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 $\mathcal{ B}_C(R)$.
Categories: math.RA math.RT