On the Gorenstein and $\mathfrak{F}$-cohomological dimensions

We prove that for any discrete group $G$ with finite $\mathfrak{F}$-cohomological dimension, the Gorenstein cohomological dimension equals the $\mathfrak{F}$-cohomological dimension. This is achieved by constructing a long exact sequence of cohomological functors, analogous to that constructed by Avramov and Martsinkovsky, containing the $\mathfrak{F}$-cohomology and complete $\mathfrak{F}$-cohomology. As a corollary we improve upon a theorem of Degrijse concerning subadditivity of the $\mathfrak{F}$-cohomological dimension under group extensions.