Extensions of cocycles, Cohen–Macaulay geometries, and a vanishing theorem for cohomology of alternating groups

A new approach to the computation of the cohomology of groups acting on Buekenhout geometries is described. It is shown that in many cases, including BN-pairs and groups acting on dimensional linear spaces, to compute the kth cohomology group Hk(G,M) it is sufficient, roughly speaking, to consider only parabolic subgroups of rank k+1. As an application it is shown that Hk(An,Fp)=0 for k 1, p>k+2 (where An is the alternating group).