內容大鋼
本書是Springer《Graduate Texts in Mathematics》系列叢書第150卷。為了更好的理解交換代數,運用幾何的觀點去研究交換代數，也就是代數幾何學觀點，是本書的一大特色。作者從基本觀點——局部化以及自分解理論出發，通過對維數理論、微分理論、同調方法、自由解理論和對偶性的研究,強調該理論的出發點以及它們與數學其他部分的聯繫，練習中大量的引用強化了對該理論的理解。本書的還專門運用了一章來講述Grobner基本觀點以及基於這個關點的對交換代數以及代數幾何很有建設性的方法。
目錄
Introduction
Advice for the Beginner
Information for the Expert
Prerequisites
Sources
Courses
A First Courses
A Second Courses
Acknowledgements
0 Elementary Definitions
0.1 Rings and Ideals
0.2 Unique Factorization
0.3 Modules
Ⅰ Basic Constructions
1 Roots of Commutative Algebra
1.1 Number Theory
1.2 Algebraic Curves and Fhnction Theory
1.3 Invariant Theory
1.4 The Basis Theorem
1.4.1 Finite Generation of Invariants
1.5 Graded Rings
1.6 Algebra and Geometry： The Nullstellensatz
1.7 Geometric Invariant Theory
1.8 Projective Varieties
1.9 Hilbert Functions and Polynomials
1.10 Free Resolutions and the Syzygy Theorem
1.11 Exercises
Noetherian Rings and Modules
An Analysis of Hilbert's Finiteness Argument
Some Rings of Invariants
Algebra and Geometry
Graded Rings and Projective Geometry
Hilbert Functions
Free Resolutions
Spec, max-Spec, and the Zariski Topology
2 Localization
2.1 Fractions
2.2 Horn and Tensor
2.3 The Construction of Primes
2.4 Rings and Modules of Finite Length
2.5 Products of Domains
2.6 Exercises
Z-graded Rings and Their Localizations
Partitions of Unity
Gluing
Constructing Primes
Idempotents, Products, and Connected Components
3 Associated Primes and Primary Decomposition
3.1 Associated Primes
3.2 Prime Avoidance
3.3 Primary Decomposition
3.4 Primary Decomposition and Factoriality
3.5 Primary Decomposition in the Graded Case
3.6 Extracting Information from Primary Decomposition
3.7 Why Primary Decomposition Is Not Unique
3.8 Geometric Interpretation of Primary Decomposition
3.9 Symbolic Powers and Functions Vanishing to High Order
3.9.1 A Determinantal Example
3.10 Exercises
General Graded Primary Decomposition
Primary Decomposition of Monomial Ideals
The Question of Uniqueness
Determinantal Ideals
……
Ⅱ Dimension Theory
Ⅲ Homological Methods
Appendix 1 Field Theory
Appendix 2 Multilinear Algebra
Appendix 3 Homological Algebra
Appendix 4 A Sketch of Local Cohomology
Appendix 5 Category Theory
Appendix 6 Limits and Colimits
Appendix 7 Where Next
Hints and Solutions for Selected Exercises
References
Index of Notation
Index
[