Table of 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
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 Flat Modules 79
Exercises 85
5 Tensor, Symmetric and Exterior Algebras 89
5.1 Tensor Product of Algebras 89
5.2 Tensor Algebras 92
5.3 Symmetric 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
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 Hilbert 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
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
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 Regular 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
20 Regular Rings 347
20.1 Regular Local Rings 347
20.2 A Differential Criterion for Regularity 350
20.3 A Homological Criterion for Regularity 352
20.4 Regular Rings 353
20.5 A Regular Local Ring is a UFD 354
20.6 The Jacobian Criterion for Geometric 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 Case of Galois Descent 373
21.5 Galois Descent in the Local Case 377
Exercises 381
Bibliography 383
Index 385
