  This paper is devoted to study the relationship between two important notions
in ring theory, category theory, and representation theory of Artin algebras;
namely, Gabriel topologies and higher Auslander(-Gorenstein) algebras. It is
well-known that the class of all torsionless modules over a higher
Auslander(-Gorenstein) algebra is a torsion-free class of a hereditary torsion
theory that is cogenerated by its injective envelope and so by Gabriel-Maranda
correspondence we can define a Gabriel topology on it. We show that higher
Auslander(-Gorenstein) algebras can be characterized by this Gabriel topology.
This characterization can be considered as a higher version of the
Auslander-Buchsbaum-Serre theorem that is considered as one of the most
important achievements of the use of homological algebra in the theory of
commutative rings. Among some applications, the results also reveal a relation
between the projective dimension of a module and the projective dimensions of
the annihilator ideals of its objects over higher Auslander algebras.
