A closedness theorem and applications in geometry of rational points
over Henselian valued fields
release_xzpiisrspjfhdd2ipfyfj33ugy
by
Krzysztof Jan Nowak
2020
Abstract
We develop geometry of algebraic subvarieties of K^n over arbitrary
Henselian valued fields K. 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 that the projections K^n×P^m(K) → K^n are definably closed maps. It enables application
of resolution of singularities in much the same way as over locally compact
ground fields. As before, the proof of that theorem uses i.a. 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 applied in this paper 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 definable functions 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 preprint, we established a version of the
closedness theorem over Henselian valued fields with analytic structure along
with some applications.
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