A characterization of IE-closed subcategories via canonical twin support $\tau$-tilting modules

TL;DR

This paper extends the classification of IE-closed subcategories from hereditary to all finite-dimensional algebras using canonical twin support τ-tilting modules and provides a constructive algorithm for canonicalization.

Contribution

It introduces canonical twin support τ-tilting modules and Ext-pairs, generalizing prior classifications to arbitrary finite-dimensional algebras.

Findings

Established bijections among IE-closed subcategories, τ-tilting modules, and Ext-pairs.

Provided a constructive algorithm to canonicalize twin support τ-tilting modules.

Generalized classification beyond hereditary algebras.

Abstract

Enomoto and Sakai classified IE-closed subcategories over hereditary algebras via twin rigid modules. However, this classification inherently relies on the vanishing of second extension spaces, thus failing for arbitrary finite-dimensional algebras. In this paper, we generalize their classification to arbitrary finite-dimensional algebras by introducing the notions of canonical twin support $\tau$-tilting modules and canonical Ext-pairs. By utilizing functorially finite torsion pairs, we provide a homological characterization of these modules. Furthermore, we establish explicit bijections up to isomorphism among functorially finite IE-closed subcategories, canonical twin support $\tau$-tilting modules, and canonical Ext-pairs. Finally, we provide a constructive algorithm to canonicalize any given twin support $\tau$-tilting module.