  Let $\Lambda$ be a finite dimensional Auslander algebra. For a
$\Lambda$-module $M$, we prove that the projective dimension of $M$ is at most
one if and only if the projective dimension of its socle soc\,$M$ is at most
one. As an application, we give a new characterization of Auslander algebra
$\Lambda$, and prove that a finite dimensional algebra $\Lambda$ is an
Auslander algebra provided its global dimension gl.d\,$\Lambda\leq2$ and an
injective $\Lambda$-module is projective if and only if the projective
dimension of its socle is at most one.
