This monograph is devoted to a new class of non-commutative rings, skew Poincaré–Birkhoff–Witt (PBW) extensions. Beginning with the basic definitions and ring-module theoretic/homological properties, it goes on to investigate finitely generated projective modules over skew PBW extensions from a matrix point of view. To make this theory constructive, the theory of Gröbner bases of left (right) ideals and modules for bijective skew PBW extensions is developed. For example, syzygies and the Ext and Tor modules over these rings are computed. Finally, applications to some key topics in the noncommutative algebraic geometry of quantum algebras are given, including an investigation of semi-graded Koszul algebras and semi-graded Artin–Schelter regular algebras, and the noncommutative Zariski cancellation problem.
The book is addressed to researchers in noncommutative algebra and algebraic geometry as well as to graduate students and advanced undergraduate students.
Author(s): William Fajardo, Claudia Gallego, Oswaldo Lezama, Armando Reyes, Héctor Suárez, Helbert Venegas
Series: Algebra and Applications, 28
Publisher: Springer
Year: 2020
Language: English
Pages: 584
City: Cham
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
