Profinite groups, profinite completions and a conjecture of Moore

Abstract

Let R be any ring (with 1), Γa group and RΓthe corresponding group ring. Let H be a subgroup of Γof finite index. Let M be an RΓ-module, whose restriction to RH is projective. Moore's conjecture: Assume for every nontrivial element x in Γ, at least one of the following two conditions holds: M1) the subgroup generated by x intersects H non-trivially (in particular this holds if Γis torsion free). M2) ord(x) is finite and invertible in R. Then M is projective as an RΓ-module. More generally, the conjecture has been formulated for crossed products R*Γand even for strongly graded rings R(Γ). We prove the conjecture for new families of groups, in particular for groups whose profinite completion is torsion free. The conjecture can be formulated for profinite modules M over complete groups rings [[RΓ]] where R is a profinite ring and Γa profinite group. We prove the conjecture for arbitrary profinite groups. This implies Serre's theorem on cohomological dimension of profinite groups.