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 $\mathcal{C}$, there exists a localization sequence $\mathrm{def}\mathcal{C}\to \mathrm{mod}\mathcal{C}\to \mathrm{lex}\mathcal{C}$, where $\mathrm{lex}\mathcal{C}$ denotes the full subcategory of finitely presented left exact functors and $\mathrm{def}\mathcal{C}$ 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 $\mathrm{def}\mathcal{U}\to \mathrm{mod}\mathcal{U}\to \mathrm{lex}\mathcal{U}$.

25 pages, Minor corrections, Published in J. Math. Soc. Japan

Cited by in corpus (6)

• Classifying substructures of extriangulated categories via Serre subcategories
• On the lattices of exact and weakly exact structures
• A resolution theorem for extriangulated categories with applications to the index
• Cut cotorsion pairs
• Localization of triangulated categories with respect to extension-closed subcategories
• Higher Auslander's defect and classifying substructures of n-exangulated categories