A Course in Commutative Algebra

by Kemper, Gregor

• ISBN13: 9783642035449
• ISBN10: 3642035442
• Format: Hardcover
• Copyright: 2010-12-10
• Publisher: Springer Nature

Summary

This textbook offers a thorough, modern introduction into commutative algebra. It is intented mainly to serve as a guide for a course of one or two semesters, or for self-study. The carefully selected subject matter concentrates on the concepts and results at the center of the field. The book maintains a constant view on the natural geometric context, enabling the reader to gain a deeper understanding of the material. Although it emphasizes theory, three chapters are devoted to computational aspects. Many illustrative examples and exercises enrich the text, and an included CD provides solutions to the exercises.

Table of Contents

Introduction	p. 1
The Algebra-Geometry Lexicon
Hilbert's Nullstellensatz	p. 7
Maximal Ideals	p. 8
Jacobson Rings	p. 12
Coordinate Rings	p. 16
Exercises	p. 19
Noetherian and Artinian Rings	p. 23
The Noether and Artin Properties for Rings and Modules	p. 23
Noetherian Rings and Modules	p. 28
Exercises	p. 30
The Zariski Topology	p. 33
Affine Varieties	p. 33
Spectra	p. 36
Noetherian and Irreducible Spaces	p. 38
Exercises	p. 42
A Summary of the Lexicon	p. 45
True Geometry: Affine Varieties	p. 45
Abstract Geometry: Spectra	p. 46
Exercises	p. 48
Dimension
Krull Dimension and Transcendence Degree	p. 51
Exercises	p. 60
Localization	p. 63
Exercises	p. 70
The Principal Ideal Theorem	p. 75
Nakayama's Lemma and the Principal Ideal Theorem	p. 75
The Dimension of Fibers	p. 81
Exercises	p. 87
Integral Extensions	p. 93
Integral Closure	p. 93
Lying Over, Going Up, and Going Down	p. 99
Noether Normalization	p. 104
Exercises	p. 111
Computational Methods
Gröbner Bases	p. 117
Buchberger's Algorithm	p. 118
First Application: Elimination Ideals	p. 127
Exercises	p. 133
Fibers and Images of Morphisms Revisited	p. 137
The Generic Freeness Lemma	p. 137
Fiber Dimension and Constructible Sets	p. 142
Application: Invariant Theory	p. 144
Exercises	p. 148
Hilbert Series and Dimension	p. 151
The Hilbert-Serre Theorem	p. 151
Hilbert Polynomials and Dimension	p. 157
Exercises	p. 161
Local Rings
Dimension Theory	p. 167
The Length of a Module	p. 167
The Associated Graded Ring	p. 170
Exercises	p. 176
Regular Local Rings	p. 181
Basic Properties of Regular Local Rings	p. 181
The Jacobian Criterion	p. 185
Exercises	p. 193
Rings of Dimension One	p. 197
Regular Rings and Normal Rings	p. 197
Multiplicative Ideal Theory	p. 201
Dedekind Domains	p. 206
Exercises	p. 212
Solutions of Some Exercises	p. 217
References	p. 235
Notation	p. 239
Index	p. 241
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