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

David Barnes, John Greenlees, Magdalena Kedziorek

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Abstract

quipping a non-equivariant topological E∞E∞-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)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 E∞E∞-ring spectrum. Moreover, the category of modules over that E∞E∞-ring spectrum is equivalent to the derived category of sheaves over the elliptic curve C with the Zariski torsion point topology.
Original languageEnglish
Number of pages25
JournalMathematische Zeitschrift
Early online date10 Aug 2020
DOIs
Publication statusEarly online date - 10 Aug 2020

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