Realising π∗eR–algebras by global ring spectra

Davies, Jack Morgan

doi:10.2140/agt.2021.21.1745

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Realising $\pi_\ast^eR$–algebras by global ring spectra

Jack Morgan Davies

Algebraic & Geometric Topology 21 (2021) 1745–1790

DOI: 10.2140/agt.2021.21.1745

Abstract

We approach a problem of realising algebraic objects in a certain universal equivariant stable homotopy theory, the global homotopy theory of Schwede (2018). Specifically, for a global ring spectrum $R$ , we consider which classes of ring homomorphisms $η_{*} : π_{*}^{e} R \to S_{*}$ can be realised by a map $η : R \to S$ in the category of global $R$ –modules, and what multiplicative structures can be placed on $S$ . If $η_{*}$ witnesses $S_{*}$ as a projective $π_{*}^{e} R$ –module, then such an $η$ exists as a map between homotopy commutative global $R$ –algebras. If $η_{*}$ is in addition étale or $S_{0}$ is a $ℚ$ –algebra, then $η$ can be upgraded to a map of $𝔼_{\infty}$ –global $R$ –algebras or a map of $𝔾_{\infty}$ – $R$ –algebras, respectively. Various global spectra and $𝔼_{\infty}$ –global ring spectra are then obtained from classical homotopy-theoretic and algebraic constructions, with a controllable global homotopy type.

Keywords

global homotopy theory, stable homotopy theory, higher algebra, equivariant homotopy theory, étale morphisms, realising algebra

Mathematical Subject Classification

Primary: 55P43, 55P91, 55P92, 55Q91

References

Publication

Received: 12 September 2019

Revised: 20 June 2020

Accepted: 27 July 2020

Published: 18 August 2021

Authors

Jack Morgan Davies
Mathematisch Instituut Universiteit Utrecht Utrecht Netherlands

Volume 21, issue 4 (2021)