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Zbl 1061.14045
Zink, Thomas
On the slope filtration. (English)
[J] Duke Math. J. 109, No. 1, 79-95 (2001). ISSN 0012-7094

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 $r\ge 0$ 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.\par It is stated in a letter of {\it 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 $1\ge\lambda_1>\cdots>\lambda_m\ge 0$. 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.\par 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:\par 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 $1\ge\lambda_1>\cdots>\lambda_m\ge 0$.\par 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 2000:: *14L05 Formal groups
14F30 p-adic cohomology
14F20 Grothendieck cohomology and topology

Citations: Zbl 0331.14021

Cited in: Zbl 1022.14013

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