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Edition: 95

Copyright: 1995

Publisher: Springer-Verlag New York

Published: 1995

International: No

Copyright: 1995

Publisher: Springer-Verlag New York

Published: 1995

International: No

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Commutative Algebra is best understood with knowledge of the geometric ideas that have played a great role in its formation, in short, with a view towards algebraic geometry. The author presents a comprehensive view of commutative algebra, from basics, such as localization and primary decomposition, through dimension theory, differentials, homological methods, free resolutions and duality, emphasizing the origins of the ideas and their connections with other parts of mathematics. Many exercises illustrate and sharpen the theory and extended exercises give the reader an active part in complementing the material presented in the text. One novel feature is a chapter devoted to a quick but thorough treatment of Grobner basis theory and the constructive methods in commutative algebra and algebraic geometry that flow from it. Applications of the theory and even suggestions for computer algebra projects are included. This book will appeal to readers from beginners to advanced students of commutative algebra or algebraic geometry. To help beginners, the essential ideals from algebraic geometry are treated from scratch. Appendices on homological algebra, multilinear algebra and several other useful topics help to make the book relatively self- contained. Novel results and presentations are scattered throughout the text.

**Eisenbud, David : University of California**

Elemenary Definitions

**Part I: Basic Constructions **

Roots of Commutative Algebra

Localization

Associated Primes and Primary Decomposition

Integral Dependence and the Nullstellensatz

Filtrations and the Artin-Rees Lemma

Flat Families

Completion's and Hensel's Lemma

**Part II: Dimension Theory **

Fundemental Definitions of Dimensional Theory

The Principal Ideal Theorem and Systems of Parameters

Dimension and Codimension One

Dimension and Hilbert-Samuel Polynomials

The Dimension of Affine

Elimination Theory, Generic Freeness, and the Dimension of Fibers

Grobner Bases

Modules of Differentials

**Part III: Homological Methods **

Regular Sequences and the Koszul Complex

Depth, Codimension, and Cohen-Macaulay Rings

Homological Theory of Regular Local Rings

Resolutions and Fitting Invariants

Duality, Canonical Modules, and Gerstein Rings

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Summary

Commutative Algebra is best understood with knowledge of the geometric ideas that have played a great role in its formation, in short, with a view towards algebraic geometry. The author presents a comprehensive view of commutative algebra, from basics, such as localization and primary decomposition, through dimension theory, differentials, homological methods, free resolutions and duality, emphasizing the origins of the ideas and their connections with other parts of mathematics. Many exercises illustrate and sharpen the theory and extended exercises give the reader an active part in complementing the material presented in the text. One novel feature is a chapter devoted to a quick but thorough treatment of Grobner basis theory and the constructive methods in commutative algebra and algebraic geometry that flow from it. Applications of the theory and even suggestions for computer algebra projects are included. This book will appeal to readers from beginners to advanced students of commutative algebra or algebraic geometry. To help beginners, the essential ideals from algebraic geometry are treated from scratch. Appendices on homological algebra, multilinear algebra and several other useful topics help to make the book relatively self- contained. Novel results and presentations are scattered throughout the text.

Author Bio

**Eisenbud, David : University of California**

Table of Contents

Elemenary Definitions

**Part I: Basic Constructions **

Roots of Commutative Algebra

Localization

Associated Primes and Primary Decomposition

Integral Dependence and the Nullstellensatz

Filtrations and the Artin-Rees Lemma

Flat Families

Completion's and Hensel's Lemma

**Part II: Dimension Theory **

Fundemental Definitions of Dimensional Theory

The Principal Ideal Theorem and Systems of Parameters

Dimension and Codimension One

Dimension and Hilbert-Samuel Polynomials

The Dimension of Affine

Elimination Theory, Generic Freeness, and the Dimension of Fibers

Grobner Bases

Modules of Differentials

**Part III: Homological Methods **

Regular Sequences and the Koszul Complex

Depth, Codimension, and Cohen-Macaulay Rings

Homological Theory of Regular Local Rings

Resolutions and Fitting Invariants

Duality, Canonical Modules, and Gerstein Rings

Publisher Info

Publisher: Springer-Verlag New York

Published: 1995

International: No

Published: 1995

International: No

**Eisenbud, David : University of California**

Elemenary Definitions

**Part I: Basic Constructions **

Roots of Commutative Algebra

Localization

Associated Primes and Primary Decomposition

Integral Dependence and the Nullstellensatz

Filtrations and the Artin-Rees Lemma

Flat Families

Completion's and Hensel's Lemma

**Part II: Dimension Theory **

Fundemental Definitions of Dimensional Theory

The Principal Ideal Theorem and Systems of Parameters

Dimension and Codimension One

Dimension and Hilbert-Samuel Polynomials

The Dimension of Affine

Elimination Theory, Generic Freeness, and the Dimension of Fibers

Grobner Bases

Modules of Differentials

**Part III: Homological Methods **

Regular Sequences and the Koszul Complex

Depth, Codimension, and Cohen-Macaulay Rings

Homological Theory of Regular Local Rings

Resolutions and Fitting Invariants

Duality, Canonical Modules, and Gerstein Rings