# 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.