Proceedings of Symposia in Pure Mathematics

Volume 47 (1987)

Cohomology Is Representation Theory

J. L. ALPERIN

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

Hl(G,

C) and

if2(G,

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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http://dx.doi.org/10.1090/pspum/047.1/933344