Crystalline Dieudonné theory over excellent schemes

by A.J. de Jong, Massachusetts Institute of Technology and W. Messing, University of Minnesota

In respectful memory of Bernard Dwork (1923 - 1998)

In this paper we add to the faithfulness results that have been obtained for the crystalline Dieudonné functor, see [3] and [4]. We pay particular attention to complete intersections and obtain fairly good results. The strongest results can be phrased loosely as follows: (a) we remove the hypothesis of finite p-basis from [4], (b) for p-divisible groups the functor is fully faithful up to isogeny over excellent local rings, and (c) for p-divisible groups 𝔻 is fully faithful over excellent schemes which are locally complete intersections.

There are two new tools used in this paper. First, we make full use of the strong Néron desigularization of Popescu [8]. Second, we use the extension theorem for homomorphisms of p-divisble groups that was obtained in [5].

1 Finite group schemes

Let S be a scheme in characteristic p. Let us write FG_{S} for the category of finite locally free group schemes over S, commutative and annihilated by some power of p. On the other hand, we denote C_{S} the category of Dieudonné crystals of finite presentation over S (see [3, 2.4]). We consider the functor [2, 3.1.5]

(1.1)  𝔻 : FG_{S} ⟶ C_{S}.

We want to know for which base schemes S this functor is faithful. This is not true in general; a counter example of Koblitz and Ogus can be found in [3, 4.4.1].

Note that if A → B is an injective 𝔽_{p}-algebra homomorphism and (1.1) is faithful over S = Spec B then (1.1) is faithful over Spec A. The same holds for injective families of ring homomorphisms A → B_{i}. In this way we see that (1.1) is faithful over any reduced scheme S, since 𝔻 is (fully) faithful over a field, see [3, 4.1.1].

The other case we can handle is that of locally complete intersections. We recall that a Noetherian local ring A is said to be a complete intersection, if its completion A^{∧} is of the form R/I, where R is a regular local ring and I is generated by a regular sequence in R.

1.2 Proposition. Suppose the scheme S is locally Noetherian and all its local rings are complete intersections. The functor 𝔻 (1.1) is faithful.

Proof. Suppose that φ : G_{1} → G_{2} is a morphism of FG_{S} and that 𝔻(φ) = 0. We want to prove that φ = 0. The question is local on S, hence we may assume there exists an n ∈ ℕ such that both G_{1} and G_{2} are killed by p^{n}. Consider the scheme

S′ = G_{1} ×_{S} G_{2}^{D}

over S. Here G_{2}^{D} is the Cartier dual of G_{2}, see [7] for example. Note that the scheme S′ also satisfies the hypothesis of the proposition, since S′ → S is a locally complete intersection morphism, see [10, III 4.15]. Over S′ we have the morphism of group schemes ℤ/p^{n}ℤ_{S′} → G_{1,S′} determined by pr_{1} : S′ → G_{1}. Similarly, we have G_{2,S′} → μ_{p^{n},S′} given by pr_{2} : S′ → G_{2}^{D}. Composing these with φ_{S′} we get an element

ζ_{φ} ∈ Hom_{S′}(ℤ/p^{n}ℤ_{S′}, μ_{p^{n},S′}) = μ_{p^{n}}(S′).

By functoriality, we have 𝔻(ζ_{φ}) = 0. Finally, we remark that φ = 0 if and only if ζ_{φ} = 1. Indeed, φ ≠ 0 if and only if there is an S-scheme T, x ∈ G_{1}(T), μ ∈ G_{2}^{D}(T) such that μ(φ(x)) ≠ 1 in O_{T}^{∗}. The universal pair (x, μ) is (pr_{1}, pr_{2}). For this pair pr_{2}(φ(pr_{1})) = ζ_{φ}. Hence if ζ_{φ} = 1, then φ = 0 and conversely.

In this way we are reduced to the case G_{1} = (ℤ/p^{n}ℤ)_{S} and G_{2} = μ_{p^{n},S}. Thus we are given ζ ∈ μ_{p^{n}}(S) with ℓ_{n}(ζ) = 0, see [3, 2.2.4] for explanation. We want to show that ζ = 1.

To do this we may replace S by the spectrum of one of its local rings: S = Spec A. Since A → A^{∧} is injective, we reduce to the case of a complete local ring. Hence by the Cohen structure theorem we have A = k[[x_{1}, …, x_{N}]]/I. By our assumption on S we know that I can be generated by a regular sequence f_{1}, …, f_{c} in k[[\underline{x}]]. The ring k[[\underline{x}]]/(f_{i}) is the projective limit of Artinian complete intersection rings of the form

k[[x_{1}, …, x_{N}]]/(f_{1}, …, f_{c}, g_{c+1}, …, g_{N}).

Thus, by the remarks preceding the proposition, it suffices to do the case of an Artinian complete intersection

A = k[x_{1}, …, x_{N}]/(f_{1}, …, f_{N}).

Clearly, we may suppose k is algebraically closed.

Let W = W (k) be the ring of Witt vectors of k. Choose lifts f̃_{i} ∈ W [x_{1}, …, x_{N}] of the polynomials f_{i}. Let

D := W [\underline{x}][\frac{f̃_{i}^{m}}{m!} ] ⊂ W [\underline{x}] ⊗ ℚ.

The ring D is the divided power envelope of (f̃_{1}, …, f̃_{N}) in W [\underline{x}], divided powers compatible with the canonical ones on (p). The argument now runs as in [3, 4.1.5]. Let D_{n} = D/p^{n}D. Take a lift ζ̂ ∈ W [\underline{x}] of ζ ∈ k[\underline{x}]/(f_{i}). The element ℓ_{n}(ζ) is given by

ℓ_{n}(ζ) = log(ζ̂^{p^{n}}) ∈ D_{n}.

The PD-completion of D_{n} is denoted D_{n}^{∧}. The nullity of ℓ_{n}(ζ) implies

1 = exp log(ζ̂^{p^{n}}) = ζ̂^{p^{n}} ∈ D_{n}^{∧}.

However, the map D_{n} → D_{n}^{∧} is injective (see loc. cit.), hence ζ̂^{p^{n}} = 1 in D_{n} also. Note, as in [3, 4.1.4] that the map Φ : A ⟶ D/pD is injective. Thus ζ̂^{p^{n}} = 1 implies ζ^{p^{n−1}} = 1 in A. In this case ℓ_{n}(ζ) = pℓ_{n−1}(ζ). As the ring D is without p-torsion, we see that ℓ_{n}(ζ) = 0 implies ℓ_{n−1}(ζ) = 0. We win by induction on n.

1.3 Remark. (a) Using the proposition we may conclude that 𝔻 is faithful for more general base schemes S. For example if S allows a locally complete intersection morphism f : S → T to a reduced scheme T. Indeed, we may assume S and T are affine and that f is given by the ring homomorphism R → A. By a limit argument we reduce to the case that R is Noetherian. In this case the family of homomorphisms A → A ⊗_{R} k, indexed by homomorphisms R → k into fields, is injective. To each of the A ⊗_{R} k we may apply our proposition. This result will not be used in the sequel.

(b) If R is a reduced ring with only finitely many minimal primes (e.g. a Noetherian ring or an integral domain) and A is a syntomic R-algebra, then the map A → ∏_{𝔭∈Spec R} A/𝔭A is obviously injective. We ask: Is this still true for an arbitrary reduced R? If so, then one can avoid the passage to the limit argument just given.

1.4 Remark. Over S as in the proposition the functor (1.1) is not fully faithful in general, see [3, 4.4.1]. In addition, over such S, the functor 𝔻 is not fully faithful on the category of truncated Barsotti-Tate groups. An explicit example of this, answering [3, question 4.4.3] in the negative, is given as follows. With regard to homomorphisms of ℤ/pℤ into μ_{p}, the question is whether the map

ℓ_{1} : μ_{p}(S) ⟶ Γ(S/Σ_{1}, O_{S/Σ_{1}})^{F =0}

is surjective (Σ_{1} = Spec 𝔽_{p}). If S = Spec 𝔽_{p}[t]/(t^{p}) the right hand side contains the infinity of elements (t^{p})^{[n]}, n ≥ 1, compare [3, 4.4.1]. But μ_{p}(S) is finite in this case.

2 Removing the hypothesis of finite p-basis from [4]

Let k be a field of characteristic p. For a scheme S of finite type over Spec k, we consider the question whether 𝔻 is fully faithful up to isogeny on p-divisible groups over S. In [4] it was shown that if k has a finite p-basis, then the answer is yes. We will show here that the general case follows from the special case, more precisely from [4, 5.1.1].

As in [4] it suffices to do the case that S is reduced. Let us choose a p-basis {x_{i}}_{i∈I} of k. For any subset J ⊂ I we put

k_{J} := ⋃_{n∈ℕ}k[y_{j,n}, j ∈ J]/(y_{j,n}^{p^{n}} − x_{j}).

This is a purely inseparable field extension of k with p-basis {x_{i}}_{i∈I\J}. Hence, if #(I \J) < ∞, then k_{J} has a finite p-basis. Furthermore, the fields k_{J_{1}} and k_{J_{2}} are linearly disjoint if J_{1} ∩ J_{2} = ∅.

We claim there is a finite subset V ⊂ I such that S ⊗k_{J} is a reduced scheme whenever J ∩ V = ∅. To see this, let {η_{ℓ}} be the (finite) set of generic points of S. For each ℓ, take a finite subset V_{ℓ} ⊂ I such that k(η_{ℓ}) ∩ k^{perf} ⊂ k_{V_{ℓ}}. The field k(η_{ℓ}) is a finitely generated field extension of k, hence this is possible. Put V = ∪_{ℓ}V_{ℓ}; this is still finite. For any J ⊂ I with J ∩ V = ∅ the tensor products k(η_{ℓ}) ⊗_{k} k_{J} are fields. For U ⊂ S open, the map

Γ(U ⊗ k_{J}, O) ⟶ ∏_{η_{ℓ}∈U} k(η_{ℓ}) ⊗_{k} k_{J}

is injective since it is the flat base change of Γ(U, O) → ∏ k(η_{ℓ}). This proves the claim.

Let G_{1}, G_{2} be p-divisible groups over S and let φ : 𝔻(G_{2}) → 𝔻(G_{1}) be given. As in [4, 5.1] we are going to prove the stronger statement: there exists a unique ψ : G_{1} → G_{2} such that 𝔻(ψ) − φ is torsion.

Put J := I \ V. We denote by an index _{J} the base change of an object over S to S ⊗ k_{J}. By [4, 5.1.1] there exists a unique ψ_{1} : G_{1,J} → G_{2,J} such that 𝔻(ψ_{1}) − φ_{J} is torsion. For any n ∈ ℕ the map ψ_{1}[p^{n}] : G_{1}[p^{n}]_{J} → G_{2}[p^{n}]_{J} exists over k_{J′} for some finite J′ ⊂ I. Next, choose J_{2} = I \ (V ∪ J′). As before we get a unique ψ_{2} : G_{1,J_{2}} → G_{2,J_{2}} such that 𝔻(ψ_{2}) − φ_{J_{2}} is torsion. By uniqueness we see that ψ_{2,J} = ψ_{1}. Therefore the morphisms ψ_{1}[p^{n}] and ψ_{2}[p^{n}] are defined over the linearly disjoint fields k_{J′} and k_{J_{2}}, and agree over the compositum k_{J} · k_{J_{2}} ⊂ k^{perf}. Clearly this gives G_{1}[p^{n}] → G_{2}[p^{n}] over S.

Putting these together for varying n gives a homomorphism ψ : G_{1} → G_{2}. We know that 𝔻(ψ)−φ becomes torsion over S⊗k_{J}. By Lemma 4.1 for example we see that 𝔻(ψ)−φ is torsion. This completes the proof of the following theorem.

2.1 Theorem. The crystalline Dieudonné module functor is fully faithful up to isogeny for p-divisible groups over any scheme of finite type over any field of characteristic p.

3 Locally complete intersections over fields

We fix a field k of characteristic p. Let S → Spec k be a locally complete intersection morphism. Consider p-divisible groups G_{1}, G_{2} over S and a morphism of Dieudonné crystals φ : 𝔻(G_{2}) → 𝔻(G_{1}). Assume in addition that S is quasi-compact, then by the theorem above we can find ψ : G_{1} → G_{2} over S such that 𝔻(ψ) = p^{n}φ for some n ∈ ℕ. This means that ψ[p^{n}] : G_{1}[p^{n}] → G_{2}[p^{n}] satisfies 𝔻(ψ[p^{n}]) = 0. By Proposition 1.2 we get ψ[p^{n}] = 0. This implies that ψ is divisible by p^{n}, i.e., ψ = p^{n}ψ′. We derive the relation p^{n}𝔻(ψ′) = p^{n}φ. This in turn implies 𝔻(ψ′) = φ, since the universal divided power envelopes of affine opens of S are p-torsion free by our assumption on S (compare Lemma 4.7). For a not necessarily quasi-compact S, we can glue the local ψ′ because of the faithfulness given in Proposition 1.2. We have proved the following theorem.

3.1 Theorem. If S → Spec k is a locally complete intersection morphism then 𝔻 is fully faithful on p-divisible groups over S.

While this theorem will be made obsolete by the stronger Theorem 4.6 below, the above proof is (much) more elementary.

4 Barsotti-Tate groups over excellent schemes

Let A be an 𝔽_{p}-algebra. Let I be the kernel of the canonical surjection ℤ_{p}[A] → A. We write D(A) = D_{I}(ℤ_{p}[A]) for the divided power algebra of ℤ_{p}[A] with respect to the ideal I. Furthermore, we let D̂(A) denote its p-adic completion. Recall that any quasi-coherent crystal E on Spec A gives rise to a D̂(A)-module M(E) defined by

M(E) = lim_{⟵} Γ((Spec D(A)/p^{n}D(A), Spec A, [ ]), E).

The functor E ↦ M(E) is faithful.

4.1 Lemma. Suppose that A → B is a homomorphism of 𝔽_{p}-algebras such that B = lim_{⟶} B_{λ}, with A-algebras B_{λ} which are faithfully flat and relative complete intersections over A. Then the pull back functor is faithful:

quasi-coherent crystals/ Spec A ⟶ quasi-coherent crystals/ Spec B

Proof. The kernel of D(A)/p^{n}D(A) → A is a nil ideal. Hence the homomorphism A → B_{λ} lifts to a faithfully flat homomorphism D(A)/p^{n}D(A) → B̃_{λ} in view of the conditions on A → B_{λ}. By flatness there is a unique divided power structure on Ker(B̃_{λ} → B_{λ}) compatible with [ ] on the divided power ideal of D(A)/p^{n}D(A). Hence we get a factorization D(A)/p^{n}D(A) → D(B_{λ})/p^{n}D(B_{λ}) → B̃_{λ}. We conclude that the functor from D(A)/p^{n}D(A)-modules to D(B_{λ})/p^{n}D(B_{λ})-modules is faithful. The result of the lemma follows as D(B)/p^{n}D(B) = lim_{⟶} D(B_{λ})/p^{n}D(B_{λ}).

4.2 Here is another principle that can be used to study crystals. Let B be a ring, I ⊂ B an ideal with p ∈ I, and A = B/I. Assume that B is written as an inductive limit B = lim_{⟶} B_{λ} with B_{λ} flat over ℤ_{p} and such that B_{λ}/pB_{λ} has a p-basis for all λ. Let I_{λ} be the inverse image of I in B_{λ}, and set A_{λ} = B_{λ}/I_{λ}. Set D_{λ} = D_{I_{λ}}(B_{λ}), and D = D_{I}(B). In this situation the natural functor

quasi-coherent crystals of finite presentation/ Spec A ⟶ D̂-modules

is faithful.

Proof. Let us write C_{A} for the category on the left and C(p^{n})_{A} for the full subcategory of crystals annihilated by p^{n}. Since A = lim_{⟶} A_{λ} we see that for each n ∈ ℕ there is an equivalence of categories

C(p^{n})_{A} ⟶ lim_{⟶} C(p^{n})_{A_{λ}}.

The assumption that B_{λ}/pB_{λ} has a p-basis implies that the natural functor

C(p^{n})_{A_{λ}} ⟶ D̂_{λ}-modules

is faithful. This follows from [3, 1.2.6]: If u : M → N is a morphism of C(p^{n})_{A}, then for any divided power thickening C of A_{λ} with p^{n}C = 0, we can find a homomorphism B_{λ}/p^{n} → C (of thickenings of A_{λ}). Hence D_{λ} will map to C. So, if the value of u on D_{λ} is zero, then so is its value on all such C.

We also have D = lim_{⟶} D_{λ}.

The result follows formally from the assertions above. Indeed, let φ : E_{1} → E_{2} be a morphism in C_{A} such that φD̂ = 0. Pick n ∈ ℕ. There exists a λ and φ_{λ} : E_{λ,1} → E_{λ,2} in C(p^{n})_{A_{λ}} such that φ_{λ} represents φ mod p^{n} in C(p^{n})_{A}. As D/p^{n}D = lim_{⟶} D_{λ}/p^{n}D_{λ} we may assume by increasing λ that (φ_{λ})D̂_{λ} = 0. Hence φ_{λ} = 0, and thus φ mod p^{n} = 0. This holds for all n and we are done.

In the proof of the following proposition we are going to use the result of Popescu that any regular ring homomorphism is the limit of smooth homomorphisms. See [8], see also [9].

4.3 Proposition. Let A be an excellent local ring of characteristic p. Put S = Spec A and let s ∈ S be the closed point. If φ : E_{1} → E_{2} is a morphism of nondegenerate F -crystals with φ|_{CRIS(s/Σ)} = 0, then φ is torsion.

Proof. The proof is in three steps. The first step is to reduce to the case of a complete local ring. Indeed, let A → A^{∧} be the map of A into its completion. This is a regular ring homomorphism as A is excellent. According to the references above A^{∧} is a limit of smooth A-algebras. Hence by Lemma 4.1 we may replace A by A^{∧}.

The second step is to reduce to the case of algebraically closed residue field. We may write A = Λ[[x_{1}, …, x_{N}]]/I, where Λ is a Cohen ring for the residue field k of A. By the references given above, the ring B = Λ[[\underline{x}]] is the limit of essentially smooth local Λ-algebras B_{λ}. For each λ, the quotient B_{λ}/pB_{λ} has a p-basis, see [6]. Therefore we may apply the principle explained in subsection 4.2. We conclude that the functor from F -crystals to D̂ = D̂_{I}(B)-modules is faithful.

Let Λ̄ = W (k̄), where k̄ is an algebraic closure of k. Choose any map Λ → Λ̄ reducing to k → k̄ modulo p; this is automatically flat. Let Ī = I · Λ̄[[\underline{x}]], and set Ā = Λ̄[[\underline{x}]]/Ī. Write D̄ = DĪ(Λ̄[[\underline{x}]]). Just as in the case of A, there is a faithful functor comparing crystals over Ā to D̂Ī(Λ̄[[\underline{x}]])-modules. There is also a commutative diagram of functors:

crystals on A ⟶ D̂-modules
↓  ↓
crystals on Ā ⟶ ˆ D̄-modules

Thus if the right vertical arrow is faithful on p-adically separated modules, then we have performed our reduction step. For this it suffices to show that the ring maps D/p^{n}D → D̄/p^{n}D̄ are faithfully flat. But Λ[[\underline{x}]] → Λ̄[[\underline{x}]] is faithfully flat (as the completion of the map of the corresponding polynomial rings localized with respect to the maximal ideal (p, x_{1}, …, x_{n})). This implies that

D̄ ≅ D ⊗_{Λ[[\underline{x}]]} Λ̄[[\underline{x}]]

and we are done with the second step.

The third step. Here we write A = k[[x_{1}, …, x_{N}]]/I. Let W = W (k) be the ring of Witt vectors of k (k is algebraically closed). Choose f_{1}, …, f_{r} ∈ W [[\underline{x}]] mapping to generators of I in k[[\underline{x}]] = W [[\underline{x}]]/(p). Let D̂ be the p-adic completion of the divided power algebra of W [[\underline{x}]] with respect to the ideal (p, f_{1}, …, f_{r}). We write (as usual) σ : W [[\underline{x}]] → W [[\underline{x}]] for the Frobenius linear map with σ(x_{i}) = x_{i}^{p}. It induces a map σ on D̂ as well.

We apply [4, 2.2.2]; in order to do so we need that k has a finite p-basis, which is true since k is algebraically closed. The F -crystals E_{i} are given by triples (M_{i}, ∇_{i}, F_{i}), where M_{i} is a finite free D̂-module, ∇_{i} is an integrable connection and F_{i} is a horizontal linear map M_{i} ⊗_{σ} D̂ → M_{i}. Nondegeneracy of E_{i} means that the kernel and cokernel of F_{i} are annihilated by a power of p. The morphism φ is given by φ : M_{1} → M_{2} which is horizontal and commutes with the F_{i}. There is a unique map D̂ → W which maps x_{i} to zero and the restriction φ|_{CRIS(s/Σ)} corresponds to the induced map M_{1} ⊗ W → M_{2} ⊗ W.

Let X be the rigid analytic n-dimensional open unit ball over K, the fraction field of W, with coordinates x_{1}, …, x_{N}. More precisely, X = Spf(W [[\underline{x}]])^{rig}, see [4, Section 7]. The space X may be considered as a “tube” for the scheme s = Spec k. Recall that σ induces a general morphism σ^{rig} : X → X, see [4, 7.2.6].

We introduce some more notation. We set

V = {x ∈ X, |f_{i}(x)| ≤ |p| }.

We let B be the quotient of the ring C = W [[\underline{x}]]\{y_{1}, …, y_{r}\}/(py_{i} − f_{i}) by its p-power torsion. Thus Spf B is a formal scheme in FS_{W} (see [4, Section 7]) to which we may associate a rigid analytic space by Berthelot’s construction. The continuous ring homomorphism W [[\underline{x}]] → B gives rise to a canonical morphism (Spf B)^{rig} → X, which maps (Spf B)^{rig} into V. We leave it to the reader to show that this is in fact an isomorphism of (Spf B)^{rig} with V. In particular, the canonical map B → Γ(V, O_{V}) is injective.

There is a W [[\underline{x}]]-algebra homomorphism ρ : D̂ → B, given by f_{i}^{[n]} ↦ \frac{f_{i}^{n}}{n!} = (\frac{f_{i}}{p})^{n} · \frac{p^{n}}{n!} . Thus φ induces a map of vector bundles over V by tensoring

φ_{V} : M_{1} ⊗_{D̂,ρ} O_{V} ⟶^{φ⊗_{ρ}1D̂} M_{2} ⊗_{D̂,ρ} O_{V} .

These locally free O_{V} -modules have integrable connections and the map φ_{V} is horizontal. By assumption, φ_{V} is zero in the point \underline{0} given by (x_{1}, …, x_{N}) = (0, …, 0) of V. Let V′ be the connected component of V containing the point \underline{0}. Since V′ is smooth and φ_{V} horizontal it follows that φ_{V} is zero on V′.

Note that σ^{rig}(V) ⊂ V and that F_{i} induces an isomorphism of M_{i,V} = M_{i} ⊗ O_{V} with (σ^{rig})^{∗}(M_{i,V}). Furthermore, φ_{V} commutes with these identifications. There is a small open ball around \underline{0} which is contained in V′, as V′ is an admissible open of X. This implies that any point x ∈ X is mapped into V′ by some power of σ^{rig}. In particular, any connected component V′′ ⊂ V of V is mapped into V′ by some power of σ^{rig}. Now we can use the compatibility of φ_{V} with F_{1} and F_{2} to deduce that the restriction of φ_{V} to V′′ is zero as well. We conclude that φ_{V} = 0, hence that φ ⊗_{ρ} 1_{B} is zero (see above).

We want to show this implies that φ is torsion. To do this we will define a homomorphism of rings

τ : B ⟶ D̂/p-power torsion

such that τ ∘ ρ = σ. Once we have τ, we immediately deduce that φ ⊗_{σ} 1D̂ is torsion. However, this implies that φ is p-power torsion as p^{m}φ equals

M_{1} ⟶^{V_{1}} M_{1} ⊗_{σ} D̂ ⟶^{φ⊗1} M_{2} ⊗_{σ} D̂ ⟶^{F_{2}} M_{2},

for a suitable m and V_{1} such that F_{1} ∘ V_{1} = p^{m} (such a Verschiebung V_{1} exists as E_{1} is a nondegenerate F -crystal).

We come to the construction of τ. Recall that B is the quotient of

C = W [[\underline{x}]]\{y_{1}, …, y_{r}\}/(py_{i} − f_{i})

by its p-power torsion. Thus it suffices to define a suitable map from C to D̂. Let us write σ(f_{i}) = f_{i}^{p} + pu_{i} for certain u_{i} = u(f_{i}) ∈ W [[\underline{x}]]. Computing formally, we should have

τ (\frac{f_{i}}{p} ) = \frac{σ(f_{i})}{p} = \frac{f_{i}^{p} + pu_{i}}{p} = (p − 1)!f_{i}^{[p]} + u_{i}.

Hence we would like to map y_{i} to (p − 1)!f_{i}^{[p]} + u_{i}. (We remark that in [4, 5.5] there is a mistake in the formula for τ(i/p): the correction term u(i) was omitted.) A general element of C can be written in the form

f = ∑_{I} a_{I}y_{1}^{i_{1}} ⋯ y_{r}^{i_{r}},

where a_{I} → 0 in the (p, \underline{x})-adic topology of W [[\underline{x}]]. By the above, the image of f should be the formal expression

∑_{I} σ(a_{I}) ( ∑_{j_{1},...,j_{r}} \binom{i_{1}}{j_{1}} ⋯ \binom{i_{r}}{j_{r}} \frac{(j_{1}p)! ⋯ (j_{r}p)!}{p^{j_{1}} ⋯ p^{j_{r}}}f_{1}^{[j_{1}p]} ⋯ f_{r}^{[j_{r}p]}u_{1}^{i_{1}−j_{1}} ⋯ u_{r}^{i_{r}−j_{r}} ) .

Let us make sense out of this. For each fixed multi-index J = (j_{1}, …, j_{r}) we let

b_{J} = ∑_{I} σ(a_{I})\binom{i_{1}}{j_{1}} ⋯ \binom{i_{r}}{j_{r}}u_{1}^{i_{1}−j_{1}} ⋯ u_{r}^{i_{r}−j_{r}}.

This is a well defined element of W [[\underline{x}]] as σ(a_{I}) → 0 in the complete ring W [[\underline{x}]]. Then we form

τ(f) := ∑_{J} b_{J} \frac{(j_{1}p)! ⋯ (j_{r}p)!}{p^{j_{1}} ⋯ p^{j_{r}}}f_{1}^{[j_{1}p]} ⋯ f_{r}^{[j_{r}p]}.

This converges p-adically as one sees by looking at the p-adic valuations of \frac{(jp)!}{p^{j}}.

Here is an observation which allows us to conclude that the construction above is well defined and is a homomorphism. Modulo p^{n} we are mapping into the W [[\underline{x}]]-submodule of D̂/p^{n} generated by the elements f_{1}^{[j_{1}p]} ⋯ f_{r}^{[j_{r}p]}, with 0 ≤ j_{i} ≤ p^{n}. Since this is a finite module, it is separated and complete in the (p, \underline{x})-topology and our map is continuous for this topology (by construction). Thus we need only to check that our map is well defined and a homomorphism on the elements of W [[\underline{x}]][y_{1}, …, y_{r}]/(py_{i} − f_{i}). This we leave to the reader. The property τ ∘ ρ = σ was built into the construction. This ends the proof of Proposition 4.3.

4.4 Proposition. Let A be an excellent local ring of characteristic p. The functor 𝔻 from the category of Barsotti-Tate groups over A to the category of Dieudonné crystals over A is fully faithful up to isogeny.

Proof. We will argue by induction on the dimension of A. We remark that the statement for A is equivalent to the statement for A_{red} = A/ Nil(A), see [4, proof of 5.1.2]. The case of dim A = 0, i.e., the case of a field, is [3, 4.1.1].

Let us assume that A is reduced. Let A′ be the product of the normalizations of A/𝔭, where 𝔭 runs through the minimal primes of A. The morphism Spec A′ → Spec A is finite and is an isomorphism outside a nowhere dense closed subset. The annihilator I, of A′/A, defines this closed subset, and

A = A′ ×_{A′/IA′} A/I.

If we have the result of the proposition for the rings A′, A/I and A′/IA′, then the result for A follows. Indeed, let G_{1} and G_{2} be Barsotti-Tate groups over Spec A and let φ : 𝔻(G_{2}) → 𝔻(G_{1}) be a homomorphism of Dieudonné crystals. By our assumption we get ψ_{A′}, ψ_{A/I}. Again by assumption, there will be n, m ∈ ℕ such that p^{n}ψ_{A′}|_{A′/IA′} = p^{m}ψ_{A/I}|_{A′/IA′}. The displayed formula shows that this implies there exists a ψ : G_{1} → G_{2} such that p^{n}ψ_{A′} = ψ|_{A′} and p^{m}ψ_{A/I} = ψ|_{A/I}. We still have to check that 𝔻(ψ) − φ is torsion. This follows from Proposition 4.3, as we have the desired equality over Spec A/𝔪_{A} by definition of ψ_{A/I}. Finally, observe that dim A/I < dim A and dim A′/IA′ < dim A, hence we reduce to the case where A is a normal domain.

The functor of the proposition is faithful as A is reduced, see Section 1. Let G_{1} and G_{2} be Barsotti-Tate groups over Spec A and let φ : 𝔻(G_{2}) → 𝔻(G_{1}) be a homomorphism of Dieudonné crystals. By fully faithfulness over fields we get a morphism ψ_{K} : G_{1,K} → G_{2,K} over the fraction field K of A such that 𝔻(ψ_{K}) = φ|_{CRIS(Spec K/Σ)}. By [5, Introduction] we can extend ψ_{K} to a homomorphism ψ : G_{1} ⟶ G_{2} over Spec A. Note that δ = 𝔻(ψ) − φ is a map of F -crystals such that δ_{K} = 0. We want to prove that δ is torsion.

Let A′ be the perfect closure of A (i.e., the integral closure of A in the perfect closure of K). The fact that δ_{K} = 0 implies that the action of δ on the values of 𝔻(G_{i}) over (Spec A′, Spec W_{n}(A′), [ ]) are zero. Hence, δ|_{CRIS(s′/Σ)} = 0, where s′ is the unique closed point of Spec A′. This implies δ|_{CRIS(s/Σ)} = 0, as the morphism s′ → s induces a faithful pull-back functor on crystals (by Lemma 4.1 for example). We conclude by the previous proposition.

4.5 Remark. What is the obstruction to proving this for a general excellent scheme? Over any normal excellent scheme one can argue by restricting to the generic points and then extending using [5]. However, it is problematic to characterize locally morphisms of F -crystals that are torsion.

4.6 Theorem. Let S be an excellent scheme all of whose local rings are complete intersections. Then the crystalline Dieudonné functor is fully faithful over S.

Proof. Let φ : 𝔻(G_{2}) → 𝔻(G_{1}) be a morphism of Dieudonné crystals of two p-divisible groups over S. By Proposition 4.4 we get for all points s ∈ S a morphism

ψ_{s} : G_{1,Spec O_{S,s}} ⟶ G_{2,Spec O_{S,s}}

such that 𝔻(ψ_{s}) = p^{n_{s}}φ|_{CRIS(O_{S,s}/Σ)} for some integers n_{s}. By the arguments preceding the proof of Theorem 3 and using Lemma 4.7 below, we see that we may actually choose ψ_{s} such that n_{s} = 0. The morphisms ψ_{s} and ψ_{s′} agree over

Spec O_{S,s} ×_{S} Spec O_{S,s′}

by faithfulness of 𝔻 over this scheme (Proposition 1.2). The usual descent gives ψ : G_{1} → G_{2} over S. Finally, we have to show that 𝔻(ψ) equals φ. For this it suffices to show that the functor

quasi-coherent crystals over S ⟶ ∏_{s∈S} quasi-coherent crystals over Spec O_{S,s}

is faithful. To see this it suffices to work with crystals annihilated by p^{n} (all n). If S is affine, say Spec A, then Spec D(A)/p^{n} → Spec A is bijective. If 𝔭 is a prime ideal of A, and 𝔭′ is the corresponding prime ideal of D(A)/p^{n}, then a morphism u of quasi-coherent crystals over S whose restriction to Spec A_{𝔭} is zero, gives a map of D(A)/p^{n}-modules whose localization at 𝔭′ is zero. If this holds for all primes 𝔭 ∈ Spec A, then the map of D(A)/p^{n}-modules is zero, i.e. u = 0.

4.7 Lemma. Let A be an excellent local ring of characteristic p which is a complete intersection (see Section 1). Then the p-adically completed universal divided power algebra D̂(A) (see beginning of this section) is p-torsion free.

Proof. We leave some of the details to the reader. Let A → A^{∧} be the map of A into its completion. We know that A^{∧} is the limit of smooth A algebras B_{λ}. Therefore D(A^{∧})/p^{n} is the limit of the faithfully flat D(A)/p^{n}-algebras D(B_{λ})/p^{n}. (Exercise: A → B syntomic implies D(A)/p^{n} → D(B)/p^{n} is flat.) This implies that the map D(A)/p^{n} → D(A^{∧})/p^{n} is a faithfully flat map. Thus we may replace A by A^{∧}.

Write A^{∧} = k[[x_{1}, …, x_{N}]]/(f_{1}, …, f_{c}), where f_{1}, …, f_{c} is a regular sequence. The map k[x_{1}, …, x_{n}] → k[[x_{1}, …, x_{n}]] is regular. Hence we can write k[[x_{1}, …, x_{n}]] as the limit, lim_{⟶} A_{λ} of smooth k[x_{1}, …, x_{n}]-algebras. We may also assume that for each λ there are f_{i,λ} ∈ A_{λ} mapping to f_{i} in k[[x_{1}, …, x_{n}]]. Of course, we may localize each A_{λ} and assume that A_{λ} is an essentially smooth local k[x_{i}]-algebra with residue field k. Obviously we have

A^{∧} = lim_{⟶} A_{λ}/(f_{1,λ}, …, f_{c,λ}).

We will show that f_{1,λ}, …, f_{c,λ} is a regular sequence in A_{λ}. Thus we have written A^{∧} as a limit of complete intersection rings B_{λ} essentially of finite type over k. If B is a complete intersection ring essentially of finite type over k, then D(B)/p^{n} is flat over ℤ/p^{n}ℤ (by the above exercise). Hence D(A^{∧})/p^{n} = lim_{⟶} D(B_{λ})/p^{n} is flat over ℤ/p^{n}ℤ. This implies that D̂(A^{∧}) is flat over ℤ_{p}.

To check that f_{1,λ}, …, f_{c,λ} is a regular sequence in A_{λ} we may pass to the completion. The reader checks immediately that

A_{λ}^{∧} ≅ k[[x_{1}, …, x_{n}, y_{1}, …, y_{N}]],

where N depends on λ, and where the y_{i} are chosen in the kernel of the canonical map A_{λ}^{∧} → k[[x_{i}]]. Thus A_{λ}^{∧}/(y_{1}, …, y_{N}, f_{1,λ}, …, f_{c,λ}) is isomorphic to k[[x_{1}, …, x_{n}]]/(f_{1}, …, f_{c}). We conclude by a dimension count that y_{1}, …, y_{N}, f_{1,λ}, …, f_{c,λ} is a regular sequence in A_{λ}^{∧}. Hence f_{1,λ}, …, f_{c,λ} is a regular sequence in A_{λ}^{∧} as desired.

References

[1] P. Berthelot, Cohomologie rigide et cohomologie rigide a support propre, Version provisoire du 9-08-1991.

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

[3] P. Berthelot and W. Messing, Théorie de Dieudonné cristalline III, in The Grothendieck Festschrift I, Progress in mathematics 86, Birkhäuser (1990), pp. 171-247.

[4] A.J. de Jong, Crystalline Dieudonné module theory via formal and rigid geometry, Publications Mathematiques 82 (1995), pp. 5-96.

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

[6] T. Kimura and N Niitsuma, Regular local rings of characteristic p and p-basis, Journal of the Mathematical Society of Japan 32 (1980), pp. 363–371.

[7] F. Oort, Commutative group schemes, Lecture Notes in Mathematics 15, Springer 1966.

[8] D. Popescu, General Néron desingularization, Nagoya Mathematics Journal 100 (1985), pp. 97–126.

[9] R.G. Swan, Néron-Popescu desingularization, Lecture Notes of the university of Chicago, pp. 1–44 (1995). To appear in: Proceedings of the International Conference on Algebra and Geometry, Taipei, Taiwan (1995).

[10] SGA3, Schémas en groupes I, by M. demazure and A. Grothendieck, Lecture Notes in Mathematics 151, Springer Verlag 1970.