An algebraic model for rational na\xefve-commutative ring SO(2)-spectra and equivariant elliptic cohomology

Abstract

Equipping a non-equivariant topological text {E}_infty -operad with the trivial G-action gives an operad in G-spaces. For a G-spectrum, being an algebra over this operad does not provide any multiplicative norm maps on homotopy groups. Algebras over this operad are called naïve-commutative ring G-spectra. In this paper we take G=SO(2) and we show that commutative algebras in the algebraic model for rational SO(2)-spectra model rational naïve-commutative ring SO(2)-spectra. In particular, this applies to show that the SO(2)-equivariant cohomology associated to an elliptic curve C of Greenlees (Topology 44(6):1213–1279, 2005) is represented by an text {E}_infty -ring spectrum. Moreover, the category of modules over that text {E}_infty -ring spectrum is equivalent to the derived category of sheaves over the elliptic curve C with the Zariski torsion point topology.