Abstract
Let [Formula: see text] be a finite dimensional Auslander algebra. For a [Formula: see text]-module [Formula: see text], we prove that the projective dimension of [Formula: see text] is at most one if and only if the projective dimension of its socle soc[Formula: see text][Formula: see text] is at most one. As an application, we give a new characterization of Auslander algebras [Formula: see text] and prove that a finite dimensional algebra [Formula: see text] is an Auslander algebra provided its global dimension gl.d[Formula: see text][Formula: see text] and an injective [Formula: see text]-module is projective if and only if the projective dimension of its socle is at most one.
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