The perfectoid Tate algebra has uncountable Krull dimension

Abstract

Let \(K\) be a perfectoid field with pseudo-uniformizer \(\pi\). We adapt an argument of Du in \cite{DuUncountable} to show that the perfectoid Tate algebra \(K\langle x^{1 / p^{\infty}} \rangle\) has an uncountable chain of distinct prime ideals. First, we conceptualize Du's argument, defining the notion of a \textit{Newton polygon formalism} on a ring. We prove a version of Du's theorem in the prescence of a sufficiently nondiscrete Newton polygon formalism. Then, we apply our framework to the perfectoid Tate algebra via a "nonstandard" Newton polygon formalism (roughly, the roles of the series variable \(x\) and the pseudo-uniformizer \(\pi\) are switched). We conclude a similar statement for multivatiate perfectoid Tate algebras using the one-variable case.