Purity of Crystalline Strata

TL;DR

This paper proves that certain stratifications of schemes associated with F^n-crystals, based on Newton polygon break points and p-rank, are always pure, refining previous results in the theory of crystalline structures.

Contribution

It establishes the purity of Newton polygon break point loci and p-rank strata for F^n-crystals over schemes, generalizing and refining prior results by de Jong--Oort, Vasiu, Yang, and Zink.

Findings

The loci of points with a fixed break point of the Newton polygon are pure in the scheme.

The p-rank strata are pure in the scheme for all m in natural numbers.

The results generalize previous purity theorems for n=1 to arbitrary n.

Abstract

Let $p$ be a prime. Let $n\in\mathbb N-\{0\}$. Let $\mathcal C$ be an $F^n$-crystal over a locally noetherian $\mathbb F_p$-scheme $S$. Let $(a,b)\in\mathbb N^2$. We show that the reduced locally closed subscheme of $S$ whose points are exactly those $x\in S$ such that $(a,b)$ 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--Oort, Vasiu, and 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S$ for which the $p$-rank of $\mathcal C_x$ is $m$ is pure in $S$; the case $n=1$ was previously obtained by Deligne (unpublished) and the general case $n\ge 1$ refines and reobtains a result of Zink.