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of the material here has appeared in some form in the literature, the main exception being Chapter 14 which is based on unpublished results of Milnor, Kervaire, and Steinberg.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
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