Relative Completions and the Cohomology of Linear Groups Over Local Rings

For a discrete group G there are two well known completions. The first is the Malcev (or unipotent) completion. This is a prounipotent group U, defined over Q, together with a homomorphism ψ : G → U that is universal among maps from G into prounipotent Q‐groups. To construct U, it suffices for us to consider the case where G is nilpotent; the general case is handled by taking the inverse limit of the Malcev completions of the G /Γ r G , where Γ • G denotes the lower central series of G . If G is abelian, then U = G ⊗ Q. We review this construction in Section 2.