Crystalline boundedness principle

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