Commutative Ring Theory (Cambridge Studies in Advanced Mathematics, Series Number 8)

✍ Scribed by H. Matsumura

• Publisher: Cambridge University Press
• Year: 1989
• Tongue: English
• Leaves: 336
• Edition: 1
• Category: Library

✦ Synopsis

In addition to being an interesting and profound subject in its own right, commutative ring theory is important as a foundation for algebraic geometry and complex analytical geometry. Matsumura covers the basic material, including dimension theory, depth, Cohen-Macaulay rings, Gorenstein rings, Krull rings and valuation rings. More advanced topics such as Ratliff's theorems on chains of prime ideals are also explored. The work is essentially self-contained, the only prerequisite being a sound knowledge of modern algebra, yet the reader is taken to the frontiers of the subject. Exercises are provided at the end of each section and solutions or hints to some of them are given at the end of the book.

✦ Table of Contents

Cover
Cambridge Studies in Advanced Mathematics
Title Page
Copyright Page
Contents
Preface
Introduction
Conventions and terminology
Chapter 1. Commutative rings and modules
1 Ideals
2 Modules
3 Chain conditions
Chapter 2. Prime ideals
4 Localisation and Spec of a ring
5 The Hilhert Nullstellensatz and first steps in dimension theory
6 Associated primes and primary decomposition
Appendix to 6. Secondary representations of a module
Chapter 3. Properties of extension rings
7 Flatness
Appendix to 7. Pure submodules
8 Completion and the Artin–Rees lemma
9 Integral extensions
Chapter 4. Valuation rings
10 General valuations
11 DVRs and Dedekind rings
12 Krull rings
Chapter 5. Dimension theory
13 Graded rings, the Hilhert function and the Samuel function
Samuel functions
Appendix to 13. Determinantal ideals (after Eagon–Northcott [1])
14 Systems of parameters and multiplicity
Multiplicity
15 The dimension of extension rings
1. Fibres
2. Polynomial and formal power series rings
3. The dimension inequality
4. The Rees ring and grsub(I)
Chapter 6. Regular sequences
16 Regular sequences and the Koszul complex
The Koszul complex
Grade
17 Cohen Macaulay rings
18 Gorenstein rings
Chapter 7. Regular rings
19 Regular rings
20 UFDs
21 Complete intersection rings
Chapter 8. Flatness revisited
22 The local flatness criterion
23 Flatness and fibres
24 Generic freeness and open loci results
Chapter 9. Derivations
25 Derivations and differentials
26 Separability
Differential bases
Imperfection modules and the Cartier equality
27 Higher derivations
Chapter 10. I-smoothness
28 I-smoothness
29 The structure theorems for complete local rings
30 Connections with derivations
Chapter 11. Applications of complete local rings
31 Chains of prime ideals
32 The formal fibre
33 Some other applications
Dimension of intersection
Integral closure of a Noetherian integral domain
Appendix A. Tensor products, direct and inverse limits
Tensor products
Change of coefficient ring
Tensor product of A-algebras
Direct limits
Inverse limits
Appendix B. Some homological algebra
Complexes
Double complexes
Projective and injective modules
The Tor functors
The Ext functors
Projective and injective dimensions
Derived functors
Injective hull
The five lemma
The snake lemma
Tensor product of complexes
Appendix C. The exterior algebra
Solutions and hints for the exercises
1-2
3-5
6-8
9
10-11
12-14
15-17
18-19
20-24
25-26, 28-29
30
References
Books
Research papers
Index
Back Cover