Locally finitely presented and coherent hearts

Abstract

Starting with a Grothendieck category $\mathcal{G}$ and a torsion pair $\mathbf{t}=(\mathcal{T},\mathcal{F})$ in $\mathcal{G}$, we study the local finite presentability and local coherence of the heart ${\mathcal{H}}_{\mathbf{t}}$ of the associated Happel–Reiten–Smalø $t$-structure in the derived category $\mathbf{D}(\mathcal{G})$. We start by showing that, in this general setting, the torsion pair $\mathbf{t}$ 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 $\mathbf{t}$ for which ${\mathcal{H}}_{\mathbf{t}}$ is locally finitely presented, obtaining a complete answer under some additional assumptions on the ground category $\mathcal{G}$, 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 $\mathbf{t}=(\mathcal{T},\mathcal{F})$ in a locally finitely presented $\mathcal{G}$ for which ${\mathcal{H}}_{\mathbf{t}}$ is locally coherent in two cases: when the tilted $t$-structure in ${\mathcal{H}}_{\mathbf{t}}$ is assumed to restrict to finitely presented objects, and when $\mathcal{F}$ is cogenerating. In the last part of the paper, we concentrate on the case when $\mathcal{G}$ 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.