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

Let p be a prime. Let n . Let C be an Fn-crystal over a locally noetherian Fp-scheme S. Let (a,b) 2. We show that the reduced locally closed subscheme of S whose points are exactly those x S such that (a,b) is a break point of the Newton polygon of the fiber Cx of 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 the reduced locally closed subscheme of S whose points are exactly those x S for which the p-rank of Cx is m is pure in S; the case n = 1 was previously obtained by Deligne (unpublished) and the general case n 1 refines and reobtains a result of Zink.

$\mathbb F_p$-scheme, $F$-crystal, Newton polygon, $p$-rank, purity
Mathematical Subject Classification 2010
Primary: 11G18, 14F30, 14L05
Secondary: 11G10, 14G35, 14K10, 14K99, 14L15
Received: 28 January 2018
Revised: 8 August 2018
Accepted: 5 September 2018
Published: 14 December 2018
Jinghao Li
Sequoia Capital Global Equities
Menlo Park, CA
United States
Adrian Vasiu
Department of Mathematical Sciences
Binghamton University
Binghamton, NY
United States