Torsion pairs and filtrations in abelian categories with tilting objects

TL;DR

This paper explores the relationship between a noetherian abelian category with a tilting object and module categories over endomorphism rings, providing explicit torsion pair descriptions and generalizing filtrations to higher homological dimensions.

Contribution

It offers an explicit description of torsion pairs in categories with tilting objects and generalizes existing filtrations to categories of any finite homological dimension.

Findings

Established a sequence of tilts relating categories and module categories.

Provided a simplified proof of a known filtration theorem.

Generalized filtrations to categories with arbitrary finite homological dimension.

Abstract

Given a noetherian abelian category $\mathcal Z$ of homological dimension two with a tilting object $T$, the abelian category $\mathcal Z$ and the abelian category of modules over $\text{End} (T)^{\textit{op}}$ are related by a sequence of two tilts; we give an explicit description of the torsion pairs involved. We then use our techniques to obtain a simplified proof of a theorem of Jensen-Madsen-Su, that $\mathcal Z$ has a three-step filtration by extension-closed subcategories. Finally, we generalise Jensen-Madsen-Su's filtration to a noetherian abelian category of any finite homological dimension.