Geometric Model Theory in Separably Closed Valued Fields

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Abstract

Let p be a fixed prime number and let SCVFp be the theory of separably closed non-trivially valued fields of characteristic p. In the talk, we will see that, in many ways, the step from ACVFp,p to SCVFp is not more complicated than the one from ACFp to SCFp. At a basic level, this is true for quantifier elimination (Delon), for which it suffices to add parametrized p-coordinate functions to any of the usual languages for valued fields. It follows that all completions are NIP. At a more sophisticated level, in finite degree of imperfection, when a p-basis is named or when one just works with Hasse derivations, the imaginaries of SCVFp are not more complicated than the ones in ACVFp,p, i.e., they are classified by the geometric sorts of Haskell-Hrushovski-Macpherson. The latter is proved using prolongations. One may also use these to characterize the stable part and the stably dominated types in SCVFp, and to show metastability.

Speakers from the University of Münster

Hils, Martin

Professorship for Mathematical Logic (Prof. Hils)