A characterization of IE-closed subcategories via $τ$-tilting theory

Abstract

Enomoto and Sakai classified functorially finite IE-closed subcategories over hereditary algebras in terms of twin rigid modules. Their approach uses the hereditary assumption essentially and therefore does not extend directly to arbitrary finite-dimensional algebras. In this paper, we introduce canonical twin support $τ$-tilting modules and prove that, for an arbitrary finite-dimensional algebra, they are in bijection with left-and-right finite IE-closed subcategories, namely those whose generated torsion and torsion-free classes are both functorially finite. We further give a characterization of canonicality via the torsion-pair decompositions associated with $\operatorname{Fac} M$ and $\operatorname{Sub} N$, which yields a canonicalization procedure whenever the associated IE-closed subcategory is left-and-right finite. We also introduce canonical Ext-pairs. If the algebra is hereditary or $τ$-tilting finite, then functorially finite IE-closed subcategories are in bijection with isomorphism classes of canonical Ext-pairs, where the corresponding pair is given by the basic Ext-progenerator and the basic Ext-injective cogenerator. In the hereditary case, this recovers the twin rigid classification of Enomoto and Sakai.