Six Lectures on Commutative Algebra. Modern Birkhäuser Classics (eBook)

Table of Contents 6
Preface 11
Infinite Free Resolutions 12
Introduction 12
1. Complexes 15
1.1. Basic constructions 15
1.2. Syzygies 18
1.3. Differential graded algebra 21
2. Multiplicative Structures on Resolutions 25
2.1. DG algebra resolutions 25
2.2. DG module resolutions 29
2.3. Products versus minimality 32
3. Change of Rings 35
4. Growth of Resolutions 45
4.1. Regular presentations 45
4.2. Complexity and curvature 49
4.3. Growth problems 52
5. Modules over Golod Rings 55
5.1. Hypersurfaces 55
5.2. Golod rings 57
5.3. Golod modules 61
6. Tate Resolutions 64
6.1. Construction 65
6.2. Derivations 69
6.3. Acyclic closures 72
7. Deviations of a Local Ring 75
7.1. Deviations and Betti numbers 76
7.2. Minimal models 77
7.3. Complete intersections 82
7.4. Localization 83
8. Test Modules 86
8.1. Residue field 86
8.2. Residue domains 88
8.3. Conormal modules 94
9. Modules over Complete Intersections 98
9.1. Cohomology operators 98
9.2. Betti numbers 103
9.3. Complexity and Tor 106
10. Homotopy Lie Algebra of a Local Ring 110
10.1. Products in cohomology 111
10.2. Homotopy Lie algebra 114
10.3. Applications 117
References 121
Generic Initial Ideals 130
Introduction 130
1. The Initial Ideal 131
2. Regularity and Saturation 148
3. The Macaulay-Gotzmann Estimates on the Growth of Ideals 158
4. Points in P2 and Curves in P3 168
5. Gins in the Exterior Algebra 183
6. Lexicographic Gins and Partial Elimination Ideals 188
References 196
Tight Closure, Parameter Ideals,and Geometry 198
Foreword 198
1. An Introduction to Tight Closure 198
2. How Does Tight Closure Arise? 204
3. The Test Ideal I 211
4. The Test Ideal II: the Gorenstein Case 216
5. The Tight Closure of Parameter Ideals 221
6. The Strong Vanishing Theorem 226
7. Plus Closure 230
8. F-Rational Rings 233
9. Rational Singularities 236
10. The Kodaira Vanishing Theorem 239
References 242
On the Use of Local Cohomology in Algebra and Geometry 252
Introduction 252
1. A Guide to Duality 254
1.1. Local Duality. 254
1.2. Dualizing Complexes and Some Vanishing Theorems. 260
1.3. Cohomological Annihilators. 267
2. A Few Applications of Local Cohomology 270
2.1. On Ideal Topologies. 270
2.2. On Ideal Transforms. 274
2.3. Asymptotic Prime Divisors. 276
2.4. The Lichtenbaum-Hartshorne Vanishing Theorem. 283
2.5. Connectedness Results. 284
3. Local Cohomology and Syzygies 287
3.1. Local Cohomology and Tor’s. 287
3.2. Estimates of Betti Numbers. 292
3.3. Castelnuovo-Mumford Regularity. 293
3.4. The Local Green Modules. 297
References 301
Problems and Results on Hilbert Functions of Graded Algebras 304
Introduction 304
1. Macaulay’s Theorem 307
2. The Perfect Codimension Two and Gorenstein Codimension Three Case 312
3. The EGH Conjecture 322
4. Hilbert Function of Generic Algebras 328
5. Fat Points: Waring’s Problem and Symplectic Packing 330
6. The HF of a CM Local Ring 340
References 352
Cohomological Degrees of Graded Modules 356
Introduction 356
1. Arithmetic Degree of a Module 360
Multiplicity 360
Castelnuovo–Mumford regularity 361
Arithmetic degree of a module 362
Stanley–Reisner rings 363
Computation of the arithmetic degree of a module 364
Degrees and hyperplane sections 365
Arithmetic degree and hyperplane sections 365
2. Reduction Number of an Algebra 368
Castelnuovo–Mumford regularity and reduction number 368
Hilbert function and the reduction number of an algebra 369
The relation type of an algebra 370
Cayley–Hamilton theorem 371
The arithmetic degree of an algebra versus its reduction number 372
Reduction equations from integrality equations 374
3. Cohomological Degree of a Module 375
Big degs 375
Dimension one 376
Homological degree of a module 376
Dimension two 377
Hyperplane section 378
Generalized Cohen–Macaulay modules 381
Homologically associated primes of a module 382
Homological degree and hyperplane sections 383
Homological multiplicity of a local ring 387
4. Regularity versus Cohomological Degrees 388
Castelnuovo regularity 389
5. Cohomological Degrees and Numbers of Generators 391
6. Hilbert Functions of Local Rings 392
Bounding rules 393
Maximal Hilbert functions 394
Gorenstein ideals 396
General local rings 396
Bounding reduction numbers 397
Primary ideals 398
Depth conditions 399
7. Open Questions 400
Bounds problems 400
Cohomological degrees problems 401
References 401
Index 404