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 $\tau$-tilting reduction of Jasso, we mainly focus on intervals $[\mathcal{U},\mathcal{T}]$ in the lattice $\mathrm{tors}\mathcal{A}$ 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 $\mathrm{tors}\mathcal{W}$ 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.

17 pages