On Ext and Hyperext groups in the category of functors

Let R be a commutative noetherian ring with unit and let f denote the category of functors from the category of finitely generated R-modules to R-modules. Let I in f denote the inclusion functor. We study homological algebra of I in the category f (Ext-groups) and its generalization when we allow coefficients to be chain complexes in f (Hyperext-groups). We compare the Ext-groups of I with coefficients in arbitrary F in f with Ext-groups of I with coefficients in stable derived functors of F. The latter groups are relatively easily calculable because the stable derived functors are linear. On the other hand, known calculations of Ext-groups of I with coefficients in F can shed light on the stable derived functors of F which are hard to approach.