## Vanishing cycles for formal schemes.(English)Zbl 0791.14008

Let $$k$$ be a non-Archimedean field, and let $${\mathfrak X}$$ be a formal scheme locally finitely presented over the ring of integers $$k^ 0$$. In this work one constructs and studies the vanishing cycles functor from the category of étale sheaves on the generic fibres $${\mathfrak X}_ \eta$$ of $${\mathfrak X}$$ (which is a $$k$$-analytic space) to the category of étale sheaves on the closed fibre $${\mathfrak X}_{\overline s}$$ of $${\mathfrak X}$$ (which is a scheme over the residue field of the separable closure of $$k)$$. One proves that if $${\mathfrak X}$$ is the formal completion $$\hat {\mathcal X}$$ of a scheme $${\mathcal X}$$ finitely presented over $$k^ 0$$ along the closed fibre, then the vanishing cycles sheaves of $$\hat {\mathcal X}$$ are canonically isomorphic to those of $${\mathcal X}$$ [as defined by P. Deligne in Sémin. Géométrie algébrique, 1967-1969, SGA7 II, Lect. Notes Math. 340, Exposé XIII, 82-115 (1973; Zbl 0266.14008)]. In particular, the vanishing cycles sheaves of $${\mathcal X}$$ depend only on $$\hat {\mathcal X}$$, and any morphism $$\varphi:\hat {\mathcal Y} \to \hat {\mathcal X}$$ induces a homomorphism from the pullback of the vanishing cycles sheaves of $${\mathcal X}$$ under $$\varphi_{\overline s}:{\mathcal Y}_{\overline s} \to {\mathcal X}_{\overline s}$$ to those of $${\mathcal Y}$$. Furthermore, one proves that, for each $$\hat {\mathcal X}$$, there exists a nontrivial ideal of $$k^ 0$$ such that if two morphisms $$\varphi,\psi:\hat {\mathcal Y} \to \hat {\mathcal X}$$ coincide modulo this ideal, then the homomorphisms between the vanishing cycles sheaves induced by $$\varphi$$ and $$\psi$$ coincide. These facts were conjectured by P. Deligne.
The second fact is deduced from a theorem on the continuity of the action of the set of morphisms between two analytic spaces on their étale cohomology groups. Its particular case states the following. Let $$X={\mathcal M} ({\mathcal A})$$ be a $$k$$-affinoid space, and let $$f_ 1,\dots,f_ n$$ be a $$k$$-affinoid generating system of elements of $${\mathcal A}$$. Then for any discrete $$\text{Gal} (k^ s/k)$$-module $$\Lambda$$ and any element of $$\alpha \in H^ q (X,\Lambda)$$ there exist $$t_ 1, \dots,t_ n>0$$ such that, for any pair of morphisms $$\varphi,\psi:Y \to X$$ over $$k$$ with $$\max_{y \in Y} | (\varphi^* f_ i-\psi^*f_ i)(y) | \leq t_ i$$, $$1 \leq i \leq n$$, one has $$\varphi^*(\alpha)=\psi^*(\alpha)$$ in $$H^ q(Y,\Lambda)$$. The essential ingredient of the proof is a generalization of the classical Krasner lemma. This result implies, in particular, the following fact. If a $$k$$-analytic group $$G$$ acts on a $$k$$-analytic space $$X$$, then the étale cohomology groups of $$X$$ with compact support are discrete $$G(k)$$-modules. The present paper is based on the previous works of the author [“Spectral theory and analytic geometry over non-Archimedean fields”, Math. Surveys Monographs 33 (1990; Zbl 0715.14013) and “Étale cohomology for non-Archimedean analytic spaces”, Publ. Math., Inst. Hautes Étud. Sci. 78, 5-171 (1993)].

### MSC:

 14F20 Étale and other Grothendieck topologies and (co)homologies 14F99 (Co)homology theory in algebraic geometry 18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) 14G20 Local ground fields in algebraic geometry 14C25 Algebraic cycles

### Citations:

Zbl 0266.14008; Zbl 0715.14013
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### References:

 [1] [Ber1] Berkovich, V. G.: Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33, American Mathematical Society, 1990 [2] [Ber2] Berkovich, V. G.: Étale cohomology for non-Archimedean analytic spaces, Publ. Math. IHES78, 5-161 (1994) [3] [Ber3] Berkovich, V. G.: Vanishing cycles for non-Archimedean analytic spaces, (submitted to Journal of the AMS) [4] [BGR] Bosch, S.; Güntzer, U.; Remmert, R.: Non-Archimedean analysis. A systematic approach to rigid analytic geometry, Grundlehren der Mathematischen Wissenschaften, Bd. 261, Springer, Berlin-Heidelberg-New York, 1984 · Zbl 0539.14017 [5] [EGA1] Grothendieck, A.; Dieudonné, J.: Eléments de Géométrie Algébrique. I. Le langage des schémas, Springer, Berlin-Heidelberg-New York, 1971 [6] [God] Godement, R.: Topologie Algébrique et Théorie des Faisceux, Hermann, Paris, 1958 [7] [Kel] Kelley, J. L.: General Topology, D. Van Nostrand Company, Toronto-London-New York, 1957 [8] [Ray] Raynaud, M.: Anneaux Locaux Henséliens, Lecture Notes in Math.169, Springer, Berlin-Heidelberg-New York, 1970 [9] [SGA1] Grothendieck, A.: Seminaire de Géométrie Algébrique. I. Revêtements étales et Groupe Fondemental, Lecture Notes in Math.224, Springer, Berlin-Heidelberg-New York, 1971 [10] [SGA4] Artin, M., Grothendieck, A., Verdier, J.-L.: Théorie des Topos et Cohomologie Étale des Schémas, Lecture Notes in Math.269, 270, 305, Springer, Berlin-Heidelberg-New York, 1972-1973 [11] [SGA41/2] Deligne, P. et al.: Cohomologie Étale, Lecture Notes in Math.569, Springer, Berlin-Heidelberg-New York, 1977 · Zbl 0349.14008 [12] [SGA7] Grothendieck, A., Deligne, P., Katz, N.: Groupes de Monodromie en Géométrie Algébrique, Lecture Notes in Math.288, 340 Springer, Berlin-Heidelberg-New York, 1972-1973
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