${\mathbf{A}}_{\text{inf}}$ has uncountable Krull dimension

Abstract

Let ${\mathcal{O}}_E$ be a complete discrete valuation ring and $R$ be a perfect ring in characteristic $p$, we also assume $R$ is a complete valuation ring whose valuation group is of rank one and non-discrete, we prove the Krull dimension of the ring $W_{{\mathcal{O}}_E}(R)$ of ${\mathcal{O}}_E$-Witt vectors over $R$ is at least the cardinality of the continuum.

8 pages; the chain of subsets we constructed in our first version might not be a chain of ideals. We managed to fix the proof using a completely different argument. Published version