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Abstract
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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was previously obtained by Deligne (unpublished) and the general case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.
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Keywords
$\mathbb F_p$-scheme, $F$-crystal, Newton polygon,
$p$-rank, purity
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Mathematical Subject Classification 2010
Primary: 11G18, 14F30, 14L05
Secondary: 11G10, 14G35, 14K10, 14K99, 14L15
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Milestones
Received: 28 January 2018
Revised: 8 August 2018
Accepted: 5 September 2018
Published: 14 December 2018
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