Valuation rings of dimension one as limits of smooth algebras

TL;DR

This paper demonstrates that certain one-dimensional valuation rings of characteristic p>0 can be expressed as limits of smooth algebras, extending the understanding of their structure and approximation properties.

Contribution

It shows that valuation rings of dimension one are limits of smooth algebras under specific conditions, generalizing aspects of Zariski's Uniformization Theorem.

Findings

Valuation rings of dimension one are filtered direct limits of smooth algebras.

Immediate extensions of valuation rings are dense if they are filtered limits of smooth morphisms.

Results hold under mild conditions related to transcendence degree.

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 ${\bf F}_p$-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.