Structured Real Snaith Equivalences

Abstract

We give a short proof of the Real Snaith equivalences and multiplicative refinements thereof. The key ingredient is control over structured Real orientations, which we manage through Wilson space theory. In particular, we develop a theory that produces $\mathbb{E}_6$-complex orientations of even periodic $\mathbb{E}_{\infty}$-ring spectra. This machinery can be used to recover an $\mathbb{E}_{2ρ}$-algebra structure on Real Brown-Peterson theory. We apply the Real Snaith theorems to compute $\mathrm{THR}(\mathrm{KU}_{\mathbb{R}})$ and $\mathrm{THR}(\mathrm{MUP}_{\mathbb{R}})$. This requires a norm inverted variant of the Real Snaith theorems, which we prove via the nilpotence theorem.