On locally coherent hearts

TL;DR

This paper investigates conditions under which t-structures in derived categories lead to hearts that are locally coherent Grothendieck categories, providing new insights into their structure and equivalences.

Contribution

It establishes that certain t-structures restrict to hearts that are locally coherent Grothendieck categories, especially in the context of finitely presented objects and compactly generated t-structures.

Findings

Hearts are locally coherent Grothendieck categories under specific conditions.

T-structures associated with torsion pairs satisfy these conditions.

In commutative noetherian rings, hearts are equivalent to finitely presented objects of some locally coherent category.

Abstract

We show that, under particular conditions, if a t-structure in the unbounded derived category of a locally coherent Grothendieck category restricts to the bounded derived category of its category of finitely presented objects, then its heart is itself a locally coherent Grothendieck category. Those particular conditions are always satisfied when the Grothendieck category is arbitrary and one considers the t-structure associated to a torsion pair in the category of finitely presented objects. They are also satisfied when one takes any compactly generated t-structure in the derived category of a commutative noetherian ring which restricts to the bounded derived category of finitely generated modules. As a consequence, any t-structure in this latter bounded derived category has a heart which is equivalent to the category of finitely presented objects of some locally coherent Grothendieck category.