GALOIS MODULES ARISING FROM

FALTINGS’S STRICT MODULES

Victor Abrashkin

0. Introduction.

Throughout all this paper O is the valuation ring of a complete discrete valuation field K with perfect residue field k of characteristic p > 0, Γ_{K} — the absolute Galois group of K and π — a uniformising element of K.

If characteristic of K is 0, denote by FGr′(ℤ_{p})_{O} the category of finite flat commutative group schemes G over O such that the order |G| is a power of p. Any such group scheme appears as a kernel of an isogeny of abelian schemes defined over O and reflects important properties of these abelian schemes.

The classification of objects of the category FGr′(ℤ_{p})_{O} was done in [Fo1] under the restriction e = 1 on the absolute ramification index e = e(K) of the extension K/ℚ_{p} in terms of finite Honda systems (this classification was not complete for p = 2, for an improved version cf. [Ab1]). Further progress was done in papers [Ab3] for e ⩽ p − 1 (group schemes killed by p), [Co] for e < p − 1 and, finally, in [Br] for an arbitrary e.

Most interesting number theoretic application of the theory of finite flat group schemes come from the study of the structure of the Γ_{K}-module H = G(K_{sep}) of geometric points of G ∈ FGr′(ℤ_{p})_{O}. We mention the following three results:

A. Serre’s Conjecture (proved in [Ra]).

This result describes the action of the inertia subgroup I_{K} ⊂ Γ_{K} on the semisimple envelope of H. It is given by characters χ : I_{K} ⟶ k^{∗} such that for some N ∈ ℕ, χ = χ_{N}^{a}, where χ_{N}(τ) = τ(π_{N})/π_{N}, π_{N}^{p^{N} −1} = π, and a = a_{0} + a_{1}p + ⋯ + a_{N−1}p^{N−1} with p-digits a_{0}, a_{1}, …, a_{N−1} ∈ [0, e];

B. Ramification estimates.

If p^{M} id_{G} = 0 then the ramification subgroups Γ_{K}^{(v)} of Γ_{K} act trivially on H if v > e(M − 1 + 1/(p − 1)), cf. [Fo2];

C. Complete description of Γ_{K}-module H (in the case e = 1, p > 3 and pH = 0);

If e = 1, p > 3 and 𝔽_{p}[Γ_{K}]-module H satisfies the above Serre’s Conjecture and the ramification estimates (i.e. the ramification subgroups Γ_{K}^{(v)} act trivially on H

if v > \frac{1}{p − 1}), then there is an G ∈ FGr′(ℤ_{p})_{O} such that H = G(K_{sep}), cf.[Ab3].

Suppose now that characteristic of K is p. In this case a reasonable analogue of the concept of finite flat group scheme would give a way to study kernels of isogenies

Typeset by AMS-TEX

of Drinfeld modules. This analogue should appear as a finite flat commutative group scheme G with continuous action of a closed subring O_{0} = 𝔽_{q}[[π_{0}]] ⊂ O. The notion of O_{0}-module scheme was not very helpful until Faltings introduced in [Fa] a concept of strict O_{0}-action. The main idea of Faltings’s definition can be explained as follows.

Suppose G = Spec A is a finite flat O_{0}-module over O. Present A as a quotient O[X_{1}, …, X_{n}]/I of a ring of polynomials by an ideal I and define a deformation A^{♭} as O[X_{1}, …, X_{n}]/(I · I_{0}), where I_{0} = (X_{1}, …, X_{n}). Faltings requires that O_{0}- module structure on G, which is given by endomorphisms [o] : A ⟶ A, o ∈ O_{0}, should have an extension to an O_{0}-module structure of the deformation (A, A^{♭}) of the algebra A and this extension must satisfy the condition of strictness. This means that if [o]^{♭} : A^{♭} ⟶ A^{♭} is an extension of [o] for o ∈ O_{0}, then [o]^{♭} must induce multiplication by o on I_{0}/I_{0}^{2} and I/(I ·I_{0}). This definition gives the category FGr′(O_{0})_{O} of strict O_{0}-modules and its objects have many interesting properties discussed in [Fa].

In this paper we study number theoretic properties of strict O_{0}-modules. In n.1 we present a concept of strict O_{0}-module in a slightly different but an equivalent to the original definition by Faltings way. In n.2 we describe the category of strict 𝔽_{q}-modules over O and apply this in n.3 to the classification of objects of the category FGr′(O_{0})_{O}. This classification does not depend on the ramification index of K over Frac O_{0} = K_{0} and requires only the study of primitive elements (i.e. the elements a ∈ A(G) such that Δa = a ⊗ 1 + 1 ⊗ a where Δ is comultiplication) of the O-algebra A(G) of G ∈ FGr′(O_{0})_{O}. We apply then this classification to prove that any object of FGr′(O_{0})_{O} can be embedded into a π_{0}-divisible group over O. This section contains also a comparison of our antiequivalence with parallel results in the theory of finite flat group schemes and p-adic representations in the mixed characteristic case from papers [Ab1,3], [Br] and [Fo4]. In n.4 we establish precise analogues of the above properties A, B and C in the category FGr′(O_{0})_{O}.

The author expresses his gratitude for hospitality to the Max-Planck-Institute in Mathematics where a part of this paper was being written.

1. Definition and simplest properties.

Let O be the valuation ring of a complete discrete valuation field K with perfect residue field k of characteristic p > 0. All O-algebras are assumed usually to be finite, i.e. to be free O-modules of finite rank.

1.1. Deformations of augmented O-algebras.

For an augmented O-algebra A, we agree to use the following notation: ε_{A} : A ⟶ O — the morphism of augmentation; Ker ε_{A} = I_{A} — the augmentation ideal; A^{loc} = \underleftarrow{lim}_{n}A/I_{A}^{n} — the I_{A}-completion of A; η_{A} : A ⟶ A^{loc} — a natural

projection. Notice, the correspondence A ↦ A^{loc} is functorial.

The objects of the category DAlg_{O} are the triples A = (A, A^{♭}, i_{A}), where A is a finite O-algebra, A^{♭} is an O-algebra such that there is a polynomial ring O[X̄] = O[X_{1}, …, X_{n}], n ⩾ 0, and its ideal I such that A^{loc} = O[X̄]/I and A^{♭} = O[X̄]/(I ▪ I_{0}) with I_{0} := (X_{1}, …, X_{n}) ⊃ I, and i_{A} : A^{♭} ⟶ A^{loc} is a natural epimorphism of O-algebras.

A morphism f̄ = (f, f^{♭}) : A ⟶ B = (B, B^{♭}, i_{B}) in DAlg_{O} is given by a morphism of augmented O-algebras f : A → B and an O-algebra morphism f^{♭} : A^{♭} ⟶ B^{♭} such that i_{A} ∘ f^{loc} = f^{♭} ∘ i_{B}.

In the category DAlg_{O}, O = (O, O, id_{O}) is an initial object and any A = (A, A^{♭}, i_{A}) has a natural augmentation to O, ε_{A} = (ε_{A}, ε_{A^{♭}}) : A ⟶ O, where Ker ε_{A^{♭}} = Ann(Ker i_{A}) := I_{A^{♭}}.

Notice that the O-modules t_{A}^{∗} = I_{0}/I_{0}^{2} (this module is free) and N_{A} = I/(I · I_{0}) do not depend on the choice of the covering O[X̄] ⟶ A^{♭}. Indeed, the first coincides with I_{A^{♭}}/I_{A^{♭}}^{2} and the second — with Ker i_{A}.

If A = (A, A^{♭}, i_{A}) and B = (B, B^{♭}, i_{B}) are objects of DAlg_{O} and f : A ⟶ B is a morphism of augmented O-algebras, then the set of all f^{♭} such that (f, f^{♭}) ∈ Hom_{DAlg_{O}}(A, B) is not empty and has a natural structure of a principal homogeneous space over the group Hom_{O -mod}(t_{A}^{∗}, N_{B}).

The deformation A = (A, A^{♭}, i_{A}) of A will be called minimal if i_{A} induces isomorphism of t_{A}^{∗} ⊗ k onto I_{A}/I_{A}^{2} ⊗ k. In other words, f is minimal if minimal systems of generators of O-algebras A^{♭} and A^{loc} contain the same number of elements. It is easy to see that if A is minimal, A′ = (A, A′^{♭}, i_{A′}) ∈ DAlg_{O} and f̄ = (id_{A}, f^{♭}) ∈ Hom_{DAlg_{O}}(A, A′) then there is an ḡ ∈ Hom_{DAlg_{O}}(A′, A) such that f̄ ∘ ḡ = id_{A}. In particular, all minimal deformations of a given O-algebra A are isomorphic in DAlg_{O}.

Let DAlg_{R}^{∗} be a quotient category for DAlg_{O}: it has the same objects but its morphisms are equivalence classes of morphisms from Hom_{DAlg_{O}}(A, B) arising from the same O-algebra morphisms f : A ⟶ B. Then the forgetful functor A = (A, A^{♭}, i_{A}) ↦ A is an equivalence of DAlg_{O}^{∗} and the category of augmented finite O-algebras.

1.2. Deformations of affine group schemes.

Let DSch_{O} be the dual category for DAlg_{O}. Its objects appear in the form H = Spec A = (H, H^{♭}, i_{H}), where H = Spec A and H^{♭} = Spec A^{♭} are finite flat O-schemes, A = (A, A^{♭}, i_{A}) ∈ DAlg_{O}, and i_{H} : H → H^{♭} is a closed embedding of O-schemes. This category has direct products: if for i = 1, 2, A_{i} = (A_{i}, A_{i}^{♭}, i_{A_{i}}) with A_{i}^{loc} = O[X̄_{i}]/I, A_{i}^{♭} = O[X̄_{i}]/(I_{i} · I_{0i}), then the product Spec A_{1} × Spec A_{2} is given by Spec(A_{1} ⊗A_{2}), where A_{1} ⊗A_{2} := (A_{1} ⊗_{O} A_{2}, (A_{1} ⊗_{O} A_{2})^{♭}, κ), (A_{1} ⊗_{O} A_{2})^{♭} is the quotient of O[X̄_{1} ⊗ 1, 1 ⊗ X̄_{2}] by the product of ideals I_{1} ⊗ 1 + 1 ⊗ I_{2} and I_{01} ⊗ 1 + 1 ⊗ I_{02} and κ is the natural projection. Notice that for i = 1, 2, two projections pr_{i} from this product to its components Spec A_{i} come from the natural embeddings of O[X̄_{i}] into O[X̄_{1} ⊗ 1, 1 ⊗ X̄_{2}].

Let FGr_{O} be the category of group objects in DSch_{R}. If G = Spec A ∈ FGr_{O} then its group structure is given via comultiplication Δ̄ = (Δ, Δ^{♭}) : A ⟶ A ⊗ A, counit ε̄ = (ε, ε^{♭}) : A ⟶ O and coinversion ī = (i, i^{♭}) : A ⟶ A morphisms, which satisfy usual axioms. The morphisms in FGr_{O} are morphisms of group objects. As usually, FGr_{O} is an additive category.

Notice that

a) G = Spec A is a finite flat group scheme over O with comultipilication Δ, counit ε and coinversion i;

b) ε̄ = ε_{A}, cf. n.1.1.

c) the counit axiom gives for i = 1, 2, Δ_{i}^{♭} ∘ pr_{i} = id_{A^{♭}} and implies a uniqueness of Δ^{♭} as a lifting of Δ;

d) if A = (A, A^{♭}, i_{A}) ∈ DAlg_{O} and G = Spec A is an affine group scheme then there is a unique structure of group object on Spec A compatible with that of G;

e) if f : G ⟶ H is a morphism of group schemes and (f, f^{♭}) ∈ Hom_{DSch_{O}}(G, H) then (f, f^{♭}) ∈ Hom_{FGr_{O}}(G, H).

The above properties have the following interpretation. Define the quotient category FGr_{O}^{∗} as the category consisiting of the objects of the category FGr_{O} but where Hom_{FGr}_{O}^{∗}(G, H) consists of equivalence classes of morphisms from the category FGr_{O} which induce the same morphisms of affine group schemes G → H. Then the natural functor G ↦ G is an equivalence of categories. We can use this equivalence to define the abelian groups of equivalence classes of short exact sequences Ext(G, H) in FGr_{O}. These groups are functorial by both arguments and there are standard 6-terms exact Hom − Ext-sequences.

1.3. The categories of strict R-modules.

Suppose R is a ring and O is an R-algebra.

Suppose G is an R-module object in the category DSch_{O}. Then G is an object of FGr_{O} and there is a map R ⟶ End_{FGr_{O}}(G) satisfying the usual axioms from the definition of R-module. For r ∈ R and G = Spec A, denote by ¯[r] = ([r], [r]^{♭}) the morphism of action of r on A = (A, A^{♭}, i_{A}). Clearly, G = Spec A is an Rmodule in the category of finite flat schemes over O. For any such G, the R-module structure on the deformation (G, G^{♭}, i) ∈ FGr_{O} is given by liftings [r]^{♭} : A^{♭} ⟶ A^{♭} of morphisms [r] : A ⟶ A, r ∈ R. Notice that [r]^{♭} are morphisms of augmented algebras. All such liftings are automatically compatible with the group structure on this deformation, i.e. for any r ∈ R, it holds [r] ∘ Δ^{♭} = Δ ∘ ([r] ⊗ [r]). So, the above system gives an R-module structure if and only if for any r_{1}, r_{2} ∈ R,

[r_{1} + r_{2}]^{♭} = Δ^{♭} ∘ ([r_{1}] ⊗ [r_{2}])^{♭}, [r_{1}r_{2}]^{♭} = [r_{1}] ∘ [r_{2}]^{♭}  (1)

where ([r_{1}] ⊗ [r_{2}])^{♭} is induced by [r_{1}]^{♭} ⊗ [r_{2}]^{♭}.

We denote by FGr(R)_{O} the category of such R-module objects G where Raction is strict, i.e. if G = Spec A then any r ∈ R acts on t_{A}^{∗} and N_{A} via scalar multiplication by r. This is a basic definition from Faltings’ paper [Fa].

Suppose G_{1} = (G, G_{1}^{♭}, i_{1}) ∈ FGr_{O} and G_{2} = (G, G_{2}^{♭}, i_{2}) ∈ FGr_{O} are two deformations of a finite flat group scheme G over O. By 1.2, G_{1} and G_{2} are isomorphic in the category FGr_{O}^{∗}. Suppose G_{1} is equiped with the strict R-action. Then there is a unique (strict) R-action on G_{2} such that any (id_{G}, ϕ) ∈ Hom_{FGr_{O}}(G_{1}, G_{2}) and any (id_{G}, ψ) ∈ Hom_{FGr_{O}}(G_{2}, G_{1}) are, actually, morphisms in the category FGr(R)_{O}.

Denote by FGr^{∗}(R)_{O} the quotient category of FGr(R)_{O} where morphisms are equivalence classes of morphisms (G, G^{♭}, i) ⟶ (H, H^{♭}, j) in the category FGr(R)_{O} inducing the same morphism G ⟶ H. By the above property, all isomorphism classes of objects in FGr(R)_{O} appear as R-module finite flat schemes G together with a lifting of its R-action to some chosen deformation G^{♭} which satisfies the above conditions (1).

For example, if q = p^{n} with n ∈ ℕ, the objects of the category FGr(𝔽_{q})_{O} appear as Spec A, where A = (A, A^{♭}, i_{A}), A^{loc} = O[X̄]/I and A^{♭} = O[X̄]/(I · I_{0}), and 𝔽_{q}- action is induced by the scalar action of 𝔽_{q} on O[X̄] (i.e. [α](X̄) = αX̄, α ∈ 𝔽_{q}) and there are generators j_{1}, …, j_{n} of the ideal I such that [α]j_{i} = αj_{i} for all i = 1, …, n and α ∈ 𝔽_{q}.

If R = 𝔽_{q}[π] with an indeterminate π and G ∈ FGr(R)_{O} then G ∈ FGr(𝔽_{q})_{O} and (in addition to the above assumptions) R-action will be determined completely by the action of π given by the correspondence

X̄ ↦ [π](X̄) = πX̄ + F̄(X̄)

where F̄ is any vector power series with coefficients in O from I_{0}^{2} such that F̄(αX̄) = αF̄(X̄) for all α ∈ 𝔽_{q}. This action is strict iff [π]j_{i} ≡ πj_{i} mod(I · I_{0}) for the above generators j_{1}, …, j_{n} of I.

1.4. Suppose i : G_{1} ⟶ G is a closed embedding of strict R-modules. Then the quotient G_{2} = G/G_{1} has a natural structure of a strict R-module scheme and the projection j : G ⟶ G_{2} is a morphism in the category FGr(R)_{O}. Similarly, if j : G ⟶ G_{2} is a fully faithful morphism of strict R-module schemes then its kernel G_{1} is a strict R-module and its (closed) embedding i : G_{1} ⟶ G is a morphism from FGr(R)_{O}.

Notice that for any G = (G, G^{♭}, i_{G}) ∈ FGr(R)_{O}, the short exact sequence of group schemes

0 ⟶ G^{loc} ⟶ G ⟶ G^{et} ⟶ 0  (2)

(here, G^{loc} is the maximal local subgroup scheme of G and G^{et} is the maximal etale quotient of G) induces a short exact sequence in FGr(R)_{O}

0 ⟶ G^{loc} ⟶ G ⟶ G^{et} ⟶ 0  (3)

where G^{loc} = (G^{loc}, G^{♭}), G^{et} = (G^{et}, Spec O).

Finally, notice that over the valution ring O_{1} of some unramified extension K_{1} of K, G^{et} ⊗ O_{1} is constant, i.e. A(G^{et}) ⊗ O_{1} = Map(H^{et}, O_{1}) with H^{et} = G^{et}(K_{1}). This also implies the existence of a finite extension K′_{1} of K such that the short exact sequence (2) (as well as (3)) splits over the valuation ring of K′_{1}.

2. Group schemes with strict 𝔽_{q}-action.

As earlier, O is the valuation ring of a complete discrete valuation field K of characteristic p with perfect residue field k, q = p^{N_{0}} with N_{0} ∈ ℕ. In this section we assume that O is an 𝔽_{q}-algebra (then 𝔽_{q} ⊂ k and O ≃ k[[π]], where π is a uniformiser in K) and study the full subcategory FGr′(𝔽_{q})_{O} of the category FGr(𝔽_{q})_{O} consisting of strict finite 𝔽_{q}-modules G over O with etale generic fibre. As it was noticed in n.1, its objects appear as finite flat group schemes G = Spec A(G) over O with 𝔽_{q}-action such that A(G^{loc}) (or even A(G)) can be presented in the form O[X̄]/I in such a way that 𝔽_{q}-action on G is induced by a scalar 𝔽_{q}-action on coordinates of X̄ and on some system of generators of the ideal I.

2.1. The category Mod_{O,σ} and the functor Gr.

Let σ : O → O be the endomorphism of q-th power. For any O-module M, set M_{σ} = M ⊗_{O} O where an O-module structure on O is given via σ. Objects of the category Mod_{O,σ} are free O-modules M of finite rank with an O-linear morphism Φ : M_{σ} → M such that its K-linear extension Φ_{K} is an isomorphism of K-vector spaces M_{σ,K} = M_{σ}⊗_{O}K and M_{K} = M⊗_{O}K. Morphisms in Mod_{O,σ} are morphisms of O-modules commuting with Φ-action.

For an M ∈ Mod_{O,σ}, introduce the O-algebra

A[M] = Sym_{O} M/{Φm − m^{⊗q} | m ∈ M}

In other words, if m_{1}, …, m_{n} is an O-basis of M and Φm_{i} = ∑_{j} o_{ij}m_{j} (where det(o_{ij}) ≠ 0), then A[M] = O[X_{1}, …, X_{n}]/(ψ_{1}, …, ψ_{n}), where ψ_{i} = X_{i}^{q}−∑_{j} o_{ij}X_{j}, i = 1, …, n. This O-algebra A[M] does not depend on the choice of a basis m_{1}, …, m_{n} and gives rise to the strict finite flat 𝔽_{q}-module scheme G[M] = Spec A[M] with comultiplication Δ such that Δ(X_{i}) = X_{i} ⊗ 1 + 1 ⊗ X_{i} and an 𝔽_{q}-action such that [α](X_{i}) = αX_{i}, where α ∈ 𝔽_{q}, i = 1, …, n. The equality Φ(M_{σ,K}) = M_{K} implies that the generic fibre of G[M] is etale, and therefore, G[M] ∈ FGr′(𝔽_{q})_{O}.

Identify M with its image in A[M] via the natural map m_{i} ↦ X_{i}, i = 1, …, n.

Proposition 1.

a) M = {m ∈ A[M] | Δm = m ⊗ 1 + 1 ⊗ m, [α]m = αm, α ∈ 𝔽_{q}};

b) the natural inclusions M_{i} ⊂ A[M_{i}], i = 1, 2, induce the identification

Hom_{Mod_{O,σ}}(M_{1}, M_{2}) = Hom_{FGr(𝔽_{q})_{O}}(G[M_{2}], G[M_{1}]).

This proposition implies that the correspondence M ↦ G[M] induces a fully faithful functor Gr : Mod_{O,σ} ⟶ FGr′(𝔽_{q})_{O}. The rest of this section will be devoted to the proof of the following theorem.

Theorem 1. The functor Gr is an antiequivalence of categories.

Notice that the correspondence Gr ⊗K : M_{K} ↦ G[M](K_{sep}) is an antiequivalence of the category of σ-etale K-modules and the category of 𝔽_{q}[Γ_{K}]-modules, cf. [Fo4].

2.2. Later we need the following property of the Hochschild cohomology of group schemes G[M].

Lemma 1. With the above notation suppose that a ∈ A[M]_{K} is such that δ^{+}a := Δ(a) − a ⊗ 1 − 1 ⊗ a ∈ A[M] ⊗ A[M] and [α]a = αa for all α ∈ 𝔽_{q}. Then there is an a′ ∈ A[M] such that δ^{+}a = δ^{+}a′ and a − a′ ∈ M_{K}.

Proof of Lemma 1. Let δ_{k}^{+} = δ^{+}⊗k : A[M]_{k} ⟶ A[M]_{k}^{⊗2}, where A[M]_{k} = A[M]⊗k. It will be sufficient to prove that

{b ∈ A[M]_{k} | δ_{k}^{+}b = 0, [α]b = αb, ∀α ∈ 𝔽_{q}} = M_{k}  (4)

Note that

{X_{1}^{i_{1}} … X_{n}^{i_{n}} | 0 ⩽ i_{1}, …, i_{n} < q}

is a k-basis of A[M]_{k} and

{X_{1}^{j_{1}} … X_{n}^{j_{n}} ⊗ X_{1}^{l_{1}} … X_{n}^{l_{n}} | 0 ⩽ j_{1}, …, j_{n}, l_{1}, …, l_{n} < q}

is a k-basis of A[M]_{k}^{⊗2}. Now the equality (4) is implied by the following 3 observations:

a) for any 0 ⩽ i_{1}, …, i_{n} < q, δ_{k}^{+}(X_{1}^{i_{1}} … X_{n}^{i_{n}}) is a linear combination of X_{1}^{j_{1}} … X_{n}^{j_{n}} ⊗ X_{1}^{l_{1}} … X_{n}^{l_{n}} with j_{1}, …, j_{n}, l_{1}, …, l_{n} ⩾ 0 such that j_{1} + l_{1} = i_{1}, …, j_{n} + l_{n} = i_{n};

b) if ∑_{0⩽i_{1},...,i_{n}<q}α_{i_{1}...i_{n}}X_{1}^{i_{1}} … X_{n}^{i_{n}} ∈ Ker δ_{k}^{+} then for each multi index (i_{1}, …, i_{n}),

either α_{i_{1}...i_{n}} = 0, or δ_{k}^{+}(X_{1}^{i_{1}} … X_{n}^{i_{n}}) = 0;

c) for any 0 ⩽ i_{1}, …, i_{n} < q, δ_{k}^{+}(X_{1}^{i_{1}} … X_{n}^{i_{n}}) = 0 iff X_{1}^{i_{1}} … X_{n}^{i_{n}} ∈ {X_{s}^{p^{t}} | 1 ⩽ s ⩽ n, 0 ⩽ t < N_{0}}.

2.3. For G ∈ FGr(𝔽_{q})_{O}, let

L(G) = {a ∈ A(G) | Δa = a ⊗ 1 + 1 ⊗ a, [α]a = αa, ∀α ∈ 𝔽_{q}}.

Clearly, the above lemma implies that if M ∈ Mod_{O,σ}, then L(G[M]) = M. The following lemma can be proved directly from the above definition of L(G).

Lemma 2.

a) If O_{1} is a complete discrete valuation ring containing O, then L(G ⊗ O_{1}) = L(G) ⊗ O_{1};

b) For any two objects G_{1} and G_{2} of FGr(𝔽_{q})_{O}, it holds L(G_{1} × G_{2}) = L(G_{1}) ⊕ L(G_{2}).

The q-th power map on A(G) induces Φ : L(G)_{σ} ⟶ L(G) which provides L(G) with the structure of an object of the category Mod_{O,σ}. Indeed, if H = G(K_{sep}) with the natural structure of 𝔽_{q}[Γ_{K}]-module, then L(G)_{K} = Hom_{𝔽_{q}[Γ_{K}] -mod}(H, K_{sep}) and

Im Φ_{K} = Hom_{𝔽_{q}[Γ_{K}] -mod}(H, σ(K_{sep})) ⊗_{σ(K)} K = L(G)_{K}

because σ(K_{sep}) ⊗_{σ(K)} K = K_{sep}.

The correspondence L : G ↦ L(G) is the functor from FGr′(𝔽_{q})_{O} to Mod_{O,σ}. Clearly, the composition Gr ∘L is equivalent to the identity functor on Mod_{O,σ}. So, Gr is an antiequivalence (with the quasi-inverse functor L) if any G ∈ FGr′(𝔽_{q})_{O} is isomorphic to G[M] for a suitable M ∈ Mod_{O,σ}. When proving this property we can enlarge (if necessary) the basic ring O by the following lemma.

Lemma 3. If O′ is a complete discrete valuation ring containing O and G ∈ FGr′(𝔽_{q})_{O} then G ≃ G[M] with M ∈ Mod_{O,σ} if and only if G ⊗ O_{1} ≃ G[M_{1}] for some M_{1} ∈ Mod_{O_{1},σ}.

Proof. Clearly, if G ≃ G[M] with M ∈ Mod_{O,σ}, then G ⊗ O_{1} ≃ G[M ⊗ O_{1}].

If G ⊗ O_{1} ≃ G[M_{1}] with M_{1} ∈ Mod_{O,σ}, then M_{1} ≃ L(G ⊗ O_{1}) = L(G) ⊗ O_{1}, therefore, G ⊗ O_{1} ≃ G[L(G)] ⊗ O_{1} and the natural inclusion G[L(G)] ⊂ G is an isomorphism. Lemma 3 is proved

2.4. 𝔽_{q}-module schemes of order q.

Proposition 2. Suppose G is an 𝔽_{q}-module scheme of order q such that G ⊗ K is etale and 𝔽_{q} acts on the cotangent space t_{G}^{∗} via scalar multiplication. Then G = G[M], where M = Om and Φm = λm with some λ ∈ O, λ ≠ 0 (in particular, G is a strict 𝔽_{q}-module).

Proof. Let I_{G} = Ker e_{G} (where e_{G} is the counit morphism). If I_{G} = I_{G}^{2} then G is etale and over some bigger (unramified over O) basic ring O_{1}, G ⊗ O_{1} is constant, i.e. G ⊗ O_{1} ≃ G[M_{1}], where M_{1} = O_{1}m_{1} and Φ(m_{1}) = m_{1}. Therefore, G ≃ G[M] where M ⊗ O_{1} ≃ M_{1}, i.e. M = Om, m = ξm_{1} with ξ ∈ O_{1}^{∗} and Φm = λm with λ = ξ^{q−1} ∈ O ∩ O_{1}^{∗} = O^{∗}.

Suppose now that I_{G} ≠ I_{G}^{2}. Consider the decomposition I_{G} = ⊕_{1⩽n<q}I_{G,n}, where an α ∈ 𝔽_{q} acts on I_{G,n} via multiplication by α^{n}.

Because t_{G}^{∗} = I_{G}/I_{G}^{2} ≠ 0, rk_{O} I_{G,1} ⩾ 1 and, therefore, for all 1 ⩽ n < q, rk_{O} I_{G,n} ⩾ 1 (if x ∈ I_{G,1}, x ≠ 0, then x^{n} ≠ 0 and x^{n} ∈ I_{G,n}). So, all these ranks are equal to 1 and, if I_{G,1} = xO, then the equality I_{G} = I_{G,1} + I_{G}^{2} implies for all 1 ⩽ n < q, I_{G,n} = x^{n}O.

Finally, x^{q} ∈ I_{G,1} and x^{q} = λx for some λ ∈ O. Clearly, λ ≠ 0 because G ⊗ K is etale. This proves that A(G) = A[M] with M = Om, Φm = λm.

Suppose Δx = x ⊗ 1 + 1 ⊗ x + δ. Then δ^{q} = λδ and δ ∈ (x ⊗ x)A(G) ⊗ A(G). This implies easily that δ = 0. It remains to note that by the choice of x, [α]x = αx for any α ∈ 𝔽_{q}, and this action of 𝔽_{q} is strict.

Definition. We shall denote by μ_{λ} the 𝔽_{q}-module scheme G[M] from the above proposition.

2.5. Suppose G ∈ FGr′(𝔽_{q})_{O} satisfies the following four assumptions:

1) G = G^{loc};

2) there is a short exact sequence in the category FGr′(𝔽_{q})_{O}

0 ⟶ μ_{λ} ⟶^{i} G ⟶ G_{1} ⟶ 0

where λ ∈ O, λ ≠ 0;

3) the above exact sequence splits over K, i.e. there is a Γ_{K}-invariant section of the projection of 𝔽_{q}[Γ_{K}]-modules G(K_{sep}) ⟶ G_{1}(K_{sep});

4) there is an M_{1} ∈ Mod_{O,σ} such that G_{1} ≃ G[M_{1}].

Proposition 3. With the above assumptions 1)-4), there is an M ∈ Mod_{O,σ} such that G ≃ G[M].

Proof. Let A(μ_{λ}) = O[u] with u^{q} = λu, Δu = u ⊗ 1 + 1 ⊗ u, [α]u = αu for α ∈ 𝔽_{q}. Suppose the closed embedding i : μ_{λ} ⟶ G is given by the O-algebra morphism i^{∗} : A ⟶ A(μ_{λ}). Set A = A(G) and B = A(G_{1}). Then

s = Δ_{G} ∘ (i^{∗} ⊗ id_{A}) : A ⟶ A(μ_{λ}) ⊗ A

is the coaction of μ_{λ} on A and B = A^{μ_{λ}} = {a ∈ A | s(a) = 1 ⊗ a}.

Choose an a_{0} ∈ A such that i^{∗}(a_{0}) = u^{q−1} and [α]a_{0} = a_{0} for all α ∈ 𝔽_{q}. Set

s(a_{0}) = 1 ⊗ a_{0} + u ⊗ a_{1} + ⋯ + u^{q−1} ⊗ a_{q−1}

where a_{1}, …, a_{q−1} ∈ A. Then for any α ∈ 𝔽_{q} and m = 1, …, q−1, [α]a_{m} = α^{−m}a_{m} and the identity

(id ⊗i^{∗})(s(a_{0})) = Δ(i^{∗}a_{0}) = Δ(u^{q−1}) = (u ⊗ 1 + 1 ⊗ u)^{q−1}

implies that i^{∗}(a_{q−1}) = 1 and i^{∗}(a_{q−2}) = −u. Therefore, a_{q−1} ∈ A^{∗} (because A = A^{loc}) and A = B[a_{q−2}] (because a_{q−2} generates A modulo a nilpotent ideal I_{B}A).

The equality

1 ⊗ s(a_{0}) + u ⊗ s(a_{1}) + ⋯ + u^{q−1} ⊗ s(a_{q−1}) = (id ⊗s)(s(a_{0}))

= (Δ ⊗ id)(s(a_{0})) = 1 ⊗ 1 ⊗ a_{0} + (u ⊗ 1 + 1 ⊗ u) ⊗ a_{1} + ⋯ + (u ⊗ 1 + 1 ⊗ u)^{q−1} ⊗ a_{q−1}

implies that s(a_{q−1}) = 1 ⊗ a_{q−1} and s(a_{q−2}) = 1 ⊗ a_{q−2} − u ⊗ a_{q−1}.

Therefore, a_{q−1} ∈ B ∩ A^{∗} = B^{∗} and for θ = −a_{q−2}/a_{q−1}, we have A = B[θ], s(θ) = 1 ⊗ θ + u ⊗ 1, [α]θ = αθ for α ∈ 𝔽_{q}, and θ^{q} − λθ = b ∈ B.

Notice also that δ^{+}θ = Δθ − θ ⊗ 1 − 1 ⊗ θ ∈ B ⊗ B, because this is a (μ_{λ} × μ_{λ})- invariant element of A ⊗ A.

From condition 3) it follows the existence of v ∈ A_{K} such that i^{∗}(v) = u, v^{q} = λv, Δv = v ⊗ 1 + 1 ⊗ v and [α]v = αv for α ∈ 𝔽_{q}. Therefore, denoting the K-linear extension of s by s_{K}, we have s_{K}(v) = 1 ⊗ v + u ⊗ 1. This implies that θ − v = b_{0} ∈ B ⊗ K, and δ^{+}b_{0} = δ^{+}θ.

Therefore, by the property of Hochshild cohomology from the end of n.2.1, there is a b_{1} ∈ B (with the property [α]b_{1} = αb_{1}, ∀α ∈ 𝔽_{q}) such that δ^{+}b_{1} = δ^{+}θ and replacing θ by θ − b_{1} we can assume that Δθ = θ ⊗ 1 + 1 ⊗ θ. Therefore, M = M_{1} + Oθ ∈ Mod_{O,σ}, A = A[M] and G = G[M]. The proposition is proved.

2.6. Now we can finish the proof of Theorem 1.

Suppose G ∈ FGr′(𝔽_{q})_{O} is of order q^{N}, N ∈ ℕ. As it was noticed in n.2.3, it will be sufficient to prove that G ≃ G[M] for a suitable M ∈ Mod_{O,σ}. When proving this property we can enlarge (if necessary) the basic ring O.

Apply induction on N.

The case N = 1 follows from Proposition 2 in n.2.4.

Suppose that G ∈ FGr′(𝔽_{q})_{O} is of order q^{N} and any 𝔽_{q}-module scheme of order q^{N−1} appears in the form G[M_{1}] with M_{1} ∈ Mod_{O,σ}. By n.1.4 and Lemma 2b), it will be sufficient to consider the following two cases:

a) G = G^{et}; In this case we can enlarge O to assume that G is a constant group scheme, where the property G ≃ G[M] with M ∈ Mod_{O,σ} is obviously true.

b) G = G^{loc}. In this case enlarge O to be able to assume that G⊗K is constant, i.e. H = G(K_{sep}) has a trivial Γ_{K}-action. Choose a 1-dimensional 𝔽_{q}-subspace H_{2} in H. It gives rise to a short exact sequence of 𝔽_{q}-module group schemes

0 ⟶ G_{2} ⟶ G ⟶ G_{1} ⟶ 0

where G_{2}(K_{sep}) = H_{2}.

Because G is a strict 𝔽_{q}-module, 𝔽_{q} acts on t_{G}^{∗} and, therefore, on t_{G_{2}}^{∗}, via scalar multiplication. By Proposition 2, G_{2} ≃ μ_{λ}, λ ∈ O, λ ≠ 0, belongs to FGr′(𝔽_{q})_{O} and, therefore, G_{1} ∈ FGr′(𝔽_{q})_{O}, cf. n.1.4. By induction, we have G_{1} ≃ G[M_{1}], M_{1} ∈ Mod_{O,σ}.

Finally, G satisfies the properties 1)-4) from n.2.5 and by Proposition 3, G ≃ G[M] for M ∈ Mod_{O,σ}. The Theorem is proved.

3. Group schemes with strict O_{0}-action.

In this section R = O_{0} is the valuation ring of a closed subfield K_{0} of K with finite residue field 𝔽_{q}, q = p^{n}, n ∈ ℕ. Fix a choice of uniformising element π_{0} in K_{0}. So, O_{0} = 𝔽_{q}[[π_{0}]].

3.1. Denote by FGr′(O_{0})_{O} a full subcategory in FGr(O_{0})_{O} consisting of objects killed by some power of [π_{0}]. This is an analogue of the classical category of p-torsion finite flat group schemes over the valuation ring of a complete discrete valuation field of mixed characteristic.

Let Mod(O_{0})_{O} be the category, consisting of M ∈ Mod_{O,σ} with O_{0}-module structure given for o ∈ O_{0} by endomorphisms [o] ∈ End_{Mod_{O,σ}}(M) such that for all m ∈ M, [α]m = αm if α ∈ 𝔽_{q} ⊂ k and [π_{0}^{N}]m = 0 for sufficiently large N. Morphisms in the category Mod(O_{0})_{O} are morphisms of the category Mod_{O,σ} which commute with all endomorphisms [o], o ∈ O_{0}. In other words, any M ∈ Mod(O_{0})_{O} is a free O-module M of finite rank equiped with

a) an O-linear map Φ : M_{σ} ⟶ M such that Φ ⊗ K is isomorphism;

b) an O-linear nilpotent endomorphism [π_{0}] : M ⟶ M commuting with Φ.

We shall denote by Mod′(O_{0})_{O} a full subcategory of Mod(O_{0})_{O} consisting of objects M such that

c) for any m ∈ M, [π_{0}]m ≡ π_{0}m mod Φ(M_{σ}).

The category Mod(O_{0})_{O} is not generally abelian. But similarly to the category FGr′(O_{0})_{O}, one can work with short exact sequences in Mod(O_{0})_{O}. The sequence of objects and morphisms in Mod(O_{0})_{O}

0 ⟶ M_{1} ⟶^{i} M ⟶^{j} M_{2} ⟶ 0

is exact if and only if i is a pure morphism of O-modules (i.e. M/i(M_{1}) has no O-torsion) and j is epimorphism of O-modules with kernel i(M_{1}). This allows to define O-modules of equivalence classes of short exact sequences which satisfy the usual functorial properties.

3.2. Let i : FGr(O_{0})_{O} ⟶ FGr(𝔽_{q})_{O} be the forgetful functor.

Proposition 4. If G ∈ FGr′(O_{0})_{O}, then i(G) ∈ FGr′(𝔽_{q})_{O}.

Proof. Indeed, if G = (G, G^{♭}) then O_{0} acts via a scalar multiplication on t_{G^{♭}}^{∗}. This implies that t_{G}^{∗} is killed by some power of π_{0} and, therefore, t_{G⊗K}^{∗} = t_{G}^{∗} ⊗ K = 0, i.e. G ⊗ K is etale. The proposition is proved.

Let L : FGr′(𝔽_{q})_{O} ⟶ Mod_{O,σ} be the functor from n.2. For any G ∈ FGr′(O_{0})_{O}, L(i(G)) has a natural structure of an object of the category Mod(O_{0})_{O}. We denote by L the functor from FGr′(O_{0})_{O} to Mod(O_{0})_{O} induced by the above correspondence G ↦ L(i(G)).

Theorem 2. The functor L induces antiequivalence of the categories FGr′(O_{0})_{O} and Mod′(O_{0})_{O}.

Proof. Let G ∈ FGr′(O_{0})_{O} and M = L(G) ∈ Mod(O_{0})_{O}. Considering M as an object of Mod_{O,σ} we can recover the 𝔽_{q}-module scheme i(G) in the form G[M]. This reduces the proof of our theorem to the following statement:

Proposition 5. G is a strict O_{0}-module iff M satisfies the condition c) from n.3.1.

Let G^{loc} be the maximal local subobject in G and M^{loc} = L(G^{loc}). Then the natural projection M ⟶ M^{loc} induces isomorphism of (O_{0}, O)-modules M/Φ(M_{σ}) and M^{loc}/Φ(M_{σ}^{loc}). So, it will be sufficient to prove the above proposition for a local G. Under this assumption choose a vector column m̄ = (m_{1}, …, m_{n})^{t} such that its coordinates create an O-basis of M = L(G) and set Φm̄ = Cm̄ and [π_{0}]m̄ = Dm̄, where C, D ∈ 𝕄_{n}(O) are n × n-matrices with coefficients from O.

Lemma 4. The above matrices C and D give a structure of an object of the category Mod′(O_{0})_{O} on M if and only if

1) det C ≠ 0;

2) D is nilpotent and σ(D)C = CD;

3) there is a B ∈ 𝕄_{n}(O) such that D − π_{0}E_{n} = BC.

Proof. Indeed, 1) means that Φ(M_{σ}) is a lattice in the O-module M.

Property 2) means that M is [π_{0}]-torsion and the morphisms Φ and [π_{0}] commute one with another. Indeed, (Φ ∘ [π_{0}])(m̄) = [π_{0}](Φm̄) = CDm̄ and ([π] ∘ Φ)(m̄) = σ(D)Cm̄.

Notice that the coordinates of the vector Cm̄ generate the O-module Φ(M_{σ}). So, the coordinates of [π_{0}](m̄) − π_{0}m̄ belong to Φ(M_{σ}) if and only if there is an B ∈ 𝕄_{n}(O) such that D − π_{0}E_{n} = BC.

The lemma is proved.

Remark. a) M = M^{loc} if and only if C is σ-nilpotent, i.e. there is an N ∈ ℕ such that σ^{N}(C) … σ(C)C = 0;

b) the above properties 1) and 3) imply that σ(D) − π_{0}E_{n} = CB.

With the above notation and agreements let G = (G, G^{♭}), where A(G) = O[X̄]/I and A(G^{♭}) = O[X̄]/(I · I_{0}), where X̄ = (X_{1}, …, X_{n}) and the ideal I is generated by the coordinates of the vector X̄^{q} − CX̄.

Then A(G) has an O-basis {X_{1}^{a_{1}} … X_{n}^{a_{n}} | 0 ⩽ a_{1}, …, a_{n} < q} and this basis can be completed to an O-basis of A(G^{♭}) by joining the elements of the set {X_{i}^{q} | 1 ⩽ i ⩽ n}.

The action of O_{0} is given by [π_{0}]X̄ = DX̄ on A(G). This action is strict if and only if it can be extended to A(G^{♭}) via the relation

[π_{0}]X̄ = DX̄ + B(X̄^{q} − CX̄)

with some B ∈ 𝕄_{n}(O) in such a way that it induces the scalar multiplication by π_{0} on t_{A(G^{♭})}^{∗} and N_{A(G^{♭})}.

The first condition is equivalent to the equality D − BC = π_{0}E_{n}.

The second can be analysed as follows,

[π_{0}]^{♭}(X̄^{q} − CX̄) ≡ ([π_{0}]^{♭}X̄)^{q} − C([π_{0}]^{♭}X̄) ≡ (σ(D) − CB)(X̄^{q} − CX̄)

and is equivalent to the matrix equality σD − CB = π_{0}E_{n}.

So, Proposition 5 and Theorem 2 follow from Lemma 4 and the above Remark b).

3.3. Clearly, the antiequivalence L transforms short exact sequences in FGr′(O_{0})_{O} to short exact sequences in the category Mod′(O_{0})_{O,σ}. In particular, we have the following property.

Proposition 6. Suppose G_{1}, G_{2} ∈ FGr′(O_{0})_{O} and f ∈ Hom_{FGr′(O_{0})_{O}}(G_{1}, G_{2}). Then

a) f is a closed embedding if and only if L(f) is surjective;

b) f is fully faithful if and only if L(f) is a pure embedding.

Suppose for n ∈ ℕ, G^{(n)} ∈ FGr′(O_{0})_{O}, i_{n} : G^{(n)} ⟶ G^{(n+1)} is a closed immersion, j_{n} : G^{(n+1)} ⟶ G^{(n)} is a fully faithful morphism. For m > n, set i_{nm} = i_{n} ∘ ⋯ ∘ i_{m−1} and j_{mn} = j_{m−1} ∘ ⋯ ∘ j_{n}. Then (following the original deinition of Tate) {G^{(n)}, i_{n}, j_{n}} is a π_{0}-divisible group in the category FGr′(O_{0})_{O} if for any m > n, (G^{(n)}, i_{nm}) = Ker[π_{0}^{m}] id_{G^{(m)}} and [π_{0}^{m−n}] id_{G^{(m)}} = j_{mn} ∘ i_{nm}.

Similar definitions can be done for the category Mod′(O_{0})_{O}, where a π_{0}-divisible group appears as the collection {M^{(n)}, i_{n}, j_{n}} where i_{n} is a pure embedding of underlying O-modules M^{(n)} ⟶ M^{(n+1)} and j_{n} : M^{(n+1)} ⟶ M^{(n)} is a surjection of O-modules.

3.4. Clearly, the functor L from Theorem 2 transforms π_{0}-divisible groups in the category FGr′(O_{0})_{O} to π_{0}-divisible groups in the category Mod′(O_{0})_{O}.

Theorem 3. For any object G of the category FGr′(O_{0})_{O}, there is a π_{0}-divisible group {H^{(n)}, i_{n}, j_{n}}_{n⩾1} and a closed embedding G ⟶ H^{(N)}, where N ∈ ℕ is such that [π_{0}^{N}] id_{G} = 0.

Remark. The statement of the above theorem is equivalent to the existence of a π_{0}-divisible group {H′^{(n)}, i_{n}, j_{n}}_{n⩾1} and a fully faithful morphism H′^{(N)} ⟶ G.

Proof. The antiequivalence L allows us to prove the dual version of this theorem in the category Mod′(O_{0})_{O}.

Let M ∈ Mod′(O_{0})_{O} and let N ∈ ℕ be such that [π_{0}^{N}]M = 0. Use induction on N.

Suppose first, that N = 1.

Then M is a free O-module with σ-linear Φ : M_{σ} ⟶ M such that π_{0}M ⊂ Φ(M_{σ}) ⊂ M. Choose an O-basis m_{1}, …, m_{n} in M and take a vector column m̄ = (m_{1}, …, m_{n})^{t}. Then Φm̄ = Cm̄, where C ∈ 𝕄_{n}(O) is a divisor of π_{0}E_{n}. Let C̃ ∈ 𝕄_{n}(O) be such that CC̃ = π_{0}E_{n}.

For l ∈ ℕ, introduce free O-modules M^{(l)} with free generators

{m_{1i}^{(k)}, m_{2i}^{(k)} | 1 ⩽ i ⩽ n, 1 ⩽ k ⩽ l}

Let m̄_{1}^{(0)} = m̄_{2}^{(0)} = ¯0 and for k ⩾ 1, let m̄_{1}^{(k)} = (m_{11}^{(k)}, …, m_{1n}^{(k)})^{t} and m̄_{2}^{(k)} = (m_{21}^{(k)}, …, m_{2n}^{(k)})^{t}. Define O-linear morphisms Φ : M_{σ}^{(l)} ⟶ M^{(l)} and [π_{0}] : M ⟶ M by setting for 1 ⩽ k ⩽ l,

Φm̄_{1}^{(k)} = C̃m̄_{1}^{(k)} + m̄_{2}^{(k−1)}, Φm̄_{2}^{(k)} = Cm̄_{2}^{(k)} + m̄_{1}^{(k)}

and [π_{o}]m̄_{1}^{(k)} = m̄_{1}^{(k−1)}, [π_{0}]m̄_{2}^{(k)} = m̄_{2}^{(k−1)}.

It is easy to see that we defined a structure of objects of the category Mod′(O_{0})_{O} on all M^{(l)}, l ∈ ℕ, and the system {M^{(l)}}_{l⩾1} together with the natural inclusions i_{n} : M^{(n} ⟶ M^{(n+1)} and projections j_{l} : M^{(l+1)} ⟶ M^{(l)} gives a π_{0}-divisible group in the category Mod′(O_{0})_{O}. Clearly, the correspondences m̄_{1}^{(1)} ↦ ¯0 and m̄_{2}^{(1)} ↦ m̄ give an epimorphic map from M^{(1)} to M. The case N = 1 has been considered.

Suppose N > 1 and Theorem 3 has been proved for all M′ ∈ Mod′(O_{0})_{O} such that [π_{0}]^{N−1}(M′) = 0.

Let M_{1} = Ker[π_{0}]^{N−1} id_{M} and M_{2} = [π_{0}]^{N−1}(M) ⊂ M. Then M_{1} and M_{2} have natural structures of objects of the category Mod′(O_{0})_{O} and in this category we have a natural short exact sequence

ε : 0 ⟶ M_{1} ⟶ M ⟶ M_{2} ⟶ 0

(Notice that, generally, the embedding M_{2} ⊂ M is not pure). By induction, there is a π_{0}-divisible group {T^{(n)}, i_{n}, j_{n}}_{n⩾1} and pure embeddings α : M_{1} ⟶ T^{(N−1)}, β : M_{2} ⟶ T^{(1)}. Consider the short exact sequences

α_{∗}ε : 0 ⟶ T^{(N−1)} ⟶ α_{∗}M ⟶^{j_{α}} M_{2} ⟶ 0

and

β^{∗}η_{N} : 0 ⟶ T^{(N−1)} ⟶ β^{∗}T^{(N)} ⟶^{j_{β}} M_{2} ⟶ 0

where the second sequence is obtained via β from the standard short exact sequence

η_{N} : 0 ⟶ T^{(N−1)} ⟶^{i_{N}} T^{(N)} ⟶^{j_{N1}} T^{(1)} ⟶ 0

Notice that there is a pure embedding M ⟶ α_{∗}M. Notice also that in the both sequences α_{∗}ε and β^{∗}η_{N}, the epimorphic maps j_{α} and j_{β} are induced by multiplication by [π_{0}^{N−1}]. In other words, the endomorphism [π_{0}^{N−1}] on α_{∗}M and, resp., on β^{∗}T^{(N)} is a composition of j_{α} and, resp., of j_{β} with the natural inclusion of M_{2} into α_{∗}M and, resp., into β^{∗}T^{(N)} (which is induced by the embedding M_{2} ⊂ T^{(1)}).

Let

0 ⟶ T^{(N−1)} ⟶ M̃ ⟶ M_{2} ⟶ 0

be the difference of α_{∗}ε and β^{∗}η_{N} considered as elements of the group Ext_{Mod′(O_{0})_{O}}(M_{2}, T^{(N−1)}). It is easy to see that [π_{0}^{N−1}](M̃) = 0 and there is an M̃_{1} ∈ Mod′(O_{0})_{O}, a surjective map j̃ and a pure embedding ĩ such that

β^{∗}T^{(N)} ⊕ M̃ ⟶^{j̃} M̃_{1} ⟵^{ĩ} α_{∗}(M)

By inductive assumption, there is a pure embedding of β^{∗}T^{(N)} ⊕ M̃ into a π_{0}- divisible group in the category Mod′(O_{0})_{O}. This implies easily the existence of such an embedding for α_{∗}M and, therefore, for M.

Theorem 3 is proved.

3.5. Relation to the mixed characteristic case.

In nn.3.5.1-3.5.3 below we need a full subcategory FGr′_{1}(O_{0})_{O} in FGr′(O_{)})_{O}. It consists of G = (G, G^{♭}) such that [π_{0}] id_{G} = 0. Then the functor L induces an antiequivalence of this category and the full subcategory Mod′_{1}(O_{0})_{O} of Mod′(O_{0})_{O} consisting of free O-modules M such that π_{0}M ⊂ Φ(M_{σ}) ⊂ M. As usually, e = e(K/K_{0}) is the ramification index of K = Frac O over K_{0} = Frac O_{0}.

3.5.1. Suppose e(K/K_{0}) = 1.

Introduce the category SH_{1}(𝔽_{q})_{O} with objects (M^{0}, M^{1}, φ_{0}, φ_{1}), where M^{0} is an O-module of finite length such that π_{0}M = 0, φ_{0} : M^{0} ⟶ M^{0} is a σ-linear morphism, M^{1} = Ker φ_{0}, φ_{1} : M^{1} ⟶ M^{0} is a σ-linear morphism and φ_{0}(M^{0}) + φ_{1}(M^{1}) = M. This is an analogue of Fontaine’s category of filtered modules.

Consider the functor SH : Mod′_{1}(O_{0})_{O} ⟶ SH_{1}(𝔽_{q}) defined by the correspondence M ↦ (M^{0}, M^{1}, φ_{0}, φ_{1}), where M^{0} = M mod π_{0}M, φ_{0} = Φ mod π_{0} and φ_{1} is

induced by \frac{1}{π_{0}} Φ. The functor SH is an equivalence of categories if q > 2 and is “very

close” to an equivalence if q = 2. This shows that strict O_{0}-modules have similar description as conventional group schemes if e = 1. Actually, one can develop the Dieudonne theory in the context of strict modules and realise the approach from [Ab1] to recover directly an analogue of Fontaine’s classification of group schemes over Witt vectors.

3.5.2. Suppose e ⩽ q − 1.

Define the category SH_{1}(𝔽_{q})_{O} of the collections (M, M^{0}, M^{1}, φ_{0}, φ_{1}), where M is an O-module of finite length killed by π_{0}, M^{0} = Ker π|_{M}, φ_{0} : M^{0} ⟶ M is a σ-linear map, M^{1} = Ker φ_{0}, φ_{1} : M^{1} ⟶ M is a σ-linear map, and φ_{0}(M^{0}) ⊗_{k} O + φ_{1}(M^{1}) ⊗_{k} O = M. This category is an analogue of the category SH_{O} from [Ab3]. Consider the functor SH : Mod′_{1}(O_{0})_{O} ⟶ SH(𝔽_{q})_{O} defined by the correspondence M ↦ (M̄, M̄^{0}, M̄^{1}, φ_{0}, φ_{1}), where M̄ is the Osubmodule in \frac{π}{π_{0}} O⊗M mod πM generated by the images of the elements of the sets

\{\frac{π}{π_{0}} Φ(m | m ∈ M\} and \{\frac{1}{π_{0}} Φ(m) | m ∈ M, Φ(m) = 0\}. Then M̄^{0} = Ker π|M̄ =

M mod πM, φ_{0} : M̄^{0} ⟶ M is induced by \frac{π}{π_{0}} Φ, M̄^{1} = Ker φ_{0} and φ_{1} : M̄^{1} ⟶ M

is induced by \frac{1}{π_{0}} Φ.

If e < q − 1 then SH is an equivalence of categories and, if e = q − 1 then SH is “very close” to an equivalence of categories. Again, the methods from [Ab3] can be used to obtain directly the classification of objects from FGr′_{1}(O_{0})_{O} via objects of the category SH(𝔽_{q})_{O}. (The details will appear in W.Gibbons’s thesis.)

Notice that when working with equal characteristic case the category SH(𝔽_{q})_{O} can be replaced by a simpler category SH′(𝔽_{q})_{O} consisting of the triples (M̄, M̄^{0}, φ), where M̄ is an O-module of finite length killed by π_{0}, M̄^{0} = Ker π|M̄ and φ_{0} : M̄^{0} ⟶ M̄ is a σ-linear morphism such that φ(M̄^{0}) ⊗ O = M̄. The functor SH′ : FGr′_{1}(O_{0})_{O} ⟶ SH′(𝔽_{q})_{O} is defined by the correspondence M ↦ (M̄, M̄^{0}, φ_{0}),

where M̄ is an O-submodule in \frac{1}{π_{0}} O ⊗ M mod πM generated by elements of the

form \frac{1}{π_{0}} Φm with m ∈ M, M̄^{0} = Ker π|_{M} = M mod πM and φ_{0} is induced by \frac{1}{π_{0}} Φ

(this map is still σ-linear because of the equal characteristic situation).

3.5.3. Suppose that e(K/K_{0}) is arbitrary.

Introduce the category BR_{1}(𝔽_{q})_{O}. Its objects are the triples (M, M^{1}, φ), where M is an O-module of finite length such that π_{0}M = 0, M^{1} is an O_{1}-submodule in M ⊗_{O} O_{1}, where O_{1} = O[π_{1}] with π_{1}^{p} = π, and φ : M^{1} ⟶ M is a σ-linear map (σ is still the q-th power map) such that φ(M^{1}) = M. This category is an equicharacteristic version of Breuil’s category Mod_{/S_{1}} appeared in his classification of period p group schemes in the mixed characteristic case, cf. [Br].

Again there is a natural functor from Mod′_{1}(O_{0})_{O} to BR_{1}(𝔽_{q})_{O} defined by the

correspondence M ↦ (M̄, M̄^{1}, φ), where M̄ = M mod π_{0} and φ is induced by \frac{1}{π_{0}} Φ.

But one can’t expect that Breuil’s method works in the equal characteristic case, because it is based very heavily on crystalline technique.

3.5.4. Again e(K/K_{0}) is arbitrary.

Introduce an equal characteristic analogue of the concept of p-etale φ-module of q-heighjt 1 over S from [Fo4].

First, introduce 2nd copy \underline{O}_{0} = 𝔽_{q}[[\underline{π}_{0}]] of O_{0} (actually, \underline{π}_{0} is an analogue of [π_{0}]). Then \underline{O}_{0} will be considered as an analogue of ℤ_{p} and \underline{Õ}_{0} = k[[\underline{π}_{0}]] is an analogue of Witt vectors W (k). Then consider S = \underline{Õ}_{0}[[π]], where k[[π]] = O appears as a subring of S. Introduce σ : S ⟶ S such that σ|_{\underline{O}_{0}} = id and σ|_{O} is (as earlier) the qth power map.

Suppose that M is an S-module of finite type together with an \underline{O}_{0}-linear map Φ : M_{σ} := M ⊗_{(O,σ)} O ⟶ M. Then M is called a \underline{π}_{0}-etale φ-module of q-height 1 (with q = \underline{π}_{0} − π_{0}), if Coker Φ is killed by multiplication by \underline{π}_{0} − π_{0}.

With this notation Theorem 2 establishes a description of the category of finite flat strict O_{0}-modules over O in terms of \underline{π}_{0}-torsion \underline{π}_{0}-etale φ-modules of q-height 1.

4. Properties of arising Galois modules.

In this section G = (G, G^{♭}) ∈ FGr′(O_{0})_{O}, H = G(K_{sep}) is O_{0}[Γ_{K}]-module of geometric points of G and e = e(K/K_{0}) — is the ramification index of K over K_{0}. We also set Γ_{K} = Gal(K_{sep}/K) and denote by I_{K} the inertia subgroup of Γ_{K}.

4.1. Characters of the semisimple envelope of H.

Suppose k̄ is an algebraic closure of k and the character χ : I_{K} ⟶ k̄^{∗} appears with a nonzero multiplicity in the semisimple envelope of the O_{0}[Γ_{K}]-module H. An analogue of the Serre Conjecture for H can be stated as follows.

Theorem 4. For the above character χ, there are a, N ∈ ℕ\pℕ such that χ = χ_{N}^{a}, where a = a_{0} + a_{1}q + ⋯ + a_{N−1}q^{N−1} with 0 ⩽ a_{i} ⩽ e and χ_{N} : I_{K} ⟶ k̄^{∗} is such that for any τ ∈ I_{K}, χ_{N}(τ) = τ(π_{N})/π_{N}, where π_{N} ∈ K_{sep} and π_{N}^{q^{N} −1} = π.

Proof. This can be deduced in the same way as it has been obtained in the case of usual group schemes in [Ra]. First we can assume that k = k̄ and e < q − 1. Then any simple object of the category Mod′(O_{0})_{O} appears in the form M = ⊕_{0⩽i<n}Om_{i}, where Φm_{0} = π^{a_{0}}m_{1}, …, Φm_{N−1} = π^{a_{N−1}}m_{0} and [π_{0}]m_{0} = ⋯ = [π_{0}]m_{N−1} = 0 with 0 ⩽ a_{i} ⩽ e, 0 ⩽ i < N.

Then the corresponding Galois module consists of K_{sep}-points of the O-algebra O[T_{0}, …, T_{N−1}], where T_{0}^{q} = π^{a_{0}}T_{1}, …, T_{N−1}^{q} = π^{a_{N−1}}T_{0}. It can be naturally identified with the O_{0}[Γ_{K}]-module {απ_{N}^{a} | α ∈ 𝔽_{q^{N}} }, where a = a_{0} + a_{1}q + ⋯ + a_{N−1}q^{N−1}. Clearly, I_{K} acts on it via the conjugacy class of characters {σ^{i}χ_{N}^{a} | 0 ⩽ i < N}.

Remark. Following Raynaud’s method one can deduce from the above description of simple objects in Mod′(O_{0})_{O} that if e < q − 1 then the functor G ↦ G(K_{sep}) is a fully faithful functor from Mod′(O_{)})_{O} to the category of O_{0}[Γ_{K}]-modules.

4.2. Ramification estimates.

These estimates are given in Theorem 5 below and are completely similar to the known estimates in the case of conventional group schemes, cf. [Fo2]. The proof is based on the knowledge of “equations” of the strict module G and is done below by the methods of the paper [Ab4]. Notice that the methods from [Fo3] also can be adjusted to obtain the same estimates.

Theorem 5. If H is killed by [π_{0}^{N}] then the ramification subgroups Γ_{K}^{(v)} act trivially

on H for v > e (N + \frac{1}{q − 1}) − 1.

Proof. We can assume that there is a π_{0}-divisible group {G^{(i)}}_{i⩾1} of a height h in FGr′(O_{0})_{O} such that G^{(N)} = G.

4.2.1. By the above assumption, L(G) is a free O-module of rank hN and we can choose its O-basis in the form

m_{1}, …, m_{h}, [π_{0}]m_{1}, …, [π_{0}]m_{h}, …, [π_{0}]^{N−1}m_{1}, …, [π_{0}^{N−1}]m_{h}

Introduce vector-columns

m̄_{1} = (m_{1}, …, m_{h})^{t}, …, m̄_{N} = [π_{0}^{N−1}]m̄_{1} = ([π_{0}^{N−1}]m_{1}, …, [π_{0}^{N−1}]m_{h})^{t}

Then, in the obvious notation, the structure of M = L(G) can be given in the following form

\frac{1}{π_{0}} C_{1}Φm̄_{N} + \frac{1}{π_{0}} C_{2}Φm̄_{N−1} + ⋯ + \frac{1}{π_{0}} C_{N}Φm̄_{1} = m̄_{N} − \frac{1}{π_{0}} m̄_{N−1}

\frac{1}{π_{0}} C_{1}Φm̄_{N−1} + ⋯ + \frac{1}{π_{0}} C_{N−1}Φm̄_{1} = m̄_{N−1} − \frac{1}{π_{0}} m̄_{N−2}

...............................

\frac{1}{π_{0}} C_{1}Φm̄_{1} = m̄_{1}

where all C_{i} ∈ GL(h, O) and det C_{1} ≠ 0.

Consider vector columns X̄_{i} = (X_{i1}, …, X_{ih})^{t} of independent variables X_{ij}, 1 ⩽ i ⩽ N, 1 ⩽ j ⩽ h. Then the algebra A(G) appears as a quotient of O[X̄_{1}, …, X̄_{N}] by the ideal generated by equations

∑_{1⩽i⩽s} C_{i}X̄_{s+1−i}^{q} = π_{0}X̄_{s} − X̄_{s−1}  (5)

where 1 ⩽ s ⩽ N and by definition X̄_{0} = ¯0.

Consider the points of G(K_{sep}) as solutions ā = (ā_{1}, …, ā_{N}) of the system (5). The following lemma can be easily proved by induction on N.

Lemma 5. If ā = (ā_{1}, …, ā_{N}) and ā′ = (ā′_{1}, …, ā′_{N}) are solutions of (4.1) such

that ā ≡ ā′ mod π0^{\frac{1}{q−1}} m_{sep}, where m_{sep} is the maximal ideal of the valuation ring of K_{sep}, then ā = ā′.

4.2.2. Suppose α ∈ ℚ_{>0} has the p-adic valuation 0. Then α = \frac{m}{q^{M} − 1} with

suitable m, M ∈ ℕ, (m, p) = 1. For any such α there is an extension K_{α} of K with [K_{α} : K] = q^{M} and the Herbrand function

φ_{K_{α}/K}(x) = \{ \begin{matrix}x, & for 0 ⩽ x ⩽ α \\ α + \frac{x − α}{q^{M}}, & for x ⩾ α. \end{matrix}

16 Notice that φ_{K_{α}/K} has only one edge point in x = α.

Explicit construction of K_{α} can be found in [Ab4], where the field K_{α} appears as an extension of K of degree q^{M} in L_{α} = K(π_{M})(T), where π_{M}^{q^{M} −1} = π and T^{q^{M}} − T = π_{M}^{−m}. Clearly, K_{α} is totaly ramified over K and, therefore, there is a field isomorphism

h_{α} : K ⟶ K_{α}

From the above construction of K_{α}, it follows easily that h_{α} can be chosen in such a way that for any a ∈ m_{K} (m_{K} is the maximal ideal in O),

a = h_{α}(a)^{q^{M}} + ã,

with ã ∈ K_{α} such that v_{K}(ã) ⩾ v_{K}(a) + α (v_{K} is the normalized valuation in K).

4.2.3. Denote by the same symbol an extension of h_{α} to an isomorphism of K_{sep} onto K_{α,sep} = K_{sep}. Clearly, X̄ = (X̄_{1}, …, X̄_{s}) ↦ h_{α}(X̄) = (h_{α}(X̄_{1}), …, h_{α}(X̄_{s})) is a one-one correspondence between solutions of the system (5) and solutions Ȳ = (Ȳ_{1}, …, Ȳ_{s}) of the similar system

∑_{1⩽i⩽s} h_{α}(C_{i})Ȳ_{s+1−i}^{q} = h_{α}(π_{0})Ȳ_{s} − Ȳ_{s−1}  (6)

where 1 ⩽ s ⩽ N and by definition Ȳ_{0} = ¯0.

Lemma 6. If α > e (N + \frac{1}{q − 1}) − 1, then for any solution X̄^{(0)} of (5) there is

a unique solution Ȳ^{(0)} of (6) such that X̄^{(0)} ≡ Ȳ^{(0)q^{M}} mod π0^{\frac{1}{q−1}} m_{sep}

Proof of lemma. the correspondence Ȳ ↦ Z̄ = Ȳ^{q^{M}} − X̄^{(0)} establishes a one-one correspondence between solutions Ȳ of (5) and solutions Z̄ = (Z̄_{1}, …, Z̄_{s}) of the system of equations

∑_{1⩽i⩽s} C_{i}Z̄_{s+1−i}^{q} = π_{0}Z̄_{s} − Z̄_{s−1} + F̄_{s}  (7)

where 1 ⩽ s ⩽ N, Z̄_{0} = 0 and

F̄_{s} = π̃_{0}Ȳ_{s}^{q^{M}} − ∑_{1⩽i⩽s} C̃_{i}Ȳ_{s+1−i}^{q^{M+1}} ∈ π_{0}^{1+ \frac{1}{q−1}} m_{sep}

because v_{K}(π̃_{0}), v_{K}(C̃_{i}) ⩾ 1 + α > e (1 + \frac{1}{q − 1}).

Now induction on s shows that the system (7) has a unique solution Z̄ with

coordinates in π0^{\frac{1}{q−1}} m_{sep}.

Lemma is proved.

With the above notation and assumptions we have the following corollary.

Corollary. Suppose E, resp. E_{α}, is obtained by joining to K, resp. K_{α}, all coordinates of all solutions of the system of equations (4.1), resp. (4.2), in K_{sep}. Then EK_{α} = E_{α}.

4.2.4. For any finite extension F ⊂ L in K_{sep}, let v(L/F) be the minimal rational number such that the ramification groups Γ_{F}^{(v)} act trivially on L for v > v(L/F).

Proposition 7. With the above notation there is the following inequality

v(E/K) ⩽ e (N + \frac{1}{q − 1}) − 1

Proof. Suppose this inequality does not hold. Then there is a rational number α satisfying the assumptions from the beginning of n.4.2.2 and the inequalities

v(E/K) > α > e (N + \frac{1}{q − 1}) − 1.

Notice that E_{α} = EK_{α} implies that

v(E_{α}/K) = max{v(E/K), v(K_{α}/K)} = v(E/K)

On the other hand, looking at the maximal edge points of Herbrand functions from the identity φ_{E_{α}/K} = φ_{E_{α}/K_{α}} ∘ φ_{K_{α}/K}, we obtain that

v(E_{α}/K) = max {v(K_{α}/K), φ_{K_{α}/K}(v(E_{α}/K_{α}))} =

max \{α, \frac{v(E/K) − α}{q} + α\} < v(E/K)

because v(E/K) = v(E_{α}/K_{α}). Contradiction.

Theorem 5 is proved.

4.3. As it was noticed in n.3.5.1, if e = 1 then killed by [π_{0}] strict O_{0}-modules behave very similarly to group schemes of period p over Witt vectors. For this reason, one can apply directly methods from [Ab2] to prove the following result.

Theorem 6. Suppose H is an 𝔽_{q}[Γ_{K}]-module such that

a) the action of inertia subgroup of Γ_{K} on the semisimple envelope of H is given by characters, which satisfy Serre’s Conjecture, cf. Theorem 4;

b) the ramification subgroups Γ_{K}^{(v)} act trivially on H if v > \frac{1}{q − 1} (i.e. the ramifi-

cation estimate from Theorem 5 holds for H).

Then there is an G = (G, G^{♭}) ∈ FGr′(O_{0})_{O} such that H ≃ G(K_{sep}).

References

[Ab1] V.Abrashkin, Honda systems of group schemes of a period p, Izv. Akad. Nauk SSSR Ser. Mat. 51 /yr 1987, no. 3, 451-484; Engl. transl. in, Math. USSR Izv. 30 (1988), no. 3, 419-453.

[Ab2] V.Abrashkin, Galois modules of period p group schemes over the ring of Witt vectors, Izv. Akad. Sci. SSSR Ser. Mat. 51 (1987), no. 4, 691-736; Engl. transl. in, Math. USSR Izv. 31 (1988), 1-46.

[Ab3] V.Abrashkin, Group schemes over a discrete valuation ring with small ramification, Algebra i Analiz 1 (1989), no. 1, 60-95; Engl. transl. in, Leningrad Math.J. 1 (1990), no. 1, 57-97.

[Ab4] V.Abrashkin, The ramification filtration of the Galois group of a local field. III, Izv. Ross. Akad. Nauk Ser. Mat. 62 (1998), no. 5, 3-48; Engl. transl. in, Izv. Math. 62 (1998), no. 5, 857-900.

[Br] C.Breuil, Groupes p-divisibles, groupes finis et modules filtres, Ann. of Math. 152 (2000), no. 2, 489-549.

[Co] B.Conrad, Finite group schemes over bases with low ramification, Composito Math. 119 (1999), no. 3, 239-320.

[Fa] G.Faltings, Group schemes with strict O-action, Moscow Math. J. 2 (2002), no. 2, 249-279.

[Fo1] J.M.Fontaine, Groupes finis commutatifs sur les vecteurs de Witt, C.R.Acad.Sci. Paris Sér. A-B 280 (1975), A1423-1425.

[Fo2] J.-M.Fontaine, Il n’y a pas de variete abelienne sur ℤ, Inv. Math. 81 (1985), no. 3, 515-538.

[Fo3] J.-M.Fontaine, Schemas propres et lisses sur ℤ, Proc. of Indo-French Conference on Geometry (Bombay, 1989), Hindustan book agency, Delhi, 1993, p. 43-56.

[Fo4] J.-M.Fontaine, Représentations p-adiques des corps locaux.I., The Grothendieck Festschrift, Progr.Math., 87, Birkhauser Boston, Boston, MA, 1990, vol. II, p. 249-309.

[Ra] M.Raynaud, Schemas en groupes de type (p, ˙,p), Bull. Soc. Math. France 102 (1974), 241-280.

Dept.of Maths., Univ. of Durham, Sci. Laboratories, Durham, DH1 3LE, UK

   E-mail address: victor.abrashkin@ dur.ac.uk