Homomorphisms of Barsotti-Tate groups and crystals in positive characteristic

by A.J. de Jong, Princeton University

CONTENTS

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Applications of 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3. Preliminary reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4. Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 5. Splitting an equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 6. Entire versus convergent power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 7. Some crystals over power series rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 8. Equality of kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 9. The theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 10. Appendix : proof of Lemma 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1 Introduction

Let G, H be p-divisible groups over a discrete valuation ring R. Set K equal to the fraction field of R. There is a natural map

(1)  Hom_{R}(G, H) ⟶ Hom_{K}(G_{K}, H_{K}).

In [13, Theorem 4] Tate proved that (1) is a bijection when the characteristic of K is zero.

From now on we assume that the characteristic of K is p > 0. Set S = Spec R and η = Spec K. There is an open immersion j : η → S. Hence there is a natural functor

F -crystals over S ⟶^{j^{∗}} F -crystals over η.

Here, an F -crystal is a nondegenerate F -crystal of [12, 3.1.1]. We recall that any characteristic discrete valuation ring essentially of finite type over a field has a p-basis, see e.g. [10]. A complete discrete valuation ring of characteristic p has a p-basis if its residue field has a finite p-basis.

1.1 Theorem. Assume R has a p-basis. The functor j^{∗} is fully faithful.

1.2 Corollary. (No assumption on R.) The map (1) is bijective.

The theorem implies the corollary by [1] or [2, §1] (there is an immediate reduction to the case where R is complete with algebraically closed residue field, see [13, page 181]). To establish the bijectivity of (1) in the characteristic p case was mentioned as a problem in the introduction of [6, Exposé IX]. The corollary implies a result on extensions of homomorphisms of p-divisible groups over normal base schemes, [13, page 180] and [2, §1].

In [1] it was shown, using a result of [9], that 1.2 holds in case the Newton polygons of G and H are constant. The results of [1] are now well understood, as the main difficulty in [1] was to produce a suitable Dieudonné module theory, which we have due to [3] and [4]. (See [7, 8] for further results.)

The author sees Theorem 1.1 as an indication that (overconvergent) F -crystals over schemes of characteristic p are the p-adic analogue of the lisse ℓ-adic sheaves. The author hopes that the methods of this article may be used to study the bad reduction behavior of overconvergent F -crystals.

In Section 2 we indicate some known applications of 1.2. First, one can deduce a good reduction criterion for abelian varieties in terms of associated p-divisible groups, [6, Exposé IX]. Second, the result that the natural map

Hom(X, Y) ⊗ ℤ_{p} ⟶ Hom(X[p^{∞}], Y [p^{∞}])

for abelian varieties X, Y over a field finitely generated over 𝔽_{p}.

In Section 3 we reduce Theorem 1.1 to Theorem 9.1. The proof of Theorem 9.1 in Sections 4-9 takes up most of this paper. In Section 4 all notations are introduced and fixed till the end of the paper. The idea of the proof of Theorem 9.1 is roughly the following. First one writes any F -module (see 4.4) in a standard form 5.5 over a ring of convergent power series Γ_{2,c}; this is a ring which is not p-adically complete. The important point is that the slopes decrease in 5.5. Going up to this ring preserves enough information so that one can deduce results about slopes from the existence of φ : M → Γ, see 5.7. Then one descends back to Ω = W [[t]] using a linear independence result (Section 8) and Dwork’s trick (Section 6). The final step is in Section 7 where a rank 1 submodule of highest slope is split off.

Acknowledgements. The author thanks P. Berthelot, C. Deninger, B. Gross, N. Katz, B. Mazur, W. Messing, and T. Scanlon for discussions on the problem and its applications. He thanks B. Edixhoven for some pertinent remarks concerning Section 2. He especially thanks U. Zannier, who first proved Theorem 9.1 in the case that M has rank 2; the idea to consider the equations over the ring Γ_{c} of convergent power series is due to U. Zannier.

2 Applications of 1.2

In this section R is a henselian discrete valuation ring of characteristic p > 0. We have K, S, η as above and s ∈ S is the closed point. We will use Corollary 1.2 a lot, sometimes without explicitly referring to it.

Some remarks on p-divisible groups. If G → H is a homomorphism of p-divisible groups over a field, then the image of G → H is a p-divisible group. The height of this image might be called the rank of the homomorphism; it is actually the rank of the induced map on Dieudonné modules. The kernel K of G → H has the following structure: there is a canonical exact sequence 0 → K′ → K → K′′ → 0, where K′ is a p-divisible group and K′′ is a finite group scheme. The p-divisible group K′ is the image of p^{n} on K for sufficiently large n. The following lemma says that the rank of a homomorphism of p-divisible groups over S is constant over S.

2.1 Lemma. With the notations above.

(i) If G is a p-divisible group over S and H ⊂ G is an fppf subsheaf which is a p-divisible group, then G/H is a p-divisible group. (ii) If G → H is a morphism of p-divisible groups over S, then the height of the image of G_{η} → H_{η} is equal to the height of the image of G_{s} → H_{s}. (iii) Let α : G → H be a morphism of p-divisible groups over S and suppose that the kernel of the generic fibre α_{η} of α is a p-divisible group. Then the kernel of α is a p-divisible group and so is the cokernel.

Proof. The proof of (i) is left to the reader. We defer the proofs of (ii) and (iii) to the appendix.

2.2 Definition. Let G_{η} be a p-divisible group over η.

(i) We say that G_{η} has good reduction over S if it extends to a p-divisible group G over S (uniquely by 1.2). (ii) We say that G_{η} has semi-stable reduction over S if there exists a filtration (0) ⊂ G_{η}^{μ} ⊂ G_{η}^{f} ⊂ G_{η} by p-divisible groups such that the following conditions hold:

a) Both G_{η}^{f} and G_{η}/G_{η}^{μ} extend to p-divisible groups G_{1} and G_{2} over S. By 1.2 there is a unique morphism G_{1} → G_{2} extending G_{η}^{f} → G_{η}/G_{η}^{μ}. Note that G_{1} → G_{2} satisfies the conditions of Lemma 2.1 (iii). Thus G^{μ} = Ker(G_{1} → G_{2}) and G^{et} = Coker(G_{1} → G_{2}) are p-divisible groups over S.

b) The sheaf G^{μ}, resp. G^{et} is a multiplicative, resp. étale p-divisible group.

2.3 Remark. The condition on semi-stable reduction of a p-divisible group G_{η} over η implies that G_{η} has ”virtuellement bonne reduction d’echelon 2” [6, Exposé IX 5.12]. The author does not know whether the concepts are equivalent in general. If G_{η} = X_{η}[p^{∞}] for some abelian variety X_{η} over η, then the concepts do coincide. We won’t use this; the proof is left to the reader as an exercise. (Hint: Use 2.5 and consider a finite Galois extension of K over which X_{η} becomes semi-stable, compare the two filtrations and use the Galois action to show that the semi-stable filtration comes from a filtration over K.)

2.4 Lemma.

(i) If G_{η} has semi-stable reduction over S, then (0) ⊂ G_{η}^{μ} ⊂ G_{η}^{f} ⊂ G_{η} can be chosen such that G^{μ} ⊗ κ(s) is the multiplicative part of G_{1} and G^{et} is the étale part of G_{2}. (ii) Suppose both G_{η} and H_{η} are p-divisible groups having semi-stable reduction over S. Let φ_{η} : G_{η} → H_{η} be a morphism. If we choose the filtrations as in (i) then φ_{η} induces morphisms G_{i} → H_{i} fitting into a commutative square with the morphisms G_{1} → G_{2} and H_{1} → H_{2}.

The first application of Corollary 1.2 is a criteriom for good reduction of Abelian varieties in terms of their p-divisible groups. Below we sketch the argument; for more precise information the reader should consult [6, Exposé IX].

2.5 Criterion for good reduction. Let X_{η} be an abelian variety over η with p-divisible group G_{η}. Then X_{η} has good reduction if and only if G_{η} has good reduction. Same for semi-stable reduction.

Proof. The direct implication is well known and follows from the description of the semistable model of X_{η}, the description of the semi-stable model of the dual abelian variety X_{η}^{t}, and the fact that X_{η}^{t}[p^{∞}] is dual to X_{η}[p^{∞}]. See [6, Exposé IX]

Suppose G_{η} has semi-stable reduction. Let K ⊂ K′ be a finite separable Galois extension such that X_{η′} has a semi-abelian model X′ over S′. Let G′ ⊂ X′[p^{∞}] be the union of the finite parts of the quasi-finite group schemes X′[p^{n}]. It is a p-divisible group. If we take the filtration (0) ⊂ G_{η}^{μ} ⊂ G_{η}^{f} ⊂ G_{η} as in Lemma 2.4 (i), then there is a morphism G_{1} ×_{S} S′ → G′. Note that this induces an isomorphism on formal groups (look at étale parts). There is an action of Gal(K′/K) on the scheme X′. The inertia subgroup I ⊂ Gal(K′/K) acts trivially on the special fibre of the formal group Ĝ′ over S′ as Ĝ′ ≅ Ĝ_{1} ×_{S} S′. Hence I acts trivially on X′_{s′}, as X̂′ ≅ Ĝ′. This implies that X_{η} has semi-stable reduction over S: For example one concludes this by looking at the action of inertia on the ℓ-adic Tate module of X_{η} for some prime ℓ not equal to p. (See [6, Exposé IX 3.5].)

Next, assume that G_{η} has good reduction. We already know that X_{η} has a semiabelian model X over S. Let X → X^{Néron} be the morphism of X to the Néron model of X_{η}. Let X[p^{n}]^{f} be the finite part of X[p^{n}]. The cokernel of the inclusion morphism X[p^{n}]^{f} → G[p^{n}] is an étale group scheme. Thus Y_{n} = (X ⊕ G[p^{n}])/X[p^{n}]^{f} is a smooth group scheme over S (we divide out by the anti-diagonal). As G[p^{n}]_{η} = X_{η}[p^{n}], we have a map Y_{n,η} → X_{η} induced by a ⊕ b ↦ a + b. This extends to a homomorphism of group schemes Y_{n} → X^{Néron} by the Néron property. This implies that X^{Néron}[p^{n}] ⊗ κ(s) has order p^{2n dim X_{η}} for all n. It follows that X^{Néron} is an abelian scheme over S.

The second application of Corollary 1.2 is a theorem relating endomorphisms of Abelian varieties to endomorphisms of their p-divisible groups. For the ideas in the proof we refer to [5] and the references therein.

2.6 Theorem. Let F be a field finitely generated over 𝔽_{p}. Let X_{F}, Y_{F} be abelian varieties over F. Then

Hom(X_{F} , Y_{F}) ⊗ ℤ_{p} ≅ Hom(X_{F} [p^{∞}], Y_{F} [p^{∞}])

Proof. It is well known that the map is injective, and identifies the left hand side with a saturated submodule of the right hand side.

There is a standard reduction to the case that the transcendence degree of F over 𝔽_{p} is 1: The case of a finite field is known [14], hence we may assume the transcendence degree is more than 1. Both X and Y have models X_{U} and Y_{U} defined over a variety U with R(U) = F. We can find an irreducible divisor Z ⊂ U such that the specialization mapping Hom(X_{F} , Y_{F}) → Hom(X_{R(Z)}, Y_{R(Z)}) is an isomorphism. (E.g. by looking at the ℓ-adic representations and using Bertini.) On the other hand, Corollary 1.2 insures there is a specialization mapping Hom(X_{F} [p^{∞}], Y_{F} [p^{∞}]) → Hom(X_{R(Z)}[p^{∞}], Y_{R(Z)}[p^{∞}]). Thus if we prove the theorem for R(Z) then the theorem for F follows. By induction we may assume that the transcendence degree of F over 𝔽_{p} is 1.

Let C be the smooth projective curve over 𝔽_{p} defined by F. By Galois descent, it suffices to prove the equality after a finite separable extension of F. Therefore, we may assume that both X_{F} and Y_{F} have semi-abelian models X and Y over C.

We first give the argument in case both X and Y are actually abelian schemes over C. Take γ_{F} ∈ Hom(X_{F} [p^{∞}], Y_{F} [p^{∞}]). By 1.2, we get γ : X[p^{∞}] → Y [p^{∞}] over C with generic fibre equal to γ_{F}. Let

Γ ⊂ X[p^{∞}] ⊕ Y [p^{∞}]

be the graph of γ. Put Z_{n} = (X ⊕ Y)/Γ[p^{n}]. This is an abelian scheme over C; we have an exact sequence of truncated Barsotti-Tate group schemes of level 1 over C as follows

0 → Γ[p] → X[p] ⊕ Y [p] → Z_{n}[p] → Γ[p] → 0.

And hence an exact sequence

(∗)  0 → ω_{Γ} → ω_{Z_{n}} → ω_{X} ⊕ ω_{Y} → ω_{Γ} → 0.

We conclude that det(ω_{Z_{n}}) ≅ det(ω_{X} ⊕ ω_{Y}) independent of n. It is known [15] that this implies there are only a finite number of isomorphism classes of abelian schemes among the Z_{n} (this is explained in [5, §7]). This implies that Γ = Im(γ′ : X[p^{∞}] ⊕ Y [p^{∞}] → X[p^{∞}] ⊕ Y [p^{∞}]) for some γ′ ∈ End(X ⊕ Y) ⊗ ℤ_{p} by known arguments, see the proof of Proposition 1 in [14]. This statement for all γ gives the equality of the theorem.

In the case X and Y are only semi-abelian schemes one has to be a little more careful. We still have Γ_{U} ⊂ X[p^{∞}]_{U} ⊕ Y [p^{∞}]_{U}, where U ⊂ C is the open subscheme over which X and Y are abelian. Let R_{v} be the complete local ring of C at a place v of F not in U. The abelian schemes Z_{n,U} = (X_{U} ⊕ Y_{U})/Γ_{U}[p^{n}] have semi-abelian models Z_{n} over C. The p-divisible group Γ_{U} has semi-stable reduction over R_{v}. Let Γ_{1,v} over R_{v} be as in Lemma 2.4 (i). Remark that there is a morphism Γ_{1,v} → X[p^{∞}]_{R_{v}} ⊕ Y [p^{∞}]_{R_{v}}. Then there is a morphism

(X_{R_{v}} ⊕ Y_{R_{v}})/Γ_{1,v}[p^{n}] ⟶ Z_{n,R_{v}}

and this morphism is étale, as follows from the definitions. Using this one checks that (*) remains true, taking for ω_{Γ} the sheaf ωΓ̂, where Γ̂ is the formal group over C with Γ̂ = Γ̂_{U} over U and Γ̂ = Γ̂_{1,v} over R_{v}.

3 Preliminary reduction

Let R, K, S and η be as in Theorem 1.1. Let R → R′ be an injective local homomorphism of discrete valuation rings with e(R′/R) = 1 and assume that R′ is complete and that the residue field k′ of R′ is algebraically closed. We have assumed R has a p-basis and R′ has a p-basis as it is complete with algebraically closed residue field. Hence R (resp. R′) has a lift (Ω, σ) (resp. (Ω′, σ′)), see [4, Section 1] and [8, 1.2]. We can choose a lift Ω → Ω′ of the map R → R′ [4]. We do not assume that it is compatible with σ and σ′ (although we could choose it that way with some effort). An F -crystal E over Spec R corresponds to a triple (M, F, ∇) over Ω, see [4, Proposition 1.3.3]. Then E_{S′} corresponds to a triple (M ⊗_{Ω} Ω′, F′, ∇′). Here ∇′ is induced from ∇. But since we did not assume Ω → Ω′ to be compatible with our σ’s the F′ is not just deduced from F by linear extension, but also involves ∇ and ∇′. Let us put Γ equal to the p-adic completion of Ω[t^{−1}], where t maps to an uniformizer of R. Similar for Γ′. Then E_{η} corresponds to the triple (M ⊗ Γ, F ⊗ σ, ∇ ⊗ 1 + 1 ⊗ d). Similar for E_{η′}.

We claim that if E_{1}, E_{2} are F -crystals over S, then

Hom_{S}(E_{1}, E_{2}) = Hom_{η}(E_{1,η}, E_{2,η}) ∩ Hom_{S′}(E_{1,S′}, E_{2,S′}).

The intersection is taken in Hom_{η′}(E_{1,η′}, E_{2,η′}). This follows from the remarks above, by looking at the coefficients of the linear maps over Γ and Ω′ in terms of a basis of M_{1} and M_{2} over Ω and the remark that

Ω = Ω′ ∩ Γ.

The intersection takes place in Γ′. Thus it suffices to prove Theorem 1.1 in case R ≅ k[[t]] with k algebraically closed.

By [4, 1.3.3] the category of F -crystals over Spec k[[t]] is equivalent to a full subcategory of the category of (F, θ)-modules (M, F, θ) (see Definitions 4.4 and 4.9) over Ω = W [[t]] (it is unimportant which full subcategory this is). This is true as {t} is a p-basis of the ring k[[t]], hence a connection on an Ω-module M corresponds to a \frac{d}{dt}-derivation on the module M. Thus we have to show that any horizontal linear map of (F, θ)-modules M_{1} and M_{2}

φ : M_{1} ⊗_{Ω} Γ ⟶ M_{2} ⊗_{Ω} Γ

compatible with F_{1} and F_{2} maps M_{1} into M_{2}. Of course φ defines a map

φ′ : (M_{1} ⊗_{Ω} M_{2}^{∗}) ⊗_{Ω} Γ → Γ.

There is a standard way to get a connection θ_{2}^{∗} on M_{2}^{∗} and for some ℓ ∈ ℤ_{≥0} the map p^{ℓ}(F_{2}^{−1})^{∗} defines the structure of an F -module on M_{2}^{∗}. Thus M_{1} ⊗ M_{2}^{∗} has the structure of (F, θ)-module over Ω. Then φ′ is horizontal (using \frac{d}{dt} on Γ) and has the property that φ′(F (m)) = p^{ℓ}σ(φ′(m)) for all m ∈ M_{1} ⊗ M_{2}^{∗}. We see that Theorem 9.1 implies Theorem 1.1.

4 Notations

In this section we introduce notations and give some definitions that will be fixed in the rest of the article. p A prime number. k An algebraically closed field of characteristic p. t An indeterminate. O = k[[t]] The ring of power series over k. K = k((t)) The fraction field of O. K_{1} A perfect closure of K. O_{1} The integral closure of O = k[[t]] in K_{1}. K_{2} An algebraic closure of K_{1}.

O_{2} The integral closure of O_{1} in K_{2}. ν : K_{2}^{∗} → ℚ The valuation on K_{2} with ν(t) = 1. We will also consider ν as a valu-

ation on the fields K and K_{1}. Note that O (resp. O_{i}) is the valuation ring of ν on K (resp. K_{i}).

{s_{n}}_{n∈ℕ} A sequence of elements in K_{2} with s_{nm}^{m} = s_{n} and s_{1} = t. We will also write s_{n} = t^{1/n}, s_{n}^{m} = t^{m/n}, etc. Note that s_{n} ∈ O_{2}. Remark that O ⊂ O_{1} ⊂ O_{2} and that O_{1} = ⋃_{n∈ℕ} O[t^{1/p^{n}}]. Also remark that O_{1} and O_{2} are perfect domains of characteristic p.

W = W (k) The Witt ring of the field k. The reduction map W → k will be denoted

a ↦ ā. A similar notation will be used for other (Witt) rings as well.

σ : W → W Frobenius. We will use σ to denote Frobenius on all rings of Witt

vectors; it will denote a lift of Frobenius on other rings as well.

W_{b} This is the ring W [π]/(π^{b}−p), where b is a positive integer. We will also write π = p^{1/b}. The map σ is extended to W_{b} by setting σ(p^{1/b}) = p^{1/b}. The reduction modulo π is a map W_{b} → k and is denoted a ↦ ā.

L_{0}, L_{0,b} These denote the fraction field of W and of W_{b}. Ω = W [[t]] The ring of power series in t over W. Again we use a ↦ ā to denote

the reduction map Ω → O.

σ : Ω → Ω The unique lift of Frobenius on Ω such that σ(t) = t^{p}. D = (Spf Ω)^{rig} This is the rigid analytic space over L_{0} associated to the formal scheme

Spf Ω, see [8, §7]. It is the open unit disc with coordinate t: D = {t : |t| < 1} and can be characterized as the strong Stein space D over L_{0} such that Γ(D, O^{∘}) ≅ Ω. Associated to σ : Ω → Ω there is a general morphism of analytic spaces σ : D → D over σ : L_{0} → L_{0}, see [8, 7.2.6]. It can be viewed as a morphism D → D ˆ⊗_{σ}L_{0}.

Γ(D, O) The global rigid analytic functions on D. These can be represented as power series f = ∑_{n≥0} a_{n}t^{n}, with a_{n} ∈ L_{0} such that for any ϵ > 0 the function n ↦ v_{p}(a_{n}) − ϵn is bounded. The map σ maps f to σ(f) = ∑_{n≥0}(a_{n})t^{pn}.

Ω_{i} = W (O_{i}) The rings of p-Witt vectors for O_{1} and O_{2}. The map σ : Ω_{i} → Ω_{i} is Frobenius. By functoriality of the Witt vectors there is unique a map Ω_{1} → Ω_{2} compatible with σ and reducing to the inclusion O_{1} ⊂ O_{2} modulo p. This map is an inclusion; we think of Ω_{1} as a subring of Ω_{2}.

t^{1/n} ∈ Ω_{2} Denotes the Teichmüller lift of s_{n} ∈ O_{2}. The elements t^{1/p^{n}} are in Ω_{1}.

The element t^{1/1} is also written t again.

Ω → Ω_{1} The unique homomorphism j of rings, with t ↦ t, compatible with σ and reducing to the inclusion O ⊂ O_{1} modulo p. Such a map exists; a construction can be given as follows. Fix n ∈ ℕ. Any element x ∈ Ω = W [[t]] can be written

x = ∑_{j=0}^{n−1} p^{j} ∑_{i=0}^{p^{n−1−j}−1}(a_{i,j})^{p^{n−1−j}}t^{i}

with a_{i,j} ∈ Ω and we set

j(x) mod p^{n} = ∑_{i,j} p^{j}[ā_{i,j}]^{p^{n−1−j}}t^{i} mod p^{n}.

Here [a] ∈ Ω_{1} for a ∈ O_{1} is the Teichmüller lift of a. Taking the limit for n → ∞ gives the map j. Compare [4, Section 1]. This map is an injection of W -algebras; we think of Ω as a sub W -algebra of Ω_{1}.

Ω_{2,b} Ω_{2}[π]/(π^{b} − p). We will also write π = p^{1/b}. Here b ∈ ℕ. We extend σ to Ω_{2,b} by σ(π) = π. Note that if b divides b′, there is a finite free map Ω_{2,b} → Ω_{2,b′}.

Γ, Γ_{i}, Γ_{2,b} The p-adic completions of the rings Ω[1/t], Ω_{i}[1/t], Ω_{2,b}[1/t]. Note that Γ (resp. Γ_{i}, resp. Γ_{2,b}) is a complete discrete valuation ring (as Γ_{i} is p-adically complete, flat over ℤ_{p} and Γ_{i}/pΓ_{i} = K_{i}) with residue field K (resp. K_{i}, resp. K_{2}) and uniformizer p (resp. p, resp. p^{1/b}). It follows that there are canonical identifications Γ_{1} = W (K_{1}) and Γ_{2} = W (K_{2}). Note that Γ_{2,b} = Γ_{2}[p^{1/b}]. There are reduction maps Γ → K, Γ_{i} → K_{i} (i = 1, 2) and Γ_{2,b} → K_{2}; these are denoted by a ↦ ā.

L_{i} The fraction field of Γ_{i}, where the index i is 1, 2, or 2, b. ν(f, −), ν_{i}(f, −) The “pole order” function of an element f ∈ Γ or f ∈ Γ_{i} (the index i

is 1, 2, or 2, b). For a precise definition see 4.1.

Γ_{c} ⊂ Γ, Γ_{i,c} ⊂ Γ_{i} These will denote the subrings of “convergent” power series in Γ and Γ_{i} (the index i is 1, 2, or 2, b). More precisely, if f ∈ Γ, then by definition f ∈ Γ_{c} if there exist constants C_{1}, C_{2} such that ν(f, n) ≤ C_{1}+nC_{2}. The same definition is used in the case of Γ_{i} using ν_{i}(f, −). It is proved in 4.3 that the maps Γ_{c} → Γ and Γ_{i,c} → Γ_{i} are local morphisms of discrete valuation rings with relative ramification index e = 1 and inducing an isomorphism on residue fields.

v_{p} The p-adic valuation on the discrete valuation rings Γ, Γ_{c}, Γ_{i} and Γ_{i,c}. It is normalized such that v_{p}(p) = 1 and v_{p}(p^{1/b}) = 1/b.

4.1 Let f be an element of Γ. It will be convenient to introduce the function ν(f, −) : ℕ ∪ {0} ⟶ ℤ ∪ {−∞} defined by

ν(f, n) = min{a ∈ ℤ | t^{a}f ∈ Ω + p^{n+1}Γ}.

For convenience we set ν(f, −1) = −∞. We explain this in more down to earth terms. We can write f uniquely in the form f = ∑_{m∈ℤ} a_{m}t^{m}, with a_{m} ∈ W and a_{m} → 0 p-adically when m → −∞. Thus we can also write (but not uniquely)

f = ∑_{n≥0} p^{n}t^{−ν_{n}}ω_{n}

with ω_{n} ∈ Ω. We may take ν_{0} = … = ν_{n−1} = ∞ and ω_{0} = … = ω_{n−1} = 0 if f is divisible by p^{n} in Γ. Our definition of ν(f, n) implies that we can find such a presentation of f with ν_{n} ≤ ν(f, n) for all n. In fact we can write f as

(4.1.1)  f = ∑_{n such that ν(f,n)>ν(f,n−1)} p^{n}t^{−ν(f,n)}ω_{n}

with ω_{n} ∈ Ω^{∗}. We leave the proof of this fact to the reader.

In the same manner we define for f ∈ Γ_{1}, resp. f ∈ Γ_{2}, resp. f ∈ Γ_{2,b} a function

ν_{1}(f, −) :  ℕ ∪ {0}  ⟶ ℤ[1/p] ∪ {−∞}, resp.
ν_{2}(f, −) :  ℕ ∪ {0}  ⟶ ℚ ∪ {−∞}, resp.
ν_{2,b}(f, −) : {0, 1/b, 2/b, …} ⟶ ℚ ∪ {−∞}.

We remark that these behave well under restriction to subrings (e.g. ν_{2}|_{Γ_{1}} = ν_{1}).

4.2 Lemma. Properties of the functions ν(f, −).

(i) The function ν(f, −) is increasing, i.e., we have ν(f, n) ≥ ν(f, n − 1). (ii) If f, g ∈ Γ, then ν(f + g, n) ≤ max{ν(f, n), ν(g, n)} with equality if ν(f, n) ≠ ν(g, n). (iii) If f, g ∈ Γ, then ν(fg, n) ≤ max_{ℓ=0,...,n} ν(f, ℓ) + ν(g, n − ℓ) with equality if there is exactly one ℓ for which the expression ν(f, ℓ) + ν(g, n − ℓ) is maximal. (iv) If f ∈ Γ and m ∈ ℕ, then ν(p^{m}f, n) = ν(f, n − m).

(v) If f ∈ Γ, then ν(σ(f), n) = pν(f, n). Similar properties hold for the functions ν_{i}(−, −), where the index i is 1, 2, or i = 2, b.

Proof. Left to the reader.

4.3 The rings Γ_{c}, Γ_{1,c}, Γ_{2,c} and Γ_{2,b,c} are discrete valuation rings whose p-adic completions are Γ, Γ_{1}, Γ_{2} and Γ_{2,b}. Let us prove this in the case of Γ_{c}. It is easy to show that Γ_{c}/pΓ_{c} → Γ/pΓ = K is an isomorphism. So we just need to check that pΓ_{c} is the unique maximal ideal of Γ_{c}. To do this we will show that any f ∈ Γ_{c} with f ∉ pΓ_{c} is a unit in Γ_{c}.

We have f ∉ pΓ, hence f ∈ Γ^{∗}, i.e., there is an element g ∈ Γ such that fg = 1. In particular, ν(fg, n) = 0 for all n ≥ 0. By Lemma 4.2 (iii) we get for all n ∈ ℕ the following inequality

ν(g, n) + ν(f, 0) ≤ max{0; ν(f, ℓ) + ν(g; n − ℓ), n ≥ ℓ ≥ 1}.

Assume that C > 0 is a constant such that ν(f, n) ≤ nC for all n ≥ 1, and such that |ν(f, 0)| ≤ C, ν(g, 0) ≤ C and ν(g, 1) ≤ C. Such a constant exists as f ∈ Γ_{c}. We prove by induction on n the statement H_{n} : ν(g, ℓ) ≤ 2ℓC, ∀ℓ, 1 ≤ ℓ ≤ n. Clearly this implies that g ∈ Γ_{c}. The assertion H_{1} is true by our choice of C. We assume H_{n−1} and prove H_{n} for n ≥ 2. The inequalities above and H_{n−1} imply

\begin{matrix}ν(g, n) ≤ & − ν(f, 0) + max\{0; nC + C; ℓC + 2(n − ℓ)C, n − 1 ≥ ℓ ≥ 1\} \\ ≤ & |ν(f, 0)| + max\{0; (n + 1)C; (2n − ℓ)C, n − 1 ≥ ℓ ≥ 1\} \\ ≤ & 2nC \end{matrix}

Thus we get H_{n}.

4.4 Definition. Let (R, σ) be an integral domain with an endomorphism σ : R → R. An F -module over R is a pair (M, F), where M is a finitely generated torsion free R-module and F : M → M is a σ-linear map such that the kernel and cokernel of the R-linear map

M ⊗_{R,σ} R ⟶ M

are annihilated by some power of p.

Our use of this definition will be restricted to cases where p is not zero in R. In case R is a perfect field of characteristic p and σ is the Frobenius of R, the definition above does not correspond to what is usually called an F -crystal over Spec R. (Such an F -crystal over Spec R corresponds rather to an F -module over W (R).)

There is an obvious notion of morphisms of F -modules. A sub F -module N of (M, F) is a submodule N ⊂ M such that F (N) ⊂ N and such that the pair (N, F |_{N}) is itself an F -module. We say that N is a saturated sub F -module if M/N is torsion free. In this case (M/N, F |_{M/N}) is an F -module. An isogeny of F -modules is a morphism (M, F) → (M′, F′) such that both the kernel and cokernel of the map M → M′ are killed by a power of p.

Assume that σ is flat. (This condition will be satisfied in the case of the rings we will consider in this article.) In this case M ⊗_{R,σ} R is torsion free for any finitely generated R-module M which is torsion free. (Use that there is an injection M ↪ R^{n}.) Thus given a pair (M, F), with M torsion free and F σ-linear, to see that the pair defines an F -module it suffices to check that the cokernel of M ⊗_{R,σ} R → M is killed by some power of p. Any finitely generated sub module N ⊂ M such that F (N) ⊂ N and all torsion of M/N is p-power torsion will be a sub F -module. In particular any finitely generated saturated sub module N of M such that F (N) ⊂ N will be a sub F -module.

If (R, σ) → (R′, σ′) is a morphism of integral domains with endomorphisms then there is a base change functor (M, F) ↦ (M ⊗_{R} R′/T, F ⊗ σ′). Here T ⊂ M ⊗_{R} R′ is the R′-torsion submodule.

4.5 Slopes

Let R be a complete discrete valuation ring of mixed characteristic with algebraically closed residue field κ of characteristic p. Let L be the quotient field of R. Let σ : R → R be an automorphism inducing the Frobenius map x ↦ x^{p} on κ. Let (V, F) be an F -module over L. (Also called an isocrystal over L in the literature.) Note that V is a finite dimensional L-vector space and that F is bijective.

There exists a basis e_{1}, …, e_{r} of V over L and rational numbers s_{1}, …, s_{r} ∈ ℚ such that F^{n}(e_{i}) = p^{ns_{i}}e_{i} for sufficiently divisible n ∈ ℕ (with ns_{i} ∈ ℤ). See [9]. The rational numbers s_{1}, …, s_{r} (listed with multiplicities) are unique and are called the slopes of the F -module (V, F). The slope-decomposition

V = ⨁_{s∈ℚ} V^{s} with V^{s} = ⟨e_{i}; s_{i} = s⟩

can be defined intrinsically (interms of V, F and the topology on V). We have

⨁_{s>s_{0}} V^{s} = {v ∈ V | lim_{n→∞} p^{−ns_{0}}F^{n}(v) = 0}

and

⨁_{s<s_{0}} V^{s} = {v ∈ V | lim_{n→∞} p^{ns_{0}}F^{−n}(v) = 0}.

Two remarks on slopes. If there exists an R-lattice M ⊂ V with F (M) ⊂ M then all the slopes of V are ≥ 0. ( This is clear from the fact that lim p^{nϵ}F^{n}(m) = 0 for any m ∈ M and any ϵ > 0. ) Suppose {v_{1}, …, v_{n}} is any basis of V, and F (v_{i}) = ∑ a_{ij}v_{j}. Then one has s_{1} + … + s_{n} = v_{p}(det(a_{ij})). See [9].

Let α : R → R be an automorphism of R commuting with σ. It induces a continuous automorphism α of L. If V is a vector space over L we write V_{α} = V ⊗_{L,α} L. If (V, F) is an isocrystal over L, then (V_{α}, F ⊗ σ) is an isocrystal over L. It follows from the above that the slope decomposition V = ⨁ V^{s} of (V, F) determines the slope decomposition of (V_{α}, F ⊗ σ). The slope s submodule of V_{α} is equal to V^{s} ⊗_{L,α} L. In particular the set of slopes with multiplicities of V_{α} is equal to that of V.

4.6 Suppose that (M, F) is an F -module over one of the rings R = Ω, Ω_{i}, Γ, Γ_{c}, Γ_{i}, Γ_{i,c}, L_{i} (the index i is 1, 2, or 2, b). Then we can look at the F -module (M ⊗_{R} L_{2,b}, F ⊗ σ) over L_{2,b}. By the above it has slopes.

4.7 Definition. The slopes s_{1} ≤ … ≤ s_{r} of a rank r F -module (M, F) over one of the rings R listed above are the slopes of the isocrystal M ⊗ L_{2,b} over L_{2,b} as defined in 4.5.

We note that (after making b more divisible) M ⊗ L_{2,b} has a basis e_{1}, …, e_{r} such that F (e_{i}) = p^{s_{i}}e_{i}. We say that (M, F) is isoclinic of slope s if all the slopes s_{i} are equal to s.

We remark that if R = Ω, Ω_{i} then there is another F -module associated to M over R, namely the F -module ((M ⊗_{R} W_{b})/torsion, F ⊗ σ) over W_{b}. This gives rise to an isocrystal over L_{0,b} with slopes s′_{i}. The slopes s_{i} are in general different from the slopes s′_{i}. A fundamental result, see [9], is that the Newton polygon of the slopes s′_{i} lies above the Newton polygon defined by the slopes s_{i}. In particular, if M is isoclinic of slope s, then all s′_{i} are equal to s as well. Although we do not use this fact it is one of the ideas behind the arguments in Section 6.

4.8 In case of the ring R is equal to Ω, Γ, Γ_{c} or Γ(D, O_{D}) we have the derivation \frac{d}{dt} : R → R. In any of these rings an element f has a unique expansion f = ∑_{n∈ℤ} a_{n}t^{n}, with a_{n} ∈ W or L_{0}. We put \frac{d}{dt} f = ∑_{n} na_{n}t^{n−1}, which again lies in R. The map \frac{d}{dt} : R → R is a derivation.

4.9 Definition. Let R be as above. An (F, θ)-module over R is a triple (M, F, θ), where (M, F) is an F -module over R and θ : M → M is an additive map such that θ(fm) = fθ(m) + \frac{d}{dt}(f)m and pt^{p−1}F (θ(m)) = θ(F (m)) for all m ∈ M and f ∈ R.

Note that if we take M = R, F = p^{ℓ}σ and θ = \frac{d}{dt}, then (M, F, θ) is an (F, θ)-module.

5 Splitting an equation

We will work in the noncommutative polynomial algebra Γ_{2,b,c}[F], where F γ = σ(γ)F for all γ ∈ Γ_{2,b,c}. It is a subring of the ring Γ_{2,b}[F].

5.1 Proposition. Let F^{n} + a_{1}F^{n−1} + … + a_{n} ∈ Γ_{2,b,c}[F] be a monic polynomial in F. There exists a b′ ∈ ℕ, b|b′ and elements λ_{1}, …, λ_{n} ∈ Γ_{2,b′,c} such that we have

(i) F^{n} + a_{1}F^{n−1} + … + a_{n} = (F − λ_{1})(F − λ_{2}) ⋯ (F − λ_{n}). (ii) v_{p}(λ_{1}) ≥ v_{p}(λ_{2}) ≥ … ≥ v_{p}(λ_{n}).

Proof. We will prove this by induction on n; the case n = 1 is trivial. We will try to solve

F^{n} + a_{1}F^{n−1} + … + a_{n} = (F^{n−1} + b_{1}F^{n−2} + … + b_{n−1})(F − λ)

with b_{i}, λ ∈ Γ_{2,b,c}. This equation is equivalent to the set of equations:

\begin{matrix}b_{1} = & a_{1} + σ^{n−1}(λ) \\ b_{2} = & a_{2} + σ^{n−2}(λ)b_{1} \\ ⋯ & ⋯ \\ b_{n−1} = & a_{n−1} + σ(λ)b_{n−2} \\ 0 = & a_{n} + λb_{n−1} \end{matrix}

If we solve b_{1}, …, b_{n−1} in terms of a_{1}, …, a_{n−1}, λ from the first n − 1 equations then the last equation becomes

(1) 0 = a_{n} + λa_{n−1} + λσ(λ)a_{n−2} + … + λσ(λ) ⋯ σ^{n−2}(λ)a_{1} + λσ(λ) ⋯ σ^{n−1}(λ).

Put

α = min_{i=1,...,n} \frac{v_{p}(a_{i})}{i}.

We may assume that α = a/b for some a ∈ ℤ_{≥0} if we make b more divisible. We are going to solve (1) for some λ ∈ Γ_{2,b,c} of the form λ = p^{α}u with u ∈ Γ_{2,b,c}^{∗}. If we can do this then the proposition follows. Indeed, if we have such a solution λ, then b_{1} = a_{1} +σ^{n−1}(λ) will be an element of Γ_{2,b,c} with v_{p}(b_{1}) ≥ α. And b_{2} = a_{2} +σ^{n−2}(λ)b_{1} implies b_{2} ∈ Γ_{2,b,c} has v_{p}(b_{2}) ≥ 2α, etc. Continuing we get v_{p}(b_{i}) ≥ iα, so that the invariant min v_{p}(b_{i})/i ≥ α has gone up. By induction we get a splitting F^{n−1} + b_{1}F^{n−2} + … + b_{n−1} = (F − λ_{1}) ⋯ (F − λ_{n−1}), with v_{p}(λ_{n−1}) = min v_{p}(b_{i})/i and we are done.

We re-write (1), by setting λ = p^{α}u and dividing by p^{nα}:

(2)  0 = p^{−nα}a_{n} + p^{−(n−1)α}a_{n−1}u + … + uσ(u) ⋯ σ^{n−1}(u).

Note that all the elements a′_{i} = p^{−iα}a_{i} ∈ Γ_{2,b,c} and for some i we have a′_{i} ∈ Γ_{2,b,c}^{∗}. Recall that a ↦ ā denotes the map Γ_{2,b,c} ⊂ Γ_{2,b} → K_{2}. The equation (2) reduces to the equation

0 = ā′_{n} + … + ā′_{i}ū^{1+p+...+p^{n−i−1}} + … + ū^{1+p+...+p^{n−1}}

in K_{2}. This has a non zero solution ū ∈ K_{2}, as it is a polynomial equation with at least two nonzero coefficients and K_{2} is algebraically closed. (We remark for later reference that if v_{p}(a_{n}) = nα and n ≥ 2, then there are at least two distinct nonzero solutions.)

We will first show that (2) has a solution u ∈ Γ_{2,b} which is congruent to the solution ū we found above; after this we will prove that actually u ∈ Γ_{2,b,c}. (Then necessarily u ∈ Γ_{2,b,c}^{∗}.) Suppose that u_{r} ∈ Γ_{2,b}, r ≥ 1 is a solution of (2) modulo p^{r/b} and that ū_{r} = ū ≠ 0. In order that u_{r+1} = u_{r} + p^{r/b}δ_{r} is a solution of (2) modulo p^{(r+1)/b}, we have to solve a polynomial equation in terms of δ̄_{r} with leading term

ū^{1+...+p^{n−2}}(δ̄_{r})^{p^{n−1}}.

As K_{2} is algebraically closed, we can solve this equation and we get u_{r+1}. The limit of the elements u_{r} will be an element u ∈ Γ_{2,b}^{∗} solving (2). We are going to prove such a solution lies in Γ_{2,b,c}. (If n = 2 and v_{p}(a_{2}) = 2α there are at least 2 solutions, see previous remark.)

Let us write ν̃(f, m) = ν_{2,b}(f, m/b) for f ∈ Γ_{2,b} and m ∈ ℤ_{≥0}. We choose a constant C > 0 such that ν(a′_{i}, m) ≤ mC for all m ≥ 1, all i and such that C is much larger then |ν̃(a′_{i}, 0)| and |ν̃(u, 0)|. (A factor p^{n+1} will do.) This is possible as all a′_{i} are elements of Γ_{2,b,c}. We are going to prove by induction on m the statement H_{m} : ν̃(u, ℓ) ≤ ℓC, 1 ≤ ℓ ≤ m. The assertion H_{0} is empty. We assume H_{m−1} and m ≥ 1 and prove H_{m}. By Lemma 4.2 we have

(3)  ν̃(uσ(u) ⋯ σ^{n−1}(u), m) ≤ max_{i_{1},...,i_{n}} ν̃(u, i_{1}) + pν̃(u, i_{2}) + … + p^{n−1}ν̃(u, i_{n})

where the maximum is taken over all n-tuples (i_{1}, …, i_{n}) with i_{j} ≥ 0 and ∑ i_{j} = m. One of the terms on the right hand side is

(4)  (1 + … + p^{n−2})ν̃(u, 0) + p^{n−1}ν̃(u, m).

It occurs if (i_{1}, …, i_{n}) = (0, …, 0, m). If this term does not strictly exceed all others, then there exists an n-tuple (i_{1}, …, i_{n}) with i_{j} ≥ 0 and ∑ i_{j} = m and i_{n} < m such that

(5) (1 + … + p^{n−2})ν̃(u, 0) + p^{n−1}ν̃(u, m) ≤ ν̃(u, i_{1}) + pν̃(u, i_{2}) + … + p^{n−1}ν̃(u, i_{n}).

The case that i_{j} = m for some j < n in (5), which implies i_{j′} = 0 for j′ ≠ j, leads to the inequality

(p^{n−1} − p^{j})ν̃(u, m) ≤ −p^{j}ν̃(u, 0).

By our choice of C and as m ≥ 1, we conclude ν̃(u, m) ≤ mC in this case. Thus we may assume i_{j} < m for all j in (5). This gives using H_{m−1} the following inequality:

\begin{matrix}ν̃(u, i_{1}) + pν̃(u, i_{2}) + … + p^{n−1}ν̃(u, i_{n}) & ≤ ∑_{i_{j}=0} p^{j−1}ν̃(u, 0) + ∑_{i_{j}≠0} p^{j−1}i_{j}C \\ ≤ & (1 + … p^{n−1})|ν̃(u, 0)| + p^{n−2}C + p^{n−1}(m − 1)C. \end{matrix}

The last inequality holds because ∑ i_{j} = m and i_{n} < m. Combining this with (5) gives

ν(u, m) ≤ \frac{2(1 + … p^{n−1})|ν̃(u, 0)|}{p^{n−1}} + (m − 1)C + p^{−1}C ≤ mC,

where the last inequality uses C >> |ν̃(u, 0)| (e.g. C ≥ p^{n+1}|ν̃(u, 0)|). This proves H_{m} in case the term (4) does not dominate in the right hand side of (3).

Therefore we may assume that (4) does dominate in the right hand side of (3) and hence we have that ν̃(uσ(u) ⋯ σ^{n−1}(u), m) equals (4) by Lemma 4.2 (iii). Hence, by Lemma 4.2 (ii), we see that (4) is dominated by the maximum of the values of ν̃(−, m) evaluated on the other terms that appear in (2). We have for ℓ ≥ 1 the following bound

ν̃(a′_{ℓ}uσ(u) ⋯ σ^{n−ℓ−1}(u), m) ≤ max_{i_{0},i_{1},...,i_{n−ℓ}} ν̃(a′_{ℓ}, i_{0}) + ν̃(u, i_{1}) + … + p^{n−ℓ−1}ν̃(u, i_{n−ℓ}),

where the maximum is taken over all (n − ℓ + 1)-tuples (i_{0}, …, i_{n−ℓ}), with i_{j} ≥ 0 and ∑ i_{j} = m. Again the assumption that a term with i_{j} = m for some j ∈ {1, …, n − ℓ} is biggest leads to an inequality of the form ν̃(u, m) ≤ C_{2}, where C_{2} is a constant depending only on ν̃(a′_{i}, 0), p, n and ν̃(u, 0). Hence we get H_{m} as m ≥ 1 and C was chosen large enough. For the other terms one gets a majoration by

i_{0}C + p^{n−ℓ−1}(m − i_{0})C + |ν̃(a′_{ℓ}, 0)| + (1 + … + p^{n−ℓ−1})|ν̃(u, 0)| ≤ p^{n−ℓ−1}mC + C_{3}.

The fact that (4) is bounded by this suffices to imply ν̃(u, m) ≤ mC, as ℓ ≥ 1 and C was chosen large enough.

5.2 Corollary. Let (M, F) be an F -module over Γ_{2,b,c}. There exists a b′ ∈ ℕ, b|b′ such that M ⊗ Γ_{2,b′,c} has a filtration 0 ⊂ N_{1} ⊂ N_{2} ⊂ … ⊂ N_{r} = M ⊗ Γ_{2,b′,c} by saturated sub F -modules N_{i} such that rank N_{i} = i.

Proof. Take a non zero element m ∈ M. There is a monic polynomial P (F) = F^{n} + … + a_{n} ∈ Γ_{2,b,c}[F] such that P (F)m = 0 in M. Say P has minimal degree. After replacing b by b′ as in the proposition we get P (F) = (F − λ_{1}) ⋯ (F − λ_{n}). The submodule N′_{1} of M generated by (F − λ_{2}) ⋯ (F − λ_{n})m has rank 1. We have F (N′_{1}) ⊂ N′_{1} as P (F)m = 0. The same hold for its saturation N_{1} = {x ∈ M | p^{a}x ∈ N′_{1} some a}. Thus N_{1} is a sub F - module and M/N_{1} is an F -module with stricly smaller rank than M. We win by applying induction (on the rank of M) to M/N_{1}.

5.3 Corollary. Let 0 → N_{1} → M → N_{2} → 0 be a short exact sequence of F -modules over Γ_{2,b,c}. Assume N_{i} has rank 1 and let s_{i} be its slope 4.5. If s_{1} ≤ s_{2}, then there is a rank 1 sub F -module N′ ⊂ M such that N′ → N_{2} is an isogeny.

Proof. Choose a generator n_{i} of N_{i}. We have F (n_{i}) = μ_{i}n_{i} for some μ_{i} ∈ Γ_{2,b,c}. Note that s_{i} = v_{p}(μ_{i}) by the remarks in 4.5, hence v_{p}(μ_{1}) ≤ v_{p}(μ_{2}) by assumption. Choose ñ_{2} ∈ M lifting n_{2}. We have

(F − μ_{2})ñ_{2} = μn_{1}

for some μ ∈ Γ_{2,b,c}. If μ = 0, then the extension is split; if not then we have v_{p}(σ(μ)μ^{−1}) = 0. (This computation takes place in L_{2,b,c}; note that v_{p} ∘σ = v_{p}.) Thus we have σ(μ)μ^{−1} ∈ Γ_{2,b,c}^{∗}. Since (F − σ(μ)μ^{−1}μ_{1})(μn_{1}) = 0, we see that

\begin{matrix}0 = & (F − σ(μ)μ^{−1}μ_{1})(F − μ_{2})ñ_{2} \\ = & (F^{2} − (σ(μ)μ^{−1}μ_{1} + σ(μ_{2}))F + σ(μ)μ^{−1}μ_{1}μ_{2})(ñ_{2}). \end{matrix}

By the proposition we get a splitting of the degree 2 polynomial as (F − λ_{1})(F − λ_{2}), with v_{p}(λ_{1}) ≥ v_{p}(λ_{2}). Note that we do not need to enlarge b, as the invariant α used in the proof of Proposition 5.1 equals v_{p}(μ_{1}). If s_{1} < s_{2} then v_{p}(λ_{2}) ≠ v_{p}(μ_{2}), since λ_{1}λ_{2} = σ(μ)μ^{−1}μ_{1}μ_{2}. In case v_{p}(μ_{1}) = v_{p}(μ_{2}), we also do not need to enlarge b and v_{p}(σ(μ)μ^{−1}μ_{1}μ_{2}) = 2α so that we may choose λ_{2} ≠ μ_{2} according to the remarks made on the number of solutions in the proof of the proposition. The element (F − λ_{2})ñ_{2} will generate the rank 1 sub F -module N′ of M we are looking for: we have N′ ∩ N_{1} = (0) as λ_{2} ≠ μ_{2}.

5.4 Lemma. Let N be a rank 1 F -module over Γ_{2,b,c}. There is a generator n ∈ N such that F (n) = p^{s}n, where s = a/b is the slope of N.

Proof. Take any generator n of N. Then F (n) = p^{s}ηn for some unit η ∈ Γ_{2,b,c}^{∗} and some s ∈ 1/bℤ. We have to solve σ(ϵ) = ηϵ with ϵ ∈ Γ_{2,b,c} a unit. Arguing as in the proof of Proposition 5.1 a solution ϵ ∈ Γ_{2,b}^{∗} is found. A consideration using the functions ν_{2,b}(ϵ, −) and ν_{2,b}(η, −) shows that ϵ is in Γ_{2,b,c} (use Lemma 4.2 (ii) and (iii)).

5.5 Proposition. Let (M, F) be an F -module over Γ_{2,b,c}.

(i) There exists a unique filtration by saturated sub F -modules 0 ⊂ M_{1} ⊂ M_{2} ⊂ … ⊂ M_{a} = M such that M_{i}/M_{i−1} is an isoclinic F -module of slope s_{i} and s_{1} > s_{2} > … > s_{a}. (ii) If M is isoclinic of slope s, then for some b′ ∈ ℕ, b|b′ the F -module M ⊗ Γ_{2,b′,c} over Γ_{2,b′,c} is isogenous to a direct sum of copies of the rank 1 module (Γ_{2,b′,c} · e, F), with F (e) = p^{s}e.

Proof. Note that if b divides b′, then Γ_{2,b′,c} = Γ_{2,b,c} ⊕ πΓ_{2,b,c} ⊕ … ⊕ π^{r−1}Γ_{2,b,c} with r = b′/b and π = p^{1/b′} as F -modules over Γ_{2,b,c} (with F = σ since σ(π) = π). Suppose M_{1} ⊂ M ⊗_{Γ_{2,b,c}} Γ_{2,b′,c} is a sub F -module. We get Γ_{2,b,c}-linear F -compatible maps

M_{1} ⟶ M ⊗_{Γ_{2,b,c}} Γ_{2,b′,c} = ⨁_{i=0}^{r−1} π^{i}M⟶^{pr_{i}}M.

Also, if M_{1} has rank r_{1} over Γ_{2,b′,c}, then M_{1} has rank rr_{1} over Γ_{2,b,c}, so at least one of these composite maps has rank at least r_{1}. Furthermore, if the slopes of the F -module M_{1} over Γ_{2,b′,c} are s_{1}, …, s_{r_{1}}, then the slopes of M_{1} considered as an F -module over Γ_{2,b,c} are s_{1}, …, s_{1}, s_{2}, …, s_{2}, …, s_{r_{1}}, …, s_{r_{1}} (each s_{i} is repeated r times). Thus if M_{1} is a maximal isoclinic Γ_{2,b′,c}-subspace of M ⊗_{Γ_{2,b,c}} Γ_{2,b′,c} with slope s_{1} and rank r_{1}, we get a Γ_{2,b,c}-subspace of M of slope s_{1} and of rank at least r_{1}. We conclude by uniqueness of slope decomposition that M_{1} is defined over Γ_{2,b,c}. Arguing by induction we see that a slope filtration over Γ_{2,b′,c} descends to a slope filtration over Γ_{2,b,c}.

Therefore we may assume b is sufficiently divisible and we get a filtration of M as in Corollary 5.2. By Corollary 5.3 we may assume the filtration 0 ⊂ N_{1} ⊂ N_{2} ⊂ … ⊂ N_{r} = M is such that the slopes of N_{i}/N_{i−1} decrease when i increases. This proves (i). The second statement is proved in the same manner using Lemma 5.4.

5.6 Remark. In fact, if (M, F) over Γ_{2,b,c} is isoclinic of slope s = a/b′, gcd(a, b′) = 1 then it is isogenous over Γ_{2,b,c} to a direct sum of copies of the module M(s) described below. Let r = b′/gcd(b, b′) and let M(s) be the free Γ_{2,b,c} module with basis e_{1}, …, e_{r}. Put F (e_{i}) = e_{i+1}, i < r and F (e_{r}) = p^{rs}e_{1}. Note that rs ∈ (1/b)ℤ so that this makes sense. We will not use this result.

5.7 Corollary. Let (M, F) be an F -module over Γ_{2,b,c}. Say M has a slope filtration 0 ⊂ M_{1} ⊂ … ⊂ M_{a} = M with slopes s_{1} > s_{2} > … > s_{a} as in the proposition. Let

φ : M ⟶ Γ_{2,b}

be Γ_{2,b,c}-linear and such that for some ℓ ∈ (1/b)ℤ, ℓ ≥ 0 we have φ(F (m)) = p^{ℓ}σ(φ(m)) for all m ∈ M. Then (i) The kernel of φ contains M_{i} whenever s_{i} > ℓ. (ii) If φ is injective and M ≠ (0), then s_{1} = ℓ, rank M_{1} = 1 and φ(M_{1}) ⊂ Γ_{2,b,c} ⊂ Γ_{2,b}. (iii) If ℓ > s_{1}, then φ = 0.

Proof. Since Ker(φ) is a saturated submodule of M and stable under F, we may replace M by M/Ker(φ) and it suffices to prove (ii) for this module. Thus we assume that φ is injective. We are allowed to make b more divisible in proving (ii), as the extension Γ_{2,b,c} ⊂ Γ_{2,b′,c} is finite free and Γ_{2,b′} = Γ_{2,b} ⊗_{Γ_{2,b,c}} Γ_{2,b′,c}. Hence we may assume by Proposition 5.5 that there is an isogeny

N_{1} ⊕ … ⊕ N_{r} ⟶ M_{1}

with rank N_{i} = 1. In fact we can find nonzero n_{i} ∈ N_{i} such that F (n_{i}) = p^{s_{1}}n_{i} for each i. Thus φ(n_{i}) = f_{i} ∈ Γ_{2,b} is an element with

p^{ℓ}σ(f_{i}) = p^{s_{1}}f_{i}.

This relation implies ℓ = s_{1} as v_{p}(σ(f_{i})) = v_{p}(f_{i}). Note that f_{i} ≠ 0 as φ is injective. Thus we have σ(f_{i}) = f_{i}. We apply ν_{2,b}(−, m) and we get

pν_{2,b}(f_{i}, m) = ν_{2,b}(f_{i}, m),

whence ν_{2,b}(f_{i}, m) ∈ {−∞, 0} for all m. Thus f_{i} ∈ Γ_{2,b,c}. This means that f_{1}, …, f_{r} are linearly dependent over Γ_{2,b,c}. As we assumed that φ is injective it follows that r = 1.

5.8 Proposition. Suppose (M, F) is an F -module over Γ_{1,c}. The slope filtration of M ⊗_{Γ_{1,c}} Γ_{2,c} given by Proposition 5.5 is defined over Γ_{1,c}. More precisely, there exists a unique filtration by saturated sub F -modules

0 ⊂ M_{1} ⊂ … ⊂ M_{a} = M

with M_{i}/M_{i−1} isoclinic of slope s_{i} and s_{1} > s_{2} > … > s_{a}.

Proof. Let G = Gal(K_{2}/K_{1}). For any τ ∈ G we get a continuous automorphism W (τ) : W (K_{2}) → W (K_{2}) by definition of the ring of Witt vectors. Note that τ commutes with Frobenius σ. The automorphism τ also preserves the subring Γ_{2,c} ⊂ Γ_{2} = W (K_{2}). Indeed, it preserves the functions ν(f, −) as τ(t^{1/n}) = ζt^{1/n} for some root of unity ζ ∈ K_{2} and ζ ∈ O_{2}^{∗}. By the remarks made in the Section 4.5 we see that the map

M ⊗_{Γ_{1,c}} Γ_{2,c} ⟶^{id_{M} ⊗W (τ)}M ⊗_{Γ_{1,c}} Γ_{2,c}

preserves the filtration by slopes 0 ⊂ M′_{1} ⊂ … ⊂ M′_{a} = M ⊗ Γ_{2,c} given by Proposition 5.5.

Note that M is a free Γ_{1,c}-module of finite rank. Let F be the partial flag scheme of M over Spec Γ_{1,c} parametrizing flags 0 ⊂ F_{1} ⊂ … ⊂ F_{a} = M in M with rank F_{i} = rank M′_{i}. A morphism of a scheme S over Γ_{1,c} into F is given by a filtration of the sheaf O_{S} ⊗_{Γ_{1,c}} M. Then F is a smooth projective scheme over Spec Γ_{1,c}. The filtration M′_{∗} defines a morphism x : Spec Γ_{2,c} → F over Spec Γ_{1,c}. Take any affine open U ⊂ F such that Im(x) ⊂ U. Take any element f ∈ Γ(U, O_{U}). The fact proven above that the filtration M′_{∗} is stable for all W (τ), τ ∈ G means that the element x^{∗}(f) ∈ Γ_{2,c} is fixed by all automorphisms W (τ) of Γ_{2,c}. Then it is easy to see that x^{∗}(f) ∈ Γ_{1,c}. This means that x and hence M′_{∗} are defined over Γ_{1,c}.

6 Entire versus convergent power series

Let (M, F, θ) be an (F, θ)-module over Ω = W [[t]].

6.1 Lemma. There is an isogeny M → M′ to a free (F, θ)-module (M′, F, θ) over Ω (i.e., M′ is a finite free Ω-module).

Proof. The dual of a finitely generated Ω-module W is finite free, since Ω is a regular local ring of dimension 2. Thus we take for M′ the double dual of M:

M′ = Hom_{Ω}( Hom_{Ω}(M, Ω) , Ω ).

The evaluation map ev : M → M′ has a finite length cokernel. If m′ ∈ M′, then t^{n}m′ ∈ M and p^{n}m′ ∈ M for some large n ∈ ℕ. Define F (m′) to be the element in M′ such that p^{n}F (m′) = F (p^{n}m′) and t^{pn}F (m′) = F (t^{n}m′). Such an element F (m′) ∈ M′ exists as t^{pn}F (p^{n}m′) = p^{n}F (t^{n}m′). Clearly F (m′) is independent of the choice of n.

To define θ, we take such an n as above. Then we define θ(m′) ∈ M′ as the element such that t^{n}θ(m′) = −nt^{n−1}m′ + θ(t^{n}m′) and p^{n}θ(m′) = θ(p^{n}m′). Existence is proved as above. It is readily proved that F and θ on M′ have the desired properties.

6.2 Lemma. Classification of rank 1 modules over Ω.

(i) Let (M, F) be a rank 1 F -module over Ω. Then M is isogenous to (Ω, p^{ℓ}σ) for some ℓ ∈ ℤ_{≥0}. If M is free then M is isomorphic to (Ω, p^{ℓ}σ). (ii) Let (M, F, θ) be a rank 1 (F, θ)-module over Ω. Then M is isogenous to (Ω, p^{ℓ}σ, \frac{d}{dt}) for some ℓ ∈ ℤ_{≥0}. If M is free of rank one then M is isomorphic to (Ω, p^{ℓ}σ, \frac{d}{dt}).

Proof. Proof of (i). We may assume that M is free by Lemma 6.1 (without θ). Let m ∈ M be a basis element. We have F (m) = λm and we have Ω/λΩ killed by some power of p (axiom of F -modules). Since Ω is a UFD in which p is irreducible, λ = p^{ℓ}η for some unit η ∈ Ω^{∗} and some ℓ ∈ ℤ_{≥0}. If we replace m by ϵm, with ϵ ∈ Ω^{∗}, then η is replaced by ησ(ϵ)ϵ^{−1}. Thus we have to solve the equation ησ(ϵ)ϵ^{−1} = 1 for a unit ϵ ∈ Ω^{∗}. It is well-known (classification of rank 1 F -modules over W) that one can solve this equation in W^{∗}. Hence we may assume we have ϵ such that ησ(ϵ)ϵ^{−1} = 1 + tω, some ω ∈ Ω. In this case ησ(ϵ+tδ)(ϵ+tδ)^{−1} ≡ 1+t(ω−ϵ^{−1}δ) mod t^{2}Ω (recall that σ(t) = t^{p} and p ≥ 2). Hence we can find a solution modulo t^{2}. By induction one finds compatible solutions modulo any power of t, hence a solution.

Proof of (ii). Again we may assume that M is free. By (i) there exists a basis element e ∈ M such that F (e) = p^{ℓ}e. It is easy to see that also θ(e) = 0, as we have p^{ℓ}θ(e) = θ(F (e)) = pt^{p−1}F (θ(e)) in this case, see Definition 4.9.

Let (M, F, θ) be an arbitrary (F, θ)-module and let M → M′ be the isogeny of Lemma 6.1. Let r = rank M. In the following we will use the map Ω = W [[t]] → W with t ↦ 0. Choose elements e_{1}, …, e_{r} ∈ M′ ⊗_{Ω} W linearly independent over W such that

F^{n}(e_{i}) = p^{a_{i}}e_{i}.

This is possible if n is the common denominator of the slopes of the F -module M′ ⊗_{Ω} W and its slopes are a_{i}/n. See 4.5, it was remarked there that a_{i}/n ≥ 0 in this case so that a_{i} ∈ ℤ_{≥0}. We lift e_{i} to an element m_{i} ∈ M′. Note that the quotient M′/⟨m_{1}, …, m_{r}⟩ is p-power torsion. We have

F^{n}(m_{i}) = p^{a_{i}}m_{i} + tδ_{i}

for certain δ_{i} ∈ M′. If we apply F^{n} once again we get F^{2n}(m_{i}) = p^{2a_{i}}m_{i} + p^{a_{i}}tδ_{i} + t^{p^{n}}F^{n}(δ_{i}). Proceeding by induction we see that

F^{Nn}(m_{i}) = p^{Na_{i}}m_{i} + ∑_{j=1}^{N} p^{(N−j)a_{i}}t^{p^{n(j−1)}}F^{n(j−1)}(δ_{i}).

In particular we have F^{Nn}(m_{i}) − p^{a_{i}}F^{(N−1)n}(m_{i}) ∈ t^{p^{n(N−1)}}M′ for N ≥ 1. We conclude that

m_{i,∞} = lim_{N→∞} p^{−Na_{i}}F^{Nn}(m_{i})

is a well defined element of M′ ⊗_{Ω} Γ(D, O_{D}). (To see this more explicitly: note that m_{i,∞} = m_{i} + ∑_{j=1}^{∞} p^{−j}t^{p^{n(j−1)}}F^{n(j−1)}(δ_{i}) and writing this in terms of a basis for M′ will give coefficients which are elements of Γ(D, O_{D}). See Section 4 for an explicit description of elements of Γ(D, O_{D}).) Note that M′ ⊗_{Ω} Γ(D, O_{D}) = M ⊗_{Ω} Γ(D, O_{D}) as M → M′ is an isogeny and p^{−1} ∈ Γ(D, O_{D}). The m_{i,∞} are elements of the (F, θ)-module M ⊗_{Ω} Γ(D, O_{D}) over Γ(D, O_{D}) such that F^{n}(m_{i,∞}) = p^{a_{i}}m_{i,∞}. Furthermore,

θ(p^{Na_{i}}m_{i,∞}) = θ(F^{Nn}(m_{i,∞})) = p^{Nn}t^{p^{Nn}−1}F^{Nn}(θ(m_{i,∞})).

We conclude that the elements θ(m_{i,∞}) of the module M ⊗_{Ω} O_{D} have vanishing power series expansions around t = 0 (with respect to any basis of M′). Thus they are zero. Therefore the elements m_{i,∞} are horizontal for the integrable connection on M ⊗O_{D} given by letting θ act as θ ⊗ 1 + 1 ⊗ \frac{d}{dt}. They also give a basis of the fiber at t = 0 of M ⊗ O_{D}, as m_{i,∞} ≡ e_{i} mod t. The exterior power m_{1,∞} ∧ ⋯ ∧ m_{r,∞} is a horizontal element of Λ^{r}(M′) ⊗ Γ(D, O_{D}). By Lemma 6.2 we know that Λ^{r}(M′) has a horizontal basis element e. Thus m_{1,∞} ∧⋯∧m_{r,∞} = e⊗f with f ∈ Γ(D, O_{D}) and \frac{d}{dt}(f) = 0, hence f ∈ W. By the above f ≢ 0 mod t, so f ≠ 0. Therefore the elements m_{i,∞} form a basis of M ⊗_{Ω}Γ(D, O_{D}). This proves Dwork’s trick.

6.3 Lemma. (Dwork’s trick.) For any (F, θ)-module (M, F, θ) the (F, θ)-module M ⊗_{Ω} Γ(D, O_{D}) over Γ(D, O_{D}) has a basis of elements d_{i}, which are horizontal and satisfy F^{n}(d_{i}) = p^{a_{i}}d_{i} for certain n ∈ ℕ and a_{i} ∈ ℤ_{≥0}.

6.4 Proposition. Notations as in previous lemma. If N′ ⊂ M ⊗_{Ω} Γ_{c} is a saturated sub (F, θ)-module then N′ = N ⊗ Γ_{c} for some saturated sub (F, θ)-module N ⊂ M.

Proof. We may replace M by M′ as in Lemma 6.1 and N′ by its saturation in M′ ⊗ Γ_{c}. Thus we may assume that M is free over Ω (and N′ & M ⊗ Γ_{c}/N′ are free over Γ_{c}). Let r = rank N′. It suffices to prove the proposition for ∧^{r}N′ ⊂ ∧^{r}(M)⊗Γ_{c}, so we may assume rank N′ = 1.

Say r = rank M and let m_{1}, …, m_{r} be a basis of M over Ω. Write a generator n′ ∈ N′ in the form

n′ = ∑ m_{i} ⊗ g_{i}, g_{i} ∈ Γ_{c}.

We may assume g_{1} ∈ Γ_{c}^{∗} as Γ_{c} is a discrete valuation ring and N′ is saturated in M ⊗ Γ_{c}. Thus we may assume g_{1} = 1. There is a constant C > 0 such that for all i

ν(g_{i}, m) ≤ mC, ∀m ≥ 1.

Using the discussion in Section 4.1 (especially equation 4.1.1) this implies that g_{i} can be viewed as a rigid analytic function on the annulus A : {t; |p|^{ϵ} < |t| < 1} for some small ϵ ∈ ℚ_{>0}. Therefore we can (and we will) view n′ as an element of M ⊗ Γ(A, O_{A}). As A ⊂ D we may write n′ = ∑ d_{i} ⊗ h_{i} with h_{i} ∈ Γ(A, O_{A}) and d_{i} as in Lemma 6.3. (We are using that M ⊗ Γ(A, O_{A}) = M ⊗ Γ(D, O_{D}) ⊗ Γ(A, O_{A}).) It is still true that n′Γ(A, O_{A}) ⊂ M ⊗ Γ(A, O_{A}) is a locally direct summand. (Use that g_{1} = 1.) The fact that θ(N′) ⊂ N′ implies that \frac{d}{dt} h_{i} = hh_{i} for some h ∈ Γ(A, O_{A}). Intuitively this means that \frac{d}{dt} log(h_{i}) = h and it implies that h_{i}/h_{j} is a constant (whenever h_{j} is not zero; the computations are formal). Thus we see that there exist λ_{1}, …, λ_{r} ∈ W (not all zero) such that n′ = (λ_{1}d_{1} + … + λ_{r}d_{r}) ⊗ f for some nonvanishing f ∈ Γ(A, O_{A}). We have shown that N′ is defined over Γ(D, O_{D}) as well, as it corresponds to the submodule spanned by λ_{1}d_{1} + … + λ_{r}d_{r} which is defined over Γ(D, O_{D}).

More precisely, this means that we have

m_{1} + m_{2} ⊗ g_{2} + … + m_{r} ⊗ g_{r} = (m_{1} ⊗ q_{1} + … + m_{r} ⊗ q_{r}) ⊗ f

for certain q_{i} ∈ Γ(D, O_{D}) and f ∈ Γ(A, O_{A}) as above. (Just write d_{i} = ∑ m_{j} ⊗ q_{ij} with q_{ij} ∈ Γ(D, O_{D}) by Lemma 6.3 and put q_{j} = ∑ λ_{i}q_{ij}.) We conclude that f is a meromorphic function on D, invertible on A. Hence it has only finitely many poles on D. (The polelocus of f is a closed analytic subset of some affinoid subdomain of D.) Therefore, we can find a polynomial P (t) ∈ W [t], P (t) ∉ pW [t] such that P (t)f ∈ Γ(D, O_{D}). This implies that P (t)g_{i} = P (t)fh_{i} (i ≥ 2) can be viewed as an element of Γ_{c} and as an element of Γ(D, O_{D}). However, the elements in Γ(D, O_{D}) are power series in t with coefficients in L_{0} = W [1/p], and the elements of Γ_{c} are power series in t and t^{−1} with coefficients in W. We conclude that P (t)g_{i} ∈ Ω for all i. Thus the subspace generated by n′ over Γ_{c} (equal to the subspace generated by P (t)n′ over Γ_{c}, since P (t) ∉ pW [t]) is defined over Ω: we have shown 0 ≠ P (t)n′ ∈ (M ⊗ 1) ∩ N′. Let N be this intersection, seen as a submodule of M. We clearly have F (N) ⊂ N and θ(N) ⊂ N, by the corresponding properties of N′. It is a saturated Ω-submodule, hence it is a sub (F, θ)-module. Also, N ⊗ Γ_{c} ⊂ N′ is an inclusion of saturated rank 1 Γ_{c} modules, so it is an equality.

6.5 Remark. The proposition holds in the following more general situation: Suppose (M, θ) is a finitely generated torsion free Ω-module with a \frac{d}{dt}-derivation θ such that M ⊗ Γ(D, O_{D}) has an horizontal basis. Any θ-invariant subspace N′ ⊂ M ⊗ Γ_{c} comes from an Ω-submodule N ⊂ M. The only point in the proof of the proposition where we used the existence of F, was to ensure that the local system on D was trivial, i.e., we used Dwork’s trick.

7 Some crystals over power series rings

We claim that we may “split off” an F -submodule N of an F -module M over Ω if all its slopes are bigger than all the slopes occurring on M/N. This follows from the following proposition using some tensor algebra. We will not use the claim in the sequel.

7.1 Proposition. Let (M, F) be an F -module over Ω. Assume N ⊂ M is a rank 1 sub F -module of slope ℓ and assume that all other slopes on M are strictly smaller than ℓ. Then there is a complement for N, i.e., a sub F -module N′ ⊂ M such that N ⊕ N′ → M is an isogeny.

Proof. We may assume that N is a saturated submodule of M. We dualize and we get an homomorphism M^{∗} → N^{∗} of Ω-modules whose cokernel has finite length. (Since N_{𝔭} ⊂ M_{𝔭} is a locally direct summand for all height 1 primes 𝔭 of Ω = W [[t]].) Let Q be the torsion free, saturated kernel of this map. For a module T over Ω, write T^{(σ)} = T ⊗_{Ω,σ} Ω. If T is finitely generated we have (T^{∗})^{(σ)} ≅ (T^{(σ)})^{∗}, as σ : Ω → Ω is finite free [11, 3.E]. By definition of F -modules the linear maps F_{lin} : M^{(σ)} → M and F_{lin} : N^{(σ)} → N induced by F have an “almost inverse” p^{n}F_{lin}^{−1} for some n large enough. Dualizing this, we get a linear map (M^{∗})^{(σ)} = (M^{(σ)})^{∗} → M^{∗} and similar for N^{∗}. Thus M^{∗} and N^{∗} are F -modules in a natural way (up to the choice of n), and so is Q (and Q is a sub F -module of M^{∗}). The slope ℓ^{∗} of N^{∗} is strictly smaller then all other slopes of M^{∗}. (Note that ℓ^{∗} = n − ℓ and any slope s^{∗} of M^{∗} is of the form s^{∗} = n − s for some slope s of M.)

By Lemma 6.2 we may choose an isogeny (Ω, p^{ℓ^{∗}}σ) → (N^{∗}, F), and scale by a power of p so that the image of Ω → N^{∗} is contained in the image of M^{∗} → N^{∗}. Further we choose an isogeny (Q, F) → (Q′, F), with Q′ a free module over Ω (see 6.1). We define the F -module M′ by the following commutative diagram with exact rows:

0 ⟶ Q ⟶ M^{∗} ⟶ N^{∗}
|| ↑ ↑
0 ⟶ Q ⟶ M′′ ⟶ Ω ⟶ 0
↓ ↓ ||
0 ⟶ Q′ ⟶ M′ ⟶ Ω ⟶ 0

(We leave it to the reader to verify that the pushout and the pullback give F -modules.) The Ω-module M′ is free. Let s_{1} be the smallest slope occurring on Q′. By assumption s_{1} > ℓ^{∗}. The extension of isocrystals (the last one having slope ℓ^{∗})

0 ⟶ Q′ ⊗_{Ω} L_{2} ⟶ M′ ⊗_{Ω} L_{2} ⟶ Ω ⊗_{Ω} L_{2} ⟶ 0

is canonically split by the theory of isocrystals over L_{2}, see 4.5. Thus the extension class of the exact sequence

0 ⟶ Q′ ⊗_{Ω} W (K_{2}) ⟶ M′ ⊗_{Ω} W (K_{2}) ⟶ Ω ⊗_{Ω} W (K_{2}) ⟶ 0

is p-power torsion. (The category of isocrystals over L_{2} with nonnegative slopes is equivalent to the category of F -modules over W (K_{2}) up to isogeny.) Replacing Ω by p^{m}Ω for m sufficiently large, i.e., doing another pullback, we may assume that

0 ⟶ Q′ ⟶ M′ ⟶ Ω ⟶ 0

becomes split after tensoring with W (K_{2}).

By the structure theory over L_{2} (and W (K_{2})) we see that the map

F^{N} : Q′ ⊗ W (K_{2}) → Q′ ⊗ W (K_{2})

for N large becomes divisible by p^{Nℓ^{∗}}, as all slopes on Q′ are ≥ s_{1} > ℓ^{∗}. (Compare Section 4.5.) The same divisibility holds in M′ ⊗ W (K_{2}) ≅ (Q′ ⊗ W (K_{2})) ⊕ (Ω ⊗ W (K_{2})). But then the map F^{N} : M′ → M′ is also so divisible: We remark that p divides an element m ∈ M′ if and only if it divides the element m ⊗ 1 ∈ M′ ⊗ W (K_{2}). Let us write

ψ = p^{−Nℓ^{∗}}F^{N} : M′ → M′.

It preserves Q′ ⊂ M′ and induces σ^{N} on Ω = M′/Q′.

Let e ∈ M′ be an element mapping to 1 ∈ Ω. Consider the sequence of elements

e_{a} = ψ^{a}(e) ∈ M′, a ≥ 1.

Of course e_{a} maps to the element 1 in Ω for all a. Further, if in the decomposition M′ ⊗ W (K_{2}) ≅ (Q′ ⊗ W (K_{2})) ⊕ (Ω ⊗ W (K_{2})) we have e = q ⊕ 1, then

e_{a} = p^{−aNℓ^{∗}}F^{aN}(q) ⊕ 1.

The terms in the first factor Q′ ⊗ W (K_{2}) converge to zero p-adically as all slopes of Q′ are ≥ s_{1} > ℓ^{∗}. So e_{a} converges p-adically to the element 0 ⊕ 1 ∈ M′ ⊗ W (K_{2}). The p-adic topology on M′ is induced from the p-adic topology on M′ ⊗ W (K_{2}), as is clear from the remark on divisibility above, hence e_{a} converges to an element e_{∞} ∈ M′. This element obviously gives the desired splitting of M′; then dualize and “isogenize” back to M, keeping track of things.

8 Equality of kernels

8.1 Proposition. The map Γ_{2,b,c} ⊗_{Γ_{c}} Γ → Γ_{2,b}, g ⊗ f ↦ gf is injective.

Proof. Note that we have Γ_{2,b,c} ⊗_{Γ_{2,c}} Γ_{2} ≅ Γ_{2,b}. Further the map Γ_{2,c} → Γ_{2,b,c} is finite free, hence flat. Thus we reduce to the case b = 1.

Write M = Γ_{2,c} ⊗_{Γ_{c}} Γ. We may consider M as a Γ_{c} or Γ_{2,c} or Γ-module. Note that M⟶^{p}M is injective as Γ_{2,c}⟶^{p}Γ_{2,c} is injective and Γ_{c} → Γ is flat. Hence M injects into

(1)  (Γ_{2,c} ⊗_{Γ_{c}} Γ)[1/p] ≅ Γ_{2,c}[1/p] ⊗_{Γ_{c}[1/p]} Γ[1/p].

Let us write μ : M → Γ_{2} for the multiplication map of the lemma.

Assume that x = ∑_{i=1}^{r} g_{i} ⊗ f_{i} ∈ M is a non zero element such that μ(x) = ∑_{i=1}^{r} g_{i}f_{i} is zero. We may assume r is minimal among all r ∈ ℕ occurring in this fashion. We claim the elements f_{1}, …, f_{r} ∈ Γ are linearly independent over Γ_{c}. If not, then one of them may be expressed as a linear combination of the others with coefficients from Γ_{c}. (Here we use as always that Γ_{c} is a discrete valuation ring.) Say f_{1} = ∑_{i=2}^{r} a_{i}f_{i}, a_{i} ∈ Γ_{c}. Thus

x = ∑_{i=1}^{r} g_{i} ⊗ f_{i} = g_{1} ⊗ ( ∑_{i=2}^{r} a_{i}f_{i}) + ∑_{i=2}^{r} g_{i} ⊗ f_{i} = ∑_{i=2}^{r}(a_{i}g_{1} + g_{i}) ⊗ f_{i}

in contradiction with the minimality of r.

After renumbering if necessary, we have v_{p}(g_{1}) ≤ v_{p}(g_{i}), for all i. Then x′ = ∑_{i=1}^{r}(g_{i}/g_{1}) ⊗ f_{i} ∈ M is nonzero, as g_{1}x′ = x and x ≠ 0. Further, μ(x′) = ∑(g_{i}/g_{1})f_{i} = (1/g_{1}) ∑ g_{i}f_{i} = 0 (this computation takes place in Γ_{2}[1/p]). We see that we may assume g_{1} = 1.

Let α : Γ_{2} → Γ_{2} be an injective ring map such that α|_{γ} = id_{Γ} and α(Γ_{2,c}) ⊂ Γ_{2,c}. In this case

μ( ∑_{i=1}^{r} α(g_{i}) ⊗ f_{i}) = ∑_{i=1}^{r} α(g_{i})f_{i} = α( ∑_{i=1}^{r} g_{i}f_{i}) = 0.

Therefore the element ∑_{i=1}^{r} g_{i} ⊗ f_{i} − ∑_{i=1}^{r} α(g_{i}) ⊗ f_{i} = ∑_{i=2}^{r}(g_{i} − α(g_{i})) ⊗ f_{i} of M lies in the kernel of μ. Hence it is zero as r was assumed minimal. Since f_{1}, f_{2}, …, f_{r} are linearly independent over Γ_{c} (with f_{1} = 1 fixed by α) we get by minimality of r that g_{i} = α(g_{i}), i = 2, …, r. (Use that f_{2}, …, f_{r} are part of a basis of the Γ_{c}[1/p]-vector space Γ[1/p] and that M injects into the space (1).) By taking α = W (τ) with τ ∈ G = Gal(K_{2}/K_{1}), we conclude that g_{i} ∈ Γ_{1,c}. (Compare with the proof of Proposition 5.8.)

Note that Γ_{1} is the p-adic completion of the ring ⋃_{n} Γ[t^{1/p^{n}}]. (This can be seen by looking at the mod p^{N} quotients of both rings.) Thus we can write any element g of Γ_{1} uniquely in the form

(2)  g = ∑_{α∈ℤ[1/p], 1>α≥0}t^{α}ω_{α}

with ω_{α} ∈ Γ. Not all sequences {ω_{α}}_{α} occur in this way: the sets {α|v_{p}(ω_{α}) ≤ C} are finite for all C > 0. If this condition holds, then the sum (2) converges in Γ_{1}. If g ∈ Γ_{1,c}, then all ω_{α} ∈ Γ_{c}, as is easy to prove using the properties of the function ν_{1}(−, −). We write our g_{i} ∈ Γ_{1,c} in this manner

g_{i} = ∑_{α} t^{α}ω_{i,α}, ω_{i,α} ∈ Γ_{c}.

We conclude that

0 = f_{1} + ω_{2,0}f_{2} + … + ω_{r,0}f_{r} + ∑_{α>0} t^{α}( ∑_{i=2}^{r} ω_{i,α}f_{i}).

By uniqueness in (2) for g = 0, we get f_{1} + ω_{2,0}f_{2} + … + ω_{r,0}f_{r} = 0 in contradiction with the linear independence of the f_{i} over Γ_{c}.

8.2 Corollary. Assume (M, F) is a nonzero F -module over Γ_{c} and φ : M → Γ is Γ_{c}- linear and injective such that φ(F (m)) = p^{ℓ}σ(φ(m)) for some ℓ ∈ ℤ_{≥0}. The largest slope (see Definition 4.7) of M is ℓ and it has multiplicity 1. In fact N = φ^{−1}(Γ_{c}) is a rank 1 sub F -module of M having slope ℓ.

Proof. Consider the composition φ_{2,b,c}

M ⊗_{Γ_{c}} Γ_{2,b,c} ⟶^{φ⊗id}Γ ⊗_{Γ_{c}} Γ_{2,b,c}⟶^{μ}Γ_{2,b}.

It is injective being the composite of two injective maps (Γ_{c} → Γ_{2,b,c} is flat). We take b so divisible that M ⊗ Γ_{2,b,c} has a slope filtration in the sense of Proposition 5.5 (ii). The result on the largest slope and its multiplicity as a slope of M follows from Corollary 5.7 (ii). By Proposition 5.8 the slope ℓ subspace of M ⊗ Γ_{2,b,c} is defined over Γ_{1,c}. Thus there is a saturated sub F -module N′ ⊂ M ⊗_{Γ_{c}} Γ_{1,c} of rank 1 and slope ℓ with

N′ = φ_{1,c}^{−1}(Γ_{1,c}).

Here the map φ_{1,c} is defined by a composition as φ_{2,b,c} above. To prove the corollary it will suffice to find one nonzero element n ∈ M such that φ(n) ∈ Γ_{c}.

Let m_{1}, …, m_{r} be a basis of M over Γ_{c}. Take a generator n′ ∈ N′ and write n′ = ∑ g_{i}m_{i}, g_{i} ∈ Γ_{1,c}. As in the proof of the previous proposition we may write

g_{i} = ∑_{α∈ℤ[1/p], 1>α≥0}t^{α}ω_{i,α}

with ω_{i,α} ∈ Γ_{c}. If φ(m_{i}) = f_{i} ∈ Γ, then we get

φ_{1,c}(n′) = ∑_{α} t^{α}( ∑_{i} ω_{i,α}f_{i}) ∈ Γ_{1,c}.

Hence ∑_{i} ω_{i,α}f_{i} ∈ Γ_{c} for all α. For some α the element n = ∑ ω_{i,α}m_{i} ∈ M will be nonzero and have φ(n) ∈ Γ_{c}.

9 The theorem

9.1 Theorem. Let (N, F, θ) be an (F, θ)-module over Ω. Let

φ : M ⟶ Γ

be an Ω-linear map such that

(i) for some ℓ ∈ ℤ_{≥0} we have φ(F (m)) = p^{ℓ}σ(φ(m)) for all m ∈ M, and (ii) we have φ(θ(m)) = \frac{d}{dt} φ(m) for all m ∈ M. Then φ(M) ⊂ Ω.

Proof. Consider the extension φ_{c} of φ to M ⊗ Γ_{c} defined by the composition

M ⊗_{Ω} Γ_{c} ⟶ Γ ⊗_{Ω} Γ_{c} ⟶ Γ.

It is easy to check that φ_{c} satisfies the analogues of (i) and (ii). Let N′ = Ker(φ_{c}) ⊂ M ⊗_{Ω} Γ_{c}. Clearly, F (N′) ⊂ N′ and θ(N′) ⊂ N′ and N′ is a saturated submodule of M ⊗_{Ω} Γ_{c}. Therefore, by Proposition 6.4 we get N′ = N ⊗ Γ_{c} for some (saturated) sub (F, θ)-module N ⊂ M of M. It is clear that N = Ker(φ) and replacing M by M/N we may assume that φ_{c} is injective.

By Corollary 8.2 we conclude that N′ = φ_{c}^{−1}(Γ_{c}) is a rank 1 slope ℓ sub F -module of M ⊗ Γ_{c} and that all other slopes on M are strictly smaller than ℓ. Clearly, N′ is also preserved by θ, hence it is a sub (F, θ)-module. We apply Proposition 6.4 and we get a saturated rank 1 sub F -module N ⊂ M over Ω with N ⊗ Γ_{c} = N′. By Lemma 6.2 we have that there is an isogeny (Ω, p^{ℓ}σ) → (N, F) of F -modules, i.e., a non zero element n ∈ N with F (n) = p^{ℓ}n such that N/Ωn is p-power torsion. The elements g ∈ Γ_{c} (or even g ∈ Γ) that satisfy σ(g) = g are g ∈ W (𝔽_{p}) ⊂ Ω. Thus φ(N) ⊂ Ω[1/p] ∩ Γ = Ω.

We have seen above that the assumptions of Proposition 7.1 are satisfied for (M, F) and N ⊂ M. The proposition gives us an isogeny

N ⊕ N′ ⟶ M.

The map N′ → M → Γ is zero, since the slopes on N′ are all strictly smaller than ℓ (which implies that N′ ⊗ Γ_{2,b,c} → Γ_{2,b} is zero by Corollary 5.7) (iii). This contradicts the injectivity of φ unless N′ = (0), so N → M is an isogeny. Since N is saturated in M we have N = M. We have won.

10 Appendix : proof of Lemma 2.1

The author does not know how to prove Lemma 2.1 (ii) and (iii) without using crystalline Dieudonné module theory. In both cases we can reduce to the case where R = k[[t]] with k algebraically closed (details left to the reader). Hence we may apply the theory of the preceding sections.

Proof of (ii). Let

(M(H), F, V, ∇) ⟶^{M(α)}(M(G), F, V, ∇)

be the associated morphism of crystalline Dieudonné modules over Ω = W [[t]]. See e.g. [8, Section 2] for definitions and notations. Say the rank of α_{η} is a, i.e., that the height of the image of α_{η} is a. Thus

Λ^{a} : Λ^{a}M(H) ⟶ Λ^{a}M(G)

is not zero. Therefore the horizontal map

Λ^{a}M(H) ⊗ O_{D} ⟶ Λ^{a}M(G) ⊗ O_{D}

of O_{D} modules with connections is not zero. Hence it cannot vanish at any point, in particular not at t = 0. We conclude that the map Λ^{a}M(H_{s}) ⟶ Λ^{a}M(G_{s}) induced by α_{s} is not zero. Hence the height of the image of α_{s} is at least a. Vanishing of the map Λ^{a+1}M(α) shows that it is not more that a. The equality of heights has been proved.

Proof of (iii). The kernel of a map of finite free W [[t]]-modules is finite free. Consider the map of Dieudonné modules

M(H) ⟶ M(G)

induced by α as above. Let M_{1} be the kernel. The operations F, V, ∇ on M(H) and M(G) induce F, V and ∇ on M_{1}. This gives a Dieudonné module (M_{1}, F, V, ∇). Similarly the kernel of the dual map

M(H)^{∧} = M(H^{t}) ⟶ M(G)^{∧} = M(G^{t})

gives a Dieudonné module, whose dual we write (M_{2}, F, V, ∇). Putting everything back together we get a complex of Dieudonné modules

(∗)  0 → M_{1} → M(H) → M(G) → M_{2} → 0.

Our assumption on α_{η} implies that the complex (∗) becomes exact after taking the tensor product with Γ.

Let us write

L = λ(M_{1}) ⊗ λ(M(G)) ⊗ (λ(M(H)) ⊗ λ(M_{2}))^{⊗−1} ,

where λ(−) indicates highest exterior power. We note that L is a free rank 1 (F, θ)-module over Ω. By the above (∗) gives an element e of L ⊗ Γ which is horizontal, which satisfies F (e) = p^{ℓ}e for some ℓ and which is a generator of L. By our main theorem e ∈ L. By a classification of rank 1 (F, θ)-modules over Ω (compare Lemma 6.2) we know that (L, θ) ≅ (Ω, \frac{d}{dt}). Therefore e generates L, and hence (∗) is exact. By the equivalence of categories [8, 2.4.4, 2.4.8, 4.1.1] the Dieudonné modules M_{1} and M_{2} correspond to pdivisible groups over S. These p-divisible groups are the kernel and cokernel of α since (∗) is exact. This ends the proof.

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