Minimal inclusions of torsion classes

Barnard, Emily; Carroll, Andrew; Zhu, Shijie

Algebraic Combinatorics, Tome 2 (2019) no. 5, pp. 879-901.

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

Classification : 05E10, 06B15
Mots-clés : lattice theory, torsion classes, canonical join representations

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