Rigidity of valuative trees under henselization

Enric Nart

Vol. 319 (2022), No. 1, 189–211

DOI: 10.2140/pjm.2022.319.189

Abstract

Let $(K,v)$ be a valued field and let $(K^h,v^h)$ be the henselization determined by the choice of an extension of $v$ to an algebraic closure of $K$. Consider an embedding $v(K^{\ast })\hookrightarrow \Lambda$ of the value group into a divisible ordered abelian group. Let $\mathcal{𝒯}(K,\Lambda )$, $\mathcal{𝒯}(K^h,\Lambda )$ be the trees formed by all $\Lambda$-valued extensions of $v$, $v^h$ to the polynomial rings $K[x]$, $K^h[x]$, respectively. We show that the natural restriction mapping $\mathcal{𝒯}(K^h,\Lambda )\to \mathcal{𝒯}(K,\Lambda )$ is an isomorphism of posets.

As a consequence, the restriction mapping ${\mathcal{𝒯}}_{v^h}\to {\mathcal{𝒯}}_v$ is an isomorphism of posets too, where ${\mathcal{𝒯}}_v$, ${\mathcal{𝒯}}_{v^h}$ are the trees whose nodes are the equivalence classes of valuations on $K[x]$, $K^h[x]$ whose restrictions to $K$, $K^h$ are equivalent to $v$, $v^h$, respectively.

Keywords

henselization, key polynomial, valuation, valuative tree

Mathematical Subject Classification

Primary: 13A18

Secondary: 12J20, 13J10, 14E15

Milestones

Received: 4 February 2022

Revised: 15 April 2022

Accepted: 16 April 2022

Published: 28 August 2022