DIEUDONNE CRYSTALS ASSOCIATED TO LUBIN-TATEFORMAL GROUPS

MATTHEW ANDOTHE UNIVERSITY OF VIRGINIA

Contents

Preface 3

1. Summary 3

2. Formal geometry, formal groups, and formal group laws 8

2.1. Formal varieties 8

2.2. Formal groups 10

3. Elementary calculus on formal varieties: the inverse function theoremand the logarithm of a formal group 12

3.1. The ring of functions on a formal variety, the tangent space of a formalvariety 12

3.2. Invariant differentials and the logarithm 14

4. Lazards classification of commutative 1dimensional formal group laws 18

4.1. Statement of Lazards Theorem 18

4.2. Proof of Lazards theorem 19

4.3. The symmetric two-cocycle lemma 21

5. Witt vectors and the Witt formal group 25

5.1. p-typical Witt vectors 26

5.2. Frobenius, Verschiebung 28

5.3. Witt vectors of perfect fields and p-adic arithmetic 30

5.4. Global Witt vectors 32

5.5. The Witt formal group 37

6. Classification of formal groups via the Dieudonne-Cartier module 37

6.1. The curves functor C 37

6.2. Endomorphisms of the curves functor 37

6.3. W represents the curves functor 416.4. C takes values in modules over the Witt vectors 43

6.5. p-typical curves over a Z(p)-algebra 461

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6.6. The Dieudonne modules associated to formal groups are uniform andreduced 49

6.7. The p-typical parameter lemma and the evaluation map 53

6.8. The functor from Dieudonne modules to formal groups 55

6.9. Example: the multiplicative group and the additive group 57

6.10. Example: Dieudonne modules over perfect fields 60

6.11. Example: a Dieudonne module of each finite height 63

6.12. Endomorphisms 64

7. Lubin and Tates deformation theory 66

7.1. Various formulations of the problem 67

7.2. The action of the automorphism group 69

7.3. Deformations and cohomology 70

7.4. Calculation of H2(; k) 73

7.5. Proof of Lubin and Tates theorem 76

8. Preliminary remarks about crystals 77

8.1. Calculus: 1-forms, connections, and curvature 77

8.2. Connections and descent 81

8.3. Connections and descent II: divided powers 82

8.4. Connections and descent III: crystals 84

8.5. Examples I 87

8.6. Examples II: de Rham cohomology 88

8.7. Base change 93

9. Classification of formal groups via the Dieudonne crystal 93

9.1. Introduction 93

9.2. p-divisible formal groups 94

9.3. Definition of Dieudonne crystal 94

9.4. Frobenius determines the connection. 96

9.5. A Dieudonne crystal is an FV -module admitting a Hodge structure 99

9.6. The functor from FV -modules to formal groups 102

9.7. The logarithm of the group GM 106

9.8. The functor from Dieudonne crystals to formal groups 107

9.9. The case of a perfect field 111

10. The functor from formal groups to Dieudonne crystals 112

10.1. Introduction 112

10.2. The universal additive extension of a p-divisible formal group 112

10.3. The equivalence of categories 114

FORMAL GROUPS 3

10.4. The relationship to de Rham cohomology 116

11. Application: the period map 119

11.1. Translation of the Lubin-Tate moduli problem into the language ofDieudonne crystals 119

11.2. Construction of the period map 120

11.3. Equivariance with respect to the action of the automorphism groupof 122

11.4. An explicit form for the Dieudonne crystals associated to theLubin-Tate formal groups 124

References 126

Preface

These notes are a companion to the papers of Gross and Hopkins and of Devinatzand Hopkins on the period mapping for Lubin-Tate space [GH94, HG94, DH93].

Gross and Hopkinss paper [GH94] constructs the period mapping without directappeal to the theory of crystals. The main purpose of these notes is to elaborateon the first few pages of [HG94], and so give a geometric description of the periodmap from the crystalline point of view.

These notes are largely based, especially chapters 2 through 9, on a course givenby Hopkins at MIT. I have prepared this document because many of the sourcesfor the material covered here are either illegible (my notes from Hopkinss course)or unpublished (e.g. [Blo]; [Car69] gives an indication of only a portion of theauthors many results on the subject). Another reason is that by keeping in mindthe rings relevant to the period mapping, the discussion of crystals can proceed ata relatively elementary level.

I am grateful to Mike Hopkins for teaching me about this material, and forallowing me to make these notes available to others. I am solely responsible for anyerrors here.

1. Summary

Formal geometry, formal groups, and formal group laws. The first section intro-duces a formal group over a ring R as a group in the category of formal varieties.This category is set up so that the morphisms between two formal varieties are, upto non-canonical isomorphism, a set of formal power series. An isomorphism is achoice of coordinates, and a formal group law is a formal group together with achoice of coordinates.

Elementary calculus on formal varieties: the inverse function theorem and the log-arithm of a formal group. Because the maps between formal varieties are given (up

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to isomorphism) by power series, it is easy to develop for formal varieties the ana-logues of many concepts from calculus, and so for formal groups the analogues ofconcepts from the theory of Lie groups. Two important examples are the following.

Theorem (3.1.8 ). A map f : V W of formal varieties over a ring R is anisomorphism if and only if

TVdf TW

is an isomorphism of R-modules.

Theorem (3.2.11). If G is a formal group over a Q-algebra R, then there is aunique isomorphism of formal groups

GlogG Lie(G)Ga,

inducing the identity on tangent spaces.

Lazards classification of commutative 1dimensional formal group laws. Mostly weshall be study properties of formal groups which are independent of any particularchoice of coordinate. It is nevertheless useful to have Lazards theorem available. IfR is a ring, let FGLR denote the set of commutative, one-dimensional formal grouplaws over R.

Theorem (4.1.2 ). There is a formal group law GL over the ring L = Z[t1, t2, . . . ]such that

Hom[L,R] FGLRf 7 fGL

is an isomorphism.

The main importance of Lazards theorem for the sequel is

Corollary (4.1.3 ). If R R0 is a surjective map of rings, then a commutativeone-dimensional formal group law over R0 lifts to a formal group law over R.

Witt vectors and the Witt formal group. The classifications of formal groups insections 6 and 9 make extensive use of various incarnations of Witt vectors: p-typicalWitt vectors, global or big Witt vectors, and their associated formal groups.Section 5 introduces these objects, together with their Frobenius and Verschiebungmaps.

Classification of formal groups via the Dieudonne-Cartier module. Section 6 studiesan analogue of the Lie algebra which is due to Dieudonne and Cartier: if G is aformal group over R, then the group of curves in G is the group

CR(G)def= FVarR[A1R, G]

of all maps of formal varieties from the formal line to G.

The first major point is that C is represented by the Witt formal group.

Theorem ( 6.3.1 ). There is a natural isomorphism

CR(G) = FGpsR[W, G]

This is used to prove

FORMAL GROUPS 5

Theorem (6.4.3 ). The functor CR takes values in modules over the Witt vectorsWR of R.

When R is a Z(p)-algebra, the Witt vectors split as a product of copies of the p-typical Witt vectors WpR. There is an analogous splitting of CR(G); the subgroupcorresponding to one copy of WpR is the group of p-typical curves,

D(G) = FGpsR[Wp, G]What sort of group can occur as D(G) for some formal group G? There are

some obvious endomorphismshomotheties, Frobenius, and Verschiebungand aDieudonne module is defined to be a module with this extra structure. When R = kis a perfect field, the definition is

Definition (6.6.9 ). A Dieudonne module over k is aWpk-moduleM together withoperators Frobenius, F , and Verschiebung,V , which are linear as maps

M FMM

V M,and which satisfy

FV = p = V F.MoreoverM is required to be uniform and reduced with respect to V (see Definition6.6.1).

The classification for perfect fields is given by

Theorem (6.6.10). Let k be a perfect field of characteristic p > 0. The functor Dis an equivalence of categories

FGpskD= (Dieudonne modules over k)

Section 6.8 is devoted to the construction of an inverse to D, which is due toCartier. It serves as a model for the construction in chapter 9.

Actually, the functor C gives a complete classification of formal groups over anyring, and D gives a complete classification for formal groups over a Z(p)-algebra;see [Haz78]. However, the category of Dieudonne modules over a perfect field turnsout to be particularly nice. For one-dimensional formal groups the result is

Proposition (6.10.1). If G is a one-dimensional formal group over a perfect fieldfield k of characteristic p > 0, then either

(i) there is a non-zero element DG such that p = 0, in which case G isisomorphic to the additive group (G has infinite height), or

(ii) DG is free of finite rank over the ring Wpk (in which case the rank h isthe height of the formal group G).

Lubin and Tates deformation theory. The Dieudonne module classifies formal groupsover perfect fields in terms of modules over the discrete valuation ringWpk, indeedoften finite free modules. Geometrically, a field is a point.

Lubin and Tate study the following problem. Suppose given a formal group over k, and a complete local ring A with residue field k (a family of infinitesimal

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neighborhoods of k). What are the formal groups G over A which reduce modulothe maximal ideal to ? Such a formal group is called a deformation or lift of; there is an associated functor from complete local rings with residue field k tosets, called Lifts .

Theorem (7.1.1). Suppose that is a formal group of finite height n over k. LetEn be the ring Wpk[[u1, . . . , un1]]. There is a formal group G over En such thatthe map

Homcts[En, A] Lifts(A)f 7 fG

is an isomorphism for all Noetherian A with residue field k.

Preliminary remarks about crystals. Section 8 is a summary of the small part of thetheory of crystals and connections used from section 9 and on. We often restrictour attention to the rings A0 = k[[u1, . . . , un1]] and A =Wpk[[u1, . . . , un1]] whicharise in the study of the Lubin-Tate formal group. With these rings in mind, onemay sketch an elementary and hopefully illuminating proof of the following basicresult.

Theorem (8.5.4). There is an equivalence of categories{A-modules withintegrable, quasi-nilpotent connection

} {Crystals on A0}.

We also discuss the example of de Rham cohomology, with the Gauss-Maninconnection.

Theorem ( 8.6.9). Let G0 be a formal group over A0. If G is a lift of G0 to A,then the crystal PH1DR(G/A) of primitives in the de Rham cohomology of G, withits Gauss-Manin connection, is independent of G up to canonical isomorphism. Itdefines a contravariant functor

{Formal groups over A0} {Crystals over A0}.

In section 10.4 we shall see that this is the (A-linear) dual of the Dieudonnecrystal studied in section 9.

Classification of formal groups over imperfect rings. Section 9 introduces Dieudonnecrystals. The main point is the construction of a functor from Dieudonne crys-tals to formal groups, extending the construction described in section 6.8. Theconstruction for crystals is due to Bloch [Blo].

The period map arises from the study of the Lubin Tate group G over En interms of Dieudonne theory. As a first approximation, consider the group G0 = kGover

A0 = En/p = k[[u1, . . . , un1]].One could study G0 via the Dieudonne module D(G0) over Wp(A0). This is amessy module: D(G0) is no longer finite or free, and Wp(A0) isnt Noetherian.

The idea of the Dieudonne crystal is to think of G as a family of formal groups,parametrized by the ui. The classifying object which results is a family of modulesover Wpk, parametrized by the ui, in other words, a (Wpk)[[u1, . . . , un1]]-module.

FORMAL GROUPS 7

It turns out that one must also have a connection , which allows you to comparethe fibers at different points in the family. The resulting object is the Dieudonnecrystal. There is then a classification, at least for p-divisible formal groups (seesection 9.2).

Definition (9.3.4 ). Let be a lift to A of the Frobenius of A0. A Dieudonnecrystal for A0 is a quadruple (M,, F, V ), where M is a finite free A-module, isan integrable, quasi-nilpotent connection on M , and F and V are horizontal maps

M FM andM

V M,with FV = p = V F . The kernel

Ker[0M0F0M0]

must be a split submodule (the analogue of uniform), and V must be topologicallynilpotent (the analogue of reduced).

Let an FV -module be an A-module with operators F and V as in the definition,without the requirement of horizontality.

Definition (9.5.2 ). A Hodge structure on an FV -module is an A0-submoduleH0 M0 such that

0H0 = KerF0 = ImV0 0M0.Theorem (9.5.4). There is an equivalence of categories

(Diedonne crystals) = (FV modules with Hodge structure)

Theorem (9.8.3). There is an equivalence of categories G

(FV modules with Hodge structure) G= (p-divisible formal groups)

The functor from formal groups to Dieudonne crystals. In this section we describethe inverse of the functor G, namely, the Lie algebra of the universal additiveextension, due to Grothendieck, Messing, and Mazur-Messing [Gro70a, Gro70b,Mes72, MM74].

Theorem ( 10.2.6). Let G be a lift of G0 to A, and let E(G) be the universaladditive extension of G. Its Lie algebra LieE(G) has the structure of a Dieudonnecrystal. It is independent of the lift G, up to canonical isomorphism.

The period map. The period map arises from expressing the Lubin-Tate moduliproblem in terms of Dieudonne crystals. By the results of chapters 9 and 10, a p-divisible formal group G0 over A0 = k[[u1, . . . , un1]] is equivalent to an FV -moduleM with Hodge structure H0. The Lubin-Tate formal group is a formal group Gover En = A =Wpk[[u1, . . . , un1]].

Theorem (10.3.7 ). Let G0 be a formal group over A0. There is a bijection betweenlifts H of the Hodge structure to M and lifts of G0 to A.

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When A0 is a perfect field, this theorem is due to Cartier [Car69]. The extensionto more general A0 is part of Blochs paper [Blo]. The association of lifts G tosubmodules H M gives rise to the period map.

At the end of section we give an explicit formula for the Dieudonne crystal ofa Lubin-Tate formal group associated to lifts of formal group law of height n forn 1.

2. Formal geometry, formal groups, and formal group laws

2.1. Formal varieties. A formal group is a group in the category of (pointedaffine) formal varieties. The category of formal varieties over a ring R is set upso that, up to non-canonical isomorphism, the maps between formal varieties areformal power series with constant term zero. It is a stripped-down version of aformal scheme; see for example [Gro70b, Kat81] .

Definition 2.1.1. If R is a ring, then an adic Ralgebra is an augmented R-algebra

A R

whose augmentation ideal is nilpotent. If A is an adic Ralgebra, then its augmen-tation ideal will be written I(A).

An adic Ralgebra is complete and separated in the topology defined its aug-mentation ideal.

Definition 2.1.2. The category AdicR of adic Ralgebras is the sub-category ofR-algebras consisting of adic Ralgebras and continuous maps.

The ring R[[x1, . . . , xn]] is a topological R-algebra, and there is an isomorphism

HomctsR [R[[x1, . . . , xn]], A] = I(A)n.This motivates the definition

Definition 2.1.3. The formal affine plane of dimension n over R is the functor

AdicRAnR (pointed sets)

A 7 HomctsR [R[[x1, . . . , xn]], A] = I(A)n.

We also define the infinite affine space A by settingA(A) = colim

nAn(A).

Definition 2.1.4. The category FVarR of (pointed affine) formal varieties over Ris the category whose objects are functors

AdicRV (pointed sets)

isomorphic to AnR for some 1 n , and whose maps are natural transformationsof functors. The dimension of a formal variety V is n if there is an isomorphism

V = AnR.Such an isomorphism is called a system of parameters or coordinates. Generally,parameters will refer to maps

An = V ;

FORMAL GROUPS 9

while coordinates will refer to maps

V = An.

Exercise 2.1.5. Check that if V is a formal variety of dimension n < over R,and W is a formal variety of dimension k

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Example 2.1.9. If V is a smooth variety over a field k, and

P V (k)is a point, then the completion of V at P is a formal variety over k. By definition,this is the functor which assigns to an adic k-algebra A the set

V P (A)def= {v V (A)|(v) = P},

where is the map V (A) V (k) induced by the augmentation.Example 2.1.10. A ring homomorphism

Rc S

gives rise to a base-change functor

AdicSc AdicR :

If A is an adic S-algebra with augmentation ideal I(A), then cA is just R I(A),with the obvious multiplication and augmentation.

By composition, there is a base change functor

FVarRc FVarS :

for V FVarR, the variety cV is the compositeAdicS

c AdicR V (sets).It is easy to check that

cAnR = AnS ,and the effect of c on maps is to apply the homomorphism c to the coefficients ofthe power series fj .

2.2. Formal groups.

Definition 2.2.1. A (not-necessarily commutative) formal group over R is a groupG in the category of formal varieties over R. A formal group law over R is a formalgroup, together with a system of parameters, i.e. an isomorphism

G = AnRof set-valued functors.

Example 2.2.2. The additive formal group Ga is the functor

Ga(A) = I(A),

where I(A) is considered as a group with its usual addition. The identity

I(A) I(A)is a coordinate. More generally, if M if a free Rmodule of finite rank k, then thefunctor

A 7M ZGa(A) =M

RI(A)

is a formal group of dimension k. A basis ofM determines a system of coordinates.

FORMAL GROUPS 11

Example 2.2.3. The multiplicative formal group is the group

Gm(A) = (1 + I(A))

of units of the form 1 + a for a I(A), with the multiplicative structure from A.One possible coordinate is

Gm(A) 3 1 a 7 a I(A).Example 2.2.4. If A is an abelian variety over k, then the completion of A at theidentity is a formal group over k.

A formal group law of dimension n over R is equivalent to an n-tuple of powerseries in 2n variables

F FVarR[A2n, An] (R[[x, y]])nsatisfying the conditions

F (x, 0) = x = F (0, x) and (unit)

F (x, F (y, z)) = F (F (x, y), z), (associative) (2.2.5)

together with an inverse series

[1]F (x) FVarR[An, An]such that

F ([1]F (x), x) = 0 = F (x, [1]F (x)).It turns out that a power series F (x, y) satisfying axioms (2.2.5) always has a uniqueinverse series [1]F (x). This is a consequence of the inverse function theorem (seeCorollary 3.1.10, below).

When F is a formal group law, we shall often use the notation

x+Fy = F (x, y).

For example, the additive formal group law is

x +Gay = x+ y

while the formal group law consisting of the multiplicative formal group with thecoordinate in example 2.2.3 is

x +Gm

y = x+ y xy.

These formal group laws are commutative. A formal group law F is commutativeif

F (x, y) = F (y, x).

We shall always take our formal groups to be commutative, that is, they will becommutative groups in the category of formal varieties. It turns out that formalgroups are often commutative anyway.

Theorem 2.2.6. There is a non-commutative one-dimensional formal group overa ring R if and only if R contains an element r which is simultaneously nilpotentand torsion.

Proof: [Laz54]. See also [Haz78], p. 38.

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3. Elementary calculus on formal varieties: the inverse functiontheorem and the logarithm of a formal group

3.1. The ring of functions on a formal variety, the tangent space of aformal variety. Let U denote the forgetful functor from adic Ralgebras to (sets)(without base point). Let V be a pointed formal variety over R of dimension n.

Definition 3.1.1. The ring of formal function on V is the ring R[[V ]] of naturaltransformations

V Uof functors to (sets).

Exercise 3.1.2. As an exercise, check that R[[V ]] is a ring, indeed an R-algebra.Show that

R[[An]] = R[[x1, . . . , xn]](canonically), and that a map of formal varieties

VfW

induces a ring homomorphism

R[[W ]]f R[[V ]].

The point 0 V (R) determines a ring homomorphismR[[V ]] R

7 (0).Its kernel, IV , consists precisely of natural transformations f : V U which factorthrough the natural transformation

A1 U .In other words, there is an isomorphism

IV = FVarR[V, A1].When V = An, this is the ideal (x1, . . . , xn).

Definition 3.1.3. The cotangent space at the origin is the R-module

T V def= IV /I2V .

If V has dimension n then it is a free module of rank n. If f R[[V ]] is a formalfunction, we denote by df0 the image in T V of f f(0).

If f R[[V ]] and v is an element of the set V (R[t]/t2), then we define an elementv(f) R by

fR[t]/t2(v) = f(0) + v(f)t.It is easy to check that

(i) v(f) = v(f f(0)).(ii) v(f) depends only on the image of f f(0) in T V .(iii) This evaluation gives a natural isomorphism

V (R[t]/t2) = ModulesR[T V,R].

FORMAL GROUPS 13

Definition 3.1.4. The tangent space to V at the origin is the Rmodule

TVdef= V (R[t]/t2) = ModulesR[T V,R].

It is a free Rmodule of rank dimV . If G is a formal group, then the tangent spaceof G at the origin is denoted Lie(G). If

VfW

is a map of formal varieties, its derivative is the homomorphism

TVdf=fR[t]/t2 TW.

Example 3.1.5. If M is a free R module, then the Lie algebra of M Ga (2.2.2)is canonically isomorphic to M by the map

M M Ga(R[t]/t2)m 7 m t. (3.1.6)

Example 3.1.7. If f, g R[[V ]] and v TV , thenv(fg) = f(0)v(g) + g(0)v(f).

Theorem 3.1.8 (Inverse function theorem). A map of finite-dimensional formalvarieties

VfW

over a ring R is an isomorphism if and only if its derivative at the origin

TVdf TW

is an isomorphism.

Proof. First one reduces to the case V = An =W , so f = (f1, . . . , fn) is an n-tupleof power series in n variables, each without constant term, and df is an nn matrixwith entries in R. Second, it is clear that one can construct the inverse of any linearisomorphism, so one reduces to the case that df is the identity matrix, that is

fi(x) = xi + o(2) (3.1.9)

for 1 i n.Of course, the inverse isomorphism g is constructed by induction on the degree.

Let g(1)(x) = x. Suppose that r 2, and that g(r1)i (x) is a polynomial of degreer 1, such that

g(r1)i (f(x)) = xi +

|J|=r

cJxJ + o(r + 1).

for some cJ in R. Set

g(r)i (x) = g

(r1)i (x)

|J|=r

cJxJ .

Equation (3.1.9) implies that

f(x)I = xI + o(|I|+ 1),so

g(r)(f(x)) = x+ o(r + 1).

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We promised the following application of the inverse function theorem.

Corollary 3.1.10. If F (x, y) is a n-tuple of power series in 2n variables, satisfyingthe conditions of a formal group law (2.2.5), then it is a formal group law: there isa unique n-tuple of power series [1]F (x) satisfying

F ([1]F (x), x) = 0 = F (x, [1]F (x)).

Proof. By the axioms (2.2.5), the map

An An s An An(x, y) 7 (x, x+

Fy)

has derivative [In In0 In

],

hence is an isomorphism. So there is a unique [1]F (x) such thats(x, [1]F (x)) = (x, 0).

3.2. Invariant differentials and the logarithm.

Definition 3.2.1. The module 1V/R of Kahler differentials or one-forms on V isthe R[[V ]]module generated by symbols da, for a R[[V ]], subject to the relations

d(ab) = adb+ bda a, b R[[V ]]dr = 0 r R.

We will discuss Kahler differentials at greater length in chapter 8.

Proposition 3.2.2. The assignment

bda 7 b(0)da0determines a restriction to the origin

1V/Re T V

map of Rmodules.

When G is a Lie group, a one-form is left-invariant if

Lggh = hfor all g, h G. A left-invariant differential on a Lie group is determined by itsvalue at the origin, since

Lyy = 0, (3.2.3)and the map 7 0 is an isomorphism from the left-invariant differentials to theone-forms at the origin. The same is true of a formal group.

Lemma 3.2.4. Any differential on A1 is of the form p(x)dx.

Proof. Any formal function on A1 is of the form f(x) R[[x]]. The Leibniz rulegives

df(x) = f (x)dx.

FORMAL GROUPS 15

Let G be a one-dimensional formal group, and let x be a coordinate on G. Let

F (x, y) R[[x, y]]denote the resulting group law.

Definition 3.2.5. A differential on G is invariant if it is invariant under leftmultiplication, i.e., writing = p(x)dx,

p(y +Fx)d(y +

Fx) = p(x)dx. (3.2.6)

Let F2 be defined as

F2(x, y)def=

F (x, y)y

= 1 + higher terms R[[x, y]].

Then equation (3.2.6) becomes

p(y +Fx)F2(y, x)dx = p(x)dx.

Setting x = 0 yields the analogue of equation (3.2.3),

p(y)F2(y, 0) = p(0).

So a necessary condition for dp(x) to be invariant is that

p(x) = a/F2(x, 0). (3.2.7)

More generally, suppose that G is a formal group law of finite dimension d, andthat x1, . . . , xd is a system of coordinates. Let F2(y, x) be the d d Jacobian ofF (y, x) with respect to the xi. By the inverse function theorem, F2(x, 0) is aninvertible matrix. Let dx denote the column vector whose i entry is dxi.

Proposition 3.2.8. The submodule of 1G/R consisting of invariant differentials onG is a free module of rank dimG, denoted (G). Restriction to the origin inducesan isomorphism

(G) = T G.If x1, . . . , xd is a system of coordinates on G, then the map

Rn (G)[a1, . . . , ad] 7 [a1, . . . , ad]F2(x, 0)1dx

is an isomorphism. In terms of this isomorphism restriction to the origin is themap

(G) e TG

aF2(x, 0)1dx 7i

ai(dxi)0.

Proof. We give the one-dimensional case. All thats left to prove is that dx/F2(x, 0)is indeed invariant. The point is that equation (3.2.7) guarantees that

Lyy = 0,

and invariance follows from associativity. To be precise, differentiate the equation

F (F (y, x), w) = F (y, F (x,w))

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with respect to w and set w = 0 to obtain the equation

F2(F (y, x), 0) = F2(y, F (x, 0))F2(x, 0) = F2(y, x)F2(x, 0).

Now let p(x) = 1/F2(x, 0). Then

p(y +Fx)F2(y, x) =

F2(y, x)F2(y +

Fx, 0)

= p(x).

Now suppose that G is a one-dimensional formal group over R, and R is a Qalgebra. Suppose also that x is a coordinate on G, and let G denote also theresulting formal group law. Suppose that p(x)dx is an invariant differential. Let

l(x) =p(x)dx

be the power series obtained by integrating p(x)dx as a power series, and requiringl(0) = 0.

Lemma 3.2.9. The power series l(x) is a homomorphism from G to the additivegroup, i.e.

l(x+Gy) = l(x) + l(y).

Its derivative at the origin

Lie(G) dl Lie(Ga) = R,considered as an element of T G, is p(0)dx0.

Proof. The part about the derivative is true by definition. To see that l is a homo-morphism, differentiate l(y +

Fx) with respect to x, to get

dl(y +Fx)

dx= l(y +

Fx)F2(y, x)

= p(y +Fx)F2(y, x)

= p(x)

= dl(x)/dx.

Thus l(y +Fx) and l(x) differ by a constant; setting x = 0 shows the constant is

l(y).

Proposition 3.2.10. Let G be a one-dimensional formal group over a Q-algebra.For every invariant differential (G), there is a unique homomorphism

Gl Ga

whose derivative at the origin is .

FORMAL GROUPS 17

Proof. Choosing a coordinateG

x A1allows you to write = p(x)dx and proceed as in (3.2.9). Let l(x) denote theresulting power series. We must show that the homomorphism

Gx A1 l Ga

is independent of the coordinate x. If y is another coordinate, then x and y arerelated by

x = (y)

dx = (y)dy

= p(x)dx = p((y))(y)dy.

Let m(y) denote the power series obtained from (3.2.9) using the coordinate y; bythe chain rule,

m(y) =p((y))(y)dy

= l((y)).

Thus the diagram

G A1

A1 Ga

wy

u

x

um

wl

commutes.

There are analogous statements for formal groups of higher dimensions; see e.g.[Haz78, section 9.6]. It will be convenient to state the result in an invariant form.There are isomorphisms

Lie(G) = ModulesR[T G,R] = ModulesR[(G), R],Via this isomorphism, Proposition 3.2.10 is equivalent in the one-dimensional caseto the following.

Corollary 3.2.11. If G is a formal group over a Q-algebra, then there is a uniqueisomorphism of formal group laws

GlogG Lie(G)Ga

which induces the identity on Lie algebras via the isomorphism (3.1.6).

Definition 3.2.12. If G is a formal group over a Q-algebra, then the isomorphismlogG of Corollary 3.2.11 is called the logarithm of G. The inverse isomorphism,denoted expG, is called the exponential. If G is a one-dimensional formal grouplaw, then we shall also call the logarithm of G the isomorphism

GlogG Ga

obtained from the invariant differential

dx/G2(x, 0).

18 ANDO

The group law may be recovered from the formula

x+Gy = expG(logG(x) + logG(y)).

Corollary 3.2.13. Let G1 and G2 be formal groups over a Qalgebra R, and let

Lie(G1)s Lie(G2)

be a homomorphism. There is a unique homomorphism

G1exp(s) G2

withd exp(s) = s.

4. Lazards classification of commutative 1dimensional formalgroup laws

4.1. Statement of Lazards Theorem. The classification of (one-dimensionalcommutative) formal group laws is due to Lazard. Recall that a one-dimensionalformal group law over a ring R is a power series

F (x, y) =i,j

ai,jxiyj R[[x, y]]

satisfying the axioms

F (x, 0) = 0 = F (0, y) (identity)

F (x, y) = F (y, x) (commutativity) (4.1.1)

F (x, F (y, z)) = F (F (x, y), z). (associativity)

More generally, an r-bud is a power series F (x, y) of degree r in two variablessatisfying the axioms (4.1.1) modulo polynomial degree r + 1: a formal group lawis an -bud. The identity axiom says that F is actually of the form

F (x, y) = x+ y +1i,j

i+j

FORMAL GROUPS 19

is an isomorphism.

Moreover the ring L(r) has a natural grading, coming from an action of themultiplicative group on the group of r-buds: if F is an r-bud over R and is a unitof R, then there is a new r-bud

F(x, y) =i,j

aijxiyj

defined by the equation

F (x, y) = F(x, y)

F (x, y) = 1ij

aij(x)i(y)j

=ij

(i+j1aij)xiyj ,

so the degree of aij is 2(i+j1) (the even grading is for consistence with topology).Finally, it is clear that there are maps of graded rings

L(r) L(r+1)L(r) L(),

such thatL() = colim

r

20 ANDO

Let log(x) U [[x]] be its compositional inverselog(exp(x)) = x;

one haslog(x) = x+

i1

mixi+1

with mi U . Let F be the formal group law over U defined by the formulaF (x, y) = exp(log(x) + log(y)).

Its classified by a mapL

u U.Thinking of exp and log as endomorphisms of A1, U acquires a grading in which|bi| = 2i = |mi|, and u becomes a map of graded rings. It is not hard to calculatethe image of the aij modulo decomposables. Let I be the ideal (b1, b2, . . . ) of U .

Lemma 4.2.1.bi mi mod I2.

Proof. Calculating modulo I2 one has

x = exp(log(x))

= x+j1

mjxj+1 +

i1

bi

x+j1

mjxj+1

i+1

= x+j1

mjxj+1 +

i1

bixi+1

= x+j

(bj +mj)xj+1.

Lemma 4.2.2.

u(aij) (i+ ji

)bi+j1 mod I2.

Proof. Set m0 = 1 = b0. Calculating again modulo I2 and using Lemma 4.2.1, onehas

i,j

u(aij)xiyj = F (x, y)

= exp(log(x) + log(y))

=i0

bi(j0

mj(xj+1 + yj+1))i+1

=j0

mj(xj+1 + yj+1) +i1

bi(x+ y)i+1

= x+ y +n1

bn((x+ y)n+1 xn+1 yn+1).

(4.2.3)

The result follows from a comparison of the coefficients of xiyj in the first and lastexpressions.

FORMAL GROUPS 21

Let

dn = GCD{(

n

k

), 0 < k < n

}.

The preceding lemma may be phrased as follows.

Proposition 4.2.4. For n < r the image of

QL2nQu QU2n = Zbn.

is the subgroup generated by dn+1bn.

Let T2n QU2n be the subgroup Zdn+1bn.The hardest part of the proof is the following.

Proposition 4.2.5. For n < r there is a canonical splitting

QL2n T2n

QL2n.

wQu

[[ [[ [[ [[u

This is a form of the so-called symmetric two-cocycle lemma. It is of funda-mental importance also in the deformation theory of Lubin and Tate; see section7. We shall give several formulations and a proof in the next section.

Proof of Theorem 4.1.2. For r > n 1 choose tn L2n such that Qu(tn) is agenerator of T2n, and consider the resulting maps

M = Z[ti|i < r] v L u U = Z[b1, b2, . . . ].Then v is an isomorphism. It is injective because

u(tn) = dn+1bn + decomposables,

so uv is injective. It is surjective modulo decomposables by Lemma 4.2.5. But asgroups, M and L are both the colimits as n of their summands of degree lessthan or equal to n; it follows by induction on n that v is an isomorphism.

4.3. The symmetric two-cocycle lemma. To calculate dn, we define for p primeand N a positive integer

vp(N) = kif N = pkm with m prime to p. For i 0 we define numbers

0 ap,i(N) p 1by the formula

N =i0

ap,i(N)pi.

Finally we definep(N) =

ap,i(N).

Lemma 4.3.1.

vp(n!) =n p(n)p 1 .

22 ANDO

Proof. The number of integers between 2 and n which are divisible by p at least ktimes is is bn/pkc. It follows that

vp(n!) = bn/pc+ bn/p2c+ . . . .It is easy to check that the right hand side is equal to

j

ap,j(N)(j1i=0

pi),

and that this quantity is equal right hand side in the statement of the lemma.

Lemma 4.3.2.

dn =

{p n = pf for some prime p1 otherwise.

Proof. By Lemma 4.3.1, dn is equal to

min1kn1

p(k) + p(n k) p(n)p 1 .

If n = pf , then this quantity is at least 1, as the addition n = k + (n k) in basep involves carrying at least once. The case k = pf1 shows that dpf = 1.

If n is not a prime power, then n there is a k such that n can be written as asum that does not involve carrying in base p. This yields dn = 0.

In the following lemma, A is an arbitrary abelian group; and L, QL2n, and T2nare as in the preceding section. For n 2 let

cn(x, y) = 1dn((x+ y)n xn yn));

by the definition of dn it has coefficients in Z[x, y]. Notice that cn(x, y) satisfies theequations

cn(x, y) = cn(y, x) (4.3.3)

cn(y, z) + cn(x, y + z) = cn(x+ y, z) + cn(x, y). (4.3.4)

Lemma 4.3.5 (Symmetric two-cocycle lemma). The following statements are equiv-alent and true.

(i) The homogeneous polynomials f(x, y) A Z[x, y] of degree n+ 1 whichsatisfy equations (4.3.3) and (4.3.4) are precisely the a cn+1 for a A.

(ii) If F (x, y) R[x, y] is an nbud, and G is any extension of F to an n+1bud, then the extensions of F to an n+1bud are precisely the polynomials

G(x, y) + rcn+1(x, y)

for r R.(iii) For n < r, any homomorphism

QL2n Afactors through the map Qu : QL2n T2n.

FORMAL GROUPS 23

(iv) Proposition 4.2.5 is true: for n < r there is a canonical splitting

QL2n T2n

QL2n.

wQu

[[ [[ [[ [[u

Proof. i. ii. First observe that ifG is an (n+1)bud and f(x, y) is a homogeneouspolynomial of degree n+ 1, then G(x, y) + f(x, y) is an (n+ 1)bud if and only iff satisfies the axioms (4.3.3) and (4.3.4), so i. ii. Next, over any ring the grouplaw

Ga(x, y) = x+ y + 0may be considered as an n+1bud extending the nbud Ga(x, y) = x+ y, so ii. i.

ii. iii. If A is an abelian group, let ZA refer to the graded ring with(ZA)0 = Z(ZA)2n = A

and a b = 0 for a, b A. The ring homomorphismL ZQL2n

induces an isomorphism

GrRgs[L,ZA] = GrRgs[ZQL2n,ZA], (4.3.6)and there is a natural isomorphism

GrRgs[ZQL2n,ZA] = AbGps[QL2n, A]. (4.3.7)It is clear from (4.3.6) and (4.3.7) that AbGps[QL2n, A] is naturally isomorphic tothe set of all extensions of Ga from an nbud to an (n+ 1)bud of the form

G(x, y) = x+ y + g(x, y)

with g(x, y) A Z[x, y] homogeneous of degree n+ 1.Claim ii. holds if and only if these extensions are precisely the polynomials

G(x, y) = x+ y + acn+1(x, y).

On the other hand, the map

ZQL2n ZQu Z T2n = Z Zdn+1bnclassifies the group law or (n+ 1)-bud

F (x, y) = x+ y + bn((x+ y)n+1 xn+1 yn+1))

= (dn+1bn)cn+1(x, y).

according to equation (4.2.3). An (n+ 1)-bud

G(x, y) = x+ y + g(x, y)

on ZA is of the form hF for a maph : Z T2n ZA

24 ANDO

precisely if g is of the form acn+1(x, y).

iv. iii. because iv. is the universal case of iii.It remains to verify version i. First, one may reduce to the case that A is finitely

generated, since there are only finitely many coefficients in the module of symmetricpolynomials of degree n. And so one reduces to the cases A = Z and A = Z/pk.

Next we reduce to the cases A = Q and A = Z/p. It is easy to check that theresult for A = Q implies the result for A = Z. If k 2 and the result holds forZ/pl with l < k, then it holds for Z/pk as well, as one checks by means of the shortexact sequence

Z/pk1 Z/pk Z/p.Thus it suffices to to verify the result for the fields A = Q and A = Z/p.

Let P [x] denote the polynomial Hopf algebra over A = Q or A = Z/p on aprimitive generator x, and let P [x1, x2, . . . , xk] denote the polynomial ring in kvariables. The complex

P [x1] P [x1, x2] P [x1, x2, x3] . . .with differential

df(x1, x2) =f(x2) f(x1 + x2) f(x1) f P [x1]df(x1, x2, x3) =f(x2, x3) f(x1 + x2, x3)+

f(x1, x2 + x3) f(x1, x2) f P [x1, x2]df(x1, . . . , xk+1) =f(x2, . . . , xk+1)+

ki=1

(1)if(x1, . . . , xi + xi+1, . . . , xk+1)

+ (1)k+1f(x1, . . . , xk) f P [x1, . . . , xk]is the cobar complex which calculates ExtP [x]comodules[A,A]. The symmetric 2-cocycle lemma in the form i. says that the part of Ext2P [x][A,A] represented bysymmetric elements of this cobar complex is

0 if A = QZ/pcp, cp2 , . . . if A = Z/p.

Here is a calculation of this Ext. The dual Hopf algebra to P [x] is the dividedpolynomial algebra [x], giving an isomorphism

ExtP [x]comodules[A,A] = Ext[x]modules[A,A].Now we may use a smaller resolution of A as a [x]module. When A = Q, onehas [x] = P [x] as algebras, and

Q P [x]a P [x]bwith

a 7 1b 7 xa

is a resolution with only a 0 and a 1 term, so Ext2P [x][Q,Q] = 0.

FORMAL GROUPS 25

If A = Z/p then there is an isomorphism of rings

[x] =r0

T [x(r)]

whereT [u] def= P [u]/(up)

and x(r) is the class of degree pr|x| inspired by xpr/(pr!) in [x]. There is then aKunneth isomorphism

Ext[x]modules[A,A] =r0

ExtT [x(r)][A,A]. (4.3.8)

To calculate ExtT [u], we construct a minimal resolution R of A in which thepieces assemble into a differential graded T [u]-algebra, namely

A R = ET [u][a] T [u]

T [u][b],

with differential determined by

d1 = 1 ( the generator of A)da = u 1dbr = up1abr1

(here br is the class inspired by br/r!).

Taking the homology of HomT [u][R, A] one finds that

Ext1T [u][A,A] = Aa |a| = 2|u|Ext2T [u][A,A] = Ab |b| = 2p|u|.

Comparing with (4.3.8), and giving x degree 2, one finds that

Ext2,n[x][A,A] =

Abr1 n = 2prAar as n = 2(pr + ps)0 otherwise.

The class br1 corresponds to the class cr1 in the cobar complex (cr1 is a cocycleby inspection; it has order p in Ext, and the calculation shows that thats all therecan be). The class ar as is represented in the cobar complex by xprxps ; it is notsymmetric. Its symmetrization is a boundary: one may check easily that

d(xpr+s

) = xpr

yps

+ xps

ypr

.

5. Witt vectors and the Witt formal group

One way to classify formal groups is via an analogue of the Lie algebra calledthe Dieudonne-Cartier module. In chapter 6 we attach to a formal group G overany ring R a module CG over the ring WR of Witt vectors of R. One descriptionof this module is

FGpsR[W, G],where W is the Witt formal group. When R is a Z(p)-algebra, then it is usefulto replace W and W with their p-typical counterparts, Wp and Wp; we call the

26 ANDO

resulting module DG. When R is a perfect field of characteristic p > 0 and G isone-dimensional, then either G is the additive group, or DG is a finite free moduleover WpR, so this is a particularly effective classification.

The purpose of this chapter is to introduce these various Witt vectors. Theuncompleted Witt vectors are affine ring schemes, that is, representable functorsfrom rings to rings. As a ring each is isomorphic to the polynomial ring Z[x1, x2, . . . ]on a countable set of generators.

We shall treat the p-typical Witt vectors first. When k is a perfect field ofcharacteristic p, the ring Wpk is to k as the ring of p-adic integers are to Fp: Wpkis a torsion-free, complete Noetherian local ring with maximal ideal generated byp, and with an isomorphism Wpk/p = k.

5.1. p-typical Witt vectors. Let An be the scheme

An def= SpecZ[x0, . . . , xn1].

Z[x0, . . . , xn1] has two Hopf algebra structures

Z[x0, . . . , xn1] Z[x0, . . . , xn1, y0, . . . , yn1],namely

xi 7 xi + yi

and

xi 7 xiyi,which correspond to products

An ZAn An.

These products describe how the functor on rings

R 7 Rn

represented by An takes its value in rings.The Witt scheme Wp,n is another ring scheme whose underlying scheme is An.

It comes with a morphism

Wp,nw An

which is given on coordinates by the Witt polynomials

w0(x0, . . . , xn1) = x0w1(x0, . . . , xn1) = x

p0 + px1

w2(x0, . . . , xn1) = xp2

0 + pxp1 + p

2x2

wj(x0, . . . , xn1) =j

d=0

pdxpjdd

FORMAL GROUPS 27

Proposition 5.1.1. There are unique maps

Wp,n ZWp,n

+WWp,n and

Wp,n ZWp,n

WWp,n,

giving Wp,n the structure of a ring scheme such that the Witt map w is a map ofring schemes.

The existence of both the sum and product on Wp,n follow from the followinguseful lemma, which describes the image of the Witt map. Let A = Z[a1, a2, . . . ]be a polynomial ring on any number of generators, and let

f(a1, a2, . . . ) = f(ap1, a

p2, . . . )

denote its p-Frobenius endomorphism.

Lemma 5.1.2 (Image of Witt). An element (b0, b1, b2, . . . ) of An(A) is of the formw(c0, c1, c2) for some (c0, c1, . . . ) in Wp,n(A) if and only if

bi (bi1) (mod pi) (5.1.3)for i 1. Moreover the ci are unique.

Proof. The ci are unique, if they exist, because for the ring A = Z[ai] the Wittmap is injective.

Note that wi(c0, c1, . . . ) depends only on cj for j i. Suppose given polyno-mials (b0(a), b1(a), b2(a) . . . , ) An(A), and that we have constructed polynomialsc0(a), c1(a), . . . , ck1(a) such that

bi(a) = wi(c0(a), c1(a), . . . )

for i < k. We shall attempt to construct ck(a) such that bk = wk(c0, . . . , ck); inother words, to solve the equation

bk(a) = c0(a)pk

+ + prcr(a)pkr + + pkck(a)(a).We may do this if and only if

bk(a) c0(a)pk + + prcr(a)pkr + + pk1cpk1(a) (mod pk). (5.1.4)Consider the equation

bk1(a) = c0(a)pk1

+ + prcr(a)pk1r + + pk1ck1(a).Applying to this equation yields

bk1(a) = (c0 (a))

pk1 + + pr(cr (a))pk1r

+ + pk1ck1(a).From the equation

cr cpr (mod p)one has

(cr )pk1r cpkrr (mod pkr)

pr(cr )pk1r prcpkrr (mod pk).

It follows that

bk1(a) c0(a)pk

+ pc1(a)pk1

+ + pk1ck1(a)p (mod pk).

28 ANDO

Substituting into equation (5.1.4) gives (5.1.3).

Proof of the Proposition. Let us treat the case of addition. We must construct adotted arrow so that the diagram

Wp,n Wp,n An An

Wp,n An.

www

u

+W

u

+

ww

By representability, it suffices to treat the case of the element

(x0, . . . , xn1, y0, . . . , yn1) Wp,n Wp,n(Z[x0, . . . , xn1, y0, . . . , yn1]).In other words, we must construct polynomials

(x+Wy)0, (x+

Wy)1, Z[x, y]

such thatw((x +

Wy)0, (x +

Wy)1, . . . ) = w(x) + w(y).

According to Lemma 5.1.2, we have

wi(x) wi1(x) (mod pi),and similarly for the y, for all i. It follows that

wi(x) + wi(y) (wi1(x) + wi1(y)) (mod pi)for all i. Lemma 5.1.2 yields the desired polynomials.

There are restriction maps

Wp,n+1 Wp,n,and the diagram

Wp,n+1 Wp,nw

y ywAn+1 An

commutes. The Witt scheme Wp is defined as the limit

Wpdef= limWp,n; (5.1.5)

it is isomorphic as a scheme to AN.

5.2. Frobenius, Verschiebung. If

(a0, a1, . . .), : ai Ris an R-valued point of Wp (= homomorphism from Z[x0, . . .] to R), then thecomponents of its image under w

(w0(a), w1(a), . . .)

are called the ghost or phantom components. As we have already done in study-ing the ring structure, we shall repeatedly use the ghost components and Lemma

FORMAL GROUPS 29

5.4.6 to examine the structure of the Witt vectors. If X denotes an operation onthe Witt vectors, then its ghostly image will be denoted Xw.

As another illustration of this technique, we study the operators Frobenius andVerschiebung.

The operator Verschiebung on Wp is given by

V (x0, x1, . . .) = (0, x0, x1, . . .);

in terms of the ghost components,

V w(w0, w1, . . .) = (w0(0, x0, x1, . . .), w1(0, x0, x1, . . .), . . .)

= (0, px0, . . . ,j

d=1

pdxpjdd1 , . . .)

= (0, px0, . . . ,j1d=0

pd+1xpjd1d , . . .)

= (0, pw0, pw1, . . .).

From the calculation in terms of ghost components we learn that V is additive.

The Frobenius operator we even define in terms of the ghost components; towit,

w(w0, w1, w2, . . .) = (w1, w2, . . .).

That w induces an operation onWp follows by the image-of-Witt condition (5.1.3).The Frobenius is both additive and multiplicative. It is difficult to write down on Witt vectors in the general case, but in characteristic p > 0 inspection of theWitt polynomials shows that

(a0, a1, . . .) = (ap0, a

p1, . . .). (5.2.1)

The ghost formulae show that

V = [p]Wp

where [p]Wp is p-fold addition in the Witt ring scheme. Moreover, the formula for in characteristic p shows that

[p]Wp(a0, a1, . . .) = V (a0, a1, . . . ) (5.2.2)

= (0, ap0, ap1, . . .) (5.2.3)

= V (a0, a1, . . .).

So at least in characteristic p, we also have V = p; one way to express this mightbe

V = [p](Wp)Fp ,

where (Wp)Fp denotes the Witt scheme pulled back over

SpecFp SpecZ.

30 ANDO

5.3. Witt vectors of perfect fields and p-adic arithmetic. Let k be a perfectfield of characteristic p > 0. The Frobenius x 7 xp is an isomorphism of k, so theideal

(0, a1, a2, . . .) Wpkis by equation (5.2.3) exactly the ideal

pWpk Wpk.Thus the map (a0, . . .) 7 a0 provides an isomorphism

Wpk/p= k, (5.3.1)

and indeedWpk/pn

=Wp,n(k).By definition (5.1.5), we have

Wpk = limn

Wpk/pn. (5.3.2)

Now suppose that k is the finite field Fpn . We could have tried to make a ring Rhaving the properties (5.3.1) and (5.3.2) of Wpk directly: suppose R is a completelocal ring with residue field k. The Teichmuller construction [Ser68, II,5, Prop. 8]provides a canonical multiplicative section

kf R.

In particular, for a 6= 0 in k, f(a) satisfies f(a)pn1 = 1, and has canonical pi rootsfor all i. The smallest ring R we could expect to get is

R = Zp(),where is a primitive (pn 1)st root of unity.Theorem 5.3.3. The map

Wpk Rgiven by

(a0, a1, . . .) 7 f(a0) + f(ap11 )p+ f(a

p22 )p

2 + . . .is an isomorphism of rings.

Remark 5.3.4. The argument shows that Wpk is initial for complete local ringsS with residue field k. This will be important when we get to crystals.

Proof. The map is an isomorphism of sets; it remains to show that its a ringhomomorphism. Its inverse is

RsWpk

f(ai)pi 7 (a0, ap1, ap2

2 , . . .).

The map s factors through the reduction

Wp(R) Wp(k)by the map

RsWp(R)

f(ai)pi 7 (f(a0), f(a1)p, . . .),

FORMAL GROUPS 31

because f is multiplicative. Then we have a diagram

R Wp(R) R

Wpk

ws

[[

[ []s

wwn

u

Each of the wn is a ring homomorphism, and the collection of them is an injectivemap

Wp(R) AN(R).(this is not true with R replaced by k, which is why it is helpful to lift the problemto R), so it suffices to show that the failure of the composite wn s to be a ringhomomorphism lies in the kernel of the reduction map. We use the following, withproof left to the reader.

Lemma 5.3.5. The Witt vector (b0, b1, . . .) Wp(R) is in the kernel of

Wp(R) Wpk

if and only if its ghost components satisfy

wm(b) 0 (mod pm+1).

The composite

RsWp(R) wn R R/pm+1

is given by

r =

bipi 7 (b0, bp1, . . .)7 bpn0 + pbp

n

1 + . . .+ pmbp

n

m

n(r) (mod pm+1).

Here is the Frobenius automorphism of R induced by the Frobenius on k, andits a ring homomorphism. It follows by the Lemma that the failure of wns to be aring homomorphism is in the kernel of the reduction to Wpk.

Remark 5.3.6. From the preceding it appears that the Teichmuller section k Wpk is the map

r(a) = (a, 0, 0, . . . )

Indeed, in terms of the ghost components,

rw(a) = (a, ap, ap2, . . . ),

which shows that r is multiplicative.

32 ANDO

5.4. Global Witt vectors. One place to start is with this simple result aboutpower series.

Lemma 5.4.1. Any power series

p(t) = 1 + b1x+ b2x2 + . . .

can be written in a unique way in the form

p(t) =n1

(1 anxn).

Proof. By induction. Suppose

p(x) =

(k1n=1

1 anxn)(1 + bkxk + o(k + 1)).

Since

1 + bkxk + o(k + 1)1 + bkxk

= 1 + o(k + 1),

we obtain

p(x) = (1 + o(k + 1))k

n=1

(1 anxn)

by setting ak = bk.

The global Witt scheme W is a ring scheme, isomorphic as a scheme to AN =limn

An, whose sum is defined in terms of this Lemma. Let the infinite sequence(a1, a2, . . .) W(A) of elements a ring A secretly correspond to the power series

n1(1 anxn).

Then the sum of

a = (a1, a2, . . .) and b = (b1, b2, . . .)

in W(A) is given byn1

(1 (a+Wb)nxn) =

n1

(1 anxn)n1

(1 bnxn), (5.4.2)

using the Lemma.

Were calling this the global Witt scheme or the ring of global Witt vectors, soitd better have something to do with the Witt vectors as we defined them earlier,and itd better be a ring scheme. To see this, well study the sum +

Wmore closely,

FORMAL GROUPS 33

and obtain analogs of the Witt polynomials. Write

n1

(1 cnxn) = explog

n1(1 cnxn)

= exp

n1

log(1 cnxn)

= exp

n1

i1

(cnxn)i

i

= exp

N1

xN

N

d|N

dcN/dd

= exp

N1

w(N)(c)N

xN

,where

w(N)(c)def=d|N

dcN/dd .

The same analysis shows

(1 anxn)

(1 bnxn) = exp

N1

w(N)(a) + w(N)(b)N

xN

.Comparing coefficients, we see that

w(N)(c) = w(N)(a) + w(N)(b).

To summarize, let PS be the scheme which associates to a ring R the group ofpower series p R[[x]] with p(0) = 1, under multiplication, and let PSQ be thesame group, restricted to Q-algebras.

Proposition 5.4.3. The diagram

W(a) 7Qn1(1anxn)= PS

(a) 7(w(1)(a),w(2)(a),... )y yAN (c) 7exp(

Pcnx

n/n) PSQ

(5.4.4)

is a commutative diagram of group schemes; the marked arrow is an isomorphism.

Corollary 5.4.5. If R is torsion free and un R, the power seriesexp(

unnxn) QR[[t]]

has coefficients in R if and only if un = w(n)(a) for some sequence of elementsa1, a2, . . . in R.

34 ANDO

Proof. Use the fact that the map

ANQ(c) 7exp(P cnxn/n) PSQ

is an isomorphism.

For the w(N), there is a generalization of the Image-of-Witt Lemma (5.1.2).

Lemma 5.4.6 (Image of Witt II). The sequence (c1, c2, . . .) with ci Z[x1, . . .] isin the image of w if and only if for all n 1 and all primes p,

cn p(cnp ) (mod pj),where j = p(n), i.e. n = pjm with (p,m) = 1.

Using the new Image-of-Witt Lemma, one checks that there is always a vector(c) such that

w(N)(c) = w(N)(a)w(N)(b),

and so

Corollary 5.4.7. There is a unique product

WZWWW

which combines with the sum +W

to give W the structure of a ring scheme, in sucha way that the Witt map

W w ANis a homomorphism of ring schemes.

As an exercise, the reader might try to work out the product in terms of theisomorphism W = PS.

Frobenius and Verschiebung. As another application, one may check exactly as inthe case of the p-typical Witt vectors that there are operators Fr and Vr on W.The Verschiebung Vr is given by the formula

Vr(x1, x2, . . . ) = (0, . . . , 0, x1r, . . . , x2

2r, . . . );

and one may check this is equivalent to the formula

V wr (w(1), w(2), . . . ) = (0, . . . , 0, rw(1)r

, . . . , rw(2)2r

)

so Vr is additive. Once again, Fr is given in terms of ghost components by

Fwr (w(1), w(2), . . . ) = (w(r), w(2r), . . . ),

and the Image-of-Witt lemma in the form (5.4.6) shows that this defines an oepra-tion on W. The formulae in terms of ghost components show that

FrVr = rW.

FORMAL GROUPS 35

Exercise 5.4.8. Show that the effect of Fr on the Witt vector (a, 0, 0, . . . ) is givenby the formula

Fr(a, 0, 0, . . . ) = (ar, 0, . . . ).

Show that when r = 0 in A, Fr is given by the formula

Fr(a1, a2, . . . ) = (ar1, ar2, . . . ). (5.4.9)

Show that when r = 0 in A,VrFr = rW.

Relationship with p-typical Witt vectors: the Artin-Hasse exponential. Now supposethat in a = (a1, a2, . . .) we have aj = 0 unless j is a power of a prime p. Thenw(n) = 0 unless n is a power of p, and we have

w(pk)(a) =ki=0

piapki

pi

which up to renumbering is the earlier Witt polynomial wk. Notice that for any a,the Witt polynomial w(pk) does not depend on aj for j prime to p.

More precisely, the p-typical Witt scheme is a quotient of W, exhibited by thediagram

W AN

Wp AN

(a) (w(1)(a), w(2)(a), . . .)

(a1, ap, ap2 , . . .) (w(1), w(p), w(p2), . . .).

ww

u uw

w

v wv

u

v

uv w

(5.4.10)

Now when R is a Z(p)-algebra, it turn out that there are lots of sections

Wp(R) W(R).Wed like to define a section

WpsW

by insisting that the diagram

Wps W

w

y ywAN

swAN

(5.4.11)

commute, with sw defined by

sw(c0, c1, c2, . . .) = (c0, 0, . . . , 0, c1p, 0, . . . , 0, c2

p2, . . .),

36 ANDO

but we need to check that the right-hand-side is in the image of w. The Image-of-Witt Lemma requires first that for c = w(a) and q 6= p,

0 = sw(c)qpk q(sw(c)pk) = q(ck) (mod q).and second that for all k,

ck p(ck1) (mod pk).The first condition wont be true in general, but it will be true trivially in a Z(p)-algebra. The second is just the p-typical image-of-Witt condition for c. We haveproved

Proposition 5.4.12. Over Z(p), there is a unique map s of ring schemes makingthe diagram (5.4.11) commute.

In fact the argument shows that for each n with (n, p) = 1 there is a section

WpsnW

given on ghost coordinates by

(c0, c1, . . .) 7 (0, . . . , c0n, . . . , c1

np, . . .).

Corollary 5.4.13. There is an isomorphism of ring schemes over SpecZ(p)

WZ(p) =

(n,p)=1

(Wp)Z(p) .

Now the construction of the section wasnt too hard, but it has the followingimportant consequence.

Corollary 5.4.14. The Artin-Hasse exponential

exp

n0

xpn

pn

has coefficients in Z(p).

This power series will provide a natural p-typical coordinate on the multiplica-tive group over Z(p); see section 6.9.

Proof. Recall from Corollary 5.4.5 that

exp

n1

cnxn

n

has coefficients in Z(p) if c is in the image of

W(Z(p))w AN(Z(p)).

The work weve just done shows that

(1, 0, . . . , 0, 1p, . . . , 1

p2, . . .) = sw(1, 1, . . .)

= sww(1, 0, . . .)

= ws(1, 0, . . . )

is in the image of w.

FORMAL GROUPS 37

5.5. The Witt formal group. The Witt schemeWp,n with just its additive struc-ture is a group scheme, isomorphic as a scheme to An. Its completion at the origin,given on the category of adic R-algebras by

Wp,n(A) = Ker[Wp,n(A) Wp,n(R)],is a formal group, isomorphic as a formal variety to AnR.

The p-typical Witt formal group is the direct limit

Wp(A) = colimn

Wp,n(A).

Thus(a0, a1, . . .)

is considered a point of Wp(A) if and only if ai I(A) for each i, and moreoverthere exists an N such that ai = 0 for i N .

Similarly there is a global Witt formal group W, whose formal sum comes from(5.4.2). Its A-valued points, for an adic R-algebra A, are sequences

(a1, a2, . . .)

such that ai I(A) for each i, and moreover there exists an N such that ai = 0for i N .

These formal groups play a major role in the study of formal groups, becausethe Witt formal group is the free formal group on the formal line: there is a naturalisomorphism

(Formal Varieties)[A1, G] = (Formal Groups)[W, G].That is the subject of the next section.

6. Classification of formal groups via the Dieudonne-Cartier module

6.1. The curves functor C. Proceeding by analogy with Lie theory, one mighttry to study a formal group in terms of its Lie algebra. Thats not enough structure,but notice that a vector v in the Lie algebra of a Lie group G yields a curve exp(tv)in G. If G is a formal group over R, one considers the full abelian group of curves.

Definition 6.1.1. If G is a formal group then the group of curves in G is the group

C(G) = CR(G)def= (Formal varieties over R)[A1R, G].

One finds for more or less formal reasons that the curves functor is faithful. Oneis led 1) to determine the endomorphisms of C, and 2) to compute its image.

6.2. Endomorphisms of the curves functor. There are three basic types ofendomorphisms of Curves, homotheties, Verschiebung, and Frobenius.

The homothety and Verschiebung operators both come from endomorphisms ofthe affine line. Homotheties come from dilations: given a curve

A1R G

and an element a R, we can form the new curve [a]() given byA1R

a A1R G.

38 ANDO

For r 0 the curve Vr is given by

A1Rt7tr A1R G.

More explicitly, if = (1, . . . , n)

with respect to some coordinate system

G= AnR,

then we can writei(t) =

k1

biktk

with bik R. Then([a])i(t) =

k1

bikaktk,

and(Vr)i(t) =

k1

biktkr.

Frobenius has a slightly more complicated description, using Newtons Lemma.

Lemma 6.2.1. The map

R[[x1, . . . , xr]]r= R[[1, . . . , r]]

i(x1, . . . , xr) iis an isomorphism, where i(x) are the elementary symmetric polynomials.

Corollary 6.2.2. This induces an isomorphism

(A1)r/r= (A1)r.

For a curve , the r-Frobenius Fr is given by the diagram

(A1)r Gr

(A1)r/r G

(A1)r

A1

wr

u u

+G

w

u'

u

(0,0,...,(1)rr)

Fr

FORMAL GROUPS 39

To compute Fr explicitly, observe that after possible base extension to a ring con-taining a primitive rthroot of unity , the diagram may be extended by a commu-tative triangle

(A1)r Gr

(A1)r/r G

(A1)r

A1 A1

wr

u u

+G

w

u'

(x,x,...,r1x)

wx7xr

u

Fr

so

(Fr)(t) =r1Gi=0

(t1r i). (6.2.3)

Example 6.2.4 (Ga). If(t) =

n

bntn

then we have

([a])(t) =

bnantn,

(Vr)(t) =n

bntnr, and (6.2.5)

(Fr)(t) =n

r1i=1

bntnr ni

=n

rbnrtn.

Example 6.2.6 (The Witt formal group). To compute Fr, consider first the path

1(t) = (t, 0, 0, . . .).

Recall that the Witt sum corresponds to multiplication of power series, where

(a1, a2, . . .)

corresponds to the power series

(1 a1x)(1 a2x2) . . . .Then we have

(Fr1)(t) = (t1r , 0, . . .) +

W(t

1r , 0, . . .) +

W. . . (t

1r r1, 0, . . .)

= (1 t 1r x) . . . (1 t 1r r1x)= (1 txr) = (0, . . . , 0, t

r, 0, . . .).

40 ANDO

For a general curve(t) = (1(t), 2(t), . . .)

we get

(Fr)(t) = (0, . . . , 1(t)r, . . . , 2(t)

2r, . . .). (6.2.7)

([a])(t) = (1(at), 2(at), . . .), (6.2.8)

and

(Vr)(t) = (1(tr), 2(tr), . . .). (6.2.9)

Proposition 6.2.10. The various homothety, Frobenius, and Verschiebung opera-tors interact according to the rules

(i) FrVs = VsFr if (r, s) = 1.(ii) FrVr = r where r denotes multiplication by r on curves, i.e. r-fold formal

sum on the formal group.(iii) FrFs = Frs and VrVs = Vrs(iv) Fr[a] = [ar]Fr(v) [a]Vr = Vr[ar]

Proof. The only hard part is to figure out in which order to apply the operations.For example, VsFr(t) is given by

A1 t7ts

A1 Fr G.Let denote a primitive rth root of unity. Then

(i) if (r, s) = 1,

(VsFr)(t) = Vs(r1Gi=0

(t1r i))

=r1Gi=0

(tsr i)

=r1Gi=0

(tsr si) if (r, s) = 1

= Fr(t 7 (ts))= (FrVs)(t).

(ii) On the other hand if s = r we get

FrVr(t) = Fr(t 7 (tr))

=r1Gi=0

(trr ri)

= [r]G((t)) = (r)(t).

(iii) exercise

FORMAL GROUPS 41

(iv)

(Fr[a])(t) =r1Gi=0

(at1r i)

=r1Gi=0

((art)1r i)

= [ar]Fr(t).

(v) exercise.

6.3. W represents the curves functor. Let W be the Witt formal group, andlet 1 be the curve

A1 1 Wt 7 (t, 0, 0, . . .).

A fundamental result of Dieudonne is

Theorem 6.3.1. For a formal group G over a ring R, restriction to 1 induces anisomorphism

FGpsR[W, G]= CG.

Under this isomorphism, the operator Fr on curves corresponds to the operator Vron the Witt formal group. If r = 0 in R, then the operator Vr on curves correspondsto the operator Fr on the Witt formal group.

Proof. First we show that the map is injective. Let pii be the projections

W A1a 7 ai.

Then, using the formulas (6.2.8,6.2.9,6.2.7), we can writei1

Fi1pii = 1 Hom[W, W]

(the sum is well-defined by the vanishing condition on W). Now suppose thatg Hom[W, G] restricts to zero, that is,

g1 = 0.

Then, since Fn is an endomorphism of the curves functor and g is a homomorphismof formal groups, we get

g = g i1

Fi1pii

=i1

Fig1pii = 0.

Now we have to show that the restriction is surjective. For a curve

A1 G

42 ANDO

we define a map

W g() Gof formal varieties by

g() =i1

Fi pii.

Clearlyg()1 =

as a map of formal varieties, so it remains to check that g is a homomorphism.

Let us suppose for simplicity that G is one-dimensional. Then we can use thetechnique of prolongation of algebraic identities: by Corollary 4.1.3 of Lazardstheorem, we can lift G to a formal group G over a torsion-free ring S, and thenstudy the problem over S Q, where the problem becomes isomorphic to theanalogous problem for the additive group via the logarithm.

G G G Q Ga

WR WS WSQ

SpecR SpecS SpecS Qu

w u wlogG

*g()

[[[^

w

u

*g()

[[[^ u

*

g()

[[[^

u

ug(logG

)

w u

Then since homomorphisms add, it suffices to check that g() is a homomorphismwhen is the curve

(t) = tn

in Ga. The formula (6.2.5) for Frobenius is

Fd(t) =d1j=0

tn/dnj =

{dtn/d d|n0 otherwise

,

so we have

g(a) =d1

(Fd)(ad)

=d|n

dand

d

= w(n)(a).

But the Witt polynomial w(n) was constructed to be a homomorphism

Ww(n) Ga.

It remains to check the claims about Frobenius and Verschiebung. Given theisomorphism already established, it suffices to check study the effect of Fr and Vron the curve 1. From example 6.2.6 we have

Fk1(t) = (0, . . . , 0, tk, 0, . . .).

FORMAL GROUPS 43

Fix an element (a1, a2, . . . ) W(A). ThenFiFr1pii(a1, a2, . . . ) =

Fir1pii(a1, a2, . . . )

= (0, . . . , a1r, . . . , a2

2r, . . . )

= Vr(a1, a2, . . . ).

In other words, FiFr1 = Vr.

Similarly,

Vr1(t) = (tr, 0, . . . )

FiVr1(t) = (0, . . . , trr, 0, . . . ).

So FiVr1pii(a1, a2, . . . ) = (ar1, a

r2, . . . ).

According to (5.4.8), this last quantity coincides with Fr(a1, a2, . . . ) when r = 0 inR.

Remark 6.3.2. There is a version of the lifting result (Corollary 4.1.3) for higher-dimensional formal groups [Haz78, section 9.6].

6.4. C takes values in modules over the Witt vectors. Weve learned that

CR(G) = FGpsR[W, G],so End[CR] = FGpsR[W, W] = CR(W). In this section we use this fact to give CRthe structure of functor to modules over W(R).

Recall that (a1, a2, . . .) W(R) corresponds to the power seriesm1

(1 amxm).

Thus the curve

A1R WR

t 7 (0, . . . , btnm, 0, . . .)

corresponds to the curve

t 7 (1 btnxm) = Vn[b]Fm1(t).This observation leads us to the following analogue of (5.4.1).

Lemma 6.4.1. Any power series

p(x, t) =i,j0

ai,jtixj

with a0,0 = 1 can be written in a unique way in the form

p(x, t) =

n,m1(1 bn,mtnxm).

44 ANDO

Were going to use this Lemma to represent curves in W, but first we need tomake an technical observation about what it means to be an element of C(W).

Although any curve C(W) can be written as = (1, 2, . . .)

withi(t) =

n1

ai,ntn Hom[A1, A1],

not every collection of i yields a curve inW: if is a curve, A is an adic R-algebra,and

a Ker[A R] = A1R(A)is an A-valued point of A1R, then when we apply

(a) = (1(a), 2(a), . . .)

we must obtain an A-valued point of WR(A). But by definition, we required ofpoints (b1, b2, . . .) WR(A) that there exist an N such that bj = 0 for j N .

Now N can depend on A, and indeed on the element (a), but with fixed someN must exist for any adic R-algebra A. This amounts to the saying that

= (1, 2, . . .)

is a curve if and only if

for all j theres an N such that for i N ,i(t) 0 (mod tj).

For example, the sequence of power series

t 7 (t, t2, t3, . . .)is a curve in C(W), even though it doesnt factor through any finite Wn. On theother hand the map

t 7 (t, t, t, . . .)isnt a curve.

Now the curve(1(t), 2(t), . . .)

corresponds to a product m1

(1 m(t)xm)

and so by Lemma 6.4.1 to a product of the formm,n1

(1 bm,ntnxm).

In order for such a product to be a curve, it is necessary and sufficient that

for all n theres an M such that for m M ,we have bn,m = 0.

Using the formulas (6.2.8,6.2.9,6.2.7), we get

FORMAL GROUPS 45

Proposition 6.4.2. The mapn,m0

Vn[bn,m]Fm 7

n,m0Vn[bn,m]Fm1

establishes a bijective correspondence between curves in CR(W) and sums of theform

n,m0Vn[bn,m]Fm

satisfying the condition

for all n theres an M such that for m M ,we have bn,m = 0.

In particular, the curve

(t) = (a1t, a2t2, a3t3, . . .)

can be written =

n1

Vn[an]Fn1.

Theorem 6.4.3. The map

(a1, a2, . . .)En1

Vn[an]Fn

is a homomorphism of rings

W(R) E End[CR].

Proof. For elements a and b, we need to show that

E(a+Wb) = E(a) + E(b) and

E(a Wb) = E(a) E(b)

in End[C(G)], where G is a formal group over R. Let us suppose as in the proofof Theorem 6.3.1 that G is one-dimensional (or see Remark 6.3.2). Applying againthe technique of prolongation of algebraic identities, it suffices to check that theseequations hold for the additive group.

To check that E is a homomorphism for the additive group, it suffices to checkon the curve (t) = tn. We have

Fd(t) =d1j=0

tn/dnj =

{dtn/d d|n0 otherwise

[ad]Fd(t) =

{da

n/dd t

n/d d|n0 otherwise

Vd[ad]Fd(t) =

{da

n/dd t

n d|n0 otherwise

46 ANDO

So(E(a))(t) = w(n)(a)tn.

But w(n) is a ring homomorphism, so

E(a+Wb)(t) = w(n)(a+

Wb)tn

= (w(n)(a) + w(n)(b))tn

= (E(a) + E(b))(t), and

E(a Wb)(t) = w(n)(a

Wb)tn

= (w(n)(a) w(n)(b))tn= (E(a) E(b))(t).

Recall (Remark 5.3.6) that r : RWR is the assignmentr(a) = (a, 0, 0, . . . ).

In terms of ghost components, r is given by

rw(a) = (a, a2, a3, . . . );

it follows that r is multiplicative, but not additive. For n 1 letrn(x, y) Z[x, y]

be the polynomial such that

r(a) +Wr(b) = (r1(a, b), r2(a, b), r3(a, b), . . . )

Corollary 6.4.4. For a and b in R, the homotheties [a], [b], and [a+b] in End[CR]are related by the formula

[a+ b] =n=1

Vn[rn(a, b)]Fn

= [a] + [b] +n=2

Vn[rn(a, b)]Fn.

(6.4.5)

6.5. p-typical curves over a Z(p)-algebra. From now on, we suppose that ourformal groups are formal groups over Z(p)-algebras. The curves functor has evenmore structure over a Z(p) algebra, because of the isomorphism (5.4.13)

WZ(p) =

(n,p)=1

(Wp)Z(p) .

In view of Dieudonnes Theorem (6.3.1), there is an isomorphism

CG =

(n,p)=1

FGps[Wp, G]

when G is a formal group law over a Z(p)-algebra. Thus the group C(G) of curvesin G is determined by the subgroup

FGps[Wp, G].

FORMAL GROUPS 47

Definition 6.5.1. A curve CG is p-typical ifFn = 0

when n is not a power of p.

Proposition 6.5.2. The set of p-typical curves is a subgroup D(G) of C(G). It isisomorphic to the group

Hom[Wp, G]via the map

g 7 g0,where 0 is the composite

A1 1 W Wp;in particular the inclusion

DG CGis represented by the projection W Wp of (5.4.10).

Proof. It suffices to show that the homomorphism

W g G,representing a curve , factors through Wp if and only if is p-typical. In Dieudonnestheorem, we learned that g is actually given by the formula

g =n0

Fnpin

which indeed factors through the projection if and only if is p-typical.

By construction, the functor D has only iterates of F def= Fp from among itsFrobenius operators: for (m, p) = 1, Fmpk = 0 by definition, and

Fpk = (Fp)k = F k.

By the relations among the endomorphisms, it is clear that homotheties preserve p-typical curves. So, it turns out, does the Verschiebung operator Vp: if is p-typical,and k = mpj with (m, p) = 1 and m 6= 1 then

FkVp = FpjVpFm = 0.

So in addition to homotheties, there are two operators, F and V , on D. We alreadylearned that in general, FV = p.

Proposition 6.5.3. We have p = 0 in R if and only if V F = p.

Proof. The point is, V F = p if and only if VpFp = p applied to the curve 1(t) inC(W). There we have

VpFp1(t) = 1 tpxp[p]W(t) = (1 tx)p.

These quantities are equal if and only if p = 0 in R.

Theorem 6.4.3 becomes

48 ANDO

Theorem 6.5.4. The group DG is a module over WpR by

(a) 7n0

V n[an]Fn.

Notice that the operations F and V are not WpR-linear. Instead we have

F (a) = Fn0

V n[an]Fn

=n0

V n[apn]Fn+1.

Recall (5.2.1) that if R is an Fp-algebra and (a0, a1, . . . ) WpR, thenF (a0, a1, . . . ) = (a

p0, a

p1, . . . ).

So if R is an Fp-algebra thenF (a) = (a)F.

Similarly, if R is an Fp-algebra then

V ((a)) = (a)V.

Suppose that f : A B is a ring homomorphism. There are natural transfor-mations

ModulesBff

ModulesA

given as usual byfN = N,

considered as an A-module via f ; and

fM def= B f,A

M.

Of course f is the left adjoint of f: there is a natural isomorphism

ModulesB [fM,N ] = ModulesA[M,fN ].

Thus if R is an Fp-algebra, then the operation F may be viewed as a homomor-phism of WpR-modules

DGF DG

or equivalently

DG F DG.The Verschiebung may be viewed as a homomorphism of WpR-modules

DGV DG.

When R = k is a perfect field, it turns out that V may equivalently be viewed asa homomorphism of WpR-modules

DGV DG.

FORMAL GROUPS 49

Lemma 6.5.5. If k is a perfect field, and M is aWpk-module, then the assignment

m 7 1mis an isomorphism of groups M = M . This isomorphism establishes an isomor-phism

Hom[M,M ] = Hom[M,M ].

Proof. If k is perfect then is an isomorphism, so in M we have

bm = 1 b1m;this shows that M = M as abelian groups.

In general if f : A B is an isomorphism, S is an A-module, and T is a B-module, then the adjunction maps

ffS ST ffT

are isomorphisms. Applying this to A =Wpk = B, and f = , we have

Hom[S, T ] = Hom[S, T ]= Hom[S, T ]= Hom[S, T ].

6.6. The Dieudonne modules associated to formal groups are uniformand reduced. Let M be an abelian group, and V : M M be a homomorphism.Definition 6.6.1. M is reduced with respect to V if

M = limr

M/V rM.

It is uniform if

M/VMV k V kM/V k+1M

is an isomorphism for k 1.

In this section we shall prove

Theorem 6.6.2. If G is a formal group over a Z(p)-algebra R, then the WpR-module DG is uniform and reduced (with respect to the Verschiebung).

We shall prove this when G is finite-dimensional. The main point is the following.

Lemma 6.6.3 (Taylor series). If G is a d-dimensional formal group over a Z(p)-algebra, and 1, . . . , d is a system of p-typical parameter, then p-typical curves arein 1-1 correspondence with curves of the form

(t) =Gn0

dGj=1

(V n[aj,n]j)(t).

50 ANDO

Remark 6.6.4. We could just as well written this curve using addition in DG:Gn0

(V n[an]0)(t) =

n0

V n[an]0

(t).Proof. For simplicity, let us treat the case d = 1. In terms of an isomorphism offormal varieties G = A1, a curve is by definition a sum of the form

(t) =n1

antn.

The first point is that one may replace the sum with a formal sum: if 1 is aparameter, the reader may check that any curve may be written in a unique wayin the form

(t) =

m0

Vm[am]1

(t).Now suppose that 1 is a p-typical parameter. It remains to show that is p-typicalif and only if am = 0 for m not a power of p. The if part is easy: For m = rps with(p, r) = 1 and r 6= 1, we find

Fm

n0

Vpn [apn ]1

=n0

FpsVpn [arpn ]Fr1 = 0.

For the only if part, suppose that m = rps is the smallest number not a power ofp for which am 6= 0, so

=kj=0

Vpj [apj ]1 + Vm[am]1 + higher terms.

Then

Fr( kj=0

Vpj [apj ]1) = Fr(Vm[am]1 + higher terms)

= rVps [am]1 + higher terms6= 0.

But the p-typical curves are a group, so if

kj=0

Vpj [apj ]1

isnt p-typical, neither is .

Corollary 6.6.5. A system i of p-typical parameters for G, considered as elementsof DG, determines a basis for DG/V DG. In particular, DG/V DG is canonicallyisomorphic to the tangent space to G at the origin, Lie(G). Remark 6.6.6. In particular, homotheties induce the structure of an R-moduleon DG/V DG, as may be seen directly from Corollary 6.4.4.

Corollary 6.6.7. If G is a finite-dimensional formal group over a Z(p)algebra,then the module DG of p-typical curves is reduced.

FORMAL GROUPS 51

Proposition 6.6.8. If G is a finite-dimensional formal group over a Z(p)algebraR, then the module DG of p-typical curves is uniform.

Proof. Let 1, . . . , d be a system of p-typical parameters. By the Taylor serieslemma, any element of

V rDG/V r+1DG

can be represented by a curve of the form

= V r[a1]1 + + V r[ad]d + V r+1DGwith the ai determined by the element it represents, i.e. the map

V rDG/V r+1DG Rdj

V r[aj ]j + V r+1DG 7 (a1, . . . ad)

is an isomorphism. The proposition follows from the obvious commutativity of thediagram

V rDG/V r+1DG V r+kDG/V r+k+1DG

Rd.

wV k

'44447 '

Definition 6.6.9. Suppose that k is an perfect field of characteristic p. ADieudonnemodule over k is a Wpk-module M equipped with operators Frobenius, F , andVerschiebung,V , which are Wpklinear maps

M FMM

V M,and which satisfy

FV = p = V F.Moreover M is required to be uniform and reduced with respect to V . The moduleDG of p-typical curves of a formal group G over R is the Dieudonne module of G.

The classification of formal groups via the Dieudonne module is given by thefollowing.

Theorem 6.6.10. Let k be a perfect field of characteristic p > 0. The functor Dinduces an equivalence of categories

FGpskD= (Dieudonne modules over k)

The next two sections are devoted to the construction of an inverse to the functorD. Before continuing, we ought to describe how the discussion fits into a much moregeneral situation; for more information the reader may consult [Haz78]. For anyZ(p)-algebra R, the p-typical curves functor D is a faithful functor

FGpsR AbGps .One is led to study its endomorphisms, namely, homotheties, Frobenius, and Ver-schiebung, related by the formulae in Proposition 6.2.10 and Corollary 6.4.4. Let

52 ANDO

CartpR to be the ring generated by symbols V , F , and [a] for a R, subject tothese relations. As above, one finds that there is an embedding WpR CartpR.

The definition of a Dieudonne module is essentially the same as the case of aperfect field, but there is a slight wrinkle. Recall (Corollary 6.6.5) that if G is aformal group over R, then DG/V DG is the tangent space LieG: in particular it isa free R-module. This structure is incorporated into the definition of a Dieudonnemodule.

Lemma 6.6.11. If k is a perfect field, and M is a Dieudonne module over k inthe sense of Definition 6.6.9, then the action of Wpk on M induces on M/VM thestructure of a k-vector space. If M and a k, we use the notation [a] for theclass of r(a) in M/VM .

Proof. If k is a perfect field, thenWpk/p = k. But if a = b+pc inWpk, and M ,then

a = (b+ pc) = b + V Fc.

In general, if R is a Z(p)-algebra, and M is a CartpR-module, then there is stillan action of R on M/VM :

Lemma 6.6.12. If M is a CartpR-module, then M/VM has the structure of anR-module.

Proof. The relation (6.4.5) implies that

[a+ b] = [a] + [b]

on M/VM .

If M/VM is to be the tangent space of a formal group over R, it had better bea free R-module.

Definition 6.6.13. Let R be a Z(p)algebra. A Dieudonne module over R is aCartp(R)-module which is reduced in the sense that the natural map

M limM/VkM

is an isomorphism; and uniform in the sense that

(i) M/VM is a free R-module(ii) the map

M/VMV k V kM/V k+1M

is an isomorphism for all k > 0.

We have already shown that D is a functor

FGpsR (Dieudonne modules over R).In fact D is an equivalence of categories for R a Z(p)-algebra. Indeed the discussionof the inverse functor given here makes sense essentially without change for anFp-algebra.

FORMAL GROUPS 53

However, when k is a perfect field the classification by the Dieudonne module isparticularly effective. For then, as we shall see in section 6.10, Dieudonne moduleshave a relatively tractable structure.

6.7. The p-typical parameter lemma and the evaluation map. The followingresults (Lemma 6.7.1 and Proposition 6.7.2) are used in section 6.8.

Lemma 6.7.1 (p-typical parameter lemma). Let R be a Z(p) algebra. A formalgroup G over R admits a system of parameters

An = A1 . . . A1 1...n Gsuch that

(i) each i is p-typical;

(ii) idG =G

i pii

Proof. Recall that in section 5.4 we constructed a section s making the diagram

Wps W

w

y ywAN

swAN

commute, with sw given by the formula

sw(c0, c1, c2, . . .) = (c0, 0, . . . , 0, c1p, 0, . . . , 0, c2

p2, . . .).

Let

An 1...n Gbe any system of parameters for G. Each i determines a homomorphism

WP

Fripir G.Let i be the p-typical parameter given by

A1 i G0

y xPFripirWp

sW,

and in terms of these parameters define the map to be

An G(x1, . . . , xn) 7

Gi(xi).

We must show that is invertible; it suffices to show that

(x1, . . . , xn) = (x1, . . . , xn) + o(2).

By construction, then, it suffices to show that

i(t) = i(t) + o(2).

54 ANDO

The universal case is

(t) = 1(t) = (t, 0, 0, . . .) CWfor which the corresponding map

W Wis the identity. To compute in that case, attach the definition of the section s tothe construction just described:

A1 Wp W

A A

t (t, 0, . . .) exp( tpnxpn

pn)

(t, tp, tp2, . . .) (t, 0, . . . , t

p

p, . . .).

w0

uwp

ws

uw

wsw

v w v wv

uvu

v w

In the upper right corner the Witt vector has been recorded as a power series usingProposition 5.4.3. It is then easy to calculate that

(t) = exp[ tpnxpn

pn

]= 1 tx+ o(t2)= (t) + o(t2).

Finally, these parameters also satisfy our second requirement: for any parametersystem,

g = (pi1(g), . . . , pin(g)),

and by construction we have

(pi1(g), . . . , pin(g)) =G

i(pii(g)).

This lemma is very useful; as an illustration we give a first hints at how thefunctor D determines the group. Recall that Dieudonnes theorem provides anisomorphism

D(G) = Hom[Wp, G].It follows that there is an evaluation map

Wp D(G) G.Theorem 6.7.2. If G is a finite dimensional formal group over a Z(p)-algebra R,then the evaluation map is onto.

FORMAL GROUPS 55

Proof. Let

An Gbe parameters provided by the p-typical parameter lemma, and suppose that

g = (a1, . . . , an) G(A).Under the evaluation map, the element

ni=1

(ai, 0, . . . , 0) i Wp(A) CG

goes to Gi(ai) = g.

6.8. The functor from Dieudonne modules to formal groups. We describenext a functor from Dieudonne modules to formal groups which is due to Cartier(see [Car69]). Chapter 9 is devoted to an extension of this construction.

Fix a perfect field k. One way to motivate the construction is to observe (6.7.2)that the evaluation map

Wp DG Gis surjective. We also know that there are relations, namely that Frobenius on curvescorresponds to Verschiebung on the Witt formal group, and vice versa. Supposethat M is a uniform, reduced Dieudonne module, and A is an adic k-algebra. LetG(M)(A) be the abelian group

G(M)(A) def=Wp(A)

WpkM

V am = a FmFam = a V m

.

The first point is that this is, as claimed, an abelian group: we have

F : M MV : M MF : Wp WpV : Wp Wp.

If f : A B is a ring homomorphism, M is an A-module, and N is a B-module,then

fN AM N

BfM

nm 7 n 1mis an isomorphism of abelian groups. So F 1 and 1 V may be viewed as mapsof abelian groups

Wp(A)M = Wp(A) M Wp(A)M ;while V 1 and 1 F may be viewed as maps of abelian groups

Wp(A)M = Wp(A) M Wp(A)M.

56 ANDO

Theorem 6.8.1. G(M) is a formal group, and the functor(Dieudonne modules) G FGpsk

is an inverse of D.

For simplicity we restrict attention to finite-dimensional formal groups, in otherwords Dieudonne modules such that M/VM is finitely generated.

Lemma 6.8.2. LetM be a Dieudonne module over k, and let 1, . . . , n be elementsof M which project to a k-basis of M/VM . The map of functors

An s G(M)given by

s(a1, . . . , an) =ni=1

(ai, 0, 0, . . .) i

is an isomorphism.

Proof. Let A be an adic kalgebra with augmentation ideal I(A). Recall that onWitt vectors, Verschiebung is given by

V (a0, a1, . . .) = (0, a0, a1, . . .),

so we can picture Wp(A) as

Wp(A) =n0

V nI(A).

In this pictureWp(A)M

V am = a Fm = I(A)M.Now A is an adic k-algebra; in particular it has characteristic p. On Witt vectorsin characteristic p, Frobenius is given by

F (a0, a1, . . .) = (ap0, a

p1, . . .).

By definition, I(A) is nilpotent. It follows that the map

I(A)n s G(M)(A) = I(A)Ma V m = Fam

(a1, . . . , an) 7

ai iis an isomorphism.

Thus G(M) is a formal group over k. Now suppose that G is a formal group overk. The evaluation map

Wp DG e Gfactors through G(DG).Lemma 6.8.3. The natural map

G(DG) Ginduced by evaluation is an isomorphism.

FORMAL GROUPS 57

Proof. Let = (1, . . . , n) be a choice of p-typical parameters for G as providedby Lemma 6.7.1, determining an isomorphism

An = G.

By the Taylor series lemma, in particular Corollary 6.6.5, the elements 1, . . . , nof DG satisfy the hypotheses of Lemma 6.8.2. By construction of the map s, thediagram

An

G(DG) G[[^ s '

) '

w

commutes.

It remains to show that there is a natural isomorphism

M D(G(M)).By construction, there is a map

Wp M G(M)whose adjoint is a map of Dieudonne modules

M FGps[Wp,G(M)] = D(G(M))which induces an isomorphism

M/VM D(G(M))/V (D(G(M))).The proof of the theorem is completed by the easy lemma.

Lemma 6.8.4. A map

Mf N

of uniform reduced Dieudonne modules is an isomorphism if and only if it inducesan isomorphism

M/VMf N/V N.

Proof. If

M/VMf N/V N

is an isomorphism, then by uniformity, the middle arrow of the diagram

V kM/V k+1M M/V k+1M M/V kM=y y y=

V kN/V k+1N N/V k+1N N/V kN

of short exact sequences is an isomorphism for all k. Then Mf N is an isomor-

phism by reducedness.

6.9. Example: the multiplicative group and the additive group.

58 ANDO

The multiplicative group. For an adic algebra A, the multiplicative group is

Gm(A) = (1 + I(A)).

With respect to the parameter

A1 Gma 7 1 a

we get

(s) +Gm

(t) = (1 s)(1 t)= (s+ t st).

Then, letting denote a primitive nth root of unity and using (6.2.3),

Fn(t) =n1i=0

(1 it 1n ) = (t).

So is represented by the homomorphism g() =Fnpin which is computed in

the diagram

W Gm

PS

a

n1

(1 an) = p(1)

p(x) =n1

(1 anxn)

wg()

u

'

A AA A

A AA A

AC

v wv

u C AA A

AAC

FORMAL GROUPS 59

Now isnt a p-typical curve, so we follow the standard procedure (see Propo-sition 5.4.12) to produce one:

A1 Gm0

y xg()Wp

s Ww

y ywA s

w

A

t exp[ tpnpn ]y xp(1)

(t, 0, . . .) exp[ tpnxpnpn ]y x

(t, tp, . . .) (t, . . . , tpp, . . .)

So the Artin-Hasse exponential, which we already determined (5.4.14) has integralcoefficients over a Z(p)-algebra, is a p-typical parameter for the multiplicative formalgroup! Let denote this parameter.

Proposition 6.9.1. F =

Proof. Because s and g() are homomorphisms, the diagram

Wps W

F0

x yg()A1

FGm

commutes. SinceF0(t) = (