Algebraic K-theory

From the Introduction: "These notes are taken from a course on algebraic K-theory [given] at the University of Chicago in 1967. They also include some material from an earlier course on abelian categories, elaborating certain parts of Gabriel's thesis. Th

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76

R. G. Swan University of Chicago, Chicago, Illinois

1968

Algebraic K-Theory

Springer-Verlag Berlin· Heidelberg· New York

Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich

76

R. G. Swan University of Chicago, Chicago, Illinois

1968

Algebraic K-Theory

Springer-Verlag Berlin· Heidelberg· New York

All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer Verlag. © by Springer- Verlag Berlin' Heidelberg 1968 Library of Congress Catalog Card Number 68 • 59063 Printed in Germany. Title No. 3682

INTRODUCTION These notes are taken from a course on algebraic K-theory which I gave at the University of Chicago in 1967.

They also

include some material from an earlier course on abelian categories, elaborating certain parts of Gabriel's thesis.

The results on

K-theory are mostly of a very general nature.

I hope to treat

some of the deeper parts of the theory, in particular, the case of finite groups, in a subsequent set of notes. taken in these notes is not always consistent.

The point of view I generalized a

number of results on modules to abelian categories but did not hesitate to return to modules when the going got a bit rough.

Most

of the material here has appeared in some form in the literature, the main exception being Chapter 14 which is based on unpublished results of Milnor, Kervaire, and Steinberg. I would like to thank J. Burroughs, G. Evans, and M. Schacher for taking the notes.

In particular, I would like to thank

G. Evans who collected the notes, rewrote them all in readable form, and proofread the final version.

Special thanks are due to

S. Mac Lane who suggested to us the idea of publishing these notes and who arranged for their typing, which was done by M. Benson, and their publication.

TABLE OF CONTENTS

PART I.

CATEGORY THEORY Quotient Categories

PART II.

K-THEORY

1

40

66

Chapter 1.

Definition of KOC!) and Some Examples

66

2.

Krull-Schmidt Theorems and Applications

75

3.

Definition of G(R) and Examples

92

4.

The Connection Between KOCR) and GO(R)

100 109

6.

Localization and Relation Between GO(R) and GO(RS) KO of Graded Rings

7.

Spec(R) and H(R)

132

8.

Picard Group and the Determinant

146

9.

Basic Topological Remarks

155

10.

Chain Complexes and the Nilpotence of KO(R)

161

11.

Serre's Theorem

171

12.

Cancelation Theorems

183

124

13.

193

14.

204

15.

The Exact Sequence of K.1. 's

211

16.

Further Results on K and KO l Relations Between Algebraic and Topological K Theory

224

17.

247

BIBLIOGRAPHY

257

INDEX

258

LIST OF SYMBOLS

261

PART I CATEGORY THEORY The purpose of this section is to provide basic information about abelian categories and Serre subcategories. Definition.

Let A be an abelian category.

A subcategory

Q

is a Serre subcategory of ! if

1)

Q is a full subcategory of A

2)

If 0 -+ AI -+A --.A"

0 is exact in A, then A £ C

if and only if AI and A" are in C.

3)

Q