Universal valued fields and lifting points in local tropical varieties

Abstract

Let $k$ be a field with a real valuation $\nu$ and $R$ a $k$-algebra. We show that there exist a $k$-algebra $K$ and a real valuation $\mu$ on $K$ extending $\nu$ such that any real ring valuation of $R$ is induced by $\mu$ via some homomorphism from $R$ to $K$; $K$ can be chosen to be a field. Then we study the case when $\nu$ is trivial and $R$ a complete local Noetherian ring with the residue field $k$. Let $K$ be the ring $\bar{k}[[t^\R]]$ of Hahn series with its natural valuation $\mu$; $\bar{k}$ is an algebraic closure of $k$. Despite $K$ is not universal in the strong sense defined above, it has 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 $f\colon R\to K$ such that $v(x_i)=\mu(f(x_i))$, $i=1,...,n$. If $R=k[[x_1,...,x_n]]/I$ for an ideal $I$, this property implies that every point of the local tropical variety of $I$ lifts to a $K$-point of $R$. Similarly, if $R=k[x_1,...,x_n]/I$ is a finitely generated algebra over $k$, lifting points in the tropical variety of $I$ can be interpreted as the weak universality property of the field $\bar{k}((t^\R))$ of Hahn series.