Finite injective dimension over rings with Noetherian cohomology

We study rings that have Noetherian cohomology over a ring of cohomology operators.Examples of such rings include commutative complete intersection rings and finite-dimensional cocommutative Hopf algebras.The main result is a criterion for a complex of modules over a ring with Noetherian cohomology to have finite injective dimension.The criterion implies in particular that for any module over such a ring, if all higher self-extensions of the module vanish, then it must have finite injective dimension.This generalizes a theorem of Avramov and Buchweitz for complete intersection rings, and a well-known theorem in the representation theory of finite groups from finitely generated to arbitrary modules.