An algebraic model for free rational G-spectra

We show that for any compact Lie group $G$ with identity component $N$ and component group $W=G\mathrm{/}N$, the category of free rational $G$-spectra is equivalent to the category of torsion modules over the twisted group ring $H^{\ast }(BN)[W]$. 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. This uses the methods of arXiv:1101.2511, and some readers may find the simpler context of the present paper highlights the main thread of the argument.