Volume 21, issue 4 (2021)

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

Jack Morgan Davies

Algebraic & Geometric Topology 21 (2021) 1745–1790
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 ${\eta }_{\ast }:{\pi }_{\ast }^{e}R\to {S}_{\ast }$ can be realised by a map $\eta :R\to S$ in the category of global $R$–modules, and what multiplicative structures can be placed on $S\phantom{\rule{-0.17em}{0ex}}$. If ${\eta }_{\ast }$ witnesses ${S}_{\ast }$ as a projective ${\pi }_{\ast }^{e}R$–module, then such an $\eta$ exists as a map between homotopy commutative global $R$–algebras. If ${\eta }_{\ast }$ is in addition étale or ${S}_{0}$ is a $ℚ$–algebra, then $\eta$ can be upgraded to a map of ${\mathbb{𝔼}}_{\infty }$–global $R$–algebras or a map of ${\mathbb{𝔾}}_{\infty }$$R$–algebras, respectively. Various global spectra and ${\mathbb{𝔼}}_{\infty }$–global ring spectra are then obtained from classical homotopy-theoretic and algebraic constructions, with a controllable global homotopy type.

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