Filtrations in abelian categories with a tilting object of homological dimension two

TL;DR

This paper explores filtrations in abelian categories induced by a tilting object of homological dimension two, extending classical torsion pair filtrations and utilizing derived equivalences for refined analysis.

Contribution

It introduces a new filtration framework for objects in abelian categories with tilting objects of dimension two, generalizing classical two-step filtrations and incorporating derived category techniques.

Findings

Defined three disjoint subcategories with no maps between them.

Established a unique filtration with factors in these subcategories.

Provided a refined filtration using derived equivalences and quasi-isomorphisms.

Abstract

We consider filtrations of objects in an abelian category $\catA$ induced by a tilting object $T$ of homological dimension at most two. We define three disjoint subcategories with no maps between them in one direction, such that each object has a unique filtation with factors in these categories. This filtration coincides with the the classical two-step filtration induced by torsion pairs in dimension one. We also give a refined filtration, using the derived equivalence between the derived categories of $\catA$ and the module category of $End_\catA (T)^{op}$. The factors of this filtration consist of kernel and cokernels of maps between objects which are quasi-isomorphic to shifts of $End_\catA (T)^{op}$-modules via the derived equivalence $\mathbb{R}Hom_\catA(T,-)$.