Lecture Notes in Mathematics 2138
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1 Lecture Notes in Mathematics 2138 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
2 More information about this series at
3 Ihsen Yengui Constructive Commutative Algebra Projective Modules Over Polynomial Rings and Dynamical Gröbner Bases 123
4 Ihsen Yengui Fac. of Science, Dept. of Mathematics University of Sfax Sfax, Tunisia ISSN ISSN (electronic) Lecture Notes in Mathematics ISBN ISBN (ebook) DOI / Library of Congress Control Number: 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 (
5 Contents 1 Introduction 1 2 Projective Modules Over Polynomial Rings Quillen s Proof of Serre s Problem Finitely-Generated Projective Modules Finitely-Generated Stably Free Modules ConcreteLocal-GlobalPrinciple The Patchings of Quillen and Vaserstein Horrocks Theorem Quillen Induction Theorem Suslin sproofofserre sproblem MakingtheUseofMaximalIdealsConstructive A Reminder About the Resultant ALemmaofSuslin AMoreGeneralStrategy(By Backtracking ) Suslin s Lemma for Rings Containing an Infinite Field Suslin salgorithm Suslin ssolutiontoserre sproblem A Simple Result About Coherent Rings Constructive Definitions of Krull Dimension Ideals and Filters Zariski Lattice Krull Boundary Pseudo-Regular Sequences and Krull Dimension Krull Dimension of a Polynomial Ring Over adiscretefield ApplicationtotheStableRangeTheorem Serre s Splitting Theorem and Forster-Swan Theorem Support on a Ring and n-stability V
6 VI CONTENTS 2.4 Projective Modules Over R[X 1,...,X n ], R anarithmeticalring A Constructive Proof of Brewer-Costa-Maroscia Theorem TheTheoremofLequain,SimisandVasconcelos Suslin s Stability Theorem Obvious Syzygies E 2 (R) as a Subgroup of SL 2 (R) Suslin s Normality Theorem Unimodular Rows and Elementary Operations Local-Global Principle for Elementary Polynomial Matrices A Realization Algorithm for SL 3 (R[X]) Elementary Unimodular Completion TheHermiteRingConjecture TheHermiteRingConjectureinDimensionOne Stably Free Modules Over R[X] of Rank > dim R arefree TwoNewConjectures Dynamical Gröbner Bases Dickson slemmaandthedivisionalgorithm Gröbner Rings Gröbner Bases Over Strongly Discrete Coherent Arithmetical Rings Gröbner Bases Over a Coherent Valuation Ring Gröbner Bases Over Z/p α Z When a Valuation Domain Is Gröbner? When a Coherent Valuation Ring with Zero-Divisors is Gröbner? Dynamical Gröbner Bases Over Gröbner Arithmetical Rings A Parallelisable Algorithm for Computing Dynamical Gröbner Bases Over Z/mZ via the Chinese Remainder Theorem A Parallelisable Algorithm for Computing Gröbner Bases Over (Z/p α Z) (Z/p α Z) Dynamical Gröbner Bases Over F 2 [a,b]/ a 2 a,b 2 b Dynamical Gröbner Bases Over the Integers Gröbner Bases Over the Integers via Prime Factorization A Branching-Free Algorithm for Computing Gröbner Bases OvertheIntegers...153
7 CONTENTS VII 3.4 Computing Syzygy Modules with Polynomial Rings Over Gröbner Arithmetical Rings Computing Syzygy Modules with Polynomial Rings Over Gröbner Valuation Rings Computing Dynamically a Generating Set for Syzygies of Polynomials Over Gröbner Arithmetical Rings Syzygies in Polynomial Rings Over Valuation Domains PreliminaryTools Saturation of Finitely-Generated Sub-V-Modules of V[X 1,...,X k ] m Saturation of a Finitely-Generated V[X]-Module, with V a Valuation Domain Computing Syzygies Over R[X], with R aprüferdomain TheCaseofaValuationDomain The Case of a PrüferDomain TheMultivariateCase HilbertSeries TheSaturationDefectSeries ASaturationAlgorithmintheMultivariateCase Exercises Detailed Solutions to the Exercises 221 Notation List 255 Bibliography 259 Index 269
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