On locally coherent hearts

Manuel Saorín

Vol. 287 (2017), No. 1, 199–221

DOI: 10.2140/pjm.2017.287.199

Abstract

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

Milestones

Received: 19 May 2016

Revised: 20 September 2016

Accepted: 1 October 2016

Published: 6 February 2017