Vanishing cycles for formal schemes
type: Item
instance of: scholarly article
title: Vanishing cycles for formal schemes
zbMATH Open document ID: 0791.14008
DOI: 10.1007/BF01231772
author: Vladimir G. Berkovich
published in: Inventiones Mathematicae
publication date: 1994-07-07T00:00:00Z
full work available at URL: https://eudml.org/doc/144181
review text: 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 \textit{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 le…
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 14F99
Mathematics Subject Classification ID: 18F20
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 14C25
zbMATH DE Number: 556340
zbMATH Keywords: analytic group
zbMATH Keywords: non-Archimedean field
zbMATH Keywords: formal scheme
zbMATH Keywords: vanishing cycles functor
zbMATH Keywords: étale sheaves
MaRDI profile type: Publication
cites work: Q3325851
cites work: Q4119314
OpenAlex ID: W4255330161
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description: scientific article; zbMATH DE number 556340