$\mathrm{RO}(G)$-graded norms for de Rham-Witt forms

Abstract

We make two case studies of $\mathrm{RO}(G)$-graded homotopy in connection with $p$-adic cohomology. First, we identify the $\mathrm{RO}(\mathbf{T})$-graded ${\mathrm{TC}}^{-}$ and $\mathrm{TP}$ of quasiregular semiperfectoid rings as multi-Rees algebras for the Nygaard and ${\varphi }^{\bullet }(I)$-adic filtrations. We additionally compute $E_{★}^{h\mathbf{T}}$ and $E_{★}^{t\mathbf{T}}$ when $E$ is a general even ${\mathbf{E}}_{\infty }$-ring with $\mathbf{T}$-action, and compute $({\mathrm{TR}}^n{)}_{★}^{h\mathbf{T}/C_{p^n}}$ in terms of iterated Nygaard filtrations. Next, we compute the $\mathrm{RO}({\mathbf{Z}}_p{)}^{\wedge }$-graded $\mathrm{TR}$ of smooth algebras over a perfectoid base, providing a pushout of the works of Hesselholt and Angeltveit-Gerhardt along the base case ${\mathbf{F}}_p$. Using this, we extend Angeltveit's Norm map of Witt vectors to the de Rham-Witt complex, which participates in a "Witt coordinates" decomposition of the $p$-primary Adams operations.

arXiv comment45 pages. Preliminary version, does not include Breuil-Kisin or prismatic norms