On Kaplansky's Embedding Theorem

2025/09/08 by Teissier, Bernard
#Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.06379

Abstract

Let R be a complete equicharacteristic noetherian local domain with an algebraically closed residue field k. Let ν be a zero dimensional valuation of rank one centered in R with value group Φ. We show that there is a valuation preserving embedding of R in the ring k[[t^Φ_{≥ 0}]] of Hahn series. The method relies on torific local uniformization for rational valuations with finitely generated semigroup and the approximation of rational valuations by semivaluations with finitely generated semigroup.

Citations

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• On the construction of valuations and generating sequences on hypersurface singularities
• Defect of an extension, key polynomials and local uniformization
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Related

• Reduction of local uniformization to the case of rank one valuations for rings with zero divisors
• Valuations, Deformations, and Toric Geometry
• Completions of valuation rings
• Constructing examples of semigroups of valuations
• Semigroups of valuations on local rings, II