Relative PGF modules and dimensions

Inspired in part by recent work of Šaroch and Šťov\'ıček 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 \(\mathcal{B}_C(R)\).