Minimal inclusions of torsion classes

Let $\mathrm{\Lambda }$ be a finite-dimensional associative algebra. The torsion classes of $mod\mathrm{\Lambda }$ form a lattice under containment, denoted by $tors, \Lambda$\mathrm{.}Inthispaper,wecharacterizethecoverrelationsin$tors, \Lambda$ 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 $tors\mathrm{\Lambda }$ in terms of representation theory. Finally, we show that, in general, the algebra $\mathrm{\Lambda }$ is not characterized by its lattice $tors\mathrm{\Lambda }$. 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 $A_n$.