Galois theory and projective geometry

Galois theory and projective geometry

FEDOR BOGOMOLOVCourant Institute, NYU

AND

YURI TSCHINKELCourant Institute, NYU, and Simons Foundation

To the Courant Institute, on the occasion of its 75th anniversary

Abstract

We explore connections between birational anabelian geometry and abstract pro-jective geometry. One of the applications is a proof of a version of the birationalsection conjecture.c© 2000 Wiley Periodicals, Inc.

1 Introduction

What is the difference between a regular 13-gon and a 17-gon?

The discovery by Gauss in 1796 that the 17-gon can be constructed with com-pass and straightedge, and the 13-gon cannot, and thatinternal, hiddensymmetriesof number fields are ultimately responsible for this discrepancy, triggered a majortransformation of mathematics, shifting the emphasis fromthe mechanical, Descar-tiancoordinatizationof space to the study of its internalsymmetries. Around 1830,Galois developed this idea, giving a criterion for solvability of polynomial equa-tions in one variable in radicals. Both Gauss and Galois solved 2000 year old prob-lems; but more importantly, both realized that rather than untangling how thingsreally work, i.e., the specifics of the construction, or which radicals to use, one hasto focus on thegroupof symmetries. This insight from plane geometry and numbertheory slowly penetrated mathematics and physics, via Klein’s 1872 unification ofgeometry and group theory in his Erlangen Program, works by Poincare, Hilbert,Minkowski, and many others. One of its culminations is Einstein’s thesis: natural

Communications on Pure and Applied Mathematics, Vol. 000, 0001–0026 (2000)c© 2000 Wiley Periodicals, Inc.

2 FEDOR BOGOMOLOV AND YURI TSCHINKEL

symmetries in many physical problems provide such strong constraints on possibletypes of dynamic equations that it becomes a matter of elementary mathematicaland physical considerations to find their exact form.

These developments stimulated investigations in abstractgroup theory, whichby itself is very rich, with many open problems. Only some instances of this the-ory are developed to an extent that they can be applied: Lie groups, finite groups,some discrete groups. One of the most transparent chapters is the theory of abeliangroups, and even here there are unpleasant features arisingfrom passage to projec-tive or injective limits, taking duals, and combinations ofthe above.

A related algebraic object is afield, a structure that combinestwo compati-ble abelian group laws, the basic example being the field of rational numbersQ.Other examples arefinite fieldsFp, function fieldsof algebraic varietiesk(X) oversome field of definitionk, and their completions with respect tovaluations. Galoisgroupsare symmetries of fields respecting these structures: by definition the Ga-lois group Gal(E/K) of a field extensionE/K is the group of automorphisms ofE fixing every element ofK. We writeGK := Gal(K/K) for the absoluteGaloisgroup of a fieldK. For example,

GR = Z/2Z,

generated by complex conjugation. In general, Galois groups areprofinitegroups,i.e., projective limits of their finite index quotients. We have

GFp = Z,

the (uncountable) profinite completion of the additive group of integersZ, topo-logically generated by theFrobenius

Fr : x 7→ xp.

Typically, Galois groups are large, noncommutative, and complicated objects. Forexample, the Galois groupGC(t) is a free profinite group on infinitely many gener-ators, and every finite group appears as its quotient. Very little is known aboutGQ,in particular, theinverse Galois problem, i.e., the realization of every finite groupas a Galois group overQ, is still open. On the other hand, the abelianization

GaQ = GQ/[GQ,GQ]

is well-understood: the corresponding field

Qab = ∪nQ(n√

1)⊂ Q

is obtained by adjoining all roots of unity. By Kummer theory, given any fieldK, containing all roots of unity, its extensions with cyclic Galois groupZ/nZ

are given by adjoiningn√

f , for some f ∈ K×, the multiplicative subgroup ofK.This recipe gives a constructive solution to the inverse Galois problem for abeliangroups. More precisely, by Kummer theory, we have a canonical pairing

[·, ·]n : GaK/n×K×/(K×)n → µn

(γ , f ) 7→ [γ , f ]n := γ( n√

f )/ n√

f ,

GALOIS THEORY AND PROJECTIVE GEOMETRY 3

for everyn∈ N. Hereµn is the (multiplicative) group of roots of unity of ordern.This pairing extends to a nondegenerate pairing

[·, ·] : GaK× K×→ lim←−µn, µn := { n

√1},

whereK× is the profinite completion ofK× and the target can be identified withZ.Thus, taking the functor Hom in the category of locally compact groups, we obtain

(1.1) GaK = Hom(K×, Z), and K× = Hom(Ga

K , Z),

so that thedouble-dualof K× is actuallyK×. We find thatGaK “encodes”, in the

weak sense, the multiplicative structure ofK.A major open problem today is to identify classes of fieldscharacterizedby

their absolute Galois groups. There exist genuinely different fields with isomorphicGalois groups, e.g.,Fp andC((t)). However, Neukirch and Uchida showed thatGalois groups of maximalsolvableextensions of number fields or function fieldsof curves over finite fields determine the corresponding field, up-to isomorphism[23], [32].

This result is the first instance ofbirational anabelian geometry, which is, insome sense, an algebraic incarnation of Einstein’s postulate: it aims to show thatGalois groups of function fields of algebraic varieties overan algebraically closedground field determine the function field, in a functorial way. The version proposedby Grothendieck in [13] introduces a class ofanabelianvarieties, functorially char-acterized by their etale fundamental groups; with prime examples being hyperboliccurves and varieties successively fibered into hyperbolic curves. For representativeresults, see [21], [34], [33], [30], as well as [15], [22], [26], [25], [20].

However, absolute Galois groups are simply too large. It turns out that there areintermediate groups, whose description involves someprojective geometry, mostimportantly, geometry of lines and points in the projectiveplane, bridging Gaussand Galois. These groups are just minimally different from abelian groups; theyencode the geometry of simple configurations. On the other hand, their structureis already sufficiently rich so that the corresponding objects in the theory of fieldsallow to captureall invariants and individual properties of large fields, i.e.,functionfields of transcendence degree at least two over algebraically closed ground fields.This insight of the first author [3], [5], [4], was developed in the series of paperswritten at the Courant Institute [6], [7], [8], [9] over the last decade. One of ourmain results is that function fieldsK = k(X) overk = Fp are determined by

GcK := GK/[GK , [GK ,GK ]],

whereGK is the maximal pro-ℓ-quotient ofGK, andG c is the canonical centralextension of its abelianizationG a

K (see also [27]).In [10] we survey the development of the main ideas merging into thisalmost

abelian anabelian geometryprogram. Here we formulate our vision of the futuredirections of research, inspired by this work. In particular, in Section 4 we prove anew result, a version of thebirational section conjecture. In Sections 5 and 6 we


discuss cohomological properties of Galois groups closelyrelated to the Bloch–Kato conjecture, proved by Voevodsky, Rost, and Weibel, andfocus on connectionsto anabelian geometry.

2 Projective geometry andK-theory

The introduction of the projective plane essentially trivialized plane geometryand provided simple proofs for many results concerning configurations of lines andpoints, considered difficult before that. More importantly, the axiomatization ef-forts in late 19th and early 20th century revealed thatabstractprojective structurescapturecoordinates, a “triumph of modern mathematical thought” [35, p. v]. Theaxioms can be found in many books, including Emil Artin’s lecture notes from acourse he gave at the Courant Institute in the Fall of 1954 [2,Chapters VI and VII].The classical result mentioned above is that an abstract projective space (subjectto Pappus’ axiom) is a projectivization of a vector space over a field. This can bestrengthened as follows:

Theorem 2.1. Let K/k be an extension of fields. Then K×/k× is simultaneouslyan abelian group and a projective space. Conversely, an abelian group with acompatible projective structure corresponds to a field extension.

Proof. See [16] or [10, Section 1]. �

In Algebraic geometry, projective spaces are the most basic objects. Overnonclosed fieldsK, they admit nontrivial forms, calledBrauer-Severivarieties.These forms are classified by theBrauer groupBr(K), which admits a Galois-cohomological incarnation:

Br(K) = H2(GK ,Gm).

The theory of Brauer groups and the local-global principle for Brauer–Severiva-rieties over number fields are cornerstones of arithmetic geometry. Much lessis known over more complicated ground fields, e.g., functionfields of surfaces.Brauer groups, in turn, are closely related to Milnor’s K2-groups, and more gener-ally K-theory, which emerged in algebra in the study of matrix groups.

We recall the definition of Milnor K-groups. LetK be a field. Then

KM1 (K) = K×

and the higher K-groups are spanned by symbols:

KMn (K) = (K×)⊗

n/〈· · ·x⊗ (1−x) · · · 〉,

the relations being symbols containingx⊗ (1− x). For i = 1,2, Milnor K-groupsof fields coincide with those defined by Quillen, and we will often omit the super-script.


Throughout, we work with function fields of algebraic varieties over alge-braically closed ground fields; by convention, the dimension of the field is its tran-scendence degree over the ground field.

Theorem 2.2. [8, Theorem 4]Assume that K and L are function fields of algebraicvarieties of dimension≥ 2, over algebraically closed fields k and l, and that thereexist compatible isomorphisms of abelian groups

K1(K)ψ1−→ K1(L) and K2(K)

ψ2−→ K2(L).

Then there exists an isomorphism of fields

ψ : K→ L

such that the induced map on K× coincides withψ±11 .

The proof in [8] assumes characteristic zero, but it extendsto positive char-acteristic after minor modifications. The main point is thatK2(K) encodes thecanonical projective structure onPk(K) = K×/k×. This is based on the followingobservations:

• The multiplicative groupsk× andl× are characterized asdivisibleelementsin K1(K), resp. K1(L). This leads to an isomorphism of abelian groups(denoted by the same symbol):

Pk(K)ψ1−→ Pl(L).

• rational functionsf1, f2 ∈ K× are algebraically dependent overk if andonly if their symbol( f1, f2) is divisible in K2(K). This allows to charac-terizePk(E) ⊂ Pk(K), for one-dimensionalE ⊂ K and we obtain afan ofinfinite-dimensional projective subspaces inPk(K). The compatibility ofψ1 with ψ2 implies that the corresponding structures onPk(K) andPl (L)coincide.• By Theorem 2.1, it remains to show thatψ1 (or 1/ψ1) maps projective lines

P1 ⊂ Pk(K) to projective lines inPl (L). It turns out that projective linescan be intrinsically characterized asintersectionsof well-chosen infinite-dimensionalPk(E1) and Pk(E2), for 1-dimensional subfieldsE1,E2 ⊂ K(see [8, Theorem 22] or [10, Proposition 9]).

The theorem proved in [8] is stronger, it addresses the case when ψ1 is aninjectivehomomorphism.

3 Projective geometry and Galois groups

Let K be a function field overk = Fp. In Section 2 we considered the abeliangroup / projective spacePk(K) and its relationship to the K-theory of the fieldK.Here we focus on adual picture.

Let Rbe one of the following rings:

R= Z, Z/ℓn, Zℓ, ℓ 6= p.


In fact, the theory below extends almostverbatim to quite general commutativerings such that the order of all torsion elementsr ∈ R is coprime top and withoutnonzero divisible elements.

Define

(3.1) WaK(R) := Hom(K×/k×,R) = Hom(K×,R),

theR-module of continuous homomorphisms, wherePk(K) = K×/k× is endowedwith discrete topology. We call WaK(R) theabelian Weil groupof K with values inR.

The abelian Weil group carries a collection of distinguished subgroups, corre-sponding to variousvaluations. Recall that a valuation is a surjective homomor-phism

ν : K×→ Γν

onto an ordered abelian group, subject to a nonarchimedean triangle inequality

(3.2) ν( f +g)≥min(ν( f ),ν(g)).

LetVK be the set of all nontrivial valuations ofK, for ν ∈ VK let oν be the valuationring, mν ⊂ oν its maximal ideal, andKν the corresponding residue field. We havethe following fundamental diagram:

1 // o×ν // K×ν // Γν // 1

1 // (1+mν)× // o×νρν // K×ν // 1

Every valuation ofk = Fp is trivial, and valuation theory over function fieldsK = k(X) is particularlygeometric: all ν ∈ VK are trivial onk× and defineΓν -valued functions onPk(K). Throughout, we restrict our attention to such groundfields. In this context, nontrivial valuations onk(t) are in bijection with pointsp ∈ P1, they measure the order of a function atp. There are many more valuationson higher-dimensional varieties.

We call

Iaν(R) := {γ ∈Wa

K(R) |γ is trivial on o×ν } ⊆WaK(R)

theabelian inertia groupand

Daν(R) := {γ ∈Wa

K(R) |γ is trivial on (1+mν)×} ⊆WaK(R)

theabelian decomposition group, we have

Daν(R)/Ia

ν(R)⊆WaKν

(R),

with equality when Ext1(Γν ,R) = 0, e.g., whenν is an algebraic valuation.


Example3.1. Let GK be the absolute Galois group of a fieldK, GaK its abelianiza-

tion, andG aK theℓ-completion ofGa

K . By Kummer theory,

GaK = Wa

K(Zℓ).

Moreover, Iaν(Zℓ) and Daν(Zℓ) are the standard abelian inertia and decomposition

subgroups corresponding toν (see, e.g., [6]).

A valuationν defines a simple geometry on the projective spacePk(K); equa-tion (3.2) implies that each finite dimensional subspacePn⊂ Pk(K) admits aflag

(3.3) P1⊂ P2⊂ ...

of projective subspaces, such that

ν : Pk(K)→ Γν

is constant onP j \P j−1, for all j, and this flag structure is preserved under multi-plicative shifts by anyf ∈ K×/k×.

Let P be a projective space overk, e.g.,P = Pk(K). We say that a mapι : P→Rto anarbitrary ring R is aflag mapif every finite-dimensionalPn⊂ P admits a flagas in (3.3) such thatι is constant on eachP j \P j−1; a subsetS⊂ P will be called aflag subsetif its set-theoretic characteristic function is a flag map onP.

Example3.2. A nonempty flag subset ofP1(k) is either a point, the complementto a point, or all ofP1(k). Nonempty proper flag subsets ofP2(k) are one of thefollowing:

• a pointp, a linel, l◦ := l\p,• P2\p, P2\ l, P2\ l◦.

Proposition 3.3. [6, Section 2]Let P be a projective space over k= Fp. A mapι : P→ R is a flag map if and only if its restriction to everyP1 ⊂ P is a flag map,i.e., is constant on the complement to one point.

Example3.4. This fails whenk = F2 andP = P2(k) is the Fano plane. The mapι takes two values as indicated in the picture: the restriction of ι to every line is aflag map, whileι is not a flag map onP2(F2).

(1:0:0)

(1:1:1)

(0:1:1)

(1:1:0)

(1:0:1) (0:0:1)

(0:1:0)


By [6, Section 6.3], flag maps are closely related to valuations: given a flaghomomorphismι : Pk(K)→ R there exists auniqueν ∈ VK and a homomorphismχ : Γν → R such that

(3.4) ι = χ ◦ν .

This means thatι ∈ Iaν(R).

We now describe the theory ofcommuting pairsdeveloped in [6]: We say thatR-linearly independentγ ,γ ′ ∈Wa

K(R) form a c-pair if for every normally closedone-dimensional subfieldE ⊂ K the image of the subgroupRγ⊕Rγ ′ in Wa

E(R) iscyclic; ac-subgroup is a noncyclic subgroupσ ⊂Wa

K(R) whose image in WaE(R)

is cyclic, for allE as above. We define thecentralizerZγ ⊂WaK(R) of an element

γ ∈WaK(R) as the subgroup of all elements forming ac-pair with γ .

The main result of [6] says:

Theorem 3.5. Assume that R is one of the following:

Z, Z/ℓn, Zℓ.

Then

• every c-subgroupσ is generated by≤ trdegk(K) elements;• for every c-subgroupσ there exists a valuationν ∈ VK such that

– σ is trivial on (1+mν)× ⊂ K×

– for σ ′ := σ ∩Hom(Γν ,R) ⊂ Hom(K×,R) = WaK(R), the R-module

σ/σ ′ is cyclic.

The groupσ ′ is, in fact, the inertia subgroup Iaν(R) corresponding toν . The

union of all σ containing an inertia subgroup Iaν(R) is the corresponding decom-

position group Daν(R). If I aν(R) is cyclic, generated byι , then Da

ν(R) = Zι , thecentralizer ofι .

The proof of Theorem 3.5 is based on the following geometric observation,linking the abelian Weil group with affine/projective geometry: Let γ ,γ ′ ∈Wa

K(R)be nonproportional elements forming ac-pair and let

Pk(K)φ−→ R2

f 7→ (γ( f ),γ ′( f ))

be the induced map to the affine planeA2(R). If follows that

(*) the image of everyP1⊂ Pk(K) satisfies a linear relation, i.e., is containedin an affine line inA2(R).

Classically, this is called acollineation. A simple model is a map

P2(Fp)φ−→ A2(F2),

wherep > 2 is a prime and where the target is a set with 4 elements. It turns outthat when the mapφ satisfies condition(∗) then the target contains only 3 points.


Furthermore, on every lineP1⊂ P2 the map is constant on the complement of onepoint! This in turn implies that there is a stratification

p⊂ l⊂ P2,

wherep is a point andl = P1 andr, r ′ ∈ F2, (r, r ′) 6= (0,0), such thatι : rγ + r ′γ ′is constanton l\p, andP2 \ l, i.e., ι is a flag map onP2. This last property failsfor p = 2, as noted in Example 3.4. Nevertheless, one can extract thefollowinggeneral fact:

• if γ ,γ ′ satisfy(∗) then there exists a nontrivialR-linear combinationι :=rγ + r ′γ ′ such that

ι : Pk(K)→ R

is a flag map, i.e.,ι ∈ Iaν(R), for someν ∈ VK .

The proof is based on a lemma from projective geometry:

Lemma 3.6. LetP2(k) = S1⊔⊔ j∈JSj be a set-theoretic decomposition into at leastthree nonempty subsets such that for everyP1(k)⊂ P2(k) we have

P1(k)⊆ S1⊔Sj , for some j∈ J, or P1⊆ ⊔ j∈JSj

Then one of the Sj is a flag subset ofP2(k).

These considerations lead to the following characterization of multiplicativegroups of valuation rings, one of the main results of [6]:

Proposition 3.7. Let o× ⊂ K×/k× be a subgroup such that its intersection withany projective planeP2 ⊂ Pk(K) = K×/k× is a flag subset, i.e., its set-theoreticcharacteristic function is a flag function. Then there exists a ν ∈ VK such thato× = o×ν /k×.

We now turn to more classical objects, namely Galois groups of fields. LetGK be the absolute Galois group of a fieldK, Ga

K its abelianization andGcK =

GK/[GK , [GK ,G]] its canonical central extension. LetK be a function field of tran-scendence degree≥ 2 overk = Fp. Fix a primeℓ 6= p and replaceGa

K andGcK by

their maximal pro-ℓ-quotients

GaK = Hom(K×/k×,Zℓ) = Wa

K(Zℓ), and GcK .

Note thatG aK is a torsion-freeZℓ-module of infinite rank.

A commuting pairis a pair of nonproportionalγ ,γ ′ ∈ G aK which lift to commut-

ing elements inG cK (this property does not depend on the choice of the lift). The

main result of the theory of commuting pairs in [6] says that if

(**) γ ,γ ′ ∈ G aK form a commuting pair


then theZℓ-linear span ofγ ,γ ′ ∈ G aK contains an inertia element of some valuation

ν of K.A key observation is that Property(∗∗) implies(∗), for eachP2

k⊂Pk(K), whichleads to a flag structure onPk(K), which in turn gives rise to a valuation.

Commuting pairs are part of an intricatefan ΣK on G aK , which is defined as the

set of all topologically noncyclic subgroupsσ ⊂ G aK such that the preimage ofσ ,

under the natural projectionG cK → G a

K , is abelian. We have:

• rkZℓ(σ)≤ trdegk(K), for all σ ∈ ΣK ;

• everyσ ∈ ΣK contains a subgroup of corank one which is the inertia sub-group of some valuation ofK.

Intersections of subgroups inΣK reflect subtle dependencies among valuations ofK. Let I a

ν := Iaν(Zℓ) ⊂ G a

K be the subgroup of inertia elements with respect toν ∈ VK andDa

ν = Daν(Zℓ) the correspondingdecomposition group. In our context,

Daν is also the group of all elementsγ ∈ G a

K forming a commuting pair with everyι ∈I a

ν . As noted above, the definitions of inertia and decomposition groups givenhere are equivalent to the classical definitions in Galois theory. We have

GaKν

= Daν/I a

ν ,

the Galois group of the residue fieldKν of ν , andΣKν is the set of projectionsof subgroups ofσ ∈ ΣK which are contained inDa

ν . WhenX is a surface,K =k(X), andν adivisorial valuation ofK, we haveI a

ν = Zℓ andDaν is a large group

spanned by subgroupsσ of rank two consisting of all element commuting with thetopological generatorδν of I a

ν . The picture is:

To summarize, the Galois groupG cK encodes information about affine and pro-

jective structures onG aK , in close parallel to what we saw in the context of K-theory

in Section 2. These structures are so individual that they allow to recover the field,via the reconstruction of the abstract projective structure onPk(K) (see [14] for theaxiomatic foundations of projective geometry):

Theorem 3.8. [7], [9] Let K and L be function fields of transcendence degree≥ 2over k= Fp andℓ 6= p a prime. Let

ψ∗ : GaL → G

aK


be an isomorphism of abelian pro-ℓ-groups inducing a bijection of sets

ΣL = ΣK.

Then there exists an isomorphism of perfect closures

ψ : K→ L,

unique modulo rescaling by a constant inZ×ℓ .

4 Z-version of the Galois group

In this section, we introduce a functorial version of thereconstruction / recog-nition theories presented in Sections 2 and 3. This version allows to recover notonly field isomorphisms from data of K-theoretic or Galois-theoretic type, but alsosections, i.e., rational points on varieties over higher-dimensional function fields.

We work in the following setup: letX be an algebraic variety overk = Fp, withfunction fieldK = k(X). We use the notation:

• for x,y∈ K× we let l(x,y) ⊂ Pk(K) be the projective line through the im-ages ofx andy in K×/k×;• for planesP2(1,x,y) ⊂ Pk(K) (the span of linesl(1,x), l(1,y)) we set

P2(1,x,y)◦ := P2(1,x,y)\{1}.We will say that ¯x, y∈ K×/k× are algebraically dependent, and write ¯x≈ y, if thisholds for any of their liftsx,y to K×.

Define theabelian Weil group

WaK := Hom(K×/k×,Z) = Hom(K×,Z),

with discrete topology. SinceK×/k× is a free abelian group, WaK is a torsion-free,infinite rankZ-module. The functor

K 7→WaK

is contravariant; it is aZ-version ofG aK , since it can be regarded as aZ-sublattice

of

GaK = Hom(K×/k×,Zℓ), ℓ 6= p,

via the natural embeddingZ → Zℓ.

We proceed to explore a functorial version of Theorem 3.8 for(WaK ,ΣK), where

ΣK is the correspondingfan, i.e., the set ofc-subgroupsσ ⊂WaK . We work with

the following diagram


Kψ // L

K×/k×ψ1 // L×/l×

WaK Wa

Lψ∗oo

whereK andL are function fields over algebraically closed ground fieldsk and l .We are interested in situations whenψ1 maps subgroups of the formE×/k×, whereE⊂ K is a subfield with trdegk(E) = 1, into similar subgroups inL×/l×. Thedualhomomorphisms

(4.1) ψ∗ : WaL→Wa

K

to suchψ1 respect the fans, in the following sense: for allσ ∈ΣL eitherψ∗(σ)∈ΣK

or ψ∗(σ) is cyclic.

Example4.1. The following examples of homomorphismsψ∗ as in Equation 4.1arise from geometry:

(1) If X→Y is a dominant map of varieties overk thenk(Y) = L⊂ K = k(X)and the induced homomorphism

ψ∗ : WaK →Wa

L

respects the fans.(2) Letπ : X→Y be a dominant map of varieties overk ands:Y→X a section

of π. There exists a valuationν ∈ VK with center the generic point ofs(Y)such that

L× ⊂ o×ν ⊂ K×

and the natural projection

ρν : o×ν → K×ν

onto the multiplicative group of the residue field induces anisomorphism

L× ≃ K×ν

which extends to an isomorphism of fields

L≃ Kν .

Let ρν : K× → L× be any multiplicative extension ofρν to K× (when itexists, for example, it exists for algebraicν). Suchρν map multiplicativesubgroups of the formk(x)× to similar subgroups ofL×, i.e., the dual map

WaL→Wa

K

preserves the fans.


(3) More generally, letν ∈ VK be a valuation, with valuation ringoν , maximalidealmν and residue fieldKν . Combining the exact sequences

1 // o×ν // K×ν // Γν // 1

1 // (1+mν)× // o×νρν // K×ν // 1

we have an exact sequence

1→ K×ν → (1+mν)×\K×→ Γν → 1.

Let Kν = k(Y), for some algebraic varietyY overk, and letL be any func-tion field containingKν = k(Y). Assume that there is a diagram

o×ν /k×

ρν��

�

� // K×/k×

ψ1

��K×ν /k×

φ// L×/k×

whereφ is injective andψ1 is an extension ofφ . Such extensions existprovided we are given a splitting ofν : K× → Γν . In this case, we willsay thatψ1 is defined by a valuation. The dual map to suchψ1 respectsthe fans. Indeed, this can be checked on multiplicative groups E∗/k∗ ofone-dimensional subfieldsE ⊂ K: either E∗/k∗ embedds intoK∗ν/k∗ orthe restriction ofν to E∗/k∗ is nontrivial and hence the image ofE∗/k∗

is cyclic. Theorem 4.2 shows a converse result:any ψ1 respecting one-dimensional subfields can be obtained via this construction.

We proceed to describe the restriction ofψ1 to projective subspaces ofPk(K), whenψ1 is obtained from the geometric construction (3); in partic-ular, ψ1 is not injective. In this case, for anyE = k(x) ⊂ K the restrictionof ψ1 to E×/k× is either• injective, or• its restriction tol(1,x) is constant on the complement of one point,

i.e., it factors through a valuation homomorphism.On planesP2(1,x,y) there are more possibilities: an inertia element

ι = ιν ∈WaK = Hom(K×/k×,Z)

restricts to anyP2 = P2k(1,x,y) as a flag map, in particular, it takes at most

three values. This leads to the following cases:(a) ι is constant onP2(1,x,y); thenP2(1,x,y) ⊂ o×ν /k× since the corre-

sponding linear space does not intersect the maximal idealmν andcontains 1. The projection

ρν : P2(1,x,y)→ K×ν /k×


is injective. If x,y are algebraically independent the values ofψ1 onP2(1,x,y)◦ are algebraically independent inK×ν .

(b) ι takes two values and (after an appropriate multiplicative shift) isconstant onP2(1,x,y) \ {x}; in this caseψ1 on P2(1,x,y) \ {x} is acomposition of a projection fromx onto l(1,y) and an embedding ofl(1,y) → K×ν /k× with the mapx→ ι(x).

(c) ι takes two values and is constant on the complement of a projectiveline, sayl(x,y); thenψ1 ≡ 1 on the complement tol(x,y). Note that1/x· l(x,y) ⊂ o×ν and hence embeds intoK×ν /k×.

(d) ι takes three values.The proof of Theorem 4.2 below relies on a reconstruction of these andsimilar flag structures from fan data, more precisely, it involves a construc-tion of a special multiplicative subseto× ⊂ K×/k× which will be equal too×ν /k×, for some valuationν ∈ VK .

Theorem 4.2. Assume that K,L are function fields over algebraic closures of finitefields k, l, respectively. Assume that

(a)ψ1 : K×/k×→ L×/l×

is a homomorphism such that for any one-dimensional subfieldE ⊂ K,there exists a one-dimensional subfield F⊂ L with

ψ1(E×/k×)⊆ F×/l×,

(b) ψ1(K×/k×) contains at least two algebraically independent elements ofL×/l×.

If ψ1 is not injective then

(1) there is aν ∈ VK such thatψ1 is trivial on (1+mν)×/k× ⊂ o×ν /k×;(2) the restriction ofψ1 to

(4.2) K×ν /k× = o×ν /k×(1+mν)×→ L×/l×

is injective and satisfies (a).

If ψ1 is injective, then there exists a subfield F⊂ L, a field isomorphism

φ : K∼−→ F ⊂ L,

and an integer m∈Z, coprime to p, such thatψ1 coincides with the homomorphisminduced byφm.

The case of injectiveψ1 has been treated in [8]. The remainder of this sectionis devoted to the proof of Theorem 4.2 in the case whenψ1 is not injective.

We introduce an equivalence relation onnonconstantrational functions:f ≈ f ′,i.e., f , f ′ are algebraically dependent. LetF be the set of equivalence classes. An


immediate application of Condition (a) is: iff ≈ f ′ then ψ1( f ) ≈ ψ1( f ′). Theconverse does not hold, in general, and we are lead to introduce the followingdecomposition:

(4.3) Pk(K) = S1⊔⊔ f∈FSf ,

whereS1 := ψ−1

1 (1)

and, for f /∈ S1,

Sf := { f ′ ∈ Pk(K)\S1 |ψ1( f ′)≈ ψ1( f )} ⊂ Pk(K)

are equivalence classes of elements whoseimagesare algebraically dependent. Werecord some properties of this decomposition:

Lemma 4.3.

(1) For all f , the set S1⊔Sf is closed under multiplication,

f ′, f ′′ ∈ S1⊔Sf ⇒ f ′ · f ′′ ∈ S1⊔Sf .

(2) Every projective linel(1, f )⊂ Pk(K), with ψ1( f ) 6= 1, is contained in S1⊔Sf .

(3) Assume that f,g are such thatψ1( f ),ψ1(g) are nonconstant and distinct.If ψ1( f )≈ ψ1(g) thenl( f ,g) ∈ S1⊔Sf . Otherwise,l( f ,g)∩S1 = /0.

(4) Let Π ⊂ Pk(K) be a projective subspace such that there exist x,y,z∈ Πwith distinct images and such thatψ1(x/z) 6≈ ψ1(y/z).

Then, for any h∈ K×/k×, the projective subspaceΠ′ := h ·Π satisfiesthe same property.

Proof. The first property is evident. Condition (a) of Theorem 4.2 implies that forevery f ∈ K×/k× we have

(4.4) ψ1(l(1, f )) ⊂ Pl (F), F = l(ψ1( f )),

this implies the second property.Considering the shiftl( f ,g) = f · l(1,g/ f ) we see that

ψ1(l( f ,g) ⊆ ψ1( f ) ·ψ1(l(1,g/ f )) ⊂ ψ1( f ) ·Pl (F),

whereF = l(ψ1(g/ f )). If ψ1( f )≈ψ1(g/ f ) thenψ1(l( f ,g))⊂S1⊔Sf . Otherwise,ψ1(l( f ,g)) is disjoint from 1.

To prove the last property is suffices to remark thatψ1(Π′) containsψ1(hx),ψ1(hy), ψ1(hz), andψ1(hx/hz) 6≈ ψ1(hy/hz). �

Lemma 4.4. LetΠ = P2⊂ Pk(K) be a projective plane satisfying condition (4) ofLemma 4.3 and such that the restriction ofψ1 to Π is not injective. Then

(1) there exists a linel⊂Π such thatψ1 is constant onΠ\ l or(2) there exists a g∈Π such that

– ψ1 is constant on every punctured linel(g, f )\g;– ψ1(g) 6≈ ψ1( f ), for every f 6= g;


– ψ1( f )≈ ψ1( f ′) for all f , f ′ /∈ g.

Proof. After an appropriate shift and relabeling, and using Lemma 4.3, we mayassume thatΠ = P2(1,x,y) and thatψ−1

1 (1) contains a nontrivial elementz ∈P2(1,x,y), i.e.,z∈ S1. Let

P2(1,x,y) = S1⊔⊔ f∈F Sf

be the decomposition induced by (4.3).Step 1.Neither of the setsS1 nor Sf contains a line. Indeed, assume that there

is a projective linel⊆ S1 and letg∈ P2(1,x,y)\S1. Every f ′ ∈ P2(1,x,y) lies on aline throughg andl( f ′,g) which intersectsl, and thusS1, i.e., all f ′ lie in S1⊔Sg, byLemma 4.3. Assume thatl⊂Sf . Everyg∈ P2(1,x,y)\S1 lies on a line of the forml(1,g), which intersectsl⊆ Sf . It follows thatg∈ Sf , contradicting our assumptionthatψ1(P

2(1,x,y)) contains at least two algebraically independent elements.

Step 2.Split F = F ′⊔F ′′ into nonempty subsets, arbitrarily, and let

P2(1,x,y) = S1⊔S′⊔S′′, S′ := ⊔ f ′∈F ′Sf ′ , S′′ := ⊔ f ′′∈F ′′Sf ′′

be the induced decomposition. By Lemma 4.3, everyl is in either

S1⊔S′, S1⊔S′′, or S′⊔S′′.

By Lemma 3.6, one of these subsets is a flag subset ofP2(1,x,y).

Step 3.Assume thatS1 is a flag subset. Since it contains at least two elementsand does not contain a projective line, by Step 1, we have:

• S1 = P2\ l, for some linel, and we are in Case (1), or• S1 = l(1,z)◦ = l(1,z)\g, for someg∈ S′ (up to relabeling).

Choose ag′′ ∈ S′′, so thatψ1(g′′) 6≈ ψ1(g), and letl be a line throughg′′, l 6= l(g,g′′). Thenl intersectsl(1,z)◦ = S1, which implies that the com-plementl\ (l∩ l(1,z)◦) ⊆ S′′. It follows thatS′′ contains the complementof l(1,z)∩ l(g,g′′). Considering projective lines through 1 and elements inS′′ and applying Lemma 4.3 we find thatS′′ ⊇ P2(1,x,y) \ l(1,z). Sinceg /∈ S′′, we have equality. Thus all elements inP2(1,x,y) \ g have alge-braically dependent images. Ifψ1 were not constant on a linel throughg,with l 6= l(1,z), let f1, f2∈ l be elements withψ1( f1) 6= ψ1( f2). Lemma 4.3implies thatg ∈ Sf1, contradicting our assumption thatψ1(g) 6≈ ψ1( f1).Thus we are in Case (2).

Step 4.Assume thatS′ is a flag subset andS1 is not. We have the followingcases:

• S′ = {g}. ThenS′ = Sg andl(1,g)◦ := l(1,g) \g⊆ S1. Assume that thereexist f1, f2 ∈ P2(1,x,y)\ l(1,g) with ψ1( f1) 6= ψ1( f2). If f2 /∈ l(g, f1) thenψ1( f1) ≈ ψ1( f2), asl( f1, f2) intersectsl(1,g)◦. If there exists at least one


f2 /∈ l(g, f1) with nonconstantψ1( f2), then by the argument above, all el-ements on the complement tol(1,g) are algebraically dependent, on everyline l throughg, ψ1 is constant onl \g, and we are in Case (2). Ifψ1 isidentically 1 onP2(1,x,y) \ l(g, f1) thenψ1 6= 1 on l(g, f1) (otherwiseS1

would contain a projective line). ThusS1 is a flag subset, contradiction.• S′ = l◦ = l\g, for some linel andg∈ l. SinceS1 has at least two elements,

there is az′ ∈ (P2\ l)∩S1. Lemma 4.3 implies thatψ1 equals 1 onl(z′,g′)\g′, for all g′ ∈ S′ = l◦. Similarly, ψ1 equals 1 onl(g,z′)◦ := l(g,z′) \g, asevery point on this punctured line lies on a line passing through S1 andintersectingS′. ThusS1 ⊇ P2(1,x,y) \ l and sinceS1 does not contain aline, these sets must be equal. It follows that we are in Case (1).• Assume thatS′ = P2(1,x,y)\ l and we are not in the previous case. Thenl

contains at least two points inS1 and the complementS′′ = l\ (l∩S1) alsohas at least two points,f ′′1 , f ′′2 . ThusS′′ = Sf ′′ for somef ′′, by Lemma 4.3.Since we were choosing the splittingF = F ′ ⊔F ′′ arbitrarily, we con-clude thatS′ = Sf ′ for somef ′.

The same argument as in Step 3. implies thatψ1 is constant onl(g′, f ′′i )\f ′′i for anyg′ ∈S′. Henceψ1 is constant onP2(1,x,y)\ l and we are in Case(1).

�

Lemma 4.5. Letu := ∪ l(1,x) ⊆ K×/k× = Pk(K)

be union over all lines such thatψ1 is injective onl(1,x). Assume that there existnonconstant x,y∈ u such thatψ1(x) 6≈ ψ1(y). Then

o× := u ·u⊂ Pk(K)

is a multiplicative subset.

Proof. First of all, if x∈ u thenx−1 ∈ u, sincel(1,x) = x· l(1,x−1).It suffices to show thatu · o× = o×, i.e., for allx,y,z∈ u one hasxyz= tw, for

somet,w∈ u.Assume first thatψ1(x) 6≈ψ1(y) and considerP2(1,x,y−1). A shift of this plane

contains the linel(1,xy). If xy /∈ u thenψ1 is not injective on this plane and wemay apply Lemma 4.4. We are not in Case (1) and not in Case (2), sinceψ1 injectsl(1,x) andl(1,y−1), contradiction. Thusψ1 is injective onP2(1,x,y−1) andxy= t,for somet ∈ u, which proves the claim.

Now assume thatψ1(x),ψ1(y),ψ1(z) ∈ F×/l×, for some 1-dimensionalF ⊂ L.By assumption, there exists aw ∈ K×/k× such thatψ1(w) is algebraically inde-pendent ofF×/l× and such thatl(1,w) injects. By the previous argument,ψ1 isinjective on the linesl(1,xw) andl(1,w−1y), so thatxw,w−1y∈ u. By our assump-tions,ψ1 is injective onl(1,z). Now we repeat the previous argument forxwandz:there is at ∈ u such thatxw·z= t. �


Proposition 4.6. Letψ1 : K×/k×→ L×/l×

be a homomorphism satisfying the assumptions of Theorem 4.2and such thatu ⊂Pk(K) contains x,y with ψ1(x) 6≈ ψ1(y). Let o× = u · u ⊂ K×/k× be the subsetdefined above. Then

ν : K×/k×→ K×/o×

is a valuation homomorphism, i.e., there exists an ordered abelian groupΓν withK×/o× ≃ Γν andν is the valuation map.

Proof. By Lemma 4.5, we know thato× is multiplicative. We claim that the re-striction ofν to everyP1 ⊂ Pk(K) is a flag map. Indeed, we haveP1 = x · l(1,y),for somex,y∈ K×/k×. On every linel(1,y) ⊂ u, the value ofν is 1, and on everyline l(1,x) 6⊂ u, the value ofν is constant on the complement to one point.

By Proposition 3.3, this implies thatν is a flag map, i.e., defines a valuation onK×/k× with values onΓν := K×/o×. �

To complete the proof of Theorem 4.2 we need to treat homomorphismsψ1

such that for most one-dimensionalE ⊂ K, the imageψ1(E×/k×) is cyclic, oreven trivial. We have two cases:

(A) there exists a linel = (1,x) such thatψ1 is not a flag map on this line.(B) ψ1 is a flag map onevery linel⊂ Pk(K).

In Case (A), letu′ be the union of linesl(1,x) such thatψ1 is not a flag mapon l(1,x), i.e., is not constant on the line minus a point. By the arguments inLemma 4.4, Lemma 4.5, and Proposition 4.6, there exists a unique 1-dimensionalnormally closed subfieldF ⊂ L such thatψ1(u

′)⊆ F×/l×. Define

o× := ψ−11 (F×/l×)⊂ K×/k×.

By assumption onψ1, o× is a proper multiplicative subset ofK×/k×. The inducedhomomorphism

ν : K×/k×→ (K×/k×)/o× =: Γν

is a valuation map, since it is a flag map on every linel ⊂ Pk(K). Indeed, it isconstant on alll(1,x) ⊆ u and a valuation map on alll(1,y) 6⊂ o×. Hence the sameholds for any projective line inPk(K) which implies the result.

In Case (B), we conclude thatψ1 is a flag map onPk(K) (see, e.g., Lemma 4.16of [7]), i.e., there exists a valuationν ∈ VK with value groupΓν = ψ1(K×/k×)such that the valuation homomorphismν = ψ1.

Remark4.7. The main steps of the proof (Lemmas 4.3, 4.4, and 4.5) are validfor more general fields: the splitting ofP2 into subsets satisfying conditions (1)and (2) of Lemma 4.3 is related to a valuation, independentlyof the ground field(see [6]). Here we restricted tok = Fp since in this case the proof avoids sometechnical details which appear in the theory of general valuations. Furthermore,


the condition thatK,L are function fields is also not essential. The only essentialproperty is the absence of an infinite tower of roots for the elements ofK,L whichare not contained in the ground field.

5 Galois cohomology

By duality, the main result of Section 4 confirms the general concept that bira-tional properties of algebraic varieties are functoriallyencoded in the structure ofthe Galois groupGc

K . On the other hand, it follows from the proof of the Bloch–Kato conjecture thatGc

K determines the full cohomology ofGK. Here we discussgroup-theoretic properties ofGK and its Sylow subgroups which we believe areultimately responsible for the validity of the Bloch–Kato conjecture.

Let G be a profinite group, acting continuously on a topologicalG-moduleM,and let Hi(G,M) be the (continuous)i-cohomology group. These groups are con-travariant with respect toG and covariant with respect toM; in most of our applica-tionsM eitherZ/ℓ or Q/Z, with trivial G-action. We recall some basic properties:

• H0(G,M) = MG, the submodule ofG-invariants;• H1(G,M) = Hom(G,M), providedM has trivialG-action;• H2(G,M) classifies central extensions

1→M→ G→G→ 1,

up to homotopy.• if G is abelian andM finite with trivial G-action then

Hn(G,M) = ∧n(H1(G,M)), for all n≥ 1,

• if M = Z/ℓm then

Hn(G,M) → Hn(Gℓ,M), for all n≥ 0,

whereGℓ is theℓ-Sylow subgroup ofG.

See [1] for further background on group cohomology and [29],[24] for backgroundon Galois cohomology. Let

G(n) := [G(n−1),G(n−1)]

then-th term of itsderivedseries,G(1) = [G,G]. We will write

Ga = G/[G,G], and Gc = G/[[G,G],G]

for the abelianization, respectively, the secondlower central seriesquotient ofG.Consider the diagram, connecting the first terms in the derived series ofG withthose in the lower central series:


1 // G(1)

��

// G

��

// Ga // 1

1 // Z // Gc // Ga // 1

We have a homomorphism between E2-terms of the spectral sequences com-puting Hn(G,Z/ℓm) and Hn(Gc,Z/ℓm), respectively. Suppressing the coefficients,we have

Hp(Ga,Hq(Z))

��

Ep,q2 (Gc)

��

+3 Ep+q(Gc)

��

Hp+q(Gc)

��Hp(Ga,Hq(G(1))) Ep,q

2 (G) +3 Ep+q(G) Hp+q(G)

We have H0(G,Z/ℓm) = Z/ℓm, for all m∈ N, and

H0(Ga,H1(Z)) = H1(Z).

The diagram of corresponding five term exact sequences takesthe form:

H1(Ga) = H1(Gc) // H1(Z)

≃��

d2 //

��

H2(Ga) // H2(Gc)

��H1(Ga) = H1(G) // H0(Ga,H1(G(1)))

d′2 // H2(Ga) // H2(G)

where the left arrows map H1(G) = H1(Ga) to zero and the isomorphism in themiddle follows from the description ofGc as a maximal quotient ofG that is acentral extension ofGa.

Let K beanyfield containingℓm-th roots of 1, for allm∈ N, and letGK be itsabsolute Galois group. We apply the cohomological considerations above toGK .By Kummer theory,

(5.1) H1(GK ,Z/ℓm) = H1(GaK ,Z/ℓm) = K1(K)/ℓm

and we obtain a diagram

H1(GaK ,Z/ℓm)⊗H1(Ga

K ,Z/ℓm)sK // // ∧2(H1(Ga

K ,Z/ℓm)) = H2(GaK ,Z/ℓm)

K1(K)/ℓm⊗K1(K)/ℓm σK // // K2(K)/ℓm

wheresK is the skew-symmetrization homomorphism (cup product) andσK is thesymbol map, and Ker(σK) is generated by symbols of the formf ⊗ (1− f ). The


Steinberg relations imply that the kernel ofsK , i.e., the symmetric tensors, is con-tained in the kernel ofσK (see, e.g., [19, Section 11]).

PuttingZK := Ker(GcK →Ga

K), we obtain a diagram

H1(ZK,Z/ℓm)�

� d2 // H2(GaK ,Z/ℓm)

π∗a // H2(GcK ,Z/ℓm)

IK(2)/ℓm �

� // ∧2(H1(GaK ,Z/ℓm)) // // K2(K)/ℓm

hK

OO

wherehK is theGalois symbol(cf. [31] or [24, Theorem 6.4.2]) andIK(2) is definedby the exact sequence

1→ IK(2)→∧2(K×)→ K2(K)→ 1.

A theorem of Merkurjev–Suslin [18] states thathK is an isomorphism

(5.2) H2(GK ,Z/ℓm) = K2(K)/ℓm.

This is equivalent to:

• π∗a : H2(GaK ,Z/ℓm)→ H2(GK) is surjective and

• H1(ZK ,Z/ℓm) = IK(2)/ℓm.

The Bloch–Kato conjecture, proved by Voevodsky, Rost, and Weibel, general-izes (5.1) and (5.2) to alln. This theorem is of enormous general interest, withfar-reaching applications to algebraic and arithmetic geometry. It states that forany fieldK and any primeℓ, one has an isomorphism between Galois cohomologyand the modℓ Milnor K-theory:

(5.3) Hn(GK ,µ⊗nℓ ) = KM

n (K)/ℓ.

It substantially advanced our understanding of relations between fields and theirGalois groups, in particular, their Galois cohomology. Below we will focus onGalois-theoretic consequences of (5.3).

We have canonical central extensions

GK

�� !!BBB

BBBB

B

1 // ZK//

��

GcK

//

��

GaK

//

��

1

1 // ZK// G c

Kπc

a // G aK

// 1

and the diagram


GK

πc

��

��

�πa

��???

????

??

G cK πc

a

// G aK .

The following theorem relates the Bloch–Kato conjecture tostatements in Galois-cohomology (see also [12], [11], [28]).

Theorem 5.1. [10, Theorem 11]Let k= Fp, p 6= ℓ, and K= k(X) be the functionfield of an algebraic variety of dimension≥ 2. The Bloch–Kato conjecture for K isequivalent to:

(1) The mapπ∗a : H∗(G a

K ,Z/ℓn)→ H∗(GK ,Z/ℓn)

is surjective and(2) Ker((πc

a)∗) = Ker(π∗a).

This implies that the Galois cohomology of the pro-ℓ- quotientGK of the abso-lute Galois groupGK encodes important birational information ofX. For example,in the case above,G c

K , and henceK, modulo purely-inseparable extensions, can berecovered from the cup-products

H1(GK ,Z/ℓn)×H1(GK ,Z/ℓn)→ H2(GK ,Z/ℓn), n∈ N.

6 Freeness

Let K be a function field over an arbitrary ground fieldk andGK the absoluteGalois group ofK. The pro-ℓ-quotientGK of K is highly individual: fork = Fp itdeterminesK up to purely-inseparable extensions. On the other hand, letGℓ(GK)be anℓ-Sylow subgroup ofGK . This group isuniversal, in the following sense:

Proposition 6.1. [5] Assume that X has dimension n and that X contains a smoothk-rational point. Then

Gℓ(GK) = Gℓ(Gk(Pn)).

In particular, whenk is algebraically closed, theℓ-Sylow subgroups dependonly on the dimension ofX. This universal group will be denoted byGℓ. Thefollowing Freenessconjecture captures an aspect of this universality. It impliesthe (proved) Bloch–Kato conjecture; but more importantly,it provides a structuralexplanation for its truth.

Conjecture 6.2(Bogomolov). Let k be an algebraically closed field of character-istic 6= ℓ, X an algebraic variety overk of dimension≥ 2, K = k(X), and write

G(1)ℓ := [Gℓ,Gℓ]


for the commutator of anℓ-Sylow subgroup ofGK. Then

(6.1) Hi(G(1)ℓ ,Z/ℓm) = 0, for all i ≥ 2,m∈ N.

Remark6.3. For profiniteℓ-groups, the vanishing in Equation (6.1) fori = 2 im-plies the vanishing forall i ≥ 2 (see [17]).

We now return to the cohomological considerations in Section 5. The standardspectral sequence associated with

1→G(1)→G→Ga→ 1,

gives

Hp(Ga,Hq(G(1),Z/ℓm))⇒ Hn(G,Z/ℓm).

We apply this toG = Gℓ; suppressing the coefficients we obtain:

0 0 0 · · ·

H0(Ga,H2(G(1))) 0 0 · · ·

H0(Ga,H1(G(1)))d2

,,YYYYYYYYYYYYYYYYYYYYYYYYYYYH1(Ga,H1(G(1))) H2(Ga,H1(G(1))) · · ·

H0(Ga,H0(G(1))) H1(Ga,H0(G(1))) H2(Ga,H0(G(1))) · · ·

Conjecture 6.2 would imply that, forG = Gℓ, and also forG = GK, H2(G(1)) andconsequentlyall entries above the second line vanish. In this case, we have a longexact sequence (see, e.g., [24, Lemma 2.1.3])

0→ H1(Ga)→ H1(G)→ H0(Ga,H1(G(1)))→ H2(Ga)→ ···

· · · → Hn(Ga,H1(G(1)))d2−→ Hn+2(Ga)→ Hn+2(G)→ ···

In Section 5 we saw that forn = 0 the homomorphismd′2 in the sequence

Hn(GaK ,H1(G(1)

K ))d′2−→ Hn+2(Ga

K)→ Hn+2(GK) = Kn+2(K)/ℓm

can be interpreted as the embedding of theskew-symmetricrelations:

H0(Ga,H1(Z)) = IK(2)/ℓm d′2−→∧2(H1(GaK)) = ∧2(K×/ℓm)→ K2(K)/ℓm.

This relied on Kummer theory and the Merkurjev–Suslin theorem. We proceed tointerpret the differentiald2 for highern.


We work withG = Gℓ, anℓ-Sylow subgroup of the absolute Galois group of afunction fieldK over an algebraically closed field. We have an exact sequenceofcontinuousGa-modules

1→ [G,G(1)]→G(1)/G(2)→ Z→ 1.

Note thatGa acts trivially onZ, and H1(Z), and viax 7→ gxg−1− x on G(1)/G(2).Dually we have a sequence ofGa-modules:

Hom([G,G(1)],Z/ℓm)← Hom(G(1)/G(2),Z/ℓm)← Hom(Z,Z/ℓm)← 1.

Define

M := Hom(G(1)/G(2),Z/ℓm),

then

MGa= Hom(Z,Z/ℓm) = H1(Z) = IK(2)/ℓm.

We have a homomorphism

(6.2) Hn(MGa)→ Hn(M)

via the natural embedding. Since H0(MGa) embeds into H2(Ga) via d2 we obtain a

natural homomorphism

tn : Hn(MGa)→ Hn+2(Ga)

and the differentiald2 on the image of Hn(MGa)→Hn(M) coincides withtn. Thus

the fact that the kernel of Hn(Ga)→ Hn(G) is generated by trivial symbols willfollow from the surjectivity of the homomorphism in (6.2). This, in turn, wouldfollow if the projectionM→M/MGa

defined a trivial map on cohomology. Thuswe can formulate the following conjecture which complements the Freeness con-jecture 6.2:

Conjecture 6.4. The projectionM→M/MGacan be factored as

M → D ։ M/MGa,

whereD is a cohomologically trivialGa-module.

We hope that a construction of a natural moduleD can be achieved via algebraicgeometry.

Acknowledgment.The first author was partially supported by NSF grants DMS-0701578, DMS-

1001662, and by the AG Laboratory GU-HSE grant RF ag. 11 11.G34.31.0023.The second author was partially supported by NSF grants DMS-0739380 and 0901777.We are grateful to M. Rovinski for useful comments.


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