#### Vol. 1, No. 4, 2019

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Purity of crystalline strata

### Jinghao Li and Adrian Vasiu

Vol. 1 (2019), No. 4, 519–538
##### Abstract

Let $p$ be a prime. Let $n\in {ℕ}^{\ast }$. Let $\mathsc{C}$ be an ${F}^{n}$-crystal over a locally noetherian ${\mathbb{F}}_{p}$-scheme $S$. Let $\left(a,b\right)\in {ℕ}^{2}$. 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 ${\mathsc{C}}_{x}$ of $\mathsc{C}$ 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 $m\in ℕ$ the reduced locally closed subscheme of $S$ whose points are exactly those $x\in S$ for which the $p$-rank of ${\mathsc{C}}_{x}$ is $m$ is pure in $S$; the case $n=1$ was previously obtained by Deligne (unpublished) and the general case $n\ge 1$ refines and reobtains a result of Zink.

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$\mathbb F_p$-scheme, $F$-crystal, Newton polygon, $p$-rank, purity