Espaces de Berkovich, polytopes, squelettes et théorie des modèles
Confluentes Mathematici, Volume 4 (2012) no. 4.

Let X be an analytic space over a non-Archimedean, complete field k and let f = (f1,…,fn) be a family of invertible functions on X. Let us recall two results. (1) If X is compact, the compact set |f|(X) is a polytope of the ℝ-vector space (R+)n (we use the multiplicative notation); this is due to Berkovich in the locally algebraic case (his proof made use of de Jong's alterations), and has been extended to the general case by the author. The locally algebraic case could also have been deduced quite formally from a former result by Bieri and Groves, based upon explicit computations on Newton polygons. (2) If moreover X is Hausdorff and n-dimensional, and if ϕ denotes the morphism XGm,kn,an induced by f, then the pre-image of the skeleton Sn of Gm,kn,an under ϕ has a piecewise-linear structure making ϕ-1(Sn) → Sn a piecewise immersion; this is due to the author, and his proof also made use of de Jong's alterations. In this article, we improve (1) and (2), and give new proofs of both of them. Our proofs are based upon the model theory of algebraically closed, nontrivially valued fields and do not involve de Jong's alterations. Let us quickly explain what we mean by improving (1) and (2).

  • Concerning (1), we also prove that if x ∈ X, there exists a compact analytic neighborhood U of x, such that for every compact analytic neighborhood V of x in X, the germs of polytopes (|f|(V), |f|(x)) and (|f|(U), |f|(x)) coincide.
  • Concerning (2), we prove that the piecewise linear structure on ϕ-1(Sn) is canonical, that is, does not depend on the map we choose to write it as a pre-image of the skeleton; we thus answer a question which was asked to us by Temkin.

Moreover, we prove that the pre-image of the skeleton "stabilizes after a finite, separable ground field extension", and that if ϕ1,…,ϕm are finitely many morphisms from X to Gm,kn,an, the union φj1(Sn) also inherits a canonical piecewise-linear structure.

Published online:
DOI: 10.1142/S1793744212500077
Antoine Ducros 1

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Antoine Ducros. Espaces de Berkovich, polytopes, squelettes et théorie des modèles. Confluentes Mathematici, Volume 4 (2012) no. 4. doi : 10.1142/S1793744212500077.

[1] V. Berkovich, Spectral Theory and Analytic Geometry over Non-Archimedean Fields, Mathematical Surveys and Monographs, Vol. 33 (Amer. Math. Soc., 1990).

[2] V. Berkovich, Étale cohomology for non-archimedean analytic spaces, Inst. Hautes Etudes Sci. Publ. Math. 78 (1993) 5–161.

[3] V. Berkovich, Smooth p-adic spaces are locally contractible II, in Geometric Aspects of Dwork Theory (Walter de Gruyter Co., 2004), pp. 293–370.

[4] R. Bieri and J. R. J. Groves, The geometry of the set of characters induced by valuations, J. Reine Angew. Math. 347 (1984) 168–195.

[5] S. Bosch, S. Güntzer and U. Remmert, Non-Archimedean Analysis. A Systematic Approach to Rigid Analytic Geometry, Grundlehren der Mathematischen Wis- senschaften, Vol. 261 (Springer-Verlag, 1984).

[6] A. Chambert-Loir et A. Ducros, Formes différentielles et courants sur les espaces de Berkovich, travail en cours.

[7] B. Conrad and M. Temkin, Non-Archimedean analytification of algebraic spaces, preprint.

[8] A. Ducros, Les espaces de Berkovich sont modérés, exposé 1056 du séminaire Bourbaki.

[9] A. Ducros, Image réciproque du squelette par un morphisme entre espaces de Berkovich de même dimension, Bull. Soc. Math. France 131 (2003) 483–506.

[10] A. Ducros, Parties semi-algébriques d’une variété algébrique p-adique, Manuscripta Math. 111 (2003) 513–528.

[11] A. Ducros, Variation de la dimension relative en géométrie analytique p-adique, Compositio. Math. 143 (2007) 1511–1532.

[12] A. Ducros, Toute forme modérément ramifiée d’un polydisque ouvert est triviale, à paraˆıtre dans Math. Z.

[13] A. Ducros, Flatness in non-Archimedean analytic geometry, preprint.

[14] D. Haskell, E. Hrushovski and D. Macpherson, Definable sets in algebraically closed valued fields : Elimination of imaginaries, J. Reine Angew. Math. 597 (2006) 175–236.

[15] E. Hrushovski and F. Loeser, Non-archimedean tame topology and stably dominated types, preprint.

[16] J. Poineau, Les espaces de Berkovich sont angéliques, prépublication.

[17] M. Temkin, On local properties of non-Archimedean analytic spaces, Math. Ann. 318 (2000) 585–607.

[18] M. Temkin, On local properties of non-Archimedean analytic spaces. II, Israel J. Math. 140 (2004) 1–27.

[19] M. Temkin, A new proof of the Gerritzen–Grauert theorem, Math. Ann. 333 (2005) 261–269.

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