{"title":"Purity of crystalline strata","authors":"Jinghao Li, A. Vasiu","doi":"10.2140/tunis.2019.1.519","DOIUrl":null,"url":null,"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S$ for which the $p$-rank of $\\mathcal C_x$ is $m$ is pure in $S$; the case $n=1$ was previously obtained by Deligne (unpublished) and the general case $n\\ge 1$ refines and reobtains a result of Zink.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2018-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/tunis.2019.1.519","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Tunisian Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2140/tunis.2019.1.519","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
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S$ for which the $p$-rank of $\mathcal C_x$ is $m$ is pure in $S$; the case $n=1$ was previously obtained by Deligne (unpublished) and the general case $n\ge 1$ refines and reobtains a result of Zink.