Locally finitely presented and coherent hearts

  • Carlos E. Parra

    Universidad Austral de Chile, Valdivia, Chile
  • Manuel Saorín

    Universidad de Murcia, Spain
  • Simone Virili

    Universitat Autònoma de Barcelona, Bellaterra, Spain
Locally finitely presented and coherent hearts cover
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Starting with a Grothendieck category and a torsion pair in , we study the local finite presentability and local coherence of the heart of the associated Happel–Reiten–Smalø -structure in the derived category . We start by showing that, in this general setting, the torsion pair 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 for which is locally finitely presented, obtaining a complete answer under some additional assumptions on the ground category , 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 in a locally finitely presented for which is locally coherent in two cases: when the tilted -structure in is assumed to restrict to finitely presented objects, and when is cogenerating. In the last part of the paper, we concentrate on the case when 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.

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Carlos E. Parra, Manuel Saorín, Simone Virili, Locally finitely presented and coherent hearts. Rev. Mat. Iberoam. 39 (2023), no. 1, pp. 201–268

DOI 10.4171/RMI/1404