  Let k be an algebraically closed field of characteristic p>0. Let W(k) be the
ring of Witt vectors with coefficients in k. We prove a motivic conjecture of
Milne that relates, in the case of abelian schemes, the \'etale cohomology with
$\dbZ_p$ coefficients to the crystalline cohomology with integral coefficients,
in the more general context of p-divisible groups endowed with {\it arbitrary}
families of crystalline tensors over a finite, discrete valuation ring
extension of W(k). This extends a result of Faltings in [Fa2]. As a main new
tool we construct global deformations of p-divisible groups endowed with
crystalline tensors over certain regular, formally smooth schemes over W(k)
whose special fibers over k have a Zariski dense set of k-valued points.
