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

TL;DR

This paper characterizes all quotient fields of perfectoid Tate algebras over perfectoid fields using valuation theory, revealing their structure as semi-immediate extensions and connecting to topologically simple valuations.

Contribution

It introduces the notion of topologically simple valuations, generalizes Abhyankar valuations, and describes quotient fields of perfectoid Tate algebras in terms of valuation-theoretic properties.

Findings

All quotient fields are semi-immediate extensions of certain completed perfections.

At least one radius must be irrational if a certain intersection is non-zero.

Topologically simple valuations coincide with Pr"ufer-Manis valuations.

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.