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Preprint Number 1938

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1938. Erik Walsberg and Jinhe Ye
Topological properties of definable sets in tame perfect fields

Submission date: 11 March 2021


Let K be a field. We continue to study the recently introduced étale open topology on the K-points V(K) of a K-variety V. The étale open topology is non-discrete if and only if K is large. If K is separably, real, p-adically closed then the étale open topology agrees with the Zariski, order, valuation topology, respectively. We show that existentially definable sets in perfect large fields behave well with respect to this topology: such sets are finite unions of étale open subsets of Zariski closed sets. This implies that existentially definable sets in arbitrary perfect large fields enjoy some of the well-known topological properties of definable sets in algebraically, real, and p-adically closed fields. We introduce and study the class of éz fields: K is éz if K is large and every definable set is a finite union of étale open subsets of Zariski closed sets. This should be seen as a generalized notion of model completeness for large fields. Algebraically closed, real closed, p-adically closed, and bounded PAC fields are éz. (In particular pseudofinite fields and infinite algebraic extensions of finite fields are éz.) We develop the basics of a theory of definable sets in éz fields. This gives a uniform approach to the theory of definable sets across all characteristic zero local fields and a new topological theory of definable sets in bounded PAC fields. We also show that some prominent examples of possibly non-model complete model-theoretically tame fields (characteristic zero Henselian fields and Frobenius fields) are éz.

Mathematics Subject Classification: 03C60, 14G27

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Full text arXiv 2103.06919: pdf, ps.

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