×

zbMATH — the first resource for mathematics

Formal and rigid geometry. IV: The reduced fibre theorem. (English) Zbl 0839.14014
[For part III of this paper see Math. Ann. 302, No. 1, 1-29 (1995; see the preceding review).]
In the classical theory of rigid geometry one has the following result: Let \(R\) be a complete height 1 valuation ring with field of fractions \(K\) and with residue field \(k\). Let \(A\) be a \(K\)-affinoid algebra and let \(A^0\) be the corresponding algebra of elements having supremum norm \(\leq 1\). Assume that \(A\) is geometrically reduced in the sense that it remains reduced after tensoring with a finite extension of \(K\). Then there exists a finite extension \(K'\) of \(K\) such that, writing \(R'\) for the algebraic closure of \(R\) in \(K'\) and \(k'\) for the residue field of \(R'\) and \(A' = A \otimes_K K'\), \(A'{}^0\) is topologically of finite type and \(A'{}^0 \otimes_{R'} k'\) is also geometrically reduced.
The aim of this paper is to prove the following relative version of the result. Let \(S\) be a noetherian formal scheme and let \({\mathcal I}\) be a coherent ideal on \(S\) such that \(S\) is complete for the \({\mathcal I}\)-adic topology. Let \(U\) be the associated rigid space on \(S - V ({\mathcal I})\). Let \(f : X \to S\) be a morphism of \({\mathcal I}\)-adic formal schemes, topologically of finite type and such that, over \(U\), \(f\) is flat and has geometrically reduced fibres. Then there exists a rig-étale cover \(S' \to S\) and a finite morphism of formal schemes \(X' \to X \times_SS'\) which is an isomorphism over \(U\) and such that \(X'\) is flat over \(S'\) and has geometrically reduced fibres. (In this terminology a formal situation is rig-true if it is true in the associated rigid situation.)

MSC:
14G20 Local ground fields in algebraic geometry
14L05 Formal groups, \(p\)-divisible groups
11G25 Varieties over finite and local fields
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] [A] Bourbaki, N.: Algèbre. Chap. 4-7, Masson, Paris (1981) · Zbl 0498.12001
[2] [AC] Bourbaki, N.: Algèbre Commutative. Chap. 1-9, Hermann, Paris (1961-1965); Masson, Paris (1980-85)
[3] [BGR] Bosch, S., Güntzer, U., Remmert, R.: Non-Archimedean analysis. Grundlehren Band 261, Berlin Heidelberg New York: Springer 1984 · Zbl 0539.14017
[4] [BLR] Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron models. Ergebnisse der Mathematik, 3. Folge, Band 21, Berlin Heidelberg New York: Springer 1990
[5] [E] Elkik, R., Solutions d’équations à coefficients dans un anneau hensélien. Ann. Scient. Ec. Norm. Sup., 4 série,6, 553-604 (1973)
[6] [EGA] Grothendieck, A.: Eléments de Géométrie Algébrique. Publ. Math. IHES 4, 8, 11, 17, 20, 24, 28, 32 (1960-67); Springer, Grundlehren Band 166, Berlin Heidelberg New York (1971)
[7] [FI] Bosch, S., Lütkebohmert, W.: Formal and rigid geometry I. Rigid spaces. Math. Ann.295, 291-317 (1993) · Zbl 0808.14017
[8] [FII] Bosch, S., Lütkebohmert, W.: Formal and rigid geometry II. Flattening techniques. Math. Ann.296, 403-429 (1993) · Zbl 0808.14018
[9] [FIII] Bosch, S., Lütkebohmert, W., Raynaud, M.: Formal and rigid geometry III. The relative maximum principle. To appear in Math. Ann · Zbl 0839.14013
[10] [GR] Grauert, H., Remmert, R.: Über die Methode der diskret bewerteten Ringe in der nicht-archimedischen Analysis. Invent. math.2, 87-133 (1966) · Zbl 0148.32401
[11] [G] Gruson, L.: Fibrés vectoriels sur un polydisque ultramétrique. Ann. scient. Ec. Norm. Sup. 4 e série,1, 45-89 (1968)
[12] [H] Hartshorne, R.: Algebraic geometry. Graduate Texts in Mathematics 52. New York Heidelberg Berlin: Springer 1977 · Zbl 0367.14001
[13] [M1] Mumford, D.: Abelian varieties. London: Oxford University Press 1970
[14] [M2] Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B.: Toroidal embeddings I. Lect. Notes Math. 339, Springer, Berlin Heidelberg New York (1973) · Zbl 0271.14017
[15] [RG] Raynaud, M., Gruson, L.: Criteres de platitude et de projectivité. Invent. math.13, 1-89 (1971) · Zbl 0227.14010
[16] [SGA2] Grothendieck, A.: Cohomologie Locale des Faisceaux Cohérents et Théorèmes de Lefschetz Locaux et Globaux. North-Holland, Amsterdam (1968)
[17] [SGA3] Demazure, M., Grothendieck, A.: Séminaire de Géométrie Algébrique 1962/64. Schémas en Groupes. Lecture Notes in Math. 151 (1970)
[18] [T] Tate, J.: Rigid analytic spaces. Invent. math.12, 257-289 (1971) · Zbl 0212.25601
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.