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Exactness of direct limits for abelian categories with an injective cogenerator |
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| Authors: | Leonid Positselski Jan Šťovíček |
| Affiliation: | 1. Institute of Mathematics, Czech Academy of Sciences, ?itná 25, 115 67 Prague 1, Czech Republic;2. Laboratory of Algebraic Geometry, National Research University Higher School of Economics, Moscow 119048, Russia;3. Sector of Algebra and Number Theory, Institute for Information Transmission Problems, Moscow 127051, Russia;4. Department of Mathematics, Faculty of Natural Sciences, University of Haifa, Mount Carmel, Haifa 31905, Israel;5. Charles University in Prague, Faculty of Mathematics and Physics, Department of Algebra, Sokolovská 83, 186 75 Praha, Czech Republic |
| 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. |
| Keywords: | 18E15 18C20 18A30 |
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