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

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