Minimal Inclusions of Torsion Classes

Let Λ be a finite-dimensional associative algebra. The torsion classes of mod Λ form a lattice under containment, denoted by tors Λ. In this paper, we characterize the cover relations in tors Λ by certain indecomposable modules. We consider three applications: First, we show that the completely join-irreducible torsion classes (torsion classes which cover precisely one element) are in bijection with bricks. Second, we characterize faces of the canonical join complex of tors Λ in terms of representation theory. Finally, we show that, in general, the algebra Λ is not characterized by its lattice tors Λ. In particular, we study the torsion theory of a quotient of the preprojective algebra of type An. We show that its torsion class lattice is isomorphic to the weak order on An.