Skew PBW Extensions: Ring and Module-theoretic Properties, Matrix and Gröbner Methods, and Applications (Algebra and Applications, 28)

Description

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. 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