N\'{e}ron desingularization of extensions of valuation rings with an Appendix by K\k{e}stutis \v{C}esnavi\v{c}ius

TL;DR

This paper proves that extensions of valuation rings containing rational numbers can be expressed as filtered direct limits of smooth algebras, correcting previous assumptions using model-theoretic ultrapower techniques.

Contribution

It establishes that valuation ring extensions are filtered direct limits of smooth algebras, correcting earlier misconceptions by employing ultrapower constructions from model theory.

Findings

Valuation ring extensions are filtered direct limits of smooth algebras.

Correction of previous assumptions regarding finitely generated value groups.

Application of ultrapower techniques from model theory.

Abstract

Zariski's local uniformization, a weak form of resolution of singularities, implies that every valuation ring containing $\bf Q$ is a filtered direct limit of smooth $\bf Q$-algebras. Given an immediate extension of valuation rings $V\subset V'$ containing $\bf Q$ we show that $V'$ is a filtered direct limit of smooth $V$-algebras. This corrects a paper of us \cite{Po1} where we thought that we may reduce to the case when the value groups are finitely generated. For this correction we use an infinite tower of ultrapowers construction that rests on results from model theory. .