The varieties and the cohomology ring of a module

Abstract

Let G be a finite group and let K be a field of characteristic p > 0. Recently several studies have focused on the homological invariants of finitely generated KG-modules. In [l] Alperin proposed the study of the complexity of a module. The complexity is related to the degree of the polynomial rate of growth of the terms in a projective resolution of the module (see Definition 2.4). In [2] Alperin and Evens proved that the complexity of a KG-module M is equal to the maximum of the complexities of the restrictions of M to elementary abelian p-subgroups of G. One of the roots of the Alperin-Evens Theorem is Quillen’s Dimension Theorem (see [21] or [22]), which with some difficulty can be interpreted as saying the same thing for the special case of the trivial KG-module. Alperin and Evens [3] and Avrunin [4] have further studied the annihilator in Ext&(K, K) of the cohomology of M, and have produced theorems of a similar nature. These are related to the work in [9] on the structure of the cohomology ring Ext&(M, M). The purpose of this paper is to investigate the relationship between these homological invariants and the structure of the module. We concentrate on the case in which G is an elementary abelian p-group. However, using such results as those mentioned above, we can apply the present work to more general groups. Even as the earlier results indicate that much of the structure and cohomology of a module is revealed in the restrictions to elementary abelian p-subgroups, we show here that, in the case of an elementary abelian group, much of this information can be found by looking at the restrictions to cyclic p-subgroups of the group of units of KG. Some of the results in this paper were announced in [S]. Let G = (xl ,..., xn) be an elementary abelian group of order p”. Suppose that K is algebraically closed. To any finitely generated KG-module M we