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THÈSEprésentée à

L’ UNIVERSITÉ BORDEAUX IÉCOLE DOCTORALE DE MATHÉMATIQUES ET INFORMATIQUE

par Michael LASKE

POUR OBTENIR LE GRADE DE

DOCTEUR

SPÉCIALITÉ: MATHÉMATIQUES PURES

Le K1 des Courbes sur les Corps Globaux.Conjecture de Bloch et Noyaux Sauvages

Thèse dirigée par Karim BELABAS

Devant la commission d’examen formée de:

M. BELABAS Karim, Professeur, IMB - Université Bordeaux

M. DE JEU Rob, Professeur, Afdeling Wiskunde - Vrije Universiteit Amsterdam

M. ELBAZ-VINCENT Philippe, Professeur, Institut Fourier - Université Grenoble

M. JAULENT Jean-Francois, Professeur, IMB - Université Bordeaux

Soutenue le 19 novembre 2009

Contents

Introduction v

1 Milnor K-theory of varieties 11.1 Somekawa Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Comparison Theorems . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Further Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Special KM-theory . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Heights 92.1 Proofs of Mordell-Weil type . . . . . . . . . . . . . . . . . . . . . 92.2 Heights on Abelian Groups . . . . . . . . . . . . . . . . . . . . . 102.3 Linear Heights . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.5 Heights in Algebraic Geometry . . . . . . . . . . . . . . . . . . . 27

3 Decomposable Elements 313.1 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Point Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3 Factor Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.4 Bounds for Multiplication . . . . . . . . . . . . . . . . . . . . . . 41

4 Bloch’s Conjecture 434.1 Heights on K-groups . . . . . . . . . . . . . . . . . . . . . . . . . 434.2 An Approach via Heights . . . . . . . . . . . . . . . . . . . . . . 47

References 51

iii

iv

Acknowledgments

The author wishes to express his sincere gratitude to Wayne Raskind andMichael Spiess for their interest in this work and thoughtful advise how to im-prove it. Further it is a pleasure to thank Bruno Kahn and Jean-Louis Colliot-Thélène for helpful discussions and comments.

The author is greatly indebted to the referees Rob de Jeu and PhilippeElbaz-Vincent for their support and patience during numerous corrections ofthe manuscript.

Special thanks are due to Karim Belabas for his encouragement and guidancethroughout the preparation of this thesis.

INTRODUCTION v

Introduction

This thesis studies the first algebraic K-group of curves over global fields.

K-theory of Curves

Let X be a smooth projective geometrically connected curve over a perfectfield k.

K1 in a nutshell. The first K-group of a curve splits along the Adamseigenspaces,

K1(X) ∼= K1(k)⊕KM1 (X) , (1)

where K1(k) = k×, and KM1 (X) will be referred to as the first Milnor K-groupof X (since this is not standard notation, see 1.1.2 for the precise definition).This group is described by a short exact sequence, provided X(k) �= ∅,

0 −→ SKM1

(X) −→ KM1

(X) N−→ KM1

(k) −→ 0

where KM1

(k) = k×, and N denotes the transfer in Milnor K-theory for smoothprojective varieties (generalized via Somekawa groups [28], [1]). If X has a k-rational point, this sequence splits. Thus the nontrivial part of K1(X) is thespecial Milnor K1-group SKM1 (X).

Ad Hoc definition of SKM1

(X). Denote by k(X) the function field of X;its second Milnor K-group is the quotient of k(X)× ⊗ k(X)× by the Steinbergrelations,

KM

2(k(X)) := k(X)× ⊗ k(X)×

��a⊗ (1− a) | a ∈ k(X)× − {1}� .

For f, g ∈ k(X)×, let {f, g} denote the equivalence class of f ⊗ g in KM2

(k(X)).Identifying Weil divisors X(1) of the curve X with its Zariski-closed points X(0),the residue field k(x), for x ∈ X(1), is a finite extension of k. The boundarymap

∂ : KM2

(k(X)) →�

x∈X(1)k(x)×,

{f, g} �→�

x∈X(1)

�(−1)ordx(f)ordx(g) f

ordx(g)

gordx(f)

�(x) ,

given by the tame symbols is well-defined, i. e. factors through the Steinbergrelations. The norm

N :�

x∈X(1)k(x)× → k× , ⊕x∈X(1) ax �→

�x∈X(1)

Nk(x)/k(ax)

satisfies the Weil reciprocity formula N ◦ ∂ = 1, hence yields a map N :K

M

1(X) := coker ∂ → k×. We define

SKM

1(X) := ker

�N : coker ∂ → k×

�.

vi INTRODUCTION

Ground Field. This explicit description of SKM1

(X) involves the underlyingground field which we are now going to specify.

(i) Let k be a finite field. Then SKM1

(X) = 0.(ii) Let k be a local field. In this case, the group SKM

1(X), under the name

V (X), has been subject to intense study concerning generalizations of class fieldtheory, initiated by Kato and Saito [15], [26], Bloch [5], Katz and Lang [16] inthe beginning of the 1980ies. Briefly, there is a reciprocity map

τ : SKM1

(X) → πab1

(X)geo

which is surjective on the torsion subgroup. Here πab1

(X) is the abelianized étalefundamental group π1(X) classifying finite étale coverings of X, with geometricpart πab

1(X)geo := ker

�π

ab

1(X) → Gabk

�.

(iii) Let k be a global field. Unlike the situation of local ground field as in(ii), the global case is not well-understood.

• The local case suggests that SKM1

(X) carries geometrical informationabout the curve X. Which information precisely and how is it encoded?

• What is the structure of SKM1

(X) as abelian group? This has been subjectto a conjecture of Bloch [5] dating back to the 1980ies. So far it has beenknown, by a result of Raskind [24, Cor. 0.3 & Lemmas 2.1, 2.2] from 1990,that

SKM

1(X)⊗Q/Z = 0 . (2)

Bloch’s Conjecture

An Analogy. A heuristic guideline for our investigation of K1 of curves, isthat SKM

1of a curve “should resemble” KM

2of the underlying ground field.

As first confirmation of this principle we observe that Matsumoto’s Theorem,stating that K2(F ) ∼= KM2 (F ) for a field F , finds in (1) its counterpart: MilnorK-theory captures all of K1(X).

The Milnor K-theory of fields is well-understood, and one of its fundamentaltheorems states that KM

2of a global field is a torsion group. One can ask the

analog question in a geometrical context.

Conjecture (Bloch, [5, 1.24]). For a smooth projective geometrically connectedcurve X over a global field k, the group SKM

1(X) is torsion.

This thesis provides a strategy for the demonstration of Bloch’s Conjecture,though we are not able to complete the proof.

As suggested by analogy, it is instructive to review the classical approachesto determine the structure of KM

2of a global field F .

(i) First, KM2

(F ) is decomposed into smaller pieces via the tame symbols,

∂ : KM2

(F ) →�

v finiteK

M

1(F (v)) =

�vF (v)×

with F (v) denoting the residue field of F at the finite place v. The homomor-phism ∂ is surjective, and the tame kernel ker ∂ equals K2(OF ) if F is a numberfield and K2(Y ) if F = F(Y ) is a function field. Thus KM2 (F ) is an extensionof the tame kernel by a torsion group.

INTRODUCTION vii

We will resume the idea to decompose SKM1

(X) later. Unlike in the classicalsituation of fields, a“useful”decomposition, i. e. into small, possibly finite pieces,requires the use of finite coefficients. So, in dealing with Bloch’s Conjecture, thiswill be of no help.

(ii) The structure of the tame kernel of F is then determined as follows. IfF is a number field, the only known proof of the finiteness of K2(OF ) is viatranscendental methods by Garland [9]. It is not clear what would correspondto this method in a geometric context.

Our strategy of proof of Bloch’s Conjecture begins with a generalization ofa result of Bass-Tate [3] which originally states that the tame kernel of a globalfield is finitely generated, shown by constructing a finite factor basis for thisgroup. The analog for SKM

1of a curve becomes weaker.

Theorem ([3.0.1]). Let X be a smooth projective geometrically connected curveover a field k, admitting a k-rational point, and let Jk� be the Jacobian of Xk�for any finite k-extension k�. There exists a finite constant B such that thehomomorphism

µ :�

k�/k[k�:k]≤B

Jk� ⊗ k�× −→ SKM1 (X) ,

is surjective, with µ as in Lemma 3.1.1.

At this point, a new idea is needed.

Heights. Heights are a classical tool of Diophantine Geometry, traditionallyused to “measure the complexity” of a point on a variety, in other words of azero-cycle. The Milnor K-theory of a variety is a theory of zero-cycles. Thisraises the hope that a theory of heights for KM may be developed based on theclassical theory, and indeed we are able to construct a height on SKM

1(X) with

hopefully “nice” properties. Our motivation for this approach is founded in thevaguely formulated intuition that a bound on height and on degree should yieldsome finiteness result (like the theorem of Northcott, cp. 2.5.2).

Definition. A canonical linear height h on an abelian group A is a nonnegativefunction

h : A −→ R+

with the following properties:

(i) h is even, i. e. h(a) = h(−a) ∀a ∈ A.

(ii) h satisfies the triangle inequality, i. e. h(a + b) ≤ h(a) + h(b) ∀a, b ∈ A .

(iii) h is N-homogeneous, i. e. h(ma) = mh(a) ∀a ∈ A,∀m ∈ Z, m > 1.

The requirements of the definition are not restrictive enough, e. g. the zerofunction is a height. Classically a height is defined with the built-in property(F) that the set of elements of bounded height is finite (as on abelian varietiesover global fields). This however would be too exclusive since it would ruleout to consider any group with torsion subgroup of infinite cardinality, such asSK

M

1(X). Thus we attach the following properties to a height h on A, saying

that h satisfies:

viii INTRODUCTION

(FF) if h�{a ∈ A | h(a) < C}

�is a finite set in R for every constant C ≥ 0.

(T) if h−1(0) = Ator.

These notions then lead to the following argument

Descent Lemma ([2.4.19]). Let A be an abelian group equipped with a canon-ical linear height h satisfying the finiteness properties (FF) and (T). Then themaximal divisible subgroup Adiv of A is torsion.

We develop a theory of heights for tensor products.

Theorem. Let (A, hA), (B, hB) be abelian groups equipped with canonical lin-ear heights.

(i) ([2.3.5]) Then hA and hB induce a canonical linear height hA ⊗ hB onA⊗Z B.

(ii) ([2.4.15]) Suppose that both groups modulo torsion are free of countablerank. If hA and hB both satisfy (FF) and (T), then there exists an inducedcanonical linear height h on A⊗Z B satisfying (FF) and (T).

Heights on K-groups. Translated into this language of heights, the DescentLemma and (2) yield:

Lemma. Bloch’s Conjecture is equivalent to the existence of a canonical linearheight on SKM

1(X) satisfying the properties (FF) and (T).

We are able to construct a height suspect to possess these properties. Herewe state a version for X̄ := X ×Spec k Spec k̄ denoting the geometric curve overa fixed separable closure k̄ of k.

Proposition ([4.1.2]). Let X be a smooth projective geometrically connectedcurve over a number field k. There exists a canonical linear height H onSK

M

1(X̄).

Unfortunately, we are not able to show that properties (FF) and (T) aresatisfied by H on SKM

1(X̄)Gk . We give a condition for their verification [4.2.4],

summarizing the gap in our proof of Bloch’s Conjecture. This essentially in-volves the problem of controlling the height of a representative of an element inSK

M

1(X) when “factoring through” a basis of elements of bounded degree.

Outlook. Wild Kernel

After looking at the structure of SKM1

(X) as an abelian group, one might con-sider how to calculate it explicitly. We would like to give a short outlook.

For this purpose we resume the idea of decomposing SKM1

(X) into smallerpieces inspired by the tame symbol ∂ reducing the study of KM

2of a global field

essentially to its tame kernel. For a number field F , the norm residue symbolinduces “Moore’s exact sequence”

0 → WK(F ) → KM2

(F ) →�

all places vnon-complex

µ(Fv) → µ(F ) → 0

INTRODUCTION ix

with WK(F ) being defined as kernel of this map, and Fv denoting the completionof F at v. The group WK(F ) is known as the “wild kernel” of F , and is asubgroup of the tame kernel, hence finite.

For a global field k, the canonical injection k �→ kv into the completion of kat a prime v induces a localization homomorphism

SKM

1(X) → SKM

1(Xv)

with Xv := X ×Spec k Spec kv denoting the localized curve over kv.

0.0.1 Definition. For smooth projective geometrically connected curve X overa global field k, and a rational prime �, we define the �-part of the wild kernelWK�(X) of X as

WK�(X) := ker

�lim←−n

λ�n : lim←−n

SKM

1(X)/�n −→ lim←−

n

�

all v

SKM

1(Xv)/�n

�,

and the wild kernel WK(X) := ⊕�WK�(X) as sum over all rational primes.

We expect WK�(X) to be related to the Selmer group of the Jacobian ofthe curve, and we think it is reasonable to ask whether it is finite. A tougherquestion would be whether it is nearly always zero, i. e. whether the wild kernelis finite.

x INTRODUCTION

Chapter 1

Milnor K-theory of smoothprojective varieties

We present a Milnor K-theory for smooth projective varieties over a field.

The K-groups attached to a variety are expected to be universal cohomologygroups, and it is certainly no exaggeration to say that they are difficult tocalculate. Milnor K-theory, claimed in [30] to be the “simplest part of algebraicK-theory”, is originally a functor based on an algebraic construction for rings.In this chapter, we fix the notation for a Milnor K-theory of varieties. All resultsin this chapter can be found in the literature.

1.1 Somekawa Groups

In [28], Somekawa proposed a broad generalization of Milnor K-groups. We willonly need a special case, presented in detail in [1], [25].

Let k be a field. Denote by SmProj/k the category of smooth projectivevarieties over k.1

1.1.1 Definition. For X ∈ SmProj/k, q > 0 a positive integer, the groupKq(k; CH0(X), Gm) is defined to be the quotient of the free abelian group

�

k�/k

CH0(Xk�)⊗ k�× ⊗ . . .⊗ k�×� �� q-times

,

with Xk� := X ×k k�, and sum running through all finite field extensions k�/k,modulo the relations

(M1) For any finite field extension Spec k2j→ Spec k1 → Spec k,

1We remark that X ∈ SmProj/k is by definition noetherian, further that X is of finite

type and separated (cf. [12, II.4.9]). In particular, X is equi-dimensional (cf. [12, III.10.2]).The faithful functor mapping each variety over k to its associated scheme gives a bijectionbetween the nonsingular k-varieties and the integral smooth quasi-projective k-schemes (cf.[12, II.4.10]).

1

2 CHAPTER 1. MILNOR K-THEORY OF VARIETIES

• for z ∈ CH0(Xk2), ai ∈ k×1 for all i = 1, . . . , q,

(j∗(z)⊗ a1 ⊗ . . .⊗ aq)k1/k − (z ⊗ j∗(a1)⊗ . . .⊗ j∗(aq))k2/k ;

• for z ∈ CH0(Xk1), ai0 ∈ k×2 , ai ∈ k×1 , with i �= i0 for some i0,

(z⊗a1⊗. . .⊗j∗(ai0)⊗. . .⊗aq)k1/k−(j∗(z)⊗j∗(a1)⊗. . .⊗ai0⊗. . .⊗j∗(aq))k2/k ;

(Note that the pushforward j∗ on k×2 becomes the field norm Nk2/k1 , andthe pullback j∗ on k×

1becomes the canonical inclusion.)

(M2) For every field K finitely generated of transcendence degree 1 over k, andall choices of h ∈ K×, y ∈ CH0(XK), and bi ∈ K× with i = 1, . . . , qsubject to the condition that for every place v of K/k (i. e. v is trivial onk) there exists i(v) such that bi ∈ O×v for all i �= i(v) (with Ov denotingthe valuation ring of K at v):

�

all places v of K/k

(sv(y)⊗ b1(v)⊗ . . .⊗ Tv(bi(v), h)⊗ . . .⊗ bq(v))k(v)/k ,

where sv : CH0(XK) → CH0(Xk(v)) is the specialization map for Chowgroup (cf. [8, 20.3]), further bi(v) denotes the reduction O×v → k(v)× fori �= i(v) and

Tv : K× ⊗K× → k(v)× , (b, h) �→ (−1)v(b)v(h)bv(h)

hv(b)(v) .

The image of (z ⊗ a1 ⊗ . . . ⊗ aq)k�/k modulo the relations (M1), (M2) isdenoted {z ⊗ a1 ⊗ . . .⊗ aq}k�/k. 2

We propose the following definition.

1.1.2 Definition. We define Milnor K-theory as a family of contravariant func-tors to the category of abelian groups

KM

q (−) : SmProj/k → Ab

indexed by the nonnegative integer q, given for objects by assigning

X �→�

CH0(X) if q = 0 ,Kq(k; CH0(X), Gm) if q > 0 .

Functoriality. Let X,Y ∈ SmProj/k. We only treat the case q > 0, sincethe remaining case q = 0 comprises the functoriality of Chow groups which iswell-known (cf. [8]).

• Base Change. Let Spec k1j→ Spec k be a finite field extension.

2To be consistent with the notation of the papers cited, we referred implicitly in Def. 1.1.1

by the denomination of Kq(k; CH0(X), Gm) to the functor CH0(X) defined by assigning toeach field extension k� of k the group CH0(Xk� ). But this additional information is dispensablefor the mere formulation of the definition.

1.2. COMPARISON THEOREMS 3

– Covariant.

j∗ : KMq (Xk1) → KMq (Xk){z, a1, . . . , aq}k�1/k1 �→ {z, a1, . . . , aq}k�1/k

– Contravariant.

j∗ : KMq (Xk) → KMq (Xk1)

{z, a1, . . . , aq}k�/k �→�m

r=1e(r){ir∗(z), ir∗(a1), . . . , ir∗(aq)}k(r)1 /k1

where k� ⊗k k1 = ⊕mr=1A(r) with A(r) an Artin algebra of dimensione(r) over k(r)

1, and ir : Spec k

(r)1→ Spec k�.

– The map j∗ ◦ j∗ : KMq (Xk) → KMq (Xk1) → KMq (Xk) is multiplicationby [k1 : k].

• Proper Pushforward. Let φ : X → Y be a proper morphism over k.

φ∗ : KMq (X) → KMq (Y ){z, a1, . . . , aq}k�/k �→ {φ∗(z), a1, . . . , aq}k�/k

• Flat Pullback. Let ψ : X → Y be a flat morphism of relative dimension 0over k.

ψ∗ : KMq (Y ) → KMq (X)

{z, a1, . . . , aq}k�/k �→ {ψ∗(z), a1, . . . , aq}

Motivation. To justify the denomination, a basic comparison isomorphism of[28] assures that, for F a field, the classical notion of Milnor K-theory KMq (F )as in (1.3) coincides with our definition, i. e.

KM

q (SpecF ) ∼= KMq (F ) .

Let X ∈ SmProj/k with d := dim X. We will see soon in Thm 1.2.5 thatin terms of motivic cohomology

KM

q (X)∼−→ H2d+q(X, Z(d + q)) .

Comparing Quillen K-theory with motivic cohomology (cf. [18, Cor. 8.2]),

Hq+2d(X, Z(q + d))⊗ Z

�1

(d + q − 1)!

�∼= grd+qγ Kq(X)⊗ Z

�1

(d + q − 1)!

�,

(1.1)we see that the Milnor K-theory of X is rationally the highest-graded nontrivialpart of the γ-filtration of the Quillen K-theory of X. Thus

KM

q (X)Q∼−→ K(d+q)q (X) . (1.2)

This motivates our definition of Milnor K-theory of smooth projective k-varieties.

1.2 Comparison Theorems

It will turn out to be useful to establish links to alternative descriptions.


Comparison Theorem I

KM of Rings. Let R be a commutative ring. Define KM

0(R) := Z, and, for

any integer q ≥ 1,

KM

q (R) =�q

m=1R×

��a1 ⊗ · · ·⊗ aq | ai + aj = 1 or 0 for some i < j� , (1.3)

The image of (a1 ⊗ · · ·⊗ aq) under the quotient map is denoted {a1, . . . , aq}.For a discrete valuation v on a field F , with associated residue field F (v),

there is a boundary homomorphism ∂qv : KMq+1(F ) → KMq (F (v)), for any integerq ≥ 0, called the tame symbol, satisfying

if u1, . . . , uq ∈ O×v , a ∈ F×, then ∂qv({u1, . . . , uq, a}) = v(a){u1(v), . . . , uq(v)}.

with Ov denoting the valuation ring of v in F . This relation completely deter-mines ∂qv on KMq+1(F ). (For more details on this classical subject cf. [11], [32],[3].)

KM of Varieties. On the Zariski site of a scheme X, the presheaf (of abelian

groups) U �→ KMq (OX(U)) induces a sheaf KMq , for each positive integer q.

1.2.1 Proposition ([1]). Let X ∈ SmProj/k. There is an exact sequence�

x∈X(1)K

M

q+1(k(x))∂→

�x∈X(0)

KM

q (k(x)) → KMq (X) → 0 ,

where X(r) denotes the set of k-closed points of X of dimension r.

1.2.2 Theorem ([14, Thm 3]). Let X be a smooth scheme of finite type ofdimension d over a field k, then

Hd(X,KMd+q) ∼= coker

�∂ :

�x∈X(1)

KM

q+1(k(x)) →�

x∈X(0)K

M

q (k(x))�

.

1.2.3 Corollary. Let X ∈ SmProj/k, d := dimX.

KM

q (X) ∼= Hd(X,KMq+d)

1.2.4 Remark. These results may be considered as a partial affirmation ofthe Gersten resolution for Milnor K-theory. For any x ∈ X(r), i. e. x is a k-closed point of X of codimension r, let ix : Spec k(x) → X denote the canonicalinclusion. Then, for any integer q ≥ 0, the following sequence of sheaves on theZariski site of X is exact,

0 −→ KMq /X −→�

x∈X(0)

(ix)∗KMq /Spec k(x)∂−→

�

x∈X(1)

(ix)∗KMq−1/Spec k(x)∂−→ · · ·

· · · ∂−→�

x∈X(d−1)

(ix)∗KMq−d+1/Spec k(x)∂−→

�

x∈X(d)

(ix)∗KMq−d/Spec k(x) −→ 0 ;

(1.4)

hence providing a flasque resolution of KMq /X .

1.3. FURTHER PROPERTIES 5

Comparison Theorem II

In [20, Lect. 3], there is defined, for any integer q ≥ 0, a complex Z(q) ofsheaves of abelian groups on the (small) Zariski site of a smooth scheme X/k.The hypercohomology of this complex, known as motivic cohomology, coincidesin the highest degree with Milnor K-theory.

1.2.5 Theorem. Let X ∈ SmProj/k, with k perfect. Let d := dim X.

KM

q (X)∼−→ Hq+2d(X, Z(q + d))

In term of higher Chow groups we obtain via Zariski-descent.

1.2.6 Corollary. Let X ∈ SmProj/k, with k perfect. Let d := dim X.

KM

q (X)∼−→ CHd+q(X, q)

A demonstration of Thm. 1.2.5 can be based on the following result in motiviccohomology.

1.2.7 Theorem. Let X ∈ SmProj/k, with k a perfect field. Then there existsan isomorphism of Zariski sheaves on X,

Hq(Z(q)) ∼−→ KMq .

1.2.8 Corollary.H

d+q(X, Z(q)) ∼= Hd(X,KMq )

for any integer q ≥ d = dim X.

Proof. To calculate the hypercohomology of the complex Z(q), use

Er,s2

= Hr(X,Hs(Z(q)) ⇒ Hr+s(X, Z(q)) . (1.5)

Choose r + s = d + q. If s < q, then r > d and Hr(X,F) = 0 for any sheafF of abelian groups by Grothendieck’s vanishing theorem (cf. [12, III.2.7]).If s > q, then Hs(Z(q)) = 0, so (1.5) degenerates to Hd(X,Hq(Z(q))) ∼−→H

d+q(X, Z(q)).

Thm 1.2.5 then obtained by replacing q by q + d, and applying 1.2.3.

1.2.9 Remark. See also [1] for a proof with explicit construction of the com-parison maps, which yields a version of 1.2.6 valid over any ground field.

1.3 Further Properties

Comparison with Quillen K-theory

1.3.1 Lemma. Let X ∈ SmProj/k, with k perfect. Let d := dimX. Thereexists a canonical morphism ϕ : KMq (X) → Kq(X).


Proof. Let Kq be the Zariski-sheaf on X associated to U �→ Kq(OX(U)), withKq the q-th K-group functor of Quillen. There exists an isomorphism

KM

1(OX(U)) ∼−→ K1(OX(U)) ,

since K1 of a semilocal commutative ring is isomorphic to its units (cf. [32]).This induces via the ring structure of K-theory, after sheafification, a morphismof sheaves KMq → Kq on X. Thus finally a morphism

Hd(X,KMq+d) → Hd(X,Kq+d) . (1.6)

According to [7], there exists a spectral sequence

Er,s2

= Hr(X,K−s) ⇒ K−r−s(X) . (1.7)

By the Gersten resolution of Kq given in [23, §7 5.11], we know that the righthand side becomes trivial for r > s + d, which yields an edge morphism

Hd(X,Kq+d) → Kq(X) . (1.8)

To obtain ϕ, compose KMq (X)∼−→ Hd(X,KMq+d), given in 1.2.3, with (1.6) and

(1.8).

1.4 Special KM-theory

Let k be a field and X ∈ SmProj/k. In this section, we are interested in thequestion how the Milnor K-theory of the base field k is reflected in that of thescheme X. The approaches via Somekawa groups and via the boundary mapoffer an intuitive answer to this question.

Via Somekawa Groups. The kernel of the zeroth Chow group with respectto the degree map fits into the exact sequence

0 → �CH0(Xk�) → CH0(Xk�)deg→ Z

�ini[xi] �→

�ini[k�([xi]) : k�]

for any (not necessarily finite) k-extension k� ⊆ k̄.1.4.1 Definition. Let k be a field. For X ∈ SmProj/k, q a positive integer,the group Kq(k; �CH0(X), Gm) is defined to be the quotient of the free abeliangroup �

k�/k

�CH0(Xk�)⊗ k�× ⊗ . . .⊗ k�×� �� q-times

,

with sum running through all finite field extensions k� over k, modulo the rela-tions (M1) and (M2) as in Def. 1.1.1 with CH0 replaced by �CH0. 3

1.4.2 Definition. We define the special Milnor K-group

SKM

q (X) :=

��CH0(X) if q = 0 ,Kq(k; �CH0(X), Gm) if q > 0 .

3Again, for reasons of consistency with Def. 1.1.1 and loc. cit., we mentioned implicitly

the functor fCH0(Xk� ) assigning to each finite field extension k�/k, the group gCH0(Xk� ).

1.4. SPECIAL KM-THEORY 7

Via the boundary map. In the Milnor K-theory of fields, there exists atransfer map Nq : KMq (E) → KMq (F ) for finite field extensions E/F (cf. [32,III.7.5]), which coincides in Milnor K-theory of varieties with covariant basechange for X = SpecE. We apply the transfer to residue fields to obtain

Nq :

�x∈X(0)

KM

q (k(x)) → KMq (k) .

Due to generalized Weil reciprocity, i. e.

Nq ◦ ∂q = 0 in KMq (k) ,

this induces a mapN

q : coker ∂q → KMq (k) . (1.9)

1.4.3 Corollary (of Prop 1.2.1). Let X ∈ SmProj/k.

SKM

q (X) ∼= ker�N

q : coker ∂q −→ KMq (k)�

.

We summarize the construction of the special KM-theory as follows.

1.4.4 Lemma. Let X ∈ SmProj/k. The sequence

0 −→ SKMq (X) −→ KMq (X)N−→ KMq (k) −→ 0

is exact and splits if and only if X admits a zero-cycle of degree 1 over k.

Chapter 2

Heights

2.1 Proofs of Mordell-Weil type

Consider a class of problems in algebraic geometry of the following form:

Determine the structure of a group A attached functorially to a geometricobject X.

A basic example of such a problem is a group scheme where the group at-tached is provided by the underlying scheme itself (see below). Further examplescan be found in algebraic-geometric K-theory, where the major problems are ofthis form. To tackle one of the latter (namely X a smooth projective curve overa global field, and A = SKM

1(X)), we review the method of proof applied to the

first example.

Review. Mordell-Weil Theorem

Consider the k-rational points A = X(k) of an abelian variety X over a numberfield k. The Mordell-Weil Theorem describes the structure of the group A. Itsproof proceeds in two steps.

(A) “weak Mordell-Weil” states:A/m is finite (for some integer m ≥ 2).

(B) “Descent”, described by

2.1.1 Lemma. Let A be an abelian group equipped with a height

h : A → R

satisfying the finiteness property

{a ∈ A | h(a) < C}

is a finite set for every constant C > 0. Suppose that A/m is finite forsome integer m ≥ 2. Then A is finitely generated.

(We specify in Def. 2.2.1 the notion of a height on a abelian group.)

9

10 CHAPTER 2. HEIGHTS

The Mordell-Weil Theorem concludes that A = X(k) is of finite type (cf. [13]).

Suppose we are interested in the structure of an abelian group A. If thegroup A is not of finite type anymore, the argumentation above does not applyof course. But just suppose that our knowledge about the abelian group A underconsideration is somehow limited, so that we do not know a priori whether A isof finite type or not. Maybe this is hard to determine, so a weaker statementabout the structure of A would already be desirable. More modestly, we mightask whether its maximal divisible subgroup is torsion. Since the conclusion wedraw is weaker, we may therefore hope to relax the conditions needed to arrivethere.

In this chapter we develop such an argumentation (leading to the DescentLemma 2.4.19), inspired by the line of arguments used in the proof of theMordell-Weil Theorem,

2.2 Heights on Abelian Groups

We will follow the philosophy that a “height” on an abelian group should be areal-valued function measuring the “complexity” its elements.

We propose the following definition.

2.2.1 Definition. A canonical linear height h on an abelian group A is a func-tion

h : A −→ R


(i) h is bounded below by 0, i. e.

h(a) ≥ 0 ∀a ∈ A .

(ii) h is even, i. e.h(a) = h(−a) ∀a ∈ A .

(iii) h satisfies the triangle inequality, i. e.

h(a + b) ≤ h(a) + h(b) ∀a, b ∈ A .

(iv) h is N-homogeneous, i. e. ∀m ∈ Z, m > 1,

h(ma) = mh(a) ∀a ∈ A .

Various Notions

Our discussion will rely on the notion of “height” as canonical linear height. Asmight be suspected from this nomination, there do exist alternatives. We brieflymention some for the interested reader.

2.2. HEIGHTS ON ABELIAN GROUPS 11

Linear Heights

2.2.2 Definition. A linear height h on an abelian group A is a function

h : A −→ R


(i) h is bounded below, i. e. ∃B ∈ R such that h(a) ≥ B for all a ∈ A.

(ii) h is quasi-even, i. e. there exists a constant C ≥ 0, depending on A, suchthat for all a ∈ A,

|h(a)− h(−a)| ≤ C .

(iii) Let a ∈ A. There exists a constant D ≥ 0, depending on A and a, so thatfor all a� ∈ A,

h(a� + a) ≤ h(a�) + D .

(iv) There exists a constant E ≥ 0, depending on A, so that for all a ∈ A andall integers m > 1

h(ma) ≥ mh(a)− E .

We remark that the Néron-Tate normalization ĥ(a) := limN→∞m−Nh(mNa)of a linear height h yields canonical linear height ĥ.

Quadratic Heights

2.2.3 Definition. A quadratic height h on an abelian group A is a function

h : A −→ R

satisfying the properties:

(i) h is bounded below, i. e. ∃B ∈ R such that h(a) ≥ B for all a ∈ A.

(ii) h is quasi-even, i. e. ∃C ≥ 0 such that |h(a)− h(−a)| ≤ C for all a ∈ A.

(iii) Let a ∈ A. There exists a constant D ≥ 0, depending on A and a, so thatfor all a� ∈ A,

h(a� + a) ≤ 2h(a�) + D .

(iv) There exists a constant E ≥ 0, depending on A, so that for all a ∈ A andfor all integer m > 1,

h(ma) ≥ m2h(a)− E .

A quadratic height h on an abelian group A is called canonical if B = C = 0,D ≤ 2h(a) for all a ∈ A, and for all integer m > 1

h(ma) = m2h(a) ∀a ∈ A .

This resembles the use of the term “height” as in [27] (see loc. cit. VII§3,Prop.1), or [13]. In particular, a quasi-even quasi-quadratic function is a quadraticheight, and a quadratic form (i. e. an even quadratic function) is a canonicalquadratic height (cf. [17, Ch.5, §1] for the definitions). If further the abeliangroup A in question is an abelian variety, this seems to be the common notionof “height” to which most textbooks restrict.


Heights of Higher Order A suitable definition for a generalization is givenin [17, Ch.3, §4] under the name “function of quasi-degree d”.

Elementary Properties

2.2.4 Lemma. Let (A, h) be an abelian group with canonical linear height.

(i) The height of 0 is 0. The elements of height 0 are a subgroup of A.

(ii) Torsion elements are of height 0, i. e.

Ator ⊆ h−1(0) ,

and h induces a canonical linear height on A/Ator, i. e.

h(a + b) = h(a) ∀a ∈ A, b ∈ Ator .

Proof. (i) For a positive integer m > 2,

mh(0) = h(m · 0) = h(0) = h(0 + 0) ≤ h(0) + h(0) = 2h(0)

implies that h(0) = 0. h−1(0) is a subgroup by the triangle inequality.(ii) Let b be a nonzero torsion element of A, i. e. ∃m ∈ Z>1 such that mb = 0.

Thenmh(b) = h(mb) = h(0) = 0

implies that h(b) = 0. Furthermore, for any a ∈ A,

h(a + b) =1m

·m · h(a + b) = 1m

h(m(a + b)) =1m

h(ma) = h(a) .

2.3 Linear Heights

Pre-heights

2.3.1 Definition. A canonical linear pre-height h on an abelian group A is anonnegative function

h : A −→ R+


(i) h is even, i. e. h(a) = h(−a) ∀a ∈ A.

(ii) h satisfies the triangle inequality, i. e. h(a + b) ≤ h(a) + h(b) ∀a, b ∈ A .

(iii) h is zero on torsion elements, i. e. h(a) = 0 ∀a ∈ Ator.

2.3.2 Proposition (Pushforward). Let

φ : A � B

2.3. LINEAR HEIGHTS 13

be an epimorphism of abelian groups. Let hA be a canonical linear pre-height onA. Then

hB : B → R

b �→ inf�

1m

hA(a)�� φ(a) = mb, m ∈ Z, m ≥ 1

�

= inf(m,a)∈Z≥1×A

φ(a)=mb

1m

hA(a)

is a canonical linear height on B. We call hB the pushforward of hA along φ,denoted hB = φ∗hA.

Proof. The function hB is well-defined and ≥ 0 due to the surjectivity of φ andsince hA ≥ 0, so we may verify the defining properties of a canonical linearheight.(i) Since hA is even, so is hB .(ii) We use the following general observation. Let Φ : X � Y be an epimorphismin the category of abelian groups.

{x ∈ X | Φ(x) = y1 + y2}� �� S1

⊇ {x1 + x2 ∈ X | Φ(x1) = y1,Φ(x2) = y2}� �� S2

for any y1, y2 ∈ Y . TheninfS1

f ≤ infS2

f (2.1)

for any real function f on X. Apply this argument to the case Φ = φ, i. e.X = A, Y = B. For b1, b2 ∈ B, we thus obtain the first inequality below.

hB(b1 + b2) = infm≥1�

infφ(a)=mb1+mb21m

hA(a)�

≤ infm≥1

�infφ(a1)=mb1

φ(a2)=mb2

1m

hA(a1 + a2)

�

≤ infm≥1�

infφ(a1)=mb11m

hA(a1) + infφ(a2)=mb21m

hA(a2)�

The second inequality above uses the triangle inequality for hA. Clearly,

infm≥1�

infφ(a1)=mb11m

hA(a1) + infφ(a2)=mb21m

hA(a2)�

≥�

infm≥1 infφ(a1)=mb11m

hA(a1)�

+�

infm�≥1 infφ(a2)=m�b21m�

hA(a2)�

= hB(b1) + hB(b2) , (2.2)

so that we actually have to show equality above. Choose a sequence (m(i)1

, a(i)1

)i∈Nin Z≥1 × A calculating hB(b1), i. e. φ(a(i)1 ) = m

(i)1

b1 for all i and hB(b1) =limi→∞ 1

m(i)1hA(a

(i)1

); analogous for b2. The triangle inequality for hA implies

that hA(ma) ≤ mhA(a) for any a ∈ A and positive integer m. Hence1

m(i)1

m(i)2

hA(m(i)2

a(i)1

) ≤ 1m

(i)1

m(i)2

m(i)2

hA(a(i)1

) =1

m(i)1

hA(a(i)1

) ∀i ,


and φ(m(i)2

a(i)1

) = m(i)1

m(i)2

b1 ∀i since φ is a homomorphism. Thus

inf(m,a)∈Z≥1×A

φ(a)=mb1

1m

hA(a) ≤ limi→∞

1

m(i)1

m(i)2

hA(m(i)2

a(i)1

) ≤ limi→∞

1

m(i)1

hA(a(i)1

)

= hB(b1) ,

hence equality holds above by definition of the pushforward. In other words,(m(i)

1m

(i)2

, m(i)2

a(i)1

)i∈N calculates hB(b1). Similarly, (m(i)1

m(i)2

, m(i)1

a(i)2

)i∈N cal-culates hB(b2). Hence (2.2) holds with equality, which establishes the triangleinequality for hB .(iii) Let n be a positive integer, and b ∈ B. The triangle inequality for hBshown in (ii) implies that hB(nb) ≤ nhB(b). Choose a sequence (m(i), a(i))i∈Nin Z≥1 × A calculating hB(nb), i. e. φ(a(i)) = m(i)nb for all i and hB(nb) =limi→∞ 1m(i) hA(a

(i)). Then

inf(m,a)∈Z≥1×A

φ(a)=mb

1m

hA(a) ≤ limi→∞

1nm(i)

hA(a(i)) =1n

hB(nb) ≤ hB(b) ,

hence equality holds above by definition of the pushforward. The sequence(nm(i), a(i))i∈N calculates hB(b), and consequently nhB(b) = h(nb).

2.3.3 Lemma. In the situation of Prop. 2.3.2, let hA be a canonical linearheight. Suppose that A is divisible and that B is torsionfree. Then

φ∗hA = inf hAφ−1 .

Proof. Let b ∈ B and n ≥ 1 an integer. We claim that the divisibility of Aimplies

φ−1(nb) = nφ−1(b + Btor) . (2.3)

Generally, the inequality of set φ−1(nb) ⊇ nφ−1(b) holds solely due to φ beinga homomorphism (thus being homogeneous in the sense that nφ(a) = φ(na)for all a ∈ A). Conversely, let a ∈ φ−1(nb). Since A is divisible, there exists1

na ∈ A, and hence nφ(1

na) = φ(n1

na) = φ(a) = nb. From this we can concludethat φ( 1na) ≡ b mod Btor. Thus

1

na ∈ φ−1(b + Btor), and a ∈ nφ−1(b + Btor).Since hA is a height and since Btor = 0, it follows that

hB(b) = inf(m,a)∈Z≥1×A

a∈φ−1(mb)

1m

hA(a) = inf(m,a)∈Z≥1×A

a∈φ−1(b)

1m

hA(ma) = infa∈A

a∈φ−1(b)

hA(a) ,

Tensor Products

Recall the construction of the tensor product of two abelian groups A, B (viewedas Z-modules). Denote by A◦Z B the free Z-module with generators all symbolsa ◦ b with a ∈ A, b ∈ B. Then A ⊗Z B is the quotient of A ◦Z B modulo therelations

(T1) (a + a�) ◦ b− a ◦ b− a� ◦ b,

2.3. LINEAR HEIGHTS 15

(T2) a ◦ (b + b�)− a ◦ b− a ◦ b�,(T3) na ◦ b− a ◦ nb for n ∈ Z,

for all a, a� ∈ A and b, b� ∈ B. Denote by

ρ : A ◦Z B � A⊗Z B

the quotient map.

2.3.4 Lemma. Let (A, hA), (B, hB) be abelian groups equipped with canonicallinear heights. The function

h◦ : A ◦Z B → R+,

�

a∈A,b∈B

na,b a ◦ b �→�

|na,b| · hA(a) · hB(b)

is a canonical linear height on A ◦Z B.

Proof. Since na,b is almost always zero, all sums appearing are finite, and h◦ iswell-defined. To show that it is a height, we verify the defining properties.(i) Since the absolute value | |, hA and hB are bounded below by 0, so is h◦.

h◦

��na,ba ◦ b

�=

�|ra,b|hA(a) · hB(b) ≥ 0

(ii) Since | | is an even function, and so is h◦.

h◦

�(−1) ·

�na,ba ◦ b

�= |−1|

�|na,b|hA(a) · hB(b) = h◦

��na,ba ◦ b

�

(iii) The triangle inequality for | | induces the triangle inequality of h◦.

h◦

��na,ba ◦ b +

�n�

a,ba ◦ b�

= h◦��

na,b + n�a,b�a ◦ b

�

=� ��na,b + n�a,b

�� hA(a)hB(b) ≤�

(|na,b|+ |n�a.b|)hA(a)hB(b)

= h◦��

na,ba ◦ b�

+ h◦��

n�

a,ba ◦ b�

.

(iv) For any positive integer m, the homogeneity of abs induces those of h◦.

h◦

�m ·

�na,ba ◦ b

�=

�|mna,b|hA(a)hB(b) = m · h◦

��na,ba ◦ b

�.

This establishes h◦ as a canonical linear height.

2.3.5 Proposition. Let (A, hA), (B, hB) be abelian groups equipped with canon-ical linear heights. Then hA and hB induce a canonical linear height h on A⊗ZB,denoted h = hA ⊗ hB.

Proof. Lemma 2.3.4 provides a canonical linear height h◦ on A ◦Z B, whichwe push forward along the quotient map ρ : A ◦Z B → A ⊗Z B, inducing byProp. 2.3.2 the canonical linear height h := ρ∗h◦ on A⊗Z B.


2.4 Properties

New Properties

2.4.1 Definition. Let (A, h) be an abelian group with canonical linear pre-height.

(T) We say that the pre-height h satisfies the property (T), called torsionproperty, if zero pre-height implies torsion, i. e.

h−1(0) = Ator .

(F) We say that the pre-height h satisfies the property (F), called strong finite-ness property, if the set of elements in A of bounded height is finite, i. e.

{a ∈ A | h(a) < C}

is a finite set of elements of A for every real constant C ≥ 0.

(FF) We say that the pre-height h satisfies the property (FF), called weak finite-ness property, if the set of values of elements in A of bounded height isfinite, i. e.

h�{a ∈ A | h(a) < C}

�

is a finite set in R for every real constant C ≥ 0.

Interdependencies

2.4.2 Lemma. Let (A, h) be an abelian group with canonical linear pre-heightsatisfying (F). Then

Ator = h−1(0) .

Proof. The inclusion Ator ⊆ h−1(0) is provided by the defining property Def. 2.3.1(iii). Conversely, if a ∈ A is of height 0, then so are all elements in the set{na | n ∈ Z≥1} as follows from the triangle inequality. Since h satisfies (F), thisset is finite, hence a is of finite order.

The relation between these newly defined properties for pre-heights can con-veniently be summarized as

(F ) =⇒ (FF ), (T ) .

It is important to notice that the converse does not hold. So (FF) and (T) takentogether constitute an essentially weaker property than (F), which we will seeto be satisfied by a wider class of pre-heights, for which this properties is still“strong enough” to be useful.

Inherited Properties

2.4.3 Lemma. Let (A, hA) be an abelian group with canonical linear pre-height,and let φ : A � B be an epimorphism inducing a canonical linear heighthB := φ∗hA on the abelian group B. Assume that, for all b ∈ B,

hB(b) = min(m,a)∈Z≥1×A

φ(a)=mb

1m

hA(a) , (2.4)

2.4. PROPERTIES 17

i. e. the height of each b ∈ B can be calculated by a tuple (m, a) in Z≥1 × A(instead of a sequence of tuples). Then, if hA satisfies (T), so does hB.

Proof. Let b ∈ B be a non-torsion element. To show that hB satisfies (T), wehave to show that hB(b) �= 0. Due to (2.4), we may choose a tuple (m, a) inZ≥1×A calculating the height of b, i. e. φ(a) = mb and 1mhA(a) = hB(b). Sinceb is non-torsion, so is mb. The pre-image of a non-torsion element is non-torsion,so a is non-torsion. If hA satisfies (T), hA(a) �= 0. Hence 1mhA(a) = hB(b) �=0.

2.4.4 Lemma. Let (A, hA) be an abelian group with canonical linear pre-height,and let φ : A � B be an epimorphism inducing a canonical linear heighthB := φ∗hA on the abelian group B. Assume that

∃m ∈ Z≥1 : ∀b ∈ B hB(b) = infa∈A

φ(a)=mb

1m

hA(a) , (2.5)

i. e. the height of each b ∈ B can be calculated by a fixed m ∈ Z≥1 and a sequence(a(i))i∈N in A (instead of a sequence of tuples). Then, if hA satisfies (FF), sodoes hB.

Proof. If hA satisfies (FF) and (2.5), then infimum is attained uniformly, i. e.

∃m ∈ Z≥1 : ∀b ∈ B hB(b) = mina∈A

φ(a)=mb

1m

hA(a) . (2.6)

Let C ≥ 0 be a real constant. We base our argumentation on the followingargument: hB satisfies (FF) if and only if every sequence (bi) ∈ BN satisfiesthat

�i∈N hB(bi) ∩ [0, C[ is finite. This refines to: hB satisfies (FF) if and only

if every sequence (bi) ∈ BN with strictly monotone (increasing or decreasing)heights satisfies that

�i∈N hB(bi) ∩ [0, C[ is finite.

So, let (bi) ∈ BN be a sequence with strictly monotone heights. Due to (2.6),we may choose, for each i, (m, ai) ∈ Z≥1 × AN calculating the height of bi, i. e.φ(ai) = mbi and 1mhA(ai) = hB(bi).

If hB(bi) decreases strictly monotonically, then hA(ai) decreases strictlymonotonically. Note the here uniformity, i. e. m does not vary, is essentialhere. Since hA satisfies (FF), hA(ai) must become stationary. This contradictsthe strict monotony of hA(ai).

If hB(bi) increases, then hA(ai) increases, both strictly monotonically. SincehA satisfies (FF), the strictly monotonically increasing sequence hA(ai) satisfieslimi→∞ hA(ai) = ∞. Again by uniformity, limi→∞ hB(bi) = ∞. Hence hB(bi)must exceed the constant C for some index, i0 say, and

�i∈N hB(bi) ∩ [0, C[=�

i


induced height on A◦Z B (as in Lemma 2.3.4), and ρ the quotient map onto thetensor product. The condition stated in Lemma 2.4.4 for the property (FF) tobe inherited by the pushforward suggests to search for a bound on the integerm appearing in the definition of hA⊗hB . The existence of such a bound wouldimply (FF) for hA⊗hB whenever the same holds both for hA and hB . However,controlling m is not the only possibility to establish (FF), as can be seen in thefollowing.

2.4.6 Example. (This instructive example was communicated to us by Rob deJeu.) Let A := Z ⊕ Z with canonical linear height hA((x, y)) := max{|x| , |y|}.hA satisfies (F), hence (FF) and (T). Consider h := hA⊗hA on A⊗Z A. Definee1 := (1, 0), e2 := (0, 1), and β0 := e1 ⊗ e1 + e2 ⊗ e2. The calculation of h(β0)illustrates the dependence on m.

Consider m = 1. Since e1 ◦ e1 + e2 ◦ e2 is a pre-image of β0 under ρ,we obtain inf h◦ρ−1(β0) ≤ 2. Here actually equality holds. This can be seenas follows. Let V1 := {a ∈ A | hA(a) = 1} = {±e1,±e2,±e1 ± e2}, andW0 := {α =

�na,a� a ◦ a� ∈ A ◦ A | na,a� = 0 or a = 0 or a� = 0}. Then

W1 := {α ∈ A ◦ A | h◦(α) ≤ 1} = W0 ∪ {±(a ◦ a�) + α0 | a, a� ∈ V1, α0 ∈ W0}.We find that ρ(W0) = 0 and β0 /∈ ρ(W1), hence inf h◦ρ−1(β0) > 1. Since theimage of h◦ is contained in Z, we conclude that inf h◦ρ−1(β0) = 2.

Considering m = 2, we take into account the equality

(a1 + a2)⊗ (a1 + a2) + (a1 − a2)⊗ (a1 − a2) = 2 (a1 ⊗ a1 + a2 ⊗ a2)for any a1, a2 ∈ A. Choosing a1 = e1, a2 = e2, we obtain inf 12h◦ρ−1(2β0) ≤ 1.Since 2β0 /∈ ρ(W1), we have inf h◦ρ−1(2β0) > 1. We conclude inf 12h◦ρ−1(2β0) =1.

2.4.7 Lemma. Let A be the direct sum of r copies of Z equipped with hA beingthe maximum of the componentwise absolute value, i. e.

hA((x1, . . . , xr)) := max{|x1| , . . . , |xr|} .The height h := hA ⊗ hA on A⊗Z A satisfies (F), hence (FF) and (T)Proof. Consider the canonical linear height �hA((x1, . . . , xr)) :=

�ri=1 |xi| on the

same group A. Then �h := �hA ⊗ �hA is a height on A⊗Z A. Lemmas 2.4.11 willprovide the tools to calculate �h explicitly. More precisely, we obtain

�h =�r

i=1�hA ◦ φi (2.7)

with canonical isomorphism φ : A⊗A ∼−→ ⊕ri=1A and φi denoting compositionof φ with the projection to the i-th component. Due to the absence of torsion,and since the sum (2.7) is finite, we can conclude that the finiteness property(F) for �hA implies the analog for �h. Note that in particular �h(β) := inf �h◦ρ−1(β)∀β ∈ A⊗A. For r = 2 applying to the situation of the previous example 2.4.6,we then find that �h(β0) = 2, calculated by (m, α) = (1, e1 ◦ e1 + e2 ◦ e2).

Comparing the two heights, we find the inequality 1r�hA ≤ hA ≤ �hA. Weshow in Lemma 2.4.13 that this yields 1r2 �h ≤ h ≤ �h on the tensor productA⊗A. Hence, for any real constant C > 0,

{α ∈ A⊗A | h(α) < C} ⊆ {α ∈ A⊗A | �h(α) < r2C}

Since �h satisfies (F), the right hand side is finite, and so is the left.

2.4. PROPERTIES 19

Conclusion. Resuming the initial question 2.4.5, it is our unproven opinionthat in general (FF) is not inherited from hA and hB to hA ⊗ hB . Thus wepresent two modifications of the initial question allowing an affirmative answer.

• We fix m. Then we gain the weak finiteness property (FF) inherited onthe tensor product, but we lose homogeneity (with respect to positiveintegers), so that we actually obtain a pre-height instead of a height (cf.Prop. 2.4.10).

• We modify the initial height in such a way that the induced height onthe tensor product satisfies (FF) and (T) whenever the initial heights doso individually. On the downside, this construction is not intrinsic (cf.Lemma 2.4.14 and Cor. 2.4.15).

The case of A = k× being the units of a number field k is of particular interestto us and is considered in detail in 2.4.16 - 2.4.18.

Induced Pre-heights

2.4.8 Lemma. Let φ : A → B be an epimorphism of abelian groups, and lethA be a canonical linear height on A. Denote by τ : B → B/Btor the canonicalprojection. Then

h : B → R+, b �→ inf(τ◦φ)(a)=τ(b)

hA(a)

is a canonical linear pre-height on B.

Proof. The defining properties (i) and (ii) of a pre-height are verified as in theproof of Prop. 2.3.2 with m = 1. Concerning property (iii), a torsion elementb ∈ Btor is mapped to 0 under τ , whose inverse image under τ ◦ φ contains0 in A. As seen in Lemma 2.2.4, hA(0) = 0, which is clearly minimal, hencehB(b) = 0.

2.4.9 Lemma. Let (A, hA), (B, hB) be abelian groups equipped with canonicallinear heights. If hA and hB both satisfy (FF), then the induced height h◦ onA ◦Z B (as in Lemma 2.3.4) satisfies (FF).Proof. Let C > 0 be a real constant. The image of A under hA is discrete dueto the weak finiteness property (FF), so that there exists a minimal nonzeroelement mA := min{hA(A) − {0}}. Analogously, mB := min{hB(B) − {0}}.Define

W0 := {0} ,W1 := {|n| · hA(a) · hB(b) : a ∈ A, b ∈ B,n ∈ Z}∩ ]0, C[ ,

Wi+1 := {v + w : v ∈ W1, w ∈ Wi}∩ ]0, C[ for integers i ≥ 1 .

Then #W1 ≤ CmAmB ·#{hA(a) : hA(a) < C/mB}·#{hB(b) : hB(b) < C/mA} isfinite. It follows inductively, that all Wi are finite. In particular, each nonemptyWi has a minimal element, wi say, satisfying wi − wi−1 ≥ w1 �= 0. Thus thereexists an index i0 such that ∀i ≥ i0: Wi = ∅. Now

h◦(A ◦Z B) ∩ [0, C[ =

�i0i=0

Wi

establishes (FF) for h◦.


2.4.10 Proposition. Let (A, hA), (B, hB) be abelian groups equipped with canon-ical linear heights. If hA and hB both satisfy (FF) and (T), then there exists aninduced canonical linear pre-height h on A⊗Z B satisfying (FF) and (T).

Proof. We define h as in Lemma 2.4.8 along the maps

A ◦Z Bρ� A⊗Z B

τ� (A⊗Z B) /torsion

with h◦ the canonical linear height on A ◦Z B, i. e.

h(β) = infα∈A◦ZB

(τ◦ρ)(α)=τ(β)

h◦(α) for β ∈ A⊗Z B.

For any real constant C ≥ 0, the construction of h yields

h◦(A ◦Z B) ∩ [0, C[ ⊇ h(A⊗Z B) ∩ [0, C[ ,

with W denoting the closure of a real subset W ⊆ R w.r.t. the standardtopology. Since h◦ satisfies (FF) by Lemma 2.4.9, the set h◦(A ◦Z B) ∩ [0, C[is finite, hence equals its closure h◦(A ◦Z B) ∩ [0, C[. It follows that the seth(A⊗Z B) ∩ [0, C[ is finite, establishing (FF) for h.

Let β ∈ A ⊗Z B such that h(β) = 0. For h to satisfy (T), we have to showthat β is torsion. Since h◦ satisfies (FF), the infimum in the definition of hbecomes a minimum, i. e. there exists α =

�na,b a ◦ b ∈ A ◦Z B such that

τ ◦ ρ(α) = τ(β) and h◦(α) = h(β). Both hA and hB satisfy (T), thus it followsfrom h(β) = 0 that

h◦

��na,b a ◦ b

�= 0 ⇔ h◦(a ◦ b) = 0 ∀na,b �= 0

⇔ hA(a) = 0 or hB(b) = 0 ∀na,b �= 0 ⇔ a or b is torsion ∀na,b �= 0

Hence a⊗ b is torsion ∀na,b �= 0, so�

na,b a⊗ b = β is torsion.

LinearizationWe investigate the following approach. Given abelian groups with heights

(A, hA) and (B, hB), the group structure of A⊗B is easy to determine. Hencedisposing of an isomorphism φ : A⊗B ∼−→ C, does there exist a canonical linearheight hC on C such that hA ⊗ hB = hC ◦ φ ?

(i) In Lemma 2.4.11, we specify cases where such is true. Lemma 2.4.14 thenproduces these cases.

(ii) Example 2.4.12 answers this question negatively. In words, the informa-tion of a height on a tensor product cannot be captured by the groupstructure alone.

2.4.11 Lemma. Let (B, hB) be an abelian group with canonical linear height.

(i) Let hZ be a canonical linear height on Z, which is necessarily a multiple ofthe absolute value, i. e. hZ = D | · | for some constant D := hZ(1) ∈ R≥0.Via the canonical isomorphism φ : Z ⊗Z B ∼−→ B sending 1 ⊗ b �→ b forb ∈ B, the induced height hZ ⊗ hB satisfies

hZ ⊗ hB = D · hB ◦ φ .

2.4. PROPERTIES 21

(ii) Let (Ai, hi)i∈I be a countable family of abelian groups with canonical linearheight, and denote by

⊕hi : ⊕i∈IAi → R+,�

iai �→

�ihi(ai)

the canonical linear height on the direct sum. Via the canonical isomor-phism φ : (⊕iAi)⊗B ∼−→ ⊕i (Ai ⊗B) sending (

�i ai)⊗ b �→

�i(ai ⊗ b),

the induced heights satisfy

(⊕hi)⊗ hB =�

i(hi ⊗ hB) ◦ φi

with φi denoting φ composed with the projection to the i-th component.

Proof. (i) To an element α ∈ Z ◦Z B, written uniquely as α =�

k,b nk,b (k ◦ b),we associate �α := 1 ◦

��k,b nk,b · k · b

�∈ Z ◦Z B. Writing ρ : Z ◦Z B → Z⊗Z B

for the quotient map, we have ρ(α) = ρ(�α). Denote by h◦ the height on Z ◦B,then

h◦(α) ≥ h◦(�α) = hZ(1) · hB(φ ◦ ρ(�α)) (2.8)

To see this, we write according to the definition h◦(α) =�

k,b |nk,b|hZ(k)hB(b) =�k,b |nk,b| · |k|hZ(1)hB(b) =

�k,b hZ(1)hB(nk,b kb) which, by the triangle in-

equality for hB , is ≥ hZ(1)hB� �

k,b nk,b kb�

= h◦(�α). Further, φ ◦ ρ(�α) =�k,b nk,b kb.Let β ∈ Z⊗ B. Choose a sequence (m(j), α(j))j∈N calculating the height of

β, i. e. ρ(α(j)) = m(j)β ∀j and limj→∞ 1m(j) h◦(α(j)) = (hZ ⊗ hB) (β). Replace

α(j) by �α(j) ∀j. It follows from (2.8), that 1

m(j)h◦(�α(j)) converges to the infimum

defining (hZ ⊗ hB) (β), thus (m(j), �α(j))j∈N is a sequence calculating the heightof β. Further (2.8) yields

h◦(�α(j)) = D · hB(φ ◦ ρ(�α(j))) = D · hB(φ(m(j)β)) = m(j)D · hB (φ(β)) .

Hence (hZ ⊗ hB) (β) = limj→∞ 1m(j) h◦(�α(j)) = D · hB ◦ φ(β).

(ii) To an element α ∈ (⊕Ai) ◦ B, written uniquely as α =�

a,b na,b (a ◦ b)with a =

�i ai, we associate αi :=

�ai,b

nai,b (ai◦b) ∈ Ai◦B with nai,b := na,b.Conversely, to an element αi :=

�ai,b

nai,b (ai ◦ b) ∈ Ai ◦ B for some i ∈ I, weassociate �αi :=

�a,b na,b (a ◦ b) with na,b := nai,b and a := ai. For the quotient

maps we write ρ : (⊕Ai) ◦ B → (⊕Ai) ⊗ B, and ρi : Ai ◦ B → Ai ⊗ B. Withrespect to the canonical isomorphism φ, it can be seen that

φ ◦ ρ(α) =�

iρi(αi) for α ∈ (⊕Ai) ◦B. (2.9)

Note that starting with a given α ∈ (⊕Ai) ◦B,�

i �αi is a well defined elementin (⊕Ai) ◦ B but not necessarily equal to α (e. g. suppose {1, 2} ⊆ I andα = (a1 +a2)◦b−(a1−a2)◦b with a1 ∈ A1, a2 ∈ A2−{0}, b ∈ B, then α1 = 0).However, it follows from (2.9) that ρ(α) = ρ(

�i �αi).

Denote by h◦ the height on (⊕Ai) ◦ B, and by h◦i the height on Ai ◦ B foreach i ∈ I. We claim that

h◦(α) ≥

�ih◦

i (αi) , h◦��

i�αi

�=

�ih◦

i (αi) . (2.10)


To see this, we write according to the definition h◦(α) =�

a,b |na,b| (⊕hi)(a)hB(b)=

�(P

i ai),b

��n(Pi ai),b�� (⊕hi)(

�i ai)hB(b) =

�(P

i ai),b

�i

��n(Pi ai),b�� hi(ai)hB(b)

which, by a cardinality argument, is ≥�

i

�ai,b

|nai,b|hi(ai)hB(b) =�

ih◦i (αi).

For the right hand side of (2.10), equality holds since each summation index�i ai is of the form ai.Let β ∈ (⊕Ai)⊗B. Choose a sequence (m(j), α(j))j∈N calculating the height

of β, i. e. ρ(α(j)) = m(j)β ∀j and limj→∞ 1m(j) h◦(α(j)) = ((⊕hi)⊗ hB) (β).

Then for each i ∈ I, ρi(α(j)i ) = m(j)βi ∀j with βi := φi(β) by (2.9). Hence byconstruction of the height, 1

m(j)h◦i (α

(j)i ) ≥ (hi ⊗ hB) (βi) ∀j. By virtue of (2.10),

the sequence�

1

m(j)h◦i (α

(j)i )

�j∈N is bounded by the converging and bounded se-

quence�

1

m(j)h◦(α(j))

�j∈N, hence the former sequence is bounded. Thus there

exists a subsequence (α(jki )i )ki∈N of (α(j)i )j∈N such that

1

m(jki

) h◦i (α(jki )) con-

verges. The limit of the subsequence is not unique, so that we choose a subse-quence with minimal limit for each i, which is unique. Then the left hand sideof (2.10) applies, providing

limj→∞1

m(j)h◦(α(j)) ≥

�ilimki→∞

1m

(jki )h◦

i (α(jki )) .

This shows that (⊕hi)⊗ hB ≥�

i (hi ⊗ hB) ◦ φi.Conversely, the set I0 := {i ∈ I | φi(β) �= 0} is finite. For each i ∈ I0, choose

a sequence (m(j)i , α(j)i )j∈N calculating the height of βi. Denote by �m(j) the least

common multiple of (m(j)i | i ∈ I0) ∀j. Writing �α(j) :=�

i∈I0bm(j)

m(j)i�α(j)i ∀j, the

right hand side of (2.10) yields

1�m(j)

h◦(�α(j)) =

�i∈I0

1

m(j)i

h◦(α(j)i )

Note that the sequence (�m(j), bm(j)

m(j)iα

(j)i )j∈N calculates the height of βi for each

i ∈ I0. Since 1m(j)i

h◦i (α

(j)i ) converges for each i ∈ I0, it follows from the equa-

tion above that 1bm(j) h◦(�α(j)) converges. Hence by construction of the height,

limj→∞ 1bm(j) h◦(�α(j)) ≥ ((⊕hi)⊗ hB) (β). This shows that (⊕hi) ⊗ hB ≤�

i (hi ⊗ hB) ◦ φi.

2.4.12 Example. Let L := {�k

j=1 njxj | nj ∈ Z} ⊆ Rn be a Z-lattice with(xj)j=1,...,k a set of R-linearly independent vectors in Rn, for some positiveintegers k ≤ n. The L1-Norm on Rn provides a canonical linear height on theabelian group L,

hL(x) =�

i=1,...,n

|xi|

for any lattice point x = (x1, . . . , xn) ∈ L. Denote by πi : Rn → R the canonicalprojection to the i-th coordinate. Then πi(L) ⊆ R is an abelian group, free ofrank at most k. Let (B, hB) be an abelian group with height. Further πi inducesa projection πi ⊗ id : L ⊗ B → πi(L) ⊗ B by sending x ⊗ b �→ xi ⊗ b. WritinghL =

�i hR ◦ πi with hR := | · |, application of Lemma 2.4.11(ii) yields

hL ⊗ hB =�

i(hR ⊗ hB) ◦ (πi ⊗ id) .

2.4. PROPERTIES 23

For each i, we have πi(L) = Zci,1 ⊕ . . .⊕ Zci,k with ci,j ∈ R≥0 (possibly zero).Writing φi,j for the canonical isomorphism composed with the projection to thej-th component, and setting hi,j : Z → R+, n �→ ci,j |n|, then hR ≤

�j hi,j◦φi,j .

If at least two of the cij are nonzero, equality cannot hold. Application ofLemma 2.4.11(i) provides

hR ⊗ hB ≤�

j(hi,j ⊗ hB) ◦ (φi,j ⊗ id) = ci,jhB ◦ ψ ◦ (φi,j ⊗ id)

with ψ : Z ⊗ B ∼−→ B the canonical isomorphism. We conclude that it is notpossible to express hL ⊗ hB in terms solely of hB and group isomorphisms.

Concerning questions of finiteness properties one might ask for this example,we would just like to point out the phenomenon which makes these difficult here:The lattice L is discrete (in Rn) by definition, but its projection πi(L) need notbe discrete (in R), it may be dense.

2.4.13 Lemma. Let A be an abelian group with canonical linear heights hA,1and hA,2 such that

hA,1 ≤ hA,2 .Then, for any canonical linear height hB on an abelian group B,

hA,1 ⊗ hB ≤ hA,2 ⊗ hB .

Proof. There are induced heights h◦i on A◦B, with i = 1, 2, sending�

na,b a◦ b �→�|na,b| · hA,i(a)hB(b). Hence h◦1 ≤ h◦2. Let β ∈ A ⊗ B and choose a se-

quence (m(i), α(i))i∈N calculating hA,2 ⊗ hB of β. Then (hA,2 ⊗ hB)(β) =limi→∞ 1m(i) h

◦2(α(i)) ≥ limi→∞ 1m(i) h

◦1(α(i)) ≥ (hA,1 ⊗ hB)(β).

2.4.14 Lemma (“Linearization”). Let (A, hA) be an abelian group with canon-ical linear height. Assume that A/Ator is free of countable rank.

(i) A choice of generators A = Ator ⊕i∈I⊆N Zei provides a canonical linearheight

�hA : A → R+, a0 +�

iniei �→

�i|ni| ci

with a0 ∈ Ator and (ci)i∈I a family of nonnegative real constants. If ci > 0∀i ∈ I, then �hA satisfies (T). If limi→∞ ci = ∞, then �hA satisfies (FF).Choosing ci := hA(ei), then �hA satisfies (T), (FF), if hA satisfies (T),(FF) respectively.

(ii) Let (B, hB) be an abelian group with canonical linear height. If hB satisfiesboth (FF) and (T), then there exists an induced canonical linear heighth = �hA ⊗ hB on A⊗Z B satisfying (FF) and (T).

Proof. (i) Assume ci > 0 ∀i ∈ I. Then �h(a) = 0 if and only if all of the associatedcoefficient ni are of absolute value 0, which is equivalent to a being torsion.Assume limi→∞ ci = ∞. Let C > 0 be a real constant. Then there exists anindex i0 ∈ I such that ci ≥ C for all i > i0, and �hA(A)∩[0, C[= ⊕i≤i0Zci∩[0, C[,which is finite.

(ii) Choose a family of positive unbounded constants (ci)i∈I . Denote by φthe canonical isomorphism A ⊗ B ∼−→ (Ator ⊗ B) ⊕

�i∈I(Zei ⊗ B). Then, by

Lemma 2.4.11(ii),�hA ⊗ hB =

�i∈I

(h(i)A ⊗ hB) ◦ φi


with height h(i)A : A → R+ sending a0 +�

iniei �→ |ni| ci for i ∈ I, and withφi denoting the composition of φ with the projection to the i-th component.Let ψi be the canonical isomorphism Zei ⊗ B ∼−→ B. By Lemma 2.4.11(i),h

(i)A ⊗ hB = ci · hB ◦ ψi.

We conclude that, since hB satisfies (T), so does �hA ⊗ hB . Let C > 0 be areal constant.

(�hA ⊗ hB)(A⊗B) ∩ [0, C[ =�

i≤i0

ci · hB(B) ∩ [0, C[ ,

with i0 as in (i). Thus (FF) for hB implies the analog property for �hA⊗hB .

It might be desirable to be more faithful to the arithmetic, which can bedone as follows.

2.4.15 Corollary. Let (A, hA), (B, hB) be abelian groups equipped with canoni-cal linear heights. Suppose that both groups modulo torsion are free of countablerank. If hA and hB both satisfy (FF) and (T), then there exists an inducedcanonical linear height h on A⊗Z B satisfying (FF) and (T).

Proof. Choose a height �hA on A, resp. �hB on B as in Lemma 2.4.14, in suchthat way that both satisfy (FF) and (T). Define

h : A⊗Z B → R+, α �→12

�(�hA ⊗ hB)(α) + (hA ⊗ �hB)(α)

�.

By Lemma. 2.4.14, �hA ⊗ hB and hA ⊗ �hB satisfy (FF) and (T), hence so doesh.

2.4.16 Example. Let k be a number field, and k× its units. Then 2.4.14 aswell as 2.4.15 apply to the construction of a non-intrinsic canonical linear heighton k×⊗Z k× which possesses the properties (FF) and (T). However, for the casek = Q, no choices have to be made.

For any number field k, there is a canonical linear height hk satisfying (F),

hk : k× → R+, a �→�

all places v

log max{|a|v , 1} , (2.11)

where |a|v := |τv(a)| for v real, and |a|v := |τv(a)|2 for v complex, with τv : k →

C denoting an associated embedding; and where |a|v := N(v)−ηv(a), with ηvdenoting the associated exponential valuation normalized by ηv(k×) = Z, andN(v) denoting the cardinality of the finite residue field k(v).

In the case of the rationals, there is an isomorphism Q× = �±1�⊕v:finite Zevand hQ(a) =

�v:finite log max{|a|v ,

��a−1��v} due to the product formula. By

Lemma. 2.4.14, the height hQ ⊗ hQ satisfies (FF) and (T).2.4.17 Example. In the initial example 2.4.6, the property (F) was establishedfor a height h by giving a lower bound for h in terms of another height �h forwhich (F) could be shown more easily. We would like to continue Example 2.4.16where it is not clear whether such a strategy is feasible.

Let S be a finite set of places of a number field k such that S contains thearchimedean places, and let O×k,S be the S-units, i. e.

O×k,S = {a ∈ k× : |a|v = 1 ∀v /∈ S} .

2.4. PROPERTIES 25

Let A := O×k,S/torsion, equipped with the canonical linear height hA induced(via restriction and passage to the quotient modulo torsion) by hk as in (2.11).Thus, hA = hk

��O×k,S

on A reads

hA : A → R+, a �→�

v∈S

log max{|a|v , 1} .

We define a second height on A:

�hA : A → R+, a �→�

v∈S

log max{|a|v ,��a−1

��v} .

For each v ∈ S: log max{|a|v ,��a−1

��v} ≤

�v∈S log max{|a|v , 1}. Hence

hA ≤ �hA ≤ shA

with s := |S| which we assumed finite.Further, let (B, hB) be an abelian group with canonical linear height. As-

sume that B is torsionfree. Passing to the tensor product A⊗B, Lemma 2.4.13applies to the above inequality, providing

hA ⊗ hB ≤ �hA ⊗ hB ≤ s · (hA ⊗ hB) . (2.12)

Hence, for any real constant C > 0,

{α ∈ A⊗B | (hA ⊗ hB)(α) < C} ⊆ {α ∈ A⊗B | (�hA ⊗ hB)(α) < sC}

In other words, (2.12) provides the conclusion: If �hA ⊗ hB satisfies (F), thenhA ⊗ hB satisfies (F). Fixing of any order of the v ∈ S provides a well-definedmap

λ : A → Rs, a �→ (log |a|v)v∈S .

We find that λ is a monomorphism of abelian groups and that �hA = | · |1 ◦ λwith | (x1, . . . , xn) |1 =

�i=1,...,n |xi| denoting the L1-Norm in Rn. Here we

encounter the same problem as pointed out in Example 2.4.12.

We summarize this approach in a parametrized version by the followinglemma. For any c ∈ R>0 define

S(c) := {finite places v of k such that N(v) < c} ∪ {infinite places of k} .

2.4.18 Lemma. Let (B, hB) be an abelian group with canonical linear heightsuch that hB on B/tor satisfies (F). Then the finiteness property (F) for �hk⊗hBon

�O×k,S(c)/tor

�⊗

�B/tor

�for any c ∈ R>0 implies that hk⊗hB on

�k×

/tor�⊗�

B/tor�

satisfies (F).

Proof. Since hB satisfies (FF), the constant mB := min(hB(B) − {0}) is well-defined. Let C > 0 be a real constant. It is possible to choose a real constantCk such that S(Ck) contains generators of the class group of k. Set C0 :=max{C, Ck}; then for any C � ≥ C0,

k× ∼= O×k,S(C�) ⊕

�v/∈S(C�)

Z . (2.13)


We claim that the height hk as in (2.11) is compatible to restriction in thefollowing sense:

(hk ⊗ hB)��O×k,S(C�)⊗B

= (hk��O×k,S(C�)

)⊗ hB .

Analogously for �hk defined in the obvious way. To see this, note that the in-equality (hk ⊗ hB)

��O×k,S(C�)⊗B

≤ (hk��O×k,S(C�)

) ⊗ hB follows from the definitionof the heights, and the converse is a consequence of the direct sum (2.13).

Let α ∈ (k×/tor)⊗(B/tor). Assume that α /∈ (O×k,S(C0)/tor)⊗(B/tor). Thenα ∈ (O×k,S(C�)/tor) ⊗ (B/tor) for some C � ∈ R>0 where C0 < C �. The finiteplaces of S(C �) not contained in S(C0) are described by (2.13), so that for thispart of �hk⊗hB we may apply Lemma 2.4.11. More precisely, if a ∈ O×k,S(C�)/torbut a /∈ O×k,S(C0)/tor, then �hk(a) ≥ C0. If b ∈ B/tor, then hB(b) ≥ mB . We canthus conclude by Lemma 2.4.11 that (�hk ⊗ hB)(α) ≥ C0mB . We summarize:

{α ∈ (k×/tor)⊗ (B/tor) | (�hk ⊗ hB)(α) < C0mB}⊆ {α ∈ (O×k,S(C0)/tor)⊗ (B/tor) | (�hk ⊗ hB)(α) < C0mB} . (2.14)

Inspecting the argument concerning α as above, we may choose C � minimally toobtain the stronger inequality (�hk ⊗hB)(α) ≥ C �mB . The application of (2.12)yields (�hk ⊗ hB)(α) ≤ |S(C �)| (hk ⊗ hB)(α). Hence (hk ⊗ hB)(α) ≥ C

�

|S(C�)|mB .

Define the function ω : R>0 → R, c �→ c|S(c)| . The asymptotical growth of ω

is known (e. g. Narkiewicz), in particular there exists a real C1 > 0 such thatω(C �) ≥ C1 ∀C � ≥ C0. Thus (hk ⊗ hB)(α) ≥ C1mB . Now, assume that α ∈(O×k,S(C0)/tor) ⊗ (B/tor). Then application of (2.12) yields the implication: if(hk⊗hB)(α) < C1mB then (�hk⊗hB)(α) < |S(C0)|C1mB ≤ |S(C0)|ω(C0)mB =C0mB . Hence, we have shown that

{α ∈ (k×/tor)⊗ (B/tor) | (hk ⊗ hB)(α) < C1mB}⊆ {α ∈ (k×/tor)⊗ (B/tor) | (�hk ⊗ hB)(α) < C0mB} . (2.15)

Rescaling of the constants in (2.14) and (2.15) then establishes the lemma.

Finally, the finiteness property (F) for hk⊗hB on�k×

/tor�⊗

�B/tor

�would

imply the properties (FF) and (T) for hk ⊗ hB on k× ⊗B.

Descent

2.4.19 Lemma (Descent Lemma). Let A be an abelian group equipped with acanonical linear height h satisfying the weak finiteness property (FF), i. e.

h�{a ∈ A | h(a) < C}

�

is a finite set for every constant C ≥ 0, and satisfying the property (T), i. e.

h−1(0) = Ator .

Then the maximal divisible subgroup Adiv of A is torsion.

2.5. HEIGHTS IN ALGEBRAIC GEOMETRY 27

Proof. Choose a nonzero non-torsion element a ∈ Adiv. Since h satisfies (T),h(a) �= 0. Since a belongs to maximal divisible subgroup of A, the element 1maexists for every positive integer m. Then

h

� 1m

a

�=

1m

·m · h� 1

ma

�=

1m

h(a) .

The sequence ( 1ma)m∈N is mapped by h to the strictly monotonically decreasingsequence ( 1mh(a))m∈N of points in R. This contradicts (FF).

Recall that, for an abelian group A, the maximal divisible subgroup Adiv ofA splits off as a direct summand. We define the free rank of A as

frk A := dimQ(A/Adiv ⊗Z Q) . (2.16)

If Adiv is torsion, then frkA = dimQ A⊗Z Q.

2.5 Heights in Algebraic Geometry

A field F is said to be a field with product formula if there exists a set MFcontaining one valuation of each equivalence class of valuations of F such that,for all a ∈ F , a �= 0,

(i) |a|v = 1 for all but finitely many v ∈ MF , and

(ii)�

v∈MF|a|v = 1 .

The class of these fields comprises number fields and function fields in one vari-able over any ground field (i. e. finite algebraic extension of a field of puretranscendence degree 1 over some ground field), cf. [2].

For a field F with product formula, the function

HF : Pn(F ) → RP = [a0, . . . , an] �→

�v∈MF

max {|a0|v, . . . , |an|v}

serves as the point of departure in many textbooks for the treatment of thetheory of heights in algebraic geometry.

Recall that a “global field” denominates exclusively a number field or a func-tion field in one variable over a finite field. In particular the residue field of anon-archimedean place of a global field is finite. Note that, according to thisdefinition, we do not consider a function field in one variable over the algebraicclosure of a finite field to be global field.

2.5.1 Proposition. Let F be a global field. The function

h : F× → R+, a �→ log HF ([a, 1])

is a canonical linear height satisfying the strong finiteness property (F).

Proof. We have to verify the defining properties.(i) h is bounded below by 0.(ii) |− a|v = |a|v for every a ∈ F× and any valuation v, hence h is even.(iv) since max{|am|v, 1} = max{|a|mv , 1} = max{|a|v, 1}m.


(iii) We notice that the triangle inequality of h does not result from the triangleinequalities of v, since we have generally for a0, a�0, a1, a�1 ∈ F×, that

max{a0a�0, a1a�1} = aia�i ≤ max{a0, a1} · a�i ≤ max{a0, a1}max{a�0, a�1} (2.17)

with i denoting either 0 or 1. In particular this holds for a1 = a�1 = 1.In the case char(F ) = 0, the finiteness property (F) for F = Q is known

as the theorem of Northcott [13, B.2.3] (actually a corollary thereof). For anyfinite extension, choose a finite set of generators and use the triangle equality toestablish the property (F). In the case char(F ) > 0, the finiteness property (F)can be seen easily for F = F(T ) with F a finite field and T an indeterminate.For any finite extension proceed as before.

The function HF is extended by passing to the algebraic closure F̄ of F .

H : Pn(F̄ ) → R, P �→�

v∈MF �max {|a0|v, . . . , |an|v}[F

�v :Fv| ]/[F

�:F ]

,

where F � is any field with P ∈ P(F �) and v| denotes the restriction of v to F .It is common to normalize with respect to a fixed prime field.

For a point P = (y0, . . . , yn) ∈ PnF (F̄ ), the field of definition is

F (P ) := F (y0/yj , . . . , yn/yj) for any j with yj �= 0 .

Its degree over F is called the degree of P , deg(P ) := [F (P ) : F ], which is afinite integer.

2.5.2 Proposition (Northcott). Let F be a number field. For any real constantsC, D ≥ 0, the set

�P ∈ Pn(F̄ ) | H(P ) < C and deg(P ) ≤ D

�

is finite.

After choosing coordinates for a closed point x of a smooth projective varietyX over some field k, the field generated over k by the coordinates of x is uniquelydefined. Its degree over k, called the degree of x, denoted deg(x), is independentof the choice of coordinates.

2.5.3 Proposition. Let A/k be an abelian variety over a number field k. Thereexists a canonical linear height

h : A(k̄) −→ R+

satisfying the finiteness property that, for any real constants C, D ≥ 0, the set�a ∈ A(k̄) | h(a) < C and deg(a) ≤ D

�

is finite. In particular, the restriction of h to A(k) satisfies the strong finitenessproperty (F).

Proof. To construct the height, choose an effective symmetric divisor (class) Don A, i. e. [−1]∗D ∼ D. Let φD : A → Pnk denote the rational map associatedto D. Define

hA,D := log H ◦ φD ,

2.5. HEIGHTS IN ALGEBRAIC GEOMETRY 29

and ĥA,D(a) := limN→∞m−2NhA,D(mNa) the Néron-Tate normalization, forsome integer m > 1. Then ĥA,D is a quadratic form by [13, B.5.5]. Also acanonical quadratic height in the sense of Def. 2.2.3. Define h(a) := (ĥA,D(a))1/2with positive sign, for a ∈ A(k̄). The triangle inequality for h is implied by theCauchy-Schwarz inequality. Now apply theorem of Northcott, 2.5.1 to deducethe finiteness property.

Chapter 3

Decomposable Elements

We will show that

3.0.1 Theorem. Let X be a smooth projective geometrically connected curveover a field k, admitting a k-rational point, and let Jk� be the Jacobian of Xk�for any finite k-extension k�. There exists a finite constant B such that thehomomorphism �

k�/k[k�:k]≤B

Jk� ⊗ k�× −→ SKM1 (X)

is surjective.

3.1 Multiplication

Let k be a field. There exists a product structure on the higher Chow groups(cf. [6]), in particular, for X ∈ SmProj/k, there is a homomorphism

νk : CHd(X)⊗ CH1(X, 1)⊗q −→ CHd+q(X, q) ,

for positive integers d, q.If k is algebraically closed, the subgroup CHd+q(X, q)dec := im νk is called

the decomposable elements of CHd+q(X, q). The indecomposable elements aredefined as CHd+q(X, q)ind := coker νk.

For an arbitrary field k, there is a map

ν :�

k�/k

CHd(Xk�)⊗CH1(Xk� , 1)⊗q⊕νk�−→

�

k�/k

CHd+q(Xk� , q)⊕(φk�/k)∗−→ CHd+q(X, q) ,

with sum over all finite field extensions k�/k, and with φk�/k : Spec k� →Spec k denoting the structure morphism. Analogously, we define decompos-able elements CHd+q(X, q)dec := im ν, as well as indecomposable elementsCHd+q(X, q)ind := coker ν.

For Milnor K-groups of curves, we will describe this map explicitly, to seethat every elements is decomposable.

31

32 CHAPTER 3. DECOMPOSABLE ELEMENTS

Curves

Let X be a smooth projective geometrically connected curve over a field k withX(k) �= ∅. For any k-extension k� ⊆ k̄, let Jk� be the Jacobian associated toXk� := X×Spec kSpec k�. There exists a morphism defined over k (and dependingon the choice of a k-rational point x0)

j : Xk� → Jk� ,

which is an embedding if g > 0 with g denoting the genus of X, and which isthe zero map if g = 0. Linear Extension of j to the divisors on Xk� induces anisomorphism

ι : Jk�∼−→ �CH

1

(Xk�) = �CH0(Xk�) .

We write J := Jk̄ and

J(k�) := H0(Gk� , Jk̄) = J(k̄)Gk� .

Since �CH0(Xk�) �→ �CH0(X̄) (cf. [13, A.2.2.10] for injectivity), J(k�) correspondsvia ι to the group of k�-rational zero-cycle classes. Whereas Jk� corresponds tothe subgroup of zero-cycle classes that are actually represented by a k�-rationalk�-zero-cycle, thus there is an injection

Jk� �→ J(k�) .

Comparison I. An alternative Description of SKM1

(X) can be obtained frompart I of the Comparison Theorems, in particular 1.4.3 provides

SKM

1(X) ∼−→ ker

coker

KM2

(k(X)) ∂−→�

x∈X(1)

k(x)×

N−→ k×

=: coker0 ∂ .

Sending {[x], a}k(x)/k �→ [a|x], provides an isomorphism KM1 (X)∼−→ coker ∂.

The inverse morphism coker ∂ ∼−→ KM1

(X) sends [a|x] �→ {[x], a}k(x)/k.The comparison isomorphisms are given explicitly by

ϑ : SKM1

(X) → coker0 ∂�z, a

�k(x1,...,xm)/k

�→��m

i=1Nk(x1,...,xm)/k(xi)(a)

ni |xi� (3.1)

for some representation [�m

i=1 nixi] = z, and with brackets [ · ] denoting thecorresponding residue classes. Conversely,

ρ : coker0 ∂ → SKM1

(X)��

ax|x��→

��ι ◦ j(x), ax

�k(x)/k

.(3.2)

Note that {z, 1} = 0 in SKM1

(X) for z ∈ �CH0(X), so that the sum actuallycomprises only finitely many summands.

3.1. MULTIPLICATION 33

Comparison II. An alternative description of the quotient map modulo theSomekawa relations can be obtained from part II of the Comparison Theorems(cf. 1.2.6).

It is a classical result ([21], [30]) that for any field k (including finite extensionof some ground field and algebraic closure) there is an isomorphism

σ : k× ∼−→ CH1(Xk, 1) .

Then the comparison isomorphism is given explicitly by

SKM

1(X) ∼−→ �CH

2

(X, 1)] , {z, a}k�/k �→�φk�/k

�∗νk�(z, σ(a)) ,

where φk�/k : Spec k� → Spec k denotes the structure morphism for a finitek-extension k�, and ν : CH1(Xk�)⊗CH1(Xk� , 1) → CH2(Xk� , 1) is the multipli-cation map on higher Chow groups (as in [6]).

3.1.1 Lemma. Let X be a smooth projective geometrically connected curve overa field k, J its Jacobian. The quotient map modulo the Somekawa relations(M1), (M2) (combined with ι) is given by

µ :�

k�/k

Jk� ⊗ k�× � SKM1 (X) ,

(y ⊗ a)k�/k �→ {ι(y), a}k�/k ,(3.3)

with sum over all finite field extensions k�/k, makes the following diagram com-mute

�k�/k Jk

�

ι∼=

��

⊗ k�×

σ∼=

��

µ �� SKM1

(X)

φ∼=

��k�/k

�CH1

(Xk�) ⊗ CH1(Xk� , 1)

⊕(φk�/k)∗◦νk�� CH2

(X, 1) .

Proof. Obvious from explicit description.

In words, µ coincides (via comparison isomorphism) with a summed-up mul-tiplication ν on the appropriate higher Chow groups.


3.2 Point Degree

As before, let X be a smooth projective geometrically connected curve definedover a perfect field k, admitting a k-rational point. To avoid trivialities in thefollowing definitions, we assume additionally that g > 0 for the genus g of X.

The degree map arises from the map

deg : X(0) → Z>0, x �→ [k(x) : k] .

Additive continuation of deg on the zero-cycles z0(X) = Z[X(0)] factors throughrational equivalence, resulting in the homomorphism CH0(X) → Z which isknown as the degree map and also denoted deg. However, we would like to usea kind of “degree function” as a measure of complexity of elements.

3.2.1 Definition. The point degree deg∗ w.r.t. to the base field k is defined

• on �CH0(X̄): We say that z ∈ �CH0(X̄) is defined over the k-extensionk� ⊆ k̄ if z has a pre-image under �CH0(Xk�) �→ �CH0(X̄). The minimal

degree over k of the fields over which z is defined is denoted deg∗gCH0(X̄)(z),or deg∗(z) for short.

This coincides with the minimal k-degree of the k-extensions k� such thatz has a representation

�i nixi where each xi is Gk� -invariant.

Further, deg∗ is defined on �CH0(Xk��) for any k-extension k�� ⊆ k̄ viathe inclusion �CH0(Xk��) �→ �CH0(X̄). Note that degk as defined above isidentically zero on �CH0(Xk��).

• on k̄× mutatis mutandis. In this case, deg∗(a) for a ∈ k̄× coincides withthe degree of the minimal polynomial of a with coefficients in k, usuallydenoted deg(a).

• on SKM1

(X) by

deg∗SKM1 (X)(β) := minPi{zi,ai}ki/k=β

maxi

degk(ki) ,

with degk(ki) = [ki : k] denoting the degree over k.

• on coker0 ∂, i. e. on coker�∂ : KM

2(k(X)) →

�x∈X(0)

k(x)×�

restricted tothe elements of norm 1, by

deg∗coker0 ∂(α) := min

[P

i ai|xi ]=αmax

idegk(xi) ,

with [�

i ai|xi ] denoting the residue class of�

i ai|xi ∈�

x∈X(0)k(x)× in

coker0 ∂.

3.2.2 Lemma (comparison of point degrees).

(i)deg∗SKM1 (X)(β) = minP

i{zi,ai}ki/k=βmax

i{deg∗(zi),deg(ai)}

3.3. FACTOR BASIS 35

(ii) Via the isomorphism SKM1

(X) ∼−→ coker0 ∂,

deg∗SKM1 (X) = deg∗

coker0 ∂ .

Proof. (i) Clearly, deg∗SKM1 (X),k(β) ≥ minP

i{zi,ai}ki/k=βmaxi{deg∗k(zi),degk(ai)}.

Consider the following identities due to the first Somekawa relations (M1) inDef. 1.1.1.

• Let k��/k� be finite k-extensions. If z ∈ �CH0(Xk�) and a ∈ k�×, then

n{z, a}k�/k = {z,Nk��/k�(a)}k�/k = {z, a}k��/k with n = [k�� : k�] .

• Let j1 : Spec k(z, a) → Spec k(z), where k(z, a) denotes a field generatedby z and a, i. e. a field of minimal k-degree over which both z and a aredefined, analog for k(z). Then

{z, a}k(z,a)/k = {z, j1∗(a)}k(z)/k = {z,Nk(z,a)/k(z)(a)}k(z)/k .

• Let j2 : Spec k(z, a) → Spec k(a), then

{z, a}k(z,a)/k = {j2∗(z), a}k(a)/k =� �

σ∈G

zσ, a

�k(a)/k

with G = Gal(k(z, a)/k(a)).

A straightforward argumentation using these identities shows that it is alwayspossible to establish equality in the inequality above.

(ii) For α ∈ coker0 ∂, choose a representation [�

ai|xi ] = α attaining deg∗(α),i. e. deg∗

coker0 ∂(α) = maxi degk(xi). Recall the comparison morphism ρ given in(3.2) to obtain Supp(ρ(α)) = {k(xi) | i}. Then maxi degk(xi) ≥ deg∗SKM1 (X)(ρ(α))by minimality.

For β ∈ SKM1

(X), choose a representation�{zi, ai}ki/k = β attaining

deg∗(β), i. e. deg∗SKM1 (X)(β) = maxi degk(ki). For each of these zi, choose arepresentation [

�mij nijxij ] = zi such that k(zi) = k(xi,1, . . . , ximi). Then, to-

gether with part (i) of the Lemma, maxi degk(ki) = maxi{deg∗(zi),deg(ai)} ≥maxi deg∗(zi) ≥ maxi,j degk(xi,j) Recall the comparison morphism ϑ given in(3.1) to obtain Supp(ϑ(β)) = {xij | i, j}. Then maxi,j degk(xi,j) ≥ deg∗coker0 ∂(ϑ(β))by minimality.

3.3 Factor Basis

In the classical paper [3] of Bass-Tate, the calculation of the tame kernel overnumber fields and over function fields of regular curves over finite fields is carriedout by a method which could be called a factor basis 1. We would like to presentthese ideas in a more general setting, so that this useful tool of computationbecomes applicable to the questions we are investigating.

1This expression refers to a common technique of computational number theory to con-

struct a filtration adapted to a given problem and to extract thereof a finite subfiltration on

which solutions of the problem can be calculated. For a prototypical factor base method see

Dixon’s random squares algorithm (e. g. [31, 19.5]).


Construction

Let k be any field. Let X be a smooth projective curve of genus g > 0 over k.Denote by F := k(X) its function field.

To a set S ⊆ X(1) = X(0) of closed points of X, we associate the ring ofS-integers AS in the function field of X,

AS := {f ∈ F | ordx(f) ≥ 0 for all x /∈ S} ,

whose invertible elements US are called S-units,

US := A×S =�f ∈ F× | ordx(f) = 0 for all x /∈ S

�.

For any positive integer q, we define

KM

q (AS) := U⊗qS

��a1 ⊗ · · ·⊗ aq | ai + aj = 1 or 0 for some i < j� .

There is a natural map KMq (AS) → KMq (F ). Denote by KMq (F |S) the subgroupof KMs (F ) generated by symbols of the form {a1, . . . , aq} with aj ∈ A×S for allj = 1, . . . , q. Then KMq (F |S) = im

�K

M

q (AS) → KMq (F )�.

For some closed point x /∈ S, the tame symbol ∂x : KMq (F ) → KMq−1(k(x))vanishes on KMq (F |S) since A×S =

�y/∈S O×X,y ⊆ O×X,x, which induces a “trun-

cated tame symbol”

∂S :=�

x/∈S

∂x : KMq (F )�K

M

q (F |S) −→�

x/∈S

KM

q−1(k(x)) . (3.4)

3.3.1 Definition. A set S ⊆ X(1) such that the truncated tame symbol ∂S isan isomorphism is called a factor basis for KMq (F ).

Note that if S is a factor basis, the map KMq (AS) → KMq (F ) becomes anepimorphism. Hence, a factor basis S ⊆ X(1) has the properties that

ker ∂ = ker�K

M

q (F |S) −→�

x∈SK

M

q−1(k(x))�

, (3.5)

coker ∂

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