Un résultat de connexité pour les variétés analytiques <i>p</i>-adiques: privilège et noethérianité

Poineau, Jérôme

doi:10.1112/s0010437x07003016

Cited by 11 publications

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References 13 publications

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“…Let us give another application of our results, in the spirit of results of Abbes and Saito [1] 5.1 and Poineau [35] Théorème 2. Theorem 14.4.4.…”

Section: More Tame Topological Propertiesmentioning

confidence: 81%

Non-archimedean tame topology and stably dominated types

Hrushovski¹,

Loeser²

2010

Preprint

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Let V be a quasi-projective algebraic variety over a nonarchimedean valued field. We introduce topological methods into the model theory of valued fields, define an analogue V of the Berkovich analytification V an of V , and deduce several new results on Berkovich spaces from it. In particular we show that V an retracts to a finite simplicial complex and is locally contractible, without any smoothness assumption on V . When V varies in an algebraic family, we show that the homotopy type of V an takes only a finite number of values. The space V is obtained by defining a topology on the pro-definable set of stably dominated types on V . The key result is the construction of a pro-definable strong retraction of V to an o-minimal subspace, the skeleton, definably homeomorphic to a space definable over the value group with its piecewise linear structure. ContentsChapter 1. Introduction Chapter 2. Preliminaries 2.1. Definable sets 2.2. Pro-definable and ind-definable sets 2.3. Definable types 2.4. Stable embeddedness 2.5. Orthogonality to a definable set 2.6. Stable domination 2.7. Review of ACVF 2.8. Γ-internal sets 2.9. Orthogonality to Γ 2.10. " V for stable definable V 2.11. Decomposition of definable types 2.12. Pseudo-Galois coverings Chapter 3. The space "V of stably dominated types 3.1. "V as a pro-definable set 3.2. Some examples 3.3. The notion of a definable topological space 3.4. "V as a topological space 3.5. The affine case 3.6. Simple points 3.7. v-open and g-open subsets, v+g-continuity 3.8. Canonical extensions 3.9. Paths and homotopies 3.10. Good metrics 3.11. Zariski topology 3.12. Schematic distance Chapter 4. Definable compactness 4.1. Definition of definable compactness 4.2. Characterization of definable compactness Chapter 5. A closer look at the stable completion iii iv CONTENTS 5.1. " A n and spaces of semi-lattices 5.2. A representation of " P n 5.3. Relative compactness Chapter 6. Γ-internal spaces 6.1. Preliminary remarks 6.2. Topological structure of Γ-internal subsets 6.3. Guessing definable maps by regular algebraic maps 6.4. Relatively Γ-internal subsets Chapter 7. Curves 7.1. Definability of " C for a curve C 7.2. Definable types on curves 7.3. Lifting paths 7.4. Branching points 7.5. Construction of a deformation retraction Chapter 8. Strongly stably dominated points 8.1. Strongly stably dominated points 8.2. A Bertini theorem 8.3. Γ-internal sets and strongly stably dominated points 8.4. Topological properties of V # Chapter 9. Specializations and ACV 2 F 9.1. g-topology and specialization 9.2. v-topology and specialization 9.3. ACV 2 F 9.4. The map R 20 21 : " V 20 → " V 21 9.5. Relative versions 9.6. g-continuity criterion 9.7. Some applications of the continuity criteria 9.8. The v-criterion on " V 9.9. Definability of v-and g-criteria Chapter 10. Continuity of homotopies 10.1. Preliminaries 10.2. Continuity on relative P 1 10.3. The inflation homotopy 10.4. Connectedness and the Zariski topology Chapter 11. The main theorem 11.1. Statement 11.2. Proof of Theorem 11.1.1: Preparatio...

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“…Let us give another application of our results, in the spirit of results of Abbes and Saito [1] 5.1 and Poineau [35] Théorème 2. Theorem 14.4.4.…”

Section: More Tame Topological Propertiesmentioning

confidence: 81%

Non-archimedean tame topology and stably dominated types

Hrushovski¹,

Loeser²

2010

Preprint

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“…There exists a finite partition P of R + in intervals such that for every I ∈ P and every (ε, ε ′ ) ∈ I 2 with ε ε ′ , the natural map π 0 (X ε ′ ) → π 0 (X ε ) is bijective. This has been established by Poineau in [18] (it had already been proved in the particular case where f is invertible by Abbes and Saito in [1]).(0.4) Hrushovski and Loeser's work. As far as tameness is concerned, there has been in 2009-2010 a major breakthrough, namely the work [16], by Hrushovski and Loeser.…”

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confidence: 72%

About Hrushovski and Loeser’s Work on the Homotopy Type of Berkovich Spaces

Ducros

2016

Simons Symposia

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(0.1) At the end of the eighties, V. Berkovich developed in [2] and [3] a new theory of analytic geometry over non-archimedean fields, coming after those by Krasner, Tate [22] and Raynaud [20]. One of the main advantages of this approach is that the resulting spaces enjoy very nice topological properties: they are locally compact and locally pathwise connected (although nonarchimedean fields are totally disconnected, and often not locally compact). Moreover, Berkovich spaces have turned out to be "tame" objects -in the informal sense of Grothendieck's Esquisse d'un programme. Before illustrating this rather vague assertion by several examples, let us fix some terminology and notations.(0.2) Let k be a field which is complete with respect to a non-archimedean absolute value |.| and let k o be the valuation ring {x ∈ k, |x| 1}.(0.2.1) A (Berkovich) analytic space over k is a (locally compact, locally pathconnected) topological space X equipped with some extra-data, among which:-a sheaf of k-algebras whose sections are called analytic functions, which makes X a locally ringed space; -for every point x of X, a complete non-archimedean field (H (x), |.|) endowed with an isometric embedding k ֒→ H (x), and an "evaluation" morphism from O X,x to H (x), which is denoted by f → f (x).(0.2.2) To every k-scheme of finite type X is associated in a functorial way a k-analytic space X an , which is called its analytification ([3], §2.6); it is provided with a natural morphism of locally ringed spaces X an → X If f is a section of O X , we will often still denote by f its pull-back to O X an (X an ).Let us recall what the underlying topological space of X an is. As a set, it consists of couples (ξ, |.|) where ξ ∈ X and where |.| is a non-archimedean absolute value on κ(ξ) extending the given absolute value of k. If x = (ξ, |.|) is a point of X an , then H (x) is the completion of the valued field (κ(ξ), |.|)). The map X an → X is nothing but (ξ, |.|) → ξ; note that the pre-image of a * During this work the author was partially supported by ValCoMo (ANR-13-BS01-0006), and by a Rosi and Max Varon visiting professorship at Weizmann Institute. 1Zariski-open subset U of X on X an can be identified with U an . The topology X an is equipped with is the coarsest such than:• X an → X is continuous (otherwise said, U an is an open subset of X an for every Zariski-open subset U of X);• for every Zariski-open subset U of X and every f ∈ O X (U ), the map (ξ, |.|) → |f (ξ)| from U an to R + is continuous.Let us now assume that X is affine, say X = Spec A. The topological space underlying X an can then be given another description: it is the set of all multiplicative maps ϕ : A → R + that extend the absolute value of k and that satisfy the inequality ϕ(a + b) max(ϕ(a), ϕ(b)) for every (a, b) ∈ A 2 (such a map will be simply called a multiplicative semi-norm); its topology is the one inherited from the product topology on R A + . This is related as follows to the former description:• to any couple (ξ, |.|) as above corresponds the multiplica...

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“…Nous nous contenterons ici d'esquisser les preuves en renvoyant à [16] pour de plus amples détails. COROLLAIRE 1.7.…”

Section: Variation De Connexitéunclassified

“…Indiquons encore que nous avons proposé une première démonstration de ce théorème, pour des ensembles définis par des équations de la forme | f | ε, de nouveau, dans [16]. Dans la méthode utilisée alors, on interprète les espaces étudiés comme les fibres d'un morphisme analytique.…”

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Sur Les Composantes Connexes D’une Famille D’espaces Analytiques -Adiques

Poineau

2014

Forum math. Sigma

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RésuméSoit X = M(A ) un espace affinoïde et soient f, g ∈ A . Nous étudions les ensembles de composantes connexes des espaces définis par une inégalité de la forme | f | r |g|, avec r 0. Nous montrons qu'il existe une partition finie de R + en intervalles sur lesquels ces ensembles sont canoniquement en bijection et que les bornes de ces intervalles appartiennent à ρ(A ). AbstractOn the connected components of a family of p-adic analytic spaces. Let X = M(A ) be an affinoid space and let f, g ∈ A . We study the sets of connected components of the spaces defined by an inequality of the form | f | r |g|, with r 0. We prove that there exists a finite partition of R + into intervals where those sets are canonically in bijection and that the bounds of those intervals belong to ρ(A ).

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Un résultat de connexité pour les variétés analytiques p-adiques: privilège et noethérianité

Cited by 11 publications

References 13 publications

Non-archimedean tame topology and stably dominated types

Non-archimedean tame topology and stably dominated types

About Hrushovski and Loeser’s Work on the Homotopy Type of Berkovich Spaces

Sur Les Composantes Connexes D’une Famille D’espaces Analytiques -Adiques

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