Realising πe∗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 [Sch18]. Specifically, for a global ring spectrum R, we consider which classes of ring homomorphisms η∗:π^e_∗R → S_∗ can be realised by a map η : R → 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∞- global R-algebras or a map of G∞-R-algebras, respectively. Various global spectra and E∞-global ring spectra are then obtained from classical homotopy theoretic and algebraic constructions, with a controllable global homotopy type.