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
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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
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ISBN 978-0-8176-4736-0 Layout and typesetting by Martin Stock, Cambridge, MA
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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
