Contemporary Mathematics

Volume 158, 1994

SYSTEMS OF PARAMETERS AND THE STRUCTURE OF

COHOMOLOGY RINGS OF FINITE GROUPS

JoN

F.

CARLSoN*

This paper is a report of the survey lecture presented by the author at the

meeting on representation theory and topology at Northwestern University. Let

G be a finite group and let k denote a field of characteristic p 0. For the

purposes of this paper all kG-modules are finitely generated. Several recent

works by Dave Benson and the author [3,4,5,6,7,9] have focused on connections

between representation theory and cohomology of groups. Many of the results are

valid for larger classes of groups, such as groups with finite virtual cohomological

dimension, and for coefficients other than fields of finite characteristic. While it is

often assumed that the field k is algebraically closed, even this is not necessary.

However for this survey we will consider only G and k as stated above. In

general the major concern of the works has been the investigation of the following

questions:

A) What is the algebraic structure of cohomology rings?

and

B) What are the implications of that structure for the representation theory?

It is well known (Evens [11], Venkov [15,16]) that the cohomology ring H*(G, k)

is a finitely generated graded commutative k-algebra and if

M

is a kG-module

then H*(G,

M)

is a finitely generated module over H*(G, k). As a result, the

maximal ideal spectrum Vc ( k) of H* ( G, k) is a finite dimensional affine variety

(see Benson [2] or Evens [12] for a general reference). Notice that the cohomology

ring H* ( G, k) is commutative modulo its Jacobson radical. Hence the spectrum

is an affine variety because every maximal ideal contains the Jacobson radical. A

fundamental theorem of Quillen [14] says that the dimension of this variety, which

1991 Mathematics Subject Classification. Primary 20C20, 20J06.

Partially supported by a grant from the NSF.

This paper is in final form and no version of it will be submitted for publication elsewhere.

1

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1994 American Mathematical Society

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http://dx.doi.org/10.1090/conm/158/01450