Algebraic K-Theory
Algebraic K-Theory has become an increasingly active area of research. With its connections to algebra, algebraic geometry, topology, and number theory, it has implications for a wide variety of researchers and graduate students in mathematics. The book i
- PDF / 28,057,050 Bytes
- 357 Pages / 438.903 x 665.373 pts Page_size
- 99 Downloads / 230 Views
Series Editors HymanB ass Joseph Oesterle Alan Weinstein
V. Srinivas
Algebraic K-Theory Second Edition
Springer Science+Business Media, LLC
V. Srinivas School of Mathematics Tata Institute of Fundamental Research Bombay, India
Library of Congress Cataloging-in-Publication Data Srinivas, V. Algebraic K-theory 1 V. Srinivas. -- 2nd ed. p. cm. -- (Progress in mathematics ; v. 90) Includes bibliographical references. ISBN 978-0-8176-4736-0 ISBN 978-0-8176-4739-1 (eBook) DOI 10.1007/978-0-8176-4739-1 1. K-theory. 1. Title. ll. Series: Progress in mathematics 93-9417 QA612.33.S67 1993 512'.55--dc20 CIP
Printed on acid-free paper
Springer Science+Business Media New York Originally published by Birkhăuser Boston in 1996 Softcover reprint ofthe hardcover 2nd edition 1996
@
Copyright is not claimed for works of U.S. Govemment employees. AU rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mecbanical, pbotocopying, recording, or otherwise, without prior permission of the copyright owner. Permission to pbotocopy for internat or personal use of specific clients is granted by Springer Science+Business Media, LLC. libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of $6.00 per copy, plus $0.20 per page is paid directly to CCC, 222 Rosewood Drive, Danvers, MA 01923, U.S.A. Special requests sbould be addressed directly to Springer Science+Business Media, LLC.
ISBN 978-0-8176-4736-0 Layout and typesetting by Martin Stock, Cambridge, MA
9 8 7 6 5 4 3 2 1
Dedicated to my parents.
Contents
Preface to the First Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
Preface to the Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
1. "Classical" K- Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Review of parts of Milnor's book: definitions of Ko, K1, K2 of rings; computation of K 1 of a noncommutative local ring; definition of symbols; statement of Matsumoto's theorem; examples of symbols (norm residue symbol, Galois symbol, differential symbol); presentation for K 2 of a commutative local ring.
2. The Plus Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
The plus construction; computation that 1r2(BGL(R)+) ~ K2(R); H-space structure of BGL(R)+ and products in K-theory (following Loday); statement of Quillen's theorem on Ki of a finite field. 3. The Classifying Space of a Small Category . . . . . . . . . . . . . . . . . . .
31
Simplicial sets; geometric realization; classifying space of a small category; elementary theorems about classifying spaces (compatibility with products, natural transformations give homotopies, adjoint functors give homotopy inverses, filtering categories are contractible); example of the classifying space of a discrete group as the classifying space of the category with one object, whose endomorphisms equal the gr
