论文标题
完美的泰特代数具有无数的krull尺寸
The perfectoid Tate algebra has uncountable Krull dimension
论文作者
论文摘要
令\（k \）是具有伪均匀器\（π\）的完美体面。我们在\ cite {duuncountable}中调整了du的论点，以表明完美的泰特代数\（k \ langle x^{1 / p^{\ infty}}} \ rangle \ \ rangle \）具有一个独特的主要理想链。首先，我们概念化了杜的论点，在戒指上定义了\ textit {newton polygon形式主义}的概念。我们证明了DU的定理的一种版本，这是一个足够无差异的牛顿多边形形式主义的预示。然后，我们通过“非标准”牛顿多边形形式主义将框架应用于完美的泰特代数（大致，串联变量的作用\（x \）和伪统一器\（π\）被切换）。我们结论了使用单变量案例对多生tate代数的类似陈述。
Let \(K\) be a perfectoid field with pseudo-uniformizer \(π\). 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 \(π\) are switched). We conclude a similar statement for multivatiate perfectoid Tate algebras using the one-variable case.
