Profinite groups, profinite completions and a conjecture of Moore

Abstract

Let R be any ring (with 1), Γ a group and R Γ the corresponding group ring. Let H be a subgroup of Γ of finite index. Let M be an R Γ -module, whose restriction to RH is projective. Moore's conjecture (J. Pure Appl. Algebra 7(1976)287): Assume for every nontrivial element x in Γ , at least one of the following two conditions holds: ( M 1 ) 〈 x 〉 ∩ H ≠ { e } (in particular this holds if Γ is torsion free) ( M 2 ) ord ( x ) is finite and invertible in R. Then M is projective as an R Γ -module. More generally, the conjecture has been formulated for crossed products R * Γ and 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 [ [ R Γ ] ] where R is a profinite ring and Γ a profinite group. We prove the conjecture for arbitrary profinite groups. This implies Serre's theorem on cohomological dimension of profinite groups.