Vanishing cycles for formal schemes. II

Abstract

Let k be a non-Archimedean field, and let X be a formal scheme locally finitely presented over the ring of integers k◦ (see §1). In this work we construct and study the vanishing cycles functor from the category of etale sheaves on the generic fibre Xη of X (which is a k-analytic space) to the category of etale sheaves on the closed fibre Xs of X (which is a scheme over the residue field of k). We prove that if X is the formal completion X of a scheme X finitely presented over k◦ along the closed fibre, then the vanishing cycles sheaves of X are canonically isomorphic to those of X (as defined in [SGA7], Exp. XIII). In particular, the vanishing cycles sheaves of X depend only on X , and any morphism φ : Ŷ → X induces a homomorphism from the pullback of the vanishing cycles sheaves of X under φs : Ys → Xs to those of Y. Furthermore, we prove that, for each X , one can find a nontrivial ideal of k◦ such that if two morphisms φ, ψ : Ŷ → X coincide modulo this ideal, then the homomorphisms between the vanishing cycles sheaves induced by φ and ψ coincide. These facts were conjectured by P. Deligne. In §1 we associate with a formal scheme X locally finitely presented over k◦ a k-analytic space Xη (in the sense of [Ber1] and [Ber2]). In §2 we find that the morphism φη : Yη → Xη, which is induced by an etale morphism of formal schemes φ : Y → X, possesses a certain property. Morphisms of k-analytic spaces with this property are called quasi-etale, and they give rise to a quasi-etale site Xqet of a k-analytic space X. There is a canonical morphism of sites Xqet → Xet, where Xet is the etale site introduced in [Ber2]. We show that the inverse image functor identifies the category of etale sheaves X et with a full subcategory of