of the trivial -module : e.g., one may always take the resolution of group rings, with morphisms Recall that for -modules N and M, HomG(N, M) is an abelian group consisting of -homomorphisms from N to M. Since is a contravariant functor and reverses the arrows, applying to F termwise and dropping produces a cochain complex : The cohomology groups of G with coefficients in the module M are defined as the cohomology of the above cochain complex: This construction initially leads to a coboundary operator that acts on the "homogeneous" cochains. These are the elements of , that is, functions that obey The coboundary operator is now naturally defined by, for example, The relation to the coboundary operator d that was defined in the previous section, and which acts on the "inhomogeneous" cochains , is given by reparameterizing so that and so on. Thus as in the preceding section. Group homology Dually to the construction of group cohomology there is the following definition of group homology: given a G-module M, set DM to be the submodule generated by elements of the form g·m − m, g ∈ G, m ∈ M. Assigning to M its so-called coinvariants, the quotient is a right exact functor. Its left derived functors are by definition the group homology The covariant functor which assigns MG to M is isomorphic to the functor which sends M to where is endowed with the trivial G-action. Hence one also gets an expression for group homology in terms of the Tor functors, Note that the superscript/subscript convention for cohomology/homology agrees with the convention for group invariants/coinvariants, while which is denoted "co-" switches: superscripts correspond to cohomology H* and invariants XG while subscripts correspond to homology H∗ and coinvariants XG := X/G. Specifically, the homology groups Hn(G, M) can be computed as follows. Start with a projective resolution F of the trivial -module as in the previous section. Apply the covariant functor to F termwise to get a chain complex : Then Hn(G, M) are the homology groups of this chain complex, for n ≥ 0. Group homology and cohomology can be treated uniformly for some groups, especially finite groups, in terms of complete resolutions and the Tate cohomology groups. The group homology of abelian groups G with values in a principal ideal domain k is closely related to the exterior algebra . Low-dimensional cohomology groups H 1 The