Relative PGF modules and dimensions

Inspired in part by recent work of Šaroch and Šťovíček in the setting of Gorenstein homological algebra, we extend the notion of Foxby-Golod ${\mathrm{G}}_{\mathrm{C}}$-dimension of finitely generated modules with respect to a semidualizing module $C$ to arbitrary modules over arbitrary rings, with respect to a module $C$ that is not necessarily semidualizing. We call this dimension $\mathrm{P}{\mathrm{G}}_{\mathrm{C}}\mathrm{F}$ dimension and show that it can serve as an alternative definition of the ${\mathrm{G}}_{\mathrm{C}}$-projective dimension introduced by Holm and Jørgensen. Modules with $\mathrm{P}{\mathrm{G}}_{\mathrm{C}}\mathrm{F}$ dimension zero are called $\mathrm{P}{\mathrm{G}}_{\mathrm{C}}\mathrm{F}$ modules. When the module $C$ is nice enough, we show that the class $\mathrm{P}{\mathrm{G}}_{\mathrm{C}}\mathrm{F}(R)$ of these modules is projectively resolving. This enables us to obtain good homological properties of this new dimension. We also show that $\mathrm{P}{\mathrm{G}}_{\mathrm{C}}\mathrm{F}(R)$ is the left-hand side of a complete hereditary cotorsion pair. This yields, from a homotopical perspective, a hereditary Hovey triple where the cofibrant objects coincide with the $\mathrm{P}{\mathrm{G}}_{\mathrm{C}}\mathrm{F}$ modules and the fibrant objects coincide with the modules in the well-known Bass class ${\mathcal{B}}_C(R)$.