Cyclic Dieudonne Modules

New York J. Math. (1998) 127{136.

Lifting Witt Subgroups to Characteristic Zero

Alan Koch

Abstract. Let^kbe a perfect eld of characteristic^p>0^:Using Dieudonne modules, we describe the exact conditions under which a Witt subgroup, i.e., a nite subgroup scheme of^Wn, lifts to the ring of Witt Vectors^W(^k).

Contents

1. Cyclic Dieudonne Modules 128

2. Finite Honda Systems 130

3. Thep-rank 1 Case 131

4. Lifts of Witt Subgroups 133

References 135

Letkbe a perfect eld, chark=p >0:LetR be a complete discrete valuation ring of characteristic 0 with residue eldk:SupposeGis a nite ane commutative k-group scheme of p-power rank. Under what conditions does G \lift" to R? In other words, when does there exist anR-group scheme ~Gwhich is a free commuta- tive group scheme ofp-power rank overR(hereafter referred to as a nite p-group as in F2]) so that ~G ^Spec(R⁾Spec (k)=G? There are instances where the answer to this lifting question is clear. If G is etale, for example, then G Spec (k) is isomorphic to a direct sum of pⁿ's for various n, where pⁿ is the group scheme that gives thep^nthroots of unity for a givenk-algebra. pⁿ clearly lifts toRfor all R: it lifts to thep^nthroots of unity functor overR:Since the question of lifting is preserved under base change OM, 2.2] we have that Glifts. As another example, ifGis of multiplicative type,Gwill always lift toRsince thenG is etale (where G = Homk^;gr(G

G

m) is the linear dual of G) and lifting is preserved by duality.

Any nite ane commutativek-group scheme decomposes into a direct sum of an etale scheme and a connected scheme. The connected group scheme decomposes further into a group scheme of multiplicative type and a group scheme that is unipotent W]. Thus the question of lifting is only of interest when G is both connected and unipotent. In the language of Hopf algebras, this simply means that H and its dual Hopf algebraH are localk-algebras, whereG= Spec (H):

Received January 19, 1998.

Mathematics Subject Classication. 14L.

Key words and phrases. Group Schemes, Dieudonne Modules, Witt Vectors.

c1998StateUniversityofNewYork

ISSN1076-9803/98

127

In 1968, Oort and Mumford OM] were the rst to show that, for all such group schemes G, there is a complete discrete valuation ring R so that G lifts toR. In other words, they showed that all nite ane commutative group schemes lift to characteristic zero. However, it is known that not every group scheme lifts to every suchR: the best known example being p the unique connected unipotent group scheme of rank pover k. p will lift only to rings which admit a factorization of pinto elements in the maximal ideal TO]. Thus this group scheme can not lift to

Z

p the ring ofp-adic integers, or for that matter any unramied extension of

Z

p. More generally, it was shown in 1992 by Roubaud R, p. 72] that, forp5anyG will lift to anyR with ramication index 1< ep^;1:

We shall focus our attention on the case e= 1. k-group schemes that can lift when e = 1 lift in the strongest possible sense, i.e., such group schemes will lift to any discrete valuation ringR with residue eld`k: These discrete valuation rings arise as the ring of Witt Vectors over somek, which shall be denotedW(k).

The issue we address is the following: forGa connected subgroup scheme of Wn

(the group scheme of Witt Vectors of nite length n), when doesGlift to W(k)?

The collection of subgroups that do lift to W(k) is surprisingly easy to describe when the question is described in terms of the Dieudonne module associated to the group scheme and we shall see that the question ofGlifting is equivalent to being able to identify the structure of much smaller group schemes.

The connected subgroups ofWn (called the Witt subgroups) correspond to the subclass of Dieudonne modules that are cyclic that is, modules that are of the form E=I for some idealI E, whereE is the non-commutative ringW(k)FV] modulo some relations. We start by recalling a classication of cyclic Dieudonne modules, paying special attention to the modules that are killed byp. The process we shall use to lift these Witt subgroups was developed by Fontaine in F2] using what are called \Finite Honda Systems." Then, we determine exactly which of the modules killed by pcorrespond to group schemes that lift. Finally, we answer the lifting question for all Witt subgroups.

Throughout this paper, letpbe a xed odd prime. Unless otherwise specied, all group schemes overkwill be nite, ane, commutative, connected, and unipotent.

The author would like to thank the referee for many helpful suggestions.

1.

Cyclic Dieudonne Modules

LetGbe ak-group scheme. LetEbe the Dieudonne ring associated tok, that is Eis the non-commutative ringW(k)FV] with the relationsFV =V F =p Fw= wF andwV =V w with w²W(k) andw dened by raising each component of w to thep^th power. ToG we can associate an E-moduleD (G) via D (G) = Homk^;gr(GC) whereC is the E-module functor of Witt Covectors as described in F1, p. 1273]. D induces an anti-equivalence between connected unipotent group schemes and E-modules killed by a power ofF andV. These modules will be called Dieudonne modules. If we do not insist on G being nite or connected (but still ane, commutative, and unipotent), we still have a correspondence, now between group schemes and E-modules killed by a power of V. Details on this correspondence can be found in DG, V ^x1 4.3]. Since D is an exact functor and D (Wn) = E=E(Vⁿ) DG, V ^x1 4.2], it is easy to see that Witt subgroups correspond precisely to cyclic Dieudonne modules. Note that Wn is viewed as a

unipotent group scheme via

Wn(A) =^f(a⁰a¹:::an^;1) ^j ai²A^g

for anyk-algebraA, with group operation induced from the law of addition of Witt vectors.

We begin with a survey of the results in K]. The general structure of a cyclic Dieudonne module begins with the classication of cyclic Dieudonne modules killed byp:Each of these modules ts one of the following two forms:

E=E(F^n;V^mp)

(1) E=E(FⁿpV^m)

(2)

where ²k: (Moreover,E=E(F^n;1V^mp)=E=E(F^n;2V^mp) if and only if there is ana²k such that¹=a^pn+m;12but this will not be needed for the results that follow.)

We will call these two forms type 1 and type 2 respectively. One major dierence between the two types is the following:

Lemma 1.1.

A cyclic Dieudonne module killed by p is of type 1 if and only if kerV = imF:

Proof.

Let M be a cyclic Dieudonne module killed by p and x = 1M so M is generated as an E-module byx: It is clear that imF kerV asV Fx =px = 0: SupposeM is of type 1. ThenM=E=E(F^n;V^mp) for somemn >0 ²k: M has ak-basis^fxFxF²x:::FⁿxV xV²x:::V^m;1x^g:Lety²kerV:We can write

y=^Xn

i⁼⁰aiFⁱx+^mX;1

j⁼¹bjV^jx with all of theai's and bj's in k. ApplyingV gives

V y=a^p0;1V x+^mX;1

j⁼¹ b^p_j^;1V^j+1x

=a^p0;1V x+^Xm

j⁼²b^p_j^;1;1V^jx

=a^p0;1V x+^mX;1

j⁼² b^p_j^;1;1V^jx+b^p_m^;1;1;1Fⁿx= 0

Byk-linear independence, this means a⁰=b¹=b²=b³ ==bm^;1 = 0: Thus we must have

y=^Xn

i⁼¹aiFⁱx:

hencey ²imF:

Conversely, if M=E=E(FⁿpV^m), i.e M is of type 2, it is clear that kerV ⁶= imF as V^m;1x²kerV butV^m;1x =²imF:

More generally, let M be a cyclic Dieudonne module of p-rank h. The term p-rank will be used to signify the smallest positive integer hsuch that p^hM = 0: M can be decomposed into a short exact sequence

0 ^;;;;! M^{0 ;;;i;!} M ^;;;;! M^{00 ;;;;!} 0

whereM⁰ =p^h;1M, M⁰⁰=M=p^h;1Mand is the natural projection. Note that M⁰ and M⁰⁰ are cyclic ofp-ranks 1 andh^;1 respectively. From this we can see that the construction of cyclic modules ofp-rankhcan be obtained by nding cyclic Dieudonne modulesM⁰ andM⁰⁰ ofp-ranks 1 andh^;1 so that there is a sequence

0 ^;;;;! M^{0 ;;;f;!} M ^;;;g;! M^{00 ;;;;!} 0

so thatf(z) =p^h;1x andg(x) =y, wherex y, andz generateM M⁰⁰and M⁰ respectively asE-modules.

Given cyclic modulesM⁰ and M⁰⁰ ofp-ranks 1 andh^;1 respectively, it is not always true that we can construct anM to t into the short exact sequence above.

The following gives a necessary (but not sucient) condition onM⁰ andM⁰⁰:

Lemma 1.2.

^{LetM0 andM00}be cyclic Dieudonne modules ofp-ranks1 andh^;1 respectively, h2:Suppose there is a short exact sequence

0 ^;;;;! M^{0 ;;;f;!} M ^;;;g;! M^{00 ;;;;!} 0

so that M has p-rankh,f(z) =p^h;1xandg(x) =y, wherex y, and z generate M M⁰⁰ andM⁰ respectively. If F^{`</sup>y=V<sup>r</sup>y, thenF<sup>`}z=V^rz:

Proof.

IfF^{`</sup>y=V<sup>r</sup>y then (F<sup>`;}V^r)x ²kerg= imf: Thus there is ane²E such thatF^`x^;V^rx=ep^h;1x:Thus

f(F^{`</sup>z<sup>;</sup>V<sup>m</sup>z) =p<sup>h;1</sup>(F<sup>`}x^;V^mx) =ep^2h;2x= 0 since 2h^;2hforh2:ThusF^`z=V^mz:

We can categorize cyclic Dieudonne modules by picking modules killed bypthat satisfy the above short exact sequence. If we pick an M⁰ and an M⁰⁰ killed by p we get a moduleM killed by p²: If we then pick a dierentM⁰ and setM⁰⁰ =M, we get a new module M killed by p³, and so on. By the repeated selection of cyclic modules killed bypin this manner we can obtain a complete classication of cyclic Dieudonne modules. (Note that, for a givenM⁰ andM⁰⁰, theMconstructed is usually not unique.) Thus we can associate to each cyclic Dieudonne module ofp-rankha sequenceM⁰M¹:::Mh^;1 of cyclic Dieudonne modules killed byp. Each of theseM_i's can be recovered fromM: M_i =pⁱM=pⁱ⁺¹M:A consequence of the above lemma is that ifMi=E=E(F^n;V^rp) andMj =E=E(F^n0;0V^r0p) withi < j, thennn⁰ andrr⁰:This observation will be important in Section 4.

2.

Finite Honda Systems

Having described the construction of a cyclic Dieudonne module, we now focus on the tool used for nding lifts of group schemes toW(k)namely the nite Honda systems. Finite Honda systems were rst developed by Fontaine in F2] in a manner analogous to (and relying heavily on) Honda's method to liftp-divisible groups.

Denition 2.1.

A nite Honda system overW(k) consists of a pair (ML), where M is a Dieudonne module andLis aW(k)-submodule ofM so that

i) kerV ^\L= 0

ii) The canonical map L^! M ^!cokerF is an isomorphism, whereL and M denote reduction modp.

By a slight abuse of notation, we shall often identifyLwith its image in cokerF and write condition (ii) as L=pL = M=FM: A morphism (M¹L¹) ^! (M²L²) consists of an E-module map ' : M^{1 !} M² such that '(L¹) L²: Thus the collection of nite Honda systems over kforms a category, which we shall denote FH(W(k)k).

The lifting theory works as follows. Suppose ~Gis aW(k)-group scheme lifting the k-group schemeG= Spec (H):LetM=D (G) = Homk^;gr(GC):Then elements ofM are in one-to-one correspondence with HomHopf^;alg(DH)the Hopf algebra homomorphisms D ^! H, where C = Spec (D): The set of all such maps is a subgroup of Homk^;alg(DH)=C(H) so we can embedM ,^!C(H):Now forK the fraction eld ofW(k) we have a mapwH :C(H)^!(H^N_W⁽_k⁾K)=H dened by

wH(:::h^;2h^;1h⁰) =^X1

i⁼⁰

~h^p;i_i pⁱ⁺¹

where ~h^;_i is a lift ofh^;_itoW(k). (It is easy to see that the map does not depend on the choice of lift.) LetL= kerwH^jM:Then (ML) is a nite Honda system.

Conversely, given a nite Honda system (ML) the nitep-group ~GoverW(k) it determines is given by, for any niteW(k)-algebraA,

G~(A) =^f2G(A=pA)^jC()(L)kerwA^g

whereM=D (G):

It can be shown that morphisms between nite Honda systems induce morphisms on the W(k)-group schemes associated to them, and hence the correspondence outlined above determines a categorical anti-equivalence betweenFH(W(k)k) and the category of nitep-groups overW(k):As (FH(W(k)k) is an abelian category F2, Cor. 1], so is this category of W(k)-group schemes. Thus the kernel and cokernel of any morphism of two nite p-groups overW(k) must also be a nite p-group.

Note that these systems are a special case of a more general systemFH(Rk) over any discrete valuation ringRof characteristic zero with residue eldk. The objects inFH(Rk) consist of quintuples (MM⁰fvL) withf :M^!M⁰ v :M^0!M so thatfv =p1M⁰ and vf =p1M and L M⁰:The system described above correponds to the caseM =M⁰ =D (G) f =F v=V:See R] for a complete description of these modules.

3.

The

^p

-rank 1 Case

We start the application of Fontaine's theory to cyclic Dieudonne modules by dealing with the simplest type of cyclic modules, namely the p-rank 1 case. Here we can quickly determine which of the modules lift toW(k):

Lemma 3.1.

Let M be a cyclic Dieudonne module killed by p. Then M lifts to W(k) if and only if M is of type 1.

Proof.

We shall explicitly either construct theLnecessary to have a nite Honda system, hence to have a lifting ofG, or show that no suchLcan exist.

Type 1: Let M=E=E(F^n;V^mp), and letx= 1Mi.e.,x is a generator of M. We can quickly nd cokerF: M=FM=E=E(FV^m):LetLbe generated overW(k) by^fxV xV²x:::V^m;1x^g:AsL^\FM= 0 and imF = kerV,L^\kerV = 0, and it is clear by the denition of Lthat L=L=pL=M=FM. Thus (ML) satises the properties of a nite Honda system, soGlifts toW(k).

Type 2: Suppose we have an L so that (ML) is a nite Honda system. Write M=E=E(FⁿpV^m): ThenM=FM =E=E(FV^m) Clearly dimkM =n+m^;1 and dimkM=FM = m: Thus dimkL=pL= dimkL = m: But kerV has a k-basis

fFxF²x:::F^n;1xV^m;1x^gand hence dimkkerV =n:Thus, dimk(L+ kerV) =n+m >dimkM

which is absurd. Thus noLcan exist to make (ML) a nite Honda system, hence the Witt subgroup corresponding toM does not lift.

In the type 1 case, theW(k)-submodule is not unique { in fact there are many other possible choices forL.

Corollary 3.2.

^{Let M = E=E(Fn ;Vmp) x = 1}M: Let L⁰ be the W(k)- submodule generated by

f(1^;Fe⁰)x(V ^;Fe¹)x(V^2;Fe²)x:::(V^m;1;Fem^;1)x^g ei²E Then (ML⁰) is a nite Honda system.

Proof.

^{If we take L to be the W(k})-submodule generated by ^fxV xV²x:::

V^m;1x^g then by the lemma (ML) is a nite Honda system. As Vⁱx (V^i;Fei)x ( modFM ) it is clear that L⁰ =M=FM. Since FV x = px = 0 it follows thatV L⁰=V Lso kerV ^\L⁰ must be zero.

We shall refer to this corollary in the proof of Theorem 4.1.

Example 3.3.

It was stated in the introduction that the group schemepdoes not lift toW(k): pis a Witt subgroup aspembeds naturally in

G

a=W¹:Lemma 3.1 provides a quick proof that it does not lift. As p is the unique k-group scheme of rank p,D (p) must be the unique simple object in the category ofE-modules, henceD (p)=E=E(FV)=k: SinceE=E(FV) is of type 2,p does not lift to W(k):

Example 3.4.

On the other hand, the simplest Witt subgroupGthat does lift is the one so that D (G) = E=E(F ^;Vp): This group scheme is characterized as follows: for anyk-algebraAwe have

G(A) =^fa^ja²A a^p2 = 0^g with a+Gb=a+b^;(a^p+b^p)^p

p

with the addition on the right-hand side determined by the addition in A. The group scheme it lifts to is given by, for any niteW(k)-algebraR,

G~(R) =^fr^j r²R=pR r~^p2+pr~²p²Rfor ~ra lift ofr^g with addition dened in the exact same way.

4.

Lifts of Witt Subgroups

Finally, we are in a position to completely answer the question of lifting Witt subgroups to W(k): We shall show that the question of lifting M is answered by examining the structure of theMi's.

The following theorem shows not only which Witt subgroups lift, it also provides a nite Honda system.

Theorem 4.1.

LetGbe a Witt subgroup,M =D (G):Lethdenote thep-rank of M, and set Mi =pⁱM=pⁱ⁺¹M i= 012:::h^;1:Then G lifts toW(k) if and only if Mi lifts for all0ih^;1:

This, when proved, will immediately give

Corollary 4.2.

^Glifts if and only if all of the Mi's are of type1.

Proof of 4.1.

We can separate all cyclic Dieudonne modules into two distinct cases:

Case 1: M is constructed by a series of cyclic modules killed by p, at least one of which is type2. Pickiso thatM_i is a type 2 module.

We shall show that ifM lifts, then so mustpⁱM=pⁱ⁺¹M:IfMlifts, then there is anL so that (ML) is a nite Honda system. We shall denote the corresponding W(k)-group scheme by ~G: Dene the morphism pⁱ] of p-groups over W(k) by pⁱ]A(g) =g+g++g (pⁱ times) forAaW(k)-algebra andg²G(A):Since the category of nitep-groups is abelian, pⁱ] induces the following short exact sequence of nitep-groups overW(k)

0 ^;;;;! pⁱ] ~G ^;;;;! G^{~ ;;;;!} G=^~ pⁱ] ~G ^;;;;! 0: This corresponds to a short exact sequence of nite Honda systems

0 ^;;;;! (pⁱML⁰) ^;;;;! (ML) ^;;;;! (M=pⁱML⁰⁰) ^;;;;! 0 for some choice ofW(k)-modulesL⁰L⁰⁰:Applying a base change to group schemes from W(k) to k commutes with pⁱ] and under this base change (ML) (resp.

(pⁱML⁰), (M=pⁱML⁰⁰)) corresponds to M (resp. pⁱM M=pⁱM). Thus we have nite Honda systems for pⁱM and M=pⁱM hence they correspond to liftable k- group schemes.

If we replaceM withpⁱMand leti= 1we get thatpⁱM=pⁱ⁺¹Mcorresponds to a liftable group scheme. AsMi is of type 2, it does not lift, hence neither doesM:

Case2: Mis constructed by a series of type1 modules killed byp. We will construct a specic nite Honda system forM after rst setting down some notation.

Letx= 1M:SinceM is constructed of type 1's, we have Mi=E=E(F^ni;iV^mip)

i ²k , 0ih^;1 withmh^;1mh^;2mh^;3m⁰=m. For notational convenience, we shall also dene mh = 0. Let mi =mi^;mi⁺¹. Now, for all i, (pⁱiV^mi;pⁱFⁿⁱ)x0 ( modpⁱ⁺¹ ), hencepⁱ(iV^mi;F^ni;pi)x= 0 for some i ² E. Dene fi = V^{mi ;}_i^;1(Fⁿⁱ +pi), 0 i h^;1, and fh = 1. Thus pⁱfi = 0 but p^i;1fi ⁶= 0, and the elements p^i;1V^jfi for 0 j mi^;1 form a k-basis forM_i^;1=FM_i^;1.

LetLbe theW(k)-submodule consisting of all elements of the form

h^X;1 i⁼⁰

mXh^;i^;1

j⁼¹ aijV^j;1fh^;ix aij ²W(k) p^h;i+1 not dividingaij for allj:

We shall show that (ML) is a nite Honda system. We shall use the term V-degree on a monomial to give its power of V modulop. It is easy to check that theV-degree of the termaijV^j;1fh^;ixisj^;1+mh^;i:We claim that each term in this double sum has one power ofV less than the next term (when we order in the obvious way): clearly this is true for the terms withj < mh^;i^;1:Ifj= mh^;i^;1 then this term hasV-degree

m_h^;_i^;1;1 +m_h^;_i=m_h^;_i^;1;m_h^;_i^;1 +m_h^;_i=m_h^;_i^;1;1: Let s be the smallest positive integer such that mh^;s >0: Then the following term is ai⁺s⁰V⁰fh^;i^;s which has V-degreemh^;i^;s = mh^;i^;1 and the claim is proved.

The smallestV-degree is 0 and the largest ism^0;1 =m^;1:ThusLis generated as aW(k)-module by

f(1^;Fe⁰)x(V ^;Fe¹)x(V^2;Fe²)x:::(V^m;1;Fem^;1)x^g

for the appropriate choice of ei: Since M=FM =M=FMwhere M =M=pM, it follows from Corollary 3.2 thatM=FM =L=pL:

To show kerV^\L= 0suppose there exists a nonzero²LwithV = 0:Write =^hX;1

i⁼⁰

mXh^;i^;1

j⁼¹ aijV^j;1fh^;ix:

Then

V =^hX;1

i⁼⁰

m^Xh^;i^;1

j⁼¹ bijV^jfh^;ix= 0

wherebij =a_ij^;1:Since thebij are not all zero, we can nd a nonnegative integer` so thatp<sup>`j</sup>bij for allijand is the largest`with this property. Of course,`h^;1 by the denition of the aij's. Writingbij =p^`cij gives us

V =^hX;1

i⁼⁰

mXh^;i^;1

j⁼¹ cijV^jp^`fh^;ix= 0:

Ifih^;`we have seen thatV<sup>j;1</sup>p<sup>`</sup>fh^;ix= 0so we may write this sum as V =^h;X`;1

i⁼⁰

m^Xh^;i^;1

j⁼¹ cijV^jp^`fh^;ix= 0:

This is an element ofp<sup>`</sup>M so we may project it ontoM` and we obtain =^h;X`;1

i⁼⁰

mXh^;i^;1

j⁼¹ cijV^jfh^;iz= 0

wherez= 1M`:The highestV-degree inis theV-degree ofch<sup>;</sup>`^;1m`V<sup>m`</sup>f`<sup>+1</sup> which is m`: Since the collection of allV^jfh^;iz's arek-linearly independent, 0 i h^;`<sup>;</sup>1 1 j mh<sup>;</sup>i<sup>;1</sup> (all of the terms have a dierent V-degree and V<sup>m</sup>M` ⁶= 0), and it is clear that cij = 0 for all i and j, i.e., p divides cij contradicting our choice of`:Thus =²kerVand the theorem is proved.

Remark 1.

While the statement of the theorem is quite simple, the constructed L is rather complicated. One might hope that theW(k)-submodule L⁰ generated by ^fxV xV²x:::V^m;1x^g might also lead to a nite Honda system. It can be shown that (ML⁰) is a nite Honda system when all of the Mi are isomorphic, however the following example shows that this result does not hold for more general M.

Example 4.3.

Let M = E=E(F^3;V³pF ^;pVp²): Here L⁰ is generated by

fxV xV²x^g:While it is clear that M=FM =L⁰=pL⁰ we have thatpV x ²L^0\ kerV:

However, sinceM=pM =E=E(F^3;V³p) is of type 1, we can construct a lift.

By the construction given in the theorem,Lis generated by^fx(V^;F)x(V^2;p)x^g: Notice how the problem withL⁰is cleared up withL: instead ofpV xwe now have p(V ^;F)xwhich is already zero. In fact,Lis constructed by starting withL⁰and adjusting terms in such a way so that anything that could be in the kernel of V is already zero. It is because of this that we believe that this L is the \simplest"

general formula for constructing a lift.

References

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280, 1975, 1423{1425.

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Department of Mathematics, Hope College, P.O. Box 9000, Holland, MI 49424- 9000Current address:Department of Mathematics, St. Edward's University, 3001 S. Con- gress Ave., Austin, TX 78704-6489

[email protected] http://www.cs.stedwards.edu/~koch/

This paper is available via http://nyjm.albany.edu:8000/j/1998/4-9.html.