On the module category of generalized preprojective algebras of Dynkin types

For a symmetrizable GCM $C$ and its symmetrizer $D$, Geiss-Leclerc-Schroer [Invent. Math. 209 (2017)] has introduced a generalized preprojective algebra $Pi$ associated to $C$ and $D$, that contains a class of modules, called locally free modules. We show that any basic support $tau$-tilting $Pi$-module is locally free and gives a classification theorem of torsion-free classes in $operatorname{mathbf{rep}}{Pi}$ as the generalization of the work of Mizuno [Math. Z. 277 (2014)].