#### Vol. 287, No. 1, 2017

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On locally coherent hearts

### Manuel Saorín

Vol. 287 (2017), No. 1, 199–221
##### Abstract

Let $\mathsc{G}$ be a locally coherent Grothendieck category. We show that, under particular conditions, if a t-structure $\tau$ in the unbounded derived category $\mathsc{D}\left(\mathsc{G}\right)$ restricts to the bounded derived category of its category of finitely presented (i.e, coherent) objects, then its heart ${\mathsc{ℋ}}_{\tau }$ is a locally coherent Grothendieck category on which is the class of finitely presented objects. Those particular conditions are always satisfied when $\mathsc{G}$ is arbitrary and $\tau$ is the Happel–Reiten–Smalø t-structure in $\mathsc{D}\left(\mathsc{G}\right)$ associated to a torsion pair in or when is the category of quasicoherent sheaves on a noetherian affine scheme $\mathbb{X}$ and $\tau$ is any compactly generated t-structure in which restricts to . In particular, the heart of any t-structure in ${\mathsc{D}}^{b}\left(\mathbb{X}\right)$ is the category of finitely presented objects of a locally coherent Grothendieck category.

##### Keywords
locally coherent Grothendieck category, triangulated category, derived category, t-structure, heart of a t-structure
##### Mathematical Subject Classification 2010
Primary: 18E15, 18E30, 13DXX, 14AXX, 16EXX