Exactness of direct limits for abelian categories with an injective cogenerator

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...

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Bibliographic Details
Published in:Journal of pure and applied algebra 2019-08, Vol.223 (8), p.3330-3340
Main Authors: Positselski, Leonid, Šťovíček, Jan
Format: Article
Language:English
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Summary: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.
ISSN:0022-4049
1873-1376
DOI:10.1016/j.jpaa.2018.11.004