  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.
