Exactness of direct limits for abelian categories with an injective cogenerator

Abstract

We prove that the exactness of direct limits in an abelian category with products and an injective cogenerator J is equivalent to a condition on J which is well-known to characterize pure-injectivity in module categories, and we describe an application of this result to the tilting theory. We derive our result as a consequence of a more general characterization of when inverse limits in the Eilenberg-Moore category of a monad on the category of sets preserve regular epimorphisms.

13 pages; version 2: examples added (Examples 2.5); version 3: minor corrections