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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2019
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1910.04616 |
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Table of Contents:
- 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 $\mathrm{K}(\mathbb{Z},n+1)$.