An application of alterations to Dieudonné modules

by A. J. de Jong

In this note we indicate a non-cohomological application of the results of [2] and [3]. The application concerns the crystalline Dieudonné module functor.

Let S be a scheme of characteristic p > 0. A Barsotti-Tate group, or p-divisible group G over S is a sheaf of abelian groups on the fppf-site of S which is p-divisible, p-power torsion, and such that G[p] is representable by a finite locally free group scheme over S. The basic example is the p^{∞}-torsion of an abelian scheme over S, i.e., G = A[p^{∞}]. It is very analogous to the ℓ-adic Tate module of A, which can be seen as a ℤ_{ℓ}-sheaf on the étale site of S. The category of p-divisible groups over S is denoted BT_{S}. For more information on p-divisible groups, see [6], [7], and [8].

The crystalline Dieudonné module functor transforms a p-divisible group over S into a Dieudonné crystal over S. Let us explain what a crystal over S = Spec 𝔽_{p}[X] amounts to: It is given by a quadruple (M, ∇, F, V), where

M is a finite free ℤ_{p}{X}-module, where ℤ_{p}{X} is the p-adic completion of ℤ_{p}[X],

∇ is a connection M → M ⊗ Ω̂_{ℤ_{p}{X}}^{1} (continuous differentials),

F is a σ-linear map M → M, i.e., a linear map M ⊗_{σ} ℤ_{p}{X}, where σ : ℤ_{p}{X} →

ℤ_{p}{X} is the ring map such that X ↦ X^{p}, and
V is a linear map M → M ⊗_{σ} ℤ_{p}{X}.

There are conditions, expressing that F anf V are compatible with ∇, and that F V = p and V F = p. For general schemes S, the category of Dieudonné crystals can be described in a similar, but more involved manner, see [9], [10] and [4, 2.3.4] for example. The category of Dieudonné crystals over S is denoted DC_{S}. An important point is that, even though Dieudonné crystals are hard to describe and hard to get hold of, they are easier to deal with than p-divisible groups. Also, to follow the gist of this note, the reader does not need to understand precisely what such a crystal amounts to.

We recall the crystalline Dieudonné module functor

𝔻 : BT_{S} ⟶ DC_{S}

from the category of Barsotti-Tate groups to the category of Dieudonné crystals over S. The relation between Barsotti-Tate groups and modules with connections evidenced by 𝔻 is analogous to the Riemann-Hilbert correspondence between perverse sheaves and Dmodules in characteristic 0. We do not give the definition of 𝔻 here, which is delicate and not easy to explain. For references we suggest the original works [7], [8]; here Grothedieck raises several questions concerning the faithfulness and surjectivity of 𝔻. For a definition of 𝔻 that is currently in vogue we refer to [11]. There has been a lot of work on this functor, we refer to the introduction of [4] for some history and results. See [12] for an application of 𝔻 to homomorphisms of p-divisible groups and Abelian varieties.

The application of alterations

Let k be a field with a finite p-basis and let S → Spec k be a morphism of finite type. In this case it was proved in [4] that

(i) 𝔻 is an equivalence if S is regular [4, Main Theorem 1], and (ii) 𝔻 is fully faithful up to isogeny for any S [4, Main Theorem 2].

The proof of (ii) in [4] was quite complicated and necessitated proving some nontrivial results on formal and rigid geometry. In this note we will sketch how to deduce (ii) from (i) using the alterations theorem [2, Theorem 4.1]. The proof given below still uses some of the rigid analytic techniques of [4, Chapters 5, 6 & 7], but they will be isolated in the proof of the lemma below. In addition, at the end of this note we discuss the question whether the functor 𝔻 might even be essentially surjective up to isogeny.

Full faithfulness up to isogeny

Let S of finite type over k as above be given. Let G_{1}, G_{2} ∈ BT_{S} and let ψ : 𝔻(G_{2}) → 𝔻(G_{1}) be a morphism of the associated Dieudonné crystals. We have to show that there is an integer n ∈ ℕ and a morphism φ : G_{1} → G_{2} such that 𝔻(φ) − p^{n}ψ is torsion in Hom(𝔻(G_{2}), 𝔻(G_{1})).

Lemma. Suppose we have n and φ : G_{1} → G_{2} such that for all points s of S the map 𝔻(φ_{s}) − p^{n}ψ|_{CRIS(s/Σ)} is zero. Then 𝔻(φ) − p^{n}ψ is torsion. Proof. (Sketch.) The idea is that the assumption implies that the induced map 𝔻(φ)^{rig} − p^{n}ψ^{rig} of convergent F -isocrystals is zero. It is known that this implies that 𝔻(φ) − p^{n}ψ is torsion, see [1, Theorem 2.4.2]. See also [4, Section 5.5] where a similar argument is given. QED

Remark. A more general lemma will be proved in the forthcoming paper [5]. In [5] it is proven that the hypothesis on k (finite p-basis) is actually superfluous for the results of [4].

We return to the problem of finding n and φ. We remark that 𝔻 is faithful over any reasonable scheme. We also remark that we may replace S by its reduction; the category of p-divisible groups up to isogeny is unchanged by this, and similar for the category of Dieudonné crystals (see [4, proof of 5.1.2]).

Thus we may assume that S is reduced. Let π : S′ → S be the normalization mapping (i.e., S′ is the disjoint union of the normalizations of the irreducible components of S). Let i : T → S be a proper closed subscheme of S and let i′ : T′ → S′ be the closed immersion of T′ = π^{−1}(T) into S′. There exists a T such that the following diagram of sheaves on S is cartesian (i.e., it is a fibre product diagram):

O_{S} ⟶ i_{∗}O_{T}
↓  ↓
π_{∗}O_{S′} ⟶ π_{∗}i′_{∗}O_{T′}

Exercise. Construct such a T.

Assume, for the moment, that (ii) is proved in the case of normal schemes of finite type over k. Then we can find n′ and φ′ : G_{1}×_{S} S′ → G_{2}×_{S} S′ such that 𝔻(φ′)−p^{n′}ψ|_{CRIS(S′/Σ)} is torsion. By Noetherian induction we have n_{T} and φ_{T} : G_{1,T} → G_{2,T} adapted to ψ|_{CRIS(T/Σ)}. Let n be the maximum of n′ and n_{T}. By faithfulness of 𝔻 we get that p^{n−n′}φ′|_{T′} = p^{n−n_{T}} φ_{T} |_{T′}.

The cartesian diagram above will imply (see below) that p^{n−n′}φ′ and p^{n−n_{T}} φ_{T} “glue” to a morphism φ : G_{1} → G_{2}. Once this is established, it follows from the lemma that p^{n}ψ − 𝔻(φ) is torsion.

Let us write A_{i,N} for the locally free sheaf of O_{S}-modules (or Hopf algebras) corresponding to the finite flat group scheme G_{i}[p^{N}] over S. There are maps

A_{2,N} ⊗ O_{T} ⟶ A_{1,N} ⊗ O_{T}

induced by p^{n−n_{T}} φ_{T} acting on G_{1,T} [p^{N}] and maps

A_{2,N} ⊗ O_{S′} ⟶ A_{1,N} ⊗ O_{S′}

induced by p^{n−n′}φ acting on G_{1,S′}[p^{N}]. These agree as maps on A_{2,N} ⊗ O_{T′} and by the cartesian diagram (which stays cartesian after tensoring with the locally free sheaf A_{i,N}) this induces maps A_{2,N} → A_{1,N}. This is a compatible system of maps of Hopf algebras giving a morphism φ : G_{1} → G_{2} as desired.

In this manner we have reduced our problem to the case of integral normal schemes S of finite type over k. Let a : S′ → S be an alteration with S′ regular [2, Theorem 4.1]. By (i) [4, Main Theorem 1] we get a morphism φ′ : G_{1} ×_{S} S′ → G_{2} ×_{S} S′ corresponding to ψ|_{CRIS(S′/Σ)}. On the other hand, let U ⊂ S denote the open dense regular locus of the scheme S. Again by (i) there is a morphism φ_{U} : G_{1,U} → G_{2,U} corresponding to the restriction of ψ to U. By functoriality we have a^{∗}φ_{U} = φ′|_{a^{−1}(U)}.

We claim that this implies that there is a unique φ : G_{1} → G_{2} over S that restricts back to φ_{U} and φ′. This is proven in exactly the same way as above using the following cartesian diagram of sheaves on S:

O_{S} ⟶  j_{∗}O_{U}
↓  ↓
a_{∗}O_{S′} ⟶ a_{∗}j′_{∗}O_{a^{−1}(U)}

Notations used: U⟶^{j}S inclusion, a^{−1}(U)⟶^{j′}S′ inclusion.

Exercise. Prove that the diagram is cartesian. (Hint: use that S is normal.)

Essential surjectivity up to isogeny

We can try to prove more. One can ask whether 𝔻 is essentially surjective up to isogeny for certain S.

A more precise question is the following: given an object E ∈ DC_{S} does there exist an n ∈ ℕ and a G ∈ BT_{S} such that 𝔻(G) ≅ E^{(n)}? Here E^{(n)} is the pull back of E under the nth iterate of the absolute Frobenius morphism of S. This formulation is suggested by the following two facts: (1) any Dieudonné crystal E is isogenous to E^{(1)}, and (2) the question has a positive answer if S is the spectrum of an Artinian ring. The result (2) is a consequence of the fact that if (A, 𝔪, k) is Artinian local of characteristic p, then for some n the nth iterate of Frobenius factors as A → k → A (and the fact that 𝔻 is an equivalence over k, see [4, Main Theorem 1] and the remark following the lemma above).

Let us consider an integral normal scheme S separated and of finite type over the field k with a finite p-basis. Let U ⊂ S be the dense open regular subscheme of S and let a : S′ → S be an alteration with S′ regular [2, Theorem 4.1]. Let E be a Dieudonné crystal over S. Using (i) we deduce the existence of G_{U} ∈ BT_{U} corresponding to E|_{U} and G′ ∈ BT_{S′} corresponding to E|′_{S} which have isomorphic restrictions over a^{−1}(U). Furthermore, for every point s ∈ S the restriction of G′ to the reduced fibre T := a^{−1}(s)_{red} is constant. This follows from [4, 5.1.1] and the fact that E|_{CRIS(T/Σ)} is constant (being the pull back of E_{s} = E|_{CRIS(s/Σ)}). We want to prove that this implies that (G′)^{(n)} descends to a p-divisible group G over S for some n. If we consider the sheaves of Hopf algebras associated to G′[p^{N}] and G_{U}[p^{N}] then we see that we get a triple as described below. Thus we want to know for which special situations S′ → S ⊃ U the answer to the question below is positive.

Let S be an integral scheme of characteristic p, let U ⊂ S be open dense, and let a : S′ → S be an alteration. We consider triples (F′, F_{U}, α), where

(a) F′ is a locally free O_{S′}-module,

(b) F_{U} is a locally free O_{U}-module,

(c) α : a^{∗}F_{U} → F|_{a^{−1}(U)} is an isomorphism, and

(d) for all s ∈ S the restriction of F′ to the scheme T := a^{−1}(s)_{red} is free (i.e.,

F′|_{T} ≅ O_{T}^{m} for some m). In the following we use F to denote the Frobenius morphism of both S and S′.

Question. Under what conditions on S, a : S′ → S and U does there exist an n ∈ ℕ with the following property: for every triple (F′, F_{U}, α) as above there exists a locally free sheaf F on S such that

a^{∗}F ≅ (F^{n})^{∗}(F′) and F|_{U} ≅ (F^{n})^{∗}F_{U} ?

Remarks. (a) In general we cannot find such an integer n. For example, when p ≠ 2, take a : S′ = Spec(k[x, y]) → S = Spec(k[x^{2}, xy, y^{2}]) and U = S \ {(x^{2}, xy, y^{2})} to get a counter example. Indeed, take F′ = O_{S′}, let G be the coherent sheaf on S corresponding to the ideal (x^{2}, xy) in the ring k[x^{2}, xy, y^{2}], and let F_{U} = G|_{U}. Then F′ and F_{U} are locally free of rank 1, and we leave it as an exercise to show the existence of a map α. We also leave it as an exercise to show that F_{U}^{⊗n} ≅ F_{U} is n is odd: the Picard group of U has order 2 and F_{U} is a generator. Finally, note that G = j_{∗}F_{U}. Hence for every extension of (F^{n})^{∗}(F_{U}) = F_{U}^{⊗p^{n}} ≅ F_{U} to a locally free sheaf F on S we have F ≅ j_{∗}(F|_{U}) ≅ j_{∗}(F_{U}) ≅ G. As G is not free, F is not free, a contradiction.

(b) The answer to the question is yes if S′ → S is the resolution of singularities of a normal surface in characteristic p, this the reader can prove by working through the set of exercises at the end of this note. Thus it seems quite likely that the above considerations will lead to a proof of essential surjectivity up to isogeny for the functor 𝔻 in the case of normal surfaces of finite type over k.

(c) Using (b) we can try to bootstrap (e.g. by trying to use (b) to prove the existence of a G outside of codimension 3 and then working on the question above in the case where S \U has codimension at least 3). We do not want to get into this here, but this demonstrates a general principle in trying to apply the alteration theorems: some work remains relating upstairs to downstairs. This work usually involves a trace map (in the case of cohomological applications) or some kind of descent question (like the question above).

Exercises. Let a : S′ → S be the resolution of singularities of a normal surface of finite type over a field of characeristic p. Let U = S^{reg} be the regular locus of S. Prove that there exists an n ∈ ℕ as in the question by completing the following exercises:

(i) Show that it suffices to prove this when S has a unique singular point s ∈ S. (ii) Let E ⊂ S′ be the reduced exceptional fibre, and let I ⊂ O_{S′} be its ideal sheaf. Show that there exists a natural number n ∈ ℕ such that for all m ≥ n we have H^{1}(E, I^{m}/I^{m+1}) = (0). (iii) Let E_{n} be the nth infinitesimal neigbourhood of E in S′, i.e., the closed subscheme defined by I^{n+1}. Show that the nth iterate of the Frobenius morphism of E_{n} factorizes as E_{n} → E → E_{n}, where E → E_{n} is the natural closed immersion. (iv) Let F′ be a locally free sheaf of rank r over S′ such that F′|_{E_{n}} ≅ O_{E_{n}}^{r}. Show that a_{∗}F′ is locally free, and that F′ ≅ a^{∗}a_{∗}F′.

(v) Show that the integer n found in (ii) works. Hint: apply (iv) to the sheaf (F^{n})^{∗}(F′).

References

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[2] A.J. de Jong, Smoothness, semi-stability and alterations, Publ. Math. I.H.E.S. 83, 51–93 (1996).

[3] A.J. de Jong, Families of curves and alterations, Ann. de l’institut Fourier 47, 599– 621 (1997).

[4] A.J. de Jong, Crystalline Dieudonné module theory via formal and rigid geometry, Publ. Math. I.H.E.S. 82, 5–96 (1995).

[5] A.J. de Jong and W. Messing, A few remarks on the crystalline Dieudonné module functor, in preparation.

[6] J. Tate, p-divisible groups, Proceedings of a conference on local fields, Driebergen (1966), Springer-Verlag.

[7] A. Grothendieck, Groupes de Barsotti-Tate et cristaux, Actes, Congrès intern. math., 1970. Tome 1, pp. 431-436.

[8] A. Grothendieck, Groupes de Barsotti-Tate et cristaux de Dieudonné, Séminaire de mathématiques supérieures, Université de Montréal, Les presses de l’université de Montréal, 1974.

[9] P. Berthelot, Cohomologie cristalline des schémas de charactéristique p > 0, Springer Lecture Notes, 407, 1974.

[10] P. Berthelot and A. Ogus, Notes on crystalline cohomology, Princeton University Press, 1978.

[11] P. Berthelot, L. Breen, W. Messing, Théorie de Dieudonné cristalline II, Springer Lecture Notes, 930, 1982.

[12] A.J. de Jong, Homomorphisms of Barsotti-Tate groups and crystals in positive characteristic, Inventiones Mathematicae 134, 1998.