Realising \(\pi_\ast^e\)R-algebras by global ring spectra (Q1983579)

The author identifies conditions ensuring that certain ring homomorphisms can be realized by suitable algebra maps in Schwede's setup of global stable homotopy theory. To describe this in more detail, recall that given an orthogonal spectrum \(X\) and a compact Lie group \(G\), one can view \(X\) as a \(G\)-spectrum by evaluation on \(G\)-representations and therefore obtain \(G\)-equivariant stable homotopy groups of \(X\) that are denoted by \(\pi_*^G(R)\). For \(G = e\) being the trivial group, these are the ordinary stable homotopy groups of \(X\). A map of orthogonal spectra is called a \textit{global equivalence} if it induces isomorphisms on the \(G\)-equivariant stable homotopy groups for all compact Lie groups \(G\). The resulting global stable homotopy theory has been extensively studied by \textit{S. Schwede} [Global homotopy theory. Cambridge: Cambridge University Press (2018; Zbl 1451.55001)]. An \textit{ultracommutative ring spectrum} is a commutative orthogonal ring spectrum viewed as a global homotopy type. One main result of the article under review states that given an ultracommutative ring spectrum \(R\) and a map of graded commutative rings \(\pi_*^{e}R \to S_*\) exhibiting \(S_*\) as a projective \(\pi_*^{e}R\)-module, the ring \(S_*\) can be realized by a ``globally flat'' homotopy commutative global \(R\)-algebra \(S\) that is unique in a suitable sense. Another main results states that under an étaleness assumption, the multiplication on \(S\) can be refined to what is called an \textit{\(E_{\infty}\) global \(R\)-algebra structure}. The corresponding non-equivariant result can be derived from work of \textit{A. Baker} and \textit{B. Richter} [Trans. Am. Math. Soc. 359, No. 2, 827--857 (2007; Zbl 1111.55009)]. The realizability results outlined above are used to construct various global homotopy types of interest, including localizations of ultracommutative ring spectra, module spectra over a global variant of periodic complex \(K\)-theory, and a global version of height \(n\) Morava \(K\)-theory at a prime \(p\).