Three results for tau-rigid modules

Abstract

$\tau$-rigid modules are essential in the $\tau$-tilting theory introduced by Adachi, Iyama and Reiten. In this paper, we give equivalent conditions for Iwanaga-Gorenstein algebras with self-injective dimension at most one in terms of $\tau$-rigid modules. We show that every indecomposable module over iterated tilted algebras of Dynkin type is $\tau$-rigid. Finally, we give a $\tau$-tilting theorem on homological dimension which is an analog to that of classical tilting modules.