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Vanishing cycles for formal schemes. II. (English) Zbl 0852.14002
Let $$k$$ be a complete discrete valuation field and $$k^0$$ its ring of integers. In part I of this work [ibid. 115, No. 3, 539-571 (1994; Zbl 0791.14008)], the author constructed and studied the vanishing cycles functor for formal schemes of locally finite type over $$k^0$$. In this part II the construction is extended to a broader class of formal schemes that includes, for example, formal completions of the above formal schemes along arbitrary subschemes of their closed fibres. The main result is a comparison theorem which states that if $${\mathcal X}$$ is a scheme of finite type over a Henselian discrete valuation ring with the completion $$k^0$$ and $${\mathcal Y}$$ is a subscheme of the closed fibre $${\mathcal X}_s$$, then the vanishing cycles sheaves of the formal completion $$\widehat {\mathcal X}_{/{\mathcal Y}}$$ of $${\mathcal X}$$ along $${\mathcal Y}$$ are canonically isomorphic to the restrictions of the vanishing cycles sheaves of $${\mathcal X}$$ to the subscheme $${\mathcal Y}$$. In particular, the restrictions of the vanishing cycles sheaves of $${\mathcal X}$$ to $${\mathcal Y}$$ depend only on $$\widehat {\mathcal X}_{/{\mathcal Y}}$$, and any morphism $$\varphi: \widehat {\mathcal X}'_{/{\mathcal Y}'}\to \widehat {\mathcal X}_{/{\mathcal Y}}$$ induces a homomorphism from the pullback of the restrictions of the vanishing cycles sheaves of $${\mathcal X}$$ to $${\mathcal Y}$$ to those of $${\mathcal X}'$$ to $${\mathcal Y}'$$. – One also proves that, given $$\widehat {\mathcal X}_{/{\mathcal Y}}$$ and $$\widehat {\mathcal X}'_{/{\mathcal Y}'}$$, one can find an ideal of definition of $$\widehat {\mathcal X}'_{/{\mathcal Y}'}$$ such that if two morphisms $$\varphi, \psi: \widehat {\mathcal X}'_{/{\mathcal Y}'}\to \widehat {\mathcal X}_{/{\mathcal Y}}$$ coincides modulo this ideal, then the homomorphisms between the vanishing cycles sheaves induced by $$\varphi$$ and $$\psi$$ coincide.
These facts generalize results of part I as well as results of G. Laumon [“Charactéristique d’Euler-Poincaré et sommes exponentielles” (Thèse, Université de Paris-Sud, Orsay 1983)], and the author [“Vanishing cycles for non-Archimedean analytic spaces”, J. Am. Math. Soc. 9, No. 4, 1187-1209 (1996)], where certain cases when $${\mathcal Y}$$ is a closed point of $${\mathcal X}_s$$ were considered. The main new ingredient in the proof of the comparison theorem is the recent stable reduction theorem of A. J. de Jong [“Smoothness, semi-stability and alterations” (preprint 1995)]. Furthermore, one proves a vanishing theorem which states that the $$q$$-dimensional étale cohomology groups of certain analytic spaces of dimension $$m$$ are trivial for $$q> m$$. This class of analytic spaces induces, for example, the finite étale coverings $$\Sigma^{d,n}$$ of the Drinfeld half-plane $$\Omega^d$$ [V. G. Drinfel’d, Funct. Anal. Appl. 10, 107-115 (1976); translation from Funkts. Anal. Prilozh. 10, No. 2, 29-40 (1976; Zbl 0346.14010)].

##### MSC:
 14F20 Étale and other Grothendieck topologies and (co)homologies 14C25 Algebraic cycles 18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) 14G20 Local ground fields in algebraic geometry 14F99 (Co)homology theory in algebraic geometry
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