 
Summary: Advances in Mathematics 201 (2006) 6376
www.elsevier.com/locate/aim
Profinite groups, profinite completions and a
conjecture of Moore
Eli Aljadeff
Department of Mathematics, Technion Israel Institute of Technology, 32000 Haifa, Israel
Received 21 April 2004; accepted 16 November 2004
Communicated by Mark Hovey
Available online 1 February 2005
Abstract
Let R be any ring (with 1), a group and R the corresponding group ring. Let H be a
subgroup of of finite index. Let M be an R module, whose restriction to RH is projective.
Moore's conjecture (J. Pure Appl. Algebra 7(1976)287): Assume for every nontrivial element
x in , at least one of the following two conditions holds:
(M1) x H = {e} (in particular this holds if is torsion free)
(M2) ord(x) is finite and invertible in R.
Then M is projective as an R module.
More generally, the conjecture has been formulated for crossed products R and even for
strongly graded rings R( ). We prove the conjecture for new families of groups, in particular
for groups whose profinite completion is torsion free.
