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Basic Commutative Algebra - Balwant Singh

Basic Commutative Algebra

Paperback Published: 1st May 2011
ISBN: 9789814313629
Number Of Pages: 404

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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.

The book will be useful to beginners and experienced researchers alike. The material is so arranged that the beginner can learn through self-study or by attending a course. For the experienced researcher, the book may serve to present new perspectives on some well-known results, or as a reference.

The text covers in a reasonable number of pages a wealth of important topics of commutative algebra. Each section contains illustrative examples and is followed by a set of exercises. The text is suitable for graduate and postgraduate students, as well as for experienced researchers interested in commutative algebra. -- Mathematical Reviews "Mathematical Reviews"

Prefacep. vii
Rings and Idealsp. 1
Recollection and Preliminariesp. 1
Prime and Maximal Idealsp. 2
Sums, Products and Colonsp. 6
Radicalsp. 8
Zariski Topologyp. 9
Exercisesp. 10
Modules and Algebrasp. 13
Modulesp. 13
Homomorphismsp. 17
Direct Products and Direct Sumsp. 19
Free Modulesp. 23
Exact Sequencesp. 25
Algebrasp. 27
Fractionsp. 30
Graded Rings and Modulesp. 35
Homogeneous Prime and Maximal Idealsp. 38
Exercisesp. 40
Polynomial and Power Series Ringsp. 45
Polynomial Ringsp. 45
Power Series Ringsp. 47
Exercisesp. 53
Homological Tools Ip. 55
Categories and Functorsp. 55
Exact Functorsp. 58
The Functor Homp. 61
Tensor Productp. 65
Base Changep. 74
Direct and Inverse Limitsp. 76
Injective, Projective and Flat Modulesp. 79
Exercisesp. 85
Tensor, Symmetric and Exterior Algebrasp. 89
Tensor Product of Algebrasp. 89
Tensor Algebrasp. 92
Symmetric Algebrasp. 94
Exterior Algebrasp. 97
Anticommutative and Alternating Algebrasp. 101
Determinantsp. 106
Exercisesp. 109
Finiteness Conditionsp. 111
Modules of Finite Lengthp. 111
Noetherian Rings and Modulesp. 115
Artinian Rings and Modulesp. 120
Locally Free Modulesp. 123
Exercisesp. 126
Primary Decompositionp. 129
Primary Decompositionp. 129
Support of a Modulep. 135
Dimensionp. 138
Exercisesp. 139
Filtrations and Completionsp. 143
Filtrations and Associated Graded Rings and Modulesp. 143
Linear Topologies and Completionsp. 147
Ideal-adic Completionsp. 151
Initial Submodulesp. 153
Completion of a Local Ringp. 154
Exercisesp. 156
Numerical Functionsp. 159
Numerical Functionsp. 159
Hilbert Function of a Graded Modulep. 162
Hilbert-Samuel Function over a Local Ringp. 163
Exercisesp. 167
Principal Ideal Theoremp. 169
Principal Ideal Theoremp. 169
Dimension of a Local Ringp. 171
Exercisesp. 172
Integral Extensionsp. 175
Integral Extensionsp. 175
Prime Ideals in an Integral Extensionp. 178
Integral Closure in a Finite Field Extensionp. 182
Exercisesp. 184
Normal Domainsp. 187
Unique Factorization Domainsp. 187
Discrete Valuation Rings and Normal Domainsp. 192
Fractionary Ideals and Invertible Idealsp. 198
Dedekind Domainsp. 199
Extensions of a Dedekind Domainp. 203
Exercisesp. 207
Transcendental Extensionsp. 209
Transcendental Extensionsp. 209
Separable Field Extensionsp. 212
L├╝roth's Theoremp. 217
Exercisesp. 220
Affine Algebrasp. 223
Noether's Normalization Lemmap. 223
Hilbert's Nullstellensatzp. 226
Dimension of an Affine Algebrap. 230
Dimension of a Graded Ringp. 234
Dimension of a Standard Graded Ringp. 236
Exercisesp. 239
Derivations and Differentialsp. 241
Derivationsp. 241
Differentialsp. 247
Exercisesp. 253
Valuation Rings and Valuationsp. 255
Valuations Ringsp. 255
Valuationsp. 258
Extensions of Valuationsp. 262
Real Valuations and Completionsp. 265
Hensel's Lemmap. 274
Discrete Valuationsp. 276
Exercisesp. 280
Homological Tools IIp. 283
Derived Functorsp. 283
Uniqueness of Derived Functorsp. 286
Complexes and Homologyp. 291
Resolutions of a Modulep. 296
Resolutions of a Short Exact Sequencep. 300
Construction of Derived Functorsp. 303
The Functors Extp. 308
The Functors Torp. 312
Local Cohomologyp. 314
Homology and Cohomology of Groupsp. 315
Exercisesp. 320
Homological Dimensionsp. 323
Injective Dimensionp. 323
Projective Dimensionp. 325
Global Dimensionp. 327
Projective Dimension over a Local Ringp. 328
Exercisesp. 330
Depthp. 331
Regular Sequences and Depthp. 331
Depth and Projective Dimensionp. 336
Cohen-Macaulay Modules over a Local Ringp. 338
Cohen-Macaulay Rings and Modulesp. 344
Exercisesp. 346
Regular Ringsp. 347
Regular Local Ringsp. 347
A Differential Criterion for Regularityp. 350
A Homological Criterion for Regularityp. 352
Regular Ringsp. 353
A Regular Local Ring is a UFDp. 354
The Jacobian Criterion for Geometric Regularityp. 356
Exercisesp. 362
Divisor Class Groupsp. 365
Divisor Class Groupsp. 365
The Case of Fractionsp. 369
The Case of Polynomial Extensionsp. 371
The Case of Galois Descentp. 373
Galois Descent in the Local Casep. 377
Exercisesp. 381
Bibliographyp. 383
Indexp. 385
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9789814313629
ISBN-10: 9814313629
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 404
Published: 1st May 2011
Publisher: World Scientific Publishing Co Pte Ltd
Country of Publication: SG
Dimensions (cm): 22.61 x 14.99  x 2.03
Weight (kg): 0.6

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