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We approach a problem of realising algebraic objects in a certain universalequivariant stable homotopy theory; the global homotopy theory of Schwede.Specifically, for a global ring spectrum $R$, we consider which classes of ringhomomorphisms $\eta_\ast\colon\pi_\ast^e R\rightarrow S_\ast$ can be realisedby a map $\eta\colon R\rightarrow S$ in the category of global $R$-modules, andwhat 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 amap between homotopy commutative global $R$-algebras. If $\eta_\ast$ is inaddition \'{e}tale or $S_0$ is a $\mathbb{Q}$-algebra, then $\eta$ can beupgraded 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 classicalhomotopy theoretic and algebraic constructions, with a controllable globalhomotopy type.

Realising $\pi_\ast^e R$-algebras by global ring spectra