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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.

14G20 Local ground fields in algebraic geometry
11G25 Varieties over finite and local fields
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[1] [A] Artin, M.: Grothendieck topologies Notes on a seminar by M. Artin, Harvard University (1962)
[2] [AC] Bourbaki, N.: Alg?bre commutative. Chap. 1-9. Paris: Hermann, 1961-65; Paris: Masson, 1980-85
[3] [B] Bosch, S.: Orthonormalbasen in der nichtarchimedischen Funktionentheorie. Manuscr. Math.1, 35-57 (1969) · Zbl 0164.21202 · doi:10.1007/BF01171133
[4] [BGR] Bosch, S.: G?ntzer, U.; Remmert, R.: Non-Archimedean analysis. (Grundlehren Band 261) Berlin Heidelberg New York: Springer 1984
[5] [EGA] Grothendieck, A.: El?ments de g?om?trie alg?brique. Publ. Math. IHES, 4, 8, 11, 17, 20, 24, 28, 32, (1960-67); (Grundlehren Band 166), Berlin Heidelberg New York: Springer 1971
[6] [FII] Bosch, S.; L?tkebohmert, W.: Formal and rigid geometry. II. Flattening techniques. To appear
[7] [GG] Gerritzen, L.; Grauert, H.: Die azyklizit?t der affinoiden ?berdeckungen. Global Analysis, Papers in Honor of K. Kodaira, 159-184, Tokyo: University of Tokyo Press, Princeton: Princeton University Press 1969
[8] [H] Hartshorne, R.: Residues and duality (Lect. Notes Math. vol. 20) Berlin Heidelberg New York. Springer 1966 · Zbl 0212.26101
[9] [K] Kiehl, R.: Der Endlichkeitssatz f?r eigentliche Abbildungen in der nichtarchimedischen Funktionentheorie. Invent. Math.2, 191-214 (1967) · Zbl 0202.20101 · doi:10.1007/BF01425513
[10] [R] Raynaud, M.: G?om?trie analytique rigide d’apr? Tate, Kiehl, ... Table ronde d’analyse non archimedienne. Bull. Soc. Math. Fr. M?m.39/40, 319-327 (1974)
[11] [RG] Raynaud, M.; Gruson, L.: Crit?res de platitude et de projectivit?. Invent. Math.13 1-89 (1971) · Zbl 0227.14010 · doi:10.1007/BF01390094
[12] [T] Tate, J.: Rigid analytic spaces. Invent. Math.12, 257-289 (1971) · Zbl 0212.25601 · doi:10.1007/BF01403307
[13] [V] Verdier, J.-L.: Cat?gories deriv?es. Notes multigraphi?es de l’I.H.E.S.
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