In 1966, Auslander introduced the notion of the G-dimension of a finitely generated module over a Cohen-Macaulay noetherian ring and found the basic properties of these dimensions. His results were valid over a local Cohen-Macaulay ring admitting a dualizing module (also see Auslander and Bridger (Mem. Amer. Math. Soc., vol. 94, 1969)). Enochs and Jenda attempted to dualize the notion of G-dime...

Let [Formula: see text] be an extriangulated category with a proper class of text]-triangles. We study complete cohomology objects in by applying text]-projective resolutions and text]-injective coresolutions constructed text]. Vanishing detects finite dimension dimension. As consequence, we obtain some criteria for the validity Wakamatsu tilting conjecture give necessary sufficient condition v...

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 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.