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

TL;DR

The paper proves that the Krull dimension of the mixed-characteristic ring $\mathbf{A}_{\mathrm{inf}} = W_{\mathcal{O}_E}(R)$ is at least the cardinality of the continuum for a perfect, non-discrete rank-one valuation ring $R$. Building on the Fargues–Fontaine viewpoint, it treats elements as holomorphic in a parameter and uses vanishing orders on the space $|Y|$ together with Newton polygons and a p-adic Jensen framework. A continuum of ideals $I_r$ is produced via prescribed vanishing orders, together with elements $g_r$, enabling an application of a dimension-lower-bound lemma to obtain the main result. The work extends Kang–Park’s equal-characteristic results to the mixed-characteristic setting and deepens understanding of Krull dimension in associated Tate algebras.

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.

Theorems & Definitions (15)

• Theorem 1
• Lemma 2: KP
• Remark 3
• Definition 4: FF
• Definition 5: FF
• Theorem 6
• proof
• Definition 7
• Definition 8
• Lemma 9