Auslander's defects over extriangulated categories: an application for the General Heart Construction

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 \(\mathcal{C}\), there exists a localization sequence \(\operatorname{\mathsf{def}}\mathcal{C}\to\operatorname{\mathsf{mod}}\mathcal{C}\to\operatorname{\mathsf{lex}}\mathcal{C}\), where \(\operatorname{\mathsf{lex}}\mathcal{C}\) denotes the full subcategory of finitely presented left exact functors and \(\operatorname{\mathsf{def}}\mathcal{C}\) 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 \((\mathcal{U},\mathcal{V})\) in a triangulated category, which was provided by Abe and Nakaoka, is same as the construction of a localization sequence \(\operatorname{\mathsf{def}}\mathcal{U}\to\operatorname{\mathsf{mod}}\mathcal{U}\to\operatorname{\mathsf{lex}}\mathcal{U}\).