Local Coherence of Hearts Associated with Thomason Filtrations

TL;DR

This paper investigates the local coherence of hearts associated with Thomason filtrations in derived categories, establishing conditions under which these hearts are locally finitely presented Grothendieck categories, especially for finite length filtrations.

Contribution

It proves the coincidence of hearts for weakly bounded below Thomason filtrations and characterizes their local coherence, especially in finite length cases, using module-theoretic conditions.

Findings

Hearts of the two t-structures coincide for weakly bounded below filtrations.

Hearts are locally finitely presented Grothendieck categories under certain conditions.

Finite length cases relate to hereditary torsion classes and their Happel-Reiten-Smalo hearts.

Abstract

Any Thomason filtration of a commutative ring yields (at least) two t-structures in the derived category of the ring, one of which is compactly generated [Hrb20,HHZ21]. We study the hearts of these two t-structures and prove that they coincide in case of a weakly bounded below filtration. Prompted by [SS20], in which it is proved that the heart of a compactly generated t-structure in a triangulated category with coproduct is a locally finitely presented Grothendieck category, we study the local coherence of the hearts associated with a weakly bounded below Thomason filtration, achieving a useful recursive characterisation in case of a finite length filtration. Low length cases involve hereditary torsion classes of finite type of the ring, and even their Happel-Reiten-Smalo hearts; in these cases, the relevant characterisations are given by few module-theoretic conditions.