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 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 '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. We also answer a question of Heitmann, showing that if $R$ is a complete local noetherian domain of mixed characteristic $(0,p)$, the $p$-adic completion of it's absolute integral closure $R^{+}$ has uncountable Krull dimension.Comment: 15 pages, 2 figures, comments welcom