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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.

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 |