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Purity of the stratification by Newton polygons. (English) Zbl 0954.14007
Let \(S\) be a locally noetherian scheme of characteristic \(p\) and \(\mathcal E\) be a non-degenerate \(F\)-crystal of rank \(r\) over \(S\). There is a stratification of \(S\) by reduced and locally closed subschemes \(\{U_\beta\}_{\beta\in{\mathcal B}}\), indexed by the set \(\mathcal B\) of the Newton polygons on \([0,r]\), with the property that a point \(s\) of \(S\) lies in the stratum \(U_\beta\) if the Newton polygon of \(\mathcal E\) at \(s\) is exactly \(\beta\). Let \(\overline{U}_\beta\) be the closure of a stratum and \(\eta\) a generic point of \(\overline{U}_\beta\setminus U_\beta\), the authors prove that \(\dim {\mathcal O}_{\overline{U}_\beta, \eta}=1\). If \(S=\text{Spec } A\) is the spectrum of a complete local noetherian ring with algebraically closed residue field, they prove that an isoclinic (all slopes of the Newton polygon are equal) \(F\)-crystal over \(S\) is isogenous to a constant \(F\)-crystal. This result implies an analogous isogeny theorem for \(p\)-divisible groups (cf. theorem 2.17). These results are used to describe all deformations of simple \(p\)-divisible groups which do not change the Newton polygon. The methods used in the proofs of the above results are also applied to prove a result about resolution of singularities of surfaces in characteristic \(p\). Precisely, the authors prove the following fact.
Let \(A\) be a local complete Noetherian ring, normal of dimension \(2\), with algebraically closed residue field \(k\) of characteristic \(p\) and with \(k\subset A\). Let \(S=\text{Spec } A\) and denote by \(0\) its closed point. Having chosen a resolution of singularities \(\pi:\widetilde S\to S\) [cf. J. Lipman, Ann. Math., II. Ser. 107, 151-207 (1978; Zbl 0349.14004)] and identifying \(U=S\setminus\{0\}\) with \(\pi^{-1}(U)\), then the natural map \(H^1_{\text{ét}}(\widetilde S,{\mathbb{Q}}_p)\to H^1_{\text{ét}}(U,{\mathbb{Q}}_p)\) is an isomorphism. For a more detailed description of the results and their consequences, we refer to the introduction of the paper.

MSC:
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14F30 \(p\)-adic cohomology, crystalline cohomology
14L05 Formal groups, \(p\)-divisible groups
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14G15 Finite ground fields in algebraic geometry
14B05 Singularities in algebraic geometry
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