- Jun 16, 10:00-11:00, 2015
- Room -101, BGU
Speaker: Antoine Ducros (Paris 6)
Title: Stability of Gauss valuations
A valued field $(k,|.|)$ is said to be stable (this terminology has no link with model-theoretic stability theory) if every finite extension $L$ of $k$ is defectless, i.e., satisfies the equality $\sum e_vf_v=[L:k]$, where $v$ goes through the set of extensions of $|.|$ to $L$, and where $e_v$ and $f_v$ are the ramification and inertia indexes of $v$. The purpose of my talk is to present a new proof (which is part of current joint reflexions with E. Hrushovski and F. Loeser) of the following classical fact (Grauert, Kuhlmann, Temkin,…) : let $(k,|.|)$ be a stable valued field, and let $(r_1,\dots,r_n)$ be elements of an ordered abelian group $G$ containing $|k^*|$. Let $|.|’$ be the $G$-valued valuation on $k(T_1,\dots,T_n)$ that sends $\sum a_I T^I$ to $\max_I |a_I|\cdot r^I$. Then $(k(T_1,\dots,T_n),|.|’)$ is stable too.
Our general strategy is purely geometric, but the proof is based upon model-theoretic tools coming from model theory (which I will first present; no knowledge of model theory will be assumed). In particular, it uses in a crucial way a geometric object defined in model-theoretic terms that Hrushovski and Loeser attach to a given $k$-variety $X$, which is called its stable completion; the only case we will have to consider is that of a curve, in which the stable completion has a very nice model-theoretic property, namely the definability, which makes it very easy to work with.