In order to pursue our study of Brauer groups, we need some basic notions from the cohomology theory of groups with abelian coefficient modules. This is a theory which is well documented in the literature; we only establish here the facts we shall need in what follows, for the ease of the reader. In particular, we establish the basic exact sequences, construct cup-products and study the maps relating the cohomology of a group to that of a subgroup or a quotient. In accordance with the current viewpoint in homological algebra, we emphasize the use of complexes and projective resolutions, rather than that of explicit cocycles and the technique of dimension-shifting (though the latter are also very useful).

As already said, the subject matter of this chapter is fairly standard and almost all facts may already be found in the first monograph written on homological algebra, that of Cartan and Eilenberg [1]. Some of the constructions were first developed with applications to class field theory in view. For instance, Shapiro's lemma first appears in a footnote to Weil [1], then with a (two-page) proof in Hochschild–Nakayama [1].

Definition of cohomology groups

Let G be a group. By a (left) G-module we shall mean an abelian group A equipped with a left action by G. Notice that this is the same as giving a left module over the integral group ring Z[G]: indeed, for elements ∑ nσσ∈ Z[G] and a ∈ A we may define (∑ nσσ)a := ∑ nσσ(a) and conversely, a Z[G]- module structure implies in particular the existence of “multiplication-by-σ” maps on A for all σ ∈ G.