A new characterization of Auslander algebras

Let $\mathrm{\Lambda }$ be a finite dimensional Auslander algebra. For a $\mathrm{\Lambda }$-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 $\mathrm{\Lambda }$, and prove that a finite dimensional algebra $\mathrm{\Lambda }$ is an Auslander algebra provided its global dimension gl.d,$\mathrm{\Lambda }\le 2$ and an injective $\mathrm{\Lambda }$-module is projective if and only if the projective dimension of its socle is at most one.