Valuation rings of dimension one as limits of smooth algebras

TL;DR

The paper addresses a positive-characteristic, dimension-one analog of Zariski's Uniformization by showing that a one-dimensional valuation ring $V$ containing a perfect field of characteristic $p>0$ can be expressed as a filtered direct limit of smooth ${\bf F}_p$-algebras under specific transcendence-degree conditions. It develops defect theory, proving that separably defectless and algebraic-separable extensions yield immediate extensions $V\subset V'$ that are filtered unions of smooth subalgebras, with density ensured via Néron-Schappacher-type arguments. A key dichotomy is established for algebraic vs transcendental pseudo convergent sequences: transcendental cases yield filtered unions of one-variable polynomial subalgebras, while algebraic cases do not, guiding how density and smoothing can be achieved. Collectively, these results advance a Zariski-like uniformization program in positive characteristic for dimension-one valuation rings by representing them as colimits of smooth algebras and clarifying density properties of immediate extensions.

Abstract

As in Zariski's Uniformization Theorem we show that a valuation ring $V$ of characteristic $p>0$ of dimension one is a filtered direct limit of smooth ${\bf F}_p$-algebras under some conditions of transcendence degree. Under mild conditions, the algebraic immediate extensions of valuation rings are dense if they are filtered direct limit of smooth morphisms.

Theorems & Definitions (37)

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