Valuation rings of mixed characteristic as limits of complete intersection rings

TL;DR

The paper investigates when mixed characteristic valuation rings $V$ with residue characteristic $p>0$ and value group $\\Gamma$ can be expressed as filtered colimits of complete intersection algebras over ${\bf Z}$ or ${\bf Z}_{(p)}$. For finitely generated $\\Gamma$ and under the torsion-free condition on $\\Gamma/{\\mathbb{Z}}\operatorname{val}(p)$ with $V$ Henselian, it constructs a DVR subring $R\subset V$ with $pR$ maximal and uses Néron desingularization to show $V$ is a filtered union of complete intersection $R$-algebras, hence of complete intersection algebras over ${\bf Z}_{(p)}$. For general $\\Gamma$, a model-theoretic ultrafilter construction and cross-sections are employed to produce a limiting valuation ring with the needed properties, and Theorem T2 then yields that $V$ is a filtered colimit of complete intersection ${\bf Z}_{(p)}$-algebras. These results extend Zariski-type uniformization phenomena to mixed characteristic, providing structural decompositions of valuation rings via complete intersections and linking to desingularization frameworks.

Abstract

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

Theorems & Definitions (27)

• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 4
• Theorem 5
• Theorem 6
• Remark 7
• Remark 8
• Corollary 9
• Theorem 10