Equipping a non-equivariant topological $E_\infty$-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 na\"{i}ve-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 na\"{i}ve-commutative ring $SO(2)$-spectra. In particular, this applies to show that the $SO(2)$-equivariant cohomology associated to an elliptic curve $C$ from previous work of the second author is represented by an $E_\infty$-ring spectrum. Moreover, the category of modules over that $E_\infty$-ring spectrum is equivalent to the derived category of sheaves over the elliptic curve $C$ with the Zariski torsion point topology.

中文翻译：

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有理朴素交换环 SO(2)-谱和等变椭圆上同调的代数模型

用简单的 $G$-action 装备一个非等变拓扑 $E_\infty$-operad 给出 $G$-spaces 中的操作数。对于$G$-spectrum，作为该操作数的代数不提供任何关于同伦群的乘法范数映射。这个操作数上的代数被称为 na\"{i}ve-commutative ring $G$-spectra。在本文中，我们取 $G=SO(2)$ 并且我们证明了有理 $SO( 2)$-spectra 模型有理 na\"{i}ve-commutative ring $SO(2)$-spectra。特别是，这适用于表明与第二作者先前工作中的椭圆曲线 $C$ 相关的 $SO(2)$-等变上同调由 $E_\infty$-环谱表示。而且，