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

Abstract

Equipping a non-equivariant topological $$\text {E}_\infty $$ 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 = S O ( 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 $$\text {E}_\infty $$ E ∞ -ring spectrum. Moreover, the category of modules over that $$\text {E}_\infty $$ E ∞ -ring spectrum is equivalent to the derived category of sheaves over the elliptic curve C with the Zariski torsion point topology.

Flatness and Shipley’s algebraicization theorem

“…We give a concrete zig-zag of Quillen equivalences which lands naturally in commutative DGAs, bypassing the need for the rectification step. We expect that this direct approach will enable a better understanding of algebraic models for naive-commutative rational G-spectra as studied by Barnes-Greenlees-K ֒ edziorek [3,4]. White-Yau [41] give an alternative approach to this zig-zag of Quillen equivalences by using the stable model structure and their theory of lifting Quillen equivalences to categories of coloured operads.…”

An Introduction to Algebraic Models for Rational G–Spectra

“…Since [GS18] was published, there have been significant developments in the field. This includes extending the existence of algebraic models to profinite groups (see [BS20] and [Sug19]) as well as taking various complexities with monoidal structure into account (see [BGK18], [BGK20] and [PW20]). We refer the reader to [BG20] for a related result stating that a nice stable, monoidal model category has a model built from categories of modules over completed rings in an adelic fashion.…”