Néron desingularization of extensions of valuation rings with an Appendix by Kęstutis Česnavičius

2019/10/21 by Dorin Popescu, Popescu, Dorin
Mathematics · #Rings, Modules, and Algebras #Advanced Topology and Set Theory

paper · pdf · doi:10.48550/arxiv.1910.09123

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⊂ V' containing \bf Q we show that V' is a filtered direct limit of smooth V-algebras. This corrects a paper of us \citePo1 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. .

Related

• Valuation rings of dimension one as limits of smooth algebras
• Extensions of valuation rings containing \bf Q as limits of smooth algebras
• Pure transcendental, immediate valuation ring extensions as limits of smooth algebras
• A special form of Zariski's Uniformization Theorem in positive characteristic
• Immediate extensions of valuation rings and ultrapowers