Deformation Theory

of Formal Modules

by Yu Yasufuku

Harvard UniversityCambridge, Massachusetts

April 3, 2000

To my parents

Acknowledgement

I would like to express my great appreciation to my thesis adviser Professor Richard Taylor for all themathematical help, for his generosity, and for his kind support. He always welcomed me into his office witha warm smile. I truly appreciate his great advisorship.

I would also like to express my thanks to Professor Brian Conrad, who kindly helped me with profinitemodules and commutative formal group schemes. I would not have been able to complete Section 4.3 withouthis help.

I thank Max Lieblich, a great roommate and a friend, for many interesting and exciting conversation,including helpful math conversations; Dan Lee for being a great friend of mine and for checking grammar inthis thesis (though of course I still take full responsibility for all the mistakes); Nils Barth for all the LATEXhelp; and suitemates Henry Fu, Kaiwen Kam, Liam McAllister, and Albert Pan for all the laughs I shared.

Finally, I would like to express my deep and heartfelt thanks to my parents, who have greatly supportedme throughout my life. I would like to especially thank my mother, who has given me tremendous mentalsupport and constant encouragement.

Thank you all!

Contents

1 Introduction 31.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Introduction to Formal Groups and Formal Modules 52.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Properties of Formal Groups: Lazard’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Generalized Formal R-Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Formal R-Modules for Various R 173.1 Universal Formal R-Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 R is a Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 R is the Valuation Ring of Local Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3.1 A Special Case: A Is A Field Over R/(π) . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 Deformation Theory 294.1 Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.1.1 Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.1.2 Universal Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2 Deformations With Level Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.3 The Deformations of Divisible Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5 Applications of Deformation Theory 515.1 Applications to Local Class Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2 Applications to the Arithmetic of Function Fields . . . . . . . . . . . . . . . . . . . . . . . . . 53

A Profinite Modules and Commutative Formal Group Schemes 55A.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55A.2 Formal Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57A.3 Duality and Commutative Group Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57A.4 Formal Etaleness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58A.5 Formal Smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

1

2

Chapter 1

Introduction

In [5], Drinfel’d proved the analogues of the Fundamental Theorem of Complex Multiplication and theLanglands Conjecture for global function fields (for d = 1, 2). As discussed in [10], this paper has inspiredmuch work in the arithmetic of global function fields, including some explicit class field theory. It has alsobeen directly influential in number theory, for example, in the theory of moduli spaces of elliptic curves.

To prove these analogues, Drinfel’d introduced the notion of elliptic modules (now called Drinfel’d modulesin the literature), an analogue of elliptic curves. Further, he studied their moduli problems, determiningthe ‘parameter space’ of all Drinfel’d modules with some additional structures, namely the level structures.Before studying ‘all’ Drinfel’d modules simultaneously as required for the determination of moduli spaces,he first analyzed them ‘locally,’ that is, analyzing Drinfel’d modules which are ‘close’ to a given one. This‘local’ analysis of Drinfel’d modules led to deformations of formal modules. So the deformation theory offormal modules lies at a foundation of [5].

Despite its importance and beauty, however, the state of the literature on this deformation theory offormal modules leaves something to be desired. The original source [5, §1 and 4] certainly has all the resultsand main ideas of proofs, but it is extremely dense (Section 4.3 is condensed into 20 lines) and lacks someimportant details that one might want to see to better understand the theory. It also assumes knowledge ofearlier results, such as the basic theory of formal groups as covered in [8] and their deformations [19]. Theonly other literature on deformation of formal modules that the author is aware of are [2] and [30]. Thesegive many of the important details that are missing in [5], but as lecture notes, they are not full accounts ofthe theory, either.

To remedy the situation, this thesis is intended to give a thorough treatment of this deformation theoryof formal modules for the reader at or around the level of beginning graduate students. As such, it coversnecessary facts from [8] and [19], and develops the theory with details. In order to convey the big picture aswell, however, we will give overviews of chapters, sections, and proofs.

We will now give an overview of this thesis. Chapter 2 discusses definitions and basic properties offormal groups and formal modules, together with examples. Chapter 3 analyzes the possibilities of formalR-modules when R is a field and when R is the valuation ring of a local field. These results about all formalmodules are obtained by determining and carefully analyzing one formal module, namely the universal one.This category-theoretic technique will be used throughout this thesis. In Chapter 4, deformation theory offormal modules (without and with level structures) is presented, using results from Chapters 2 and 3. Thelast chapter is devoted to applications of the deformation theory developed in Chapter 4. In addition to abrief description of how it is used in the study of global function fields in [5], we will include an applicationto local class field theory.

Naturally the main reference of this thesis is [5]. The notes [2] and [30] were extremely helpful, especially[2] for Section 4.3 and [30] for Section 3.3.1. Other references will be given for each section.

We will assume that the reader is familiar with basic category theory [15], with commutative algebra[1], and with some valuation theory of local fields [22] [23]. We will also assume some familiarity with thelanguage of modern algebraic geometry [11]. Everything else will be either developed or cited explicitly.In particular, there is an appendix on profinite modules and commutative formal group schemes which

3

summarizes results that we need from [3].

1.1 Notation

We now set forth standard notations which are used throughout this thesis. As usual, Z and Q indicate theintegers and the rational numbers. p and ` will always denote a prime number (while l is sometimes usedfor index, but only in places where ` is not around), q is always a power of p. We will denote the finite fieldwith pn elements Fpn . All rings are commutative with 1 unless otherwise specified. If B is an A-algebraby f : A → B, we will often write a ∈ B instead of f(a), even if f is not injective. This does clarify theexposition significantly, and whenever there is a possibility of confusion, we will write · after a, just likemodules. We will use a bar over a field to denote its algebraic closure, and we will use a bar over a ring toindicate the ring modulo an ideal. If F is a power series (in several variables), we often refer to the sum ofall monomials in F with total degree strictly less than n as ‘F modulo polynomial degree n’ and write thisas (mod poly. deg. n). The standard notation for this seems to be (mod deg n), but since we will haveanother notion of degree (on the universal formal R-module ring) we need to distinguish the two.

Unfortunately, we must use two different definitions of ‘formal modules’ in this thesis, the one in Section2.3 more general than the one in Section 2.1. We will use definition of Section 2.3 only in Sections 2.3 and 4.3,and in these sections, we will call those satisfying the definition of Section 2.1 the CFS formal modules (nameto be explained later). We could have referred to the formal modules of Section 2.1 as CFS formal moduleseverywhere, but since this thesis is mostly about CFS formal modules, it only clutters up the writing. Wewill remind the reader about this change of definition in the corresponding sections.

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Chapter 2

Introduction to Formal Groups andFormal Modules

This chapter discusses definitions and elementary properties of formal groups and formal modules. Section2.1 is mostly definitions and examples. Among the examples, the image (Example 2.1.10) and push-forwardwill be used throughout this thesis.

The second section describes an elementary formal group theory [8, III §1]. In particular, we show Lazard’sTheorem which gives the possibilities of F (X,Y )−G(X,Y ) when F and G are formal groups with F ≡ G(mod poly. deg. n). We will also prove an easy way of characterizing formal groups which are isomorphicto the additive group. Although this thesis is about formal modules, knowing the underlying formal groupstructure sometimes severely limits the possibilities of the module structures, so this section will be useful.

Section 2.3 gives a more general notion of formal R-modules that we will need for Section 4.3. This sectionassumes knowledge of Appendix A. After giving this general definition, we will show that this definition isindeed more general than the one given in Section 2.1, and we will show an example (the etale formal R-module) which does not satisfy the definition in Section 2.1. The rest of the section is devoted to short exactsequences of formal R-modules, in particular to the connected-etale sequences and to the group structure onthe extension classes of formal R-modules. Main reference for this section is [3], but [27], [29], and [31] wereall used.

2.1 Basic Properties

In this section, we will introduce the notions of formal groups and modules, with some examples. First, wedefine formal groups:

Definition 2.1.1. A one-dimensional commutative formal group over a ring A is a power series F (X,Y ) ∈A[[X,Y ]] satisfying the following four conditions:

(1) F (X,F (Y, Z)) = F (F (X,Y ), Z)(2) F (X,Y ) = F (Y,X)(3) F (X, 0) = X and F (0, Y ) = Y(4) There exists a unique G(X) ∈ (X) ⊆ A[[X]] such that F (X,G(X)) = F (G(X), X) = 0

It is helpful to think of a formal group as a formal group law, by viewing the variables X and Y as thehypothetical ‘group elements’ and the power series F as the law of ‘adding’ X and Y (in fact, [13] and[25] use the terminology formal group law). Although a formal group does not give us any specific groupelements, this view enables us to see condition (1) as associativity, (2) as commutativity, (3) as existence of0, and (4) as existence and uniqueness of inverses. Taking this one step further, if for example A is a-adicallycomplete (thus also separated in the a-adic topology), then for elements x, y ∈ a, F (x, y) is a well-definedelement of a, and F (x, y) = x+

Fy defines a (usual) abelian group structure on a. We will use F (a) to denote

this abelian group structure on a.

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Example 2.1.2. As an example of how a formal group naturally arises, we will briefly discuss elliptic curves[28]. They can be thought of as (smooth) curves E of genus 1 over a ring A with a specified base point, oras the locus in P2 of the equation (x = X/Z, y = Y/Z are the non-homogeneous coordinates)

y2 + a1xy + a3y = x3 + a2x2 + a4x+ a6 (2.1)

together with the ‘point of infinity’ O = [0 : 1 : 0] ∈ P2. We can define an addition on the points on ellipticcurves, with O serving as the identity. The completion of the localization of the coordinate ring at the originis a complete discrete valuation ring, with z = −x

y as a uniformizer. Letting w = − 1y , (z, w) gives a local

coordinate around O. Using (2.1) recursively, we can obtain an expression for w as an element g(z) ∈ A[[z]],so we can define F (z1, z2) ∈ A[[z]] to be the z-coordinate of the sum of two points (z1, g(z1)) and (z2, g(z2)).This F is a formal group. Note that F arises by analyzing the addition law in an ‘infinitesimal’ neighborhoodaround O.

We can generalize Definition 2.1.1 to higher dimensions: writing X = X1, . . . , Xn and Y = Y1, . . . , Yn,an n-dimensional formal group F over A is a set of n power series Fi(X,Y ) ∈ A[[X,Y ]] such thatF = (F1, . . . , Fn) satisfies the conditions of Definition 2.1.1. In this thesis, however, we will only need one-dimensional commutative formal groups, so from now on, a formal group will always mean a one-dimensionalcommutative formal group.

Though we need all four conditions in Definition 2.1.1 to get a parallel of abelian groups, it is actuallyonly necessary to check the first three conditions:

Proposition 2.1.3. If F ∈ A[[X,Y ]] satisfies (1) through (3) of Definition 2.1.1, then it satisfies (4).

Proof. To avoid infinite sums of elements of A, we must pick G(X) from the ideal (X) ⊆ A[[X]]. Wewill inductively construct Gn(X) =

∑ni=1 λiX

i such that F (X,Gn(X)) ≡ 0 (mod poly. deg. n + 1). Thiskind of ‘successive approximation’ will be used very frequently. λ1 = −1 starts us off, so suppose we haveconstructed up to Gn−1(X). We can write F (X,G(X)) = X + G(X) +

∑aijX

i(G(X))j , where the sumruns over i+ j ≥ 2. Now, modulo Xn+1, there is no contribution of terms involving λn from inside the sum,so we can solve for λn to continue induction. Commutativity implies F (G(X), X) = 0 for the same G, andif G1 and G2 both satisfy (4), then we have (just like the usual group theory case)

G1(X) = G1(X) +F

0 = G1(X) +F

(X +

FG2(X)

)=(G1(X) +

FX

)+FG2(X) = G2(X).

There are two simple examples of formal groups. First, Ga(X,Y ) = X + Y defines the additive formalgroup, corresponding to the addition in A. Secondly, Gm(X,Y ) = X + Y +XY defines the multiplicativeformal group, corresponding to the multiplication on the set 1 + a for a proper ideal a of A.

Next, we introduce the notion of homomorphisms.

Definition 2.1.4. Given formal groups F,G over A, α(X) ∈ (X) ⊆ A[[X]] is a formal group homomorphismfrom F to G if G(α(X), α(Y )) = α(F (X,Y )). An endomorphism of the formal group F is a formal grouphomomorphism from F to F , and the set of all endomorphisms is denoted End(F ). By the derivative ofα, we mean the coefficient of X in α. As usual, we call α an isomorphism if there exists a formal grouphomomorphism β : G → F such that α(β(X)) = X and β(α(X)) = X. Here, α(β(X)) = α β denotes thepower series

a1

(∑biX

i)

+ a2

(∑biX

i)2

+ · · · ,

where α(X) =∑ajX

j and β(X) =∑biX

i. In this case, we denote β = α−1.

Note that α does not have a constant term, so α−1 cannot possibly refer to the inverse of α as anelement of A[[X]], so there should be no confusion. By viewing formal groups as giving laws of addition,the above condition says α(X +

FY ) = α(X) +

Gα(Y ), which makes sense. As in the case of abelian group

homomorphisms, we have the following fact:

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Proposition 2.1.5. End(F ) is a (not necessarily commutative) ring for any formal group F .

Proof. Given α1, α2 ∈ End(F ), we define the ring structure by

(α1 +Fα2)(X) = F (α1(X), α2(X)) and (α1 · α2)(X) = α1(α2(X)).

Using (1) through (4) of the definition of formal groups, it is straight-forward to check that these operationsproduce elements of End(F ) and that this forms a ring.

In particular, we have a ring homomorphism Z −→ End(F ), sending m 7→ [m]F (X) = X +F· · ·+

FX︸ ︷︷ ︸

m

. We

also have an easy way of detecting isomorphisms:

Proposition 2.1.6. Suppose that α is a formal group homomorphism from F to G over A. Then α is anisomorphism if and only if the derivative is in A∗.

Proof. ⇐=: Using the successive approximation method as in the proof of Proposition 2.1.3, because of thederivative hypothesis on α, we can construct β(X) such that α(β(X)) = X. By the same argument, sincethe derivative of β is a unit, there exists α2(X) such that β(α2(X)) = X. But then α(X) = α(β(α2(X))) =α2(X), so α and β are inverses. Also, β is a formal group homomorphism, since

β(G(X,Y )) = β(G(α(β(X)), α(β(Y ))

))= F (β(X), β(Y )).

The converse is clear by looking at the derivative of α(β(X)) = X.

Given the above proposition, we will define an element α ∈ A[[X]] to be -invertible if its derivative isin A∗, and denote its -inverse by α−1 (this is not a standard terminology).

Example 2.1.7. Suppose pA = 0. If∑aiX

i is an endomorphism of Ga, then aiXi + aiY

i = ai(X + Y )i.If i = pνs with (p, s) = 1 and s > 1, then

((i

pν

), p)

= 1, so ai = 0. On the other hand∑aiX

pi

is an

endomorphism of Ga. Thus, an element of End(Ga) is of the form∑aiτ

i, where each ai corresponds to theendomorphism aiX and τ corresponds to Xp, with the ring structure given by the relation apτ = τa. Wewill denote this ring by Aτ. Automorphisms are elements of Aτ whose constant term is in A∗.

We will now define formal R-modules over A, where R is a ring and A is an R-algebra.

Definition 2.1.8. A formal R-module over A is a pair (F, [ ]F ) such that F ∈ A[[X,Y ]] is a formalgroup, and the map [ ]F : R → End(F ) sending r 7→ [r]F is a ring homomorphism satisfying the condition[r]F ≡ r ·X (mod poly. deg. 2).

Again this parallels the definition of usual modules. Note that the additive formal group over A can bemade into the additive formal R-module Ga, just by defining [r]F = r · X. Similarly as before, if A isa-adically complete, then we can view the set a as an R-module via F ; we denote this module by F (a).

To facilitate successive approximation of formal R-modules, it is useful to define the notion of a germ: aformal R-module germ of degree n over A is (F, [ ]F ) with F ∈ A[[X,Y ]]/(X,Y )n and [r]F ∈ A[[X]]/(X)n

such that all the relations of a formal R-module are satisfied modulo polynomial degree n.

Definition 2.1.9. α : F → G is a formal R-module homomorphism if it is a formal group homomorphismand α([r]F (X)) = [r]G(α(X)) for all r ∈ R.

Given two formal R-module homomorphisms αi, both the ‘addition’ α1 +Gα2 = G(α1(X), α2(X)) and the

composition α1 α2 result in a formal R-module homomorphism. With these operations, the set EndR(F ) ofall formal R-module endomorphisms of F becomes a (not necessarily commutative) ring, just as in Propo-sition 2.1.5. We will continue to use End(F ) to denote the set of formal group homomorphisms of theunderlying formal group. By employing the same argument as in the proof of Proposition 2.1.6, we candetect formal R-module isomorphisms using the derivative. We will now discuss a simple but importantexample of formal R-modules and their homomorphisms.

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Example 2.1.10. Let (F, [ ]) be a formal R-module over A and let α ∈ (X) ⊆ A[[X]] be -invertible. LetG(X,Y ) = α(F (α−1(X), α−1(Y ))) and [r]G(X) = α([r]F (α−1(X))). This makes G into a formal R-moduleover A, with α as a formal R-module isomorphism from F to G. We will call G the image of F via theisomorphism α (this is not a standard terminology). In particular, if α(X) ≡ X+aXn (mod poly. deg. n+1)for some a ∈ A, then α−1(X) ≡ X − aXn (mod poly. deg. n+ 1), and so we obtain

G(X,Y ) ≡ F (X,Y ) + a((X + Y )n −Xn − Y n) (mod poly. deg. n+ 1)[r]G(X) ≡ [r]F (X) + (rn − r) · aXn (mod poly. deg. n+ 1).

These formulae will be useful for successive approximation later on.

Another important operation is the following. Given a formal R-module F over A and an R-algebrahomomorphism f : A → B, the push-forward f∗(F ), defined by changing all the coefficients of F (X,Y )and [r]F (X) via f , is a formal R-module over B. A special case is reduction of a formal R-module over Amodulo an ideal a ⊆ A. With push-forwards, the association sending an R-algebra A to the set of formalR-modules over A becomes a functor. This functor will be studied for various R’s and A’s in Chapter 3.

More sophisticated examples of the formal modules will be given throughout this thesis. From now on,just as we do with modules, we will slightly abuse the notation by using F to refer to both the entire formalR-module structure and F (X,Y ) in particular, depending on context.

2.2 Properties of Formal Groups: Lazard’s Theorem

Although our primary focus in this thesis is formal modules, we first need to understand the possibilities andlimitations of formal groups. In particular, we will present Lazard’s Theorem which is useful for successiveapproximation. As a consequence of this, we will be able to characterize formal groups which are isomorphicto Ga.

Before stating Lazard’s theorem, we need to introduce a notation. Let ε(n) be 1 if n is not a power of aprime, and let it be ` when n is a power of a prime `. Then define

Cn(X,Y ) =1ε(n)

((X + Y )n −Xn − Y n) .

Cn(X,Y ) has coefficients in Z, and moreover it is primitive. Indeed, suppose on the contrary that a prime pdivided all the coefficients. If n is a power of p, then p2 divides all the coefficients of (X + Y )n −Xn − Y n,so by induction, mn ≡ m (mod p2) for all positive integers m, a contradiction. If n = pνr with (p, r) = 1and r > 1, then p -

(npν

), contradiction.

Theorem 2.2.1 (Lazard). If F and G are formal groups over A such that F ≡ G (mod poly. deg. n), thenF ≡ G+ aCn(X,Y ) (mod poly. deg. n+ 1) for some a ∈ A.

This is obvious for n ≤ 1, so let us assume that n ≥ 2. Let h(X,Y ) be the homogeneous polynomialof degree n such that F − G ≡ h (mod poly. deg. n + 1). Because F and G are formal groups, h(X,Y ) =h(Y,X), and h(X, 0) = h(0, X) = 0. Letting G2(X,Y ) = G(X,Y ) − X − Y (i.e. the higher order terms),modulo polynomial degree n+ 1, we have

F (F (X,Y ), Z)) ≡ G(F (X,Y ), Z) + h(F (X,Y ), Z) ≡ F (X,Y ) + Z +G2(F (X,Y ), Z) + h(X + Y,Z)≡ G(X,Y ) + h(X,Y ) + Z +G2(G(X,Y ), Z) + h(X + Y,Z)≡ G(G(X,Y ), Z) + h(X,Y ) + h(X + Y, Z).

Similarly, F (X,F (Y, Z)) = G(X,G(Y,Z)) + h(Y,Z) + h(X,Y + Z), so h(X,Y ) + h(X + Y, Z) = h(Y, Z) +h(X,Y + Z). Therefore, for the theorem, it suffices to prove

Proposition 2.2.2. A homogeneous polynomial h(X,Y ) of degree n with coefficients in A such that h(X,Y ) =h(Y,X), h(X, 0) = h(0, X) = 0, and h(X,Y ) + h(X + Y,Z) = h(Y,Z) + h(X,Y + Z), must be of the formaCn(X,Y ) for some a ∈ A.

8

Proof. We will first prove this when A is a field. Let h(X,Y ) =∑aiX

iY n−i. Then the conditions put on hin this proposition are all linear in terms of ai. Moreover, by primitivity of Cn, the coefficients for Cn(X,Y )form a nonzero solution of these linear conditions, so it suffices to show that the space of (ai) satisfyingthese linear conditions is a vector space of dimension at most 1 over A. By the first two equations, we knowa0 = an = 0 and ai = an−i. Comparing the coefficients of XiY jZn−i−j in the third equation with i > 0 andn− i− j > 0, we obtain ai+j

(i+jj

)= ai

(n−i

j

). We note two special cases: when i = 1 and j = k− 1 for some

1 ≤ k < n, we get

akk = a1

(n− 1k − 1

), (2.2)

and when j = 1 and i = k for some 0 ≤ k < n− 1, we get

ak+1(k + 1) = ak(n− k). (2.3)

If the characteristic of A is zero, then (2.2) finishes the proof. So let us now assume that the characteristicof R is p. If n = p or (n, p) = 1, then for all 1 ≤ k ≤ n− 1, we have kak =

(n−1k−1

)a1 and (n−k)ak =

(n−1

n−k−1

),

so we are done in this case as well. So we are now left with the case n = mp with m > 1; we prove this byinduction. Note that in this case, if p|k, then n − k = 0 and k + 1 is invertible, so (2.3) shows ak+1 = 0.Similarly, if ak = 0 and p - (k + 1), then ak+1 = 0. Combined with an−k = ak, we conclude that ak = 0for all (k, p) = 1, so h(X,Y ) = h′(Xp, Y p). Then h′(X,Y ) must satisfy the conditions of the proposition.Hence, h′(X,Y ) = aCm(X,Y ) by induction, so it now suffices to check the following to finish the case of Rbeing a field.

Claim. When A is a field of characteristic p, Cm(Xp, Y p) = bCn(X,Y ) for some b ∈ A.

Proof. If m is not a power of p, then Cm = u · ((X + Y )m −Xm − Y m) with u ∈ A a unit. ThenuCn(X,Y ) = u((X + Y )n −Xn − Y n) = u

(((X + Y )p

)m −Xn − Y n)

= Cm(Xp, Y p). If m is apower of p, working over Z, we have

pCm(Xp, Y p) = (Xp + Y p)m −Xn − Y n = ((X + Y )p − pCp(X,Y ))m −Xn − Y n

= pCn(X,Y ) +m∑

i=1

(m

i

)(−1)ipi(Cp(X,Y ))i(X + Y )p(m−i),

where each term inside the summation has coefficient divisible by p2. Thus, dividing the aboveformula, Cm(Xp, Y p) ≡ Cn(X,Y ) (mod p).

Amazingly, the above case for a field is really all the content of this theorem. The important observationis that the three conditions that are put on h(X,Y ) and the final conclusion that we draw about h(X,Y )only use the additive structure of A, and not the ring structure. So we can completely forget about the ringstructure. Since h(X,Y ) has a finite number of coefficients, it suffices to show the proposition for a finitelygenerated abelian group. If h has coefficients in A1 ⊕ A2 and satisfies these three conditions, these threeconditions are satisfied for each component Ai. So by the structure theorem for finitely-generated abeliangroups, it suffices to check the proposition for A = Z and A = Z/pν .

Suppose A = Z. By above, h(X,Y ) = aCn(X,Y ) for a ∈ Q, but by primitivity of Cn, a ∈ Z.Now suppose that A = Z/pν . The ν = 1 case is a field, so we will prove this by induction. Taking lifts

to Z, what we need is h(X,Y ) ≡ aCn(X,Y ) (mod pν+1) for some a ∈ Z, given that h satisfies the threeconditions modulo pν+1. By the inductive hypothesis, we can write h(X,Y ) ≡ aCn(X,Y ) + pνh1(X,Y )(mod pν+1) for some h1 with coefficients in Z. Then h1(X,Y ) must satisfy the three conditions modulo p,so h1(X,Y ) ≡ a1Cn(X,Y ) (mod p) for some a1 ∈ Z. Hence, h(X,Y ) ≡ (a + pνa1)Cn(X,Y ) (mod pν+1).

9

Before proving a theorem characterizing formal groups isomorphic to the additive group, we need apreliminary lemma.

Lemma 2.2.3. If F,G are formal groups and F (X,Y ) ≡ G(X,Y ) + aCn(X,Y ) (mod poly. deg. n + 1),then for any natural number m, we have

[m]F (X) ≡ [m]G(X) +mn −mε(n)

aXn (mod poly. deg. n+ 1).

Proof. By induction, we can show that [m]F (X) and [m]G(X) both start out with mX. When m = 1,[m]F = [m]G = X, so assume that we have proven the lemma for m. Working modulo polynomial degreen+ 1,

[m+ 1]F (X) = F ([m]F (X), X) ≡ G(

[m]G(X) +mn −mε(n)

aXn, X

)+ aCn([m]F (X), X)

= G([m]G(X), X) +mn −mε(n)

aXn + aCn(mX,X)

= [m+ 1]G(X) +mn −mε(n)

aXn +1ε(n)

a((m+ 1)n −mn − 1)Xn

= [m+ 1]G(X) +(m+ 1)n − (m+ 1)

ε(n)aXn.

Theorem 2.2.4. A formal group F over A is isomorphic to the additive group if and only if for all primesp, [p]F (X) has coefficients in pA.

Proof. If α : F → G is an isomorphism of formal groups, then [m]F = α−1 [m]G α by induction, so inparticular, [p]bGa

= pX shows the necessity.For sufficiency, we use the method of successive approximations. Specifically, we will successively construct

-invertible power series βn such that

βn−1 ≡ βn (mod poly. deg. n), and βn F β−1n ≡ X + Y (mod poly. deg. n+ 1).

Start with β1 = X. Suppose that we have constructed up to βn−1. By Example 2.1.10, we know thatH = βn−1 F βn−1

−1 is a formal group with βn−1 a formal group isomorphism from F to H, satisfyingH ≡ X + Y (mod poly. deg. n). By composing with βn−1, it suffices to find a power series α ∈ A[[X]] suchthat α ≡ X (mod poly. deg. n) and α H α−1 ≡ X + Y (mod poly. deg. n + 1). By Lazard’s Theorem,H ≡ X + Y + aCn(X,Y ) (mod poly. deg. n+ 1) for some a ∈ A. Note that when n is a power of a primep, Lemma 2.2.3 shows [p]H(X) ≡ pX + (pn−1 − 1)aXn (mod poly. deg. n + 1). The first paragraph shows[p]H(X) ∈ (pA)[[X]], so in particular, a = pa′ for some a′ ∈ A.

Hence (including the case when n is not a power of a prime), H ≡ X + Y + a′ ((X + Y )n −Xn − Y n)(mod poly. deg. n+ 1). Then α = X − a′Xn works by Example 2.1.10.

Corollary 2.2.5. Suppose F is a formal group over A.(1) If A is a Q-algebra (e.g. a field of characteristic zero), F is isomorphic to Ga.(2) If pA = 0 for some prime p, F is isomorphic to Ga if and only if [p]F is identically equal to zero.(3) If A is a local ring with residue field characteristic p, F is isomorphic to Ga if and only if [p]F (X) hascoefficients in pA.

Proof. (1) follows from the fact that every p is a unit of A. The necessities of (2) and (3) are clear, and forsufficiency, note that a prime ` 6= p is a unit in A.

These results are useful even in the analysis of formal R-modules, since knowing that a formal R-moduleis isomorphic as a formal group to the additive formal group severely limits the potential actions of R (cfthe proof of Theorem 3.2.1).

10

2.3 Generalized Formal R-Modules

It turns out that we can generalize the notion of formal R-modules to something that does not arise frompower series. For Section 4.3 only, we will need this more general version, so we will cover the basicdefinitions and results of formal R-modules in this general sense to clarify the proofs later. This sectionassumes knowledge of profinite modules and commutative formal group schemes, as discussed in AppendixA, and we will also be using the same notations.

Throughout this section, we let A be a pseudocompact local ring with maximal ideal m. For example, Acan be a complete noetherian local ring. We consider the category of formal schemes over A, i.e. those ofthe form SpfA(B) for some B ∈ PAA. It is tempting to define a formal R-module over A to be an R-moduleobject in this category. But recalling that commutative formal group schemes over A are defined to betopologically flat over A to obtain a well-behaved concept of short exact sequences (see Section A.3), we willdefine formal R-modules as follows.

Definition 2.3.1. A formal scheme F = SpfA(B) over A is said to be a formal R-module over A if it is an R-module object in the category of formal schemes over A and if B is topologically flat over A. We will denoteB = O(F ). A map F1 −→ F2 between two formal R-modules is said to be an R-module homomorphism ifit is an R-module object homomorphism.

When a terminology that can be applied to a profinite A-module or a profinite A-algebra is applied to aformal R-module F , we mean that the underlying ring O(F ) satisfies the terminology over A. For example,we will talk about formally etale formal R-modules or topologically faithfully flat formal R-modules.

Although it is often better in algebraic geometry to state everything in terms of geometry and not interms of its underlying rings, in this thesis we will usually have to be working directly with underlying ringsof formal R-modules. There are several reasons for this. First, the easiest way to establish that the formalR-modules defined in Section 2.1 are formal R-modules as defined in this section, is via the underlying rings(Proposition 2.3.2). Second, most of the constructions we do in this section and Section 4.3.4 will be directlyfrom the rings, though sometimes motivated by geometry.

Therefore, as a reference, we will now list various properties of formal R-modules in terms of underlyingrings. First, F = SpfA(B) with B topologically flat over A is a formal R-module if and only if there existcontinuous A-algebra maps µ∗, ε∗, inv∗ and [r]∗F for each r ∈ R such that [1]F = id on B and the followingdiagrams commute:

B⊗B⊗B B⊗(B⊗B) B⊗Bid b⊗µ∗oo

(B⊗B)⊗B

B⊗B

µ∗ b⊗ id

OO

B

µ∗

OO

µ∗oo

B B⊗A B⊗Bid b⊗ε∗oo

A⊗B

B⊗B

ε∗ b⊗ id

OO

B

idGGGGGGGGGGG

ccGGGGGGGGGGGµ∗

OO

µ∗oo

(2.4)

B⊗B

∆∗

B⊗Binv∗ b⊗ idoo

B Bε∗

oo

µ∗

OO B⊗B B⊗Bswitchoo

B

µ∗

<<zzzzzzzzµ∗

bbDDDDDDDD

B⊗B B⊗B[r]∗F b⊗[r]∗Foo

B

µ∗

OO

B[r]∗F

oo

µ∗

OO

B⊗B

∆∗

B⊗B[r1]

∗F

b⊗[r2]∗Foo

B B[r1+r2]

∗F

oo

µ∗

OO B B[r2]

∗Foo

B

[r1r2]∗F

ffNNNNNNNNNNNNN[r1]

∗F

OO

11

(where ∆∗ is the diagonal map, is the unique map A→ B, and ‘switch’ switches the two components).Secondly, a map α : F1 = SpfA(B1) −→ F2 = SpfA(B2) between formal R-modules is an R-module

homomorphism if and only if α∗ is a continuous A-algebra map which makes

B1⊗B1 B2⊗B2

α∗ b⊗α∗oo

B1

µ∗1

OO

B2α∗

oo

µ∗2

OOB1 B2

α∗oo

B1

[r]∗F1

OO

B2α∗

oo

[r]∗F2

OO (2.5)

commute for all r ∈ R. For example, F1

F1−−→ SpfA(A)εF2−−→ F2 is an R-module homomorphism, called

the zero map. The set of all R-module homomorphisms is denoted by HomR(F1, F2), and the set of allhomomorphisms of the underlying commutative formal group schemes (i.e. having the commutativity of thefirst diagram of (2.5)) will be denoted HomZ(F1, F2) (and similarly with End). In contrast, the set of allformal scheme maps from F1 to F2 (i.e. no commutativity of diagram) will be denoted by MorA(F1, F2).

Finally, for a formal R-module F over A and a formal scheme T over A, the evaluation F (T ) = MorA(T, F )is an R-module, with the actions given by

ϕ1 +F (T )

ϕ2 : O(F )µ∗−→ O(F )⊗AO(F )

ϕ1 b⊗ϕ2−−−−→ O(T )⊗AO(T ) ∆∗

−−→ O(T ) (2.6)

r · ϕ : O(F )[r]∗F−−→ O(F )

ϕ−→ O(T ). (2.7)

Now we are ready to discuss some examples. First, we will show that this new definition of formalR-module does include the formal R-modules that we have been dealing with so far.

Proposition 2.3.2. Let A be a complete noetherian local ring. If (F, [ ]) is a formal R-module over A inthe sense of Definition 2.1.8, then it is a formal R-module. Its underlying formal scheme is SpfA(A[[T ]]), andthe evaluation SpfA(A[[T ]])(A) is isomorphic to F (mA) as R-modules.

Proof. A[[T ]] is clearly topologically free over A. Let us define PAA-maps µ∗ : A[[T ]] → A[[T ]]⊗AA[[T ]] ∼=A[[X,Y ]] by T 7→ F (X,Y ), [r]∗ : A[[T ]]→ A[[X]] by T 7→ [r](X), inv∗ : A[[T ]]→ A[[X]] by T 7→ g(X) whereg(X) is the unique power series with the property F (X, g(X)) = 0, and ε∗ : A[[T ]] → A by T 7→ 0. Theconditions required on F (X,Y ), g(X), and [r](T ) by the definition of formal R-module are exactly what weneed for the commutativity of the seven diagrams in (2.4). For the last statement, note that SpfA(A[[T ]])(A)is in one-to-one correspondence with elements of m, so using (2.6), we are done.

We call a formal R-module F connected if O(F ) is local. Thus, the example above is a connectedformal R-module. It turns out that a formal R-module over a complete noetherian local ring which isone-dimensional, connected, and formally smooth always has the form discussed in Proposition 2.3.2 ([3,Theorem 2.3.5]). Since we will never discuss anything which is not one-dimensional, we will refer to formalR-modules of the form in Proposition 2.3.2 as CFS (connected formally smooth) formal R-modules inthis section and Section 4.3.

We will now go back to a more general pseudocompact local ring A, and discuss examples of formalR-modules that we have not seen.

Proposition 2.3.3. For any R-module S, we can define a formal R-module structure on F = SpfA(AS),where AS is the direct product of S copies of A with the product topology. This formal R-module is formallyetale. F (A) is canonically isomorphic to S as R-modules, and for any formal R-module H, there is a one-to-one correspondence via the evaluation at A between the set of all formal R-module homomorphisms F → Hand the set of all R-module homomorphisms S → H(A).

Proof. Intuitively speaking, F is the ‘disjoint union’ of S copies of ‘a single point’ SpfA(A) with the formal R-module structure defined directly from the R-module structure of S, so all the statements in this propositionare certainly reasonable.

12

More precisely, by viewing an element of B = AS as a set map S → A and an element of B⊗AB asa set map S × S → A, we can define (µ∗(b))(s, s′) = b(s + s′), ε∗(b)(s) = b(0), (inv∗(b))(s) = b(−s), and([r]∗(b))(s) = b(rs). Since B is topologically free over A, we conclude that F is a formal R-module bychecking (2.4). Theorem A.4.1 immediately tells us that this is etale.

Note that SpfA(AS)(A) =∐

SpfA(A)(A) by the same proof used in obtaining (A.2), so we now know thatevery element of SpfA(AS)(A) is a projection onto the value of b ∈ B at s for some s ∈ S. Using (2.6), wecan check that SpfA(B)(A) ∼= S as R-modules.

Now we show the last statement. A map α : F −→ H of formal schemes over A corresponds to acontinuous A-algebra map α∗ : O(H) −→ AS , which in turn corresponds to a family of continuous A-algebramaps α∗s : O(H) −→ A for all s ∈ S. Thus, α corresponds to a (set-theoretic) map S → H(A). Because ofthe way we defined the formal R-module structure on F , we can check using (2.5) and (2.6) that α is a formalR-module homomorphism if and only if the corresponding S → H(A) is an R-module homomorphism.

Because SpfA(AS) can be thought of as a ‘disjoint union of a single point’, it is far from the connectedformally smooth case that we have been dealing with previously. In fact, connected and etale formal R-modules can be considered as ‘opposite extremes.’ More precisely,

Proposition 2.3.4. If SpfA(B) is a connected formal R-module, then Bet = A, identified via the algebrastructure map. If SpfA(B) is a connected formal R-module and SpfA(B′) is an etale formal R-module, thenthere are no non-trivial formal R-module homomorphisms from SpfA(B) to SpfA(B′).

Proof. By Theorem A.4.1, A is a formally etale closed subalgebra of B, so it is contained in Bet. On theother hand, Bet⊗A(A/m) = (B⊗A(A/m))

et→ B⊗A(A/m) ∼= B/mB. Since B is local and Bet⊗A(A/m) is a

product of finite separable extensions of A/m, Bet⊗A(A/m) is a field extension of A/m. But Bet⊗A(A/m)→B/mB → A/m via the identity section, so Bet⊗A(A/m) ∼= A/m. By Nakayama’s lemma, Bet = A.

For the second statement, note that B ← B′ factors through Bet = A since B′ is etale (see Section A.4).But since SpfA(B)→ SpfA(B′) sends the identity section to the identity section, it must factor through theidentity section of B′.

In Section 4.3, both connected and etale formal R-modules will be important. Indeed, the main objectsof study in Section 4.3 are formal R-modules which can be constructed by ‘combining’ these two extremes offormal R-modules. This combination process is achieved through the use of short exact sequences of formalR-modules, so for the rest of this section, we will discuss some properties of short exact sequences of formalR-modules. We define

Definition 2.3.5. A sequence of formal R-modules and R-module homomorphisms 0→ F ′ → F → F ′′ → 0is said to be a short exact sequence if the underlying commutative formal group schemes form a short exactsequence.

From Appendix A.3, this means that (as commutative formal group schemes) F ′ → F is a closed immer-sion, F → F ′′ is topologically faithfully flat, F ′ −→ F ′′ vanishes, and we can identify F ′ as the kernel ofF → F ′′. Thus, applying a continuous (i.e. local) base change to a short exact sequence of formal R-modulesresults in a short exact sequence.

To justify this definition, we want to be able to complete to a short exact sequence given either a closedimmersion F ′ → F or topologically faithfully flat map F F ′′ of formal R-modules. In the first case, notethat the construction of the quotient discussed in Section A.3 creates a formal R-module if we start withformal R-modules, because the duality process is functorial. We then get a short exact sequence 0→ F ′ →F → F ′′ → 0. In the second case, the kernel F ′ = F ×F ′′ SpfA(A) has a natural formal R-module structurecoming from F , and we then get a short exact sequence of formal R-modules 0→ F ′ → F → F ′′ → 0.

As discussed in Section A.3, this definition of short exactness is equivalent to requiring F ′ −→ F ′′ to vanishand the sequence to form a short exact sequence over the closed fiber A/m. In particular, if we construct amap over general A in a manner compatible with base change, monomorphisms and epimorphisms can bechecked over the closed fiber. Since commutative formal group schemes over a field form an abelian categoryby Theorem A.3.2, this definition often allows us to reduce many statements of short exactness to the caseover an abelian category. We will use this technique repeatedly in the following discussion.

13

The notion of short exact sequences of formal R-modules has properties that we expect short exactsequences to have; we will prove these properties in the process of proving the next proposition. Beforestating it, we need to introduce the notion of extensions in this setting. Fix two formal R-modules F ′, F ′′

over A. Let ExtR,A(F ′′, F ′) denote the set of short exact sequences

0→ F ′ → F → F ′′ → 0

of formal R-modules over A modulo the equivalence which identifies extensions fitting into a commutativediagram

0 // F ′ // F1//

α

F ′′ // 0

0 // F ′ // F2// F ′′ // 0

where α is a formal R-module homomorphism. As in the case of modules, α must then be a formal R-module isomorphism, since over the closed fiber we have an abelian category (so the five lemma holds). Inparticular, we have defined an equivalence relation. Note that we have a well-defined base change map ofsets ExtR,A(F ′′, F ′) −→ ExtR, eA(F ′′, F ′).

We are now ready to prove

Proposition 2.3.6. ExtR,A(F ′′, F ′) is a group in a manner compatible with base change on A (by the ‘same’operations as in the case of R-modules).

Proof. Recall that we can define the group structure on extensions of objects in abelian categories by a processcalled Yoneda Ext ([4], [6]). This agrees with the usual homological method in the case of modules ([15,page 353]), and it has the advantage of not requiring projectives and injectives. Unfortunately, we cannotquote this result in the present setting, as formal R-modules over A do not form an abelian category. Butwe will be able to prove the proposition by the technique of reduction to the closed fiber, where we do havean abelian category. Essentially, we will make all the requisite constructions by category-theoretic methods,then check that these constructions create formal R-modules and formal R-module homomorphisms, andfinally use base change compatibility of our constructions to reduce to checking the necessary properties overthe closed fiber. For notational shorthand, we will write F ′ → F F ′′ for a short exact sequence.

There are three main constructions in the definition of the addition on ExtR,A: taking products, pull-back,and push-out. We will now discuss how each of them adapts the Yoneda Ext construction in an abeliancategory to the present setting.

(1) Taking the product of two exact sequences F ′ → Fi F ′′: then the product diagram

0→ F ′ × F ′ → F1 × F2 → F ′′ × F ′′ → 0

is a diagram of formalR-modules with the composite of the two maps zero, and this construction is compatiblewith base change. Thus, short exactness may be checked on the closed fiber, where the usual abelian categoryargument works.

(2) Pull-back: suppose F ′ → F F ′′ is short exact and G′′ → F ′′ is a formal R-module map. ConsiderF ×F ′′ G

′′. This is topologically flat over A, as F → F ′′ and G′′ → SpfA(A) are topologically flat, so it is aformal R-module. The maps F ′ → F and F ′ → SpfA(A) ε−→ G′′ induce F ′ → F ×F ′′ G

′′, which is a formalR-module homomorphism. So we have a sequence of formal R-modules F ′ → F ×F ′′ G

′′ → G′′. Since thecomposite of the two maps is zero and this construction is compatible with base change, we can again reduceto checking over the closed fiber.

(3) Push-out: suppose F ′ → F F ′′ is short exact and F ′ → G′ is a formal R-module map. SinceF ′ → G′ × F is a closed immersion of formal R-modules, let G′ ×F ′ F be the quotient formal R-module.Since F ′ → G′ × F → F → F ′′ is zero, G′ ×F ′ F → F ′′ is a well-defined formal R-module homomorphism.Thus, we have a sequence of formal R-modules G′ → G′ ×F ′ F → F ′′ whose composite is zero, and thisconstruction is compatible with base change. Similarly, as above, we are done by the usual abelian categoryargument.

14

Note that all three constructions send equivalent extensions to equivalent extensions. Further, the oper-ations of pull-back and push-out make ExtR,A a bifunctor: if F ∈ ExtR,A(F ′′, F ′) represents F ′ → F → F ′′

and α′ : F ′ → G′ and α′′ : G′′ → F ′′ are formal R-module homomorphisms, then (α′)∗ (α′′)∗(F ) isrepresented by

G′ →((

(F ×F ′′ G′′)×G′)/(image of F ′)

) G′′

and (α′′)∗ (α′)∗(F ) is represented by

G′ →((F ×G′/image of F ′)×F ′′ G

′′) G′′,

so we have a map of formal R-module from the first to the second making the diagram commute.Now given two sequences F ′ → Fi F ′′, we can define their sum to be the element

(∆F ′′)∗ (∇F ′)∗(F ′ × F ′ → F1 × F2 → F ′′ × F ′′) = (∇F ′)∗ (∆F ′′)∗(F ′ × F ′ → F1 × F2 → F ′′ × F ′′)

in ExtR,A, where ∆F ′′ : F ′′ → F ′′ × F ′′ is the diagonal and ∇F ′ : F ′ × F ′ → F ′ corresponds to the additionof the formal R-module F ′. To check that this defines a group structure with the split extension class as theidentity, we can just follow [4]: to check for associativity and existence of inverses, we need to define certainformal R-modules and homomorphisms using bifunctoriality and coproducts and products. Once we do thisin a manner compatible with base change, checking that these constructions really work reduces to checkingit over the closed fiber, where we have an abelian category, so the arguments in [4] go through.

We will prove one more general property of a short exact sequence:

Proposition 2.3.7. If 0→ F ′ → F → F ′′ → 0 is an exact sequence of formal R-modules over A, then forany formal scheme X over A, the evaluation at X results in an exact sequence 0→ F ′(X)→ F (X)→ F ′′(X)as R-modules.

Proof. By definition of short exact sequence, the injectivity of F ′(X) → F (X) and vanishing of F ′(X) →F ′′(X) follow. If ϕ : X → F is such that X → F → F ′′ factors through the identity section, then we have awell-defined map of formal schemes X → F ×F ′′ SpfA(A) ∼= F ′.

For the rest of this section, we will briefly discuss an important class of short exact sequences, called theconnected-etale sequences. [3] covers this for finite flat group schemes but does not explicitly cover it forformal group schemes. Since we need a corresponding result for the formal R-modules, we will construct andprove fundamental properties about this sequence.

Suppose that F = SpfA(B) is a formal R-module over A. Since Bet → B is topologically faithfully flat andB is topologically flat over A, Bet is topologically flat over A. Note that the restriction of µ∗ to Bet factorsthrough (B⊗AB)

et= Bet⊗AB

et (see Appendix A.4). Similarly, the structure map, the identity section,the inverse, and [r]∗F for all r ∈ R all factor through Bet, so we can canonically put a formal R-modulestructure on Bet in a way so that SpfA(B) → SpfA(Bet) is a topologically faithfully flat formal R-modulehomomorphism. We can now complete the exact sequence of formal R-modules by taking the kernel F 0.

We will show that F 0 is connected, i.e. B0 = O(F 0) is local. It certainly suffices to show locality afterbase changing to the algebraic closure of A/m, so we assume that everything is over an algebraically closedfield k. We write B =

∏Bn over all open maximals, and using Theorem A.4.1 and locality of Bn, Bet =

∏k

indexed by open maximals of B. Thus B0 ∼= B⊗Qkk ∼= Bn0 , where n0 is the unique component which is not

killed by the identity section∏k → k. Hence F 0 is connected, and we have constructed an exact sequence

of formal R-modules called the connected-etale sequence 0→ SpfA(B0)→ SpfA(B)→ SpfA(Bet)→ 0 ofthe formal R-module SpfA(B). We call SpfA(B0) and SpfA(Bet) the connected component and the etalecomponent, respectively. In view of Proposition 2.3.4, we are ‘decomposing’ a formal R-module into twoopposite extremes.

Since the construction of the maximal etale subalgebra is compatible with local base change and takingproducts, the connected-etale sequence is compatible with local base change and taking products. Thefollowing will be used in Section 4.3

15

Proposition 2.3.8. Suppose A = k is an algebraically closed field. Then for a formal R-module G over k,the connected-etale sequence splits. Further, there are no non-trivial homomorphisms from an etale formalR-module to a connected formal R-module. Thus the connected-etale sequence splits uniquely (in a mannercompatible with base change and taking products).

Proof. Let G = Spfk(B), and let τ be the set of topological nilpotents of B. As discussed in Appendix A.1,B/τ ∼=

∏Bn/nBn. k = k implies Bn/nBn

∼= k, and so (B/τ)⊗k(B/τ) ∼=∏k have no topological nilpotents.

Thus, B → B⊗kB → (B/τ)⊗k(B/τ) factors through B/τ . Similarly, inv∗, ε∗, [r]∗G can be defined on B/τ ,so Spfk(B/τ) is a formal R-module. Since Bn is local and k = k, Theorem A.4.1 shows Bn

et ∼= k. Thus,Bn

et → Bn → Bn/nBn is an isomorphism. Taking products, Bet → B → B/τ is an isomorphism, inducingan isomorphism of formal R-modules. This gives the splitting.

For the second statement, on the level of rings, it suffices to check that there is only one PAk map from alocal B′ to an etale B′′. But by Theorem A.4.1 and k = k, B′′ =

∏k, so it now suffices to show that there

is only one PAk map from B′ → k. But any such B′ → k factors through B′/mB′ ∼= k, so we are done. Ifwe have two splittings, i.e. two formal R-module sections SpfkBet → Spfk(B), then taking the ‘difference’ ofthe sections, we get a formal R-module homomorphism from an etale formal R-module to a connected one.

16

Chapter 3

Formal R-Modules for Various R

Our goal in this chapter is to determine the possibilities of formal R-modules when R is the valuation ringof a local field. This lies at the foundation of the deformation theory of Chapter 4.

Our main technique will be the use of universal objects. Because any object can be viewed as the imageof the universal object (if such a thing exists), the universal object often contains useful information about‘all’ objects simultaneously. Thus, determination of the universal object and its properties is extremelyimportant. As an example from this chapter, knowledge of the universal formal R-module allows us tosystematically construct formal R-modules of a given height. The use of universal objects continues intoChapter 4.

In Section 3.1, as a starting point, we will talk in general about universal formal R-modules. We willrecall the definition of universal objects, and discuss properties of universal formal R-modules that hold forany R. We will see that the ring ΛR over which the universal formal R-module is defined turns out to be aquotient of a polynomial ring, having a natural gradation.

In Section 3.2, we prove that for an infinite field R, ΛR is simply R[Ω1,Ω2, . . .] where each Ωi has degreei in ΛR.

In Section 3.3, we discuss many specific properties of formal R-modules when R is the valuation ringof a local field. These will be important, as Chapter 4 is entirely on formal modules with this R. Wewill show that ΛR is again the polynomial ring R[Υ1,Υ2, . . .] with deg Υi = i in ΛR, using the result fromSection 3.2. Moreover, we will characterize formal R-modules which are isomorphic to the additive moduleGa (an analogue of Theorem 2.2.4). Furthermore, we will show an extremely useful formula, which tellsus the degree n part of F − G given (F, [ ]F ) ≡ (G, [ ]G) (mod poly. deg. n). This formula enables us tosuccessively approximate differences of formal R-modules, going up one polynomial degree at a time. Finally,subsection 3.3.1 deals with the case of formal R-modules over a separably closed field of characteristic p.This case turns out to be particularly nice, as the height of a formal R-module determines the isomorphismclass. This fact will be essential in Chapter 4.

3.1 Universal Formal R-Modules

In this section, we will discuss properties of universal formal R-modules which hold for any R. This sets upthe stage for the rest of this chapter.

Let us first recall the notion of a universal and representing object, since we will need this conceptthroughout this thesis (cf. [20]). A covariant representable functor is a covariant functor F : C → Setsuch that there exist Cuniv ∈ C and a set map φC : homC (Cuniv, C)

∼=−→ F (C)) for each C ∈ C making the

17

diagram

homC (Cuniv, C1)φC1 //

F (C1)

homC (Cuniv, C2) φC2

// F (C2)

commutative (i.e. the functor F is naturally isomorphic to the Hom functor). For such a functor, (Cuniv, φ)is called a representing object. The same concept, put in a different language, is the notion of theuniversal element. We call (Cuniv, u) an universal object of F if for any C ∈ C and x ∈ F (C), thereexists a unique map f ∈ homC (Cuniv, C) such that F (f)(u) = x. For a fixed Cuniv, there is a bijectionbetween representing objects (Cuniv, φ) and universal elements (Cuniv, u), given by u = φCuniv(idCuniv) andφC(f) = F (f)(u). And if (Cuniv, φ, u) and (Cuniv′ , φ′, u′) both represent F , then there is an isomorphismCuniv → Cuniv′ with compatible representing and universal objects.

Let C be the category of all R-algebras, and consider the functor F : C → Set which sends A to the setof all formal R-modules over A and which sends f : A1 → A2 to the push-forward f∗ from the set of formalR-modules over A1 to the set of formal R-modules over A2. Note that a formal R-module over A is just(F (X,Y ), [r]F (X)) with coefficients in A such that these coefficients satisfy certain polynomial conditions (bycomparing coefficients of each monomial in X and Y ). Therefore, F is represented by R[Γi,j ,Θk,r] moddedout by relations imposed by the definition of formal R-modules, where Γi,j corresponds to the coefficient ofXiY j in F (X,Y ) and Θk,r corresponds to the coefficient of Xk in [r]F (X). In other words, the universalformal R-module is F univ(X,Y ) =

∑Γi,jX

iY j and [r]Funiv(X) =∑

Θk,rXk, and the representing object

ΛR is the polynomial ring R[Γi,j ,Θk,r] modded out by relations which make F univ into a formal R-module.We will now analyze ΛR.

Proposition 3.1.1. ΛR can be given a gradation in such a way that Γi,j has degree i+ j − 1 and Θk,r hasdegree r − 1, and that every degree 0 element can be identified with an element of R.

Proof. We need to show that the polynomial relations by which we mod out to obtain ΛR from R[Γi,j ,Θk,r]are all homogeneous with the degree assignment in the statement of the proposition. Note that this degreeassignment is reasonable, as Γ1,0 = Γ0,1 = 1 and Θ1,r = r can be homogeneous only if these variables havedegree 0. It certainly suffices to show that in these polynomial relations by which we mod out to obtain ΛR,the degree of the coefficient (i.e. involving Γ’s and Θ’s) is always one less than the polynomial degree of thevariables (i.e. involvingX’s and Y ’s) in any monomial. We will not check everything here, but as an example,let us pick out Γi,jX

iY j from the inside F and Γi′,j′ from the outside F in the expansion F (F (X,Y ), Z).Then we get Γi′,j′Γi′

i,jXii′Y ji′Zj′ . Note that the degree of the coefficient in ΛR is (i′+ j′−1)+ i′(i+ j−1) =

ii′ + ji′ + j′ − 1, which is indeed one less than the polynomial degree.Once we have the gradation, the last statement is clear since the elements of degree 0 among Γ and Θ

can be identified with elements of R, as noted above.

So we have two different notions of degree. We will distinguish them by always saying ‘polynomial degree’whenever we mean the total degree of the monomial in X and Y and saying degree to refer to the degreeof ΛR. We will denote the n-th degree component by Λn

R. Knowing this gradation is extremely useful. Forexample, to obtain a formal R-module germ G of degree n over A, we need to specify up to coefficients ofmonomials of polynomial degree n− 1 in such a way that it forms a formal R-module. Thus, by above, thisis same as giving an R-module map ϕ : ⊕n−2

i=0 ΛiR → A such that it is a ring homomorphism wherever it is

defined.Note that ΛR is in general huge, and in order to construct the formal R-module corresponding to f :

ΛR → A, all we need to know are the images of various Γ’s and Θ’s, and not their products (of course it isimportant that f is a ring map, but if we somehow know this already then all we need are the images of Γ’sand Θ’s). This motivates us to define a smaller ring (a quotient of ΛR) in which various Γ’s and Θ’s are notidentified with each other but in which we can ignore all products of two such variables. Specifically, let I1be the ideal of ΛR generated by homogeneous elements of degree at least 1, and let ΛR = ΛR/I

21 . This is a

18

much smaller ring with gradation. Using Lazard’s Theorem, we actually have a pretty good grasp of whatthis ring is:

Proposition 3.1.2. Let n ≥ 2. As an R-module, Λn−1R is exactly the R-module generated by γn−1 and

θn−1,r (for each r ∈ R) satisfying the relations

(rn − r) · γn−1 = ε(n)θn−1,r (3.1)θn−1,r1+r2 − θn−1,r1 − θn−1,r2 = Cn(r1, r2) · γn−1 (3.2)

r1θn−1,r2 + rn2 θn−1,r1 = θn−1,r1r2 , (3.3)

where ε(n) and Cn(X,Y ) are as defined in Section 2.2, γn−1 here is an element such that∑

Γi,n−iXiY n−i =

γn−1Cn(X,Y ) in Λn−1R , and θn−1,r is the image of Θn,r in Λn−1

R .

Proof. In ΛR, if a monomial of total degree n−1 in Γ and Θ do not contain a variable of degree n−1, then itmust be a product of at least two positive-degreed variables. These monomials of degree n− 1 vanish when

we mod out by I21 , so as an R-module, Λn−1

R is generated by images of Γi,n−i and Θn,r satisfying the formalR-module identities (with the assumption that any product of two positive-degreed Γ’s or Θ’s are zero). Let

θn−1,r be the image of Θn,r in Λn−1R .

In order to obtain γn−1, let F univn =

∑Γi,n−iX

iY n−i over ΛR. Clearly, F univn (X,Y ) = F univ

n (Y,X) andF univ

n (X, 0) = F univn (0, X) = 0. Now, look at the polynomial degree n part of F univ(F univ(X,Y ), Z) =

F univ(X,F univ(Y,Z)). On the LHS, in order to get a nonzero expression, if we use a positive-degreedcoefficient from the outside F univ, then we cannot use any positive-degreed coefficient from the inside F univ.Thus, the only nonzero monomial contribution comes from either using a degree (n − 1) coefficient fromthe outside F univ or using a degree (n − 1) coefficient from the inside F univ. In other words, we getF univ

n (X + Y, Z) + F univn (X,Y ) from the LHS. By a similar computation for the RHS and equating both

sides we getF univ

n (X + Y,Z) + F univn (X,Y ) = F univ

n (X,Y + Z) + F univn (Y,Z).

By Proposition 2.2.2, F univn (X,Y ) = γn−1Cn(X,Y ) for some γn−1 ∈ ΛR. Thus, γn−1 has degree n − 1 in

ΛR (∵ Cn(X,Y ) has Z-coefficients) and all Γi,n−i’s are generated by γn−1 as an R-module. All the formalgroup identities (viewed inside ΛR) hold now, so in order to determine all the relations that γn−1 and θn−1,r

satisfy, we now only need to look at the identities about the R actions.Denoting [r] for [r]Funiv , looking at polynomial degree n terms of F univ([r](X), [r](Y )) = [r](F univ(X,Y ))

in ΛR, we obtain F univn (rX, rY ) + θn−1,rX

n + θn−1,rYn = θn−1,r(X + Y )n + rF univ

n (X,Y ). So

(rn − r)γn−1Cn(X,Y ) = θn−1,r((X + Y )n −Xn − Y n) = θn−1,rε(n)Cn(X,Y ),

resulting in the first identity (∵ Cn(X,Y ) is a Z-coefficient primitive polynomial). Similarly, comparing Tn

coefficients of [r1 +r2](T ) = F univ([r1](T ), [r2](T )) and [r1]([r2](T )) = [r1r2](T ), we get the second and thirdequations, respectively.

3.2 R is a Field

We will now analyze the possibilities for the formal R-modules, when R is a field. Actually, as in the followingtheorem, the possibilities are extremely limited in this case: they are all isomorphic to the additive formalmodule. This case will serve as a stepping stone for the case of R being the ring of integers of local fields.

Theorem 3.2.1. Let R be a field, and A be an R-algebra. Then every formal R-module over A is isomorphicto Ga. If further |R| = ∞, then there exists a unique isomorphism αF (X) ∈ A[[X]] from F to Ga withderivative 1, and ΛR

∼= R[Ω1,Ω2, . . .] where Ωi has degree i and corresponds to the coefficients of Xi+1 inαF (X).

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Proof. We first assume that R is characteristic 0. Then by Corollary 2.2.5 (1), there exists an -invertibleα ∈ A[[X]] such that α(F (X,Y )) = α(X) + α(Y ). Let G be the image of F via α (cf. Example 2.1.10).Then G(X,Y ) = X + Y , so in particular, [r]G(X) + [r]G(Y ) = [r]G(X + Y ). By looking at the coefficientsof XY i−1 on both sides, we conclude that [r]G(X) = r · X. Thus, G must be the additive module. If α1

and α2 are both isomorphisms from F to Ga with derivative 1, then α1 α−12 is an isomorphism of Ga with

derivative 1, but by a similar argument as above, this can only be X. Thus, we have a unique isomorphism.For the last statement, given an R-algebra map f : R[Ω1, . . .]→ A, let

F = (X + f(Ω1)X2 + f(Ω2)X3 + · · · )−1 Ga (X + f(Ω1)X2 + f(Ω2)X3 + · · · )

be the formal R-module corresponding to f . This association is a natural isomorphism by above, so we haveΛR∼= R[Ω1,Ω2, . . .].

Now assume that R is characteristic p. Since R −→ End(F ) is a ring homomorphism, [p]F = 0. Thus,pA = 0 together with Corollary 2.2.5 (2) implies that there exists an isomorphism α : F → Ga of formalgroups. Let G be the image of F via α; as above, G(X,Y ) = X+Y . From Example 2.1.7, End(G) = Aτ.We will successively construct isomorphisms to eventually obtain the isomorphism with the additive module:

Claim. Fix k ≥ 1. Suppose [r]G ≡ r + ω(r)τk (mod τk+1) inside Aτ for all r ∈ R withω(r) ∈ A. Then there is an element a ∈ A such that in the ring Aτ, (1+aτk)[r]G(1 + aτk)−1 ≡r (mod τk+1). Moreover, this choice of a is unique if R is infinite.

Proof. Note that in End(G), (1+aτk)(r+ω(r)τk)(1 + aτk)−1 ≡ (1+aτk)(r+ω(r)τk)(1−aτk) ≡r+ arpk

τk + ω(r)τk − raτk (mod τk+1), so what we desire is being able to find a ∈ A such that(rpk − r)a = ω(r). Because [ ]G : R → Aτ is a ring homomorphism, we certainly haveω(r1 + r2) = ω(r1) + ω(r2). Moreover,

[r1]G[r2]G ≡ (r1 + ω(r1)τk)(r2 + ω(r2)τk) ≡ r1r2 + rpk

2 ω(r1)τk + r1ω(r2)τk (mod τk+1),

and using r1r2 = r2r1 and rearranging terms, we have

rpk

2 ω(r1)− r2ω(r1) = rpk

1 ω(r2)− r1ω(r2).

Thus, if R 6⊆ Fpk , then we have uniqueness and existence of the desired a, as a = ω(r)

rpk−rfor some

r ∈ R\Fpk .

Now suppose that R ⊆ Fpk . Then the above calculation shows that ω(r1 + r2) = ω(r1) + ω(r2)and ω(r1r2) = r1ω(r2)+ r2ω(r1), i.e. ω is a derivation. By induction, we can show that ω(rm) =mrm−1ω(r) (just like derivatives), so for any r ∈ R, we have ω(r) = ω(rpk

) = pkrpk−1ω(r) = 0.Thus we are done.

We know that [r]G ≡ r+ ω1(r)τ (mod τ2) for all r ∈ R in Aτ, so by the claim there exists a1 ∈ A suchthat (1 + a1τ)[r]G(1 + a1τ)

−1 ≡ r mod τ2 for all r ∈ R. Since 1 + a1τ ∈ End(F ), the image G1 of G viathis map keeps the additive formal group structure while it now has the property that [r]G1 ≡ r+ω2(r)τ2 ≡(mod τ3) for all r ∈ R. So we can continue by induction, and the product · · · (1 + a3τ

3)(1 + a2τ2)(1 + a1τ)

is the desired isomorphism of G with the additive R-module with derivative 1 (this infinite product makessense, since the relation apτ = τa keeps the power of τ).

Now suppose that R is infinite, and let∑a′iτ

i be a formal R-module isomorphism from G to Ga witha′0 = 1. Then note that (1 + a′1τ)[r]G(1 + a′1τ)

−1 ≡ r (mod τ2), so a′1 is uniquely determined by the claim.Similarly, (1 + a′1τ + a′2τ

2)[r]G(1 + a′1τ + a′2τ2)−1 ≡ r (mod τ3) and 1 + a′1τ + a′2τ

2 ≡ (1 + a′2τ2)(1 + a′1τ)

(mod τ3), so again we can use the claim above to conclude the uniqueness of a′2. We continue by induction,and obtain uniqueness of the isomorphism with the additive module with derivative 1. ΛR

∼= R[Ω1, . . .]follows by the same argument as in the characteristic zero case.

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It is not true that every formal group over a field of finite characteristic is isomorphic to the additivegroup. For instance, if R is a field of characteristic 2 and A is an R-algebra, F (X,Y ) = X + Y +XY overA is not isomorphic to the additive group, because of [2]F (X) = F (X,X) = X2 6= 0 and Corollary 2.2.5 (2).So this is an example of a formal group which cannot be made into a formal R-module.

3.3 R is the Valuation Ring of Local Field

Now we come to what we are most interested in, namely formal R-modules when R is the valuation ring ofa local field. Since we will be working with this set-up for the rest of this thesis, we will fix the followingnotations from now on: K is a local field with the valuation ring R (i.e. R is a discrete valuation ring withfinite residue field), π is a uniformizer of R, and k is the residue field R/(π) of order q which is a power of aprime p.

It turns out that we can use the results from the field case in analyzing ΛR. The first step is analyzingthe closely-related ring ΛR:

Proposition 3.3.1. Let n ≥ 2. Then Λn−1R is isomorphic to R as R-modules. More specifically, using the

notation from Proposition 3.1.2, we can pick a generator η of Λn−1R so that

(1) if n is not a power of q, (rn−r)η = θn−1,r for all r ∈ R and γn−1 ·Cn(X,Y ) = η · ((X+Y )n−Xn−Y n).(2) if n is a power of q, θn−1,r = rn−r

π · η for all r ∈ R and γn−1 = pπ · η.

Proof. (1a) Suppose n is not a power of p. Let η = γn−1. From (3.1), θn−1,r ∈ Rγn−1 = η for all r, since

any other prime is a unit in R. So by Proposition 3.1.2, Λn−1R is generated by η. Plugging θn−1,r = (rn−r)

ε(n) η

into (3.2) and (3.3), we get 0 = 0 in both cases, so by the same proposition, Λn−1R∼= R by η 7→ 1.

(1b) Suppose n is a power of p, but not a power of q. If rn − r ∈ (π) for all r ∈ R, then R/(π) = Fq ⊆ Fn,which is a contradiction. Thus, let r ∈ R be such that rn − r /∈ (π) (so a unit). Let η = (rn − r)−1

θn−1,r.By (3.1), γn−1 = pη ∈ Rη. Using (3.3) and rr1 = r1r, we obtain (rn − r)θn−1,r1 = (rn

1 − r1)θn−1,r, so

θn−1,r1 ∈ Rη for all r1 ∈ R. Thus, Λn−1R is generated by η. Plugging γn−1 = pη and θn−1,r = (rn − r)η into

(3.1), (3.2), and (3.3), we get 0 = 0, so we conclude that Λn−1R∼= R via η 7→ 1.

(2) For this part, it is helpful to think of each θn−1,r as evaluation of the function θ : R → Λn−1R at r. We

will first show Λn−1R is generated by θ(π) as an R-module by showing M = Λn−1

R /(θ(π)) = 0. For any r ∈ R,by (3.3), rθ(π) + πnθ(r) = θ(rπ) = θ(πr) = πθ(r) + rnθ(π), so in M we must have θ(πr) = πnθ(r) = πθ(r).Hence (πn − π)θ(r) = 0, so

πθ(r) = θ(rπ) = 0 for all r ∈ R. (3.4)

Now (3.3) shows θ(1) = 0. Further, using (3.2),

θ(p) = Cn(p− 1, 1)γn−1 + θ(p− 1) + θ(1) = Cn(p− 1, 1)γn−1 + Cn(p− 2, 1)γn−1 + θ(p− 2) + θ(1)

= · · · = (Cn(p− 1, 1) + Cn(p− 2, 1) + · · ·+ Cn(1, 1)) γn−1 = (pn−1 − 1)γn−1.

Thus by (3.4), we have 0 = θ(p) = (pn−1− 1)γn−1, so γn−1 = 0. Using this together with (3.2) and (3.4), we

see that θ : R θ−→ Λn−1R →M is an additive map factoring through R→ R/(π). By assumption, rn− r ∈ (π)

for all r ∈ R, so by (3.3) and (3.4), θ : R/(π) → M is a derivation. As before, this means θ = 0. On theother hand, by definition of the map and by γn−1 = 0, θ is surjective, so M = 0.

We can check by the three equations in Proposition 3.1.2 that there is a well-defined R-module homomor-

phism Λn−1R → R by γn−1 7→ p

π and θ(r) 7→ rn−rπ . Since θ(π) is sent to πn−1−1 (a unit), this homomorphism

is surjective. Since Λn−1R is generated by θ(π) as above and πn−1− 1 is a unit, we have the injectivity of this

homomorphism, so we get the desired isomorphism. Thus η = 1πn−1−1θ(π) is sent to 1 via this R-module

isomorphism, and we have now proved everything.

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The fact that we know explicitly what each of Λn−1R is in the above proposition will be useful later when

we need successive approximation techniques for formal R-modules. Moreover, we can use the above andTheorem 3.2.1 to obtain ΛR:

Theorem 3.3.2. ΛR∼= R[Υ1,Υ2, . . .] as graded R-algebras, with deg Υi = i.

Proof. For each n ≥ 2, letting Υn−1 map to a lift of the generator of Λn−1R given in Proposition 3.3.1 to an

element of Λn−1R , we obtain a graded R-algebra homomorphism f : R[Υ1,Υ2, . . .]→ ΛR. The degree-0 part

of this map is surjective, so assume by induction that everything up to the degree-(n− 1) part of this mapis surjective. Then since all elements of degree n can be written as a sum of r · f(Υn) and an element inI21 ∩Λn

R for some r ∈ R by construction, we get the surjectivity for the degree-n-part by induction. Thereforef is surjective, and in particular, R[Υ1, . . .]⊗R K ΛR ⊗R K.

Now let F be the image of Ga via the isomorphismX+Ω1X2+· · · defined over R[Ω1, . . .]. This corresponds

to a graded R-algebra homomorphism g : ΛR → R[Ω1, . . .]. Thus, g ⊗ 1 : ΛR ⊗R K → R[Ω1, . . .] ⊗R K isa graded K-algebra homomorphism sending Γ(R)

i,j ⊗ 1 to the coefficient of XiY j in F and Θ(R)l,r ⊗ 1 to the

coefficient of X l in [r]F . By Proposition 3.2.1, composing g ⊗ 1 with the isomorphism K[Ω1, . . .] ∼= ΛK , wesend Γ(R)

i,j ⊗ 1 to Γ(K)i,j and Θ(R)

l,r ⊗ 1 to Θ(K)l,r . By the proof of Proposition 2.1.6, Θ(K)

l,r−1 can be expressed as

a finite sum of products of elements of K and Θ(K)l′,r for l′ ≤ l, so we conclude that g ⊗ 1 is surjective.

Hence we have graded R-algebra homomorphisms R[Υ1, . . .]f // // ΛR

g // R[Ω1, . . .] such that bothf ⊗ 1 and g ⊗ 1 are surjective. But the corresponding graded components of K[Υ1, . . .] and K[Ω1, . . .]have the same dimension, so surjectivity of K-vector space map between them implies injectivity. Hence(g ⊗ 1) (f ⊗ 1) is injective. By freeness of R[Υ1, . . .] as an R-module, we can recover g f as

R[Υ1, . . .]⊗R R // R[Υ1, . . .]⊗R K (gf)⊗1 // R[Ω1, . . .]⊗R K ,

thus obtaining injectivity of g f . Hence, f is indeed an isomorphism as graded R-algebras.

Because ΛR is simple, we will be able to get a lot of information about formal R-modules. For example,we have the following:

Corollary 3.3.3. (1) Any formal R-module germ over an R-algebra A actually arises from a formal R-module over A.

(2) If A→ B is a surjective R-algebra homomorphism, then any formal R-module over B is the image ofa formal R-module over A.

Proof. (1) As remarked earlier, a germ of degree n corresponds to anR-module homomorphism f : ⊕n−2i=0 Λi

R →A such that f(1) = 1 and f is a ring homomorphism wherever it is defined. There is an R-algebra homo-morphism f : ΛR

∼= R[Υ1, . . .]→ A defined by f(Υi) = f(Υi) for 1 ≤ i ≤ n− 2 and f(Υi) = 0 for i ≥ n− 1,and by the assumption f agrees with f on ⊕n−2

i=0 ΛiR.

(2) Given R[Υ1, . . .] → B, we can lift each of the image of Υi to A and then extend this unqiuely to anR-algebra homomorphism R[Υ1, . . .]→ A. By functoriality, this is what we need.

One more consequence of Proposition 3.3.1 is that we can prove an analogue of Lazard’s Theorem in thesetting of formal R-modules. It turns out that equality of two formal R-modules modulo polynomial degreen puts a severe restriction on the difference of two formal R-modules. This is extremely useful when weemploy the successive approximation techniques.

Proposition 3.3.4. Suppose that F and G are two formal R-modules over A such that (F, [ ]F ) ≡ (G, [ ]G)(mod poly. deg. n). If n is not a power of q, then there exists a ∈ A such that

F (X,Y ) ≡ G(X,Y ) + a ((X + Y )n −Xn − Y n) (mod poly. deg. n+ 1)[r]F (X) ≡ [r]G(X) + (rn − r) · aXn (mod poly. deg. n+ 1) ∀r ∈ R.

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If n is a power of q, then there exists a ∈ A such that

F (X,Y ) ≡ G(X,Y ) +p

π· aCn(X,Y ) (mod poly. deg. n+ 1)

[r]F (X) ≡ [r]G(X) +rn − rπ

· aXn (mod poly. deg. n+ 1) ∀r ∈ R.

Proof. Suppose F,G correspond to R-algebra homomorphisms f, g : ΛR → A. By restriction, we obtainR-module homomorphisms ⊕n−1

i=0 ΛiR → A which are ring homomorphisms wherever it is defined. Since

the coefficients of monomials in F and G and [ ]F and [ ]G agree up to polynomial degree n − 1, by thering homomorphism property of f and g, f and g agree on all of ⊕n−2

i=0 ΛiR. Using the ring homomorphism

property of f and g again, f and g must agree on I21 ∩ Λn−1

R . Thus, if we let h : ⊕n−1i=0 Λi

R → A be theR-module homomorphism created by taking the difference of f and g, h must be zero on I2

1 ∩Λn−1R . Hence,

we have an R-module homomorphism h : Λn−1R = Λn−1

R /(I21 ∩Λn−1

R )→ A. Using the notation in Proposition3.3.1, let a = h(η). What we are interested in are f(Γi,n−i) − g(Γi,n−i) = h(Γi,n−i) = h(Γi,n−i) andf(Θn,r) − g(Θn,r) = h(Θn,r) = h(Θn,r). Therefore, using the fact that h is an R-module homomorphism,we are done by Propositions 3.1.2 (for description of γn−1) and 3.3.1.

Just as Lazard’s Theorem led to Theorem 2.2.4 in the formal group case, the above result gives an easycriterion for when formal R-modules are isomorphic to the additive module:

Corollary 3.3.5. If a formal R-module F over A satisfies (F, [ ]F ) ≡ Ga (mod poly. deg. n) and the coef-ficient of Xn in [π]F (X) is in π · A, then there exists a power series α ≡ X (mod poly. deg. n) such thatthe image of F via the isomorphism α is congruent to the additive module Ga (mod poly. deg. n+ 1). If aformal R-module F over A is such that the coefficients of X, . . . ,Xn in [π]F (X) are in π ·A, then (F, [ ]F )is isomorphic to a formal R-module which is congruent to Ga modulo polynomial degree n+ 1 (from now on,we will write this as ‘F is isomorphic to Ga modulo polynomial degree n+1’). In particular, F is isomorphicto the additive module Ga if and only if all the coefficients of [π]F (X) are in π ·A.

Proof. Suppose (F, [ ]F ) ≡ Ga (mod poly. deg. n). We will use Proposition 3.3.4. If n is not a power of q,then F ≡ X + Y + a((X + Y )n − Xn − Y n) (mod poly. deg. n + 1) and [r]F (X) = rX + (rn − r) · aXn

(mod poly. deg. n + 1) for some a ∈ A, so if we let α = X − aXn in Example 2.1.10, we conclude thatα F α−1 ≡ X + Y (mod poly. deg. n + 1). If n is a power of q, then F ≡ X + Y + p

π · aCn(X,Y )(mod poly. deg. n+ 1) and [r]F (X) ≡ rX + rn−r

π · aXn (mod poly. deg. n+ 1). Since πn−ππ is a unit, the

assumption implies that a = π · a′ for some a′ ∈ A. Using α = X − a′Xn in Example 2.1.10, we obtainαF α−1 ≡ F (X,Y )−a′((X+Y )n−Xn−Y n) ≡ X+Y + p

π ·π ·a′Cn(X,Y )−a′((X+Y )n−Xn−Y n) ≡ X+Y

(mod poly. deg. n+ 1), and similarly α [r]F α−1(X) ≡ rX (mod poly. deg. n+ 1).We always have (F, [ ]F ) ≡ Ga (mod poly. deg. 2), so by above, we obtain α2 = X − a2X

2 such that theimage F2 of F via α2 is congruent to Ga modulo polynomial degree 3. [π]F2 = (X−a2X

2)[π]F (X−a2X2)−1

has coefficients of X, . . . ,Xn in π ·A, so we now apply the first part to F2. We can thus continue by induction,ending with αn = X − anX

n such that the image of Fn via αn is congruent to X + Y modulo polynomialdegree n+1. αn αn−1 · · · α2 is the desired isomorphism. The ‘if’ part of the last statement holds by thesame process, since the infinite composition · · · (X−anX

n) (X−an−1Xn−1) · · · (X−a2X

2) convergesin A[[X]]. The ‘only if’ part is clear, since if Ga

α−→ F is an isomorphism, then [π]F = α (π ·X) α−1.

In particular, we get a parallel of Corollary 2.2.5 (2): if image of π in A is zero, then a formal R-moduleF is isomorphic to the additive module if and only if [π]F (X) = 0.

3.3.1 A Special Case: A Is A Field Over R/(π)

We continue the discussion of formal R-modules for R the valuation ring of local fields, but now we willspecialize our R-algebra A and give more specific results. Some results such as Theorem 3.3.11 below willbe crucial in the deformation theory of Chapter 4.

23

The ‘specialized’ R-algebras that we ultimately want to study are separably closed fields A = E overR/(π), where the R-algebra map is defined by R→ R/(π) → E. In this case, a number called the height ofa formal module determines its isomorphism class (Proposition 3.3.9, Theorem 3.3.11). Formal R-modulesof height∞ are isomorphic to the additive formal module, and the endomorphism rings of formal R-modulesof finite height have a very nice property. But in proving these theorems for separably closed fields, we willneed to restrict our attention to some subfield which is not separably closed, so we will first develop sometheory for the case when E is just a field over R/(π). Thus for now, let us assume that E is a field overR/(π) = k, with the R-algebra map R→ R/(π) → E. In this setting, we have the following:

Proposition 3.3.6. If F and G are formal R-modules over E and α : F → G is a nonzero formal R-modulehomomorphism, then we can rewrite α(X) as β(Xqh

) with β′(0) 6= 0 for some h ≥ 0.

Proof. First, let us focus on the fact that α is a formal group homomorphism. If α′(0) 6= 0, then we are done,so let us assume that α′(0) = 0. By applying the ‘chain rule’ (which can be proven directly in this algebraicsetting) to G(α(X), α(Y )) = α(F (X,Y )), we obtain ∂G

∂X (α(X), α(Y )) · α′(X) = α′(F (X,Y )) · ∂F∂X (X,Y ).

Setting X = 0, we get

0 = α′(Y ) · ∂F∂X

(0, Y ) = α′(Y ) · (1 + higher terms)

as elements in E[[Y ]], so α′(Y ) = 0. Because any integer which is not divisible by p is a unit in E, it nowfollows that α(X) = α1(Xp). Because the Frobenius map E → E taking every element to its p-th power is aring homomorphism (not an R-algebra homomorphism), F (p) defined by raising every coefficient to its pthpower is a formal group (not in general a formal R-module). Moreover, we have

α1(F (p)(Xp, Y p)) = α1((F (X,Y ))p) = α(F (X,Y )) = G(α(X), α(Y )) = G(α1(Xp), α1(Y p)),

so α1 is a formal group homomorphism from F (p) to G, thus we can continue the process by induction. Sinceα 6= 0, we end at some point, that is, α = β(Xpl

) with β′(0) 6= 0 for some l.Now we use the fact that α is a formal R-module homomorphism. Note that α([r]F (X)) = [r]G(α(X))

implies β(([r]F (X))pl

) = [r]G(β(Xpl

)). Comparing Xpl

coefficients, we obtain rpl · β′(0) = r · β′(0) in E.Because E is a field and β′(0) 6= 0, we have rpl

= r in E for all r ∈ R. This happens if and only if Fq ⊆ Fpl ,i.e. if and only if pl is a power of q.

Using this proposition, we define

Definition 3.3.7. The height htα of a nonzero formal R-module homomorphism α between formal R-modules over E is the integer h such that α(X) = β(Xqh

) with β′(0) 6= 0. The height of the zero homomor-phism is defined to be ∞.

Next, we put together easy computational results about height of formal R-module homomorphisms.

Proposition 3.3.8. (1) ht(α1 α2) = ht(α1) + ht(α2).(2) ht(α) = 0 if and only if α is a formal R-module isomorphism.(3) ht(α1 +

Gα2) ≥ min(ht(α1),ht(α2)), for α1, α2 : F → G.

Proof. All the statements become trivial with the zero map, so we will only check these for nonzero maps.By definition, the height is logq of the first exponent which has a nonzero coefficient.

(1) (a1Xqh1 + · · · ) (a2X

qh2 + · · · ) = a1aqh1

2 Xqh1+h2 + · · · .(2) ht(α) = 0 iff the coefficient of X is nonzero (i.e. -invertible), so we are done by Proposition 2.1.6.(3) G(α1(X), α2(X)) = α1(X) + α2(X) + higher order terms.

Given a formal R-module F over E, every [r]F is a formal R-module endomorphism. For a unit u ∈ R,Proposition 3.3.8 (2) shows that ht([u]F (X)) = 0, so by (1), we know ht([πu]F (X)) = ht([π]F (X)). Thus,we can now define the height ht(F ) of a formal R-module to be ht([π]F (X)).

As an example of a formal R-module with finite height, consider Gm as a Zp-module over Fp. Then wecan prove by induction that [p](X) = Xp, so it has height 1.

24

One property of the height of a formal R-module is immediate: α([π]F (X)) = [π]G(α(X)) together withProposition 3.3.8 (1) shows ht(α) + ht(F ) = ht(G) + ht(α), so if α 6= 0, then ht(F ) = ht(G). In other words,there exists a nonzero formal R-module homomorphism only if the two formal R-modules are of the sameheight, and in particular, isomorphic formal R-modules have the same height. Using knowledge of universalformal R-modules we can prove the following:

Proposition 3.3.9. (1) Given h ≥ 0, we can construct a formal R-module over E of height h.(2) A formal R-module over E of height h is isomorphic to the additive module modulo polynomial degree

qh.(3) A formal R-module over E is isomorphic to the additive module if and only if the height is ∞.

Proof. (1) Because ΛR∼= R[Υ1, . . .] by Proposition 3.3.2, let F be the formal R-module over E corresponding

to the R-algebra homomorphism R[Υ1, . . .]→ E defined by sending Υqh−1 to 1 and everything else to zero.Since Γi,j and Θl,r with 1 ≤ i + j − 1, l − 1 ≤ qh − 2 are R-linear combinations of products of Υ’s ofdegree between 1 and qh − 2, it follows that the coefficients in (F, [ ]F ) of polynomial degree between 2 andqh − 1 are all zero. In other words, F ≡ Ga (mod poly. deg. qh), so by Proposition 3.3.4, there exists a ∈ Esuch that F (X,Y ) ≡ X + Y + p

π · aCqh(X,Y ) (mod poly. deg. qh + 1) and [r]F (X) ≡ r ·X + rqh−r

π · aXqh

(mod poly. deg. qh + 1). If a = 0, then these formula imply that every Γi,j and Θl,r up to and includingdegree qh− 1 are sent to 0 via f , so as a sum of products of these elements, Υqh−1 must have been sent to 0by f , contradiction. So a 6= 0, and [π]F (X) ≡ (πqh−1 − 1) · aXqh

(mod poly. deg. qh + 1) shows that F hasheight exactly h.

(2) We know that the coefficients of X, . . . ,Xqh−1 in the expression [π]F (X) are 0 ∈ (0) = π · E, so weare done by the second statement in Corollary 3.3.5.

(3) π = 0 in E shows ht(Ga) = ∞, so every formal R-module isomorphic to Ga also has height ∞.Conversely, if ht(F ) =∞, then [π]F (X) = 0 ∈ π · E, so by Corollary 3.3.5, F is isomorphic to Ga.

So we understand the formal R-modules over E of height ∞ very well. To get a similar nice result forthe possibilities for the finite height case, we will need to put more conditions on E, namely separableclosure. This special case is all we need in our usage in the deformation theory. Thus, for the rest of thesection, we will further assume that E is separably closed. Let us now fix a non-negative integer h, andwe will now analyze formal R-modules of height h over a separably closed field E, with the R-algebra mapR→ R/(π) → E.

Since E is separably closed, Fqh ⊆ E as the subset of elements satisfying xqh − x = 0. We now define thecentral player of the formal R-modules over E of height h:

Definition 3.3.10. A formal R-module F over E satisfying(1) [π]F (X) = Xqh

(2) F (X,Y ) ∈ Fqh [[X,Y ]] and [r]F (X) ∈ Fqh [[X]].(3) (F, [ ]F ) ≡ Ga (mod poly. deg. qh)

is said to be π-normal of height h.

Here is the analogue of Proposition 3.3.9 (3) in the finite height case:

Theorem 3.3.11. Every formal R-module over E of height h is isomorphic to a π-normal one, and givenany two π-normal formal R-modules over E of height h, there is an isomorphism between them which is≡ X (mod poly. deg. qh +1). In particular, all formal R-modules over E of height h are isomorphic to eachother.

Proof. Let n = qh. By height consideration, [π]F ≡ aXn (mod poly. deg. 2n) for some a ∈ E. Because ofseparable closure, there exists a1 ∈ E such that an−1

1 = a. Then the image F2 of F via the isomorphism α1 =a1X satisfies [π]F2(X) ≡ Xn (mod poly. deg. 2n). Now assume by induction that for a formal R-moduleFi, [π]Fi

(X) ≡ Xn (mod poly. deg. in) for some i ≥ 2. Let [π]Fi(X) ≡ Xn + biX

in (mod poly. deg. in+1)for some bi ∈ E. There exist ai ∈ E such that an

i − ai + bi = 0, and the image Fi+1 of Fi via theisomorphism (X + aiX

i)−1 has the property that [π]Fi+1(X) ≡ Xn (mod poly. deg. in + 1). By height

25

consideration, [π]Fi+1(X) ≡ Xn (mod poly. deg. (i + 1)n). Thus, the image F∞ of F via the isomorphismmade from the infinite composition · · ·(X+a4X

4)−1 (X+a3X3)−1 (X+a2X

2)−1 a1X has the propertythat [π]F∞(X) = Xn. So we may now assume that our original F has the property that [π]F (X) = Xn.But now, F ([π]F (X), [π]F (Y )) = [π]F (F (X,Y )) and [r]F [π]F (X) = [π]F [r]F (X) with coefficients incharacteristic p, imply that all the coefficients of F and [r]F (X) are in Fn. Therefore, F is actually a formalR-module over Fn. By Proposition 3.3.9 (2), there exists an -invertible α ∈ Fn[[X]] such that the image Gof F via the isomorphism α is congruent to the additive module modulo polynomial degree qh. If we letα−1(X) =

∑ciX

i, [π]G(X) = α [π]F (∑ciX

i) = α(∑

cni Xin)

= α α−1(Xn) = Xn. Therefore, we getthe desired isomorphism with a π-normal formal R-module over E of height h.

Now suppose that F and G are two π-normal formal R-modules over E of height h. Since F ≡ G(mod poly. deg. qh) and [π]F = [π]G, Proposition 3.3.4 shows that F ≡ G (mod poly. deg. qh+1). Fromhere, we can use the successive approximation, as in the proof of Corollary 3.3.5. That is, if we know F ≡ G(mod poly. deg. m) with m not a power of q, it is the same proof. If m is a power of q, [π]F = [π]G = Xqh

shows that F ≡ G (mod poly. deg. m+ 1). The important point is that by thinking of F and G as formalR-modules over Fqh , the isomorphism that we obtain by this process have coefficients in Fqh , so as before,the [π]F (X) = Xqh

property (thus normality) is preserved via isomorphisms.

The above theorem often enables us to reduce studying formal R-modules of height h to studying π-normalformal R-modules of height h, if we are over a separably closed field. We will frequently use this method inChapter 4. Moreover, we can prove the following about End(F ) when F has height h, which is an analogueof [8, page 72]. Although we will not need this in our discussion of deformation theory, it is a beautiful andimportant result so we include here. For this proof only, we will assume some knowledge of central divisionalgebra and Brauer groups in connection with local fields [23].

Theorem 3.3.12. If F is a π-normal formal R-module of height h, then EndR(F ) is isomorphic to theintegral closure of R in a central division algebra D of dimension h2 over K whose invariant is 1

h .

Proof. Let E be EndR(F ). In this proof only, given αi ∈ E , α1 + α2 and α1α2 indicate the addition andmultiplication in this (not necessarily commutative) ring, unless otherwise specified (so we are not using theusual addition and multiplication of power series rings). We have a ring homomorphism R → E defined byr 7→ [r]F (X), and we will often write the image of r in E as simply r. By definition, rα = αr for all α ∈ E

and r ∈ R, and in particular, (α(X))qh

= πα = απ = α(Xqh

) over a field of characteristic p shows that allcoefficients of α lie in Fqh . The same equation also shows that πα = 0 if and only if α = 0.

If htα ≥ nh, then Proposition 3.3.6 enables us to write α(X) = β(Xqnh

) = βπn = πnβ, and this β is anelement of E . So we conclude πnE = α ∈ E : htα ≥ nh. Thus,

⋂πnE = (0). Using this and Proposition

3.3.8, the infinite sum∑πnαn is a well-defined power series with coefficients in Fqh , and checking that this

is an element of E reduces to checking for various finite sums. Hence, E is a π-adically complete R-module.

Lemma 3.3.13. The map ϕ : E → (Fqh)⊕h sending α to the coefficients of X,Xq, Xq2, . . . , Xqh−1

is an R-module homomorphism (R-module action is given via R → R/(π) → Fqh), inducing anisomorphism E /πE → (Fqh)⊕h. From this, we obtain an R-module isomorphism E → R⊕h2

.

Proof. F ≡ Ga (mod poly. deg. qh) shows that ϕ is an R-module homomorphism. Proposition3.3.6 shows that any α ∈ E must start from a monomial with exponent a power of q, so ϕ(α) = 0⇐⇒ htα ≥ h ⇐⇒ α ∈ πE . For surjectivity of ϕ, by the additivity it suffices to check theexistence of α such that ϕ(α) = (λ0, . . . , λh−1) ∈ (Fqh)⊕h for λ0 6= 0. Now, β = λ0X +λ1X

q + · · ·+ λh−1Xqh−1

is -invertible, so as in Example 2.1.10, β is an R-module isomorphismfrom F to G, where G is the image of F under β. As before, [π]G(X) = Xqh

, and G ≡ Ga

(mod poly. deg. qh) because of (X + Y )q = Xq + Y q. In other words, G is π-normal of heighth, and Theorem 3.3.11 (2) gives us a formal R-module isomorphism β2 : G → F with β2 ≡ X(mod poly. deg. qh + 1). Composition β2 β gives the desired element.

26

For the last statement, note that as R-modules, (Fqh)⊕h ∼= (Fq)⊕h2. For i = 1, . . . , h2, let αi ∈ E

be a pull-back via ϕ of a generator of the i-th component of (Fq)⊕h2. Given α ∈ E , there exist

r0,i ∈ R such that α−∑r0,iαi = πβ1 for some β1 ∈ E . We can then start from β1 and find r1,i

and β2 with the similar property, and continue by induction. Note that

α−∑

i

(∑n

πnrn,i

)αi ∈

∞⋂i=0

πnE = (0),

because πnα′n +πn+1α′n+1 + · · · ∈ πnE by the completeness of E for α′i ∈ E . By the completenessof R, each

∑n π

nrn,i converges, so we now know that αi’s generate E . Suppose∑riαi =

0. Modding out by πE , we see that each ri = πr1,i. Thus π(∑r1,iαi) = 0, implying that∑

r1,iαi = 0. Continuing this process, we conclude that each ri = 0.

Let D = K ⊗R E , a K-algebra of dimension h2. By transferring all high powers of π to the side of E , wesee that every element of D is of the form πn⊗α for some n ∈ Z and α ∈ E . Because E ∼= R⊕h2

, πn⊗α = 0implies α = 0 for any n. Thus, the fact that E has no zero-divisor (checked by height consideration) impliesthat D has no zero-divisors. Hence, by finite-dimensionality, D is a division algebra over K. Note also thatby definition, K is contained in the center of D.

By freeness of E over R, E → K⊗RE . Moreover, since R → K → K⊗RE agrees with R→ E → K⊗RE ,R → E is injective. Identifying E and R with their images in D, we will now prove that E is the integralclosure of R in D. By defining ht(πn ⊗ α) = hn + ht(α), we get a well-defined function on D, and bytransferring the powers of π across ⊗ and using Proposition 3.3.8, we can show that this is a valuation onD. If ht(π−n⊗α) ≥ 0, then as before, α = β(Xqnh

) = πnβ for some β ∈ E , and π−n⊗α = 1⊗β ∈ E . Thus,E is exactly the subset of D where ht is non-negative. Now, for d ∈ D, K(d)/K must be a (commutative)finite extension of local fields. Since ht(r) = h · v(r) for all r ∈ R where v is the normalized π-adic valuationon K, and by valuation theory [23, Theorem 8.17], ht(d) ≥ 0 if and only if d is integral over R. Thus, E isthe integral closure of R inside D.

To prove that the center L of D is K, we will first need the following computational lemma:

Lemma 3.3.14. For α, β ∈ E , if ht(α) = n then ht(αβ − βqn

α) ≥ n+ 1.

Proof. If ht(β) > 0, then this follows by Proposition 3.3.8. Otherwise, since both αβ and βqn

αstart with the same Xqn

monomial, αβ − βqn

α ≡ 0 (mod poly. deg. qn + 1). By Proposition3.3.6, we are done.

Let M be the maximal unramified extension of K inside L. Let β ∈ OM , the valuation ring of M . Since βis integral over R, β ∈ E . Let α be a height 1 element of E , which exists by Lemma 3.3.13. By Lemma 3.3.14and centrality, ht(βα−βqα) ≥ 2, so multiplying on the right by α−1 in D, we conclude that ht(β−βq) ≥ 1.Thus β ≡ βq (mod mOM

) for all β ∈ OM . So by unramifiedness, M must be K. Thus, L/K is totallyramified, so let us write OL = OK(Π) for some uniformizer Π in L. Suppose on the contrary that L 6= K sothat ht(Π) < ht(π) = h by Proposition 3.3.8.

In order to get a contradiction from this, we need to first show that D contains a copy of Kh, an unramifiedextension of K of degree h (this is unique if we work inside an algebraic closure, but inside D we cannot dothis). Letting mE be the (two-sided) ideal of E consisting of elements of height at least 1, we have an R-algebra isomorphism E /mE → Fqh defined by sending α to its derivative (use Lemma 3.3.13 for surjectivity).Lifting a primitive element of Fqh/Fq via this isomorphism to d ∈ E , we know that K(d)/K is a finiteextension of local fields. Since mOK(d) = OK(d) ∩mE , the residue field of K(d) is Fqh . By [23, Theorem 8.26],K(d)/K contains an unramified extension Kh of degree h over K, so in particular, D does contain a copy ofKh.

So now let β ∈ OKh. Then similarly as before, we have ht(βΠ − βqht Π

Π) ≥ ht(Π) + 1 by Lemma 3.3.14and centrality of Π, so by multiplying on the left by Π−1 in D, we obtain ht(β − βqht Π

) ≥ 1. This implies

27

that the residue field of OKhhas at most qht Π elements, contradiction. Thus, the center L of D must be K,

and so we have finally proved that E is the integral closure of R inside the central division algebra D overK of dimension h2.

Before proving the last statement, note that we have a group homomorphism ϕ : D∗ −→ Gal((E /mE )/k

)by letting ϕ(d) be the map x 7→ dxd−1. Since E /mE

∼= Fqh is commutative, any element of E ∗ (i.e. height0 elements) is sent to the identity map by ϕ. Moreover, π is in K which is the center of D, so E ∗ × 〈π〉is in the kernel of ϕ. On the other hand, given any element of Gal(Fqh/Fq) there exists a (unique) lift toσ ∈ Gal(Kh/K). By Noether-Skolem Theorem, σ : Kh −→ Kh → D is given by a conjugation by somed ∈ D∗. By construction, ϕ(d) is the element of Gal(Fqh/Fq) that we started out with, so we now haveproved that ϕ is surjective. Since D∗/

(E ∗ × 〈π〉

)is just Z/(ht(π)Z), counting shows that the kernel of ϕ is

exactly E ∗ × 〈π〉. Therefore, d1, d2 ∈ D∗ induce the same conjugation map on Gal((E /mE )/k

)if and only

if ht(d1) − ht(d2) is a multiple of h = ht(π). In other words, there exists a unique coset mod Z formed byv(d)v(π) , where v is a valuation on D and d ∈ D∗ is an element satisfying

dd1d−1 ≡ (d1)q (mod mD) for all d1 ∈ OD. (3.5)

By the same argument as above, there exists some d such that x 7→ dxd−1 defines the unique elementof Gal(Kh/K) which reduces to the Frobenius map σ ∈ Gal(Fqh/Fq). Such a d satisfies (3.5). In this case,dh commutes with every element of Kh, so dh ∈ Kh. Moreover, σ(dh) = ddhd−1 = dh shows that dh ∈ K.Thus D is isomorphic to the cyclic algebra [Kh/K, σ, d

h], by dimension comparison and simplicity of cyclicalgebras. Now the invariant of such a cyclic algebra over a local field is defined by v(dh)/h where v is thenormalized valuation of K ([23]). Therefore, the previous paragraph shows that the invariant of D is theunique coset mod Z formed by v(d)

v(π) with d satisfying (3.5).By Lemma 3.3.14, any height 1 element d ∈ D satisfies (3.5), so the invariant must be 1

h . We are finallydone with the proof.

28

Chapter 4

Deformation Theory

We are finally ready to begin discussing deformation theory. Before giving an overview of the chapter, wewill set forth the notations that are used throughout. As before, K is a local field, R the valuation ring,π a uniformizer, k = R/(π) the residue field of order q, a power of p. After fixing an algebraic closure ofK, we can speak of the maximal unramified extension Knr. The completion of Knr with π-adic topologyis denoted Knr with valuation ring Rnr. Here, π still generates the maximal ideal, and Rnr/(π) ∼= k, thealgebraic closure of k. Let C be the category whose objects are complete noetherian local Rnr-algebra A withmaximal ideal mA such that Rnr/(π) ∼= A/mA via the algebra map : Rnr → A and whose morphisms arelocal Rnr-algebra homomorphisms. We will often implicitly identify Rnr/(π) with A/mA via : for instance,a formal R-module over one of them is automatically considered a formal R-module over the other viapush-forward by (or its inverse).

A deformation over A ∈ C of a formal R-module G over k refers to a formal R-module F over A whichreduces modulo mA to G. We identify two deformations F1 and F2 of G over A whenever there is anisomorphism of formal R-modules which reduces modulo mA to the identity. The first section of the chapterstudies such deformation classes. More specifically, we are interested in knowing the universal deformation,i.e. the universal object of the functor which associates A ∈ C with the set of deformation classes over A. Itturns out that the universal deformation ring is a power series ring over Rnr with the number of variablesone less than the height of G, with a specific description of the universal deformation F univ (at least we knowwhat [π]Funiv(X) looks like). To prove this theorem, we will need to study a connection between cohomologyand formal R-modules.

The second section introduces the notion of level structures to formal R-modules. This definition of levelstructures was a great insight of Drinfel’d’s. For a formal R-module F over A, the structure of level n (forn ≥ 1) is a homomorphism ϕ : (π−nR/R)h → F (mA) such that

Pϕ(X) =∏

s∈(π−1R/R)h

(X − ϕ(s))

∣∣∣∣∣∣ [π]F (X),

where h is the height of F . The main theorem of the section determines the universal deformation withstructure of level n. The universal deformation ring Dh,n is very nice; it is a regular local ring of dimensionh with a system of local parameters given by the images of the standard basis of (π−nR/R)h under theuniversal level structure, and the natural map Dh,n → Dh,n+1 is finite and flat. The proof uses the factthat we know the universal deformation ring from Section 4.1. At the end, we will use some of these niceproperties of Dh,n to construct a level structure on a ‘quotient’ formal R-module.

In the final section, we will be using the more general notion of formal R-modules as discussed in Section2.3. We define a divisible formal R-module of type (h, j) to be a formal R-module F whose connectedcomponent F 0 is a CFS formal R-module (i.e. those satisfying Definition 2.1.8) with ht(F 0) = h and whoseetale component is SpfA(A(K/R)j

). We will then define the notion of level structures for each n in this setting.Unlike the deformations of the previous section, there is more than one choice of level structure on G, so we

29

will discuss deformations of (G,φ), where φ is a fixed level structure on G. The universal deformation ringEh,n,j of (G,φ) with structure of level n turns out to be a regular local ring again (in fact, it is simply thepower series ring over Dh,n with j variables). So Eh,n,j → Eh,n+1,j is finite and flat.

The deformation theory of Section 4.3 is what Drinfel’d was really after in [5], as the ‘local’ analysis ofDrinfel’d modules results in deformations of divisible formal R-modules. On the other hand, both divisibleformalR-modules and level structures on them are defined using the corresponding notions for the CFS formalR-modules, so the deformation theory of divisible formal R-modules is heavily based on the deformationtheory discussed in the first two sections of this chapter.

The descriptions of the universal formal R-module that we obtained in the last chapter will be usedthroughout this chapter. Moreover, we will frequently use the fact that every formal R-module of a givenfinite height is isomorphic to a π-normal one over a separably closed field.

4.1 Deformations

Let us fix A ∈ C and let m = mA. Geometrically speaking, a deformation of a geometric object X0 isa ‘continuous family’ of geometric objects including X0. In the language of modern algebraic geometry,‘continuous family’ corresponds to a flat morphism X → T for some scheme T and the notion of X0 beinga part of this family corresponds to the statement that X0 is isomorphic to a fiber of this morphism atsome point t ∈ T . Recall that we can always view a formal R-module F (in the sense of Definition 2.1.8) asSpfA(A[[X]]) by Proposition 2.3.2. Since the structure map SpfA(A[[X]]) → SpfA(A) is topologically flat, itmakes sense to say that the formal R-module F is a deformation of a formal R-module G over A/m if G isisomorphic to the unique closed fiber, i.e. G is isomorphic to F (F reduced modulo m). This motivates usto define deformation of a formal R-module G over Rnr/(π) as follows:

Definition 4.1.1. The pair (F, ι) is said to be a deformation of G over A if F is a formal R-module F over Aand ι is an isomorphism over Rnr/(π) from F modulo m (identified via ) to G. We define two deformations(Fi, ιi) to be isomorphic if there exists a formal R-module isomorphism α : F1 → F2 over A such that

F1 mod m

ι1$$JJJJJJJJJJα // F2 mod m

ι2zztttttttttt

G

commutes.

We will also use the following:

Definition 4.1.2. F is said to be a deformation of G over A if F is a formal R-module over A such thatreducing modulo m (and identifying via ) results in G. We say two deformations F1 and F2 over A areisomorphic if there exists an isomorphism α : F1 → F2 of formal R-modules which reduces to the identitymap modulo m.

We will classify deformations up to isomorphisms, and this concept is the same no matter which definitionwe use. Given a deformation F in the sense of 4.1.2, (F, id) is a deformation in the sense of 4.1.1. Given adeformation (F, ι) in the sense of 4.1.1, we can lift ι ∈

(Rnr/(π)

)[[X]] to a -invertible element ι of A[[X]], and

then letting F be the image of F via ι, (F , id) is isomorphic to (F, ι) as deformations. These processes sendisomorphic deformations to isomorphic deformations, and we now see that the two definitions of deformationsup to isomorphism indeed agree.

Both definitions have advantages. Definition 4.1.2 treats fewer things as deformations, and this is anadvantage in proving a general fact about deformations but it is a disadvantage whenever we want to showsomething is a deformation. So we will use both of these definitions in this chapter, using more convenientone for each occasion. We always deal with deformations up to isomorphism, so this should not cause anyconfusion.

30

We already have high hopes for classifying deformations in the present setting, because formal R-modulesover Rnr/(π) are well-understood by Proposition 3.3.9 and Theorem 3.3.11, and deformations are supposedto be in some sense ‘close’ to these well-understood ones. Furthermore, the following makes this situationeven better:

Proposition 4.1.3. Suppose that a formal R-module F over A reduces to F over A/m which has a finiteheight, and that F ′ is a formal R-module over A. Then the reduction modulo m map HomR(F, F ′) −→HomR(F , F ′) is injective.

Proof. Because HomR(F, F ′) −→ HomR(F , F ′) is an abelian group homomorphism, we need to prove α ∈HomR(F, F ′) with α = 0 implies α = 0. If m = 0, then we are done. Otherwise, suppose on the contrarythat α 6= 0, so there exists i ≥ 1 such that α ≡ 0 (mod mi) and α 6≡ 0 (mod mi+1). We then obtain anonzero formal R-module homomorphism α : F → F ′ over A = A/mi+1 whose coefficients all lie inside mi.As (mi)2 = 0, α(F (X,Y )) = F ′(α(X), α(Y )) = α(X) + α(Y ) and α([r] eF (X)) = [r]fF ′(α(X)) = r · α(X).Let a ∈ A be a coefficient of α which is nonzero. Since mi ⊆ A is an (A/m)-vector space, there existsan (A/m)-linear map f : mi → A/m which is nonzero at a. Letting β(X) denote f∗(α)(X), we see thatβ ∈ (A/m)[[X]] is a nonzero formal R-module homomorphism from F modulo m to the additive module Ga

over the field A/m, contradicting Proposition 3.3.8.

In particular, if F1 and F2 are isomorphic deformations of a finite height G, then there exists a uniqueisomorphism of deformations between F1 and F2.

Our goal is the determination of the universal object, i.e. the universal deformation up to isomorphism.Before moving on, however, we will first develop some cohomology theory for formal R-modules to clarifyproofs later.

4.1.1 Cohomology

In this subsection, we will prove a relationship between ‘cohomology’ and formal R-modules which differfrom a fixed one by a power of the maximal ideal. Here, to simplify notation, we will assume that A ∈ Chas the property that mi+1 = 0 for some i ≥ 0. We fix a formal R-module G over Rnr/(π) of finite height h.

Definition 4.1.4. A pair (Ξ, ξ) is said to be a 2-cocycle of G with coefficients in mi if Ξ(X,Y ) ∈ (X,Y ) ⊆mi[[X,Y ]] and ξr(X) ∈ (X) ⊆ mi[[X]] for each r ∈ R satisfying the following five equations:

Ξ(Y, Z) + Ξ(X,G(Y, Z)) = Ξ(X,Y ) + Ξ(G(X,Y ), Z)Ξ(X,Y ) = Ξ(Y,X)

ξr(X) + ξr(Y ) + Ξ([r]G(X), [r]G(Y )) = r · Ξ(X,Y ) + ξr(G(X,Y ))ξr1(X) + ξr2(X) + Ξ([r1]G(X), [r2]G(X)) = ξr1+r2(X)

r1 · ξr2(X) + ξr1([r2]G(X)) = ξr1r2(X).

These equations make sense, since the different lifts of coefficients of G to A differ by m, which becomesnegligible because the coefficients of Ξ and ξ are in mi. We denote the set of all 2-cocycles by Z2(G,mi).We say a pair (Ξ, ξ) is a 2-coboundary of G with coefficients in mi if there exists ϕ ∈ mi[[X]] such that

Ξ(X,Y ) = ϕ(G(X,Y ))− ϕ(X)− ϕ(Y ) and ξr(X) = ϕ([r]G(X))− r · ϕ(X).

These make sense for the same reasoning, and we denote the set of all 2-coboundaries d(ϕ) by B2(G,mi).

Note that Z2(G,mi) and B2(G,mi) form A/m-vector spaces. Also, by direct computation using formalR-module conditions on G, one can check that every 2-coboundary is a 2-cocycle. Thus, we can define the2-cohomology H2(G,mi) to be the A/m-vector space Z2(G,mi)/B2(G,mi).

Recall that in the definition of the usual cohomology of a group G with coefficients in a G-module M ,Ξ : G × G → M is a 2-cocycle if it satisfies g0 · Ξ(g1, g2) + Ξ(g0, g1g2) = Ξ(g0g1, g2) + Ξ(g0, g1) and Ξ is a2-coboundary if there exists ϕ : G → M such that Ξ(g0, g1) = ϕ(g0g1) − g0 · ϕ(g1) − ϕ(g0). Thus, the first

31

equations for the definition of both cocycle and coboundary really are cohomological conditions, when weview the formal module G as giving the ‘group structure’ and take the trivial action of the variables X,Y, Z.

We get the equations of Definition 4.1.4 by ‘linear approximation’ of differences between formal R-modules.We will see this more specifically in the proof of the following proposition, which links cohomology withcoefficients in mi with isomorphic deformations up to mi. By (∂1F )(X,Y ), we mean the formal partialderivative of F (X,Y ) with respect to X, i.e. given F (X,Y ) =

∑aijX

iY j , (∂1F )(X,Y ) =∑iaijX

i−1Y j .Note that (∂1F )(0, Y ) starts from the constant term 1, so it is invertible with the usual multiplication inA[[Y ]] (not -invertibility here). We have an analogue of [19, Proposition 2.4]:

Proposition 4.1.5. Let i ≥ 1. Suppose F1 and F2 are two deformations of G over A, and suppose α ∈ A[[X]]is such that α ∈ (A/m)[[X]] is X, α(F1(X,Y )) ≡ F2(α(X), α(Y )) (mod mi), and α([r]F1(X)) ≡ [r]F2(α(X))(mod mi) for all r ∈ R. Using usual multiplication in power series rings,

Ξ(X,Y ) =α(F1(X,Y ))− F2(α(X), α(Y ))

(∂1G)(0, G(X,Y ))

ξr(X) =α([r]F1(X))− [r]F2(α(X))

(∂1G)(0, G(X,Y ))

defines Ξ ∈ mi[[X,Y ]] and ξr ∈ mi[[X]], and further (Ξ, ξ) is a 2-cocycle. Moreover, (Ξ, ξ) ∈ B2(G,mi) if andonly if there exists β ∈ A[[X]] such that β ≡ α (mod mi) and β(F1(X,Y )) = F2(β(X), β(Y )) in A[[X,Y ]].

Proof. Since the numerators of the expressions for Ξ and ξ are in mi, the other coefficients only matter modulom, so the expressions for Ξ and ξ make sense and create elements in mi[[X,Y ]] and mi[[X]], respectively. Forany formal R-module F , by differentiating F (X,F (Y, Z)) = F (F (X,Y ), Z) with respect to X and thenevaluating at X = 0, we obtain (∂1F )(0, F (Y, Z)) = (∂1F )(Y, Z) · (∂1F )(0, Z) by the ‘chain rule.’ Also,F (X + Y, Z) ≡ F (X,Z) + (∂1F )(X,Z) · Y (mod Y 2). Using these identities and noting that product ofelements in mi are 0 (so we are ‘linear approximating’) and that coefficients matter only modulo m wheneveran mi-valued coefficient is present, we have

α(F1(F1(X,Y ), Z)) = F2

(α(F1(X,Y )), α(Z)

)+ Ξ(F1(X,Y ), Z) · (∂1G)

(0, G(F1(X,Y ), Z)

)= F2

(F2(α(X), α(Y )) + Ξ(X,Y ) · (∂1G)(0, G(X,Y )), α(Z)

)+ Ξ(G(X,Y ), Z) · (∂1G)

(0, G(G(X,Y ), Z)

)= F2

(F2

(α(X), α(Y )

), α(Z)

)+ (∂1G)

(G(X,Y ), Z

)· Ξ(X,Y ) · (∂1G)(0, G(X,Y ))

+ Ξ(G(X,Y ), Z) · (∂1G)(0, G(G(X,Y ), Z)

)= G

(G(α(X), α(Y )), α(Z)

)+ (∂1G)

(0, G(G(X,Y ), Z)

)·(Ξ(X,Y ) + Ξ(G(X,Y ), Z)

).

By symmetry, we get the first identity for a cocycle. Other relations can be similarly proved.If (Ξ, ξ) = d(ϕ) for some ϕ ∈ mi[[X]], we can define β(X) = α(X) + (∂1G)(0, X) · ϕ(X); conversely given

β with β ≡ α (mod mi), we can solve for ϕ ∈ mi[[X]] satisfying the same equation. For these β and ϕ, onecan show by a similar technique that

(∂1G)(0, G(X,Y )

)· Ξ(X,Y ) = ψ(F1(X,Y ))− F2(ψ(X), ψ(Y )) + (∂1G)(0, X) · (ϕ(X) + ϕ(Y )− ϕ(G(X,Y )))

(∂1G)(0, [r]G(X)

)· ξr(X) = ψ([r]F1(X))− [r]F2(ψ(X)) + ϕ([r]G(X))− r · ϕ(X).

Thus, the last statement is now proved.

Since cohomology can be viewed as the ‘first approximation’ of the difference between formal R-modules,we can use the ‘successive approximation’ results from Section 3.3:

Proposition 4.1.6. Let i ≥ 0. Suppose (Ξ, ξ) ∈ Z2(G,mi) such that (Ξ, ξ) ≡ 0 (mod poly. deg. n). Thenthere exists a ∈ A such that

(Ξ, ξ) ≡

(a((X + Y )n −Xn − Y n

), (rn − r) · aXn

)n is not a power of q(

pπ · aCn(X,Y ), rn−r

π · aXn)

n is a power of q(mod poly. deg. n+ 1).

32

Proof. Fix a deformation F over A of G, and define H(X,Y ) = F (Ξ(X,Y ), F (X,Y )) and [r]H(X) =F (ξr(X), [r]F (X)). We can check that the cocycle conditions are exactly those necessary for (H, [ ]H) to bea formal R-module germ of degree n+1 (by the similar arguments as the beginning of the proof of Theorem2.2.1). But Corollary 3.3.3 tells us that H arises from a formal R-module H over A, so Proposition 3.3.4and (H, [r] eH) ≡ (H, [r]H) ≡ (F, [r]F ) + (Ξ, ξr) (mod poly. deg. n+ 1) give us the desired result.

4.1.2 Universal Deformation

Theorem 4.1.7. Suppose G is a formal R-module over Rnr/(π) of height h. The functor from C to Setdefined by A the set of deformations of G over A up to isomorphism is represented by Rnr[[i1, . . . ,ih−1]],the power series ring in h− 1 variables. If G is π-normal, the universal deformation F univ can be chosen tohave the property that the coefficient of Xqj

in [π]Funiv(X) is equal to (πqj−1 − 1) · ij for all 1 ≤ j ≤ h− 1.

Proof. By Theorem 3.3.11, there exists a formal R-module isomorphism β : G → Gnor over Rnr/(π) whereGnor is π-normal of height h, so given a deformation (F, ι) of G, (F, βι) is a deformation of Gnor, preserveingisomorphism classes. Therefore, once we prove that (F univ, ιuniv) defined over Rnr[[i1, . . . ,ih−1]] is a universalobject for the above functor for Gnor, (F univ, β−1 ιuniv) is a universal object for the functor for G. Thus, wecan assume that G is normal, so the corresponding R-algebra homomorphism g : ΛR = R[Υ1, . . .]→ Rnr/(π)(obtained from Theorem 3.3.2) sends Υi 7→ 0 for all i < qh − 1. For the rest of the proof, we will be dealingwith deformations in the sense of Definition 4.1.2.

Let us define the R-algebra homomorphism funiv : R[Υ1, . . . , ]→ Rnr[[i1, . . . ,ih−1]] by

funiv(Υqi−1) = ii 1 ≤ i ≤ h− 1

funiv(Υj) = 0 1 ≤ j < qh − 1 and j + 1 not a power of q

funiv(Υj) = a lift of g(Υj) to Rnr otherwise.

Let F univ be the corresponding formal R-module over Rnr[[i1, . . . ,ih−1]]. Since ΛR is the universal formal R-module, F univ is a deformation of G by construction. Also, note that by chasing constructions in Propositions3.1.2, 3.3.1, and 3.3.2, we can choose the universal formalR-module so that Υqj−1 is sent to (πqj−1−1)−1Θqj ,π

in ΛR. In other words, in this case, we know that the coefficient of Xqj

in [π]Funiv(X) is (πqj−1 − 1)ij .We will now show that F univ is a universal object. In other words, given a deformation F of G over A,

we will show the existence and uniqueness of a local Rnr-algebra map f : Rnr[[i1, . . . ,ih−1]]→ A such thatF is isomorphic to f∗(F univ) as deformations.

Suppose that for each i ≥ 1, we have proven

There exists a unique local Rnr-algebra map f (i) : Rnr[[i1, . . . ,ih−1]]→ A/mi such that (4.1)

f(i)∗ (F univ) ∼= F as deformations of G over A/mi.

By Proposition 4.1.3, there exists a unique isomorphism α(i) : F → f(i)∗ (F univ) of deformations over A/mi for

each i. Reducing α(i)(F (X,Y )) = f(i)∗ (F univ)(α(i)(X), α(i)(Y )) mod mi−1 and using uniqueness of f (i−1) and

α(i−1), we see that f (i) and α(i) are compatible with the inverse system A/mi. Thus, by completeness ofA, we obtain a unique local Rnr-algebra map f : Rnr[[i1, . . . ,ih−1]]→ A and α ∈ A[[X]] such that f∗(F univ)is isomorphic to F via α, finishing the proof.

We will prove (4.1) by induction on i. When i = 1, the only way to get f (i) to be a homomorphism isby sending all ij to 0. Since this is same as reducing modulo the maximal ideal, this f (i) works with theidentity isomorphism.

So suppose that we have proved (4.1) until i. For the rest of the proof, it is helpful think of F univ(X,Y )and [r]Funiv(X) as F univ(i1, . . . ,ih−1)(X,Y ) and [r]Funiv(i1, . . . ,ih−1)(X). For the i + 1 step of (4.1),we may assume mi+1 = 0 and prove existence and uniqueness of m(i+1) = (m(i+1)

1 , . . . ,m(i+1)h−1 ) ∈ m⊕(h−1)

such that the formal R-module(F univ(m(i+1))(X,Y ), [r]Funiv(m(i+1))(X)

)is isomorphic to F , assuming by

induction that there exists m(i) = (m(i)1 , . . . ,m

(i)h−1) ∈ m⊕(h−1) (defined by an arbitrary lift of the solution

33

from step i) such that the above isomorphism holds modulo mi with m(i) unique modulo mi. By uniquenessat step i, what we need to show is the existence and the uniqueness of δ = (δ1, . . . , δh−1) ∈ (mi)⊕(h−1) suchthat the formal R-module

(F univ(m(i) + δ)(X,Y ), [r]Funiv(m(i) + δ)(X)

)is isomorphic to F .

Suppose α(i) ∈ A[[X]] is a lift of the isomorphism obtained in the i-th step, so that we have α(i)(F (X,Y )) ≡F univ(m(i))(α(i)(X), α(i)(Y )) (mod mi) and similar statements concerning each [r](X). These equations stillhold when we use F univ(m(i) +δ)(α(i)(X), α(i)(Y )) on the RHS, so by Proposition 4.1.5, we obtain a cocycle(Ξ(δ), ξ(δ)) by

Ξ(δ)(X,Y ) =α(i)(F (X,Y ))− F univ(m(i) + δ)(α(i)(X), α(i)(Y ))

(∂1G)(0, G(X,Y ))

ξ(δ)(X) =α(i)([r]F (X))− [r]Funiv(m(i) + δ)(α(i)(X))

(∂1G)(0, [r]G(X)).

Since (mi)2 = 0, linear approximation is exact, so

F univ(m(i) + δ)(α(i)(X), α(i)(Y )) = F univ(m(i))(α(i)(X), α(i)(Y )) +∑

j

∂F univ

∂ij(m(i))(α(i)(X), α(i)(Y ))δj

[r]Funiv(m(i) + δ)(α(i)(X)) = [r]Funiv(m(i))(α(i)(X)) +∑

j

∂[r]Funiv

∂ij(m(i))(α(i)(X))δj .

δj ∈ mi implies that in the above equation, we need to look at other coefficients inside the sum modulo m,so combining with α(i) = X and m =

−→0 modulo m (we know this from induction), we conclude

Ξ(0)(X,Y )− Ξ(δ)(X,Y ) =∑

j

∂Funiv

∂ij(0)(X,Y )

(∂1G)(0, G(X,Y ))δj (4.2)

ξ(0)(X)− ξ(δ)(X) =∑

j

∂[r]Funiv

∂ij(0)(X)

(∂1G)(0, [r]G(X))δj .

Claim.

(Ξj , ξj) =

∂Funiv

∂ij(0)(X,Y )

(∂1G)(0, G(X,Y )),

∂[r]Funiv

∂ij(0)(X)

(∂1G)(0, [r]G(X))

is a cocycle with coefficients in Rnr/(π) (so i = 0 case of Section 4.1.1) for each j = 1, . . . , h− 1,and they form a basis of the cohomology.

Proof. Let us apply Proposition 4.1.5 with k[ε]/ε2 as the ring, i = 1, identity as the isomorphismmod the maximal, and F univ(0)(X,Y ) and Fj = F univ(0, . . . , 0, ε, 0, . . . , 0)(X,Y ) with ε in thej-th spot as formal R-modules (without the bar, these formal modules are defined over Rnr bydefinition, so no i terms exist even before modding out). Since ε2 = 0, the linear approximationis precise, so

Fj = F univ(0)(X,Y ) +∂F univ

∂ij(0)(X,Y )ε

and similarly for [r], so by Proposition 4.1.5, ε(Ξj , ξj) is a cocycle with coefficients in k[ε]/ε2.Therefore, we see that (Ξj , ξj) is a cocycle over k. As noted above, Υqj−1 is sent to (πqj−1−1)−1

times the coefficient of Xqj

in [π]Funiv(X). So in this case,

∂[π]Funiv

∂ij(0)(X) ≡ (πqj−1 − 1)−1Xqj

6≡ 0 (mod poly. deg. qj + 1).

34

In particular, by Proposition 4.1.6, (Ξj , ξj) ≡ ( pπ · a

′jCn(X,Y ), rn−r

π · a′jXn) with a′j invertible inRnr/(π).

Now we are ready to prove that these form a basis of the cohomology. If∑λj(Ξj , ξj) = 0,

then using ∂[π]Funiv

∂ij(0)(X) ≡ 0 (mod poly. deg. qj) and 6≡ 0 (mod poly. deg. qj + 1) for each

j, we get λj = 0 by induction. For span, let us prove by induction that a cocycle (Ξ, ξ) ≡∑λj(Ξj , ξj) + d(ϕn−1) (mod poly. deg. n). The n = 1 case is automatic, so suppose we have

done up to n. If n is not a power of q, then by Proposition 4.1.6,

(Ξ, ξ) ≡∑

λj(Ξj , ξ(j)) + d(ϕn−1) +

(a · ((X + Y )n −Xn − Y n), a · (rn − r)Xn

)=∑

λj(Ξj , ξj) + d(ϕn−1 +Xn) (mod poly. deg. n+ 1).

If n = q1, . . . , qh−1, then Proposition 4.1.6 and invertibility of a′j show what we need. Supposenow n = qi for i ≥ h. We can write G(X,Y ) ≡ X+Y + p

π ·a′′Cqh(X,Y ) (mod poly. deg. qh +1)

and [r]G(X) ≡ rX+ rqh−r

π ·a′′Xqh

(mod poly. deg. qh+1) by Proposition 3.3.4. Working modulopolynomial degree qi + 1,

d(Xqi−h

) ≡( pπ· (a′′)qi

Cqh(Xqi−h

, Y qi−h

),rqi − rπ

· (a′′)qi

Xqi)

=( pπ· (a′′)qi

Cqi(X,Y ),rqi − rπ

· (a′′)qi

Xqi),

because elements in R/(π) are fixed under taking q-th powers and Cqh(Xqi−h

, Y qi−h

) = Cqi(X,Y )(see the proof of the claim in Proposition 2.2.2). Thus, we are done by Proposition 4.1.6.

Now, because mi is an A/m-vector space, (4.2) and the above claim show the existence and uniqueness ofδ such that (Ξ(δ)(X,Y ), ξ(δ)(X)) is zero as an element in the cohomology. Therefore, by Proposition 4.1.5,we obtain (4.1).

By this theorem, we now know that there is only one deformation class of height 1 over any A ∈ C . Thisfact will be important in our application to local class field theory (Section 5.1). Also, if G is a π-normalformal R-module of height h over Rnr/(π), two deformations F1 and F2 of G over A are isomorphic if andonly if [π]F1(X) and [π]F2(X) agree on the coefficients of X,Xq, . . . , Xqh−1

. This is an extremely convenientway of checking for isomorphic deformations.

4.2 Deformations With Level Structure

In this section, we consider deformations with level structure. As mentioned in the introduction, Drinfel’d’sused analogues of elliptic curves and modular curves to prove analogues of the Fundamental Theorem ofComplex Multiplication and the Langlands Conjecture for global function fields (d = 1, 2). Just as we havemodular curves X1(n) for each level n in the case of elliptic curves, Drinfel’d creates modular varieties foreach level structure (which he defines) on Drinfel’d modules. The level structure on formal modules discussedin this section is the one coming from the level structure on Drinfel’d modules via the process of going fromthe ‘global’ moduli problem to the ‘local’ deformation theory, so the result in this section is the ‘local’ versionof the moduli problem of Drinfel’d modules with level structure.

Coming up with the appropriate level structure on formal modules was one of the many great insights ofDrinfel’d’s in [5]. A naive notion of level structure on a formal R-module F over A ∈ C with ht(F ) = h canbe given by an R-module homomorphism

ϕ :(π−nR/R

)h −→ F (mA)

35

which is an isomorphism with the πn-torsion in F (mA). But this does not work very well. For example,if we take F to be the Zp-module Gm over a field of characteristic p, then we can check by induction that[p](X) = Xp, i.e. it is height 1. It is clear that there is no level structure (according to the above definition)on this Zp-module Gm. Drinfel’d’s great idea was to define a level structure to be just a homomorphismbut with the additional condition that [π](X) to be a unit multiple of

∏(X − ϕ(s)), where s runs through(

π−1R/R)h. This makes the zero homomorphism a level n structure of the Zp-module Gm over a field of

characteristic p for any n.Our goal of the section is determination of the universal deformation ring with level n structure. After we

determine the universal deformation with level structure, we will be able to explicitly see what Drinfel’d’sdefinition of level-n structure captures (Example 4.2.6).

As before, assume that G is a formal R-module over Rnr/(π) of finite height h, and that A ∈ C withmaximal ideal m.

Definition 4.2.1. Let n ≥ 1. For a formal R-module F over A, a structure of level n on F is an R-modulehomomorphism ϕ : (π−nR/R)h → F (m) such that the monic polynomial of degree qh

Pϕ(X) =∏

s∈(π−1R/R)h

(X − ϕ(s))

divides [π]F (X).

In this case, reducing [π]F (X) = g(X) ·Pϕ(X) modulo m, we obtain Xqh ·unit = g(X) ·Xqh

in (A/m)[[X]].Therefore, [π]F (X) and Pϕ are unit multiples of each other.

If α is an isomorphism and ϕ is structure of level n for a formal R-module F (n ≥ 1), then α ϕ doesdefine a level structure for the image of F via α. Indeed,

∏(π−1R/R)h

(X − α(ϕ(s)))

∣∣∣∣∣∣α [π]F α−1 ⇐⇒∏

(π−1R/R)h

(α(X)− α(ϕ(s))

)∣∣∣∣∣∣α [π]F ,

and noting that α(X) − α(ϕ(s)) = (X − ϕ(s)) · unit (because α is an isomorphism and ϕ(s) ∈ m), we aredone.

Definition 4.2.2. A deformation of G of level 0 over A simply refers to a deformation F of G over A. Forn ≥ 1, a deformation of G of level n over A is a pair (F,ϕ), where F is a deformation of G over A (in thesense of Definition 4.1.2) and ϕ is a structure of level n on F .

Two deformations (F1, ϕ1) and (F2, ϕ2) of G of level n over A to are said to be isomorphic if there existsan isomorphism α : F1 → F2 as deformations such that (π−nR/R)h ϕ1−→ F1(m) α−→ F2(m) is ϕ2.

As in Section 4.1, we can also define a deformation of level n as a triple (F, ι, ϕ), where ι is the isomorphismidentifying F modulo m with G, and define the isomorphic deformations by a commutative diagram as inDefinition 4.1.1. By lifting the isomorphism over Rnr/(π) to that over A, we again see that these twodefinitions agree when we discuss deformations up to isomorphism.

When A = Rnr/(π), there is only one homomorphism (π−nR/R)h → F (m), namely the zero map. This isindeed a level structure, since we are assuming that the height of G is h. Because of unicity of level structurein this case, we do not need to specify in Definition 4.2.2 what the level structure ϕ over A becomes overthe residue field (cf. Definition 4.3.3).

The goal is the determination of the universal object of the functor which associates A ∈ C to the setof deformations of G of level n over A up to isomorphisms. Before launching into this discussion, we willdigress a little and discuss properties of power series over A ∈ C .

Proposition 4.2.3. (1) α ∈ A[[X]] such that α ≡ Xν (mod m) for some ν. Then we can write α(X) =β(X)γ(X), where β(X) is a monic polynomial of degree ν, γ(X) is a unit in A[[X]], and β ≡ Xν and γ ≡ 1modulo m.

36

(2) If β(X) is a monic polynomial of degree ν such that β ≡ Xν (mod m), then A′ = A[[X]]/(β(X)) is afinite free A-module with basis 1, X, . . . ,Xν−1.(3) If α ∈ A[[X]] such that α(m) = 0 for some m ∈ m, then α(X) is a product of (X−m) and another powerseries.

Proof. (1) Starting from β1(X) = Xν and γ1(X) = 1, we create monic polynomials βi(X) of degree ν andpower series γi(X), such that βi+1− βi ∈ mi[[X]], γi+1− γi ∈ mi[[X]], and α ≡ βi · γi (mod mi). If we haveconstructed until the i-th step, then define

βi+1 = βi + β, and γi+1 = γi +α− βiγi − βγi

Xν,

where β is the polynomial obtained by taking up to degree ν − 1 of the power series α−βiγi

γi.

(2) Suppose β(X) = Xν + β(X), so that β has coefficients in m. Then Xν = −β(X) in A′. Thus, fori ≥ ν, if we can write Xi in terms of 1, . . . , Xν−1 with coefficients in mj , then we can write Xi+1 in terms of1, . . . , Xν−1 with coefficients in mj . Moreover, Xjν can be written with coefficients in mj , so it now followsthat any power series in A[[X]] can be spanned by these ν elements, using completeness of A. For linearindependence, if a polynomial of degree strictly less than ν can be written as β(X)γ(X), then modding outby m, we see that γ ≡ 0 (mod m). Plugging this back in and writing β(X) = β(X) +Xν , we then obtainγ ≡ 0 (mod m2). We can continue by induction.(3) Writing α =

∑aiX

i, α(X) = α(X)−α(a) = a1(X−a)+a2(X−a)(X+a)+a3(X−a)(X2+aX+a2)+· · · ,where the coefficient of Xi on the RHS converges for each i.

Now we are ready to determine the universal deformation of level n. For convenience, let e1, . . . , eh

denote the standard R/(πn)-module basis of (π−nR/R)h, and let D0 = Rnr[[i1, . . . ,ih−1]] be the universaldeformation ring. As a piece of notation, let i0 = π ∈ Rnr.

Theorem 4.2.4. A universal element of the functor which sends A ∈ C to the set of deformations of Gof level n over A up to isomorphism is given by (F univ

n , ϕunivn ) defined over a regular local ring Dh,n ∈ C .

For n ≥ 1, ϕunivn is injective, and the images of ei’s under ϕuniv

n form a system of local parameters of Dh,n.Further, the natural homomorphism Dh,n → Dh,n+1 (obtained from Yoneda’s lemma) is finite and flat.

Proof. By the same argument as in the proof of Theorem 4.1.7, we may assume that G is a π-normal formalR-module of height h. The case when n = 0 was completely done in Theorem 4.1.7. Because the condition ofPϕ(X) | [π]F only involves looking at s ∈

(π−1R/R

)h, n = 1 turns out to be the main case of this theorem.Note that a D0-algebra A ∈ C corresponds to (an isomorphism class representative of) a deformation FA ofG over A.

Lemma 4.2.5. Let j = 0, . . . , h. The universal object of the functor which sends a D0-algebra A ∈ C to theset of R-module homomorphisms ϕ :

(π−1R/R

)j → FA(mA) satisfying

Pj,ϕ(X) =∏

s∈(π−1R/R)j

(X − ϕ(s))

∣∣∣∣∣∣ [π]FA(X) (4.3)

is (Lj , ϕunivj,1 ), where ϕuniv

j,1 is injective and Lj is a regular local ring with a system of parameters given byϕuniv

j,1 (e1), . . . , ϕunivj,1 (ej) in Lj together with images of ij , . . . ,ih−1. The natural map Lj−1 → Lj is finite

and flat for all j = 1, . . . , h.

Proof. Observe first that this functor is well-defined, since the condition (4.3) does not depend on therepresentative of the deformation class. When j = 0, only the zero homomorphism works as ϕ, so L0 = D0

and ϕuniv1,0 = 0. Suppose that the lemma has been proved up to j, 0 ≤ j < h. Let α(X) ∈ Lj [[X]] such that

α(X) ·Pj,ϕunivj,1

(X) = [π]FLj(X), and let Lj+1 = Lj [[x]]/(α(x)). Before constructing ϕuniv

j+1,1 and showing (4.3),we will establish that Lj+1 is a regular local ring.

37

Since G is normal, α(X) ≡ Xqh−qj

(mod mLj). Writing α = βγ as in Proposition 4.2.3, we see that

Lj+1 = Lj [[x]]/(β(x)). So by the same proposition, Lj+1 is finite and free over Lj , so in particular, thedimension of these two rings are the same. Therefore, by the inductive hypothesis, we get the regularity ofLj+1 if we show that I = (ϕuniv

j,1 (e1), . . . , ϕunivj,1 (ej), x,ij+1, . . . ,ih−1) is equal to the maximal ideal mLj+1 =

I + (ij).By Theorem 4.1.7, the coefficient of Xqj

in [π]FD0(X) is (πqj−1 − 1)ij , so by definition, the coefficient of

Xqj

in [π]Lj (X) is the image of this under D0 → Lj . Using Pj,ϕunivj,1≡ Xqj

, β(X) ≡ Xqh−qj

, and γ(X) ≡ 1

modulo mLj , we can check that the coefficient of Xqj

in Pj,ϕ(X)α(X) is β(0) modulo m2Lj

. Hence,

(πqj−1 − 1)ij ≡ β(0) (mod m2Lj

). (4.4)

Sinceα(X) = α(0) +Xqh−qj

+X ·((mLj

)-coefficient power series),

upon multiplying (4.4) by γ(0), we have ij ≡ (πqj−1 − 1)γ(0)−1xqh−qj

(mod mLjmLj+1). Thus,

mLj+1 = I + mLjmLj+1 = I + m2

LjmLj+1 = · · · = I

by the Krull intersection theorem (applied to the module mLj+1/I with mLj-adic topology), so we now know

that Lj+1 is a regular local ring.Define

ϕunivj+1,1(

∑λiei) = FLj+1(ϕ

univj,1 (λ1e1 + · · ·+ λjej), [λj+1]FLj+1

(x)),

for λi ∈ R/(π). Since [π]Lj (x) = α(x)Pj,ϕunivj,1

(x) = 0 in Lj+1 and FLj+1 is the push-forward of FLj by

Lj → Lj+1, ϕunivj+1,1 is a well-defined R-module homomorphism

(π−1R/R

)j+1 → FLj+1(mLj+1). Also, if∑λiei is in kernel of ϕuniv

j+1,1,

0 = ϕunivj+1,1(

∑λiei) = [λ1](ϕuniv

j,1 (e1)) +FLj+1

· · · +FLj+1

[λj ](ϕunivj,1 (ej)) +

FLj+1

[λj+1](x)

≡ λj+1x+j∑

i=1

λiϕunivj,1 (ej) (mod m2

Lj+1),

but a system of parameters of a regular local ring form a basis of mLj+1/m2Lj+1

, so this implies λi ∈ (π).Thus, ϕuniv

j+1,1 is injective. Now, [π]FLj+1(ϕuniv

j+1,1(∑λiei)) = 0 and Proposition 4.2.3 shows that [π]FLj+1

(X)is divisible by each X − ϕuniv

j+1,1(∑λiei), so by injectivity of ϕuniv

j+1,1 and domainness of Lj+1, we obtain (4.3)for ϕuniv

j+1,1.

Finally, let us show that ϕunivj+1,1 is the universal element of this functor. Suppose ϕ :

(π−1R/R

)j+1 →FA(mA) is an R-module homomorphism satisfying (4.3). Then ϕ|(π−1R/R)j gives rise to a unique map Lj → A

sending ϕunivj,1 to ϕ|(π−1R/R)j . Thus, the only candidate of the requisite Lj+1 → A is obtained by using this

Lj → A and sending x to ϕ(ej+1). In order for this to work, α(ϕ(ej+1)) = 0. But by functoriality, viaLj → A, α(X) is sent to

[π]FA(X)∏

s∈(π−1R/R)j (X − ϕ(s)),

so α(ϕ(ej+1)) = 0.

We now continue the proof of Theorem 4.2.4. Let Dh,1 = Lh and ϕuniv1 = ϕuniv

1,h . If (F,ϕ) is a deformationof G of level 1 over A, then there exists a unique map D0 → A which pushes-forward F univ to the deformationclass of F , and given this, the above lemma shows that there exists a unique D0-algebra homomorphismDh,1 → A pushing-forward ϕuniv

1 to ϕ, so we have now proven the theorem for n = 1.

38

So suppose that we have proven the theorem up to n, n ≥ 1. Let ei be the standard basis of(π−n+1R/R

)hand bi = ϕuniv

n (πei) ∈ mDh,n. We claim that

Dh,n+1 = Dh,n[[k1, . . . ,kh]]/([π]FDh,n

(k1)− b1, . . . , [π]FDh,n(kh)− bh

)and ϕuniv

n+1 sending ei to ki form the desired universal element. Because ϕunivn+1 agrees with ϕuniv

n on (π−nR/R)h

(in particular on(π−1R/R

)h), ϕunivn+1 is a well-defined R-module homomorphism and a structure of level n+1.

By adding one ki at a time, Proposition 4.2.3 tells us that Dh,n+1 is a noetherian local ring which is finiteand free as a Dh,n-module. Since the maximal ideal is (b1, . . . , bh,k1, . . . ,kh) = (k1, . . . ,kh), Dh,n+1 is aregular local ring. By the same argument as in proof of the lemma, ϕuniv

n+1(∑λiei) = 0 shows each λi ∈ (π),

and then by the injectivity of ϕunivn , we obtain injectivity of ϕuniv

n+1 . Checking universality is similar as before:given (F,ϕ) over A, restrict ϕ to (π−nR/R)h and get Dh,n → A, and then note that there is only oneway of making the requisite map Dh,n+1 → A, namely by sending ki to ϕ(ei), and this is well-defined byconstruction.

Example 4.2.6. We will now trace the proof to obtain an explicit description of the universal deformationof level n for small h. First, let h = 1. For n = 1, by the proof of Lemma 4.2.5, L1 = D1,0[[x]]/

([π]Funiv (x)

x

).

Note also that ϕuniv1 sends 1

π to x by construction. For n = 2, tracing the construction yields (by functoriality,[π] for different formal modules are all images of the universal one over D1,0, so they are compatible)

D1,2∼=

D1,1[[k1]]([π](k1)− ϕuniv

1 ( 1π )) ∼= D1,0[[x,k1]](

[π](x)x , [π](k1)− ϕuniv

1 ( 1π )) ∼= D1,0[[k1]]

[π2](k1)[π](k1)

.

Similarly, we can continue by induction to show that D1,n∼= D1,0[[k1]]/

[πn](k1)[πn−1](k1)

. Thus, for Zp-module Gm

over Fp, deformation classes of level n over A correspond to maps Rnr[[k]]/(kpn−pn−1)→ A in C .

Now let h = 2. For n = 1, tracing through Lemma 4.2.5 yields

L2 =

(D2,0[[x]] /

( [π](x)x

))[[y]](

[π](y)Qλ∈(π−1R/R)1

(y−[λ](x))

) ∼= D2,0[[x, y]]([π](x)

x , [π](y)Qλ∈(π−1R/R)1

(y−[λ](x))

) .Similarly to the h = 1 case, we can show by induction that D2,n is D2,0[[k1,k2]] modded out by the samerelations as in the last ring above except we replace x by [πn−1](k1) and y by [πn−1](k2).

We can now explicitly see the connection between Drinfel’d’s level structures and πn-torsions.

We will next use the nice properties of the universal deformation of level n to create a deformation of leveln′ < n which is a ‘quotient’ deformation of what we started out with. To deal with quotients in general, weneed to expand the notion of formal R-modules; we will do so in Section 4.3. But this particular exampleof a quotient can be concretely constructed while staying in the realm of the formal R-modules discussed sofar. This construction will only work over regular local rings, but every deformation of level n over A is apush-forward of the universal one defined over a regular local ring by Theorem 4.2.2, so we can simply takethe push-forward of the quotient constructed over regular local ring to be the quotient in general. This is anexample of why nice properties of universal deformation rings are important. This construction does playan important role in Drinfel’d’s paper (see Section 5.2).

Let S ⊆ (π−nR/R)h be an R-submodule. We will construct the quotient F/ϕunivn (S), i.e. the formal

R-module in which the S part of the deformation is forced to be 0. Let α(X) =∏

s∈S Funivn (X,ϕuniv

n (s)).Since α(X) ≡ X |S| (mod mDh,n

) and Dh,n[[α]] is a complete noetherian local ring, Proposition 4.2.3 tells usthat Dh,n[[α]] −→ Dh,n[[X]] ∼= Dh,n[[α]][[X]] is injective and free of rank |S|. In other words, two power seriesin one variable which agree upon evaluation at α(X) must actually be the same. By induction (using thefact that Dh,n[[X]] is also regular local), two power series in a finite number of variables which agree uponevaluation at α(X)’s must agree. This will be important.

39

For each s ∈ S, let σs : Dh,n[[X]]→ Dh,n[[X]] be a ring homomorphism defined by X 7→ F univn (X,ϕuniv

n (s)),which is well-defined since ϕuniv

n (s) ∈ mDh,n)). Note that σs1 σs2 sends X to

F univn (F univ

n (X,ϕunivn (s2)), ϕuniv

n (s1)) = F univn (X,F univ

n (ϕunivn (s2), ϕuniv

n (s1))) = F univn (X,ϕuniv

n (s1 + s2)),

so σs1 σs2 = σs1+s2 . Also, σ0 = id. Therefore, the additive group S acts on Dh,n[[X]]. The same calculationalso shows that σs(α(X)) = α(X) for all s ∈ S, so Dh,n[[α]] is fixed by this action of S.

We will now prove that Dh,n[[α]] = (Dh,n[[X]])S . Since Dh,n is a domain, we can extend this action of Son Dh,n[[X]] to Frac(Dh,n[[X]]), where Frac denotes the field of fractions. We have already proved

Frac(Dh,n[[X]]) ⊇ (Frac(Dh,n[[X]]))S ⊇ Frac(Dh,n[[X]]S) ⊇ Frac(Dh,n[[α]]).

By Artin’s Theorem in Galois theory, the extension degree of the first two fields is |S|. Since Dh,n is aregular local ring, it is a UFD [21, Theorem 20.3], so it is integrally closed. This implies that any basis ofDh,n[[X]] over Dh,n[[α]] is actually a basis for the field extension, so the extension degree of the extreme twofields is also |S|. Therefore, Frac(Dh,n[[X]]S) = Frac(Dh,n[[α]]). Now, dimDh,n[[α]] = dimDh,n[[X]] and thenumber of generators of the maximal ideals is the same, so Dh,n[[α]] is also a regular local ring, thus integrallyclosed. Therefore, we now conclude that Dh,n[[X]]S = Dh,n[[α]]. By the same argument, we can show thatDh,n[[X,Y ]]S×S = Dh,n[[α(X), α(Y )]].

Note that

σs1

(α([r]Funiv

n(X)

))= α

([r]Funiv

n

(F univ

n (X, ϕunivn (s1))

))= α

(F univ

n

([r]Funiv

n(X), [r]Funiv

n(ϕuniv

n (s1))))

=∏s

F univn

(F univ

n

([r]Funiv

n(X), [r]Funiv

n(ϕuniv

n (s1))), ϕuniv

n (s))

=∏s

F univn

([r]Funiv

n(X), ϕuniv

n (rs1 + s))

= α([r]Funivn

(X)),

for all s1 ∈ S, so α([r]Funivn

(X)) = [r]H(α(X)) for a unique [r]H ∈ Dh,n[[X]]. Similarly, we can checkthat α(F univ

n (X,Y )) is H × H-invariant, so we must have α(F univn (X,Y )) = H(α(X), α(Y )) for a unique

H ∈ Dh,n[[X,Y ]]. Furthermore, we can show the following:

Proposition 4.2.7. (H, [ ]H) defines a formal R-module over Dh,n, and α is a formal R-module homomor-

phism F univn → H. The map (π−nR/R)h ϕuniv

n−−−→ F univn (mDh,n

) α−→ H(mDh,n) factors through (π−nR/R)h

/S,

and if(π−n′R/R

)h

→ (π−nR/R)h/S for some n′ ≥ 1, then

ψ :(π−n′R/R

)h

→( (π−nR/R

)h/S)→ H(mDh,n

)

is a structure of level n′ on H.

Proof. Using Hα = αF univn and [r]H α = α[r]Funiv

n, we get identities such as H(H(α(X), α(Y )), α(Z)) =

H(α(X),H(α(Y ), α(Z))), so H must be a formal R-module. For s ∈ S,

α(ϕunivn (s)) =

∏s′

F univn (ϕuniv

n (s), ϕunivn (s′)) =

∏s′

ϕunivn (s+ s′) = 0,

so α ϕunivn factors through (π−nR/R)h

/S. We now prove the last statement. Since α(ϕunivn (t)) =∏

s(ϕunivn (t) − ϕuniv

n (s)) for t ∈(π−n′R/R

)h

→ (π−nR/R)h and Dh,n is an integral domain, injectivity of

ϕunivn (Theorem 4.2.4) shows injectivity of ψ. Moreover, for t ∈ (π−nR/R)h, [π]H(ψ(t)) = [π]H(α(ϕuniv

n (t))) =α([π]Funiv

n(ϕuniv

n (t))) = 0, so similarly as before, ψ is a structure of level n′ on H.

40

Remark. As mentioned earlier, we could construct this H as a quotient of more generalized notion of formalR-modules, discussed in Section 2.3. Just as a reference, we will briefly describe the above results in thislanguage. We need to prove that

0 −→(F ′ = SpfDh,n

(Dh,n[[X]]/

∏s∈S

(X − ϕunivn (s))

))−→

(F = SpfDh,n

(Dh,n[[X]]))

−→(H = SpfDh,n

(Dh,n[[α]]))−→ 0.

is a short exact sequence of formal R-modules. Indeed, all of these rings are topologically flat over Dh,n (useProposition 4.2.3), Dh,n[[X]]← Dh,n[[α]] is topologically faithfully flat (it is free), and

Dh,n[[X]]/∏s∈S

(X − ϕunivn (s)) ← Dh,n[[X]]

is a surjection. Further, injectivity of ϕunivn and α(ϕuniv

n (s)) = 0 as derived above show that α is a multipleof∏

s∈S(X − ϕunivn (s)). So these must differ by a unit in Dh,n[[X]], because α ≡

∏(X − ϕuniv

n (s)) ≡ X |S|

(mod mDh,n). This shows

Dh,n[[X]]/∏s∈S

(X − ϕunivn (s))

∼=←− Dh,n[[X]]⊗Dh,n[[α]]Dh,n,

showing exactness, as desired. By chasing maps, we also get identities such as H(α(X), α(Y )) = α(F (X,Y )).So H can be viewed as the quotient F/F ′ with this broader definition of formal R-modules.

4.3 The Deformations of Divisible Modules

We are finally ready to discuss deformation theory of formal R-modules that Drinfel’d was after, namelydeformations of divisible formal R-modules. Although this is a more general notion of formal R-modules,they are restricted enough so that we can ‘extend’ the notion of the level structures and the results ofdeformation theory that we have developed in the previous two sections. Again, our goal is determining theuniversal deformation ring with level structures and deriving their properties.

In this section, formal R-modules refer to those satisfying Definition 2.3.1, and the formal R-modules ofthe form discussed previously (the power series form) will be called CFS formal R-modules. We will freelyuse the results and notations from both Section 2.3 and Appendix A.

Let A ∈ C , j be a fixed non-negative integer, and let S be the R-module (K/R)j (j will never be changedin this section, so this notation should not cause confusion). Here is the central object of study in thissection:

Definition 4.3.1. A divisible formal R-module over A of type (h, j) is a formal R-module over A such that(1) the connected component F 0 of F is a CFS formal R-module over A such that F 0 over Rnr/(π) is ofheight h, and (2) there is a short exact sequence of formal R-modules

0 −→ F 0 −→ F −→ SpfA(AS) −→ 0.

By Proposition 2.3.3, SpfA(AS) is etale and it can be viewed as the ‘discrete’ formal R-module over Amade from the R-module S. So by putting condition (2) on the connected-etale sequence of a divisibleformal R-module F , we are in some sense forcing F to be S disjoint copies of the CFS formal R-moduleswith ‘twists’. This statement can be made precise (see the construction of the divisible formal R-moduleF in the proof of Proposition 4.3.5 below), and explains why it is not surprising to get a nice deformationtheory of divisible formal modules by generalizing the results proven already for the CFS formal R-modules.

Note that divisiblity of F does not come with a specified short exact sequence in (2) above. We only knowthat F can be fit into such a sequence. From Section 2.3, we do know that quotients are well-defined in thissetting, so we get every possibility of F → SpfA(AS) by starting from one F → SpfA(AS) and compose with

41

all formal R-module isomorphisms SpfA(AS) → SpfA(AS). Whenever we use this sequence of F , we mustalways remember to make sure that we are not making a choice for the quotient map.

We will now give the definition of level structure on a divisible formal R-module F . Recall by Propositions2.3.3, 2.3.2, and 2.3.7, SpfA(AS)(A) ∼= S canonically, F 0(A) is F 0(m), and evaluating a connected-etalesequence at A gives a left-exact sequence of R-modules.

Definition 4.3.2. For n ≥ 0, a structure of level n on a divisible formal R-module F of type (h, j) isan R-module homomorphism Φ : (π−nR/R)h+j → F (A) such that (1) The image of Φ|(π−nR/R)h lies inF 0(m) = F 0(A) → F (A), giving a structure of level n for the CFS formal R-module F 0, and (2) The imageof the composition (π−nR/R)h+j Φ−→ F (A) −→ SpfA(AS)(A) ∼= S is exactly equal to (π−nR/R)j .

Note that although we are using the quotient map F → SpfA(AS) in condition (2) for the level structure,another choice would result in composing with an isomorphism S → S. Since this isomorphism does notchange the πn-torsions, we see that (2) is a well-defined condition for divisible modules. From now on, we willnot always go through this kind of reasoning every time we use the quotient map part of the connected-etalesequence for F .

If j = 0, F is a CFS formal R-module, and this definition of level structure does agree with that ofSection 4.2. Moreover, this definition is relatively restrictive. For example, the two conditions above andthe exactness of F 0(A) → F (A) → SpfA(AS)(A) together with finiteness of (π−nR/R)j imply that thecomposition (

π−nR/R)j→(π−nR/R

)h+j Φ−→ F (A) −→(π−nR/R

)jis an isomorphism. This fact will be used later for constructing level structures. In this section, a levelstructure will always be denoted by one of φ,Φ, ψ.

When A = k, the connected-etale sequence splits (Proposition 2.3.8), so a divisible formal R-module Gof type (h, j) has the form G0 × Spfk(kS). As the maximal ideal is 0, G(k) −→ Spfk(kS)(k) = S is anisomorphism. This shows that level structures on G correspond to R-module automorphisms of (π−nR/R)j ,since the level structure must be defined 0 on (π−nR/R)h.

We next define a deformation of level n, essentially by following what we did for the CFS case. Thereis one significant difference, however. As noted above, there is more than one level structure on a divisibleformal R-module G over k of type (h, j) when j > 0, in contrast to the case j = 0. Therefore it is cleanerto develop the deformation theory of divisible formal R-modules by fixing a level structure φ on G over theresidue field. So we will discuss deformations of (G,φ), where G is a divisible formal R-module over k oftype (h, j) and φ is a structure of level n on G. The deformations should have the property that the formalmodule and the level structure ‘reduce mod mA’ to (G,φ). More precisely,

Definition 4.3.3. A triple (F, ι,Φ) is said to be a deformation of (G,φ) over A if (1) F is a divisible formalR-module over A of type (h, j) and ι is an isomorphism from F (i.e. base change by (A/mA) and identifyRnr/(π) ∼= A/mA) to G over k, and (2) Φ is a structure of level n on F such that the composition(

π−nR/R)h+j Φ−→ F (A) −→ F (A) ι−→ G(k)

is φ.We call two deformations (Fi, ιi,Φi) of (G,φ) of level n over A isomorphic if there exists an isomorphism

α : F1 −→ F2 of formal R-modules such that (π−nR/R)h+j Φ1−−→ F1(A) α−→ F2(A) is Φ2 and

F1 mod mA

ι1%%KKKKKKKKKKα // F2 mod mA

ι2yyssssssssss

G

commutes.

42

In this section, we will not use the definition of deformation which takes ι to be id. Also, note that thisdefinition agrees with the definition given in the last section when j = 0.

As usual, our goal is the determination of the universal deformation (up to isomorphism) of (G,φ) of leveln:

Theorem 4.3.4. Suppose (G,φ) is a divisible formal R-module G of type (h, j) over k with a structureφ of level n. The functor which associates A ∈ C to the set of all deformations of (G,φ) over A up toisomorphism is represented by Eh,n,j = Dh,n[[f1, . . . ,fj ]], where Dh,n is as in Theorem 4.2.4. Eh,n,j is aregular local ring of dimension h+ j, and the natural map Eh,n,j −→ Eh,n+1,j is finite and flat.

We will spend the rest of this section proving this theorem. The proof takes advantage of the relativelystrong conditions put on divisible modules and their level structures so that we can use deformation theoryof the CFS formal R-modules developed in Section 4.2. More specifically, given a deformation (F, ι,Φ) of(G,φ), it follows by definition that (F 0, ι|F 0 ,Φ|(π−nR/R)h) is a deformation of level n of the CFS formal

R-module G0 (∵ ι induces an isomorphism η from F 0 = F0

to G0). Therefore to prove the theorem (whichrequires knowing what the possibilities are for deformations), we will first classify all the deformations of(G,φ) that restrict to a certain fixed deformation of level n of G0, and then use the deformation theory ofthe CFS case to ‘naturally’ change the deformation on the connected component.

We use the notion of ‘extensions’ to classify all the deformations (F, ι,Φ) of (G,φ) which restrict to agiven deformation (H, η, ψ) of G0. This may be a bit concerning, because the deformation F does not comewith the information of identifying F/F 0 with SpfA(AS). However we will get around this problem by fixinga quotient map G/G0 → Spfk(kS) over the residue field. A complete justification on why we can use thenotion of extensions to prove Theorem 4.3.4 will be given at the very end of the proof.

Studying ‘extensions’ will be divided into two big parts. First we will deal with extensions of formalR-modules (Proposition 4.3.5). That is, we will ignore the level structure and just analyze formal R-modulesF with connected component H. This sets up the stage for the second part, which add consideration of thelevel structures (Proposition 4.3.7). Before stating this proposition we will precisely define the notion of an‘extension’ of a deformation of G0 to a deformation of G.

Proving these two propositions in detail actually require a lot of work. To clarify the main ideas of theproofs, we will have a brief overview section at the beginning of each proof.

We will now begin the proof of Theorem 4.3.4. Our first task as noted above is analyzing the possibilitiesof divisible formal R-modules when we fix the connected component, using the notion of extensions of formalR-modules (cf. Proposition 2.3.6).

Proposition 4.3.5. There is a natural bijection between ExtR,A(SpfA(AS),H) and ExtR(S,H(mA)) givenby evaluation at A.

Proof. We will first give an overview of the proof. First, we will show that evaluation at A produces awell-defined element of ExtR(S,H(mA)) if we start from an element in ExtR,A(SpfA(AS),H). We will thenconstruct an element of ExtR,A from an element of ExtR, after we describe each element of ExtR in a formsuitable for motivating this construction. Next comes the tedious verification that this construction ‘works’.Then we show that ExtR → ExtR,A → ExtR is the identity. Finally, after showing that ExtR,A → ExtR is agroup homomorphism (ExtR,A is a group by Proposition 2.3.6), we will show that the kernel of ExtR,A →ExtR is 0.

Given an extension of the form0→ H → F → SpfA(AS)→ 0,

we now show that the evaluation on the base

0→ H(A)→ F (A)→ SpfA(AS)(A)→ 0

is an exact sequence of R-modules. By definition, these are R-modules and R-module homomorphisms. Theleft-exactness follows from Proposition 2.3.7. The right-exactness comes from the fact that F is formallysmooth, but we will include a more explicit proof here since this will be useful later. Let F = SpfA(B), andlet B =

∏Bn where n runs through open maximals of B. Base changing by A/m and using Proposition

43

2.3.8, we must have∏

(Bn⊗Ak) ∼= k[[T ]]S . Because rings inside the product sign are all local for both sides,this isomorphism induces an isomorphism on each local factor (by the same argument as given in SectionA.5). In other words, k[[T ]] ∼= Bn⊗Ak for each n, and n’s are in a bijective correspondence with elements ofS. Lifting the image of T under this isomorphism to Bn, we get a map A[[T ]] → Bn of PAA which reducesto an isomorphism modulo mA. Because Bn is topologically flat (∵ B is topological flat and B =

∏Bn),

the Nakayama/projectivity argument discussed in Appendix A gives us A[[T ]] ∼= Bn as elements of PAA, soA[[T ]]S ∼= B. Moreover, since products and (−)et commute, we obtain (A[[T ]]S)et = AS by Proposition 2.3.4.Now, by definition of connected-etale sequences, F → SpfA(AS) corresponds to B ∼= A[[T ]]S ← AS , so it isnow clear that F → SpfA(AS) has a formal A-scheme section. Hence, F (A)→ SpfA(AS)(A) is surjective, asdesired.

Since every map of formal R-modules functorially induces a map of R-modules upon evaluation, thisprocess sends equivalent extensions of formal R-modules to equivalent extensions of R-modules. Therefore,we have now created ExtR,A → ExtR.

So we are now in the second stage of our proof, where we will reinterpret an R-module extension

0→ H(mA)f−→ E

g−→ S → 0 (4.5)

in a way suitable for motivating our construction of the corresponding element of ExtR,A. We choose aset-theoretic section g′ of the map E → S with the property that g′(−s) = −g′(s) for every s ∈ S. Then wedenote how off this g′ is from being an R-module homomorphism by letting

Ξ(s, s′) = g′(s) + g′(s′)− g′(s+ s′), and ξr(s) = rg′(s)− g′(rs) ∀s, s′ ∈ S, r ∈ R.

Note that Ξ(s1, s2) and ξr(s) take values inH(mA) → E. Also, by the assumption on g′, Ξ(s, 0) = Ξ(s,−s) =ξr(0) = 0 for all r ∈ R and s ∈ S. Using these, we can check directly that the diagram of exact sequences

0 // H(mA) // S ×H(mA) //

S // 0

0 // H(mA)f

// E g// S // 0

(4.6)

commutes, where we give S ×H(mA) the structure of an R-module by

(s1,m1) + (s2,m2) = (s1 + s2,m1 +m2 + Ξ(s1, s2)) and r(s,m) = (rs, rm+ ξr(s)), (4.7)

H(mA) → S ×H(mA) by m 7→ (0,m), S ×H(mA) → S by (s,m) 7→ s, and S ×H(mA) → E by (s,m) 7→g′(s) + f(m).

So there is a representative of the R-module extension class of (4.5) such that it is S ×H(mA) (as a set)with the R-module structure ‘twisted’ by the section g′. This serves as a good motivation for our constructionof the corresponding element of ExtR,A, with middle term F . In other words, we should take F to be ‘Scopies of H’ with the formal R-module structure ‘twisted’ by the section g′ as in (4.7). So let us defineF = SpfA(A[[T ]]S) and

(µ∗(b)) (s, s′) = b(s+ s′)(X +HY +

HΞ(s, s′)) and ([r]∗F (b)) (s) = b(rs)([r]H(T ) +

Hξr(s)),

where we have viewed an element b ∈ B = A[[T ]]S as a (set) map from S → A[[T ]] and an element ofB⊗AB ∼= (A[[X,Y ]])S×S as a (set) map from S × S → A[[X,Y ]]. We also need maps α : H −→ F andβ : F −→ SpfA(AS). Again, looking at the re-interpretation of module extensions above, α should be a‘closed immersion’ into the s = 0 part (so on rings, take A[[T ]] ← A[[T ]]S defined by evaluation at 0), andβ should be the projection onto the S-component (so on rings, take A[[T ]]S ← AS defined by the naturalinclusion on each component).

Now, we have come to the stage of tedious verification. Here is what we need to check:(1) F is a formal R-module.

44

(2) α and β are formal R-module homomorphisms.(3) 0→ H → F → SpfA(AS)→ 0 is an exact sequence of formal R-modules.(4) Different choices of section g′ result in equivalent extensions of formal R-modules.(5) Equivalent extensions in ExtR are sent to equivalent extensions in ExtR,A.

In this detail-checking process, we will often have to check commutativity of diagrams. We will use arrowssuch as

oo ↑ to indicate which way we are tracing a diagram, and given these arrows, we will note the imageof an element b via the first map by b† and its image via the second map by b†† etc. This allows us to tracethrough diagrams without having to label all the maps.

(1) Note that B is topologically flat. Let

ε∗(b) = b(0)(0), and (inv∗(b)) (s) = b(−s)(invH(T )),

where invH(T ) is the power series satisfying H(T, invH(T )) = 0. Since µ∗, [r]∗F , ε∗, inv∗ are all maps of PAA

and ([1]F (b))(s) = b(s)(T ), it suffices to check the commutativity of the seven diagrams (2.4). Here we willonly check the first diagram (associativity); other diagrams are checked by the same methods.

We will first trace byoo ↑. By definition, b†(s, s′) = b(s+ s′)(X +

HY +

HΞ(s, s′)). In other words, this map

µ sends every occurrence of T in (s+ s′)-th component to X +HY +

HΞ(s, s′) in (s, s′)-th component (thinking

this way helps computing id ⊗µ etc). Denoting B⊗B⊗B ∼= A[[U, V,W ]]S×S×S , we have

b††(t, t′, t′′) = b†(t, t′ + t′′)(U, (V +

HW +

HΞ(t′, t′′))

)= b(t+ t′ + t′′)

(U +

HV +

HW +

H(Ξ(t′, t′′) +

HΞ(t, t′ + t′′))

).

By assumption, H(m) → E as R-modules, so identifying H(m) as a submodule of E,

Ξ(t′, t′′)+H

Ξ(t, t′ + t′′) = (g′(t′) + g′(t′′)− g′(t′ + t′′)) + (g′(t) + g′(t′ + t′′)− g′(t+ t′ + t′′))

= g′(t) + g′(t′) + g′(t′′)− g′(t+ t′ + t′′)

This expression of b†† does not show that we traced the diagramoo ↑ rather than ↑ oo , so we get the

commutativity of the first diagram.Thus, F = SpfA(B) is indeed a formal R-module.

(2) Since α and β are maps of PAA, we need commutativity of two diagrams (2.5). Here we will explicitly showthis for α; we can use the same methods for β. Going

oo ↑ in the first diagram, we obtain b†† = b†(0, 0) =b(0)(H(X,Y )). Going ↑ oo , we obtain b†† = b(0)(H(X,Y )), so the first diagram commutes. For the seconddiagram, going

oo ↑ results in b†† = b†(0) = b(0)([r]H(T )). Going ↑ oo results in b†† = b(0)([r]H(T )), sothe second diagram commutes.

(3) Since A[[T ]] ← A[[T ]]S ← AS is given by evaluation at 0, β α is indeed the zero map. α is clearly aclosed immersion. As shown in the first stage of the proof, AS is the maximal etale subalgebra of A[[T ]]S , soas discussed in Appendix A, β is topologically faithfully flat. Finally, A[[T ]]

∼=←− A[[T ]]S⊗ASA follows, since(0, (1, 1, . . .))⊗1 is 0 in (A× AS−0)⊗ASA. Because AS is the maximal etale subalgebra of A[[T ]]S , we canconclude by the exactness of this sequence that H is the connected component of F .

(4) Suppose g1 and g2 are two sections of g (both satisfying gi(−s) = −gi(s)), resulting in formal R-modulesF1 and F2. Since g1(s) − g2(s) ∈ H(m) for all s ∈ S, we can define γ∗ : B = O(F2) → O(F1) = B by(γ∗(b))(s) = b(s)

(T +

H(g1(s)− g2(s))

). By a direct computation, we can check that

A[[T ]] O(F1)α1oo AS

β1oo

β2||yyyy

yyyy

y

O(F2)

α2

ccHHHHHHHHHγ∗

OO

45

is commutative. Now, we show that γ : F1 → F2 is a formal R-module homomorphism by checking commu-tativity of the two diagrams (2.5). We will only check the second diagram here. Going ↑ oo , we have

b††(s) = b†(rs)([r]H(T ) +Hξ1(r)(s)) = b(rs)

([r]H(T ) +

Hξ1(r)(s) +

H(g1(rs)− g2(rs))

).

Goingoo ↑, we have

b††(s) = b†(s)(T +

H(g1(s)− g2(s))

)= b(rs)

([r]H

(T +

H(g1(s)− g2(s))

)+Hξ2(r)(s)

).

Using the same H(mA) → E argument (we get b(rs)([r](T ) +

H(rg1(s)− g2(rs))

)in both cases), we have the

commutativity. Thus, we now see that the element in ExtR,A created by this process is independent of thechoice of sections.

(5) Suppose we have a commutative diagram of extensions

0 // H(m) //

""EEEE

EEEE

E1g1 //

j

S // 0

E2

g2

??

If g′1 is a section of g1 with g′1(−s) = −g′1(s), then g′2 = j g′1 is a section of g2 with g′2(−s) = −g′2(s).Moreover, because we identify H(m) → Ei, (Ξ, ξ)’s agree for g′1 and g′2. Thus, clearly the constructedextensions of formal R-modules involving F1 and F2 are equivalent.

This finally finishes the tedious checking stage, proving that we have a well-defined process ExtR → ExtR,A.Next stage is checking that ExtR → ExtR,A → ExtR is the identity. This essentially follows from our

motivation for the construction of F . We will now go through the details. We already know that H(A) =H(m) and SpfA(AS) = S as R-modules. By the same argument as given in Section A.5, locality of Aimplies SpfA(A[[T ]]S)(A) = S × SpfA(A[[T ]]) = S × H(m) set-theoretically (where (s,m) corresponds tothe projection onto the s-th coordinate and then sending T to m). Using (2.6), the sum of (s,m) and(s′,m′) in F (A) corresponds to B → A defined by b 7→ b(s + s′)(m+

Hm′ +

HΞ(s, s′)), which is the element

(s + s′,m +m′ + Ξ(s, s′)) ∈ S ×H(m). This agrees with the addition structure put on S ×H(m) in (4.6).Similarly, using the second diagram of (2.6), we can check that the R actions are the same. It only remains toshow that the maps agree. For the first map, the element m (corresponding to the map A[[T ]]→ A sending Tto m) is sent to A← A[[T ]]← A[[T ]]S , defined by projection to the 0-th component followed by sending T tom. Thus, the first map sends m to (0,m), agreeing with (4.6). We can check similarly that the second mapagrees, so we have now shown that ExtR → ExtR,A → ExtR is the identity. In particular, ExtR,A → ExtR issurjective.

Now we have come to the last stage of the proof. By Proposition 2.3.6, ExtR,A(SpfA(AS),H) is a group.Moreover, because the group operations on both ExtR,A and ExtR are defined by category-theoretic methodsof Yoneda Ext, ExtR,A → ExtR must be a group homomorphism. Therefore, it now suffices to check thatthe kernel of ExtR,A → ExtR is 0, i.e. an extension of formal R-modules which splits upon evaluation at Amust be a split exact sequence.

So suppose a short exact sequence of formal R-modules 0 → Hα−→ K = SpfA(C)

β−→ SpfA(AS) → 0

is such that 0 → H(m)f−→ K(A)

g−→ S → 0 splits, i.e. there exists an R-module section g′ : S → K(A)(we know from the first stage of the proof that C ∼= A[[T ]]S , but we won’t need this here). By Proposition2.3.3, there exists a formal R-module homomorphism β′ : SpfA(AS)→ K such that s-th component (β′)∗s of(β′)∗ : C → AS is g′(s) for all s ∈ S. To show that β′ is a section of β, by the same proposition, it suffices toshow that β β′ corresponds to the identity map S → SpfA(AS)(A). We have (β β′)∗s = β′

∗s β∗ = g′(s)β∗,

and by definition, g′(s) β∗ is g(g′(s)) = s, so we are done.

46

The next stage is adding consideration of level structures. Before going on any further, we will define theconcept of ‘extension’ of a deformation of G0 to a deformation of G and rigorously establish the connectionbetween classification of such ‘extensions’ and classifications of deformations with a fixed deformation on theconnected component.

As remarked right after the definition of divisible modules, the quotient map part of the connected-etalesequence is not fixed. Let us start from one such G −→ Spfk(kS). The evaluation at the base G(k) →Spfk(kS) = S is an isomorphism, whose inverse create a formal R-module isomorphism Spfk(kS)→ Spfk(kS)by Proposition 2.3.3. Composing this with the original quotient map, we now have a new quotient mapG −→ Spfk(kS) such that the evaluation at k is the identity map. From now on until the end of the proof,we will always use this particular short exact sequence 0→ G0 → G→ Spfk(kS)→ 0 over the residue field.

Now suppose that F is a divisible formal R-module and ι is an isomorphism from F to G. Then thereexists a unique quotient map F −→ SpfA(AS) such that the induced map makes the diagram

F //

ι

Spfk(kS)

id

G // Spfk(kS)

commutative. Indeed starting from one quotient map F −→ SpfA(AS), we obtain every other one bycomposing an isomorphism SpfA(AS) → SpfA(AS). Since this isomorphism is uniquely determined by themap on the evaluation at A (Proposition 2.3.3) and since F (A) → SpfA(AS)(A) = S is same as F (A) →F (A)→ Spfk(kS)(k) = S, we do indeed have the existence and the uniqueness of such quotient map. Withthis choice of the quotient, we have the commutativity of

F (A)

// SpfA(AS)(A) = S

id

F (k) //

ι

Spfk(kS)(k) = S

id

G(k)

id// Spfk(kS)(k)

In particular, with this choice, F (A) → SpfA(AS)(A) = S is the same map as F (A) → F (k) ι−→ G(k) = S.This will be extremely useful in our construction of level structure below.

Because there is a well-defined quotient map given F and ι, it is safe to go into the language of ‘extensions.’

Definition 4.3.6. Suppose (H, η, ψ) is a deformation of level n over A of G0. Then a deformation (F, ι,Φ)of (G,φ) is called an extension of (H, η, ψ) if F 0 = H, ι|F 0 = η, and Φ|(π−nR/R)h = ψ. As discussed above,this set-up gives us a well-defined short exact sequence 0→ H → F → SpfA(AS)→ 0.

We say two extensions (Fi, ιi,Φi) of (H, η, ψ) are equivalent if there exists a formal R-module homomor-phism α : F1 → F2 such that

0 // H // F1//

α

SpfA(AS) // 0

0 // H // F2// SpfA(AS) // 0

commutes, and ιi’s and Φi’s are compatible via α in an obvious sense (α is then an isomorphism by the sameargument as presented immediately before Proposition 2.3.6).

Proposition 4.3.7. Let (H,ψ, η) be a deformation of level n of G0 over A. Then the extension classes of(H,ψ, η) to a deformation (F,Φ, ι) of (G,φ) over A are naturally classified by the set ExtR

((K/π−nR)j ,H(mA)

).

47

Proof. We will give a brief overview of the proof. We will explicitly present two processes, one going from theextension classes of deformations to ExtR((K/π−nR)j ,H(m)) and the other going the opposite way. Bothprocesses will use the bijective correspondence given in Proposition 4.3.5. We will then prove that these twoprocesses are indeed inverses of each other.

Let (F, ι,Φ) be an extension of (H, η, ψ) to a deformation of (G,φ). By ignoring the level structureinformation and using the corresponding short exact sequence, Proposition 4.3.5 tells us that this correspondsto an extension of R-modules of the form 0 −→ H(m)→ F (A)

f−→ S → 0. Using the level structure, we willshow that this extension of R-modules is a pull-back of a well-defined element in ExtR((K/π−nR)j ,H(m)).In other words, what we need is a commutative diagram of exact sequences

0 // H(m) // E′ // (K/π−nR)j // 0

0 // H(m) // F (A)

OO

f// S

OO

// 0

(4.8)

Let E′ = F (A)/Φ((π−nR/R)j). We will now construct H(m)→ E′ and E′ → (K/π−nR)j . As remarked

after Definition 4.3.2, the map g : (π−nR/R)j Φ−→ F (A)f−→ (π−nR/R)j is an isomorphism. This implies ker f∩

Φ((π−nR/R)j) = (0), so we have (ker f) = H(m) → F (A)/Φ((π−nR/R)j). Moreover, since f(Φ((π−nR/R)j))lies inside πn-torsions of S, we have a well-defined R-module homomorphism E′ −→ (K/π−nR)j . Byconstruction, the above diagram commutes. We will now check exactness of the top row. E′ −→ (K/π−nR)j

is surjective, as f and S −→ (K/π−nR)j are. The composition H(m) → E′ → (K/π−nR)j is clearly zero.So now let e′ ∈ E′ be such that its image in (K/π−nR)j is zero. By taking a lift e′ ∈ F (A), we must havef(e′) ∈ (π−nR/R)j . Then Φ

(g−1(f(e′))

)∈ e′ + ker f , but this is also in Φ((π−nR/R)j), so we conclude

e′ ∈ ker f → E′. Thus we have now constructed a desired element of ExtR((K/π−nR)j ,H(m)) correspondingto an extension (F, ι,Φ).

If we start from two equivalent extensions (Fi, ιi,Φi) of (H, η, ψ), then by Proposition 4.3.5, we obtain thesame element of ExtR(S,H(m)), and by compatibility of the level structures, F1(A)→ F2(A) induces E′

1 →E′

2 which fit into a commutative diagram of module extensions, so we can trace through the constructionsand conclude that we obtain the same element of ExtR((K/π−nR)j ,H(m)).

Conversely, given an extension of R-modules 0 → H(m) → E′ → (K/π−nR)j → 0, we will create anextension (F, ι,Φ) of (H, η, ψ). Pulling back the R-module extension by the quotient map S → (K/π−nR)j ,we get an exact sequence 0 → H(m) → E → S → 0. By Proposition 4.3.5, we get the corresponding shortexact sequence of formal R-modules 0 → H → F → SpfA(AS) → 0. Because the connected-etale sequencecanonically splits over k (Proposition 2.3.8), there exists a unique isomorphism ι : F → G such that

0 // H //

η

F //

ι

Spfk((k)S) //

id

0

0 // G0 // G // Spfk((k)S) // 0

(4.9)

commutes.By construction, we have commutativity of (4.8) in this case. Therefore F (A) is identified as an R-module

as (e′, s) ∈ E′ ⊕ S : images of e′ and s agree in (K/π−nR)j. Using this identification, we define

Φ(t1, t2) = (ψ(t1), φ(t1, t2)),

where t1 ∈ (π−nR/R)h and t2 ∈ (π−nR/R)j . Note that both ψ(t1) and φ(t1, t2) vanish in (K/π−nR)j , soΦ is indeed a well-defined R-module homomorphism. Since φ is a level structure over k, φ(t1, 0) = 0, soΦ(t1, 0) = ψ(t1) as required. Moreover, as (π−nR/R)h+j Φ−→ F (A) → S is φ, surjectivity to (π−nR/R)j isalso clear. Thus, Φ is a level structure for F . As remarked before, with a choice of ι making (4.9) commute,

48

F (A) → S is the same map as F (A) → F (A) ι−→ G(k), so Φ does reduce to φ on G. Hence, we have nowcreated an extension (F, ι,Φ) of (H, η, ψ).

Note that in the above construction of (F, ι,Φ), we had no choice at all after we obtained the short exactsequence for F . Indeed ι must make (4.9) commutative, as ι|H = η and the constructed short exact sequence0→ H → F → SpfA(AS)→ 0 must be the specified one corresponding to F and ι. Once ι is fixed, becauseΦ must reduce to φ (via ι) and Φ|(π−nR/R)h = ψ, we have no choice but to define it as above. This is in facta key reason for going into the language of extensions to prove Theorem 4.3.4.

If H(m) → E′i (K/π−nR)j are equivalent, then their pull-backs to H(m) → Ei S are equivalent.

By Proposition 4.3.5, H → Fi SpfA(AS) are equivalent formal R-module extensions. By uniqueness ofιi as noted above, the isomorphism F1 → F2 and ι’s are compatible. By the commutativity of the initialdiagram of extensions (involving E′

i), the constructed level structures Φ1 and Φ2 are compatible.Now that we have two operations, we will check that these are inverses of each other. The key is the fact

that we have no choice for ι and Φ once we have the short exact sequence. Suppose we have an extension(F,Φ, ι). Tracing the construction, we create the exact sequence 0 → H(m) → F (A) → S → 0 and lettingE′ = F (A)/Φ((π−nR/R)j), we get the exact sequence 0 → H(B) → E′ → (K/π−nR)j → 0. Now to goback, we pull-back, but we can certainly use 0→ H(m)→ F (A)→ S → 0. So by Proposition 4.3.5, we canuse the original short exact sequence involving F . But by the fact that we have no choice after fixing shortexact sequence, the original ι and Φ must be the ones that we get by the construction above.

Now we compose these processes in the opposite way. Starting from an exact sequence 0 → H(m) →E′ → (K/π−nR)j → 0, we create the pull-back with middle term E. The level structure Φ put on thecorresponding formal R-module is Φ(t1, t2) = (ψ(t1), φ(t1, t2)). Note that under the map E → E′, Φ(0, t2)is sent to ψ(0) = 0, so we do indeed have a map E/Φ((π−nR/R)j) → E′. By following the construction ofmaps H(m)→ E/Φ((π−nR/R)j) and E/Φ((π−nR/R)j)→ (K/π−nR)j , it is clear that these two extensionsare equivalent.

Therefore, we get the desired one-to-one correspondence.

Before finishing the proof of Theorem 4.3.4, we need the following fact:

Proposition 4.3.8. ExtR(K/R,H(mA)) is naturally isomorphic to H(mA). Therefore, we have a natu-ral (on the second factor) isomorphism between ExtR((K/π−nR)j ,H(mA)) and H(mA)j, once we fix anisomorphism K/R ∼= K/π−nR.

Proof. We will prove two things: H(mA) is a π-adically completely R-module, and ExtR-mod(K/R,M) ∼= Mfor all π-adically complete R-modules. For the first statement, it suffices to check that [πm]H(y) ∈ mm+1

A forall m and y ∈ mA, since A is mA-adically complete. The m = 0 case is clear and if we know it for m, thenbecause [π]H(X) = πX + a2X

2 + · · · and π ∈ mA, we have

[πm+1]H(y) = [π]H ([πm]H(y)) = π ([πm]H(y)) + a2 ([πm]H(y))2 + · · · ∈ mm+2A .

For the second statement, we note that 0→ R→ K → K/R→ 0 induces an exact sequence

HomR(K,M)→ HomR(R,M)→ ExtR(K/R,M)→ ExtR(K,M).

So it suffices to prove that both the first and the last terms of the above sequence are 0. The first termis zero, since the image of any element of K under such an R-module homomorphism must lie in ∩πmM ,which is 0 by completeness. For the last term, we need to show that

0 −→Mf−→ N

g−→ K −→ 0

splits. To create a section g′ of g, we need to assign g′(π−i) for all i ≥ 0 such that πg′(π−i−1) = g′(π−i).Pick ni ∈ N such that g(ni) = π−i and let mi = ni − πni+1 ∈M . Then by completeness, we can define

g′(π−i) = ni − (mi + πmi+1 + π2mi+2 + · · · ).

for all i ≥ 0. This is a section of g.

49

We will now finish the proof of Theorem 4.3.4. Let (F univn , ϕuniv

n , id) be the universal deformation of leveln of the connected component G0. Let (F, ι,Φ) be a deformation of (G,φ) of level n over A. This is an exten-sion of (F 0, ι|F 0 ,Φ(π−nR/R)h), so we have the corresponding element E(F,ι,Φ) ∈ ExtR((K/π−nR)j , F 0(mA)) byProposition 4.3.7. There also exists a unique map f : Dh,n → A of objects in C such that (F 0, ι|F 0 ,Φ(π−nR/R)h)is isomorphic as a deformation of G0 to f∗(F univ

n , ϕunivn , id). This isomorphism induces a map

ExtR((K/π−nR)j , F 0(mA))→ ExtR

((K/π−nR)j ,

(f∗(F univ

n ))(mA)

),

and let E be the image of E(F,ι,Φ) under this map. Because equivalent extensions of the same deformation ofG0 are clearly isomorphic as deformations, using Proposition 4.3.7 again, the extension of f∗(F univ

n , ϕunivn , id)

corresponding to E is isomorphic as a deformation to (F, ι,Φ). Note that because of the way we fixed thequotient map F → SpfA(AS) (given ι) we force the commutativity of

F 01

//

∼=

F1//

α

SpfA(AS)

F 02

// F2// SpfA(AS)

whenever α : F1 → F2 is an isomorphism of deformations. Thus isomorphic deformations are indeed gettingsent to the same E by construction. Using Proposition 4.3.8, E is naturally sent to an element of mj

A. Becauseisomorphic deformations with the same connected component are certainly equivalent (again because of theway we fix the quotient map) as extensions, different elements of mj

A do correspond to different isomorphismclasses of deformations.

Therefore we have now shown that the set of deformations of level n of G over A is naturally in bijectionwith the set of maps Dh,n[[f1, . . . ,fj ]] → A in C . Thus Eh,n,j

∼= Dh,n[[f1, . . . ,fj ]], which is a regular localring of dimension h+ j by Theorem 4.2.4. We are finally done with proving Theorem 4.3.4.

In particular note that Eh,0,j∼= Rnr[[i1, . . . ,ih−1,f1, . . . ,fj ]]. This is formally smooth over Rnr.

50

Chapter 5

Applications of Deformation Theory

In this chapter we discuss two applications of deformation theory. The first application is that to local classfield theory, in particular to the Lubin-Tate theory. It turns out that we can prove the two main propositionsof the Lubin-Tate theory by the deformation theory of Chapter 4. Although it is not very difficult to provethese two propositions directly, this approach has the advantage that we can see them as the ‘height = 1’special case of the formal module theory.

The second application is Drinfel’d’s original purpose of developing this deformation theory of formalR-modules, namely that to the global function fields. Although we will not discuss this application in detail,we will try to give a feeling of how knowledge of various universal deformation rings that we constructed inChapter 4 is helpful in this application.

5.1 Applications to Local Class Field Theory

In this section, we will consider how Drinfel’d’s deformation theory of formal modules can be used to deriveLubin-Tate theory of the local class field theory. We continue the notations of Chapter 4: K for a local field,R for the valuation ring, π a uniformizer of R, k = R/(π) is a field with q elements (a power of p), Knr isthe maximal unramified extension (after fixing the algebraic closure) with valuation ring Rnr, and Knr isthe completion of Knr whiose valuation ring is Rnr.

There are two fundamental theorems in local class field theory: the isomorphism theorem which statesGal(L/K) ∼= K∗/NL/K(L∗) for every abelian extension L/K of local fields, and the existence theorem whichstates that every open and finite-indexed subgroup of K∗ is NL/K(L∗) for some abelian extension L/K. Bythese, we have a one-to-one correspondence between the abelian extensions of K and open and finite-indexedsubgroups of K∗. The problem is, though, if we start from an open and finite-indexed subgroup of K∗, itis often extremely difficult to construct the corresponding abelian extension of K. Neither the cohomologymethod (as in [25] or [26]) nor the character-theoretic method (as in [23]) gives us these concrete constructionsof abelian extensions.

To fix this problem, the Lubin-Tate theory was developed [18]. With this theory, we can explicitlyconstruct abelian extensions Kn

π of K corresponding to the norm subgroup 〈π〉 × U (n)K , where U (n)

K is then-th higher unit group. From this, we can show (

⋃Kn

π )Knr is the maximal abelian extension of K, thusobtaining a satisfactory explicit description of the maximal abelian extension. Moreover, Lubin-Tate theoryalso gives an explicit description of the Artin reciprocity map.

For construction of Knπ and proof of the above properties, Lubin and Tate exploits a general principle

of class field theory, which is to establish a connection between prime elements and Frobenius maps. Thisconnection is done through the notion of formal R-modules, as follows:

Proposition 5.1.1. If f ∈ R[[X]] such that f ≡ πX (mod poly. deg. 2) and f ≡ Xq (mod π), then thereexists a unique formal R-module Ff over R such that [π]Ff

(X) = f .

51

f satisfying the conditions in the above proposition is called a Lubin-Tate series, and the set of all suchseries is denoted by Fπ. The proofs of Lubin-Tate theory are essentially done by using one Lubin-Tate series,namely f = πX +Xq, but in specific examples (such as over Qp, we would want to use (1 +X)p − 1 to getthe cyclotomic field), it is useful to be able to choose from Fπ.

For construction of the Artin reciprocity map, we need to know that the construction does not dependon the choice of the uniformizer. The following is the essential:

Proposition 5.1.2. Let σ be the continuous extension of the Frobenius automorphism of Gal(Knr/K) toKnr. Let π, ω be two uniformizers of R such that ω = πu, and let f ∈ Fπ and g ∈ Fω. Then there exists anisomorphism α ∈ Rnr[[X]] from Ff to Fg of formal R-modules such that σ∗(α) = α [u]Ff

.

These are the two main propositions of the Lubin-Tate theory. The usual method of proving these usesuccessive approximation together with surjectivity of u 7→ σ(u)

u in U bKnr ([22], [25]). We will now give theproof of these propositions using Drinfel’d’s theory of deformation of formal R-modules.

Proof of Proposition 5.1.1. By Proposition 3.3.2, a normal formal R-module over k of height 1 correspondsto some R[Υ1, . . .]→ Fq, so by lifting image of each Υi, we get a formal R-module F2 over R such that

[π]F2(X) ≡ f (mod poly. deg. 2), and [π]F2(X) ≡ Xq (mod π).

Suppose we have constructed up to Fn such that

[π]Fn(X) ≡ f (mod poly. deg. n) and [π]Fn(X) ≡ Xq (mod π).

The image Fn+1,r of Fn via the isomorphism ϕ = (X + rXn)−1 with r ∈ R has the property that[π]Fn+1,r (X) ≡ Xq (mod π) and [π]Fn+1,r (X) ≡ [π]Fn(X) + (rπ − rπn)Xn (mod poly. deg. n + 1), sof ≡ [π]Fn (mod π) from the inductive hypothesis gives us the suitable r to continue induction. We aredone by taking limit of the formal R-modules Fn.

Uniqueness is successive application of Proposition 3.3.4 and the fact that R is an integral domain.

Proof of Proposition 5.1.2. Both Ff and Fg are height 1 formal R-modules over k, so by Theorem 3.3.11,there exists an isomorphism α : Ff → Fg. Thus, (Ff , α) and (Fg, id) are both deformations of Fg in thesense of Definition 4.1.1, so by Theorem 4.1.7 applied to h = 1, there must exist an isomorphism of formalR-modules α : Ff → Fg over Rnr which reduces to α modulo m bRnr . Clearly, α [u]Ff

defines a formalR-module homomorphism from Ff to Fg, and because σ fixes all elements of R and Ff is defined overR, it follows that σ∗(α)(Ff (X,Y )) = σ∗(α(Ff (X,Y ))) = σ∗(Fg(α(X), α(Y ))) = Fg(σ∗(α)(X), σ∗(α)(Y )).Similarly, σ∗(α)([r]Ff

(X)) = [r]Fg(σ∗(α)(X)), so σ∗(α) is a formal R-module homomorphism from Ff to Fg.

On the residue field, we have

σ∗(α)(Xq) = (α(X))q = [ω]Fg(α(X)) = α([ω]Ff

) = α([u]Ff(Xq)),

so σ∗(α)(X) = α [u]Ff(X) on the residue field, so by Proposition 4.1.3, we obtain σ∗(α) = α [u]Ff

.

Once we have these two propositions, we can simply follow [22] and [25], as below. Let us denote m = mK .Since we have R-modules Ff (mL) for each finite extension L/K in a compatible manner, Mf = Ff (m) isalso an R-module. Let En

f denote the πn-division points of the module Mf , i.e.

Enf = λ ∈ m : [πn]Ff

(λ) = 0,

and let Knf = K(En

f ). By Proposition 5.1.2 applied with ω = π, for f, g ∈ Fπ, there is an isomorphismα of formal R-modules Ff and Fg over ObKnr with σ∗(α) = α (so α has coefficients in R). Thus, α sendsπn-division points of Ff to that of Fg, so we see Kn

f = Kng . We will denote this common field by Kn

π . Usingthe two propositions above, we have

52

Theorem 5.1.3. (a) Enf∼= R/(πn) as R-modules. So we have Aut(En

f ) ∼= UK/U(n)K .

(b) Knπ is a totally ramified abelian extension of K. If f is of the form Xq +π(aq−1X

q−1 + · · ·+a2X2)+πX,

every element in Enf \E

n−1f is a prime element of Kn

π whose minimal polynomial is an Eisenstein polynomial

gn =f (n)(X)f (n−1)(X)

= (f (n−1)(X))q−1 + π(aq−1(f (n−1)(X))q−2 + · · ·+ a2(f (n−1)(X))

)+ π ∈ OK [X],

where f (n)(X) = f · · · f n times. Moreover, Gal(Knπ/K) ∼= Aut(En

f ) by the natural restriction map.

(c) NKnπ /K((Kn

π )∗) contains 〈π〉 × U (n)K .

Proof. For (a) and (b), see [22] or [25]. For (c), see [14, Lemma 5.15].

Using the isomorphism theorem and counting degrees, we now see thatNKnπ /K((Kn

π )∗) = 〈π〉×U (n)K . Using

this and NKn/K(K∗n) = 〈πn〉 for the unramified extension Kn/K of degree n, corollaries of the isomorphism

and existence theorems (such as N(L1 ·L2) = N(L1)∩N(L2)) show that (⋃Kn

π )Knr is the maximal abelianextension Kab of K (see [22, Corollary V.5.7] for details).

We can also express the Artin reciprocity map in terms of Knπ . Note that Gal(Kab/K) ∼= Gal(Kπ/K)×

Gal(Knr/K), where Kπ =⋃Kn

π . For any f satisfying the conditions of Theorem 5.1.3 (b), let [u]f denotethe unique element of Gal(Kπ/K) whose restriction to En

f agrees with [u]Fffor all n. We can now define a

well-defined group homomorphism rf : K∗ −→ Gal(Kab/K) by(1) rf (π) = id on Kπ and Frobenius σ on Knr,(2) For u ∈ UK , rf (u) = [u−1]f on Kπ and id on Knr.

We will now check that if we use another uniformizer ω = πu and g ∈ Fω, rf = rg. Since uniformizers generateK∗, it suffices to check rf (ω) = rg(ω). Thus we need to show rf (ω) = id on Kω, so let λ ∈ Eg =

⋃En

g .Letting α be the formal R-module isomorphism over Rnr from Ff to Fg obtained from Proposition 5.1.2, weknow that λ = α(µ) for some µ ∈ Ef . So what we need to check is rf (ω)(α(µ)) = α(µ). But since α hascoefficients in Rnr, we have

rf (ω)(α(µ)) = σ∗(α)(rf (ω)(µ)) = σ∗(α)(rf (π) rf (u)(µ)) = σ∗(α)([u−1]f (µ)) = α(µ).

So indeed, rf = rg, and because of the property that r(π) is identity on corresponding Kπ and Frobenius onKnr for each uniformizer π, this must agree with the reciprocity map.

Remark. If we did not assume the isomorphism and existence theorems of local class field theory, wecould actually prove them using Kn

π ([12], [14]). After establishing Theorem 5.1.3, we can obtain N(Knπ ) =

〈π〉 × U (n)K by Coleman’s Norm operator [14] or by ‘denseness’ of separable polynoamials [12]. We can also

prove the Hasse-Arf theorem by elementary congruences as in [24]. Then using Gal(Knπ/K)n = 1 as in [9]

and a general fact that Gal(L/K)n = 1 =⇒ |Gal(L/K)0| ≤ qn−1(q − 1) for an abelian extension L/K,we can show that any finite abelian extension is contained in KπK

nr. From these information, we can provethe two fundamental theorems of local class field theory, using the above description of the Artin reciprocitymap ([14, Chapter VI]).

5.2 Applications to the Arithmetic of Function Fields

Now we will briefly discuss Drinfel’d’s original purpose for developing the deformation theory of formalR-modules, namely the study of global function fields. As discussed in the introduction, Drinfel’d wasinterested in proving the analogues of the fundamental theorem of Complex Multiplication and the LanglandsConjecture for a global function field K. For this, Drinfel’d introduces the notion of elliptic OK-modulesof rank d over a scheme S (now in the literature, these are called Drinfel’d modules). Further, Drinfel’ddefines a level structure for each ideal I ⊆ OK , and defines a modular variety Md

I to be the one whichrepresents (i.e. ‘parametrizes’) the functor S the set of Drinfel’d OK-modules (up to isomorphism) of rankd with structure of level I over S.

53

Since it is hard to directly study the modular variety (which contains information about ‘all’ the Drinfel’dmodules simultaneously), it is helpful to study locally at first at Drinfel’d modules which are ‘close’ to agiven one. The local analysis of Drinfel’d OK-modules leads to studying deformations of divisible formal(OK)v-modules for each v ∈ Spec OK (here each v is viewed as a place), which have already been discussed indetail in Section 4.3. In essence, any ‘local’ properties of the modular varieties that one wants to prove canbe proved using the local deformation theory. For example, as smoothness and flatness are local properties,the proof of the smoothness of Md

I over SpecA−V (I) (where V (I) as usual denotes primes which contain I)and flatness of Md

J →MdI for J ⊆ I do reduce to the nice properties of the universal deformation ring that we

have obtained in Theorem 4.3.4 ([5, Corollary of Proposition 5.4]). Furthermore, using a similar reductionprocess to the local deformation theory, Drinfel’d defines a group action on the (global) modular varietylocally by using Proposition 4.2.7. So the local deformation theory lies at the foundation of Drinfel’d’s studyof global function fields.

As discussed in [10], the arithmetic of function fields developed dramatically since this Drinfeld’s paper.It was particularly influential in the explicit class field theory for the function fields which Hayes and othersdeveloped. Moreover, even outside the strict realm of global function fields, Drinfel’d’s ideas of the ‘full levelstructures’ over a general scheme gave a new interpretation to reduction modulo p of the moduli space ofelliptic curves with N -level structure where p | N , and in fact, [16] uses this idea of level structures to thestudy of moduli spaces of elliptic curves.

Thus, [5] is an extremely important paper, proving analogues of theorems of number fields in the equi-characteristic global function fields case, and significantly contributing to the development of study of func-tion fields and other areas of math. At the basis of this is the local deformation theory of formal R-moduleswhich has been presented in detail in the previous chapters, and it is the author’s hope that both beautyand the importance of this theory have been conveyed to the reader.

54

Appendix A

Profinite Modules and CommutativeFormal Group Schemes

In this appendix, we will briefly discuss some properties and proof techniques concerning profinite modulesand commutative formal group schemes. We will omit many proofs here. Since we pick out only thingsthat are needed for this thesis, this is not necessarily the best approach to take (for instance, many of thefollowing hold in more generality, where proofs are simplified). For complete proofs and for a more coherentand thorough approach, see [3] (which is based on SGA3).

A.1 Preliminary

Suppose A is a topological ring which has a base of opens around 0 consisting of ideals. We define A =lim←−A/a, where a runs through all the open ideals with the directed system being the reverse inclusion. Suchan A is said to be pseudocompact if the natural continuous map A −→ A is a topological isomorphismand A/a is an artinian ring for all open ideals a. Similarly, given a topological A-module M with base ofopens around 0 given by A-submodules, we define M = lim←−M/M ′, where M ′ runs through all the openA-submodules. We say such an M is profinite if the natural continuous map M −→ M is a topologicalisomorphism and each M/M ′ is a finite-length A-module. We denote the category of profinite A-modules byPFA. We call an A-algebra B profinite if B is profinite as an A-module; the category of all such A-algebrasis denoted by PAA.

From now on, A will denote a pseudocompact ring, and all modules are assumed to be profinite. Theaim of the notion of pseudocompactness is to preserve some nice properties of artin rings in a frameworkwhich is stable under formation of inverse limits. Therefore, properties which hold for artin rings usuallyhold for pseudocompact rings if we insert the word ‘open’ in the appropriate places. For instance, every openprime of a pseudocompact ring is maximal. On the other hand, pseudocompact rings are often huge (suchas arbitrary products of complete noetherian local rings with the product topology). Such rings naturallyarise in the context of etale formal groups.

For a profinite A-module M and an element m of an open submodule M ′, 0 ·m ∈M ′ implies that thereexists an open ideal a of A such that a · m ⊆ M ′. Since M/M ′ is a finite A-module, by taking a finiteintersection of such a’s, we conclude that there exists an open ideal b such that bM ⊆ M ′. Thus, eachquotient of M by an open submodule is finite over an artinian ring (thus it is artinian). By this argument,we see that every object of PAA is pseudocompact.

In general, inverse limits are only left-exact. But if Mα −→ Nα is a map between two systems ofA-modules such that Mα −→ Nα is surjective with kernel an artinian module for each α, we can show thatlim←−Mα −→ lim←−Nα is also surjective by the use of Zorn’s lemma (this is one of the places where the artinianassumption is crucial, as it ensures a kind of “Mittag-Leffler” property). From this, it follows that givenany closed submodule N of M ∈ PFA, N and M/N are in PFA with the subspace and quotient topologies,

55

respectively. Further, any map between objects in PFA is closed and PFA is an abelian category wheremonomorphisms are injective maps and epimorphisms are surjective maps ([3, Theorem 1.2.2] for all of theabove).

Since A ∼= A implies that the intersection of all open ideals is (0), when A is artinian, it follows that (0)is open (i.e. the topology is discrete). We would like to have a localization which behaves well topologically,so it makes sense to define localization of an artinian ring as the algebraic localization with the discretetopology and to define localization of a psueudocompact ring as their inverse limit. In other words, for apseudocompact ring A and an open maximal ideal m, we define

Am = lim←−(A/a)m/a,

where the limit is taken over all open ideals a ⊆ m. Since the inverse limit of profinite A-algebras is again aprofinite A-algebra ([3, Corollary 1.2.3]), Am is a profinite A-algebra (thus, it is pseudocompact). Moreover,using the structure theorem of artinian rings, it follows that A is isomorphic to

∏Am where the product is

taken over open maximals. Moreover, by the universal property (and using that Am is local), it follows thatAm is the algebraic localization Am if we ignore the topology ([3, Theorem 1.2.5]). From now on, we willwrite Am for the topological localization as well. This isomorphism A ∼=

∏Am enables us to reduce many

proofs to the local case. Using this decomposition, we can show that the set τ of topological nilpotents(a ∈ A with an → 0) satisfy A/τ ∼=

∏Am/mAm where the product is over the open maximals ([3, Corollary

1.2.6]).For any topological A-modules M,N , we can define completed tensor product

M⊗AN = lim←−(M/M ′)⊗A (N/N ′)

where triples of opens (a,M ′, N ′) : aM ⊆ M ′, aN ⊆ N ′ are made into a directed system by reverseinclusion. ⊗ has the usual universal property (with the additional continuity assumption). ⊗ also has theusual base change properties: for a pseudocompact ring A, a pseudocompact A-algebra B, a pseudocompactB-algebra C, and M,N ∈ PFA, we have M⊗AN ∈ PFA, M⊗AB ∈ PFB , (M⊗AB)⊗BC ∼= M⊗AC ([3,Theorem 1.1.8 and Corollary 1.3.8]). As we expect, the functor M⊗A : PFA → PFA is right-exact and wehave natural isomorphisms

M⊗A

(∏i∈I

Ni

)∼=∏i∈I

(M⊗ANi

), M⊗A

(lim←−Ni

)∼= lim←−(M⊗ANi),

M ∼=∏

(M⊗AAm), and HomPFA(M,N) ∼=

∏HomPFAm

(M⊗AAm, N⊗AAm),

where the product in the last line is over all open maximals of A ([3, Theorem 1.3.1]). When B ∈ PAA isused instead of M , we get the similar decomposition of HomPAA

. This shows that M⊗AAm is the algebraiclocalization as modules (by universal property). All of these results are crucial, since they enable us toanalyze each local part separately.

One advantage of going to the local case, as in the usual commutative ring case, is Nakayama’s lemma.For a pseudocompact local A with the unique (thus, open by A ∼= A) maximal ideal m, mM = M impliesM = 0 for a profinite A-module ([3, Theorem 1.3.2]). Thus, M⊗A(A/m) ∼= M/mM = 0 implies M = 0.This is a powerful tool which allows us to analyze our situation over formal closed fiber. For instance, ifM → N is a map of PFA with N a projective object and the induced map M⊗A(A/m)→ N⊗A(A/m) is anisomorphism, then M ∼= N ([3, the proof of Theorem 1.3.6]). Also, if B is a local ring which is a profiniteA-algebra, then the inverse image in A of mB is an open prime in A, so the map A→ B is local, and B/mB

is a finite extension field of A/mA.We say M ∈ PFA is topologically free if M ∼=

∏Aei, M is topologically flat if M⊗A is exact on PFA,

and M is topologically faithfully flat if a sequence of PFA is exact if and only if the sequence ⊗AM isexact. Using the above properties of ⊗ together with Nakayama/projectivity arguments and Zorn’s lemma,we can show that M ∈ PFA is a projective object ⇐⇒ M is topologically flat ⇐⇒ Mm is topologically freeover Am for each open maximal m ([3, Theorem 1.3.6]). Further, M is topologically faithfully flat if and

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only if it is topologically flat and Mm 6= 0 for all open maximals; in particular, B ∈ PAA is topologicallyfaithfully flat over A if and only if it is topologically flat and the map A → B is injective ([3, Corollary1.3.7]). Thus, these conditions can be checked locally. We can also show by the usual method that B ∈ PAA

is topologically faithfully flat if and only if it is topologically flat and B⊗AM = 0 implies M = 0 for everyM ∈ PFA. Finally, using the Nakayama/projectivity argument, we can prove the ‘fiber-by-fiber’ criterionfor topological flatness ([3, Theorem 2.3.2]):

Theorem A.1.1. A map B1 −→ B2 between topologically flat profinite A-algebras is topologically flat if andonly if it is topologically flat on the formal fiber over all open maximals of A (i.e. after applying ⊗A(A/m)for all open maximals m).

A.2 Formal Schemes

Let FA be the category of profinite A-algebras which are artinian as a ring (this is the ‘finite quotient part’of PAA). Given a covariant functor X : FA → Set, by letting X ′(B) = lim←−X(B/b) where inverse limit runsthrough all open ideals of B ∈ PAA, we obtain a functor X ′ : PAA → Set. We can show ([3, Theorem1.4.2]) that this functor commutes with inverse limits (i.e. X ′(lim←−Bi) = lim←−X

′(Bi) for every inverse systemsof elements in PAA). This X X ′ defines an equivalence between the category of covariant functorsFA → Set and the category of covariant functors PAA → Set which are ‘continuous’ in the sense thatthey commute with inverse limits, where the inverse process is given by the restriction (in both categories,the morphisms are the natural transformations). Note that the representable functors HomPAA

(B,−) are‘continuous’. We define a formal scheme over A to be a covariant functor SpfA(B) : FA → Set, which isthe restriction of HomPAA

(B,−) to FA for some B ∈ PAA. In other words, a formal scheme is a functor onFA which is representable by a profinite object. The above discussion together with Yoneda’s lemma showsthat HomPAA

(B2, B1) is naturally isomorphic to Hom(SpfA(B1),SpfA(B2)). By the universal property ofcompleted tensor product and Yoneda’s lemma, SpfA(B1)× SpfA(B2) is naturally isomorphic to the functorSpfA(B1⊗AB2), and for A′ ∈ PAA, the restriction of the functor SpfA(B) to FA′ is naturally isomorphic tothe functor SpfA′(A′⊗AB). Thus, the formal schemes over A form a “category” in which products and thefinal object (i.e. SpfA(A)) exist.

One of the reasons for wanting to deal with profinite objects (with large index set) is to have the followingnice representability theorem of Grothendieck. We won’t need this, but mention it in passing ([3, Theorem1.4.7]).

Theorem A.2.1. A covariant functor X : FA → Set is a formal scheme over A if and only if it is left exactand commutes with products.

A.3 Duality and Commutative Group Schemes

We will first talk about the duality of profinite modules. Suppose A is an artin local ring (which is pseudocom-pact with the discrete topology) with maximal ideal m andM is a topologically flat profinite A-module. SinceA = Am, M must be topologically free (see section A.1), soM ∼=

∏Aei. For any element ϕ ∈ HomPFA

(M,A),ϕ−1(0) is open, so ϕ factors through a finite product of A’s. Hence, if we write MD = HomPFA

(M,A), thenMD is an A-module which is the direct sum of A’s with the same index set as the original product forM . Conversely, starting from a free A-module N , we can define ND = HomA(N,A), where we define opensubmodules to be the annihilators of finite-length submodules of N . This makes ND ∼=

∏Aei with the same

index set as that for N , so ND is an object of PFA which is topologically free. Because free objects in thecategory of A-modules (resp. topologically flat objects in PFA) are projective in their respective categories(see Section A.1), every exact sequence splits in each category. Moreover, (resp. completed) tensor productcommutes with direct sums (resp. products). Hence, we have the following ([3, Theorem 1.3.4]):

Theorem A.3.1. Assume A is an artin local ring. For a topologically flat M ∈ PFA and a free A-moduleN , M −→ MDD and N −→ NDD are natural isomorphisms. Moreover, the functors (−)D take exactsequences of free A-modules (resp. topologically flat profinite A-modules) to exact sequences of topologically

57

flat profinite A-modules (resp. free A-modules). (−)D also interchange direct products with direct sums.Further, (M ′)D ⊗A MD −→ (M ′⊗AM)D and (N ′)D⊗AN

D −→ (N ′ ⊗A N)D are natural isomorphisms.

As a consequence of this duality, we can prove that the functors (−)D take commutative group objectswhich are free (resp. topologically flat) to commutative group objects which are topologically flat (resp.free) by the same argument as in the proof of Cartier duality ([31, page 17]).

When A = k is a field, we call a commutative group object in the category of affine (resp. formal) schemesover k to be a commutative affine (resp. formal) group scheme. This is the simplest case, and thequestions about group objects over more general A will often be reduced to the case over fields by basechange, so it is fortunate that we have the following theorem ([3, Theorem 2.1.4]):

Theorem A.3.2. The category of affine (resp. formal) commutative group schemes is abelian, with monomor-phisms being the surjections on rings and epimorphisms being the (resp. topologically) faithfully flat mapson rings.

Now, we assume that A is a pseudocompact local ring, with the maximal ideal m. This is more complicatedthan the field case, since in general we will not be able to conclude that the category is abelian. But we dowant to have a concept of a short exact sequence, so we will now define this with a bit more care. Note thatin our discussion of duality, having flatness was important. So we actually define commutative formalgroup scheme over A to be a commutative group object in the category of formal schemes such that it isalso topologically flat over A. We then say that a sequence of commutative formal group schemes

0→ SpfA(B1)→ SpfA(B2)→ SpfA(B3)→ 0 (A.1)

is a short exact sequence if the composition of the two maps is zero, SpfA(B1) → SpfA(B2) is a closedimmersion (i.e. B1 ← B2 is surjective), SpfA(B2) → SpfA(B3) is topologically faithfully flat, and SpfA(B1)is the kernel of SpfA(B2)→ SpfA(B3) (i.e. B1 ← B2⊗B3A is an isomorphism).

We will now show that (A.1) is a short exact sequence if and only if the composition of the two maps iszero and it is a short exact sequence modulo the maximal ideal m (i.e. after applying ⊗A(A/m), where weare now working inside an abelian category). The formal Nakayama shows that SpfA(B1) → SpfA(B2) is aclosed immersion if and only if it is so over the closed fiber. If B2 ← B3 is topologically faithfully flat, thenso is the map over the closed fiber. Conversely, by Theorem A.1.1, we know topological flatness of B2 ← B3

knowing topological flatness over the closed fiber, so we just need to show that B2⊗B3M = 0 implies M = 0for M ∈ PFB3 . But this follows from 0 = (A/m)⊗AB2⊗B3M

∼=((A/m)⊗AB2

)⊗A/mb⊗AB3

(A/m⊗AM

)and

Nakayama’s lemma. Finally, because B1 is topologically flat, Nakayama/projectivity argument shows thatB1 ← B2⊗B3A is an isomoprphism if and only if it is so over the closed fiber. Hence by Theorem A.3.2,we have now proved the equivalence of these two notions of a short exact sequence of commutative formalgroup schemes. We will use the more convenient definition for each occasion in this thesis.

Of course, to justify the terminology ‘short exact sequence,’ we need to prove several properties. Forexample, we can extend a closed immersion SpfA(B1) → SpfA(B2) of commutative formal group schemesto a short exact sequence. To construct the quotient, we first observe that we only have to do this overartin local rings, since we can define the quotient as the inverse limit of the quotients over A/a’s in general.When A is an artin local ring, we have the duality. In particular, we can show that BD

1 → BD2 is faithfully

flat by the fiber-by-fiber criterion, so we can base change by the identity section to obtain a faithfully flataffine commutative group scheme over A. The Cartier dual of this is the quotient formal commutative groupscheme (SpfA(B2)) / (SpfA(B1)) (for justification, see [3, page 54]).

We can also extend a topologically faithfully flat SpfA(B2)→ SpfA(B3) to a short exact sequence, simplyby taking the kernel (which is a commutative formal group scheme over A). This notion of short exactsequence also satisfies stability under products and stability under base change. We will not need all of theseproperties, but what we will need is proven in Section 2.3, together with important examples.

A.4 Formal Etaleness

Given a pseudocompact local ring A, and R ∈ PAA, we can have the notion of the topologized moduleof differentials. Specifically, we let IR be the closed ideal ker(R⊗AR → R), and define Ω1

R/A to be

58

IR/I2R. We can show the usual properties of the module of differentials in this setting by the same proof,

taking care with continuity: we have first and the second fundamental sequences, a natural isomorphismΩR/A⊗AA

′ ∼= Ω1Rb⊗AA′/A′

whenever A → A′ a (continuous) map of pseudocompact rings, and Ω1Rb⊗AS/A

∼=Ω1

Rb⊗AS/R⊕ Ω1

Rb⊗AS/Sfor R,S ∈ PAA ([3, Theorem 1.5.7]). We say a profinite A-algebra R is formally etale

if R is topologically flat over A and Ω1R/A = 0. We immediately see that taking completed tensor product

of two formally etale algebras is again formally etale, and that etaleness is preserved via base change. Byanalyzing what happens for each artinian quotient, we can prove the following useful theorem ([3, Theorem1.5.9]):

Theorem A.4.1. For R ∈ PAA, R is formally etale if and only if R is topologically flat over A andR⊗A(A/m) is a product of finite separable field extensions of A/m.

Because a pseudocompact local ring is henselian (page 27), it follows that for B ∈ PAA which is topo-logically flat, there exists a unique maximal formally etale closed A-subalgebra Bet. By construction, themap Bet → B is topologically faithfully flat and (

∏Bi)

et =∏Bi

et ([3, Theorem 1.5.10]). By reducing tothe analysis of non-formal etaleness of artin local rings, we can also show that for a local map of pseudo-compact local rings A→ A′, Bet⊗AA

′ → B⊗AA′ is the maximal etale subalgebra ([3, Theorem 1.5.12]) and

B1et⊗AB2

et → B1⊗AB2 is the maximal etale subalgebra ([3, Theorem 1.5.11]). Moreover, any map from aformally etale A-algebra to B factors through Bet.

A.5 Formal Smoothness

Again, for simplicity, let us assume that A is a local pseudocompact ring with maximal ideal m. Let B ∈ PAA.Then B is said to be formally smooth over A if for every profinite artinian A-algebra R and an ideal I ⊆ Rwith I2 = 0, the map SpfA(B)(R) → SpfA(B)(R/I) is surjective. Because we can write R =

∏Rn where

n runs through open maximals of R and (R/I)n∼= Rn/In, it suffices to check the above condition for artin

local R with the maximal ideal n. If ϕ ∈ SpfA(∏Bi)(R), then ϕ−1(n) is an open prime, so it is an open

maximal by pseudocompactness. But the only idempotent inside n is 0 by locality of R, so ϕ factors throughone of Bi’s. Hence,

SpfA(∏

Bi)(R) =∐

SpfA(Bi)(R). (A.2)

Thus, it is now clear that∏Bi is formally smooth over A if and only if each Bi is formally smooth over A.

In particular, B is formally smooth over A if and only if localization of B at each of the open maximals isformally smooth over A.

So now the important question about formal smoothness is determination of all the local profinite A-algebras which are formally smooth. We have the following result to this effect, but we won’t need it: if Bis a local profinite A-algebra such that A/m ∼= B/mB by the structure map, then B is formally smooth ifand only if B ∼= A[[Xi]] for some index set ([3, Theorem 1.5.15]).

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