Gorenstein $\mathrm{FP}_n$-flat modules and weak global dimensions

Abstract

In this paper we characterize the relative Gorenstein weak global dimension of the generalized Gorenstein $\mathrm{FP}_n$-flat $R$-modules and Projective Coresolved $\mathrm{FP}_n$-flat $R$-modules recently studied by S. Estrada, A. Iacob, and M. A. P\'erez. As application we prove that the weak global dimension that comes from the Gorenstein $\mathrm{FP}_n$-flat modules is finite over a Gorenstein $n$-coherent ring and coincide with the flat dimension of the right $\mathrm{FP}_n$-injective $R$-modules. This result extends the known for Gorenstein flat modules over Iwanaga-Gorenstein and Ding-Chen rings. We also show that there is a close relationship between the global dimensions of the generalized Gorenstein $\mathrm{FP}_n$-projectives and $\mathrm{FP}_n$-injectives and the relative Gorenstein weak global dimension presented here, obtaining in the process a balanced pair.