Frames and Finite Group Schemes over

Complete Regular Local Rings

Eike Lau

Received: November 10, 2009 Revised: May 21, 2010

Communicated by Peter Schneider

Abstract. Let p be an odd prime. We show that the classification of p-divisible groups by Breuil windows and the classification of commutative finite flat group schemes of p-power order by Breuil modules hold over every complete regular local ring with perfect residue field of characteristic p. We set up a formalism of frames and windows with an abstract deformation theory that applies to Breuil windows.

2010 Mathematics Subject Classification: 14L05, 14F30

Keywords and Phrases: Finite group scheme, p-divisible group, frame, window, deformation theory, Dieudonné display

1 Introduction

Let R be a complete regular local ring of dimension r with perfect residue field k of odd characteristic p. Let W (k) be the ring of Witt vectors of k. One can write R = 𝔖/E𝔖 with

𝔖 = W (k)[[x_{1}, …, x_{r}]]

such that E ∈ 𝔖 is a power series with constant term p. Let σ be the endomorphism of 𝔖 that extends the Frobenius automorphism of W (k) by σ(x_{i}) = x_{i}^{p}. Following Vasiu and Zink, a Breuil window relative to 𝔖 → R is a pair (Q, ϕ) where Q is a free 𝔖-module of finite rank, and where

ϕ : Q → Q^{(σ)}

is an 𝔖-linear map with cokernel annihilated by E.

Theorem 1.1. The category of p-divisible groups over R is equivalent to the category of Breuil windows relative to 𝔖 → R.

If R has characteristic p, this follows from more general results of A. de Jong [dJ]; this case is included here only for completeness. If r = 1 and E is an Eisenstein polynomial, Theorem 1.1 was conjectured by Breuil [Br] and proved by Kisin [K1]. When E is a deformation of an Eisenstein polynomial the result is proved in [VZ1]. Like in these cases one can deduce a classification of commutative finite flat group schemes of p-power order over R: A Breuil module relative to 𝔖 → R is a triple (M, φ, ψ) where M is a finitely generated 𝔖-module annihilated by a power of p and of projective dimension at most one, and where

φ : M → M^{(σ)}, ψ : M^{(σ)} → M

are 𝔖-linear maps with φψ = E and ψφ = E. If R has characteristic zero, such triples are equivalent to pairs (M, φ) such that the cokernel of φ is annihilated by E.

Theorem 1.2. The category of commutative finite flat group schemes over R annihilated by a power of p is equivalent to the category of Breuil modules relative to 𝔖 → R. ^{1}

This result is applied in [VZ2] to the question whether abelian schemes or pdivisible groups defined over the complement of the maximal ideal in Spec R extend to Spec R.

Frames and windows

To prove Theorem 1.1 we show that Breuil windows are equivalent to Dieudonné displays over R, which are equivalent to p-divisible groups over R by [Z2]; the same route is followed in [VZ1]. So the main part of this article is purely module theoretic. We introduce a notion of frames and windows (motivated by [Z3]) which allows to formulate a deformation theory that generalises the deformation theory of Dieudonné displays developed in [Z2] and that also applies to Breuil windows. Technically the main point is the formalism of σ_{1} in Definition 2.1; the central result is the lifting of windows in Theorem 3.2. This is applied as follows. Let 𝔪_{R} be the maximal ideal of R. For each positive integer a we consider the rings 𝔖_{a} = 𝔖/(x_{1}, …, x_{r})^{a}𝔖 and R_{a} = R/𝔪_{R}^{a}. There is an obvious notion of Breuil windows relative to 𝔖_{a} → R_{a} and a functor

κ_{a} : (Breuil windows relative to 𝔖_{a} → R_{a}) → (Dieudonné displays over R_{a}).

Here κ_{1} is trivially an equivalence because 𝔖_{1} = W (k) and R_{1} = k. The deformation theory implies that on both sides lifts from a to a+1 are classified by lifts of the Hodge filtration in a compatible way. Thus κ_{a} is an equivalence for all a by induction, and Theorem 1.1 follows.

1Recently, Theorems 1.1 and 1.2 have been extended to the case p = 2. See: A relation between Dieudonné displays and crystalline Dieudonné theory (in preparation).

Documenta Mathematica 15 (2010) 545–569

Complements

There is some freedom in the choice of the Frobenius lift on 𝔖. Namely, let σ be a ring endomorphism of 𝔖 which preserves the ideal J = (x_{1}, …, x_{r}) and which induces the Frobenius on 𝔖/p𝔖. If the endomorphism σ/p of J/J^{2} is nilpotent modulo p, Theorems 1.1 and 1.2 hold without change. All of the above equivalences of categories are compatible with the natural duality operations on both sides. If the residue field k is not perfect, there is an analogue of Theorems 1.1 and 1.2 for connected groups. Here p = 2 is allowed. The ring W (k) is replaced by a Cohen ring of k, and the operators ϕ and φ must be nilpotent modulo the maximal ideal of 𝔖. In the first version of this article [L3] the formalism of frames was introduced only to give an alternative proof of the results of Vasiu and Zink [VZ1]. In response, they pointed out that both their and this approach apply in greater generality, e.g. in the case where E ∈ 𝔖 takes the form E = g + pϵ such that ϵ is a unit and g divides σ(g) for a general Frobenius lift σ as above. However, the method of loc. cit. seems not to give Theorem 1.1 completely.

All rings in this article are commutative and have a unit. All finite flat group schemes are commutative.

Acknowledgements. The author thanks A. Vasiu and Th. Zink for valuable discussions, in particular Th. Zink for sharing his notion of κ-frames and for suggesting to include section 10, and the referee for many helpful comments.

2 Frames and windows

Let p be a prime. The following notion of frames and windows differs from [Z3]. Some definitions and arguments could be simplified by assuming that the relevant rings are local, which is the case in our applications, but we work in greater generality until section 4. If S is a ring equipped with a ring endomorphism σ, for an S-module M we write M^{(σ)} = S ⊗_{σ,S} M, and for a σ-linear map of S-modules g : M → N we denote by g^{♯} : M^{(σ)} → N its linearisation, g^{♯}(s ⊗ m) = sg(m). If g^{♯} is invertible, g is called a σ-linear isomorphism.

Definition 2.1. A frame is a quintuple

ℱ = (S, I, R, σ, σ_{1})

consisting of a ring S, an ideal I of S, the quotient ring R = S/I, a ring endomorphism σ : S → S, and a σ-linear map of S-modules σ_{1} : I → S, such that the following conditions hold:

i. I + pS ⊆ Rad(S),

ii. σ(a) ≡ a^{p} mod pS for a ∈ S,

iii. σ_{1}(I) generates S as an S-module.

We do not assume here that R is the specific ring considered in the introduction. In our examples σ_{1}(I) contains the element 1.

Lemma 2.2. For every frame ℱ there is a unique element θ ∈ S such that σ(a) = θσ_{1}(a) for a ∈ I.

Proof. Condition iii means that the homomorphism σ_{1}^{♯} : I^{(σ)} → S is surjective. Let us choose b ∈ I^{(σ)} such that σ_{1}^{♯}(b) = 1. Then necessarily θ = σ^{♯}(b). For a ∈ I we compute σ(a) = σ_{1}^{♯}(b)σ(a) = σ_{1}^{♯}(ba) = σ^{♯}(b)σ_{1}(a) as desired.

Definition 2.3. Let ℱ be a frame. A window over ℱ, also called an ℱ - window, is a quadruple

𝒫 = (P, Q, F, F_{1})

where P is a finitely generated projective S-module, Q ⊆ P is a submodule, F : P → P and F_{1} : Q → P are σ-linear map of S-modules, such that the following conditions hold:

1. There is a decomposition P = L ⊕ T with Q = L ⊕ IT,

2. F_{1}(ax) = σ_{1}(a)F (x) for a ∈ I and x ∈ P,

3. F_{1}(Q) generates P as an S-module.

A decomposition as in 1 is called a normal decomposition of (P, Q) or of 𝒫.

Remark 2.4. The operator F is determined by F_{1}. Indeed, if b ∈ I^{(σ)} satisfies σ_{1}^{♯}(b) = 1, then condition 2 implies that F (x) = F_{1}^{♯}(bx) for x ∈ P. In particular we have F (x) = θF_{1}(x) when x lies in Q.

Remark 2.5. Condition 1 implies that

1′. P/Q is a projective R-module.

If finitely generated projective R-modules lift to projective S-modules, necessarily finitely generated because I ⊆ Rad(S), condition 1 is equivalent to 1′. In all our examples, this lifting property holds because S is either local or I-adic.

Lemma 2.6. Let ℱ be a frame, let P = L ⊕ T be a finitely generated projective S-module, and let Q = L ⊕ IT. The set of ℱ -window structures (P, Q, F, F_{1}) on these modules is mapped bijectively to the set of σ-linear isomorphisms

Ψ : L ⊕ T → P

by the assignment Ψ(l + t) = F_{1}(l) + F (t) for l ∈ L and t ∈ T.

The triple (L, T, Ψ) is called a normal representation of (P, Q, F, F_{1}).

Proof. If (P, Q, F, F_{1}) is an ℱ -window, by conditions 2 and 3 of Definition 2.3 the linearisation of Ψ is surjective, thus bijective since P and P^{(σ)} are projective S-modules of equal rank by conditions i and ii of Definition 2.1. Conversely, if Ψ is given, one gets an ℱ -window by F (l + t) = θΨ(l) + Ψ(t) and F_{1}(l + at) = Ψ(l) + σ_{1}(a)Ψ(t) for l ∈ L, t ∈ T, and a ∈ I.

Example. The Witt frame of a p-adic ring R is

𝒲_{R} = (W (R), I_{R}, R, f, f_{1})

where W (R) is the ring of p-typical Witt vectors of R, f is its Frobenius endomorphism, and f_{1} : I_{R} → W (R) is the inverse of the Verschiebung homomorphism. Here θ = p. We have I_{R} ⊆ Rad(W (R)) because W (R) is I_{R}-adic; see [Z1, Proposition 3]. Windows over 𝒲_{R} are 3n-displays over R in the sense of [Z1], called displays in [M2], which is the terminology we follow.

Functoriality

Let ℱ = (S, I, R, σ, σ_{1}) and ℱ′ = (S′, I′, R′, σ′, σ′_{1}) be frames.

Definition 2.7. A homomorphism of frames α : ℱ → ℱ′, also called a frame homomorphism, is a ring homomorphism α : S → S′ with α(I) ⊆ I′ such that σ′α = ασ and σ′_{1}α = u · ασ_{1} for a unit u ∈ S′. If u = 1, then α is called strict.

Remark 2.8. The unit u is unique because ασ_{1}(I) generates S′ as an S′-module. We have α(θ) = uθ′. If we want to specify u, we say that α is a u-homo-morphism. There is a unique factorisation of α into frame homomorphisms

ℱ ⟶^{α′} ℱ′′ ⟶^{ω} ℱ′

such that α′ is strict and ω is an invertible u-homomorphism. Here ℱ′′ is the u^{−1}-twist of ℱ′ defined as ℱ′′ = (S′, I′, R′, σ′, u^{−1}σ′_{1}).

Let α : ℱ → ℱ′ be a u-homomorphism of frames.

Definition 2.9. Let 𝒫 be an ℱ -window and let 𝒫′ be an ℱ′-window. A homomorphism of windows g : 𝒫 → 𝒫′ over α, also called an α-homomorphism, is an S-linear map g : P → P′ with g(Q) ⊆ Q′ such that F′g = gF and F′_{1}g = u · gF_{1}. A homomorphism of ℱ -windows is an id_{𝒫} -homomorphism in the previous sense.

Lemma 2.10. For each ℱ -window 𝒫 there is a base change window α_{∗}𝒫 over ℱ′ together with an α-homomorphism of windows 𝒫 → α_{∗}𝒫 that induces a bijection Hom_{ℱ′} (α_{∗}𝒫, 𝒫′) = Hom_{α}(𝒫, 𝒫′) for all ℱ′-windows 𝒫′.

Proof. Clearly this requirement determines α_{∗}𝒫 uniquely. It can be constructed explicitly as follows: If (L, T, Ψ) is a normal representation of 𝒫, a normal representation of α_{∗}𝒫 is (S′ ⊗_{S} L, S′ ⊗_{S} T, Ψ′) where Ψ′ is defined by Ψ′(s′ ⊗ l) = uσ′(s′) ⊗ Ψ(l) and Ψ′(s′ ⊗ t) = σ′(s′) ⊗ Ψ(t).

If α_{∗}𝒫 = (P′, Q′, F′, F′_{1}), then P′ = S′ ⊗_{S} P, and Q′ ⊆ P′ is the S′-submodule generated by I′P′ and by the image of Q. Remark 2.11. As suggested in [VZ2], the above definitions of frames and windows can be generalised as follows. Instead of condition iii of Definition 2.1, the element θ given by Lemma 2.2 is taken as part of the data. For a uhomomorphism α : ℱ → ℱ′ of generalised frames in this sense it is necessary to require that α(θ) = uθ′. For a window over a generalised frame the relation F (x) = θF_{1}(x) of Remark 2.4 becomes part of the definition, and condition 3 of Definition 2.3 is replaced by the requirement that F_{1}(Q) + F (P) generates P. Then the results of sections 2–4 hold for generalised frames and windows as well. Details are left to the reader.

Limits

Windows are compatible with projective limits of frames in the following sense. Assume that for each positive integer n we have a frame

ℱ_{n} = (S_{n}, I_{n}, R_{n}, σ_{n}, σ_{1n})

and a strict frame homomorphism π_{n} : ℱ_{n+1} → ℱ_{n} such that the involved maps S_{n+1} → S_{n} and I_{n+1} → I_{n} are surjective and Ker(π_{n}) is contained in Rad(S_{n+1}). We obtain a frame \underleftarrow{lim} ℱ_{n} = (S, I, R, σ, σ_{1}) with S = \underleftarrow{lim} S_{n} etc. By definition, an ℱ_{∗}-window is a system 𝒫_{∗} of ℱ_{n}-windows 𝒫_{n} together with isomorphisms π_{n∗}ℱ_{n+1} ≅ ℱ_{n}.

Lemma 2.12. The category of (\underleftarrow{lim} ℱ_{n})-windows is equivalent to the category of ℱ_{∗}-windows.

Proof. The obvious functor from (\underleftarrow{lim} ℱ_{n})-windows to ℱ_{∗}-windows is fully faithful. We have to show that for an ℱ_{∗}-window 𝒫_{∗}, the projective limit \underleftarrow{lim} 𝒫_{n} = (P, Q, F, F_{1}) defined by P = \underleftarrow{lim} P_{n} etc. is a window over \underleftarrow{lim} ℱ_{n}. The condition Ker(π_{n}) ⊆ Rad(S_{n+1}) implies that P is a finitely generated projective S-module and that P/Q is projective over R. In order that P has a normal decomposition it suffices to show that each normal decomposition of 𝒫_{n} lifts to a normal decomposition of 𝒫_{n+1}. Assume that P_{n} = L′_{n} ⊕ T′_{n} and P_{n+1} = L_{n+1} ⊕ T_{n+1} are normal decompositions and let P_{n} = L_{n} ⊕ T_{n} be induced by the second. Since T_{n} ⊗ R_{n} ≅ P_{n}/Q_{n} ≅ T′_{n} ⊗ R_{n} and L_{n} ⊗ R_{n} ≅ Q_{n}/IP_{n} ≅ L′_{n} ⊗ R_{n}, we have T_{n} ≅ T′_{n} and L_{n} ≅ L′_{n}. Hence the two decompositions of P_{n} differ by an automorphism of L_{n} ⊕T_{n} of the type ω = ( \begin{matrix}a & b \\ c & d \end{matrix} ) with c : L_{n} → I_{n}T_{n}. Now ω lifts to an endomorphism ω′ = ( \begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix} ) of L_{n+1} ⊕ T_{n+1} with c′ : L_{n+1} → I_{n+1}T_{n+1}, and ω′ is an automorphism since Ker(π_{n}) ⊆ Rad(S_{n+1}). The required lifting of normal decompositions follows. All remaining window axioms for \underleftarrow{lim} 𝒫_{n} are easily checked.

Remark 2.13. Assume that S_{1} is a local ring. Then all S_{n} and S are local too. Hence \underleftarrow{lim} ℱ_{n} satisfies the lifting property of Remark 2.5, so the normal decomposition of P in the preceding proof is automatic.

Duality

Let 𝒫, 𝒫′, and 𝒫′′ be windows over a frame ℱ. A bilinear form of ℱ - windows β : 𝒫 × 𝒫′ → 𝒫′′ is an S-bilinear map β : P × P′ → P′′ such that β(Q × Q′) ⊆ Q′′ and

β(F_{1}(x), F′_{1}(x′)) = F′′_{1}(β(x, x′))  (2.1)

for x ∈ Q and x′ ∈ Q′. Let ℱ also denote the ℱ -window (S, I, σ, σ_{1}). For every ℱ -window 𝒫 there is a unique dual ℱ -window 𝒫^{t} together with a bilinear form 𝒫 × 𝒫^{t} → ℱ which induces for each ℱ -window 𝒫′ an isomorphism Hom(𝒫′, 𝒫^{t}) ≅ Bil(𝒫 × 𝒫′, ℱ). Explicitly we have 𝒫^{t} = (P^{∨}, Q^{t}, F^{t}, F_{1}^{t}) where P^{∨} = Hom_{S}(P, S) and

Q^{t} = {x′ ∈ P^{∨} | x′(Q) ⊆ I}.

The operators F_{1}^{t} and F^{t} are determined by (2.1) with σ_{1} in place of F′′_{1}. If (L, T, Ψ) is a normal representation for 𝒫, a normal representation for 𝒫^{t} is given by (T^{∨}, L^{∨}, Ψ^{t}) where (Ψ^{t})^{♯} is equal to ((Ψ^{♯})^{−1})^{∨}. This shows that F_{1}^{t} and F^{t} are well-defined. There is a natural isomorphism 𝒫^{tt} ≅ 𝒫. For a more detailed exposition of the duality formalism in the case of (Diedonné) displays we refer to [Z1, Definition 19] or [L2, Section 3].

Lemma 2.14. Let α : ℱ → ℱ′ be a u-homomorphism of frames and let c ∈ S′ be a unit such that c^{−1}σ′(c) = u. For all ℱ -windows 𝒫 there is a natural isomorphism (depending on c)

α_{∗}(𝒫^{t}) ≅ (α_{∗}𝒫)^{t}.

Proof. We consider the ℱ′-window ℱ′_{u} = (S′, I′, uσ′, uσ′_{1}). The given bilinear form 𝒫 × 𝒫^{t} → ℱ induces a bilinear form α_{∗}𝒫 × α_{∗}(𝒫^{t}) → ℱ′_{u}; this is easily verified using that under base change by α each of the operators F_{1}, F′_{1}, and F′′_{1} = σ_{1} accounts for one factor of u in (2.1). Multiplication by c is an isomorphism of ℱ′-windows ℱ′_{u} ≅ ℱ′. The resulting bilinear form α_{∗}𝒫 × α_{∗}(𝒫^{t}) → ℱ′ induces an isomorphism α_{∗}(𝒫^{t}) ≅ (α_{∗}𝒫)^{t}.

3 Crystalline homomorphisms

Definition 3.1. A homomorphism of frames α : ℱ → ℱ′ is called crystalline if the functor α_{∗} : (ℱ -windows ) → (ℱ′-windows ) is an equivalence of categories.

Theorem 3.2. Let α : ℱ → ℱ′ be a strict frame homomorphism that induces an isomorphism R ≅ R′ and a surjection S → S′ with kernel 𝔞 ⊂ S. We assume that there is a finite filtration 𝔞 = 𝔞_{0} ⊇ ⋯ ⊇ 𝔞_{n} = 0 with σ(𝔞_{i}) ⊆ 𝔞_{i+1} and σ_{1}(𝔞_{i}) ⊆ 𝔞_{i} such that σ_{1} is elementwise nilpotent on 𝔞_{i}/𝔞_{i+1}. We assume that finitely generated projective S′-modules lift to projective S-modules. Then α is crystalline.

In many applications the lifting property of projective modules holds because 𝔞 is nilpotent or S is local. The proof of Theorem 3.2 is a variation of the proofs of [Z1, Theorem 44] and [Z2, Theorem 3].

Proof. The homomorphism α factors into ℱ → ℱ′′ → ℱ′ where the frame ℱ′′ is determined by S′′ = S/𝔞_{1}, so by induction we may assume that σ(𝔞) = 0. The functor α_{∗} is essentially surjective because normal representations (L, T, Ψ) can be lifted from ℱ′ to ℱ. In order that α_{∗} is fully faithful it suffices to show that α_{∗} is fully faithful on automorphisms because a homomorphism g : 𝒫 → 𝒫′ can be encoded by the automorphism ( \begin{matrix}1 & 0 \\ g & 1 \end{matrix} ) of 𝒫 ⊕ 𝒫′. Since for a window 𝒫 over ℱ an automorphism of α_{∗}𝒫 can be lifted to an S-module automorphism of P, it suffices to prove the following assertion. Assume that 𝒫 = (P, Q, F, F_{1}) and 𝒫′ = (P, Q, F′, F′_{1}) are two ℱ -windows such that F ≡ F′ and F_{1} ≡ F′_{1} modulo 𝔞. Then there is a unique ℱ -window isomorphism g : 𝒫 ≅ 𝒫′ with g ≡ id modulo 𝔞. We write F′_{1} = F_{1} + η and F′ = F + ε and g = 1 + ω, where the σ-linear maps η : Q → 𝔞P and ε : P → 𝔞P are given, and where ω : P → 𝔞P is an arbitrary S-linear map. The induced g is an isomorphism of windows if and only if gF_{1} = F′_{1}g on Q, which translates into the identity

η = ωF_{1} − F′_{1}ω.  (3.1)

We fix a normal decomposition P = L ⊕ T, thus Q = L ⊕ IT. For l ∈ L, t ∈ T, and a ∈ I we have

η(l + at) = η(l) + σ_{1}(a)ε(t),
ω(F_{1}(l + at)) = ω(F_{1}(l)) + σ_{1}(a)ω(F (t)),
F′_{1}(ω(l + at)) = F′_{1}(ω(l)) + σ_{1}(a)F′(ω(t)).

Here F′ω = 0 because for a ∈ 𝔞 and x ∈ P we have F′(ax) = σ(a)F′(x), and σ(𝔞) = 0. As σ_{1}(I) generates S we see that (3.1) is equivalent to:

\{ \begin{matrix}ε = ωF  on T, \\ η = ωF_{1} − F′_{1}ω on L. \end{matrix}  (3.2)

Since Ψ : L ⊕ T ⟶^{F_{1}+F} P is a σ-linear isomorphism, to give ω is equivalent to give a pair of σ-linear maps

ω_{L} = ωF_{1} : L → 𝔞P, ω_{T} = ωF : T → 𝔞P.

Let λ : L → L^{(σ)} be the composition L ⊆ P ⟶^{(Ψ^{♯})^{−1}} L^{(σ)} ⊕ T^{(σ)} ⟶^{pr_{1}} L^{(σ)} and

let τ : L → T^{(σ)} be analogous with pr_{2} in place of pr_{1}. Then the restriction ω|_{L} is equal to ω_{L}^{♯}λ + ω_{T}^{♯} τ, and (3.2) becomes:

\{ \begin{matrix}ω_{T} = ε|_{T} , \\ ω_{L} − F′_{1}ω_{L}^{♯}λ = η|_{L} + F′_{1}ω_{T}^{♯} τ. \end{matrix}  (3.3)

Let ℋ be the abelian group of σ-linear maps L → 𝔞P. We claim that the endomorphism U of ℋ given by U(ω_{L}) = F′_{1}ω_{L}^{♯}λ is elementwise nilpotent, which implies that 1 − U is bijective, and (3.3) has a unique solution in (ω_{L}, ω_{T}) and thus in ω. The endomorphism F′_{1} of 𝔞P is elementwise nilpotent because F′_{1}(ax) = σ_{1}(a)F′(x) and because σ_{1} is elementwise nilpotent on 𝔞 by assumption. Since L is finitely generated it follows that U is elementwise nilpotent.

Remark 3.3. The same argument applies if instead of σ_{1} being elementwise nilpotent one demands that λ is (topologically) nilpotent, which is the original situation in [Z1, Theorem 44]; see section 10.

4 Abstract deformation theory

Definition 4.1. The Hodge filtration of a window 𝒫 is the submodule

Q/IP ⊆ P/IP.

Lemma 4.2. Let α : ℱ → ℱ′ be a strict homomorphism of frames such that S = S′; thus R → R′ is surjective and we have I ⊆ I′. Then ℱ -windows 𝒫 are equivalent to pairs consisting of an ℱ′-window 𝒫′ = (P′, Q′, F′, F′_{1}) and a lift of its Hodge filtration to a direct summand V ⊆ P′/IP′.

Proof. The equivalence is given by the functor 𝒫 ↦ (α_{∗}𝒫, Q/IP), which is easily seen to be fully faithful. We show that it is essentially surjective. Let an ℱ′-window 𝒫′ and a lift of its Hodge filtration V ⊆ P′/IP′ be given and let Q ⊆ P′ be the inverse image of V; thus Q ⊆ Q′. We have to show that 𝒫 = (P′, Q, F′, F′_{1}|_{Q}) is an ℱ -window. First we need a normal decomposition for 𝒫; this is a decomposition P′ = L ⊕ T such that V = L/IL. Since 𝒫′ has a normal decomposition, 𝒫 has one too for at least one choice of V. By modifying the isomorphism P′ ≅ L ⊕ T with an automorphism ( \begin{matrix}1 & 0 \\ c & 1 \end{matrix} ) of L ⊕ T for some homomorphism c : L → I′T one reaches every lift of the Hodge filtration. It remains to show that F′_{1}(Q) generates P′. In terms of a normal decomposition P′ = L ⊕ T for 𝒫 this means that F′_{1} + F′ : L ⊕ T → P′ is a σ-linear isomorphism, which holds because 𝒫′ is an ℱ′-window.

Assume that a strict homomorphism of frames α : ℱ → ℱ′ is given such that S → S′ is surjective with kernel 𝔞, and I′ = IS′. We want to factor α into strict frame homomorphisms

(S, I, R, σ, σ_{1}) ⟶^{α_{1}} (S, I′′, R′, σ, σ′′_{1}) ⟶^{α_{2}} (S′, I′, R′, σ′, σ′_{1})  (4.1)

such that α_{2} satisfies the hypotheses of Theorem 3.2. Necessarily I′′ = I + 𝔞. The main point is to define σ′′_{1} : I′′ → S, which is equivalent to defining a σ-linear map σ′′_{1} : 𝔞 → 𝔞 that extends the restriction of σ_{1} to I ∩ 𝔞 and satisfies the hypotheses of Theorem 3.2. Once this is achieved, Theorem 3.2 and Lemma 4.2 will show that ℱ -windows are equivalent to ℱ′- windows 𝒫′ plus a lift of the Hodge filtration of 𝒫′ to a direct summand of P/IP, where 𝒫′′ = (P, Q′′, F, F′′_{1}) is the unique lift of 𝒫′ under α_{2}.

5 Dieudonné frames

Let R be a noetherian complete local ring with maximal ideal 𝔪_{R} and with perfect residue field k of characteristic p. If p = 2, we assume that p annihilates R. Let Ŵ (𝔪_{R}) ⊂ W (R) be the ideal of all Witt vectors whose coefficients lie in 𝔪_{R} and converge to zero 𝔪_{R}-adically. There is a unique subring 𝕎(R) of W (R) which is stable under the Frobenius f such that the projection 𝕎(R) → W (k) is surjective with kernel Ŵ (𝔪_{R}), and the ring 𝕎(R) is also stable under the Verschiebung v; see [Z2, Lemma 2]. Let 𝕀_{R} be the kernel of the projection to the first component 𝕎(R) → R. Then v : 𝕎(R) → 𝕀_{R} is bijective.

Definition 5.1. The Dieudonné frame associated to R is

𝒟_{R} = (𝕎(R), 𝕀_{R}, R, f, f_{1})

with f_{1} = v^{−1}.

Here θ = p. Windows over 𝒟_{R} are Dieudonné displays over R in the sense of [Z2]. We note that 𝕎(R) is a local ring, which guarantees the existence of normal decompositions; see Remark 2.5. The inclusion 𝕎(R) → W (R) is a strict homomorphism of frames 𝒟_{R} → 𝒲_{R}. If R′ has the same properties as R, a local ring homomorphism R → R′ induces a strict frame homomorphism 𝒟_{R} → 𝒟_{R′}. Assume that R′ = R/𝔟 for an ideal 𝔟 which is equipped with elementwise nilpotent divided powers γ. Then 𝕎(R) → 𝕎(R′) is surjective with kernel Ŵ (𝔟) = W (𝔟) ∩ Ŵ (𝔪_{R}). In this situation, a factorisation (4.1) of the homomorphism 𝒟_{R} → 𝒟_{R′} can be defined as follows. We recall that the γ-divided Witt polynomials are defined as

w′_{n}(X_{0}, …, X_{n}) = (p^{n} − 1)!γ_{p^{n}}(X_{0}) + (p^{n−1} − 1)!γ_{p^{n−1}}(X_{1}) + ⋯ + X_{n}.

Thus p^{n}w′_{n} is the usual Witt polynomial w_{n}(X_{0}, …, X_{n}) = X_{0}^{p^{n}} + ⋯ + p^{n}X_{n}. Let 𝔟^{<∞>} be the W (R)-module of all sequences [b_{0}, b_{1}, …] with elements b_{i} ∈ 𝔟 that converge to zero 𝔪_{R}-adically, such that x ∈ W (R) acts on 𝔟^{<∞>} by [b_{0}, b_{1}, …] ↦ [w_{0}(x)b_{0}, w_{1}(x)b_{1}, …]. We have an isomorphism of W (R)- modules

log : Ŵ (𝔟) ≅ 𝔟^{<∞>}; b ↦ (w′_{0}(b), w′_{1}(b), …);

see the remark after [Z1, Cor. 82]. For b ∈ Ŵ (𝔟) we call log(b) the logarithmic coordinates of b. Let

𝕀_{R/R′} = 𝕀_{R} + Ŵ (𝔟).

In logarithmic coordinates, the restriction of f_{1} to 𝕀_{R} ∩ Ŵ (𝔟) is given by

f_{1}([0, b_{1}, b_{2}, …]) = [b_{1}, b_{2}, …].

Thus f_{1} : 𝕀_{R} → 𝕎(R) extends uniquely to an f-linear map

f̃_{1} : 𝕀_{R/R′} → 𝕎(R)

with f̃_{1}([b_{0}, b_{1}, …]) = [b_{1}, b_{2}, …] on Ŵ (𝔟), and we obtain a factorisation

𝒟_{R} ⟶^{α_{1}} 𝒟_{R/R′} = (𝕎(R), 𝕀_{R/R′}, R′, f, f̃_{1}) ⟶^{α_{2}} 𝒟_{R′}.  (5.1)

Proposition 5.2. The frame homomorphism α_{2} is crystalline.

This is a reformulation of [Z2, Theorem 3] if 𝔪_{R} is nilpotent, and the general case is an easy consequence. As explained in section 4, it follows that deformations of Dieudonné displays from R′ to R are classified by lifts of the Hodge filtration; this is [Z2, Theorem 4].

Proof of Proposition 5.2. When 𝔪_{R} is nilpotent, α_{2} satisfies the hypotheses of Theorem 3.2; the required filtration of 𝔞 = Ŵ (𝔟) is 𝔞_{i} = p^{i}𝔞. In general, these hypotheses are not fulfilled because f_{1} : 𝔞 → 𝔞 is only topologically nilpotent. However, one can find a sequence of ideals R ⊃ I_{1} ⊃ I_{2} ⋯ which define the 𝔪_{R}-adic topology such that each 𝔟 ∩ I_{n} is stable under the divided powers of 𝔟. Indeed, for each n there is an l with 𝔪_{R}^{l} ∩ 𝔟 ⊆ 𝔪_{R}^{n}𝔟; for I_{n} = 𝔪_{R}^{n}𝔟 + 𝔪_{R}^{l} we have 𝔟 ∩ I_{n} = 𝔪_{R}^{n}𝔟. The proposition holds for each R/I_{n} in place of R, and the general case follows by passing to the projective limit, using Lemma 2.12.

6 κ-frames

The results in this section are essentially due to Th. Zink (private communication); see also [Z3, Section 1] and [VZ1, Section 3].

Definition 6.1. A κ-frame is a frame ℱ = (S, I, R, σ, σ_{1}) such that

iv. S has no p-torsion,

v. W (R) has no p-torsion,

vi. σ(θ) − θ^{p} = p · unit in S.

The numbering extends i–iii of Definition 2.1. In the following we refer to conditions i–vi without explicitly mentioning Definitions 2.1 and 6.1.

Remark 6.2. If ii and iv hold, we have a (non-additive) map

τ : S → S, τ(x) = \frac{σ(x) − x^{p}}{p} ,

and vi says that τ(θ) is a unit. Condition v is satisfied if and only if the nilradical 𝒩 (R) has no p-torsion, for example if R is reduced, or flat over ℤ_{(p)}.

Proposition 6.3. To each κ-frame ℱ one can associate a u-homomorphism of frames κ : ℱ → 𝒲_{R} lying over id_{R} for a well-defined unit u of W (R). The homomorphism κ and the unit u are functorial in ℱ with respect to strict frame homomorphisms.

Proof. Conditions iv and ii imply that there is a well-defined ring homomorphism δ : S → W (S) with w_{n}δ = σ^{n}; see [Bou, IX.1, proposition 2]. We have fδ = δσ. Let κ be the composite ring homomorphism

κ : S ⟶^{δ} W (S) → W (R).

Then fκ = κσ and κ(I) ⊆ I_{R}. Clearly κ is functorial in ℱ. To define u we write 1 = ∑ y_{i}σ_{1}(x_{i}) in S with x_{i} ∈ I and y_{i} ∈ S. This is possible by iii. Recall that θ = ∑ y_{i}σ(x_{i}); see the proof of Lemma 2.2. Let

u = ∑ κ(y_{i})f_{1}κ(x_{i}).

Then pu = κ(θ) because pf_{1} = f and fκ = κσ. We claim that f_{1}κ = u · κσ_{1}. By condition v this is equivalent to the relation p · f_{1}κ = pu · κσ_{1}, which holds since pf_{1} = f and pu = κ(θ) and θσ_{1} = σ. It remains to show that u is a unit in W (R). Let pu = κ(θ) = (a_{0}, a_{1}, …) as a Witt vector. By Lemma 6.4 below, u is a unit if and only if a_{1} is a unit in R. In W_{2}(S) we have δ(θ) = (θ, τ(θ)) because (w_{0}, w_{1}) applied to both sides gives (θ, σ(θ)). Hence a_{1} is a unit by vi. We conclude that κ : ℱ → 𝒲_{R} is a u-homomorphism of frames. Finally, u is functorial in ℱ by its uniqueness, see Remark 2.8.

Lemma 6.4. Let R be a ring with p ∈ Rad(R) and let u ∈ W (R) be given. For an integer r ≥ 0 let p^{r}u = (a_{0}, a_{1}, a_{2}, …). The element u is a unit in W (R) if and only if a_{r} is a unit in R.

Proof. Assume first that r = 0. It suffices to show that an element ū of W_{n+1}(R) that maps to 1 in W_{n}(R) is a unit. If ū = 1 + v^{n}(x) with x ∈ R, then ū^{−1} = 1 + v^{n}(y) where y ∈ R is determined by the equation x + y + p^{n}xy = 0, which has a solution since p ∈ Rad(R). For general r, by the case r = 0 we may replace R by R/pR. Then we have p(b_{0}, b_{1}, …) = (0, b_{0}^{p}, b_{1}^{p}, …) in W (R), which reduces the assertion to the case r = 0.

Corollary 6.5. Let ℱ be a κ-frame with S = W (k)[[x_{1}, …, x_{r}]] for a perfect field k of odd characteristic p. Assume that σ extends the Frobenius automorphism of W (k) by σ(x_{i}) = x_{i}^{p}. Then u is a unit in 𝕎(R), and κ induces a u-homomorphism of frames κ : ℱ → 𝒟_{R}.

Proof. We claim that δ(S) lies in 𝕎(S). Indeed, δ(x_{i}) = [x_{i}] because w_{n} applied to both sides gives x_{i}^{p^{n}}. Thus for each multi-exponent e = (e_{1}, …, e_{r}) the element δ(x^{e}) = [x^{e}] lies in 𝕎(S). Let 𝔪_{S} be the maximal ideal of S. Since 𝕎(S) = \underleftarrow{lim} 𝕎(S/𝔪_{S}^{n}) and since for each n all but finitely many x^{e} lie in 𝔪_{S}^{n}, the claim follows. Hence the image of κ : S → W (R) is contained in 𝕎(R). By its construction the unit u lies in 𝕎(R); it is invertible in 𝕎(R) because the inclusion 𝕎(R) → W (R) is a local homomorphism of local rings.

7 The main frame

Let R be a complete regular local ring with perfect residue field k of characteristic p ≥ 3. We choose a ring homomorphism

𝔖 = W (k)[[x_{1}, …, x_{r}]] ⟶^{π} R

such that x_{1}, …, x_{r} map to a regular system of parameters of R. Since the graded ring of R is isomorphic to k[x_{1}, …, x_{r}], one can find a power series E_{0} ∈ 𝔖 with constant term zero such that π(E_{0}) = −p. Let E = E_{0} + p and I = E𝔖. Then R = 𝔖/I. Let σ : 𝔖 → 𝔖 be the ring endomorphism that extends the Frobenius automorphism of W (k) by σ(x_{i}) = x_{i}^{p}. We have a frame

ℬ = (𝔖, I, R, σ, σ_{1})

where σ_{1} is defined by σ_{1}(Ey) = σ(y) for y ∈ 𝔖.

Lemma 7.1. The frame ℬ is a κ-frame.

Proof. Let θ ∈ 𝔖 be the element given by Lemma 2.2. The only condition to be checked is that τ(θ) is a unit in 𝔖. Let E′_{0} = σ(E_{0}). Since σ_{1}(E) = 1, we have θ = σ(E) = E′_{0} + p. Hence

τ(θ) = \frac{σ(E′_{0}) + p − (E′_{0} + p)^{p}}{p} ≡ 1 + τ(E′_{0}) mod p.

Since the constant term of E_{0} is zero, the same is true for τ(E′_{0}), which implies that τ(θ) is a unit as required.

Thus Proposition 6.3 and Corollary 6.5 give a ring homomorphism κ from 𝔖 to 𝕎(R), which is a u-homomorphism of frames

κ : ℬ → 𝒟_{R}.

Here the unit u ∈ 𝕎(R) is determined by the identity pu = κσ(E).

Theorem 7.2. The frame homomorphism κ is crystalline (Definition 3.1).

To prove this we consider the following auxiliary frames. Let J ⊂ 𝔖 be the ideal J = (x_{1}, …, x_{r}), and let 𝔪_{R} be the maximal ideal of R. For each positive integer a let 𝔖_{a} = 𝔖/J^{a}𝔖 and R_{a} = R/𝔪_{R}^{a}. Then R_{a} = 𝔖_{a}/E𝔖_{a}, where E is not a zero divisor in 𝔖_{a}. There is a well-defined frame

ℬ_{a} = (𝔖_{a}, I_{a}, R_{a}, σ_{a}, σ_{1a})

such that the projection 𝔖 → 𝔖_{a} is a strict frame homomorphism ℬ → ℬ_{a}. Indeed, σ induces an endomorphism σ_{a} of 𝔖_{a} because σ(J) ⊆ J, and for y ∈ 𝔖_{a} one can define σ_{1a}(Ey) = σ_{a}(y).

For simplicity, the image of u in 𝕎(R_{a}) is denoted by u as well. The uhomomorphism κ induces a u-homomorphism

κ_{a} : ℬ_{a} → 𝒟_{R_{a}}

because for e ∈ ℕ^{r} we have κ(x^{e}) = [x^{e}], which maps to zero in 𝕎(R_{a}) when e_{1} + ⋯ + e_{r} ≥ a. We note that ℬ_{a} is again a κ-frame, so the existence of κ_{a} can also be viewed as a consequence of Proposition 6.3.

Theorem 7.3. For each positive integer a the homomorphism κ_{a} is crystalline.

To prepare for the proof, for each a ≥ 1 we will construct the following commutative diagram of frames, where vertical arrows are u-homomorphisms and where horizontal arrows are strict.

\xymatrix{ℬ_{a+1} \ar[r]^{ι} \ar[d]^{κ_{a+1}} & ℬ̃_{a+1} \ar[r]^{π} \ar[d]^{κ̃_{a+1}} & ℬ_{a} \ar[d]^{κ_{a}} \\ 𝒟_{R_{a+1}} \ar[r]^{ι′} & 𝒟_{R_{a+1}/R_{a}} \ar[r]^{π′} & 𝒟_{R_{a}}}  (7.1)

The upper line is a factorisation (4.1) of the projection ℬ_{a+1} → ℬ_{a}. This means that the frame ℬ̃_{a+1} necessarily takes the form

ℬ̃_{a+1} = (𝔖_{a+1}, Ĩ_{a+1}, R_{a}, σ_{a+1}, σ̃_{1(a+1)})

with Ĩ_{a+1} = E𝔖_{a+1} + J^{a}/J^{a+1}. We define σ̃_{1(a+1)} : Ĩ_{a+1} → 𝔖_{a+1} to be the extension of σ_{1(a+1)} : E𝔖_{a+1} → 𝔖_{a+1} by zero on J^{a}/J^{a+1}. This is well-defined because

E𝔖_{a+1} ∩ J^{a}/J^{a+1} = E(J^{a}/J^{a+1})

and because for x ∈ J^{a}/J^{a+1} we have σ_{1(a+1)}(Ex) = σ_{a+1}(x), which is zero since σ(J^{a}) ⊆ J^{ap}. The lower line of (7.1) is the factorisation (5.1) with respect to the trivial divided powers on the kernel 𝔪_{R}^{a}/𝔪_{R}^{a+1}. In order that the diagram commutes it is necessary and sufficient that κ̃_{a+1} is given by the ring homomorphism κ_{a+1}. It remains to show that κ̃_{a+1} is a u-homomorphism of frames. The only nontrivial condition is that f̃_{1}κ_{a+1} = u · κ_{a+1}σ̃_{1(a+1)} on Ĩ_{a+1}. This relation holds on E𝔖_{a+1} because κ_{a+1} is a u-homomorphism of frames. On J^{a}/J^{a+1} we have κ_{a+1}σ̃_{1(a+1)} = 0 by definition. For y ∈ 𝔖_{a+1} and e ∈ ℕ^{r} with e_{1} + ⋯ + e_{r} = a we compute

f̃_{1}(κ_{a+1}(x^{e}y)) = f̃_{1}([x^{e}]κ_{a+1}(y)) = f̃_{1}([x^{e}])f(κ_{a+1}(y)) = 0

because log([x^{e}]) = [x^{e}, 0, 0, …] and thus f̃_{1}([x^{e}]) = 0. As these x^{e} generate J^{a}, the required relation on Ĩ_{a+1} follows. Thus the diagram is constructed.

Proof of Theorem 7.3. We use induction on a. The homomorphism κ_{1} is crystalline because it is invertible. Assume that κ_{a} is crystalline for some positive integer a and consider the diagram (7.1). The homomorphism π′ is crystalline by Proposition 5.2, while π is crystalline by Theorem 3.2; the required filtration of J^{a}/J^{a+1} is trivial. Hence κ̃_{a+1} is crystalline. By Lemma 4.2, lifts of windows under ι or under ι′ are classified by lifts of the Hodge filtration. Since κ_{a+1} lies over the identity of R_{a+1} and since κ̃_{a+1} lies over the identity of R_{a}, it follows that κ_{a+1} is crystalline too.

Proof of Theorem 7.2. The frame homomorphism κ : ℬ → 𝒟_{R} is the projective limit of the frame homomorphisms κ_{a} : ℬ_{a} → 𝒟_{R_{a}}. By Lemma 2.12, ℬ-windows are equivalent to compatible systems of ℬ_{a}-windows for a ≥ 1, and 𝒟_{R}-windows are equivalent to compatible systems of 𝒟_{R_{a}}-windows for a ≥ 1. Thus Theorem 7.2 follows from Theorem 7.3.

8 Classification of group schemes

The following consequences of Theorem 7.2 are analogous to [VZ1]. Recall that we assume p ≥ 3. Let ℬ = (𝔖, I, R, σ, σ_{1}) be the frame defined in section 7.

Definition 8.1. A Breuil window relative to 𝔖 → R is a pair (Q, ϕ) where Q is a free 𝔖-module of finite rank and where ϕ : Q → Q^{(σ)} is an 𝔖-linear map with cokernel annihilated by E.

Lemma 8.2. Breuil windows relative to 𝔖 → R are equivalent to ℬ-windows in the sense of Definition 2.3.

Proof. This is similar to [VZ1, Lemma 1]. For a ℬ-window (P, Q, F, F_{1}) the module Q is free over 𝔖 because I = E𝔖 is free. Hence F_{1}^{♯} : Q^{(σ)} → P is bijective, and we can define a Breuil window (Q, ϕ) where ϕ is the inclusion Q → P composed with the inverse of F_{1}^{♯}. Conversely, if (Q, ϕ) is a Breuil window, Coker(ϕ) is a free R-module. Indeed, ϕ is injective because it becomes bijective over 𝔖[E^{−1}], so Coker(ϕ) has projective dimension at most one over 𝔖, which implies that it is free over R by using depth. Thus one can define a ℬ-window as follows: P = Q^{(σ)}, the inclusion Q → P is ϕ, F_{1} : Q → Q^{(σ)} is given by x ↦ 1 ⊗ x, and F (x) = F_{1}(Ex). The two constructions are mutually inverse.

By [Z2], p-divisible groups over R are equivalent to Dieudonné displays over R. Together with Theorem 7.2 and Lemma 8.2 this implies:

Corollary 8.3. The category of p-divisible groups over R is equivalent to the category of Breuil windows relative to 𝔖 → R.

Let us use the following abbreviation: An admissible torsion 𝔖-module is a finitely generated 𝔖-module annihilated by a power of p and of projective dimension at most one.

Definition 8.4. A Breuil module relative to 𝔖 → R is a triple (M, φ, ψ) where M is an admissible torsion 𝔖-module together with 𝔖-linear maps φ : M → M^{(σ)} and ψ : M^{(σ)} → M such that φψ = E and ψφ = E.

When R has characteristic zero, each of the maps φ and ψ determines the other one; see Lemma 8.6 below.

Theorem 8.5. The category of (commutative) finite flat group schemes over R annihilated by a power of p is equivalent to the category of Breuil modules relative to 𝔖 → R.

This follows from Corollary 8.3 by the arguments of [K1] or [VZ1]. For completeness we give a detailed proof here.

Proof of Theorem 8.5. In this proof, all finite flat group schemes are of p-power order over R, and all Breuil modules or windows are relative to 𝔖 → R. A homomorphism g : (Q_{0}, ϕ_{0}) → (Q_{1}, ϕ_{1}) of Breuil windows is called an isogeny if it becomes invertible over 𝔖[1/p]. Then g is injective, and its cokernel is naturally a Breuil module; the required ψ is induced by the 𝔖-linear map Eϕ_{1}^{−1} : Q_{1}^{(σ)} → Q_{1}. A homomorphism γ : G_{0} → G_{1} of p-divisible groups is called an isogeny if it becomes invertible in Hom(G_{0}, G_{1}) ⊗ ℚ. Then γ is a surjection of fppf sheaves, and its kernel is a finite flat group scheme. We denote isogenies by X_{∗} = [X_{0} → X_{1}]. A homomorphism of isogenies q : X_{∗} → Y_{∗} is called a quasi-isomorphism if its cone is a short exact sequence. In the case of p-divisible groups this means that q induces an isomorphism of finite flat group schemes on the kernels; in the case of Breuil windows this means that q induces an isomorphism of Breuil modules on the cokernels. The equivalence between p-divisible groups and Breuil windows preserves isogenies and short exact sequences, and thus also quasi-isomorphisms of isogenies. We note the following two facts.

(a) Each finite flat group scheme over R of p-power order is the kernel of an isogeny of p-divisible groups over R. See [BBM, Théorème 3.1.1].

(b) Each Breuil module is the cokernel of an isogeny of Breuil windows. This is analogous to [VZ1, Proposition 2]; a proof is also given below.

Let us define an additive functor H ↦ M(H) from finite flat group schemes to Breuil modules. We write each H as the kernel of an isogeny of p-divisible groups G_{0} → G_{1} and define M(H) as the cokernel of the associated isogeny of Breuil windows. Assume that h : H → H′ is a homomorphism of finite flat group schemes, and H′ is written as the kernel of an isogeny of p-divisible groups G′_{0} → G′_{1}. We embed H into G′′_{0} = G_{0} ⊕ G′_{0} by (1, h) and define G′′_{1} = G′′_{0}/H. The coordinate projections G_{0} ← G′′_{0} → G′_{0} induce homomorphisms of isogenies G_{∗} ← G′′_{∗} → G′_{∗} such that the first map is a quasi-isomorphism, and the composition induces h on the kernels. Let Q_{∗} ← Q′′_{∗} → Q′_{∗} be the associated homomorphisms of isogenies of Breuil windows. The first map is a quasiisomorphism, and the composition induces a homomorphism M(h) : M(H) → M(H′) on the cokernels.

One has to show that the construction is independent of the choice and defines an additive functor. This is an easy verification based on the following observation: If a homomorphism of isogenies of p-divisible groups q : G_{∗} → G′_{∗} induces zero on the kernels, then q is null-homotopic. The construction of an additive functor M ↦ H(M) from Breuil modules to finite flat group schemes is analogous. Each M is written as the cokernel of an isogeny of Breuil windows Q_{0} → Q_{1}, and H(M) is defined as the kernel of the associated isogeny of p-divisible groups. If m : M → M′ is a homomorphism of Breuil modules and if M′ is written as the cokernel of an isogeny of Breuil windows Q′_{0} → Q′_{1}, let Q′′_{0} be the kernel of the surjection Q′′_{1} = Q_{1}⊕Q′_{1} → M′ given by (m, 1). The coordinate inclusions Q_{1} → Q′′_{1} ← Q′_{1} induce homomorphisms of isogenies Q_{∗} → Q′′_{∗} ← Q′_{∗}, where the second map is a quasi-isomorphism. The associated homomorphisms of isogenies of p-divisible groups induce a homomorphism of finite flat group schemes H(m) : H(M) → H(M′) on the kernels. Again, it is easy to verify that this construction is independent of the choice and defines an additive functor, using that a homomorphism of isogenies of Breuil windows is null-homotopic if and only if it induces zero on the cokernels. Clearly the two functors are mutually inverse. Finally, let us prove (b). If (M, φ, ψ) is a Breuil module, one can find free 𝔖-modules P and Q together with surjective 𝔖-linear maps ξ : Q → M and ξ′ : P → M^{(σ)} and 𝔖-linear maps φ̃ : Q → P and ψ̃ : P → Q which lift φ and ψ such that φ̃ψ̃ = E and ψ̃φ̃ = E. Next one can choose an isomorphism α : P ≅ Q^{(σ)} compatible with the projections ξ′ and ξ^{(σ)} to M^{(σ)}. Let ϕ = αφ̃. Then (Q, ϕ) is a Breuil window, and (M, φ, ψ) is the cokernel of the isogeny of Breuil windows (Ker ξ, ϕ′) → (Q, ϕ), where ϕ′ is the restriction of ϕ.

Lemma 8.6. If R has characteristic zero, the category of Breuil modules relative to 𝔖 → R is equivalent to the category of pairs (M, φ) where M is an admissible torsion 𝔖-module and where φ : M → M^{(σ)} is an 𝔖-linear map with cokernel annihilated by E.

Proof. Cf. [VZ1, Proposition 2]. For a non-zero admissible torsion 𝔖-module M the set of zero divisors on M is equal to 𝔭 = p𝔖 because every associated prime of M has height one and contains p. In particular, M → M_{𝔭} is injective. The hypothesis of the lemma means that E ∉ 𝔭. For a given pair (M, φ) as in the lemma this implies that φ_{𝔭} : M_{𝔭} → M_{𝔭}^{(σ)} is surjective, thus bijective because both sides have the same finite length. It follows that φ is injective, and (M, φ) is extended uniquely to a Breuil module by ψ(x) = φ^{−1}(Ex).

Duality

The dual of a Breuil window (Q, ϕ) is the Breuil window (Q, ϕ)^{t} = (Q^{∨}, ψ^{∨}) where Q^{∨} = Hom_{𝔖}(Q, 𝔖) and where ψ : Q^{(σ)} → Q is the unique 𝔖-linear map with ψϕ = E. Here we identify (Q^{(σ)})^{∨} and (Q^{∨})^{(σ)}. For a p-divisible group G over R let G^{∨} be the Serre dual of G, and let 𝕄(G) be the Breuil window associated to G by the equivalence of Corollary 8.3.

Proposition 8.7. There is a functorial isomorphism λ_{G} : 𝕄(G^{∨}) ≅ 𝕄(G)^{t}.

Proof. The equivalence between p-divisible groups over R and Dieudonné displays over R is compatible with duality by [L2, Theorem 3.4]. It is easy to see that the equivalence of Lemma 8.2 preserves duality, so it remains to show that the functor κ_{∗} preserves duality as well. By Lemma 2.14 it suffices to find a unit c ∈ 𝕎(R) with c^{−1}f(c) = u. Since E has constant term p, u maps to 1 in W (k) and thus lies in 1 + Ŵ (𝔪_{R}). Hence we can define c^{−1} by the infinite product uf(u)f^{2}(u) ⋯, which converges in 𝕎(R) = \underleftarrow{lim} 𝕎(R/𝔪^{n}) in the sense that for each n, all but finitely many factors map to 1 in 𝕎(R/𝔪^{n}).

The dual of a Breuil module 𝕄 = (M, φ, ψ) is defined as the Breuil module 𝕄^{t} = (M^{⋆}, ψ^{⋆}, φ^{⋆}) where M^{⋆} = Ext_{𝔖}^{1}(M, 𝔖). Here we identify (M^{(σ)})^{⋆} and (M^{⋆})^{(σ)} using that ( )^{(σ)} preserves projective resolutions as σ is flat. For a finite flat group scheme H over R of p-power order let H^{∨} be the Cartier dual of H and let 𝕄(H) be the Breuil module associated to H by the equivalence of Theorem 8.5.

Proposition 8.8. There is a functorial isomorphism λ_{H} : 𝕄(H^{∨}) ≅ 𝕄(H)^{t}.

Proof. Choose an isogeny of p-divisible groups G_{0} → G_{1} with kernel H. Then 𝕄(H) is the cokernel of 𝕄(G_{0}) → 𝕄(G_{1}), which implies that 𝕄(H)^{t} is the cokernel of 𝕄(G_{1})^{t} → 𝕄(G_{0})^{t}. On the other hand, H^{∨} is the kernel of G_{1}^{∨} → G_{0}^{∨}, so 𝕄(H^{∨}) is the cokernel of 𝕄(G_{1}^{∨}) → 𝕄(G_{0}^{∨}). The isomorphisms λ_{G_{i}} of Proposition 8.7 give an isomorphism λ_{H} : 𝕄(H^{∨}) ≅ 𝕄(H)^{t}. One easily checks that λ_{H} is independent of the choice of G_{∗} and functorial in H.

9 Other lifts of Frobenius

One may ask how much freedom we have in the choice of σ for the frame ℬ. Let R = 𝔖/E𝔖 be as in section 7; in particular we assume that p ≥ 3. Let J = (x_{1}, …, x_{r}). To begin with, let σ : 𝔖 → 𝔖 be an arbitrary ring endomorphism such that σ(J) ⊂ J and σ(a) ≡ a^{p} modulo p𝔖 for a ∈ 𝔖. We consider the frame

ℬ = (𝔖, I, R, σ, σ_{1})

with σ_{1}(Ey) = σ(y). Again this is a κ-frame; the proof of Lemma 7.1 uses only that σ preserves J. Thus Proposition 6.3 gives a homomorphism of frames

κ : ℬ → 𝒲_{R}.

By the assumptions on σ we have σ(J) ⊆ J^{p} + pJ, which implies that the endomorphism σ : J/J^{2} → J/J^{2} is divisible by p.

Proposition 9.1. The image of κ : 𝔖 → W (R) lies in 𝕎(R) if and only if the endomorphism σ/p of J/J^{2} is nilpotent modulo p.

We have a non-additive map τ : J → J given by τ(x) = (σ(x) − x^{p})/p. Let 𝔪 be the maximal ideal of 𝔖. We write gr_{n}(J) = 𝔪^{n}J/𝔪^{n+1}J.

Lemma 9.2. For n ≥ 0 the map τ preserves 𝔪^{n}J and induces a σ-linear endomorphism of k-modules gr_{n}(τ) : gr_{n}(J) → gr_{n}(J). We have gr_{0}(τ) = σ/p as an endomorphism of gr_{0}(J) = J/(J^{2} + pJ). There is a commutative diagram of the following type with πi = id

\xymatrix{gr_{n}(J) \ar[r]^{gr_{n}(τ)} \ar[d]_{π} & gr_{n}(J) \\ gr_{0}(J) \ar[r]^{gr_{0}(τ)} & gr_{0}(J). \ar[u]_{i}}

Proof. Let J′ = p^{−1}𝔪J as an 𝔖-submodule of J ⊗ ℚ. Then J ⊂ J′, and gr_{n}(J) is an 𝔖-submodule of gr_{n}(J′) = 𝔪^{n}J′/𝔪^{n+1}J′. The composition J ⟶^{τ} J ⊂ J′ can be written as τ = σ/p − φ/p, where φ(x) = x^{p}. One checks that φ/p : 𝔪^{n}J → 𝔪^{n+1}J′ (which requires p ≥ 3 when n = 0) and that σ/p : 𝔪^{n}J → 𝔪^{n}J′. Hence σ/p and τ induce the same map 𝔪^{n}J → gr_{n}(J′). This map is σ-linear and zero on 𝔪^{n+1}J because this holds for σ/p, and its image lies in gr_{n}(J) because this is true for τ. We define i : gr_{0}(J) → gr_{n}(J) by x ↦ p^{n}x. For n ≥ 1 let K_{n} be the image of 𝔪^{n−1}J^{2} → gr_{n}(J). Then i maps gr_{0}(J) bijectively onto gr_{n}(J)/K_{n}, so there is a unique homomorphism π : gr_{n}(J) → gr_{0}(J) with kernel K_{n} such that πi = id. Clearly i commutes with gr(τ). Thus, in order that the diagram commutes, it suffices that gr_{n}(τ) vanishes on K_{n}. We have σ(J) ⊆ 𝔪J, which implies that (σ/p)(𝔪^{n−1}J^{2}) ⊆ 𝔪^{n+1}J′, and the assertion follows.

Proof of Proposition 9.1. Recall that κ = πδ, where δ : 𝔖 → W (𝔖) is defined by w_{n}δ = σ^{n} for n ≥ 0, and where π : W (𝔖) → W (R) is the obvious projection. For x ∈ J and n ≥ 1 let

τ_{n}(x) = (σ(x)^{p^{n−1}} − x^{p^{n}})/p^{n},

thus τ_{1} = τ. It is easy to see that

τ_{n+1}(x) ∈ J · τ_{n}(x),

in particular we have τ_{n} : J → J^{n}. If δ(x) = (y_{0}, y_{1}, …), the coefficients y_{n} are determined by y_{0} = x and w_{n}(y) = σw_{n−1}(y) for n ≥ 1, which translates into the equations

y_{n} = τ_{n}(y_{0}) + τ_{n−1}(y_{1}) + ⋯ + τ_{1}(y_{n−1}).

Assume now that σ/p is nilpotent on J/J^{2} modulo p. By Lemma 9.2 this implies that gr_{n}(τ) is nilpotent for every n ≥ 0. We will show that for x ∈ J the element δ(x) lies is 𝕎(𝔖), which means that the above sequence (y_{n}) converges to zero. Assume that for some N ≥ 0 we have y_{n} ∈ 𝔪^{N}J for all but finitely many n. The last two displayed equations give that y_{n} − τ(y_{n−1}) ∈ 𝔪^{N+1}J for all but finitely many n. As gr_{N}(τ) is nilpotent it follows that y_{n} ∈ 𝔪^{N+1}J for all but finitely many n. Thus δ(x) ∈ 𝕎(𝔖) and in particular κ(x) ∈ 𝕎(R). Conversely, if σ/p is not nilpotent on J/J^{2} modulo p, then gr_{0}(τ) is not nilpotent by Lemma 9.2, so there is an x ∈ J such that τ^{n}(x) ∉ 𝔪J for all n ≥ 0. For δ(x) = (y_{0}, y_{1}, …) we have y_{n} ≡ τ^{n}x modulo 𝔪J. The projection 𝔖 → R induces an isomorphism J/𝔪J ≅ 𝔪_{R}/𝔪_{R}^{2}. It follows that κ(x) lies in W (𝔪_{R}) but not in Ŵ (𝔪_{R}), thus κ(x) ∉ 𝕎(R).

Now we assume that σ/p is nilpotent on J/J^{2} modulo p. Then we have a homomorphism of frames

κ : ℬ → 𝒟_{R}.

As earlier let ℬ_{a} = (𝔖_{a}, I_{a}, R_{a}, σ_{a}, σ_{1a}) with 𝔖_{a} = 𝔖/J^{a} and R_{a} = R/𝔪_{R}^{a}. The proof of Lemma 7.1 shows that ℬ_{a} is a κ-frame. Since 𝕎(R_{a}) is the image of 𝕎(R) in W (R_{a}), we get a homomorphism of frames compatible with κ:

κ_{a} : ℬ_{a} → 𝒟_{R_{a}}.

Theorem 9.3. The homomorphisms κ and κ_{a} are crystalline.

Proof. The proof is similar to that of Theorems 7.2 and 7.3. First we repeat the construction of the diagram (7.1). The restriction of σ_{1(a+1)} to E(J^{a}/J^{a+1}) = p(J^{a}/J^{a+1}) is given by σ_{1} = σ/p = τ, which need not be zero in general, but still σ_{1} extends uniquely to J^{a}/J^{a+1} by the formula σ_{1} = σ/p. In order that κ̃_{a+1} is a u-homomorphism of frames we need that f̃_{1}κ_{a+1} = u · κ_{a+1}σ̃_{1(a+1)} on J^{a}/J^{a+1}. Here u acts on J^{a}/J^{a+1} as the identity. By the proof of Proposition 9.1, for x ∈ J^{a}/J^{a+1} we have in W (J^{a}/J^{a+1})

δ(x) = (x, τ(x), τ^{2}(x), …).

Since σ̃_{1(a+1)}(x) = τ(x), the required relation follows easily. To complete the proof we have to show that π : ℬ̃_{a+1} → ℬ_{a} is crystalline. Now σ/p is nilpotent modulo p on J^{n}/J^{n+1} for n ≥ 1. Indeed, for n = 1 this is our assumption, and for n ≥ 2 the endomorphism σ/p of J^{n}/J^{n+1} is divisible by p^{n−1} since σ(J) ⊆ pJ + J^{p}. In order to apply Theorem 3.2 we need another sequence of auxiliary frames: For c ∈ ℕ let 𝔖_{a+1,c} = 𝔖_{a+1}/p^{c}J^{a}𝔖_{a+1} and let ℬ̃_{a+1,c} = (𝔖_{a+1,c}, I_{a+1,c}, R_{a}, …) be the obvious quotient frame of ℬ̃_{a+1} . Then ℬ_{a} is isomorphic to ℬ̃_{a+1,0} , and ℬ̃_{a+1} is the projective limit of ℬ̃_{a+1,c} for c → ∞. Theorem 3.2 shows that each projection ℬ̃_{a+1,c+1} → ℬ̃_{a+1,c} is crystalline, which implies that π is crystalline by Lemma 2.12.

If σ/p is nilpotent on J/J^{2} modulo p, then Corollary 8.3, Theorem 8.5, and the duality Propositions 8.7 and 8.8 follow as before.

10 Nilpotent windows

All results in this article have a nilpotent counterpart where only connected p-divisible groups and nilpotent windows are considered; in this case k need not be perfect and p need not be odd. The necessary modifications are standard, but for completeness we work out the details.

10.1 Nilpotence condition

Let ℱ = (S, I, R, σ, σ_{1}) be a frame. For an ℱ -window 𝒫 = (P, Q, F, F_{1}) there is a unique S-linear map

V^{♯} : P → P^{(σ)}

with V^{♯}(F_{1}(x)) = 1 ⊗ x for x ∈ Q. In terms of a normal representation Ψ : L ⊕ T → P of 𝒫 we have V^{♯} = (1 ⊕ θ)(Ψ^{♯})^{−1} for θ as in Lemma 2.2. For simplicity, the composition

P ⟶^{V^{♯}} P^{(σ)} ⟶^{(V^{♯})^{(σ)}} P^{(σ^{2})} → ⋯ → P^{(σ^{n})}

is denoted (V^{♯})^{n}. The nilpotence condition depends on the choice of an ideal J ⊂ S such that σ(J) + I + θS ⊆ J, which we call an ideal of definition for ℱ.

Definition 10.1. Let J ⊂ S be an ideal of definition for ℱ. An ℱ -window 𝒫 is called nilpotent (with respect to J) if (V^{♯})^{n} ≡ 0 modulo J for sufficiently large n.

Remark 10.2. For an ℱ -window 𝒫 we consider the composition

λ : L ⊆ L ⊕ T ⟶^{(Ψ^{♯})^{−1}} L^{(σ)} ⊕ T^{(σ)} → L^{(σ)}.

Then 𝒫 is nilpotent if and only if λ is nilpotent modulo J.

10.2 Nil-crystalline homomorphisms

If α : ℱ → ℱ′ is a homomorphism of frames and J ⊂ S and J′ ⊂ S′ are ideals of definition with α(J) ⊆ J′, the functor α_{∗} preserves nilpotent windows. We call α nil-crystalline if it induces an equivalence between nilpotent ℱ -windows and nilpotent ℱ′-windows. The following variant of Theorem 3.2 formalises [Z1, Theorem 44].

Theorem 10.3. Let α : ℱ → ℱ′ be a homomorphism of frames that induces an isomorphism R ≅ R′ and a surjection S → S′ with kernel 𝔞 ⊂ S. We assume that there is a finite filtration 𝔞 = 𝔞_{0} ⊇ ⋯ ⊇ 𝔞_{n} = 0 such that σ(𝔞_{i}) ⊆ 𝔞_{i+1} and σ_{1}(𝔞_{i}) ⊆ 𝔞_{i}. We assume that finitely generated projective S′-modules lift to projective S-modules. If J ⊂ S is an ideal of definition for ℱ such that J^{n}𝔞 = 0 for large n, then α is nil-crystalline with respect to J ⊂ S and J′ = J/𝔞 ⊂ S′.

Proof. The assumptions imply that 𝔞 ⊆ I ⊆ J, in particular J′ is well-defined. An ℱ -window 𝒫 is nilpotent if and only if α_{∗}𝒫 is nilpotent. Using this, the proof of Theorem 3.2 applies with the following modification in its final paragraph. We claim that the endomorphism U of ℋ is nilpotent, which again implies that 1−U is bijective. Since 𝒫 is nilpotent, λ is nilpotent modulo J, so λ is nilpotent modulo J^{n} for each n ≥ 1 as J is stable under σ. Since J^{n}𝔞 = 0 by assumption, the claim follows from the definition of U.

10.3 Nilpotent displays

Let R be a ring which is complete and separated in the 𝔠-adic topology for an ideal 𝔠 ⊂ R containing p. We consider the Witt frame

𝒲_{R} = (W (R), I_{R}, R, f, f_{1}).

Here I_{R} ⊆ Rad R as required since W (R) = \underleftarrow{lim} W_{n}(R/𝔠^{n}) and the successive kernels in this projective system are nilpotent. The inverse image of 𝔠 is an ideal of definition J ⊂ W (R). Nilpotent windows over 𝒲_{R} with respect to J are displays over R which are nilpotent over R/𝔠. By [Z1] and [L1] these are equivalent to p-divisible groups over R which are infinitesimal over R/𝔠. (Here one uses that displays and p-divisible groups over R are equivalent to compatible systems of the same objects over R/𝔠^{n} for n ≥ 1; cf. Lemma 2.12 above and [M1, Lemma 4.16].) Assume that R′ = R/𝔟 for a closed ideal 𝔟 ⊆ 𝔠 equipped with (not necessarily nilpotent) divided powers. One can define a factorisation

𝒲_{R} ⟶^{α_{1}} 𝒲_{R/R′} = (W (R), I_{R/R′}, R′, f, f̃_{1}) ⟶^{α_{2}} 𝒲_{R′}

of the projection of frames 𝒲_{R} → 𝒲_{R′} as follows. Necessarily I_{R/R′} = I_{R} + W (𝔟). The divided Witt polynomials define an isomorphism

log : W (𝔟) ≅ 𝔟^{∞},

and f̃_{1} : I_{R/R′} → W (R) extends f_{1} such that f̃_{1}([b_{0}, b_{1}, …]) = [b_{1}, b_{2}, …] in logarithmic coordinates on W (𝔟). Let J′ ⊂ W (R′) be the image of J. This is an ideal of definition for 𝒲_{R′}, and J is an ideal of definition for 𝒲_{R/R′}. We assume that the 𝔠-adic topology of R can be defined by a sequence of ideals R ⊃ I_{1} ⊃ I_{2} ⋯ such that 𝔟 ∩ I_{n} is stable under the divided powers of 𝔟 for each n. This is automatic when 𝔠 is nilpotent or when R is noetherian; see the proof of Proposition 5.2.

Proposition 10.4. The homomorphism α_{2} is nil-crystalline with respect to the ideals of definition J for 𝒲_{R/R′} and J′ for 𝒲_{R′}.

This is essentially [Z1, Theorem 44].

Proof. By a limit argument the assertion is reduced to the case where 𝔠 ⊂ R is a nilpotent ideal; see Lemma 2.12. Then Theorem 10.3 applies: The required filtration of 𝔞 = W (𝔟) is 𝔞_{i} = p^{i}𝔞. The condition J^{n}𝔞 = 0 for large n is satisfied because J^{n} ⊆ I_{R} for some n and I_{R}^{n+1} ⊆ p^{n}W (R) for all n, and W (𝔟) ≅ 𝔟^{∞} is annihilated by some power of p.

10.4 The main frame

Let now R be a complete regular local ring with arbitrary residue field k of characteristic p. Let C be a complete discrete valuation ring with maximal ideal pC and residue field k. We choose a surjective ring homomorphism

𝔖 = C[[x_{1}, …, x_{r}]] → R

that lifts the identity of k such that x_{1}, …, x_{r} map to a regular system of parameters for R. There is a power series E ∈ 𝔖 with constant term p such that R = 𝔖/E𝔖. Let σ : 𝔖 → 𝔖 be a ring endomorphism which induces the Frobenius on 𝔖/p𝔖 and preserves the ideal (x_{1}, …, x_{r}). Such σ exist because C has a Frobenius lift; see [Gr, Chap. 0, Théorème 19.8.6]. We consider the frame

ℬ = (𝔖, I, R, σ, σ_{1})

where σ_{1}(Ey) = σ(y). Here θ = σ(E). The proof of Lemma 7.1 shows that ℬ is again a κ-frame, so we have a u-homomorphism of frames

κ : ℬ → 𝒲_{R}.

Let 𝔪 ⊂ 𝔖 and 𝔫 ⊂ W (R) be the maximal ideals.

Theorem 10.5. The homomorphism κ is nil-crystalline with respect to the ideals of definition 𝔪 of ℬ and 𝔫 of 𝒲_{R}.

Proof. The proof of Theorem 9.3 applies with the following modification: The initial case a = 1 is not trivial because C is not isomorphic to W (k) if k is not perfect, but one can apply [Z3, Theorem 1.6]. In the diagram (7.1) the frame homomorphisms π′ and π are only nil-crystalline in general; whether π is crystalline depends on the choice of σ.

10.5 Connected group schemes

One defines Breuil windows relative to 𝔖 → R and Breuil modules relative to 𝔖 → R as before. A Breuil window (Q, ϕ) or a Breuil module (M, φ, ψ) is called nilpotent if ϕ or φ is nilpotent modulo the maximal ideal of 𝔖. The proof of Lemma 8.2 shows that nilpotent Breuil windows are equivalent to nilpotent ℬ-windows. Hence Theorem 10.5 implies:

Corollary 10.6. Connected p-divisible groups over R are equivalent to nilpotent Breuil windows relative to 𝔖 → R.

Similarly we have:

Theorem 10.7. Connected finite flat group schemes over R of p-power order are equivalent to nilpotent Breuil modules relative to 𝔖 → R.

This is proved like Theorem 8.5, using two additional remarks:

Lemma 10.8. Every connected finite flat group scheme H over R is the kernel of an isogeny of connected p-divisible groups.

Proof. We know that H is the kernel of an isogeny of p-divisible groups G → G′. There is a functorial exact sequence 0 → G_{0} → G → G_{1} → 0 of p-divisible groups where G_{0} is connected and G_{1} is etale. Since Hom(H, G_{1}) is zero, H is the kernel of the isogeny G_{0} → G′_{0}.

Lemma 10.9. Every nilpotent Breuil module (M, φ, ψ) relative to 𝔖 → R is the cokernel of an isogeny of nilpotent Breuil windows.

Proof. See also [K2, Section 1.3]. As in the proof of Theorem 8.5 we see that (M, φ, ψ) is the cokernel of an isogeny of Breuil windows (Q, ϕ) → (Q′, ϕ′). There is a functorial exact sequence 0 → Q_{0} → Q → Q_{1} → 0 of Breuil windows where Q_{0} is nilpotent and where Q_{1} is etale in the sense that ϕ : Q_{1} → Q_{1}^{(σ)} is bijective. Indeed, by [Z2, Lemma 10] it suffices to construct the sequence over k. Let ϕ_{k} : Q ⊗_{𝔖} k → Q^{(σ)} ⊗_{𝔖} k be the special fibre of ϕ. Then Q_{0} ⊗_{𝔖} k is the kernel of the obvious iterate (ϕ_{k})^{n} : Q ⊗_{𝔖} k → Q^{(σ^{n})} ⊗_{𝔖} k for large n. We claim that the free 𝔖-modules Q_{1} and Q′_{1} have the same rank. Let us identify C with 𝔖/(x_{1}, …, x_{r}). Since Q → Q′ becomes bijective over 𝔖[1/p], the homomorphism Q ⊗_{𝔖} C → Q′ ⊗_{𝔖} C becomes bijective over C[1/p]. Hence the etale parts (Q ⊗_{𝔖} C)_{1} and (Q′ ⊗_{𝔖} C)_{1} have the same rank. The claim follows since (Q ⊗_{𝔖} C)_{1} = Q_{1} ⊗_{𝔖} C and similarly for Q′. Let us consider M̄ = Q′_{1}/Q_{1}. Here ϕ′ induces a homomorphism φ̄ : M̄ → M̄^{(σ)}, which is surjective as Q′_{1} is etale. The natural surjection π : M → M̄ satisfies π^{(σ)}φ = φ̄π. Since φ_{k} is nilpotent it follows that φ̄_{k} is nilpotent, thus M̄ = 0 by Nakayama’s lemma. Hence Q_{1} → Q′_{1} is bijective because both sides are free of the same rank, and consequently M = Q′_{0}/Q_{0} as desired.

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Eike Lau

Fakultät für Mathematik Universität Bielefeld D-33501 Bielefeld lau@math.uni-bielefeld.de 570

Documenta Mathematica 15 (2010)