Direct limits in the heart of a t-structure: the case of a torsion pair

Carlos E. Parra; Manuel Saorín

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arXiv · 1311.6166

Direct limits in the heart of a t-structure: the case of a torsion pair

Carlos E. Parra·Manuel Saorín

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Abstract

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.

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BibTeX RIS

Carlos E. Parra, Manuel Saorín. 2014-09-21. Direct limits in the heart of a t-structure: the case of a torsion pair. https://arxiv.org/abs/1311.6166

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