Classifying torsion classes for algebras with radical square zero via sign decomposition

Abstract

To study the set of torsion classes of a finite dimensional basic algebra, we use a decomposition, called sign-decomposition, parametrized by elements of $\{\pm 1{\}}^n$ where $n$ is the number of simple modules. If $A$ is an algebra with radical square zero, then for each $\epsilon \in \{\pm 1{\}}^n$ there is a hereditary algebra $A_{\epsilon }^{!}$ with radical square zero and a bijection between the set of torsion classes of $A$ associated to $\epsilon$ and the set of faithful torsion classes of $A_{\epsilon }^{!}$. Furthermore, this bijection preserves the property of being functorially finite. As an application in $\tau$-tilting theory, we prove that the number of support $\tau$-tilting modules over Brauer line algebras (resp. Brauer odd-cycle algebras) having $n$ edges is $\left(\genfrac{}{}{0ex}{}{2n}{n}\right)$ (resp. $2^{2n-1}$).

25 pages. Change title. Many improvements and changes compared to the first version (in particular, mainly study torsion classes and $\tau$-tilting theory. The results in the first version is appeared in Section 3.4 and Section 5)