Valuation rings of dimension one as limits of smooth algebras

Abstract

As in Zariski's Uniformization Theorem we show that a valuation ring $V$ of characteristic $p>0$ of dimension one is a filtered direct limit of smooth ${\mathbf{F}}_p$-algebras under some conditions of transcendence degree. Under mild conditions, the algebraic immediate extensions of valuation rings are dense if they are filtered direct limit of smooth morphisms.

arXiv commentTheorem 19 is easier in a restrictive case. arXiv admin note: text overlap with arXiv:2004.11004