let $k$ be an algebraically closed field of characteristic $p>0$. let $w(k)$ be the ring of witt vectors with coefficients in $k$. a motivic conjecture of milne relates, in the case of abelian schemes over $w(k)$, the \'etale cohomology with $\bbb z_p$ coefficients to the crystalline cohomology with coefficients in $w(k)$. in this talk we report on the proof of this conjecture in the more general context of $p$-divisible groups over $w(k)$ endowed with arbitrary families of crystalline tensors. if $v$ is a discrete valuation ring which is a finite extension of $w(k)$ of index of ramification $e>1$, we provide examples which show that the conjecture is not true in general over $v$ and we also mention some general cases in which the conjecture does hold over $v$. our results extend previous results of faltings. 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.