Gorenstein \(\mathcal{F}\mathcal{P}_n\)-Flat Modules and Weak Global Dimensions

In this paper we characterize the relative Gorenstein weak global dimension of the Gorenstein \(\mathcal {B}\)-flat R-modules and projectively coresolved Gorenstein \(\mathcal {B}\)-flat R-modules recently studied by S. Estrada, A. Iacob, and M. A. Pérez, which are a relativisation of the ones introduced by J. Šaroch and J. Št’ovíchěk. As application we prove that the weak global dimension with respect to the Gorenstein \(\textrm{FP}_n\)-flat R-modules is finite over a Gorenstein n-coherent ring R and in this case coincides with the flat dimension of the right \(\textrm{FP}_n\)-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 relative global dimension of the Gorenstein \(\textrm{FP}_n\)-projectives and the Gorenstein weak global dimension respect to the class of Gorenstein \(\textrm{FP}_n\)-flat R-modules. We also get an hereditary and complete cotorsion triple and consequently a balanced pair.