Flattening and Subanalytic Sets in Rigid Analytic Geometry |
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Authors: | Gardener, T. S. Schoutens, Hans |
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Affiliation: | Mathematical Institute, University of Oxford 2429 St. Giles', Oxford OX1 3LB, gardener{at}maths.ox.ac.uk Department of Mathematics, Rutgers University Hill CenterBush Campus, Piscataway, NJ 08854, USA, hschoute{at}math.rutgers.edu |
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Abstract: | Let K be an algebraically closed field endowed with a completenon-archimedean norm with valuation ring R. Let f: Y X be amap of K-affinoid varieties. In this paper we study the analyticstructure of the image f(Y) X; such an image is a typical exampleof a subanalytic set. We show that the subanalytic sets areprecisely the D-semianalytic sets, where D is the truncateddivision function first introduced by Denef and van den Dries.This result is most conveniently stated as a Quantifier Eliminationresult for the valuation ring R in an analytic expansion ofthe language of valued rings. To prove this we establish a Flattening Theorem for affinoidvarieties in the style of Hironaka, which allows a reductionto the study of subanalytic sets arising from flat maps, thatis, we show that a map of affinoid varieties can be renderedflat by using only finitely many local blowing ups. The caseof a flat map is then dealt with by a small extension of a resultof Raynaud and Gruson showing that the image of a flat map ofaffinoid varieties is open in the Grothendieck topology. Using Embedded Resolution of Singularities, we derive in thezero characteristic case, a Uniformization Theorem for subanalyticsets: a subanalytic set can be rendered semianalytic using onlyfinitely many local blowing ups with smooth centres. As a corollarywe obtain the fact that any subanalytic set in the plane R2is semianalytic. 2000 Mathematical Subject Classification: 32P05,32B20, 13C11, 12J25, 03C10. |
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Keywords: | rigid analytic geometry flatness Quantifier Elimination subanalytic sets |
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