Rigidity of valuative trees under henselization

TL;DR

This paper proves that the structure of valuative trees remains unchanged under henselization, showing an isomorphism between trees of valuations before and after henselization for valued fields.

Contribution

It establishes that the valuative trees associated with a valued field and its henselization are isomorphic, revealing a rigidity property of these trees under henselization.

Findings

The natural restriction map between valuative trees is an isomorphism.

Valuative trees for a valued field and its henselization are structurally identical.

The result applies to trees formed by all Lambda-valued extensions of valuations.

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 $v(K^*)\hookrightarrow\Lambda$ of the value group into a divisible ordered abelian group. Let $T(K,\Lambda)$, $T(K^h,\Lambda)$ 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 $T(K^h,\Lambda)\to T(K,\Lambda)$ is an isomorphism of posets. As a consequence, the restriction mapping $T_v\to T_{v^h}$ is an isomorphism of posets too, where $T_v$, $T_{v^h}$ are the trees whose nodes are the equivalence classes of valuations on $K[x]$, $K^h[x]$ whose restriction to $K$, $K^h$ are equivalent to $v$, $v^h$, respectively.