Almost Higher Auslander Correspondence

In this paper,we investigate the Gorenstein projective dimensions of modules over n-minimal Auslander-Gorenstein algebras,the relative dominant dimension with respect to an injective module and the almost n-precluster tilting modules.In 2007,O.Iyama proved that there is a one-to-one correspondence between the equivalence classes of finite n-cluster tilting subcategories and the Morita e-quivalence classes of n-Auslander algebras.This is called the higher Auslander correspondence.The n-almost split sequences in the n-cluster tilting subcate-gories give rise to the minimal projective resolutions of simple modules over the corresponding n-Auslander algebras.In 2018,O.Iyama and(?).Solberg generalized the higher Auslander correspondence.They proved that there is a one-to-one correspondence between the equivalence classes of finite n-precluster tilting sub-categories and the Morita equivalence classes of n-minimal Auslander-Gorenstein algebras.First,we use the n-almost split extensions in the finite n-precluster tilting subcategories to calculate the Gorenstein projective dimensions of sim-ple modules over the corresponding n-minimal Auslander-Gorenstein algebras.Then we investigate the relations between the Gorenstein projective dimensions of modules and their socles for n-minimal Auslander-Gorenstein algebras.Final-ly,we show that n-minimal Auslander-Gorenstein algebras can be characterised by these relations.As an important homological dimension,the dominant dimension is widely used in the representation theory of algebras.Inspired by the(m,n)-condition,we investigate the relative dominant dimension with respect to an injective module whose projective dimension is finite.Many results of the dominant dimension can be generalized to those of the relative dominant dimension.In particular,we discuss the algebras with.finite relative dominant dimension and show how to construct this class of algebras.Finally,we prove that all algebras with finite relative dominant dimension arise from this method of construction.In 2019,T.Adachi and M.Tsukamoto replaced the dominant dimension in the definition of n-minimal Auslander-Gorenstein algebras with the relative dominant dimension and introduced the almost n-minimal Auslander-Gorenstein algebras.They characterized this class of algebras via the existence of certain tilting mod-ules.In this paper,we investigate almost n-minimal Auslander-Gorenstein al-gebras from the viewpoint of the higher Auslander correspondence.First,we define the almost n-precluster tilting modules and introduce some basic prop-erties.Then we prove the almost higher Auslander correspondence,that is,there is a one-to-one correspondence between the equivalence classes of almost n-precluster tilting modules and the Morita equivalence classes of almost n-minimal Auslander-Gorenstein algebras.Finally,we describe the Gorenstein projective modules over almost n-minimal Auslander-Gorenstein algebras in terms of the corresponding almost n-precluster tilting modules.