Groups, Trees and Projective Modules
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790 Warren Dicks
Groups, Trees and Projective Modules
Springer-Verlag
Berlin Heidelberg New York 1980
Author Warren Dicks Department of Mathematics Bedford College London NWt 4NS England
AMS Su bject Classifications (1980): 16 A 50, 16 A ?2, 20 E 06, 20 J 05 ISBN 3-540-099?4-3 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-09974-3 Springer-Verlag NewYork Heidelberg Berlin Library of Congress Cataloging in PublicationData.Dicks, Warren, 1947- Groups, trees, and projectivemodules.(Lecture notes in mathematics;790) Bibliography:p. Includes indexes.1. Associativerings. 2. Groups, Theoryof. 3. Trees (Graph theory)4. Projective modules (Algebra) I. Title. II. Series: Lectures notes in mathematics(Berlin); ?90. O.A3.L28 no. 790. [QA251.5], 510s. [512'.4]. 80-13138 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1980 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 214113140-543210
To the memory of my mother
PREFACE
For 1978/9 the Ring Theory Study Group at Bedford College rather naively set out to learn what had been done in the preceding decade on groups of cohomological dimension One. attractive subject,
This is a particularly
that has witnessed substantial success,
essentially beginning in 1968 with results of Serre, Stallings and Swan,
later receiving impetus from the introduction of the concept
of the fundamental group of a connected graph of groups by Bass and Serre,
and recently culminating in Dunwoody's contribution which
completed the characterization.
Without going into definitions,
one can state the result simply enough: (associative, with 1) and group group ring
RIG]
is right
G,
For any nonzero ring
R
the augmentation ideal of the
R[G]-projective if and only if
G
is
the fundamental group of a graph of finite groups having order invertible in
R.
These notes,
a (completely) revised version of those prepared
for the Study Group,
collect together material from several
sources to present a self-contained proof of this fact,
assuming
at the outset only the most elementary knowledge - free groups, projective modules, etc.
By making the role of derivations even
more central to the subject than ever before, simplify some of the existing proofs,
we were able to
and in the process obtain
a more general "relativized" version of Dunwoody's result, IV.2.10.
~
cf
amusing outcome of this approach is that we here have
a proof of one of the major results in the theory of cohomology of groups that nowhere mentions cohomology - which should make this account palatable to hard-line ring t
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