Setting: Let $p$ be a prime number and let $S$ be a scheme such that $p$ is locally nilpotent on $\mathcal O_S$ ($p^N=0$). Let $X$ be a $p$-divisible group over $S$. Let $X[p^n] $ be the kernel of multiplication by $p^n$ on $X$. Then it is a finite locally free group scheme. Let $e$ denote the identity section.
Question: why do we have that $\omega_X:=e^*\Omega^1_{X[p^k]/ S} $ does not depend on $k$ for $k>>0$ ($k\geq N$)? The independence is stated in most sources on $p$-divisible groups but I can't think of an argument for this.
