Pure transcendental, immediate valuation ring extensions as limits of smooth algebras

TL;DR

This work proves that a pure transcendental, immediate extension $V\subset V'$ of valuation rings containing a field $V'$ can be realized as a filtered union of smooth $V$-subalgebras of $V'$. The approach combines extensions of Ostrowski's lemma, multi-variable polynomial refinements, and pseudo-convergent sequence techniques to construct smooth and, in positive characteristic, complete intersection subalgebras that approximate $V'$. Key results include a reduction to finite-type purely transcendental extensions and corollaries that parallel Zariski's Uniformization in positive characteristic. The findings provide a structural, uniformization-type description of valuation-ring extensions and contribute to Artin-type questions by linking immediate extensions to smooth or complete intersection approximations.

Abstract

We show that a pure transcendental, immediate extension of valuation rings $V\subset V'$ containing a field is a filtered union of smooth $V$-subalgebras of $V'$.

Theorems & Definitions (39)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Corollary 5
• Theorem 6
• Corollary 7
• Lemma 8
• proof
• Lemma 9