Commutative Algebra : Expository Papers Dedicated to David Eisenbud on the Occasion of His 75th Birthday

This contributed volume is a follow-up to the 2013 volume of the same title, published in honor of noted Algebraicist David Eisenbud's 65th birthday. It brings together the highest quality expository papers written by leaders and talented junior mathematicians in the field of Commutative Algebra. Contributions cover a very wide range of topics, including core areas in Commutative Algebra and also relations to Algebraic Geometry, Category Theory, Combinatorics, Computational Algebra, Homological Algebra, Hyperplane Arrangements, and Non-commutative Algebra. The book aims to showcase the area and aid junior mathematicians and researchers who are new to the field in broadening their background and gaining a deeper understanding of the current research in this area. Exciting developments are surveyed and many open problems are discussed with the aspiration to inspire the readers and foster further research. Biosketch of David Eisenbud Mostly Mathematical Fragments of Autobiography Contents Bernstein-Sato Polynomials in Commutative Algebra 1 Introduction 2 Preliminaries 2.1 Differential Operators 2.2 Differentiably Admissible K-Algebras 2.3 Log-Resolutions 2.4 Methods in Prime Characteristic 3 The Classical Theory for Regular Algebras in Characteristic Zero 3.1 Definition of the Bernstein-Sato Polynomial of an Hypersurface 3.2 The D-Modules DA|K[s] fs and Af[s] fs 3.2.1 Direct Construction of Af[s] fs 3.2.2 Local Cohomology Construction of Af[s] fs 3.2.3 Constructions of the Module DA|K[s] fs 3.3 Existence of Bernstein-Sato Polynomials for Polynomial Rings via Filtrations 3.4 Existence of Bernstein-Sato Polynomials for Differentiably Admissible Algebras via Homological Methods 3.5 First Properties of the Bernstein-Sato Polynomial 4 Some Families of Examples 4.1 Quasi-Homogeneous Singularities 4.2 Irreducible Plane Curves 4.3 Hyperplane Arrangements 5 The Case of Nonprincipal Ideals and Relative Versions 5.1 Bernstein-Sato Polynomial for General Ideals in Differentiably Admissible Algebras 5.2 Bernstein-Sato Polynomial of General Ideals Revisited 5.2.1 Monomial Ideals 5.2.2 Determinantal Varieties 5.3 Bernstein-Sato Ideals 5.3.1 Hyperplane Arrangements 5.4 Relative Versions 5.5 V-Filtrations 6 Bernstein-Sato Theory in Prime Characteristic 6.1 Bernstein-Sato Roots: p-Adic Version 6.2 Bernstein-Sato Roots: Base p Expansion Version 6.3 Nonprincipal Case 7 An Extension to Singular Rings 7.1 Nonexistence of Bernstein-Sato Polynomials 7.2 Existence of Bernstein-Sato Polynomials 7.3 Differentiable Direct Summands 8 Local Cohomology 9 Complex Zeta Functions 10 Multiplier Ideals 11 Computations via F-Thresholds References Lower Bounds on Betti Numbers 1 Introduction 2 What Is a Free Resolution? 3 Why Study Resolutions? 3.1 Betti Numbers Encode Geometry 3.2 Resolutions for Ideals with Few Generators 3.3 How Small Can the Ranks of Syzygies Be? 3.4 Other Possible Directions 4 The Buchsbaum–Eisenbud–Horrocks Conjecture and the Total Rank Conjecture 4.1 General Purpose Tools 4.2 Other Results 4.3 The Total Rank Conjecture 5 Stronger Bounds 5.1 The Multigraded Case 5.2 Low Regularity Case References The Simplest Minimal Free Resolutions in P1 P1 1 Introduction 1.1 Motivation from Geometric Modeling 1.2 Mathematical Background 1.2.1 Bigraded Betti Numbers 1.2.2 Bigraded Algebra and Line Bundles on P1 P1 1.2.3 Koszul Homology and Bigraded Hilbert Series 1.3 Roadmap of This Chapter 2 Koszul Homology H1(K•(f,R)) and the Generic Case 2.1 Koszul Homology 2.2 Determining H1(K•(f,R)) 2.3 Understanding (H1)a 2.4 The Generic Case 2.5 The Fröberg-Lindqvist Conjecture on Bigraded Hilbert Series 3 Koszul Homology H1(K•(f,R)) for d=(1,n) 3.1 Tautological First Syzygies: Degrees (1,*) and (2,*) 3.2 First Syzygies of Degree (3,*) 3.3 Computing Betti Numbers, the General Setting 4 Factorization of Sections of OP1 P1(1,n) and the Segre Variety 1,n 4.1 Intersection with 1,n 4.2 Minimal free Resolutions Determined by the Geometry of W 1,n 5 Higher Segre Varieties 5.1 Intersection with 3, n-1 5.2 Connections to the Hurwitz Discriminant and Sylvester Map 5.3 Concluding Remarks References Castelnuovo–Mumford Regularity and Powers 1 Castelnuovo–Mumford Regularity Over General Base Rings 2 Bigraded Castelnuovo–Mumford Regularity 3 A Non-standard Z2-Grading 4 Regularity and Powers 5 Linear Powers References The Eisenbud-Green-Harris Conjecture 1 An Introduction to the Conjecture 2 Monomial Regular Sequences and the Clements–Lindström Theorem 3 Artinian Reduction and Linkage 4 Results on the EGH Conjecture 5 Applications and Examples References Fibers of Rational Maps and Elimination Matrices: An Application Oriented Approach 1 Introduction 2 Graph of a Rational Map 2.1 Graph and Rees Algebra 2.2 Symmetric Algebra 3 Elimination Matrices and Fibers of Projections 3.1 Elimination Ideal 3.2 Finite Fibers 4 When the Source Is P1 4.1 Matrix Representations 4.2 Regularity Estimate for Rational Curves 4.2.1 The Four Equivalent Definitions of Castelnuovo-Mumford Regularity 4.2.2 The Regularity Estimate for Rational Curves 5 Morphisms from Pn-1k to Pnk 6 When the Source Is of Dimension 2 6.1 Fitting Ideals Associated to ψ 6.2 One Dimensional Fibers 7 When the Base Locus Is of Positive Dimension 7.1 The Base Locus 7.2 Fibers 7.3 The Main Theorem 7.4 Idea of the Proof of the Main Theorem 7.5 Curve with No Section in Negative Degree References Three Takes on Almost Complete Intersection Ideals of Grade 3 1 Introduction 2 Almost Complete Intersections Following Avramov and Brown 3 Generic Almost Complete Intersections 3.1 Quotients of Even Type 3.2 Quotients of Odd Type 4 The Equivariant Form of the Format (1,4,n,n-3) 4.1 Quotients of Even Type 4.2 Quotients of Odd Type 5 Schubert Varieties in Orthogonal Grassmannians vs. Almost Complete Intersection and Gorenstein Ideals of Codimension 3 5.1 The Action of the Weyl Group 5.2 Schubert Varieties 5.3 Spinor Coordinates 5.4 The Case of Even n 5.5 The Case of Odd n 5.6 Minimal Free Resolutions Appendix A Pfaffian Identities Following Knuth B Minors via Pfaffians Following Brill C Generic Almost Complete Intersections: The Proofs C.1 Quotients of Even Type C.2 Quotients of Odd Type References Stickelberger and the Eigenvalue Theorem 1 Introduction 2 Ludwig Stickelberger 3 The Eigenvalue Theorem 4 Scheja and Storch 1988 5 Counting Real Solutions 6 Conclusion References Multiplicities and Mixed Multiplicities of Filtrations 1 Filtrations 2 Multiplicity of mR-Primary Ideals and of mR-Filtrations 3 Rees's Theorem 4 Outline of the Proof of Rees's Theorem for Filtrations 4.1 Multiplicities of Filtrations 4.2 The Integral Closure of a Filtration I and the Convex Sets (I) 4.3 The Invariant γμ(I) 4.4 Rees's Theorem for Divisorial mR-Filtrations 5 Mixed Multiplicities of mR-Primary Ideals and of mR-Filtrations 6 The Minkowski Inequalities 7 An Overview of the Proof of the Characterization of the Minkowski Equality 8 Examples References Stanley-Reisner Rings 1 Introduction 2 Simplicial Complexes and Stanley-Reisner Rings 2.1 Stanley-Reisner Rings 2.2 Edge Rings and Clutters 2.3 Facet Rings 3 Hilbert Series and Hochster's Formula 3.1 Hilbert Series 3.2 Homology 3.3 Hochster's Formula 4 Reisner's Criteria 5 Gorenstein, Buchsbaum Rings, and Serre's Condition Sr 5.1 Gorenstein Rings 5.2 Buchsbaum Rings 5.3 Rings Satisfying Sr 5.4 (Locally) Complete Intersection 6 Shellability 6.1 Pure Shellability 6.2 Non-pure Shellability 7 Eagon-Reiner's Theorem 8 Polarization 9 Resolutions, Betti Numbers, Regularity 9.1 Resolutions 9.2 Linear and Pure Resolutions 9.3 Betti Numbers and Regularity 9.4 Infinite Resolutions 10 Linear Quotients 11 Componentwise Linear Ideals and Sequentially CM Complexes 12 Powers and Symbolic Powers of Stanley-Reisner Ideals 13 Shifting 14 Edge Ideals, Path Ideals, Facet Ideals 14.1 Edge Ideals of Graphs and Clutters 14.2 Facet Ideals 14.3 Vertex Decomposability 14.4 Path Ideals 15 Books References Symbolic Rees Algebras 1 Introduction 2 Symbolic Rees Algebras 2.1 A Brief History 3 Criteria for Noetherianity 3.1 Noetherianity 3.2 Generation Type and Standard Veronese Degree 4 Applications to Containment Problems and Asymptotic Invariants 4.1 The Containment Problem 4.2 Noetherian Symbolic Rees Algebras and the Containment Problem 4.3 Asymptotic Invariants References The Alexander–Hirschowitz Theorem and Related Problems 1 Introduction: The Alexander–Hirschowitz Theorem 2 The General Case (d≥4 and n≥3) 3 The Exceptional Cases 4 The Case of P2 (n=2) 5 The Case of Cubics (d=3) 6 Open Problems A Appendix: Secant Varieties and the Waring Problem B Appendix: Symbolic Powers C Appendix: Hilbert Function D Appendix: Semi-continuity of the Hilbert Function and Reduction to Special Configurations E Appendix: Hilbert Schemes of Points and Curvilinear Subschemes References Depth Functions and Symbolic Depth Functions of HomogeneousIdeals 1 Introduction 2 Ordinary Depth Functions 3 Symbolic Depth Functions 4 Open Questions References Algebraic Geometry, Commutative Algebra and Combinatorics: Interactions and Open Problems 1 Introduction 2 Semi-effectivity 2.1 Waldschmidt Constants 2.2 Computing and Bounding Waldschmidt Constants 2.3 Index of Semi-effectivity 2.4 Bounded Negativity Conjecture 3 Containment Problems 3.1 Related Containment Problems 4 Splitting Types 4.1 Ascenzi Curves and the SHGH Conjecture 4.2 Unexpected Curves References Maximal Cohen-Macaulay Complexes and Their Uses: A PartialSurvey 1 Introduction 2 Local Cohomology and Derived Completions 2.1 Derived I-torsion 2.2 Derived I-completion 2.3 Koszul Complexes 2.4 Depth 3 Complexes of Maximal Depth and the Intersection Theorems 3.1 Complexes of Maximal Depth 4 MCM Complexes 4.1 Big Cohen-Macaulay Complexes 4.2 Dualizing Complexes 4.3 Via Resolution of Singularities 5 Applications to Birational Geometry References Subadditivity of Syzygies of Ideals and Related Problems 1 Introduction 2 Background 3 Effective Bounds on Regularity 3.1 Ullery's Designer Ideals via Idealizations 3.2 Graded Bourbaki Ideals 4 Subadditivity of Syzygies 5 General Syzygy Bounds 6 Quadratic Ideals and Linear Syzygies 7 Questions and Conjectures 7.1 Subadditivity of Syzygies 7.2 Weak Convexity of Syzygies 7.3 Syzygy Bounds on Regularity 7.4 Syzygies of Quadratic Ideals References Applications of Liaison 1 Introduction 2 Background 3 Stick Figures, Zeuthen's Problem and Configurations of Linear Subvarieties 3.1 Stick Figure Curves in P3 3.2 Arithmetically Gorenstein Generalized Stick Figures of Codimension Three 3.3 Arithmetically Gorenstein (Generalized) Stick Figures of Any Codimension 4 The Singular Locus of a Hyperplane Arrangement 5 The Eisenbud-Green-Harris Conjecture and Cayley-Bacharach 6 The Genus of Space Curves 7 Liaison and Graded Betti Numbers 8 Gröbner Bases and Rees Algebras 9 Vertex Decomposability 10 Unprojections 11 Open Questions References Survey on Regularity of Symbolic Powers of an Edge Ideal 1 Introduction 2 Castelnuovo-Mumford Regularity, Symbolic Powers and Degree Complexes 2.1 Graph Theory 2.2 Simplicial Complex 2.3 Stanley-Reisner Correspondence 2.4 Castelnuovo-Mumford Regularity 2.5 Symbolic Powers 2.6 Edge Ideals and Their Symbolic Powers 2.7 Degree Complexes 3 Intermediate Ideals for Second and Third Powers 4 Intermediate Ideals for Edge Ideals of Small Dimensions 5 Bounds on Regularity of Powers/Symbolic Powers 6 Mixed Sum and Fiber Product References Applications of Differential Graded Algebra Techniques in Commutative Algebra 1 Introduction 2 Growth of Bass and Betti Numbers 3 Friendliness and Persistence of Local Rings 4 Bass Series of Local Ring Homomorphisms of Finite Flat Dimension 5 Ascent Property of pd-test Modules 6 A Conjecture of Vasconcelos on the Conormal Module 7 A Conjecture of Vasconcelos on Semidualizing Modules 8 Complete Intersection Maps and the Proxy Small Property 9 Conjectures of Quillen on André-Quillen Homology 10 Finite Generation of Hochschild Homology Algebras References Regularity Bounds by Projection 1 Introduction 2 Construction of Projection 3 Complexity of Fibers of Projections 4 Regularity Bounds References The Zariski-Riemann Space of Valuation Rings 1 Introduction 2 Projective Models 3 Topology of the Zariski-Rieman Space 4 The Patch Topology 5 Schemes in Zar(F/k) 6 Affine Schemes in Zar(F/k) 7 Example: Two-dimensional Noetherian Domains 8 Example: Holomorphy Rings References Rational Points and Trace Forms on a Finite Algebra over a Real Closed Field 1 Introduction 2 Type, Signature and Classification of Hermitian Forms 3 Trace Forms and Rational Points 4 Counting Rational Points of Finite Affine Algebraic Sets References Hermite Reciprocity and Schwarzenberger Bundles 1 Introduction 2 Hermite Reciprocity 2.1 An Algebraic Construction 2.2 Via Schwarzenberger Bundles 2.3 The Isomorphisms Agree 2.4 Compatibility of Hermite Isomorphisms 3 Exterior Powers of Schwarzenberger Bundles 4 Secant Varieties of Rational Normal Curves 5 Self-Duality for the Rank One Ulrich Module 6 Syzygies of Canonical Curves 6.1 Rational Cuspidal Curves 6.2 Ribbon Curves References Generation in Module Categories and Derived Categories of Commutative Rings 1 Generation Problem 2 Preliminaries 3 Classification of Subcategories 4 Dimensions of Subcategories References Existence and Constructions of Totally Reflexive Modules 1 Introduction 2 Totally Reflexive Modules Over Rings That Have an Embedded Deformation 3 Yoshino's Conditions for Rings with m3=0 4 Exact Zero Divisors 5 Constructing Totally Reflexive Modules from Exact Zero Divisors 6 Lifting Totally Reflexive Modules 7 When Does a Generic Matrix Give Rise to a Totally Reflexive Module? 8 Other Constructions of Totally Reflexive Modules 9 G-Regular Rings References Local Cohomology—An Invitation 1 Introduction 1.1 Koszul Cohomology 1.2 The Čech Complex 1.3 Limits of Ext-modules 1.4 Local Duality 2 Finiteness and Vanishing 2.1 Finiteness Properties 2.2 Vanishing 2.3 Annihilation of Local Cohomology 3 D- and F-Structure 3.1 D-Modules 3.1.1 Characteristic 0 3.1.2 D-Modules and Group Actions 3.1.3 Coefficient Fields of Arbitrary Characteristic 3.2 F-Modules 3.2.1 F-Modules 3.2.2 A{f}-Modules: Action of Frobenius 3.2.3 The Lyubeznik Functor HR,A 3.3 Interaction Between D-Modules and F-Modules 4 Local Cohomology and Topology 4.1 Arithmetic Rank 4.1.1 Some Examples and Conjectures 4.1.2 Endomorphisms of Local Cohomology 4.2 Relation with de Rham and étale Cohomology 4.2.1 The Čech–de Rham Complex 4.2.2 Algebraic de Rham Cohomology 4.2.3 Lefschetz and Barth Theorems 4.2.4 Results via étale Cohomology 4.3 Other Applications of Local Cohomology to Geometry 4.3.1 Bockstein Morphisms 4.3.2 Variation of Hodge Structures and GKZ-Systems 4.3.3 Milnor Fibers and Torsion in the Jacobian Ring 4.4 Lyubeznik Numbers 4.4.1 Combinatorial Cases and Topology 4.4.2 Projective Lyubeznik Numbers References Which Properties of Stanley–Reisner Rings and Simplicial Complexes are Topological? 1 Introduction 2 Dimension 3 Minimal Free Resolution and Depth 4 Munkres' Proof of Theorems 3.4 and 3.5 5 Local Cohomology Proof of Theorems 3.4 and 3.5 6 Cohen–Macaulay, Gorenstein, Buchsbaum 7 n-Purity, n-Cohen–Macaulay and n-Buchsbaum 8 Other Properties References This contributed volume brings together the highest quality expository papers written by leaders and talented junior mathematicians in the field of Commutative Algebra. Contributions cover a very wide range of topics, including core areas in Commutative Algebra and also relations to Algebraic Geometry, Algebraic Combinatorics, Hyperplane Arrangements, Homological Algebra, and String Theory. The book aims to showcase the area, especially for the benefit of junior mathematicians and researchers who are new to the field; it will aid them in broadening their background and to gain a deeper understanding of the current research in this area. Exciting developments are surveyed and many open problems are discussed with the aspiration to inspire the readers and foster further research.