Commutative Algebra

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.

Table of Contents

0 Elementary Definitions

0.1 Rings and Ideals

0.2 Unique Factorization

0.3 Modules

I Basic Constructions

1 Roots of Commutative Algebra

2 Localization

3 Associated Primes and Primary Decomposition

4 Integral Dependence and the Nullstellensatz

5 Filtrations and the Artin-Rees Lemma

6 Flat Families

7 Completions and Hensel’s Lemma

II Dimension Theory

8 Introduction to Dimension Theory

9 Fundamental Definitions of Dimension Theory

10 The Principal Ideal Theorem and Systems of Parameters

11 Dimension and Codimension One

12 Dimension and Hilbert-Samuel Polynomials

13 The Dimension of Affine Rings

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

15 Grobner Bases

16 Modules of Differentials

Ill Homological Methods

J 17 Regular Sequences and the Koszul Complex

J 18 Depth, Codimension, and Cohen-Macaulay Rings

J 19 Homological Theory of Regular Local Rings

20 The Resolutions and Fitting Invariants

21 Duality, Canonical Modules, and Gorenstein Rings

Review

D. Eisenbud

Commutative Algebra with a View Toward Algebraic Geometry

“This text has personality―Those familiar with Eisenbud”s own research will recognize its traces in his choice of topics and manner of approach. The book conveys infectious enthusiasm and the conviction that research in the field is active and yet accessible.“―MATHEMATICAL REVIEWS