L’INSTITUTFOURIER DE ANNALES

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A L

E S

E D L ’IN IT ST T U

F O U R

ANNALES

DE

L’INSTITUT FOURIER

Ioan BERBEC

Group Schemes over artinian rings and Applications Tome 59, n^o6 (2009), p. 2371-2427.

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GROUP SCHEMES OVER ARTINIAN RINGS AND APPLICATIONS

by Ioan BERBEC (*)

Abstract. — Letnbe a positive integer andA⁰a complete characteristic zero discrete valuation ring with maximal idealm, absolute ramification indexe < p−1 and perfect residue fieldkof characteristicp >2. In this paper we classify smooth finite dimensional formal p-faithful groups over A⁰_n = A⁰/mⁿA⁰,i.e. groups on which the “multiplication byp” morphism is faithfully flat, in particularp-divisible groups. As applications, we prove thatp-divisible groups overk, and the morphisms between them, lift canonically toA⁰/pA⁰, and we study liftings to characteristic zero of certain connectedp-divisible groups of dimensiondand heighthoverk=k, with dandhcoprime. Whene= 1, we classify finite flat group schemes overA⁰/p²A⁰of p-power order and prove that a finite flat group scheme overA⁰/pⁿA⁰ofp-power order, having flatpⁱ-torsion for everyi>1, lifts toA⁰.

Résumé. — Soitnun entier positif etA⁰un anneau de valuation discrète com- plet de caractéristique zéro avec idéal maximal m, indice de ramification absolu e < p−1et corps résiduel parfaitkde caractéristiquep >2. Dans cet article nous classifions les groupes formels lissesp-fidèlesde dimension finie surA⁰_n=A⁰/mⁿA⁰, i.e. les groupes sur lesquels le morphisme “multiplication parp” est fidèlement plat, en particulier les groupesp-divisibles. Comme application, nous prouvons que les groupesp-divisibles surk, et les morphismes entre eux, se relèvent canonique- ment àA⁰/pA⁰, et nous étudions les relèvements en caractéristique zéro de certains groupes p-divisibles connexes de dimensiondet hauteur hsur k=k, oudeth sont étrangers. Quande= 1, nous classifions les schémas en groupes finis et plats surA⁰/p²A⁰d’ordre une puissance depet nous prouvons que tous les schémas en groupes finis et plats surA⁰/pⁿA⁰d’ordre une puissance dep, avecpⁱ-torsion plate pour chaquei>1, se relèvent àA⁰.

Introduction

Letp >2 be a prime. LetA⁰ be a complete characteristic0discrete valuation ring with absolute ramification indexe=e(A⁰)< p−1and perfect

Keywords:Group scheme,p-divisible group, almost canonical lifting.

Math. classification:14L15, 14L05.

(*) The author wishes to thank the Holy Putna Monastery for providing the best working conditions while this article was revised.

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residue field of characteristicp. Letm be its maximal ideal,k=A⁰/m and n a positive integer. In this paper we classify smooth finite dimensional (commutative) formal p-faithful groups, i.e. groups on which the “multiplication byp” morphism is faithfully flat, in particularp-divisible groups, over A⁰_n = A⁰/mⁿA⁰, and use it to derive other classification results and some applications.

Fontaine classified smoothp-groups overA⁰,cf.Theorem 1.11 below. We use his work to achieve the classification of smoothp-groups over A⁰_n,cf.

Theorem 2.8. We associate to any such group a so-called smooth Honda system overA⁰_n, i.e. linear algebra data constructed from the Dieudonné module of the special fiber of the group. In general, we can prove that this correspondence is essentially surjective and full. In order to prove that this correspondence is also faithful, and thus achieve our classification, we have to restrict top-faithful groups. While essential surjectivity follows more or less easily from Fontaine’s work, fully faithfulness is nontrivial, reflecting phenomena specific to groups over A⁰_n, cf. Lemma 2.10. In the end, we prove that our classification is compatible with Fontaine’s. More precisely, we prove that if ap-faithful groupΓoverA⁰is classified, via Fontaine, by the pair(L, M)then its base changeΓn toA⁰_n is classified by(L/mⁿ⁻¹L, M), cf.Proposition 2.12.

In the casen=e, from the algebraic properties of the Honda system as- sociated to ap-divisible group overA⁰_e, we deduce the following proposition.

It is implied by Proposition 2.17 and Corollary 2.18.

Proposition. — For every p-divisible group Γ over k there exists a canonical p-divisible group Γ^can over A⁰_e such that Γ^can ×Speck ' Γ.

Moreover, any morphismf: Γ→Γ⁰ betweenp-divisible groups overklifts canonically to a morphismf^can: Γ^can→(Γ⁰)^can of p-divisible groups over A⁰_e. In particular, any abelian variety (resp. finite group scheme) overklifts canonically to an abelian scheme (resp. finite flat group scheme) overA⁰_e.

We apply our classification to the study of liftings to characteristic zero of Manin’s groupsG_d,h−d,cf.Remark 4.1, whered < hare coprime andk is algebraically closed. The groupGd,h−d is a connected p-divisible group of dimension d and absolute height h over k. All connected p-divisible groups of heighthand dimension1are isomorphic toG_1,h. In general, all p-divisible groups of dimension dand height hare isogeneous to G_d,h−d, cf.[13], p. 3. We prove,cf. Theorem 4.4, the following result.

Theorem 1. — Letd < hbe two coprime positive integers, letkbe an algebraically closed field of characteristicp, letObe the ring of integers in

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a degreehextensionKofQp, with absolute ramification indexe < p−1, and letA⁰ be the ring of integers in a degreee, totally ramified extension of the fraction field of the Witt ring W(k), which contains the maximal unramified extension ofO.

There exists ap-divisible groupΓoverA⁰such thatΓ×Speck'G_d,h−d andEndA⁰−gr(Γ) =Oif and only if h>ed. In this case:

(i)There are exactlye^d/gisomorphism classes of suchΓ’s, wheregis the number of automorphisms ofK which fix its maximal unramified subex- tension.

(ii)For every suchΓ and everyn>1 End_A⁰

n−gr(Γ_A⁰

n) =O+πⁿ⁻¹End_k−gr(Γ_k)

where ΓA⁰_n (resp. Γk) is the base change to A⁰_n (resp. k) of Γ and π is a uniformizer ofO.

We refer the reader to Section 4 for details concerning this result. This Theorem, via Honda systems, becomes a beautiful, yet nontrivial, exercise in semilinear algebra. Over bases with low ramification, Part (i) of the Theorem generalizes to arbitrary dimension results of Lubin, [10], Theo- rem 4.3.2, and Part (ii) generalizes results of Gross, [6], §3, and Yu, [15], Section 14.

Another application of our classification of p-divisible groups over A⁰_n is the study of finite flat group schemes over A⁰_n of p-power order, finite groups in the sequel, in the casee= 1. Our main tool is Oort’s result,cf.

Theorem 3.1, which states that any finite group embeds into ap-divisible one.

Fontaine associated to a finite group overA⁰ a so-calledfinite Honda sys- tem(L, M)over A⁰ that classifies the group, withM being the Dieudonné module of the special fiber of the group,cf.[3], Theorem 1.4. We associate to a finite group over A⁰_n a finite Honda system over A⁰_n, consisting of a triple (Lⁿ, L_n, M), with (Lⁿ, M[pⁿ⁻¹]) and (L_n, M/pⁿ⁻¹M) being finite Honda systems overA⁰,cf.Definition 3.6 and Proposition 3.7. In the case n= 2we are able to prove that this correspondence classifies finite groups overA⁰/p²A⁰,cf. Corollary 3.10. For generalnwe can prove that this correspondence classifies a certain class of finite groups overA⁰_n among which are the truncated Barsotti-Tate groups of levels>1,cf.Remark 3.11 (2).

One of the main differences between the situation overA⁰ and the situation overA⁰_n is that the “special fiber” functor from finite groups overA⁰_n to finite groups overk is not faithful. This implies, in particular, that the

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category of finite groups overA⁰_n is not abelian,cf.Remark 3.11 (1). Nev- ertheless, we are able to prove,cf. Theorem 3.16, that the “special fiber”

functor is faithful on the morphisms that lift toA⁰_n+mformlarge enough.

We also prove, cf.Theorem 3.13, the following result.

Theorem 2. — SupposeA⁰ is unramified. LetGbe a finite flat group scheme overA⁰/pⁿA⁰ ofp-power order.

(i) If the pⁱ-torsion subgroup G[pⁱ] is flat for every i > 1 then G lifts toA⁰.

(ii)The torsion subgroupG[pⁱ]is flat for ibetween1and some positive integerrif and only ifGlifts toA⁰/p^n+rA⁰.

A future generalization of this paper would be to include higher (e >

p−1) ramification on the base. We think that Breuil’s techniques,cf. [2], can be used to achieve this.

Here is the structure of this paper: in Section 1 we introduce notations and we review concepts and results of Fontaine and Conrad that we will use in our paper. In Section 2 we classify smooth formalp-groups. In Section 3 we study finite groups. In Subsection 3.1 we show how most of the results can be carried out mutatis mutandis in the case of finite groups overA⁰_n with e > 2 and n of the form qe+ 1. In Section 4 we study liftings to characteristic zero of Manin’s groups and their endomorphismsmodmⁿ.

Acknowledgement. — I would like to thank Professor Robert Coleman for suggesting the motivating problem behind this paper to me and for his guidance throughout my graduate studies. I also thank Brian Conrad for helpful suggestions, which lead to some of the applications in this paper, and the referee for very useful comments.

1. Notations and Preliminaries

The main references for this paper are Fontaine’s book [5] and Conrad’s article [3]. For the convenience of the reader we review here all the defini- tions and results we use from the above papers.

Throughout this paper:

n>2is a positive integer.

p>3 is a fixed prime number.

kis a perfect field of characteristicp.

A=W(k)is the ring of Witt vectors ofk. We let(A⁰,m)be the valuation ring of a finitetotally ramified extensionK⁰ of the fraction field

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K of A, with e = e(A⁰) the absolute ramification index of A⁰, and A⁰_n=A⁰/mⁿ. We fix a uniformizerπofA⁰.

If R(resp. R) is an A⁰ (respA⁰_n) algebra, we denote byRk (resp.R_k) the special fiberR ⊗A⁰k(resp.R⊗A⁰_nk) ofR(resp.R) and byRK⁰ =RK

the generic fiberR ⊗_A⁰K ofR.

Dk = A[F, V] is the Dieudonné ring, i.e. the variables satisfy F V = V F =p, F α=σ(α),V α=σ⁻¹(α), for allα∈A, whereσ:A→A is the Frobenius morphism.

For us, agroup is agroup scheme, formal or finite.

A pseudo-compact ring S is a separated and complete linearly topologized ring such that the ring S/I is artinian for all open ideals I of S.

Obviouslyk with the discrete topology, A⁰ andA⁰_n with the p-adic topol- ogy are pseudo-compact.

Definition 1.1. — Let (S,m) be a local pseudo-compact ring with residue characteristicp.

1. Aformal S-group functorF is a functor defined on finiteS-algebras with values in abelian groups. Thus all our groups arecommutative.

2. A formal S-group is a pro-representable formal S-group functor. A formalp-group Gover S is a formal S-group G such thatG 'lim

−→G[pⁱ].

We say that a formalS-groupGissmoothif for all finiteS-algebrasRand all square zero ideals I of R the canonical map from G(R)to G(R/I) is surjective.

3. We say that a smooth formal p-group G over S is p-faithful if the

“multiplication byp” morphism[p] :G→Gis faithfully flat.

4. We say that a p-faithful groupG over S is p-divisibleof height h if G[pⁱ]has orderp^ih for alli>1.

5. Afinite flat group scheme ofp-power order overS is a formalp-group which is a finite flat scheme overS.

A profinite S-module M is a linearly topologized S-module such that for any open submodule M⁰ the quotientM/M⁰ is an S-module of finite length. Aprofinite S-algebraB is anS-algebra such that B is a profinite S-module.

We briefly review the theory of Witt covectors from [5], Chapter II, §§1-4.

For any commutative ring S (the reader should have in mind k, A or A_n finite algebras as a typical example) we define theS-valued Witt cov- ectors CW(S) to be the set of sequences a = (. . . , a_−i, . . . , a0) of elements a_−i ∈ S verifying the condition: there is an integer r > 0 such that the ideal ofS generated by the a−i’s for i >r is nilpotent. Letting

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S_m ∈ Z[X₀, . . . , X_m, Y₀, . . . , Y_m] denote the m^th addition polynomial for Witt vectors,cf.[5], pp. 71–72, and choosingaandbinCW(S), the nilpo- tence condition ensures that the sequence

Sm(a_−i−m, . . . , a_−i, b_−i−m, . . . , b_−i) _m

>0

is stationary. Denoting the limit by c−i it is true that c = (c−i) is in CW(S),cf. [5], Chapter II, Proposition 1.1. Defining

a+b=c

makesCW(S)into a commutative group with identity(. . . ,0, . . . ,0),cf.[5], Chapter II, Proposition 1.4.

We refer the reader to [5], Chapter II, §1.6 for the natural topology ofCW(S). We note that CW(S)is complete and separable with respect to this topology and that CW^u(S) ={a; a_−i = 0for largei} is a dense subgroup. Moreover, for every morphism of commutative ringsϕ: S→S⁰ the map

CW(ϕ) :CW(S)→CW(S⁰) defined by

CW(ϕ) (. . . , a_−i, . . . , a₀)

= . . . , ϕ(a_−i), . . . , ϕ(a₀)

is continuous. Thus, CW is a functor from the category of commutative rings to the category of topological groups. It can be extended in an obvious way to the category of separable, complete linearly topologized commutative rings,cf. [5], Chapter II, §1.7.

Now we specialize to k-algebras S. In this case, CWk(S) = CW(S) admits a unique structure of topological module overA, such that for all xin k, with Teichmüller lift[x] = (x,0, . . . ,0, . . .)∈A, we have

[x]·a= . . . , x^p⁻ⁱa_−i, . . . , x^p⁻¹a₋₁, a₀ . The operations F, V:CWk(S)→CWk(S)given by

F(a) = (. . . , a^p_−i, . . . , a^p₀), V(a) = (. . . , a−i−1, . . . , a−1)

are additive, continuous and satisfy the relationsF V = V F = p, F α = σ(α),V α=σ⁻¹(α), withα∈A. In other words,CWk(S)is a topological Dk-module. This is all functorial inS.

The group functorCW_k on finitek-algebras is pro-representable.We denote byCW[_k the group scheme that represents it,cf.[5], Chapter II, §4.2.

For any formalp-groupGoverkwe define itsDieudonné module M(G) = Hom_k−gr(G,CW[k)

as the group of formalk-group morphisms from Gto CW[k.

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Remark 1.2. — By viewing morphisms of formal group schemes between GandCW[_k as morphisms of schemes, we obtain an embedding

M(G),→Hom_k−sch(G,CW[k)'CWk(R)

whereR is the affine algebra of G. Moreover, if∆ is the comultiplication ofRthen

M(G) =

a∈CWk(R); CW(∆)(a) =a⊗1 + 1b ⊗ab .

This allows us to view M(G) as a closed topological Dk-submodule of CW_k(R).

All of the standard properties of the classical Dieudonné module theory are proven in [5], Chapter III based on this definition. The main result of this theory, [5], Theorem 1, p. 127, is comprised in the following theorem.

Theorem 1.3. — The functorM sets up a duality of abelian categories between formalp-groups overkand certain topologicalD_k-modules.

Now, for any separable, complete linearly topologized A-algebra S, in particular forA_n-algebras, CW(S) has a natural structure of topological A-module, which is uniquely determined by

[x]·a=

. . . , σ⁻ⁱ([x])a_−i, . . . , σ⁻¹([x])a₋₁,[x]a0

for everyx∈kand everya = (. . . , a_−i, . . . , a0)∈CW^u(S), cf. [5], Chap- ter II, §2.4. We denote thisA-module byCW_A(S).

Recall from [5], Chapter II, §5 and Chapter IV, §3 the following defini- tions and notations:

Definition 1.4.

(i) A p-adicA⁰-algebra R is a separable, complete linearly topologized A⁰-algebra, with the topology being the p-adic one, such that p is not a zero divisor inR.

(ii) A special A⁰-algebra R is a profinite formally smooth A⁰-algebra locally of finite dimension, i.e. a profinite A⁰-algebra whose every local component is isomorphic to a power series ring in a finite number of in- determinates with coefficients in the ring of integers of a finite unramified extension ofK⁰.

For a p-adicA-algebraRone can construct a continuousA-linear map

(1.1) wbR:CWA(R)→ RK

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the topology onRK being thep-adic one, defined by wbR (. . . ,ba−i, . . . ,ba0)

=

∞

X

i=0

p⁻ⁱ(ba−i)^pⁱ which induces anA-linear continuous map

(1.2) w_R:CW_k(R_k)→ R_K

pR defined by

wR (. . . , a−i, . . . , a0)

=

∞

X

i=0

p⁻ⁱ(ba−i)^pⁱ

whereba_−i∈ Ris an arbitrary lift ofa_−i,cf. [5], Chapter II, §§5.1-2.

If (R,m) is a local specialA⁰-algebra then we let Rb^an_K be the separable completion ofRK with respect to the idealsJs=P∞

i=1p⁻ⁱ⁺¹m^is, fors>1, i.e.Rb^an_K = lim

←−RK/Js. IfR is an arbitrary specialA⁰-algebra and if R= QRm is the decomposition of R into local components, we let Rbân_K be Q(Rm)ân_K, where Rm is lim←−(R/I)m/I, with I running through all open ideals ofRcontained in the open maximal idealm. Let us denote byΩA⁰(R) (resp.ΩA⁰(Rbân_K)) the module of continuousA⁰-differentials ofR(resp.Rbân_K).

We let

(1.3) P(R) =n

α∈Rbân_K;d(α)∈ΩA⁰(R)o whered:Rbân_K →ΩA⁰(Rbân_K)is the canonical morphism.

For a specialA-algebraRone can construct anA-linear continuous map (1.4) wb_R:CW_A(R)→Rb^an_K

defined by the same formula as (1.1) above, whose image isP(R), cf. [5], Chapter II, Proposition 5.5, and which induces anA-linear continuous isomorphism

(1.5) wR:CWk(Rk)→ P(R)

pR defined as (1.2) above.

1.1. Group schemes over discrete valuation rings

In this Subsection we review Fontaine’s classification of smoothp-groups and Conrad’s classification of finite flat groups over discrete valuation rings with ramification indexe < p−1.

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Definition 1.5. — LetM be aD_k-module.

(i) LetM⁽¹⁾ be the D_k-module, whose underlying space is M, with A- action given bya·x:=σ⁻¹(a)x, for every a∈Aandx∈M, and withF andV acting as before. ThusF andV can be seen asA-linear maps

M ^V //M⁽¹⁾ , M oo ^F M⁽¹⁾.

(ii) Define MA⁰ to be the direct limit of the following diagram of A⁰- modules

m⊗_AM ^V¹ //

ϕ₀

p⁻¹m⊗AM⁽¹⁾

A⁰⊗_AM A⁰⊗_AM⁽¹⁾

F1

oo

ϕ₁

OO

where the vertical maps are the obvious “inclusions”,V₁(λ⊗x) =p⁻¹λ⊗ V(x)andF1(λ⊗x) =λ⊗F(x), withF,V the usual operators.

(iii) It is obvious how to associate to a Dk-morphism ϕ:M → M⁰ an A⁰-morphismϕ_A⁰:M_A⁰ →M_A⁰0.

Remark 1.6.

(i) More explicitly,M_A⁰ is the quotient ofA⁰⊗_AM⊕p⁻¹m⊗_AM⁽¹⁾ by the submodule

{(ϕ₀(u)−F₁(w), ϕ₁(w)−V₁(u));u∈m⊗_AM, w∈A⁰⊗_AM⁽¹⁾}.

In particular, it is easy to see that any element inMA⁰ can be written as (1⊗m₀,Pe−1

i=1p⁻¹πⁱ⊗m_i),cf. also [3], Lemma 2.2.

(ii) We denote the image of the natural morphismp⁻¹m⊗_AM⁽¹⁾→M_A⁰ byMA⁰[1].

(iii) In the case A⁰ = A there is a canonical isomorphism between M andM_A⁰, via whichMA⁰[1]corresponds toF M. The reader should readM instead ofM_AandF M instead ofM_A[1]in this case, in all the statements we make.

We have the following basic result, cf.[5], Chapter IV, Proposition 2.3.

Proposition 1.7. — The natural map M/F M → MA⁰/MA⁰[1], given by[x]7→[(1⊗x,0)], is an isomorphism ofk-vector spaces.

Let Rbe anA⁰-special algebra. We defined, cf. (1.4), an A-linear mor- phismwb_R:CW_A(R)→P(R). It is clear thatwb_R(CW_A(mR))⊂mR(e<p−1), hence wbR induces a morphism w⁰_R: CWk(Rk) → P(R)/mR, which in

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turn, via extension of scalars, induces a morphismw⁰⁰_R:A⁰⊗ACW_k(Rk)→ P(R)/mR. Now,A⁰⊗ACWk(Rk)surjects onto

CWk(Rk)

A⁰ =:CWk,A⁰(Rk),

cf. [5], Chapter IV, Proposition 2.5. Fontaine proved that w_R⁰⁰ induces an A⁰-linear map

(1.6) w_R:CWk,A⁰(Rk)→ P(R) mR

which is an isomorphism,cf.[5], Chapter IV, Proposition 3.2.

For future reference, we note that for ap-adicA⁰-algebraS we can construct, as in the case ofA⁰-special algebras, anA⁰-linear map

(1.7) w_S:CWk,A⁰(Sk)→ SK

mS starting withwb_S:CW_A(S)→ S_K,cf.(1.1) above.

Now let Gbe a smooth formal p-group over A⁰ and letR be its affine algebra. Then Gk = G×Speck, its special fiber, has affine algebra Rk. LetM = M(G_k) be the Dieudonné module of G_k. We denote by ∆ the comultiplication of R and by ∆b the extension of ∆ to Rb^an_K. Let δ(α) :=

α⊗1 + 1b ⊗αb −∆(α)b forα∈Rb^an_K. Let:

(1.8) L₁=

α∈P(R); δ(α)∈mR⊗b_A⁰R andL=

α∈P(R); δ(α) = 0 . Fontaine proved the following result.

Proposition 1.8.

(i)The natural morphismM_A⁰ →CW_k,A⁰(R_k), induced by the inclusion M ⊂CWk(Rk), is an injection.

(ii) The map wR, cf. (1.6) above, induces an isomorphism w between MA⁰ andL₁/mR.

Remark 1.9. — Conrad proved, cf. [3] last part of Lemma 2.7, that M_A⁰ is an A⁰-submodule of CW_k,A⁰(R_k) also in the case whenM is the Dieudonné module of a finite group.

We now take some time to describe the map w_R on the less obvious part of MA⁰, namely on MA⁰[1] = Im(p⁻¹m⊗AM⁽¹⁾ → MA⁰). Let x= (0, p⁻¹πⁱ⊗a) be in M_A⁰[1], where a = (. . . , a_−j, . . . , a₀) ∈ M⁽¹⁾ ⊂ CWk(Rk)⁽¹⁾andiis between1ande−1. Then, as an element ofCWk,A⁰(Rk) via the natural inclusion M_A⁰ ,→ CW_k,A⁰(R_k), x is equal to (πⁱ⊗b,0), whereb= (. . . , a−j, . . . , a0, a1)witha1∈ Rk an arbitrary element. This is

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so becausep⁻¹πⁱ⊗a=V₁(πⁱ⊗b)inp⁻¹m⊗ACW_k(Rk)⁽¹⁾,cf.Definition 1.5 (ii) and Remark 1.6 (i). Therefore

(1.9) w_R(x) =πⁱ

∞

X

j=0

p^−j−1(ba_−j)^p^j+1 =πⁱβ∈ L1

mR whereba−j∈ Ris any lift ofa−j.

Before we introduce the category of classifying objects, we note that a profiniteD_k-moduleM is aD_k-moduleM which isA-profinite and is such that its openDk-submodules form a fundamental system of neighborhoods of0. Now, the category of classifying objectsΛ^l_A0,cf.[5], Chapter IV, §4.3, is defined as follows:

Definition 1.10.

1)

a. The objects are triples(L, M, ρ), where

i) M is a profiniteDk-module on which the action ofF is injective such that the quotient M/F M is a finite dimensional k-vector space,

ii) Lis a free A⁰-module of finite rank,

iii) ρ: L → M_A⁰ is A⁰-linear such that the induced morphism ρ: L/mL → MA⁰/MA⁰[1] −→^∼ M/F M is an isomorphism of k- vector spaces.

b. A morphism u: (L, M, ρ) → (L⁰, M⁰, ρ⁰) is a couple (uL, uM) with u_L:L → L⁰ (resp.uM:M →M⁰) anA⁰ (resp.Dk) linear morphism such thatuM,A⁰◦ρ=ρ⁰◦uL.

2) The category of smooth Honda systems H^d_A0 over A⁰ has as objects pairs(L, M) withM and L as in 1(a) and with L included in M_A⁰. The morphisms are the obvious ones.

3)We denote bySFA⁰ the category of smooth finite dimensional formal p-groups overA⁰.

Fontaine defined a functor LMA⁰ (resp.LM_A^d0) from the category SFA⁰

(resp. of p-divisible groups over A⁰) to the category Λ^l_A0 (resp. H^d_A0) by LMA⁰(G) = (L, M, ρ)(resp.LM_A^d0(G) = (ρ(L), M)), withρthe composition

(1.10) L,→ L1→ L1

mR

−→∼ MA⁰

whereL,L1andM are as in (1.8) above.

Then he proved, cf.[5], Chapter IV, Theorem 2 (resp. Proposition 5.1), the following theorem.

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Theorem 1.11. — The functorLMA⁰ (resp.LM_A^d0) induces a duality of categories between the categorySFA⁰ (resp. ofp-divisible groups over A⁰) and the categoryΛ^l_A0 (resp.H^d_A0).

We now review Conrad’s classification of finite flat group schemes overA⁰. Definition 1.12.

1) Let SH^f_A0 be the category of finite Honda systems over A⁰ whose objects are triples(L, M, j)where:

i) M is aD_k-module with finiteA⁰-length,

ii) L→^j MA⁰is a morphism ofA⁰-modules such that the naturalk-linear mapL/mL→MA⁰/MA⁰[1]is an isomorphism ofk-vector spaces and such thatV ◦j is injective, where

V:MA⁰ →A⁰⊗AM⁽¹⁾ is induced by the maps

id⊗V:A⁰⊗_AM →A⁰⊗_AM⁽¹⁾, p⊗id :p⁻¹m⊗AM⁽¹⁾ →A⁰⊗AM⁽¹⁾.

A morphism u: (L, M, j) → (L⁰, M⁰, j⁰) is a pair u = (uL, uM), with u_M:M →M⁰ a continuousD_k-linear morphism and u_L: L→ L⁰ anA⁰- linear morphism such thatuM,A⁰◦j=j⁰◦uL.

2) We denote byF FA⁰ the category of finite flat group schemes overA⁰ ofp-power order.

LetGbe a finite flat group scheme overA⁰ ofp-power order, with affine A⁰-algebra R. Let M = M(G_k) be the Dieudonné module of G_k and let L⊆MA⁰ denote the kernel of theA⁰-linear composite map

(1.11) MA⁰ ,→CWk,A⁰(Rk)−→^w^R RK

mR wherewR is the map (1.7) above.

Conrad defined,cf. [3], §3, a functor

LM_A⁰:F FA⁰ →SH^f_A0

by LMA⁰(G) := (L, M). Moreover, he proved, cf. [3], Theorem 3.6, the following theorem.

Theorem 1.13. — The functor LM_A⁰ is fully faithful and essentially surjective.

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2. Smooth p-groups

In this Section we classify smoothp-faithful groups overA⁰_n. We start by defining the category of classifying objects.

Definition 2.1.

1) LetΛA⁰_n be the category whose objects are triples(Ln, M, ρ)where:

i) M is a profiniteD_k-module on which the action ofFis injective such that the quotientM/F M is a finite dimensionalk-vector space;

ii) Ln is a freeA⁰_n−1-module;

iii) ρ: Ln→MA⁰/mⁿ⁻¹MA⁰ isA⁰_n−1-linear such that the induced mor- phism

ρ: Ln

mLn

→ MA⁰

mMA⁰

→ MA⁰

MA⁰[1] → M F M is an isomorphism ofk-vector spaces.

A morphismu: (Ln, M, ρ)→(L⁰_n, M⁰, ρ⁰)is a pair(uL, uM), withuL:Ln→ L⁰_n (resp. u_M:M →M⁰) anA⁰_n−1 (resp.D_k) linear morphism, for which the following diagram is commutative

Ln

u_L

−→ L⁰_n

 y

ρ 

y

ρ⁰ M_A0

mⁿ⁻¹M_A0

u_M,A0

−→ ^M

0 A0

mⁿ⁻¹M⁰

A0

whereu_M,A⁰ is induced by u_M,A⁰. 2)LetΛ^f_A0

nbe the full subcategory ofΛA⁰_nof objects(Ln, M, ρ)such that the “multiplication byp” map [p] :M →M is injective.

3) We denote by SFA⁰_n (resp. SF FA⁰_n) the category of smooth finite dimensional formal p-groups (resp. p-faithful groups) over A⁰_n. Recall, cf.

Definition 1.1, that a smooth formalp-group isp-faithful if the “multipli- cation byp” morphism is faithfully flat.

Now we want to construct a functor

LMn:SFA⁰_n→ΛA⁰_n.

For this, letGbe a smooth formalp-group overA⁰_nand letRbe its affine algebra. ThenG_k=G×Speck, its special fiber, has affine algebraR_k. Let M =M(Gk) be the Dieudonné module ofGk. Let Rbe a smooth A⁰-lift ofR. We know it is unique up to non-unique isomorphism. We denote by

∆(resp.∆k) the comultiplication ofR(resp.Rk). Let∆ :b R → R⊗bA⁰Rbe

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anA⁰-algebra morphism that lifts ∆. We also denote by∆b the extension of∆b toRb^an_K. Forn>r>1 define:

(2.1) Lr=

α∈P(R); δ(α)∈m^rRb⊗A⁰R withP(R)andδas in (1.3) and (1.8) above, respectively.

Remark 2.2. — OurL1is Fontaine’sMHA⁰(G),cf.[5], pp. 166–167 and p. 202.

Lemma 2.3. — The setsLrare independent of the lift∆, i.e. they onlyb depend onGand the liftR.

Proof. — Suppose∆b and ∆b₁:R → R⊗b_A⁰Rare twoA⁰-lifts of∆ which are uniquely extended to Rb^an_K. Let δ and δ1 be the corresponding morphisms. For everyα∈ R we have ∆(α)b ≡∆b1(α) (modmⁿRb⊗R), that is

∆ (α) =b ∆b1 (α) +πⁿxwith x∈ Rb⊗A⁰R. Therefore, since∆b and ∆b1 are morphisms of algebras,

(2.2) ∆ (αb ^pⁱ) = (b∆ (α))^pⁱ = (b∆1(α))^pⁱ+pⁱπⁿy=∆b1(α^pⁱ) +pⁱπⁿy with y ∈ R⊗bA⁰R. So p⁻ⁱ∆ (αb ^pⁱ) ≡ p⁻ⁱ∆b1(α^pⁱ) (modmⁿRb⊗R) for all integers i > 0. In the unramified setting every element of P(R) is an infinite sum of elements of the formp⁻ⁱβ^pⁱ with β ∈ R, cf. (1.4). In the ramified setting the situation is similar, becauseA⁰⊗ACWA(R)surjects onP(R), cf. (1.6). Hence every element ofP(R)is an A⁰-combination of infinite sums of elements of the formp⁻ⁱβ^pⁱ withβ∈ R. Therefore we get

thatδ(α)−δ1(α)∈mⁿRb⊗R.

Letρebe the composition

P(R)^proj.−→P(R)/mR^w

−1

−→R CWk,A⁰(Rk) wherew_R is the map (1.6) above.

The following Lemma follows directly from [5], Chapter IV, Lemmas 1.2 and 4.3.

Lemma 2.4. — Letrbe an integer between1andn−1and letα∈ Lr. There exists an elementγ∈ L1such thatρ(γ)e ∈MA⁰[1]and(α−π^r−1γ)∈ L_r+1.

Letρe₀ be the composition

(2.3) Ln,→ L1

proj.

−→ L1/mR^w

−1

−→MA⁰

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where,wis the map in Proposition 1.8 (ii) above, and letρ₀be the induced map

(2.4) ρ₀: Ln

mⁿ⁻¹L1

→ MA⁰

mⁿ⁻¹M_A⁰. Note that these maps depend only onGand the liftR.

We have the following key result.

Lemma 2.5.

(i)The morphismρ0induces a map

ρ₀: Ln

mLn+mⁿ⁻¹L1

'

Ln

mⁿ⁻¹L1

m· _mn−1^LⁿL1

→

M_A0

mⁿ⁻¹M_A0

m·_mn−1^M^AM⁰_A0

proj.

−→ MA⁰

MA⁰[1]

−→∼ M F M

which is an isomorphism ofk-vector spaces.

(ii)L_n/mⁿ⁻¹L₁is a freeA⁰_n−1-module.

Proof.

(i) Since the surjectivity ofρ₀:Ln/(mLn+mⁿ⁻¹L1)→MA⁰/MA⁰[1]follows from Lemma 2.4,cf.also [5], Chapter IV, Proposition 1.1, all we have to prove is that it is injective. For this, we need to prove thatρe⁻¹(MA⁰[1])∩

Ln = mLn+mⁿ⁻¹L1 (we can view ρ₀ as being induced by ρe:P(R) → CW_k,A⁰(Rk)).

“⊆” Let α ∈ Ln such that ρ(α)e ∈ MA⁰[1]. This means, cf. (1.9), that α=πβ for someβ ∈P(R). Sinceαis inL_n it follows thatβ is inL_n−1. On the other hand, from Lemma 2.4 we know there is aγ∈ L1such that ρ(γ)e ∈M_A⁰[1]and(β−πⁿ⁻²γ)∈ L_n. Hence

α=πβ=π(β−πⁿ⁻²γ+πⁿ⁻²γ) =π(β−πⁿ⁻²γ)+πⁿ⁻¹γ∈mLn+mⁿ⁻¹L₁. This proves the first inclusion. Since the other inclusion “⊇” is obvious, it follows thatρ₀is injective.

For future reference, we observe that ρeinduces a surjective morphism, also denotedρe

(2.5) ρ:e Ln M_A⁰

MA⁰[1]

−→∼ M F M.

(ii) First of all, from (i) and some linear algebra follows thatLn/mⁿ⁻¹L1

is a finite A⁰ (hence A⁰_n−1)-module. From (2.5) above, it follows that ρe induces a surjective morphism ρ1: Ln/mLn−1 → M/F M. Since mLn +