## Dorin Popescu: Valuation rings of dimension one as limits of smooth algebras, 63-73

### Abstract:

As in Zariski's Uniformization Theorem we show that a valuation ring of characteristic 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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