[Paper Review] An algebraic model for rational naive-commutative equivariant ring spectra

David Barnes, J. P. C. Greenlees+1 more|arXiv (Cornell University)|Aug 29, 2017

Homotopy and Cohomology in Algebraic Topology14 references4 citations

TL;DR

This paper establishes a Quillen equivalence between commutative algebras in the algebraic model for rational $G$-spectra and rational naïve-commutative ring $G$-spectra—those governed by the trivial $G$-action on the non-equivariant $E_∞$ operad. The key result shows that symmetric monoidal Quillen equivalences between rational $G$-spectra and their algebraic models do not lift to Quillen equivalences of commutative algebras, highlighting a fundamental obstruction due to the failure of norm structures to be preserved under idempotent splitting and Bousfield localization.

ABSTRACT

Equipping a non-equivariant topological E_\infty operad with the trivial G-action gives an operad in G-spaces. The algebra structure encoded by this operad in G-spectra is characterised homotopically by having no non-trivial multiplicative norms. Algebras over this operad are called naive-commutative ring G-spectra. In this paper we let G be a finite group and we show that commutative algebras in the algebraic model for rational G-spectra model the rational naive-commutative ring G-spectra.

Motivation & Objective

• To determine which level of commutativity in equivariant ring spectra is captured by the algebraic model for rational $G$-spectra.
• To resolve the question of whether Quillen equivalences between rational $G$-spectra and their algebraic models lift to Quillen equivalences of commutative algebras.
• To analyze the interaction between $N_{∞}$-operads, Bousfield localizations, and the preservation of algebraic structures under idempotent splitting.
• To construct a model structure on commutative ring $G$-spectra with rational equivalences as weak equivalences, independent of $N_{∞}$-operad theory.

Proposed method

• Lifts model structures for $G$-spectra and their algebraic models using isotropic combinatorics and left Bousfield localization techniques.
• Identifies conditions under which $N_{∞}$-operads and localizations commute, ensuring that fibrant replacement in localized categories preserves algebraic structures.
• Employs the rational sphere spectrum’s commutative multiplication to construct a cofibrantly generated model structure on ${\mathsf{Comm}}$-algebras in rational $G$-spectra.
• Uses equivariant homotopy colimits and $\Sigma_n$-equivariant weak equivalences to show that $S_{\mathbb{Q}}^{\wedge n}/\Sigma_n \to S_{\mathbb{Q}}$ is a weak equivalence of $G$-spectra.
• Applies the Barratt-Priddy-Quillen theorem and universal spaces $E_G\Sigma_n$ to relate symmetric powers to homotopy orbits and establish rational equivalences.
• Relies on the fact that the algebraic model for rational $G$-spectra arises via complete idempotent splitting and left Bousfield localization, which do not preserve norms.

Experimental results

Research questions

• RQ1 Does the symmetric monoidal Quillen equivalence between rational $G$-spectra and their algebraic models lift to a Quillen equivalence of commutative algebras?
• RQ2 Which level of commutativity in equivariant ring spectra is captured by the algebraic model for rational $G$-spectra?
• RQ3 Can a model structure on commutative ring $G$-spectra be constructed where weak equivalences are rational equivalences, independent of $N_{∞}$-operad structure?
• RQ4 How do Bousfield localizations interact with $N_{∞}$-operads and the structure of $\mathcal{O}$-algebras in the rational setting?

Key findings

• Commutative algebras in the algebraic model for rational $G$-spectra are Quillen equivalent to rational naïve-commutative ring $G$-spectra, i.e., those with no non-trivial multiplicative norms.
• The Quillen equivalence between rational $G$-spectra and their algebraic model does not lift to a Quillen equivalence of ${\mathsf{Comm}}$-algebras, demonstrating a failure of symmetric monoidal functors to preserve commutative monoids in the equivariant setting.
• The algebraic model does not capture higher commutative structures (e.g., those with norms) because idempotent splitting and Bousfield localization do not preserve norm data.
• A cofibrantly generated model structure exists on ${\mathsf{Comm}}$-algebras in rational $G$-spectra, with weak equivalences defined as rational equivalences, constructed via the commutative multiplication on the rational sphere spectrum.
• The fibrant replacement functor in the localized category $L_E G\text{-}\mathrm{Sp}$ preserves $\mathcal{O}$-algebras under suitable conditions on $E$ and $\mathcal{O}$, enabling the lifting of model structures.
• The result provides a foundational step toward modeling rational $SO(2)$-equivariant elliptic cohomology via $E_∞^1$-ring spectra, enabling algebraic-geometric descriptions of associated module categories.