Crystalline boundedness principle

Details Crystalline boundedness principle Crystalline boundedness principle

2002-05-17

arxiv.org/abs/math/0205199v5

Ann. Scient. \'Ec. Norm. Sup. 39 (2006), no. 2, pp. 245--300

Mathematics

Number Theory

Final version (63 pages) accepted for publication in Ann. Sci. Ec. Norm. Sup

Scientific paper

10.1016/j.ansens.2005.12.003

We prove that an $F$-crystal $(M,\vph)$ over an algebraically closed field $k$ of characteristic $p>0$ is determined by $(M,\vph)$ mod $p^n$, where $n\ge 1$ depends only on the rank of $M$ and on the greatest Hodge slope of $(M,\vph)$. We also extend this result to triples $(M,\vph,G)$, where $G$ is a flat, closed subgroup scheme of ${\bf GL}_M$ whose generic fibre is connected and has a Lie algebra normalized by $\vph$. We get two purity results. If ${\got C}$ is an $F$-crystal over a reduced ${\bf F}_p$-scheme $S$, then each stratum of the Newton polygon stratification of $S$ defined by ${\got C}$, is an affine $S$-scheme (a weaker result was known before for $S$ noetherian). The locally closed subscheme of the Mumford scheme ${\Ma_{d,1,N}}_k$ defined by the isomorphism class of a principally quasi-polarized $p$-divisible group over $k$ of height 2d, is an affine ${\Ma_{d,1,N}}_k$-scheme.

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