Flattening and subanalytic sets in rigid analytic geometry
| dc.creator | Gardener, T. S. | |
| dc.creator | Schoutens, Hans | |
| dc.date | 2000-12-07 | |
| dc.date.accessioned | 2026-07-07T04:39:06Z | |
| dc.date.available | 2026-07-07T04:39:06Z | |
| dc.description | 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. Using Embedded Resolution of Singularities, we derive in the zero characteristic case a Uniformization Theorem for subanalytic sets: after finitely many local blowing ups with smooth centres, a subanalytic set becomes semi-analytic. 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. Specifically we show that a map of affinoid varieties can be rendered flat by using only finitely many local blowing ups. The case of an image under a flat map is then dealt with by a small extension of a result of Raynaud. Our result can be conveniently stated as a Quantifier Elimination theorem for the valuation ring $R$ in an analytic expansion of the language of valued fields. This formulation is in the style of Denef and van den Dries. | |
| dc.description | 27 Pages | |
| dc.identifier | https://arxiv.org/abs/math/0012049 | |
| dc.identifier | http://arxiv.org/abs/math/0012049 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/60524 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Commutative Algebra | |
| dc.subject | Logic | |
| dc.subject | 32P05, 32B20, 13C11, 12J25, 03C10 | |
| dc.title | Flattening and subanalytic sets in rigid analytic geometry | |
| dc.type | text |