Relative singularity categories with respect to gorenstein flat modules

Abstract

Let R be a right coherent ring and D b (R-Mod) the bounded derived category of left R-modules. Denote by \({D^b}{\left( {R - Mod} \right)_{\widehat {\left[ {GF,C} \right]}}}\) the subcategory of D b (R-Mod) consisting of all complexes with both finite Gorenstein flat dimension and cotorsion dimension and K b (F ∩ C) the bounded homotopy category of flat cotorsion left R-modules. We prove that the quotient triangulated category \({D^b}{\left( {R - Mod} \right)_{\widehat {\left[ {GF,C} \right]}}}\)/K b (F ∩ C) is triangle-equivalent to the stable category GF ∩ C of the Frobenius category of all Gorenstein flat and cotorsion left R-modules.