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 over a field, we use a decomposition, called sign-decomposition, parameterized by elements of {±1}n where n is the number of simple modules. If A is an algebra with radical square zero, then for each ϵ∈{±1}n there is a hereditary algebra Aϵ! with radical square zero and a bijection between the set of torsion classes of A associated to ϵ and the set of faithful torsion classes of Aϵ!. Furthermore, this bijection preserves the property of being functorially finite. From a point of view of tilting theory, it implies that there is a bijection between the set of isomorphism classes of basic two-term silting complexes for A associated to ϵ and the set of isomorphism classes of basic tilting Aϵ!-modules. As an application, we prove that the number of two-term tilting complexes over Brauer line algebras (respectively, Brauer cycle algebras) having n edges is (2nn) (respectively, 22n−1 if n is odd, and ∞ if n is even).