De Jong-Oort Purity for p-Divisible Groups

Abstract

De Jong-Oort purity states that for a family of p-divisible groups X → S over a noetherian scheme S the geometric fibres have all the same Newton polygon if this is true outside a set of codimension bigger than 2. A more general result was first proved in [JO] and an alternative proof is given in [V1]. We present here a short proof which is based on the fact that a formal p-divisible group may be defined by a display ([Z1], [Me2]). There are two other ingredients of the proof which are known for a long time. One is the boundedness principal for crystals over an algebraically closed field ([O], [V1], [V2]) and the other is the existence of a slope filtration for a p-divisible group over a non-perfect field ([Z2]). The last fact was already mentioned in a letter of Grothendieck to Barsotti [G]. The boundedness property is also an important ingredient in the proof given by Vasiu in [V1]. We discuss in detail some elementary consequences of the display structure. The other two ingredients can be found in the literature above. Therefore we discuss them only briefly. I thank B.Messing for pointing out the correct formulation of Proposition 3 below.