Determinants, Gröbner Bases and Cohomology

Description

This book offers an up-to-date, comprehensive account of determinantal rings and varieties, presenting a multitude of methods used in their study, with tools from combinatorics, algebra, representation theory and geometry. After a concise introduction to Gröbner and Sagbi bases, determinantal ideals are studied via the standard monomial theory and the straightening law. This opens the door for representation theoretic methods, such as the Robinson–Schensted–Knuth correspondence, which provide a description of the Gröbner bases of determinantal ideals, yielding homological and enumerative theorems on determinantal rings. Sagbi bases then lead to the introduction of toric methods. In positive characteristic, the Frobenius functor is used to study properties of singularities, such as F-regularity and F-rationality. Castelnuovo–Mumford regularity, an important complexity measure in commutative algebra and algebraic geometry, is introduced in the general setting of a Noetherian base ring and then applied to powers and products of ideals. The remainder of the book focuses on algebraic geometry, where general vanishing results for the cohomology of line bundles on flag varieties are presented and used to obtain asymptotic values of the regularity of symbolic powers of determinantal ideals. In characteristic zero, the Borel–Weil–Bott theorem provides sharper results for GL-invariant ideals. The book concludes with a computation of cohomology with support in determinantal ideals and a survey of their free resolutions. Determinants, Gröbner Bases and Cohomology provides a unique reference for the theory of determinantal ideals and varieties, as well as an introduction to the beautiful mathematics developed in their study. Accessible to graduate students with basic grounding in commutative algebra and algebraic geometry, it can be used alongside general texts to illustrate the theory with a particularly interesting and important class of varieties. Preface Contents 1 Gröbner Bases, Initial Ideals and Initial Algebras 1.1 Monomial Orders 1.2 Gröbner Bases 1.3 Sagbi Bases 1.4 Initial Subspaces and Gradings 1.5 Initial Subspaces for Weights. Homogenization 1.6 Deformation to the Initial Object 2 More on Gröbner Deformations 2.1 Minimal and Associated Primes 2.2 Connectedness 2.3 Optimal Gröbner Deformations 2.4 Square-Free Gröbner Deformations 3 Determinantal Ideals and the Straightening Law 3.1 Minors and Determinantal Ideals 3.2 Standard Bitableaux and the Straightening Law 3.3 Refinement of the Straightening Law 3.4 Determinantal Rings 3.5 Powers and Products of Determinantal Ideals 3.6 Representation Theory 4 Gröbner Bases of Determinantal Ideals 4.1 An Instructive Case: the Ideal of 2-Minors 4.2 The Robinson–Schensted–Knuth Correspondence 4.3 RSK and Gröbner Bases of Ideals 4.4 Cohen–Macaulayness and Hilbert Series of Determinantal Rings 4.5 Gröbner Bases via Secants 4.6 Multigraded Structures and the Multidegree 5 Universal Gröbner Bases 5.1 Universal Gröbner Bases 5.2 Universal Gröbner Bases for Maximal Minors 5.3 Universal Gröbner Bases for 2-Minors 6 Algebras Defined by Minors 6.1 A Recap of Toric Algebra 6.2 Algebras Defined by Maximal Minors 6.3 Rees Algebras of Determinantal Ideals 6.4 The Algebra of Minors 6.5 Algebraic Relations between Minors 6.6 Exterior Powers of Linear Maps 6.7 Sagbi Deformations of Determinantal Rings 7 F-singularities of Determinantal Rings 7.1 F-purity 7.2 F-regularity 7.3 F-rationality and Gröbner Deformations 7.4 Equivariant Deformations 7.5 F-pure Thresholds 8 Castelnuovo–Mumford Regularity 8.1 Castelnuovo–Mumford Regularity over General Base Rings 8.2 Castelnuovo–Mumford Regularity and Multigraded Structures 8.3 Vanishing over Nonstandard Multigraded Rings 8.4 Regularity of Powers and Products of Ideals 8.5 Linear Powers and Linear Products 8.6 A Family of Determinantal Ideals with Linear Products 8.7 Maximal Minors of Matrices of Linear Forms and Linear Powers 9 Grassmannians, Flag Varieties, Schur Functors and Cohomology 9.1 Fundamental Multilinear Algebra Constructions 9.2 Affine and Projective Bundles, Grassmannians, Flag Varieties 9.3 Basic Calculus with Direct Images 9.4 The Kempf Vanishing Theorem 9.5 Characteristic-Free Vanishing for some Nondominant Weights 9.6 A Sharp Characteristic-Free Vanishing on Projective Bundles 9.7 Algebraic Groups and Representations 9.8 Schur Functors 9.9 Grothendieck Duality 9.10 Cohomology Vanishing on Grassmannians 10 Asymptotic Regularity for Symbolic Powers of Determinantal Ideals 10.1 Quotients of Ideals Defined by Shape 10.2 Filtrations Associated with Ideals Defined by Shape 10.3 An algorithm to Compute Z(Σ) 10.4 The Filtration for Symbolic Powers of Determinantal Ideals 10.5 Sheaves Defined by Shape and Their Quotients 10.6 A Geometric Realization of the Modules J(σ)l 10.7 Regularity of Symbolic Powers of Determinantal Ideals 10.8 Some Geometric Properties of Determinantal Varieties 11 Cohomology and Regularity in Characteristic Zero 11.1 The Borel–Weil–Bott Theorem 11.2 The Optimality of Theorem 9.5.1摥映數爠eflinkthm:charspsfreespsvanishing9.5.19 11.3 Borel–Weil–Bott on Grassmannians 11.4 Effective Vanishing of Cohomology in Characteristic Zero 11.5 Cauchy's Formula Revisited 11.6 Explicit Description of `3́9`42`"̇613A``45`47`"603AExt Modules for J(σ)l 11.7 `3́9`42`"̇613A``45`47`"603AExt Modules for Symbolic Powers 11.8 The Basics on `3́9`42`"̇613A``45`47`"603AGL-Invariant Ideals 11.9 `3́9`42`"̇613A``45`47`"603AExt Modules for `3́9`42`"̇613A``45`47`"603AGL-Invariant Ideals 11.10 Local Cohomology with Determinantal Support 11.11 Syzygies of `3́9`42`"̇613A``45`47`"603AGL-Invariant Ideals Appendix List of Symbols References Index