Classifying $t$-structures via ICE-closed subcategories and a lattice of torsion classes

TL;DR

This paper explores the relationship between ICE-closed subcategories and $t$-structures in triangulated categories, introducing ICE sequences and establishing a bijection with homology-determined preaisles, with applications to derived categories of $ au$-tilting finite algebras.

Contribution

It introduces ICE sequences and establishes a bijection with homology-determined preaisles, providing new insights into $t$-structure classification in triangulated categories.

Findings

Established a bijection between ICE sequences and homology-determined preaisles.

Provided a sufficient condition for ICE sequences to induce $t$-structures.

Described ICE sequences in the derived category of $ au$-tilting finite algebras.

Abstract

In a triangulated category equipped with a $t$-structure, we investigate a relation between ICE-closed (=Image-Cokernel-Extension-closed) subcategories of the heart of the $t$-structure and aisles in the triangulated categories. We introduce an ICE sequence, a sequence of ICE-closed subcategories satisfying a certain condition, and establish a bijection between ICE sequences and homology-determined preaisles. Moreover we give a sufficient condition that an ICE sequence induces a $t$-structure via the bijection. In the case of the bounded derived category $D^b({\mathsf{mod}}\Lambda)$ of a $\tau$-tilting finite algebra $\Lambda$, we give a description of ICE sequences in ${\mathsf{mod}}\Lambda$ which induce bounded $t$-structures on $D^b({\mathsf{mod}}\Lambda)$ from the viewpoint of a lattice consisting of torsion classes in ${\mathsf{mod}}\Lambda$.