Bosch, Siegfried; Lütkebohmert, Werner Formal and rigid geometry. I: Rigid spaces. (English) Zbl 0808.14017 Math. Ann. 295, No. 2, 291-317 (1993). 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. Reviewer: G.van Steen (Antwerpen) Cited in 4 ReviewsCited in 99 Documents MSC: 14G20 Local ground fields in algebraic geometry 11G25 Varieties over finite and local fields Keywords:formal schemes; rigid analytical spaces Citations:Zbl 0808.14018 × Cite Format Result Cite Review PDF Full Text: DOI EuDML References: [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. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.