Let $k$ be a perfect field of characteristic $p>0$, and consider $X=(X_m,i_m)$ a $p$-divisible group of height $h$ over $\operatorname{Spec}(k)$: it is an inductive system where $X_m$ is a finite group scheme over $k$ of order $p^{mh}$, such that $X_m$ is identified via $i_m:X_m\rightarrow X_{m+1}$ with the $p^m$-torsion of $X_{m+1}$.
The « classical » Dieudonné module of $X$ is defined as the inverse limit $\mathbb D(X):=\varprojlim\mathbb D(X_m)$, where $\mathbb D(X_m)$ is the contravariant Dieudonné module of $X_m$ and the transition maps are induced by the $i_m$.
I want to understand $\mathbb D(^tX)$, where $^tX$ is the Serre dual of $X$. This is the $p$-divisible group induced by Cartier duality applied to each $X_m$, with $^tX_m\rightarrow\,^tX_{m+1}$ being dual to $p:X_{m+1}\rightarrow X_m$.
It would be nice if $\mathbb D(^tX)$ were related to $\mathbb D(X)$ under some sort of duality property.
I know from Chai-Conrad-Oort's book – theorem 1.4.1.1 (5) – that $\mathbb D(^tX_m)\cong \operatorname{Hom}_{W}(\mathbb D(X_m),W[1/p]/W)$ where $W=W(k)$ is the ring of Witt vectors over $k$. Now, $\operatorname{Hom}$ transforms the inverse limit into a direct limit, so that
$$\mathbb D(^tX)=\varprojlim\mathbb D(^tX_m)=\varprojlim\operatorname{Hom}_{W}(\mathbb D(X_m),W[1/p]/W)=\operatorname{Hom}_W(\varinjlim \mathbb D(X_m),W[1/p]/W)$$
Unfortunately, it's the direct limit of the $\mathbb D(X_m)$ with transition maps induced by $p:X_{m+1}\rightarrow X_m$ which appears. Is there any way to relate this direct limit with $\mathbb D(X)$ ?
Motivation: I have been thinking about the above discussion because I wish to understand why a polarization $\lambda:X\rightarrow\, ^tX$ should induce a (non-degenerate skew-symmetric) bilinear pairing $\mathbb D_{cov}(X)\times \mathbb D_{cov}(X)\rightarrow W$, where $\mathbb D_{cov}(X):=\mathbb D(^tX)$ is the covariant Dieudonné module.
Best Answer
This is not answer maybe, but rather a longer comment, hoping that someone can confirm or correct this. I will address your motivation and not your actual question.
My idea is that rather than using the morphism $\lambda:X\to X^t$, which level-wise are of the form $$ \lambda_n :X_n\to (X^t)_n=(X_n)^t$$ we can use Cartier duality, which says that such a morphism $\lambda_n$ is equivalent to a bilinear pairing $$ X_n\times X_n\to \mathbb{G}_m.$$ Since $X_n$ is $p^n$-torsion, the bilinear pairing factors through $\mu_{p^n}$ $$ b_n: X_n\times X_n\to \mu_{p^n}.$$ Now applying the Dieudonne functor should yield a bilinear morphism $$ D(b_n): D(X_n)\times D(X_n)\to D(\mu_{p^n})=W_n.$$ (Note that while $b_n$ is not a group homomorphism, so that it is not possible to literally apply the functor $D$, nonetheless the functor $D$ can be extended to all scheme morphisms.) Now take the inverse limit over all these maps and you get $$ D(b):D(X)\times D(X)\to W. $$ This is almost what you wanted. To get the covariant version, maybe
Maybe these two should yield the same thing, but I can't prove it right now.