Cohomology of Groups

In this chapter we shall apply the theory of derived functors to the important special case where the ground ring Λ is the group ring ℤG of an abstract group G over the integers. This will lead us to a definition of cohomology groups Hn(G, A) and homology groups H _n (G, B) n ≧ 0, where A is a left and B a right G-module (we speak of “G-modules” instead of “ℤG-modules”). In developing the theory we shall attempt to deduce as much as possible from general properties of derived functors. Thus, for example, we shall give a proof of the fact that H2(G, A) classifies extensions which is not based on a particular (i.e. standard) resolution.