Une construction d'extensions faiblement non ramifi\'ees d'un anneau de valuation

TL;DR

The paper provides conditions for constructing valuation rings with specified properties, especially when extending a given valuation ring by a field extension, focusing on cases where the residue field or extension is separable.

Contribution

It introduces a sufficient condition for a local ring dominating a valuation ring to be a valuation ring with the same value group, and applies this to construct valuation rings with prescribed extensions.

Findings

A sufficient condition for a local dominating ring to be a valuation ring with the same value group.

Construction of valuation rings containing a given valuation ring and a prescribed field extension.

Applicability when the extension or residue field is separable over the base field.

Abstract

\'Etant donn\'e un anneau de valuation $V$, de corps r\'esiduel $F$ et de groupe des valeurs $\Gamma$, on donne une condition suffisante pour qu'un anneau local dominant $V$ soit un anneau de valuation de groupe $\Gamma$. Lorsque $V$ contient un corps $k$, ce r\'esultat est appliqu\'e \`a la construction d'un anneau de valuation contenant $V$ et une extension donn\'ee $k'$ de $k$, de groupe $\Gamma$ et de corps r\'esiduel engendr\'e par $k'$ et $F$. Cela s'av\`ere possible, notamment, lorsque $k'$ ou $F$ est s\'eparable sur $k$. Given a valuation ring $V$, with residue field $F$ and value group $\Gamma$, we give a sufficient condition for a local ring dominating $V$ to be a valuation ring with the same value group. When $V$ contains a field $k$, we apply this result to the problem of constructing a valuation ring $W$ containing $V$ and a prescribed extension $k'$ of $k$, with value group $\Gamma$ and residue field generated by $k'$ and $F$; this is possible in particular when either $k'$ or $F$ is separable over $k$.