TY - JOUR
T1 - An algebraic model for rational naive-commutative ring SO(2)--spectra and equivariant elliptic cohomology
AU - Barnes, David
AU - Greenlees, John
AU - Kedziorek, Magdalena
PY - 2020/8/10
Y1 - 2020/8/10
N2 - 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.
AB - 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.
U2 - 10.1007/s00209-020-02554-0
DO - 10.1007/s00209-020-02554-0
M3 - Article
SN - 0025-5874
JO - Mathematische Zeitschrift
JF - Mathematische Zeitschrift
ER -