Group Cohomology - Formal Constructions

Formal Constructions

In this article, G is a finite group. The collection of all G-modules is a category (the morphisms are group homomorphisms f with the property f(gx) = g(f(x)) for all g in G and x in M). This category of G-modules is an abelian category with enough injectives (since it is isomorphic to the category of all modules over the group ring ℤ).

Sending each module M to the group of invariants MG yields a functor from this category to the category of abelian groups. This functor is left exact but not necessarily right exact. We may therefore form its right derived functors; their values are abelian groups and they are denoted by Hn(G,M), "the n-th cohomology group of G with coefficients in M". H0(G,M) is identified with MG.