An Analogue of Brumer’s Criterion for Hypercohomology Groups and Class Formations

The aim of this paper is to obtain a criterion determining the strict cohomological dimension of a profinite group. The strict cohomological dimension of a profinite group G, written scdpG, is the smallest integer n such that the p-primary component of Hn+1(G,M) vanishes for all discrete G-modules M , where p is a prime number. There is also the notion of cohomological dimension of G, denoted by cdpG. This is the smallest integer n such that Hn+1(G,A) = 0 for all discrete p-primary G-modules A. It is well-known that the strict cohomological dimension of G is equal to cdpG or cdpG + 1. This result is often useful. In many cases, in fact, it enables us to obtain sufficient information on the cohomology groups of a given profinite group. It is, nevertheless, quite difficult to determine the strict cohomological dimension of a given profinite group. Brumer gave a useful criterion determining it [1]: