De Jong-Oort Purity for p-Divisible Groups

Thomas Zink

Faculty of Mathematics, University of Bielefeld, POB 100131, D 33501 Bielefeld

zink@math.uni-bielefeld.de

Dedicated to Professor Yuri Manin

1 Introduction

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.

2 Frobenius Modules

We fix a prime number p. Let R be a commutative ring, such that p is nilpotent in R. The ring of Witt vectors with respect to p is denoted by W (R). We write I_{R} = V W (R) for the Witt vectors whose first component is 0. The Witt polynomials are denoted by w_{n} : W (R) → R. The truncated Witt vectors of length n are denoted by W_{n}(R). If pR = 0 the Frobenius endomorphism F of the ring W (R) induces an endomorphism F : W_{n}(R) → W_{n}(R).

Definition 1. A Frobenius module over R is a pair (M, F), where M is a projective finitely generated W (R)-module of some fixed rank h and F : M → M is a Frobenius-linear homomorphism, such that det F = p^{d}ϵ locally for the Zariski topology on R, where ϵ : det M → det M is a Frobenius-linear isomorphism and d ≥ 0 is some integer. We call h the height of the Frobenius module and d the dimension.

This definition implies that the factorization det F = p^{d}ϵ exists even globally, but we will never use this. Since the kernel of w_{0} : W (R) → R is in the radical of W (R), there is always a covering Spec R = ⋃_{i} Spec R_{f_{i}}, such that W (R_{f_{i}}) ⊗_{W (R)} M is a free W (R_{f_{i}})-module for each i. Therefore we will often consider the case where M is a free W (R)-module. If we choose a basis of M we may view det F as an element of W (R). Then (M, F) is a Frobenius module iff det F = p^{d}η for some unit η ∈ W (R). In a question which is local on Spec R we will consider det F as an element of W (R) without futher notice.

In this article a display over R is a 3n-display in the sense of [Z1]. The displays of [Z1] are called nilpotent displays. If P = (P, Q, F, F_{1}) is a display over R then (P, F) is a Frobenius module over R.

Let X be a p-divisible over R and assume that p is nilpotent in R. If we evaluate the Grothendieck-Messing crystal of X at W (R) we obtain a finitely generated locally free W (R)-module M_{X}, which is endowed with a Frobenius linear map F : M_{X} → M_{X}. If X is the formal p-divisible group associated to a nilpotent display P then (M_{X}, F) = (P, F) is a Frobenius module. The pair (M_{Y} , F) is also a Frobenius module, if Y is an extension of an étale p-divisible group by X.

If we assume moreover that R is a complete local noetherian ring (M_{X}, F) is a Frobenius module for an arbitrary p-divisible group X over R. Indeed if the special fibre of X has no étale part then (M_{X}, F) comes from a display and is therefore a Frobenius module. Since X is an extension of an étale pdivisible group by a p-divisible group with no étale part in the special fibre, we see that (M_{X}, F) is a Frobenius module in general.

By these remarks any (M_{X}, F) appearing in this work are Frobenius modules.

We add that Lau [L] in a forthcoming paper will associate a display to any p-divisible group over a ring R, where p is nilpotent. Thereby he obtains a functor from p-divisible groups to Frobenius modules. If we could use this functor it would be more satisfying then the remark above.

The following lemma is mainly a motivation for the definitions we are going to make:

Lemma 2. Let P and P′ be displays over a ring R of the same height and dimension. Let α : P → P′ be a homomorphism.

Locally on Spec R the element det α ∈ W (R) satisfies an equation:

F det α = ε · det α,

where ε ∈ W (R)^{∗} is a unit.

Proof. We choose normal decompositions

\begin{matrix}P = L ⊕ T, & Q = L ⊕ I_{R}T \\ P′ = L′ ⊕ T′, & Q′ = L′ ⊕ I_{R}T′. \end{matrix}

Without loss of generality we may assume that L, L′, T, T′ are free W (R)- modules. We choose identifications

L ≃ W (R)^{l} ≃ L′, T ≃ W (R)^{t} ≃ T′.

Then operators F_{1} and F′_{1} are given by invertible block-matrices with coefficient in W (R):

F_{1} ( \underline{x} ; ^{V} \underline{y} ) = ( \begin{matrix}X & Y \\ Z & W \end{matrix} ) ( ^{F} \underline{x} ; \underline{y} )

F′_{1} ( \underline{x} ; ^{V} \underline{y} ) = ( \begin{matrix}X′ & Y′ \\ Z′ & W′ \end{matrix} ) ( ^{F} \underline{x} ; \underline{y} )

The block-matrices are invertible by the definition of a display. We also represent α by a block matrix

α ( \underline{x} ; ^{V} \underline{y} ) = ( \begin{matrix}A & B \\ ^{V} C & D \end{matrix} ) ( \underline{x} ; ^{V} \underline{y} )

Since α commutes with the operators F_{1} and F′_{1} we find

( \begin{matrix}X′ & Y′ \\ Z′ & W′ \end{matrix} ) ( \begin{matrix}^{F} A & p ^{F} B \\ C & ^{F} D \end{matrix} ) = ( \begin{matrix}A & B \\ ^{V} C & D \end{matrix} ) ( \begin{matrix}X & Y \\ Z & W \end{matrix} )  (1)

We see that

F ( \begin{matrix}A & B \\ ^{V} C & D \end{matrix} ) = ( \begin{matrix}^{F} A & ^{F} B \\ pC & ^{F} D \end{matrix} )

has the same determinant as

( \begin{matrix}^{F} A & p ^{F} B \\ C & ^{F} D \end{matrix} )

But then taking determinants in (1) gives the result. ⊓ ⊔

Proposition 3. Let R be a noetherian ring such that Spec R is connected. We assume that pR = 0. Let α : P → P′ be a homomorphism of displays of the same height h and the same dimension d.

If det α ≠ 0 then there is a nonnegative integer u, such that locally on Spec R the following equation holds:

det α = p^{u}ε, where ε ∈ W (R)^{∗}, u ∈ ℤ_{≥0}.

Proof. If the number u exists locally it is clearly a locally constant function. Therefore the question is local. We may replace Spec R by a small affine connected neighbourhood.

We set η = det α. By the last proposition we find:

F η = ζ · η for some ζ ∈ W (R)^{∗}.  (2)

We write η = ^{V^{t}}ξ, such that w_{0}(ξ) ≠ 0. We claim that (2) implies:

F ξ = ^{F^{t}}ζ · ξ.  (3)

To verify this we may assume that t > 0. We obtain:

F V^{t}ξ = ζ ^{V^{t}}ξ = ^{V^{t}}( ^{F^{t}}ζξ)

Since pR = 0 the operators F and V acting on W (R) commute. Therefore we deduce (3)

Let w_{0}(ξ) = x and w_{0}( ^{F^{t}}ζ) = e ∈ R^{∗}. We apply w_{0} to the equation (3) and obtain:

x^{p} = ex.  (4)

Since the product

x(x^{p−1} − e) = 0

has relatively prime factors, it follows that

\begin{aligned}D(x) ∪ D(x^{p−1} − e) & = & Spec R \\ D(x) ∩ D(x^{p−1} − e) & = & ∅. \end{aligned}

Hence by connectedness either D(x) = Spec R or D(x) = ∅. In the first case x is nilpotent. But then we find x = 0, by iterating the equation (4). This is a contradiction to our choices. Therefore D(x) = Spec R and x is a unit. Then ξ is a unit too. We find

F^{t}η = ^{F^{t}V^{t}}ξ = p^{t}ξ.

But by (2) ^{F^{t}}η may be expressed as the product of η by a unit. This proves the result. ⊓ ⊔

Definition 4. A homomorphism as in the proposition is called an isogeny of displays.

Let R be a ring such that pR = 0. Assume that the ideal of nilpotent elements of R is nilpotent. Let α : P → P′ be a homomorphism of nilpotent displays of the same height and dimension. By the functor from the category of nilpotent displays to the category of formal p-divisible groups ([Z1] 3.1) we obtain from α a morphism ϕ : X → X′ of p-divisible groups. It follows from loc.cit. Prop. 66 and Prop. 99 that α is an isogeny, iff ϕ is an isogeny of p-divisible groups.

Since pR = 0 the Frobenius endomorphism on W (R) induces a Frobenius endomorphism on the truncated Witt vectors F : W_{n}(R) → W_{n}(R). Therefore we may consider truncated Frobenius modules. We are going to prove a version of Proposition 3 for truncated Frobenius modules.

Definition 5. Let R be a ring such that pR = 0. A truncated Frobenius module of level n, dimension d, and height h over R is a finitely generated projective W_{n}(R)-module M of rank h equipped with a Frobenius linear operator F : M → M, such that locally on Spec R the determinant has the form

det F = p^{d}ε

where ε : det M → det M is a Frobenius linear isomorphism.

A Frobenius module M over R induces a truncated Frobenius module, if we tensor it by W_{n}(R).

Definition 6. Let M and N be truncated Frobenius modules of level n and of the same dimension d and height h. A morphism of Frobenius modules α : M → N is called an isogeny if there is a natural number u < n such that the determinant of α has locally on Spec R the form:

F^{d} det α = p^{u}ε, ε ∈ W_{n}(R)^{∗}.

The number u is called the height of the isogeny.

Proposition 7. Let M and N be truncated Frobenius modules of level n and of the same dimension d and height h over a ring R such that Spec R is connected and pR = 0.

Let u ≥ 0 be an integer, such that n > u + d. Let α : M → N be a homomorphism of Frobenius modules such that

F^{d} det α ∉ V^{u+1}W_{n−u−1}(R).  (5)

Then α becomes an isogeny if we truncate it to level n − d:

α[n − d] : M[n − d] → N[n − d].

Proof. We may assume that M and N are free W_{n}(R)-modules. We choose isomorphisms

det M ≃ W_{n}(R) ≃ det N

and view θ := det α as an element of W (R). Then we obtain a commutative diagram

det M ⟶^{θ} det N
p^{d}τ_{M} F ↓  ↓_{p^{d}τ_{N} F}
det M ⟶^{θ} det N,

where τ_{M}, τ_{N} ∈ W_{n}(R)^{∗} are units. We obtain

p^{d}τ_{N} ^{F} θ = θp^{d}τ_{M}.  (6)

Using p^{d} = V^{d}F^{d} in W_{n}(R), we can divide (6) by V^{d}. We then obtain an equality in W_{n−d}(R):

F^{d+1}θ[n − d] = ^{F^{d}}θ[n − d]ρ.  (7)

Here θ[n − d] denotes the image of θ by the natural restriction W_{n}(R) → W_{n−d}(R) and ρ ∈ W_{n−d}(R)^{∗} is a unit.

On the other hand we may write by assumption:

F^{d}θ = ^{V^{u_{1}}} σ,  (8)

where u_{1} ≤ u, and w_{0}(σ) = s_{0} ≠ 0. Clearly we may assume u = u_{1}. Since n − d > u we obtain from equation (7)

s_{0}^{p} = s_{0}e

for some unit e ∈ R^{∗}. As in the proof of Proposition 3 (see: (4)) we conclude that s_{0} is a unit. Then σ is a unit too. From (8) we obtain

F^{d+u}θ = p^{u}σ.

We truncate this equation to W_{n−d}(R) and use (7) to obtain

F^{d}θ[n − d] = p^{u}ε

for some unit ε ∈ W_{n−d}(R)^{∗}. ⊓ ⊔

Let n > u be natural numbers. It is clear that a morphism of displays α : P → P′ is an isogeny of height u, iff the map of the truncated Frobenius modules α[n] : (P [n], F) → (P′[n], F) is an isogeny of height u.

3 Proof of Purity

For the proof of the purity theorem of de Jong and Oort for p-divisible groups we need to recall a few facts on completely slope divisible p-divisible groups (abbreviated: c.s.d. groups) from [Z2] and [OZ] Definition 1.2. We will use truncated Frobenius modules of p-divisible groups over any scheme U. These are locally free W_{n}(O_{U})-modules.

Lemma 8. Let Y be a c.s.d. group over a normal noetherian scheme U over 𝔽̄_{p}. Let n be a natural number. Then there is a finite morphism U′ → U, such that the truncated Frobenius module M_{Y} [n] of Y over U′ is obtained by base change from a truncated Frobenius module over 𝔽̄_{p}, i.e. we can find a Frobenius module N over 𝔽̄_{p} such that there is an isomorphism of Frobenius modules

W_{n}(O_{U′}) ⊗_{W_{n}(O_{U})} M_{Y} [n] ≃ W_{n}(O_{U′}) ⊗_{W (𝔽̄_{p})} N  (9)

Proof. This is an immediate consequence of [OZ] Proposition 1.3, since it says that this is true if we take for U′ the perfect hull of the universal pro-étale cover of U. Another proof is obtained by substituting in the proof of loc.cit. Frobenius modules. ⊓ ⊔

Proposition 9. Let T be a regular connected 1-dimensional scheme over 𝔽_{p}. Then any p-divisible group X with constant Newton polygon over T is isogenous to a c.s.d. group.

Proof. This follows from the main result of [OZ] Thm. 2.1. for any normal noetherian scheme T. But under under the assumptions made the proof is much easier (compare [Z2] proof of Thm. 7). Indeed let K = K(T) be the function field of T. Then we find by over K an isogeny to a c.s.d. group:

X_{K} → Y^{∘}  (10)

Let G^{∘} be the finite group scheme which is the kernel of (10) and let G ⊂ X be its scheme theoretic closure. We set Y = X/G. Using the fact that X has constant Newton polygon one proves that Y is c.s.d. ⊓ ⊔

The third ingredient is the boundedness principle, which seems to have been known for a long time [M].

Proposition 10. Let k be an algebraic closed of characteristic p field. Let h be a natural number. Then there is a constant c ∈ ℕ with the following property:

Let M_{1} and M_{2} be Frobenius modules of height ≤ h over k. Let n ∈ ℕ be arbitrary and let ᾱ : M_{1}/p^{n}M_{1} → M_{2}/p^{n}M_{2} be a morphism of truncated Frobenius modules which lifts to a morphism of truncated Frobenius modules M_{1}/p^{n+c}M_{1} → M_{2}/p^{n+c}M_{2}. Then ᾱ lifts to a morphism of Frobenius modules α : M_{1} → M_{2}.

A weaker version of this is contained in [O], where the existence of the constant c is only asserted for given modules M_{1} and M_{2}. But one can show that for given modules N_{1} resp. N_{2} in the isogeny class of M_{1} resp. M_{2}, there are always isogenies N_{1} → M_{1} resp. N_{1} → M_{1} whose degrees are bounded by a constant only depending on h. This is another well-known boundedness principle. As an alternative to this proof the reader may use the much stronger results discussed in the introduction of [V2].

Theorem 11. (de Jong-Oort) Let R be a noetherian local ring of Krull dimension ≥ 2 with p · R = 0. Let U = Spec R \ {𝔪}, the complement of the closed point. A p-divisible group X over SpecR, which has constant Newton polygon over U has constant Newton polygon over Spec R.

Proof. It is not difficult to reduce to the case where R is complete, normal of Krull dimension 2 with algebraically closed residue class field k = R/𝔪 ([JO]).

Then U is a 1-dimensional regular scheme. We find by Proposition 9 a c.s.d. group Y over U and an isogeny

α : Y → X_{|U},  (11)

Let d be the dimension of X let u be the height of α and let c be the number from Proposition 10. We choose a natural number n > c+u+d.. After a finite extension of R we may assume by Lemma 8 that the truncated Frobenius module of Y is constant

M_{Y} [n] ≃ W_{n}(O_{U}) ⊗_{W (𝔽̄_{p})} N  (12)

where N is a Frobenius module over 𝔽̄_{p}. In particular the Newton polygons of N and Y must be the same by the boundedness principle applied to the field K̄, where K is the field of fractions of R.

Combining (11) and (12) we find an isogeny of height u of truncated Frobenius modules

W_{n}(O_{U}) ⊗_{W (𝔽̄_{p})} N → W_{n}(O_{U}) ⊗_{R} M_{X}[n].  (13)

By the normality of R we find Γ (U, W_{n}(O_{U})) = W_{n}(R). Taking the global section of (13) over U we obtain a morphism of truncated Frobenius modules

W_{n}(R) ⊗_{W (𝔽̄_{p})} N → M_{X}[n].  (14)

We know that (14) is an isogeny over K of height u. Therefore Proposition 3 is applicable to the morphism (14). We obtain therefore an isogeny of height u of truncated Frobenius modules over R:

W_{n−d}(R) ⊗_{W (𝔽̄_{p})} N → M_{X}[n − d],

It is clear that the base change of an isogeny of truncated Frobenius modules is again an isogeny. Making the base change R → k we obtain an isogeny:

W_{n−d}(k) ⊗_{W (𝔽̄_{p})} N → W_{n−d}(k) ⊗_{W (R)} M_{X}[n − d] = M_{X_{k}}[n − d].

The boundedness principle shows that X_{k} and N have the same Newton polygon. ⊓ ⊔

References

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