arXiv:0808.2792v3 [math.NT] 3 Sep 2009 Breuil’s classification of p-divisible groups over regular local rings of arbitrary dimension Adrian Vasiu and Thomas Zink November 7, 2018 To appear in Advanced Studies in Pure Mathematics, Algebraic and Arithmetic Structures of Moduli Spaces (Sapporo 2007) Abstract. Let k be a perfect field of characteristic p ≥ 3. We classify pdivisible groups over regular local rings of the form W (k)[[t1 , . . . , tr , u]]/(ue + pbe−1 ue−1 + . . . + pb1 u + pb0 ), where b0 , . . . , be−1 ∈ W (k)[[t1 , . . . , tr ]] and b0 is an invertible element. This classification was in the case r = 0 conjectured by Breuil and proved by Kisin. MSC 2000: 11G10, 11G18, 14F30, 14G35, 14K10, and 14L05. 1 Introduction Let p ∈ N be an odd prime. Let k be a perfect field of characteristic p. Let W (k) be the ring of Witt vectors with coefficients in k. Let r ∈ N ∪ {0}. We consider the ring of formal power series S := W (k)[[t1 , . . . , tr , u]]. We extend the Frobenius endomorphism σ of W (k) to S by the rules σ(ti ) = tpi and σ(u) = up . If M is a S-module we define M (σ) := S ⊗σ,S M. 1 (1) Let e ∈ N. Let E(u) = ue + ae−1 ue−1 + · · · + a1 u + a0 be a polynomial with coefficients in W (k)[[t1 , . . . , tr ]] such that p divides ai for all i ∈ {0, . . . , e − 1} and moreover a0 /p is a unit in W (k)[[t1 , . . . , tr ]]. We define R := S/ES; it is a regular local ring of dimension r+1 with parameter system t1 , . . . , tr , u. The following notion was introduced in [B] for the case r = 0. Definition 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 a S-linear map whose cokernel is annihilated by E. As φ[ E1 ] : Q[ E1 ] → Q(σ) [ E1 ] is a S[ E1 ]-linear epimorphism between free S[ E1 ]modules of the same finite rank, it is an injection. This implies that φ itself is an injection. We check that C := Coker(φ) is a free R-module. For this, we can assume that C 6= 0 and thus that the S-module C has projective dimension 1. Since S is a regular ring, we have depth C = dim S−1 = dim R. As C has the same depth viewed as an R-module or as a S-module, we conclude that C is a free R-module. The goal of the paper is to prove the following result whose validity is suggested by previous works of Breuil and Kisin (see [B] and [K]). Theorem 1 The category of p-divisible groups over R is equivalent to the category of Breuil windows relative to S → R. This theorem was proved by Kisin [K] in the case r = 0. We prove the generalization by a new method which is based on the theory of Dieudonné displays [Z2]. This theory works only for a perfect field k of characteristic p ≥ 3. Our method yields results for more general fields if we restrict ourselves to formal p-divisible groups over R, i.e. p-divisible groups over R whose special fibers over k are connected. Then we can substitute Dieudonné displays by nilpotent displays. To state the result we have to define nilpotent Breuil windows relative to S → R. Let (Q, φ) be a Breuil window relative to S → R. Then we define a σ-linear map F : Q(σ) → Q(σ) by F (x) = id ⊗φ−1 (Ex) 2 for x ∈ Q(σ) . If we tensor (Q(σ) , F ) by the W (k)-epimorphism S → W (k) which maps the variables ti and u to 0, we obtain a Dieudonné module over W (k). In the theorem above this is the covariant Dieudonné module of the special fibre of the p-divisible group over R which corresponds to (Q, σ). We say that (Q, φ) is a nilpotent Breuil window relative to S → R if (Q(σ) , F ) ⊗S W (k) is the covariant Dieudonné module of a connected pdivisible group over k. Then the arguments of this paper show that for p = 2 the category of nilpotent Breuil windows is equivalent to the category of formal p-divisible groups over R. Exactly the same statement holds for non perfect fields of arbitrary characteristics p > 0 if we replace the ring W (k) by a Cohen ring Ck and S by Ck [[t1 , . . . tr , u]]. One can view Theorem 1 as a ramified analogue of Faltings deformation theory over rings of the form W (k)[[t1 , . . . , tr ]] (see [F, Thm. 10]). The importance of Theorem 1 stems from its potential applications to modular and moduli properties and aspects of Shimura varieties of Hodge type (see [VZ] for applications with r = 1). As in the case r = 0 (see [K]), Theorem 1 implies a classification of finite flat, commutative group schemes of p power order over R. Definition 2 A Breuil module relative to S → R is a pair (M, ϕ), where M is a S-module of projective dimension at most one and annihilated by a power of p and where ϕ : M → M (σ) is a S-linear map whose cokernel is annihilated by E. Theorem 2 The category of finite flat, commutative group schemes of p power order over R is equivalent to the category of Breuil modules relative to S → R. The first author would like to thank MPI Bonn, Binghamton University, and Bielefeld University for good conditions to work on the paper. The second author would like to thank Eike Lau for helpful discussions. Both authors thank the referee for some valuable remarks. 2 Breuil windows modulo powers of u We need a slight variant of Breuil windows, which was also considered by Kisin in his proof of Theorem 1 for r = 0. 3 For a ∈ N we define Sa := S/(uae ); it is a p-adic ring without p-torsion. Clearly E is not a zero divisor in Sa . The Frobenius endomorphism σ of S induces naturally a Frobenius endomorphism σ of Sa . We write E = E(u) = ue + pǫ, (2) where ǫ := (ae−1 /p)ue−1 + . . . + (a1 /p)u + (a0 /p) is a unit in S. As uae and pa (−ǫ)a are congruent modulo the ideal (E), we have identities Sa /(E) = S/(E, pa ) = R/pa R. Definition 3 A Breuil window relative to Sa → R/pa R is a pair (Q, φ), where Q is a free Sa -module of finite rank and where φ : Q → Q(σ) is a Slinear map whose cokernel is annihilated by E and is a free R/pa R-module. We will call this shortly a Sa -window, even though this is not a window in the sense of [Z3]. To avoid cases, we define S∞ := S and R/p∞ R := R and we will allow a to be ∞. Thus from now on a ∈ N ∪ {∞}. We note down that a S∞ -window will be a Breuil window relative to S → R. Next we relate Sa -windows to the windows introduced in [Z3]. We will use the following convention from [Z1]. Let α : M → N be a σ-linear map of Sa -modules. Then we denote by α♯ : M (σ) = Sa ⊗σ,Sa M → N its linearization. We say that α is a σ-linear epimorphism (etc.) if α♯ is an epimorphism. We consider triples of the form (P, Q, F ), where P is a free Sa -module of finite rank, Q is a Sa -submodule of P , and F : P → P is a σ-linear map, such that the following two properties hold: (i) E · P ⊂ Q and P/Q is a free R/pa R-module. (ii) F (Q) ⊂ σ(E) · P and F (Q) generates σ(E) · P as a Sa -module. As σ(E) is not a zero divisor in Sa , we can define F1 := (1/σ(E))F : Q → P . Any triple (P, Q, F ) has a normal decomposition. This means that there exist Sa -submodules J and L of P such that we have: P = J ⊕ L and Q = E · J ⊕ L. 4 (3) This decomposition shows that Q is a free Sa -module. The map F ⊕ F1 : J ⊕ L → P (4) is a σ-linear isomorphism. A normal decomposition of (P, Q, F ) is not unique. If P̃ is a free Sa -module of finite rank and if P̃ = L̃ ⊕ J˜ is a direct sum decomposition, then each arbitrary σ-linear isomorphism J˜ ⊕ L̃ → P̃ defines naturally a triple (P̃ , Q̃, F̃ ) as above. We can often identify the triple (P̃ , Q̃, F̃ ) with an invertible matrix with coefficients in Sa which is a matrix representation of the σ-linear isomorphism J˜ ⊕ L̃ → P̃ . Each triple (P, Q, F ) is isomorphic to a triple constructed as (P̃ , Q̃, F̃ ). Lemma 1 The category of triples (P, Q, F ) as above is equivalent to the category of Sa -windows. Proof: Assume we are given a triple (P, Q, F ). By definition F1 induces a Sa -linear epimorphism F1♯ : Q(σ) = Sa ⊗σ,Sa Q ։ P. (5) Due to the existence of normal decompositions of (P, Q, F ), Q is a free Sa module of the same rank as P . Therefore F1♯ is in fact an isomorphism. To the triple (P, Q, F ) we associate the Sa -window (Q, φ), where φ : Q → Q(σ) is the composite of the inclusion Q ⊂ P with (F1♯ )−1 . Conversely assume that we are given a Sa -window (Q, φ). We set P := (σ) Q and we consider Q as a submodule of P via φ. We denote by F1 : Q → P the σ-linear map which induces the identity Q(σ) = P . Finally we set F (x) := F1 (Ex) for x ∈ P . Then (P, Q, F ) is a triple as above.  Henceforth we will not distinguish between triples and Sa -windows i.e., we will identify (Q, φ) ≡ (P, Q, F ). We can describe a normal decomposition directly in terms of (Q, φ). Indeed, we can identify Q with J ⊕ L via (1/E) idJ ⊕ idL . Then a normal decomposition of (Q, φ) is a direct sum decomposition Q = J ⊕ L which induces a normal decomposition P = Q(σ) = (1/E)φ(J) ⊕ φ(L). If a ∈ N, then each Sa -window lifts to a Sa+1 -window (this is so as each invertible matrix with coefficients in Sa lifts to an invertible matrix with coefficients in Sa+1 ). 5 3 The p-divisible group of a Breuil window We relate Sa -windows to Dieudonné displays over R/pa R as defined in [Z2], Definition 1. Let S be a complete local ring with residue field k and maximal ideal n. We denote by Ŵ (n) the subring of all Witt vectors in W (n) whose components converge to zero in the n-adic topology. From [Z2], Lemma 2 we get that there exists a unique subring Ŵ (S) ⊂ W (S), which is invariant under the Frobenius F and Verschiebung V endomorphisms of W (S) and which sits in a short exact sequence: 0 → Ŵ (n) → Ŵ (S) → W (k) → 0. It is shown in [Z2] that the category of p-divisible groups over S is equivalent to the category of Dieudonné displays over Ŵ (S). For a ∈ N ∪ {∞} there exists a unique homomorphism δa : Sa → Ŵ (Sa ) (6) such that for all x ∈ Sa and for all n ∈ N we have wn (δa (x)) = σ n (x) (here wn is the n-th Witt polynomial). It maps ti 7→ [ti ] = (ti , 0, 0, . . .) and u 7→ [u] = (u, 0, 0, . . .). If we compose δa with the canonical W (k)homomorphism Ŵ (Sa ) → Ŵ (R/pa R) we obtain a W (k)-homomorphism κa : Sa → Ŵ (R/pa R). (7) We note that p is not a zero divisor in Ŵ (R). Lemma 2 The element κ∞ (σ(E)) ∈ Ŵ (R) is divisible by p and the fraction τ := κ∞ (σ(E))/p is a unit in Ŵ (R). Proof: We have κ∞ (E) ∈ V Ŵ (R). Since κ∞ is equivariant with respect to σ and the Frobenius F of Ŵ (R) we get: κ∞ (σ(E)) = F (κ∞ (E)) ∈ pŴ (R). We have to verify that w0 (τ ) is a unit in R. We have: w0 (τ ) = w0 (κ∞ (σ(E)))/p = σ(E)/p. With the notation of (2) we have σ(E) = uep +pσ(ǫ). Since uep ≡ (pǫ)p mod (E) we see that σ(E)/p is a unit in R. We note that this proof works for all primes p (i.e., even if p = 2).  6 For a ∈ N ∪ {∞} we will define a functor: Sa -windows −→ Dieudonné displays over R/pa R. (8) Let (Q, φ) ≡ (P, Q, F ) be a Sa -window. Let F1 : Q → P be as in section 2. To (P, Q, F ) we will associate a Dieudonné display (P ′ , Q′ , F ′ , F1′ ) over R/pa R. Let P ′ := Ŵ (R/pa R) ⊗κa ,Sa P . Let Q′ be the kernel of the natural Ŵ (R/pa R)-linear epimorphism: P ′ = Ŵ (R/pa R) ⊗κa ,Sa P ։ P/Q. We define F ′ : P ′ → P ′ as the canonical F -linear extension of F . We define F1′ : Q′ → P ′ by the rules: F1′ (ξ ⊗ y) = ′ V F1 ( ξ ⊗ x) = F ξ ⊗ τ F1 (y), for ξ ∈ Ŵ (R/pa R), y ∈ Q ξ ⊗ F (x), for ξ ∈ Ŵ (R/pa R), x ∈ P. Using a normal decomposition of (P, Q, F ), one checks that (P ′, Q′ , F ′, F1′ ) is a Dieudonné display over R/pa R. Since the category of Dieudonné displays over R/pa R is equivalent to the category of p-divisible groups over R/pa R (see [Z2]) we obtain from (8) a functor Sa -windows −→ p-divisible groups over R/pa R. (9) In particular, for the p-divisible group G associated to (Q, φ) ≡ (P, Q, F ) we have identifications of R/pa R-modules Lie(G) = P ′ /Q′ = P/Q = Coker(φ). (10) In S1 the elements E and p differ by a unit. Therefore the notion of a S1 -window is the same as that of a Dieudonné S1 -window over R/pR introduced in [Z3], Definition 2. By Theorem 6 (or 3.2) of loc. cit. we get that the functor (9) is an equivalence of categories in the case a = 1. We would like to mention that the contravariant analogue of this equivalence for a = 1 also follows from [dJ], Theorem of Introduction and Proposition 7.1. The faithfulness of the functors (8) and (9) follows from the mentioned equivalence in the case a = 1 and from the following rigidity property: Lemma 3 Let a ≥ 1 be a natural number. Let P = (P, Q, F ) and P ′ = (P ′, Q′ , F ′) be Sap -windows. By base change we obtain windows P̄ and P̄ ′ over Sa . Then the natural map 7 HomSap (P, P ′ ) → HomSa (P̄, P̄ ′ ) is injective. Proof: Let α : P → P ′ be a morphism, which induces 0 over Sa . We have α(P ) ⊂ uae P ′ . To prove that α = 0 it is enough to show that α(F1 y) = 0 for each y ∈ Q. We have α(y) ∈ uae P ′ ∩ Q′ . We choose a normal decomposition P ′ = J ′ ⊕ L′ and we write α(y) = j ′ + l′ . Then we have j ′ ∈ uae J ′ ∩ EJ ′ = uae EJ ′ and l′ ∈ uae L′ . In Sap we have σ(uae ) = uape = 0. We conclude that F1′ j ′ = 0 and F1′ l′ = 0. Finally we obtain α(F1 y) = F1′ (αy) = 0.  Lemma 4 The functors (8) and (9) are essentially surjective on objects. Proof: We will first prove the lemma for a ∈ N. We will use induction on a ∈ N. We already know that this is true for a = 1. The inductive passage from a to a + 1 goes as follows. It suffices to consider the case of the functor (8). Let P̃ ′ = (P̃ ′ , Q̃′ , F̃ ′ , F̃1′ ) be a Dieudonné display over R/pa+1 R. We denote by P ′ = (P ′ , Q′ , F ′ , F1′ ) its reduction over R/pa R. Then we find by induction a Sa -window P which is mapped to P ′ by the functor (8). We lift P to a Sa+1 -window P̃ = (Q̃, φ̃) ≡ (P̃ , Q̃, F̃ ), cf. end of section 2. Let F̃1 : Q̃ → P̃ be obtained from F̃ as in section 2. We apply to P̃ the functor (8) and we obtain a Dieudonné display P̃ ′′ = (P̃ ′′ , Q̃′′ , F̃ ′′ , F̃1′′ ) over R/pa+1 R. By [Z2], Theorem 3 we can identify (P̃ ′, F̃ ′ , Φ1 ) = (Ŵ (R/pa+1 R) ⊗κa+1 ,Sa+1 P̃ = P̃ ′′ , F̃ ′′ , Φ1 ). (11) Here Φ1 : Q̆′ → P̃ ′ is a Frobenius linear map from the inverse image Q̆′ of Q′ in P̃ ′ = P̃ ′′ which extends both F̃1′ and F̃1′′ and which satisfies the identity Φ1 ([pa ]P̃ ′ ) = 0 (this identity is due to the fact that we use the trivial divided power structure on the kernel of the epimorphism R/pa+1 R ։ R/pa R). The composite map: F̃ τ 1 Q̃ −→ P̃ → P̃ ′ → P̃ ′ , coincides with the composite map Φ 1 Q̃ → Q̆′ −→ P̃ ′. 8 We define Q̃∗ ⊂ P̃ as the inverse image of the natural map P̃ → P̃ ′/Q̃′ deduced from the identity P̃ ′ = Ŵ (R/pa+1 R) ⊗κa+1 ,Sa+1 P̃ . The images of Q̃ and Q̃∗ by the canonical map P̃ → P are the same. Therefore for each y ∗ ∈ Q̃∗ there exists an y ∈ Q̃ such that we have y ∗ = y + uae x for some x ∈ P̃ . Since F̃ (uae x) = 0 we conclude that F̃ (y ∗ ) = F̃ (y) ∈ σ(E) · P̃ . This proves that P̃ ∗ = (P̃ , Q̃∗ , F̃ ) is a Sa+1 -window which lifts the Sa -window P. Let F̃1∗ : Q̃∗ → P̃ be obtained from F̃ as in section 2. We claim that the image of P̃ ∗ via the functor (8) coincides with the Dieudonné display P̃ ′ . For this we have to show that the composite map F̃ ∗ τ 1 Q̃∗ −→ P̃ → P̃ ′ → P̃ ′, coincides with the composite map Φ 1 Q̃∗ → Q̆′ −→ P̃ ′ . This follows again from the decomposition y ∗ = y + uae x and the facts that: (i) we have F̃1∗ (y ∗ ) = F̃1 (y) (as we have F̃ (y ∗ ) = F̃ (y)) and (ii) the image of uae x in Q̆′ is mapped to zero by Φ1 . We conclude that P̃ ′ is in the essential image of the functor (8). This ends the induction. The fact that the lemma holds even if a = ∞ follows from the above induction via a natural limit process.  4 Extending morphisms between S1-windows In this section we prove an extension result for an isomorphism between S1 windows. We begin by considering for a ∈ N ∪ {∞} an extra W (k)-algebra: Ta := S[[v]]/(pv − ue , v a ) with the convention that v ∞ := 0. This ring is without p-torsion. It is elementary to check that the canonical ring homomorphism Sa → Ta is injective. For a = 1 this is an isomorphism S1 ∼ = T1 . We set T = T∞ . In Ta we have E = p(v + ǫ) and thus the elements p and E differ by a unit. We have an isomorphism: Ta /pTa ∼ = (R/pR)[[v]]/(v a ). 9 We extend the Frobenius endomorphism σ to Ta by the rule: σ(v) = ue(p−1) v = pp−1 v p We note that the endomorphism σ on Ta no longer induces the Frobenius modulo p. But the notion of a window over Ta still makes sense as follows. Definition 4 A window over Ta is a triple (P, Q, F ), where P is a free Ta module, Q is a Ta -submodule of P such that P/Q is a free Ta /pTa -module, and F : P → P is a σ-linear endomorphism. We require that F (Q) ⊂ pP and that this subset generates pP as a Ta -module. We define a σ-linear map F1 : Q → P by pF1 (y) = F (y) for y ∈ Q. Its linearisation F1♯ is an isomorphism. Taking the composite of the inclusion Q ⊂ P with (F1♯ )−1 we obtain a Ta -linear map φ : Q → Q(σ) , whose cokernel is a free Ta /pTa -module. If we start with a triple (P, Q, F ) as in Lemma 1 an tensor it with Ta ⊗Sa we obtain a window over Ta . A window over Ta is not a window in sense of [Z3] because σ on Ta /pTa is not the Frobenius endomorphism. We have still the following lifting property. Proposition 1 Let (Q1 , φ1 ) and (Q2 , φ2 ) be two Breuil windows relative to S → R. Let (Q̆1 , φ̆1 ) and (Q̆2 , φ̆2 ) be the S1 -windows which are the reduction modulo ue of (Q1 , φ1 ) and (Q2 , φ2 ) (respectively). Let ᾰ : Q̆1 → Q̆2 be an isomorphism of windows relative to S1 → R/pR i.e., a S1 -linear isomorphism such that we have φ̆2 ◦ ᾰ = (1 ⊗ ᾰ) ◦ φ̆1 . Then there exists a unique isomorphism α : T ⊗S Q1 → T ⊗S Q2 which commutes in the natural sense with φ1 and φ2 and which lifts ᾰ with respect to the S-epimorphism T → S1 that maps v to 0. Proof: We choose a normal decomposition Q̆1 = L̆1 ⊕ J˘1 . Applying ᾰ we obtain a normal decomposition Q̆2 = L̆2 ⊕ J˘2 . We lift these normal decompositions to S: Q1 = J1 ⊕ L1 and 10 Q2 = J2 ⊕ L2 . We find an isomorphism γ : Q1 → Q2 which lifts ᾰ and such that γ(L1 ) = L2 and γ(J1 ) = J2 . We identify the modules Q1 and Q2 via γ and we write: Q = Q1 = Q2 , J = J1 = J2 , L = L1 = L2 . We choose a S-basis {e1 , . . . , ed } for J and a S-basis {ed+1 , . . . er } for L. Then {1 ⊗ e1 , . . . , 1 ⊗ er } is a S-basis for Q(σ) . For i ∈ {1, 2} we write φi : Q → Q(σ) as a matrix with respect to the mentioned S-bases. It follows from the properties of a normal decomposition that this matrix has the form:   E · Id 0 , Ai 0 Ic where A1 and A2 are invertible matrices in GL r (S) and where c := r − d. By the construction of γ, the S-linear maps φ1 and φ2 coincide modulo (ue ). From this and the fact that E modulo (ue ) is a non-zero divisor of S/(ue ), we get that we can write (A2 )−1 A1 = Ir + ue Z, (12) where Z ∈ Mr (S). We set C :=  E · Id 0 0 Ic  . To find the isomorphism α is the same as to find a matrix X ∈ GL r (T ) which solves the equation A2 CX = σ(X)A1 C (13) and whose reduction modulo the ideal (v) of T is the matrix representation of ᾰ i.e., it is the identity matrix. Therefore we set: X = Ir + vY (14) for a matrix Y ∈ Mr (T ). As E/p = v + ǫ is a unit of T , the matrix pC −1 has coefficients in T . From the equations (13) and (14) we obtain the equation: p(Ir + vY ) = pC −1 A−1 2 (Ir + σ(v)σ(Y ))A1 C. 11 We insert (12) in this equation. With the notation D := pC −1 ZC we find: ue Y = ue D + (uep /p)pC −1 A−1 2 σ(Y )A1 C. Since ue is not a zero divisor in T we can write: Y − (ue(p−1) /p)pC −1 A−1 2 σ(Y )A1 C = D. (15) We have (ue(p−1) /p) = vue(p−2) . Thus the σ-linear operator Ψ(Y ) = (ue(p−1) /p)pC −1 A−1 2 σ(Y )A1 C on the T -module Mr (T ) is topologically P∞ nilpotent. Therefore the equation (15) has a unique solution Y = n=0 Ψ(D) ∈ Mr (T ). Therefore X = Ir + vY ∈ GL r (T ) exists and is uniquely determined.  5 Proof of Theorem 1 In this section we prove Theorem 1. In section 3 we constructed a functor (cf. (9) with a = ∞) Breuil windows relative to S → R −→ p-divisible groups over R (16) which is faithful and which (cf. Lemma 4) is essentially surjective on objects. Thus to end the proof of Theorem 1, it suffices to show that this functor is essentially surjective on isomorphisms (equivalently, on morphisms). This will be proved in Lemma 5 below. For the proof of Lemma 5, until the end we take a ∈ N and we will begin by first listing some basic properties of the rings Sa and Ta . 5.1 On Sa and Ta There exists a canonical homomorphism Sa → R/pa R whose kernel is the principal ideal of Sa generated by E modulo (uae ). Let Sa ⊂ Sa ⊗ Q 12 be the subring generated by all elements une /n! over Sa with n ∈ {0, . . . , a}. Then Sa is a p-adic ring without p-torsion. There exists a commutative diagram Sa J  Sa JJ JJ JJ JJ $$ // R/pν(a) R, n where ν(a) := inf{ordp ( pn! )| for n ≥ a}. With the notation of (2), the horizontal homomorphism maps une /n! to (pn /n!)(−ǫ)n . By [Z3], Theorem 6 or 3.2, the epimorphism Sa → R/pν(a) R is a frame which classifies p-divisible groups over R/pν(a) R. In what follows, by a Sa -window we mean a Dieudonné Sa -window over R/pν(a) R in the sense of [Z3], Definition 2. Both Sa and Ta are subrings of Sa ⊗ Q. Because of the identity  n une pn u e pn = = vn n! n! p n! we have an inclusion of W (k)-algebras: Sa ⊂ Ta . The Frobenius endomorphism σ of S induces a homomorphism Sa → Spa . It maps the subalgebra Ta ⊂ Sa ⊗ Q to Spa ⊂ Spa ⊗ Q. We denote this homomorphism by τa : Ta → Spa . Lemma 5 Let (Q1 , φ1 ) and (Q2 , φ2 ) be two Breuil windows relative to S → R. Let G1 and G2 be the corresponding p-divisible groups over R, cf. the functor (16). Let γ : G1 → G2 be an isomorphism. Then there exists a unique morphism (automatically isomorphism) α0 : (Q1 , φ1 ) → (Q2 , φ2 ) of Breuil windows relative to S → R which maps to γ via the functor (16). Proof: For i ∈ {1, 2} let (Q̆i , φ̆i ) ≡ (P̆i , Q̆i , F̆i ) be the S1 -window which is the reduction of (Qi , φi ) ≡ (Pi , Qi , Fi ) modulo ue . Since the category of S1 -windows is equivalent to the category of p-divisible groups over R/pR, the reduction of γ modulo p is induced via the functor (9) by an isomorphism 13 ᾰ : (Q̆1 , φ̆1) → (Q̆2 , φ̆2 ). Due to Proposition 1, the last isomorphism extends to an isomorphism α : T ⊗S (Q1 , φ1 ) → T ⊗S (Q2 , φ2) of windows over T . Here and in what follows we identify Ta ⊗S (Qi , φi ) ≡ Ta ⊗S (Pi , Qi , Fi ) and therefore we will refer to Ta ⊗S (Qi , φi ) as a window over Ta (here a can be also ∞). As in the proof of Proposition 1 we can identify normal decompositions Q1 = J1 ⊕ L1 = J2 ⊕ L2 = Q2 and we can represent the mentioned isomorphism of windows over T by an invertible matrix X ∈ GL r (T ). Let Xa ∈ GL r (Ta ) be the reduction of X modulo v a . The matrix X has the following crystalline interpretation. The epimorphism Ta ։ (R/pR)[[v]]/(v a ) is a pd-thickening. (We emphasize that it is not a frame in the sense of [Z3], Definition 1 because σ modulo p is not the Frobenius endomorphism of Ta /pTa .) We have a morphism of pd-thickenings Sa //  R/p ν(a) R Ta (17)  // (R/pR)[[v]]/(v a ). By the crystal associated to a p-divisible group  over R/pν(a) R we mean the Lie algebra crystal of the universal vector extension crystal of  as defined in [M]. The crystal of Gi evaluated at the pd-thickening Sa → R/pν(a) R (σ) coincides in a functorial way with Sa ⊗Sa Qi = Sa ⊗Sa Pi (if Gi is a formal p-divisible group, this follows from either [Z1], Theorem 6 or from [Z3], Theorem 1.6; the general case follows from [L]). Let Ği,a be the push forward of Gi via the canonical homomorphism R → (R/pR)[[v]]/(v a ). The diagram (σ) (17) shows that Ta ⊗S Qi = Ta ⊗S Pi is the crystal of Ği,a evaluated at the pd-thickening Ta → (R/pR)[[v]]/(v a ). The isomorphism γ : G1 → G2 induces an isomorphism αa of Sa -windows which is defined by a Sa -linear isomorphism Sa ⊗S P1 → Sa ⊗S P2 . Via base change of αa through the diagram (17), we get an isomorphism βa : Ta ⊗S (Q1 , φ1) → Ta ⊗S (Q2 , φ2 ) of windows over Ta defined by an isomorphism Ta ⊗S (P1 , Q1 , F1 ) → Ta ⊗S (P2 , Q2 , F2 ). We note that after choosing a normal decomposition a window is simply an invertible matrix (to be compared with end of section 2) and base change applies to the coefficients of this matrix the homomorphism Sa → Ta . The system of isomorphisms βa induces in the limit an isomorphism of windows over T : β : T ⊗S (Q1 , φ1 ) → T ⊗S (Q2 , φ2 ). (18) 14 We continue the base change (17) using the following diagram: Ta //  (R/pR)[[v]]/(v a ) // S1 (19)  R/pR. The vertical arrows are thickenings with divided powers. From αa we obtain by base change the isomorphism ᾰ since windows associated to p-divisible groups commute with base change. But this shows that the isomorphism β coincides with the isomorphism α : T ⊗S (Q1 , φ1 ) → T ⊗S (Q2 , φ2 ), cf. the uniqueness part of Proposition 1. We will show that the assumption that X has coefficients in S implies the Lemma. This assumption implies that α is the tensorization with T of an isomorphisms α0 : (Q1 , φ1 ) → (Q2 , φ2 ). Let γ0 : G1 → G2 be the isomorphism associated to the isomorphism α0 via the functor (16). As the functor (9) is an equivalence of categories for a = 1, we get that γ0 and γ coincide modulo p. Therefore γ = γ0 and thus the Lemma holds. Thus to end the proof of the Lemma it suffices to show by induction on a ∈ N that the matrix Xa has coefficients in Sa . The case a = 1 is clear. The inductive passage from a to a+1 goes as follows. We can assume that Xa has coefficients in Sa . Therefore the invertible matrix τa (Xa ) ∈ GL r (τ (Ta )) ⊂ GL r (Spa ) defines a Spa -linear isomorphism (σ) (σ) Spa ⊗S P1 = Spa ⊗S Q1 → Spa ⊗S Q2 = Spa ⊗S P2 which respects the Hodge filtration i.e., it is compatible with the R/pν(pa) Rlinear map Lie G1,R/pν(pa) R → Lie G2,R/pν(pa) R induced by γ. Since τa (Xa ) has coefficients in Spa , we obtain a commutative diagram of Spa -modules: (σ) // Lie G1,R/pν(pa) R Spa ⊗S Q1  (σ) Spa ⊗S Q2  // 15 Lie G1,R/pν(pa) R . Since ν(pa) ≥ a + 1, from (10) we obtain a commutative diagram 0 // Sa+1 ⊗S Q1 // S Sa+1 ⊗S Q2 // S (σ) // Lie G1,R/p(a+1) R (σ) a+1 ⊗S Q2 // Lie G1,R/p(a+1) R a+1 ⊗S Q1  0 // (20)  with exact rows. The left vertical arrow is induced by σ(Xa+1 ). On the kernels of the horizontal maps we obtain a Sa+1 -linear isomorphism Sa+1 ⊗S Q1 → Sa+1 ⊗S Q2 . (21) As E is not a zero divisor in Ta+1 , by tensoring the short exact sequences of (20) with Ta+1 we get short exact sequences of Ta+1 -modules. This implies that the tensorization of the Sa+1 -linear isomorphism (21) with Ta+1 is given by the matrix Xa+1 . Therefore Xa+1 has coefficients in Sa+1 . This completes the induction and thus the proofs of the Lemma and of Theorem 1.  6 Breuil modules To prove Theorem 2 we first need the following basic result on Breuil modules relative to S → R. Proposition 2 Let (M, ϕ) be a Breuil module relative to S → R. Then the following four properties hold: (i) The S-linear map ϕ is injective. (ii) There exists a short exact sequence 0 → (Q′ , φ′ ) → (Q, φ) → (M, ϕ) → 0, where (Q′ , φ′ ) and (Q, φ) are Breuil windows relative to S → R. (iii) If (M, ϕ) → (M̃ , ϕ̃) is a morphism of Breuil modules relative to S → R, then it is the cokernel of a morphism between two exact complexes 0 → (Q′ , φ′ ) → (Q, φ) and 0 → (Q̃′ , φ̃′ ) → (Q̃, φ̃) of Breuil windows relative to S → R. (iv) The quotient M (σ) /ϕ(M) is an R-module of projective dimension at most one. 16 Proof: Let (p) := pS; it is a principal prime ideal of S. Then σ induces an endomorphism of the discrete valuation ring S(p) which fixes p. Thus the length of a S(p) -module remains unchanged if tensored by σ(p) : S(p) → S(p) . One easily checks that (M(p) )(σ) = S(p) ⊗σ(p) ,S(p) M(p) ∼ = (M (σ) )(p) . Let x, p be a regular sequence in S. As the S-module M has projective dimension at most one and as M is annihilated by a power of p, the multiplication by x is a S-linear monomorphism x : M ֒→ M. Since no element of S \ pS is a zero divisor in M, we conclude that M ⊂ M(p) . The S-linear map ϕ : M → M (σ) becomes an epimorphism when tensored with S(p) . We obtain an epimorphism of S(p) -modules of the same length: ϕ(p) : M(p) → (M(p) )(σ) . As M is a finitely generated S-module annihilated by a power of p, the S(p) module M(p) has finite length. From the last two sentences we get that ϕ(p) is injective. Thus ϕ is also injective i.e., (i) holds. We consider free S-modules J and L of finite ranks and a S-linear epimorphism τ J ⊕ L −→ M (σ) which maps the free S-submodule EJ ⊕ L surjectively to ϕ(M). Let τ1 : J ⊕ L → M be the unique S-linear map such that we have a commutative diagram EJ ⊕ L −−−→ Im(ϕ) x x ϕ  E idJ + idL   J ⊕ L −−−→ M whose vertical maps are isomorphisms. We consider a S-linear isomorphism γ : J ⊕ L → J (σ) ⊕ L(σ) which makes the following diagram commutative τ // (σ) J ⊕ LM 88 M MMM q q q MMM q qqq(σ) γ MMMM q q q τ1 && J (σ) ⊕ L(σ) . The existence of γ is implied by the following general property. Let N be a finitely generated module over a local ring A. Let F1 and F2 be two free Amodules of the same rank equipped with A-linear epimorphisms τ1 : F1 → N 17 and τ2 : F2 → N. Then there exists an isomorphism γ12 : F1 → F2 such that we have τ2 ◦ γ12 = τ1 . We set Q := J ⊕ L and φ := γ ◦ (E idJ + idL ) : J ⊕ L → J (σ) ⊕ L(σ) . Then the pair (Q, φ) is a Breuil window relative to S → R. We have a commutative diagram τ1 Q −−− → M     ϕy φy τ (σ) Q(σ) −−1−→ M (σ) . Hence τ1 is a surjection from (Q, φ) to (M, ϕ). It is obvious that the kernel of τ1 : (Q, φ) → (M, ϕ) is again a Breuil module (Q′ , φ′ ) relative to S → R. We obtain a short exact sequence: 0 → (Q′ , φ′ ) → (Q, φ) → (M, ϕ) → 0. Thus (ii) holds. Next we prove (iii). We have seen above that for any Breuil module (M, ϕ) relative to S → R there is a Breuil window (Q, φ) relative to S → R and an epimorphism (Q, φ) → (M, ϕ). We remark that our argument uses only the properties that ϕ : M → M (σ) is injective and that its cokernel is annihilated by E. In the situation (iii) we choose a surjection (Q̃, φ̃) → (M̃, ϕ̃) from a Breuil window (Q̃, φ̃) relative to S → R. We form the fibre product of S-modules N = M ×M̃ Q̃. The functor which associates to an S-module L the S-module L(σ) is exact and therefore respects fibre products. We obtain a S-linear map ψ : N → N (σ) which is compatible with ϕ and φ̃. Clearly ψ is injective and its cokernel is annihilated by E. Therefore there is a surjection (Q, φ) → (N, ψ) from a Breuil window (Q, φ) relative to S → R. We deduce the existence of a commutative diagram (Q, φ) −−−→ (M, ϕ)     y y (Q, φ̃) −−−→ (M̃, ϕ̃). As remarked above the kernels of the horizontal arrows are Breuil modules relative to S → R. This implies that (iii) holds. To prove (iv) we consider the short exact sequence of (ii). As Coker(φ) and Coker(φ′ ) are free R-modules and as we have a short exact sequence 18 0 → Coker(φ′ ) → Coker(φ) → Coker(ϕ) → 0 of R-modules, we get that (iv) holds as well.  6.1 Proof of Theorem 2 We prove Theorem 2. Let H be a finite flat, commutative group scheme of p power order over R. Due to a theorem of Raynaud (see [BBM], Theorem 3.1.1), H is the kernel of an isogeny of p-divisible groups over R 0 → H → G′ → G → 0. Let (Q′ , φ′ ) and (Q, φ) be the Breuil windows relative to S → R which correspond by Theorem 1 to the p-divisible groups G′ and G. Let (Q′ , φ′ ) → (Q, φ) be the morphism that corresponds to the isogeny G′ → G. This morphism is an isogeny i.e., it is a monomorphism and its cokernel is annihilated by a power of p (as G′ → G is an isogeny). Then it is immediate that the cokernel (M, ϕ) of (Q′ , φ′ ) → (Q, φ) is a Breuil module relative to S → R. One can check that (M, φ) is independent of the chosen resolution of H and that the association H 7→ (M, ϕ) is a functor. Conversely let (M, φ) be a Breuil module relative to S → R. By Proposition 2 (ii) we have a short exact sequence 0 → (Q′ , φ′ ) → (Q, φ) → (M, ϕ) → 0, where (Q′ , φ′ ) and (Q, φ) are Breuil windows relative to S → R. By Theorem 1, the monomorphism (Q′ , φ′ ) → (Q, φ) gives rise to an isogeny of p-divisible groups G′ → G. Based on this and Proposition 2 (iii) we obtain a functor which associates to (M, φ) the kernel of the isogeny G′ → G. This is a quasi-inverse to the functor of the previous paragraph. Thus Theorem 2 holds.  References [BBM] Berthelot, P., Breen, L., Messing, W.: Théorie de Dieudonné Cristalline II, Lecture Notes in Mathematics, Vol. 930, Springer-Verlag, Berlin-New York, 1982. [B] Breuil, Ch.: Schémas en groupes et corps de normes, 13 pages, unpublished manuscript (1998). 19 [dJ] de Jong, J.: Finite locally free group schemes in characteristic p and Dieudonn modules, Invent. Math. 114 (1993), no. 1, pp. 89–137. [F] Faltings, G.: Integral crystalline cohomology over very ramified valuation rings, J. Amer. Math. Soc. 12 (1999), no. 1, pp. 117–144. [K] Kisin, M.: Crystalline representations and F -crystals, Algebraic geometry and number theory, pp. 459–496, Progr. Math., Vol. 253, Birkhäuser Boston, Boston, MA, 2006. [L] Lau, E.: A comparison of crystals for Dieudonné displays, personal communication to Thomas Zink. [M] Messing, W.: The crystals associated to Barsotti-Tate groups: with applications to abelian schemes, Lecture Notes in Mathematics, Vol. 264, Springer-Verlag, Berlin-New York, 1972. [VZ] Vasiu, A. and Zink, Th.: Purity results for finite flat group schemes over ramified bases, manuscript July 2008. [Z1] Zink, Th.: The display of a formal p-divisible group, Cohomologies p-adiques et applications arithmétiques. I, J. Astérisque, No. 278 (2002), pp. 127–248. [Z2] Zink, Th.: A Dieudonné theory for p-divisible groups, Class field theory—its centenary and prospect (Tokyo, 1998), pp. 139–160, Adv. Stud. Pure Math., Vol. 30, Math. Soc. Japan, Tokyo, 2001. [Z3] Zink, Th.: Windows for displays of p-divisible groups. Moduli of abelian varieties (Texel Island, 1999), pp. 491–518, Progr. Math., Vol. 195, Birkhäuser, Basel, 2001. Adrian Vasiu, Email: adrian@math.binghamton.edu Address: Department of Mathematical Sciences, Binghamton University, Binghamton, New York 13902-6000, U.S.A. Thomas Zink, Email: zink@math.uni-bielefeld.de Address: Fakultät für Mathematik, Universität Bielefeld, P.O. Box 100 131, D-33 501 Bielefeld, Germany. 20