On band modules and $$\tau $$-tilting finiteness

In this paper, motivated by a $$\tau $$ τ -tilting version of the Brauer-Thrall Conjectures, we study general properties of band modules and their endomorphisms in the module category of a finite dimensional algebra. As an application we describe properties of torsion classes containing band modules. Furthermore, we show that a special biserial algebra is $$\tau $$ τ -tilting finite if and only if no band module is a brick. We also recover a criterion for the $$\tau $$ τ -tilting finiteness of Brauer graph algebras in terms of the Brauer graph.

Moduli Spaces of Representations of Special Biserial Algebras

We show that the irreducible components of any moduli space of semistable representations of a special biserial algebra are always isomorphic to products of projective spaces of various dimensions. This is done by showing that irreducible components of varieties of representations of special biserial algebras are isomorphic to irreducible components of products of varieties of circular complexes and therefore normal, allowing us to apply recent results of the second and third authors on moduli spaces.