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Basic commutative algebra:

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Bibliographic Details
Main Author:	Singh, Balwant 1940- (Author)
Format:	Book
Language:	English
Published:	Singapore [u.a.] World Scientific 2011
Subjects:	Kommutative Algebra Lehrbuch
Links:	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
Physical Description:	XIV, 390 S.
ISBN:	9789814313612 9789814313629 9814313610 9814313629

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Record in the Search Index

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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
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spellingShingle	Singh, Balwant 1940- Basic commutative algebra Kommutative Algebra (DE-588)4164821-3 gnd
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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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