An Introduction to Homological Algebra
With a wealth of examples as well as abundant applications to Algebra, this is a must-read work: a clearly written, easy-to-follow guide to Homological Algebra. The author provides a treatment of Homological Algebra which approaches the subject in t
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Joseph J. Rotman
An Introduction to Homological Algebra Second Edition
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Joseph J. Rotman Department of Mathematics University of Illinois at Urbana-Champaign Urbana IL 61801 USA [email protected]
Editorial board: Sheldon Axler, San Francisco State University Vincenzo Capasso, Universit`a degli Studi di Milano Carles Casacuberta, Universitat de Barcelona Angus MacIntyre, Queen Mary, University of London Kenneth Ribet, University of California, Berkeley ´ Claude Sabbah, CNRS, Ecole Polytechnique Endre S¨uli, University of Oxford Wojbor Woyczynski, Case Western Reserve University
ISBN: 978-0-387-24527-0 DOI 10.1007/978-0-387-68324-9
e-ISBN: 978-0-387-68324-9
Library of Congress Control Number: 2008936123 Mathematics Subject Classification (2000): 18-01 c Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper springer.com
To the memory of my mother Rose Wolf Rotman
Contents
Preface to the Second Edition . . . . . . . . . . . . . . . . . . .
x
How to Read This Book . . . . . . . . . . . . . . . . . . . . . . . xiii Chapter 1 1.1 1.2 1.3
Introduction
Simplicial Homology . . . . . . . . . . . . . . . . . . . . . . 1 Categories and Functors . . . . . . . . . . . . . . . . . . . . . 7 Singular Homology . . . . . . . . . . . . . . . . . . . . . . . 29
Chapter 2
Hom and Tensor
2.1 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.2 Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . 69 2.2.1 Adjoint Isomorphisms . . . . . . . . . . . . . . . . . . . . . . 91
Chapter 3 3.1 3.2 3.3 3.3.1
Projective Modules Injective Modules . Flat Modules . . . Purity . . . . . . .
Chapter 4 4.1 4.2 4.3
Special Modules . . . .
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Specific Rings
Semisimple Rings . . . . . . . . . . . . . . . . . . . . . . . . 154 von Neumann Regular Rings . . . . . . . . . . . . . . . . . . 159 Hereditary and Dedekind Rings . . . . . . . . . . . . . . . . . 160 vii
viii
4.4 4.5 4.6 4.7 4.8
Contents
Semihereditary and Pr¨ufer Rings Quasi-Frobenius Rings . . . . . Semipe
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