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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Abstract

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.

Cite this article

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