Constructive Commutative Algebra Projective Modules Over Polynomial
The main goal of this book is to find the constructive content hidden in abstract proofs of concrete theorems in Commutative Algebra, especially in well-known theorems concerning projective modules over polynomial rings (mainly the Quillen-Suslin theorem)
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Ihsen Yengui
Constructive Commutative Algebra Projective Modules Over Polynomial Rings and Dynamical Gröbner Bases
Lecture Notes in Mathematics Editors-in-Chief: J.-M. Morel, Cachan B. Teissier, Paris Advisory Board: Camillo De Lellis, Zurich Mario di Bernardo, Bristol Alessio Figalli, Austin Davar Khoshnevisan, Salt Lake City Ioannis Kontoyiannis, Athens Gabor Lugosi, Barcelona Mark Podolskij, Aarhus Sylvia Serfaty, Paris and NY Catharina Stroppel, Bonn Anna Wienhard, Heidelberg
2138
More information about this series at http://www.springer.com/series/304
Ihsen Yengui
Constructive Commutative Algebra Projective Modules Over Polynomial Rings and Dynamical Gr¨obner Bases
123
Ihsen Yengui Fac. of Science, Dept. of Mathematics University of Sfax Sfax, Tunisia
ISSN 0075-8434 Lecture Notes in Mathematics ISBN 978-3-319-19493-6 DOI 10.1007/978-3-319-19494-3
ISSN 1617-9692 (electronic) ISBN 978-3-319-19494-3 (eBook)
Library of Congress Control Number: 2015956600 Mathematics Subject Classification (2010): 13Cxx, 13Pxx, 14Qxx, 03Fxx Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, 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. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.springer.com)
Contents 1
Introduction
1
2 Projective Modules Over Polynomial Rings 2.1 Quillen’s Proof of Serre’s Problem . . . . . . . . . . . . 2.1.1 Finitely-Generated Projective Modules . . . . . 2.1.2 Finitely-Generated Stably Free Modules . . . . . 2.1.3 Concrete Local-Global Principle . . . . . . . . 2.1.4 The Patchings of Quillen and Vaserstein . . . . . 2.1.5 Horrocks’ Theorem . . . . . . . . . . . . . . . . 2.1.6 Quillen Induction Theorem . . . . . . . . . . . . 2.2 Suslin’s Proof of Serre’s Problem . . . . . . . . . . . . . 2.2.1 Making the Use of Maximal Ideals Constructive 2.2.2 A Reminder About the Resultant . . . . . . . . 2.2.3 A Lemma of Suslin . . .
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