Algebraic K-Theory Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands Volume 311 Algebraic K-Theory by Hvedri Inassaridze Georgian Academy of Sciences, Tbilisi, Georgia Springer-Science+Business Media, B.V. Library of Congress Cataloging-in-Publication Data Inassaridze. H. (Hvedri). 1932- Algebraic K-theory / by Hvedri Inassacidze. p. cm. -- (Mathematics and Its applications v.311> Inc 1u des bib 1 i ograph i ca 1 references and index. (acid-free) 1. K-theory. I. Title. II. Series: Mathematics and its applications (Kluwer Academic Publishers) ; v. 311. QA612.33.I53 1995 512' .55--dc20 94-35621 Printed on acid-free paper All Rights Reserved ISBN 978-90-481-4479-2 ISBN 978-94-015-8569-9 (eBook) DOI 10.1007/978-94-015-8569-9 © 1995 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1995. Softcover reprint of the hardcover 1st edition 1995 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. To Indo Inassari Contents Introduction 1 Chapter I. Classical Algebraic K-functors 3 § 1. The Grothendieck functor Ko 3 A. Definition, examples and some properties 3 B. Spec(R) and H(R) 12 C. Serre's theorem 18 § 2. The Bass-Whitehead functor Kf 29 § 3. The Milnor functor Kf 35 Chapter II. Higher K -functors 43 § 1. K -theory of Quillen for exact categories 43 § 2. The Quillen plus construction 72 A. Definition and properties 72 B. Computation of K~ (k) for a finite field k 93 C. Quillen's K-groups for free algebras 96 D. Negative algebraic K-theory 102 E. Finite generation of Quillen's K -groups of the rings of algebraic integers 108 F. Transfer map in the localization theorem 117 § 3. K-Theory of Swan 127 § 4. K -theory of Karoubi-Villamayor 140 § 5. K-theory of Waldhausen 149 Chapter III. Properties of algebraic K-functors 163 VII VlU CONTENTS § 1. Exactness, excision and the Mayer-Vietoris sequence 163 A. Exactness 163 B. Excision 170 C. The Mayer-Vietoris sequence 181 § 2. The localization theorem 184 § 3. The fundamental theorem 197 § 4. Products in algebraic K -theory 222 § 5. Stability 248 Chapter IV. Relations between algebraic K-theories 253 § 1. Isomorphism of Quillen's algebraic K-theories. Agreement of plus construction and Q-construction 253 § 2. Connection of Quillen's plus construction with Swan's algebraic 2m ~t~ory § 3. Comparison of Swan's and Karoubi-Villamayor's algebraic K theories 278 Chapter V. Relation between algebraic and topological K-theories 289 § 1. Equivalence of categories of finitely generated projective mod ules and vector bundles over a compact space for C*-algebras 289 § 2. K-theory of special normed algebras and Zrgraded C*-algebras305 A. K-theory of special normed algebras 305 B. K-theory of Z2-graded C*-algebras 328 § 3. Isomorphism of Swan '8 and Karoubi-Villamayor's J{ -theories with topological J( -theory for real Banach algebras 353 Chapter VI. The problem of Serre for polynomial and monoid alge- bras 361 § 1. Proof of Anderson's conjecture 361 A. Normal and seminormal monoids 361 B. Projective modules over normal monoid rings 368 C. On the triviality of the Picard group for monoid algebras over a principal ideal domain 380 § 2. The algebraic proof of Swan 387 Chapter VII. Connection with cyclic homology 423 References 429 Index 433 Introduction I dedicate this book to my father Niko Inassaridze, Georgian writer. Algebraic K -theory is a modern and perspective branch of algebra having many important applications in fundamental areas of mathematics connected with alge bra, topology, algebraic geometry, functional analysis and algebraic number the ory. Methods of algebraic K-theory are actively used in algebra and related fields getting interesting results. The aim of this book is to present elements of algebraic K-theory and it can be considered as an essay on foundations of algebraic K-theory (Chapters I-IV). So my intention is to help mathematicians (graduate students and researchers) mainly working in algebra, topology, functional analysis and algebraic geometry who want to learn about K-theory. The book is essentially based on the fundamental works of Milnor, Swan, Bass, Quillen, Karoubi, Gersten, Loday and Waldhausen. The results of exposed works are given with minor modifications and complete proofs (with exception of the stability theorems and the cyclic homology). Following Swan and Milnor Chapter I is devoted to the classical algebraic K Kr theory, i.e. to the Grothendieck functor [(0, the Whitehead-Bass functor and the Milnor functor K~1. The functor Ko is treated in detail. In Chapter II the construction of basic algebraic K-theories is given. We expose Quillen's fundamental work on higher [(-theory K. of exact categories (with the use of Quillen's Q-construction), the plus construction of Quillen and the corre sponding K-theory K~ following Loday and Gersten, Swan's K-theory J{~ general ized by the author in the category of Banach algebras over a commutative Banach ring k with unit, Karoubi-Villamayor's K-theory of Banach K-algebras, the K. S-construction of Waldhausen and the corresponding K-theory K;:' (the functor A( -) for topological spaces), the negative algebraic [(-theory following Wagoner 2 INTRODUCTION and computations of Quillen's K-groups for free rings due to Gersten, for finite fields and for rings of algebraic integers in a finite number field due to Quillen, and the computation of the transfer map in the localization theorem following Gersten. In Chapter III basic properties of algebraic K-functors are investigated, includ ing exactness, excision, the Mayer-Vietoris sequence, the localization theorem, the algebraic periodicity (fundamental theorem) following Grayson's publication, which implies Bott periodicity (Swan's proof is given), the products (Loday's prod uct, Karoubi's product and Waldhausen's product), the stability (without proofs). We use works of Swan, Quillen, Loday, Karoubi, Gersten, Weibel and Keune. Chapter IV contains the relationships among algebraic K-theories constructed in Chapter II, namely the relations between Quillen's Q-construction and plus con struction, following Grayson's publication, between Swan's K-theory and Quillen's plus construction following D. W. Anderson, between Swan's K-theory and Ka roubi-Villamayor's K-theory due to the author. The topological K -theory can be viewed as Swan's algebraic K -theory of Banach algebras and Chapter V is devoted to the relation between algebraic and topolog ical K -theories of normed algebras (special normed algebras and C* -algebras). The proof, due to Kandelaki, of the equivalence between the category of Hilbert A-fibrations over a compact Hausdorff space X and the category of finitely gene rated projective A(X)-modules, where A is a C*-algebra with unit, is given. This result generalizes Swan's well-known theorem for A the field of complex numbers. The topological K-theory of special normed algebras over a commutative special normed ring with unit, any Banach algebra being a special normed algebra, is treated by the author, and the topological K -theory of ZTgraded C* -algebras due to Kandelaki. Finally, for real Banach algebras the isomorphism of Swan's and Karoubi-Villamayor's K-theories with topological K-theory is established. Chapter VI is devoted to D. F. Anderson's conjecture generalizing Serre's prob lem on the freeness of finitely generated projective modules over polynomial rings over a field to the case of affine normal rings generated by monomials. Gubeladze's proof of Anderson's conjecture and Swan's algebraic version of this proof are given. In Chapter VII a brief review of cyclic homology and its relation with algebraic K-theory realized by the Chern characteristic classes is exposed. I want to thank the young Georgian algebraists I. Gubeladze, M. Jibladze and T. Kandelaki for helpful conversations and suggestions on topics concerning Quil len's fundamental work, Chapter V and Chapter VI. CHAPTER I Classical Algebraic K-functors In this chapter we give the definition and some properties of the three basic algebraic I<-functors I<o, I<~ and I<!j which form the Classical Algebraic I< theory. Other important properties of these functors will be given in Chapter III. § 1. The Grothendieck functor I<o A. Definition, examples and some properties. DEFINITION 1.1. Let A be a small abelian category. I<o(A) is the abelian group with generators [A], A E Ob(A), and relations [A] = [A'] + [A"] for each short exact sequence o (1) --t A' --t A --t A" --t 0 in the category A. The canonical injection Ob(A) --t I<o(A) has the universal property with respect to f : Ob(A) --t abelian groups such that f(A) = f(A') + f(A") for each short exact sequence (1). The original example is the Grothendieck construction: A is the category of coherent sheaves on an algebraic variety X and f(A) is the Euler characteristic X(X, A) of the sheaf A. PROPOSITION 1.2. We have (1) [0] = 0, (2) If A is isomorphic to B, then [A] = [B], (3) [A EEl B] = [A] + [B]. PROOF. Follows from (1) 0 --t 0 --t 0 --t 0 --t 0 is exact, (2) 0 --t 0 --t A --t B --t 0 is exact, 3