Wilson Spaces, Snaith Constructions, and Elliptic Orientations

TL;DR

This paper constructs a family of even periodic _{\u221e}-ring spectra for each prime and height, revealing a redshift pattern linking lower height theories to higher height elliptic and Morava theories.

Contribution

It introduces a new construction method for _{\u221e}-ring spectra across all primes and heights, generalizing Snaith's height 1 case to higher heights with elliptic orientations.

Findings

Constructed a canonical family of spectra for all primes and heights.

Established a redshift pattern linking _{\u221e} theories to Morava E-theories.

Connected the spectra to known localization and homotopy fixed point constructions.

Abstract

We construct a canonical family of even periodic $\mathbb{E}_{\infty}$-ring spectra, with exactly one member of the family for every prime $p$ and chromatic height $n$. At height $1$ our construction is due to Snaith, who built complex $K$-theory from $\mathbb{CP}^{\infty}$. At height $2$ we replace $\mathbb{CP}^{\infty}$ with a $p$-local retract of $\mathrm{BU} \langle 6 \rangle$, producing a new theory that orients elliptic, but not generic, height $2$ Morava $E$-theories. In general our construction exhibits a kind of redshift, whereby $\mathrm{BP}\langle n-1 \rangle$ is used to produce a height $n$ theory. A familiar sequence of Bocksteins, studied by Tamanoi, Ravenel, Wilson, and Yagita, relates the $K(n)$-localization of our height $n$ ring to work of Peterson and Westerland building $E_n^{hS\mathbb{G}^{\pm}}$ from $\mathrm{K}(\mathbb{Z},n+1)$.