Abhyankar valuations, Pr\"ufer-Manis valuations, and perfectoid Tate algebras

Abstract

Let $K$ be a perfectoid field. We describe all quotient fields of the perfectoid Tate algebra\begin{equation*}T_{n,K}^{\text{perfd}}=K\langle X_{1}^{1/p^{\infty}},\dots, X_{n}^{1/p^{\infty}}\rangle\end{equation*}in any number $n\geq1$ of variables in terms of (completed perfections of) the nonarchimedean fields $K_{r_1,\dots,r_l}$ occuring in Berkovich geometry. We prove that every quotient field\begin{equation*}L=T_{n,K}^{\text{perfd}}/\mathfrak{m}\end{equation*}is a so-called \textit{semi-immediate} extension of $K_{r_1,\dots,r_l}^{\text{perfd}}$ for some\begin{equation*}l\leq\min(n-\text{ht}(\mathfrak{m}^{\flat}\cap (T_{n,K^{\flat}})^{\text{coperf}}),n-1), \end{equation*}which pins down the value groups and the residue fields of the possible quotient fields $L$. Moreover, we show that if\begin{equation*}\mathfrak{m}^{\flat}\cap(T_{n,K^{\flat}})^{\text{coperf}}\neq 0,\end{equation*} at least one of the radii $r_{i}$ has to be irrational, i.e.,\begin{equation*}r_{i}\not\in\sqrt{|K^{\times}|}.\end{equation*} The main ingredient in our proof is the notion of \textit{topologically simple} valuations, which generalize type (IV) points in the classification of points on $\text{Spa}(K\langle T\rangle)$ to the case of higher-dimensional polydisks. We also consider \textit{rational Abhyankar} valuations and \textit{irrational Abhyankar} valuations, which generalize type (II) and (III) points, respectively. We deduce our main result from a description of topologically simple absolute values and of Abhyankar absolute values on usual Tate algebra. Along the way, we also show that our topologically simple valuations are the same as Pr\"ufer-Manis valuations in the sense of Knebusch-Zhang. Finally, we also show that all allowed possibilities for the quotient fields $L$ do indeed occur (i.e., the above bound $l\leq n-1$ is optimal) by generalizing an example of Gleason.