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Perversity and variation. (Perversité et variation.) (English) Zbl 1036.14008

This paper is devoted to various special, largely technical results in the cohomology theory of schemes, as well as to generalizations of some known theorems in local cohomology, étale cohomology, \(\ell\)-adic cohomology, and the theory of perverse sheaves. Based upon the deep and extensive theories developed in A. Grothendieck’s “Séminaire de Géométrie Algébrique” (SGA 2 – SGA 7, 1962–1977) and by many authors thereafter, including the author himself, P. Deligne, O. Gabber, N. Katz, W. Messing, K. Fujiwara, and others, two types of promoting results are established in this paper under review. The general context of this work is related to the so-called “absolute purity conjecture” (A. Grothendieck) and its recent proof by O. Gabber [cf. K. Fujiwara, in: Algebraic Geometry 2000, Azumino, Proc. Symp. Nagano 2000, Adv. Stud. Pure Math. 36, 153–183 (2002; Zbl 1059.14026)].
More precisely, the author studies a scheme \(X\) of finite type over the spectrum \(S\) of a strictly Henselian discrete valuation ring with residue field of characteristic \(p > 0\) and a group \(\Lambda = \mathbb{Z}/\ell^{\nu}\cdot\mathbb{Z}\), where \(\ell\) is a prime number different from the characteristic \(p\) and \(\nu\) is a natural number. Then he derives an estimate for the étale depth of the constant sheaf \(\Lambda\) and proves a cohomological vanishing theorem for the fibres in \(X\). This leads to an affine Lefschetz-type theorem (à la SGA 4) which appears to be closely related to the absolute purity conjecture. In fact, the author provides here many details needed in Gabber’s spectacular proof (still unpublished) of Grothendieck’s old conjecture.
In the second part of this paper, the author studies the complex of vanishing cycles for the datum \((X,S,\Lambda )\), together with the so-called variation morphism of complexes established in SGA 7 I and SGA 7 II [Lect. Notes Math. 288 and 340 (1972/1973; see Zbl 0237.00013 and Zbl 0258.00005)]. This variation morphism depends on an element \(\sigma\) in the tame inertia group \(I_t\) of the given set-up, and the author shows under what conditions such a variation morphism is an isomorphism. Also this result is related to O. Gabber’s proof of the absolute purity conjecture, as the author points out, and it might be seen as an (\(p\)-adic) analogue of a corresponding result in the transcendental theory of hypersurface singularities (à la J. Milnor, 1968).
No doubt, the paper under review is a very important contribution towards a better understanding of O. Gabber’s approach to Grothendieck’s problem of absolute cohomological purity.

MSC:

14F20 Étale and other Grothendieck topologies and (co)homologies
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
14D06 Fibrations, degenerations in algebraic geometry
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