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arXiv · 2509.06379

On Kaplansky's Embedding Theorem

Bernard Teissier
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Abstract

Let $R$ be a complete equicharacteristic noetherian local domain with an algebraically closed residue field $k$. Let $\nu$ be a zero dimensional valuation of rank one centered in $R$ with value group $\Phi$. We show that there is a valuation preserving embedding of $R$ in the ring $k[[t^{\Phi_{\geq 0}}]]$ 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.

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Bernard Teissier. 2025-09-08. On Kaplansky's Embedding Theorem. https://arxiv.org/abs/2509.06379

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