Equivariant homology of representation spheres and computations indexed by Picard groups

In G-equivariant (Bredon) cohomology the integer grading is often extended to a grading over the real representations of G. We consider the general question of what other indices we could grade cohomology on. In a precise sense we will show that the universal group to grade on is the Picard group of the G-equivariant stable category. When G is a finite cyclic group we identify this group and show that it is generated by representation spheres. We then calculate the Pic(G) graded cohomology of a point which turns out to already be an interesting calculation.