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

Hanpeng Gao, Dajun Liu, Yu-zhe Liu

Abstract

For Artin algebras, we establish a bijective between IE-closed subcategories and canonical intervals in the lattice of torsion classes. Enomoto and Sakai previously achieved a classification of IE-closed subcategories over hereditary algebras using twin rigid modules. However, this result fails for the non-hereditary algebras. In this paper, we generalize this classification to arbitrary $τ$-tilting finite algebras by replacing twin rigid modules with canonical twin support $τ$-tilting modules. We provide a homological characterization of these modules via relative torsion theory, and then obtain a constructive algorithm to canonicalize an arbitrary twin support $τ$-tilting module while preserving its associated heart.

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

Abstract

For Artin algebras, we establish a bijective between IE-closed subcategories and canonical intervals in the lattice of torsion classes. Enomoto and Sakai previously achieved a classification of IE-closed subcategories over hereditary algebras using twin rigid modules. However, this result fails for the non-hereditary algebras. In this paper, we generalize this classification to arbitrary -tilting finite algebras by replacing twin rigid modules with canonical twin support -tilting modules. We provide a homological characterization of these modules via relative torsion theory, and then obtain a constructive algorithm to canonicalize an arbitrary twin support -tilting module while preserving its associated heart.
Paper Structure (5 sections, 11 theorems, 16 equations, 3 tables)

This paper contains 5 sections, 11 theorems, 16 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Lattice of IE-closed Subcategories
  4. Twin support $\tau$-tilting modules
  5. Examples

Key Result

Theorem 2.1

Let $\Lambda$ be an Artin algebra. There is a bijection between the set of isomorphism classes of basic support $\tau$-tilting $\Lambda$-modules and the set of functorially finite torsion classes in $\operatorname{mod}\Lambda$. The bijection is explicitly given by $M \mapsto \mathsf{Fac} M$.

Theorems & Definitions (28)

  • Theorem 2.1: AIR14, Theorem 2.7
  • Theorem 2.2: DIJ19, Theorem 3.8
  • Definition 2.3: ES23
  • Proposition 2.4: ES23, Proposition 2.3
  • Definition 3.1
  • Lemma 3.2
  • Definition 3.3
  • Lemma 3.4
  • proof
  • Definition 3.5
  • ...and 18 more