Minimal inclusions of torsion classes

Abstract

Let $\Lambda$ be a finite-dimensional associative algebra. The torsion classes of $mod\, \Lambda$ form a lattice under containment, denoted by $tors\, \Lambda$. In this paper, we characterize the cover relations in $tors\, \Lambda$ 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 $tors\, \Lambda$ in terms of representation theory. Finally, we show that, in general, the algebra $\Lambda$ is not characterized by its lattice $tors\, \Lambda$. In particular, we study the torsion theory of a quotient of the preprojective algebra of type $A_n$. We show that its torsion class lattice is isomorphic to the weak order on $A_n$.