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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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