We define truncated displays over rings in which a prime $p$ is nilpotent, we associate crystals to truncated displays, and we define functors from truncated displays to truncated Barsotti–Tate groups.

Let X/R be a smooth scheme over a ring R. Consider the category of locally free crystals of finite rank on the situs Crys(X/Wt (R)). We show that it is equivalent to the category of locally free -modules of finite rank endowed with a nilpotent, integrable de Rham-Witt connection. In the case where R is a perfect field this was shown by Etesse [E] and Bloch [Bl]. We use the result for a construction of the Gauß-Manin connection as a de Rham-Witt connection.

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