Let be a prime. Let . Let be an -crystal over a locally noetherian -scheme . Let . We show that the reduced locally closed subscheme of whose points are exactly those such that is a break point of the Newton polygon of the fiber of at is pure in , i.e., it is an affine -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 the reduced locally closed subscheme of whose points are exactly those for which the -rank of is is pure in ; the case was previously obtained by Deligne (unpublished) and the general case refines and reobtains a result of Zink.
"Purity of crystalline strata." Tunisian J. Math. 1 (4) 519 - 538, 2019. https://doi.org/10.2140/tunis.2019.1.519