Introduction to Homotopic Algebra
Let C be an arbitrary category, L and R two classes of morphisms in C. The class L is said to be left complementary to R (and R is said to be right complementary to L) if the following condition is satisfied: for any solid arrow commutative square with l
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Springer-Verlag Berlin Heidelberg GmbH
Sergei I. Gelfand
Yuri I. Manin
Methods of Homological Algebra Second Edition
123
Sergei 1. Gelfand American Mathematical Society 201 Charles Street Providence, RI 02904 USA e-mail: [email protected]
Yuri 1. Manin Max Planck Institute for Mathematics Vivatsgasse 7 53111 Bonn Germany e-mail: [email protected]
Library ofCongress Cataloging-in-Publication Data Gel'fand, S. 1. (Sergei Izrailevich) (Metody gomologicheskoi algebry. English] Methods ofhomological algebra / Sergei 1. Gelfand, Yuri 1. Manin.-- 2nd ed. p. cm. -- (Springer monographs in mathematics, ISSN 1439-7382) Includes bibliographical reference, and index. ISBN 978-3-642-07813-2 ISBN 978-3-662-12492-5 (eBook) DOI 10.1077/978-3-662-12492-5
1. Algebra, Homological. 1. Maoin, ID. 1. II. Title. III. Series. QA169 .G4513 2003 512'.55--dc21
2002190826
Mathematics Subject Classification (2000): 12D15, 12J15, 12JI0, 14PI0
ISSN 1439-7382 ISBN 978-3-642-07813-2 This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concemed, specifically the rights of trans1ation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfihn 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 Berlin Heidelberg GmbH. Violations are liable for prosecution under the German Copyright Law. http://www.springer.de © Springer-Verlag Berlin Heidelberg 1997, 2003 Originally published by Springer-Verlag Berlin Heidelberg New York in 2003 Softcover reprint of the hardcover 2nd edition 2003
The use of general descriptive narnes, registered narnes, trademarks etc. in this publication does not imply, even in the absence of a specific statement, that such narnes are exempt from the relevant protective laws and regn1ations and therefore free for general use. Cover design: Erich Kirchner, Heidelberg Typesetting: Le-TeX Jelonek, Schmidt & VOckler GbR, Leipzig SPlN 11616542 Printed on acid-free paper
41/3111ck-5432
Preface to the Second Edition
In the second edition of our book, many of the typos, errors, and imprecise statements that found their way into the first edition, are corrected. The authors are extremely grateful to Professor Sergei Merkulov who took upon himself the difficult task of carefully reading the text and incorporating appropriate changes. Without his efforts, most of the errors in the first edition would probably have remained unnoticed. Of course, the authors retain full responsibility for any mistakes that may still remain in the text.
September 2002
S. Gelfand, Yu. Manin
Foreword
... utinam intelligere possim rationacinationes pulcherrimas quae e propositione concisa DE QUADRATUM NIHILO EXAEQUARI fluunt. (... if I could only understand the beautiful consequence following from the concise proposition d2 = 0.) From Henri Cartan
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