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Minor Thesis: Basic Dieudonn´e Theory

Jonathan Pottharst

May 28, 2004

Contents

1 Introduction1

2 Review and Conventions2

2.1 The Base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.2 Group Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.3 Finite Group Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3

2.4 Formal Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Dieudonn´e Modules, and the Dictionary5

3.1 Prologue: Gabriel"s Duality Theory . . . . . . . . . . . . . . . . . .. . . . . 5

3.2 Frobenius and Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6

3.3 Witt Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.4 The Dieudonn´e Module Functor . . . . . . . . . . . . . . . . . . . . . . .. . 8

3.5 The Dictionary, Special Case . . . . . . . . . . . . . . . . . . . . . . . .. . . 10

3.6 The Dictionary, More Generally . . . . . . . . . . . . . . . . . . . . . .. . . 11

4 Classification of Dieudonn´e Modules13

4.1 Classification Up To Isogeny . . . . . . . . . . . . . . . . . . . . . . . .. . . 13

4.2 Classification Up To Isomorphism . . . . . . . . . . . . . . . . . . . .. . . . 15

5 Abelian Varieties and Barsotti-Tate Groups 16

5.1 Barsotti-Tate Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 16

5.2 The Barsotti-Tate Group of an Abelian Variety . . . . . . . . .. . . . . . . 17

5.3 The Dieudonn´e Module of an Abelian Variety . . . . . . . . . . . .. . . . . 18

5.4 Consequences Over Finite Fields . . . . . . . . . . . . . . . . . . . .. . . . . 19

1 Introduction

The aim of this minor thesis is to gain some understanding of the basics of Dieudonn´e theory: what background information goes into it, what the construction looks like, and what its applications are. For simplicity, the scope is limited to the base being a perfect field of positive characteristic. The primary reference for this material has been [5], and we have 1 followed it closely, especially in the last two sections; moral support was provided by [2]. Background material on group schemes was found in [9], [10],[6], and the encyclopedic [1]. As the story goes, the study of group schemes in characteristic 0 is fairly predictable. One has the Lefschetz principle for algebraically closed bases, and Galois/descent theory for the rest. Smooth, connected group schemes may be analyzed effectively by their Lie algebras, while finite groups scheme are always ´etale. In characteristicp, the situation changes considerably. Even over a perfect field, the Lie algebra does not characterize up to local isomorphism, and there are non-´etale finite group schemes. An interesting substitute for the Lie algebra, found by J. Dieudonn´e, is a module over a torsion-free, noncommutative ring of operators. The association to a group of its Dieudonn´e module ends up being an anti-equivalence ofcategories in several cases, allowing classification and investigation of certain aspects from new viewpoints. It is this construction, and some immediate applications, that we view in this paper. In the second section, we review finite and formal group schemes over a perfect field, and in general lay out the language we use to describe them in thispaper. It is recommended that the reader at least glance over this, so that no confusion arises. In the third section, describe each of the ingredients of the Dieudonn´e module functor. We sketch the proof of the equivalence of categories that it provides in a particular case, and state the most general result that applies to our setup. In the fourth section, we describe Dieudonn´e"s classification of modules up to isogeny, and Manin"s work on classification up to isomorphism. Most of the work in this section applies only when the base is algebraically closed. In the fifth section, we describe Barsotti-Tate groups, and describe how they andtheir Dieudonn´e modules can be used to glean information about abelian varieties.

2 Review and Conventions

In this section we list our conventions and assumptions, anddescribe the more common facts about group schemes that we use in the course of the paper.

2.1 The Base

For the entirety of this paper,kdenotes aperfectfield of characteristicp >0. Often we requirekto be algebraically closed, but this assumption will be madeexplicit when it is needed. We keepS= Speck, and writeSch/kfor the category ofS-schemes and morphisms overS.

2.2 Group Schemes

As is customary, agroup schemeis defined to be a group object inSch/k. We shall write Grp/k(resp.Ab/k) for the category of group schemes overk(resp. commutative ones), with morphisms beingS-homomorphisms. There is an easy-to-formulate notion of kernel, but the notion of cokernel inGrp/kis more subtle. AlthoughAb/kis not an abelian category, a general method of analysis is to embed it the abelian categoryAb(Sfppf) of abelian sheaves over the fppf site ofS; here we may talk about cokernels, exactness, etc. 2 It turns out that the quotient of sheaves represented by group schemes is representable by a group scheme, and we call this representing object the quotient group scheme. When we omit the category from the subscript of Hom ?(G,G?), it is safe to assume that the morphisms take place inAb(Sfppf). We shall tacitly assume that every group scheme we refer to isoffinite type, unless noted otherwise. We also deal exclusively withcommutativegroup schemes. Since our base is a field, all finite group schemes will be flat, and so we usually avoid mention of this property. An affine group scheme is the Spec of a finite-typek-Hopf algebra. We writeAffAb/k for the category of (commutative) affine group schemes. For anyG= SpecA?AffAb/k, we write?A(or?G, or even?for short) for the augmentationA→k, andIA= ker?A(orIG orI) for the augmentation ideal. The augmentation ideal definesthe trivial closed subgroup scheme ofG, which is of course SpecA/IA= Speck=S. Resultingly, if?:G→G?is a homomorphism of affine group schemes induced byf:A?→A, then ker?is defined by the idealA·f(IA?).

2.3 Finite Group Schemes

We call an affine group schemeG= SpecAfinitewhenAis finite-dimensional overk, and writeFinAb/kfor the corresponding category. SupposeG= SpecA?FinAb/k. We define theorderofGto be dimkA. IfAsep?Adenotes the maximal separablek-algebra in A, then the idealI0=A·IAsepdefines the maximal connected subgroup scheme ofG. We have the exactconnected-´etale sequence,

0→G0→G→Get→0,

whereG0= SpecA/I0as above, andGet= SpecAsepis the maximal ´etale quotient. Fortunately,FinAb/kis an abelian category. SettingA?= Homk-mod(A,k), andG?= SpecA?, the Hopf algebra structure onAgives rise to a dual Hopf algebra structure onA?, and thus a group scheme structure onG?. This procedure, called Cartier duality, defines an exact anti-equivalence ofFinAb/kwith itself. We also have the reduced-local structure theory. The connected-´etale sequence forG splits, and we change the notation slightly. We can uniquelywriteA=Ared?kAloc(or G=Gred×SGloc), withAredreduced andAloclocal; thenGet=GredandG0=Gloc. One has Hom k-alg(Ared,Bloc) = Homk-alg(Aloc,Bred) = 0, and we can split up our category as

FinAb/k=FinAbred/k?FinAbloc/k.

In fact, using Cartier duality, we can further subdivide ourcategory into where, e.g.,FinAbrl/kconsists of thoseG?FinAb/kwithGreduced (´etale) andG?local (connected). It is easy to describe the first three direct summands in eqn. 2.1 above. To do this, fix a separable closureksepofk, and putπ= Gal(ksep/k). Then basic facts about ´etale morphisms show that there is an equivalence of categories betweenFinAbred/kand finite (discrete)π- modules, given byG?→G(ksep), with inverse given byM?→Spec Homπ-set(M,ksep). 3 The dissection ofFinAbred/kinto parts with reduced and local duals corresponds to the dissection of the category of finiteπ-modules into itsp-unit andp-primary components, respectively. Thus,FinAbrr/kandFinAbrl/kconsist of groups of the above form, of prime- to-pandp-power order, respectively. The categoryFinAblr/kis obtained fromFinAbrl/kby taking Cartier duals, so that we have a description of it too. We go one step further, and apply Pontryagin duality (given by Hom Ab(·,Q/Z)) to the category of finite abelian groups to get a covariant equvalence of FinAb lr/kwith the category of finite,p-power orderπ-modules. In particular, whenkis algebraically closed, we arrive at the following conclusion. Proposition 2.1LetGbe a reduced, commutative, finite group scheme over an algebraically closed field. ThenGis nothing other than a finite group. If, instead of assumed reduced,G is connected with reduced dual, thenGis just the Pontryagin dual of a finite abelianp-group, and hence a abelianp-group. Whenkis not algebraically closed, the above classification boilsdown to the study of finite-image Galois representations. We have said nothing about how to classify the objects ofFinAbll/k, of course. One basic result along these lines, which will be an important input into the Dieudonn´e theory later on, is the following consequence of the general theoryof unipotent algebraic groups, and is valid even forknot algebraically closed. Proposition 2.2There is a unique simple (nonzero with no nontrivial subobjects) element ofFinAbll/k. It is given byαp= Speck[x]/(xp),Δ(x) =x?1 + 1?x,ι(x) =-x.

2.4 Formal Groups

Let (A,mA) be a (noetherian) localk-algebra, andX= SpecA. We define theformal schemerepresented by its completion (?A,mbA), written?X= Spf?A, to be the restriction of the functor of points ofSto the category of localk-algebrasBof finite length. One finds easily that (Spf?A)(B) = lim-→Homk-alg(A/miA,B).

There is a unique extension of Spf

?Ato complete noetherian localk-algebrasB, via the formula (Spf?A)(B) = lim←-lim-→Homk-alg(A/miA,B/mjB). We shall generally consider a formal scheme to be a functor onsuch rings. We writeFmlSch/kfor the category ofaffineformal schemes overk. As is standard, we define aformal groupto be a commutative group object inFmlSch/k; we writeFmlAb/k for the category of (commutative) formal groups overk. A result of Gabriel shows that FmlAb/kis an abelian category; clearly it containsFinAbloc/kas a full subcategory (since any artinian local ring is complete). Moreover, it is easy to verify that every formal group is the direct limit of its finite subgroups, giving the embedding FmlAb/k?Ind(FinAbloc/k) = Ind(FinAblr/k)?Ind(FinAbll/k).(2.2) 4 In particular, this allows us to view the completion?Gof an arbitrary group schemeGas the direct limit of its finite, connected subgroup schemes. It is worth pointing out that Ind(FinAbloc/k) contains all formal groups coming from nonnoetherianA, as well as many other group ind-schemes. A formal group that lies in the first (resp. second) direct summand eqn. 2.2 is called a toroidal group(resp.Dieudonn´e group); to denote thatGis a group of this type, we write G?FmlAblr/k(resp.G?FmlAbll/k). We have, clearly,FmlAb/k=FmlAblr/k? FmlAb ll/k. (We will generally abuse language use to the terms "toroidal group" and "Dieudonn´e group" to refer to group ind-schemes.) Suppose thatkis algebraically closed. Then, in the notation of the preceding section, we haveπ= 1, so thatFinAblris dual to the category of finite abelianp-groups. Using this fact, one can prove the following result of Dieudonn´e. Proposition 2.3LetGbe a toroidal group over an algebraically closed field. ThenGmay be embedded as a closed subgroup of??Gm? nfor somen≥0. If moreoverGis reduced, then

Gmust be isomorphic to??Gm?

n. In the not algebraically closed case, one can imagine that the classification of toroidal groups is dual to the study of continuous representations ofGalois groups on profinite groups (with a special subclass coming from ind-groups that are "actual" formal groups). As in the finite case, we have done nothing to describeFmlAbll/k. In fact, producing a satisfactory description requires the delicate methods ofDieudonn´e theory, as developed in chs. 3 and 4 of this paper. Lastly, we point out that Cartier duality extends naturallyto an anti-equivalence between Ind(FinAb/k) and Pro(FinAb/k), preserving the "ll" and "rr" parts, and swapping the "lr" and "rl" parts.

3 Dieudonn´e Modules, and the Dictionary

In this section we view the Dieudonn´e correspondence for finite (as well as ind-finite and pro-finite) commutativep-group schemes. We draw the picture carefully, because all other instances of Dieudonn´e theory build on this basic one, and fundamentals must always be treated respectfully. We begin with a fairly abstract theorem of Gabriel, which provides the mechanism of our theory. The rest of the section is more concrete, carrying out the specific computations of group schemes necessary to make the correspondence precise.

3.1 Prologue: Gabriel"s Duality Theory

In this subsection we spoil the plot, so to speak. One could set up the entire theory first, and then pull this result out of a hat, thus miraculously saving the day. However, I think it is nice to keep this in mind as the goal the whole time, at least to motivate all the work with group schemes that is to follow. 5 For this subsection,Cdenotes an abelian category that satisfies the following three ax- ioms. • Cis closed under direct limits, and taking lim-→is exact. •The class of isomorphism classes of finite-length objects ofCis aset. •The smallest abelian subcategory ofCcontaining this set, and closed under direct limit, is equivalent toCitself. The structure of such a categoryChas been reduced to noncommutative algebra, by use of the following method due to Gabriel. Choose a set{Si}i?Iof isomorphism class representatives for all simple objects inC, and, for eachSi, choose an injective hullIi. (Yes, such a category has enough injectives, and in fact injective hulls.) LetIbe the direct sum of theIi. We defineDbethe functor represented byI, i.e.D(C) = HomC(C,I), and we writeD=D(I) for the endomorphism ring ofI. Note that for anyC? C, the abelian hypothesis onCmakesD(C) ito an additive group. Moreover,D(C) is naturally a leftD-module by postcomposition. In this paper, we say that a (left) module over a (possibly noncommutative) ring is completeif, in its unique linear topology generated by its submodules of finite colength, it is complete. This criterion is equivalent to the module being an inverse limit of finite- length modules. In our present subsection, we writeD-complfor the category of complete

D-modules.

The following is Gabriel"s key "duality" theorem.

Theorem 3.1With the preceding notations, for anyC? C,D(C)is complete. (In partic- ular, theDis complete.) Moreover,Dgives an anti-equivalence of categories,

D:C≂=D-compl.

3.2 Frobenius and Shift

To the number theorist, the importance of the Frobenius morphism Frob on a ring is an old story. In this section, we recall its manifestationFon a group scheme. We also introduce the shift operatorV(for the German "verschiebung"; in French, it is often called "d´ecalage"). These will be useful, since, in a certain sense, they generate the Dieudonn´e ringD, which takes the place of the ringDin subsec. 3.1. Since we are assumingkto be perfect, the Frobenius endomorphism Frob:x?→xpis an automorphism. For anyk-algebraA, we write Frob for the ring homomorphism onAgiven by the same formula. TheFrobenius morphismofTis the Frob-linear morphismF:T→Tassociated to Frob on coordinate rings. WhenTis a gorup schemeG, one easily shows thatFis a homomorphism.

The shift morphism is a Frob

-1-linear mappingV:T→Twhich is defined for any scheme using some messy tensor algebra. (See, for instance, [1, IV.3.4].) WhenTis a group scheme G, the mappingVis a homomorphism. In the case of afinitegroup schemeG, having Cartier duality in hand,Vcan be defined as the Cartier dual of the mappingFonG?. A simple calculation (see [5, I.3.3] for a shortcut in the finite case) yields the following fact. 6 Lemma 3.2For any group schemeG, the compositionsV◦FandF◦Vare multiplication bypin the group law onG. (respectively). As a matter of notation, for an idealI?A, we writeI(a)for the ideal generted byia fori?I. Note that whileI(a)?Ia, the containment is sometimes strict. Also,I(pn) G= A·Frobn(IG) is the ideal of definition of the kernel ofFonG= SpecA. This little fact will be used in the proof of the following proposition, which tells us thatFandVpick up some interesting information about a group scheme. Proposition 3.3LetGbe a finite group scheme. Then for allnsufficiently large,G0= kerFnandGet= cokerFn. Proof.By the connected-´etale sequence, the two claims are equivalent and thus we only need to prove the first one. LetG= SpecA, and writeA=?si=1Ai, where eachAiis a local ring with respective maximal idealmi. We number the factors so thatIG=m1?A2?···?As.

For alln, it is easy to see thatI(pn)

G=m(pn)

1?A2?···?As.For allnlarge enough thatmpn

1= 0, the idealI(pn) Gis evidently generated by the idempotents of the nonidentity components.

Therefore, for suchn, we have kerFn= SpecA/I(pn)

G=G0.?

We now give a sequence of immediate corollaries to this proposition. Corollary 3.4LetGbe a commutative finite group scheme. Then the following claims hold.

1.Gis connected if and only ifFis nilpotent.

2.Gis ´etale if and only ifFis an isomorphism.

3.G?is connected if and only ifVis nilpotent.

4.G?is ´etale if and only ifVis an isomorphism.

The preceding corollary may be expanded to include the pairwise consideration ofFand V, and the categoriesFinAb??/k, but this is straightforward and hence omitted.

3.3 Witt Vectors

We now recall a collection of group schemes on whichFandVhave very simple descriptions. The heart of Dieudonn´e theory is to use them (or, the functorsthey represent) to extract information concerning the action ofFandVon all groups. For most of this material, we follow (and refer for more details to) [7, II.6] and [2, ch. I]. We define thelengthnWitt vectorsto be the unique ring schemeWnwith underlying schemeAn, and such that the morphism

φ:W

n-→An? x

1, xp1+px1, ... ,xpn-1

1+pxpn-2

2+···+pn-1xn?

←-(x1,...,xn) is a ring scheme homomorphism. In fact, a morphism of schemesschemesψ:G→Wnis a ring (resp. group) homomorphism if and only ifφ◦ψis. 7 The group homomorphisms Frobenius and shift admit the simple descriptions onWnas follows: (Notice that sinceFis defined as a morphism of schemes, it acts the same onWnas it does onAn.) Frobenius is even a ring scheme homomorphism overFp. One finds then that p=FV(1,0,0,...,0) = (0,1,0,...,0)?= 0?Wnforn >1. The Witt vectors can be compiled both projectively and injectively. The two systems of maps are R T We let the(full) Witt vectors, writtenW, be the inverse limit of theWnwith resepct to theRn. The mapsRnare ring scheme homomorphisms, and henceWis a ring pro-scheme too. We clearly haveWn=W/VnW=W/pnW. The inductive variant may be formed in two ways. One may use the system of maps T n, or one may use the system of mapsTnF. We call the respective direct limitsW∞and W ∞. OverFp, these are both group ind-schemes, and there is a natural homomorphism u:W∞→W?∞compiling theFn-1. For every perfect ringAof characteristicp,u(A) is an isomorphism. We haveWn= ker(Vn|W∞). We fix once and for all the notationW=W(k). This ring is the unique (up to unique isomorphism,kbeing perfect) complete DVR with uniformizerpand residue fieldk. The fieldK= FracW=W[p-1] has characteristic zero, andK/Wis canonically isomorphic to W ∞(k). More generally, the restriction of theWn,W, andW∞toS-schemes have canonical structures ofW-module scheme (resp. pro-scheme, ind-scheme), withWacting onWn throughF1-n.

3.4 The Dieudonn´e Module Functor

Recall thatW=W(k) is the ring of Witt vectors ofk, an integral domain.Note carefully the following convention. The automorphism Frob ofkinduces an automorphism ofWthat was calledF(more precisely,F(k)) in the preceding section. From now on, when applied to the ringW, we call this automorphism Frob. This map induces automorphisms ofKand K/W, which are also henceforth called Frob. These conventions doe not intersect the old use of the name Frob, which applied only to rings of characteristicp. We define theDieudonn´e ringDto be the noncommutative polynomial ringW[F,V], modulo the relations FV=V F=p, F·a= (Froba)·F, a·V=V·(Froba), for alla?W. Sincekis perfect, the map Frob is invertible onW, and using this one shows that every element ofDmay be uniquely written in the form a+N? n=1(bnFn+cnVn). 8 ThusDis a fairly manageable noncommutative ring. It has no zero-divisors, and is left and right Noetherian. By the invertibility of Frob onW, the left and right ideals generated by FandVare all actually two-sided ideals, and one has canonical identifications

D/(F) =k[V],D/(V) =k[F].

(Note that the coefficient ring on the right hand side isk, notW.) IfGis a finite commutative group scheme, then we have natural operatorsFandVon GsatisfyingFV=V F=p. In order to make aD-module out of the endomorphism ring of

G, we need an action ofW(subject to relations).

Fortunately, we already have a system of groups, namely theWn, that admitW-module scheme structures. Thus for eachn,m, we have that HomGrp/k(Wn,Wm) is a (left and right)D-module. Moreover, by the eqns. 3.3, we know thatVm= 0 onWm. The following proposition says that this is essentially the only possiblerelation whenn=m. Proposition 3.5The natural mapD/(Vn)→HomGrp/k(Wn,Wn)is an isomorphism. Sketch of proof.One proceeds by induction onn. The casen= 1 is the claim that Hom the commutative diagram

0→(Vm)/(Vm+1)→ D/(Vm+1)→ D/(Vm)→0

and shows that it has exact rows. Then, one produces the vertical arrow on the left (making the left square commutative), and shows this arrow to be an isomorphism, concluding via the five-lemma.? Finally, we define the contravariant Dieudonn´e module functorD, in pieces. We denote byFinAbp/kthe category of (commutative) finite group schemes ofp-power order overk, i.e. those with no "rr" component. Our functorDwill be defined on the categoriesFinAbp/k, Ind(FinAbp/k), and Pro(FinAbp/k), with values in the categories ofD-modules that have finite length, that are complete, and are torsion, ind-finnite, respectively. (We say a module is ind-finite if each element spans a finite-length submodule.) First, For aD-moduleM, we define thePontryagin dualofM, writtenM?, to be the D-module HomW-mod(M,K/W), withFandWoperating onφ:M→K/Wby the formulas (Fφ)(m) = Frob[φ(V m)] and (V φ)(m) = Frob-1[φ(Fm)], m?M. Pontryagin duality interchanges complete and ind-small modules, and preserves finite-length ones. Now on to the definition. ForG?FinAb?l/k(where?= l or r), we put D(G) = Hom(G,W∞) = lim-→Hom(G,Wn)? D-mod.

Next, forG?FinAblr/k, we set

D(G) =D(G?)?,

9 the first wedge being Cartier dual, and the second being Pontryagin dual. By makingD distribute over direct sums, we obtain a definition on all ofFinAbp/k. This definition is extended to the Ind and Pro categories in the standard contravariant manner: ifG= lim-→Gα andG?= lim←-G?β, respectively, then we put D(G) = lim←-D(Gα) andD(G?) = lim-→D(G?β). The ringDis an "all-purpose" ring: various types of its modules classify various types of groups. These classifications each require different tools.However, several can be attacked using Gabriel"s duality theory. We investigate the functorin detailonly as it applies to (finite) "ll" group schemes, which is a case where Gabriel"s duality applies, and sketch the analogous results on more general types of groups.

3.5 The Dictionary, Special Case

First we construct a special subgroup ofW∞, which plays the role ofW∞in the "ll" domain. For eachn, letLn= ker(Fn|Wn). Just as in the definition ofW∞, these compile via theTn, and we set L ∞= lim-→Ln. We note that, for any "ll" finite groupG, any homomorphismG→Wnfactors through L n?→Wn, grantednis large enough thatFn= 0 onG. (This is because the homomorphism Frob commutes with any homomorphism ofk-algebras.) Thus for any suchG, we have

D(G) = Hom(G,W∞) = Hom(G,L∞).

The corresponding ring in the "ll" domain is

?D, the (F,V)-adic completion ofD. One shows easily that ?D=W[[F,V]] (noncommutative power series ring), modulo the same relations given in the definition of the Dieudonn´e ringD. Then a complete?D-module is nothing other than a completeD-module on whichFandVoperate topologically nilpotently. With this notation in mind, the main result we have been holding out for is as follows. Theorem 3.6The Dieudonn´e module functorDdefines an anti-equivalence of categories betweenInd(FinAbll/k)and the category of complete?D-modules, i.e. completeD-modules on whichFandVare topologically nilpotent. This equivalence specializes to one between FinAb ll/kand the category of all finite-lengthD-modules, on whichFandVare nilpotent. By Gabriel"s abstract duality theorem, and the comments preceding the theorem, it suffices to notice (and possibly prove) the following facts.

1. The category Ind(FinAbll/k) is of the type viewed in subsec. 3.1.

2. In this category, the unique simple object isαp.

3. The injective hull ofαpisL∞.

4. The endomorphism ring ofL∞is?D.

10 The first of the above claims is clear, and the second claim follows from prop. 2.2. The dual of the third claim is proved using proalgebraic group schemes, and calculations involving universal profinite extensions, and the profiniteanalogues of component group and fundamental group. We refer to [1, V.2-V.3] for the numerousdetails. The fouth claim results from a calculation, which we carry out here. Notably, the endo- morphism ring ofL∞is lim ←-lim-→HomAffAb/k(Ln,Lm) = lim←-HomAffAb/k(Ln,Ln) = lim←-D/(Vn,Fn) =?D.(3.4)

We justify each equality in turn.

The first equality results from the following proposition, which is proved via a short calculation. (See [2, II.6.2].) Proposition 3.7LetGbe a group scheme that is killed byVn, andφ:G→Wma homo- morphism, wherem≥n. Thenφfactors uniquely throughTn:Wn?→Wm. Accordingly, in the first expression of eqns. 3.4, a homomorphismφ:Ln→ Lm?→Wm must arise from a homomorphism?φ:Ln→Wn. But by the comments at the beginning of this section, sinceFn= 0 onLn, we find that?φfactors uniquely throughLn?→Wn. This establishes the first equality. The second equality of eqns. 3.4 is an analogue of prop. 3.5, and it is proved in a similar manner. The third and final equality of eqns. 3.4 follows from the standard fact that (F2n,V2n)?(F,V)2n?(Fn,Vn)?(F,V)n, whence the two systems of ideals{(Fn,Vn)}and{(F,V)n}are cofinal, and induce identical completions.

3.6 The Dictionary, More Generally

This subsection is complementary to the preceding. Whereaspreviously we viewed the "inside" of the Dieudonn´e correspndence, here we view the "outside" of it. We state various classification results involving the functorD, and we also give some examples auxillary results which allow us to use the classifications more transparently. A first auxillary result concerns the embodiment of Cartier duality. The reader will probably have already guessed its validity, in no smally waydue to the suggestively chosen notation. Proposition 3.8IfGis a finite, ind-, or pro-p-group scheme, then there is a canonical isomorphismD(G?)≂=D(G)?. Here is a list of classification theorems. Their proofs use the same methods as seen in the preceding subsection, but more calculations and facts fromthe theory of unipotent algebraic groups. 11 Theorem 3.9The Dieudonn´e module functorDdefines an anti-equivalence of categories between each of the following pairs.

1. finitep-group schemes, and finite lengthD-modules

2. finite ´etale (resp. connected, co´etale, coconnected)p-group schemes, and finite length

D-modules on whichFis an isomorphism (resp.Fis nilpotent,Vis an isomorphism,

Vis nilpotent)

3. ind-p-group-schemes, and completeD-modules

4. ind-p-group-schemes of certain type, and completeD-modules with isomorphism or

topological nilpotence conditions onForV

5. (noetherian) Dieudonn´e formal groups, and complete

?D-modulesMthat are finitely generated and haveM/FMof finite length

6. profinitep-group schemes, and ind-smallD-modules

7. profinitep-group schemes of certain type, and ind-smallD-modules with isomorphism

or effaceability conditions onForV Remarks.IfLis an endomorphism of a moduleM, we say thatMisL-effaceableif for everym?M, there is an integernwithLnm= 0. Embedding Dieudonn´e formal groups into group ind-schemes,and writingG= SpfA, asking thatAbe noetherian boils down to precisely the finiteness criteria imposed in item 5 the above theorem. Using slightly more general methods, one can classify (not necessarily finite type) affine group schemes of "unipotent" type as corresponding viaDto thoseD-modules that are V-effaceable (i.e., every element is killed by a power ofV). The finite type unipotent affine group schemes correspond to finitely generated modules of this sort. There are likely other basic correspondences for other types of affine group schemes, and these can probably be souped-up in a manner similar to the above theorem, though wedo not pursue them here. As a consequence of thm. 3.6, the functor on Ind(FinAbp/k) defined by D ll(G) = Hom(G,L∞) embodies theprojectiononto the "ll" part. In the special case wherekis algebraically closed, the dual "lr" and "rl" categories become especiallysimple (since the Galois group is trivial). The finite group schemeμpofpth roots of unity becomes the unique simple object ofFinAblr/k, and hence of Ind(FinAblr/k). The injective hull ofμpin the latter category is Gm, and the endomorphism ring of?GmisZp. Using these data, we can employ Gabriel"s duality theorem all over again. After some simple calculations to make the comparison, we get the following result. 12 Theorem 3.10Supposekis algebraically closed, and letG?Ind(FinAbloc/k)with decom- position into components given byG=Gll×Glr. Then one may recover the corresponding components of its Dieudonn´e module as

D(Gll) = Hom(G,L∞),and

D(Glr) =W?ZpHom(G,?Gm)

(Compare this theorem and its preceding discussion with prop. 2.3!) Here is an example of how we can extract information about a group from its Dieudonn´e module.quotesdbs_dbs45.pdfusesText_45

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