# Three results for tau-rigid modules

## TL;DR

This paper explores $ au$-rigid modules within $ au$-tilting theory, providing new characterizations for certain Gorenstein algebras, classifying modules over Dynkin-type algebras, and establishing a $ au$-tilting homological dimension theorem.

## Contribution

It offers equivalent conditions for Gorenstein algebras via $ au$-rigid modules, classifies indecomposable modules over Dynkin-type algebras as $ au$-rigid, and proves a $ au$-tilting homological dimension theorem.

## Key findings

- Equivalent conditions for Gorenstein algebras in terms of $ au$-rigid modules.
- All indecomposable modules over iterated tilted algebras of Dynkin type are $ au$-rigid.
- A $ au$-tilting theorem on homological dimension analogous to classical tilting 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.