Resolution Dimension Relative To Resolving Subcategories In Extriangulated Categories

In the classical homological theory,homological dimensions are important invariants,and every homological dimension is defined relative to some certain resolving subcategory.Resolving subcategories play an important roles in homological theory.Exact categories and triangulated categories are two important structures in the theory of categories.The notion of extriangulated categories is a simultaneous generalization of exact categories and triangulated categories.Many results on exact categories and triangulated categories can be unified in the same framework.In this paper,we devote to further studying homological dimensions relative to a resolving subcategory in extriangulated categories which recovers lots of known results in abelian and triangulated categories,and is new in exact categories.Let((?),E,s)be an extriangulated category with a proper class ? of E-triangles and(?)a resolving subcategory of C.We introduce the notion of(?)-resolution dimension relative to the subcategory(?)in(?).In particular,we obtain Auslander-Buchweitz approximations for these objects.As application of finite(?)—resolution dimension,we construct adjoint pairs for two kinds of inclusion functors,and construct a new resolving subcategory GP(?)(?).The subcategory GT(?)(?)generalizes the Gorenstein projective subcategory GP(?).