Almost Ring Theory
This book develops thorough and complete foundations for the method of almost etale extensions, which is at the basis of Faltings' approach to p-adic Hodge theory. The central notion is that of an "almost ring". Almost rings are the commutative unita
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Ofer Gabber Lorenzo Ramero
Almost Ring Theory
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Authors Ofer Gabber IHES Le Bois-Marie 35, route de Chartres 91440 Bures-sur-Yvette, France e-mail: [email protected]
Lorenzo Ramero Institut de Math´ematiques Universit´e de Bordeaux I 351, cours de la Lib´eration 33405 Talence, France e-mail: [email protected]
Cataloging-in-Publication Data applied for Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de
The cover figure entitled "Prescano" is reproduced by kind permission of Michel Mend`es France
Mathematics Subject Classification (2000): 13D10, 13B40, 12J20, 14G22, 18D10, 13D03 ISSN 0075-8434 ISBN 3-540-40594-1 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm 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-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH http://www.springer.de c Springer-Verlag Berlin Heidelberg 2003 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 specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10896419
41/3142/du-543210 - Printed on acid-free paper
C ONTENTS 1. Introduction 1.1. Motivations and a little history 1.2. The method of almost e´ tale extensions 1.3. Contents of this book 1.4. The view from above 1.5. Acknowledgements 2. Homological theory 2.1. Some ring-theoretic preliminaries 2.2. Categories of almost modules and algebras 2.3. Uniform spaces of almost modules 2.4. Almost homological algebra 2.5. Almost homotopical algebra 3. Almost ring theory 3.1. Flat, unramified and e´ tale morphisms 3.2. Nilpotent deformations of almost algebras and modules 3.3. Nilpotent deformations of torsors 3.4. Descent 3.5. Behaviour of e´ tale morphisms under Frobenius 4. Fine study of almost projective modules 4.1. Almost traces m. 4.2. Endomorphisms of G 4.3. Modules of almost finite rank 4.4. Localization in the flat site 4.5. Construction of quotients by flat equivalence relations 5. Henselization and completion of almost algebras 5.1. Henselian pairs 5.2. Criteria for unramified morphisms 5.3. Topological algebr
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