Abstract
Let (C,E,s) be an extriangulated category with a proper class ξ of E-triangles and X a resolving subcategory of C. In this paper, we introduce the notion of X-resolution dimension relative to the subcategory X in C, and then give some descriptions of objects with finite X-resolution dimension. In particular, we obtain Auslander-Buchweitz approximations for these objects. As applications, we construct adjoint pairs for two kinds of inclusion functors, and construct a new resolving subcategory from a given resolving subcategory which reformulates some known results.
Highlights
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In [12], Nakaoka and Palu introduced the notion of extriangulated categories as a simultaneous generalization of exact categories and extension-closed subcategories of triangulated categories
The subcategory consisting of Gorenstein projective objects is a resolving subcategory, the aim of this paper is to introduce a notion of resolving subcategories in extriangulated categories, which regards the subcategory consisting of ξ-G projective objects as a special example
Summary
Huang investigated homological dimensions relative to (pre)resolving subcategories in triangulated categories with a proper class of triangles. In [12], Nakaoka and Palu introduced the notion of extriangulated categories as a simultaneous generalization of exact categories and extension-closed subcategories of triangulated categories. We devote to further studying homological dimensions relative to a resolving subcategory in extriangulated categories which. We introduce the notion of resolving subcategories in extriangulated categories with a proper class of E-triangles. Throughout this paper, all subcategories are full, additive and closed under isomorphisms
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