作者：（美）艾森巴德
页数：785
出版社：世界图书出版公司
出版日期：2008
ISBN：9787506292450
电子书格式：pdf/epub/txt
内容简介
本书是Springer“数学研究生教材”第150卷，为了更好的理解交换代数,运用几何的观点，即代数几何学观点去研究交换代数，是本书的一大特色。作者从基本概念——局部化以及自分解理论出发，通过对维数理论、微分理论、同调方法、自由解理论和对偶性的研究,强调这些理论的出发点以及它们与数学其它分支的联系，书中的大量练习更强化了对读者对这些理论的理解。
作者简介
D.Eisenbud，加利福尼亚大学(University of California)数学系教授。加州大学，位于美国加州的一个由数所公立大学组成的大学系统，也是世界上影响较大力的公立大学系统，被誉为”世界最好的公立大学”和”公立高等教育的典范”。
目录
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
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
