The ramification tree and almost Dedekind domains of prescribed SP-rank

Balint Rago and Dario Spirito

Vol. 337 (2025), No. 2, 273–295

DOI: 10.2140/pjm.2025.337.273

Abstract

Given a valuation $v$ on a field $K$ and a chain $\mathcal{𝒦}:K=K_0\subseteq K_1\subseteq \cdots$ of finite extensions of $K$, we construct a weighted tree $\mathcal{𝒯}(v,\mathcal{𝒦})$ encoding information about the ramification of $v$ in the extensions $K_i$; conversely, when $v$ is a discrete valuation, we show that a weighted tree $\mathcal{𝒯}$ can be expressed as $\mathcal{𝒯}(v,\mathcal{𝒦})$ under some mild hypotheses on $v$ or on $\mathcal{𝒯}$. We use this correspondence to construct, for every countable successor ordinal number $\alpha$ and every discrete valuation ring $V$, an almost Dedekind domain $D$ integral over $V$ whose SP-rank is $\alpha$. Subsequently, we extend this result to countable limit ordinal numbers by considering integral extensions of Dedekind domains with countably many maximal ideals.

Keywords

almost Dedekind domains, SP-domains, SP-rank, extensions of valuations, ramification

Mathematical Subject Classification

Primary: 13F05

Secondary: 05C05, 12J20, 13A15, 13F30

Milestones

Received: 27 June 2024

Revised: 7 May 2025

Accepted: 7 May 2025

Published: 3 June 2025