Valuation rings of mixed characteristic as limits of complete intersection rings

TL;DR

This paper proves that certain mixed characteristic valuation rings can be expressed as filtered colimits of complete intersection $f Z$-algebras under specific conditions involving the value group and Henselian property.

Contribution

It establishes a new structural description of mixed characteristic valuation rings as limits of complete intersection algebras, expanding understanding of their algebraic properties.

Findings

Valuation rings are limits of complete intersection $f Z$-algebras.

The result applies when the value group quotient has no $p$-torsion.

Henselian property is essential for the main theorem.

Abstract

We show that a mixed characteristic valuation ring with a value group $\Gamma$, $\val$ its valuation and a residue field of characteristic $p>0$, is a filtered colimit of complete intersection $\bf Z$-algebras if $\Gamma/{\bf Z}\val(p)$ has no $p$-torsion and $V$ is Henselian.