zbMATH — the first resource for mathematics

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)].

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
Full Text: DOI EuDML
[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
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.