On τ-Tilting Theory and Perpendicular Categories

Abstract

We extend the classification of finitely generated modules and the combinatorial description of Auslander-Reiten sequences for finite-dimensional string algebras to infinite dimensional completed string algebras. The proof of the classification of the finitely generated modules is based on work by Crawley-Boevey. Using the combinatorics of strings we prove that mutation of finitely generated support tau-tilting pairs is possible for completed string algebras. <br /> Furthermore, we study perpendicular categories for 1-Iwanaga-Gorenstein algebras. We prove that for finite-dimensional 1-Iwanaga-Gorenstein algebras the perpendicular category of an indecomposable partial tilting module satisfying certain conditions is again equivalent to a module category over a 1-Iwanaga-Gorenstein algebra. We apply and generalize this result for a special class of algebras which were recently introduced by Geiß, Leclerc and Schöer. These algebras are 1-Iwanaga-Gorenstein algebras defined via quivers with relations associated with symmetrizable Cartan matrices.