\tau-rigid modules over Auslander algebras

We give a characterization of $\tau$-rigid modules over Auslander algebras in terms of projective dimension of modules. Moreover, we show that for an Auslander algebra $\Lambda$ admitting finite number of non-isomorphic basic tilting $\Lambda$-modules and tilting $\Lambda^{op}$-modules, if all indecomposable $\tau$-rigid $\Lambda$-modules of projective dimension $2$ are of grade $2$, then $\Lambda$ is $\tau$-tilting finite.