A Course in Commutative Algebra

by Gregor Kemper

Overview

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.

Product Details

ISBN-13:	9783642035449
Publisher:	Springer Berlin Heidelberg
Publication date:	12/10/2010
Series:	Graduate Texts in Mathematics , #256
Edition description:	2011
Pages:	248
Product dimensions:	6.40(w) x 9.20(h) x 1.10(d)

About the Author

The author is professor of algorithmic algebra at the Technische Universität München, in Munich. He regularly teaches courses in commutative algebra, invariant theory, and computer algebra. In 2007 he received an award from the state of Bavaria for excellence in teaching.

Table of Contents

Introduction 1

Part I The Algebra-Geometry Lexicon

1 Hilbert's Nullstellensatz 7

1.1 Maximal Ideals 8

1.2 Jacobson Rings 12

1.3 Coordinate Rings 16

Exercises 19

2 Noetherian and Artinian Rings 23

2.1 The Noether and Artin Properties for Rings and Modules 23

2.2 Noetherian Rings and Modules 28

Exercises 30

3 The Zariski Topology 33

3.1 Affine Varieties 33

3.2 Spectra 36

3.3 Noetherian and Irreducible Spaces 38

Exercises 42

4 A Summary of the Lexicon 45

4.1 True Geometry: Affine Varieties 45

4.2 Abstract Geometry: Spectra 46

Exercises 48

Part II Dimension

5 Krull Dimension and Transcendence Degree 51

Exercises 60

6 Localization 63

Exercises 70

7 The Principal Ideal Theorem 75

7.1 Nakayama's Lemma and the Principal Ideal Theorem 75

7.2 The Dimension of Fibers 81

Exercises 87

8 Integral Extensions 93

8.1 Integral Closure 93

8.2 Lying Over, Going Up, and Going Down 99

8.3 Noether Normalization 104

Exercises 111

Part III Computational Methods

9 Gröbner Bases 117

9.1 Buchberger's Algorithm 118

9.2 First Application: Elimination Ideals 127

Exercises 133

10 Fibers and Images of Morphisms Revisited 137

10.1 The Generic Freeness Lemma 137

10.2 Fiber Dimension and Constructible Sets 142

10.3 Application: Invariant Theory 144

Exercises 148

11 Hilbert Series and Dimension 151

11.1 The Hilbert-Serre Theorem 151

11.2 Hilbert Polynomials and Dimension 157

Exercises 161

Part IV Local Rings

12 Dimension Theory 167

12.1 The Length of a Module 167

12.2 The Associated Graded Ring 170

Exercises 176

13 Regular Local Rings 181

13.1 Basic Properties of Regular Local Rings 181

13.2 The Jacobian Criterion 185

Exercises 193

14 Rings of Dimension One 197

14.1 Regular Rings and Normal Rings 197

14.2 Multiplicative Ideal Theory 201

14.3 Dedekind Domains 206

Exercises 212

Solutions of Some Exercises 217

References 235

Notation 239

Index 241

Editorial Reviews

From the reviews:

“This recent addition to Springer’s famous Graduate Texts in Mathematics (GTM) series comprises a thorough, modern introduction to commutative algebra with the central concepts and results almost exclusively motivated by their applications in algebraic geometry. … The author writes in an engaging, reader-friendly manner. … The main concepts are deftly presented and well-motivated and key ideas and methods are clearly highlighted. … This textbook will be a very useful entrée for beginning graduate students … .” (Nick Lord, The Mathematical Gazette, Vol. 97 (539), July, 2013)

“This book is an introductory text to commutative algebra with the idea also of being a guide to the algorithmic branch of the subject. … this is a valuable and readable textbook on modern commutative algebra. It contains a huge number of exercises and it appeals to geometric intuition whenever possible. It can be highly recommended for independent reading or as material for preparation of courses.” (Alejandro Melle Hernandez, The European Mathematical Society, July, 2012)

“This book gives a modern introduction to commutative algebra for students who had a first course in abstract algebra and who are familiar with the most basic notions of topology. … The book can also serve as a first introduction to algebraic geometry. A strong feature of the book are the interesting exercises which nicely complement and illustrate the theory. Altogether this a stimulating book on a classical subject with an emphasis on the connection to algebraic geometry.” (J. Mahnkopf, Monatshefte für Mathematik, Vol. 164 (3), November, 2011)

“A student of algebraic geometry might naturally wish to make quick work of the algebraic prerequisites and progress. This volume’s focused selection of material and geometrically informed viewpoint are tailored to this purpose. … this is a useful, practical, and unintimidating book, especially suitable for advanced undergraduate students. Summing Up: Recommended. Upper-division undergraduates.” ( D. V. Feldman, Choice, Vol. 48 (11), August, 2011)

“This is a well-written book that goes right away to the core of the subject: Commutative algebra as an introduction to algebraic geometry … . This being said, this is a fine book, in the class of many other well established and fine books on the subject … . I am sure it could become a text of choice for an introductory course in commutative algebra. Both the lecturer and the student would benefit from its balanced and novel approach to the subject.” (Felipe Zaldivar, The Mathematical Association of America, March, 2011)

“This book provides a selection of basic material for an introductory graduate course in commutative algebra. … it is intended to serve as a useful source for both instructors and students in the field, in particular also for the purpose of profound self-study. … the exercises often come with directing hints helping the reader to tackle them successfully. … Being really tailor-made for effective teaching and self-study, the text stands out by its expository mastery, didactic skill, versatility, expediency, and absolute reader-friendliness.” (Werner Kleinert, Zentralblatt MATH, Vol. 1210, 2011)

“This book is a pleasant elementary course in commutative algebra with views toward algebraic geometry, computer algebra and invariant theory. … It also studies Noetherian topological spaces and their irreducible components with an application to the spectrum of a Noetherian ring. … Each chapter contains many exercises, and some of them are solved at the end of the book.” (D.-M. Popescu, Mathematical Reviews, Issue 2011 j)

From the Publisher