  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.
