Complete cohomology for extriangulated categories

Abstract

Let (C,E,s) be an extriangulated category with a proper class of E-triangles. In this paper, we study complete cohomology of objects in (C,E,s) by applying -projective resolutions and -injective coresolutions constructed in (C,E,s). Vanishing of complete cohomology detects objects with finite -projective dimension and finite -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 -Gprojective dimension. As an application, the relationships between -projective dimensions and -Gprojective dimensions for objects in (C,E,s) are given.