Finite injective dimension over rings with Noetherian cohomology

TL;DR

This paper provides a new criterion for determining when a complex over rings with Noetherian cohomology has finite injective dimension, generalizing previous results and applicable to various algebraic structures.

Contribution

It introduces a generalized criterion for finite injective dimension over rings with Noetherian cohomology, removing previous finiteness restrictions.

Findings

Modules with vanishing higher self-extensions have finite injective dimension.

The criterion applies to rings like complete intersections and Hopf algebras.

Unifies and extends previous results in the area.

Abstract

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.