Rational torus-equivariant stable homotopy I: Calculating groups of stable maps

We construct an abelian category image of sheaves over a category of closed subgroups of the r-torus G and show that it is of finite injective dimension. It can be used as a model for rational G-spectra in the sense that there is a homology theory image on rational G-spectra with values in image and the associated Adams spectral sequence converges for all rational G-spectra and collapses at a finite stage.