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

Abstract In this paper we characterize the relative Gorenstein weak global dimension of the Gorenstein $$\mathcal {B}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>B</mml:mi> </mml:math> -flat R -modules and projectively coresolved Gorenstein $$\mathcal {B}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>B</mml:mi> </mml:math> -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 $$\textrm{FP}_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>FP</mml:mtext> <mml:mi>n</mml:mi> </mml:msub> </mml:math> -flat R -modules is finite over a Gorenstein n -coherent ring R and in this case coincides with the flat dimension of the right $$\textrm{FP}_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>FP</mml:mtext> <mml:mi>n</mml:mi> </mml:msub> </mml:math> -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 $$\textrm{FP}_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>FP</mml:mtext> <mml:mi>n</mml:mi> </mml:msub> </mml:math> -projectives and the Gorenstein weak global dimension respect to the class of Gorenstein $$\textrm{FP}_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>FP</mml:mtext> <mml:mi>n</mml:mi> </mml:msub> </mml:math> -flat R -modules. We also get an hereditary and complete cotorsion triple and consequently a balanced pair.