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Degeneration of $$l$$-adic weight spectral sequences. (English) Zbl 1033.14012
From the introduction: Let $$A$$ be a henselian discrete valuation ring and let $$l$$ be a prime number invertible in $$A$$. We prove:
Theorem 1. For a proper semistable family over $$A$$ whose special fiber has smooth irreducible components, the Steenbrink-Rapoport-Zink $$l$$-adic weight spectral sequence degenerates in $$E_2$$-terms.
The degeneracy was proved in cases where
(i) $$A=\mathbb{C}\{t\}$$ by J. H. M. Steenbrink [Invent. Math. 31, 229–257 (1976; Zbl 0303.14002)]; and
(ii) The residue field of $$A$$ is finite by a result by M. Rapoport and Th. Zink [Invent. Math. 68, 21-101 (1982; Zbl 0498.14010)].
The second case was proved by the Weil conjecture on Frobenius weights. L. Illusie [Astérisque 223, 9-57 (1994; Zbl 0837.14013)] conjectured that the degeneracy holds over an arbitrary A. Theorem 1 gives an affirmative answer to it.

##### MSC:
 14G20 Local ground fields in algebraic geometry 14F30 $$p$$-adic cohomology, crystalline cohomology 14D06 Fibrations, degenerations in algebraic geometry
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