RELATIVE COMPLETIONS AND THE COHOMOLOGY OF LINEAR GROUPS OVER LOCAL RINGS

Abstract

For a discrete group G there are two well known completions. The first is the Malcev (or unipotent) completion. This is a prounipotent group [Uscr ], defined over ℚ, together with a homomorphism ψ : G → [Uscr ] that is universal among maps from G into prounipotent ℚ-groups. To construct [Uscr ], 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^rG, where Γ^{[bull ]}G denotes the lower central series of G. If G is abelian, then [Uscr ] = G [otimes ] ℚ. We review this construction in Section 2.