Families of p-Divisible Groups

with Constant Newton Polygon

Frans Oort and Thomas Zink

Received: May 2, 2002 Revised: September 19, 2002

Communicated by Peter Schneider

Abstract. Let X be a p-divisible group with constant Newton polygon over a normal Noetherian scheme S. We prove that there exists an isogeny X → Y such that Y admits a slope filtration. In case S is regular this was proved by N. Katz for dim S = 1 and by T. Zink for dim S ≥ 1.

2000 Mathematics Subject Classification: 14L05, 14F30

Introduction

In this paper we work over base fields, and over base schemes over 𝔽_{p}, i.e. we work entirely in characteristic p. We study p-divisible groups X over a base scheme S (and, colloquially, a p-divisible group over a base scheme of positive dimension will be called a “family of p-divisible groups”), such that the Newton polygon of a fiber X_{s} is independent of s ∈ S. We call X a p-divisible group with constant Newton polygon. A p-divisible group over a field has a slope filtration, see [Z1], Corollary 13; for the definition of a slope filtration, see Definition 1.1. Over a base of positive dimension a slope filtration can only exist if the Newton polygon is constant. In Example 4.1 we show that even in this case there are p-divisible groups which do not admit a slope filtration. The main result of this paper has as a corollary that for a p-divisible group with constant Newton polygon over a normal base up to isogeny a slope filtration does exist, see Corollary 2.2. We have access to this kind of questions by the definition of a completely slope divisible p-divisible group, see Definition 1.2, which implies a structure finer than a slope filtration. The main theorem of this paper, Theorem 2.1, says that over a normal base this structure on a p-divisible group exists up to isogeny.

In [Z1], Theorem 7, this was shown to be true over a regular base. In 4.2 we show that without the condition “normal” the conclusion of the theorem does not hold. Here is a motivation for this kind definition and of results:

• A p-divisible group over an algebraically closed field is isogeneous with a p-divisible group which can be defined over a finite field.

• A p-divisible group over an algebraically closed field is completely slope divisible, if and only if it is isomorphic with a direct sum of isoclinic p-divisible groups which can be defined over a finite field, see 1.5.

We see that a completely slope divisible p-divisible group comes “as close as possible” to a constant one, in fact up to extensions of p-divisible groups annihilated by an inseparable extension of the base, and up to monodromy. From Theorem 2.1 we deduce constancy results which generalize results of Katz [K] and more recently of de Jong and Oort [JO]. In particular we prove, Corollary 3.4 below: Let R be a Henselian local ring with residue field k. Let h be a natural number. Then there exists a constant c with the following property. Let X and Y be isoclinic p-divisible groups over S = Spec R whose heights are smaller than h. Let ψ : X_{k} → Y_{k} be a homomorphism. Then p^{c}ψ lifts to a homomorphism X → Y.

1 Completely slope divisible p-divisible groups

In this section we present basic definitions and methods already used in the introduction. Let S be a scheme over 𝔽_{p}. Let Frob : S → S be the absolute Frobenius morphism. For a scheme G/S we write:

G^{(p)} = G ×_{S,Frob} S.

We denote by Fr = Fr_{G} : G → G^{(p)} the Frobenius morphism relative to S. If G is a finite locally free commutative group scheme we write Ver = Ver_{G} : G^{(p)} → G for the “Verschiebung”. Let X be a p-divisible group over S. We denote by X(n) the kernel of the multiplication by p^{n} : X → X. This is a finite, locally free group scheme which has rank p^{nh} if X is of height h. Let s = Spec k the spectrum of a field of characteristic p. Let X be a p-divisible group over s. Let λ ≥ 0 be a rational number. We call X isoclinic of slope λ, if there exists integers r ≥ 0, s > 0 such that λ = r/s, and a p-divisible group Y over S, which is isogeneous to X such that

p^{−r} Fr^{s} : Y → Y^{(p^{s})}

p-Divisible Groups 185

is an isomorphism. A p-divisible group X over S is called isoclinic of slope λ, if for each point s ∈ S the group X_{s} is isoclinic of slope λ.

1.1 Definition. Let X/S be a p-divisible group over a scheme S. A filtration

0 = X_{0} ⊂ X_{1} ⊂ X_{2} ⊂ … ⊂ X_{m} = X

consisting of p-divisible groups contained in X is called a slope filtration of X if there exists rational numbers λ_{1}, …, λ_{m} satisfying 1 ≥ λ_{1} > … > λ_{m} ≥ 0 such that every subquotient X_{i}/X_{i−1}, 1 < i ≤ m, is isoclinic of slope λ_{i}.

A p-divisible group X over a field admits a slope filtration, see [Z1], Corollary 13. The slopes λ_{i} and the heights of X_{i}/X_{i−1} depend only on X. The height of X_{i}/X_{i−1} is called the multiplicity of λ_{i}. Over connected base scheme S of positive dimension a slope filtration of X can only exist if the slopes of X_{s} and their multiplicities are independent of s ∈ S. In this case we say that X is a family of p-divisible groups with constant Newton polygon. Even if the Newton polygon is constant a slope filtration in general does not exist, see Example 4.1 below.

1.2 Definition. Let s > 0 and r_{1}, …, r_{m} be integers such that s ≥ r_{1} > r_{2} > … > r_{m} ≥ 0. A p-divisible group Y over a scheme S is said to be completely slope divisible with respect to these integers if Y has a filtration by p-divisible subgroups:

0 = Y_{0} ⊂ Y_{1} ⊂ … ⊂ Y_{m} = Y

such that the following properties hold:

• The quasi-isogenies

p^{−r_{i}} Fr^{s} : Y_{i} → Y_{i}^{(p^{s})}

are isogenies for i = 1, …, m.

• The induced morphisms:

p^{−r_{i}} Fr^{s} : Y_{i}/Y_{i−1} → (Y_{i}/Y_{i−1})^{(p^{s})}

are isomorphisms.

Note that the last condition implies that Y_{i}/Y_{i−1} is isoclinic of slope λ_{i} := r_{i}/s. A filtration described in this definition is a slope filtration in the sense of the previous definition.

Remark. Note that we do not require s and r_{i} to be relatively prime. If Y is as in the definition, and t ∈ ℤ_{>0}, it is also completely slope divisible with respect to t·s ≥ t·r_{1} > t·r_{2} > … > t·r_{m} ≥ 0.

We note that the filtration Y_{i} of Y is uniquely determined, if it exists. Indeed, consider the isogeny Φ = p^{−r_{m}} Fr^{s} : Y → Y^{(p^{s})}. Then Y_{m}/Y_{m−1} is necessarily the Φ-étale part Y^{Φ} of Y, see [Z1] respectively 1.6 below. This proves the uniqueness by induction. We will say that a p-divisible group is completely slope divisible if it is completely slope divisible with respect to some set integers and inequalities s ≥ r_{1} > r_{2} > … > r_{m} ≥ 0.

Remark. A p-divisible group Y over a field K is completely slope divisible iff Y ⊗_{K} L is completely slope divisible for some field L ⊃ K. - Proof. The slope filtration on Y/K exists. We have Ker(p_{Y}^{r}) ⊂ Ker(Fr_{Y}^{s}) iff Ker(p_{Y_{L}}^{r}) ⊂ Ker(Fr_{Y_{L}}^{s}), and the same for equalities. This proves that the conditions in the definition for completely slope divisibility hold over K iff they hold over L ⊃ K.

1.3 Proposition. Let Y be a completely slope divisible p-divisible group over a perfect scheme S. Then Y is isomorphic to a direct sum of isoclinic and completely slope divisible p-divisible groups.

Proof. With the notation of Definition 1.2 we set Φ = p^{−r_{m}} Fr^{s}. Let Y (n) = Spec A(n) and let Y (n)^{Φ} = Spec L(n) be the Φ-étale part (see Corollary 1.7 below). Then Φ induces a Frob^{s}-linear endomorphism Φ^{∗} of A(n) and L(n) which is by definition bijective on L(n) and nilpotent on the quotient A(n)/L(n). One verifies (compare [Z1], page 84) that there is a unique Φ^{∗}- equivariant section of the inclusion L(n) → A(n). This shows that Y_{m}/Y_{m−1} is a direct factor of Y = Y_{m}. The result follows by induction on m. Q.E.D.

Although not needed, we give a characterization of completely slope divisible p-divisible groups over a field. If X is a p-divisible group over a field K, and k ⊃ K is an algebraic closure, then X is completely slope divisible if and only if X_{k} is completely slope divisible; hence it suffices to give a characterization over an algebraically closed field. Convention: We will work with the covariant Dieudonné module of a p-divisible group over a perfect field ([Z2], [Me]). We write V, respectively F for Verschiebung, respectively Frobenius on Dieudonné modules. Let K be a perfect field, and let W (K) be its ring of Witt vectors. A Dieudonné module M over K is the Dieudonné module of an isoclinic p-divisible group of slope r/s, iff there exists a W (K)-submodule M′ ⊂ M such that M/M′ is annihilated by a power of p, and such that p^{−r}V^{s}(M′) = M′. For later use we introduce the p-divisible group G_{m,n} for coprime positive integers m and n. Its Dieudonné module is generated by one element, which is stable under F^{m} − V^{n}. G_{m,n} is isoclinic of slope λ = m/(m + n) (in the terminology of this paper). The height of G_{m,n} is h = m + n, and this pdivisible group is completely slope divisible with respect to h = m + n ≥ m. The group G_{m,n} has dimension m, and its Serre dual has dimension n. We have G_{1,0} = μ_{p^{∞}}, and G_{0,1} = \underline{ℚ_{p}/ℤ_{p}}.

1.4 Proposition. Let k be an algebraically closed field. An isoclinic pdivisible group Y over k is completely slope divisible iff it can be defined over a finite field; i.e. iff there exists a p-divisible group Y′ over some 𝔽_{q} and an isomorphism Y ≅ Y′ ⊗_{𝔽_{q}} k. Proof. Assume that Y is slope divisible with respect to s ≥ r ≥ 0. Let M be the covariant Dieudonné module of Y. We set Φ = p^{−r}V^{s}. By assumption this is a semilinear automorphism of M. By a theorem of Dieudonné (see 1.6 below) M has a basis of Φ-invariant vectors. Hence M = M_{0} ⊗_{W (𝔽_{p^{s}})} W (k) where M_{0} ⊂ M is the subgroup of Φ-invariant vectors. Then M_{0} is the Dieudonné module of a p-divisible group over 𝔽_{p^{s}} such that Y ≅ Y′ ⊗_{𝔽_{q}} k. Conversely assume that Y is isoclinic over a finite field k of slope r/s. Let M be the Dieudonné module of Y. By definition there is a finitely generated W (k)- submodule M′ ⊂ M ⊗ ℚ such that p^{m}M′ ⊂ M ⊂ M′ for some natural number m, and such that p^{−r}V^{s}(M′) = M′. Then Φ = p^{−r}V^{s} is an automorphism of the finite set M′/p^{m}M′. Hence some power Φ^{t} acts trivially on this set. This implies that Φ^{t}(M) = M. We obtain that p^{−rt}F^{st} induces an automorphism of Y. Therefore Y is completely slope divisible. Q.E.D.

1.5 Corollary. Let Y be a p-divisible group over an algebraically closed field k. This p-divisible group is completely slope divisible iff Y ≅ ⊕Y_{i} such that every Y_{i} is isoclinic, and can be defined over a finite field.

Proof. Indeed this follows from 1.3 and 1.4. Q.E.D.

1.6 The Φ-étale part.

For further use, we recall a notion explained and used in [Z1], Section 2. This method goes back to Hasse and Witt, see [HW], and to Dieudonné, see [D], Proposition 5 on page 233. It can be formulated and proved for locally free sheaves, and it has a corollary for finite flat group schemes. Let V be a finite dimensional vector space over a separably closed field k of characteristic p. Let f : V → V be a Frob^{s}-linear endomorphism. The set C_{V} = {x ∈ V | f(x) = x} is a vector space over 𝔽_{p^{s}}. Then V^{f} = C_{V} ⊗_{𝔽_{p^{s}}} k is a subspace of V. The endomorphism f acts as a Frob^{s}-linear automorphism on V^{f} and acts nilpotently on the quotient V/V^{f}. This follows essentially form Dieudonné loc.cit.. Moreover if k is any field of characteristic p we have still unique exact sequence of k-vector spaces

0 → V^{f} → V → V/V^{f} → 0

such that f acts as a a Frob^{s}-linear automorphism on V^{f} and acts nilpotently on the quotient V/V^{f}. This can be applied in the following situation: Let S be a scheme over 𝔽_{p}. Let G be a locally free group scheme over S endowed with a homomorphism

Φ : G → G^{(p^{s})}.

In case S = Spec(K), where K is a field we consider the affine algebra A of G. The Φ induces a Frob^{s}-linear endomorphism f : A → A. The vector subspace A^{f} inherits the structure of a bigebra. We obtain a finite group scheme G^{Φ} = Spec A^{f}, which is called the Φ-étale part of G. Moreover we have an exact sequence of group schemes:

0 → G^{Φ−nil} ⟶ G ⟶ G^{Φ} → 0.

The morphism Φ induces an isomorphism G^{Φ} → (G^{Φ})^{(p^{s})}, and acts nilpotently on the kernel G^{Φ−nil}. Let now S be an arbitrary scheme over 𝔽_{p}. Then we can expect a Φ-étale part only in the case where the rank of G_{x}^{Φ} is independent of x ∈ S:

1.7 Corollary. Let G → S be a finite, locally free group scheme; let Φ : G → G^{(p^{t})} be a homomorphism. Assume that the function

S → ℤ, defined by x ↦ rank((G_{x})^{Φ_{x}})

is constant. Then there exists an exact sequence

0 → G^{Φ−nil} ⟶ G ⟶ G^{Φ} → 0

such that Φ is nilpotent on G^{Φ−nil} and an isomorphism on the Φ-étale part G^{Φ}.

The prove is based on another proposition which we use in section 3. Let M be a finitely generated, locally free O_{S}-module. Let t ∈ ℤ_{>0}

f : O_{S} ⊗_{Frob_{S}^{t},S} M = M^{(p^{t})} → M

be a morphism of O_{S}-modules. To every morphism T → S we associate

C_{M}(T) = {x ∈ Γ(T, M_{T}) | f(1 ⊗ x) = x}.

1.8 Proposition (see [Z1], Proposition 3). The functor C_{M} is represented by a scheme that is étale and affine over S. Suppose S to be connected; the scheme C_{M} is finite over S iff for each geometric point η → S the cardinality of C_{M}(η) is the same.

Let X be a p-divisible group over a field K. Suppose Φ : X → X^{(p^{t})} is a homomorphism. Then the Φ-étale part X^{Φ} is the inductive limit of X(n)^{Φ}. This is a p-divisible group.

1.9 Corollary. Let X be a p-divisible group over S. Assume that for each geometric point η → S the height of the Φ-étale part of X_{η} is the same. Then a p-divisible group X^{Φ} exists and commutes with arbitrary base change. There is an exact sequence of p-divisible groups:

0 → X^{Φ−nil} → X → X^{Φ} → 0.

The following proposition can be deduced from proposition 1.8.

1.10 Corollary. Assume that G → S is a finite, locally free group scheme over a connected base scheme S. Let Φ : G ⟶^{∼} G^{(q)}, q = p^{s} be an isomorphism. Then there exists a finite étale morphism T → S, and a morphism T → Spec(𝔽_{q}), such that G_{T} is obtained by base change from a finite group scheme H over 𝔽_{q}:

H ⊗_{Spec 𝔽_{q}} T ⟶^{∼} G_{T} .

Moreover Φ is induced from the identity on H.

Remark. If S is a scheme over 𝔽̄_{p} the Corollary says in particular that G_{T} is obtained by base change from a finite group scheme over 𝔽̄_{p}. In this case we call G_{T} constant (compare [K], (2.7)). This should not be confused with the étale group scheme associated to a finite Abelian group A. We will discuss “constant” p-divisible groups, see Section 3 below.

2 The main result: slope filtrations

In this section we show:

2.1 Theorem. Let h be a natural number. Then there exists a natural number N(h) with the following property. Let S be an integral, normal Noetherian scheme. Let X be a p-divisible group over S of height h with constant Newton polygon. Then there is a completely slope divisible p-divisible group Y over S, and an isogeny:

φ : X → Y over S with deg(φ) ≤ N(h).

In Example 4.2 we see that the condition “S is normal” is essential. By this theorem we see that a slope filtration exists up to isogeny:

2.2 Corollary. Let X be a p-divisible group with constant Newton polygon over an integral, normal Noetherian scheme S. There exists an isogeny φ : X → Y, such that Y over S admits a slope filtration.

Q.E.D.

2.3 Proposition. Let S be an integral scheme with function field K = κ(S). Let X be a p-divisible group over S with constant Newton polygon, such that X_{K} is completely slope divisible with respect to the integers s ≥ r_{1} > r_{2} > … > r_{m} ≥ 0. Then X is completely slope divisible with respect to the same integers.

Proof. The quasi-isogeny Φ = p^{−r_{m}} Fr^{s} : X → X^{(p^{s})} is an isogeny, because this is true over the general point. Over any geometric point η → S the Φ-étale part of X_{η} has the same height by constancy of the Newton polygon. Hence the Φ-étale part of X exists by Corollary 1.9. We obtain an exact sequence:

0 → X^{Φ−nil} → X → X^{Φ} → 0.

Assuming an induction hypothesis on X^{Φ−nil} gives the result. Q.E.D.

A basic tool in the following proofs is the moduli scheme of isogenies of degree d of a p-divisible group (compare [RZ] 2.22): Let X be a p-divisible group over a scheme S, and let d be a natural number. Then we define the following functor M on the category of S-schemes T. A point of M(T) consists of a p-divisible group Z over T and an isogeny α : X_{T} → Z of degree d up to isomorphism. The functor M is representable by a projective scheme over S. Indeed, to each finite, locally free subgroup scheme G ⊂ X_{T} there is a unique isogeny α with kernel G. Let n be a natural number such that p^{n} ≥ d. Then G is a finite, locally free subgroup scheme on X(n)_{T}. We set X(n) = Spec_{S} A. The affine algebra of G is a quotient of the locally free sheaf A_{T}. Hence we obtain a point of the Grassmannian of A. This proves that M is representable as a closed subscheme of this Grassmannian.

2.4 Lemma. For every h ∈ ℤ_{>0} there exists a number N(h) ∈ ℤ with the following property. Let S be an integral Noetherian scheme. Let X be a pdivisible group of height h over S with constant Newton polygon. There is a non-empty open subset U ⊂ S, and a projective morphism π : S^{∼} → S of integral schemes which induces an isomorphism π : π^{−1}(U) → U such that there exist a completely slope divisible p-divisible group Y over S^{∼}, and an isogeny X_{S^{∼}} → Y, whose degree is bounded by N(h).

Proof. Let K be the function field of S. We know by [Z1], Prop. 12, that there is a completely slope divisible p-divisible group Y^{0} over K, and an isogeny β^{0} : X_{K} → Y^{0}, whose degree is bounded by a constant which depends only on the height of X. The kernel of this isogeny is a finite group scheme G^{0} ⊂ X_{K}(n), for some n. Let Ḡ be the scheme-theoretic image of G^{0} in X(n), see EGA, I.9.5.3. Then Ḡ is flat over some nonempty open set U ⊂ S, and inherits there the structure of a finite, locally free group scheme G ⊂ X(n)_{U}. We form the p-divisible group Z = X_{U}/G. By construction there are integers s ≥ r_{m}, such that

p^{−r_{m}} Fr^{s} : Z_{K} → Z_{K}^{(p^{s})}

is an isogeny, and r_{m}/s is a smallest slope in the Newton polygon of X. Therefore Φ = p^{−r_{m}} Fr^{s} : Z → Z^{(p^{s})} is an isogeny too. As in the proof of the last proposition the constancy of the Newton polygon implies that the Φ-étale part Z^{Φ} exists. We obtain an exact sequence of p-divisible groups on U:

0 → Z^{Φ−nil} → Z → Z^{Φ} → 0.

By induction we find a non-empty open subset V ⊂ U and a completely slope divisible p-divisible group Y_{m−1} which is isogeneous to Z_{V}^{Φ−nil}. Taking the push-out of the last exact sequence by the isogeny Z_{V}^{Φ−nil} → Y_{m−1} we find a completely slope divisible p-divisible group Y over V which is isogeneous to X_{V}. Let d be the degree of the isogeny ρ : X_{V} → Y. We consider the moduli scheme M of isogenies of degree d of X defined above. The isogeny X_{V} → Y defines an S-morphism V → M. The scheme-theoretic image S^{∼} of V is an integral scheme, which is projective over S. Moreover the morphism π : S^{∼} → S induces an isomorphism π^{−1}(V) → V. The closed immersion S^{∼} → M corresponds to an isogeny ρ^{∼} : X_{S^{∼}} → Y^{∼} to a p-divisible group Y^{∼} on S^{∼}. Moreover the restriction of ρ^{∼} to V is ρ. Since Y^{∼} has constant Newton polygon, and since Y^{∼} is completely slope divisible in the generic point of S^{∼} it is completely slope divisible by Proposition 2.3. Q.E.D.

2.5 Lemma. Let k be an algebraically closed field of characteristic p. Let s ≥ r_{1} > r_{2} > … > r_{m} ≥ 0 and d > 0 be integers. Let X be a p-divisible group over k. Then there are up to isomorphism only finitely many isogenies X → Z of degree d to a p-divisible group Z, which is completely slope divisible with respect to s ≥ r_{1} > r_{2} > … > r_{m} ≥ 0.

Proof. It suffices to show this in case also X is completely slope divisible with respect to s ≥ r_{1} > r_{2} > … > r_{m} ≥ 0. Then X and Z are a direct product of isoclinic slope divisible groups. Therefore we assume that we are in the isoclinic case m = 1. In this case we consider the contravariant Dieudonné modules M of X, and N of Z. Let σ be the Frobenius on W (k). Then N ⊂ M is a submodule such that length M/N = log_{p} d. By assumption Φ = p^{−r_{1}}F^{s} induces a σ^{s}-linear automorphism of M respectively N. Let C_{N} respectively C_{M} be the invariants of Φ acting of N respectively M. Hence C_{N} is a W (𝔽_{p^{s}})-submodule of N such that W (k)⊗_{W (𝔽_{p^{s}})}C_{N} = N (e.g. [Z2] 6.26). The same holds for M. We see that C_{N} is a W (𝔽_{p^{s}})-submodule of C_{M}, such that length C_{M}/C_{N} = log_{p} d. Since there are only finitely many such submodules, the assertion follows. Q.E.D.

2.6 Lemma. Let f : T → S be a proper morphism of schemes such that f_{∗}O_{T} = O_{S}. Let g : T → M be a morphism of schemes. We assume that for any point ξ ∈ S the set-theoretic image of the fiber T_{ξ} by g is a single point in M. Then there is a unique morphism h : S → M such that hf = g.

Proof. For ξ ∈ S we set h(ξ) = f(T_{ξ}). This defines a set-theoretic map h : S → M. If U ⊂ M is an open neighborhood of h(ξ) then g^{−1}(U) is an open neighborhood of T_{ξ}. Since f is closed we find an open neighborhood V of ξ with f^{−1}(V) ⊂ g^{−1}(U). Hence h(V) ⊂ U; we see that h is continuous. Then h_{∗}O_{S} = h_{∗}f_{∗}O_{T} = g_{∗}O_{T}. We obtain a morphism of ringed spaces O_{M} → g_{∗}O_{T} = h_{∗}O_{S}. Q.E.D.

Theorem 2.1 follows from the following technical variant which is useful if we do not know that the normalization is finite. We will need that later on.

2.7 Proposition. Let h be a natural number. Then there exists a natural number N(h) with the following property. Let S be an integral Noetherian scheme. Let X be a p-divisible group over S of height h with constant Newton polygon. Then there is a finite birational morphism T → S, a completely slope divisible p-divisible group Y over T, and an isogeny:

X_{T} → Y

over T whose degree is smaller than N(h).

Proof. Consider the proper birational map π : S^{∼} → S, and the isogeny ρ : X_{S^{∼}} → Y given by Lemma 2.4. Take the Stein factorization S^{∼} → T → S. It is enough to find over T an isogeny to a completely slope divisible p-divisible group. Therefore we assume S = T, i.e. π_{∗}O_{S^{∼}} = O_{S}.

Let M → S be the moduli scheme of isogenies of X of degree d = degree ρ. We will show that the S-morphism g : S^{∼} → M defined by Y factors through S → M.

Let ξ ∈ S. We write S_{ξ}^{∼} = S^{∼} ×_{S} Spec(κ(ξ)). By Lemma 2.6 it is suffices to show that the set-theoretic image of S_{ξ}^{∼} by g is a single point of M. Clearly M_{ξ} classifies isogenies starting at X_{ξ} of degree d. Over the algebraic closure ξ̄, by Lemma 2.5, there are only finitely many isogenies of Xξ̄ → Z of degree d to a completely slope divisible group (for fixed s and r_{i}). This shows that the image of S_{ξ}^{∼}(ξ̄) → M_{ξ}(ξ̄) is finite. Since S_{ξ}^{∼} is connected, see [EGA], III^{1}.4.3.1, the image of S_{ξ}^{∼} is a single point of M_{ξ}.

Hence we have the desired factorization S → M. It defines an isogeny X → Z over S. Finally Z is completely slope divisible since it is completely slope divisible in the general point of S, and because its Newton polygon is constant.

Q.E.D. 2.7 & 2.1

3 Constancy results

Let T be a scheme over 𝔽̄_{p}. We study the question if a p-divisible group X over T is constant up to isogeny, i.e. there exists a p-divisible group Y over 𝔽̄_{p} such that X is isogeneous to Y ×𝔽̄_{p} T.

3.1 Proposition. Let S be a Noetherian integral normal scheme over 𝔽̄_{p}. Let K be the function field of S and let K̄ be an algebraic closure of K. We denote by L ⊂ K̄ the maximal unramified extension of K with respect to S. Let T be the normalization of S in L. Let X be an isoclinic p-divisible group over S. Then there is a p-divisible group X_{0} over 𝔽̄_{p} and an isogeny X ×_{S} T → X_{0} ×_{Spec} 𝔽̄_{p} T such that the degree of this isogeny is smaller than an integer which depends only on the height of X.

Proof: We use Theorem 2.1: there exists an isogeny φ : X → Y, where Y over S is completely slope divisible. There are natural numbers r and s, such that

Φ = p^{−r} Fr^{s} : Y → Y^{(p^{s})}

is an isomorphism. Applying Corollary 1.10 to Y (n) and Φ we obtain finite group schemes X_{0}(n) over 𝔽_{p^{s}} and isomorphisms

Y (n)_{T} ≅ X_{0}(n) ×_{𝔽_{p^{s}}} T

The inductive limit of the group schemes X_{0}(n) is a p-divisible group X_{0} over 𝔽_{p^{s}}. It is isogeneous to X over T. Q.E.D.

3.2 Corollary. Let S and T be as in the proposition. Let T^{perf} → T be the perfect hull of T. Let X be a p-divisible group over S with constant Newton polygon. Then there is a p-divisible group X_{0} over 𝔽̄_{p} and an isogeny X_{0} ×_{Spec} 𝔽̄_{p} T^{perf} → X ×_{S} T^{perf}, whose degree is smaller than an integer which depends only on the height of X.

Proof. This follows using Proposition 1.3. Q.E.D.

Finally we prove constancy results without the normality condition.

3.3 Proposition. Let R be a strictly Henselian reduced local ring over 𝔽̄_{p}. Let X be an isoclinic p-divisible group over S = Spec R. Then there is a pdivisible group X_{0} over 𝔽̄_{p} and an isogeny X_{0} ×_{Spec} 𝔽̄_{p} S → X, whose degree is smaller than an integer which depends only on the height of X. 3.4 Corollary. To each natural number h there is a natural number c with the following property: Let R be a Henselian reduced local ring over 𝔽_{p} with residue field k. Let X and Y be isoclinic p-divisible groups over S = Spec R whose heights are smaller than h. Let ψ : X_{k} → Y_{k} be a homomorphism. Then p^{c}ψ lifts to a homomorphism X → Y.

A proof of the proposition, and of the corollary will be given later.

Remark. In case the R considered in the previous proposition, or in the previous corollary, is not reduced, but satisfies all other properties, the conclusions still hold, except that the integer bounding the degree of the isogeny, respectively the integer c, depend on h and on R.

If R is strictly Henselian the corollary follows from Proposition 3.3. Indeed, assume that X and Y are isogeneous to constant p-divisible groups X_{0} and Y_{0} by isogenies which are bounded by a constant which depends only on h. The corollary follows because:

Hom((X_{0})_{k}, (Y_{0})_{k}) = Hom((X_{0})_{R}, (Y_{0})_{R}).

Conversely the corollary implies the proposition since by Proposition 3.1 over a separably closed field an isoclinic p-divisible group is isogeneous to a constant p-divisible group.

Remark. Assume Corollary 3.4. An isoclinic slope divisible p-divisible group Y over k can be lifted to an isoclinic slope divisible p-divisible group over R. Indeed the étale schemes associated by 1.8 to the affine algebra of Y (n) and the isomorphism p^{−r} Fr^{s} lift to R. Hence the categories of isoclinic p-divisible groups up to isogeny over R respectively k are equivalent.

3.5 Lemma. Consider a commutative diagram of rings over 𝔽_{p}:

\begin{aligned}R & → & A \\ ↓ & & ↓ \\ R_{0} & → & A_{0}. \end{aligned}

Assume that R → R_{0} is a surjection with nilpotent kernel 𝔞, and that A → A_{0} is a surjection with nilpotent kernel 𝔟. Moreover let R → A be a monomorphism. Let X and Y be p-divisible groups over R. Let φ_{0} : X_{R_{0}} → Y_{R_{0}} be a morphism of the p-divisible groups obtained by base change. Applying base change with respect to R_{0} → A_{0} we obtain a morphism ψ_{0} : X_{A_{0}} → Y_{A_{0}}. If ψ_{0} lifts to a morphism ψ : X_{A} → Y_{A}, then φ_{0} lifts to a morphism φ : X → Y.

Proof. By rigidity, liftings of homomorphisms of p-divisible groups are unique. Therefore we may replace R_{0} by its image in A_{0} and assume that R_{0} → A_{0} is injective. Then we obtain 𝔞 = 𝔟 ∩ R. Let n be a natural number such that 𝔟^{n} = 0. We argue by induction on n. If n = 0, we have 𝔟 = 0 and therefore 𝔞 = 0. In this case there is nothing to prove. If n > 0 we consider the commutative diagram:

R  →  A
↓  ↓
R/(𝔟^{n−1} ∩ R) → A/𝔟^{n−1}
↓  ↓
R_{0}  → A_{0}.

We apply the induction hypothesis to the lower square. Hence it is enough to show the lemma for the upper square. We assume therefore without loss of generality that 𝔞^{2} = 0, 𝔟^{2} = 0. Let D_{X} and D_{Y} be the crystals associated to X and Y by Messing [Me]. The values D_{X}(R) respectively D_{Y} (R) are finitely generated projective R-modules which are endowed with the Hodge filtration F il_{X} ⊂ D_{X}(R) respectively F il_{Y} ⊂ D_{Y} (R). We put on 𝔞 respectively 𝔟 the trivial divided power structure. Then φ_{0} induces a map D_{X}(R) → D_{Y} (R). By the criterion of Grothendieck and Messing φ_{0} lifts to a homomorphism over R, iff D(φ_{0})(F il_{X}) ⊂ F il_{Y}. Since the construction of the crystal commutes with base change, see [Me], Chapt. IV, 2.4.4, we have canonical isomorphisms:

\begin{matrix}D_{X_{A}}(A) = A ⊗_{R} D_{X}(R), & F il_{X_{A}} = A ⊗_{R} F il_{X}, \\ D_{Y_{A}}(A) = A ⊗_{R} D_{Y} (R), & F il_{Y_{A}} = A ⊗_{R} F il_{Y} . \end{matrix}

Since ψ_{0} lifts we have id_{A} ⊗ D(φ)(A ⊗_{R} F il_{X}) ⊂ A ⊗_{R} F il_{Y}. Since R → A is injective this implies D(φ_{0})(F il_{X}) ⊂ F il_{Y}. Q.E.D.

Proof of Proposition 3.3. We begin with the case where R is an integral domain. By Proposition 2.7 there is a finite ring extension R → A such that A is contained in the quotient field of R, and such that there is an isogeny X_{A} → Y to a completely slope divisible p-divisible group Y over A. The degree of this isogeny is smaller than a constant which depends only on the height of X. Since A is a product of local rings we may assume without loss of generality that A is local. The ring A is a strictly Henselian local ring, see [EGA] IV 18.5.10, and has therefore no non-trivial finite étale coverings. The argument of the proof of Proposition 3.1 shows that Y is obtained by base change from a p-divisible group X_{0} over 𝔽̄_{p}. Therefore we find an isogeny

φ : (X_{0})_{A} → X_{A}

Let us denote by k the common residue field of A and R. Then φ induces an isogeny φ : (X_{0})_{k} → X_{k}. The last lemma shows that φ_{0} lifts to an isogeny φ̂ : (X_{0})R̂ → XR̂ over the completion R̂ of R. We apply the following fact: Claim. Consider a fiber product of rings:

\begin{aligned}R & → & A_{1} \\ ↓ & & ↓ \\ A_{2} & → & B. \end{aligned}

Let X and Y be p-divisible groups over R. Let ψ_{i} : X_{A_{i}} → Y_{A_{i}} for i = 1, 2 be two homomorphisms of p-divisible groups which agree over B. Then there is a unique homomorphism ψ : X → Y which induces ψ_{1} and ψ_{2}. In our concrete situation we consider the diagram:

R →  A
↓  ↓
R̂ → Â = R̂ ⊗_{R} A.

The morphisms φ̂ and φ agree over Â because they agree over the residue field k. This proves the case of an integral domain R. In particular we have shown the Corollary 3.4 in the case where R is a strictly Henselian integral domain. To show the corollary in the reduced case we consider the minimal prime ideals 𝔭_{1}, … 𝔭_{s} of R. Let ψ : X_{k} → Y_{k} be a homomorphism. Then p^{c}ψ lifts to a homomorphims over each of the rings R/𝔭_{i}, for i = 1, …, s. But then we obtain a homomorphism over R using the Claim above. This proves the Corollary 3.4 and hence the Proposition 3.3 in the case where R is reduced and strictly Henselian. If R is not reduced one applies standard deformation theory to R → R_{red}, [Z2], 4.47. Q.E.D.

Proof of Corollary 3.4. Consider the diagram:

\begin{aligned}R & → & R^{sh} \\ ↓ & & ↓ \\ R̂ & → & R̂^{sh}. \end{aligned}

The upper index “sh” denotes the strict henselization. Using the fact that the categories of finite étale coverings of R, k, respectively R̂ are equivalent it is easy to see that the last diagram is a fiber product. We have already proved that p^{c}ψ lifts to a homomorphism over R^{sh}. Applying Lemma 3.5 to the following diagram we see that p^{c}ψ lifts to R̂. This is enough to prove the corollary (compare the Claim above).

R/𝔪^{n} → R^{sh}/𝔪^{n}R^{sh}
↓  ↓
R/𝔪 → R^{sh}/𝔪R^{sh}

In this diagram n is a positive integer and 𝔪 is the maximal ideal of R. Q.E.D.

3.6 Corollary. Let R be a strictly Henselian reduced local ring over 𝔽̄_{p}. Let R^{perf} be the perfect hull of R. Let X be a p-divisible group over S = Spec R with constant Newton polygon. We set S^{perf} = Spec(R^{perf}). Then there is a p-divisible group X_{0} over 𝔽̄_{p} and an isogeny X_{0} ×_{Spec} 𝔽̄_{p} S^{perf} → X ×_{S} S^{perf} such that the degree of this isogeny is bounded by an integer which depends only on the height of X.

Proof: This follows using Proposition 1.3. Q.E.D.

4 Examples

In this section we use the p-divisible groups either Z = G_{1,n}, with n ≥ 1, or Z = G_{m,1}, with m ≥ 1 as building blocks for our examples. These have the property to be iso-simple, they are defined over 𝔽_{p}, they contain a unique subgroup scheme N ⊂ Z isomorphic with α_{p}, and Z/N ≅ Z. Indeed, for Z = G_{1,n} we have an exact sequence of sheaves:

0 → α_{p} → Z ⟶^{Fr} Z → 0.

For Z = G_{m,1} we have the exact sequence

0 → α_{p} → Z ⟶^{Ver} Z → 0.

Moreover every such Z has the following property: if Z_{K} → Z′ is an isogeny over some field K, and k an algebraic closed field containing K, then Z_{k} ≅ Z′_{k}.

4.1 Example. In this example we produce a p-divisible group X with constant Newton polygon over a regular base scheme which does not admit a slope filtration.

Choose Z_{1} and Z_{2} as above, with slope(Z_{1}) = λ_{1} > λ_{2} = slope(Z_{2}); e.g. Z_{1} = G_{1,1} and Z_{2} = G_{1,2}. We choose R = K[t], where K is a field. We write S = Spec(R) and Z_{i} = Z_{i} × S for i = 1, 2. We define

(id, t) : α_{p} → α_{p} × α_{p} ≅ N_{1} × N_{2}; this defines ψ : α_{p} × S → Z_{1} × Z_{2}.

Claim: X := (Z_{1} × Z_{2})/ψ(α_{p} × S) is a p-divisible group over S which does not admit a slope filtration. Indeed, for the generic point we do have slope filtration, where X = X ⊗ K(t), and 0 ⊂ X_{1} ⊂ X is given by: X_{1} is the image of

ξ_{K} : (Z_{1} ⊗ K(t) → (Z_{1} × Z_{2}) ⊗ K(t) → X).

However the inclusion ξ_{K} extends uniquely a homomorphism ξ : Z_{1} → X, which is not injective at t ↦ 0. This proves the claim.

4.2 Example. In this example we construct a p-divisible group X with constant Newton polygon over a base scheme S which is not normal, such that there is no isogeny ϕ : X → Y to a completely slope divisible p-divisible group. (i.e. we show the condition that S is normal in Theorem 2.1 is necessary).

We start again with the exact sequences over 𝔽_{p}:

\begin{aligned}0 & → α_{p} & → G_{2,1} & ⟶^{Ver} G_{2,1} & → 0, \\ 0 & → α_{p} & → G_{1,2} & ⟶^{Fr} G_{1,2} & → 0. \end{aligned}  (1)

We fix an algebraically closed field k. We write T = ℙ_{k}^{1}, and:

Z_{1} = G_{2,1} ×_{𝔽_{p}} T, Z_{2} = G_{1,2} ×_{𝔽_{p}} T, Z = Z_{1} × Z_{2}, A = α_{p} ×_{𝔽_{p}} T.

By base change we obtain sequences of sheaves on the projective line T = ℙ_{k}^{1}:

\begin{aligned}0 & → A & → Z_{1} & ⟶^{Ver} Z_{1} & → 0, \\ 0 & → A & → Z_{2} & ⟶^{Fr} Z_{2} & → 0. \end{aligned}

Lemma. Consider Z → T = ℙ_{k}^{1} as above. Let β : Z → Y be an isogeny to a completely slope divisible p-divisible group Y over T. Then Y = Y_{1} × Y_{2} is a product of two p-divisible groups and β = β_{1} ×β_{2} is the product of two isogenies β_{i} : Z_{i} → Y_{i}.

Proof. The statement is clear if we replace the base T by a perfect field, see Proposition 1.3. In our case we show first that the kernel of the morphisms Z_{i} → Y, i = 1, 2 induced by β are representable by a finite, locally free group schemes G_{i}. Indeed, let G_{i} the kernel in the sense of f.p.p.f sheaves. Let us denote by G the kernel of the isogeny β. Choose a number n such that p^{n} annihilates G. Then p^{n} annihilates G_{i}. Therefore G_{i} coincides with the kernel of the morphism of finite group schemes Z_{i}(n) → Y (n). Hence G_{i} is representable by a finite group scheme G_{i}. We prove that G_{i} is locally free. It suffices to verify that the rank of G_{i} in any geometric point η of T is the same. But we have seen that over η the p-divisible group Y splits into a product Y_{η} = (Y_{η})_{1} × (Y_{η})_{2}. This implies that

G_{η} = (G_{1})_{η} × (G_{2})_{η}.

We conclude that the ranks of (G_{i})_{η} are independent of η since G is locally free. We define p-divisible groups Y_{i} = Z_{i}/G_{i}. We obtain a homomorphism of pdivisible groups Y_{1} × Y_{2} → Y which is an isomorphism over each geometric point η. Therefore this is an isomorphism. Q.E.D.

Next we construct a p-divisible group X on T = ℙ_{k}^{1}. Let L be a line bundle on ℙ_{k}^{1}. We consider the associated vector group

V(L)(T′) = Γ(T′, L′_{T}),

where T′ → T = ℙ_{k}^{1} is a scheme and L′_{T} is the pull-back. The kernel of the Frobenius morphism Fr : V(L) → V(L)^{(p)} is a finite, locally free group scheme α_{p}(L) which is locally isomorphic to α_{p}. We set A(−1) = α_{p}(O_{ℙ_{k}^{1}}(−1)). There are up to multiplication by an element of k^{∗} unique homomorphisms ι_{0} respectively ι_{∞} : O_{ℙ_{k}^{1}}(−1) → O_{ℙ_{k}^{1}} whose unique zeroes are 0 ∈ ℙ_{k}^{1} respectively ∞ ∈ ℙ_{k}^{1}. This induces homomorphisms of finite group schemes ι_{0} : A(−1) → A respectively ι_{∞} : A(−1) → A which are isomorphisms outside 0 respectively outside ∞. We consider the embeddings

A(−1) ⟶^{(ι_{0},ι_{∞})} A × A ⊂ Z_{1} × Z_{2} = Z.

We define X = Z/A(−1):

ψ : Z ⟶ Z/A(−1) = X.

Note that

ψ_{0} : G_{2,1} × G_{1,2} = Z_{0} ⟶ G_{2,1} × (G_{1,2}/α_{p}) = X_{0},

and

ψ_{∞} : G_{2,1} × G_{1,2} = Z_{∞} ⟶ (G_{2,1}/α_{p}) × G_{1,2} = X_{∞}.

We consider the quotient space S = ℙ_{k}^{1}/{0, ∞}, by identifying 0 and ∞ into a normal crossing at P ∈ S, i.e.

O_{S,P} = {f ∈ O_{T,0} ∩ O_{T,∞} | f(0) = f(∞)};

S is a nodal curve, and

ℙ_{k}^{1} = T ⟶ S, 0 ↦ P, ∞ ↦ P,

is the normalization morphism. A finite, locally free scheme G over S is the same thing as a finite, locally free scheme G over ℙ_{k}^{1} endowed with an isomorphism

G_{0} ≅ G_{∞}.

It follows that the category of p-divisible groups Y over S is equivalent to the category of pairs (Y, γ_{Y}), where Y is a p-divisible group on ℙ_{k}^{1} and γ_{Y} is an isomorphism

γ_{Y} : Y_{0} ≅ Y_{∞}

of the fibers of Y over 0 ∈ ℙ_{k}^{1} and ∞ ∈ ℙ_{k}^{1}. We call γ_{Y} the gluing datum of Y. We construct a p-divisible X over S by defining a gluing datum on the pdivisible group X. In fact, the exact sequences in (1) give:

X_{0} = G_{2,1} × (G_{1,2}/α_{p}) ≅ (G_{2,1}/α_{p}) × G_{1,2} = X_{∞};

this gluing datum provides a p-divisible group X over S. Claim: This p-divisible group X → S satisfies the property mentioned in the example. Let us assume that there exists an isogeny ϕ : X → Y to a completely slope divisible p-divisible group Y over S. We set Y = Y ×_{S} T and consider the induced isogeny

ϕ ×_{S} T = φ : X → Y.

and the induced isogeny:

β = φ·ψ : (Z ⟶^{ψ} X ⟶^{φ} Y).

By the lemma we see that

β = β_{1} × β_{2} : Z ⟶ Y_{1} × Y_{2} = Y.

Note that

φ_{∞} = ϕ_{P} = φ_{0} : X_{∞} = X_{P} = X_{0} ⟶ Y_{∞} = Y_{P} = Y_{0};

these p-divisible groups both have a splitting into isoclinic summands:

X_{∞} = X_{P} = X_{0} = X′ × X′′, Y_{∞} = Y_{P} = Y_{0} = Y′ × Y′′,

and

ϕ_{P} = φ′ × φ′′ : X′ × X′′ ⟶ Y′ × Y′′

is in diagonal form. On the one hand we conclude from

((β_{1})_{0} : (Z_{1})_{0} ⟶ (Y_{1})_{0}) = ((Z_{1})_{0} →^{∼} X′ → Y′)

that deg(β_{1}) = deg(β_{1})_{0} = deg(φ′); on the other hand

((β_{1})_{∞} : (Z_{1})_{∞} ⟶ (Y_{1})_{∞}) = ((Z_{1})_{∞} → (G_{2,1}/α_{p}) = X′ → Y′) ;

hence

deg(β_{1}) = deg(β_{1})_{∞} = p· deg(φ′).

We see that the assumption that the isogeny ϕ : X → Y to a completely slope divisible Y → S would exist leads to a contradiction. This finishes the description and the proof of Example 4.2. Q.E.D.

4.3 Example. For every positive integer d there exists a scheme S′ of dimension d, a point P′ ∈ S′ such that S′ is regular outside P′, and a p-divisible group X′ → S′ which does not admit an isogeny to a completely slope divisible group over S′.

This follows directly form the previous example. Indeed choose T as in the previous example, and let T′ → T smooth and surjective with T′ of dimension d. Pull back X/T to X′/T′; choose geometric points 0′ and ∞′ ∈ T′ above 0 and ∞ ∈ T; construct S′ by “identifying 0′ and ∞′”: outside P′ ∈ S′, this scheme is T′\{0′, ∞′}, and the local ring of P′ ∈ S′ is the set of pairs of elements in the local rings of 0′ and ∞′ having the same residue value. We can descend X′ → T′ to X′ → S′, and this has the desired property.

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[Z1] Th. Zink: On the slope filtration. Duke Math. J. Vol.109 (2001), 79-95.

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Frans Oort Mathematisch Instituut P.O. Box. 80.010 NL - 3508 TA Utrecht The Netherlands oort@math.uu.nl Thomas Zink Fakultät für Mathematik Universität Bielefeld Postfach 100131 D-33501 Bielefeld Deutschland zink@mathematik.uni-bielefeld.de 202

Documenta Mathematica 7 (2002)