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Eisenbud, English, Paperback, 2008

Product Details

Author David Eisenbud

Press Beijing World Publishing Corporation

Isbn 9787506292450

Publication Date May 2008

Pages 797

Size 24k

Edition First

Cover Paperback

Language English

Page material gelatine plate paper

Product Description:

It has seemed to me for a long time that commutative algebra is best practiced with knowledge of the geometric ideas that played a great role in its formation: in short, with a view toward algebraic geometry.

Most texts on commutative algebra adhere to the tradition that says a subject should be purified until it references nothing outside itself. There are good reasons for cultivating this style; it leads to generality, elegance, and brevity, three cardinal virtues. But it seems' to me unnecessary and undesirable to banish, on these grounds, the motivating and fructifying ideas on which the discipline is based.

Content:

Introduction

Advice for the Beginner

Information for the Expert

Prerequisites

Sources

Courses

Acknowledgements

0 Elementary Definitions

0.1 Rings and Ideals

0.2 Unique Factorization

0.3 Modules

1 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

1.5 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

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

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

3.10 Exercises

4 Integral Dependence and the Nullstellensatz

4.1 The Cayley-Hamilton Theorem and Nakayama's Lemma

4.2 Normal Domains and the Normalization Process

4.3 Normalization in the Analytic Case

4.4 Primes in an Integral Extension

4.5 The Nullstellensatz

4.6 Exercises

5 Filtrations and the Artin-Rees Lemma

5.1 Associated Graded Rings and Modules

5.2 The Blowup Algebra

5.3 The Krull Intersection Theorem

5.4 The Tangent Cone

5.5 Exercises

6 Flat Families

6.1 Elementary Examples

6.2 Introduction to Tor

6.3 Criteria for Flatness

6.4 The Local Criterion for Flatness

6.5 The Rees Algebra

6.6 Exercises

7 Completions and Hensel's Lemma

7.1 Examples and Definitions

7.2 The Utility of Completions

7.3 Lifting Idempotents

7.4 Cohen Structure Theory and Coefficient Fields

7.5 Basic Properties of Completion

7.6 Maps from Power Series Rings

7.7 Exercises

2 Dimension Theory

8 Introduction to Dimension Theory

8.1 Axioms for Dimension

8.2 Other Characterizations of Dimension Fundamental Definitions of Dimension Theory

9.1 Dimension Zero

9.2 Exercises

10 The Principal Ideal Theorem and Systems of Parameters

10.1 Systems of Parameters and Ideals of Finite Colength

10.2 Dimension of Base and Fiber

10.3 Regular Local Rings

10.4 Exercises

11 Dimension and Codimension One

11.1 Discrete Valuation Rings

11.2 Normal Rings and Serre's Criterion

11.3 Invertible Modules

11.4 Unique Factorization of Codimension-One Ideals

11.5 Divisors and Multiplicities

11.6 Multiplicity of Principal Ideals

11.7 Exercises

12 Dimension and Hilbert-Samuel Polynomials

12.1 Hilbert-Samuel Functions

12.2 Exercises

13 The Dimension of Affine Rings

13.1 Noether Normalization

13.2 The Nullstellensatz

13.3 Finiteness of the Integral Closure

13.4 Exercises

14 Elimination Theory, Generic Freeness, and the Dimension of Fibers

14.1 Elimination Theory

14.2 Generic Preeness

14.3 The Dimension of Fibers

14.4 Exercises

15 GrSbner Bases

15.1 Monomials and Terms

15.2 Monomial Orders

15.3 The Division Algorithm

15.4 Gr5bner Bases

15.5 Syzygies

15.6 History of Gr5bner Bases

15.7 A Property of Reverse Lexicographic Order

15.8 Gr5bner Bases and Flat Families

15.9 Generic Initial Ideals

15.10 Applications

15.11 Exercises

15.12 Appendix: Some Computer Algebra Projects

16 Modules of Differentials

16.1 Computation of Differentials

16.2 Differentials and the Cotangent Bundle

16.3 Colimits and Localization

16.4 Tangent Vector Fields and Infinitesimal Morphisms

16.5 Differentials and Field Extensions

16.6 Jacobian Criterion for Regularity

16.7 Smoothness and Generic Smoothness

16.8 Appendix: Another Construction of Kahler Differentials

16.9 Exercises

3 Homological Methods

17 Regular Sequences and the Koszul Complex

17.1 Koszul Complexes of Lengths I and 2

17.2 Koszul Complexes in General

17.3 Building the Koszul Complex from Parts

17.4 Duality and Homotopies

17.5 The Koszul Complex and the Cotangent Bundle of Projective Space

17.6 Exercises

18 Depth, Codimension, and Cohen-Macaulay Rings

18.1 Depth

18.2 Cohen-Macaulay Rings

18.3 Proving Primeness with Serre's Criterion

18.4 Flatness and Depth

18.5 Some Examples

18.6 Exercises

19 Homological Theory of Regular Local Rings

19.1 Projective Dimension and Minimal Resolutions

19.2 Global Dimension and the Syzygy Theorem

19.3 Depth and Projective Dimension: The Auslander-Buchsbaum Formula

19.4 Stably Free Modules and Factoriality of Regular Local Rings

19.5 Exercises

20 Free Resolutions and Fitting Invariants

20.1 The Uniqueness of Free Resolutions

20.2 Fitting Ideals

20.3 What Makes a Complex Exact?

20.4 The Hilbert-Burch Theorem

20.5 Castelnuovo-Mumford Regularity

20.6 Exercises

21 Duality, Canonical Modules, and Gorenstein Rings

21.1 Duality for Modules of Finite Length

21.2 Zero-Dimensional Gorenstein Rings

21.3 Canonical Modules and Gorenstein Rings in Higher Dimension

21.4 Maximal Cohen-Macaulay Modules

21.5 Modules of Finite Injective Dimension

21.6 Uniqueness and (Often) Existence

21.7 Localization and Completion of the Canonical Module

21.8 Complete Intersections and Other Gorenstein Rings

21.9 Duality for Maximal Cohen-Macaulay Modules

21.10 Linkage

21.11 Duality in the Graded Case

21.12 Exercises

Appendix 1 Field Theory

A1.1 Transcendence Degree

A1.2 Separability

A1.3 p-Bases

Appendix 2 Multilinear Algebra

A2.1 Introduction

A2.2 Tensor Products

A2.3 Symmetric and Exterior Algebras

A2.4 Coalgebra Structures and Divided Powers

A2.5 Schur Functors

A2.6 Complexes Constructed by Multilinear Algebra

Appendix 3 Homological Algebra

A3.1 Introduction

Part 1: Resolutions and Derived Functors

A3.2 Free and Projective Modules

A3.3 Free and Projective Resolutions

A3.4 Injective Modules and Resolutions

A3.5 Basic Constructions with Complexes

A3.6 Maps and Homotopies of Complexes

A3.7 Exact Sequences of Complexes

A3.8 The Long Exact Sequence in Homology

A3.9 Derived Functors

A3.10 Tor

A3.11 Ext

Part 2: From Mapping Cones to Spectral Sequences

A3.12 The Mapping Cone and Double Complexes

A3.13 Spectral Sequences

A3.14 Derived Categories

Appendix 4 A Sketch of Local Cohomology

A4.1 Local Cohomology and Global Cohomology

A4.2 Local Duality

A4.3 Depth and Dimension

Appendix 5 Category Theory

A5.1 Categories, Functors, and Natural Transformations

A5.2 Adjoint Functors

A5.3 Representable Functors and Yoneda's Lemma

Appendix 6 Limits and Colimits

A6.1 Colimits in the Category of Modules

A6.2 Flat Modules as Colimits of Free Modules

A6.3 Colimits in the Category of Commutative Algebras

A6.4 Exercises

Appendix 7 Where Next?

Hints and Solutions for Selected Exercises

References

Index of Notation

Index

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