2022/12/26 by Jack J. Garzella, Garzella, Jack J · 2 citations
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2212.13315
openalex publication_date 2022/12/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \(K\) be a perfectoid field with pseudo-uniformizer \(π\). We adapt an argument of Du in \citeDuUncountable to show that the perfectoid Tate algebra \(K⟨ x^1 / p∞ ⟩\) 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 \(π\) are switched). We conclude a similar statement for multivatiate perfectoid Tate algebras using the one-variable case.