An algebraic model for rational ultracommutative rings

TL;DR

This paper develops an algebraic framework for rational ultracommutative rings using geometric fixed points and spans of groupoids, establishing equivalences for rational objects and connecting to existing algebraic models.

Contribution

It introduces a functorial algebraic model for rational ultracommutative rings via geometric fixed points and span categories, extending and unifying previous results.

Findings

Constructs a functor from ultracommutative rings to span categories of finite groupoids.

Establishes an equivalence between rational objects in the algebraic model.

Recovers known theorems on algebraic models for rational global spectra and normed G-commutative rings.

Abstract

Given a global equivariant ultracommutative ring spectrum $E$ and inclusion $H\hookrightarrow G$ of finite groups, one may apply geometric fixed points to the norm $N_H^G E_H \to E_G$ to obtain what we call a \emph{geometric norm} $\Phi^H E \to \Phi^G E$. We prove that, together with inflations, these assemble into a functor $\Phi\colon\mathrm{UCom}_{\mathrm{fin}} \to \mathrm{Fun}(\mathrm{Span}(\mathcal{G},\mathcal{E},\mathcal{O}),\mathrm{CAlg})$, where $\mathrm{Span}(\mathcal{G},\mathcal{E},\mathcal{O})$ is the span category of finite connected groupoids with full backwards maps and faithful forwards maps, and that $\Phi$ restricts to an equivalence between full subcategories of rational objects. Central to our construction is a refinement of geometric fixed points to a natural transformation $\Phi\colon \mathrm{Sp}_\bullet\to\mathrm{Fun}(\mathrm{Orb}_\bullet^\simeq,\mathrm{Sp})$ which is compatible with restrictions and norms, and which restricts to an equivalence on full subcategories of rational objects. We explain how this may also be used to recover theorems of Barrero--Barthel--Pol--Strickland--Williamson and Wimmer on algebraic models for rational global spectra and normed $G$-commutative ring spectra respectively.