Universal valued fields and lifting points in local tropical varieties

TL;DR

This paper constructs universal valued fields and explores lifting points in local tropical varieties, showing how valuations extend and how points in tropical varieties can be lifted to points over Hahn series fields.

Contribution

It introduces a universal $k$-algebra with an extending valuation and demonstrates a weak universality property for Hahn series fields in lifting tropical points.

Findings

Existence of a universal $k$-algebra with an extending valuation.

Weak universality property of Hahn series fields for local valuations.

Every point in local tropical varieties lifts to a $K$-point in the constructed framework.

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.