Minimal resolutions for finite groups

Abstract

(1) ~~*+Fz-+F,+F,+Z--+O where G acts trivially on Z and each F, is ZG-free. For computation of the cohomology of G it is convenient to choose a resolution (1) in which the Fi do not have too many generators. I will consider here the following general problem: Suppose we are given a sequence of integers fO, fi, .... What conditions must the fi satisfy in order that there exists a resolution (1) in which Fi is ZG-free on fi generators? The main result will be an almost complete solution of this problem for the case in which G is finite. The conditions on thefi will be a set of inequalities connecting them with cohomological invariants of G. If G is a finite p-group, our theorem implies that there is a resolution (1) in whichfi = dim H’(G, Z,). Therefore G has a resolution which is minimal in an obvious sense. Before proving the main theorem I will give a number of necessary conditions which thefi must satisfy. These conditions do not require the group G to be finite. Since the conditions are closely related to the Morse inequalities, it will be convenient to make the following definition DEFINITION. Let F be a resolution ~~~-+F2--+F1-+F,,--+Z--+0