Progress in Commutative Algebra 1 (eBook)

Preface 6
Boij-Söderberg Theory: Introduction and Survey 14
1 The Boij-Söderberg Conjectures 18
1.1 Resolutions and Betti Diagrams 18
1.2 The Positive Cone of Betti Diagrams 19
1.3 Herzog-Kühl Equations 20
1.4 Pure resolutions 21
1.5 Linear Combinations of Pure Diagrams 22
1.6 The Boij-Söderberg Conjectures 24
1.7 Algorithmic Interpretation 25
1.8 Geometric Interpretation 25
2 The Exterior Facets of the Boij-Söderberg Fan and Their Supporting Hyperplanes 26
2.1 The Exterior Facets 26
2.2 The Supporting Hyperplanes 28
2.3 Pairings of Vector Bundles and Resolutions 34
3 The Existence of Pure Free Resolutions and of Vector Bundles with Supernatural Cohomology 37
3.1 The Equivariant Pure Free Resolution 37
3.2 Equivariant Supernatural Bundles 42
3.3 Characteristic Free Supernatural Bundles 42
3.4 The Characteristic Free Pure Resolutions 43
3.5 Pure Resolutions Constructed from Generic Matrices 46
4 Cohomology of Vector Bundles on Projective Spaces 48
4.1 Cohomology Tables 48
4.2 The Fan of Cohomology Tables of Vector Bundles 50
4.3 Facet Equations 51
5 Extensions to Non-Cohen-Macaulay Modules and to Coherent Sheaves 54
5.1 Betti Diagrams of Graded Modules in General 55
5.2 Cohomology of Coherent Sheaves 56
6 Further Topics 58
6.1 The Semigroup of Betti Diagrams of Modules 58
6.2 Variations on the Grading 62
6.3 Poset Structures 63
6.4 Computer Packages 63
6.5 Three Basic Problems 64
Hilbert Functions of Fat Point Subschemes of the Plane: the Two-fold Way 68
1 Introduction 68
2 Approach I: Nine Double Points 71
3 Approach I: Points on Cubics 80
4 Approach II: Points on Cubics 87
Edge Ideals: Algebraic and Combinatorial Properties 98
1 Introduction 98
2 Algebraic and Combinatorial Properties of Edge Ideals 99
3 Invariants of Edge Ideals: Regularity, Projective Dimension, Depth 103
4 Stability of Associated Primes 117
Three Simplicial Resolutions 140
1 Introduction 140
2 Background and Notation 141
2.1 Algebra 141
2.2 Combinatorics 142
3 The Taylor Resolution 143
4 Simplicial Resolutions 145
5 The Scarf Complex 148
6 The Lyubeznik Resolutions 149
7 Intersections 151
8 Questions 152
A Minimal Poset Resolution of Stable Ideals 156
1 Introduction 156
2 Poset Resolutions and Stable Ideals 160
3 The Shellability of PN 163
4 The topology of PN and properties of D (PN) 168
5 Proof of Theorem 2.4 173
6 A Minimal Cellular Resolution of R/N 176
Subsets of Complete Intersections and the EGH Conjecture 180
1 Introduction 180
2 Preliminary Definitions and Results 181
2.1 The Eisenbud-Green-Harris Conjecture and Complete Intersections 181
2.2 Some Enumeration 184
3 Rectangular Complete Intersections 186
4 Some Key Tools 190
4.1 Pairs of Hilbert Functions and Maximal Growth 190
4.2 Ideals Containing Regular Sequences 191
5 Subsets of Complete Intersections in P2 192
6 Subsets of C.I. (2, d2,d3) with d2 = d3 194
7 Subsets of C.I. (3, d2, d3) with d3 = d2 199
8 An Application: The Cayley-Bacharach Property 207
The Homological Conjectures 212
1 Introduction 212
2 The Serre Multiplicity Conjectures 214
2.1 The Vanishing Conjecture 217
2.2 Gabber’s Proof of the Nonnegativity Conjecture 218
2.3 The Positivity Conjecture 219
3 The Peskine-Szpiro Intersection Conjecture 220
3.1 Hochsterzs Metatheorem 222
4 Generalizations of the Multiplicity Conjectures 224
4.1 The Graded Case 224
4.2 The Generalized Rigidity Conjecture 227
5 The Monomial, Direct Summand, and Canonical Element Conjectures 228
6 Cohen-Macaulay Modules and Algebras 230
6.1 Weakly Functorial Big Cohen-Macaulay Algebras 231
7 The Syzygy Conjecture and the Improved New Intersection Conjecture 232
8 Tight Closure Theory 233
9 The Strong Direct Summand Conjecture 235
10 Almost Cohen-Macaulay Algebras 236
11 A Summary of Open Questions 238
11.1 The Serre Positivity Conjecture 238
11.2 Partial Euler Characteristics 238
11.3 Strong Multiplicity Conjectures 239
11.4 Cohen-Macaulay Modules and Related Conjectures 239
11.5 Almost Cohen-Macaulay Algebras 240
The Compatibility, Independence, and Linear Growth Properties 244
1 Introduction 244
2 Primary decomposition 245
3 Compatibility of Primary Components 249
4 Maximal Primary Components, Independence 250
5 Linear growth of primary components 252
6 Linear Growth of {TorRc (M/ImM, N/JnN)} 254
7 Secondary Representation 260
8 Compatibility of Secondary Components 263
9 Applying a Result of Sharp on Artinian Modules 264
10 Independence 272
11 Minimal Secondary Components 275
12 Linear Growth of Secondary Components 277
Recent Progress in Coherent Rings: a Homological Perspective 284
1 Introduction 284
2 Coherent Rings and Grade 285
2.1 Coherent Rings and (FP)R8 Modules 285
2.2 Non-Noetherian Grade 285
3 Cohen-Macaulay Rings 287
4 Gorenstein Dimensions and the Auslander-Bridger Property 290
4.1 Gorenstein Dimensions 290
4.2 The Auslander-Bridger Formula 293
5 Gorenstein Rings and Injective Dimensions 294
6 Foundations for Coherent Complete Intersections 301
Non-commutative Crepant Resolutions: Scenes from Categorical Geometry 306
1 Introduction 306
2 Morita Equivalence 309
3 (Quasi)coherent Sheaves 311
4 Derived Categories of Modules 312
5 Derived Categories of Sheaves 314
6 Example: Tilting on Projective Space 316
7 The Non-existence of Non-commutative Spaces 319
8 Resolutions of Singularities 323
9 The Minimal Model Program 326
10 Categorical Desingularizations 329
11 Example: the McKay Correspondence 331
12 Non-commutative Crepant Resolutions 336
13 Example: Normalization 341
14 MCM Endomorphism Rings 342
15 Global Dimension of Endomorphism Rings 344
16 Rational Singularities 347
17 Examples: Finite Representation Type 349
18 Example: the Generic Determinant 351
19 Non-commutative Blowups 356
20 Omissions and Open Questions 358