Relative PGF modules and dimensions

Inspired in part by recent work of Saroch and Stov\'cek in the setting of Gorenstein homological algebra, we extend the notion of Foxby-Golod GC-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 PGCF dimension and show that it can serve as an alternative definition of the GC-projective dimension introduced by Holm and J rgensen. Modules with PGCF dimension zero are called PGCF modules. When the module C is nice enough, we show that the class PGCF (R) of these modules is projectively resolving. This enables us to obtain good homological properties of this new dimension. We also show that PGCF (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 PGCF modules and the fibrant objects coincide with the modules in the well-known Bass class BC (R).