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Formal-algebraic and rigid-analytic geometry. (English) Zbl 0716.32022
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 (should also be assigned at least one other classification number from Section 32-XX describing the type of problem)
14B20 Formal neighborhoods in algebraic geometry
14G20 Local ground fields in algebraic geometry
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References:
[1] Artin, M.: Algebraization of formal moduli. I. Global analysis (in honor of K. Kodaira), pp. 21-71, Princeton: Princeton Univ. Press 1970 · Zbl 0213.47203
[2] Bourbaki, N.: Algèbre commutative. Chaps. 1-9, Paris: Hermann 1961-1965, Paris: Masson 1980-1985 · Zbl 0108.04002
[3] Andreotti, A., Grauert, H.: Théorèmes de finitude pour la cohomologie des espaces complexes. Bull. Soc. Math. Fr.90, 193-259 (1962) · Zbl 0106.05501
[4] Chai, C.-L., Faltings, G.: Semiabelian degeneration and compactification. Forthcoming book (1990)
[5] Grothendieck, A.: Eléments de la géométrie algébrique. Publ. Math. IHES4, 8, 11, 17, 20, 24, 28, 32 (1960-1967)
[6] Faltings, G.:p-adic Hodge theory. J. AMS1 (1988)
[7] Grothendieck, A.: Fondements de la géometrie algébrique. Extraits du Séminaire Bourbaki (1957-1962) no 149, 182, 190, 195, 212, 221, 232, 236. New York: Benjamin 1966
[8] Gerritzen, L., Grauert, H.: Die Azyklizität der affinoiden Überdeckungen. Global analyis, Papers in Honor of K. Kodaira, 159-184. Tokyo: University of Tokyo Press; Princeton: Princeton University Press 1969 · Zbl 0197.17303
[9] Goodman, J., Hartshorne, R.: Schemes with finite-dimensional cohomology groups. Am. J. Math.91, 258-266 (1969) · Zbl 0176.18303 · doi:10.2307/2373281
[10] Grauert, H., Remmert, R.: Nichtarchimedische Funktionentheorie. Weierstraß-Festschrift, Wissenschaftl. Abh. Arbeitsgemeinschaft für Forschung des Landes Nordrhein-Westfalen33, 393-476 (1966)
[11] Grauert, H., Remmert, R.: Über die Methode der diskret bewerteten Ringe in der nichtarchimedischen Analysis. Invent. Math.2, 87-133 (1966) · Zbl 0148.32401 · doi:10.1007/BF01404548
[12] Hartshorne, R.: Ample subvarieties of algebraic varieties. (Lect. Notes Math., vol. 156). Berlin Heidelberg New York: Springer 1970 · Zbl 0208.48901
[13] Kiehl, R.: Theorem A und Theorem B in der nichtarchimedischen Funktiönentheorie. Invent. Math.2, 256-273 (1967) · Zbl 0202.20201 · doi:10.1007/BF01425404
[14] Kiehl, R.: Der Endlichkeitssatz für eigentliche Abbildungen in der nichtarchimedischen Funktionentheorie. Invent. Math.2, 191-214 (1967) · Zbl 0202.20101 · doi:10.1007/BF01425513
[15] Kiehl, R.: Analytische Familien affinoider Algebren. Heidelb. Sitzungsber. math. nat. Kl. 25-49 (1968) · Zbl 0177.06101
[16] Mehlmann, F.: Flache Homomorphismen affinoider Algebren. Schriftenr. Math. Inst. Univ. Münster, 2. Ser.19 (1981) · Zbl 0455.14014
[17] Nagata, M.: Imbedding of an abstract variety in a complete variety. J. Math. Kyoto Univ.2, 1-10 (1962) · Zbl 0109.39503
[18] Nagata, M.: A generalization of the imbedding problem of an abstract variety in a complete variety. J. Math. Kyoto Univ.3, 89-102 (1962) · Zbl 0223.14011
[19] Gruson, L., Raynaud, M.: Critères de platitude et de projectivité. Invent. Math.13, 1-89 (1971) · Zbl 0227.14010 · doi:10.1007/BF01390094
[20] Raynaud, M.: Géométrie analytique rigide. Table ronde d’analyse non archimedienne, Mém. Soc. Math. Fr. Nouv. Ser.39-40, 319-327 (1974)
[21] Tate, J.: Rigid analytic spaces. Invent. Math.12, 257-289 (1971) · Zbl 0212.25601 · doi:10.1007/BF01403307
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