The perfectoid Tate algebra has uncountable Krull dimension
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866929377461141504 |
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| author | Garzella, Jack J |
| author_facet | Garzella, Jack J |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_13315 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The perfectoid Tate algebra has uncountable Krull dimension Garzella, Jack J Number Theory Commutative Algebra Algebraic Geometry 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. |
| title | The perfectoid Tate algebra has uncountable Krull dimension |
| topic | Number Theory Commutative Algebra Algebraic Geometry |
| url | https://arxiv.org/abs/2212.13315 |