An improvement of de Jong-Oort's purity theorem (Q2883479)

This paper is of interest to those studying number theory and algebraic geometry. It studies \(F\)-crystals on a Noetherian scheme \(S\) of characteristic \(p >0\) and their associated Newton polygons. A crystal on \(S\) is a sheaf of \(\mathcal{O}\)-modules on the big crystalline site of \(S\) satisfying a certain condition analogous to quasi-coherence. An \(F\)-crystal on \(S\) is a crystal of finite locally free \(\mathcal{O}\)-modules on the big crystalline site of \(S\) equipped with a map \(F: \mathcal{E}^{(1)} \rightarrow \mathcal{E}\) of crystals.NEWLINENEWLINEIf \(S\) is a field and \(\sigma\) a lift of Frobenius to \(\Lambda\) (the Cohen ring of \(S\)) then an \(F\)-crystal on \(S\) is given by a triple of data \((\mathcal{E},\nabla,F)\) where \(\mathcal{E}\) is a finite free \(\Lambda\)-module, \(\nabla\) is an integrable topologically quasi-nilpotent connection and \(F\) is a horizontal \(\sigma\)-linear map \(F: \mathcal{E} \rightarrow \mathcal{E}\). After restricting to the generic point of the Witt ring of the algebraic closure of \(S\) and formally adding some rational exponents of \(p\), there is a basis \(\{e_1,\dots,e_n\}\) of \(\mathcal{E}\) and rational numbers \(\lambda_i\) such that \(F(e_i)=p^{\lambda_i}e_i\). The rational numbers \(\lambda_1,\dots,\lambda_n\) are called the Newton slopes and are the data used to construct the Newton polygon. The points on the Newton polygon where the slope changes are called break points. More generally, when \(S\) is a scheme, one defines the Newton polygons at each point \(s \in S\) by pulling back \(\mathcal{E}\) to the residue field at \(s\). Denote by \(NP(\mathcal{E})_s\) the Newton polygon at \(s \in S\).NEWLINENEWLINE\textit{A. J. de Jong} and \textit{F. Oort} [J. Am. Math. Soc. 13, No. 1, 209--241 (2000; Zbl 0954.14007)] proved if \(S\) is connected and the function \(s \mapsto NP(\mathcal{E})_s\) is constant on a subset whose complement is of codimension at least \(2\) then the function is a constant. The reviewed paper proves that this theorem extends to common break points for Newton polygons. Namely, if \(Q\) is a break point in \(NP(\mathcal{E})_s\) for all \(s\) in a subset whose complement is of codimension at least \(2\) then \(Q\) is a break point in \(NP(\mathcal{E})_s\) for all \(s \in S\).