Imaginaries in Separably Closed Valued Fields

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Abstract

Let p be a fixed prime number and let SCVFp be the first order theory of separably closed non-trivially valued fields of characteristic p. In the talk, we will see that, in many ways, from a model-theoretic point of view, the step from algebraically closed VALUED fields in characteristic p to SCVFp is not more complicated than the one from algebraically closed fields to separably closed fields in characteristic p. 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. At a more sophisticated level, in finite degree of imperfection, when a p-basis is named by constants or when one just works with Hasse derivations, the imaginaries (i.e. definable quotients) are classified by so-called the geometric sorts of Haskell-Hrushovski-Macpherson, certain higher-dimensional analogs of the residue field and the value group. This classification is proved by a reduction to the algebraically closed case, using prolongations. This is joint work with Moshe Kamensky and Silvain Rideau.

Speakers from the University of Münster

Hils, Martin

Professorship for Mathematical Logic (Prof. Hils)