On Kaplansky’s embedding theorem
Abstract
Let $R$ be a complete equicharacteristic noetherian local domain with an algebraically closed residue field $k$. Let $\nu $ be a zero-dimensional valuation centered in $R$ with value group $\Phi $. We prove that if the value semigroup $\Gamma =\nu (R\setminus \{0\})$ of the valuation is finitely generated or the valuation is of rank $1$, the valuation $\nu $ is the restriction, via an embedding $R\subset k[[t^{\Phi _{\geq 0}}]]$, of the $t$-adic valuation $\nu _t$ of the ring of Hahn series with coefficients in $k$ and exponents in $\Phi _{\geq 0}$. This embedding is given by the series $$\xi _i\mapsto \xi _i(t)=\rho _i t^{\gamma _i}+\sum_{\delta \gt \gamma _i}c^{(i)}_\delta t^\delta \quad\ \text{with } c^{(i)}_\delta \in k,\, \rho _i\in k^*, $$ where the $(\gamma _i)_{i\in I}$ constitute a minimal system of generators, indexed by an ordinal $I\leq \omega $, of the semigroup $\Gamma =\nu (R\setminus \{0\})$. The $\xi _i\in R$ are representatives of a minimal system $(\overline \xi _i)_{i\in I}$ of homogeneous generators of the graded $k$-algebra ${\rm gr}_\nu R$ associated to $\nu $. The $\rho _i$ parametrize an isomorphism of graded $k$-algebras between ${\rm gr}_\nu R$ and $k[t^\Gamma ]$.
