Commutative ring theory

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Описание

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