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# The perfectoid Tate algebra has uncountable Krull dimension

Dec 2022

Let $K$ be a perfectoid field with pseudo-uniformizer $\pi$. We adapt anargument 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 polygonformalism' on a ring. We prove a version of Du's theorem in the prescence of asufficiently nondiscrete Newton polygon formalism. Then, we apply our frameworkto 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 Tatealgebras using the one-variable case. We also answer a question of Heitmann,showing that if $R$ is a complete local noetherian domain of mixedcharacteristic $(0,p)$, the $p$-adic completion of it's absolute integralclosure $R^{+}$ has uncountable Krull dimension.

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