  We study rings which have Noetherian cohomology under the action of a ring of
cohomology operators. The main result is a criterion for a complex of modules
over such a ring to have finite injective dimension. This criterion
generalizes, by removing finiteness conditions, and unifies several previous
results. In particular we show that for a module over a ring with Noetherian
cohomology, if all higher self-extensions of the module vanish then it must
have finite injective dimension. Examples of rings with Noetherian cohomology
include commutative complete intersection rings and finite dimensional
cocommutative Hopf algebras over a field.
