Finiteness theorems for vanishing cycles of formal schemes

Abstract

Let k be a non-Archimedean field with nontrivial valuation, and k • its ring of integers. In this paper we prove constructibility of vanishing cycles sheaves for arbitrary formal schemes locally finitely presented over k • as well as special formal schemes over k • (for discretely valued k). This allows us to extend continuity results, established earlier for locally algebraic formal schemes, to the whole classes of formal schemes.

Monodromy and the Lefschetz fixed point formula

“…-We shall only make use of Theorem 5.1.1 when X = A n and K = k((t)) with k a field of characteristic zero. Note also that, though in subsequent arXiv versions of [28] the proof of Theorem 5.1.1 relies on Theorem 1.1 of [6] which uses de Jong's results on alterations and Gabber's weak uniformization theorem, the first version is based on Corollary 5.5 of [4], which does not use any of these results.…”

Constructibility and Reflexivity in non-Archimedean geometry

“…Firstly, the class of constructible sheaves is large enough to include both Huber's notion of constructibility (cf. Remark 3.2) and those introduced by Berkovich in [9]. In Proposition 3.7, we prove that this class is stable for pushforwards and the lower shriek functor.…”

“…We now turn our attention to specific hypercoverings. We need the following analogue of [9,Theorem 1.3.1]. Lemma 4.16.…”

“…In fact it was only recently shown that if X is a compact strictly k-analytic space then the groups H q (X, Z/ℓZ) ∼ = H q c (X, Z/ℓZ) are of finite dimension (cf. [7], [8], [9]) where k is an algebraically closed complete non-Archimedean real valued field and ℓ is a prime number different from the characteristic of the residue field k. In the language of adic spaces, if f : X → Y is smooth and quasi-compact then there is a theorem of stability for R q f ! with respect to a certain class of constructible sheaves, cf.…”

“…This allows us to define in a natural way the nearby cycles functor cf. §2.4.3 which was studied in [19] (in the adic context) as well as [7], [8] and [9] (in the Berkovich context). More precisely, given a formal model X of X, we have a functor of derived categories…”