Formal-algebraic and rigid-analytic geometry. (English) Zbl 0716.32022

Lütkebohmert, Werner

Math. Ann. 286, No. 1-3, 341-371 (1990).

With a rigid analytic space X (over a discretely valued field K) one associates a formal scheme \({\mathcal X}\) over the valuation ring R of K, and conversely, X is the analytification of the special fibre of \({\mathcal X}\). More precisely; the category of K-analytic spaces is equivalent to the category of formal schemes over R, localized with respect to certain blowing-ups. This result, due to Raynaud, is proved in the first section of this paper. It permits to obtain basic results in rigid geometry as corollaries of Grothendieck’s theory of formal schemes in EGA. This is carried out in the second section. But the main result of the paper concerns the concept of properness: Kiehl has given a geometric definition of proper maps of rigid-analytic spaces, which proved useful but remained somewhat miraculous for a long time.
The present author now shows that Kiehl’s notion agrees, via the above mentioned equivalence of categories, with the usual concept of proper morphisms of formal schemes. He also examines Kiehl’s geometric condition of “enlarging” certain affinoid subspaces: the result here is that on a proper K-analytic space, and more generally on the analytification of a K-variety, any open affinoid subspace can be enlarged.

Reviewer: F.Herrlich

MSC:

32P05	Non-Archimedean analysis
14B20	Formal neighborhoods in algebraic geometry
14G20	Local ground fields in algebraic geometry

Keywords:

modification; rigid analytic space; formal scheme; rigid geometry; proper morphisms

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