Characterizing higher Auslander(-Gorenstein) Algebras

Abstract

It is well known that for Auslander algebras, the category of all (finitely generated) projective modules is an abelian category and this property of abelianness characterizes Auslander algebras by Tachikawa theorem in 1974. Let $n$ be a positive integer. In this paper, by using torsion theoretic methods, we show that $ n $-Auslander algebras can be characterized by the abelianness of the category of modules with projective dimension less than $ n $ and a certain additional property, extending the classical Auslander-Tachikawa theorem. By Auslander-Iyama correspondence a categorical characterization of the class of Artin algebras having $ n $-cluster tilting modules is obtained. Since higher Auslander algebras are a special case of higher Auslander-Gorenstein algebras, the results are given in the general setting as extending previous results of Kong. Moreover, as an application of some results, we give categorical descriptions for the semisimplicity and selfinjectivity of an Artin algebra. Higher Auslander-Gorenstein Algebras are also studied from the viewpoint of cotorsion pairs and, as application, we show that they satisfy in two nice equivalences.