Vanishing cycles for non-archimedean analytic spaces

Abstract

In this work we develop a formalism of vanishing cycles for non-Archimedean analytic spaces which is an analog of that for complex analytic spaces from [SGA7], Exp. XIV. As an application we prove that in the equicharacteristic case the stalks of the vanishing cycles sheaves of a scheme X at a closed point x E X, depend only on the formal completion Spf(Ox,,) of X at x. In particular, any continuous homomorphism OX,x -+ y,y induces a homorphism from the stalks of the vanishing cycles sheaves of X at x to those of Y at y. Furthermore, we prove that, given Ox,x and O y, , there exists n > 1 such that, for any pair of continuous homomorphisms OX,x -Oy,y that coincide modulo the n-th power of the maximal ideal of Oy,y, the induced homomorphisms between the stalks of the vanishing cycles sheaves coincide. These facts generalize a result of G. Laumon from [Lau] (see Remark 7.6). Throughout the paper we fix a non-Archimedean field k (whose valuation is not assumed to be nontrivial). In ?1 we study etale Galois sheaves on k-analytic spaces. To define the vanishing cycles functor and to work with it, we use the language of pro-analytic spaces, i.e., pro-objects of the category of analytic spaces ([SGA4], Exp. I). Examples of such objects are the germs of analytic spaces as in [Ber2], ?3.4. Another example is considered in ?3. In ?4 we define the vanishing cycles functor and establish its basic properties. In ?5 we show that the vanishing cycles sheaves are trivial for smooth morphisms. In ?6 we prove a comparison theorem for vanishing cycles. This theorem is more general than its analog over C from [SGA7], Exp. XIV, and its proof does not use Hironaka's theorem on resolution of singularities. In ?7 we apply the comparison theorem to prove the properties of the vanishing cycles sheaves of schemes formulated above. It is worthwhile to note that this application is obtained by considering non-Archimedean analytic geometry over fields with trivial valuation. Like [Ber3], this work arose from a suggestion of P. Deligne to apply the etale cohomology theory from [Ber2] to the study of the vanishing cycles sheaves of schemes. I am very grateful to him for useful discussions on the subject. I also