Proceedings of the Conference on Orders, Group Rings and Related Topics

234 Pages·1973·4.276 MB·English

by  John S. Hsia, Manohar L. Madan, Thomas G. Ralley (auth.)

Lecture Notes ni
Mathematics
detidE yb A ,dtoD Heidelberg dna B ,nnamkcE hcirJ~Z
353
Proceedings fo eht Conference
no Orders, Group Rings dna
Related Topics
Organized yb John .S Hsia, Manohar .L Madan dna Thomas .G hlaR
Ohio State Columbus, University, OH/USA
galreV-regnirpS
Berlin-Heidelberg. New York 73 19
AMS Subject Classifications (1970): 10C05, 12A90, 13D15, 14G10, 51 A63, 16A18, 16A26, 16A36,
16A46, 16A50, 16A 52, 16A54, 16A64, 18F25, 20C05, 20C10,
20KI5, 20K20, 20K45
ISBN 3~540-06518-0 Springer-Verlag Berlin - Heidelberg • New York
ISBN 0-387-06518-0 Springer-Verlag New York • Heidelberg - Berlin
This work is subject to copyright. 1IA rights are reserved, whether the whole or part of the material is concerned,
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the amount of the eef to be determined by agreement with the publisher.
© by Springer-Verlag Berlin . Heidelberg .3791 Library of Congress Catalog Card Number .15451-37 Printed in Germany.
Julius Beltz, Offsetdruck: .rtsgreB/hcabsmeH
Dedicated to
Professor Hans J. Zassenhaus
on the occasion of his sixtieth birthday
FOREWORD
It gives me great pleasure to extend my warm good wishes and those of
cclleag~Aes to Our distinguished colleague Professor Hans Zassenhaus upon the
occasion of h/s sixtieth birthday.
Professor Zassenhaus has contributed important ideas and methods to many
and diverse branches of mathematics. Among these the range of ideas concerned
ht_#~ Orders# Group Rings ~nd P.elated Topics have held his attention for many
years. His fundamental contributions to these fields have been kno~m, appreciated
dn.8 used very .yled'_~ It ~as thought appropriate to mark Professor Zassenhaus'
sixtieth birthday by bringing together a group of fellow mathematicians who
share sih_ interest in Orders and GrOup Rings for a discussion of the progress
to date and the prospects for the future of this difficult and important field
of mathematics2_ endeavor.
The conference ~¢hich ~zrked Professor Zassenhaus ~ birthday brought together
a large nttmber of able and accomglished mathemzticians and resulted in much
stimulating discussion. It ~¢ould be difficult to describe fully the exciting
and invigorating atmosphere pre~ent in a gathering of a group of creative
people re~resenting a broad ~pectru_m of experience and of generations~ who
nevertheless share many interests in common. We have been able to put together
a record of formal presentations which we are happy to offer in the present
volume of essays.
Arnold .E Ross
PREFACE
This volume records th~ talks given at the Conference on Orders# Group
Rings and Related Topics held at Ohio State in ~s~, 1972, to honor our
distinguished colleague Hans Zassenhaus whose contributions to mathematics in
the theory of orders are widely known and appreciated. Our original intention,
to convene an international gathering of mathematiclans who have worked in this
or some related area, had tob e scaled down when it became clear that the funds
available were not sufficient to support such an undert~aklng. None the less~
as the reader examines the contents~ we believe that he will share our feeling
that it presents an interesting picture of areas of current interest in mathematles,
a ~efreshin~ blend of expository ~th technical material and that the meeting is
to be counted a ~lecess.
The planning and organization involved many individuals and it is appropriate
that mention of their efforts be made here.
Professor Arnold .E Ross ~s instrumental in bringing the conference into
being. From inception he encouraged its planning and ~hen we ~ere unable to
obtain outside support, he n~naged to find resources for its funding. His
efforts are deeply appreciated.
We thank the authors for their co-operation in the preparation of this report.
Their original manuscripts were carefully done and in our hands by the agreed
time. They showed great patience when various matters prevented us from holding
to our original schedule.
At a meeting where current ~reetions and new ideas for mathematical research
were to be discussed, ~e felt it important for the forthcoming generation of
mathematicisns to have an opportunity to attend and participate. The Ohio State
Graduate School made availab~e money which permitted us to provide some measure
of support for graduate students and wm are grateful to Deans Ar~iss Roaden and
Elmer Baumer for this.
Any cc~ference requires attention to a large number of details hcih;.,~ are
unexciting but necessary for the smooth flc~g of events. Responsibility for
these matters fell upon the org~nizino~ committee. As designated chairman of
that conmLitte% i want to express ym. gratitude to its r~bers, .nhoJ Hsia and
Mauohar Madan~ for their service. For his major role in the preparation of
this volume John deserves special cc~mendation. The cc~mittee also ~.~shes to
express its thanks to Professor Zassenhaus for his ad~ice and assistance in
n~tters relating to the conference.
The final typing ~s done by Liss l Dodie Huffmau. She carried out that
responsibility with skill and dedication and she has our deepest thanks for
her fine work~
Finslly# w~ are grateful to Springer Ver!ag, to the editors of the Lecture
Note Series Professors Albrecht Do!d and Beno Eckmann and to .rD k~aus Peters
and Bernd Grossmann for providing this opportunity to present the record of the
conference to the general mathematical eo~nunity.
.T Ralley
Table of Contents
I. ROGER C.ALPERIN r R.KEITH DENNIS I MICHAEL R.STEIN:
The Non-triviality of SKI (~) . . . . . . . . . . . . . . . .
II. ~AURICE AUSLANDER~ I DUN REITEN:
Stable Equivalence of Artin Algebras . . . . . . . . . . . . . .
III. GORO AZU~YA:
Some Properties of TTF-classes . . . . . . . . . . . . . . . . . 72
IV. HYmN BASS:
Is SKI (~)= 0 for~( a Finite Abelian Group . . . . . . . . . . 84
V. R.KEITH DENNIS:
Stability for 2 K . . . . . . . . . . . . . . . . . . . . . . . . 85
VI. WILLIAM H.GUSTAFSON:
The Theory of Relative Grothendieck Rings . . . . . . . . . . . 59
VII. T.A.HANNULA:
The Morita Context and the Construction of QF Rings ...... 311
VIII. STAN KLASA:
On Steinberg Groups . . . . . . . . . . . . . . . . . ..... 131
IX. I.RETNER, S.ULLOM: '
Class Groups of Orders and a Mayer-Vietoris Sequence ...... 139
X. S.K.SEHGAL:
Lie Properties in Modular Group Algebras ............. 152
XI. RICHARD G.SWAN:
K-theory and Algebraic Correspondences . . . . . . . . . . . . . 161
XII. ROGER WARE:
The Structure of the Witt Rings and Quotients of Abelian Group
Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
VIII
XIII. TOSHIHIK0 YAMADA:
The Schur Subgroup of the Brauer Group . . . . . . . . . . . . 187
XIV. HANS ZASSENHAUS:
On the Embedding of an Order into a Maximal Order . . . . . . . 204
LIST O~ PARTICIPANTS
A~en~ Endoj Lawrence
Ohio State University University of Illinois
Auslander, Igaurice Ferrar# Joe
Brandeis University Ohio State University
Azumaya 3 Goro Fossum, .T V.
Indiana University University of Illinois
Bass# Hymau G~. ovich I Steve
Columbia University Carlet on College
Borror3 Jeffry A. Cillam 3 John D.
Case-Western Reserve University Ohio University
Brc~n I Robert GI over 3 Henry
Ohio State University Ohio State University
Carlson3 Jon F. Green~ Ed~ard .L
University of Georgia Brandeis University
Chang ~ Kuang-I Gustafson~ William .H
Ohio State University Indiana University
Changj Morgan F. H. Hannula~ Thomas
C ol~mbla University University of Maine
Chi~ 3 .W Hill t Walter
Ohio State University Princeton University
Chittenden, Charles Hsia3 JoMn
Ohio State University Ohio State University
Cliff 3 Gerald Hughe 3 s lan
University of ILlinois Queen s ' Univer sity
Coleman 3 Donald Jain~ S. K.
University of Kentuek~ Ohio University
Cunningham 3 Joel James# Donald °G
University of Kentuc~ Penn State University
i_~neD # s Kelth Johnson, Robert .P
Cornel! University Ohio State University
Divis 3 Bohu slav Ki~el 3 Davld
Ohio State University Ohio State University
Eldridge# KLaus .E Elasa3 J.
Ohio University McGill University
E!klns~ Bryce Klasa~ .S
Ohio State University McGIII University
X
Liang 3 Joseph Sehgal, .S .K
University of South Florida University of Alberta
,nad~41 .M Siu, Man Ketmg
Ohio State University Columbia University
Nartin, Robert Swan 3 Richard .G
Columbia University University of Chicago
McCulloh 3 Leon Symonds, Robin .G
University of Illinois Ohio University
Merklen 3 Hector A. Ull o'm, Stephen
Ohio State University University of Illinois
Miller, Len Upton, John
Ohio State University University of Maryland
Mislin, Guido Wsng3 Chaug-Yeau
Ohio State University University of Illinois
Moreno# Carlos .S ,eraW Roger
University of Illinois North~e ts ern University
Ost erburg, James Wei, Chc~'Hsis~ng
University of Cincinnati Queen s ' University
Pet erson# Roger Yamada 3 To ahihlko
Ohio State University Queen's University
Pollack I David Yang, John
Queen s ' University Ohio State University
Queen, Clifford .S Zas senhaus~ Hans
Ohio State University Ohio State University
Ealley3 .T Zilber, Joseph
Ohio State University Ohio State University
Relner, Irving
University of Illinois
en~ Reit Id~m
Brandeis University
Ross3 A. .E
Ohio State University
Sallee, William
University of Kentucky
Santa Pietro, John
Rutgers University