  For a smooth projective scheme $X$ over a ring $R$ on which $p$ is nilpotent
that meets some general assumptions we prove that the crystalline cohomology is
equipped with the structure of a higher display which is a relative version of
Fontaine's strongly divisible lattices. Frobenius-divisibilty is induced by the
Nygaard filtration on the relative de Rham-Witt complex. For a nilpotent
PD-thickening $S/R$ we also consider the associated relative display and can
describe it explicitly by a relative version of the Nygaard filtration on the
de Rham-Witt complex associated to a lifting of $X$ over $S$. We prove that
there is a crystal of relative displays if moreover the mod $p$ reduction of
$X$ has a smooth and versal deformation space.
