Let $M$ be the Dieudonne module of a p-divisible group $G_0$ over $k$, and let a lift of $G_0$ to $A$ be a p-divisible group $G$ over $A$ such that $G \otimes_A k \simeq G_0$. Let $\omega_G$ be the sheaf of invariant differentials of the p-divisible group $G$. Weinstein states in his notes on The Geometry of Lubin Tate Spaces (bottom of pg 10):
Even though the module $M$ (and the endomorphism $F$) only depend on $G_0$, the line spanned by $\omega_G$ in $M$ really does depend on the lift $G$. Any lift of $G_0$ gives rise via its invariant differential to a line $Fil \subset M$ having the property that the image of $Fil$ spans $M/FM$. Different lifts could (and indeed do) give rise to different lines in $M$ having this property. These ideas were made precise by Fontaine, that the set of lifts of $G_0$ is canonically the same as the set of lines $Fil \subset M$ whose image spans $M/FM$.
I looked through groupes p-divisibles sur les corps locaux and could not find this result, nor could I find in Demazure-Gabriel why $G$ and $G'$ are isogenous lifts of $G_0$ iff $\omega_G$ and $\omega_{G'}$ define the same $k$-line in $M/FM$. (I assume that the set of lifts of $G_0$ in the statement above is taken up to isogeny, though I may be mistaken, there must be some equivalence relation!) I am desperate to have a reference for this result, as I'm perplexed attempting to rederive it.
