A new characterization of Auslander algebras

Let $Λ$ be a finite dimensional Auslander algebra. For a $Λ$-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 $Λ$, and prove that a finite dimensional algebra $Λ$ is an Auslander algebra provided its global dimension gl.d\,$Λ\leq2$ and an injective $Λ$-module is projective if and only if the projective dimension of its socle is at most one.

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