Duality, refined partial Hasse invariants and the canonical filtration (Q1720111)

The author defines refined partial Hasse invariants for a $p$-divisible group over an arbitrary scheme of characteristic $p$ under a certain condition on sheaves of direct summand of the Dieudonné crystal of $G$. More specifically, let $k$ be an algebraically closed field of characteristic $p$, let $S$ be a $k$-scheme and let $G$ be a $p$-divisible group over $S$. Assume that $G$ has an action of $O_F$, and fix the type of this type of action. Let $\mathcal{E}$ be the Dieudonné crystal of $G$ evaluated at $S$. Recall that $\mathcal{E}$ is a $k$-vector space of dimension the height of $G$ and $\mathcal{E} = \bigoplus_{i=1}^f \mathcal{E}_i$. \par Theorem. Assume the existence of adequate filtrations on the sheaves $\mathcal{E}_i$. Then there exist sections $h_j^{[i]}(G)$ of invertible sheaves for $1 \leq j \leq f$ and $1 \leq i \leq f$, such that \[ Ha_i(G) = h_i^{[i]} \cdot (h_{i-1}^{[i]})^p \cdot \cdots \cdot (h_{i+1}^{[i]})^{p^{f-1}}. \] Moreover, the author shows that the construction of these refined Hasse invariants is compatible with duality and that these invariants can be used to compute the partial degrees of the canonical filtration if it exists.