On locally coherent hearts

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.