Complete cohomology for extriangulated categories

Abstract

Let (C, E, s) be an extriangulated category with a proper class ξ of E-triangles.In this paper, we study complete cohomology of objects in (C, E, s) by applying ξ-projective resolutions and ξ-injective coresolutions constructed in (C, E, s). Vanishing of complete cohomology detects objects with finite ξ-projective dimension and finite ξ-injective dimension. As a consequence, we obtain some criteria for the validity of the Wakamatsu Tilting Conjecture and give a necessary and sufficient condition for a virtually Gorenstein algebra to be Gorenstein. Moreover, we give a general technique for computing complete cohomology of objects with finite ξ-Gprojective dimension. As an application, the relationships between ξ-projective dimensions and ξ-Gprojective dimensions for objects in (C, E, s) are given.

$ξ$-tilting objects in extriangulated categories

“…Throughout this paper, let C be an additive category. We recall some basics on extriangulated categories from [15,8,9,10]. Suppose C is equipped with a biadditive functor E : C op × C → Ab, where Ab is the category of abelian groups.…”

“…The following concepts are quoted verbatim from [8,9,10]. A class of E-triangles ξ is closed under base change if for any E-triangle…”

Avramov-Martsinkovsky type exact sequences for extriangulated categories

“…In [15], we introduced the notion of ξ-complete cohomology in an extriangulated category. In this section, we will give an Avramov-Martsinkovsky type exact sequence which connects ξ-cohomology, ξ-Gorenstein cohomology and ξ-complete cohomology.…”

“…By specifying a class of E-triangles, which is called a proper class ξ of E-triangles, the authors introduced ξ-projective dimensions and ξ-Gprojective dimensions, and discussed their properties. Recently, we studied ξ-cohomology in [14] and developed a ξ-complete cohomology theory for an extriangulated category in [15], which extends Tate cohomology defined in the category of modules or in a triangulated category. The aim of this paper is to study Avramov-Martsinkovsky type exact sequences for extriangulated categories.…”