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.