We prove that if a positively-graded ring R is Gorenstein and the associated torsion functor has finite cohomological dimension, then the corresponding noncommutative projective scheme Tails(R) is a Gorenstein category in the sense of [10]. Moreover, under this condition, a (right) recollement relating Gorensteininjective sheaves in Tails(R) and (graded) Gorenstein-injective R-modules is given.

In 1966 [1], Auslander introduced a class of finitely generated modules having a certain complete resolution by projective modules. Then using these modules, he defined the G-dimension (G ostensibly for Gorenstein) of finitely generated modules. It seems appropriate then to call the modules of G-dimension 0 the Gorenstein projective modules. In [4], Gorenstein projective modules (whether finite...

Let Λ be an Auslander’s 1-Gorenstein Artinian algebra with global dimension 2. If Λ admits a trivial maximal 1-orthogonal subcategory of modΛ, then for any indecomposable module M ∈ modΛ, we have that the projective dimension of M is equal to 1 if and only if so is its injective dimension and that M is injective if the projective dimension of M is equal to 2, which implies that Λ is almost here...

For an (n− 1)-Auslander algebra Λ with global dimension n, we give some necessary conditions for Λ admitting a maximal (n − 1)-orthogonal subcategory in terms of the properties of simple Λ-modules with projective dimension n − 1 or n. For an almost hereditary algebra Λ with global dimension 2, we prove that Λ admits a maximal 1orthogonal subcategory if and only if for any non-projective indecom...

In this paper, we introduce and investigate the notions of ξ-strongly copure projective objects in a triangulated category. This extends Asadollahi’s notion of ξ-Gorenstein projective objects. Then we study the ξ-strongly copure projective dimension and investigate the existence of ξ-strongly copure projective precover.

Let Λ be an Auslander’s 1-Gorenstein Artinian algebra with global dimension two. If Λ admits a trivial maximal 1-orthogonal subcategory of modΛ, then for any indecomposable module M ∈ modΛ, we have that the projective dimension of M is equal to one if and only if so is its injective dimension and that M is injective if the projective dimension of M is equal to two. In this case, we further get ...

Let R be a ring with identity and C(R) denote the category of complexes of R-modules. In this paper we study the homotopy categories arising from projective (resp. injective) complexes as well as Gorenstein projective (resp. Gorenstein injective) modules. We show that the homotopy category of projective complexes over R, denoted K(Prj C(R)), is always well generated and is compactly generated p...

Let R be a ring and n a fixed positive integer, we investigate the properties of n-strongly Gorenstein projective, injective and flat modules. Using the homological theory , we prove that the tensor product of an n-strongly Gorenstein projective (flat) right R -module and projective (flat) left R-module is also n-strongly Gorenstein projective (flat). Let R be a coherent ring ,we prove that the...

In this paper, we investigate the relative dominant dimension with respect to an injective module and characterize algebras finite dimension. As application, introduce almost n-precluster tilting establish a correspondence between modules n-minimal Auslander-Gorenstein algebras. Moreover, give description of Gorenstein projective over in terms corresponding modules.

Since Eilenberg and Moore [EM], the relative homological algebra, especially the Gorenstein homological algebra ([EJ2]), has been developed to an advanced level. The analogues for the basic notion, such as projective, injective, flat, and free modules, are respectively the Gorenstein projective, the Gorenstein injective, the Gorenstein flat, and the strongly Gorenstein projective modules. One c...