Relative PGF modules and dimensions

Inspired in part by recent work of \v{S}aroch and \v{S}\v{t}ov\'{\i}\v{c}ek in the setting of Gorenstein homological algebra, we extend the notion of Foxby-Golod ${\rm G_C}$-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 ${\rm PG_CF}$ dimension and show that it can serve as an alternative definition of the ${\rm G_C}$-projective dimension introduced by Holm and J\o rgensen. Modules with ${\rm PG_CF}$ dimension zero are called ${\rm PG_CF}$ modules. When the module $C$ is nice enough, we show that the class ${\rm PG_CF}(R)$ of these modules is projectively resolving. This enables us to obtain good homological properties of this new dimension. We also show that ${\rm PG_CF}(R)$ 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 ${\rm PG_CF}$ modules and the fibrant objects coincide with the modules in the well-known Bass class $\mathcal{B}_C(R)$.