Hi,

Let $k$ be a perfect field of characteristic $p > 0$. Let $A/k$ be an abelian variety. Then the first crystalline cohomology group of $A$ with respect to $W(k)$ (= Witt vectors) is canonically isomorphic to the (contravariant) Dieudonne functor of the $p$-divisible group of $A$.

This is well-known result and I am wondering if there is a simple proof of it. Specifically,
I am wondering if there is a proof that doesn't require defining the Dieudonne *crystal* for a $p$-divisible group as an intermediary.

Thanks!