Finite locally free group schemes in characteristic p

and Dieudonné modules

A. J. de Jong*

Introduction

This article stems from an attempt of the author to understand commutative finite locally free p-group schemes in characteristic p and in particular whether or not it is possible to describe them in terms of Dieudonné modules.

Let S_{0} → Spec(𝔽_{p}) be a scheme of characteristic p. Let C(1)_{S_{0}} be the category of commutative finite locally free group schemes over S_{0} killed by p.

If S_{0} is the spectrum of a perfect field k then classical Dieudonné theory gives us a contravariant functor G ⇝ M(G) of C(1)_{S_{0}} into the category of finite dimensional k-vectorspaces. Since there is a canonical isomorphism M(G^{(p)}) = M(G)^{(p)}, the module M(G) is endowed with the action of Frobenius F and Verschiebung V. The functor G ⇝ (M(G), F, V) is an anti equivalence of C(1)_{S_{0}} with the category of triples (M, F, G), where M is a finite dimensional k-vectorspace and F : M^{(p)} → M, resp. V : M → M^{(p)} are linear maps such that F ∘ V = 0 and V ∘ F = 0.

For arbitrary schemes S_{0} → Spec(𝔽_{p}) a natural candidate for the target category of a similar functor is the category M(1)_{S_{0}} of triples (M, F, V), where M is a locally free O_{S_{0}}- module of finite rank and F : M^{(p)} → M, resp. V : M → M^{(p)} are O_{S_{0}}-linear maps such that F ∘ V = 0 and V ∘ F = 0. Indeed, for any S_{0} there already exist anti-equivalences of categories α : C(V =0)_{S_{0}} → M(V =0)_{S_{0}} and ω : C(F =0)_{S_{0}} → M(F =0)_{S_{0}}; here C(V =0)_{S_{0}} (resp. C(F =0)_{S_{0}}) denotes the full subcategory of C(1)_{S_{0}} consisting of group schemes whose Verschiebung (resp. Frobenius) is zero and similar for M(V =0)_{S_{0}} (resp. M(F =0)_{S_{0}}) (see §2).

Hence our aim is to construct a functor M : C(1)_{S_{0}} → M(1)_{S_{0}} which is compatible with α and ω. We do not construct such a functor for arbitrary schemes S_{0}. It is problably impossible to do so.

Suppose we are given a scheme S, flat over Spec(ℤ/p^{2}ℤ), such that S_{0} = V (p) ⊂ S. In §3 we construct a functor M_{S} from C(1)_{S_{0}} into the category of O_{S_{0}}-modules of finite presentation using the scheme S. It is exact (Theorem 3.10), in §5 we show that it is compatible with the functors α and ω and in §6 we show that there is a natural isomorphism

M_{S}(G^{D}) = Hom_{O_{S_{0}}} (M_{S}(G), O_{S_{0}}).

* Supported by the Netherlands Organisation for Scientific Research (N.W.O.), in the project Arithmetic Algebraic Geometry The construction of M_{S} is functorial in the pair (S, S_{0}) and hence we can define a canonical isomorphism M_{S}(G^{(p)}) = M_{S}(G)^{(p)} if the Frobenius endomorphism of S_{0} can be lifted to a morphism of schemes f_{S} : S → S. In §8 it is proven that M_{S}(G) is a locally free O_{S_{0}}-module for any G ∈ ob C(1)_{S_{0}} if such a lift f_{S} exists.

Therefore we get our functor M_{S} : C(1)_{S_{0}} → M(1)_{S_{0}} if a pair (S, f_{S}) as above exists. Suppose that S_{0} is the spectrum of a Noetherian complete local ring whose residue field has a

finite p-basis. In this case the functor M_{S} induces an equivalence of the full subcategory of C(1)_{S_{0}} consisting of group schemes which have multiplicative part equal to zero and a corresponding subcategory of M(1)_{S_{0}} (Theorem 9.3). If the residue field is perfect then we can show under some additional hypotheses on (S, f_{S}) that M_{S} : C(1)_{S_{0}} → M(1)_{S_{0}} is an equivalence (Theorem 10.2).

In the article, we do not restrict ourselves to the case of group schemes killed by p. One of the results is the classification of p-divisible groups over certain complete local rings of characteristic p. To describe this, let k be a perfect field of characteristic p, let R = W (k)[[x_{1}, ..., x_{n}]]/I, and suppose that R is flat over ℤ_{p}. We assume given an endomorphism σ : R → R such that σ(r) = r^{p} mod pR.

Theorem. If \overline{γ} and \overline{u} are nilpotent (see §10) then the category of p-divisible groups over Spec(R/pR) is equivalent to the category of triples (M, F, V), where M is a free R-module of finite rank and F : M ⊗_{σ} R → M, resp. V : M → M ⊗_{σ} R are R-linear maps such that F ∘V = p and V ∘ F = p.

This theorem is a corollary of Theorem 10.2. The conditions on \overline{γ} and \overline{u} are satisfied for example if Spec(R/pR) is reduced and all σ(x_{i}) can be represented by powerseries whose constant and linear terms are zero.

The methods employed in this article are those of crystalline cohomology. We use extensively the results and methods developped by P. Berthelot, L.Breen and W. Messing in their book Théorie de Dieudonné Crystalline II ([1]). For example our functor M_{S} is defined as

M_{S}(G) := Ext_{S_{0}/Σ}^{1}(\underline{G}, 𝔍_{S_{0}/Σ})_{(S_{0},S,γ)} .

The results of this article can therefore be interpreted as results on the crystalline Dieudonné functor G ⇝ 𝔻(G) (or at least on the filtered Dieudonné modules) although we did not formulate them in that way.

This article was part of the author’s thesis. The author would like to thank his advisors Prof. F. Oort and Prof. J. Steenbrink for their constant support during its writing.

Notation and conventions

– All group schemes will be commutative group schemes.

– We use the sign = to denote canonical isomorphisms.

– The prime number p is fixed throughout the article.

– We say that S is a scheme of characteristic p if there exists a morphism S → Spec(𝔽_{p}).

– A sheaf of O_{S}-modules is called finite locally free if it is locally free of finite rank.

– The rank of a finite locally free sheaf or a finite locally free group scheme over S is a locally constant function on S.

– A lot of notation and conventions are taken from [1], which is our main reference.

1. Finite locally free group schemes

Let S be a scheme. In this section we recall some definitions and results on finite flat group schemes over S. Remark that a group scheme G over S is finite locally free if and only if it is finite flat and of finite presentation over S. If S is locally Noetherian then any finite flat group scheme is finite locally free.

We will work in the fppf topology on the category of schemes over S. A group scheme over S will be identified with the sheaf of abelian groups it gives rise to. A sequence G′ → G → G′′ of group schemes over S will be called exact if the associated sequence of sheaves is exact.

Let ϕ : G → G′ be a morphism of finite locally free group schemes over S. The sheaves of abelian groups Ker(ϕ), Im(ϕ) and Coker(ϕ) are in general not representable by finite locally free group schemes over S. However, if one of them is so representable then they all are. Such a ϕ will be called admissible.

Proposition 1.1. A sequence of finite locally free group schemes

G′⟶^{ϕ}G⟶^{ψ}G′′

is exact if and only if ψϕ = 0 and for every closed point s ∈ S the sequence G′_{s} → G_{s} → G′′_{s} is exact. The morphisms ϕ and ψ are then admissible. Proof. The kernel of ψ is representable by a scheme finite and of finite presentation over S. Consider the morphism of S-schemes G′ → Ker(ψ). For any closed point s ∈ S the morphism G′_{s} → Ker(ψ)_{s} is flat as it is a surjection of group schemes over a field. Hence we can apply the criterion of flatness per fibre ([3] IV 11.3.10) to see that the morphism G′ → Ker(ψ) is flat and that Ker(ψ) is flat over S. Thus ψ is admissible and as Im(ϕ) = Ker(ψ) follows from the above, the sequence is exact and ϕ is admissible also. □

Duality 1.2. (See [9].) For a finite locally free group scheme G over S we define its dual G^{D} by the formula

G^{D} = Hom(G, 𝔾_{m,S}).

This sheaf is representable by a finite locally free group scheme over S. The evaluation morphism

ev : G ⟶ G^{DD} = Hom(Hom(G, 𝔾_{m,S}), 𝔾_{m,S})

is an isomorphism. The functor G ⇝ G^{D} defines a duality of the category of finite locally free group schemes over S. The dual sequence of an exact sequence is exact; consequently the dual of an admissible morphism is admissible.

Suppose S is a scheme of characteristic p. We denote by f_{S} the Frobenius endomorphism of S. For any group scheme G over S we write G^{(p)} for the pullback of G by f_{S}. The Frobenius morphism of G is the morphism of group schemes F_{G} : G → G^{(p)} which is simply the Frobenius morphism of the scheme G over S. If G is finite locally free over S then the Verschiebung of G is the morphism V_{G} : G^{(p)} → G is defined by the formula V_{G} = (F_{G^{D}})^{D}. We have F_{G} ∘ V_{G} = p · id_{G^{(p)}} and V_{G} ∘ F_{G} = p · id_{G}. Any morphism of finite locally free group schemes ϕ : G → G′ induces a commutative diagram

\begin{matrix}G & ⟶^{F_{G}} & G^{(p)} & ⟶^{V_{G}} & G \\ ↓^{ϕ} & & ↓_{ϕ^{(p)}} & & ↓^{ϕ} \\ G′ & ⟶^{F_{G′}} & (G′)^{(p)} & ⟶^{V_{G′}} & G′ \end{matrix} .

Let S be an arbitrary scheme. We denote by C(n)_{S} the category of finite locally free group schemes over S which are killed by p^{n}.

We recall the definition of truncated Barsotti-Tate groups (see [7] I 3.4 and [4] 1.1).

Definition 1.3. A truncated Barsotti-Tate group of level n on S is a finite locally free group scheme G over S such that: (1) G is killed by p^{n} and ℤ/p^{n}ℤ-flat, (2) (if n = 1) if S_{0} := V (p) ⊂ S and G_{0} = G ×_{S} S_{0} then the sequence G_{0} ⟶^{F} G_{0}^{(p)} ⟶^{V} G_{0} is exact.

We will denote by BT (n)_{S} the category of truncated Barsotti-Tate groups of level n on S.

Remarks 1.4.

1) The kernel of multiplication by p^{n} on an abelian scheme over S is a truncated Barsotti-Tate group of level n on S (see [7] I 3.4).

2) The kernel of p^{k} on a truncated Barsotti-Tate group of level n with n ≥ k is a truncated level k Barsotti-Tate group (see [7] II 3.3.11).

3) By a theorem of Raynaud ([1] Theorem 3.1.1) any finite locally free group scheme over S can locally in the Zariski topology on S be embedded into an abelian scheme. It follows that if S is the spectrum of a local ring then the category C(n)_{S} is generated by the category BT (n)_{S}.

4) The dual of a truncated Barsotti-Tate group is a truncated Barsotti-Tate group of the same level.

To a finite locally free group scheme G over S there are associated a number of sheaves of O_{S}-modules. Let us recall two of these:

α_{G} := Hom(G, 𝔾_{a,S}) = {x ∈ O_{G} | μ(x) = x ⊗ 1 + 1 ⊗ x}

where μ : O_{G} → O_{G} ⊗_{O_{S}} O_{G} is the comultiplication and

ω_{G} := o^{∗}(Ω_{G/S}^{1}) = I_{G}/I_{G}^{2}

the sheaf of invariant 1-forms of G over S. Here o : S → G is the zero section and I_{G} ⊂ O_{G} is the augmentation ideal.

Both α and ω give contravariant functors of the category of finite locally free group schemes over S into the category of quasi-coherent O_{S}-modules. The sheaf ω_{G} is an O_{S}-module of finite presentation.

Proposition 1.5.

(1) There is a canonical isomorphism α_{G} = Hom_{O_{S}}(ω_{G^{D}}, O_{S}). Assume S lies over Spec(𝔽_{p}).

(2) If F_{G} = 0 then ω_{G} is finite locally free. Dually, if V_{G} = 0 then α_{G} is finite locally free.

(3) If G ∈ ObBT (1)_{S} then α_{G} and ω_{G} are finite locally free. Proof. See [6] 14.4 for a proof of (1). See [2], exposé VII_{A} 7.4 for a proof of (2). The proof of (3) can be found in [7] II, see also [4] 2.2.1. □

2. Group schemes killed by F or V

In this section S will be a scheme over 𝔽_{p}. We recall the description of group schemes G over S with F_{G} = 0 as given in [2], exposé VII_{A}.

For any O_{S}-module M we write M^{(p)} for the pullback of M via the Frobenius morphism: M^{(p)} = f_{S}^{∗}M. If M is finite locally free then we have an injection

M^{(p)} ↪ Sym^{p}(M).

It sends 1 ⊗ m ∈ O_{S} ⊗_{f_{S}^{−1}O_{S}} M = M^{(p)} to m^{p} ∈ Sym^{p}(M).

Suppose we have a pair (M, F) where M is a finite locally free sheaf of O_{S}-modules and F : M^{(p)} → M a homomorphism of O_{S}-modules. Put

A(M, F) = Sym^{∗}(M)/𝔍

where 𝔍 is the ideal generated by elements x−F (x) for local sections x in M^{(p)} ⊂ Sym^{p}(M). The sheaf of O_{S}-algebras A(M, F) can be made into a sheaf of Hopf algebras with comultiplication

μ : A(M, F) → A(M, F) ⊗_{O_{S}} A(M, F)

which is defined by μ(m) = m ⊗ 1 + 1 ⊗ m for m a local section of M.

If S = Spec(R) and Γ(S, M) is a free R-module with basis {x_{1}, …, x_{m}} and F (1 ⊗ x_{i}) = Σa_{i}^{j}x_{j} then the Hopf algebra Γ(S, A(M, F)) is simply R[x_{1}, …, x_{m}]/(x_{i}^{p} − Σa_{i}^{j}x_{j}) with μ(x_{i}) = x_{i} ⊗ 1 + 1 ⊗ x_{i}.

Let us agree to denote the group scheme over S defined by A(M, F) by G(M, F).

Lemma 2.1. The group scheme G(M, F) is finite locally free of rank p^{d} where d = rank_{O_{S}}(M). The Verschiebung of it is zero and there exists a canonical isomorphism α_{G(M,F)} = M. Proof. This is easily verified using the explicit description of A(M, F) given above. □

Much more is true. If G ∈ ObC(1)_{S} has V_{G} = 0 then we know that α_{G} is finite locally free (Proposition 1.5). The canonical isomorphism α_{G}^{(p)} = α_{G^{(p)}} composed with the morphism α(F_{G}) : α_{G^{(p)}} → α_{G} defines a map α_{G}^{(p)} → α_{G}.

Proposition 2.2. There exists a canonical isomorphism

G = G(α_{G}, α_{G}^{(p)} ⟶ α_{G}).

The above constructions yield an anti-equivalence of categories:

The category of ⟷ The category of pairs
G ∈ ObC(1)_{S} with V_{G} = 0  (M, F) as above.

Proof. This is a special case of [2], exposé VII_{A}, Theorem 7.4 . □

Example 2.3. The Verschiebung of the group scheme ℤ/pℤ_{S} is equal to zero. We can identify Hom(ℤ/pℤ_{S}, O_{S}) with O_{S} via ϕ ⟼ ϕ(\overline{1}) ∈ O_{S}. Then O_{S}^{(p)} = O_{S} ·1⊗1 and ℤ/pℤ_{S} corresponds to the pair (O_{S}, 1 ⊗ 1 ⟼ 1).

Dually, we get that G ∈ ObC(1)_{S} with F_{G} = 0 are classified by the pairs (ω_{G}, ω_{G} → ω_{G}^{(p)}). Here the map ω_{G} → ω_{G}^{(p)} is the composition of the map ω(V_{G}) : ω_{G} → ω_{G^{(p)}} with the canonical isomorphism ω_{G^{(p)}} = ω_{G}^{(p)}.

Example 2.4. The group scheme μ_{p,S} is killed by Frobenius. We see that ω_{μ_{p,S}} = O_{S} · \frac{dt}{t} if we write μ_{p,S} =Spec(O_{S}[t]/(t^{p} − 1)) with comultiplication t ↦ t ⊗ t. Then ω_{μ_{p,S}}^{(p)} = O_{S}^{(p)} · \frac{dt}{t} and μ_{p,S} corresponds to the pair (O_{S} · \frac{dt}{t}, \frac{dt}{t} ↦ 1 ⊗ \frac{dt}{t}).

Remark 2.5. The canonical isomorphism ℤ/pℤ_{S}^{D} = μ_{p,S} gives rise to an isomorphism

α_{ℤ/pℤ_{S}} ⟶ Hom_{O_{S}}(ω_{μ_{p,S}}, O_{S})

via Proposition 1.5. Under the identifications O_{S} = α_{ℤ/pℤ_{S}} and O_{S} = ω_{μ_{p,S}} made in examples 2.3 and 2.4 this map is transformed into the canonical map

O_{S} ⟶ Hom_{O_{S}}(O_{S}, O_{S}).

3. Some properties of Ext_{S_{0}/Σ}^{1}(\underline{G}, 𝔍_{S_{0}/Σ})

In this section and in the rest of the article we will use the notations and results of Berthelot, Breen and Messing from their book Théorie de Dieudonné Cristalline II. In fact we will use the methods developed there to define a Dieudonné module theory and prove some theorems on it.

Situation 3.1. Fix n ∈ ℕ. In the following we will consider diagrams

S_{0} ⟶ S
↓ ↓
Σ_{1} ⟶ Σ_{n+1}

where Σ_{k} = Spec(ℤ/p^{k}ℤ) and the morphism S → Σ_{n+1} is flat. Further we demand that S_{0} = Σ_{1} ×_{Σ_{n+1}} S, in other words S_{0} := V (p) ⊂ S.

Let Σ = Spec(ℤ_{p}). There is a unique divided power structure γ on the ideal pO_{S} ⊂ O_{S} which is compatible with the canonical divided power structure on pO_{Σ}. In this way we get an object (S_{0}, S, γ) of CRIS(S_{0}/Σ).

Definition 3.2. We define the contravariant functor

M_{S} : C(n)_{S_{0}}^{o} ⟶ Mod(O_{S}/p^{n}O_{S})

by

M_{S}(G) := Ext_{S_{0}/Σ}^{1}(\underline{G}, 𝔍_{S_{0}/Σ})_{(S_{0},S,γ)}

for G ∈ ObC(n)_{S_{0}}.

Remarks 3.3.

1) A priori M_{S}(G) is an O_{S}-module but as p^{n} is zero on G it is an O_{S}/p^{n}O_{S}-module.

2) We use the fppf topology, but another topology would give the same result. See [1] Corollaire 2.3.11.

3) If U ⊂ S is an open subscheme then there are canonical identifications M_{U}(G ×_{S_{0}} U_{0}) = M_{S}(G)_{|U}. (Use 1.3.3.8 of [1]).

Proposition 3.4.

1) If f : A → S_{0} is an abelian scheme then M_{S}(A[p^{k}]) (k ≤ n) is a locally free O_{S}/p^{k}O_{S}-module of rank 2 · dim_{S_{0}} A.

2) If G is a p-divisible group over S_{0} then M_{S}(G[p^{k}]) (k ≤ n) is a locally free O_{S}/p^{k}O_{S}-module of rank equal to the height of G.

Proof. The exact sequence on S_{0}

0 ⟶ A[p^{k}] ⟶ A ⟶^{p^{k}} A ⟶ 0

gives rise to an exact sequence of sheaves on CRIS(S_{0}/Σ)

0 ⟵ Ext_{S_{0}/Σ}^{1}(\underline{A[p^{k}]}, 𝔍_{S_{0}/Σ}) ⟵ Ext_{S_{0}/Σ}^{1}(\underline{A}, 𝔍_{S_{0}/Σ}) ⟵^{p^{k}} Ext_{S_{0}/Σ}^{1}(\underline{A}, 𝔍_{S_{0}/Σ})

by [1] 1.1.7 and 2.5.8(i). We deduce from this the exact sequence of sheaves on S

0 ⟵ M_{S}(A[p^{k}]) ⟵ Ext_{S_{0}/Σ}^{1}(\underline{A}, 𝔍_{S_{0}/Σ})_{(S_{0},S,γ)} ⟵^{p^{k}} Ext_{S_{0}/Σ}^{1}(\underline{A}, 𝔍_{S_{0}/Σ})_{(S_{0},S,γ)}.

From the exact sequence

0 ⟶ 𝔍_{S_{0}/Σ} ⟶ O_{S_{0}/Σ} ⟶ \underline{𝔾_{a}} ⟶ 0

we get (use again 2.5.8)

0 ⟶ Ext_{S_{0}/Σ}^{1}(\underline{A}, 𝔍_{S_{0}/Σ}) ⟶ 𝔻(A) ⟶ i_{S_{0}/Σ^{∗}}(Ext_{S_{0}}^{1}(A, 𝔾_{a,S_{0}})) ⟶ 0.

In particular

Ext_{S_{0}/Σ}^{1}(\underline{A}, 𝔍_{S_{0}/Σ})_{(S_{0},S,γ)} = Ker(𝔻(A)_{(S_{0},S,γ)} ⟶ i_{∗}R^{1}f_{∗}O_{A})

where i denotes the inclusion S_{0} ↪ S. Because on S_{0} we have the locally split exact sequence

0 ⟶ ω_{A} ⟶ 𝔻(A)_{S_{0}} ⟶ R^{1}f_{∗}O_{A} ⟶ 0

we can choose locally on S a basis \{s_{1}, …, s_{2g}\} of 𝔻(A)_{(S_{0},S,γ)} such that (\overline{s}_{i} is the image of s_{i} in 𝔻(A)_{S_{0}})

a) \overline{s}_{1}, \overline{s}_{2}, …, \overline{s}_{g} generate ω_{A} ⊂ 𝔻(A)_{S_{0}}

b) \overline{s}_{g+1}, …, \overline{s}_{2g} project to a basis of R^{1}f_{∗}O_{A}. Then we see that M_{S}(A[p^{k}]) has locally the basis {s_{1}, …, s_{g}, ps_{g+1}, …, ps_{2g}} mod p^{k}⟨s_{1}, …, s_{g}, ps_{g+1}, …, ps_{2g}⟩.

The proof in the case of a p-divisible group is the same except that one has to use Propositions 3.3.2, 3.3.4 and 3.3.5 of [1]. □

Proposition 3.5.

1) If G ∈ ObC(n)_{S_{0}} is embedded into an abelian scheme A over S_{0} and B = A/G then the sequence

0 ⟵ M_{S}(G) ⟵ M_{S}(A[p^{n}]) ⟵ M_{S}(B[p^{n}])

is exact.

2) The module M_{S}(G) is an O_{S}-module of finite presentation. Its formation commutes with base change π : T → S where T is flat over Σ_{n+1}:

M_{T} (G ×_{S_{0}} T_{0}) = π^{∗}M_{S}(G).

3) If G ∈ ObC(k)_{S_{0}} and k < n then for S′ = V (p^{k+1}) ↪^{i} S we have i_{∗}M_{S′}(G) = M_{S}(G). Proof. The exact sequence ([1] 1.1.7)

0 ⟶ \underline{G} ⟶ \underline{A} ⟶ \underline{B} ⟶ 0

gives rise to the following commutative diagram with exact rows

0 ⟵ M_{S}(G) ⟵ Ext_{S_{0}/Σ}^{1}(\underline{A}, 𝔍_{S_{0}/Σ})_{(S_{0},S,γ)} ⟵ Ext_{S_{0}/Σ}^{1}(\underline{B}, 𝔍_{S_{0}/Σ})_{(S_{0},S,γ)}
∥  ↓  ↓
M_{S}(G) ⟵  M_{S}(A[p^{n}])  ⟵  M_{S}(B[p^{n}]) .

From this we see the result.

Locally on S_{0} we can embed G into an abelian scheme. Therefore, by 1) and Proposition 3.4, M_{S}(G) is of finite presentation. Let us now define the morphism π^{∗}M_{S}(G) → M_{T} (G_{T_{0}}). For any G we have

M_{T} (G_{T_{0}}) = Ext_{T_{0}/Σ}^{1}(\underline{G_{T_{0}}}, 𝔍_{T_{0}/Σ})_{(T_{0},T,γ)} ≃ Ext_{T_{0}/T}^{1} (\underline{G_{T_{0}}}, 𝔍_{T_{0}/T})_{(T_{0},T,γ)}
≃ Ext_{S_{0}/Σ}^{1}(\underline{G}, 𝔍_{S_{0}/Σ})_{(T_{0},T,γ)}

(twice [1] 1.3.3.8). We get a map

π^{−1}Ext_{S_{0}/Σ}^{1}(\underline{G}, 𝔍_{S_{0}/Σ})_{(S_{0},S,γ)} ⟶ Ext_{S_{0}/Σ}^{1}(\underline{G}, 𝔍_{S_{0}/Σ})_{(T_{0},T,γ)}

because Ext_{S_{0}/Σ}^{1}(\underline{G}, 𝔍_{S_{0}/Σ}) is a sheaf on CRIS(S_{0}/Σ). (See [1] 1.1.3.) We compose these maps to get

π^{−1}M_{S}(G) ⟶ M_{T} (G_{T_{0}})

which induces

π^{∗}M_{S}(G) ⟶ M_{T} (G_{T_{0}}).

Claim. If A is an abelian scheme over S_{0} then the map

π^{∗}M(A[p^{n}]) ⟶ M_{T} (A_{T_{0}}[p^{n}])

is an isomorphism.

To prove this we consider the diagram

0 ⟶ π^{∗}Ext_{S_{0}/Σ}^{1}(\underline{A}, 𝔍_{S_{0}/Σ})_{(S_{0},S,γ)} ⟶ π^{∗}Ext_{S_{0}/Σ}^{1}(\underline{A}, O_{S_{0}/Σ})_{(S_{0},S,γ)} ⟶ π^{∗}R^{1}f_{∗}O_{A} ⟶ 0
↓ ↓ ↓
0 ⟶ Ext_{T_{0}/Σ}^{1}(\underline{A_{T_{0}}}, 𝔍_{T_{0}/Σ})_{(T_{0},T,γ)} ⟶ Ext_{T_{0}/Σ}^{1}(\underline{A_{T_{0}}}, O_{T_{0}/Σ})_{(T_{0},T,γ)} ⟶ R^{1}f_{T_{0}∗}O_{A_{T_{0}}} ⟶ 0.

Here the vertical arrows are defined with the same method as we used for M_{S}(G). It is easily seen that the upper row is exact. The middle vertical arrow is an isomorphism because Ext_{S_{0}/Σ}^{1}(\underline{A}, O_{S_{0}/Σ}) = 𝔻(A) is a crystal in O_{S_{0}/Σ}-modules ([1] 2.5.6) and the right vertical arrow for example by [1] 2.5.2. So the left vertical arrow is an isomorphism too. Our claim follows from the diagram with exact rows:

0 ⟵ π^{∗}(M_{S}(A[p^{n}])) ⟵ π^{∗}Ext_{S_{0}/Σ}^{1}(\underline{A}, 𝔍_{S_{0}/Σ})_{(S_{0},S,γ)} ⟵^{p^{n}} π^{∗}Ext_{S_{0}/Σ}^{1}(\underline{A}, 𝔍_{S_{0},Σ})_{(S_{0},S,γ)}
↓  ↓^{≃}  ↓^{≃}
0 ⟵ M_{T} (A_{T_{0}}[p^{n}]) ⟵ Ext_{T_{0}/Σ}^{1}(\underline{A_{T_{0}}}, 𝔍_{T_{0}/Σ})_{(T_{0},T,γ)} ⟵^{p^{n}} Ext_{T_{0}/Σ}^{1}(\underline{A_{T_{0}}}, 𝔍_{T_{0}/Σ})_{(T_{0},T,γ)}.

10

The question whether π^{∗}M_{S}(G) → M_{T} (G_{T_{0}}) is an isomorphism for general G ∈ ObC(n)_{S_{0}} is local in the Zariski topology on S, so we may assume that G is embedded into an abelian scheme A over S_{0}. Put B = A/G. The result follows from the above and the commutative diagram with exact rows

0 ⟵ π^{∗}M_{S}(G) ⟵ π^{∗}M_{S}(A[p^{n}]) ⟵ π^{∗}M_{S}(B[p^{n}])
↓ ↓^{≃} ↓^{≃}
0 ⟵ M_{T} (G_{T_{0}}) ⟵ M_{T} (A_{T_{0}}[p^{n}]) ⟵ M_{T} (B_{T_{0}}[p^{n}]).

The third statement of the proposition is proved in the same manner as the second statement of the proposition. Let us simply define the arrow M_{S}(G) → i_{∗}M_{S′}(G) and leave the rest of the argument to the reader. The morphism (id_{S_{0}}, i) : (S_{0}, S′, γ) → (S_{0}, S, γ) induces a morphism ([1] 1.1.3)

ρ_{id_{S_{0}} ,i} : i^{−1}Ext_{S_{0}/Σ}^{1}(\underline{G}, 𝔍_{S_{0}/Σ})_{(S_{0},S,γ)} ⟶ Ext_{S_{0}/Σ}^{1}(\underline{G}, 𝔍_{S_{0}/Σ})_{(S_{0},S′,γ)}.

Via adjunction we get the desired map. □

Proposition 3.6. If G ∈ ObBT (k)_{S_{0}} (k ≤ n) then M_{S}(G) is a locally free O_{S}/p^{k}O_{S} module of rank equal to the height of G. Proof. We only have to prove this when S is the spectrum of a local ring. Let us assume this is so and put h = height(G).

Let X → Σ_{n+1} be the scheme representing the functor

Sch/Σ_{n+1} ⟶ (Sets)
T  ⟼ The set of Hopf algebra structures on O_{T}^{p^{hk}} such that the associated

group scheme is an object of BT (k)_{T}.

The scheme X is of finite type over Σ_{n+1}. It is even smooth over Σ_{n+1} because by a theorem of Grothendieck there are no obstructions to deforming a truncated Barsotti-Tate group (see [4] Theorem 4.4).

Let X_{0} = Spec(𝔽_{p}) ×_{Σ_{n+1}} X and let G_{0} be the universal group scheme over X restricted to X_{0}. If we choose a basis of π_{∗}O_{G} (π : G → S_{0} the structural morphism) then we get a morphism f_{0} : G_{0} → X_{0} and an isomorphism f_{0}^{∗}G_{0} ⟶^{∼} G. Also, because X → Σ_{n+1} is smooth, there exists a lift f : S → X of f_{0}. Thus by Proposition 3.5 we only have to prove the proposition for the sheaf M_{X}(G_{0}).

Since the morphism X_{0} → Spec(𝔽_{p}) is of finite type, the completions of the local rings of X_{0} at its closed points are Noetherian complete local rings with perfect residue fields. Over these rings we can find a Barsotti-Tate group H with H[p^{k}] ≃ G_{0} restricted to these rings (see [4] Theorem 4.4). Hence it follows from Proposition 3.4 that M_{X}(G_{0}) is locally free over O_{X}/p^{k}O_{X} of rank h = height(G) at all the closed points of X. □

The rest of this section is devoted to the proof of the following.

Theorem 3.7. The functor

M_{S} : C(n)_{S_{0}}^{o} ⟶ Mod(O_{S}/p^{n}O_{S})

is exact. That is: it transforms exact sequences into exact sequences. □

By Proposition 1.1 we only have to show that a short exact sequence of C(n)_{S_{0}}

0 ⟶ G′ ⟶ G ⟶ G′′ ⟶ 0

is transformed into a short exact sequence of O_{S}-modules by our functor M_{S}. The sequence

0 ⟶ M_{S}(G′′) ⟶ M_{S}(G) ⟶ M_{S}(G′) ⟶ 0

is exact in the middle by the very definition of the functor M_{S}.

Let us show that the arrow on the right is surjective. The question is local in the Zariski topology on S so we may assume that G is embedded into an abelian scheme A over S_{0}. The composition G′ → G → A[p^{n}] is injective. Hence, by Proposition 3.5, we get that the composition M_{S}(A[p^{n}]) → M_{S}(G) → M_{S}(G′) is surjective. The result follows.

To show that the arrow on the left is injective is more difficult. Let us look at the long exact sequence of ext-sheaves:

… ⟶Hom_{S_{0}/Σ}(\underline{G}, 𝔍_{S_{0}/Σ})_{(S_{0},S)}⟶^{γ}Hom_{S_{0}/Σ}(\underline{G}′, 𝔍_{S_{0}/Σ})_{(S_{0},S)}⟶M_{S}(G′′)⟶M_{S}(G)⟶ …

We have to prove that γ is surjective.

Lemma 3.8. For all G ∈ ObC(n)_{S_{0}} the canonical map

Hom_{S_{0}/Σ}(\underline{G}, 𝔍_{S_{0}/Σ})_{(S_{0},S)} ⟶ Hom_{S_{0}/Σ}(\underline{G}, O_{S_{0}/Σ})_{(S_{0},S)}

is an isomorphism. Proof. We only have to prove that the morphism

Hom_{S_{0}/Σ}(\underline{G}, O_{S_{0}/Σ})_{(S_{0},S)} ⟶ Hom_{S_{0}/Σ}(\underline{G}, \underline{𝔾_{a}})_{(S_{0},S)}

is zero.

Suppose ϕ ∈ Γ(U, Hom_{S_{0}/Σ}(\underline{G}, O_{S_{0}/Σ})) = Hom_{U_{0}/U}(\underline{G}, O_{U_{0}/U}) where U ⊂ S is open affine. Take a lift G_{U} over U of G_{U_{0}} over U_{0} as a finite locally free scheme. This is possible as G_{U_{0}} → U_{0} is a finite locally free locally complete intersection morphism (see [2] III 4.15) and there are no obstructions to deforming such over our affine U (use [5] III 2.1.7 and III 3.2.6; see also [1] Lemma 1.1.9).

The element id_{G_{U_{0}}} ∈ Γ((G_{U_{0}}, G_{U}), \underline{G}) is mapped to an element x:=ϕ(id_{G_{U_{0}}} )∈ Γ(G_{U}, O_{G_{U}}). Under the identifications

Hom_{U_{0}/U}(\underline{G}, \underline{𝔾_{a}}) = Hom_{U_{0}}(G_{U_{0}}, 𝔾_{a,U_{0}}) = \{x ∈ Γ(G_{U_{0}}, O_{G_{U_{0}}} ) | μ(x) = 1 ⊗ x + x ⊗ 1\}

(see [1] Proposition 2.3.12) ϕ is mapped to the class of x in O_{G_{U}} /pO_{G_{U}} = O_{G_{U_{0}}} .

However, as G ∈ ObC(n)_{S_{0}} we have p^{n}id_{G_{U_{0}}} = 0 and this implies p^{n}x = 0. Since O_{G_{U}} is ℤ/p^{n+1}ℤ-flat it follows that x ∈ pO_{G_{U}}. Hence the image of ϕ is zero. □

Let us put S′ := V (p^{n}) ⊂ S. For any sheaf F of O_{S_{0}/Σ}-modules on CRIS(S_{0}/Σ) we see F_{(S_{0},S_{0})}, F_{(S_{0},S′)} and F_{(S_{0},S)} as sheaves of O_{S}-modules on S. Let us write F_{S_{0}}, F_{S′} and F_{S} for these sheaves. We always have the canonical maps (see [1], page 11)

F_{S} ⟶ F_{S′} ⟶ F_{S_{0}}.

Lemma 3.9. For any G ∈ ObC(n)_{S_{0}} the map

𝔻(G)_{S} ⟶ 𝔻(G)_{S′}

is an isomorphism. Proof. As usual we may assume that G is embedded into an abelian scheme A over S_{0}. We put B = A/G and look at the commutative diagram

p^{n}𝔻(A)_{S} ⟵ p^{n}𝔻(B)_{S}
↓ ↓
0 ⟵ 𝔻(G)_{S} ⟵ 𝔻(A)_{S} ⟵ 𝔻(B)_{S}
↓ ↓ ↓
0 ⟵ 𝔻(G)_{S′} ⟵ 𝔻(A)_{S′} ⟵ 𝔻(B)_{S′}
↓ ↓
0  0 .

This diagram has exact rows and columns. The map p^{n}𝔻(A)_{S} → 𝔻(G)_{S} is zero because G ∈ ObC(n)_{S_{0}}. The result follows by a diagram chase. □

Let us denote temporarily by ℍ(G) the sheaf Hom_{S_{0}/Σ}(\underline{G}, O_{S_{0}/Σ}) for G ∈ ObC(n)_{S_{0}}.

Proposition 3.10. For any G ∈ ObC(n)_{S_{0}} the morphism

′′p^{n}′′ : 𝔻(G)_{S′} ⟶ ℍ(G)_{S′}

of [1], pp.173-174 is an isomorphism.

Theorem 3.7 follows from this. Indeed, we have to show that the map γ of page 54 is surjective. By Lemma 3.8 this is equivalent to the surjectivity of

ℍ(G)_{S} ⟶ ℍ(G′)_{S}.

However, by the long exact sequence of ext-sheaves this is equivalent to injectivity of the map

𝔻(G′′)_{S} ⟶ 𝔻(G)_{S}.

We apply Lemma 3.9 and Proposition 3.10 to see that this is equivalent to injectivity of

ℍ(G′′)_{S′} ⟶ ℍ(G)_{S′}

and this is true by the definition of ℍ.

To prove Proposition 3.10 we first need another lemma.

Lemma 3.11. For any G ∈ ObC(n)_{S_{0}} there exists a canonical isomorfism κ : ℍ(G)_{S′} → ℍ(G)_{S}. The composition

ℍ(G)_{S} ⟶ ℍ(G)_{S′} ⟶ ℍ(G)_{S}

is equal to multiplication by p. Proof. For any open subscheme U′ ⊂ S′, let U ⊂ S denote the corresponding open subscheme of S. A section ϕ ∈ Γ(U′, ℍ(G)_{S′}) can be interpreted as a homomorphism of \underline{G} into O_{U_{0}/U′} on CRIS(U_{0}/U′). Let (V, T, δ) be an object of CRIS(U_{0}/U). Then (V, T ×_{U} U′, δ) will be an object of CRIS(U_{0}/U′), thus for any x ∈ G(V) we have ϕ(x) ∈ Γ(T ×_{U} U′, O_{U′}). Take a lift ϕ̃(x) ∈ Γ(T, O_{T}) then p · ϕ̃(x) will be an element in Γ(T, O_{T}) independent of the coice of ϕ̃(x) and in this way we get a homomorphism G(V) → Γ(T, O_{T}). This construction gives us an element κ(ϕ) ∈ Γ(U, ℍ(G)_{S}).

We remark that the second statement is an immediate consequence of this constuction. An analogous construction can be used to construct a map F_{S′} → F_{S} for the sheaf F =

Hom_{O_{S_{0}/S}}(𝔻(G^{D}), O_{S_{0}/S}). As p^{n} is zero on G^{D} and S is flat over Σ_{n+1} it is easily seen that this map will be an isomorphism.

To finish the proof we simply remark that the isomorphism Φ_{G^{D}}^{0} of [1] 5.2.7.1 induces a commutative diagram

F_{S′}  ⟶^{∼}  F_{S}
↓^{Φ_{G^{D}}^{0}}  ↓^{Φ_{G^{D}}^{0}}
ℍ(G)_{S′} ⟶ ℍ(G)_{S}.
□

Remark 3.12. The fact that κ is an isomorphism can also be proved without using the isomorphism Φ_{G^{D}}^{0}. Choosing Zariski locally on S_{0} an exact sequence A_{2}[p^{n}] → A_{1}[p^{n}] → G → 0, we see that we are reduced to the case G = A[p^{n}] for some abelian scheme A over S_{0}. Let M = 𝔻(A)_{S}. We have ℍ(G)_{S} = Ker(p^{n}|_{M}) and ℍ(G)_{S′} = M/p^{n}M. The map κ corresponds to the map ′′p′′ : M/p^{n}M → Ker(p^{n}|_{M}) which is an isomorphism since O_{S} is ℤ/p^{n+1}ℤ-flat and M is a locally free O_{S}-module. Proof of Proposition 3.10. We may assume that G is the kernel of a homomorphism α : A → B of abelian schemes over S_{0}. Let β : B → A be the unique homomorphism such that α ∘ β = p^{n} · id_{B} and β ∘ α = p^{n} · id_{A}. Claim. In this situation we have an exact sequence

𝔻(A)_{S′} ⟶^{𝔻(β)} 𝔻(B)_{S′} ⟶^{𝔻(α)} 𝔻(A)_{S′} ⟶^{𝔻(β)} 𝔻(B)_{S′} .

Consider the following commutative diagram with exact rows

0 ⟶ ℍ(G)_{S′} ⟶ 𝔻(B)_{S′}  ⟶^{𝔻(α)_{S′}} 𝔻(A)_{S′} ⟶ 𝔻(G)_{S′} ⟶ 0
π↑  ↑  ↑  ↑
0 ⟶ ℍ(G)_{S} ⟶ 𝔻(B)_{S} ⟶^{𝔻(α)_{S}} 𝔻(A)_{S} ⟶ 𝔻(G)_{S} ⟶ 0

Suppose x ∈ ℍ(G)_{S′} = Ker(𝔻(α)_{S′}). Choose x̃ ∈ 𝔻(B)_{S} lifting x. Choose y ∈ 𝔻(A)_{S} such that p^{n}y = 𝔻(α)_{S}(x̃). This is possible because 𝔻(A)_{S} is a locally free O_{S}-module. We have x′ = x̃−𝔻(β)_{S}(y) ∈ Ker(𝔻(α)_{S}). Hence x = π(x′)+𝔻(β)_{S′}(y′). In other words x = px′′ +𝔻(β)_{S′}(y′) for a certain x′′ ∈ ℍ(G)_{S′} by Lemma 3.11. The same holds for x′′ hence x = p^{2}x′′′ + Im(𝔻(β)). Continuing like this we see that Im(𝔻(β)_{S′}) = Ker(𝔻(α)_{S′}). The same arguments show that Im(𝔻(α)_{S′}) = Ker(𝔻(β)_{S′}).

From this we see that we get an isomorphism

𝔻(G)_{S′} ⟶ ℍ(G)_{S′}

by the following procedure: for any s ∈ 𝔻(G)_{S′} take a lift s̃ ∈ 𝔻(A)_{S′}. Then 𝔻(β)(s̃) ∈ Ker(𝔻(α)) = ℍ(G)_{S′}. To prove our proposition we have to show that ′′p^{n}′′ is equal to this map.

To do this we proceed as follows. A section s in 𝔻(G)_{S′} corresponds to an extension

0 ⟶ O_{S_{0}/S′} ⟶ E ⟶ \underline{G} ⟶ 0

on CRIS(S_{0}/S). A lift s̃ in 𝔻(A)_{S′} of s corresponds to an extension

0 ⟶ O_{S_{0}/S′} ⟶ Ẽ ⟶ \underline{A} ⟶ 0

which gives E after pullback with the map \underline{G} → \underline{A}. Multiplication by p^{n} on Ẽ induces a homomorphism ψ : \underline{A} → Ẽ. Then we see that we have a commutative diagram

0 ⟶  \underline{G}  ⟶ \underline{A} ⟶^{α} \underline{B} ⟶ 0
↓  ↓^{ψ}  ↓^{β}
0 ⟶ O_{S_{0}/S′} ⟶ Ẽ ⟶ \underline{A} ⟶ 0

We leave it to the reader to deduce from this that ′′p^{n}′′s = 𝔻(β)(s̃). □

4. Invariant differentials and Ext_{S_{0}/Σ}^{1}(\underline{G}, 𝔍_{S_{0}/Σ})

Let S be a scheme where p is locally nilpotent. In [1] there is constructed a canonical isomorphism

ω_{G} ⟶^{∼} Ext_{S/Σ}^{1}(\underline{G}, 𝔍_{G/Σ})_{S}

where G is a smooth group scheme over S and ω_{G} is the sheaf of translation invariant differentials on G. In this section we will give a second construction of such an isomorphism.

Let f : G → S be a smooth group scheme. We have the isomorphism

Ext_{S/Σ}^{1}(\underline{G}, 𝔍_{S/Σ})_{S} = Ext_{S/S}^{1}(\underline{G}, 𝔍_{S/S})_{S}

so it suffices to work on the site CRIS(S/S). On this site we have the sheaf π_{S/S}^{∗}(G) described by

Γ((U, T, δ), π_{S/S}^{∗}(G)) = G(T).

The natural map of sheaves

π_{S/S}^{∗}(G) ⟶ \underline{G}

induced by the maps G(T) → G(U) is surjective as the immersions U ↪ T are nilimmersions. Let us denote its kernel by G(𝔍_{S/S}).

Suppose η ∈ Γ(S, ω_{G}) is an invariant differential 1-form on G over S. We are going to construct a map

ϕ_{η} : G(𝔍_{S/S}) ⟶ 𝔍_{S/S}.

Locally on S the ideal I_{G} ⊂ O_{G} of the zero section of G is generated by a regular sequence t_{1}, …, t_{d}. Here we can write

O_{S}[[t_{1}, …, t_{d}]] = lim_{\overleftarrow{k}} f_{∗}(O_{G}/I_{G}^{k})

and η = Σf_{i}dt_{i} (continuous differentials). As η is closed, there exists a unique expression

g = ∑_{I} a_{I} · t^{(I)}

with a_{I} ∈ O_{S} and t^{(I)} = Π_{α=1}^{d}(t_{α})^{i_{α}}/i_{α} such that dg = η. Any element x of Γ((U, T, δ), G(𝔍_{S/S})) defines a morphism O_{G}/I_{G}^{k} → O_{T} for some k and so we get j_{1}, …, j_{d} ∈ 𝔍_{T/U}, the images of t_{1}, …, t_{d}. We define

ϕ_{η}(x) = g(j_{1}, …, j_{d}) = ∑_{I} a_{I} · Π_{α=1}^{d}(i_{α} − 1)!γ_{i_{α}}(j_{α}) ∈ 𝔍_{T/U}

this sum is locally finite as p is locally nilpotent.

Lemma 4.1. The map ϕ_{η} is a homomorphism. In this way we get an injective morphism of O_{S}-modules

ω_{G} ⟶ Hom_{S/S}(G(𝔍_{S/S}), 𝔍_{S/S})_{S}.

Proof. In the ring O_{S}[[t_{i} ⊗ 1, 1 ⊗ t_{i}]] we have the equality μ(η) = η ⊗ 1 + 1 ⊗ η as η is a translation invariant differential. (μ is the comultiplication O_{S}[[t_{i}]] → O_{S}[[t_{i} ⊗1, 1⊗t_{i}]] induced by the addition on G.) Hence by unicity of g we have μ(g) = g ⊗ 1 + 1 ⊗ g. The rest is left to the reader. □

The exact sequence

0 ⟶ G(𝔍_{S/S}) ⟶ π_{S/S}^{∗}(G) ⟶ \underline{G} ⟶ 0

gives us the desired morphism

ω_{G} ⟶ Hom_{S/S}(G(𝔍_{S/S}), 𝔍_{S/S})_{S} ⟶ Ext_{S/S}^{1}(\underline{G}, 𝔍_{S/S})_{S}.

Proposition 4.2. This map is an isomorphism. Proof. We already have an isomorphism ω_{G} ≃ Ext_{S/S}^{1}(\underline{G}, 𝔍_{S/S})_{S} by [1] Proposition 2.2.3. The morphism ω_{G} → ω_{G} so constructed is injective for any S as Hom_{S/S}(π_{S/S}^{∗}(G), 𝔍_{S/S})_{S} = (0). Since it commutes with base change we have that the maps ω_{G} ⊗ k(s) → ω_{G} ⊗ k(s) for s ∈ S injective too and hence isomorphisms. By Nakayama’s lemma ω_{G} → ω_{G} is an isomorphism. □

Proposition 4.3. If f : A → S is an abelian scheme the isomorphism constructed above is the same as the one constructed in Proposition 2.5.8 of [1]. Proof. Let us describe the isomorphism constructed in [1] Proposition 2.5.8. Any extension of \underline{A} by 𝔍_{S/S} defines an 𝔍_{A/S}-torsor on CRIS(A/S). This is an element η ∈ R^{1}f_{cris∗}(𝔍_{A/S}) = f_{∗}Ω_{A/S}^{1} = ω_{A}.

The isomorfism R^{1}f_{cris∗}(𝔍_{A/S}) = f_{∗}Ω_{A/S}^{1} can be described as follows: let Δ_{A/S}^{1} be the first infinitesimal neighborhood of A in A×_{S} A. Any 𝔍_{A/S}-torsor on CRIS(A/S) has a unique section over (A, A). Let us call it s. The difference p_{1}^{∗}(s) − p_{2}^{∗}(s) on (A, Δ_{A/S}^{1}) lies in Γ(Δ_{A/S}^{1}, 𝔍_{A/S}) = Γ(A, Ω_{A/S}^{1}).

If we apply this procedure to the extension obtained from η ∈ ω_{A} via push out with the map ϕ_{η} then we simply get η back. □

For any finite locally free group scheme G over S we get a canonical isomorphism

z_{G} : ω_{G} ⟶^{∼} Ext_{S/Σ}^{1}(\underline{G}, 𝔍_{S/Σ})_{S}

using the isomorphisms constructed above. In fact for any embedding G ↪ H of G into a smooth group scheme we get a diagram with exact rows

0 ⟵  ω_{G}  ⟵  ω_{H}  ⟵  ω_{H/G}
↓^{≃}  ↓^{≃}
0 ⟵ Ext_{S/S}^{1}(\underline{G}, 𝔍_{S/S})_{S} ⟵ Ext_{S/S}^{1}(\underline{H}, 𝔍_{S/S})_{S} ⟵ Ext_{S/S}^{1}(\underline{H/G}, 𝔍_{S/S})_{S}

and z_{G} will be the dotted arrow. One checks that this does not depend on the choice of G ↪ H in the usual manner.

Remark 4.4. The morphism z_{G} constructed here is equal to −α_{G}^{1}. (See [1] Corollaire 3.2.6.) We will not use this fact in the sequel.

5. Ext_{S_{0}/Σ}^{1}(\underline{G}, 𝔍_{S_{0}/Σ}) and group schemes killed by p

In this section we place ourselves in Situation 3.1 with n = 1. We will study our functor

M_{S} : C(1)_{S_{0}} ⟶ Mod(O_{S}/pO_{S}) = Mod(O_{S_{0}})

in more detail. Sheaves of O_{S}-modules killed by p are identified with sheaves of O_{S_{0}}-modules. The exact sequence

0 ⟶ 𝔍_{S_{0}/Σ} ⟶ O_{S_{0}/Σ} ⟶ \underline{𝔾_{a}} ⟶ 0

defines a transformation

Hom( · , \underline{𝔾_{a}}) ⟶ Ext_{S_{0}/Σ}^{1}( · , 𝔍_{S_{0}/Σ}).

In particular for G ∈ ObC(1)_{S_{0}} we get

α_{G} = Hom(\underline{G}, \underline{𝔾_{a}})_{(S_{0},S,γ)} ⟶ M_{S}(G).

The identification on the left side is the one of [1] Proposition 2.3.12.

Proposition 5.1. The natural transformation of functors α → M_{S} is injective: for any G ∈ ObC(1)_{S_{0}} the map α_{G} → M_{S}(G) is injective. Proof. We have the exact sequence

Hom_{S_{0}/Σ}(\underline{G}, O_{S_{0}/Σ})_{(S_{0},S,γ)} ⟶ Hom_{S_{0}/Σ}(\underline{G}, \underline{𝔾_{a}})_{(S_{0},S,γ)} ⟶ M_{S}(G).

In the proof of Lemma 3.8 it is shown that the first map is zero. □ The canonical morphism

Ext_{S_{0}/Σ}^{1}(\underline{G}, 𝔍_{S_{0}/Σ})_{(S_{0},S)} ⟶ Ext_{S_{0}/Σ}^{1}(\underline{G}, 𝔍_{S_{0}/Σ})_{S_{0}}

composed with the natural isomorphism of §4

z_{G}^{−1} : Ext_{S_{0}/Σ}^{1}(\underline{G}, 𝔍_{S_{0}/Σ})_{S_{0}} ⟶ ω_{G}

gives us a natural transformation of functors M_{S} → ω: for any G ∈ ObC(1)_{S_{0}} we get

M_{S}(G) ⟶ ω_{G}.

Proposition 5.2. For any G ∈ C(1)_{S_{0}} the map M_{S}(G) → ω_{G} is surjective; if x ∈ M_{S}(G) has image zero in ω_{G} then it lies in α_{G} ⊂ M_{S}(G). Proof. The question is local on S so we may assume G is embedded into an abelian scheme A over S_{0}. The commutative diagram

0 ⟵  ω_{G}  ⟵  ω_{A[p]}
↑  ↑
0 ⟵ M_{S}(G) ⟵ M_{S}(A[p])

shows that is is enough to prove surjectivity for A[p]. We see that it is enough also to prove that the map

Ext_{S_{0}/Σ}^{1}(\underline{A}, 𝔍_{S_{0}/Σ})_{(S_{0},S)} ⟶ Ext_{S_{0}/Σ}^{1}(\underline{A}, 𝔍_{S_{0}/Σ})_{S_{0}}

is surjective. This is the case as one sees easily from the proof of Proposition 3.4. To prove the second statement consider the following commutative diagram with exact rows

Ext_{S_{0}/Σ}^{1}(\underline{G}, O_{S_{0}/Σ})_{(S_{0},S)} ⟵ Ext_{S_{0}/Σ}^{1}(\underline{G}, 𝔍_{S_{0}/Σ})_{(S_{0},S)} ⟵ Hom_{S_{0}/Σ}(\underline{G}, \underline{𝔾_{a}})_{(S_{0},S)}
↓  ↓  ↓
Ext_{S_{0}/Σ}^{1}(\underline{G}, O_{S_{0}/Σ})_{S_{0}}  ⟵  Ext_{S_{0}/Σ}^{1}(\underline{G}, 𝔍_{S_{0}/Σ})_{S_{0}}  ⟵  Hom_{S_{0}/Σ}(\underline{G}, \underline{𝔾_{a}})_{S_{0}}

The two outer vertical arrows are isomorphisms: the one on the right by [1] Proposition 2.3.12, the one on the left by Proposition 4.3.1. Thus any element in the upper middle group which maps to zero in the group below it also maps to zero in the group on the left. Hence it comes from an element on the right. Quod erat demonstrandum. □

Remark 5.3. The composition α_{G} → M_{S}(G) → ω_{G} is in general not zero. It can be described by

{x ∈ O_{G} | μ(x) = x ⊗ 1 + 1 ⊗ x} =α_{G} ⟶ ω_{G} = invariant diff. forms on G
x ⟼ dx.

Indeed, an element x ∈ α_{G} defines a map G → 𝔾_{a} and by functoriality we only have to check the formula if G = 𝔾_{a} and x = id_{𝔾_{a}}. Then x gives rise to the extension

0 ⟶ 𝔍_{S_{0}/Σ} ⟶ O_{S_{0}/Σ} ⟶ \underline{𝔾_{a}} ⟶ 0

and we have to see that its restriction to CRIS(S_{0}/S_{0})

0 ⟶ 𝔍_{S_{0}/S_{0}} ⟶ O_{S_{0}/S_{0}} ⟶ \underline{𝔾_{a}} ⟶ 0

is the same as the extension constructed in §4 starting with the differential form dT. We can identify π_{S_{0}/S_{0}}^{∗}(𝔾_{a}) with O_{S_{0}/S_{0}} and 𝔾_{a}(𝔍_{S_{0}/S_{0}}) with 𝔍_{S_{0}/S_{0}}. The map ϕ_{dT} will simply be the identity and so we are done. Proposition 5.4.

1) If G ∈ ObC(1)_{S_{0}} has F_{G} = 0 then the canonical map M_{S}(G) → ω_{G} is an isomorphism.

2) If G ∈ ObC(1)_{S_{0}} has V_{G} = 0 then the canonical map α_{G} → M_{S}(G) is an isomorphism.

3) If G ∈ ObBT (1)_{S_{0}} then M_{S}(G) is an extension of ω_{G} by α_{G}. Proof. We first prove the second statement. Locally on S_{0} we can write O_{G} = O_{S_{0}}[x_{1}, …, x_{k}]/ (x_{i}^{p} − Σα_{i}^{j}x_{j}) with α_{G} = O_{S_{0}} · x_{1} ⊕ … ⊕ O_{S_{0}} · x_{k} (see §2). We see that ω_{G} = O_{S_{0}}dx_{1} ⊕ … ⊕ O_{S_{0}}dx_{k}/⟨Σα_{i}^{j}dx_{j}⟩ and that the map α_{G} → ω_{G} is surjective. The statement follows.

Dually, if G ∈ ObC(1)_{S_{0}} has F_{G} = 0 then the morphism α_{G} → ω_{G} is injective. The first statement follows from this.

If G ∈ ObBT(1)_{S_{0}} then the map α_{G} → ω_{G} is zero. Namely, G is the extension

0 ⟶ Ker(F_{G}) ⟶ G ⟶ Ker(V_{G}) ⟶ 0

and so M_{S}(Ker(V_{G})) = α_{Ker(V_{G})} ⟶^{∼} α_{G} → ω_{G} ⟶^{∼} ω_{Ker(F_{G})} = M_{S}(Ker(F_{G})) is zero. □ Example 5.5. In particular we have shown that M_{S}(G) is a finite locally free O_{S_{0}}-module in these cases (without additional hypothesis on S). However, this is not true in general.

As an example take p = 2 and S = Spec((ℤ/4ℤ)[ε]/(ε^{2} − 2)). Let G ∈ ObC(1)_{S_{0}} be given by the Hopf algebra O_{S_{0}}[x]/(x^{2} − εx) with comultiplication μ(x) = x ⊗ 1 + 1 ⊗ x + εx ⊗ x. Then we have ω_{G} ≅ O_{S_{0}}/εO_{S_{0}} · dx and α_{G} ≅ O_{S_{0}}/εO_{S_{0}} · εx.

Consider the following sheaf on CRIS(S_{0}/S):

(U, T, δ) ⟶ {(x̃, i) | x̃ ∈ O_{T} , i ∈ I_{T/U} and x^{2} = εx where x is the image of x̃ in O_{U}} On this sheaf we can define an addition:

(x̃_{1}, i_{1}) + (x̃_{2}, i_{2}) := (x̃_{1} + x̃_{2} + εx̃_{1}x̃_{2}, i_{1} + i_{2}).

Let us denote this sheaf of abelian groups by F. Claim. The subsheaf (U, T, δ) → {(i, i−εδ_{2}(i)) | i ∈ I_{T/U}} of F is a subsheaf of abelian groups. Proof. We simply have to prove that if i_{1}, i_{2} ∈ I_{T/U} then (i_{1}+i_{2}+εi_{1}i_{2}, i_{1}−εδ_{2}(i_{1})+i_{2}−εδ_{2}(i_{2})) is of the form (i, i − εδ_{2}(i)). We compute

i_{1} + i_{2} + εi_{1}i_{2} − εδ_{2}(i_{1} + i_{2} + εi_{1}i_{2}) =

=i_{1} + i_{2} + εi_{1}i_{2} − ε(δ_{2}(i_{1}) + δ_{2}(i_{2}) + δ_{2}(εi_{1}i_{2}) + i_{1}i_{2} + εi_{1}^{2}i_{2} + εi_{1}i_{2}^{2})
=i_{1} − εδ_{2}(i_{1}) + i_{2} − εδ_{2}(i_{2}) − ε^{3}i_{1}^{2}δ_{2}(i_{2}) − ε^{2}i_{1}^{2}i_{2} − ε^{2}i_{1}i_{2}^{2}
=i_{1} − εδ_{2}(i_{1}) + i_{2} − εδ_{2}(i_{2}).

The last terms are zero: for example ε^{2}i_{1}^{2}i_{2} = 2i_{1}^{2}i_{2} = 4δ_{2}(i_{1})i_{2} = 0 as we are working over S and 0 = 4 ∈ O_{S}. □

Let E denote the quotient of F by this subsheaf. Then E is an extension

0 ⟶ 𝔍_{S_{0}/S} ⟶ E ⟶ \underline{G} ⟶ 0

via the maps i ↦ class of (0, i) and the class of (x̃, i) ↦ x ∈ G(U). Also it is not difficult to see that the class of E in M_{S}(G) maps to the element dx ∈ ω_{G}. From this it follows that if we prove that ε · [E] = 0 in M_{S}(G) then ε acts as zero on the whole of M_{S}(G). In particular M_{S}(G) is not a locally free O_{S_{0}}-module.

Thus we want to show the existence of the dotted arrow in the following diagram

0 ⟶ 𝔍_{S_{0}/S} ⟶ E ⟶ \underline{G} ⟶ 0
·ε↓
𝔍_{S_{0}/S}

The morphism (x̃, i) ↦ εi − εx̃ + x̃^{2} + (−εx̃ + x̃^{2})^{2} works.

The ’same’ example can be given for arbitrary p.

Example 5.6. Let us compute for any ϕ : ℤ/pℤ_{S_{0}} → μ_{p,S_{0}} the morphism M_{S}(ϕ) : M_{S}(μ_{p,S_{0}}) → M_{S}(ℤ/pℤ_{S_{0}}). As we have the isomorphisms ω_{μ_{p}} ⟶^{∼} M_{S}(μ_{p}) and M_{S}(ℤ/pℤ) → α_{ℤ/pℤ} we simply have to determine a λ_{ϕ} ∈ O_{S_{0}} such that M_{S}(ϕ) (\frac{dt}{t}) = λ_{ϕ} · e where \{e\} is the basis of α_{ℤ/pℤ} constructed in Example 2.3.

Let E be an arbitrary extension of \underline{ℤ/pℤ} by 𝔍_{S_{0}/S} and suppose E corresponds to the element λ·e in M_{S}(ℤ/pℤ). The element λ ∈ Γ(S_{0}, O_{S_{0}}) can be computed as follows: suppose we have an element α ∈ Γ((S_{0}, S), E) which maps to the element \overline{1} ∈ Γ((S_{0}, S), \underline{ℤ/pℤ}_{S_{0}}) = Γ(S_{0}, ℤ/pℤ_{S_{0}}). Then pα lies in H^{0}((S_{0}, S), 𝔍_{S_{0},S}) = pO_{S} = p·O_{S_{0}}. The element λ is the element′′ \frac{1}{p} ′′·pα ∈ O_{S_{0}}.

Consider the exact sequence of sheaves on CRIS(S_{0}/S)

0 ⟶ (1 + 𝔍_{S_{0}/S})^{∗} ⟶ O_{S_{0}/S}^{∗} ⟶ \underline{𝔾_{m}} ⟶ 0.

Via the homomorphism log: (1 + 𝔍_{S_{0}/S})^{∗} ⟶ 𝔍_{S_{0}/S} we get an extension

0 ⟶ 𝔍_{S_{0}/S} ⟶ U′_{S_{0}/S} ⟶ \underline{𝔾_{m}} ⟶ 0.

Finally if we pullback this extension with the morphism \underline{μ_{p}} ↪ \underline{𝔾_{m}} we get the element in M_{S}(μ_{p}) which corresponds to the element \frac{dt}{t} ∈ ω_{μ_{p}} ≃ M_{S}(μ_{p}).

Let us return to our element ϕ ∈ Hom_{S_{0}}(ℤ/pℤ_{S_{0}}, μ_{p,S_{0}}). Put x = ϕ(\overline{1}) ∈ μ_{p}(O_{S_{0}}). Using the remarks above it is now easy to see that λ_{ϕ} = ′′ \frac{1}{p} ′′ · log(x̃^{p}) where x̃ ∈ O_{S}^{∗} is an arbitrary lift of x ∈ O_{S_{0}}^{∗}.

The group scheme μ_{p,S_{0}} can also be described by the Hopf algebra O_{S_{0}}[y]/(y^{p}) with comultiplication μ(y) = y ⊗ 1 + 1 ⊗ y − B_{p}(y ⊗ 1, 1 ⊗ y) where B_{p}(x, y) ∈ 𝔽_{p}[x, y] is the polynomial

B_{p}(x, y) = \frac{1}{p}[(x + y)^{p} − x^{p} − y^{p}].

(See [10], pp. 8,9.) If we describe μ_{p,S_{0}} in this way then μ_{p}(O_{S_{0}}) = {y ∈ O_{S_{0}} | y^{p} = 0} with addition: y_{1} ∗ y_{2} = y_{1} + y_{2} − B_{p}(y_{1}, y_{2}). The element y ∈ O_{S_{0}} with y^{p} = 0 corresponds to the element

x = 1 + y + \frac{y^{2}}{2!} + \frac{y^{3}}{3!} + … + \frac{y^{p−1}}{(p − 1)!} ∈ O_{S_{0}}^{∗}

if we see μ_{p}(O_{S_{0}}) ⊂ O_{S_{0}}^{∗} as usual. Let us now compute ′′ \frac{1}{p} ′′ log(x̃^{p}) for this element. Take a lift ỹ ∈ O_{S} of the element y ∈ O_{S_{0}}. As a lift of x we take

x̃ = 1 + ỹ + … + \frac{ỹ^{p−1}}{(p − 1)!}.

Then x̃^{p} = (1 + ỹ + … + \frac{ỹ^{p−1}}{(p−1)!})^{p} = 1 + pỹ + ỹ^{p}. This we can see as follows: write out the expression (1 + ỹ+ \frac{ỹ^{2}}{2} + … + \frac{ỹ^{p−1}}{(p−1)!})^{p} as a polynomial in ỹ. For the terms of degree ≤ p − 1 in ỹ this is equal to the terms of degree ≤ p − 1 in the powerseries (exp ỹ)^{p} = exp(pỹ) and so equal to 1 + pỹ. The coefficient for ỹ^{p} is only important modulo p as ỹ^{p} ∈ pO_{S}; it is easily seen that it can be taken to be 1. The terms of degree ≥ p + 1 all have either a coefficient divisible by p (and so are zero) or have degree ≥ 2p (and so are zero).

Let us define

γ_{S} : {y ∈ O_{S_{0}} | y^{p} = 0 } ⟶ O_{S_{0}}
y  ⟼ \frac{1}{p} · ỹ^{p} for some lift ỹ of y.

Then x̃^{p} = 1 + p(y + γ_{S}(y)) and log(x̃^{p}) = log(1 + p(y + γ_{S}(y))) = p(y + γ_{S}(y)) ∈ pO_{S}. Our conclusions are thus as follows: If ϕ : ℤ/pℤ_{S_{0}} → μ_{p,S_{0}} corresponds to the element y ∈ O_{S_{0}} with y^{p} = 0 then M_{S}(ϕ) (\frac{dt}{t}) = (y + γ_{S}(y)) · e.

6. Duality and Ext_{S_{0}/Σ}^{1}(\underline{G}, 𝔍_{S_{0}/Σ})

Let S_{0} ↪ S be a morphism of schemes as in Situation 3.1.

Definition 6.1. For any O_{S}-module F put

F^{∨} := Hom_{O_{S}}(F, O_{S}).

If F is a finite locally free O_{S}/p^{k}O_{S}-module (k ≤ n) then F^{∨} is a finite locally free O_{S}/p^{k}O_{S}-module too. The purpose of this section is to define a canonical map

γ_{G} : M_{S}(G) ⟶ M_{S}(G^{D})^{∨}

for G ∈ C(n)_{S_{0}} and to derive some of its properties.

To do this we will use the map

ϕ_{A} : 𝔻(A)^{∨} ⟶ 𝔻(A^{t})

constructed in [1] 5.1.7.2. Here A is an abelian scheme over S_{0} and A^{t} is its dual. If we use the fact that ϕ_{A} preserves the Hodge filtration (l.c. Proposition 5.1.10) we see that we get the following commutative diagram:

Hom_{O_{S}}(M_{S}(A[p^{n}]), O_{S})
∥
Hom_{O_{S}}(M_{S}(A[p^{n}]), pO_{S})
↓^{≃}
Hom_{O_{S}}(Ext_{S_{0}/Σ}^{1}(\underline{A}, 𝔍_{S_{0}/Σ})_{(S_{0},S)}, pO_{S})  ⟶^{∼}  M_{S}(A^{t}[p^{n}])
↑  ↑
\{ϕ ∈ Hom_{O_{S}}(𝔻(A)_{(S_{0},S)}, O_{S}) | ϕ(Ext_{S_{0}/Σ}^{1}(\underline{A}, 𝔍_{S_{0}/Σ})_{(S_{0},S)}) ⊂ pO_{S}\}⟶^{∼}Ext_{S_{0}/Σ}^{1}(\underline{A}^{t}, 𝔍_{S_{0}/Σ})_{(S_{0},S)}
∩  ∩
Hom_{O_{S}}(𝔻(A)_{(S_{0},S)}, O_{S})  ⟶^{ϕ_{A}}  𝔻(A^{t})_{(S_{0},S)}

Dualising the uppermost isomorphism we get an isomorphism

M_{S}(A[p^{n}]) ⟶ M_{S}(A^{t}[p^{n}])^{∨}.

Let us agree to make the identification

∂_{A,n} : A[p^{n}]^{D} ⟶ A^{t}[p^{n}]

which comes from the exact sequence

0 ⟶ A[p^{n}] ⟶ A ⟶^{p^{n}} A ⟶ 0

by applying RHom( · , 𝔾_{m}):

0 ⟶ Hom(A[p^{n}], 𝔾_{m}) ⟶^{∂_{A,n}} Ext^{1}(A, 𝔾_{m}) ⟶^{p^{n}} Ext^{1}(A, 𝔾_{m})
∥  ∥  ∥
0 ⟶  A[p^{n}]^{D}  ⟶^{∂_{A,n}}  A^{t}  ⟶^{p^{n}}  A^{t}.

We get γ_{A} : M_{S}(A[p^{n}]) → M_{S}(A[p^{n}]^{D})^{∨} by composition of the map M_{S}(A[p^{n}]) → M_{S}(A^{t}[p^{n}])^{∨} with the morphism M_{S}(∂_{A,n}^{−1})^{∨}.

The morphism γ_{A} is functorial in A : if u : A → B is a morphism of abelian schemes over S_{0} then γ_{A} and γ_{B} fit into the following commutative diagram:

γ_{B} : M_{S}(B[p^{n}]) ⟶ M_{S}(B[p^{n}]^{D})^{∨}
↓^{M_{S}(u)} ↓^{M_{S}(u^{D})^{∨}}
γ_{A} : M_{S}(A[p^{n}]) ⟶ M_{S}(A[p^{n}]^{D})^{∨}.

This is proved using the functoriality of ϕ_{A} and ∂_{A,n}.

Suppose now we have a group scheme G ∈ ObC(n)_{S_{0}} and an embedding of G into an abelian scheme A over S_{0}. Let B = A/G. Then the following commutative diagram

0 ⟵ M_{S}(G) ⟵ M_{S}(A[p^{n}]) ⟵ M_{S}(B[p^{n}])
↓^{γ_{G}} ↓^{γ_{A}} ↓^{γ_{B}}
M_{S}(G^{D})^{∨} ⟵ M_{S}(A[p^{n}]^{D})^{∨} ⟵ M_{S}(B[p^{n}]^{D})^{∨}

defines the map γ_{G} as the upper row is exact (see Proposition 3.5). In the usual manner it is shown that γ_{G} does not depend on the choice of the embedding G ↪ A. In particular we can define γ_{G} for arbitrary G ∈ ObC(n)_{S_{0}}: locally in the Zariski topology on S_{0} we can embed G into an abelian scheme, on these open sets we get maps as before, on the intersections of these open sets they will coïncide and so we can glue to get a global map γ_{G}.

Also it is easy to see that γ_{G} is functorial in G: for any G_{1} ⟶^{ϕ} G_{2} in C(n)_{S_{0}} the diagram

M_{S}(G_{2}) ⟶^{γ_{G_{2}}} M_{S}(G_{2}^{D})^{∨}
↓^{M_{S}(ϕ)} ↓^{M_{S}(ϕ^{D})^{∨}}
M_{S}(G_{1}) ⟶^{γ_{G_{1}}} M_{S}(G_{1}^{D})^{∨}

is commutative.

Remarks 6.2.

1) If we have G ∈ ObC(n)_{S_{0}} then we also have G^{D} and γ_{G^{D}} : M_{S}(G^{D}) → M_{S}(G^{DD})^{∨}. The maps γ_{G} and γ_{G^{D}} are compatible: the following diagram is commutative:

M_{S}(G)  ⟶^{γ_{G}}  M_{S}(G^{D})^{∨}
↓^{M_{S}(ev)}  ^{γ_{G^{D}}^{∨}} ↑
M_{S}(G^{DD}) ⟶^{ev} M_{S}(G^{DD})^{∨∨}.

This can be proved using [1] Proposition 5.1.9 and [8] Corollary 1.3.

2) If G ∈ C(k)_{S_{0}} with k < n, let S′ = V (p^{k+1}) ↪^{i} S. By Proposition 3.5 we have i_{∗}M_{S′}(G) = M_{S}(G). There is an obvious compatibility between the homomorphisms γ_{G,S′} and γ_{G,S}. We leave it to the reader to describe this compatibility exactly.

3) The formation of γ_{G} commutes with base change as in Proposition 3.5.2.

Proposition 6.3. For all G ∈ ObC(n)_{S_{0}} the map γ_{G} is an isomorphism. Proof. Locally on S_{0} we choose an embedding of G^{D} into an abelian scheme A over S_{0}. Putting B = A/G^{D} we get an exact sequence 0 → G^{D} → A[p^{n}] → B[p^{n}] and dualizing this we get an exact sequence

0 ⟵ G ⟵ A^{t}[p^{n}] ⟵ B^{t}[p^{n}].

By Theorem 3.7 we get a commutative diagram with exact rows

0 ⟶ M_{S}(G) ⟶ M_{S}(A^{t}[p^{n}]) ⟶ M_{S}(B^{t}[p^{n}])
↓^{γ_{G}} ↓^{γ_{A^{t}}} ↓^{γ_{B^{t}}}
0 ⟶ M_{S}(G^{D})^{∨} ⟶ M_{S}(A^{t}[p^{n}]^{D})^{∨} ⟶ M_{S}(B^{t}[p^{n}]^{D})

As the two right vertical arrows are isomorphisms, γ_{G} is an isomorphism too. □

Remark 6.4. In particular this proposition implies that the canonical map M_{S}(G) → M_{S}(G)^{∨∨} is an isomorphism always (use Remark 6.2). In other words, the sheaves M_{S}(G) for G ∈ ObC(n)_{S_{0}} are reflexive O_{S}-modules.

Let us now take n = 1 and G ∈ C(1)_{C_{0}}. Then by Proposition 5.2 we have the surjection M_{S}(G^{D}) → ω_{G^{D}}. Hence an injection

α_{G} ↪ M_{S}(G^{D})^{∨}  (1)

via the identifications Hom_{O_{S}}(ω_{G^{D}}, O_{S}) = Hom_{O_{S}}(ω_{G^{D}}, pO_{S}) ≃ Hom_{O_{S_{0}}} (ω_{G^{D}}, O_{S_{0}}) = α_{G} (see Proposition 1.5).

Proposition 6.5. For any G ∈ ObC(1)_{S_{0}} the diagram

α_{G}  =  α_{G}
↓  ↓^{(1)}
M_{S}(G) ⟶^{γ_{G}} M_{S}(G^{D})^{∨}

is commutative. Proof. As usual it is enough to prove this in the case that G = A[p] for some abelian scheme A over S_{0}. Then the exact sequence

0 ⟶ A[p] ⟶ A ⟶ A ⟶ 0

gives us a map

∂ : Hom(A[p], 𝔾_{a}) ⟶ Ext^{1}(A, 𝔾_{a}) = Lie(A^{t}).

The morphism α_{G} → α_{G} induced by γ_{G} is the composition

\begin{matrix}α_{G} = & Hom(A[p], 𝔾_{a}) ⟶^{∂} Ext^{1}(A, 𝔾_{a}) = Lie(A^{t}) = \\ ω_{A^{t}}^{∨} = & ω_{A^{t}[p]}^{∨} ⟶^{(∂_{A,1})^{∨}} ω_{A[p]^{D}}^{∨} = α_{A[p]} = α_{G}. \end{matrix}

This can be seen by chasing through a lot of diagrams, using [1] Proposition 5.1.10 and our Proposition 4.3. To say that this map is the identity is the same as saying that the map

Lie(G^{D}) = Hom(A[p], 𝔾_{a}) ⟶^{∂} Ext^{1}(A, 𝔾_{a}) = Lie(A^{t})

is the differential of the map

G^{D} = Hom(A[p], 𝔾_{m}) ⟶^{∂_{A,1}} Ext^{1}(A, 𝔾_{m}) = A^{t}.

This is proved by the standard argument relating 𝔾_{a} and 𝔾_{m}. □

Remark 6.6. This proposition gives another proof of Proposition 6.3 for G ∈ ObBT(1)_{S_{0}}. Indeed, we get an isomorphism of exact sequences (see Proposition 5.4)

0 ⟶ α_{G} ⟶ M_{S}(G) ⟶ ω_{G} ⟶ 0
↓^{≃} ↓^{γ_{G}} ↓^{≃}
0 ⟶ ω_{G^{D}}^{∨} ⟶ M_{S}(G^{D})^{∨} ⟶ α_{G^{D}}^{∨} ⟶ 0.

By the proposition above the isomorphisms on the left and right are those of Proposition 1.5.1 (use Remark 6.2.1 for the one on the right).

7. More properties of M_{S}

In this section we relate M_{S} to the crystalline Dieudonné module and the classical Dieudonné module. We deduce some consequences of this relation.

Let us place ourselves in Situation 3.1. The relation of M_{S} and 𝔻( )_{S} is the following.

Proposition 7.1. There is a functorial isomorphism

M_{S}(G^{(p)}) = 𝔻(G)_{(S_{0},S,γ)}

for G ∈ C(n)_{S_{0}}. Proof. If A is an abelian scheme over S_{0} then the Verschiebung V_{A} of A gives us a map

𝔻(V_{A}) : 𝔻(A)_{(S_{0},S)} ⟶ 𝔻(A^{(p)})_{(S_{0},S)}.

The Cokernel of this is 𝔻(Ker(V_{A}))_{(S_{0},S)} = Lie(A^{(p)t}) and thus the image of 𝔻(V_{A}) is exactly Ext_{S_{0}/Σ}^{1}(\underline{A^{(p)}}, 𝔍_{S_{0}/Σ})_{(S_{0},S)}. Hence we see that this induces an isomorphism

𝔻(A[p^{n}])_{(S_{0},S)} =𝔻(A)_{(S_{0},S)}/p^{n}𝔻(A)_{(S_{0},S)} →Ext_{S_{0}/Σ}^{1}(\underline{A^{(p)}}, 𝔍_{S_{0}/Σ})_{(S_{0},S)}/p^{n}· idem=M_{S}(A^{(p)}[p^{n}]).

Obviously this isomorphism is functorial in the abelian scheme A over S_{0} and therefore we can extend this isomorphism to general G ∈ ObC(n)_{S_{0}} in the usual manner. □ Proposition 7.2. If S = Spec(W_{n+1}(k)), where k is a perfect field, then there is a canonical isomorphism for G ∈ C(n)_{S_{0}}

M_{S}(G) = M(G),

where M(G) denotes the classical Dieudonné module of G. Proof. Let σ denote the Frobenius automorphism of S. We have

M_{S}(G) = (σ^{−1})^{∗}M_{S}(G^{(p)}) = (σ^{−1})^{∗}𝔻(G)_{(S_{0},S)} = M(G)

where the last isomorphism is [1] Theorem 4.2.14. □

Let I ⊂ O_{S} be a ℤ/p^{n+1}ℤ-flat ideal.

Lemma 7.3. For any G ∈ ObC(n)_{S_{0}} we have

Tor_{O_{S}}^{i}(O_{S}/I, M_{S}(G)) = 0 i ≥ 1.

Proof. Locally on S_{0} in the Zariski topology we can choose a “resolution”

0 ⟶ G ⟶ G_{0} ⟶ G_{1} ⟶ G_{2} ⟶ … ⟶ G_{N}

of G by truncated Barsotti-Tate groups G_{i} of level n. By Theorem 3.7 we get a resolution

M_{S}(G_{N}) ⟶ … ⟶ M_{S}(G_{1}) ⟶ M_{S}(G_{0}) ⟶ M_{S}(G) ⟶ 0

of M_{S}(G) by locally free O_{S}/p^{n}O_{S}-modules. Proposition 3.5 and Theorem 3.7 imply that the sequence

M_{S}(G_{N}) ⊗_{O_{S}} O_{S}/I ⟶ … ⟶ M_{S}(G_{0}) ⊗_{O_{S}} O_{S}/I ⟶ M_{S}(G) ⊗_{O_{S}} O_{S}/I ⟶ 0

is still exact. As Tor_{O_{S}}^{i}(O_{S}/p^{n}O_{S}, O_{S}/I) = (0) for i ≥ 1 by our assumption on I we get the result. □

In this way the existence of an ideal I as above implies that the modules M_{S}(G) have good properties. In the following situation we can be more precise.

Situation 7.4. Here R is a ℤ/p^{n+1}ℤ-flat local algebra which has the following property: There exists a ℤ/p^{n+1}ℤ-flat ideal m̃ ⊂ R such that m̃ + pR is the maximal ideal of R. If we have this we put S = Spec(R) and S_{0} = Spec(R/pR).

In this situation suppose

G′ ⟶^{ϕ} G ⟶^{ψ} G′′

is a sequence of C(n)_{S_{0}} with ψϕ = 0. We write M = Γ(S, M_{S}(G)), M′ = Γ(S, M_{S}(G′)) and M′′ = Γ(S, M_{S}(G′′)).

Lemma 7.5. The sequence G′ → G → G′′ is exact if and only if the sequence

M′′/m̃M′′ ⟶ M/m̃M ⟶ M′/m̃M′

is exact. Proof. Let T = Spec(R/m̃). By Proposition 3.5 M/m̃M = M_{T} (G_{T_{0}}) and the same for G′ and G′′. By Proposition 1.1 we prove our lemma if we prove the lemma for M_{T} .

The ring Λ = R/m̃ is a local ring with maximal ideal pΛ. Let k be a perfect closure of the field Λ/pΛ. We can find a lift Λ → W_{n+1}(k) of the injection Λ/pΛ ↪ k (because T → Σ_{n+1} is formally smooth). This induces a faithfully flat morphism T′ = Spec(W_{n+1}(k)) → T. Hence we only need to prove the lemma for M_{T′} . However, this is a corollary of Proposition 7.2. □

In addition to this lemma we can say the following.

Proposition 7.6. In Situation 7.4 we have

1) If G ∈ ObC(1)_{S_{0}} then M_{S}(G) is a locally free O_{S}/pO_{S}-module of rank equal to log_{p} rankG.

2) If M_{S}(G) is a locally free O_{S}/p^{k}O_{S}-module for G ∈ ObC(k)_{S_{0}} with n ≥ k ≥ 2 then G is a truncated Barsotti-Tate group of level k and height equal to the O_{S}/p^{k}O_{S}-rank of M_{S}(G).

3) A complex in C(n)_{S_{0}}

0 ⟶ G′ ⟶ G ⟶ G′′ ⟶ 0

is exact if and only if the complex

0 ⟶ M_{S}(G′′) ⟶ M_{S}(G) ⟶ M_{S}(G′) ⟶ 0

is exact. Proof. The first statement is a direct consequence of Lemma 7.3. Indeed the lemma implies that the module M_{S}(G) over the local ring R/pR satisfies Tor_{R/pR}^{i}(R/m, M_{S}(G)) = (0), i ≥ 1 and hence is a free R/pR module. The statement on the rank follows from Proposition 7.2 by an argument as in the proof of Lemma 7.5.

Suppose G ∈ ObC(k)_{S_{0}} satisfies the conditions of the second statement. Then by Lemma 7.5 we see that the sequences G ⟶^{p^{i}} G ⟶^{p^{k−i}} G, 1 ≤ i ≤ k −1 are exact. Thus G is a truncated Barsotti-Tate group (see [4] Remark 1.3a). The statement on the rank is in Proposition 3.6.

The last statement follows from a combination of Lemmas 7.3 and 7.5 □

Remark 7.7. In the situation of Lemma 7.5 it is not always true that G′ → G → G′′ is exact if M′′ → M → M′ is exact. As an example we can take R = ℤ/p^{2}ℤ[ε]/(ε^{2}), G′ = G = G′′ = α_{p,S_{0}} and the maps ϕ and ψ given by multiplication by ε on α_{p,S_{0}}. The induced sequence of R-modules will be R/pR ⟶^{·ε} R/pR ⟶^{·ε} R/pR which is exact, whereas the sequence G′ → G → G′′ is not exact since both ϕ and ψ are zero modulo ε. It is true that G′ → G → G′′ is exact if M′′ → M → M′ is exact and Tor_{R}^{1}(R/m̃, M′/M) = (0).

8. The functor M_{S} in case there is a lift of Frobenius

In this section we suppose that we are in Situation 3.1 and further we suppose that we are given a morphism of schemes f_{S} : S → S which lifts the Frobenius map S_{0} → S_{0}. In this case the functor M_{S} becomes equipped with extra structure. Indeed, for any G ∈ ObC(n)_{S_{0}} we have (see Proposition 3.5)

f_{S}^{∗}M_{S}(G) = M_{S}(G^{(p)})

and the Frobenius and Verschiebung of G act via this identification:

f_{S}^{∗}M_{S}(G) ⟶^{M_{S}(F_{G})} M_{S}(G) ⟶^{M_{S}(V_{G})} f_{S}^{∗}M_{S}(G^{(p)}).

But before we even use this extra structure let us note that the existence of f_{S} already implies that M_{S} has a number of good properties.

Proposition 8.1. If there exists a lift of Frobenius f_{S} then M_{S} satisfies the conclusions of Proposition 7.6. Proof. If there exists a faithfully flat morphism of schemes T → S such that M_{T} satisfies the conclusions of Proposition 7.6 then this is true for M_{S} also. Thus we may assume S is the spectrum of a local ring and the next lemma finishes the proof. □

Lemma 8.2. Let R be a ℤ/p^{n+1}ℤ-flat local algebra with an endomorphism F : R → R such that F (x) ≡ x^{p} mod pR.

1) If the residue field k of R is perfect then there is a unique section of the canonical map W_{n+1}(k) → R compatible with F on R and Frobenius on the Witt vectors.

2) If the residue field of R is not perfect then there exists a faithfully flat homomorphism of local rings R → R′ such that R′ allows a lift of Frobenius also and its residue field is perfect. Proof. Suppose the residue field k of R is perfect and let \overline{x}_{i}, i ∈ I be a minimal set of generators for the maximal ideal of R/pR. Take lifts x_{i} ∈ R of the elements \overline{x}_{i} ∈ R/pR. Then F (x_{i}) = x_{i}^{p} + pu_{i} and if we change x_{i} into x′_{i} = x_{i} + p^{k}λ_{i} then we get u′_{i} ≡ u_{i} + p^{k−1}λ_{i}^{p} mod p^{k}R (here 1 ≤ k ≤ n). Hence, by repeatedly changing x_{i} we may assume u_{i_{0}} ∈ ⟨x_{i} | i ∈ I⟩ for all i_{0} ∈ I. Claim. The ideal m̃ = ⟨x_{i} | i ∈ I⟩ is ℤ/p^{n+1}ℤ-flat.

To prove this we have to show that any relation Σ\overline{a}_{i}\overline{x}_{i} = 0 in R/pR lifts to a relation Σa_{i}x_{i} = 0 in R. We are going to prove by induction that for any k ∈ {1, …, n + 1} we can choose a_{i} ∈ R such that

Σa_{i}x_{i} ∈ p^{k}R.

For k = 1 this is true. Assume that we already have a_{i} ∈ R such that

Σa_{i}x_{i} = p^{N}h (N ≤ n).

The obstruction to choosing a_{i} ∈ R such that Σa_{i}x_{i} ∈ p^{N+1}R is the class of h in R/m̃ + pR.

For any x ∈ R write F (x) = x^{p} + pu(x). Remark that F^{k}(x) ≡ p^{k}u^{k}(x) mod p^{k+1}R if x ∈ p^{k}R and that u(m̃) ⊂ m̃. Thus if we apply F^{N} to the above relation we see that

\begin{aligned}& F^{N}(Σa_{i}x_{i}) = F^{N}(p^{N}h) \\ ⟹ & p^{N}u^{N}(Σa_{i}x_{i}) ≡ p^{N}h^{p^{N}} mod p^{N+1}R. \end{aligned}

But Σa_{i}x_{i} ∈ m̃ ⇒ u^{N}(Σa_{i}x_{i}) ∈ m̃ and hence the obstruction mentioned above is zero.

The ring Λ = R/m̃ is now a ℤ/p^{n+1}ℤ-flat ring such that Λ/pΛ = k. Hence it is canonically isomorphic to W_{n+1}(k) and the map R → Λ = W_{n+1}(k) defines the section we were looking for. The assertion on the uniqueness of this map is left to the reader.

If the residue field k of R is not perfect let \{\overline{x}_{i} | i ∈ I\} be a p-basis of k (see [3] O_{IV} 21.4.2). Choose elements x_{i} ∈ R lifting the elements \overline{x}_{i} ∈ k. We let

R′ = R [\{y_{i,k}\}_{i∈I,k∈ℕ}] / ( \begin{matrix}y_{i,k}^{p} & = y_{i,k−1}, k ≥ 2 \\ y_{i,1}^{p^{n}} & = x_{i}^{p^{n}} \end{matrix} )

This ring is free over R so it is faithfully flat over R. If m_{R} denotes the maximal ideal of R then the unique maximal ideal of R′ is

m_{R′} = m_{R} · R′ + ⟨y_{i,1} − x_{i} | i ∈ I⟩

and the quotient R′/m_{R′} is a perfect closure of k. We can extend the morphism F on R to R′ simply by putting F′(y_{i,k}) = y_{i,k}^{p}. □

It seems rather difficult to describe exactly which O_{S}-modules can occur as M_{S}(G) for some G ∈ ObC(n)_{S_{0}}. To avoid this difficulty we only define the categories M(k)_{S} which “should correspond” to C(1)_{S_{0}} if k = 1 and BT (k)_{S_{0}} if 2 ≤ k ≤ n.

Definition 8.3. The category M(k)_{S} will be the category whose objects are triples (M, F, V) where

- M is a finite locally free O_{S}/p^{k}O_{S}-module

- F and V are O_{S}-linear maps

F : f_{S}^{∗}M ⟶ M

and

V : M ⟶ f_{S}^{∗}M

such that F ∘ V = p · id_{M} and V ∘ F = p · id_{f{∗ \atop S}M}. The morphisms of (M, F, V) to (M′, F′, V′) will be the set of linear maps ϕ : M → M′ compatible with F and V: i.e. such that V′ ∘ ϕ = f_{S}^{∗}ϕ ∘ V and F′ ∘ f_{S}^{∗}ϕ = ϕ ∘ F.

As described at the beginning of this section we get functors

\begin{matrix}BT (k)_{S_{0}} & ⟶ M(k)_{S} \\ G & ⟼ (M_{S}(G), M_{S}(F_{G}), M_{S}(V_{G})) \end{matrix}

for 2 ≤ k ≤ n and a functor

\begin{matrix}C(1)_{S_{0}} & ⟶ M(1)_{S} \\ G & ⟼ (M_{S}(G), M_{S}(F_{G}), M_{S}(V_{G})). \end{matrix}

These functors will be denoted by M_{S} also.

Definition 8.4. A morphism (M, F, V) → (M′, F′, V′) in M(1)_{S} will be called admissible if and only if M/M′ is a locally free O_{S}/pO_{S}-module. A sequence in M(1)_{S}

(M_{1}, F_{1}, V_{1}) ⟶ (M_{2}, F_{2}, V_{2}) ⟶ (M_{3}, F_{3}, V_{3})

will be called exact if and only if M_{1} → M_{2} → M_{3} is an exact sequence of O_{S}/pO_{S}-modules and (M_{2}, F_{2}, V_{2}) → (M_{3}, F_{3}, V_{3}) is admissible (so M_{2}/M_{1} will be locally free too).

From this definition and the exactness of M_{S} it is clear that admissible maps in C(1)_{S_{0}} are mapped to admissible maps in M(1)_{S}. Similarly exact sequences in C(1)_{S_{0}} go over in exact sequences in M(1)_{S}. Conversely a sequence G′ →^{ϕ} G →^{ψ} G′′ in C(1)_{S_{0}} with ψϕ = 0 is exact if the sequence induced from it in M(1)_{S} is exact (this follows via the same reasoning as in the proof of Proposition 8.1 from Remark 7.7).

For any (M, F, V) ∈ ObM(1)_{S} we have f_{S}^{∗}(M, F, V) := (f_{S}^{∗}M, f_{S}^{∗}F, f_{S}^{∗}V) ∈ ObM(1)_{S} and F and V induce morphisms

(M, F, V) ⟶^{V} f_{S}^{∗}(M, F, V) ⟶^{F} (M, F, V).

Let us denote by BM(1)_{S} the full subcategory of M(1)_{S} of objects (M, F, V) such that the above sequence is exact. By the remark above, the image of any G ∈ ObC(1)_{S_{0}} lies in BM(1)_{S} if and only if G ∈ ObBT (1)_{S_{0}}.

In C(1)_{S_{0}} (resp. M(1)_{S}) we can define for objects A, B in C(1)_{S_{0}} (resp. M(1)_{S}) the group

Ext_{C(1)_{S_{0}}}^{1} (A, B) (resp. Ext_{M(1)_{S}}^{1}(A, B))

as the set of isomorphism classes of exact sequences

0 ⟶ B ⟶ E ⟶ A ⟶ 0

with the usual addition. It is proved in the standard way that short exact sequences induce six term exact sequences involving Hom and Ext^{1}. Of course the functor M_{S} induces homomorphisms of groups

Ext_{C(1)_{S_{0}}}^{1} (A, B) ⟶ Ext_{M(1)_{S}}^{1}(M_{S}(B), M_{S}(A))

functorial in A and B and compatible with the boundary operator in the six term exact sequences.

As an application of these ideas let us prove the following.

Theorem 8.5. Suppose S is the spectrum of a Noetherian complete local ring with perfect residue field. If the functor

C(1)_{S_{0}}^{o} ⟶ M(1)_{S}

is an equivalence of categories then the same is true for the functors (n ≥ k ≥ 2)

BT (k)_{S_{0}}^{o} ⟶ M(k)_{S}.

Proof. This is proved by induction on k. Let us prove it for k = 2.

- The functor is faithful. Let G_{1}, G_{2} ∈ ObBT (2)_{S_{0}} and let ϕ : G_{1} → G_{2} be a morphism. If M_{S}(ϕ) = 0 then it follows that G_{1}[p] → G_{2}[p] is zero by our assumption. Consequently ϕ is a composition G_{1} → G_{1}/G_{1}[p] ⟶^{ψ} G_{2}[p] → G_{2}. But M_{S}(ψ) = 0 also. Hence ψ = 0 and ϕ = 0.

- The functor is fully faithful. Let G_{1}, G_{2} ∈ ObBT (2)_{S_{0}} and α : M_{S}(G_{2}) → M_{S}(G_{1}) in M(2)_{S}. By assumption we already can a find a ϕ′ : G_{1}[p] → G_{2}[p] such that M_{S}(ϕ′) = α mod p : M_{S}(G_{2})/pM_{S}(G_{2}) → M_{S}(G_{1})/pM_{S}(G_{1}). Consider the following diagram

0 ⟶ G_{1}[p] ⟶ G_{1} ⟶^{′′p′′} G_{1}[p] ⟶ 0
↓^{ϕ′}  ↓^{ϕ}  ↓^{ϕ′}
0 ⟶ G_{2}[p] ⟶ G_{2} ⟶^{′′p′′} G_{2}[p] ⟶ 0

It shows that the obstruction to find any ϕ fitting into it lies in the group Ext_{C(1)_{S_{0}}}^{1} (G_{1}[p], G_{2}[p]). But this obstruction is mapped to a similar obstruction in Ext_{M(1)_{S}}^{1}(M_{S}(G_{2}[p]), M_{S}(G_{1}[p])) which is derived from α mod p and hence zero. By assumption the above Ext^{1}-groups are isomorphic so we can choose such a ϕ. The set of all these ϕ is a homogeneous space under Hom(G_{1}[p], G_{2}[p]) and thus by an analogous argument we can alter ϕ such that M_{S}(ϕ) = α.

- Essentially surjective. Suppose (M, F, V) ∈ ObM(2)_{S}. The object (\overline{M}, \overline{F} , \overline{V}) = (M/pM, F mod p, V mod p) is actually in BM(1)_{S}: if \overline{x} ∈ \overline{M} is in the kernel of \overline{V} then, for any \overline{y} ∈ f_{S}^{∗}\overline{M} such that py = V x for lifts x ∈ M, y ∈ f_{S}^{∗}M, we have \overline{F} \overline{y} = \overline{x}. Thus the group scheme G′ ∈ ObC(1)_{S_{0}} corresponding to (\overline{M}, \overline{F} , \overline{V}) actually is an object of BT (1)_{S_{0}}.

From [4] Theorem 4.4 we get a truncated Barsotti-Tate group H of level 2 over S_{0} such that H[p] = G′. The set of isomorphism classes of all such groups is a homogeneous space under the group Ext_{C(1)_{S_{0}}}^{1} (G′, G′). The set of all isomorphism classes of all M′ in M(2)_{S} with \overline{M}′ = \overline{M} is a homogeneous space under Ext_{M(1)_{S}}^{1}(\overline{M}, \overline{M}). Hence we can alter our choice of H by an element of Ext_{C(1)_{S_{0}}}^{1} (G′, G′) such that M_{S}(H) ≅ (M, F, V).

The induction step is completely analogous to the above and is left to the reader. □

Let us return to the general setting as introduced at the beginning of this section. From §2 we already know that the functor M_{S} induces equivalences of certain subcategories of C(1)_{S_{0}} and M(1)_{S}. We can extend this result in the following way.

Proposition 8.6.

1) If G ∈ ObC(1)_{S_{0}} has V_{G} = 0 then M_{S} induces isomorphisms

Hom_{C(1)_{S_{0}}} (H, G) ⟶^{∼} Hom_{M(1)_{S}}(M_{S}(G), M_{S}(H))

and

Ext_{C(1)_{S_{0}}}^{1} (H, G) ⟶^{∼} Ext_{M(1)_{S}}^{1}(M_{S}(G), M_{S}(H))

for all H ∈ ObC(1)_{S_{0}}.

2) Dually, if G ∈ ObC(1)_{S_{0}} has F_{G} = 0 then M_{S} induces isomorphisms

Hom_{C(1)_{S_{0}}} (G, H) ⟶^{∼} Hom_{M(1)_{S}}(M_{S}(H), M_{S}(G))

and

Ext_{C(1)_{S_{0}}}^{1} (G, H) ⟶^{∼} Ext_{M(1)_{S}}^{1}(M_{S}(H), M_{S}(G))

for all H ∈ ObC(1)_{S_{0}}. Proof. We only prove the second statement as the first one then follows from it by duality.

Let G ∈ ObC(1)_{S_{0}} have F_{G} = 0 and H ∈ ObC(1)_{S_{0}} arbitrary. As we only have to show the assertion Zariski locally on S we may assume that we have an exact sequence 0 → H → H_{1} → H_{2} with H_{i} ∈ ObBT (1)_{S_{0}}. We see from the commutative diagram with exact rows

Hom_{C(1)_{S_{0}}} (G, H)  ↪  Hom_{C(1)_{S_{0}}} (G, H_{1})  ⟶  Hom_{C(1)_{S_{0}}} (G, H_{2})
↓  ↓  ↓
Hom_{M(1)_{S}}(M_{S}(H), M_{S}(G)) ↪ Hom_{M(1)_{S}}(M_{S}(H_{1}), M_{S}(G)) ⟶ Hom_{M(1)_{S}}(M_{S}(H_{2}), M_{S}(G))

that it is enough to prove the statement on Hom if H ∈ ObBT (1)_{S_{0}}. In this case put K := Ker(F_{H}). Then obviously

Hom_{C(1)_{S_{0}}} (G, K) ⟶^{∼} Hom_{C(1)_{S_{0}}} (G, H)

and similarly

Hom_{M(1)_{S}}(M_{S}(K), M_{S}(G)) ⟶^{∼} Hom_{M(1)_{S}}(M_{S}(H), M_{S}(G)).

By Proposition 5.4 we have M_{S}(G) ≃ ω_{G} and M_{S}(K) ≃ ω_{K}. Thus the result follows from the dual of Proposition 2.2.

Now let us prove the statement on Ext’s. A consideration of the six term exact sequences of Hom’s and Ext’s associated to the exact sequence 0 → H → H_{1} → H_{1}/H → 0 reveals that it is enough to prove the statement when H ∈ ObBT (1)_{S_{0}}.

Again we let K := Ker(F_{H}) . Further we let C(F =0)_{S_{0}} be the full subcategory of C(1)_{S_{0}} consisting of objects whose Frobenius morphism is zero. Then in this case

Ext_{C(F =0)_{S_{0}}}^{1} (G, K) ⟶ Ext_{C(1)_{S_{0}}}^{1} (G, H)

will be an isomorphism: the inverse of it is given as follows. For any extension E of G by H the kernel of F_{E} will be an extension of G by K:

0 ⟶ K ⟶ Ker(F_{E}) ⟶ G
↓  ↓  ∥
0 ⟶ H ⟶  E  ⟶ G ⟶ 0
↓^{F_{H}}  ↓^{F_{E}}  ↓^{F_{G}=0}
0 ⟶ H^{(p)} ⟶  E^{(p)}  ⟶ G^{(p)} ⟶ 0.

Indeed, take x ∈ G, take a lift x̃ ∈ E then F_{E}(x̃) ∈ Ker(V_{H}). Thus ∃y ∈ H such that F_{H}(y) = F_{E}(x̃) as H ∈ ObBT (1)_{S_{0}}. The element x̃ − y ∈ Ker(F_{E}) lifts x ∈ G = Ker(F_{G}).

If we denote M(F =0)_{S} the corresponding full subcategory of M(1)_{S} then the statement

Ext_{M(1)_{S}}^{1}(M_{S}(H), M_{S}(G)) ≅ Ext_{M(F =0)_{S}}^{1}(M_{S}(K), M_{S}(G))

can be proved in exactly the same way.

Finally, the isomorphism

Ext_{C(F =0)_{S_{0}}}^{1} (G, K) ⟶^{∼} Ext_{M(F =0)_{S}}^{1}(M_{S}(K), M_{S}(G))

follows from (the dual of) Proposition 2.2. □

As an example of extension groups which are more difficult to handle with our theory we will describe the map

Ext_{C(1)_{S_{0}}}^{1} (ℤ/pℤ_{S_{0}}, μ_{p,S_{0}}) ⟶ Ext_{M(1)_{S}}^{1}(M_{S}(μ_{p,S_{0}}), M_{S}(ℤ/pℤ_{S_{0}})).

First let us describe the group on the left.

Lemma 8.7. There exists a natural injection

Γ(S_{0}, O_{S_{0}}^{∗})/Γ(S_{0}, O_{S_{0}}^{∗})^{p} ⟶ Ext_{C(1)_{S_{0}}}^{1} (ℤ/pℤ_{S_{0}}, μ_{p,S_{0}})

which is an isomorphism if H_{fppf}^{1}(S_{0}, 𝔾_{m,S_{0}})[p] = 0. Proof. An extension of ℤ/pℤ_{S_{0}} by μ_{p,S_{0}} with p = 0 is completely determined by the μ_{p,S_{0}} torsor which lies over the section \overline{1} ∈ Γ(S_{0}, ℤ/pℤ_{S_{0}}). The result follows from the cohomology sequence associated to the exact sequence of fppf sheaves on S_{0}:

1 ⟶ μ_{p,S_{0}} ⟶ 𝔾_{m,S_{0}} ⟶ 𝔾_{m,S_{0}} ⟶ 1.

However, let us describe more precisely the extension associated to a section g ∈ Γ(S_{0}, O_{S_{0}}^{∗}). The sheaf associated to g will be E(g):

E(g)(U) = {(f, n) | f ∈ Γ(U, O_{U}^{∗}), n ∈ ℤ, f^{p} = g^{n}} /⟨(g, p)⟩

for any scheme U over S_{0}. Further

μ_{p}(U) ∋ f ⟼ (f, 0) ∈ E(g)(U)

and

E(g)(U) ∋ (f, n) ⟼ \overline{n} ∈ ℤ/pℤ(U)

are the maps which determine the structure of an extension on E(g). □

Let us write O_{S_{0}}^{(p)} = f_{S}^{∗}O_{S_{0}}. Of course it is canonically isomorphic to O_{S_{0}} and we can thus identify Hom_{O_{S}}(O_{S_{0}}, O_{S_{0}}^{(p)}) = O_{S_{0}} and Hom_{O_{S}}(O_{S_{0}}^{(p)}, O_{S_{0}}) = O_{S_{0}}. The following objects of M(1)_{S} are isomorphic to M_{S}(μ_{p,S_{0}}) and M_{S}(ℤ/pℤ_{S_{0}}) respectively

M_{1} = (O_{S_{0}}, 0, 1) and M_{2} = (O_{S_{0}}, 1, 0).

(See Proposition 5.4 and Examples 2.3 & 2.4.)

The group on the right is now described as follows.

Lemma 8.8. There exists a natural injection

Γ(S_{0}, O_{S_{0}}) mod Γ(S_{0}, O_{S_{0}})^{p} ⟶ Ext_{M(1)_{S}}^{1}(M_{1}, M_{2})

which is an isomorphism if H_{fppf}^{1}(S_{0}, O_{S_{0}})[F] = 0.

Proof. An extension of M_{1} by M_{2} is a triple (E, F_{E}, V_{E}) and a diagram (with exact rows)

0 ⟶ O_{S_{0}}  ⟶^{i} E ⟶^{π} O_{S_{0}} ⟶ 0
↓^{0}  ↓^{V_{E}}  ↓^{1}
0 ⟶ O_{S_{0}}^{(p)} ⟶^{i^{(p)}} E^{(p)} ⟶^{π^{(p)}} O_{S_{0}}^{(p)} ⟶ 0
↓^{1}  ↓^{F_{E}}  ↓^{0}
0 ⟶ O_{S_{0}}  ⟶^{i} E ⟶^{π} O_{S_{0}} ⟶ 0.

If we have a section s of π then the maps V_{E} and F_{E} are given by

\{ \begin{matrix}V_{E}(s(1)) = s^{(p)}(1) & + a · i^{(p)}(1) \\ F_{E}(s^{(p)}(1)) = & − a · i(1) \end{matrix}

for some a ∈ Γ(S_{0}, O_{S_{0}}). If we had chosen s′ = s − b · i for some b ∈ Γ(S_{0}, O_{S_{0}}) then we would have gotten a′ = a + b^{p}. In this way we get the injection

Γ(S_{0}, O_{S_{0}}) mod Γ(S_{0}, O_{S_{0}})^{p} ⟶ Ext_{M(1)_{S}}^{1}(M_{1}, M_{2}).

This map is surjective if for any such E as above we can find such a section. The obstruction to this lies in Ext_{O_{S_{0}}}^{1} (O_{S_{0}}, O_{S_{0}}) = H^{1}(S_{0}, O_{S_{0}}). However, the image of V_{E} determines a splitting of the extension E^{(p)} of O_{S_{0}}^{(p)} by O_{S_{0}}^{(p)}. Hence, if H^{1}(S_{0}, O_{S_{0}})[F] = 0, we get an isomorphism. □

For any section x of O_{S} write f_{S}(x) = x^{p} + pu(x). Then u becomes a map
u : O_{S} ⟶ O_{S}/p^{n}O_{S}

which satisfies

\{ \begin{matrix}u(x + y) & = u(x) − B_{p}(x, y) + u(y) \\ u(xy) & = x^{p}u(y) + y^{p}u(x) + pu(x)u(y). \end{matrix}

Hence we can define a homomorphism

Γ(S_{0}, O_{S_{0}}^{∗})/Γ(S_{0}, O_{S_{0}}^{∗})^{p} ⟶ Γ(S_{0}, O_{S_{0}}) mod Γ(S_{0}, O_{S_{0}})^{p}

by the formula

class of g ⟼ class of (\frac{u(g̃)}{g̃^{p}} mod p) .

where g̃ ∈ Γ(S, O_{S}^{∗}) is any lift of g.

Proposition 8.9. The following diagram is commutative:

Γ(S_{0}, O_{S_{0}}^{∗})/Γ(S_{0}, O_{S_{0}}^{∗})^{p} ⟶ Γ(S_{0}, O_{S_{0}}) mod Γ(S_{0}, O_{S_{0}})^{p}
↓ ↓
Ext_{C(1)_{S_{0}}}^{1} (ℤ/pℤ_{S_{0}}, μ_{p,S_{0}}) ⟶  Ext_{M(1)_{S}}^{1}(M_{1}, M_{2})

Proof. Let g be an element of Γ(S_{0}, O_{S_{0}}^{∗}) and g̃ ∈ Γ(S, O_{S}^{∗}) any lift of g. Consider the following two sheaves on CRIS(S_{0}/S):

V  : V((U, T, δ)) := \{f ∈ Γ(T, O_{T}^{∗}) | the image \overline{f} of f in O_{U}^{∗} satisfies \overline{f}^{p} = 1\}
V(g̃) : V(g̃)((U, T, δ)) := \{(f, n) | f ∈ Γ(T, O_{T}^{∗}), n ∈ ℤ and the image \overline{f} of f
in O_{U}^{∗} satisfies \overline{f}^{p} = g^{n}\}/⟨(g̃, p)⟩.

These sheaves fit in an obvious way into the following commutative diagram

0 ⟶ (1 + 𝔍_{S_{0}/S})^{∗} ⟶ V  ⟶ \underline{μ_{p,S_{0}}} ⟶ 0
↓ ↓ ↓
0 ⟶ (1 + 𝔍_{S_{0}/S})^{∗} ⟶ V(g̃) ⟶ \underline{E(g)} ⟶ 0.

Via the homomorphism log : (1 + 𝔍_{S_{0}/S})^{∗} → 𝔍_{S_{0}/S} we get

0 ⟶ 𝔍_{S_{0}/S} ⟶ U  ⟶ \underline{μ_{p,S_{0}}} ⟶ 0
↓ ↓ ↓
0 ⟶ 𝔍_{S_{0}/S} ⟶ U(g̃) ⟶ \underline{E(g)} ⟶ 0.

In this way we get a splitting of M_{S}(E(g)) as in Lemma 8.8; the extension U corresponds the canonical generator \frac{dt}{t} ∈ M_{S}(μ_{p,S_{0}}) (see Example 5.6).

Remark that if we choose another lift g̃′ of g then the difference U(g̃′) − U(g̃) will be an element of M_{S}(ℤ/pℤ_{S_{0}}). Using the remarks in Example 5.6 it is seen that this element is equal to \frac{1}{p} log(g̃/g̃′) = \frac{1}{p} \frac{g̃−g̃′}{g̃′} mod p.

The splitting s^{(p)} = f_{S}^{∗}(s) is determined by the extension (f_{S})_{crys}^{∗}U(g̃) = U(f_{S}(g̃)). Finally the splitting determined by V_{E(g)} is given by U(g̃^{p}): it is easily seen that we can lift the homomorphism V_{E(g)} : \underline{E(g)}^{(p)} → \underline{E(g)} to a homomorphism U(g̃^{p}) → U(g̃).

Therefore, the element a ∈ Γ(S_{0}, O_{S_{0}}) induced from the splitting s as in Lemma 8.8 is equal to U(g̃^{p}) − U(f_{S}(g̃)) = \frac{1}{p} · \frac{pu(g̃)}{g̃^{p}} = \frac{u(g̃)}{g̃^{p}}. □

9. The local complete case

In this section we study the functor

M_{S} : C(1)_{S_{0}} ⟶ M(1)_{S}

in the case where S is the spectrum of a Noetherian complete local ring. In fact we will show that it induces an equivalence of the subcategory of C(1)_{S_{0}} consisting of group schemes which have no multiplicative part and a corresponding subcategory of M(1)_{S}.

First we define exactly the setting for this section.

Situation 9.1. We have S as in Situation 3.1, S = Spec(R) where we assume:

- R is a Noetherian complete local ring.

- The residue field k of R has a finite p-basis (for example if k is perfect or of finite type over the prime field 𝔽_{p} ⊂ k).

- There is given a lift of Frobenius

f_{S} : S ⟶ S.

We will write \overline{R} = R/pR and hence S_{0} = Spec(\overline{R}).

To any finite flat group scheme G over S_{0} there is canonically associated a sequence of finite flat group schemes

G^{μ} ⟶ G ⟶ G^{ét}.

Here G^{ét}, the étale part of G, is the group of connected components of G and G^{μ}, the multiplicatif part of G, is the dual group scheme of the group of connected components of G^{D}. If G ∈ ObC(1)_{S_{0}} then locally in the étale topology G^{μ} ≃ (μ_{p})^{r_{1}} and G^{ét} ≃ (ℤ/pℤ)^{r_{2}}.

We let C′_{S_{0}} (resp. C′′_{S_{0}}) be the full subcategory of C(1)_{S_{0}} consisting of group schemes G with G^{μ} = 0 (resp. G^{ét} = 0). We have G ∈ ObC′_{S_{0}} ⟺ G^{D} ∈ ObC′′_{S_{0}}.

In the same vein we can define the étale and multiplicative part of an object (M, F, V) of M(1)_{S}. Here we see M simply as a free \overline{R} module, F and V as homomorphisms of \overline{R}-modules:

M ⟶^{V} M^{(p)} ⟶^{F} M.

It is elementary and easy to show that ⋂_{k} F^{k}(M^{(p^{k})}) is a free direct summand M^{ét} ⊂ M; it is the largest free direct summand M^{ét} ⊂ M such that F ((M^{ét})^{(p)}) = M^{ét}. From the dual object (M^{∨}, V^{∨}, F^{∨}) we get a free quotient M ↠ M^{μ} characterised by the fact that it is the largest free quotient such that V induces an isomorphism V_{μ} : M^{μ} → M^{μ(p)}. Hence to any object (M, F, V) there is canonically associated the sequence

(M^{ét}, F |_{M^{ét}}, 0) ⟶ (M, F, V) ⟶ (M^{μ}, 0, V_{μ}).

As before we write M′_{S} (resp. M′′_{S}) for the full subcategory of M(1)_{S} consisting of objects (M, F, V) such that M^{μ} = (0) (resp. M^{ét} = (0)). Another way to state this is

(M, F, V) ∈ ObM′′_{S} ⟺ ⋂_{k} Im(F^{k}) = (0)

where F^{k} denotes the composition

M^{(p^{k})} ⟶^{F^{(p^{k−1})}} M^{(p^{k−1})} ⟶ … ⟶ M^{(p)} ⟶^{F^{(p)}} M .

Also (M, F, V) ∈ ObM′_{S} ⟺ (M^{∨}, V^{∨}, F^{∨}) ∈ ObM′′_{S}.

Lemma 9.2. ∀G ∈ ObC(1)_{S_{0}}

G ∈ ObC′_{S_{0}} ⟺ M_{S}(G) ∈ ObM′_{S}.

Proof. It is enough to prove this after replacing R by the ring R′ constructed in Lemma 8.2, so we may assume that k is perfect. Let T = Spec(W_{n+1}(k)) ↪ S be the closed subscheme of S constructed in Lemma 8.2. Remark that G ∈ ObC′_{S_{0}} ⟺ G_{T_{0}} ∈ ObC′_{T_{0}} and similar for M_{S}(G). Hence, by Proposition 7.2, the result follows from the classical Dieudonné module theory. □

Here is the main theorem of this section.

Theorem 9.3. In Situation 9.1 the functor M_{S} induces equivalences of categories

C′_{S_{0}}^{o} ⟶^{∼} M′_{S}

and

C′′_{S_{0}}^{o} ⟶^{∼} M′′_{S}.

Remark 9.4.a) As we are only concerned now with group schemes killed by p we may assume that n = 1. Also the definition of the categories M(1)_{S}, M′_{S} and M′′_{S} only involve the ring \overline{R} and do not depend on the ring R or the morphism f_{S}. Thus we could have written M′_{S} = M′_{S_{0}} and we could have stated Theorem 9.3 as follows: Given \overline{R} if there exists a ring R as in 9.1 (with n = 1) then there exists an anti-equivalence of categories C′_{S_{0}} ⟶^{∼} M′_{S_{0}}.

b) We can formulate an analog of Theorem 8.5 involving categories BT′(k)_{S_{0}} and M′(k)_{S} in stead of the categories BT (k)_{S_{0}} and M(k)_{S}. Here BT′(k)_{S_{0}} is the full subcategory of those G ∈ ObBT (k) such that G[p] ∈ ObC′_{S_{0}} and a similar definition for M′(k)_{S}. We can then prove this analog in exactly the same way as we proved Theorem 8.5. Hence we get equivalences of categories BT′(k)_{S_{0}} ⟶^{∼} M′(k)_{S} for 2 ≤ k ≤ n whenever the residue field of R is perfect. However, we can remove this last assumption by the following arguments: Let {x_{1}, …, x_{m}} ⊂ R be a set of elements projecting to a p-basis of the residue field of R. For all N ∈ ℕ we put

R_{N} = R[y_{1,N}, …, y_{m,N}]/((y_{i,N})^{p^{N+n−1}} − x_{i}^{p^{n}})

we have R_{N−1} ⊂ R_{N} by the map y_{1,N−1} ↦ y_{1,N}^{p}. We can extend the lift of frobenius on R to a lift of Frobenius on R_{N} by sending y_{i,N} to y_{i,N}^{p}. Then

R′ = ⋃_{N} R_{N}

is the ring constructed in Lemma 8.2. Put S′ = Spec(R′) and S_{N} = Spec(R_{N}). All of the schemes S′, S_{N} satisfy the conditions of 9.1 and S′ has a perfect residue field. The morphisms S_{N} → S are finite flat and the morphism S′ → S is faithfully flat. Also it is easily seen that the schemes S_{N} ×_{S} S_{N} and S_{N} ×_{S} S_{N} ×_{S} S_{N} satisfy the conditions of 9.1.

The existence of the faithfully flat morphism S′ → S implies that the functor BT′(k)_{S_{0}}^{o} → M′(k)_{S_{0}} is faithful for any S as in 9.1. Then we can show that it is fully faithful: given ψ : M_{S}(G_{2}) → M_{S}(G_{1}) we get a ϕ′ : G_{1S′_{0}} → G_{2S′_{0}} by the equivalence on S′. This ϕ′ is already defined on one of the S_{N}: we get a ϕ_{N} : G_{1(S_{N})_{0}} → G_{2(S_{N})_{0}}; but our functor on S_{N} ×_{S} S_{N} is faithfull so the fact that ψ_{N} descends implies that ϕ_{N} descends to a ϕ : G_{1} → G_{2} with M_{S}(ϕ) = ψ. A similar argument shows that the functor is essentially surjective also. Hence we get the following.

Corollary of Theorem 9.3. In Situation 9.1 the functor M_{S} induces equivalences of categories

BT′(k)_{S_{0}}^{o} ⟶ M′(k)_{S}

for 2 ≤ k ≤ n.

The proof of Theorem 9.3 runs in several steps.

Reduction step 1. To prove Theorem 9.3 we only need to prove that M_{S} induces an equivalence

C′_{S_{0}}^{o} ⟶ M′_{S}.

Proof. This follows by duality, the remarks made about the categories C′_{S_{0}}, C′′_{S_{0}}, M′_{S} and M′′_{S} and Lemma 9.2. □

To make the next step let us define

BT′_{S_{0}} = C′_{S_{0}} ∩ BT (1)_{S_{0}} ⊂ C(1)_{S_{0}}

and

BM′_{S} = M′_{S} ∩ BM(1)_{S} ⊂ M(1)_{S}.

Proposition 9.5.

1) For all G ∈ ObC′_{S_{0}} there exist H_{1}, H_{2} ∈ ObBT′_{S_{0}} and an injection of G into H_{1} and a surjection of H_{2} onto G.

2) For all (M, F, V) ∈ ObM′_{S} there exist (M_{1}, F_{1}, V_{1}), (M_{2}, F_{2}, V_{2}) ∈ ObBM′_{S} and an admissible (see Definition 8.4) injection of (M, F, V) into (M_{1}, F_{1}, V_{1}) respectively a surjection of (M_{2}, F_{2}, V_{2}) onto (M, F, V). Proof. By the theorem of Raynaud there is an injection of G into an object H ∈ ObBT (1)_{S_{0}}. Then let H_{1} = H/H^{μ}. As the multiplicative part of G is zero the morphism G → H → H/H^{μ} = H_{1} is an injection still. The multiplicative part of H_{1} is zero and it is a truncated Barsotti-Tate group since H^{μ} and H are truncated Barsotti-Tate groups (see [1] 3.3.9).

To get H_{2} we take an injection G^{D} ↪ H for some H ∈ ObBT (1)_{S_{0}}. Then the same arguments as above show that G^{D} ↪ Ker(H → H^{ét}) =: K and that K ∈ ObBT′′_{S_{0}}. We can take H_{2} = K^{D} ↠ G.

We can use exactly the same arguments to prove the second statement if we can prove the analog of Raynaud’s theorem: for any object (M, F, V) of M(1)_{S} there exists an admissible injection into an object of BM(1)_{S}.

To construct such an object let \{e_{1}, …, e_{l}\} be a basis for the \overline{R}-module M and denote by {e_{1}^{p}, …, e_{l}^{p}} the corresponding basis of M^{(p)}. We have F e_{i}^{p} = Σa_{i}^{j}e_{j} and V e_{i} = Σb_{i}^{j}e_{j}^{p} for certain a_{i}^{j}, b_{i}^{j} ∈ \overline{R}. Let M′ be the \overline{R}-module with basis \{e_{1}, …, e_{l}, f_{1}, …, f_{l}\}. We define \overline{R}-linear maps V′ : M′ → M′^{(p)} and F′ : M′^{(p)} → M′ by the formulas

\begin{aligned}F′(e_{i}^{p}) & = Σa_{i}^{j} & e_{j} \\ F′(f_{i}^{p}) & = & e_{i} − Σb_{i}^{j}f_{j} \\ V′(e_{i}) & = Σb_{i}^{j} & e_{j}^{p} \\ V′(f_{i}) & = & e_{i}^{p} − Σa_{i}^{j}f_{j}^{p}. \end{aligned}

Then F′ ∘ V′ = 0, V′ ∘ F′ = 0 and rank(F′) = rank(V′) = l so (M′, F′, V′) is an object of BM(1)_{S}. It is clear that we have an admissible injection (M, F, V) ↪ (M′, F′, V′). □

Reduction step 2. To prove Theorem 9.3 it is enough to prove that M_{S} induces an equivalence

BT′_{S_{0}}^{o} ⟶^{∼} BM′_{S}.

Proof. Suppose we have G_{1} and G_{2} ∈ ObC′_{S_{0}}. By the proposition above we can choose exact sequences

\begin{aligned}0 ⟶ G_{2} ⟶ & H_{0} ⟶ H_{1} \\ & E_{1} ⟶ E_{0} ⟶ G_{1} ⟶ 0 \end{aligned}

with H_{i}, E_{i} ∈ ObBT′_{S_{0}}. We get a commutative diagram with exact rows

0 →  Hom_{C′_{S_{0}}} (G_{1}, G_{2})  →  Hom_{C′_{S_{0}}} (E_{0}, H_{0})  →  Hom_{C′_{S_{0}}} (E_{1}, H_{1})
↓  ↓  ↓
0 → Hom_{M′_{S}}(M_{S}(G_{2}), M_{S}(G_{1}) → Hom_{M′_{S}}(M_{S}(H_{0}), M_{S}(E_{0})) → Hom_{M′_{S}}(M_{S}(H_{1}), M_{S}(E_{1}))

Hence if the functor BT′_{S_{0}} → BM′_{S} is fully faithful then the functor C′_{S_{0}} → M′_{S} is fully faithful. Let (M, F, V) ∈ ObM′_{S}. By the proposition above we can choose an exact sequence

0 ⟶ (M, F, V) ⟶ (M_{1}, F_{1}, V_{1}) ⟶ (M_{2}, F_{2}, V_{2}) ⟶ (M_{3}, F_{3}, V_{3})

with (M_{i}, F_{i}, V_{i}) ∈ ObBM′_{S}. If the functor BT′_{S_{0}} → BM′_{S} is an equivalence then we can find a corresponding sequence of BT′_{S_{0}}:

G_{1} ⟵^{α} G_{2} ⟵^{β} G_{3}

such that αβ = 0. By the remark at the bottom of page 73 we see that this sequence is exact. Then by Proposition 1.1 G =Cokern(α) is a finite flat group scheme. As the functor M_{S} is exact we get that

M_{S}(G) = Ker((M_{1}, F_{1}, V_{1}) ⟶ (M_{2}, F_{2}, V_{2})) = (M, F, V).  □

Reduction step 3. To prove Theorem 9.3 it is enough to prove that M_{S} induces an isomorphism

Hom_{C′_{S_{0}}} (Q, K) ⟶^{∼} Hom_{M′_{S}}(M_{S}(K), M_{S}(Q))

for all Q, K ∈ ObC′_{S_{0}} with V_{Q} = 0 and F_{K} = 0.

To prove this we may assume that S lies over Σ_{2}, i.e. that n = 1. Indeed, if this is not the case then we can replace R by R/p^{2}R and prove it for this ring. The compatibility 3.5.3 will imply the result for R.

Let us assume that for all S as in Situation 9.1 (with n = 1) and for all Q, K ∈ ObC′_{S_{0}} with V_{Q} = 0 and F_{K} = 0 we get an isomorphism

Hom_{C′_{S_{0}}} (Q, K) ⟶^{∼} Hom_{M′_{S}}(M_{S}(K), M_{S}(Q)).

Under this assumption let us prove that M_{S} : BT′_{S_{0}} → BM′_{S} is faithful. Choose any morphism ϕ : H_{1} → H_{2} of BT′_{S_{0}} such that M_{S}(ϕ) = 0. Let K_{i} = Ker(F_{H_{i}}) and Q_{i} = Im(F_{H_{i}}). Functoriality of the Frobenius morphism gives us a commutative diagram

0 ⟶ K_{1} ⟶ H_{1} ⟶ Q_{1} ⟶ 0
↓^{ϕ_{K}} ↓^{ϕ} ↓^{ϕ_{Q}}
0 ⟶ K_{2} ⟶ H_{2} ⟶ Q_{2} ⟶ 0 .

But ω_{K_{i}} = ω_{H_{i}} and α_{Q_{i}} = α_{H_{i}} and hence M_{S}(ϕ) = 0 implies ϕ_{K} = 0 and ϕ_{Q} = 0 (see Propositions 5.4 & 2.2). Thus ϕ must be a composition H_{1} → Q_{1} →^{ψ} K_{2} → H_{2}. As M_{S}(H_{2}) ↠ M_{S}(K_{2}) and M_{S}(Q_{1}) ↪ M_{S}(H_{1}) the fact that M_{S}(ϕ) is zero implies that M_{S}(ψ) = 0. By our assumption we get ψ = 0.

To prove that M_{S} : BT′_{S_{0}} → BM′_{S} is fully faithful (under our assumption) we use the following lemma.

Lemma 9.6. In Situation 9.1 assume n = 1. There exists a finite flat morphism π : S′ → S where S′ satisfies the conditions of 9.1 and is such that:

- The diagram \begin{matrix}S′ & ⟶^{f_{S′}} & S′ \\ ↓^{π} & & ↓^{π} \\ S & ⟶^{f_{S}} & S \end{matrix} commutes.

- For all G ∈ ObBT (1)_{S_{0}} ∃H ∈ ObBT (1)_{S′_{0}} such that π_{0}^{∗}G ≃ H^{(p)}. Proof. The assumptions on R imply that there exists a finite set of elements \{\overline{x}_{1}, …, \overline{x}_{l}\} of \overline{R} such that any element r ∈ \overline{R} can be written as r = Σ(a_{i})^{p}\overline{x}_{i} for certain a_{i} ∈ \overline{R}. Choose \{x_{1}, …, x_{l}\} ⊂ R which projects onto such a set of elements of \overline{R}. Write f_{S}(x_{i}) = x_{i}^{p} + pu_{i} and u_{i} = Σ(a_{i}^{j})^{p}\overline{x}_{j} for certain a_{i}^{j} ∈ \overline{R}. The ring R′ we are looking for is the ring

R′ := R[z_{1}, …, z_{l}]/(z_{i}^{p} − x_{i} + p(Σa_{i}^{j}z_{j}))

with f_{S′} determined by f_{S′}(z_{i}) = z_{i}^{p} = x_{i} − p(Σa_{i}^{j}z_{j}).

It is easily seen that R ⊂ R′ is finite flat and that R′ has only one maximal ideal. The field extension k ⊂ k′ is such that k = (k′)^{p}. Thus k′ ≃ k and k′ has a finite p-basis also. The ring R′ is Noetherian and complete because the ring extension R ⊂ R′ is finite flat.

The only thing left to prove is the last statement of the lemma. To this end we consider the morphism of schemes S_{0} →^{i} S′_{0} induced by the ringhomomorphism

\begin{matrix}\overline{R}′ & ⟶ \overline{R} \\ r & ⟼ r^{p} ∈ \overline{R} ⊂ \overline{R}′. \end{matrix}

The morphism S_{0} →^{i} S′_{0} is a closed immersion as by construction \overline{R} ⊂ \overline{R}′ is exactly the set of p^{th}-powers of \overline{R}′. Let I ⊂ \overline{R}′ be the kernel of the homomorphism \overline{R}′ → \overline{R}. Then I is a nilpotent ideal as it is generated by finitely many elements (\overline{R}′ Noetherian!) whose p^{th}-power is zero.

From [4] Theorem 4.4 we get a truncated Barsotti-Tate group H on S′_{0} such that i^{∗}H ≃ G. The commutative diagram

\begin{matrix}S′_{0} & ⟶^{f_{S′_{0}}} & S′_{0} \\ ↓^{π_{0}} & ↗^{i} & ↓^{π_{0}} \\ S_{0} & ⟶^{f_{S_{0}}} & S_{0} \end{matrix}

shows that H^{(p)} = f_{S′_{0}}^{∗}H = π_{0}^{∗}i^{∗}H ≅ π_{0}^{∗}G. □

Remark 9.7. It follows from the proof of the lemma that the schemes S′×_{S} S′ and S′×_{S} S′×_{S} S′ satisfy the conditions of 9.1. Further we get various commutative diagrams involving lifts of Frobenii and projection maps. This will allow us to use descent arguments.

Let ψ : M_{S}(G_{2}) → M_{S}(G_{1}) be a morphism of BM′_{S} for certain G_{1}, G_{2} ∈ ObBT′_{S_{0}}. We want to construct a ϕ : G_{1} → G_{2} such that M_{S}(ϕ) = ψ.

Suppose that we have a ϕ′ : π_{0}^{∗}G_{1} → π_{0}^{∗}G_{2} such that M_{S′}(ϕ′) = π^{∗}ψ for π : S′ → S as in Lemma 9.6. The difference ϕ_{12} = pr_{1}^{∗}(ϕ′) − pr_{2}^{∗}(ϕ′) : (π_{0} × π_{0})^{∗}G_{1} → (π_{0} × π_{0})^{∗}G_{2} has M_{S′×_{S}S′}(ϕ_{12}) = pr_{1}^{∗}π^{∗}ψ − pr_{2}^{∗}π^{∗}ψ = 0. Hence ϕ_{12} = 0 since we already know that M_{S′×_{S}S′} is faithful. Thus ϕ′ = π_{0}^{∗}ϕ for some morphism ϕ : G_{1} → G_{2} which automatically satisfies M_{S}(ϕ) = ψ.

Therefore, we can work on S = S′ and we may assume that G_{i} = H_{i}^{(p)} for certain H_{i} ∈ ObBT′_{S_{0}}. If we put K_{i} = Ker(F_{H_{i}}) and Q_{i} = Im(F_{H_{i}}) then we see that G_{i} is an extension

0 ⟶ Q_{i} ⟶ G_{i} ⟶ K_{i} ⟶ 0.

Two of the associated exact sequences are

0 ⟶ Hom(G_{1}, Q_{2}) ⟶ Hom(G_{1}, G_{2}) ⟶ Hom(G_{1}, K_{2}) ⟶ Ext^{1}(G_{1}, Q_{2})

and

0 ⟶ Hom(K_{1}, K_{2}) ⟶ Hom(G_{1}, K_{2}) ⟶ Hom(Q_{1}, K_{2}) ⟶ Ext^{1}(K_{1}, K_{2}).

(We have dropped the subscript C′_{S_{0}}.) Further we have corresponding exact sequences involving M_{S}(K_{i}), M_{S}(G_{i}), M_{S}(Q_{i}) and M_{S} defines maps between those exact sequences. By the assumption and by Proposition 8.6 we see that all of the maps induced by M_{S} are already isomorphisms except for possibly Hom(G_{1}, K_{2}) →^{α} Hom(M_{S}(K_{2}), M_{S}(G_{1})) and Hom(G_{1}, G_{2}) →^{β} Hom(M_{S}(G_{2}), M_{S}(G_{1})). However, the second of the exact sequences above shows that α is an isomorphism; the first then allows us to see that the same is true for β.

To finish the proof of reduction step 3 let us prove (under our assumption) that M_{S} : BT′_{S_{0}} → BM′_{S} is essentially surjective.

Let (M, F, V) ∈ ObBM′_{S} and let π : S′ → S be as in Lemma 9.6. We only need to constuct a G′ ∈ ObBT′_{S′_{0}} such that M_{S′}(G′) ≃ π^{∗}(M, F, V). Indeed, we already know that M_{S′×_{S}S′} and M_{S′×_{S}S′×_{S}S′} are fully faithful restricted to the appropriate categories. In this way the natural descent data on π^{∗}(M, F, V) are translated into descent data on G′ and we will get our group G on S_{0}.

Choose an \overline{R}-basis \{e_{1}, …, e_{l}\} for the \overline{R}-module M and write \{e_{1}^{p}, …, e_{l}^{p}\} for the corresponding basis of M^{(p)}. As (M, F, V) ∈ ObBT′_{S_{0}} the morphism F : M^{(p)} → M has a kernel which is a locally free direct summand of M^{(p)}. Hence we can choose an \overline{R}-basis \{f_{1}, …, f_{l}\} of M^{(p)} such that Ker(F) = ⟨f_{1}, …, f_{r}⟩. We have f_{i} = Σa_{i}^{j}e_{j}^{p} for certain a_{i}^{j} ∈ \overline{R}. Choose elements b_{i}^{j} ∈ \overline{R}′ such that (b_{i}^{j})^{p} = a_{i}^{j} (this is possible; see proof of Lemma 9.6). The set \{Σb_{1}^{j}e_{j}, …, Σb_{l}^{j}e_{j}\} will be a basis for the \overline{R}′-module π^{∗}M = \overline{R}′ ⊗\overline{R} M. Let M′ ⊂ π^{∗}M be the (free) submodule of π^{∗}M generated by Σb_{1}^{j}e_{j}, …, Σb_{r}^{j}e_{j} and let M′′ = π^{∗}M/M′ the (free) quotient. Remark that we have M′^{(p)} = Ker(π^{∗}F) by construction and hence Im(π^{∗}V) = Ker(π^{∗}F) = M′^{(p)}. From all of this we get an exact sequence in M′_{S′}

0 ⟶ (M′, 0, π^{∗}(V)|_{M′}) ⟶ (π^{∗}M, π^{∗}F, π^{∗}V) ⟶ (M′′, π^{∗}F mod M′, 0) ⟶ 0.

By Proposition 2.2 and its dual we get finite flat group schemes Q and K on S′_{0} such that V_{Q} = 0, F_{K} = 0, M_{S′}(Q) ≃ (M′′, π^{∗}F mod M′, 0) and M_{S′}(K) ≃ (M′, 0, π^{∗}(V)|_{M′}). By Proposition 8.6 we have Ext_{C(1)_{S′_{0}}}^{1} (K, Q) ≃ Ext_{M(1)_{S′}}^{1} (M_{S′}(Q), M_{S′}(K)) and hence we get a group scheme G′ ∈ ObC(1)_{S′_{0}} corresponding to the extension defined by (π^{∗}M, π^{∗}F, π^{∗}V). Since M_{S′}(G′) ∈ ObBM′_{S′} it follows that G′ ∈ ObBT′_{S′_{0}} (see Lemma 9.2 and remark on page 74). This concludes the proof of reduction step 3.

Let us take K, Q ∈ ObC′_{S_{0}} with V_{Q} = 0 and F_{K} = 0. To prove Theorem 9.3 we have to show that M_{S} induces an isomorphism

(∗)  Hom_{C′_{S_{0}}} (Q, K) ⟶^{∼} Hom_{M′_{S}}(M_{S}(K), M_{S}(Q)).

Let us introduce some notation. We write Q_{1} = Q and Q_{2} = K^{D}. Then Q_{1} = Spec(A_{1}) and Q_{2} = Spec(A_{2}) where

A_{1} = \overline{R}[x_{1}, …, x_{k}]/(x_{i}^{p} − Σa_{i}^{j}x_{j})

and

A_{2} = \overline{R}[y_{1}, …, y_{l}]/(y_{i}^{p} − Σb_{i}^{j}y_{j})

for certain a_{i}^{j} ∈ \overline{R} and b_{i}^{j} ∈ \overline{R} (see §2). The Hopf algebra structure on A_{1} (resp. A_{2}) is given by x_{i} ↦ x_{i} ⊗ 1 + 1 ⊗ x_{i} (resp. y_{i} ↦ y_{i} ⊗ 1 + 1 ⊗ y_{i}); the augmentation ideal I_{A_{1}} of A_{1} is the ideal generated by x_{1}, …, x_{k}, similar for the augmentation ideal I_{A_{2}} of A_{2}.

The fact that K ∈ ObC′_{S_{0}}, i.e. that Q_{2} has no étale part, implies that A_{2} has exactly one maximal ideal m_{A_{2}}. Hence we must have m_{A_{2}} = \sqrt{\overline{m} · A_{2}} where \overline{m} denotes the maximal ideal of \overline{R} and we must have I_{A_{2}} ⊂ m_{A_{2}}. Therefore, for all i = 1, …, l, there exists a natural number N_{i} such that

y_{i}^{N_{i}} ∈ \overline{m} · A_{2}.

From this we infer that there exists a natural number N such that

I_{A_{2}}^{N} ⊂ \overline{m} · A_{2}.

For any 𝔽_{p}-algebra B we put I_{p}(B) = {x ∈ B | x^{p} = 0}. It is an ideal of B. Let I ⊂ A_{1} ⊗\overline{R} A_{2} be the following ideal

I := I_{A_{1}} ⊗\overline{R} I_{A_{2}} ∩ I_{p}(A_{1}) ⊗\overline{R} I_{A_{2}} ∩ I_{A_{1}} ⊗\overline{R} I_{p}(A_{2}).

In particular we have I ⊂ I_{p}(A_{1} ⊗\overline{R} A_{2}). Suppose we have two finite flat R-algebras Ã_{1} and Ã_{2} which lift A_{1} and A_{2}, i.e. such that A_{1} = Ã_{1}/pÃ_{1} and A_{2} = Ã_{2}/pÃ_{2}. For any x ∈ I we can choose a lift x̃ ∈ Ã_{1} ⊗_{R} Ã_{2}. The element x̃^{p} will lie in pA_{1} ⊗\overline{R} A_{2} ⊂ Ã_{1} ⊗_{R} Ã_{2}. In this way we get an element

γ_{_{A_{1}}⊗_{R}_{A_{2}}}(x) := \frac{1}{p} · x̃^{p} ∈ A_{1} ⊗\overline{R} A_{2}.

As usual this element does not depend on the choice of the lift x̃. (Compare with Example 5.6.)

Lemma 9.8. Let x ∈ I.

1) The element γ_{_{A_{1}}⊗_{R}_{A_{2}}}(x) does not depend on the choice of the lifts Ã_{1} and Ã_{2}. We will write γ(x) for this element.

2) γ(x) ∈ I.

3) lim_{n→∞} γ^{n}(x) = 0 in the \overline{m}-adic topology on A_{1} ⊗\overline{R} A_{2}. Proof. We can write

x = ∑ i_{α} ⊗ w_{α}

for certain i_{α} ∈ I_{p}(A_{1}) and w_{α} ∈ I_{A_{2}}. If we choose

x̃ = ∑ ĩ_{α} ⊗ w̃_{α}

where ĩ_{α} ∈ Ã_{1} lifts i_{α} and w̃_{α} ∈ Ã_{2} lifts w_{α} then

x̃^{p} = (∑ ĩ_{α} ⊗ w̃_{α})^{p} = ∑ ĩ_{α}^{p} ⊗ w̃_{α}^{p} + p · expression in i_{α} ⊗ w_{α}.

Thus we get

\frac{1}{p}x̃^{p} = ∑ \frac{1}{p}(ĩ_{α})^{p} ⊗ w_{α}^{p} + expression in i_{α} ⊗ w_{α}.

It is clear that this expression does not depend on the lift Ã_{2}. Similarly we may change Ã_{1} without changing γ_{_{A_{1}}⊗_{R}_{A_{2}}}(x).

Choose lifts ã_{i}^{j} and b̃_{i}^{j} ∈ R of the elements a_{i}^{j} and b_{i}^{j} ∈ \overline{R}. We can take

Ã_{1} = R[x_{1}, …, x_{k}]/(x_{i}^{p} − Σã_{i}^{j}x_{j})

and

Ã_{2} = R[y_{1}, …, y_{l}]/(y_{i}^{p} − Σb̃_{i}^{j}y_{j}).

The ideals I_{A_{1}} = x_{1}Ã_{1} + … + x_{k}Ã_{1} and I_{A_{2}} = y_{1}Ã_{2} + … + y_{l}Ã_{2} are R-flat. If we choose our element x̃ ∈ I_{A_{1}} ⊗_{R} I_{A_{2}} then we see that \frac{1}{p}x̃^{p} lies in I_{A_{1}} ⊗\overline{R} I_{A_{2}}.

Let us take A_{1} ⊂ B_{1}, B̃_{1} and F : B̃_{1} → B̃_{1} as in Sublemma 9.9. Further we take any lift Ã_{2} of A_{2}. We can compute γ(x) also as follows: for any lift x̃ ∈ B̃_{1} ⊗_{R} Ã_{2} of the element x ∈ A_{1} ⊗\overline{R} A_{2} ⊂ B_{1} ⊗\overline{R} A_{2} we will have \frac{1}{p}x̃^{p} = γ(x) ∈ A_{1} ⊗\overline{R} A_{2} ⊂ B_{1} ⊗\overline{R} A_{2}. This can be proved with exactly the same proof as we used to prove that γ_{_{A_{1}}⊗_{R}_{A_{2}}}(x) does not depend on the choice of Ã_{1} or Ã_{2}.

Again we write

x = ∑_{α=1}^{m} i_{α} ⊗ w_{α}

with i_{α} ∈ I_{p}(B_{1}) and w_{α} ∈ I_{A_{2}}. We choose lifts ĩ_{α} ∈ B̃_{1} and w̃_{α} ∈ Ã_{2} of the elements i_{α} and w_{α}. As a lift of x we take

x̃ = ∑ ĩ_{α} ⊗ w̃_{α} ∈ B̃_{1} ⊗_{R} Ã_{2}.

The exact formula for \frac{1}{p}x̃^{p} is then

\frac{1}{p}x̃^{p} = ∑_{α=1}^{m} \frac{1}{p}(ĩ_{α})^{p} ⊗ w_{α}^{p} + ∑_{{j_{1}+...+j_{m}=p \atop 0≤j_{i}≤p−1}} \frac{1}{p}\binom{p}{j_{1}, …, j_{m}}(i_{1})^{j_{1}} … (i_{m})^{j_{m}} ⊗ (w_{1})^{j_{1}} … (w_{m})^{j_{m}}.

By Sublemma 9.10 the elements \frac{1}{p}(ĩ_{α})^{p} are in I_{p}(B_{1}). Hence it follows that \frac{1}{p}x̃^{p} ∈ I_{p}(B_{1}) ⊗\overline{R} I_{A_{2}} ∩ A_{1} ⊗\overline{R} A_{2} = I_{p}(A_{1}) ⊗\overline{R} I_{A_{2}}. Reversing the roles of A_{1} and A_{2} we derive in the same manner that γ(x) ∈ I_{A_{1}} ⊗\overline{R} I_{p}(A_{2}). Hence we have shown that γ(x) ∈ I.

If we look more closely at the formula for \frac{1}{p}x̃^{p} above then we see that in fact γ(x) ∈ I_{p}(B_{1}) ⊗\overline{R} (I_{A_{2}})^{p}. If we use the same expression to calculate γ(γ(x)) then we see that γ(γ(x)) = γ^{2}(x) ∈ I_{p}(B_{1}) ⊗\overline{R} (I_{A_{2}})^{p^{2}}. Continuing this we get γ^{n}(x) ∈ I_{p}(B_{1}) ⊗\overline{R} (I_{A_{2}})^{p^{n}}. We know that there exists a natural number N such that I_{A_{2}}^{N} ⊂ \overline{m} · A_{2} (see page 87) and hence γ^{n}(x) ∈ I_{p}(B_{1}) ⊗\overline{R} \overline{m}^{⌊p^{n}/N⌋} · A_{2}. But γ^{n}(x) is also in A_{1} ⊗\overline{R} A_{2} and we get γ^{n}(x) ∈ \overline{m}^{⌊p^{n}/N⌋} · A_{1} ⊗\overline{R} A_{2}. This proves the last statement of Lemma 9.8. □

Sublemma 9.9. For any \overline{R}-algebra A of the form

A = \overline{R}[x_{1}, …, x_{k}]/(x_{i}^{p} − Σa_{i}^{j}x_{j})

there exists a finite, flat extension A ⊂ B and a R-flat lift B̃ of B which has a lift of Frobenius F : B̃ → B̃. That is F is a ring endomorphism of B̃ such that F (b) = b^{p} mod pB̃ ∀b ∈ B̃. Proof. Let R′ be the ring constructed in Lemma 9.6. Choose c_{i}^{j} ∈ R′ such that (c_{i}^{j})^{p} ≡ a_{i}^{j} mod pR′. Put

B̃ = R′[y_{1}, …, y_{k}]/(y_{i}^{p^{2}} − Σ(c_{i}^{j})^{p}y_{j}^{p} − pe_{i})

where e_{i} is the expression:

e_{i} = ∑_{{i_{1}+...+i_{k}=p \atop 0≤i_{α}≤p−1}} \frac{1}{p}\binom{p}{i_{1}, …, i_{k}}(c_{i}^{1}y_{1})^{i_{1}} … (c_{i}^{k}y_{k})^{i_{k}}.

It is seen that putting F (y_{i}) = y_{i}^{p} will define a lift of Frobenius on B̃. Further we have

B = B̃/pB̃ = \overline{R}′[y_{1}, …, y_{k}]/(y_{i}^{p^{2}} − Σa_{i}^{j}y_{j}^{p})
≃ \overline{R}′[x_{1}, …, x_{k}, y_{1}, …, y_{k}]/(y_{i}^{p} − x_{i}, x_{i}^{p} − Σa_{i}^{j}x_{j})
≃ \overline{R}′ ⊗\overline{R} A[y_{1}, …, y_{k}]/(y_{i}^{p} − x_{i}).

This proves that A ⊂ B is finite flat. □

Sublemma 9.10. Let B̃ be a ℤ/p^{2}ℤ-flat ring which has a lift of frobenius F : B̃ → B̃. For any x̃ ∈ B̃ such that x̃ mod p lies in I_{p}(B̃/pB̃) the element \frac{1}{p}x̃^{p} lies in I_{p}(B̃/pB̃) also. Proof. Write x̃^{p} = py and F (x̃) = x̃^{p} + pu.

x̃^{p} = py ⇒ F (x̃^{p}) = F (py) ⇒ (F (x̃))^{p} = py^{p}
⇒ (x̃^{p} + pu)^{p} = py^{p} ⇒ (p(y + u))^{p} = py^{p} ⇒ y^{p} = 0.  □

Suppose we have a homomorphism

ϕ : Q ⟶ K.

It induces M_{S}(ϕ) : ω_{K} = M_{S}(K) → M_{S}(Q) = α_{Q}. By Proposition 1.5 we can see this as an element

M_{S}(ϕ) ∈ α_{Q_{1}} ⊗\overline{R} α_{Q_{2}} ⊂ A_{1} ⊗\overline{R} A_{2}.

The homomorphism ϕ also gives rise to a homomorphism

e_{ϕ} : ℤ/pℤ_{Q_{1}×_{S_{0}} Q_{2}} ⟶ μ_{p,Q_{1}×_{S_{0}} Q_{2}}

as follows:

ℤ/pℤ_{Q_{1}×_{S_{0}} Q_{2}} ⟶^{id_{Q_{1}}} (Q_{1})_{Q_{1}×_{S_{0}} Q_{2}} ⟶^{ϕ} (Q_{2})_{Q_{1}×_{S_{0}} Q_{2}}^{D} ⟶^{(id_{Q_{2}} )^{D}} μ_{p,Q_{1}×_{S_{0}} Q_{2}}.

Let us choose finite flat schemes Q̃_{1} and Q̃_{2} over S which lift Q_{1} and Q_{2}. As in Example 5.6 we identify the element M_{_{Q_{1}}×_{S} _{Q_{2}}}(e_{ϕ}) with an element of A_{1} ⊗\overline{R} A_{2}.

Claim 9.11. M_{S}(ϕ) = M_{_{Q_{1}}×_{S} _{Q_{2}}}(e_{ϕ}).

Proof. Let π : Q̃_{1} ×_{S} Q̃_{2} → S denote the structural morphism. The above equality is a consequence of the following commutative diagram: (we write X = Q̃_{1} ×_{S} Q̃_{2})

M_{X}(ℤ/pℤ_{X_{0}}) ⟵  M_{X}(Q_{1,X_{0}})  ⟵  M_{X}(Q_{2,X_{0}}^{D})  ⟵ M_{X}(μ_{p,X_{0}})
≃↑  ^{≃}↑  ^{≃}↑  ^{≃}↑
A_{1} ⊗\overline{R} A_{2} ⟵ α_{Q_{1}} ⊗\overline{R} A_{1} ⊗\overline{R} A_{2} ⟵^{M_{S}(ϕ)⊗id} ω_{Q_{2}^{D}} ⊗\overline{R} A_{2} ⊗\overline{R} A_{1} ⟵ A_{1} ⊗\overline{R} A_{2}
a ⊗ 1  ⟵  a ⊗ 1 ⊗ 1  Σ_{i}x_{i}^{∗} ⊗ 1 ⊗ x_{i} ⟵  1

({x_{i}^{∗}} is the dual basis of the basis {x_{i}} of α_{Q_{2}}). The commutativity of the left and right squares are is a consequence of the functoriality of the transformations α → M_{S} → ω. The middle square is commutative simply by Proposition 3.5.2. □

Recall from Example 5.6 that the homomorphism e_{ϕ} is determined by an element y_{ϕ} ∈ I_{p}(A_{1} ⊗\overline{R} A_{2}).

Claim 9.12. y_{ϕ} ∈ I. Proof. Let us denote by σ : Q_{1} ×_{S_{0}} Q_{2} → ℤ/pℤ_{Q_{1}×_{S_{0}} Q_{2}} the section corresponding to \overline{1} ∈ ℤ/pℤ. The morphism e_{ϕ} ∘ σ : Q_{1} ×_{S_{0}} Q_{2} → μ_{p} is the bilinear pairing

(x, y) ⟼ y(ϕ(x)) ∀x ∈ Q_{1}, y ∈ Q_{2}

where we have used Q_{2} = Hom(K, μ_{p,S_{0}}). Thus we see that y_{ϕ} ∈ I_{A_{1}} ⊗\overline{R} I_{A_{2}} as e_{ϕ} ∘ σ(x, 0) = 1 and e_{ϕ} ∘ σ(0, y) = 1. In the same manner y_{ϕ} ∈ I_{p}(A_{1}) ⊗\overline{R} A_{2} and y_{ϕ} ∈ A_{1} ⊗\overline{R} I_{p}(A_{2}) is a consequence of the fact that the diagram

Q_{1}  ⟶^{F_{Q_{1}}} Q_{1}^{(p)}  ⟶^{0} Q_{1}
↓^{ϕ}  ↓_{ϕ^{(p)}}  ↓^{ϕ}
Q_{2}^{D}  ⟶^{0} Q_{2}^{(p)D}  ⟶^{F_{Q_{2}}^{D}} Q_{2}^{D}

is commutative. □

We are now able to show that the map (∗) is injective. First we remark that ϕ = 0 ⟺ e_{ϕ} = 0 ⟺ y_{ϕ} = 0. If M_{S}(ϕ) = 0 then by Claim 9.11 M_{_{Q_{1}}×_{S} _{Q_{2}}}(e_{ϕ}) = 0. Example 5.6 shows that M_{_{Q_{1}}×_{S} _{Q_{2}}}(e_{ϕ}) = y_{ϕ} + γ_{_{Q_{1}}×_{S} _{Q_{2}}}(y_{ϕ}) and this is equal to y_{ϕ} + γ(y_{ϕ}) by Claim 9.12. Thus M_{S}(ϕ) = 0 ⇒ y_{ϕ} = −γ(y_{ϕ}) = −γ(−γ(y_{ϕ})) = γ^{2}(y_{ϕ}) = … = (−1)^{N}γ^{N}(y_{ϕ}). By Lemma 9.8.3 this implies y_{ϕ} is zero.

Finally let us show that the map (∗) is surjective. Suppose ψ ∈ Hom_{C′_{S_{0}}} (M_{S}(Q_{2}^{D}), M_{S}(Q_{1})). As we did with M_{S}(ϕ) we can see ψ as an element of α_{Q_{1}} ⊗\overline{R} α_{Q_{2}} . Thus we can write

ψ = Σs^{ij}x_{i} ⊗ y_{j}.

The fact that ψ ∈ Hom_{C′_{S_{0}}} (M_{S}(Q_{2}^{D}), M_{S}(Q_{1})) is equivalent to the fact that

Σ(s^{ij})^{p}a_{i}^{m} = 0 and Σ(s^{ij})^{p}b_{j}^{m} = 0

in other words that ψ is an element of I.

Lemma 9.13. The map I → I : x ↦ x + γ(x) is a bijection of sets. Proof. We saw above that it is injective. For any x ∈ I we define a sequence of elements x_{0}, x_{1}, … in I inductively by

\{ \begin{matrix}x_{0} & = x \\ x_{n+1} & = x − γ(x_{n}), n ≥ 0. \end{matrix}

If this is a Cauchy sequence for the \overline{m}-adic topology on A_{1} ⊗\overline{R} A_{2} then the limit x_{∞} ∈ I will satisfy x_{∞} + γ(x_{∞}) = x. (The \overline{m}-adic topology on A_{1} ⊗\overline{R} A_{2} induces the \overline{m}-adic topology on I and γ is continuous.)

Note the following identity

x_{n+1} − x_{n} = x − γ(x_{n}) − x_{n}
= x − γ(x_{n}) − x + γ(x_{n−1})
= γ(x_{n−1}) − γ(x_{n})
= γ(x_{n−1} − x_{n}) − B_{p}(x_{n−1}, −x_{n}).

Here we used that γ(x + y) = γ(x) + B_{p}(x, y) + γ(y) for all x, y ∈ I.

Take N ∈ ℕ. We have to show that there exists a M ∈ ℕ such that ∀n ≥ M : x_{n+1} − x_{n} ∈ \overline{m}^{N} · A_{1} ⊗\overline{R} A_{2}. The identity above shows that

x_{n+1} − x_{n} ≡ −γ(x_{n} − x_{n−1}) ≡ (−1)^{n}γ^{n}(x_{1} − x_{0}) ≡ (−1)^{n+1}γ^{n+1}(x)

modulo I^{p}. By Lemma 9.8 we see that there exists a natural number M_{p} such that ∀n ≥ M_{p}

x_{n+1} − x_{n} ∈ I^{p} + \overline{m}^{N} · A_{1} ⊗\overline{R} A_{2}.

Thus for all n ≥ M_{p} + 1 we see that

B_{p}(x_{n−1}, −x_{n}) ≡ B_{p}(x_{n−1}, −x_{n−1}) ≡ 0

mod I^{p+1} + \overline{m}^{N} · A_{1} ⊗\overline{R} A_{2}. Hence ∀n ≥ M_{p} + 1

x_{n+1} − x_{n} ≡ ±γ^{n−M_{p}}(x_{M_{p+1}} − x_{M_{p}}) mod I^{p+1} + \overline{m}^{N} · A_{1} ⊗\overline{R} A_{2}.

Again by applying Lemma 9.8 we see that there exists a natural number M_{p+1} such that ∀n ≥ M_{p+1}

x_{n+1} − x_{n} ∈ I^{p+1} + \overline{m}^{N} · A_{1} ⊗\overline{R} A_{2}.

Repeating this argument we see that for every natural number m there exists a M_{m} such that ∀n ≥ M_{m}

x_{n+1} − x_{n} ∈ I^{m} + \overline{m}^{N} · A_{1} ⊗\overline{R} A_{2}.

However, the ideal I is nilpotent as it is generated by finitely many elements (A_{2} ⊗\overline{R} A_{2} is Noetherian) whose p^{th}-powers are zero. Hence taking m large enough gives the desired result. □

Let us take an element y_{ψ} ∈ I such that y_{ψ} + γ(y_{ψ}) = ψ.

Claim 9.13. The element y_{ψ} defines a bilinear paring Q_{1} ×_{S_{0}} Q_{2} → μ_{p,S_{0}}, hence we get a homomorphism ϕ : Q_{1} → Q_{2}^{D} which satisfies M_{S}(ϕ) = ψ. Proof. The element y_{ψ} defines a ringhomomorphism

\begin{matrix}O_{μ_{p}} = 𝔽_{p}[y]/(y^{p}) & ⟶ & A_{1} ⊗\overline{R} A_{2} \\ y & ⟼ & y_{ψ} \end{matrix}

as in Example 5.6. We have to show that the diagram

𝔽_{p}[y]/(y^{p})  ⟶  A_{1} ⊗\overline{R} A_{2}
↓^{μ}  ↓^{μ⊗id_{A_{2}}}
𝔽_{p}[y]/(y^{p}) ⊗_{𝔽_{p}} 𝔽_{p}[y]/(y^{p}) ⟶ A_{1} ⊗\overline{R} A_{1} ⊗\overline{R} A_{2}
y ⊗ 1  ⟼  (y_{ψ})_{13}
1 ⊗ y  ⟼  (y_{ψ})_{23}

commutes; here we have written (x)_{13} = Σa_{α} ⊗1⊗b_{α} for any element x = Σa_{α} ⊗b_{α} of A_{1} ⊗\overline{R} A_{2} and similarly (x)_{23} = Σ1 ⊗ a_{α} ⊗ b_{α}.

If we replace in the definition of the ideal I the ring A_{1} by the ring A_{1} ⊗\overline{R} A_{1} then we get an ideal I′ ⊂ A_{1} ⊗\overline{R} A_{1} ⊗\overline{R} A_{2} and a map γ′ : I′ → I′ having the properties expressed in Lemma 9.8. Since the element γ(y_{ψ}) can be computed using a lift Ã_{2} of A_{2} only (see proof of Lemma 9.8) we see that (μ ⊗ id_{A_{2}})(γ(y_{ψ})) = γ′((μ ⊗ id_{A_{2}})(y_{ψ})). Hence we see that (μ ⊗ id_{A_{2}})(y_{ψ}) is the unique element of I′ such that

(μ ⊗ id_{A_{2}})(y_{ψ}) + γ′((μ ⊗ id_{A_{2}})(y_{ψ})) = (μ ⊗ id_{A_{2}})(ψ)
=(μ ⊗ id_{A_{2}})(Σs^{ij}x_{i} ⊗ y_{j}) = Σs^{ij}(x_{i} ⊗ 1 ⊗ y_{j} + 1 ⊗ x_{i} ⊗ y_{j}) = (ψ)_{13} + (ψ)_{23}

However, it is also easy to see that

(y_{ψ})_{13} + γ′((y_{ψ})_{13}) = (ψ)_{13}

and

(y_{ψ})_{23} + γ′((y_{ψ})_{23}) = (ψ)_{23}.

Now look at

A : = (y_{ψ})_{13} + (y_{ψ})_{23} − B_{p}((y_{ψ})_{13}, (y_{ψ})_{23}) + γ′((y_{ψ})_{13} + (y_{ψ})_{23} − B_{p}((y_{ψ})_{13}, (y_{ψ})_{23})))
= (y_{ψ})_{13} + (y_{ψ})_{23} − B_{p}((y_{ψ})_{13}, (y_{ψ})_{23}) + γ′((y_{ψ})_{13} + (y_{ψ})_{23})
+ B_{p}((y_{ψ})_{13} + (y_{ψ})_{23}, B_{p}((y_{ψ})_{13}, (y_{ψ})_{23})) + γ′(B_{p}((y_{ψ})_{13}, (y_{ψ})_{23}))

The last two terms are zero: the first one because it is a homogeneous polynomial of degree bigger than 2p − 1 in (y_{ψ})_{13} and (y_{ψ})_{23} and so any momonial occuring in it must contain either of them to a power bigger or equal to p, the second one because γ′(i_{1} · i_{2}) = 0 for all i_{1}, i_{2} ∈ I′ and B_{p}((y_{ψ})_{13}, (y_{ψ})_{23}) = (y_{ψ})_{13} (a polynomial in (y_{ψ})_{13} and (y_{ψ})_{23} of degree ≥ 1). Thus we go on:

\begin{aligned}A & = (y_{ψ})_{13} + (y_{ψ})_{23} + γ′((y_{ψ})_{13}) + γ′((y_{ψ})_{23}) \\ & = (ψ)_{13} + (ψ)_{23} \end{aligned}

and we conclude that

μ ⊗ id_{A_{2}}(y_{ψ}) = (y_{ψ})_{13} + (y_{ψ})_{23} − B_{p}((y_{ψ})_{13}, (y_{ψ})_{23}).

However, this is exactly equivalent to the commutativity of the diagram above:

μ(y) = y ⊗ 1 + 1 ⊗ y − B_{p}(y ⊗ 1, 1 ⊗ y)

(see Example 5.6).

The commutativity of the diagram where the roles of A_{1} and A_{2} are interchanged is proved in exactly the same way. □

10. Perfect residue field

In this section S will be as in Situation 9.1 and moreover we assume that R has a perfect residue field. We will find conditions on (S, f_{S}) which imply that the functor

C(1)_{S_{0}}^{o} ⟶ M(1)_{S}

is an equivalence.

As before we write \overline{m} for the maximal ideal of \overline{R} and write \overline{m}(p) for the set of p^{th}-powers in \overline{m}. For any x ∈ R we write f_{S}(x) = x^{p} + pu(x). Thus u is a map

u : R ⟶ R/p^{n}R

which has the following properties

\{ \begin{matrix}u(x + y) & = u(x) + u(y) − B_{p}(x, y) \\ u(x · y) & = x^{p}u(y) + y^{p}u(x) + pu(x)u(y). \end{matrix}

By Lemma 8.2 and its proof there is a unique ℤ/p^{n+1}ℤ-flat ideal m̃ ⊂ R such that u(m̃) ⊂ m̃. Hence we can define a p-linear map

\overline{u} : \overline{m}/\overline{m}^{2} ⟶ \overline{m}/\overline{m}^{2}

by the following procedure: for any \overline{x} ∈ \overline{m} take a lift x ∈ m̃ and put \overline{u} (class of x̃) := class of u(x) mod p. It is easy to check that this is well-defined.

From the map

γ_{S} : I_{p}(\overline{R}) ⟶ I_{p}(\overline{R})

(see Example 5.6 and Sublemma 9.10) we derive a p-linear map

\overline{γ} : I_{p}(\overline{R})/\overline{m} · I_{p}(\overline{R}) ⟶ I_{p}(\overline{R})/\overline{m} · I_{p}(\overline{R}).

Using the maps \overline{u} and \overline{γ} we can say something about Ext_{C(1)_{S_{0}}}^{1} (ℤ/pℤ_{S_{0}}, μ_{p,S_{0}}) and Hom_{S_{0}}(ℤ/pℤ_{S_{0}}, μ_{p,S_{0}}).

Lemma 10.1.

1) If \overline{γ} is nilpotent then

Hom_{C(1)_{S_{0}}} (ℤ/pℤ_{S_{0}}, μ_{p,S_{0}}) ⟶ Hom_{M(1)_{S}}(M_{S}(μ_{p,S_{0}}), M_{S}(ℤ/pℤ_{S_{0}}))

is an isomorphism.

2) If \overline{u} is nilpotent then

Ext_{C(1)_{S_{0}}}^{1} (ℤ/pℤ_{S_{0}}, μ_{p,S_{0}}) ⟶ Ext_{M(1)_{S}}^{1}(M_{S}(μ_{p,S_{0}}), M_{S}(ℤ/pℤ_{S_{0}}))

is an isomorphism. Proof. A homomorphism of \overline{R}-modules ψ : M_{S}(μ_{p,S_{0}}) → M_{S}(ℤ/pℤ_{S_{0}}) is in Hom_{M(1)_{S}}(M_{S}(μ_{p,S_{0}}), M_{S}(ℤ/pℤ_{S_{0}})) if and only if ψ(\frac{dt}{t}) = λ · e with λ ∈ I_{p}(\overline{R}). Hence by Example 5.6

Hom_{C(1)_{S_{0}}} (ℤ/pℤ_{S_{0}}, μ_{p,S_{0}}) ⟶ Hom_{M(1)_{S}}(M_{S}(μ_{p,S_{0}}), M_{S}(ℤ/pℤ_{S_{0}}))

can be identified with the map

\begin{matrix}I_{p}(\overline{R}) & ⟶ & I_{p}(\overline{R}) \\ x & ⟼ & x + γ_{S}(x). \end{matrix}

The condition that \overline{γ} is nilpotent is equivalent with the condition that lim_{n→∞} γ^{n}(x) = 0 in the \overline{m}-adic topology on \overline{R}.Therefore, the first statement follows from the same reasoning as in the proof of Lemma 9.13.

Since H_{fppf}^{1}(S_{0}, 𝔾_{m}) = 0 and H_{fppf}^{1}(S_{0}, 𝔾_{a}) = 0 the map
Ext_{C(1)_{S_{0}}}^{1} (ℤ/pℤ_{S_{0}}, μ_{p,S_{0}}) ⟶ Ext_{M(1)_{S}}^{1}(M_{S}(μ_{p,S_{0}}), M_{S}(ℤ/pℤ_{S_{0}}))

can be identified with the map

\overline{R}^{∗}/\overline{R}^{∗p}  ⟶ \overline{R} mod \overline{R}^{p}
class of \overline{x} ∈ \overline{R}^{∗} ⟼ \frac{u(x)}{\overline{x}^{p}} mod \overline{R}^{p}

(See Lemma’s 8.7 and 8.8 and Proposition 8.9). As the residue field of \overline{R} is perfect we have \overline{R}^{∗}/\overline{R}^{∗p} ≃ (1 + \overline{m})^{∗}/(1 + \overline{m})^{∗p} and \overline{R} mod \overline{R}^{p} ≃ \overline{m} mod \overline{m}(p). Hence we have to show that if \overline{u} is nilpotent then the map

α : (1 + \overline{m})^{∗}/(1 + \overline{m})^{∗p} ⟶ \overline{m} mod \overline{m}(p)
∥  ↗
α : (1 + \overline{m})^{∗}/(1 + \overline{m}(p))^{∗}

is an isomorphism. A computation shows that

α(class of 1 + \overline{x}) = class of \frac{−B_{p}(1, \overline{x}) + u(x)}{1 + \overline{x}^{p}}.

Put V = (1 + \overline{m})^{∗}/(1 + \overline{m}(p))^{∗} and W = \overline{m} mod \overline{m}(p) and consider the decreasing filtrations

\begin{aligned}V^{i} & := (1 + \overline{m}^{i})^{∗}/(1 + \overline{m}^{i} ∩ \overline{m}(p))^{∗} & ⊂ V, \\ W^{i} & := \overline{m}^{i} mod \overline{m}^{i} ∩ \overline{m}(p) & ⊂ W \end{aligned}

on V and W. The formula for the image of 1 + \overline{x} under α shows that α(V^{i}) ⊂ W^{i}. Indeed, if \overline{x} ∈ \overline{m}^{i} then we choose x ∈ m̃^{i} and get

x = Σ_{α}m_{α1} … m_{αi}, m_{αj} ∈ m̃.

Hence if i ≥ 2:

u(x) = u(Σ_{α}m_{α1} … m_{αi})
≡ Σ_{α}u(m_{α1} … m_{αi}) mod m̃^{p·i}
≡ Σ_{α,j}m_{α1}^{p} … m̂_{αj}^{p} … m_{αi}^{p}u(m_{αj})
≡ 0 mod m̃^{p(i−1)+1}.

In this way we also see that the map

V^{1}/V^{2} ≃ \overline{m}/\overline{m}^{2} ⟶ \overline{m}/\overline{m}^{2} = W^{1}/W^{2}

is given by \overline{m} ↦ \overline{x} + \overline{u}(\overline{x}) and that the maps

V^{i}/V^{i+1} ≃ \overline{m}^{i}/(\overline{m}^{i+1} + \overline{m}^{i} ∩ \overline{m}(p)) ⟶ \overline{m}^{i}/(\overline{m}^{i+1} + \overline{m}^{i} ∩ \overline{m}(p)) = W^{i}/W^{i+1}

are given by \overline{x} ↦ \overline{x} for i ≥ 2.

Since V = lim_{←} V/V^{m} and W = lim_{←} W/W^{m} we see that α is an isomorphism if and only if id_{\overline{m}/\overline{m}^{2}} + \overline{u} is an isomorphism. This is the case if \overline{u} is nilpotent. □

Theorem 10.2. If \overline{γ} and \overline{u} are nilpotent then

M_{S} : C(1)_{S_{0}}^{o} ⟶ M(1)_{S}

is an equivalence of categories, hence by Theorem 8.3 we also have equivalences

M_{S} : BT (k)_{S_{0}}^{o} ⟶ M(k)_{S}

for all 2 ≤ k ≤ n. Proof. Let x ∈ S_{0} ⊂ S be the closed point of S. Hence x = Spec(k) and k is a perfect field. Remark that the category of finite étale schemes over S is (canonically) equivalent to the category of finite étale schemes over S_{0} and is (canonically) equivalent to the category of finite étale schemes over x. Furthermore each finite étale scheme S′ over S automatically has a lift of frobenius f_{S′} compatible with f_{S} and is a finite disjoint union of schemes satisfying the hypotheses of the theorem.

Suppose ϕ : G_{1} → G_{2} is a map in C(1)_{S_{0}} with M_{S}(ϕ) = 0. The induced maps

Ker(G_{1} ⟶ G_{1}^{ét}) ⟶ Ker(G_{2} ⟶ G_{2}^{ét})

and

G_{1}/G_{1}^{μ} ⟶ G_{2}/G_{2}^{μ}

will be zero by Theorem 9.3. Hence ϕ must be a composition G_{1} ↠ G_{1}^{ét} ⟶^{ψ} G_{2}^{μ} ↪ G_{2}. By the remark above we may replace S by a finite étale cover of S and we may assume that G_{1}^{ét} ≃ (ℤ/pℤ_{S_{0}})^{r} and G_{2}^{μ} ≃ (μ_{p,S_{0}})^{s}. We still have M_{S}(ψ) = 0 and hence ψ = 0 by Lemma 10.1. This proves that M_{S} is faithful for any S satisfying the conditions of the theorem.

Suppose ψ : M_{S}(G_{2}) → M_{S}(G_{1}) is a morphism of M(1)_{S} for certain G_{1}, G_{2} ∈ ObC(1)_{S_{0}}. By Theorem 9.3 we already have morphisms

ϕ′ : G_{1}/G_{1}^{μ}  ⟶  G_{2}/G_{2}^{μ}
ϕ′′ : Ker(G_{1} → G_{1}^{ét}) ⟶ Ker(G_{2} → G_{2}^{ét})

which are equal on Ker(G_{1} → G_{1}^{ét})/G_{1}^{μ} and which correspond to the maps defined by ψ on the corresponding objects of M′(1)_{S} and M′′(1)_{S}. The obstruction to find a ϕ inducing ϕ′ and ϕ′′ is an element η of

Ext_{C(1)_{S_{0}}}^{1} (G_{1}^{ét}, G_{2}^{μ})

and M_{S}(η) = 0 since we know that ψ exists.

Suppose S′ → S is a finite étale morphism such that G_{1,S′_{0}}^{ét} ≃ (ℤ/pℤ_{S′_{0}})^{r} and G_{2,S′_{0}}^{μ} ≃ (μ_{p,S′_{0}})^{s}. By Lemma 10.1 we see that η_{S′_{0}} = 0 and we get a ϕ_{S′_{0}} : G_{S′_{0}} → G_{S′_{0}} which induces ϕ′_{S′_{0}} and ϕ′′_{S′_{0}}. The set of all such ϕ_{S′_{0}} is homogeneous under

Hom_{C(1)_{S′_{0}}} (G_{1,S′_{0}}^{ét}, G_{2,S′_{0}}^{μ}) ≃ Hom_{M(1)_{S′_{0}}} (M_{S′}(G_{1,S′_{0}}^{ét}), M_{S′}(G_{2,S′_{0}}^{μ}))

by Lemma 10.1. Hence we can alter ϕ_{S′_{0}} by an element of this group to get M_{S′}(ϕ_{S′}) = ψ_{S′}. The usual argument shows that ϕ_{S′_{0}} descends to give a ϕ with M_{S}(ϕ) = ψ. (See for example the proof of reduction step 3 in §9.)

Finally, we have to show that M_{S} is essentially surjective. Take an object M ∈ ObC(1)_{S_{0}}. We have the canonical sequence

M^{ét} ⟶ M ⟶ M^{μ}

(see §9). Let M_{12} be its cohomology, M_{1} = Ker(M → M^{μ}) and M_{2} = M/M^{ét}. We have M_{1} ∈ M′_{S}, M_{2} ∈ M′′_{S} and a commutative diagram:

0 ↑

M^{μ} = M^{μ}
↑ ↑
M^{ét} ⟶ M ⟶ M_{2}
∥ ↑ ↑
0 ⟶ M^{ét} ⟶ M_{1} ⟶ M_{12} ⟶ 0
↑ ↑
0  0

From Theorem 9.3 we get correspondingly:

G^{μ} = G^{μ}
↓ ↓
G^{ét} ⟵ ? ⟵ G_{2}
∥ ↓ ↓
G^{ét} ⟵ G_{1} ⟵ G_{12}

with M_{S}(G_{i}) = M_{i} etc. We want to fill in the question mark by a group G such that M_{S}(G) ≅ M. The obstruction to fill in the diagram with any sheaf of 𝔽_{p}-modules G lies in Ext^{2}(G^{ét}, G^{μ}), where this is the Ext-group computed in the category of sheaves of 𝔽_{p}-modules on the big fppf site of S_{0}.

Since we only need to prove the existence of G after a finite étale base extension S′ → S we may assume G^{ét} ≃ (ℤ/pℤ_{S_{0}})^{r} and G^{μ} ≃ (μ_{p,S_{0}})^{s}. However,

Ext^{2}(ℤ/pℤ_{S_{0}}, μ_{p,S_{0}}) ≃ H_{fppf}^{2}(S_{0}, 𝔾_{m,S_{0}})[p]

and this last group is zero since S_{0} is the spectrum of a complete local ring with perfect residue field of characteristic p. Thus we can choose a sheaf of groups G fitting into the diagram.

Finally, G will be an object of C(1)_{S_{0}} as it is an extension 0 → G^{μ} → G → G_{1} → 0 of representable sheaves and hence representable. The set of isomorphism classes of all these G is homogeneous under the group Ext_{C(1)_{S_{0}}}^{1} (G^{ét}, G^{μ}) and hence by Lemma 10.1 we can alter the choice of G by an element of this group to get M_{S}(G) ≃ M. □

Remark 10.3. As examples of rings satisfying the hypotheses of Theorem 10.2 we can take

W_{n+1}(k)[[x_{1}, …, x_{l}]]

(k a perfect field of characteristic p) or any quotient of one of these by an ideal generated by monomials. In this case as a lift of Frobenius we take the endomorphism determined by x_{i} ↦ x_{i}^{p}.

References

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3. Grothendieck, A., Dieudonné, J.: Eléments de géométrie algébriques I,II,III,IV. Publ. Math. I.H.E.S. 4, 8, 11, 17, 20, 24, 28, 32 (1961-1967)

4. Illusie, L.: Déformations de groupes de Barsotti-Tate. In: Szpiro, L.: Séminaire sur les pinceaux arithmétiques: la conjecture de Mordell (Astérisque 127, pp. 151-198) Paris: Société mathématique de France 1985

5. Illusie, L.: Complexe cotangent et déformations I,II (Lect. Notes Math. 239 & 283) Berlin Heidelberg New York: Springer 1971, 1972

6. Mazur, B., Messing, W.: Universal extensions and one-dimensional crystalline cohomology (Lect. Notes Math. 370) Berlin Heidelberg New York: Springer 1974

7. Messing, W.: The crystals associated to Barsotti-Tate groups: with applications to abelian schemes (Lect. Notes Math. 264) Berlin Heidelberg New York: Springer 1972

8. Oda, T.: The first de Rham cohomology groups and Dieudonné modules. Ann. Sci. Ec. Norm. Superieur 2, 63-135 (1959)

9. Oort, F.: Commutative group schemes (Lect. Notes Math. 15) Berlin Heidelberg New York: Springer 1966

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