Topological transcendence degree and the dimension problem for perfectoid spaces

Abstract

Let $K$ be a perfectoid field of characteristic $p$ with countable value group (for example, $K=\mathbb{C}_{p}^{\flat}$). We study Zariski-closed subsets \begin{equation*}Z\subseteq \mathbb{B}_{K}^{n,\mathrm{perfd}}\end{equation*}of the $n$-dimensional perfectoid closed unit ball $\mathbb{B}_{K}^{n,\mathrm{perfd}}$ over $K$. We use an example of Gleason, and its generalization obtained in the authors' previous work, to show that two natural notions of dimension for such Zariski-closed subsets, the Krull dimension $\dim(Z)$ and Huber's iterated transcendence degree $\mathrm{dim.tr}(Z)$, do not coincide, and that the gap between them can be arbitrarily large.