Profinite groups, profinite completions and a conjecture of Moore

Eli Aljadeff

doi:10.1016/j.aim.2004.11.005

Profinite groups, profinite completions and a conjecture of Moore

Eli Aljadeff^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

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.

Original language	English
Pages (from-to)	63-76
Number of pages	14
Journal	Advances in Mathematics
Volume	201
Issue number	1
DOIs	https://doi.org/10.1016/j.aim.2004.11.005
State	Published - 20 Mar 2006
Externally published	Yes

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abstract = "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.",

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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.

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Profinite groups, profinite completions and a conjecture of Moore

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