Abhyankar valuations, Prüfer-Manis valuations, and perfectoid Tate algebras
Dimitri Dine, Jack J Garzella
TL;DR
The paper analyzes quotient fields of the perfectoid Tate algebra $T_{n,K}^{\operatorname{perfd}}$ over a perfectoid field $K$, proving that every quotient is a semi-immediate extension of a completed-perfection field $K_{r_{1},\dots,r_{l}}^{\operatorname{perfd}}$, with an optimal bound on the number of radii $l \le \min\left(n-\operatorname{ht}(\frak m^{\flat}\cap (T_{n,K^{\flat}})^{\operatorname{coperf}}),\ n-1\right)$. The authors develop a valuation-theoretic framework centered on topologically simple valuations and their Prüfer-Manis interpretation, distinguishing rational vs irrational Abhyankar valuations and connecting these to Berkovich-type points via Gauss norms and Temkin’s reduction. A key methodological pillar is tilting to characteristic $p$ to transfer the problem to Abhyankar valuation theory, combined with Gleason-style division arguments to realize surjections onto completed perfections and Noether normalization to control dimensions. The results illuminate the structure of perfectoid finite-type geometry by clarifying which quotient fields can arise and how they sit inside semi-immediate towers, with implications for Berkovich geometry and the interplay between valuation theory and perfectoid algebra. The work also provides a bridge between topological simplicity and valuation-theoretic finiteness, showing that topologically simple valuations coincide with Prüfer-Manis valuations and thereby constrain the landscape of possible quotient fields.
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 Prüfer-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.