Higher Auslander-Gorenstein Algebras And Gabriel Topologies (2402.17803v2)

Abstract: 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.