A new characterization of Auslander algebras

Abstract

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 ≤ 2 and an injective Λ-module is projective if and only if the projective dimension of its socle is at most one.

Gorenstein projective dimensions of modules over minimal Auslander-Gorenstein algebras

“…Our results in this section generalise the main theorems in [10], where the theorems were proved for the special case of Auslander algebras.…”

“…Then Λ is a minimal n-Auslander-Gorenstein algebra if and only if it satisfies GP ďn pΛq " tN P Λ-mod | GpdpsocN q ď nu " subΛ. Theorem 2 and Theorem 3 generalise the main results in [10] where the results are proved for the special case of Auslander algebras. This paper is arranged as follows.…”

“…One might ask whether in a higher Auslander algebra it holds that a module N has projective dimension 2 if and only if its socle has projective dimension 2, as a generalisation of Theorem B in [10] from Auslander algebras to higher Auslander algebras. The next example shows that this is not true in general.…”

Three Results Concerning Auslander Algebras

“…Definition 1.18. Let Λ be an algebra and let Another characterization was given by Li and Zhang in [6]. Recall, Q is the direct sum of representatives of the isomorphism classes of all indecomposable projectiveinjective Λ-modules.…”