On Kaplansky’s embedding theorem

Let R be a complete equicharacteristic noetherian local domain with an algebraically closed residue field k. Let ν be a zero-dimensional valuation centered in R with value group Φ. We prove that if the value semigroup Γ=ν(R∖{0}) of the valuation is finitely generated or the valuation is of rank 1, the valuation ν is the restriction, via an embedding R⊂k[[tΦ≥0]], of the t-adic valuation νt of the ring of Hahn series with coefficients in k and exponents in Φ≥0. This embedding is given by the series ξi↦ξi(t)=ρitγi+∑δ>γic(i)δtδ with c(i)δ∈k,ρi∈k∗, where the (γi)i∈I constitute a minimal system of generators, indexed by an ordinal I≤ω, of the semigroup Γ=ν(R∖{0}). The ξi∈R are representatives of a minimal system (ξ−i)i∈I of homogeneous generators of the graded k-algebra grνR associated to ν. The ρi parametrize an isomorphism of graded k-algebras between grνR and k[tΓ].