ICE-closed subcategories and wide $\tau$-tilting modules

TL;DR

This paper explores the structure of ICE-closed subcategories in abelian length categories, linking them to torsion classes and wide $ au$-tilting modules, and extends existing bijections in representation theory.

Contribution

It introduces the concept of wide $ au$-tilting modules, establishes a bijection with ICE-closed subcategories, and characterizes ICE-closed subcategories via lattice theory.

Findings

ICE-closed subcategories are exactly torsion classes in some wide subcategories.

A bijection is established between wide $ au$-tilting modules and doubly functorially finite ICE-closed subcategories.

The paper discusses the mutation of rigid modules in the hereditary case.

Abstract

In this paper, we study ICE-closed (= Image-Cokernel-Extension-closed) subcategories of an abelian length category using torsion classes. To each interval $[\mathcal{U},\mathcal{T}]$ in the lattice of torsion classes, we associate a subcategory $\mathcal{T} \cap \mathcal{U}^\perp$ called the heart. We show that every ICE-closed subcategory can be realized as a heart of some interval of torsion classes, and give a lattice-theoretic characterization of intervals whose hearts are ICE-closed. In particular, we prove that ICE-closed subcategories are precisely torsion classes in some wide subcategories. For an artin algebra, we introduce the notion of wide $\tau$-tilting modules as a generalization of support $\tau$-tilting modules. Then we establish a bijection between wide $\tau$-tilting modules and doubly functorially finite ICE-closed subcategories, which extends Adachi--Iyama--Reiten's bijection on torsion classes. For the hereditary case, we discuss the Hasse quiver of the poset of ICE-closed subcategories by introducing a mutation of rigid modules.