Proceedings of Symposia in Pure Mathematics
Volume 47 (1987)
Cohomology Is Representation Theory
1. Introduction. G roup theorists find cohomology of groups a confusing
and difficult subject: a great deal of formalism is necessary to formulate the
subject while the applications to group theory involve only a fraction of the
machinery; connections with topology are exciting but mysterious at the same
time. The history of the subject is not much help either. Early work by Schur,
at the turn of the century, studying projective complex representations, that
is, homomorphisms of a finite group G into PGL(n, C), introduced cohomology
groups we now know as
C) and
C). Later work by Schreier on
group extensions led to other second cohomology groups. The third cohomology
groups were interpreted by Eilenberg-Mac Lane. The full machinery appears as
a very formal calculus of many-valued functions on groups and rather arbitrary
boundary operators with only low-dimensional cases actually used.
However, cohomology of groups is part of a much broader subject, homological
algebra, which has a much different history. Hilbert's work on syzygies is usually
taken as the starting point of this subject and this involves rings and modules.
In the case of finite groups, modules are a way of looking at representations of
groups. Hence, cohomology can be viewed as part of the representation theory
of groups. The interpretations of the low-dimensional cohomology groups in
group-theoretical terms are now nothing more than another use of representation
theory in group-theoretical problems. This paper will show how accepting this
view leads to a whole new area of representation theory while explaining, at the
same time, some seemingly isolated parts of cohomology of groups.
Let us begin by making this more explicit. The central idea of homological
algebra is that of resolutions. To see what this means in our context, we fix a
finite group G and a field k of prime characteristic p. All our fcG-modules will
be finite-dimensional and our notation is standard [1]. A projective resolution of
the kG module U is an exact sequence
Pn - - Px - P0 - U - 0
1980 Mathematics Subject Classification (1985 Revision). Primary 20C20, 20J06.
Supported in part by National Science Foundation Grant DMS 8421367.
©1987 American Mathematical Society
0082-0717/87 $1.00 + $.25 per page
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