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Six lectures on commutative algebra

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Interest in commutative algebra has surged over the past decades. In order to survey and highlight recent developments in this rapidly expanding field, the Centre de Recerca Matematica in Bellaterra organized a ten-days Summer School on Commutative Algebra in 1996. Lectures were presented by six high-level specialists, L. Avramov (Purdue), M.K. Green (UCLA), C. Huneke (Purdue), P. Schenzel (Halle), G. Valla (Genova) and W.V. Vasconcelos (Rutgers), providing a fresh and extensive account of the results, techniques and problems of some of the most active areas of research. The present volume is a synthesis of the lectures given by these authors. Research workers as well as graduate students in commutative algebra and nearby areas will find a useful overview of the field and recent developments in it.

Reviews

"All six articles are at a very high level; they provide a thorough survey of results and methods in their subject areas, illustrated with algebraic or geometric examples." - Acta Scientiarum Mathematicarum

Avramov lecture: "... it contains all the major results [on infinite free resolutions], it explains carefully all the different techniques that apply, it provides complete proofs (…). This will be extremely helpful for the novice as well as the experienced." - Mathematical reviews

Huneke lecture: "The topic is tight closure, a theory developed by M. Hochster and the author which has in a short time proved to be a useful and powerful tool. (…) The paper is extremely well organized, written, and motivated." - Zentralblatt MATH

Schenzel lecture: "… this paper is an excellent introduction to applications of local cohomology." - Zentralblatt MATH

Valla lecture: "… since he is an acknowledged expert on Hilbert functions and since his interest has been so broad, he has done a superb job in giving the readers a lively picture of the theory." - Mathematical reviews

Vasconcelos lecture: "This is a very useful survey on invariants of modules over noetherian rings, relations between them, and how to compute them." - Zentralblatt MATH

Author(s): J. Elias, J. M. Giral, Rosa M. Miró-Roig, Santiago Zarzuela
Series: Modern Birkhäuser Classics
Publisher: Birkhauser
Year: 2010

Language: English
Pages: 408

Six Lectures on
Commutative Algebra......Page 2
Title page......Page 4
Copyright Page......Page 5
Table of Contents......Page 6
Preface......Page 11
Introduction......Page 12
1.1. Basic constructions......Page 15
1.2. Syzygies......Page 18
1.3. Differential graded algebra......Page 21
2.1. DG algebra resolutions......Page 25
2.2. DG module resolutions......Page 29
2.3. Products versus minimality......Page 32
3.1. Universal resolutions......Page 35
3.2. Spectral sequences......Page 40
3.3. Upper bounds......Page 42
4.1. Regular presentations......Page 45
4.2. Complexity and curvature......Page 49
4.3. Growth problems......Page 52
5.1. Hypersurfaces......Page 55
5.2. Golod rings......Page 57
5.3. Golod modules......Page 61
6. Tate Resolutions......Page 64
6.1. Construction......Page 65
6.2. Derivations......Page 69
6.3. Acyclic closures......Page 72
7. Deviations of a Local Ring......Page 75
7.1. Deviations and Betti numbers......Page 76
7.2. Minimal models......Page 77
7.3. Complete intersections......Page 82
7.4. Localization......Page 83
8.1. Residue field......Page 86
8.2. Residue domains......Page 88
8.3. Conormal modules......Page 94
9.1. Cohomology operators......Page 98
9.2. Betti numbers......Page 103
9.3. Complexity and Tor......Page 106
10. Homotopy Lie Algebra of a Local Ring......Page 110
10.1. Products in cohomology......Page 111
10.2. Homotopy Lie algebra......Page 114
10.3. Applications......Page 117
References......Page 121
Introduction......Page 130
1. The Initial Ideal......Page 131
2. Regularity and Saturation......Page 148
3. The Macaulay-Gotzmann Estimates on the Growth of Ideals......Page 158
4. Points in P2 and Curves in P3......Page 168
5. Gins in the Exterior Algebra......Page 183
6. Lexicographic Gins and Partial Elimination Ideals......Page 188
References......Page 196
1. An Introduction to Tight Closure......Page 198
2. How Does Tight Closure Arise?......Page 204
3. The Test Ideal I......Page 211
4. The Test Ideal II: the Gorenstein Case......Page 216
5. The Tight Closure of Parameter Ideals......Page 221
6. The Strong Vanishing Theorem......Page 226
7. Plus Closure......Page 230
8. F-Rational Rings......Page 233
9. Rational Singularities......Page 236
10. The Kodaira Vanishing Theorem......Page 239
References......Page 242
Introduction......Page 251
1.1. Local Duality......Page 253
1.2. Dualizing Complexes and Some Vanishing Theorems......Page 259
1.3. Cohomological Annihilators......Page 266
2.1. On Ideal Topologies......Page 269
2.2. On Ideal Transforms......Page 273
2.3. Asymptotic Prime Divisors......Page 275
2.4. The Lichtenbaum-Hartshorne Vanishing Theorem......Page 282
2.5. Connectedness Results......Page 283
3.1. Local Cohomology and Tor’s......Page 286
3.2. Estimates of Betti Numbers......Page 291
3.3. Castelnuovo-Mumford Regularity......Page 292
3.4. The Local Green Modules......Page 296
References......Page 300
Introduction......Page 303
1. Macaulay’s Theorem......Page 306
2. The Perfect Codimension Two and Gorenstein Codimension Three Case
......Page 311
3. The EGH Conjecture......Page 321
4. Hilbert Function of Generic Algebras......Page 327
5. Fat Points: Waring’s Problem and Symplectic Packing......Page 329
6. The HF of a CM Local Ring......Page 339
References......Page 351
Introduction......Page 355
Multiplicity......Page 359
Castelnuovo–Mumford regularity......Page 360
Arithmetic degree of a module......Page 361
Stanley–Reisner rings......Page 362
Computation of the arithmetic degree of a module......Page 363
Arithmetic degree and hyperplane sections......Page 364
Castelnuovo–Mumford regularity and reduction number......Page 367
Hilbert function and the reduction number of an algebra......Page 368
The relation type of an algebra......Page 369
Cayley–Hamilton theorem......Page 370
The arithmetic degree of an algebra versus its reduction number......Page 371
Reduction equations from integrality equations......Page 373
Big degs......Page 374
Homological degree of a module......Page 375
Dimension two......Page 376
Hyperplane section......Page 377
Generalized Cohen–Macaulay modules......Page 380
Homologically associated primes of a module......Page 381
Homological degree and hyperplane sections......Page 382
Homological multiplicity of a local ring......Page 386
4. Regularity versus Cohomological Degrees......Page 387
Castelnuovo regularity......Page 388
5. Cohomological Degrees and Numbers of Generators......Page 390
6. Hilbert Functions of Local Rings......Page 391
Bounding rules......Page 392
Maximal Hilbert functions......Page 393
General local rings......Page 395
Bounding reduction numbers......Page 396
Primary ideals......Page 397
Depth conditions......Page 398
Bounds problems......Page 399
References......Page 400
Index......Page 403