New results related to a conjecture of Moore

Abstract

Let Γ be a group, Γ′ be a subgroup of Γ of finite index, and R be a ring with identity. Assume that M is an RΓ-module whose restriction to RΓ′ is projective. Moore's conjecture: Assume that, for all x ∈ (Γ-Γ′), either there is an integer n such that 1 ≠ xn ∈ Γ′ or x has finite order and is invertible in R. Then M is also projective over RΓ. In this paper, we consider an analogue of this conjecture for injective modules. It turns out that the validity of the conjecture for injective modules implies the validity of it on projective and flat modules. It is also shown that the conjecture for injective modules is true whenever Γ belongs to Kropholler's hierarchy LHF. In addition, assume that M is an RΓ-module whose restriction to RΓ′ is Gorenstein projective (resp. injective), it is proved that M is Gorenstein projective (resp. injective) over RΓ whenever Γ′ is a subgroup of Γ of finite index. © 2013 Springer Basel.