Crystalline Dieudonné module theory via formal and rigid geometry

by A.J. de Jong*

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 Some algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Crystals, formal schemes and p-divisible groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Deformations of Dieudonné crystals and p-divisible groups . . . . . . . . . . . . . . . . . . . . . . 22 4 Proof of Main Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5 Proof of Main Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6 Connected rigid analytic varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 7 Formal schemes and rigid analytic geometry; Berthelot’s construction . . . . . . . . . . . . 66

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Introduction

Let S be a base scheme in characteristic p. Consider the crystalline Dieudonné module functor 𝔻 on the category of p-divisible groups over S to the category of Dieudonné crystals over S. We ask whether 𝔻 is fully faithful or fully faithful up to isogeny over S. If so, we can ask whether 𝔻 is even an equivalence.

The idea of associating Dieudonné crystals to p-divisible groups goes back to Grothendieck. We refer to his letter to J. Tate of 1966, his Montréal lectures in 1970 [G1] and his talk at the Nice congress [G2]. In Section 3 of [G2] Grothendieck mentions two constructions of the Dieudonné crystal; one using the exponential, another using the method of ♮-extensions. The first approach is developed in [M], the second in [MM]. In [M] the Dieudonné crystal is constructed using universal extensions by vector groups, and Messing proves the deformation theorem. In [MM] the authors prove the comparison theorem, which compares the de Rahm cohomology of an abelian scheme with the Dieudonné crystal of its p-divisible group. Open questions in the theory at that time were the full faithfulness/equivalence questions mentioned above and the problem of extending 𝔻 to the category of finite, locally free p-groups. These questions were raised in [G1] and [MM].

In [B] and [BBM] Grothendieck’s formalism of crystalline sites and crystals is developed and extended. Using this the Dieudonné crystal is defined in [BBM] as an Ext-sheaf on the crystalline site. This definition generalizes to give the crystalline Dieudonné module functor on the category of finite, locally free p-groups.

There are partial results on the full faithfulness/equivalence problem. Berthelot and Messing [BM] prove that 𝔻 is fully faithful on schemes having locally a p-basis. There are

* The research of Dr. A.J. de Jong has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences. unpublished results of Kato proving that 𝔻 is an equivalence over schemes smooth over perfect fields of characteristic p > 2. This continues work of Bloch on p-divisible formal Lie groups (1974, unpublished). Messing proved (unpublished) that the result of Kato for arbitrary characteristic follows from the results of [dJ]. However, the full faithfulness question does not have a positive answer in general; there are counterexamples given in [BM].

In this paper we extend the results of Berthelot, Bloch, Kato and Messing. The first result is an equivalence theorem.

Main Theorem 1. Let 𝔛 be a formal scheme in characteristic p, formally smooth over Spec(𝔽_{p}), such that 𝔛_{red} is locally of finite type over a field with a finite p-basis. The crystalline Dieudonné module functor 𝔻 is an equivalence over 𝔛. □

In particular 𝔻 is an equivalence over regular schemes which are of finite type over a field with a finite p-basis.

Our second main result is that 𝔻 is fully faithful up to isogeny over schemes of finite type over a field with a finite p-basis. This assertion (Corollary 5.1.2) follows from the following slightly stronger theorem.

Main Theorem 2. Let S be a reduced scheme of finite type over a field with a finite p-basis. Let G_{1}, G_{2} be p-divisible groups over S. We have the following equality:

Hom_{DC/S}(𝔻(G_{2}), 𝔻(G_{1})) = torsion subgroup ⊕ 𝔻(Hom_{S}(G_{1}, G_{2})).  □

For certain schemes S it is known that the torsion subgroup is zero, for example if S is a locally complete intersection. If S is such a scheme and satisfies the conditions of the theorem, then the crystalline Dieudonné module functor is fully faithful.

We turn to an overview of the contents of the chapters. Chapter 1 contains some algebraic preliminaries. In Chapter 2 the crystalline Dieudonné module theory is extended to formal schemes. We start with the definition of 𝔻 from [BBM]. We analyse what it means to have Dieudonné crystals over formal schemes. After this the fully faithfulness in the Main Theorem 1 follows from the fully faithfulness result of [BM] by a rather formal argument.

In Chapter 3 we deform Dieudonné crystals and p-divisible groups. An important technical point is the introduction of special Dieudonné modules. This allows us to deform Dieudonné crystals. We want to deform a p-divisible group given a deformation of its Dieudonné crystal. Here we use the precise description of the deformations of a p-divisible group G in terms of filtrations on its Dieudonné crystal. We can do this since 𝔻(G) agrees with the crystal defined defined in [M], as is shown in [BM].

In Chapter 4 we prove essential surjectivity in Main Theorem 1. We use induction on the dimension of 𝔛_{red}. The crucial step is the case dim(𝔛_{red}) = 0. Here we have to prove that 𝔻 is essentially surjective over a field with a finite p-basis. The proof uses two ingredients: the description, in [dJ], of formal p-divisible groups in terms of Dieudonné modules, and a result on extensions of étale by multiplicative p-divisible groups.

The Main Theorem 2 is proved in Chapter 5. The proof uses ideas of Berthelot on convergent isocrystals ([B2]). By Main Theorem 1 we already know the result over the regular part of S. Over the singular part of S we get the result by induction on the dimension. Hence, we need to show that the two resulting homomorphisms glue. Using the ideas of Berthelot it is proved that they glue rigid analytically over a tube U of S. See the beginning of Chapter 5 for the definition of a tube. We are able to descend from U back to S using certain results, proved in Chapter 7, relating rigid geometry and formal geometry.

The result of Chapter 6 is that on a connected rigid analytic variety X over a discretely valued field any two points may be connected by curves on X. This result is used in Chapter 5. In Chapter 7 we describe Berthelot’s functor 𝔛 ↦ 𝔛^{rig}. Here 𝔛 is a formal scheme over a complete discrete valuation ring R, where 𝔛 need not be of finite type over R. The result 𝔛^{rig} is a rigid analytic variety over the quotient field of R. In Theorem 7.4.1 we compare bounded rigid analytic functions on 𝔛^{rig} with formal functions on 𝔛. In Proposition 7.5.2 we prove a result on rigid descent of closed formal subschemes.

This work is strongly influenced by the work of Prof. Berthelot, indeed, this paper might not have existed had I not attended a talk by Berthelot on convergent isocrystals. I would like to thank him for stimulating conversations. Furthermore, I would like to thank Prof. Messing, Prof. Zink and R. Huber for discussions on subjects related to this paper.

1 Some algebra

In this chapter we introduce a class of formally smooth 𝔽_{p}-algebra’s and give some of their properties. Further, we recall some notations and definitions. All rings considered are assumed commutative with 1.

1.1 Rings with p-bases.

1.1.1 Definition. A subset {x_{α}} of an 𝔽_{p}-algebra A is called a p-basis of A if A, considered as an A-module via the Frobenius of A, is a free A-module which admits as a basis the set of monomials x^{I}, where I runs over multi-indices I = (i_{α}), 0 ≤ i_{α} < p and almost all i_{α} are zero.

This definition is taken from [BM] (Definition 1.1.1). It implies that A is reduced and is a formally smooth 𝔽_{p}-algebra. If A has a p-basis then any localization S^{−1}A has a p-basis and any polynomial ring A[T_{i}] has a p-basis. Any field of characteristic p has a p-basis.

1.1.2 Lemma. For any 𝔽_{p}-algebra A having a p-basis the natural augmentation of the cotangent complex of A over 𝔽_{p} :

L_{A/𝔽_{p}} ⟶ Ω_{A/𝔽_{p}}^{1}

(see [Ill, II 1.2.4.1]) is a quasi-isomorphism. (Here L_{A/𝔽_{p}} is viewed as an object of the derived category D(A).) Proof. We first remark that for any ring A in characteristic p the Frobenius endomorphism A → A induces an endomorphism of L_{A/𝔽_{p}} which is homotopic to the zero morphism. Indeed, the augmentation P_{𝔽_{p}}(A) → A [Ill, I 1.5.5.6] is a homotopy equivalence. Hence any endomorphism of P_{𝔽_{p}}(A) lifting the Frobenius endomorphism of A is homotopic to the canonical one (the one from [Ill, II 1.2]). Therefore, we may take the Frobenius morphism of P_{𝔽_{p}}(A) (i.e., the Frobenius endomorphism on each P_{𝔽_{p}}(A)_{n}) to compute the action of Frobenius on L_{A/𝔽_{p}}. As (L_{A/𝔽_{p}})_{n} = Ω_{P(A)_{n}/𝔽_{p}}^{1}, this induces the zero morphism on L_{A/𝔽_{p}}.

Let us write for A^{p} ⊂ A the subring of p-powers of A. Frobenius induces an isomorphism A → A^{p}, hence by the remark above the inclusion A^{p} ⊂ A induces the zero morphism L_{A^{p}/𝔽_{p}} → L_{A/𝔽_{p}}. Consider the distinghuised triangle associated to 𝔽_{p} ⊂ A^{p} ⊂ A (see [Ill, II 2.1.2.1]):

L_{A^{p}/𝔽_{p}} ⊗_{A^{p}} A ⟶^{(1)} L_{A/𝔽_{p}} ⟶^{(2)} L_{A/A^{p}} ⟶^{(3)} L_{A^{p}/𝔽_{p}} ⊗_{A^{p}} A[1].

We have seen that (1) induces the zero map on homology. It is easy to see that (2) induces an isomorphism Ω_{A/𝔽_{p}}^{1} → Ω_{A/A^{p}}^{1}. To finish the proof we have to show that (3) is injective on H_{−1}(L_{A/A^{p}}) and that H_{−i}(L_{A/A^{p}}) = 0 for all i ≥ 2.

Let {x_{j}}_{j∈J} be a p-basis of A. Consider the ideal I generated by the elements T_{j}^{p} − x_{j}^{p}, j ∈ J in the polynomial ring P := A^{p}[T_{j} ; j ∈ J]. It is easily shown to be Koszul ([Ill, III 3.3]). Since A ≅ P/I we get by [Ill, Proposition 3.3.6] that

L_{A/A^{p}} ≅ (0 ⟶ I/I^{2} ⟶^{d_{P/A^{p}}} Ω_{P/A^{p}}^{1} ⊗_{P} A ⟶ 0).

Of course in our situation d_{P/A^{p}} = 0. We leave it to the reader to show by an explicit calculation that (3) induces an isomorphism

I/I^{2} ⟶ Ω_{A^{p}/𝔽_{p}}^{1} ⊗_{A^{p}} A.

(It maps T_{j}^{p} − x_{j}^{p} to −dx_{j}^{p} ⊗ 1.) □

1.1.3 Lemma. Let A be an 𝔽_{p}-algebra with a finite p-basis, I ⊂ A a finitely generated ideal. The completion Â of A with respect to I has a finite p-basis also. Proof. Choose a p-basis {x_{1}, …, x_{N}} of A and generators f_{1}, …, f_{M} of I. We denote by x ↦ x̂ the map A → Â. We want to write any element a = lim a_{n} of Â (with a_{n} ∈ A/I^{n}) in the form

a = ∑_{J=(j_{1},...,j_{N}), 0≤j_{α}<p}(a_{J})^{p}x̂^{J}, a_{J} ∈ Â.

By assumption we can write a_{n} = ∑(a_{J,n})^{p}x^{J} mod I^{n}. We see that

∑(a_{J,n+1} − a_{J,n})^{p}x^{J} ∈ I^{n}.

Considering the effect of

(\frac{∂}{∂x_{1}} )^{n_{1}} … (\frac{∂}{∂x_{N}} )^{n_{N}}

on elements of the form ∑ b_{J}^{p}x^{J} and on I^{n}, it becomes clear that

(a_{J,n+1} − a_{J,n})^{p} ∈ I^{n−pN}.

Next, we can write any element of I^{n−pN} (in particular the element (a_{J,n+1} − a_{J,n})^{p}) as

∑_{|K|≥n/p−N−(p−1)M} α_{K}(f^{K})^{p}, α_{K} ∈ A.

Here the sum runs over multi-indices K = (k_{1}, …, k_{M}) with total degree |K| := k_{1} + … + k_{M} at least c(n) := n/p − N − (p − 1)M. Writing α_{K} = ∑(α_{K,J})^{p}x^{J} we see that

(a_{J,n+1} − a_{J,n})^{p} = ∑_{J} ( ∑_{|K|≥c(n)} α_{K,J}f^{K})^{p}x^{J}.

The only non vanishing term on the right is the one with J = 0. Therefore, we conclude that

a_{J,n+1} − a_{J,n} ∈ I^{c(n)}.

Thus, putting

a_{J} := lim_{⟵} a_{J,n} mod I^{c(n)} ∈ Â

we get

a = ∑(a_{J})^{p}x̂^{J}

as desired. Unicity is proved in the same manner. □

1.2 Lifts of rings in characteristic p to characteristic 0.

1.2.1 Definition. Let A be a ring of characteristic p. A lift of A (resp. a lift of A modulo p^{n}) is a p-adically complete ring Ã, flat over ℤ_{p} (resp. flat over ℤ/p^{n}ℤ) endowed with an isomorphism Ã/pÃ ≅ A. A lift of Frobenius on such a lift Ã is a ring endomorphism σ : Ã → Ã such that σ(a) ≡ a^{p} mod pÃ. In this situation we will also call the pair (Ã, σ) a lift of A (resp. a lift of A modulo p^{n}).

1.2.2 Lemma. Suppose A is an 𝔽_{p}-algebra whose cotangent complex L_{A/𝔽_{p}} is quasiisomorphic to Ω_{A/𝔽_{p}}^{1}. If, in addition, Ω_{A/𝔽_{p}}^{1} is a projective A-module then a lift (Ã, σ) of A exists. Proof. We are going to find a lift (A_{n}, σ) of A modulo p^{n}. We argue by induction on n (the case n = 1 is OK). The obstruction to find A_{n+1} given A_{n} lies in

𝔼xt_{A_{n}}^{2}(L_{A_{n}/ℤ/p^{n}ℤ}, A)

by [Ill, III 2.1.3.3]. Since L_{A_{n}/ℤ/p^{n}ℤ} ⊗ A ≅ L_{A/𝔽_{p}} (see [Ill, II 2.3.10]) we get

𝔼xt_{A_{n}}^{2}(L_{A_{n}/ℤ/p^{n}ℤ}, A) ≅ 𝔼xt_{A}^{2}(L_{A/𝔽_{p}}, A) ≅ 𝔼xt_{A}^{2}(Ω_{A/𝔽_{p}}^{1}, A)

and the last group is zero by assumption. Similarly, given A_{n+1}, the obstruction to lift σ : A_{n} → A_{n} to A_{n+1} lies in

𝔼xt_{A_{n}}^{1}(L_{A_{n}/ℤ/p^{n}ℤ} ⊗_{σ} A_{n}, A).

(Use [Ill, III 2.2.2], we remark that a subscript _{0} is missing from [Ill, III Formula 2.2.1.4].) In this case we have the quasi-isomorphisms

L_{A_{n}/ℤ/p^{n}ℤ} ⊗_{σ} A_{n} ⊗ A ≅ L_{A_{n}/ℤ/p^{n}ℤ} ⊗ A ⊗_{σ} A ≅ L_{A/𝔽_{p}} ⊗_{σ} A ≅ Ω_{A/𝔽_{p}}^{1} ⊗_{σ} A

and we conclude in the same manner that the obstruction is zero. □

1.2.3 Remarks. a) Combining Lemmata 1.1.2 and 1.2.2 we see that any 𝔽_{p}-algebra A with a p-basis has a lift (Ã, σ). In [BM] an explicit construction of such a lift is given.

b) Although the lift Ã of the lemma is always unique up to isomorphism, this is not true in general for the pair (Ã, σ).

1.3 Formally smooth rings in characteristic p.

1.3.1 Let A be an 𝔽_{p}-algebra and I ⊂ A an ideal of A. We consider the following condition on the pair (A, I): 1.3.1.1 – A is Noetherian and complete with respect to the I-adic topology on A.

– The ring A is formally smooth over 𝔽_{p} (see EGA 0_{IV} 19.3.1 or [Mat, 28.C], we use the I-adic topology on A).

– The ring A/I contains a field with a finite p-basis and is a finitely generated algebra over this field.

We will say that an 𝔽_{p}-algebra A satisfies 1.3.1.1 if there is an ideal I ⊂ A such that (A, I) satisfies 1.3.1.1 above. Usually, we will assume that the ideal I is as big as possible, i.e., I = \sqrt{I}. We remark that this assumption and 1.3.1.1 uniquely determine I ⊂ A. This follows from the fact that a ring finitely generated over a field has only one adic topology for which it is complete: the discrete topology.

1.3.2 Examples. Let k be a field of characteristic p with [k : k^{p}] < ∞, i.e., k has a finite p-basis. 1.3.2.1 Any completion of a polynomial algebra k[x_{1}, …, x_{n}] with respect to an ideal

satisfies 1.3.1.1. By Lemma 1.1.3 such a ring has a finite p-basis. 1.3.2.2 Any finitely generated k-algebra A = A/I which is a regular ring satisfies 1.3.1.1.

(To see that A is formally smooth over 𝔽_{p} use EGA 0_{IV} 22.6.7.)

From Lemma 1.1.3 we see that rings of the type described in 1.3.2.1 have a finite p-basis. Hence by [BM] we know that crystals over such rings can be described in terms of modules with connections over lifts. The next lemma will imply the same for rings as in 1.3.1.

1.3.3 Lemma. Let A be a ring satisfying 1.3.1.1.

a) There exists a ring B of the form described in 1.3.2.1 and a surjection π : B → A such that the inverse image of an ideal of definition of A is an ideal of definition of B. We may then choose i : A → B such that π ∘ i = id_{A}.

b) The homomorphism Ω_{A}^{1} → Ω̂_{A}^{1} (continuous differentials) is an isomorphism; Ω_{A}^{1} is a finite projective A-module.

c) The morphism L_{A/𝔽_{p}} → Ω_{A}^{1} is a quasi-isomorphism.

d) If we have B, i and π as in a) then we can find an isomorphism

B ≅ A[[M]] = ∏_{n≥0} Sym_{A}^{n}(M)

where M is a finite projective A-module. We can find a lift (Ã, σ) of A (1.2.2); as a lift of B we can take

B̃ = Ã[[M̃]],

for some lift M̃ of the module M. Furthermore, we can find a lift of Frobenius σ on B̃ satisfying σ(M̃) ⊂ Sym^{p}(M̃) and inducing σ on Ã. Proof. Let I ⊂ A be an ideal as in 1.3.1.1 and let k ⊂ A/I be the field which is supposed to exist by 1.3.1.1. Take a lift k → A of the homomorphism k → A/I (this is possible, k is formally smooth over 𝔽_{p}). Next we take a homomorphism

φ : k[x_{1}, …, x_{r}, x_{r+1}, …, x_{n}] ⟶ A

such that the images of x_{1}, …, x_{r} generate A/I as a k-algebra and φ(x_{r+1}), …, φ(x_{n}) are generators of the ideal I. Let B be the completion of k[x_{1}, …, x_{n}] in the kernel of k[x_{1}, …, x_{n}] → A/I. The homomorphism φ induces a surjection π : B → A. Let J ⊂ B (resp. K ⊂ B) be the kernel of B → A/I (resp. B → A/I^{2}). The ring A is formally smooth over 𝔽_{p} by 1.3.1.1, hence we can find a ring homomorphism ψ_{2} : A → B/J^{2} fitting into the diagram:

A ⟶ A/I^{2} ⟶ A/I
↓^{ψ_{2}}  ↑≅  ↑≅
B/J^{2} ⟶ B/K ⟶ B/J.

(Remark that J^{2} ⊂ K ⊂ J and that ψ_{2} need not be k-linear.) We see that I^{2} ⊂ Ker(ψ_{2}), as ψ_{2}(I) ⊂ J/J^{2}, therefore we get ψ_{2} : A/I^{2} → B/J^{2}. By induction on n we choose homomorphisms ψ_{n} : A/I^{n} → B/J^{n} (n ≥ 3), such that ψ_{n} mod I^{n−1} = ψ_{n−1} (use that A is formally smooth over 𝔽_{p}). We put

ψ_{∞} = lim_{⟵} ψ_{n} : A ⟶ B.

Our choice of ψ_{2} implies that π ∘ ψ_{∞} ≡ id_{A} mod I^{2}, so it must be an automorphism of A. If we take i := ψ_{∞} ∘ (π ∘ ψ_{∞})^{−1} then we see that π ∘ i = id_{A}. This proves a).

The usual argument shows that, since B is formally smooth also, we must have B ≅ A[[M]] with M a finite (B Noetherian) projective A-module (cf. EGA 0_{IV} 19.5.3). This implies that the ideal MB = Ker(B → A) is regular and that the sequence

0 ⟶ M ⟶ Ω_{B}^{1} ⊗ A ⟶ Ω_{A}^{1} ⟶ 0

is split exact. Our result L_{B/𝔽_{p}} ≅ Ω_{B}^{1} (Lemma 1.1.2) gives c) by [Ill, III 3.3.6]. Statement b) follows from the fact that Ω_{B}^{1} is a finite free B-module and is isomorphic to the module Ω̂_{B}^{1} of continuous differentials (with respect to the J-adic topology). This fact is easily proved using the finite p-basis of B (compare with the proof of Lemma 1.1.3).

We already proved the first part of d). There are no obstructions to lifting finite projective modules, so we can find the lift M̃ of M. This is a finite projective Ã-module; let us take a finite free module ⨁ Ã·x_{i} and homomorphisms ⨁ Ã·x_{i} ⟶^{s} M̃ and ⨁ Ã·x_{i} ⟵^{t} M̃ with s ∘ t = idM̃. For σ we take the composition

Ã[[M̃]]  ⟶  Ã[[x_{i}]]  ⟶  Ã[[x_{i}]]  ⟶  Ã[[M̃]]
m̃ ⟼ t(m̃)  x_{i} ⟼ x_{i}^{p}  x_{i} ⟼ s(x_{i})
idÃ  σ on Ã  idÃ.

This concludes the proof of Lemma 1.3.3. □

1.3.4 Remarks. a) The lift Ã of the lemma above is also complete for the Ĩ-adic topology; here Ĩ denotes the inverse image in Ã of an ideal of definition I of A.

b) The module

Ω̂1 ; Ã = differentials continuous for the p-adic topology on Ã := lim_{⟵} Ω_{Ã/p^{n}Ã}^{1}

is a finite projective Ã-module; it is flat over Ã and we have Ω̂1 ; Ã/pΩ̂1 ; Ã ≅ Ω_{A}^{1}. To see this, use 1.3.3 d) to reduce to the case that A has a finite p-basis and apply [BM, 1.3.1]. More generally for any lift Ã as constructed in Lemma 1.2.2 the module Ω̂1 ; Ã is a flat lift of Ω_{A}^{1}. The reader may enjoy proving this general result for himself.

c) The natural homomorphism

Ω̂1 ; Ã = Ω̂_{(Ã,pÃ)}^{1} ⟶ Ω̂_{(Ã,Ĩ)}^{1}

is an isomorphism. The Ã-module on the right denotes differentials continuous in the Ĩ-adic topology (see a). This follows from 1.3.3 b).

2 Crystals, formal schemes and p-divisible groups

In this chapter we describe crystals in terms of modules with connections. We are a bit more general then [BM]. In addition we do the same for Dieudonné crystals. Finally, we define a Kodaira-Spencer map for Dieudonné crystals on certain formal schemes which measures in a certain sense how versal the crystal is.

2.1 Crystalline sites and crystals.

2.1.1 We always work in the absolute case, with base scheme (Σ, I, γ) where Σ = Spec(ℤ_{p}), I = pO_{Σ} and γ denotes the canonical divided powers on I. For any Σ-scheme T we also denote by γ the canonical divided powers on the ideal pO_{T} . If S is a Σ-scheme such that p is locally nilpotent on S then we write CRIS(S/Σ) for the big fppf-crystalline site of S over (Σ, I, γ) (see [BBM, 1.1]). We will always endow it with the fppf-topology.

We refer to [BBM] for the notations and conventions we use regarding these sites, sheaves on them, etc. For example, recall [BBM, 1.1.3] that a sheaf E on CRIS(S/Σ) is given by the following data: 2.1.1.1 a sheaf E_{(U,T,δ)} (usually denoted E_{T}) on the small fppf-site T_{fppf} of T for any

object (U, T, δ) of CRIS(S/Σ), 2.1.1.2 for any morphism (u, v) : (U′, T′, δ′) → (U, T, δ) of CRIS(S/Σ) a sheaf homomor-

phism v^{−1}(E_{T}) → E_{T′} These have to satisfy a number of obvious conditions.

Let C be a site, A a sheaf of rings on C and F a sheaf of A-modules. We say that F is a quasi-coherent sheaf of A-modules if F is locally on C isomorphic to the cokernel of a map A^{(I)} → A^{(J)}. More precisely, for any X ∈ ObC, there exists a covering {U_{i} → X} of X such that F|_{U_{i}} is isomorphic to the cokernel of a morphism A^{(I)}|_{U_{i}} → A^{(J)}|_{U_{i}} of A-modules. Compare [EGA, O_{I} 5.1.3]; see also [B, Section 1.1] for this and the following definition.

2.1.2 Definition. Suppose A is a sheaf of rings on CRIS(S/Σ). We say that a sheaf of A-modules E is a crystal of quasi-coherent A-modules if the following conditions hold: 2.1.2.1 E_{T} is a quasi-coherent sheaf of A_{T} -modules on T_{fppf} for any object (U, T, δ) of

CRIS(S/Σ), 2.1.2.2 the transition homomorphisms 2.1.1.2 induce isomorphisms

v^{−1}(E_{T}) ⊗_{v^{−1}(A_{T})} A_{T′} ⟶ E_{T′}.

We get the definition of crystals of finite locally free A-modules by replacing in 2.1.2.1 the word quasi-coherent by finite locally free. In the same manner one can also define crystals of A-modules of finite presentation over CRIS(S/Σ).

2.1.3 Let X, Y be Σ-schemes on which p is locally nilpotent. Let i : Y → X be a closed immersion. To i is associated a morphism of functoriality

i_{CRIS} = (i_{CRIS}^{∗}, i_{CRIS∗}) : (Y/Σ)_{CRIS} ⟶ (X/Σ)_{CRIS}

of crystalline topoi, see [B, III 4.2.1] and [BBM, 1.1.10]. We want to study the effect of i_{CRIS∗} on crystals of O_{Y/Σ}-modules.

Let us describe i_{CRIS∗}. For any object (U, T, δ) of CRIS(X/Σ) we write (V, D_{V} (T),^{[ ]}) for the divided power envelope of V := Y ×_{X}U in T, the divided powers^{[ ]} taken compatible with those on the ideal of U in T (i.e., δ) and with γ on pO_{T}. The triple (V, D_{V} (T),^{[ ]}) is an object of CRIS(Y/Σ). For a sheaf E on CRIS(Y/Σ) we have the following isomorphism

2.1.3.1  i_{CRIS∗}(E)_{T} ≅ (p_{T})_{∗} (E_{D_{V} (T)}) .

Here p_{T} is the canonical morphism D_{V} (T) → T; it induces a functor (p_{T})_{∗} : Sheaves on D_{V} (T)_{fppf} → Sheaves on T_{fppf}. This is proved exactly as in [B, IV 1.3]. We remark that for T′ → T flat we have D_{V′}(T′) ≅ D_{V} (T) ×_{T} T′ (see [B, I 2.7.1]). Clearly, if V = U, that is, if the morphism U → X factors through Y, then (V, D_{V} (T),^{[ ]}) = (U, T, δ).

The description of i_{CRIS}^{∗} is easier. For any object of CRIS(Y/Σ), say (U, T, δ), we denote by i_{!}(U, T, δ) the triple (U, T, δ) considered as an object of CRIS(X/Σ), i.e., U is considered as a scheme over X via the composition U → Y → X. For a sheaf E on CRIS(X/Σ) we have

2.1.3.2  i_{CRIS}^{∗}(E)_{(U,T,δ)} = E_{i_{!}(U,T,δ)}.

(See [BBM, 1.1.10.3].) Thus, for any sheaf E on CRIS(Y/Σ) we have

2.1.3.3  i_{CRIS}^{∗}i_{CRIS∗}(E) ≅ E.

2.1.4 Lemma. In the situation of 2.1.3, the functor i_{CRIS∗} induces an equivalence of the following categories :

C1. The category of crystals of quasi-coherent (resp. finite locally free) O_{Y/Σ}-modules

on CRIS(Y/Σ). C2. The category of crystals of quasi-coherent (resp. finite locally free) i_{CRIS∗}(O_{Y/Σ})-

modules on CRIS(X/Σ). A similar statement holds for crystals of finite presentation.

Proof. Suppose E is an object of C1. Take any object (U, T, δ) of CRIS(X/Σ). The sheaf E_{D_{V} (T)} is a quasi-coherent sheaf of O_{D_{V} (T)}-modules on D_{V} (T)_{fppf}. By descent theory, it arises from a quasi-coherent sheaf of O_{D_{V} (T)}-modules on D_{V} (T)_{Zar}. The morphism p_{T} : D_{V} (T) → T is affine, hence covering T by affines we may assume there is an exact sequence

2.1.4.1  O_{D_{V} (T)}^{(I)} ⟶ O_{D_{V} (T)}^{(J)} ⟶ E_{D_{V} (T)} ⟶ 0

on D_{V} (T)_{fppf}. (If E is of finite presentation then the index sets I and J may be chosen finite; if E is finite locally free then we may choose I = ∅ and J finite. Adding such remarks, the rest of the proof will go through in each of the three cases.) Since the morphism p_{T} : D_{V} (T) → T is affine this gives an exact sequence

2.1.4.2  i_{CRIS∗} (O_{Y/Σ})_{T}^{(I)} ⟶ i_{CRIS∗} (O_{Y/Σ})_{T}^{(J)} ⟶ i_{CRIS∗}(E)_{T} ⟶ 0

on T_{fppf}. Therefore, we have proven that i_{CRIS∗}(E) satisfies 2.1.2.1 with A = i_{CRIS∗}(O_{Y/Σ}). Let (u, v) : (U′, T′, δ′) → (U, T, δ) be a morphism of CRIS(X/Σ). The crystal property of E implies that E_{D_{V ′} (T′)} is the pullback of E_{D_{V} (T)} via the morphism D_{V′}(T′) → D_{V} (T). Hence we also have a sequence 2.1.4.1 over D_{V′}(T′) and a sequence 2.1.4.2 over T′. Thus, the sheaf i_{CRIS∗}(E) satisfies 2.1.2.2 with A = i_{CRIS∗}(O_{Y/Σ}) since this is clearly true for i_{CRIS∗}(O_{Y/Σ}). We conclude that i_{CRIS∗}(E) is an object of C2. By 2.1.3.3 the functor i_{CRIS∗} : C1 → C2 is fully faithful.

Suppose E ∈ ObC2. Take any object (U, T, δ) of CRIS(Y/Σ). By assumption we can find a covering family

(U_{i}, T_{i}, δ_{i}) ⟶ i_{!}(U, T, δ)

in CRIS(X/Σ) such that on each T_{i,fppf} there is an exact sequence

i_{CRIS∗} (O_{Y/Σ})_{T_{i}}^{(I)} ⟶ i_{CRIS∗} (O_{Y/Σ})_{T_{i}}^{(J)} ⟶ E_{T_{i}} ⟶ 0.

Since U_{i} = U ×_{T} T_{i}, the given map U_{i} → X factors through Y ↪ X. The system (U_{i}, T_{i}, δ_{i}) → (U, T, δ) can thus be viewed as a covering of (U, T, δ) in CRIS(Y/Σ). By 2.1.3.2 we get exact sequences

(O_{Y/Σ})_{T_{i}}^{(I)} ⟶ (O_{Y/Σ})_{T_{i}}^{(J)} ⟶ i_{CRIS}^{∗}(E)_{T_{i}} ⟶ 0.

We conclude that i_{CRIS}^{∗}(E) is in C1 (verification of 2.1.2.2 is trivial). To finish the proof we need only to show that the natural homomorphism

E ⟶ i_{CRIS∗}i_{CRIS}^{∗}(E)

is an isomorphism. This may be checked locally on CRIS(X/Σ), hence on objects (U, T, δ) of CRIS(X/Σ) where we have an exact sequence

i_{CRIS∗} (O_{Y/Σ})_{T}^{(I)} ⟶ i_{CRIS∗} (O_{Y/Σ})_{T}^{(J)} ⟶ E_{T} ⟶ 0.

The description of i_{CRIS∗} implies that this is the exact sequence

((p_{T})_{∗}O_{D_{V} (T)})^{(I)} ⟶ ((p_{T})_{∗}O_{D_{V} (T)})^{(J)} ⟶ E_{T} ⟶ 0

on T_{fppf}. Hence E_{T} ≅ (p_{T})_{∗}F for some quasi-coherent sheaf of O_{D_{V} (T)}-modules F on D_{V} (T)_{fppf}. By the crystal property 2.1.2.2 for E we must have F ≅ E_{D_{V} (T)} (here (V, D_{V} (T),^{[ ]}) is considered as an object of CRIS(X/Σ)). Since it is also true that

i_{CRIS∗}i_{CRIS}^{∗}(E)_{T} ≅ (p_{T})_{∗}E_{D_{V} (T)},

we have won. □

The sheaf i_{CRIS∗}O_{Y/Σ} of Lemma 2.1.4 is not in general a crystal of O_{X/Σ}-modules. For instance if i is the closed immersion Spec(𝔽_{p}) → Spec(𝔽_{p}[T]) the sheaf i_{CRIS∗}O_{Y/Σ} satisfies 2.1.2.1 but not 2.1.2.2. The following proposition shows that we may replace i_{CRIS∗}O_{Y/Σ} by a crystal.

2.1.5 Proposition. In the situation 2.1.3 there is a sheaf of O_{X/Σ}-algebra’s A on CRIS(X/Σ) and a homomorphism (of O_{X/Σ}-algebra’s)

2.1.5.1  A ⟶ i_{CRIS∗}O_{Y/Σ}

such that:

a) The sheaf A is a crystal of quasi-coherent O_{X/Σ}-modules.

b) For any object (U, T, δ) of CRIS(X/Σ) such that the morphism U → X is flat the homomorphism 2.1.5.1 induces an isomorphism

A_{T} ⟶^{≅} (i_{CRIS∗}O_{Y/Σ})_{T} .

The category of crystals of quasi-coherent O_{Y/Σ}-modules is equivalent to the category of crystals of quasi-coherent A-modules. This equivalence induces an equivalence of the subcategories consisting of crystals of finite presentation (resp. the subcategories consisting of crystals of finite locally free modules). Proof. We remark that a) and b) determine A up to unique isomorphism, hence it suffices to construct A in the case that X is affine. Say X = Spec(R) and Y ↪ X given by the ideal J ⊂ R.

Let us take a surjection P := ℤ_{p}[x_{α}] → R with kernel I ⊂ P and let us denote by (D, \overline{I}, δ) the divided power envelope of I in P, the divided powers taken compatible with γ on pP. Similarly for the exact sequence

0 ⟶ I(2) ⟶ P (2) := P ⊗_{ℤ_{p}} P ≅ ℤ_{p}[x_{α}, y_{α}] ⟶ R ⟶ 0

we get (D(2), \overline{I}(2), δ(2)). Next, we write J̃:= Ker(D → R/J) and put

D(J̃) = divided power envelope of J̃ in D, divided powers taken compatible with δ on \overline{I} and γ on pD.

In the same way we get J̃(2) ⊂ D(2) and its divided power envelope D(2, J̃(2)). Suppose (U′, T′, δ′) is an affine object of CRIS(X/Σ):

(U′, T′, δ′) = (Spec(B′/I′), Spec(B′), δ′_{n}: I′ → I′).

By construction of D we can find a homomorphism of PD-algebra’s φ : (D, \overline{I}, δ) ⟶ (B′, I′, δ′) lifting the given homomorphism R → B′/I′. We define A_{T} as the sheaf on T′_{fppf} associated to the B′-module B′ ⊗_{D} D(J̃). To prove this does not depend on the choice of φ we remark that the natural maps

D(J̃) ⊗_{D} D(2) ⟶ D(2, J̃(2)) ⟵ D(2) ⊗_{D} D(J̃)

are isomorphisms, see [B, I Proposition 2.8.2]. These isomorphisms satisfy an obvious cocycle condition. By construction, the sheaf A so defined is a crystal of quasi-coherent O_{X/Σ}-modules.

The value of i_{CRIS∗}(O_{Y/Σ}) on T′_{fppf} is the sheaf associated to the B′ module D_{B′}(J′) = the divided power envelope of J′ := Ker(B′ → (B′/I′)/J(B′/I′)) in B′, the divided powers taken compatible with δ′ on I′ and γ on pB′. The map 2.1.5.1 is defined as the sheaf homomorphism which on T_{fppf} is determined by the homomorphism of B′-algebra’s

2.1.5.2  B′ ⊗_{D} D(J̃) ⟶ D_{B′}(J′)

Therefore [B, I Proposition 2.8.2] gives b)′: assertion b) for (U, T, δ), where U → X is an open immersion. A slight generalization, Proposition 2.1.7 below, gives b) in general. However, for the proof of the last assertions of 2.1.5 we do not need 2.1.7. Indeed, we use 2.1.4 to go from O_{Y/Σ}-modules to i_{CRIS∗}O_{Y/Σ}-modules. Since these crystals are determined by their values on (U, T, δ)’s with U → X an open immersion (for example T = Spec(D/p^{n}D) we derive the equivalence from b)′. □

2.1.6 In this section the conventions and notations are those of [B, I Section 2.8.1], excepting (2.8.1) of [B], which we replace by:

2.1.6.1  The ring homomorphism B/I → B′/I′ is flat and J′/I′ = J/I · B′/I′.

Thus, we are given a commutative diagram

(B, I, δ)  ⟶  (B′, I′, δ′)
↖  ↗
(A, I_{0}, γ)

of algebra’s, δ and δ′ compatible with γ, B → B′ is a PD-homomorphism and ideals J ⊂ B (resp. J′ ⊂ B′) containing I (resp. I′). The divided power envelope D_{B}(J) of J in B (divided powers taken compatible with δ and γ) and similar D_{B′}(J′) are given.

2.1.7 Proposition. In the situation 2.1.6, the homomorphism

D_{B}(J) ⊗_{B} B′ ⟶ D_{B′}(J′)

is an isomorphism. Proof. The proof of this is the same as the proof of [B, I Proposition 2.8.2] except that one has to replace the sentence beginning on line 2 of page 55 by the following arguments: “Suppose a section x of J′_{2} ∩ Im(K ⊗ B′) is of the form

x = ∑_{i}(x_{i}^{[1]} − x_{i})a_{i} ⊗ α_{i}

with x_{i} ∈ J, a_{i} ∈ B and α_{i} ∈ B′. Transferring the elements a_{i} to the other side of the ⊗-sign we may assume a_{i} = 1 for all i. The degree zero component of x lies in the degree zero component of J′_{2}, which is I′. Hence we see that ∑_{i} x_{i}α_{i} = 0 in B′/I′. Flatness of B′/I′ over B/I implies that there are elements b_{ik} ∈ B and α′_{k} ∈ B′ such that

c_{i} := α_{i} − ∑ b_{ik}α′_{k} ∈ I′

and

d_{k} := ∑ x_{i}b_{ik} ∈ I.

(See [Mat, Theorem 1].) Thus we see that

x = ∑_{i}(x_{i}^{[1]} − x_{i}) ⊗ α_{i}

= ∑_{i}(x_{i}^{[1]} − x_{i}) ⊗ c_{i} + ∑_{i,k}(x_{i}^{[1]} − x_{i}) ⊗ b_{ik}α′_{k}

= ∑_{i}(x_{i}^{[1]} − x_{i}) ⊗ c_{i} + ∑_{k}(d_{k}^{[1]} − d_{k}) ⊗ α′_{k}.

Each of the terms lies in J′_{2} ∩ Im(K ⊗ B′).” □

2.2 Crystals and modules with connections.

2.2.1 Suppose A is an 𝔽_{p}-algebra satisfying 1.3.1.1. Let Ã be a lift of A and let J ⊂ Ã be an ideal of Ã such that p^{n}Ã ⊂ J for some n ∈ ℕ. We are going to describe crystals of O_{S/Σ}-modules, with S = Spec(Ã/J) in terms of modules with connections. We put

2.2.1.1  (D̂, Ĵ,^{[ ]}) = the p-adic completion of D_{Ã,γ}(J).

We denote by Ω̂1 ; Ã the module of continuous differentials of Ã (cf. Remark 1.3.4). There is a natural connection

2.2.1.2  ∇ : D̂ ⟶ D̂ ˆ⊗Ã Ω̂1 ; Ã ≅ D̂ ⊗Ã Ω̂1 ; Ã

compatible with d : Ã → Ω̂1 ; Ã and such that

2.2.1.3  ∇(j^{[n]}) = j^{[n−1]} ⊗ dj ∀j ∈ J, n ∈ ℕ.

(See Remark 2.2.4 d) for the construction of ∇.)

2.2.2 Proposition. The category of crystals of quasi-coherent O_{S/Σ}-modules is equivalent to the category of (p-adically) complete D̂-modules M endowed with an integrable, topologically quasi-nilpotent connection

∇ : M ⟶ M ⊗Ã Ω̂1 ; Ã

compatible with the connection 2.2.1.2 on D̂.

2.2.3 Corollary. The category of crystals of quasi-coherent O_{Spec(A)/Σ}-modules is equivalent to the category of (p-adically) complete Ã-modules M endowed with an integrable, topologically quasi-nilpotent connection

∇ : M ⟶ M ⊗Ã Ω̂1 ; Ã.

2.2.4 Remarks. a) By our conventions a complete module is separated.

b) In the above equivalences crystals of finite locally free O_{S/Σ}-modules correspond to finite locally free D̂-modules and vice-versa. Similarly, crystals of O_{S/Σ}-modules of finite presentation correspond to D̂-modules locally of finite presentation.

c) The condition of topological quasi-nilpotence means the following: given any derivation ϑ ∈ HomÃ(Ω̂1 ; Ã, Ã) the induced endomorphism ∇_{ϑ} : M → M is topologically quasinilpotent, i.e., for any m ∈ M, ∃n ∈ ℕ such that ∇_{ϑ}^{n}(m) ∈ pM.

d) We would like to add that, if E is such a crystal of quasi-coherent O_{S/Σ}-modules, the corresponding module M is defined as:

M := lim_{\overleftarrow{n}} Γ((S, Spec(D̂/p^{n}D̂),^{[ ]}), E).

As usual the connection comes from the identification

ε : M ˆ⊗D̂ D̂(2) ⟶^{≅} D̂(2) ˆ⊗D̂ M

which we get from the crystal property of E. Here D̂(2) is the p-adic completion of

D_{Ã⊗_{ℤp} Ã,γ}(Ker(Ã ⊗_{ℤ_{p}} Ã → Ã/J))

and we remark that if K := Ker(D̂(2) → D̂) then K is a sub PD-ideal and K/K^{[2]} ≅ D̂ ⊗Ã Ω̂1 ; Ã. If we apply this to E = O_{S/Σ} we get M ≅ D̂ and in this manner we derive the connection 2.2.1.2. We leave it to the reader to prove 2.2.1.3.

e) It follows from the preceding remark that if M (resp. M′) corresponds to E (resp. E′) then M ˆ⊗D̂ M′ corresponds to E ⊗_{O_{S/Σ}} E′.

f) Let J′ ⊂ Ã be a second ideal and D̂′ the corresponding divided power algebra 2.2.1.1. Assume that J ⊂ J′ ⊂ Ã, so that we get a closed immersion

i : S′: = Spec(Ã/J′) ⟶ S

Let A be the sheaf of O_{S/Σ}-algebra’s constructed in Proposition 2.1.5 for the closed immersion i. There is a diagram (commutative up to isomorphism of functors)

2.2.4.1 crystals of q.c. O_{S′/Σ}-modules

⟶^{2.1.5}

crystals of q.c. A-modules

⟶^{(1)}

crystals of q.c. O_{S/Σ}-modules

↓^{d)}  ↓^{d)}

complete D̂′-modules with i.t.q.n. connection

−

induced ; by D̂ → D̂′
⟶

complete D̂-modules with i.t.q.n. connection

The functor (1) is the forgetful functor induced by the homomorphism O_{S/Σ} → A giving the structure of O_{S/Σ}-algebra on A (Proposition 2.1.5).

g) If in f) we take J′ = J + pÃ then D̂ ≅ D̂′ and the map O_{S/Σ} → A is an isomorphism. (With the notations of the proof of 2.1.5, in this case D ≅ D(J̃).) Thus, in this situation, all horizontal arrows in 2.2.4.1 are equivalences. Hence to prove 2.2.2 we may assume pÃ ⊂ J.

h) The construction E ↦ M is functorial with respect to the pair (Ã, J). If (Ã′, J′) is another pair as in 2.2.1 and φ : Ã → Ã′ is a homomorphism with φ(J) ⊂ J′ then φ induces a homomorphism D̂ → D̂′ and f : S′ → S. The module M ˆ⊗D̂ D̂′ corresponds to f_{crys}^{∗}(E) if M corresponds to E.

Proof of 2.2.2 and 2.2.3. Let us assume 2.2.3 and deduce 2.2.2 from it. By remark 2.2.4 part g) we may assume there is a closed immersion i : S ↪ Spec(A). The crystal of quasi-coherent O_{Spec(A)/Σ}-modules A of Proposition 2.1.5 is determined by the Ã-module D̂ with its connection ∇ (2.2.1.1): the value of A on the triple

(Spec(A), Spec(Ã/p^{n}Ã), γ)

is D̂/p^{n}D̂ by 2.1.5 b). Its algebra structure is determined by the horizontal homomorphism

D̂ ˆ⊗Ã D̂ ⟶ D̂.

By Proposition 2.1.5, to prove 2.2.2 for S, we have to characterize crystals of quasi-coherent A-modules as D̂-modules with a connection. However, such a crystal E can be seen as a crystal of quasi-coherent O_{Spec(A)/Σ}-modules E together with a morphism of O_{Spec(A)/Σ}- modules

A ⊗_{O_{Spec(A)/Σ}} E ⟶ E

satisfying the usual conditions. Using 2.2.3 this translates into a complete Ã-module M, with a topologically quasi-nilpotent and integrable connection ∇ and a horizontal homomorphism

D̂ ˆ⊗Ã M ⟶ M

(of Ã-modules) satisfying certain conditions; these conditions are exactly that M becomes a D̂-module and the ‘horizontal’ implies that ∇ on M is compatible with ∇ on D̂. In this way we get 2.2.2 from 2.2.3.

To prove 2.2.3 we use lemma 1.3.3. It produces B, B̃ and Ã ← B̃, Ã → B̃ such that B has a finite p-basis. The result for S = Spec(B) is [BM, Proposition 1.3.3]. Using the functoriality 2.2.4 h) for Spec(A) ← Spec(B), Spec(A) → Spec(B) we deduce 2.2.3 in general. □

2.3 Dieudonné crystals and Dieudonné modules.

2.3.1 Let S → Spec(𝔽_{p}) be a scheme of characteristic p. If E is a crystal of O_{S/Σ}-modules we denote by E^{σ} its inverse image under the absolute Frobenius morphism of S.

2.3.2 Definition. A Dieudonné crystal over S is a triple (E, f, v) where:

(1) E is a crystal of finite locally free O_{S/Σ}-modules.

(2) f : E^{σ} → E and v : E → E^{σ} are homomorphisms of O_{S/Σ}-modules such that f ∘ v = p · id_{E} and v ∘ f = p · id_{E^{σ}}.

It is clear what homomorphisms of such Dieudonné crystals should be. We point out that this definition is different from [BM, 2.4] where E is only supposed to be of finite presentation. Since we are mainly interested in p-divisible groups, we prefer Definition 2.3.2. (But see also Remark 2.4.10.)

2.3.3 In [BBM, 3.3.6 & 3.3.10] there is constructed a crystalline Dieudonné module functor:

𝔻 : Category of p-divisible groups over S ⟶ Category of Dieudonné crystals over S

functorial with respect to the base scheme S.

2.3.4 Definition. In the situation 2.2.1, suppose given a lift of Frobenius σ : Ã → Ã such that σ(J) ⊂ J; in particular it induces σ on the algebra D̂ (2.2.1.1). A Dieudonné module over D̂ is a quadruple

(M, ∇, F, V)

where:

– M is a finite locally free D̂-module

– ∇ : M → M ⊗Ã Ω̂1 ; Ã is an integrable, topologically quasi-nilpotent connection – F : M ˆ⊗_{σ} D̂ → M and V : M → M ˆ⊗_{σ} D̂ are D̂-linear, horizontal and such that

F ∘ V = p · id_{M} and V ∘ F = p · id_{M ˆ⊗_{σ} D̂}.

Clearly, the category of Dieudonné crystals over S = Spec(Ã/J) is equivalent to the category of Dieudonné modules over D̂ (use 2.2.2 and 2.2.4 h). By Definition 2.3.2 this statement makes sense only for S with pO_{S} = 0, but we can extend Definition 2.3.2 to general S by relating crystals on S to crystals on S × Spec(𝔽_{p}) (compare 2.2.4 g).

2.4 Formal schemes, p-divisible groups and Dieudonné crystals.

2.4.1 Our formal schemes will always be adic, locally Noetherian formal schemes. Hence such a formal scheme 𝔛 has a biggest ideal of definition J ⊂ O_{𝔛}. We can write

𝔛 = lim_{⟶} 𝔛_{n}

where 𝔛_{n} is the scheme Spec(O_{𝔛}/J^{n}). The reduction 𝔛_{red} of 𝔛 is the reduced scheme 𝔛_{1} = Spec(O_{𝔛}/J).

In order to have a Dieudonné module functor over formal schemes, we simply adapt our definitions.

2.4.2 Definition. Let 𝔛 be a formal scheme.

a) A p-divisible group G over 𝔛 is a system (G_{n})_{n≥1} of p-divisible groups G_{n} over 𝔛_{n} endowed with isomorphisms G_{n+1}|_{𝔛_{n}} ≅ G_{n}.

b) Suppose 𝔛 → Spec(𝔽_{p}) lies in characteristic p. A Dieudonné crystal E over 𝔛 is a system (E_{n})_{n≥1} of Dieudonné crystals E_{n} over 𝔛_{n} endowed with isomorphisms E_{n+1}|_{CRIS(𝔛_{n}/Σ)} ≅ E_{n}.

2.4.3 It is clear that 2.3.3 extends to a functor

𝔻 : Category of p-divisible groups over 𝔛 ⟶ Category of Dieudonné crystals over 𝔛

(𝔛 → Spec(𝔽_{p}) in characteristic p). For a morphism of formal schemes 𝔛 ⟶^{φ} 𝔜 there are pullback functors φ^{∗} and there is an isomorphism 𝔻 ∘ φ^{∗} ≅ φ^{∗} ∘ 𝔻.

2.4.4 Lemma. If 𝔛 = Spf(A) is an affine formal scheme then the functor

Category of p-divisible groups over Spec(A) ⟶ Category of p-divisible groups over 𝔛 = Spf(A)

is an equivalence. Proof. Suppose G is a p-divisible group over 𝔛; we construct a p-divisible group over Spec(A). The group G is given by p-divisible groups G_{n} over 𝔛_{n} = Spec(A/I^{n}). The schemes G_{n}[p^{k}] := Ker(p^{k} : G_{n} → G_{n}) are given as Spec(B_{n,k}), where B_{n,k} is a finite locally free A/I^{n}-algebra of rank p^{k·height(G)}. The maps B_{n+1,k} → B_{n,k} being surjective (see Definition 2.4.2), we see that

B_{k} := lim_{\overleftarrow{n}} B_{n,k}

is a finite locally free A-algebra of rank p^{k·height(G)}. The p-divisible group we are looking for is ⋃_{k} Spec(B_{k}). □

The corresponding assertion for crystals of quasi-coherent O_{𝔛/Σ}-modules (defined as in 2.4.2) is not true. A counterexample can be given with 𝔛 = Spf(𝔽_{p}[[t]]). The ring A = 𝔽_{p}[[t]] has {t} as a p-basis. The free rank 1 A-module M = A with operator ∇ given by a ↦ t^{p+1}∂a/∂t ⊗ dt does not define a crystal on Spec(A), since t^{p+1}∂/∂t is not quasi-nilpotent on A, see 2.2.4 c). However, for each n, the pair (M, ∇) defines a crystal on Spec(𝔽_{p}[T]/(T^{n})). We prove the corresponding assertion for Dieudonné crystals only in the special case where A satisfies the condition of smoothness that was studied in 1.3. Let us give the corresponding definition for formal schemes.

2.4.5 Definition. We say that a formal scheme 𝔛 → Spec(𝔽_{p}) of characteristic p has property (†) if it satisfies one of the following two equivalent conditions: 2.4.5.1 There is a covering 𝔛 = ⋃ Spf(A_{i}) by affines of 𝔛 and each 𝔽_{p}-algebra A_{i} satisfies

1.3.1.1. 2.4.5.2 The morphism 𝔛 → Spec(𝔽_{p}) is formally smooth and the scheme 𝔛_{red} is locally

of the form Spec(Ā), where Ā is an algebra of finite type over a field with a finite p-basis.

2.4.6 Remark. We leave it to the reader to find the definition of a formally smooth morphism of formal schemes (used in 2.4.5.2) generalizing both [EGA, IV 17.1.1] and [EGA, 0_{IV} 19.3.1].

2.4.7 Examples. Let k be a field with char(k) = p and [k : k^{p}] < ∞. 2.4.7.1 Any formal scheme which is the completion of a scheme smooth over Spec(k) in

a closed subscheme satisfies (†). 2.4.7.2 Any scheme X which is locally of finite type over Spec(k) and regular satisfies (†).

(See 1.3.2.2; X is considered as a formal scheme with X = X_{red}.) In particular, we can take X = Spec(k).

2.4.8 Proposition. If 𝔛 = Spf(A) with (A, I) as in 1.3.1.1 then the functor

2.4.8.1  Category of Dieudonné crystals over Spec(A) ⟶ Category of Dieudonné crystals over Spf(A)

is an equivalence of categories. Proof. Suppose (Ã, σ) is a lift of A (1.3.3 & 1.2.2). Let Ĩ ⊂ Ã denote the inverse image of I in A. Let D̂_{n} be the divided power algebra constructed in 2.2.1.1 for the ideal J = Ĩ^{n} ⊂ Ã; it is isomorphic to the divided power algebra D̂′_{n} constructed for the ideal J′ = Ĩ^{n} + pÃ. Therefore, D̂_{n} is the ring figuring in 2.2.2 in the description of crystals on S_{n} = Spec(Ã/J′) = Spec(A/I^{n}). The homomorphism σ induces σ : D̂_{n} → D̂_{n}. Finally, we denote by i_{n} : S_{n} → S_{n+1} the closed immersion induced by A/I^{n+1} → A/I^{n}; there is also a corresponding homomorphism D̂_{n+1} → D̂_{n}.

A Dieudonné crystal on 𝔛 is given by Dieudonné crystals (E_{n}, f_{n}, v_{n}) on S_{n}, plus isomorphisms i_{n,CRIS}^{∗}(E_{n+1}) ≅ E_{n} of Dieudonné crystals. By 2.2.2 this translates into Dieudonné modules (M_{n}, ∇_{n}, F_{n}, V_{n}) over D̂_{n}, plus isomorphisms of Dieudonné modules

2.4.8.2  M_{n+1} ˆ⊗ D̂_{n} ⟶ M_{n}.

Let us consider the surjection

2.4.8.3  D̂_{n} ⟶ Ã/Ĩ^{n}.

Tensoring with it we get from the D̂_{n}-module M_{n} a finite locally free Ã/Ĩ^{n}-module

N_{n} := M_{n} ⊗ Ã/Ĩ^{n}.

On N_{n} there is an integrable connection ∇_{n} : N_{n} → N_{n} ⊗ Ω_{Ã/Ĩ^{n}}^{1} coming from ∇_{n} on M_{n} since by 2.2.1.3 we have

∇(Ker(2.4.8.3)) ⊂ Ker(D̂_{n} ⊗ Ω̂1 ; Ã ⟶ Ω_{Ã/Ĩ^{n}}^{1}).

The maps F_{n} and V_{n} induce horizontal homomorphisms

F_{n} : N_{n} ⊗_{σ} Ã/Ĩ^{n} ⟶ N_{n} and V_{n} : N_{n} ⟶ N_{n} ⊗_{σ} Ã/Ĩ^{n}

satisfying F_{n}V_{n} = p and V_{n}F_{n} = p. Finally, the isomorphisms 2.4.8.2 give homomorphisms N_{n+1} → N_{n}, compatible with ∇, F and V, inducing isomorphisms N_{n+1} ⊗ Ã/Ĩ^{n} ≅ N_{n}.

Thus the module

N := lim_{⟵} N_{n}

is a finite locally free Ã-module (Ã is complete for the Ĩ-adic topology, Remark 1.3.4 a). It is endowed with an integrable connection ∇ = lim ∇_{n} ( lim Ω_{Ã/Ĩ^{n}}^{1} = Ω̂1 ; Ã, Remark 1.3.4 c). There are horizontal homomorphisms

F = lim_{⟵} F_{n} : N^{σ} = N ⊗_{σ} Ã ⟶ N

and

V = lim_{⟵} V_{n} : N ⟶ N^{σ} = N ⊗_{σ} Ã

which satisfy F V = p and V F = p. To conclude that (N, ∇, F, V) is a Dieudonné module over Ã we still have to prove that ∇ is topologically quasi-nilpotent. The maps F an V give a complex

2.4.8.4  … ⟶ N^{σ}/pN^{σ} ⟶^{F} N/pN ⟶^{V} N^{σ}/pN^{σ} ⟶^{F} N/pN ⟶ …

which is automatically exact: for example, if x ∈ N/p^{2}N is such that V (x) = py for some y ∈ N^{σ}/p^{2}N^{σ} then x ≡ F (y) mod pN. For any ϑ ∈ HomÃ(Ω̂1 ; Ã, Ã) the operator ∇_{ϑ} is nilpotent of order at most p on N^{σ}/pN^{σ}, as this module has generators of the form x ⊗ 1 and ∇_{ϑ}(x ⊗ 1) = 0. By exactness of the complex above ∇_{ϑ} is nilpotent of order at most 2p on N/pN.

Conclusion: we have constructed a functor which goes from right to left in 2.4.8.1. We claim that it is quasi-inverse to 2.4.8.1. We only show how to get a functorial isomorphism of Dieudonné modules

2.4.8.5  N ˆ⊗Ã D̂_{k} ⟶ M_{k} , ∀k ∈ ℕ

and leave it to the reader to deduce the claim from it. To construct 2.4.8.5 it is enough to give a homomorphism N → M_{k} compatible with ∇, F and V fitting in a commutative diagram:

N ⟶ M_{k}
↘  ↙
N_{k}

Let f_{1}, …, f_{r} ∈ Ã be generators of the ideal Ĩ^{k}. Take x ∈ N:

x = lim_{⟵} x_{n} ∈ lim_{⟵} N_{n} = N

2.4.8.6 For any l, if n ≥ (r(p − 1) + pl)k then the element

y_{l} = image of x̃_{n} in M_{k}/p^{l}M_{k}

does not depend on the lift x̃_{n} ∈ M_{n} of the element x_{n} ∈ N_{n}. (The image is taken via the maps

M_{n} ⟶ M_{n−1} ⟶ … ⟶ M_{k}

in particular y_{l} also does not depend on n ≫ 0.)

Clearly, N → M_{k}, x = (x_{n}) ↦ (y_{l}) ∈ lim M_{k}/p^{l}M_{k} = M_{k} is the desired map.

Proof of 2.4.8.6. Since Ĩ^{k} is generated by the r elements f_{1}, …, f_{r}, Ĩ^{n} is generated by elements of the form

j = i · f_{1}^{α_{1}} · . . . · f_{r}^{α_{r}} · (f_{1}^{β_{1}} · . . . · f_{r}^{β_{r}})^{p}

with i ∈ Ĩ^{n−(r(p−1)+pl)k}, 0 ≤ α_{i} < p, 0 ≤ β_{i} and Σα_{i} + p(Σβ_{i}) = r(p − 1) + pl. This implies Σβ_{i} ≥ l. The image of such an element j in D̂_{k} is divisible by p^{l}:

j = (p!)^{Σβ_{i}} · i · f_{1}^{α_{1}} · . . . · f_{r}^{α_{r}} · (f_{1}^{[p]})^{β_{1}} · . . . · (f_{r}^{[p]})^{β_{r}} ∈ D̂_{k}

Therefore, the image in D̂_{k} of Ĵ_{n} = Ker(D̂_{n} → Ã/Ĩ^{n}) is contained in p^{l}D̂_{k}. (The elements j^{[m]}, with j as above generate Ĵ_{n}.) This implies 2.4.8.6 since

Ker(M_{n} ⟶ N_{n}) = Ĵ_{n} · M_{n}.  □

2.4.9 Corollary. Over a formal scheme 𝔛 of characteristic p which satisfies (†) (Definition 2.4.5) the crystalline Dieudonné module functor 2.4.3 is fully faithful. Proof. The question is local on 𝔛 so we may assume that 𝔛 = Spf(A) with A as in 1.3.1.1. By 2.4.4 and 2.4.8 we may replace 𝔛 by X = Spec(A) and the functor 2.4.3 by 2.3.3. Next, we choose an 𝔽_{p}-algebra B as in Lemma 1.3.3. A formal argument shows that it suffices to demonstrate fully faithfulness of 𝔻 over Spec(B). In this case we can apply [BM, Theorem 4.1.1]. □

2.4.10 Remark. With the appropriate definitions, the functor 𝔻 2.4.3 is also fully faithful on finite flat group schemes over 𝔛 as in 2.4.5. That is, in the definition (2.3.2) of Dieudonné crystals we have to work with crystals of O_{S/Σ}-modules of finite presentation and in the definition (2.3.4) of Dieudonné modules we have to work with D̂-modules of finite presentation. The proof of 2.4.4 works for finite flat group schemes as well and the proof of 2.4.8 still works to show that the functor 2.4.8.1 on Dieudonné crystals of finite presentation is fully faithful. Hence, the proof of 2.4.9 shows that 𝔻 is fully faithful on finite flat group schemes over 𝔛.

In the situation of Dieudonné crystals of finite presentation the proof of 2.4.8 no longer shows that 2.4.8.1 is an equivalence. Indeed, the argument to prove that ∇ on N is topologically nilpotent fails: it is no longer clear that 2.4.8.4 is exact if N is just some Ã-module of finite presentation. To remedy this, we introduce truncated Dieudonné crystals of level n: these are triples (E, f, v), where E is a crystal of finite locally free O_{S/Σ}/p^{n}O_{S/Σ}-modules and f & v are as usual. In the case n = 1 we have to add the condition that the sequence

2.4.10.1  … ⟶ E^{σ} ⟶^{f} E ⟶^{v} E^{σ} ⟶^{f} E ⟶ …

be exact. The reduction modulo p: (E/pE, f, v) of a truncated Dieudonné crystal of level n is a truncated Dieudonné crystal of level 1. By Nakayama’s lemma exactness of 2.4.10.1 need only be checked in the case that S is the spectrum of a field and here we can use the same argument as was used to prove exactness of 2.4.8.4. Hence for these truncated Dieudonné crystals proposition 2.4.8 will be true.

2.5 Filtration and the Kodaira-Spencer map.

2.5.1 In this section 𝔛 will be a formal scheme over Spec(𝔽_{p}) satisfying (†) and E will be a Dieudonné crystal over 𝔛. In this situation we put

E_{𝔛} := lim_{⟵} E_{𝔛_{n}} := lim_{⟵} E_{n,𝔛_{n}}

if {(E_{n}, f_{n}, v_{n})}_{n≥1} is the system defining E. The sheaf E_{𝔛} is a finite locally free O_{𝔛}-module. There is an integrable (quasi-nilpotent) connection

2.5.1.1  ∇ : E_{𝔛} ⟶ E_{𝔛} ⊗_{O_{𝔛}} Ω̂_{𝔛}^{1}.

Here we have put

Ω̂_{𝔛}^{1} = lim_{⟵} Ω_{𝔛_{n}/𝔽_{p}}^{1}.

It is a finite locally free O_{𝔛}-module (compare 1.3.3). The connection 2.5.1.1 may be constructed locally on 𝔛 by 2.2.2 or by using the connections ∇_{n} on E_{n,𝔛_{n}} coming from the fact that E_{n} is a crystal. Finally, there is an exact sequence (compare 2.4.8.4):

2.5.1.2  … ⟶ E^{(p)} ⟶^{f} E ⟶^{v} E^{(p)} ⟶^{f} E ⟶ …

with E_{𝔛}^{(p)} = Frob_{𝔛}^{∗}(E_{𝔛}) and where f (resp. v) is deduced from (f_{n})_{n≥0} (resp. (v_{n})_{n≥0}. In particular the homomorphisms f & v are horizontal, for example f(∇^{(p)}(s)) = ∇(f(s)) for a local section s of E_{𝔛}^{(p)}.

2.5.2 Proposition. There exists a locally free, locally direct summand ω ⊂ E_{𝔛} such that ω^{(p)} ⊂ E_{𝔛}^{(p)} is equal to Im(v) = Ker(f). It is uniquely determined by these conditions. Proof. For any closed point x ∈ 𝔛 the sequence 2.5.1.2 restricted to x is also exact. Indeed, this sequence is just the sequence associated to the Dieudonné crystal E|_{CRIS(x/Σ)} over x. Thus Im(v) is a locally direct summand of E_{𝔛}^{(p)}. As v is horizontal, Im(v) is invariant under ∇^{(p)} and hence is equipped with a connection whose p-curvature [K] is zero (the p-curvature of ∇^{(p)} is zero). Thus we are done if we prove that the functor Frob_{𝔛}^{∗} induces an equivalence of categories

finite locally free O_{𝔛}-modules ⟶ finite locally free O_{𝔛}-modules with a connection whose p-curvature is zero.

This is [K, Theorem 5.1] in the case of schemes smooth over 𝔽_{p}; exactly the same proof works for schemes having locally a finite p-basis. But we can reduce our case to this case by Lemma 1.3.3. □

2.5.3 We always denote by ω the subsheaf of E_{𝔛} found in 2.5.2 and write α for the quotient: α := E_{𝔛}/ω. It is a finite locally free O_{𝔛}-module. The connection induces an O_{𝔛}-linear homomorphism

2.5.3.1  ω ⟶ α ⊗_{O_{𝔛}} Ω̂_{𝔛}^{1}

We write Θ_{𝔛} := Hom_{O_{𝔛}}(Ω̂_{𝔛}^{1}, O_{𝔛}) for the sheaf of vector fields on 𝔛.

2.5.4 Definition. The Kodaira-Spencer map of the Dieudonné crystal E is the homomorphism of O_{𝔛}-modules

κ_{E} : Θ_{𝔛} ⟶ Hom_{O_{𝔛}}(ω, α)

deduced from 2.5.3.1. We say that E is versal if κ_{E} is a surjection.

2.5.5 If E = 𝔻(G), the Dieudonné crystal associated to the p-divisible group G, then the filtration

0 ⟶ ω ⟶ E_{𝔛} ⟶ α ⟶ 0

is nothing but the Hodge-filtration of 𝔻(G)_{𝔛} (see [BBM, 3.3.5 & 4.3.10]). We will say that G is versal if 𝔻(G) is versal. It follows from [BM, Corollary 3.2.11] and [M, V Theorem 1.6] that if G is versal then the restrictions of G to complete local rings of 𝔛 are versal deformations.

3 Deformations of Dieudonné crystals and p-divisible groups

In this chapter we construct versal deformations of Dieudonné crystals and p-divisible groups, at least over formal schemes 𝔛 satisfying (†).

3.1 Deformations of Dieudonné modules.

3.1.1 Notations. In this section (A_{0}, I_{0}) is a pair as in 1.3.1.1 and (Ã_{0}, σ) is a lift of A_{0}. We are going to look at A := A_{0}[[x_{1}, …, x_{n}]] and Ã := Ã_{0}[[x_{1}, …, x_{n}]]. We extend σ to Ã by putting σ(x_{i}) = x_{i}^{p}. Further we fix a natural number m ∈ ℕ. Put

B := A/(x_{1}^{m}, …, x_{n}^{m}) = Ã/(p, x_{1}^{m}, …, x_{n}^{m})

and

B_{+} := A/(x_{1}^{m+1}, …, x_{n}^{m+1}) = Ã/(p, x_{1}^{m+1}, …, x_{n}^{m+1})

The divided power algebra 2.2.1.1 associated to the ideal J = (p, x_{1}^{m}, …, x_{n}^{m}) of Ã (resp. J_{+} = (p, x_{1}^{m+1}, …, x_{n}^{m+1}) of Ã) is denoted by D̂ (resp. D̂_{+}).

For a multi-index I = (i_{1}, …, i_{n}) we define

x^{[I/m]} := ∏_{α=1}^{n} x_{α}^{i_{α}−m⌊i_{α}/m⌋} · (x_{α}^{m})^{[⌊i_{α}/m⌋]}.

(The symbol ⌊ ⌋ denotes the integer part function ℝ → ℤ.) It is easily seen that D̂ has the following description:

D̂ = {∑_{I} a_{I} · x^{[I/m]} | a_{I} ∈ Ã_{0} and lim_{|I|→∞} a_{I} = 0 in the p-adic topology}

A similar description can be given for D̂_{+}. From this it follows that both D̂ and D̂_{+} are p-torsion free.

We introduce a (large) number of ring homomorphisms.

1. The maps D̂ → B and D̂_{+} → B_{+}. (Reduction modulo the divided power ideals.)

2. The maps B_{+} → B and D̂_{+} → D̂. (Induced by J_{+} ⊂ J.)

3. A map π : D̂ → B_{+}. It is the unique Ã_{0}-linear map, such that π(x_{i}) = class of x_{i} and π((x_{i}^{m})^{[l]}) = 0 for l ≥ 2. The universal property of divided power envelopes implies that π is defined by the trivial divided power structure on the square zero ideal J/J_{+}.

4. The maps induced by σ on D̂ and D̂_{+}, also denoted σ, i.e., σ : D̂ → D̂ and σ : D̂_{+} → D̂_{+}.

5. A homomorphism τ : D̂ → D̂_{+}. It is defined as the σ-linear Ã_{0}-module morphism such that τ(x_{i}) = x_{i}^{p} and such that it is a divided power homomorphism. Thus τ((x_{i}^{m})^{[l]}) = x_{i}^{{(p−1)m−1}l} · (x_{i}^{m+1})^{[l]}.

6. The maps (cf. [BBM, 4.3.4.1]) Φ : B → D̄ := D̂/pD̂ and Φ_{+} : B_{+} → D̄_{+} := D̂_{+}/pD̂_{+}. These are the σ-linear Ã_{0}-module morphisms such that Φ(x_{i}) = class of x_{i}^{p} and Φ_{+}(x_{i}) = class of x_{i}^{p}.

With these definitions and notations we have the following commutative diagrams: (the bar ¯ stands for reduction modulo p.)

3.1.1.1  \begin{aligned}D̂_{+} & ⟶ & D̂ & ⟶^{τ} & D̂_{+} & D̂_{+} & ⟶ & D̂ \\ & & ↓^{π} & & ↓ & & σ ↘ & ↓^{τ} \\ & & B_{+} & ⟶^{Φ_{+}} & D̄_{+} & & & D̂_{+} \end{aligned}

and

3.1.1.2  \begin{matrix}D̄_{+} & ⟶ & D̄ & D̄ & ⟶ & B & D̄_{+} & ⟶ & B_{+} \\ ↓ & π ↙ & ↓ & & σ̄ ↘ & ↓^{Φ} & & σ̄_{+} ↘ & ↓^{Φ_{+}} \\ B_{+} & ⟶ & B & & & D̄ & & & D̄_{+} \\ ↓^{Φ_{+}} & & ↓^{Φ} \\ D̄_{+} & ⟶ & D̄ \end{matrix}

In diagram 3.1.1.1 the composition D̂_{+} → D̂⟶^{π}B_{+} is the reduction map 1. Finally, we remark that the connection ∇ on D̂ satisfies

∇(x_{i}^{m})^{[n]} = (x_{i}^{m})^{[n−1]}dx_{i}^{m} = mx_{i}^{m−1}(x_{i}^{m})^{[n−1]}dx_{i}

and that similar holds for ∇_{+}. From this it is easily seen that

3.1.1.3  ∇(Ker(π)) ⊂ Ker(D̂ → B) ⊗ Ω̂1 ; Ã.

It does not seem possible to describe deformations of a Dieudonné module from D̂ to D̂_{+}, or even to deform them at all in some cases. Therefore we introduce the notion of a special Dieudonné module.

3.1.2 Definition. A special Dieudonné module over D̂ is a six-tuple (N, ∇, F, V, α, χ) where

– (N, ∇, F, V) is a Dieudonné module over D̂,

– α is a finite locally free B-module,

– χ : N ⊗D̂ B → α is a surjection, these data being subject to the condition that the sequence

3.1.2.1  0 ⟶ N ⟶^{V} N ⊗_{σ} D̂ ⟶^{χ⊗idD̂} α ⊗_{Φ} D̄ ⟶ 0

is exact.

Note that in 3.1.2.1 the map V is automatically injective (since D̂ is p-torsion free) and that χ⊗idD̂ is automatically surjective. Of course, the same definition gives the notion of a special Dieudonné module over D̂_{+}.

We will say that a homomorphism of B_{+}-modules

χ′ : N ⊗_{π} B_{+} ⟶ α_{+}

lifts the homomorphism χ of Definition 3.1.2 if the following conditions are satisfied: 3.1.2.2 α_{+} is a finite locally free B_{+}-module, 3.1.2.3 χ′ is surjective, 3.1.2.4 there is a surjection α_{+} → α inducing an isomorphism α_{+} ⊗ B ≅ α and making

the following diagram commutative:

3.1.2.5  \begin{matrix}N ⊗_{π} B_{+} & ⟶^{χ′} & α_{+} \\ ↓ & & ↓ \\ N ⊗ B & ⟶^{χ} & α \end{matrix}

We remark that 3.1.2.3 follows from 3.1.2.2 & 3.1.2.4. It is important to note that given χ we can always find such a lift χ′. This is so since N ⊗_{π} B_{+} is a locally free B_{+}-module lifting N ⊗ B, hence the filtration defined by χ can be lifted.

3.1.3 Proposition. a) Given any special Dieudonné module (N, ∇, F, V, α, χ) over D̂ and a lift χ′ : N ⊗_{π}B_{+} ⟶ α_{+} of χ as above, there exists a special Dieudonné module (N_{+}, ∇_{+}, F_{+}, V_{+}, α_{+}, χ_{+}) over D̂_{+} and an isomorphism of Dieudonné modules over D̂:

N_{+} ˆ⊗ D̂ ⟶ N

such that χ_{+} is equal to the composition

3.1.3.1  N_{+} ⊗ B_{+} ≅ N_{+} ⊗ D̂ ⊗_{π} B_{+} ≅ N ⊗_{π} B_{+}⟶^{χ′}α_{+}.

The special Dieudonné module (N_{+}, ∇_{+}, F_{+}, V_{+}, α_{+}, χ_{+}) is determined up to unique isomorphism by this.

b) Any lift of (N, ∇, F, V, α, χ) to a special Dieudonné module (N_{+}, ∇_{+}, F_{+}, V_{+}, α_{+}, χ_{+}) is determined by a lift χ′ of χ as in part a).

Proof. Given (N, ∇, F, V, α, χ) and χ′ as in a), we define N_{+} as the kernel of the homomorphism:

3.1.3.2  \begin{matrix}N ˆ⊗_{τ} D̂_{+} & ⟶ & α_{+} ⊗_{Φ_{+}} D̄_{+} \\ 3.1.1.1↘ & ↗χ′ ⊗ id \\ (N ⊗_{π} B_{+}) ⊗_{Φ_{+}} D̄_{+} \end{matrix}

It is a finite locally free D̂_{+}-module by the following three facts: 1) α_{+} is a finite locally free B_{+}-module, 2) the map 3.1.3.2 is surjective, 3) D̂_{+} is p-torsion free. The homomorphism τ is compatible with the connections ∇ and ∇_{+} on D̂ and D̂_{+}, hence the connection ∇ on N induces a connection ∇′_{+} on N ˆ⊗_{Φ_{+}} D̄_{+} compatible with ∇_{+}. If we define a connection on α_{+} ⊗_{Φ_{+}} D̄_{+} by requiring that elements a ⊗ 1 are horizontal then it follows that 3.1.3.2 is compatible with these connections. In particular, we get a connection ∇_{+} : N_{+} ⟶ N_{+} ⊗ Ω̂1 ; Ã. Tensoring the defining exact sequence

3.1.3.3  0 ⟶ N_{+} ⟶ N ˆ⊗_{τ} D̂_{+} ⟶ α_{+} ⊗_{Φ_{+}} D̄_{+} ⟶ 0

with D̂ we get a commutative diagram:

3.1.3.4 \begin{aligned}0 & ⟶ & N_{+} ⊗ D̂ & ⟶ & N ⊗_{τ} D̂_{+} ⊗ D̂ & ⟶ & α_{+} ⊗_{Φ_{+}} D̄_{+} ⊗ D̄ & ⟶ & 0 \\ & & ↑? & & ↑≅ & & ↑≅ \\ 0 & ⟶ & N & ⟶^{V} & N ⊗_{σ} D̂ & ⟶ & α ⊗_{Φ} D̄ & ⟶ & 0 \end{aligned}

The upper line is exact since D̂ is p-torsion free. The isomorphisms of the right square and its commutativity are consequences of 3.1.1.1, 3.1.1.2 and 3.1.2.5. (The lower line is of course 3.1.2.1.) Thus we derive the existence of the left vertical arrow: N ≅ N_{+} ⊗ D̂. Using this we can write (use 3.1.1.1)

N ˆ⊗_{τ} D̂_{+} ≅ N_{+} ˆ⊗ D̂ ˆ⊗_{τ} D̂_{+} ≅ N_{+} ˆ⊗_{σ} D̂_{+}.

Fitting this into 3.1.3.3 we get

0 ⟶ N_{+} ⟶ N ˆ⊗_{τ} D̂_{+} ⟶ α_{+} ⊗_{Φ_{+}} D̄_{+} ⟶ 0
V_{+} ↘  ↓≅  ↗
N_{+} ˆ⊗_{σ} D̂_{+}

The south-east arrow in this diagram is our definition of V_{+}. It is left to the reader to check that the north-east arrow is χ_{+} ⊗ idD̂_{+} if we define χ_{+} by 3.1.3.1. Hence we immediately get (3.1.2.1)_{+} for (N_{+}, ∇_{+},?, V_{+}, α_{+}, χ_{+}). A consequence of this is that we can define F_{+} as the map x ↦ V_{+}^{−1}(px). It is easy to see that the various morphisms written above all are compatible with the corresponding connections. Hence, F_{+} and V_{+} are horizontal and (N_{+}, ∇_{+}, F_{+}, V_{+}) is a Dieudonné module . Finally, we have to check that N → N_{+} ⊗ D̂ is an isomorphism of Dieudonné modules. Since this involves writing down a large number of commutative diagrams, we feel this must be left to the reader.

The statement of unicity is easily proved, noting that for any solution (N_{+}, ∇_{+}, F_{+}, V_{+}, α_{+}, χ_{+}) of the problem, N_{+} must be (canonically) isomorphic to the kernel of 3.1.3.2. This proves a).

Part b) is proved in the same manner as the proof of unicity in part a). □

3.1.4 Remark. In the situation of 3.1.3 a) let

ω_{+} := Ker(χ′ : N ⊗_{π} B_{+} ⟶ α_{+}).

By 3.1.1.3 we get a Kodaira-Spencer map induced by ∇ on N:

κ : ω_{+} ⟶ N ⊗_{π} B_{+} ⟶^{∇} N ⊗ B ⊗ Ω̂_{A}^{1} ⟶ α_{+} ⊗ B ⊗ Ω̂_{A}^{1}

On the other hand, we get from ∇_{+} a Kodaira-Spencer map

κ_{+} : ω_{+} ⟶ α_{+} ⊗ Ω_{B_{+}}^{1}.

It follows from the proof of 3.1.3 that κ and κ_{+} agree as maps to α_{+} ⊗ B ⊗ Ω̂_{A}^{1}.

3.2 Deformations of Dieudonné crystals and p-divisible groups.

In this section we use 3.1.3 to deform a p-divisible group given a deformation of its crystal. Further we deform any Dieudonné crystal to a versal one, at least locally on formal schemes satisfying (†). If 𝔛 ↪ 𝔜 is a closed immersion of formal schemes and E is a (Dieudonné) crystal on 𝔜, we denote by E|_{𝔛} the restriction of E to 𝔛, i.e., the pullback of E via 𝔛 ↪ 𝔜.

3.2.1 Proposition. Suppose 𝔛 ↪ 𝔜 is a closed immersion of formal schemes satisfying (†), which induces an isomorphism 𝔛_{red} ≅ 𝔜_{red}. Given any p-divisible group G′ on 𝔛, a Dieudonné crystal E on 𝔜 and an isomorphism 𝔻(G′) ≅ E|_{𝔛}, there exists a p-divisible group G on 𝔜 with 𝔻(G) ≅ E. Proof. Since 𝔻 is fully faithful on 𝔜 by 2.4.9, the problem is local. Hence we may assume that 𝔛 = Spf(A_{0}), 𝔜 = Spf(A), 𝔛 ↪ 𝔜 is given by a surjection A → A_{0} and that A_{0} satisfies 1.3.1.1. Exactly the same argument as used in the proof of Lemma 1.3.3 shows that there exists an isomorphism A ≅ A_{0}[[M]] (compatible with A → A_{0}) where M is a finite projective A_{0}-module. Localizing a bit more we may assume that A ≅ A_{0}[[x_{1}, …, x_{n}]]. Thus we are in the situation of 3.1.1. Let us use the notations introduced there.

By Lemma 2.4.4, G′ comes from a p-divisible group (denoted G_{0}) over Spec(A_{0}) and by 2.4.8 E comes from a Dieudonné crystal (also denoted E) over Spec(A). By 2.5.2 the Dieudonné module (M, ∇, F, V) over Ã associated to E over 𝔜 has a unique structure of a special Dieudonné module (3.1.2) 𝕄 = (M, ∇, F, V, α, χ) over Ã.

Next, we remark that any p-divisible group G over Spec(B) (B as in 3.1.1, with m arbitrary) gives rise to a special Dieudonné module over D̂; we write 𝕄(G) = (M(G), ∇(G), F (G), V (G), α(G), χ(G)) for it. The Dieudonné module (M(G), ∇(G), F (G), V (G)) is just the Dieudonné module associated to the Dieudonné crystal of G. The B-module α(G) is the module which gives rise to the sheaf Lie(G^{∗}) on S_{fppf} = Spec(B)_{fppf} and χ(G) is the map induced by 𝔻(G)_{S} → Lie(G^{∗}), the Hodge filtration, see [BBM, 3.3.5]. That 𝕄(G) satisfies 3.1.2.1 is a consequence of [BBM, 4.3.10].

Suppose we have such a G and an isomorphism of special Dieudonné modules:

3.2.1.1  𝕄(G) ≅ 𝕄 ⊗ D̂

(It is clear how 𝕄 gives rise to a special Dieudonné module 𝕄⊗D̂ over D̂: 𝕄⊗D̂ = (M ˆ⊗ D̂, ∇ ⊗ 1, F ⊗ 1, V ⊗ 1, α ⊗ B, χ ⊗ id_{B}) .)

Claim. There exists a lift G_{+} over Spec(B_{+}) of the p-divisible group G over Spec(B), with an isomorphism

𝕄(G_{+}) ≅ 𝕄 ⊗ D̂_{+}

lifting 3.2.1.1.

The proof of this is a consequence of [M, V Theorem 1.6], [Ill2] and [BM, Corollary 3.2.11]. Indeed, since the ideal of S ↪ S_{+} = Spec(B_{+}) has a nilpotent divided power structure δ (with δ_{n} = 0 for n ≥ 0), we know that lifts of G are determined by lifts

ω_{+} ⊂ 𝔻(G)_{(S,S_{+},δ)}

of ω_{G} ⊂ 𝔻(G)_{S} (use references above). We remark that the lift G_{+} determined by ω_{+} will be such that ω_{+} = ω_{G_{+}} in 𝔻(G)_{(S,S_{+},δ)} = 𝔻(G_{+})_{S_{+}}.

By Proposition 3.1.3 b) our lift 𝕄 ⊗ D̂_{+} determines a lift

χ′ := χ ⊗ id_{B_{+}} : (M ⊗ D̂) ⊗_{π} B_{+} ⟶ α_{+} = α ⊗ B_{+}

of χ ⊗ id_{B}. Next, we use the isomorphism 3.2.1.1 to translate this into a filtration ω_{+} ⊂ M(G) ⊗_{π} B_{+}, which is the module associated to the sheaf 𝔻(G)_{(S,S_{+},δ)}. It determines G_{+} whose special Dieudonné module 𝕄(G_{+}) must be isomorphic to 𝕄 ⊗ D̂_{+} by the remark above and Proposition 3.1.3. This concludes the proof of the claim above.

Thus, if we show that for m = 1 we have

3.2.1.2  𝕄(G_{0}) ≅ 𝕄 ⊗ D̂

then by the claim above we can find, by induction, for all m a p-divisible group G_{m} over Spec(B_{m}) = Spec(A_{0}[x_{1}, …, x_{n}]/(x_{1}^{m}, …, x_{n}^{m})) with G_{m+1} ⊗ B_{m} ≅ G_{m} and such that 𝔻(G_{m}) corresponds to M ⊗ D̂_{m}. Taking the limit G = lim G_{m} over Spec(A) (see Lemma 2.4.4) gives the desired p-divisible group .

To prove 3.2.1.2 we note that the isomorphism M(G_{0}) → M ⊗ D̂, which is supposed to exist in the proposition, must automatically be an isomorphism of special Dieudonné module . Indeed, in the case m = 1, the kernel of χ : N ⊗D̂ A_{0} → α (see Definition 3.1.2) is the set of elements x ∈ N ⊗ A_{0} such that x ⊗ 1 ∈ N ⊗ A_{0} ⊗_{σ} A_{0} ≅ N ⊗_{σ} A_{0} lies in the image of V : N ⊗ A_{0} ⟶ N ⊗_{σ} A_{0}. Comparing this with Proposition 2.5.2 gives the result. □

3.2.2 Proposition. Let E be a Dieudonné crystal over an affine formal scheme 𝔛 = Spf(A_{0}). If A_{0} satisfies 1.3.1.1 then there exists a closed immersion 𝔛 ↪ 𝔜 = Spf(A) and a Dieudonné crystal E′ over 𝔜 such that

– 𝔛 ↪ 𝔜 induces an isomorphism 𝔛_{red} ≅ 𝔜_{red},

– E ≅ E′|_{𝔛},

– A satisfies 1.3.1.1,

– the Dieudonné crystal E′ is versal (Definition 2.5.4).

Proof. By abuse of notation we write α (resp. ω) for the finite locally free A_{0}-module corresponding to the sheaf α (resp. ω) of 2.5.3. Take generators k_{1}, …, k_{N} of the finite locally free A_{0}-module

K := Hom_{A_{0}}(ω, α)

and let

A = A_{0}[[x_{1}, …, x_{N}]].

We are in the situation of 3.1.1, the special Dieudonné module (N, ∇, F, V, α, χ) coming from E over 𝔛; it is a special Dieudonné module over D̂ (with m = 1): indeed, it is of the form

N ≅ N_{0} ˆ⊗Ã_{0} D̂

where (N_{0}, ∇_{0}, F_{0}, V_{0}) is the Dieudonné module over Ã_{0} associated to E over 𝔛. By Proposition 2.5.2 we have the exact sequence

3.2.2.1  0 ⟶ ω ⟶ N_{0} ⊗ A_{0} ⟶^{χ_{0}} α ⟶ 0

and χ is defined as the composition

N ⊗ A_{0} ≅ N_{0} ˆ⊗ D̂ ⊗ A_{0} ≅ N_{0} ⊗ A_{0} ⟶^{χ_{0}} α

It satisfies 3.1.2.1 by definition of ω (see 2.5.2).

In this way we also see that

3.2.2.2  N ⊗_{π} B_{+} ≅ N_{0} ˆ⊗Ã_{0} D̂ ⊗_{π} B_{+} ≅ N_{0} ⊗ A_{0}[x_{i}]/(x_{i}^{2}),

as B_{+} = A_{0}[x_{i}]/(x_{i}^{2}). Let us take a splitting of 3.2.2.1:

N_{0} ⊗ A_{0} ≅ ω ⊕ α.

As a lift ω_{+} ⊂ N_{0} ⊗ B_{+} of ω ⊂ N_{0} ⊗ A_{0} we take:

ω_{+} = {The submodule of N_{0} ⊗ B_{+} ≅ ω ⊗ B_{+} ⊕ α ⊗ B_{+} generated by elements of the form ξ ⊗ 1 + ∑_{i} k_{i}(ξ) ⊗ x_{i} ∈ ω ⊗ B_{+} ⊕ α ⊗ B_{+} where ξ runs through ω.

For α_{+} we take (N_{0} ⊗ B_{+})/ω_{+}. Next, we apply 3.1.3 to χ′ : N_{0} ⊗ B_{+} → α_{+}, the lift of χ. This gives a special Dieudonné module (N_{+}, ∇_{+}, F_{+}, V_{+}, α_{+}, χ_{+}) and an isomorphism N_{+} ⊗ D̂ ≅ N. By applying 3.1.3 again to a certain lift χ′_{+} of χ_{+} (it exists, see remark before Proposition 3.1.3) we get N_{2} over D̂_{2}, a special Dieudonné module over D̂_{2} (D̂_{+} of 3.1.1 with m = 2). Repeating this ad infinitum we get a special Dieudonné module (N_{∞}, ∇_{∞}, F_{∞}, V_{∞}, α_{∞}, χ_{∞}) over Ã_{0}[[\underline{x}]] (use 2.4.8), plus an isomorphism

3.2.2.3  N_{+} ≅ N_{∞} ⊗ D̂_{+}

of special Dieudonné modules.

We claim that the Kodaira-Spencer map

κ_{∞} : Hom_{Ã_{0}[[\underline{x}]]}(Ω_{Ã_{0}[[\underline{x}]]}^{1}, Ã_{0}[[\underline{x}]]) ⟶ Hom_{Ã_{0}[[\underline{x}]]}(ω_{∞}, α_{∞})

is surjective. By 3.2.2.3 and Remark 3.1.4 we see that this map is congruent to the map derived from:

ω_{+} ⟶ N ⊗_{π} B_{+} ⟶^{∇} N ⊗ B ⊗ Ω̂_{A}^{1} ⟶ α_{+} ⊗ B ⊗ Ω̂_{A}^{1}.

The isomorphism 3.2.2.2 is horizontal, hence

∇(ξ ⊗ 1 ⊕ ∑_{i} k_{i}(ξ) ⊗ x_{i}) ≡ ∇_{0}(ξ) ⊗ 1 + ∑_{i} k_{i}(ξ)dx_{i}.

It follows that κ_{∞}(∂/∂x_{i}) ≡ k_{i} mod (x_{1}, …, x_{N}). Here we used that

Hom_{Ã_{0}[[\underline{x}]]}(ω_{∞}, α_{∞}) mod (x_{1}, …, x_{N}) ≅ K

and that ∂/∂x_{i} : A_{0}[[\underline{x}]] → A_{0}[[\underline{x}]] is the derivation such that ∂/∂x_{i}(a) = 0 ∀a ∈ A_{0} and ∂/∂x_{i}(x_{j}) = δ_{ij} ∀i, j ∈ \{1, …, N\}. As A_{0}[[\underline{x}]] is complete with respect to the ideal (x_{1}, …, x_{N}) it follows that κ_{∞} is surjective. □

3.2.3 Remark. We proved something that is slightly stronger than 3.2.2. Namely, that one can take A = A_{0}[[x_{1}, …, x_{N}]] such that the elements ∂/∂x_{i} ∈ Θ_{𝔜} are mapped to a generating set of Hom_{O_{𝔜}}(ω′, α′). It is also clear that we can do this simultaneously for two crystals E_{1}, E_{2}, i.e., such that the elements ∂/∂x_{i} generate the module Hom_{O_{𝔜}}(ω′_{1}, α′_{1}) ⊕ Hom_{O_{𝔜}}(ω′_{2}, α′_{2}).

3.2.4 Corollary. The conclusion of 3.2.2 also holds for a p-divisible group G over 𝔛: We can deform G to a versal p-divisible group G′ over 𝔜. Proof. Combine 3.2.1 with 3.2.2. □

4 Proof of Main Theorem 1

This chapter is entirely devoted to the proof of the first Main Theorem. In short it asserts that the crystalline Dieudonné module functor is an equivalence over certain smooth formal schemes.

4.1 Statement of Main Theorem 1.

4.1.1 Theorem. If the formal scheme 𝔛 over 𝔽_{p} satisfies (†) then the functor 2.4.3

𝔻 : Category of p-divisible groups over 𝔛 ⟶ Category of Dieudonné crystals over 𝔛

is an anti-equivalence. □

We already know that 𝔻 is fully faithful (Corollary 2.4.9). It therefore suffices to construct G given a Dieudonné crystal E over 𝔛. The problem is local on 𝔛 and thus we may assume 𝔛 = Spf(A), with A as in 1.3.1.1.

The construction of G is done in a number of steps. We begin with the most difficult case, the case that A = k, a field with a finite p-basis.

4.2 Existence of G over fields.

This means that we are in the situation 𝔛 = 𝔛_{red} = Spec(k), where k is a field with a finite p-basis, i.e., [k : k^{p}] < ∞. The result will be proven in three steps. First we reduce to the case that k is separably closed. Next, we construct G in the case that E corresponds to a formal group G or the dual of a formal group. Finally, we do the general case over a separably closed field, using a result on extensions from 4.3.

Let (Λ, σ) be a lift of k. The Dieudonné crystal E determines and is determined by a Dieudonné module (M, ∇, F, V) over Λ. By 2.5.2 we also have a k-subvectorspace ω ⊂ M̄ = M/pM such that ω^{(p)} ⊂ M̄^{(p)} is the image of V : M̄ → M̄^{(p)}.

4.2.1 Reduction to the case: k is separably closed. Let {x_{1}, …, x_{n}} ⊂ k be a p-basis of k. We choose a separable closure k^{sep} of k. Choose a set I indexing all field extensions of k contained in k^{sep}. For each i ∈ I let k_{i} ⊂ k^{sep} denote the corresponding subfield and say that i = sep gives k_{sep} = k^{sep}. Choose a lift Λ_{i} of k_{i} and let x̃_{i,1}, …, x̃_{i,n} ∈ Λ_{i} be elements lifting x_{1}, …, x_{n} ∈ k ⊂ k_{i}. The elements x_{1}, …, x_{n} form a p-basis for k_{i}, ∀i ∈ I. As a lift of Frobenius on each Λ_{i} we take the unique lift σ such that σ(x̃_{i,l}) = (x̃_{i,l})^{p}, l = 1, …, n (see [BM,1.2.7]). If k_{i} ⊂ k_{j}, i, j ∈ I then the homomorphism

4.2.1.1  \begin{aligned}α_{i,j} & : & Λ_{i} & ⟶ & Λ_{j} \\ & & x̃_{i,l} & ↦ & x̃_{j,l} & , l = 1, …, n \end{aligned}

lifting k_{i} ↪ k_{j} (see [BM, 1.2.6]) commutes with σ. We note that

4.2.1.2  k^{sep} = k_{sep} = lim_{\overrightarrow{[k_{i}:k]<∞}} k_{i}

and for any n ∈ ℕ:

4.2.1.3  Λ_{sep}/p^{n}Λ_{sep} = lim_{\overrightarrow{[k_{i}:k]<∞}} Λ_{i}/p^{n}Λ_{i}

This is a consequence of [BM, 1.1.5] and 4.2.1.2.

Let E be our given Dieudonné crystal over Spec(k) and let (M, ∇, F, V) be the Dieudonné module over Λ associated to it. Suppose we have a p-divisible group G′ over Spec(k^{sep}) with

𝔻(G′) ≅ E|_{CRIS(Spec(k^{sep})/Σ)}.

By 4.2.1.2 the finite group scheme G′[p^{n}] is of the form H ⊗ k^{sep} where H is a finite group scheme over Spec(k_{i}) for some i ∈ I such that [k_{i} : k] < ∞. The isomorphism

𝔻(H)|_{CRIS(Spec(k^{sep})/Σ)} ≅ 𝔻(G′[p^{n}]) ≅ E/p^{n}E|_{CRIS(Spec(k^{sep})/Σ)}

is defined by an isomorphism of truncated Dieudonné modules of level n (compare Remark 2.4.10)

M(H) ⊗_{Λ_{i}} Λ_{sep} ⟶ (M/p^{n}M) ⊗_{Λ} Λ_{sep}.

From 4.2.1.3 we see that this map is already defined over some Λ_{j}, with j ∈ I such that k_{i} ⊂ k_{j} and [k_{j} : k] < ∞. The resulting isomorphism over Λ_{j} will be automatically an isomorphism of truncated Dieudonné modules over Λ_{j} since the maps α_{i,j} and α_{i,sep} (4.2.1.1) are compatible with σ’s and Ω_{Λ_{j}/p^{n}Λ_{j}}^{1} → Ω_{Λ_{∞}/p^{n}Λ_{∞}}^{1} is injective [BM, 1.3.1].

We conclude that for all n ∈ ℕ there exists a finite separable extension k ⊂ k′, a finite group scheme H over Spec(k′) and an isomorphism

4.2.1.4  𝔻(H) ≅ (E/p^{n}E)|_{CRIS(Spec(k′)/Σ)}.

Since 𝔻 is fully faithful over Spec(k′ ⊗_{k} k′) and Spec(k′ ⊗_{k} k′ ⊗_{k} k′) on finite group schemes [BM, Theorem 4.1.1] we see that H descends to a group scheme G_{n} over Spec(k) with 𝔻(G_{n}) ≅ E/p^{n}E (the isomorphism 4.2.1.4 descends too). By fully faithfulness over Spec(k) we conclude that the system (G_{n})_{n≥1} gives a p-divisible group over Spec(k).

4.2.2 The case that F (or V) is topologically nilpotent. This means that the composition

F^{n} = F ∘ F^{σ} ∘ … ∘ F^{σ^{n−1}} : M^{σ^{n}} ⟶ M

lands in pM for some n big enough. (M^{σ} := M ⊗_{σ} Λ.) By duality it suffices to treat the case that F is topologically nilpotent (see [BBM, 5.3]). We remark that in this subsection we do not need to assume that k is separably closed.

Let us define M_{1} ⊂ M by

M_{1} := {m ∈ M|m mod p ∈ ω ⊂ M̄}.

By our choice of ω ⊂ M̄ the map M_{1}^{σ} → M^{σ} gives an isomorphism of M_{1}^{σ} with V M. Inverting V gives us an isomorphism M_{1}^{σ} → M. The association

(M, ∇, F, V) ↦ (M_{1}, the isomorphism M_{1}^{σ} ≅ M)

is functorial both in the Dieudonné module (M, ∇, F, V) and for homomorphisms Λ → Λ′ commuting with σ&σ′ (in particular σ itself). Thus we get F_{1} : M_{1}^{σ} → M_{1} and V_{1} : M_{1} → M_{1}^{σ} from F and V and an isomorphism

(M_{1}^{σ}, F_{1}^{σ}, V_{1}^{σ}) ≅ (M, F, V).

We put S = Spec(Λ); it has a lift of Frobenius f_{S} = Spec(σ). If G is a p-divisible group over k, then we put

M_{S}(G) = lim_{\overleftarrow{n}} M_{S}(G[p^{n}]),

where M_{S} is the contravariant functor on truncated Barsotti-Tate groups defined in [dJ, Section 8]. By [dJ, Corollary of Theorem 9.3] we can find a p-divisible group G over Spec(k) with an isomorphism

M_{S}(G) ≅ (M_{1}, F_{1}, V_{1}).

By [dJ, 7.1 & 3.5.2] we get

lim_{\overleftarrow{n}} 𝔻(G)_{(Spec(Λ/p^{n}Λ),Spec(k),γ)} ≅ M_{S}(G^{(p)}) ≅ M_{S}(G)^{σ} ≅ M_{1}^{σ} ≅ M.

All the isomorphisms occurring in this line are compatible with the actions of Frobenius and Verschiebung of G. Hence we see that

4.2.2.1  (M_{G}, F_{G}, V_{G}) ≅ (M, F, V)

where (M_{G}, ∇_{G}, F_{G}, V_{G}) is the Dieudonné module over Λ associated to the Dieudonné crystal 𝔻(G) of G. By transport of structure, 4.2.2.1 gives a second connection ∇′ on M (coming from ∇_{G}) such that F and V are horizontal. If we show that ∇′ = ∇ then E ≅ 𝔻(G) and we are done.

We remark that

p^{n}(∇ − ∇′) = F^{n}V^{n}(∇ − ∇′) = F^{n}(∇ − ∇′)^{σ^{n}}V^{n}.

Since (∇ − ∇′)^{σ^{n}} is divisible by p^{n} (σ : Ω̂_{Λ}^{1} → Ω̂_{Λ}^{1} is divisible by p) we get

∇ − ∇′ = F^{n} \frac{(∇ − ∇′)^{σ^{n}}}{p^{n}} V^{n}.

As F is topologically nilpotent, the right hand side tends to zero as n goes to infinity.

4.2.3 Neither F nor V topologically nilpotent. We assume given a separably closed field k of characteristic p, with a finite p-basis and a Dieudonné crystal E over Spec(k). As before (Λ, σ) is a lift of k and (M, ∇, F, V) is the Dieudonné module over Λ associated to E. The proof of the existence of G will be completely analogous to the proof of Theorem 10.3 in [dJ].

First we claim there is a commutative diagram of Dieudonné modules over Λ:

Diagram I  \begin{aligned}& & & & 0 & & 0 \\ & & & & ↑ & & ↑ \\ & & & & M^{μ} & = & M^{μ} \\ & & & & ↑ & & ↑ \\ 0 & ⟶ & M^{ét} & ⟶ & M & ⟶ & M_{1} & ⟶ & 0 \\ & & ∥ & & ↑ & & ↑ \\ 0 & ⟶ & M^{ét} & ⟶ & M_{2} & ⟶ & M_{12} & ⟶ & 0 \\ & & & & ↑ & & ↑ \\ & & & & 0 & & 0 \end{aligned}

with exact rows and columns and such that – F is topologically nilpotent on M_{12}, M_{1}, M^{μ},

– V is topologically nilpotent on M_{12}, M_{2}, M^{ét},

– F is an isomorphism on M^{ét},

– V is an isomorphism on M^{μ}. Here M_{1} is the first member of (M_{1}, ∇_{1}, F_{1}, V_{1}), etc. The construction of the diagram is left to the reader, we only point out that

M^{ét} = ⋂_{n≥1} F^{n}(M^{σ^{n}}) ⊂ M,

the quotient M → M^{μ} is defined dually, M_{1} = M/M^{ét}, M_{2} = Ker(M → M^{μ}) and M_{12} = Ker(M_{1} → M^{μ}). See also [dJ, proof of theorem 10.3].

By 4.2.2 we already have p-divisible groups G_{12}, G_{1}, G_{2}, G^{μ} and G^{ét} corresponding to (M_{12}, ∇_{12}, …), … The p-divisible group G^{ét} is étale since F^{ét} is an isomorphism. As k is separably closed we can choose an isomorphism G^{ét} ≅ (ℚ_{p}/ℤ_{p})^{s} some s ∈ ℕ. Similarly, G^{μ} ≅ (𝔾_{m}[p^{∞}])^{t} some t ∈ ℕ.

Fully faithfulness of 𝔻 gives us an incomplete diagram corresponding to diagram I:

Diagram II  \begin{aligned}& & & & 0 & & 0 \\ & & & & ↓ & & ↓ \\ & & & & G^{μ} & = & G^{μ} \\ & & & & ↓ & & ↓ \\ 0 & ⟵ & G^{ét} & ⟵ & ? & ⟵ & G_{1} & ⟵ & 0 \\ & & ∥ & & ↓ & & ↓ \\ 0 & ⟵ & G^{ét} & ⟵ & G_{2} & ⟵ & G_{12} & ⟵ & 0 \\ & & & & ↓ & & ↓ \\ & & & & 0 & & 0 \end{aligned}

(The lower line and right column are exact sequences of p divisible groups.) For all n ∈ ℕ we derive from diagram II a diagram by taking the kernel of multiplication by p^{n}:

Diagram n  \begin{aligned}& & & & 0 & & 0 \\ & & & & ↓ & & ↓ \\ & & & & G^{μ}[p^{n}] & = & G^{μ}[p^{n}] \\ & & & & ↓ & & ↓ \\ 0 & ⟵ & G^{ét}[p^{n}] & ⟵ & ? & ⟵ & G_{1}[p^{n}] & ⟵ & 0 \\ & & ∥ & & ↓ & & ↓ \\ 0 & ⟵ & G^{ét}[p^{n}] & ⟵ & G_{2}[p^{n}] & ⟵ & G_{12}[p^{n}] & ⟵ & 0 \\ & & & & ↓ & & ↓ \\ & & & & 0 & & 0 \end{aligned}

The obstruction to find a sheaf of ℤ/p^{n}ℤ-modules G_{n} on Spec(k)_{fppf} fitting into diagram n (i.e., sitting in place of ? and making the middle line and middle column exact) lies in the group

Ext_{ℤ/p^{n}ℤ}^{2}(G^{ét}[p^{n}], G^{μ}[p^{n}])

(extensions of sheaves of ℤ/p^{n}ℤ-modules on Spec(k)_{fppf}). It is easy to prove that

Ext_{ℤ/p^{n}ℤ}^{2}(ℤ/p^{n}ℤ, μ_{p^{n}}) ≅ H_{fppf}^{2}(Spec(k), 𝔾_{m})[p^{n}] = 0.

Therefore this obstruction is zero. The set of all choices of such sheaves G_{n} is homogeneous under the group

Ext_{ℤ/p^{n}ℤ}^{1}(G^{ét}[p^{n}], G^{μ}[p^{n}]) ≅ Ext_{ℤ/p^{n}ℤ}^{1}(ℤ/p^{n}ℤ, μ_{p^{n}})^{s·t} ≅ (k^{∗}/(k^{∗})^{p^{n}})^{s·t}.

Since the homomorphism k^{∗}/k^{∗}^{p^{n+1}} → k^{∗}/k^{∗}^{p^{n}} is surjective, given any G_{n} fitting into diagram n we can find a G_{n+1} fitting into diagram n + 1 such that G_{n} ≅ G_{n+1}[p^{n}] (an isomorphism of sheaves fitting into diagram n, this isomorphism is unique: μ_{p^{n}}(k) = {1}).

Clearly, this proves that we can find at least a sheaf G′ = ⋃ G_{n} fitting into the diagram II. Since the sheaves G_{n} are representable, as they are extensions of representable sheaves, it follows that G′ is a p-divisible group. The set of isomorphism classes of such G′ fitted into diagram II is homogeneous under the group

Extp-div. gr.^{1}(G^{ét}, G^{μ}) ≅ Extp-div. gr.^{1}(ℚ_{p}/ℤ_{p}, 𝔾_{m}[p^{∞}])^{s·t}

and the corresponding group for diagram I is:

Ext_{DC}^{1}(𝔻(G^{μ}), 𝔻(G^{ét})) ≅ Ext_{DC}^{1}(𝔻(𝔾_{m}[p^{∞}]), 𝔻(ℚ_{p}/ℤ_{p}))^{s·t}

To produce G it therefore suffices to prove Proposition 4.3.6 below.

4.3 Extensions of étale p-groups by multiplicative p-groups.

In this section we prove the analogs of [dJ] 8.7, 8.8 and 8.9 in the crystalline setting.

Let S be a scheme of characteristic p. We introduce the following categories: For any n ∈ ℕ we put

BT(n)_{S} = the category of truncated Barsotti-Tate groups of level n over S

and

DC(n)_{S} = the category of truncated Dieudonné crystals of level n over S (see Remark 2.4.10).

In both categories it is clear what a short exact sequence is and there are corresponding extension groups. The crystalline Dieudonné module functor

𝔻 : BT(n)_{S} ⟶ DC(n)_{S}

(see [BBM, 3.3.6 & 3.3.10]) transforms short exact sequences into short exact sequences; this follows from [BBM, 4.3.1].

4.3.1 Lemma. There exists a natural injection

Γ(S, O_{S}^{∗})/Γ(S, O_{S}^{∗})^{p^{n}} ⟶ Ext_{BT(n)_{S}}^{1}(ℤ/p^{n}ℤ_{S}, μ_{p^{n},S})

which is an isomorphism if H_{fppf}^{1}(S, 𝔾_{m})[p] = (0). Proof. This is well known (cf. [dJ, Lemma 8.7]). Let us describe the extension E_{n}(g) associated to a section g ∈ Γ(S, O_{S}^{∗}). The sheaf E_{n}(g) is defined as follows:

E_{n}(g)(U) = {(f, m) ∈ Γ(U, O_{U}^{∗} × ℤ_{U})| f^{p^{n}} = g^{m}}/⟨(g, p^{n})⟩

for U ∈ ObSch/S. The maps which define the structure of extension on E_{n}(g) are

μ_{p^{n}}(U) ∋ f ↦ (f, 0) ∈ E_{n}(g)(U)

and

E_{n}(g) ∋ (f, m) ↦ m̄ ∈ ℤ/p^{n}ℤ(U).  □

In [BM, 2.1 & 2.2] the Dieudonné crystals of μ_{p^{n},S} and ℤ/p^{n}ℤ_{S} are determined. The result is as follows:

𝔻(ℤ/p^{n}ℤ_{S}) ≅ (O_{S/Σ}/p^{n}O_{S/Σ}, 1, p)

and

𝔻(μ_{p^{n},S}) ≅ (O_{S/Σ}/p^{n}O_{S/Σ}, p, 1).

The identifications are given by 2.1.3.1 and 2.2.3.1 of [BM].

Suppose that S = Spec(A) and A is a ring with a p-basis. Let (Ã, σ) be a lift of A, put A_{n} = Ã/p^{n}Ã and Ω_{n} = Ω_{A_{n}}^{1}. For any n ∈ ℕ we define an abelian group W_{n} as follows:

W_{n} := {(η, a) ∈ Ω_{n} × A_{n} | dη = 0, σ(η) = da + pη}/{(db, σ(b) − pb)|b ∈ A_{n}}

4.3.2 Lemma. There is natural isomorphism

W_{n} ⟶ Ext_{DC(n)_{S}}^{1}(𝔻(μ_{p^{n},S}), 𝔻(ℤ/p^{n}ℤ_{S})).

Proof. For each pair (η, a) ∈ Ω_{n} × A_{n} with dη = 0 and σ(η) = da + pη we define an extension of truncated Dieudonné modules of level n:

0 ⟶ (A_{n}, d_{A_{n}/ℤ}, 1, p) ⟶ M(η, a) ⟶ (A_{n}, d_{A_{n}/ℤ}, p, 1) ⟶ 0.

It is defined by the following formulae:

4.3.2.1  \{ \begin{matrix}M(η, a) & = A_{n} · 1 ⊕ A_{n} · ˜1 \\ ∇1 & = 0 \\ F 1^{σ} & = 1 \\ V 1 & = p · 1^{σ} \end{matrix}  \begin{aligned}∇˜1 & = 1 ⊗ η \\ F ˜1^{σ} & = a · 1 + p · ˜1 \\ V ˜1 & = −a · 1^{σ} + ˜1^{σ} \end{aligned}

The reader can easily convince himself that this defines indeed an extension of Dieudonné modules and that any such extension can be written in this form. Also, it is clear that all possible isomorphisms of such extensions are the isomorphisms (with b ∈ A_{n})

\begin{matrix}M(η + db, a + σ(b) − pb) & ⟶ & M(η, a) \\ 1 & ⟼ & 1 \\ ˜1 & ⟼ & b · 1 + ˜1. \end{matrix}

Finally, it is also easy to show that

[M(η, a)] + [M(η′, a′)] = [M(η + η′, a + a′)].

(Addition of extension classes.)

Therefore our lemma follows from the description of crystals over S in terms of Ãmodules with connections [BM, 1.3.3] and the description of 𝔻(μ_{p^{n}}) and 𝔻(ℤ/p^{n}ℤ) given above. □

4.3.3 Remark. A similar result holds in the situation of Definition 2.3.4.

We define a homomorphism

4.3.4  A^{∗}/(A^{∗})^{p^{n}} ⟶ W_{n}

by the rule:

class of g ∈ A^{∗} ⟼ class of (dlog(g̃), log(σ(g̃)g̃^{−p}))

Here g̃ ∈ A_{n}^{∗} is an element mapping to g ∈ A^{∗}. Since σ(g̃) ≡ g̃^{p} mod pA_{n}, the product σ(g̃)g̃^{−p} can be written as 1 + pδ and we can take its logarithm:

log(1 + pδ) = pδ − \frac{1}{2}(pδ)^{2} + \frac{1}{3}(pδ)^{3} − … ∈ A_{n}

It is easy to prove that the element of W_{n} so defined does not depend on the lift g̃ (use that any other lift g̃′ can be written as g̃ · (1 + pδ)).

4.3.5 Lemma. The following diagram is commutative:

A^{∗}/(A^{∗})^{p^{n}}  ⟶^{4.3.4}  W_{n}
↓^{4.3.1}  ↓^{4.3.2}
Ext_{BT(n)_{S}}^{1}(ℤ/p^{n}ℤ_{S}, μ_{p^{n},S}) ⟶^{𝔻} Ext_{DC(n)_{S}}^{1}(𝔻(μ_{p^{n},S}), 𝔻(ℤ/p^{n}ℤ_{S}))

Proof. Suppose (U, T, δ) is an object of CRIS(S/Σ) and g ∈ A^{∗} is lifted to g̃ ∈ Γ(T, O_{T}^{∗}). To this we will associate an extension of sheaves on CRIS(U/T):

0 ⟶ O_{U/T} ⟶ U(g̃) ⟶ \underline{E_{n}(g)_{U}} ⟶ 0

We recall that for a sheaf F on the big fppf-site of S, we write \underline{F} for the sheaf on CRIS(S/Σ) defined by \underline{F}(U, T, δ) := F(U); see [BBM, page 15] for this notation. The extension U(g̃) is defined as the push-out by log : (1 + I_{U/T})^{∗} → O_{U/T} of an extension

0 ⟶ (1 + I_{U/T})^{∗} ⟶ V(g̃) ⟶ \underline{E_{n}(g)_{U}} ⟶ 0.

Here the sheaf V(g̃) is defined as follows: for (U′, T′, δ′) ∈ Ob CRIS(U/T) the sections of V(g̃) over (U′, T′, δ′) are:

{(f, m) ∈ Γ(T′, O_{T′}^{∗} × ℤ_{T′})|the image f̄ of f in O_{U′}^{∗} satisfies f̄^{p^{n}} = g^{m}}/⟨(g̃, p^{n})⟩

It is clear how V(g̃) becomes an extension as indicated above.

The extension U(g̃) defines a section ˜1(g̃) of the sheaf

Ext_{U/T}^{1} (\underline{E_{n}(g)_{U}}, O_{U/T}) ≅ 𝔻(E_{n}(g)_{U})|_{CRIS(U/T)}.

Under the map

𝔻(E_{n}(g)_{U})_{T} ⟶ 𝔻(μ_{p^{n}})_{T} ≅ O_{T} /p^{n}O_{T}

the element ˜1(g̃) maps to 1; this is a consequence of our definitions and the definition of [BM, 2.2.3.1].

In a similar way the sheaf X (a), defined for a ∈ Γ(T, O_{T}) on CRIS(U/T) by

Γ((U′, T′, δ′), X (a)) := {(f, m) ∈ Γ(T′, O_{T′} × ℤ_{T′})}/⟨(a, p^{n})⟩

is an extension

0 ⟶ O_{U/T} ⟶ X (a) ⟶ \underline{ℤ/p^{n}ℤ_{U}} ⟶ 0

and corresponds to the section a of

𝔻(ℤ/p^{n}ℤ)_{U/T} = Ext_{U/T}^{1} (\underline{ℤ/p^{n}ℤ_{U}}, O_{U/T}) ≅ O_{U/T} /p^{n}O_{U/T} .

Suppose g̃_{1} and g̃_{2} ∈ Γ(T, O_{T}^{∗}) are two sections lifting the image of g in Γ(U, O_{U}^{∗}). We claim that the difference ˜1(g̃_{1}) − ˜1(g̃_{2}) is given by the class of the extension X (log(g̃_{1}g̃_{2}^{−1})) in

𝔻(ℤ/p^{n}ℤ)_{T} ⊂ 𝔻(E_{n}(g))_{T} .

To prove this we consider the pullback

0 ⟶ O_{U/T} ⊕ O_{U/T} ⟶  P  ⟶  \underline{E_{n}(g)_{U}}  ⟶ 0
∥  ↓  ↓^{(1,−1)}
0 ⟶ O_{U/T} ⊕ O_{U/T} ⟶ U(g̃_{1}) ⊕ U(g̃_{2}) ⟶ \underline{E_{n}(g)_{U}} ⊕ \underline{E_{n}(g)_{U}} ⟶ 0

and we construct a homomorphism

α : P ⟶ X (log(g̃_{1}g̃_{2}^{−1}))

which proves the equality (we leave it to the reader to check this). If (U′, T′, δ′) ∈ Ob CRIS(U/T) and (f_{i}, m_{i}) ∈ Γ(T′, O_{T′}^{∗}) × ℤ is such that f̄_{i}^{p^{n}} = g^{m_{i}} for i = 1, 2 and l ∈ ℤ is such that f̄_{1} · g^{l} = f̄_{2}^{−1}, m_{1} + lp^{n} = −m_{2} then we associate to this the pair

(log(f_{1}f_{2}g̃_{1}^{l}), m_{1} + lp^{n}) ∈ Γ(T′, O_{T′}) × ℤ.

(And hence a section of X (log(g̃_{1}g̃_{2}^{−1}).) We again leave it to the reader to check this induces a homomorphism α as desired using the definitions of V(g̃_{i}), etc; the crucial point is that to the pairs (f_{1}, m_{1}) and (f_{2} · g̃_{2}^{−1}, m_{2} − p^{n}) is associated the pair

(log(f_{1}f_{2}g̃_{2}^{−1}g̃_{1}^{l+1}), m_{1} + lp^{n} + p^{n}) = (log(g̃_{1}g̃_{2}^{−1}), p^{n}) + (log(f_{1}f_{2}g̃_{1}^{l}), m_{1} + lp^{n}).

Finally, we are able to prove the commutativity of the square of the lemma. Choose a lift g̃ ∈ A_{n}^{∗} of g as in the definition of 4.3.4. We have to check that the element ˜1(g̃) ∈ 𝔻(E_{n}(g))_{(S,Spec(A_{n}),γ)} satisfies formulae 4.3.2.1 with η = dlog(g̃) and a = log(σ(g̃)g̃^{−p}). This is checked using the above; for example the property that ∇˜1(g̃) = dlog(g̃) · 1 follows from the fact that

pr_{1}^{−1}(˜1(g̃)) − pr_{2}^{−1}(˜1(g̃)) = log(pr_{1}^{∗}(g̃) · pr_{2}^{∗}(g̃)^{−1}) · 1,

here pr_{1} and pr_{2} are the projection morphisms

Spec(A_{n} ⊗_{ℤ} A_{n}/I^{2}) ⟶ ⟶ Spec(A_{n})

of the first infinitesimal neighbourhood of the diagonal embedding of Spec(A_{n}). (See also the proof of [dJ, Proposition 8.9].) □

We note that the obvious homomorphisms W_{n} → W_{n−1} are induced by taking the kernel of p^{n−1} on an extension of 𝔻(μ_{p^{n}}) by 𝔻(ℤ/p^{n}ℤ). Let us write

W_{n} ⊃ W′_{n} := {x ∈ W_{n}|∃y ∈ W_{n+1} whose image in W_{n} is x}

Since any element of A^{∗}/(A^{∗})^{p^{n}} can be lifted to an element of A^{∗}/(A^{∗})^{p^{n+1}} it follows that the image of 4.3.4 lies in W′_{n}.

4.3.6 Proposition. a) If A is a field of characteristic p the map 4.3.4

A^{∗}/(A^{∗})^{p^{n}} ⟶ W′_{n}

is an isomorphism for all n ∈ ℕ.

b) Over a field k of characteristic p the crystalline Dieudonné module functor 𝔻 induces a bijection of extension groups

Extp-div. gr.^{1}(ℚ_{p}/ℤ_{p}, 𝔾_{m}[p^{∞}]) ≅ Ext_{DC}^{1}(𝔻(𝔾_{m}[p^{∞}]), 𝔻(ℚ_{p}/ℤ_{p})).

Proof. Part b) is a consequence of part a) since the map of Ext-groups is just the map (k = A)

lim_{⟵} A^{∗}/(A^{∗})^{p^{n}} ⟶ lim_{⟵} W_{n} ≅ lim_{⟵} W′_{n}.

Injectivity of A^{∗}/(A^{∗})^{p^{n}} → W′_{n} is a consequence of fully faithfulness of 𝔻 on Spec(A), but is also easy to prove directly: if (g̃^{−1}dg̃, dlog(σ(g̃)g̃^{−p})) = (db, σ(b) − pb) then b ∈ pA_{n} (look at second factor), hence dg̃ ∈ pA_{n}, hence g̃ = h^{p}(1 + pf), hence b ∈ p^{2}A_{n}, etc.

Surjectivity is proved as follows. Since the kernel of the map W′_{n} → W′_{1} can be identified with W′_{n−1}, we only need to prove surjectivity for n = 1. Let us take (η, a) ∈ Ω_{2} ×A_{2} with dη = 0 and σ(η) = da+pη. We can write a = a_{1}^{p} +pa_{2} since da = σ(η)−pη ∈ pΩ_{2}. Changing (η, a) by (da_{1}, σ(a_{1}) − pa_{1}) we may assume that a ∈ pA_{2}: a = pa′. We can now divide the equation σ(η) = da + pη by p to get

\frac{1}{p}σ(η) = da′ + η ∈ Ω_{1}.

This equation can be viewed as a condition on the 1-form η ∈ Ω_{1} = Ω_{A}^{1} without referring to the lift (A_{2}, σ): the equation says that the relation

∑_{i} f_{i}^{p}g_{i}^{p−1}dg_{i} ≡ ∑_{i} f_{i}dg_{i} (mod dA)

holds in Ω_{A}^{1}/dA for any presentation η = ∑ f_{i}dg_{i} of η. We have to prove that any η ∈ Ω_{A}^{1} satisfying this is a logarithmic differential: η = g^{−1}dg, for some g ∈ A^{∗}.

A relation of the type above is realised by a relation

∑_{i} f_{i}^{p}g_{i}^{p−1}dg_{i} = ∑_{i} f_{i}dg_{i} + ∑_{j} dα_{j} + ∑_{l} β_{l}(d(γ_{l} + δ_{l}) − dγ_{l} − dδ_{l})

+ ∑_{m} ϵ_{m}(dζ_{m}ϑ_{m} − ζ_{m}dϑ_{m} − ϑ_{m}dζ_{m})

in A[dA]. Clearly, there exists a subfield k ⊂ A finitely generated over 𝔽_{p} containing all the elements f_{i}, g_{i}, α_{j}, β_{l}, γ_{l}, δ_{l}, ϵ_{m}, ζ_{m}, ϑ_{m}. We have reduced the problem to the case of a field k finitely generated over 𝔽_{p}. The 1-form η comes from a 1-form η ∈ Γ(X, Ω_{X}^{1}) of some variety X smooth over 𝔽_{p} with 𝔽_{p}(X) ≅ k. By assumption dη = 0 and η lies in the kernel of the map (see [Ill3, §2])

W^{∗} − C : ZΩ_{X/𝔽_{p}}^{1} ⟶ Ω_{X/𝔽_{p}}^{1}.

Hence by [Ill3, 2.1.17] it is a logarithmic differential on an open part of X. □

4.4 Construction of G in the reduced case.

In this section we assume that 𝔛 = 𝔛_{red}. Hence, we may assume that 𝔛 = S = Spec(A) is affine and of finite type over a field k with a finite p-basis. Since S is smooth over Spec(𝔽_{p}), it is a regular scheme. Also we may and do assume that S is irreducible (i.e., connected). We argue by induction on dim S, the case dim S = 0 is 4.2.

Let there be given a Dieudonné crystal (E, f, v) over S. By 4.2 we have a p-divisible group G_{K} over Spec(K), where K is the function field of S (it is finitely generated over k, hence it also has a finite p-basis). This p-divisible group is such that

4.4.1  𝔻(G_{K}) ≅ (E, f, v)|_{CRIS(Spec(K)/Σ)}.

Fix n ∈ ℕ. There exists a nonempty Zariski open U ⊂ S and a finite locally free group scheme G_{n} over U with G_{n} ×_{U} Spec(K) ≅ G_{K}[p^{n}]. Suppose (A_{n}, σ) is a lift of A modulo p^{n}. A lift of K modulo p^{n} is:

lim_{\overrightarrow{f∈A_{n},f modp≠0}} A_{n}[\frac{1}{f}]

with the natural action of σ on it. Using the description of crystals in terms of modules with connection 2.2.3 and arguing as in 4.2.1 we see that we may assume that the isomorphism 4.4.1 modulo p^{n} exists over U (after possibly shrinking U a bit):

4.4.2  𝔻(G_{n}) ≅ (E/p^{n}E)|_{CRIS(U/Σ)}

Using the fully faithfulness of 𝔻 ([BBM, 4.2.6] or Remark 2.4.10) we may even suppose U is the biggest Zariski open subscheme of S such that a finite locally free group scheme G_{n} over U exists, endowed with an isomorphism 4.4.2.

Let T = S \ U with the reduced closed subscheme structure. The regular locus T^{reg} ⊂ T is open dense in T. We are going to extend G_{n} to U ∪ T^{reg} = S \ Sing(T) together with the isomorphism 4.4.2. This contradicts the maximality of U unless T = ∅, which is what we wanted to prove. Indeed, then we have G_{n} over S with 4.4.2 for all n and hence by fully faithfulness ([BBM, 4.2.6] or Remark 2.4.10) a p-divisible group G = (G_{n})_{n≥1}.

Again using the fully faithfulness of 𝔻 on finite locally free group schemes, the problem of extending G_{n} is a local one. Thus we may assume that T is a regular scheme and that

T = Spec(A/I) ⟶ S = Spec(A)

is defined by an ideal I generated by a regular sequence f_{1}, …, f_{c} ∈ A. We put 𝔜 = Spf(Â) where

Â := lim_{⟵} A/I^{n} ≅ A/I[[f_{1}, …, f_{c}]].

(To find such an isomorphism compare with Lemma 1.3.3 and recall that A/I is regular.)

By our induction hypothesis (dim T < dim S), we can find a p-divisible group H′′ over T with 𝔻(H′′) ≅ E|_{CRIS(T/Σ)}. By Proposition 3.2.1 we can find a p-divisible group H′ over 𝔜 with 𝔻(H′) ≅ E|_{𝔜}. By Lemma 2.4.4 this comes from a p-divisible group H over Y := Spec(Â) with

4.4.3  𝔻(H) ≅ E|_{CRIS(Spec(Â)/Σ)}.

4.4.4 Lemma. The functor 𝔻 is fully faithful on finite locally free group schemes on each of the schemes

Y_{i} := Spec(Â[1/f_{i}]) and Y_{ij} := Spec(Â[1/f_{i}f_{j}]).

Proof. Since we have the isomorphism Â ≅ A/I[[f_{1}, …, f_{c}]] we may use Lemma 1.3.3 to replace A/I by a ring B with a finite p-basis. The ring B[[f_{1}, …, f_{c}]] has a finite p-basis by Lemma 1.1.3. Hence so does B[[f_{1}, …, f_{c}]][1/f_{i}] and B[[f_{1}, …, f_{c}]][1/f_{i}f_{j}]. It suffices to apply 4.1.1 of [BM]. □

The morphisms Y_{i} → S and Y_{ij} → S factor through U ⊂ S and by the lemma we get isomorphisms

4.4.5  α_{i} : H[p^{n}]_{Y_{i}} ≅ G_{n,Y_{i}}

from the isomorphisms

𝔻(H[p^{n}]_{Y_{i}}) ≅ E/p^{n}E|_{CRIS(Y_{i}/Σ)} ≅ 𝔻(G_{n,Y_{i}})

These isomorphisms agree on Y_{ij} after pullback with Y_{ij} → Y_{i} and Y_{ij} → Y_{j}. By Proposition 4.6.2, we get a unique finite locally free group scheme G′_{n} over S with isomorphisms G′_{n,U} ≅ G_{n} and G′_{n,Y} ≅ H[p^{n}] and inducing 4.4.5.

Suppose T_{n} ↪ S_{n} is a lift of T ↪ S modulo p^{n}: S_{n} = Spec(A_{n}), A_{n} is a lift of A modulo p^{n} and T_{n} = Spec(A_{n}/(f̃_{1}, …, f̃_{c})) where f̃_{i} ∈ A_{n} lifts f_{i} ∈ A. We also get Y_{i,n} and Y_{ij,n} in a similar manner. Crystals killed by p^{n} on T (resp. S, Y_{i} or Y_{ij}) are described by O_{T_{n}}-modules (resp. O_{S_{n}}, O_{Y_{i,n}} or O_{Y_{ij,n}}-modules) with connections, hence Proposition 4.6.2 now gives us an isomorphism of crystals

𝔻(G′_{n}) ≅ E/p^{n}E

agreeing with 4.4.2 and 4.4.3. It is an isomorphism of Dieudonné crystals since this was true of 4.4.2. This concludes the case 4.4.

4.5 Existence of G in the general case.

We proceed as in 4.4 by induction on the dimension of 𝔛_{red}. The case that dim 𝔛_{red} = 0 is a combination of 4.2 and Proposition 3.2.1.

As noted in 4.1 we may assume that 𝔛 = Spf(A) is affine and that A is of type 1.3.1.1. Using Lemma 1.3.3 we can replace A by a complete ring of type 1.3.2.1, which has a finite p-basis (this does not change the dimension of 𝔛_{red}!). Suppose I ⊂ A is the biggest ideal of definition of A. The scheme Spec(A/I) is a reduced scheme of finite type over a field, hence its regular locus is a dense open subset. Thus we can choose an element f ∈ A, f ∉ I such that Spec(A/I[1/f]) is regular and such that dim Spec(A/I + fA) < dim Spec(A/I). We put

A{1/f} = lim_{⟵} A/I^{n}[1/f]

and

Â = lim_{⟵} A/f^{n}A , Î = I · Â.

Suppose we are given a Dieudonné crystal E′ on 𝔛 = Spf(A). By 4.4 we get a pdivisible group G′′_{1} over Spec(A/I[1/f]) and by Proposition 3.2.1 it can be deformed to a p-divisible group G′_{1} over Spf(A{1/f}) with

4.5.1  𝔻(G′_{1}) ≅ E′|_{Spf(A{1/f})}.

By our induction hypothesis we get a p-divisible group G′_{2} over Spf(Â) with

4.5.2  𝔻(G′_{2}) ≅ E′|_{Spf(Â)}.

However, using Lemma 2.4.4 and Proposition 2.4.8 we may assume E′ comes from a Dieudonné crystal E over Spec(A), G′_{1} (resp. G′_{2}) comes from a p-divisible group G_{1} (resp. G_{2}) over Spec(A{1/f}) (resp. Spec(Â)) and the isomorphism 4.5.1 (resp. 4.5.2) is defined over Spec(A{1/f}) (resp. Spec(Â)).

By Lemma 1.1.3 the ring Â has a finite p-basis. Hence Â[1/f] has a finite p-basis, hence the ring

B := lim_{⟵} (Â/Î^{n})[1/f]

has a finite p-basis (Lemma 1.1.3 again). Clearly, we have homomorphisms Â → B and A{1/f} → B. By fully faithfulness over Spec(B) ([BBM, 4.1.1]) and 4.5.1 & 4.5.2 we get an isomorphism

4.5.3  G_{1,Spec(B)} ≅ G_{2,Spec(B)}.

Reducing everything modulo a fixed power n of I we get G_{1,n} over Spec(A/I^{n}[1/f]), G_{2,n} over Spec(Â/Î^{n}) and

4.5.4  G_{1,n} × Spec(Â/Î^{n}[1/f]) ≅ G_{2,n} × Spec(Â/Î^{n}[1/f]).

By Proposition 4.6.2 we get a p-divisible group H_{n} over Spec(A/I^{n}) agreeing with G_{1,n} over Spec(A/I^{n}[1/f]), agreeing with G_{2,n} over Spec(Â/Î^{n}) and inducing the isomorphism 4.5.4. Taking the limit for n → ∞ (Lemma 2.4.4) gives a p-divisible group H over Spec(A), isomorphic to G_{1} over Spec(A{1/f}), isomorphic to G_{2} over Spec(Â) and inducing 4.5.3.

To prove that 𝔻(H) ≅ E we use the description of crystals in terms of modules with connections ([BM, 1.3.3] and 2.2.2) over suitable lifts of A, Â, … modulo p^{n}. The argument is similar to the last part of the proof of 4.4 and is left to the reader.

4.6 A remark on descent. Only included for quick reference.

Suppose A is a Noetherian ring and I ⊂ A is an ideal. Put X = Spec(A), Z = Spec(A/I), U = X \ Z, Y = Spec(Â) where

Â = lim_{⟵} A/I^{n},

and U′ = Y \ Z = U ×_{X} Y. We have the morphisms

p_{1} : U′ ⟶ U and p_{2} : U′ ⟶ Y.

We consider the following category: 4.6.1 Objects are triples (E_{U}, E_{Y} , α) where

– E_{U} (resp. E_{Y}) is a coherent sheaf of O_{U} (resp. O_{Y}) modules,

– α is an isomorphism

α : p_{1}^{∗}(E_{U}) ⟶^{∼} p_{2}^{∗}(E_{Y})

Morphisms (f_{U}, f_{Y}) : (E_{U}, E_{Y} , α) → (E_{U′}, E_{Y′}, α′) are module homomorphisms f_{U} : E_{U} → E_{U′} and f_{Y} : E_{Y} → E_{Y′} compatible with α and α′.

4.6.2 Proposition. The natural functor

F : coherent O_{X}-modules ⟶ 4.6.1

is an equivalence of categories. Proof. For any (E_{U}, E_{Y} , α) ∈ Ob 4.6.1 put

Γ((E_{U}, E_{Y} , α)) := {(s_{U}, s_{Y}) ∈ Γ(U, E_{U}) × Γ(Y, E_{Y})|α(p_{1}^{∗}(s_{U})) = p_{2}^{∗}(s_{Y})}

It is easy to show that for a coherent O_{X}-module E we have Γ(X, E) = Γ(F (E)). Thus F is fully faithful. Returning to our object (E_{U}, E_{Y} , α) of 4.6.1, it is easy to find a finitely generated submodule M ⊂ Γ((E_{U}, E_{Y} , α)) such that M̃_{U} → E_{U} is an isomorphism (M̃ is the sheaf on X associated to the A-module M). Then E_{Y} /M̃_{Y} will be a coherent sheaf of O_{Y} -modules with zero restriction on U′ ⊂ Y, hence killed by some power of Î = I ·Â. Thus (0, E_{Y} /M̃_{Y} , 0) is in the essential image of F. To show that (E_{U}, E_{Y} , α) is in the essential image of F also we note that F identifies Ext-groups. If N is a finitely generated A-module annihilated by some power of I then the natural map

Ext_{A}^{1}(N, M) ⟶ Ext1 ; Â(N, M ⊗ Â)

is an isomorphism. □

5 Proof of Main Theorem 2

In this chapter we prove the second main theorem. It implies that the crystalline Dieudonné module functor is fully faithful up to isogeny over schemes of finite type over a field with a finite p-basis.

The proof of the theorem relies on ideas of Berthelot on convergent isocrystals. Let me try to indicate briefly two of the main ideas of the article [B2]. See also [B3].

Take a reduced scheme S of finite type over a field k of characteristic p > 0. Choose a Cohen ring R for the field k. Consider a formal scheme 𝔛, formally smooth over Spf(R), whose underlying reduced scheme 𝔛_{red} is equal to S. Note that we do not assume that 𝔛 is of finite type over Spf(R). For example we can construct 𝔛 by taking a closed immersion of S into a scheme X which is smooth over Spec(R) and taking 𝔛 to be the completion of X along S. To the formal scheme 𝔛 we associate a smooth rigid analytic space 𝔛^{rig} (see Chapter 7 or [B2, 0.2.6]). If 𝔛 is the completion of X as above, this is just the admissible open subvariety of the rigid analytic variety X^{rig} consisting of points x ∈ X^{rig} specializing to points of S. Such a rigid analytic variety U = 𝔛^{rig} is called a tube of S. Any two tubes U_{1}, U_{2} of S are dominated by a third tube U and we may assume that the morphisms U → U_{i} are fibrations whose fibres are open unit balls. Therefore we get a cohomology theory of varieties S in characteristic p by associating to S the cohomology of one of its tubes.

Berthelot constructs a functor from the category of Dieudonné crystals over S to the category of convergent F -isocrystals over U; these are finite, locally free sheaves of O_{U}- modules endowed with an integrable connection and an action of F (suitably defined). This last category is independent of the tube U chosen. Berthelot shows that this functor is fully faithful up to isogeny.

5.1 Statement of Main Theorem 2.

5.1.1 Theorem. Let k be a field with a finite p-basis. Suppose that S is a reduced scheme of finite type over Spec(k). For any two p-divisible groups G_{1}, G_{2} over S we have

5.1.1.1  Hom_{DC/S}(𝔻(G_{2}), 𝔻(G_{1})) = torsion subgroup ⊕ 𝔻(Hom_{S}(G_{1}, G_{2})).

That is, for any φ : 𝔻(G_{2}) → 𝔻(G_{1}) there is a unique ψ : G_{1} → G_{2} over S such that φ − 𝔻(ψ) is torsion. □

The proof of this theorem is long and complicated and will occupy most of the rest of this chapter. But first let us deduce a corollary of it.

5.1.2 Corollary. Let S be a scheme of characteristic p which has a finite open covering S = ⋃_{i} U_{i} such that each U_{i} is of finite type over a field k_{i} which has a finite p-basis. The crystalline Dieudonné module functor 𝔻 (2.3.3) is fully faithful up to isogeny over S. Proof. Suppose G_{1}, G_{2} are p-divisible groups over S. We have to show that 𝔻 induces a bijection:

5.1.2.1  Hom_{S}(G_{1}, G_{2}) ⊗ ℚ ≅ Hom_{DC/S}(𝔻(G_{2}), 𝔻(G_{1})) ⊗ ℚ

Here DC/S stands for the category of Dieudonné crystals over S (2.3.2). Of course we may replace S by any of the U_{i}; thus the reduction S′ of S satisfies the assumptions of 5.1.1. On the other hand, it is well known that the maps

Hom_{S}(G_{1}, G_{2}) ⊗ ℚ ⟶ Hom_{S′}(G_{1,S′}, G_{2,S′}) ⊗ ℚ

and

Hom_{DC/S}(𝔻(G_{2}), 𝔻(G_{1})) ⊗ ℚ ⟶ Hom_{DC/S′}(𝔻(G_{2,S′}), 𝔻(G_{1,S′})) ⊗ ℚ

are isomorphisms. In both cases this can be seen using that some power of the Frobenius morphism of S factors through S′. Consequently 5.1.2 follows from 5.1.1. □

5.2 Formal setup.

Take S, G_{1} and G_{2} as in the theorem. If we prove equality in 5.1.1.1 over the members of a finite open cover of S then equality in 5.1.1.1 follows for S. Thus we may and do assume that S is connected, affine and of finite type over a field k with a finite p-basis. The proof of 5.1.1 is by induction on dim(S). The case dim(S) = 0 follows from Theorem 4.1.1. Our induction hypothesis is now that we have proven 5.1.1 for all schemes T as in 5.1.1 with dim T < dim S.

Suppose we are given a homomorphism of Dieudonné crystals over S:

5.2.1  φ : 𝔻(G_{2}) ⟶ 𝔻(G_{1})

We are looking for an element

ψ ∈ Hom_{S}(G_{1}, G_{2})

such that φ − 𝔻(ψ) is torsion. If we solve this problem for

φ′ : 𝔻(G_{1} ×_{S} G_{2}) ⟶ 𝔻(G_{1} ×_{S} G_{2})
↓ ≅  ↓ ≅
𝔻(G_{1}) ⊕ 𝔻(G_{2}) ⟶ 𝔻(G_{1}) ⊕ 𝔻(G_{2})

given by the matrix

( \begin{matrix}𝔻(id_{G_{1}}) & φ \\ 0 & 𝔻(id_{G_{2}}) \end{matrix} )

then we have also solved the problem for φ (just put ψ equal to G_{1} → G_{1} ×_{S} G_{2} →^{ψ′} G_{1} ×_{S} G_{2} → G_{2}). Thus we may assume that φ is an isomorphism.

Let us choose a closed immersion S ↪ 𝔸_{k}^{n} for some n ∈ ℕ. We consider the formal scheme 𝔛_{0} which is the completion of 𝔸_{k}^{n} along S. Thus 𝔛_{0} = Spf(A) is an affine formal scheme, A is of type 1.3.2.1 and S ≅ (𝔛_{0})_{red}. By [Ill2, Theorem 4.4] we can deform G_{1} (resp. G_{2}) to a p-divisible group H′_{1,0} (resp. H′_{2,0}) over 𝔛_{0} (see 2.4.2). By Corollary 3.2.4 (see also Remark 3.2.3) we can find a natural number N, such that we can deform H′_{1,0} and H′_{2,0} over the closed immersion

𝔛_{0} = Spf(A) ⟶ 𝔜_{0} = Spf(A[[x_{1}, …, x_{N}]])

to p-divisible groups H_{1,0} and H_{2,0} over 𝔜_{0} which are simultaneously versal: the Kodaira-Spencer maps (see 2.5.4) of 𝔻(H_{1,0}) and 𝔻(H_{2,0}) induce a surjection

5.2.2  ⨁_{i=1}^{N} O_{𝔜_{0}} \frac{∂}{∂x_{i}} ⟶ Hom_{O_{𝔜_{0}}} (ω_{1,0}, α_{1,0}) ⊕ Hom_{O_{𝔜_{0}}} (ω_{2,0}, α_{2,0})

Here we have written ω_{i,0} and α_{i,0} for the sheaves ω and α constructed in section 2.5.3 for the Dieudonné crystal 𝔻(H_{i,0}) over 𝔜_{0}.

Let us take a lift (O, σ) of k and put K equal to the quotient field of O. Next, we take a lift (Ã, σ) of A over (O, σ); i.e., such that there is a homomorphism O → Ã compatible with σ. (This is easy to find by lifting 𝔸_{k}^{n} to 𝔸_{O}^{n} and taking the completion of 𝔸_{O}^{n} along S ⊂ 𝔸_{k}^{n} ⊂ 𝔸_{O}^{n}.) As a ‘lift’ of 𝔜_{0} we take the formal scheme

𝔜 := Spf(Ã[[x_{1}, …, x_{N}]]).

We have a lift of Frobenius σ : 𝔜 → 𝔜 given by σ on Ã and σ(x_{i}) = x_{i}^{p}. Finally, we again use [Ill2, Theorem 4.4] to lift H_{1,0} (resp. H_{2,0}) to a p-divisible group H_{1} (resp. H_{2}) over 𝔜. (This can be done as follows: H_{i,0} may be considered as a p-divisible group over Spec(A[[\underline{x}]]) by Lemma 2.4.4, then lift H_{i,0} successively to Spec(Ã[[\underline{x}]]/p^{l}Ã[[\underline{x}]]) using the theorem of Illusie.)

The value 𝔻(H_{i})_{𝔜} of the Dieudonné crystal 𝔻(H_{i}) over 𝔜 (defined by a suitable inverse limit as in section 2.5) is endowed with a canonical integrable, topologically quasi-nilpotent connection (compare 2.2.3)

5.2.3  ∇ : 𝔻(H_{i})_{𝔜} ⟶ 𝔻(H_{i})_{𝔜} ⊗_{O_{𝔜}} Ω̂_{𝔜}^{1}.

By [BBM, 3.3.5] there is a Hodge filtration

0 ⟶ ω_{i} ⟶ 𝔻(H_{i})_{𝔜} ⟶ α_{i} ⟶ 0

reducing to the filtration ω_{i,0} ⊂ 𝔻(H_{i,0})_{𝔜_{0}} modulo pO_{𝔜}. Hence, we see that 5.2.3 still induces a surjection:

5.2.4  ⨁_{i=1}^{N} O_{𝔜} \frac{∂}{∂x_{i}} ⟶ Hom_{O_{𝔜}}(ω_{1}, α_{1}) ⊕ Hom_{O_{𝔜}}(ω_{2}, α_{2})

We also have canonical horizontal isomorphisms

σ^{∗}𝔻(H_{i})_{𝔜} ≅ 𝔻(σ^{∗}H_{i})_{𝔜} ≅ 𝔻(Frob_{𝔜_{0}}^{∗}(H_{i,0}))_{(𝔜_{0},𝔜,γ)},

hence Frobenius and Verschiebung induce horizontal homomorphisms

F : σ^{∗}𝔻(H_{i})_{𝔜} → 𝔻(H_{i})_{𝔜} and V : 𝔻(H_{i})_{𝔜} → σ^{∗}𝔻(H_{i})_{𝔜}

satisfying the usual relations.

5.3 Rigid analytic setup.

To the formal scheme 𝔜 there is associated a rigid analytic space Y = 𝔜^{rig} over K. See [B2, 0.2.6], see also chapter 7. It is a smooth rigid analytic variety over K, since it is an open subspace of 𝔸_{K}^{n+N,rig} (see above). To the finite locally free sheaves ω_{i}, α_{i} and 𝔻(H_{i})_{𝔜} over 𝔜 there are associated finite locally free sheaves of O_{Y} -modules ω_{i}^{rig}, α_{i}^{rig} and 𝔻(H_{i})^{rig} on Y. See 7.1.11. The lift σ on 𝔜 induces a general morphism σ : Y → Y over the continuous homomorphism σ : K → K. See 7.2.6. Frobenius and Verschiebung on 𝔻(H_{i})_{𝔜} induce

F : σ^{∗}𝔻(H_{i})^{rig} → 𝔻(H_{i})^{rig} and V : 𝔻(H_{i})^{rig} → σ^{∗}𝔻(H_{i})^{rig}.

In addition, the connection ∇ (5.2.3) induces an integrable connection (see 7.1.12)

∇^{rig} : 𝔻(H_{i})^{rig} ⟶ 𝔻(H_{i})^{rig} ⊗_{O_{Y}} Ω_{Y} .

Again we have that surjectivity of 5.2.4 implies surjectivity of the ‘Kodaira-Spencer’ maps of 0 → ω_{i}^{rig} → 𝔻(H_{i})^{rig} → α_{i}^{rig} → 0 induced by ∇^{rig}, i.e., the following homomorphism is surjective:

5.3.1  ⨁_{i=1}^{N} O_{Y} \frac{∂}{∂x_{i}} ⟶ Hom_{O_{Y}} (ω_{1}^{rig}, α_{1}^{rig}) ⊕ Hom_{O_{Y}} (ω_{2}^{rig}, α_{2}^{rig})

We claim that 𝔻(H_{i})^{rig} is canonically isomorphic to the convergent isocrystal 𝔻(G_{i})^{an} over Y associated to the Dieudonné crystal 𝔻(G_{i}) on CRIS(S/Σ). See [B2, 2.4.1]. This is easy to prove and left to the reader. In fact we only need this to get the homomorphism of sheaves of O_{Y} -modules

φ^{an} : 𝔻(H_{2})^{rig} = 𝔻(G_{2})^{an} ⟶ 𝔻(G_{1})^{an} = 𝔻(H_{1})^{rig}.

constructed in [B2, 2.4]; it is deduced from φ by functoriality of the construction of an associated convergent isocrystal. Let us recall its construction.

The Dieudonné crystal 𝔻(H_{i,0}) is given by a Dieudonné module (M_{i}, ∇_{i}, F_{i}, V_{i}) over Ã[[\underline{x}]] (see 2.2.3). The closed immersion S ↪ 𝔜_{0} ↪ 𝔜 is given by an ideal I ⊂ Ã[[\underline{x}]]. We remark that I is the biggest ideal of definition of Ã[[\underline{x}]] and that pÃ[[\underline{x}]] + x_{1}Ã[[\underline{x}]] + … + x_{N} Ã[[\underline{x}]] ⊂ I. Consider the divided power algebra (D̂, Î,^{[ ]}) constructed for the pair I ⊂ Ã[[\underline{x}]] in 2.2.1.1. The Dieudonné module defined by G_{i} is the Dieudonné module (M_{i} ˆ⊗ D̂, ∇_{i} ⊗ 1 + 1 ⊗ ∇, F_{i} ⊗ 1, V_{i} ⊗ 1) over D̂. Thus our φ comes from an isomorphism

5.3.2  φ : M_{2} ˆ⊗ D̂ ⟶ M_{1} ˆ⊗ D̂

of Dieudonné modules.

Let us consider the ring B_{n} of 7.1.1 constructed starting out with I ⊂ Ã[[\underline{x}]]. It is the p-adic completion of Ã[[\underline{x}]][I^{n}/p]. As in 7.1.3 the space Y = 𝔜^{rig} is the increasing union Y = ⋃ V_{n} of the affinoid subvarieties V_{n} = Sp(C_{n}) = Sp(B_{n} ⊗_{O} K). Berthelot [B2, 2.4.1.2] constructs a homomorphism of Ã[[\underline{x}]]-algebra’s

5.3.3  ρ : D̂ ⟶ B_{1}

mapping i^{[n]} to i^{n}/n! for all i ∈ I. To prove that it exists take generators f_{1}, …, f_{r} of I ⊂ Ã[[\underline{x}]] and recall that any element x of D̂ can be written as

x = ∑_{M=(m_{1},...,m_{r})} a_{M}f_{1}^{[m_{1}]}f_{2}^{[m_{2}]} … f_{r}^{[m_{r}]}

with a_{M} ∈ Ã[[\underline{x}]] and a_{M} → 0 p-adically if |M| → ∞. Thus the sum

ρ(x) := ∑_{M} a_{M} \frac{1}{m_{1}! m_{2}! … m_{r}!}f_{1}^{m_{1}} … f_{r}^{m_{r}}

converges in B_{1} since f^{m}/m! ∈ B_{1} for f ∈ I. We also remark that since σ(I) ⊂ I^{p} +pÃ[[\underline{x}]] it induces homomorphisms

5.3.4  σ : B_{n} ⟶ B_{pn} , n = 1, 2, 3, …

We conclude that σ(Sp(C_{n+1})) ⊂ Sp(C_{n}), i.e., σ(V_{n+1}) ⊂ V_{n}.

The construction of φ^{an} is as follows. Using 5.3.2 and ρ we get an isomorphism

5.3.5  M_{2} ˆ⊗_{Ã[[\underline{x}]]} B_{1} ≅ M_{2} ˆ⊗ D̂ ˆ⊗_{ρ} B_{1} ⟶ M_{1} ˆ⊗ D̂ ˆ⊗_{ρ} B_{1} ≅ M_{1} ˆ⊗_{Ã[[\underline{x}]]} B_{1}.

Hence, we get an isomorphism M_{2} ˆ⊗_{Ã[[\underline{x}]]} C_{1} → M_{2} ˆ⊗_{Ã[[\underline{x}]]} C_{1}; or what is the same, an isomorphism

φ_{1} : 𝔻(H_{2})^{rig}|_{V_{1}} ⟶ 𝔻(H_{1})^{rig}|_{V_{1}}.

This isomorphism is horizontal for ∇^{rig} and compatible with F_{i}^{rig} and V_{i}^{rig} since φ is so. Next we define

φ_{n} : 𝔻(H_{2})^{rig}|_{V_{n}} ⟶ 𝔻(H_{1})^{rig}|_{V_{n}}

by induction on n as follows:

5.3.6  φ_{n+1} := \frac{1}{p}(F_{1}^{rig} on 𝔻(H_{1})^{rig})|_{V_{n+1}} ∘ σ^{∗}(φ_{n}) ∘ (V_{2}^{rig} on 𝔻(H_{2})^{rig})|_{V_{n+1}}

Thus it is 1/p times the composition:

𝔻(H_{2})^{rig}|_{V_{n+1}} ⟶^{V_{2}^{rig}} σ^{∗}𝔻(H_{2})^{rig}|_{V_{n+1}} ⟶^{σ^{∗}(φ_{n})} σ^{∗}𝔻(H_{1})^{rig}|_{V_{n+1}} ⟶^{F_{1}^{rig}} 𝔻(H_{1})^{rig}|_{V_{n+1}}

We have used σ(V_{n+1}) ⊂ V_{n} to get σ^{∗}(φ_{n}) over V_{n+1}. Remark that since φ_{n} is compatible with F and V we get φ_{n+1}|_{V_{n}} = φ_{n}. It follows easily that φ_{n+1} is horizontal for ∇^{rig} and compatible with F_{i}^{rig} and V_{i}^{rig} since φ_{n} is so. The compatible system of maps (φ_{n})_{n≥1} defines our isomorphism

5.3.7  φ^{an} := lim_{⟶} φ_{n} : 𝔻(H_{2})^{rig} ⟶ 𝔻(H_{1})^{rig}

It is horizontal and compatible with F and V. We remark that φ^{an} is the unique (horizontal) map 𝔻(H_{2})^{rig} → 𝔻(H_{1})^{rig} extending φ_{1}. (If χ where a second such map then φ^{an} − χ would be a morphism of finite locally free sheaves over the connected smooth rigid analytic space Y (see 7.4) zero on a nonempty open subset V_{1} of Y, hence zero.) In particular, φ^{an} does not depend on the choice of σ on 𝔜.

5.4 Relation between rigid points and φ.

5.4.1 Situation. Here O ⊂ O′ is a finite extension of complete discrete valuation rings, K′ = O′ ⊗ K is the quotient field of O′. Further, x : Spf(O′) → 𝔜 is a morphism of formal schemes over Spf(O). To this there is associated a morphism of rigid analytic spaces over K (see §7):

5.4.1.1  x^{rig} : Sp(K′) ⟶ Y

Its image consists of one point, denoted x^{rig} ∈ Y by abuse of notation. Any point of Y occurs as x^{rig} for some x as above (see 7.1.10). Let x_{0} = Spec(k′) = Spf(O′)_{red} be the unique point of Spf(O′). The morphism x induces

5.4.1.2  x_{red} : x_{0} = Spec(k′) ⟶ 𝔜_{red} = S.

This induces a morphism of crystalline topoi (x_{0}/Σ)_{CRIS} → (S/Σ)_{CRIS}. Hence we get, putting G_{i,x_{0}} := x_{red}^{∗}(G_{i}), an isomorphism of Dieudonné crystals over CRIS(x_{0}/Σ):

x_{red,CRIS}^{∗}(φ) : 𝔻(G_{2,x_{0}}) ⟶ 𝔻(G_{1,x_{0}})

By Theorem 4.1.1 (k′ is a finite extension of k, hence has a finite p-basis also) we get a corresponding isomorphism of p-divisible groups

ψ_{x_{0}} : G_{1,x_{0}} ⟶ G_{2,x_{0}}.

Let us take a uniformizer π ∈ O′. If ℓ ∈ ℕ is large enough then π^{ℓ}O′ has a divided power structure δ. Put U = Spec(O′/π^{ℓ}O′) and T = Spf(O′). The triple (U, T, δ) can be considered as an ind-object of CRIS(U/Σ). By obstruction theory and since x_{0} ↪ U is a thickening of x_{0}, the map p^{ℓ}ψ_{x_{0}} lifts (uniquely) to a homomorphism

p̃^{ℓ}ψ_{x_{0}} : H_{1,U} ⟶ H_{2,U}.

It induces a homomorphism of O′-modules:

5.4.1.3  χ := 𝔻(p̃^{ℓ}ψ_{x_{0}})_{(U,T,δ)} : 𝔻(H_{2,U})_{(U,T,δ)} ⟶ 𝔻(H_{1,U})_{(U,T,δ)}

Note that we have the identification of O′-modules:

5.4.1.4  𝔻(H_{i,U})_{(U,T,δ)} ≅ M_{i} ⊗_{Ã[[\underline{x}]]} O′

On the other hand we have a canonical isomorphism of K′-vectorspaces

5.4.1.5  (x^{rig})^{∗}𝔻(H_{i})^{rig} ≅ M_{i} ⊗_{Ã[[\underline{x}]]} K′.

If x^{rig} ∈ V_{n} then it is given by x^{#} : Ã[[\underline{x}]] → B_{n} → O′ and thus (x^{rig})^{∗}𝔻(H_{i})^{rig} ≅ M_{i} ˆ⊗_{Ã[[\underline{x}]]} C_{n} ⊗_{C_{n}} K′ ≅ M_{i} ⊗_{Ã[[\underline{x}]]} K′. We claim that we have the following equality

5.4.1.6  χ ⊗ id_{K′} = p^{ℓ}(x^{rig})^{∗}(φ^{an})

as homomorphisms M_{2} ⊗ K′ → M_{1} ⊗ K′, where we used the identifications 5.4.1.4 and 5.4.1.5.

We only need to prove 5.4.1.6 for the ℓ ∈ ℕ such that π^{ℓ}O′ = pO′. Indeed, if ℓ′ > ℓ then χ′ = p^{ℓ′−ℓ}χ. We remark that, as x^{#}(I) ⊂ πO′ and σ(I) ⊂ I^{p} + pÃ[[\underline{x}]], we get x^{#}σ^{ℓ}(I) ⊂ pO′ and hence a commutative diagram

Ã[[\underline{x}]] ⟶^{σ^{ℓ}}  Ã[[\underline{x}]]
↓  ↓_{x^{#}}
D̂ ⟶^{τ} O′

where τ : (D̂, Î,^{[ ]}) → (O′, pO′, γ) is a PD-homomorphism. We also have x^{#}(I^{ℓ}) ⊂ pO′, hence x^{#} factorises as Ã[[\underline{x}]] → B_{ℓ} → O′. We remark finally that the map τ and B_{ℓ} → O′ fit into a commutative diagram:

5.4.1.7  \begin{matrix}D̂ & ⟶^{ρ} & B_{1} \\ ↓^{τ} & & ↓_{σ^{ℓ}} \\ O′ & ⟵ & B_{ℓ} \end{matrix}

The ℓ^{th}-iterate Frob_{U}^{ℓ} of the Frobenius morphism of U factors through x_{0} as U →^{s}

x_{0} ↪^{i} U. Thus we see that p̃^{ℓ}ψ_{x_{0}} is the composition:

H_{1,U} ⟶^{(F_{H_{1}} )^{ℓ}} (Frob_{U}^{ℓ})^{∗}H_{1,U} ⟶^{s^{∗}(ψ_{x_{0}} )} (Frob_{U}^{ℓ})^{∗}H_{2,U} ⟶^{(V_{H_{2}} )^{ℓ}} H_{2,U}

Since 𝔻(ψ_{x_{0}}) = φ|_{CRIS(x_{0}/Σ)} we see that χ is equal to the composition :

M_{2} ⊗_{Ã[[\underline{x}]]} O′ ⟶^{V_{2}^{ℓ}} M_{2}^{σ^{ℓ}} ⊗_{Ã[[\underline{x}]]} O′  M_{1}^{σ^{ℓ}} ⊗_{Ã[[\underline{x}]]} O′ ⟶^{F_{1}^{ℓ}} M_{1} ⊗_{Ã[[\underline{x}]]} O′
∥  ∥
M_{1} ⊗ D̂ ⊗_{τ} O′ ⟶^{φ⊗id_{O′}} M_{2} ⊗ D̂ ⊗_{τ} O′

Working through the inductive procedure used to define φ_{ℓ}, we see that p^{ℓ}φ_{ℓ} on M_{2} ⊗ B_{ℓ} is defined as the composition:

M_{2}⊗_{Ã[[\underline{x}]]} B_{ℓ} ⟶^{V_{2}^{ℓ}} M_{2}^{σ^{ℓ}}⊗_{Ã[[\underline{x}]]} B_{ℓ}  M_{1}^{σ^{ℓ}}⊗_{Ã[[\underline{x}]]} B_{ℓ} ⟶^{F_{1}^{ℓ}} M_{1}⊗_{Ã[[\underline{x}]]} B_{ℓ}
∥  ∥
M_{2}⊗ D̂ ⊗_{ρ} B_{1}⊗_{σ^{ℓ}} B_{ℓ} ⟶^{φ⊗id⊗id} M_{1}⊗ D̂ ⊗_{ρ} B_{1}⊗_{σ^{ℓ}} B_{ℓ}

The equality 5.4.1.6 is now a consequence of the description of φ_{ℓ} = φ^{an}|_{V_{ℓ}} and χ given above and the commutativity of 5.4.1.7.

5.4.2 Lemma. In Situation 5.4.1.

a) The following two statements are equivalent:

(i) There exists a morphism ψ_{x} : H_{1,O′} = x^{∗}(H_{1}) → x^{∗}(H_{2}) = H_{2,O′} of p-divisible groups over O′ lifting ψ_{x_{0}}.

(ii) There exists a morphism α : H_{1,O′} → H_{2,O′} of p-divisible groups over O′ such

that

(x^{rig})^{∗}(φ^{an}) = 𝔻(α)_{O′} ⊗ id_{K′}.

If these conditions are fulfilled then we have ψ_{x} = α. b)The following two statements are equivalent:

(i) For some n ∈ ℕ there exists a homomorphism α : H_{1,O′} → H_{2,O′} of p-divisible

groups over O′ lifting p^{n}ψ_{x_{0}}.
(ii) The homomorphism (x^{rig})^{∗}(φ^{an}) maps (x^{rig})^{∗}(ω_{2}^{rig}) into (x^{rig})^{∗}(ω_{1}^{rig}).

Proof. We choose an ℓ ∈ ℕ so big that π^{ℓ}O′ has a nilpotent divided power structure.

Let us prove b). “(ii)⇒(i)”: By 5.4.1.6 we know that the homomorphism χ = 𝔻(p̃^{ℓ}ψ_{x_{0}})_{(U,T,δ)} maps ω_{H_{2,O′}} into ω_{H_{1,O′}}. By [BM, 3.2.11], and [M, V Theorem 1.6] this

implies that p̃^{ℓ}ψ_{x_{0}} lifts to a homomorphism over O′. “(i)⇒(ii)”: If α lifts p^{n}ψ_{x_{0}} then p^{ℓ−n}α lifts p^{ℓ}ψ_{x_{0}}, hence also lifts p̃^{ℓ}ψ_{x_{0}} (we may take ℓ large enough so that ℓ − n is positive). By 5.4.1.6 and the references above, we now see that (x^{rig})^{∗}(φ^{an}) maps (x^{rig})^{∗}(ω_{2}^{rig}) into (x^{rig})^{∗}(ω_{1}^{rig}).

Let us prove a). “(i)⇒(ii)”: If ψ_{x} lifts ψ_{x_{0}} then p^{ℓ}ψ_{x} lifts p^{ℓ}ψ_{x_{0}} and p̃^{ℓ}ψ_{x_{0}}. Hence, by 5.4.1.6, we get p^{ℓ}(x^{rig})^{∗}(φ^{an}) = 𝔻(p^{ℓ}ψ_{x})_{O′} ⊗ id_{K′} = p^{ℓ}𝔻(ψ_{x})_{O′} ⊗ id_{K′}. Dividing by p^{ℓ} on both sides and putting α = ψ_{x} gives (ii). “(ii)⇒(i)”: If we have α as in (ii) then the conditions of b)(ii) are satisfied. Hence, there is an α′ : H_{1,O′} → H_{2,O′} lifting p^{n}ψ_{x_{0}} with 𝔻(α′)_{O′} ⊗id_{K′} = p^{n}(x^{rig})^{∗}(φ^{an}) (see proof of part b). Thus we see that 𝔻(α′ −p^{n}α)_{O′} = 0, hence α′ = p^{n}α. Since α′ lifts p^{n}ψ_{x_{0}} it follows that α lifts ψ_{x_{0}}. □

5.4.3 Claim. There exists a closed formal subscheme ℨ ⊂ 𝔜, which is formally smooth over Spf(O) (2.4.6), with ℨ_{red} = 𝔜_{red} = S and there exists an isomorphism of p-divisible groups over ℨ

ψ : H_{1}|_{ℨ} ⟶ H_{2}|_{ℨ}

such that the homomorphism of O_{ℨ^{rig}}-modules

𝔻(ψ)^{rig} : 𝔻(H_{2})^{rig}|_{ℨ^{rig}} = 𝔻(H_{2}|_{ℨ})^{rig} ⟶ 𝔻(H_{1}|_{ℨ})^{rig} = 𝔻(H_{1})^{rig}|_{ℨ^{rig}}

is equal to φ^{an}|_{ℨ^{rig}}. The closed formal subscheme ℨ ⊂ 𝔜 is characterized by the following property: 5.4.3.1 Suppose x : Spf(O′) → 𝔜 is as in 5.4.1. It factorizes through ℨ if and only if the

equivalent conditions of 5.4.2 a) hold.

In 5.5 we show that this implies the theorem. We will prove the claim in two steps. First, in 5.6, we prove it in the case that we already know 5.1.1 is true. This produces a closed formal subscheme ℨ_{1} in the formal open subscheme of 𝔜 whose underlying reduced scheme is the regular locus of S and a closed formal subscheme ℨ_{2} in the formal scheme which is the completion of 𝔜 along the singular locus of S, by Theorem 4.1.1 resp. induction. Using Proposition 7.5.2 and some analytic geometry we get ℨ ⊂ 𝔜 by gluing ℨ_{1} and ℨ_{2}.

5.5 The claim implies the theorem.

Let us prove that our theorem follows from 5.4.3. The isomorphism ψ induces an isomorphism over ℨ_{red} = S:

ψ|_{S} : G_{1} ⟶ G_{2}

The difference δ := φ − 𝔻(ψ|_{S}) ∈ Hom_{DC/S}(𝔻(G_{2}), 𝔻(G_{1})) has

δ^{an}|_{ℨ^{rig}} = φ^{an}|_{ℨ^{rig}} − 𝔻(ψ|_{ℨ_{red}})|_{ℨ^{rig}} = φ^{an}|_{ℨ^{rig}} − 𝔻(ψ)^{rig} = 0.

It is now a consequence of [B2, Theorem 2.4.2] that δ is torsion. This is seen as follows. Since δ^{an} is the unique horizontal extension of δ^{an}|_{ℨ^{rig}} we also have δ^{an} = 0. We let τ : B_{1} → D̂/p-torsion be the homomorphism which is σ-linear with respect to Ã[[\underline{x}]] such that τ(i/p) = (p − 1)! i^{[p]}, for all i ∈ I. This is well defined as B_{1} has no p-torsion. The homomorphism σ : D̂ → D̂ (D̂ as in 5.3) can be factored modulo p-torsion as

D̂/p-torsion⟶^{ρ}B_{1}⟶^{τ}D̂/p-torsion.

The element

δ ∈ Hom_{Ã[[\underline{x}]]}(M_{2}, M_{1}) ⊗_{Ã[[\underline{x}]]} D̂

satisfies pδ = F_{1} ∘ σ_{∗}(δ) ∘ V_{2} (it is a homomorphism of Dieudonné modules). Hence

pδ = F_{1} ∘ τ_{∗}ρ_{∗}(δ) ∘ V_{2} = 0 ∈ Hom_{Ã[[\underline{x}]]}(M_{2}, M_{1}) ⊗_{Ã[[\underline{x}]]} D̂/p-torsion

since δ^{an}|_{V_{1}} = 0 implies ρ_{∗}(δ) = 0. Thus δ is torsion.

5.6 Proof of 5.4.3 in case 5.1.1 is known.

The assumption that 5.1.1 holds is true for example if S is a regular scheme (Theorem 4.1.1) or if S is replaced by a scheme T with smaller dimension (this is our induction hypothesis). In this case we have a homomorphism of p-divisible groups over S

ψ_{S} : G_{1} ⟶ G_{2}

such that 𝔻(ψ_{S}) − φ is torsion. It follows that 𝔻(ψ_{S})^{an} = φ^{an}.

We define a closed formal subscheme ℨ ⊂ 𝔜 as follows. Let 𝔜_{n} denote the n^{th} infinitesimal neighbourhood of S in 𝔜. By obstruction theory, the homomorphism p^{n}ψ_{S} lifts to a homomorphism

p̃^{n}ψ_{S} : H_{1,𝔜_{n}} ⟶ H_{2,𝔜_{n}}.

We put

ℨ_{n} = closed subscheme of 𝔜_{n} defined by the vanishing of the morphism H_{1}[p^{n}]_{𝔜_{n}} \underrightarrow{_{p^{n}ψ_{S}}}H_{2,𝔜_{n}}.

It is easy to see that any morphism of schemes f : T → 𝔜_{n} factors through ℨ_{n} if and only if f^{∗}(ψ_{S}) over T ×_{𝔜_{n}} S lifts to a homomorphism over T. Thus we have ℨ_{n} = 𝔜_{n} ∩ ℨ_{n+1} (scheme theoretic intersection) and therefore we get

ℨ := lim_{⟶} ℨ_{n} ⊂ lim_{⟶} 𝔜_{n} = 𝔜.

By the very definition of ℨ the homomorphism ψ_{S} lifts to

ψ : H_{1}|_{ℨ} ⟶ H_{2}|_{ℨ}.

In addition, 5.4.3.1 is clear from the definition of ℨ. Also if x : Spf(O′) → ℨ is such a morphism then x^{∗}(ψ) is a lift of ψ_{x_{0}}. Hence, we get (x^{rig})^{∗}𝔻(ψ)^{rig} = (x^{rig})^{∗}(φ^{an}) for all points x^{rig} of ℨ^{rig} (see 5.4.2). Thus, we see that we will have 𝔻(ψ)^{rig} = φ^{an}|_{ℨ^{rig}} if we can prove that ℨ is formally smooth over Spf(O), for in that case ℨ^{rig} will be a smooth rigid analytic space (see 7.1.12) and hence homomorphisms of locally free O_{ℨ^{rig}}-modules are determined by their values at points of ℨ^{rig}.

Let J_{ℨ} ⊂ O_{𝔜} denote the ideal sheaf of ℨ in 𝔜. The obstruction to lift ψ to the first infinitesimal neighbourhood of ℨ in 𝔜 lies in

Hom_{O_{𝔜}}(ω_{2}, α_{1}) ⊗_{O_{𝔜}} J_{ℨ}/J_{ℨ}^{2}

by [Ill2, Theorem 4.4]; here we can also use [M, V Theorem 1.6]. By a standard argument, it follows that ℨ ⊂ 𝔜 is locally given by (at most) dd^{∗} equations, i.e., J_{ℨ} is locally generated by (at most) dd^{∗} sections. Here d = dim(G_{1}) = dim(G_{2}) = rk(ω_{2}) and d^{∗} = dim(G_{1}^{t}) = dim(G_{2}^{t}) = rk(α_{1}).

Let us show that in any closed point s ∈ S these equations define a linear subspace J̄ of 𝔪_{s}/(𝔪_{s}^{2} + pO_{𝔜,s}) of at least dimension dd^{∗}. Here O_{𝔜,s} is the local ring of 𝔜 at the point s ∈ S = 𝔜_{red} and 𝔪_{s} ⊂ O_{𝔜,s} is its maximal ideal. Put R = O_{𝔜,s}/(𝔪_{s}^{2} + pO_{𝔜,s} + J̄), thus Spec(R) → 𝔜 is a closed immersion factoring through ℨ ⊂ 𝔜. Our ψ over ℨ gives a homomorphism H_{1,R} → H_{2,R} and hence a horizontal homomorphism of Dieudonné modules over R compatible with filtrations. Thus we get a commutative diagram

5.6.1  \begin{aligned}0 & ⟶ & ω_{2} ⊗ R & ⟶ & M_{2} ⊗ R & ⟶ & α_{2} ⊗ R & ⟶ & 0 \\ & & ↓^{ω(ψ)⊗R} & ↓^{𝔻(ψ)⊗R} & ↓^{α(ψ)⊗R} \\ 0 & ⟶ & ω_{1} ⊗ R & ⟶ & M_{1} ⊗ R & ⟶ & α_{1} ⊗ R & ⟶ & 0 \end{aligned}

and 𝔻(ψ) ⊗ R is horizontal for ∇_{i} : M_{i} ⊗ R → M_{i} ⊗ Ω_{R}^{1}. By our choice of 𝔜 = Spf(Ã[[\underline{x}]]) there is a homomorphism

Ω_{R}^{1} ⟶ Ω_{R/Ã}^{1}

and it is clear that k(s) ⊗ Ω_{R/Ã}^{1} is generated by the elements 1 ⊗ dx_{i}. All the relations between these elements are determined as follows: write any element f ∈ J̄ as f ≡ ∑ a_{i}x_{i}+ g , a_{i} ∈ k(s) with g ∈ 𝔪_{s} ∩ Ã, then the element ∑ a_{i} ⊗ dx_{i} is zero in k(s) ⊗ Ω_{R/Ã}^{1}. Recall that we have the Kodaira-Spencer map:

κ_{i}(∂/∂x_{j}) : ω_{i} ⊗ k(s) ⟶ α_{i} ⊗ k(s)

for j = 1, …, N and i = 1, 2. The composition

ω_{i} ⊗ R ⟶ M_{i} ⊗ R⟶^{∇_{i}}M_{i} ⊗ Ω_{R}^{1} ⟶ α_{i} ⊗ Ω_{R}^{1} ⟶ α_{i} ⊗ k(s) ⊗ Ω_{R/Ã}^{1}

is a map ω_{i} ⊗ k(s) → α_{i} ⊗ k(s) ⊗ Ω_{R/Ã}^{1}. Using the definition of κ_{i} it is easy to see that under this map we have:

η ⟼ class of ∑_{j} κ_{i}(∂/∂x_{j})(η) ⊗ dx_{j}.

At this point the commutativity of 5.6.1 gives that the homomorphism ω_{2} ⊗ k(s) → α_{1} ⊗ k(s) ⊗ Ω_{R/Ã}^{1} given by

η ⟼ ∑_{j} (α(ψ)(κ_{2}(∂/∂x_{j})(η)) − κ_{1}(∂/∂x_{j})(ω(ψ)(η))) ⊗ dx_{j}

must be zero. Taking a basis for ω_{2} and a basis for α_{1} we see that this gives dd^{∗} elements in ⊕k(s)dx_{j} which should be zero in Ω_{R/Ã}^{1} ⊗ k(s). Our assumption that 5.2.4 is surjective implies that these dd^{∗} elements are k(s)-linearly independent. Hence, dim_{k(s)} J̄ ≥ dd^{∗}.

However, we have proved something which is even slightly stronger, namely: there exist dd^{∗} elements f_{1}, …, f_{dd^{∗}} ∈ Ô_{𝔜,s} ≅ Ã^{∧}[[x_{1}, …, x_{N}]] generating the ideal J^{∧} of ℨ in Ã^{∧}[[x_{1}, …, x_{N}]] such that modulo (𝔪_{s}^{∧})^{2} the linear terms in the x_{j} of the f_{i} are linearly independent. The superscript ^{∧} refers to 𝔪_{s}-adic completion. We leave it to the reader to show that this implies that

Ã^{∧}[[x_{1}, …, x_{N}]]/J^{∧} ≅ Ã^{∧}[[y_{1}, …, y_{N−dd^{∗}}]].

Thus this algebra is formally smooth over O (since Ã^{∧} is so). We conclude that the algebra Γ(ℨ, O_{ℨ}) = Ã[[\underline{x}]]/J is formally smooth over O since all its complete local rings at maximal ideals are so.

5.7 Construction of the analytic closed subvariety Z.

To start we define Z ⊂ Y as a set as follows:

5.7.1  Z = {x^{rig} ∈ Y such that the morphism x : Spf(O′) → 𝔜 corresponding to it (see 7.1.10 and 5.4.1) satisfies the equivalent conditions of Lemma 5.4.2 part a).}

On the other hand we have an analytic closed subvariety W ⊂ Y whose points x^{rig} are those such that x satisfies the equivalent conditions of Lemma 5.4.2 part b). It is defined as follows:

5.7.2  W = The analytic closed subvariety of Y defined by the vanishing of the homomorphism ω_{2}^{rig} → 𝔻(H_{1})^{rig}/ω_{1}^{rig} ≅ α_{1}^{rig} induced by φ^{an}.

Thus it is clear from Lemma 5.4.2 that Z ⊂ W as a set.

5.7.3 Lemma. The rigid analytic closed subvariety W ⊂ Y is smooth and all components of it have codimension dd^{∗} in Y. Proof. It is immediately clear from 5.7.2 that W is locally defined by (at most) dd^{∗} equations in Y. The argument to prove that it is locally smooth of codimension dd^{∗} is exactly the same as the proof of the corresponding fact for ℨ in 5.6. (Use that φ^{an} is horizontal and that 5.3.1 is surjective.) □

Let us take a point z_{0} ∈ Z ⊂ W. Recall that Y was defined as the countable union Y = ∪_{n}V_{n} of affinoid varieties V_{n} over K. Let us decompose the affinoid variety W ∩ V_{n} into its connected (hence irreducible) components:

5.7.4  W ∩ V_{n} = W_{n,1} ∪ W_{n,2} ∪ … ∪ W_{n,i_{n}}

Each W_{n,i} is a connected smooth affinoid variety and the cover 5.7.4 is admissible. We define

5.7.5  Z′ = ⋃_{(n,i) with z_{0}∈W_{n,i}} W_{n,i},

we only take those W_{n,i} which contain z_{0}. It is a connected rigid analytic variety since it is the increasing union of connected affinoids. Its complement W′ in W can be written as

5.7.6  W′ = ⋃_{(n,i)} W_{n,i},

where the union runs over those pairs (n, i) such that W_{n,i} ⊄ Z′. (We remark that this is not equivalent to z_{0} ∉ W_{n,i} in general.) Since the covering

5.7.7  W = ⋃_{n} W ∩ V_{n} = ⋃_{(n,i)} W_{n,i}

is an admissible affinoid cover of W, it follows that both Z′ and W′ are admissible open subvarieties of W and that 5.7.5 and 5.7.6 define admissible affinoid coverings of Z′ and W′. It also follows that W = Z′ ∪ W′ is an admissible open cover of W (it can be refined to 5.7.7). Thus, we finally conclude that Z′ is an analytic closed subvariety of W. Since W is analytically closed in Y, we also get that Z′ ⊂ Y is analytically closed.

By our definition of Z′, any point z ∈ Z′ can be connected by curves in Z′ to z_{0} (see chapter 6).

5.7.8 Proposition. a) For any morphism f : C → Z′ of a reduced normal connected 1-dimensional affinoid variety C to Z′ we have:

∃c ∈ C, f(c) ∈ Z ⇔ Im(f : C → W) ⊂ Z

b) Any two points z_{1}, z_{2} of Z can be connected by curves in Z, i.e., there exist morphisms f_{i} : C_{i} → Y of connected affinoid curves C_{i} to Y such that Im(f_{i} : C_{i} → Y) ⊂ Z (set theoretically) and connecting z_{1} to z_{2} as in chapter 6.

5.7.9 Corollary. As subsets of Y we have Z = Z′. Thus Z has the natural structure of a smooth closed subvariety of codimension dd^{∗} of Y.

Proof of the corollary. By the remark above the proposition we see that 5.7.8 part a) implies Z′ ⊂ Z. Part b) implies that any point z ∈ Z may be connected by curves in Z to z_{0} ∈ Z′ ⊂ Z. Of course these curves then map into W and hence they must lie in the connected component of W containing z_{0}. This is Z′, hence z ∈ Z′. □

Proof of 5.7.8 a). The morphism of rigid analytic spaces C → Z′ → Y is given by a morphism

5.7.10  𝔣 : ℭ → 𝔜

of FS_{O} where ℭ is a model for C (see 7.1.6 and 7.1.7). Thus ℭ is an affine formal scheme, flat and of finite type over Spf(O), say ℭ = Spf(R). Since C is normal, we may normalize ℭ and hence assume that R is normal. Since dim(C) = 1, we see that T = Spec(R) is an excellent normal two dimensional scheme. Its singular points lie in T_{0} = Spec(R/pR) ⊂ T. By resolution of singularities of two dimensional schemes [L], we see there exists a blow up T′ of T such that T′ is a regular scheme. This blow up is done in a subscheme of codimension two of T contained in T_{0} ⊂ T (Remark C on page 155 of [L]). We can blow up T′ some more in closed points to reach the situation where all (reduced) irreducible components T′_{0,i} of T′_{0} = V (p) ⊂ T′ are regular (1-dimensional) schemes and the intersections of T_{0,i} with T_{0,j} are transversal for all i ≠ j, i.e., such that T_{0,i} ∩ T_{0,j} (scheme theoretic intersection) is a reduced zero dimensional scheme.

Let us consider the completion ℭ′ of the scheme T′ along its special fibre T′_{0}. It is a formal scheme of finite type over Spf(O) and it is well known that the morphism ℭ′ → ℭ induces an isomorphism ℭ^{′rig} → ℭ^{rig} of rigid analytic varieties over K. (See [R] or [BL, Theorem 4.1]). Hence, we may replace ℭ by ℭ′ and hence assume that: ℭ_{red} = ∪_{i}ℭ_{red,i} where each ℭ_{red,i} is a regular 1-dimensional scheme of finite type over Spec(k) and the intersections of these are transversal. (Of course ℭ is no longer an affine formal scheme.)

These assumptions imply that the restriction

φ|_{CRIS(ℭ_{red,i}/Σ)} : 𝔻(G_{2})|_{CRIS(ℭ_{red,i}/Σ)} ⟶ 𝔻(G_{1})|_{CRIS(ℭ_{red,i}/Σ)}

(5.7.10 induces ℭ_{red,i} → 𝔜_{red} = S) comes from a homomorphism of p-divisible groups (by Theorem 4.1.1) G_{1}|_{ℭ_{red,i}} → G_{2}|_{ℭ_{red,i}}. Since the intersections are reduced, 𝔻 is fully faithful over ℭ_{red,i} ∩ ℭ_{red,j} (Theorem 4.1) and hence these homomorphisms glue to give

ψ_{ℭ_{red}} : G_{1}|_{ℭ_{red}} → G_{2}|_{ℭ_{red}}.

As the closed immersion ℭ_{red} → Spec(O_{ℭ}/p^{2}O_{ℭ}) =: ℭ_{2} is nilpotent, there exists an ℓ ∈ ℕ such that p^{ℓ}ψ_{ℭ_{red}} lifts to a homomorphism

p̃^{ℓ}ψ_{ℭ_{red}} : H_{1}|_{ℭ_{2}} → H_{1}|_{ℭ_{2}}.

Let us consider the triple △ := (ℭ_{2}, ℭ, γ) as an ind-object of CRIS(ℭ_{2}/Σ). Thus p̃^{ℓ}ψ_{ℭ_{red}} induces a homomorphism

5.7.11  𝔻(p̃^{ℓ}ψ_{ℭ_{red}})_{△} : 𝔻(H_{2})_{△} ≅ 𝔻(H_{2,ℭ})_{ℭ} ⟶ 𝔻(H_{1,ℭ})_{ℭ} ≅ 𝔻(H_{1})_{△}

For any x : Spf(O′) → ℭ → 𝔜 the pullback x^{∗}(p̃^{ℓ}ψ_{ℭ_{red}}) is the unique lift of p^{ℓ}ψ_{x_{0}} over Spec(O′/p^{2}O′). Hence, by 5.4.1.6, we see that

(x^{rig})^{∗}( (5.7.11)^{rig}) = ( (𝔣 ∘ x)^{rig})(φ^{an}).

Recall that the morphism f : ℭ^{rig} → Y factors through Z′ ⊂ W. Hence by the equation just proved we see that 5.7.11 preserves the Hodge filtrations defined by H_{i,ℭ}. Thus by [BM, Corollary 2.3.11] and [M, V Theorem 1.6] we get a morphism of p-divisible groups (the divided power structure on p^{2}O_{ℭ} is nilpotent)

α : H_{1,ℭ} ⟶ H_{2,ℭ}

which lifts p^{ℓ}ψ_{red}.

The scheme H_{i}[p^{ℓ}]_{ℭ} is a finite flat group scheme over ℭ. Thus the associated rigid analytic variety N_{i} = H_{i}[p^{ℓ}]_{ℭ}^{rig} is a group variety over C = ℭ^{rig}, which is finite flat over C. Since C lives in characteristic 0, the structure morphism N_{i} → C must be étale. Of course α induces α^{rig} : N_{1} → N_{2}. Now note that for all c ∈ C we have

f(c) ∈ Z ⇔ Ker(α^{rig})_{c} = N_{1,c}.

Also Ker(α^{rig}) is a closed analytic subvariety of N_{1}, hence finite étale over C. To prove the assertion a) of Proposition 5.7.8 it suffices to note that for a finite étale morphism of rigid analytic spaces the cardinality of the fibres is locally constant on the base. □

Proof of 5.7.8 part b). We first prove that any two points z_{1}, z_{2} of Z with sp(z_{1}) = sp(z_{2}) may be connected by curves in Z. Here sp is the specialization morphism Y → 𝔜_{red} = S (see 7.1.10). To see this, we use that by 7.2.5 we have

sp^{−1}(s) ≅ (𝔜_{s}^{∧})^{rig} , s ∈ S closed point.

Since s = Spec(k(s)) is a regular scheme of finite type over k we get by 5.6 a solution ℨ_{s} ⊂ 𝔜_{s}^{∧} of the problem posed in 5.4.3; by 5.4.3.1 we have

sp^{−1}(s) ∩ Z = ℨ_{s}^{rig}.

(We remark that all the properties of the data S, G_{1}, G_{2}, H_{1}, H_{2}, φ, 𝔜, κ_{i}, etc used in 5.6 still hold over the completion 𝔜_{s}^{∧} of 𝔜. For example 𝔜_{s}^{∧} = Spf(Ã^{∧}[[x_{1}, …, x_{N}]]) and Ã^{∧} is a formally smooth O-algebra, (5.2.4)_{s}^{∧} is surjective over 𝔜_{s}^{∧} as it is the completion of 5.2.4, and so on.)

Thus ℨ_{s} ≅ Spf(R) where R is a formally smooth, local complete O-algebra. By Proposition 7.3.6 ℨ_{s}^{rig} is connected. Since it is a countable increasing union of affinoids, it is also the countable increasing union of connected affinoids. Applying Proposition 6.1.1 gives that any two points in sp^{−1}(s) ∩ Z can be connected by curves in Z.

To finish the proof of 5.7.8 b) let us take two closed points s_{1}, s_{2} of S. We have to show that we may connect sp^{−1}(s_{1}) ∩ Z to sp^{−1}(s_{2}) ∩ Z by curves in Z. Since S is connected (see assumption in 5.2), we can connect s_{1} to s_{2} by curves in S: we can find a sequence of connected regular affine 1-dimensional schemes C_{i} of finite type over Spec(k) and morphisms

f_{i} : C_{i} ⟶ S , i = 1, …, n

which connect s_{1} to s_{2}:

1) s_{1} ∈ Im(C_{1} → S), s_{2} ∈ Im(C_{n} → S)
2) Im(C_{i} → S) ∩ Im(C_{i+1} → S) ≠ ∅ i = 1, …, n − 1

(We leave it to the reader to construct C_{i}, f_{i}, the hypothesis of regularity is trivial to establish: just take normalizations.) Therefore it suffices to connect the sets sp^{−1}(f(c))∩Z by curves in Z if we are given a single such curve f : C → S as above.

To do this write C = Spec(R). The morphism f is given by a k-algebra homomorphism A → R. Choose a lift R̃ of R (see 1.2.2, 1.3.3 and 1.3.2.2) and a lift Ã → R̃ of A → R (this is possible since Ã is formally smooth over ℤ_{p}). We remark that Ã → R̃ defines an O-algebra structure on R̃, compatible with the k-algebra structure on R. We extend this map to a homomorphism

5.7.12  Ã[[x_{1}, …, x_{N}]] ⟶ R̃[[x_{1}, …, x_{N}]]

in the obvious manner. This gives a morphism

ϵ : 𝔜′ = Spf(R̃[[x_{1}, …, x_{N}]]) ⟶ Spf(Ã[[x_{1}, …, x_{N}]]) = 𝔜.

At this point we have to check that all the assumptions on the data ϵ^{∗}H_{1}, ϵ^{∗}H_{2}, ϵ_{red}^{∗}G_{1}, ϵ_{red}^{∗}G_{2}, ϵ_{red}^{∗}(φ), 𝔜′ which are needed in 5.6 for the proof of 5.4.3 hold: By construction 𝔜′ → Spf(O) is formally smooth and 𝔜′_{red} is regular. Surjectivity of 5.2.4 over 𝔜′ follows from our choice of 5.7.12. We also remark that we have (ϵ^{rig})^{∗}(φ^{an}) = ϵ_{red,CRIS}^{∗}(φ)^{an}, this is just the functoriality of the construction (·)^{an}. Applying 5.6 to the situation over 𝔜′ we get a closed formal subscheme ℨ′ ⊂ 𝔜′, with ℨ′_{red} = 𝔜′_{red} = C, which is formally smooth over Spf(O). Thus ℨ′ = Spf(R̃[[x_{1}, …, x_{N}]]/J) and R̃[[x_{1}, …, x_{N}]]/J is a regular, hence normal, ring without nontrivial idempotents (since R has none). By 7.4.1 we see that Γ( (ℨ′)^{rig}, O^{∘}) has no nontrivial idempotents, hence (ℨ′)^{rig} is connected. Thus any two points in (ℨ′)^{rig} may be connected by curves in (ℨ′)^{rig}. (Argument as before using Proposition 6.1.1.)

It follows from the characterisation of ℨ′ ⊂ 𝔜′ in 5.4.3 that the composition (ℨ′)^{rig} ⊂ (𝔜′)^{rig} → 𝔜^{rig} = Y maps into Z (set theoretically). Since we have the diagram

(ℨ′)^{rig} ⟶ (𝔜′)^{rig} ⟶ 𝔜^{rig}
↓^{sp_{ℨ′}}  ↓^{sp_{𝔜′}}  ↓^{sp_{𝔜}}
C  ⟶  C  ⟶^{f} S

and since sp_{ℨ′}^{−1}(c) ≠ ∅∀c ∈ C, we get the desired result. □

5.8 End of the proof of 5.4.3.

Let us take an element f̄ ∈ Γ(S, O_{S}) such that 1) dim V (f̄) < dim S,

2) S \ V (f̄) is a regular scheme. This is possible as S is an affine reduced scheme of finite type over a field. We put T = V (f̄)_{red} equal to the reduced closed subscheme of S underlying V (f̄). Further we take U = S \ T the complement of T in S and regular by our choice of f̄. Take f ∈ Ã lifting f̄ ∈ Γ(S, O_{S}). As in 7.5.1 we put

𝔗 = 𝔜_{T}^{∧} = completion of 𝔜 along T,
𝔘 = open formal subscheme of 𝔜 with 𝔘_{red} = U.

It is clear from our choice of f ∈ Ã that

𝔗 = Spf(Ã^{∧}[[x_{1}, …, x_{N}]]) , Ã^{∧} = lim_{⟵} Ã/f^{n}Ã

and

𝔘 = Spf(Ã{1/f}[[x_{1}, …, x_{N}]]).

As was argued above (in the proof of 5.7.8 b), the essential properties of G_{i}, H_{i}, φ, 𝔜 used in 5.6 hold for G_{i}|_{𝔗}, H_{i}|_{𝔗}, … and G_{i}|_{𝔘}, H_{i}|_{𝔘}, … By our induction hypothesis 5.1.1 holds for T and 5.1.1 holds for U since it is regular (Theorem 4.1.1). Thus we get pairs:

(ℨ_{T} ⊂ 𝔗, ψ_{T} : H_{1}|_{ℨ_{T}} → H_{2}|_{ℨ_{T}}) and (ℨ_{U} ⊂ 𝔘, ψ_{U} : H_{1}|_{ℨ_{U}} → H_{2}|_{ℨ_{U}})

as in 5.4.3. Hence by our definition of Z ⊂ Y above, we see that

ℨ_{T}^{rig} = Z ∩ 𝔗^{rig} (resp. ℨ_{U} = Z ∩ 𝔘^{rig})

as rigid analytic closed subvarieties of 𝔗^{rig} (resp. 𝔘^{rig}). Hence we may now apply 7.5.2 to get a closed formal subscheme ℨ ⊂ 𝔜 with ℨ ∩ 𝔗 = ℨ_{T} and ℨ ∩ 𝔘 = ℨ_{U}. It follows that ℨ is formally smooth over Spf(O) since this was true of both ℨ_{T} and ℨ_{U}.

Therefore, if we write ℨ = Spf(R) and argue as in the proof of Lemma 1.2.2, we see that we can choose an isomorphism

Ã[[\underline{x}]] ≅ R[[y_{1}, …, y_{dd^{∗}}]].

By Lemma 1.3.3 we can choose a lift of Frobenius σ_{R} on R and use it to get a new lift of Frobenius σ′ on Ã[[\underline{x}]] compatible with Ã[[\underline{x}]] → R. Since the new choice of the lift of Frobenius σ′ on 𝔜 does not change φ^{an}, we see that we may suppose that σ fixes ℨ ⊂ 𝔜. Thus M_{i} ⊗_{Ã[[\underline{x}]]} R has the natural structure of a Dieudonné module over R (since M_{i}^{σ} ⊗ R = (M_{i} ⊗ R)^{σ_{R}}).

We claim that the homomorphism of O_{ℨ^{rig}} = O_{Z}-modules

φ^{an}|_{Z} : 𝔻(H_{2})^{rig}|_{Z} ⟶ 𝔻(H_{2})^{rig}|_{Z}

has matrix coefficients lying in Γ(Z, O_{Z}^{∘}); these matrix coefficients have to be computed with respect to bases of M_{1} and M_{2}. (Recall that 𝔻(H_{i})^{rig} = M_{i} ⊗ O_{Y}.) To see that our claim is true we use that φ^{an}|_{ℨ_{T}^{rig}} = 𝔻(ψ_{T})^{rig} and φ^{an}|_{ℨ_{U}^{rig}} = 𝔻(ψ_{U})^{rig}, so that the absolute

values of these matrix coefficients is ≤ 1 in points lying in ℨ_{T}^{rig} ∪ ℨ_{U}^{rig} = ℨ^{rig}. Therefore, we can use that

Γ(Z, O_{Z}^{∘}) = Γ(ℨ^{rig}, O_{ℨ^{rig}}^{∘}) = Γ(ℨ, O_{ℨ})

by Theorem 7.4.1 to see that φ^{an}|_{Z} comes from a homomorphism

φ′ : 𝔻(H_{2})_{ℨ} ⟶ 𝔻(H_{1})_{ℨ}.

Since φ^{an} is horizontal, we get that φ^{an}|_{ℨ^{rig}} is horizontal, which implies that φ′ must be horizontal (see 7.1.12). Since φ^{an} commutes with F and V, we get that φ^{an}|_{ℨ^{rig}} commutes with F and V (here we use that σ(ℨ) ⊂ ℨ) which gives that φ′ must commute with F and V. In this way we see that φ′ comes from a homomorphism of Dieudonné modules φ′ : M_{2} ⊗R → M_{1} ⊗R. By Theorem 4.1.1 therefore, we get a homomorphism of p-divisible groups ψ_{0} : H_{1}|_{Spec(R/pR)} → H_{2}|_{Spec(R/pR)}. Note that φ′ is also compatible with the Hodge filtrations defined by H_{i}|_{ℨ} (by definition of Z), hence ψ_{0} lifts to a homomorphism

ψ : H_{1}|_{ℨ} → H_{2}|_{ℨ}

of p-divisible groups over ℨ such that 𝔻(ψ)_{R} = φ′ and hence 𝔻(ψ)^{rig} = φ^{an}|_{ℨ^{rig}}. (If p = 2 one has to argue as in the proof of 5.7.8 a).) This concludes the proof of 5.4.3.

6 Connected rigid analytic varieties

Let O be a complete discrete valuation ring, K its quotient field, π ∈ O a uniformizer and k the residue field: k = O/πO. In this section we will prove that connected rigid analytic varieties are also path-connected in a certain sense. For the general definitions and notations concerning rigid analytic varieties we refer to [BGR].

6.1 Formulation of the result.

For any rigid analytic variety X over K we will call a curve in X a nonconstant morphism C → X of a connected 1-dimensional affinoid variety C over K into X. We will say that two points x_{0}, x_{1} ∈ X can be connected by curves in X, if there exists a sequence of curves in X

C_{i} ⟶ X , i = 1, …, n

such that:

1) x_{0} ∈ Im(C_{1} → X), x_{1} ∈ Im(C_{n} → X)

2) Im(C_{i} → X) ∩ Im(C_{i+1} → X) ≠ ∅ i = 1, …, n − 1

6.1.1 Proposition. Let X be a quasi-compact rigid analytic variety. If X is connected then any two points in X can be connected by curves in X. □

The proof of this will occupy the rest of this chapter.

There is an obvious reduction to the case that X is a reduced, irreducible, normal affinoid variety. Say X = Sp(A) has dimension d. We argue by induction on d. The case d = 1 is OK, thus we may assume that d ≥ 2.

Let us take a finite surjective morphism

f : X ⟶ 𝔹^{d} = Sp(T_{d})

of X onto the unit ball of dimension d (see [BGR, Corollary 6.1.2/2]).

6.2 Reduction to the case that f is generically étale.

We claim that we may assume that f is generically étale, i.e., that there exists a Zariski open U ⊂ 𝔹^{d} such that f : f^{−1}(U) → U is flat and unramified. (Unramified is equivalent to Ω_{f^{−1}(U)/U}^{1} = (0).) Since f is generically flat by [Mat, Theorem 53], we see that this is equivalent to the assertion that the extension of quotient fields Q(T_{d}) ⊂ Q(A) is separable. Hence, if char(K) = 0 then f is automatically generically étale. If char(K) = p then there exists a field L,

Q(T_{d}) ⊂ L ⊂ Q(A)

such that Q(T_{d}) ⊂ L is separable and L ⊂ Q(A) is purely inseparable. Let A′ be the normal closure of T_{d} in L, it is equal to A ∩ L and it is an affinoid algebra over K, see [BGR, 6.1.2/4 & 6.1.1/6]. The morphism f factorizes as

X⟶^{i}X′ = Sp(A′)⟶^{f′}𝔹^{d}.

The map i has the property that O_{X′} ⊂ i_{∗}O_{X} is purely inseparable in the sense that for any s ∈ i_{∗}O_{X} there exists an n ∈ ℕ such that s^{p^{n}} ∈ O_{X′}. This implies that i induces a bijection on points and that for any fibre product

C = C′ ×_{X′} X ⟶ X
↓ ↓
C′  ⟶ X′

the same holds for C → C′. Thus, if C′ → X′ is a curve in X′, then C → X is a curve in X. Hence we may replace X by X′ and f by f′ and in this way get the situation that f is generically étale.

6.3 Reduction to the case f̃ generically étale.

Let us write O\{\underline{x}\} or O\{x_{1}, …, x_{d}\} for the π-adic completion of O[x_{1}, …, x_{d}]. The homomorphism T_{d} → A induces a finite homomorphism

O{x_{1}, …, x_{d}} = T_{d}^{∘} ⟶ A^{∘}

(see [BGR, Corollary 6.4.1/6]). To this situation we associate some numerical invariants:

n(A^{∘}/T_{d}^{∘}) = n(A/T_{d})
= the multiplicity of (π) ⊂ O\{\underline{x}\} in the discriminant of A^{∘} over T_{d}^{∘}

and

deg(A^{∘}/T_{d}^{∘}) = deg(A/T_{d}) = the degree of T_{d} → A = [Q(A) : Q(T_{d})].

The discriminant of A^{∘} over T_{d}^{∘} is defined: T_{d}^{∘} → A^{∘} is finite and generically étale, T_{d}^{∘} is a regular ring and A^{∘} is normal as X is normal. Further, let η denote the generic point of Spec(k[x_{1}, …, x_{d}]) and let η_{1}, …, η_{r} denote the generic points of Spec(Ã). (Recall that Ã := A^{∘}/A^{∘∘} = A^{∘}/\sqrt{πA^{∘}}.) The inverse image of the point η ∈ Spec(k[x_{1}, …, x_{d}]) ⊂ Spec(T_{d}^{∘}) in Spec(A^{∘}) is {η_{1}, …, η_{r}}. The residue fields k(η_{1}), …, k(η_{r}) are finite extensions of the field k(η) = k(x_{1}, …, x_{d}). We want to reduce to the case that all the field extensions k(η) ⊂ k(η_{i}) are separable. Of course if char(k) = 0 then there is nothing to prove, so let us assume for the moment that char(k) = p.

Note that if K′ is a finite extension of K and Y is a rigid analytic variety over K′ then Y can be viewed as a rigid analytic variety over K also. If Y is connected then Y seen as a rigid analytic variety over K is connected also. Thus if we can find a surjective morphism of K-varieties Y → X and the assertion of the proposition is true for Y over K′, then the assertion is true for X (over K) also.

Suppose that the field extensions k(η) ⊂ k(η_{i}), i < s are separable for some s ∈ {1, …, r}. There exists a purely inseparable finite field extension k′ of k and a number ℓ ∈ ℕ such that the field extension

\begin{matrix}k(η) = k(x_{1}, …, x_{d}) & ⊂ & k′(y_{1}, …, y_{d}) \\ x_{i} & ↦ & y_{i}^{p^{ℓ}} \end{matrix}

makes k(η) ⊂ k(η_{s}) separable; this is meant to signify that the residue field of the local ring

k′(y_{1}, …, y_{d}) ⊗_{k(η)} k(η_{s})

is a finite separable extension of k′(y_{1}, …, y_{d}).

6.3.1 Lemma. There exists an extension of complete discrete valuation rings O ⊂ O′ with ramification index 1, such that the extensions of quotient fields K ⊂ K′ is separable and such that O′/πO′ ≅ k′. Proof. It suffices to do the case that k′ ≅ k[α]/(α^{p} − a). Take O′ = O[α]/(α^{p} − ã) if char(K) = 0 and O′ = O[α]/(α^{p} − πα − ã) if char(K) = p. □

We consider the ring extension (O′ as in 6.3.1)

6.3.2

O{x_{1}, …, x_{d}} ⟶ O′{y_{1}, …, y_{d}}
x_{i}  ⟼  y_{i}^{p^{ℓ}}  if char(K) = 0,
x_{i}  ⟼  y_{i}^{p^{ℓ}} + πy_{i}  if char(K) = p.

It defines a finite flat morphism Spec(()O′\{\underline{y}\}) → Spec(()O\{\underline{x}\}) which is generically étale. Let us put

B′ := K′⟨y_{1}, …, y_{d}⟩ ⊗_{K⟨x_{1},...,x_{d}⟩} A.

It is an affinoid algebra over K′, reduced as K⟨\underline{x}⟩ ⊂ K′⟨\underline{y}⟩ is generically étale. Any component of Spec(()B′) dominates Spec(()A) as A → B′ is flat. The map K′⟨\underline{y}⟩ → B′ is étale since it is a base change of K⟨\underline{x}⟩ → A. Let B be the normalization of B′.

If Sp(B) is not connected, say Sp(B) = X_{1} ∪ X_{2} with X_{1} connected then we have a diagram of rigid analytic spaces over K:

X_{1} ⟶ X_{1} ∪ X_{2} ⟶ 𝔹_{K′}^{d}
↘ ↓ ↓
X  ⟶ 𝔹_{K}^{d}

Since X_{1} → X is finite étale, it is surjective. Hence we may replace X by X_{1}, seen as a rigid analytic variety over K′ (see remark above). In this case the degree of X_{1} → 𝔹_{K′}^{d} is smaller than the degree of X → 𝔹^{d}.

Suppose that Sp(B) is connected. We note that we have an inclusion:

6.3.3  O′{y_{1}, …, y_{d}} ⊗_{O{x_{1},...,x_{d}}} A^{∘} ⊂ B^{∘}

This gives

6.3.4  n(B^{∘}/O′\{\underline{y}\}) ≤ n(A^{∘}/O\{\underline{x}\}).

(Use that formation of discriminant commutes with base extension.) If equality holds in 6.3.4 then 6.3.3 must be an isomorphism generically along V (π). More precisely, let η′ be the generic point of Spec(k′[y_{1}, …, y_{d}]) and let C_{η′} be the local ring of η′ in the ring O′\{\underline{y}\}; if equality holds in 6.3.4 then 6.3.3⊗C_{η′} is an isomorphism. In particular, in that case, the generic points of Spec(B̃) can be numbered η′_{1}, …, η′_{r} and have residue fields:

k′(η′_{i}) ≅ residue field of the local ring k′(η′) ⊗_{k(η)} k(η_{i}) = k′(y_{1}, …, y_{d}) ⊗_{k(x_{1},...,x_{d})} k(η_{i})

Thus we see by our choice of k′ and ℓ that the fields k′(η′_{1}), …, k′(η′_{s}) are separable over k′(η′). In other words, we have increased s by 1.

Continuing in this manner, we see that either deg(A/T_{d}) decreases to 1, or n(A/T_{d}) decreases to zero, or s becomes equal to r. In all cases we will have s = r, i.e., all the field extensions k(η) ⊂ k(η_{i}) are separable.

6.4 Rest of the proof.

Before we proceed, let us prove the proposition holds for X = 𝔹^{d}. We have a projection onto the d^{th} factor pr_{d} : 𝔹^{d} → 𝔹, which has a section s : 𝔹 → 𝔹^{d}. For any point x ∈ 𝔹^{d} the fibre F := pr_{d}^{−1}(pr_{d}(x)) ⊂ 𝔹^{d} is a connected (d−1)-dimensional variety. Hence, by our induction hypothesis, x may be connected by curves in F to s(pr_{d}(x)). Since s : 𝔹 → 𝔹^{d} is a curve in 𝔹^{d}, we get the desired result. The proof for general X follows the same pattern.

First we construct the morphism X → 𝔹 replacing pr_{d}. To this end we consider the morphism of schemes

f̃: X̃ = Spec(Ã) ⟶ Spec(k[x_{1}, …, x_{d}]) = 𝔸_{k}^{d}

induced by f. We remark that X̃ is equidimensional of dimension d. Let us decompose X̃ into its irreducible components:

X̃ = Z_{1} ∪ Z_{2} ∪ … ∪ Z_{r}

Let U ⊂ 𝔸_{k}^{d} be the Zariski open subset over which f̃ is étale. By our result that the extensions k(η) ⊂ k(η_{i}) are separable we know that U is not empty. The schemes f̃^{−1}(U)∩ Z_{i} are connected étale coverings of U, hence determined by π_{1}(U)-sets with transitive π_{1}(U)-actions.

For any partitioning {1, …, r} = I ∪ J, I ≠ ∅, I ≠ ∅ and I ∩ J = ∅ we put

Z_{I,J} = ( ⋃_{i∈I} Z_{i}) ⋂ ( ⋃_{j∈J} Z_{j}).

Note that the admissible open subvariety V of X defined by (sp is the specialization mapping, see [BGR, 7.1.5] or our 7.1.10)

X ⊃ V := {x ∈ X such that sp(x) ∉ Z_{I,J}}

is disconnected. This is so since V is the rigid analytic variety associated to a disconnected formal scheme over Spf(O), namely the open formal subscheme U of Spf(A^{∘}) whose reduction is equal to X̃ \ Z_{I,J}, which is disconnected by construction of Z_{I,J}. This implies that

6.4.1  codimX̃Z_{I,J} = 1.

Indeed, if this codimension were bigger then any function g ∈ Γ(V, O_{V}) would extend uniquely to an analytic function on the whole of X, by [Bart, Satz 3.5] or [Lü2, Satz 2]. In particular a nontrivial idempotent on V would extend to a nontrivial idempotent on X, in contradiction with the assumption that X is connected.

Let S ⊂ Spec(A^{∘}) be the Zariski closed subset of primes 𝔭 ⊂ A^{∘} such that the local ring A_{𝔭}^{∘} is not Cohen-Macaulay. To see that S is closed, we remark that by [Mat, Theorem 46] A_{𝔭}^{∘} is Cohen-Macaulay if and only if A_{𝔭}^{∘} is flat over O\{\underline{x}\}_{𝔮}, 𝔮 = 𝔭 ∩ O\{\underline{x}\}. Hence, the complement of S is the set of primes 𝔭 for which A_{𝔭}^{∘} is flat over O\{\underline{x}\}_{𝔮}, which is an open set by [Mat, Theorem 53]. Since A^{∘} is normal, we conclude that the codimension of S in Spec(A^{∘}) is at least 3 (by Serre’s criterium: normal ⇔ (S_{2}) and (R_{1}) [Mat, Theorem 39]). This implies that the codimension of S̃ := S ∩ X̃ is at least 2.

In the same manner the singular locus T of Spec(A^{∘}) is closed of codimension at least 2. Thus T̃ := T ∩ X̃ has codimension at least 1 in X̃.

Let us consider a general hyperplane

𝔸_{k}^{d} ⊃ H = {(x_{1}, …, x_{d}) ∈ 𝔸_{k}^{d} | ∑ a_{i}x_{i} = a}

for a_{i}, a ∈ k. We claim that if we choose a_{i}, a ∈ k sufficiently general then 6.4.2 f̃^{−1}(H) ∩ Z_{i} is irreducible, generically reduced and not contained in T̃, 6.4.3 for any partitioning {1, …, r} = I ∪J, I ≠ ∅, I ≠ ∅ and I ∩J = ∅ the intersection

f̃^{−1}(H) ∩ Z_{I,J} has a component of dimension d − 2 not contained in S̃. Of course this might not be possible if k is finite, in that case it will be possible after a finite extension k ⊂ k′, which is harmless for our purposes (replace O by O′, K by K′, etc). Condition 6.4.3 is easy to satisfy since the codimension of S̃ in X̃ is 2 and the codimension of Z_{I,J} in X̃ is 1. The schemes f̃^{−1}(H) ∩ Z_{i} are generically reduced as soon as H ∩ U is nonempty. The irreducibility of f̃^{−1}(H)∩Z_{i} in condition (6.4.2) follows from [J]. Avoiding T̃ is no problem since its codimension is 1 and hence so is the codimension of its image in 𝔸_{k}^{d}.

Next, we take any lifts ã_{i}, ã ∈ O of the elements a_{i}, a ∈ k. Write h = ∑ ã_{i}x_{i} − ã ∈ O{x_{1}, …, x_{d}} ⊂ A^{∘}. I claim that the affinoid space

Sp(A/hA) = {x ∈ X | h(x) = 0}

is connected. If not then the scheme Spec(A^{∘}/hA^{∘}) is reducible, say Spec(A^{∘}/hA^{∘}) = T_{1} ∪ T_{2}, dim(T_{1}) = dim(T_{2}) = d (recall that dim(Spec(A^{∘})) = d + 1 !) and (T_{1} ∩ T_{2})_{red} ⊂ V (π) (otherwise Sp(A/hA) would still be connected). By the construction of h we know that we must have

(T_{1} ∩ V (π))_{red} = ⋃_{i∈I} f̃^{−1}(H) ∩ Z_{i}

for a certain subset I ⊂ {1, …, r} and similarly for some J ⊂ {1, …, r}

(T_{2} ∩ V (π))_{red} = ⋃_{j∈J} f̃^{−1}(H) ∩ Z_{j}.

Clearly, I ∪ J = {1, …, r}. Let ξ_{i} be the generic point of f̃^{−1}(H) ∩ Z_{i}. The local ring A_{ξ_{i}}^{∘} of Spec(A^{∘}) at ξ_{i} is regular (6.4.2) and has dimension 2. Let t ∈ A_{ξ_{i}}^{∘} be the irreducible element of A_{ξ_{i}}^{∘} defining Z_{i} ∩ Spec(A_{ξ_{i}}^{∘}), it is also the generator of the kernel of A_{ξ_{i}}^{∘} → Ã_{ξ_{i}}. Since f̃^{−1}(H) ∩ Z_{i} is generically reduced for all i (6.4.2), we get that

A_{ξ_{i}}^{∘}/(t, h) ≅ k(ξ_{i}).

Thus {t, h} is a regular system of parameters for A_{ξ_{i}}^{∘} ([Mat, 12.J]) and hence (A^{∘}/hA^{∘})_{ξ_{i}} = A_{ξ_{i}}^{∘}/hA_{ξ_{i}}^{∘} is a regular ring. This implies that only one component of Spec(A^{∘}/hA^{∘}) passes through ξ_{i}, hence I ∩ J = ∅.

Let 𝔮 ⊂ A^{∘} be the prime ideal corresponding to a generic point of a component of f̃^{−1}(H) ∩ Z_{I,J}. Remark that 𝔮 ∈ T_{1} and 𝔮 ∈ T_{2}. By our assumption 6.4.3 above, we can choose this 𝔮 such that A_{𝔮}^{∘} is Cohen-Macaulay of dimension 3. Since h ∈ 𝔮A_{𝔮}^{∘} is not a zero divisor, the local ring R := A_{𝔮}^{∘}/hA_{𝔮}^{∘} is Cohen-Macaulay too. Thus dim(R) = depth(R) = 2. By [Hart, Proposition 2.1], we see that Spec(R) \ {𝔪} is connected (𝔪 is the maximal ideal of R). Thus the decomposition of Spec(R) as

Spec(R) = (T_{1} ∩ Spec(R)) ∪ (T_{2} ∩ Spec(R)) = T′_{1} ∪ T′_{2}

cannot have T′_{1} ∩ T′_{2} = {𝔪}. However, T′_{1} ∩ T′_{2} ⊂ Spec(R) ∩ V (π) = Spec(R/πR) consists of 𝔪 and a number of prime ideals corresponding to the generic points of f̃^{−1}(H) ∩ Z_{i}. Since I ∩ J = ∅ we get T′_{1} ∩ T′_{2} = {𝔪} a contradiction.

Conclusion: the morphism

h : X ⟶ 𝔹

has a connected fibre h^{−1}(0) of dimension smaller than d. By our induction hypothesis, we can connect any two points in h^{−1}(0) by curves in h^{−1}(0). Therefore, it suffices to connect any point x ∈ X to a point of h^{−1}(0) via curves in X. To do this, remark that the morphism h factorizes as

X ⟶^{f} 𝔹^{d} ⟶^{h′} 𝔹

by the very definition of h. It is also clear from the definition of our h that we may assume that h′ = pr_{d} by changing coordinates on 𝔹^{d}. Given our point x ∈ X with f(x) = (x_{1}, …, x_{d}) we put

C := {(y_{1}, …, y_{d}) ∈ 𝔹^{d} | y_{1} = x_{1}, …, y_{d−1} = x_{d−1}}.

It is an irreducible curve in 𝔹^{d}, the map pr_{d} : C → 𝔹 is surjective. The inverse image f^{−1}(C) ⊂ X is a finite union of irreducible curves in X: f^{−1}(C) = ⋃ C_{i}. For each of these curves the morphism f : C_{i} → C is finite hence surjective, hence h : C_{i} → 𝔹 is surjective. Since one of these curves contains the point x we are done.

7 Formal schemes and rigid analytic geometry Berthelot’s construction

Let O be a complete discrete valuation ring, K its quotient field, π ∈ O a uniformizer and k the residue field: k = O/πO. In [B2], Berthelot has constructed a functor which associates to a formal scheme over Spf(O) a rigid analytic space over K. This construction is more general than that of Raynaud [R] since the formal schemes are not necessarily of finite type over Spf(O). Indeed, the source category for this functor is defined as follows.

7.0.1 Definition. We write FS_{O} for the category of locally Noetherian adic formal schemes 𝔛 over Spf(O) whose reduction 𝔛_{red} is a scheme locally of finite type over Spec(k). Morphisms are morphisms of formal schemes over Spf(O).

The target category for the functor is simply the category of rigid analytic varieties over K (see [BGR, 9.3]); this category will be denoted Rig_{K}, the subcategory of affinoids over K is written Aff_{K}. Berthelot’s functor [B2, 0.2.6] will be denoted (·)^{rig} : FS_{O} → Rig_{K}, 𝔛 ↦ 𝔛^{rig}.

7.1 Berthelot’s construction for affine formal schemes.

Let 𝔛 = Spf(A) be an affine formal scheme over Spf(O). Let I ⊂ A be the biggest ideal of definition of A. (Recall that by our conventions on formal schemes the topological ring A is Noetherian and adic.) Thus 𝔛 is an object of FS_{O} if and only if A/I is a finitely generated k-algebra. We remark that this is also equivalent to the fact that A is the quotient of the ring A_{n,m} := O{x_{1}, …, x_{n}}[[y_{1}, …, y_{m}]] for some n, m ∈ ℕ. Let us assume that A satisfies these conditions.

It is clear that the rigid space associated to the formal scheme Spf(A_{n,m}) should be 𝔹^{n} × D^{m}, where 𝔹 is the closed unit disc and D is the open unit disc over K. If we take as coordinates x_{1}, …, x_{n} on 𝔹^{n} and as coordinates y_{1}, …, y_{m} on D^{m} then we can view any element f of A_{n,m} as a (bounded) analytic function on 𝔹^{n} × D^{m}, also denoted f by abuse of notation. Suppose that g_{1}, …, g_{r} ∈ A_{n,m} generate the kernel of the surjection A_{n,m} → A. Again it is clear that 𝔛^{rig} ⊂ 𝔹^{n} × D^{m} should be defined as the closed analytic subvariety given by the vanishing of the analytic functions g_{1}, …, g_{r}. Although this gives a definition of 𝔛^{rig} which works, we proceed in a somewhat more functorial manner below. It can be shown using 7.1.7 below that both descriptions give isomorphic rigid analytic spaces.

7.1.1 Let A be as above. Take n ∈ ℕ. Let us write A[I^{n}/π] for the subring of A ⊗_{O} K generated by the image of A → A ⊗_{O} K and the elements i/π, i ∈ I^{n}. We define the ring B_{n} = B_{n}(A) as the I-adic completion of A[I^{n}/π], i.e., the completion of A[I^{n}/π] with respect to the ideal IA[I^{n}/π]. Further, we introduce the notation C_{n} = C_{n}(A) := B_{n}⊗_{O}K. There are continuous homomorphisms B_{n+1} → B_{n} induced by the inclusions A[I^{n+1}/π] ↪ A[I^{n}/π]. This gives A ⊗_{O} K-algebra homomorphisms C_{n+1} → C_{n}. Furthermore the construction that associates to A the direct system {B_{n}}_{n≥1} (resp. {C_{n}}_{n≥1}) is a functor on the category of O-algebra’s as above.

7.1.2 Lemma. With notations as above.

a) The ring homomorphism A ⊗_{O} K → C_{n} is flat.

b) The K-algebra C_{n} is an affinoid algebra over K.

c) The morphism Sp(C_{n}) → Sp(C_{n+1}) identifies Sp(C_{n}) with an affinoid subdomain of Sp(C_{n+1}). Proof. Since A[I^{n}/π] is a Noetherian ring, the homomorphism A[I^{n}/π] → B_{n} is flat, hence (by base extension) A ⊗_{O} K = A[I^{n}/π] ⊗_{O} K → C_{n} is flat.

It is clear that IA[I^{n}/π] ⊂ πA[I^{n}/π] ⊂ I^{n}A[I^{n}/π], thus B_{n} is also the π-adic completion of A[I^{n}/π]. Suppose f_{1}, …, f_{N} ∈ I^{n} are generators of I^{n}, then there is a surjection

\begin{matrix}A/I[x_{1}, …, x_{N}] & ⟶ & A[I^{n}/π]/IA[I^{n}/π] ≅ B_{n}/IB_{n} \\ x_{i} & ⟼ & class of f_{i}/π. \end{matrix}

We conclude that B_{n}/IB_{n} is a k-algebra of finite type as A/I was assumed to be so. It follows that B_{n} is an O-algebra topologically of finite type and that C_{n} ≅ B_{n} ⊗_{O} K is an affinoid algebra over K.

The obvious inclusion A[I^{n+1}/π] ⊂ A[I^{n}/π] induces a continuous homomorphism B_{n+1} → B_{n}. We get a surjection

B_{n+1}{x_{1}, …, x_{N}} ⟶ B_{n}
x_{i} ⟼ image of f_{i}/π in B_{n}

and this induces a surjection C_{n+1}⟨x_{1}, …, x_{N}⟩ ⟶ C_{n}. In its kernel are certainly the elements πx_{i} − f_{i}. It induces an isomorphism

C_{n+1}⟨x_{1}, …, x_{N}⟩/(πx_{i} − f_{i}) ≅ C_{n}.

This follows from the fact that the map

A[I^{n+1}/π][x_{1}, …, x_{N}]/(πx_{i} − f_{i}) ⟶ A[I^{n}/π]

is an isomorphism modulo π-torsion. □

7.1.3 Definition. The rigid analytic space 𝔛^{rig} associated to the affine formal scheme 𝔛 = Spf(A) is defined as the (increasing) admissible union of the affinoid spaces Sp(C_{n}). In a formula: 𝔛^{rig} := ⋃_{n} Sp(C_{n})

Thus 𝔛^{rig} is a separated K-analytic space (see [BGR, 9.6.1/7]). If φ : 𝔜 = Spf(A′) → 𝔛 is a morphism of affine objects of FS_{O} then we get a morphism of rigid analytic spaces φ^{rig} : 𝔜^{rig} ⟶ 𝔛^{rig}. This is clear from 7.1.1. The next lemma shows that our functor agrees with Raynaud’s functor (see [R] or [BL, Theorem 4.1]) in the case that 𝔛 is of finite type over Spf(O).

7.1.4 Lemma. Let 𝔛 = Spf(A) as above.

a) The rigid analytic variety 𝔛^{rig} depends only on A′ = A/π-torsion; more precisely, the morphism φ : 𝔜: = Spf(A′) → 𝔛 induces an isomorphism φ^{rig} : 𝔜^{rig} → 𝔛^{rig}.

b) If the ideal πA is an ideal of definition of A or equivalently, if 𝔛 → Spf(O) is of finite type [EGA, I 10.13.3], then 𝔛^{rig} = Sp(A ⊗_{O} K). Proof. Trivial. □

We would like to show that the rigid analytic space 𝔛^{rig} represents a functor, at least on the category of affinoid spaces. Before we do so let us show that any rigid analytic space is determined by the functor it represents on Aff_{K}.

7.1.5 Lemma. The functor

Rig_{K} ⟶ Funct(Aff_{K}^{∘}, Sets)

which maps X to h_{X}(·) := Mor_{Rig_{K}} (·, X) is fully faithful.

Proof. Suppose that X, Y are rigid analytic varieties and α : h_{X} → h_{Y} is a morphism of functors. By substituting Sp(K′) in h_{X}(·), where K′ runs through all finite extensions of K we find a map of sets β : X → Y. Let i : U → X define an affinoid subdomain of X and consider the morphism α(i) : U → Y. As α is a transformation of functors the map α(i) agrees with β on U ⊂ X. A similar compatibility holds in case we have a composition V → U → X of open immersions of affinoids. Therefore β comes from a unique morphism X → Y, which agrees with α(i) on U, for any i : U → X as above. □

7.1.6 Suppose that Y is an affinoid rigid analytic variety over K. A model for Y is an affine formal scheme 𝔜, flat and of finite type over Spf(O) endowed with an isomorphism 𝔜^{rig} ≅ Y. Such a model always exists. A morphism of models 𝔜_{1} and 𝔜_{2} is a morphism φ : 𝔜_{1} → 𝔜_{2} of formal schemes over Spf(O) compatible with the given isomorphisms 𝔜_{i} ≅ Y. For any two models 𝔜_{1}, 𝔜_{2} of Y there is a third 𝔜 lying over both of them, i.e., such that there are maps of models 𝔜 → 𝔜_{1} and 𝔜 → 𝔜_{2}. In addition, if ψ : Y′ → Y is a morphism of affinoids and 𝔜 is a model of Y then there exists a model 𝔜′ of Y′ and a morphism φ : 𝔜′ → 𝔜 such that φ^{rig} = ψ.

7.1.7 Proposition. By the functoriality of (·)^{rig} we get a map

7.1.7.1  lim_{\overrightarrow{models 𝔜 of Y}} Mor_{FS_{O}}(𝔜, 𝔛) ⟶ Mor_{Rig_{K}} (Y, 𝔛^{rig}).

Both sides are in a natural way contravariant functors on Aff_{K}; the above is an isomorphism of functors. This property determines 𝔛^{rig} up to unique isomorphism in view of Lemma 7.1.5. Proof. The remarks made in 7.1.6 show that the left hand side of 7.1.7.1 defines a functor and that 7.1.7.1 is a transformation of functors. We will construct the inverse to 7.1.7.1. Suppose that Y = Sp(B) and that Y → 𝔛^{rig} is given by Y → Sp(C_{n}) for some n ∈ ℕ. Take a surjection O{x_{1}, …, x_{N}} → B_{n} and extend the composition O{x_{1}, …, x_{N}} → B_{n} → B to a surjection K⟨x_{1}, …, x_{N}, x_{N+1}, …, x_{M}⟩ → B. The image of O{x_{1}, …, x_{M}} in B is a subring R ⊂ B which is flat and of topologically finite type over O and is such that R ⊗ K = B. Hence 𝔜 := Spf(R) is a model of 𝔜. The composition

𝔜 = Spf(R) ⟶ Spf(B_{n}) ⟶ Spf(A) = 𝔛

gives an element of the left hand side of 7.1.7.1. We leave it to the reader to prove that this defines an inverse to 7.1.7.1. □

7.1.8 It is clear from our construction of 𝔛^{rig} (𝔛 as above) that there is a canonical homomorphism

7.1.8.1  A ⊗_{O} K = Γ(𝔛, O_{𝔛}) ⊗_{O} K ⟶ Γ(𝔛^{rig}, O_{𝔛^{rig}}).

The image of A under this homomorphism lies in fact in the sheaf O_{𝔛^{rig}}^{∘} of functions whose absolute value is bounded by 1:

Γ(U, O_{𝔛^{rig}}^{∘}) = {f ∈ Γ(U, O_{𝔛^{rig}}) | |f(x)| ≤ 1 ∀x ∈ U}

This is clear from the fact that the homomorphism A → C_{n} factors as A → B_{n} → B_{n}⊗K ≅ C_{n} and that B_{n} ⊂ C_{n}^{∘} =powerbounded elements of C_{n}. (See [BGR, 6.2.3/1].) Thus we get a homomorphism

7.1.8.2  Γ(𝔛, O_{𝔛}) ⟶ Γ(𝔛^{rig}, O_{𝔛^{rig}}^{∘}).

7.1.9 Lemma. Let 𝔛 = Spf(A) as above. There is a bijection functorial in A between the following two sets:

1. Maximal ideals 𝔪 ⊂ A ⊗_{O} K.

2. Points x of 𝔛^{rig}.

If the point x(𝔪) ∈ 𝔛^{rig} corresponds to 𝔪 ⊂ A ⊗_{O} K then there exists a canonical homomorphism of local rings

(A ⊗_{O} K)_{𝔪} ⟶ O_{𝔛^{rig},x(𝔪)}

compatible with 7.1.8.1. This homomorphism induces an isomorphism on completions. Proof. To establish the bijection one might use the bijection 7.1.7.1 and argue as in [BL, 3.4]. (See also 7.1.10 below.) However, we also give another argument.

We may assume that A is O-flat. Take a point x ∈ 𝔛^{rig}. It corresponds to a maximal ideal 𝔭 ⊂ C_{n} for some n big enough. Consider the homomorphism

φ : A ⊗_{O} K ⟶ C_{n}/𝔭.

The field on the right is a finite extension of K; hence it is finite algebraic over Im(φ). We also have that the image of φ is dense, since A ⊗ K is dense in C_{n}. Thus we get that φ is surjective, in other words that 𝔪 := Ker(φ) ⊂ A ⊗ K is a maximal ideal with the same residue field as 𝔭 in C_{n}.

On the other hand, suppose 𝔪 ⊂ A⊗K is a maximal ideal. The prime ideal 𝔮 = A∩𝔪 of A is maximal among the prime ideals of A not containing π. This implies that \sqrt{𝔮 + πA} is a maximal ideal of A. We see from this that A/𝔮 is a local ring of dimension 1 and that its residue field is a finite extension of k (by our assumption on A). Thus O → A/𝔮 is a finite flat ring homomorphism. The normalization O′ of the ring A/𝔮 is a complete discrete valuation ring finite over O. Let us denote the valuation by | | : O′ → ℝ_{+}. The image of I ⊂ A in O′ lands in the maximal ideal of O′, hence there exists an n ∈ ℕ such that |i| ≤ |π|^{1/n} ∀i ∈ I (use that I is finitely generated). This implies that |i| ≤ |π| ∀i ∈ I^{n}. This divisibility property allows us to define a ring homomorphism

7.1.9.1 A[I^{n}/π] ⟶  O′ i/π ⟼ (image of i in O′)/π.

Again by the inequality above, this extends to a homomorphism B_{n} → O′ and finally to a homomorphism

C_{n} = B_{n} ⊗_{O} K ⟶ O′ ⊗_{O} K = A/𝔮 ⊗_{O} K = (A ⊗_{O} K)/𝔪.

This homomorphism is surjective since it extends the homomorphism A ⊗ K → A/𝔮 ⊗ K. Hence, its kernel is a maximal ideal 𝔭 ⊂ C_{n}.

We leave it to the reader to show that these constructions define mutually inverse bijections as indicated in the lemma.

Suppose 𝔪 ⊂ A⊗K, 𝔮 ⊂ A and 𝔭 ⊂ C_{n} correspond to each other in the sense explained above. Take generators g_{1}, …, g_{s} ∈ A of 𝔮. We claim that 𝔭 = g_{1}C_{n} + … + g_{s}C_{n}. If we prove this then we have shown: the homomorphism

A_{𝔮} = (A ⊗_{O} K)_{𝔪} ⟶ (C_{n})_{𝔭}

is local, identifies residue fields (see above), is flat (Lemma 7.1.2) and unramified. Thus it induces an isomorphism on completions. By [BGR, 7.3.2/3] this gives the last assertion of the lemma.

Let us take generators f_{1}, …, f_{N} ∈ A of the ideal I^{n}. Since O′ is the normalization of A/𝔮, any element of O′ satisfies a monic equation with coefficients in A/𝔮. In particular we can find such an equation for the image of f_{i}/π in O′. This implies that we can find equations

7.1.9.2  (f_{i})^{d} = a_{i,1}π(f_{i})^{d−1} + a_{i,2}π^{2}(f_{i})^{d−2} + … + a_{i,d}π^{d} + ∑ λ_{i}^{j}g_{j}

with a_{i,ℓ} ∈ A, λ_{i}^{j} ∈ A.

The ring C_{n}/g_{1}C_{n} +…+g_{s}C_{n} is an affinoid algebra over A/𝔮⊗K = (A⊗K)/𝔪. The elements f_{i}/π give an affinoid generating system of C_{n}/(g_{1}, …, g_{s}) over A/𝔮 ⊗ K (they generate A[I^{n}/π] over A). The equations 7.1.9.2 give that these elements are integral over A/𝔮 ⊗ K, hence by [BGR, 6.3.2/2] C_{n}/(g_{1}, …, g_{s}) is finite over A/𝔮 ⊗ K. As A ⊗ K is dense in C_{n}, we must have that A/𝔮 ⊗ K is dense in C_{n}/(g_{1}, …, g_{s}). Thus we must have equality. □

7.1.10 Let us look at morphisms of FS_{O} of the form Spf(O′) → 𝔛, where O ⊂ O′ is a finite extension of discrete valuation rings. A second such morphism Spf(O′′) → 𝔛 is said to be equivalent to Spf(O′) → 𝔛 if there exists a commutative diagram

Spf(O′′′) ⟶ Spf(O′)
↓  ↓
Spf(O′′) ⟶  𝔛

where O ⊂ O′′′ is also a finite extension of discrete valuation rings. It follows from the above that points of 𝔛^{rig} correspond bijectively with equivalence classes of such Spf(O′) → 𝔛.

There is a specialization mapping

sp : 𝔛^{rig} ⟶ 𝔛.

It maps the point x ∈ 𝔛^{rig} corresponding to 𝔪 ⊂ A ⊗ K and 𝔮 = 𝔪 ∩ A ⊂ A to the unique maximal ideal of A containing π an 𝔮. If the point x corresponds to the equivalence class of φ : Spf(O′) → 𝔛 as above then sp(x) = φ(unique point of Spf(O′)). The map sp can be viewed as a morphism of ringed sites, see [B2, 0.2.6]. We remark that if Z ⊂ 𝔛_{red} is locally closed then sp^{−1}(Z) is an admissible open subvariety of 𝔛^{rig}. For example, if Z is closed, defined by the ideal (g_{1}, g_{2}, …, g_{s}) of A then

sp^{−1}(Z) = {x ∈ 𝔛^{rig} | |g_{i}(x)| < 1 ∀i = 1, …, s}.

(Use 7.1.8.1 to view g_{i} as a function on 𝔛^{rig}.) This is an admissible open subset by [BGR, 9.1.4/5].

7.1.11 Suppose 𝔛 = Spf(A) as above. There is a functor from coherent O_{𝔛}-modules to coherent O_{𝔛^{rig}}-modules, let us denote this functor by 𝔉 ↦ 𝔉^{rig}. If 𝔉 is defined by the finite A-module M [EGA, I 10.10.5] then the sheaf 𝔉^{rig} is defined by the sequence of modules M ⊗_{A} C_{n} ≅ M ˆ⊗_{A} C_{n}. It is clear that if 𝔉 is finite locally free then 𝔉^{rig} is finite locally free. There is an obvious map Γ(𝔛, 𝔉) → Γ(𝔛^{rig}, 𝔉^{rig}) extending 7.1.8.1.

7.1.12 In this subsection we assume that [k : k^{p}] < ∞. On any 𝔛 as above we have the sheaf of continuous differentials defined as follows

Ω̂_{𝔛}^{1} := lim_{⟵} Ω_{𝔛_{n}}^{1}.

It is a coherent sheaf of O_{𝔛}-modules. This can be seen by writing A as a quotient of O{x_{1}, …, x_{n}}[[y_{1}, …, y_{m}]] for some n, m and using the assumption that [k : k^{p}] < ∞.

Let us define the sheaf of differentials on an affinoid space Y = Sp(B) over K. Take any model 𝔜 = Spf(R) of Y. We put Ω_{Y} := (Ω̂_{𝔜}^{1})^{rig}. We leave it to the reader to prove that this the sheaf associated to the module of continuous differentials Ω̂_{B/ℤ}^{1} of B (see [EGA, 0_{IV} 20.4]). Thus the result is independent of the choice of the model and formation of it commutes with taking affinoid open subdomains of Y. By gluing we define the sheaf of differentials Ω_{Y} for any rigid analytic variety Y over K. We remark that the sheaf of differentials on Sp(K) is nontrivial in general: it has rank [k : k^{p}] if char(K) = 0 and rank [k : k^{p}] + 1 if char(K) = p.

At this point we can state in more generality that for any formal scheme 𝔛 as above there is a canonical isomorphism

(Ω̂_{𝔛}^{1})^{rig} ≅ Ω_{𝔛^{rig}}.

This follows from the fact that the module Ω̂_{B_{n}}^{1} is equal to the completion of Ω_{A[I^{n}/π]}^{1} and that this is equal to Ω_{A}^{1} up to π-torsion.

Finally, suppose the formal scheme 𝔛 is formally smooth over Spf(O) (see 2.4.3). In this case Ω̂_{𝔛}^{1} is finite locally free. Let the coherent sheaf of O_{𝔛}-modules 𝔉 be endowed with an integrable connection ∇ : 𝔉 → 𝔉 ⊗ Ω̂_{𝔛}^{1}. It is now clear from the above that the resulting rigid analytic sheaf 𝔉^{rig} comes equipped with an integrable connection

∇^{rig} : 𝔉^{rig} ⟶ 𝔉^{rig} ⊗_{O_{𝔛^{rig}}} Ω_{𝔛^{rig}}.

7.1.13 Let us give a more precise description of the rings A[I^{n}/π] and B_{n}. To give it, we choose generators f_{1}, …, f_{r} ∈ A of the ideal I. Consider the polynomial ring R := A/πA[x_{1}, …, x_{r}] and the homogeneous ideal J ⊂ R generated by homogeneous polynomials P (x_{1}, …, x_{r}) ∈ R with

P (f_{1}, …, f_{r}) = 0 in A/πA.

It is generated by finitely many homogeneous polynomials P̄_{1}, …, P̄_{s} with degrees d_{1}, …, d_{s}. For any i = 1, …, s we choose a homogeneous lift P_{i} ∈ A[x_{1}, …, x_{r}] of P̄_{i} ∈ R. By our definition of J we can write P_{i}(f_{1}, …, f_{r}) = πq_{i}, for some q_{i} ∈ A. Put c := max{d_{1}, …, d_{s}}. For any n ≥ c we consider the subring

A[T_{M}; |M| = n] ⊂ A[x_{1}, …, x_{r}],

generated by monomials of degree n: For any multi-index M = (m_{1}, …, m_{r}) of total degree |M| = m_{1} + … + m_{r} = n it has one variable T_{M} and these are subject to the relations T_{M_{1}}T_{M_{2}} = T_{M_{3}}T_{M_{4}} if M_{1} + M_{2} = M_{3} + M_{4}. We define a homomorphism

φ : A[T_{M}; |M| = n] ⟶ A[I^{n}/π]

by putting φ(T_{M}) = f^{M}/π = (1/π) ∏_{i} f_{i}^{m_{i}}. It is surjective. If x^{J} is a monomial of degree n − d_{i} then x^{J}P_{i}(x_{1}, …, x_{r}) is homogeneous of degree n and hence corresponds to an element

L_{J,i} := x^{J}P_{i}(x_{1}, …, x_{r}) ∈ A[T_{M}; |M| = n].

We remark that it is linear in the variables T_{M}. The elements L_{J,i} − πq_{i}f^{J} of the ring A[T_{M}; |M| = n] are clearly in the kernel of φ.

7.1.13.1 Lemma. If A has no π-torsion then the elements L_{J,i} − πq_{i}f^{J} generate the kernel of φ as an ideal. Proof. If F (T_{M}) ∈ Ker(φ), write F (T_{M}) = F_{0} + F_{1}(T_{M}) + … + F_{d}(T_{M}) with F_{i} homogeneous of degree i in T_{M}, then it is seen that F̄_{d}(x^{M}) is in J. Thus it follows that we can reduce F to a polynomial of lower degree modulo the elements L_{J,i} − πq_{i}f^{J}. Details are left to the reader. □

We conclude that in this case

A[I^{n}/π] ≅ A[T_{M}; |M| = n]/(L_{J,i} − πq_{i}f^{J})

In particular we derive from this the existence of a surjective ring homomorphism

7.1.13.2  β_{n} : A[I^{n}/π] ⟶ A/I^{n−c}.

It is defined as the A-algebra homomorphism with β_{n}(T_{M}) = 0 ∀M. It exists:

β_{n}(L_{J,i} − πq_{i}f^{J}) = −πq_{i}f^{J} ∈ I^{n−c},

since |J| = n − d_{i} ≥ n − c. These homomorphisms induce homomorphisms β_{n} : B_{n} → A/I^{n−c} and are compatible for varying n: We have commutative diagrams

7.1.13.3  \begin{matrix}B_{n} & ⟶ & A/I^{n−c} \\ ↓ & & ↓ \\ B_{n−1} & ⟶ & A/I^{n−c−1} \end{matrix} .

7.2 Berthelot’s functor for general formal schemes.

In this section we construct 𝔛^{rig} for general 𝔛 in FS_{O}. This is done by gluing 𝔘^{rig} for affine open formal subschemes 𝔘 ⊂ 𝔛. This is possible by the proposition below.

7.2.1 Proposition. Let φ : 𝔜 → 𝔛 be a morphism of affine formal schemes of FS_{O}.

a) Suppose 𝔘 ⊂ 𝔛 is an affine open formal subscheme of 𝔛 with underlying reduced scheme U := 𝔘_{red} ⊂ 𝔛_{red}. The morphism 𝔘^{rig} → 𝔛^{rig} induces an isomorphism of 𝔘^{rig} onto the open analytic subvariety [BGR, p. 354] sp^{−1}(U) ⊂ 𝔛^{rig}.

b) Suppose 𝔛 = ⋃ 𝔘_{i} is an affine covering of 𝔛. The covering 𝔛^{rig} = ⋃ 𝔘_{i}^{rig} is admissible.

c) If φ is of finite type [EGA, I 10.13.3] then φ^{rig} is quasi-compact.

d) If φ is finite (resp. a closed immersion) then so is φ^{rig}. Proof. Suppose 𝔘 is the affine open formal subscheme Spf(A{1/f}) of 𝔛 = Spf(A). In this case sp^{−1}(U) is clearly equal to {x ∈ 𝔛^{rig}; |f(x)| ≥ 1} and by Lemma 7.2.2 below we have 𝔘^{rig} ≅ ⋃_{n} Sp(C_{n}⟨1/f⟩). Hence a) follows in this case. Next, suppose the covering 𝔛 = ⋃ 𝔘_{i} is given by affine open 𝔘_{i} of the form 𝔘_{i} = Spf(A{1/f_{i}}). Finitely many f_{i} generate the unit ideal of A, say f_{1}, …, f_{N} suffice. Thus for any affinoid subdomain V ⊂ 𝔛^{rig}, the intersections V ∩𝔘_{i}^{rig} for i = 1, …, N are affinoid and cover V. It follows that the covering 𝔛^{rig} = ⋃ 𝔘_{i}^{rig} is admissible. The general case of a) and b) follow from a formal argument using that a basis for the topology of 𝔛 is given by the subschemes Spf(A{1/f}).

Assertions c) and d) are a direct consequence of Lemma 7.2.2 below. □

7.2.2 Lemma. If A is as above and A → A′ is topologically of finite type (i.e., satisfies the conditions of [EGA, OI 7.5.5]) then

Spf(A′)^{rig} ≅ ∪_{n}Sp(C_{n} ˆ⊗_{A} A′) = ∪_{n}Sp(K ⊗_{O} B_{n} ˆ⊗_{A} A′).

More precisely, the inverse image of Sp(C_{n}) in Spf(A′)^{rig} is equal to the affinoid space Sp(C_{n} ˆ⊗_{A}A′). Proof. First note that C_{n} ˆ⊗_{A} A′ is an affinoid K-algebra, since it is of topologically finite type over C_{n}. Second, Sp(C_{n} ˆ⊗_{A} A′) → Sp(C_{n+1} ˆ⊗_{A} A′) defines an affinoid subdomain:

\begin{aligned}C_{n} ˆ⊗_{A} A′ & ≅ C_{n+1}⟨x_{1}, …, x_{N}⟩/(πx_{i} − f_{i}) ˆ⊗_{A} A′ \\ & ≅ (C_{n+1} ˆ⊗_{A} A′)⟨x_{1}, …, x_{N}⟩/(πx_{i} − f_{i}) \end{aligned}

(Notations as in the proof of Lemma 7.1.2.) Let us define Y′ := ⋃ Sp(C_{n} ˆ⊗_{A} A′). It is a separated rigid analytic space over K. There is a morphism of rigid analytic spaces Y′ → Spf(A′)^{rig}: For each n there is a morphism

Spf(B_{n} ˆ⊗_{A} A′) ⟶ Spf(A′)

hence a morphism

Sp(C_{n} ˆ⊗_{A} A′) = Spf(B_{n} ˆ⊗_{A} A′)^{rig} → Spf(A′)^{rig}.

Finally, we have to show that any morphism ψ : W → Spf(A′)^{rig}, W an affinoid K-variety, factors as W → Y′ → Spf(A′)^{rig} (see 7.1.5). We use (7.1.7) that ψ comes from a morphism of formal schemes: 𝔚 = Spf(R) ⟶ Spf(A′), where 𝔚 is a model of W. This morphism is given by a continuous ring homomorphism A′ → R. On the other hand, the composition W → Spf(A′)^{rig} → Spf(A)^{rig} comes from a continuous ring homomorphism C_{n} → R ⊗ K for some n. Of course these have to agree on A, hence we get C_{n} ˆ⊗_{A} A′ → R ⊗ K as desired. □

7.2.3 Let us construct 𝔛^{rig} for a separated [EGA, I 10.15.1] formal scheme 𝔛 ∈ ObFS_{O}. Choose a covering 𝔛 = ⋃ 𝔘_{i} by affine formal schemes 𝔘_{i}. The intersections 𝔘_{ij} = 𝔘_{i} ∩ 𝔘_{j} are affine also. Thus by Proposition 7.2.1 we see that 𝔘_{ij}^{rig} is an open analytic subvariety of both 𝔘_{i}^{rig} and 𝔘_{j}^{rig}. Therefore, by [BGR, 9.3.2], we can define 𝔛^{rig} as the pasting of the 𝔘_{i}^{rig} along 𝔘_{ij}^{rig}.

For a general 𝔛 we use the same procedure, using that we already have the definition of 𝔘_{ij}^{rig} for the rigid analytic space associated to the separated but not necessarily affine formal schemes 𝔘_{ij}. We leave the verifications to the reader.

7.2.4 Proposition. Suppose φ : 𝔛 → 𝔜 is a morphism of FS_{O}.

a) 𝔛^{rig} is a quasi-separated rigid analytic space; if 𝔛 is separated then 𝔛^{rig} is separated.

b) If 𝔛 is flat over Spf(O) and nonempty then 𝔛^{rig} is nonempty.

c) If 𝔛 is a formally reduced (resp. normal) formal scheme (see the proof for definitions) then 𝔛^{rig} is reduced (resp. normal). d) If φ is an open immersion then φ^{rig} is an open immersion [BGR, p. 354].

e) If φ is a closed immersion (resp. a finite morphism) then so is φ^{rig}.

f) If φ is a morphism of finite type then φ^{rig} is a quasi-compact.

g) If ℨ → 𝔜 is a second morphism in FS_{O} with target 𝔜 then

7.2.4.1  (𝔛 ×_{𝔜} ℨ)^{rig} ≅ 𝔛^{rig} ×_{𝔜^{rig}} ℨ^{rig}

(Here the fibre products are taken in the category of rigid analytic spaces [BGR, 9.3.5/2] and in the category of formal schemes [EGA, I 10.7.3]. We remark that 𝔛 ×_{𝔜} ℨ is in FS_{O}.)

h) If φ is a separated morphism of formal schemes [EGA, I 10.15.1] then φ^{rig} is separated. Proof. We say that 𝔛 is formally reduced (resp. normal) if it can be covered by affines Spf(A) with A reduced (resp. normal). This is equivalent to the condition that all complete local rings of 𝔛 are reduced (resp. normal), since our rings A are excellent. Thus c) follows from Lemma 7.1.9.

Part b) is left to the reader. Part d) follows from the definition. Parts e) & f) follow from 7.2.1.

The diagonal morphism △ : 𝔛 → 𝔛×_{O} 𝔛 is of finite type. Indeed, it is clearly adic and △_{red} : 𝔛_{red} → (𝔛 ×_{O} 𝔛)_{red} ≅ (𝔛 ×_{k} 𝔛)_{red} is of finite type (use that △_{red} is an immersion and that (𝔛 ×_{O} 𝔛)_{red} is locally Noetherian, next apply [EGA I 6.3.5]). Thus it follows from f) and g) that 𝔛^{rig} is quasi-separated.

We claim that a morphism φ : 𝔛 → 𝔜 in FS_{O} is separated if and only if the diagonal morphism △ : 𝔛 → 𝔛 ×_{𝔜} 𝔛 is a closed immersion. To see that △ is a closed immersion if φ is separated we use Lemma 10.14.4 of [EGA I]. First, 𝔛 ×_{𝔜} 𝔛 is locally Noetherian and second △(𝔛) ⊂ 𝔛 ×_{𝔜} 𝔛 is a closed subset (by definition). For the U_{α} occurring in the Lemma we take Spf(A ˆ⊗_{B} A) ⊂ 𝔛 ×_{𝔜} 𝔛, where Spf(A) is an affine open formal subscheme of 𝔛 mapping into the open affine Spf(B) of 𝔜. The reverse implication is trivial. We conclude that h) follows from g) and e).

Finally, we have to prove g). By functoriality of Berthelot’s construction, there are morphisms:

(𝔛 ×_{𝔜} ℨ)^{rig} → 𝔛^{rig} and (𝔛 ×_{𝔜} ℨ)^{rig} → ℨ^{rig}

agreeing as maps to 𝔜^{rig}. Thus we get the morphism

(𝔛 ×_{𝔜} ℨ)^{rig} ⟶ 𝔛^{rig} ×_{𝔜^{rig}} ℨ^{rig}.

To check that it is an isomorphism we may assume that our formal schemes are affine. In this case the fact that 7.2.4.1 is an isomorphism follows easily from 7.1.5 and 7.1.7. Indeed, by 7.1.7, a morphism of an affinoid variety W into 𝔛^{rig} ×_{𝔜^{rig}} ℨ^{rig} is given by morphisms

𝔚 → 𝔛 and 𝔚 → ℨ

agreeing as morphisms to 𝔜, of a suitable model 𝔚 of W. Hence we get 𝔚 → 𝔛 ×_{𝔜} ℨ which induces W = 𝔚^{rig} → (𝔛 ×_{𝔜} ℨ)^{rig}. Thus we have

Mor_{Rig_{K}} (W, (𝔛 ×_{𝔜} ℨ)^{rig}) ≅ Mor_{Rig_{K}} (W, (𝔛^{rig} ×_{𝔜^{rig}} ℨ^{rig})).

The result now follows from 7.1.5. □

7.2.5 Lemma. Let 𝔛 be an object of FS_{O} and suppose Z ⊂ 𝔛_{red} is a closed subscheme. The completion 𝔛_{Z}^{∧} of 𝔛 along Z is an object of FS_{O} also and the morphism (𝔛_{Z}^{∧})^{rig} → 𝔛^{rig} induced by 𝔛_{Z}^{∧} → 𝔛 is an open immersion inducing an isomorphism of (𝔛_{Z}^{∧})^{rig} with the open subvariety sp^{−1}(Z) of 𝔛^{rig}. Proof. See [B2, 0.2.7]. In the case that 𝔛 is affine this can also be seen using 7.1.7. Suppose that W is an affinoid variety over K and that a morphism f : W → sp^{−1}(Z) is given. This comes from a morphism φ : 𝔚 → 𝔛 of a model 𝔚 of W. We must have φ(𝔚_{red}) ⊂ Z, since the specialization mapping sp : W → 𝔚_{red} is surjective [BGR, 7.1.5/4]. Hence φ factors as 𝔚 → 𝔛_{Z}^{∧} → 𝔛. Thus f is a composition W → (𝔛_{Z}^{∧})^{rig} → 𝔛^{rig}. Using that sp^{−1}(Z) is admissible open (7.1.10) it follows from 7.1.5 that (𝔛_{Z}^{∧})^{rig} ≅ sp^{−1}(Z). □

7.2.6 Suppose that σ : O → O′ is a homomorphism of complete discrete valuation rings, coming from an extension of quotient fields K ⊂ K′ such that the topology on K is induced by that of K′. In this case there is a base field extension functor X ↦ X ˆ⊗K′ defined for quasi-separated analytic varieties X over K. See [BGR, 9.3.6]. Of course there is also a base change functor FS_{O} → FS_{O′}, given simply by 𝔛 ↦ 𝔛×_{Spf(O)} Spf(O′). It results easily from the definitions that there is a canonical isomorphism

(𝔛 ×_{Spf(O)} Spf(O′))^{rig} ≅ 𝔛^{rig} ˆ⊗K′.

Given an analytic variety Y over K′ and a quasi-separated analytic variety X over K we define a general morphism f : Y → X over σ to be a morphism Y → X ˆ⊗K′ of analytic varieties over K. For motivation, see [JP]. We remark that such a general morphism gives rise to pullback functors f^{∗} on sheaves and coherent sheaves. Furthermore, if the sheaves of differentials on X and Y are defined (see 7.1.12), then there is a canonical homomorphism f^{∗}Ω_{X} → Ω_{Y}.

Finally, suppose we are given formal schemes 𝔜 of FS_{O′} and 𝔛 of FS_{O}and a morphism of formal schemes φ : 𝔜 → 𝔛 lying over the morphism Spf(σ) : Spf(O) → Spf(O′). It follows from the above that φ induces a general morphism φ^{rig} : 𝔜^{rig} → 𝔛^{rig}.

7.3 Complete local rings and Berthelot’s construction.

7.3.1 Situation. Here A and B are complete local Noetherian flat O-algebra’s such that A/𝔪_{A} and B/𝔪_{B} are finite extensions of k = O/πO. We assume that A has no zero divisors. In addition, we are given a finite injective homomorphism φ : A ⟶ B. We define the degree of φ as deg(φ) = dim_{Q(A)}Q(A) ⊗_{A} B. Here Q(A) is the quotient field of the ring A.

The formal schemes Spf(A) and Spf(B) are objects of FS_{O}. Let us put X := Spf(A)^{rig} and Y := Spf(B)^{rig}. The map Spf(φ) : Spf(B) → Spf(A) induces a morphism of rigid analytic spaces φ^{rig} : Y → X. It is a finite morphism by Proposition 7.2.1.

7.3.2 Lemma. a) There is an analytically closed, nowhere dense subset Z ⊂ X such that the morphism

φ^{rig} : Y \ (φ^{rig})^{−1}(Z) ⟶ X \ Z

is finite flat of degree equal to deg(φ).

b) If A and B are normal local rings then one can choose Z such that codim_{X}Z ≥ 2.

Proof. Consider a presentation of B as an A-module:

A^{m_{1}} ⟶^{T} A^{m_{0}} ⟶ B ⟶ 0

The ideal J ⊂ A generated by the (m_{2} − deg(φ))-minors of T is nonzero by assumption. We claim that Z := Spf(A/J)^{rig} ⊂ X works. By Proposition 7.2.1 it is a closed subvariety of X and using Lemma 7.1.9 it is easy to see that Z is nowhere dense in X (since J ≠ 0). This proves a).

Let us prove the claim. Lemma 7.2.2 implies the equality Y = ⋃_{n} Sp(C_{n} ⊗_{A} B). It is clear that if g ∈ J then B[1/g] ≅ (A[1/g])^{deg(φ)} as an A[1/g]-module. Hence, (C_{n} ⊗_{A} B)[1/g] ≅ (C_{n}[1/g])^{deg(φ)}. Since Z ∩ Sp(C_{n}) is given by Sp(C_{n}/JC_{n}) (again 7.2.2) the claim is proven.

If A and B are normal and 𝔭 ⊂ A has height 1 then then A_{𝔭} (and B_{𝔮} for any 𝔮 ⊂ B lying over 𝔭) is regular. Hence by [Mat, Theorem 46] the homomorphism A_{𝔭} → B ⊗ A_{𝔭} = ∏ B_{𝔮} is finite flat. Thus 𝔭 ∉ V (J) ⊂ Spec(A). Conclusion: if A and B are normal then codim_{Spec(A)}V (J) ≥ 2 hence codim_{X}Z ≥ 2. (Use Lemma 7.1.9.) This proves b). □

7.3.3 Lemma. Suppose A and B are normal. Any element f ∈ Γ(Y, O_{Y}) satisfies an equation

T^{deg(φ)} + b_{1}T^{deg(φ)−1} + … + b_{deg(φ)} = 0

with b_{i} ∈ Γ(X, O_{X}). If f ∈ Γ(Y, O_{Y}^{∘}) then we may take b_{i} ∈ Γ(X, O_{X}^{∘}). Proof. Take Z ⊂ X as in part b) of 7.3.2. The sheaf E := φ_{∗}^{rig}(O_{Y})|_{X\Z} is a finite locally free sheaf of O_{X\Z}-modules of rank deg(φ). Hence the endomorphism T of E given by multiplication by f satisfies an equation

T^{deg(φ)} + b_{1}T^{deg(φ)−1} + … + b_{deg(φ)} = 0

with b_{i} ∈ Γ(X \ Z, O_{X\Z}). The functions b_{i} extend to analytic functions b_{i} ∈ Γ(X, O_{X}) on X by [Lü, Theorem 1.6 part II] (X is normal by Proposition 7.2.4). This proves the first statement.

Now let f ∈ Γ(Y, O_{Y}^{∘}). We have to prove b_{i} ∈ Γ(X, O_{X}^{∘}). To do this take any affinoid subvariety W ⊂ X and put W′ = (φ^{rig})^{−1}(W) ⊂ Y. It is an affinoid subvariety of Y. We consider the ring extension

R := Γ(W, O_{W}^{∘}) ⊂ R′ := Γ(W′, O_{W′}^{∘})

The extension R ⊂ R′ is finite by [BGR, 6.4.1/6]. Since X and Y are normal, R and R′ are normal O-algebra’s topologically of finite type. By the result of the lemma above we have

deg(φ) = dim_{Q(R)}Q(R) ⊗_{R} R′.

Thus an argument similar to the proof of the lemma gives us an ideal J ⊂ R with codimV (J) ≥ 2 and such that Spec(R′) → Spec(R) is finite flat of degree deg(φ) outside V (J) ⊂ Spec(R). Thus the element f (seen as an element of R′ via Γ(Y, O_{Y}^{∘}) → Γ(W′, O_{W′}^{∘}) = R′) satisfies an equation

T^{deg(φ)} + b′_{1}T^{deg(φ)−1} + … + b′_{deg(φ)} = 0

with b′_{i} ∈ Γ(Spec(R) \ V (J), O_{Spec(R)}) = Γ(Spec(R), O_{Spec(R)}) = R. Comparing these b′_{i} with the b_{i} above, we see that they must be equal as elements of Γ(W, O_{W}). This concludes the proof of the lemma. □

7.3.4 Lemma. If A = O[[x_{1}, …, x_{n}]] then the homomorphism 7.1.8.2

O[[x_{1}, …, x_{n}]] ⟶ Γ(X, O_{X}^{∘})

is an isomorphism. In this case there is an isomorphism X ≅ D^{n} of X with the ndimensional open unit ball D^{n} with coordinates x_{1}, …, x_{n}. Proof. A power-series ∑ a_{I}x^{I} is bounded by 1 on the open unit ball if and only if all |a_{I}| ≤ 1, i.e., a_{I} ∈ O. Thus we need only to find an isomorphism X → D^{n}. Remark that A = O[[x_{1}, …, x_{n}]] is the completion of R := O{x_{1}, …, x_{n}} in the maximal ideal (x_{1}, …, x_{n}). Since clearly Spf(R)^{rig} ≅ 𝔹^{n}, it follows from Lemma 7.2.5 that X ≅ D^{n} (see also 7.1.10). □

7.3.5 Lemma. If B is as in 7.3.1 and B is normal then Y = Spf(B)^{rig} is a connected rigid analytic variety. Proof. Choose a finite injective homomorphism

φ : A = O[[x_{1}, …, x_{d}]] ⟶ B

(to find it argue as in [Mat, p. 212]). Consider the maximal separable extension Q(A) ⊂ L contained in Q(B). Let B′ = B ∩ L =normal closure of A in L. It is a local ring which is a finite extension of A. The morphism Y → X = Spf(A)^{rig} factors as Y → Y′ → X with Y′ = Spf(B′)^{rig}. In case char(K) = 0, we have Y = Y′. In case char(K) = p, the morphism i : Y → Y′ is finite and purely inseparable: for any local section s of i_{∗}(O_{Y}) there exists a number n such that s^{p^{n}} lies in O_{Y′}. Thus the morphism Y → Y′ is an isomorphism on underlying Zariski-topological spaces. (It is even an isomorphism on underlying G-topological spaces.) Hence it suffices to prove that Y′ is connected. We may therefore assume that Q(A) ⊂ Q(B) is finite separable.

Let us choose a finite Galois extension Q(A) ⊂ L containing Q(B). The normal closure B′ of A in L is a local ring, finite over A (since A is Nagata [Mat, 31.C]) and contains B. Clearly, we may replace B by B′ (since Spf(B′)^{rig} → Spf(B)^{rig} is surjective). Hence, we may assume Q(A) ⊂ Q(B) finite Galois, say with group G = Gal(Q(B)/Q(A)).

The group G acts by automorphisms on B over A. Hence on Y over X. We remark that there exists a Zariski open U ⊂ Spec(A) such that the inverse image V ⊂ Spec(B) is a principal homogeneous G-space over U. In fact we can take U = Spec(A)\the discriminant locus of Spec(B) → Spec(A). Hence, the same argument as in the proof of Lemma 7.3.2 shows that there is a Zariski open U ⊂ X such that (φ^{rig})^{−1}(U) is a principal homogeneous G-space over U.

For any connected component Z ⊂ Y the morphism

φ^{rig}|_{Z} : Z ⟶ X

is finite (as the composition Z → Y → X). As it is also generically flat (by 7.3.2) its image must be a connected component of X. Since X is connected (7.3.4) we see that Z → X is surjective. Consequently, Y has only finitely many (connected) components, say Y = Z_{1} ∪ … ∪ Z_{r}. Let Z = Z_{1} and put

G ⊃ H := {g ∈ G | g(Z) = Z}.

Claim. #H = deg(φ^{rig}|_{Z} : Z → X) and the natural homomorphism

Γ(X, O_{X}^{∘}) ⟶ Γ(Z, O_{Z}^{∘})^{H}

is an isomorphism.

If we have this then we are through. Indeed, the homomorphism B ⟶ Γ(Y, O_{Y}^{∘}) ⟶ Γ(Z, O_{Z}^{∘}) is H-invariant. By the claim we would get a homomorphism

B^{H} ⟶ Γ(Z, O_{Z}^{∘})^{H} = Γ(X, O_{X}^{∘}) = A

of A-algebra’s. This contradicts Galois theory unless H = G, i.e., Z = Y.

To prove the claim consider a point x ∈ U ⊂ X in the Zariski open subset of X constructed above. The group G acts transitively on the finite set (φ^{rig})^{−1}({x}), hence it acts transitively on the components Z_{1}, …, Z_{r}, hence #H = #G/r. We must also have deg(φ^{rig}|_{Z_{i}}) = deg(φ^{rig}|_{Z}) (transitivity again) and hence

#G = deg(φ) = ∑_{i} deg(φ^{rig}|_{Z_{i}}) = r · deg(φ^{rig}|_{Z}).

Any H-invariant function on Z can be uniquely extended to a G-invariant function on Y. Therefore, it suffices to show the map

Γ(X, O_{X}^{∘}) ⟶ Γ(Y, O_{Y}^{∘})^{G}

is a bijection. Take an f ∈ Γ(Y, O_{Y}^{∘}) which is G-invariant. The morphism (φ^{rig})^{−1}U → U is a quotient morphism, hence f|_{(φ^{rig})^{−1}U} comes from a unique g ∈ Γ(U, O_{U}^{∘}). Since it is bounded, it comes from a unique g ∈ Γ(X, O_{X}^{∘}) by [Lü, Theorem 1.6 part I] (by Lemma 7.3.4 X ≅ D^{d} is absolutely normal). This proves the claim. □

Finally, we come to the main result of this subsection.

7.3.6 Proposition. If B is as in 7.3.1 and B is normal, then the homomorphism 7.1.8.2

B ⟶ Γ(Spf(B)^{rig}, O^{∘}) = Γ(Y, O_{Y}^{∘})

is bijective. Proof. Choose a finite injective homomorphism

φ : A = O[[x_{1}, …, x_{d}]] ⟶ B.

Choose f ∈ Γ(Y, O_{Y}^{∘}). From lemma 7.3.3 and Lemma 7.3.4 we see that f satisfies an equation

T^{n} + a_{1}T^{n−1} + … + a_{n} = 0

with a_{i} ∈ A = O[[\underline{x}]]. We will show that the assumption f ∉ B leads to a contradiction with Lemma 7.3.5.

The ring B[f] ⊂ Γ(Y, O_{Y}^{∘}) is reduced (Y is reduced). Since Y is irreducible (for the Zariski-topology) as it is connected and normal, we see that B[f] has no zero divisors. By the equation above we see that B[f] is a finite B-algebra. The normal closure

B[f]^{nor} ⊂ Q(B[f])

is a finite B-algebra, since B is Nagata. Any element g ∈ B[f]^{nor} can be seen as a meromorphic function on Y which is integral over B[f] ⊂ Γ(Y, O_{Y}^{∘}), hence g ∈ Γ(Y, O_{Y}^{∘}). Therefore we have B[f]^{nor} ⊂ Γ(Y, O_{Y}^{∘}). Arguing as before, we see that B[f]^{nor} has no zero divisors, thus (being a finite B-algebra) it is a local B-algebra. We consider the morphism

ψ : Y′ := Spf(B[f]^{nor})^{rig} ⟶ Y

induced by B ⊂ B[f]^{nor}. We claim that ψ has a section s : Y → Y′. This is seen as follows. Take any affinoid subdomain U ⊂ Y. Put R := Γ(U, O_{U}^{∘}). The homomorphism

B[f]^{nor} → Γ(Y, O_{Y}^{∘}) → Γ(U, O_{U}^{∘}) = R

induces a morphism Spf(R) → Spf(B[f]^{nor}) and hence U = Spf(R)^{rig} → Spf(B[f]^{nor})^{rig} = Y′. This defines s|_{U}. It is a section of ψ since the composition B → B[f]^{nor} → Γ(Y, O_{Y}^{∘}) is the map 7.1.8.2. Since Y and Y′ are reduced, normal and connected and since ψ is finite surjective, we see that s(Y) must be a connected component of Y′. Lemma 7.3.5 thus implies that Y′ = s(Y). By Lemma 7.3.2 the degree of ψ is dim_{Q(B)}Q(B) ⊗_{B} B[f]^{nor}. Thus we get a contradiction unless this degree is 1, i.e., f ∈ B. □

7.4 Analytic functions and formal functions.

In this section we prove that bounded rigid analytic functions on 𝔛^{rig} come from functions on 𝔛.

7.4.1 Theorem. Let 𝔛 be a formal scheme in FS_{O}. If 𝔛 is O-flat and (formally) normal then the homomorphism 7.1.8.2

Γ(𝔛, O_{𝔛}) ⟶ Γ(𝔛^{rig}, O_{𝔛^{rig}}^{∘})

is an isomorphism. Proof. We may assume 𝔛 = Spf(A) is affine. Let us write X = 𝔛^{rig}. It is constructed (7.1.3) as the union:

X = ⋃_{n} Sp(C_{n}) = ⋃_{n} Sp(B_{n} ⊗_{O} K)

Take f ∈ Γ(X, O_{X}^{∘}). Its restriction to Sp(C_{n}) is an element f_{n} ∈ C_{n} which is integral over B_{n}, i.e., it lies in the integral closure B′_{n} of B_{n} in C_{n} = B_{n} ⊗_{O} K: f_{n} ∈ B′_{n}.

Take a maximal ideal 𝔪 ⊂ B_{n}. Put 𝔪′ = 𝔪 ∩ A; it is a maximal ideal of A (B_{n}/𝔪 is a finite extension of k). Consider the commutative diagram of O-algebra’s:

7.4.1.1  \begin{matrix}A & ⟶ & B_{n} & ⟶ & B′_{n} \\ ↓ & & ↓ & & ↓ \\ A_{𝔪′}^{∧} & ⟶ & (B_{n})_{𝔪}^{∧} & ⟶ & (B′_{n})_{𝔪B′_{n}}^{∧} \end{matrix}

Here (·)_{𝔭}^{∧} denotes completion with respect to 𝔭. This diagram gives rise to a commutative diagram of rigid analytic spaces by applying the functor (·)^{rig}. We remark that A_{𝔪′}^{∧} is normal, so that we can apply 7.3.6 to f restricted to Spf(A_{𝔪′}^{∧})^{rig} ⊂ X. This gives f_{𝔪′} ∈ A_{𝔪′}^{∧}. The image of f_{n} and f_{𝔪′} in the ring (B′_{n})_{𝔪B′_{n}}^{∧} coincide (they give the same rigid analytic functions). Thus we conclude that f_{n} lies in B′_{n} ∩ (B_{n})_{𝔪}^{∧}. Write f_{n} as a/b, with a, b ∈ B_{n}, then we see that a ∈ b(B_{n})_{𝔪}^{∧}, hence a ∈ bB_{n,𝔪} (by faithful flatness of B_{n,𝔪} ⊂ (B_{n})_{𝔪}^{∧}). Thus f_{n} ∈ B_{n,𝔪} ∀𝔪 ∈ Ω(B_{n}). This gives that f_{n} ∈ B_{n}.

At this point we use the homomorphisms β_{n} : B_{n} → A/I^{n−c} (7.1.13.3). The element

f̂ = lim_{⟵} β_{n}(f_{n}) ∈ lim_{⟵} A/I^{n−c} = A

lies in A and in this way we get an inverse to the homomorphism A → Γ(X, O_{X}^{∘}). Finally, we have to show that if f̂ = 0 then 0 = f ∈ Γ(X, O_{X}^{∘}). This is so since

Ker(β_{n}) · B_{n_{0}} ⊂ I^{n−n_{0}}B_{n_{0}}

if n, n_{0} > c. This follows easily from the explicit description of the rings B_{n} in 7.1.13. □

7.4.2 Remark. The most general hypothesis on 𝔛 under which 7.4.1 is true are the following: 𝔛 should be O-flat and for any open affine formal subscheme Spf(A) of 𝔛 the ring A should be integrally closed in the ring A ⊗_{O} K. This fact will not be used in this paper.

7.5 Rigid descent of closed formal subschemes.

7.5.1 Suppose 𝔛 is an object of FS_{O}. Let T ⊂ 𝔛_{red} be a closed subscheme and let U = 𝔛_{red} \ T. We define

𝔗 = 𝔛_{T}^{∧} = completion of 𝔛 along T,
𝔘 = open formal subscheme of 𝔛 with 𝔘_{red} = U,
X = 𝔛^{rig}.

We remark that 𝔗^{rig} ≅ sp^{−1}(T) is an open subvariety of X by Lemma 7.2.5. Similarly, 𝔘^{rig} ≅ sp^{−1}(U) is an open subvariety of X by 7.2.1. In fact X is the disjoint union of sp^{−1}(T) and sp^{−1}(U) but the covering X = sp^{−1}(T)∪sp^{−1}(U) is not admissible in general.

Let us consider triples (ℨ_{T} , ℨ_{U}, Z) of the following kind 7.5.1.1 ℨ_{T} (resp. ℨ_{U}) is an O-flat closed formal subscheme of 𝔗 (resp. 𝔘) and Z ⊂ X is

a closed analytic subvariety of X satisfying the following condition 7.5.1.2 we have the equality (ℨ_{T})^{rig} = Z ∩ sp^{−1}(T) (resp. (ℨ_{U})^{rig} = Z ∩ sp^{−1}(U)) as

closed analytic subvarieties of 𝔗^{rig} ≅ sp^{−1}(T) (resp. 𝔘^{rig} ≅ sp^{−1}(U)). For each O-flat closed formal subscheme ℨ ⊂ 𝔛 we get a triple (ℨ ∩ 𝔗, ℨ ∩ 𝔘, ℨ^{rig}) as in 7.5.1.1 satisfying 7.5.1.2. The proposition below shows that the converse is true also. The idea is that in some sense the “covering” of 𝔛 given by

sp^{−1}(T) ⊔ sp^{−1}(U)\overrightarrow{⟶}X ⊔ 𝔘 ⊔ 𝔗 ⟶ 𝔛

is sufficiently good to allow descent for closed subvarieties.

7.5.2 Proposition. Any triple 7.5.1.1 (ℨ_{T} , ℨ_{U}, Z) satisfying 7.5.1.2 comes from an O-flat closed formal subscheme ℨ ⊂ 𝔛 as described above. Proof. We may assume that 𝔛 = Spf(A) is affine. We may assume that A has no πtorsion. Let I ⊂ A be the biggest ideal of definition of A and suppose that T ⊂ Spec(A/I) is given by the ideal I + g_{1}A + … + g_{r}A of A. (We assume that g_{i} ∉ I for all i.) We put

A^{∧} := lim_{⟵} A/(g_{1}A + … + g_{r}A)^{n} ≅ lim_{⟵} A/(I + g_{1}A + … + g_{r}A)^{n}

and for i = 1, …, r:

A{1/g_{i}} := lim_{⟵} A/I^{n}[1/g_{i}]

These are flat A-algebra’s. We note that we may assume T is reduced so that I + g_{1}A^{∧} + … + g_{r}A^{∧} is the biggest ideal of definition of A^{∧}. We have 𝔗 = Spf(A^{∧}) and 𝔘 = ⋃ Spf(A{1/g_{i}}), so that ℨ_{T} and ℨ_{U} are given by ideals

J^{∧} ⊂ A^{∧} and J_{i} ⊂ A{1/g_{i}}.

These ideals are such that A^{∧}/J^{∧} and A{1/g_{i}}/J_{i} are π-torsion free (and of course J_{i}A{1/g_{i}g_{j}} = J_{j}A{1/g_{i}g_{j}}).

In 7.1.3 we defined X as the increasing union X = ⋃_{n} V_{n} of affinoid varieties V_{n}:

V_{n} := Sp(C_{n}) = Sp(B_{n} ⊗_{O} K) = Spf(B_{n})^{rig}

The closed subvariety Z ∩ V_{n} is defined by an ideal I_{n} ⊂ B_{n}; we define I_{n} as the kernel of the homomorphism

B_{n} ⟶ Γ(Z ∩ V_{n}, O_{Z}).

Thus it is clear that B_{n}/I_{n} is π-torsion free.

Recall that in 7.1.13 we defined surjective ring homomorphisms β_{n} : B_{n} → A/I^{n−c}. We will prove that the ideals β_{n}(I_{n}) ⊂ A/I^{n−c} satisfy the following two statements (at least after replacing c by a sufficiently large constant)

1) β_{n}(I_{n})A^{∧}/(IA^{∧})^{n−c} = J^{∧} mod (IA^{∧})^{n−c},

2) β_{n}(I_{n})A{1/g_{i}}/(IA{1/g_{i}})^{n−c} = J_{i} mod (IA{1/g_{i}})^{n−c}.

If we have this then we are done: 1) and 2) imply that β_{n+1}(I_{n+1}) mod I^{n−c} = β_{n}(I_{n}), i.e., the maps β_{n+1}(I_{n+1}) → β_{n}(I_{n}) are surjective. Thus the limit

J := lim_{⟵} β_{n}(I_{n}) ⊂ lim_{⟵} A/I^{n−c} = A

is an ideal of A such that JA^{∧} = J^{∧} and JA{1/g_{i}} = J_{i}. Hence we can take ℨ = Spf(A/J). To prove 1) and 2) we put

B_{n}^{∧} := lim_{\overleftarrow{ℓ}} B_{n}/(g_{1}B_{n} + … + g_{r}B_{n})^{ℓ} ≅ B_{n} ˆ⊗_{A} A^{∧}

and

B_{n}\{1/g_{i}\} := lim_{\overleftarrow{ℓ}} B_{n}/π^{ℓ}[1/g_{i}] ≅ B_{n} ˆ⊗_{A} A\{1/g_{i}\}.

It is clear that β_{n} induces homomorphisms

β_{n}^{∧} : B_{n}^{∧} ⟶ A^{∧}/(IA^{∧})^{n−c}

and

β{1/g_{i}} : B_{n}{1/g_{i}} ⟶ A{1/g_{i}}/(IA{1/g_{i}})^{n−c}.

Furthermore it is easily seen that

β_{n}(I_{n}) · A^{∧}/(IA^{∧})^{n−c} = β_{n}^{∧}(I_{n}B_{n}^{∧})

and that

β_{n}(I_{n}) · A{1/g_{i}}/(IA{1/g_{i}})^{n−c} = β{1/g_{i}}(I_{n}B_{n}{1/g_{i}}).

By our choice of I_{n} we get the equality

Spf(B_{n}/I_{n})^{rig} = V_{n} ∩ Z

thus we also have (use 7.2.4 part g) and 7.2.5)

Spf(B_{n}^{∧}/I_{n}B_{n}^{∧})^{rig} = V_{n} ∩ sp^{−1}(T) ∩ Z = V_{n} ∩ Spf(A^{∧})^{rig} ∩ Z

and (use 7.2.1)

Spf(B_{n}{1/g_{i}}/I_{n}B_{n}{1/g_{i}})^{rig} = V_{n} ∩ {|g_{i}| ≥ 1} ∩ Z = V_{n} ∩ Spf(A{1/g_{i}})^{rig} ∩ Z.

At this point we are able to prove the inclusions ⊃ of 1) and 2). For 1), take h ∈ J^{∧}. The image of h in B_{n}^{∧} lies in I_{n}B_{n}^{∧}: it is zero as a function on V_{n} ∩ sp^{−1}(T) ∩ Z and B_{n}^{∧}/I_{n}B_{n}^{∧} ≅ (B_{n}/I_{n})^{∧} is π-torsion free. Thus β_{n}(h) ≡ h lies in β_{n}^{∧}(I_{n}B_{n}^{∧}). For 2), take h ∈ J_{i}. The image of h in B_{n}{1/g_{i}} lies in I_{n}B_{n}{1/g_{i}}: it is zero as a function on V_{n} ∩ Spf(A{1/g_{i}})^{rig} ∩ Z and B_{n}{1/g_{i}}/I_{n}B_{n}{1/g_{i}} ≅ (B_{n}/I_{n}){1/g_{i}} is π-torsion free. Thus β_{n}(h) ≡ h lies in β_{n}{1/g_{i}}(I_{n}B_{n}{1/g_{i}}).

To prove the other inclusion, we need that the construction of 7.1.13, which associates to an O-algebra A the system (B_{n}, β_{n})_{n≥1} is a functor. That is, if A → A′ is a homomorphism of such algebra’s then there exists a c ∈ ℕ and commutative diagrams:

B_{n} = B_{n}(A) ⟶ B_{n}(A′)
↓^{β_{n}} ↓^{β′_{n}}
A/I^{n−c} ⟶ A′/(I′)^{n−c}

This is easy to see using the explicit constructions in 7.1.1 and 7.1.13. Thus we have a commutative diagram (for some c ∈ ℕ independent of n):

B_{n}{1/g_{i}}  ≅  B_{n}(A{1/g_{i}})  ⟶  B_{n}(A{1/g_{i}}/J_{i})
↓^{β_{n}{1/g_{i}}}  ↓^{β_{n}}  ↓^{β_{n}}
(A/I^{n−c}){1/g_{i}} ≅ A{1/g_{i}}/(IA{1/g_{i}})^{n−c} ⟶ A{1/g_{i}}/J_{i} + (IA{1/g_{i}})^{n−c}

Also it is clear that I_{n}B_{n}{1/g_{i}} lies in the kernel of the upper horizontal arrow of this diagram; an element of I_{n} gives the zero function of Spf(B_{n}(A{1/g_{i}}/J_{i}))^{rig} ⊂ Z ∩ V_{n}. Thus we see that the inclusion ⊂ holds in 2) for some constant c independent of n.

It is more tricky to prove the inclusion ⊂ in 1). To do it we note that we have the following equalities of open subvarieties of V_{n}:

Spf(B_{n}^{∧})^{rig} = {x ∈ V_{n} | |g_{i}(x)| < 1}
= V_{n} ∩ Spf(A^{∧})^{rig}
= ⋃_{N} (V_{n} ∩ Spf(B_{N}(A^{∧}))^{rig})
= ⋃_{N} (Spf(B_{n})^{rig} ∩ Spf(B_{N}(A^{∧}))^{rig}) ⊂ ⋃_{N} Spf(B_{N})^{rig}

Thus we see that Spf(B_{n}^{∧})^{rig} may be written as the union of the rigid spaces associated to the O-algebra’s

B_{n} ˆ⊗_{B_{N}} B_{N}(A^{∧}).

The inclusions are given by the homomorphisms

B_{n}^{∧} ≅ B_{n} ˆ⊗_{A} A^{∧} ⟶ B_{n} ˆ⊗_{B_{N}} B_{N}(A^{∧}).

These fit into the following commutative diagram (for some c independent of n, N):

B_{n}^{∧}  ⟶^{β_{n}^{∧}}  A^{∧}/(IA^{∧})^{n−c}
↓ ≅  ↓ ≅
B_{n} ˆ⊗_{A} A^{∧} ⟶^{β_{n}⊗id_{A^{∧}}} (A/I^{n−c}) ⊗_{A} A^{∧}
↓  ↓
B_{n} ˆ⊗_{B_{N}} B_{N}(A^{∧})  ⟶ A/I^{n−c} ⊗_{A/I^{N−c}} A^{∧}/(IA^{∧} + g_{1}A^{∧} + … + g_{r}A^{∧})^{N−c}
↓  ↓
B_{n} ˆ⊗_{B_{N}} B_{N}(A^{∧}/J^{∧}) ⟶  A^{∧}/J^{∧} + I^{n−c} + (IA^{∧} + g_{1}A^{∧} + … + g_{r}A^{∧})^{N−c}

In the same way as before we see that I_{n}B_{n}^{∧} maps to zero under the left vertical arrows. Hence we see that

β_{n}^{∧}(I_{n}B_{n}^{∧}) ⊂ J^{∧} + I^{n−c} + (IA^{∧} + g_{1}A^{∧} + … + g_{r}A^{∧})^{N−c}

for all N > c. This proves the inclusion ⊂ in 1). The proof of 7.5.2 is complete. □

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