Purity Results for p-Divisible Groups

and Abelian Schemes over Regular Bases

of Mixed Characteristic

Adrian Vasiu and Thomas Zink

Received: January 21, 2010 Revised: March 19. 2010

Communicated by Peter Schneider

Abstract. Let p be a prime. Let (R, 𝔪) be a regular local ring of mixed characteristic (0, p) and absolute index of ramification e. We provide general criteria of when each abelian scheme over Spec R \ {𝔪} extends to an abelian scheme over Spec R. We show that such extensions always exist if e ≤ p−1, exist in most cases if p ≤ e ≤ 2p−3, and do not exist in general if e ≥ 2p − 2. The case e ≤ p − 1 implies the uniqueness of integral canonical models of Shimura varieties over a discrete valuation ring O of mixed characteristic (0, p) and index of ramification at most p − 1. This leads to large classes of examples of Néron models over O. If p > 2 and index p−1, the examples are new.

2010 Mathematics Subject Classification: 11G10, 11G18, 14F30, 14G35, 14G40, 14K10, 14K15, 14L05, 14L15, and 14J20. Keywords and Phrases: rings, group schemes, p-divisible groups, Breuil windows and modules, abelian schemes, Shimura varieties, and Néron models.

To Michel Raynaud, for his 71th birthday

1 Introduction

Let p be a prime number. We recall the following global purity notion introduced in [V1], Definitions 3.2.1 2) and 9) and studied in [V1] and [V2].

Definition 1 Let X be a regular scheme that is faithfully flat over Spec ℤ_{(p)}. We say X is healthy regular (resp. p-healthy regular), if for each open subscheme U of X which contains X_{ℚ} and all generic points of X_{𝔽_{p}}, every abelian scheme (resp. p-divisible group) over U extends uniquely to an abelian scheme (resp. a p-divisible group) over X.

In (the proofs of) [FC], Chapter IV, Theorems 6.4, 6.4′, and 6.8 was claimed that every regular scheme which is faithfully flat over Spec ℤ_{(p)} is healthy regular as well as p-healthy regular. This claim was disproved by an example of Raynaud–Gabber (see [Ga] and [dJO], Section 6): the regular scheme Spec W (k)[[T_{1}, T_{2}]]/(p − (T_{1}T_{2})^{p−1}) is neither p-healthy nor healthy regular. Here W (k) is the ring of Witt vectors with coefficients in a perfect field k of characteristic p. The importance of healthy and p-healthy regular schemes stems from their applications to the study of integral models of Shimura varieties. We have a local version of Definition 1 as suggested by Grothendieck’s work on the classical Nagata–Zariski purity theorem (see [Gr]).

Definition 2 Let R be a local noetherian ring with maximal ideal 𝔪 such that depth R ≥ 2. We say that R is quasi-healthy (resp. p-quasi-healthy) if each abelian scheme (resp. p-divisible group) over Spec R \ {𝔪} extends uniquely to an abelian scheme (resp. a p-divisible group) over Spec R.

If R is local, complete, regular of dimension 2 and mixed characteristic (0, p), then the fact that R is p-quasi-healthy can be restated in terms of finite flat commutative group schemes annihilated by p over Spec R (cf. Lemma 20). Our main result is the following theorem proved in Subsections 4.4 and 5.2.

Theorem 3 Let R be a regular local ring of dimension d ≥ 2 and of mixed characteristic (0, p). We assume that there exists a faithfully flat local R-algebra R̂ which is complete and regular of dimension d, which has an algebraically closed residue class field k, and which is equipped with an epimorphism R̂ ↠ W (k)[[T_{1}, T_{2}]]/(p − h) where h ∈ (T_{1}, T_{2})W (k)[[T_{1}, T_{2}]] is a power series whose reduction modulo the ideal (p, T_{1}^{p}, T_{2}^{p}, T_{1}^{p−1}T_{2}^{p−1}) is non-zero. Then R is quasihealthy. If moreover d = 2, then R is also p-quasi-healthy.

For instance, Theorem 3 applies if the strict completion of R is isomorphic to W (k)[[T_{1}, …, T_{d}]]/(p−T_{1}·. . .·T_{m}) with 1 ≤ m ≤ min{d, 2p−3} (cf. Subsection 5.3). The following consequence is also proved in Subsection 5.2.

Corollary 4 Let R be a regular local ring of dimension d ≥ 2 and of mixed characteristic (0, p). Let 𝔪 be the maximal ideal of R. We assume that p ∉ 𝔪^{p}. Then R is quasi-healthy. If moreover d = 2, then R is also p-quasi-healthy.

Directly from Theorem 3 and from very definitions we get:

Purity Results for p-Divisible Groups and … 573

Corollary 5 Let X be a regular scheme that is faithfully flat over Spec ℤ_{(p)}. We assume that each local ring R of X of mixed characteristic (0, p) and dimension at least 2 is such that the hypotheses of Theorem 3 hold for it (for instance, this holds if X is formally smooth over the spectrum of a discrete valuation ring O of mixed characteristic (0, p) and index of ramification e ≤ p−1). Then X is healthy regular. If moreover dim X = 2, then X is also p-healthy regular.

The importance of Corollary 5 stems from its applications to Néron models (see Section 6). Theorem 31 shows the existence of large classes of new types of Néron models that were not studied before in [N], [BLR], [V1], [V2], or [V3], Proposition 4.4.1. Corollary 5 encompasses (the correct parts of) [V1], Subsubsection 3.2.17 and [V2], Theorem 1.3. Theorem 28 (i) shows that if X is formally smooth over the spectrum of a discrete valuation ring O of mixed characteristic (0, p) and index of ramification e at least p, then in general X is neither p-healthy nor healthy regular. From this and Raynaud–Gabber example we get that Theorem 3 and Corollary 4 are optimal. Even more, if R = W (k)[[T_{1}, T_{2}]]/(p − h) with h ∈ (T_{1}, T_{2}), then one would be inclined to expect that R is p-quasi-healthy if and only if h does not belong to the ideal (p, T_{1}^{p}, T_{2}^{p}, T_{1}^{p−1}T_{2}^{p−1}); this is supported by Theorems 3 and 28 and by Lemma 12. In particular, parts (ii) and (iii) of Theorem 28 present two generalizations of the Raynaud–Gabber example. Our proofs are based on the classification of finite flat commutative group schemes of p power order over the spectrum of a local, complete, regular ring R of mixed characteristic (0, p) and perfect residue class field. For dim R = 1 this classification was a conjecture of Breuil [Br] proved by Kisin in [K1] and [K2] and reproved by us in [VZ], Theorem 1. Some cases with dim R ≥ 2 were also treated in [VZ]. The general case is proved by Lau in [L1], Theorems 1.2 and 10.7. Proposition 15 provides a new proof of Raynaud’s result [R2], Corollary 3.3.6. Subsection 5.1 disproves an additional claim of [FC], Chapter V, Section 6. It is the claim of [FC], top of p. 184 on torsors of liftings of p-divisible groups which was not previously disproved and which unfortunately was used in [V1] and [V2], Subsection 4.3. This explains why our results on p-healthy regular schemes and p-quasi-healthy regular local rings work only for dimension 2 (the difficulty is for the passage from dimension 2 to dimension 3). Implicitly, the phealthy part of [V2], Theorem 1.3 is proved correctly in [V2] only for dimension 2. The paper is structured as follows. Different preliminaries on Breuil windows and modules are introduced in Section 2. In Section 3 we study morphisms between Breuil modules. Our basic results on extending properties of finite flat group schemes, p-divisible groups, and abelian schemes are presented in Sections 4 and 5. Section 6 contains applications to integral models and Néron models.

Acknowledgement. The first author would like to thank Binghamton and Bielefeld Universities for good conditions with which to write this note; he was partially supported by the NSF grant DMS #0900967. The second author would like to thank Eike Lau for helpful discussions. Both authors would like to thank the referee for several valuable comments.

2 Preliminaries

In this paper the notions of frame and window are in a more general sense than in [Z1]. The new notions are suggested by the works [Br], [K1], [VZ], and [L1]. In all that follows we assume that a ring is unitary and commutative and that a finite flat group scheme is commutative and (locally) of p power order.

Definition 6 A frame F = (R, S, J, σ, σ̇, θ) for a ring R consists of the following data:

(a) A ring S and an ideal J ⊂ S.

(b) An isomorphism of rings S/J ≅ R.

(c) A ring homomorphism σ : S → S.

(d) A σ-linear map σ̇ : J → S.

(e) An element θ ∈ S.

We assume that pS + J is in the radical of S, that σ induces the Frobenius endomorphism on S/pS, and that the following equation holds:

σ(η) = θσ̇(η), for all η ∈ J.  (1)

Assume that

Sσ̇(J) = S.  (2)

In this case we find an equation 1 = ∑_{i} ξ_{i}σ̇(η_{i}) with ξ_{i} ∈ S and η_{i} ∈ J. From this and (1) we get that θ = ∑_{i} ξ_{i}σ(η_{i}). Therefore for each η ∈ J we have

σ(η) = ∑_{i} ξ_{i}σ(η)σ̇(η_{i}) = ∑_{i} ξ_{i}σ̇(ηη_{i}) = ∑_{i} ξ_{i}σ(η_{i})σ̇(η) = θσ̇(η).

We conclude that the equation (2) implies the existence and uniqueness of an element θ such that the equation (1) is satisfied. If M is an S-module we set M^{(σ)} := S ⊗_{σ,S} M. The linearization of a σ-linear map ϕ : M → N is denoted by ϕ^{♯} : M^{(σ)} → N.

Definition 7 A window with respect to F is a quadruple (P, Q, F, Ḟ) where:

(a) P is a finitely generated projective S-module.

(b) Q ⊂ P is an S-submodule.

(c) F : P → P is a σ-linear map.

(d) Ḟ : Q → P is a σ-linear map.

We assume that the following three conditions are satisfied:

(i) There exists a decomposition P = T ⊕ L such that Q = JT ⊕ L.

(ii) F (y) = θḞ (y) for y ∈ Q and Ḟ (ηx) = σ̇(η)F (x) for x ∈ P and η ∈ J.

(iii) F (P) and Ḟ (Q) generate P as an S-module.

If (2) holds, then from the second part of (ii) we get that for x ∈ P we have F (x) = 1F (x) = ∑_{i} ξ_{i}σ̇(η_{i})F (x) = ∑_{i} ξ_{i}Ḟ (η_{i}x); if moreover x ∈ Q, then F (x) = ∑_{i} ξ_{i}σ(η_{i})Ḟ (x) = θḞ (x). Thus if (2) holds, then we have an inclusion F (P) ⊂ SḞ (Q) and the first condition of (ii) follows from the second condition of (ii). A decomposition as in (i) is called a normal decomposition. We note that for each window (P, Q, F, Ḟ) as above, the S-linear map

F^{♯} ⊕ Ḟ^{♯} : S ⊗_{σ,S} T ⊕ S ⊗_{σ,S} L → T ⊕ L  (3)

is an isomorphism. Conversely an arbitrary isomorphism (3) defines uniquely a window with respect to F equipped with a given normal decomposition. Let R be a regular local ring of mixed characteristic (0, p). Let 𝔪 be the maximal ideal of R and let k := R/𝔪. A finite flat group scheme H over Spec R is called residually connected if its special fibre H_{k} over Spec k is connected (i.e., has a trivial étale part). If R is complete, then H is residually connected if and only if H as a scheme is connected and therefore in this case we will drop the word residually. We apply the same terminology to p-divisible groups over Spec R. In this section we will assume moreover that k is perfect and that R is complete (in the 𝔪-adic topology). Let d = dim R ≥ 1. We choose regular parameters t_{1}, …, t_{d} of R. We denote by W (†) the ring of Witt vectors with coefficients in a ring †. We set 𝔖 := W (k)[[T_{1}, …, T_{d}]]. We consider the epimorphism of rings

𝔖 ↠ R,

which maps each indeterminate T_{i} to t_{i}. Let h ∈ 𝔖 be a power series without constant term such that we have

p = h(t_{1}, …, t_{d}).

We set E := p − h ∈ 𝔖. Then we have a canonical isomorphism

𝔖/E𝔖 ≅ R.

As R is a regular local ring of mixed characteristic (0, p), we have E ∉ p𝔖. We extend the Frobenius automorphism σ of W (k) to 𝔖 by the rule

σ(T_{i}) = T_{i}^{p}.

Note that σ : 𝔖 → 𝔖 is flat. Let σ̇ : E𝔖 → 𝔖 be the σ-linear map defined by the rule

σ̇(Es) := σ(s).

Definition 8 We refer to the sextuple L = (R, 𝔖, E𝔖, σ, σ̇, σ(E)) as a standard frame for the ring R. A window for this frame will be called a Breuil window.

As σ̇(E) = 1, the condition (2) holds for L. Let Ŵ (R) ⊂ W (R) be the subring defined in [Z2], Introduction and let σ_{R} be its Frobenius endomorphism. There exists a natural ring homomorphism

κ : 𝔖 → Ŵ (R)

which commutes with the Frobenius endomorphisms, i.e. for all s ∈ 𝔖 we have

κ(σ(s)) = σ_{R}(κ(s)).

The element κ(T_{i}) is the Teichmüller representative [t_{i}] ∈ Ŵ (R) of t_{i}.

Theorem 9 (Lau [L1], Theorem 1.1) If p ≥ 3, then the category of Breuil windows is equivalent to the category of p-divisible groups over Spec R.

This theorem was proved first in some cases in [VZ], Theorem 1. As in [VZ], the proof in [L1] shows first that the category of Breuil windows is equivalent (via κ) to the category of Dieudonné displays over Ŵ (R). Theorem 9 follows from this and the fact (see [Z2], Theorem) that the category of Dieudonné displays over Ŵ (R) is equivalent to the category of p-divisible groups over Spec R. There exists a version of Theorem 9 for connected p-divisible groups over Spec R as described in the introductions of [VZ] and [L1]. This version holds as well for p = 2, cf. [L1], Corollary 10.6. The categories of Theorem 9 have natural exact structures [Me2]. The equivalence of Theorem 9 (or of its version for p ≥ 2) respects the exact structures. For a prime ideal 𝔭 of R which contains p, let κ(𝔭)^{perf} be the perfect hull of the residue class field κ(𝔭) of 𝔭. We deduce from κ a ring homomorphism κ_{𝔭} : 𝔖 → W (κ(𝔭)^{perf}) = Ŵ (κ(𝔭)^{perf}).

Proposition 10 Let G be a p-divisible group over R. If p = 2, we assume that G is connected. Let (P, Q, F, Ḟ) be the Breuil window of G. Then the classical Dieudonné module of G_{κ(𝔭)^{perf}} is canonically isomorphic (in a functorial way) to W (κ(𝔭)^{perf}) ⊗_{κ_{𝔭},𝔖} P endowed with the σ_{κ(𝔭)^{perf}} -linear map σ_{κ(𝔭)^{perf}} ⊗ F.

Proof. If G is connected, we consider its display (P′, Q′, F′, Ḟ′) over W (R). By [L1], Theorem 1.1 and Corollary 10.6 we have P′ = W (R)⊗_{κ,𝔖}P. Let us denote by 𝔻(G) the Grothendieck–Messing crystal associated to G, cf. [Me1]. By [Z1], Theorem 6 there is a canonical and functorial isomorphism 𝔻(G)_{W (R)} ≅ P′ ≅ W (R)⊗_{κ,𝔖}P. The functor 𝔻 commutes with base change. If we apply this to the morphism of pd-extensions W (R) → W (κ(𝔭)^{perf}) of R and κ(𝔭)^{perf} (respectively), we get that 𝔻(G_{κ(𝔭)^{perf}} ) ≅ 𝔻(G)_{W (κ(𝔭)^{perf} )} ≅ W (κ(𝔭)^{perf}) ⊗_{κ_{𝔭},𝔖} P. From this the proposition follows provided G is connected. In the case p ≥ 3 the same argument works if we replace (P′, Q′, F′, Ḟ′) by the Dieudonné display of G over Ŵ (R). But in this case the isomorphism 𝔻(G)_{Ŵ (R)} ≅ P′ follows from [L2], Theorem 6.9. □

All results in this paper are actually independent of the nonconnected part of the last Proposition. This is explained in the proof of Corollary 21. In what follows we will not need to keep track of the σ_{κ(𝔭)^{perf}} -linear maps σ_{κ(𝔭)^{perf}} ⊗ F and thus we will simply call W (κ(𝔭)^{perf}) ⊗_{κ_{𝔭},𝔖} P the fibre of the Breuil window (P, Q, F, Ḟ) over 𝔭. We often write a Breuil window in the form (Q, ϕ) originally proposed by Breuil, where ϕ is the composite of the inclusion Q ⊂ P with the inverse of the 𝔖-linear isomorphism Ḟ^{♯} : Q^{(σ)} ≅ P. In this notation P, F, and Ḟ are omitted as they are determined naturally by Ḟ^{♯} and thus by ϕ (see [VZ], Section 2). A Breuil window in this form is characterized as follows: Q is a finitely generated free 𝔖-module and ϕ : Q → Q^{(σ)} is a 𝔖-linear map whose cokernel is annihilated by E. We note that this implies easily that there exists a unique 𝔖-linear map ψ : Q^{(σ)} → Q such that we have

ϕ ∘ ψ = E id_{Q^{(σ)}}, ψ ∘ ϕ = E id_{Q} .

Clearly the datum (Q, ψ) is equivalent to the datum (Q, ϕ). In the notation (Q, ϕ), its fibre over 𝔭 is W (κ(𝔭)^{perf}) ⊗_{κ_{𝔭},𝔖} Q^{(σ)} = W (κ(𝔭)^{perf}) ⊗_{σκ_{𝔭},𝔖} Q. The dual of a Breuil window is defined as follows. Let M be a 𝔖-module. We set M̂ := Hom_{𝔖}(M, 𝔖). A 𝔖-linear map M → 𝔖 defines a homomorphism M^{(σ)} → 𝔖^{(σ)} = 𝔖. This defines a 𝔖–linear map:

M̂^{(σ)} = 𝔖 ⊗_{σ,𝔖} Hom_{𝔖}(M, 𝔖) → Hom_{𝔖}(M^{(σ)}, 𝔖) = { \atop M^{(σ)}}.

It is clearly an isomorphism if M is a free 𝔖-module of finite rank and therefore also if M is a finitely generated 𝔖-module by a formal argument which uses the flatness of σ : 𝔖 → 𝔖. If (Q, ϕ) is a Breuil window we obtain a 𝔖-linear map

ϕ̂: Q̂^{(σ)} = { \atop (Q^{(σ)})} → Q̂.

More symmetrically we can say that if (Q, ϕ, ψ) is a Breuil window then (Q̂, ψ̂, ϕ̂) is a Breuil window. We call (Q̂, ψ̂, ϕ̂) the dual Breuil window of (Q, ϕ, ψ). Taking the fibre of a Breuil window (Q, ϕ) over 𝔭 is compatible with duals as we have:

W (κ(𝔭)^{perf})⊗_{κ_{𝔭},𝔖}Q̂^{(σ)} ≅ Hom_{W (κ(𝔭)^{perf} )}(W (κ(𝔭)^{perf})⊗_{κ_{𝔭},𝔖}Q^{(σ)}, W (κ(𝔭)^{perf})). (4)

We recall from [VZ], Definition 2 and [L1] that a Breuil module (M, φ) is a pair, where M is a finitely generated 𝔖-module which is of projective dimension at most 1 and which is annihilated by a power of p and where φ : M → M^{(σ)} is a 𝔖-linear map whose cokernel is annihilated by E. We note that the map φ is always injective (the argument for this is the same as in [VZ], Proposition 2 (i)). It follows formally that there exists a unique 𝔖-linear map ϑ : M^{(σ)} → M such that we have

φ ∘ ϑ = E id_{M^{(σ)}}, ϑ ∘ φ = E id_{M} .

We define the dual Breuil module (M^{∗}, ϑ^{∗}, φ^{∗}) of (M, φ, ϑ) by applying the functor M^{∗} = Ext_{𝔖}^{1}(M, 𝔖) in the same manner we did for windows. It is easy to see that the 𝔖-module M^{∗} has projective dimension at most 1. The fibre of (M, φ) (or of (M, φ, ϑ)) over 𝔭 is

W (κ(𝔭)^{perf}) ⊗_{κ_{𝔭},𝔖} M^{(σ)} = W (κ(𝔭)^{perf}) ⊗_{σκ_{𝔭},𝔖} M.  (5)

The duals are again compatible with taking fibres as we have:

W (κ(𝔭)^{perf}) ⊗_{σκ_{𝔭},𝔖} M^{∗} ≅
≅ Ext_{W (κ(𝔭)^{perf} )}^{1}(W (κ(𝔭)^{perf}) ⊗_{σκ_{𝔭},𝔖} M, W (κ(𝔭)^{perf}))
= Hom_{W (κ(𝔭)^{perf} )}(W (κ(𝔭)^{perf}) ⊗_{σκ_{𝔭},𝔖} M, W (κ(𝔭)^{perf}) ⊗_{ℤ} ℚ).

(6)

Assume that p annihilates M, i.e. M is a module over 𝔖̄ = 𝔖/p𝔖. As depth M is the same over either 𝔖 or 𝔖̄ and it is d if M ≠ 0, we easily get that M is a free 𝔖̄-module. From this we get that M^{∗} = Hom𝔖̄(M, 𝔖̄) (to be compared with the last isomorphism of (6)). Thus in this case the duality works exactly as for windows. If p ≥ 3, it follows from Theorem 9 that the category of finite flat group schemes over Spec R is equivalent to the category of Breuil modules (the argument for this is the same as for [VZ], Theorem 2). We have a variant of this for p = 2 (cf. [L1], Theorem 10.7): the category of connected finite flat group schemes over Spec R is equivalent to the category of nilpotent Breuil modules (i.e., of Breuil modules (M, φ) that have the property that the reduction of φ modulo the maximal ideal of 𝔖 is nilpotent in the natural way). We recall from [VZ], Subsection 6.1 that the Breuil module of a finite flat group scheme H over Spec R is obtained as follows. By a theorem of Raynaud (see [BBM], Theorem 3.1.1) we can represent H as the kernel

0 → H → G_{1} → G_{2} → 0

of an isogeny G_{1} → G_{2} of p-divisible groups over Spec R. If p = 2, then we assume that H and G_{1} are connected. Let (Q_{1}, ϕ_{1}) and (Q_{2}, ϕ_{2}) be the Breuil windows of G_{1} and G_{2} (respectively). Then the Breuil module (M, φ) of H is the cokernel of the induced map (Q_{1}, ϕ_{1}) → (Q_{2}, ϕ_{2}) in a natural sense. From Proposition 10 we get that the classical covariant Dieudonné module of H_{κ(𝔭)^{perf}} is canonically given by (5).

3 Morphisms between Breuil modules

In this section, let R be a complete regular local ring of mixed characteristic (0, p) with maximal ideal 𝔪 and perfect residue class field k. We write R = 𝔖/E𝔖, where d = dim R ≥ 1, 𝔖 = W (k)[[T_{1}, …, T_{d}]], and E = p − h ∈ 𝔖 are as in Section 2. We use the standard frame L of the Definition 8. Let e ∈ ℕ^{∗} be such that p ∈ 𝔪^{e} \ 𝔪^{e+1}. It is the absolute ramification index of R. Let 𝔯̄:= (T_{1}, …, T_{d}) ⊂ 𝔖̄ := 𝔖/p𝔖 = k[[T_{1}, …, T_{d}]]. Let h̄ ∈ 𝔯̄ ⊂ 𝔖̄ be the reduction modulo p of h. The surjective function ord : 𝔖̄ ↠ ℕ ∪ {∞} is such that ord(𝔯̄^{i} \ 𝔯̄^{i+1}) = i for all i ≥ 0 and ord(0) = ∞. Let gr R := gr_{𝔪} R. The obvious isomorphism of graded rings

k[T_{1}, …, T_{d}] → gr R  (7)

maps the initial form of h̄ to the initial form of p. Thus e is the order of the power series h̄.

Lemma 11 We assume that R is such that p ∉ 𝔪^{p} (i.e., e ≤ p − 1). Let C be a 𝔖-module which is annihilated by a power of p. Let φ : C → C^{(σ)} be a 𝔖-linear map whose cokernel is annihilated by E. We assume that there exists a power series f ∈ 𝔖 \ p𝔖 which annihilates C. Then we have:

(a) If p ∉ 𝔪^{p−1} (i.e., if e ≤ p − 2), then C = 0.

(b) If e = p − 1, then either C = 0 or the initial form of p in gr R generates an ideal which is a (p − 1)-th power.

Proof. It suffices to show that C = 0 provided either e ≤ p − 2 or e = p − 1 and the initial form of p in gr R generates an ideal which is not a (p − 1)-th power. By the lemma of Nakayama it suffices to show that C/pC = 0. It is clear that φ induces a 𝔖-linear map C/pC → (C/pC)^{(σ)}. Therefore we can assume that C is annihilated by p. Let u be the smallest non-negative integer with the following property: for each c ∈ C there exists a power series g_{c} ∈ 𝔖̄ such that ord(g_{c}) ≤ u and g_{c} annihilates c. From the existence of f in the annihilator of C we deduce that the number u exists. If C ≠ 0, then we have u > 0. We will show that the assumption that u > 0 leads to a contradiction and therefore we have C = 0. By the minimality of u there exists an element x ∈ C such that for each power series a in the annihilator 𝔞 ⊂ 𝔖̄ of x we have ord(a) ≥ u. Consider the 𝔖̄-linear injection

𝔖̄/𝔞 ↪ C  (8)

which maps 1 to x. Let 𝔞^{(p)} ⊂ 𝔖̄ be the ideal generated by the p-th powers of elements in 𝔞. Each power series in 𝔞^{(p)} has order ≥ pu. If we tensorize the injection (8) by σ : 𝔖̄ → 𝔖̄ we obtain a 𝔖̄-linear injection

𝔖̄/𝔞^{(p)} ≅ 𝔖̄ ⊗_{σ,𝔖̄} 𝔖̄/𝔞 ↪ C^{(σ)}.

Thus each power series in the annihilator of 1 ⊗ x ∈ C^{(σ)} has at least order pu. On the other hand the cokernel of φ is by assumption annihilated by h̄. Thus h̄(1 ⊗ x) is in the image of φ. By the definition of u we find a power series g ∈ 𝔖̄ with ord(g) ≤ u which annihilates h̄(1 ⊗ x). Thus gh̄ ∈ 𝔞^{(p)}. We get

u + ord(h̄) ≥ ord(gh̄) ≥ pu.

Therefore e = ord(h̄) ≥ (p − 1)u ≥ p − 1. In the case (a) we obtain a contradiction which shows that C = 0. In the case e = p − 1 we obtain a contradiction if u > 1. Assume that u = 1. As gh̄ ∈ 𝔞^{(p)} and as ord(gh̄) = p, there exists a power series f̄ ∈ 𝔖̄ of order 1 and a non-zero element ξ ∈ k such that

gh̄ ≡ ξf̄^{p} mod 𝔯̄^{p+1}.

This shows that the initial forms of g and f̄ in the graded ring gr𝔯̄ 𝔖̄ differ by a constant in k. If we divide the last congruence by f̄ we obtain

h̄ ≡ ξf̄^{p−1} mod 𝔯̄^{p}.

By the isomorphism (7) this implies that initial form of p in gr R generates an ideal which is a (p − 1)-th power. Contradiction. □

Lemma 12 We assume that d = 2 and we consider the ring 𝔖̄ = k[[T_{1}, T_{2}]]. Let h̄ ∈ 𝔖̄ be a power series of order e ∈ ℕ^{∗}. Then the following two statements are equivalent:

(a) The power series h̄ does not belong to the ideal 𝔯̄^{(p)} + 𝔯̄^{2(p−1)} = (T_{1}^{p}, T_{2}^{p}, T_{1}^{p−1}T_{2}^{p−1}) of 𝔖̄.

(b) If C is a 𝔖̄-module of finite length equipped with a 𝔖̄-linear map φ : C → C^{(σ)} whose cokernel is annihilated by h̄, then C is zero.

Proof. We consider the 𝔖̄-linear map

τ : k = k[[T_{1}, T_{2}]]/(T_{1}, T_{2}) → k^{(σ)} = k[[T_{1}, T_{2}]]/(T_{1}^{p}, T_{2}^{p})

that maps 1 to T_{1}^{p−1}T_{2}^{p−1} modulo (T_{1}^{p}, T_{2}^{p}). If h̄ ∈ 𝔯̄^{(p)} + 𝔯̄^{2(p−1)}, then the cokernel of τ is annihilated by h̄. From this we get that (b) implies (a). Before proving the other implication, we first make some general remarks. The Frobenius endomorphism σ : 𝔖̄ → 𝔖̄ is faithfully flat. If △ is an ideal of 𝔖̄, then with the same notations as before we have σ(△)𝔖̄ = △^{(p)}. Let C be a 𝔖̄-module of finite type. Let 𝔞 ⊂ 𝔖̄ be the annihilator of C. Then 𝔞^{(p)} is the annihilator of C^{(σ)} = 𝔖̄⊗_{σ,𝔖̄}C. This is clear for a monogenic module C. If we have more generators for C, then we can use the formula

𝔞^{(p)} ∩ 𝔟^{(p)} = (𝔞 ∩ 𝔟)^{(p)}

which holds for flat ring extensions in general.

We say that a power series f ∈ 𝔖̄ of order u is normalized with respect to (T_{1}, T_{2}) if it contains T_{1}^{u} with a non-zero coefficient. This definition makes sense with respect to any regular system of parameters T̃_{1}, T̃_{2} of 𝔖̄. We now assume that (a) holds and we show that (b) holds. Thus the 𝔖̄-module C has finite length and the cokernel of φ : C → C^{(σ)} is annihilated by h̄. Let k′ be an infinite perfect field that contains k. Let 𝔖̄′ := k′[[T_{1}, T_{2}]]. To show that C = 0, it suffices to show that 𝔖̄′ ⊗𝔖̄ C = 0. Thus by replacing the role of k by the one of k′, we can assume that k is infinite. This assumption implies that for almost all λ ∈ k, the power series h̄ is normalized with respect to (T_{1}, T_{2} + λT_{1}). By changing the regular system of parameters (T_{1}, T_{2}) in 𝔖̄, we can assume that h̄ is normalized with respect to (T_{1}, T_{2}). By the Weierstraß preparation theorem we can assume that h̄ is a Weierstraß polynomial ([Bou], Chapter 7, Section 3, number 8). Thus we can write

h̄ = T_{1}^{e} + a_{e−1}(T_{2})T_{1}^{e−1} + … + a_{1}(T_{2})T_{1} + a_{0}(T_{2}),

where a_{0}(T_{2}), …, a_{e−1}(T_{2}) ∈ T_{2}k[[T_{2}]]. We note that the assumption (a) implies that e ≤ 2p − 3. Let u be the minimal non-negative integer such that there exists a power series of the form T_{1}^{u} + g ∈ 𝔖̄, with g ∈ T_{2}𝔖̄, for which we have (T_{1}^{u} + g)C = 0. By our assumptions, such a non-negative integer u exists. We will show that the assumption that C ≠ 0 leads to a contradiction. This assumption implies that u ≥ 1. The annihilator 𝔞 of the module C has a set of generators of the following form:

T_{1}^{u_{i}} + g_{i}, (with i = 1, …, l), g_{i} (with i = l + 1, …, m),

where u_{i} ≥ u for i = 1, …, l and g_{i} ∈ T_{2}𝔖̄ for i = 1, …, m. As the cokernel of φ is annihilated by h̄ we find that (T_{1}^{u} + g)h̄C^{(σ)} = 0. As the annihilator of C^{(σ)} is generated by the elements

T_{1}^{pu_{i}} + g_{i}^{p} (with i = 1, …, l), g_{i}^{p} (with i = l + 1, …, m),  (9)

we obtain the congruence

(T_{1}^{u} + g)h̄ ≡ 0 mod (T_{1}^{up}, T_{2}^{p}).  (10)

We consider this congruence modulo T_{2}. We have g ≡ 0 mod (T_{2}) and h̄ ≡ T_{1}^{e} mod (T_{2}) because h̄ is a Weierstraß polynomial. This proves that

T_{1}^{u}T_{1}^{e} ≡ 0 mod (T_{1}^{up}).

But this implies that u + e ≥ up. If u ≥ 2, then e ≥ up − u ≥ 2p − 2 and this contradicts the inequality e ≤ 2p − 3. Therefore we can assume that u = 1. By replacing (T_{1}, T_{2}) with (T_{1} +g, T_{2}), without loss of generality we can assume that T_{1}C = 0 and (cf. (10) and the equality u = 1) that

T_{1}h̄ ≡ 0 mod 𝔯̄^{(p)}.

This implies that up to a unit in 𝔖̄ we can assume that h̄ is of the form:

h̄ ≡ T_{2}^{s}T_{1}^{p−1} + ∑_{i=0}^{∞} T_{2}^{i+p}δ_{i}(T_{1}) mod (T_{1}^{p}),

where 0 ≤ s ≤ p − 2 and where each δ_{i} ∈ k[T_{1}] has degree at most p − 2. Let now v be the smallest natural number such that T_{2}^{v}C = 0. Then the annihilator 𝔞 of C is generated by T_{1}, T_{2}^{v} and the annihilator 𝔞^{(p)} of C^{(σ)} is generated by T_{1}^{p}, T_{2}^{pv}. As the cokernel of φ is annihilated by h̄, we find that T_{2}^{v}h̄C^{(σ)} = 0. Thus we obtain the congruence

T_{2}^{v}(T_{2}^{s}T_{1}^{p−1} + ∑_{i=0}^{∞} T_{2}^{i+p}δ_{i}(T_{1})) ≡ 0 mod (T_{1}^{p}, T_{2}^{pv}).

But this implies v + s ≥ pv. Thus s ≥ (p − 1)v ≥ p − 1 and (as 0 ≤ s ≤ p − 2) we reached a contradiction. Therefore C = 0 and thus (a) implies (b). □

Proposition 13 We assume that p ∉ 𝔪^{p} (i.e., e ≤ p − 1). If p ∈ 𝔪^{p−1}, then we also assume that the ideal generated by the initial form of p in gr R is not a (p − 1)-th power (thus p > 2). We consider a morphism α : (M_{1}, φ_{1}) → (M_{2}, φ_{2}) of Breuil modules for the standard frame L. Let 𝔭 be a prime ideal of R which contains p. We consider the W (κ(𝔭)^{perf})-linear map obtained from α by base change

W (κ(𝔭)^{perf}) ⊗_{σκ_{𝔭},𝔖} M_{1} → W (κ(𝔭)^{perf}) ⊗_{σκ_{𝔭},𝔖} M_{2}.  (11)

Then the following two properties hold:

(a) If (11) is surjective, then the 𝔖-linear map M_{1} → M_{2} is surjective.

(b) If (11) is injective, then the 𝔖-linear map M_{1} → M_{2} is injective and its cokernel is a 𝔖-module of projective dimension at most 1.

Proof. We prove (a). We denote by 𝔭̃ the ideal of 𝔖 which corresponds to 𝔭 via the isomorphism 𝔖/(E, p) ≅ R/(p). It follows from the lemma of Nakayama that (M_{1})𝔭̃ → (M_{2})𝔭̃ is a surjection. Let us denote by (C, φ) the cokernel of α. We conclude that C is annihilated by an element f ∉ 𝔭̃ ⊃ p𝔖. Therefore we conclude from Lemma 11 that C = 0. Thus (a) holds. Part (b) follows from (a) by duality as it is compatible with taking fibres (6). Indeed for the last assertion it is enough to note that the kernel of the surjection M_{2}^{∗} ↠ M_{1}^{∗} has clearly projective dimension at most 1. □

Remark. If e = p−1 and the ideal generated by the initial form of p in gr R is a (p − 1)-th power, then the Proposition 13 is not true in general. This is so as there exist non-trivial homomorphisms (ℤ/pℤ)_{R} → μ μ μ_{p,R} for suitable such R’s.

Proposition 14 We assume that R has dimension d = 2 and that h ∈ 𝔖 does not belong to the ideal (p, T_{1}^{p}, T_{2}^{p}, T_{1}^{p−1}T_{2}^{p−1}) of 𝔖. Let α : (M_{1}, φ_{1}) → (M_{2}, φ_{2}) be a morphism of Breuil modules for the standard frame L. We assume that for each prime ideal 𝔭 of R with p ∈ 𝔭 ≠ 𝔪, the W (κ(𝔭)^{perf})-linear map obtained from α by base change

W (κ(𝔭)^{perf}) ⊗_{σκ_{𝔭},𝔖} M_{1} → W (κ(𝔭)^{perf}) ⊗_{σκ_{𝔭},𝔖} M_{2}  (12)

is surjective (resp. is injective). Then the 𝔖-linear map M_{1} → M_{2} is surjective (resp. is injective and its cokernel is a 𝔖-module of projective dimension at most 1).

Proof. As in the last proof we only have to treat the case where the maps (12) are surjective. We consider the cokernel (C, φ) of α. As in the last proof we will argue that C = 0 but with the role of Lemma 11 being replaced by Lemma 12. It suffices to show that C̄ := C/pC is zero. Let φ̄ : C̄ → C̄^{(σ)} be the map induced naturally by φ. Lemma 12 is applicable if we verify that C̄ is a 𝔖̄-module of finite length. We denote by 𝔭̄ the ideal of 𝔖̄ = 𝔖/p𝔖 which corresponds to 𝔭 via the isomorphism 𝔖̄/(h̄) ≅ R/pR. It follows from (12) by the lemma of Nakayama that for each prime ideal 𝔭̄ ⊃ h̄𝔖̄ different from the maximal ideal of 𝔖̄ the maps α_{𝔭} : (M_{1})𝔭̄ → (M_{2})𝔭̄ are surjective. Using this one constructs inductively a regular sequence f_{1}, …, f_{d−1}, h̄ in the ring 𝔖̄ such that the elements f_{1}, …, f_{d−1} annihilate C̄. As C̄ is a finitely generated module over the 1-dimensional local ring 𝔖̄/(f_{1}, …, f_{d−1}), we get that C̄[1/h̄] is a module of finite length over the regular ring A = 𝔖̄[1/h̄]. If we can show that C̄[1/h̄] = 0, then it follows that C̄ is of finite length. As we are in characteristic p, the Frobenius σ acts on the principal ideal domain A = 𝔖̄[1/h̄]. By the definition of a Breuil module the maps φ_{i}[1/h̄] for i = 1, 2 become isomorphisms. Therefore φ gives birth to an isomorphism:

φ̄[1/h̄] : C̄[1/h̄] → (C̄[1/h̄])^{(σ)}.  (13)

As A is regular of dimension d−1, for each A-module ‡ of finite length we have

length ‡^{(σ)} = p^{d−1} length ‡.

We see that the isomorphism (13) is only possible if C̄[1/h̄] = 0. Thus C̄ has finite length and therefore from Lemma 12 we get that C̄ = 0. This implies that C = 0. □

4 Extending epimorphisms and monomorphisms

In this section let R be a regular local ring of mixed characteristic (0, p) with maximal ideal 𝔪 and residue class field k. Let K be the field of fractions of R.

4.1 Complements on Raynaud’s work

We first reprove Raynaud’s result [R2], Corollary 3.3.6 by the methods of the previous sections. We state it in a slightly different form.

Proposition 15 We assume that p ∉ 𝔪^{p−1} (thus p > 2). Let H_{1} and H_{2} be finite flat group schemes over Spec R. Let β : H_{1} → H_{2} be a homomorphism, which induces an epimorphism (resp. monomorphism) H_{1,K} → H_{2,K} between generic fibres. Then β is an epimorphism (resp. monomorphism).

Proof. We prove only the statement about epimorphisms because the case of a monomorphism follows by Cartier duality. It is enough to show that the homomorphism H_{1} → H_{2} is flat. By the fibre criterion of flatness it is enough to show that the homomorphism β_{k} : H_{1,k} → H_{2,k} between special fibres is an epimorphism. To see this we can assume that R is a complete local ring with algebraically closed residue class field k. We write R ≅ 𝔖/(p − h), with 𝔖 = W (k)[[T_{1}, …, T_{d}]]. Then the reduction h̄ ∈ k[[T_{1}, …, T_{d}]] of h modulo p is a power series of order e < p − 1. By Noether normalization theorem we can assume that h̄ contains the monom T_{1}^{e}. By replacing R with R/(T_{2}, …, T_{d}), we can assume that R is one-dimensional. We consider the morphism

(M_{1}, φ_{1}) → (M_{2}, φ_{2})

of Breuil modules associated to β. Let (C, φ) be its cokernel. Let C̄ := C/C_{0}, where C_{0} is the 𝔖-submodule of C whose elements are annihilated by a power of the maximal ideal 𝔯 of 𝔖. The 𝔖-linear map φ factors as

φ̄ : C̄ → 𝔖 ⊗_{σ,𝔖} C̄

and the cokernel of φ̄ is annihilated by h̄. The maximal ideal 𝔯 of 𝔖 is not associated to 𝔖 ⊗_{σ,𝔖} C̄. Thus either C̄ = 0 or depth C̄ ≥ 1. Therefore the 𝔖-module C̄ is of projective dimension at most 1. As C̄ is annihilated by a power of p, we conclude that (C̄, φ̄) is the Breuil module of a finite flat group scheme D over Spec R. We have induced homomorphisms

H_{1} → H_{2} → D.

The composition of them is zero and the second homomorphism is an epimorphism because it is so after base change to k. As H_{1,K} ↠ H_{2,K} is an epimorphism we conclude that D_{K} = 0. But then D = 0 and the Breuil module (C̄, φ̄) is zero as well. We conclude by Lemma 11 that C = 0. □

The next proposition is proved in [R2], Remark 3.3.5 in the case of biconnected finite flat group schemes H and D.

Proposition 16 Let R be a discrete valuation ring of mixed characteristic (0, p) and index of ramification p − 1; we have K = R[1/p]. Let β : H → D be a homomorphism of residually connected finite flat group schemes over Spec R which induces an isomorphism (resp. epimorphism) over Spec K. Then β is an isomorphism (resp. epimorphism).

Proof. It is enough to show the statement about isomorphisms. Indeed, assume that β_{K} is an epimorphism. Consider the schematic closure H_{1} of the kernel of β_{K} in H. Then H/H_{1} → D is an isomorphism. Therefore we can assume that β_{K} is an isomorphism. By extending R we can assume that R is complete and that k is algebraically closed. By considering the Cartier dual homomorphism β^{t} : D^{t} → H^{t} one easily reduces the problem to the case when H is biconnected. As the case when D is also biconnected is known (cf. [R2], Remark 3.3.5), one can easily reduce to the case when D is of multiplicative type. Then D contains μ μ μ_{p,R} as a closed subgroup scheme. We consider the schematic closure H_{1} of μ μ μ_{p,K} in H. Using an induction on the order of H it is enough to show that the natural homomorphism β_{1} : H_{1} → μ μ μ_{p,R} is an isomorphism. This follows from [R2], Proposition 3.3.2 3^{∘}. For the sake of completeness we reprove this in the spirit of the paper. We write R = 𝔖/(E), where E ∈ 𝔖 = W (k)[[T]] is an Eisenstein polynomial of degree e = p − 1. The Breuil window of the p-divisible group μ μ μ_{p^{∞},R} is given by

\begin{matrix}𝔖 & → & 𝔖^{(σ)} ≅ 𝔖 \\ f & ↦ & Ef. \end{matrix}

The Breuil module (N, τ) of μ μ μ_{p,R} is the kernel of the multiplication by p : μ μ μ_{p^{∞},R} → μ μ μ_{p^{∞},R} and therefore it can be identified with

\begin{matrix}N = k[[T]] & → & N^{(σ)} = k[[T]]^{(σ)} ≅ k[[T]] \\ f & ↦ & T^{e}f. \end{matrix}

Let (M, φ) be the Breuil module of H_{1}. As H_{1} is of height 1 we can identify M = k[[T]]. Then φ : M → M^{(σ)} ≅ M is the multiplication by a power series g ∈ k[[T]] of order ord(g) ≤ e = p − 1. To the homomorphism β_{1} : H_{1} → μ μ μ_{p,R} corresponds a morphism α_{1} : (M, φ) → (N, τ) that maps 1 ∈ M to some element a ∈ N. We get a commutative diagram:

\xymatrix{k[[T]] \ar[r]^{a} \ar[d]_{g} & k[[T]] \ar[d]^{T^{e}} \\ k[[T]] \ar[r]^{a^{p}} & k[[T]].}

We obtain the equation ga = a^{p}T^{p−1}. As a ≠ 0 this is only possible if

ord(g) = (p − 1)(ord(a) + 1).

As ord(g) ≤ p − 1, we get that ord(a) = 0 and ord(g) = p − 1. As ord(a) = 0, both α_{1} and β_{1} are isomorphisms. Thus β : H → D is an isomorphism. □

4.2 Basic extension properties

The next three results will be proved for p = 2 under certain residually connectivity assumptions. But in Subsection 4.3 below we will show how these results hold under no residually connectivity assumptions even if p = 2.

Proposition 17 We assume that dim R ≥ 2, that p ∉ 𝔪^{p}, and that the initial form of p in gr R generates an ideal which is not a (p − 1)-th power. Let β : H_{1} → H_{2} be a homomorphism of finite flat group schemes over Spec R which induces an epimorphism (resp. a monomorphism) over Spec K. If p = 2, we assume as well that H_{1} and H_{2} (resp. that the Cartier duals of H_{1} and H_{2}) are residually connected. Then β is an epimorphism (resp. a monomorphism).

Proof. As before we can assume that R is a complete regular local ring. We can also restrict our attention to the case of epimorphisms. By Proposition 15 we can assume that p ∈ 𝔪^{p−1}. Let 𝔭 be a minimal prime ideal which contains p. We show that the assumption that p ∈ 𝔭^{p−1} leads to a contradiction. Then we can write p = uf^{p−1}, where u, f ∈ R and f is a generator of the prime ideal 𝔭. It follows from our assumptions that u ∉ 𝔪 and that f ∈ 𝔪 \ 𝔪^{2}. Therefore the initial form of p in gr R generates an ideal which is a (p − 1)-th power. Contradiction. We first consider the case when k is perfect. Let α : (M_{1}, φ_{1}) → (M_{2}, φ_{2}) denote also the morphism of Breuil modules associated to β : H_{1} → H_{2}. As p ∉ 𝔭^{p−1}, we can apply Proposition 15 to the ring R_{𝔭}. It follows that β induces an epimorphism over the spectrum of the residue class field κ(𝔭) of 𝔭 and thus also of its perfect hull κ(𝔭)^{perf}. From this and the end of Section 2 we get that the hypotheses of Proposition 13 hold for α. We conclude that α is an epimorphism and thus β is also an epimorphism. Let R → R′ be a faithfully flat extension of noetherian local rings, such that 𝔪R′ is the maximal ideal of R′ and the extension of residue class fields k ↪ k′ is radical. We consider the homomorphism of polynomial rings gr R → gr R′. As gr R and gr R′ are unique factorization domains, it is easy to see that the condition that the initial form of p is not a (p − 1)-th power is stable by the extension R → R′. But β is an epimorphism if and only if β_{R′} is so. Therefore we can assume that the residue class field of R is perfect and this case was already proved. □

Proposition 18 We assume that dim R = 2. Let U = Spec R \ {𝔪}. We also assume that the following technical condition holds:

(*) there exists a faithfully flat local R-algebra R̂ which is complete, which has an algebraically closed residue class field k, and which has a presentation R̂ = 𝔖/(p − h) where 𝔖 = W (k)[[T_{1}, T_{2}]] and where h ∈ (T_{1}, T_{2}) does not belong to the ideal (p, T_{1}^{p}, T_{2}^{p}, T_{1}^{p−1}T_{2}^{p−1}).

Let β : H_{1} → H_{2} be a homomorphism of finite flat group schemes over Spec R. We also assume that for each geometric point Spec L → U such that L has characteristic p the homomorphism β_{L} is an epimorphism (resp. a monomorphism); thus β_{U} : H_{1,U} → H_{2,U} is an epimorphism (resp. a monomorphism). If p = 2, we assume that H_{1} and H_{2} (resp. that the Cartier duals of H_{1} and H_{2}) are residually connected. Then β is an epimorphism (resp. a monomorphism).

Proof. We have dim R̂/𝔪R̂ = 0 and thus Spec(R̂) \ (Spec(R̂) ×_{Spec R} U) is the closed point of Spec(R̂). Based on this we can assume that R = R̂. Thus the proposition follows from Proposition 14 in the same way Proposition 17 followed from Propositions 15 and 13. □

Corollary 19 We assume that the assumptions on R of either Proposition 17 or Proposition 18 are satisfied (thus dim R ≥ 2). We consider a complex

0 → H_{1} → H_{2} → H_{3} → 0

of finite flat group schemes over Spec R whose restriction to U = Spec R\{𝔪} is a short exact sequence. If p = 2, then we assume that H_{2} and H_{3} are residually connected. Then 0 → H_{1} → H_{2} → H_{3} → 0 is a short exact sequence.

Proof. Propositions 17 and 18 imply that H_{2} → H_{3} is an epimorphism. Its kernel is a finite flat group scheme isomorphic to H_{1} over U and thus (as we have dim R ≥ 2) it is isomorphic to H_{1}. □

4.3 Dévissage properties

In this subsection we explain how one can get the results of Subsection 4.2 under no residually connectivity assumptions for p = 2. We assume that dim R = 2 and that p is arbitrary. Let U = Spec R \ {𝔪}. Let V be a locally free O_{U}-module of finite rank over U. The R-module H^{0}(U, V) is free of finite rank. Using this it is easy to see that each finite flat group scheme over U extends uniquely to a finite flat group scheme over Spec R. This is clearly an equivalence between the category of finite flat group schemes over U and the category of finite flat group schemes over Spec R. The same holds if we restrict to finite flat group schemes annihilated by p. Next we also assume that R is complete. Each finite flat group scheme H over Spec R is canonically an extension

0 → H^{∘} → H → H^{ét} → 0,

where H^{∘} is connected and H^{ét} is étale over Spec R. In particular a homomorphism from a connected finite flat group scheme over Spec R to an étale finite flat group scheme over Spec R is zero. From this one easily checks that if H_{1} → H_{2} is a homomorphism of finite flat group schemes over Spec R which is an epimorphism over U and if H_{1} is connected, then H_{2} is connected as well.

Lemma 20 We assume that dim R = 2 and that R is complete. Then the following four statements are equivalent:

(a) Each short exact sequence of finite flat group schemes over U extends uniquely to a short exact sequence of finite flat group schemes over Spec R.

(b) Let H_{1} and H_{2} be connected finite flat group schemes over Spec R. A homomorphism H_{1} → H_{2} over Spec R is an epimorphism if its restriction to U is an epimorphism.

(c) Let H_{1} and H_{2} be connected finite flat group schemes over Spec R which are annihilated by p. A homomorphism H_{1} → H_{2} over Spec R is an epimorphism if its restriction to U is an epimorphism.

(d) The regular ring R is p-quasi-healthy.

Proof. It is clear that (a) implies (b). We show that (b) implies (a). Let

0 → H_{1} → H_{2} → H_{3} → 0

be a complex of finite flat group schemes over Spec R whose restriction to U is a short exact sequence. It suffices to show that β : H_{1} → H_{2} is a monomorphism. Indeed, in this case we can form the quotient group scheme H_{2}/H_{1} and the homomorphism H_{2}/H_{1} → H_{3} is an isomorphism as its restriction to U is so. We check that the homomorphism

β^{∘} : H_{1}^{∘} → H_{2}^{∘}

is a monomorphism. Let H_{4} be the finite flat group scheme over R whose restriction to U is H_{2,U}^{∘}/H_{1,U}^{∘}. We have a complex 0 → H_{1}^{∘} → H_{2}^{∘} → H_{4} → 0 whose restriction to U is exact. We conclude that H_{4} is connected. Thus we have an epimorphism H_{2}^{∘} ↠ H_{4} (as we are assuming that (b) holds) whose kernel is H_{1}^{∘}. Therefore (b) implies that β^{∘} is a monomorphism. As β^{∘} is a monomorphism, it suffices to show that the induced homomorphism β̄ : H_{1}/H_{1}^{∘} → H_{2}/H_{1}^{∘} is a monomorphism. In other words, without loss of generality we can assume that H_{1} = H_{1}^{ét} is étale. Let H′′_{1} be the kernel of H_{2}^{ét} → H_{3}^{ét}; it is a finite étale group scheme over Spec R. Let H′_{1} be the kernel of H_{1} → H′′_{1}. It suffices to show that H′_{1} → H_{2}^{∘} is a monomorphism. Therefore we can also assume that H_{2} is connected. This implies that H_{3} is connected. As we are assuming that (b) holds, H_{2} → H_{3} is an epimorphism. Its kernel is H_{1} and therefore β : H_{1} → H_{2} is a monomorphism. Thus (b) implies (a). We show that (c) implies (b). The last argument shows that a short exact sequence of finite flat group schemes over U annihilated by p extends to a short exact sequence of finite flat group schemes over Spec R. We start with a homomorphism H_{2} → H_{3} between connected finite flat group schemes over Spec R which induces an epimorphism over U. We extend the kernel of H_{2,U} → H_{3,U} to a finite flat group scheme H_{1} over Spec R. Then we find a complex

0 → H_{1} → H_{2} → H_{3} → 0

whose restriction to U is a short exact sequence. To show that H_{2} → H_{3} is an epimorphism it is equivalent to show that H_{1} → H_{2} is a monomorphism.

As H_{1,U} has a composition series whose factors are annihilated by p, we easily reduce to the case where H_{1} is annihilated by p. We embed H_{2} into a p-divisible group G over Spec R. To check that H_{1} → H_{2} is a monomorphism, it suffices to show that H_{1} → G[p] is a monomorphism. But this follows from the second sentence of this paragraph. It is clear that (a) implies (d). We are left to show that (d) implies (c). It suffices to show that a homomorphism β : H_{1} → H_{2} of finite flat group schemes over Spec R annihilated by p is a monomorphism if its restriction β_{U} : H_{1,U} → H_{2,U} is a monomorphism. We embed H_{2} into a p-divisible group G over Spec R. The quotient G_{U}/H_{1,U} is a p-divisible group over U which extends to a pdivisible group G′ over Spec R (as we are assuming that (d) holds). The isogeny G_{U} → G′_{U} extends to an isogeny G → G′. Its kernel is a finite flat group scheme and therefore isomorphic to H_{1}. We obtain a monomorphism H_{1} → G. Thus β : H_{1} → H_{2} is a monomorphism, i.e. (d) implies (c). □

Corollary 21 Propositions 17 and 18 (and thus implicitly Corollary 19) hold without any residually connectivity assumption for p = 2.

Proof. We can assume that R is complete. By considering an epimorphism R ↠ R′ with R′ regular of dimension 2, we can also assume that dim R = 2. Based on Lemma 20, it suffices to prove Propositions 17 and 18 in the case when connected finite flat group schemes are involved. But this case was already proved in Subsection 4.2. This also shows that for p ≥ 3 we can restrict to the connected case and avoid to apply Proposition 10 for nonconnected p-divisible groups. □

Corollary 22 We assume that k is perfect and that R = W (k)[[T_{1}, T_{2}]]/(p − h) with h ∈ (T_{1}, T_{2}). Let h̄ ∈ k[[T_{1}, T_{2}]] be the reduction of h modulo p. Then the fact that R is or is not p-quasi-healthy depends only on the orbit of the ideal (h̄) of k[[T_{1}, T_{2}]] under automorphisms of k[[T_{1}, T_{2}]].

Proof. The category of Breuil modules associated to connected finite flat group schemes over Spec R annihilated by p is equivalent to the category of pairs (M, φ), where M is a free k[[T_{1}, T_{2}]]-module of finite rank and where φ : M → M^{(σ)} is a k[[T_{1}, T_{2}]]-linear map whose cokernel is annihilated by h̄ and whose reduction modulo the ideal (T_{1}, T_{2}) is nilpotent in the natural sense. The last category depends only on the orbit of (h̄) under automorphisms of k[[T_{1}, T_{2}]]. The corollary follows from the last two sentences and the equivalence of (c) and (d) in Lemma 20. □

4.4 The p-quasi-healthy part of Theorem 3

In this subsection we show that if R is as in Theorem 3 for d = 2, then R is p-quasi-healthy. It follows from the definition of a p-divisible group and the uniqueness part of the first paragraph of Subsection 4.3, that it is enough to show that a complex 0 → H_{1} → H_{2} → H_{3} → 0 of finite flat group schemes over Spec R is a short exact sequence if its restriction to Spec R \ {𝔪} is a short exact sequence. This is a local statement in the faithfully flat topology of Spec R and thus to check it we can assume that R = R̂ = W (k)[[T_{1}, T_{2}]/(p−h) with h ∈ (T_{1}, T_{2}) but h ∉ (p, T_{1}^{p}, T_{2}^{p}, T_{1}^{p−1}T_{2}^{p−1}). By Lemma 20 we can assume that H_{2} and H_{3} are connected. Thus 0 → H_{1} → H_{2} → H_{3} → 0 is a short exact sequence, cf. Corollary 19. □

5 Extending abelian schemes

Proposition 23 Let R be a regular local ring of mixed characteristic (0, p).

(a) Let U = Spec R \ {𝔪}, where 𝔪 is the maximal ideal of R. Let Ă be an abelian scheme over U. If the p-divisible group of Ă extends to a p-divisible group over Spec R, then Ă extends uniquely to an abelian scheme A over Spec R.

(b) If R is p-quasi-healthy, then R is quasi-healthy.

Proof. The uniqueness part of (a) is well known (cf. [R1], Chapter IX, Corollary 1.4). Part (a) is a particular case of either proof of [V2], Proposition 4.1 (see remark that starts the proof) or [V2], Remark 4.2. Part (b) follows from (a). □

Lemma 24 Let S ↠ R be a ring epimorphism between local noetherian rings whose kernel is an ideal 𝔞 with 𝔞^{2} = 0 and depth_{R} 𝔞 ≥ 2. We assume that depth R ≥ 2 and that R is quasi-healthy. Then S is quasi-healthy as well.

Proof. Let 𝔪 and 𝔫 be the maximal ideals of R and S (respectively). We set U = Spec R \ {𝔪} and V = Spec S \ {𝔫}. Let B̆ be an abelian scheme over V and let Ă be its reduction over U. Then Ă extends uniquely to an abelian scheme A over Spec R. Let π : A ↠ Spec R be the projection. It is well-known that the set of liftings of A with respect to Spec R ↪ Spec S is a trivial torsor under the group H^{0}(Spec R, R^{1}π_{∗}\underline{Hom}(Ω_{A/R}, 𝔞)) and that the set of liftings of Ă with respect to V ↠ U is a trivial torsor under the group H^{0}(U, R^{1}π_{∗}\underline{Hom}(Ω_{A/R}, 𝔞)). As depth_{R} 𝔞 ≥ 2, the last two groups are equal. Thus there exists a unique abelian scheme B over Spec S which lifts A and whose restriction to V is B̆. □

Proposition 25 Let S be a complete noetherian local ring of mixed characteristic (0, p). Let S ↠ R be a ring epimorphism with kernel 𝔞 (thus R is a complete noetherian local ring). We assume that there exists a sequence of ideals

𝔞 = 𝔞_{0} ⊃ 𝔞_{1} ⊃ …

such that the intersection of these ideals is 0 and for all i ≥ 0 we have 𝔞_{i}^{2} ⊂ 𝔞_{i+1} and depth_{S} 𝔞_{i}/𝔞_{i+1} ≥ 2. We also assume that depth R ≥ 2 and that R is quasihealthy. Let 𝔫 be the maximal ideal of S. Let V := Spec S \ {𝔫}.

(a) Then each abelian scheme over V that has a polarization extends to an abelian scheme over Spec S.

(b) We assume that S is integral and geometrically unibranch (like S is normal). Then S is quasi-healthy.

Proof. By a well-known theorem we have S = lim_{←} S/𝔞_{i}.

For (b) (resp. (a)) we have to show that each abelian scheme Ă (resp. each abelian scheme Ă that has a polarization λĂ) over V extends to an abelian scheme over Spec S. We write U_{i} = (Spec S/𝔞_{i}) \ {𝔫}. The topological space underlying U_{i} is independent of i and it will be denoted by U. It is easy to see that depth S/𝔞_{i} ≥ 2. By Lemma 24 the ring S/𝔞_{i} is quasi-healthy. We denote by Ă_{i} the base change of Ă to U_{i}. Then Ă_{i} extends uniquely to an abelian scheme A_{i} over Spec S/𝔞_{i}. If S is integral and geometrically unibranch, then from [R1], Chapter XI, Theorem 1.4 we get that Ă is projective over V. Thus from now on we can assume that there exists a polarization λĂ of Ă and we will not anymore differentiate between parts (a) and (b). From the uniqueness of A_{i} we easily get that the reduction of λĂ modulo 𝔞_{i} extends uniquely to a polarization λ_{A_{i}} of A_{i} (this also follows from [R1], Chapter IX, Corollary 1.4). We get that the A_{i}’s inherit a compatible system of polarizations. From this and the algebraization theorem of Grothendieck, we get that there exists an abelian scheme A over Spec S which lifts the A_{i}’s. Next we will prove that the p-divisible group G of A restricts over V to the p-divisible group Ğ of Ă. This implies that Ă extends to an abelian scheme over Spec S (cf. Proposition 23 (a)) which is then necessarily isomorphic to A. Let G_{i}[m] be the kernel of the multiplication by p^{m} : A_{i} → A_{i}, and let G[m] be the kernel of the multiplication by p^{m} : A → A. Let C_{i}[m] be the S/𝔞_{i}-algebra of global functions on G_{i}[m]. Then B[m] = lim_{←} C_{i}[m] is the S-algebra of global functions on G[m].

We write \underline{Spec} C̆[m] = Ğ[m], where C̆[m] is a finite O_{Ğ[m]}-algebra. We have a natural homomorphism:

H^{0}(V, C̆[m]) → H^{0}(U, (C_{i}[m])^{∼}) = C_{i}[m].

Here (C_{i}[m])^{∼} is the restriction to U of the O_{Spec S/𝔞_{i}}-algebra associated to C_{i}[m]. The last equality follows from the fact that depth S/𝔞_{i} ≥ 2. This gives birth to an S-algebra homomorphism

H^{0}(V, C̆[m]) → B[m].

If we restrict it to a homomorphism between O_{V} -algebras we obtain a homomorphism of finite flat group schemes over V

G[m]_{V} → Ğ[m]

and thus a homomorphism of corresponding p-divisible groups G_{V} → Ğ over V. By construction this is an isomorphism if we restrict it to U. As V is connected, from the following proposition we conclude that G_{V} → Ğ is an isomorphism. □

Proposition 26 Let β : G → G′ be a homomorphism of p-divisible groups over a noetherian scheme X. Then the set Y of points x ∈ X such that β_{x} is an isomorphism is open and closed in X. Moreover β_{Y} is an isomorphism.

Proof. It is clear that β is an isomorphism if and only if β[1] : G[1] → G′[1] is an isomorphism. Therefore the subfunctor of X defined by the condition that β_{Y} is an isomorphism is representable by an open subscheme Z ⊂ X. By the theorems of Tate and de Jong on extensions of homomorphisms between p-divisible groups, we get that the valuative criterion of properness holds for Z → X. Thus Z is as well closed in X and therefore we can take Y = Z. □

5.1 Counterexample for the p-quasi-healthy context

Lemma 24 does not hold for the p-quasi-healthy context even in the simplest cases. Here is an elementary counterexample. We take R = W (k)[[T_{1}]]. From Subsection 4.4 we get that R is p-quasi-healthy. We take S = R[T_{2}]/(T_{2}^{2}). Let V := Spec S \ {𝔫}, where 𝔫 is the maximal ideal of S. Let O := R_{(T_{1})}; we have a natural identification V = Spec S[\frac{1}{T_{1}} ] ∪ Spec O[T_{2}]/(T_{2}^{2}). Let E_{k} be an elliptic curve over Spec k. We can identify the formal deformation space of E_{k} with Spf R. Let E be the elliptic curve over Spec R which is the algebraization of the uni-versal elliptic curve over Spf R. Let ι_{1} : R ↪ S[\frac{1}{T_{1}} ] be the W (k)-monomorphism that maps T_{1} to T_{1} + T_{2}T_{1}^{−1}; it lifts the natural W (k)-monomorphism R ↪ R[\frac{1}{T_{1}} ]. The W (k)-monomorphism ι_{1} gives birth to an elliptic curve E_{1} over Spec S[\frac{1}{T_{1}} ] whose restriction to Spec S[\frac{1}{T_{1}} ]/(T_{2}) = Spec R[\frac{1}{T_{1}} ] extends to the elliptic curve E over Spec R. We check that the assumption that E_{1} extends to an elliptic curve E_{V} over V leads to a contradiction. To E_{V} and E_{U} correspond morphisms U → V → 𝔸^{1}, where 𝔸^{1} is the j-line over Spec W (k). As the topological spaces of U and V are equal, we get that we have a natural factorization U → V → Spec R → 𝔸^{1}, where we identify Spec R with the completion of 𝔸^{1} at its k-valued point defined by E_{k}. As S = H^{0}(V, O_{V}), the image of ι_{1} is contained in S. Contradiction. But the p-divisible group G_{1} of E_{1} extends to a p-divisible group G_{V} over V. This is so as each p-divisible group over Spec O extends uniquely to an étale p-divisible group over Spec O[T_{2}]/(T_{2}^{2}). Finally we check that the assumption that G_{V} extends to a p-divisible group G_{2} over Spec S leads to a contradiction. Let E_{2} be the elliptic curve over Spec S which lifts E and whose p-divisible group is G_{2}. Let ι_{2} : R → S be the W (k)-homomorphism that defines E_{2}. We check that the resulting two W (k)- homomorphisms ι_{1}, ι_{2} : R → S[\frac{1}{T_{1}} ] are equal. It suffices to show that their composites ι_{3}, ι_{4} : R → Ŝ_{(p)} with the natural W (k)-monomorphism S[\frac{1}{T_{1}} ] ↪ Ŝ_{(p)} = R̂_{(p)}[T_{2}]/(T_{2}^{2}) are equal (here ˆ△ denotes the completion of the local ring △). But this follows from Serre–Tate deformation theory and the fact that the composites ι_{5}, ι_{6} : R → R̂_{(p)} of ι_{3}, ι_{4} with the W (k)-epimorphism Ŝ_{(p)} ↠ R̂_{(p)}, are equal. As the two W (k)-homomorphisms ι_{1}, ι_{2} : R → S[\frac{1}{T_{1}} ] are equal, the image of ι_{1} is contained in S. Contradiction. We conclude that S is not p-quasi-healthy. A similar argument shows that for all n ≥ 2, the ring R[T_{2}]/(T_{2}^{n}) is not p-quasi-healthy. This counterexample disproves the claims on [FC], top of p. 184 on torsors of liftings of p-divisible groups.

5.2 Proofs of Theorem 3 and Corollary 4

Let 𝔪 be the maximal ideal of R. Let U = Spec R \ {𝔪}. Let d = dim R. We prove Theorem 3. We first assume that d = 2. It is enough to show that R is p-quasi-healthy, cf. Proposition 23 (b). But this follows from Subsection 4.4. We next assume that d ≥ 3. We have to show that each abelian scheme over U extends uniquely to an abelian scheme over Spec R. This is a local statement in the faithfully flat topology of Spec R and thus to check it we can assume that R = R̂ is complete with algebraically closed residue class field k. We have an epimorphism R ↠ W (k)[[T_{1}, T_{2}]]/(p − h) where h is a power series in the maximal ideal of W (k)[[T_{1}, T_{2}]] whose reduction modulo (p, T_{1}^{p}, T_{2}^{p}, T_{1}^{p−1}T_{2}^{p−1}) is non-zero. As W (k)[[T_{1}, T_{2}]]/(p − h) is p-quasi-healthy (cf. the case d = 2), from Proposition 25 we get that R is quasi-healthy. This proves Theorem 3. We prove Corollary 4. Thus dim R = d ≥ 2 and p ∉ 𝔪^{p}. As in the previous paragraph we argue that we can assume that R = R̂ is complete with algebraically closed residue class field k. We write R = 𝔖/(p − h) where h ∈ (T_{1}, …, T_{d}) is such that its reduction h̄ ∈ 𝔖̄ = 𝔖/p𝔖 modulo p is a power series of order e = ord(h̄) ≤ p − 1. Due to Noether normalization theorem we can assume that h̄ contains the monom T_{1}^{e}. We set R′ := 𝔖/(p−h, T_{3}, …, T_{d}); if d = 2, then R′ = R. From Proposition 25 applied to the epimorphism R ↠ R′, we get that it suffices to show that R′ is quasi-healthy and p-quasi-healthy. Thus in order not to complicate the notations, we can assume that d = 2 (i.e., R = R′). As e ≤ p − 1, the reduction of h̄ modulo the ideal (T_{1}^{p}, T_{2}^{p}, T_{1}^{p−1}T_{2}^{p−1}) is non-zero. Thus Corollary 4 follows from Theorem 3. □

5.3 Example

Let R be a regular local of mixed characteristic (0, p) such that the strict completion of R is isomorphic to

C_{k}[[T_{1}, …, T_{d}]]/(p − T_{1}^{e_{1}} · . . . · T_{m}^{e_{m}})

where C_{k} is a Cohen ring of the field k, where 1 ≤ m ≤ d, and where the m-tuple (e_{1}, …, e_{m}) ∈ ℕ^{m} has the property that there exists a disjoint union {1, …, m} = I_{1} ⨆ I_{2} for which we have m_{1} := ∑_{i∈I_{1}} e_{i} ∈ {1, …, p − 1} and m_{2} := ∑_{i∈I_{2}} e_{i} ∈ {0, …, p − 2}. To check that R is quasi-healthy we can assume that the field k is algebraically closed and that R = W (k)[[T_{1}, …, T_{d}]]/(p − T_{1}^{e_{1}} · . . . · T_{m}^{e_{m}}) (thus C_{k} = W (k)).

We consider the ring epimorphism R ↠ R′ := W (k)[[T_{1}, T_{2}]]/(p − T_{1}^{m_{1}}T_{2}^{m_{2}}) that maps T_{i} with i ∈ I_{1} to T_{1}, that maps T_{i} with i ∈ I_{2} to T_{2}, and that maps T_{i} with i > m to 0. From Theorem 3 we get that R is quasi-healthy. Concrete example: if 1 ≤ m ≤ min{d, 2p − 3} and if the strict completion of R is C_{k}[[T_{1}, …, T_{d}]]/(p − T_{1} · . . . · T_{m}), then R is quasi-healthy. From this and Corollary 5 we get that each étale scheme over Spec O[T_{1}, …, T_{d}]/(p − T_{1} · . . . · T_{m}) is healthy regular, provided O is a discrete valuation ring of mixed characteristic (0, p) and index of ramification 1.

5.4 Regular schemes which are not (p-) healthy

Let R be a local regular ring of dimension 2 and mixed characteristic (0, p). Let 𝔪 be the maximal ideal of R. Let U = Spec R \ {𝔪}. The ring R is (p-) quasi-healthy if and only if Spec R is (p-) healthy regular. The next lemma provides an easy criterion for when R is not (p-) quasi-healthy.

Lemma 27 We assume that there exists a homomorphism H → D of finite flat group schemes over Spec R which is not an epimorphism and whose restriction to U is an epimorphism. Then R is neither quasi-healthy nor p-quasi-healthy.

Proof. We embed the Cartier dual D^{t} of D into an abelian scheme A over Spec R, cf. [BBM], Theorem 3.1.1. The Cartier dual homomorphism H_{U}^{t} → D_{U}^{t} is a closed immersion. The abelian scheme A_{U}/H_{U}^{t} over U does not extend to an abelian scheme over Spec R. Based on [R1], Chapter IX, Corollary 1.4, the argument for this is similar to the one used to prove that (d) implies (c) in Lemma 20. Thus R is not quasi-healthy. From Proposition 23 (a) we get that the p-divisible group of A_{U}/H_{U}^{t} does not extend to Spec R. Thus the fact that R is not p-quasi-healthy follows from either Lemma 20 or Proposition 23 (a). □

Based on Lemma 27, the following theorem adds many examples to the classical example of Raynaud–Gabber.

Theorem 28 We consider the ring 𝔖 = W (k)[[T_{1}, T_{2}]]. Let h ∈ (T_{1}, T_{2}) \ p𝔖. Let R := 𝔖/(p − h). Let 𝔖̄ := 𝔖/p𝔖 and let h̄ be the reduction of h modulo p. We assume that one of the following three properties hold:

(i) The element h̄ is divisible by u^{p}, where u is a power series in the maximal ideal of 𝔖̄ (the class of rings R for which (i) holds includes O[[T]], where O is a totally ramified discrete valuation ring extension of W (k) of index of ramification at least equal to p).

(ii) There exists a regular sequence u, v in 𝔖̄ such that u^{p−1}v^{p−1} divides h̄.

(iii) We can write h̄ = (aT_{1}^{p} + bT_{2}^{p} + cT_{1}^{p−1}T_{2}^{p−1})c, where a, b, c ∈ 𝔖̄.

Then R is neither p-quasi-healthy nor quasi-healthy.

Proof. It is enough to construct a homomorphism β : H → D of connected finite flat group schemes over Spec R which is not an epimorphism but whose restriction to U is an epimorphism, cf. Lemma 27. We will construct H and D by specifying their Breuil modules (M, φ) and (N, τ) (respectively) associated to a standard frame for R. We first present with full details the case (i) and then we will only mention what are the changes required to be made for the other two cases (ii) and (iii). We assume that the condition (i) holds. We set M = 𝔖̄^{3} and we identify M^{(σ)} with 𝔖̄^{3}. We choose an element t ∈ 𝔖̄ such that t and u are a regular sequence in 𝔖̄. We define the homomorphism φ by the following matrix:

Γ = ( \begin{matrix}0 & 0 & u^{p} \\ t − t^{p}u^{p−1} & u & (u − t^{p−1})(t − t^{p}u^{p−1}) \\ u^{p−1} & 0 & u^{p−1}(u − t^{p−1}) \end{matrix} ) .

It is easy to see that there exists a matrix Δ ∈ M_{3×3}(𝔖̄) such that

ΔΓ = ΓΔ = u^{p}I_{3},

where I_{3} is the unit matrix. It follows that the cokernel of φ is annihilated by u^{p} and thus also by h̄. Moreover the image of φ is contained in (t, u)M. Therefore (M, φ) is the Breuil module H of a connected finite flat group scheme over Spec R annihilated by p. We set N = 𝔖̄ and N^{(σ)} = 𝔖̄ and we define τ as the multiplication by u^{p}. This defines another connected finite flat group scheme D over Spec R annihilated by p. One easily checks the following equation of matrices:

(t^{p}, u^{p}, (tu)^{p})Γ = u^{p}(t, u, tu).

This equation shows that the 𝔖-linear map M → N defined by the matrix (t, u, tu) is a morphism of Breuil modules

α : (M, φ) → (N, τ).

As α is not surjective, the homomorphism β : H → D associated to α is not an epimorphism. Let 𝔭 ≠ 𝔪 be a prime ideal of R which contains p. The base change of α by κ_{𝔭} : 𝔖 → W (κ(𝔭)^{perf}) (of Section 2) is an epimorphism as the cokernel of α is k. This implies that β_{U} : H_{U} → D_{U} is an epimorphism. We assume that the condition (ii) holds. The proof in this case is similar to the case (i) but with the definitions of M, Γ, τ, and M → N modified as follows. Let M = 𝔖̄^{2}. Let

Γ = ( \begin{matrix}u^{p−1} & 0 \\ 0 & v^{p−1} \end{matrix} ) .

Let τ be defined by (uv)^{p−1}. Let M → N be defined by the matrix (v u).

We assume that the condition (iii) holds. Let M = 𝔖̄^{2}. Let

Γ = ( \begin{matrix}aT_{1} + cT_{2}^{p−1} & aT_{2} \\ bT_{1} & bT_{2} + cT_{1}^{p−1} \end{matrix} ) .

Let τ be defined by aT_{1}^{p} + bT_{2}^{p} + cT_{1}^{p−1}T_{2}^{p−1}. Let M → N be defined by the matrix (T_{1} T_{2}). The determinant of Γ is h̄. □

6 Integral models and Néron models

Let O be a discrete valuation ring of mixed characteristic (0, p) and of index of ramification at most p − 1. Let K be the field of fractions of O. A flat O-scheme ⋆ is said to have the extension property, if for each O-scheme X which is a healthy regular scheme, every morphism X_{K} → ⋆_{K} of K-schemes extends uniquely to a morphism X → ⋆ of O-schemes (cf. [V1], Definition 3.2.3 3)).

Lemma 29 Let Z_{K} be a regular scheme which is formally smooth over Spec K. Then there exists at most one regular scheme which is a formally smooth Oscheme, which has the extension property, and whose fibre over Spec K is Z_{K}.

Proof. Let Z_{1} and Z_{2} be two regular schemes which are formally smooth over Spec O, which satisfy the identity Z_{1,K} = Z_{2,K} = Z_{K}, and which have the extension property. Both Z_{1} and Z_{2} are healthy regular schemes, cf. Corollary 5. Thus the identity Z_{1,K} = Z_{2,K} extends naturally to morphisms Z_{1} → Z_{2} and Z_{2} → Z_{1}, cf. the fact that both Z_{1} and Z_{2} have the extension property. Due to the uniqueness part of the extension property, the composite morphisms Z_{1} → Z_{2} → Z_{1} and Z_{2} → Z_{1} → Z_{2} are identity automorphisms. Thus the identity Z_{1,K} = Z_{2,K} extends uniquely to an isomorphism Z_{1} → Z_{2}. □

Corollary 30 The integral canonical models of Shimura varieties defined in [V1], Definition 3.2.3 6) are unique, provided they are over the spectrum of a discrete valuation ring O as above.

Let d ≥ 1 and n ≥ 3 be natural numbers. We assume that n is prime to p. Let A_{d,1,n} be the Mumford moduli scheme over Spec ℤ[\frac{1}{n}] that parameterizes principally polarized abelian scheme over Spec ℤ[\frac{1}{n}]-schemes which are of relative dimension d and which have level-n symplectic similitude structures (cf. [MFK], Theorems 7.9 and 7.10). For Néron models over Dedekind domains we refer to [BLR], Chapter I, Subsection 1.2, Definition 1.

Theorem 31 Let D be a Dedekind domain which is a flat ℤ[\frac{1}{n}]-algebra. Let K be the field of fractions of D. We assume that the following two things hold:

(i) the only local ring of D whose residue class field has characteristic 0, is K; (ii) if v is a prime of D whose residue class field has a prime characteristic

p_{v} ∈ ℕ^{∗}, then the index of ramification of the local ring of v is at most p_{v} − 1.

Let N be a finite A_{d,1,n,D}-scheme which is a projective, smooth D-scheme. Then N is the Néron model over D of its generic fibre N_{K}.

Proof. Let Y be a smooth D-scheme. Let δ_{K} : Y_{K} → N_{K} be a morphism of K-scheme. Let V be an open subscheme of Y which contains Y_{K} and all generic points of fibres of Y in positive characteristic and for which δ_{K} extends uniquely to a morphism δ_{V} : V → N (cf. the projectiveness of N). Let (B_{V} , λ_{V}) be the pull back to V of the universal principally polarized abelian scheme over A_{d,1,n,D} via the composite morphism ν_{V} : V → N → A_{d,1,n,D}. From Corollary 5 we get that B_{V} extends uniquely to an abelian scheme B over Y. From [R1], Chapter IX, Corollary 1.4 we get that λ_{V} extends (uniquely) to a polarization λ of B. The level-n symplectic similitude structure of (B_{V} , λ_{V}) defined naturally by ν_{V} extends uniquely to a level-n symplectic similitude structure of (B, λ), cf. the classical Nagata–Zariski purity theorem. Thus ν_{V} extends uniquely to a morphism ν : Y → A_{d,1,n,D}. As Y is a normal scheme, as N is finite over A_{d,1,n,D}, and as ν restricted to V factors through N, the morphism ν factors uniquely through a morphism δ : Y → N which extends δ_{V} and thus also δ_{K}. Hence the Theorem follows from the very definition of Néron models. □

Remarks. (a) From Theorem 31 and [V3], Remark 4.4.2 and Example 4.5 we get that there exist plenty of Néron models over O whose generic fibres are not finite schemes over torsors of smooth schemes over Spec K. We can take N to be the pull back to O of those Néron models of Theorem 31 whose generic fibres have the above property (cf. [V3], Remark 4.5). If p > 2 and e = p − 1, then these Néron models N are new (i.e., their existence does not follow from [N], [BLR], [V1], [V2], or [V3]).

(b) One can use Theorem 28 (i) and Artin’s approximation theorem to show that Theorem 31 does not hold in general if there exists a prime v of D whose residue class field has a prime characteristic p_{v} ∈ ℕ^{∗} and whose index of ramification is at least p_{v}. Counterexamples can be obtained using integral models of projective Shimura varieties of PEL type, cf. [V3], Corollary 4.3.

References

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Adrian Vasiu P. O. Box 6000, Department of Mathematical Sciences Binghamton University Binghamton, New York 13902-6000 U.S.A. adrian@math.binghamton.edu Thomas Zink Fakultät für Mathematik Universität Bielefeld P.O. Box 100 131 D-33 501 Bielefeld Germany zink@math.uni-bielefeld.de 600

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