FLATTENING AND SUBANALYTIC SETS IN RIGID ANALYTIC GEOMETRY

Abstract

Let K be an algebraically closed field endowed with a complete non-archimedean norm with valuation ring R. Let $f\colon Y \to X$ be a map of K-affinoid varieties. In this paper we study the analytic structure of the image $f(Y) \subset X$; such an image is a typical example of a subanalytic set. We show that the subanalytic sets are precisely the $\mathbf D$-semianalytic sets, where $\mathbf D$ is the truncated division function first introduced by Denef and van den Dries. This result is most conveniently stated as a Quantifier Elimination result for the valuation ring R in an analytic expansion of the language of valued rings. To prove this we establish a Flattening Theorem for affinoid varieties in the style of Hironaka, which allows a reduction to the study of subanalytic sets arising from flat maps, that is, we show that a map of affinoid varieties can be rendered flat by using only finitely many local blowing ups. The case of a flat map is then dealt with by a small extension of a result of Raynaud and Gruson showing that the image of a flat map of affinoid varieties is open in the Grothendieck topology. Using Embedded Resolution of Singularities, we derive in the zero characteristic case, a Uniformization Theorem for subanalytic sets: a subanalytic set can be rendered semianalytic using only finitely many local blowing ups with smooth centres. As a corollary we obtain the fact that any subanalytic set in the plane R2 is semianalytic. 2000 Mathematical Subject Classification: 32P05, 32B20, 13C11, 12J25, 03C10.