The Auslander-Reiten Conjecture for Group Rings

This paper studies the vanishing of Ext modules over group rings. Let R be a commutative noetherian ring and Γ a group. We provide a criterion under which the vanishing of self extensions of a nitely generated RΓ-module M forces it to be projective. Using this result, it is shown that RΓ satis es the Auslander-Reiten conjecture, whenever R has nite global dimension and Γ is a nite acyclic group.