Abstract

Let $R$ be a local noetherian integral domain with an algebraically closed residue field. Let $R_{\nu}$ be the ring of a valuation of the field of fractions $K$ of $R$. Assume that $R_{\nu}$ dominates $R$ and $R/\frak{m}=R_{\nu}/\frak{m}_{\nu}$. The associated graded ring of $R$ with respect to $\nu$ is a quotient of a polynomial ring in a well-ordered set of indeterminates by a prime binomial ideal. It is of finite Krull dimension. A suitable completion of the ring $R$ appears as a deformation of a completion at the origin of this graded ring. Some consequences will be shown.



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