Introduction to Commutative Algebra and Algebraic Geometry

252 Pages·2013·25.81 MB·English

by  Ernst Kunz (auth.)

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Introduc tion to Commutative A lgebra
and Algebraic G eometry
Ernst K unz

translated by Michael Ackerman
with a preface by David Mumford
Reprint of the 1985 Edition
Ernst Kunz
Fakultät für Mathematik
Universität Regensburg
Regensburg, Germany
ISBN978-1-4614-5986-6 ISBN978-1-4614-5987-3 (eBook)
DOI10.1007/978-1-4614-5987-3
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ERNSTKUNZ
Introduction to Commutative Algebra
and Algebraic Geometry
translated by Michael Ackerman
with a preface by David Mumford
Birkhauser Boston • Basel • Berlin
EmstKunz
Fakultiit fiir Mathematik
Universitat Regensburg
8400 Regensburg
GERMANY
Originally published under the title
"Einfllhrung in die kommutative Algebra und algebraische Geometrie"
© Friedr, Viehweg & Sohn Verlagsgesellscbaft mbH, Braunschweig 1980
Library of Congress Cataloging-In-Publication Data
Kunz, Ernst, 1933-
Introduction to commutative algebra and
algebraic geometry.
Translation of: Einfllhrung in die
kommutative Algebra und algebraische
Geometrie.
Bibliography: p.
Includes index.
1. Comutative algebra. 2. Geometry,
Algebraic. I. Title
QA 251.3.K8613 512'.24 81-18059
ISBN 0-8176-3065-1 ISBN 3-7643-3065-1 AACR2
CIP-Kurztltelaufnahme der Deutscben BibUotbek
Kunz. Emst:
Introduction to commutative algebra and
algebraic geometry/Ernst Kunz --Boston;
Basel; Berlin: Birkhauser, 1985
Einheitssacht: Einfilhrung in die kommutative
Algebra und algebraische Geometrie (engl.)
ISBN 0-8176-3605-1 ISBN 3-7643-3065-1
m®
Printed on acid-free paper
© 1985 Birkhauser Boston Birkhiiuser ~
Copyright is not claimed for works of U.S. Government employees.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or
transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or
otherwise, without prior permission of the copyright owner.
Permission to photocopy for internal or personal use of specific clients is granted by Birkhliusrz Boston
for lilnries and other users registered with the Copyright Cearance Center (CCC), provided that the
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01970, U.S.A. Special requests should be addressed directly to Birkhiuser Boston, 675 Massachusetts
Avenue, Cambridge, MA 02139, U.S.A.
ISBN 0-8176-3065-1
ISBN 0-8176-3065-1
Printed and bOund by Edwards Brothers, Inc., Ann Arbor, Michigan.
Printed in the U.S.A.
98765
Contents
Foreword ix
Preface . xi
Preface to the English Edition xi
Terminology . . . . . . . . xii
Chapter I. Algebraic varieties 1
§1. Affine algebraic varieties . 1
§2. The Hilbert Basis Theorem. Decomposition of a variety into
irreducible components 10
§3. Hilbert's Nullstellensatz . . . . . . . . . . . . 16
§4. The spectrum of a ring . . . . . . . . . . . . 22
§5. Projective varieties and the homogeneous spectrum 29
References . . . . . . 38
Chapter II. Dimension 39
§1. The Krull dimension of topological spaces and rings 39
§2. Prime ideal chains and integral ring extensions . . . , . 44
§3. The dimension of affine algebras and affine algebraic varieties 49
§4. The dimension of projective varieties 59
References . . . . . . . . . . . . . . . . . . . . . . . . . 61
Chapter III. Regular and rational functions on algebraic varieties.
Localization . . . . . . . . . . . . . . . . . 63
§1. Some properties of the Zariski topology . . . . . . 63
§2. The sheaf of regular functions on an algebraic variety 66
§3. Rings and modules of fractions. Examples . . . . . 73
§4. Properties of rings and modules of fractions . . . . 78
§5. The fiber sum and fiber product of modules. Gluing modules 88
References . . . . . . . . . . . . . . . . . . . . . . . 92
Chapter IV. The local-global principle in commutative algebra 93
§1. The passage from local to global 93
§2. The generation of modules and ideals 104
§3. Projective modules 110
References . . . . . . . . . . . . . . 122
vii
viii TABLE OF CONTENTS
Chapter V. On the number of equations needed to describe an
algebraic variety . . . . . . . . . . . . . . . . . . 123
§1 . Any variety in n-dimensional space is the intersection of n
hypersurfaces . . . . . . . . . . . . . . . . . . . 123
§2. Rings and modules of finite length . . . . . . . . . . . 127
§3. Krull's Principal Ideal Theorem. Dimension of the intersection
of two varieties . . . . . . . . . . . . . . . . . . . . 130
§4. Applications of the Principal Ideal Theorem in Noetherian rings 141
§5. The graded ring and the conormal module of an ideal 149
References . . . . . . . . . . . . . · . . . . . . . . . . 162
Chapter VI. Regular and singular points of algebraic varieties 163
§1. Regular points of algebraic varieties. Regular local rings . 163
§2. The zero divisiors of a ring or module. Primary decomposition 176
§3. Regular sequences. Cohen-Macaulay modules and rings . . . 183
§~. A connectedness theorem for set-theoretic complete intersections
in projective space 192
References . . . . . . . . . . . 194
Chapter VII. Projective Resolutions 196
§1. The projective dimension of modules 196
§2. Homological characterizations of regular rings and local
complete intersections . . . . . . . . . . . . . . . . 205
§3. Modules of projective dimension :5 1 . . . . . . . . . . 211
§4. Algebraic curves in A3 that are locally complete intersections
can be represented as the intersection of two algebraic surfaces 219
References . 222
Bibliography 224
A. Textbooks 224
B. Research papers 225
List of symbols 229
Index . . . . 232
Foreword
It has been estimated that, at the present stage of our knowledge, one could give a
200 semester course on commutative algebra and algebraic geometry without ever
repeating himself. So any introduction to this subject must be highly selective.
I first want to indicate what point of view guided the selection of material
for this book. This introduction arose from lectures for students who had taken
a basic course in algebra and could therefore be presumed to have a knowledge
of linear algebra, ring and field theory, and Galois theory. The present text
shouldn't require much more.
In the lectures and in this text I have undertaken with the fewest possible
auxiliary means to lead up to some recent results of commutative algebra and
algebraic geometry concerning the representation of algebraic varieties as in
tersections of the least possible number of hypersurfaces and-a closely related
problem-with the most economical generation of ideals in Noetherian rings.
The question of the equations needed to describe an algebraic variety was
addressed by Kronecker in 1882. In the 1940s it was chiefly Perron who was
interested in this question; his discussions with Severi made the problem known
and contributed to sharpening the relevent concepts. Thanks to the general
progress of commutative algebra many beautiful results in this circle of questions
have been obtained, mainly after the solution of Serre's problem on projective
modules. Because of their relatively elementary character they are especially
suitable for an introduction to commutative algebra.
If one sets the goal of leading up to these results {and some still unsolved
problems), one is led into dealing with a large part of the basic concepts of
commutative algebra and algebraic geometry and to proving many facts which
can then serve as a basic stock for a deeper study of these subjects. Through
the close linking of ring-theoretic problems with those of algebraic geometry, the
role of commutative algebra in algebraic geometry becomes clear, and conversely
the algebraic inquiries are motivated by those of geometric origin.
Since the original question is classical, we begin with classical concepts of
algebraic geometry: varieties in affine or projective space. This quite naturally
presents us with an opportunity to lead up to the modern generalizations (spec
tra, schemes) and to exhibit their utility. If the detour is not too great, we shall
also pass through neighboring subjects on the way to our main goal. Yet some
elementary themes of commutative algebra have been entirely neglected, among
them: flat modules, completions, derivations and differentials, Hilbert polyno
mials and multiplicity theory. From homological algebra we use only projective
resolutions and the Snake Lemma. We do not try to derive the most general
known form of a proposition if to do so would seem to harm the readability of
the text or if the expense seems too great. The references at the end of each
ix
X FOREWORD
chapter and the many exercises, which often contain parts of recent publications,
should help the reader to become more deeply informed.
The center of gravity of this book lies more in commutative algebra than in
algebraic geometry. For a continued study of algebraic geometry I recommend
one of the excellent works which have recently appeared and for which this text
may serve as preparation.
I shall now indicate more precisely what knowledge this book assumes.
a) The most common facts of linear and multilinear algebra for modules over
commutative rings.
b) The simplest basic concepts of set-theoretic topology.
c) The basic facts of the theory of rings and ideals, including factorial rings
(unique factorization domains) and the Noether isomorphism theorems for
rings and modules.
d) The theory of algebraic extensions of fields, including Galois theory, as well
as the basic facts about transcendence degree and transcendence bases.
Most of what is needed should come up in any introductory course on algebra,
so that the book can be read in connection with such a course.
In preparing the text I have been helped by the critical remarks and many
good suggestions of H. Knebl, J. Koch, J. Rung, Dr. R. Sacher, and above all
Dr. R. Waldi. I have much to thank them for, as well as the Regensburg students
who industriously worked on the exercises. My special thanks also goes to Miss
Eva Weber for her patience in typing the manuscript.
Ernst Kunz
Regensburg, November 1978