K-Theory of Finite Groups and Orders
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149 Richard G. Swan Notes by
E. Graham Evans
K-Theory of Finite Groups and Orders
Springer-Verlag Berlin Heidelberg New York Tokyo
Author Richard G. Swan Department of Mathematics University of Chicago, 5734 University Avenue Chicago, Illinois 60637, USA
1st Edition 1970 2nd Corrected Printing 1986
ISBN 3-540-04938-X Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-04938-X Springer-Verlag New York Heidelberg Berlin Tokyo
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© by Springer-Verlag Berlin Heidelberg 1970 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
NOTE Throughout these notes, the lower case German "p" has been consistently typed as a lower case Roman "y". This results in some rather unusual notation , but should not cause any difficulty if the reader is prepared for it. The lower case German "c" is also indistinguishable from the lower case Greek tau, but again no real confusion should result.
TABLE OF CONTENTS Chapter 1. Chapter 2. Chapter 3. Chapter 4. Chapter
5.
Chapter 6. Chapter
....................... Frobenius Functors •• ....................... Finiteness Theorems••••••• ................. Results on KO and GO' •• .................... Maximal Orders••••••••• .................... Introduction••••••••
1
13 37 54 '83
Orders. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• 104
7. KO of a Maximal Order•••••••••••••••••••••• 126
Chapter 8.
K1 and G1 ••••••••••••••••••••••••••..••••• 140
Chapter 9.
Theorems•••••••••••••••••••••• 167
Appendix ••••.•••••••••••••••••••.•••••••••••...•••••.• 205
References ••••••••••••••• List of Symbols•••••••••••
............................. 233
.............................
235
Index ....•••••...•...............•••.•.•..•....••..... 236
Chapter One:
- 1 -
Introduction
Let R be a ring.
Then KO(R) is the abelian group given by
generators [p] where P is a finitely generated projective R module, with relations [p]
=
[PI] + [pIt] whenever 0
P"
P
P" --. 0
is an exact sequence of finitely generated projective R modules. KO is a covariant functor from rings to abelian groups. f: R-.R', then KO(f'): KO(R) ...... KO(R') by [P] ....
If IfR
is left noetherian, then GO(R) is the abelian group with generators [M] where M is a finitely generated left R module with relations [M] •
[M I] + [M"] whenever 0 .... M I .-.... M __ M" - - 0 is an exact se-
quence of finitely generated left R modules. since the tensor product
will preserve all the relations only
when R' is flat as a right R module. given by
GO is not a functor
There is a map KO(R) -..GO(R)
This map is called the Cartan
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