Purity Results on $F$-crystals

TL;DR

This paper proves a stronger form of purity for certain stratifications of schemes associated with $F$-crystals, extending previous results by Vasiu and Deligne to include fixed break points of Newton polygons.

Contribution

It establishes that for any fixed break point, the corresponding stratum in the scheme is pure, strengthening earlier purity results for Newton polygon and $p$-rank strata.

Findings

Proves the purity of fixed break point strata in schemes for $F$-crystals.

Extends previous purity results to a more general setting.

Strengthens understanding of the geometric structure of stratifications.

Abstract

For an $F^n$-crystal $\mathfrak C$ over a reduced locally Noetherian $\mathbb F_p$-scheme $S$, Vasiu first obtained the purity of the Newton polygon strata of $S$ defined by $\mathfrak C$. Deligne later obtained the purity of the $p$-rank strata of $S$ defined by $\mathfrak C$. We prove a stronger result that for every fixed point $P_0$ in the $xy$-coordinate plane, the reduced locally closed subscheme $S_{P_0}=\{s\in S| NP(\mathfrak C_s)\mbox{ has }P_0\mbox{ as a break point}\}$ of $S$ is pure in $S$ (A weaker result was known for the weak purity of $S_{p_0}$).