$\mathrm{RO}(\mathbb T)$-graded $\mathrm{TF}$ of perfectoid rings

Abstract

For a perfectoid ring $R$, we compute the full $\mathrm{RO}(\mathbb T)$-graded ring $\mathrm{TF}_\bigstar(R;\mathbf Z_p)$. This extends and simplifies work of Gerhardt and Angeltveit-Gerhardt. In even degrees, we find an $\mathrm{RU}(\mathbb T)$-graded version of B\"okstedt periodicity, with some additional classes in the case of perfect $\mathbf F_p$-algebras. In odd degrees, we find extremely intricate and rather mysterious torsion. We also discuss the $\mathrm{RO}(\mathbb T)$-graded homotopy Tambara functors $\underline\pi_\bigstar\mathrm{THH}(R;\mathbf Z_p)$.