p-adic Lie groups

Grundlehren dermathematischen Wissenschaften 344A Series of Comprehensive Studies in Mathematics

Series editors

M. Berger P. de la Harpe F. HirzebruchN.J. Hitchin L. Hörmander A. KupiainenG. Lebeau F.-H. Lin S. MoriB.C. Ngô M. Ratner D. SerreN.J.A. Sloane A.M. Vershik M. Waldschmidt

Editor-in-Chief

A. Chenciner J. Coates S.R.S. Varadhan

For further volumes:www.springer.com/series/138

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Peter Schneider

p-Adic Lie Groups

Peter SchneiderInstitute of MathematicsUniversity of MünsterEinsteinstrasse 62Münster [email protected]

ISSN 0072-7830ISBN 978-3-642-21146-1 e-ISBN 978-3-642-21147-8DOI 10.1007/978-3-642-21147-8Springer Heidelberg Dordrecht London New York

Library of Congress Control Number: 2011930424

Mathematics Subject Classification: 22E20, 16S34

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Introduction

This book presents a complete account of the foundations of the theory ofp-adic Lie groups. It moves on to some of the important more advancedaspects. Although most of the material is not new, it is only in recent yearsthat p-adic Lie groups have found important applications in number theoryand representation theory. These applications constitute, in fact, an increas-ingly active area of research. The book is designed to give to the advanced,but not necessarily graduate, student a streamlined access to the basics ofthe theory. It is almost self contained. Only a few technical computationswhich are well covered in the literature will not be repeated. My hope is thatresearchers who see the need to take up p-adic methods also will find thisbook helpful for quickly mastering the necessary notions and techniques.The book comes in two parts. Part A on the analytic side grew out of acourse which I gave at Munster for the first time during the summer term2001, whereas part B on the algebraic side is the content of a course givenat the Newton Institute during September 2009.

The original and proper context of p-adic Lie groups is p-adic analysis.This is the point of view in Part A. Of course, in a formal sense the notionof a p-adic Lie group is completely parallel to the classical notion of a realor complex Lie group. It is a manifold over a nonarchimedean field whichcarries a compatible group structure. The fundamental difference is thatthe p-adic notion has no geometric content. As we will see, a paracompactp-adic manifold is topologically a disjoint union of charts and therefore is,from a geometric perspective, completely uninteresting. The point insteadis that, like for real Lie groups, manifolds and Lie groups in the p-adic worldare a rich source, through spaces of functions and distributions, of interest-ing group representations as well as various kinds of important topologicalgroup algebras. We nevertheless find the geometric language very intuitiveand therefore will use it systematically. In the first chapter we recall whata nonarchimedean field is and quickly discuss the elementary analysis oversuch fields. In particular, we carefully introduce the notion of a locally an-alytic function which is at the base for everything to follow. The secondchapter then defines manifolds and establishes the formalism of their tan-gent spaces. As a more advanced topic we include the construction of thenatural topology on vector spaces of locally analytic functions. This is dueto Feaux de Lacroix in his thesis. It is the starting point for the represen-tation theoretic applications of the theory. In the third chapter we finallyintroduce p-adic Lie groups and we construct the corresponding Lie algebras.The main purpose of this chapter then is to understand how much informa-

v

vi Introduction

tion about the Lie group can be recovered from its Lie algebra. Here againlies a crucial difference to Lie groups over the real numbers. Since p-adic Liegroups topologically are totally disconnected they contain arbitrarily smallopen subgroups. Hence the Lie algebra determines the Lie group only locallyaround the unit element which is formalized by the notion of a Lie groupgerm. As the length of the chapter indicates this relation between Lie groupsand Lie algebras is technically rather involved. It requires a whole range ofalgebraic concepts which we all will introduce. As said before, only for a fewcomputations the reader will be referred to the literature. The key result iscontained in the discussion of the convergence of the Hausdorff series.

There are three existing books on the material in Part A: “Varietesdifferentielles et analytiques. Fascicule de resultats” and “Lie Groups andLie Algebras” by Bourbaki and Serre’s lecture notes on “Lie Algebras andLie groups”. The first one contains no proofs, the nature of the second one isencyclopedic, and the last one some times is a bit short on details. All threedevelop the real and p-adic case alongside each other which has advantagesbut makes a quick grasp of the p-adic case alone more difficult. The presen-tation in the present book places its emphasis instead on a streamlined butstill essentially self contained introduction to exclusively the p-adic case.

Lazard discovered in the 1960s a purely algebraic approach to p-adicLie groups. Unfortunately his seminal paper is notoriously difficult to read.Part B of this book undertakes the attempt to give an account of Lazard’swork again in a streamlined form which is stripped of all inessential general-ities and ramifications. Lazard proceeds in an axiomatic way starting fromthe notion of a p-valuation ω on a pro-p-group G. After some preliminariesin the fourth chapter this concept is explained in chapter five. It will not betoo difficult to show that any p-adic Lie group has an open subgroup whichcarries a p-valuation. Lazard realized that, vice versa, any pro-p-group witha p-valuation (and satisfying an additional mild condition of being “of finiterank”) is a compact p-adic Lie group in a natural way. The technical toolto achieve this important result is the so called completed group ring Λ(G)of a profinite group G. It is the appropriate analog of the algebraic groupring of a finite (or, more generally, discrete) group in the context of profinitegroups. In the presence of a p-valuation ω Lazard develops a technique ofcomputation in Λ(G), which as such is a highly complicated and in generalnoncommutative algebra. All of this will be presented in the sixth chapter.In the last chapter seven we go back to Lie algebras. Being a p-adic Lie groupa pro-p-group G with a p-valuation of finite rank ω has a Lie algebra Lie(G)over the field of p-adic numbers Qp. By inverting p and a further completionprocess the completed group ring Λ(G) can be enlarged to a Qp-Banach

Introduction vii

algebra ΛQp(G,ω) which turns out to be naturally isomorphic to a certaincompletion of the universal enveloping algebra of Lie(G). This is anotherone of Lazard’s important results. It provides us with a different route toconstruct Lie(G) which is independent of any analysis. In fact, it does betterthan that since it leads to a natural Lie algebra over the ring over p-adicintegers Zp associated with the pair (G,ω). This means that the algebraictheory, via this notion of a p-valuation, makes the connection between Liegroup and Lie algebra much more precise than the analytic theory was ableto do. The final question addressed in the last chapter is the question onthe possibility of varying the p-valuation on the same group G. Using thenewly established direct connection to the Lie algebra this problem can betransferred to the latter. There it eventually becomes a problem of convexitytheory which is much easier to solve. This, in particular, allows to prove thevery useful technical fact that there always exists a p-valuation with ratio-nal values. Its most important consequence is the result that the completedgroup ring Λ(G) of any (G,ω) of finite rank is a noetherian ring of finiteglobal dimension. This is why completed group rings of p-adic Lie groupshave become important in number theory (where they are applied to Galoisgroups G), and why they deserve further systematic study in noncommuta-tive algebra.

This is the first textbook in the proper sense on Lazard’s work. Thebook “Analytic Pro-p-Groups” by Dixon, du Sautoy, Mann, and Segal has acompletely different perspective. It is written entirely from the point of viewof abstract group theory. Moreover, it does not mention Lazard’s conceptof a p-valuation at all but replaces it by an alternative axiomatic approachbased on the notion of a uniformly powerful pro-p-group. This approach isvery conceptual as well but also less flexible and more restrictive than theone by Lazard which we follow.

It is a pleasure to thank J. Coates for persuading me to undertake thislecture series at the Newton Institute and to write it up in this book, theaudience for the valuable feedback, the Newton Institute for its hospitalityand support, and T. Schoeneberg for a careful reading of Part B.

Munster, February 2011 Peter Schneider

Contents

A p-Adic Analysis and Lie Groups 1

I Foundations 3

1 Ultrametric Spaces 3

2 Nonarchimedean Fields 8

3 Convergent Series 14

4 Differentiability 17

5 Power Series 25

6 Locally Analytic Functions 38

II Manifolds 45

7 Charts and Atlases 45

8 Manifolds 47

9 The Tangent Space 56

10 The Topological Vector Space Can(M,E), Part 1 74

11 Locally Convex K-Vector Spaces 79

12 The Topological Vector Space Can(M,E), Part 2 84

III Lie Groups 89

13 Definitions and Foundations 89

14 The Universal Enveloping Algebra 101

ix

x Contents

15 The Concept of Free Algebras 106

16 The Campbell-Hausdorff Formula 111

17 The Convergence of the Hausdorff Series 124

18 Formal Group Laws 132

B The Algebraic Theory of p-Adic Lie Groups 155

IV Preliminaries 157

19 Completed Group Rings 157

20 The Example of the Group Zdp 163

21 Continuous Distributions 164

22 Appendix: Pseudocompact Rings 165

V p-Valued Pro-p-Groups 169

23 p-Valuations 169

24 The Free Group on Two Generators 175

25 The Operator P 178

26 Finite Rank Pro-p-Groups 181

27 Compact p-Adic Lie Groups 192

VI Completed Group Rings of p-Valued Groups 195

28 The Ring Filtration 195

29 Analyticity 201

30 Saturation 208

Contents xi

VII The Lie Algebra 219

31 A Normed Lie Algebra 219

32 The Hausdorff Series 232

33 Rational p-Valuations and Applications 243

34 Coordinates of the First and of the Second Kind 247

References 251

Index 253

Part A

p-Adic Analysis and Lie

Groups

Chapter I

Foundations

1 Ultrametric Spaces

We begin by establishing some very basic and elementary notions.

Definition. A metric space (X, d) is called ultrametric if the strict triangleinequality

d(x, z) ≤ max(d(x, y), d(y, z)) for any x, y, z ∈ X

is satisfied.

Examples will be given later on.

Remark. i. If (X, d) is ultrametric then (Y, d |Y × Y ), for any subsetY ⊆ X, is ultrametric as well.

ii. If (X1, d1), . . . , (Xm, dm) are ultrametric spaces then the cartesian prod-uct X1 × · · · × Xm is ultrametric with respect to

d((x1, . . . , xm), (y1, . . . , ym)) := max(d1(x1, y1), . . . , dm(xm, ym)).

Let (X, d) be an ultrametric space in the following.

Lemma 1.1. For any three points x, y, z ∈ X such that d(x, y) �= d(y, z) wehave

d(x, z) = max(d(x, y), d(y, z)).

Proof. We may assume that d(x, y) < d(y, z). Then

d(x, y) < d(y, z) ≤ max(d(y, x), d(x, z)) = max(d(x, y), d(x, z)).

The maximum in question therefore necessarily is equal to d(x, z) so that

d(x, y) < d(y, z) ≤ d(x, z).

We deduce that

d(x, z) ≤ max(d(x, y), d(y, z)) ≤ d(x, z).

P. Schneider, p-Adic Lie Groups,Grundlehren der mathematischen Wissenschaften 344,DOI 10.1007/978-3-642-21147-8 1, © Springer-Verlag Berlin Heidelberg 2011

3

http://dx.doi.org/10.1007/978-3-642-21147-8_1

4 I Foundations

Let a ∈ X be a point and ε > 0 be a positive real number. We call

Bε(a) := {x ∈ X : d(a, x) ≤ ε}

the closed ball and

B−ε (a) := {x ∈ X : d(a, x) < ε}

the open ball around a of radius ε. Any subset in X of one of these two kindsis simply referred to as a ball . As the following facts show this language hasto be used with some care.

Lemma 1.2. i. Every ball is open and closed in X.

ii. For b ∈ Bε(a), resp. b ∈ B−ε (a), we have Bε(b) = Bε(a), resp. B−

ε (b) =B−

ε (a).

Proof. Obviously B−ε (a) is open and Bε(a) is closed in X. We first consider

the equivalence relation x ∼ y on X defined by d(x, y) < ε. The corre-sponding equivalence class of b is equal to B−

ε (b) and hence is open. Sinceequivalence classes are disjoint or equal this implies B−

ε (b) = B−ε (a) when-

ever b ∈ B−ε (a). It also shows that B−

ε (a) as the complement of the otheropen equivalence classes is closed in X.

Analogously we may consider the equivalence relation x ≈ y on X definedby d(x, y) ≤ ε. Its equivalence classes are the closed balls Bε(b), and weobtain in the same way as before the assertion ii. for closed balls. It remainsto show that Bε(a) is open in X. But by what we have established alreadywith any point b ∈ Bε(a) its open neighbourhood B−

ε (b) is contained inBε(b) = Bε(a).

The assertion ii. in the above lemma can be viewed as saying that anypoint of a ball can serve as its midpoint. By way of an example we will seelater on that also the notion of a radius is not well determined.

Lemma 1.3. For any two balls B and B′ in X such that B ∩ B′ �= ∅ wehave B ⊆ B′ or B′ ⊆ B.

Proof. Pick a point a ∈ B ∩ B′. As a consequence of Lemma 1.2.ii. thefollowing four cases have to be distinguished:

1. B = B−ε (a), B′ = B−

δ (a),

2. B = B−ε (a), B′ = Bδ(a),


3. B = Bε(a), B′ = B−δ (a),

4. B = Bε(a), B′ = Bδ(a).

Without loss of generality we may assume that ε ≤ δ. In cases 1, 2, and 4we then obviously have B ⊆ B′. In case 3 we obtain B ⊆ B′ if ε < δ andB′ ⊆ B if ε = δ.

Remark. If the ultrametric space X is connected then it is empty or consistsof one point.

Proof. Assuming that X is nonempty we pick a point a ∈ X. Lemma 1.2.i.then implies that X = Bε(a) for any ε > 0 and hence that X = {a}.

Lemma 1.4. Let U =⋃

i∈I Ui be a covering of an open subset U ⊆ X byopen subsets Ui ⊆ X; moreover let ε1 > ε2 > · · · > 0 be a strictly descendingsequence of positive real numbers which converges to zero; then there is adecomposition

U =⋃

j∈J

Bj

of U into pairwise disjoint balls Bj such that :

(a) Bj = Bεn(j)(aj) for appropriate aj ∈ X and n(j) ∈ N,

(b) Bj ⊆ Ui(j) for some i(j) ∈ I.

Proof. For a ∈ U we put

n(a) := min{n ∈ N : Bεn(a) ⊆ Ui for some i ∈ I}.

The family of balls J := {Bεn(a)(a) : a ∈ U} by construction has the proper-

ties (a) and (b) and covers U (observe that for any point a in the open setUi we find some sufficiently big n ∈ N such that Bεn(a) ⊆ Ui). The balls inthis family indeed are pairwise disjoint: Suppose that

Bεn(a1)(a1) ∩ Bεn(a2)

�= ∅.

By Lemma 1.3 we may assume that Bεn(a1)(a1) ⊆ Bεn(a2)

(a2). But thenLemma 1.2.ii. implies that Bεn(a2)

(a1) = Bεn(a2)(a2) and hence Bεn(a1)

(a1) ⊆Bεn(a2)

(a1). Due to the minimality of n(a1) we must have n(a1) ≤ n(a2),resp. εn(a1) ≥ εn(a2). It follows that Bεn(a1)

(a1) = Bεn(a2)(a1) = Bεn(a2)

(a2).

6 I Foundations

As usual the metric space X is called complete if every Cauchy sequencein X is convergent.

Lemma 1.5. A sequence (xn)n∈N in X is a Cauchy sequence if and only iflimn→∞ d(xn, xn+1) = 0.

For a subset A ⊆ X we call

d(A) := sup{d(x, y) : x, y ∈ A}

the diameter of A.

Lemma 1.6. Let B ⊆ X be a ball with ε := d(B) > 0 and pick any pointa ∈ B; we then have B = B−

ε (a) or B = Bε(a).

Proof. The inclusion B ⊆ Bε(a) is obvious. By Lemma 1.2.ii. the ball B isof the form B = B−

δ (a) or B = Bδ(a). The strict triangle inequality thenimplies ε = d(B) ≤ δ. If ε = δ there is nothing further to prove. If ε < δ wehave B ⊆ Bε(a) ⊆ B−

δ (a) ⊆ B and hence B = Bε(a).

Let us consider a descending sequence of balls

B1 ⊇ B2 ⊇ · · · ⊇ Bn ⊇ · · ·

in X. If X is complete and if limn→∞ d(Bn) = 0 then we claim that⋂

n∈N

Bn �= ∅.

If we pick points xn ∈ Bn then (xn)n∈N is a Cauchy sequence. Put x :=limn→∞ xn. Since each Bn is closed we must have x ∈ Bn and thereforex ∈

⋂n Bn.

Without the condition on the diameters the intersection⋂

n Bn can beempty (compare the exercise further below). This motivates the followingdefinition.

Definition. The ultrametric space (X, d) is called spherically complete ifany descending sequence of balls B1 ⊇ B2 ⊇ · · · in X has a nonemptyintersection.

Lemma 1.7. i. If X is spherically complete then it is complete.

ii. Suppose that X is complete; if 0 is the only accumulation point of theset d(X × X) ⊆ R+ of values of the metric d then X is sphericallycomplete.


Proof. i. Let (xn)n∈N be any Cauchy sequence in X. We may assume thatthis sequence does not become constant after finitely many steps. Then the

εn := max{d(xm, xm+1) : m ≥ n}

are strictly positive real numbers satisfying εn ≥ εn+1 and εn ≥ d(xn, xn+1).Using Lemma 1.2.ii. we obtain Bεn(xn) = Bεn(xn+1) ⊇ Bεn+1(xn+1). Byassumption the intersection

⋂n Bεn(xn) must contain a point x. We have

d(x, xn) ≤ εn for any n ∈ N. Since the sequence (εn)n converges to zero thisimplies that x = limx→∞ xn.

ii. Let B1 ⊇ B2 ⊇ · · · be any decreasing sequence of balls in X. Obviouslywe have d(B1) ≥ d(B2) ≥ · · · . By our above discussion we only need toconsider the case that infn d(Bn) > 0. Our assumption on accumulationpoints implies that d(Bn) ∈ D(X × X) for any n ∈ N and then in factthat the sequence (d(Bn))n must become constant after finitely many steps.Hence there exists an m ∈ N such that 0 < ε := d(Bm) = d(Bm+1) = · · · .By Lemma 1.6 we have, for any n ≥ m and any a ∈ Bn, that

Bn = B−ε (a) or Bn = Bε(a).

Moreover, which of the two equations holds is independent of the choiceof a by Lemma 1.2.ii. Case 1: We have Bn = Bε(a) for any n ≥ m andany a ∈ Bn. It immediately follows that Bn = Bm for any n ≥ m andhence that

⋂n Bn = Bm. Case 2: There is an � ≥ m such that B� = B−

ε (a)for any a ∈ B�. For any n ≥ � and any a ∈ Bn ⊆ B� we then obtainB−

ε (a) = B� ⊇ Bn ⊇ B−ε (a) so that B� = Bn and hence

⋂n Bn = B�.

Exercise. Suppose that X is complete, and let B1 ⊃ B2 ⊃ · · · be a decreas-ing sequence of balls in X such that d(B1) > d(B2) > · · · and infn d(Bn) > 0.Then the subspace Y := X\(

⋂n Bn) is complete but not spherically complete.

Lemma 1.8. Suppose that X is spherically complete; for any family (Bi)i∈I

of closed balls in X such that Bi ∩ Bj �= ∅ for any i, j ∈ I we then have⋂i∈I Bi �= ∅.

Proof. We choose a sequence (in)n∈N of indices in I such that:

– d(Bi1) ≥ d(Bi2) ≥ · · · ≥ d(Bin) ≥ · · · ,

– for any i ∈ I there is an n ∈ N with d(Bi) ≥ d(Bin).

The proof of Lemma 1.6 shows that Bi = Bd(Bi)(a) for any a ∈ Bi. Ourassumption on the family (Bi)i therefore implies that:

8 I Foundations

– Bi1 ⊇ Bi2 ⊇ · · · ⊇ Bin ⊇ · · · ,

– for any i ∈ I there is an n ∈ N with Bi ⊇ Bin .

We see that ⋂

i∈I

Bi =⋂

n∈N

Bin �= ∅.

2 Nonarchimedean Fields

Let K be any field.

Definition. A nonarchimedean absolute value on K is a function

| | : K −→ R

which satisfies :

(i) |a| ≥ 0,

(ii) |a| = 0 if and only if a = 0,

(iii) |ab| = |a| · |b|,

(iv) |a + b| ≤ max(|a|, |b|).

Exercise. i. |n · 1| ≤ 1 for any n ∈ Z.

ii. | | : K× −→ R×+ is a homomorphism of groups; in particular, |1| =

|−1| = 1.

iii. K is an ultrametric space with respect to the metric d(a, b) := |b − a|;in particular, we have |a + b| = max(|a|, |b|) whenever |a| �= |b|.

iv. Addition and multiplication on the ultrametric space K are continuousmaps.

Definition. A nonarchimedean field (K, | |) is a field K equipped with anonarchimedean absolute value | | such that :

(i) | | is non-trivial, i. e., there is an a ∈ K with |a| �= 0, 1,

(ii) K is complete with respect to the metric d(a, b) := |b − a|.


The most important class of examples is constructed as follows. We fixa prime number p. Then

|a|p := p−r if a = pr mn with r,m, n ∈ Z and p � |mn

is a nonarchimedean absolute value on the field Q of rational numbers. Thecorresponding completion Qp is called the field of p-adic numbers. Of course,it is nonarchimedean as well. We note that |Qp|p = pZ ∪ {0}. Hence Qp isspherically complete by Lemma 1.7.ii. On the other hand we see that in theultrametric space Qp we can have Bε(a) = Bδ(a) even if ε �= δ. To havemore examples we state without proof (compare [Se1] Chap. II §§1–2) thefollowing fact. Let K/Qp be any finite extension of fields. Then

|a| := [K:Qp]√|NormK/Qp

(a)|p

is the unique extension of | |p to a nonarchimedean absolute value on K.The corresponding ultrametric space K is complete and spherically completeand, in fact, locally compact.

In the following we fix a nonarchimedean field (K, | |). By the stricttriangle inequality the closed unit ball

oK := B1(0)

is a subring of K, called the ring of integers in K, and the open unit ball

mK := B−1 (0)

is an ideal in oK . Because of o×K = oK \mK this ideal mK is the only maximalideal of oK . The field oK/mK is called the residue class field of K.

Exercise 2.1. i. If the residue class field oK/mK has characteristic zerothen K has characteristic zero as well and we have |a| = 1 for anynonzero a ∈ Q ⊆ K.

ii. If K has characteristic zero but oK/mK has characteristic p > 0 thenwe have

|a| = |a|− log |p|

log pp for any a ∈ Q ⊆ K;

in particular, K contains Qp.

A nonarchimedean field K as in the second part of Exercise 2.1 is calleda p-adic field .

10 I Foundations

Lemma 2.2. If K is p-adic then we have

|n| ≥ |n!| ≥ |p|n−1p−1 for any n ∈ N.

Proof. We may obviously assume that K = Qp. Then the reader should dothis as an exercise but also may consult [B-LL] Chap. II §8.1 Lemma 1.

Now let V be any K-vector space.

Definition. A (nonarchimedean) norm on V is a function ‖ ‖ : V −→ R

such that for any v, w ∈ V and any a ∈ K we have:

(i) ‖av‖ = |a| · ‖v‖,

(ii) ‖v + w‖ ≤ max(‖v‖, ‖w‖),

(iii) if ‖v‖ = 0 then v = 0.

Moreover, V is called normed if it is equipped with a norm.

Exercise. i. ‖v‖ ≥ 0 for any v ∈ V and ‖0‖ = 0.

ii. V is an ultrametric space with respect to the metric d(v, w) := ‖w−v‖;in particular, we have ‖v +w‖ = max(‖v‖, ‖w‖) whenever ‖v‖ �= ‖w‖.

iii. Addition V × V+−−→ V and scalar multiplication K × V −→ V are

continuous.

Lemma 2.3. Let (V1, ‖ ‖1) and (V2, ‖ ‖2) let two normed K-vector spaces;a linear map f : V1 −→ V2 is continuous if and only if there is a constantc > 0 such that

‖f(v)‖2 ≤ c · ‖v‖1 for any v ∈ V1.

Proof. We suppose first that such a constant c > 0 exists. Consider anysequence (vn)n∈N in V1 which converges to some v ∈ V1. Then (‖vn − v‖1)n

and hence (‖f(vn)−f(v)‖2)n = (‖f(vn−v)‖2)n are zero sequences. It followsthat the sequence (f(vn))n converges to f(v) in V2. This means that f iscontinuous.

Now we assume vice versa that f is continuous. We find a 0 < ε < 1such that

Bε(0) ⊆ f−1(B1(0)).

Since | | is non-trivial we may assume that ε = |a| for some a ∈ K. In otherwords

‖v‖1 ≤ |a| implies ‖f(v)‖2 ≤ 1


for any v ∈ V1. Let now 0 �= v ∈ V1 be an arbitrary nonzero vector. We findan m ∈ Z such that

|a|m+2 < ‖v‖1 ≤ |a|m+1.

Setting c := |a|−2 we obtain

‖f(v)‖2 = |a|m · ‖f(a−mv)‖2 ≤ |a|m < c · ‖v‖1.

Definition. The normed K-vector space (V, ‖ ‖) is called a K-Banach spaceif V is complete with respect to the metric d(v, w) := ‖w − v‖.

Examples. 1) Kn with the norm ‖(a1, . . . , an)‖ := max1≤i≤n |ai| is aK-Banach space.

2) Let I be a fixed but arbitrary index set. A family (ai)i∈I of elements inK is called bounded if there is a c > 0 such that |ai| ≤ c for any i ∈ I.The set

�∞(I) := set of all bounded families (ai)i∈I in K

with componentwise addition and scalar multiplication and with thenorm

‖(ai)i‖∞ := supi∈I

|ai|

is a K-Banach space.

3) With I as above let

c0(I) := {(ai)i∈I ∈ �∞(I) : for any ε > 0 we have |ai| ≥ ε

for at most finitely many i ∈ I}.

It is a closed vector subspace of �∞(I) and hence a K-Banach spacein its own right. Moreover, for (ai)i ∈ c0(I) we have

‖(ai)i‖∞ = maxi∈I

|ai|.

For example, c0(N) is the Banach space of all zero sequences in K.

Remark. Any K-Banach space (V, ‖ ‖) over a finite extension K/Qp whichsatisfies ‖V ‖ ⊆ |K| is isometric to some K-Banach space (c0(I), ‖ ‖∞);moreover, all such I have the same cardinality.

12 I Foundations

Proof. Compare [NFA] Remark 10.2 and Lemma 10.3.

Let V and W be two normed K-vector spaces. From now on we denote,unless this causes confusion, all occurring norms indiscriminately by ‖ ‖. Itis clear that

L(V,W ) := {f ∈ HomK(V,W ) : f is continuous}

is a vector subspace of HomK(V,W ). By Lemma 2.3 the operator norm

‖f‖ := sup{‖f(v)‖‖v‖ : v ∈ V, v �= 0

}

= sup{‖f(v)‖‖v‖ : v ∈ V, 0 < ‖v‖ ≤ 1

}

is well defined for any f ∈ L(V,W ).

Lemma 2.4. L(V,W ) with the operator norm is a normed K-vector space.

Proof. This is left to the reader as an exercise.

Proposition 2.5. If W is a K-Banach space then so, too, is L(V,W ).

Proof. Let (fn)n∈N be a Cauchy sequence in L(V,W ). Then, on the onehand, (‖fn‖)n is a Cauchy sequence in R and therefore converges, of course.On the other hand, because of

‖fn+1(v) − fn(v)‖ = ‖(fn+1 − fn)(v)‖ ≤ ‖fn+1 − fn‖ · ‖v‖

we obtain, for any v ∈ V , the Cauchy sequence (fn(v))n in W . By assumptionthe limit f(v) := limn→∞ fn(v) exists in W . Obviously we have

f(av) = af(v) for any a ∈ K.

For v, v′ ∈ V we compute

f(v) + f(v′) = limn→∞

fn(v) + limn→∞

fn(v′) = limn→∞

(fn(v) + fn(v′))

= limn→∞

fn(v + v′) = f(v + v′).

This means that v �−→ f(v) is a K-linear map which we denote by f . Since

‖f(v)‖ = limn→∞

‖fn(v)‖ ≤ ( limn→∞

‖fn‖) · ‖v‖


it follows from Lemma 2.3 that f is continuous. Finally the inequality

‖f − fn‖ = sup{‖(f − fn)(v)‖

‖v‖ : v �= 0}

= sup{

limm→∞ ‖fm(v) − fn(v)‖‖v‖ : v �= 0

}

≤ limm→∞

‖fm − fn‖ ≤ supm≥n

‖fm+1 − fm‖

shows that f indeed is the limit of the sequence (fn)n in L(V,W ).

In particular,V ′ := L(V,K)

always is a K-Banach space. It is called the dual space to V .

Lemma 2.6. Let I be an index set ; for any j ∈ I let 1j ∈ c0(I) denote thefamily (ai)i∈I with ai = 0 for i �= j and aj = 1; then

c0(I)′∼=−−→ �∞(I)

� �−→ (�(1i))i∈I

is an isometric linear isomorphism.

Proof. We give the proof only in the case I = N. The general case followsthe same line but requires the technical concept of summability (cf. [NFA]end of §3). Let us denote the map in question by ι. Because of

|�(1i)| ≤ ‖�‖ · ‖1i‖∞ = ‖�‖

it is well defined and satisfies

‖ι(�)‖∞ ≤ ‖�‖ for any � ∈ c0(N)′.

For trivial reasons ι is a linear map. Consider now an arbitrary nonzerovector v = (ai)i ∈ c0(N). In the Banach space c0(N) we then have theconvergent series expansion

v =∑

i∈N

ai · 1i.

Applying any � ∈ c0(N)′ by continuity leads to

�(v) =∑

i∈N

ai�(1i).

14 I Foundations

We obtain

|�(v)|‖v‖∞

≤ supi |ai||�(1i)|supi |ai|

≤ supi

|�(1i)| = ‖ι(�)‖∞.

It follows that‖�‖ ≤ ‖ι(�)‖∞

and together with the previous inequality that ι in fact is an isometry andin particular is injective. For surjectivity let (ci)i ∈ �∞(N) be any vector andput ε := ‖(ci)i‖∞. We consider the linear form

� : c0(N) −→ K

(ai)i �−→∑

i

aici

(note that the defining sum is convergent). Using Lemma 2.3 together withthe inequality

|�((ai)i)| =

∣∣∣∣∣

∑

i

aici

∣∣∣∣∣≤ sup

i|ai||ci| ≤ sup

i|ai| · sup

i|ci| = ε · ‖(ai)i‖∞

we see that � is continuous. It remains to observe that

ι(�) = (�(1i))i = (ci)i.

3 Convergent Series

From now on throughout the book (K, | |) is a fixed nonarchimedean field.For the convenience of the reader we collect in this section the most basic

facts about convergent series in Banach spaces (some of which we have usedalready in the proof of Lemma 2.6).

Let (V, ‖ ‖) be a K-Banach space.

Lemma 3.1. Let (vn)n∈N be a sequence in V ; we then have:

i. The series∑∞

n=1 vn is convergent if and only if limn→∞ vn = 0;

ii. if the limit v := limn→∞ vn exists in V and is nonzero then ‖vn‖ = ‖v‖for all but finitely many n ∈ N;

3 Convergent Series 15

iii. let σ : N → N be any bijection and suppose that the series v =∑∞

n=1 vn

is convergent in V ; then the series∑∞

n=1 vσ(n) is convergent as wellwith the same limit v.

Proof. i. This is immediate from Lemma 1.5. ii. If v �= 0 then ‖v‖ �= 0 andhence ‖vn − v‖ < ‖v‖ for any sufficiently big n ∈ N. For these n Lemma 1.1then implies that

‖vn‖ = ‖(vn − v) + v‖ = max(‖vn − v‖, ‖v‖) = ‖v‖.

iii. We fix an ε > 0 and choose an m ∈ N such that∥∥∥∥∥v −

s∑

n=1

vn

∥∥∥∥∥

< ε for any s ≥ m.

Then also

‖vs‖ =

∥∥∥∥∥

(

v−s−1∑

n=1

vn

)

−(

v−s∑

n=1

vn

)∥∥∥∥∥≤ max

(∥∥∥∥∥v−

s−1∑

n=1

vn

∥∥∥∥∥,

∥∥∥∥∥v−

s∑

n=1

vn

∥∥∥∥∥

)

< ε

for any s > m. Setting � := max{σ−1(n) : n ≤ m} ≥ m we have

{σ−1(1), . . . , σ−1(m)} ⊆ {1, . . . , �}

and hence, for any s ≥ �,

{σ(1), . . . , σ(s)} = {1, . . . , m} ∪ {n1, . . . , ns−m}

with appropriate natural numbers ni > m. We conclude that∥∥∥∥∥v −

s∑

n=1

vσ(n)

∥∥∥∥∥

=

∥∥∥∥∥

(

v −m∑

n=1

vn

)

− vn1 − · · · − vns−m

∥∥∥∥∥

≤ max

(∥∥∥∥∥v −

m∑

n=1

vn

∥∥∥∥∥, ‖vn1‖, . . . , ‖vns−m‖

)

< ε

for any s ≥ �.

The following identities between convergent series are obvious:

–∑∞

n=1 avn = a ·∑∞

n=1 vn for any a ∈ K.

16 I Foundations

– (∑∞

n=1 vn) + (∑∞

n=1 wn) =∑∞

n=1(vn + wn).

Lemma 3.2. Let∑∞

n=1 an and∑∞

n=1 vn be convergent series in K and V ,respectively ; then the series

∑∞n=1 wn with wn :=

∑�+m=n a�vm is conver-

gent, and∞∑

n=1

wn =

( ∞∑

n=1

an

)( ∞∑

n=1

vn

)

.

Proof. Let A := supn |an| and C := supn ‖vn‖. The other cases being trivialwe will assume that A,C > 0. For any given ε > 0 we find an N ∈ N suchthat

|an| <ε

Cand ‖vn‖ <

ε

Afor any n ≥ N.

Then

‖wn‖ ≤ max�+m=n

|a�| · ‖vm‖ ≤ max(C · max

�≥N|a�|, A · max

m≥N‖vm‖

)< ε

for any n ≥ 2N . By Lemma 3.1.i. the series∑∞

n=1 wn therefore is convergent.To establish the asserted identity we note that its left hand side is the limitof the sequence

Ws :=s∑

n=1

wn =∑

�+m≤s

a�vm

whereas its right hand side is the limit of the sequence

W ′s :=

(s∑

n=1

an

)(s∑

n=1

vn

)

=∑

�,m≤s

a�vm.

It therefore suffices to show that the differences Ws − W ′s converge to zero.

But we have

‖Ws − W ′s‖ =

∥∥∥∥∥

∑

�,m≤s�+m>s

a�vm

∥∥∥∥∥≤ max

�,m≤s�+m>s

|a�| · ‖vm‖

≤ max(C · max

�> s2

|a�|, A · maxm> s

2

‖vm‖).

Analogous assertions hold true for series∑∞

n1,...,nr=1 vn1,...,nr indexed bymulti-indices in N× · · · ×N. But we point out the following additional fact.


Lemma 3.3. Let (vm,n)m,n∈N be a double sequence in V such that

limm+n→∞

vm,n = 0;

we then have∞∑

m=1

∞∑

n=1

vm,n =∞∑

n=1

∞∑

m=1

vm,n

which, in particular, means that all series involved are convergent.

Proof. It is immediate from the assumption and Lemma 3.1.i. that all “in-ner” series

∑∞n=1 vm,n and

∑∞m=1 vm,n are convergent. Let ε > 0. By as-

sumption we find an N ∈ N such that ‖vm,n‖ ≤ ε whenever m + n > N .This implies that ‖

∑∞n=1 vm,n‖ ≤ ε for any m > N and ‖

∑∞m=1 vm,n‖ ≤ ε

for any n > N . Again using Lemma 3.1.i. we obtain that the “outer” seriesin the asserted identity are convergent as well. But we also see that

∥∥∥∥∥

∞∑

m=1

∞∑

n=1

vm,n −∑

m+n≤N

vm,n

∥∥∥∥∥

=

∥∥∥∥∥

∞∑

m=1

∑

n>N−m

vm,n

∥∥∥∥∥≤ ε.

By symmetry we also have ‖∑∞

n=1

∑∞m=1 vm,n −

∑m+n≤N vm,n‖ ≤ ε and

hence ∥∥∥∥∥

∞∑

m=1

∞∑

n=1

vm,n −∞∑

n=1

∞∑

m=1

vm,n

∥∥∥∥∥≤ ε.

Since ε was arbitrary this implies the asserted identity.

4 Differentiability

Let V and W be two normed K-vector spaces, let U ⊆ V be an open subset,and let f : U −→ W be some map.

Definition. The map f is called differentiable in the point v0 ∈ U if thereexists a continuous linear map

Dv0f : V −→ W

such that for any ε > 0 there is an open neighbourhood Uε = Uε(v0) ⊆ U ofv0 with

‖f(v) − f(v0) − Dv0f(v − v0)‖ ≤ ε‖v − v0‖ for any v ∈ Uε.

18 I Foundations

We may of course assume in the above definition that the open neigh-bourhood Uε is of the form Uε = Bδ(ε)(v0) for some sufficiently small radiusδ(ε) > 0. We claim that the linear map Dv0f is uniquely determined. Fixan ε′ > 0, and choose a basis {v′j}j∈J of the vector space V . By scaling wemay assume that ‖v′j‖ ≤ δ(ε′). We put vj := v′j + v0. Then

vj ∈ Bδ(ε′)(v0) = Uε′ .

More generally, for any 0 < ε ≤ ε′ we pick a tε ∈ K× such that

|tε|δ(ε′) ≤ δ(ε).

Thentε(vj − v0) + v0 ∈ Bδ(ε)(v0) = Uε for any j ∈ J.

It follows that

‖f(tε(vj − v0) + v0) − f(v0) − Dv0f(tε(vj − v0))‖ ≤ ε‖tε(vj − v0)‖

and hence that∥∥∥∥f(tε(vj − v0) + v0) − f(v0)

tε− Dv0f(vj − v0)

∥∥∥∥ ≤ ε‖vj − v0‖.

By letting ε tend to zero we obtain

Dv0f(v′j) = Dv0f(vj − v0) = limt∈K×,t→0

f(t(vj − v0) + v0) − f(v0)t

for any j ∈ J . But as a linear map Dv0f is uniquely determined by its valueon the basis vectors v′j .

The continuous linear map Dv0f : V −→ W is called (if it exists) thederivative of f in the point v0 ∈ U . In case V = K we also write f ′(a0) :=Da0f(1).

Remark 4.1. i. If f is differentiable in v0 then it is continuous in v0.

ii. (Chain rule) Let V,W1, and W2 be normed K-vector spaces, U ⊆ Vand U1 ⊆ W1 be open subsets, and f : U −→ U1 and g : U1 −→ W2

be maps; suppose that f is differentiable in some v0 ∈ U and g isdifferentiable in f(v0); then g ◦ f is differentiable in v0 and

Dv0(g ◦ f) = Df(v0)g ◦ Dv0f.


iii. A continuous linear map u : V −→ W is differentiable in any v0 ∈ Vand Dv0u = u; in particular, in the situation of ii. we have

Dv0(u ◦ f) = u ◦ Dv0f.

iv. (Product rule) Let V,W1, . . . ,Wm, and W be normed K-vector spaces,let U ⊆ V be an open subset with maps fi : U −→ Wi, and letu : W1 × · · · × Wm −→ W be a continuous multilinear map; sup-pose that f1, . . . , fm all are differentiable in some point v0 ∈ U ; thenu(f1, . . . , fm) : U −→ W is differentiable in v0 and

Dv0(u(f1, . . . , fm)) =m∑

i=1

u(f1(v0), . . . , Dv0fi, . . . , fm(v0)).

Proof. These are standard arguments of which we only recall the proof ofii. Let ε > 0 and choose a δ > 0 such that

δ2, δ‖Dv0f‖, δ‖Df(v0)g‖ ≤ ε

(here ‖ ‖ refers to the operator norm of course). By assumption on g wehave

(1) ‖g(w) − g(f(v0)) − Df(v0)g(w − f(w0))‖ ≤ δ‖w − f(v0)‖

for any w ∈ Uδ(f(v0)). By the differentiability and hence continuity of f inv0 there exists an open neighbourhood U(v0) ⊆ U of v0 such that f(U(v0)) ⊆Uδ(f(v0)) and

(2) ‖f(v) − f(v0) − Dv0f(v − v0)‖ ≤ δ‖v − v0‖

for any v ∈ U(v0). In particular

‖f(v) − f(v0)‖ = ‖Dv0f(v − v0) + f(v) − f(v0) − Dv0f(v − v0)‖(3)≤ max(‖Dv0f‖ · ‖v − v0‖, δ‖v − v0‖)

for any v ∈ U(v0). We now compute

‖g(f(v)) − g(f(v0)) − Df(v0)g ◦ Dv0f(v − v0)‖= ‖g(f(v)) − g(f(v0)) − Df(v0)g(f(v) − f(v0))

+ Df(v0)g(f(v) − f(v0) − Dv0f(v − v0)

)‖

(1)

≤ max(δ‖f(v) − f(v0)‖, ‖Df(v0)g‖ · ‖f(v) − f(v0) − Dv0f(v − v0)‖

)

20 I Foundations

(2)

≤ max(δ‖f(v) − f(v0)‖, ‖Df(v0)g‖ · δ‖v − v0‖

)

(3)

≤ max(δ‖Dv0f‖, δ2, δ‖Df(v0)g‖

)· ‖v − v0‖

≤ ε‖v − v0‖

for any v ∈ U(v0).

Suppose that the vector space V = V1 ⊕ · · · ⊕ Vm is the direct sumof finitely many vector spaces V1, . . . , Vm and that the norm on V is themaximum of its restrictions to the Vi. Write a vector v0 ∈ U as v0 = v0,1 +· · ·+v0,m with v0,i ∈ Vi. For each 1 ≤ i ≤ m there is an open neighbourhoodUi ⊆ Vi of v0,i such that

U1 + · · · + Um ⊆ U.

Therefore the maps

fi : Ui −→ W

vi �−→ f(v0,1 + · · · + vi + · · · + v0,m)

are well defined. If it exists the continuous linear map

D(i)v0

f := Dv0,ifi : Vi −→ W

is called the i-th partial derivative of f in v0. We recall that differentiabilityof f in v0 implies the existence of all partial derivatives together with theidentity

Dv0f =m∑

i=1

D(i)v0

f.

Let us go back to our initial situation V ⊇ Uf−→ W .

Definition. The map f is called strictly differentiable in v0 ∈ U if thereexists a continuous linear map Dv0f : V −→ W such that for any ε > 0there is an open neighbourhood Uε ⊆ U of v0 with

‖f(v1) − f(v2) − Dv0f(v1 − v2)‖ ≤ ε‖v1 − v2‖ for any v1, v2 ∈ Uε.

Exercise. Suppose that f is strictly differentiable in every point of U . Thenthe map

U −→ L(V,W )v �−→ Dvf

is continuous.


Our goal for the rest of this section is to discuss the local invertibilityproperties of strictly differentiable maps.

Lemma 4.2. Let Bδ(v0) be a closed ball in a K-Banach space V and letf : Bδ(v0) −→ V be a map for which there exists a 0 < ε < 1 such that

(4) ‖f(v1) − f(v2) − (v1 − v2)‖ ≤ ε‖v1 − v2‖ for any v1, v2 ∈ Bδ(v0);

then f induces a homeomorphism

Bδ(v0)−−→ Bδ(f(v0)).

Proof. By Lemma 1.1 we have

(5) ‖f(v1) − f(v2)‖ = ‖v1 − v2‖ for any v1, v2 ∈ Bδ(v0).

In particular, f is a homeomorphism onto its image which satisfies f(Bδ(v0))⊆ Bδ(f(v0)). It remains to show that this latter inclusion in fact is anequality. Let w ∈ Bδ(f(v0)) be an arbitrary but fixed vector. For any v′ ∈Bδ(v0) we put

v′′ := w + v′ − f(v′).

We compute

‖v′′ − v0‖ ≤ max(‖v′′ − v′‖, ‖v′ − v0‖)≤ max(‖w − f(v′)‖, δ)≤ max(‖w − f(v0)‖, ‖f(v0) − f(v′)‖, δ)≤ max(δ, ‖f(v0) − f(v′)‖)(5)

≤ max(δ, ‖v0 − v′‖)≤ δ

which means that v′′ ∈ Bδ(v0). Hence we may define inductively a sequence(vn)n≥0 in Bδ(v0) by

vn+1 := w + vn − f(vn).

Using (4) we see that

‖vn+1 − vn‖ = ‖vn − f(vn) − (vn−1 − f(vn−1))‖= ‖f(vn−1) − f(vn) − (vn−1 − vn)‖≤ ε‖vn − vn−1‖

22 I Foundations

and therefore‖vn+1 − vn‖ ≤ εn‖v1 − v0‖ ≤ εnδ

for any n ≥ 1. It follows that (vn)n is a Cauchy sequence and, since Vis complete, is convergent. Because Bδ(v0) is closed in V the limit v :=limn→∞ vn lies in Bδ(v0). By passing to the limit in the defining equationand using the continuity of f we finally obtain that f(v) = w.

Proposition 4.3. (Local invertibility) Let V and W be K-Banach spaces,U ⊆ V be an open subset, and f : U −→ W be a map which is strictlydifferentiable in the point v0 ∈ U ; suppose that the derivative Dv0f : V

∼=−→ Wis a topological isomorphism; then there are open neighbourhoods U0 ⊆ U ofv0 and U1 ⊆ W of f(v0) such that :

i. f : U0−−→ U1 is a homeomorphism;

ii. the inverse map g : U1 −→ U0 is strictly differentiable in f(v0), and

Df(v0)g = (Dv0f)−1.

Proof. We consider the map

f1 := (Dv0f)−1 ◦ f : U −→ V.

As a consequence of the chain rule it is strictly differentiable in v0 and

Dv0f1 = (Dv0f)−1 ◦ Dv0f = idV .

Hence, fixing some 0 < ε0 < 1 we find a neighbourhood Bδ0(v0) ⊆ U of v0

such that the condition (4) in Lemma 4.2 is satisfied. The lemma then saysthat

f1 : U0 := Bδ0(v0)−−→ Bδ0(f1(v0))

is a homeomorphism. Since Dv0f is a homeomorphism by assumption U1 :=Dv0f(Bδ0(f1(v0))) is an open neighbourhood of f(v0) in W and f : U0

−→ U1

is a homeomorphism.To prove ii. let ε > 0. We have ‖(Dv0f)−1‖ > 0 since (Dv0f)−1 is bijec-

tive. Hence we find a δ > 0 such that

δ‖(Dv0f)−1‖ < 1 and δ‖(Dv0f)−1‖2 ≤ ε.

By the strict differentiability of f in v0 we have

‖f(v1) − f(v2) − Dv0f(v1 − v2)‖ ≤ δ‖v1 − v2‖ for any v1, v2 ∈ Uδ.


Applying (Dv0f)−1 gives

‖(Dv0f)−1(f(v1) − f(v2)) − (v1 − v2)‖ ≤ δ‖(Dv0f)−1‖ · ‖v1 − v2‖.

By our choice of δ and Lemma 1.1 we deduce

‖(Dv0f)−1(f(v1) − f(v2))‖ = ‖v1 − v2‖.

Combining the last two formulas we obtain

‖v1 − v2 − (Dv0f)−1(f(v1) − f(v2))‖≤ δ‖(Dv0f)−1‖ · ‖(Dv0f)−1(f(v1) − f(v2))‖≤ δ‖(Dv0f)−1‖2 · ‖f(v1) − f(v2)‖≤ ε‖f(v1) − f(v2)‖

for any v1, v2 ∈ Uδ. It follows that

‖g(w1) − g(w2) − (Dv0f)−1(w1 − w2)‖ ≤ ε‖w1 − w2‖

for any w1, w2 ∈ f(U0 ∩ Uδ). Since f(U0 ∩ Uδ) is an open neighbourhood off(v0) in U1 this establishes ii.

In regard to the assumption on the derivative in the above propositionit is useful to have in mind the open mapping theorem (cf. [NFA] Cor. 8.7)which says that any continuous linear bijection between K-Banach spacesnecessarily is a topological isomorphism. We also point out the trivial factthat any linear map between two finite dimensional K-Banach spaces iscontinuous.

Corollary 4.4. Let U ⊆ Kn be an open subset and f : U −→ Km be a mapwhich is strictly differentiable in v0 ∈ U ; suppose that Dv0f is injective;then there are open neighbourhoods U0 ⊆ U of v0 and U1 ⊆ Km of f(v0) aswell as a ball Bε(0) ⊆ Km−n around zero and linearly independent vectorsw1, . . . , wm−n ∈ Km such that the map

U0 × Bε(0) −−→ U1

(v, (a1, . . . , am−n)) �−→ f(v) + a1w1 + · · · + am−nwm−n

is a homeomorphism.

24 I Foundations

Proof. We choose the vectors wi in such a way that

Km = im(Dv0f) ⊕ Kw1 ⊕ · · · ⊕ Kwm−n.

Then the linear map

u : Kn × Km−n ∼=−−→ Km

(v, (a1, . . . , am−n)) �−→ (Dv0f)(v) + a1w1 + · · · + am−nwm−n

is a topological isomorphism. One checks that the map

f : U × Km−n −→ Km

(v, (a1, . . . , am−n)) �−→ f(v) + a1w1 + · · · + am−nwm−n

is strictly differentiable in (v0, 0) with D(v0,0)f = u. Now apply Prop. 4.3.i.

Corollary 4.5. Let U ⊆ Kn be an open subset and f : U −→ Km be a mapwhich is strictly differentiable in v0 ∈ U ; suppose that Dv0f is surjective;then there are open neighbourhoods U0 ⊆ U of v0 and U1 ⊆ Km of f(v0) aswell as a ball Bε(0) ⊆ Kn−m around zero and a linear map p : Kn −→ Kn−m

such that the map

U0−−→ U1 × Bε(0)

v �−→ (f(v), p(v) − p(v0))

is a homeomorphism; in particular, the restricted map f : U0 −→ Km isopen.

Proof. We choose a decomposition

Kn = ker(Dv0f) ⊕ C

and letp : Kn −→ ker(Dv0f) ∼= Kn−m

be the corresponding projection map. Then

u : Kn −→ Km × Kn−m

v �−→ ((Dv0f)(v), p(v))

is a topological isomorphism. One checks that

f : U −→ Km × Kn−m

v �−→ (f(v), p(v) − p(v0))

is strictly differentiable in v0 with Dv0 f = u. Now apply Prop. 4.3.i.

5 Power Series 25

We finish this section with a trivial observation. A map f : X −→ Afrom some topological space X into some set A is called locally constant iff−1(a) is open (and closed) in X for any a ∈ A.

Lemma 1.4 implies that in our standard situation of two normed K-vector spaces V and W and an open subset U ⊆ W there are plenty oflocally constant maps f : U −→ W . They all are strictly differentiable inany v0 ∈ U with Dv0f = 0.

5 Power Series

Let V be a K-Banach space. By a power series f(X) in r variables X =(X1, . . . , Xr) with coefficients in V we mean a formal series

f(X) =∑

α∈Nr0

Xαvα with vα ∈ V.

Here and in the following we use the usual conventions for multi-indices

Xα := Xα11 · . . . · Xαr

r and |α| := α1 + · · · + αr

if α = (α1, . . . , αr) ∈ Nr0 (with N0 := N ∪ {0}).

For any ε > 0 the power series f(X) =∑

α Xαvα is called ε-convergentif

lim|α|→∞

ε|α|‖vα‖ = 0.

Remark. If f(X) is ε-convergent then it also is δ-convergent for any 0 <δ ≤ ε.

The K-vector space

Fε(Kr;V ) := all ε-convergent power series f(X) =∑

α∈Nr0

Xαvα

is normed by‖f‖ε := max

αε|α|‖vα‖.

By a straightforward generalization of the argument for c0(N) it is shownthat Fε(Kr;V ) is a Banach space. By the way, in case ε = |c| for somec ∈ K× the map

c0(Nr0)

∼=−−→ F|c|(Kr;K)

(aα)α �−→∑

α

aα

c|α|Xα

is an isometric linear isomorphism.

26 I Foundations

Remark. The vector space Fε(Kr;V ) together with its topology only de-pends on the topology of V (and not on its specific norm).

Proof. Let ‖ ‖′ be a second norm on V which induces the same topology as‖ ‖. Applying Lemma 2.3 to the identity map idV we obtain two constantsc1, c2 > 0 such that

c1‖ ‖ ≤ ‖ ‖′ ≤ c2‖ ‖.

Then obviously

lim|α|→∞

ε|α|‖vα‖′ = 0 if and only if lim|α|→∞

ε|α|‖vα‖ = 0.

This means that using ‖ ‖′ instead of ‖ ‖ leads to the same vector spaceFε(Kr;V ) but which carries the two norms ‖ ‖ε and ‖f‖′ε := maxα ε|α|‖vα‖′.The above inequalities immediately imply the analogous inequalities

c1‖ ‖ε ≤ ‖ ‖′ε ≤ c2‖ ‖ε.

By Lemma 2.3 for the identity map on Fε(Kr;V ) this means that ‖ ‖ε and‖ ‖′ε induce the same topology.

Consider a convergent series

f =∞∑

i=0

fi

in the Banach space Fε(Kr;V ). Suppose that

f(X) =∑

α

Xαvα and fi(X) =∑

α

Xαvi,α.

We have∥∥∥∥∥f −

n∑

i=0

fi

∥∥∥∥∥

ε

=

∥∥∥∥∥

∑

α

Xα

(

vα −n∑

i=0

vi,α

)∥∥∥∥∥

ε

= maxα

ε|α|

∥∥∥∥∥vα −

n∑

i=0

vi,α

∥∥∥∥∥.

This shows that

vα =∞∑

i=0

vi,α for any α ∈ Nr0

which means that limits in Fε(Kr;V ) can be computed coefficientwise.

5 Power Series 27

Let Bε(0) denote the closed ball around zero in Kr of radius ε. We recallthat Kr always is equipped with the norm ‖(a1, . . . , ar)‖ = max1≤i≤r |ai|.By Lemma 3.1.i. we have the K-linear map

Fε(Kr;V ) −→ K-vector space of maps Bε(0) −→ V

f(X) =∑

α

Xαvα �−→ f(x) :=∑

α

xαvα.

Remark 5.1. For any x ∈ Bε(0) the linear evaluation map

Fε(Kr;V ) −→ V

f �−→ f(x)

is continuous of operator norm ≤ 1.

Proof. We have

‖f(x)‖ =

∥∥∥∥∥

∑

α

xαvα

∥∥∥∥∥≤ max

αε|α|‖vα‖ = ‖f‖ε.

Proposition 5.2. Let u : V1 × V2 −→ V be a continuous bilinear mapbetween K-Banach spaces; then

U : Fε(Kr;V1) ×Fε(Kr;V2) −→ Fε(Kr;V )(

∑

α

Xαvα,∑

α

Xαwα

)

�−→∑

α

Xα

(∑

β+γ=α

u(vβ, wγ)

)

is a continuous bilinear map satisfying

U(f, g)∼(x) = u(f(x), g(x)) for any x ∈ Bε(0)

and any f ∈ Fε(Kr;V1) and g ∈ Fε(Kr;V2).

Proof. By a similar argument as for Lemma 2.3 the bilinear map u is con-tinuous if and only if there is a constant c > 0 such that

‖u(v1, v2)‖ ≤ c‖v1‖ · ‖v2‖ for any v1 ∈ V1, v2 ∈ V2.

We therefore have

ε|α|

∥∥∥∥∥

∑

β+γ=α

u(vβ, wγ)

∥∥∥∥∥≤ c max

β+γ=α

(ε|β|‖vβ‖ · ε|γ|‖vγ‖

).

28 I Foundations

This shows (compare the proof of Lemma 3.2) that with f and g also U(f, g)is ε-convergent and that

‖U(f, g)‖ε ≤ c‖f‖ε‖g‖ε.

Hence U is well defined, bilinear, and continuous. The asserted identitybetween evaluations is an immediate generalization of Lemma 3.2.

Proposition 5.3. Fε(Kr;K) is a commutative K-algebra with respect tothe multiplication

(∑

α

bαXα

)(∑

α

cαXα

)

:=∑

α

(∑

β+γ

bβcγ

)

Xα;

in addition we have

(fg)∼(x) = f(x)g(x) for any x ∈ Bε(0)

as well as‖fg‖ε = ‖f‖ε‖g‖ε

for any f, g ∈ Fε(Kr;K).

Proof. Apart from the norm identity this is a special case of Prop. 5.2 (andits proof) for the multiplication in K as the bilinear map. It remains to showthat

‖fg‖ε ≥ ‖f‖ε‖g‖ε.

Let ≥ denote the lexicographic order on Nr0, and let μ and ν be lexicograph-

ically minimal multi-indices such that

|bμ|e|μ| = ‖f‖ε and |cν |ε|ν| = ‖g‖ε, respectively.

Put λ := μ + ν and consider any equation of the form λ = β + γ.Claim: β ≤ μ or γ ≤ ν.Otherwise we would have β > μ and γ > ν. This means that there are

1 ≤ i, j ≤ r such that

β1 = μ1, . . . , bi−1 = μi−1, and βi > μi

as well asγ1 = ν1, . . . , γj−1 = νj−1, and γj > νj .

5 Power Series 29

By symmetry we may assume that i ≤ j. We then obtain the contradiction

λi = μi + νi < βi + γi = λi.

This establishes the claim.We of course have

|bβ |ε|β| ≤ ‖f‖ε and |cγ |ε|γ| ≤ ‖g‖ε.

But in case (β, γ) �= (μ, ν) the fact that β < μ or γ < ν together with theminimality property of μ and ν implies that

|bβ |ε|β| < ‖f‖ε or |cγ |ε|γ| < ‖g‖ε.

It follows that

|bβcγ |ε|λ| = |bβ |ε|β| · |cγ |ε|γ| < ‖f‖ε‖g‖ε

whenever β + γ = λ but (β, γ) �= (μ, ν). We conclude that

‖fg‖ε ≥∣∣∣∣∣

∑

β+γ=λ

bβcγ

∣∣∣∣∣ε|λ| = |bμcν |ε|λ| = ‖f‖ε‖g‖ε.

Proposition 5.4. Let g ∈ Fδ(Kr;Kn) such that ‖g‖δ ≤ ε; then

Fε(Kn;V ) −→ Fδ(Kr;V )

f(Y ) =∑

β

Y βvβ �−→ f ◦ g(X) :=∑

β

g(X)βvβ

is a continuous linear map of operator norm ≤ 1 which satisfies

(f ◦ g)∼(x) = f(g(x)) for any x ∈ Bδ(0) ⊆ Kr.

Proof. Using the obvious identification

Fδ(Kr;Kn) =n∏

i=1

Fδ(Kr;K)

g = (g1, . . . , gn)

30 I Foundations

we have maxi ‖gi‖δ = ‖g‖δ ≤ ε. The Prop. 5.3 therefore implies that g(X)β ∈Fδ(Kr;K) for each β and

‖g(X)β‖δ =

∥∥∥∥∥

n∏

i=1

gβii

∥∥∥∥∥

δ

=n∏

i=1

‖gi‖βi

δ ≤ ε|β|.

It follows that g(X)βvβ ∈ Fδ(Kr;V ) for each β with

‖g(X)βvβ‖δ = ‖g(X)β‖δ‖vβ‖ ≤ ε|β|‖vβ‖.

Since the right hand side goes to zero by the ε-convergence of f the se-ries f ◦ g(X) =

∑β g(X)βvβ is convergent in the Banach space Fδ(Kr;V ).

Moreover, we have

‖f ◦ g‖δ ≤ maxβ

‖g(X)βvβ‖δ ≤ maxβ

ε|β|‖vβ‖ = ‖f‖ε.

To see the asserted identity between evaluations we first note that by Re-mark 5.1 the map g indeed maps the ball Bδ(0) ⊆ Kr into the ball Bε(0) ⊆Kn. Using Remark 5.1 together with Prop. 5.3 we obtain

(f ◦ g)∼(x) =∑

β

(gβ)∼(x)vβ =∑

β

g(x)βvβ = f(g(x)).

As a consequence of the discussion before Remark 5.1 and of Prop. 5.3the power series f ◦ g can be computed by formally inserting g into f .

The reader is warned that although, for any g ∈ Fδ(Kr;Kn), we have,by Remark 5.1, the inequality

supx∈Bδ(0)

‖g(x)‖ ≤ ‖g‖δ

it is, in general, not an equality. This means that we may have g(Bδ(0)) ⊆Bε(0) even if ε < ‖g‖δ. Then, for any f ∈ Fε(Kn;V ), the composite of mapsf ◦ g exists but the composite of power series f ◦ g ∈ Fδ(Kr;V ) may not.

Exercise. An example of such a situation is

g(X) := Xp − X ∈ F1(Qp; Qp) and f(Y ) :=∞∑

n=0

Y n ∈ F 1p(Qp; Qp).

5 Power Series 31

Corollary 5.5. (Point of expansion) Let f ∈ Fε(Kr;V ) and y ∈ Bε(0);then there exists an fy ∈ Fε(Kr;V ) such that ‖fy‖ε = ‖f‖ε and

f(x) = fy(x − y) for any x ∈ Bε(0) = Bε(y).

Proof. Let e1, . . . , er denote the standard basis of Kr. Applying Prop. 5.4to the power series g(X) := X + y =

∑ri=1 Xiei + y ∈ Fε(Kr;Kr) which

satisfies ‖g‖ε ≤ ε we obtain the existence of fy(X) := f(X + y) satisfying

‖fy‖ε ≤ ‖f‖ε and fy(x) = f(x + y) for x ∈ Bε(0).

By symmetry we also have

‖f‖ε = ‖(fy)−y‖ε ≤ ‖fy‖ε.

It will be convenient in the following to use the short notation

i := (0, . . . , 1, . . . , 0)

for the multi-index whose only nonzero entry is a 1 in the i-th place.Suppose that the power series f(X) =

∑α Xαvα is ε-convergent. Then

also, for any 1 ≤ i ≤ r, its i-th formal partial derivative

∂f

∂Xi(X) :=

∑

α

Xα−iαivα

is ε-convergent (since ‖αivα‖ ≤ ‖vα‖). In case our field K has characteristiczero it follows inductively that

vα =1

αi! · . . . · αr!

((∂

∂X1

)α1

· · ·(

∂

∂Xr

)αr

f

)∼(0) for any α.

Proposition 5.6. The map f is strictly differentiable in every point z ∈Bε(0) and satisfies

D(i)z f(1) =

(∂f

∂Xi

)∼(z).

Proof. Case 1: We assume that z = 0, and introduce the continuous linearmap

D : Kr −→ V

(a1, . . . , ar) �−→r∑

i=1

aivi.

32 I Foundations

Let δ > 0 and choose a 0 < δ′ < ε such that

δ′‖f‖ε

ε2≤ δ.

By induction with respect to |α| one checks that

|xα − yα| ≤ (δ′)|α|−1‖x − y‖ for any x, y ∈ Bδ′(0).

We now compute

‖f(x) − f(y) − D(x − y)‖ =

∥∥∥∥∥

∑

|α|≥2

(xα − yα)vα

∥∥∥∥∥

≤ max|α|≥2

|xα − yα| · ‖vα‖

≤ ‖f‖ε · max|α|≥2

|xα − yα|ε|α|

≤ ‖f‖ε · max|α|≥2

(δ′)|α|−1

ε|α|· ‖x − y‖

= ‖f‖ε ·δ′

ε2· ‖x − y‖

≤ δ‖x − y‖

for any x, y ∈ Bδ′(0). This proves that f is strictly differentiable in 0 withD0f = D and hence

D(i)0 f(1) = vi =

(∂f

∂Xi

)∼(0).

Case 2: Let z ∈ Bε(0) be an arbitrary point. By Cor. 5.5 we find a powerseries fz(X) =

∑α Xαvα(z) in Fε(Kr;V ) such that

f(x) = fz(x − z) for any x ∈ Bε(0).

Using the chain rule together with the first case we see that f is strictlydifferentiable in z with

D(i)z f(1) = D

(i)0 fz(1) = vi(z).

Since fz(X) can be computed by formally inserting X +z into f(X) we have∑

α

Xαvα(z) =∑

α

(X + z)αvα

5 Power Series 33

and hence

vi(z) =∑

α

zα−iαivα =(

∂f

∂Xi

)∼(z).

By Prop. 5.6 the map

∂f

∂xi: Bε(0) −→ V

x �−→ D(i)x f(1)

is well defined and satisfies

∂f

∂xi=

(∂f

∂Xi

)∼.

Corollary 5.7. (Taylor expansion) If K has characteristic zero then wehave

f(X) =∑

α

Xα 1α1! · . . . · αr!

((∂

∂x1

)α1

· · ·(

∂

∂xr

)αr

f

)

(0).

Corollary 5.8. (Identity theorem for power series) If K has characteristiczero then for any nonzero f ∈ Fε(Kr;V ) there is a point x ∈ Bε(0) suchthat f(x) �= 0.

In fact much stronger results than Cor. 5.8 hold true. In particular, theassumption on the characteristic of K is superfluous. But this requires adifferent method of proof (cf. [BGR] 5.1.4 Cor. 5 and subsequent comment).In any case the map

Fε(Kr;V ) −→ strictly differentiable maps Bε(0) −→ V

f �−→ f

is injective and commutes with all the usual operations as considered above.We therefore will simplify notations in the following and write very often ffor the power series as well as the corresponding map.

Proposition 5.9. (Invertibility for power series) Let f(X) ∈ Fε(Kr;Kr)such that f(0) = 0, and suppose that D0f is bijective; we fix a 0 < δ <

ε2

‖f‖ε‖(D0f)−1‖2 ; then δ < ‖f‖ε, and there is a uniquely determined g(Y ) ∈Fδ(Kr;Kr) such that

g(0) = 0, ‖g‖δ < ε, and f ◦ g(Y ) = Y ;

34 I Foundations

in particular, the diagram

Bδ(0)⊆

g

Bε(0)f

B‖f‖ε(0)

is commutative.

Proof. Case 1: We assume that D0f = idKr . Let

f = (f1, . . . , fr) and fi(X) =∑

α

ai,αXα.

Since f(0) = 0 we have ai,0 = 0. Moreover, by Prop. 5.6, the matrix of D0fin the standard basis of Kr is equal to

(∂fi

∂Xj(0)

)

i,j

= (ai,j)i,j .

But we are in the special case that this matrix is the identity matrix. Henceai,j = 0 for i �= j and ai,i = 1. We therefore see that

(6) fi(X) = Xi +∑

|α|≥2

ai,αXα.

It follows in particular that‖f‖ε ≥ ε

and hence

(7) δ <ε2

‖f‖ε‖(D0f)−1‖2=

ε2

‖f‖ε≤ ε ≤ ‖f‖ε.

In case 1 of the proof of Prop. 5.6 we have computed that

‖f(x) − f(y) − (x − y)‖ ≤ δ‖f‖ε

ε2‖x − y‖ for any x, y ∈ Bδ(0).

As δ ‖f‖ε

ε2 < 1 this is the condition (4) in Lemma 4.2. We therefore concludethat

f : Bδ(0) −−→ Bδ(0)

5 Power Series 35

is a homeomorphism. Furthermore, for |α| ≥ 2 we have

|ai,j |δ|α| ≤ ‖f‖ε

(δ

ε

)|α|≤ ‖f‖ε

(δ

ε

)2

= δ

(

δ‖f‖ε

ε2

)

< δ.

Hence it follows from (6) that

‖f‖δ = δ.

In a next step we establish the existence of a formal power series g =(g1, . . . , gr) with

gi(Y ) =∑

|β|≥1

bi,βY β

such thatf(g(Y )) = Y.

First of all let us check that formally inserting any such g into f is a welldefined operation. We formally compute

fi(g(Y )) =∑

|α|≥1

ai,αg1(Y )α1 · . . . · gr(Y )αr

=∑

|α|≥1

ai,α

(∑

|β|≥1

b1,βY β

)α1

· . . . ·(

∑

|β|≥1

br,βY β

)αr

=∑

|γ|≥1

(∑

···ai,αb1,β(1) · . . . · b1,β(α1)b2,β(α1+1) · . . . · br,β(α1+···+αr)

)

Xγ

where in the last expression the multi-indices in the inner sum run over allα, β(1), . . . , β(α1 + · · · + αr) such that |α|, |β(1)|, . . . , |β(α1 + · · · + αr)| ≥ 1and

β(1)+ · · ·+β(α1)+β(α1 +1)+ · · ·+β(α1 +α2)+ · · ·+β(α1 + · · ·+αr) = γ.

Because of |β(ν)| ≥ 1 this condition enforces |α| ≤ |γ| so that these innersums in fact are finite. We now set

Yi = fi(g(Y ))

and compare coefficients. For γ = i we obtain

1 =r∑

�=1

ai,�b�,i = bi,i.

36 I Foundations

For γ �= i we have

0 =∑

···ai,αb1,β(1) · . . . = bi,γ +

∑

2≤|α|≤|γ|ai,αC(α, γ)

where C(α, γ) is a (finite) sum of products of the form

b1,β(1) · . . . · br,β(α1+···+αr) with |β(ν)| ≥ 1 and∑

ν

β(ν) = γ.

In particular, ∑

ν

|β(ν)| = |γ| and |β(ν)| ≥ 1.

Since the number of summands |β(ν)| is equal to |α| ≥ 2 it follows that|β(ν)| < |γ|. We see that on the right hand side of the equation

bi,γ = −∑

2≤|α|≤γ

ai,αC(α, γ)

only coefficients b�,β appear with |β| < |γ|. This means that the coefficientsβi,γ can be computed recursively from these equations. Hence g exists andis uniquely determined. In addition we check inductively that

|bi,γ | ≤(‖f‖ε

ε2

)|γ|−1

holds true. For |γ| = 1 we have βi,γ = 0 or 1 and the inequality is trivial. If|γ| ≥ 2 then the induction hypothesis implies

|C(α, γ)| ≤ max···

|b1,β(1)| · . . . · |br,β(α1+···+αr)|

≤ max···

(‖f‖ε

ε2

)(|β(1)|−1)+···+(|β(α1+···+αr)|−1)

=(‖f‖ε

ε2

)|γ|−|α|.

Hence we have

|bi,γ | ≤ max2≤|α|≤|γ|

|ai,α| · |C(α, γ)| ≤ max2≤|α|≤|γ|

‖f‖ε

ε|α||C(α, γ)|

≤ max2≤|α|≤|γ|

‖f‖ε

ε|α|·(‖f‖ε

ε2

)|γ|−|α|= max

2≤|α|≤|γ|

(ε

‖f‖ε

)|α|−2

·(‖f‖ε

ε2

)|γ|−1

=(‖f‖ε

ε2

)|γ|−1

5 Power Series 37

(the last identity since ε ≤ ‖f‖ε). We deduce that

|bi,γ |δ|γ| ≤(

δ‖f‖ε

ε2

)|γ|−1

δ.

Because of δ ‖f‖ε

ε2 < 1 this shows that g is δ-convergent with ‖g‖δ ≤ δ. Butbi,i = 1 then implies that ‖g‖δ = δ. Altogether we have shown so far that:

– f is δ-convergent with ‖f‖δ = δ;

– there is a uniquely determined δ-convergent g with

g(0) = 0, ‖g‖δ = δ < ε, and f ◦ g(Y ) = Y.

Using Prop. 5.4 we see that f ◦ g = id so that

Bδ(0)f

Bδ(0)g

are homeomorphisms which are inverse to each other. But we also concludethat g ◦ f(X) exists as a δ-convergent power series as well and satisfies(g ◦ f)∼ = g ◦ f = id. The identity theorem Cor. 5.8 then implies that

g ◦ f(X) = X.

Case 2: Let D0f be arbitrary bijective. If (aij)i,j is the matrix of (D0f)−1

in the standard basis of Kr then the operator norm is given by

‖(D0f)−1‖ = maxi,j

|aij |.

Viewed as a power series (D0f)−1 is ε′-convergent for any ε′ > 0 with

‖(D0f)−1‖ε′ = ε′ · maxi,j

|ai,j | = ε′‖(D0f)−1‖.

In the following we put ε′ := ε‖(D0f)−1‖ . Then ‖(D0f)−1‖ε′ = ε, and from

Prop. 5.4 we obtain

f0 := f ◦ (D0f)−1 ∈ Fε′(Kr;Kr) and ‖f0‖ε′ ≤ ‖f‖ε.

Any δ as in the assertion then satisfies

δ <ε2

‖f‖ε‖(D0f)−1‖2=

ε′2

‖f‖ε≤ ε′2

‖f0‖ε′.

38 I Foundations

Obviously f0(0) = 0, and D0f0 = idKr by the chain rule. So we may applythe first case to f0 and obtain a uniquely determined g0 ∈ Fδ(Kr;Kr) suchthat

g0(0) = 0, ‖g0‖δ = δ, and f0 ◦ g0(Y ) = Y

as well as

δ <ε′2

‖f0‖ε′≤ ε′ =

ε

‖(D0f)−1‖ ≤ ‖f0‖ε′ ≤ ‖f‖ε

by (7). We define

g := (D0f)−1 ◦ g0 ∈ Fδ(Kr;Kr).

Then g(0) = 0 and

f(g(Y )) = f((D0f)−1 ◦ g0(Y )) = f ◦ (D0f)−1(g0(Y )) = f0(g0(Y )) = Y.

In addition, Prop. 5.4 implies

‖g‖δ ≤ ‖(D0f)−1‖δ = δ‖(D0f)−1‖ < ε.

The unicity of g easily follows from the unicity of g0.

The next result is more or less obvious.

Proposition 5.10. Let u : V −→ W be a continuous linear map betweenK-Banach spaces; then

Fε(Kr;V ) −→ Fε(Kr;W )

f(X) =∑

α

Xαvα �−→ u ◦ f(X) :=∑

α

Xαu(vα)

is a continuous linear map of operator norm ≤ ‖u‖ which satisfies

u ◦ f(x) = u(f(x)) for any x ∈ Bε(0).

6 Locally Analytic Functions

Let U ⊆ Kr be an open subset and V be a K-Banach space.

Definition. A function f : U −→ V is called locally analytic if for anypoint x0 ∈ U there is a ball Bε(x0) ⊆ U around x0 and a power seriesF ∈ Fε(Kr;V ) such that

f(x) = F (x − x0) for any x ∈ Bε(x0).


The set

Can(U, V ) := all locally analytic functions f : U −→ V

is a K-vector space with respect to pointwise addition and scalar multi-plication. For f1, f2 ∈ Can(U, V ) and x0 ∈ U let Fi ∈ Fεi(K

r;V ) suchthat fi(x) = Fi(x − x0) for any x ∈ Bεi(x0). Put ε := min(ε1, ε2). ThenF1 + F2 ∈ Fε(Kr;V ) and

(f1 + f2)(x) = (F1 + F2)(x − x0) for any x ∈ Bε(x0).

The vector space Can(U, V ) carries a natural topology which we will discusslater on in a more general context.

Example. By Cor. 5.5 we have F ∈ Can(Bε(0), V ) for any F ∈ Fε(Kr;V ).

Proposition 6.1. Suppose that f : U −→ V is locally analytic; then f isstrictly differentiable in every point x0 ∈ U and the function x �−→ Dxf islocally analytic in Can(U,L(Kr, V )).

Proof. Let F ∈ Fε(Kr;V ) such that


From Prop. 5.6 and the chain rule we deduce that f is strictly differentiablein every x ∈ Bε(x0) and

Dxf((a1, . . . , ar)) =r∑

i=1

aiD(i)x−x0

F (1) =r∑

i=1

ai

(∂F

∂Xi

)∼(x − x0).

Let∂F

∂Xi(X) =

∑

α

Xαvi,α.

For any multi-index α we introduce the continuous linear map

Lα : Kr −→ V

(a1, . . . , ar) �−→ a1v1,α + · · · + arvr,α.

Because of ‖Lα‖ ≤ maxi ‖vi,α‖ we have

G(X) :=∑

α

XαLα ∈ Fε(Kr;L(Kr, V ))

andDxf = G(x − x0) for any x ∈ Bε(x0).

40 I Foundations

Remark 6.2. If K has characteristic zero then, for any function f : U −→V , the following conditions are equivalent :

i. f is locally constant ;

ii. f is locally analytic with Dxf = 0 for any x ∈ U .

Proof. This is an immediate consequence of the Taylor formula in Cor. 5.7.

We now give a list of more or less obvious properties of locally analyticfunctions.

1) For any open subset U ′ ⊆ U we have the linear restriction map

Can(U, V ) −→ Can(U ′, V )f �−→ f |U ′.

2) For any open and closed subset U ′ ⊆ U we have the linear map

Can(U ′, V ) −→ Can(U, V )

f �−→ f!(x) :=

{f(x) if x ∈ U ′,

0 otherwise

called extension by zero.

3) If U =⋃

i∈I Ui is a covering by pairwise disjoint open subsets then

Can(U, V ) ∼=∏

i∈I

Can(Ui, V )

f �−→ (f |Ui)i.

4) For any two K-Banach spaces V and W we have

Can(U, V ⊕ W ) ∼= Can(U, V ) ⊕ Can(U,W )f �−→ (prV ◦f,prW ◦f).

In particular

Can(U,Kn) ∼=n∏

i=1

Can(U,K).

5) For any continuous bilinear map u : V1×V2 −→ V between K-Banachspaces we have the bilinear map


Can(U, V1) × Can(U, V2) −→ Can(U, V )(f, g) �−→ u(f, g)

(cf. Prop. 5.2). In particular, Can(U,K) is a K-algebra (cf. Prop. 5.3),and Can(U, V ) is a module over Can(U,K).

6) For any continuous linear map u : V −→ W between K-Banach spaceswe have the linear map

Can(U, V ) −→ Can(U,W )f �−→ u ◦ f

(cf. Prop. 5.10).

Lemma 6.3. Let U ′ ⊆ Kn be an open subset and let g ∈ Can(U,Kn) suchthat g(U) ⊆ U ′; then the map

Can(U ′, V ) −→ Can(U, V )f �−→ f ◦ g

is well defined and K-linear.

Proof. Let x0 ∈ U and put y0 := g(x0) ∈ U ′. We choose a ball Bε(y0) ⊆ U ′

and a power series F ∈ Fε(Kn;V ) such that

f(y) = F (y − y0) for any y ∈ Bε(y0).

We also choose a ball Bδ(x0) ⊆ U and a power series G ∈ Fδ(Kr;Kn) suchthat

g(x) = G(x − x0) for any x ∈ Bδ(x0).

Observing that

‖G − G(0)‖δ′ ≤δ′

δ‖G − G(0)‖δ for any 0 < δ′ ≤ δ

we may decrease δ so that

‖G − y0‖δ = ‖G − G(0)‖δ ≤ ε

(and, in particular, g(Bδ(x0)) ⊆ Bε(y0)) holds true. It then follows fromProp. 5.4 that F ◦ (G − y0) ∈ Fδ(Kr;V ) and

(F ◦ (G − y0))∼(x − x0) = F (G(x − x0) − y0)= F (g(x) − y0)= f(g(x))

for any x ∈ Bδ(x0).

42 I Foundations

The last result can be expressed by saying that the composite of locallyanalytic functions again is locally analytic.

Proposition 6.4. (Local invertibility) Let U ⊆ Kr be an open subset andlet f ∈ Can(U,Kr); suppose that Dx0f is bijective for some x0 ∈ U ; thenthere are open neighbourhoods U0 ⊆ U of x0 and U1 ⊆ Kr of f(x0) suchthat :

i. f : U0−−→ U1 is a homeomorphism;

ii. the inverse map g : U1 −→ U0 is locally analytic, i. e., g ∈ Can(U1,Kr).

Proof. According to Prop. 4.3 we find open neighbourhoods U ′0 ⊆ U of x0

and U ′1 ⊆ Kr of f(x0) such that

f : U ′0

−−→ U ′1 is a homeomorphism.

We choose a ball Bε(x0) ⊆ U ′0 and a power series F ∈ Fε(Kr;Kr) such that


The power series F1(X) := F (X) − f(x0) ∈ Fε(Kr;Kr) satisfies F1(0) = 0.Moreover, D0F1 = Dx0f is invertible. By Prop. 5.9 we therefore find, for asufficiently small 0 < δ < ε, a power series G1(Y ) ∈ Fδ(Kr;Kr) such that

G1(0) = 0, ‖G1‖δ < ε, and F1 ◦ G1(Y ) = Y.

In particular, G1 : Bδ(0) −→ Bε(0) is locally analytic. Hence the composite

g : U1 := Bδ(f(x0))−f(x0)−−−−−→ Bδ(0) G1−−→ Bε(0) +x0−−−→ Bε(x0)

is locally analytic and satisfies

f ◦ g(y) = f(G1(y − f(x0)) + x0) = F (G1(y − f(x0)))= F1(G1(y − f(x0))) + f(x0) = y − f(x0) + f(x0) = y

for any y ∈ U1. By further decreasing δ we may assume that U1 ⊆ U ′1, and

by setting U0 := g(U1) we obtain the commutative diagram

U ′0

f

U ′1

Bε(x0)

⊆

U0

⊆f

U1g

⊆


in which the two lower horizontal arrows both are locally analytic and areinverse to each other.

There are “locally analytic versions” of the Corollaries 4.4 and 4.5 theformulation of which we leave to the reader.

We have done so already and we will systematically continue to call amap f : U −→ U ′ between open subsets U ⊆ Kr and U ′ ⊆ Kn locallyanalytic if the composite U

f−→ U ′ ⊆−−→ Kn is a locally analytic function.

Chapter II

Manifolds

We continue to fix the nonarchimedean field (K, | |). But we change oursystem of notations insofar as from now on we will denote K-Banach spacesby letters like E whereas we reserve letters like U and V for open subsetsin a topological space.

7 Charts and Atlases

Let M be a Hausdorff topological space.

Definition. i. A chart for M is a triple (U,ϕ,Kn) consisting of an opensubset U ⊆ M and a map ϕ : U −→ Kn such that :

(a) ϕ(U) is open in Kn,

(b) ϕ : U�−−→ ϕ(U) is a homeomorphism.

ii. Two charts (U1, ϕ1,Kn1) and (U2, ϕ2,K

n2) for M are called compati-ble if both maps

ϕ1(U1 ∩ U2)ϕ2◦ϕ−1

1

ϕ2(U1 ∩ U2)ϕ1◦ϕ−1

2

are locally analytic.

We note that the condition in part ii. of the above definition makessense since ϕ1(U1 ∩ U2) is open in Kni . If (U,ϕ,Kn) is a chart then theopen subset U is called its domain of definition and the integer n ≥ 0 itsdimension. Usually we omit the vector space Kn from the notation andsimply write (U,ϕ) instead of (U,ϕ,Kn). If x is a point in U then (U,ϕ) isalso called a chart around x.

Lemma 7.1. Let (Ui, ϕi,Kni) for i = 1, 2 be two compatible charts for M ;

if U1 ∩ U2 �= ∅ then n1 = n2.

Proof. Let x ∈ U1∩U2 and put xi := ϕi(x). We consider the locally analyticmaps

ϕ1(U1 ∩ U2)f :=ϕ2◦ϕ−1

1

ϕ2(U1 ∩ U2).g:=ϕ1◦ϕ−1

2


45

http://dx.doi.org/10.1007/978-3-642-21147-8_2

46 II Manifolds

They are differentiable and inverse to each other, and x2 = f(x1). Hence,by the chain rule, the derivatives

Kn1

Dx1f

Kn2

Dx2g

are linear maps inverse to each other. It follows that n1 = n2.

Definition. i. An atlas for M is a set A = {(Ui, ϕi,Kni)}i∈I of charts

for M any two of which are compatible and which cover M in the sensethat M =

⋃i∈I Ui.

ii. Two atlases A and B for M are called equivalent if A ∪ B also is anatlas for M .

iii. An atlas A for M is called maximal if any equivalent atlas B for Msatisfies B ⊆ A.

Remark 7.2. i. The equivalence of atlases indeed is an equivalence re-lation.

ii. In each equivalence class of atlases there is exactly one maximal atlas.

Proof. i. Let A,B, and C be three atlases such that A is equivalent to B andB is equivalent to C. Then A is equivalent to C if we show that any chart(U1, ϕ1) in A is compatible with any chart (U2, ϕ2) in C. By symmetry itsuffices to show that the map ϕ2 ◦ ϕ−1

1 : ϕ1(U1 ∩ U2) −→ ϕ2(U1 ∩ U2) islocally analytic in a sufficiently small open neighbourhood of ϕ1(x) for anypoint x ∈ U1 ∩ U2. Since B covers M we find a chart (V, ψ) around x in B.By assumption (V, ψ) is compatible with both (U1, ϕ1) and (U2, ϕ2). Thenϕ1(U1 ∩V ∩U2) is an open neighbourhood of ϕ1(x) in ϕ1(U1 ∩U2) on whichthe map ϕ2 ◦ϕ−1

1 is the composite of the two locally analytic maps ϕ2 ◦ψ−1

and ψ ◦ ϕ−11 . Hence it is locally analytic by Lemma 6.3.

ii. If the given equivalence class consists of the atlases Aj for j ∈ J thenA :=

⋃j∈J Aj is the unique maximal atlas in this class.

Lemma 7.3. If A is a maximal atlas for M the domains of definition of allthe charts in A form a basis of the topology of M .

Proof. Let U ⊆ M be an open subset. We have to show that U is the unionof the domains of definition of the charts in some subset of A, or equivalentlythat for any point x ∈ U we find a chart (Ux, ϕx) around x in A such that

8 Manifolds 47

Ux ⊆ U . Since A covers M we at least find a chart (U ′x, ϕ′

x) around x in A.We put Ux := U ′

x ∩ U and ϕx := ϕ′x|Ux. Clearly (Ux, ϕx) is a chart around

x for M such that Ux ⊆ U . We claim that (Ux, ϕx) is compatible with anychart (V, ψ) in A. But we do have the locally analytic maps

ϕ′x(U ′

x ∩ V )ψ◦ϕ′−1

x

ψ(U ′x ∩ V )

ϕ′x◦ψ−1

which restrict to the locally analytic maps

ϕx(Ux ∩ V )ψ◦ϕ−1

x

ψ(Ux ∩ V ).ϕx◦ψ

Hence B := A ∪ {(Ux, ϕx)} is an atlas equivalent to A. The maximality ofA then implies that B ⊆ A and a fortiori (Ux, ϕx) ∈ A.

Definition. An atlas A for M is called n-dimensional if all the charts inA with nonempty domain of definition have dimension n.

Remark 7.4. Let A be an n-dimensional atlas for M ; then any atlas Bequivalent to A is n-dimensional as well.

Proof. Let (V, ψ) be any chart in B and choose a point x ∈ V . We find achart (U,ϕ) in A around x. Since A and B are equivalent these two chartshave to be compatible. It then follows from Lemma 7.1 that both have thesame dimension u.

8 Manifolds

Definition. A (locally analytic) manifold (M,A) (over K) is a Hausdorfftopological space M equipped with a maximal atlas A. The manifold is calledn-dimensional (we write dim M = n) if the atlas A is n-dimensional.

By abuse of language we usually speak of a manifold M while consideringA as given implicitly. A chart for M will always mean a chart in A.

Example. Kn will always denote the n-dimensional manifold whose maxi-mal atlas is equivalent to the atlas {(U,⊆,Kn) : U ⊆ Kn open}.

Remark 8.1. Let (U,ϕ,Kn) be a chart for the manifold M ; if V ⊆ U is anopen subset then (V, ϕ|V,Kn) also is a chart for M .

48 II Manifolds

Proof. This was shown in the course of the proof of Lemma 7.3.

Let (M,A) be a manifold and U ⊆ M be an open subset. Then

AU := {(V, ψ,Kn) ∈ A : V ⊆ U},

by Lemma 7.3, is an atlas for U . We claim that AU is maximal. Let (V0, ψ0)be a chart for U which is compatible with any chart in AU . To see that(V0, ψ0) ∈ AU it suffices, by the maximality of A, to show that (V0, ψ0)is compatible with any chart (V, ψ) in A. The Remark 8.1 implies that(V ∩U,ψ|V ∩U) is a chart in A and hence in AU . By assumption (V0, ψ0) iscompatible with (V ∩U,ψ|V ∩U). Since V0∩V ⊆ V ∩U the compatibility of(V0, ψ0) with (V, ψ) follows trivially. The manifold (U,AU ) is called an opensubmanifold of (M,A).

As a nontrivial example of a manifold we discuss the d-dimensional pro-jective space Pd(K) over K. We recall that Pd(K) = (Kd+1 \ {0})/ ∼ is theset of equivalence classes in Kd+1 \ {0} for the equivalence relation

(a1, . . . , ad+1) ∼ (ca1, . . . , cad+1) for any c ∈ K×.

As usual we write [a1 : . . . : ad+1] for the equivalence class of (a1, . . . , ad+1).With respect to the quotient topology from Kd+1 \ {0} the projective spacePd(K) is a Hausdorff topological space. For any 1 ≤ j ≤ d + 1 we have theopen subset

Uj := {[a1 : . . . : ad+1] ∈ Pd(K) : |ai| ≤ |aj | for any 1 ≤ i ≤ d + 1}

together with the homeomorphism

ϕj : Uj�−−→ B1(0) ⊆ Kd

[a1 : . . . : ad+1] −→(

a1

aj, . . . ,

aj−1

aj,aj+1

aj, . . . ,

ad+1

aj

)

.

The (Uj , ϕj ,Kd) are charts for Pd(K) such that

⋃j Uj = Pd(K). We claim

that they are pairwise compatible. For 1 ≤ j < k ≤ d + 1 the composite

f : V := {x ∈ B1(0) : |xk−1| = 1}ϕ−1

j−−−→ Uj ∩ Ukϕk−−→ {y ∈ B1(0) : |yj | = 1}

is given by

f(x1, . . . , xd) =(

x1

xk−1, . . . ,

xj−1

xk−1,

1xk−1

,xj

xk−1, . . . ,

xk−2

xk−1,

xk

xk−1, . . . ,

xd

xk−1

)

.

8 Manifolds 49

Let a ∈ V be a fixed but arbitrary point and choose a 0 < ε < 1. ThenBε(a) ⊆ V . We consider the power series

Fj(X) :=1

ak−1

∑

n≥0

(

− 1ak−1

)n

Xnk−1

and

Fi(X) := Fj(X) ·{

(Xi + ai) if 1 ≤ i < j or k ≤ i ≤ d,

(Xi−1 + ai−1) if j < i < k.

Because of |ak−1| = 1 we have F := (F1, . . . , Fd) ∈ Fε(Kd;Kd). For x ∈Bε(a) we compute

Fj(x − a) =1

ak−1

∑

n≥0

(

−xk−1 − ak−1

ak−1

)n

=1

ak−1· 11 + xk−1−ak−1

ak−1

=1

xk−1

and thenf(x) = F (x − a).

Hence f is locally analytic. In case j > k the argument is analogous. Theabove charts therefore form a d-dimensional atlas for Pd(K).

Exercise. Let (M,A) and (N,B) be two manifolds. Then

A× B := {(U × V ), ϕ × ψ,Km+n) : (U,ϕ,Km) ∈ A, (V, ψ,Kn) ∈ B}

is an atlas for M × N with the product topology. We call M × N equippedwith the equivalent maximal atlas the product manifold of M and N .

Let M be a manifold and E be a K-Banach space.

Definition. A function f : M −→ E is called locally analytic if f ◦ ϕ−1 ∈Can(ϕ(U), E) for any chart (U,ϕ) for M .

Remark 8.2. i. Every locally analytic function f : M −→ E is contin-uous.

ii. Let B be any atlas consisting of charts for M ; a function f : M −→E is locally analytic if and only if f ◦ ϕ−1 ∈ Can(ϕ(U), E) for any(U,ϕ) ∈ B.

50 II Manifolds

The set

Can(M,E) := all locally analytic functions f : M −→ E

is a K-vector space with respect to pointwise addition and scalar multipli-cation. It is easy to see that a list of properties 1)–6) completely analogousto the one given in Sect. 6 holds true. In a later section we will come backto a more detailed study of this vector space.

Let now M and N be two manifolds. The following result is immediate.

Lemma 8.3. For a map g : M −→ N the following assertions are equiva-lent :

i. g is continuous and ψ ◦ g ∈ Can(g−1(V ),Kn) for any chart (V, ψ,Kn)for N ;

ii. for any point x ∈ M there exist a chart (U,ϕ,Km) for M around xand a chart (V, ψ,Kn) for N around g(x) such that g(U) ⊆ V andψ ◦ g ◦ ϕ−1 ∈ Can(ϕ(U),Kn).

Definition. A map g : M −→ N is called locally analytic if the equivalentconditions in Lemma 8.3 are satisfied.

Lemma 8.4. i. If g : M −→ N is a locally analytic map and E is aK-Banach space then

Can(N,E) −→ Can(M,E)f −→ f ◦ g

is a well defined K-linear map.

ii. With Lf−→ M

g−→ N also g ◦ f : L −→ N is a locally analytic map ofmanifolds.

Proof. This follows from Lemma 6.3.

Examples 8.5. 1) For any open submanifold U of M the inclusion map

U⊆−−→ M is locally analytic.

2) Let g : M −→ N be a locally analytic map; for any open submanifoldV ⊆ N the induced map g−1(V )

g−→ V is locally analytic.

8 Manifolds 51

3) The two projection maps

pr1 : M × N −→ M and pr2 : M × N −→ N

are locally analytic.

4) For any pair of locally analytic maps g : L −→ M and f : L −→ Nthe map

(g, f) : L −→ M × N

x −→ (g(x), f(x))

is locally analytic.

For the remainder of this section we will discuss a certain technical butuseful topological property of manifolds. First let X be an arbitrary Haus-dorff topological space. We recall:

– Let X =⋃

i∈I Ui and X =⋃

j∈J Vj be two open coverings of X. Thesecond one is called a refinement of the first if for any j ∈ J there isan i ∈ I such that Vj ⊆ Ui.

– An open covering X =⋃

i∈I Ui of X is called locally finite if everypoint x ∈ X has an open neighbourhood Ux such that the set {i ∈ I :Ux ∩ Ui �= ∅} is finite.

– The space X is called paracompact , resp. strictly paracompact , if anyopen covering of X can be refined into an open covering which is locallyfinite, resp. which consists of pairwise disjoint open subsets.

Remark 8.6. i. Any ultrametric space X is strictly paracompact.

ii. Any compact space X is paracompact.

Proof. i. This follows from Lemma 1.4. ii. This is trivial.

Proposition 8.7. For a manifold M the following conditions are equivalent :

i. M is paracompact ;

ii. M is strictly paracompact ;

iii. the topology of M can be defined by a metric which satisfies the stricttriangle inequality.

52 II Manifolds

Proof. The implication iii. =⇒ ii. is Lemma 1.4, and the implication ii. =⇒i. is trivial.

i. =⇒ ii. We suppose that M is paracompact. From general topology werecall the following property of paracompact Hausdorff spaces (cf. [B-GT]Chap. IX §4.4 Cor. 2). Let A ⊆ U ⊆ M be subsets with A closed and Uopen. Then there is another open subset V ⊆ M such that

A ⊆ V ⊆ V ⊆ U.

Step 1: We show that the open and closed subsets of M form a basis ofthe topology. Given a point x in an open subset U ⊆ M we have to find anopen and closed subset W ⊆ M such that x ∈ W ⊆ U . By Lemma 7.3 wemay assume that U is the domain of definition of a chart (U,ϕ,Kn) for M .As recalled above there is an open neighbourhood V ⊆ M of x such thatV ⊆ U . We then have the vertical homeomorphisms

V⊆

�

V⊆

�

U⊆

�

M

ϕ(V )⊆

ϕ(V )⊆

ϕ(U)⊆

Kn.

Since ϕ(V ) is open in Kn there is a ball B := Bε(ϕ(x)) ⊆ ϕ(V ) aroundϕ(x). We put W := ϕ−1(B) ⊆ V . Clearly x ∈ W ⊆ U . The ball B is openand hence B is open in V and M . But the ball B also is closed in Kn. HenceW is closed in V and therefore in M . This finishes step 1.

Let now M =⋃

i∈I Ui be a fixed but arbitrary open covering. By Lem-ma 7.3 we may assume, after refinement, that any Ui is the domain of defi-nition of some chart for M . By the first step and Remark 8.1 we may evenassume, after a further refinement, that each Ui is open and closed in M andis the domain of definition of some chart for M . In particular, each Ui hasthe topology of an ultrametric space. By assumption we may pick a locallyfinite refinement (Vj)j∈J of (Ui)i∈I . So we have the locally finite open cover-ing M =

⋃j∈J Vj , and for each j ∈ J there is an i(j) ∈ I such that Vj ⊆ Ui(j).

Step 2: We construct a covering M =⋃

j∈J Wj by open and closedsubsets Wj ⊆ M such that Wj ⊆ Vj for any j ∈ J . For this purpose weequip J with a well-order ≤ (recall that this is a total order on J with theproperty that each nonempty subset of J has a minimal element—by theaxiom of choice such a well-order always exists). We now use transfiniteinduction to find open and closed subsets Wj ⊆ M such that

8 Manifolds 53

(a) Wj ⊆ Vj for any j ∈ J , and

(b) M =( ⋃

j≤k Wj

)∪

( ⋃j>k Vj

)for any k ∈ J .

We fix a k ∈ J and suppose that the Wj for j < k are constructed already.Claim: M =

( ⋃j<k Wj

)∪

( ⋃j≥k Vj

).

Let x ∈ M . Since the covering (Vj)j is locally finite the set

{j ∈ J : x ∈ Vj} = {j1 < · · · < jr}

is finite. If jr ≥ k then x ∈ Vjr ⊆⋃

j≥k Vj . If jr < k then x �∈ Vj forany j > jr and the induction hypothesis (property (b) for jr) implies x ∈⋃

j≤jrWj ⊆

⋃j<k Wj . This establishes the claim.

We see that the closed subset

W := M \((

⋃

j<k

Wj

)

∪(

⋃

j>k

Vj

))

of M satisfiesW ⊆ Vk ⊆ Ui(k).

Claim: Let (X, d) be an ultrametric space; for any subsets A ⊆ U ⊆ Xwith A closed and U open there exists an open and closed subset V ⊆ Xsuch that

A ⊆ V ⊆ U.

For any subset D ⊆ X and any x ∈ X we put

d(x,D) := infy∈D

d(x, y).

The strict triangle inequality implies that the function d(., D) on X is con-tinuous and that

D(ε) := {x ∈ X : d(x,D) = ε},

for any ε > 0, is open in X. Moreover, D(0) = D. The closed subsets Aand B := X \ U of X satisfy A ∩ B = ∅. By the continuity of the functionsd(., A) and d(., B) the subset

V := {x ∈ X : d(x,A) < d(x,B)}

therefore is open in X and satisfies A ⊆ V ⊆ U . Similarly V ′ := {x ∈ X :d(x,A) > d(x,B)} is open in X. It follows that V as the complement in Xof the open subset V ′ ∪

( ⋃ε>0(A(ε) ∩ B(ε))

)is closed. This establishes the

claim.

54 II Manifolds

We apply this claim to W ⊆ Vk ⊆ Ui(k) and obtain an open and closedsubset Wk ⊆ Ui(k) such that W ⊆ Wk ⊆ Vk. With Ui(k) also Wk is openand closed in M . As W ⊆ Wk the index k has the property (b). It remainsto show that the Wj for j ∈ J actually cover M . Let x ∈ M . As arguedbefore the set {j ∈ J : x ∈ Vj} = {j1 < · · · < jr} is finite. Then x �∈ Vj

for any j > jr. The property (b) for the index jr therefore implies thatx ∈

⋃j≤jr

Wj . This finishes step 2.

Step 3: At this point we have constructed a locally finite refinement(Wj)j∈J of our initial covering which consists of open and closed subsetsWj ⊆ M .

Claim: W ′L :=

⋃j∈L Wj , for any subset L ⊆ J , is open and closed in M .

Obviously W ′L is open. To see that its complement M \ W ′

L is open aswell let x ∈ M \ W ′

L be any point. In particular, x �∈ Wj for any j ∈ L.Since the covering (Wj)j is locally finite we find an open neighbourhoodUx ⊆ M of x such that the set {j ∈ L : Ux ∩Wj �= ∅} = {j1, . . . , js} is finite.Then Ux \ (Wj1 ∪ · · · ∪Wjs) is an open neighbourhood of x in M \W ′

L. Thisestablishes the claim.

We finally define a new index set P by

P := all nonempty finite subsets of J,

and for any L ∈ P we put

WL :=

(⋂

j∈L

Wj

)

\(

⋃

j∈J\LWj

)

=

(⋂

j∈L

Wj

)

\ W ′J\L.

Clearly any WL is contained in some Wj . By the above claim each WL isopen and closed in M . To check that

M =⋃

L∈P

WL

holds true let x ∈ M be any point. Then x ∈ WL for the finite set L :={j ∈ J : x ∈ Wj}. Moreover, the WL are pairwise disjoint: Let L1 �= L2

be two different indices in P . By symmetry we may assume that there is aj ∈ L1 \ L2. Then WL1 ⊆ Wj and WL2 ⊆ M \ Wj . It follows that (WL)L∈P

is a refinement of our initial covering by pairwise disjoint open subsets. Thisproves that M is strictly paracompact.

ii. =⇒ iii. We start with an open covering of M by domains of definitionof charts for M . By assumption we may refine it into a covering M =

⋃i∈I Ui

8 Manifolds 55

by pairwise disjoint open subsets. According to Remark 8.1 each Ui also isthe domain of definition of some chart for M . In particular, the topology ofUi can be defined by a metric d′i which satisfies the strict triangle inequality.We put

di(x, y) :=d′i(x, y)

1 + d′i(x, y)for any x, y ∈ Ui.

Obviously we have di(x, y) = di(y, x) and di(x, y) = 0 if and only if x = y.To see that di satisfies the strict triangle inequality we compute

di(x, z) =d′i(x, z)

1 + d′i(x, z)≤ max(d′i(x, y), d′i(y, z))

1 + max(d′i(x, y), d′i(y, z))

= max(

d′i(x, y)1 + d′i(x, y)

,d′i(y, z)

1 + d′i(y, z)

)

= max(di(x, y), di(y, z)).

Here we have used the simple fact that t ≥ s ≥ 0 implies t(1 + s) = t + ts ≥s + st = s(1 + t) and hence t

1+t ≥ s1+s . For trivial reasons we have di ≤ d′i.

On the other handd′i =

di

1 − di

and hence, for 0 < ε ≤ 1,

d′i(x, y) ≤ ε if di(x, y) ≤ ε

2.

This shows that the metrics d′i and di define the same topology on Ui. Wenote that

di(x, y) < 1 for any x, y ∈ Ui.

We now define

d : M × M −→ R≥0

(x, y) −→{

di(x, y) if x, y ∈ Ui for some i ∈ I,

1 otherwise.

This is a metric on d. The strict triangle equality

d(x, z) ≤ max(d(x, y), d(y, z))

only needs justification if not all three points lie in the same subset Ui. Butthen the right hand side is ≥ 1 whereas the left hand side is ≤ 1. We claimthat this metric d defines the topology of M . First consider any ball Bε(x)

56 II Manifolds

with respect to d in M . If ε ≥ 1 then Bε(x) = M , and if ε < 1 then Bε(x) isopen in some Ui. Hence Bε(x) is open in M . Vice versa let V ⊆ M be anyopen subset and let x ∈ V . We choose an i ∈ I such that x ∈ Ui. Then V ∩Ui

is an open neighbourhood of x in Ui. Hence, for some 0 < ε < 1, the ballBε(x) with respect to d (or equivalently di) is contained in V ∩Ui ⊆ V .

Corollary 8.8. Open submanifolds and product manifolds of paracompactmanifolds are paracompact.

9 The Tangent Space

Let M be a manifold, and fix a point a ∈ M . We consider pairs (c, v) where

– c = (U,ϕ,Km) is a chart for M around a and

– v ∈ Km.

Two such pairs (c, v) and (c′, v′) are called equivalent if we have

Dϕ(a)(ϕ′ ◦ ϕ−1)(v) = v′.

It follows from the chain rule that this indeed defines an equivalence relation.

Definition. A tangent vector of M at the point a is an equivalence class[c, v] of pairs (c, v) as above.

We define

Ta(M) := set of all tangent vectors of M at a.

Lemma 9.1. Let c = (U,ϕ,Km) and c′ = (U ′, ϕ′,Km) be two charts for Maround a; we then have:

i. The map

θc : Km ∼−−→ Ta(M)v −→ [c, v]

is bijective.

ii. θ−1c′ ◦ θc : Km

∼=−−→ Km is a K-linear isomorphism.


Proof. (We recall from Lemma 7.1 that the dimensions of two charts aroundthe same point necessarily coincide.) i. Surjectivity follows from

[c′′, v′′] = [c,Dϕ′′(a)(ϕ ◦ ϕ′′−1)(v′′)].

If [c, v] = [c, v′] then v′ = Dϕ(a)(ϕ ◦ϕ−1)(v) = v. This proves the injectivity.ii. From [c, v] = [c′, Dϕ(a)(ϕ′ ◦ ϕ−1)(v)] we deduce that

θ−1c′ ◦ θc = Dϕ(a)(ϕ

′ ◦ ϕ−1).

The set Ta(M), by Lemma 9.1.i., has precisely one structure of a topo-logical K-vector space such that the map θc is a K-linear homeomorphism.Because of Lemma 9.1.ii. this structure is independent of the choice of thechart c around a.

Definition. The K-vector space Ta(M) is called the tangent space of M atthe point a.

Remark. The manifold M has dimension m if and only if dimK Ta(M) = mfor any a ∈ M .

Let g : M −→ N be a locally analytic map of manifolds. By Lemma 8.3.ii.we find charts c = (U,ϕ,Km) for M around a and c = (V, ψ,Kn) for Naround g(a) such that g(U) ⊆ V . The composite

Ta(g) : Ta(M) θ−1c−−−→ Km Dϕ(a)(ψ◦g◦ϕ−1)

−−−−−−−−−−−→ Kn θc−−→ Tg(a)(N)

is a continuous K-linear map. We claim that Ta(g) does not depend onthe particular choice of charts. Let c′ = (U ′, ϕ′) and c′ = (V ′, ψ′) be othercharts around a and g(a), respectively. Using the identity in the proof ofLemma 9.1.ii. as well as the chain rule we compute

θc ◦ Dϕ(a)(ψ ◦ g ◦ ϕ−1) ◦ θ−1c

= θc′ ◦ Dψ(g(a))(ψ′ ◦ ψ−1) ◦ Dϕ(a)(ψ ◦ g ◦ ϕ−1) ◦ Dϕ(a)(ϕ

′ ◦ ϕ−1)−1 ◦ θ−1c′

= θc′ ◦ Dϕ′(a)(ψ′ ◦ g ◦ ϕ′−1) ◦ θ−1

c′ .

Definition. Ta(g) is called the tangent map of g at the point a.

Remark. Ta(idM ) = idTa(M).

58 II Manifolds

Lemma 9.2. For any locally analytic maps of manifolds Lf−→ M

g−→ Nwe have

Ta(g ◦ f) = Tf(a)(g) ◦ Ta(f) for any a ∈ L.

Proof. This is an easy consequence of the chain rule.

Proposition 9.3. (Local invertibility) Let g : M −→ N be a locally analyticmap of manifolds, and suppose that Ta(g) : Ta(M)

∼=−−→ Tg(a)(N) is bijectivefor some a ∈ M ; then there are open neighbourhoods U ⊆ M of a and V ⊆ Nof g(a) such that g restricts to a locally analytic isomorphism

g : U�−−→ V

of open submanifolds.

Proof. By Lemma 8.3.ii. we find charts c = (U ′, ϕ,Km) for M around a andc = (V ′, ψ,Kn) for N around g(a) such that g(U ′) ⊆ V ′. We consider thelocally analytic function

ϕ(U ′)ϕ−1

−−−→ U ′ g−→ V ′ ψ−−→ ψ(V ′) ⊆ Kn.

By assumption the derivative

Dϕ(a)(ψ ◦ g ◦ ϕ−1) = θ−1c ◦ Ta(g) ◦ θc

is bijective. Prop. 6.4 therefore implies the existence of open neighbourhoodsW0 ⊆ ϕ(U ′) of ϕ(a) and W1 ⊆ ψ(V ′) of ψ(g(a)) such that

ψ ◦ g ◦ ϕ−1 : W0�−−→ W1

is a locally analytic isomorphism. Hence

g : U := ϕ−1(W0)�−−→ V := ψ−1(W1)

is a locally analytic isomorphism as well (observe the subsequent exercise).

Exercise. Let (U,ϕ,Km) be a chart for the manifold M ; then ϕ : U�−−→

ϕ(U) is a locally analytic isomorphism between the open submanifolds U ofM and ϕ(U) of Km.


Let M be a manifold, E be a K-Banach space, f ∈ Can(M,E), and a ∈M . If c = (U,ϕ,Km) is a chart for M around a then f ◦ϕ−1 ∈ Can(ϕ(U), E).Hence

daf : Ta(M) θ−1c−−→ Km Dϕ(a)(f◦ϕ−1)

−−−−−−−−−→ E

[c, v] �−−−−−−−−−−−→ Dϕ(a)(f ◦ ϕ−1)(v)

is a continuous K-linear map. If c′ = (U ′, ϕ′,Km) is another chart arounda then

Dϕ(a)(f ◦ ϕ−1) ◦ θ−1c = Dϕ(a)(f ◦ ϕ−1) ◦ Dϕ(a)(ϕ

′ ◦ ϕ−1)−1 ◦ θ−1c′

= Dϕ′(a)(f ◦ ϕ′−1) ◦ θ−1c′ .

This shows that daf does not depend on the choice of the chart c.

Definition. daf is called the derivative of f in the point a.

Remark 9.4. For E = Kr viewed as a manifold and for the chart c0 =(Kr, id, E) for E we have

Ta(f) = θc0 ◦ daf.

Obviously the map

Can(M,E) −→ L(Ta(M), E)f −→ daf

is K-linear.

Lemma 9.5. (Product rule)

i. Let u : E1×E2 −→ E be a continuous bilinear map between K-Banachspaces; if fi ∈ Can(M,Ei) for i = 1, 2 then u(f1, f2) ∈ Can(M,E) and

da(u(f1, f2)) = u(f1(a), daf2) + u(daf1, f2(a)) for any a ∈ M.

ii. For g ∈ Can(M,K) and f ∈ Can(M,E) we have

da(gf) = g(a) · daf + dag · f(a) for any a ∈ M.

60 II Manifolds

Proof. i. It is a straightforward consequence of Prop. 5.2 that the functionu(f1, f2) is locally analytic (compare the property 5) of the list in Sect. 6).Let c = (U,ϕ) be a chart of M around a. Using the product rule in Re-mark 4.1.iv. we compute

da(u(f1, f2))([c, v]) = Dϕ(a)(u(f1, f2) ◦ ϕ−1)(v)

= Dϕ(a)(u(f1 ◦ ϕ−1, f2 ◦ ϕ−1))(v)

= u(f1 ◦ ϕ−1(ϕ(a)), Dϕ(a)(f2 ◦ ϕ−1)(v))

+ u(Dϕ(a)(f1 ◦ ϕ−1)(v), f2 ◦ ϕ−1(ϕ(a)))

= u(f1(a), daf2([c, v])) + u(daf1([c, v]), f2(a)).

ii. This is a special case of the first assertion.

Let c = (U,ϕ,Km) be a chart for M . On the one hand, by definition, wehave daϕ = θ−1

c for any a ∈ U ; in particular

daϕ : Ta(M)∼=−−→ Km

is a K-linear isomorphism. On the other hand viewing ϕ = (ϕ1, . . . , ϕm) asa tuple of locally analytic functions ϕi : U −→ K we have

daϕ = (daϕ1, . . . , daϕm).

This means that {daϕi}1≤i≤m is a K-basis of the dual vector space Ta(M)′.Let {(

∂

∂ϕi

)

(a)}

1≤i≤m

denote the corresponding dual basis of Ta(M), i. e.,

daϕi

((∂

∂ϕj

)

(a))

= δij for any a ∈ U

where δij is the Kronecker symbol. For any f ∈ Can(M,E) we define thefunctions

∂f

∂ϕi: U −→ E

a −→ daf

((∂

∂ϕi

)

(a))

.


Lemma 9.6. ∂f∂ϕi

∈ Can(U,E) for any 1 ≤ i ≤ m, and

daf =m∑

i=1

daϕi ·∂f

∂ϕi(a) for any a ∈ U.

Proof. We have

∂f

∂ϕi(a) = Dϕ(a)(f ◦ ϕ−1) ◦ θ−1

c

((∂

∂ϕi

)

(a))

= Dϕ(a)(f ◦ ϕ−1)(ei)

where e1, . . . , em denotes the standard basis of Km. Hence ∂f∂ϕi

is the com-posite

Uϕ−→ ϕ(U)

x →Dx(f◦ϕ−1)−−−−−−−−−−→ L(Km, E)D →D(ei)−−−−−−−→ E.

The function in the middle is locally analytic by Prop. 6.1. Clearly, D −→D(ei) is a continuous K-linear map. Hence the composite of the right twomaps is locally analytic by the property 6) in Sect. 6. That the full composite∂f∂ϕi

is locally analytic now follows from Lemma 8.4.i. Let

t =m∑

i=1

ci

(∂

∂ϕi

)

(a) ∈ Ta(M)

be an arbitrary vector. By the definition of the dual basis we have ci =daϕi(t). We now compute

daf(t) =m∑

i=1

ci · daf

((∂

∂ϕi

)

(a))

=m∑

i=1

daϕi(t) ·∂f

∂ϕi(a).

In a next step we want to show that the disjoint union

T (M) :=⋃

a∈M

Ta(M)

in a natural way is a manifold again. We introduce the projection map

pM : T (M) −→ M

t −→ a if t ∈ Ta(M).

Hence Ta(M) = p−1M (a).

62 II Manifolds

Consider any chart c = (U,ϕ,Km) for M . By Lemma 9.1.i. the map

τc : U × Km ∼−−→ p−1M (U)

(a, v) −→ [c, v] viewed in Ta(M)

is bijective. Hence the composite

ϕc : p−1M (U) τ−1

c−−−→ U × Km ϕ×id−−−−→ Km × Km = K2m

is a bijection onto an open subset in K2m. The idea is that

cT :=(p−1

M (U), ϕc,K2m

)

should be a chart for the manifold T (M) yet to be constructed. Clearly wehave

T (M) =⋃

c=(U,ϕ)

p−1M (U).

Let c = (V, ψ,Km) be another chart for M such that U ∩ V �= ∅. Thecomposed map

ϕ(U ∩ V )×Km ϕ−1c−−−→ p−1

M (U ∩ V ) = p−1M (U)∩ p−1

M (V )ψc−−→ ψ(U ∩ V )×Km

is given by

(8) (x, v) −→ (ψ ◦ ϕ−1(x), Dx(ψ ◦ ϕ−1)(v)).

The first component ψ◦ϕ−1 of this map is locally analytic on ϕ(U ∩V ) sinceM is a manifold. The second component can be viewed as the composite

ϕ(U ∩ V ) × Km −→ L(Km,Km) × Km −→ Km

(x, v) −→ (Dx(ψ ◦ ϕ−1), v)(u, v) −→ u(v).

The left function is locally analytic by Prop. 6.1. The right bilinear map iscontinuous. Hence the composite is locally analytic by Lemma 9.5.i. Thisshows that, once cT and cT are recognized as charts for T (M) with respectto a topology yet to be defined, they in fact are compatible, and hence thatthe set {cT : c a chart for M} is an atlas for T (M).

We have shown in particular that the composed map

(9) (U ∩ V ) × Km τc−→ p−1M (U ∩ V )

τ−1c−−−→ (U ∩ V ) × Km

is a homeomorphism.


Definition. A subset X ⊆ T (M) is called open if τ−1c (X ∩ p−1

M (U)) is openin U × Km for any chart c = (U,ϕ,Km) for M .

This defines the finest topology on T (M) which makes all composedmaps

U × Km τc−−→ p−1M (U) ⊆−−→ T (M)

continuous.

Lemma 9.7. i. The map τc : U ×Km �−−→ p−1M (U) is a homeomorphism

with respect to the subspace topology induced by T (M) on p−1M (U).

ii. The map pM is continuous.

iii. The topological space T (M) is Hausdorff.

Proof. i. The continuity of τc holds by construction. Let Y ⊆ U × Km

be an open subset. We will show that τc(Y ) is open in T (M), i. e., thatτ−1c (τc(Y )∩ p−1

M (V )) is open in V ×Km for any chart c = (V, ψ,Kn) for M .We may of course assume that U ∩V �= ∅ so that n = m. Clearly the subsetY ∩ ((U ∩ V ) × Km) is open in (U ∩ V ) × Km. By (9) the subset

τ−1c (τc(Y ∩ ((U ∩ V ) × Km))) = τ−1

c (τc(Y ) ∩ p−1M (U ∩ V ))

= τ−1c (τc(Y ) ∩ p−1

M (U) ∩ p−1M (V ))

= τ−1c (τc(Y ) ∩ p−1

M (V ))

is open in (U ∩ V ) × Km and therefore in V × Km.ii. The above reasoning for Y = U × Km shows that τc(Y ) = p−1

M (U) isopen in T (M) where U is the domain of definition of any chart for M . Itthen follows from Lemma 7.3 that pM is continuous.

iii. Let s and t be two different points in T (M). Case 1: We have pM (s) �=pM (t). Since M is Hausdorff we find open neighbourhoods U ⊆ M of pM (s)and V ⊆ M of pM (t) such that U∩V = ∅. By ii. then p−1

M (U) and p−1M (V ) are

open neighbourhoods of s and t, respectively, such that p−1M (U) ∩ p−1

M (V ) =p−1

M (U ∩ V ) = ∅. Case 2: We have a := pM (s) = pM (t). We choose a chartc = (U,ϕ,Km) for M around a. The two points s and t lie in the open (byii.) subset p−1

M (U) of T (M). But by i. the subspace p−1M (U) is homeomorphic,

via the map τc, to the Hausdorff space U ×Km. Hence p−1M (U) is Hausdorff

and s and t can be separated by open neighbourhoods in p−1M (U) and a

fortiori in T (M).

64 II Manifolds

The Lemma 9.7 in particular says that cT indeed is a chart for T (M).Altogether we now have established that {cT : c a chart for M} is an atlasfor T (M). We always view T (M) as a manifold with respect to the equivalentmaximal atlas.

Definition. The manifold T (M) is called the tangent bundle of M .

Remark. If M is m-dimensional then T (M) is 2m-dimensional.

Lemma 9.8. The map pM : T (M) −→ M is locally analytic.

Proof. Let c = (U,ϕ,Km) be a chart for M . It suffices to contemplate thecommutative diagram

T (M)

pM

p−1M (U)

⊇ϕc(p−1

M (U)) = ϕ(U) × Km

pr1

⊆K2m

M U⊇ ϕ

ϕ(U)⊆

Km.

Let g : M −→ N be a locally analytic map of manifolds. We define themap

T (g) : T (M) −→ T (N)

byT (g)|Ta(M) := Ta(g) for any a ∈ M.

In particular, the diagram

T (M)T (g)

pM

T (N)

pN

Mg

N

is commutative.

Proposition 9.9. i. The map T (g) is locally analytic.

ii. For any locally analytic maps of manifolds Lf−→ M

g−→ N we have

T (g ◦ f) = T (g) ◦ T (f).


Proof. i. We choose charts c = (U,ϕ,Km) for M and c = (V, ψ,Kn) for Nsuch that g(U) ⊆ V . The composite

ϕ(U) × Km ϕ−1c−−−→ p−1

M (U)T (g)−−−→ p−1

N (V )ψc−−→ ψ(V ) × Kn

is given by

(x, v) −→ (ψ ◦ g ◦ ϕ−1(x), Dx(ψ ◦ g ◦ ϕ−1)(v)).

It is locally analytic by the same argument as for (8).ii. This is a restatement of Lemma 9.2.

The following is left to the reader as an exercise.

Remark 9.10. i. If U ⊆ M is an open submanifold then T (⊆) inducesan isomorphism between T (U) and the open submanifold p−1

M (U).

ii. For any two manifolds M and N the map

T (pr1) × T (pr2) : T (M × N) �−−→ T (M) × T (N)

is an isomorphism of manifolds.

Now let M be a manifold and E be a K-Banach space. For any f ∈Can(M,E) we define

df : T (M) −→ E

t −→ dpM (t)f(t).

Lemma 9.11. We have df ∈ Can(T (M), E).

Proof. Let c = (U,ϕ,Km) be a chart for M . The composed map

ϕ(U) × Km ϕ−1c−−−→ p−1

M (U)df−−→ E

is given by(x, v) −→ Dx(f ◦ ϕ−1)(v)

and hence is locally analytic by the same argument as for (8).

Lemma 9.12. Let g : M −→ N be a locally analytic map of manifolds; forany f ∈ Can(N,E) we have

d(f ◦ g) = df ◦ T (g).

66 II Manifolds

Proof. This is a consequence of the chain rule.

Exercise. The map

d : Can(M,E) −→ Can(T (M), E)f −→ df

is K-linear.

Remark 9.13. If K has characteristic zero then a function f ∈ Can(M,E)is locally constant if and only if df = 0.

Proof. Let c = (U,ϕ) be any chart for M . As can be seen from the proof ofLemma 9.11 we have df |p−1

M (U) = 0 if and only if Dx(f ◦ ϕ−1) = 0 for anyx ∈ ϕ(U). By Remark 6.2 the latter is equivalent to f ◦ ϕ−1 being locallyconstant on ϕ(U) which, of course, is the same as f being locally constanton U .

Definition. Let U ⊆ M be an open subset ; a vector field ξ on U is a locallyanalytic map ξ : U −→ T (M) which satisfies pM ◦ ξ = idU .

We define

Γ(U, T (M)) := set of all vector fields on U.

It follows from Remark 9.10.i. that

Γ(U, T (M)) = Γ(U, T (U)).

Suppose that U is the domain of definition of some chart c = (U,ϕ,Km) forM . Because of the commutative diagram

p−1M (U) �

ϕc

pM

ϕ(U) × Km

(x,v)→ϕ−1(x)

U

the map

Can(U,Km) ∼−−→ Γ(U, T (M))

f −→ ξf (a) := ϕ−1c ((ϕ(a), f(a))) = τc(a, f(a))

is bijective. The left hand side is a K-vector space. On the right hand sidethis vector space structure corresponds to the pointwise addition and scalar


multiplication of maps which makes sense since each Ta(M) is a K-vectorspace. The latter we can define on any open subset U ⊆ M . For any c ∈ Kand ξ, η ∈ Γ(U, T (M)) we define

(c · ξ)(a) := c · ξ(a) and (ξ + η)(a) := ξ(a) + η(a).

Obviously the result are again maps ? : U −→ T (M) satisfying pM◦ ? = idU .But since U can be covered by domains of definition of charts for M theabove discussion actually implies that these maps are locally analytic again.We see that

Γ(U, T (M)) is a K-vector space.

We have the bilinear map

Γ(M,T (M)) × Can(M,E) −→ Can(M,E)(ξ, f) −→ Dξ(f) := df ◦ ξ.

Lemma 9.14. Let u : E1 ×E2 −→ E be a continuous bilinear map betweenK-Banach spaces; for any ξ ∈ Γ(M,T (M)) and fi ∈ Can(M,Ei) we have

Dξ(u(f1, f2)) = u(Dξ(f1), f2) + u(f1, Dξ(f2)).

Proof. This follows from the product rule in Lemma 9.5.i.

Corollary 9.15. For any vector field ξ ∈ Γ(M,T (M)) the map

Dξ : Can(M,K) −→ Can(M,K)

is a derivation, i. e.:

(a) Dξ is K-linear,

(b) Dξ(fg) = Dξ(f)g + fDξ(g) for any f, g ∈ Can(M,K).

Proposition 9.16. Suppose that M is paracompact ; then for any derivationD on Can(M,K) there is a unique vector field ξ on M such that D = Dξ.

The proof requires some preparation. In the following we always assumeM to be paracompact. At first we fix a point a ∈ M . A K-linear mapΔ : Can(M,K) −→ K will be called an a-derivation if

Δ(fg) = Δ(f)g(a) + f(a)Δ(g) for any f, g ∈ Can(M,K).

The a-derivations form a K-vector subspace

Dera(M,K)

of the dual vector space Can(M,K)∗.

68 II Manifolds

Lemma 9.17. Suppose that M is paracompact, and let Δ be an a-derivation;if f ∈ Can(M,K) is constant in a neighbourhood of the point a then Δ(f) =0.

Proof. Case 1: We assume that f vanishes in the neighbourhood U ⊆ Mof a. By Prop. 8.7 we may assume that U is open and closed in M . Thenthe function

g(x) :=

{1 if x �∈ U,

0 if x ∈ U

lies in Can(M,K) and satisfies gf = f . It follows that

Δ(f) = Δ(gf) = Δ(g)f(a) + g(a)Δ(f) = 0.

Case 2: We assume that f is constant on M with value c. Let 1M denotethe constant function with value one on M . Then f = c1M and hence

Δ(f) = cΔ(1M ) = cΔ(1M1M ) = cΔ(1M ) + cΔ(1M ) = 2cΔ(1M ) = 2Δ(f)

which means Δ(f) = 0.Case 3: In general we write

f = f(a)1M + (f − f(a)1M )

and use the K-linearity of Δ together with the first two cases.

As a consequence of the product rule Lemma 9.5.ii. we have the K-linearmap

Ta(M) −→ Dera(M,K)t −→ Δt(f) := daf(t).

(10)

Proposition 9.18. If M is paracompact then (10) is an isomorphism.

Proof. We fix a chart c = (U,ϕ,Km) for M around a point a and writeϕ = (ϕ1, . . . , ϕm). Since M is paracompact we may assume by Prop. 8.7that U is open and closed in M . Then each ϕi extends by zero to a functionϕi! ∈ Can(M,K). In the discussion before Lemma 9.6 we had seen that

daϕ = (daϕ1, . . . , daϕm) : Ta(M)∼=−−→ Km

is an isomorphism, and we had introduced the K-basis ti := ∂∂ϕi

(a) of Ta(M).


Injectivity : Let t ∈ Ta(M) such that Δt(f) = 0 for any f ∈ Can(M,K).In particular

0 = Δt(ϕi!) = daϕi!(t) = daϕi(t) for any 1 ≤ i ≤ m.

This means daϕ(t) = 0 and hence t = 0.Surjectivity : From the injectivity which we just have established we de-

duce that the Δti are linearly independent in Dera(M,K). It therefore suf-fices to write an arbitrarily given Δ ∈ Dera(M,K) as a linear combinationof the Δti . In fact, we claim that

Δ =m∑

i=1

Δ(ϕi!) · Δti

holds true. Let f ∈ Can(M,K). We find an open and closed neighbour-hood V ⊆ U of a such that ϕ(V ) = Bε(ϕ(a)) and a power series F (X) =∑

α cαXα ∈ Fε(Km;K) such that

f(x) = F (ϕ(x) − ϕ(a)) for any x ∈ V.

We may write

f(x) =∑

α

cα(ϕ(x) − ϕ(a))α = f(a) +m∑

i=1

(ϕi(x) − ϕi(a))gi(x)

for any x ∈ V where the gi ∈ Can(V,K) are appropriate functions satisfyinggi(a) = ci (recall that i = (0, . . . , 1, . . . , 0)). From the proof of Prop. 6.1 weknow that

Dϕ(a)(f ◦ ϕ−1)(ei) =∂F

∂Xi(0) = ci

where e1, . . . , em denotes, as usual, the standard basis of Km. We thereforeobtain

gi(a) = ci = Dϕ(a)(f ◦ ϕ−1)(ei) = daf(θc(ei)).

By the construction of the ti we have θc(ei) = ti. It follows that

gi(a) = daf(ti).

On the other hand we extend each gi by zero to a function gi! ∈ Can(M,K).The function

f −m∑

i=1

(ϕi! − ϕi(a))gi!

70 II Manifolds

is constant (with value f(a)) in a neighbourhood of a. Using Lemma 9.17we compute

Δ(f) = Δ

(m∑

i=1

(ϕi! − ϕi(a))gi!

)

=m∑

i=1

Δ(ϕi! − ϕi(a)

)gi(a)

=m∑

i=1

Δ(ϕi!) · daf(ti)

=m∑

i=1

Δ(ϕi!) · Δti(f).

Since f was arbitrary this establishes our claim.

Proof of Proposition 9.16: First of all we note that the relation betweenderivations and a-derivations on Can(M,K) is given by the formula

(11) Dξ(f)(a) = df(ξ(a)) = daf(ξ(a)) = Δξ(a)(f).

Therefore, if Dξ = 0 then Δξ(a) = 0 for any a ∈ M . The Prop. 9.18 thenimplies that ξ(a) = 0 for any a ∈ M , i. e., that ξ = 0. This shows that theξ in our assertion is unique if it exists. For the existence we first fix a pointa ∈ M and consider the a-derivation Δ(f) := D(f)(a). By Prop. 9.18 thereis a tangent vector ξ(a) ∈ Ta(M) such that Δ = Δξ(a). For varying a ∈ Mthis gives a map ξ : M −→ T (M) which satisfies pM ◦ξ = idM . It remains toshow that ξ is locally analytic, since D = Dξ then is a formal consequenceof (11). So let c = (U,ϕ,Km) be a chart for M . In the proof of Prop. 9.18we have seen that

D(f)(a) =m∑

i=1

D(ϕi!)(a) · Δθc(ei)(f).

It follows that

ξ(a) =m∑

i=1

D(ϕi!)(a) · θc(ei) = θc((D(ϕ1!)(a), . . . , D(ϕm!)(a))

).

Using the commutative diagram

Km v −→(a,v)

θc∼=

U × Km

τc�

Ta(M)⊆

p−1M (U)


we rewrite this as

ξ(a) = τc(a, (D(ϕ1!)(a), . . . , D(ϕm!)(a))).

This means that under the identification discussed after the definition ofvector fields we have

ξ|U = ξf with f := (D(ϕ1!), . . . , D(ϕm!)) ∈ Can(U,Km).

Hence ξ is locally analytic. �

Lemma 9.19. For any derivations B,C,D : Can(M,K) −→ Can(M,K) wehave:

i. [B,C] := B ◦ C − C ◦ B again is a derivation;

ii. [ , ] is K-bilinear ;

iii. [B,B] = 0 and [B,C] = −[C,B];

iv. (Jacobi identity) [[B,C], D] + [[C,D], B] + [[D,B], C] = 0.

Proof. These are straightforward completely formal computations.

Definition. A K-vector space g together with a K-bilinear map

[ , ] : g × g −→ g

which is antisymmetric (i. e., [z, z] = 0 for any z ∈ g) and satisfies theJacobi identity is called a Lie algebra over K.

If M is paracompact then, using Prop. 9.16 and Lemma 9.19, we maydefine the Lie product [ξ, η] of two vector fields ξ, η ∈ Γ(M,T (M)) by therequirement that

D[ξ,η] = Dξ ◦ Dη − Dη ◦ Dξ

holds true. This makes Γ(M,T (M)) into a Lie algebra over K.

Proposition 9.20. Suppose that M is paracompact, and let E be a K-Banach space and ξ, η ∈ Γ(M,T (M)) be two vector fields ; on Can(M,E) wethen have

D[ξ,η] = Dξ ◦ Dη − Dη ◦ Dξ.

72 II Manifolds

Proof. Let f ∈ Can(M,E). We have to show equality of the functions

D[ξ,η](f) = df ◦ [ξ, η]

and

(Dξ ◦ Dη − Dη ◦ Dξ)(f) = d(Dη(f)) ◦ ξ − d(Dξ(f)) ◦ η

= d(df ◦ η) ◦ ξ − d(df ◦ ξ) ◦ η.

This, of course, can be done after restriction to the domain of definition Uof any chart c = (U,ϕ,Km) for M . Since M is paracompact we furthermoreneed only to consider charts for which U is open and closed in M . Letϕ = (ϕ1, . . . , ϕm) and denote, as before, by ϕi! ∈ Can(M,K) the extensionby zero of ϕi. We now make use of the following identifications. If ? denotesthe restriction to U of any of the vector fields ξ, η, and [ξ, η] then, as discussedafter the definition of vector fields, we have a commutative diagram

p−1M (U) �

ϕcϕ(U) × Km

U

?

�ϕ ϕ(U)

x →(x,g?(x))

with g? ∈ Can(ϕ(U),Km). On the other hand, as noticed already in the proofof Lemma 9.11, we have, for any function ? ∈ Can(U,E), the commutativediagram

E

p−1M (U)

d?

�ϕc

ϕ(U) × Km.

(x,v) →Dx(?◦ϕ−1)(v)

These identifications reduce us to proving the equality of the following twofunctions of x ∈ ϕ(U) given by

(12) Dx(f ◦ ϕ−1)(g[ξ,η](x))

and

Dx(df ◦ η ◦ ϕ−1)(gξ(x)) − Dx(df ◦ ξ ◦ ϕ−1)(gη(x))(13)

= Dx

(D.(f ◦ ϕ−1)(gη(.))

)(gξ(x)) − Dx

(D.(f ◦ ϕ−1)(gξ(.))

)(gη(x)),


respectively. By viewing D.(f ◦ϕ−1), resp. gη(.) and gξ(.), as functions fromϕ(U) into L(Km, E), resp. into Km, we may apply the product rule Re-mark 4.1.iv. for the continuous bilinear map

L(Km, E) × Km −→ E

(u, v) −→ u(v)

to both summands in the last expression for (13) and rewrite it as

= [Dx(D.(f ◦ ϕ−1))(gξ(x))](gη(x)) + Dx(f ◦ ϕ−1)[Dxgη(gξ(x))](14)

− [Dx(D.(f ◦ ϕ−1)(gη(x))](gξ(x)) − Dx(f ◦ ϕ−1)[Dxgξ(gη(x))].

To simplify this further we establish the following generalClaim: For any open subset V ⊆ Km, any point x ∈ V , any vectors v =

(v1, . . . , vm) and w = (w1, . . . , wm) in Km, and any function h ∈ Can(V,E)we have

Dx(D.h(v))(w) = Dx(D.h(w))(v).

(Note that the function D.h(v) is the composite

Vy →Dyh−−−−−→ L(Km, E)

u→u(v)−−−−−→ E.)

We expand h around the point x into a power series

h(y) = H(y − x).

By the proof of Prop. 6.1 we then have

Dyh(v) =m∑

i=1

vi∂H

∂Yi(y − x)

and

Dx(D.h(v))(w) =m∑

i=1

viDx

(∂H

∂Yi(. − x)

)

(w)

=m∑

i=1

vi

m∑

j=1

wj

(∂

∂Yj

∂

∂YiH

)

(0)

=m∑

j=1

wj

m∑

i=1

vi

(∂

∂Yi

∂

∂YjH

)

(0)

...= Dx(D.h(w))(v).

74 II Manifolds

Applying this claim to (14) we see that the expression for the function(13) simplifies to

Dx(f ◦ ϕ−1)[Dxgη(gξ(x)) − Dxgξ(gη(x))].

Comparing this with (12) we are reduced to showing that the identity

(15) g[ξ,η](x) = Dxgη(gξ(x)) − Dxgξ(gη(x))

holds true in Can(ϕ(U),Km). But in case E = K our assertion and thewhole computation above holds by construction. In particular we have

Dx(ϕi! ◦ ϕ−1)(g[ξ,η](x)) = Dx(ϕi! ◦ ϕ−1)[Dxgη(gξ(x)) − Dxgξ(gη(x))]

for any 1 ≤ i ≤ m. Since

Dx(ϕi! ◦ ϕ−1) : Km −→ K

(v1, . . . , vm) −→ vi

the identity (15) follows immediately.

Remarks 9.21. i. The identity (15) shows that Can(V,Km), for anyopen subset V ⊆ Km, is a Lie algebra with respect to

[f, g](x) := Dxf(g(x)) − Dxg(f(x)).

ii. The identity (15) can be made into a definition of which one then canshow that it is compatible with any change of charts for M . In thisway a Lie product [ξ, η] can be obtained and Prop. 9.20 can be provedeven for manifolds which are not paracompact.

10 The Topological Vector Space Can(M, E), Part 1

Throughout this section M is a paracompact manifold and E is a K-Banachspace. Following [Fea] we will show that Can(M,E) in a natural way is atopological vector space.

To motivate the later construction we first consider a fixed functionf ∈ Can(M,E). Since, by Prop. 8.7, M is strictly paracompact we find afamily of charts (Uj , ϕj ,K

mj ), for j ∈ J , for M such that the Uj are pairwisedisjoint and M =

⋃j∈J Uj . According to Remark 8.2.ii. the function f is

locally analytic if and only if all f ◦ ϕ−1j : ϕj(Uj) −→ E, for j ∈ J , are

10 The Topological Vector Space Can(M, E), Part 1 75

locally analytic. For each ϕj(Uj) we find balls Bεj,ν (xj,ν) ⊆ Kmj and powerseries Fj,ν ∈ Fεj,ν (Kmj ;E) such that

(16) ϕj(Uj) =⋃

ν

Bεj,ν (xj,ν)

andf ◦ ϕ−1

j (x) = Fj,ν(x − xj,ν) for any x ∈ Bεj,ν (xj,ν).

By Lemma 1.4 the covering (16) can be refined into a covering by pair-wise disjoint balls Bδj,α

(yj,α). Consider a fixed α. We find a ν such thatBδj,α

(yj,α) ⊆ Bεj,ν (xj,ν). In fact we then have

Bmin(δj,α,εj,ν)(yj,α) = Bδj,α(yj,α) ⊆ Bεj,ν (xj,ν) = Bεj,ν (yj,α).

Hence we may assume that δj,α ≤ εj,ν . Using Cor. 5.5 we may change Fj,ν

into a power series Fj,α ∈ Fδj,α(Kmj ;E) such that

f ◦ ϕ−1j (x) = Fj,α(x − yj,α) for any x ∈ Bδj,α

(yj,α).

We put Uj,α := ϕ−1j (Bδj,α

(yj,α)). The (Uj,α, ϕj ,Kmj ) again are charts for M

such that the Uj,α cover M and are pairwise disjoint.

Resume: Given f ∈ Can(M,E) there is a family of charts (Ui, ϕi,Kmi),

for i ∈ I, for M together with real numbers εi > 0 such that :

(a) M =⋃

i∈I Ui, and the Ui are pairwise disjoint ;

(b) ϕi(Ui) = Bεi(xi) for one (or any) xi ∈ ϕi(Ui);

(c) there is a power series Fi ∈ Fεi(Kmi ;E) with

f ◦ ϕ−1i (x) = Fi(x − xi) for any x ∈ ϕi(Ui).

We note that by Cor. 5.5 the existence of Fi as well as its norm ‖Fi‖εi

is independent of the choice of the point xi.Let (c, ε) be a pair consisting of a chart c = (U,ϕ,Km) for M and a

real number ε > 0 such that ϕ(U) = Bε(a) for one (or any) a ∈ ϕ(U). As aconsequence of the identity theorem for power series Cor. 5.8 the K-linearmap

Fε(Km;E) −→ Can(U,E)F −→ F (ϕ(.) − a)

76 II Manifolds

is injective. Let F(c,ε)(E) denote its image. It is a K-Banach space withrespect to the norm

‖f‖ = ‖F‖ε if f(.) = F (ϕ(.) − a).

By Cor. 5.5 the pair (F(c,ε)(E), ‖ ‖) is independent of the choice of the pointa.

Definition. An index for M is a family I = {(ci, εi)}i∈I of charts ci =(Ui, ϕi,K

mi) for M and real numbers εi > 0 such that the above conditions(a) and (b) are satisfied.

For any index I for M we have

FI(E) :=∏

i∈I

F(ci,εi)(E) ⊆∏

i∈I

Can(Ui, E) = Can(M,E).

Our above resume says that

Can(M,E) =⋃

IFI(E)

where I runs over all indices for M . Hence Can(M,E) is a union of directproducts of Banach spaces. This is the starting point for the construction ofa topology on Can(M,E).

But first we have to discuss the inclusion relations between the subspacesFI(E) for varying I. Let I = {(ci = (Ui, ϕi,K

mi), εi)}i∈I and J = {(dj =(Vj , ψj ,K

nj ), δj)}j∈J be two indices for M .

Definition. The index I is called finer than the index J if for any i ∈ Ithere is a j ∈ J such that :

(i) Ui ⊆ Vj,

(ii) there is an a ∈ ϕi(Ui) and a power series Fi,j ∈ Fεi(Kmi ;Knj ) with

‖Fi,j − Fi,j(0)‖εi ≤ δj and

ψj ◦ ϕ−1i (x) = Fi,j(x − a) for any x ∈ ϕi(Ui).

We observe that if the condition (ii) holds for one point a ∈ ϕi(Ui) thenit holds for any other point b ∈ ϕi(Ui) as well. This follows from Cor. 5.5which implies that Gi,j(X) := Fi,j(X + b − a) ∈ Fεi(K

mi ;Knj ) with

ψj ◦ ϕ−1i (x) = Gi,j(x − b) for any x ∈ ϕi(Ui)


and

‖Gi,j − Gi,j(0)‖εi = ‖(Fi,j − Fi,j(0))(X + b − a) + Fi,j(0) − Gi,j(0)‖εi

≤ max(‖(Fi,j − Fi,j(0))(X + b − a)‖εi , δj)= max(‖Fi,j − Fi,j(0)‖εi , δj)= δj .

Lemma 10.1. If I is finer than J then we have FJ (E) ⊆ FI(E).

Proof. Let f ∈ FJ (E). We have to show that f |Ui ∈ F(ci,εi)(E) for any i ∈ I.In the following we fix an i ∈ I. We have ϕi(Ui) = Bεi(a). By assumptionwe find a j ∈ J and an Fi,j ∈ Fεi(K

mi ;Knj ) such that

– Ui ⊆ Vj ,

– ‖Fi,j − Fi,j(0)‖εi ≤ δj , and

– ψj ◦ ϕ−1i (x) = Fi,j(x − a) for any x ∈ ϕi(Ui).

We putb := ψj ◦ ϕ−1

i (a) = Fi,j(0) ∈ ψj(Vj).

Since f ∈ FJ (E) we also find a Gj ∈ Fδj(Knj ;E) such that

f ◦ ψ−1j (y) = Gj(y − b) for any y ∈ ψj(Vj) = Bδj

(b).

As a consequence of Prop. 5.4 then the power series

Fi := Gj ◦ (Fi,j − Fi,j(0)) ∈ Fεi(Kmi ;E)

exists and satisfies

Fi(x − a) = Gj(Fi,j(x − a) − b) = f ◦ ψ−1j (ψj ◦ ϕ−1

i (x)) = f ◦ ϕ−1i (x)

for any x ∈ ϕi(Ui).

The relation of being finer only is a preorder. If the index I is finer thanthe index J and J is finer than I one cannot conclude that I = J . But itdoes follow that FI(E) = FJ (E) which is sufficient for our purposes.

Lemma 10.2. For any two indices J1 and J2 for M there is a third indexI for M which is finer than J1 and J2.

78 II Manifolds

Proof. Let J1 = {((Ui, ϕi,Kni), εi)}i∈I and J2 = {((Vj , ψj ,K

mj ), δj)}j∈J .We have the covering

M =⋃

i,j

Ui ∩ Vj

by pairwise disjoint open subsets. For any pair (i, j) ∈ I × J the function

ψj ◦ ϕ−1i : ϕi(Ui ∩ Vj) −→ Kmj

is locally analytic. Hence we may cover ϕi(Ui ∩ Vj) by a family of ballsBi,j,k = Bβi,j,k

(ai,j,k) such that

– βi,j,k ≤ min(εi, δj), and

– there is a power series Fi,j,k ∈ Fβi,j,k(Kni ;Kmj ) with

ψj ◦ ϕ−1i (x) = Fi,j,k(x − ai,j,k) for any x ∈ Bi,j,k.

Using the fact that

‖Fi,j,k − Fi,j,k(0)‖α ≤ α

βi,j,k‖Fi,j,k − Fi,j,k(0)‖βi,j,k

for any 0 < α ≤ βi,j,k

together with Cor. 5.5 we may, after possibly decreasing the βi,j,k, assumein addition that

– ‖Fi,j,k − Fi,j,k(0)‖βi,j,k≤ δj .

After a possible further refinement based on Lemma 1.4 (compare the argu-ment for the resume at the beginning of this section) we finally achieve thatthe Bi,j,k are pairwise disjoint. We put

Wi,j,k := ϕ−1i (Bi,j,k)

and obtain the index I := {((Wi,j,k, ϕi,Kni), βi,j,k)}i,j,k for M . By construc-

tion I is finer than J2. Moreover, observing that ϕi◦ϕ−1i : ϕi(Wi,j,k)

⊆−−→ Kni

is the inclusion map and that βi,j,k ≤ εi we see that I is finer than J1 fortrivial reasons.

Given any index I for M we consider FI(E) =∏

i∈I F(ci,εi)(E) fromnow on as a topological K-vector space with respect to the product topol-ogy of the Banach space topologies on the F(ci,εi)(E). Obviously FI(E) isHausdorff. But it is not a Banach space if I is infinite. Suppose that thetopology of FI(E) can be defined by a norm. The corresponding unit ball


B1(0) is open. By the definition of the product topology there exist finitelymany indices i1, . . . , ir ∈ I such that

∏

i�=i1,...,ir

F(ci,εi)(E) × {0} × · · · × {0} ⊆ B1(0).

As a vector subspace the left hand side then necessarily is contained in anyball Bε(0) for ε > 0. The intersection of the latter being equal to {0} itfollows that I is finite.

Lemma 10.3. If I is finer than J then the inclusion map FJ (E) ⊆−−→FI(E) is continuous.

Proof. For any i ∈ I there exists, by assumption, a j(i) ∈ J such that theconditions (i) and (ii) in the definition of “finer” are satisfied. The inclusionmap in question can be viewed as the map

∏

j

F(dj ,δj)(E) −→∏

i

F(ci,εi)(E)

(fj)j −→ (fj(i)|Ui)i.

Hence it suffices to show that each individual restriction map

F(dj(i),δj(i))(E) −→ F(ci,εi)(E)

is continuous. But we even know from Prop. 5.4 that the operator norm ofthis map is ≤ 1.

We point out that, for I finer than J , the topology of FJ (E) in generalis strictly finer than the subspace topology induced by FI(E).

In the present situation there is a certain universal procedure to constructfrom the topologies on all the FI(E) a topology on their union Can(M,E) =⋃

I FI(E). Since this construction takes place within the class of locallyconvex topologies we first need to review this concept in the next section.

11 Locally Convex K-Vector Spaces

This section serves only as a brief introduction to the subject. The readerwho is interested in more details is referred to [NFA]. Let E be any K-vectorspace.

Definition. A (nonarchimedean) seminorm on E is a function q : E −→ R

such that for any v, w ∈ E and any a ∈ K we have:

80 II Manifolds

(i) q(av) = |a| · q(v),

(ii) q(v + w) ≤ max(q(v), q(w)).

It follows immediately that a seminorm q also satisfies:

(iii) q(0) = |0| · q(0) = 0;

(iv) q(v) = max(q(v), q(−v)) ≥ q(v − v) = q(0) = 0 for any v ∈ E;

(v) q(v + w) = max(q(v), q(w)) for any v, w ∈ E such that q(v) �= q(w)(compare the proof of Lemma 1.1);

(vi) −q(v − w) ≤ q(v) − q(w) ≤ q(v − w) for any v, w ∈ E.

Let (qi)i∈I be a family of seminorms on E. We consider the coarsesttopology on E such that:

(1) All maps qi : E −→ R, for i ∈ I, are continuous,

(2) all translation maps v + . : E −→ E, for v ∈ E, are continuous.

It is called the topology defined by (qi)i∈I . For any finitely many qi1 , . . . , qir

and any w ∈ E and ε > 0 we define

Bε(qi1 , . . . , qir ;w) := {v ∈ E : qir(v − w), . . . , qir(v − w) ≤ ε}.

The following properties are obvious:

(a) Bε(qi1 , . . . , qir ;w) = Bε(qi1 ;w) ∩ · · · ∩ Bε(qir ;w);

(b) Bε1(qi1 ;w1) ∩ Bε2(qi2 ;w2) =⋃

w Bmin(ε1,ε2)(qi1 , qi2 ;w) where w runsover all points in the left hand side;

(c) Bε(qi1 , . . . , qir ;w) = w + Bε(qi1 , . . . , qir ; 0);

(d) a · Bε(qi1 , . . . , qir ;w) = B|a|ε(qi1 , . . . , qir ; aw) for any a ∈ K×.

Lemma 11.1. The subsets Bε(qi1 , . . . , qir ;w) form a basis for the topologyon E defined by (qi)i∈I .

Proof. The Bε(qi1 , . . . , qir ;w), by (a) and (b), do form a basis for a (unique)topology T ′ on E. On the other hand let T denote the topology defined by(qi)i∈I . We first show that T ′ ⊆ T . By (a), (c), and (2) it suffices to check


that Bε(qi; 0) ∈ T for any i ∈ I and ε > 0. As a consequence of (1) and (2)we certainly have that

B−δ (qi;w) := {v ∈ E : qi(v − w) < δ} ∈ T

for any w ∈ E and δ > 0. But using (ii) we see that

Bε(qi; 0) = B−ε (qi; 0) ∪

⋃

qi(w)=ε

B−ε (qi;w).

To conclude that actually T ′ = T holds true it now suffices to show that T ′

satisfies (1) and (2). The continuity property (2) follows immediately from(c). To establish (1) for T ′ we have to show that q−1

i ((α, β)) ∈ T ′ for anyi ∈ I and any open interval (α, β) ⊆ R. Because of (iv) we may assume thatβ > 0. Let w ∈ q−1

i ((α, β)) be any point. Case 1: We have qi(w) > 0. Chooseany 0 < ε < qi(w). It then follows from (v) that Bε(qi;w) ⊆ q−1

i (qi(w)) ⊆q−1i ((α, β)). Case 2: We have qi(w) = 0. Choose any 0 < ε < β. We obtain

Bε(qi;w) ⊆ q−1i ([0, ε]) ⊆ q−1

i ((α, β)) since necessarily α < 0 in this case.

Lemma 11.2. E is a topological K-vector space, i. e., addition and scalarmultiplication are continuous, with respect to the topology defined by (qi)i∈I .

Proof. Using Lemma 11.1 this easily follows from the following inclusions:

– Bε(qi1 , . . . , qir ;w1) + Bε(qi1 , . . . , qir ;w2) ⊆ Bε(qi1 , . . . , qir ;w1 + w2);

– Bδ(a) · B|a|−1ε(qi1 , . . . , qir ;w) ⊆ Bε(qi1 , . . . , qir ; aw) provided δ ≤ |a|and δ · max(qi1(w), . . . , qir(w)) ≤ ε;

– Bδ(0) · Bε(qi1 , . . . , qir ;w) ⊆ Bε(qi1 , . . . , qir ; 0) provided δ ≤ 1 and δ ·max(qi1(w), . . . , qir(w)) ≤ ε.

The details are left to the reader as an exercise.

Exercise. The topology on E defined by (qi)i∈I is Hausdorff if and only iffor any vector 0 �= v ∈ E there is an index i ∈ I such that qi(v) �= 0.

Definition. A topology on a K-vector space E is called locally convex if itcan be defined by a family of seminorms. A locally convex K-vector space isa K-vector space equipped with a locally convex topology.

Obviously any normed K-vector space and in particular any K-Banachspace is locally convex.

82 II Manifolds

Remark 11.3. Let {Ej}j∈J be a family of locally convex K-vector spaces;then the product topology on E :=

∏j∈J Ej is locally convex.

Proof. Let (qj,i)i be a family of seminorms which defines the locally convextopology on Ej . Moreover, let prj : E −→ Ej denote the projection maps.Using Lemma 11.1 one checks that the family of seminorms (qj,i ◦ prj)i,j

defines the product topology on E.

Exercise 11.4. Let {Ej}j∈J be a family of locally convex K-vector spacesand let E :=

∏j∈J Ej with the product topology ; for any continuous semi-

norm q on E there is a unique minimal finite subset Jq ⊆ J such that

q

(∏

j∈J\Jq

Ej × {0} × · · · × {0})

= {0}.

For our purposes the following construction is of particular relevance.Let E be a any K-vector space, and suppose that there is given a family{Ej}j∈J of vector subspaces Ej ⊆ E each of which is equipped with a locallyconvex topology.

Lemma 11.5. There is a unique finest locally convex topology T on E suchthat all the inclusion maps Ej

⊆−−→ E, for j ∈ J , are continuous.

Proof. Let Q be the set of all seminorms q on E such that q|Ej is continuousfor any j ∈ J , and let T be the topology on E defined by Q. It followsimmediately from Lemma 11.1 that all the inclusion maps Ej

⊆−−→ (E, T )are continuous. On the other hand, let T ′ be any topology on E definedby a family of seminorms (qi)i∈I such that Ej

⊆−−→ (E, T ′) is continuous forany j ∈ J . Obviously we then have (qi)i∈I ⊆ Q. This implies, using againLemma 11.1, that T ′ ⊆ T .

The topology T on E in the above Lemma is called the locally convexfinal topology with respect to the family {Ej}j∈J . Suppose that the family{Ej}j∈J has the additional properties:

– E =⋃

j∈J Ej ;

– the set J is partially ordered by ≤ such that for any two j1, j2 ∈ Jthere is a j ∈ J such that j1 ≤ j and j2 ≤ j;

– whenever j1 ≤ j2 we have Ej1 ⊆ Ej2 and the inclusion map Ej1⊆−−→ Ej2

is continuous.


In this case the locally convex K-vector space (E, T ) is called the locallyconvex inductive limit of the family {Ej}j∈J .

Lemma 11.6. A K-linear map f : E −→ E into any locally convex K-vector space E is continuous (with respect to T ) if and only if the restrictionsf |Ej, for any j ∈ J , are continuous.

Proof. It is trivial that with f all restrictions f |Ej are continuous. Let ustherefore assume vice versa that all f |Ej are continuous. Let (qi)i∈I be afamily of seminorms which defines the topology of E. Then all seminormsqi := qi◦f , for i ∈ I, lie in the set of seminorms Q which defines the topologyT of E. It follows that

f−1(Bε(qi1 , . . . , qir ; f(w)) = Bε(qi1 , . . . , qir ;w)

is open in E. Because of Lemma 11.1 this means that f is continuous.

Lemma 11.7. Let {Ej}j∈J be a family of locally convex K-vector spacesand let E :=

∏j∈J Ej with the product topology ; suppose that each Ej has

the locally convex final topology with respect to a family of locally convexK-vector spaces {Ej,k}k∈Ij

and that Ej =⋃

k Ej,k; for any k = (kj)j ∈ I :=∏j∈J Ij we put Ek :=

∏j∈J Ej,kj

with the product topology; then the topologyof E is the locally convex final topology with respect to the family {Ek}k∈I .

Proof. By the proof of Lemma 11.5 the locally convex topology of Ej isdefined by the set Qj of all seminorms q such that q|Ej,k is continuous for anyk ∈ Ij . Let prj : E −→ Ej denote the projection maps. By Remark 11.3 thetopology of E is defined by the set of seminorms Q :=

⋃j∈J{q◦prj : q ∈ Qj}.

For any q ∈ Qj and any k ∈ I we have the commutative diagram

Ek⊆

prj

E

prj

q◦prj

Ej,kj

⊆Ej

qR.

Hence the restriction of any seminorm in Q to any Ek is continuous. Thismeans that the locally convex final topology on E with respect to the family{Ek}k is finer than the product topology. Vice versa, let q be any seminormon E such that q|Ek, for any k ∈ I, is continuous. We have to show that q iscontinuous. By Exercise 11.4 we find, for any k ∈ I, a unique minimal finite

84 II Manifolds

subset Jq,k ⊆ J such that the restriction q|Ek factorizes into

Ekpr

q

∏j∈Jq,k

Ej,kj

R .

In particular, q|Ej,kj�= 0 for any j ∈ Jq,k. We claim that the set

Jq :=⋃

k∈I

Jq,k

is finite. We define � = (�j)j ∈ I in the following way. If j ∈ Jq we choosea k ∈ I such that j ∈ Jq,k and we put �j := kj ; in particular, q|Ej,�j

=q|Ej,kj

�= 0. For j ∈ J \ Jq we pick any �j ∈ Ij . By construction we haveJq ⊆ Jq,� so that Jq necessarily is finite. This means that the seminorm q onE factorizes into

Epr

q

∏j∈Jq

Ej

R .

It follows that

q(v) ≤ maxj∈Jq

(q|Ej) ◦ prj(v) for any v ∈ E.

Since each q|Ej is continuous by assumption we conclude that q is continu-ous.

12 The Topological Vector Space Can(M, E), Part 2

As in Sect. 10 we let M be a paracompact manifold and E be a K-Banachspace. We have seen that

Can(M,E) =⋃

IFI(E)

where I runs over all indices for M . Each FI(E) by Remark 11.3 is locallyconvex as a product of Banach spaces. By Lemmas 10.1–10.3 we may andalways will view Can(M,E) as the locally convex inductive limit of the family{FI(E)}I (where I ≤ J if J is finer than I). All our earlier constructionsinvolving Can(M,E) are compatible with this topology. In the following webriefly discuss the most important ones.


Proposition 12.1. For any a ∈ M the evaluation map

δa : Can(M,E) −→ E

f −→ f(a)

is continuous.

Proof. It suffices, by Lemma 11.6, to show that the restriction δa|FI(E) iscontinuous for any index I for M . Let I = {(ci = (Ui, ϕi,K

mi), εi)}i∈I .There is a unique i ∈ I such that a ∈ Ui. Then ϕi(Ui) = Bεi(ϕi(a)), and wehave the commutative diagram

FI(E)δa

pr

E

F(ci,εi)(E) Fεi(Kmi ;E).

∼=F (ϕi(.)−ϕi(a))←�F

F →F (0)

The left vertical projection map clearly is continuous. The lower horizontalmap is a topological isomorphism by construction. By Remark 5.1 the rightvertical evaluation map is continuous of operator norm ≤ 1.

Corollary 12.2. The locally convex vector space Can(M,E) is Hausdorff.

Proof. Let f �= g be two different functions in Can(M,E). We find a pointa ∈ M such that f(a) �= g(a). Since E is Hausdorff there are open neighbour-hoods Vf of f(a) and Vg of g(a) in E such that Vf ∩Vg = ∅. Using Prop. 12.1we see that Uf := δ−1

a (Vf ) and Ug := δ−1a (Vg) are open neighbourhoods of f

and g, respectively, in Can(M,E) such that Uf ∩ Ug = ∅.

Remark 12.3. With M also its tangent bundle T (M) is paracompact.

Proof. Since M is strictly paracompact by Prop. 8.7 we find a family ofcharts {ci = (Ui, ϕi,K

mi)}i∈I for M such that the Ui are pairwise disjointand M =

⋃i Ui. Then the ci,T = (p−1

M (Ui), ϕi,ci ,K2mi) form a family of

charts for T (M) such that T (M) is the disjoint union of the open subsetsp−1

M (Ui). Each p−1M (Ui) being homeomorphic to an open subset in K2mi car-

ries the topology of an ultrametric space. The construction in the proof ofimplication ii. =⇒ iii. in Prop. 8.7 then shows that the topology of T (M) canbe defined by a metric which satisfies the strict triangle inequality. HenceT (M) is paracompact by Lemma 1.4.

86 II Manifolds

Proposition 12.4. i. The map d : Can(M,E) −→ Can(T (M), E) iscontinuous.

ii. For any locally analytic map of paracompact manifolds g : M −→ Nthe map

Can(N,E) −→ Can(M,E)f −→ f ◦ g

is continuous.

iii. For any vector field ξ on M the map Dξ : Can(M,E) −→ Can(M,E)is continuous.

Proof. i. By Lemma 11.6 we have to show that d|FI(E) is continuous forany index I = {(ci = (Ui, ϕi,K

mi), εi)}i∈I for M . Let f ∈ FI(E). We havethe commutative diagrams

E

p−1M (Ui)

df

ϕi,ci

pM

ϕi(Ui) × Kmi

pr

(x,v)→Dx(f◦ϕ−1i )(v)

Uiϕi

ϕi(Ui)

(cf. the proof of Lemma 9.11). We also have power series Fi ∈ Fεi(Kmi ;E)

such that

f ◦ ϕ−1i (x) = Fi(x − ai) for any x ∈ ϕi(Ui) = Bεi(ai).

From the proof of Prop. 6.1 we recall the formula

Dx(f ◦ ϕ−1i )(v) =

mi∑

j=1

vj∂Fi

∂Xj(x − ai)

for any x ∈ ϕi(Ui) and any v = (v1, . . . , vmi) ∈ Kmi . We now cover Kmi bypairwise disjoint balls Bεi(w

(i)k ) where w

(i)k = (w(i)

k,1, . . . , w(i)k,mi

) runs over anappropriate family of vectors in Kmi , and we put

Gi,k(X1, . . . , Xmi , Y1, . . . , Ymi) :=mi∑

j=1

(Yj + w(i)k,j)

∂Fi

∂Xj∈ Fεi(K

2mi ;E).


Thendf ◦ ϕ−1

i,ci(x, v) = Dx(f ◦ ϕ−1

i )(v) = Gi,k(x − ai, v − w(i)k )

for any (x, v) ∈ ϕi(Ui) × Bεi(w(i)k ) = Bεi(ai, w

(i)k ). This means that df ∈

FJ (E) ⊆ Can(T (M), E) for the index

J := {((ϕ−1i,ci

(Bεi(ai, w(i)k )), ϕi,ci ,K

2mi), εi)}i,k.

In other words we have the commutative diagram

Can(M,E) dCan(T (M), E)

FI(E)

⊆

FJ (E).

⊆

Since the vertical inclusion maps are continuous by construction this reducesus to showing the continuity of the lower horizontal map FI(E) −→ FJ (E).But this easily follows from the inequalities

‖Gi,k‖εi ≤ max(

1,|w(i)

k,1|εi

, . . . ,|w(i)

k,mi|

εi

)

· ‖Fi‖εi .

ii. We only sketch the argument and leave the details to the reader. LetI = {((Ui, ϕi,K

ni), εi)}i∈I be an index for N . We refine the covering M =⋃i g

−1(Ui) into a covering M =⋃

j∈J Vj which underlies an appropriateindex J = {((Vj , ψj ,K

mj ), δj)}j∈J and such that, for any i ∈ I and j ∈J with Vj ⊆ g−1(Ui), there is a power series Gi,j ∈ Fδj

(Kmj ;Kni) with‖Gi,j − Gi,j(0)‖δj

≤ εi and

ϕi ◦ g ◦ ψ−1j (x) = Gi,j(x − aj) for any x ∈ ψj(Vj) = Bδj

(aj).

In this situation we have the commutative diagram

Can(N,E)f →f◦g

Can(M,E)

FI(E)

⊆

FJ (E)

⊆

where the lower horizontal arrow in terms of power series is given by themaps

Fεi(Kni ;E) −→ Fδj

(Kmj ;E)

F −→ F ◦ (Gi,j − Gi,j(0))

88 II Manifolds

whose continuity was established in Prop. 5.4.iii. follows from i. and ii.

Proposition 12.5. For any covering M =⋃

i∈I Ui by pairwise disjoint opensubsets Ui we have

Can(M,E) =∏

i∈I

Can(Ui, E)

as topological vector spaces.

Proof. Using Lemma 10.2 one checks that in the construction of Can(M,E)as a locally convex inductive limit it suffices to consider indices for M whoseunderlying covering of M refines the given covering M =

⋃i Ui. Then the

assertion is a formal consequence of Lemma 11.7.

Chapter III

Lie Groups

As before we fix the nonarchimedean field (K, | |).

13 Definitions and Foundations

Definition. A Lie group G (over K) is a manifold (over K) which alsocarries the structure of a group such that the multiplication map

m = mG : G × G −→ G

(g, h) �−→ gh

is locally analytic.

In the following let G be a Lie group, and let e ∈ G denote the unitelement.

Lemma 13.1. For any h ∈ G the maps

�h : G�−−→ G and rh : G

�−−→ G

g �−→ hg g �−→ gh

are locally analytic isomorphisms (of manifolds).

Proof. By symmetry we only need to consider the case of the left multipli-cation �h. This map can be viewed as the composite

G −→ G × Gm−−→ G

g �−→ (h, g).

The left arrow is locally analytic by Example 8.5.4 and the right arrowby assumption. Hence the map �h is locally analytic by Lemma 8.4.ii. Weobviously have �h ◦ �h−1 = �hh−1 = �e = idG and then also �h−1 ◦ �h = idG.It follows that �−1

h = �h−1 is locally analytic as well.

Corollary 13.2. For any two elements g, h ∈ G the map

Tg(�hg−1) : Tg(G)∼=−−→ Th(G)

is a K-linear isomorphism; in particular,

Te(�g) : Te(G)∼=−−→ Tg(G)

is an isomorphism for any g ∈ G.


89

http://dx.doi.org/10.1007/978-3-642-21147-8_3

90 III Lie Groups

Proof. Use Lemma 9.2.

Corollary 13.3. Every Lie group is n-dimensional for some n ≥ 0.

Proof. We have dim G = dimK Te(G) = dimK Tg(G) for any g ∈ G.

Examples. 1) Kn and more generally any ball Bε(0) (as an open sub-manifold of Kn) with the addition is a Lie group.

2) K× and more generally B−1 (1) and Bε(1) for any 0 < ε < 1 (as open

submanifolds of K) with the multiplication (observe that ab − 1 =(a − 1)(b − 1) + (a − 1) + (b − 1)) are Lie groups.

3) GLn(K) viewed as the open submanifold in Kn2defined by “det �= 0”

with the matrix multiplication is a Lie group.

Let g, h ∈ G. We know from Remark 9.10.ii. that the map

T (pr1) × T (pr2) : T(g,h)(G × G)∼=−−→ Tg(G) × Th(G)

is a K-linear isomorphism. In order to describe its inverse we introduce themaps

ih : G −→ G × G and jg : G −→ G × G

x �−→ (x, h) x �−→ (g, x)

which are locally analytic by Example 8.5.4). We have

pr1 ◦ ih = idG and pr2 ◦ ih = constant map with value h

and henceT (pr1) ◦ T (ih) = T (idG) = idT (G)

andT (pr2) ◦ T (ih) = T (constant map) = 0.

This means that the composed map

Tg(G)T (ih)−−−−→ T(g,h)(G × G)

T (pr1)×T (pr2)−−−−−−−−−−→ Tg(G) × Th(G)

sends t to (t, 0). Analogously the composed map

(17) Th(G)T (jg)−−−−→ T(g,h)(G × G)

T (pr1)×T (pr2)−−−−−−−−−−→ Tg(G) × Th(G)


sends t to (0, t). We conclude that

Tg(ih) + Th(jg) : Tg(G) × Th(G) −→ T(g,h)(G × G)

(t1, t2) �−→ Tg(ih)(t1) + Th(jg)(t2)

is the inverse of T(g,h)(pr1) × T(g,h)(pr2).

Lemma 13.4. The diagram

T(g,h)(G × G)T (m)

∼=T (pr1)×T (pr2)

Tgh(G)

Tg(G) × Th(G)Tg(rh)+Th(lg)

is commutative for any g, h ∈ G.

Proof. We compute

T(g,h)(m) ◦ (T(g,h)(pr1) × T(g,h)(pr2))−1 = T(g,h)(m) ◦ (Tg(ih) + Th(jg))

= Tg(m ◦ ih) + Th(m ◦ jg)= Tg(rh) + Th(lg).

Corollary 13.5. The diagram

T(e,e)(G × G)T (m)

∼=T (pr1)×T (pr2)

Te(G)

Te(G) × Te(G)

+

is commutative.

Proposition 13.6. The map

ι = ιG : G −→ G

g �−→ g−1

is a locally analytic isomorphism (of manifolds).

92 III Lie Groups

Proof. Because of ι2 = idG it suffices to show that the map ι is locallyanalytic. To do so we use the bijective locally analytic map

μ : G × G −→ G × G

(x, y) �−→ (xy, y).

We claim that the tangent map T(g,h)(μ), for any g, h ∈ G, is bijective. ByLemma 13.4 the diagram

T(g,h)(G × G)T(g,h)(μ)

∼=T (pr1)×T (pr2)

T(gh,h)(G × G)

∼= T (pr1)×T (pr2)

Tg(G) × Th(G) Tgh(G) × Th(G)

in which the lower horizontal arrow is given by

(t1, t2) �−→ (Tg(rh)(t1) + Th(lg)(t2), t2)

is commutative. Suppose that (t1, t2) lies in the kernel of this latter map.Then t2 = 0 and hence 0 = Tg(rh)(t1) + Th(lg)(t2) = Tg(rh)(t1). The analogof Cor. 13.2 for the right multiplication implies that t1 = 0. We see thatthis lower horizontal map and therefore T(g,h)(μ) are injective. But all vec-tor spaces in the diagram have the same finite dimension. Our claim thatT(g,h)(μ) is bijective follows. We now may apply the criterion for local invert-ibility in Prop. 9.3 and we conclude that the inverse μ−1 is locally analyticas well. It remains to note that ι is the composite

Gje−−→ G × G

μ−1

−−−→ G × Gpr1−−−→ G.

Corollary 13.7. For any g ∈ G the diagram

Te(G)Te(lg)

−1

Tg(G)

Tg(ι)

Te(G)Te(rg−1 )

Tg−1(G)

is commutative; in particular, the map Te(ι) coincides with the multiplicationby −1.


Proof. The composed map

Glg−−→ G

ι−→ Grg−−→ G

ι−→ G

is the identity. Hence the diagram

Te(G)Te(lg)

Te(ι)

Tg(G)

Tg(ι)

Te(G)Te(rg−1 )

Tg−1(G)

is commutative. This reduces us to showing the special case in our assertion.We consider the diagram

Te(G)t�→(0,t)T (je)

T(e,e)(G × G)

T(e,e)(μ−1)=T(e,e)(μ)−1

T (pr1)×T (pr2)Te(G) × Te(G)

(t1,t2) �→(t1−t2,t2)

T(e,e)(G × G)T (pr1)×T (pr2)

T (pr1)

Te(G) × Te(G)

pr1

Te(G) .

In the proof of Prop. 13.6 we have seen that the map μ−1(x, y) = (xy−1, y)is locally analytic and that the central square in the above diagram is com-mutative. The top triangle is commutative by (17). The commutativity ofthe bottom triangle is trivial. It remains to observe that passing from topto bottom along the left, resp. right, hand side is equal to Te(ι), resp. to themultiplication by −1.

Corollary 13.8. For every n ∈ Z the map

fn : G −→ G

g �−→ gn

is locally analytic, and Te(fn) coincides with the multiplication by n.

94 III Lie Groups

Proof. Case 1: For n = 0 the map f0 is the constant map with value e andTe(f0) = 0.

Case 2: Let n ≥ 1. We may view fn as the composite

Gdiag−−−→ G × · · · × G

mult−−−→ Gg �−→ (g, . . . , g)

(g1, . . . , gn) �−→ g1 . . . gn.

Both maps are locally analytic, the left diagonal map by Example 8.5.4) andthe right multiplication map by assumption. Hence in the diagram

Te(G × · · · × G)

Q

i T (pri)

Te(mult)

Te(G)

Te(diag)

diag

Te(G)

Te(G) × · · · × Te(G)

(t1,...,tn)�→t1+···+tn

the top, resp. bottom, composed map is equal to Te(fn), resp. the multipli-cation by n. But this diagram is commutative, the left triangle for trivialreasons and the right triangle as a consequence of Cor. 13.5.

Case 3: Let n ≤ −1. Since fn = f−n ◦ ι we obtain, using the previouscase and Cor. 13.7, that

Te(fn) = Te(f−n) ◦ Te(ι) = (−n · id) ◦ (− id) = n · id .

Cor. 13.2 already indicates that the tangent space Te(G) in the unitelement of G plays a distinguished role. We want to investigate this in greaterdetail.

Proposition 13.9. The maps

rT : Te(G) × G�−−→ T (G) and lT : G × Te(G) �−−→ T (G)

(t, g) �−→ Te(rg)(t) (g, t) �−→ Te(lg)(t)

are locally analytic isomorphisms (of manifolds); the diagram

Te(G) × GrT

pr2

T (G)

pG

G × Te(G)lT

pr1

G

is commutative.


Proof. By symmetry it suffices to discuss the map rT . We choose a chartc = (U,ϕ,Kn) for G around e. By Lemma 9.1.i. the map

θc : Kn ∼=−−→ Te(G)v �−→ [c, v]

is a K-linear isomorphism. We equip Te(G) with the unique structure of amanifold such that θc becomes a locally analytic isomorphism of manifolds.By Lemma 9.1.ii. this structure does not depend on the choice of the chartc. Of course, we then view Te(G)×G as the product manifold of Te(G) and

G. The inclusion map Te(G) ⊆−−→ T (G) is locally analytic since it can beviewed as the composite of the locally analytic maps

Te(G) θ−1c−−−→ Kn v �→(e,v)−−−−−−→ U × Kn τc−−→ p−1

G (U) ⊆−−→ T (G).

We recall that τc((g, v)) = [c, v] ∈ Tg(G) is locally analytic by the construc-tion of T (G) as a manifold. Let

ξ0 : G −→ T (G)g �−→ 0 ∈ Tg(G)

denote the “zero vector field”, i. e., the zero vector in the vector spaceΓ(G,T (G)). Using Lemma 13.4 we see that the composed locally analyticmap

Te(G) × G⊆× id−−−−→ T (G) × G

id×ξ0−−−−→ T (G) × T (G)(T (pr1)×T (pr2))

−1

−−−−−−−−−−−−−→ T (G × G)T (m)−−−−→ T (G)

sends (t, g) to Te(rg)(t) + Tg(le)(0) = Te(rg)(t) and hence coincides with rT .This shows that the map rT is locally analytic. It is easy to check that themap

T (G) −→ Te(G) × G

t �−→ (TpG(t)(rpG(t)−1)(t), pG(t))

is inverse to rT . Its second component pG is locally analytic by Lemma 9.8.It therefore remains to prove that the map

f : T (G) −→ Te(G)t �−→ TpG(t)(rpG(t)−1)(t)

96 III Lie Groups

is locally analytic. Using again Lemma 13.4 we compute that the composedlocally analytic map

T (G)id×pG−−−−→ T (G) × G

id×ι−−−−→ T (G) × Gid×ξ0−−−−→ T (G) × T (G)

(T (pr1)×T (pr2))−1

−−−−−−−−−−−−−→ T (G × G)T (m)−−−−→ T (G)

sends t to TpG(t)(rpG(t)−1)(t). It follows that the left vertical composite inthe commutative diagram

T (G)

f

T (G)

f

=

f

Te(G)

⊆

Te(G)=

t�→(e,θ−1c (t))

=Te(G)

T (G) U × Knτc (g,v) �→θc(v)

is locally analytic. Since τc is an open embedding we conclude that the rightvertical composite is locally analytic. With the lower oblique arrow thereforealso the upper oblique arrow f is locally analytic.

Corollary 13.10. The maps

Γ(G,T (G))∼=←−− Can(G,Te(G))

∼=−−→ Γ(G,T (G))ξlf (g) := lT ((g, f(g))) ←− � f �−→ ξr

f (g) := rT ((f(g), g))

are isomorphisms of K-vector spaces.

Proof. The maps ξ �−→ pr2 ◦(lT )−1◦ξ and ξ �−→ pr1 ◦(rT )−1◦ξ, respectively,are inverses.

In Can(G,Te(G)) we have, for any t ∈ Te(G), the constant map

constt(g) := t.

We put

ξlt(g) := ξl

constt(g) = Te(lg)(t) and ξr

t (g) := ξrconstt

(g) = Te(rg)(t).

Definition. A vector field ξ ∈ Γ(G,T (G)) is called left invariant, resp. rightinvariant, if ξ(g) = Te(lg)(ξ(e)), resp. ξ(g) = Te(rg)(ξ(e)), holds true for anyg ∈ G.


Corollary 13.11. The maps

Te(G)∼=−−→ {ξ ∈ Γ(G,T (G)) : ξ is left invariant}

t �−→ ξlt

and

Te(G)∼=−−→ {ξ ∈ Γ(G,T (G)) : ξ is right invariant}

t �−→ ξrt

are K-linear isomorphisms.

Proof. The map ξ �−→ ξ(e) is the inverse in both cases.

Let E be a K-Banach space. With any vector field ξ on G we had asso-ciated the K-linear map

Dξ : Can(G,E) −→ Can(G,E)f �−→ Dξ(f) = df ◦ ξ.

If ξ is left or right invariant what consequence does this have for the mapDξ? We observe that as a consequence of Lemma 13.1 we have a left K-linearaction by “left translation”

G × Can(G,E) −→ Can(G,E)

(h, f) �−→ hf(g) := f(h−1g)

of the group G on the vector space Can(G,E) as well as a right K-linearaction by “right translation”

Can(G,E) × G −→ Can(G,E)

(f, h) �−→ fh(g) := f(gh−1).

Lemma 13.12. If ξ ∈ Γ(G,T (G)) is right, resp. left, invariant then we have

Dξ(fh) = Dξ(f)h, resp. Dξ(hf) = hDξ(f),

for any f ∈ Can(G,E) and h ∈ G. In the case E = K the converse holdstrue as well.

98 III Lie Groups

Proof. By symmetry we only consider the “right” case. First we supposethat ξ is right invariant, i. e., ξ(g) = Te(rg)(ξ(e)). It follows that

T (rh−1) ◦ ξ(g) = Tg(rh−1) ◦ Te(rg)(ξ(e)) = Te(rgh−1)(ξ(e)) = ξ(gh−1).

We now compute

Dξ(fh)(g) = dfh ◦ ξ(g) = d(f ◦ rh−1) ◦ ξ(g)= df ◦ T (rh−1) ◦ ξ(g)

= df ◦ ξ(gh−1) = Dξ(f)(gh−1)

= Dξ(f)h(g)

where for the second line we have used Lemma 9.12. If vice versa Dξ satisfiesthe asserted identity (for some E) then we have

df ◦ T (rh−1) ◦ ξ(g) = Dξ(fh)(g) = Dξ(f)h(g) = Dξ(f)(gh−1) = df ◦ ξ(gh−1)

for any f and any g, h. We rewrite this as

df ◦ T (rh−1) ◦ ξ(gh) = df ◦ ξ(g).

With ξ alsoξh(g) := T (rh) ◦ ξ(gh−1)

is a vector field on G. Hence we obtain the identity

Dξh = Dξ for any h ∈ G.

Later on (Cor. 18.8) we will see that G is paracompact. In the case E = Kthe map ξ �→ Dξ therefore is injective by Prop. 9.16. It follows that ξh = ξ,i. e., that

T (rh) ◦ ξ(gh−1) = ξ(g)

holds true for any g, h ∈ G. In particular, for g = h we obtain

T (rg)(ξ(e)) = ξ(g) for any g ∈ G

which means that ξ is right invariant.

The Lie product of vector fields is characterized by the identity

Dξ ◦ Dη − Dη ◦ Dξ = D[ξ,η]

(which even holds for general E by Prop. 9.20 and Cor. 18.8).


Corollary 13.13. If the vector fields ξ and η on G both are left or rightinvariant then so, too, is the vector field [ξ, η].

Proof. With Dξ and Dη obviously also D[ξ,η] satisfies the identity assertedin Lemma 13.12.

Using Lemma 13.11 and Cor. 13.13 we see that for any s, t ∈ Te(G) thereare uniquely determined tangent vectors [s, t]l and [s, t]r in Te(G) such that

ξl[s,t]l

= [ξls, ξ

lt] and ξr

[s,t]r= [ξr

s , ξrt ].

Then(Te(G), [ , ]l)

t�→ξlt−−−−→ (Γ(G,T (G)), [ , ])

and(Te(G), [ , ]r)

t�→ξrt−−−−→ (Γ(G,T (G)), [ , ])

are injective maps of Lie algebras. Is there a relation between the two Lieproducts [ , ]l and [ , ]r on Te(G)? For any ξ ∈ Γ(G,T (G)) also

ιξ(g) := Tg−1(ι) ◦ ξ(g−1)

is a vector field on G. This provides us with an involutory K-linear auto-morphism

ι. : Γ(G,T (G)) −→ Γ(G,T (G)).

Remark 13.14. The diagram

Te(G)ξl.

−1

Γ(G,T (G))

ι.

Te(G)ξr.

Γ(G,T (G))

is commutative.

Proof. Using Cor. 13.7 we compute

ι(ξlt)(g) = T (ι) ◦ ξl

t(g−1) = T (ι) ◦ T (lg−1)(t) = −T (rg)(t) = −ξr

t (g).

Lemma 13.15. Any vector fields ξ and η on G satisfy

[ιξ, ιη] = ι[ξ, η].

100 III Lie Groups

Proof. Using Lemma 9.12 we compute

Dιξ(f)(g) = df ◦ ιξ(g) = df ◦T (ι) ·ξ(g−1) = d(f ◦ι)◦ξ(g−1) = Dξ(f ◦ι)(g−1).

This amounts to the identity

Dιξ(f) ◦ ι = Dξ(f ◦ ι).

We continue computing

(Dιξ ◦ Dιη)(f)(g) = Dξ(Dιη(f) ◦ ι)(g−1)

= Dξ(Dη(f ◦ ι))(g−1)

= (Dξ ◦ Dη)(f ◦ ι)(g−1)

and consequently

D[ιξ,ιη](f)(g) = [Dιξ, Dιη](f)(g)

= [Dξ, Dη](f ◦ ι)(g−1)

= D[ξ,η](f ◦ ι)(g−1)

= Dι[ξ,η](f)(g).

Corollary 13.16. We have

[s, t]r = −[s, t]l for any s, t ∈ Te(G).

Proof. Using Remark 13.14 and Lemma 13.15 we compute

ξr[s,t]r

= [ξrs , ξ

rt ] = [−ξr

s ,−ξrt ] = [ξr

−s, ξr−t]

= [ιξls,

ιξlt] = ι[ξl

s, ξlt] = ιξl

[s,t]l

= ξr−[s,t]l

.

From now on we simplify the notation by setting

[s, t] := [s, t]r and Dt := Dξrt

for any s, t ∈ Te(G). We then have the identity

D[s,t] = Ds ◦ Dt − Dt ◦ Ds

on Can(G,E).

Definition. Lie(G) := (Te(G), [ , ]) is called the Lie algebra of G.


We obviously have

dimK Lie(G) = dimG.

The guiding question for the rest of this book is how much informationthe Lie algebra Lie(G) retains about the Lie group G. The answer requiresseveral purely algebraic concepts which we discuss in the next few sections.

Definition. Let G1 and G2 be two Lie groups over K; a homomorphism ofLie groups f : G1 −→ G2 is a locally analytic map which also is a grouphomomorphism.

Definition. If (g1, [ , ]1) and (g2, [ , ]2) are two Lie algebras over K thena homomorphism (of Lie algebras) σ : g1 −→ g2 is a K-linear map whichsatisfies

[σ(x), σ(y)]2 = σ([x, y]1) for any x, y ∈ g1.

We write HomK((g1, [ , ]1), (g2, [ , ]2)) for the set of all homomorphismsof Lie algebras σ : g1 −→ g2.

Exercise. For any homomorphism of Lie groups f : G1 −→ G2 the mapLie(f) := Te(f) : Lie(G1) −→ Lie(G2) is a homomorphism of Lie algebras.

14 The Universal Enveloping Algebra

In this section K is allowed to be a completely arbitrary field.

Exercise. i. Let A be an associative K-algebra with a unit element.Then (A, [ , ]A) with

[x, y]A := xy − yx

is a Lie algebra over K. In the case of a matrix algebra A = Mn×n(K)the corresponding Lie algebra is denoted by gln(K).

ii. If the field K is nonarchimedean then we have gln(K) = Lie(GLn(K)).

How general are the Lie algebras in this exercise? Obviously (A, [ , ]A)may have Lie subalgebras which do not correspond to associative subalge-bras. We want to show that any Lie algebra in fact arises as a subalgebraof an associative algebra. A K-linear map σ : g −→ A from a Lie algebra g

into an associative algebra A, of course, will be called a homomorphism ifit satisfies

σ([x, y]) = σ(x)σ(y) − σ(y)σ(x) for any x, y ∈ g.

102 III Lie Groups

At this point we need to recall the following general construction from mul-tilinear algebra. Let E be any K-vector space. Then

T (E) := ⊕n≥0E⊗n where E⊗n := E ⊗K · · · ⊗K E (n factors)

is an associative K-algebra with unit (note that E⊗0 = K). The multiplica-tion is given by the linear extension of the rule

(v1 ⊗ · · · ⊗ vn)(w1 ⊗ · · · ⊗ wm) := v1 ⊗ · · · ⊗ vn ⊗ w1 ⊗ · · · ⊗ wm.

This algebra T (E) is called the tensor algebra of the vector space E. It hasthe following universal property.

Any K-linear map σ : E −→ A into any associative K-algebra with unitA extends in a unique way to a homomorphism of K-algebras with unitσ : T (E) −→ A. In fact, this extension satisfies

σ(v1 ⊗ · · · ⊗ vn) = σ(v1) · . . . · σ(vn).

Let g be a Lie algebra over K. Viewed as a K-vector space we mayform the tensor algebra T (g). In T (g) we consider the two sided ideal J(g)generated by all elements of the form

x ⊗ y − y ⊗ x − [x, y] for x, y ∈ g.

Note that x ⊗ y − y ⊗ x ∈ g⊗2 whereas [x, y] ∈ g⊗1. Then

U(g) := T (g)/J(g)

is an associative K-algebra with unit and

ε : g −→ U(g)x �−→ x + J(g)

is a homomorphism.

Definition. U(g) is called the universal enveloping algebra of the Lie alge-bra g.

This construction has the following universal property. Let σ : g −→ A beany homomorphism into any associative K-algebra with unit A. It extendsuniquely to a homomorphism σ : T (g) −→ A of K-algebras with unit.Because of

σ(x ⊗ y − y ⊗ x − [x, y]) = σ(x)σ(y) − σ(y)σ(x) − σ([x, y]) = 0


we haveJ(g) ⊆ ker(σ).

Hence there is a uniquely determined homomorphism of K-algebras withunit

σ : U(g) −→ A with σ ◦ ε = σ,

i. e., the diagram

g

ε

σ

⊆A

T (g)

σ

pr

U(g)

σ

is commutative.The tensor algebra T (E) has the increasing filtration

T0(E) ⊆ T1(E) ⊆ · · · ⊆ Tm(E) ⊆ · · ·

defined byTm(E) := ⊕0≤n≤mE⊗n.

The Tm(E) do not form ideals in T (E). But they satisfy

Tl(E) · Tm(E) ⊆ Tl+m(E) for any l,m ≥ 0.

Correspondingly we obtain an increasing filtration

U0(g) ⊆ U1(g) ⊆ · · · ⊆ Um(g) ⊆ · · ·

in U(g) defined byUm(g) := Tm(g) + J(g)/J(g)

and which satisfies

(18) Ul(g) · Um(g) ⊆ Ul+m(g) for any l,m ≥ 0.

For example, we have U0(g) = K · 1 and U1(g) = K · 1 + ε(g). We define

gr• U(g) := ⊕m≥0 grm U(g) with grm U(g) := Um(g)/Um−1(g)

104 III Lie Groups

(and the convention that U−1(g) := {0}). Because of (18) the K-bilinearmaps

grl U(g) × grm U(g) −→ grl+m U(g)(y + Ul−1(g), z + Um−1(g)) �−→ yz + Ul+m−1(g)

are well defined. Together they make gr• U(g) into an associative K-algebrawith unit.

Theorem 14.1. (Poincare-Birkhoff-Witt) The algebra gr• U(g) is isomor-phic to a polynomial ring over K in possibly infinitely many variables Xi

and, in particular, is commutative. More precisely, let {xi}i∈I be a K-basisof g; then

K[{Xi}i∈I ]∼=−−→ gr• U(g)

Xi �−→ ε(xi) + U0(g) ∈ gr1 U(g)

is an isomorphism of K-algebras with unit.

Proof. Compare [B-LL] Chap. I §2.7 or [Hum] §17.3.

Corollary 14.2. The map ε : g −→ U(g) is injective.

Because of this fact the map ε usually is viewed as an inclusion and isomitted from the notation. We see that g indeed is a Lie subalgebra of anassociative algebra.

Corollary 14.3. Let d := dimK g < ∞; if x1, . . . , xd is an (ordered) K-basis of g then {xi1 · . . . · xim : m ≥ 0, 1 ≤ i1 ≤ · · · ≤ im ≤ d} is a K-basisof U(g).

Proof. The Theorem 14.1 of Poincare-Birkhoff-Witt implies that, for anym ≥ 0, the set

{xi1 · . . . · xim + Um−1(g) : 1 ≤ i1 ≤ · · · ≤ im ≤ d}

is a K-basis of Um(g)/Um−1(g) (recall the convention that the empty pro-duct, in the case m = 0, is equal to the unit element).

This last corollary obviously remains true, by choosing a total orderingof a K-basis of g, even if g is not finite dimensional.

Let τ : g1 −→ g2 be a homomorphism of Lie algebras. Applying theuniversal property gives a homomorphism of K-algebras with unit

U(τ) : U(g1) −→ U(g2)


such that the diagram

g1τ

⊆

g2

⊆

U(g1)U(τ)

U(g2)

is commutative. We want to apply this in two specific situations. First letg1 and g2 two Lie algebras. Obviously, g1 × g2 again is a Lie algebra withrespect to the componentwise Lie product. There are the correspondingmonomorphisms of Lie algebras

i1 : g1 −→ g1 × g2 and i2 : g2 −→ g1 × g2

x �−→ (x, 0) y �−→ (0, y).

Lemma 14.4. The map

U(g1) ⊗K U(g2)∼=−−→ U(g1 × g2)

a ⊗ b �−→ U(i1)(a) · U(i2)(b)

is an isomorphism of K-algebras with unit.

Proof. Obviously, we have the K-bilinear map

U(g1) × U(g2) −→ U(g1 × g2)(a, b) �−→ U(i1)(a) · U(i2)(b).

By the universal property of the tensor product it induces the map in theassertion as a K-linear map. The latter is bijective by a straightforwardapplication of Cor. 14.3. Since we have

[i1(x), i2(y)] = [(x, 0), (0, y)] = ([x, 0], [0, y]) = (0, 0)

for any x ∈ g1 and any y ∈ g2 it follows easily that U(i1)(a) and U(i2)(b), forany a ∈ U(g1) and any b ∈ U(g2), commute with one another. This impliesthat the asserted map is a homomorphism and hence an isomorphism ofK-algebras with unit.

We point out that under the isomorphism in the above lemma the ele-ments

x ⊗ 1 + 1 ⊗ y ←→ (x, y)

106 III Lie Groups

correspond to each other.Secondly, for any Lie algebra g the diagonal map

Δ : g −→ g × g

x �−→ (x, x)

is a homomorphism of Lie algebras. We obtain the commutative diagram

gΔ

⊆

g × g

⊆

U(g)U(Δ)

U(g × g)∼=

U(g) ⊗K U(g).

Definition. The composed map U(g) −→ U(g) ⊗K U(g) in the lower lineof the above diagram is denoted (by abuse of notation) again by Δ and iscalled the diagonal (or comultiplication) of the algebra U(g).

We note that for x ∈ g ⊆ U(g) we have

Δ(x) = x ⊗ 1 + 1 ⊗ x.

Proposition 14.5. If the field K has characteristic zero then we have

g = {a ∈ U(g) : Δ(a) = a ⊗ 1 + 1 ⊗ a}.

Proof. Compare [B-LL] Chap. II §1.5 Cor.

15 The Concept of Free Algebras

In this section K again is an arbitrary field. We will discuss the followingproblem. Let A be a specific class (or category) of K-algebras. We have inmind the following list of examples:

– ComK := all commutative and associative K-algebras with unit;

– AssK := all associative K-algebras with unit;

– LieK := all Lie algebras over K;

– AlgK := all K-algebras, i. e., all K-vector spaces A equipped with aK-bilinear “multiplication” map A × A −→ A.


We suppose given a finite set X = {X1, . . . , Xd}, and we ask for an algebraAX in the class A together with a map X −→ AX which have the followinguniversal property: For any map X −→ A from the set X into any algebraA in the class A there is a unique homomorphism AX −→ A of algebras inA such that the diagram

X A

AX

is commutative. If it exists AX is called the free A-algebra on X.The case ComK : The polynomial ring AX := K[X1, . . . , Xd] over K in

the variables X1, . . . , Xd has the requested universal property.The case AssK : As we have recalled in Sect. 14 the tensor algebra

AX := AsX := T (Kd)

of the standard K-vector space Kd together with the map

X −→ Kd ⊆ T (Kd)Xi �−→ i-th standard basis vector ei

satisfies the requested universal property. It sometimes is useful to view AsX

as the ring of all “noncommutative” polynomials

P (X1, . . . , Xd) =∑

(i1,...,im)

a(i1,...,im)Xi1 · . . . · Xim

with coefficients a(i1,...,im) ∈ K where the sum runs over finitely many tu-ples (i1, . . . , im) with entries from the set {1, . . . , d} (including possibly theempty tuple). The multiplication is determined by the rule that the variablescommute with the coefficients but not with each other. The algebra AsX ina natural way is graded by As

(n)X := Kd ⊗K · · · ⊗K Kd (n factors) which

means that

AsX = ⊕n≥0As(n)X with As

(l)X · As

(m)X ⊆ As

(l+m)X for any l,m ≥ 0.

The case AlgK : Here we have to preserve the information about theorder in which the multiplications in a “monomial” Xi1 · . . . · Xim are per-formed (and we have to omit the unit element). This can be done in thefollowing way. We inductively define sets X(n) for n ≥ 1 by X(1) := X and

X(n) := disjoint union of all X(p) × X(q) for p + q = n,

108 III Lie Groups

and we putMX := disjoint union of all X(n).

The obvious inclusion maps X(m) × X(n) ⊆−−→ X(m + n) combine into a“multiplication” map

μX : MX × MX −→ MX .

We now form the K-algebra

AX := the K-vector space on the basis MX

in which the multiplication is given by the linear extension of the map μX .There are the obvious inclusions X ⊆ MX ⊆ AX .

Let γ : X −→ A be any map into any K-algebra A. We inductivelyextend γ to a map γ : MX −→ A by

γ : X(n) ⊇ X(p) × X(q) −→ A

(x, y) �−→ γ(x)γ(y).

This extension by construction is multiplicative in the sense that the diagram

MX × MXμX

γ×γ

MX

γ

A × A·

A

is commutative. Hence it further extends by linearity to a homomorphismof K-algebras

γ : AX −→ A.

We stress that the algebra AX is graded by

A(n)X := the K-vector space on the basis X(n),

i. e., we have

AX = ⊕n∈NA(n)X with A

(l)X · A(m)

X ⊆ A(l+m)X for any l,m ≥ 1.

The case LieK : In AX we consider the two sided ideal JX which isgenerated by all expressions of the form

aa and (ab)c + (bc)a + (ca)b for a, b, c ∈ AX .


ThenLX := AX/JX with [a + JX , b + JX ] := ab + JX

is a Lie algebra over K.Let γ : X −→ g be any map into any Lie algebra g over K. As discussed

above it extends to a homomorphism of K-algebras γ : AX −→ g. Weobviously have

JX ⊆ ker(γ).

Hence there is a uniquely determined homomorphism of Lie algebras γ :LX −→ g such that the diagram

Xγ

⊆g

AX

γ

pr

LX

γ

is commutative.

Exercise. i. We have JX = ⊕n∈NJX ∩ A(n)X and hence

LX = ⊕n∈NL(n)X with [L(l)

X , L(m)X ] ⊆ L

(l+m)X for any l,m ≥ 1

if we define L(n)X := A

(n)X /JX∩A

(n)X (i. e., the Lie algebra LX is graded).

ii. The set X is (more precisely, maps bijectively onto a) K-basis of L(1)X .

iii. The set {[Xi, Xj ] : i < j} is a K-basis of L(2)X .

The inclusion map X⊆−−→ AsX extends uniquely to a homomorphism of

Lie algebrasφ : LX −→ (AsX , [ , ]AsX

).

By the universal property of the universal enveloping algebra this map φfurther extends uniquely to a homomorphism of associative K-algebras withunit

Φ : U(LX) −→ AsX .

Proposition 15.1. The map Φ is an isomorphism of algebras.

110 III Lie Groups

Proof. The composed map

X −→ LX −→ U(LX)

extends uniquely to a homomorphism of associative K-algebras with unit

Ψ : AsX −→ U(LX).

We have the commutative diagram

X

⊆ ⊆LX

φ

AsXΨ

U(LX)Φ

AsX .

It therefore follows from the unicity in the universal property of AsX thatΦ ◦ Ψ = id. We also have the commutative diagram

X

⊆LX

φ

LX

U(LX)Φ

AsXΨ

U(LX) .

The universal property of LX then implies that even the diagram

LX

U(LX)Φ

AsXΨ

U(LX)

is commutative. Using the universal property of U(.) we conclude that Ψ ◦Φ = id as well.

Corollary 15.2. The map φ : LX −→ AsX is injective.

Proof. Combine Cor. 14.2 and Prop. 15.1.


Exercise. i. If we view AsX as the ring of noncommutative polynomi-als P (X1, . . . , Xd) over K in the variables X1, . . . , Xd then φ(LX) isthe smallest K-vector subspace of AsX which contains the variablesX1, . . . , Xd and is closed under the operation (P,Q) �−→ P ·Q−Q ·P .

ii. φ(L(n)X ) ⊆ As

(n)X for any n ∈ N, i. e., the homomorphism φ is graded.

16 The Campbell-Hausdorff Formula

Again K is an arbitrary field and X = {X1, . . . , Xd} is a fixed finite set. Werecall that the free associative K-algebra with unit AsX on X is graded:

AsX = ⊕n≥0As(n)X and As

(l)X · As

(m)X ⊆ As

(l+m)X for any l,m ≥ 0.

ThereforeAsX :=

∏

n≥0

As(n)X

with the multiplication

(an)n · (bn)n :=

(n∑

i=0

aibn−i

)

n

also is an associative K-algebra with unit (containing AsX as a subalgebra).It is called the Magnus algebra on X. Similarly as for AsX it is useful toview AsX as the ring of all “noncommutative” formal power series over Kin the variables X1, . . . , Xd. In AsX we have the two sided maximal ideal

mX := {(an)n ∈ AsX : a0 = 0}.

Lemma 16.1. i. As×X = {(an)n ∈ AsX : a0 �= 0}.

ii. 1 + mX is a subgroup of As×X .

Proof. i. The map

AsX −→ K

(an)n �−→ a0

is a homomorphism of K-algebras with unit. The group of multiplicativeunits As

×X therefore must be contained in the complement of the kernel of

112 III Lie Groups

this map. Vice versa let a = (an)n ∈ AsX be an element such that a0 �= 0.We have

a = a0 · (1 − u) where u := (0,−a−10 · a1, . . . ,−a−1

0 · an+1, . . .).

Since um ∈ {0} × · · · × {0} ×∏

n≥m As(n)X the sum

∑m≥0 um is well defined

in AsX . For b := a−10 ·

( ∑m≥0 um

)we then obtain ab = ba = 1.

ii. This is obvious from i. Note that 1 + mX is the kernel of the homo-morphism of groups

As×X −→ K×

(an)n �−→ a0.

In the last proof we have used a special case of the following generalprinciple. For each m ≥ 0 let u(m) ∈ {0}× · · · × {0}×

∏n≥m As

(n)X be some

element. Then the sum∑

m≥0 u(m) ∈ AsX is well defined. In particular, forany u ∈ mX we have the well defined homomorphism of K-algebras withunit

εu : K[[T ]] −→ AsX

F (T ) �−→ F (u).

Proposition 16.2. If the field K has characteristic zero then the maps

exp : mX −→ 1 + mX and log : 1 + mX −→ mX

u �−→∑

n≥0

un

n!1 + u �−→

∑

n≥1

(−1)n+1 un

n

are well defined and inverse to each other.

Proof. Because of

exp(u) = εu(exp(T )) and log(1 + u) = εu(log(1 + T ))

the maps in the assertion are well defined. Applying εu to the identities

exp(log(1 + T )) = 1 + T and log(exp(T )) = T

in the ring Q[[T ]] shows that they are inverse to each other.


Exercise. If a, b ∈ mX commute with each other (multiplicatively) then wehave

exp(a + b) = exp(a) · exp(b).

Using Cor. 15.2 and the subsequent Exercise we may view LX as the Liesubalgebra of AsX “generated” by the elements X1, . . . , Xd. Moreover, wehave

LX = L(1)X ⊕ L

(2)X ⊕ · · ·

∩ ∩ ∩AsX = K⊕ As

(1)X ⊕ As

(2)X ⊕ · · ·

We now defineLX :=

∏

n≥1

L(n)X ⊆ mX ⊆ AsX .

Lemma 16.3. LX is a Lie subalgebra of AsX .

Proof. Let a = (an)n and b = (bn)n be any two elements of LX . We have toshow that ab − ba ∈ LX holds true. For any m ≥ 1 we put

a(m) := (0, a1, . . . , am, 0, . . .), v(m) := (0, . . . , 0, am+1, am+2, . . .),

b(m) := (0, b1, . . . , bm, 0, . . .), u(m) := (0, . . . , 0, bm+1, bm+2, . . .).

Then a(m), b(m) ∈ LX and hence a(m)b(m) − b(m)a(m) ∈ LX . Moreover

ab − ba = (a(m) + v(m))(b(m) + u(m)) − (b(m) + u(m))(a(m) + v(m))

= a(m)b(m) − b(m)a(m) + (0, . . . , 0, cm+1, . . .).

It follows that for n ≤ m we have

n-th component of ab− ba = n-th component of a(m)b(m) − b(m)a(m) ∈ L(n)X .

Since m was arbitrary we conclude that ab − ba ∈ LX .

Since U(LX) = AsX by Prop. 15.1 we may view the comultiplication Δof U(LX) as a homomorphism of K-algebras with unit

Δ : AsX −→ AsX ⊗K AsK .

It satisfies Δ(Xi) = Xi ⊗ 1 + 1⊗Xi for any 1 ≤ i ≤ d. Since the X1, . . . , Xd

form a K-basis of As(1)X it follows that

Δ(As(1)X ) ⊆ As

(1)X ⊗K As

(0)X + As

(0)X ⊗K As

(1)X

114 III Lie Groups

and then inductively that

Δ(As(n)X ) ⊆

⊕

l+m=n

[As

(l)X ⊗K As

(m)X

]

for any n ≥ 0. This makes it possible to extend Δ to the homomorphism ofK-algebras with unit

Δ : AsX =∏

n≥0

As(n)X −→ AsX⊗KAsX :=

∏

l,m≥0

[As

(l)X ⊗K As

(m)X

]

=∏

n≥0

⊕

l+m=n

[As

(l)X ⊗K As

(m)X

]

(an)n �−→(Δ(an)

)n.

Lemma 16.4. If the field K has characteristic zero then we have

LX = {a ∈ AsX : Δ(a) = a ⊗ 1 + 1 ⊗ a}.

Proof. Let a = (an)n ∈ AsX be any element. We have Δ(a) = a⊗ 1 + 1⊗ aif and only if Δ(an) = an⊗1+1⊗an for any n ≥ 0. By Prop. 14.5 the latteris equivalent to an ∈ LX ∩ As

(n)X = L

(n)X for any n ≥ 0 which exactly is the

condition that a ∈ LX .

Theorem 16.5. (Campbell-Hausdorff ) Suppose that K has characteristiczero; then the map

exp : LX∼−−→ {b ∈ 1 + mX : Δ(b) = b ⊗ b}

is a well defined bijection; moreover, the right hand side is a subgroup of1 + mX .

Proof. The second part of the assertion follows immediately from Δ beinga ring homomorphism. Since LX ⊆ mX the map exp is defined on LX andis injective by Prop. 16.2. For the subsequent computations we observe thatthe componentwise construction of the ring homomorphism Δ implies thatΔ commutes with the maps exp and log. First let a ∈ LX . Then Δ(a) =a ⊗ 1 + 1 ⊗ a, and we compute

Δ(exp(a)) = exp(Δ(a)) = exp(a ⊗ 1 + 1 ⊗ a)= exp(a ⊗ 1) · exp(1 ⊗ a)= (exp(a) ⊗ 1) · (1 ⊗ exp(a))= exp(a) ⊗ exp(a).


This shows that exp(a) indeed lies in the target of the asserted map. Viceversa let b ∈ 1 + mX such that Δ(b) = b ⊗ b. By Prop. 16.2 we may definea := log(b) ∈ mX so that b = exp(a). We compute

Δ(a) = Δ(log(b)) = log(Δ(b))= log(b ⊗ b) = log((b ⊗ 1) · (1 ⊗ b))= log(b ⊗ 1) + log(1 ⊗ b)= log(b) ⊗ 1 + 1 ⊗ log(b)= a ⊗ 1 + 1 ⊗ a.

Hence Lemma 16.4 implies that a ∈ LX . We see that the asserted map issurjective.

Corollary 16.6. Suppose that the field K has characteristic zero; then LX

equipped with the multiplication

a � b := log(exp(a) · exp(b))

is a group whose neutral element is the zero vector 0 and such that −a isthe inverse of a.

Proof. Since exp(0) = 1 the neutral element for � must be the zero vector0. Furthermore, since a and −a commute with respect to the usual multi-plication in AsX we have exp(a) · exp(−a) = exp(−a) · exp(a) = exp(0) = 1.Hence a � (−a) = (−a) � a = log(1) = 0.

Definition. For the field K = Q and the two-element set {Y,Z} we call

H(Y,Z) := Y � Z ∈ L{Y,Z} ⊆ As{Y,Z}

the Hausdorff series (in Y and Z).

As alluded to earlier we should view H(Y,Z) as a noncommutative formalpower series in the variables Y,Z with coefficients in the field Q. We have

exp(Y ) · exp(Z) = 1 + W with W =∑

r+s≥1

Y r

r!· Zs

s!

and hence

116 III Lie Groups

H(Y,Z) =∑

m≥1

(−1)m+1 Wm

m

=∑

m≥1

(−1)m+1

m

(∑

r+s≥1

Y r

r!· Zs

s!

)m

=∑

n≥1

∑

r+s=n

n∑

m=1

(−1)m+1

m

∑

r1+···+rm=rs1+···+sm=s

r1+s1≥1,...,rm+sm≥1

m∏

i=1

Y ri

ri!· Zsi

si!.

Here and in the following the product sign∏m

i=1 always has to be under-stood in such a way that the corresponding multiplications are carried outin the order of the enumeration i = 1, . . . ,m. It is convenient to use theabbreviations

Hr,s :=r+s∑

m=1

(−1)m+1

m

∑

r1+···+rm=rs1+···+sm=s

r1+s1≥1,...,rm+sm≥1

m∏

i=1

Y ri

ri!· Zsi

si!

andHn :=

∑

r+s=n

Hr,s.

We note that Hr,s is a sum of noncommutative monomials of degree r in Yand s in Z. As a sum of noncommutative monomials of total degree n theelement Hn lies in As

(n){Y,Z}. We have

H =∑

n≥1

Hn or, more formally, H = (Hn)n.

From the theory we know that Hn ∈ L(n){Y,Z} for each n ≥ 1 but this is not

visible from the above explicit formula.

Examples. 1) H1,0 = Y , H0,1 = Z, and H1 = Y + Z.

2) Hr,0 = H0,r = 0 for any r ≥ 2 (observe, for example, that Hr,0 is theterm of degree r in log(exp(Y )) = Y ).

3) H2 = H2,0 + H1,1 + H0,2 = H1,1 = Y Z − 12(Y Z + ZY ) = 1

2 [Y,Z].

If g is any Lie algebra over (any) K then the K-linear map

ad(z) : g −→ g

x �−→ [z, x],


for any z ∈ g, is a derivation in the sense that

ad(z)([x, y]) = [ad(z)(x), y] + [x, ad(z)(y)] for any x, y ∈ g

holds true. This is just a reformulation of the Jacobi identity in g.

Proposition 16.7. (Dynkin’s formula) For r + s ≥ 1 we have

Hr,s =1

r + s(H ′

r,s + H ′′r,s)

with H ′r,s defined as

∑

m≥1

(−1)m−1

m

∑

r1+···+rm=rs1+···+sm−1=s−1

r1+s1≥1,...,rm−1+sm−1≥1

((m−1∏

i=1

ad(Y )ri

ri!◦ad(Z)si

si!

)

◦ad(Y )rm

rm!

)

(Z)

and

H ′′r,s :=

∑

m≥1

(−1)m−1

m

∑

r1+···+rm−1=r−1s1+···+sm−1=s

r1+s1≥1,...,rm−1+sm−1≥1

(m−1∏

i=1

ad(Y )ri

ri!◦ ad(Z)si

si!

)

(Y ).

Proof. Compare [B-LL] Chap. II §6.4. We note that the above defining sumsare finite and make it visible that Hn lies in L

(n){Y,Z}.

Remark 16.8. Suppose that K has characteristic zero; then we have

a � b = H(a, b) for any a, b ∈ LX .

Proof. The above explicit computations including Dynkin’s formula werecompletely formal and therefore are valid for any a, b (instead of Y,Z). Theexpression H(a, b), of course, has to be calculated componentwise in AsX

using the observation before Prop. 16.2.

The exploitation of these “universal” considerations is based upon thefollowing technique. For any finite dimensional K-vector space V let

Map(V × V ;V ) := K-vector space of all maps f : V × V −→ V.

We pick a K-basis e1, . . . , ed of V .

118 III Lie Groups

Definition. A map f : V ×V −→ V is called polynomial (of degree (r, s)) ifthere are (homogeneous) polynomials Pi(X1, . . . , Xd, Y1, . . . , Yd) over K (ofdegree r in X1, . . . , Xd and degree s in Y1, . . . , Yd), for 1 ≤ i ≤ d, such that

f

(d∑

i=1

aiei,

d∑

i=1

biei

)

=d∑

i=1

Pi(a1, . . . , ad, b1, . . . , bd)ei for any ai, bi ∈ K.

In Map (V × V ;V ) we have the vector subspace Pol(V × V ;V ) of allpolynomial maps. It decomposes into

Pol(V × V ;V ) = ⊕n≥0 Poln(V × V ;V ) = ⊕n≥0 ⊕r+s=n Polr,s(V × V ;V )

where Polr,s(V ×V ;V ) denotes the subspace of all polynomial maps of degree(r, s) and

Poln(V × V ;V ) := ⊕r+s=n Polr,s(V × V ;V )

is the subspace of all polynomial maps of total degree n.

Lemma 16.9. Given any f ∈ Polr,s(V × V ;V ) and gi ∈ Polli,mi(V × V ;V )

for i = 1, 2 the map

(v, w) �−→ f(g1(v, w), g2(v, w))

lies in Polrl1+sl2,rm1+sm2(V × V ;V ).

Proof. Straightforward.

Corollary 16.10. The property of a map f : V × V −→ V of being poly-nomial (of a certain degree) does not depend on the choice of the K-basisof V .

Proof. View the change of bases as a polynomial map and apply Lemma 16.9.

Suppose now that the vector space V is a Lie algebra g of finite dimensiond := dimK g. Then also the vector space Map(g× g; g) is a Lie algebra withrespect to the Lie product

[f, g](x, y) := [f(x, y), g(x, y)].

Corollary 16.11. Pol(g × g; g) is a Lie subalgebra of Map(g × g; g); moreprecisely, we have

[Polr,s(g × g; g),Polr′,s′(g × g; g)] ⊆ Polr+r′,s+s′(g × g; g).


Proof. Apply Lemma 16.9 to the map f := [, ] lying in Pol1,1(g× g; g).

We identify the two-element set {Y,Z} with the subset of Pol(g × g; g)consisting of the two projection maps pri : g × g −→ g by sending Y to pr1and Z to pr2. By the universal property of free Lie algebras this extendsuniquely to a homomorphism of graded Lie algebras

θ : L{Y,Z} −→ Pol(g × g; g).

It satisfies

(19) θ([Y, a])(y, z) = [y, θ(a)(y, z)] and θ([Z, a])(y, z) = [z, θ(a)(y, z)]

for any a ∈ L{Y,Z}.We define

Pow(g × g; g) :=∏

n≥0

Poln(g × g; g)

as a K-vector space. The elements of Pow(g × g; g) can be viewed (if K isinfinite, and after the choice of a K-basis of g) as d-tuples of usual formalpower series in the variables Y1, . . . , Yd, Z1, . . . , Zd with coefficients in K. Asa consequence of Cor. 16.11 the Lie product on Pol(g × g; g) extends by

[(fn)n, (gn)n] :=

(∑

l+m=n

[fl, gm]

)

n

to a Lie product on Pow(g × g; g). Being graded θ extends to the K-linearmap

θ : L{Y,Z} −→ Pow(g × g; g)

(fn)n �−→ (θ(fn))n.

Using the trick in the proof of Lemma 16.3 we obtain for any m ∈ N, withthe notations in this proof, that

θ([a, b]) ≡ θ([a(m), b(m)]) ≡ [θ(a(m)), θ(b(m))]

≡ [θ(a), θ(b)] mod∏

n>m

Poln(g × g; g)

for any a, b ∈ L{Y,Z}. Since m is arbitrary this means that θ also is a homo-morphism of Lie algebras.

From now on we assume for the rest of this section that the field K hascharacteristic zero. We put

120 III Lie Groups

H := Hg := θ(H) ∈ Pow(g × g; g).

More precisely, we have

H =∑

r+s≥1

Hr,s with Hr,s := θ(Hr,s) ∈ Polr,s(g × g; g).

Using (19) Dynkin’s formula in Prop. 16.7 implies that

Hr,s =1

r + s(H ′

r,s + H ′′r,s)

with

H ′r,s : g × g −→ g

(y, z) �−→∑

m≥1

(−1)m−1

m

∑

···

((m−1∏

i=1

ad(y)ri

ri!◦ ad(z)si

si!

)

◦ ad(y)rm

rm!

)

(z)

and

H ′′r,s : g × g −→ g

(y, z) �−→∑

m≥1

(−1)m−1

m

∑

···

(m−1∏

i=1

ad(y)ri

ri!◦ ad(z)si

si!

)

(y).

Examples 16.12. H1,0 = pr1, H0,1 = pr2, and

H1,1 : g × g −→ g

(y, z) �−→ 12[y, z].

What else do we know about H? First of all we recall from Cor. 16.6and Remark 16.8 that we have

(20)H(a,H(b, c)) = H(H(a, b), c),

H(a, 0) = H(0, a) = a, and H(a,−a) = 0

for any a, b, c ∈ L{Y,Z}. In order to use this we reinterpret the evaluation ofH in a and b in the following way.

Let a, b ∈ L{Y,Z} be any two elements. By the universal property of freeLie algebras there is a unique homomorphism of Lie algebras

εa,b : L{Y,Z} −→ L{Y,Z}


mapping Y to a and Z to b. By construction it satisfies

εa,b(L(m){Y,Z}) ⊆ {0} × · · · × {0} ×

∏

n≥m

L(n){Y,Z}

for any m ≥ 1 and therefore extends, by the observation before Prop. 16.2,to the K-linear map

εa,b : L{Y,Z} −→ L{Y,Z}

(cn)n �−→∑

n≥1

εa,b(cn).

The same reasoning as for θ shows that εa,b in fact is a homomorphismof Lie algebras. On the other hand of course, εa,b is the restriction of acorresponding unique homomorphism of associative K-algebras with unit

εa,b : As{Y,Z} −→ As{Y,Z}.

Viewing an element in As{Y,Z} as a noncommutative polynomial G(Y,Z) itis clear that

εa,b(G) = G(a, b)

holds true. It follows that

εa,b(H) =∑

n≥1

εa,b(Hn) =∑

n≥1

Hn(a, b) = H(a, b).

There is an analogous construction for the Lie algebra Pow(g × g; g).Quite generally, given any g1, g2 ∈ Map(V × V ;V ) there is the homomor-phism (of Lie algebras in case V = g)

Map(V × V ;V ) −→ Map(V × V ;V )f �−→ f(g1, g2)(v, w) := f(g1(v, w), g2(v, w)).

If g1, g2 ∈ Pol(V × V ;V ) then Lemma 16.9 says that it restricts to

Pol(V × V ;V ) −→ Pol(V × V ;V )

and satisfies

f(g1, g2) ∈ Polrn1+sn2(V × V ;V )if f ∈ Polr,s(V × V ;V ) and gi ∈ Polni(V × V ;V ).

122 III Lie Groups

Hence for g1, g2 in

Pow0(g × g; g) := {0} ×∏

n≥1

Poln(g × g; g)

we obtain, by the usual componentwise procedure, a homomorphism of Liealgebras

Pow(g × g; g) −→ Pow(g × g; g)f −→ f(g1, g2).

Indeed, this is just a reformulation of the fact that a formal power serieswithout constant term can be inserted into any formal power series. Wenote that

pri(g1, g2) = gi.

As before let now a, b ∈ L{Y,Z} be any two elements. Then θ(a), θ(b) liein Pow0(g × g; g), and we have the commutative diagram

(21)

L{Y,Z}εa,b

θY,Z �→pr1,pr2Y,Z �→θ(a),θ(b)

L{Y,Z}

θ

Pow(g × g; g)f �→f(θ(a),θ(b))

Pow(g × g; g).

Lemma 16.13. Suppose that K has characteristic zero; for any y ∈ g andr + s ≥ 1 we have:

i. Hr,s(y,−y) = 0;

ii. Hr,s(y, 0) =

{y if r = 1, s = 0,0 otherwise;

iii. Hr,s(0, y) =

{y if r = 0, s = 1,0 otherwise.

Proof. We apply the commutative diagram (21) to the Hausdorff series H ∈L{Y,Z} and various choices for the elements a and b. For a := Y and b := −Ywe have θ(a) = pr1 and θ(b) = −pr1 and we obtain from (21) that

H H(Y,−Y )

H H(pr1,−pr1).


Since H(Y,−Y ) = 0 by (20) the assertion i. follows. For a := Y and b := 0we similarly obtain

H H(Y, 0)

H H(pr1, 0).

Again by (20) we have H(Y, 0) = Y and hence H(pr1, 0) = pr1 which isthe assertion ii. The last assertion iii. comes symmetrically from the choicea := 0 and b := Y .

The discussion leading to the commutative diagram (21) can easily begeneralized to the three-element set {U, Y, Z} and the Lie algebra

Pow(g × g × g; g)

of d-tuples of formal power series over K in 3d variables. We leave the detailsto the reader. This leads to the homomorphism of Lie algebras

θ : L{U,Y,Z} −→ Pow(g × g × g; g)

which sends U, Y, and Z to pr1,pr2, and pr3, respectively. For any choiceof elements a, b ∈ L{U,Y,Z} we obtain, analogously to (21), the commutativediagram

(22)

L{Y,Z}Y,Z �→a,b

θ

L{U,Y,Z}

θ

Pow(g × g; g)f �→f(θ(a),θ(b))

Pow(g × g × g; g).

Lemma 16.14. Suppose that K has characteristic zero; we then have

H(pr1, H(pr2,pr3)) = H(H(pr1,pr2),pr3).

Proof. Apply (22) to the Hausdorff series H ∈ L{Y,Z} and the two choicesa := U, b := H(Y,Z) and a := H(U, Y ), b := Z, respectively, and use thefirst part of (20).

124 III Lie Groups

17 The Convergence of the Hausdorff Series

We fix a Lie algebra g of finite dimension d over a field K of characteristiczero. We also pick a K-basis e1, . . . , ed of g.

Definition. The elements γkij ∈ K, for 1 ≤ i, j, k ≤ d, defined by the

equations

[ei, ej ] =d∑

k=1

γkijek

are called the structure constants of g with respect to the basis {ei}1≤i≤d.

If we define the Lie product [ , ]′ on Kd by

(23) [(v1, . . . , vd), (w1, . . . , wd)]′ :=

(∑

i,j

γ1ijviwj , . . . ,

∑

i,j

γdijviwj

)

then the isomorphism g ∼= Kd becomes an isomorphism of Lie algebras.Using this same isomorphism we also may view the element

H = Hg ∈ Pow0(g × g; g)

as a d-tuple

H(Y ,Z) := Hg(Y ,Z) = (H(1)(Y ,Z), . . . , H(d)(Y ,Z))

of formal power series H(i)(Y ,Z) over K in the variables Y = (Y1, . . . , Yd)and Z = (Z1, . . . , Zd). The Examples 16.12 tell us that

H(i)(Y ,Z) = Yi + Zi +12

∑

j,k

γijkYjZk + · · · .

Lemma 17.1. i. H(Y , 0) = Y , H(0, Z) = Z.

ii. H(Y ,−Y ) = 0.

iii. H(U,H(Y ,Z)) = H(H(U, Y ), Z).

Proof. This is a restatement of Lemma 16.13 and Lemma 16.14.

From now on let (K, | |) be a nonarchimedean field of characteristic zero.Via the linear isomorphism g ∼= Kd we may view g as a manifold over K(but which structure does not depend on the choice of the basis).


Let us suppose at this point that there is an ε > 0 such that

(24) H(Y ,Z) ∈ Fε(Kd × Kd;Kd) and ‖H‖ε ≤ ε

(where Kd is equipped with the usual maximum norm). We then considerthe open submanifold

Gε := Bε(0) ⊆ Kd ∼= g.

Obviously

Gε × Gε −→ Gε

(g, h) �−→ gh := H(g, h)

is a well defined locally analytic map. Prop. 5.4 and Lemma 17.1 togetherimply that we have

g10 = 0g1 = g1, g1(−g1) = 0, and g1(g2g3) = (g1g2)g3

for any g1, g2, g3 ∈ Gε. This proves the first half of the following fact provided(24) holds true.

Proposition 17.2. Gε is a d-dimensional Lie group over K whose neutralelement is the zero vector 0 and such that −g is the inverse of g ∈ Gε;moreover, if g, h ∈ Gε satisfy [g, h] = 0 then gh = g + h.

Proof. The second half of the assertion is immediate from the fact that,by Dynkin’s formula, all maps Hr,s for (r, s) �= (1, 0), (0, 1) are iteratedcommutators and therefore vanish on (g, h).

If two ε ≥ ε′ > 0 satisfy (24) then Gε′ of course is an open subgroup Gε.

Definition. {Gε}ε is called the Campbell-Hausdorff Lie group germ of theLie algebra g.

What is the Lie algebra of Gε (still assuming (24))? We have the “global”chart c := (Gε,⊆,Kd) for the manifold Gε and correspondingly the locallyanalytic isomorphism

τc : Gε × Kd �−−→ T (Gε)(g, v) �−→ [c, v] ∈ Tg(Gε)

126 III Lie Groups

as well as the linear isomorphism

Can(Gε,Kd)

∼=−−→ Γ(Gε, T (Gε))f �−→ ξf (g) = τc(g, f(g)) = [c, f(g)] ∈ Tg(Gε).

From Remark 9.21.i. we know that the Lie product of vector fields corre-sponds on the left hand side to the Lie product

[f1, f2](g) = Dgf1(f2(g)) − Dgf2(f1(g)).

On the other hand the Lie product on Lie(Gε) = T0(Gε) is induced via theinclusion

T0(Gε) −→ Γ(Gε, T (Gε))t �−→ ξt(g) = T0(rg)(t)

by the Lie product of vector fields. By the construction of the tangent mapT0(rg) in Sect. 9 we have the commutative diagram

Kd D0rg

v �→[c,v] ∼=

Kd

v �→[c,v]∼=

T0(Gε)T0(rg)

Tg(Gε).

We define fv(g) := D0(rg)(v) and compute

ξ[c,v](g) = T0(rg)([c, v]) = [c,D0rg(v)] = ξfv(g)

which means that the diagram

Kd v �→fv

v �→[c,v] ∼=

Can(Gε,Kd)

f �→ξf∼=

T0(Gε)t�→ξt

Γ(Gε, T (Gε))

is commutative. Let [ , ]′′ denote the Lie product on Kd which under theleft perpendicular arrow corresponds to the Lie product of the Lie algebraLie(Gε) = T0(Gε). The commutativity of the diagram then says that wehave

f[v,w]′′ = [fv, fw] for any v, w ∈ Kd.


Observing thatfv(0) = D0 idGε(v) = v

we deduce that

[v, w]′′ = [fv, fw](0) = D0fv(fw(0)) − D0fw(fv(0))= D0fv(w) − D0fw(v)

(25)

for any v, w ∈ Kd.We summarize that we have identified both Lie algebras, g and Lie(Gε),

with Kd thereby obtaining the two Lie products [ , ]′ and [ , ]′′ on Kd.

Proposition 17.3. Lie(Gε) ∼= g as Lie algebras.

Proof. By the above discussion it suffice to show that

[ , ]′ = [ , ]′′

holds true. To further compute the Lie product [ , ]′′ we start from theidentity

rg(h) = H(h, g).

Since, by Lemma 17.1.i., H does not contain monomials of degree (0, s) in(Y ,Z) with s ≥ 2 we may write

H(i)(Y ,Z) = Zi +d∑

j=1

P(i,j)(Z)Yj + terms of degree ≥ 2 in Y .

Using Prop. 5.6 we deduce that

D0rg =(

∂H(i)(Y , g)∂Yj

∣∣Y =0

)

i,j

=(P(i,j)(g)

)i,j

and hence that

fv(g) = D0(rg)(v) =

(d∑

j=1

vjP(1,j)(g), . . . ,d∑

j=1

vjP(d,j)(g)

)

for any v = (v1, . . . , vd) ∈ Kd. To derive the function fv in 0 we must derivethe P(i,j)(Z) in Z and subsequently set Z = 0. By the Examples 16.12 wemay write

P(i,j)(Z) = δij +12

d∑

k=1

γijkZk + terms of degree ≥ 2 in Z

128 III Lie Groups

where δij denotes the Kronecker symbol. It follows that

∂P(i,j)(Z)∂Zk

∣∣Z=0

=12γi

jk

and hence that

D0fv =

(12

d∑

j=1

γijkvj

)

i,k

and

D0fv(w) =

(12

d∑

j,k=1

γ1jkvjwk, . . . ,

12

d∑

j,k=1

γdjkvjwj

)

=12[v, w]′

for any v = (v1, . . . , vd), w = (w1, . . . , wd) ∈ Kd. We conclude that

[v, w]′′ = D0fv(w) − D0fw(v) =12[v, w]′ − 1

2[w, v]′ = [v, w]′.

Having seen the interesting consequences of a possible convergence ofthe Hausdorff series we now must address the main question of this sectionwhether any ε satisfying (24) exists.

Using the isomorphism g ∼= Kd any element f ∈ Pol(g × g; g) can beviewed as a d-tuple f of polynomials in the variables Y and Z and hence,in particular, as an element f ∈ Fε(Kd × Kd;Kd) for any ε > 0. Since thepolynomials in Hr,s := Hr,s are homogeneous of total degree r + s we have

‖Hr,s‖ε = ‖Hr,s‖1εr+s.

Suppose that there is a 0 < ε0 ≤ 1 such that

(26) ‖Hr,s‖1 ≤ ε−(r+s−1)0 for any r + s ≥ 1.

It follows that for any 0 < ε < ε0 we have

‖Hr,s‖ε = ‖Hr,s‖1εr+s ≤ ‖Hr,s‖1ε

r+s−10 ε ≤ ε for any r + s ≥ 1

and


limr+s→∞

‖Hr,s‖ε ≤ ε · limr+s→∞

‖Hr,s‖1εr+s−1

= ε · limr+s→∞

‖Hr,s‖1εr+s−10

(ε

ε0

)r+s−1

≤ ε · limr+s→∞

(ε

ε0

)r+s−1

= 0.

AsH =

∑

r+s≥1

Hr,s

we conclude that (26) implies (24) for any 0 < ε < ε0.The coefficients of the Hausdorff series H are explicitly known and their

absolute values therefore can easily be estimated. But in order to translatethis knowledge into an estimate for the norms ‖Hr,s‖1 we need a particularlywell behaved basis of the K-vector space L{Y,Z}.

The free K-algebra A{Y,Z} by construction has the K-basis M{Y,Z} =⋃n≥1{Y,Z}(n). For any x ∈ M{Y,Z} we let ex denote its image in the factor

algebra L{Y,Z}. These ex obviously generate L{Y,Z} as a K-vector space.Hence there exist subsets B ⊆ M{Y,Z} such that {ex}x∈B is a K-basis ofL{Y,Z}. In the following we have to make a particularly clever choice ofsuch a subset B. But first we note that also the free associative K-algebrawith unit As{Y,Z} has an obvious K-basis which is the set Mon{Y,Z} ofall noncommutative monomials in Y and Z. All of this is valid over anarbitrary field K. Since our K is nonarchimedean we may introduce theoK-submodules

As≤1{Y,Z} :=

∑

μ∈Mon{Y,Z}

oKμ

of As{Y,Z} andL≤1{Y,Z} := L{Y,Z} ∩ As≤1

{Y,Z}

of L{Y,Z}.

Proposition 17.4. i. (K arbitrary) There is a subset B ⊆ M{Y,Z} suchthat we have

– {ex}x∈B is a K-basis of L{Y,Z},

– {Y,Z} ⊆ B, and

– for any x ∈ B \ {Y,Z} there are x′, x′′ ∈ B with x = x′x′′ and, inparticular, ex = [ex′ , ex′′ ].

130 III Lie Groups

ii. (K nonarchimedean) There is a subset B ⊆ M{Y,Z} as in i. and suchthat

L≤1{Y,Z} =

∑

x∈B

oKex.

Proof. Apply [B-LL] Chap. II §2.10 Prop. 11, §2.11 Thm. 1, and §3.1 Re-mark 1) over K and oK , respectively.

We now define the constant ε0 by

ε0 :=

{|p|

1p−1 ε−1

1 if K is p-adic for some p,

ε−11 otherwise

whereε1 := max(1,max

i,j,k|γk

ij |).

We note that 0 < ε0 ≤ 1. The constant ε1 has the property that

‖[f, g]‖1 ≤ ε1‖f‖1‖g‖1 for any f, g ∈ Pol(g × g; g).

Lemma 17.5. Let {ex}x∈B be any K-basis of L{Y,Z} as in Prop. 17.4.i.; wethen have

‖θ(ex)‖1 ≤ εn−11 for any x ∈ B(n) := B ∩ {Y,Z}(n).

Proof. We proceed by induction with respect to n. For x = Y we haveθ(eY ) = pr1 and hence θ(eY ) = (Y1, . . . , Yd) so that ‖θ(eY )‖1 = 1 = ε0

1. Thecase x = Z is analogous. Any x ∈ B(n) with n ≥ 2 can be written as

x = x′x′′ with x′ ∈ B(l), x′′ ∈ B(m), and l + m = n.

Since l,m < n we may apply the induction hypothesis to x′ and x′′ andobtain

‖θ(ex)‖1 = ‖θ([ex′ , ex′′ ])‖1 = ‖[θ(ex′), θ(ex′′)]‖1

≤ ε1‖θ(ex′)‖1‖θ(ex′′)‖1

≤ ε1εl−11 εm−1

1 = εn−11 .

Proposition 17.6. For any 0 < ε < ε0 we have H ∈ Fε(Kd ×Kd;Kd) and‖H‖ε ≤ ε.


Proof. As discussed after (26) it suffices to show that

‖Hr,s‖1 ≤ ε−(r+s−1)0 for any r + s ≥ 1.

We fix n := r + s ≥ 1. We also pick a basis {ex}x∈B as in Prop. 17.4.i. andwrite

Hr,s =∑

x∈B

cxex.

Since Hr,s ∈ L(n){Y,Z} we in fact have

Hr,s =∑

x∈B(n)

cxex where B(n) = B ∩ {Y,Z}(n).

Lemma 17.5 then implies that

‖Hr,s‖1 ≤ maxx∈B(n)

|cx|‖θ(ex)‖1 ≤ εn−11 max

x∈B(n)|cx|.

In order to estimate the |cx| we have to distinguish cases. But we emphasizethat this is a question solely about the Hausdorff series (and not the Liealgebra g) and therefore, in principle, can be treated over the field Q.

Case 1: K is not p-adic for any p. Since Q ⊆ K we may choose the basisalready over the field Q. Then all coefficients cx lie in Q. By Exercise 2.1.i.we have |cx| = 0 or 1 and hence ‖Hr,s‖1 ≤ εn−1

1 = ε−(n−1)0 .

Case 2: K is p-adic for some p. By Exercise 2.1.ii. we have Qp ⊆ K and

|a| = |a|− log |p|

log pp for any a ∈ Qp.

Hence we may assume without loss of generality that (K, | |) = (Qp, | |p),and we choose B as in Prop. 17.4.ii. We want to show that

maxx∈B(n)

|cx|p ≤ pn−1p−1 .

Since the left hand side is an integral power of p this amounts to showingthat

plcx ∈ Zp := oQp for any x ∈ B(n)

where l is the unique integer such that

l ≤ n − 1p − 1

< l + 1.

132 III Lie Groups

By our particular choice of the set B this is equivalent to

plHr,s ∈ L≤1{Y,Z} and hence to plHr,s ∈ As≤1

{Y,Z}.

The explicit form of the coefficients of Hr,s then reduces us to showing that∣∣∣∣∣

pl

m

m∏

i=1

1ri!si!

∣∣∣∣∣p

≤ 1, or equivalently,

∣∣∣∣∣m

m∏

i=1

ri!si!

∣∣∣∣∣p

≥ p−l

whenever 1 ≤ m ≤ n, r1 + · · · + rm = r, s1 + · · · + sm = s, and ri + si ≥ 1.But Lemma 2.2 implies

∣∣∣∣∣m

m∏

i=1

ri!si!

∣∣∣∣∣p

≥ p−1

p−1((m−1)+(r1+s1−1)+···+(rm+sm−1))

= p−n−1p−1 .

Since the left hand side is an integral power of p it indeed must be ≥ p−l. (Foran alternative argument compare the proof of the later Lemma 32.4.)

18 Formal Group Laws

Let K be any field of characteristic zero. We fix a natural number d, and letR := K[[Y1, . . . , Yd, Z1, . . . , Zd]] denote the ring of formal power series overK in the variables Y = (Y1, . . . , Yd) and Z = (Z1, . . . , Zd).

Definition. A formal group law (of dimension d over K) is a d-tuple F =(F1, . . . , Fd) of power series Fi ∈ R such that we have:

(i) F (Y , 0) = Y and F (0, Z) = Z,

(ii) F (U,F (Y ,Z)) = F (F (U, Y ), Z).

We observe that the condition (i) implies that

(27) Fi(Y ,Z) = Yi + Zi + terms of degree ≥ 1 both in Y and Z.

Hence the two sides in the condition (ii) are well defined.

Examples. 1) Fi(Y ,Z) = Yi + Zi.

2) F (Y,Z) = Y + Z + Y Z (for d = 1).


3) (Lemma 17.1) F := Hg for a finite dimensional Lie algebra g over K(and some choice of K-basis of g).

The last example has a converse. Let F be any formal group law. Wehave

Fi(Y ,Z) = Yi + Zi +∑

j,k

cijkYjZk + terms of degree ≥ 3.

We define a bilinear map bF : Kd × Kd −→ Kd by

bF ((v1, . . . , vd), (w1, . . . , wd)) :=

(∑

j,k

c1jkvjwk, . . . ,

∑

j,k

cdjkvjwk

)

,

and we put

[v, w]F := bF (v, w) − bF (w, v) for v, w ∈ Kd.

Lemma 18.1. [ , ]F satisfies the Jacobi identity.

Proof. Compare [Haz] §14.1 or [Se2] Part II, Chap. IV §7 formula 6).

We see that [ , ]F is a Lie product on Kd. In the case of the formal grouplaw Hg it follows from the Examples 16.12 that [ , ]Hg coincides (up to theisomorphism g ∼= Kd) with the Lie product on g.

Next we discuss a seemingly very different construction of a formal grouplaw from a finite dimensional Lie algebra g over K by using the universalenveloping algebra U(g). We have the following list of K-linear maps:

• (multiplication) m = mg : U(g) ⊗K U(g) −→ U(g),

• (unit) e = eg : K −→ U(g) sending a to a · 1,

• (comultiplication) Δ = Δg : U(g)U(Δ)−−−−→ U(g × g) ∼= U(g) ⊗K U(g),

• (counit) c = cg : U(g) = T (g)/J(g)pr−−→ g⊗0 = K.

Of course, the maps m and e satisfy the axioms for a (noncommutative) asso-ciative K-algebra with unit, and Δ and c are homomorphisms of K-algebraswith unit. In addition, the maps Δ and c have the following properties:

(28) (counit property) (c ⊗ id) ◦ Δ = id = (id⊗ c) ◦ Δ;

(29) (coassociativity) (id⊗Δ) ◦ Δ = (Δ ⊗ id) ◦ Δ;

134 III Lie Groups

(30) (cocommutativity) the diagram

U(g)ΔΔ

U(g) ⊗K U(g)x⊗y �→y⊗x

U(g) ⊗K U(g)

is commutative.

They easily follow, by applying the universal property of U(g), from thecorresponding properties of the diagonal map Δ : g −→ g × g. We nowconsider the K-linear dual

U(g)∗ := HomK(U(g),K)

together with the K-linear map

μ : U(g)∗ ⊗K U(g)∗ can−−−→ [U(g) ⊗K U(g)]∗ Δ∗−−−→ U(g)∗.

l1 ⊗ l2 �−→ [x ⊗ y �→ l1(x) · l2(y)]

Proposition 18.2. (U(g)∗, μ, c) is a commutative and associative K-algebrawith unit.

Proof. The required axioms are exactly dual to the properties (28)–(30).

In order to determine the algebra U(g)∗ explicitly we pick an (ordered)K-basis e1, . . . , ed of g. We know from Cor. 14.3 of the Poincare-Birkhoff-Witt theorem that the

eα :=eα11

α1!· . . . ·

eαdd

αd!for α = (α1, . . . , αd) ∈ Nd

0

form a K-basis of U(g).


U(g)∗∼=−−→ K[[U1, . . . , Ud]]

� �−→ F (U) :=∑

α∈Nd0

�(eα)Uα

is an isomorphism of K-algebras with unit onto the ring of formal powerseries over K in the variables U = {U1, . . . , Ud}.


Proof. The fact that {eα}α is a K-basis of U(g) immediately implies thatthe asserted map is a K-linear isomorphism. The unit element c of U(g)∗

is the projection map onto Ke0∼= K which is mapped to Fc = 1. For the

multiplicativity we first recall that

Δ(emk ) = Δ(ek)m = (ek ⊗ 1 + 1 ⊗ ek)m

=m∑

i=0

(m

i

)

(eik ⊗ 1)(1 ⊗ em−i

k )

=m∑

i=0

(m

i

)

eik ⊗ em−i

k

for any 1 ≤ k ≤ d and any m ≥ 0. By induction one deduces that

(31) Δ(eα) =∑

β+γ=α

eβ ⊗ eγ for any α ∈ Nd0

holds true. We now compute

Fμ( 1, 2)(U) =∑

α

μ(�1, �2)(eα)Uα =∑

α

(�1 ⊗ �2)(Δ(eα))Uα

=∑

α

∑

β+γ=α

�1(eβ)�2(eγ)Uβ+γ

=

(∑

β

�1(eβ)Uβ

)(∑

γ

�2(eγ)Uγ

)

= F 1(U)F 2(U).

By dualizing the multiplication map

U(g × g) ∼= U(g) ⊗K U(g) m−−→ U(g)

we obtain a K-linear map

U(g)∗ m∗−−−→ U(g × g)∗.

Applying Prop. 18.3 to both sides (with (e1, 0), . . . , (ed, 0), (0, e1), . . . , (0, ed)as an ordered K-basis for g × g) we may view the latter as a K-linear map

K[[U1, . . . , Ud]]m∗

−−−→ K[[Y1, . . . , Yd, Z1, . . . , Zd]] = R.

136 III Lie Groups

We define

F(i) := m∗(Ui) ∈ R and F g := (F(1), . . . , F(d)).

At this point we have to recall a few basic facts about formal powerseries rings. First of all, the formal power series ring K[[U1, . . . , Ur]] has aunique maximal ideal mU which is the ideal generated by U1, . . . , Ur. This isan immediate consequence of the fact that any formal power series F overK with F (0) �= 0 is invertible.

Definition. i. A commutative ring with unit is called local if it has aunique maximal ideal.

ii. A homomorphism of local rings is called local if it maps the maximalideal into the maximal ideal.

Let K[[U1, . . . , Ur]] and K[[V1, . . . , Vs]] be two formal power series rings.For any F = (F1, . . . , Fr) ∈ mV × · · · × mV the map

εF : K[[U1, . . . , Ur]] −→ K[[V1, . . . , Vs]]G �−→ G(F ) := G(F1, . . . , Fr)

is a well defined local homomorphism of local rings. We have εF (Ui) = Fi.

Lemma 18.4. Let ε : K[[U1, . . . , Ur]] −→ K[[V1, . . . , Vs]] be any homomor-phism of K-algebras with unit which is local ; we then have

ε = εF with Fi := ε(Ui).

Proof. Since ε is local we have Fi ∈ mV so that εF is well defined. Both εand εF are homomorphisms of K-algebras with unit. Hence the identitiesε(Ui) = εF (Ui) imply that

ε(G) = εF (G) for any polynomial G ∈ K[U1, . . . , Ur].

We now write an arbitrary formal power series G ∈ K[[U1, . . . , Ur]] as

G =∑

n≥0

Gn

where Gn is a homogeneous polynomial of degree n. In particular, Gn liesin mn

U . Since ε is local we obtain ε(Gn) ∈ mnV . Therefore the element

∑

n≥0

ε(Gn) ∈ K[[V1, . . . , Vs]]


is well defined. We have

ε(G) −∑

n≥0

ε(Gn) = ε(G0) + · · · + ε(Gk) + ε

(∑

n>k

Gn

)

−∑

n≥0

ε(Gn)

= ε

(∑

n>k

Gn

)

−∑

n>k

ε(Gn)

∈ mk+1V

for any k ≥ 0. Since⋂

k≥0 mk+1V = {0} we conclude that

ε(G) =∑

n≥0

ε(Gn).

The same reasoning, of course, applies to εF . Hence

ε(G) =∑

n≥0

ε(Gn) =∑

n≥0

εF (Gn) = εF (G).

Proposition 18.5. i. m∗ is a local homomorphism.

ii. m∗ = εF g.

iii. F g is a formal group law.

iv. [ , ]F g coincides (up to the isomorphism g ∼= Kd given by the basise1, . . . , ed) with the Lie product on g.

Proof. i. The multiplicativity of m∗ amounts to the commutativity of theouter square in the diagram

U(g)∗ ⊗K U(g)∗ m∗⊗m∗

can

U(g × g)∗ ⊗K U(g × g)∗

can

[U(g) ⊗K U(g)]∗(m⊗m)∗

Δ∗g

[U(g × g) ⊗K U(g × g)]∗

Δ∗g×g

U(g)∗ m∗U(g × g)∗.

The upper square is commutative for trivial reasons. The lower square is thedual of the diagram

138 III Lie Groups

U(g × g)Δg×g

∼=

U(g × g) ⊗K U(g × g)

∼=⊗∼=

U(g) ⊗K U(g)

m

[U(g) ⊗K U(g)] ⊗K [U(g) ⊗K U(g)]

m⊗m

U(g)Δg

U(g) ⊗K U(g).

We check its commutativity on the K-basis

eα,β :=(e1, 0)α1

α1!· . . . · (ed, 0)αd

αd!· (0, e1)β1

β1!· . . . · (0, ed)βd

βd!,

for α = (α1, . . . , αd), β = (β1, . . . , βd) ∈ Nd0, of U(g × g). Under the algebra

isomorphism U(g×g) ∼= U(g)⊗K U(g) an element (x, y) ∈ g×g correspondsto x⊗1+1⊗y. It follows that eα,β corresponds to eα⊗eβ . Using the formula(31) we obtain

eα,βΔg×g ∑

(γ,λ)+(δ,μ)=(α,β)

eγ,λ ⊗ eδ,μ

eα ⊗ eβ

m

∑

γ+δ=αλ+μ=β

(eγ ⊗ eλ) ⊗ (eδ ⊗ eμ)

m⊗m

eαeβΔg ( ∑

γ+δ=α

eγ ⊗ eδ

)( ∑

λ+μ=β

eλ ⊗ eμ

)=

∑

γ+δ=αλ+μ=β

eγeλ ⊗ eδeμ.

This establishes the multiplicativity of m∗. The unit element in U(g)∗, resp.in U(g×g)∗, is the linear form cg, resp. cg×g, which is the projection map ontoKe0

∼= K, resp. onto Ke0,0∼= K. Since cg is a homomorphism of algebras

we have

m∗(cg)(eα,β) = cg(eαeβ) = cg(eα)cg(eβ) =

{1 if (α, β) = (0, 0),0 otherwise.

It follows that m∗(cg) = cg×g.The dual of the map e : K −→ U(g) is the linear form � �−→ �(e0) on

U(g)∗, resp. the linear form F �−→ F (0) on K[[U1, . . . , Ud]]. We therefore


see that under the isomorphism K[[U1, . . . , Ud]] ∼= U(g)∗ in Prop. 18.3 themaximal ideal mU is the orthogonal complement of the subspace im(e) =K · 1 ⊆ U(g). But the multiplication map m respects unit elements, i. e.,m◦eg×g = eg. Hence m∗ respects the corresponding orthogonal complementswhich means that m∗ is local.

ii. This is immediate from i. and Lemma 18.4.iii. The associativity of the multiplication in U(g) can be expressed as

the identitym ◦ (id⊗m) = m ◦ (m ⊗ id).

Dually we obtain

(id⊗m∗) ◦ m∗ = (m∗ ⊗ id) ◦ m∗.

By ii. the evaluation on U gives the identity

F g(U,F g(Y ,Z)) = F g(F g(U, Y ), Z).

The commutative diagram

U(g) e⊗id

=

U(g) ⊗K U(g) ∼= U(g × g)

m

U(g)

gives rise on the dual side to the commutative diagram

K[[Y1, . . . , Yd, Z1, . . . , Zd]]Yi �→0,Zi �→Ui

K[[U1, . . . , Ud]]

K[[U1, . . . , Ud]].

m∗ =

This means that F g(0, Z) = Z. Analogously we obtain F g(Y , 0) = Y .iv. We write

m∗(Ui) = F(i)(Y ,Z) = Yi + Zi +∑

j,k

cijkYjZk + terms of degree ≥ 3.

The linear form

�i : U(g) −→ K∑

α∈Nd0

cαeα �−→ ci

140 III Lie Groups

(where as before i = (. . . , 0, 1, 0, . . .) with 1 in the i-th place) satisfies F i=

Ui. Hence Fm∗( i) = m∗F i= m∗(Ui) = F(i) and therefore

cijk = m∗(�i)(ej,0e0,k) = �i(ejek).

It follows that

[ej , ek] =d∑

i=1

�i([ej , ek])ei =d∑

i=1

�i(ejek − ekej)ei

=d∑

i=1

(cijk − ci

kj)ei.

Hence the structure constants cijk − ci

kj of the Lie product [ , ]F g coincidewith the structure constants of g.

Example. For commutative g we have F g = (Y1 + Z1, . . . , Yd + Zd).

As a last construction we will associate a formal group law with any Liegroup. Let K be a nonarchimedean field of characteristic zero, and let G bea d-dimensional Lie group over K with Lie algebra g := Lie(G). We pick achart c = (U,ϕ,Kd) for G around the unit element e ∈ G such that ϕ(e) = 0.(The latter, of course, can always be achieved by translating ϕ by ϕ(e).)Since the multiplication mG is continuous we find an open neighbourhoodV ⊆ U of e such that V · V ⊆ U . Then also (V, ϕ|V,Kd) is a chart of Garound e, by Remark 8.1, and the map

ϕ(V ) × ϕ(V )ϕ◦mG◦(ϕ−1×ϕ−1)−−−−−−−−−−−−→ ϕ(U)

is locally analytic. Hence there exists, for sufficiently small ε > 0, a d-tupleof power series FG,c ∈ Fε(Kd × Kd;Kd) such that

FG,c(v, w) = ϕ(ϕ−1(v)ϕ−1(w)) for any v, w ∈ Bε(0).

By the identity theorem Cor. 5.8 the d-tuple FG,c(Y ,Z) of formal powerseries does not depend on the choice of ε.

Proposition 18.6. i. FG,c is a formal group law.

ii. [ , ]F G,ccoincides, modulo the isomorphism θ−1

c : g = Te(G)∼=−−→ Kd,

with the Lie product on g.


Proof. i. Because of ϕ(e) = 0 we have

FG,c(v, 0) = v and FG,c(0, w) = w for any v, w ∈ Bε(0).

Again using the identity theorem Cor. 5.8 this translates into the identitiesof formal power series

FG,c(Y , 0) = Y and FG,c(0, Z) = Z.

In particular, the formation of

FG,c(U,FG,c(Y ,Z)) and FG,c(FG,c(U, Y ), Z)

is well defined. For sufficiently small δ > 0 these formations commute, byProp. 5.4, with the evaluation in any points u, v, w ∈ Bδ(0). But by theassociativity of the multiplication in G we have

FG,c(u, FG,c(v, w)) = FG,c(FG,c(u, v), w) for any u, v, w ∈ Bδ(0).

By a third application of Cor. 5.8 this translates into the identity of formalpower series

FG,c(U,FG,c(Y ,Z)) = FG,c(FG,c(U, Y ), Z).

ii. This is exactly the same computation as in the proof of Prop. 17.3.

Lemma 18.7. The Lie group G has a family {H|a|}|a| of open subgroupsindexed by the sufficiently big |a| ∈ |K| which forms a fundamental systemof open neighbourhoods of e ∈ G and such that each H|a| is isomorphic, viaϕ, to (B 1

|a|(0), FG,c).

Proof. With ε > 0 as above we put ε0 := ‖FG,c‖ε, and we choose any|a| ≥ max(1

ε , ε0ε2 ) (so that, in particular, 1

|a| ≤ ε). We claim that

‖FG,c‖ 1|a|

≤ 1|a|

holds true. Let

FG,c(Y ,Z) = Y + Z +∑

|α|,|β|≥1

vα,βY αZβ with vα,β ∈ Kd.

142 III Lie Groups

We have ‖vα,β‖ ≤ ε0ε−|α|−|β| and hence

‖FG,c‖ 1|a|

= max(

1|a| , max

|α|,|β|≥1‖vα,β‖

1|a||α|+|β|

)

≤ max(

1|a| , max

|α|,|β|≥1ε0

(1

|a|ε

)|α|+|β|)

≤ max(

1|a| , ε0

(1

|a|ε

)2)

≤ 1|a| .

By possibly enlarging the lower bound for |a| we can make exactly the sameargument for the power series expansion of the map g �−→ g−1 on G in asufficiently small neighbourhood of ϕ(e) = 0 (cf. Prop. 13.6). The familyH|a| := ϕ−1(B 1

|a|(0)) then has the required properties.

Corollary 18.8. Every Lie group is paracompact.

Proof. By Lemma 18.7 we find an open subgroup H ⊆ G which as a manifoldis isomorphic to a ball Bε(0). Any coset gH, for g ∈ G, then is isomorphic,as a manifold, to Bε(0) as well. By Lemma 1.4 the ultrametric space Bε(0)and therefore any coset gH is strictly paracompact. As a disjoint union ofcosets gH the Lie group G also is strictly paracompact.

Remark 18.9. If Gε is the Campbell-Hausdorff Lie group germ of a Liealgebra g then we have

Hg = FGε,c

for the chart c := (Gε,⊆,Kd).

In the present situation of a Lie group G and with the choice of theK-basis of g which corresponds to the standard basis of Kd under the iso-morphism θ−1

c : g∼=−−→ Kd we now have the three formal group laws

Hg, F g, and FG,c

whose Lie products[ , ]Hg = [ , ]F g = [ , ]F G,c

coincide and coincide with the Lie product on g (Example 16.12, Lem-ma 17.1, Prop. 18.5, Prop. 18.6).

In order to compare formal group laws we need the following concept.


Definition. Let F and F ′ be formal group laws over K of dimension dand d′, respectively. A formal homomorphism Φ : F −→ F ′ is a d′-tupleΦ = (Φ1, . . . ,Φd′) of formal power series Φi ∈ K[[U1, . . . , Ud]] such thatΦi(0) = 0 and

Φ(F (Y ,Z)) = F ′(Φ(Y ),Φ(Z)).

The formal group laws F and F ′ are called isomorphic if d = d′ and if thereare formal homomorphisms Φ : F −→ F ′ and Φ′ : F ′ −→ F such thatΦ(Φ′(U)) = U = Φ′(Φ(U)).

We write Hom(F , F ′) for the set of all formal homomorphisms Φ : F −→F ′, and we consider the linear map

Hom(F , F ′) −→ HomK(Kd,Kd′)

Φ �−→ σΦ :=(

∂Φi(U)∂Uj

∣∣U=0

)

i,j

.

The chain rule implies that

σΦ′◦Φ = σΦ′ ◦ σΦ

holds true for any two formal homomorphisms FΦ−−→ F ′ Φ′

−−→ F ′′.In a first step we study the particular situation where F is a formal group

law over K of dimension d and where F g is the formal group law whichwe have constructed above for the Lie algebra g := (Kd, [ , ]F ) (and its(ordered) standard basis e1, . . . , ed). We want to show that these two formalgroup laws F and F g are isomorphic. This will be achieved by comparing theuniversal enveloping algebra U(g) to an algebra U(F ) which is constructeddirectly from F . Given a local K-algebra R with maximal ideal m we letR∗ := HomK(R,K) denote its K-linear dual and we introduce the K-vectorsubspace

Rd := {� ∈ R∗ : �|mn = 0 for some n ≥ 0}.

Any local homomorphism of K-algebras α : R0 −→ R1 induces a K-linearmap

αd : Rd1 −→ Rd

0

� �−→ � ◦ α.

We apply this to the local homomorphism

εF : K[[U1, . . . , Ud]] −→ K[[Y1, . . . , Yd, Z1, . . . , Zd]] = R

144 III Lie Groups

and obtain the K-linear map

εdF : Rd −→ U(F ) := K[[U1, . . . , Ud]]d.

Exercise. The inclusion of K-algebras

K[[U1, . . . , Ud]] ⊗K K[[U1, . . . , Ud]] −→ R

Ui ⊗ 1 �−→ Yi

1 ⊗ Ui �−→ Zi

induces a K-linear isomorphism Rd∼=−→ U(F ) ⊗K U(F ).

We therefore may introduce the composite map

m : U(F ) ⊗K U(F ) ∼= RdεdF−−→ U(F ).

Moreover, we let e ∈ U(F ) denote the linear form which sends a formalpower series to its constant coefficient.

Proposition 18.10. (U(F ),m, e) is an associative K-algebra with unit.

Proof. See [Haz] (36.1.5) or [Se2] Part II, Chap. V §6 Lemma 1.

Let ψ : g −→ U(F ) denote the K-linear map which maps the stan-dard basis vector ei to the linear form sending a formal power series to thecoefficient of the monomial Ui.

Lemma 18.11. ψ([x, y]F ) = ψ(x)ψ(y) − ψ(y)ψ(x) for any x, y ∈ g.

Proof. See [Haz] Lemma 36.2.3 or [Se2] Part II, Chap. V §6 Thm. 1.

By the universal property of the universal enveloping algebra the map ψtherefore extends uniquely to a homomorphism of K-algebras with unit

ψ : U(g) −→ U(F ).

Theorem 18.12. The map ψ : U(g)∼=−−→ U(F ) is an isomorphism.

Proof. See [Haz] Thm. 37.4.7 or [Se2] Part II, Chap. V §6 Thm. 2.

We deduce from this theorem the commutative diagram of local homo-morphisms

K[[U ]]

εF

U(F )∗

m∗

∼=ψ∗ U(g)∗

m∗

∼=K[[U ]]

εFg

R = Rd∗ (U(F ) ⊗K U(F ))∗(ψ⊗ψ)∗

(U(g) ⊗K U(g))∗∼=

R.


Proposition 18.13. If g := (Kd, [ , ]F ) then there is an isomorphism of

formal group laws Φ : F g∼=−−→ F such that σΦ = idKd.

Proof. According to Lemma 18.4 the composed isomorphism in the upperrow of the above diagram is of the form εΦ for some d-tuple Φ = (Φ1, . . . ,Φd)of formal power series Φi ∈ K[[U1, . . . , Ud]] with Φi(0) = 0. The commuta-tivity of the diagram then amounts to the equation

Φ(F g(Y ,Z)) = F (Φ(Y ),Φ(Z)).

This says that Φ : F g −→ F is an isomorphism of formal group laws. Wecompute

∂Φi(U)∂Uj

∣∣U=0

= ψ∗(Ui)(ej) = ψ(ej)(Ui) =

{1 if i = j,

0 if i �= j.

Hence σΦ = idKd .

Theorem 18.14. For any two formal group laws F and F ′ over K of di-mension d and d′, respectively, the map

Hom(F , F ′)∼=−−→ HomK((Kd, [ , ]F ), (Kd′ , [ , ]F ′))

Φ �−→ σΦ

is well defined and bijective; in particular, the formal group laws F and F ′

are isomorphic if and only if the corresponding Lie products [ , ]F and [ , ]F ′

are isomorphic.

Proof. Well defined : Let

Fi(Y ,Z) = Yi + Zi +d∑

j,k=1

cijkYjZk + terms of degree ≥ 3

F ′i (Y ,Z) = Yi + Zi +

d′∑

j,k=1

dijkYjZk + terms of degree ≥ 3

Φi(U) =d∑

=1

ai U +

d∑

m,n=1

bimnUmUn + terms of degree ≥ 3.

The terms of degree ≤ 2 in Φi(F (Y ,Z)) are

146 III Lie Groups

∑

ai

(

Y + Z +∑

j,k

c jkYjZk

)

+∑

m,n

bimn(Ym + Zm)(Yn + Zn)

=∑

ai (Y +Z )+

∑

m,n

bimn(YmYn+ZmZn)+

∑

j,k

(

bijk+bi

kj+∑

ai c

jk

)

YjZk

whereas in F ′i (Φ(Y ),Φ(Z)) they are

∑

ai (Y +Z )+

∑

m,n

bimn(YmYn+ZmZn)+

∑

m,n

dimn

(∑

j

amj Yj

)(∑

k

ankZk

)

=∑

ai (Y + Z ) +

∑

m,n

bimn(YmYn + ZmZn) +

∑

j,k

(∑

m,n

dimnam

j ank

)

YjZk.

The equation Φi(F (Y ,Z)) = F ′i (Φ(Y ),Φ(Z)) therefore implies

bijk + bi

kj +∑

ai c

jk =

∑

m,n

dimnam

j ank .

We finally compute

[σΦ(ej), σΦ(ek)]F ′ =

[∑

m

amj em,

∑

n

anken

]

F ′

=∑

m,n

amj an

k [em, en]F ′

=∑

m,n

amj an

k

∑

i

(dimn − di

nm)ei =∑

i

(∑

m,n

(dimn − di

nm)amj an

k

)

ei

=∑

i

(∑

ai (c

jk − c

kj)

)

ei =∑

(c jk − c

kj)

(∑

i

ai ei

)

=∑

(c jk − c

kj)σΦ(e ) = σΦ

(∑

(c jk − c

kj)e

)

= σΦ([ej , ek]F )

which shows that σΦ is a homomorphism of Lie algebras.Injectivity : Differentiating the equation Φ(F (Y ,Z)) = F ′(Φ(Y ),Φ(Z))

gives the matrix equation(

∂Φi(U)∂Uj

∣∣U=F (Y ,Z)

)

i,j

·(

∂Fi(Y ,Z)∂Yj

)

i,j

=(

∂F ′i (Y ,Z)∂Yj

∣∣(Y ,Z)=(Φ(Y ),Φ(Z))

)

i,j

·(

∂Φi(U)∂Uj

∣∣U=Y

)

i,j

.


Setting Y = 0 and remembering that F (0, Z) = Z and Φ(0) = 0 we obtain

(∂Φi(U)

∂Uj∣∣U=Z

)

i,j

·(

∂Fi(Y ,Z)∂Yj

∣∣Y =0

)

i,j

=(

∂F ′i (Y ,Z)∂Yj

∣∣(Y ,Z)=(0,Φ(Z))

)

i,j

· σΦ.

Assuming that σΦ = 0 we then have(

∂Φi(U)∂Uj

∣∣U=Z

)

i,j

·(

∂Fi(Y ,Z)∂Yj

∣∣Y =0

)

i,j

= 0.

Equation (27) shows that (∂Fi(Y ,Z)∂Yj

∣∣Y =Z=0

)i,j is the identity matrix. Hence

the determinant of the matrix (∂Fi(Y ,Z)∂Yj

∣∣Y =0

)i,j is a formal power series with

nonzero constant term, i. e., is a unit the ring K[[Z1, . . . , Zd]]. It follows thatthe matrix (∂Fi(Y ,Z)

∂Yj

∣∣Y =0

)i,j is invertible and consequently that

(∂Φi(U)

∂Uj

)

i,j

= 0.

Since Φ(0) = 0 (and K has characteristic zero) this implies that Φ = 0.Surjectivity : We abbreviate g := (Kd, [ , ]F ) and g′ := (Kd′ , [ , ]F ′). As

a consequence of Prop. 18.13 we have a commutative diagram of the form

Hom(F , F ′)∼=

Φ �→σΦ

Hom(F g, F g′)

Φ�→σΦ

Hom(g, g′).

It therefore suffices to establish the surjectivity of the right oblique arrow.Let σ : g −→ g′ be any homomorphism of Lie algebras. It extends uniquelyto a homomorphism of K-algebras U(σ) : U(g) −→ U(g′) such that

(U(σ) ⊗ U(σ)) ◦ Δg = Δg′ ◦ U(σ), cg′ ◦ U(σ) = cg, and eg′ = U(σ) ◦ eg.

Hence U(σ) dualizes to a local homomorphism

U(σ)∗ : U(g′)∗ ∼= K[[V1, . . . , Vd′ ]] −→ U(g)∗ ∼= K[[U1, . . . , Ud]].

148 III Lie Groups

As such it is, by Lemma 18.4, of the form

U(σ)∗ = εΦ

for a unique d′-tuple Φ of formal power series in K[[U1, . . . , Ud]]. The identity

U(σ × σ)∗ ◦ m∗g′ = m∗

g ◦ U(σ)∗

implies that Φ : F g −→ F g′ , in fact, is a formal homomorphism. If �i ∈ U(g′)∗

are the linear forms corresponding to the variables Vi then the restrictions�1|g′, . . . , �d′ |g′ form the K-basis dual to the standard basis e1, . . . , ed′ ofg′. The formal power series Φi(U) then corresponds to the linear formU(σ)∗(�i) = �i ◦ U(σ) ∈ U(g)∗. It follows that

Φi(U) =∑

α∈Nd0

�i ◦ U(σ)(eα)Uα

and hence that∂Φi(U)

∂Uj∣∣U=0

= �i(σ(ej)).

We conclude that σΦ is the matrix of σ with respect to the standard basis.

Corollary 18.15. The three formal group laws Hg, F g, and FG,c are mu-tually isomorphic.

Proposition 18.16. Let G1 and G2 be two Lie groups over K and let ci =(Ui, ϕi,K

di), for i = 1, 2, be a chart for Gi around the unit element ei ∈ Gi

such that ϕi(ei) = 0; for any formal homomorphism Φ : FG1,c1 −→ FG2,c2

there is an ε > 0 such that Φ ∈ Fε(Kd1 ;Kd2).

Proof. In a first step we consider the special case that G1 = (K,+) is theadditive group of the field K and the chart is c1 = (K, id,K). The FG1,c1 =Y + Z. We abbreviate d := d2 and F := FG2,c2 . The formal homomorphismΦ is a d-tuple of formal power series in one variable U which satisfies

Φ(0) = 0 and Φ(Y + Z) = F (Φ(Y ),Φ(Z)).

Deriving the last identity with respect to Z and then setting Z equal to zeroleads to

Φ′(U) =∂F

∂Z(Φ(U), 0) · Φ′(0).


We defineG(Y ) :=

∂F

∂Z(Y , 0) · Φ′(0)

and obtain the system of differential equations

Φ′(U) = G(Φ(U)) with Φ(0) = 0.

We write

G(Y ) =∑

α∈Nd0

Y α(Mα · Φ′(0)) with Mα ∈ Md×d(K)

andΦ(U) =

∑

n≥1

Un wn

n!with wn = (wn,1, . . . , wn,d) ∈ Kd.

Our system of differential equations now reads

∑

n≥0

Un wn+1

n!=

∑

α∈Nd0

(∑

m≥1

Um wm,1

m!

)α1

· . . . ·(

∑

m≥1

Um wm,d

m!

)αd

(Mα ·Φ′(0)).

By comparing coefficients we obtain the equations

wn+1 =∑

α∈Nd0

(∑

m1,1+···+md,αd=n

∏

i,j

wmi,j ,i · n!mi,j !

)

(Mα · Φ′(0))

for n ≥ 0, where the second summation runs over all |α|-tuples

(m1,1, . . . , m1,α1 ,m2,1, . . . ,m2,α2 , . . . , md,1, . . . ,md,αd)

of integers ≥ 1 whose sum is equal to n. Since each n!m1,1!·...·md,αd

! is an integerit follows that

‖wn+1‖≤ max

{(∏

i,j

|wmi,j ,i|)

·‖Mα·Φ′(0)‖ : α ∈ Nd0,m1,1+· · ·+md,αd

= n

}

≤ max

{(∏

i,j

‖wmi,j‖)

·‖Mα‖·‖Φ′(0)‖ : α ∈ Nd0,m1,1+· · ·+md,αd

= n

}

.

In the proof of Lemma 18.7 we have seen that

‖F‖ 1|a|

≤ 1|a|

150 III Lie Groups

holds true for any sufficiently big |a| ∈ |K|. This implies the existence ofsome |a| ≥ 1 such that

‖Mα‖ ≤ |a||α| for any α ∈ Nd0.

We claim that

‖wn+1‖ ≤ |a|n · ‖Φ′(0)‖n+1 for any n ≥ 0.

The case n = 0 is obvious from w1 = Φ′(0). We now proceed by inductionwith respect to n. Since 1 ≤ mi,j < n the induction hypothesis gives

‖wmi,j‖ ≤ |a|mi,j−1 · ‖Φ′(0)‖mi,j .

We deduce ∏

i,j

‖wmi,j‖ ≤ |a|n−|α| · ‖Φ′(0)‖n

and therefore

‖wn+1‖ ≤ maxα

|a|n−|α| · ‖Φ′(0)‖n · |a||α| · ‖Φ′(0)‖ = |a|n · ‖Φ′(0)‖n+1.

Using Exercise 2.1.i. and Lemma 2.2 we conclude that there are appropriateε0, ε1 > 0 such that

‖wn

n!‖ ≤ ε0ε

n1 for any n ≥ 1.

It follows that Φ ∈ Fε(K;Kd) for any 0 < ε < ε−11 .

We now consider the general case, and we fix a K-basis x1, . . . , xd1 ofg1 (where gi := Lie(Gi)). For any x ∈ g1 we may apply Thm. 18.14 to thehomomorphism of Lie algebras

K −→ g1

a �−→ ax

and obtain a unique formal homomorphism

Φx : F(K,+),id −→ FG1,c1

such thatΦ′

x(0) = σΦx(1) = θ−1

c1 (x).

We introduce the homomorphism of Lie algebras

σ := θc2 ◦ σΦ ◦ θ−1c1 : g1 −→ g2.


The unicity part of Thm. 18.14 implies in addition that we must have

(32) Φσ(x)(U) = Φ(Φx(U)).

By the special case which we have treated already we find an ε > 0 suchthat

Φxi∈ Fε(K;Kd1) and Φσ(xi) ∈ Fε(K;Kd2) for any 1 ≤ i ≤ d1.

Hence, for sufficiently small ε > 0, the maps

G1

Kd1 ⊇ Bε(0)

f1((a1,...,ad1)):=ϕ−1

1 (Φx1(a1))·...·ϕ−1

1 (Φxd1(ad1

))

f2((a1,...,ad1)):=ϕ−1

2 (Φσ(x1)(a1))·...·ϕ−12 (Φσ(xd1

)(ad1))

G2

are well defined and locally analytic. Using Cor. 13.5 we see that the tangentmap at 0 of the upper map is equal to (a1, . . . , ad1) �−→ a1x1 + · · · + ad1xd1

which is a bijection. Hence by Prop. 9.3 the upper map can be inverted asa locally analytic map in a sufficiently small open neighbourhood V1 ⊆ U1

of e1 ∈ G1. Because of (32) the resulting composed locally analytic mapf2 ◦ f−1

1 : V1 −→ Bε(0) −→ G2 has Φ as its power series expansion (withrespect to the charts ϕ1|V1 and ϕ2) around ϕ1(e1) = 0.

Proposition 18.17. Let G1 and G2 be Lie groups over K, and let σ :Lie(G1) −→ Lie(G2) be a homomorphism of Lie algebras; we then have:

i. There exist open subgroups Hi ⊆ Gi as well as a homomorphism ofLie groups f : H1 −→ H2 such that Lie(f) = σ;

ii. if (H ′1,H

′2, f

′) is another triple as in i. then f |H = f ′|H on some opensubgroup H ⊆ H1 ∩ H ′

1.

Proof. i. We choose, for i = 1, 2, a chart ci = (Ui, ϕi,Kdi) for Gi around ei

satisfying ϕi(e) = 0. By Lemma 18.7 there is an open subgroup Hi ⊆ Gi

such that ϕi restricts to an isomorphism of Lie groups

Hi∼=−−→ (B 1

|ai|(0), FGi,ci

)

152 III Lie Groups

for some sufficiently big |ai| ∈ K. On the other hand, according to Thm.18.14, there is a formal homomorphism

Φ : FG1,c1 −→ FG2,c2

such that the diagram

Lie(G1)σ Lie(G2)

(Kd1 , [ , ]F G1,c1)

θc1∼=

σΦ(Kd2 , [ , ]F G2,c2

)

θc2∼=

is commutative (note that the perpendicular arrows are isomorphisms ofLie algebras by Prop. 18.6.ii.). After possibly enlarging |a1| we have Φ ∈F 1

|a1|(Kd1 ;Kd2), by Prop. 18.16, with ‖Φ‖ 1

|a1|≤ 1

|a2| . Hence Φ defines a

homomorphism of Lie groups

f : (B 1|a1|

(0), FG1,c2) −→ (B 1|a2|

(0), FG2,c2)

x �−→ Φ(x)

such that Lie(f) = σΦ. We put f := ϕ−12 ◦ f ◦ ϕ1. Using Lemma 9.2 we see

that

Lie(f) = Lie(ϕ2)−1 ◦ Lie(f) ◦ Lie(ϕ1) = θc2 ◦ σΦ ◦ θ−1c1 = σ.

ii. By shrinking the involved open subgroups we may assume:

– f and f ′ both are homomorphisms from H1 to H2;

– there are charts c1 and c2 with domains of definition H1 and H2,respectively, with respect to which f and f ′ are given by tuples Φ andΦ′, respectively, of convergent power series;

– with respect to ci the multiplication in Hi is given by the formal grouplaw FGi,ci

.

The identity theorem Cor. 5.8 then implies that

Φ,Φ′ : FG1,c1 −→ FG2,c2

both are formal homomorphisms. But

σΦ = θ−1c2 ◦ Lie(f) ◦ θc1 = θ−1

c2 ◦ σ ◦ θc1 = θ−1c2 ◦ Lie(f ′) ◦ θc1 = σΦ′ .

Hence Thm. 18.14 implies that Φ = Φ′ and a fortiori that f = f ′.


Corollary 18.18. Let G1 and G2 be Lie groups over K; if Lie(G1) ∼=Lie(G2) then there are open subgroups Hi ⊆ Gi such that H1

∼= H2.

Proof. We fix an isomorphism σ : Lie(G1)∼=−−→ Lie(G2). By applying Prop.

18.17 to σ as well as σ−1 we find open subgroups H1 ⊆ H ′1 ⊆ G1 and

H ′2 ⊆ H ′′

2 ⊆ G2 together with homomorphisms of Lie groups

H ′2

g−→ H ′1

f−→ H ′′2

such that f ◦ g = idH′2, f(H1) ⊆ H ′

2, and g ◦ (f |H1) = idH1 . Hence bothmaps

H1f−→ H ′

2g−→ H ′

1

are injective with their composite being the identity of H1. It follows that forany open subset V ⊆ H1 we have f(V ) = g−1(V ). This shows that f restrictsto an isomorphism between H1 and the open subgroup H2 := g−1(H1).

Corollary 18.19. Let G be a Lie group over K, and let {Gε}ε>0 be theCampbell-Hausdorff Lie group germ of Lie(G); for any sufficiently small εthere is a homomorphism of Lie groups expG,ε : Gε −→ G such that :

i. expG,ε(Gε) is open in G;

ii. expG,ε is an isomorphism of Lie groups onto its image;

iii. Lie(expG,ε) = idLie(G).

For any two such homomorphisms expG,ε1and expG,ε2

there is a 0 < ε ≤min(ε1, ε2) such that

expG,ε1|Gε = expG,ε2

|Gε.

Proof. We fix a sufficiently small ε′ > 0 and view Gε′ as an open submanifoldof Lie(G). Then Lie(Gε′) = Lie(G) by Prop. 17.3. Hence the assertion followsfrom Prop. 18.17 and Cor. 18.18.

Definition. expG,ε is called an exponential map for the Lie group G.

Part B

The Algebraic Theory of

p-Adic Lie Groups

Chapter IV

Preliminaries

Throughout this Part B we fix a complete discrete valuation ring O with fieldof fractions K. We also pick a prime element π in O. Then O = lim←−m

O/πmOas a topological ring.

19 Completed Group Rings

Let G be any profinite group. We denote by N (G) the set of open normalsubgroups in G. Then

(33) G = lim←−N∈N (G)

G/N

is the projective limit, as a topological group, of the finite groups G/N .By functoriality, the algebraic group rings O[G/N ] form, for varying N inN (G), a projective system of rings. Its projective limit

Λ(G) := O[[G]] := lim←−N∈N (G)

O[G/N ]

is called the completed group ring or the Iwasawa algebra of G over O. Asan immediate consequence of (33) the natural map

O[G] −→ Λ(G)

is injective. We therefore will always view O[G] as a subring of Λ(G).Each O[G/N ] is finitely generated and free as an O-module and therefore

is a complete topological O-algebra for the π-adic topology. We equip Λ(G)with the projective limit topology of these π-adic topologies and make it inthis way into a complete topological O-algebra. By construction the subringO[G] is dense in Λ(G).

If the residue field O/πO is finite then all rings O, O[G/N ], and henceΛ(G) are compact. For general O/πO we contemplate the identity

Λ(G) = lim←−N∈N (G)

lim←−m≥1

(O/πmO)[G/N ] .

Each factor ring (O/πmO)[G/N ] obviously is (left and right) artinian. Wesee that Λ(G), as a left or right module over itself, is the projective limit of


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158 IV Preliminaries

modules of finite length. This means that Λ(G) is a pseudocompact topolog-ical ring. In the Appendix we will collect the basic properties of this notionwhich, for our purposes, retains sufficiently many features of compact rings.We also see that the two-sided ideals

Jm,N (G) := ker(Λ(G)

pr−−→ (O/πmO)[G/N ]),

for m ≥ 1 and N ∈ N (G), form a fundamental system of open neighbour-hoods of zero in Λ(G). The ideal m(G) := J1,G(G) satisfies Λ(G)/m(G) =O/πO and therefore is a maximal (left or right) ideal in Λ(G).

The group G is contained as a subgroup in the group of units O[G]× ofO[G] and a fortiori in the group of units Λ(G)× of Λ(G).

Lemma 19.1. The inclusion map G ↪→ Λ(G) is a homeomorphism onto itsimage.

Proof. The map in question is the projective limit of the inclusion mapsG/N ↪→ O[G/N ] for N ∈ N (G). Hence it is continuous and injective. SinceG is compact and Λ(G) is Hausdorff it then necessarily is a homeomorphismonto its image.

Proposition 19.2. (Universal property) Let M be any complete (and Haus-dorff ) topological O-module in which the open O-submodules form a funda-mental system of neighbourhoods of zero, and let f : G −→ M be any con-tinuous map; then there is a unique continuous O-module homomorphismfΛ : Λ(G) −→ M such that fΛ|G = f .

Proof. It is clear that f extends uniquely to an O-module homomorphismfO : O[G] −→ M by fO(

∑g cgg) :=

∑g cgf(g). Since O[G] is dense in Λ(G)

the continuous extension fΛ of fO to Λ(G) is unique if it exists. To establishthe existence let (Mi)i∈I be the family of all open O-submodules in M . Wefirst consider an arbitrary but fixed index i ∈ I. By the continuity of f wefind, for any g ∈ G, an Ng ∈ N (G) such that f(gNg) ⊆ f(g) + Mi. Becauseof the compactness of G finitely many cosets g1Ng1 , . . . , gsNgs cover G. Weput Ni := Ng1 ∩ · · · ∩ Ngs ∈ N (G). Then

f(gNi) ⊆ f(g) + Mi for any g ∈ G.

On the other hand one easily checks that the kernel of the projection mapO[G] −→ O[G/Ni] is generated, as an O-module, by the elements gh− g forg ∈ G and h ∈ Ni. This shows the existence of the commutative diagram of


O-module homomorphisms

O[G]

pr

fOM

pr

O[G/Ni]f ′

iM/Mi.

Since G/Ni is finite the map fi is continuous for the π-adic topology onO[G/Ni] and the discrete topology on M/Mi. Hence we even have a com-mutative diagram

O[G]

pr

fOM

pr

(O/πmiO)[G/Ni]fi

M/Mi

for some sufficiently large mi ≥ 1. We now vary i ∈ I and obtain in the limitthe continuous extension

fΛ : Λ(G) −→ lim←−i∈I

O[G]/(O/πmiO)[G/Ni]lim←− fi

−−−−→ lim←−i∈I

M/Mi = M

of fO.

Corollary 19.3. Let A be any complete (and Hausdorff ) topological O-algebra in which the open O-submodules form a fundamental system of neigh-bourhoods of zero, and let f : G −→ A× be any group homomorphism suchthat the composed map G

f−→ A× ⊆−→ A is continuous; then there is a uniquecontinuous unital O-algebra homomorphism fΛ : Λ(G) −→ A such thatfΛ|G = f .

Proof. Applying Prop. 19.2 to M := A we obtain the O-module homomor-phisms fO and fΛ. The assumption that f is a group homomorphism im-mediately implies that fO is a unital ring homomorphism and by continuityfΛ then has to be a ring homomorphism as well.

Corollary 19.4. Let f : G −→ H be a continuous homomorphism of profi-nite groups; we then have:

i. f extends uniquely to a continuous unital O-algebra homomorphismΛ(f) : Λ(G) −→ Λ(H);


ii. if f is injective then Λ(f) is a homeomorphism onto its image;

iii. if f is surjective then Λ(f) is a topological quotient map;

iv. if f is the inclusion map of an open subgroup G in H then Λ(H) isfinitely generated free of rank [H : G] as a left or right Λ(G)-module(via Λ(f)).

Proof. i. This is immediate from applying Cor. 19.3 to A := Λ(H).ii. By compactness f is a homeomorphism onto its image. Hence {G∩N :

N ∈ N (H)} is cofinal in N (G). This implies that Λ(f) is the projective limitof the maps

O[G/G ∩ N ] −→ O[H/N ]

induced by f which clearly are homeomorphisms onto their images (for theπ-adic topologies).

iii. We may view Λ(H) via Λ(f) as a pseudocompact (left) Λ(G)-moduleso that Λ(f) becomes a continuous homomorphism of pseudocompact Λ(G)-modules. In this situation we quite generally know, by Thm. 22.3.ii., thatim(Λ(f)) is closed in Λ(H) and that on im(Λ(f)) the subspace topologyinduced by Λ(H) coincides with the quotient topology induced by Λ(f). Butim(Λ(f)) obviously contains O[H] which is dense in Λ(H). Hence im(Λ(f)) =Λ(H) and its topology is the quotient topology induced by Λ(f).

iv. The set N of all open normal subgroups in H which are contained inG is cofinal in both N (H) and N (G). Hence Λ(f) is the projective limit ofthe inclusion maps O[G/N ] ↪→ O[H/N ] for N ∈ N . If h1, . . . , hs ∈ H is anyset of representatives for the right or left cosets of G in H then each O[H/N ]is free as a left or right O[G/N ]-module with basis h1, . . . , hs. It follows thath1, . . . , hs also is a basis of Λ(H) as a left or right Λ(G)-module.

Next we establish a few straightforward facts about the ideal structureof Λ(G). Let

JN (G) := ker(Λ(G)

pr−−→ O[G/N ])

for any N ∈ N (G).

Proposition 19.5. Suppose that the subgroup N ∈ N (G) is topologicallygenerated by the finitely many elements h1, . . . , hr; we then have:

i. JN (G) is generated, as a left or as a right ideal, by the finitely manyelements h1 − 1, . . . , hr − 1;

ii. Jm,N (G), as a left or as a right ideal, is generated by the elementsπm, h1 − 1, . . . , hr − 1.


Proof. The second assertion is a trivial consequence of the first one. We willprove only the left version of i. the argument for the right version beingcompletely analogous. Suppose first that G is finite in which case the hi

generate algebraically. We have used already the simple fact that then JN (G)as a left ideal in Λ(G) = O[G] is generated by the elements h− 1 for h ∈ N .Using the identities

h′h − 1 = h′(h − 1) + (h′ − 1) and h−1 − 1 = −h−1(h − 1)

for any h, h′ ∈ N we conclude that JN (G) in fact is generated by h1 −1, . . . , hr − 1. In general we have

JN (G) = lim←−N ′∈N (G),N ′⊆N

JN/N ′(G/N ′).

Since each finite group N/N ′ is algebraically generated by the elementsh1N

′, . . . , hrN′ it follows that the maps

O[G/N ′]r −→ JN/N ′(G/N ′)

(λ1, . . . , λr) �−→r∑

i=1

λi(hi − 1)

are surjective. Passing to the projective limit with respect to N ′ gives themap

Λ(G)r −→ JN (G)

(λ1, . . . , λr) �−→r∑

i=1

λi(hi − 1)

which, by Thm. 22.3.iv., remains surjective.

We recall that, quite generally for any unital ring R, the Jacobson radicalJac(R) is defined to be the intersection of all maximal left ideals (cf. [Lam]§4). It is a two-sided ideal which equivalently can be characterized as theintersection of all maximal right ideals, and it contains any left or right nilideal. We put J(G) := Jac(Λ(G)). Obviously, J(G) ⊆ m(G). We also recallthat a nonzero unital ring R is called local if the subset of nonunits R \R×

is additively closed; in this case one has R \ R× = Jac(R) (cf. [Lam] §19).

Lemma 19.6. Suppose that char(O/πO) = p > 0 and that G is a finitep-group; then, for any m ≥ 1, the ring (O/πmO)[G] is local and its maximalideal is nilpotent.


Proof. Let m denote the kernel of the augmentation map

(O/πmO)[G] −→ O/πO∑

g

cgg �−→∑

g

cg mod π.

This obviously is a maximal (left) ideal and therefore contains the Jacobsonradical. We first consider the case m = 1 and show, by induction with respectto the order |G| of G, that then m|G|2 = 0. Let Z ⊆ G be the center of Gwhich is nontrivial. By the induction hypothesis we have

m|G/Z|2 ⊆ ker(O/πO[G] −→ O/πO[G/Z]).

This right hand kernel is generated, as an ideal, by the central elements h−1for h ∈ Z. Since the order of Z is a power of the characteristic of O/πO weobtain (h − 1)|Z| = h|Z| − 1 = 1 − 1 = 0. Hence

m|G|2 ⊆ ker(O/πO[G] −→ O/πO[G/Z])|Z|2 = 0.

For general m we deduce that m|G|2 ⊆ π(O/πmO)[G] and hence iterativelythat m|G|2·m = 0. It follows that m is contained in the Jacobson radical.Therefore m is the unique maximal left ideal which means that the ring(O/πmO)[G] is local.

Proposition 19.7. Suppose that char(O/πO) = p > 0 and that G is apro-p-group; we then have:

i. Λ(G) is a local ring with residue field O/πO;

ii. m(G) = J(G);

iii. if G is topologically finitely generated then the J(G)i, for i ≥ 1, arefinitely generated as left or right ideals and form a fundamental systemof open neighbourhoods of zero (i. e., the topology of Λ(G) is the J(G)-adic one).

Proof. It is a direct consequence of Lemma 19.6 that each ideal Jm,N (G)contains a suitable power of the ideal m(G). This means that the m(G)-adictopology is finer than the pseudocompact topology. The lemma also says thatany element λ ∈ Λ(G) \ m(G) is a unit modulo each ideal Jm,N (G). Henceλ is a unit in Λ(G). This shows that Λ(G) is local, that m(G) = J(G),and that the J(G)-adic topology is finer than the pseudocompact one. Wenow suppose that G is topologically generated by the finitely many elements

20 The Example of the Group Zdp 163

g1, . . . , gr. We then know from Prop. 19.5.ii. that m(G) = J(G) is generatedby π, g1 − 1, . . . , gr − 1 as a left as well as a right ideal. This implies, byinduction with respect to i, that J(G)i as a left as well as a right ideal isgenerated by the finitely many elements {π�(gj1 − 1) · · · (gjm − 1) : �,m ≥0, � + m = i, 1 ≤ j1, . . . , jm ≤ r}. As a finitely generated left ideal J(G)i

is closed in Λ(G) by Cor. 22.4. Moreover, each subquotient J(G)i/J(G)i+1

is a finite dimensional vector space over Λ(G)/J(G) = O/πO. It followsinductively that Λ(G)/J(G)i with the quotient topology is a pseudocompactΛ(G)-module of finite length (cf. Thm. 22.3.i.). The topology of any suchmodule necessarily is discrete. We conclude that J(G)i is open in Λ(G).

Lemma 19.8. Suppose that the ring Λ(G) is noetherian; then its topologyis the J(G)-adic one, and any left or right ideal is closed in Λ(G).

Proof. By Prop. 22.5 and Cor. 22.4 this is a general fact about noetherianpseudocompact rings.

20 The Example of the Group Zdp

Let G = Zdp, for some d ≥ 1, be the d-fold direct product of the additive

group of p-adic integers Zp with itself. Its completed group ring can becomputed explicitly by comparing it with the ring O[[X1, . . . , Xd]] of formalpower series in the variables X1, . . . , Xd over O. The latter is a local ring withmaximal ideal m generated by π,X1, . . . , Xd. The m-adic topology makes itinto a pseudocompact ring. Let gi := (. . . , 0, 1, 0, . . .) with the entry 1 in thei-th place, for 1 ≤ i ≤ d, denote the standard topological generators of thegroup Zd

p.

Proposition 20.1. Suppose that char(O/πO) = p > 0; there is a uniquetopological isomorphism of pseudocompact O-algebras O[[X1, . . . , Xd]]

∼=−→Λ(Zd

p) which sends Xi to gi − 1.

Proof. We obviously have the unique O-algebra homomorphism

f0 : O[X1, . . . , Xd] −→ Λ(Zdp)

Xi �−→ gi − 1

where the left hand side is the polynomial ring over O in the variablesX1, . . . , Xd. From Prop. 19.7 we know that Λ(Zd

p) is a local ring with max-imal ideal m(Zd

p) generated by π, g1 − 1, . . . , gd − 1, and that the pseu-docompact topology of Λ(Zd

p) is the m(Zdp)-adic one. Since clearly f0(m ∩


O[X1, . . . , Xd]) ⊆ m(Zp) it follows that f0 extends uniquely to a continuousring homomorphism

f : O[[X1, . . . , Xd]] −→ Λ(Zdp).

We observe that

O[[X1, . . . , Xd]]/πO[[X1, . . . , Xd]] = O/πO[[X1, . . . , Xd]]

andO[[X1, . . . , Xd]] = lim←−

n

O[[X1, . . . , Xd]]/πnO[[X1, . . . , Xd]]

as well as, by Thm. 22.3.iv. and Lemma 22.1, that

Λ(Zdp)/πΛ(Zd

p) = O/πO[[Zdp]]

andΛ(Zd

p) = lim←−n

Λ(Zdp)/πnΛ(Zd

p).

Hence the bijectivity of f can be shown modulo πn for any n ≥ 1. SinceO[[X1, . . . , Xd]] and Λ(Zd

p), both by construction, are O-torsion free thisreduces, by an inductive argument, to the bijectivity of f modulo π. But fmod π is the projective limit of the corresponding ring homomorphisms

O/πO[[X1, . . . , Xd]]/〈Xpn

1 , . . . , Xpn

d 〉 −→ O/πO[Zdp/pnZd

p]

which are well defined because of the identities 0 = gpn

i − 1 = (gi − 1)pnin

O/πO[Zdp/pnZd

p]. These are easily seen to be isomorphisms.

21 Continuous Distributions

The completed group ring Λ(G) of a profinite group G also has an importantfunctional analytic description. We let C∞(G), resp. C(G), denote the O-module of all O-valued locally constant, resp. continuous, functions on G.Clearly C∞(G) is a submodule of C(G).

If G is finite we have the natural O-module isomorphism

O[G]∼=−→ HomO(C(G),O)

which sends a group element g to the linear form which evaluates a functionin g. By passing to the projective limit we then obtain for a general G a


corresponding O-module isomorphism

Λ(G) ∼= lim←−N∈N (G)

HomO(C(G/N),O)(34)

= HomO(lim−→N

C(G/N),O)

= HomO(C∞(G),O).

We always equip C(G) with the π-adic topology. Because of

C(G) = lim←−m

{functions G −→ O/πmO} = lim←−m

C(G)/πmC(G)

the submodule C∞(G) is dense in C(G). Moreover,

C∞(G) ∩ πmC(G) = πmC∞(G)

which implies that the π-adic topology on C∞(G) is the subspace topologyinduced by the π-adic topology on C(G).

Lemma 21.1. Λ(G) = HomO(C(G),O).

Proof. Because of (34) it suffices to see that any O-linear form on C∞(G)extends uniquely to an O-linear form on C(G). But this follows from thediscussion before the lemma once one observes that any O-linear form isautomatically continuous for the π-adic topology.

In this sense we may view Λ(G) as the space of O-valued continuousdistributions on G.

22 Appendix: Pseudocompact Rings

A left pseudocompact ring R is a unital topological ring which is Hausdorffand complete and which has a fundamental system (Li)i∈I of open neigh-bourhoods of zero consisting of left ideals Li ⊆ R such that R/Li is of finitelength as an R-module.

A pseudocompact left R-module M is a unital topological R-modulewhich is Hausdorff and complete and which has a fundamental system(Mi)i∈I of open neighbourhoods of zero consisting of submodules Mi ⊆ Msuch that the R-module M/Mi is of finite length.

In the following let R be a left pseudocompact ring. The two technicalkey facts about pseudocompact R-modules are the following.


Lemma 22.1. Let M be a pseudocompact R-module and (Mi)i∈I be a de-creasingly filtered family of closed submodules; then the natural map M −→lim←−i

M/Mi is surjective.

Proof. [Gab] IV.3 Prop. 10.

Lemma 22.2. Let M be a pseudocompact R-module, N a closed submodule,and (Mi)i∈I a decreasingly filtered family of closed submodules; then

N +⋂

i

Mi =⋂

i

(N + Mi).

Proof. [Gab] IV.3 Prop. 11.

Theorem 22.3. i. Let M be a pseudocompact R-module and N ⊆ Mbe a closed submodule; then N , with the subspace topology, and M/N ,with the quotient topology, are pseudocompact R-modules.

ii. Let α : M −→ N be a continuous homomorphism of pseudocompactR-modules; we then have:

a) im(α) is closed in N ;

b) on im(α) the subspace topology induced by N coincides with thequotient topology induced by α.

iii. The R-module projective limit of any projective system of pseudocom-pact R-modules is pseudocompact for the projective limit topology.

iv. The formation of filtered projective limits of pseudocompact R-modulesis exact.

Proof. [Gab] IV.3 Thm. 3.

Corollary 22.4. Any finitely generated submodule N in a pseudocompactR-module M is closed.

Proof. Write N as the image of some R-module homomorphism α : Rm −→M . Since α automatically is continuous we may apply Thm. 22.3.ii.

Suppose that R is left noetherian, and let M be an abstract finitelygenerated left R-module. We may write M as a quotient Rm � M of somefinitely generated free R-module. The latter carries the product topologyand M then the corresponding quotient topology.


Exercise. This quotient topology on M is independent of the choice of thepresentation Rm � M .

Since the kernel of Rm � M is finitely generated, by our assumption on R,it is closed in Rm by Cor. 22.4. Hence Thm. 22.3.i. implies that M is pseudo-compact in the quotient topology. In fact, this is the unique pseudocompacttopology on M .

Proposition 22.5. If R is left noetherian then the pseudocompact topologyon R is the Jac(R)-adic topology.

Proof. According to [Gab] IV.3 Prop. 13.b the ring R/ Jac(R) is a product ofendomorphism rings of vector spaces over division rings. But R/ Jac(R) byassumption is left noetherian. Hence it must be left artinian. Each quotientJac(R)i/ Jac(R)i+1 is a finitely generated left module over the left artinianring R/ Jac(R) and hence is of finite length. It follows inductively that eachR/ Jac(R)i is a left R-module of finite length. On the other hand Jac(R)i isclosed in R by Cor. 22.4. Hence, by Thm. 22.3.i., R/ Jac(R)i is a pseudocom-pact R-module of finite length. Its topology therefore must be the discreteone which means that Jac(R)i is open.

Again by [Gab] IV.3 Prop. 13.b we have⋂

i Jac(R)i = 0. Let L ⊆ R beany open left ideal. Then Lemma 22.2 implies that L = L +

⋂i Jac(R)i =⋂

i(L + Jac(R)i). Since R/L is of finite length it follows that Jac(R)i ⊆ Lfor any sufficiently big i.

There are obvious “right” versions of everything above. A unital topo-logical ring R which is left as well as right pseudocompact will be calledpseudocompact.

Chapter V

p-Valued Pro-p-Groups

From now on we fix once and for all a prime number p.

23 p-Valuations

Let G be any abstract group. A p-valuation ω on G is a real valued function

ω : G \ {1} −→ (0,∞)

which, with the convention that ω(1) = ∞, satisfies

(a) ω(g) > 1p−1 ,

(b) ω(g−1h) ≥ min(ω(g), ω(h)),

(c) ω([g, h]) ≥ ω(g) + ω(h),

(d) ω(gp) = ω(g) + 1

for any g, h ∈ G. Here the commutator is normalized to be [g, h] = ghg−1h−1;as usual, for any two subsets A,B ⊆ G we will write [A,B] for the subgroupof G generated by the set of commutators {[g, h] : g ∈ A, h ∈ B}. Thereason for the somewhat mysterious axiom (a) will only become clear later.Applying (b) with h = 1 gives ω(g−1) ≥ ω(g) and hence by symmetry

ω(g−1) = ω(g) for any g ∈ G.

From (b) and (c) we deduce

ω(ghg−1) = ω([g, h]h) ≥ min(ω([g, h]), ω(h))≥ min(ω(g) + ω(h), ω(h)) = ω(h)

and therefore, by symmetry,

(35) ω(ghg−1) = ω(h) for any g, h ∈ G.

Finally, if ω(g) > ω(h) then

ω(h) = ω(g−1(gh)) ≥ min(ω(g), ω(gh)) ≥ min(ω(g), ω(h)) = ω(h)


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170 V p-Valued Pro-p-Groups

and hence ω(gh) = ω(h). It follows that

(36) ω(gh) = min(ω(g), ω(h)) if ω(g) �= ω(h).

For any real number ν > 0 we put

Gν := {g ∈ G : ω(g) ≥ ν} and Gν+ := {g ∈ G : ω(g) > ν}.

These are subgroups by (b), and they are normal by (35).

Remark 23.1. Gν/Gν+, for any ν > 0, is a central subgroup of G/Gν+.

Proof. By (c) we have ω(ghg−1h−1) > ω(h) for any g, h ∈ G. This meansthat ghg−1h−1 ∈ Gω(h)+ and hence that ghGω(h)+ = hgGω(h)+.

The subgroups Gν form a decreasing exhaustive and separated filtrationof G with the additional properties

Gν =⋂

ν′<ν

Gν′ and [Gν , Gν′ ] ⊆ Gν+ν′ .

There is a unique (Hausdorff) topological group structure on G for whichthe Gν form a fundamental system of open neighbourhoods of the identityelement. It will be called the topology defined by ω.

Example 23.2. Let E be a finite extension of the field Qp of p-adic numbers,oE its ring of integers, πE ∈ oE a prime element, and v its additive valuationnormalized by v(p) = 1. We fix an n ∈ N, and we consider the algebraMn×n(E) of n × n-matrices over E. For any nonzero matrix A = (aij) weput

w(A) := mini,j

v(aij),

and w(0) := ∞. This function w on Mn×n(E) has the following straightfor-ward properties:

(i) w(A + B) ≥ min(w(A), w(B));

(ii) w(A + B) = min(w(A), w(B)) if w(A) �= w(B);

(iii) w(AB) ≥ w(A) + w(B);

(iv) w(A) = 0 for any A ∈ GLn(oE).

23 p-Valuations 171

Now let

G :={

g ∈ GLn(E) : w(g − 1) >1

p − 1

}

andω(g) := w(g − 1) for g ∈ G.

Then G is an open neighbourhood of 1 in GLn(E). By its defining conditionany g ∈ G satisfies g ∈ Mn×n(oE) and g ≡ 1 mod πE , hence det g ∈ o×E andtherefore g ∈ GLn(oE). We now compute:

1) For g ∈ G we have

ω(g−1) = w(g−1 − 1) = w(−g−1(g − 1)) ≥ w(g − 1) = ω(g).

In particular, also g−1 lies in G, and then by symmetry ω(g−1) = ω(g).Using (iii) it moreover follows that G is a subgroup of GLn(oE).

2) For g, h ∈ G we have

ω(g−1h) = w(g−1h − 1)

= w((g−1 − 1)(h − 1) + (g−1 − 1) + (h − 1))

≥ min(w(g−1 − 1), w(h − 1))

= min(ω(g−1), ω(h))= min(ω(g), ω(h)).

3) For g, h ∈ G we also have

ω([g, h]) = w([g, h] − 1)

= w(((g − 1)(h − 1) − (h − 1)(g − 1))g−1h−1)≥ w((g − 1)(h − 1) − (h − 1)(g − 1))≥ min(w((g − 1)(h − 1)), w((h − 1)(g − 1)))≥ w(g − 1) + w(h − 1)= ω(g) + ω(h).

4) For g �= 1 in G we have in Mn×n(E) the equation

gp − 1 = ((g − 1) + 1)p − 1 =p∑

j=1

(p

j

)

(g − 1)j

= p(g − 1) + p(g − 1)p−2∑

j=1

aj(g − 1)j + (g − 1)p


with appropriate integers aj ∈ Z. Since w(∑p−2

j=1 aj(g − 1)j) > 0 andw((g − 1)p) ≥ pw(g − 1) > 1 + w(g − 1) we obtain

ω(gp) = w(gp − 1) = w(p(g − 1)) = 1 + w(g − 1) = ω(g) + 1.

Altogether these computations tell us that ω is a p-valuation on G. It isclear from the definition of G that ω defines the subspace topology on theopen subgroup G of GLn(E).

Example 23.3. Retaining the notations of the previous example we let Gn ⊆GLn(Zp), for any n ≥ 2, denote the open pro-p-subgroup of all matrices(aij) such that

v(aij) ≥ 1 for any 1 ≤ j < i ≤ n and v(aii − 1) ≥ 1 for any 1 ≤ i ≤ n.

If n ≥ p − 1 then Gn contains an element of order p (exercise!) and hencecannot carry a p-valuation. We therefore assume in the following that

2 ≤ n < p − 1.

Then we may pick a rational number 1p−1 < α < p−2

(p−1)(n−1) . For g = (aij) ∈Gn we define

ωα(g) := min( min1≤i�=j≤n

((j − i)α + v(aij)), min1≤i≤n

v(aii − 1)).

In order to see that ωα indeed is a p-valuation on Gn we will embed (Gn, ωα)into a group of the form discussed in the previous example. We pick a finiteextension E of Qp which contains an element c of valuation v(c) = α. LetC ∈ GLn(E) denote the diagonal matrix with entries cn, cn−1, . . . , c andconsider the monomorphism of groups

f : Gn −→ GLn(E)

g −→ CgC−1.

For g = (aij) we have f(g) = (cj−iaij). This immediately implies

ωα(g) = w(f(g) − 1) for any g ∈ Gn.

Furthermore, by the choice of α, we have

v(aii − 1) ≥ 1 >1

p − 1for any 1 ≤ i ≤ n,

(j − i)α + v(aij) ≥ α + v(aij) ≥ α >1

p − 1for any 1 ≤ i < j ≤ n,

23 p-Valuations 173

and

(j − i)α + v(aij) ≥ (1− n)α + v(aij) > (1− n)p − 2

(p − 1)(n − 1)+ 1 =

1p − 1

for any 1 ≤ j < i ≤ n

if (aij) ∈ Gn. It follows that f maps Gn into the subgroup of GLn(E) onwhich w(. − 1) is a p-valuation by the previous example. It is clear that ωα

defines the topology of Gn.

For later use we recall the following commutator identities in G.

Exercise. For any g, h, k ∈ G we have:

(A) [gh, k] = g[h, k]g−1[g, k],

(B) [g, hk] = [g, h]h[g, k]h−1,

(C) [[g, h], hkh−1][[h, k], kgk−1][[k, g], ghg−1] = 1.

We now form, for each ν > 0, the subquotient group

grν G := Gν/Gν+.

It is commutative by (c) and therefore will be denoted additively. One of ourimportant goals in the following is to investigate the graded abelian group

grG :=⊕

ν>0

grν G.

An element ξ ∈ gr G is called, as usual, homogeneous (of degree ν) if it liesin grν G. Furthermore, in this case any g ∈ Gν such that ξ = gGν+ is calleda representative of ξ.

We immediately observe that gr G has considerably more structure. Firstof all, by (d) we have pξ = 0 for any homogeneous element ξ ∈ grG. HencegrG in fact is an Fp-vector space.

Lemma 23.4. For any ν, ν ′ > 0 the map

grν G × grν′ G −→ grν+ν′ G

(ξ, η) −→ [ξ, η] := [g, h]G(ν+ν′)+,

where g and h are representatives of ξ and η, respectively, is a well definedbi-additive map; we have [ξ, ξ] = 0 and [ξ, η] = −[η, ξ].


Proof. First of all, by (c), we have [g, h] ∈ Gν+ν′ so that [ξ, η] indeed is ahomogeneous element of degree ν + ν ′. Let now k ∈ Gν′ be any element.Then [g, hk] = [g, h]h[g, k]h−1 by (B). Using Remark 23.1 we obtain

[g, hk]G(ν+ν′)+ = [g, h]h[g, k]h−1G(ν+ν′)+ = [g, h][g, k]G(ν+ν′)+.

If k ∈ Gν′+ then [g, k] ∈ G(ν+ν′)+ and hence

[g, hk]G(ν+ν′)+ = [g, h]G(ν+ν′)+.

This proves the independence of the choice of the representative as well as theadditivity in the second component. For the first component the argument iscompletely analogous but using (A). It remains to note the trivial identities[g, g] = 1 and [g, h] = [h, g]−1.

By bilinear extension we therefore obtain a graded Fp-bilinear map

[ , ] : gr G × gr G −→ gr G

which satisfies[ξ, ξ] = 0 for any ξ ∈ gr G.

Lemma 23.5. Any ζ, ξ, η ∈ grG satisfy the Jacobi identity

[[ζ, ξ], η] + [[ξ, η], ζ] + [[η, ζ], ξ] = 0.

Proof. We may assume that ζ, ξ, η are homogeneous of degrees ν, ν ′, ν ′′ withrepresentatives g, h, and k, respectively. According to (C) we have

[[ζ, ξ], hkh−1Gν′′+] + [[ξ, η], kgk−1Gν+] + [[η, ζ], ghg−1Gν′+] = 0.

But as a consequence of Remark 23.1 the set of representatives of any givenhomogeneous element is invariant under conjugation. It follows that

hkh−1Gν′′+ = η, kgk−1Gν+ = ζ, and ghg−1Gν′+ = ξ.

Altogether we see that gr G is a graded Lie algebra over Fp. Due to thelast axiom (d) there is another piece of additional structure, though. Itsconstruction requires less straightforward arguments which we will preparefor in the next section.


24 The Free Group on Two Generators

Let M denote the set of all noncommutative monomials in two variables Xand Y , i. e., the free monoid generated by X and Y . If μ ∈ M is μ = μ1 · · ·μd

with each μi being equal to X or Y then deg μ := d is called the degree ofμ. For any d ≥ 0 let M(d) be the free abelian group on the subset of M ofall monomials of degree d.

The Magnus algebra M in the variables X and Y is defined to be ring ofall associative (but not commutative) formal power series in X and Y withcoefficients in Z. Each element F ∈ M can uniquely be written as

F =∑

μ∈M

cμμ with cμ ∈ Z.

Equivalently we may view M as the direct product M =∏

d≥0 M(d) whereF corresponds to (F (d))d with F (d) :=

∑deg μ=d cμμ. If F �= 0 then the

nonnegative integer ordF := min{deg μ : cμ �= 0} is called the order of F .By convention, ord 0 = ∞. Instead of ord(F − F ′) ≥ we often will write

F ≡ F ′ mod deg .

If in the difference F−F ′ all monomials of degree < occur with a coefficientdivisible by p we write

F ≡ F ′ mod (p,deg ).

Using the geometric series we see that the elements 1+X and 1+Y areunits in the ring M. More generally we have the following.

Exercise 24.1. Let a ∈ Z and F, U ∈ M with ord U ≥ 1; we then have:

i. F is invertible in M if and only if F (0) = ±1;

ii. (1 + U)a ≡ 1 + aU (ord U) mod deg((ord U) + 1);

iii. (1 + U)pa ≡ (1 + Up)a ≡ 1 mod (p,deg(p · ord U)).

We let F denote the subgroup of the group of units M× generated by1 + X and 1 + Y .

Proposition 24.2. F is the free group generated by 1 + X and 1 + Y .

Proof. [B-LL] Chap. II §5.3 Thm. 1.


Next we putFn := {F ∈ F : ord(F − 1) ≥ n}

for any n ≥ 1.

Theorem 24.3. F1 = F and Fn+1 = [Fn,F] for n ≥ 1.

Proof. [B-LL] Chap. II §5.4 Thm. 2.

Corollary 24.4. Fn/Fn+1, for any n ≥ 1, is a finitely generated abeliangroup.

For example, F/F1 is a free abelian group of rank two. In fact, this corol-lary can be made much more precise. First of all, by completely analogousarguments as for Lemma 23.4 and Lemma 23.5 one shows that the gradedabelian group

grF :=⊕

n≥1

Fn/Fn+1

is a graded Lie algebra over Z with the Lie bracket being the sum of themaps

F�/F�+1 × Fn/Fn+1 −→ F�+n/F�+n+1

(EF�+1, FFn+1) −→ EFE−1F−1F�+n+1.

On the other hand the associative algebra M is a Lie algebra in the usual wayfor the Lie bracket (E,F ) −→ EF − FE. Let L ⊆ M be the Lie subalgebragenerated by the variables X and Y . This L is known to be the free Liealgebra on the set {X,Y }, and it is naturally graded by

L =⊕

d≥1

L ∩ M(d);

in addition each abelian group L∩M(d) is a direct summand of M(d) ([B-LL]Chap. II §3.1 Thm. 1.a and Remark 3).

Theorem 24.5. The unique homomorphism of graded Lie Z-algebras

φ : L∼=−−→ grF

sending X to (1 + X)F2 and Y to (1 + Y )F2 is an isomorphism. Its inverseis given by

FFn+1 −→ F (n).


Proof. [B-LL] Chap. II §5.4 Thm. 3 (in particular part C) in the proof).

For our purposes we need the following fact.

Notation. If H is any group then Hpndenotes the subgroup generated the

pn-th powers of all the elements of H. For any two subsets A,B ⊆ H letAB := {ab : a ∈ A, b ∈ B}.

Proposition 24.6. For any pm ≥ ≥ m ≥ 1 we have

{F ∈ Fm : F ≡ 1 mod (p,deg )} ⊆ FpmF�.

Proof. As a consequence of Cor. 24.4 we find a strictly increasing sequenceof integers d0 = 0 < d1 = 2 < d2 < · · · < dn < · · · and a sequenceE1, . . . , Ei, . . . in F such that the Ei for dn−1 < i ≤ dn lie in Fn with theircosets modulo Fn+1 forming a Z-basis of Fn/Fn+1. We then may write

F = Eadm−1+1

dm−1+1 · · ·Ead�−1

d�−1F1

with F1 ∈ F� and adm−1+1, . . . , ad�−1∈ Z. We claim that each integer adm−1+1

up to ad�−1is divisible by p. Suppose we already have shown this to hold

for all adm−1+1, . . . , adn with some m − 1 ≤ n < − 1. We then obtain fromEx. 24.1.iii. that

Eadm−1+1

dm−1+1 · · ·Eadndn

≡ 1 mod (p,deg pm).

On the other hand we have

F ≡ Eadm−1+1

dm−1+1 · · ·Eadn+1

dn+1mod deg(n + 2).

With the help of Ex. 24.1.ii., and observing that pm ≥ ≥ n+2, we deduce

F ≡ Eadn+1

dn+1 · · ·Eadn+1

dn+1mod (p,deg(n + 2))

≡ (1 + adn+1E(n+1)dn+1 ) · · · (1 + adn+1E

(n+1)dn+1

) mod (p,deg(n + 2))

≡ 1 + adn+1E(n+1)dn+1 + · · · + adn+1E

(n+1)dn+1

mod (p,deg(n + 2)).

Since ≥ n + 2 our assumption on F therefore implies

adn+1E(n+1)dn+1 + · · · + adn+1E

(n+1)dn+1

∈ pM(n+1).

But according to Thm. 24.5 the E(n+1)dn+1 , . . . , E

(n+1)dn+1

form a Z-basis of L ∩M(n+1). Since L∩M(n+1) is a direct summand of M(n+1) this forces the co-efficients adn+1, . . . , adn+1 to be divisible by p. Our claim follows by applyingthis argument iteratively.


Corollary 24.7. i. (1 + Y )−p(1 + X)−p((1 + X)(1 + Y ))p ∈ Fp2Fp.

ii. [1 + X, 1 + Y ]−p[(1 + X)p, 1 + Y ] ∈ Fp3Fp+1 (here [ , ] denotes the

commutator in the group F).

Proof. i. It suffices to show that

F := (1 + Y )−p(1 + X)−p((1 + X)(1 + Y ))p.

satisfies

F ≡ 1 mod deg 2 and F ≡ 1 mod (p,deg p).

But these congruences are immediate from Ex. 24.1.ii. and iii., respectively.ii. Here it suffices to show that

F := [1 + X, 1 + Y ]−p[(1 + X)p, 1 + Y ]

satisfies

F ≡ 1 mod deg 3 and F ≡ 1 mod (p,deg(p + 1)).

One easily sees that

[1 + X, 1 + Y ] ≡ 1 + XY − Y X mod deg 3.

By Ex. 24.1.ii. and iii., respectively, this implies

[1 + X, 1 + Y ]−p ≡ 1 − p(XY − Y X) mod deg 3,

[1 + X, 1 + Y ]−p ≡ 1 mod (p,deg 2p).

On the other hand, by direct calculation we obtain

[(1 + X)p, 1 + Y ] ≡ 1 + p(XY − Y X) mod deg 3,

[(1 + X)p, 1 + Y ] ≡ [1 + Xp, 1 + Y ] ≡ 1 mod (p,deg(p + 1)).

25 The Operator P

Again ω is a fixed p-valuation on the abstract group G. The last axiom (d)says that

ω(gp) = ω(g) + 1 for any g ∈ G.

25 The Operator P 179

Proposition 25.1. For any two elements g, h ∈ G we have:

i. ω(h−pg−p(gh)p) > max(ω(g), ω(h)) + 1;

ii. ω(g−pnhpn

) = ω(g−1h) + n for any n ≥ 1.

Proof. i. We may assume that the group G is generated by the two elementsg and h (otherwise replace G by the subgroup generated by g and h and ωby its restriction this subgroup). There is a unique epimorphism of groupsψ : F −→ G which sends 1+X to g and 1+Y to h. Cor. 24.7.i. implies thath−pg−p(gh)p lies in ψ(F2)pψ(Fp). Since ψ(F2) is the commutator subgroupof G it follows from the commutator identities (A) and (B) in Sect. 23 thatit is the smallest normal subgroup of G which contains the commutators[g, h], [g−1, h], [g, h−1], and [g−1, h−1] (in fact, [g, h] would suffice becauseof relations like [g−1, h] = g−1[g, h]−1g). The axioms (b) – (d) for ω thenimply that ω has values ≥ ω(g) + ω(h) > max(ω(g), ω(h)) on ψ(F2) and> max(ω(g), ω(h)) + 1 on ψ(F2)p. Similarly ψ(Fp) is the smallest normalsubgroup of G which contains all iterated commutators with p entries fromthe set {g, g−1, h, h−1}. So axioms (b) and (c) imply that the values of ω onψ(Fp) are ≥ (p−1) min(ω(g), ω(h))+max(ω(g), ω(h)). Using now for the firsttime the axiom (a) we see that this latter number is > max(ω(g), ω(h))+1as well.

ii. By induction it suffices to consider the case n = 1. We may assumethat ω(g) ≤ ω(h). Applying the inequality in i. to the pair g and g−1h andusing the axioms (b) and (d) gives

ω((g−1h)−pg−php) > max(ω(g), ω(g−1h)) + 1 = ω(g−1h) + 1 = ω((g−1h)p).

Hence (36) implies

ω(g−php) = min(ω((g−1h)p), ω((g−1h)−pg−php))

= ω((g−1h)p) = ω(g−1h) + 1.

Let ν > 0 and g, h ∈ G such that ω(h) ≥ ν = ω(g). As a consequence ofProp. 25.1.i. we have

(gh)pG(ν+1)+ = gphpG(ν+1)+.

If ω(h) > ν then ω(hp) > ν + 1 and hence

(gh)pG(ν+1)+ = gpG(ν+1)+.


This shows that the map

grν G −→ grν+1 G

gGν+ −→ gpG(ν+1)+

is well defined and Fp-linear. For varying ν the direct sum of these mapsthen is an Fp-linear map of degree one

P : gr G −→ gr G.

We therefore may and always will view grG as a graded module over thepolynomial ring Fp[P ] in one variable over Fp.

Remark 25.2. The Fp[P ]-module gr G is torsionfree.

Proof. Let q(P ) be a nonzero polynomial over Fp of degree d with highestterm aP d and let ξ =

∑ν ξν ∈ grG be a finite nonzero sum with homoge-

neous components ξν such that q(P )(ξ) = 0. By the axiom (d) the operatorP on grG is injective and of degree one. The latter property implies thatP d(aξν0) = aP d(ξν0) = 0 where ν0 := max{ν : ξν �= 0}. But this contradictsthe injectivity of P .

Proposition 25.3. ω([g, h]−p[gp, h]) > ω(g) + ω(h) + 1 for any g, h ∈ G.

Proof. The proof is exactly analogous to the proof of Prop. 25.1.i. by usingCor. 24.7.ii. Observe that, with the notations in the proof of Prop. 25.1.i.the values of ω on ψ(F3)p are

≥ 2 min(ω(g), ω(h)) + max(ω(g), ω(h)) + 1 > ω(g) + ω(h) + 1

and on ψ(Fp+1) are

≥ p min(ω(g), ω(h)) + max(ω(g), ω(h))= (p − 1) min(ω(g), ω(h)) + ω(g) + ω(h)> 1 + ω(g) + ω(h)

where the last inequality uses axiom (a).

Prop. 25.3 implies that for g, h ∈ G \ {1} we have

[gp, h]G(ω(g)+ω(h)+1)+ = [g, h]pG(ω(g)+ω(h)+1)+.

This means that the Lie bracket on gr G is bilinear for the Fp[P ]-modulestructure. In other words, grG is a Lie algebra over the ring Fp[P ]. Thiscompletes the discussion of the formal structure of grG.


26 Finite Rank Pro-p-Groups

Let G be a profinite group and ω be a p-valuation on G which we assume todefine the topology of G. Then the Gν are open subgroups, hence the G/Gν

are finite, and G = lim←−νG/Gν . The axiom (d) implies that each G/Gν is a p-

group. We see that in the presence of a “defining” p-valuation G necessarilyis a pro-p-group.

Lemma 26.1. ω(G \ {1}) is a discrete subset of the interval (0,∞).

Proof. This is an immediate consequence of the finiteness of the G/Gν .

In the last section we have seen that grG is a graded Lie algebra overthe polynomial ring Fp[P ] and is torsionfree as an Fp[P ]-module.

Definition. The pair (G,ω) is called of finite rank if grG is finitely gener-ated as an Fp[P ]-module.

Later on we will show that the property of being of finite rank in factdoes not depend on the choice of the p-valuation. From that point onwardswe will simply speak of G being of finite rank.

For the rest of this section we suppose (G,ω) to be of finite rank. By theelementary divisor theorem a finitely generated torsionfree module over theprincipal ideal domain Fp[P ] is free. We call

rank(G,ω) := rankFp[P ] grG

the rank of the pair (G,ω).

Exercise 26.2. For any closed subgroup H ⊆ G we have

rank(H,ω|H) ≤ rank(G,ω);

if H is open in G then rank(H,ω|H) = rank(G,ω).

For any 0 < ν ≤ 1 we let

(grG)(ν) := grν G ⊕ grν+1 G ⊕ · · · ⊕ grν+n G ⊕ · · ·

Since the operator P is of degree one the obvious decomposition

gr G =⊕

0<ν≤1

(grG)(ν)


is a decomposition of Fp[P ]-modules. Because of the finite generation itfollows that at most finitely many of the (grG)(ν) can be nonzero. This, inparticular, shows that there are finitely many real numbers ν1, . . . , νs > 1

p−1such that

ω(G \ {1}) =⋃

1≤j≤s

νj + N0.

Before we continue to explore the structure of gr G we need to make asimply observation on p-adic powers in the pro-p-group G. Let g ∈ G be anyelement. We then have the group homomorphism

c : Z −→ G

m −→ gm.

Since G/N , for any N ∈ N (G), is a p-group we obtain c−1(N) = paN Z

for some aN ≥ 0. It follows that c extends uniquely to a continuous grouphomomorphism

c : Zp −→ lim←−N

Z/paN Zc−−−−→ lim←−

N

G/N = G

which we always will write as gx := c(x). More generally, for any finitelymany elements g1, . . . , gr ∈ G, we have the continuous map

Zrp −→ G

(x1, . . . , xr) −→ gx11 · . . . · gxr

r

(37)

but which, in general, depends on the order of the gi and therefore is nota group homomorphism. Nevertheless we introduce the following notion,where v denotes the usual p-adic valuation on Qp (i. e., v(pmx) = m for anym ∈ Z and x ∈ Z×

p , and v(0) = ∞).

Definition. The sequence of elements (g1, . . . , gr) in G is called an orderedbasis of (G,ω) if the map (37) is a bijection (and hence, by compactness, ahomeomorphism) and

ω(gx11 · · · gxr

r ) = min1≤i≤r

(ω(gi) + v(xi)) for any x1, . . . , xr ∈ Zp.

Notation. For any element g �= 1 in G we put σ(g) := gGω(g)+ ∈ gr G.

Remark 26.3. Let g �= 1 in G and x �= 0 in Zp; further let a ∈ F×p denote

the image of p−v(x)x; we then have

ω(gx) = ω(g) + v(x) and σ(gx) = (aP v(x))(σ(g)).


Proof. This is straightforward from the definitions and axiom (d) once weconvince ourselves that ω(gx) = ω(g) holds whenever x ∈ Z×

p is a p-adic unit.Since Gω(g) is open and hence closed in G we certainly have ω(gx) ≥ ω(g).Hence ω(g) = ω((gx)x−1

) ≥ ω(gx) ≥ ω(g).

Lemma 26.4. For any sequence (g1, . . . , gr) in G \ {1} the following asser-tions are equivalent :

i. The elements σ(g1), . . . , σ(gr) ∈ gr G are linearly independent overFp[P ];

ii. ω(gx11 · · · gxr

r ) = min1≤i≤r(ω(gi) + v(xi)) for any x1, . . . , xr ∈ Zp;

iii. ω(g−1h) = min1≤i≤r(ω(gi) + v(xi − yi)) for g := gx11 · · · gxr

r , h :=gy11 · · · gyr

r , and any x1, . . . , xr, y1, . . . , yr ∈ Zp.

Proof. i. =⇒ ii. By axiom (b) and Remark 26.3 we have ω(gx11 · · · gxr

r ) ≥ν := min1≤i≤r(ω(gi) + v(xi)). Let us therefore suppose that gx1

1 · · · gxrr ∈

Gν+. If {i1 < · · · < is} = {1 ≤ i ≤ r : ω(gxii ) = ω(gi) + v(xi) = ν}

then already gxi1i1

· · · gxisis

∈ Gν+ since Gν+ is normal in G. It follows thatσ(g

xi1i1

) + · · · + σ(gxisis

) = 0 in grν G. By Remark 26.3 the latter is in fact anontrivial linear relation in grG of the form

(ai1Pv(xi1

))(σ(gi1)) + · · · + (aisPv(xis ))(σ(gis)) = 0

which contradicts i.ii. =⇒ i. We consider any relation

q1(P )(σ(g1)) + · · · + qr(P )(σ(gr)) = 0

in grG with polynomials qi(P ) =∑

j aijPj in Fp[P ] where the coefficients

aij are the images in Fp of integers aij ∈ Z. Let νi := ω(gi). Inserting thedefinitions we may rewrite this relation more explicitly as

r∑

i=1

∑

j

gaijpj

i G(νi+j)+ = 0.

If we collect, for any ν, the terms in grν G then we obtain

r∏

i=1

gai,ν−νi

pν−νi

i ∈ Gν+


where we set ai,ν−νi := 0 if ν − νi is not a nonnegative integer. But by ii.the value of ω on this latter product has to be equal to

min1≤i≤r

(ω(gi) + ν − νi + v(ai,ν−νi)) = ν + min1≤i≤r

v(ai,ν−νi)

which therefore is strictly bigger than ν. It follows that ai,ν−νi = 0 for anyi and then, since ν was arbitrary, that all aij = 0. Hence q1(P ) = · · · =qr(P ) = 0.

iii. =⇒ ii. This implication is trivial.ii. =⇒ iii. Let ν := min1≤i≤r(ω(gi)+v(xi−yi)). Since Gν/Gν+ is central

in G/Gν+ by Remark 23.1 we compute iteratively

g−1hGν+ = g−yrr · · · g−y1

1 gx11 · · · gxr

r Gν+

= g−yrr · · · g−y2

2 gx1−y11 gx2

2 · · · gxrr Gν+

= gx1−y11 (g−yr

r · · · g−y33 gx2−y2

2 gx33 · · · gxr

r )Gν+

...

= gx1−y11 · · · gxr−yr

r Gν+.

It follows that ω(g−1h) = ω((gx1−y11 · · · gxr−yr

r )k) with ω(k) > ν and

ω(gx1−y11 · · · gxr−yr

r ) = ν

by ii. Hence (36) implies that ω(g−1h) = ν.

Proposition 26.5. If (G,ω) is of finite rank then for any sequence of ele-ments (g1, . . . , gr) in G \ {1} the following assertions are equivalent :

i. (g1, . . . , gr) is an ordered basis of (G,ω);

ii. σ(g1), . . . , σ(gr) is a basis of the Fp[P ]-module grG.

Proof. First we assume i. to hold true. Then σ(g1), . . . , σ(gr) are linearlyindependent by Lemma 26.4. To see that they generate gr G as an Fp[P ]-module let g = gx1

1 · · · gxrr be any element �= 1 in G, and let {i1 < · · · <

is} = {1 ≤ i ≤ r : ω(gxii ) = ω(g)}. Then

σ(g) = σ(gxi1i1

· · · gxisis

) = σ(gxi1i1

) + · · · + σ(gxisis

)

= (ai1Pv(xi1

))(σ(gi1)) + · · · + (aisPv(xis ))(σ(gis))

by Remark 26.3.


We now assume, vice versa, that ii. holds true. Because of Lemma 26.4it only remains to be seen that any g ∈ G can be written in the formg = gx1

1 · · · gxrr . In other words, if X ⊆ G denotes the image of the continuous

map

c : Zrp −→ G

(x1, . . . , xr) −→ gx11 · . . . · gxr

r

then we have to show that X = G. As the continuous image of a compactspace, X is closed in G. Hence it suffices to establish that, given any g �= 1in G and any ν > 0, there exists a g′ ∈ X such that gGν = g′Gν .

Step 1: By ii. we may write σ(g) = q1(P )(σ(g1))+ · · ·+qr(P )(σ(gr)) withappropriate polynomials qi(P ) ∈ Fp[P ]. By comparing degrees we see that,more precisely, there is a sequence 1 ≤ i1 < · · · < is ≤ r and p-adic unitsa1, . . . , as ∈ Z×

p with images a1, . . . , as ∈ F×p such that all ω(g) − ω(gij ) are

nonnegative integers and we have

σ(g) = (a1Pω(g)−ω(gi1

))(σ(gi1)) + · · · + (asPω(g)−ω(gis))(σ(gis))

= σ(gxi1i1

· · · gxisis

) with xij := ajpω(g)−ω(gij

).

In this way we have found an element g(1) := gxi1i1

· · · gxisis

∈ X such thatgGω(g)+ = g(1)Gω(g)+. But note that in addition we have

(38) ω(g) = ω(g(1)) = ω(gxi1i1

) = · · · = ω(gxisis

).

Step 2: We write g = g(1)h with ω(h) > ω(g). If h = 1 we have g ∈ X.Otherwise we apply the first step to h and obtain the element h(1) ∈ Xsuch that gGω(h)+ = g(1)hGω(h)+ = g(1)h(1)Gω(h)+. Using the additionalproperty (38) for h(1) and the centrality of Gω(h)/Gω(h)+ in G/Gω(h)+, byRemark 23.1, we conclude that

g(1)h(1)Gω(h)+ = g(2)Gω(h)+ for some g(2) ∈ X.

Hence g = g(2)k with ω(k) > ω(h). Again, if k = 1 we have g ∈ X, andotherwise we apply the first step to k. Proceeding inductively in this way weeither arrive at g ∈ X after finitely many steps or we construct an infinitesequence g(1), . . . , g(m), . . . in X as well as a strictly increasing sequence ofreal numbers ω(g) =: ν1 < · · · < νm < · · · such that

gGνm+ = g(m)Gνm+ for any m ≥ 1.

It remains to note that, since the set of values of ω is discrete by Lemma 26.1,we have νm ≥ ν for any sufficiently big m.


We point out that the implication from i. to ii. in the above propositiondoes not require the assumption that (G,ω) is of finite rank. It thereforeshows that the existence of an ordered basis implies the finite rank property.

Proposition 26.6. Any (G,ω) of finite rank has an ordered basis (of lengthequal to rank(G,ω)).

Proof. The graded Fp[P ]-module gr G is free of some finite rank r. In viewof Prop. 26.5 all that remains to be shown is that we find a basis of grGconsisting of homogeneous elements. For this it suffices to consider any of thedirect summands M := (grG)(ν). Let ν1 := min{ν + : ≥ 0, grν+� G �= 0}.Then M = Mν1 ⊕Mν1+1⊕· · · with Mν1+� := grν1+� G. We define M (1) ⊆ Mto be the Fp[P ]-submodule generated by Mν1 which, in fact, is equal to

M (1) = Mν1 ⊕ P (Mν1) ⊕ P 2(Mν1) ⊕ · · ·

Since M is torsionfree this shows that any Fp-basis of Mν1 is an Fp[P ]-basis ofM (1). It also shows that on the quotient module M/M (1) the multiplicationby P again is injective. Hence, by the argument in the proof of Remark25.2, the module M/M (1) remains torsionfree. We therefore may repeat thisprocess by defining M (2) ⊆ M to be the submodule such that M (2)/M (1) isgenerated by its homogeneous part of smallest degree ν2 > ν1. In this waywe construct an increasing sequence of submodules

M (1) ⊆ M (2) ⊆ · · · ⊆ M (s) = M

(which necessarily is finite by the finite generation of M) with the propertythat each step has an Fp[P ]-basis consisting of homogeneous elements. Bylifting all this basis elements to homogeneous elements of M we obtain sucha basis for M .

We point out that the method of proof of Prop. 26.6 in fact shows thatgrG always is a free Fp[P ]-module and that it has a basis consisting ofhomogeneous elements; the only difference is that the increasing sequenceM (i) of finitely generated free submodules in the above proof is exhaustivebut possibly not finite.

Corollary 26.7. If (G,ω) is of finite rank then G is topologically finitelygenerated.

Corollary 26.8. The members of an ordered basis of (G,ω) cannot be p-thpowers in G.


Proof. Let (g1, . . . , gr) be any ordered basis and suppose that gi0 = hp withh ∈ G for some 1 ≤ i0 ≤ r. By Prop. 26.5 we have σ(h) = q1(P )(σ(g1)) +· · · + qr(P )(σ(gr)) with appropriate polynomials qi(P ) ∈ Fp[P ]. Hence

σ(gi0) = P (σ(h)) = (Pq1(P ))(σ(g1)) + · · · + (Pqr(P ))(σ(gr)).

This contradicts the fact that, by Prop. 26.5, the σ(g1), . . . , σ(gr) are linearlyindependent over Fp[P ].

There is an important additional property a p-valuation can have. Notethat ω(gp) > p

p−1 for any g ∈ G.

Definition. (G,ω) is called saturated if any g ∈ G such that ω(g) > pp−1 is

a p-th power in G.

Remark 26.9. If (G,ω) is saturated then {gpn: g ∈ G} = G(n+ 1

p−1)+ is a

subgroup for any n ≥ 0.

Lemma 26.10. (G,ω) is saturated if and only if P (grG) = ⊕ν> pp−1

grν G.

Proof. The necessity of the condition is obvious. Vice versa let g ∈ G be anyelement with ν0 := ω(g) > p

p−1 . By assumption we have

g = gp1k1 with ν1 := ω(k1) > ν0

as well as k1 ∈ hp1Gν1+. Hence g ∈ gp

1hp1Gν1+. Note that ω(gp

1) = ν0 < ν1 =ω(hp

1). Prop. 25.1.i. therefore implies gp1h

p1Gν1+ = (g1h1)pGν1+. We see that

g = gp2k2 with g2 := g1h1 and ν2 := ω(k2) > ν1.

We now repeat this reasoning for k2 and continue inductively. In this way weobtain a sequence g1, . . . , gm, . . . in G as well as a strictly increasing sequenceof real numbers ω(g) = ν0 < · · · < νm < · · · such that

gGνm−1+ = gpmGνm−1+ for any m ≥ 1.

Using Prop. 25.1.ii. we deduce

ω(g−1m+1gm) + 1 = ω(g−p

m+1gpm) > νm−1 for any m ≥ 1.

Since the sequence (νm)m is unbounded by Lemma 26.1 it follows that thesequence (gm)m converges to some h ∈ G and that hp = g.


Proposition 26.11. Suppose that (G,ω) is of finite rank with ordered basis(g1, . . . , gr); then it is saturated if and only if

1p − 1

< ω(gi) ≤p

p − 1for any 1 ≤ i ≤ r.

Proof. By Cor. 26.8 the gi are not p-th powers. Hence, if (G,ω) is saturated,they must satisfy ω(gi) ≤ p

p−1 . Let us now assume, vice versa, that the latterinequalities are satisfied. Then the Fp[P ]-basis σ(g1), . . . , σ(gr) of grG (cf.Prop. 26.5) lies in ⊕ν≤ p

p−1grν G. Therefore each grν G for ν > p

p−1 must liein the image of the operator P . It follows that the assumption of Lemma26.10 is satisfied, and (G,ω) consequently is saturated.

Corollary 26.12. Suppose that (G,ω) is of finite rank and saturated andlet (g1, . . . , gr) be an ordered basis; for any n ≥ 0 we then have:

i. Gpn= {gx1

1 · · · gxrr : v(x1), . . . , v(xr) ≥ n};

ii. two elements gx11 · · · gxr

r and gy11 · · · gyr

r lie in the same coset moduloGpn

if and only if v(x1 − y1), . . . , v(xr − yr) ≥ n;

iii. [G : Gpn] = pn rank(G,ω).

Proof. By Remark 26.9 we have gx11 · · · gxr

r ∈ Gpnif and only if v(xi) >

n + 1p−1 − ω(gi) for any 1 ≤ i ≤ r. But −1 ≤ 1

p−1 − ω(gi) < 0 according toProp. 26.11. Hence, since v(xi) is a nonnegative integer, the above inequal-ities are equivalent to v(xi) ≥ n. This shows the first assertion. The secondone follows by exactly the same argument based on Lemma 26.4.iii. The lastassertion is an immediate consequence of the second one.

We now turn to the question of variation of the p-valuation on the pro-p-group G. But we do keep our initial p-valuation which defines the topologyof G and is of finite rank. To avoid confusion we write grω′

ν G and grω′G

for the associated graded Lie algebra when it is formed with respect to anyother p-valuation ω′ on G.

There is one obvious possibility of modifying ω. Let C be any real numberwhich satisfies

0 < C <(ming �=1

ω(g))− 1

p − 1.

We note that by Lemma 26.1 this minimum exists and is strictly bigger than1

p−1 so that such constants C do exist. It is straightforward to see that

ωC(g) := ω(g) − C


is another p-valuation on G which defines the topology of G. But, in fact,the axiom (c) holds in the stronger form

(39) ωC([g, h]) > ωC(g) + ωC(h)

for any g, h ∈ G. This, of course, means that the Lie algebra grωC G isabelian. This simple observation will later on turn out to be very useful sothat we record it as part of the subsequent lemma.

Lemma 26.13. Suppose that (G,ω) is of finite rank ; for any 0 < C <(ming �=1 ω(g)) − 1

p−1 we have:

i. grωC G is an abelian Lie algebra;

ii. rank(G,ωC) = rank(G,ω);

iii. suppose that there is an ordered basis (g1, . . . , gr) of (G,ω) such that

maxi

ω(gi) − mini

ω(gi) < 1;

then (G,ωC) is saturated provided in addition C ≥ (maxi ω(gi))− pp−1

(and such C exists).

Proof. i. This was the observation (39).ii. It is clear that the Fp[P ]-modules grω G and grωC G coincide up to a

shift of the degrees of their homogeneous components.iii. It also is clear that (g1, . . . , gr) is an ordered basis of (G,ωC) as well.

We have

ωC(gj) = ω(gj) − C ≤ ω(gj) − maxi

ω(gi) +p

p − 1≤ p

p − 1.

Hence Prop. 26.11 applies and shows that (G,ωC) is saturated.

Lemma 26.14. Suppose that (G,ω) is of finite rank and pick any orderedbasis (g1, . . . , gr); for any ν ≥ maxi ω(gi) and any sufficiently small ε > 0,

ω′(g) := ω(g) − ν +1

p − 1+ ε

is a p-valuation on Gν which defines the topology of Gν and such that (Gν , ω′)

is of finite rank (equal to rank(G,ω)) and is saturated.


Proof. We have gx11 · · · gxr

r ∈ Gν if and only if v(xi) ≥ ν − ω(gi) for any1 ≤ i ≤ r. It follows that (h1, . . . , hr) is an ordered basis of (Gν , ω|Gν)where hi := gpni

i and ni is the smallest integer greater or equal to ν −ω(gi).Note that ni ≥ 0 since ν − ω(gi) ≥ 0 by assumption. We have

ν ≤ ω(gi) + ni = ω(hi) < ν + 1

so that ω|Gν satisfies the assumption in Lemma 26.13.iii. By this lemma (forC := ν − ε − 1

p−1) it therefore remains to observe that ν ≥ 1p−1 + ε and

(max

iω(hi)

)− p

p − 1< ν + 1 − ε − p

p − 1

= ν − ε − 1p − 1

<(min

iω(hi)

)− 1

p − 1

whenever ε > 0 is sufficiently small.

Formulated a little less technical the above lemma says that if (G,ω) is offinite rank then there is an open normal subgroup H ⊆ G and a p-valuationω′ on H which defines the topology of H and such that (H,ω′) is of finiterank and saturated.

The above two lemmas, which in themselves are rather simple, haveimportant consequences.

Proposition 26.15. If (G,ω) is of finite rank then the subgroups Gpn, for

n ≥ 1, form a fundamental system of open neighbourhoods of the identityelement in G, and

rank(G,ω) = limn→∞

v([G : Gpn])

n.

Proof. Step 1: We assume in addition that (G,ω) is saturated. Then theassertion is an immediate consequence of Remark 26.9 and Cor. 26.12.iii.In fact the second part of the assertion holds in the stronger form thatrank(G,ω) = v([G : Gpn

])/n for any n ≥ 1.Step 2: According to Lemma 26.14 we find an open subgroup H ⊆ G

together with a p-valuation ω′ on H defining the topology and such that(H,ω′) is saturated of rank equal to rank(G,ω). If [G : H] = pe then Gpn+e ⊆Hpn

for any n ≥ 1. This together with the first step for (H,ω′) implies the


first part of the assertion for G as well as, for any n ≥ e,

v([G : Gpn]) = v([H : Hpn

]) + v([G : H]) − v([Gpn: Hpn

])

= n rank(H,ω′) + e − v([Gpn: Hpn

])

= n rank(G,ω) + e − v([Gpn: Hpn

])

≥ n rank(G,ω) + e − v([Hpn−e: Hpn

])= n rank(G,ω) + e − e rank(H,ω′)

and hence

rank(G,ω) +e

n− e

nrank(H,ω′) ≤ v([G : Gpn

])n

≤ rank(G,ω) +e

n.

Passing to the limit with respect to n therefore results in the asserted equa-tion.

Proposition 26.16. Suppose that (G,ω) is of finite rank ; then any otherp-valuation ω′ on G defines the topology of G and satisfies rank(G,ω′) =rank(G,ω).

Proof. By Prop. 26.15 the Gpnform a fundamental system of open neigh-

bourhoods of the identity element. The axiom (d) therefore implies that thetopology defined by ω′ necessarily is coarser than the topology of G. ButG is compact and the topology defined by ω′ is Hausdorff. Hence the lattermust be equal to the topology of G. In particular, there is a ν0 > 1

p−1 suchthat

(40) Gω′ν0

⊆ Gp

where Gω′ν0

denotes the filtration of G with respect to the p-filtration ω′.If we establish that grω′

G is finitely generated over Fp[P ] then the limitformula in Prop. 26.15 shows that rank(G,ω′) = rank(G,ω). By the remarkafter Prop. 26.6, whether grω′

G is finitely generated or not, there is a family{hi : i ∈ I} ⊆ G \ {1} such that the σ(hi) form a basis of grω′

G over Fp[P ].The argument in the proof of Cor. 26.8 shows that none of the hi is a p-thpower in G. Hence it follows from (40) that ω′(hi) < ν0 for any i ∈ I. We seethat all σ(hi) are contained in

⊕0<ν<ν0

grω′ν G, which is finite since G/Gω′

ν0

is finite. It follows that I has to be finite.

As promised earlier we now have seen that the rank of (G,ω) actuallyis independent of the choice of ω. Hence, from now on we simply call G offinite rank and write rankG. The meaning of this invariant of G will becomeclearer in the next section. We also introduce the following terminology.


Definition. A pro-p-group G is called p-valuable if there exists a p-valuationω on G which defines the topology on G and such that the rank of G is finite.

27 Compact p-Adic Lie Groups

The question of the existence of p-valuable groups is overdue. It will beanswered in this section with the help of the theory of p-adic Lie groups.By a p-adic Lie group G we always will mean a Lie group G over the fieldQp of p-adic numbers (cf. Sect. 13). We recall that this means that G is a(locally analytic) manifold over Qp as well as a group such that the groupmultiplication

mG : G × G −→ G

(g, h) −→ gh

is a map of manifolds. As a manifold G has a certain finite dimension dimGwhich also can be characterized as the dimension as a Qp-vector space ofthe tangent space Th(G) of G at any point h ∈ G. In the following it will beconvenient to some times denote the unit element in G by e instead of 1.

Theorem 27.1. Let G be any p-adic Lie group; then there exist a compactopen subgroup G′ ⊆ G and an integral valued p-valuation ω on G′ definingthe topology of G′ such that we have:

i. (G′, ω) is saturated ;

ii. rankG′ = dim G.

Proof. Let d := dimG. We pick a chart c = (U,ϕ, Qdp) for G around the unit

element e ∈ G such that ϕ(e) = 0. By the continuity of the multiplicationmap mG there is an open neighbourhood V ⊆ U of e such that mG(V ×V ) ⊆U . Then (V, ϕ|V, Qd

p) is a chart around e as well, and the map

ϕ ◦ mG ◦ (ϕ−1 × ϕ−1) : ϕ(V ) × ϕ(V ) −→ ϕ(U)

is locally analytic. By expanding this map around 0 we find a d-tuple(F1, . . . , Fd) of power series

Fi(X,Y ) =∑

α,β

ci,α,βXαY β

in variables X = (X1, . . . , Xd) and Y = (Y1, . . . , Yd) with coefficients ci,α,β ∈Qp (here, as usual, α, β run over all multi-indices and Xα = Xα1

1 · · ·Xαdd if

α = (α1, . . . , αd)) such that, for any sufficiently big n > 0, we have

27 Compact p-Adic Lie Groups 193

– pnZdp ⊆ ϕ(V ),

– lim|α|+|β|→∞(v(ci,α,β) + n(|α| + |β|)

)= ∞ for any 1 ≤ i ≤ d, and

– (F1(x, y), . . . , Fd(x, y)) = ϕ(ϕ−1(x)ϕ−1(y)) for any x = (x1, . . . , xd)and y = (y1, . . . , yd) in pnZd

p

(here |α| = α1 + · · ·+αd). By possibly increasing n further one has (Lemma18.7)

v(ci,α,β) + n(|α| + |β|) ≥ n for any α, β, and i.

This implies that the ϕ−1(pnZdp), for sufficiently big n, are compact open

subsets in G which are multiplicatively closed. By repeating this reasoningfor the power series expansion of the inverse map g −→ g−1 around e one seesthat these ϕ−1(pnZd

p) for large n actually are subgroups of G. We now pickany of these large n and replace the chart ϕ by the chart ψ := p−nϕ. Thishas the effect that the new coefficients ci,α,β as well as the new coefficients ofthe power series for g −→ g−1 lie in Zp. By composition of power series thenalso the maps g −→ gp and (g, h) −→ [g, h] have power series expansions(with respect to the chart ψ) converging on all of Zd

p with coefficients in Zp

(Prop. 5.4). We putG′ := ψ−1(p2Zd

p)

andω(g) := + δ if g ∈ ψ−1(p�+1Zd

p) \ ψ−1(p�+2Zdp)

with δ := 1 for p = 2 and δ := 0 for p �= 2. Obviously ω satisfies theaxioms (a) and (b) of a p-valuation, the latter since we already know thatG′

�+δ = ψ−1(p�+1Zdp) is a subgroup of G′ for any ∈ N. We note that

Fi(X,Y ) = Xi + Yi + terms of degree ≥ 1 both in X and Y

for any 1 ≤ i ≤ d (Prop. 18.6.i.). This implies that ψ actually inducesisomorphisms of groups

gr�+δ G′ ∼= p�+1Zdp/p�+2Zd

p

and, in particular, that

[G′ : G′�+δ] = pd(�−1)

for any ∈ N. It also shows that

power series expansion of g −→ gp = pX + terms of degree ≥ 2


from which one deduces immediately the axiom (d) for ω. Finally, the factthat the commutator map (g, h) −→ [g, h] has value e if one of the com-ponents is equal to e translates into its power series expansion consistingentirely of terms which are of degree ≥ 1 in both X and Y . This implies theaxiom (c). (We point out that it was here in case p = 2 that the p2 in thedefinition of G′ is needed.) Hence we have established that ω is an integralvalued p-valuation on G′. It obviously defines the topology of G′.

For any ∈ N the operator P : gr�+δ G′ −→ gr�+δ+1 G′ is an injec-tive map between finite abelian groups of the same cardinality. Hence it isbijective. This shows that gr G′ is finitely generated by gr1+δ G′ as an Fp[P ]-module. It also allows for applying Lemma 26.10 to conclude that (G′, ω) issaturated. At last, we use Prop. 26.15 in order to see that

rankG = limn→∞

v([G′ : G′pn])

n

= limn→∞

v([G′ : G′n+1+δ])

n

= limn→∞

dn

n= d = dim G.

This result gives us, of course, an abundance of p-valuable groups. Infact, it gives all of them. We will show in Sect. 29 that any p-valuable grouphas a natural structure of a p-adic Lie group.

Chapter VI

Completed Group Ringsof p-Valued Groups

Throughout this part we fix a p-valuable group G together with a p-valuationω on it. We recall that O is our complete discrete valuation ring. We alwayswill assume that p is a prime element of O. In particular, Zp ⊆ O and thediscrete valuation of K extends v. It therefore will also be denoted by v.

28 The Ring Filtration

Let us pick an ordered basis (g1, . . . , gr) of (G,ω). We then have the home-omorphism

c : Zrp

∼−−→ G

(x1, . . . , xr) �−→ gx11 · . . . · gxr

r .

It induces, by pulling back functions, an isomorphism of O-modules

c∗ : C(G) �−−→ C(Zrp)

and hence dually, by Lemma 21.1, an isomorphism of O-modules

c∗ : Λ(Zrp)

�−−→ Λ(G).

Of course, this is not a ring isomorphism in general. Alternatively it canbe obtained by applying the universal property Prop. 19.2 to the maps cand c−1. This shows that c∗ is a topological isomorphism of pseudocompactO-modules.

We have computed Λ(Zrp) explicitly in Prop. 20.1 as the formal power

series ring in variables X1, . . . , Xr over O. This leads to the following descrip-tion of Λ(G). We introduce the elements bi := gi−1 and bα := bα1

1 · . . . ·bαrr ,

for any multi-index α = (α1, . . . , αr), in Λ(G). Then

c : O[[X1, . . . , Xr]]�−−→ Λ(G)

∑

α

cαXα �−→∑

α

cαbα


195

http://dx.doi.org/10.1007/978-3-642-21147-8_6

196 VI Completed Group Rings of p-Valued Groups

is a topological isomorphism of pseudocompact O-modules. We emphasizethat both series constitute convergent expansions in the respective pseudo-compact ring. This allows us to define a function

ω : Λ(G) \ {0} −→ [0,∞)

by

ω

(∑

α

cαbα

)

:= infα

(

v(cα) +r∑

i=1

αiω(gi)

)

.

With the convention that ω(0) := ∞ it obviously satisfies

ω(λ1 + λ2) ≥ min(ω(λ1), ω(λ2))

for any λ1, λ2 ∈ Λ(G). The values of ω lie in the additive submonoid of[0,∞) generated by 1 and the values of ω. Because of Lemma 26.1 this setof values is a discrete subset of [0,∞). In particular, we have

ω

(∑

α

cαbα

)

= minα

(

v(cα) +r∑

i=1

αiω(gi)

)

.

We introduce, for any real number ν ≥ 0, the O-submodules

Λ(G)ν := {λ ∈ Λ(G) : ω(λ) ≥ ν}

andΛ(G)ν+ := {λ ∈ Λ(G) : ω(λ) > ν}

in Λ(G).

Remark 28.1. The Λ(G)ν form a fundamental system of open neighbour-hoods of zero in the pseudocompact topology of Λ(G).

Proof. Let m denote the maximal ideal in the ring O[[X1, . . . , Xr]], and recallthat its pseudocompact topology is the m-adic one. We pick any integerM ≥ maxi ω(gi). Then

M

(

v(cα) +∑

i

αi

)

≥ v(cα) +∑

i

αiω(gi) ≥1

p − 1

(

v(cα) +∑

i

αi

)

.

It follows that

c(mM(p−1)n) ⊆ Λ(G)Mn ⊆ c(mn) for any n ∈ N.


In order to understand this filtration we compare it with a less explicitbut seemingly more conceptual filtration (Jν)ν≥0 where Jν is defined to bethe smallest closed O-submodule of Λ(G) which contains all elements

p�(h1 − 1) · · · (hs − 1)

for any �, s ≥ 0 and h1, . . . , hs ∈ G with

� + ω(h1) + · · · + ω(hs) ≥ ν.

The density of O[G] in Λ(G) implies

J0 = Λ(G).

We obviously have

Jν ⊇ Jν′ , if ν ≤ ν ′, and Jν · Jν′ ⊆ Jν+ν′ .

In particular, each Jν is a two-sided ideal in Λ(G) which depends on ω butnot on the choice of the ordered basis (g1, . . . , gr). It also is clear that

Λ(G)ν ⊆ Jν .

In particular, by Remark 28.1, the Jν are open in Λ(G). By defining

Jν+ :=⋃

ν′>ν

Jν′ and grν Λ(G) := Jν/Jν+

we obtain the graded algebra

grΛ(G) :=⊕

ν≥0

grν Λ(G)

over the graded ringgrO :=

⊕

n≥0

pnO/pn+1O.

We point out that the discreteness of the set of values of ω implies that theset {ν : grν Λ(G) = 0} is discrete as well. Next we observe that for g, h ∈ Gν

we have g − 1, h − 1 ∈ Jν as well as

(gh − 1) + Jν+ = (g − 1) + (h − 1) + (g − 1)(h − 1) + Jν+

= (g − 1) + (h − 1) + Jν+.


Hence

Lν : grν G −→ grν Λ(G)gGν+ �−→ (g − 1) + Jν+

is a homomorphism of abelian groups. We put

L :=⊕

ν

Lν : grG −→ gr Λ(G).

Secondly, for g = 1, we observe that

gp − 1 + J(ω(g)+1)+ = ((g − 1) + 1)p − 1 + J(ω(g)+1)+

=p∑

j=1

(p

j

)

(g − 1)j + J(ω(g)+1)+

= p(g − 1) + p(g − 1)p−2∑

j=1

Z(g − 1)j + (g − 1)p + J(ω(g)+1)+

= p(g − 1) + J(ω(g)+1)+,

where, for the last identity, we have used that pω(g) > ω(g) + 1 by axiom(a). This shows that L is a homomorphism of graded Fp[P ]-modules withrespect to the Fp-algebra map Fp[P ] ↪→ grO which sends P to p + p2O ∈gr1 O. Thirdly we consider, for any g, h = 1, the identity

(g − 1)(h − 1) − (h − 1)(g − 1) = gh − hg = ([g, h] − 1)hg.

Since

([g, h] − 1)(hg − 1) ∈ Jω([g,h]) · Jω(hg) ⊆ Jω(g)+ω(h) · Jmin(ω(g),ω(h))

⊆ J(ω(g)+ω(h))+

we deduce that

(g − 1)(h − 1) − (h − 1)(g − 1) + J(ω(g)+ω(h))+

= ([g, h] − 1)hg + J(ω(g)+ω(h))+ = [g, h] − 1 + J(ω(g)+ω(h))+.

This finally implies that L is a homomorphism of graded Lie algebras overFp[P ]. By the universal property of the universal enveloping algebra U(grG)([B-LL] Chap. I §2.1 Prop. 1 or §14) the map L therefore extends uniquelyto a homomorphism of graded associative grO-algebras

L : grO ⊗Fp[P ] U(grG) −→ gr Λ(G).


Lemma 28.2.⋂

ν≥0 Jν = 0.

Proof. By Prop. 19.7 the pseudocompact topology of Λ(G) coincides withthe m(G)-adic topology. Let n ∈ N be arbitrary but fixed. We find an N ∈N (G) such that Jn,N (G) ⊆ m(G)n. We note that h − 1 ∈ Jn,N (G) for anyh ∈ N . Next we choose a ν0 > 0 such that Gν0 ⊆ N . Then h − 1 ∈ m(G)n

whenever ω(h) ≥ ν0. We put ν := n(ν0 +1) and claim that Jν ⊆ m(G)n. Letp�(h1−1) · · · (hs−1) be any element with �+ω(h1)+· · ·+ω(hs) ≥ ν. We mayassume that � < n and ω(h1), . . . , ω(hs) < ν0. But then n + sν0 > ν = n +nν0. Hence we have s > n and therefore p�(h1 − 1) · · · (hs − 1) ∈ m(G)n.

Theorem 28.3. i. The map

L : grO ⊗Fp[P ] U(grG)∼=−−→ gr Λ(G)

is an isomorphism.

ii. Jν = Λ(G)ν for any ν ≥ 0.

Proof. We first establish the surjectivity of L. By Thm. 22.3.ii. the sum oftwo closed submodules in a pseudocompact module is closed. This impliesthat we have

Jν = Jν+ +∑

�+ω(h1)+···+ω(hs)=ν

Op�(h1 − 1) · · · (hs − 1)

for any ν ≥ 0. It follows that grΛ(G) as a grO-algebra is generated by theelements (g − 1) + Jω(g)+, for g ∈ G \ {1}, which all lie in the image of L.

The injectivity will be shown alongside with the assertion ii. By Prop.26.5 the elements σ(g1), . . . , σ(gr) form a basis of the Fp[P ]-module gr G. ThePoincare-Birkhoff-Witt theorem (cf. [B-LL] Chap. I §2.7 Cor. 3) then impliesthat the (ordered) monomials σ(g1)α1 · . . . · σ(gr)αr with α = (α1, . . . , αr)running over all multi-indices form an Fp[P ]-basis of U(grG). This implies,by the surjectivity which we know already, that grν Λ(G) is generated as aO/pO-vector space by the finitely many elements

p�(g1 − 1)α1 · · · (gr − 1)αr + Jν+ for � + α1ω(g1) + · · · + αrω(gr) = ν.

For the injectivity we therefore must see that these elements are O/pO-linearly independent. But first we draw, from this generation property, theconclusion that

Jν = Jν+ +∑

�+α1ω(g1)+···+αrω(gr)=ν

Op�(g1 − 1)α1 · · · (gr − 1)αr


and henceJν = Jν+ + Λ(G)ν

holds true for any ν ≥ 0. But the set of all ν such that grν Λ(G) = 0 isdiscrete in [0,∞). We therefore get by an inductive argument that

Jν = Jν′ + Λ(G)ν for any ν ′ > ν.

Since Λ(G)ν is closed in Λ(G) by Remark 28.1 we may apply Lemma 22.2and, moreover using Lemma 28.2, obtain

Jν =

(⋂

ν′>ν

Jν′

)

+ Λ(G)ν = Λ(G)ν .

Now suppose that we have a relation of the form∑

�+α1ω(g1)+···+αrω(gr)=ν

cαp�(g1 − 1)α1 · · · (gr − 1)αr ∈ Jν+ = Λ(G)ν+

for some cα ∈ O. Then

ν < ω

(∑

α1ω(g1)+···+αrω(gr)=ν−�

cαp�(g1 − 1)α1 · · · (gr − 1)αr

)

= minα1ω(g1)+···+αrω(gr)=ν−�

(

v(cα) + � +r∑

i=1

αiω(gi)

)

= ν + minα

v(cα).

It follows that all cα lie in pO which is the linear independence over O/pOwhich we wanted.

Corollary 28.4. i. The function ω only depends on ω and not on thechoice of the ordered basis (g1, . . . , gr).

ii. ω(g − 1) = ω(g) for any g ∈ G.

Proof. i. This follows from Thm. 28.3.ii.ii. The map L : grG −→ ⊕νΛ(G)ν/Λ(G)ν+ is injective as a consequence

of Thm. 28.3.

Corollary 28.5. i. gr Λ(G) is an integral domain.

ii. ω(λ1λ2) = ω(λ1) + ω(λ2) for any λ1, λ2 ∈ Λ(G).

29 Analyticity 201

iii. Λ(G) is an integral domain.

Proof. i. Using Thm. 28.3.i. this follows from the fact that the universalenveloping algebra of a Lie algebra which is free over the base ring is anintegral domain (cf. [B-LL] Chap. I §2.7 Cor. 7) provided the base ring is anintegral domain.

ii. Because, by Thm. 28.3.ii., the Λ(G)ν satisfy Λ(G)ν ·Λ(G)ν′ ⊆ Λ(G)ν+ν′

we have ω(λ1λ2) ≥ ω(λ1) + ω(λ2). The equality then is an immediate con-sequence of the first assertion.

iii. This is a direct consequence of the second assertion.

Corollary 28.6. If the p-valuation ω is such that the Lie algebra gr G isabelian then gr Λ(G) is a polynomial ring in 1+rankG variables over O/pO.

Proof. By construction, the universal enveloping algebra of an abelian Liealgebra g is the symmetric algebra over g. In our case gr G is free overFp[P ] of rank equal to rankG. Hence U(grG) is a polynomial ring in rankGvariables over Fp[P ]. We now apply Thm. 28.3.i.

29 Analyticity

As an application of the methods of computation in the completed groupring Zp[[G]], which we have established in the last section, we now will beable to see that our p-valuable group G naturally is a p-adic Lie group.

As before we pick an ordered basis (g1, . . . , gr) of (G,ω) so that we havethe homeomorphism

c : Zrp

∼−−→ G

(x1, . . . , xr) �−→ gx11 · . . . · gxr

r .

If ϕ denotes the inverse of c then we may view the triple (G,ϕ, Qrp), which

by abuse of notation we also denote by c, as a “global” chart for G. Byequipping G with the maximal atlas equivalent to the atlas consisting of thesingle chart c we obtain the structure of an r-dimensional manifold over Qp

on G (cf. Sects. 7 and 8). Equivalently this structure is characterized by thefact that it makes the original map c into an isomorphism of manifolds. Wealso will need the individual “coordinate functions”

ϕi : G −→ Qp

g �−→ xi if g = gx11 · . . . · gxr

r

for 1 ≤ i ≤ r. They, by construction, are locally analytic.


Remark 29.1. For any h ∈ G and any y ∈ Zp we have the convergentexpansion

hy =∑

n≥0

(y

n

)

(h − 1)n

in Zp[[G]].

Proof. We recall the definition(

y

n

)

:=y(y − 1) · · · (y − n + 1)

n!for n ≥ 1

and(y0

):= 1. Because of the integrality of the usual binomial coefficients

and the density of Z in Zp we have(

yn

)∈ Zp. We write y = limj→∞ mj as

the limit of a sequence of integers mj ∈ Z. Then

hy = hlim mj = limj→∞

hmj (in G) = limj→∞

hmj (in Zp[[G]])

= limj→∞

((h − 1) + 1)mj

= limj→∞

∑

n≥0

(mj

n

)

(h − 1)n

=∑

n≥0

(

limj→∞

(mj

n

))

(h − 1)n

=∑

n≥0

(y

n

)

(h − 1)n

where, in order to exchange the limit and the sum, we have used thatlimn→∞ ω((h − 1)n) = ω(h) limn→∞ n = ∞.

Proposition 29.2. Let h1, . . . , hs be any elements in G; then, for any1 ≤ i ≤ r, there is a power series Fi(Y ) =

∑α ci,αY α in s variables

Y = (Y1, . . . , Ys) with coefficients in Qp such that we have:

i. lim|α|→∞ v(ci,α) = ∞;

ii. Fi(y1, . . . , ys) ∈ Zp for any (y1, . . . , ys) ∈ Zsp;

iii. hy11 · · ·hys

s = gF1(y1,...,ys)1 · · · gFr(y1,...,ys)

r for any (y1, . . . , ys) ∈ Zsp.

In particular, the functions

29 Analyticity 203

Zsp −→ Qp

(y1, . . . , ys) �−→ ϕi(hy11 · · ·hys

s ),

for 1 ≤ i ≤ s, are locally analytic.

Proof. We note that the condition i. assures that the power series Fi con-verges on Zs

p so that the assertion in ii. makes sense.By multiplying the expansions in Remark 29.1 we obtain

(41) gx11 · · · gxr

r =∑

α

(x1

α1

)

· · ·(

xr

αr

)

bα

and in the same way

(42) hy11 · · ·hys

s =∑

β

(y1

β1

)

· · ·(

ys

βs

)

(h1 − 1)β1 · · · (hs − 1)βs .

We now insert into the right hand side of (42) the “standard” expansions

(43) (h1 − 1)β1 · · · (hs − 1)βs =∑

α

cα,βbα

in Zp[[G]] and obtain

(44) hy11 · · ·hys

s =∑

α

(∑

β

(y1

β1

)

· · ·(

ys

βs

)

cα,β

)

bα.

By comparing (41) and (44) for xi = ϕi(hy11 · · ·hys

s ) we deduce(

ϕ1(hy11 · · ·hys

s )α1

)

· · ·(

ϕr(hy11 · · ·hys

s )αr

)

=∑

β

(y1

β1

)

· · ·(

ys

βs

)

cα,β.

Moreover, applying Cors. 28.4.ii. and 28.5.ii. to (43) gives

minα

(

v(cα,β) +r∑

i=1

αiω(gi)

)

= ω((h1 − 1)β1 · · · (hs − 1)βs) =s∑

j=1

βjω(hj).

In particular, we see that, for any fixed α, we indeed have lim|β|→∞ v(cα,β) =∞. We now specialize to the multi-index α = i := (. . . , 0, 1, 0, . . .) which hasa one in the i-th place, and we get

ϕi(hy11 · · ·hys

s ) =∑

β

(y1

β1

)

· · ·(

ys

βs

)

ci,β


and

v(ci,β) ≥ −ω(gi)+s∑

j=1

βjω(hj) = −ω(gi)+s∑

j=1

βj

p − 1+

s∑

j=1

βj

(

ω(hj)−1

p − 1

)

.

Using that v(n!) ≤ np−1 (cf. Lemma 2.2) we obtain the estimate

v

(ci,β

β1! · · ·βs!

)

≥ −ω(gi) +s∑

j=1

βj

(

ω(hj) −1

p − 1

)

.

It therefore follows from axiom (a) that

lim|β|→∞

v

(ci,β

β1! · · ·βs!

)

= ∞.

On the other hand

Fβ(Y1, . . . , Ys) :=s∏

j=1

Yj(Yj − 1) · · · (Yj − βj + 1)

is a polynomial of degree |β| with coefficients in Z satisfying

Fβ(y1, . . . , ys) =(

y1

β1

)

· · ·(

ys

βs

)

β1! · · ·βs!.

We conclude that the power series

Fi :=∑

β

ci,β

β1! · · ·βs!Fβ

has all the asserted properties.

Remark 29.3. If (G,ω) is saturated then the power series F1, . . . , Fr inProp. 29.2 have coefficients in Zp.

Proof. From the proof of Prop. 29.2 (and with its notations) we know that

v

(ci,β

β1! · · ·βs!

)

≥ −ω(gi) −s∑

j=1

v(βj !) +s∑

j=1

βjω(hj)

and that it suffices to show that

v

(ci,β

β1! · · ·βs!

)

≥ 0.

29 Analyticity 205

But the left hand side is an integer so that we, in fact, need only to establishthat

v

(ci,β

β1! · · ·βs!

)

> −1.

Moreover, by looking at (43) we see that ci,0 = 0. Hence we may assume thatβj0 = 0 for some 1 ≤ j0 ≤ s. Axiom (a) implies M := minj ω(hj) > 1

p−1 . ByProp. 26.11 we have ω(gi) ≤ p

p−1 . Therefore showing that

− p

p − 1−

∑

j

v(βj !) + M∑

j

βj > −1

or, equivalently, that

− 1p − 1

−∑

j

v(βj !) + M∑

j

βj > 0

is sufficient. The left hand side is equal to(

M − 1p − 1

) ∑

j

βj +∑

j =j0

(βj

p − 1− v(βj !)

)

+(

βj0 − 1p − 1

− v(βj0 !))

which is greater or equal to (M − 1p−1)βj0 > 0 since 1

p−1 < M , v(βj !) ≤ βj

p−1 ,

and v(βj0 !) ≤βj0

−1

p−1 (cf. Lemma 2.2).

Corollary 29.4. The structure of G as a manifold over Qp is independentof the choice of ω and the choice of an ordered basis (g1, . . . , gr).

Proof. Apply Prop. 29.2 to any ordered basis (h1, . . . , hr) with respect to apossibly different p-valuation. It follows that the coordinate changes betweenany two ordered bases are locally analytic maps. Hence the correspondingtwo global charts belong to the same maximal atlas.

Corollary 29.5. The multiplication map mG is a morphism of manifolds.

Proof. This follows by applying Prop. 29.2 to the sequence of elements(h1, . . . , h2r) := (g1, . . . , gr, g1, . . . , gr).

Corollary 29.6. Any p-valuable pro-p-group G carries a natural structureof a compact p-adic Lie group, and dimG = rankG.

In fact, as we will see presently, a much stronger statement holds true.


Lemma 29.7. Let G be a p-adic Lie group which is pro-p; for any g ∈ Gwe have:

i. The map

cg : Zp −→ G

x �−→ gx

is a homomorphism of Lie groups;

ii. if g is the image of y ∈ Lie(G) under some exponential map for G (cf.Cor. 18.19) then

T0(cg) : T0(Zp) = Qp −→ Lie(G)x �−→ xy.

Proof. In Sect. 26 we have seen that cg is a well defined continuous homo-morphism of groups. It therefore suffices to show that its restriction cg|pnZp,for some n ≥ 0, is a locally analytic map. Equivalently it suffices to showthat cg is a locally analytic map after we have replaced g by gpn

for somen ≥ 0. The continuity of cg implies

limn→∞

gpn= cg( lim

n→∞pn) = cg(0) = 1.

Using Cor. 18.19 we therefore may replace G by Gε, for some sufficientlysmall ε > 0, where {Gε}ε denotes the Campbell-Hausdorff Lie group germ ofLie(G). But in this case it follows from Prop. 17.2 by induction that gm = mgfor any integer m ≥ 0. Since N0 is dense in Zp we obtain by continuity that,for Gε, the map cg coincides with the obviously locally analytic map

Zp −→ Gε ⊆ Lie(G)x �−→ xg.

It remains to note that the associated map between tangent spaces at 0simply is

Qp −→ Lie(G)x �−→ xg.

Theorem 29.8. Any p-valuable pro-p-group G carries a unique structureof manifold over Qp which makes it into a p-adic Lie group.

Proof. Let G1 denote G viewed as a p-adic Lie group according to Cor. 29.6.We write G2 for G equipped with any other (fixed) structure of a p-adic Lie

29 Analyticity 207

group. We emphasize that our only assumption is that the identity mapG1

idG−−−→ G2 is an isomorphism of abstract groups. The assertion claimsthat idG then necessarily is an isomorphism of Lie groups.

We begin by picking an ordered basis (g1, . . . , gr) of G1. Then, by con-struction,

c1 : Zrp

∼−−→ G1

(x1, . . . , xr) �−→ gx11 · . . . · gxr

r

is an isomorphism of manifolds. Viewed as a map c1 : Zrp −→ G2 it at

least is locally analytic as a consequence of Lemma 29.7.i. This implies thatG1

idG−−−→ G2 is a homomorphism of p-adic Lie groups. Next we will showthat the tangent map Lie(idG) is bijective.

We also pick a Qp-basis y1, . . . , ys of Lie(G2). By replacing the yj byappropriate nonzero scalar multiples we may assume that this basis lies inthe domain of definition of some exponential map expG2,ε for G2. We puthj := expG2,ε(yj). According to Lemma 29.7 the map

c2 : Zsp −→ G2

(y1, . . . , ys) �−→ hy11 · . . . · hys

s

is locally analytic with associated tangent map (cf. Cor. 13.5)

T0(c2) : T0(Zsp) = Qs

p −→ Lie(G2)

(y1, . . . , ys) �−→ y1y1 + · · · + ysys.

In particular, T0(c2) is bijective. Since, by the same lemma, c2 also can beviewed as a locally analytic map Zs

p −→ G1 there exists, by Prop. 29.2, alocally analytic map ψ : Zs

p −→ Zrp such that the diagram

Zrp

c1G1

idG

Zsp

ψ

c2G2

is commutative. On tangent spaces we obtain

Qrp

∼= Lie(G1)

Lie(idG)

Qsp

T0(ψ)

∼= Lie(G2)

which shows that Lie(idG) is surjective.


On the other hand, by Cor. 29.6 we have r = dimG1 = rankG1. More-over, by Thm. 27.1, there exists an open subgroup G′ ⊆ G which is p-valuablesuch that s = dimG2 = rankG′. Using Prop. 26.16 and Ex. 26.2 we con-clude that s = rank G′ = rank G1 = r. This establishes that Lie(idG) is anisomorphism of Lie algebras.

We finally use Prop. 18.17 (consult the proof of Cor. 18.18) in order to seethat there is an open subgroup U ⊆ G such that idG |U is an isomorphismof Lie groups. It follows immediately that idG has to be an isomorphism ofLie groups as well.

30 Saturation

Our goal in this section is to show that the pair (G,ω) always can be embed-ded as an open subgroup into another pair (G′, ω′) which is saturated. Forthis purpose it suffices to consider Zp as the basic discrete valuation ring.Therefore, throughout this section we let O = Zp.

To motivate the later strategy we begin with the following observation.We have three continuous group homomorphisms

Δ : G −→ G × G, ι1 : G −→ G × G, ι2 : G −→ G × G.

g �−→ (g, g) g �−→ (g, 1) g �−→ (1, g)

They induce corresponding continuous monomorphisms between completedgroup algebras

Δ∗, ι1∗, and ι2∗ : Λ(G) −→ Λ(G × G).

Remark 30.1. G = {λ ∈ Λ(G) \ {0} : Δ∗(λ) = ι1∗(λ) · ι2∗(λ)}.

Proof. Obviously G is contained in the right hand side. In order to showequality we may assume, by a projective limit argument, that G is finite.Let 0 = λ =

∑g∈G cgg be any element in the right hand side. We then have

∑

g

cg(g, g) =

(∑

g

cg(g, 1)

)(∑

g

cg(1, g)

)

=∑

g,h

cgch(g, h).

Comparing the coefficients gives cg = c2g and cgch = 0 if g = h. Since λ = 0

there must be at least one nonzero coefficient cg0 . The first identity thenimplies that cg0 = 1 and the second that ch = 0 for any h = g0. Henceλ = g0.

30 Saturation 209

The strategy in the following will be to embed Λ(G) into a bigger algebraand, by turning, for this bigger algebra, the Remark 30.1 into a definition,to obtain a bigger group. We start with the Qp-algebra Qp ⊗Zp Λ(G) anyelement of which can be written as a ⊗ λ with a ∈ Q×

p and λ ∈ Λ(G). Thisallows us to extend our function ω to Qp ⊗Zp Λ(G) by the rule

ω(a ⊗ λ) := v(a) + ω(λ).

Using for a moment multiplicative notation we set ‖a ⊗ λ‖ω := p−eω(a⊗λ).Then the pair (Qp⊗Zp Λ(G), ‖ ‖ω) is a normed Qp-algebra whose norm ‖ ‖ω,by Cor. 28.5.ii., in fact is multiplicative. We let (ΛQp(G,ω), ‖ ‖ω) denotethe corresponding completion which then is a Qp-Banach algebra with mul-tiplicative norm. We emphasize that, by Remark 28.1, the norm topologyon ΛQp(G,ω) induces on Λ(G) the compact topology. Having said this werevert to additive notation and we keep writing ω also for its continuousextension to ΛQp(G,ω).

We may visualize ΛQp(G,ω) as a normed vector space more concretelyby picking an ordered basis (g1, . . . , gr) of (G,ω). Then ΛQp(G,ω) is theQp-vector space of all expansions

∑

α

cαbα with cα ∈ Qp and lim|α|→∞

(

v(cα) +r∑

i=1

αiω(gi)

)

= ∞,

and

ω

(∑

α

cαbα

)

= minα

(

v(cα) +r∑

i=1

αiω(gi)

)

.

We point out that, in general, the set of values of ω on ΛQp(G,ω) is nolonger discrete.Example. Let G = Zp, for some p = 2, be the additive group of p-adicintegers with the p-valuation ω(pnu) := n + e for n ≥ 0 and u ∈ Z×

p , wheree ≥ 1 is a fixed integer. According to Prop. 20.1 we may view Λ(Zp) as theformal power series ring Zp[[X]] in one variable X over Zp. In this picture wehave ω(

∑n≥0 cnXn) = minn(v(cn)+ne). Phrased more analytically Zp[[X]]

is the ring of rigid Qp-analytic functions on the open unit disk which arebounded by one. On the other hand

ΛQp(Zp, ω) =

{∑

n

cnXn ∈ Qp[[X]] : limn→∞

(v(cn) + ne) = ∞}

=

{∑

n

c′n

(1pe

X

)n

∈ Qp

[[1pX

]]

: limn→∞

v(c′n) = ∞}


can be seen as the ring of all rigid Qp-analytic functions on the smaller closeddisk of radius 1

pe .

We equip the group G × G with the p-valuation

ω2(g, h) := min(ω(g), ω(h)).

Exercise. G × G is of finite rank equal to twice the rank of G.

The group homomorphisms Δ, ι1, and ι2 are isometric with respect to ωand ω2. We claim that the ring homomorphisms Δ∗, ι1∗, and ι2∗ are isometricfor ω and ω2. To see this let (g1, . . . , gr) be an ordered basis of (G,ω). Wewill take ((g1, 1), . . . , (gr, 1)(1, g1), . . . , (1, gr)) as an ordered basis for G×G.It then is immediately visible that ι1∗ and ι2∗ are isometries. To treat Δ∗ wefirst have to compute it explicitly. For this purpose we recall the followingnotational conventions about multi-indices:

β + α := (β1 + α1, . . . , βr + αr),α ≤ β if αi ≤ βi for any 1 ≤ i ≤ r,β − α := (β1 − α1, . . . , βr − αr) if α ≤ β, andα! := α1! . . . αr!.

(45)

Lemma 30.2. We have

Δ∗(bα) =∑

β,γ≤α

β+γ≥α

α!(α − β)!(α − γ)!(β + γ − α)!

ι1∗(bβ)ι2∗(bγ)

for any α.

Proof. First of all, for any g ∈ G, we compute

Δ∗(g − 1) = (g, g) − 1 = (g, 1)(1, g) − 1= ((g, 1) − 1)((1, g) − 1) + ((g, 1) − 1) + ((1, g) − 1)= ι1∗(g − 1)ι2∗(g − 1) + ι1∗(g − 1) + ι2∗(g − 1).

Next, quite generally, one checks, either by applying the binomial formulatwice or by induction with respect to n, that

(46) (xy + x + y)n =∑

0≤j,�≤n

j+�≥n

n!(n − j)!(n − �)!(j + � − n)!

xjy�

30 Saturation 211

holds true for any n ≥ 0 (the coefficients being integers of course). Wededuce the formula

Δ∗(bαi) =∑

0≤βi,γi≤n

βi+γi≥n

αi!(αi − βi)!(αi − γi)!(βi + γi − αi)!

ι1∗(bβi)ι2∗(bγi)

for any 1 ≤ i ≤ r. Multiplying these latter formulae together gives theassertion. Observe that we constantly are using that any element in theimage of ι1∗ commutes with any element in the image of ι2∗.

By this lemma we obtain the explicit formula

(47) Δ∗

(∑

α

cαbα

)

=∑

β,γ

(∑

β,γ≤α≤β+γ

mβ,γ,αcα

)


for any λ =∑

α cαbα ∈ Λ(G) where the mα,β,γ are certain integers (notdepending on λ). We see that

ω2(Δ∗(λ)) = minβ,γ

(

v

(∑

β,γ≤α≤β+γ

mβ,γ,αcα

)

+r∑

i=1

(βi + γi)ω(gi)

)

≥ minα

(

v

(

cα +r∑

i=1

αiω(gi)

))

= ω(λ).

It remains to observe that for β = 0 the only possibility for α is α = γ inwhich case the coefficient is m0,γ,γ = 1. Hence we actually have

ω2(Δ∗(λ)) = min(ω(λ), minβ =0,γ

(· · · )) ≤ ω(λ)

which shows thatω2(Δ∗(λ)) = ω(λ).

We now have established that the ring homomorphism Δ∗ is isometric aswell.

This enables us to extend Δ∗, ι1∗, and ι2∗ first by linearity to Qp⊗ZpΛ(G)and then by density to isometric homomorphisms of Qp-Banach algebrasfrom ΛQp(G,ω) into ΛQp(G×G,ω2) which we will denote by the same sym-bols. The identity (47) remains valid for λ ∈ ΛQp(G).


Definition. We put

σω(G) := {λ ∈ ΛQp(G,ω) \ {0} : Δ∗(λ) = ι1∗(λ)ι2∗(λ)}.

Let us examine the expansions of elements in σω(G) more closely. Forthis we will need the following elementary identity.

Remark 30.3. For any integers j, � ≥ 0 and any x ∈ Qp we have

∑

j,�≤n≤j+�

n!(n − j)!(n − �)!(j + � − n)!

(x

n

)

=(

x

j

)(x

�

)

.

Proof. First we assume that x ≥ 0 is an integer. Then, using the binomialformula together with (46), we compute the identity

∑

0≤j,�≤x

(x

j

)(x

�

)

Y jZ� = (Y + 1)x(Z + 1)x = (Y Z + Y + Z + 1)x

=∑

n≥0

(x

n

)

(Y Z + Y + Z)n

=∑

n≥0

(x

n

) ∑

j,�≤n≤j+�

n!(n − j)!(n − �)!(j + � − n)!

Y jZ�

between polynomials in two variables Y and Z. Comparing coefficients weobtain

∑

j,�≤n≤j+�

n!(n − j)!(n − �)!(j + � − n)!

(x

n

)

=(

x

j

)(x

�

)

for any integer x ≥ 0. But this means we actually have the identity∑

j,�≤n≤j+�

n!(n − j)!(n − �)!(j + � − n)!

(X

n

)

=(

X

j

)(X

�

)

between polynomials in one variable X with rational coefficients where(

X

n

)

:=X(X − 1) · · · (X − n + 1)

n!.

Of course, we may insert into this latter identity now any x ∈ Qp.

We recall that i = (. . . , 0, 1, 0, . . .) denotes the multi-index with theentry 1 in the i-th place and zero elsewhere. Let e1, . . . , er ≥ 0 be the uniqueintegers such that

1p − 1

< ω(gi) − ei ≤ 1 +1

p − 1.

30 Saturation 213


{(y1, . . . , yr) ∈ Qrp : v(yi) ≥ −ei for any i} −→ σω(G)

y = (y1, . . . , yr) �−→ λy :=∑

α

(y1

α1

)

· · ·(

yr

αr

)

bα

is a bijection which satisfies

ω(λy − 1) = min1≤i≤r

(v(yi) + ω(gi)) >1

p − 1.

Proof. We recall once more (cf. Lemma 2.2) that v(n!) ≤ n−1p−1 for any n ∈

N. Let y = (y1, . . . , yr) be a point in the left hand side, and put cα :=(y1

α1

)· · ·

(yr

αr

). If v(yi) < 0 then

v

((yi

αi

))

+ αiω(gi) = αi(v(yi) + ω(gi)) − v(αi!)

=αi − 1p − 1

− v(αi!) + (αi − 1)(

v(yi) + ω(gi) −1

p − 1

)

+ v(yi) + ω(gi)

≥ (αi − 1)(

ω(gi) − ei −1

p − 1

)

+ v(yi) + ω(gi)

for any αi ≥ 1. If v(yi) ≥ 0 then

v

((yi

αi

))

+ αiω(gi) ≥ v(yi) + αiω(gi) − v(αi!)

= v(yi) + ω(gi) +αi − 1p − 1

− v(αi!) + (αi − 1)(

ω(gi) −1

p − 1

)

≥ (αi − 1)(

ω(gi) − ei −1

p − 1

)

+ v(yi) + ω(gi)

for any αi ≥ 1. Since ω(gi) − ei − 1p−1 > 0 this implies that

lim|α|→∞

(

v(cα) +r∑

i=1

αiω(gi)

)

= ∞

and that

v(cα) +r∑

i=1

αiω(gi) ≥ min1≤i≤r

(v(yi) + ω(gi)) >1

p − 1for any α = 0.


Hence λy is a well defined nonzero (as c0 = 1) element in ΛQp(G,ω) with

ω(λy − 1) = min1≤i≤r

(v(yi) + ω(gi)) >1

p − 1.

The Remark 30.3 together with (47) implies Δ∗(λy) = ι1∗(λy)ι2∗(λy). Henceλy lies in σω(G). The injectivity of the map is clear since ci = yi.

For the surjectivity of the map let λ =∑

α cαbα be any element inσω(G). Because of (47) the coefficients cα satisfy the relation

∑

β,γ≤α≤β+γ

α!(α − β)!(α − γ)!(β + γ − α)!

cα = cβcγ .

For β = γ = 0 we in particular obtain c20 = c0. Moreover, for β = 0 the only

possibility for α in the left hand sum is α = γ. It follows that cγ = c0cγ forany γ. Since λ = 0 by assumption we cannot have c0 = 0. Therefore c0 = 1.For β = i the above identity reads

γicγ + (γi + 1)cγ+i = cicγ

or, equivalently,

cγ+i =ci − γi

γi + 1cγ .

From this it follows inductively that we have

cα =(

c1

α1

)

· · ·(

cr

αr

)

for any α. In order to verify that v(ci) ≥ −ei we observe that(cin

)is the

coefficient for the multi-index (. . . , 0, n, 0, . . .) with n in the i-th place. Wetherefore have

limn→∞

v

((ci

n

))

+ nω(gi) = ∞.

The asserted inequality being trivial otherwise we may assume that v(ci) <0. Then v(

(cin

)) = nv(ci) − v(n!) and hence

limn→∞

n(v(ci) + ω(gi)) − v(n!) = ∞.

For n = pm we have v(pm!) = pm−1p−1 and therefore obtain

∞ = limm→∞

pm(v(ci) + ω(gi)) −pm − 1p − 1

= limm→∞

pm

(

v(ci) + ω(gi) −1

p − 1

)

.

30 Saturation 215

It follows that v(ci) > 1p−1 − ω(gi). Since v(ci) is an integer we must have

v(ci) ≥ −ei. This shows that λ = λy for y := (c1, . . . , cr).

Since Δ∗, ι1∗, and ι2∗ are ring homomorphisms the set σω(G) is invari-ant under multiplication and under passing to the multiplicative inverse (ifit exists). If ω(λ − 1) > 1

p−1 then the geometric series∑

n≥0(1 − λ) con-verges in ΛQp(G,ω) and provides a multiplicative inverse λ−1 for λ. Sincethe λ ∈ σω(G) satisfy this assumption by Prop. 30.4 we see that σω(G) isa subgroup of the group of units in ΛQp(G,ω). We equip σω(G) with thetopology induced by the norm topology of ΛQp(G,ω). The multiplicationin σω(G) is continuous since it is the multiplication in the Banach algebraΛQp(G). As a consequence of Cor. 28.5.ii. any unit λ ∈ ΛQp(G,ω)× satisfiesω(λ−1) = −ω(λ) = 0. Hence ω(λ−1−λ′−1) = ω(λ−1)+ ω(λ′−1)+ ω(λ−λ′) =ω(λ−λ′) for these units. We see that σω(G) is a Hausdorff topological group.Of course it contains G as a compact subgroup.

We define the function ω on σω(G) by

ω(λ) := ω(λ − 1).

By Prop. 30.4 it satisfies axiom (a) for a p-valuation. The other axioms (b)to (d) follow by a computation of exactly the same type as we have donealready twice, in Example 23.2 and before Lemma 28.2, and which we willnot repeat here. That ω defines the topology of σω(G) is clear. Finally, byCor. 28.4.ii., this function ω restricts to the original p-valuation ω on G.

We collect the results of this discussion so far.

Lemma 30.5. σω(G) is a Hausdorff topological group carrying the p-valua-tion ω which defines its topology and which restricts to the p-valuation ω onthe compact subgroup G.

By Prop. 30.4 we have in σω(G) the elements

hi :=∑

n≥0

(p−ei

n

)

bni for 1 ≤ i ≤ r.

Lemma 30.6. For any 1 ≤ i ≤ r we have:

i. 1p−1 < ω(hi) ≤ p

p−1 ;

ii. hpei

i = gi.


Proof. i. This is immediate form the definition of the ei and either theformula for ω in Prop. 30.4 or the second assertion by axiom (d). ii. ByProp. 30.4 the element hpei

i is completely determined by the coefficient of bi

in its expansion. This coefficient is easily computed to be one which also isthe coefficient belonging to gi. Hence the asserted equality.

Theorem 30.7. i. σω(G) is p-valuable of finite rank equal to the rankof G.

ii. h1, . . . , hr is an ordered basis of (σω(G), ω).

iii. G is an open subgroup of σω(G).

iv. (σω(G), ω) is saturated.

v. If (G,ω) is saturated then σω(G) = G.

Proof. If y ∈ Zrp then, by definition, λy lies in Λ(G) and hence, by Re-

mark 30.1, in G. The formula for ω in Prop. 30.4 then implies that

σω(G)ν ⊆ G

for any sufficiently big ν. In particular, G is open in σω(G). Let now Hi ⊆σω(G), for any 1 ≤ i ≤ r, denote the closed subgroup topologically generatedby hi. On the one hand Hi∩G is open in the commutative group Hi. On theother hand Hi/Hi ∩G then is a finite cyclic group generated by the coset ofhi (cf. Lemma 30.6). Therefore Hi is profinite and the map

Zp −→ Hi⊆−→ σω(G)

x �−→ hxi

is well defined and continuous. In combination this gives the continuous map

c : Zrp −→ σω(G)

(x1, . . . , xr) �−→ hx11 · . . . · hxr

r .

We claim thatc((x1, . . . , xr)) = λ(p−e1x1,...,p−erxr)

holds true for any (x1, . . . , xr) ∈ Zrp. Since, by definition, we have

λ(y1,...,yr) = λ(y1,0,...,0) · . . . · λ(0,...,0,yr)

30 Saturation 217

it suffices to show that

hxi = λ(...,0,p−eix,0,...) for any x ∈ Zp.

By Prop. 30.4 this amounts to showing that 0, . . . , 0, p−eix, 0, . . . , 0 are thecoefficients of b1, . . . ,br, respectively, in the expansion of hx

i . We observethat the map

∑α cαbα �−→ cβ , for any fixed β, is a continuous linear form

on the Banach space ΛQp(G,ω). Hence we are reduced to computing thesecoefficients in the case when x ∈ Z is an integer. But then the assertedanswer is clear from the defining expansion of hi. This establishes our claim.It now follows from Prop. 30.4 that c is a continuous bijection. In particular,σω(G) is compact with the open profinite subgroup G and therefore itself isprofinite. Moreover, it follows that

ω(hx11 · · ·hxr

r ) = ω(λ(p−e1x1,...,p−erxr) − 1)

= min1≤i≤r

(v(xi) − ei + ω(gi))

= min1≤i≤r

(v(xi) + ω(hi)).

Hence (h1, . . . , hr) is an ordered basis of (σω(G), ω). Because of 1p−1 <

ω(hi) ≤ pp−1 Prop. 26.11 says that (σω(G), ω) is saturated. Finally, if (G,ω)

already is saturated then ei = 0 and hence hi = gi which implies σω(G) =G.

Chapter VII

The Lie Algebra

We keep fixing a p-valuable group G with a p-valuation ω on it. We alsocontinue to assume that O = Zp.

31 A Normed Lie Algebra

In Sect. 30 we have introduced the Qp-Banach algebra ΛQp(G,ω) with the(additively written) norm ω on it. Inside the group of units of ΛQp(G,ω) wefound the subgroup of group-like elements

σω(G) = {λ ∈ ΛQp(G,ω) \ {0} : Δ∗(λ) = ι1∗(λ)ι2∗(λ)}.

In this section we will study the vector subspace

Lω(G) := {λ ∈ ΛQp(G,ω) : Δ∗(λ) = ι1∗(λ) + ι2∗(λ)}

of primitive elements. The associative algebra ΛQp(G,ω) is a Lie algebraover Qp in the usual way by the Lie bracket being the additive commutator

[λ, μ] := λμ − μλ.

We have

ω([λ, μ]) ≥ ω(λ) + ω(μ) for any λ, μ ∈ ΛQp(G,ω).

Since elements in the image of ι1∗ and ι2∗, respectively, commute witheach other a one line computation shows that Lω(G) is a Lie subalgebraof ΛQp(G,ω). Of course, Lω(G) comes equipped with the norm ω.

In order to investigate this normed Lie algebra we will use the fact thatthe exponential and logarithm power series converge on large parts of theBanach algebra ΛQp(G,ω). Let λ ∈ ΛQp(G,ω).

First we suppose that ω(λ) > 1p−1 . Then

ω

(1n!

λn

)

= nω(λ) − v(n!) ≥ n

(

ω(λ) − 1p − 1

)

for any n ≥ 0

and, if λ �= 0,

ω

(1n!

λn

)

= nω(λ) − v(n!) ≥ nω(λ) − n − 1p − 1

= n

(

ω(λ) − 1p − 1

)

+1

p − 1> ω(λ) for any n ≥ 2.


219

http://dx.doi.org/10.1007/978-3-642-21147-8_7

220 VII The Lie Algebra

Henceexp(λ) :=

∑

n≥0

1n!

λn

converges in ΛQp(G,ω) and satisfies

ω(exp(λ) − (1 + λ)) > ω(λ)

(if λ �= 0). In particular,

ω(exp(λ) − 1) = ω(λ).

On the other hand suppose that ω(λ − 1) > 0. Then (exercise!)

ω

(1n

(λ − 1)n

)

= nω(λ − 1) − v(n) n→∞−−−−→ ∞.

Hence

log(λ) :=∑

n≥1

(−1)n+1

n(λ − 1)n

converges in ΛQp(G,ω). If λ �= 1 and ω(λ − 1) > 1p−1 then

ω

(1n

(λ − 1)n

)

= nω(λ − 1) − v(n!) + v((n − 1)!) ≥ nω(λ − 1) − n − 1p − 1

= n

(

ω(λ − 1) − 1p − 1

)

+1

p − 1> ω(λ − 1)

for any n ≥ 2 and hence

(48) ω(log(λ) − (λ − 1)) > ω(λ − 1).

In particular,

ω(log(λ)) = ω(λ − 1) whenever ω(λ − 1) >1

p − 1.

Exercise. Where they are defined the maps exp and log are continuous and,if ω(λ), ω(λ′), ω(μ − 1), ω(μ′ − 1) > 1

p−1 , they satisfy:

– log(exp(λ)) = λ, and exp(log(μ)) = μ;

– exp(λ+λ′) = exp(λ) exp(λ′) and log(μμ′) = log(μ)+ log(μ′) wheneverλλ′ = λ′λ and μμ′ = μ′μ, respectively;

– exp(mλ) = exp(λ)m and log(μm) = m log(μ) for any m ∈ Z.


Altogether, we see that exp and log define, for any ν > 1p−1 , homeomor-

phisms which are inverse to each other between the subsets

{λ : ω(λ) ≥ ν}exp

{λ : ω(λ − 1) ≥ ν}log

in ΛQp(G,ω).

Lemma 31.1. The maps exp and log restrict to homeomorphisms

Lω(G) 1p−1

+ :={

λ ∈ Lω(G) : ω(λ) >1

p − 1

} exp

σω(G)log

which are inverse to each other ; we have

log(hx) = x log(h) for any h ∈ σω(G) and x ∈ Zp.

Proof. By the above discussion we have

{

λ : ω(λ) >1

p − 1

} exp {

λ : ω(λ − 1) >1

p − 1

}

.log

The maps exp and log, of course, exist in the same way for ΛQp(G×G,ω2).By their very construction they commute with any isometric algebra homo-morphism from ΛQp(G,ω) into ΛQp(G × G,ω2). Consider first an elementλ ∈ Lω(G) 1

p−1+. We compute

Δ∗(exp(λ)) = exp(Δ∗(λ)) = exp(ι1∗(λ) + ι2∗(λ))= exp(ι1∗(λ)) · exp(ι2∗(λ)) = ι1∗(exp(λ)) · ι2∗(exp(λ))

where the third identity uses the fact that ι1∗(λ) and ι2∗(λ) commute witheach other. Hence λ ∈ σω(G). If, on the other hand, we start with an elementλ ∈ σω(G) then we know from Prop. 30.4 that ω(λ−1) > 1

p−1 . We thereforemay compute, similarly as above,

Δ∗(log(λ)) = log(Δ∗(λ)) = log(ι1∗(λ) · ι2∗(λ))= log(ι1∗(λ)) + log(ι2∗(λ)) = ι1∗(log(λ)) + ι2∗(log(λ)),

and we see that log(λ) ∈ Lω(G) 1p−1

+.The second part of the assertion follows, by continuity, from the case

x ∈ Z.


We note that Lω(G) 1p−1

+ is closed under the Lie bracket and thereforecan be viewed as a Lie algebra over Zp.

Proposition 31.2. For any ordered basis (g1, . . . , gr) of (G,ω) the elementslog(g1), . . . , log(gr) form a basis of the Qp-vector space Lω(G) satisfying

ω

(r∑

i=1

yi log(gi)

)

= min1≤i≤r

(v(yi) + ω(gi))

for any y1, . . . , yr ∈ Qp. In particular, we have dimQp Lω(G) = rankG.

Proof. By Lemma 31.1 the log(gi) and therefore any linear combination∑ri=1 yi log(gi) lie in Lω(G). Using (48) we have

ω

(r∑

i=1

yi log(gi)

)

= min1≤i≤r

minn∈N

(v(yi) − v(n) + nω(gi))

= min1≤i≤r

(v(yi) + minn∈N

(nω(gi) − v(n)))

= min1≤i≤r

(v(yi) + ω(gi)).

This in particular shows that the log(gi) are Qp-linearly independent. Letnow λ =

∑α cαbα be any element in Lω(G). From our explicit formula (47)

we know that Δ∗(λ) is equal to

∑

β,γ

(∑

β,γ≤α≤β+γ

α!(α − β)!(α − γ)!(β + γ − α)!

cα

)


= ι1∗(bβ) + ι2∗(bγ) − c0

+∑

β,γ =0

(∑

β,γ≤α≤β+γ

α!(α − β)!(α − γ)!(β + γ − α)!

cα

)

ι1∗(bβ)ι2∗(bγ).

It follows that c0 = 0 and that∑

β,γ≤α≤β+γ

α!(α − β)!(α − γ)!(β + γ − α)!

cα = 0 for any β, γ �= 0.

For β = i and γ �= 0 this identity becomes

γicγ + (γi + 1)cγ+i = 0

or, equivalently,cγ+i = − γi

γi + 1cγ .

We inductively conclude that


cα = 0 whenever at least two different entries of the multi-index α arenonzero, and

c(...,0,αi,0,...) = (−1)αi−1

αici for any 1 ≤ i ≤ r and αi ≥ 1.

This amounts to λ =∑

1≤i≤r ci log(gi).

Corollary 31.3. There is an ordered basis (h1, . . . , hr) of (σω(G), ω) suchthat the elements log(h1), . . . , log(hr) form a Zp-basis of Lω(G) 1

p−1+ satis-

fying

ω

(r∑

i=1

xi log(hi)

)

= min1≤i≤r

(v(xi) + ω(hi)) = ω(hx11 . . . hxr

r )

for any x1, . . . , xr ∈ Zp.

Proof. We pick an ordered basis (g1, . . . , gr) of (G,ω). In Thm. 30.7.ii. wehave seen the existence of an ordered basis (h1, . . . , hr) of (σω(G), ω) suchthat hpei

i = gi for the unique integer ei ≥ 0 such that 1p−1 < ω(gi)−ei ≤ p

p−1 .By using Prop. 31.2 we obtain that

∑i yip

ei log(hi) =∑

i yi log(gi) lies inLω(G) 1

p−1+ if and only if v(yi) > 1

p−1−ω(gi), resp. v(yipei) > 1

p−1−ω(gi)+ei,

for any 1 ≤ i ≤ r. Since v(.) always is an integer and −1 ≤ 1p−1 −ω(gi)+ei <

0 the latter is equivalent to v(yipei) ≥ 0.

Corollary 31.4. The map

Lω(G) 1p−1

+ ⊕ Lω(G) 1p−1

+

∼=−−−−→ι1∗+ι2∗

Lω2(G × G) 1p−1

+

is an isomorphism of Lie algebras over Zp.

Proof. We pick an ordered basis (g1, . . . , gr) of (G,ω), and we define the inte-gers ei ≥ 0 by 1

p−1 < ω(gi)−ei ≤ pp−1 . In the proof of Cor. 31.3 we have seen

that 1pe1 log(g1), . . . , 1

per log(gr) is a Zp-basis of Lω(G) 1p−1

+. The exactly same

reasoning applies to the ordered basis ((g1, 1), . . . , (gr, 1), (1, g1), . . . , (1, gr))of (G × G,ω2) and the corresponding integers (e1, . . . , er, e1, . . . , er). Hencewe see that

1pe1

log(ι1(g1)) = ι1∗

(1

pe1log(g1)

)

, . . . ,1

perlog(ι1(gr)) = ι1∗

(1

perlog(gr)

)

,

1pe1

log(ι2(g1)) = ι2∗

(1

pe1log(g1)

)

, . . . ,1

perlog(ι2(gr)) = ι2∗

(1

perlog(gr)

)

is a Zp-basis of Lω2(G × G) 1p−1

+.


By the universal property of the universal enveloping algebra (cf. Sect. 14)the inclusion Lω(G) ⊆ ΛQp(G,ω) extends uniquely to a homomorphism ofassociative Qp-algebras

I : U(Lω(G)) −→ ΛQp(G,ω).

Theorem 31.5. i. The map I is injective with dense image.

ii. Let (g1, . . . , gr) be an ordered basis of (G,ω); any λ ∈ ΛQp(G,ω) hasa unique convergent expansion of the form

λ =∑

α

dα log(g1)α1 · · · log(gr)αr with dα ∈ Qp

such that

lim|α|→∞

(

v(dα) +r∑

i=1

αiω(gi)

)

= ∞,

and

ω(λ) = minα

(

v(dα) +r∑

i=1

αiω(gi)

)

;

conversely, any series as above converges in ΛQp(G,ω).

Proof. According to Prop. 31.2 the elements log(g1), . . . , log(gr) form a basisof Lω(G). Hence, by the Poincare-Birkhoff-Witt theorem (cf. [B-LL] Chap. I§2.7 Cor. 3), the (ordered) monomials

log(g•)α := log(g1)α1 · · · log(gr)αr ,

with α = (α1, . . . , αr) running over all multi-indices, form a basis of theuniversal enveloping algebra U(Lω(G)). For the injectivity of I we thereforehave to show that these monomials, when viewed in ΛQp(G,ω), are Qp-linearly independent. We have the expansions

log(gi) =∑

n≥1

(−1)n+1

nbn

i

and hencelog(g•)α = bα + b-terms of degree > |α|.

Now suppose that there is a finite relation of the form∑

α cα log(g•)α = 0with m := min{|α| : cα �= 0} < ∞. We then have

0 =∑

α

cα log(g•)α =∑

|α|=m

cαbα + b-terms of degree > m,


and we deduce the contradiction that cα = 0 whenever |α| = m. This estab-lishes the claimed linear independence and hence the injectivity of I.

Let B ⊆ ΛQp(G,ω) denote the closure of the image of I. With log(g) ∈Lω(G) ⊆ B, for any g ∈ G, by Lemma 31.1 we have g = exp(log(g)) =∑

n≥01n! log(g)n ∈ B. Hence Zp[G] ⊆ B. Since Zp[G] is dense in Λ(G) we

obtain Qp ⊗Zp Λ(G) ⊆ B and therefore B = ΛQp(G,ω).Knowing now that ΛQp(G,ω) is the completion of U(Lω(G)) with respect

to ω it suffices, for the proof of ii., to check that

ω

(∑

α

dα log(g•)α

)

= minα

(

v(dα) +r∑

i=1

αiω(gi)

)

holds true for any finite sum∑

α dα log(g•)α. We have

ω(log(gi) − bi) > ω(bi) = ω(gi)

for any 1 ≤ i ≤ r by (48). It follows inductively that

ω(log(g•)α − bα) > ω(bα) =r∑

i=1

αiω(gi)

for any α. We now compute

ω

(∑

α

dα log(g•)α −∑

α

dαbα

)

= ω

(∑

α

dα(log(g•)α − bα)

)

≥ minα

(v(dα) + ω(log(g•)α − bα)

)

> minα

(v(dα) + ω(bα)

)

= ω

(∑

α

dαbα

)

.

This implies

ω

(∑

α

dα log(g•)α

)

= ω

(∑

α

dαbα

)

= minα

(

v(dα) +r∑

i=1

αiω(gi)

)

.

We now can prove a converse to Cor. 31.3.

Proposition 31.6. Let λ1, . . . , λr be a Zp-basis of Lω(G) 1p−1

+ satisfying

ω

(r∑

i=1

xiλi

)

= min1≤i≤r

(v(xi) + ω(λi))


for any x1, . . . , xr ∈ Zp; we then have:

i. Any λ ∈ ΛQp(G,ω) has a unique convergent expansion of the form

λ =∑

α

dαλα11 · · ·λαr

r with dα ∈ Qp

and

ω(λ) = minα

(

v(dα) +r∑

i=1

αiω(λi)

)

;

ii. (exp(λ1), . . . , exp(λr)) is an ordered basis of (σω(G), ω).

Proof. i. By Cor. 31.3 and Thm. 31.5.ii. there exists at least one Zp-basisμ1, . . . , μr of Lω(G) 1

p−1+ which satisfies the assertion. According to the

Poincare-Birkhoff-Witt theorem both sets of (ordered) monomials

μα• := μα1

1 · · ·μαrr and λα

• := λα11 · · ·λαr

r

form a Qp-basis of the universal enveloping algebra U(Lω(G)).Since U(Lω(G)), by Thm. 31.5.i., is dense in the Banach space ΛQp(G,ω)

it suffices to show that any λ =∑

α dαλα• ∈ U(Lω(G)) satisfies

(49) ω(λ) = minα

(

v(dα) +r∑

i=1

αiω(λi)

)(

= minα

(v(dα) + ω(λα• ))

).

For trivial reasons the left hand side is ≥ the right hand side. We thereforehave to establish the reverse inequality ≤. By assumption it holds for allelements λ ∈ Lω(G). We also make the easy observation that if λ = c1ν1 +· · · + cmνm is a linear combination of elements νj ∈ U(Lω(G)) such thatω(λ) ≤ min1≤j≤m(v(cj) + ω(νj)) then λ satisfies (49) if ν1, · · · , νm satisfy(49). This applies, for example, to any λ written as a linear combination ofthe μα

• . We consequently are reduced to showing that (49) holds true for allmonomials λ := μα1

1 · · ·μαrr . If

μi = ai1λ1 + · · · + airλr with aij ∈ Qp

we haveω(μi) = min

j(v(aij + ω(λj))


by assumption. It follows that

λ = μα11 · · ·μαr

r =

(r∑

j=1

a1jλj

)α1

· · ·(

r∑

j=1

arjλj

)αr

=r∑

j11,··· ,jrαr=1

a1j11 · · · a1j1α1a2j21 · · · a2j2α2

· · · arjrαrλj11 · · ·λjrαr

with

ω(λ) =r∑

i=1

αiω(μi) =r∑

i=1

αi minj

(v(aij) + ω(λj))

= minj11,...,jrαr

(v(a1j11) + · · · + v(arjrαr

) + ω(λj11) + · · · + ω(λjrαr))

= minj11,...,jrαr

(v(a1j11 · · · arjrαr

) + ω(λj11 · · ·λjrαr))

(recall that ω is “multiplicative”). Hence our easy observation applies andfurther reduces us to showing that arbitrary products of the form λ :=λi1 · · ·λin satisfy (49).

We will argue by induction with respect to n ≥ 2 that the following twoassertions hold true:

a) For any 1 ≤ i1, . . . , in ≤ r and any permutation σ of {1, . . . , n}, if

λiσ(1)· · ·λiσ(n)

− λi1 · · ·λin =∑

α

dαλα• ,

thenmin

α(v(dα) + ω(λα

• )) ≥ ω(λi1) + · · · + ω(λin).

b) For any 1 ≤ i1, . . . , in ≤ r the product λ := λi1 · · ·λin satisfies (49).

First of all we note that a) implies b). If we reorder the product λi1 · · ·λin

in such a way that the indices are increasing we obtain an element λβ• for

some appropriate multi-index β. Let

λi1 · · ·λin − λβ• =

∑

α

dαλα• .

By a) we haveω(λi1 · · ·λin) ≤ min

α(v(dα) + ω(λα

• )).


Since ω(λi1 · · ·λin) = ω(λi1) + · · · + ω(λin) = ω(λβ• ) it follows that

ω(λi1 · · ·λin) ≤ min(v(dβ + 1) + ω(λβ

• ), minα=β

(v(dα) + ω(λα• ))

).

For n = 2 we have λi2λi1 −λi1λi2 = [λi2 , λi1 ] ∈ Lω(G) and ω([λi2 , λi1 ]) ≥ω(λi1) + ω(λi2) so that a) holds by assumption.

For general n we may assume by another induction with respect to thelength of the permutation σ that σ is an elementary transposition. In thiscase we are looking at a difference of the form

λ := λi1 · · ·λim−1(λim+1λim)λim+2 · · ·λin − λi1 · · ·λin

= λi1 · · ·λim−1 [λim+1 , λim ]λim+2 · · ·λin .

Let [λim+1 , λim ] =∑r

j=1 djλj . Then ω([λim+1 , λim ]) = minj(v(dj) + ω(λj)),and we obtain

λ =r∑

j=1

djλi1 · · ·λim−1λjλim+2 · · ·λin

with

ω(λ) = ω(λi1 · · ·λim−1) + ω([λim+1 , λim ]) + ω(λim+2 · · ·λin)= min

j

(ω(λi1 · · ·λim−1) + v(dj) + ω(λj) + ω(λim+2 · · ·λin)

)

= minj

(v(dj) + ω(λi1 · · ·λim−1λjλim+2 · · ·λin)).

By the induction hypothesis all products λi1 · · ·λim−1λjλim+2 · · ·λin satisfy(49). Hence our easy observation at the beginning shows that λ satisfies (49)and a fortiori a).

ii. Let g ∈ σω(G) be an arbitrary element. According to i. we have anexpansion

g =∑

α

dαλα• ,

of which we will compute the image Δ∗(g) in two different ways. First of allthe definition of σω(G) says that

Δ∗(g) = ι1∗(g)ι2∗(g) =

(∑

β

dβι1∗(λ•)β

)(∑

γ

dγι2∗(λ•)γ

)

=∑

β,γ

dβdγι1∗(λ•)βι2∗(λ•)γ


with the notational convention that ιj∗(λ•)β := ιj∗(λ1)β1 · · · ιj∗(λr)βr . Sec-ondly, since Δ∗ is a continuous algebra homomorphism we compute (recallthe conventions (45))

Δ∗(g) =∑

α

dαΔ∗(λ1)α1 · · ·Δ∗(λr)αr

=∑

α

dα(ι1∗(λ1) + ι2∗(λ1))α1 · · · (ι1∗(λr) + ι2∗(λr))αr

=∑

α

dα

∑

β+γ=α

α!β!γ!

ι1∗(λ•)βι2∗(λ•)γ

=∑

β,γ

(β + γ)!β!γ!

dβ+γι1∗(λ•)βι2∗(λ•)γ .

As a consequence of Cor. 31.4 and the unicity part in i. (applied to theBanach algebra ΛQp(G × G,ω2)) the above two expansions of Δ∗(g) mustcoincide termwise. Hence

dβ+γ =β!γ!

(β + γ)!dβdγ for any β, γ.

In particular,

dβ = dβd0 and dβ+i =1

βi + 1dβdi for any β.

Since g �= 0 the first identity implies d0 = 1. The second identity then canbe used to obtain inductively the formula

dα =1α!

r∏

i=1

dαii for any α.

Since ω(g − 1) = minα=0(v(dα) + ω(λα• )) by i. and ω(g − 1) > 1

p−1 byLemma 31.1 we have

v(di) + ω(λi) >1

p − 1for any 1 ≤ i ≤ r.

On the other hand the members of a Zp-basis of Lω(G) 1p−1

+ necessarily

satisfy 1p−1 < ω(λi) ≤ 1 + 1

p−1 . It follows that v(di) > −1 and hence that

yi := di ∈ Zp for any 1 ≤ i ≤ r.


Inserting this information into the initial expansion of g gives

g =∑

α

1α1!

(y1λ1)α1 · · · 1αr!

(yrλr)αr

=

(∑

n≥1

1n!

(y1λ1)n

)

· · ·(

∑

n≥1

1n!

(yrλr)n

)

= exp(y1λ1) · · · exp(yrλr)= exp(λ1)y1 · · · exp(λr)yr .

The last identity follows by continuity from the case where the yi are integers.Moreover, we have

ω(g) = ω(g − 1) = minα=0

(v(dα) + ω(λα• ))

= minα=0

(

ω

((y1λ1)α1

α1!

)

+ · · · + ω

((yrλr)αr

αr!

))

= mini

ω(yiλi) = mini

(v(yi) + ω(λi))

= mini

(v(yi) + ω(exp(λi) − 1))

= mini

(v(yi) + ω(exp(λi))).

Here the fourth identity is a consequence of the fact, which we have discussedearlier, that ω(λn

n! ) > ω(λ) whenever n ≥ 2 and ω(λ) > 1p−1 .

Corollary 31.7. If (G,ω) is saturated then the map log induces a bijectionbetween the set of all ordered bases of (G,ω) and the set of all ordered Zp-bases (λ1, . . . , λr) of Lω(G) 1

p−1+ satisfying

ω

(r∑

i=1

xiλi

)

= min1≤i≤r

(v(xi) + ω(λi)) for any x1, . . . , xr ∈ Zp.

Proof. This is Cor. 31.3 (which, by its proof, applies to any ordered basis ofthe saturated (G,ω)) and Prop. 31.6.ii.

If we define a multiplication • on Lω(G) 1p−1

+ by

λ • μ := log(exp(λ) exp(μ))


then, for the restriction to Lω(G) 1p−1

+ of the natural topology of the finite

dimensional Qp-vector space Lω(G), the pair Lω(G) 1p−1

+ is a topologicalgroup such that

(Lω(G) 1

p−1+, •

) exp

σω(G)log

are isomorphisms of topological groups (inverse to each other). Letting λ•x,for λ ∈ Lω(G) 1

p−1+ and x ∈ Zp, denote the p-adic power formed with respect

to the multiplication • then we have

λ•x = log(exp(λ)x) = log(exp(xλ)) = xλ.

In particular, λ•(−1) = −λ. Moreover, since

ω(g) = ω(g − 1) = ω(log(g)) for any g ∈ σω(G)

the maps(Lω(G) 1

p−1+, •

) exp

σω(G)log

in fact are isomorphisms of topological groups with a p-valuation defining thetopology and hence (cf. Thm. 29.8) are isomorphisms of p-adic Lie groups.As such they induce an isomorphism between Lie algebras

Lie((

Lω(G) 1p−1

+, •)) ∼= Lie(σω(G)) = Lie(G)

(the latter identification holds since G is open in σω(G)). On the otherhand it follows from Prop. 17.6 that (Lω(G) 1

p−1+, •) is nothing else but the

Campbell-Hausdorff Lie group germ {Gε}ε of the Lie algebra Lω(G). Moreprecisely, let us choose a Zp-basis of Lω(G) 1

p−1+ as a basis for Lω(G). The

constant ε0 in Prop. 17.6 then is equal to |p|1

p−1 and(Lω(G) 1

p−1+, •

)⊇

⋃

ε<ε0

Gε.

We therefore may apply Prop. 17.3 to get

Lie((

Lω(G) 1p−1

+, •)) ∼= Lω(G).

Together we obtain an isomorphism of Lie algebras

Lω(G) ∼= Lie(G).


Viewing the latter as an identification the map

exp : Lω(G) 1p−1

+ −→ σω(G)

becomes an exponential map for σω(G) in the sense of Cor. 18.19.

32 The Hausdorff Series

Of course, the maps

Lω(G) 1p−1

+

exp

σω(G)log

do not, in general, respect the algebraic structures on the two sides, whichare of a different nature anyway: Lie algebra versus group. Nevertheless, in arather subtle way they can be used to transport important information fromone side to the other. This comes from a careful application of the p-adicproperties of the Hausdorff series.

Let MQ =∏

d≥0 M(d)Q denote the Magnus algebra of associative formal

power series in the variables X and Y with coefficients in Q. We will use thesame notations as in Sect. 24 except indicating by a subscript Q that nowwe work with rational coefficients. Inside MQ we have the Lie subalgebraLQ which is free on {X,Y } and which is graded by LQ =

⊕d≥1 LQ ∩ M

(d)Q .

We view the formal power series exp(X) =∑

d≥01d!X

d and exp(Y ) =∑

d≥01d!Y

d as elements of MQ and form their product exp(X) exp(Y ) ∈ MQ.It is straightforward to see (cf. Prop. 16.2) that then

H(X,Y ) =∑

n≥1

(−1)n+1

n(exp(X) exp(Y ) − 1)n

is a well defined element in MQ called the Hausdorff series. If

H(X,Y ) = H1(X,Y ) + · · · + Hd(X,Y ) + · · ·

the calculation of its degree d component Hd ∈ M(d)Q only involves finitely

many components of exp(X) and exp(Y ). It follows from the Campbell-Hausdorff formula (cf. Thm. 16.5) that

Hd ∈ LQ ∩ M(d)Q for any d ≥ 1.

For example,

H1 = X + Y and H2 =12(XY − Y X).


Each noncommutative polynomial Q(X,Y ) ∈ M(d)Q , for any d ≥ 0, in-

duces, in an obvious way, a well defined map

ΛQp(G,ω) × ΛQp(G,ω) −→ ΛQp(G,ω)(λ, λ′) �−→ Q(λ, λ′).

Theorem 32.1. For any λ, λ′ ∈ ΛQp(G,ω) such that ω(λ), ω(λ′) > 1p−1 we

have the convergent expansion

log(exp(λ) exp(λ′)) =∑

d≥1

Hd(λ, λ′)

in ΛQp(G,ω).

Proof. First of all we note that

ω(exp(λ) exp(λ′) − 1)= ω

((exp(λ) − 1)(exp(λ′) − 1) + (exp(λ) − 1) + (exp(λ′) − 1)

)

≥ min(ω(exp(λ) − 1) + ω(exp(λ′) − 1), ω(exp(λ) − 1), ω(exp(λ′) − 1)

)

= min(ω(λ) + ω(λ′), ω(λ), ω(λ′))

>1

p − 1.

Hence the left hand side in the assertion is well defined.Let C := min(ω(λ), ω(λ′)) − 1

p−1 > 0. We more precisely will establishthat

ω

(

log(exp(λ) exp(λ′)) −D∑

d=1

Hd(λ, λ′)

)

≥ (D + 1)C

holds true for any D ≥ 1. In a first step we use the convergent expansion


n≥1

(−1)n+1

n(exp(λ) exp(λ′) − 1)n.

By the first inequality above and Lemma 2.2 we have

ω

((−1)n+1

n(exp(λ) exp(λ′) − 1)n

)

≥ n min(ω(λ), ω(λ′)) − n

p − 1= nC.

This reduces us to showing that

ω

(D∑

n=1

(−1)n+1

n(exp(λ) exp(λ′) − 1)n −

D∑

d=1

Hd(λ, λ′)

)

≥ (D + 1)C.


By multiplying together the convergent expansions exp(λ) =∑

r≥01r!λ

r andexp(λ′) =

∑s≥0

1s!λ

′s we obtain the convergent expansions

(−1)n+1

n(exp(λ) exp(λ′)−1)n =

∑

r1,...,rn≥0s1,...,sn≥0

r1+s1≥1,...,rn+sn≥1

(−1)n+1

n

λr1

r1!λ′s1

s1!· · · λrn

rn!λ′sn

sn!.

On the other hand

Hd(λ, λ′) =∑

1≤e≤dr1,...,re,s1,...,se≥0

r1+s1≥1,...,re+se≥1r1+s1+···+re+se=d

(−1)e+1

e

λr1

r1!λ′s1

s1!· · · λre

re!λ′se

se!.

Hence

D∑

n=1

(−1)n+1

n(exp(λ) exp(λ′) − 1)n −

D∑

d=1

Hd(λ, λ′)

=∑

1≤n≤Dr1,...,rn,s1,...,sn≥0

r1+s1≥1,...,rn+sn≥1r1+s1+···+rn+sn>D

(−1)n+1

n

λr1

r1!λ′s1

s1!· · · λrn

rn!λ′sn

sn!.

It remains to notice that

ω

((−1)n+1

n

λr1

r1!λ′s1

s1!· · · λrn

rn!λ′sn

sn!

)

≥ −n +∑n

i=1(ri + si − 1)p − 1

+n∑

i=1

(ri + si) min(ω(λ), ω(λ′))

≥n∑

i=1

(ri + si)C

(compare the computation in the proof of Prop. 17.6).

For Q(X,Y ) ∈ LQ ∩ M(d)Q the map (λ, λ′) �−→ Q(λ, λ′) restricts to a

map Lω(G) × Lω(G) −→ Lω(G) (cf. the discussion after Cor. 16.11). Inparticular, if λ, λ′ ∈ Lω(G) 1

p−1+ then the convergent expansion in Thm. 32.1

is an expansion in the finite dimensional Qp-vector space Lω(G).


To state finer p-adic properties of the Hd we need to recall certain tech-nical facts about L. Let M(d), for any d ≥ 1 denote the set of nonassociativeand noncommutative monomials of degree d in the variables X and Y . It isdefined inductively by M(1) := {X,Y } and

M(d) := disjoint union of all M(a) ×M(b) for a + b = d.

We inductively construct maps

M(d) −→ L ∩ M(d)

x �−→ ex

by eX := X, eY := Y , and ex := [ey, ez] if x = (y, z) ∈ M(a) × M(b) ⊆M(a + b). Since L is the free Lie algebra on eX and eY each L ∩ M(d),as an abelian group, is generated by {ex}x∈M(d). The important technicalpoint is (cf. [B-LL] Chap. II §2.10-11 and §8.1 Prop. 1, resp. Prop. 17.4 andthe proof of Prop. 17.6) that one can choose subsets B(d) ⊆ M(d) suchthat:

(H1) {ex}x∈B(d) is a Z-basis of L∩M(d) and hence a Q-basis of LQ ∩M(d)Q ;

(H2) if Hd =∑

x∈B(d) ad,xex then v(ad,x) ≥ −d−1p−1 for any x ∈ B(d).

We now consider any function

w : Lω(G) 1p−1

+ \ {0} −→(

1p − 1

,∞)

which satisfies (with the usual convention that w(0) := ∞)

(b+) w(λ − λ′) ≥ min(w(λ), w(λ′)),

(c+) w([λ, λ′]) ≥ w(λ) + w(λ′), and

(d+) w(cλ) = v(c) + w(λ)

for any λ, λ′ ∈ Lω(G) 1p−1

+ and c ∈ Zp. Of course, w = ω is a possiblechoice. But for our later applications it is crucial to allow an arbitrary suchw. Clearly, w extends uniquely to a function w : Lω(G) \ {0} −→ R whichsatisfies (b+) – (d+) for any λ, λ′ ∈ Lω(G) and any c ∈ Qp.

Lemma 32.2. For any x ∈ M(d) and any λ, λ′ ∈ Lω(G) 1p−1

+ we have:


i. w(ex(λ, λ′)) > w(λ + λ′) + d−1p−1 if d ≥ 2;

ii. w(ex(λ, λ′)) > w([λ, λ′]) + d−2p−1 if d ≥ 3.

Proof. i. We proceed by induction with respect to d. If d = 2 then ex = 0or ±ex = XY − Y X, and we have

w(λλ′ − λ′λ) = w(λ(λ + λ′) − (λ + λ′)λ)= w([λ, λ + λ′])≥ w(λ) + w(λ + λ′)

> w(λ + λ′) +1

p − 1.

Now suppose that a + b = d > 2 and x = (y, x) ∈ M(a)×M(b). Then ex =[ey, ez]. Since a, b < d we may assume the assertion to hold, by induction,for ey and ez. In case a, b ≥ 2 we compute

w(ex(λ, λ′)) = w([ey(λ, λ′), ez(λ, λ′)])≥ w(ey(λ, λ′)) + w(ez(λ, λ′))

> w(λ + λ′) +a − 1p − 1

+ w(λ + λ′) +b − 1p − 1

> w(λ + λ′) +a − 1p − 1

+1

p − 1+

b − 1p − 1

= w(λ + λ′) +d − 1p − 1

.

In case a = d − 1 ≥ 2 and b = 1 we have


> w(λ + λ′) +a − 1p − 1

+1

p − 1

= w(λ + λ′) +d − 1p − 1

,

and for a = 1 the computation is entirely analogous.ii. Again proceeding by induction let x = (y, z) ∈ M(a) × M(b) with

a + b = d ≥ 3. In case a ≥ 3 and b ≥ 2 we have

w(ex(λ, λ′)) = w([ey(λ, λ′), ez(λ, λ′)])


≥ w(ey(λ, λ′)) + w(ez(λ, λ′))

> w([λ, λ′]) +a − 2p − 1

+ w(λ + λ′) +b − 1p − 1

> w([λ, λ′]) +a − 2p − 1

+1

p − 1+

b − 1p − 1

= w([λ, λ′]) +d − 2p − 1

,

where in the second inequality we have used i. for the second summand. Incase a ≥ 3 and b = 1 we have


> w([λ, λ′]) +a − 2p − 1

+1

p − 1

= w([λ, λ′]) +d − 2p − 1

.

The case b ≥ 3 follows by symmetric computations. If a = b = 2 thennecessarily ex = 0 and there is nothing to prove. By symmetry it remainsto consider the case a = 2 and b = 1. If ex �= 0 then ey(λ, λ′) = ±[λ, λ′] andhence


> w([λ, λ′]) +1

p − 1

= w([λ, λ′]) +d − 2p − 1

.

Proposition 32.3. For any λ, λ′ ∈ Lω(G) 1p−1

+ we have

w(log(exp(λ) exp(λ′))

)= w(λ + λ′).

Proof. By Thm. 32.1 we have the convergent expansion


d≥1

Hd(λ, λ′)

in Lω(G). Since H1(λ, λ′) = λ + λ′ it suffices to show that


w

(∑

d≥2

Hd(λ, λ′)

)

> w(λ + λ′).

The function w necessarily is continuous on the finite dimensional Qp-vectorspace Lω(G) (cf. [NFA] Prop. 4.13). This further reduces us to the inequal-ities

w(Hd(λ, λ′)) > w(λ + λ′) for any d ≥ 2.

They follow from (H2) and Lemma 32.2.i.

Lemma 32.4. If Hd(X,Y ) − Hd(Y,X) =∑

x∈B(d) cd,xex then v(cd,x) ≥−d−2

p−1 for any x ∈ B(d).

Proof. Let K := Qp(π) where πp−1 = −p. In particular, v(π) = 1p−1 . We

consider the Hausdorff series H(X,Y ) in the Magnus algebra MK with co-efficients in K. As a consequence of Lemma 2.2 the formal power seriese(X) :=

∑n≥1

πn−1

n! Xn has coefficients in the ring of integers O of K. InMK we have

1π

H(πX, πY ) =1π

∑

n≥1

(−1)n+1

n(exp(πX) exp(πY ) − 1)n

=∑

n≥1

(−π)n−1

n

(exp(πX) exp(πY ) − 1

π

)n

=∑

n≥1

(−π)n−1

n

(πe(X)e(Y ) + e(X) + e(Y )

)n.

Exercise. v(n) < n−1p−1 for any n ≥ 2 such that n �= p.

It follows that the associative formal power series 1πH(πX, πY ) has coeffi-

cients in O and satisfies

1π

H(πX, πY )

≡ πe(X)e(Y ) + e(X) + e(Y ) − (πe(X)e(Y ) + e(X) + e(Y ))p mod πO≡ e(X) + e(Y ) + (e(X) + e(Y ))p mod πO.

The symmetry in X and Y of this latter expression shows that

1π

H(πX, πY ) − 1π

H(πY, πX) ≡ 0 mod πO.


This means that the associative polynomials πd−2(Hd(X,Y ) − Hd(Y,X))have coefficients in O. Since

(L ∩ M(d)) ⊗Z K ∩ M

(d) ⊗Z O = (L ∩ M(d)) ⊗Z O

by [B-LL] Chap. II §3.1 Remark 1) this implies πd−2cd,x ∈ O.

Proposition 32.5. For any λ, λ′ ∈ Lω(G) 1p−1

+ we have

w(log

(exp(λ) exp(λ′) exp(−λ) exp(−λ′)

))= w([λ, λ′]).

Proof. Using Prop. 32.3 and Thm. 32.1 we obtain

w(log

(exp(λ) exp(λ′) exp(−λ) exp(−λ′)

))

= w(log

(exp

(log(exp(λ) exp(λ′))

)exp

(log(exp(−λ) exp(−λ′))

)))

= w(log(exp(λ) exp(λ′)) + log(exp(−λ) exp(−λ′))

)

= w(log(exp(λ) exp(λ′)) − log(exp(λ′) exp(λ))

)

= w

(∑

d≥1

(Hd(λ, λ′) − Hd(λ′, λ)

))

= w

(

[λ, λ′] +∑

d≥3

(Hd(λ, λ′) − Hd(λ′, λ)

))

By the continuity of w it therefore suffices to show that

w(Hd(λ, λ′) − Hd(λ′, λ)) > w([λ, λ′]) for any d ≥ 3.

This follows from Lemma 32.4 and Lemma 32.2.ii.

This analysis of the Hausdorff series now allows us to transport thefunctions w on Lω(G) 1

p−1+ to p-valuations on σω(G).

Proposition 32.6. For any function w : Lω(G) 1p−1

+ \ {0} −→ ( 1p−1 ,∞)

which satisfies (b+), (c+), and (d+) the function ω′(g) := w(log(g)) is a p-valuation on σω(G). Moreover, if in addition w−1(( p

p−1 ,∞)) ⊆ pLω(G) 1p−1

+

holds true then (σω(G), ω′) is saturated.


Proof. The axiom (a) for ω′ holds by construction. The axiom (d) translates(d+). Using Prop. 32.3 and (b+) we compute

ω′(g−1h) = w(log(g−1h)) = w(log(exp(log(g))−1 exp(log(h)))

)

= w(log(exp(− log(g)) exp(log(h)))

)

= w(− log(g) + log(h))≥ min(w(log(g)), w(log(h)))= min(ω′(g), ω′(h)).

The axiom (c) follows in an analogous way from (c+) and Prop. 32.5. For thesecond part of the assertion let g ∈ σω(G) be any element such that ω′(g) >

pp−1 . Then, by assumption, log(g) = pλ for some λ ∈ Lω(G) 1

p−1+. Setting

h := exp(λ) ∈ σω(G) we obtain hp = exp(λ)p = exp(pλ) = exp(log(g)) = g.Hence (σω(G), ω′) is saturated.

For technical reasons we later will need the following generalization ofThm. 32.1. Let r ≥ 2 be an integer. We consider the iterated Hausdorffseries

H(X1, . . . , Xr) := log(exp(X1) · · · exp(Xr))= H(· · · (H(H(X1, X2), X3), . . .), Xr)

as an associative formal power series in the variables X1, . . . , Xr with coef-ficients in Q. We write

H(X1, . . . , Xr) =∑

β =0

Hβ(X1, . . . , Xr)

where Hβ(X1, . . . , Xr), for any multi-index β = (β1, . . . , βr), denotes thecomponent of H(X1, . . . , Xr) which is homogeneous of degree βi in the vari-able Xi, for any 1 ≤ i ≤ r.

Proposition 32.7. i. For any β �= 0 the coefficients of the associative(noncommutative) polynomial Hβ(X1, . . . , Xr) have a p-adic valuation≥ − |β|−1

p−1 .

ii. For any λ1, . . . , λr ∈ Lω(G) 1p−1

+ we have the convergent expansion

log(exp(λ1) · · · exp(λr)) =∑

β =0

Hβ(λ1, . . . , λr)

such that Hβ(λ1, . . . , λr) ∈ Lω(G) 1p−1

+ for any β �= 0.


Proof. First of all we establish by induction with respect to r that eachHβ(X1, . . . , Xr)

a) is a finite sum of associative monomials of the form cXj1 · · ·Xj|β| , withc ∈ Q and 1 ≤ j1, . . . , j|β| ≤ r, which are homogeneous of degree βi inXi and such that v(c) ≥ − |β|−1

p−1 , and

b) lies in the free Lie algebra L{X1,...,Xr} with coefficients in Q (which,by Cor. 15.2, is naturally contained in the free associative algebraAs{X1,...,Xr}).

For r = 2 this is (H1) and (H2) (the latter remains valid for Hβ—comparethe proof of Prop. 17.6). The induction step is carried out by applying thefollowing simple observation to the substitution

H(H(X1, . . . , Xr−1), Xr) = H(X1, . . . , Xr).

Let cZ1 · · ·Zm be an associative monomial in two variables X and Y(i. e., each Zi

is equal to either X or Y ) with coefficient c ∈ Q such thatv(c) ≥ −m−1

p−1 . Moreover, for any 1 ≤ i ≤ m, let Mi = biXji1 . . . Xjidibe an

associative monomial in the variables X1, . . . , Xr with coefficient in Q suchthat v(bi) ≥ −di−1

p−1 . Substituting Mi for Zithen gives the monomial

cM1 · · ·Mm = cb1 · · · bmXj11 · · ·Xjmdm

of degree d := d1 + · · · + dm in the variables X1, . . . , Xr with

v(cb1 · · · bm) = v(c) + v(b1) + · · · + v(bm)

≥ − 1p − 1

(m − 1 + d1 − 1 + · · · + dm − 1)

= −d − 1p − 1

.

In a completely analogous way the substitution of Lie monomials for thevariables in a given Lie monomial leads again to a Lie monomial. By defini-tion, a Lie monomial is an element in L{X1,...,Xr} of the form cex with c ∈ Q

and ex the image of any x ∈ M{X1,...,Xr} under the composite map

M{X1,...,Xr}⊆−−→ A{X1,...,Xr}

pr−−→ L{X1,...,Xr}

in Sect. 15. By construction, any element in L{X1,...,Xr} is (not uniquely) afinite sum of such Lie monomials.


The assertion i. follows from a), and b) implies that Hβ(λ1, . . . , λr) ∈Lω(G) ⊆ ΛQp(G,ω) for any λ1, . . . , λr ∈ Lω(G). But as a consequence of i.we have

ω(Hβ(λ1, . . . , λr)) ≥ −|β| − 1p − 1

+ β1ω(λ1) + · · · + βrω(λr).

If λ1, . . . , λr ∈ Lω(G) 1p−1

+ then the right hand side is > 1p−1 and hence

Hβ(λ1, . . . , λr) ∈ Lω(G) 1p−1

+.It remains to establish the expansion in ii. In case r = 2 this is a slightly

more precise version of Thm. 32.1 which has the same proof. For general rwe, in particular, have the convergent expansion

H(λ1, . . . , λr) =∑

(α1,α2)

H(α1,α2)(H(λ1, . . . , λr−1), λr).

We write the associative formal power series

H(α1,α2)(H(X1, . . . , Xr−1), Xr) =∑

(α1,α2)

H(α1,α2)β (X1, . . . , Xr)

as the sum of its homogeneous components H(α1,α2)β of degree βi in Xi. The

induction step in the first part of this proof applies as well and gives thateach H

(α1,α2)β (X1, . . . , Xr) satisfies a). It follows that

ω(H(α1,α2)β (λ1, . . . , λr)) ≥ −|β| − 1

p − 1+ β1ω(λ1) + · · · + βrω(λr)

≥ −|β| − 1p − 1

+ |β| min1≤i≤r

ω(λi)

> |β|(

min1≤i≤r

ω(λi) −1

p − 1

)

.

Since the Hausdorff series has no constant term there are, for any β, at mostfinitely many (α1, α2) such that H

(α1,α2)β (X1, . . . , Xr) �= 0, and

Hβ(X1, . . . , Xr) =∑

(α1,α2)

H(α1,α2)β (X1, . . . , Xr).

This, in particular, shows that

(50) limα1+α2+|β|→∞

H(α1,α2)β (λ1, . . . , λr) = 0.


Again by induction with respect to r we now assume that the assertedexpansion holds for H(λ1, . . . , λr−1). An appropriate version of Lemma 3.2then implies that, for any (α1, α2), we have the convergent expansion

H(α1,α2)(H(λ1, . . . , λr−1), λr) =∑

(α1,α2)

H(α1,α2)β (λ1, . . . , λr).

Because of (50) we finally may apply Lemma 3.3 and conclude that

H(λ1, . . . , λr) =∑

(α1,α2)

∑

β

H(α1,α2)β (λ1, . . . , λr)

=∑

β

∑

(α1,α2)

H(α1,α2)β (λ1, . . . , λr)

=∑

β

Hβ(λ1, . . . , λr).

33 Rational p-Valuations and Applications

In this section we will show that on any p-valuable group there exists a p-valuation with values in Q. The technique to do this actually is to constructrational valued functions w on Lω(G) 1

p−1+ and then to transfer them to

σω(G) and hence by restriction to G. This important existence result willenable us to establish various fundamental ring theoretic properties of thecompleted group ring Λ(G).

Lemma 33.1. There is a function w : Lω(G) 1p−1

+ \ {0} −→ ( 1p−1 ,∞) ∩ Q

which satisfies (b+), (c+), (d+) and has the property that w−1(( pp−1 ,∞)) ⊆

pLω(G) 1p−1

+.

Proof. We fix a Zp-basis λ1, . . . , λr of Lω(G) 1p−1

+ as constructed in Cor. 31.3.It necessarily satisfies

1p − 1

< ω(λi) ≤p

p − 1for 1 ≤ i ≤ r.

We let

(51)1

p − 1< τi ≤

p

p − 1for 1 ≤ i ≤ r


be arbitrary but fixed real numbers, and we introduce the function

w : Lω(G) 1p−1

+ \ {0} −→(

1p − 1

,∞)

r∑

i=1

xiλi �−→ min1≤i≤r

(v(xi) + τi).

It is straightforward to see that w satisfies (b+) and (d+) and has theadditional property stated in the assertion. Let cijk ∈ Zp, for 1 ≤ i, j, k ≤ r,be such that

[λi, λj ] =r∑

k=1

cijkλk.

If w also satisfies (c+) then we, in particular, have

mink

(v(cijk) + τk) = w

(∑

k

cijkλk

)

= w([λi, λj ]) ≥ w(λi) + w(λj) = τi + τj .

Hence the τi satisfy the further inequalities

(52) τi + τj ≤ v(cijk) + τk for any 1 ≤ i, j, k ≤ r.

Suppose, vice versa, that these inequalities hold. We then compute

w

([∑

i

xiλi,∑

j

yjλj

])

= w

(∑

i,j

xiyj [λi, λj ]

)

= w

(∑

i,j,k

cijkxiyjλk

)

= mink

(

v

(∑

i,j

cijkxiyj

)

+ τk

)

≥ mini,j,k

(v(cijk) + v(xi) + v(yj) + τk)

≥ mini,j

(v(xi) + v(yj) + τi + τj)

= mini

(v(xi) + τi) + minj

(v(yj) + τj)

= w

(∑

i

xiλi

)

+ w

(∑

j

yjλj

)


which means that w satisfies (c+). In order to prove our assertion we there-fore have to find rational numbers τ1, . . . , τr which satisfy the inequalities(51) and (52), which both have rational coefficients. Note that these inequal-ities do have the real solutions τi := ω(λi).

We choose rational numbers 1p−1 < qi ≤ ω(λi) for 1 ≤ i ≤ r and consider

the slightly more restrictive system of inequalities

−τi ≤ −qi for 1 ≤ i ≤ r,

τi ≤p

p − 1for 1 ≤ i ≤ r,

τi + τj − τk ≤ v(cijk) for 1 ≤ i, j, k ≤ r.

(53)

The set S ⊆ Rr of real solutions of (53) still is nonempty and, of course, isconvex. We want to show that it contains a rational solution. Let E denotethe set of r3 + 2r inequalities in the system (53). For any e ∈ E and anys ∈ S we write s < e, resp. s = e, if s satisfies the strict inequality, resp. theequality, corresponding to e. We divide up E into the subsets

E< := {e ∈ E : s < e for some s ∈ S}

andE= := {e ∈ E : s = e for any s ∈ S}.

The equations corresponding to the e ∈ E= define a rational affine subspaceA ⊆ Rr which contains S. We claim that S as a subset of A has nonemptyinterior, and consequently contains a rational point. Consider the set

S0 := {s ∈ A : s < e for any e ∈ E<}

which obviously is open in A and is contained in S. It suffices to show thatS0 is nonempty. If E< = ∅ then S0 = S(= A) is nonempty. Otherwise thereis, for any e ∈ E<, a solution se ∈ S such that se < e. Then

s :=1

|E<|∑

e∈E<

se

is, by convexity, a solution in S as well and lies in S0 by construction.

Theorem 33.2. There always is a p-valuation ω′ on σω(G) with values inQ and such that (σω(G), ω′) is saturated.

Proof. Combine Lemma 33.1 and Prop. 32.6.


Corollary 33.3. Any p-valuable group G carries a p-valuation ω with valuesin 1

N Z for some N ∈ N.

Proof. By the theorem we find an ω with values in Q. On the other hand weknow that the set of values always is of the form ω(G)\{1} =

⋃1≤j≤s νj+N0.

Since the νj have to be rational numbers we can take for N their smallestcommon denominator.

To draw the conclusions for the completed group ring Λ(G) we from nowon assume that the p-valuation ω on our p-valuable group G has values insome 1

N Z. We also allow O to be again a general complete discrete valuationring in which p is a prime element.

Theorem 33.4. For any p-valuable group G the completed group ring Λ(G)is noetherian, regular of global dimension ≤ 1 + rankG, and is Auslanderregular.

Proof. According to Lemma 26.13.i. we may assume, by possibly replacingω by ω − C for some sufficiently small rational number C > 0, that the Liealgebra gr G is abelian. We then know from Cor. 28.6 that the graded ringgr Λ(G) is a polynomial ring in 1+rankG variables over O/pO. In particular,gr Λ(G) is noetherian and regular of global dimension 1+rankG. Let N ∈ N

be a common denominator for the values of ω which then also is a commondenominator for the values of ω on Λ(G). We introduce the exhaustive andseparated decreasing filtration

Film Λ(G) := Λ(G)mN

indexed by integers m ≥ 0 on Λ(G). It is a ring filtration in the sense that

Fil Λ(G) · Film Λ(G) ⊆ Fil+m Λ(G)

holds true for any ,m ≥ 0. Moreover, Λ(G) is complete with respect to thisfiltration, i. e.,

Λ(G) = lim←−m

Λ(G)/ Film Λ(G).

Finally, for the trivial reason that N + m

N = +mN , the associated graded

ring⊕

m Film Λ(G)/ Film+1 Λ(G) coincides, as an ungraded O/pO-algebra,with gr Λ(G). In such a situation it is a general fact that the three ringtheoretic properties in our assertion lift from the associated graded ring tothe complete filtered ring Λ(G). A detailed discussion of this lifting techniqueand its application to our three properties can be found in [LvO] (or [NkA]).The lifting of being noetherian also is treated in [B-CA] Chap. III §2.9Cor. 2.


34 Coordinates of the First and of the SecondKind

Throughout this section we assume that the pair (G,ω) is saturated. Weview G as a p-adic Lie group as constructed in Cor. 29.6. We recall that thismeans the following. Given any fixed ordered basis (g1, . . . , gr) of (G,ω) thecoordinate functions

ϕi : G −→ Zp ⊆ Qp

g �−→ xi if g = gx11 · . . . · gxr

r

for 1 ≤ i ≤ r are locally analytic. These (ϕ1, . . . , ϕr) are called coordinatesof the second kind . On the other hand it follows from Cor. 31.3 (more pre-cisely, its proof) that the elements log(g1), . . . , log(gr) form a Zp-basis ofLω(G) 1

p−1+ satisfying

ω

(r∑

i=1

yi log(gi)

)

= min1≤i≤r

(v(yi) + ω(gi)) = ω(gy11 · · · gyr

r )

for any y1, . . . , yr ∈ Zp. we therefore obtain functions

ψi : G −→ Zp ⊆ Qp

g �−→ yi if log(g) =r∑

i=1

yi log(gi).

These (ψ1, . . . , ψr) are called coordinates of the first kind . We have seen atthe end of Sect. 31 that the maps

Lω(G) 1p−1

+

exp

Glog

are isomorphisms of manifolds inverse to each other. This immediately im-plies that the functions ψi are locally analytic. But there is a more precisestatement which we want to establish.

Proposition 34.1. There are formal power series Fi(Y ) =∑

β ai,βY β andEi(Y ) =

∑β bi,βY β , for any 1 ≤ i ≤ r, in r variables Y = (Y1, . . . , Yr) with

coefficients in Zp such that we have:

i. lim|β|→∞ v(ai,β) = ∞ and lim|β|→∞ v(bi,β) = ∞;


ii. ψi(g) = Fi(ϕ1(g), . . . , ϕr(g)) for any g ∈ G;

iii. ϕi(g) = Ei(ψ1(g), . . . , ψr(g)) for any g ∈ G.

Proof. The assertion i. guarantees that the Fi and Ei converge on Zrp so that

the assertions ii. and iii. make sense. There is nothing to prove for r = 1, sowe assume in the following that r ≥ 2.

Let g ∈ G and put xi := ϕi(g) and yi := ψ(gi).For the construction of the power series F1(Y ), . . . , Fr(Y ) we start from

the identity

g = gx11 · · · gxr

r

= exp(log(g1))x1 · · · exp(log(gr))xr

= exp(x1 log(g1)) · · · exp(xr log(gr)),

where the last line follows by continuity from the case where the xi areintegers. We obtain

r∑

i=1

yi log(gi) = log(g) = log(exp(x1 log(g1)) · · · exp(xr log(gr))

).

Applying Prop. 32.7 to the right hand side and using the homogeneity prop-erty of the Hβ give rise to the expansion

r∑

i=1

yi log(gi) =∑

β =0

xβ11 · · ·xβr

r Hβ(log(g1), . . . , log(gr))

in Lω(G) 1p−1

+. In particular, lim|β|→∞ ω(Hβ(log(g1), . . . , log(gr))) = ∞. Welet

Hβ(log(g1), . . . , log(gr)) =r∑

i=1

ai,β log(gi)

with ai,β ∈ Zp and lim|β|→∞ v(ai,β) = ∞, and we obtain

r∑

i=1

yi log(gi) =r∑

i=1

∑

β

xβ11 · · ·xβr

r ai,β log(gi).

It remains to define Fi(Y ) :=∑

β ai,βY β .The construction of the power series E1(Y ), . . . , Er(Y ) proceeds along

the same lines as the proof of Prop. 29.2. We have the expansion

(54) g = gx11 · · · gxr

r =∑

α

(x1

α1

)

· · ·(

xr

αr

)

bα


in Zp[[G]]. On the other hand the identity

g = exp(y1 log(g1) + · · · + yr log(gr))

leads to the expansion

g =∑

n≥0

1n!

(r∑

i=1

yi log(gi)

)n

=∑

β

yβ11 · · · yβr

r

1|β|!Mβ(log(g1), . . . , log(gr))

in ΛQp(G,ω) where Mβ(Z1, . . . , Zr) denotes the sum of all noncommutative(associative) monomials in the variables Zi which, for any 1 ≤ i ≤ r, haveexactly βi factors Zi. We have

ω

(1|β|!Mβ(log(g1), . . . , log(gr))

)

≥ −v(|β|!) +r∑

i=1

βiω(log(gi))

= −v(|β|!) +r∑

i=1

βiω(gi)

≥ |β|(

min(ω(g1), . . . , ω(gr)) −1

p − 1

)

.

Next we use the “standard” expansions

1|β|!Mβ(log(g1), . . . , log(gr)) =

∑

α

cα,βbα

with cα,β ∈ Qp and lim|α|→∞ v(cα,β)+∑r

i=1 αiω(gi) = ∞ for any β. We have

v(cα,β) +r∑

i=1

αiω(gi) ≥ ω

(1|β|!Mβ(log(g1), . . . , log(gr))

)

≥ |β|(

min(ω(g1), . . . , ω(gr)) −1

p − 1

)

.

It follows in particular that

lim|α|+|β|→∞

v(cα,β) +r∑

i=1

αiω(gi) = ∞.

Hence we obtain the expansion

(55) g =∑

α

cαbα with cα :=∑

β

cα,βyβ11 · · · yβr

r .


By comparing (54) and (55) we deduce(

x1

α1

)

· · ·(

xr

αr

)

=∑

β

cα,βyβ11 · · · yβr

r

and, in particular,xi =

∑

β

ci,βyβ11 · · · yβr

r

for the multi-index α = i. We define Ei(Y ) :=∑

β ci,βY β . The coeffi-cients satisfy v(ci,β)+ω(gi) ≥ |β|(min(ω(g1), . . . , ω(gr))− 1

p−1) which implieslim|β|→∞ v(ci,β) = ∞. It remains to show that ci,β ∈ Zp or, equivalently, thatv(ci,β) > −1. By construction ci,0 = 0. For β �= 0 we have the more preciseestimate

v(ci,β) + ω(gi) ≥ |β|min(ω(g1), . . . , ω(gr)) −|β| − 1p − 1

.

Since (G,ω) is saturated we have ω(gi) ≤ pp−1 and hence

v(ci,β) + 1 ≥ |β|(

min(ω(g1), . . . , ω(gr)) −1

p − 1

)

> 0.

Since a change of basis of Lω(G) is a linear and hence locally analyticmap we deduce that, for any Qp-basis λ1, . . . , λr of Lω(G), the functions

ψi : G −→ Qp

g �−→ yi if log(g) =r∑

i=1

yiλi

are locally analytic. Moreover, this remains true even if (G,ω) is not satu-rated (since it is open in its saturation). It gives us a more general kind ofcoordinates of the first kind .

References

[BGR] Bosch S., Guntzer U., Remmert R.: Non-Archimedean Analysis.Berlin - Heidelberg - New York: Springer 1984

[B-CA] Bourbaki N.: Commutative Algebra. Berlin - Heidelberg - NewYork: Springer 1989

[B-GT] Bourbaki N.: General Topology. Berlin - Heidelberg - New York:Springer 1989

[B-LL] Bourbaki N.: Lie Groups and Lie Algebras. Berlin - Heidelberg -New York: Springer 1998

[B-VA] Bourbaki N.: Varietes differentielles et analytiques. Fascicule deresultats. Berlin - Heidelberg - New York: Springer 2007, reprint of1982 printing

[DDMS] Dixon J.D., du Sautoy M.P.F., Mann A., Segal D.: Analytic Pro-p-Groups. Cambridge Univ. Press 1999

[Fea] Feaux de Lacroix C.T.: Einige Resultate uber die topologischenDarstellungen p-adischer Liegruppen auf unendlich dimensionalenVektorraumen uber einem p-adischen Korper. Thesis, Koln 1997.Schriftenreihe Math. Inst. Univ. Munster, 3. Serie, Heft 23, pp. 1–111 (1999)

[Gab] Gabriel P.: Des Categories Abeliennes. Bull. Soc. Math. France 90,323–448 (1962)

[Haz] Hazewinkel M.: Formal Groups and Applications. New York: Aca-demic Press 1978

[Hum] Humphreys J.: Introduction to Lie Algebras and RepresentationTheory. Berlin - Heidelberg - New York: Springer 1972

[Lam] Lam T.Y.: A First Course in Noncommutative Rings. Berlin - Hei-delberg - New York: Springer 2001

[La1] Lazard M.: Quelques calculs concernant la formule de Hausdorff.Bull. Soc. Math. France 91, 435–451 (1963)

[La2] Lazard M.: Groupes analytiques p-Adiques. Publ. Math. IHES 26,389–603 (1965)

P. Schneider, p-Adic Lie Groups,Grundlehren der mathematischen Wissenschaften 344,DOI 10.1007/978-3-642-21147-8, © Springer-Verlag Berlin Heidelberg 2011

251

http://dx.doi.org/10.1007/978-3-642-21147-8

252 References

[LvO] Li Huishi, van Oystaeyen F.: Zariskian Filtrations. Dordrecht:Kluwer 1996

[NkA] Schneider P.: Ausgewahlte Kapitel aus der nichtkommutativen Al-gebra. Vorlesungsskriptum, Munster 2000

[NFA] Schneider P.: Nonarchimedean Functional Analysis. Berlin - Hei-delberg - New York: Springer 2002

[Se1] Serre J-P.: Local Fields. Berlin - Heidelberg - New York: Springer1979

[Se2] Serre J-P.: Lie Algebras and Lie Groups. Lect. Notes Math. 1500.Berlin - Heidelberg - New York: Springer 1992

Index

a-derivation, 67algebra

free A-, 107Lie, 71Magnus, 111tensor, 102universal enveloping, 102

atlas, 46equivalent, 46maximal, 46n-dimensional, 47

ball, 4closed, 4open, 4

Banach space, 11

Campbell-Hausdorfftheorem, 114Lie group germ, 125

chain rule, 18, 58, 66chart, 45

around x, 45compatible, 45dimension, 45domain of definition, 45

completed group ring, 157comultiplication, 106, 133coordinates

of the first kind, 247, 250of the second kind, 247

coveringlocally finite, 51refinement, 51

derivation, 67derivative, 18, 59diagonal, 106

diameter, 6distribution, 165dual space, 13Dynkin’s formula, 117

ε-convergent, 25expansion, 31exponential map, 153exponential power series, 219

finite rank, 181formal group law, 132formal homomorphism, 143

group-like element, 219

Hausdorff series, 115, 232homomorphism

formal, 143into associative algebra, 101local, 136of Lie algebras, 101of Lie groups, 101

identity theorem, 33index for M , 76invertibility

for power series, 33local, 22, 42, 58

Iwasawa algebra, 157

Jacobi identity, 71

Lie algebra, 71graded, 174of G, 100, 219, 222

Lie group, 89p-adic, 192

P. Schneider, p-Adic Lie Groups,Grundlehren der mathematischen Wissenschaften 344,DOI 10.1007/978-3-642-21147-8, © Springer-Verlag Berlin Heidelberg 2011

253

http://dx.doi.org/10.1007/978-3-642-21147-8

254 Index

localhomomorphism, 136ring, 136

local invertibility, 22, 42, 58locally analytic

function, 38, 49manifold, 47map, 50

locally convexfinal topology, 82inductive limit, 83topology, 81vector space, 81

logarithm power series, 219

Magnus algebra, 111, 175, 232manifold, 47

n-dimensional, 47open sub-, 48product, 49

mapdifferentiable, 17exponential, 153locally analytic, 50locally constant, 25polynomial, 118strictly differentiable, 20tangent, 57

nonarchimedeanabsolute value, 8field, 8norm, 10seminorm, 79

norm, 10

open mapping theorem, 23operator norm, 12ordered basis, 182

p-adic field, 9p-adic valuation, 182paracompact, 51partial derivative, 20Poincare-Birkhoff-Witt theorem, 104primitive element, 219product rule, 19, 59pseudocompact

module, 165ring, 165

p-valuable, 192p-valuation, 169

rational, 243

rank, 181residue class field, 9ring of integers, 9

saturated, 187saturation, 208seminorm, 79strictly paracompact, 51

tangentbundle, 64map, 57space, 57vector, 56

Taylor expansion, 33

ultrametric space, 3complete, 6spherically complete, 6

universal enveloping algebra, 102

vector field, 66left, right invariant, 96

vector spacedual, 13locally convex, 81normed, 10

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