Homological systems in triangulated categories

TL;DR

This paper introduces homological systems in triangulated categories, linking them to stratified algebras and exact categories, and explores their connections with cotorsion pairs and derived equivalences.

Contribution

It generalizes stratifying systems and exceptional sequences, establishing derived equivalences between associated stratified algebras and describing the structure of filtered objects.

Findings

Existence of derived equivalent standardly stratified algebras A and B.

The category of filtered objects admits a natural exact structure.

Connections with cotorsion pairs and cluster tilting categories.

Abstract

We introduce the notion of homological systems $\Theta$ for triangulated categories. Homological systems generalize, on one hand, the notion of stratifying systems in module categories, and on the other hand, the notion of exceptional sequences in triangulated categories. We prove that, attached to the homological system $\Theta,$ there are two standardly stratified algebras $A$ and $B,$ which are derived equivalent. Furthermore, it is proved that the category $\F(\Theta),$ of the $\Theta$-filtered objects in a triangulated category $\T,$ admits in a very natural way an structure of an exact category, and then there are exact equivalences between the exact category $\F(\Theta)$ and the exact categories of the $\Delta$-good modules associated to the standardly stratified algebras $A$ and $B.$ Some of the obtained results can be seen also under the light of the cotorsion pairs in the sense of Iyama-Nakaoka-Yoshino (see \ref{CotorsionThetaSys1} and \ref{CotorsionThetaSys2}). We recall that cotorsion pairs are studied extensively in relation with cluster tilting categories, $t$-structures and co-$t$-structures.