Minimal inclusions of torsion classes

Abstract

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.

On sign-coherence of c-vectors

“…The first one is that the arrows of the exchange graph of τ -tilting pairs of every algebra can be labeled with c-vectors as follows. Note that the previous labelling coincides with the brick labelling studied for instance in [12,5].…”

Wall and chamber structure for finite-dimensional algebras

“…As a immediate consequence of Theorem 4.3 we obtain the following. This labeling by bricks appeared independently in [2,16,7]. Moreover, based on the results developed here, the brick labeling was studied further in [37], showing that dimension vectors of these bricks correspond to the c-vectors of mod A.…”

The Canonical Join Complex

“…Recently, canonical join representations have played a role in the study of functorially finite torsion classes for the preprojective algebra of Dynkintype W , when W is a simply laced Weyl group (see for example [15,21]). In the forthcoming [11], the authors study the canonical join complex for any finite dimensional associative algebra Λ of finite representation type. Since the weak order on any finite Coxeter group W and the lattice of torsion classes for Λ of finite representation type are both examples of finite semidistributive lattices (see [7,Lemma 9] and [15,Theorem 4.5]), we obtain the following two applications of Theorem 1.1: Corollary 1.6.…”