Wilson spaces, snaith constructions, and elliptic orientations

We construct a canonical family of even periodic ${\mathbb{E}}_{\mathrm{\infty }}$ $\mathbb{E}_{\infty}$ -ring spectra, with exactly one member of the family for every prime $p$ $p$ and chromatic height $n$ $n$ . At height 1 our construction is due to Snaith, who built complex $K$ $K$ -theory from ${\mathbb{CP}}^{\mathrm{\infty }}$ $\mathbb{CP}^{\infty}$ . At height 2 we replace ${\mathbb{CP}}^{\mathrm{\infty }}$ $\mathbb{CP}^{\infty}$ with a $p$ $p$ -local retract of $\mathrm{BU}〈6〉$ $\mathrm{BU} \langle 6 \rangle $ , producing a new theory that orients elliptic, but not generic, height 2 Morava $E$ $E$ -theories.

In general our construction exhibits a kind of redshift, whereby $\mathrm{BP}〈n-1〉$ $\mathrm{BP}\langle n-1 \rangle $ is used to produce a height $n$ $n$ theory. A familiar sequence of Bocksteins, studied by Tamanoi, Ravenel, Wilson, and Yagita, relates the $K(n)$ $K(n)$ -localization of our height $n$ $n$ ring to work of Peterson and Westerland building $E_n^{hS{\mathbb{G}}^{\pm }}$ $E_{n}^{hS\mathbb{G}^{\pm}}$ from $\mathrm{K}(\mathbb{Z},n+1)$ $\mathrm{K}(\mathbb{Z},n+1)$ .