Filtrations of torsion classes in proper abelian subcategories

TL;DR

This paper generalizes a filtration method for objects in abelian categories using extension-closed classes, extending previous work from tilting objects of projective dimension 1 to broader contexts.

Contribution

It extends the filtration framework from tilting objects of projective dimension 1 to proper abelian subcategories, broadening the applicability of the method.

Findings

Established a generalized filtration technique for proper abelian subcategories.

Extended the class of objects that can be filtered beyond tilting objects of projective dimension 1.

Provided a new perspective on torsion class filtrations in abelian categories.

Abstract

In an abelian category $\mathscr{A}$, we can generate torsion pairs from tilting objects of projective dimension $\leq 1$. However, when we look at tilting objects of projective dimension $2$, there is no longer a natural choice of an associated torsion pair. Instead of trying to generate a torsion pair, Jensen, Madsen and Su generated a triple of extension closed classes that can filter any objects of $\mathscr{A}$. We generalize this result to proper abelian subcategories.