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{\Large Department of Mathematics, BGU}

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{\Huge Special Seminar}\\[0.2\baselineskip]

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\textbf{Speaker}: \emph{Antoine Ducros} (Paris 6)

\textbf{Title}: \emph{Stability of Gauss valuations}

\textbf{Abstract}:

A valued field \$(k,\textbar{}.\textbar{})\$ is said to be \emph{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 \$\textbackslash{}sum e\_vf\_v={[}L:k{]}\$, where \$v\$ goes through the set of extensions of \$\textbar{}.\textbar{}\$ 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,\ldots{}) : let \$(k,\textbar{}.\textbar{})\$ be a stable valued field, and let \$(r\_1,\textbackslash{}dots,r\_n)\$ be elements of an ordered abelian group \$G\$ containing \$\textbar{}k\^{}*\textbar{}\$. Let \$\textbar{}.\textbar{}'\$ be the \$G\$-valued valuation on \$k(T\_1,\textbackslash{}dots,T\_n)\$ that sends \$\textbackslash{}sum a\_I T\^{}I\$ to \$\textbackslash{}max\_I \textbar{}a\_I\textbar{}\textbackslash{}cdot r\^{}I\$. Then \$(k(T\_1,\textbackslash{}dots,T\_n),\textbar{}.\textbar{}')\$ 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 \emph{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.




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\textbf{Time}: \emph{Jun 16, 10:00—11:00, 2015}

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\textbf{Location}: \emph{Room -101, BGU}

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\textbf{Web}: \emph{/en/research/events/25}


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