Elias / Giral / Miró-Roig Six Lectures on Commutative Algebra

1998
ISBN: 978-3-0346-0329-4
Verlag: Birkhäuser Basel
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 409 Seiten

Reihe: Modern Birkhäuser Classics

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 TOC:Preface.- Infinite Free Resolutions.- Generic Initial Ideals.- Tight Closure, Parameter Ideals, and Geometry.- On the Use of Local Cohomology in Algebra and Geometry.- Problems and Results on Hilbert Functions of Graded Algebras.- Cohomological Degrees of Graded Modules.

Autoren/Hrsg.

Elias, J.

Giral, J. M.

Miró-Roig, Rosa M.

Zarzuela, Santiago

Inhaltsverzeichnis

1;Table of Contents ;6
2;Preface;11
3;Infinite Free Resolutions;12
3.1;Introduction;12
3.2;1. Complexes;15
3.2.1;1.1. Basic constructions;15
3.2.2;1.2. Syzygies;18
3.2.3;1.3. Differential graded algebra;21
3.3;2. Multiplicative Structures on Resolutions;25
3.3.1;2.1. DG algebra resolutions;25
3.3.2;2.2. DG module resolutions;29
3.3.3;2.3. Products versus minimality;32
3.4;3. Change of Rings;35
3.5;4. Growth of Resolutions;45
3.5.1;4.1. Regular presentations;45
3.5.2;4.2. Complexity and curvature;49
3.5.3;4.3. Growth problems;52
3.6;5. Modules over Golod Rings;55
3.6.1;5.1. Hypersurfaces;55
3.6.2;5.2. Golod rings;57
3.6.3;5.3. Golod modules;61
3.7;6. Tate Resolutions;64
3.7.1;6.1. Construction;65
3.7.2;6.2. Derivations;69
3.7.3;6.3. Acyclic closures;72
3.8;7. Deviations of a Local Ring;75
3.8.1;7.1. Deviations and Betti numbers;76
3.8.2;7.2. Minimal models;77
3.8.3;7.3. Complete intersections;82
3.8.4;7.4. Localization;83
3.9;8. Test Modules;86
3.9.1;8.1. Residue field;86
3.9.2;8.2. Residue domains;88
3.9.3;8.3. Conormal modules;94
3.10;9. Modules over Complete Intersections;98
3.10.1;9.1. Cohomology operators;98
3.10.2;9.2. Betti numbers;103
3.10.3;9.3. Complexity and Tor;106
3.11;10. Homotopy Lie Algebra of a Local Ring;110
3.11.1;10.1. Products in cohomology;111
3.11.2;10.2. Homotopy Lie algebra;114
3.11.3;10.3. Applications;117
3.12;References;121
4;Generic Initial Ideals;130
4.1;Introduction;130
4.2;1. The Initial Ideal;131
4.3;2. Regularity and Saturation;148
4.4;3. The Macaulay-Gotzmann Estimates on the Growth of Ideals;158
4.5;4. Points in P2 and Curves in P3;168
4.6;5. Gins in the Exterior Algebra;183
4.7;6. Lexicographic Gins and Partial Elimination Ideals;188
4.8;References;196
5;Tight Closure, Parameter Ideals,and Geometry;198
5.1;Foreword;198
5.2;1. An Introduction to Tight Closure;198
5.3;2. How Does Tight Closure Arise?;204
5.4;3. The Test Ideal I;211
5.5;4. The Test Ideal II: the Gorenstein Case;216
5.6;5. The Tight Closure of Parameter Ideals;221
5.7;6. The Strong Vanishing Theorem;226
5.8;7. Plus Closure;230
5.9;8. F-Rational Rings;233
5.10;9. Rational Singularities;236
5.11;10. The Kodaira Vanishing Theorem;239
5.12;References;242
6;On the Use of Local Cohomology in Algebra and Geometry;252
6.1;Introduction;252
6.2;1. A Guide to Duality;254
6.2.1;1.1. Local Duality.;254
6.2.2;1.2. Dualizing Complexes and Some Vanishing Theorems.;260
6.2.3;1.3. Cohomological Annihilators.;267
6.3;2. A Few Applications of Local Cohomology;270
6.3.1;2.1. On Ideal Topologies.;270
6.3.2;2.2. On Ideal Transforms.;274
6.3.3;2.3. Asymptotic Prime Divisors.;276
6.3.4;2.4. The Lichtenbaum-Hartshorne Vanishing Theorem.;283
6.3.5;2.5. Connectedness Results.;284
6.4;3. Local Cohomology and Syzygies;287
6.4.1;3.1. Local Cohomology and Tor’s.;287
6.4.2;3.2. Estimates of Betti Numbers.;292
6.4.3;3.3. Castelnuovo-Mumford Regularity.;293
6.4.4;3.4. The Local Green Modules.;297
6.5;References;301
7;Problems and Results on Hilbert Functions of Graded Algebras;304
7.1;Introduction;304
7.2;1. Macaulay’s Theorem;307
7.3;2. The Perfect Codimension Two and Gorenstein Codimension Three Case;312
7.4;3. The EGH Conjecture;322
7.5;4. Hilbert Function of Generic Algebras;328
7.6;5. Fat Points: Waring’s Problem and Symplectic Packing;330
7.7;6. The HF of a CM Local Ring;340
7.8;References;352
8;Cohomological Degrees of Graded Modules;356
8.1;Introduction;356
8.2;1. Arithmetic Degree of a Module;360
8.2.1;Multiplicity;360
8.2.2;Castelnuovo–Mumford regularity;361
8.2.3;Arithmetic degree of a module;362
8.2.4;Stanley–Reisner rings;363
8.2.5;Computation of the arithmetic degree of a module;364
8.2.6;Degrees and hyperplane sections;365
8.2.7;Arithmetic degree and hyperplane sections;365
8.3;2. Reduction Number of an Algebra;368
8.3.1;Castelnuovo–Mumford regularity and reduction number;368
8.3.2;Hilbert function and the reduction number of an algebra;369
8.3.3;The relation type of an algebra;370
8.3.4;Cayley–Hamilton theorem;371
8.3.5;The arithmetic degree of an algebra versus its reduction number;372
8.3.6;Reduction equations from integrality equations;374
8.4;3. Cohomological Degree of a Module;375
8.4.1;Big degs;375
8.4.2;Dimension one;376
8.4.3;Homological degree of a module;376
8.4.4;Dimension two;377
8.4.5;Hyperplane section;378
8.4.6;Generalized Cohen–Macaulay modules;381
8.4.7;Homologically associated primes of a module;382
8.4.8;Homological degree and hyperplane sections;383
8.4.9;Homological multiplicity of a local ring;387
8.5;4. Regularity versus Cohomological Degrees;388
8.5.1;Castelnuovo regularity;389
8.6;5. Cohomological Degrees and Numbers of Generators;391
8.7;6. Hilbert Functions of Local Rings;392
8.7.1;Bounding rules;393
8.7.2;Maximal Hilbert functions;394
8.7.3;Gorenstein ideals;396
8.7.4;General local rings;396
8.7.5;Bounding reduction numbers;397
8.7.6;Primary ideals;398
8.7.7;Depth conditions;399
8.8;7. Open Questions;400
8.8.1;Bounds problems;400
8.8.2;Cohomological degrees problems;401
8.9;References;401
9;Index;404