Direct limits in the heart of a t-structure: the case of a torsion pair
| Autor: | Parra, Carlos E., Saorín, Manuel |
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| Rok vydání: | 2013 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We study the behavior of direct limits in the heart of a t-structure. We prove that, for any compactly generated t-structure in a triangulated category with exact coproducts, countable direct limits are exact in its heart. Then, for a given Grothendieck category G and a torsion pair t = (T ;F) in G, we show that if the heart of the associated t-structure in the derived category D(G) is AB5, then F is closed under taking direct limits. The reverse implication is true, even implying that the heart is a Grothendieck category, for a wide class of torsion pairs which include the hereditary ones, those for which T is a cogenerating class and those for which F is a generating class. The results allow to extend well-known results by Buan-Krause, Bazzoni and Colpi-Gregorio to the general context of Grothendieck categories and to improve some results of (co)tlting theory of modules. Comment: New proposition 4.7 added and subsequent improvement of the main result of the paper |
| Databáze: | arXiv |
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