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

TL;DR

This paper proves that pure transcendental, immediate valuation ring extensions containing a field can be expressed as filtered unions of smooth subalgebras, providing a structural understanding of such extensions.

Contribution

It demonstrates that these valuation ring extensions are limits of smooth algebras, offering new insights into their algebraic structure.

Findings

Extension is a filtered union of smooth subalgebras

Provides a structural description of pure transcendental, immediate extensions

Enhances understanding of valuation ring extension limits

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 $V'$.