Gorenstein (ℒ,𝒜)-flat dimension of complexes and relative singularity categories

Abstract

Assume that the class of all Gorenstein [Formula: see text]-flat modules is closed under extensions. We define a notion of Gorenstein [Formula: see text]-flat dimension for complexes and consider equivalent characterizations of the finiteness of Gorenstein [Formula: see text]-flat dimension of complexes. Furthermore, in addition, assume that [Formula: see text] is a right coherent ring. We construct the relative singularity category with respect to Gorenstein [Formula: see text]-flat modules, as the triangulated quotient of the triangulated subcategory of [Formula: see text] consisting of all complexes with both finite Gorenstein [Formula: see text]-flat dimension and cotorsion dimension by the bounded homotopy category of flat-cotorsion modules, and show that the relative singularity category is triangulated equivalent to the stable category of the Frobenius category consisting of all Gorenstein [Formula: see text]-flat and cotorsion modules.