Homological algebra for the representation Green functor for abelian groups

In this paper we compute some derived functors ${Ext}$ of the internal homomorphism functor in the category of modules over the representation Green functor. This internal homomorphism functor is the left adjoint of the box product.

When the group is a cyclic $2$-group, we construct a projective resolution of the module fixed point functor, and that allows a direct computation of the graded Green functor ${Ext}$.

When the group is $G=\mathbb{Z} /2\times\mathbb{Z} /2$, we can still build a projective resolution, but we do not have explicit formulas for the differentials. The resolution is built from long exact sequences of projective modules over the representation functor for the subgroups of $G$ by using exact functors between these categories of modules. This induces a filtration which gives a spectral sequence which converges to the desired ${Ext}$ functors.