Auslander–Reiten components over pure-semisimple hereditary rings

Let R be a hereditary, indecomposable, left pure-semisimple ring. We show that R has finite representation type if and only if a certain finitely presented module is endofinite, namely, the tilting and cotilting module W studied in L. Angeleri Hügel (2007) [2]. We then apply the tilting and the cotilting functors to study the endomorphism ring of W and its Auslander–Reiten components. Finally, we transfer this information to the category of right R-modules.