Extensions of valuation rings containing $\bf Q$ as limits of smooth algebras

TL;DR

This paper characterizes when extensions of valuation rings containing the rationals are expressible as filtered direct limits of smooth algebras, providing a precise criterion.

Contribution

It establishes a necessary and sufficient condition for such valuation ring extensions to be limits of smooth algebras, advancing understanding of their structure.

Findings

Provides a clear criterion for valuation ring extensions to be limits of smooth algebras

Enhances the theoretical framework for valuation rings over $f Q$

Bridges valuation theory and algebraic geometry concepts

Abstract

We give a necessary and sufficient condition for an extension of valuation rings containing $\bf Q$ to be a filtered direct limit of smooth algebras.