Basic Commutative Algebra

by Singh, Balwant

• ISBN13: 9789814313629
• ISBN10: 9814313629
• Format: Paperback
• Copyright: 2010-12-31
• Publisher: World Scientific Pub Co Inc

Summary

This textbook, set for a one or two semester course in commutative algebra, provides an introduction to commutative algebra at the postgraduate and research levels. The main prerequisites are familiarity with groups, rings and fields. Proofs are self-contained.

Table of Contents

Preface	p. vii
Rings and Ideals	p. 1
Recollection and Preliminaries	p. 1
Prime and Maximal Ideals	p. 2
Sums, Products and Colons	p. 6
Radicals	p. 8
Zariski Topology	p. 9
Exercises	p. 10
Modules and Algebras	p. 13
Modules	p. 13
Homomorphisms	p. 17
Direct Products and Direct Sums	p. 19
Free Modules	p. 23
Exact Sequences	p. 25
Algebras	p. 27
Fractions	p. 30
Graded Rings and Modules	p. 35
Homogeneous Prime and Maximal Ideals	p. 38
Exercises	p. 40
Polynomial and Power Series Rings	p. 45
Polynomial Rings	p. 45
Power Series Rings	p. 47
Exercises	p. 53
Homological Tools I	p. 55
Categories and Functors	p. 55
Exact Functors	p. 58
The Functor Hom	p. 61
Tensor Product	p. 65
Base Change	p. 74
Direct and Inverse Limits	p. 76
Injective, Projective and Flat Modules	p. 79
Exercises	p. 85
Tensor, Symmetric and Exterior Algebras	p. 89
Tensor Product of Algebras	p. 89
Tensor Algebras	p. 92
Symmetric Algebras	p. 94
Exterior Algebras	p. 97
Anticommutative and Alternating Algebras	p. 101
Determinants	p. 106
Exercises	p. 109
Finiteness Conditions	p. 111
Modules of Finite Length	p. 111
Noetherian Rings and Modules	p. 115
Artinian Rings and Modules	p. 120
Locally Free Modules	p. 123
Exercises	p. 126
Primary Decomposition	p. 129
Primary Decomposition	p. 129
Support of a Module	p. 135
Dimension	p. 138
Exercises	p. 139
Filtrations and Completions	p. 143
Filtrations and Associated Graded Rings and Modules	p. 143
Linear Topologies and Completions	p. 147
Ideal-adic Completions	p. 151
Initial Submodules	p. 153
Completion of a Local Ring	p. 154
Exercises	p. 156
Numerical Functions	p. 159
Numerical Functions	p. 159
Hilbert Function of a Graded Module	p. 162
Hilbert-Samuel Function over a Local Ring	p. 163
Exercises	p. 167
Principal Ideal Theorem	p. 169
Principal Ideal Theorem	p. 169
Dimension of a Local Ring	p. 171
Exercises	p. 172
Integral Extensions	p. 175
Integral Extensions	p. 175
Prime Ideals in an Integral Extension	p. 178
Integral Closure in a Finite Field Extension	p. 182
Exercises	p. 184
Normal Domains	p. 187
Unique Factorization Domains	p. 187
Discrete Valuation Rings and Normal Domains	p. 192
Fractionary Ideals and Invertible Ideals	p. 198
Dedekind Domains	p. 199
Extensions of a Dedekind Domain	p. 203
Exercises	p. 207
Transcendental Extensions	p. 209
Transcendental Extensions	p. 209
Separable Field Extensions	p. 212
Lüroth's Theorem	p. 217
Exercises	p. 220
Affine Algebras	p. 223
Noether's Normalization Lemma	p. 223
Hilbert's Nullstellensatz	p. 226
Dimension of an Affine Algebra	p. 230
Dimension of a Graded Ring	p. 234
Dimension of a Standard Graded Ring	p. 236
Exercises	p. 239
Derivations and Differentials	p. 241
Derivations	p. 241
Differentials	p. 247
Exercises	p. 253
Valuation Rings and Valuations	p. 255
Valuations Rings	p. 255
Valuations	p. 258
Extensions of Valuations	p. 262
Real Valuations and Completions	p. 265
Hensel's Lemma	p. 274
Discrete Valuations	p. 276
Exercises	p. 280
Homological Tools II	p. 283
Derived Functors	p. 283
Uniqueness of Derived Functors	p. 286
Complexes and Homology	p. 291
Resolutions of a Module	p. 296
Resolutions of a Short Exact Sequence	p. 300
Construction of Derived Functors	p. 303
The Functors Ext	p. 308
The Functors Tor	p. 312
Local Cohomology	p. 314
Homology and Cohomology of Groups	p. 315
Exercises	p. 320
Homological Dimensions	p. 323
Injective Dimension	p. 323
Projective Dimension	p. 325
Global Dimension	p. 327
Projective Dimension over a Local Ring	p. 328
Exercises	p. 330
Depth	p. 331
Regular Sequences and Depth	p. 331
Depth and Projective Dimension	p. 336
Cohen-Macaulay Modules over a Local Ring	p. 338
Cohen-Macaulay Rings and Modules	p. 344
Exercises	p. 346
Regular Rings	p. 347
Regular Local Rings	p. 347
A Differential Criterion for Regularity	p. 350
A Homological Criterion for Regularity	p. 352
Regular Rings	p. 353
A Regular Local Ring is a UFD	p. 354
The Jacobian Criterion for Geometric Regularity	p. 356
Exercises	p. 362
Divisor Class Groups	p. 365
Divisor Class Groups	p. 365
The Case of Fractions	p. 369
The Case of Polynomial Extensions	p. 371
The Case of Galois Descent	p. 373
Galois Descent in the Local Case	p. 377
Exercises	p. 381
Bibliography	p. 383
Index	p. 385
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