A closedness theorem and applications in geometry of rational points over Henselian valued fields

Krzysztof Jan Nowak

Journal of Singularities, Tome 21 (2020), pp. 212-233

Résumé

We develop geometry of algebraic subvarieties of K^n over arbitrary Henselian valued fields K of equicharacteristic zero. This is a continuation of our previous article concerned with algebraic geometry over rank one valued fields. At the center of our approach is again the closedness theorem to the effect that the projections K^n x P^m(K) -> K^n are definably closed maps. It enables, in particular, application of resolution of singularities in much the same way as over locally compact ground fields. As before, the proof of that theorem uses, among others, the local behavior of definable functions of one variable and fiber shrinking, being a relaxed version of curve selection. But now, to achieve the former result, we first examine functions given by algebraic power series. All our previous results will be established here in the general settings: several versions of curve selection (via resolution of singularities) and of the Łojasiewicz inequality (via two instances of quantifier elimination indicated below), extending continuous hereditarily rational functions as well as the theory of regulous functions, sets and sheaves, including Nullstellensatz and Cartan's theorems A and B. Two basic tools are quantifier elimination for Henselian valued fields due to Pas and relative quantifier elimination for ordered abelian groups (in a many-sorted language with imaginary auxiliary sorts) due to Cluckers--Halupczok. Other, new applications of the closedness theorem are piecewise continuity of definable functions, Hölder continuity of functions definable on closed bounded subsets of K^n, the existence of definable retractions onto closed definable subsets of K^n and a definable, non-Archimedean version of the Tietze--Urysohn extension theorem. In a recent paper, we established a version of the closedness theorem over Henselian valued fields with analytic structure along with several applications.

Détail

DOI : 10.5427/jsing.2020.21m

Classification : 12J25, 14B05, 14P10, 13J15, 14G27, 03C10