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# Mathematical Research Letters

## Volume 19 (2012)

### Number 4

### Finite injective dimension over rings with Noetherian cohomology

Pages: 741 – 752

DOI: http://dx.doi.org/10.4310/MRL.2012.v19.n4.a1

#### Author

#### Abstract

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.