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Flattening and Subanalytic Sets in Rigid Analytic Geometry
Author(s) -
Gardener T. S.,
Schoutens Hans
Publication year - 2001
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/83.3.681
Subject(s) - mathematics , valuation ring , resolution of singularities , quantifier elimination , uniformization theorem , gravitational singularity , function field , pure mathematics , geometry , mathematical analysis , discrete mathematics , field (mathematics) , geometric function theory , riemann surface , riemann–hurwitz formula
Let K be an algebraically closed field endowed with a complete non‐archimedean norm with valuation ring R. 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. 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.