As in Zariski's Uniformization Theorem we show that a valuation ring
of dimension one is a filtered direct limit of smooth
-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.
Key Words: Immediate extensions of valuations rings, pseudo convergent sequences, pseudo limits, smooth morphisms, Henselian rings.
2020 Mathematics Subject Classification: Primary 13F30; Secondary 13A18, 13L05, 13B40.
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