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

Zariski’s local uniformization, a weak form of resolution of singularities, implies that every valuation ring containing $\mathbf{Q}$ is a filtered direct limit of smooth $\mathbf{Q}$-algebras. Given an immediate extension of valuation rings $V\subset V^{\mathrm{\prime }}$ containing $\mathbf{Q}$ we show that $V^{\mathrm{\prime }}$ 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. .