Commutative Algebra : Expository Papers Dedicated to David Eisenbud on the Occasion of His 65th Birthday

Cover......Page 1
Commutative Algebra......Page 4
Cohomological Degrees and Applications......Page 6
Contents......Page 8
1.1 Definition and First Properties......Page 12
1.2 Simple and Non-simple Lazarsfeld–Mukai Bundles......Page 13
1.3 The Petri Conjecture Without Degenerations......Page 14
1.4 Mukai Manifolds of Picard Number One......Page 15
2.1 Constancy of the Gonality. I......Page 17
2.2 Constancy of the Clifford Index......Page 18
2.3 Constancy of the Gonality. II......Page 19
2.4 Parameter Spaces of Lazarsfeld–Mukai Bundles and Dimension of Brill–Noether Loci......Page 20
3.1 Koszul Cohomology......Page 21
3.2 Statement of Green\'s Conjecture......Page 23
3.3 Voisin\'s Approach......Page 24
3.4 The Role of Lazarsfeld–Mukai Bundles in the Generic Green Conjecture and Consequences......Page 26
3.5 Green\'s Conjecture for Curves on K3 Surfaces......Page 28
4 Counterexamples to Mercat\'s Conjecture in Rank Two......Page 30
References......Page 31
1 Introduction......Page 33
2.1 Closed String Theories......Page 34
2.2 Open–Closed Strings......Page 36
2.4 Hochschild Cohomology......Page 38
3 Toric Geometry and Phases......Page 40
3.1 Tilting Collections......Page 42
3.2 Complete Intersections......Page 44
3.3 Landau–Ginzburg Theories......Page 48
3.4 Hochschild Cohomology......Page 53
4 Monodromy......Page 54
4.1 K-Theory......Page 56
4.2 The GKZ System......Page 57
4.3 A Calabi–Yau and Landau–Ginzburg Example......Page 58
4.4 Monodromy of the Structure Sheaf......Page 60
References......Page 62
1 Introduction......Page 65
2.1 Analytic Approach......Page 67
2.2 Computing Complex Singularity Exponent by Monomializing......Page 69
2.3 Algebro-Geometric Approach......Page 71
2.4 The Canonical Divisor of a Map......Page 72
2.5 Computations of Log Canonical Thresholds......Page 75
2.6 Multiplier Ideals and Jumping Numbers......Page 76
3 Positive Characteristic: The Frobenius Map and F-Thresholds......Page 79
3.1 The Frobenius Map......Page 80
3.2 F-Threshold......Page 81
3.3 Comparison of F-Threshold and Multiplicity......Page 85
3.4 Computing F-Thresholds......Page 86
3.5 Comparison of F-Thresholds and Log Canonical Thresholds......Page 88
3.6 Test Ideals and F-Thresholds......Page 89
3.7 An Interpretation of F-Thresholds and Test Ideals Using Differential Operators......Page 95
4 Unifying the Prime Characteristic and Zero Characteristic Approaches......Page 98
4.1 Idea of Reduction Modulo p......Page 100
4.2 Trace......Page 101
References......Page 103
1 Introduction......Page 106
2 Preliminaries......Page 108
3 Extremal Betti Tables in the Graded Case......Page 109
4 Extremal Betti Tables in the Local Case......Page 120
5 Extremal Betti Tables in the Multigraded Case......Page 122
References......Page 127
1 Introduction......Page 129
2.1 Prerequisites and Notation......Page 131
2.2 Frobenius and Pushforward......Page 132
2.4 Exercises......Page 133
3 p-e-Linear Maps: Definition and Examples......Page 135
3.1 The Cartier Isomorphism......Page 137
3.2 Grothendieck Trace of Frobenius......Page 139
3.3 The Trace Map for Singular Varieties......Page 140
3.4 Exercises......Page 142
4 Connections with Divisors......Page 143
4.1 A Generalization with Line Bundles......Page 146
4.2 Exercises......Page 148
5 Frobenius Splittings......Page 150
5.1 Local Properties of Frobenius Split Varieties......Page 151
5.2 Global Properties of Frobenius Split Varieties......Page 154
5.3 Tools for Proving Proper Varieties Are Frobenius Split......Page 156
5.4 Exercises......Page 158
6 Frobenius Non-splittings......Page 160
6.1 Global Considerations......Page 161
6.2 Fedder\'s Lemma......Page 163
6.3 Exercises......Page 166
7.1 Closed Subschemes......Page 170
7.2 Birational Maps......Page 171
7.3 Finite Maps......Page 173
7.4 Exercises......Page 175
8 Cartier Modules......Page 177
8.1 Finiteness Results for Cartier Modules......Page 181
8.2 Cartier Crystals......Page 185
8.3 Arithmetic Aspects of p-e-Linear Maps......Page 187
8.4 Exercises......Page 188
9.1 Cartier Modules and Local Cohomology......Page 190
9.2 Contracting Property of p-e-Linear Maps......Page 194
9.3 Algebras of Maps and the Test Ideal......Page 197
9.4 Exercises......Page 202
References......Page 207
1 Introduction......Page 212
2 Some Preliminaries......Page 214
3 Characteristic Varieties of D-Modules and the Regularity of Annihilators......Page 218
4 A Regularity Bound for Ext-Modules......Page 225
5 Regularity Bounds for Tor-Modules......Page 231
References......Page 240
1 Introduction......Page 242
2 Some Background......Page 243
2.1 Generalized Koszul Complexes......Page 244
2.2 Hooks......Page 245
2.3 Determinantal Ideals......Page 246
2.4 Z-Forms......Page 247
3 Weyl and Schur Modules......Page 248
3.1 Shape Matrices and Tableaux......Page 249
3.2 Associating Weyl and Schur Modules to Shape Matrices......Page 253
4.1 Illustration of Letter-Place for Two Places......Page 256
4.2 Examples of Place Polarization Maps......Page 258
4.3 The Two-Rowed Resolution......Page 259
5.1 Positive Places and the Divided Power Algebra......Page 261
5.2 Negative Places and the Exterior Algebra......Page 265
5.3 Almost Full Generality......Page 268
5.4 Place Polarization Maps and Capelli Identities......Page 269
5.5 Return to Weyl and Schur Maps......Page 270
5.6 Some Kernel Elements of Weyl and Schur Maps......Page 272
5.7 Tableaux, Straightening, and the Straight Basis Theorem......Page 275
5.7.1 Tableaux for Weyl and Schur Modules......Page 276
5.7.2 Straightening Tableaux......Page 277
5.7.3 Taylor-Made Tableaux, or a Straight-Filling Algorithm......Page 279
6.1 Theorem 2, Part 1: The Double Standard Tableaux Generate......Page 281
6.2 Theorem 2 Part 2: Linear Independence of Double Standard Tableaux......Page 285
References......Page 288
1 Introduction......Page 290
2 Generalities......Page 292
3 How to Prove that an Algebra is Koszul?......Page 299
3.1 Gröbner Basis of Quadrics......Page 300
3.2 Transfer of Koszulness......Page 301
3.3 Filtrations......Page 304
4.1 Universally Koszul......Page 307
4.2 Absolutely Koszul......Page 308
5 Regularity and Koszulness......Page 310
6.1 Koszul Rings......Page 314
6.2 Koszul Modules and Linear Defect......Page 317
References......Page 318
1 Introduction......Page 321
3 The Eventual Linearity......Page 323
4 The m-Primary Case......Page 326
5 The Constant......Page 328
6 The Stabilization Index......Page 330
7 Powers of Ideal Sheaves and Symbolic Powers......Page 335
References......Page 336
1 Introduction......Page 338
2.1 Generalities on Rings and Modules......Page 340
2.3 Serre\'s Condition (Sn)......Page 341
2.5 Change of Rings Sequences......Page 343
2.6 Chow and Grothendieck Groups......Page 344
3.1 Tor Rigidity......Page 345
4.1 Tor Rigidity and the UFD Property......Page 347
4.2 Torsion-Freeness of Tensor Products and a Flatness Criterion......Page 348
4.3 An Equivalent Condition for Vanishing of Tor and Intersection Multiplicities......Page 349
5.1 Periodic Resolutions......Page 351
5.2 Hochster\'s Theta Pairing and Tor Rigidity......Page 352
5.3 Vanishing of θR(-,-)......Page 353
6 Asymptotic Behavior of Ext and Tor Over Complete Intersections......Page 356
6.1 Cohomology Operators and Support Varieties......Page 357
6.2 Length of Tor and a Generalized Version of Hochster\'s Pairing......Page 358
7 Applications: Class Groups and Picard Groups......Page 360
7.2 The Grothendieck–Lefschetz theorem......Page 361
7.3 Gabber\'s Conjecture......Page 363
8.1 Local Rings of Cones of Projective Varieties......Page 365
8.2 Intersections of Subvarieties......Page 366
8.3 Splitting of Vector Bundles......Page 367
8.4 Non-commutative Crepant Resolutions......Page 369
9.1 Some Open Questions......Page 370
References......Page 372
1 Introduction......Page 375
2 Preliminaries......Page 376
3 Chromatic Number and Odd Cycles in Graphs......Page 379
4 Associated Primes and Perfect Graphs......Page 384
5 Equality of Symbolic and Ordinary Powers and Linear Programming......Page 388
References......Page 393
1 Bounding Heights of Order Ideals: The Main Theorems......Page 397
2 The Syzygy Theorem and Applications......Page 405
3 Serre Intersection Theorem and Order Ideals of Consecutive Syzygy Modules......Page 409
4 The State of Mixed Characteristic......Page 412
References......Page 418
1.1 Basic Definitions......Page 423
1.2 Examples......Page 424
1.3 The Geometric Technique......Page 427
2.1 Schur Functors......Page 428
2.2 Descriptions of Some Homogeneous Spaces......Page 429
2.3 Borel–Weil–Bott Theorem......Page 430
2.4 Vinberg θ-Representations......Page 432
2.5 The Vinberg Method for Classifying Orbits......Page 433
2.6 Example: 3 C7......Page 436
3.1 Abelian Varieties......Page 441
3.3 Degeneracy Loci......Page 442
4.1 Modules Over OU......Page 444
4.3 Examples of Singular Quintic Curves......Page 445
4.4 Secant and Tangential Varieties......Page 447
4.5 Chow Forms......Page 448
4.6 Projective Duality......Page 449
5 3 C9......Page 450
5.2 Geometric Data from a Section......Page 451
5.3 Macaulay2 Code......Page 452
6.1 Modules Over OU......Page 454
6.2 Geometric Data from a Section......Page 455
6.3 Flag Variety......Page 457
6.5 Doing Calculations......Page 459
7 40 C8......Page 460
7.1 Modules Over OU......Page 461
8 spin(16)......Page 462
9 C4 spin(10)......Page 464
9.1 Modules Over OU......Page 465
9.3 Quadratic Complexes......Page 466
9.4 Doing Calculations......Page 467
References......Page 468
2 Pure and Split Extensions......Page 471
3 Review of Local Cohomology......Page 474
4 Proving that Rings Are Cohen–Macaulay......Page 476
5 Some Results of Peskine and Szpiro......Page 479
6 Small Cohen–Macaulay Modules......Page 480
7 Tight Closure and Splinters......Page 481
References......Page 483
1 Introduction......Page 485
2 Early History......Page 487
3 Basics......Page 489
4 Hilbert–Kunz Multiplicity Equal to One......Page 497
5 Hilbert–Kunz Multiplicity and Tight Closure......Page 501
6 F-Signature......Page 503
7 A Second Coefficient......Page 508
8 Estimates on Hilbert–Kunz Multiplicity......Page 515
References......Page 522
1 Introduction......Page 526
2 Monomial Level Algebras......Page 527
3 Pure O-Sequences and Combinatorics......Page 536
4 Enumerations of Pure O-Sequences......Page 542
5 Open Problems......Page 545
References......Page 546
1 Introduction......Page 550
2 Background and an Equivalent Problem......Page 551
3 Upper Bounds and Special Cases......Page 556
3.1 Three-Generated Ideals......Page 557
3.2 Ideals Generated by Quadratic Polynomials......Page 560
4 Lower Bounds and Examples......Page 568
4.1 Ideals with Large Projective Dimension......Page 569
4.2 Ideals with Larger Projective Dimension......Page 570
5 Related Bounds......Page 572
6 Questions......Page 573
References......Page 574
1 Introduction......Page 576
2 Dimension One......Page 578
3 Brauer–Thrall I for Hypersurfaces......Page 584
4 Brauer–Thrall I for Excellent Isolated Singularities......Page 585
5 Brauer–Thrall II......Page 587
References......Page 589
1 Introduction: Brenner\'s Insight......Page 592
2 Some Identities Involving bold0mu mumu PPequationPPPP......Page 594
3 The Spaces bold0mu mumu XXequationXXXX and bold0mu mumu YYequationYYYY: The Case of Transcendental bold0mu mumu ααequationαααα......Page 595
4 Matrix Calculations: The Case of Algebraic bold0mu mumu ααequationαααα......Page 599
5 Test Elements......Page 602
6 The Module bold0mu mumu H2H2equationH2H2H2H2: Completion of the Proof......Page 603
References......Page 606
1 Introduction......Page 607
2 Multidimensional Matrices and the Local Geometry of Segre Varieties......Page 608
3 The Biduality Theorem and the Contact Loci in the Segre Varieties......Page 611
4 Degenerate Matrices and the Hyperdeterminant......Page 614
5 Schläfli Technique of Computation......Page 617
6 Multilinear Systems, Matrices of Boundary Format......Page 621
7 Link with Geometric Invariant Theory in the boundary Format Case......Page 626
8 The Symmetric Case......Page 628
9 Weierstrass Canonical Form and Kac\'s Theorem......Page 631
10 The Rank and the k-Secant Varieties......Page 633
11 Open Problems......Page 634
References......Page 635
1 Introduction......Page 637
2.1 The Ideal of a Subspace Arrangement......Page 638
2.2 Numerical Invariants of Generic Subspace Arrangements......Page 639
2.3.1 Coordinate Subspace Arrangements......Page 641
2.3.2 Arrangements Defined by Binomial Linear Forms......Page 643
2.3.3 Graph Curves......Page 645
2.3.4 Secant Varieties of Graph Curves......Page 647
3 Hyperplane Arrangements......Page 648
3.1 Freeness of the Module of Derivations D(A)......Page 650
3.2 Resonance Varieties and Orlik–Solomon Algebra......Page 653
3.3 The Orlik–Terao Algebra, Nets and Multinets......Page 657
References......Page 659
1.1 The Ordinary Multiplicity......Page 666
1.2 Extended Degree Functions......Page 669
1.3 General Properties of Extended Degrees......Page 671
2 Degs and Castelnuovo–Mumford Regularity......Page 674
2.2 Hyperplane Sections......Page 676
3 Homological Degree......Page 679
3.1 Construction......Page 680
3.2 Buchsbaum Modules......Page 686
3.3 Cohomological Degrees and Samuel Multiplicities......Page 688
3.4 Homological Torsion......Page 689
3.6 Specialization and Torsion......Page 691
4 Bdeg: The Extreme Cohomological Degree......Page 693
4.1 Rules of Computation of bdeg......Page 694
4.2 Monomial Ideals......Page 697
4.3 Multiplicity-Based Complexity of Derived Functors......Page 700
5 Some Open Questions......Page 703
References......Page 704