Vanishing cycles for formal schemes. II.

*(English)*Zbl 0852.14002Let \(k\) 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)].

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)].

Reviewer: V.G.Berkovich (Rehovot)