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Universal valued fields and lifting points in local tropical varieties | | D. A. Stepanov
; | | Date: |
29 Apr 2013 | | Abstract: | Let $k$ be a field with a real valuation $
u$ and $R$ a $k$-algebra. We show
that there exist a $k$-algebra $K$ and a real valuation $mu$ on $K$ extending
$
u$ 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 $
u$ is trivial and $R$ a complete local Noetherian ring with
the residue field $k$. Let $K$ be the ring $ar{k}[[t^R]]$ of Hahn series
with its natural valuation $mu$; $ar{k}$ 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 $fcolon R o
K$ 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 $ar{k}((t^R))$ of Hahn series. | | Source: | arXiv, 1304.7726 | | Services: | Forum | Review | PDF | Favorites | |
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