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- W4313304789 abstract "Let $K$ be a perfectoid field with pseudo-uniformizer $pi$. We adapt an argument of Du to show that the perfectoid Tate algebra $Klangle x^{1 / p^{infty}} rangle$ 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." @default.
- W4313304789 created "2023-01-06" @default.
- W4313304789 creator A5048852158 @default.
- W4313304789 date "2022-12-26" @default.
- W4313304789 modified "2023-09-23" @default.
- W4313304789 title "The perfectoid Tate algebra has uncountable Krull dimension" @default.
- W4313304789 doi "https://doi.org/10.48550/arxiv.2212.13315" @default.
- W4313304789 hasPublicationYear "2022" @default.
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