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