Realising π∗eR–algebras by global ring spectra

We approach a problem of realising algebraic objects in certain universal equivariant stable homotopy theory; the global theory Schwede. Specifically, for ring spectrum $R$, we consider which classes homomorphisms $\eta_\ast\colon\pi_\ast^e R\rightarrow S_\ast$ can be realised by map $\eta\colon S$ category $R$-modules, and what multiplicative structures placed on $S$. If $\eta_\ast$ witnesses $S_\ast$ as projective $\pi_\ast^e R$-module, then such an $\eta$ exists between commutative $R$-algebras. is addition \'{e}tale or $S_0$ $\mathbb{Q}$-algebra, upgraded to $\mathbb{E}_\infty$-global $R$-algebras $\mathbb{G}_\infty$-$R$-algebras, respectively. Various spectra are obtained from classical theoretic constructions, with controllable type.