Universal valued fields and lifting points in local tropical varieties
D. A. Stepanov · Communications in Algebra · 2016
Let k be a field with a real valuation ν and R a k-algebra. We show that there exist a k-algebra K and a valuation μ on K extending ν such that any real valuation of R is induced by μ through some homomorphism from R to K. Let ν be trivial and R a complete local Noetherian ring with the residue field k. Let K be the ring k̄[[tℝ]] of Hahn series with its natural valuation μ and coefficients in k̄. We prove the following weak universality property: For any local valuation v and a finite set of elements x1,…,xn of R, there exists a homomorphism f:R→K such that v(xi)=μ(f(xi)), i = 1,…,n. This implies that if R=k[[x1,…,xn]]∕I for an ideal I, then every point of the local tropicalization of I lifts to a K-point of R.