Rigid Subanalytic Sets

Let K be an algebraically closed field endowed with a complete non-archimedean norm. Let f : Y → X be a map of K-affinoid varieties. In this paper we study the analytic structure of the image f(Y ) ⊂ X; such an image is a typical example of a subanalytic set. We show that the subanalytic sets are precisely the D-semianalytic sets, where D is the truncated division function first introduced by Denef and van den Dries. 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. More precisely, 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 that any subanalytic set in the plane is semianalytic. Let K be an algebraically closed field with a complete non-archimedean norm. The free Tate algebra of all formal power series f = ∑ ν aνS ν over K for which |aν | tends to zero as ν goes to infinity, is denoted by K〈S1, . . . , Sn〉 and its elements 1991 Mathematics Subject Classification. Primary 32P05, 32B20; Secondary 13C11, 12J25, 03C10. The authors want to thank Jan Denef, for sharing with them his idea that Raynaud’s Theorem could be used in conjunction with a rigid analog of Hironaka’s work. c ©0000 American Mathematical Society