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De Rham-Witt cohomology for a proper and smooth morphism. (English) Zbl 1100.14506
Summary: We construct a relative de Rham-Witt complex \(W\varOmega^{\cdot}_{X/S}\) for a scheme \(X\) over a base scheme \(S\). It coincides with the complex defined by L. Illusie [Ann. Sci. Éc. Norm. Supér., IV. Sér. 12, 501–661 (1979; Zbl 0436.14007)] if \(S\) is a perfect scheme of characteristic \(p>0\). The hypercohomology of \(W\varOmega^{\cdot}_{X/S}\) is compared to the crystalline cohomology if \(X\) is smooth over \(S\) and \(p\) is nilpotent on \(S\). We obtain the structure of a \(3n\)-display on the first crystalline cohomology group if \(X\) is proper and smooth over \(S\).

MSC:
14F30 \(p\)-adic cohomology, crystalline cohomology
14F40 de Rham cohomology and algebraic geometry
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