Ideals and Reality Projective Modules and Number of Generators of Id
This monograph tells the story of a philosophy of J-P. Serre and his vision of relating that philosophy to problems in affine algebraic geometry. It gives a lucid presentation of the Quillen-Suslin theorem settling Serre's conjecture. The central topic of
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Friedrich Ischebeck Ravi A. Rao
Ideals and Reality Projective Modules and Number of Generators of Ideals
GI - Springer
Friedrich Ischebeck Institut fur Mathematik Universitat Munster Einsteinstr. 62 48149 Munster, Germany e-mail: [email protected]
Ravi A. Rao School of Mathematics Tata Institute of Fundamental Research Dr Homi Bhabha Road 400005 Mumbai, India e-mail: [email protected]
Library of Congress Control Number: 2004114476
Mathematics Subject Classification (2000): 11C99,13A99,13Cio,13C40,13D15,55R25 ISSN 1439-7382 ISBN 3-540-23032-7 Springer Berlin Heidelberg New York This workis subject to copyright. All rights are reserved, whether the whole or part ofthe 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 for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com O Springer-Verlag Berlin Heidelberg 2005 Printed in Germany 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. Typeset by the authors using a Springer !+VEXmacro package Production: LE-TEX Jelonek, Schmidt & Vockler GbR, Leipzig Cover design: Erich Kirchner, Heidelberg Printed on acid-free paper
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Dedicated t o our close friend and former colleague Hartmut Lindel who left us much too early
Preface
Besides giving an introduction to Commutative Algebra - the theory of commutative rings - this book is devoted to the study of projective modules and the minimal number of generators of modules and ideals. The notion of a module over a ring R is a generalization of that of a vector space over a field k. The axioms are identical. But whereas every vector space possesses a basis, a module need not always have one. Modules possessing a basis are called free. So a finitely generated free R-module is of the form Rn for some n E IN,equipped with the usual operations. A module is called projective, iff it is a direct summand of a free one. Especially a finitely generated R-module P is projective iff there is an R-module Q with P @ Q S Rn for some n. Remarkably enough there do exist nonfree projective modules. Even there are nonfree P such that P @ Rm S Rn for some m and n. Modules P having the latter property are called stably free. On the other hand there are many rings, all of whose projective modules are free, e.g. local rings and principal ideal domains. (A commutative ring is called local iff it has exactly one maximal ideal.) For two decades it w
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