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 Illusie (Annls Sci. Ec. Norm. Super.12 (1979), 501–661) 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 $S$.

AMS 2000 Mathematics subject classification: Primary 14F30; 14F40