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

• Carlos E. Parra ^[1] ; Manuel Saorín ^[2]
  1. [1] Universidad de Los Andes (Venezuela)

     Universidad de Los Andes (Venezuela)

     Venezuela

     
  2. [2] Universidad de Murcia

     Universidad de Murcia

     Murcia, España

     
• Localización: Journal of pure and applied algebra, ISSN 0022-4049, Vol. 219, Nº 9 (September 2015), 2015, págs. 4117-4143
• Idioma: inglés
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• Resumen

  • 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 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 the heart Ht of the associated t-structure in the derived category D(G) is AB5 if, and only if, it is a Grothendieck category. If this is the case, then F is closed under taking direct limits. The reverse implication is true 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 results by Buan–Krause and Colpi-Gregorio to the general context of Grothendieck categories and to improve some results of (co)tilting theory of modules.