Contents
Preface
Introduction
Conventions and terminology
1 Commutative rings and modules
1 Ideals
2 Modules
3 Chain conditions
2 Prime ideals
4 Localisation and Spec of a ring
^ The Hilbert Nullstellensatz and first steps in dimension theory
6 Associated primes and primary decomposition Appendix to §6. Secondary representations of a module
3 Properties of extension rings
7 Flatness
Appendix to §7. Pure submodules
8 Completion and the Artin—Rees lemma
9 Integral extensions
4 Valuation rings
10 General valuations
11 DVRs and Dedekind rings
12 Krull rings
5 Dimension theory
^ Graded rings, the Hilbert function and the Samuel function Appendix to §13. Determinantal ideals
14 Systems of parameters and multiplicity
15 The dimension of extension rings
6 Regular sequences
16 Regular sequences and the Koszul complex
17 Cohen-Macaulay rings
18 Gorenstein rings
7 Regular rings
19 Regular rings
20 UFDs
21 Complete intersection rings
vii
22 The local flatness criterion 17g
23 Flatness and fibres 185
24 Generic freeness and open loci results ^
9 Derivations 190
25 Derivations and differentials
26 Separability 207
27 Higher derivations 213
10 I-smoothness 213
28 1-smoothness 223
29 The structure theorems for complete oca rings 230
30 Connections with derivations 246
11 Applications of complete local rings 246
31 Chains of prime ideals 255
32 The formal fibre 261
33 Some other applications
Appendix Л. Tensor products, direct and inverse limits ^66
Appendix B. Some homological algebra 283
Appendix C. The exterior algebra 2g7
Solutions and hints for exercises 298
References 315
Index