , g t ). Suppose that there exists a u = (u1 , . . . , ut ) ∈ Syz(g 1 , . . . , g t ) − hSi. We can choose u with X δ := max {lm(lm(ui )lm(g i ))} minimal with respect to . Let 1≤i≤t

J := {j ∈ {1, . . . , t} | lm(lm(uj )lm(g j )) = X δ }. Pt P Since i=1 ui g i = 0, in particular we have j∈J lc(uj )σ αj (lc(g j ))cαj ,βj = 0, where αi := exp(ui ) for 1 ≤ i ≤ t. If X J := lcm{lm(g j ) | j ∈ J}, then X J | X δ and therefore there is a θ ∈ Nn with θ + exp(X J ) = δ. But αj +βj = δ and γj +βj = exp(X J ) for all j ∈ J, then θ+γj +βj = αj +βj , i.e., θ + γj = αj . Thus, (lc(uj ))j∈J ∈ SθJ := SyzR [σ θ+γj (lc(gj ))cθ+γj ,βj | j ∈ J]. If J 0 is the saturation of J, then XJ = XJ 0 and w = (wj )j∈J 0 given by ( lc(uj ), if j ∈ J, wj = 0, if j ∈ J 0 − J 0

is an element of SθJ . According to Remark 13.4.2, there exists (bj )j∈J 0 ∈ 0 S J := SyzR [σ γj (lc(gj ))cγj ,βj | j ∈ J 0 ] such that wj = σ θ (bj )cθ,γj for j ∈ J 0 . P lJ 0 0 J 0 This implies that bj = 0 for j ∈ J 0 − J. Now, (bj )j∈J 0 = v=1 rv bv , with 0 0 {bJv | 1 ≤ v ≤ lJ 0 } a system of generators for S J and rv0 ∈ R for 1 ≤ v ≤ lJ 0 . P lJ 0 0 J 0 P lJ 0 θ 0 θ J 0 Hence, bj = v=1 rv bvj and thus wj = v=1 σ (rv )σ (bvj )cθ,γj for all j ∈ J 0 . P lJ 0 0 θ J0 Define u := u − v=1 rv x sv , with rv := σ θ (rv0 ) for 1 ≤ v ≤ lJ 0 ; then P lJ 0 0 u0 ∈ Syz(G) since v=1 rv xθ sJv ∈ hSi. Note that lJ 0 X

0

0

0

X

bJ1j xγj e j − (h11 , . . . , h1t )] + · · ·

rv xθ sJv = r1 xθ sJ1 + · · · + rlJ 0 xθ sJlJ 0

v=1

= r1 x θ [

0

j∈J 0

+ rl J 0 x θ [

X

0

0

l

J0 bJlJ 0 j xγj e j − (hlJ 1 , . . . , ht )]

j∈J 0

= r1 [

X j∈J 0

θ

0

σ (bJ1j )cθ,γj xθ+γj + p1j e j − (h11 , . . . , h1t )] + · · ·

302

15 Elementary Applications of Gr¨ obner Theory

+ rlJ 0 [

X

0

l

l

l

σ θ (bJlJ 0 j )cθ,γj xθ+γj + pjJ 0 e j − (h1J 0 , . . . , htJ 0 )]

j∈J 0

Thus, for j ∈ J we have that u0j = uj − [

lJ 0 X

0

rv σ θ (bJvj )cθ,γj xθ+γj +

v=1

X

pvj −

v=1

lJ 0

= uj − [

lJ 0 X

0

v=1

= uj − lc(uj )xαj −

X

pvj −

v=1

X

hvj ]

v=1

lJ 0

σ θ (rv0 )σ θ (bJvj )cθ,γj xαj + lJ 0

lJ 0 X

lJ 0 X

hvj ]

v=1

lJ 0

pvj +

v=1

X

hvj

v=1

P lJ 0 θ 0 θ J 0 since for j ∈ J, γj + θ = αj and wj = lc(uj ) = v=1 σ (rv )σ (bvj )cθ,γj . Here pvj = 0 or deg(pvj ) < |θ + γj | for every 1 ≤ v ≤ lJ 0 . Then lm(lm(uj − lc(uj )xαj )lm(g j )) ≺ lm(lm(uj )lm(g j )) = X δ , lm(pvj gj ) ≺ xθ+γj +βj = X δ , and X 0 lm(lm(hvj )lm(g j ))  lm( bJvj xγj g j ) ≺ XJ 0 = X J  X δ j∈J 0

and, therefore, lm(lm(u0j )lm(g j )) ≺ X δ . Now, if j ∈ J 0 − J, then wj = P lJ 0 θ 0 θ J 0 v=1 σ (rv )σ (bvj )cθ,γj = 0, and lm(lm(uj )lm(gj )) ≺ X δ , and thus P lJ 0 v lm(lm(u0j )lm(g j )) ≺ X δ . Finally, if j ∈ / J 0 , then u0j = uj + v=1 hj and lm(lm(u0j )lm(g j )) ≺ X δ . So, lm(lm(u0i )lm(g i )) ≺ X δ for every 1 ≤ i ≤ t and, by minimality of u, we have that u0 ∈ hSi and hence, u ∈ hSi, a contradiction. Thus hSi = Syz(g 1 , . . . , g t ). t u We return to the task of calculating a system of generators for Syz(f 1 , . . . , f s ), where {f 1 , . . . , f s } is a collection of nonzero vectors, which do not necessarily form a Gr¨ obner basis for M = hf 1 , . . . , f s i. From Theorem 15.1.1 (for modules), there exist H ∈ Ms×t (A)and Q ∈ Mt×s (A) such that GT = H T F T and F T = QT GT , where G := g 1 · · · g t , F := f1 · · · f s and G is a Gr¨obner basis for hf 1 , . . . , f s i. By Theorem 15.2.8, we compute a set of generators {s1 , . . . , sl } for Syz(g 1 , . . . , g t ). Thus, for each 1 ≤ i ≤ l we have si H T F T = si GT = 0, and therefore, hsi H T | 1 ≤ i ≤ li ⊆ Syz(f 1 , . . . , f s ). If Syz(G) := Z(G) :=   s 1 · · · s l , then Syz(g 1 , . . . , g t ) is the module generated by columns of Z(G) and this last equation may be written as Z(G)T H T F T = Z(G)T GT = 0. Further,

(15.2.2)

15.2 Computing Syzygies

303







  f1 f1 f1 0  ..   ..  T T  ..  T T  ..  [Is − Q H ]  .  =  .  − Q H  .  =  .  , fs





fs



fs

0

and thereby the rows r1 , . . . , r s of Is − QT H T also belong to Syz(f 1 , . . . , f s ).

Theorem 15.2.9. With the above notation, we have Syz(f1 , . . . , fs ) = hs1 H T , . . . , sl H T , r1 , . . . , rs i ≤ As . In the matrix notation, Syz(F ) coincides with   the column module of the extended matrix (Z(G)T H T )T Is − (QT H T )T , i.e.,   Syz(F ) = (Z(G)T H T )T |Is − (QT H T )T . (15.2.3) Proof. Let s T = (a1 , . . . , as ) be an element in Syz(f 1 , . . . , f s ), then 0 = s T F T = s T Q T GT , and therefore sT QT ∈ Syz(g 1 , . . . , g t ). Thus, s T QT = Pl pi ∈ A. Thereby, s T QT H T = i=1 pi (si H T ), and thus

Pl

i=1

pi si for some

s T = s T − s T QT H T + s T QT H T = sT (Is − QT H T ) +

l X

pi (si H T )

i=1

=

s X i=1

ai ri +

l X

pi (si H T );

i=1

whence, s T ∈ hs1 H T , . . . , sl H T , r 1 , . . . , r s i and we obtain the required equality. t u Example 15.2.10. We consider again the additive analogue of the Weyl algebra A = A2 ( 21 , 13 ), used in Example 14.4.7, with the same monomial order on Mon(A) and on Mon(A2 ). For this example, we want to find a finite set of generators for Syz[f 1 , f 2 ], where f 1 = x1 y12 e 1 + x2 y2 e 2 and f 2 = x2 y22 e 1 + x1 y1 e 2 . As we saw in Example 14.4.7, G = {f 1 , f 2 , f 3 }, with f 3 = − 14 x21 y13 e 2 + 19 x22 y23 e 2 − 32 x1 y12 e 2 + 43 x2 y22 e 2 is a Gr¨obner basis for M. Now, according to Theorem 15.2.8, to compute a system of generators for Syz(G) = Syz[f 1 , f 2 , f 3 ], we must compute the saturated subsets J of {1, 2, 3} with respect to {y12 e1 , y22 e1 , y13 e2 }. We have: • For J1 = {1} we compute a system BJ1 of generators of SyzR [σ γ1 (lc(f 1 ))cγ1 ,β1 ],

304

15 Elementary Applications of Gr¨ obner Theory

where β1 := exp(lm(f 1 )) and γ1 = exp(X J1 ) − β1 = (0, 0). Then BJ1 = {0}, and hence we have only one generator bJ1 1 = (bJ111 ) = 0 and s J1 1 = bJ111 xγ1 e e1 − (0, 0, 0) = (0, 0, 0), with e e1 = (0, 0, 0)T . • For J2 = {2} and J3 = {3} the situation is similar. • For J1,2 = {1, 2}, a system of generators of SyzR [σ γ1 (lc(f 1 ))cγ1 ,β1 σ γ1 (lc(f 2 ))cγ2 ,β2 ], where β1 = exp(lm(f 1 )), β2 = exp(lm(f 2 )), γ1 = (0, 2), γ2 = (2, 0), cγ1 ,β1 = 1 and cγ2 ,β2 = 1, is BJ1,2 = {(x2 , −x1 )}. J Thus, we only have one generator b1 1,2 = (x2 , −x1 ). Since x2 y22 f 1 − x1 y12 f 2 = f 3 , we have J

s1 1,2 = x2 y22 e e1 − x1 y12 e e2 − (0, 0, 1)   2 x 2 y2 = −x1 y12  . −1 • For J1,3 = {1, 3} and J2,3 = {2, 3}, we have X J1,3 = X J2,3 = 0. Hence,    x2 y22  Syz(G) = −x1 y12  . −1  T Finally, we compute a generator set for Syz(M ): Let s = x2 y22 −x1 y12 −1 ; from Theorem 15.1.2 there exist matrices H and Q such that GT = H T F T and F T = QT GT ; in this case,     10 1 0 x2 y22   H= 2 and Q = 0 1 . 0 1 −x1 y1 00     00 Hence, sT H T = 0 0 and I2 − QT H T = . Then Syz(f 1 , f 2 ) = 0 and 00 therefore, M is a free left module of rank two. Remark 15.2.11. The matrix right version of Theorem 15.2.9 is as follows: r Syz (F ) coincides with the column module of the extended matrix  HZ(G) Is − QH , i.e.,   Syzr (F ) = HZ(G)|Is − HQ . (15.2.4) For right modules (see Remark 9.1.2) we will denote by Syzr the module of syzygies.

15.3 Intersections

305

15.3 Intersections In this section, using syzygies we will compute the intersection of left ideals of A and submodules of Am . Thus, let I = hf1 , . . . , fs } and J = hg1 , . . . , gt } be left ideals of A; for h ∈ I ∩ J there exist some elements a1 , . . . , as and b1 , . . . , bt in A such that h = a1 f1 + · · · + as fs = b1 g1 + · · · + bt gt . The above can be reformulated as saying that     1 1 f1  g1         −h a1 . . . as  .  = 0 and −h b1 . . . bt  .  = 0,  ..   ..  gt fs i.e., (−h.a1 , . . . , as )T ∈ Syz(1, f1 , . . . , fs ) and (−h, b1 , . . . , bt )T ∈ Syz(1, g1 , . . . , gt ). Setting i := (1, 1)T , f 1 := (f1 , 0)T , . . . , f s := (fs , 0)T , g 1 := (0, g1 )T , . . . , g t := (0, gt )T , these two conditions may be rewritten as the following single condition: there exist polynomials a1 , . . . , as , b1 , . . . , bt ∈ A such that the vector (−h, a1 , . . . , as , b1 , . . . , bt )T is a syzygy of H, where   H := i f 1 · · · f s g 1 · · · g t . Since h ∈ I ∩J if and only if −h ∈ I ∩J, we may rephrase the above condition as (h, a1 , . . . , as , b1 , . . . , bt )T ∈ Syz(H). We have proved the following result. Theorem 15.3.1. The elements in I ∩ J are polynomials h ∈ A such that there exist a1 , . . . , as , b1 , . . . , bt ∈ A with (h, a1 , . . . , as , b1 , . . . , bt )T ∈ Syz(H). A system of generators for the intersection is given in the following corollary. Corollary 15.3.2. Let {h1 , . . . , hl } be a generating set for Syz(H). If h1j is the first coordinate of hj , for 1 ≤ j ≤ l, then L = {h11 , . . . , h1l } generates I ∩ J. Proof. Let h ∈ I ∩ J, then there exist a1 , . . . , as , b1 , . . . , bt ∈ A such that h = a1 f1 +· · ·+as fs = b1 g1 +· · ·+bt gt ; thus, (h, a1 , . . . , as , b1 , . . . , bt )T ∈ Syz(H),

15 Elementary Applications of Gr¨ obner Theory

306

Pl and hence (h, a1 , . . . , as , b1 , . . . , bt )T = j=1 rj h j for certain r1 , . . . , rl ∈ A. Pl From this we get that h = j=1 rj h1j , i.e., I ∩ J ⊆ hL}. The other inclusion t u follows from the definition of Syz(H). Now we consider the intersection of an arbitrary finite family of left ideals of A, Ij = hf1j , . . . , ftj j }, 1 ≤ j ≤ r. We define f 11

i := (1, 1, . . . , 1)T , = (f11 , 0, . . . , 0) , f 21 = (f21 , 0, . . . , 0)T . . . , f t1 1 = (ft1 1 , 0, . . . , 0)T , . . . , f 1r = (0, . . . , 0, f1r )T , . . . , f tr r = (0, . . . , 0, ftr r )T , T

and   H = i f 11 f 21 · · · f t1 1 · · · f 1r f 2r · · · f tr r ∈ Mr×l (A), Pr where l = 1 + j=1 tj . Thus, if s ∈ Syz(H), then s T H T = 0. As we observed above, the first coordinates of a generating set for Syz(H) turns out to be a generating set for I1 ∩ · · · ∩ Ir . We can extend the previous results computing the intersection of submodules. For this, let M and N be two submodules of Am , with m ≥ 1. Suppose that M = hf 1 , . . . , f s i and N = hg 1 , . . . , g r i. Thus, h ∈ M ∩ N if and only if there exist a1 , . . . , as , b1 , . . . , bt ∈ A such that h = a 1 f 1 + · · · + a s f s = b1 g 1 + · · · + bt g t . If h = h1 · · · hm


T

, then

−h1 · · · −hm a1 · · · as


T

and −h1 · · · −hm b1 · · · bt


T

are syzygies of the matrices     Im f 1 · · · f s and Im g 1 · · · g t , respectively, where Im is the identity matrix of order m. Mimicking the reasoning for the ideal case, we define the matrix H by   I f ··· f s 0 ··· 0 , H := m 1 Im 0 · · · 0 g 1 · · · g t and it is easy to prove the following result. Proposition 15.3.3. With the above notation, M ∩ N consists exactly of vectors h whose coordinates are precisely the first m elements of vectors of Syz(H). Moreover, the set of vectors which consist of the first m coordinates of each element of a set of generators for Syz(H) is a system of generators for M ∩ N . The previous result can be extended to more than two modules; let M1 , . . . , Mr be submodules of Am , with r ≥ 3. Suppose that each Mi is generated by the columns of some matrix F i ∈ Mm×ti (A), and we define

15.3 Intersections

307



Im Im  H :=  .  .. Im

F1 0 .. . 0

0 F2 .. .

··· ··· .. .

0 0 .. .

   . 

0 ··· Fr

Tr Proposition 15.3.4. With the previous notation, the intersection i=1 Mi is the set of all vectors h which are the first m coordinates of vectors in Syz(H). Furthermore, the set of vectors that consist of the first m entries of each of the vectors of a generator set for Syz(H) is a system of generators for the intersection. Example 15.3.5. Let A = σ(Q)hx, yi be defined by the relation yx = −xy + 1. Over Mon(A) we consider the deglex order, with x  y. Let I := hxy, y 2 } and J := hy} be left ideals of A. We will compute a system of generators of I ∩ J. In this case   1 xy y 2 0 H= . 1 0 0 y Considering the TOPREV order on Mon(A2 ), with e1 < e2 , we  can prove that Syz(H) = (xy, −1, 0, −x)T , (0, −x, y, 0)T , (−x2 y, 0, y, x2 )T . Hence, I ∩ J = hxy, x2 y} = hxy}. Example 15.3.6. We consider Example 6.2.5 in [74]. We will verify the calculations developed therein, but using our algorithms. Let A = σ(Q)hx, yi, with yx = −xy and the deglex order on Mon(A). Let M , N be submodules of A2 , where M := h(x, x), (y, 0)i and N := h(0, y 2 ), (y, x)i. In this case, the matrix H is given by   10xy 0 0 0 1 x 0 0 0   H :=  1 0 0 0 0 y . 0 1 0 0 y2 x So, if we consider the TOP order on Mon(A4 ), with e 4 > e 3 > e 2 > e 1 , then a Gr¨obner basis for the left A-module generated by the columns of H is G = {f i }8i=1 , where f i is the i-th column of H for 1 ≤ i ≤ 6, f 7 = y 2 e 2 and f 8 = −xe 1 − ye 3 . A set of generators for Syz(G) is {y 2 e 2 − e 5 − e 7 , xe 2 − e 3 − e 6 − e 8 , y 2 e 3 − xye 4 − xe 7 , −y 2 e 1 + (x + y)e 4 − ye 8 , xy 2 e 2 − xe 5 − xe 7 , y 3 e 1 + xy 2 e 2 − y 2 e 4 − y 2 e 6 − xe 7 }. Making the computations involved in Theorem 15.2.9 we obtain that Syz(H) = h(0, −xy 2 , y 2 − xy, x, 0)T , (−y 2 , xy, y, x + y, 0, y)T , (y 3 , 0, 0, −y 2 , x, −y 2 )T i. Thus, M ∩ N is generated by (0, −xy 2 ), (−y 2 , xy), (y 3 , 0), but (y 3 , 0) = −y(−y 2 , xy) + (0, −xy 2 ), hence M ∩ N = h(0, xy 2 ), (−y 2 , xy)i.

15 Elementary Applications of Gr¨ obner Theory

308

15.4 Quotients We can use syzygies to compute a set of generators for the quotient of left ideals and modules. Recall that A represents a bijective skew P BW extension of an LGS ring R, so A is left noetherian. Let I be a left ideal of A, say I = hf1 , . . . , fs }, and let G be an arbitrary subset of A. Recall that (I : G) consists of elements h ∈ A such that hg ∈ I for all g P ∈ G, in other words, given s g ∈ G there exist a1g , . . . , asg ∈ A such that hg = i=1 aig fi . Furthermore, (I : G) =

\

(I : g).

g∈G

So, if G = {g1 . . . , gt }, then (I : G) =

t \

(I : gi ).

i=1

In this particular situation we consider the matrix H defined by   g1 f1 · · · fs 0 · · · 0   H :=  ... ... · · · · · · ... · · · ...  . gt 0 · · · 0 f1 · · · fs It follows that (I : G) is the left ideal of A consisting of all elements in A that are the first coordinates of vectors in Syz(H), and a generator set is given by the first coordinates of the vectors in a generator system for Syz(H). Example 15.4.1. Let A be the ring σ(Q)hx, yi, where yx = xy + x. Given I := hx2 y, xy} and G = {x2 , y}, we will compute a generator set for (I : G). For this, we consider the following matrix  2 2  x x y xy 0 0 H := . y 0 0 x2 y xy Now, if Mon(A) is ordered by deglex order, with x  y, and Mon(A2 ) is ordered by TOPREV order, with e 1 > e 2 , then a Gr¨ obner basis for the left 6 A-module generated by columns of H is G = {f i }i=1 , where f i is the i-th column of H and f 6 = y 2 e 2 − 2ye 2 . Further, Syz(G) = h(y − 2)e 1 − e 2 − e 6 , (y − 2)e 1 − xe 3 − e 6 , e 4 − xe 5 , (y − 3)e 5 − xe 6 , (y − 1)e 4 − xye 5 , −3e 4 , xye 5 − x2 e 6 i. From this we compute a system of generators for Syz(H): {(0, 1, −x, 0, 0)T , (0, 0, 0, 1, −x)T , (−xy + 2x, x, 0, 0, y − 3)T , (0, 0, 0, y − 1, −xy)T , 2 (−x y + 2x2 , x2 , 0, −3, xy)T }. Consequently, (I : G) = h−xy + 2x}.

15.5 Presentation of a Module

309

15.5 Presentation of a Module Let M = hf 1 , . . . , f s i be a submodule of Am . There exists a natural surjective homomorphism πM : As −→ M defined by πM (e i ) := f i , where {e i }1≤i≤s is the canonical basis of As . We have the isomorphism πM : As / ker(πM ) ∼ = M, defined by πM (e i ) := f i , where e i := e i + ker(πM ). Since As is a noetherian A-module, ker(πM ) is also a finitely generated module, ker(πM ) := hh 1 , . . . , h s1 i, and hence, we have the exact sequence δ

π

M As1 −− → As −−M → M −→ 0,

(15.5.1)

0 0 with δM := lM ◦ πM , where lM is the inclusion of ker(πM ) in As and πM s1 is the natural surjective homomorphism from A to ker(πM ). We note that ker(πM ) = Syz(M ) = Syz(F ), where F = [f 1 · · · f s ] ∈ Mm×s (A).

Definition 15.5.1. We say that As /Syz(M ) is a presentation of M , the sequence (15.5.1) is a finite presentation of M , and M is a finitely presented module. Theorem 15.2.9 gives a method for computing a presentation of M when A is a bijective skew P BW extension. Moreover, let ∆M be the matrix of δM in the canonical bases of As1 and As ; since Im(δM ) = ker(πM ), then   h11 · · · h1s1    ..  ∈ M ∆M = h1 · · · hs1 =  ... s×s1 (A), .  hs1 · · · hss1 and hence, the columns of ∆M are the generators of Syz(F ). With the notation of Section 15.2, ∆M = Z(F ). Definition 15.5.2. With the previous notation, we say that ∆M is a matrix presentation of M . As we just saw, ∆M is computable when A is a bijective skew P BW extension. We can also compute presentations of quotient modules. Indeed, let N ⊆ M be submodules of Am , where M = hf 1 , . . . , f s i, N = hg 1 , . . . , g t i and M/N = hf 1 , . . . , f s i, then we have a canonical surjective homomorphism As −→ M/N such that a presentation of M/N is given by M/N ∼ = As /Syz(M/N ). But Syz(M/N ) can be computed in the following way: h = (h1 , . . . , hs )T ∈ Syz(M/N ) if and only if h1 f 1 + · · · + hs f s ∈ hg 1 , . . . , g t i if and only if there exist hs+1 , . . . , hs+t ∈ A such that h1 f 1 + · · · + hs f s + hs+1 g 1 + · · · + hs+t g t = 0 if and only if (h1 , . . . , hs , hs+1 , . . . , hs+t )T ∈ Syz(H), where H := [f 1 · · · f s g 1 · · · g t ]. Theorem 15.5.3. With the notation above, a presentation of M/N is given by As /Syz(M/N ), where a set of generators of Syz(M/N ) are the first s coordinates of generators of Syz(H). Thus, a finite presentation of M/N is effective computable.

310

15 Elementary Applications of Gr¨ obner Theory

Example 15.5.4. Let again A be the ring σ(Q)hx, yi, where yx = xy+x. Given M := h(1, 1), (xy, 0), (y 2 , 0), (0, x)i, we will compute a finite presentation for M . For this consider the deglex order on Mon(A), with x  y, and the TOP order over Mon(A2 ), with e 2 > e 1 . A straightforward calculation shows that G = {(1, 1), (xy, 0), (y 2 , 0), (0, x), (x, 0)} is a Gr¨obner basis for M . Moreover, a set of generators for Syz(G) is given by {(x, 0, 0, −1, −1)T , (0, 1, 0, 0, −y + 1)T , (0, −y + 1, x, 0, 0)T , (0, −y − 1, 0, 0, y 2 − 1)T }, and using this, the computations involved in Theorem 15.2.9 show that Syz(M ) = hs 1 = (0, −y + 1, x, 0)T , s 2 = (−xy, 1, 0, y − 1)T , s 3 = (xy 2 + 2xy, −y − 1, 0, 1 − y 2 )T i. Thus, we have obtained the following presentation for M : M∼ = A4 /hs 1 , s 2 , s 3 i.

15.6 Computing Free Resolutions In this section we will compute free resolutions for submodules of Am . Let M be a submodule of Am . We recall that a free resolution of M is an exact sequence of free modules and homomorphisms fr+1

fr−1

fr

f2

f1

f0

· · · −−−→ Asr −→ Asr−1 −−−→ · · · −→ As1 −→ As0 −→ M −→ 0, with si ≥ 0 for each i ≥ 0. We assume that A0 = 0. r is the length of this sequence if sr 6= 0 and si = 0 for i ≥ r + 1. The following theorem describes a simple procedure for constructing a free resolution of M . (0)

Theorem 15.6.1. Let M = hf1 , . . . , f(0) s0 i be a submodule of the free module (0) m A . Let F0 be the matrix whose columns are f1 , . . . , f(0) s0 , and for i ≥ 1 let (i)

Fi := Syz(Fi−1 ) = [f1 · · · f(i) si ]. Then, fr+1

fr−1

fr

f2

f1

f0

· · · −−−→ Asr −→ Asr−1 −−−→ · · · −→ As1 −→ As0 −→ M −→ 0, is a free resolution of M , where (i)

(i)

(i)

fi (eji ) = fji = ((eji )T FiT )T (i)

and {eji }1≤ji ≤si is the canonical basis of Asi .

15.7 The Kernel and Image of a Homomorphism

311

Proof. Observe that the matrix of fi in the canonical bases is Fi , and hence, a resolution of M can be described as a sequence of matrices {Fi }i≥0 , where the columns of Fi are the generators of Syz(Fi−1 ), i ≥ 1. The columns of F0 are the generators of M . We note that Im(fi ) = hFi i = Syz(Fi−1 ) = ker(fi−1 ) for each i ≥ 1, and moreover F0 is a surjective homomorphism. t u We can illustrate this procedure with the following example. Example 15.6.2. Let A be the ring σ(Q)hx, yi, where yx = xy+x. We will calculate a free resolution for the left module M := h(1, 1), (xy, 0), (y 2 , 0), (0, x)i given in Example 15.5.4. There we saw that M ∼ = A4 /hs 1 , s 2 , s 3 i, where s 1 = (0, −y +1, x, 0), s 2 = (−xy, 1, 0, y −1), s 3 = (xy 2 +2xy, −y −1, 0, 1−y 2 ). Now, we must compute a generator set for Syz(s 1 , s 2 , s 3 ). For such task, we consider the deglex order on Mon(A), with x  y, and the TOP order over Mon(A2 ), with e 2 > e 1 . Is not difficult to see that {s 1 , s 2 , s 3 } is a Gr¨obner basis and Syz(s 1 , s 2 , s 3 ) = h(0, y + 1, 1)i. Finally, SyzA (s) = 0, where s = (0, y + 1, 1). Therefore,     0 −xy xy 2 + 2xy   0 2   1 xy y 0 −y + 1 1 −y − 1    F0 = , F1 =   x  , F2 = y + 1 1 0 0 x 0 0 1 0 y − 1 1 − y2 and a free resolution for M is given by −xy xy 2 + 2xy  1 1 −y − 1       1 xy y 2 0  0 0   1 0 0 x 0 y − 1 1 − y2 −−−−−−−−−−−−−−−−−−−−−→ A4 −−−−−−−−−−→ M −−−−−→ 0. 





0  + 1  1 0 −→ A −−−−−−→ A3  y 

0



 −y +    x 

15.7 The Kernel and Image of a Homomorphism Let M ⊆ Am and N ⊆ Al be modules, with M = hf 1 , . . . , f s i, N = hg 1 , . . . , g t i, and let φ : M −→ N be a homomorphism. Then, there exists a matrix Φ = [φji ] of size t × s with entries in A given by φ(f i ) = φ1i g 1 + · · · + φti g t , for each 1 ≤ i ≤ s. In this section, we will calculate a system of generators and presentations for ker φ and Im(φ) using the matrix Φ induced by the homomorphism φ. Let As /Syz(M ) and At /Syz(N ) be presentations of M and N , respectively. We consider the canonical isomorphisms πM : As /Syz(M ) −→ M , πN : At /Syz(N ) −→ N defined by πM (e i ) = f i , for 1 ≤ i ≤ s, and πN (e 0j ) = g j , for 1 ≤ j ≤ t, where {e i }1≤i≤s is the canonical basis of As and {e 0j }1≤j≤t is the canonical basis of At . Thus, we have the following commutative diagram

312

15 Elementary Applications of Gr¨ obner Theory φ

−−−−→

M   y

N   y

(15.7.1)

φ

As /Syz(M ) −−−−→ At /Syz(N ) where the vertical arrows are the isomorphisms (πM )−1 and (πN )−1 . Hence, φ(e i ) = (πN )−1 ◦φ◦πM (e i ) = φ1i e 01 +· · ·+φti e 0t , for each 1 ≤ i ≤ s. Note that ker(φ) ∼ = ker(φ) and Im(φ) ∼ = Im(φ): in fact, is enough to see that (πM )−1 restricted to ker(φ) is an isomorphism between ker(φ) and ker(φ); analogously for Im(φ) and Im(φ). Let m ∈ ker(φ), then m = a1 f 1 + · · · + as f s and thus (πN )−1 (φ(h1 f 1 + · · · + hs f s )) = 0 = φ((πM )−1 (h1 f 1 + · · · + hs f s )) = φ(h1 e 1 + · · · + hs e s ) = h1 φ(e 1 ) + · · · + hs φ(e s ) = h1 (φ11 e 01 + · · · + φt1 e 0t ) + · · · + hs (φ1s e 01 + · · · + φts e 0t ) = (h1 φ11 + · · · + hs φ1s )e 01 + · · · + (h1 φt1 + · · · + hs φts )e 0t . This implies that (h1 φ11 + · · · + hs φ1s )e 01 + · · · + (h1 φt1 + · · · + hs φts )e 0t ∈ Syz(N ). With Theorem 15.2.9 we can compute a system of generators for Syz(N ) = hs 1 , . . . , s t1 i ⊆ At . Hence, there exist as+1 , . . . , as+t1 ∈ A such that     φ11 φ1s  ..   ..  a1  .  + · · · + as  .  + as+1 s 1 + · · · + as+t1 s t1 = 0. φt1

φts

Conversely, if (a1 , . . . , as ) ∈ ker(φ), the above calculations allow us to conclude that a1 f 1 + · · · + as f s ∈ ker(φ). Thus, we have obtained that a1 f 1 + · · · + as f s ∈ ker(φ) ⇔ (a1 , . . . , as ) ∈ ker(φ). We have proved the following theorem. Theorem 15.7.1. With the above notation, let   H = Φ 1 · · · Φ s s 1 · · · s t1 , where Φi is the i-th column of the matrix Φ, for 1 ≤ i ≤ s. Then, (a1 , . . . , as , as+1 , . . . , as+t1 )T ∈ Syz(H) ⇔ a1 f1 + · · · + as fs ∈ ker(φ). Thus, if {z1 , . . . , zv } ⊂ As+t1 is a system of generators of Syz(H), let z0k ∈ As be the vector obtained from zk by omitting the last t1 components, 1 ≤ k ≤ v, then {z01 , . . . , z0v } is a system of generators for ker(φ). Moreover, if z01 = (h11 , . . . , h1s )T , . . . , z0v = (hv1 , . . . , hvs )T , then {h11 f1 + · · · + h1s fs , . . . , hv1 f1 + · · · + hvs fs } is a system of generators for ker(φ).

15.7 The Kernel and Image of a Homomorphism

313

A presentation of ker(φ) is given in the following way. Corollary 15.7.2. In the notation of this section, a presentation of ker(φ) is given by Av /K, where   K = Syz(ker(φ)) = Syz h11 f1 + · · · + h1s fs · · · hv1 f1 + · · · + hvs fs . Now we also want to compute an explicit presentation of ker(φ). We assume that we have computed a system of generators for Syz(M ) = hw 1 , . . . , w s1 i ⊆ As . We know that a presentation of ker(φ) is given by ker(φ) ∼ = Av /K 0 , where K 0 = Syz(ker(φ)) = Syz(hz 01 , . . . , z 0v i). But (l1 , . . . , lv ) ∈ Syz(hz 01 , . . . , z 0v i) if and only if there exist lv+1 , . . . , lv+s1 ∈ A such that l1 z 01 + · · · + lv z 0v + lv+1 w 1 + · · · + lv+s1 w s1 = 0. Thus, we have proved the following corollary. Corollary 15.7.3. In the above notation, let   L = z01 · · · z0v w1 · · · ws1 . If {l1 , . . . , lq } ⊆ Av+s1 is a system of generators of Syz(L), let l0k ∈ Av be the vector obtained from lk by omitting the last s1 components, 1 ≤ k ≤ q, then {l01 , . . . , l0q } is a system of generators for K 0 , and hence, a presentation of ker(φ) is Av /K 0 . We consider now the image of the homomorphism φ : M −→ N in (15.7.1). Then the following result is clear from the above discussion. Corollary 15.7.4. A system of generators for Im(φ) is given by Im(φ) = hφ11 g1 + · · · + φt1 gt , . . . , φ1s g1 + · · · + φts gt i. A presentation of Im(φ) is As /I, where   I = Syz φ11 g1 + · · · + φt1 gt . . . φ1s g1 + · · · + φts gt . Many of the theoretical results of the present chapter will be illustrated by concrete examples from the last chapter. We conclude this section by giving an explicit presentation of Im(φ). We know that Im(φ) = hφ11 e 01 + · · · + φt1 e 0t , . . . , φ1s e 01 + · · · + φts e 0t i, thus a presentation of Im(φ) is given by Im(φ) ∼ = As /Syz(Im(φ)). Let T (h1 , . . . , hs ) ∈ Syz(Im(φ)), then there exist hs+1 , . . . , hs+t1 ∈ A such that     φ11 φ1s  ..   ..  h1  .  + · · · + hs  .  + hs+1 u 1 + · · · + hs+t1 u t1 = 0. φt1

φts

Thus, we have proved the following corollary. Corollary 15.7.5. Let H be the matrix in Theorem 15.7.1. If {z1 , . . . , zv } ⊆ As+t1 is a system of generators of Syz(H), let z0k ∈ As be the vector obtained from zk by omitting the last t1 components, 1 ≤ k ≤ v. Then, {z01 , . . . , z0v } is a system of generators for Syz(Im(φ)) and As /Syz(Im(φ)) is a presentation of Im(φ).

314

15 Elementary Applications of Gr¨ obner Theory


Example 15.7.6. Let A := σ(Q[x1 ])hx2 , x3 i = O3 2, 12 , 3 . Let M := hf 1 , f 2 i ⊆ A2 , where f 1 = x21 x22 e1 + x2 x3 e2 and f 2 = 2x1 x2 x3 e1 + x2 e2 . In a similar way as was done in Example 15.2.10, we can prove that Syz(M ) = 0 and hence M is free with basis {f 1 , f 2 }. Let N := hg 1 , g 2 i ⊆ A2 , where g 1 = (2x1 + 1)x22 e1 + x2 x3 e2 and g 2 = (4x21 + x1 )e1 + x1 x22 x3 e2 . We consider the homomorphism φ : M −→ N given by φ(f 1 ) := g 1 + 2g 2 φ(f 2 ) := x1 g 1 + g 2 . The matrix Φ induced by φ is 

 1 x1 Φ= . 2 1 Using the results of Section 15.2 we verify that   x1 x2 Syz(N ) = , −1 so the matrix H of Theorem 15.7.1 is   1 x1 x1 x2 H= . 2 1 −1 Again, by the results of Section 15.2, a system of generators of Syz(H) is  2 1  1  1 2 2 2  2x1 − 2 x1 + x21 x2 − 12 x1 x2 2 x1 + 2x1 x2 + 2 x1 x2 + x1 x2   −2x1 + 1 − 2x21 x2 + x1 x2  ,   , − 12 − 2x21 x22 − 32 x1 x2 2   1 1 1 2 2 4x1 − 3x1 + 2 4x1 x2 − 2 x1 x2 + x1 − 2 and by Theorem 15.7.1, a system of generators of ker(φ) is
{ 2x21 − 12 x1 + x21 x2 − 12 x1 x2 f 1 + (−2x1 + 12 − 2x21 x2 + x1 x2 )f 2 , ( 12 x1 + 2x21 x2 + 12 x1 x2 + x21 x22 )f 1 + (− 12 − 2x21 x22 − 32 x1 x2 )f 2 }, and a system of generators of Im(φ) is {φ(f 1 ), φ(f 2 )} = {g 1 +2g 2 , x1 g 1 +g 2 }. We finish with some key remarks about the right version of the results presented in the present chapter. Remark 15.7.7. From Remark 14.5.7 (and using also Remark 9.1.2) we conclude that the applications established for left ideals and submodules of left free modules also have their right versions for right skew P BW extensions of RGS rings. In particular, if A is a bijective left skew P BW extension, then A is a right P BW extension; the right A-module of syzygies of a right A-submodule M of the free right A-module Am will be denoted by Syzr (M ), and all results about right syzygies are valid assuming that R is RGS. For example, if M = hf 1 , . . . , f s i is a submodule of the right A-module Am and

15.7 The Kernel and Image of a Homomorphism

315

πM : As → M is the canonical homomorphism defined by πM (e i ) := f i , 1 ≤ i ≤ s, then   ker(πM ) = Syzr (M ) = Syzr (F ) = ker(F ), with F := f 1 · · · f s . With these remarks, the following corollary is established. Corollary 15.7.8. Let R be an RGS ring. If A = σ(R)hx1 , . . . , xn i is a bijective (right) skew P BW extension of R, then A is RGS.

Chapter 16

Computing Tor and Ext

In this chapter we compute the Tor and Ext modules over skew P BW extensions. By computing we mean to give presentations of TorA r (M, N ), where M is a finitely generated centralizing subbimodule of Am , m ≥ 1, and N is a left A-submodule of Al , l ≥ 1. For ExtrA (M, N ), M is a left A-submodule of Am and N is a finitely generated centralizing subbimodule of Al . The technique we will use for computing the modules Tor and Ext is very simple: we compute presentations of submodules of free modules using syzygies and Gr¨obner bases as we saw in the previous chapters, and then, we compute free resolutions and the corresponding homology modules. As applications, we will test stably-freeness, reflexiveness, and we will compute the torsion, the dual and the grade of a given submodule of a free module.

16.1 Centralizing Bimodules For the computations of the present chapter we have to consider a special type of subbimodule and also the following notion. Definition 16.1.1. Let R be a ring. We will say that R is Gr¨ obner soluble (GS) if R is both LGS and RGS. From now on in this chapter we will assume that R is a GS ring and A = σ(R)hx1 , . . . , xn i is a bijective skew P BW extension of R. Definition 16.1.2 ([74]). Let M be an A-bimodule. The centralizer of M is defined by CenA (M ) := {f ∈ M | f a = af, for every a ∈ A}. We say that M is centralizing if M is generated as a left A-module (and, equivalently, as a right A-module) by its centralizer. M is a finitely generated centralizing A-bimodule if there exists a finite set of elements in CenA (M ) that generates M . © Springer Nature Switzerland AG 2020 W. Fajardo et al., Skew PBW Extensions, Algebra and Applications 28, https://doi.org/10.1007/978-3-030-53378-6_16

317

318

16 Computing Tor and Ext

Example 16.1.3. Let M :=A hf 1 , . . . , f s iA be a finitely generated centralizing A-subbimodule of Am ; note that every f i has the form f i = (a1 , . . . , am )T , with aj ∈ A, 1 ≤ j ≤ m. Therefore, aaj = aj a, for any a ∈ A, i.e., aj ∈ Z(A). Observe that the center of most nontrivial skew P BW extensions is too small, so it is not easy to give enough non-examples of this type of bimodule. However, for the applications we will consider, the centralizing bimodule we need is A = A h1iA . Nevertheless, we now present an example of a finitely generated centralizing subbimodule: Consider the enveloping algebra A := U (G) of the 3-dimensional Lie algebra G over a field K with [x1 , x2 ] = 0 = [x1 , x3 ] and [x2 , x3 ] = x1 . Observe that x1 ∈ Z(U (G)), and hence, 2 A h(x1 , x1

+ 1, 1), (1, x1 − 1, x21 )iA

is a centralizing subbimodule of A3 . Since we have to consider subbimodules, we will use the following special notation in this chapter: A hXi denotes the left A-module generated by X, hXiA the right A-module generated by X and A hXiA the bimodule generated by X; if X is contained in the centralizer, then A hXiA = A hXi = hXiA .

16.2 Computation of M ⊗ N In this second section we compute a presentation of the left A-module M ⊗A N , where M :=A hf 1 , . . . , f s iA is a finitely generated centralizing Asubbimodule of Am , i.e., f i ∈ CenA (M ), 1 ≤ i ≤ s, and N :=A hg 1 , . . . , g t i is a finitely generated left A-submodule of Al . This computation has also been considered in [160] over commutative rings. For the presentation we will compute an explicit set of generators of Syz(M ⊗A N ). We simplify the notation by writing M ⊗ N instead of M ⊗A N . We start with the following preliminary well-known result. Proposition 16.2.1. Let S be an arbitrary ring, M be a right module over S and N be a left S-module. Let mj ∈ M, gj ∈ N , 1 ≤ j ≤ t, such that N =A hg1 , . . . , gt i. Then, m1 ⊗ g1 + · · · + mt ⊗ gt = 0 if and only if there exist elements m0v ∈ M , 1 ≤ v ≤ r, and a matrix H := [hjv ] ∈ Mt×r (S), such that [m1 · · · mt ] = [m01 · · · m0r ]H T and H T [g1 · · · gt ]T = 0. Thus, the left S-module expanded by the columns of H is contained in Syz(N ). Proof. ⇒) We consider the following presentation of N F

F

1 0 L −→ S t −→ N → 0,

where F0 (e j ) := g j and L is a free left S-module. We get the exact sequence

16.2 Computation of M ⊗ N i

319 ⊗F

i

⊗F

1 0 −−−→ M ⊗ N → 0, M ⊗ L −M −−−→ M ⊗ S t −M

and since m 1 ⊗ g 1 + · · · + m t ⊗ g t = 0, we have m 1 ⊗ e 1 + · · · + m t ⊗ e t ∈ ker(iM ⊗ F0 )= Im(iM ⊗ F1 ). Thus, there exist elements m 0v ∈ M and l v ∈ L, 1 ≤ v ≤ r, such that (iM ⊗F1 )(m 10 ⊗l 1 +· · ·+m 0r ⊗l r ) = m 1 ⊗e 1 +· · ·+m t ⊗e t , and hence, if F1 (l v ) := h v := (h1v , . . . , htv )T , 1 ≤ v ≤ r, then m 1 ⊗ e 1 + · · · + m t ⊗ e t = m 01 ⊗ h 1 + · · · + m 0r ⊗ h r r r X X =( m 0v htv ) ⊗ e t . m v0 h1v ) ⊗ e 1 + · · · + ( v=1

v=1

0 Since M ⊗ S t ∼ = M t , we have m j = v=1 m v hjv , 1 ≤ j ≤ t. Moreover, Pt j=1 hjv g j = 0 since h v ∈ Im(F1 ) = ker(F0 ) = Syz(N ), for each 1 ≤ v ≤ r. ⇐) We have Pr Pr m 1 ⊗ g 1 + · · · + m t ⊗ g t = ( v=1 m 0v h1v ) ⊗ g 1 + · · · + ( v=1 m v0 htv ) ⊗ g t ,

Pr

from this we get m 1 ⊗g 1 +· · ·+m t ⊗g t = m 10 ⊗(

Pt

j=1

hjv g j )+· · ·+m r0 ⊗(

Pt

j=1

hjv g j ) = 0. u t

Remark 16.2.2. If S is a left noetherian ring, then in the previous proof we can take H = Syz(N ) and L = S r . Now we are able to prove the main result of the present section. Theorem 16.2.3. Let M :=A hf1 , . . . , fs iA be a finitely generated centralizing A-subbimodule of Am , with fi ∈ CenA (M ), 1 ≤ i ≤ s, and N :=A hg1 , . . . , gt i be a finitely generated left A-submodule of Al . Then, M ⊗N ∼ = Ast /Syz(M ⊗ N ), where Syz(M ⊗ N ) =A h[Syz(M ) ⊗ It | Is ⊗ Syz(N )]i. Proof. Since every f i ∈ CenA (M ), it is clear that M ⊗ N =A hf 1 ⊗ g 1 , . . . , f 1 ⊗ g t , . . . , f s ⊗ g 1 , . . . , f s ⊗ g t i. Let Syz(M ) := hf 01 , . . . , f r0 i be the left A-module of syzygies of the left Amodule M , and Syz(N ) := hg 10 , . . . , g 0p i be the left A-module of syzygies of the left A-module N , with f 01 := (f11 , . . . , fs1 )T , . . . , f 0r := (f1r , . . . , fsr )T and g 01 := (g11 , . . . , gt1 )T , . . . , g p0 := (g1p , . . . , gtp )T . In matrix notation,

320

16 Computing Tor and Ext



f11 . . .  .. Syz(M ) =  . . . . fs1 . . .





f1r g11 . . . ..  , Syz(N ) =  ..  . ... .  gt1 . . . fsr

 g1p .  .. . gtp

Then, f11 f 1 + · · · + fs1 f s = 0 . .. f1r f 1 + · · · + fsr f s = 0 and g11 g 1 + · · · + gt1 g t = 0 .. . g1p g 1 + · · · + gtp g t = 0. We note that any of the following tr vectors has st entries and belongs to Syz(M ⊗ N ) (f11 , 0, . . . , 0, . . . , fs1 , 0, . . . , 0)T . .. (0, 0, . . . , f11 , . . . , 0, 0, . . . , fs1 )T .. . (f1r , 0, . . . , 0, . . . , fsr , 0, . . . , 0)T . .. (0, 0, . . . , f1r , . . . , 0, 0, . . . , fsr )T . Since for every 1 ≤ i ≤ s, f i ∈ CenA (M ), any of the following ps vectors has st entries and belongs to Syz(M ⊗ N ) (g11 , . . . , gt1 , . . . , 0, . . . , 0)T . .. (0, . . . , 0, . . . , g11 , . . . , gt1 )T .. . (g1p , . . . , gtp , . . . , 0, . . . , 0)T ... (0, . . . , 0, . . . , g1p , . . . , gtp )T . We can dispose these tr + ps vectors as columns in a matrix [C | B] of size st × (tr + ps), where

16.2 Computation of M ⊗ N



f11     0   C :=   fs1    0 

g11  ..  .   gt1   B :=  ...   0   .  ..

321



. . . 0 . . . f1r .. . ... . . . f11 . . . 0 .. . ... . . . 0 . . . fsr .. . ... . . . fs1

... 0  ..  .  . . . f1r    ..  = Syz(M ) ⊗ It .  ... 0    ..  . . . . 0 . . . fsr

... .. .

0 ... .. . ... ... 0 ... . . .. . . ... . . . g11 . . . .. .. . . ...

g1p .. .

... .. .

gtp . . . .. . . . . 0 ... .. .. . .

0 . . . gt1 . . . 0 . . . But B can be changed by  g11 . . . g1p  .. .. ..  . . .   gt1 . . . gtp   .. .. ..  . . .   0 ... 0   . . .  .. .. ..

... 0 ... . . . . . .. .. ... 0 ... . . .. .. . . . . . . g11 . . . . ..

... .. .

0 . . . 0 . . . gt1 . . .

 0 ..  .   0   ..  . .   g1p   ..  .  gtp

 0 ..  .   0   ..  = I ⊗ Syz(N ). s .   g1p   ..  .  gtp

Thus, we have proved that A h[Syz(M ) ⊗ It | Is ⊗ Syz(N )]i ⊆ Syz(M ⊗ N ). Now we assume that h := (h11 , . . . , h1t , . . . , hs1 , . . . , hst )T ∈ Syz(M ⊗ N ), then h11 (f 1 ⊗ g 1 ) + · · · + h1t (f 1 ⊗ g t ) + · · · + hs1 (f s ⊗ g 1 ) + · · · + hst (f s ⊗ g t ) = 0. From this we get that m 1 ⊗ g 1 + · · · + m t ⊗ g t = 0, where m j = h1j f 1 + · · · + hsj f s ∈ M , with 1 ≤ j ≤ t. From Proposition 16.2.1, there exist polynomials ajv ∈ A and Pr Pt vectors m 0v ∈ M such that m j = v=1 ajv m 0v and j=1 ajv g j = 0, for each 1 ≤ v ≤ r. This means that (a1v , . . . , atv )T ∈ Syz(N ) for each 1 ≤ v ≤ r. Since m 0v ∈ M there exist quv ∈ A such that m 0v = q1v f 1 + · · · + qsv f s , and then

322

16 Computing Tor and Ext

Ps

hi1 f i = a11 (q11 f 1 + · · · + qs1 f s ) + · · · + a1r (q1r f 1 + · · · + qsr f s ) .. .

Ps

hit f i = at1 (q11 f 1 + · · · + qs1 f s ) + · · · + atr (q1r f 1 + · · · + qsr f s ).

i=1

i=1

From this we get that Ps

− (a11 qi1 + · · · + a1r qir ))f i = 0 .. .

Ps

− (at1 qi1 + · · · + atr qir ))f i = 0

i=1 (hi1

i=1 (hit

i.e., (h11 − (a11 q11 + · · · + a1r q1r ), . . . , hs1 − (a11 qs1 + · · · + a1r qsr ))T ∈ Syz(M ) .. . (h1t − (at1 q11 + · · · + atr q1r ), . . . , hst − (at1 qs1 + · · · + atr qsr ))T ∈ Syz(M ). This implies that (h11 , . . . , hs1 )T − (a11 q11 + · · · + a1r q1r , . . . , a11 qs1 + · · · + a1r qsr )T ∈ Syz(M ) .. . T (h1t , . . . , hst ) − (at1 q11 + · · · + atr q1r , . . . , at1 qs1 + · · · + atr qsr )T ∈ Syz(M ).

Then, (hi1 )si=1 = (a11 q11 + · · · + a1r q1r , . . . , a11 qs1 + · · · + a1r qsr ) + (f11 , . . . , fs1 ) ... s = (at1 q11 + · · · + atr q1r , . . . , at1 qs1 + · · · + atr qsr ) + (f1t , . . . , fst ), (hit )i=1

with (f11 , . . . , fs1 )T , . . . , (f1t , . . . , fst )T ∈ Syz(M ). From this we get h11 = f11 + a11 q11 + · · · + a1r q1r . .. h1t = f1t + at1 q11 + · · · + atr q1r .. . hs1 = fs1 + a11 qs1 + · · · + a1r qsr .. . hst = fst + at1 qs1 + · · · + atr qsr , and hence h is a linear combination of the columns of the following matrix [Syz(M ) ⊗ It | D], where

16.3 Computation of Tor

323



a11  ..  .   at1   D :=  ...   0   .  ..



. . . a1r . . . .. .. . . ... . . . atr . . . .. .. . . . . .

0 ... 0 .. .. ..  . . .   0 ... 0   .. .. ..  . . . .   . . . 0 . . . a11 . . . a1r   .. .. .. .. ..  . . ... . . .  0 . . . 0 . . . at1 . . . atr

But A hDi ⊆ hIs ⊗ Syz(N )i, and hence, h ∈A h[Syz(M ) ⊗ It | Is ⊗ Syz(N )]i. This completes the proof of the theorem. t u

16.3 Computation of Tor Using syzygies we present next an easy procedure for computing the left A-modules TorA r (M, N ) for r ≥ 0, where M :=A hf 1 , . . . , f s iA is a finitely generated centralizing A-subbimodule of Am , i.e., f i ∈ CenA (M ), 1 ≤ i ≤ s, and N :=A hg 1 , . . . , g t i is a finitely generated left A-submodule of Al . By computing we mean to find a presentation and a system of generators of TorA r (M, N ). Our computations of course extend the well-known results on commutative polynomial rings, see for example [160], Proposition 7.1.3 (see also [239]). For r = 0, the computation was given in Theorem 16.2.3. So we assume that r ≥ 1. Presentation of TorA r (M, N ), r ≥ 1. Step 1. We first compute left presentations of M and N , M∼ = As /Syz(M ), N ∼ = At /Syz(N ); recall that Syz(M ) and Syz(N ) are the kernels of the natural left A-homomorphisms πM : As −→ M and πN : At −→ N defined by πM (e i ) := f i , πN (e 0j ) := g j , 1 ≤ i ≤ s, 1 ≤ j ≤ t, where {e i }1≤i≤s is the canonical basis of Am and {e 0j }1≤j≤t is the canonical basis of At (see Section 15.5). However, since every generator f i of M is in CenA (M ), πM is a A-bimodule homomorphism and hence Syz(M ) is a subbimodule of As . Thus, As /Syz(M ) is an A-bimodule. Step 2. We compute a free resolution of the left A-module At /Syz(N ) using Theorem 15.6.1, gr+2

gr+1

gr−1

gr

g2

g1

g0

· · · −−−→ Atr+1 −−−→ Atr −→ Atr−1 −−−→ · · · −→ At1 −→ At0 −→ At /Syz(N ) −→ 0. Step 3. We consider the complex i⊗gr+2

i⊗gr+1

i⊗gr

i⊗g2

· · · −−−−→ As /Syz(M ) ⊗ Atr+1 −−−−→ As /Syz(M ) ⊗ Atr −−−→ · · · −−−→ i⊗g1

As /Syz(M ) ⊗ At1 −−−→ As /Syz(M ) ⊗ At0 −→ 0,

324

16 Computing Tor and Ext

where i is the identical homomorphism of As /Syz(M ); then we have the isomorphism of left A-modules A s t ∼ TorA r (M, N ) = Torr (A /Syz(M ), A /Syz(N )) = ker(i ⊗ gr )/Im(i ⊗ gr+1 ).

If Gr is the matrix of gr in the canonical bases, then ker(i⊗gr ) = Syz(Is ⊗Gr ) and we get ∼ TorA (16.3.1) r (M, N ) = Syz(Is ⊗ Gr )/A hIs ⊗ Gr+1 i. Step 3. Let qr be the number of generators of Syz(Is ⊗ Gr ), then by Theorem 15.5.3, a presentation of TorA r (M, N ) is given by ∼ qr TorA r (M, N ) = A /Syz(Syz(Is ⊗ Gr )/A hIs ⊗ Gr+1 i),

(16.3.2)

where a set of generators of Syz(Syz(Is ⊗ Gr )/A hIs ⊗ Gr+1 i) is given by the first qr coordinates of generators of Syz[Syz[Is ⊗ Gr ]|Is ⊗ Gr+1 ].

A system of generators of TorA r (M, N ), r ≥ 1. By (16.3.1), a system of generators of TorA r (M, N ) is given by a system of generators of Syz(Is ⊗ Gr ). Thus, if Syz[Is ⊗ Gr ] := [h 1 · · · h qr ], then f g TorA r (M, N ) =A hh 1 , . . . , h qr i, fv := h v +A hIs ⊗ Gr+1 i, 1 ≤ v ≤ qr . where h

16.4 Computation of Hom In this section we will compute the right A-module HomA (M, N ), where M :=A hf 1 , . . . , f s i is a finitely generated left A-submodule of Am and N :=A hg 1 , . . . , g t iA is a finitely generated centralizing A-subbimodule of Al , i.e., g i ∈ CenA (N ), 1 ≤ i ≤ s. By computing HomA (M, N ) we mean to find a presentation of HomA (M, N ) and find a specific set of generators for HomA (M, N ). We divide the procedure into four steps. Step 1. Presentations of M and N . In order to compute a presentation of HomA (M, N ) we first compute presentations of M and N . Thus, we have M∼ = As /Syz(M ), N ∼ = At /Syz(N ), where Syz(M ) and Syz(N ) are the kernels of the natural left A-homomorphisms

16.4 Computation of Hom

325

πM : As −→ M and πN : At −→ N defined by πM (e i ) := f i , πN (e 0j ) := g j , 1 ≤ i ≤ s, 1 ≤ j ≤ t, where {e i }1≤i≤s is the canonical basis of As and {e 0j }1≤j≤t is the canonical basis of At (see Section 15.5). Since every generator g j of N is in CenA (N ), πN is a bimodule homomorphism, Syz(N ) is a subbimodule of At and hence At /Syz(N ) is an A-bimodule. Thus, HomA (M, N ) ∼ = HomA (As /Syz(M ), At /Syz(N )), and we can compute a presentation of HomA (As /Syz(M ), At /Syz(N )) instead of HomA (M, N ). We recall that Syz(M ) and Syz(N ) are computed by the syzygies of the matrices     FM = f 1 · · · f s , F N = g 1 · · · g t , i.e., Syz(M ) = Syz(FM ), Syz(N ) = Syz(FN ) (see Section 15.5). Step 2. HomA (As /Syz(M ), At /Syz(N )) as a kernel. Recall that As and l A are left noetherian A-modules, so Syz(M ) is generated by a finite set of s1 elements and Syz(N ) is generated by t1 elements. Therefore, we have 0 surjective homomorphisms of left A-modules, πM : As1 −→ Syz(M ) and 0 t1 πN : A −→ Syz(N ), hence the following sequences of left A-modules are exact jM δ (16.4.1) As1 −−−M−→ As −−−−→ As /Syz(M ) −−−−→ 0 δ

jN

At1 −−−N−→ At −−−−→ At /Syz(N ) −−−−→ 0 0 0 where δM := lM ◦πM , δN := lN ◦πN , lM , lN denote inclusions, and jM , jN are the canonical A-homomorphisms. Since At /Syz(N ) is an A-bimodule, from (16.4.1) we get the exact sequence of right A-modules p

0 → HomA (As /Syz(M ), At /Syz(N )) − → HomA (As , At /Syz(N )) d

− → HomA (As1 , At /Syz(N )), where d(α) := α ◦ δM ,

for α ∈ HomA (As , At /Syz(N ));

p is defined in a similar way. Since Im(p) = ker(d), we have the following isomorphism of right A-modules: HomA (As /Syz(M ), At /Syz(N )) ∼ = ker(d).

(16.4.2)

Step 3. Computation of HomA (As , At /Syz(N )) & HomA (As1 , At /Syz(N )) by presentations. According to (16.4.2), we must compute presentations of the right A-modules HomA (As , At /Syz(N )) and HomA (As1 , At /Syz(N )). Next we show the details for the first case, the second one is similar. We have the isomorphism of right A-modules (actually, of A-bimodules): HomA (As , At /Syz(N )) ∼ = [At /Syz(N )]s s t f ∈ HomA (A , A /Syz(N )) 7→ (f (e 1 ), . . . , f (e s ))T .

326

16 Computing Tor and Ext

Let ∆N ∈ Mt×t1 (A) be the matrix of δN in the canonical bases of At1 and At ; we have the following isomorphism of right A-modules: [At /Syz(N )]s ∼ = Ats /hIs ⊗ ∆N iA ((a11 , . . . , at1 ), . . . , ((a1s , . . . , ats ))T 7→ (a11 , . . . , at1 , . . . , a1s , . . . , ats )T . Hence, a presentation of HomA (As , At /Syz(N )) is HomA (As , At /Syz(N )) ∼ = Ats /hIs ⊗ ∆N iA ,

(16.4.3)

where the isomorphism is defined in the following way: if f ∈ HomA (As , At /Syz(N )) and f (e i ) := (a1i , . . . , ati )T , 1 ≤ i ≤ s, then θs,t

HomA (As , At /Syz(N )) −−→ Ats /hIs ⊗ ∆N iA f 7→ (a11 , . . . , at1 , . . . , a1s , . . . , ats )T . In the same way we have the isomorphism of right A-modules θs

,t

1 HomA (As1 , At /Syz(N )) −−− → Ats1 /hIs1 ⊗ ∆N iA .

(16.4.4)

Step 4. Presentation of HomA (As /Syz(M ), At /Syz(N )). From (16.4.3) and (16.4.4) we define the homomorphism d of right A-modules by the following commutative diagram d

HomA (As , At /Syz(N )) −−−−→ HomA (As1 , At /Syz(N ))    θ θs,t y y s1 ,t Ats /hIs ⊗ ∆N iA

d

−−−−→

Ats1 /hIs1 ⊗ ∆N iA

−1 i.e., d := θs1 ,t ◦ d ◦ θs,t . Hence, ker(d) ∼ = ker(d), and from (16.4.2), a presentation of ker(d) gives a presentation of HomA (As /Syz(M ), At /Syz(N )). We can give the explicit definition of d: if a := (a11 , . . . , at1 , . . . , a1s , . . . , ats )T ∈ Ats /hIs ⊗ ∆N iA , then

d(a) = (

Ps

k=1 δk1 a1k , . . . ,

Ps

Ps Ps k=1 δk1 atk , . . . , k=1 δks1 a1k , . . . , k=1 δks1 atk ) (∆T M ⊗ It )a,

=

where ∆M := [δij ] ∈ Ms×s1 (A) is the matrix of δM in the canonical bases. In order to apply the right version of results of the previous chapter, we have to interpret A as a right bijective skew P BW extension of the GS ring R and rewrite the entries of matrices over A with the coefficients on the right side. With this, note that a ∈ ker(d) ⇐⇒ d(a) = 0 ⇐⇒ (∆TM ⊗ It )a ∈ hIs1 ⊗ ∆N iA ⇐⇒

16.5 Computation of Ext

327

the coordinates of a are the first st entries of some element of Syzr ([(∆M ⊗ It )T |Is1 ⊗ ∆N )]) ⇐⇒ a ∈ hU iA /hIs ⊗ ∆N iA , where hU iA is the right A-module generated by the columns of the matrix U defined by columns of U := first st coordinates of generators of Syzr ([(∆M ⊗ It )T | Is1 ⊗ ∆N )]). Thus, we have proved that ker(d) = hU iA /hIs ⊗ ∆N iA , and we get the following theorem. Theorem 16.4.1. With the notation above, HomA (M, N ) ∼ = hU iA /hIs ⊗ ∆N iA ,

(16.4.5)

and a presentation of hU iA /hIs ⊗ ∆N iA is a presentation for HomA (M, N ). Observe that a system of generators of hU iA gives a system of generators for HomA (M, N ).

16.5 Computation of Ext Using syzygies we now describe an easy procedure for computing the right A-modules ExtrA (M, N ), for r ≥ 0, where M :=A hf 1 , . . . , f s i is a finitely generated left A-submodule of Am and N :=A hg 1 , . . . , g t iA is a finitely generated centralizing A-subbimodule of Al , i.e., g i ∈ CenA (N ), 1 ≤ i ≤ s. By computing ExtrA (M, N ) we mean to find a presentation of ExtrA (M, N ) and a system of generators. For r = 0, Ext0A (M, N ) = HomA (M, N ) and the computation was given in the previous section. So we assume that r ≥ 1. Presentation of ExtrA (M, N ), r ≥ 1. Step 1. As in the previous section, we compute presentations of M and N , M∼ = As /Syz(M ), N ∼ = At /Syz(N ). Step 2. Using Theorem 15.6.1 we compute a free resolution of the left A-module As /Syz(M ), fr+2

fr+1

fr

fr−1

f2

f1

f0

· · · −−−→ Asr+1 −−−→ Asr −→ Asr−1 −−−→ · · · −→ As1 −→ As0 −→ As /Syz(M ) −→ 0. For every r ≥ 1, let Fr be the matrix of fr in the canonical bases. Step 3. As we observed in the previous section, At /Syz(N ) is an Abimodule, so we get the complex of right A-modules

328

16 Computing Tor and Ext f∗

f∗

1 r 0 −→ HomA (As0 , At /Syz(N )) −→ · · · −→ HomA (Asr , At /Syz(N )) ∗ fr+1

∗ fr+2

−−−→ HomA (Asr+1 , At /Syz(N )) −−−→ · · · According to Remark 15.7.7, we can interpret A as a right bijective skew P BW extension of the GS ring R, and hence, we can apply any of results of the previous chapter, but in the right version. Recall that ∗ ExtrA (M, N ) ∼ )/Im(fr∗ ). = ExtrA (As /Syz(M ), At /Syz(N )) = ker(fr+1 (16.5.1) By (16.4.3), for each r ≥ 1, a presentation of HomA (Asr , At /Syz(N )) is given by the following isomorphism of right A-modules HomA (Asr , At /Syz(N )) ∼ = Atsr /hIs ⊗ Syz(N )iA . r

Fr∗

Step 4. We can associate a matrix to the homomorphism fr∗ , as follows: as we saw in (15.7.1), we have the following commutative diagram ∗ fr+1

HomA (Asr , At /Syz(N )) −−−−→ HomA (Asr+1 , At /Syz(N ))     y y ∗ fr+1

Atsr /Kr

Atsr+1 /Kr+1

−−−−→

where Kr := hIsr ⊗ Syz(N )iA and the vertical arrows are isomorphisms of right A-modules obtained by concatenating the columns of matrices of ∗ HomA (Asr , At /Syz(N )); moreover, the matrices of homomorphisms fr+1 and ∗ fr+1 coincide (see Section 15.7). So, we can replace the above complex with the following equivalent complex f∗

f∗

∗ fr+1

∗ fr+2

1 r 0 −→ Ats0 /K0 −→ · · · −→ Atsr /Kr −−−→ Atsr+1 /Kr+1 −−−→ · · · .

∗ We will compute the matrix Fr+1 : let {e 1 , . . . , e tsr } be the canonical basis of tsr A , then for each 1 ≤ i ≤ tsr , the element e i = e i + Kr can be replaced by ∗ its corresponding canonical matrix Gi , and since fr+1 (g) = gfr+1 for every sr t g ∈ HomA (A , A /Syz(N )), we conclude that ∗ T Fr+1 = It ⊗ Fr+1 .

Step 5. By (16.5.1), a presentation of ExtrA (M, N ) is given by a presenta∗ tion of ker(fr+1 )/Im(fr∗ ). Hence, we can apply Theorem 15.5.3. Let pr be the ∗ ∗ T number of generators of ker(fr+1 ) = Syzr (Fr+1 ) = Syzr (It ⊗ Fr+1 ), where the entries of the matrix Fr+1 should be rewritten with coefficients on the right side; we know how to compute this syzygy, and hence, we know how to compute pr . We also know how to compute the matrix Fr∗ = It ⊗ FrT . Then, a presentation of ExtrA (M, N ) is given by ∗ ExtrA (M, N ) ∼ )/Im(fr∗ )), = Apr /Syz(ker(fr+1

(16.5.2)

∗ where a set of generators of Syz(ker(fr+1 )/Im(fr∗ )) is given by the first pr coordinates of the generators of

16.5 Computation of Ext

329 T Syzr [Syzr [It ⊗ Fr+1 ] | It ⊗ FrT ].

(16.5.3)

A system of generators of ExtrA (M, N ), r ≥ 1. By (16.5.1), a system of generators for ExtrA (M, N ) is defined by a system ∗ ∗ T of generators of ker(fr+1 ) = Syzr (Fr+1 ) = Syzr (It ⊗ Fr+1 ), hence, if T Syzr [It ⊗ Fr+1 ] := [h 1 · · · h pr ],

then f1 , . . . , h g ExtrA (M, N ) = hh pr i A , where hfu := h u + Im(Fr∗ ), 1 ≤ u ≤ pr . Remark 16.5.1. (i) Observe that if we take M = A, then M ⊗ N ∼ = N and this agrees with the conclusion of Theorem 16.2.3: in fact, in this trivial case, s = 1 and we have Syz(M ⊗ N ) ∼ = Syz(N ); Syz(M ⊗ N ) =A h[Syz(M ) ⊗ It | Is ⊗ Syz(N )]i =A h[0 ⊗ It | I1 ⊗ Syz(N )]i ∼ = Syz(N ). Thus, a presentation of M ⊗ N is N∼ =M ⊗N ∼ = Ast /Syz(M ⊗ N ) = At /Syz(A ⊗ N ) ∼ = At /Syz(N ). (ii) For M = A and r ≥ 1, TorA r (M, N ) = 0 and this agrees with (16.3.1) and (16.3.2) since in this case s = 1. (iii) Taking M = A in HomA (M, N ) we have HomA (M, N ) ∼ = N (isomorphism of right A-modules); on the other hand, from Theorem 16.4.1 we get the isomorphism of right A-modules HomA (M, N ) ∼ = hU iA /hI1 ⊗ ∆N iA = hU iA /h∆N iA , where the columns of the matrix U are defined by columns of U = first t coordinates of generators of Syzr ([(∆M ⊗ It )T |Is1 ⊗ ∆N )]) = Syzr ([(0 ⊗ It )T |I0 ⊗ ∆N )]) = Syzr (∆N ) = Syzr ([(0 ⊗ It )T | 0 ⊗ ∆N )]) = Syzr (0) = At . Thus, N ∼ = HomA (M, N ) ∼ = At /h∆N iA is the finite presentation of N as a right A-module. (iv) If M = A, then ExtrA (M, N ) = 0 for r ≥ 1 and this agrees with (16.5.2) and (16.5.3) since in this case fi = 0 and Fi = 0 for i ≥ 1.

330

16 Computing Tor and Ext

16.6 Some Applications As applications of the computations of Hom and Ext we will test stablyfreeness, reflexiveness, and we will also compute the torsion, the dual and the grade of a given submodule of Am . For these applications the centralizing bimodule involved is trivial, A = A h1iA . We will illustrate these applications with examples, and for this we will use some computations we found in [133]. Example 16.6.1. The first application is a method for testing stably-freeness. Recall that if S is a ring and M is an S-module with exact sequence f1

f0

0 → S s −→ S r −→ M → 0, M is stably-free if and only if Ext1S (M, S) = 0 (see Theorem 9.2.15). We can illustrate the method with the following example: consider the skew P BW extension A := σ(Q)hx, yi, with yx = −xy. We will check if the following left A-module is stably-free M := he 3 + e 1 , e 4 + e 2 , xe 2 + xe 1 , ye 1 , y 2 e 4 , xe 4 + ye 3 i. On Mon(A) we consider the deglex order with x  y and the T OP order on Mon(A4 ) with e 4 > e 3 > e 2 > e 1 . According to [133], a system of generators for Syz(M ) is S = {(0, −xy 2 , y 2 , −xy, x, 0), (−y 2 , xy, y, x + y, 0, y), (y 3 , 0, 0, −y 2 , x, −y 2 )}. From this we get that a finite presentation of M is given by F

F

1 0 A3 −−−− → A6 −−−− → M −→ 0,

where  0 −y 2 y 3  −xy 2 xy 0  1  2   y 0 y 0     F1 :=  2  and F0 :=  1  −xy x + y −y   x 0 x  0 0 y −y 2 

0 1 0 1

x x 0 0

y 0 0 0

0 0 0 y2

 0 0 . y x

Applying again the method for computing syzygies we get that Syz(F1 ) = 0 and hence we obtain F

F

1 0 0 −→ A3 −−−− → A6 −−−− → M −→ 0.

From (16.5.2), a presentation for Ext1A (M, A) is given by Ext1A (M, A) ∼ = Apr /Syz(ker(f2∗ )/Im(f1∗ )), and from (16.5.3) a system of generators for Syz(ker(f2∗ )/Im(f1∗ )) is given by the first pr coordinates of the generators of

16.6 Some Applications

331

Syzr [Syzr [It ⊗ F2T ] | It ⊗ F1T ], and pr is the number of generators of Syzr [It ⊗F2T ]. In our particular situation t = 1 and   0 F2 = 0 , 0 whence Syzr [It ⊗ F2T ] = I3 and pr = 3. Therefore,   1 0 0 0 −xy 2 y 2 −xy x 0 Syzr [Syzr [It ⊗ F2T ] | It ⊗ F1T ] = Syzr 0 1 0 −y 2 xy y x + y 0 y  . 0 0 1 y3 0 0 −y 2 x −y 2 In order to compute this syzygy we first have to compute a Gr¨obner basis G for   1 0 0 0 −xy 2 y 2 −xy x 0 F := 0 1 0 −y 2 xy y x + y 0 y  , 0 0 1 y3 0 0 −y 2 x −y 2 but this task is trivial since the first three vectors are the canonical vectors of A3 , so G = {e 1 , e 2 , e 3 }, and hence, Syzr (G) = 0. According to the procedure for computing Syzr (F ) (see Remark 15.2.11 for the right version of the procedure for computing syzygies, see also [133]), we have G = F H, F = GQ, with   100 0 1 0   0 0 1   0 0 0    H= 0 0 0 , Q = F, 0 0 0   0 0 0   0 0 0 000 and   Syzr (F ) = HSyz r (G) | Is − HQ = Is − HQ. Thus, 0 0 0 0 xy 2 0 0 0 y 2 −xy  0 0 0 −y 3 0  0 0 0 1 0  0 0 0 0 1 I9 − HQ =   0 0 0 0 0  0 0 0 0 0  0 0 0 0 0 000 0 0 

and

−y 2 xy −y −x − y 0 y2 0 0 0 0 1 0 0 1 0 0 0 0

−x 0 −x 0 0 0 0 1 0

 0 −y   y2   0   0  , 0   0   0  1

332

16 Computing Tor and Ext

Ext1A (M, A) ∼ = A3 /hSiA , where S := {(0, y 2 , −y 3 ), (xy 2 , −xy, 0), (−y 2 , −y, 0), (xy, −x − y, y 2 ), (−x, 0 − x), (0, −y, y 2 )}.

This proves that Ext1A (M, A) 6= 0, and hence, M is not stably-free. Example 16.6.2. The second application consists in using Theorem 16.4.1 in order to compute a presentation for the dual right A-module M ∗ = HomA (M, A), where M is a left A-submodule of Am . Let M ⊆ A4 be as in the previous example. By (16.4.5), HomA (M, A) ∼ = hU iA /hI6 ⊗ ∆A iA , where hU iA is the right A-module generated by the columns of the matrix U , and the columns of U are the first 6 coordinates of the generators of Syzr ([(∆M ⊗ I1 )T | I3 ⊗ ∆A )]), with ∆M = F1 and ∆A = [1]. Thus, we have to compute Syzr ([F1T |I3 ]). In a similar manner as in the previous example, Syzr ([F1T |I3 ]) is generated by the columns of I9 − HQ, where   0 −xy 2 y 2 −xy x 0 1 0 0 Q = F := −y 2 xy y x + y 0 y 0 1 0 y3 0 0 −y 2 x −y 2 0 0 1 and



0 0  0  0  H :=  0 0  1  0 0

0 0 0 0 0 0 0 1 0

 0 0  0  0  0 . 0  0  0 1

Therefore, 

1 0  0 1   0 0   0 0  0 0 I9 − HQ =    0 0   0 xy 2  2  y −xy −y 3 0

0 0 0 0 1 0 0 1 0 0 0 0 −y 2 xy −y −x − y 0 y2

Hence, U = I6 and M ∗ = HomA (M, A) = 0.

0 0 0 0 1 0 −x 0 −x

0 0 0 0 0 1 0 −y y2

 000 0 0 0  0 0 0  0 0 0  0 0 0 . 0 0 0  0 0 0  0 0 0 000

16.6 Some Applications

333

Example 16.6.3. Our next application concerns the grade of a module. Let S be a ring, the grade j(M ) of a left S-module M is defined by j(M ) := min{i | ExtiS (M, S) 6= 0}, or ∞ if no such i exists. Therefore, the grade of a given submodule M ⊆ Am can be computed. For example, take again A = σ(Q)hx, yi, with yx = −xy and M = he 3 + e 1 , e 4 + e 2 , xe 2 + xe 1 , ye 1 , y 2 e 4 , xe 4 + ye 3 i ⊆ A4 , from the previous two examples we have that Ext0A (M, A) = 0 and Ext1A (M, A) 6= 0. Hence, j(M ) = 1. Example 16.6.4. Now suppose that the ring R of coefficients of the extension A is a noetherian domain (left and right), then A is also a noetherian domain. F

F

1 0 Let M be a left A-module given by the presentation As −→ Ar −→ M → 0. The Ext modules and some results of [92] can be used for computing the torsion submodule t(M ), and they can also be applied for testing reflexiveness of M in the particular case when M ⊆ Am . In fact, Theorems 5 of [92] states that t(M ) ∼ = Ext1A (M T , A), where M T is the transposed module of M , M T =

FT

As /Im(F1T ), i.e., M T is given by the presentation Ar −−1→ As → M T → 0. From this we get that M is torsion-free if and only if Ext1A (M T , A) = 0, and also, M is a torsion module if and only if Ext1D (M T , A) = M . Let M and A again be as in Example 16.6.1. For M T we have the finite presentation FT

0 −→ A6 −−−1−→ A3 −−−−→ M T −→ 0; moreover, Syzr [I1 ⊗ F2T ] = Syzr [0] = I6 , so Syzr [Syzr [It ⊗ F2T ] | I1 ⊗ (F1T )T ] = Syzr [I6 | F1 ]. Reasoning   as in Example 16.6.1, in this case we have Q = F := [I6 | F1 ] and I H := 6 , where 0 is a null matrix of size 3 × 6, so we compute 0  000000 0 y 2 −y 3 0 0 0 0 0 0 xy 2 −xy 0    0 0 0 0 0 0 −y 2 −y 0    0 0 0 0 0 0 xy −x − y y 2    0 −x  I9 − HQ =  0 0 0 0 0 0 −x . 2  0 0 0 0 0 0 0 −y y   0 0 0 0 0 0 1 0 0    0 0 0 0 0 0 0 1 0  000000 0 0 1 

Syzr (F ) is generated by the columns of I9 − HQ and Ext1A (M T , A) ∼ = A6 /J, where J is the right A-module generated by the first six coordinates of the generators of Syzr (F ), so J = Syzr (M ), and hence

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16 Computing Tor and Ext

Ext1A (M T , A) ∼ = A6 /Syzr (M ) ∼ = M. Observe that another interesting method for checking whether M is a torsion module is Corollary 1 of [92]: M is a torsion module if and only if M ∗ = 0. Thus, for our particular situation, in Example 16.6.2 we showed that M ∗ = 0, so M is a torsion module, and this agrees with what we have just proved. Finally, recall that a module M is reflexive if (M ∗ )∗ ∼ = M . Theorem 6 of [92] proves that M is reflexive if and only if ExtiA (M T , A) = 0, for i = 1, 2. Thus, for our particular example, M is not reflexive.

Chapter 17

Matrix Computations Using Gr¨ obner Bases

In this chapter we will make some matrix computations over skew P BW extensions using Gr¨ obner bases. Recall that we are assuming that A = σ(R)hx1 , . . . , xn i is a bijective skew P BW extension of R, with R an LGS ring (see Definition 13.2.1) and Mon(A) endowed with some monomial order (see Definition 13.1.1).

17.1 Computing the Inverse of a Matrix We will present an algorithm that determines if a given rectangular matrix over a bijective skew P BW extension is left invertible, and in such case, the algorithm that computes its left inverse. A similar algorithm will be constructed for the right side case. The algorithm is supported by the following elementary fact about left invertible matrices. Proposition 17.1.1. Let F be a rectangular matrix of size r × s with entries in a ring S. If F has a left inverse, then r ≥ s. Moreover, F has a left inverse if and only if the module generated by the rows of F coincides with S s . Proof. The first statement follows from the fact that we are assuming the S is RC (Proposition 9.1.4 and Remark 9.1.9). Now, suppose that F has a left inverse L ∈ Ms×r (S), i.e., LF = Is . Define the following S-homomorphisms f t : Sr → Ss a 7→ (aT F )T

lt : S s → S r b 7→ (b T L)T ,

then m(f t ) = F T and m(lt ) = LT (for the notation see Chapter 9). Whence, m(f t ◦lt ) = (LF )T = IsT = Is , i.e., f t is an epimorphism. Hence, Im(f t ) = S s , i.e., the left submodule generated by the rows of F coincides with the free module S s . Conversely, suppose that the module generated by the rows of F coincides with S s , then for f t defined as above, there exist a 1 . . . , a s ∈ S r such that f t (a i ) = e i for each 1 ≤ i ≤ s, and where e 1 , . . . , e s denote the © Springer Nature Switzerland AG 2020 W. Fajardo et al., Skew PBW Extensions, Algebra and Applications 28, https://doi.org/10.1007/978-3-030-53378-6_17

335

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17 Matrix Computations Using Gr¨ obner Bases

 T canonical vectors of S s . Thus, if a i = a1i a2i · · · ari , we have   a Ti F = a1i a2i · · · ari F = a1i F(1) + · · · + ari F(r) = e i , where F(j) denotes the j-th row of F , 1 ≤ j ≤ r. Therefore, if L is the matrix whose rows are the vectors a Ti , then LF = Is , i.e., F has a left inverse. t u Corollary 17.1.2. Let F ∈ Mr×s (A) be a rectangular matrix over A. The algorithm below determines if F is left invertible, and in the positive case, it computes the left inverse of F . Algorithm for the left inverse of a matrix INPUT: A rectangular matrix F ∈ Mr×s (A) OUTPUT: A matrix L ∈ Ms×r (A) satisfying LF = Is if it exists, and 0 otherwise INITIALIZATION: IF r < s RETURN 0 IF r ≥ s, let G := {g 1 , . . . , g t } be a Gr¨obner basis for the left submodule generated by rows of F and {e i }si=1 be the canonical basis of As . Use the division algorithm to verify if e i ∈ hGi for each ≤ i ≤ s. IF there exists some e i such that e i ∈ / hGi, RETURN 0 IF hGi = As , let H ∈ Mr×t (A) with the property GT = H T F , and consider K := [kij ] ∈ Mt×s , where the kij ’s are such that e i = k1i g 1 + k2i g 2 + · · · + kti g t for 1 ≤ i ≤ s. Thus, Is = K T GT RETURN L := K T H T Example 17.1.3. Let A = σ(Q)hx, yi be defined by the relation yx = −xy + 1. Given the matrix   1 1 xy 0  F =  x2 0  , 1 y we apply the above algorithm in order to verify if F has a left inverse. For this, we compute a Gr¨ obner basis of the left module generated by the rows of F . Considering the deglex order on Mon(A), with x  y, and the TOPREV order on Mon(A2 ), with e 1 > e 2 , a Gr¨ obner basis for hF T i is {e 1 , e 2 } (here, we also used Corollary 13.3.4). In consequence, F has a left inverse and, from calculations obtained during the process of Buchberger’s algorithm, we have that   xy 2 − y y + 1 0 −xy + 1 L= −xy 2 + y + 1 −y − 1 0 xy − 1 is a left inverse for F .

17.1 Computing the Inverse of a Matrix

337

Corollary 17.1.4. Let F be a square matrix of size r × r with entries in a ring S. Then, F is invertible if and only if the rows of F conform a basis of Ss. Proof. Let L ∈ Mr (A) such that LF = Ir = F L. From LF = Ir it follows that the rows of F generate S r . Let f t and lt be as in the proof of Proposition 17.1.1; since F L = Ir , we have lt ◦ f t = iS r and, therefore, f t is a monomorphism, i.e., Syz(F T ) = 0. Thus, the rows of F are linearly independent, and this complete the first implication. Conversely, since the rows of F generate S r , by Proposition 17.1.1, F has a left inverse. Let L be such an inverse, then LF = Ir . We have F LF = F . This implies that (F L − Ir )F = 0r , but Syz(F T ) = 0, then F L = Ir , i.e., F −1 = L. t u Corollary 17.1.5. Let F ∈ Mr (A) be a square matrix over A. The algorithm below determines if F is invertible, and in the positive case, it computes the inverse of F . Example 17.1.6. We consider again the additive analogue of the Weyl algebra A = A2 ( 21 , 13 ) used in Example 14.4.7, with the same monomial order on Mon(A) and on Mon(A2 ). For this example, let F be the following matrix   x y2 x y2 F = 1 1 2 2 . x 2 y2 x 1 y 1 Algorithm for the inverse of a square matrix INPUT: A square matrix F ∈ Mr (A) OUTPUT: A matrix L ∈ Mr (A) satisfying LF = Ir = F L if it exists, and 0 otherwise INITIALIZATION: Use the algorithm in Corollary 17.1.2 to determine if F is left invertible IF F is not left invertible RETURN 0 ELSE Compute Syz(F T ) IF Syz(F T ) 6= 0 RETURN 0 ELSE Compute the matrices H and K in the algorithm of Corollary 17.1.2 RETURN L := K T H T We want to check if the columns of F conform a basis for A2 . From Section 9.1 we know that this is true if and only if F T is invertible. Using the above algorithm, we start by verifying if F T has a left inverse; for this, we compute

338

17 Matrix Computations Using Gr¨ obner Bases

a Gr¨obner basis of the left A-module generated by the rows of F T , i.e., of Im(F ). By Example 14.4.7, G = {f 1 , f 2 , f 3 } is a Gr¨obner basis for this module, where f 1 = x1 y12 e 1 + x2 y2 e 2 , f 2 = x2 y22 e 1 + x1 y1 e 2 and f 3 = − 14 x21 y13 e 2 + 19 x22 y23 e 2 − 32 x1 y12 e 2 + 43 x2 y22 e 2 . Using the division algorithm we check that e 1 ∈ / hGi, hence hGi 6= A2 . Thus, F T does not have a left inverse, so the columns of F do not form a basis for A2 . Remark 17.1.7. According to Proposition 1.13 in [217], if S is a left (or right) noetherian ring, then S is WF (see Remark 11.1.6). Therefore, to test if F ∈ Mr (S) is invertible, it is enough to show that F has a right or a left inverse. So, in the above algorithm, since we are assuming that A is a bijective P BW extension of an LGS ring, it is not necessary to compute SyzS (F T ) to test whether the matrix is invertible, it is sufficient to apply the algorithm of Corollary 17.1.2 for computing the left inverse. Now we will consider the right inverse of a rectangular matrix. We start with the following theoretic results. Proposition 17.1.8. Let F be a rectangular matrix of size r × s with entries in the ring S. If F has a right inverse, then s ≥ r and the module of syzygies of the submodule generated by the rows of F is zero, i.e., Syz(F T ) = 0. In other words, if F has a right inverse then the rows of F are linearly independent. Proof. s ≥ r since we are assuming that S is RC (see Proposition 9.1.4 and Remark 9.1.9). Let L ∈ Ms×r (S) such that F L = Ir . Consider the homomorphisms f t and lt as in Proposition 17.1.1, then f t is a monomorphism. Hence, ker(f t ) = 0, i.e., Syz(F T ) = 0. t u Proposition 17.1.9. Let F be a rectangular matrix of size r × s with entries in the ring S. If F has a right inverse, then s ≥ r. Moreover, F has a right inverse if and only if Syz(F T ) = 0 and Im(F T ) is a summand direct of S s , where Im(F T ) denotes the module generated by the columns of F T i.e., the module generated by the rows of F . Proof. As in the proof of the previous proposition, s ≥ r and if L ∈ Ms×r (S) is such that F L = Ir , then considering the homomorphisms f t and lt as in Proposition 17.1.1, lt ◦ f t = iS r , i.e, f t is a split monomorphism. Thus, S s = Im(f t ) ⊕ ker(lt ), and Im(f t ) is a direct summand of S s . Conversely, let M be a submodule of S s such that S s = Im(f t ) ⊕ M . So, given f ∈ S s there exist unique elements f 1 ∈ Im(f t ) and f 2 ∈ M such that f = f 1 + f 2 . Define the homomorphism lt : S s → S r as lt (f ) := h f , where h f ∈ S r is such that f t (h f ) = f 1 . By hypothesis, Syz(F T ) = 0, so f t is injective and we get that lt is well defined. It is not difficult to show that lt is an S-homomorphism. Consequently, lt ◦ f t = iS r and if LT := m(lt ), then F L = Ir , i.e., F has a right inverse. t u Let F := [fij ] ∈ Mr×s (A), with s ≥ r (recall that A = σ(R)hx1 , . . . , xn i is a bijective skew P BW extension of an LGS ring R), following [91] and [323] assume that A is endowed with an involution θ, i.e., a function θ : A → A such

17.1 Computing the Inverse of a Matrix

339

that θ(a + b) = θ(a) + θ(b), θ(ab) = θ(b)θ(a) and θ2 = iA , for all a, b ∈ A. Note that θ(1) = 1, and hence, θ is an anti-isomorphism of A. We define θ(F ) := [θ(fij )]. Observe that if K ∈ Ms×r (A), then θ(F K)T = θ(K)T θ(F )T .

(17.1.1)

Proposition 17.1.10. Let A be endowed with an involution θ and let F := [fij ] ∈ Mr×s (A), with s ≥ r. Then, F has a right inverse if and only if for G0

each 1 ≤ j ≤ r, ej −→+ 0, where G0 is a Gr¨ obner basis of the left A-module generated by the columns of θ(F ) and {ej }rj=1 is the canonical basis of Ar . Proof. G := [gij ] ∈ Ms×r (A) is a right inverse of F if and only if F G = Ir , and this is equivalent to saying that     f11 f1s f21  f2s      e j =  .  · g1j + · · · +  .  · gsj , 1 ≤ j ≤ r;  ..   ..  fr1

frs

applying θ we obtain 

   θ(f11 ) θ(f1s ) θ(f21 ) θ(f2s )     e j = θ(g1j ) ·  .  + · · · + θ(gsj ) ·  . . . .  .   .  θ(fr1 )

θ(frs )

Thus, G is a right inverse of F if and only if the canonical vectors of Ar belong to the left A-module generated by the columns of θ(F ), i.e., e 1 , . . . , e r ∈ hθ(F )i. Let G0 be a Gr¨ obner basis of hθ(F )i, then by Theorem 14.3.2, G is a G0

right inverse of F if and only if for each j, e j −→+ 0.

t u

Corollary 17.1.11. Let F ∈ Mr×s (A) be a rectangular matrix over A. The algorithm below determines if F is right invertible, and in the positive case, it computes the right inverse of F . Proof. Applying (17.1.1) we get Ir = K T G0T = K T J T θ(F )T = θ(θ(K))T θ(θ(J))T θ(F )T = θ(θ(J)θ(K))T θ(F )T = θ(F θ(J)θ(K))T , so Ir = θ(F θ(J)θ(K)) = θ(Ir ), and from this we get Ir = F θ(J)θ(K).

t u

Example 17.1.12. Let us consider the ring A = σ(Q)hx, yi, with yx = −xy+1. Using the algorithm below, we will compute a right inverse for   x 0 1 F = y−1 x−1 x−y

340

17 Matrix Computations Using Gr¨ obner Bases

provided that it exists. For this, we consider the involution θ of A given by θ(x) = −x and θ(y) = −y. With this involution, we have that θ(xy) = −xy + 1. Algorithm 1 for the right inverse of a matrix INPUT: An involution θ of A; a rectangular matrix F ∈ Mr×s (A) OUTPUT: A matrix H ∈ Ms×r (A) satisfying F H = Ir if it exists, and 0 otherwise INITIALIZATION: IF s < r RETURN 0 IF s ≥ r, let G0 := {g 1 , . . . , g t } be a Gr¨obner basis for the left submodule generated by columns of θ(F ) and {e j }rj=1 be the canonical basis of Ar . Use the division algorithm to verify if e j ∈ hG0 i for each ≤ j ≤ r. IF there exists some e j such that e j ∈ / hG0 i, RETURN 0 IF hG0 i = Ar , let J ∈ Ms×t (A) with the property G0T = J T θ(F )T , and consider K := [kij ] ∈ Mt×r , where the kij ’s are such that e j = k1j g 1 + k2j g 2 + · · · + ktj g t for 1 ≤ j ≤ r. Thus, Ir = K T G0T RETURN H := θ(J)θ(K) Thus,  θ(F ) =

 −x 0 1 . −y − 1 −x − 1 −x + y

We start by computing a Gr¨ obner basis for the left module generated by the columns of θ(F ). From Corollaries 14.3.4 and 14.4.4, we get G0 = {e 1 , e 2 } is a Gr¨obner basis for A hθ(F )i. In this case, F has a right inverse and   −x + y −1 J = x2 + 2xy − y 2 − x + y − 1 x + y − 1 is such that G0T = J T θ(F )T . −x2 − xy + 2 −x Since G0T = I2 , we have K = I2 and L := θ(J) is a right inverse for F , where   x−y −1 θ(J) = x2 − 2xy − y 2 + x − y + 1 −x − y − 1. −x2 + xy + 1 x To find involutions of skew P BW extensions is a difficult task, so the above algorithm is not practical. A second algorithm for testing the existence and computing a right inverse of a matrix uses the theory of Gr¨obner bases for right modules developed in Section 14.5, assuming additionally that the ring R of coefficients of the bijective skew P BW extension A is RGS (see Remark 14.5.7). For the second algorithm we will first make a simple adaptation of Proposition 17.1.1 and Corollary 17.1.2 for right submodules, using the right notation in Remark 9.1.2.

17.1 Computing the Inverse of a Matrix

341

Proposition 17.1.13. Let F be a rectangular matrix of size r×s with entries in a ring S. If F has a right inverse, then s ≥ r. Moreover, F has a right inverse if and only if the right module generated by the columns of F coincides with S r . Proof. The first statement follows from Proposition 9.1.4 and Remark 9.1.9. Now suppose that F has a right inverse and let L be a matrix such that F L = Ir . Define the following homomorphism of right free S-modules: f : Ss → Sr a 7→ F a

l : Sr → Ss b 7→ Lb,

then m(f ) = F and m(l) = L. Whence, m(f ◦ l) = F L = Ir , i.e, f is an epimorphism. Therefore, Im(f ) = S r , i.e., the right submodule generated by columns of F coincides with the free module S r . Conversely, if Im(F ) = S r , then for f defined as above, there exist a 1 . . . , a r ∈ S s such that f (a j ) = e j for each 1 ≤ j ≤ r, where e 1 , . . . , e r denote the canonical vectors of S r .  T Thus, if a j = a1j a2j · · · asj , we have F a j = F (1) a1j + · · · + F (s) asj = e j , where F (j) denotes the j-th column of F , 1 ≤ j ≤ r. So, if L is the matrix whose columns are the vectors a j , then F L = Ir , i.e., F has a right inverse. t u In the algorithm on the following page we additionally assume that R is RGS. Corollary 17.1.14. Let F ∈ Mr×s (A) be a rectangular matrix over A. The algorithm below determines if F is right invertible, and in the positive case, it computes a right inverse of F . Example 17.1.15. Consider the ring A = σ(Q)hx, yi, with yx = −xy + 1, and let F be the matrix given by  y 2 −xy y F = . xy − 1 x2 x 

Applying the right versions of Buchberger’s algorithm and Corollary 14.3.4, we have that a Gr¨ obner basis for the right module generated by the columns of F is G = {e 1 , e 2 }. From Corollary 17.1.14 we can show that F has a right inverse; moreover, one right inverse for F is given by 

 0 −1 L = −1 0 . x y

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17 Matrix Computations Using Gr¨ obner Bases

Algorithm 2 for the right inverse of a matrix INPUT: A rectangular matrix F ∈ Mr×s (A) OUTPUT: A matrix L ∈ Ms×r (A) satisfying F L = Ir if it exists, and 0 otherwise INITIALIZATION: IF s ≤ r RETURN 0 IF s ≥ r, let G := {g 1 , . . . , g t } be a right Gr¨obner basis for the right submodule generated by columns of F and let {e j }rj=1 be the canonical basis of ArA . Use the right version of the division algorithm to verify if e i ∈ hGiA for each 1 ≤ i ≤ r. IF there exists some e j such that e j ∈ / hGiA , RETURN 0 IF hGiA = Ar , let H ∈ Ms×t (A) with the property G = F H (see Remark 15.1.4), and consider K := [kij ] ∈ Mt×s , where the kij ’s are such that e j = g 1 k1j + g 2 k2j + · · · + g t ktj for 1 ≤ i ≤ r. Thus, Ir = GK RETURN L := HK

17.2 Computing the Projective Dimension Theorem 9.4.2 holds for any projective resolution of M , thus we can consider a free resolution {fi }i≥0 computed using the results of Section 15.6. Hence, by Theorem 9.4.1 and results of Section 16.5, we obtain the following algorithm Projective dimension of a module over a bijective skew P BW extension Algorithm 1 INPUT: lgld(A) < ∞, M = hf 1 , . . . , f s i ⊆ Am , with f k 6= 0, 1≤k≤s OUTPUT: pd(M ) INITIALIZATION: Compute a free resolution {fi }i≥0 of M i := 0 WHILE i ≤ lgld(A) DO IF Im(fi ) is projective THEN pd(M ) = i ELSE i = i + 1

which computes the projective dimension of a module M ⊆ Am given by a finite set of generators, where A is a bijective skew P BW extension of an

17.2 Computing the Projective Dimension

343

LGS ring R with finite left global dimension: note that A is left noetherian (Theorem 3.1.5) and lgld(A) < ∞ (Theorem 7.1.4). Observe that in the previous algorithm we do not need to compute finite free resolutions of M , any free resolution computed using syzygies is enough. We present next another algorithm for computing the left projective dimension of a module M ⊆ Am given by a finite free resolution: fm−1

fm

fm−2

f2

f1

f0

0 → Asm −−→ Asm−1 −−−→ Asm−2 −−−→ · · · −→ As1 −→ As0 −→ M −→ 0. (17.2.1) This algorithm is supported by Corollary 9.4.4 and Theorem 9.4.5. Projective dimension of a module over a bijective skew P BW extension Algorithm 2 INPUT: An A-module M defined by a finite free resolution (17.2.1) OUTPUT: pd(M ) INITIALIZATION: Set j := m and Hj := Fm , with Fm the matrix of fm in the canonical bases WHILE j ≤ m DO Step 1. Check whether or not HjT admits a right inverse GTj : (a) If no right inverse of HjT exists, then pd(M ) = j T (b) If there exists a right inverse GT j of Hj and

(i) if j = 1, then pd(M ) = 0 (ii) if j = 2, then compute (9.4.6) (iii) if j ≥ 3, then compute (9.4.5)

Step 2. j := j − 1 Example 17.2.1. Let A be the extension σ(Q)hx, yi, where yx = xy + x. We will calculate the projective dimension of the left module M = h(1, 1), (xy, 0), (y 2 , 0), (0, x)i given in Example 15.5.4. As we saw in Example 15.6.2, a free resolution for M is given by: −xy xy 2 + 2xy  1 1 −y − 1       1 xy y 2 0  0 0   1 0 0 x 0 y − 1 1 − y2 −−−−−−−−−−−−−−−−−−−−−→ A4 −−−−−−−−−−→ M −−−−−→ 0. 





0  + 1  1 0 −→ A −−−−−−→ A3  y 

0



 −y +    x 

 T To apply the above algorithm, we start by checking whether F2 = 0 y + 1 1 has a right inverse. A straightforward calculation shows that a right inverse  T for F2 is G2 = 0 1 −y , so we compute (9.4.6):

344

17 Matrix Computations Using Gr¨ obner Bases H

H

1 0 0 −→ A3 −−−− → A5 −−−− → M −−−−→ 0,

(17.2.2)

where  0 −xy xy 2 + 2xy −y + 1 1   −y − 1  2    and H0 := 1 xy y 0 . x 0 0 H1 :=    1 0 0 x  0 y − 1 1 − y2  0 1 −y 

To verify if H1T has a right inverse, we must calculate a Gr¨obner basis for the right module generated by the columns of H1T . Since the ring A considered is a bijective skew P BW extension, we can use the right version of Buchberger’s algorithm. For this, we consider the deglex order on Mon(A), with x  y, and the TOP order over Mon(A3 ), with e 1 < e 2 < e 3 . Applying this algorithm, together with the right version of Corollary 14.3.4, we obtain the following Gr¨obner basis for hH1T iA , G = {(x, 0, 0), (1 − y, 0, −1), (0, −1, 1), (0, −x, 0), (0, y − 1, 0)}. Note that e 1 is not reducible by G, thus hGiA 6= A3 and hence H1T does not have a right inverse. Therefore, pd(M ) = 1. Remark 17.2.2. The above algorithms can be used for testing if a given module M is projective: we can compute its projective dimension, thus, M is projective if and only if pd(M ) = 0.

17.3 Test for Stably-freeness Theorem 9.2.15 gives a procedure for testing stably-freeness for a module M ⊆ Am given by an exact sequence f1

f0

0 → As −→ Ar −→ M → 0, where A is a bijective skew P BW extension. Example 17.3.1. Let A = σ(Q)hx, yi, with yx = −xy. We want to know if the left A-module M given by M = he 3 + e 1 , e 4 + e 2 , xe 2 + xe 1 , ye 1 , y 2 e 4 , xe 4 + ye 3 i is stably free or not. To answer this question, we start by computing a finite presentation of M . Considering the deglex order on Mon(A) with x  y, the TOP order on Mon(A4 ) with e 4 > e 3 > e 2 > e 1 , and using the methods of Section 15.2, we get that a system of generators for Syz(M ) is given by S = {(0, −xy 2 , y 2 , −xy, x, 0)T , (−y 2 , xy, y, x + y, 0, y)T , (y 3 , 0, 0, −y 2 , x, −y 2 )T }.

17.3 Test for Stably-freeness

345

Algorithm 2 for testing stably-freeness INPUT: M a A-module with exact sequence f1

f0

0 → As −→ Ar −→ M → 0 OUTPUT: TRUE if M is stably free, FALSE otherwise INITIALIZATION: Compute the matrix F1 of f1 IF F1T has right inverse THEN RETURN TRUE

ELSE RETURN FALSE

Therefore, we get the following finite presentation for M : F

F

(17.3.1)

1 0 A3 −−−− → A6 −−−− → M −→ 0,

where,  0 −y 2 y 3  −xy 2 xy 0  1  2   y  0 y 0   F1 :=   −xy x + y −y 2  and F0 := 1    x 0 x  0 0 y −y 2 

0 1 0 1

x x 0 0

y 0 0 0

0 0 0 y2

 0 0 . y x

But, applying again the method for computing the syzygy module, we get that SyzA (F1 ) = 0, so the presentation obtained in (17.3.1) takes the form F

F

1 0 0 −→ A3 −−−− → A6 −−−− → M −→ 0.

Finally, we must test if F1T has a right inverse. For this, we calculate a Gr¨obner basis for the right module generated by the columns of F1T . Using the TOP order on Mon(A3 ), with e 3 > e 2 > e 1 , a Gr¨obner basis for hF1T iA is given by G = {f i }7i=1 , where f i is the i-th column of F1T for 1 ≤ i ≤ 6, and f 7 = −e 2 xy 2 + e 1 xy 2 . Since A6 6= hGiA , F1T has no right inverse and hence M is not stably free. Remark 17.3.2. From Theorem 9.2.15, if M is a left A-module with exact f1 f0 sequence 0 → As −→ Ar −→ M → 0, then M T ∼ = Ext1A (M, A), where T s T T r s M = A /Im(f1 ) and f1 : A → A is the homomorphism of right free Amodules induced by the matrix F1T . Thus, for testing if M is stably free, we can use the results in Section 14.5 and compute a Gr¨obner basis for the right module generated by columns of F1T . Using the right version of the division algorithm, it is possible to check whether As = Im(F1T ). If this last equality holds, then M T = 0 and M is stably free.

346

17 Matrix Computations Using Gr¨ obner Bases

Corollary 9.4.4 gives another procedure for testing stably-freeness for a module M ⊆ Am given by a finite free resolution (9.4.4) with S = A: indeed, if m ≥ 3 and fm has no left inverse, then M is not stably free; if fm has a left inverse, we then compute the new finite free resolution (9.4.5) and we check if hm−1 has a left inverse. We can repeat this procedure until (9.4.6); if h1 has no left inverse, then M is not stably free. If h1 has a left inverse, then M is stably free. Example 17.3.3. Let A be the extension σ(Q)hx, yi, where yx = xy + x and consider the left module M = h(1, 1), (xy, 0), (y 2 , 0), (0, x)i given in Example 15.5.4. As we saw in Example 17.2.1, a finite presentation for M is given by: H

H

1 0 0 −→ A3 −−−− → A5 −−−− → M −−−−→ 0,

(17.3.2)

where  0 −xy xy 2 + 2xy −y + 1 1   −y − 1  2     and H0 := 1 xy y 0 . 0 0 H1 :=  x  1 0 0 x  0 y − 1 1 − y2  0 1 −y 

In that example we showed that H1T does not have a right inverse, hence M is not a stably free module.

17.4 Computing Minimal Presentations If M ⊆ Am is a stably free module given by the finite free resolution (9.4.4) with S = A, then Corollary 9.4.4 gives a procedure for computing a minimal presentation of M . In fact, if m ≥ 3, then fm has a left inverse (if not, pd(M ) = m, but this is impossible by Theorem 9.4.5 since M is projective). Hence, we compute the finite presentation (9.4.5) and we will repeat the procedure until we get a finite presentation as in (9.4.6), which is a minimal presentation of M . Example 17.4.1. Let us consider again the ring A = σ(Q)hx, yi, with yx = −xy+1. Let M be the left A-module given by presentation A2 /Im(F1 ), where  2  y xy − 1 F1 = . −xy x2 Regarding the deglex order on Mon(A), with y  x, and the TOP order over Mon(A2 ) with e 2 > e 1 , we have that Syz(F1 ) is generated by (x, y)T . So, the following exact sequence is obtained:  





x y 2 xy − 1    y −xy x2 F0 0 −→ A −−−−→ A2 −−−−−−−−−−−→ A2 −−−− → M −→ 0.

17.5 Computing Free Bases

347

    y T T Note that F2 := x y has a right inverse: G2 = . Thus, from Corollary x 9.4.4 we get the following finite presentation for M : h

h

0 −→ A2 −−−1−→ A3 −−−0−→ M −→ 0,

(17.4.1)

where H1T

=



F1T

GT2



and

H0T

 F0T = . 0 

Applying the algorithm in Corollary 17.1.14, we can show that H1T has a right inverse:   0 −1 LT1 = −1 0 . x y Therefore, (17.4.1) is a minimal presentation for M , and hence, M is stably free.

17.5 Computing Free Bases Let S be a ring, Theorem 9.3.6 shows that if M is stably free with rank(M ) ≥ sr(S), then M is free with dimension equals to rank(M). The proof of this theorem and Lemma 9.3.5 allow us to establish two algorithms to compute a basis for M , when M is given by a minimal presentation f1

f0

0 → S s −→ S r −→ M → 0,

(17.5.1)

with g1 : S r → S s such that g1 f1 = iS s , and rank(M ) = r − s ≥ sr(S). To compute a basis of M with the first algorithm, we start with the following preliminary proposition. Algorithm for computing U in Proposition 9.3.3  T INPUT: A unimodular stable column vector v = v1 · · · vr over S. OUTPUT: An elementary matrix U ∈ Mr (S) such that U v = e1 . DO: Compute a1 , . . . , ar−1 ∈ S such that (9.3.1) holds. Compute the matrix E1 given in (9.3.2). Calculate b1 , . . . , br−1 ∈ S with the property that Pr−1 0 the elements 0 b v = 1, with v = vi + ai vr for 1 ≤ i ≤ r − 1. i i=1 i i 4. Define vi00 := (vi0 − 1 − vr )bi for 1 ≤ i ≤ r − 1, and compute the matrices E2 , E3 and E4 given in (9.3.3)-(9.3.5). 1. 2. 3.

RETURN: U := E4 E3 E2 E1 .

348

17 Matrix Computations Using Gr¨ obner Bases

Proposition 17.5.1. The matrix U of Proposition 9.3.3 can be computed with the above algorithm. Next, we will illustrate the algorithm. Example 17.5.2. In this example we consider the Jordanian quantum Weyl algebra A2 (Ja,b ) defined by Alev and Dumas in [12] (see also [328]), with K = Q, a = 0 and b = −1. Thus, the relations in this ring are given by: x1 x2 = x2 x1 ∂2 ∂1 ∂ 1 x1 ∂ 1 x2 ∂ 2 x1 ∂ 2 x2

= ∂1 ∂2 − ∂22 = 1 + x1 ∂ 1 = x2 ∂ 1 − x2 ∂ 2 = x1 ∂ 2 = 1 + x1 ∂ 2 + x2 ∂ 2 .

Observe that A2 (J0,−1 ) = σ(Q[x1 , ∂2 ])hx2 , ∂1 i is a bijective skew P BW extension. E4 (A2 (J0,−1 )) will denote the group generated by all elementary  T ∂ + x ∂ + ∂ x ∂ matrices of size 4 × 4 over A (J ). Let v = , then 2 1 2 1 2 1 2 0,−1   u = ∂1 −∂2 0 −x1 is such that uv = 1, where v ∈ Umc (4, A2 (J0,−1 )) (see  T 0 ∂ + x ∂ x Definition 9.2.11). Moreover, the column vector v = has a 2 1 2 2   0 0 left inverse u = 0 x2 − x1 ∂2 , so v is a stable unimodular column. In this case, a1 = 0, a2 = −1, a3 = 0 and the matrix E1 is given by   100 0 0 1 0 −1  E1 =  0 0 1 0 . 000 1  T With this elementary matrix we get E1 v = ∂2 + x1 ∂2 x2 ∂1 . If we define v100 := 0, v200 := (∂2 + x1 − 1 − ∂1 )(x2 − x1 ), v300 = (∂2 + x1 − 1 − ∂1 )∂2 and   1 0 0 0 0 1 0 −1  E2 =  0 0 1 0 , 0 v200 v300 1  T we obtain E2 E1 v = ∂2 + x1 ∂2 x2 ∂2 + x1 − 1 . Finally, if we define   1 0 0 −1 0 1 0 0   E3 =  0 0 1 0  ∈ E4 (A2 (J0,−1 )), 000 1   1 000  −∂2 1 0 0  ∈ E4 (A2 (J0,−1 )) E4 =   −x2 0 1 0 −∂2 − x1 + 1 0 0 1

17.5 Computing Free Bases

349

and U := E4 E3 E2 E1 ∈ E4 (A2 (J0,−1 )), then we have U v = e 1 . Applying the proof of Theorem 9.3.6, we present below the first algorithm for computing free bases. In the second algorithm we will compute the matrix U of Lemma 9.3.5 (if it exists), and with this, a basis for the stably free module given by (17.5.1). For this, recall that U ∈ Mr (S) is invertible if and only if the rows of U conform a basis for S r . Since g1 f1 = iS s , we have that f1 is injective and therefore Syz(F1 ) = 0, where F1 := m(f1 ) is the matrix of f1 in the canonical bases. Thus, to construct the matrix U , we must find vectors g 1 , . . . , g r−s ∈ Syz(G1 ) (1) (s) (j) such that {F1 , . . . , F1 , g 1 , . . . , g r−s } is a basis for S r , with F1 denoting the j-th column of F1 and G1 the matrix of g1 . We also have the algorithm below in which the computation of a basis of M depends on the existence of the matrix U . Next we present an example that illustrates the first algorithm. Example 17.5.3. Let A := A2 (Ja,b ) be the algebra considered in Example 17.5.2. Take M = A6 /Im(F1 ), where 

0 x2  0 F1 =   ∂1  x1 ∂2

 ∂1 ∂2   −x1  . 0   1  −1

Applying the algorithm described in Corollary 17.1.14, with the deglex order on Mon(A), x2 > ∂1 , the TOPREV order on Mon(A6 ) and e 1 > e 2 , it is possible to show that F1T has a right inverse given by 

x1 ∂ 1  0  2  ∂1 GT1 =   x1   −∂1 0

 x1 0  ∂1  . 0  0 0

Hence, we have the following minimal presentation for M : F

F

1 0 0 → A2 −→ A6 −→ M → 0,

(17.5.2)

where F0 is the canonical projection. Thus, M is a stably free A-module with rank(M) = 4. Since lKdim(A) = 3 (see [125], Theorem 2.2), sr(A) ≤ 4, and by Theorem 9.3.6, M is free with dimension equal to rank(M ). We will use algorithm 1 for computing a basis of M .

350

17 Matrix Computations Using Gr¨ obner Bases

Algorithm 1 for computing bases INPUT: F1 = m(f1 ) such that F1T ∈ Ms×r (S) has a right inverse GT1 ∈ Mr×s , and satisfies r − s ≥ sr(S).  T OUTPUT: A matrix U ∈ Mr (S) such that U GT1 = Is 0 ; by Lemma 9.3.5 the set {(U T )(s+1) , . . . , (U T )(r) } is a basis for M , where (U T )(j) denotes the j-th column of U T for s + 1 ≤ j ≤ r. INITIALIZATION: i = 1, V = Ir . WHILE i < r DO: 1. Denote by v i ∈ S r−i+1 the column vector given by taking the last r − i + 1 entries of the i-th column of V GT1 . 2. Apply the algorithm of Proposition 17.5.1 to compute Ei ∈ Er−i+1 (S) such that Ei vi = e 1 .  I 0 3. Define the matrix Ui := i−1 ∈ Er (S) for i > 1, and U1 := 0 Ei E1 . 4. i = i + 1 RETURN U := P Us V , where P is an adequate elementary matrix.

• Step 1. Let V = I6 and v 1 be the first column of V GT1 , i.e.,  T v 1 = x1 ∂1 0 ∂12 x1 −∂1 0 ,   then v 1 ∈ Umc (6, A) and u 1 = 0 x2 0 ∂1 x1 −∂1 is such that u 1 v 1 = 1.  T Note that v 01 = x1 ∂1 0 ∂12 x1 −∂1 is trivially unimodular. Applying to v 1 the algorithm of Proposition 17.5.1, we have that E1 = I6 , 

 1 0 0 0 0 0 0 1 0 0 0 0   0 0 1 0 0 0 , E2 =  0 0 0 1 0 0   0 0 0 0 1 0  0 (x1 ∂1 − 1)x  2 0 (x1 ∂1 −1)∂1 (x1 ∂1 − 1)x1 1  1 0 0 0 0 −1 1 00000 0 1 0 0 0 0   0 1 0 0 0 0     0 0 1 0 0 0   −∂12 0 1 0 0 0    . E3 =   and E4 =  −x1 0 0 1 0 0 0 0 0 1 0 0    0 0 0 0 1 0   ∂1 0 0 0 1 0 00000 1 −x1 ∂1 + 1 0 0 0 0 1 We can check that

17.5 Computing Free Bases

351

Algorithm 2 for computing bases INPUT: A nonzero stably free module M given by the minimal presentation (17.5.1).   I OUTPUT: A matrix U ∈ GLr (S) satisfying U GT1 = s , where G1 = 0 m(g1 ), and such that the last r − s columns of U T conform a basis for M and the first s columns of U T conform the matrix F1 = m(f1 ), if U exists. Otherwise, return 0. INITIALIZATION: Compute a finite set of generators X = {g 1 , . . . , g l } for Syz(G1 ). IF l < r − s RETURN 0 IF l ≥ r − s, for each subset {g ji1 , . . . , g jir−s } from X verify if Uj := h iT F g ji1 . . . g jir−s has a left inverse using the Corollary 17.1.2. IF none of these matrices Uj has a left inverse RETURN 0 IF the matrices Uj1 , . . . , Ujt have left inverse, compute Syz(UjTk ) for 1 ≤ k ≤ t. IF Syz(UjTk ) 6= 0 for all 1 ≤ k ≤ t RETURN 0 IF there exists Uj is such that Syz(UjT ) = 0 RETURN U := Uj .

U1 := E4 E3 E2 E1 = 1 0    −∂12   −x1  ∂1 −x1 ∂1 + 1



−(x1 ∂1 − 1)x2 1 ∂12 (x1 ∂1 − 1)x2 x1 (x1 ∂1 − 1)x2 −∂1 (x1 ∂1 − 1)x2 x1 ∂1 (x1 ∂1 − 1)x2

0 −(x1 ∂1 − 1)∂1 −(x1 ∂1 − 1)x1 0 0 0 1 ∂12 (x1 ∂1 − 1)∂1 ∂12 (x1 ∂1 − 1)x1 0 x1 (x1 ∂1 − 1)∂1 + 1 x1 (x1 ∂1 − 1)x1 0 −∂1 (x1 ∂1 − 1)∂1 −∂1 (x1 ∂1 − 1)x1 + 1 0 x1 ∂1 (x1 ∂1 − 1)∂1 x1 ∂1 (x1 ∂1 − 1)x1

−1 0   ∂12   x1  −∂1  x1 ∂ 1



∈ E6 (A)

and 

 1 x1 0  0   2  0 −x1 ∂1 − ∂1  . U1 GT1 =  0  −x21   0 x1 ∂ 1 + 1  0 −x21 ∂1 • Step 2. Make V := U1 and let v 2 be the column vector given by taking the last five entries of the 2-th column of V GT1 ; i.e.,  T v 2 = 0 −x1 ∂12 − ∂1 −x21 x1 ∂1 + 1 −x21 ∂1 .

352

17 Matrix Computations Using Gr¨ obner Bases

  Note that u 2 = 0 −x1 ∂12 3 0 satisfies u 2 v 2 = 1, thus v2 ∈ Umc (5, A). Moreover, v02 = 0 −x1 ∂12 − ∂1 −x21 x1 ∂1 + 1 is unimodular, and u 02 =  2 0 −x1 ∂1 3 is such that u 02 v 02 = 1, hence v 2 is stable. Using again the algorithm of Proposition 17.5.1, we get that E1 = I5 ,   1 0 0 0 0 0 1 0 0 0   0 0 1 0 0 E2 =   , 0 0 0 1 0 0 −(−1 + x21 ∂1 )x1 (−1 + x21 ∂1 )∂12 3(−1 + x21 ∂1 ) 1     1 0000 1 0 0 0 −1 0 1 0 0 0   x1 ∂12 + ∂1 1 0 0 0       x21 0 1 0 0 E3 = 0 0 1 0 0  and E4 =   . 0 0 0 1 0  −x1 ∂1 − 1 0 0 1 0 1 0001 0000 1 Making the respective computations, we have that L2 := E4 E3 E2 E1 = 

1

(−1 + x21 ∂1 )x1

−(−1 + x21 ∂1 )∂12

0

0

 x1 ∂12 + ∂1 1 + (x1 ∂12 + ∂1 )(−1 + x21 ∂1 )x1 −(x1 ∂12 + ∂1 )(−1 + x21 ∂1 )∂12  x2 x2 (−1 + x2 ∂1 )x1 1 − x21 (−1 + x21 ∂1 )∂12   −(x ∂1 + 1) −(x ∂1 + 1)(−11 + x2 ∂ )x 2 2 (x 1 1 1 1 1 1 1 ∂1 + 1)(−1 + x1 ∂1 )∂1 1 1

−3(−1 + x21 ∂1 ) −1 −3(x1 ∂12 + ∂1 )(−1 + x21 ∂1 ) −(x1 ∂12 + ∂1 )   −3x21 (−1 + x21 ∂1 ) −x21  2 1 + 3(x1 ∂1 + 1)(−1 + x1 ∂1 ) x1 ∂ 1 + 1  0 0



 1 0 ; then 0 L2   1 x1 0 1    0 0   U2 U1 GT1 =  0 0 .   0 0  0 0

and L2 v 2 = e 1 ∈ A5 . Define U2 :=



Finally, if 

1 −x1 0 1  0 0 P1 :=  0 0  0 0 0 0

   0000 10 0 1 0 0 0 0      1 0 0 0 , then U GT1 = 0 0,  0 0 0 1 0 0    0 0 0010 0001 00

17.5 Computing Free Bases

353

where U := P1 U2 U1 . Thus, a basis for M is given by T T T T )}, {F0 (U(3) ), F0 (U(5) ), F0 (U(6) ), F0 (U(4) T with U(i) denoting the transpose of i-th row of the matrix U , for i = 3, 4, 5, 6; i.e.,

−x13 ∂12 + x1 ∂13 − 4x12 ∂1 − 2x1 + ∂1 )(1 − x1 ∂12 x2 + x13 ∂13 x2 + ∂1 x2 )     1 + (x1 ∂12 + ∂1 )(−1 + x12 ∂1 )x1   = , 4 3 3 2 2 3 (x1 ∂ + ∂1 )(x ∂ − x1 ∂ + 2∂ − x1 ∂ )   (x ∂ 2 + ∂ 1)(∂ x −1x 1∂ 2 x +1x3 ∂ 3 x1 − 3x21∂ + 3) 1 1 1 1 1 1 1 1 1 1 1 1 1 (x1 ∂12 + ∂1 )(−∂1 + x12 ∂12 − x1 ∂1 ) + ∂12





(x1 ∂12

T U(3)

x12 ∂1 − x41 ∂12 + x13 ∂1 − x21 − x1 + (−x21 ∂1 + x41 ∂12 − x31 ∂1 + x1 )(x1 ∂1 − 1)x2   −x31 + x51 ∂1 + x14   = , 3 2 3 4 4 2 3 5  −x1 ∂1 + x1 ∂1 + 2x1 ∂1 − x1 ∂1 − x1 ∂1 + 1   −x4 ∂ 2 − x3 ∂ + x6 ∂ 3 + 3x5 ∂ 2 − 3x4 ∂ + 3x2  1 1 1 1 1 1 1 1 1 1 1 −x21 ∂1 + x41 ∂12 − x13 ∂1 + x1





T U(4)

 T U(5)

 x2  1

−x1 ∂12 + x13 ∂13 + 2x12 ∂12 − x1 ∂1 + 1



x1 ∂1 (−1 + x1 ∂12 x2 − x13 ∂13 x2 ) − x13 ∂13 x2 − 1     −(x1 ∂1 + 1)(−1 + x12 ∂1 )x1   = , (x1 ∂1 + 1)(x1 ∂13 − x31 ∂14 + x21 ∂13 − ∂12 )   (x ∂ + 1)(x ∂ 2 x − x3 ∂ 3 + 3x2 ∂ − 3) − x2 ∂ 2 + 2x ∂ + 1 1 1 1 1 1 1 1 1 1 1 1 1 1 −(x1 ∂1 + 1)(−∂1 + x21 ∂12 − x1 ∂1 ) − ∂1 0

  T U(6)

1    0  =  . 0  0  0

Part IV Applications: The Noncommutative Algebraic Geometry of Skew P BW Extensions

Chapter 18

Semi-graded Rings

In this first chapter of applications we study some algebraic and categorical topics of the noncommutative algebraic geometry of skew P BW extensions. For this task we will follow the approach presented by M. Artin and J.J. Zhang in [28], by A.B. Verevkin in [398], [399] and by D. Rogalski in [344]. The geometry presented in the cited works is done for finitely N-graded algebras, however, observe that skew P BW extensions in general are not N-graded. For this reason we will introduce a new class of rings called semi-graded, and for them we will develop and adapt the most basic ideas and techniques of the noncommutative algebraic geometry of finitely graded algebras. We will see that finitely semi-graded rings generalize finitely graded algebras as well as skew P BW extensions. In order to understand better the results presented here for semi-graded rings, we include in Appendix A a quick review of the noncommutative algebraic geometry of finitely graded algebras.

18.1 Semi-graded Rings and Modules In this section we introduce the semi-graded rings and modules, we will prove some of their elementary properties, and we will show that N-graded rings, finitely graded algebras and skew P BW extensions are particular cases of this new type of noncommutative ring. Definition 18.1.1. Let B be a ring. We say that B is semi-graded (SG) if there exists a collection {Bn }n≥0 of subgroups Bn of the additive group B + such that the following conditions hold: L (i) B = n≥0 Bn . (ii) For every m, n ≥ 0, Bm Bn ⊆ B0 ⊕ · · · ⊕ Bm+n . (iii) 1 ∈ B0 . The collection {Bn }n≥0 is called a semi-graduation of B and we say that the elements of Bn are homogeneous of degree n. Let B and C be semi© Springer Nature Switzerland AG 2020 W. Fajardo et al., Skew PBW Extensions, Algebra and Applications 28, https://doi.org/10.1007/978-3-030-53378-6_18

357

358

18 Semi-graded Rings

graded rings and let f : B → C be a ring homomorphism. We say that f is homogeneous if f (Bn ) ⊆ Cn for every n ≥ 0. Definition 18.1.2. Let B be an SG ring and let M be a B-module. We say that M is Z-semi-graded, or simply semi-graded, if there exists a collection {Mn }n∈Z of subgroups Mn of the additive group M + such that the following conditions hold: L (i) M = n∈Z Mn . L (ii) For every m ≥ 0 and n ∈ Z, Bm Mn ⊆ k≤m+n Mk . We say that M is positively semi-graded, also called N-semi-graded, if Mn = 0 for every n < 0. Let f : M → N be a homomorphism of B-modules, where M and N are semi-graded B-modules. We say that f is homogeneous if f (Mn ) ⊆ Nn for every n ∈ Z. As for the case of rings, the collection {Mn }n∈Z is called a semi-graduation of M and we say that the elements of Mn are homogeneous of degree n. Let B be a semi-graded ring and let M be a semi-graded B-module, let N P be a submodule of M , let Nn := N ∩ Mn , n ∈ Z; observe that the sum n Nn is direct. This induces the following definition. Definition 18.1.3. Let B be an SG ring and M be a semi-graded module over B. Let N be a submodule of M . We say that N is a semi-graded L submodule of M if N = n∈Z Nn . L Note that if N is semi-graded, then Bm Nn ⊆ k≤m+n Nk , forLevery n ∈ Z and m ≥ 0: in fact, let b ∈ Bm and z ∈ Nn , then bz ∈ Bm Mn ⊆ k≤m+n Mk and bz = z1 + · · · + zl , with zi ∈ Nni ⊆ Mni , but since the sum is direct, ni ≤ m + n for every 1 ≤ i ≤ l. Finally, we introduce an important class of semi-graded rings that includes finitely graded algebras and skew P BW extensions. Definition 18.1.4. Let B be a ring. We say that B is finitely semi-graded (F SG) if B satisfies the following conditions: (i) B is SG. (ii) There exists finitely many elements x1 , . . . , xn ∈ B such that the subring generated by B0 and x1 , . . . , xn coincides with B. (iii) For every n ≥ 0, Bn is a free B0 -module of finite dimension. Moreover, if M is a B-module, we say that M is finitely semi-graded if M is semi-graded, finitely generated, and for every n ∈ Z, Mn is a free B0 -module of finite dimension. Remark 18.1.5. Observe that if B is F SG, then B0 Bp = Bp for every p ≥ 0, and if M is finitely semi-graded, then B0 Mn = Mn for all n ∈ Z. From the definitions above we get the following conclusions.

18.1 Semi-graded Rings and Modules

359

L Proposition 18.1.6. Let B = n≥0 Bn be an SG ring and I be a proper two-sided ideal of B semi-graded as left ideal. Then, (i) B0 is a subring of B. Moreover, for any n ≥ 0, B0 ⊕ · · · ⊕ Bn is a B0 -B0 -bimodule, as well as B. (ii) B has a standard N-filtration given by Fn (B) := B0 ⊕ · · · ⊕ Bn .

(18.1.1)

(iii) The associated graded ring Gr(B) satisfies ∼ Bn , for every n ≥ 0 (isomorphism of abelian groups). Gr(B)n = L (iv) Let M = n∈Z Mn be a semi-graded B-module and N a submodule of M . The following conditions are equivalent: (a) N is semi-graded. (b) For every z ∈ N , the homogeneous components of z are in N . (c) M/N is semi-graded with semi-graduation given by (M/N )n := (Mn + N )/N , n ∈ Z. (v) B/I is SG. (vi) If B is F SG and I ∩ Bn ⊆ IBn for every n, then B/I is F SG. Proof. (i) and (ii) are obvious. ∼ Bn for every n ≥ 0 For (iii) observe that Gr(B)n = Fn (B)/Fn−1 (B) = (isomorphism of abelian groups); in addition, note how the product acts: let z := b0 + · · · + bm ∈ Gr(B)m , z 0 := c0 + · · · + cn ∈ Gr(B)n , and bm cn = d0 + · · · + dm+n , then zz 0 = bm cn = d0 + · · · + dm+n = dm+n ∈ Gr(B)m+n ∼ = Bn+m . (iv) (a)⇔(b) is obvious. (b)⇒(c) L Let M n := (M/N )n := (Mn + N )/N , n ∈ Z, then M := M/N = n∈Z M n . In fact, let z ∈ M , then z ∈ M can be written as z = P z1 + · · · + zl = z1 + · · · + zl , with zk ∈ Mnk , 1 ≤ k ≤ l, thus, z ∈ n∈Z M n , P and hence, M = n∈Z M n . This sum is direct since if z1 + · · · + zl = 0, then z1 + · · · + zl ∈ N , so by (b) zk ∈ N , i.e., zk = 0 for every 1 ≤ k ≤ l. Now, let bm ∈ Bm and zn ∈ M n , then bm zn = bm zn = Ld1 + · · · + dp , with di ∈ Mni and ni ≤ m + n, so bm zn = d1 + · · · + dp ∈ k≤m+n M k . We have proved that M is semi-graded. (c)⇒(b) Let z = z1 +L· · · + zl ∈ N , with zi ∈ Mni , 1 ≤ i ≤ l, then 0 = z1 + · · · + zl ∈ M = n∈Z M n , therefore, zi = 0, and hence zi ∈ N for every i. (v) The proof is similar to (b)⇒(c) in (iv), but we include it anyway, for completeness. Let B n := (B/I)n := (Bn + I)/I, n ≥ 0, then B := L B/I = B . n≥0 n In fact, let b ∈ B, then b ∈ B can be written as b = P b0 + · · · + bn = b0 + · · · + bn , with bk ∈ Bk , 0 ≤ k ≤ n, thus, b ∈ n≥0 B n , P and hence, B = n≥0 B n . This sum is direct since if b0 + · · · + bn = 0, then

18 Semi-graded Rings

360

L b0 +· · ·+bn ∈ I = n≥0 In , with In = I ∩Bn , so b0 +· · ·+bn = c0 +· · ·+cn (up to zero summands), with ck ∈ Ik , and hence bk = ck ∈ I, i.e., bk = 0 for every 0 ≤ k ≤ n. Now, let bm ∈ B m and bn ∈ B n , then bm bn = c0 + · · · + cm+n , with ci ∈ Bi , 0 ≤ i ≤ m + n, so bm bn = c0 + · · · + cm+n ∈ B 0 ⊕ · · · ⊕ B m+n . We have proved that B is SG. (vi) By (v), B is SG. Let x1 , . . . , xn ∈ B such that the subring generated by B0 and x1 , . . . , xn coincides with B, then it is clear that the subring of B generated by B 0 and x1 , . . . , xn coincides with B. Let n ≥ 0 and {z1 , . . . , zl } be a basis of the free left B0 -module Bn , then {z1 , . . . , zl } is a basis of B n : in fact, let z ∈ B n with z ∈ Bn , then z = c1 z1 +· · ·+cl zl , with ci ∈ B0 , 1 ≤ i ≤ l, and hence, z = c1 z1 + · · · + cl zl , i.e., B n is generated by z1 , . . . , zl over B 0 ; now, if c1 z1 +· · ·+cl zl = 0, with ci ∈ B0 , then c1 z1 +· · ·+cl zl ∈ I ∩Bn ⊆ IBn and hence we can write c1 z1 + · · · + cl zl = d1 z1 + · · · + dl zl , with di ∈ I, 1 ≤ i ≤ l. From this we get that ci = di , so ci = 0 for every i.

t u

Remark 18.1.7. (i) According to (iv)(b) in the previous proposition, if N is a semi-graded submodule of M , then N can be generated by homogeneous elements; however, if N is a submodule of M generated by homogeneous elements, then we cannot assert that N is semi-graded. In fact, consider the Weyl algebra B = A1 (K), with semi-graduation B = K ⊕K ht, xi ⊕K ht2 , tx, x2 i ⊕ · · · Let N := Bt be the principal left ideal of B generated by t; then t is homogeneous of degree 1; if N were semi-graded, then we would have xt = tx+1 ∈ N and 1 ∈ N , but this is a contradiction: in fact, suppose that 1 ∈ N , then d ]; therefore, takN = B, and hence, x ∈ N , so x = pt with p ∈ B = K[t][x; dt ing the degree with respect to x in the skew polynomial ring B, we conclude that p = p0 (t) + p1 (t)x, whence p1 (t)t = 1, a contradiction. (ii) Let B be an SG ring. As we saw in (ii) of the previous proposition, B is N-filtered. Conversely, if we assume that B is an N-filtered ring with filtration {Fn (B)}n≥0 such that for any n ≥ 0, Fn (B)/Fn−1 (B) is F0 (B)-projective, then it is easy to prove that B is SG with semi-graduation {Bn }n≥0 given by B0 := F0 (B) and Bn is such that Fn−1 (B) ⊕ Bn = Fn (B), n ≥ 1. (iii) If B is an F SG ring, then for every n ≥ 0, Gr(B)n ∼ = Bn as B0 modules. Proposition 18.1.8. (i) Any N-graded ring is SG. (ii) Let K be a field. Any finitely graded K-algebra is an F SG ring. (iii) Any skew P BW extension is an F SG ring. Proof. (i) and (ii) follow directly from the definitions. P (iii) Let A = σ(R)hx1 , . . . , xn i be a skew P BW extension, then A = k≥0 ⊕Ak , where

18.2 Generalized Hilbert Series and Hilbert Polynomials

361

Ak :=R hxα ∈ Mon(A)| deg(xα ) = ki. Thus, Ak is a free left R-module with     n+k−1 n+k−1 dimR Ak = = k n−1 (recall that R is an RC ring, and hence, IBN ).

(18.1.2) t u

Remark 18.1.9. (i) Note that the class of SG rings properly includes the class of N-graded rings: indeed, if B = A1 (K) has an N-graduation, let x = b0 + · · · + bp , t = c0 + · · · + cq , with bp , cq 6= 0, then since B is a domain, from xt = tx + 1 and 1 ∈ B0 we get that p + q = 0, so p = q = 0, thus the only N-graduation is the trivial one. Other examples are U 0 (so(3, K)), the Dispin algebra U (osp(1, 2)) and U (sl(2, K)). (ii) Part (i) also proves that the class of F SG rings properly includes the class of finitely graded algebras. (iii) The class of F SG rings properly includes the skew P BW extensions: let K be a field, then B := K[x]/hx2 i = K ⊕ Kx is a finite-dimensional K-algebra, so A is artinian, but no skew P BW extension is an artinian ring. Another example is the algebra defined by the following relations: yx = xy + z 2 , zy = yz + x2 , xz = zx + y 2 . Observe that this algebra is a particular case of a Sklyanin algebra, which in general is defined by the following relations (see [344], Example 1.14): ayx + bxy + cz 2 = 0, azy + byz + cx2 = 0, axz + bzx + cy 2 = 0, a, b, c ∈ K. (iv) The class of F SG rings properly includes the multiple Ore extensions introduced in [406]: the proof of this statement is as in part (iii) of Proposition 18.1.8 and also considering the K-algebra B = K[x]/hx2 i of part (iii).

18.2 Generalized Hilbert Series and Hilbert Polynomials In this section we introduce the notions of generalized Hilbert series and generalized Hilbert polynomials for finitely semi-graded rings. As in the classical case of finitely graded algebras, these notions depend on the semi-graduation, in particular, they depend on the ring B0 . We will compute these tools for skew P BW extensions. P L Definition 18.2.1. Let B = n≥0 ⊕Bn be an F SG ring and M = n∈Z Mn be a finitely semi-graded B-module. The generalized Hilbert series of M is defined by P GhM (t) := n∈Z (dimB0 Mn )tn . In particular,

362

18 Semi-graded Rings

GhB (t) :=

P∞

n=0 (dimB0

Bn )tn .

We say that B has a generalized Hilbert polynomial if there exists a polynomial GpB (t) ∈ Q[t] such that dimB0 Bn = GpB (n), for all n ≫ 0. In this case GpB (t) is called the generalized Hilbert polynomial of B. Remark 18.2.2. (i) Note that if K is a field and B is a finitely graded Kalgebra, then the generalized Hilbert series coincides with the usual Hilbert series, i.e., GhB (t) = hB (t); the same is true for the generalized Hilbert polynomial (see Definitions A.1.5 and A.1.10). (ii)LObserve that if a semi-graded ring B has another semi-graduation B = n≥0 Cn , then its generalized Hilbert series and its generalized Hilbert polynomial can change, i.e., the notions of generalized Hilbert series and generalized Hilbert polynomial depend on the semi-graduation, in particular on B0 . For example, consider the usual real polynomial ring in two variables B := R[x, y], then GhB (t) =

1 (1−t)2

and GpB (t) = t + 1;

but if we view this ring as B = (R[x])[y] then C0 = R[x], its generalized 1 Hilbert series is 1−t and its generalized Hilbert polynomial is 1. However, in Appendix A we will see that for finitely graded algebras over fields generated in degree 1, the Hilbert series is unique; this nice result was proved recently by Jason Bell and James J. Zhang ([51]). For skew P BW extensions the generalized Hilbert series and the generalized Hilbert polynomial can be computed explicitly. Theorem 18.2.3. Let A = σ(R)hx1 , . . . , xn i be an arbitrary skew P BW extension. Then, (i) GhA (t) =

1 . (1 − t)n

(18.2.1)

(ii) 1 [tn−1 − s1 tn−2 + · · · + (−1)r sr tn−r−1 + · · · + (n − 1)!], (n − 1)! (18.2.2) where s1 , . . . , sk , . . . , sn−1 are the elementary symmetric polynomials in the variables 1 − n, 2 − n, . . . , (n − 1) − n. GpA (t) =

Proof. (i) We have GhA (t) =

∞ X k=0

k

(dimR Ak )t =

 ∞  X n+k−1 k=0

k

tk =

1 . (1 − t)n

18.3 Gelfand–Kirillov Dimension for F SG Rings

363

(ii) Note that dimR Ak = =

 n + k − 1 k

=

(n + k − 1)! k!(n − 1)!

(k + n − 1)(k + n − 2)(k + n − 3) · · · (k + n − (n − 1))(k + n − n)! k!(n − 1)!

(k + n − 1)(k + n − 2)(k + n − 3) · · · (k + n − (n − 1)) (n − 1)! 1 n−1 n−2 = [k − s1 k · · · + (−1)r sr kn−r−1 + · · · + (−1)n−1 sn−1 kn−n ] (n − 1)! 1 [kn−1 − s1 kn−2 · · · + (−1)r sr kn−r−1 + · · · + (n − 1)!], = (n − 1)! =

where s1 , . . . , sr , . . . , sn−1 are the elementary symmetric polynomials in the variables 1 − n, 2 − n, . . . , (n − 1) − n. Thus, we have found a polynomial GpA (t) ∈ Q[t] of degree n − 1 such that dimR Ak = GpA (k) for all k ≥ 0.

(18.2.3) t u

Example 18.2.4. From Theorem 18.2.3 we can compute the generalized Hilbert series and the generalized Hilbert polynomial for all examples of skew P BW extensions described in Chapter 2. In addition, for the skew quantum polynomials introduced in Section 4.4, we can interpreted some of them as quasi-commutative bijective skew P BW extensions of the r-multiparameter quantum torus. These computations are presented in Tables 18.1 and 18.2 at the end of the chapter.

18.3 Gelfand–Kirillov Dimension for F SG Rings The classical Gelfand–Kirillov dimension was studied in Section 7.4 for skew P BW extensions where the ring R of coefficients is a finitely generated Kalgebra, K a field. Now we will introduce the notion of generalized Gelfand– Kirillov dimension for F SG rings, and we will compute it for skew P BW extensions where the ring R of coefficients is a left noetherian domain. The classical definition in this case does not apply since, in general, for a ring R, a finitely generated left R-module is not free. Whence, we have to replace the classical dimension of free modules with another invariant. Next we will show that for our purposes the Goldie dimension works properly, assuming that R is a left noetherian domain. A similar problem was considered in [44] for algebras over commutative noetherian domains replacing the vector space dimension with the reduced rank (see also [52] for arbitrary commutative domains). The following two remarks induce our definition.

364

18 Semi-graded Rings

(i) If R is a left noetherian domain, then R is a left Ore domain and from this we get that udim(R R) = 1. In fact, we should prove that R R is a uniform module. Firstly, 0 6= R; secondly, let 0 6= I be any left ideal of R. We have to show that I is essential; let 0 6= J be a left ideal of R; there exist 0 6= x ∈ I and 0 6= y ∈ J, and from the left Ore condition, there exist u, v ∈ R − {0} such that 0 6= ux = vy ∈ I ∩ J. Thus, we have the following conclusion: let V be a free R-module of finite dimension, i.e., dimR V = k, then udim(V ) = k: in fact, V ∼ = Rk , and from this we obtain udim(V ) = udim(R R ⊕ · · · ⊕R R) = udim(R R) + · · · + udim(R R) = k. (ii) Let R be a left noetherian domain and A = σ(R)hx1 , . . . , xn i be a skew P BW extension of R, then (18.2.1) takes the following form: GhA (t) = =

∞ X

(udimAk )tk =

k=0 ∞  X k=0

∞ X

(dimR Ak )tk

k=0

 n+k−1 k 1 t = . k (1 − t)n

Definition 18.3.1. Let B be an F SG ring such that B0 is a left noetherian domain. The generalized Gelfand–Kirillov dimension of B is defined by GGKdim(B) := supV limk→∞ logk udimV k , where V ranges over all frames of B and V k := B0 hv1 · · · vk |vi ∈ V i (a frame of B is a finite-dimensional B0 -free submodule of B such that 1 ∈ V ). Remark 18.3.2. (i) Note that B has at least one frame: B0 is a frame of dimension 1. We say that V is a generating frame of B if the subring of B generated by V and B0 is B. For example, if R is a left noetherian domain and A = σ(R)hx1 , . . . , xn i is a skew P BW extension of R, then V :=R h1, x1 , . . . , xn i is a generating frame of A. (ii) In a similar way as was observed in Remark 18.2.2, the notion of generalized Gelfand–Kirillov dimension of a finitely semi-graded ring B depends on the semi-graduation, in particular, it depends on B0 . Note that this type of consideration was made in [44] for an alternative notion of the Gelfand– Kirillov dimension using the reduced rank. (iii) If K is a field and B is a finitely graded K-algebra, then the classical Gelfand–Kirillov dimension of B coincides with the notion just defined above, i.e., GGKdim(B) = GKdim(B) (see (7.4.1)). Proposition 18.3.3. Let B be an F SG ring such that B0 is a left noetherian domain. Let V be a generating frame of B, then GGKdim(B) = lim logk (udimV k ). k→∞

(18.3.1)

Moreover, this equality does not depend on the generating frame V . Proof. It is clear that limk→∞ logk (udimV k ) ≤ GGKdim(B). Let W be any frame of B; since dimB0 W < ∞, there exists an m such that W ⊆ V m , and

18.3 Gelfand–Kirillov Dimension for F SG Rings

365

hence for every k we have W k ⊆ V km , but observe that V km is a finitely generated left B0 -module, and since B0 is left noetherian, V km is a left noetherian B0 -module, so udimV km < ∞. From this, udimW k ≤ udimV km . Therefore, logk (udimW k ) ≤ logk (udimV km ) = (1 + logk m) logkm (udimV km ). Since limk→∞ (1 + logk m) = 1, we get that limk→∞ logk (udimW k ) ≤ limk→∞ logkm (udimV km ). But observe that lim logkm (udimV km ) ≤ lim logk (udimV k ),

k→∞

k→∞

whence GGKdim(B) = sup lim logk (udimW k ) ≤ lim logk (udimV k ). W k→∞

k→∞

t u

The proof of the second statement is completely similar.

Remark 18.3.4. Let K be a field and let B be a K-algebra; suppose that B has a generating frame V :=K h1, x1 , . . . , xn i, then observe that {V k }k≥0 is an N-filtration of B, so B is an F SG ring: in fact, B is SG with semi-graduation given in Remark 18.1.7 (ii), in particular, B0 := F0 (B) = V 0 = K, K ⊕ B1 = F1 (B) = V , so B1 =K hx1 , . . . , xn i and hence B is generated as a K-algebra by x1 , . . . , xn , finally, for every k ≥ 0, Bk has finite dimension over B0 = K since Bk ⊆ Fk (V ) = V k . Thus, we have shown that Definition 18.3.1 generalizes the notion of Gelfand–Kirillov dimension for K-algebras having a generating frame. Next we present the main result of the present subsection. Theorem 18.3.5. Let R be a left noetherian domain and A = σ(R)hx1 , . . . , xn i be a skew P BW extension of R. Then, GGKdim(A) = lim logk ( k→∞

k X

dimR Ai ) = 1 + deg(GpA (t)) = n.

i=0

Proof. According to (18.3.1), GGKdim(A) = limk→∞ logk (udimV k ), with V :=R h1, x1 , . . . , xn i = A0 ⊕ A1 ; note that V k = A0 ⊕ A1 ⊕ · · · ⊕ Ak . From this and using (18.2.3) we get GGKdim(A) = lim logk (udim( k→∞

= lim logk ( k→∞

k X i=0

k X

⊕Ai )) = lim logk ( k→∞

i=0

dimR Ai ) = lim logk ( k→∞

k X i=0

GpA (i))

k X i=0

udimAi )

366

18 Semi-graded Rings

= lim logk (GpA (0) + GpA (1) + · · · + GpA (k)), k→∞

but according to (18.2.2), every coefficient in GpA (t) is positive, so GpA (i) is positive for every 0 ≤ i ≤ k, moreover, GpA (i) ≤ GpA (k), so GpA (0) + GpA (1) + · · · + GpA (k) ≤ (k + 1)GpA (k) and hence GGKdim(A) ≤ lim logk ((k + 1)GpA (k)) k→∞

= lim logk (k + 1) + lim logk (GpA (k)) = 1 + lim logk (GpA (k)). k→∞

k→∞

k→∞

Observe that every summand of GpA (k) in the bracket of (18.2.2) is ≤ k n−1 n for k large enough, so GpA (k) ≤ (n−1)! k n−1 for k ≫ 0 and this implies that n + lim log k n−1 = 1 + 0 + n − 1 k→∞ (n − 1)! k→∞ k = 1 + deg(GpA (t)) = n.

GGKdim(A) ≤ 1 + lim logk

n−1

Now we have to prove that GGKdim(A) ≥ n. Note that W := V k is n n a frame of A and udimW k = udimV k ≥ k n , therefore, logk (udimV k ) ≥ logk k n = n, and hence n

GGKdim(A) ≥ limk→∞ logk (udimW k ) = limk→∞ logk (udimV k ) ≥ limk→∞ n = n. t u Example 18.3.6. (i) With respect to the GGKdim, for all examples of skew P BW extensions described in Chapter 2, Theorem 18.3.5 applies. In addition, recall that for the skew quantum polynomials introduced in Section 4.4, if 1 ≤ k ≤ n − 1, then Qk,n q,σ (R) is a quasi-commutative bijective skew P BW extension of the k-multiparameter skew quantum torus, so in this case Theorem 18.3.5 also applies. (ii) In Section 7.4 the classical Gelfand–Kirillov dimension was computed for many important examples of skew P BW extensions, assuming that the ring R is a finitely generated K-algebra, K a field (see also [329]). In some of these examples R = K (see also Example 19.2.4), so the GGKdim in Theorem 18.3.5 coincides with the values computed in Tables 8.3 and 8.4. This is the case, for example, for the classical polynomial algebra K[x1 , . . . , xn ], the universal enveloping algebra of a Lie algebra, the quantum algebra U 0 (so(3, K)), the dispin algebra, the Woronowicz algebra, the q-Heisenberg algebra, the algebra of shift operators, the algebra of discrete linear systems, the multiplicative analogue of the Weyl algebra, the algebra of linear partial shift operators, the algebra of linear partial q-dilation operators, and the algebra of quantum polynomials. (iii) Recall that the Jordan plane is the K-algebra defined by J := K{x, y}/hyx − xy − x2 i.

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367

J is a skew P BW extension of K[x], J = σ(K[x])hyi, with yx = xy + x2 , hence, from Theorem 18.3.5, GGKdim(J ) = 1. However, K[x] is a finitely generated K-algebra and we can apply the results of Section 7.4, so GKdim(J ) = 2. This difference happens because J is not a skew P BW extension of K. A similar situation occurs for the generalized Hilbert series and the usual 1 Hilbert series: from Theorem 18.2.3, GhJ (t) = 1−t , but considering J as a finitely graded K-algebra, a direct computation (see Definition A.1.5) shows 1 that hJ (t) = (1−t) 2.

18.4 Noncommutative Schemes Associated to SG Rings The purpose of this section is to extended the notion of a noncommutative projective scheme (see [30]) to the case of semi-graded rings. We will assume that the ring B satisfies the following conditions: (C1) (C2) (C3) (C4)

B is left noetherian SG. B0 is left noetherian. For every n, Bn is a finitely generated left B0 -module. B0 ⊂ Z(B).

Remark 18.4.1. (i) From (C4) we have that B0 is a commutative noetherian ring. (ii) All important examples of skew P BW extensions satisfy (C1) and (C2). Indeed, let A = σ(R)hx1 , . . . , xn i be a bijective skew P BW extension of R, assuming that R is left noetherian, then A is also left noetherian (Theorem 3.1.5); in addition, by Proposition 18.1.8, A also satisfies (C3). (iii) With respect to condition (C4), it is satisfied for finitely graded K-algebras since in such case B0 = K. On the other hand, let A = σ(R)hx1 , . . . , xn i be a skew P BW extension of R, then in general R = A0 * Z(A), unless A is a K-algebra, with A0 = K a commutative ring. (iv) We remark that some results below can be proved without assuming all conditions (C1)–(C4). For example, in Definition 18.4.7 we only need (C1). (v) If B is an arbitrary semi-graded ring, the collection SGR-B of semigraded modules over B is an abelian category where the morphisms are the homogeneous B-homomorphisms. In the next proposition we consider a subcategory of SGR-B. Proposition 18.4.2. Let sgr-B be the collection of all finitely generated semi-graded B-modules, then sgr-B is an abelian category where the morphisms are the homogeneous B-homomorphisms. Proof. It is clear that sgr-B is a category. sgr-B has kernels and co-kernels: let M, M 0 be objects of sgr-B and let f : M → M 0 be a homogeneous Bhomomorphism. Let L := ker(f ). Since B is left noetherian and M is finitely generated, L is a finitely generated semi-graded B-module. Let M 0 /Im(f )

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be the co-kernel of f . Note that Im(f ) is semi-graded, so M 0 /Im(f ) is a semi-graded finitely generated B-module. sgr-B is normal and co-normal: Let f : M → M 0 be a monomorphism in sgr-B, then f is the kernel of the canonical homomorphism j : M 0 → M 0 /Im(f ). Now, let f : M → M 0 be an epimorphism in sgr-B, then f is the co-kernel of the inclusion ι : ker(f ) → M 0 . sgr-B is additive: The trivial module 0 is an object L of sgr-B; if {Mi } is a finite family of objects of sgr-B, then its co-product Mi in the category of left B-modules is an object of sgr-B, with semi-graduation given by L L ( Mi )p := (Mi )p , p ∈ Z. Thus, sgr-B has finite co-products. Finally, for any objects M, M 0 of sgr-B, Mor(M, M 0 ) is an abelian group and the composition of morphisms is bilinear with respect to the operations in these groups. t u Definition 18.4.3. Let M be an object of sgr-B. (i) For s ≥ 0, B≥s is the least two-sided ideal of B that satisfies the following conditions: L (a) B≥s contains p≥s Bp . (b) B≥s is semi-graded as left ideal of B. n (ii) An element x ∈ M is torsion if there exist s, n ≥ 0 such that B≥s x = 0. The set of torsion elements of M is denoted by T (M ). M is torsion if T (M ) = M and torsion-free if T (M ) = 0. L Remark 18.4.4. (i) Observe that if B is N-graded, then B≥s = p≥s Bp . (ii) Let J be a two-sided ideal of B; if J is semi-graded as a left ideal with semi-graduation {Jn }n≥0 , then J is also semi-graded as a right ideal with the same semi-graduation. In fact, for every m, n ≥ 0, Jn Bm ⊆ (B0 ⊕ · · · ⊕ Bm+n ) ∩ J, but since J is semi-graded as a left ideal, (B0 ⊕ · · · ⊕ Bm+n ) ∩ J ⊆ J0 ⊕ · · · ⊕ Jm+n . The converse is also true. Thus, B≥s coincides with the least two-sided ideal of B that satisfies (a) and the right version of (b). (iii) T (M ) is a submodule of M : indeed, let x, y ∈ T (M ), then there exist m n r, s, n, m ≥ 0 such that B≥s x = 0 and B≥r y = 0; observe that B≥r+s ⊆ n+m n+m n m y ⊆ B≥r y = 0, whence B≥s , B≥r , so B≥r+s x ⊆ B≥s x = 0 and B≥r+s n+m n n , so B≥r+s (x + y) = 0, i.e., x + y ∈ T (M ); if b ∈ B, then B≥s b ⊆ B≥s n n B≥s bx ⊆ B≥s x = 0, i.e., bx ∈ T (M ). (iv) If we assume that B is a domain, and hence, a left Ore domain, an alternative notion of torsion can be defined as in the classical case of commutative domains: an element x ∈ M is torsion if there exists a b 6= 0 in B such that bx = 0; the set t(M ) of torsion elements of M is in this case also a submodule of M . In addition, note that T (M ) ⊆ t(M ): since B≥s 6= 0, let b 6= 0 in B≥s , then bn x = 0 and bn 6= 0, i.e., x ∈ t(M ). (v) It is clear that the collection T of modules M in sgr-B such that t(M ) = M conforms a full subcategory of sgr-B. Moreover, let 0 → M 0 → M → M 00 → 0 be a short exact sequence in sgr-B; it is obvious that t(M ) =

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M if and only if t(M 0 ) = M 0 and t(M 00 ) = M 00 , i.e., the collection T is a Serre subcategory of sgr-B. The next lemma shows that this property is also satisfied by the torsion modules introduced in Definition 18.4.3. Theorem 18.4.5. The collection stor-B of torsion modules is a Serre subcategory of sgr-B, and the quotient category qsgr-B := sgr-B/stor-B is abelian. ι

j

Proof. Clearly stor-B is a full subcategory of sgr-B. Let 0 → M 0 → − M − → M 00 → 0 be a short exact sequence in sgr-B. Suppose that M is in stor-B and let x0 ∈ M 0 , then ι(x0 ) ∈ M and there n 0 n exist s, n ≥ 0 such that ι(B≥s x ) = B≥s ι(x0 ) = 0, but since ι is injective, n 0 0 0 B≥s x = 0. This means that x ∈ T (M ), so T (M 0 ) = M 0 , i.e., M 0 is in stor-B. Now let x00 ∈ M 00 , then there exists an x ∈ M such that j(x) = x00 ; m n 00 there exist r, m ≥ 0 such that B≥r x = 0, whence B≥s x = 0, which implies 00 00 00 00 00 that x ∈ T (M ). Thus, T (M ) = M , i.e., M is in stor-B. Conversely, suppose that M 0 and M 00 are in stor-B; let x ∈ M , then n n there exist s, n ≥ 0 such that B≥s j(x) = 0, i.e., j(B≥s x) = 0. Therefore, n 0 B≥s x ⊆ ker(j) = Im(ι), but since M is torsion, Im(ι) is also a torsion n module. Because B is left noetherian, there exist a1 , . . . , al ∈ B≥s such that mi n B≥s = Ba1 + · · · + Bal ; there exist ri , mi ≥ 0, 1 ≤ i ≤ l, such that B≥r ax= i i 0. Without loss of generality we can assume that r1 ≥ ri for every i, so ml ml m1 m1 m2 m2 B≥r1 ⊆ B≥ri and hence B≥r ⊆ B≥r , B≥r ⊆ B≥r , . . . , B≥r ⊆ B≥r ; 1 1 1 2 1 l ml m1 m2 from this we get that B≥r1 a1 x = 0, B≥r1 a2 x = 0, . . . , B≥r1 al x = 0, let m m := max{m1 , . . . , ml }, then B≥r ai x = 0 for every 1 ≤ i ≤ l, with r := r1 . Therefore, m n m m m B≥r B≥s x = B≥r (Ba1 + · · · + Bal )x = B≥r a1 x + · · · + B≥r al x = 0, m n m m n n i.e., B≥r B≥s x = 0, but observe that B≥r+s ⊆ B≥r and B≥r+s ⊆ B≥s , so m+n m+n m n B≥r+s ⊆ B≥r B≥s and hence B≥r+s x = 0, i.e., x ∈ T (M ). We have proved that T (M ) = M , i.e., M is in stor-B. The second statement of the theorem is a well-known property of abelian categories (see Theorem 2.13.8 in [366]). We want to recall only that the objects of qsgr-B are the objects of sgr-B; moreover, given M, N objects of qsgr-B the set of morphisms from M to N in the category qsgr-B is defined by 0 0 Homqsgr−B (M, N ) := lim (18.4.1) −→Homsgr-B (M , N/N ),

where the direct limit is taken over all M 0 ⊆ M , N 0 ⊆ N in sgr-B with M/M 0 ∈ stor-B and N 0 ∈ stor-B (see [162], [129], or also [366] Proposition 2.13.4). More exactly, the limit is taken over the set P of all pairs (M 0 , N 0 ) in sgr-B such that M 0 ⊆ M , N 0 ⊆ N , M/M 0 ∈ stor-B and N 0 ∈ stor-B. P is partially ordered with order defined by (M 0 , N 0 ) ≤ (M 00 , N 00 ) if and only if M 00 ⊆ M 0 and N 0 ⊆ N 00 .

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P is directed: Let (M 0 , N 0 ), (M 00 , N 00 ) ∈ P. From Proposition 18.1.6 and since B is left noetherian, we conclude that (M 0 ∩ M 00 , N 0 + N 00 ) ∈ P, and this couple satisfies (M 0 , N 0 ), (M 00 , N 00 ) ≤ (M 0 ∩ M 00 , N 0 + N 00 ). t u Remark 18.4.6. (i) Recall from the general theory of abelian categories (see [366]) that the quotient functor π : sgr-B → qsgr-B is exact and defined by π(M ) := M and π(f ) := f , f

where M and M − → N ∈ sgr-B (observe that in (18.4.1) the pair (M, 0) ∈ P). (ii) If B is a left noetherian finitely graded K-algebra, then in Proposition 5.3.7 in [366] it is proved additionally that Homqgr-B (M, N ) = lim −→Homgr−B (M≥s , N ), where the direct limit is taken over maps of abelian groups Homgr−B (M≥s , N ) → Homgr−B (M≥s+1 , N ) induced by the inclusion homomorphism M≥s+1 → M≥s . We have all the ingredients needed in order to define noncommutative schemes associated to semi-graded rings. Definition 18.4.7. We define sproj(B) := (qsgr-B, π(B)) and we call it the noncommutative semi-projective scheme associated to B.

18.5 The Serre–Artin–Zhang–Verevkin theorem for semi-graded rings Now we will investigate the noncommutative version of the Serre–Artin– Zhang–Verevkin theorem for semi-graded rings. For this goal some preliminaries are needed. L Definition 18.5.1. Let M be a semi-graded B-module, M = n∈Z Mn . Let i ∈ Z. The semi-graded module M (i) defined by M (i)n := Mi+n is called a shift of M , i.e., L L M (i) = n∈Z M (i)n = n∈Z Mi+n . Remark 18.5.2. Note that for every i ∈ Z, M = M (i) as B-modules, but they are different objects in the category SGR-B.

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The next proposition shows that the shift of degrees is an autoequivalence. Proposition 18.5.3. Let s : sgr-B → sgr-B be defined by M 7→ M (1), f (1)

f

M− → N 7→ M (1) −−−→ N (1), f (1)(m) := f (m), m ∈ M (1). Then, (i) s is an autoequivalence. (ii) For every d ∈ Z, sd (M ) = M (d). (iii) s induces an autoequivalence of qsgr-B also denoted by s. Proof. For (i), observe that s−1 : sgr-B → sgr-B defined by s−1 (M ) := M (−1) and s−1 (f ) := f (−1) is a functor such that ss−1 = isgr-B and s−1 s = isgr-B . (ii) is evident. For (iii), we define s

qsgr-B − → qsgr-B s(M ) := M (1) s Homqsgr-B (M, N ) − → Homqsgr-B (s(M ), s(N )) s(f ) := f (1) where f ∈ Homsgr-B (M 0 , N/N 0 ) with (M 0 , N 0 ) ∈ P (see (18.4.1)). We have to show that s is well-defined over the morphisms. Observe that Homqsgr-B (s(M ), s(N )) = Homqsgr-B (M (1), N (1)) and f (1) ∈ Homsgr-B (M 0 (1), (N/N 0 )(1)), but note that M 0 (1) ⊆ M (1), N 0 (1) ⊆ N (1) in sgr-B, M (1)/M 0 (1) and N 0 (1) ∈ stor-B and (N/N 0 )(1) = N (1)/N 0 (1). In fact, as B-modules, M 0 (1) = M 0 ⊆ M = M (1), but since M 0 is a semi-graded submodule of M , the homogeneous components of every element of M 0 are in M 0 (Proposition 18.1.6), so the homogeneous components of every element of M 0 (1) are in M 0 (1), i.e., M 0 (1) is a semi-graded submodule of M (1). A similar argument shows that N 0 (1) is a semi-graded submodule of N (1). On the other hand, since the torsion of a semi-graded B-module does not depend on its semi-graduation (it only depends of the semi-graduation of B), it is clear that M (1)/M 0 (1) and N 0 (1) ∈ stor-B. Finally, as abstract B-modules, (N/N 0 )(1) = N/N 0 = N (1)/N 0 (1), and for every n ∈ Z Nn+1 + N 0 N0 0 N (1)n + N N (1)n + N 0 (1) = = = (N (1)/N 0 (1))n . 0 N N 0 (1)

[(N/N 0 )(1)]n = (N/N 0 )n+1 =

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Thus, we have proved that f (1) ∈ Homqsgr-B (M (1), N (1)). Now suppose that f = g in Homqsgr-B (M, N ), where g ∈ Homsgr-B (M 00 , N/N 00 ) with (M 00 , N 00 ) ∈ P; we have to show that f (1) = g(1) in Homqsgr-B (M (1), N (1)). There exists (M 000 , N 000 ) ∈ P such that (M 0 , N 0 ), (M 00 , N 00 ) ≤ (M 000 , N 000 ); taking k := (M 000 , N 000 ), i := (M 0 , N 0 ) and j := (M 00 , N 00 ) we have in the lim −→ h

Homsgr-B (M 0 , N/N 0 ) −−ik → Homsgr-B (M 000 , N/N 000 ), hjk

Homsgr-B (M 00 , N/N 00 ) −−→ Homsgr-B (M 000 , N/N 000 ), hik (f ) = hjk (g), j1

ι

1 where hik (f ) := j1 ◦f ◦ι1 , with M 000 −→ M 0 the inclusion and N/N 0 −→ N/N 000 the canonical homogeneous homomorphism. In a similar way, hjk (f ) := j2 ◦f ◦

j2

ι

2 ι2 , with M 000 −→ M 00 the inclusion and N/N 00 −→ N/N 000 the canonical homogeneous homomorphism. From this we get (M 0 (1), N 0 (1)), (M 00 (1), N 00 (1)) ≤ (M 000 (1), N 000 (1)), (j1 ◦ f ◦ ι1 )(1) = (j2 ◦ f ◦ ι2 )(1), i.e., j1 (1) ◦ f (1) ◦ ι1 (1) = j2 (1) ◦ f (1) ◦ ι2 (1), and

g h

Homsgr-B (M 0 (1), N (1)/N 0 (1)) −−ik → Homsgr-B (M 000 (1), N (1)/N 000 (1)), hjk

Homsgr-B (M 00 (1), N (1)/N 00 (1)) −−→ Homsgr-B (M 000 (1), N (1)/N 000 (1)), f hf ik (f (1)) = hjk (g(1)), i.e., f (1) = g(1) in Homqsgr-B (M (1), N (1)). s is an equivalence since we have the functor s−1 : qsgr-B → qsgr-B defined by s−1 (M ) := M (−1) and s−1 (f ) := f (−1), with f (−1)(m) := f (m), for m ∈ M (−1). As in (i), it is clear that ss−1 = iqsgr-B and s−1 s = iqsgr-B . t u Proposition 18.5.4. sπ = πs. Proof. We have π

s

s

π

sgr-B − → qsgr-B − → qsgr-B and sgr-B − → sgr-B − → qsgr-B, so, for M in sgr-B, sπ(M ) = s(π(M )) = π(M )(1) = M (1) and πs(M ) = f

π(M (1)) = M (1); for M − → N in sgr-B, sπ(f ) = s(f ) = f (1) and πs(f ) = π(f (1)) = f (1). t u Definition 18.5.5. Let M, N be objects of sgr-B. Then L (i) HomB (M, N ) := d∈Z Homsgr-B (M, N (d)). (ii) ExtiB (M, N ) is defined by taking a projective resolution of M in sgr-B, applying the functor HomB ( , N ) and taking the ith homology. Remark 18.5.6. (i) Note that HomB (M, N ) ,→ HomB (M, N ). In fact, we have the group homomorphism ι : HomB (M, N ) → HomB (M, N )

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given by (. . . , 0, fd1 , . . . , fdt , 0, . . . ) 7→ fd1 + · · · + fdt ; observe that fd1 + · · · + fdt = 0 if and only if fd1 = · · · = fdt = 0. Indeed, let m ∈ M be homogeneous of degree p, then 0 = (fd1 + · · · + fdt )(m) = fd1 (m) + · · · + fdt (m) ∈ Nd1 +p ⊕ · · · ⊕ Ndt +p , whence, for every j, fdj (m) = 0. This means that fdj = 0 for 1 ≤ j ≤ t, and hence, ι is injective. (ii) Since N (d) = N as abstract B-modules (see Remark 18.5.2), (i) can be interpreted as Homsgr−B P(M, N (d)) ⊆ HomB (M, N (d)) = HomB (M, N ), therefore HomB (M, N ) = d∈Z ⊕Homsgr−B (M, N (d)) ⊆ HomB (M, N ). i (iii) L In [30],i p. 240 (see also [366], p. 257) the group ExtB (M, N ) is defined by d∈Z Extsgr-B (M, N (d)), but observe that L ExtiB (M, N ) ∼ = d∈Z Extisgr-B (M, N (d)). Proposition 18.5.7. Let M and N be semi-graded B-modules such that each of its homogeneous components are B0 -modules. Then, (i) Homsgr-B (M, N ) is a B0 -module. (ii) HomB (M, N ) is a B0 -module. (iii) Extisgr-B (M, N ) is a B0 -module for every i ≥ 1. (iv) ExtiB (M, N ) is a B0 -module for every i ≥ 1. Proof. (i) If f ∈ Homsgr-B (M, N ) and b0 ∈ B0 , then the product b0 ·f defined by (b0 · f )(m) := b0 · f (m), m ∈ M , is an element of Homsgr-B (M, N ): in fact, b0 · f is obviously additive; let b ∈ B, then (b0 · f )(b · m) = b0 · f (b · m) = b0 [b · f (m)] = (b0 b) · f (m) = (bb0 ) · f (m) = b · (b0 · f (m)) = b · (b0 · f )(m); b0 · f is homogeneous: let m ∈ Mp , then (b0 · f )(m) = b0 · f (m) ∈ b0 · Np ⊆ Np , for every p ∈ Z. It is easy to check that Homsgr-B (M, N ) is a B0 -module with the defined product. (ii) This follows from (i). (iii) Taking a projective resolution of M in the abelian category sgr-B and applying the functor Homsgr-B (−, N ), it is easy to verify using (i) that in the complex defining Extisgr-B (M, N ) the kernels and the images are B0 -modules, i.e., every abelian group Extisgr-B (M, N ) is a B0 -module. (iv) This follows from (ii). t u Definition 18.5.8. Let i ≥ 0; we say that B satisfies the s-χi condition if for every finitely generated semi-graded B-module N and for any j ≤ i, ExtjB (B/B≥1 , N ) is finitely generated as a B0 -module. The ring B satisfies the s-χ condition if it satisfies the s-χi condition for all i ≥ 0. Remark 18.5.9. (i) Since B/B≥1 is a B-B-bimodule, then from the inclusion B0 ,→ B we get that ExtjB (B/B≥1 , N ) is a B0 -module. (ii) In the theory of graded rings and modules the conditions defined above are usually denoted simply by χi and χ. In this situation, B/B≥1 ∼ = B0 . (iii) Observe that in the case of finitely graded K-algebras, B0 = K, j B/B≥1 ∼ = K and the s-χi condition means that dimK ExtB (K, N ) < ∞ for any j ≤ i.

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Definition 18.5.10. Let s be the autoequivalence of qsgr-B defined by the shifts of degrees. We define L∞ Γ (π(B))≥0 := d=0 Homqsgr-B (π(B), sd (π(B))). Following the ideas in the proof of Theorem 4.5 in [30] and Proposition 4.11 in [344], we have the following key lemma. Lemma 18.5.11. Let B be a ring that satisfies (C1)–(C4). (i) Γ (π(B))≥0Lis an N-graded ring. ∞ (ii) Let B := d=0 Homsgr-B (B, sd (B)). Then, B is an N-graded ring and there exists a ring homomorphism B → Γ (π(B))≥0 . (iii) For any object M of sgr-B L∞ Γ (M )≥0 := d=0 Homsgr-B (B, sd (M )) is a graded B-module, and L∞ Γ (π(M ))≥0 := d=0 Homqsgr-B (π(B), sd (π(M ))) is a graded Γ (π(B))≥0 -module. (iv) B has the following properties: (a) (B)0 ∼ = B0 and B satisfies (C2). (b) B satisfies (C3). More generally, let N be a finitely generated graded B-module, then every homogeneous component of N is finitely generated over (B)0 . (c) B satisfies (C1). (v) If B satisfies X1 , then (a) Γ (π(B))≥0 satisfies (C2). (b) Γ (π(B))≥0 satisfies (C3). More generally, let N be a finitely generated graded Γ (π(B))≥0 -module, then every homogeneous component of N is finitely generated over (Γ (π(B))≥0 )0 . (c) Γ (π(B))≥0 satisfies (C1). Proof. (i) Since qsgr-B is an abelian category, Homqsgr-B (π(B), sd (π(B))) is an abelian group; the product in Γ (π(B))≥0 is defined by bilinearity and the following rule: if f ∈ Homqsgr-B (π(B), sn (π(B))) and g ∈ Homqsgr-B (π(B), sm (π(B))), then f ? g := sn (g) ◦ f ∈ Homqsgr-B (π(B), sm+n (π(B))). We have to prove that ? is well-defined: let f1 , f2 ∈ Homqsgr-B (π(B), sn (π(B))) = Homqsgr-B (B, B(n)), with f1 = f2 , g1 , g2 ∈ Homqsgr-B (π(B), sm (π(B))) = Homqsgr-B (B, B(m)), with g1 = g2 . Since sn is well-defined (see the proof of Proposition 18.5.3), sn (g1 ) = sn (g2 ), i.e., g1 (n) = g2 (n), but the composition of morphisms in a quotient category is well-defined (see [366], Proposition 13.4), so g1 (n) ◦ f1 = g2 (n) ◦ f2 , i.e., f1 ? g1 = f2 ? g2 . ? is associative: in fact, if h ∈ Homqsgr-B (π(B), sp (π(B))), then

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(f ? g) ? h = [sn (g) ◦ f ] ? h = sm+n (h) ◦ sn (g) ◦ f = f ? (g ? h). With the given product, Γ (π(B))≥0 is an N-graded ring; the unity of Γ (π(B))≥0 is iB (by (18.4.1), we have 0 0 Homqsgr-B (B, B) = lim −→Homsgr-B (I , B/J ),

where I 0 , J 0 are left semi-graded ideals of B and B/I 0 , J 0 ∈ stor-B, so we take I 0 := B and J 0 = 0). (ii) The proof that B is an N-graded ring is as in (i). For the second assertion we can apply the quotient functor π to define the function ρ

B− → Γ (π(B))≥0 f0 + · · · + fd 7→ π(f0 ) + · · · + π(fd ),

(18.5.1)

which is a ring homomorphism: in fact, π is exact (see Remark 18.4.6 (ii)), so π is additive, and hence, ρ is additive. Note that (B)0 = Homsgr-B (B, B) and 1B = iB , so ρ(1B ) = ρ(iB ) = π(iB ) = iB = 1Γ (π(B))≥0 . We have to show that ρ is multiplicative, i.e, ρ(f ? g) = ρ(f ) ? ρ(g), where we can suppose that f and g are homogeneous of degrees p and q, respectively. Since the functors s and π commute, ρ(f ? g) = ρ(sp (g) ◦ f ) = π(sp (g) ◦ f ) = π(sp (g)) ◦ π(f ) = sp (π(g)) ◦ π(f ) = ρ(f ) ? ρ(g). (iii) The proof of both assertions are as in (i), we only illustrate the product in the first case: if f ∈ Homsgr-B (B, sn (B)) and g ∈ Homsgr-B (B, sm (M )), then f ? g := sn (g) ◦ f ∈ Homsgr-B (B, sm+n (M )). (iv) (a) Recall that (B)0 = Homsgr-B (B, B). Consider the function α

B0 − → Homsgr-B (B, B), α(x) = αx , αx (b) := bx, x ∈ B0 , b ∈ B;

(18.5.2)

since B0 ⊂ Z(B) this function is a ring homomorphism, moreover, it is bijective. Thus, (B)0 is a commutative noetherian ring, so (B)0 satisfies (C2). In addition, observe that the B0 -module structure of Homsgr-B (B, B) induced by α coincides with the structure defined in Proposition 18.5.7. λ (b) Note that the function Homsgr-B (B, B(d)) − → Bd defined by f 7→ f (1) is an injective B0 -homomorphism. Since B0 is noetherian and B satisfies (C3), Homsgr-B (B, B(d)) is finitely generated over B0 ∼ = (B)0 . For the second part, let N be generated by x1 , . . . , xr , with xi ∈ Ndi , 1 ≤ i ≤ r. Let x ∈ Nd , then there exist f1 , . . . , fr ∈ B such that x = f1 · x1 + · · · + fr · xr . From this we can assume that fi ∈ (B)d−di ; by the just proved property (C3) of B we obtain that every (B)d−di is finitely generated as a (B)0 -module. This implies that Nd is finitely generated over (B)0 . (c) By (ii), B is not only SG but N-graded.

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18 Semi-graded Rings

B is left noetherian: we will adapt a proof given in [30]. Recall that an N-graded ring is left noetherian if and only if it is left graded noetherian, i.e., every left ideal is finitely generated if and only if every left graded ideal is finitely generated (see [292], Theorem II.3.5). Thus, let I be a graded left ideal of B. We will show that I is finitely generated. Let f ∈ B be homogeneous of degree df , i.e., f ∈ Homsgr-B (B, B(df )), then f induces a f− morphism s−df (B) B; thus, given aPfinite set F of homogeneous elements L −−→−d of I, let PF := f ∈F s f (B), fF := f ∈F f− : PF → B (the sum is taken in B) and let NF := Im(fF ). Since B is left noetherian we can choose a finite set F0 such that NF0 is maximal among L∞such images. Let N := NF0 and P := PF0 ; we define N 00 := Γ (N )≥0 := d=0 Homsgr-B (B, sd (N )), according to (iii), N 00 is an N-graded B-module. Given any element f ∈ I homogeneous f

of degree df , i.e., B − → B(df ), we have the morphism f− : s−df (B) → B, but since F0 ∪ {f } ⊇ F0 and N is maximal, Im(f− ) ⊆ N , and this implies f−

f

sdf (s−df (B) −−→ N ) = B − → N (df ), i.e., f ∈ N 00 , so I ⊆ N 00 . On the other hand, given f ∈ I homogeneous of degree df , the homogeneous Bf−

homomorphism s−df (B) −→ B given by f− (h) := hf has its image in I. fF

0 Therefore, N 0 ⊆L I, where N 0 is the image ofP the induced morphism P 00 −−→ 00 −df 0 B, with P := f ∈F0 s (B) and f F0 := f ∈F0 f − . Thus, we have N ⊆ N 00 , where both are N-graded B-modules, whence we have the N-graded Bmodule N 00 /N 0 . If we prove that N 00 /N 0 is noetherian, then since I/N 0 ⊆ N 00 /N 0 we get that I/N 0 is also noetherian, so I/N 0 is finitely generated, but N 0 is a finitely generated left ideal of B, so I is finitely generated. N 00 /N 0 is noetherian: note first that N 00 /N 0 is a module over (B)0 ; if we prove that N 00 /N 0 is noetherian over (B)0 , then it is also noetherian over B. According to (a), we only need to show that N 00 /N 0 is finitely generated over (B)0 . But this follows from (b) since, by the construction, N 00 /N 0 is finitely generated as a B-module and right bounded (i.e., there exists an n  0 such that the homogeneous component of N 00 /N 0 of degree k ≥ n is zero, see [30]). (v) We set Γ := Γ (π(B))≥0 . Then, (a) Γ satisfies (C2): from (18.4.1) we have Γ0 = Homqsgr-B (π(B), π(B)) = 0 0 Homqsgr-B (B, B) = lim ), where the direct limit is taken −→Homsgr-B (I , B/N 0 0 0 over all pairs (I , N ) in sgr-B, with I , N 0 ⊆ B, B/I 0 ∈ stor-B and N 0 ∈ stor-B. Since π is a covariant functor, we obtain a ring homomorphism (taking in particular I 0 = B and N 0 = 0)

γ

(B)0 = Homsgr-B (B, B) − → Homqsgr-B (B, B) = Γ0 γ(f ) := π(f ) = f . Since B0 ∼ = (B)0 , then Γ0 , Γ and B are B0 -modules. Actually, they are B0 -algebras: we check this for B, the proof for Γ (π(B))≥0 is similar, and from this we get also that Γ0 is a B0 -algebra. If f ∈ Homsgr-B (B, B(n)), g ∈ Homsgr-B (B, B(m)), x ∈ B0 and b ∈ B, then [x · (f ? g)](b) = x · (sn (g) ◦ f )(b) = xg(n)(f (b)); [f ? (x · g)](b) = [sn (x · g) ◦ f ](b) = (x · g)(n)(f (b)) = xg(n)(f (b)).

18.5 The Serre–Artin–Zhang–Verevkin theorem for semi-graded rings

377

Since B0 is noetherian, in order to prove that Γ0 is a noetherian ring, the idea is to show that Γ0 is finitely generated as a B0 -module, but since B satisfies X1 , this follows from Proposition 3.1.3 (3) in [30]. Thus, Γ0 is a commutative noetherian ring, and hence, Γ satisfies (C2). (b) Γ satisfies (C3): since Γ is graded, Γd is a Γ0 -module for every d, but by γ in (a), the idea is to prove that Γd is finitely generated over B0 , but again we apply Proposition 3.1.3 (3) in [30]. For the second part of (b), let N be a Γ -module generated by a finite set of homogeneous elements x1 , . . . , xr , with xi ∈ Ndi , 1 ≤ i ≤ r. Let x ∈ Nd , then there exist f1 , . . . , fr ∈ Γ such that x = f1 · x1 + · · · + fr · xr , from this we can assume that fi ∈ Γd−di , but as was observed before, every Γd−di is finitely generated as a Γ0 -module, so Nd is finitely generated over Γ0 for every d. (c) Γ satisfies (C1): by (iii), Γ is not only SG but N-graded. The proof of (C1) is exactly as in (c) of (iv). t u Proposition 18.5.12. Let S be a commutative noetherian ring and ρ : C → D be a homomorphism of N-graded left noetherian S-algebras. If the kernel and cokernel of ρ are right bounded, then D⊗ C − defines an equivalence of categories qgr-C ' qgr-D, where ⊗ denotes the graded tensor product. Proof. Proposition 2.5 in [30] applies since it is independent of the notion of torsion. t u We are prepared for the proof of the main theorem of the present section. Theorem 18.5.13. Let B be an SG ring that satisfies (C1)–(C4) and assume that B satisfies the condition X1 , then there exists an equivalence of categories qgr-B ' qgr-Γ (π(B))≥0 .

(18.5.3)

Proof. The ring homomorphism in (18.5.1) satisfies the conditions of Proposition 18.5.12, with S = B0 , C = B and D = Γ (π(B))≥0 . In fact, from the proof of Lemma 18.5.11 we know that B and Γ (π(B))≥0 are N-graded left noetherian B0 -algebras. Since B satisfies the condition X1 , we can apply the proof of part S10 in Theorem 4.5 in [30] to conclude that the kernel and cokernel of ρ are right bounded. t u Corollary 18.5.14. Let B be an SG ring that satisfies (C1)–(C4). Then, (i) There is an injective homomorphism of N-graded B0 -algebras η : B → Gr(B). (ii) If B0 = K is a field and Gr(B) is left noetherian and satisfies X1 , then B satisfies X1 and the following equivalences of categories hold: qgr-Gr(B) ' qgr-B ' qgr-Γ (π(B))≥0 . Proof. (i) η is defined by (see Proposition 18.1.6)

(18.5.4)

378 ∞ M

18 Semi-graded Rings η

Homsgr-B (B, B(d)) = B − → Gr(B) =

d=0

∞ M

Gr(B)d =

d=0

∞ M B0 ⊕ · · · ⊕ Bd B0 ⊕ · · · ⊕ Bd−1 d=0

f0 + · · · + fd 7→ f0 (1) + · · · + fd (1), with fi ∈ Homsgr-B (B, B(i)), 0 ≤ i ≤ d. It is clear that η is additive and η(1) = 1; η is multiplicative: η(fn ? gm ) = η(sn (gm ) ◦ fn ) = (sn (gm ) ◦ fn )(1) = sn (gm )(fn (1)) = gm (fn (1)) = fn (1)gm (1) = fn (1) gm (1) = η(fn )η(gm ). η is a B0 -homomorphism: let x ∈ B0 and fd ∈ Homsgr-B (B, B(d)), from (18.5.2) η(x · fd ) = η(fd ◦ αx ) = (fd ◦ αx )(1) = fd (αx (1)) = fd (x) = x · fd (1) = x · fd (1) = x · fd (1) = x · η(fd ). η is injective: if f0 (1) + · · · + fd (1) = 0, then fk (1) = 0 for every 0 ≤ k ≤ d, so fk (1) ∈ (B0 ⊕ · · · ⊕ Bk−1 ) ∩ Bk since fk (1) ∈ B(k)0 = Bk . (ii) Since B and Gr(B) are N-graded left noetherian K-algebras (K a field) and the kernel and cokernel of η are right bounded, we apply Lemma 8.2 in [30] to conclude that B satisfies X1 . Thus, from Theorem 18.5.13 we get the second equivalence of (18.5.4). Applying Proposition 18.5.12 to η we obtain the first equivalence. t u Example 18.5.15. The examples of skew P BW extensions below are semigraded (most of them not N-graded) rings and satisfy the conditions (C1)– (C4); in each case we will prove that B satisfies the condition X1 ; therefore, for these algebras Theorem 18.5.13 is true. In every example B0 = K is a field, we indicate the relations defining B (see Chapter 2 or also [248]) and the associated graded ring Gr(B): (i) The enveloping algebra of a Lie K-algebra G of dimension n, U (G): xi k − kxi = 0, k ∈ K; xi xj − xj xi = [xi , xj ] ∈ G = Kx1 + · · · + Kxn , 1 ≤ i, j ≤ n; Gr(B) = K[x1 , . . . , xn ]. (ii) The quantum algebra U 0 (so(3, K)), with q ∈ K − {0}: x2 x1 − qx1 x2 = −q 1/2 x3 ,

x3 x1 − q −1 x1 x3 = q −1/2 x2 , −q 1/2 x1 ;

x3 x2 − qx2 x3 =

in this case Gr(B) = Kq [x1 , x2 , x3 ] is the 3-multiparametric quantum space, i.e., a quantum polynomial ring in 3 variables, with   1 q q −1 q = q −1 1 q . q q −1 1

18.5 The Serre–Artin–Zhang–Verevkin theorem for semi-graded rings

379

(iii) The dispin algebra U (osp(1, 2)): x1 x2 − x2 x1 = x1 ,

x3 x1 + x1 x3 = x2 , x2 x3 − x3x2 = x3 ; 1 1 −1 Gr(B) = Kq [x1 , x2 , x3 ], with q =  1 1 1 . −1 1 1

(iv) The Woronowicz algebra W ν (sl(2, K)), where ν ∈ K − {0} is not a root of unity: x1 x3 − ν 4 x3 x1 = (1 + ν 2 )x1 ,

x1 x2 − ν 2 x2 x1 = νx3 , (1 + ν 2 )x2 ;  1 ν −2  Gr(B) = Kq [x1 , x2 , x3 ], with q = ν 2 1 ν 4 ν −4

x 3 x2 − ν 4 x2 x3 =  ν −4 ν 4 . 1

(v) Eight types of 3-dimensional skew polynomial algebras (see Section 2.5): x2 x3 − x3 x2 = x3 , x3 x1 − βx1 x3 = x2 , x1 x2 − x2 x1 = x1 ; x2 x3 − x3 x2 = 0, x3 x1 − βx1 x3 = x2 , x1 x2 − x2 x1 = 0; x2 x3 − x3 x2 = x3 , x3 x1 − βx1 x3 = 0, x1 x2 − x2 x1 = x1 ; x2 x3 − x3 x2 = x3 , x3 x1 − βx1 x3 = 0, x1 x2 − x2 x1 = 0; x2 x3 − x3 x2 = x1 , x 3 x1 − x1 x3 = x2 , x 1 x2 − x2 x1 = x3 ; x2 x3 − x3 x2 = 0, x3 x1 − x1 x3 = 0, x1 x2 − x2 x1 = x3 ; x2 x3 − x3 x2 = −x2 , x3 x1 − x1 x3 = x1 + x2 , x1 x2 − x2 x1 = 0; x2 x3 − x3 x2 = x3 , x3 x1 − x1 x3 = x3 , x1 x2 − x2 x1 = 0; Gr(B) = Kq [x1 , x2 , x3 ], where q is an appropriate matrix in every case. Observe that in every example, Gr(B) is a noetherian Artin–Schelter regular algebra, and hence, Gr(B) satisfies the X1 condition (Theorem 5.12.6 in [366]; see also Remark B.2.2). From Corollary 18.5.14 we conclude that B also satisfies X1 . Finally, from (18.5.4) we get qgr − K[x1 , x2 , x3 ] ' qgr − Γ (π(B))≥0 , with B = U (G); qgr − Kq [x1 , x2 , x3 ] ' qgr − Γ (π(B))≥0 , with B = U 0 (so(3, K)), U (osp(1, 2)), W ν (sl(2, K)) or any of eight types of 3-dimensional skew polynomial algebras above, and q an appropriate matrix in every case. Remark 18.5.16. Applying the theory developed in the present section to the particular case of graded rings, we have B ∼ = B. In fact, we define θ by ∞ M d=0

θ

Homsgr-B (B, B(d)) = B − →B=

∞ M

Bd

d=0

f0 + · · · + fd 7→ f0 (1) + · · · + fd (1), with fi ∈ Homsgr-B (B, B(i)), 0 ≤ i ≤ d. Similarly as in the proof of Corollary 18.5.14, we can prove that θ is an isomorphism of B0 -algebras. Thus, we get as a particular case the Serre–Artin–Zhang–Verevkin equivalence qgr-B ' qgrΓ (π(B))≥0 .

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18 Semi-graded Rings

18.6 Point Modules and the Point Functor In the previous section we studied the collection of all finitely generated semigraded B-modules from a categorical point of view. Now we will consider the particular subclass of point modules for F SG rings. A standard Zariski topology will be defined for them as well as the point functor. We will follow [244]. L Definition 18.6.1. Let B = n≥0 Bn be an F SG ring that is generated in degree 1, i.e., the generating elements x1 , . . . , xn in Definition 18.1.4 are in B1 . (i) A L point module for B is a finitely N-semi-graded B-module M = n∈N Mn such that M is cyclic, generated in degree 0, i.e., there exists an element m0 ∈ M0 such that M = Bm0 , and dimB0 (Mn ) = 1 for all n ≥ 0. (ii) Two point modules M and M 0 for B are isomorphic if there exists a homogeneous B-isomorphism between them. (iii) P (B) is the collection of isomorphism classes of point modules for B. 1 Observe that if M is a point module, then GhM (t) = 1−t . Moreover, if {mn } is a B0 -basis of Mn , then {mn }n∈N is a B0 -basis of M . The following result is the first step in the construction of the geometric structure for P (B). L Theorem 18.6.2. Let B = n≥0 Bn be an F SG ring generated in degree 1. Then, P (B) has a Zariski topology generated by finite unions of sets V (J) defined by

V (J) := {M ∈ P (B) | Ann(M ) ⊇ J},

where J ranges over the semi-graded left ideals of B. Proof. Taking J = B we get from Definition 18.6.1 that V (B) = ∅; for J = 0 we have V (0) = P (B). Let {Ji }i∈C be a family P of semi-graded as left ideals of B, then from (iv) of Proposition 18.1.6, i∈C Ji is semi-graded as a left ideal and we have T P i∈C V (Ji ) = V ( i∈C Ji ). t u L Definition 18.6.3. Let B = n≥0 Bn be an F SG ring generated in degree 1 such that B0 is commutative and B is a B0 -algebra. Let S be a commutative B0 -algebra. An S-point module for B is an N-semi-graded S ⊗B0 B-module M which is cyclic and generated in degree 0. Mn is a locally free S-module with rankS (Mn ) = 1 for all n ≥ 0, and M0 = S. P (B; S) will denote the set of S-point modules for B. Remark 18.6.4. (i) Note that S ⊗B0 B is an F SG ring generated in degree 1 and with S in degree 0:

18.6 Point Modules and the Point Functor

381

P P S ⊗B0 B = S ⊗B0 ( n≥0 ⊕Bn ) = n≥0 ⊕(S ⊗B0 Bn ), so (S ⊗B0 B)n := S ⊗B0 Bn ; (S ⊗B0 B)n (S ⊗B0 B)m ⊆ (S ⊗B0 B)0 ⊕ · · · ⊕ (S ⊗B0 B)n+m ; 1 ⊗ 1 ∈ (S ⊗B0 B)0 = S ⊗B0 B0 = S; dimS ((S ⊗B0 B)n ) = dimB0 (Mn ). S ⊗B0 B is a S-algebra, moreover, if x1 , . . . , xm ∈ B1 generate B as a B0 algebra, then 1 ⊗ x1 , . . . , 1 ⊗ xm ∈ (S ⊗B0 B)1 generate S ⊗B0 B as an S-algebra. (ii) Taking S = B0 we get that P (B) ⊆ P (B; B0 ). If B0 = K is a field, then clearly P (B) = P (B; K). L Theorem 18.6.5. Let B = n≥0 Bn be an F SG ring generated in degree 1 such that B0 is commutative and B is a B0 -algebra. Let B0 be the category of commutative B0 -algebras and let Set be the category of sets. Then, P defined by P

B0 − → Set S 7→ P (B; S) S → T 7→ P (B; S) → P (B; T ), given by M 7→ T ⊗S M is a covariant functor called the point functor for B. Proof. (i) Firstly note that if M is an S-point module, then T ⊗S M is a T -point module: (a) M is an S-module because of the homomorphism S → S ⊗B0 B, s 7→ s ⊗ 1. In addition, by the hypothesis, every homogeneous component Mn is an S-module; M is a left B-module because of the homomorphism B → S ⊗B0 B, b 7→ 1 ⊗ b; T ⊗S M is a T ⊗B0 B-module with a product given by (t ⊗ b) · (t0 ⊗ m) := tt0 ⊗ b · m = tt0 ⊗ (1 ⊗ b) · m; in a similar way, T ⊗S M is a T ⊗S (S ⊗B0 B)-module with product t ⊗ (s ⊗ b) · (t0 ⊗ m) := tt0 ⊗ (s ⊗ b) · m. (b) Since M is S ⊗B0 B-cyclic, generated in degree zero, there exists an m0 ∈ M0 such that M = (S ⊗B0 B) · m0 , whence T ⊗S M = T ⊗S [(S ⊗B0 B) · m0 ] = [T ⊗S (S ⊗B0 B)] · (1 ⊗ m0 ) = ((T ⊗S S) ⊗B0 B) · (1 ⊗ m0 ) = (T ⊗B0 B) · (1 ⊗ m0 ), i.e., T ⊗S M is T ⊗B0 B-cyclic with generator 1 ⊗ m0 . (c) T ⊗S M is N-semi-graded with respect to T ⊗B0 B with semi-graduation (T ⊗S M )n := T ⊗S Mn , n ∈ N. P P In fact, T ⊗S M = T ⊗S ( n∈N ⊕Mn ) = n∈N ⊕(T ⊗S Mn ); (T ⊗B0 Bm )(T ⊗S Mn ) ⊆ T ⊗S Bm · Mn , but Bm · Mn = (1 ⊗ Bm )Mn ⊆ M0 ⊕ · · · ⊕ Mm+n since M is S ⊗B0 B-semi-graded. Thus, (T ⊗B0 Bm )(T ⊗S Mn ) ⊆ (T ⊗S M0 ) ⊕ · · · ⊕ (T ⊗S Mm+n ). (d) 1 ⊗ m0 ∈ T ⊗S M0 ∈ (T ⊗S M )0 . (e) T ⊗S M0 = T ⊗S S = T , i.e., (T ⊗S M )0 = T .

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18 Semi-graded Rings

(f) It is clear that (T ⊗S M )n = T ⊗S Mn is a T -module; let L be a prime ideal of T and Q := f −1 (L), where f : S → T is the given homomorphism, (r) then it is easy to check that SQ → TL , ur 7→ ff (u) , is a ring homomorphism, and from this we get (T ⊗S Mn )L ∼ = TL ⊗T (T ⊗S Mn ) ∼ = (TL ⊗T T ) ⊗S Mn ∼ = T L ⊗ S Mn ∼ (TL ⊗S SQ ) ⊗S Mn ∼ T ⊗ (S ⊗ M = = L SQ Q S n) Q ∼ ∼ = T L ⊗ S Q SQ = T L . This proves that T ⊗S Mn is locally free of rank 1. φ

ϕ

(ii) P is a covariant functor: it is clear that P (iR ) = iP (R) ; if R − →S− →T are morphisms in B0 , then P (ϕφ) = P (ϕ)P (φ). In fact, T ⊗S (S ⊗R M ) ∼ = T ⊗R M . t u Next we recall some basic facts about schemes (see [359] or also [243]). A scheme is a local ringed space (X, F ) for which every point x ∈ X has a neighborhood Ux such that the induced local ringed space (Ux , F|Ux ) is isomorphic as a local ringed space to (Spec(Rx ), Rx ), where Rx is some commutative ring. Let B0 be a commutative ring; recall that a B0 -scheme is a scheme (X, F ) such that F (U ) is a B0 -algebra for every open U ⊆ X. For example, if R is a commutative B0 -algebra, then the affine scheme (Spec(R), R) is a B0 -scheme (see Example A.4.2). The category of B0 -schemes is a subcategory of the category of schemes, and in turn, this last one is a subcategory of the category of local ringed spaces. A morphism between B0 -schemes is a morphism of the corresponding local ringed spaces such that the ring homomorphisms are B0 -algebra homomorphisms. Given two B0 -schemes (X, F ), (Y, G) the set of morphisms from (X, F ) to (Y, G) will be denoted by HomB0 -schemes (X, Y ). Fixing a B0 -scheme (X, F ), which we will denote simply by X, we have the representable functor hX := HomB0 -schemes (−, X) defined in the following way, where as above, B0 is the category of commutative B0 -algebras, Set is the category of sets and Af f is the category of the affine schemes: h

B0 −−X → Set, R 7→ HomB0 -schemes (Spec(R), X), φ

R− → S 7→ hX (φ) HomB0 -schemes (Spec(R), X) −−−−→ HomB0 -schemes (Spec(S), X) e hX (φ):=αφ

α

Spec(R) − → X 7→ Spec(S) −−−−−−−→ X, where φe ∈ HomK -schemes (Spec(S), Spec(R)) is the image of φ under the Spec functor, Spec

B0 −−−→ Af f R 7→ (Spec(R), R) φ

e φ

R− → S 7→ Spec(S) − → Spec(R).

18.6 Point Modules and the Point Functor

383

L Definition 18.6.6. Let B = n≥0 Bn be an F SG ring generated in degree 1 such that B0 is commutative and B is a B0 -algebra. We say that a B0 scheme X parametrizes the point modules of B if the point functor P is naturally isomorphic to hX . L Proposition 18.6.7. Let B = n≥0 Bn be an F SG ring generated in degree 1 such that B0 = K is a field and B is a K-algebra. Let X be a K-scheme that parametrizes P (B). Then, there exists a bijective correspondence between the closed points of X and P (B). Proof. According to Remark 18.6.4, P (B; K) = P (B); moreover, since Spec(K) = {0}, every morphism of HomK -schemes (Spec(K), X) determines one closed point of X, and vice versa. Thus, we have the bijective correspondence HomK -schemes (Spec(K), X) ↔ closed points of X. Now, since X parametrizes P (B), the point functor P is naturally isomorphic to hX , so we have a bijective function between HomK -schemes (Spec(K), X) and P (B). Therefore, we get a bijective function between P (B) and the closed points of X. t u L Theorem 18.6.8. Let B = n≥0 Bn be an F SG ring generated in degree 1 such that B0 = K is a field and B is a K-algebra. Then, (i) There is an injective function α

(P (B)/ ∼) − → P (Gr(B)), where ∼ is the relation in P (B) defined by [M ] ∼ [M 0 ] ⇔ Gr(M ) ∼ = Gr(M 0 ), with [M ] the class of point modules isomorphic to the point module M . (ii) Let X be a K-scheme that parametrizes P (Gr(B)). Then there is an injective function from (P (B)/ ∼) to the closed points of X: α e

(P (B)/ ∼) − → X. Proof. (i) We divide the proof of this part into three steps. Step 1. Note that Gr(B) is a finitely graded K-algebra generated in degree 1: according to Proposition 18.1.6, Gr(B) is N-graded with Gr(B)n ∼ = Bn for all n ≥ 0 (isomorphism of K-vector spaces), in particular, Gr(B)0 = K; if x1 , . . . , xm ∈ B1 generate B as a K-algebra, then x1 , . . . , xm ∈ Gr(B)1 generate Gr(B) as a K-algebra. L Step 2. Let [M ] ∈ P (B), with M = n≥0 Mn , M = Bm0 , with m0 ∈ M0 , and dimK (Mn ) = 1 for all n ≥ 0. As in Proposition 18.1.6, we define L Gr(M ) := n≥0 Gr(M )n , 0 ⊕···⊕Mn ∼ Gr(M )n := MM0 ⊕···⊕M = Mn (isomorphism of K-vector spaces). n−1 The Gr(B)-module structure for Gr(M ) is given by bilinearity and the product

384

18 Semi-graded Rings

bp · mn := bp mn ∈ Gr(M )p+n . This product is well-defined: if bp = cp , then bp − cp ∈ B0 ⊕ · · · ⊕ Bp−1 and hence (bp − cp )mn ∈ M0 ⊕ · · · ⊕ Mp−1+n , so in Gr(M )p+n we have (bp − cp )mn = 0, i.e., bp mn = cp mn ; now, if mn = ln , then mn − ln ∈ M0 ⊕ · · ·⊕Mn−1 , whence bp (mn −ln ) ∈ M0 ⊕· · ·⊕Mp+n−1 , so in Gr(M )p+n we have bp (mn − ln ) = 0, i.e., bp mn = bp ln . The associative law for this product holds: bq · (bp · mn ) = bq · (bp mn ) = bq (bp mn ) = (bq bp )mn = (c0 + · · · + cp+q )mn = cp+q mn = (bq bp )·mn . Finally, 1 · mn = mn . Observe that Gr(M ) = Gr(B) · m0 : Let mn ∈ Gr(M )n , with mn ∈ Mn , we have mn = bm0 = (b0 + · · · + bn )m0 = b0 m0 + · · · + bn−1 m0 + bn m0 = bn m0 = bn · m0 . Step 3. We define α0

P (B) −→ P (Gr(B)) [M ] 7→ [Gr(M )] It is clear that α0 is well-defined, i.e., if M ∼ = M 0 , then Gr(M ) ∼ = Gr(M 0 ). Note that the relation ∼ defined in the statement of the theorem is an equivalence relation, let [[M ]] be the class of [M ] ∈ P (B), then we define α

(P (B)/ ∼) − → P (Gr(B)) [[M ]] 7→ α0 ([M ]) = [Gr(M )]. It is clear that α is a well-defined injective function. (ii) This follows from (i) and applying Proposition 18.6.7 to Gr(B).

t u

With the scheme Xm0 defined in Proposition A.3.8, we can prove the following geometric property of skew P BW extensions. Corollary 18.6.9. Let K be a field and A = σ(K)hx1 , . . . , xn i be a bijective skew P BW extension such that A is a K-algebra. Then, there exists an m0 ≥ 1 such that there is an injective function from P (A)/ ∼ to the closed points of Xm0 : (P (A)/ ∼) −→ Xm0 . Proof. From Proposition 18.1.8, A is an F SG ring generated in degree 1 with A0 = K, and by hypothesis, A is a K-algebra. Observe that the filtration in Theorem 3.1.2 coincides with the one in (18.1.1), thus, Gr(A) is the nmultiparametric quantum space Kq [x1 , . . . , xn ] (in A, xi r = rxi , for all r ∈ K and 1 ≤ i ≤ n). It is clear that Gr(A) is a finitely graded algebra with finite presentation as in Proposition A.3.8, moreover, Gr(A) is strongly noetherian (Theorem 8.5.5), so the result follows from Corollary A.3.9 and Theorem 18.6.8. t u Example 18.6.10. The following examples of skew P BW extensions satisfy the hypothesis of Corollary 18.6.9, and hence, for them there is an m0 ≥ 1

18.6 Point Modules and the Point Functor

385

such that there exists an injective function from P (A)/ ∼ to the closed points of Xm0 . 1. The enveloping algebra of a Lie K-algebra G of dimension n, U (G): in this case Gr(A) = K[x0 , . . . , xn−1 ], so m0 = 1 and X1 = Pn−1 (Proposition A.3.3). We have a similar situation for the algebra Sh of shift operators: for this algebra Gr(A) = K[x0 , x1 ], so m0 = 1 and X1 = P1 . 2. The quantum algebra U 0 (so(3, K)), with q ∈ K − {0}: since in this case Gr(A) is a 3-multiparametric quantum affine space, from Example A.3.10 we get that m0 = 2 and X2 is the same as there. We have similar descriptions for the dispin algebra U (osp(1, 2)), the Woronowicz algebra W ν (sl(2, K)), where ν ∈ K − {0} is not a root of unity, nine types of 3dimensional skew polynomial algebras (see Section 2.5 and also Example 19.2.1 in the next chapter), and the multiplicative analogue of the Weyl algebra with 3 variables. 3. For the following algebras, Gr(A) = Kq [x1 , . . . , xn ], so from Example A.3.11 we conclude that m0 = 2 and X2 is the same as there: the qHeisenberg algebra, the algebra of linear partial q-dilation operators, the algebra D for multidimensional discrete linear systems, the algebra of linear partial shift operators. Example 18.6.11. The following algebras are not skew P BW extensions of the base field K, however, they satisfy the hypothesis of Corollary A.3.9, and hence, for them there is an m0 ≥ 1 such that Xm0 parametrizes P (A) and there exists a bijective correspondence between the closed points of Xm0 and P (A): the algebras of diffusion type, the algebra U, the Manin algebra, or more generally, the algebra Oq (Mn (K)) of quantum matrices, some quadratic algebras in 3 variables, and the quantum symplectic space Oq (sp(K 2n )).

18 Semi-graded Rings

386 Ring

GhA (t)

GpA (t)

Usual polynomial ring R[x1 , . . . , xn ] Ore extension of bijective type R[x1 ; σ1 , δ1 ] · · · [xn ; σn , δn ] Weyl algebra An (K) Extended Weyl algebra Bn (K) Enveloping algebra of a Lie algebra G of dimension n, U (G) Tensor product R ⊗K U (G) Crossed product R ∗ U (G) Algebra of q-differential operators Dq,h [x, y] Algebra of shift operators Sh Mixed algebra Dh

1 (1−t)n 1 (1−t)n 1 (1−t)n 1 (1−t)n 1 (1−t)n 1 (1−t)n 1 (1−t)n 1 1−t 1 1−t 1 (1−t)2 1 (1−t)n 1 (1−t)m 1 (1−t)m 1 (1−t)n 1 (1−t)n 1 (1−t)m 1 (1−t)m 1 (1−t)m 1 (1−t)m 1 (1−t)m 1 (1−t)m 1 (1−t)n 1 (1−t)n 1 (1−t)n−1 1 (1−t)3 1 (1−t)3 1 (1−t)3 1 (1−t)3 1 (1−t)6 1 (1−t)2n 1 (1−t)3 1 (1−t)3 1 (1−t)2n 1 (1−t)2 1 (1−t)2n 1 (1−t)n 1 1−t 1 (1−t)2 1 (1−t)2 1 (1−t)2 1 (1−t)2 1 1−t

1 [tn−1 (n−1)! 1 [tn−1 (n−1)! 1 [tn−1 (n−1)! 1 [tn−1 (n−1)! 1 [tn−1 (n−1)! 1 [tn−1 (n−1)! 1 [tn−1 (n−1)!

Discrete linear systems K[t1 , . . . , tn ][x1 , σ1 ] · · · [xn ; σn ] Linear partial shift operators K[t1 , . . . , tn ][E1 , . . . , Em ] Linear partial shift operators K(t1 , . . . , tn )[E1 , . . . , Em ] L. P. Differential operators K[t1 , . . . , tn ][∂1 , . . . , ∂n ] L. P. Differential operators K(t1 , . . . , tn )[∂1 , . . . , ∂n ] L. P. Difference operators K[t1 , . . . , tn ][∆1 , . . . , ∆m ] L. P. Difference operators K(t1 , . . . , tn )[∆1 , . . . , ∆m ] (q)

(q)

L. P. q-dilation operators K[t1 , . . . , tn ][H1 , . . . , Hm ] L. P. L. P.

(q) (q) q-dilation operators K(t1 , . . . , tn )[H1 , . . . , Hm ] (q) (q) q-differential operators K[t1 , . . . , tn ][D1 , . . . , Dm ] (q) (q) q-differential operators K(t1 , . . . , tn )[D1 , . . . , Dm ]

L. P. Algebras of diffusion type Additive analogue of the Weyl algebra An (q1 , . . . , qn ) Multiplicative analogue of the Weyl algebra O n (λji ) Quantum algebra U 0 (so(3, K)) 3-dimensional skew polynomial algebras Dispin algebra U (osp(1, 2)) Woronowicz algebra W ν (sl(2, K)) Complex algebra Vq (sl3 (C)) Algebra U Manin algebra O q (M2 (K)) Coordinate algebra of the quantum group SLq (2) q-Heisenberg algebra Hn (q) Quantum enveloping algebra of sl(2, K), U q (sl(2, K)) Hayashi algebra Wq (J)

Differential operators on a quantum space Sq , Dq (Sq ) Witten’s Deformation of U (sl(2, K) Quantum Weyl algebra of Maltsiniotis Aq,λ n Quantum Weyl algebra An (q, pij ) Multiparameter Weyl algebra AQ,Γ (K) n Quantum symplectic space O q (sp(K 2n )) Quadratic algebras in 3 variables

+ · · · + 1] + · · · + 1] + · · · + 1] + · · · + 1] + · · · + 1] + · · · + 1] + · · · + 1]

1 1 t+1 1 [tn−1 + · · · + 1] (n−1)! 1 [tm−1 + · · · + 1] (m−1)! 1 [tm−1 + · · · + 1] (m−1)! 1 [tn−1 + · · · + 1] (n−1)! 1 [tn−1 + · · · + 1] (n−1)! 1 [tm−1 + · · · + 1] (m−1)! 1 [tm−1 + · · · + 1] (m−1)! 1 [tm−1 + · · · + 1] (m−1)! 1 [tm−1 + · · · + 1] (m−1)! 1 [tm−1 + · · · + 1] (m−1)! 1 [tm−1 + · · · + 1] (m−1)! 1 n−1 + · · · + 1] [t (n−1)! 1 [tn−1 + · · · + 1] (n−1)! 1 [tn−2 + · · · + 1] (n−2)! 1 2 [t + 3t + 1] 2 1 2 [t + 3t + 1] 2 1 2 [t + 3t + 1] 2 1 2 [t + 3t + 1] 2 1 [t5 + 15t4 + 85t3 + 217t2 120 1 [t2n−1 + · · · + 1] (2n−1)! 1 2 [t + 3t + 1] 2 1 2 [t + 3t + 1] 2 1 [t2n−1 + · · · + 1] (2n−1)!

+ 274t + 120]

t+1 1 [t2n−1 + · · · + 1] (2n−1)! 1 n−1 + · · · + 1] [t (n−1)!

1 t+1 t+1 t+1 t+1 1

Table 18.1 Hilbert series and Hilbert polynomials for some examples of bijective skew P BW extensions. Ring n-Multiparametric n-Multiparametric n-Multiparametric n-Multiparametric

skew quantum space Rq,σ [x1 , . . . , xn ] quantum space Rq [x1 , . . . , xn ] skew quantum space Kq,σ [x1 , . . . , xn ] quantum space Kq [x1 , . . . , xn ]

±1 Ring of skew quantum polynomials Rq,σ [x±1 1 , . . . , xr , xr+1 , . . . , xn ] ±1 Ring of quantum polynomials Rq [x±1 1 , . . . , xr , xr+1 , . . . , xn ] ±1 Algebra of skew quantum polynomials Kq,σ [x±1 1 , . . . , xr , xr+1 , . . . , xn ] ±1 Algebra of quantum polynomials O q = Kq [x±1 1 , . . . , xr , xr+1 , . . . , xn ]

GhA (t)

GpA (t)

1 (1−t)n 1 (1−t)n 1 (1−t)n 1 (1−t)n 1 (1−t)n−r 1 (1−t)n−r 1 (1−t)n−r 1 (1−t)n−r

1 [tn−1 + · · · + 1] (n−1)! 1 [tn−1 + · · · + 1] (n−1)! 1 [tn−1 + · · · + 1] (n−1)! 1 [tn−1 + · · · + 1] (n−1)! 1 [tn−r−1 + · · · (n−r−1)! 1 [tn−r−1 + · · · (n−r−1)! 1 [tn−r−1 + · · · (n−r−1)! 1 [tn−r−1 + · · · (n−r−1)!

+ 1] + 1] + 1] + 1]

Table 18.2 Hilbert series and Hilbert polynomials for some skew quantum polynomials.

Chapter 19

Semi-graded Algebras

Motivated by Theorem 18.5.13, Example 18.5.15 and Theorem 18.6.8 of the previous chapter, we will introduce semi-graded versions of Koszul and Artin– Schelter regular algebras. We will classify many of the skew P BW extensions considered in the present monograph according to these two types of algebras. In order to understand better the results presented here for the semi-graded case, we include in Appendix B a quick review of Koszul and Artin–Schelter regular algebras.

19.1 Definition In the present section we define the finitely semi-graded algebras. Let R be a commutative ring and let B be a finitely generated R-algebra, so there exist finitely many elements g1 , . . . , gn ∈ B that generate B as an R-algebra and an R-algebra homomorphism f : R{x1 , . . . , xn } → B, with f (xi ) := gi , 1 ≤ i ≤ n; let I := ker(f ), then we get a presentation of B: B∼ = R{x1 , . . . , xn }/I.

(19.1.1)

Recall that B is said to be finitely presented if I is finitely generated. In the previous chapter we defined the finitely semi-graded rings and we observed that they generalize finitely graded algebras over fields (Definition A.1.1) and skew P BW extensions. In this chapter we will focus on a particular class of these rings which have a finite presentation and satisfy some other extra natural conditions. Definition 19.1.1. Let R be a commutative ring and B be an R-algebra. We say that B is finitely semi-graded (F SG) if the following conditions hold: L (i) B is an F SG ring with semi-graduation B = p≥0 Bp . (ii) For every p, q ≥ 1, Bp Bq ⊆ B1 ⊕ · · · ⊕ Bp+q . (iii) B is connected, i.e., B0 = R. © Springer Nature Switzerland AG 2020 W. Fajardo et al., Skew PBW Extensions, Algebra and Applications 28, https://doi.org/10.1007/978-3-030-53378-6_19

387

388

19 Semi-graded Algebras

(iv) B is generated in degree 1. The most important examples of F SG algebras are when R = K is a field. If nothing contrary is said, from now on in this chapter we will assume that K is a field and B is an algebra over K. In all of the examples that we will study, additionally B is finitely presented.

Remark 19.1.2. Let B be an F SG K-algebra; (i) Of course B0 = K ⊂ Z(B). (ii) Since B is locally finite, i.e., for every n ≥ 0, dimK Bn < ∞, and B is finitely generated in degree 1, any K-basis of B1 generates B as a K-algebra. (iii) The canonical projection ε : B → K is a homomorphism of KL algebras, called the augmentation map, with ker(ε) = n≥1 Bn . Therefore, the class of F SG algebras is contained in the class of augmented algebras, i.e., algebras with augmentation (see [311]), however, a semi-graduation is a nice tool for defining some invariants useful for the study of the algebra. L From Definition 18.4.3 we get that n≥1 Bn = B≥1 , called the augmentation ideal. Thus, K becomes a B-bimodule with products given by b · λ := b0 λ, λ · b := λb0 , with b ∈ B, λ ∈ K and b0 is the homogeneous component of b of degree zero. On the other hand, note that K is not a submodule of B with the product of B (bλ ∈ / K for b ∈ B and λ ∈ K), moreover, the decomposition B = K ⊕ B≥1 is as vector subspaces but not as B-modules. (iv) It is well known that B is finitely graded if and only if the ideal I in (19.1.1) is homogeneous ([344]). In addition, any finitely graded algebra generated in degree 1 is F SG. On the other hand, all skew P BW algebras in Example 18.5.15 are F SG but not finitely graded generated in degree 1. Another example is B := K{x, y}/hxy − xi with semi-graduation Bn :=K hy k xn−k |0 ≤ k ≤ ni, n ≥ 0. Thus, the class of F SG algebras properly includes all finitely graded algebras generated in degree 1. (v) Any F SG algebra is N-filtered (see Proposition 18.1.6), but note that the Weyl algebra A1 (K) = K{t, x}/hxt − tx − 1i is N-filtered but not F SG, i.e., the class of F SG algebras do not coincide with the class of N-filtered algebras. Proposition 19.1.3. Let B be an F SG algebra over K. Then B≥1 is the unique two-sided maximal ideal of B semi-graded as left ideal. Proof. From Remark 19.1.2, B≥1 is a two-sided maximal ideal of B, and of course, semi-graded as a left ideal. Let I be another two-sided maximal ideal of B semi-graded as a left ideal; since I is proper, I ∩ B0 = I ∩ K = 0; let x ∈ I, then x = x0 + x1 + · · · + xn , with xi ∈ Bi , 1 ≤ i ≤ n, but since I is semi-graded, xi ∈ I for every i, so x0 = 0, and hence, x ∈ B≥1 . Thus, I ⊆ B≥1 , but I is maximal, so I = B≥1 . t u

19.2 Examples of F SG Algebras

389

19.2 Examples of F SG Algebras In this section we will present a lot of examples of F SG algebras, many of them within the class of skew P BW extensions. For this we first want to classify the examples of skew P BW extensions given in Chapter 2, in particular, the 3-dimensional skew polynomial algebras. Example 19.2.1. Let K be a field. The 3-dimensional skew polynomial algebras were introduced in Section 2.5; recall that this type of K-algebra, denoted by A, is generated by the variables x, y, z satisfying the relations yz − αzy = λ,

zx − βxz = µ,

xy − γyx = ν,

such that λ, µ, ν ∈ K + Kx + Ky + Kz, and α, β, γ ∈ K ∗ ; if the standard monomials {xi y j z l | i, j, l ≥ 0} conform a K-basis of the algebra, then A is a skew P BW extension of the field K (see the identities of Section 2.5). We present next the classification of these algebras given in [347], Theorem C.4.3.1: (a) If |{α, β, γ}| = 3, then A is defined by yz − αzy = 0,

zx − βxz = 0,

xy − γyx = 0.

(19.2.1)

(b) If |{α, β, γ}| = 2 and β 6= α = γ = 1, then A is one of the following algebras: (i) (ii) (iii) (iv) (v) (vi)

yz − zy yz − zy yz − zy yz − zy yz − zy yz − zy

= z, = z, = 0, = 0, = az, = z,

zx − βxz = y, zx − βxz = b, zx − βxz = y, zx − βxz = b, zx − βxz = 0, zx − βxz = 0,

xy − yx = x; xy − yx = x; xy − yx = 0; xy − yx = 0; xy − yx = x; xy − yx = 0.

Here a, b are any elements of K. All nonzero values of b give isomorphic algebras. (c) If |{α, β, γ}| = 2 and β 6= α = γ 6= 1, then A is one of the following algebras: (i) yz − αzy = 0, (ii) yz − αzy = 0,

zx − βxz = y + b, xy − αyx = 0; zx − βxz = b, xy − αyx = 0.

In these cases b is an arbitrary element of K. Again, any nonzero values of b give isomorphic algebras. (d) If α = β = γ 6= 1, then A is the algebra yz − αzy = a1 x + b1 ,

zx − αxz = a2 y + b2 ,

xy − αyx = a3 z + b3 .

If ai = 0, i = 1, 2, 3, all nonzero values of bi give isomorphic algebras. (e) If α = β = γ = 1, A is isomorphic to one of the following algebras (i) yz − zy = x,

zx − xz = y,

xy − yx = z;

390

19 Semi-graded Algebras

(ii) (iii) (iv) (v)

yz − zy yz − zy yz − zy yz − zy

= 0, zx − xz = 0, xy − yx = z; = 0, zx − xz = 0, xy − yx = b; = −y, zx − xz = x + y, xy − yx = 0; = az, zx − xz = z, xy − yx = 0.

The parameters a, b ∈ K are arbitrary and all nonzero values of b generate isomorphic algebras. In previous chapters we defined the quasi-commutative and bijective skew P BW extensions as well as the skew P BW extensions of derivation or endomorphism type (see Definitions 1.1.4, 5.1.3). Now we present some other special types introduced in [378], and we will classify the examples of Chapter 2 according to them. Definition 19.2.2. Let A = σ(R)hx1 , . . . , xn i be a skew P BW extension of R. (a) A is called constant if condition (iii) in Definition 1.1.1 is replaced by: for every 1 ≤ i ≤ n and r ∈ R, xi r = rxi .

(19.2.2)

(b) A is called pre-commutative if condition (iv) in Definition 1.1.1 is replaced by: for any 1 ≤ i, j ≤ n there exists ci,j ∈ R \ {0} such that xj xi − ci,j xi xj ∈ Rx1 + · · · + Rxn .

(19.2.3)

(f) A is called semi-commutative if A is quasi-commutative and constant. Example 19.2.3. We classify the examples of skew P BW extensions as constant (C), bijective (B), pre-commutative (P), quasi-commutative (QC) and semi-commutative (SC); the classification is presented in the following tables, where the symbols ? and X denote negation and affirmation, respectively. Example 19.2.4 (Skew P BW extensions that are F SG algebras). Note that a skew P BW extension of the field K is an F SG algebra if and only if it is constant and pre-commutative. Thus, we have: (i) According to the classifications presented in the tables the following skew P BW extensions of the field K are F SG algebras: the classical polynomial algebra; the universal enveloping algebra of a Lie algebra; the quantum algebra U 0 (so(3, K)); the dispin algebra; the Woronowicz algebra; the q-Heisenberg algebra; nine types of 3-dimensional skew polynomial algebras (Example 19.2.1: (a); (b) items (i), (iii), (v), (vi); (e) items (i), (ii), (iv), (v)).

19.2 Examples of F SG Algebras

391

Skew PBW extension

C B P QC SC

Classical polynomial ring Ore extensions of bijective type Weyl algebra Universal enveloping algebra of a Lie algebra Tensor product Crossed product Algebra of q-differential operators Algebra of shift operators Mixed algebra Algebra of discrete linear systems Linear partial differential operators Linear partial shift operators Algebra of linear partial difference operators Algebra of linear partial q-dilation operators Algebra of linear partial q-differential operators Algebras of diffusion type Additive analogue of the Weyl algebra Multiplicative analogue of the Weyl algebra Quantum algebra U 0 (so(3, K)) Dispin algebra Woronowicz algebra Complex algebra Algebra U Manin algebra q-Heisenberg algebra Quantum enveloping algebra of sl(2, K) Hayashi’s algebra The algebra of differential operators on a quantum space Sq Witten’s deformation of U (sl(2, K)) Quantum Weyl algebra of Maltsiniotis Quantum Weyl algebra Multiparameter quantized Weyl algebra Quantum symplectic space Quadratic algebras in 3 variables

XX ? X ? X XX XX ? X ? X ? X ? X ? X ? X ? X ? X ? X ? X XX XX XX XX XX XX ? X ? X ? X XX ? X ? X ? X ? X ? X ? X ? X ? X ? X

X X X X X ?

X

X

? ? ? ? ? ?

? ? ?

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

X

X

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

X X X ?

?

X X X X X X X

X

?

X X X X ? ?

X X ? ? ? ? ? ? ? ? ?

?

X ?

X

(ii) Many skew P BW extensions in the table above are marked as nonconstant, however, reconsidering the ring of coefficients, some of them can be also viewed as skew P BW extensions of the base field K; in this way, they are F SG algebras over K: the algebra of shift operators; the algebra of discrete linear systems; the multiplicative analogue of the Weyl algebra; the algebra of linear partial shift operators; the algebra of linear partial q-dilation operators. (iii) In the class of skew quantum polynomials (see Section 4.4) the multi-parameter quantum affine n-space is another example of a skew P BW

392 Cardinality

19 Semi-graded Algebras 3-dimensional skew polynomial algebras

|{α, β, γ}| = 3 yz − αzy = 0, zx − βxz = 0, xy − γyx = 0 yz − zy = z, zx − βxz = y, xy − yx = x yz − zy = z, zx − βxz = b, xy − yx = x |{α, β, γ}| = 2, yz − zy = 0, zx − βxz = y, xy − yx = 0 β 6= α = γ = 1 yz − zy = 0, zx − βxz = b, xy − yx = 0 yz − zy = az, zx − βxz = 0, xy − yx = x yz − zy = z, zx − βxz = 0, xy − yx = 0 |{α, β, γ}| = 2, yz − αzy = 0, zx − βxz = y + b, xy − αyx = 0 β 6= α = γ 6= 1 yz − αzy = 0, zx − βxz = b, xy − αyx = 0 α = β = γ 6= 1 yz − αzy = a1 x + b1 , zx − αxz = a2 y + b2 , xy − αyx = a3 z + b3 yz − zy = x, zx − xz = y, xy − yx = z yz − zy = 0, zx − xz = 0, xy − yx = z α = β = γ = 1 yz − zy = 0, zx − xz = 0, xy − yx = b yz − zy = −y, zx − xz = x + y, xy − yx = 0 yz − zy = az, zx − xz = x, xy − yx = 0

C B P QC SC

X X X X X X X X X X X X X X X

X X X X X X X X X X X X X X X

X X X ? ?

X ?

X X ? ? ?

X X ?

X X

? ? ? ? ? ? ? ? ? ? ? ? ?

X ? ? ? ? ? ? ? ? ? ? ? ? ? ?

extension of the field K that is an F SG (actually finitely graded) algebra. In particular, this is the case for the quantum plane. (iv) The following skew P BW extensions of the field K are F SG but not finitely graded: the universal enveloping algebra of some Lie algebras; the quantum algebra U 0 (so(3, K)); the dispin algebra; the Woronowicz algebra; the q-Heisenberg algebra; eight types of 3-dimensional skew polynomial algebras (Example 19.2.1: (b) items (i), (iii), (v), (vi); (e) items (i), (ii), (iv), (v)). Example 19.2.5 (F SG algebras that are not skew P BW extensions of K). The following algebras are F SG but not skew P BW extensions of the base field K (however, in every example below the algebra is a skew P BW extension of some other subring): (i) Recall that the Jordan plane J is the K-algebra generated by x, y with relation yx = xy + x2 (see Example 18.3.6), i.e., A = K{x, y}/hyx − xy − x2 i. A is not a skew P BW extension of K, but of course, it is an F SG algebra over K. However, A can be viewed as a skew P BW extension of K[x], i.e., A = σ(K[x])hyi, but note that in this interpretation B0 = K[x] * Z(A). Thus, we will consider A as an F SG algebra, actually, as a finitely graded algebra over K. (ii) The finitely graded K-algebra in Example 1.18 of [344] is not a skew PBW extension of K: A = K{x, y, z}/hz 2 − xy − yx, zx − xz, zy − yzi. However, A can be viewed as a skew P BW extension of K[z], i.e., A = σ(K[z])hx, yi, but note that in this interpretation A1 A1 * A1 ⊕ A2 . Thus, we will consider A as an F SG algebra, actually, as a finitely graded algebra over K. (iii) The algebra A of diffusion type is a skew P BW extension of the polynomial ring K[x1 , . . . , xn ] but is not a skew P BW extension of K. Note that A is F SG algebra over K. (iv) The following examples are similar to the previous ones: the Manin algebra, or more generally, the algebra Oq (Mn (K)) of quantum matrices;

19.3 Koszulity

393

the complex algebra Vq (sl3 (C)); the algebra U; Witten’s deformation of U (sl(2, K)); the quantum symplectic space Oq (sp(K 2n )); some quadratic algebras in 3 variables. Example 19.2.6 (F SG algebras that are not skew P BW extensions). The following F SG algebras are not skew P BW extensions: (i) The particular Sklyanin algebra in Remark 18.1.9 (iii) is not a skew PBW extension, but clearly it is an F SG algebra over K. (ii) The finitely graded K-algebra in Example 1.17 of [344]: B = K{x, y}/hyx2 − x2 y, y 2 x − xy 2 i. (iii) Any monomial quadratic algebra B = K{x1 , . . . , xn }/hxi xj , (i, j) ∈ Si, with S any finite set of pairs of indices ([315]). (iv) B = K{x, y}/hx2 − xy, yx, y 3 i ([78]).

19.3 Koszulity Koszul algebras were defined by Stewart B. Priddy in [322], these algebras are N-graded, connected, finitely generated in degree one and quadratic. Later in 2001, Roland Berger in [57] introduced a generalization of Koszul algebras which are called generalized Koszul algebras or N -Koszul algebras. The 2Koszul algebras of Roland Berger are the Koszul algebras of Priddy. N -Koszul algebras are finitely graded where all generators of the ideal I of relations are homogeneous and have the same degree N ≥ 2. In 2008 Thomas Cassidy and Brad Shelton ([78]) generalized the N -Koszul algebras introducing the K2 algebras; this type of algebra accepts that the generators of the ideal I of relations have different degrees, but again all generators are homogeneous since the K2 algebras are graded. Later, Phan in [311] extended this notion to Km algebras for any m ≥ 1. For more details, see Appendix B. In this section we study the semi-graded version of Koszulity, and for this purpose we will follow the lattice interpretation of this notion (see [34], [57], [124] and [315]). Next below we will follow the presentation given in [34] (see also [246] and [150]). Recall that a lattice is a collection L endowed with two idempotent commutative and associative binary operations ∧, ∨ : L × L → L satisfying the following absorption identities: a ∧ (a ∨ b) = a, (a ∧ b) ∨ b = b. A sublattice of a lattice L is a nonempty subset of L closed under ∧ and ∨. A lattice is called distributive if it satisfies the following distributivity identity: a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c). If X ⊆ L, the sublattice generated by X, denoted [X], consists of all elements of L that can be obtained from the elements of X by the operations ∧ and ∨. We will say that X is distributive if [X] is a distributive. The (direct) product of the family of lattices {Lω }ω∈Ω is defined as follows:

394

19 Semi-graded Algebras

Y Ω

Lω := (

Y

Lω , ∧, ∨),

Ω

which is the cartesian product, with ∧ and ∨ operating component-wise. A Q semidirect product of the family {Lω }ω∈Ω is Q a sublattice L of Ω Lω such that for every ω0 ∈ Ω, the composition L ,→ Ω Lω  Lω0 is surjective. Proposition 19.3.1. If L is a semidirect product of the family {Lω }ω∈Ω , then L is distributive if and only if for all ω ∈ Ω, Lω is distributive. Proof. See [34], Lemma 1.1.

t u

Let K be a field and V be a K-vector space. The set L(V ) of all its linear subspaces is a lattice with respect to the operations of sum and intersection. Proposition 19.3.2. Let V be a vector space and X1 , . . . , Xn ⊆ V be a finite collection of subspaces of V . The following conditions are equivalent: (i) The collection X1 , . . . , Xn is distributive. (ii) There exists a basis B := {ωi }i∈C of V such that each of the subspaces Xi is the linear span of a set of vectors ωi . (iii) There exists a basis B of V such that B ∩ Xi is a basis of Xi , for every 1 ≤ i ≤ n. Proof. See [315], [34], Lemma 1.2, or also [377], Proposition 3.2.11.

t u

With the previous elementary facts about lattices, we have the following notions (compare with [34]) associated to any F SG algebra presented as in (19.1.1). Definition 19.3.3. Let B = K{x1 , . . . , xn }/I be an F SG algebra. The lattice associated to B is the sublattice L(B) of subspaces of the free algebra s h F := K{x1 , . . . , xn } generated by {F≥1 I g F≥1 |s, g, h ≥ 0}. For any integer j ≥ 2, the j-th lattice associated to B is defined by Lj (B) := [{Fs Ig Fh |s, h ≥ 0, g ≥ 2, s + g + h = j}] ⊂ {subspaces of Fj ; ∩, +}, where Fs Ig Fh is the subspace of Fj consisting of finite sums of elements of the form abc, with a ∈ Fs , b ∈ Ig , c ∈ Fh , and Ig := {ag ∈ Fg |ag is the g-th component of some element in I}. For any two-sided ideal H of F , the K-subspace Hg is defined similarly. From now on in this section we will denote K{x1 , . . . , xn } by F . Theorem 19.3.4. Let B = K{x1 , . . . , xn }/I be an F SG algebra with I = hb1 , . . . , bm i such that bi ∈ F≥1 for 1 ≤ i ≤ m. Then L(B) is a semidirect product of the family of lattices {Lj (B) ∪ {0, Fj }j≥2 } ∪ {{0, K}, {0, F1 }}. In particular, L(B) is distributive if and only if for all j ≥ 2, Lj (B) is distributive.

19.3 Koszulity

395

Proof. The proof of Lemma 2.4 in [34] can be easy adapted. Step 1. For any j ≥ 2 and any X ∈ Lj (B) we have 0 ⊆ X ⊆ Fj . So Lj (B) ∪ {0, Fj } is in fact a lattice. Step 2. If s ≥ 0, g ≥ 1, h ≥ 0 and j ≥ 2 + s + h, then s h (F≥1 I g F≥1 )j = Fs (I g )j−s−h F. s h We only have to prove that (F≥1 I g F≥1 )j ⊆ Fs (I g )j−s−h Fh since the other s h containment is trivial. Recall that one element of (F≥1 I g F≥1 )j is the j-th s h s h component of some element of F≥1 I g F≥1 ; let zj ∈ (F≥1 I g F≥1 )j , then there s h exists a y ∈ F≥1 I g F≥1 such that zj is the j-th component of y; the element s y is a finite sum of elements of the form abc, with a ∈ F≥1 = F≥s , b ∈ I g h and c ∈ F≥1 = F≥h , so the j-th component of y is a sum of the j − th components of elements of the form ak bat , with k ≥ s, b ∈ I g and t ≥ h, but since Fk = Fs Fk−s for k ≥ s and Ft = Ft−h Fh for t ≥ h, the j-th component of ak bat is the j-th component of as (ak−s bat−h )ah , i.e., it is an element of Fs (I g )j−s−h Fh . Step 3. For g ≥ 1 and j ≥ 2, X (I g )j = Fk0 Il1 Fk1 Il2 · · · Fkg−1 Ilg Fkg ,

where the Psum is taken over all relevant k0 , . . . , kg , l1 , . . . , lg such that P g k m m + n ln = j. Indeed, if p ∈ I , then p is a finite sum of elements (0) (1) (g−1) (g) of the form a p1 a p2 · · · a pg a , with a(r) ∈ F , pi ∈ {b1 , . . . , bm }, 0 ≤ r ≤ g, 1 ≤ i ≤ g. Step 4. For any g ≥ 2 and any 2g + 1 non-negative integers k0 , . . . , kg , l1 , . . . , lg we have Fk0 Il1 Fk1 Il2 · · · Fkg−1 Ilg Fkg =

g \

Fk0 +l1 +···+ka−1 Ila Fka +···+kg .

a=1

In fact, let q = a0 p1 a1 · · · pg ag ∈ Fk0 Il1 Fk1 Il2 · · · Fkg−1 Ilg Fkg , with ar ∈ Fkr , pi ∈ Ili , 0 ≤ r ≤ g, 1 ≤ i ≤ g, then q ∈ Fk0 +l1 +···+ka−1 Ila Fka +···+kg for every 1 ≤ a ≤ g; the converse follows from the fact that for any a ∈ F − {0} homogeneous with a = bc = de, then b, c, d, e are homogeneous; in addition, if b ∈ Fk , d ∈ Ft with t ≥ s, then there is an f such that a = bf e, d = bf and c = f e. s h Step 5. For any s ≥ 0, g ≥ 1, h ≥ 0 and j < 1+s+h we have (F≥1 I g F≥1 )j = g 0 since bi ∈ F≥1 for 1 ≤ i ≤ m; likewise, for j < g, (I )j = 0. From these steps, L(B) is a sublattice of the product of the given family, i.e., Y L(B) ,→ {0, K} × {0, F1 } × ( Lj (B) ∪ {0, Fj }). j≥2

Finally, fix j ≥ 2, then L(B) → Lj (B) ∪ {0, Fj } is a lattice surjective map s since: (a) (I g )j = 0 if j < g; (b) (F≥1 )j = Fj if j ≥ s; (c) if s, h ≥ 0, g ≥ 2

396

19 Semi-graded Algebras

s h and s + g + h = j, then Fs Ig Fh = (F≥1 I g F≥1 )j . The cases j = 0, 1 can be proved by the same method. Thus, L(B) is a semidirect product of the given family. t u

Definition 19.3.5. Let B = K{x1 , . . . , xn }/I be an F SG algebra. We say that B is semi-graded Koszul, denoted SK, if B satisfies the following conditions: (i) B is finitely presented with I = hb1 , . . . , bm i and bi ∈ F≥1 for 1 ≤ i ≤ m. (ii) L(B) is distributive. Remark 19.3.6. (i) We are adopting the following definition of Koszul algebras (see [34], [35], [57], [124], [315]; see also Proposition B.1.3). Let B be a K-algebra, B is Kozul if B satisfies the following conditions: (a) B is Ngraded, connected, finitely generated in degree one; (b) B is quadratic, i.e., the ideal I in (19.1.1) is finitely generated by homogeneous elements of degree 2; (c) L(B) is distributive. (ii) From (i) it is clear that any Koszul algebra is SK. Many examples of skew P BW extensions are actually Koszul algebras. In [341] and [378] it was proved that the following skew P BW extensions are Koszul algebras: the classical polynomial algebra; the multiplicative analogue of the Weyl algebra; the algebra of linear partial q-dilation operators; the multi-parameter quantum affine n-space, in particular, the quantum plane; the 3-dimensional skew polynomial algebra with |{α, β, γ}| = 3; the Jordan plane; algebras of diffusion type; the algebra U; the Manin algebra, more generally, the algebra Oq (Mn (K)) of quantum matrices; and some quadratic algebras in 3 variables. The following theorem gives a wide list of SK algebras within the class of (k ) skew P BW extensions. If at least one of the constants aij i,j is nonzero, then the algebra is not Koszul but it is SK. Theorem 19.3.7. If A is a skew P BW extension of a field K with presentation A = K{x1 , . . . , xn }/I, where (k

)

(k

)

I = hxj xi − ci,j xi xj − aij i,j xki,j | ci,j , aij i,j ∈ K, ci,j 6= 0, 1 ≤ j < i ≤ ni, then A is SK. Proof. Note that A is an F SG algebra. Let F := K{x1 , . . . , xn }, N := {x1 , . . . , xn }, and J := {ki,j ∈ {1, . . . , n}|aki,j 6= 0, 1 ≤ i < j ≤ n}. We are going to show that Lm (A) is a distributive lattice for m ≥ 2. If |J| = n, we define

Bm :=

m [ r=1

where

! Dr(m)

,

19.3 Koszulity

397

Dr(m) := {a1 · · · ar−1 xi ar+1 · · · am |at ∈ N, t = 1, . . . , r − 1, r + 1, . . . , n; 1 ≤ i ≤ n}; Bm is a basis of Fm . Now, consider Fs Ig Fh ≤ Fm with s, h ≥ 0, g ≥ 2 (m) (m) and s + g + h = m. Since Fs Ig Fh is generated by Ds+1 , . . . , Ds+g , we have Ss+g (m) Fs Ig Fh ∩ Bm = r=s+1 Dr , which is a basis of Fs Ig Fh . If |J| = n − 1, define m [

Bm :=

! Dr(m)

∪ {xm l },

r=1

where l ∈ / J, and Dr(m) := {a1 · · · ar−1 xi ar+1 · · · am |at ∈ N, t = 1, . . . , r − 1, r + 1, . . . , n; i ∈ J}; again Bm is a basis of Fm . As before, consider Fs Ig Fh ≤ Fm with s, h ≥ 0, (m) (m) g ≥ 2 and s + g + h = m; since Fs Ig Fh is generated by Ds+1 , . . . , Ds+g , we Ss+g (m) have Fs Ig Fh ∩ Bm = r=s+1 Dr , which is a basis of Fs Ig Fh . If |J| ≤ n − 2, we define

Bm :=

m−1 [ r=1

! Br(m)

∪

m−1 [

! Cr(m)

r=1

∪

m [

! Dr(m)

∪ E,

r=1

where (m)

Br

:= {a1 · · · ar−1 xj xi ar+2 · · · am |at ∈ N ; t = 1, 2, . . . , r − 1, r + 2, . . . , m; i, j ∈ / J; i < j},

(m) Cr

:= {a1 · · · ar−1 (xi xj − cij xj xi )ar+2 · · · am |at ∈ N ; t = 1, 2, . . . , r − 1, r + 2, . . . , m; i, j ∈ / J; i < j},

(m) Dr

:= {a1 · · · ar−1 xl ar+1 · · · am |at ∈ N, t = 1, . . . , r − 1, r + 1, . . . , n; l ∈ J},

E = {xm / J}. i |i ∈

Bm is a basis of Am ; consider Fs Ig Fh ≤ Fm with s, h ≥ 0, g ≥ 2 and (m) (m) (m) (m) s+g+h = m; since Fs Ig Fh is generated by Cs+1 , . . . , Cs+g−2 , Ds+1 , . . . , Ds+g ,   Ss+g Ss+g−2 (m) we have Fs Ig Fh ∩ Bm = r=s+1 Cr ∪ r=s+1 Dr , which is a basis of Fs Ig Fh . t u Example 19.3.8 ([150]). (i) The following algebras satisfy the conditions of the previous theorem, and hence, they are SK (but not Koszul): the dispin algebra U (osp(1, 2)); the q-Heisenberg algebra; the quantum algebra U 0 (so(3, K)); the Woronowicz algebra Wν (sl(2, K)); the algebra Sh of shift operators; the algebra D for multidimensional discrete linear systems; the

398

19 Semi-graded Algebras

algebra of linear partial shift operators; and the universal enveloping algebra of some Lie algebras. (ii) The following algebras do not satisfy the conditions of the previous theorem, but by direct computation we proved that the lattice L(B) is distributive, so they are SK (but not Koszul): the algebra Vq (sl3 (C)); Witten’s deformation of U (sl(2, K); and the quantum symplectic space Oq (sp(K 2n )). Example 19.3.9. There exist F SG algebras that are not SK: the algebra A = K{x, y}/hx2 − xy, yx, y 3 i (see ([78]) ) is not a skew P BW extension, but is an F SG algebra. This algebra satisfies that L(A) is a subdirect product of the family of lattices {Lj (A) ∪ {0, Aj }}j≥2 ∪ {{0, K}, {0, A1 }}, but L3 (A) is not distributive. In fact, note that the lattice L3 (A) is generated by A1 I2 , I2 A1 , I3 and 1. A1 I2 is K-generated by D = {x3 − xyx, x2 y − xy 2 , yx2 , yxy}, which is K-linearly independent, therefore dimK (A1 I2 ) = 4. 2. I2 A1 is K-generated by C = {x3 − x2 y, yx2 − yxy, xyx, y 2 x}, which is K-linearly independent, therefore dimK (I2 A1 ) = 4. Now, let us suppose B = {a1 , a2 , . . . , a8 } is a K-basis of A3 such that X := B ∩ A1 I2 is a basis of A1 I2 and Y := B ∩ I2 A1 is a basis of I2 A1 . Without loss of generality, suppose that X = {a1 , . . . , a4 }, then yx2 = λ1 a1 +λ2 a2 +λ3 a3 +λ4 a4 and yxy = β1 a1 +β2 a2 +β3 a3 +β4 a4 with λi , βi ∈ K for 1 ≤ i ≤ 4, λ1 6= β1 , λ1 6= 0 and λj 6= βj , for some j = 2, 3, 4, otherwise, if λj = βj , for j = 2, 3, 4, then yxy − λβ11 yx2 = 0, which is impossible. So yx2 − yxy = (λ1 − β1 )a1 + (λ2 − β2 )a2 + (λ3 − β3 )a3 + (λ4 − β4 )a4 , with at least a1 , aj ∈ X ∩ Y . Consequently, a1 = α1 (x3 − xyx) + α2 (x2 y − xy 2 ) + α3 (yx2 ) + α4 (yxy) = γ1 (x2 − x2 y) + γ2 (yx2 − yxy) + γ3 (xyx) + γ4 (y 2 x), aj = η1 (x3 − xyx) + η2 (x2 y − xy 2 ) + η3 (yx2 ) + η4 (yxy) = µ1 (x2 − x2 y) + µ2 (yx2 − yxy) + µ3 (xyx) + µ4 (y 2 x), with αi , γi , ηi , µi ∈ K for 1 ≤ i ≤ 4. Thus there exist two different nontrivial K-combinations of C ∪ D equal to 0, hence the base B does not exist. Thus, A is an F SG algebra but is not SK.

19.4 Artin–Schelter Regularity

399

19.4 Artin–Schelter Regularity Artin–Schelter regular algebras (denoted AS, for short) were introduced by Michael Artin and William Schelter in [27]. In noncommutative algebraic geometry these algebras play the role that K[x1 , . . . , xn ] plays in commutative algebraic geometry, thus, in particular, the noetherian AS algebras satisfy the condition X of Definition A.5.3 (Theorem 12.6 in [366]), and hence, for them the noncommutative version of Serre’s theorem about coherent sheaves is true (Corollary A.5.6). Nowadays AS algebras are being intensively investigated. In Appendix B we present the definition and we summarize some important properties and results on these algebras. AS algebras are N-graded and connected (see Definition B.2.1), however, recently Gaddis in [130] and [131] introduced the technique of homogenization and the notion of essentially regular algebras in order to study the Artin–Schelter condition for non-N-graded algebras. On the other hand, with the purpose of giving new examples of AS algebras, in [265] and [266] the Zs -graded Artin–Schelter regular algebras are defined and therein some results on the classification of AS algebras of dimension 5 with two generators are proved. The semi-graded Artin–Schelter regular algebras that we will introduce and study in the present section is a new different way of extending the notion of an Artin–Schelter regular algebra defined originally in [27]. Many nontrivial examples of these new algebras are given within the skew P BW extensions. Definition 19.4.1. Let K be a field and B be a K-algebra. We say that B is a left semi-graded Artin–Schelter regular algebra (SAS) if the following conditions hold: L (i) B is an F SG ring with semi-graduation B = p≥0 Bp . (ii) B is connected, i.e., B0 = K. (iii) lgld(B) := d < ∞. ( i ∼ 0 if i 6= d (iv) ExtB (B (B/B≥1 ),B B) = (B/B≥1 )B if i = d Remark 19.4.2. (i) The right semi-graded Artin–Schelter regular algebras are defined in a similar way.. (ii) If K ∩ B≥1 6= 0, then B/B≥1 = 0; if K ∩ B≥1 = 0 then B/B≥1 ∼ = K, where this is an isomorphism of K-algebras induced by the canonical projection  : B → K (recall that B≥1 is semi-graded). Moreover, B (B/B≥1 ) ∼ = BK and (B/B≥1 )B ∼ = KB . (iii) Our definition extends (except for the GKdim) the classical notion of an Artin–Schelter algebra since in such case B/B≥1 ∼ = K, lgld(B) = pd(B K) = pd(KB ) = rgld(B) = gld(B) and ExtiB (K, B) = ExtiB (K, B) (see Propositions A.1.20 and A.2.2). Thus, every Artin–Schelter regular algebra is SAS. From now on in this section, for the matrix representation of homomorphisms of left modules we will use the left row notation and for right modules the right column notation (see Remark 9.1.2). In order to compute

400

19 Semi-graded Algebras

free resolutions we will apply Theorem 15.6.1, which has been implemented in the library SPBWE.lib developed in Maple by W. Fajardo in [117] (see also [118]). This library contains the packages SPBWETools, SPBWERings and SPBWEGrobner with utilities to define and perform calculations with skew P BW extensions (see Appendix D). Next we present some examples of left SAS algebras that are not Artin– Schelter, in all cases, the algebra is left-right noetherian, whence lgld(B) = rgld(B) = gld(B). Example 19.4.3. The Weyl algebra A1 (K) is not Artin–Schelter, but it is an SAS algebra. In fact, the canonical semi-graduation of A1 (K) is A1 (K) = K ⊕

K hx, yi

⊕

K hx

2

, xy, y 2 i ⊕ · · ·

Recall that gld(A1 (K)) ≤ 2 (Example 7.2.5); moreover, A1 (K)≥1 = A1 (K), so A1 (K)/A1 (K)≥1 = 0. Hence, the condition (iv) in Definition 19.4.1 trivially holds. This argument can also be applied to the general Weyl algebra An (K) and the generalized Weyl algebra Bn (K). Example 19.4.4. The quantum deformation Aq1 (K) of the Weyl algebra is not Artin–Schelter, but it is an SAS algebra. Indeed, recall that Aq1 (K) is the K-algebra defined by the relation yx = qxy+1, with q ∈ K−{0} (Aq1 (K) coincides with the additive analog of the Weyl algebra in two variables, as well as with the linear algebra of q-differential operators in two variables). The semigraduation of Aq1 (K) is as in the previous example, moreover gld(Aq1 (K)) ≤ 2 and Aq1 (K)≥1 = Aq1 (K), so Aq1 (K)/Aq1 (K)≥1 = 0 and the condition (iv) in Definition 19.4.1 holds. Example 19.4.5. Consider the algebra J1 := Khx, yi/hyx − xy + y 2 + 1i. According to Corollary 2.3.14 in [130], gld(J1 ) = 2. The semi-graduation of J1 is as in the previous example, so (J1 )≥1 = J1 . Thus, J1 is SAS but is not Artin–Schelter. Example 19.4.6. Now we consider the dispin algebra U (osp(1, 2)), which in this example we will be denoted simply as B (see Example 18.5.15). Recall that B is defined by the relations x1 x2 − x2 x1 = x1 ,

x 3 x1 + x1 x3 = x2 ,

x 2 x3 − x3 x2 = x3 .

Since B is not finitely graded, B is not Artin–Schelter. We will show that B is SAS. We know that gld(B) = 3 (Table 8.1); the semi-graduation of B is given by B=K⊕

K hx1 , x2 , x3 i

⊕

2 2 2 K hx1 , x1 x2 , x1 x3 , x2 , x2 x3 , x3 i

⊕ ···

Note that B≥1 = ⊕p≥1 Bp and this ideal coincides with the two-sided ideal of B generated by x1 , x2 , x3 ; moreover, B (B/B≥1 ) ∼ =B K, where the left Bmodule structure for K is given by the canonical projection  : B → K (the same is true for the right structure, (B/B≥1 )B ∼ = KB , see Remark 19.4.2). With SPBWE we get the following free resolution of B K (see Subsection C.2.2 and Appendix D):

19.4 Artin–Schelter Regularity

401 







1 + x2 −x1 0 x1     φ0 =  x3 −1 x1    x2  φ2 = −x3 x2 x1 0 x 1 − x x 3 2 3  0 → B −−−−−−−−−−−−−→ B 3 −−−−−−−−−−−−−−−−−−−→ B 3 −−−−−−−→ B − → K → 0. h

 φ 1 = 

i

Now we apply HomB (−,

B B)

and we get the complex of right B-modules φ∗

∗

0 0 → HomB (K, B) −→ HomB (B, B) −→ HomB (B 3 , B)

φ∗

φ∗

1 2 −→ HomB (B 3 , B) −→ HomB (B, B) → 0.

Note that HomB (K, B) = 0: in fact, let α ∈ HomB (K, B) and α(1) := b ∈ B, then α(x1 1) = α(0) = 0 = x1 α(1) = x1 b, so b = 0 (recall that B is a domain) and from this α(k) = 0 for every k ∈ K, i.e., α = 0. Moreover, from the isomorphisms of right B-modules HomB (B, B) ∼ = B and HomB (B 3 , B) ∼ = B3 we obtain the complex 1 + x2 −x1 0  x3 −1 x1   h i φ∗ x3 0 x 1 − x 2 = −x3 x2 x1 3 2 0 → B −−−−−−→ B 3 −−−−−−−−−−−−−−−−−−−→ B 3 −−−−−−−−−−−−→ B → 0. 

 x1     φ∗ 0 =x2 





  φ∗ 1 =

Actually, with SPBWE we have checked that this complex is exact. So, Ext0B (K, B) = HomB (K, B) = 0, Ext1B (K, B) = 0 = Ext2B (K, B) and Ext3B (K, B) = B/Im(φ∗2 ) = B/B≥1 ∼ = KB . This shows that B is SAS. Example 19.4.7. The next examples are similar to the previous ones. In every case gld(B) = 3 and B is SAS. The free resolutions have been computed with SPBWE. (a) Consider the universal enveloping algebra of the Lie algebra sl(2, K), B := U (sl(2, K)). B is the K-algebra generated by the variables x, y, z subject to the relations [x, z] = −2x,

[x, y] = z,

[y, z] = 2y.

According to Remark 18.1.9 (i), U (sl(2, K)) is not N-graded, so it is not Artin–Schelter. The free resolutions are: 



 

y −x 1 x     φ0 =  − 2 0 −x  y  h i φ2 = −z y −x 0 z + 2 −y z  0 → B −−−−−−−−−−−→ B 3 − −−−−−−−−−−−−−−−− → B 3 −−−−−−→ B − → K → 0.  φ1 =  z

y −x 1  − 2 0 −x  h i φ∗ z 0 z + 2 −y 2 = −z y −x 3 3 0 → B −−−−−−→ B −−−−−−−−−−−−−−−−→ B −−−−−−−−−−−→ B → 0.  

x

    φ∗ 0 = y 





  φ∗ 1 =z

(b) Now let B := U (so(3, K)) be the K-algebra generated by the variables x, y, z subject to the relations

19 Semi-graded Algebras

402

[x, z] = −y,

[x, y] = z,

[y, z] = x.

In this case we have 

 



y −x 1 x    y  φ0 =  −1 −x   h i φ2 = −z y −x 1 z −y z  0 → B −−−−−−−−−−−→ B 3 −−−−−−−−−−−−→ B 3 −−−−−−→ B − → K → 0,  z φ1 = 

y −x 1  −1 −x  i h φ2∗ = −z y −x 1 z z −y 0 → B −−−−−−→ B 3 −−−−−−−−−−−→ B 3 −−−−−−−−−−−→ B → 0. 

 

x

   φ0∗ = y 



 φ1∗ = z

(c) The quantum algebra B := U 0 (so(3, K)), with q ∈ K − {0}: x3 x1 − q −1 x1 x3 = q −1/2 x2 , −q 1/2 x1 . In this case the free resolutions are: x2 x1 − qx1 x2 = −q 1/2 x3 ,

x3 x2 − qx2 x3 =

−qx1 q 1/2 x1    1/2 −x   φ0 =  −q 1   x2  i h φ2 = −x3 x2 −x1 x x −qx 2 3 3  → K → 0, 0 → B −−−−−−−−−−−−−−→ B 3 −−−−−−−−−−−−−−−−−−→ B 3 −−−−−−−→ B − 



x2   qx φ1 =  3 q 1/2









x −qx1 q 1/2  x1  2     ∗ ∗   φ1 = qx3 −q 1/2 −x1  φ0 =x2  i  h  1/2 φ∗ x3 −qx2 q x3 2 = −x3 x2 −x1 3 3 0 → B −−−−−−→ B −−−−−−−−−−−−−−−−−→ B −−−−−−−−−−−−−→ B → 0. 



(d) The Woronowicz algebra B := W ν (sl(2, K)), where ν ∈ K − {0} is not a root of unity: x1 x3 − ν 4 x3 x1 = (1 + ν 2 )x1 ,

x1 x2 − ν 2 x2 x1 = νx3 , (1 + ν 2 )x2 .

x 3 x2 − ν 4 x2 x3 =

The free resolutions are: h φ2 = −ν 4 x3

i

ν 6 x2 −x1 0 → B −−−−−−−−−−−−−−−−−→ B 3 ν 2 x2 −x1 ν   4 2  φ1 = x + 1) 0 ν + (ν −x 1  3  2 4 0 x3 − (ν + 1) −ν x2 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ B 3 

 x1   x  φ0 = 2 

x3  − K → 0, −−−−−−→ B →



19.4 Artin–Schelter Regularity

403

ν 2 x2 −x1 ν  + (ν 2 + 1) 0 −x1   x3 0 x3 − (ν 2 + 1) −ν 4 x2 3 0 → B −−−−−−→ B −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ B 3 

 x1     φ∗ 0 =x2 





 4  φ∗ 1 =ν x3

h 4 φ∗ 2 = −ν x3

i

ν 6 x2 −x1 −−−−−−−−−−−−−−−−−→ B → 0. Example 19.4.8. In this example we study the semi-graded Artin–Schelter condition for the eight types of 3-dimensional skew polynomial algebras considered in Example 18.5.15, five of them are SAS and the other three are not. The first type coincides with the dispin algebra taking β = −1, and the fifth type corresponds to U (so(3, K)), thus they are SAS. For the other six types we compute next the free resolutions with SPBWE: x2 x3 − x3 x2 = 0, x3 x1 − βx1 x3 = x2 , x1 x2 − x2 x1 = 0 (SAS): 







−x1 0 x1     φ0 =  −1 −βx x2  1    h i φ2 = −x3 x2 −βx1 0 x −x x 3 2 3  3 3 0 → B −−−−−−−−−−−−−−−→ B −−−−−−−−−−−−−−−−→ B −−−−−−−→ B − → K → 0, x2  φ1 =   x3

−x1 0  −1 −βx1   h i φ∗ x3 0 x −x 2 = −x3 x2 −βx1 3 2 3 3 0 → B −−−−−−→ B −−−−−−−−−−−−−−−→ B −−−−−−−−−−−−−−→ B → 0. 

 x1     φ∗ 0 =x2 





x2   φ∗ 1 =x3

x2 x3 − x3 x2 = x3 , x3 x1 − βx1 x3 = 0, x1 x2 − x2 x1 = x1 (SAS): 







x2 + 1 −x1 0 x1     φ0 =  x3 0 −βx1    x2  h i φ2 = −x3 x2 −βx1 0 x −x + 1 x 3 2 3  0 → B −−−−−−−−−−−−−−−→ B 3 −−−−−−−−−−−−−−−−−−−−−→ B 3 −−−−−−−→ B − → K → 0,  φ1 =  









x1 x2 + 1 −x1 0     φ∗ x2  0 −βx1  1 = x 3   h i φ∗ x3 0 x3 −x2 + 1 2 = −x3 x2 −βx1 3 3 0 → B −−−−−−−→ B −−−−−−−−−−−−−−−−−−−−−→ B −−−−−−−−−−−−−−−→ B → 0.  φ∗ 0 = 

x2 x3 − x3 x2 = x3 , x3 x1 − βx1 x3 = 0, x1 x2 − x2 x1 = 0 (not SAS) :

19 Semi-graded Algebras

404 







0 x1 −x1     φ0 =  0 −βx1    x2  h i φ2 = −x3 x2 − 1 −βx1 + 1 x 0 x −x 2 3 3  → K → 0, 0 → B −−−−−−−−−−−−−−−−−−→ B 3 −−−−−−−−−−−−−−−−−−→ B 3 −−−−−−−→ B − x2  φ 1 =  x3









0 x2 −x1 x1     φ∗ x2  0 −βx1  1 = x3   i h φ∗ 0 x3 −x2 + 1 x3 2 = −x3 x2 − 1 −βx1 0 → B −−−−−−−→ B 3 −−−−−−−−−−−−−−−−−−→ B 3 −−−−−−−−−−−−−−−−−−→ B → 0.  φ∗ 0 =



x2 x3 − x3 x2 = 0, x3 x1 − x1 x3 = 0, x1 x2 − x2 x1 = x3 (SAS):  x2  φ1 =   x3







−x1 1 x1      φ0 =  0 −x1   x2  i h φ2 = −x3 x2 −x1 0 x3 −x2 x3  3 3 → K → 0, 0 → B −−−−−−−−−−−−−−→ B −−−−−−−−−−−−−−−→ B −−−−−−−→ B −









x1 x2 −x1 1       φ∗ 0 −x1  x2  1 = x3 h i φ∗ x3 0 x −x 3 2 2 = −x3 x2 −x1 3 3 0 → B −−−−−−−→ B −−−−−−−−−−−−−−−→ B −−−−−−−−−−−−−−→ B → 0. ∗  = φ0



x2 x3 − x3 x2 = −x2 , x3 x1 − x1 x3 = x1 + x2 , x1 x2 − x2 x1 = 0 (not SAS) : 







x2 x1 0 −x1    x  φ0 =  − 1 −1 −x1  2  i h φ2 = −x3 + 2 x2 −x1 x3 − 1 −x2 0 x3  → K → 0, 0 → B −−−−−−−−−−−−−−−−−→ B 3 −−−−−−−−−−−−−−−−−−−→ B 3 −−−−−−−→ B −  x φ1 =  3

 x1   ∗  φ0 = x2   





x2 −x1 0   − 1 −1 −x1  i h φ∗ 0 1 −x − x3 x 2 3 2 = −x3 + 2 x2 −x1 3 3 0 → B −−−−−−−→ B −−−−−−−−−−−−−−−−−−−→ B −−−−−−−−−−−−−−−−−→ B → 0.  ∗  φ1 = x3

x2 x3 − x3 x2 = x3 , x3 x1 − x1 x3 = x3 , x1 x2 − x2 x1 = 0 (not SAS): 



0 −x1   0 −x1 − 1 h i φ2 = −x3 x2 − 1 −x1 − 1 0 x + 1 −x 2 3 0 → B −−−−−−−−−−−−−−−−−−−−→ B 3 −−−−−−−−−−−−−−−−−−→ B 3 x2  φ1 =   x3

19.4 Artin–Schelter Regularity 

405



x1    φ0 =   x2 

x3  −−−−−−−→ B − → K → 0,









x1 x2 −x1 0     φ∗ x2  0 −x1 − 1 1 =  x3   x3 0 x3 −x2 + 1 0 → B −−−−−−−→ B 3 −−−−−−−−−−−−−−−−−−→ B 3  φ∗ 0 = 

φ∗ 2=

h

i

−x3 x2 − 1 −x1 − 1 −−−−−−−−−−−−−−−−−−−−→ B → 0.

Next we present an algebra which is essentially regular in the sense of Gaddis (see [130] and [131]) but not SAS. Recall that an N-filtered algebra A is essentially regular if and only if Gr(A) is AS (see Proposition 2.3.7 in [130]). Example 19.4.9. Consider the algebra U := K{x, y}/hyx − xy + yi, by Corollary 2.3.14 in [130] U is essentially regular of global dimension 2. Clearly U is not Artin–Schelter, actually we will show that U is not SAS. The semigraduation of U is as in Example 19.4.3, so U≥1 = ⊕p≥1 Up and this ideal coincides with the two-sided ideal of U generated by x and y. Observe that ∼ U (U /U≥1 ) =U K, where the left U -module structure for K is given by the canonical projection  : U → K (the same is true for the right structure, (U /U≥1 )U ∼ = KU ). The following sequence is a free resolution of U K:  

x y 1−x  0 → U −−−−−−−−−−→ U 2 −−−−−→ U → − K → 0. h φ1 = y

i

φ0 = 

This statement can be proved using SPBWE or simply by hand. In fact,  is  clearly surjective; φ1 is injective since if φ1 (u) = 0 for u ∈ U , then u y 1 − x = 0, so u = 0 since U is a domain. Im(φ0 ) = ker() = U≥1 since     x uv = ux + vy, with u, v ∈ U . y Im(φ1 ) ⊆ ker(φ0 ) since φ1 φ0 = 0:     x y 1−x = yx + (1 − x)y = 0. y   Now, ker(φ0 ) ⊆ Im(φ1 ): in fact, let u v ∈ ker(φ0 ), then ux + vy = 0; let u = u0 + u1 x + u2 y + u3 x2 + u4 xy + u5 y 2 + · · · , v = v0 + v1 x + v2 y + v3 x2 + v4 xy + v5 y 2 + · · · ,

406

19 Semi-graded Algebras

from ux + vy = 0 we conclude that all terms of u involving only x’s have coefficient equal to zero, i.e., u = py for some p ∈ U , hence vy = −pyx = −p(xy − y) = p(1 − x)y, but since A is a domain, v = p(1 − x), whence,     u v = p y 1 − x ∈ Im(φ1 ). Now we apply HomU (−, U U ) and we get the complex of right U -modules φ∗

∗

φ∗

0 1 0 → HomU (K, U ) −→ HomU (U , U ) −→ HomU (U 2 , U ) −→ HomU (U , U ) → 0.

As in Example 19.4.6, HomU (K, U ) = 0, HomU (U , U ) ∼ = U and HomU (U 2 , U ) 2 ∼ = U , so we obtain the complex  

x h i φ∗ y 1= y 1 − x 2 0 → U −−−−−−→ U −−−−−−−−−−→ U → 0,   φ∗ 0=

with φ∗0 is injective. Moreover, Im(φ∗0 ) = ker(φ∗1 ): indeed, it is clear that   u Im(φ∗0 ) ⊆ ker(φ∗1 ); let ∈ ker(φ∗1 ), then yu + (1 − x)v = 0, from this by a v direct computation we get that v = q 0 y for some q 0 ∈ U , but again by a direct computation it is easy to show that given a polynomial q 0 there exists a q ∈ U such that q 0 y = yq, whence v = yq for some q ∈ U . Hence, yu + (1 − x)yq = 0 implies y(u − xq) = 0, so u = xq. Therefore, ker(φ∗1 ) ⊆ Im(φ∗0 ) and we have proved the claimed equality. Thus, Ext0U (K, U ) = HomU (K, U ) = 0, Ext1U (K, U ) = 0 and Ext2U (K, U ) = U /Im(φ∗1 )  KU . In fact, suppose there exists a right U -module isomorphism α U /Im(φ∗1 ) − → KU , let α(1) := λ, then α(1)·(1−x) = λ·(1−x), so α(1·(1−x)) = λ, i.e., α(0) = 0 = λ, whence α = 0, a contradiction. We conclude that U is not SAS. Now we will give an example of an algebra that is SAS but is not essentially regular. Example 19.4.10. Let S be the algebra of Theorem 4.0.7 in [130] defined by B := K{x, y}/hyx − 1i. B has a semi-graduation as in Example 19.4.3. It is known that gld(B) = 1 (see [130], Proposition 4.1.1 and also [42]) and it is clear that B/B≥1 = 0. Thus, the condition (iv) in Definition 19.4.1 trivially holds and B is SAS. Now observe that Gr(B) ∼ = Ryx , where Ryx is defined in [130] by Ryx := K{x, y}/hyxi. According to Corollary 2.3.14 in [130], B is not essentially regular. We will give a direct proof of this in our next example. We conclude the list of examples with an algebra that is neither essentially regular nor SAS.

19.4 Artin–Schelter Regularity

407

Example 19.4.11. Let B := Ryx . Observe that B is a finitely graded algebra with graduation as in Example 19.4.3; by a direct computation we get the following exact sequences  

x y 0  0 → B −−−−−−→ B 2 −−−−−→ B → − K → 0, h φ1 = y

i

φ0 = 

 

x h i φ∗ y 1= y 0 2 0 → B −−−−−−→ B −−−−−−→ B → 0.   φ∗ 0=

According to [130], gld(B) = 2, but note that B/yB B K. In fact, suppose α there exists a right B-module isomorphism B/yB − → KB , let α(1) := λ, then α(1) · x = λ · x, so α(x) = 0, i.e., x = 0 but clearly x ∈ / yB. This says that B is neither SAS nor essentially regular. The previous examples induce the following general result. Theorem 19.4.12. Let K be a field and A := σ(R)hx1 , . . . , xn i be a bijective skew P BW extension that satisfies the following conditions: (i) R and A are K-algebras. (ii) lgld(R) < ∞. L (iii) R is an F SG ring with semi-graduation R = p≥0 Rp . (iv) R is connected, i.e., R0 = K. (v) For 1 ≤ i ≤ n, σi , δi in Proposition 1.1.3 are homogeneous, and there exist i, j such that the parameter dij in (1.2.1) satisfies dij ∈ K − {0}. Then, B is an SAS algebra. Proof. First note that A is F SG and connected with semi-graduation A0 := K, Ap :=K hRq xα | q + |α| = pi for p ≥ 1, with xα as in Definition 1.1.1 and |α| as in Definition 1.1.7. Observe that if r1 , . . . , rm generate R as a K-algebra, then r1 , . . . , rm , x1 , . . . , xn generate A as a K-algebra. Moreover, dimK Ap < ∞ for every p ≥ 0. From Theorem 7.1.4 we know that lgld(A) < ∞. Condition (v) in the statement of the theorem says that A/A≥1 = 0, so condition (iv) in Definition 19.4.1 trivially holds. t u The next theorem extends the condition X of Definition A.5.3 introduced in [27] for finitely graded left noetherian algebras. Theorem 19.4.13. Let B be a left noetherian SAS algebra. Then, for every finitely generated left B-module N , dimK (ExtjB (B/B≥1 , N )) < ∞ for every j ≥ 0. Proof. Let N be a finitely generated left B-module, we will assume first that N is free, i.e, N ∼ = B n , therefore

408

19 Semi-graded Algebras j ExtjB (B/B≥1 , N ) ∼ = ExtB (B/B≥1 , B ⊕ · · · ⊕ B) ∼ = ExtjB (B/B≥1 , B) ⊕ · · · ⊕ ExtjB (B/B≥1 , B),

but for d := lgld(B) we have ( ExtjB (B/B≥1 , B)

∼ =

0 if j 6= d B/B≥1 if j = d

and dimK (B/B≥1 ) ≤ 1 (see Remark 19.4.2). Thus, dimK (ExtjB (B/B≥1 , N )) < ∞. Now assume that N is projective, i.e., pd(N ) = 0, then there exist a finitely generated free B-module F and a B-module N 0 such that N ⊕ N 0 = F , so ExtjB (B/B≥1 , F ) = ExtjB (B/B≥1 , N ) ⊕ ExtjB (B/B≥1 , N 0 ), and from this we get dimK (ExtjB (B/B≥1 , N )) < ∞. Let pd(N ) > 0. Consider the exact sequence 0 → L → F → N → 0, with F free finitely generated; since B is left noetherian, L is finitely generated, moreover, pd(L) = pd(N ) − 1, so from the long exact sequence we get the exact portion ExtiB (B/B≥1 , F ) → ExtiB (B/B≥1 , N ) → ExtiB (B/B≥1 , L), and by induction on the projective dimension of N we obtain dimK (ExtjB (B/B≥1 , N )) < ∞. t u Example 19.4.14. Let B be a left noetherian SAS algebra and B be the associated graded ring introduced in Lemma 18.5.11. Next we compute B for some concrete examples of SAS algebras of global dimension ≤ 3 (Examples 19.4.6–19.4.10), assuming that char(K) = 0. From our computations below, for these algebras the equivalence (18.5.3) says coh(Pn ) ' qgr − B ' qgr − Γ (π(B))≥0 , with n = 1, 2. (a) For the dispin algebra B := U (osp(1, 2)), B ∼ = K[x]. In fact, let f ∈ (B)1 = Homsgr-B (B, B(1)), then f (1) = ax1 + bx2 + cx3 , with a, b, c ∈ K, so f (x3 ) = x3 f (1) = −ax1 x3 + ax2 + bx2 x3 − bx3 + cx23 , and hence, a = 0 = b. This says that given f ∈ (B)1 , there exists a cf ∈ K such that f (1) = cf x3 , moreover, f = 0 if and only if cf = 0. We fix the homomorphism corresponding to cf = 1, and we denote it by x. Hence, given any element g ∈ (B)1 we have g = cg x, whence (B)1 = K hxi. Now let g ∈ (B)2 , then g(1) = ax21 + bx1 x2 + cx1 x3 + dx22 + ex2 x3 + kx23 ∈ B3 , with a, b, c, d, e, k ∈ K, from the relations defining B we conclude that a = b = c = d = e = k = 0, so g(1) = kx23 . Thus, given g ∈ (B)2 , there exists a kg ∈ K such that g(1) = kg x23 . This implies that (B)2 = K hx2 i: in fact, g = kg x2 since kg x2 (1) = kg (x ? x)(1) = kg [s1 (x) ◦ x](1) = kg s1 (x)(x3 ) = kg x23 , where ? is the product in (B) defined in the proof of Lemma 18.5.11. In general, (B)n = K hxn i for every n ≥ 1. Indeed, by induction on n it is easy to show that

19.4 Artin–Schelter Regularity

409

(

x12s−1 (x1 x3 − s) if n = 2s, s ≥ 1 x2s 1 (−x1 x3 + x2 − s) if n = 2s + 1, s ≥ 0,
Pn x3 . x3 xn2 = i=0 (−1)i ni xn−i 2 P Let g ∈ (B )n = Homsgr-B (B, B(n)), then g(1) = i+j+l=n ai,j,l xi1 xj2 xl3 ∈ Bn P i j l and for g(x3 ) = i+j+l=n ai,j,l x3 x1 x2 x3 ∈ Bn+1 we analyze two cases: if i = 2s, then
Pj j l x3 (xi1 xj2 xl3 ) = x2s−1 (x1 x3 − s)xj2 xl3 = x2s−1 [x1 t=0 (−1)t jt xj−t 1 1 2 x3 − sx2 ]x3 , x3 xn1 =

and if i = 2s + 1, then j l x3 (xi1 xj2 xl3 ) = x2s 1 x3 + x2 − s)x2 x3 = 1 (−x
P j j−t j+1 t j x2s − sxj2 ]xl3 , 1 [−x1 t=0 (−1) t x2 x3 + x2 j l hence ai,j,l sx2s−1 xj2 xl3 = 0 in the first case, and ai,j,l sx2s 1 x2 x3 = 0 in the 1 second case. Thus, ai,j,l = 0 when i 6= 0 or j 6= 0. This means that g = kg xn3 for some kg ∈ K, and as before, g = kg xn . This completes the proof that B∼ = K[x]. (b) For B defined by x2 x3 −x3 x2 = x3 , x3 x1 −βx1 x3 = 0, x1 x2 −x2 x1 = x1 , we obtain B ∼ = K[x]. Indeed, the proof is as in (a), we include only the main details. In this case we have the following identities that can be proved by induction on n ≥ 1:

x2 xn1 = xn1 x2 − nxn1 , x3 xn1 = β n xn1 x3 , x3 xn2 = (x2 − 1)n x3 . P Let f ∈ (B)n , then f (1) = i+j+l=n ai,j,l xi1 xj2 xl3 ∈ Bn , and hence P P f (x3 ) = i+j+l=n ai,j,l x3 xi1 xj2 xl3 = i+j+l=n ai,j,l β i xi1 (x2 −1)j xl+1 ∈ Bn+1 . 3 If j 6= 0, the term ai,j,l β i xi1 xl+1 ∈ Bi+l+1 , whence ai,j,l = 0. Therefore, 3 P P f (x2 ) = i+l=n ai,0,l x2 xi1 xl3 = i+l=n ai,0,l (xi1 x2 − ixi1 )xl3 = P i l i l i+l=n ai,0,l x1 x2 x3 − ai,0,l ix1 x3 ∈ Bn+1 , so for i 6= 0, ai,0,l ixi1 xl3 = 0. From this we conclude that f (1) = kf xn3 for some kf ∈ K. Thus, as in part (a), B ∼ = K[x]. (c) For B defined by x2 x3 −x3 x2 = 0, x3 x1 −βx1 x3 = x2 , x1 x2 −x2 x1 = 0, where β is not a root of unity, we have B ∼ = K[x, y]. Indeed, let f ∈ (B)1 , then there exists a, b, c ∈ K such that f (1) = ax1 +bx2 +cx3 , whence f (x3 ) = aβx1 x3 + ax2 + bx2 x3 + cx23 ∈ B2 , so a = 0. Thus, given f ∈ (B)1 , there exists bf , cf ∈ K such that f (1) = bf x2 + cf x3 . We fix two homomorphisms x and y corresponding to bx = 1, cx = 0 and by = 0, cy = 1, respectively. Therefore (B)1 = K hx, yi. In fact, let g ∈ (B)1 , then g = bg x + cg y since (bg x + cg y)(1) = bg x(1) + cg y(1) = bg x2 + cg x3 = g(1). Now let g ∈ (B)2 , then g(1) = ax21 + bx1 x2 + cx1 x3 + dx22 + ex2 x3 + kx23 , so g(x3 ) = aβ 2 x21 x3 +a(β+1)x1 x2 +bβx1 x2 x3 +bx22 +cβx1 x23 +cx2 x3 +dx22 x3 +ex2 x23 +kx33 ,

410

19 Semi-graded Algebras

whence a = b = c = 0. This shows that (B)2 = K hx2 , xy, y 2 i. Since x2 x3 = x3 x2 , we have x ? y = y ? x in (B). By induction on n ≥ 1 we have proved that Pn−1 x3 xn1 = β n xn1 x3 + ( i=0 β i )xn−1 x2 . 1 P Hence, if g ∈ (B)n , then g(1) = i+j+l=n ai,j,l xi1 xj2 xl3 ∈ Bn and P g(x3 ) = i+j+l=n ai,j,l x3 xi1 xj2 xl3 = P Pi−1 s i−1 j l i i i+j+l=n ai,j,l (β x1 x3 + ( s=0 β )x1 x2 )x2 x3 = P P i−1 j j+1 l l+1 i i + ai,j,l ( s=0 β s )xi−1 1 x2 x3 ]. i+j+l=n [ai,j,l β x1 x2 x3 Pi−1 l Observe that for i ≥ 1, the term ai,j,l ( s=0 β s )xi−1 xj+1 2 x3 ] ∈ Bn but g(x3 ) ∈ Pi−1 s Pi−1 s 1 Bn+1 , so ai,j,l ( s=0 β ), and since s=0 β 6= 0, then ai,j,l = 0 for i ≥ 1. This says that (B)n = K hxj y l | j + l = ni. This completes that proof that B∼ = K[x, y]. (d) For B defined by x2 x3 −x3 x2 = 0, x3 x1 −x1 x3 = 0, x1 x2 −x2 x1 = x3 , we have B ∼ = K[x, y]. This isomorphism can be proved as in (c) using the identity xn2 x1 = x1 xn2 − nxn−1 x3 , n ≥ 1. 2 (e) For B = Aq1 (K) (where q is not a root of unity) we have B ∼ = K[x]. This isomorphism can be proved using the identity Pn−1 yxn = q n xn y + ( i=0 q i )xn−1 , n ≥ 1. Similarly, when q = 1 we can prove that B ∼ = K[x] for B = A1 (K). (f) Using similar identities as in the previous examples, it can be proved that if B is either J1 , U (sl(2, K)), W ν (sl(2, K) or K{x, y}/hyx − 1i, then B∼ = K[x]. Problem 19.4.15. Is any SAS algebra with finite GKdim a left noetherian domain? Example 19.4.16. Using the well-known results on Artin–Schelter regular algebras available in the literature (see also Appendix B), we list next some examples of skew P BW extensions that are Artin–Schelter, and hence, SAS algebras. Let K be a field, the following K-algebras are Artin–Schelter regular: The classical polynomial algebra K[x1 , . . . , xn ] ([344], Example 2.9). (q) (q) Linear partial q-dilation operators K[t1 , . . . , tn ][H1 , . . . , Hm ] and (q) (q) K(t1 , . . . , tn )[H1 , . . . , Hm ] ([249], Example 4.6). Algebras of diffusion type ([249], Example 4.7). The multiplicative analogue of the Weyl algebra On (λji ) ([249], Example 4.6). The algebra U ([249], Example 4.7). The Manin algebra Oq (M2 (K)) ([249], Example 4.7).

19.5 Classification of Skew P BW Algebras

411

The coordinate algebra of the quantum group SLq (2) ([249], Example 4.7). Some quadratic algebras in 3 variables ([249], Example 4.7). Multi-parameter quantum affine n-space ([344], Example 2.10). In particular, the quantum plane and some 3-dimensional skew polynomial algebras (when |{α, β, γ}| = 3). The Jordan plane ([344], Example 2.10). Example 19.4.17. From the results of the present section, we list next some examples of skew P BW extensions that are not Artin–Schelter, but are SAS algebras. Let K be a field, the following K-algebras are not Artin–Schelter, but they are SAS: The Weyl algebra An (K) and extended Weyl algebra Bn (K) The algebra of q-differential operators Dq,h [x, y] The mixed algebra Dh The algebra of linear partial differential operators The algebra of linear partial difference operators The algebra of linear partial q-differential operators The additive analogue of the Weyl algebra The dispin algebra U (osp(1, 2)) The universal enveloping algebra of the Lie algebra sl(2, K), U (sl(2, K)) U (so(3, K)) The quantum algebra U 0 (so(3, K)) Some 3-dimensional skew polynomial algebras The Woronowicz algebra W ν (sl(2, K)) The quantum Weyl algebra of Maltsiniotis Aq,λ n The algebra of differential operators Dq (Sq ) on a quantum space Sq The quantum Weyl algebra An (q, pij ) The multiparameter quantized Weyl algebra AQ,Γ n (K)

19.5 Classification of Skew P BW Algebras In the next table we resume the classification of the examples of skew P BW extensions, according to the semi-graded K-algebras introduced in the present chapter:

412

19 Semi-graded Algebras

Skew PBW algebra

K SK AS SAS

Classical polynomial algebra K[x1 , . . . , xn ] Weyl algebra An (K) Extended Weyl algebra Bn (K) Universal enveloping algebra of some Lie algebras G, U (G) Algebra of q-differential operators Dq,h [x, y] Mixed algebra Dh L. P. Differential operators K[t1 , . . . , tn ][∂1 , . . . , ∂n ] L. P. Differential operators K(t1 , . . . , tn )[∂1 , . . . , ∂n ] L. P. Difference operators K[t1 , . . . , tn ][∆1 , . . . , ∆m ] L. P. Difference operators K(t1 , . . . , tn )[∆1 , . . . , ∆m ] (q) (q) L. Partial q-dilation operators K[t1 , . . . , tn ][H1 , . . . , Hm ] (q) (q) L. Partial q-dilation operators K(t1 , . . . , tn )[H1 , . . . , Hm ] (q) (q) L. P. q-differential operators K[t1 , . . . , tn ][D1 , . . . , Dm ] (q) (q) L. P. q-differential operators K(t1 , . . . , tn )[D1 , . . . , Dm ] Algebras of diffusion type Additive analogue of the Weyl algebra An (q1 , . . . , qn ) Multiplicative analogue of the Weyl algebra O n (λji ) Quantum algebra U 0 (so(3, K)) Some 3-dimensional skew polynomial algebras Dispin algebra U (osp(1, 2)) Woronowicz algebra W ν (sl(2, K)) Algebra U Manin algebra O q (M2 (K)) Coordinate algebra of the quantum group SLq (2) Differential operators on a quantum space Sq , Dq (Sq ) Quantum Weyl algebra of Maltsiniotis Aq,λ n Quantum Weyl algebra An (q, pij ) Multiparameter quantized Weyl algebra AQ,Γ (K) n Some quadratic algebras in 3 variables Multi-parameter quantum affine n-space Jordan plane

X X

X

× × × × × × × × × × × × ×

× × × × × ×

× × × × × × × × ×

X X

X X

× ×

× ×

X X

X

× ×

×

X X × X X X × X × X X X X X X X

X

× × × ×

× × × ×

× × × ×

X X X X X X

X X X

× ×

X

×

X × ×

X X X

X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

Table 19.1 Koszul (K), semi-graded Koszul (SK), Artin Schelter regular (AS), semigraded Artin–Schelter regular (SAS)

Chapter 20

The Zariski Cancellation Problem for Skew P BW Extensions

The Zariski cancellation problem (ZCP) arises in commutative algebra and can be formulated in the following way: Let K be a field, n ≥ 1 and B be a commutative K-algebra, then does it follow that if K[t1 , . . . , tn ][t] ∼ = B[t], then K[t1 , . . . , tn ] ∼ = B? The commutative ZCP has very interesting connections with some classical problems in commutative algebraic geometry: the Automorphism Problem, the Characterization Problem, the Linearization Problem, the Embedding Problem, and the Jacobian Conjecture, see the discussion in [52] and [208]. In the noncommutative setting, the previous research suggests that it is closely related to the Automorphism Problem of noncommutative algebras, but it is unclear how to formulate a noncommutative version of the Characterization Problem, the Linearization Problem, the Embedding Problem, and so on. Recently the ZCP has been considered for noncommutative algebras. In [52] Bell and Zhang studied the Zariski cancellation problem for some noncommutative Artin–Schelter regular algebras; other works on this problem are [50], [82], [83], [84], [85], [234], [255], [267], [382], [383], [385]. In this chapter we want to apply the techniques used in the specialized literature to investigate the Zariski problem for skew P BW extensions. In this chapter R is a commutative domain and A is an R-algebra. In some places R = K is a field, or R = Z, i.e., we will study also the Zariski problem for rings.

20.1 The Zariski Cancellation Problem We start by presenting the general formulation of the Zariski cancellation problem. Definition 20.1.1. Let R be a commutative domain and let A be an Ralgebra. (i) A is cancellative if for every R-algebra B, © Springer Nature Switzerland AG 2020 W. Fajardo et al., Skew PBW Extensions, Algebra and Applications 28, https://doi.org/10.1007/978-3-030-53378-6_20

413

414

20 The Zariski Cancellation Problem for Skew P BW Extensions

A[t] ∼ = B[t] ⇒ A ∼ = B. (ii) A is strongly cancellative if for any d ≥ 0 and every R-algebra B, A[t1 , . . . , td ] ∼ = B[t1 , . . . , td ] ⇒ A ∼ = B. (iii) A is universally cancellative if for any R-flat finitely generated commutative domain S such that S/I ∼ = R for some ideal I of S, and any R-algebra B A⊗S ∼ =B⊗S ⇒A∼ = B, where the tensor product is over R. Remark 20.1.2. (i) All isomorphisms in the previous definition are isomorphisms of R-algebras, and hence, these notions depend on the ring R. (ii) Observe that the commutative classical Zariski cancellation problem asks if the K-algebra of polynomials K[t1 , . . . , tn ] is cancellative. Abhyankar, Eakin and Heinzer in [1] proved that K[t1 ] is cancellative (actually, they proved that every commutative finitely generated domain of Gelfand–Kirillov dimension one is cancellative, see Corollary 3.4 in [1]); Fujita in [126] and Miyanishi and Sugie in [280] proved that if char(K) = 0, then K[t1 , t2 ] is cancellative; if char(K) > 0, Russell in [353] proved that K[t1 , t2 ] is cancellative. Recently, in 2014, Gupta proved that if n ≥ 3 and char(K) > 0 then K[t1 , . . . , tn ] is not cancellative (see [164], [165]); the problem remains open for n ≥ 3 and char(K) = 0. Proposition 20.1.3 ([52], Remark 1.2). For any R-algebra A, universally cancellative ⇒ strongly cancellative ⇒ cancellative. Proof. Universally cancellative ⇒ strongly cancellative: this follows from the fact that A[t1 , . . . , td ] ∼ = A ⊗ R[t1 , . . . , td ], B[t1 , . . . , td ] ∼ = B ⊗ R[t1 , . . . , td ], R[t1 , . . . , td ] is an R-flat finitely generated commutative domain and R[t1 , . . . , td ]/ht1 , . . . , td i ∼ = R. Strongly cancellative ⇒ cancellative: evident. t u

20.2 The Center and the Zariski Cancellation Problem The first subalgebra considered in [52] for the investigation of the cancellation problem is the center. In this section K is a field and A is a K-algebra; recall that considering the canonical injective homomorphism of K-algebras K → A, α 7→ α · 1A , we can say that K ⊂ A. Theorem 20.2.1 ([52], Proposition 1.3). Let K be a field and A be a K-algebra. If Z(A) = K, then A is universally cancellative, and hence, cancellative.

20.2 The Center and the Zariski Cancellation Problem

415

Proof. Let S be a commutative domain finitely generated as a K-algebra and with an ideal I such that S/I ∼ = K (S is K-flat since in this case R := K is a field); let B be a K-algebra and φ : A ⊗ S → B ⊗ S an isomorphism of K-algebras. We will construct an isomorphism of K-algebras θ : A → B. We divide the proof into four steps. Step 1. Z(A ⊗ S) = 1A ⊗ S: in fact, it is well known that Z(A ⊗ S) = Z(A) ⊗ Z(S), so Z(A ⊗ S) = K ⊗ S = 1A ⊗ S := {1A ⊗ s|s ∈ S}. Step 2. Z(B ⊗ S) = 1B ⊗ S: we know that Z(B ⊗ S) = Z(B) ⊗ Z(S) = Z(B) ⊗ S, but observe that Z(B) = K: in fact, φ induces an isomorphism φ0 of K-algebras between the centers of A ⊗ S and B ⊗ S, namely, φ0 is the φ0

restriction of φ to the centers, thus we have K ⊗ S ∼ = Z(B) ⊗ S, and from this we get the isomorphism of K-algebras S ∼ = Z(B) ⊗ S. This implies that Z(B) is a commutative domain. Taking the GKdim we get that GKdim(Z(B)) = 0, so Z(B) is a field. From S ∼ = K: in = K we get Z(B) ∼ = Z(B) ⊗ S and S/I ∼ ∼ ∼ ∼ ∼ Z(B) ⊗ K ∼ fact, K = S/I = (Z(B) ⊗ S)/(Z(B) ⊗ I) = Z(B) ⊗ S/I = = Z(B). Thus, dimK Z(B) = 1, but dimK K = 1 and K ⊆ Z(B), then Z(B) = K. Therefore, Z(B ⊗ S) = K ⊗ S = 1B ⊗ S. Step 3. According to the previous steps, we have the isomorphism of Kalgebras φ0

1A ⊗ S ∼ = 1B ⊗ S. In addition, we have the following sequence of isomorphisms of K-algebras: ιA

ι−1

φ0

B ∼ 1A ⊗ S = ∼ 1B ⊗ S ∼ S= = S, ιA (s) := 1A ⊗ s, ιB (s) := 1B ⊗ s;

−1 0 let ρ := ιB φ ιA ; note that I 0 := ρ(I) is an ideal of S, and 1B ⊗I 0 = φ0 (1A ⊗I) (this is because ιB ρ(I) = φ0 ιA (I)); moreover, S/I 0 ∼ = K since we have the following induced isomorphism of K-algebras, ρ

S/I − → S/ρ(I), ρ(s) := ρ(s), s ∈ S. Step 4. It is easy to check that the following functions are well-defined isomorphisms of K-algebras: α

β

φ

A∼ = A ⊗ (S/I) ∼ = (A ⊗ S)/h1A ⊗ Ii ∼ = (B ⊗ S)/φ(h1A ⊗ Ii) = γ

δ

(B ⊗ S)/hφ(1A ⊗ I)i = (B ⊗ S)/hφ0 (1A ⊗ I)i = (B ⊗ S)/h1B ⊗ I 0 i ∼ = B ⊗ (S/I 0 ) ∼ = B, α(a) := a ⊗ 1S ; β(a ⊗ s) := a ⊗ s; φ(a ⊗ s) := φ(a ⊗ s); γ(b ⊗ s) := b ⊗ s; δ(b ⊗ s) := bs.

Therefore, the desired isomorphism of K-algebras is θ := δγφβα.

t u

Next we present some elementary examples of skew P BW extensions that are K-algebras and have trivial center, so they are universally cancellative, and hence cancellative. In the last section of the chapter some other interesting examples of cancellative skew P BW extensions will be presented.

416

20 The Zariski Cancellation Problem for Skew P BW Extensions

Example 20.2.2 ([52], Example 1.4). (i) If K is a field of characteristic zero, then the Weyl algebra An (K) is universally cancellative. In fact, Z(An (K)) = K (see also Proposition 3.3.1). (ii) Let K be a field and q ∈ K ∗ ; consider the skew P BW extension A := Kq [x1 , . . . , xn ] defined by xj xi = qxi xj for all 1 ≤ i < j ≤ n. If n ≥ 2 and q is not a root of unity, then Z(A) = K, and hence, A is universally cancellative (see Proposition 3.3.13). In particular, the quantum plane is, in this case, universally cancellative. Example 20.2.3. If K is a field of characteristic zero, then the algebra of shift operators Sh := K[t][xh ; σh , δh ], where σh (p(t)) := p(t − h) and δh := 0, is universally cancellative: in fact, Z(Sh ) = K (see Proposition 3.3.7). Example 20.2.4. Let K be a field with char(K) = 0 and h ∈ K ∗ . Then, Z(Dh ) = K (see Proposition 3.3.9), and hence, the mixed algebra Dh is universally cancellative. Example 20.2.5. If K is a field of characteristic zero, then the Jordan plane is universally cancellative: in fact, in [362] it is proved that the center of the Jordan plane is K (see also Proposition 3.3.11, or [188]). Example 20.2.6. Let K be a field of characteristic zero, and let G be a three-dimensional completely solvable Lie algebra with basis x, y, z such that [y, x] = y, [z, x] = λz and [y, z] = 0, λ ∈ K − Q. Then, U (G) is universally cancellative since Z(U (G)) = K (see [278], Example 14.4.2; see also Proposition 3.3.3). Example 20.2.7. Assuming the conditions of Theorem 3.3.21, the following algebras have trivial center, and hence are cancellative: the algebra of linear partial differential operators; the algebra of linear partial shift operators; the algebra of linear partial difference operators; the algebra of linear partial q-dilation operators; the algebra of linear partial q-differential operators; the algebra for multidimensional discrete linear systems; and the additive analogue of the Weyl algebra. Example 20.2.8. Assuming that char(K) = 0, in [251] it has been proved that the following algebras have trivial center, and hence, they are cancellative: (a) The Woronowicz algebra W ν (sl(2, K)). (b) The algebra U generated over the field C by the set of variables xi , yi , zi , 1 ≤ i ≤ n, subject to the relations: xj xi = xi xj , yj yi = yi yj , zj zi = zi zj , 1 ≤ i, j ≤ n, yj xi = q δij xi yj , zj xi = q −δij xi zj , 1 ≤ i, j ≤ n, zj yi = yi zj , i 6= j, zi yi = q 2 yi zi − q 2 x2i , 1 ≤ i ≤ n, where q ∈ C − {0} is not a root of unity. (c) When K is an ordered field, the quadratic algebra A2 generated by x, y, z with relations yx = xy + axz + bz 2 , zx = xz, zy = yz + cz 2 , a, b, c ∈ K − {0} and ac < 0 .

20.3 Gelfand–Kirillov Dimension for Rings

417

(d) The following particular Witten deformation of U (sl(2, K) generated by x, y, z subject to the relations yx = cxy + bz 2 + z, 1 1 zx = xz − x, a a zy = ayz + y, with a, b, c ∈ K − {0}. We conclude this second section with the following property of finitely graded algebras generated in degree 1. Proposition 20.2.9 ([51], Theorem 3.1). Let K be a field and let A and B be two connected graded K-algebras finitely generated in degree one. Suppose that Z(A) ∩ A1 = 0. For every n ≥ 0, if A[t1 , . . . , tn ] ∼ = B[t1 , . . . , tn ] as K-algebras, then A ∼ = B. Proof. For n = 0 the result is trivial. Assume L that n ≥ 1. Let L C := A[t1 , . . . , tn ] and D := B[t1 , . . . , tn ]; if A = A and B = i i≥0 i≥0 Bi are the N-graduations of A and B, respectively, then C and D are connected graded K-algebras finitely generated in degree 1, with graduations Cp :=K har tα | r + |α| = pi, Dp :=K hbr tα | r + |α| = pi, p ≥ 0, αn 1 with tα := tα 1 · · · tn and |α| := α1 + · · · + αn . Since C ∼ = D as K-algebras, there exists a graded algebra isomorphism φ : C → D (Theorem A.1.6); from Z(A) ∩ A1 = 0 we obtain Z(C) C1 = Ln L∩ n −1 since for every j, t ∈ Z(D), we have φ (t ) ∈ Kti , j j i=1 Kti , and L i=1 Ln n and hence φ−1 ( i=1 Kti ) = i=1 Kti . Therefore, φ

A∼ = D/ht1 , . . . , tn i ∼ = B, = C/ht1 , . . . , tn i ∼ where φ is the induced isomorphism of K-algebras and ht1 , . . . , tn i is the two-sided ideal generated by t1 , . . . , tn . t u

20.3 Gelfand–Kirillov Dimension for Rings In Section 18.3 we studied a generalization of the classical Gelfand–Kirillov dimension for finitely-semigraded rings. There is another generalization that we will use in the present chapter for the investigation of the Zariski cancellation problem. This generalization was introduced by Bell and Zhang in [52], however, they did not provide a detailed investigation of the behavior of this new notion for the classical algebraic constructions such as polynomial rings, matrix rings, localizations, filtered-graded rings and skew P BW extensions, etc. We will do this next as background material for the sequel.

418

20 The Zariski Cancellation Problem for Skew P BW Extensions

Let R be a commutative domain and Q be the field of fractions of R. Let B be an R-algebra, then Q ⊗ B is a Q-algebra and its classical Gelfand–Kirillov dimension is denoted by GKdim(Q ⊗ B) (see [210]). Recall that if M is a finitely generated R-module, then the rank of M is defined by rankM := dimQ (Q ⊗R M ) < ∞. From now on in this section all tensors are over R. Definition 20.3.1 ([52], Section 1). Let R be a commutative domain and B be an R-algebra. The Gelfand–Kirillov dimension of B is defined to be GKdim(B) := sup lim logn rankV n , (20.3.1) V

n→∞

where V varies over all frames of B and V n := R hv1 · · · vn |vi ∈ V, 1 ≤ i ≤ ni; a frame of B is a finitely generated R-submodule of B containing 1. A frame V generates B if B is generated by V as an R-algebra. Proposition 20.3.2 ([52], Section 3). Let B be an R-algebra. Then, GKdim(B) = GKdim(Q ⊗ B). Proof. Let V be a frame of B, then Q ⊗ V is a frame of the Q-algebra Q ⊗ B. In fact, if V = R hv1 , . . . , vm i, then Q ⊗ V = Q h1 ⊗ v1 , . . . , 1 ⊗ vm i and 1 ⊗ 1 ∈ Q ⊗ V . Observe that for every n ≥ 0, Q ⊗ V n = (Q ⊗ V )n , hence GKdim(B) = sup lim logn rankV n V

n→∞

= sup lim logn (dimQ (Q ⊗ V n )) V

n→∞

= sup lim logn (dimQ (Q ⊗ V )n ) ≤ GKdim(Q ⊗ B). V

n→∞

Now let W be a frame of Q⊗B, then there exist finitely many v1 , . . . , vm ∈ B such that VW := R h1, v1 , . . . , vm i is a frame of B and W ⊆ Q ⊗ VW . Indeed, let W = Q hz1 , . . . , zk i, then for every 1 ≤ i ≤ k, zi = qi1 ⊗ vi1 + · · · + qimi ⊗ vimi , with qij ∈ Q and vij ∈ B; hence, W ⊆ Q h1 ⊗ v11 , . . . , 1 ⊗ vkmk i and we have found elements v1 , . . . , vm ∈ B such that W ⊆ with VW :=

Q h1

⊗ 1, 1 ⊗ v1 , . . . , 1 ⊗ vm i ⊆ Q ⊗ VW ,

R h1, v1 , . . . , vm i.

This proves the claim. Therefore,

GKdim(Q ⊗ B) = sup lim logn dimQ W n W n→∞

≤ sup lim logn dimQ (Q ⊗ VW )n VW n→∞

20.3 Gelfand–Kirillov Dimension for Rings

419

n = sup lim logn dimQ (Q ⊗ VW ) VW n→∞

n ≤ GKdim(B). = sup lim logn rankVW VW n→∞

t u

Observe that B is finitely generated if and only if B has a generator frame. In fact, if x1 , . . . , xm generate B as an R-algebra, then V := R h1, x1 , . . . , xm i is a generator frame of B; the converse is trivial from Definition 20.3.1. Proposition 20.3.3. Let B be an R-algebra and V be a frame that generates B. Then, GKdim(B) = lim logn (rankV n ). n→∞

Moreover, this equality is independent of the generator frame. Proof. Notice that Q ⊗ V generates Q ⊗ B, so from Proposition 20.3.2 GKdim(B) = GKdim(Q ⊗ B) = lim logn (dimQ (Q ⊗ V )n ) n→∞

= lim logn (dimQ (Q ⊗ V n )) n→∞

= lim logn rankV n . n→∞

The second part is trivial since the proof was independent of the chosen frame. t u Next we will compute the Gelfand–Kirillov dimension of the classical algebraic constructions. For this we will apply the corresponding properties of the Gelfand–Kirillov dimension over fields (see [210]). Theorem 20.3.4. Let B be an R-algebra. (i) If B is finitely generated, then GKdim(B) = 0 if and only if rankB < ∞. Moreover, if B is a domain with GKdim(B) < ∞, then B is an Ore domain. (ii) GKdim(B[x1 , . . . , xm ]) = GKdim(B) + m. (iii) For m ≥ 2, GKdim(R{x1 , . . . , xm }) = ∞. (iv) GKdim(Mn (B)) = GKdim(B). (v) If I is a proper two-sided ideal of B, then GKdim(B/I) ≤ GKdim(B). (vi) Let C be a subalgebra of B. Then, GKdim(C) ≤ GKdim(B). Moreover, if C ⊆ Z(B) and B is finitely generated as a C-module, then GKdim(C) = GKdim(B). (vii) Let C be an R-algebra. Then, GKdim(B × C) = max{GKdim(B), GKdim(C)}. (viii) Let C be an R-algebra. Then,

20 The Zariski Cancellation Problem for Skew P BW Extensions

420

max{GKdim(B), GKdim(C)} ≤ GKdim(B ⊗ C) ≤ GKdim(B) + GKdim(C). In addition, suppose that C contains a finitely generated subalgebra C0 such that GKdim(C0 ) = GKdim(C), then GKdim(B ⊗ C) = GKdim(B) + GKdim(C). (ix) Let S be a multiplicative system of B consisting of central regular elements. Then, GKdim(BS −1 ) = GKdim(B). (x) If G is a finite group, then GKdim(B[G]) = GKdim(B). (xi) Let B be N-filtered and locally finite, i.e., for every p ∈ N, Fp (B) is a finitely generated R-submodule of B. If Gr(B) is finitely generated, then GKdim(Gr(B)) = GKdim(B). (xii) Suppose that B has a generator frame V . If σ is an R-linear automorphism of B such that σ(V ) ⊆ V and δ is an R-linear σ-derivation of B, then GKdim(B[x; σ, δ]) = GKdim(B) + 1. (xiii) Let σ be an R-linear automorphism of B that is locally algebraic, i.e., for any b ∈ B the set {σ m (b) | m ∈ N} is contained in a finitely generated R-submodule of B. Then, GKdim(B[x; σ]) = GKdim(B) + 1 = GKdim(B[x±1 ; σ]). (xiv) Suppose that B has a generator frame V and let A = σ(B)hx1 , . . . , xt i be a bijective skew P BW extension of B. If for 1 ≤ i ≤ t, σi , δi are R-linear and σi (V ) ⊆ V , then GKdim(A) = GKdim(B) + t. Proof. (i) Notice that Q ⊗ B is finitely generated, so GKdim(Q ⊗ B) = 0 if and only if dimQ (Q ⊗ B) < ∞. Since rankB = dimQ (Q ⊗ B), we get GKdim(B) = 0 if and only if GKdim(Q ⊗ B) = 0 if and only if rankB < ∞. For the second statement, we will show that B is a right Ore domain. The proof on the left side is similar. Suppose on the contrary that there exist 0 6= s ∈ B and b ∈ B such that sB ∩ bB = 0. Since B is a domain, the following sum is direct bB + sbB + s2 bB + s3 bB + · · · . Let V be a frame of B, then WV := WV2n ⊇

R hV, s, bi

n

R hV, s, bi

is a frame of B and

V n ⊇ bV n + sbV n + s2 bV n + · · · + sn−1 bV n

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and since the last sum is direct, GKdim(B) ≥ sup lim logn rankWV2n ≥ sup lim logn nrankV n WV n→∞

V

n→∞

= 1 + GKdim(B), but this is impossible since GKdim(B) < ∞. (ii) It is enough to consider the case m = 1. We have the isomorphism of R-algebras B[x] ∼ = B ⊗ R[x], whence we have the following isomorphism of Q-algebras: Q ⊗ B[x] ∼ = Q ⊗ (B ⊗ R[x]) ∼ = (Q ⊗ B) ⊗ R[x] ∼ = (Q ⊗ B)[x]. Therefore, GKdim(B[x]) = GKdim(Q ⊗ B[x]) = GKdim((Q ⊗ B)[x]) = GKdim(Q ⊗ B) + 1 = GKdim(B) + 1. (iii) We have the isomorphism of Q-algebras Q ⊗ (R{x1 , . . . , xm }) ∼ = Q{x1 , . . . , xm } X X α q⊗ rα x 7→ (qrα )xα . α

α

Therefore, GKdim(R{x1 , . . . , xm }) = GKdim(Q ⊗ (R{x1 , . . . , xm })) = GKdim(Q{x1 , . . . , xm }) = ∞. (iv) We have the isomorphism of R-algebras Mn (B) ∼ = B ⊗ Mn (R), whence we have the following isomorphism of Q-algebras: Q ⊗ Mn (B) ∼ = Q ⊗ (B ⊗ Mn (R)) ∼ = (Q ⊗ B) ⊗ Mn (R) ∼ = Mn (Q ⊗ B). Therefore, GKdim(Mn (B)) = GKdim(Q ⊗ Mn (B)) = GKdim(Mn (Q ⊗ B)) = GKdim(Q ⊗ B) = GKdim(B). (v) Let W be a frame of B/I. We can assume that W = R h1, w1 , . . . , wt i, then VW := R h1, w1 , . . . , wt i is a frame of B, and hence, Q ⊗ W is a frame of Q ⊗ (B/I) and Q ⊗ VW is a frame of Q ⊗ B. Observe that for every n ≥ 0, Wn =

n, · · · win | wij ∈ {1, w1 , . . . , wt }i = VW n n, Q ⊗ W = Q ⊗ VW n ) ≤ dim (Q ⊗ V n ). dimQ (Q ⊗ W n ) = dimQ (Q ⊗ VW Q W R hwi1

The inequality can be justified in the following way. Both Q-vector spaces n and Q ⊗ V n have finite dimension and we have the surjective homoQ ⊗ VW W

20 The Zariski Cancellation Problem for Skew P BW Extensions

422

n n , q ⊗ z 7→ q ⊗ z, with q ∈ Q, → Q ⊗ VW morphism of Q-vector spaces Q ⊗ VW n z ∈ VW . Therefore, GKdim(B/I) ≤ GKdim(B). (vi) Since Q is R-flat, Q ⊗ C is a Q-subalgebra of Q ⊗ B, hence

GKdim(C) = GKdim(Q ⊗ C) ≤ GKdim(Q ⊗ B) = GKdim(B). For the second statement, Q ⊗ C is a Q-subalgebra of Q ⊗ Z(B) ⊆ Z(Q ⊗ B); moreover, if B = C hb1 , . . . , bt i, with bi ∈ B, 1 ≤ i ≤ t, then Q⊗B =

Q⊗C h1

⊗ b1 , . . . , 1 ⊗ bt i.

Therefore, GKdim(C) = GKdim(Q ⊗ C) = GKdim(Q ⊗ B) = GKdim(B). (vii) We have the following isomorphism of Q-algebras Q ⊗ (B × C) ∼ = (Q ⊗ B) × (Q ⊗ C) q ⊗ (b, c) 7→ (q ⊗ b, q ⊗ c). Hence, GKdim(B × C) = GKdim(Q ⊗ (B × C)) = GKdim((Q ⊗ B) × (Q ⊗ C)) = max{GKdim(Q ⊗ B), GKdim(Q ⊗ C)} = max{GKdim(B), GKdim(C)}. (viii) We have the following isomorphisms of Q-algebras (Q ⊗ B) ⊗ (Q ⊗ C) ∼ = Q ⊗ (B ⊗ Q) ⊗ C ∼ = (Q ⊗ Q) ⊗ (B ⊗ C) ∼ = Q ⊗ (B ⊗ C). Therefore, GKdim(B ⊗ C) = GKdim(Q ⊗ (B ⊗ C)) = GKdim((Q ⊗ B) ⊗ (Q ⊗ C)) ≤ GKdim(Q ⊗ B) + GKdim(Q ⊗ C) = GKdim(B) + GKdim(C); max{GKdim(B), GKdim(C)} = max{GKdim(Q ⊗ B), GKdim(Q ⊗ C)} ≤ GKdim((Q ⊗ B) ⊗ (Q ⊗ C)) = GKdim(Q ⊗ (B ⊗ C)) = GKdim(B ⊗ C). For the second part, if C contains a finitely generated subalgebra C0 such that GKdim(C0 ) = GKdim(C), then Q ⊗ C contains the finitely generated Q-algebra Q ⊗ C0 , GKdim(Q ⊗ C0 ) = GKdim(Q ⊗ C), and since Q ⊗ Q ∼ = Q, we obtain GKdim(B ⊗ C) = GKdim(Q ⊗ (B ⊗ C)) = GKdim((Q ⊗ B) ⊗ (Q ⊗ C)) = GKdim(Q ⊗ B) + GKdim(Q ⊗ C) = GKdim(B) + GKdim(C). (ix) Let W := R h ws11 , . . . , wstt i be a frame of BS −1 ; taking a common denominator s we can assume W = R h ws1 , . . . , wst i. Then sW ⊆ R hw1 , . . . , wt i ⊆ B and observe that VW := R h1, s, w1 , . . . , wt i is a frame of B. For every n ≥ 0, n 1 n n 1 ∼ n W n ⊆ VW ⊆ Q ⊗ VW sn and since Q is R-flat, Q ⊗ W sn = Q ⊗ V W

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n (isomorphism of Q-vector spaces), so dimQ (Q ⊗ W n ) ≤ dimQ (Q ⊗ VW ), n i.e., rank(W n ) ≤ rank(VW ), whence GKdim(BS −1 ) ≤ GKdim(B). Since B ⊆ BS −1 , (vi) completes the proof. (x) We have the following isomorphism of R-algebras

B ⊗ R[G] ∼ = B[G] X X b⊗( rg g) 7→ (brg ) · g g∈G

g∈G

and from this we obtain the isomorphism of Q-algebras Q ⊗ B[G] ∼ = Q ⊗ (B ⊗ R[G]) ∼ = (Q ⊗ B) ⊗ R[G] ∼ = (Q ⊗ B)[G]. Therefore, GKdim(B[G]) = GKdim(Q ⊗ B[G]) = GKdim((Q ⊗ B)[G]) = GKdim(Q ⊗ B) = GKdim(B). (xi) Q ⊗ B has an induced natural N-filtration given by {Q ⊗ Fp (B)}p∈N , moreover, Q⊗B is locally finite. Since Gr(B) is finitely generated, Q⊗Gr(B) is finitely generated. We have the following isomorphism of Q-algebras: Q ⊗ Gr(B) ∼ = Gr(Q ⊗ B), q ⊗ bp 7→ q ⊗ bp , with q ∈ Q and bp ∈ Fp (B) Fp (B) ∼ (this isomorphism is induced by the isomorphisms Q⊗ Fp−1 (B) = Hence, Gr(Q ⊗ B) is finitely generated and we have

Q⊗Fp (B) Q⊗Fp−1 (B) ).

GKdim(Gr(B)) = GKdim(Q ⊗ Gr(B)) = GKdim(Gr(Q ⊗ B)) = GKdim(Q ⊗ B) = GKdim(B). (xii) We know that Q ⊗ V generates the Q-algebra Q ⊗ B. Observe that Q ⊗ (B[x; σ, δ]) ∼ = (Q ⊗ B)[x; σ, δ] is an isomorphism of Q-algebras, where σ, δ : Q ⊗ B → Q ⊗ B are defined in the following natural way, σ := iQ ⊗ σ, δ := iQ ⊗ δ. It is easy to check that σ is a bijective homomorphism of Q-algebras, δ is a Q-linear σ-derivation and σ(Q ⊗ V ) ⊆ Q ⊗ V . Therefore, GKdim(B[x; σ, δ]) = GKdim((Q ⊗ B)[x; σ, δ]) = GKdim(Q ⊗ B) + 1 = GKdim(B) + 1. (xiii) As in part (xii), we have the isomorphism of Q-algebras Q ⊗ (B[x; σ]) ∼ = (Q ⊗ B)[x; σ], where σ = iQ ⊗ σ is a bijective homomorphism of Q-algebras. Let z be an element of Q ⊗ B, then there exists s 6= 0 in R and b ∈ B such that z = 1s ⊗ b, whence σ m (z) = 1s ⊗ σ m (b). By the hypothesis, {σ m (z) | m ∈ N} is contained in a finite-dimensional subspace of Q ⊗ B. From [230], Proposition 1 (see also Lemma 7.4.2) we get

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20 The Zariski Cancellation Problem for Skew P BW Extensions

GKdim(B[x; σ]) = GKdim(Q ⊗ B[x; σ]) = GKdim((Q ⊗ B)[x; σ]) = GKdim(Q ⊗ B) + 1 = GKdim(B) + 1. Moreover, using the universal property of the skew polynomial ring (Q ⊗ B)[x; σ] it is easy to prove that the function f : (Q ⊗ B)[x; σ] → Q ⊗ (B[x±1 ; σ]) defined by f (q ⊗ b) := q ⊗ 1b , f (x) := 1 ⊗ x1 , with q ∈ Q and b ∈ B, is a ring homomorphism that satisfies the conditions of the ring of fractions of (Q ⊗ B)[x; σ] with respect to the system {xk | k ≥ 0}, i.e., Q ⊗ B[x±1 ; σ] ∼ = (Q ⊗ B)[x±1 ; σ]. Therefore, GKdim(B[x±1 ; σ]) = GKdim(Q ⊗ B[x±1 ; σ]) = GKdim((Q ⊗ B)[x±1 ; σ]) = GKdim(Q ⊗ B) + 1 = GKdim(B) + 1. (xiv) Note first that A is an R-algebra. Let V := R h1, v1 , . . . , vl i be a generator frame S of B, then {V n }n≥0 is a locally finite N-filtration of B, in particular, B = n≥0 V n . Let m := max{m11 , . . . , mtl } ≥ 1 with δi (vj ) ∈ V mij , 1 ≤ i ≤ t and 1 ≤ j ≤ l. Then, δi (V ) ⊆ V m for every 1 ≤ i ≤ t. In addition, m can be chosen such that all parameters in (1.2.1) belong to V m . By induction we can show that for n ≥ 0, δi (V n ) ⊆ V n+m . In fact, δi (V 0 ) = δi (R) = 0 ⊆ V m ; δi (V ) ⊆ V m ⊆ V m+1 ; assume that δi (V n ) ⊆ V n+m and let z ∈ V n and v ∈ V , then δi (zv) = σi (z)δi (v) + δi (z)v, but σi (z) ∈ V n , whence δi (zv) ∈ V n+1+m . Thus, δi (V n+1 ) ⊆ V n+1+m . From this we obtain in particular that δi (V m ) ⊆ V 2m . Since V m is also a generator frame, we can assume that the generator frame V satisfies δi (V ) ⊆ V 2 and all parameters in (1.2.1) belong to V . From this, in turn, we conclude that this generator frame V = R h1, v1 , . . . , vl i satisfies δi (V n ) ⊆ V n+1 , for every n ≥ 0. Let X :=

R h1, x1 , . . . , xt i,

Xn :=

R hx

α

and for every n ≥ 1 let

∈ Mon(A)| deg(xα ) ≤ ni (see Definition 1.1.1).

Note that for every n ≥ 1, Xn ⊆ X n ⊆ V n−1 Xn . The first inclusion is trivial and the second can be proved by induction. Indeed, X 1 = X = X1 = RX1 = V 1−1 X1 ; assume that X n−1 ⊆ V n−2 Xn−1 and let z ∈ X n , we can suppose that z has the form z = z1 · · · zn with zi ∈ {1, x1 , . . . , xt }, 1 ≤ i ≤ n. If at least one zi is equal 1, then z ∈ X n−1 , and by induction z ∈ V n−2 Xn−1 ⊆ V n−1 Xn . Thus, we can suppose that every zi ∈ {x1 , . . . , xt }. If z ∈ Mon(A), then z ∈ Xn ⊆ V n−1 Xn . Assume that z ∈ / Mon(A), then at least one factor of z should be moved in order to represent z through the B-basis Mon(A) of A. But the maximum number of permutations in order to do this is ≤ n−1 (and this is true for every factor to be moved). Notice that the parameters of A arise in every permutation, and as was observed above, these parameters belong to V . Hence, once the factor

20.3 Gelfand–Kirillov Dimension for Rings

425

is in the right position, we can apply induction and get that z, represented in the standard form through the basis Mon(A), belongs to V n−1 Xn . Xn is a free left R-module with  n  X t+i−1 , dimR Xn = i i=0 and for n ≫ 0

(n/n − 1)t ≤ dimR Xn ≤ (n + 1)t .

Now we can complete the proof, dividing it into two steps. For this, let W := V + X, then W is a generating frame of A. Step 1. GKdim(A) ≤ GKdim(B) + t. We will show first that for every n ≥ 0, W n ⊆ V nX n. For n = 0, W 0 = R = V 0 X 0 ; for n = 1, W 1 = V + X ⊆ V X. Suppose that W n ⊆ V n X n and let w ∈ W and z ∈ W n , then denoting by δ any of the elements of {δ1 , . . . , δt }, we get wz ∈ (V + X)V n X n ⊆ V n+1 X n + XV n X n ⊆ V n+1 X n+1 + (V n X + δ(V n ))X n ⊆ V n+1 X n+1 + V n X n+1 + V n+1 X n ⊆ V n+1 X n+1 . Thus, W n+1 ⊆ V n+1 X n+1 . Hence, rankW n ≤ rankV n X n ≤ rankV n V n−1 Xn = rankV 2n−1 Xn ≤ rankV 2n Xn , but since V 2n ⊆ B and the R-basis of Xn is conformed by standard monomials with dimR Xn ≤ (n + 1)t , we have rankV 2n Xn ≤ (n + 1)t rankV 2n . In fact, let l := dimR Xn and {xα1 , . . . , xαl } be an R-basis of Xn , then we have the R-isomorphism V 2n ⊕ · · · ⊕ V 2n → V 2n Xn (b1 , . . . , bl ) 7→ b1 xα1 + · · · + bl xαl , and hence rankV 2n Xn = rank(V 2n ⊕ · · · ⊕ V 2n ) = lrankV 2n ≤ (n + 1)t rankV 2n . Therefore, GKdim(A) = lim logn (rankW n ) n→∞

≤ lim logn (rankV 2n Xn ) n→∞

≤ lim logn ((n + 1)t dimQ (Q ⊗ V 2n )) n→∞

= t + lim logn (dimQ (Q ⊗ V n )) n→∞

= t + lim logn (rankV n ) n→∞

= t + GKdim(B). Step 2. GKdim(A) ≥ GKdim(B) + t. Observe that for every n ≥ 0, V n X n ⊆ W 2n .

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20 The Zariski Cancellation Problem for Skew P BW Extensions

In fact, V 0 X 0 = R = W 0 and for n ≥ 1, V n X n ⊆ (V +X)n (V +X)n = W 2n . Therefore, W 2n ⊇ V n X n ⊇ V n Xn , and as in step 1, we get rankW 2n ≥ rankV n Xn ≥ (n/n − 1)t rankV n , whence GKdim(A) = lim logn (rankW 2n ) n→∞

≥ lim logn (rankV n Xn ) n→∞

≥ lim logn ((n/n − 1)t dimQ (Q ⊗ V n )) n→∞

= t + lim logn (dimQ (Q ⊗ V n )) n→∞

= t + lim logn (rankV n ) n→∞

= t + GKdim(B).

t u

Remark 20.3.5. Comparing with the classical Gelfand–Kirillov dimension over fields (see [210]), we have the followings remarks about Definition 20.3.1. (i) If V is a generator frame of B, then {V n }n≥0 is an N-filtration of B. Note that GKdim(B) measures the asymptotic behavior of the sequence {rankV n }n≥0 . (ii) From Proposition 20.3.2 we get that GKdim(B) = r if and only if r ∈ {0} ∪ {1} ∪ [2, ∞]. Indeed, let r := GKdim(B) = GKdim(Q ⊗ B). Then, it is well known (see [210]) that r ∈ {0} ∪ {1} ∪ [2, ∞]. Conversely, suppose that r is in this union, then there exists a Q-algebra A such that GKdim(A) = r. Notice that A is an R-algebra. Let X be a Q-basis of A and let B be the R-subalgebra of A generated by X. We have the surjective homomorphism of Q-algebras α

Q⊗B − → A, q ⊗ b 7→ q · b, with q ∈ Q and b ∈ B. Hence, A ∼ = (Q ⊗ B)/ ker(α), so r = GKdim(A) ≤ GKdim(Q ⊗ B) = GKdim(B). On the other hand, since Q is R-flat we have the injective homomorphism of Q-algebras Q ⊗ B ,→ Q ⊗ A, and moreover, Q ⊗ A ∼ = A, with q ⊗ a 7→ q · a (isomorphism of Q-algebras). Therefore, Q ⊗ B is a Qsubalgebra of A, and hence, GKdim(B) = GKdim(Q ⊗ B) ≤ GKdim(A) = r. Thus, GKdim(B) = r. (iii) If B is finitely generated and commutative, then GKdim(B) is a nonnegative integer. Indeed, this property is well-known (see [210], Theorem 4.5) for commutative finitely generated algebras over fields since in such situation GKdim = CLKdim. Hence, since Q ⊗ B is finitely generated and commutative, GKdim(B) = GKdim(Q ⊗ B) is a nonnegative integer. (iv) If B is a domain and torsion-free as an R-module, then Q ⊗ B is a domain. In fact, the statement follows from the Q-isomorphism valid for every R-algebra B: b 1 S0−1 B ∼ 7 ⊗ b. = Q ⊗ B, with S0 := R − {0} and → s s

(20.3.2)

20.3 Gelfand–Kirillov Dimension for Rings

427

(v) Suppose that B is a domain, torsion-free as an R-module and GKdim(B) = 1, then B is commutative. In fact, from (iv), Q ⊗ B is a domain with GKdim(Q ⊗ B) = 1, whence Q ⊗ B is commutative (see [364]). This implies that B is commutative. (vi) In [144] the Gelfand–Kirillov transcendence degree for algebras over fields was defined (see also [416]). This notion can be extended to algebras over commutative domains. Let B be an R-algebra, then the Gelfand–Kirillov transcendence degree of B is defined by Tdeg(B) := sup inf GKdim(R[bV ]), V

b

where V ranges over all frames of B and b ranges over all regular elements of B. Since R[bV ] is an R-subalgebra of B, we have GKdim(R[bV ]) ≤ GKdim(B) for all V and b, whence Tdeg(B) ≤ GKdim(B). If B is commutative, then Q ⊗ B is commutative and it is known that for commutative algebras over fields equality holds, therefore Tdeg(Q ⊗ B) = GKdim(Q ⊗ B) (see [416], Proposition 2.2). In addition, if B is torsion-free as an R-module, then Tdeg(Q ⊗ B) ≤ Tdeg(B). In fact, Tdeg(Q ⊗ B) := sup inf GKdim(Q[zW ]) ≤ sup inf GKdim(Q[uW ]), W

z

W

u

where W ranges over all frames of Q ⊗ B, z ranges over all regular elements of Q ⊗ B and u over all regular elements of Q ⊗ B of the form 1 ⊗ b, with b any regular element of B (if b is a regular element of B, then from the general isomorphism (20.3.2), 1 ⊗ b is a regular element of Q ⊗ B). As we saw in the proof of Proposition 20.3.2, given a frame W there exists a frame VW of B such that W ⊆ Q ⊗ VW , hence uW ⊆ u(Q ⊗ VW ) = Q ⊗ bVW ⊆ Q ⊗ R[bVW ], whence Q[zW ] ⊆ Q ⊗ R[bVW ], and from this Tdeg(Q ⊗ B) ≤ sup inf GKdim(Q ⊗ R[bVW ]) VW

b

= sup inf GKdim(R[bVW ]) ≤ Tdeg(B). VW

b

This proves the claimed. Thus, if B is commutative and torsion-free as an R-module, then Tdeg(B) = GKdim(B). Definition 20.3.1 can be extended to modules. Let M be a right B-module, then M is an R-B-bimodule: Indeed, for r ∈ R, m ∈ M and b ∈ B we define r · m := m · (r · 1B ), and from this it is easy to check that (r · m) · b = r · (m · b).

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20 The Zariski Cancellation Problem for Skew P BW Extensions

Definition 20.3.6. Let B an R-algebra and M be a right B-module. The Gelfand–Kirillov dimension of M is defined by GKdim(M ) := sup lim logn rankF V n , V,F n→∞

where V varies over all frames of B and F over all finitely generated Rsubmodules of M . In addition, GKdim(0) := −∞. Notice that F V n is a finitely generated R-submodule of M : This follows from the identity (r · m) · (r0 · b) = (rr0 ) · (m · b), with r, r0 ∈ R, m ∈ M and b ∈ B. Moreover, Q ⊗ M is a right module over the Q-algebra Q ⊗ B, with product (q ⊗ m) · (q 0 ⊗ b) := (qq 0 ) ⊗ (m · b), where q, q 0 ∈ Q, m ∈ M and b ∈ B. Proposition 20.3.7. Let B be an R-algebra and M be a right B-module. Then, GKdim(M ) = GKdim(Q ⊗ M ). Proof. Let V be a frame of B and F be a finitely generated R-submodule of M , then Q ⊗ F is a finitely generated Q-vector subspace of the right Q ⊗ Bmodule Q⊗M , and Q⊗V is a frame of Q⊗B. In addition, (Q⊗F )(Q⊗V )n = Q ⊗ F V n is a finitely generated Q-subspace of Q ⊗ M . Therefore, GKdim(M ) = sup lim logn rankF V n V,F n→∞

= sup lim logn (dimQ (Q ⊗ F V n )) V,F n→∞

= sup lim logn (dimQ ((Q ⊗ F )(Q ⊗ V )n )) V,F n→∞

≤ GKdim(Q ⊗ M ).

Now let W be a frame of Q ⊗ B. We showed in the proof of Proposition 20.3.2 that there exist finitely many v1 , . . . , vm ∈ B such that VW := R h1, v1 , . . . , vm i is a frame of B and W ⊆ Q ⊗ VW . Similarly, if G is a Q-subspace of Q ⊗ M of finite dimension, then there exist finitely many m1 , . . . , mt ∈ M such that FG := R hm1 , . . . , mt i satisfies G ⊆ Q ⊗ FG . Moreover, for every n ≥ 0, n n GW n ⊆ (Q ⊗ FG )(Q ⊗ VW )n = (Q ⊗ FG )(Q ⊗ VW ) = Q ⊗ FG V W .

Therefore, GKdim(Q ⊗ M ) = sup lim logn dimQ GW n W,G n→∞

≤ sup

n lim logn dimQ (Q ⊗ FG VW )

VW ,FG n→∞

20.3 Gelfand–Kirillov Dimension for Rings

= sup VW ,FG

429 n lim logn rankFG VW

n→∞

≤ GKdim(M ).

t u

In the next theorem we present some basic properties of the Gelfand– Kirillov dimension of modules. Theorem 20.3.8. Let B be an R-algebra and M be a right B-module. Then, (i) GKdim(BB ) = GKdim(B). (ii) GKdim(M ) ≤ GKdim(B). (iii) Let 0 → K → M → L → 0 be an exact sequence of right B-modules, then GKdim(M ) ≥ max{GKdim(K), GKdim(L)}. (iv) Let I be a two-sided ideal of B and M I = 0, then GKdim(MB ) = GKdim(MB/I ). (v) GKdim(

Pn

i=1

Mi ) = max{GKdim(Mi )}ni=1 = GKdim(

Ln

i=1

Mi ).

Proof. (i) Let V be a frame of B, then for every n ≥ 0, V n ⊆ V n+1 = V V n , hence GKdim(B) = sup lim logn rankV n V

n→∞

≤ sup lim logn rankV V n ≤ GKdim(BB ). V,V n→∞

On the other hand, if F is a finitely generated R-submodule of B, then for every frame V of B we have that WV := V + F is a frame of B, moreover, for every n, WVn+1 = (V + F )n+1 ⊇ F V n , so GKdim(BB ) = sup lim logn rankF V n V,F n→∞

≤ sup lim logn rankWVn+1 ≤ GKdim(B). WV n→∞

(ii) From Proposition 20.3.7 and (i) we get GKdim(M ) = GKdim(Q ⊗ M ) ≤ GKdim(Q ⊗ B) = GKdim(B). (iii) Since Q is an R-flat module we get the following exact sequence of Q⊗Bmodules 0 → Q ⊗ K → Q ⊗ M → Q ⊗ L → 0, whence GKdim(M ) = GKdim(Q ⊗ M ) ≥ max{GKdim(Q ⊗ K), GKdim(Q ⊗ L)} = max{GKdim(K), GKdim(L)}. (iv) As in part (v) of Theorem 20.3.4, let W be a frame of B/I. We can assume that W = R h1, w1 , . . . , wt i, then VW := R h1, w1 , . . . , wt i is a frame of B. Let G be a finitely generated R-submodule of MB/I , then FG := G is also a finitely generated R-submodule of MB . For every n ≥ 0 we have

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20 The Zariski Cancellation Problem for Skew P BW Extensions

n ) = dim (Q ⊗ F V n ). The last equaldimQ (Q ⊗ GW n ) = dimQ (Q ⊗ GVW Q G W ity in this case can be justified in the following way. The Q-vector spaces n and Q ⊗ F V n have finite dimension and we have the following Q ⊗ GVW G W homomorphisms of Q-vector spaces: n n , q ⊗ g · z 7→ q ⊗ g · z, with q ∈ Q, g ∈ F and Q ⊗ FG VW → Q ⊗ GVW G n z ∈ VW , n → Q ⊗ F V n , q ⊗ g · z 7→ q ⊗ g · z, with q ∈ Q, g ∈ G and z ∈ V n . Q ⊗ GVW G W W

The last homomorphism is well-defined since M I = 0. It is clear that the composites of these homomorphisms give the identities. Hence, GKdim(MB/I ) = sup lim logn rankGW n W,G n→∞

= sup lim logn dimQ (Q ⊗ GW n ) W,G n→∞

= sup

n lim logn dimQ (Q ⊗ FG VW )

= sup

n lim logn rankFG VW

VW ,FG n→∞ VW ,FG n→∞

≤ GKdim(MB ). Conversely, let V := R h1, v1 , . . . , vt i be a frame of B, then WV := R h1, v1 , . . . , vt i is a frame of B/I; let F be a finitely generated R-submodule of MB , then GF := F is a finitely generated R-submodule of MB/I . As above, for every n ≥ 0 we have dimQ (Q ⊗ F V n ) = dimQ (Q ⊗ GF WVn ). Hence, GKdim(MB ) = sup lim logn rankF V n V,F n→∞

= sup lim logn dimQ (Q ⊗ F V n ) V,F n→∞

= sup

lim logn dimQ (Q ⊗ GF WVn )

WV ,GF n→∞

= sup WV ,GF

lim logn rankGF WVn

n→∞

≤ GKdim(MB/I ). Therefore, GKdim(MB ) = GKdim(MB/I ). (v) The equalities can be proved tensoring by Q. (vi) From Proposition 20.3.7 and since Gr(Q ⊗ M ) ∼ = Q ⊗ Gr(M ), we get GKdim(Gr(M )Gr(B) ) = GKdim[(Q ⊗ Gr(M ))Q⊗Gr(B) ].

t u

Remark 20.3.9. (i) Taking R = Z in Definition 20.3.1 we get the notion of Gelfand–Kirillov dimension for arbitrary rings, and hence, all properties in Theorems 20.3.4, 20.3.8 and Remark 20.3.5 hold in this particular situation.

20.4 Makar-Limanov Invariants

431

(ii) The proof of Theorem 4.6.3 can be easy adapted to the case of algebras over commutative domains. Indeed, we can replace the vector subspace W of Z(A) and its dimension by a finitely generated R-submodule of Z(A) and its respective rank. Thus, Theorem 4.6.3 can be extended in the following way: let R be a commutative domain and A be a right Ore domain. If A is a finitely generated R-algebra such that GKdim(A) < GKdim(Z(A)) + 1, then Z(Qr (A)) = { pq | p, q ∈ Z(A), q 6= 0} ∼ = Q(Z(A)).

20.4 Makar-Limanov Invariants In order to investigate the Zariski cancellation problem for algebras having nontrivial center, we now study the Makar-Limanov invariants (actually, subalgebras) considered in [52] and [269]. We first recall some definitions. Definition 20.4.1. Let R be a commutative domain and A be an R-algebra. (i) Der(A) denotes the collection of R-derivations of A and LND(A) the collection of locally nilpotent R-derivations of A; δ ∈ Der(A) is locally nilpotent if given a ∈ A there exists an n ≥ 1 such that δ n (a) = 0. (ii) A higher derivation on A is a sequence of R-linear endomorphisms ∂ := (∂i )i≥0 of A such that Pn ∂0 = iA and ∂n (ab) = i=0 ∂i (a)∂n−i (b), for all a, b ∈ A and for all n ≥ 0. The collection of higher derivations is denoted by DerH (A). (iii) A higher derivation is iterative if
∂i ∂j = i+j i ∂i+j for all i, j ≥ 0. (iv) A higher derivation is locally nilpotent if (a) Given a ∈ A there exists an n ≥ 1 such that ∂i (a) = 0 for all i ≥ n. (b) The function G∂,t defined by P∞ G∂,t : A[t] → A[t], t 7→ t, a 7→ i=0 ∂i (a)ti is an algebra automorphism of A[t]. The collection of locally nilpotent higher derivations is denoted by LNDH (A) and the collection of locally nilpotent iterative higher derivations is denoted by LNDI (A). (v) For any ∂ ∈ DerH (A), the kernel of ∂ is defined to be \ ker(∂) := ker(∂i ). (20.4.1) i≥1

From the previous definitions we have that

432

20 The Zariski Cancellation Problem for Skew P BW Extensions

LNDI (A) ⊆ LNDH (A) ⊆ DerH (A).

(20.4.2)

The first two elementary results presented in [52] are the following. Proposition 20.4.2. (i) If ∂ = (∂i )i≥0 ∈ DerH (A), then ∂1 ∈ Der(A), and hence, there is a function α

DerH (A) − → Der(A), ∂ = (∂i )i≥0 7→ ∂1 . (ii) If char(R) = 0 and δ ∈ Der(A), then there is only one iterative higher derivation ∂ = (∂i )i≥0 on A such that ∂1 = δ, and it is given by ∂i :=

δi i! ,

for all i ≥ 0 (the canonical higher derivation associated to δ).

Moreover, the function β

i

Der(A) − → DerH (A), δ 7→ ∂ := ( δi! )i≥0 , is the right inverse of α, i.e., αβ = iDer(A) . t u

Proof. See [52] or also [346].

Before defining the Makar-Limanov invariants we present another lemma. Lemma 20.4.3. Let ∂ = (∂i )i≥0 ∈ DerH (A). (i) If ∂ ∈ LNDH (A), then for any c ∈ R, the function Gc,∂ defined by P∞ Gc,∂ : A → A, a 7→ i=0 ci ∂i (a) is an algebra automorphism of A. (ii) If ∂ is iterative and satisfies the condition (a) of Definition 20.4.1, i.e., given a ∈ A there exists an n ≥ 1 such that ∂i (a) = 0 for all i ≥ n, then ∂ is locally nilpotent, i.e., G∂,t is an algebra automorphism of A[t]. (iii) Let G : A[t] → A[t] be an R[t]-algebra automorphism such that G(a) ≡ a (mod t) for all a ∈ A, then G = G∂,t for some ∂ ∈ LNDH (A). t u

Proof. See [52], Lemma 2.2.

Now we will present the Makar-Limanov invariants for A introduced in [52]. Definition 20.4.4. Let A be an R-algebra. (i) The Makar-Limanov invariant of A is defined to be \ ML(A) := ker(δ). δ∈LND(A)

(ii) The Makar-LimanovH invariant of A is defined to be \ MLH (A) := ker(∂). ∂∈LNDH (A)

20.4 Makar-Limanov Invariants

433

(iii) The Makar-LimanovI invariant of A is defined to be \ MLI (A) := ker(∂). ∂∈LNDI (A)

A is LND-rigid if ML(A) = A, or equivalently, LND(A) = {0}; LNDH rigidity and LNDI -rigidity are similarly defined. In addition, A is strongly LND-rigid if ML(A[t1 , . . . , td ]) = A, for all d ≥ 0. Strongly LNDH rigidity and Strongly LNDI -rigidity are defined in a similar way. It is clear that Strongly LND-rigid ⇒ LND-rigid, Strongly LNDH -rigid ⇒ LNDH -rigid, Strongly LNDI -rigid ⇒ LNDI -rigid. Remark 20.4.5. (a) Note that Der(R) = {0} since for every r ∈ R and δ ∈ Der(R), δ(r) = δ(r1R ) = rδ(1R ) = 0. From this, ML(R) = R, and hence, R is LND-rigid. On the other hand, let ∂ = (∂i )i≥0 ∈ DerH (R), then ∂i = 0 for i ≥ 1. In fact, we know that ∂1 = 0; for r ∈ R, ∂2 (r) = ∂2 (r1R ) = r∂2 (1R ) = 0 since ∂2 (1R ) = ∂2 (1R 1R ) = ∂0 (1R )∂2 (1R ) + ∂1 (1R )∂1 (1R ) + ∂2 (1R )∂0 (1R ), i.e., ∂2 (1R ) = 0. We complete the proof by induction. From this we obtain that LNDI (R) = LNDH (R) = DerH (R) = {(iR , 0, 0, . . . )}, so MLI (R) = R = MLH (R), and hence, R is LNDH -rigid and LNDI -rigid. (b) If char(R) = 0, then the induced function (the restriction of β in Proposition 20.4.2 to locally nilpotent derivations) β

i

LND(A) − → LNDH (A), δ 7→ ∂ := ( δi! )i≥0 is injective. In fact, let δ ∈ LND(A), then given a ∈ A there exists an n ≥ 1 such that δ n (a) = 0, so δ i (a) = 0 for all i ≥ n, i.e., β(δ) ∈ LNDH (A), moreover, β is injective since it is the restriction of an injective function. Thus, we can write LND(A) ⊆ LNDH (A), and from this we get that char(R) = 0 ⇒ MLH (A) ⊆ ML(A). Indeed, ML(A) =

\

ker(δ) =

\

[

\

ker(δ n )]

δ∈LND(A) n≥1

δ∈LND(A)

⊇

\ ∂∈LNDH (A)

ker(∂) = MLH (A).

434

20 The Zariski Cancellation Problem for Skew P BW Extensions

In particular, LNDH -rigidity ⇒ LND-rigidity. (c) If char(R) = 0, then LNDI (A) = LND(A). In fact, by Proposition 20.4.2, in (b) we actually have β(LND(A)) = LNDI (A). Therefore, there is a bijective correspondence between LNDI (A) and LND(A), and hence, char(R) = 0 ⇒ MLI (A) = ML(A). Indeed, ML(A) =

\

ker(δ) =

\

[

\

ker(δ n )]

δ∈LND(A) n≥1

δ∈LND(A)

=

\

ker(∂) = MLI (A).

∂∈LNDI (A)

In particular, LNDI -rigidity ⇔ LND-rigidity. (d) Since LNDI (A) ⊆ LNDH (A), MLH (A) ⊆ MLI (A), in particular, LNDH -rigidity ⇒ LNDI -rigidity. Thus, if char(R) = 0, then LNDH -rigidity ⇒ LNDI -rigidity ⇔ LND-rigidity.

(20.4.3)

(e) If char(R) = 0, it is an open question if MLH (A) = ML(A). In particular, it is an open question if LND-rigidity is equivalent to LNDH -rigidity. (f) If char(R) 6= 0, then there are examples (see [82], Example 3.9, and also [346]) such that MLH (A) = MLI (A) ! ML(A), in particular, LND-rigidity < LNDH -rigidity. In the following proposition we emphasize that the Makar-Limanov invariants are actually subalgebras of A, and we summarize some statements of the previous remark. Proposition 20.4.6. Let A be an R-algebra. Then, ML(A), MLH (A) and MLI (A) are subalgebras of A and satisfy MLH (A) ⊆ MLI (A),

(20.4.4)

20.4 Makar-Limanov Invariants

435

char(R) = 0 ⇒ MLI (A) = ML(A).

(20.4.5)

char(R) = 0 ⇒ MLH (A) ⊆ ML(A).

(20.4.6)

Proof. By Remark 20.4.5 we only have to show that ML(A), MLH (A) and MLI (A) are subalgebras of A. Let δ ∈ Der(A), in particular, if δ ∈ LND(A), then ker(δ) is a subalgebra of A: we know that ker(δ) is an R-submodule of A; if a, b ∈ ker(δ), then δ(ab) = δ(a)b + aδ(b) = 0, i.e., ab ∈ ker(δ), and from this we also get that 1A ∈ ker(δ). Since the intersection of subalgebras is a subalgebra, ML(A) is a subalgebra of A. Let ∂ ∈ DerH (A), in particular, if ∂ ∈ LND(A)H , or ∂ ∈ LND(A)I , then ker(∂) is a subalgebra of A: by (20.4.1), ker(∂) is an R-submodule of A; if a, b ∈ ker(∂), then for every i ≥ 1, a, b ∈ ker(∂i ), therefore ∂n (ab) = 0 for every n ≥ 1, so ab ∈ ker(∂n ) for every n ≥ 1, i.e., ab ∈ ker(∂); note that ∂1 (1A ) = 0, and by induction, as in the proof of (a) of Remark 20.4.5, ∂n (1A ) = 0 for every n ≥ 1, so 1A ∈ ker(∂). t u In item (a) of Remark 20.4.5 we observed that R is always rigid. Now we consider the strong conditions. Proposition 20.4.7. If char(R) = 0, then for every d ≥ 0, ML(R[t1 , . . . , td ]) = MLH (R[t1 , . . . , td ]) = MLI (R[t1 , . . . , td ]) = R. Therefore, R is strongly LNDH -rigid, and hence, strongly LNDI -rigid and strongly LND-rigid. Proof. For d = 0, see Remark 20.4.5. Let A := R[t]; by the previous remark, MLH (A) ⊆ MLI (A) = ML(A), but ML(A) = R: in fact, let δ ∈ LND(A), we d d know that R ⊆ ker(δ), so R ⊆ ML(A), but R = ker( dt ) and dt ∈ LND(A). H I Thus, ML (A) ⊆ ML (A) = ML(A) = R; finally, R ⊆ MLH (A) since if ∂ = (∂i )i≥0 ∈ LNDH (A), then as we observed in item (a) of the previous remark, R ⊆ ker(∂i ) for every i ≥ 1, i.e., R ⊆ ker(∂). For the general case A := R[t1 , . . . , td ] we can repeat the previous proof t u taking δ := ∂t∂1 + · · · + ∂t∂d . Note that R = ker(δ) and δ ∈ LND(A). Lemma 20.4.8. For every d ≥ 0, (i) ML(A[t1 , . . . , td ]) ⊆ ML(A). (ii) MLH (A[t1 , . . . , td ]) ⊆ MLH (A). (iii) MLI (A[t1 , . . . , td ]) ⊆ MLI (A). Proof. For d = 0 the statements are trivial. (i) If we prove the statement for d = 1, then by induction ML(A[t1 , . . . , td ]) = ML(A[t1 , . . . , td−1 ][td ]) ⊆ ML(A[t1 , . . . , td−1 ]) ⊆ ML(A). Let δ ∈ LND(A), then we can extend δ to A[t] defining P P δ( α cα tα ) := α δ(cα )tα , cα ∈ A.

20 The Zariski Cancellation Problem for Skew P BW Extensions

436

δ ∈ LND(A[t]): clearly δ is R-linear; δ(cα tα cβ tβ ) = δ(cα cβ tα+β ) = δ(cα cβ )tα+β = [cα δ(cβ ) + δ(cα )cβ ]tα+β = cα tα δ(cβ tβ ) + δ(cα tα )cβ tβ ; moreover, it is also clear that δ is locally nilpotent. From this we get that LND(A) ⊆ LND(A[t]), and hence, ML(A[t]) ⊆ ML(A). (ii) As in (i), we only have to prove the statement for d = 1. Let ∂ = e := A[t] defining ∂e := (∂ei )i≥0 by (∂i )i≥0 ∈ LNDH (A). We extend ∂ to A P P ∂ei ( α cα tα ) := α ∂i (cα )tα , with cα ∈ A. e In fact, note first that ∂e is a higher derivation: Observe that ∂e ∈ LNDH (A). ∂e0 = iAe, ∂ei is R-linear, and ∂ei (cα tα cβ tβ ) = ∂ei (cα cβ tα+β ) = ∂i (cα cβ )tα+β = Pi [ k=0 ∂k (cα )∂i−k (cβ )]tα+β = k=0 ∂k (cα )tα ∂i−k (cβ )tβ = Pi f α g β k=0 ∂k (cα t )∂i−k (cβ t ). P α e and ∈ A, We will show that ∂e is locally nilpotent: given a := α cα t considering the coefficients cα , there exists an n ≥ 1 such that ∂ei (a) = 0 for i ≥ n; moreover, we have to prove that Pi

e e G∂,z e : A[z] → A[z], z 7→ z ∞ X a 7→ ∂ei (a)z i i=0

e is an algebra automorphism of A[z]. n m e [z], ∈A G∂,z e is R-linear: let p = a0 +a1 z+· · ·+an z , q = b0 +b1 z+· · ·+bm z we can assume that n = m, so for r ∈ R we have ! n ∞ X X i ∂ei (raj + bj )z z j G e (rp + q) = ∂,z

i=0

j=0

=

n X

∞ X

j=0

i=0

=r

! i

r∂ei (aj )z + ∂ei (bj )z

n X

∞ X

j=0

i=0

i

zj

! ! ∞ n X X i j i e e ∂i (bj )z z j ∂i (aj )z z + j=0

i=0

= rG∂,z e (p) + G∂,z e (q). G∂,z is multiplicative: we only need to consider the product of two terms, e n az and bz m ; there exists a k ≥ 1 such that ∂ei (a) = 0 for i > k, and there exists an l ≥ 1 such that ∂ej (b) = 0, for j > l, then

20.4 Makar-Limanov Invariants

n m G∂,z e (az )G∂,t e (bz ) =

437 k X

! 

! ∂ei (a)z i

zn

i=0 k X

=

=

k+l X

 =

k+l X

∂ej (b)z j  z m 

j=0

i=0 j=0

j=0



 X





∂ei (a)∂ej (b) z p  z n+m = 

 p=0



   ! l k X l   X X j  n+m i+j  n+m i  e e e e  ∂i (a)z = z ∂i (a)∂j (b) z ∂j (b)z z

i=0



l X





i+j=p

k+l X

p X

p=0

i=0



! ∂ei (a)∂ep−i (b)

z p  z n+m

 ∂ep (ab)z

p  n+m

z

m n = G∂,z abz n+m = G∂,z e e ((az ) (bz )).




p=0

m1 e G∂,z + · · · + an z mn , with e is injective: let p ∈ A [z], p 6= 0, then p = a1 z ai 6= 0 for 1 ≤ i ≤ n and m1 < · · · < mn ; for every 1 ≤ i ≤ n, there exists a fk (ai ) = 0 for every k ≥ ji , let M := max{ji }n , then ji ≥ 1 such that ∂ i=1

G∂,z e (p) =

  PM  e PM  e i m1 i z ) · · · ∂ z mn . ) z ∂ (a z + + (a n i 1 i i=0 i=0


Note that (∂e0 a1 )z 0 z m1 = a1 z m1 6= 0 and the degree of this term is less than the degree of any other term of G∂,z / ker(G∂,z e (p), thus G∂,z e (p) 6= 0, i.e, p ∈ e ), so ker(G∂,z e ) = 0. e [z]. We have to find G e is surjective: let p = a0 + a1 z + · · · + an z n ∈ A ∂,z

e [z] such that G e (q) = p, but since G e is additive and multiplicative, q∈A ∂,z ∂,z e or p = z; in the second case G e (z) = z; so we can assume that p = a ∈ A ∂,z

e = A[t]; since ∂ is locally nilpotent then G∂,t : A[t] → assume that p = a ∈ A A[t] is surjective, so there exists a q := q0 + q1 t + · · · + qn tn ∈ A[t] such that p = G∂,t (q), thus, there exists an L such that p = G∂,t (q0 + q1 t + · · · + qn tn ) =

L X

∂i (q0 )ti + (

i=0

=

L X

L X

∂i (q1 )ti )t + · · · + (

i=0

L X

∂i (qn )ti )tn

i=0

(∂i (q0 )ti + ∂i (q1 )ti+1 + · · · + ∂i (qn )ti+n )

i=0

=

L X

(∂ei (q0 ti ) + ∂ei (q1 ti+1 ) + · · · + ∂ei (qn ti+n ))

i=0

= ∂ei (

L X

q0 ti + q1 ti+1 + · · · + qn ti+n )

i=0

= ∂ei (

L X i=0

q0 ti + q1 ti+1 + · · · + qn ti+n )z 0

438

20 The Zariski Cancellation Problem for Skew P BW Extensions

= G∂,z e (

L X

q0 ti + q1 ti+1 + · · · + qn ti+n ).

i=0

Thus, LNDH (A) ⊆ LNDH (A[t1 , . . . , td ]), and hence, MLH (A[t1 , . . . , td ]) ⊆ MLH (A). (iii) We only have to prove the statement for d = 1, but from (ii), we need to check only that ∂e is iterative: for i, j ≥ 0 we have
P P P α ∂ei ∂ej ( α cα tα ) = α ∂i ∂j (cα )tα = α i+j i ∂i+j (cα )t =
P i+j e α α cα t ). i ∂i+j ( Therefore, MLI (A[t1 , . . . , td ]) ⊆ ML(A) = MLI (A).

t u

Remark 20.4.9. (i) It is not known if in the previous lemma the equalities hold. (ii) Makar-Limanov conjectured in [269] that if the algebra A is a commutative domain over a field of characteristic zero, then ML(A[t1 , . . . , td ]) = ML(A); he proved the conjecture for GKdim(A) = 1. (iii) The following L property of the Gelfand–Kirillov dimension will be used below: Let Y := i≥0 Yi be an N-graded R-algebra, suppose that Y is a domain; if Z is a subalgebra of Y containing Y0 such that GKdim(Z) = GKdim(Y0 ) < ∞, then Z = Y0 (see [52], Lemma 3.2; compare also with Lemma 20.8.4 below). Theorem 20.4.10 ([52], Theorem 3.3). Let A be a finitely generated domain of finite Gelfand–Kirillov dimension. (i) If A is strongly LND-rigid, then A is strongly cancellative, and hence cancellative. If ML(A[t]) = A, then A is cancellative. (ii) If A is strongly LNDH -rigid, then A is strongly cancellative, and hence cancellative. If MLH (A[t]) = A, then A is cancellative. (iii) If A is strongly LNDI -rigid, then A is strongly cancellative, and hence cancellative. If MLI (A[t]) = A, then A is cancellative. Proof. Let B be an algebra and let φ : A [t1 , . . . , td ] → B [t1 , . . . , td ] be an isomorphism of algebras; if δ is a locally nilpotent derivation of B[t1 , . . . , td ], then it is easy to check that φ−1 ◦ δ ◦ φ is a locally nilpotent derivation of A[t1 , . . . , td ]. Similarly it can be proved that if δ 0 is a locally nilpotent derivation of A[t1 , . . . , td ], then φ ◦ δ 0 ◦ φ−1 is a locally nilpotent derivation of B[t1 , . . . , td ] (see [346]). In [346] it is also proved that if ∂ = (∂i )i≥0 is a locally nilpotent higher derivation, or a locally nilpotent iterative higher derivation of B[t1 , . . . , td ], then ∂φ = (φ−1 ◦ ∂i ◦ φ)i≥0 is a locally nilpotent iterative higher derivation or a locally nilpotent iterative higher derivation of A[t1 , . . . , td ]. In a similar way, if ∂ 0 = (∂i )i≥0 is a locally nilpotent higher derivation, or a locally nilpotent iterative higher derivation of A[t1 , . . . , td ], then ∂φ0 = (φ ◦ ∂i ◦ φ−1 )i≥0 is a locally nilpotent higher derivation, or a locally nilpotent iterative higher derivation of B[t1 , . . . , td ].

20.4 Makar-Limanov Invariants

439

Considering the previous assertions, we will show that ML∗ (A [t1 , . . . , td ]) ∼ = ML∗ (B [t1 , . . . , td ]) , where ∗ indicates blank, I or H. Since φ is an isomorphism, it is enough to prove that φ (ML∗ (A [t1 , . . . , td ])) = ML∗ (B [t1 , . . . , td ]). There exists a bijective correspondence between LN D∗ (A [t1 , ..., td ]) and LN D∗ (B [t1 , . . . , td ]) given by ∂ 0 7→ φ ◦ ∂ 0 ◦ φ−1 , with ∂ 0 ∈ LN D∗ (A [t1 , . . . , td ]); the inverse is given by ∂ 7→ φ−1 ◦ ∂ ◦ φ, with ∂ ∈ LN D∗ (B [t1 , . . . , td ]). Let a ∈ ML∗ (A [t1 , ..., td ]), i.e., ∂ 0 (a) = 0 for all ∂ 0 ∈ LN D∗ (A [t1 , . . . , td ]); let ∂ ∈ LN D∗ (B [t1 , . . . , td ]), then ∂ = φ ◦ ∂ 0 ◦ φ−1 with ∂ 0 ∈ LN D∗ (A [t1 , . . . , td ]), whence (φ ◦ ∂ 0 ◦ φ−1 )(φ(a)) = (φ ◦ ∂ 0 )(a) = 0, so φ(a) ∈ ML∗ (B [t1 , . . . , td ]). In a similar way can be proved that if b ∈ ML∗ (B [t1 , . . . , td ]) then φ−1 (b) ∈ ML∗ (A [t1 , . . . , td ]). By the hypothesis, ML∗ (A [t1 , . . . , td ]) = A, moreover, we have ML∗ (B [t1 , . . . , td ]) ⊆ ML∗ (B) ⊆ B (see above Lemma 20.4.8); since φ(A) = φ (ML∗ (A [t1 , . . . , td ])) = ML∗ (B [t1 , . . . , td ]) ⊆ B, we get A ⊆ φ−1 (B). On the other hand, we have GKdim (A [t1 , . . . , td ]) = GKdim (B [t1 , . . . , td ]), i.e., GKdim (A) + d = GKdim (B) + d, and since GKdim (A) < ∞, we have GKdim (A) = GKdim (B) < ∞; taking Y := A [t1 , . . . , td ] and A = Y0 ⊆ Z := φ−1 (B), we can apply Remark 20.4.9 and we get A = Y0 = Z = φ−1 (B), i.e., A and B are isomorphic. This proves the first statements of (i)–(iii). We can repeat the previous proof, taking d = 1, for the second statements. t u Other interesting theorems proved in [52] are the following. Theorem 20.4.11. Let A be a finitely generated domain of characteristic zero of finite Gelfand–Kirillov dimension. If A is LND-rigid, then A is cancellative. Proof. Since A is a domain of finite Gelfand–Kirillov dimension, it is an Ore domain (see Theorem 20.3.4 (i)). By Lemma 3.5 of [52], ML(A[t]) = A. The result follows from Theorem 19.4.10 (i). t u Theorem 20.4.12. Let A be a finitely generated domain of Gelfand–Kirillov dimension two over an algebraically closed field of characteristic zero. Then, (i) If A is noncommutative and P I, then A is cancellative. (ii) If A is not P I, then A is cancellative.

440

20 The Zariski Cancellation Problem for Skew P BW Extensions

Proof. See Corollary 3.7 in [52].

t u

Theorem 20.4.13. Let A := Kq [x1 , . . . , xn ] be the algebra of quantum polynomials, where K is a field, q := [qij ] and each qij is a root of unity. Then, A mn 1 is strongly LNDH -rigid if and only if Z(A) is a subalgebra of K[xm 1 , . . . , xn ] for some m1 , . . . , mn ≥ 2. Thus, under this last condition A is cancellative. Proof. This follows from Theorem 5.7 in [52] and from Theorem 20.4.10. t u Example 20.4.14. When q is a root of unity of degree m ≥ 1, it is proved in [362] that the center of the quantum plane A := Kq [x, y] is the subalgebra generated by xm and y m (see also Proposition 3.3.14). If m = 1, then Kq [x, y] = K[x, y] is cancellative (Remark 20.1.2). Let m ≥ 2; Theorem 20.4.13 shows that Kq [x, y] is strongly LNDH -rigid, so by Theorem 20.4.10, Kq [x, y] is strongly cancellative, and hence, cancellative (it is clear that Kq [x, y] is a finitely generated domain, and recall that GKdim(Kq [x, y]) = 2, see Table 8.4 or Theorem 18.3.5).

20.5 The Discriminant and the Divisor Algebra as Tools for the Cancellation Problem In this section we will consider another important subalgebra very useful for the investigation of the cancellation problem of a given algebra. We will follow the presentation given in [84] and [85]. As in the previous sections, R is a commutative domain and A is an R-algebra. Let Z be a central subalgebra of A such that A is finitely generated and free over Z, let r be the rank of A over Z (i.e., the dimension of the free Z-module A). Consider the injective ring homomorphism (injective homomorphism of Z-algebras) A ,→ EndZ (AZ ), a 7→ la , la : A → A, la (x) := ax. Recall that EndZ (AZ ) ∼ = Mr (Z); the trace function is defined by trm tr : A → EndZ (AZ ) ∼ = Mr (Z) −−→ Z,

where trm is the usual matrix trace. Observe that tr is independent of the chosen Z-basis of A. Definition 20.5.1. Let A, Z and r be as above and let {z1 , . . . , zr } be a Z-basis of A; the discriminant of A over Z is defined to be d(A/Z) := det(tr(zi zj )) ∈ Z. For x, y ∈ Z we use the notation x =Z ∗ y to indicate that x = cy for some c ∈ Z ∗ . With this notation we have the following property.

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Proposition 20.5.2. d(A/Z) is unique up to a scalar in Z ∗ . Thus, d(A/Z) =Z ∗ det(tr(zi zj )). Proof. Let Y = {y1 , . . . , yr }P be another Z-basis of A, then there exist aij ∈ Z, 1 ≤ i, j ≤ r such that yi = j aij zj , so d(A/Z) = det(tr(yi yj )) = det((aij )tr(zi zj )(aij )T ) = det((aij ))2 det(tr(zi zj )), with det((aij )) ∈ Z ∗ . Thus, d(A/Z) =Z ∗ det(tr(zi zj )).

t u

With the above notation we present next some properties and some concrete computations of the discriminant that we found in [84]. Lemma 20.5.3. Let (A0 , Z 0 ) be another pair of algebras such that Z 0 is a central subalgebra of A0 and A0 is a free Z 0 -module of rank r. Suppose that g : A → A0 is a ring homomorphism such that (i) g(Z) ⊆ Z 0 . (ii) {g(z1 ), . . . , g(zr )} is a Z 0 -basis of A0 . Then, g(d(A/Z)) =(Z 0 )∗ d(A0 /Z 0 ). 0 Proof. For every a ∈ A, we denote Pr g(a) by a , moreover, tr(g(a)) = g(tr(a)), and in addition we write azi = j=1 aij zj for all i. Then applying g, we have Pr a0 zi0 = j=1 a0ij zj0 , whence

! 0

tr(g(a)) = tr(a ) =

X i

a0ii

=g

X

aii

= g(tr(a))

i

for all a ∈ A. By Definition 20.5.1 and the previous equation, we get g(d(A/Z)) = g(det(tr(zi zj ))) = det(tr(zi0 zj0 )) =(Z 0 )∗ d(A0 /Z 0 ).

t u

If C is a multiplicative subset of Z, then AC −1 exists and ZC −1 ⊆ Z(AC −1 ). With the above notation we have. Lemma 20.5.4. AC −1 is free over ZC −1 of rank r. Moreover, d(AC −1 /ZC −1 ) =(ZC −1 )∗ d(A/Z). Proof. Let {z1 , . . . , zr } be a Z-basis of A. Then it is also a ZC −1 -basis of AC −1 . The result follows from Lemma 20.5.3. t u As above, let X := {z1 , . . . , zr } be a Z-basis of A, we can assume that z1 = 1; suppose that A is N-filtered with filtration {Fp (A)}p≥0 such that Gr(A) is a domain; note that Gr(Z) is a central subalgebra of Gr(A). For any f ∈ A, let gr(f ) denote the associated element in Gr(A), i.e., if p ∈ N is minimum such that f ∈ Fp (A), then gr(f ) := f + Fp−1 (A).

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20 The Zariski Cancellation Problem for Skew P BW Extensions

Lemma 20.5.5. Suppose that Gr(A) is finitely generated and free over Gr(Z) with basis gr(X). Then gr(d(A/Z)) =Gr(Z)∗ d(Gr(A)/Gr(Z)). Proof. See [84], Lemma 1.5, and [83], Proposition 4.10.

t u

Example 20.5.6 ([84], Proposition 1.4; [83], Proposition 2.8). Let R be a commutative domain and let A := Rq [x1 , . . . , xn ] be the R-algebra of quantum polynomials (also called the n-multiparametric quantum space, see Example 4.4.2), where q := [qij ] ∈ Mn (R); in addition, in this example we assume that qij is a nontrivial root of unity for all 1 ≤ i < j ≤ n (recall that qij ∈ R∗ and qij qji = 1, so qji is also a nontrivial root of unity; moreover, qii = 1). αn 1 If Z := R[xα 1 , . . . , xn ] is a central subalgebra of A (see Proposition 3.3.17), where αi ≥ 1, 1 ≤ i ≤ n, then Qn Qn i −1 r (i) If r := i=1 αi , then d(A/Z) =R∗ rr ( i=1 xα ) . i m m (ii) If n = 2, Z = R[x1 , x2 ] and q12 is a primitive m-th root of unity, then 2 m m(m−1) d(A/Z) =R∗ m2m (xm . 1 x2 ) (iii) If qij = −1 for all i < j and αi = 2 for all i, then d(A/Z) =R∗ n Qn n−1 2n2 ( i=1 x2i )2 . Example 20.5.7 ([84], Theorem 2.4). We consider the q-quantum Weyl Ralgebra Aq generated by x, y with rule of multiplication yx = qxy + ya, where q ∈ R∗ and a ∈ R (see Proposition 3.3.18); in this example we assume additionally that q is an n-th primitive root of unity for some n ≥ 2. Then the discriminant of Aq over its center is 2

d(Aq /Z(Aq )) =R∗ (nm)n [(1 − q)n xn y n − an ]n(n−1) , Qn−1 with m := i=2 (1 + q + · · · + q i−1 ). If n = 2, then m := 1. Next we present another useful tool for the cancellation problem. Definition 20.5.8 ([84]). Let R be a commutative domain and let A be an R-algebra. Let F ⊆ A. Then, (i) Sw(F ) := {g ∈ A|agb ∈ F − {0}, for some a, b ∈ A}. (ii) For n ≥ 0, Dn (F ) is inductively defined as the R-subalgebra of A generated by Sw(Dn−1 (F )), with S D0 (F ) := F . The F -divisor subalgebra of A is defined by D(F ) := n≥0 Dn (F ). If F = {f }, then D(f ) := D(F ). (iii) If A is finitely generated and free over Z(A) and f := d(A/Z(A)), then the discriminant-divisor subalgebra of A is defined by D(A) := D(f ). Remark 20.5.9. Observe that if F is either {0} or ∅, then Sw(F ) = ∅ and D1 (F ) is the subalgebra generated by Sw(∅) = ∅, i.e., D1 (F ) = R · 1A . Moreover, for every F 6= {0}, F ⊆ Sw(F ), whence Dn−1 (F ) ⊆ Dn (F ) for every n ≥ 1, thus D(F ) is a subalgebra of A. If F = {0}, then D0 (0) = 0, D1 (0) = R · 1A and again Dn−1 (F ) ⊆ Dn (F ) for every n ≥ 1, so D(F ) is a subalgebra of A.

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Proposition 20.5.10. Let R be a commutative domain and let A be an Ralgebra. Assume that A is a domain finitely generated and free over Z(A). (i) g(D(A)) ⊆ D(A), for every g ∈ Aut(A). (ii) If A∗ = R∗ , then for every d ≥ 0, D(A) ⊆ ML(A[t1 , . . . , td ]) ⊆ ML(A) ⊆ A. Proof. (i) Let f = d(A/Z), by Lemma 20.5.3, f is g-invariant up to a unit. So, if g ∈ Aut(A), then g maps Sw(f ) to Sw(f ) and D1 (f ) to D1 (f ). By induction, g maps Dn (f ) to Dn (f ). (ii) By Lemma 20.4.8 part (i), ML(A[t1 , . . . , td ]) ⊆ A ⊆ A[t1 , . . . , td ], by Lemma 7.7 of [84], f ∈ ML(A[t1 , . . . , td ]), then D0 (f ) ⊆ ML(A[t1 , . . . , td ]) by Lemma 7.5 of [85]. By the definition, it is clear that D(f ) ⊆ D0 (f ). The assertion follows. t u Theorem 20.5.11 ([84], Theorem 8.3.). Let A be an R-algebra that satisfies the following conditions: (i) A is a finitely generated domain. (ii) GKdim(A) < ∞. (iii) A is finitely generated and free over Z(A). (iv) A∗ = R∗ . If D(A) = A, then A is strongly cancellative, and hence, cancellative. Proof. By Proposition 20.5.10, ML(A[t1 , . . . , td ]) = A for every d ≥ 0, i.e., A is strongly LND-rigid, so the assertion follows from Theorem 20.4.10. t u Example 20.5.12 ([84], Example 8.4). Let A be the R-algebra generated by the elements x1 , x2 , x3 , x4 with relations x2 x1 = −x1 x2 , x3 x1 = −x1 x3 , x4 x1 = −x1 x4 + x23 , x3 x2 = −x2 x3 , x4 x2 = −x2 x4 , x4 x3 = −x3 x4 . Using Theorem 1.3.1, it is easy to show that A = σ(R[x3 ])hx1 , x2 , x4 i is a bijective skew P BW extension of R[x3 ]: in fact, for i = 1, 2, 4 we have the automorphisms and σ-derivations of R[x3 ], σi (x3 ) := −x3 , δi := 0; for every r ∈ R and k ≥ 0, (x2 x1 )r = x2 (x1 r), (x4 x1 )r = x4 (x1 r), (x4 x2 )r = x4 (x2 r), (x2 x1 )xk3 = x2 (x1 xk3 ), (x4 x1 )xk3 = x4 (x1 xk3 ), (x4 x2 )xk3 = x4 (x2 xk3 ), (x4 x2 )x1 = x4 (x2 x1 ). Assuming that Z ⊆ A (i.e., char(A) = 0), then from Proposition 3.2.1, A satisfies (i) of Theorem 20.5.11; considering the properties of the Gelfand– Kirillov dimension over commutative domains (see Section 20.3), we conclude that A satisfies (ii); in [84] it is proved that Z(A) = R[y1 , y2 , y3 , y4 ], with yi := 1 α2 α3 α4 x2i , 1 ≤ i ≤ 4, so (iii) holds since a Z(A)-basis of A is {xα 1 x2 x3 x4 |0 ≤ αi ≤ 1}; from Corollary 3.2.2 we obtain that the condition (iv) holds; finally, in [84] it is shown that D(A) = A, so A is cancellative.

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20 The Zariski Cancellation Problem for Skew P BW Extensions

20.6 Noncommutative Cancellative Algebras: Nondomain Examples In this section we study the noncommutative Zariski cancellation problem for some algebras that are not necessarily domains. We will follow the paper [255] but include many details omitted there in. As always in the present chapter, R is a commutative domain and A is an R-algebra. In some places R = K is a field. We start with an easy proposition and a lemma containing elementary facts. Proposition 20.6.1. Let A be an algebra. (i) If Rad(A) is nilpotent, then for every n ≥ 0, Rad(A[t1 , . . . , tn ]) is nilpotent and rad(A[t1 , . . . , tn ]) = rad(A)[t1 , . . . , tn ] = Rad(A[t1 , . . . , tn ]) = Rad(A)[t1 , . . . , tn ].

(ii) If Rad(A[t]) is nilpotent and Rad(A[t]) = Rad(A)[t], then Rad(A) is nilpotent and the equalities in (i) hold. Proof. (i) For n = 0 the result is trivial. It is enough to prove the case n = 1 since then Rad(A[t]) = Rad(A)[t] is nilpotent. By the hypothesis, rad(A) = Rad(A) is nilpotent and there exists an m ≥ 1 such that rad(A)m = 0, then rad(A)m [t] = 0, whence (rad(A)[t])m ⊆ rad(A)m [t] = 0, therefore rad(A)[t] is nilpotent; thus, if a(t) ∈ rad(A)[t], then a(t) is nilpotent, so a(t) is strongly nilpotent (see [278], p. 5), therefore a(t) ∈ rad(A[t]). We have proved that rad(A)[t] ⊆ rad(A[t]). Conversely, let a(t) ∈ rad(A[t]), then a(t) is strongly nilpotent, so nilpotent, whence a(t)m = 0 and the leading coefficient ar of a(t) is nilpotent. From this we get that 1 + ar is invertible, but for any b ∈ A, a(t)b ∈ rad(A[t]), so 1+ar b is also invertible, whence ar A ⊆ Rad(A) = rad(A), and hence ar ∈ rad(A), so ar tr ∈ rad(A)[t] ⊆ rad(A[t]); thus, a(t) − ar tr ∈ rad(A[t]), so by induction on r, all coefficients of a(t) are in rad(A). This completes the proof of the equality rad(A[t]) = rad(A)[t]. For the second equalities, Rad(A) = rad(A) is nilpotent; by Amitsur’s theorem (see [218]), Rad(A[t]) = N [t], with N := A ∩ Rad(A[t]), but it is easy to show that N ⊆ Rad(A), so N m = 0 for some m, and hence, N [t]m ⊆ N m [t] = 0, i.e., Rad(A[t]) is nilpotent, whence, Rad(A[t]) = rad(A[t]) = rad(A)[t] = Rad(A)[t]. m m (ii) We have , P (Rad(A[t])) = 0 for some m ≥ 1.mLet z ∈ (Rad(A)) m then z = a1i · · · ami , with aji ∈ Rad(A), so zt ∈ (Rad(A)[t]) = (Rad(A[t]))m = 0, i.e., z = 0. Thus, (Rad(A))m = 0. t u Lemma 20.6.2. Let n ≥ 0. Let A and B be two algebras such that A[t1 , . . . , tn ] ∼ = B[s1 . . . , sn ]. Then the following hold.

20.6 Noncommutative Cancellative Algebras: Nondomain Examples

445

(1) A is commutative iff B is. (2) A is left (right) noetherian iff B is. (3) Assume that A is left (right) noetherian. Then, lKdimA = lKdimB (rKdimA = rKdimB). (4) A is left (right) artinian iff B is. (5) GKdimA = GKdimB. (6) A is a field iff B is. (7) A is a division ring iff B is. (8) A is a finite direct sum of m fields iff B is. (9) A is a finite direct sum of m division rings iff B is. (10) A is local left (right) artinian iff B is. Proof. (1) and (2) are evident. (3) Trivial since lKdim(A[t1 , . . . , tn ]) = lKdim(A) + n ([242], Chapter 5). (4) This is a direct consequence of (3). (5) Trivial since GKdim(A[t1 , . . . , tn ]) = GKdim(A) + n ([241], Chapter 4). (6) If A is a field, then by (5) and (1) B is a field (([241], Chapter 4)). (7) If A is a division ring, then A is a left artinian domain, whence B is a left artinian domain, and hence, rad(B) = 0 = Rad(B), so B is a semisimple domain. This implies that B is a division ring. (8) From (1) and (4), B is commutative artinian; moreover, rad(A) = 0, so rad(A[t1 , . . . , tn ]) = 0 and hence rad(B[s1 , . . . , sn ]) = 0 = rad(B)[s1 , . . . , sn ], so rad(B) = 0 = Rad(B), whence, B is a finite direct sum of fields. If A = K1 ⊕ · · · ⊕ Km , with Ki a field, 1 ≤ i ≤ m, then A[t1 , . . . , tn ] ∼ = K1 [t1 . . . , tn ] ⊕ · · · ⊕ Km [t1 , . . . , tn ], taking the Goldie dimension we obtain that udim(A[t1 . . . , tn ]) = m = udim(B[s1 , . . . , sn ]), so B is also a finite direct sum of m fields. (9) By (4), B is left (right) artinian; Rad(A) = 0 = rad(A), so by Proposition 20.6.1, rad(A[t1 , . . . , tn ]) = rad(A)[t1 , . . . , tn ] = 0. Therefore, rad(B[s1 , . . . , sn ]) = 0 = rad(B)[s1 , . . . , sn ], hence rad(B) = 0 = Rad(B), whence, B is a finite direct sum of matrix rings over division rings, but since the only nilpotent element of A[t1 , . . . , tn ] is zero, B is a finite direct sum of division rings. The assertion with respect to m is proved as in (8). (10) From (4), A is left (right) artinian iff B is; in this situation, we again apply Proposition 20.6.1 and we get

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20 The Zariski Cancellation Problem for Skew P BW Extensions

Rad(A[t1 , . . . , tn ]) = Rad(A)[t1 , . . . , tn ], Rad(B[s1 , . . . , sn ]) = Rad(B)[s1 , . . . , sn ]. ∼ B[s1 , . . . , sn ] we obtain From A[t1 , . . . , tn ] = ∼ A[t1 , . . . , tn ]/Rad(A[t1 , . . . , tn ]) (A/Rad(A))[t1 , . . . , tn ] = ∼ = B[s1 , . . . , sn ]/Rad(B[s1 , . . . , sn ]) ∼ = (B/Rad(B))[s1 , . . . , sn ], so from (7), A is local iff B is local.

u t

Another easy lemma concerns central idempotents of an algebra A. Lemma 20.6.3. Let A be an algebra, Z(A) be its center and CI(A) be the set of central idempotents of A. Then, (i) CI(A) = CI(Z(A)). (ii) The following are equivalent: (a) A = A1 ⊕ A2 ⊕ · · · ⊕ An , for some integer n, where Ai is an indecomposable algebra, 1 ≤ i ≤ n. (b) |CI(A)| = 2n for some integer n. (c) CI(A) is finite. (iii) If A is N-graded, then CI(A) = CI(A0 ). (iv) CI(A[t1 , . . . , tn ]) = CI(A). Proof. (i) Evident. (ii) (a)⇒(b) If a = (a1 , . . . , an ) ∈ CI(A), then ai ∈ CI(Ai ) for every 1 ≤ i ≤ n, but since Ai is indecomposable, then either ai = 0 or ai = ei (the unit of Ai ). This implies that |CI(A)| = 2n . (b) ⇒(c) Trivial. (c) ⇒(a) If |CI(A)| = 2, then A is indecomposable; assume that |CI(A)| ≥ 3, let e ∈ CI(A)−{0, 1}, then A = A1 ⊕A2 where A1 := eA and A2 := (1−e)A are algebras. If A1 , A2 are indecomposable, we are finished. Suppose that A1 is decomposable. This implies that there exists an f ∈ Z(A1 ) such that f 2 = f , f 6= 0, e and A1 = A11 ⊕ A12 , with A11 = f A1 , A12 := (1 − f )A1 . Note that (f, 0) ∈ CI(A) − {0, 1}. We can do the same for A2 and for A11 , A12 , but this procedure of decomposition stops since CI(A) is finite, and this proves (a). (iii) Let e ∈ CI(A) and e = e0 + e1 + · · · + ep , with ei ∈ Ai , where Ai is the homogeneous component of A of degree i, 0 ≤ i ≤ p; since e is central, ei is central for every i. From e2 = e we get that e20 = e0 , so e0 ∈ CI(A). This implies that L f1 := e − e0 e and f22 := e0 − e0 e are central idempotents. Since f1 , f2 ∈ i≥1 Ai , then from fj = fj , j = 1, 2, we get that fj = 0, whence e = (f1 − f2 ) + e0 = e0 . Thus, CI(A) ⊆ CI(A0 ). It is clear that CI(A0 ) ⊆ CI(A) (note that CI(A0 ) consists of idempotent elements of A0 in Z(A)). t u (iv) This is an immediate consequence of (iii).

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Definition 20.6.4. An algebra A is called retractable if for any algebra B, an algebra isomorphism φ : A[t] ∼ = B[s] implies that φ(A) = B. An algebra A is strongly retractable if for any algebra B and any integer n ≥ 0, an algebra isomorphism φ : A[t1 , . . . , tn ] ∼ = B[s1 , . . . , sn ] implies that φ(A) = B. It is clear that SR ⇒ R ⇒ C. The polynomial ring K[x] is cancellative, but not retractable: φ

K[x][t] ∼ = K[t][x], φ(x) := x, φ(t) := t, φ(K[x]) = K[x] 6= K[t]. For a fixed n ≥ 0, A[t1 , . . . , tn ] is abbreviated as A[t] and B[s1 , . . . , sn ] as B[s]. Remark 20.6.5. In [386], Lemma 5.1, it is proved that for algebras over fields the retractable condition is equivalent to the following weak condition: a K-algebra A is retractable if for any algebra B, an algebra isomorphism φ : A[t] ∼ = B[s] implies that φ(A) ⊆ B. A is strongly retractable if for any algebra B and any integer n ≥ 0, an algebra isomorphism φ : A[t1 , . . . , tn ] ∼ = B[s1 , . . . , sn ] implies that φ(A) ⊆ B. Lemma 20.6.6. If A is a finite direct sum of division rings, then A is strongly retractable. Proof. This follows from the facts: (1) Let A = D1 ⊕ · · · ⊕ Dr , with Di a ring, 1 ≤ i ≤ r, then A[t1 , . . . , tn ] ∼ = D1 [t1 , . . . , tn ] ⊕ · · · ⊕ Dr [t1 , . . . , tn ]. (2) Assuming that every Di is a division ring, the idempotents of A[t1 , . . . , tn ] are just the idempotents of A; the same is true for the invertible elements. Therefore, let a ∈ A, then a = (d1 , . . . , dr ) = (d1 , 1, . . . , 1)(1, 0, . . . , 0) + · · · + (1, . . . , 1, dr )(0, . . . , 0, 1), with di ∈ Di , 1 ≤ i ≤ r; observe that (1, . . . 1, di , 1, . . . , 1) is either invertible or idempotent, and (0, . . . , 0, 1, 0, . . . , 0) is idempotent, therefore, if φ

A[t1 , . . . , tn ] ∼ = B[s1 , . . . , sn ], then φ(a) ∈ B, i.e., φ(A) ⊆ B. Considering φ−1 we conclude that φ(A) = B. t u Theorem 20.4.10 proved that the Makar-Limanov invariants and rigidity control cancellation, actually they control the retractable property.

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Lemma 20.6.7. Suppose A is a finitely generated domain of finite GKdimension. (1) If ML(A[t]) = A or ML(A[t])H = A or ML(A[t])I = A, then A is retractable. (2) If A is strongly LND-rigid, or strongly LNDH -rigid, or strongly LNDI rigid then A is strongly retractable. Proof. See the proof of Theorem 20.4.10.

t u

Another lemma about LND-rigidity shows that all prime algebras that are not domains are not LND-rigid. Lemma 20.6.8. Let A be an algebra. (i) If A has a non-central nilpotent element, then A is not LND-rigid. (ii) If A is a prime algebra that is not a domain, then every nilpotent element is not central. As a consequence, A is not LND-rigid. (iii) Let A be prime. If A is LND-rigid, then A is a domain. Proof. (i) Let x be a non-central nilpotent element. Then adx : a 7→ xa − ax is a nonzero LND. So A is not LND-rigid. (ii) Since A is prime, Z(A) is a domain. So every nilpotent element is not in Z(A). Since A is not a domain, there are 0 6= x, y ∈ A such that xy = 0. Since A is prime, yAx 6= 0. Let f = yax 6= 0 for some a. Then f 2 = 0. By part (i) and the assertion we just proved, A is not LND-rigid. (iii) This follows from part (ii). t u Remark 20.6.9. (i) From Lemma 20.6.7 we get that there are noncommutative domains that are retractable. The lemma also provides examples that are strongly retractable. We refer to papers [52], [82], [83] for examples that are strongly LND-rigid or strongly LNDH -rigid. (ii) On the other hand, there are noncommutative domains that are not cancellative, and hence, not retractable: in fact, if R is a commutative domain that is not cancellative (for example, not every commutative domain over C of Gelfand–Kirillov dimension 2 is cancellative, see [104] for a concrete counterexample), then taking any noncommutative C-algebra with trivial center (for example the Weyl algebra A1 (C)) we have that R ⊗ C is not cancellative: since R is not cancellative, there exists a commutative C-algebra B such that R[t] ∼ = B[s] with R  B; note that (R ⊗ C)[t] ∼ = R[t] ⊗ C ∼ = B[s] ⊗ C ∼ = (B ⊗ C)[s], but R ⊗ C  B ⊗ C: if R ⊗ C ∼ = B ⊗ C, then taking the center we get Z(R) ⊗ Z(C) ∼ = Z(B) ⊗ Z(C), i.e., R = R ⊗ C ∼ = B ⊗ C = B, a contradiction. A similar example can be given in positive characteristic.

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Next we define new invariants involving the center. Definition 20.6.10. Let A be an algebra. (i) The Makar-Limanov center of A is defined to be MLZ (A) := ML(A) ∩ H I I Z(A). Also, MLH Z (A) := ML (A)∩Z(A) and MLZ (A) := ML (A)∩Z(A). We say that A is LN DZ -rigid if MLZ (A[t]) = Z(A), and A is strongly H LN DZ -rigid if MLZ (A[t1 , . . . , tn ]) = Z(A) for all n ≥ 1. LN DZ I H rigidity, LN DZ -rigidity, strongly LN DZ -rigidity and strongly I LN DZ -rigidity are similarly defined. (ii) A is Z-retractable if for any algebra B, an algebra isomorphism φ : A[t] ∼ = B[s] implies that φ(Z(A)) = Z(B). A is strongly Z-retractable if for any algebra B and every n ≥ 0, an algebra isomorphism φ : A[t1 , . . . , tn ] ∼ = B[s1 , . . . , sn ] implies that φ(Z(A)) = Z(B). Note that we use A[t] (instead of A) in the definition of LNDZ -rigidity, which is slightly different from the LND-rigidity in Definition 20.4.4. It is obvious that R ⇒ ZR, SR ⇒ SZR. Proposition 20.6.11. For any algebra A, MLZ (A) ⊇ ML(Z(A)). Proof. Let δ ∈ LND(A), then δ induces a locally nilpotent derivation of Z(A) since if a ∈ Z(A), then δ(ax) = δ(xa) for any x ∈ A, so δ(a)x + aδ(x) = δ(x)a + xδ(a), i.e., δ(a)x = xδ(a). Hence, ML(A) ⊇ ML(Z(A)), so ML(A) ∩ Z(A) ⊇ ML(Z(A)) ∩ Z(A) = ML(Z(A)). t u Lemma 20.6.12. Let A be an algebra such that Z(A) is a finitely generated domain of finite GK-dimension. (i) If MLZ (A[t]) = Z(A), i.e., if A is LNDZ -rigid, or MLH Z (A[t]) = Z(A), i.e., if A is LNDH -rigid, then A is Z-retractable. Z (ii) If A is strongly LNDZ -rigid or strongly LNDH Z -rigid, then A is strongly Z-retractable. Proof. (i) We will prove only the case MLZ , the proof for MLH Z is similar. Let φ : A[t] ∼ B[s] be an isomorphism; by the hypothesis, Z(A) = MLZ (A[t]), so = applying the first part of the proof of Theorem 20.4.10, we have φ(Z(A)) = φ(ML(A[t]) ∩ Z(A[t])) = φ(ML(A[t])) ∩ φ(Z(A[t])) = ML(B[t]) ∩ Z(B[t]), and from Lemma 20.4.8, we get φ(Z(A)) ⊆ ML(B)∩Z(B)[t] ⊆ B ∩Z(B)[t] ⊆ Z(B). Now we can adapt the final part of the proof of Theorem 20.4.10. (ii) This proof is similar, changing A[t] to A[t1 , . . . , tn ] and B[s] to B[s1 , . . . , sn ]. t u Now we introduce another notion closely related to the cancellation property. If B is a subring of C and f1 , . . . , fm are elements of C, then the subring generated by B and the set {f1 , . . . , fm } is denoted by B{f1 , . . . , fm }.

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Definition 20.6.13. An algebra A is called detectable if for any algebra B, an algebra isomorphism φ : A[t] ∼ = B[s] implies that B[s] = B{φ(t)}, or equivalently, s ∈ B{φ(t)}. A is called strongly detectable if for any algebra B and any n ≥ 0, an algebra isomorphism φ : A[t1 , . . . , tn ] ∼ = B[s1 , . . . , sn ] implies that B[s1 , . . . , sn ] = B{φ(t1 ), . . . , φ(tn )}, or equivalently, s1 , . . . , sn ∈ B{φ(t1 ), . . . , φ(tn )}. In the above definition, we do not assume that φ(t) = s. It is clear that SD ⇒ D. The polynomial ring K[x] is cancellative, but not detectable: as above, we φ

take K[x][t] ∼ / K[t]{t} = K[t]. = K[t][x], φ(t) = t, x ∈ Lemma 20.6.14. If A is Z-retractable (respectively, strongly Z-retractable), then it is detectable (respectively, strongly detectable). Proof. We only prove the strongly version. The proof of the non-strongly version is similar. Let B be any algebra such that φ : A[t] → B[s] is an isomorphism. Since A is strongly Z-retractable, φ restricts to an isomorphism φ|Z(A) : Z(A) → Z(B). Write fi := φ(ti ) for 1 ≤ i ≤ n. Then Z(B){f1 , . . . , fn } = φ(Z(A)){φ(t1 ), . . . , φ(tn )} = φ(Z(A){t1 , . . . , tn }) = φ(Z(A)[t]) = φ(Z(A[t])) = Z(B[s]) = Z(B)[s]. Then, for every i, si ∈ Z(B)[s] = Z(B){f1 , . . . , fn } ⊆ B{f1 , . . . , fn }, as desired. t u Thus, R ⇒ ZR ⇒ D, SR ⇒ SZR ⇒ SD.

(20.6.1)

Definition 20.6.15. Let A be an R-algebra. (i) We say that A is Hopfian if every R-algebra epimorphism from A to itself is an automorphism. (ii) We say that A is strongly Hopfian if A[t1 , . . . , tn ] is Hopfian for every n ≥ 0. It is clear that SH ⇒ H. In the previous definition, the Hopfian property is dependent on the base ring R. Hopfian algebras have been studied by many authors (see [53], [210], [235], [302]). Every left noetherian algebra is strongly Hopfian, see the next lemma. Some non-noetherian examples can beLconstructed using Lemma ∞ 20.6.16 (ii)–(iii). An N-graded R-algebra A = i=0 Ai is called locally finite if each homogeneous component Ai is a finitely generated R-module.

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Lemma 20.6.16. The following algebras are strongly Hopfian. (i) Left (right) noetherian algebras. (ii) Finitely generated locally finite N-graded R-algebras that are R-flat. (iii) Prime affine algebras satisfying a polynomial identity. Proof. (i) Let A be left (right) noetherian, then A is Hopfian. Since A[t1 , . . . , tn ] is also left (right) noetherian, A is strongly Hopfian. (ii) If A is an affine locally finite N-graded algebra, so is A[t]. Thus, it suffices to show that A is Hopfian. Replacing A by its localization A ⊗R K, where K is the field of fractions of R, we can assume that R is a field K. In fact, note that the R-flatness insures that A ⊗R K being K-Hopfian implies that A is R-Hopfian: let α : A → A be a surjective homomorphism of Ralgebras, then α ⊗ iK : A ⊗ K → A ⊗ K is a surjective homomorphism of K-algebras and hence 0 = ker(α ⊗ iK ) = ker(α) ⊗ K = ker(α)S −1 , where S := K − {0}, so if x ∈ ker(α), then there exists a u ∈ K − {0} such that xu = 0, but since A is R-flat, it is torsion-free, whence x = 0. Thus, ker(α) = 0, and α is an automorphism. Next we assume that R = K and prove that A is K-Hopfian by contradiction. Suppose that φ : A → A is a surjective endomorphism of A that has s a nonzero kernel. PsFix 0 6= r ∈ ker φ. Let r ∈ ⊕i=0 Ai , for some integer s ≥ 0, and write r = i=0 ri where some ri ∈ Ai are nonzero. Choose a K-linear basis b := {bj }dj=1 of ⊕si=0 Ai of homogeneous elements so that b contains all nonzero ri ’s. We use S to denote subrings of K of the special form Z{f1 , . . . , fw } when char(K) = 0 or of the special form Fp {f1 , . . . , fw } when char(K) = p > 0. Since A is finitely generated, A is generated by a finite set of homogeneous generators, say g := {gt }dt=1 . We may assume that g is K-linearly independent and contains b. Then there is a subring S ⊆ K of the form specified as above such that φ restricts to a surjective S-algebra endomorphism of AS , where AS denotes the S-subalgebra S{g1 , . . . , gd } of A generated by g. Adding only finitely many new fi to S if necessary we can assume that every product of any two generators gt1 , gt2 ∈ g has coefficients in S in terms of the basis b if such a product has degree no more than s. By the choice of b and g, it is clear that r ∈ AS and that ⊕si=0 (AS )i is a finitely generated free S-module with S-basis b. By the construction of S, every simple factor ring F of S is a finite field. The induced map φS ⊗ F : AS ⊗S F → AS ⊗S F is still a surjective F -algebra endomorphism. Since F is finite and AS ⊗S F is locally finite over F , AS ⊗S F is residually finite in the sense that the ideals of finite index in ring have a trivial intersection. By [235, Theorem 3], AS ⊗S F is Hopfian (and then F -Hopfian). This yields a contradiction because φS ⊗S F is a surjective endomorphism such that φS ⊗S F (r ⊗S 1) = 0, but r ⊗S 1 6= 0. Therefore the assertion follows. (iii) See [53], Corollary 2.3. t u

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Lemma 20.6.17. Suppose A is strongly Hopfian. (i) If A is detectable, then A is cancellative. (ii) If A is strongly detectable, then A is strongly cancellative. Proof. We only prove (ii) since (i) is a particular case of (ii). Let φ : A[t] → B[s] be an isomorphism. Let fi := φ(ti ) for 1 ≤ i ≤ n. Then fi are central elements in B[s]. Thus B{f1 , . . . , fn } is a homomorphic image of B[s1 , . . . , sn ] by sending si 7→ fi . Suppose A is strongly detectable. Then B{f1 , . . . , fn } = B[s]. Then we have algebra homomorphisms π = B[s] − → B{f1 , . . . , fn } − → B[s] ∼ = A[t].

(20.6.2)

Since A is strongly Hopfian, A[t] and then B[s] is Hopfian. Now (20.6.2) implies that π is an isomorphism. As a consequence, B{f1 , · · · , fn } = B[f1 , . . . , fn ] considering fi as central indeterminants in B[f1 , . . . , fn ]. Now we have π A∼ → B[f1 , . . . , fn ]/hf i ∼ = A[t]/hti − = B. t u

Therefore A is strongly cancellative. Thus, SH + D ⇒ C, SH + SD ⇒ SC.

(20.6.3)

One advantage of considering the detectable property is the following. Lemma 20.6.18. Let A be an algebra with center Z(A). If Z(A) is (strongly) detectable, so is A. Proof. The proof of the strongly case includes the proof of the detectable case. Suppose B is an algebra such that φ : A[t] ∼ = B[s]. Taking the center, we have φ : Z(A)[t] ∼ Z(B)[s]. Since Z(A) is strongly detectable, si ∈ = Z(B){φ(tj )} for all i. Thus si ∈ B{φ(tj )} for all i. This means that A is strongly detectable. t u Another property concerning the detectable property is the following. Lemma 20.6.19. Let A be an algebra and J be the Jacobson radical of A, which is nilpotent. If A/J is (strongly) detectable, so is A. Proof. Let φ : A[t] → B[s] be an isomorphism. Since J m = 0 for some m, the Jacobson radical of A[t] is J[t], which is also nilpotent (Proposition 20.6.1). Therefore, Rad(B[s]) is nilpotent, and applying again Proposition 20.6.1 we get that this radical is of the form J(B)[s], where J(B) is the Jacobson radical of B, moreover, J(B) is also nilpotent. Modulo the radical, we obtain that φ0 :

(A/J)[t] ∼ = (B/J(B))[s],

where t and s are images of t and s in appropriate algebras respectively, but keeping the same notation. Since A/J is strongly detectable, si ∈

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(B/J(B)){f1 , . . . , fn } for all i, which means that si ∈ B{f1 , . . . , fn } modulo J(B)[s]. Here fi := φ(ti ) for all i. Another way of saying this is, for every i, X fi = si + bd1 ,...,dn sd11 · · · sdnn ∈ B{f1 , . . . , fn } for all bd1 ,...,dn ∈ J(B). The point is that si = fi modulo J(B)[s]. Now we re-write si as a polynomial in fj with coefficients in J(B), starting with, X si = fi − bd1 ,...,dn sd11 · · · sdnn X X d0 d0 = fi − bd1 ,...,dn f1d1 · · · fndn + b0d01 ,...,d0n s11 · · · snn , where b0d0 ,...,d0 are in J(B)2 [s]. This means that si equals a polynomial of n 1 f1 , . . . , fn modulo J(B)2 [s]. By induction, si equals a polynomial of f1 , . . . , fn modulo J(B)p [s] for every p ≥ 1. Since J(B) is nilpotent, si is a polynomial of f1 , . . . , fn when taking p  0. Therefore si ∈ B{f1 , . . . , fn }, as required. t u We also mention an easy consequence of Lemma 20.6.18 and Theorem 3.3 of [1]. Proposition 20.6.20. Let A be a K-algebra. If the center of A is a finitely generated domain of GK-dimension one that is not isomorphic to K[x], then A is strongly detectable. Proof. By Theorem 3.3 of [1], Z(A) is strongly retractable. By Lemma 20.6.14, Z(A) is strongly detectable. The assertion follows from Lemma 20.6.18. t u Now we pass to prove the main results of the present section. Theorem 20.6.21. If A is left (or right) artinian, then A is strongly detectable. As a consequence, A is strongly cancellative, and hence cancellative. Proof. Since A is noetherian and then strongly Hopfian (Lemma 20.6.16 (i)), the strongly cancellative property is a consequence of the strongly detectable property by Lemma 20.6.17 (ii). Let J be the Jacobson radical of A. Then J is nilpotent. By Lemma 20.6.19, it suffices to show that A0 := A/J is strongly detectable. Since A0 is a finite direct sum of matrix algebras over division rings, Z(A0 ) is a finite direct sum of fields. By Lemma 20.6.6, Z(A0 ) is strongly retractable, but since Z(A0 ) is commutative, it is strongly Z-retractable, so by Lemma 20.6.14, Z(A0 ) is strongly detectable. By Lemma 20.6.18, A0 is strongly detectable, as required. t u Theorem 20.6.22. Let A be an algebra with center Z. Suppose J is the prime radical of Z such that (a) J is nilpotent and (b) Z/J is a finite direct sum of fields. (i) A is strongly detectable.

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(ii) If further A is strongly Hopfian, then A is strongly cancellative, and hence cancellative. Proof. (i) Similar to the proof of Theorem 20.6.21, Z is strongly detectable. By Lemma 20.6.18, A is strongly detectable. (ii) Follows from Lemma 20.6.17 and part (1). t u Theorem 20.6.23. Suppose A is strongly Hopfian and the center of A is artinian. Then A is strongly detectable and strongly cancellative, and hence, cancellative. Proof. This is an immediate consequence of the previous theorem.

t u

Corollary 20.6.24. Let A be a left (right) noetherian algebra such that its center is artinian. Then A is cancellative. Corollary 20.6.25. Any finite-dimensional K-algebra is cancellative. Related to the implication (20.6.1) we have the following counterexample. Example 20.6.26. Let A = K[x, y]/(x2 , y 2 , xy, 0). By Theorem 20.6.21 A is strongly detectable. But A is not retractable: in fact, consider the isomorphism φ : A[t] → A[s] where φ(x) = x, φ(t) = s and φ(y) = y + sx. Remark 20.6.27. Some other important classes of cancellative algebras were considered in [50], [255], [385]: (i) Let K be a field and Q be a finite quiver. Then, the path algebra KQ is cancellative. (ii) Let K be a field with char(K) = 0. Then, any finitely generated domain algebra of GKdim one is cancellative. In [50] one can find counterexamples when char(K) = p > 0. (iii) If K is an algebraic closed field, then Mn (K[x]) is strongly cancellative, and hence, cancellative. (iv) Let A and B be two algebras, (a) If A and B are (strongly) cancellative, so is A ⊕ B. (b) If A and B are (strongly) retractable, so is A ⊕ B. (c) If A and B are (strongly) detectable, so is A ⊕ B. We only prove part (a). The proof of the other parts is similar. Suppose φ : (A⊕B)[t] ∼ = C[s] is an algebra isomorphism, and let e1 and e2 be two counital central idempotents (i.e., 1 = e1 + e2 ) corresponding to the decomposition A⊕B. Let fi := φ(ei ) for i = 1, 2. By part (iv) of Lemma 20.6.3, f1 and f2 are two counital central idempotents of C. Thus C = C1 ⊕C2 , C[s] = C1 [s]⊕C2 [s] and hence A[t] ∼ = C1 [s] and B[t] ∼ = C2 [s]: indeed, let T := (A ⊕ B)[t] = A[t] ⊕ B[t] and S := C1 [s] ⊕ C2 [s] = C[s], then A[t] = e1 T and B[t] = e2 T , whence φ(A[t]) = φ(e1 )φ(T ) = f1 S = C1 [s], i.e., A[t] ∼ = C1 [s]; similarly, B[t] ∼ = C2 [s]. Since A and B are strongly cancellative, A ∼ = C1 and B ∼ = C2 , so A ⊕ B ∼ = C1 ⊕ C2 = C. The proof of the non-“strongly” version is similar.

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(v) Let K be a field with char(K) = 0 and A be a noetherian connected graded AS algebra generated in degree 1 with gld(A) = 3. If A is not P I, then A is cancellative. (vi) Let A be a prime noetherian connected graded algebra finitely generated with gld(A) = 3. If A is not P I, then A is cancellative. (vii) Let K be a field with char(K) = 0 and A be a K-algebra. Suppose that A satisfies the following conditions: A is graded generated in degree 1, A is connected, A is noetherian, A is a domain, gld(A) < ∞, gld(A/hti) = ∞ for every homogeneous element t ∈ Z(A) of positive degree, and GKdim(Z(A)) ≤ 1. Then A is strongly cancellative, and hence, cancellative.

20.7 The Zariski Cancellation Problem for Rings Although many of the important results about the cancellation problem are for algebras over fields (for example, Theorem 20.2.1 and Corollary 20.6.25), the general formulation of this problem is for R-algebras, where R is an arbitrary commutative domain (Definition 20.1.1). Hence, the general definition includes the algebras over fields as well as the particular case of Z-algebras, i.e., arbitrary rings. Moreover, recall that the early works on the cancellation property were about isomorphic polynomial rings, i.e., about Z-cancellation, see [1], [68] and [101]. For a fixed R, let A be an R-algebra. Note that if A is a cancellative ring then A is not necessarily a cancellative R-algebra. In turn, if A is cancellative as an R-algebra, A is not necessarily a cancellative ring. Thus, the cancellation property depends on R (see also [1]). It is clear that all results about cancellation proved for arbitrary R-algebras are valid for rings (as well as for K-algebras), even those where the condition char(R) = 0 is assumed. In this section we will concentrate our attention on cancellative rings. Definition 20.7.1. Let A be an arbitrary ring, we say that A is cancellative if A is cancellative as a Z-algebra. The conditions SC, R, SR, ZR, SZR, D, SD, H and SH, for rings, are defined similarly. We will examine the cancellation property for some classical ring-theoretic algebraic constructions, and also, for some special types of rings, namely, for semiperfect, uniquely clean and centrally clean rings. We have the following elementary and known facts. Example 20.7.2. (i) Let R be a commutative domain with char(R) = 0, then R as R-algebra is cancellative: this follows from Theorem 20.4.10 and Proposition 20.4.7 (or Theorem 20.4.11 and Remark 20.4.5) since GKdim(R) = GKdim(Q(R) ⊗ R) = 0, where Q(R) is the field of fractions of R. Thus, Z is a cancellative ring. (ii) If R is a Dedekind domain, then R[x] as an R-algebra is SC ([1], Corollary 4.7). In particular, Z[x] is an SC ring, and hence, a cancellative ring. Let K be a field. We know that K[x] is cancellative as a K-algebra;

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Corollary 3.4 in [1] shows that K[x] is also a cancellative ring. Theorem 6.5 in [1] shows that every Dedekind domain containing a field of characteristic zero is SC, and hence, a cancellative ring. (iii) If R is a commutative domain with Rad(R) 6= 0, then R is an SR ring (see 1.10 in [1]). Hence, R is cancellative. For example, if P is a prime ideal of Z, then the local ring ZP is cancellative. (iv) Theorem 20.6.21 implies that any artinian ring is cancellative; in particular, division rings and finite rings are cancellative. Thus, Zn , for n ≥ 2, is cancellative. On the other hand, related to Theorem 20.6.22, in [386] it is shown that if A is a K-algebra such that either Z(A) or Z(A)/rad(Z(A)) is SR, then A is SC, and hence cancellative. Related to Theorem 20.2.1, in [101], Corollary 4, it is proved that rings whose center is a field are cancellative. More generally, related to Theorem 20.6.23, in [68], Theorem 2,it is proved that if A is a ring such that Kdim(Z(A)) = 0, then A is SC, and hence, A is cancellative. Thus, if A is a ring with artinian center, then A is cancellative. From this we get that if Z(A) is artinian, then Mn (A) is cancellative, and in addition, since the product of cancellative rings is cancellative (Remark 20.6.27), semisimple rings are cancellative. Another application of cancellation for rings with artinian center is the following: let S be a commutative artinian ring that is not a field (for example, S := K × K, where K is a field with char(K) = 0); by Proposition 3.3.1, the Weyl ring An (S) satisfies Z(An (S)) = Z(S) = S. Hence, An (S) is a cancellative ring. Another example of this type is S ⊗K U (G), where G is as in Proposition 3.3.3, (i) or (ii). (v) In [68], Corollary 1 and 2, it is shown that von Neumann regular rings and a finite product of local rings are SC, and hence, cancellative rings. Recall that a ring A is a von Neumann regular ring if every cyclic right (left) ideal is generated by an idempotent; this condition is equivalent to the following: given a ∈ A there exists an a0 ∈ A such that aa0 a = a. (vi) A ring A is perfect if every module over A has a projective cover. This condition is equivalent to Rad(A) being T -nilpotent and A/Rad(A) being semisimple ([199]). In particular, semiprimary rings are perfect (A is semiprimary if Rad(A) is nilpotent and A/Rad(A) is semisimple, [218]). Perfect rings are cancellative (see [101]). (vii) If A is a cancellative algebra and B ⊇ A is an extension of A, then B is not always cancellative: indeed, take B := K[x, y, z] with char(K) = p > 0 and A := K. (viii) If A is a cancellative algebra and B is a subalgebra of A, then B is not always cancellative: take B := K[x, y, z], with char(K) = p > 0 and A := K(x, y, z) the field of fractions of B. (ix) If A is a cancellative algebra and I is a proper two-sided ideal of A, then A/I is not always cancellative: in fact, if A := K{X} is the free algebra in the alphabet X, then Z(A) = K, so A is cancellative, but K[x, y, z], with char(K) = p > 0 is not cancellative. (x) The tensor product of cancellative algebras is not always cancellative: in fact, K[x] ⊗ K[x, y] ∼ = K[x, y, z], with char(K) = p > 0.

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(xi) Let S be a multiplicative system of A such that S −1 A exists; if S −1 A is a cancellative algebra, this does not always imply that A is cancellative: take A := K[x, y, z] with char(K) = p > 0 and AS −1 the field of fractions of A. (xii) If A is cancellative algebra and σ ∈ Aut(A) then A[x; σ] is not always cancellative: take A := K[x, y], with char(K) = p > 0, then A is cancellative but K[x, y, z] = A[z; iA ] is not cancellative. A ring A is semiperfect if every finitely generated module over A has a projective cover (see Definition 7.2.10 in [64]). Thus, perfect rings are semiperfect (see also [199], Corollary 11.6.2). The following theorem extends a result in [101] on cancellation of perfect rings. Theorem 20.7.3. The following rings are cancellative: semiperfect, uniquely clean and centrally clean. Proof. (i) Semiperfect rings. Let A be a semiperfect ring, then there exists a finite set {e1 , . . . , en } of orthogonal idempotents of A such that 1 = e1 + · · · + en and ei Aei is a local ring for every 1 ≤ i ≤ n (see [286] or also [64], Theorem 7.2.23). From this it can be proved (see for example [66]) that A is isomorphic to the ring of matrices   e1 Ae1 e1 Ae2 · · · e1 Aen  .. .. .. ..   . . . .  en Ae1 en Ae2 · · · en Aen endowed with the habitual operations of sum and product of matrices. From this it is easy to show that Z(A) ∼ = e1 Ae1 ⊕ · · · ⊕ en Aen . Theorem 3 in [68] says that A is cancellative. (ii) Uniquely clean rings. Recall that a ring A is clean if every element a ∈ A is the sum of an idempotent e ∈ A and an invertible u ∈ A∗ , a = e + u; if e is unique for a (and hence, u is also unique for a), it is said that A is uniquely clean (see [293]). Let A be a uniquely clean ring. Since every idempotent in A is central (see Lemma 1.2 in [355]), Z(A) is also uniquely clean. From this we get that Z(A)/rad(Z(A)) is clean and reduced (i.e., with trivial nilradical). If we prove that B := Z(A)/rad(Z(A)) is SR, then from Theorem 1 in [68], A is SC, and hence, A is cancellative. Let C be a ring with isomorphism φ ∼ C[s]. B[t] = If we prove that φ(B) ⊆ C, then from Lemma 2 in [68] we get that φ(B) = C, i.e., B is SR. Let b ∈ B, then there exist an idempotent e ∈ B and u ∈ B ∗ such that b = e+u, therefore φ(b) = φ(e)+φ(u), where φ(e) is an idempotent of C[s] and φ(u) is invertible in C[s]. Since C is commutative and reduced, φ(e) ∈ C and φ(u) ∈ C ∗ , i.e., φ(b) ∈ C. (iii) Centrally clean rings. Let A be a centrally clean ring, i.e., Z(A) is clean, then, as in (ii), Z(A)/rad(Z(A)) is clean and reduced, and hence, A is cancellative. t u

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20 The Zariski Cancellation Problem for Skew P BW Extensions

Remark 20.7.4. In the previous proof we used Lemma 2 in [68] which concerns a weaker version of the retractable property for rings. Compare with Remark 20.6.5 for the case of algebras over fields.

20.8 Skew P BW Cancellation The classical cancellation property has been extended to skew polynomial rings of derivation type and endomorphism type in [18] and [50], and more recently, to iterated skew polynomial rings in [386]. In this section we introduce the following natural extension of the skew cancellation given in [386]. Definition 20.8.1. Let R be a commutative domain and T be an R-algebra. T is skew PBW cancellative if for any R-algebra S and any n ≥ 1, any φ algebra isomorphism σ(T )ht1 , . . . , tn i ∼ = σ(S)hs1 , . . . , sn i implies that T ∼ = S, where σ(T )ht1 , . . . , tn i and σ(S)hs1 , . . . , sn i are bijective skew P BW extensions that are R-algebras. Taking R = K or R = Z we get the notions of skew P BW cancellative algebras over fields and skew P BW cancellative rings, respectively. We will prove in this section an analog of Theorem 0.10 of [386] about strongly skew cancellation to the case of skew P BW cancellation. Observe that iterated skew polynomial rings of injective type are particular cases of skew P BW extensions. As in [386], for the proof of our main theorem, some preliminary lemmas are needed. In the first two lemmas the divisor subalgebras of Definition 20.5.8 are used. To emphasize that the divisor subalgebra D(1) is with respect to the algebra A, we will write DA (1). Lemma 20.8.2. Let T be a domain that is an R-algebra and let A := σ(T )ht1 , . . . , tn i be a skew P BW extension of T that is an R-algebra. Then, DA (1) = DT (1). Proof. Since T is a domain, A is a domain (Proposition 3.2.1), hence SwA (1) = SwT (1), and from this, DA (1) = DT (1). t u φ

Lemma 20.8.3. Let T and S be domains that are R-algebras. If A ∼ = B is an algebra isomorphism, where A := σ(T )ht1 , . . . , tn i and B := σ(S)hs1 , . . . , sn i are skew P BW extensions, then φ(DA (1)) = DB (1) ⊆ S. Proof. Since φ is bijective, φ(DA (1)) = DB (φ(1)) = DB (1), but from the previous lemma, DB (1) = DS (1) ⊆ S. t u Lemma 20.8.4. Let Y be an N-filtered R-algebra with filtration {Fp (Y )}p≥0 such that the graded algebra Gr(Y ) is a domain. Let Z be a subalgebra of Y and let Z0 := Z ∩ F0 (Y ). If GKdim(Z) = GKdim(Z0 ) < ∞, then Z = Z0 .

20.8 Skew P BW Cancellation

459

Proof. The proof is exactly as in Lemma 3.2 in [386], but using Theorem 20.3.4. Consider the induced filtration on Z, Fp (Z) := Z ∩ Fp (Y ), p ≥ 0, so Gr(Z) is a subalgebra of Gr(Y ), and hence, Gr(Z) is a domain. On the other hand, GKdim(Z) = GKdim(Q ⊗ Z) ≥ GKdim(Gr(Q ⊗ Z)) = GKdim(Q ⊗ Gr(Z)) = GKdim(Gr(Z)) ≥ GKdim(Gr(Z)0 ) = GKdim(F0 (Z)) = GKdim(Z0 ) = GKdim(Z). Thus, GKdim(Gr(Z)) = GKdim(Z0 ). Assume that Z0 ( Z, so there exists a nonzero element a ∈ Gr(Z) of positive degree, and hence the following sum is direct Z0 + Z0 a + Z0 a2 + · · · + Z0 an , for every n ≥ 0. Since any frame of Gr(Z)0 = Z0 is a frame of Gr(Z)), from the previous sum we get that GKdim(Gr(Z)) ≥ GKdim(Z0 ) + 1, a contradiction. Therefore, Z0 = Z. t u Lemma 20.8.5. Let T be an R-algebra that is a domain. Let A := σ(T )ht1 , . . . , tn i be a skew P BW extension of T that is an R-algebra. If A0 is a subalgebra of A containing T such that GKdim(T ) = GKdim(A0 ) < ∞, then A0 = T . Proof. We will follow the proof of Proposition 3.3 in [386], but considering the properties of the GKdim for R-algebras (Theorem 20.3.4). Let m ≤ n be minimal such that A0 is contained in the subalgebra Tm generated by T and x1 , . . . , xm . The idea is to prove that m = 0, if so, A0 = T since T0 = T . Suppose that m ≥ 1 and let Tm−1 the subalgebra of A generated by T and x1 , . . . , xm−1 . Tm is N-filtered with filtration given by Pp Fp (Tm ) := i=0 Tm−1 xim , p ≥ 0. Let Z := A0 and Z0 := Z ∩ Tm−1 . Since m is minimal, Z 6= Z0 , so, from the hypothesis GKdim(Z0 ) ≥ GKdim(T ) = GKdim(A0 ) = GKdim(Z) ≥ GKdim(Z0 ). From the previous lemma, Z = Z0 , false. Therefore, m = 0 and A0 = T .

t u

Our main theorem about skew P BW cancellation is for a special class of algebras. We do not need to assume that these algebras are noetherian (compare with Theorem 0.10 in [386]). Definition 20.8.6. Let R be a commutative domain. We say that an Ralgebra S is frame stable if S is a domain finitely generated as an R-algebra and has a generator frame V such that σ(V ) ⊆ V for all σ ∈ Aut(S), where Aut(S) is the group of algebra automorphisms of S. Theorem 20.8.7. Let T be a frame stable algebra such that DT (1) = T . Then, T is skew P BW cancellative in the collection of frame stable algebras.

460

20 The Zariski Cancellation Problem for Skew P BW Extensions

Proof. We will follow the proof of Theorem 0.10 in [386]. Let A := σ(T )ht1 , . . . , tn i and B := σ(S)hs1 , . . . , sn i be bijective skew P BW extensions, with S frame stable, and let φ

σ(T )ht1 , . . . , tn i ∼ = σ(S)hs1 , . . . , sn i be an algebra isomorphism. From Theorem 20.3.4, GKdim(A) = GKdim(T )+ n and GKdim(B) = GKdim(S) + n. By the hypothesis and Lemmas 20.8.2 and 20.8.3, φ(T ) = φ(DT (1)) = φ(DA (1)) = DB (1) ⊆ S. Let A0 := φ−1 (S). Hence, T ⊆ A0 and GKdim(A0 ) = GKdin(S). Therefore, GKdim(A0 ) = GKdim(S) = GKdim(B) − n = GKdim(A) − n = GKdim(T ). By Lemma 20.8.5, A0 = T , whence φ(T ) = S, i.e., T ∼ = S.

t u

An easy corollary of the previous theorem is the following result about isomorphisms of skew P BW extensions. Corollary 20.8.8. Let R be a commutative domain and σ(R)ht1 , . . . , tn i be a bijective skew P BW extension of R that is an R-algebra. If S is a stable frame R-algebra and σ(S)hs1 , . . . , sn i is a bijective skew P BW extension of S that is an R-algebra, with an algebra isomorphism φ

σ(R)ht1 , . . . , tn i ∼ = σ(S)hs1 , . . . , sn i, then S ∼ = R. Proof. This follows from Theorem 20.8.7 since R is frame stable and DR (1) = R. t u We conclude this section by presenting some examples of frame stable algebras and computing its divisor algebra. Example 20.8.9. (i) R and the algebra of commutative polynomials R[x] are frame stable. Since DR (1) = R, from Lemma 20.8.2 we get that DR[x] (1) = R. (ii) Let K be a field and A := Kq [x, y] be the quantum plane, where q is not a root of unity. Then A is frame stable since given σ ∈ Aut(A) there exist λ, γ ∈ K ∗ such that σ(x) = λx and σ(y) = γy (see [24], or also [312]). From Lemma 20.8.2, DA (1) = DK (1) = K. (iii) Let K be a field and A := Kq [x1 , . . . , xn ] be the K-algebra of quantum polynomials (also called the n-multiparametric quantum space, see Section 4.4). If n ≥ 3 and A is generic, i.e., for 1 ≤ i < j ≤ n, the constants qij are independent in K ∗ , then A is frame stable since given σ ∈ Aut(A) there exist r1 , . . . , rn ∈ K ∗ such that σ(xj ) = rj xj , for every 1 ≤ j ≤ n (see [21], [26], [394], or also [312]). As in (ii), DA (1) = K. This example can be ±1 extended to the general case Kq [x±1 1 , . . . , xk , xk+1 , . . . , xn ] (see [26]), i.e., this algebra is frame stable. Moreover, we know that it is a skew P BW ±1 extension of the quantum torus Kq [x±1 1 , . . . , xk ] (see Example 4.4.6), hence ±1 ±1 DA (1) = DKq [x±1 ,...,x±1 ] (1) = Kq [x1 , . . . , xk ]. Observe that the quantum 1 k torus satisfies the hypothesis of Theorem 20.8.7.

20.8 Skew P BW Cancellation

461

(iv) Let Oq (Mm,n ) be the quantization of the ring of regular functions on m × n matrices with entries in C, where q is a nonzero complex number that is not a root of unity ([222]). This C-algebra is a domain generated by m × n variables Yi,α , 1 ≤ i ≤ m and 1 ≤ α ≤ n, subject to the following relations: Yi,β Yi,α = q −1 Yi,α Yi,β , −1

α 1. F1 =   0  x y−1 y2 − 1 

 F0 =

1 xy y 2 0 , 1 0 0 x 

Therefore, a free resolution for S is given by −xy −xy 2 − 2xy   1 2      1 xy y 2 0  0 x   1 0 0 x y−1 y2 − 1 −−−−−−−−−−−−−−−−→ A4 −−−−−−−−−−→ M −−−−−→ 0. 



     

0 −−−−−→ A2

Compare with Example 15.6.2. Example C.2.6. Considering the diffusion algebra A := σ(Q[x1 , x2 ])hx, yi, subject to the relation D2 D1 = 2D1 D2 +x2 D1 −x1 D2 , we will calculate a free resolution for the left ideal I := hx21 x2 D12 D2 , x22 D1 D22 }. Using the statement FreeResolution(M, gradlex, TOP, A) for the matrix

 M :=

x21 x2 D12 D2 x22 D1 D22



we get [F0 , F1 ], where     x D D − 3x22 D1 + x1 x22 F0 = x21 x2 D12 D2 x22 D1 D22 , F1 = 2 1 2 2 2 −4x1 D1 + 4x31 D1 − x41 and Fr = 0 for r > 1. Therefore, a free resolution for I is given by 

 x D D − 3x22 D1 + x1 x22   2 1 2 −4x21 D12 + 4x31 D1 − x41

0 −−−−−→ A −−−−−−−−−−−−−−−−−−−−−−→

h

A2

i

x21 x2 D12 D2 x22 D1 D22 −−−−−−−−−−−−−−−−−→ M −−−−−→ 0.

Example C.2.7. We consider the Witten algebra A := σ(Q)hx, y, zi subject to the relations zx = xz − x,

zy = z + 2y

yx = −xy.

C Implementation of Skew P BW Extensions With Maple

528

We will calculate a free resolution for the left module S := h(1, x + y), (0, z), (y, xz)i. Using the statement FreeResolution(M, gradlexrev, TOP, A) for the matrix   1 x+y M := 0 z  y xz we get [F0 , F1 , F2 ], where

 F0 =

 1 0 y , x + y z xz

 F2 =

 z−3 , −x − y

F1 =



yz 2 − yz − 2y

xyz + y 2 z + xy − 2y 2



x2 y + x2 z − 2xy 2 − xyz − y 3 − 2x2 + 5xy xyz + xz 2 − y 2 z − 3xy − 7xz − 3y 2 + 10x −z 2 + 5z − 4

−xz − yz + x + 4y

and Fr = 0 for r > 2. Thus, a free resolution for S is given by 







z−3  1 0 y  −x − y x + y z xz F 1 −−−−−−− → A2 −−−−−→ A3 −−−−−−−−−−→ M −−−−−→ 0. 0 −−−−−→ A − 

Example C.2.8. In this example we take the multiplicative analogue of the Weyl algebra A := σ(Q)hx1 , x2 , x3 i = O3 (λji ), with λ31 = 2, λ32 = 2 and λ21 = −1, subject to the relations: x3 x1 = 2x1 x3 ,

x3 x2 = 2x2 x3

x2 x1 = −x1 x2 .

We calculate a free resolution for the left module S := h(x1 , x2 + 1), (x3 , 0), (1, x2 ), (x2 , x1 x3 ), (x3 − 1, 0)i. Using the statement FreeResolution(M, gradlex, TOP, A) for the matrix   x1 x2 + 1   x3 0     x2  M :=  1  x2 x1 x3  x3 − 1 0 we get [F0 , F1 , F2 , F3 ], where   x1 x3 1 x2 x3 − 1 F0 = , x2 + 1 0 x2 x1 x3 0

C.2 Some Homological Computations −2x2 `21   2x2 + 2  0 2x1 x2 + 2x2 + 2



F1 =

529

0 2x1 x2 x3 − x2 x3 −2x1 x3 `26 `23 `22 `24 `25  0 2x1 x3 0 0 −2x1 x2 x3 + x2 x3 + 21 x3 ,  0 0 2 0 2x2 2 2 2 2 x3 4x1 x2 x3 −4x1 x3 + 2x2 4x1 x2 x3 −4x1 x2 x3 + 2x2 0

0



with `21 = −2x1 x2 − 2x2 − 2, `22 = −x3 + 1, `23 = −4x1 x2 x3 + 4x1 x2 , `24 = 4x21 x3 − 2x1 − 2x2 , `25 = −4x21 x2 x3 + 4x21 x2 and `26 = 4x21 x2 x3 − 2x22 − x2 − 12 .   0 − 14 x3 14 x3 − 14 x3 − 14 x3 − 12 x3 0 − 14 x3 m21 m22 m23 m24 m25 m26 m27 m28     −2 0 0 x1 0 0 0 0   F2 =   0 −x2 x2 −x2 −x2 −2x2 0 −x2  ,    0 1 0 0 0 0 1 0  0 1 −1 1 1 2 0 1 where m21 = 8x1 x2 , m22 = −4x21 x2 +2x1 x2 +x2 + 12 , m23 = −2x1 x2 −x2 − 12 , m24 = −4x21 x2 +2x1 x2 +x2 + 12 , m25 = 2x1 x2 +x2 + 12 , m26 = 4x1 x2 +2x2 +1, m27 == 4x21 x2 and m28 = 2x1 x2 + x2 + 12 .   0 0 12 x1 0 0  0 0 0 −1 0    0 0 1 −1 1    0 0 1 0 0  F3 =   0 1 0 0 0 ,    1 0 0 0 0    0 0 0 1 0 −2 −1 0 0 1 and Fr = 0 for r > 3. Thus, a free resolution for S is given by  0 0 1 x1 0 0 2 0 0 0 −1 0  0 0 1 −1 1  0 0 1 0 0  0 1 0 0 0  1 0 0 0 0    0 0 0 1 0 x1 x3 1 x2 x3 − 1 −2 −1 0 0 1 x 2 + 1 0 x2 x1 x 3 0 F2 F1 5 8 6 5 0 −→ A −−−−−−−−−−−−−−−−− → A − −−−−−− → A − −−−−−− → A − −−−−−−−−−−−−−−−−−−−−−−− → M −→ 0.           

C.2.3 Computing the Left Inverse of a Matrix Corollary C.2.9. Let A be a bijective skew P BW extension and F ∈ Mr×s (A) be a rectangular matrix over A. The algorithm below determines if F is left invertible, and in the positive case, it computes the left inverse of F: Next we will illustrate with examples the implementation of the algorithm in Maple. The statement that computes a left inverse matrix of a matrix M is as follows:

530

C Implementation of Skew P BW Extensions With Maple Algorithm for the left inverse of a matrix

INPUT: A rectangular matrix F ∈ Mr×s (A) OUTPUT: A matrix L ∈ Ms×r (A) satisfying LF = Is if it exists, and 0 otherwise INITIALIZATION: IF r ≤ s RETURN 0 IF r ≥ s, let G := {g 1 , . . . , g t } be a Gr¨ obner basis for the left submodule generated by rows of F and {e i }si=1 be the canonical basis of As . Use the division algorithm to verify if e i ∈ hGi for each ≤ i ≤ s. IF there exists some e i such that e i ∈ / hGi, RETURN 0 IF hGi = As , let H ∈ Mr×t (A) with the property GT = H T F , and consider K := [kij ] ∈ Mt×s , where the kij ’s are such that e i = k1i g 1 + k2i g 2 + · · · + kti g t for 1 ≤ i ≤ s. Thus, Is = K T GT RETURN L := K T H T

LeftInverseMatSkewPoly(M, ord, ORD, A). Example C.2.10. Consider the extension A := σ(K)hx, yi, subject to the relation: yx = −xy + 1. Case (K := Q). Given the matrices     1 1 1 1  xy 1  xy 0  2  (1) (2)   y x . M :=  and M :=    x2 0   x 2 1 y xy 2 y Using the statement LeftInverseMatSkewPoly(M (k) , gradlex, TOPREV, A) for k = 1, 2, we get the respective left inverse matrices   xy 2 − y y + 1 0 −xy + 1 left inverse of M (1) = , −xy 2 + y + 1 −y − 1 0 xy − 1 1 2 1  x + x + 1 y + 1 − 14 y − 14 x − 14 x − 21 0 left inverse of M (2) = 4 1 2 2 1 4 1 . − 4 x − 2 x − 4 y 14 y 14 x 41 x + 12 0 For M (1) compare with Example 17.1.3. Case (K := Z2 ). Given the matrices 

M

(3)

x y 2  y y :=   x2 −xy x+y 1



0 −x  and M (4) 1  −y



1 x + 1  :=   −xy  xy 2 y2

 x y 0 0   1 −y   y 1  1 x+y

and using a similar statement for k = 3, 4, we get the left inverse matrices

531

C.2 Some Homological Computations



left inverse of M (3)

left inverse of M (4)

(3)

m  11 (3) = m21 (3) m31  (4) m  11 = m(4) 21 (4) m31

(3)

(3)

m12 m13 (3) (3) m22 m23 (3) (3) m32 m33

 (3) m14 (3)  m24  , (3) m34

 (4) (4) (4) (4) m12 m13 m14 m15 (4) (4) (4)  (4) m22 m23 m24 m25  , (4) (4) (4) (4) m32 m33 m34 m35

where (3)

m11 = x2 y 4 + x3 y 2 + xy 4 + x2 y 2 + y 4 + x3 + y 3 + y, (3)

m12 = xy 2 + y 3 + y 2 + x, (3)

m13 = xy 4 + x2 y 2 + y 4 + xy 2 + y 3 + x2 + y 2 , (3)

m14 = xy 3 + xy 2 + y 3 + y 2 , (3)

m21 = x2 y 4 + x3 y 2 + x2 y 3 + xy 4 + x3 y + xy 3 + y 4 + xy 2 + x2 + xy + y 2 + y, (3)

m22 = xy 2 + y 3 + xy + 1, (3)

m23 = xy 4 + x2 y 2 + xy 3 + y 4 + x2 y + y, (3)

m24 = xy 3 + y 3 + y + 1, (3)

m31 = x, (3)

m32 = 0, (3)

m33 = 1, (3)

m34 = 0,

and (4)

m11 = xy 2 + xy + y 2 + y, (4)

m12 = x2 y 3 + x2 y 2 + xy 2 + y 3 + y, (4)

m13 = xy 2 + y 2 , (4)

m14 = x2 y + x2 + y + 1, (4)

m15 = xy + x + y + 1, (4)

m21 = xy 2 + xy + x + y, (4)

m22 = x2 y 3 + x2 y 2 + xy 3 + x2 y + y 3 + xy + y 2 + x + y + 1, (4)

m23 = xy 2 + x2 + xy, (4)

m24 = x2 y + x2 + xy + 1, (4)

m25 = x + y, (4)

m31 = x, (4)

m32 = x2 y + xy 2 + y 3 + xy + x + y + 1, (4) m33 (4) m34 (4) m35

= x2 + xy + y + 1, = 0, = xy + y + 1.

532

C Implementation of Skew P BW Extensions With Maple

Example C.2.11. Consider the extension σ(C)hx, y, zi := C[x, y, z; σ] subject to the relations xy = yx, xz = zx, yz = zy, with σ(λ) := λ, λ ∈ C. Given the matrices     1 + x2 z −ix i − iy 3 −ixz + 2y 2 y + i , M (1) :=  −ixz 1  and M (2) :=  −iy 2 ix3 y xy iy 3 −iy and using the statement LeftInverseMatSkewPoly(M (k) ,gradlex,TOP,A) for k = 1, 2, we get the left inverse matrices   1 ix 0 0 (1) left inverse of M = , i xz −x2 z + 1 0 " # (2) (2) (2) ` `12 `13 left inverse of M (2) = 11 (2) (2) (2) , `21 `22 `23 where (2)

`11 = −xy 3 z − ixy 2 z + iy 3 − i, (2)

`12 = ix2 y 2 z 2 + 2xy 4 z + x2 yz 2 + 3ixy 3 z + 2xy 2 z − 2iy 4 + y 3 − ixz + 2iy, (2)

`13 = −ix2 yz 2 − 3xy 3 z − x2 z 2 − 4ixy 2 z − 2xyz + 3iy 3 − xz − y 2 − 2i, (2)

`21 = −y 3 − iy 2 , (2)

`22 = ixy 2 z + 2y 4 + xyz + 3iy 3 + y 2 − i, (2)

`23 = −ixyz − 3y 3 − xz − 4iy 2 − y − 1.

Example C.2.12. Now we take the Witten algebra σ(Q)hx, y, zi subject to the relations zx = xz − x, zy = yz + 2y, and yx = 2xy. Given the matrix



x 1 M :=  y −z

 2y −z   1 0

and using the statement LeftInverseMatSkewPoly(M ,gradlex,TOP,A), we get the left inverse matrix of M  3  − 4 yz + 32 y − z + 54 − 32 y 2 − 94 x − 2y + 1 − 94 xz + 32 y + z − 15 4 xy − x + y . 1 − 38 yz + 34 y − 12 z + 58 − 34 y 2 − 98 x − y − 98 xz + 34 y + 1 − 15 8 xy − 2 x Example C.2.13. Consider the multiplicative analogue of the Weyl algebra A := σ(Q)hx1 , x2 , x3 i = O2 (λji ), with λ31 = 2, λ32 = 2 and λ21 = −1. Given the matrix

C.3 Algorithm for the Quillen–Suslin Theorem

533





x1 x2 1 −x1 x3 x3 + 1  M :=   x2 x3  1 x1 x3 and using the statement LeftInverseMatSkewPoly(M ,gradlex,TOP,A), we get the left inverse matrix of M  4 3 4 4 4 11 x1 x2 − 11 x1 x3 + 11 − 11 x1 x2 − 11 4 3 7 4 4 − 11 x1 x2 + 11 x1 x3 + 11 11 x1 x2 + 11  4 2 8 2 8 4 4 4 4 11 x1 x2 + 11 x1 x3 − 11 x1 + 11 11 x1 x2 − 11 x1 x3 − 11 x2 + 1 . 4 2 8 2 3 4 4 4 4 − 11 x1 x2 − 11 x1 x3 − 11 x1 − 11 − 11 x1 x2 + 11 x1 x3 + 11 x2

C.3 Algorithm for the Quillen–Suslin Theorem In this section we present the algorithm for computing the matrix U in the proof of the Quillen–Suslin theorem for Ore extensions (Theorem 12.6.1); the algorithm also calculates the basis of a given finitely generated projective module (Corollary 10.2.4). We present two versions of the algorithm, a constructive simplified version and a more complete computational version over fields. The computational version was implemented using Maple (see Remark C.3.6 below).

Algorithm for the Quillen–Suslin theorem Constructive version INPUT: An Ore extension A := K[x, σ, δ] (K a field, σ bijective); F ∈ Ms (A) an idempotent matrix. OUTPUT: Matrices U , U −1 and a basis X of hF i, where U F U −1 =



0 0 0 Ir

 and r = dim(hF i).

(C.3.1)

INITIALIZATION: F1 := F . FOR k from 1 to s − 1 DO 1. Follow the reduction procedures (B1) and (B2) in the proof of Theorem 12.6.1 in order to compute matrices Uk0 , Uk0−1 and Fk+1 such that Uk0 Fk Uk0−1 =







αk 0 , where αk ∈ {0, 1}. 0 Fk+1



Ik−1 0 Uk−1 ; compute Uk−1 . 0 Uk0 3. By permutation matrices modify Us−1 .

2. Uk :=

RETURN U := Un−1 , U −1 satisfying (C.3.1), and a basis X of hF i.

C Implementation of Skew P BW Extensions With Maple

534

Example C.3.1. For A := K[x, σ, δ], with K := C, σ(z) := z and δ := 0, we consider in M4 (A) the idempotent matrix    F =  

 1 − ix − x2 + (1 + i)x3 −1 + (2 − i)x2 + (−1 − i)x3 −i − x + (1 + i)x2 1 + ix + (−1 + i)x2  −ix + (1 + i)x3 ix + (1 − i)x2 + (−1 − i)x3 −i + (1 + i)x2 1 + (−1 + i)x2  .  ix2 −x − ix2 1 + ix x x 3 − x2 −ix + (1 − i)x2 − x3 x2 − x 1 + ix + ix2

We apply the constructive version of the Quillen–Suslin algorithm, i.e., following the reductions (B1) and (B2), we compute the matrices Uk and Fk , for 1 ≤ k ≤ 3:    U1 = 

 1 − ix − x2 + (1 + i)x3 −1 + (2 − i)x2 + (−1 − i)x3 −i − x + (1 + i)x2 1 + ix + (−1 + i)x2  x −i − x 1 i ,  0 1 0 0 0 0 0 1

 1 i + x + (−1 − i)x2 0 0  0 0 1 0  U1−1 =  −x 1 + ix − x2 + (1 − i)x3 i + x −i , 0 0 0 1     1 0 00 0 00   0 0 0 0 2    U1 F U1−1 =  0 −i + (1 + i)x2 1 0 , F2 = −i + 2(1 + i)x 1 0 ; x −x 01 0 x2 − x 01 

  U2 =  

 1 − ix − x2 + (1 + i)x3 −1 + (2 − i)x2 + (−1 − i)x3 −i − x + (1 + i)x2 1 + ix + (−1 + i)x2  ix + (−1 − i)x3 −ix + (−1 + i)x2 + (1 + i)x3 i + (−1 − i)x2 −1 + (1 − i)x2 ,  x −i − x 1 i 0 0 0 1

 1 0 i + x + (−1 − i)x2 0  0 −1 i + (−1 − i)x2 0 , U2−1 =  −x −i − x −ix2 −i 0 0 0 1   10 0 0    0 1 0 0 0 0 −1   U2 F U2 =  ,F = 2 ; 0 0 0 0 3 x −x 1 2 0 0 x −x 1 

   U3 =  

 1 − ix − x2 + (1 + i)x3 −1 + (2 − i)x2 + (−1 − i)x3 −i − x + (1 + i)x2 1 + ix + (−1 + i)x2   ix + (−1 − i)x3 −ix + (−1 + i)x2 + (1 + i)x3 i + (−1 − i)x2 −1 + (1 − i)x2 ,  −x3 + x2 ix + (−1 + i)x2 + x3 −x2 + x −1 − ix − ix2 x −i − x 1 i

 1 0 0 i + x + (−1 − i)x2  0 −1 0 i + (−1 − i)x2  , = −x −i − x i  −ix 2 0 0 −1 −x + x   1000    0 1 0 0  U3 F U3−1 =   0 0 1 0 , F 4 = 0 . 0000 

U3−1

Finally, using permutation matrices, we get

535

C.3 Algorithm for the Quillen–Suslin Theorem 



i 1 −i − x x  3 −1 + (2 − i)x2 + (−1 − i)x3 −i − x + (1 + i)x2 1 + ix + (−1 + i)x2  2 1 − ix − x + (1 + i)x  U =  , 3 2 3 2 −ix + (−1 + i)x + (1 + i)x i + (−1 − i)x ix + (−1 − i)x −1 + (1 − i)x2   −x3 + x2 −x2 + x −1 − ix − ix2 ix + (−1 + i)x2 + x3

i + x + (−1 − i)x2 1 0  i + (−1 − i)x2 −1 0 =  −x −i − x −ix 0 0 −x2 + x   0000  0 1 0 0 −1  UFU =   0 0 1 0 . 0001 

U −1

 0  0 , i  −1

So, r = 3 and the last three rows of U conform a basis X = {x1 , x 2 , x 3 } of hF i, x 1 = (1−ix−x2 +(1+i)x3 , −1+(2−i)x2 +(−1−i)x3 , −i−x+(1+i)x2 , 1+ix+(−1+i)x2 ), x 2 = (ix + (−1 − i)x3 , −ix + (−1 + i)x2 + (1 + i)x3 , i + (−1 − i)x2 , −1 + (1 − i)x2 ), x 3 = (−x3 + x2 , ix + (−1 + i)x2 + x3 , −x2 + x, −1 − ix − ix2 ).

Next we present a second illustration of the constructive algorithm. Example C.3.2. Let M4 (A), where A := K[x, σ, δ], K := Q(t), σ := idQ(t) d and δ := dt ; we consider the idempotent matrix F := [F (1) F (2) F (3) F (4) ], (i) F the ith column of F , where   2 + 2t + (13t2 − 5t)x + (8t3 − 6t2 )x2 + t3 (t − 1)x3 2t2 + t + (13t3 − 8t2 )x + (8t4 − 7t3 )x2 + t4 (t − 1)x3   F (1) =:   3t + 2 + (14t2 − 8t)x + (8t3 − 7t2 )x2 + t3 (t − 1)x3  , t2 + t + (t3 + 6t2 )x + 6t3 x2 + t4 x3   −t3 x3 − 5t2 x2 − 3tx + 1 t + (−3t2 + 2t)x + (−5t3 + t2 )x2 − t4 x3  , F (2) =:    −t3 x3 − 5t2 x2 − 3tx + 1 3 3 2 2 −t x − 5t x − 3tx + 1   t3 x3 + 5t2 x2 + 3tx − 1   t4 x3 + 5t3 x2 + 2t2 x − 2t  F (3) =:  −t − 1 + (−t2 + 5t)x + 6t2 x2 + t3 x3  , −t2 + t + (−t3 + 6t2 )x + 2t3 x2   0   tx . F (4) =:    tx 1 + (t2 − 2t)x − t2 x2 Applying the algorithm we obtain

C Implementation of Skew P BW Extensions With Maple

536

U (1) =:  2t + 1 + (10t2 − 5t)x + (7t3 − 6t2 )x2 + (t4 − t3 )x3   −3t − 2 + (−14t2 + 8t)x + (−8t3 + 7t2 )x2 + (−t4 + t3 )x3   ,  −2t + 2 − t(t − 1)x −2t2 + 7t − 2 − t(4t2 − 21t + 10)x − t2 (t2 − 10t + 7)x2 + t3 (t − 1)x3   −t3 x3 − 4t2 x2 − tx   t3 x3 + 5t2 x2 + 3tx − 1  , U (2) =:   tx + 1 2t(t − 3)x + t2 (t − 6)x2 − t3 x3   −t − 1 + (−t2 + 3t)x + 5t2 x2 + t3 x3   t + 2 + (t2 − 5t)x − 6t2 x2 − t3 x3  U (3) =:   , −tx − 1 −t + 1 − t(2t − 7)x − t2 (t − 6)x2 + t3 x3   tx   −tx ; U (4) =:  0  1   tx + 1   t − 2 + t(t − 1)x , (U −1 )(1) =:    0 −t + 2 − t(t − 4)x + t2 x2   tx + 1  t − 1 + t(t − 1)x  , (U −1 )(2) =:    1 2 2 2 1 + (−t + 3t)x + t x 





(U −1 )(3)

−t2 x2 − 2tx + 1   t + (−4t2 + 4t)x + (−2t3 + 5t2 )x2 + t3 x3   =:  , 1 + (−2t2 + t)x + (−t3 + 4t2 )x2 + t3 x3 1 + (−2t3 + 8t2 − 5t)x + (−t4 + 11t3 − 18t2 )x2 + (2t4 − 9t3 )x3 − t4 x4



(U −1 )(4)

 0   tx . =:    tx 1 + (t2 − 2t)x − t2 x2

With these computations we have 

U F U −1

 0000  0 0 0 0 ,  = 0 0 1 0 0001

thus, r = 2 and a base of hF i is X = {x1 , x 2 }, with

537

C.3 Algorithm for the Quillen–Suslin Theorem

x1 = (−2t + 2 − t(t − 1)x, tx + 1, −tx − 1, 0), x 2 = (−2t2 + 7t − 2 − t(4t2 − 21t + 10)x − t2 (t2 − 10t + 7)x2 + t3 (t − 1)x3 , 2t(t − 3)x + t2 (t − 6)x2 − t3 x3 , − t + 1 − t(2t − 7)x − t2 (t − 6)x2 + t3 x3 , 1).

Algorithm for the Quillen–Suslin theorem Computational version REQUIRE: A := K[x; σ, δ] and an idempotent matrix F ∈ Ms (A). 1: k := 0, F 0 := F ; 2: WHILE k < s − 1 DO 3: k := k + 1 0 ) | i = 1 or j = 1} = −∞ THEN 4: IF max{deg(fij 5: F 0 := SubM atrix(F 0 , 2..s, 2..s); 6: ELSE 7: (B): 0 = 0 THEN 8: IF f11 0 6= 0) F 0 := T (−1)F 0 T (−1)−1 ; 9: if (f1k k1 k1 0 6= 0) F 0 := T (−1)F 0 T (−1)−1 ; if (fk1 1k 1k 10: END IF 11: (B1): 0 ∈ K − {0} THEN 12: IF f11 13: Apply: OrderReduction1; 14: ELSE 15: Apply: (B2) OrderReduction2; 16: END IF 17: END IF 18: END WHILE 19: RETURN Matrices U, U −1 , U F U −1 ; a basis X of hF i; process step by step.

Example C.3.3. In this example we will illustrate the computational version of the Quillen–Suslin algorithm; let M3 (A), where A := K[x, σ, δ], K := Q(t), p(t−1) σ( p(t) q(t) ) := q(t−1) and δ := 0; we have the idempotent matrix   F =

1−

2t 1+t x

1 1+t x t 1+t x

2t(3+2t) 1+t x 3+2t 1+t x + t(3+2t) 1+t x



2t − −t

1

2t (1+t)2 x  −1 (1+t)2 x  . t − (1+t) 2x

Let F 0 := F . Throughout the example, we will replace the matrices F 0 , U and U −1 with the new versions given by the procedures of the algorithm. 2t 0 Step 1. Since f11 = 1 − 1+t x, we will apply the reduction procedure of (B2), i.e, OrderReduction2: 0 0 Step1.1. The idea is to convert f1,i = 0 for i > 2 and f1,2 6= 0. −1 Applying first T2,3 ( t(1+2t) ), then T3,2 (t(1+2t)− t(3+2t)(1+2t) x), and finally 1+t permuting the rows and columns 2 and 3, we get

C Implementation of Skew P BW Extensions With Maple

538

2t 1 − 1+t x  t(1+2t) x − 2t(1+2t) x2 =  1+t t+2 2t (1+2t)(1+t) x



U F U −1

where

2 1+2t 2t(1+2t) (3+2t)(1+t) x −2 (1+2t)2



1 0 0 t(1 + 2t) − t(3+2t)(1+2t) x U = 1+t 0 1  1 0 0 1 3+2t  U −1 = 0 t(1+2t) 1+t x 0

−t(1 + 2t) +

1

Step 1.2. Since the new F 0 is 

2t 1 − 1+t x  t(1+2t) x − 2t(1+2t) x2 0 F =  1+t t+2 2t (1+2t)(1+t) x

0

t(1+2t)  (1+t)2 x , −1 t(1+2t)

 U −1 = 

1

 0 0 , 1

 0 0 , 1

0 −t(1 +

 .

we apply T2,1 ( −t(1+2t) (1+t) x) and

0

x 1 1+t t(1+2t) t(1+2t) 1 1+t x



2 1+2t 2t(1+2t) (3+2t)(1+t) −2 (1+2t)2

where the new U and U −1 are  1 0  −t(1+2t) x t(1 + 2t) − t(3+2t)(1+2t) x U =  1+t 1+t 0 1 



t(3+2t)(1+2t) x 1+t

0 we want to reduce the degree of f1,1 ; for this we obtain  2 1 1+2t −1 0 U F U = 0 −2 0 (1+2t) 2

 0 0 , 1

0



t(1+2t)  (1+t)2 x , −1 t(1+2t)



3+2t  . 1+t x t(3+2t)(1+2t) 2t) + x 1+t

Step 2. The new F 0 is 

1 F 0 = 0 0

2 1+2t

0 −2 (1+2t)2

 0 0 ; 1

0 since f1,1 = 1 we apply (B1), i.e., OrderReduction1, for this we consider the matrices    −2  2 1 1+2t 0 1 1+2t 0 −1    S = 0 1 0 , and S = 0 1 0 , 0 0 1 0 0 1

and then

539

C.3 Algorithm for the Quillen–Suslin Theorem



1 S F 0 S −1 = 0 0



0 0 −2 (1+2t)2

0 0 . 1

Therefore, the new F 0 is F0 =







0 −2 (1+2t)2

0 , and U F U −1 1

1 = 0 0

0 0 −2 (1+2t)2

where the new U and U −1 are  2t 1 − 1+t x 2t − 2t(3+2t) 1+t x  t(3+2t)(1+2t) U =  −t(1+2t) x t(1 + 2t) − x 1+t 1+t 0 1   U −1 = 

−2 1+2t

1

 0 0 , 1



2t (1+t)2 x t(1+2t)  , (1+t)2 x −1 t(1+2t)



0

x 1 2 1+t t(1+2t) − (1+t)(3+2t) x t(1+2t) 2t(1+2t) 1 − (1+t)(3+2t) x 1+t x

−t(1 +

3+2t  . 1+t x 2t) + t(3+2t)(1+2t) x 1+t

0 Since f1,1 = 0, we apply T1,2 (−1), we get



U F U −1

1 0 = 0

0



0 2

2 −4t −4t+1  (1+2t)2 (1+2t)2  −2 4t2 +4t−1 (1+2t)2 (1+2t)2

" and F 0 =

where the new U and U −1 are  2t 1 − 1+t x 2t − 2t(3+2t) 1+t x  −t(1+2t) 2 U =  1+t x 2t + t − 1 − t(3+2t)(1+2t) x 1+t 0 1   U −1 = 

−2 1+2t

1

1 1 2 1+t x t(1+2t) − (1+t)(3+2t) x t(1+2t) 2t(1+2t) 1 − (1+t)(3+2t) x 1+t x

0 Since f1,1 =

2 (1+2t)2

1 2

,



2t (1+t)2 x t(1+2t)  1 , t(1+2t) + (1+t)2 x −1 t(1+2t)

 −2 1+2t 2 1 4t +12t+7  . t(1+2t) + (1+t)(3+2t) x 2 +12t+7) −2t2 − t + 1 + t(1+2t)(4t x (1+t)(3+2t)

"

 and T

so

−1



T F 0 T −1 Thus, the new F 0 is

#

is invertible, we apply OrderReduction1 with matrices

1 −2t2 − 2t + T = 1 1 

2 −4t2 −4t+1 (1+2t)2 (1+2t)2 −2 4t2 +4t−1 (1+2t)2 (1+2t)2

=

2 4t2 +4t−1 (1+2t)2 (1+2t)2 −2 2 (1+2t)2 (1+2t)2

 100 = 0 1 0 . 000

# ,

C Implementation of Skew P BW Extensions With Maple

540

  100   F 0 = 0 and U F U −1 = 0 1 0 , 000 where the new U and U −1 are  2t 1 − 1+t x 2t − 2t(3+2t) 1+t x  −t(1+2t) t(3+2t)(1+2t) 1 x U =  1+t x −t − 2 − 1+t −t(1+2t) t(3+2t)(1+2t) 2 x 2t + t − x 1+t 1+t   U −1 = 

1

2t (1+t)2 x t(1+2t)  1+2t 2t + (1+t)2 x , t(1+2t) (1+t)2 x



−2 1+2t 1  t(1+2t)  . 1 1+2t



0

x −2 1+t (1+t)(3+2t) x t(1+2t) 2t(1+2t) 2t 1+t x 1+2t − (1+t)(3+2t) x

Permuting, we have finally 

U F U −1

 000 = 0 1 0 , 001

where the new U and U −1 are  −t(1+2t) x 2t2 + t − t(3+2t)(1+2t) x 1+t 1+t  2t(3+2t) 2t 2t − 1+t x U =  1 − 1+t x −t(1+2t) x 1+t

−t −

−

t(3+2t)(1+2t) x 1+t

−2 1 0 1+2t 1 −2  1   t(1+2t) 1+t x . (1+t)(3+2t) x t(1+2t) 2t(1+2t) 1 2t 1+2t 1+t x 1+2t − (1+t)(3+2t) x

 U −1 =

1 2

 t(1+2t) (1+t)2 x  2t , (1+t)2 x t(1+2t) 1+2t + x 2t (1+t)2 

Therefore, r = 2 and the last two rows of U conform a basis X = {x1 , x 2 }, of hF i, 2t(3+2t) 2t 2t 1+t x, 2t − 1+t x, (1+t)2 x), t(1+2t) ( −t(1+2t) x, −t − 12 − t(3+2t)(1+2t) x, 1+2t 1+t 1+t 2t + (1+t)2 x).

x1 = (1 −

x2 =

Example C.3.4. Let M4 (A), where A := K[x, σ, δ], K := Q(t), σ(f (t)) := (t) f (qt) and δ(f (t)) := f (qt)−f t(q−1) , where q ∈ K − {0, 1}; we consider the idempotent matrix F := [F (1) F (2) F (3) F (4) ], F (i) the ith column of F and a ∈ Q, where   −t2 qx2 (−ta + 2 t) x − 2 a + 2 , F (1) =    tx + 2 −1

C.3 Algorithm for the Quillen–Suslin Theorem

541



F (2)



−2 tx + 2 −t2 qx2 + (ta − 4 t) x + 2 a − 1 , =   −tx − 2 tx + 2



F (3)

 −tx − 2
 −2 t2 qa + 3 t2 q x2 + a2 t − 8 ta + 8 t x + 2 a2 − 3 a + 1 , =   t2 qx2 + (−ta + 4 t) x − 2 a + 2 (ta − 2 t) x + 2 a − 2



 −t3 q 3 x3 + −q 2 t2 − 5 t2 q x2 − 5 tx + 2
−t3 q 3 x3 + −q 2 t2 − 3 t2 q x2 + (−ta + t) x − 2 a + 2 . =   tx + 2 2 2 t qx + 2 tx − 1 

F (4)

Applying the algorithm we obtain   tx + 1 0 t2 qx2 + 2 tx − 1 t2 qx2 + 3 tx  1 −tx − 2 (−ta + 2 t) x − 2 a + 2 −t2 qx2 − 2 tx + 2 , U = tx − 1 1 t2 qx2 + a − 1 t2 qx2 + 2 tx − 1  1 0 tx tx + 1  U

−1

tx

−1

−tx − 2

 a − 1 −tx + a − 1 −t2 qx2 + (ta − 4 t) x + 2 a − 1 =  −1 −1 −tx − 2 0

1tx + 2



0 t 3 q 3 x3

4) t2 qx2

− (−q + a − + (−3 ta + 3 t) x + 1  ,  t2 qx2 + 3 tx 2 2 −t qx − 2 tx + 1



U F U −1

0 0 = 0 0

0 0 0 0

0 0 1 0

 0 0 , 0 1

Hence, r = 2 and the last two rows of U conform a basis X = {x1 , x 2 } of hF i, x1 = (tx − 1, 1, t2 qx2 + a − 1, t2 qx2 + 2 tx − 1), x2 = (1, 0, tx, tx + 1). We conclude this section by presenting some remarks about the implementation of the computational version of the Quillen–Suslin algorithm. Remark C.3.5. OrderReduction1 is based on the implementation of procedure (B1) in the proof of Theorem 12.6.1; for OrderReduction2, the following algorithm describes its functionality:

542

C Implementation of Skew P BW Extensions With Maple

Algorithm OrderReduction2 REQUIRE: A := K[x; σ, δ] and an idempotent matrix F ∈ Ms (A) with deg(f11 ) ≥ 1. Make f1,j = 0 for j > 2 and f1,2 6= 0; Reduce degree of f1,1 ; IF f1,1 = 0 IF max{deg(fi,j ) > 0 | i = 1 or j = 1} > 0 Make f1,j = 0 for j > 2 and f1,2 6= 0; F := P12 F P12 ; Apply: OrderReduction1; ELSE F 0 := SubM atrix(F, 2..s, 2..s); ENDIF ELSE Apply: OrderReduction1; ENDIF   α 0 14: RETURN Matrices U, U −1 , F 0 and U F U −1 = , with α ∈ {0, 1}. 0 0 F 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13:

Remark C.3.6. For the implementation of the Quillen–Suslin algorithm we used Maple, and we create the library OrePolyToolKit.lib consisting in two packages: ˆ OrePolyUtility: this is a new and very useful collection of functions for operating with matrices, vectors and lists over an UnivariateOreRing K[x; σ, δ]; the UnivariateOreRing structure was taken from the library OreTools within the standard Maple libraries. ˆ OrePolyQS: this is the most important new collection of functions related to the Quillen–Suslin algorithm over K[x; σ, δ]; the main routine of the algorithm was implemented here, the following functions are fundamentals:

– GenerateIdemp: this function generates idempotent matrices over the Ore extension K[x; σ, δ]; the arguments are the matrix order and the UnivariateOreRing, and return an idempotent matrix of the given dimension over the respective UnivariateOreRing. – QSAlgKsd: this is the main function of the algorithm, it shows the sequence of all steps of the Quillen–Suslin algorithm; the arguments are the idempotent matrix and the UnivariateOreRing, and return the matrix U F U −1 in the form of Theorem 12.6.1, the matrices U and U −1 , the basis of hF i and the complete process step by step.

Appendix D

Maple Library Documentation

In this appendix we will show the content of the fundamentals packages contained in the library SPBWE.lib: SPBWETools, SPBWEGrobner, SPBWERings and RingTools.

D.1 The Package SPBWETools The package SPBWETools is a collection of functions that allows us to define and make computations with skew P BW extensions. To invoke this package we use the following sentence at the beginning of a Maple spreadsheet: with(SPBWETools); In the definition of a skew P BW extension it is also important to invoke the package RingTools in order to define the coefficient ring, for this we use the sentence: with(RingTools); Now, we will define a coefficients ring.

Calling Sequence SetCoeffsRing(Name=name, StructRing=strRing, charact=chr)

Parameters name: This parameter is a string that defines the name of the coefficient ring. If it is omitted, then name is assigned by a default name string. strRing: This parameter indicates the structure of the ring. In this implementation we have only worked with structures of commutative multivariate polynomial rings or subrings predefined in Maple, but it is feasible to define new structures of coefficient rings or to use other packages that define the structure of rings such as UnivariateOreRings or © Springer Nature Switzerland AG 2020 W. Fajardo et al., Skew PBW Extensions, Algebra and Applications 28, https://doi.org/10.1007/978-3-030-53378-6

543

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Skew Algebras; this parameter is optional, if it is omitted then the structure is defined as standard, i.e., it is defined as some subring of the fraction ring K(t1 , . . . , tm ), where K is some subring of C or the ring Zchr . chr: Characteristic of the ring; this parameter is optional, if it is omitted then the characteristic assigned is zero.

D.1.1 Skew P BW Extensions We recall first Definition 1.1.1 and Proposition 1.1.3. A skew P BW extension σ(R)hx1 , . . . , xn i is subject to the relations: (ij)

xj xi = ci,j xi xj + d1 x1 + · · · + d(ij) n xn + dij for 1 ≤ i < j ≤ n, xi r = σi (r) + δi (r) for 1 ≤ i ≤ n with r ∈ R.

(D.1.1)

The following parametric description is based on Sections 1.2 and 1.3 (see also [4]) and allows us to define a skew P BW extension.

Calling Sequence SetSkewPBWExtension(vars, rels, sigmas, deltas, R, value)

Parameters vars: list of variables [x1 , . . . , xn ], this list determines the order x1  . . .  xn . rels: list of relations [rel1 , . . . , relt ] determined by (D.1.1), where each rela(ij) (ij) tion relk has the form [[xi , xj ], cij , d1 , . . . , dn , dij ] for each i < j. R: the coefficient ring coeffsRing defined as before for the skew P BW extensions. sigmas: list of functions [σ1 , . . . , σn ] of (D.1.1). deltas: list of functions [δ1 , . . . , δn ] of (D.1.1). value: a boolean parameter that declares the structure ˆ if value is true, then the implementation checks if the structure effectively defines a skew P BW extension. In the affirmative case, the implementation returns “This structure represents a Skew P BW Extension” and the skew P BW extension is defined effectively. In the negative case, it returns “Non definite Skew P BW extension”, and the structure is not defined. ˆ if value is false the implementation allows us to realize computations with this structure, but it is not defined as a skew P BW extension.

This parameter is optional. If it is omitted then value is predefined as true.

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Remark ˆ The right definition of a skew P BW extension is subject to all σi and δi being injective endomorphisms of R and σi -derivations, respectively. ˆ The statement SetSkewPBWExtension returns a type in Maple called SPBWE.

D.1.2 Some Useful Functions With Skew P BW Extensions If a skew P BW extension is effectively defined (or even if the argument value is entered as false), it is possible to make computations with diverse functions implemented in Maple, which we will look at next.s We remark that all operations with polynomials in a skew P BW extension are assumed with the order declared in vars.

D.1.2.1

SkewProd

This function returns the product p · q or the product are polynomials in a skew P BW extension A.

Qn

i=1

pi , where p, q, pi

Calling Sequence SkewProd(p, q, A) SkewProd(L, A)

Parameters p : polynomial in A. q : polynomial in A. L : list of polynomials [p1 , . . . , pn ], with n ≥ 2 in A. A : skew P BW extension, this is a type SPBWE defined in D.1.1.

D.1.2.2

SkewSum

This function returns the sum of two polynomials in a skew P BW extension A.

Calling Sequence SkewSum(p, q, A) SkewSum(L, A)

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Parameters p : polynomial in A. q : polynomial in A. L : list of polynomials [p1 , . . . , pn ], with n ≥ 2 in A. A : skew P BW extension, this is a type SPBWE defined in D.1.1.

D.1.2.3

SkewSubs

This function returns the subtraction p − q, where p and q are polynomials in a skew P BW extension A.

Calling Sequence SkewSubs(p, q, A)

Parameters p : polynomial in A. q : polynomial in A. A : skew P BW extension, this is a type SPBWE defined in D.1.1.

D.1.2.4

SkewRelation

This function prints the relation existing between two variables in a skew P BW extension A.

Calling Sequence SkewRelation(xi , xj , A)

Parameters xi : variable of A. xj : variable of A. A : skew P BW extension, this is a type SPBWE defined in D.1.1.

D.1.2.5

deg

This function returns the degree of a polynomial p in a skew P BW extension A.

Calling Sequence deg(p, A)

Parameters p : polynomial in A. A : skew P BW extension, this is a type SPBWE defined in D.1.1.

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CanonicalVector

This function returns the n-dimensional vector e i .

Calling Sequence CanonicalVector(i, n)

Parameters i : positive integer that indicates the i-th canonical vector. n : positive integer that indicates the size of the vector.

D.1.2.7

SkewScalarProd

This function returns the product of a polynomial p with a polynomial vector v over a skew P BW extension A.

Calling Sequence SkewScalarProd(p, v , A)

Parameters p : polynomial in A. v : polynomial vector in A. A : skew P BW extension, this is a type SPBWE defined in D.1.1.

D.1.2.8

SkewPointedProd

This function returns the scalar product u · v , where u and v are polynomial vectors over a skew P BW extension A.

Calling Sequence SkewScalarProd(u, v , A)

Parameters u : polynomial vector in A. v : polynomial vector in A. A : skew P BW extension, this is a type SPBWE defined in D.1.1.

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SkewSumVector

This function returns the sum of two polynomial vectors over a skew P BW extension A.

Calling Sequence SkewScalarProd(u, v , A)

Parameters u : polynomial vector over A. v : polynomial vector over A. A : skew P BW extension, this is a type SPBWE defined in D.1.1.

D.1.2.10

SkewMinusVector

This function returns the vector −v over a skew P BW extension A.

Calling Sequence SkewMinusVector(v )

Parameters v : polynomial vector over A.

D.1.2.11

GeneratePolyMatrix

This function returns a random matrix with polynomials of degree deg and whose coefficients are integers module mod when mod is positive integer or numbers in C when mod is zero.

Calling Sequence GeneratePolyMatrix(rows, cols, vars, {mod = m}, {deg = d})

Parameters rows : positive integer that indicates the number of rows of the matrix. cols : positive integer for the number of columns of the matrix. vars: list of variables [x1 , . . . , xn ] of polynomials in the matrix. m : non-negative integer that indicates the integer mod. This parameter is optional. If it is omitted, then it assumes the value 0. d : non-negative integer for the maximum degree of polynomials in the matrix. This parameter is optional. If it is omitted, then it assumes the value 1.

D.1 The Package SPBWETools

D.1.2.12

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SkewProdMatrix

This function returns the matrix product P ·Q, where P and Q are polynomial matrices over a skew P BW extension A.

Calling Sequence SkewProdMatrix(P, Q, A)

Parameters P : polynomial matrix over A. Q : polynomial matrix over A. A : skew P BW extension, this is a type SPBWE defined in D.1.1.

D.1.2.13

SkewSumMatrix

This function returns the matrix sum of two matrices P and Q over a skew P BW extension A.

Calling Sequence SkewSumMatrix(P, Q, A)

Parameters P : polynomial matrix over A. Q : polynomial matrix over A. A : skew P BW extension, this is a type SPBWE defined in D.1.1.

D.1.2.14

SkewSubsMatrix

This function returns the matrix subtraction P − Q, where P and Q are polynomial matrices over a skew P BW extension A.

Calling Sequence SkewSubsMatrix(P, Q, A)

Parameters P : polynomial matrix over A. Q : polynomial matrix over A. A : skew P BW extension, this is a type SPBWE defined in D.1.1.

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D.2 The Package SPBWEGrobner The following functions include the main applications of this implementation: the division algorithm, Buchberger’s algorithm, the computations of syzygies, free resolutions and the left inverse of a matrix.

D.2.0.1

lcVector

This function returns the leading coefficient of a polynomial vector over a skew P BW extension A.

Calling Sequence lcVector(v , ordP oly, ord, A)

Parameters v : polynomial vector over A. ordP oly : polynomial order over A. ord : polynomial vector order over A. A : skew P BW extension, this is a type SPBWE defined in D.1.1.

D.2.0.2

ltVector

This function returns the leading term of a polynomial vector over a skew P BW extension A.

Calling Sequence ltVector(v , ordP oly, ord, A)

Parameters v : polynomial vector over A. ordP oly : polynomial order over A. ord : polynomial vector order over A. A : skew P BW extension, this is a type SPBWE defined in D.1.1.

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D.2.0.3

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lmVector

This function returns the leading monomial of a polynomial vector over a skew P BW extension A.

Calling Sequence lmVector(v , ordP oly, ord, A)

Parameters v : polynomial vector over A. ordP oly : polynomial order over A. ord : polynomial vector order over A. A : skew P BW extension, this is a type SPBWE defined in D.1.1.

D.2.0.4

PrintSkewPolyVector

This function prints an ordered polynomial vector over a skew P BW extension A according to a polynomial order and vector order.

Calling Sequence PrintSkewPolyVector(v , vars, ordP oly, ord, R)

Parameters v : polynomial vector in A. vars : list of variables [x1 , . . . , xn ], this list determines the order x1  . . .  xn . ordP oly : polynomial order over A. ord : polynomial vector order over A. R : the coefficient ring of the polynomial vector v ; coeffsRing is defined as before for the skew P BW extensions.

D.2.0.5

DivisionAlgorithm

According to Subsections C.1.2 and C.1.4, this function implements the division algorithm for a skew P BW extension A and returns a list [Q, h], where Q is a list of polynomials (vectors) and h is a polynomial (polynomial vector), satisfying the conditions of the division algorithm for ideals (modules).

Calling Sequence DivisionAlgorithm(f, L, ordP oly, A) %% version for ideals DivisionAlgorithm(f , F, ordP oly, ord, A) %% version for modules

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Parameters f : polynomial in A. L : list [f1 , . . . , fn ] of polynomials in A. F : list of polynomial vectors [f 1 , . . . , f n ] over A. f : polynomial vector over A. ordP oly : polynomial order over A. ord : polynomial vector order over A. A : skew P BW extension, this is a type SPBWE defined in D.1.1.

D.2.0.6

BuchbergerAlgSkewPoly

According to Theorems C.1.11 and 14.4.4, this function implements Buchberger’s algorithm for a bijective skew P BW extension A and returns a set of polynomial vectors that form a Gr¨ obner basis of an ideal or module.

Calling Sequence BuchbergerAlgSkewPoly(L, ordP oly, A) %% version for ideals BuchbergerAlgSkewPoly(F, ordP oly, ord, A) %% version for modules

Parameters L : list [f1 , . . . , fn ] of polynomials in A; the ideal is hf1 , . . . , fn }. F : list of polynomial vectors [f 1 , . . . , f n ] over a skew P BW extension A; the module is generated by f 1 , . . . , f n . ordP oly : polynomial order over A. ord : polynomial vector order over A. A : bijective skew P BW extension, this is a type SPBWE defined in D.1.1.

D.2.0.7

SyzModule

This function calculates the syzygy module of a finite set of polynomial vectors F over a bijective skew P BW extension A. This function returns a matrix whose rows conform the Syzygy module of F .

Calling Sequence SyzModule(M, ordP oly, ord, A,view)

Parameters M : polynomial matrix over A where each row of M corresponds to a vector of F . ordP oly : polynomial order over A. ord : polynomial vector order over A. A : bijective skew P BW extension, this is a type SPBWE defined in D.1.1.

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view : boolean value; if it is true, then the function prints preliminary results, and if it is false, then the preliminaries are not written; this parameter is optional, if it is omitted, then it is assumed false.

D.2.0.8

FreeResolution

This function calculates a free resolution of left A-module, where A is a bijective skew P BW extension. According to Theorem 15.6.1, this function returns the list of matrices F = [F0 , F1 , . . . , Fr ].

Calling Sequence FreeResolution(M, ordP oly, ord, A,view)

Parameters M : polynomial matrix over A, this matrix corresponds to F0 in Theorem 15.6.1. ordP oly : polynomial order over A. ord : polynomial vector order over A. A : bijective skew P BW extension, this is a type SPBWE defined in D.1.1. view : boolean value; if it is true, then the function prints preliminary results, and if it is false, then the preliminaries are not written; this parameter is optional, if it is omitted, then it is assumed false.

D.2.0.9

HQMatrices

This function computes a list [H, Q], where H and Q are the matrices in Theorem 15.1.1; F is a set of polynomial vectors over a bijective skew P BW extension A.

Calling Sequence HQMatrices(M, ordP oly, ord, A)

Parameters M : polynomial matrix over A whose rows are the vectors of F . ordP oly : polynomial order over A. ord : polynomial vector order over A. A : bijective skew P BW extension, this is a type SPBWE defined in D.1.1.

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LeftInverseMatrix

According to Corollary C.2.9, this function calculates a left inverse of a matrix (if it exists) with entries in a bijective skew P BW extension A.

Calling Sequence LeftInverseMatrix(M, ordP oly, ord, A)

Parameters M : polynomial matrix over A. ordP oly : polynomial order over A. ord : polynomial vector order over A. A : bijective skew P BW extension, this is a type SPBWE defined in D.1.1.

Appendix E

Examples of Skew P BW Extensions in SPBWE.lib

The following concrete examples of skew P BW extensions are predefined in the library SPBWE.lib. For completeness of the present appendix, we have included again the definition of these examples which were introduced in Chapter 2.

E.1 P BW Extensions P BW extensions are a subclass of the class of bijective skew P BW extensions σ(R)hx1 , . . . , xn i, in this case σi = iR and δi = 0 for each 1 ≤ i ≤ n and ci,j = 1 for every 1 ≤ i, j ≤ n (see Section 1.1). A P BW extension is subject to the relations: (ij)

xj xi = xi xj + d1 x1 + · · · + d(ij) n xn + dij for 1 < i < j < n.

(E.1.1)

Calling Sequence PBWExtension(vars, rels, R, value)

Parameters vars: list of variables [x1 , . . . , xn ]. rels: list of relations [rel1 , . . . , relt ] determined by (E.1.1), where each rela(ij) (ij) tion relk has the form [[xi , xj ], d1 , . . . , dn , dij ] for each i < j. R: the coefficient ring coeffsRing defined as before for the skew P BW extensions. value: boolean parameter defined as before for skew P BW extensions. © Springer Nature Switzerland AG 2020 W. Fajardo et al., Skew PBW Extensions, Algebra and Applications 28, https://doi.org/10.1007/978-3-030-53378-6

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E.2 The Dispin Algebra The dispin algebra U (osp(1, 2)) is generated by x, y, z over the commutative ring K satisfying the relations yz − zy = z,

zx + xz = y,

xy − yx = x.

Thus, U (osp(1, 2)) ∼ = σ(K)hx, y, zi.

Calling Sequence DispinAlgebra(vars, char=p)

Parameters vars: list of variables [x1 , x2 , x3 ] (the order determined by vars is x1  x2  x3 ); this parameter is optional, if it is omitted then the list of variables assigned is [x, y, z], p : characteristic of the ring K; this parameter is optional, if it is omitted then the characteristic assigned is zero.

E.3 The Manin Algebra of 2 × 2 Quantum Matrices This algebra is also known as the coordinate algebra of the quantum matrix space Mq (2) (see Section 2.5 and also [257] and [271]). By definition, Oq (M2 (K)), also denoted O(Mq (2)), is the coordinate algebra of the quantum matrix space Mq (2), it is the K-algebra generated by the variables x, y, u, v satisfying the relations xu = qux,

yu = q −1 uy,

vu = uv,

(E.3.1)

and xv = qvx,

vy = qyv,

yx − xy = −(q − q −1 )uv.

(E.3.2)

where q ∈ K − {0}. Thus, O(Mq (2)) ∼ = σ(K[u])hx, y, vi. Due to the last relation in (E.3.2), we remark that it is not possible to consider O(Mq (2)) as a skew P BW extension of K. This algebra can be generalized to n variables, Oq (Mn (K)), and coincides with the coordinate algebra of the quantum group SLq (2) (see the monograph [74] for more details).

Calling Sequence ManinAlgebra(q, vars, char=p)

Parameters q: parameter defined in relations (E.3.1) and (E.3.2), q ∈ K − {0}.

E.5 The Heisenberg Algebra

557

vars: list of variables [u1 , x1 , x2 , x3 ] (the order determined by vars is x1  x2  x3 ; u1 is assigned to variable u of relations (E.3.1) and (E.3.2)); this parameter is optional, if it is omitted then the list of variables assigned is [u, x, y, v]. p : characteristic of the ring; this parameter is optional, if it is omitted then the characteristic assigned is zero.

E.4 The Woronowicz Algebra The Woronowicz algebra, denoted by Wν (sl(2, K)) (see Section 2.5), is generated by x, y, z subject to the relations xz − ν 4 zx = (1 + ν 2 )x,

xy − ν 2 yx = νz,

zy − ν 4 yz = (1 + ν 2 )y, (E.4.1)

where ν ∈ K − {0} is not a root of unity. Then Wν (sl(2, K)) ∼ = σ(K)hx, y, zi.

Calling Sequence WoronowiczAlgebra(ν, vars, char=p)

Parameters ν: parameter defined in relations (E.4.1). vars: list of variables [x1 , x2 , x3 ] (the order determined by vars is x1  x2  x3 ); this parameter is optional, if it is omitted then the list of variables assigned is [x, y, z]. p : characteristic of the ring; this parameter is optional, if it is omitted then the characteristic assigned is zero.

E.5 The Heisenberg Algebra The K-algebra Hn (q) is generated by x1 , . . . , x n , y 1 , . . . , y n , z1 , . . . , z n and subject to the relations: xj xi = xi xj , zj zi = zi zj , yj yi = yi yj , 1 ≤ i, j ≤ n, zj yi = yi zj , zj xi = xi zj , yj xi = xi yj , i 6= j, zi yi = qyi zi , zi xi = q

−1

xi zi + yi , yi xi = qxi yi , 1 ≤ i ≤ n,

(E.5.1)

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with q ∈ K − {0}. Note that Hn (q) is not a skew P BW extension of K[x1 , . . . , xn ], however, with respect to K, Hn (q) is a bijective skew P BW extension of K: Hn (q) = σ(K)hx1 , . . . , xn ; y1 , . . . , yn ; z1 , . . . , zn i.

Calling Sequence HeisenbergAlgebra(n, q, char=p)

Parameters n: parameter defined in (E.5.1), it must be a non-negative integer. q: parameter defined in (E.5.1), q ∈ K − {0}. p : characteristic of the ring; this parameter is optional, if it is omitted then the characteristic assigned is zero.

Remark ˆ The variables are determined by the following order:

x 1  · · ·  x n  y 1  · · ·  y n  z1  · · ·  zn . ˆ To use polynomials in Hn (q), the variables must be called using the sentence: x[i], y[i] and z[i].

E.6 The Univariate Skew Polynomial Ring R[x; σ, δ] The univariate skew polynomial ring R[x; σ, δ] of injective type, i.e., with σ injective, is a skew P BW extension; in this case we have R[x; σ, δ] = σ(R)hxi.

Calling Sequence UnivariateSkewRing(var, sigma, delta, char=p)

Parameters var: variable x of the ring. sigma: injective endomorphism σ of R. delta: σ-derivation. p : characteristic of the ring; this parameter is optional, if it is omitted then the characteristic assigned is zero.

E.8 The Multiplicative Analogue of the Weyl Algebra

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E.7 The Additive Analogue of the Weyl Algebra The K-algebra An (q1 , . . . , qn ) is generated by x1 , . . . , xn , y1 , . . . , yn and subject to the following relations: xj xi = xi xj , yj yi = yi yj , 1 ≤ i, j ≤ n, yi xj = xj yi , i 6= j, yi xi = qi xi yi + 1, 1 ≤ i ≤ n,

(E.7.1)

where qi ∈ K − {0}. Recall (see Section 2.5) that An (q1 , . . . , qn ) is isomorphic to the iterated skew polynomial ring K[x1 , . . . , xn ][y1 ; σ1 , δ1 ] · · · [yn ; σn , δn ] over the commutative polynomial ring K[x1 , . . . , xn ]: σj (yi ) := yi , δj (yi ) := 0, 1 ≤ i < j ≤ n, σi (xj ) := xj , δi (xj ) := 0, i 6= j, σi (xi ) := qi xi , δi (xi ) := 1, 1 ≤ i ≤ n. An (q1 , . . . , qn ) is bijective and An (q1 , . . . , qn ) = σ(K[x1 , . . . , xn ])hy1 , . . . , yn i.

Calling Sequence AdditiveAnalogueWeylAlgebra(n, q, char=p)

Parameters n: parameter defined in (E.7.1), n must be a non-negative integer. q: list [q1 , . . . , qn ] defined in (E.7.1), for all 1 ≤ i ≤ n, qi ∈ K − {0}. p : characteristic of the ring; this parameter is optional, if it is omitted then the characteristic assigned is zero.

Remark ˆ The variables are determined by the following order:

x 1  · · ·  x n  y 1  · · ·  yn . ˆ To use polynomials in An (q1 , . . . , qn ), the variables must be invoked using the sentence: x[i] and y[i].

E.8 The Multiplicative Analogue of the Weyl Algebra The K-algebra On (λji ) is generated by x1 , . . . , xn and subject to the following relations: xj xi = λji xi xj , 1 ≤ i < j ≤ n, (E.8.1) where λji ∈ K − {0} (see Section 2.5). Recall that On (λji ) is isomorphic to the iterated skew polynomial ring K[x1 ][x2 ; σ2 ] · · · [xn ; σn ], with

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σj (xi ) := λji xi , 1 ≤ i < j ≤ n. In addition, On (λji ) can be viewed as On (λji ) = σ(K[x1 ])hx2 , . . . , xn i, and also as a skew P BW extension of K, On (λji ) = σ(K)hx1 , . . . , xn i.

Calling Sequence MultiplicativeAnalogueWeylAlgebra(n, λ, char=p)

Parameters n: parameter defined in (E.8.1), it must be a non-negative integer. λ: matrix (λji ) defined in (E.8.1), λji ∈ K − {0}. p : characteristic of the ring; this parameter is optional, if it is omitted then the characteristic assigned is zero.

E.9 The Witten Algebra Witten’s deformation of U (sl(2, K) was defined and studied by E. Witten, introducing a 7-parameter deformation of the universal enveloping algebra U (sl(2, K)) depending on a 7-tuple of parameters ξ = (ξ1 , . . . , ξ7 ) and subject to the relations xz − ξ1 zx = ξ2 x,

zy − ξ3 yz = ξ4 y,

yx − ξ5 xy = ξ6 z 2 + ξ7 z.

(E.9.1)

The algebra is denoted by W (ξ). Note that W (ξ) ∼ = σ(K[x])hy, zi (see Section 2.5).

Calling Sequence WittenAlgebra(ξ, char=p, value)

Parameters ξ: list [ξ1 , . . . , ξ7 ] defined in (E.9.1) with ξ6 = 0. p : characteristic of the ring; this parameter is optional, if it is omitted then the characteristic assigned is zero. value: a boolean parameter that declares the structure.

E.10 The σ-Multivariate Ore Extension

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E.10 The σ-Multivariate Ore Extension The Ore extension A := K[x1 , . . . , xn ; σ] introduced in Chapter 12, with K a field, and subject to the relations: xi xj = xj xi ,

xi r = σ(r)xi , r ∈ K, 1 ≤ i, j ≤ n.

(E.10.1)

Calling Sequence SigmaOreExtension(n, σ, char=p)

Parameters n: number of variables, this parameter must be a positive integer. σ: automorphism of K. p : characteristic of the ring; this parameter is optional, if it is omitted then the characteristic assigned is zero.

Remark ˆ The variables are determined by the order: x1  · · ·  xn . ˆ To use polynomials in A, each variable xi must be called using the sentence: x[i].

References

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Index

3-dimensional skew polynomial algebra, 32 abelian ring, 111 absorption identities, 393, 506 abstract Hilbert scheme, 498 additive analogue of the Weyl algebra, 9, 31, 511 admissible order, 238 affine scheme, 382, 493 affine scheme defined by a ring, 494 AG, 139 algebra for multidimensional discrete linear systems, 8, 28 algebra of differential operators on a quantum space, 37 algebra of functions on a quantum space, 37 algebra of Laurent polynomials, 84 algebra of linear partial difference operators, 29 algebra of linear partial differential operators, 28 algebra of linear partial q-differential operators, 30 algebra of linear partial q-dilation operators, 29 algebra of linear partial shift operators, 29 algebra of q-differential operators, 28 algebra of quantum polynomials, 87 algebra of shift operators, 8, 28 algebraic, 89 algebras of linear partial differential operators, 26 algorithm 1 for computing bases, 350 algorithm 1 for the right inverse of a matrix, 340 algorithm 2 for computing bases, 351 algorithm 2 for testing stably-freeness, 345

algorithm 2 for the right inverse of a matrix, 342 algorithm for computing U , 347 algorithm for the inverse of a square matrix, 337 algorithm for the left inverse of a matrix, 336, 530 algorithm for the Quillen–Suslin theorem (computational version), 537 algorithm for the Quillen–Suslin theorem (constructive version), 533 algorithm OrderReduction2, 542 AR, 139 Armendariz ring, 111 Artin–Schelter regular algebra, 506 AS algebra, 506 augmentation ideal, 388 augmentation map, 388 augmented algebras, 388 Auslander condition, 139 Auslander–Gorenstein ring, 139 Auslander regular ring, 139 autoequivalence, 500 B ring, 193 B0 -scheme, 382 B ring, 193 Bass’s theorem, 218 BD, 194 BD, 194 BDr , 194 bijective, 8 boundary condition, 215 boundary ideal, 215 Buchberger’s algorithm for bijective skew P BW extensions, 255 Buchberger’s algorithm for modules over bijective skew P BW extensions, 280 Buchberger’s algorithm in A, 516

© Springer Nature Switzerland AG 2020 W. Fajardo et al., Skew PBW Extensions, Algebra and Applications 28, https://doi.org/10.1007/978-3-030-53378-6

579

580 Buchberger’s algorithm in Am , 521 C ring, 193 C ring, 193 cancellable ring, 193 cancellative algebra, 413 cancellative ring, 455 CD, 194 centralizer, 317 centralizing, 317 centrally clean ring, 457 χ condition, 500 χi condition, 500 classical Gelfand–Kirillov dimension, 133 clean ring, 457 CM , 141 Cohen–Macaulay algebra, 141 commutative system, 99 complex algebra, 34 complexity, 20 condition (SA1), 119 constant skew P BW extension, 390 coordinate algebra of the quantum group SLq (2), 36 coordinate algebra of the quantum matrix space Mq (2), 35 Cr ring, 193 crossed product, 6, 27 d-Hermite, 210 D ring, 455 deg, 12, 262 deglex order, 15 ∆-commutative, 100 ∆-compatible ring, 118 ∆-invariant ideal, 99 ∆-prime ideal, 99 ∆-prime ring, 99 detectable algebra, 450 d-H, 210 diffusion algebra, 30, 513 direct product of lattices, 393 discriminant, 440 discriminant-divisor subalgebra, 442 dispin algebra, 33 distinguished object, 499 distributive collection of subspaces, 506 distributive lattice, 393, 506 divides, 238, 262 division algorithm in A, 244, 514 division algorithm in Am , 267, 518 double Ore extension, 508 E, 218 ED ring, 193 EDD, 194

Index elementary divisor ring, 193 essentially regular algebras, 399 exp, 12, 262 extended module, 218 extended quantized Weyl algebra, 461 extended ring, 218 extended Weyl algebra, 5, 25 F -divisor subalgebra, 442 fine moduli space, 498 finite presentation, 309 finitely generated centralizing bimodule, 317 finitely graded algebra, 465 finitely presented algebra, 387, 466 finitely presented module, 309 finitely semi-graded module, 358 finitely semi-graded ring, 358, 387 frame, 364, 418 frame stable domain, 459 free resolutions algorithm in Am , 526 F SG ring, 358, 387 G-algebra, viii, 39 Gelfand–Kirillov dimension, 418, 428 Gelfand–Kirillov transcendence degree, 427 generalized Gelfand–Kirillov dimension, 364 generalized Hilbert polynomial, 362 generalized Hilbert series, 361 generalized Koszul algebras, 393, 503 generated by frame, 418 generated sublattice, 393 generating frame, 364 generic algebra, 460 global dimension, 476 Goldie’s theorem (bijective case), 82 Goldie’s theorem (quasi-commutative case), 80 grade, 139 graded free module, 470 graded left global dimension, 474 graded projective dimension, 473 graded right global dimension, 474 Gr¨ obner basis, 248, 273 Gr¨ obner soluble ring, 317 GS ring, 317 H, 183 H ring, 455 H, 183 Hayashi algebra, 37 HD, 183 Hermite ring, 183 higher derivation, 431

Index Hilbert polynomial, 468 Hilbert series, 466 Hilbert’s basis theorem, 48 homogeneous element, 357, 358 homogeneous module homomorphism, 358 homogeneous ring homomorphism, 358 homogenization, 399 Hopfian algebra, 450 Horrocks’ theorem, 230 Hr , 183 IBN , 164 ID ring, 193 IDD, 194 invariant basis number ring, 164 isomorphic point modules, 380 iterated skew polynomial ring, 4 iterated skew polynomial ring of bijective type, 27 iterated skew polynomial ring of injective type, 8 iterative higher derivation, 431 Jordan plane, 56 Jordanian quantum Weyl algebra, 348 j-th lattice associated to an F SG algebra, 394 K-scheme, 497 kernel of a locally nilpotent higher derivation, 431 KH ring, 193 KH ring, 193 KHD, 194 KHD, 194 KHDr , 194 KHr ring, 193 Koszul algebras, 503 K¨ othe’s conjecture, 122 Kronecker’s theorem, 215 lattice, 393, 506 lattice associated to an F SG algebra, 394 Laurent polynomial ring, 84 leading coefficient, 15, 237, 263 leading monomial, 15, 237, 263 leading term, 15, 237, 263 least common multiple, 262 left B´ ezout ring, 193 left cancellable ring, 193 left Gr¨ obner soluble, 239 left K-Hermite ring, 193 left Ore condition, 65 left Ore domain, 65 left semi-graded Artin–Schelter regular algebra, 399

581 left stable rank, 173 left strongly noetherian algebra, 142 left Zariski ring, 140 left Zariskian filtration, 140 Levitzky radical, 118 LGS ring, 239 linear affine algebraic, 89 LNDH -rigid algebra, 433 LNDH Z -rigid algebra, 449 LNDI -rigid algebra, 433 LNDIZ -rigid algebra, 449 LND-rigid algebra, 433 LNDZ -rigid algebra, 449 local normal element, 141 local ringed space, 493 locally algebraic automorphism, 133 locally finite algebra, 450 locally nilpotent, 431 locally nilpotent higher derivation, 431 locally nilpotent R-derivations, 431 lower nil radical, 118 Makar-Limanov center, 449 Makar-LimanovH invariant, 432 Makar-LimanovI invariant, 433 Makar-Limanov invariant, 432 Manin algebra of 2 × 2 quantum matrices, 35 matrix presentation, 309 minimal, 240, 264 mixed algebra, 8, 28 module extension, 218 monomial in Am , 262 monomial order, 237 monomial order on Mon(Am ), 262 monomial quadratic algebra, 393 morphism of ringed spaces, 493 multilinearization, 487 multiparameter quantized Weyl algebra, 38 multiplicative analogue of the Weyl algebra, 10, 32, 512 N -Koszul algebras, 393, 503 n-multiparametric quantum space, 84 n-multiparametric quantum torus, 85 n-multiparametric skew quantum space, 84 n-multiparametric skew quantum torus, 84, 85 N-semi-graded module, 358 Nakayama’s lemma, 471 noncommutative projective scheme associated to an algebra, 499 noncommutative semi-projective scheme, 370 Ore algebra, 4

582 Ore algebra of bijective type, 28 Ore algebra of derivation type, 5, 25 Ore algebra of injective type, 8 Ore extension of bijective type, 28 Ore extension of derivation type, 5, 25 Ore extension of injective type, 8 Ore extensions, 4 Ore’s theorem (bijective case), 79 Ore’s theorem (quasi-commutative case), 78 parametrizes the point modules, 383, 498 P BW basis, 3 P BW extension, 3 P BW ring, viii perfect ring, 456 P F ring, 183 PF ring, 183 PF D, 183 P Fr ring, 183 Poincar´ e–Birkhoff–Witt extension, 3 Poincar´ e–Birkhoff–Witt theorem, 5, 26 point module, 380, 481 polynomial ring, 25 position over term order, 263 positive semi-graded module, 358 POT order, 263 POTREV order, 263 pre-commutative skew P BW extension, 390 presentation, 309 presentation of an algebra, 387 presheaf, 492 projective dimension of a module over a bijective skew P BW extension algorithm 1, 342 projective dimension of a module over a bijective skew P BW extension algorithm 2, 343 P SF ring, 183 PSF ring, 138, 183 PSF D, 183 P SFr ring, 183 q-Heisenberg algebra, 10, 36, 513 quantization of the ring of regular functions, 461 quantum algebra, 32 quantum enveloping algebra of sl(2, K), 36 quantum plane, 60, 84 quantum symplectic space, 39 quantum torus, 85 quantum Weyl algebra, 38 quantum Weyl algebra of Maltsiniotis, 37 quasi-commutative, 7 Quillen’s patching theorem, 229

Index Quillen–Suslin theorem, 231 quotient functor, 499 R-derivations, 431 R-point module, 496 R ring, 455 rank, 166, 213 rank condition, 164 RC, 164 reduced, 240, 264 reduced ring, 111 reduces, 240, 264 reduces in one step, 239, 263 reducible, 240, 264 reducible vector, 173 Rees ring, 140 regular element, 106 remainder, 240, 264 representable functor, 382, 497, 498 retractable algebra, 447 RGS ring, 286 right B´ ezout ring, 193 right cancellable ring, 193 right Gr¨ obner soluble rings, 286 right K-Hermite ring, 193 right skew P BW extension, 283 right Zariski ring, 140 right Zariskian filtration, 140 rigid endomorphism, 111 rigid endomorphisms family, 112 ring of coefficients, 6 ring of left polynomial type, 3 ring of polynomial type, 283 ring of quantum polynomials, 87 ring of right polynomial type, 283 ringed space, 493 s-χ condition, 373 s-χi condition, 373 S-point module, 380 SAS algebra, 399 saturated, 292, 300 saturation, 292, 300 SC ring, 455 scheme, 493 SD ring, 455 sections, 492 semi-commutative skew P BW extension, 390 semi-graded Artin–Schelter regular algebras, 399 semi-graded Koszul F SG algebra, 396 semi-graded module, 358 semi-graded ring, 357 semi-graded submodule, 358 semi-graduation, 357, 358

Index semidirect product of lattices, 394 semiperfect ring, 457 semiprimary ring, 456 semiprime ring, 111 Serre’s theorem, 138, 500 SG ring, 357 SH ring, 455 sheaf, 492 shift, 370, 469 Σ-commutative, 100 Σ-compatible ring, 118 (Σ, ∆)-Armendariz ring, 112 (Σ, ∆)-commutative, 100 (Σ, ∆)-compatible ring, 118 (Σ, ∆)-invariant ideal, 99 (Σ, ∆)-prime ideal, 99 (Σ, ∆)-prime radical, 121 (Σ, ∆)-prime ring, 99 (Σ, ∆)-weak Armendariz ring, 112 σ-derivation, 4 Σ-invariant ideal, 99 Σ-prime ideal, 99 Σ-prime ring, 99 σ-rigid ring, 111, 112 Σ-skew Armendariz ring, 112 similar matrices, 219 skew-Armendariz ring, 113 skew Laurent extension, 83 skew Laurent polynomial ring, 83 skew P BW cancellative domain, 458 skew P BW extension, 6 skew P BW extension of automorphism type, 100 skew P BW extension of derivation type, 100 skew P BW extension of endomorphism type, 100 skew P BW extensions, vii, 3 skew polynomial ring, 4 skew polynomial ring of bijective type, 27 skew polynomial ring of derivation type, 4, 25 skew polynomial ring of injective type, 8 skew polynomial rings of injective type, vii skew quantum plane, 84 skew quantum polynomials, 86 skew quantum torus, 85 skew-Armendariz ring, 112 Sklyanin algebra, 361 σ-P BW extension, 6 SR ring, 455 stable range theorem, 211 stable vector, 173 stably free module, 138, 166 stalk, 493 standard monomials, 3

583 strongly cancellative algebra, 414 strongly detectable algebra, 450 strongly Hopfian algebra, 450 strongly LNDH -rigid algebra, 433 strongly LNDH Z -rigid algebra, 449 strongly LNDI -rigid algebra, 433 strongly LNDIZ -rigid algebra, 449 strongly LND-rigid algebra, 433 strongly LNDZ -rigid algebra, 449 strongly retractable algebra, 447 strongly Z-retractable algebra, 449 structure sheaf, 493 sublattice, 506 sublattice generated by subspaces, 506 symmetric multiparameter quantized Weyl algebra, 461 syzygy, 291 syzygy module, 291 syzygy module algorithm in Am , 522 SZR ring, 455 term, 262 term over position order, 262 TOP order, 262 TOPREV order, 262 torsion element, 368 torsion module, 368 torsion-free module, 368 transposed module, 171 unimodular matrix, 169 uniquely clean ring, 457 univariate skew polynomial ring, 510 universal enveloping algebra, 26 universal enveloping algebra of the Lie algebra sl(2, K), 5 universal property, 18 universally cancellative algebra, 414 upper nil radical, 118 Vaserstein’s theorem, 224 weak Σ-skew Armendariz ring, 112 weak skew-Armendariz ring, 114 weakly Artin–Schelter regular algebra, 507 Wedderburn radical, 118 Weyl algebra, 5, 25 Weyl ring, 26 Witten algebra, 513 Witten’s deformation of U (sl(2, K)), 38 Woronowicz algebra, 34 Z-retractable algebra, 449

Z-semi-graded module, 358

584 Zariski lattice, 213 ZR ring, 455

Index

Zs -graded Artin–Schelter regular algebras, 399