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
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