Nilpotency of Bocksteins, Kropholler’s Hierarchy and a Conjecture of Moore

A conjecture of Moore claims that if Γ is a group and H a finite index subgroup of Γ such that Γ −H has no elements of prime order (e.g. Γ is torsion free), then a Γ-module which is projective over H is projective over Γ. The conjecture is known for finite groups. In that case, it is a direct consequence of Chouinard’s theorem which is based on a fundamental result of Serre on the vanishing of products of Bockstein operators. It was observed by Benson, using a construction of Baumslag, Dyer and Heller, that the analog of Serre’s Theorem for infinite groups is not true in general. We prove that the conjecture is true for groups which satisfy the analog of Serre’s theorem. Using a result of Benson and Goodearl, we prove that the conjecture holds for all groups inside Kropholler’s hierarchy LHF , extending a result of Aljadeff, Cornick, Ginosar, and Kropholler. We show two closure properties for the class of pairs of groups (Γ, H) which satisfy the conjecture, the one is closure under morphisms, and the other is a closure operation which comes from Kropholler’s construction. We use this in order to exhibit cases in which the analog of Serre’s theorem does not hold, and yet the conjecture is true. We will show that in fact there are pairs of groups (Γ, H) in which H is a perfect normal subgroup of prime index in Γ, and the conjecture is true for (Γ, H). Moreover, we will show that it is enough to prove the conjecture for groups of this kind only.