A Course In Commutative Algebra

243 Pages·2010·3.467 MB·English

by  Gregor Kemper

Graduate Texts in Mathematics 256
Editorial Board
S. Axler
K.A. Ribet
Forothertitlespublished inthisseries,goto
www.springer.com/series/136
Gregor Kemper
A Course
in Commutative Algebra
With 14 Illustrations
123
Gregor Kemper
Technische Universitä t
Zentrum Mathematik - M11
Boltzmannstr. 3
85748 Garching
Germany
[email protected]
Editorial Board
S.Axler K.A. Ribet
Mathematics Department Mathematics Department
San Francisco State University University of California, Berkeley
San Francisco, CA 94132 Berkeley, CA 94720-3840
USA USA
[email protected] [email protected]
ISSN 0072-5285
ISBN 978-3-642-03544-9 e-ISBN 978-3-642-03545-6
DOI 10.1007/978-3-642-03545-6
Springer Heidelberg Dordrecht London New York
© Springer-Verlag Berlin Heidelberg 2011
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is
concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,
reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication
or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965,
in its current version, and permission for use must always be obtained from Springer. Violations are
liable to prosecution under the German Copyright Law.
The use of general descriptive names, registered names, trademarks, etc. in this publication does not
imply, even in the absence of a specific statement, that such names are exempt from the relevant protective
laws and regulations and therefore free for general use.
Cover design: WMXDesign GmbH, Heidelberg, Germany
Printed on acid-free paper
Springer is part of Springer Science+Business Media (www.springer.com)
To Idaleixis, Lorenz, and Martin
•
Preface
This book has grown out of various courses in commutative algebra that
I have taught in Heidelberg and Munich. Its primary objective is to serve
as a guide for an introductory graduate course of one or two semesters, or
for self-study. I have striven to craft a text that presents the concepts at the
center of the field in a coherent, tightly knit way, with streamlined proofs
and a focus on the core results. Needless to say, for an imperfect writer like
me, such high-flying goals will always remain elusive. To introduce readers
to the more recent algorithmic branch of the subject, one part of the book
is devoted to computational methods. Virtually all concepts and results of
commutativealgebrahavenaturalgeometricinterpretations.Infact,itisthe
geometricviewpointthatbringsoutthe“truemeaning”ofthetheory.Thisis
why the first part of the book is entitled “The Algebra–Geometry Lexicon,”
and why I have tried to keep a focus on the geometric context throughout.
Ihopethatthiswillmakethetheorymorealiveforreaders,moremeaningful,
more visual, and easier to remember.
Iwelcomeanycomments,suggestionsforimprovements,anderrorreports
from readers. Please send them to [email protected].
Acknowledgments.Firstandforemost,I thankthe students whoattended
the three courses on commutative algebra that I have taught at Heidelberg
andMunich.Thisbookhasbenefitedgreatlyfromtheirparticipation.Partic-
ularlyfruitfulwasthelastcourse,givenin2008,inwhichIawardedoneeuro
foreverymistakeinthemanuscriptthatthe studentsreported.Thismethod
wassosuccessfulthatitcostmeasmallfortune.IwouldliketomentionPeter
Heinig inparticular,who broughtto my attentioninnumerablemistakesand
quite a few didactic subtleties.
IamalsogratefultoGert-MartinGreuel,BerndUlrich,RobinHartshorne,
Viet-Trung Ngo, Dale Cutkosky, Martin Kohls, and Steve Gilbert for inter-
esting conversations.
Myinterestincommutativealgebragrewoutofmymainresearchinterest,
invariant theory. In particular, the books by Sturmfels [50] and Benson [4],
althoughtheydonotconcentrateoncommutativealgebra,firstawakenedmy
vii
viii Preface
fascination for it. So my thanks go to Bernd Sturmfels and David Benson,
too.
Lastbut not least, I am grateful to David Kramerfor his outstanding job
of copyediting the manuscript, to the anonymous referees,and to the people
at Springer for the swift and efficient handling of the publication process.
Munich Gregor Kemper
November 2010
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
ix
x Contents
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 Gro¨bner 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