Wide Subcategories and Lattices of Torsion Classes

Abstract

In this paper, we study the relationship between wide subcategories and torsion classes of an abelian length category $\mathcal {A}$ from the point of view of lattice theory. Motivated by τ-tilting reduction of Jasso, we mainly focus on intervals $[\mathcal {U},\mathcal {T}]$ in the lattice $\operatorname {\mathsf {tors}} \mathcal {A}$ of torsion classes in $\mathcal {A}$ such that $\mathcal {W}:=\mathcal {U}^{\perp } \cap \mathcal {T}$ is a wide subcategory of $\mathcal {A}$; we call these intervals wide intervals. We prove that a wide interval $[\mathcal {U},\mathcal {T}]$ is isomorphic to the lattice $\operatorname {\mathsf {tors}} \mathcal {W}$ of torsion classes in the abelian category $\mathcal {W}$. We also characterize wide intervals in two ways: First, in purely lattice theoretic terms based on the brick labeling established by Demonet–Iyama–Reading–Reiten–Thomas; and second, in terms of the Ingalls–Thomas correspondences between torsion classes and wide subcategories, which were further developed by Marks–Šťovíček.