Locally finitely presented and coherent hearts

• Carlos E. Parra ^[1] ; Manuel Saorín ^[2] ; Simone Virili ^[3]
  1. [1] Universidad Austral de Chile

     Universidad Austral de Chile

     Valdivia, Chile

     
  2. [2] Universidad de Murcia

     Universidad de Murcia

     Murcia, España

     
  3. [3] Universitat Autònoma de Barcelona

     Universitat Autònoma de Barcelona

     Barcelona, España

     

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• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 39, Nº 1, 2023, págs. 201-268
• Idioma: inglés
• DOI: 10.4171/RMI/1404
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• Resumen

  • Starting with a Grothendieck category G and a torsion pair t D .T ; F / in G, we study the local finite presentability and local coherence of the heart Ht of the associated Happel–Reiten–Smalø t-structure in the derived category D.G/. We start by showing that, in this general setting, the torsion pair t is of finite type, if and only if it is quasi-cotilting, if and only if it is cosilting. We then proceed to study those t for which Ht is locally finitely presented, obtaining a complete answer under some additional assumptions on the ground category G, which are general enough to include all locally coherent Grothendieck categories, all categories of modules and several categories of quasi-coherent sheaves over schemes. The third problem that we tackle is that of local coherence. In this direction, we characterize those torsion pairs t D .T ; F / in a locally finitely presented G for which Ht is locally coherent in two cases: when the tilted t-structure in Ht is assumed to restrict to finitely presented objects, and when F is cogenerating. In the last part of the paper, we concentrate on the case when G is a category of modules over a small preadditive category, giving several examples and obtaining very neat (new) characterizations in this more classical setting, underlying connections with the notion of an elementary cogenerator.