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

Abstract

For a perfectoid ring $R$, we compute the full $\mathrm{RO}(\mathbb{T})$-graded ring ${\mathrm{TF}}_{★}(R;{\mathbf{Z}}_p)$. This extends and simplifies work of Gerhardt and Angeltveit-Gerhardt. In even degrees, we find an $\mathrm{RU}(\mathbb{T})$-graded version of Bökstedt 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 ${\underline{\pi }}_{★}\mathrm{THH}(R;{\mathbf{Z}}_p)$.

18 pages, 15 figures; see warning in §1.1