An algebraic model for rational S1-equivariant stable homotopy theory

Greenlees defined an abelian category A whose derived category is equivalent to the rational S-1-equivariant stable homotopy category whose objects represent rational S-1-equivariant cohomology theories. We show that in fact the model category of differential graded objects in A models the whole rational S-1-equivariant stable homotopy theory. That is, we show that there is a Quillen equivalence between dgA and the model category of rational S-1-equivariant spectra, before the quasi-isomorphisms or stable equivalences have been inverted. This implies that all of the higher-order structures such as mapping spaces, function spectra and homotopy (co)limits are reflected in the algebraic model. The construction of this equivalence involves calculations with Massey products. In an Appendix we show that Toda brackets, and hence also Massey products, are determined by the derived category.