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Espaces de Berkovich sur \({\mathbb {Z}}\): morphismes étales

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Résumé

Nous étudions les morphismes non ramifiés, étales et lisses entre espaces de Berkovich sur \({\mathbb {Z}}\). Nous obtenons des analogues des propriétés classiques des morphismes de schémas ainsi que des critéres par analyfication. Ces résultats sont aussi valables sur les corps valués complets, les anneaux d’entiers de corps de nombres et les anneaux de valuation discréte. Ces différents cas sont traités de façon unifiée.

Abstract

(Berkovich spaces over \({\mathbb {Z}}\): étale morphisms).— We develop properties of unramified, étale and smooth morphisms between Berkovich spaces over \({\mathbb {Z}}\). We prove that they satisfy properties analogous to those of morphisms of schemes and we provide analytification criteria. Our results hold for any valued field, rings of integers of a number field and discrete valuation rings. Those cases are treated by a unified way.

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References

  1. Berkovich, V.G.: Spectral theory and analytic geometry over non-archimedean fields. Volume 33 of Mathematical surveys and monographs. Am. Math. Soc. (1990)

  2. Berkovich, V.G.: Étale cohomology for non-Archimedean analytic spaces. Publications mathématiques de l’IHÉS 78, 5–161 (1993)

    Article  MATH  Google Scholar 

  3. Berkovich, V.G.: Vanishing cycles for formal schemes. Inventiones Mathematicae 115, 539–571 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bosch, S.: Algebraic Geometry and Commutative Algebra. Universitext. Springer (2013)

    Book  MATH  Google Scholar 

  5. Carayol, H.: Non-abelian Lubin-Tate theory. Perspectives in mathematics. Publication Title: Automorphic forms, Shimura varieties, and L-functions: proceedings of a conference held at the University of Michigan, Ann Arbor, July 6–16, 1988. Volume II. Academic Press (1990)

  6. Ducros, A.: Variation de la dimension relative en géométrie analytique p-adique. Compositio Mathematica 143 (2007), no. 6, 1511–1532

  7. Fu, L.: Etale cohomology theory. Nankai tracts in mathematics v. 13. World Scientific, (2011)

  8. Grothendieck, A.: Revêtements étales et groupe fondamental (SGA 1): Séminaire de géométrie algébrique du Bois-Marie 1960-61. EDP Sciences, (2003)

  9. Huber, R.: Continuous valuations. Mathematische Zeitschrift 212 (1993), no. 3, 455–478

  10. Huber, R.: A generalization of formal schemes and rigid analytic varieties. Mathematische Zeitschrift P217 (1994)

  11. Lemanissier, T., et Stevenson, M.: Topology of Hybrid Analytifications. (2019). arXiv:1903.01926

  12. Lemanissier, T., et Poineau, J.: Espaces de Berkovich sur Z: catégorie, topologie, cohomologie. 2022. arXiv:2010.08858

  13. Poineau, J.: La droite de Berkovich sur Z. Volume 334. Astérisque. (2010)

  14. Poineau, J.: Espaces de Berkovich sur Z: étude locale. Inventiones mathematicae 194 (2013)

  15. Poineau, J.: Dynamique analytique sur Z. I: Mesures d’équilibre sur une droite projective relative. (2022). arXiv:2201.08480

  16. Raynaud, M.: Géométrie analytique rigide d’après Tate, Kiehl... Mémoires de la Société mathématique de France 39–40 (1974)

  17. Stacks Project authors. Stacks Project. 2021. https://stacks.math.columbia.edu

  18. Tate, J.: Rigid analytic spaces. Inventiones Mathematicae 12, 257–289 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  19. Weil, A: Basic number theory. T. 144. Grundlehren der mathematischen Wissenschaften. Springer (1995)

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Remerciements

Nous remercions Jérôme Poineau pour les nombreuses discussions à l’origine de cet article, ainsi que Thibaud Lemanissier pour ses remarques.

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Correspondence to Dorian Berger.

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Berger, D. Espaces de Berkovich sur \({\mathbb {Z}}\): morphismes étales. Math. Z. 304, 66 (2023). https://doi.org/10.1007/s00209-023-03308-4

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