TY - JOUR

T1 - Profinite groups, profinite completions and a conjecture of Moore

AU - Aljadeff, Eli

N1 - Copyright:
Copyright 2006 Elsevier B.V., All rights reserved.

PY - 2006/3/20

Y1 - 2006/3/20

N2 - Let R be any ring (with 1), Γ a group and RΓ the corresponding group ring. Let H be a subgroup of Γ of finite index. Let M be an RΓ-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: (M1) 〈x〉 ∩ H ≠ {e} (in particular this holds if Γ is 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 * 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 [[RΓ]] where R is a profinite ring and Γ a profi nite group. We prove the conjecture for arbitrary profinite groups. This implies Serre's theorem on cohomological dimension of profinite groups.

AB - Let R be any ring (with 1), Γ a group and RΓ the corresponding group ring. Let H be a subgroup of Γ of finite index. Let M be an RΓ-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: (M1) 〈x〉 ∩ H ≠ {e} (in particular this holds if Γ is 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 * 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 [[RΓ]] where R is a profinite ring and Γ a profi nite group. We prove the conjecture for arbitrary profinite groups. This implies Serre's theorem on cohomological dimension of profinite groups.

UR - http://www.scopus.com/inward/record.url?scp=33144466565&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2004.11.005

DO - 10.1016/j.aim.2004.11.005

M3 - 文章

AN - SCOPUS:33144466565

VL - 201

SP - 63

EP - 76

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 1

ER -