Cohomology Theory of Finite Groups

Abstract

Homology and cohomology are usually thought of as dual to one another. We have seen in Chapter III, for example, that homology has a number of formal properties and that cohomology has “dual” properties. If G is finite, however, then homology and cohomology seem to have similar properties rather than dual ones. For example, since every subgroup H of a finite group G has finite index, we have restriction and corestriction maps for arbitrary subgroups, in both homology and cohomology. For another example, the distinction between induced modules and co-induced modules disappears, so we have a single class I of G-modules (namely, the induced modules ℤG ⊗ A) with the following properties: (a) Every M ∈ I is acyclic for both homology and cohomology. (b) For every G-module M there is a module M̅ ∈ I such that M is a quotient of M̅ and M can be embedded in M̅.