Realising ${\pi }_{\ast }^{e}R$-algebras by global ring spectra

Abstract

We approach a problem of realising algebraic objects in a certain universal equivariant stable homotopy theory; the global homotopy theory of Schwede. 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 $S$. 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 $\mathbb{Q}$-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.

40 page, v2 -- significant changes, added examples. Feedback is always welcome!

References in corpus (4)

• Global homotopy theory
• Global orthogonal spectra
• Universal Toda brackets of ring spectra
• Global group laws and equivariant bordism rings