An algebraic model for rational naïve-commutative ring \(SO(2)\)-spectra and equivariant elliptic cohomology (Q2663074)

Building on previous work, \textit{D. Barnes} et al. [Algebr. Geom. Topol. 17, No. 2, 983--1020 (2017; Zbl 1369.55005)] gave a symmetric monoidal algebraic model for the category of rational SO(2)-spectra. As a consequence, one obtains a model for rational ring SO(2)-spectra in terms of monoids in the algebraic model; however, this does not imply analogous results about commutative monoids. Recent work of \textit{A. J. Blumberg} and \textit{M. A. Hill} [Adv. Math. 285, 658--708 (2015; Zbl 1329.55012)] describes a class of commutative multiplicative structures on the category of \(G\)-spectra. Their work raises the following question: which level of commutativity is modelled by commutative monoids in the algebraic model for rational SO(2)-spectra? The goal of this paper is to answer this question. The authors show that the homotopy theory of commutative monoids in the algebraic model is equivalent to that of rational naïve-commutative ring \(\mathrm{SO}(2)\)-spectra. These ring spectra are algebras over a non-equivariant topological \(E_\infty\)-operad equipped with the trivial \(\mathrm{SO}(2)\)-action. An \(\mathrm{SO}(2)\)-spectrum which is an algebra over this operad is called naïve since its homotopy groups do not admit any multiplicative norm maps. As an application of their result, the authors show that the \(\mathrm{SO}(2)\)-equivariant cohomology associated to an elliptic curve \(C\) of [\textit{J. P. C. Greenlees}, Topology 44, No. 6, 1213--1279 (2005; Zbl 1085.55002)] is represented by an \(E_\infty\)-ring spectrum. Moreover, its category of modules is equivalent to the derived category of sheaves over the elliptic curve \(C\) with the Zariski torsion point topology.