An algebraic model for rational $S^1$-equivariant stable homotopy theory

ABSTRACT. Greenlees defined an abelian category $\mathcal{A}$ 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 $\mathcal{A}$ ($dg\mathcal{A}$) models the whole rational $S^1$-equivariant stable homotopy theory. That is, we show that there is a Quillen equivalence between $dg\mathcal{A}$ 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 new ingredients here are certain Massey product calculations and the work on rational stable model categories from Classification of stable model categories and Equivalences of monoidal model categories with Schwede. In an appendix we show that Toda brackets, and hence also Massey products, are determined by the triangulated derived category.