Universal valued fields and lifting points in local tropical varieties. Issue 2 (1st February 2017) - Stepanov, D. A.
ABSTRACT: 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 k ̄ [ [ t ℝ ] ] of Hahn series with its natural valuation μ and coeﬃcients in k ̄ . We prove the following weak universality property: For any local valuation v and a finite set of elements x 1, …, x n of R, there exists a homomorphism f : R → K such that v ( x i ) = μ ( f ( x i ) ), i = 1, …, n . This implies that if R = k [ [ x 1, …, x n ] ] ∕ I for an ideal I, then every point of the local tropicalization of I lifts to a K -point of R .