Basic commutative algebra:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Singapore [u.a.]
World Scientific
2011
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIV, 390 S. |
| ISBN: | 9789814313612 9789814313629 9814313610 9814313629 |
Internformat
MARC
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| 100 | 1 | |a Singh, Balwant |d 1940- |e Verfasser |0 (DE-588)143985639 |4 aut | |
| 245 | 1 | 0 | |a Basic commutative algebra |c Balwant Singh |
| 264 | 1 | |a Singapore [u.a.] |b World Scientific |c 2011 | |
| 300 | |a XIV, 390 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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Datensatz im Suchindex
| _version_ | 1804143823328116736 |
|---|---|
| adam_text | Titel: Basic commutative algebra
Autor: Singh, Balwant
Jahr: 2011
Contents
Preface vii
1. Rings and Ideals 1
1.0 Recollection and Preliminaries ................. 1
1.1 Prime and Maximal Ideals.................... 2
1.2 Sums, Products and Colons................... 6
1.3 Radicals.............................. 8
1.4 Zariski Topology......................... 9
Exercises................................. 10
2. Modules and Algebras 13
2.1 Modules.............................. 13
2.2 Homomorphisms......................... 17
2.3 Direct Products and Direct Sums................ 19
2.4 Free Modules........................... 23
2.5 Exact Sequences......................... 25
2.6 Algebras.............................. 27
2.7 Fractions ............................. 30
2.8 Graded Rings and Modules................... 35
2.9 Homogeneous Prime and Maximal Ideals............ 38
Exercises................................. 40
x Contents
3. Polynomial and Power Series Rings 45
3.1 Polynomial Rings......................... 45
3.2 Power Series Rings........................ 47
Exercises................................. 53
4. Homological Tools I 55
4.1 Categories and Functors..................... 55
4.2 Exact Functors.......................... 58
4.3 The Functor Hom ........................ 61
4.4 Tensor Product.......................... 65
4.5 Base Change........................... 74
4.6 Direct and Inverse Limits.................... 76
4.7 Injective, Projective and Fiat Modules............. 79
Exercises................................. 85
5. Tensor, Symmetrie and Exterior Algebras 89
5.1 Tensor Product of Algebras................... 89
5.2 Tensor Algebras ......................... 92
5.3 Symmetrie Algebras....................... 94
5.4 Exterior Algebras......................... 97
5.5 Anticommutative and Alternating Algebras.......... 101
5.6 Determinants........................... 106
Exercises................................. 109
6. Finiteness Conditions 111
6.1 Modules of Finite Length.................... 111
6.2 Noetherian Rings and Modules................. 115
6.3 Artinian Rings and Modules................... 120
6.4 Locally Free Modules....................... 123
Exercises................................. 126
Contents xi
7. Primary Decomposition 129
7.1 Primary Decomposition..................... 129
7.2 Support of a Module....................... 135
7.3 Dimension............................. 138
Exercises................................. 139
8. Filtrations and Completions 143
8.1 Filtrations and Associated Graded Rings and Modules .... 143
8.2 Linear Topologies and Completions............... 147
8.3 Ideal-adic Completions...................... 151
8.4 Initial Submodules........................ 153
8.5 Completion of a Local Ring................... 154
Exercises................................. 156
9. Numerical Functions 159
9.1 Numerical Functions....................... 159
9.2 Hubert Function of a Graded Module ............. 162
9.3 Hilbert-Samuel Function over a Local Ring.......... 163
Exercises................................. 167
10. Principal Ideal Theorem 169
10.1 Principal Ideal Theorem..................... 169
10.2 Dimension of a Local Ring.................... 171
Exercises................................. 172
11. Integral Extensions 175
11.1 Integral Extensions........................ 175
11.2 Prime Ideals in an Integral Extension ............. 178
11.3 Integral Closure in a Finite Field Extension.......... 182
Exercises................................. 184
xii Contents
12. Normal Domains 187
12.1 Unique Factorization Domains ................. 187
12.2 Discrete Valuation Rings and Normal Domains........ 192
12.3 Fractionary Ideals and Invertible Ideals ............ 198
12.4 Dedekind Domains........................ 199
12.5 Extensions of a Dedekind Domain ............... 203
Exercises................................. 207
13. Transcendental Extensions 209
13.1 Transcendental Extensions.................... 209
13.2 Separable Field Extensions ................... 212
13.3 Lüroth s Theorem ........................ 217
Exercises................................. 220
14. Affine Algebras 223
14.1 Noether s Normalization Lemma................ 223
14.2 Hilbert s Nullstellensatz..................... 226
14.3 Dimension of an Affine Algebra................. 230
14.4 Dimension of a Graded Ring .................. 234
14.5 Dimension of a Standard Graded Ring............. 236
Exercises................................. 239
15. Derivations and Differentials 241
15.1 Derivations............................ 241
15.2 Differentials............................ 247
Exercises................................. 253
16. Valuation Rings and Valuations 255
16.1 Valuations Rings......................... 255
16.2 Valuations............................. 258
16.3 Extensions of Valuations..................... 262
16.4 Real Valuations and Completions................ 265
Contents xiii
16.5 Hensel s Lemma ......................... 274
16.6 Discrete Valuations........................ 276
Exercises................................. 280
17. Homological Tools II 283
17.1 Derived Functors......................... 283
17.2 Uniqueness of Derived Functors................. 286
17.3 Complexes and Homology.................... 291
17.4 Resolutions of a Module..................... 296
17.5 Resolutions of a Short Exact Sequence............. 300
17.6 Construction of Derived Functors................ 303
17.7 The Functors Ext......................... 308
17.8 The Functors Tor......................... 312
17.9 Local Cohomology........................ 314
17.10 Homology and Cohomology of Groups............. 315
Exercises................................. 320
18. Homological Dimensions 323
18.1 Injective Dimension ....................... 323
18.2 Projective Dimension....................... 325
18.3 Global Dimension......................... 327
18.4 Projective Dimension over a Local Ring............ 328
Exercises................................. 330
19. Depth 331
19.1 Regulär Sequences and Depth.................. 331
19.2 Depth and Projective Dimension................ 336
19.3 Cohen-Macaulay Modules over a Local Ring ......... 338
19.4 Cohen-Macaulay Rings and Modules.............. 344
Exercises................................. 346
xiv Contents
20. Regulär Rings 347
20.1 Regulär Local Rings....................... 347
20.2 A Differential Criterion for Regularity............. 350
20.3 A Homological Criterion for Regularity............. 352
20.4 Regulär Rings........................... 353
20.5 A Regulär Local Ring is a UFD................. 354
20.6 The Jacobian Criterion for Geometrie Regularity....... 356
Exercises................................. 362
21. Divisor Class Groups 365
21.1 Divisor Class Groups....................... 365
21.2 The Case of Fractions...................... 369
21.3 The Case of Polynomial Extensions............... 371
21.4 The Caseof Galois Descent................... 373
21.5 Galois Descent in the Local Case................ 377
Exercises................................. 381
Bibliography 383
Index 385
|
| any_adam_object | 1 |
| author | Singh, Balwant 1940- |
| author_GND | (DE-588)143985639 |
| author_facet | Singh, Balwant 1940- |
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| classification_tum | MAT 130f |
| ctrlnum | (OCoLC)711817843 (DE-599)OBVAC08316061 |
| dewey-full | 512.44 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.44 |
| dewey-search | 512.44 |
| dewey-sort | 3512.44 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| illustrated | Not Illustrated |
| indexdate | 2024-07-09T22:53:47Z |
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| isbn | 9789814313612 9789814313629 9814313610 9814313629 |
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| spelling | Singh, Balwant 1940- Verfasser (DE-588)143985639 aut Basic commutative algebra Balwant Singh Singapore [u.a.] World Scientific 2011 XIV, 390 S. txt rdacontent n rdamedia nc rdacarrier Kommutative Algebra (DE-588)4164821-3 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Kommutative Algebra (DE-588)4164821-3 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021135594&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Singh, Balwant 1940- Basic commutative algebra Kommutative Algebra (DE-588)4164821-3 gnd |
| subject_GND | (DE-588)4164821-3 (DE-588)4123623-3 |
| title | Basic commutative algebra |
| title_auth | Basic commutative algebra |
| title_exact_search | Basic commutative algebra |
| title_full | Basic commutative algebra Balwant Singh |
| title_fullStr | Basic commutative algebra Balwant Singh |
| title_full_unstemmed | Basic commutative algebra Balwant Singh |
| title_short | Basic commutative algebra |
| title_sort | basic commutative algebra |
| topic | Kommutative Algebra (DE-588)4164821-3 gnd |
| topic_facet | Kommutative Algebra Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021135594&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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