An algebraic model for free rational G-spectra

We show that for any compact Lie group G with identity component N and component group W = G/N, the category of free rational G-spectra is equivalent to the category of torsion modules over the twisted group ring H ∗ (BN)[W]. This gives an algebraic classification of rational G-equivariant cohomology theories on free G-spaces and a practical method for calculating the groups of natural transformations between them.