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Formal and rigid geometry. I: Rigid spaces. (English) Zbl 0808.14017
Rigid geometry can be approached in two different ways. A first approach is via analytic techniques. A second method is to view rigid spaces in the framework of formal schemes.
The authors explain this latter method in a general situation. In the classical rigid case the base is a valuation ring of Krull dimension 1. Here the base \(R\) is allowed to be a noetherian ring which is complete with respect to the \({\mathcal I}\)-adic topology for some ideal \({\mathcal I} \subset R\). The basic results and constructions concerning admissible formal schemes are given. Then the correspondence between admissible formal schemes and rigid analytical spaces is explained. In a final section the authors extend their approach to M. Raynaud’s relative rigid spaces over a global noetherian ring.

MSC:
14G20 Local ground fields in algebraic geometry
11G25 Varieties over finite and local fields
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