Auslander's defects over extriangulated categories: An application for the general heart construction

Abstract

The notion of extriangulated category was introduced by Nakaoka and Palu giving a simultaneous generalization of exact categories and triangulated categories. Our first aim is to provide an extension to extriangulated categories of Auslander's formula: for some extriangulated category 𝒞, there exists a localization sequence def 𝒞 → mod 𝒞 → lex 𝒞, where lex 𝒞 denotes the full subcategory of finitely presented left exact functors and def 𝒞 the full subcategory of Auslander's defects. Moreover we provide a connection between the above localization sequence and the Gabriel–Quillen embedding theorem. As an application, we show that the general heart construction of a cotorsion pair (𝒰, 𝒱) in a triangulated category, which was provided by Abe and Nakaoka, is the same as the construction of a localization sequence def 𝒰 → mod 𝒰 → lex 𝒰.