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On the slope filtration. (English) Zbl 1061.14045
From the introduction: If $$X$$ is a $$p$$-divisible group over a perfect field, the Dieudonné classification implies that $$X$$ is isogenous to a direct product of isoclinic $$p$$-divisible groups. The author studies what remains true if the perfect field is replaced by a ring $$R$$ such that $$pR= 0$$. To that extent, let $$X$$ be a $$p$$-divisible group over $$R$$. Denote by $$\text{Fr}_X\to X^{(p)}$$ the Frobenius homomorphism. We call $$X$$ isoclinic and slope divisible if there are natural numbers $$r\geq 0$$ and $$s>0$$ such that $$p^{-r}\text{Fr}^s_X:X\to X^{(p^s)}$$ is an isomorphism. The rational number $$r/s$$ is called the slope of $$X$$, and $$X$$ is isoclinic of slope $$r/s$$; that is, it is isoclinic of slope $$r/s$$ over each geometric point of $$\text{Spec}\,R$$. If $$R$$ is a field, a $$p$$-divisible group is isoclinic if and only if it is isogenous to a $$p$$-divisible group that is isoclinic and slope divisible.
It is stated in a letter of A. Grothendieck to I. Barsotti [“Groupes de Barsotti-Tate et cristaux de Dieudonné”. Sém. Math. Sup. 45 Montreal (1974; Zbl 0331.14021)] that over a field $$K=R$$ any $$p$$-divisible group admits a slope filtration $$(1)$$ $$0= X_0\subset X_1\subset X_2\subset\cdots\subset X_m=X$$. This filtration is uniquely determined by the following properties: the inclusions are strict, and the factors $$X_i/X_{i-1}$$ are isoclinic $$p$$-divisible groups of slope $$\lambda$$ such that $$1\geq\lambda_1>\cdots>\lambda_m\geq 0$$. Moreover, the rational numbers $$\lambda_i$$ are uniquely determined. A proof of this statement was never published but can be found in the paper under review.
The heights of the factors and the numbers $$\lambda_i$$ determine the Newton polygon, and conversely. For a slope filtration over $$R$$, it must be assumed that the Newton polygon is the same in any point of $$\text{Spec}\,R$$. One says in this case that $$X$$ has a constant Newton polygon. Using Dieudonné theory over a perfect field, the author proves the following:
Theorem. Let $$R$$ be a regular ring. Then any $$p$$-divisible group over $$R$$ with constant Newton polygon is isogenous to a $$p$$-divisible group $$X$$ which admits a strict filtration $$(1)$$ such that the quotients $$X_i/X_{i-1}$$ are isoclinic and slope divisible of slope $$\lambda_i$$ with $$1\geq\lambda_1>\cdots>\lambda_m\geq 0$$.
Let $$S$$ be a regular scheme, and let $$U$$ be an open subset such that the codimension of the complement is greater than or equal to 2. The author shows that a $$p$$-divisible group over $$U$$ with constant Newton polygon extends up to isogeny to a $$p$$-divisible group over $$S$$, which one might call the Nagata-Zariski purity for $$p$$-divisible groups.

##### MSC:
 14L05 Formal groups, $$p$$-divisible groups 14F30 $$p$$-adic cohomology, crystalline cohomology 14F20 Étale and other Grothendieck topologies and (co)homologies
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