Abstract

Let $p$ be a prime. Let $n\in {\mathbb{N}}^{\ast }$. Let $\mathcal{C}$ be an $F^n$-crystal over a locally noetherian ${\mathbb{F}}_p$-scheme $S$. Let $\left(a,b\right)\in {\mathbb{N}}^2$. We show that the reduced locally closed subscheme of $S$ whose points are exactly those $x\in S$ such that $\left(a,b\right)$ is a break point of the Newton polygon of the fiber ${\mathcal{C}}_x$ of $\mathcal{C}$ at $x$ is pure in $S$, i.e., it is an affine $S$-scheme. This result refines and reobtains previous results of de Jong and Oort, of Vasiu, and of Yang. As an application, we show that for all $m\in \mathbb{N}$ the reduced locally closed subscheme of $S$ whose points are exactly those $x\in S$ for which the $p$-rank of ${\mathcal{C}}_x$ is $m$ is pure in $S$; the case $n=1$ was previously obtained by Deligne (unpublished) and the general case $n\ge 1$ refines and reobtains a result of Zink.

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Mathematical Subject Classification 2010

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