A special form of Zariski's Uniformization Theorem in positive characteristic

TL;DR

This paper discusses a specific aspect of Zariski's Uniformization Theorem in positive characteristic, focusing on valuation ring extensions as filtered unions of smooth algebras.

Contribution

It provides an exposition on how valuation rings can be expressed as filtered unions of smooth algebras in positive characteristic.

Findings

Valuation rings can be represented as filtered unions of smooth algebras.

Extension techniques in positive characteristic are clarified.

Connections to Zariski's Uniformization are elucidated.

Abstract

This is mainly a small exposition on extensions of valuation rings as a filtered union of smooth algebras.