Cohomological invariants of SK - COnnecting REpositoriestu as sugg´er´e, trouvait sa place. Merci...

Arenberg Doctoraatsschool Wetenschappen amp TechnologieFaculteit WetenschappenDepartement Wiskunde

Cohomological invariants of SK1

Tim Wouters

Promotoren Proefschrift voorgedragenProf Philippe Gille tot het behalen van deProf Joost van Hamel graad van Doctor in deProf Willem Veys Wetenschappen (Wiskunde)

Mei 2010


Tim Wouters

PromotorenProf Philippe GilleProf Joost van HamelProf Willem Veys

ExamencommissieProf Jan Denef Proefschrift voorgedragenProf Paul Igodt tot het behalen van deProf Johannes Nicaise graad van Doctor in deProf Jean-Pierre Tignol Wetenschappen (Wiskunde)Prof Jan Van Geel

Mei 2010

Gedrukt door Procopia nvAmbachtenlaan 29 B-3001 Leuven (Belgie)httpwwwprocopiabe

copy Katholieke Universiteit Leuven ndash Faculteit WetenschappenKasteelpark Arenberg 11 - bus 2100 B-3001 Leuven (Belgie)

Alle rechten voorbehouden Niets uit deze uitgave mag worden vermenig-vuldigd enof openbaar gemaakt worden door middel van druk fotokopiemicrofilm elektronisch of op welke andere wĳze ook zonder voorafgaandeschriftelĳke toestemming van de uitgever

All rights reserved No part of the publication may be reproduced in anyform by print photoprint microfilm or any other means without writtenpermission from the publisher

Wettelĳk depot D20101070533ISBN 978-90-8649-340-1

Avec tout mon respect et ma consideration pourla communaute mathematique francaise et russe

S glubokim uvaeniem i poqitaniem k

francuzskomu i russkomu matematiqeskomu

soobwestvu

Dankwoord

ldquoThe more you knowthe more you realise

how little you knowrdquomdash Daodejing

Negen jaar intensief wiskunde studeren verandert een mens Het proces gaatgeleidelĳk aan maar je merkt de effecten van het abstract redeneren in jedagelĳkse leven Je begint zowat alles dieper te analyseren Na negen jaarwiskunde besef ik dat ik zeker niet nog alles in de wiskunde gezien heb maarik wil graag met de opgedane ervaringen andere horizonten gaan verkennenDit is dan ook de uitgelezen kans om enkele mensen te danken die me degesteund hebben in mĳn trektocht doorheen de wiskunde

Wim je gaf me 5 jaar geleden de kans om het wiskundig onderzoek te lerenkennen Ik heb hierbĳ in verschillende statuten gewerkt en je hebt steedsde administratieve kant in orde gebracht De laatste jaren heb je ook hetofficiele promotorschap vervult Bedankt hiervoor ook al omdat ik weet datje geen fan bent van al die administratie Dat geldt ook om me toe te lateneen artikel in het Frans te schrĳven

Pour la realisation de ma these je dois beaucoup au soutien drsquoun grandhomme Philippe tout drsquoabord merci pour avoir accepte la tache de continuerle travail de Joost Crsquoetait un grand honneur de pouvoir travailler avec toiLes journees a Paris me manqueront Le temps que tu mrsquoas consacre nrsquoestpas mesurable Ca me prenait toujours tellement de temps pour digerer lecontenu de notre discussions crsquoetait un peu comme un oracle de Delphes Ilme fallait de longs moments pour comprendre mais finalement tout ce quetu as suggere trouvait sa place Merci pour tout

Verder dank ik ook de andere leden van de jury voor hun aanmoedigendenthousiasme en de suggesties ter verbetering van dit werk Jan DenefPaul Igodt Johannes Nicaise Jan Van Geel en Jean-Pierre Tignol Jean-Pierre merci pour mrsquoavoir introduit dans la cohomologie galoisienne pourton interesse dans ma recherche et pour tes suggestions quant a la theoriedes algebres simples centrales

iii

DANKWOORD

De voorbĳe jaren zĳn tevens voorbĳgevlogen door de gemoedelĳke sfeerop de afdeling Algebra Dank aan iedereen die hieraan bĳgedragen heeft(in een korte of een lange samenwerking) Uiteraard ook dank aan mĳn(schoon)ouders familie en vrienden die me steeds steunden alhoewel ik nooitecht uitgelegd heb waarmee ik me al die tĳd bezig hield Het was belangrĳksteeds te kunnen terugvallen op deze morele steun

Voor de praktische zaken dank ik het secretariaat van het DepartementWiskunde en van de Faculteit Wetenschappen net als de medewerkers vande bibliotheek (voor de vele aanvragen die ze voor mĳ behandelden) Ookdank aan het personeel van de NMBS SNCF en Thalys voor de tallozeverplaatsingen die ze mogelĳk maakten In het bĳzonder dank om niet (alte veel) te staken op de dagen dat ik van hun diensten gebruik maakteUiteraard ben ik de KULeuven het FWO Vlaanderen en de Ecole NormaleSuperieure (Parĳs) veel dank verschuldigd voor de financiele ondersteuning

Net als in mĳn licentiaatsthesis wil ik de brouwers danken die me voorbĳejaren van goed bier hebben laten genieten Daarnaast dank aan deBierpallieters om samen beetje bĳ beetje zotter te worden van geuze (ookdank aan Nicolas voor Russische vertalingshulp) Het is leuk om naasthet abstract wiskundige denken ook in wat meer aardse dingen te kunnenopgaan Grazie mille anche agli studenti drsquoitaliano del CLT per le numeroserisate

Tot slot wil ik Sarah danken om er steeds voor mĳ te zĳn zowel in demomenten dat ik rsquos avonds thuis kwam na een weinig nuttige dag als indie (eerder schaarse) momenten dat ik een resultaatje boekte Alhoewelik je nooit heb uitgelegd wat ik al die tĳd deed heb je me steeds volledigbegrepen Ik kan je niet uitleggen hoeveel dat voor mĳ betekend heeft maarik hoop dat je me ook nu wel zal begrĳpen

Deze thesis was nooit tot stand kunnen komen zonder Joost van Hamel Ikben ondergedompeld in dit type onderzoek door zĳn wiskundig enthousiasmeHet is een mooie wereld die hĳ me leren kennen heeft en die ik nu achter melaat Ik zal nooit weten wat hĳ van dit proefschrift zou gevonden hebbenmaar zeker ben ik wel van mĳn dankbaarheid voor de richting waarin hĳ meuitgestuurd heeft Appendix B bevat het onderwerp van mĳn eerste artikeldat hĳ in zĳn laatste levensmaanden intensief begeleidde Deze appendix isaan hem opgedragen

Tim WoutersMei 2010

iv

Abstract

The subject of this thesis is the group functor SK1(A) for a central simplealgebra A over a field k We concentrate on cohomological invariants ofSK1(A) which can - as claimed by Suslin [Sus] - help to explain Platonovrsquosexamples of non-trivial SK1 [Pla] Most of the existing ones restrict to centralsimple algebras A of indk(A) isin ktimes

A first aim of this thesis is to generalise these invariants to any centralsimple algebra (so that we can drop the condition on the index) For thatpurpose we use a lift from positive characteristic to characteristic zeroAs the condition on the index is empty in characteristic zero we can usethe existence of the invariants in characteristic zero and then specialise ina proper way to positive characteristic This involves notions of p-ringsKatorsquos logarithmic differentials and some important results by Kahn andMerkurjev

We also compare this construction with an invariant of SK1 for biquaternionalgebras as defined by Knus-Merkurjev-Rost-Tignol [KMRT sect17] Thisinvariant also does not have the condition on the index For biquaternionalgebras in characteristic 2 we prove this invariant essentially equals ageneralised invariant of Suslin We finish this thesis by proving the non-triviality of an invariant of SK1(A) recently introduced by Kahn [Kah3]We also give a formula for the value on the centre of the tensor productof two symbol algebra which generalises a formula from Merkurjev for thecentre of two biquaternions [Mer2]

In an appendix we describe the behaviour of the so-called elementaryobstruction under the Weil restriction The elementary obstruction candetermine whether a variety contains no rational points In this appendixwe prove the invariance of this elementary obstruction under taking a Weilrestriction of scalars This was the content of a first paper of the authorAlthough the subject is quite different from the core of this thesis themethods used are very similar

v

SAMENVATTING

Samenvatting

In deze doctoraatsverhandeling bestuderen we de groepsfunctor SK1(A)voor een centrale enkelvoudige algebra A Daarbĳ concentreren we onsop cohomologische invarianten van deze groepsfunctor Zoals veronderstelddoor Suslin [Sus] is de hoop dat deze (onder meer) Platonovs voorbeeldenvan niet-triviale SK1 kunnen verklaren Het merendeel van de reedsbestaande invarianten beperkt zich steeds tot centrale enkelvoudige algebrarsquosA met indk(A) isin ktimes

In deze thesis introduceren we een methode om deze invarianten teveralgemenen (zodat we de voorwaarde op de index kunnen laten vallen)Hiervoor gebruiken we een opheffing van positieve karakteristiek naarkarakteristiek nul Aangezien de voorwaarde in karakteristiek nul niet-bestaande is kunnen we het bestaan van invarianten in karakteristiek nulgebruiken om via een specialisatie invarianten in positieve karakteristiek teverkrĳgen Dit vereist het gebruik van p-ringen logaritmische differentialen(op zĳn Katorsquos) en belangrĳke hulpresultaten van Kahn en Merkurjev

We vergelĳken deze constructie ook met een invariant van SK1 voorbiquaternionen ingevoerd door Knus-Merkurjev-Rost-Tignol [KMRT sect17]Deze invariant heeft ook geen voorwaarde op de index We bewĳzen datdeze gelĳk is aan de nieuw geconstrueerde invariant Tot slot tonen weaan dat een specifieke invariant van Kahn niet triviaal is voor het productvan twee symboolalgebrarsquos op zĳn Platonovs Tevens veralgemenen we eenformule van Merkurjev voor de waarde op het centrum van biquaternionen[Mer2] naar het het tensorproduct van twee symboolalgebrarsquos

In een appendix beschrĳven we het gedrag van de elementaire obstructie vaneen varieteit onder de weilrestrictie De elementaire obstructie kan bepalendat een varieteit geen rationale punten heeft We bewĳzen dat de elementaireobstructie invariant is onder het nemen van de weilrestrictie Dit was deinhoud van een eerste artikel van de auteur Alhoewel het onderwerp opzich verschillend is van de rest van de thesis zĳn de gebruikte methodengelĳkaardig

vi

Contents

Dankwoord iii

Abstract v

Samenvatting vi

Contents vii

Notations and conventions ix

Introduction 1

I1 SK1 of a central simple algebra 4

I2 Wangrsquos theorem and Suslinrsquos conjecture 7

I3 Reductions of the problem 9

I4 Overview of the thesis 11

1 Cycle modules and invariants 13

11 Cohomology groups 14

12 Cycle modules 18

13 Invariants a la Merkurjev 23

14 Invariants of SK1 24

2 Lifting and specialising invariants 29

21 Moderate case 29

vii

CONTENTS

22 Wild case 40

23 General case 56

24 Some remarks 58

3 Comparing invariants 63

31 Invariants for biquaternion algebras 63

32 Kahnrsquos invariant 76

Conclusion 89

A Verification of cycle module rules 93

B Elementary obstruction and Weil restriction 101

B1 Introduction 101

B2 Product varieties 104

B3 Weil restriction 109

Bibliography 115

Glossary 121

Index 125

viii

Notations and conventions

Throughout this thesis we use some standard notations and conventions ofthe field of research (unless explicitly otherwise stated) The reader cancome back to these pages when he wants to recall them We also refer to theglossary for a comprehensive list of the notations in use

bull For a field k we denote by k an algebraic closure and by ks sub ka separable closure Furthermore Γk = Gal(ksk) is the absoluteGalois group cd(k) (resp cdp(k)) is the (p-)cohomological dimension(for p a prime) k((t1)) ((tn)) is the n-fold iterated Laurent seriesfield over k in variables t1 tn and Gm is the multiplicative groupSpec(Z[t tminus1])

bull We use standard notations for the following categories the categorySets of sets the category k-fields of field extensions of a field k thecategory Groups of groups and the category Ab of abelian groups

bull We always suppose k-algebras to be associative to have a multiplica-tive identity 1 and to be finite dimensional over k

bull IfA is a k-algebra and ifK is a field extension of k we denote byAK theK-algebra AotimeskK obtained from A by base extension to K Likewiseif X is a k-scheme XK is the K-scheme XtimeskK(= XtimesSpec(k)Spec(K))obtained from X by base extension to K Furthermore X(K) is theset of K-rational points of X

bull A prime factorisation pe11 middot middot perr of a (positive) integer m is alwayssupposed to be primitive (ie m = pe11 middot middot perr with pi primes ei ge 1integers for 1 le i le n and pi 6= pj for any 1 le i lt j le r)

bull For an integer m gt 0 invertible1 in a field k we denote by microm theΓk-module of m-th roots of unity in ks If one forgets about the Γk-action microm is isomorphic to ZmZ Unless k contains a primitive m-th

1We use this expression for brevity it actually comes down to requiring gcd(m p) = 1if char(k) = p gt 0 and m gt 0 arbitrary if char(k) = 0

ix

NOTATIONS AND CONVENTIONS

root of unity (so in particular if m = 1 2) the isomorphism does notcontinue to an isomorphism of Γk-modules (if ZmZ is equipped withthe trivial Γk-action) We write microm(k) for the m-th roots of unityinside k itself (so that it can be viewed as the rational points of theappropriate sheaf) We also use the latter notation for arbitrary rings

bull The cohomology groups used are Galois (or etale) cohomology groups

bull A discrete valuation v on a field F is supposed to be non-trivial andof rank 1 We denote the valuation ring by Ov and the residue fieldby κ(v) The maximal unramified extension of F with respect to vis denoted as Fnr If x isin Ov its residue in κ(v) is x This notation isalso used for other objects with natural residues (induced by a discretevaluation on a field) We also distinguish two different cases of discretevaluation fields depending on the characteristics the equicharacteristiccase if char(F ) = char(κ(v)) and the mixed characteristic case ifchar(F ) = 0 and char(κ(v)) = p

bull For any group G and integer m we denote by mG the m-torsion pointsof G

bull For any scheme X of finite dimension and integer i ge 0 we denoteby X(i) the points of codimension i of X An algebraic k-group is asmooth affine group scheme over k of finite type

As for references the author tries to include the exact reference to thetheorem in use unless the cited article lacks numbered theorems In thelatter case no further details probably means the article has one maintheorem which is the one referred to

x

Introduction

ldquoUne conjecture est drsquoautant plus utile qursquoelle estplus precise et de ce fait testable sur des exemplesrdquo

mdash Jean-Pierre Serre

In this thesis we are interested in central simple algebras over a field k Thesek-algebras have centre equal to k = (k1) (central) and have no two-sidedideals except for the trivial ones 0 and the algebra itself (simple) Unlessotherwise stated in this introduction we always consider A to be a centralsimple k-algebra

Very important examples of central simple algebras are central divisionalgebras these are central k-algebras containing a multiplicative inverse forall of its non-zero elements More generally every matrix algebra Mn(D)over a central division algebra is a central simple algebra The followingalternative definition shows that these are actually all examples of centralsimple algebras

Theorem I1 (see eg [GS sectsect21 - 22])Let A be an algebra over a field k then the following conditions areequivalent

(i) A is a central simple k-algebra

(ii) there exists a central division algebra D over k such that A sim=Mr(D) as k-algebras (r some integer)

(iii) there exists a field extension Kk such that AK sim= Mn(K) asK-algebras (n some integer)

Remark I2 ndash The equivalence (i)harr(ii) is commonly known as Wedderburnrsquostheorem as it was proved by Wedderburn in 1908 [Wed] Even morethe central division algebra is uniquely determined up to isomorphism

1

INTRODUCTION

Wedderburnrsquos theorem is used to prove the equivalence (i)harr(iii) A fieldK satisfying condition (iii) is called a splitting field of A It can be provedthat k ks and even a finite extension of k suffice The choice of this finitesplitting field depends (of course) heavily on A (and not just on k)

This theorem gives rise to the definition of the Brauer group Br(k) of afield k Two central simple k-algebras A and B are said to be Brauer-equivalent (A simBr B) if there exist two positive integers nm such thatAotimesk Mn(k) sim= B otimesk Mm(k) as k-algebras We denote the equivalence classof A by [A] the Brauer class of A For two central simple k-algebras A andB the tensor product AotimeskB is again a central simple k-algebra by TheoremI1 (iii) It can be proved that this endows Br(k) with the structure of anabelian group The identity element is the class of k (or Mn(k)) The inverseof A is the opposite algebra

Aop = aop isin A | a isin A

with addition and (scalar) multiplication defined by

aop + bop = (a+ b)op λaop = (λa)op and aop middot bop = (b middot a)op

for a b isin A and λ isin k See [GS Prop 248] for a proof By TheoremI1 (ii) every Brauer class contains a central division algebra unique up toisomorphism Another very well known description of the Brauer group isby Galois cohomology Br(k) sim= H2(k ktimess ) (ibid sect44)

For a field extension K of k there exists a morphism Br(k)rarr Br(K) sendingthe class [A] to the class [AK ] Note that because of Theorem I1 (iii) it isclear that the base extension of a central simple algebra is still a centralsimple algebra By Br(Kk) we denote ker(Br(k) rarr Br(K)) ie thesubgroup of Br(k) consisting of the classes of central simple algebras whichsplit after base extension to K So eg Br(ksk) = Br(k) For more factsand trivia about central simple algebras we refer to some standard works as[Dra Ch 1 amp 2] [GS Ch 2 amp 4] [KMRT sect1] and others

In particular all of this gives rise to the definition of three integers attachedto a central simple algebra

2

INTRODUCTION

Definition I3Let A be a central simple algebra over a field k Define the followingintegers

bull the degree of A as deg(A) =radic

dimk(A)

bull the period of A as the order perk(A) of [A] in Br(k) and

bull the index of A as indk(A) =radic

dimk(D) where D is the uniquecentral division k-algebra Brauer-equivalent to A

Remark I4 ndash The fact that dimk(A) is a square follows by Theorem I1(iii) since dimK(AK) = dimk(A) for any field extension K of k The factthat the order of [A] isin Br(k) is finite follows by the isomorphism Br(k) =H2(k ktimess ) and calculations with Galois cohomology using restrictions andcorestrictions (see eg [GS sect44]) In the notation for period and indexwe deliberately used a subscript for the base field as it is not invariantunder base extension The degree however is fixed under extensions of thebase field

It can also be proved that perk(A) divides indk(A) and that they have thesame prime factors (ibid Prop 4513) A whole field of study is dedicatedto determining the possible values of indk(A)perk(A) This problem iscommonly known as the period-index problem For sure the index and periodare not always equal (see eg Example I10) See (ibid Rem 455) forsome comments on this problem We do not go into details on this subjectwe rather study other constructions related to central simple algebras

Example I5 ndash Let us first give some important examples of central simplealgebras

(i) Cyclic algebrasSupposeK is a cyclic field extension of k of degree n (ie Gal(Kk) sim=ZnZ) Let σ be any generator of Gal(Kk) and a isin ktimes We definethe cyclic algebra (Kk σ a) as the k-algebra generated by K anda variable x satisfying the relations xn = a and xc = σ(c)x forany c isin K So we can write this cyclic algebra as oplusnminus1

i=0 Kxi with

multiplication defined as above Also deg (Kk σ a) = n and Kis a splitting field of (Kk σ a) (see [GS sect25] where also anotherdescription of cyclic algebras is given)

3

INTRODUCTION

(ii) Symbol algebrasLet n isin ktimes be an integer and suppose k contains an n-th primitiveroot of unity ξn For any a b isin ktimes we define the symbol algebra(a b)n as the central simple k-algebra generated by variables x andy satisfying xn = a yn = b and xy = ξnyx Clearly deg (a b)n = nNote that this algebra depends on the choice of the primitive root ofunity [Dra sect11 Lem 6]2

(iii) p-algebrasIf k is a field of char(k) = p gt 0 then for a isin k and b isin ktimes wedefine the p-algebra [a b)p as the central simple k-algebra generatedby u and v satisfying up minus u = a vp = b and uv = v(u + 1) Alsodeg [a b)p = p These p-algebras play the role of symbol algebras withdegree equal to char(k) = p gt 0 as in this case k lacks (non-trivial)primitive roots of unity

Both symbol division algebras and division p-algebras are a special caseof cyclic algebras [GS Cor 255 amp Rem 256] If k contains an n-thprimitive root of unity and if K = k( n

radica) for a isin ktimes then any symbol

division algebra (a b)n is k-isomorphic to (Kk σ b) for a well chosen σIn case n = p = char(k) and if K is the cyclic Galois extension defined byxpminusxminusa then any division p-algebra [a b)p is k-isomorphic to (Kk σ b)for a well chosen σAlgebras of the form (a b)2 or [a b)2 are called quaternion algebras Thename comes from the fact that Hamiltonian quaternions are retrieved fork = R and a b = minus1 As usual for quaternion algebras we drop thesubscript 2 If we want to treat both symbol and p-algebras we looselyspeak about algebras of the form [(a b)p as Draxl does in [Dra sect14] Wetrust on the readerrsquos good-will to make the proper assumptions on a b andthe characteristic of the base field k

I1 SK1 of a central simple algebra

Our interest in this thesis goes to the functor SK1(A) To define it we needthe notion of the reduced norm of A We recall the notions without giving(rigorous) proofs see eg [Dra sect22] and [GS sectsect26 amp 28] for details

2One could incorporate the chosen root of unity in the notation In this text we do notexplicitly work with symbol algebras defined with different primitive roots of unity Hencewe use this more elementary notation which actually does not show the true colours ofthe algebra

4

SK1 OF A CENTRAL SIMPLE ALGEBRA

Definition I6Let A be a central simple k-algebra A splitting field K of A defines amultiplicative map called the reduced norm NrdAk as composition of

Aidotimes1rarr Aotimesk K sim= Mn(K) detrarr K

which can be proved to be independent of the splitting field and tohave values in k Even more the elements in A with reduced norm inktimes are exactly the units of A

Using a splitting field K of A the embedding id otimes 1 A rarr A otimesk K andthe corresponding terms for matrices one can also define a reduced traceTrdAk A rarr k and a reduced characteristic polynomial Prdak(X) isin k[X]of an element a isin A Even more for any a isin A the reduced norm NrdAk(a)and trace TrdAk(a) can be expressed as coefficients of Prdak(X)

Prdak(X) = XnminusTrdAk(a)Xnminus1+bnminus2Xnminus2+ +b1X+(minus1)nNrdAk(a)

(I1)This is a generalisation of the expression of the norm NKk(x) and traceTrKk(x) of an element x of a finite extension K of k as coefficients of itsminimal polynomial [Lan Ch VI Thm 51]

The original construction of SK1(A) uses K1(A) the first K-group of A orWhitehead group of A Let R be any ring then we can consider the towerof embeddings

GL1(R) sub GL2(R) sub sub GLn(R) sub GLn+1(R) sub

where the injections are given by identifying any A isin GLn(R) with thematrix (

A 00 1

)isin GLn+1(R)

Then define

GLinfin(R) =⋃ngt0

GLn(R) and K1(R) = GLinfin(R)[GLinfin(R)GLinfin(R)]

For any positive integer n there is an isomorphism K1(R) sim= K1(Mn(R))called the Morita isomorphism This isomorphism is induced by the map

Mm(R)rarrMnm(R) A 7rarr(A 00 Inmminusm

)

5

INTRODUCTION

where m is any positive integer So using Wedderburnrsquos theorem we seethat for our central simple k-algebra A the isomorphism class of K1(A) onlydepends on the Brauer class of A

Furthermore it is also possible to define a reduced norm map NrdK1(A) K1(A)rarr ktimes using the composition

GLn(A) sim= GL1(Mn(A))NrdMn(A)minusminusrarr ktimes

This brings us to the definition of SK1(A)

Definition I7For any central simple k-algebra A the reduced Whitehead group is

SK1(A) = ker(NrdK1(A))

Suppose that D is the unique central division algebra Brauer-equivalent toA (so A sim= Mn(D) for an integer n) Then note that the isomorphismK1(A) sim= K1(D) from above also leads to an isomorphism SK1(A) sim= SK1(D)what we call the Morita invariance of SK1 (ie SK1(A) only depends on theBrauer class of A) Also by definition the composition

Atimes rarr K1(A)NrdK1(A)minusminusrarr ktimes

coincides with the reduced norm map Atimes rarr ktimes Denote

SL1(A) = a isin A |NrdAk(a) = 1

the special linear group of A If A = Mn(k) then SL1(A) coincides withSLn(k) We clearly have an injection

SL1(A)[Atimes Atimes] rarr SK1(A)

which is known to be bĳective for central division algebras The morphism

SL1(D)rarr SL1(A) B rarr(B 00 Inminus1

)

6

WANGrsquoS THEOREM AND SUSLINrsquoS CONJECTURE

induces a commutative diagram

SL1(D)[Dtimes Dtimes]sim=

SK1(D)

sim=

SL1(A)[Atimes Atimes] SK1(A)

giving us the following property

Proposition I8For any central simple k-algebra A there is an isomorphism

SK1(A) sim= SL1(A)[Atimes Atimes]

Remark I9 ndash Since NrdAk is multiplicative it is straightforward to seethat the commutators of Atimes are part of SL1(A) so that this quotient doesmake sense

In the following we use this description when we speak about SK1(A)

I2 Wangrsquos theorem and Suslinrsquos conjecture

In 1943 Tannaka and Artin independently asked whether SK1(A) is alwaystrivial or not ie whether any element of SL1(A) is always a commutatorin Atimes or not [NM Wan] In 1950 Wang proved the triviality of SK1(A) ifindk(A) is square-free [Wan] During more than 30 years one tried to solvethe Tannaka-Artin problem by proving the triviality of SK1 in full generality

Fortunately for the sake of interest of this thesis in 1976 Platonov came upwith examples of non-trivial SK1 using valuation theory [Pla] Let us recallquickly the most important of his examples

Example I10 (ibid Thms 47 amp 59) ndash Let k be local field (eg Fp((x)) orQp for a prime p) and let K1 K2 be two cyclic extensions of degree n over kwhich are linearly disjoint and set K = K1otimeskK2 = K1 middotK2 (as of [Bou A

7

INTRODUCTION

V13]) Let σ1 (resp σ2) be a generator of Gal(K1k) (resp Gal(K2k))Now let F = k((t1))((t2)) F1 = K1((t1))((t2)) and F2 = K2((t1))((t2))Then Platonov proves that

A = (F1F σ1 t1)otimesF (F2F σ2 t2)

is a division F -algebra and SK1(A) sim= Zn To prove the latter he usesan isomorphism

SK1(A) sim= Br(Kk)(Br(K1k)Br(K2k)) (I2)

Platonov also gives central simple k-algebras A with SK1(A) = 0 butSK1(AK) 6= 0 where K is a particular field extension of k (ibid Corr 63)Furthermore he also proves that for any positive integers i p one can findfields k and central simple k-algebras A such that SK1(A) sim= (ZpZ)i (ibidThm 62) The first encounter of these situations was striking

These examples inspired Suslin to refine the Tannaka-Artin problem to aconjecture he stated in 1991 For this conjecture he rather uses a functorialversion of SK1

Definition I11For a field k and a central simple k-algebra A define

SK1(A) k-fieldsrarr Ab K 7rarr SK1(A)(K) = SK1(AK)

Conjecture I12 (Suslin [Sus Intro])Let A be a central simple k-algebra then SK1(A) = 0 if and only ifindk(A) is square-free

Remark I13 ndash By SK1(A) = 0 we mean of course that SK1(A)(K) = 0 forany field extension K of k By Wangrsquos theorem it is turned into a necessitystatement as ind(AK) | ind(A) for any field extension K [Pie Prop 134]Furthermore by Wangrsquos theorem it also follows that SK1(A)(K) = 0 if Kis a splitting field of k Also if K is a finite field extension of k of degreeprime to indk(A) then SK1(A)(k) rarr SK1(A)(K) is an injection [Drasect23 Lem 3]

8

REDUCTIONS OF THE PROBLEM

Due to Proposition I8 this problem is related to the linear algebraic k-group

SL1(A) = Spec(k[X1 Xn2 ]

I)

whereX1 Xn2 are variables parametrising the coefficients of the elementsof A with respect to a k-vector space basis and I is the ideal generated bythe polynomial in the Xi defined by requiring that the reduced norm equals1 Of course SL1(A)(K) = SL1(Aotimesk K)

Suslinrsquos conjecture translates into a conjecture whether or not indk(A) issquare-free when SL1(A) is a stably k-rational variety (ie SL1(A) timesk An

k

is k-birational to an affine space for an integer n) In this setting Suslinrsquosconjecture is a special case of the Kneser-Tits problem on R-equivalence See[Gil2 sect22] for further details

I3 Reductions of the problem

There are some (well-known) reductions of Suslinrsquos Conjecture First of allone can restrict to checking Suslinrsquos conjecture for central division algebrasas the isomorphism class of SK1(A) depends only on the Brauer class ofA (and as A is Brauer-equivalent to a unique central division k-algebra byWedderburnrsquos theorem)

Furthermore suppose D is a central division k-algebra of deg(D) =indk(D) = n and let n = pe11 middot middot perr be a prime factorisation of n ThenBrauerrsquos decomposition theorem [GS Prop 4516] gives central divisionk-algebras Di for i = 1 r such that indk(Di) = peii and such that

D sim= D1 otimes otimesDr (I3)

This decomposition induces a decomposition of SK1(D) [GS Ch 4 Ex 9(a)]

SK1(D) sim= SK1(D1)oplus oplus SK1(Dr) (I4)So in order to verify Suslinrsquos conjecture one can even restrict to centraldivision algebras of primary degrees

We can even reduce further and restrict to central division algebras of indexp2 for a prime p Indeed using the index reduction formula [SVdB Thm13] Blanchet gets the following result which justifies this restriction

9

INTRODUCTION

Proposition I14 ([Bla Prop 4])Let A be a central simple k-algebra of indk(A) = n Suppose r |nthen there exists a field extension K of k such that indK(AK) = r

Remark I15 ndash This proposition would even allow us to restrict to centraldivision algebras of index p2 without using a Brauer decomposition ofthe central division algebra However it would be unfair to withhold theisomorphism (I4) from the readerrsquos knowledge

Rehmann-Tikhonov-Yanchevskiı prove that one can even restrict to checkSuslinrsquos conjecture for cyclic division algebras [RTY Thm 019] whichimmediately follows from the following theorem

Theorem I16 (ibid Thm 014)For any field k there exists a (regular) field extension K such that

(i) any central simple K-algebra is cyclic and(ii) for any central simple k-algebra A indK(AK) = indk(A)

On the other hand Prokopchuk-Tikhonov-Yanchevskiı prove that we canmake a restriction to central simple algebras of the form [(a b)p otimes [(c d)p[PTY] This follows by a theorem similar to the previous one

Theorem I17 (loc cit)Let A be a central division algebra over a field k with indk(A) = p2Then there exists a field extension K of k and a b c d isin K such thatindK(AK) = indk(A) and

AK simBr [(a b)p otimesK [(c d)p

Remark I18 ndash Note that [PTY] actually only contains an explicit proof ofthe case char(k) 6= p but their methods equally work in the case whenchar(k) = p As main tool the proof uses the index reduction formula[SVdB Thm 13] In the case char(k) 6= p and indk(A) = p2 they alsoexplain why (to prove Suslinrsquos conjecture) they can assume k to have a

10

OVERVIEW OF THE THESIS

p-th primitive root of unity so that they can surely define symbol algebras(ibid p 2) Let us recall the argument Suppose ξp isin k a primitive p-throot of unity and ξp 6isin k (so in particular p odd) Then [k(ξp) k] le pminus 1as ξp is a root of

sumpminus1i=0 X

i But then SK1(A)(k) rarr SK1(A)(k(ξp)) isinjective (Remark I13) so that it suffices to prove SK1(Ak(ξp)) 6= 0

So all in the end we have the following restriction

Proposition I19Suslinrsquos conjecture holds if and only if SK1(A) 6= 0 for all cyclicdivision algebras A of the form [(a b)p otimes [(c d)p

Merkurjev proves in two different ways that Suslinrsquos conjecture holds forcentral simple algebras of 2-primary index ie he proves the followingtheorem

Theorem I20 ([Mer1 Mer4])If A is a central simple k-algebra with 4 | indk(A) then SK1(A) 6= 0

He proves this using the reductions above Actually he does not needTheorem I16 or I17 for this reduction as it is known that any central simplealgebra of degree 4 and period 1 or 2 is a product of two quaternion algebraswhat is called a biquaternion algebra [Alb1 p369]

I4 Overview of the thesis

In this thesis we study cohomological invariants of SK1(A) It is the hopethat these invariants help to describe and understand SK1(A) in a better wayMost of the invariants found in the literature are only defined if indk(A) isinktimes

In Chapter 1 we recall the notion of invariants and cycle modules We alsogive an overview of the known invariants of SK1(A) and explain why theseinvariants can explain the examples of non-trivial SK1

11

INTRODUCTION

In Chapter 2 we generalise these invariants to any central simple algebraThis is done by a lift from positive characteristic to characteristic zero Thelift is performed in a generic way ie it does not depend on the definition ofthe invariants It rather uses the existence so that given any invariant wecan generalise it to any central simple algebra

In Chapter 3 we compare the invariants into play This allows us toprove that an invariant introduced by Kahn is non-trivial for Platonovrsquosexamples knowing that another invariant is non-trivial in the same case Forbiquaternion algebras we compare an invariant of Knus-Merkurjev-Rost-Tignol that already exists in characteristic 2 to an invariant obtained inChapter 2 We also generalise a formula of Merkurjev for the value of thecentre of a biquaternion algebra to the tensor product of two symbol algebras

12

Cycle modules and invariants

Chapter 1

ldquoScience is a wonderful thing if one doesnot have to earn onersquos living at itrdquo

mdash Albert Einstein

In this chapter we recall some notions needed in the rest of the thesis Fora field k and two functors

A k-fieldsrarr Sets and M k-fieldsrarr Sets

a natural transformation of functors ϕ Ararr M is called an invariant of Awith values in M So for every field extension K of k there exists a mapϕK A(K) rarr M(K) which is functorial to other field extensions ie if K primeis a field extension of K we have a commutative diagram

A(K)

ϕK M(K)

A(K prime)ϕKprime

M(K prime)

where the vertical maps are coming from the functors A and M In ourresults we do not work with the lsquovaguersquo category of sets Our functors havevalues in the more concrete category of groups (or abelian groups) So let

A k-fieldsrarr Groups and M k-fieldsrarr Groups

be two group functors By an invariant ϕ of A in M we mean a naturaltransformation of functors as before but we also require for every fieldextension K of k the morphism ϕK to be a group morphism If M evenhas values in Ab all invariants of A in M form an abelian group Inv(AM)When M is (some kind of) a cohomology group we say ϕ is a cohomologicalinvariant of A

13

CYCLE MODULES AND INVARIANTS

Merkurjev introduces a nice framework to work with [Mer3 sect2] He ratherconsiders M as (a component of) a cycle module and then gives a practicalalternative description of invariants when A is an algebraic group In thischapter we recall the formalism of Rostrsquos cycle modules [Ros2 sect12] andMerkurjevrsquos description Using this setting we recall the various invariantsof SK1 found in the literature We first give some introductory examples ofcohomology groups we use later on These lead us to the formal definitionof a cycle module

11 Cohomology groups

In this section we take F to be a field and m gt 0 an integer invertible in F

(a) Definition ndash Let microotimesim be the i-th tensor product of microm as ZmZ-module(i ge 0) Then consider the following Galois cohomology groups

Definition 11For any field F and integers im ge 0 with m isin Ftimes we define

H im(F ) = H i(F microotimesim (minus1)) with microotimesim (minus1) = HomΓF (microm microotimesim )

a Tate twist For i lt 0 we set H im(F ) = 0

Clearly microotimesi+1m (minus1) = microotimesim for all i ge 0 and so H i+1

m (F ) = H i+1(F microotimesim )1The short exact Kummer sequence

1rarr microm rarr Ftimessmrarr Ftimess rarr 1 (11)

then implies the well-known cohomological interpretation of the part of m-torsion of the Brauer group of F

mBr(F ) sim= H2m(F ) (12)

1We try to use as much as possible the superscript i+ 1 in stead of i to keep up withtradition (which rather defines Him(F ) as Hi(F microotimesim )) and to stay in conformity with thewild case (sect221) where it is clearly more natural to use this superscript In any caseany appearance of Him(F ) is to be interpreted as the Galois cohomology group definedover here (and not as Hi(F microotimesim ) - unless microm sub F )

14

COHOMOLOGY GROUPS

(b) Kn(F )-module structure ndash Consider Milnorrsquos K-groups2 Kn(F ) for aninteger n ge 0 Recall that

Kn(F ) = Ftimes otimesZ otimesZ Ftimes︸︷︷︸

n times

J

where J is the subgroup generated by the symbols of the form x1 otimes otimes xnsuch that xi + xj = 1 for some 1 le i lt j le n The primitive symbolsx1 otimes otimes xn are denoted as x1 xn Kummerrsquos short exact sequence(11) induces an isomorphism h1

mF as composition K1(F )mK1(F ) =Ftimes(Ftimes)m sim= H1(F microm) We retrieve the Galois symbol using the cup-product

hnmF Kn(F )mKn(F ) rarr Hn(F microotimesnm ) defined by

x1 xn 7rarr h1mF (x1) cup cup h1

mF (xn) (13)

As a matter of fact hnmF is an isomorphism (Bloch-Kato conjecture -theorem of Voevodsky-Rost-Weibel [BK Voe Ros3 Wei2]) We call thisthe Bloch-Kato isomorphism By taking the cup product with this Galoissymbol we can define a Kn(F )-module structure on (H i+1

m (F ))ige0

Kn(F )timesH i+1m (F )rarr Hn+i+1

m (F ) (a b) 7rarr hnmF (a) cup b

We denote this scalar product by a middot b = hnmF (a) cup b for a isin Kn(F ) a itsclass in Kn(F )mKn(F ) and b isin H i+1

m (F )

Remark 12 ndash Suppose F contains an m-th primitive root of unity so thatH im(F ) sim= H i(F microotimesim ) Then under the isomorphism (12) the class of a

symbol F -algebra (a b)m is mapped to h2mF (a b) [GS Prop 471]

(c) Residue maps ndash Suppose F is complete for a discrete valuation v Thevaluation v extends uniquely to a valuation on Fs which in its turn gives riseto a residue morphism ΓF rarr Γκ(v) of absolute Galois groups This inducesfor any integer i ge 0 an injection

ϕi H im(κ(v))rarr H i

m(F )2In the following we mainly use Milnor K-groups To ease notations we do not use

the superscript M of the more common notation KMn (F ) of Milnor K-groups Whenusing Quillen K-groups we use the notation KQn

15


Furthermore if π is a uniformiser with respect to v we have a map for anyi ge 0

ψi H im(κ(v))rarr H i+1

m (F ) a 7rarr h1mF (π) cup ϕi(a)

It can be proved that ϕi+1oplusψi is an isomorphism [GMS Prop 77] Hencethis gives us a morphism parti+1

v H i+1m (F ) rarr H i

m(κ(v)) called a residuemorphism So we have a split exact sequence

0rarr H i+1m (κ(v))rarr H i+1

m (F ) parti+1vrarr H i

m(κ(v))rarr 0 (14)

Suppose F is endowed with a discrete valuation v but is not complete forthe topology defined by v Then we still have a residue Indeed take F tobe the completion of F with respect to v which also has residue field κ(v)The residue is then defined as composition

parti+1v H i+1

m (F )rarr H i+1m (F )rarr H i

m(κ(v))

where obviously the last morphism is the residue for the complete field F

We refer to [Ser1 Ch II amp III] for the assertions on valuation theory

Remark 13 ndash These notions can be extended to other Galois cohomologygroups of fields with a discrete valuation There exists for example ingeneral a short exact sequence as (14) for the Galois cohomology groupsH i(F microotimesi+jn ) for any integer j They are defined in a similar way See[GMS sect7] for more information on these residue maps

(d) Relative version ndash We define a relative version of the Galois cohomologygroups H i+1

m (F )

Definition 14Let A be a central simple F -algebra with indF (A) = n isin Ftimes and withBrauer class [A] isin nBr(F ) sim= H2

n(F ) Then define for any integersi ge 1 and r

H i+1nAotimesr(F ) = H i+1

n (F )(H iminus1(F microotimesiminus1

n ) cup r[A])

Remark 15 ndash Note that if r equiv 0 mod perk(A) we find H i+1nAotimesr(F ) =

H i+1n (F ) as r[A] = 0 in Br(F ) We could hence restrict the possible values

16

COHOMOLOGY GROUPS

of r but for ease of notation we just take r any integer Allowing thecase r equiv 0 mod perk(A) to happen we cover both the relative and theabsolute version with the relative one

Remark 16 ndash Remark also that by the Bloch-Kato isomorphism and theKn(F )-module-structure we can give an equivalent definition

H i+1nAotimesr(F ) = H i+1(F microotimesin ) (Kiminus1(F ) middot r[A]) (15)

If F is complete for a discrete valuation v we can extend the residues ofH i+1n (F ) to relative residues We suppose A to be a central simple κ(v)-

algebra with indκ(v)(A) isin κ(v)times and indκ(v)(A) = n isin Ftimes

Under the injection nBr(κ(v)) rarr nBr(F ) from (14) the class of A mapsto the class of a central simple K-algebra BK called a lifted central simplealgebra In sect212 (a) we give more comments on this construction3 Thedescription in terms of explicit cocycles [GMS Ex 712] guarantees that

parti+1v (H iminus1(F microotimesiminus1

n ) cup r[BK ]) sub H iminus2(κ(v) microotimesiminus2n ) cup r[A]

Then we get a commutative diagram (for i ge 2)

0 H iminus1(κ(v) microotimesiminus1n )

cup r[A]

H iminus1(F microotimesiminus1n )

cup r[BK ]

H iminus2(κ(v) microotimesiminus2n )

cup r[A]

0

0 H i+1(κ(v) microotimesin ) H i+1(F microotimesin ) H i(κ(v) microotimesiminus1n ) 0

As the short exact sequences are split the snake lemma allows us to constructthe following short exact sequence

0rarr H i+1nAotimesr(κ(v))rarr H i+1

nBotimesrK(F )

parti+1vAotimesrrarr H i

nAotimesr(κ(v))rarr 0 (16)

The map partvAotimesr is the relative residue Furthermore as (14) is split (16)is so too

3We use the subscript K in BK as this is in conformity with the discussion in sect212(a) where we pass via Azumaya algebras

17


12 Cycle modules

The common properties of H i+1n (F ) and Milnor K-groups have inspired Rost

to define a formal structure respecting these homological properties [Ros2sectsect12] Let us briefly recall this formalism of cycle modules

(a) Definition of a cycle module ndash For a discrete valuation ring R letR-fields be the category of R-fields these are R-algebras which are fieldsso field extensions of Frac(R) or κ(v) the residue field Let us literally recallthe definition of a cycle module

Definition 17 (loc cit)For any discrete valuation ring R a cycle module M with base Rconsists of an object function

R-fieldsrarr Ab

equipped with a grading M = (Mj)jge0 and data D1-D4 satisfyingcompatibility (R1a-R3e) and geometrical rules (FD and C) as below(EF objects in R-fields and ϕ a morphism in R-fields)

D1 Any ϕ F rarr E induces ϕlowast M(F )rarrM(E) of degree 0

D2 Any finite ϕ F rarr E induces ϕlowast M(E)rarrM(F ) of degree 0

D3 For all F the group M(F ) has a Kn(F )-module structure suchthat Kn(F ) middotMm(F ) subMn+m(F ) (nm ge 0 integers)

D4 If F is an R-field with a discrete valuation v such that theresidue field κ(v) is also a R-field then there exists a residuepartv M(F )rarrM(κ(v)) of degree minus1

Remark 18 ndash Note that for obtaining his goals Rost puts more restrictionson his base R but he comments it is allowed to moderate these (ibid sect1p 328) Also in loose notation Mj for j lt 0 equals the trivial group Amorphism from a graded abelian group (Aj)jge0 to a graded abelian group(Bj)jge0 is a collection of group morphism ϕj Aj rarr Bj+d for a fixedinteger d the degree of the morphism

18

CYCLE MODULES

Let us now give the rules mentioned in the definition In all of this letEFG be arbitrary R-fields and suppose that any map between fields is amorphism in R-fields For a discrete valuation on an R-field we assume thatthe residue field is also an R-field

R1a Any ϕ F rarr Eψ E rarr G satisfy (ψ ϕ)lowast = ψlowast ϕlowastR1b Any finite ϕ F rarr Eψ E rarr G satisfy (ψ ϕ)lowast = ϕlowast ψlowastR1c Take ϕ F rarr Eψ F rarr G with ϕ finite and S = GotimesF E For any

p isin Spec(S) let ϕp G rarr Sp ψp E rarr Sp be the natural mapsand let lp be the length of the localised ring Sp Then

ψlowast ϕlowast =sump

lp middot (ϕp)lowast (ψp)lowast

R2 For ϕ F rarr E x isin KlowastF y isin KlowastE ρ isin M(F ) micro isin M(E) one has(with ϕ finite in R2b and R2c)

R2a ϕlowast(x middot ρ) = ϕlowast(x) middot ϕlowast(ρ)R2b ϕlowast(ϕlowast(x) middot micro) = x middot ϕlowast(micro) andR2c ϕlowast(y middot ϕlowast(ρ)) = ϕlowast(y) middot ρ

R3a Let ϕ E rarr F and let v be a discrete valuation on F which restrictsto a non-trivial valuation w on E with ramification index e Letϕ κ(w)rarr κ(v) be the induced map Then

partv ϕlowast = e middot ϕlowast partw

R3b Let ϕ F rarr E be finite and v a discrete valuation on F For anyextension w of v on E let ϕw κ(v) rarr κ(w) be the induced mapThen

partv ϕlowast =sumw|v

ϕlowastw partw

R3c Let ϕ E rarr F and let v be a discrete valuation on F which is trivialon E Then

partv ϕlowast = 0

R3d Let ϕ E rarr F let v be a valuation on F which is trivial on E letϕ E rarr κ(v) be the induced map and let π be an uniformiser of vDefine furthermore sπv M(F ) rarr M(κ(v)) by sπv (ρ) = partv(minusπ middot ρ)then

sπv ϕlowast = ϕlowast

19


R3e Let v be a discrete valuation on F u a v-unit and ρ isinM(F ) then

partv(minusu middot ρ) = minusu middot partv(ρ)

For any R-scheme X we denote M(x) = M(κ(x)) for x isin X with residuefield κ(x) If X is irreducible we denote its generic point by ξ If X isnormal any x isin X (1) induces partx M(ξ) rarr M(x) For x y isin X we definepartxy One sets partxy = 0 if Z = x and y 6isin Z(1) Otherwise let Z rarr Z be thenormalisation and

partxy =sumz|y

ϕlowastz partz

where z runs through all points of Z lying above y and where ϕz is the finitemorphism κ(y)rarr κ(z)

FD (Finite support of divisors) Let X be a normal R-scheme and ρ isinM(ξ)Then partx(ρ) = 0 for all but finitely many x isin X (1)

C (Closedness) Let X be an integral R-scheme local of dimension 2 andlet x0 be its closed point Then

0 =sum

xisinX (1)

partxx0 partξx M(ξ)rarrM(x0)

(b) The base and coexistence of two cycle modules ndash In the classical case acycle module has as base a field (with definition as above replacing R by afield) In this thesis however we use cycle modules with a complete discretevaluation ring R as base Let K be the fraction field of R and k its residuefield A cycle module M with base R attaches then to any field extensionL of K a graded group M(L) and likewise to any field extension L of k agraded group M(L)

Remark that one can hence restrict a cycle module with base R to a cyclemodule with base K and to one with base k by restricting either to fieldextensions of K or to field extensions of k A cycle module with base R istherefore the coexistence of two cycle modules with as base a field with anadditional link given by the data D1-D4 (in the mixed characteristic case onlyD4) So we use the notion of a cycle module with base R on the one handto ease notation and on the other hand to work in a more general settingNevertheless one could reformulate the arguments using two different cyclemodules and using the link given by the data as an additional link of thetwo cycle modules

20

CYCLE MODULES

(c) Gersten complex ndash Take as above R any complete discrete valuationring with fraction field K and residue field k Let F be an R-field X anF -variety and M a cycle module The existence of residues (D4) and therules of cycle modules induce a cycle complex called the Gersten complexClowast(XMj) [Ros2 sect33] (i j ge 0)

rarroplus

xisinX(iminus1)

Mjminusi+1(F (x)) partiminus1rarr

oplusxisinX(i)

Mjminusi(F (x)) partirarr

oplusxisinX(i+1)

Mjminusiminus1(F (x))rarr

where F (x) is the residue field of x a point of codimension i The mapparti is the sum of the residues induced by the valuations associated with thecodimension 1 points of X(i) The homology of this complex on spot i isdenoted Ai(XMj)

(d) Privileged examples ndash Let us link these cycle modules to the previoussection of Galois cohomology groups Let R be a complete discrete valuationring with fraction field K and residue field k let A be a central simple k-algebra of indk(A) = n such that n isin Ktimes and n isin ktimes and let BK be a liftedcentral simple K-algebra Then the functors

Hlowastm = (Him)ige0 R-fieldsrarr Ab F 7rarr(H im(F )

)ige0 and

HlowastnBotimesr = (HinBotimesr)ige2 R-fieldsrarr Ab F 7rarr(H inBotimesr(F )

)ige2

are cycle modules where r is any integer and H inBotimesr(F ) is to be interpreted

in the appropriate way For a field extension F of k it is H inAotimesr(F ) For a

field extension F of K it is rather H inBotimesrF

(F ) with BF = BK otimesK F If werestrict HlowastnBotimesr to field extensions of k (resp K) as in sect12 (b) we write itas HlowastnAotimesr (resp Hlowast

nBotimesrK)

The verification of the rules R1a-R3e FD and C for Hlowastm in the equichar-acteristic case was done by Rost (ibid Rem 111) The case of mixedcharacteristics follows analogously This also induces HlowastnBotimesr to be a cyclemodule as the data and rules of Hlowastm behave well under taking the quotientsinto play (see eg (16)) For R-fields endowed with a valuation but notcomplete the residue for HlowastnBotimesr is retrieved by passing via a completion (asin sect11 (c))

21


Other examples of cycle modules with as base a discrete valuation ring R (orpossibly just a field) are Milnorrsquos K-groups (Ki)ige0 Datum D1 is definedin the obvious way Let E be a finite field extension of an R-field F thendatum D2 is induced by the norm NEF applied to the primitive symbols[BT Ch I sect5] Datum D3 is defined by the multiplicative structure of theK-groups

Kn(F )timesKm(F ) 7rarr Kn+m(F ) defined by

(x1 xn y1 ym) 7rarr (x1 xn y1 ym)

Now let F be an R-field with a discrete valuation v then the residueKn(F )rarr Knminus1(κ(v)) ndash datum D4 ndash is defined by

π x2 xn 7rarr x2 xn

x1 x2 xn 7rarr 0

with x1 xn isin Otimesv and π an uniformiser of F [Mil5 Lem 21]

Furthermore if r gt is an integer then (Kir)ige0 also forms a cycle modulewith base R as the definitions above go through If r is prime to thecharacteristic of the residue field of R (and hence also to the characteristicof the fraction field of R) we have a short exact sequence similar to (14)Indeed in that case for any R-field F complete for a discrete valuation vthere is a short exact sequence for any integer i ge 0 (ibid Lem 26)

0rarr Ki+1(κ(v))r irarr Ki+1(F )r parti+1vrarr Ki(κ(v))r rarr 0 (17)

Here parti+1v is of course the residue as above and i is defined by

x0 xi (mod r) 7rarr x0 xi (mod r)

for x0 xi isin Otimesv Note that this sequence is split by the retraction ψ Ki(κ(v))r rarr Ki+1(F )r defined by

x1 xi (mod r) 7rarr π x1 xi (mod r)

where π is still the uniformiser as above Note that by the Bloch-Kato isomorphism this comes down to the short exact sequence for theH i(k microotimesin )rsquos (as in Remark 13) The similar behaviour of both groups wasactually a motivation to believe in the Bloch-Kato conjecture

22

INVARIANTS A LA MERKURJEV

13 Invariants a la Merkurjev

In this section let k be a field and M = (Mj)jge0 a cycle module withbase k and of bounded exponent (ie rM = 0 for some integer r)Merkurjev discovered a interesting deep link between the groups A0(GMj)and invariants of an algebraic k-group G in M of degree j We recall thislink but first we give the notion of the degree of an invariant with values ina cycle module

(a) Invariants with values in cycle modules ndash Suppose G k-fields rarrGroups is a group functor (eg an algebraic group) and consider furthermoreMj (for an integer j ge 0) as group functor k-fieldsrarr Groups An invariantρ of G in M of degree j is an invariant ρ GrarrMj These invariants forman abelian group which we denote by Invj(GM) We can define the sameterminology if M is any functor of graded abelian groups

(b) Merkurjevrsquos link ndash Let G be an algebraic group then Merkurjevconstructs an injective morphism

θ Invj(GM)rarr A0(GMj) ρ 7rarr ρK(ξ) (18)

where K = k(G) and ξ isin G(K) is the generic point of G He provesthat the image is the multiplicative subgroup A0(GMj)mult consisting of themultiplicative elements of A0(GMj) [Mer3 Lem 21 and Thm 23] Theseare the elements x isin A0(GMj) such that

plowast1(x) + plowast2(x) = mlowast(x)

where plowast1 plowast2 and mlowast are the morphisms A0(GMj) rarr A0(G times GMj)

induced by the two projections p1 p2 G timesG rarr G and the multiplicationm GtimesGrarr G

He also proves that A0(GMj)mult sub A0(GMj) where A0(GMj) is thereduced subgroup of A0(GMj) (ibid Lem 19) The reduced subgroup isthe kernel of the morphism ulowast A0(GMj) rarr A0(1Mj) induced by theunit morphism u 1 rarr G This morphism ulowast also induces a splittingA0(GMj) sim= A0(GMj)oplus A0(kMj) whence the equivalent definition

A0(GMj) = A0(GMj)A0(kMj)

ie ldquoA0(GMj) modulo the constantsrdquo

23


(c) What about SK1 ndash So we would like to describe invariants of SK1(A)using (18) However SK1(A) is not an algebraic group But for anyfield extension F of k we do have a canonical projection SL1(A)(F ) rarrSL1(A)(F )[AtimesF AtimesF ] sim= SK1(A)(F ) which gives us an injective morphismon invariants

Lemma 19Let k be a field A a central simple k-algebra and M a cycle moduleThe projection of k-functors π SL1(A) rarr SK1(A) induces for anyinteger j an injection

π Invj(SK1(A)M) rarr Invj(SL1(A)M)

This lemma allows us to use Merkurjevrsquos description when working withinvariants of SK1(A) We just look at the induced invariant for SL1(A)

14 Invariants of SK1

In order to explain Platonov examples of non-trivial SK1 Suslin conjecturedin 1991 the existence of an invariant for any central simple k-algebra A ofindk(A) = n isin ktimes [Sus Conj 116]

ρA isin Inv4(SK1(A)HlowastnA) (19)

Here we consider HlowastnA = (HinA)ige2 as a cycle module with base k Makingthe right hypotheses on A we could see it as a cycle module with as base acomplete discrete valuation ring R restricted to its fraction field or residuefield as in sect12 (b)

(a) Suslin 1991 ndash Let us explain why Suslin conjectured the existence ofsuch an invariant So we use now the same notation as in Example I10 Inthis case SK1(A) can be expressed in terms of Brauer groups ie secondGalois cohomology groups On the other hand F is a field equipped witha discrete valuation of rank 2 so this induces the existence of two residuespart3t1 part

4t2 in Galois cohomology (sect11 (c) amp (d)) Then using (I2) the invariant

24

INVARIANTS OF SK1

should be able to complete the diagram

SK1(A)sim=

ρAF

Br(Kk)(Br(K1k)Br(K2k))

H4n2A(F )

part3t1part4t2

H2n2(k)part3

t1 part4t2(H2(k microotimes2

n2 ) cup [A])

(110)

In 1991 Suslin was not able to define this invariant in full generality Hewas however able to define an invariant

ρS91A isin Inv4(SK1(A)HlowastnAotimes2)

satisfying a compatibility as above In particular this invariant is not trivialfor Platonovrsquos examples (see also proof of Theorem 316)

(b) Biquaternion algebras ndash In the case of biquaternion algebras Rost wasable to define a related invariant of SK1(A) Suppose A = (a b) otimes (c d) isa biquaternion algebra over a field k of char(k) 6= 2 Then Rostrsquos invariantρRostA is an invariant sitting in Inv4(SK1(A)Hlowast2) [Mer2 Thm 4] Moreoverit fits into an exact sequence

0rarr SK1(A)(k)rarr H4(kZ2Z)rarr H4(k(Y )Z2Z) (111)

where Y is a quadratic k-form defined by

ax21 + bx2

2 minus abx23 minus cx2

4 minus dx25 + cdx2

6 (112)

a so-called Albert form of A Note that microotimesi2sim= Z2 as Γk-modules for any

integer i which is used freely above (and in the following)

This invariant was generalised in [KMRT sect17] to biquaternion algebras inany characteristic using Witt groups and Witt rings The exact definitionof this generalisation requires more terminology to be introduced but afterall the definition is very concrete This contrasts sharply with the otherinvariants into play which are defined using (a lot of) homological argumentsand which are very abstract by definition We come back to this generalisedinvariant in Chapter 3 where we also recall Witt groups and Witt rings

25


(c) Suslin 2006 ndash Using Voevodskyrsquos motivic etale cohomology Suslin wasable to define his conjectured invariant (19) in 2006 It is however notclear whether (110) commutes for this invariant We denote this invariantby ρS06A It is clear that this invariant (as well as any other invariant) istrivial after base extension to the function field of the Severi-Brauer varietyX = SB(A) Indeed

SK1(A)(k)

H4nA(k)

SK1(A)(k(X)) H4nA(k(X))

commutes by definition of an invariant and furthermore SK1(A)(k(X)) = 0as k(X) is a splitting field of A (see eg [GS sect54])

Suslin also proves his invariant is essentially the same as Rostrsquos invariantρRostA for a biquaternion algebra A over a field k of char(k) 6= 2 He doesthis by proving

SK1(A)(k)

id

ρS06 ker[H4

4A(k)rarr H44A(k(X))

]rA

SK1(A)(k)ρRost

ker[(H4

2 (k)rarr H42 (k(Y ))

]

(113)

is a commutative diagram where rA is the morphism induced on Galoiscohomology by the map microotimes3

4 rarr micro2 a 7rarr a2 and where X and Y are asabove This also proves ρS06 is injective for biquaternion algebras and

SK1(A)(k) sim= ker[H4


]

Note that these statements are functorial so that we can also generalisethem to any field extension of k

(d) Kahnrsquos approach ndash Kahn revisited Suslinrsquos construction and generalisedSuslinrsquos invariant ρS06 [Kah3 sect8B] For any central simple k-algebra withn = indk(A) isin ktimes he defined for r = 1 perk(A)minus 1

ρr isin Inv4(SK1(A)H4nAotimesr)

26

INVARIANTS OF SK1

Suslinrsquos invariant ρS06 is retrieved setting r = 1 It is however not clearwhether ρS91 equals ρ2 Kahn also proves ρr is trivial after base extensionto the function field of the the generalised Severi-Brauer variety SB(r A)

He also gives a bound on the torsion of these invariants as elements ofInv4(SK1(A)HlowastnAotimesr) if l = perk(A) is a prime Indeed from (ibid Thm71(c) amp Cor 1210) it follows that the ρr have

bull l-torsion if indk(A) = perk(A) = l gt 2

bull l2-torsion if indk(A) gt perk(A) = l gt 2 and

bull 2-torsion if perk(A) = 2

For any integer n with prime factorisation pe11 middot middot perr we denote by nthe integer pe1minus1

1 middot middot perminus1r If A is a central simple k-algebra A with n =

indk(A) isin ktimes and perk(A) = nn then we get a similar bound on the torsionusing a Brauer decomposition Take a prime factorisation n = pe11 middot middot perrand let D1 otimes otimesDr be a Brauer decomposition of A as in (I3) Then putm = pf11 middot middot pfrr where fi = 1 if pi = 2 or if indk(Di) = perk(Di) = pi gt 2and fi = 2 if indk(Di) gt perk(Di) = pi gt 2 Then it is clear that ρr hasm-torsion

On the other hand Kahn also approaches invariants a la Merkurjev Bycalculations with Quillenrsquos K-theory he shows A0(SL1(A)H4

n)mult is a finitecyclic group [Kah3 Def 113] So by (18) and Lemma 19 we also findInv4(SK1(A)Hlowastn) to be a finite cyclic group Using Kahnrsquos calculations(loc cit) we can pick a canonical generator that we call Kahnrsquos invariantρKahnA of SK1(A)

Furthermore Kahn argues that the size of Inv4(SL1(A)Hlowastn) is boundedby ind(A)l if n = indk(A) is the power of a prime l (ibid Lem 121)Hence the same holds for Inv4(SK1(A)Hlowastn) by Lemma (19) Using Brauerrsquosdecomposition theorem (I3) it is easy to generalise this statement

Lemma 110Let k be a field and A a central simple algebra of indk(A) = n isin ktimesThen

|Inv4(SK1(A)Hlowastn)| le n

27


Proof Let pe11 middot middot perr be a prime decomposition of n and D1 otimes otimes Dr

a Brauer decomposition as in (I3) Recall that this gives rise to adecomposition of SK1(A) (I4) and that SK1(Di) has peii -torsion [Dra sect23Lem 3] Then the result follows immediately from the primary result ofKahn and the isomorphism

H4n(k) sim= H4

pe11

(k)oplus oplusH4perr

(k)

Remark 111 ndash As Kahn mentions this bound is sharp for biquaterniondivision algebras [Kah3 sect12] This follows from [Mer3 Prop 49 amp Thm54] In particular ρKahn is not trivial for biquaternion division algebrasIn sect321 (c) we generalise this result

28

Lifting and specialisinginvariants

Chapter 2

ldquoIf I have seen farther than others it is becauseI was standing on the shoulders of giantsrdquo

mdash Isaac Newton

In this chapter we generalise the invariants of sect14 to central simple k-algebras A with indk(A) possibly not prime to char(k) We use a lift frompositive characteristic to characteristic zero to obtain this as in characteristiczero the invariants mentioned are always defined This method is genericie it does not depend on the precise definition of any of the invariantsbut just on the existence This allows us to perform the lift for a generalinvariant and then we retrieve the generalisations for any of the invariantsmentioned before

As a warmer-up we perform such a lift for central simple k-algebras whenchar(k) = p gt 0 but still p - indk(A) In this case the invariants arealready defined but this gives us some techniques and terminology to treatthe general case where we perform a similar lift using Katorsquos logarithmicdifferentials The content of this chapter was first treated by the author in[Wou3]

21 Moderate case

In this first section we hence start off by lifting from moderate characteristicto characteristic 0 We explain our strategy (for both the moderate andthe wild case) We postpone explicit and detailed arguments to the next(sub)sections

211 Strategy

Let k be a field of char(k) = p gt 0 let A be a central simple k-algebrawith indk(A) = n isin ktimes and let r be any integer Consider k as a residue

29

LIFTING AND SPECIALISING INVARIANTS

field of a ring R which is complete for a discrete valuation v and such thatK = Frac(R) is of characteristic 0 Then A lifts to an Azumaya R-algebra Band BK = BotimesRK is a central simple K-algebra (of same period degree andindex as A) actually the lifted central simple algebra of sect11 (d) Suppose weare given an invariant ρprime isin Inv4(SK1(BK)Hlowast

nBotimesrK) The approach consists

of two steps

(i) Constructing an auxiliary invariant ndash To construct an invariant ρ isinInv4(SK1(A)HlowastnAotimesr) we first construct an auxiliary invariant ρ isinInv3(SK1(A)HlowastnAotimesr) Hence for any field extension kprime of k we haveto define a morphism

ρkprime SK1(A)(kprime)rarr H3nAotimesr(kprime)

So let K prime be a field complete for a discrete valuation w with residuefield kprime such that K prime is a field extension of K and such that w extends vDue to an isomorphism SK1(BK)(K prime) rarr SK1(A)(kprime) and the residueH4nBotimesrK

(K prime)rarr H3nAotimesr(kprime) we are able to construct the morphism ρkprime

This morphism is not necessarily an invariant as the functoriality infield extensions is not immediately obtained There exist after alldifferent possibilities of finding field extensions K prime as above We areable to resolve this aspect using p-rings which are sufficiently canonical

(ii) Deducing the required invariant ndash As the residue of cycle modulesappears in a functorial short exact sequence (16) we obtain aninvariant in Inv4(SK1(A)HlowastnAotimesr) as soon as ρ is trivial By Lemma19 to prove ρ is trivial it suffices to show that the invariant π(ρ) ofSL1(A) is trivial For that purpose we use Merkurjevrsquos morphism θ(18) So we show θ(π(ρ)) = 0 carrying out some calculations on A0-groups and using essential results obtained by Kahn and Merkurjev

We can summarise the strategy by the slogan

Lift and specialise

30

MODERATE CASE

By this we mean that in the diagram

SK1(A)(kprime)A

ED

sim= SK1(BK)(K prime)

0 H4nAotimesr(kprime) H4

nBotimesrK(K prime) H3

nAotimesr(kprime) 0

we first construct the bended arrow SK1(A)(kprime) rarr H3nAotimesr(kprime) using a lift

and the existence of ρK SK1(BK)(K) rarr H4nBotimesrK

(K prime) Then we prove it iszero so that we can specialise ρK to find the (dotted) invariant of SK1(A)

212 Lifting objects

Before lifting invariants we have to be able to lift the objects we are workingwith in a proper way We explain how to lift fields and central simplealgebras

(a) Central simple algebras ndash For any field k we can find a complete discretevaluation ring R such that k is the residue field (eg a p-ring R associatedwith k ndash see (b)) Denote by K the fraction field of R

The way of lifting central simple k-algebras to central simple K-algebras ispassing by Azumaya R-algebras (of constant rank) These are the naturalgeneralisations of central simple algebras to any ring see [KO Ch IIIsectsect56] They also come with a splitting A otimesR S sim= Mn(S) for a faithfullyflat R-algebra S and one can also define the Brauer group Br(R) of R asequivalence classes of Azumaya algebras

Now let P (R) respectively P (k) be the set of isomorphism classes ofAzumaya R-algebras respectively central simple k-algebras Then theresidue map P (R) rarr P (k) associating with the isomorphism class of anAzumaya R-algebra B the class of BotimesR k is bĳective [Gro2 Thm 61] Sogiven any central simple k-algebra A we can find a lifted Azumaya R-algebraB of A (ie such that BotimesR k sim= A) Then BK = BotimesRK is a central simpleK-algebra of same index and degree as A

The bĳection P (R) rarr P (k) induces furthermore an isomorphism Br(R) sim=Br(k) and base extension from R to K gives an injection Br(R) rarr Br(K)

31


[AG Thm 72] So in total we have an injection Br(k) rarr Br(K) HenceBK has also the same period as A For an integer n isin ktimes this coincideson the n-torsion part with the injection nBr(k)rarr nBr(K) from (14) Thisexplains why we worked in sect11 (d) with a lifted central simple algebra witha subscript K

Remark 21 ndash These morphisms can also be retrieved in a more generalway using the group scheme PGLRinfin as Br(R) sim= H1

et(RPGLRinfin) - see[KO Ch III Cor 67] and [Mil1 Ch III Cor 47 amp p134] IndeedGrothendieck proves that for any smooth R-group scheme G with specialfibre G specialisation gives an isomorphism H1

et(RG) sim= H1(kG) [SGAExp XXIV Prop 81] We refer to this result as Henselrsquos lemma a laGrothendieck Now PGLRinfin is a smooth R-scheme so we retrieve theisomorphism Br(R) sim= Br(k) Furthermore as Spec(K) can be consideredas an open of Spec(R) we get from a long exact sequence from etalecohomology Br(R) rarr H1(KPGLKinfin) = Br(K) [Mil1 Ch III Prop125]

The power of this lifting of algebras is that SK1(A)(k) and SK1(BK)(K)are isomorphic This result is essentially due to Platonov for central divisionalgebras The valuation v on K extends to any central division K-algebraD with valuation w = 1

mv NrdDK on D where m gt 0 is the generator ofv NrdDK(D) sub Z [Ser1 Ch XII sect2] Let OD be the valuation algebra ofw and PD its maximal ideal then we denote by D = ODPD the residualdivision k-algebra ndash see also [Wad sect2] We say that D is unramified overK if [D k] = [D K] and if Z(D) is separable over k The residue mapOD rarr D restricts to a residue morphism SL1(D)(K) rarr SL1(D)(k) andPlatonov proves the following rigidity property

Theorem 22 ([Pla Prop 34 Thm 312 Cor 313])Let K be a field complete for a discrete valuation v with residuefield k and D an unramified central division K-algebra The residuemorphism

SL1(D)(K)rarr SL1(D)(k)

is surjective with kernel contained in [Dtimes Dtimes] This induces anisomorphism

SK1(D)(K) sim= SK1(D)(k)

32

MODERATE CASE

From this we try to deduce an isomorphism between SK1(A)(k) andSK1(BK)(K) We use of course Wedderburnrsquos theorem and the Moritainvariance of SK1

Corollary 23Let AB kR and K as above then

SK1(A)(k) sim= SK1(BK)(K)

Proof By Wedderburnrsquos theorem BK sim= Mm(D) for a central division K-algebra D and an integer m gt 0 By the injectivity of Br(R) rarr Br(K) wefind that Mm(OD) is Brauer-equivalent to B So again by Wedderburnrsquostheorem A sim= Mm(D) and it is clear that D is unramified Hence Theorem22 and the Morita invariance of SK1 guarantee that

SK1(BK)(K) sim= SK1(D)(K) sim= SK1(D)(k) sim= SK1(A)(k)

Remark 24 ndash This isomorphism is also functorial in the following senseSuppose K prime is a field extension of K which is also complete for a discretevaluation vprime extending v Let kprime be the residue field of K prime which is a fieldextension of k Then the isomorphism from above commutes with baseextension of K to K prime and k to kprime There is of course no equivalence offunctors as there is no bĳection between field extensions of k and those ofK

(b) p-rings ndash p-rings provide a sufficiently canonical way of lifting fields ofpositive characteristic to rings of characteristic zero Let us start by recallingthe definition of these p-rings

Definition 25A p-ring is a complete discrete valuation ring whose residue field is ofcharacteristic p gt 0 and whose maximal ideal is generated by p

The name ldquop-ringrdquo is as in [Mat sect23] but we always suppose them tobe complete This is because in the sequel we only use complete p-rings

33


Starting from a field k of char(k) = p gt 0 Schoeller gives a explicitconstruction of p-rings with residue field k [Sch sect3] They are subringsof the ring of (infinite) Witt vectors over k Rings of Witt vectors aregeneralisations of the construction of the p-adic integers Zp out of ZpZSee [Wit1 sect1] or also [Ser1 ChII sect6] for more details

When k is perfect the p-ring is exactly the ring of Witt vectors over k Ingeneral the p-ring contains the ring of Witt vectors of the maximal perfectsubfield of k Also note that these p-rings are of mixed characteristic sothey indeed provide a way to perform lifts from positive characteristic tocharacteristic zero Let us recall the following important result of thesep-rings which allows to perform a lift of invariants

Theorem 26 ([Coh] see also [Gro1 Thm 1986])

(i) Let W be a p-ring C a complete local noetherian ring and Ian ideal of C not equal to C Then any local homomorphismu W rarr CI factors in W

vrarr C rarr CI where v is a localhomomorphism

(ii) Let k a field of characteristic p gt 0 Then there exists a p-ringW with residue field isomorphic to k If W prime is a second p-ringwith residue field kprime then any isomorphism u k rarr kprime descendsby quotient from an isomorphism v W rarr W prime

Remark 27 ndash Remark that property (i) induces that p-rings are initialobjects in the category of complete local noetherian rings with a fixedresidue field This theorem seems to suggest that there exists a universalproperty of p-rings However the induced morphisms do not have to beunique They are if and only if the residue field k of the p-ring is perfectSo by lack of uniqueness we call this harmed universal property a versalproperty as Serre does [GMS sect5]

Example 28 (of non-uniqueness) ndash An example of non-uniqueness of themorphism is by the previous remark to be found in non-perfect fieldsand the most standard example of a non-perfect field gives us easily suchexamplesThe Laurent series field Fp((t)) is the most common non-perfect field for aprime p Denote by F is the field consisting of those series

sumiisinZ ait

i with

34

MODERATE CASE

coefficients in Qp bounded below for the p-adic valuation and such thatlimirarrminusinfin |ai|p = 0 Then the p-adic valuation v on Qp extends to F bydefining the valuation of a series as the infimum of the p-adic valuationsof its coefficients The valuation ring Ov is given by similar series with allcoefficients in Zp Moreover Ov is clearly a p-ring of Fp((t)) (See also[Ras Ex 23])Take an element u isin Ztimesp with residue 1 isin Ftimesp Then

Ov rarr Ov defined by t 7rarr ut

is a well defined automorphism and when passing to the residue fieldFp((t)) it gives us the identity Hence the identity map on Fp((t)) induces(infinitely) many choices for lifts to an automorphism of Ov

Fortunately on the cohomological level we are not constrained by these scars

Corollary 29Let WW prime be p-rings such that the residue field kprime of W prime is a fieldextension of k the residue field of W Denote by u k rarr kprime thisinclusion Theorem 26 (i) provides a local homomorphism v W rarrW prime Let A be a central simple k-algebra with indk(A) = n isin ktimes

and lifted Azumaya W -algebra B Denote furthermore K = Frac(W )and K prime = Frac(W prime) Now v defines for any integers i n r ge 0 anhomomorphism of split exact sequences

0 H i+1nAotimesr(k)

ulowast

H i+1nBotimesrK

(K) parti

vlowast

H inAotimesr(k)

ulowast

0

0 H i+1nAotimesr(kprime) H i+1

nBotimesrK(K prime)

parti H i

nAotimesr(kprime) 0

Moreover vlowast does not depend on the choice of v If k = kprime we find inparticular an isomorphism H i+1

nBotimesrK(K) sim= H i+1

nBotimesrK(K prime)

Proof The local homomorphism v sends by definition of a morphism theuniformiser p isin W to p isin W prime So the diagram and independence of choice of

35


v follow immediately from the splitting of (16) by taking the cup productwith the class of p If u is an isomorphism v is also an isomorphism byTheorem 26 (ii) hence one finds an isomorphism of short exact sequences

To ease the notation and our discussion we introduce a notion of triples1

Definition 210If F is a (complete) field equipped with a discrete valuation v then wesay (FOv κ(v)) is a (complete) valuation triple (recall the notationsand conventions on page x) A valuation triple (KR k) where R is ap-ring (for a prime p gt 0) is called a p-triple A (finite resp separableresp Galois) p-extension (K prime Rprime kprime) of (KR k) is a p-triple such thatkprime is a (finite resp separable resp Galois) field extension of k

Remark 211 ndash Given a field k of char(k) = p gt 0 Theorem 26 (ii)gives us a (non-unique) p-triple (KR k) associated with k Even moreif (K prime Rprime kprime) is a (finite resp separable resp Galois) p-extension of(KR k) Theorem 26 (i) implies that K prime is a (finite resp unramifiedresp Galois) extension of K ndash see also [Ser1 sectIII5]If (KR k) is a p-triple F an R-field and (FOv κ(v)) a valuation triplesuch that κ(v) is also an R-field then one says that (FOv κ(v)) is anR-valuation triple

Remark 212 ndash We can reformulate the functorial property of theisomorphism of Corollary 23 as formulated in Remark 24 using p-extensions as follows For any p-extension (K prime Rprime kprime) of (KR k) wehave a commutative diagram

SK1(A)(k)sim=

SK1(BK)(K)

SK1(A)(kprime) sim= SK1(BKprime)(K prime)

1Any use of terminology is purely coincidental and has nothing to do with the authorrsquoslove for craft beer

36

MODERATE CASE

The difference in cumbrousness between Remarks 24 and 212 givesimmediately a feeling why it is useful to introduce the notion of triples

213 The lift

We have now done the necessary preparations to lift and specialise invariantsin moderate characteristic

Theorem 213Let k be a field of char(k) = p gt 0 and A a central simple k-algebra with indk(A) = n isin ktimes Denote by (KR k) a p-tripleassociated with k by B the lifted Azumaya R-algebra of A and letρprime isin Inv4(SK1(BK)Hlowast

nBotimesrK) (for r any integer) There exists a unique

ρ isin Inv4(SK1(A)HlowastnAotimesr) such that for any p-extension (K prime Rprime kprime) of(KR k) the following diagram commutes

SK1(A)(kprime)ρkprime H4

nAotimesr(kprime)

SK1(BK)(K prime)

sim=

OO

ρprimeKprime

H4nBotimesrK

(K prime)

(21)

Remark 214 ndash The cycle modules HlowastnBotimesrK

= (Hj

nBotimesrK)jge2 with base K and

HlowastnAotimesr = (HjnAotimesr)jge2 with base k are as described in sect12 (d) They are

the cycle modules obtained by restricting the cycle module HlowastnBotimesr withbaseR respectively toK and k Note also that the morphismH4

nAotimesr(kprime)rarrH4nBotimesrK

(K prime) is the injection of the short exact sequence (16)

First we carry out the second step of the general strategy explained in sect211This relies heavily on the following proposition We refer to eg [Mil3] forthe terminology related to algebraic groups

37


Proposition 215 (Merkurjev [Mer3 Lem 48 and Prop 49])Let k be a field and G a semi-simple simply connected algebraic k-group then A0(GH3

n) = 0 for any n isin ktimes In particular (by sect13(b)) Inv3(GHlowastn) = 0

We allow us to tweak this result by a couple of homological arguments tothe following helpful result

Corollary 216Let k be a field G a semi-simple simply connected algebraic k-groupand A a central simple k-algebra such that indk(A) = n isin ktimes thenInv3(GHlowastnAotimesr) = 0 for any integer r

Remark 217 ndash For r equiv 0 mod perk(A) we retrieve Proposition 215

Proof By (18) it suffices to prove A0(GH3nAotimesr) to be trivial First we

consider the commutative diagram

H1(k micron)

cup r[A]

H1(k(G) micron)part1

cup r[Ak(G)]

oplusxisinG(1) H0(k(x)ZnZ)

oplusxisinG(1)cup r[Ak(x)]

H3n(k)

H3n(k(G))

part3

oplusxisinG(1) H2

n(k(x))

H3nAotimesr(k) H3

nAotimesr(k(G))part3Aotimesr

oplusxisinG(1) H2

nAotimesr(k(x))

(22)where the rows are chain complexes the central one being exact byProposition 215 It suffices to show the exactness of the lower row Kummertheory and the properties of residues [GMS Rem 62] show that part1 a sumof residues is actually the principle divisor morphism

k(G)times(k(G)times)n rarroplus

xisinG(1)

ZnZ = Div(G)nDiv(G) f 7rarr div(f)

38

MODERATE CASE

This morphism is however surjective as Pic(G) = 0 [San Lem 69]

The exactness of the lower chain complex follows by a diagram chase Indeedsuppose x isin H3

n(k(G)) such that part3Aotimesr(x) = 0 for x the image of x in

H3nAotimesr(k(G)) Then the surjectivity of part1 gives us y isin H1(k(G) micron) such

that x minus(y cup [Aotimesrk(G)]

)isin ker part3 The exactness of the middle row gives us

then x isin H3nAotimesr(k) as required

Proof of Theorem 213 Let ρprime isin Inv4(SK1(BK)HlowastnBotimesrK

) We first constructρ isin Inv3(SK1(A)HlowastnAotimesr) (as explained in sect211) So we first have todefine ρkprime SK1(A)(kprime) rarr H3

nAotimesr(kprime) for any field extension kprime of kand then prove functoriality in field extensions So let (K prime Rprime kprime) bea p-extension of (KR k) associated with kprime Then we surely have amorphism ρprimeKprime SK1(BK)(K prime)rarr H4

nBotimesrK(K prime) Denote by π the isomorphism

SK1(BKprime)(K prime)rarr SK1(A)(kprime) of Corollary 23 then we define

ρkprime = part4Aotimesr ρprimeKprime πminus1 SK1(A)(kprime)rarr H3

nAotimesr(kprime)

Remark that this construction does not depend on the particular choiceof the p-extension Indeed if (K primeprime Rprimeprime kprime) is another p-extension associatedwith kprime Corollary 29 gives an isomorphism of split exact sequences like (16)with the identity on the factors H4

nAotimesr(kprime) and H3nAotimesr(kprime) Moreover part4

Aotimesr ρprimeKprime and π are functorial for such field extensions so this constructs indeedan invariant ρ isin Inv3(SK1(A)HlowastnAotimesr)

Corollary 216 and Lemma 19 show that ρ = 0 So for a isin SK1(A)(kprime)we get that ρprimeKprime πminus1(a) comes from a unique element in H4

nAotimesr(kprime) (bythe short exact sequence (16)) This way we again get a morphism ρkprime SK1(A)(kprime) rarr H4

nAotimesr(kprime) As before the short exact sequence (16) isfunctorial and the choice of p-ring has no influence on the definition so thisdoes define an invariant ρ isin Inv4(SK1(A)HlowastnAotimesr)

The commutative diagram (21) follows immediately by the constructionand the uniqueness follows from the injectivity of H4

nAotimesr(kprime)rarr H4nBotimesrK

(K prime)and Corollary 216

Remark 218 ndash As the exact sequence (16) is split we could have definedthe specialised invariant just using the splitting This would us not havegiven us the same diagram we have right now (21) Moreover with ourmethod we are sure not to lose information in degree 3 On the other hand

39


as a result of our method we do find that the two methods give exactlythe same invariant

Remark 219 ndash For a field k of char(k) = p gt 0 and a central simple k-algebra A of indk(A) isin ktimes the invariants from sect14 are already definedIf (KR k) is p-triple B the lifted Azumaya R-algebra and ρ any of theinvariants ρS91BK ρS06BK ρrBK or ρKahnBK then it is to be expected thatthe specialised invariant of ρ is the same as the original one for SK1(A)To obtain this compatibility one can verify that these invariants verify alifting property as in Theorem 213 (ie there is a commutative diagramas (21) with ρ the original invariant for SK1(A) and ρprime the invariant forSK1(BK)) If we refer to these specialised invariants of SK1(A) we denotethem distinctly by ρS91A ρS06A ρrA and ρKahnA to stress the (a priori)difference

22 Wild case

Let k be a field of characteristic p gt 0 and A a central simple k-algebra withindk(A) = n possibly divisible by p We enter now a new world as the cyclemodule HlowastnAotimesr is not adjusted to our goals Indeed as micropn(ks) is trivialthe Galois cohomology groups Hj+1(k microotimesjpn ) are trivial as well MoreoverKummerrsquos exact sequence (11) does not exist any more so we no longerhave an isomorphism of H2(k micropn) with pnBr(k) as in the moderate case

In this section we describe new cohomology groups (introduced by Kato[Kat1]) which give in this wild case an isomorphism with pnBr(k) We needsuch an isomorphism in order to define relative cycle modules as in sect11 (d)They are furthermore equipped with a short exact sequence comparable to(14) This gives us all the ingredients we need to lift and specialise Wecarry out this job in the case when the central simple algebra has indexpn In Section 23 we deduce the general case from it using the Brauerdecomposition of a central division algebra


In this section let (KR k) be a p-triple and F an R-field Let us first recallthe notion of logarithmic differentials of Kato (ibid) and the definition

40

WILD CASE

of Hq+1pn (k) along with (some of) its properties (for integers n q ge 0)2

Nowadays the differentials are often defined using de Rham-Witt complexes

(a) Logarithmic differentials ndash The definition ofHq+1pn (k) is the most explicit

for n = 1 and this also explains the terminology So let Ωqk =

andΩ1kZ and

let d Ωqminus1k rarr Ωq

k be the usual exterior derivative (if q = 0 we set d = 0)Then Hq+1

p (k) is defined as cokernel of the Cartier morphism

F minus 1 Ωqk rarr Ωq

kdΩqminus1k defined by

xdy1

y1and and dyq

yq7rarr (xp minus x)dy1

y1and and dyq

yqmod dΩqminus1

k

with x isin k y1 yq isin ktimes and F (x) = xp [Car Ch 2 sect6] The kernel ofthis morphism is traditionally denoted by ν1(q)k

(b) Generalisation ndash We can generalise this definition of Hq+1p (k) to a

definition of Hq+1pn (k) for any integer n gt 0 (for n = 0 set Hq+1

pn (k) = 0)This is however quite formal and it is no longer clear why we speak aboutcohomology of logarithmic differentials We start from

Dqpn(k) = Wn(k)otimes ktimes otimes otimes ktimes︸︷︷︸

q times

whereWn(k) is the group of p-Witt vectors of length n on k Now we quotientout by a subgroup generated by the exact relations so that for n = 1 we endup with the cohomology of logarithmic differentials under an identification

xdy1

y1and and dyq

yqharr xotimes y1 otimes otimes yq (23)

for x isin k and y1 yq isin ktimes So let first J primeq(k) be the subgroup of Dqpn(k)

generated by the elements of the form

(i) w otimes b1 otimes otimes bq satisfying bi = bj for 1 le i lt j le q2The superscript q + 1 is again due to tradition but is also quite natural in this case

41


Then Cqpn(k) = Dq

pn(k)J primeq(k) is a generalisation of logarithmic differentialsNote that the antisymmetry also holds for this generalisation as w otimes b1b2 otimesb1b2 otimes bq = 0 (w isin Wn(k) b1 bq isin ktimes)

Subsequently we introduce cohomology Note that these groups are equippedwith a derivative d Cqminus1

pn (k) rarr Cqpn(k) for a b2 bq isin ktimes and q gt 0

defined by

(0 0 a 0 0)otimes b2otimes otimes bq 7rarr (0 0 a 0 0)otimesaotimes b2otimes otimes bq

For q = 0 we again set d = 0 The cohomology group Hq+1pn (k) is then

defined as the cokernel of the Cartier morphism

F minus 1 Cqpn(k) rarr Cq

pn(k)dCqminus1pn (k) defined by

w otimes b1 otimes otimes bq 7rarr (w(p) minus w)otimes b1 otimes otimes bq

Here F (w) = w(p) = (ap1 apn) for w = (a1 an) For q lt 0 weset Hq+1

pn (k) = 0 It is clear that this gives us a generalisation under theidentification (23) In conformity with the case n = 1 we denote by νn(q)kthe kernel of the Cartier morphism Alternatively Hq+1

pn (k) sim= Dqpn(k)Jq(k)

where Jq(k) is the subgroup of Dqpn(k) generated by elements of the form (i)

and [Kat1 Proof of Thms 1amp 2]

(ii) (0 0 a 0 0)otimes aotimes b2 otimes otimes bq

(iii) (w(p) minus w)otimes b1 otimes otimes bq

Define dlog ktimess rarr νn(1)ks a 7rarr (1 0 0) otimes a A calculation with Wittvectors and tensor products gives a short exact sequence of Γk-modules [CarCh 2 Prop 8]

1 ktimesspn

ktimessdlog

νn(1)ks 1

The associated long exact sequence induces (using Hilbert 90) an isomor-phism on the pn-torsion part of the Brauer group H1(k νn(1)ks) sim= pnBr(k)On the other hand we have an exact sequence

0 νn(q)ks Cqpn(ks)

Fminus1 Cq

pn(ks)dCqminus1pn (ks) 0

(24)

42

WILD CASE

The surjectivity of F minus 1 follows from Theorem 221 (infra) which provesHq+1pn (ks) = 0 for any q ge 0 and n gt 0 Indeed if k is the residue field of a

field K complete for a discrete valuation then ks is the residue field of KnrAs Cq

pn(ks) is a ks-vector space such that Cqpn(ks)Γk = Cq

pn(k) we get by theadditive version of Hilbert 90 an isomorphism

H1(k νn(q)ks) sim= Hq+1pn (k) (25)

So as in the moderate case we find

H2pn(k) sim= pnBr(k) (26)

Remark 220 ndash Comparable to the moderate case (Remark 12) the classof a p-algebra [a b)p corresponds to a dbb isin H2

p (k) [GS Prop 925]

(c) Katorsquos exact sequence ndash As announced there is also an exact sequenceas (14) Katorsquos theory of cohomology of logarithmic differentials is slightlymore difficult but we still have the following result

Theorem 221 (Kato [Kat1] Izhboldin [Izh])Let (FOv κ(v)) be a complete valuation triple and let

Hq+1pnnr(F ) = ker[Hq+1

pn (F )rarr Hq+1pn (Fnr)]

Then we have a split short exact sequence

0rarr Hq+1pn (κ(v))rarr Hq+1

pnnr(F )rarr Hqpn(κ(v))rarr 0 (27)

Remark 222 ndash Let us explain the splitting and morphisms without givingproofs Depending on the characteristics of F and κ(v) there are threesituations to be discussed

bull In the case of mixed characteristic (char(F ) = 0 and char(κ(v)) = p)the splitting is obtained by morphisms due to Kato [Kat1 Proof ofThms 1amp 2] Let first i be the canonical homomorphism

Wn(κ(v))w(p) minus w|w isin Wn(κ(v))ϕsim= H1(κ(v)ZpnZ)

rarr H1(FZpnZ)

43


The last injection is defined as in the short exact sequence (14) andthe isomorphism ϕ comes from the additive version of Hilbert 90applied to the long exact sequence obtained from Wittrsquos short exactsequence [Wit1 sect5]

0 ZpnZ Wn(κ(v)s)x(p)minusx

Wn(κ(v)s) 0

Note that this short exact sequence is actually an instance of (24) (forq = 0) Then on the one hand we have an inclusion ilowast Hq+1

pn (κ(v))rarrHq+1pnnr(F ) of degree 0 defined by

w otimes b1 otimes otimes bq mod Jq(κ(v)) 7rarr i(w) cup hqpnF (b1 bq)

On the other hand we have an inclusion ψ Hqpn(κ(v))rarr Hq+1

pnnr(F )of degree 1 defined by

w otimes b2 otimes otimes bq mod Jqminus1(κ(v)) 7rarr i(w) cup hqpnF (π b2 bq)

Here w isin Wn(κ(v)) π is a fixed uniformiser of F bi isin Otimesv andhqpnF is the Galois symbol (13) Kato shows that ilowastoplusψ gives us thementioned isomorphism

Hq+1pn (κ(v))oplusHq

pn(κ(v)) sim= Hq+1pnnr(F )

The morphisms in (27) are the obvious morphisms induced by thisisomorphism

bull The case of equicharacteristic 0 (char(F ) = char(κ(v)) = 0) is likethe moderate case Indeed Hq+1

pnnr(F ) = Hq+1pn (F ) as (14) gives us

Hq+1pn (Fnr) sim= Hq+1

pn (κ(v)s)oplusHq+1pn (κ(v)s) = 0

bull The case of equicharacteristic p (char(F ) = char(κ(v)) = p) isdescribed by Izhboldin [Izh Prop 68] In this case the morphismilowast Hq+1

pn (κ(v))rarr Hq+1pnnr(F ) is defined by

w otimes b1 otimes otimes bq mod Jq(κ(v)) 7rarr w otimes b1 otimes otimes bq mod Jq(F )

On the other hand there is again a morphism ψ Hqpn(κ(v)) rarr

Hq+1pnnr(F ) defined by

wotimes b2otimes otimes bq mod Jqminus1(κ(v)) 7rarr wotimesπotimesb2otimes otimesbq mod Jq(F )

44

WILD CASE

where π is again a fixed uniformiser of F bi isin Otimesv w = (a1 an) isinWn(Ov) and w = (a1 an) its residue in Wn(κ(v)) Izhboldinshows that ilowast oplus ψ induces a splitting of Hq+1

pnnr(F ) Also in thiscase the morphisms in (27) are the obvious ones induced by thisisomorphism

(d) Definition of the R-cycle module HlowastpnL ndash Now we can define our cyclemodule needed to generalise the invariants

Definition 223Let (KR k) be a p-triple with a finite Galois p-extension (L S L)For any integer n gt 0 we define HlowastpnL = (HipnL)igt0 as the cyclemodule with base R and Hj+1

pnL(F ) = Hj+1pnL(F ) where

Hj+1pnL(F ) =

ker[Hj+1

pn (F )rarr Hj+1pn (F otimesK L)] if F isin K-fields

ker[Hj+1pn (F )rarr Hj+1

pn (F otimesk L)] if F isin k-fields

Remark 224 ndash Note that for any F isin K-fields the cohomology groupsare usual Galois cohomology groups and for F isin k-fields the cohomologygroups are the freshly introduced ones Remark that FotimesKL (or FotimeskL) isnot necessarily a field However as L is finitely separable over K F otimesK Lis a finite product of finite separable field extensions of L [Mil4 Thm118] Then the cohomology groups can be interpreted as etale cohomologygroups (in characteristic zero) or as the finite direct sum of the cohomologygroups defined before (in both characteristics)

Remark 225 ndash If (L1 S1 L1) and (L2 S2 L2) are two finite Galois p-extension of (KR k) then there exists a finite Galois p-extension (L S L)of (KR k) which is a common p-extension of both (L1 S1 L1) and(L2 S2 L2) In this case there exist injections HlowastnL1

rarr HlowastnL andHlowastnL2

rarr HlowastpnL This indicates that the choice of L does not play a bigroleThe reason why we need to fix an L at all is in order to obtain a well-defined cycle module with a nice short exact sequence as in (14) If weforget about this L it is not possible to define the residues (D4) in fullgenerality

45


Using direct limits of HlowastpnLrsquos where L runs over all finite Galois extensionsof k we can replace L by ks (and L byKnr) The data and the rules behavewell under taking direct limits the proofs of the analogous statements canalways be reduced to the finite case We leave the adding-in of directlimits as an exercise for the reader who is interested in such a result Inour construction we do not need to go to the separable closure (see Remark241)

We still have to show that this defines a cycle module So we need to definethe four data D1-D4 (see sect12 (a)) The data D1 D2 and D3 only occur inequicharacteristics while datum D4 can occur in mixed characteristics

The definition of the functoriality (D1) is straightforward For a finiteextension E of F we define datum D2 Remark that EotimesF Cq

pn(F ) sim= Cqpn(E)

One defines a trace on Cqpn(E) using the trace TrEF of E to F

Cqpn(E) sim= E otimesF Cq

pn(F )TrEFotimesidminusminusrarr F otimesF Cq

pn(F ) sim= Cqpn(F )

This extends in a natural way to a definition of D2 on the cohomology groupsHq+1pnL(F )

(e) Km(F )-module structure (D3) ndash Take the data as in Definition 223 Ifchar(F ) = 0 (ie F is an extension of K) the Km(F )-module structure isdefined as in the moderate case If char(F ) = p (ie F is an extension ofk) this structure is inspired by the differential symbol in stead of the Galoissymbol For any m ge 1

ρmF Km(F )rarr ΩmF defined by x1 xm 7rarr

dx1

x1and and dxm

xm

is an homomorphism Indeed d(ab) = bd(a) + ad(b) induces d(ab)ab = da

a + dbb

and if a + b = 1 we have daa and

dbb = 0 as da + db = 0 (a b isin ktimes) So

ρmF induces a map Km(F )pKm(F )rarr ΩmF as char(F ) = p (and so dxp = 0)

Even more the image is clearly contained in ν1(m)F The differential symbolis the morphism

hmpF Km(F )pKm(F )rarr ν1(m)F

Bloch-Kato-Gabber prove this is actually an isomorphism [BK Thm 21]

46

WILD CASE

Inspired by this definition we can propose the following Km(F )-modulestructure

ρmpnF Km(F )timesHq+1pn (K) rarr Hq+m+1

pn (F ) defined by

(x1 xm w otimes b1 otimes otimes bq) 7rarr w otimes x1 otimes otimes xm otimes b1 otimes otimes bq

The same arguments as above guarantee this is well defined For a isin Km(F )and b isin Hq+1

pn (F ) we denote the scalar multiplication by a middot b = ρmpnF (a b)This structure restricts to a Km(F )-module structure on (Hq+1

pnL(F ))qge0 for(L S L) as in Definition 223 Indeed if b isin Jq(F otimes L) we have a middot b isinJq+m(F otimes L) for any a isin Km(F )

(f) The residue and an exact sequence ndash We are left with the task to definea residue (datum D4) and we also would like to generalise the short exactsequence (14)

Proposition 226Let (KR k) be a p-triple and (L S L) a finite Galois p-extension Forany complete R-valuation triple (FOv κ(v)) and for all integers n gt 0and q ge 0 we have a split short exact sequence

0rarr Hq+1pnL(κ(v))rarr Hq+1

pnL(F )rarr HqpnL(κ(v))rarr 0 (28)

Proof We certainly have two versions of the sequence (27)

0 Hq+1pn (κ(v))

Hq+1pnnr(F )

Hqpn(κ(v))

0

0 Hq+1pn (κ(v)otimes L) Hq+1

pnnr(F otimes L) Hqpn(κ(v)otimes L) 0

As the sequences are split and the splittings respect the commutativediagram the split exact sequence follows from the snake lemma HereHq+1pnnr(F otimes L) is to be interpreted in the same way as in Remark 224

47


Remark 227 ndash The residues for an R-field F complete for a discretevaluation v are defined by this sequence Suppose F is endowed witha discrete valuation but is not complete for the topology defined by thisvaluation Then take a completion F of F with respect to v The residuefield of F is then equal to the residue field κ(v) of F and in this case theresidue is defined (in the same way as in sect11 (c)) as composition of

H i+1pnL(F )rarr H i+1

pnL(F )rarr H ipnL(κ(v))

Hence we have introduced the four required data to have a cycle modulealong with this practical short exact sequence One also has to verify allthe rules of the cycle modules We refer to Appendix A for a detailedcomputation The only non-trivial rule is actually C and this follows fromthe rule C for the Milnor K-groups using the Bloch-Kato isomorphism andthe Bloch-Kato-Gabber isomorphism

(g) Relative version ndash As in sect11 (d) we define relative cycle modules usingisomorphism (26) and the action of K-theory ndash similar to the alternativedefinition (15) of the moderate cycle module

Definition 228Let (KR k) be a p-triple A a central simple k-algebra of indk(A) =pn and B the lifted Azumaya R-algebra Let (L S L) be a finiteGalois extension of (KR k) such that L is a splitting field of A Wedefine for any integer r a cycle moduleHlowastpnLBotimesr = (HjpnLBotimesr)jge2 withbase R by

Hj+1pnLBotimesr(F ) = Hj+1

pnLBotimesr(F ) = Hj+1pnL(F )(Kjminus1(F ) middot r[BF ])

with F isin R-fields and [BF ] be the class of BF = B otimesR F in pnBr(F )

Remark 229 ndash Note that BF = AF if F is a field extension of k Inthis case we also use the notation Hj+1

pnLAotimesr(F ) For a field extension F

of K we also use the notation Hj+1pnLBotimesrK

(F ) If we restrict HlowastpnLBotimesr tofield extensions of k (resp K) as in sect12 (b) we write it similarly asHlowastpnLAotimesr (resp Hlowast

pnLBotimesrK) Note that for r equiv 0 mod perk(A) we find


pnL(F ) (cfr Remark 15)

48

WILD CASE

Remark 230 ndash The choice of L is possible by (a more enhanced version of)Wedderburnrsquos theorem which gives us a finite separable extension Lprime of ksplitting A We obtain L by taking a finite extension of Lprime such that Lkis Galois Then we associate a p-triple (L S L) with LWe can even suppose L to be a cyclic extension of k Indeed Albertrsquostheorem [Alb2 Thm 18] states that any central simple k-algebra of degreepn is Brauer-equivalent to a cyclic k-algebra (as in Example I5)The fact that we choose L to be a splitting field of A is to guarantee thatthe scalar multiplication ends up in HlowastpnL Indeed for an extension F ofk the base extension morphism Br(F ) rarr Br(F otimes L) sends the class of[AF ] to zero and hence the same holds for the subgroup Kjminus1(F ) middot r[AF ]Also for a field extension F of K the subgroup Kjminus1(F ) middot r[BF ] is trivialafter base extension by L This follows from the previous statement andsect212 (a)

We still have to verify that this relative definition gives us indeed a cyclemodule We base ourselves of course on the fact that the absolute one isa cycle module and we verify that the data are well defined modulo thesubgroups taken into account

Data D1 D2 and D3 follow more or less immediately from the definitionas the fields appearing in these data have the same characteristic DatumD4 for equicharacteristics also follows from the definition of the residue ofHlowastpnL So it suffices to verify datum D4 for the case of mixed characteristicIn addition we have to generalise the exact sequence (28) As D4 is definedusing this exact sequence it even suffices just to generalise the exact sequence(28)

Proposition 231Using the same notations as in Definition 228 we have for any R-valuation triple (FOv κ(v)) a split short exact sequence

0rarr Hq+1pnLBotimesr(κ(v))rarr Hq+1

pnLBotimesr(F )rarr HqpnLBotimesr(κ(v))rarr 0 (29)

Proof By the previous remarks it suffices to prove the proposition in thecase of mixed characteristic The goal is to verify that (28) commutes with

49


inclusions in a commutative diagram (for q ge 2 and up to a sign)

0 Hq+1pnL(κ(v))

ilowast Hq+1pnL(F )

part HqpnL(κ(v)) 0

0 Kqminus1(κ(v)) middot r[Aκ(v)]

OO

Kqminus1(F ) middot r[BF ]

OO

Kqminus2(κ(v)) middot r[Aκ(v)]

OO

0

Let us first verify that the diagram

H2pn(κ(v))

sim=

ilowast H2pnnr(F )

sim=

pnBr(κ(v))i

pnBrnr(F )

(210)

commutes where Brnr(F ) = ker(Br(F ) rarr Br(Fnr)) ilowast is the morphism ofthe short exact sequence (27) and i is the injection of sect212 (a) Theverification is a straightforward computation with cocycles Let us carrythis out First take a generator a otimes x isin H2

pn(κ(v)) with a isin Wn(κ(v)) andx isin Otimesv Then

ilowast(aotimes x) =((τ(y)y)σ(b)minusb

)στisin H2

pn(F )

with yp = x and a = bp minus b for well chosen y isin Ftimesnr and b isin Wn(Fnr) Herewe consider σ(b)minus b as an element of ZpnZ (with σ the residue of σ isin ΓFin Γκ(v)) Then the image in pnH

2(F Ftimess ) sim= pnBr(F ) is represented by thesame expression On the other hand the image of a otimes x isin H2

pn(κ(v)) inpnH

2(κ(v) κ(v)timess ) sim= pnBr(κ(v)) is c =((σ(y)y)τ(b)minusb

)στ

So

i(c) =((σ(y)y)τ(b)minusb

)στisin H2

pn(F )

As ilowast is defined by a cup product this equals minusilowast(aotimes x)

50

WILD CASE

The restriction of (210) to the subgroups gives a commutative diagram (upto a sign)

H2pnL(κ(v))

sim=

ilowast H2pnL(F )

sim=

pnBr(Lotimesk κ(v)κ(v))i

pnBr(LotimesK FF )

The proof of this proposition hence follows immediately from this fact asilowast part and ψ (see Remark 222) respect the K-theory module structure andas the sign disappears when taking quotients So

ilowast(Kqminus1(κ(v)) middot r[Aκ(v)]

)= ilowastK

(Kqminus1(κ(v))

)middot ilowast(r[Aκ(v)]

)sub Kqminus1(F ) middot r[BF ]

part(Kqminus1(F ) middot r[BF ]

)= partK

(Kqminus1(F )

)middot r[Aκ(v)]

= Kqminus2(κ(v)) middot r[Aκ(v)] and

ψ(Kqminus2(κ(v)) middot r[Aκ(v)]

)= ψK

(Kqminus2(κ(v))



Here ilowastK partK and ψK are maps in Milnorrsquos K-theory defined as in sect12 (d)

Remark that this exact sequence also satisfies a property as Corollary 29as also in this case the splittings are given by a choice of uniformiser (seeRemark 222) which is canonical for p-rings

51


Corollary 232Take the notations of Definition 228 and let (K prime Rprime kprime) be a p-extension of (KR k) Denote by u k rarr kprime the inclusion Theorem26 (i) gives a local homomorphism v R rarr Rprime which defines for anyintegers i n ge 0 an homomorphism of split exact sequences

0 H i+1pnLAotimesr(k)

ulowast

H i+1pnLBotimesrK

(K) parti

vlowast

H ipnLAotimesr(k)

ulowast

0

0 H i+1pnLAotimesr(kprime) H i+1

pnLBotimesrK(K prime) parti H i

pnLAotimesr(kprime) 0


pnLBotimesrK(K) sim= H i+1

pnLBotimesrK(K prime)

222 The lift

Before lifting we prove a result analogous to the one of Merkurjev(Proposition 215) This is an immediate consequence of a result of Kahnwhich uses Zariski cohomology groups and reduced Zariski cohomologygroups

H0Zar(GH3

pn) sim= H0Zar(GH3

pn)H3pn(k)

Here H3pn is the functor k-fields rarr Ab associated with the cohomology

of logarithmic differentials (see also sect322) This uses also notions aboutalgebraic groups we refer to eg [Mil3] for the definitions

Theorem 233 (Kahn [Kah1])Let k be a field of char(k) = p gt 0 G a semi-simple simply connectedabsolutely almost simple algebraic k-group G = G timesk ks and n gt 0an integer If CH2(G) = 0 then the base extension G rarr G inducesan injection

H0Zar(GH3

pn) rarr H0Zar(GH3

pn)

52

WILD CASE

Remark 234 ndash The proof consists of putting together various results Theauthor apologises for the non-transparency of the arguments and the plentyof references to the literature but he hopes it improves the readability ofthe whole of this passage For further details on the objects mentioned inboth the theorem and the proof the reader can find more information inthe references These are only used as auxiliary objects and therefore theyare not explained in full details

Proof Let Γ = Γk be the absolute Galois group of k Using motiviccohomology a la Lichtenbaum Kahn constructs a morphism (ibid firstcomplex after the diagram p 406)

H0Zar(GH3

pn)rarr H5(GksΓ(2))Γ (211)

with kernel contained in H1(FH1Zar(GK2)) Here H5(GksΓ(2)) is an

hypercohomology group defined by Kahn as the (fifth) etale hypercohomol-ogy of a relative complex based on the Lichtenbaum complex Γ(2) [Lic] andK2 is the Zariski sheaf obtained from the presheaf U 7rarr KQ

2 (U) (where KQ2

is Quillenrsquos K-theory) In order to define this morphism H0Zar(GK2) sim=

KQ2 (ks) has to hold this is due to Esnault-Kahn-Levine-Viehweg [EKLV

Prop 320 (i)] As H1Zar(GK2) sim= Z [Gil1 Prop 1rsquo] the morphism (211)

is injective (see [Kah1 diagram p 406]) Using CH2(G)Γ = 0 [EKLV Prop320 (iii)] and the following injection of Kahn (ibid exact sequence (18)p 404) we find a desired injective morphism

H5(GksΓ(2))Γ rarr H0Zar(GH3

pn)

It follows from the computations in [Kah1] that this morphism is indeed thenatural map induced by base extension

Corollary 235Let k be a field of characteristic p gt 0 L a finite Galois extension ofk and G a semi-simple simply connected absolutely almost simplealgebraic k-group such that CH2(G) = 0 Then Inv3(GHlowastpnL) = 0for any integer n gt 0

53


Remark 236 ndash Here H3pnL is the cycle module of Definition 223 restricted

to k-fields as in sect12 (b) To ease notation we use L in stead of L whichappears in Definition 223

Proof By (18) it suffices to show that A0(GH3pnL) = 0 As Rost proves

Ai(GMj) sim= H iZar(GMj) for a cycle module M and integers i j [Ros2

Cor 65] it suffices to show that H0Zar(GH3

pnL) = 0 So let x isinH0

Zar(GH3pnL) sub H0

Zar(GH3pn) We know that H3

pn(k(G)) rarr H3pn(ks(G))

factors through H3pn(k(G) otimes L) So x isin ker

[H3pn(k(G))rarr H3

pn(ks(G))]

as x isin H3pnL(k(G)) and hence x isin ker

[H0

Zar(GH3pn) rarr H0

Zar(GH3pn)]

Theorem 233 gives x = 0

The arguments used in the proof of Theorem 213 are purely homologicaland can be recycled in this wild case if one replaces Proposition 215 byCorollary 235 Hence we get the following theorem

Theorem 237Let k be a field of char(k) = p gt 0 A a central simple k-algebraof indk(A) = pn and L a finite Galois extension of k that splits ALet (KR k) be a p-triple associated with k and (L S L) a p-tripleassociated with L Let B be the lifted Azumaya R-algebra and ρprime isinInv4(SK1(BK)Hlowast

pnLBotimesrK) (for r any integer) There exists a unique

ρ isin Inv4(SK1(A)HlowastpnLAotimesr) such that for any p-extension (K prime Rprime kprime)of (KR k) the following diagram commutes


pnLAotimesr(kprime)

SK1(BK)(K prime)

sim=

OO

ρprimeKprime

H4pnLBotimesrK

(K prime)

Remark 238 ndash Recall that the cycle modules HlowastpnLBotimesrK

= (HjpnLBotimesrK

)jge2

with base K andHlowastpnLAotimesr = (HjpnLAotimesr)jge2 with base k are the respectiverestrictions of HlowastpnLBotimesr with base R to K and to k (Remark 229)

54

WILD CASE

Proof To generalise the proof of Theorem 213 one has to generaliseCorollary 216 So it suffices to define a diagram as (22) since the otherarguments are a diagram chase transferable to this wild setting So letG = SL1(A) We consider the following diagram with exact columns

ktimes

middotr[A]

k(G)timespart1

middotr[Ak(G)]

oplusxisinG(1) Z

oplusxisinG(1) middotr[Ak(x)]

H3pn(k)

H3pn(k(G)) part3

oplusxisinG(1) H2

pn(k(x))

H3pnAotimesr(k) H3

pnAotimesr(k(G))part3Aotimesr

oplusxisinG(1) H2

pnAotimesr(k(x))

Note that CH2(G) = 0 as G is an interior form of SLm(k) with m = degk(A)[Pan] and hence the central row in the diagram is exact by Corollary 235Again part1 is the divisor morphism and as Pic(G) = 0 [San Lem 69] part1 issurjective So the same diagram chase and a similar construction as in themoderate case finish the proof

We can now deduce generalisations of the invariants of sect14

Corollary 239Under the same conditions as in Theorem 237 the invariantsρS91BK ρS06BK ρrBK and ρKahnBK induce unique invariants ofSK1(A) satisfying the lifting property We denote them respectivelyby ρS91A ρS06A ρrA and ρKahnA and call them the respectivegeneralised invariants

Proof We have to show that if ρ is any of the given invariants for SK1(BK)then it has values in H4

pnLBotimesrK(for r the appropriate integer) This

55


immediately follows from the commutative diagram

SK1(BK)ρK

H4pnLBotimesrK

(K)

SK1(BL)ρL

H4pnLBotimesrK

(L)

and the triviality of SK1(BL) (as L splits BK)

Remark 240 ndash Note that ρKahnA and ρS06A are injective if A is abiquaternion algebra (over a field of even characteristic) Indeed thisfollows from the construction and the injectivity of the moderate invariantsfor biquaternion algebras (see (111113) and Remark 111)

Remark 241 ndash The definition of these generalised invariants does notdepend on the choice of L as in any case the invariants are trivial afterbase extension to a splitting field of the central simple algebra In thesame way as in Remark 225 we could however replace L by ks

23 General case

We conclude the lifting and specialising procedure by considering the generalcase So let k be a field of characteristic p gt 0 and A a central simple k-algebra of arbitrary index e = pnm (p - m) Wedderburnrsquos theorem gives aunique (up to isomorphism) central division k-algebra D Brauer-equivalentto A Brauerrsquos decomposition theorem gives central division k-algebras Dpn

and Dm of indk(Dpn) = pn and indk(Dm) = m such that D sim= Dpn otimes DmThis gives us an isomorphism of functors by (I4)

SK1(A) sim= SK1(D) sim= SK1(Dpn)oplus SK1(Dm)

Let us also use a slight abuse of notation and set Apn = Dpn and Am = Dm

In order to define the invariants in full generality we glue the moderatecase (Theorem 213) and the wild case (Theorem 237) together with this

56

GENERAL CASE

isomorphism of SK1(A) So we also have to glue to cycle modules togetherin the obvious way

Definition 242Let (KR k) be a p-triple A a central simple k-algebra of indk(A) =e = pnm (p - m) and B the lifted Azumaya R-algebra Let L be afinite Galois extension of k such that it is a splitting field of Apn andlet (L S L) be an associated p-triple We define for any integer r thefollowing cycle module with base R

HlowasteLBotimesr = HlowastmBotimesrm

oplusHlowastpnLBotimesr

pn

Here Bm and Bpn correspond to the Brauer decomposition of A (and BK)we use this notation to keep up with the definitions in sectsect12 (d) and 221(g) Using Theorems 213 and 237 we get the following theorem

Theorem 243Let k be a field of char(k) = p gt 0 A a central simple k-algebra ofindk(A) = e = pnm (p - m) and L a finite Galois extension of ksplitting Apn Let (KR k) a p-triple associated with k and (L S L)a p-triple associated with L Let B the lifted Azumaya R-algebraand ρprime isin Inv4

(SK1(BK)Hlowast

eLBotimesrK

)(for r any integer) There exists

a unique ρ isin Inv4(SK1(A)HlowasteLAotimesr

)such that for any p-extension

(K prime Rprime kprime) of (KR k) the following diagram commutes


eLAotimesr(kprime)

SK1(BK)(K prime)

sim=

OO

ρprimeKprime

H4eLBotimesrK

(K prime)

Then we can define the generalised invariants of SK1(A)

57


Corollary 244Under the same conditions as in Theorem 243 the invariantsρS91BK ρS06BK ρrBK and ρKahnBK induce unique invariants ofSK1(A) satisfying the lifting property We denote them respectively byρS91A ρS06A ρrA and ρKahnA we call them the respective generalisedinvariants

24 Some remarks

Let us finish this chapter by giving some remarks on our construction

241 Other possible constructions

There are a couple of points where we could tweak the construction to obtainactually the same invariant We did not mention (all of) them at the relevantpoints in order to stay focused on our aims at that time Over here we collectthem together

bull As mentioned in sect12 (b) we could have worked with two different cyclemodules This would be just a matter of notation and noting that thereare residue maps from the one cycle module (in characteristic zero) tothe other (in positive characteristic)

bull As mentioned in Remark 218 we could have used the splitting of theexact sequences (16) and (29) This a priori gives another diagram ofcompatibility of invariants From method presented it follows howeverthat both constructions give the same invariant

bull In stead of splitting up the discussion into the moderate (prime-to-p)and wild case (p-primary) we could treat them together as Kahnrsquosresults to prove Theorem 233 also hold in the moderate case If wewould have done this we had to split up some of the other constructionsand proofs into a moderate and a wild case It seems more structuredto split up the discussion at an earlier level

We can also refine the morphism of Lemma 19 to an isomorphism of interestTo do so we need the following definition

58

SOME REMARKS

Definition 245Let k be a field let A k-fields rarr Groups be a group functor andlet M be a cycle module with base k An invariant ρ isin Invj(AM) iscalled unramified if for any field extension F of k the composition

A(F ((t))

) ρrarrMj

(F ((t))

) partjrarrMjminus1(F )

is trivial The subgroup of unramified invariants is denoted byInvjnr(AM)

Remark 246 ndash Usually unramified objects are defined being trivial passingto any discrete valuation field and not just to a field of Laurent series[CT Thm 411] This definition also gives us Proposition 247 but notimmediately Corollary 248

Lemma 19 can be proved to restrict to an isomorphism

Proposition 247Let k be a field A a central simple k-algebra of e = indk(A) and L afinite Galois splitting field of A The canonical projection SL1(A) rarrSK1(A) induces an isomorphism for any integers r j ge 0

Invjnr(SK1(A)HlowasteLAotimesr)simrarr Invjnr(SL1(A)HlowasteLAotimesr)

Proof It is clear that the injection from Lemma 19 restricts well to aninjection on the unramified subgroups Hence it remains to prove thesurjectivity so take any ρ isin Invjnr(SL1(A)HlowasteLAotimesr) Let kprime be any fieldextension of k then we prove that ρ([a b]) = 0 for any commutator [a b] ofAtimeskprime Set α(t) = [t+ (1minus t)a b] a commutator of Atimeskprime((t)) As ρ is unramifiedpartj(ρ(α(t))) = 0 Glue now the short exact sequences (16) and (29) into

0rarr HjeLAotimesr(k

prime)rarr HjeLAotimesr

(kprime((t))

)rarr Hjminus1

eLAotimesr(kprime)rarr 0

We find that ρ(α(t)) is an element of HjeLAotimesr(kprime) so it is constant That

gives us0 = ρ(α(0)) = ρ(α(1)) = ρ([a b])

59


Corollary 248With the same conditions as in Proposition 247 we have anisomorphism

Inv4(SK1(A)HlowasteLAotimesr)simrarr Inv4(SL1(A)HlowasteLAotimesr)

Proof In view of Lemma 19 and Proposition 247 it suffices to prove

Inv4nr(SL1(A)HlowasteLAotimesr) sim= Inv4(SL1(A)HlowasteLAotimesr)

This follows immediately from Corollary 216 and its wild analogue provedin the proof of Theorem 237 Indeed if ρ isin Inv4(SL1(A)HlowasteLAotimesr) then

SK1(A)(F )rarr H4eLAotimesr(F )rarr H4

eLAotimesr(F ((t))

)rarr H3

eLAotimesr(F )

for F a field extension of k gives an invariant in Inv3(SL1(A)HlowasteLAotimesr) =0

In stead of using the injectivity in the construction we can actuallyjust concentrate on generalising invariants of SL1(A) and use Merkurjevrsquosdescription (18) Indeed by this corollary this amounts to defininginvariants of SK1(A) To incorporate this immediately in sectsect21 22 23 onefirst had to prove Corollary 216 and its wild analogue (proof of Theorem237) This would have taken about the same effort as now

242 Other view point

Using the groups Ai A0 and A0mult of sect12 (c) and sect13 (b) there is yet

another way of looking at the construction Let (KR k) be a p-triple Aa central simple k-algebra of indk(A) = n B the lifted Azumaya R-algebra(L S L) a finite Galois p-extension of (KR k) such that L splits A andHlowast = HlowastnLBotimesr the cycle module with base R of Definition 242 (for r anyinteger)

Denote GGG = SL1(B) It is defined like SL1(BK) as the kernel of a reducednorm on B induced by a splitting BotimesR S sim= Mm(S) ndash see [Knu Ch III sect1]

60

SOME REMARKS

for more details The generic fibre GGGK = SL1(BK) is an open of GGG Call Zthe complement the image of the special fibre G = SL1(A) in GGG under theimmersion of schemes ψ Grarr GGG For any integer i ge 0 the points of Z ofcodimension i + 1 correspond under ψ to points of codimension i in G Inthe same way Spec(K) is an open of Spec(R) with complement the imageof Spec(k) Rostrsquos localising sequence [Ros2 sect5] gives exact sequences

0 A0(RH4)

A0(KH4)

A0(kH3)

0

0 A0(GGGH4) A0(GGGK H4) A0(GH3)

(212)Corollaries 216 and 235 (generalised to Hlowast in the proof of Theorem 237)show that A0(GH3) is trivial Using diagram (212) the snake lemma givesan isomorphism

A0(GGGK H4) sim= A0(GGGH4)preserving multiplicative elements Due to Merkurjevrsquos description (sect13(b)) we get an isomorphism

Inv4(GGGK Hlowast) sim= A0(GGGH4)mult

The group on the right hand side is defined in the same way as was done foralgebraic groups in sect13 (b) As Hlowast has base R the morphism of schemesGrarrGGG gives also a morphism

A0(GGGH4)rarr A0(GH4)

giving in the same way a morphism

A0(GGGH4)mult rarr Inv4(GHlowast) (213)

In total we obtain a diagram

Inv4(SK1(BK)Hlowast) π

Inv4(GGGK Hlowast)

ϕ

Inv4(SK1(A)Hlowast) Inv4(GHlowast)

61


which induces the existence of the dotted arrow Indeed let ρ isinInv4(SK1(BK)Hlowast) and (F S F ) a p-extension of (KR k) then (ϕπ(ρ))Fsends commutators of Atimes

Fto 0 as they correspond to commutators of BtimesF

due to the isomorphism SK1(A)(F ) sim= SK1(BK)(F ) (Corollary 23)

In Theorem 243 we constructed this same dotted arrow by a more explicitconstruction

62

Comparing invariants

Chapter 3

ldquoIch habe Angst dass die Mathematik vor dem Endedes Jahrhunderts zugrunde geht wenn dem Trend

nach sinnloser Abstraktion - die Theorie der leerenMenge wie ich es nenne - nicht Einhalt geboten wirdrdquo

mdash Carl Ludwig Siegel

It is generally assumed that all defined invariants of SK1 are essentially thesame but very few results exist on this subject In this chapter we comparesome of the different existing invariants

First of all we treat the biquaternion case (Section 31) In the Book ofInvolutions [KMRT sect17] Knus-Merkurjev-Rost-Tignol construct an explicitcohomological invariant ρBI of SK1(A) when A is a biquaternion algebra overk we call it KMRTrsquos invariant They do not put any restriction on the indexIf char(k) 6= 2 they prove their invariant is essentially the same as Suslinrsquosinvariant ρS06 Using the construction of Chapter 2 we prove that for basefields of characteristic 2 ρBI essentially equals ρS06

In Section 32 we compare several of the invariants with Kahnrsquos invariantρKahn Using the fact that ρS91 is non-trivial for Platonovrsquos examples of non-trivial SK1 we also find that ρKahn is not trivial for these examples We alsoprove a formula for the value on the centre of the product of two symbolalgebras under Kahnrsquos invariant which generalises a formula of Merkurjevfor biquaternion algebras

The results obtained in this chapter were first studied by the author in[Wou2]

31 Invariants for biquaternion algebras

The aim of this section is to compare ρBI in the characteristic 2 case toρS06 We first recall the definition of ρBI which needs Witt groups and Witt

63

COMPARING INVARIANTS

rings and also recall why these invariants are essentially the same when thecharacteristic of the base field is different from 2 Then we are able to dothe comparison in the wild case proving ρBI satisfies a lifting property

311 An explicit invariant

We start by giving the concrete definition of KMRTrsquos invariant This needsthe notion of involutions on Azumaya algebras and Witt groups and rings

(a) Involutions on Azumaya algebras ndash In order to define the invariant asymplectic involution σ on the biquaternion algebra is used We recall thedefinition of a symplectic involution on an Azumaya algebra (so in particularon a central simple algebra) We treat this in this general setting of Azumayaalgebras because we need this for our purposes later on We refer to [KnuCh III sect8] for more details on involutions on Azumaya algebras

Definition 31Let R be a ring and A an Azumaya algebra over R with an R-linearinvolution σ Suppose α AotimesRS simrarrMn(S) is a faithfully flat splittingof A Then σ = α(σ otimes 1)αminus1 is an involution on Mn(S) Since x 7rarrσ(xt) is an automorphism of Mn(S) we can choose u isin GLn(S) suchthat σ(x) = uxtuminus1 for all x isinMn(S) Because σ2 = 1 we get ut = εufor ε isin micro2(S) Then ε is called the type of σ (it is well defined andindependent of the choice of faithfully flat splitting [Knu Ch III811]) If 2 6= 0 in R an involution of type 1 is called orthogonaland an involution of type minus1 is called symplectic If 2 = 0 in R aninvolution is called symplectic if u as above can be written as v minus vtfor v isinMn(S) otherwise it is called orthogonal

Remark 32 ndash If R is an integral domain then an involution on an Azumayaalgebra can only have type 1 or minus1 When k is a field a central simple k-algebra of odd degree can only have orthogonal involutions while a centralsimple algebra of even degree can have involutions of both types [KMRTCor 28]

If A is a central simple algebra over k of degree 2n with a symplecticinvolution σ we can refine the definition of reduced norm trace and

64

INVARIANTS FOR BIQUATERNION ALGEBRAS

characteristic polynomial Set first Symd(A σ) = a + σ(a) | a isin A thevector space of symmetrised elements of A under σ If a isin Symd(A σ)the reduced characteristic polynomial Prdak(X) is a square [KMRT Prop29] Take Prpσak(X) the unique monic polynomial such that Prdak(X) =(Prpσak(X))2 this is the Pfaffian characteristic polynomial The Pfaffiantrace Trpσk(a) and the Pfaffian norm Nrpσk(a) are defined as coefficientsof Prpσak(X) compatible with the expression of NrdAk(a) and TrdAk(a)as coefficients of Prdak(X) (I1)

Prpσak(X) = Xn minus Trpσk(a)Xnminus1 + + (minus1)n Nrpσk(a)

So NrdAk(a) = (Nrpσk(a))2 and TrdAk(a) = 2 Trpσk(a) For any fieldextension kprime of k we abbreviate Prpσkprime aprimekprime(X) by Prpσaprimekprime(X) for aprime isin Akprimeand σprimek = σotimesk id the base extension of σ to kprime which is a symplectic involutionon Akprime = Aotimesk kprime Likewise we use the notation Trpσkprime(aprime) and Nrpσkprime(aprime)for aprime isin Akprime

(b) Witt groups ndash To explain the value group of KMRTrsquos invariant we needWitt groups and Witt rings1 The Witt group Wq(k) is the group of Witt-equivalence classes of non-singular quadratic spaces over k with additiondefined by the orthogonal sum perp

bull Given two quadratic spaces (V q) and (V prime qprime) over k the orthogonalsum (V q) perp (V prime qprime) is given by (V oplus V prime q perp qprime) where q perp qprime isdefined by

(q perp qprime)(v vprime) = q(v) + q(vprime) (v isin V vprime isin V prime)

bull The Witt group Wq(k) consists of non-singular quadratic spaces over kup to Witt-equivalence Two non-singular quadratic spaces (V q) and(V prime qprime) are Witt-equivalent if (V q) perp M is isometric to (V prime qprime) perp M prime

for M and M prime some hyperbolic quadratic spaces An hyperbolic plane isgiven by H = (k2 [0 0]) where [0 0] stands for k2 rarr k (x y) 7rarr xyAn hyperbolic quadratic space is the orthogonal sum of hyperbolicplanes

The Witt ring W (k) is the ring of Witt-equivalence classes of non-singularsymmetric bilinear spaces with addition given by the orthogonal sum perp andmultiplication by the tensor product otimes

1Do not mix up the Witt group and Witt ring with Wn(k) consisting of the Wittvectors on a field k - see sectsect212 (b) and 221 (b)

65


bull Given two bilinear spaces (VB) and (V prime Bprime) over k the orthogonalsum (VB) perp (V prime Bprime) is given by (V oplus V prime B perp Bprime) where B perp Bprime isdefined by

(B perp Bprime)((v vprime) (wwprime)) = B(v w) +B(vprime wprime) (v w isin V vprime wprime isin V prime)

The tensor product (VB)otimes(V prime Bprime) is given by (V otimesV prime BotimesBprime) whereB otimesBprime is defined by

(BotimesBprime)((votimesvprime) (wotimeswprime)) = B(v w)middotB(vprime wprime) (v w isin V vprime wprime isin V prime)

bull The Witt ring W (k) has as elements the non-singular symmetricbilinear spaces over k up to Witt-equivalence Two non-singular bilinearspaces (VB) and (V prime Bprime) are Witt-equivalent if (VB) perp M isisometric to (V prime Bprime) perp M prime for M and M prime metabolic bilinear spacesA metabolic plane is given by H = (k2 lt a 1 0 gt) where a isin k andlt a 1 0 gt stands for the bilinear form B on k2 with B(e1 e1) = aB(e2 e2) = 0 and B(e1 e2) = 1 where e1 e2 is a k-vector space basisfor k2 A metabolic bilinear space is an orthogonal sum of metabolicplanes

Remark 33 ndash If char(k) 6= 2 we know that as groups (with the orthogonalsum) Wq(k) and W (k) are isomorphic We are however interested inthe characteristic 2 case so we have to make a clear distinction Formore information on Witt groups and Witt rings in this general case werefer to [Bae Ch I] and [Kah2 Ch 1] (including the discussion on thecharacteristic 2 case by Laghribi in [Kah2 App D])

Example 34 ndash Suppose that (V q) is a non-singular quadratic space overk (of char(k) 6= 2) and that e1 en is a orthogonal basis for V (withrespect to q) For any x =

sumni=1 xiei isin V we have q(x) = a1x

21+ +anx2

n

with ai = q(ei) isin ktimes Then we denote (V q) = 〈a1 an〉 An n-foldPfister form is given by

〈〈a1 an〉〉 = 〈1minusa1〉 otimes otimes 〈1minusan〉

for a1 an isin ktimes The tensor product of the quadratic forms is inducedby the tensor product of the corresponding bilinear forms These Pfisterforms can be generalised in characteristic 2 in a similar way See (ibidD112)

66


We can equip Wq(k) with a W (k)-module structure If (VB) is a non-singular symmetric bilinear space on k and if (V prime q) is a non-singularquadratic space on k then (V otimes V prime B otimes q) is a quadratic space on k withB otimes q defined by

(B otimes q)(v otimes vprime) = B(v v)q(vprime) for v isin V vprime isin V prime

Let I(k) be the fundamental ideal of W (k) (generated by the non-singularbilinear spaces of even dimension) For any integer n ge 0 we set In(k) =(I(k))n (with I0(k) = W (k)) and InWq(k) = In(k) otimesWq(k) This clearlydefines a filtration

Wq(k) = I0Wq(k) sup I1Wq(k) sup I2Wq(k) sup

We denote the graded quotients by InWq(k) = InWq(k)In+1Wq(k)

Remark 35 ndash Set W primeq(k) the subgroup of Wq(k) consisting of equivalenceclasses of even-dimensional non-singular quadratic spaces over k andInW primeq(k) = In(k) otimesW primeq(k) Also denote InW primeq(k) = InW primeq(k)In+1W primeq(k)If char(k) 6= 2 we have InW primeq(k) = In+1(k) by the equivalence ofsymmetric bilinear and quadratic spaces Again in general we are notable to use this fact

(c) Definition ndash Suppose A is a biquaternion algebra over k with asymplectic involution σ Knus-Merkurjev-Rost-Tignol construct an explicitmap [KMRT Def 175]

SL1(A)rarr I3W primeq(k) a 7rarr

0 if σ hyperbolicΦv + I4W primeq(k) if σ not hyperbolic

with kernel equal to [Atimes Atimes] Recall that an involution is called hyperbolicif there exists an idempotent e isin A such that σ(e) = 1minus e Furthermore Φv

is the quadratic form

Ararr k x 7rarr Φv(x) = Trpσ(σ(x)vx)

where v isin Symd(A σ) cap Atimes satisfies v(Trpσ(v) minus v)minus1 = minusσ(a)a Therealways exists a v satisfying this condition (ibid Lem 173) This definitionis well defined and independent of the choice of v and σ Moreover theconstruction is functorial so that we have an invariant

ρBIA SK1(A)rarr I3W primeq

67


where I3W primeq is the functor

k-fieldsrarr Ab F 7rarr I3W primeq(F )

Remark 36 ndash The element v isin Symd(A σ) cap Atimes in the definition abovecan be given more explicitly If σ(a)a = 1 one can take for v any unit inx isin Symd(A σ) | Trpσk(x) = 0 If σ(a)a 6= 1 the element v is uniqueand equal to 1minus σ(a)a (ibid Lem 173)

312 Comparison KMRT-Suslin moderate case

In this section we recall why ρBIA and ρS06A are equal if A is a biquaternionalgebra over k of char(k) 6= 2 This is because both Suslin and Knus-Merkurjev-Rost-Tignol prove their invariant equals ρRostA We alreadyrecalled the commutative diagram (113) giving the equality of ρS06A andρRostA

To compare ρBI to ρRost famous isomorphisms are used most of themrecently proved Indeed there are isomorphisms ψ1

F K4(F )2 rarr I4(F ) =I4(F )I5(F ) for any F of char(F ) 6= 2 (Milnorrsquos conjecture for quadraticforms [Mil5 Q 43] proved by Orlov-Vishik-Voevodsky [OVV Thm 41])and ψ2

F H4(F micro2) rarr K4(F )2 (Milnorrsquos conjecture [Mil5 sect6] or a specialcase of the Bloch-Kato isomorphism)

So the obvious way of comparing ρBI and ρRost is by the composedisomorphism ψF = ψ1

F ψ2F Indeed Knus-Merkurjev-Rost-Tignol prove

that the following diagram commutes [KMRT Notes sect17]

0 SK1(A)(F )

=

ρRostAF H4

2 (F )

ψ

H42 (F (Y ))

sim=

0 SK1(A)(F )ρBIAF

I4(F ) I4(F (Y ))

(31)for F any field extension of k and Y the Albert form defined by (112)

So combining diagrams (113) and (31) it follows that ρS06 and ρBI are thesame for biquaternion algebras in characteristic different from 2

68


313 Lifting algebras with involution

We first explain how to lift central simple algebras with a symplecticinvolution We do this for general central simple algebras and later on usethe result for biquaternion algebras

(a) Lifting generally ndash Let (KR k) be a p-triple and A an Azumaya algebraover R of degree 2n with symplectic involution σ Define the R-group schemePGSp(A σ) = Aut(A σ) defined for any R-algebra S by

Aut(A σ)(S) = Aut(AS σS) = ϕ isin AutS(AS) |ϕ σS = σS ϕ

with σS = σ otimes id the canonical extension of σ to AS = AotimesR S It is knownthat all Azumaya algebras of degree 2n with symplectic involutions up toisomorphism are classified by H1

et(RPGSp(A σ)) [KMRT 2922] SincePGSp(A σ) is a smooth group scheme (proof as in the field case - ibid p347) we can use Henselrsquos lemma a la Grothendieck to get an isomorphism

H1et(RPGSp(A σ)) sim= H1(kPGSp(A σ))

where A = AotimesR k is the reduced central simple k-algebra and σ = σotimes id isthe reduced involution on A which is also symplectic On the other handwe have an inclusion [Mil1 Ch III Prop 125]

H1et(RPGSp(A σ)) rarr H1(KPGSp(AK σK))

So in total we have an inclusion

H1(kPGSp(A σ)) rarr H1(KPGSp(AK σK))

Remark 37 ndash Note that this lift coincides with lifting central simplealgebras as explained in sect212 (a) Over there we actually used the samearguments for the smooth R-group scheme PGLRinfin (see Remark 21)

So starting with a central simple k-algebra A with symplectic involutionσ we find a lifted Azumaya algebra B over R with symplectic involutionτ and hence a central simple K-algebra BK with symplectic involutionτK In particular degk(A) = degK(BK) and perk(A) = perK(BK) Sincebiquaternion algebras are exactly the central simple algebras of degree 4 andperiod 1 or 2 we see that a biquaternion algebra over k with a symplecticinvolution lifts to a biquaternion algebra with a symplectic involution overK

69


(b) Lifting explicitly ndash We can also perform this lift more explicitly in thewild case2 The lift in the moderate case is canonical symbol algebras liftto symbol algebras by lifting the relations This follows also from Remark12 and the injection defined by (14) The wild case is a little bit morecomplicated Please be aware of an abuse of notation both in positivecharacteristic and in characteristic zero variables u and v are used

Let (KR k) be a 2-triple A = [a b) otimesk [c d) a biquaternion k-algebrawhere a c isin R and b d isin Rtimes Then the lifted Azumaya R-algebra is B =[a b)otimesR [c d) where eg [a b) is the R-algebra generated by u v satisfyingslightly different relations than usual u2+u = a v2 = b and uv = minusv(u+1)We can rewrite it as B = (4a + 1 b)R otimesR (4c + 1 d)R where (4a + 1 b)Ris the R-algebra generated by i j with i2 = 4a + 1 j2 = b and ij = minusjiIndeed an isomorphism is given by i = 2u+ 1 and j = v

bull For a symplectic involution on A it suffices by [KMRT Prop 223(1)] to take the product of an orthogonal involution σ1 on [a b) anda symplectic involution σ2 on [c d) Let σ1 be defined by σ1(u) =u σ1(v) = v (and hence σ1(uv) = uv + v) and σ2 defined by σ2(u) =u + 1 σ2(v) = v (and hence σ2(uv) = uv) By (ibid Prop 26 (2))an involution on a quaternion algebra in characteristic 2 is symplecticif and only if 1 is a symmetrised element So σ1 is indeed orthogonaland σ2 is symplectic as

Symd([a b) σ1

)= 〈v〉 and Symd

([c d) σ2

)= 〈1〉

So σ = σ1 otimes σ2 is a symplectic involution on A In total we getSymd(A σ) = 〈1otimes 1 uotimes 1 v otimes 1 uv otimes 1 + v otimes u v otimes v v otimes uv〉

bull To find a lifted symplectic involution on BK again by (ibid Prop223 (1)) it suffices to take the product of an orthogonal involution τ1on (4a+ 1 b) and a symplectic τ2 involution on (4c+ 1 d) We try tofind these involutions such that τ1 (resp τ2) is a lift of σ1 (resp σ2)We see immediately that a lift τ1 from σ1 should satisfy τ1(i) = i (asτ1(2i + 1) = 2i + 1) τ1(j) = plusmnj and hence τ1(ij) = ∓ij So we gettwo possible lifts τ1 defined by τ1(i) = i τ1(j) = j and τ1(ij) = minusijand τ prime1 defined by τ prime1(i) = i τ prime1(j) = minusj and τ prime1(ij) = ij Then

Symd ((4a+ 1 b) τ1) = 〈1 i j〉 and

Symd((4a+ 1 b) τ prime1

)= 〈1 i ij〉

2This calculation is the result of a discussion with Jean-Pierre Tignol

70


For a symplectic involution on a quaternion algebra in characteristicdifferent from 2 the vector space of symmetrised elements hasdimension 1 while for an orthogonal involution it is of dimension 3(ibid Prop 26 (1)) So we see that both τ1 and τ prime1 are orthogonalOn the other hand a lift τ2 from σ2 should clearly satisfy τ2(i) = minusiτ2(j) = plusmnj and hence τ2(ij) = plusmnij So we get again two possible liftsτ2 defined by τ2(i) = minusi τ2(j) = minusj and τ2(ij) = minusij and τ prime2 definedby τ prime2(i) = minusi τ prime2(j) = j and τ prime2(ij) = ij So

Symd ((4c+ 1 d) τ2) = 〈1〉 and

Symd((4c+ 1 d) τ prime2

)= 〈1 j ij〉

Then τ2 is a symplectic involution and τ prime2 is an orthogonal involutionSo we get two possible lifted symplectic involutions on BK namelyτ = τ1 otimes τ2 and τ prime = τ prime1 otimes τ2 (If we would have started from anothersymplectic involution on A we would have got yet different symplecticinvolutions on BK )

We haveSymd(BK τ) = 〈1otimes 1 iotimes 1 j otimes 1 ij otimes i ij otimes j ij otimes ij〉 and

Symd(BK τ prime) = 〈1otimes 1 iotimes 1 ij otimes 1 j otimes i j otimes j j otimes ij〉Furthermore it follows that

Symd(B τ)otimesR k = Symd(A σ) = Symd(B τ prime)otimesR kas under the identification i = 2u+ 1 j = v we have

Symd(BK τ)

= 〈1otimes 1 uotimes 1 v otimes 1 2uv otimes u+ v otimes u+ uv otimes 1 2uv otimes v + v otimes v

4uv otimes uv + 2v otimes uv + 2uv otimes v + v otimes v〉


2uv otimes uv + v otimes uv〉

Symd(BK τ prime)

= 〈1otimes 1 uotimes 1 2uv otimes 1 + v otimes 1 2v otimes u+ v otimes 1 v otimes v v otimes uv〉

= 〈1otimes 1 uotimes 1 2uv otimes 1 + v otimes 1 v otimes uminus uv otimes 1 v otimes v v otimes uv〉

71


This follows (if indK(BK) = 4 and so BK is a division algebra) also bya theorem of Renard-Tignol-Wadsworth [RTW Prop 313 (ii) Prop 315](Use (ibid Rem 24) to see that v is defectless)

314 Lifting the invariant

We now continue the work of sect312 in the wild case Throughout this sectionlet (KR k) be a 2-triple and A a biquaternion algebra over k with liftedAzumaya algebra B over R Now ρS06 and ρBI have different value groupsso we first give some remarks on how they relate and how we can use theuniqueness statement of Theorem 237 to compare the invariants

(a) Preparing the ingredients ndash By a theorem of Kato we have anisomorphism ψk H4

2 (k)rarr I3Wq(k) [Kat2] Similar to Suslinrsquos construction(113) we can also give a morphismH4

4A(k)rarr H42 (k) Indeed the projection

π21 W2(k)rarr W1(k) (a0 a1)rarr (a0)

gives a morphism r H44 (k)rarr H4

2 (k) Since π21 sends elements of order 2 to

0 r does exactly the same Hence we get a morphism rA H44A(k)rarr H4

2 (k)because any element of K2(k) middot [A] is of order 2 Now we can compare thedifferent groups with a commutative diagram

Proposition 38For any 2-extension (K prime Rprime kprime) of (KR k) the following diagramcommutes

H44A(kprime)

ilowast

rA H42 (kprime) sim=

ψkprime

ilowast

I3Wq(kprime)

j

H44BK (K prime)

rB H4

2 (K prime)sim=

ψKprime

I3Wq(K prime)

(32)

Remark 39 ndash The morphisms rB = rBKprime and ψKprime are as in (113) and(31) while rA = rAkprime and ψkprime are as above The morphism j on Witt

72


groups is as in [Bae Ch V Cor 15] it is the composition of a bĳectionof Wq(Rprime) sim= Wq(kprime) induced by the residual morphism Rprime rarr kprime andan injection Wq(Rprime) rarr Wq(K prime) Here Wq(Rprime) is the Witt group ofquadratic spaces of constant rank over Rprime See [Bae Ch I and V] formore information The maps ilowast are defined by Kato as in Remark 222and Proposition 231

Proof Let (K primenr Rprimenr k

primes) be a 2-triple associated with kprimes So Rprimenr is the

integral closure of Rprime in K primenr

We first prove ilowast rA = rB ilowast This follows merely by the definition of ilowastLet (a0 a1) otimes x1 otimes x2 otimes x3 isin H4

4A(kprime) and take (b0 b1) isin W2(kprimes) such that(b20 b21)minus (b0 b1) = (a0 a1) Then (a0) = (b0)2 minus (b0) isin W1(kprime) and

ilowast rA((a0 a1)otimes x1 otimes x2 otimes x3) = (σ(b0)minus b0)σisinΓKprime cup h32(x1 x2 x3)

where we consider σ(b0) minus b0 as an element of Z2Z for any σ isin ΓKprime (withresidue σ isin Γkprime) On the other hand

rB ilowast((a0 a1)otimes x1 otimes x2 otimes x3)

= rB[(σ(b0 b1)minus (b0 b1))σisinΓKprime cup h

34(x1 x2 x3)

]= (σ(b0)minus (b0))σisinΓKprime cup h

32(x1 x2 x3)

The commutativity of the right square is essentially due to Kato [Kat2 Lem11] He proves the existence of a commutative diagram

Hn2 (kprime)

sim=

ϕ

I3Wq(kprime)

j

Kn(K prime)2Kn(K prime)ψ1Kprime

sim= I3Wq(K prime)

where ψ1Kprime is the isomorphism of Milnorrsquos conjecture on quadratic forms (see

sect312) and where ϕ is defined by

bda1

a1and da2

a2and da3

a3mod I 7rarr 1 + 4b a1 a2 a3 mod 2Kn(K prime)

73


for a1 a2 a3 b isin Rprime Since the isomorphism ψKprime H42 (K prime) rarr I3Wq(K prime)

is defined as composition of ψ1Kprime with the Galois symbol h4

2Kprime it suffices tocheck i(b) = h1

2kprime(1+4b) for any b isin Rprime So take c isin kprimes such that c2minusc = bThen

i(b) = (σ(c)minus c)σisinΓKprime isin H1(K primeZ2)

Take c to be a lift of c in Rnr After change of the representant of b in Rprimewe can assume c2 minus c = b Then 1 + 4b = (2c+ 1)2 and

h12Kprime(1 + 4b) = (σ(2c+ 1)(2c+ 1))σisinΓKprime isin H

12 (K prime)

So if σ(2c + 1)(2c + 1) = 1 we have σ(c) = c On the other hand ifσ(2c+1)(2c+1) = minus1 we get σ(c) = minuscminus1 This gives indeed the desiredequality

(b) Cooking up the result ndash Using Theorem 237 and Proposition 38 wecan prove the main theorem

Theorem 310Let k be a field of characteristic 2 and A a biquaternion algebra overk then

ρBIA = ψ rA ρS06A

with ψ and rA as in (32)

Proof Let (KR k) be a 2-triple associated with k and let (K prime Rprime kprime) be any2-extension of (KR k) Suppose σ is a symplectic involution on A and takeB a lifted Azumaya R-algebra with lifted symplectic involution τ We usethe morphisms from Proposition 38 We know j is injective (Remark 39)ilowast ρS06A = ρS06BK (by definition of ρS06A) and ρBIBK = ϕ πlowast ρS06BK(sect312) So it suffices to prove that ρBIBK = j ρBIA

Suppose SK1(A)(kprime) 6= 0 This means indk(A) = indK(BK) = 4 sinceotherwise SK1(A) = 0 = SK1(BK) by Theorem I20 Also indkprime(Akprime) =indKprime(BKprime) = 4 so we get that Akprime and BKprime are division algebras Then BKprimeis equipped with a valuation w (see sect212 (a)) Recall that the associatedvaluation ring is BRprime with reduced k-algebra Akprime that SL1(BK)(K prime) is partof BRprime and that the isomorphism SK1(BK)(K prime) sim= SK1(A)(kprime) is inducedby the residue map on SL1(BK)(K prime)

74


In this case σ and τ cannot be hyperbolic due to [KMRT Prop 67 (3)]Take a isin SK1(A)(kprime) with lift b isin SK1(BK)(K prime) Then by definitionit follows that PrdAakprime(X) = PrdBbKprime(X) where the residue is thecanonical residue on Rprime[X] So we also get Prpσakprime(X) = PrpτbKprime(X)and Trpσkprime(a) = TrpτKprime(b) Now take y isin Symd(BKprime τKprime)capBtimesKprime satisfyingy(TrpτKprime(y)minus y)minus1 = minusτ(b)b We can assume w(y) ge 0 since if w(y) lt 0ie NrdBKprimeKprime(y) = λmicro isin K prime with λ micro isin Rprime then w(microy) = v(λ) ge 0 and

microy(TrpτKprime(microy)minus microy

)minus1= y(TrpτKprime(y)minus y)minus1

Hence for w(y) ge 0 we get y(Trpσkprime(y)minus y)minus1 = minusσ(a)a because b is a liftof a Moreover clearly y isin Symd(A σ)

Then

ρBIAkprime(a) = Φy Akprime rarr kprime x 7rarr Trpσkprime(σkprime(x)yx) and

ρBIBKprime Kprime(b) = Φy BKprime rarr K prime x 7rarr TrpτKprime(τKprime(x)yx)

Since for x isin B we have TrpτKprime(τKprime(x)yx) = Trpσkprime(σkprime(x)yx) we get therequired compatibility

(c) Non-triviality of the invariant ndash Because the invariants for biquater-nions in characteristic zero are injective they are also injective in character-istic 2 due to the lifting property (Theorem 237) As SK1 is not trivial forPlatonovrsquos examples (Example I10) and in general for biquaternion algebrasof index 4 (Theorem I20) we retrieve non-trivial invariants in characteristic2

Another argument for non-triviality of ρBI in characteristic different from2 is given by a formula of Merkurjev for the value on the centre of thebiquaternion algebra [Mer2 Ex p 70] ndash see also [KMRT Ex 1723] Usingthis formula and the lift from characteristic 2 to characteristic 0 one couldhope to prove the non-triviality of ρBI (and hence of ρS06) in the case whenchar(k) = 2 but this fails Let us comment on this fact

Let (KR k) be a 2-triple and let A = [a b) otimesk [c d) be a biquaternionk-algebra for a c isin R and b d isin Rtimes Then the lifted Azumaya R-algebrais B = (4a + 1 b)R otimesR (4c + 1 d)R (see sect313 (b)) Suppose K contains aprimitive fourth root of unity ζ then by (loc cit) we have

ρBIBK K([ζ]) = 〈〈4a+ 1 b 4c+ 1 d〉〉+ I4W primeq(K)

75


where [ζ] is the class of ζ in SK1(BK)(K)

Let π be the isomorphism SK1(BK)(K) sim= SK1(A)(k) then π([ζ]) = [1]because k contains no non-trivial fourth roots of unity By the proofof Theorem 310 we have j ρBIBK K([ζ]) = ρBIAk π([ζ]) = 0 isinI3W primeq(k) Because the map j from Proposition 38 is injective we get that〈〈4a+ 1 b 4c+ 1 d〉〉 = 0 isin I3W primeq(K) We can also verify this by calculatingwith Pfister forms Define Q as the symbol R-algebra (4a + 1 b) and let Xbe the natural affine R-scheme with

X (R) = x isin Q |NrdQKK(x) = 4c+ 1

where QK = QotimesRK Then X is an R-torsor under SL1(Q) where SL1(Q)is the natural affine R-scheme so that SL1(Q)(R) = SL1(QK)(K) cap QThe special fibre Xk = X timesR k clearly has a rational point so its class[Xk] isin H1(kSL1(Qk)) is trivial By Henselrsquos lemma a la Grothendieckwe get [X ] = 0 isin H1

et(RSL1(Q)) Hence X (as well as the genericfibre XK) has a rational point but then by theory of Pfister forms we get〈〈4a+ 1 b 4c+ 1〉〉 = 0 isin W primeq(K) [Kah2 Cor 2110] Indeed NrdQKK(x)corresponds with a value of 〈〈4a+ 1 b〉〉 So a fortiori 〈〈4a+ 1 b 4c+ 1 d〉〉 =0 isin I3W primeq(k)

32 Kahnrsquos invariant

We compare now all defined invariants of SK1(A) to ρKahnA in the moderatecase ie as they are originally defined The results can be generalised to thewild invariants but with some loss of information We also generalise theformula of Merkurjev (sect314 (c)) for the value on the centre of biquaternionalgebras to the tensor product of two symbol algebras

For sake of convenience we also use the following terminology

Definition 311Suppose ρ is an invariant of SK1 which is defined for any central simplealgebra A with index n not divisible by the characteristic of its basefield and which has values in the Galois cohomology group H4

nAotimesr forr a fixed integer Then we say ρ is a moderate invariant of SK1 withvalues in H4

otimesr We denote by ρA the invariant for a central simplealgebra A

76

KAHNrsquoS INVARIANT

321 Moderate case

Let A be a central simple k-algebra with indk(A) = n isin ktimes and m =perk(A) We explain two natural ways of comparing the invariant groupsInv4(SK1(A)Hlowastn) and Inv4(SK1(A)HlowastnAotimesr)

(a) Ways of looking ndash For any field extension F of k and any integer r wecan look at the composition

mr H4nAotimesr(F ) middotmrarr H4

nm(F )rarr H4n(F )

and at the projection

πr H4n(F )rarr H4

nAotimesr(F )

These induce respectively maps

mr Inv4(SK1(A)HlowastnAotimesr) rarr Inv4(SK1(A)Hlowastn) and

πr Inv4(SK1(A)Hlowastn) rarr Inv4(SK1(A)HlowastnAotimesr)

The maps πr where introduced by Kahn [Kah3 Rem 116] but we ratherconsider the maps mr to compare because of the special definition of Kahnrsquosinvariant as generator of the the target group We could also refine mr

if H2(k microotimes2n ) cup r[A] has mprime-torsion for an integer 0 le mprime lt m A good

comprehension of both maps actually relies as Kahn mentions on a goodcomprehension of the cup product with the class of A (loc cit)

By the cyclicity of Inv4(SK1(A)Hlowastn) (sect14 (d)) we certainly find thefollowing relations Recall the definition of the integer n retrieved froman integer n (sect14 (d))

Proposition 312Let A be a central simple k-algebra with indk(A) = n isin ktimes Thenfor any integer r and any ρ isin Inv4(SK1(A)HlowastnAotimesr) there exists aninteger dA isin Zn such that

mr(ρ) = dA ρKahnA isin Inv4(SK1(A)Hlowastn) sub Zn

77


Proof Use the definition of ρKahn and the bounds on Inv4(SK1(A)Hlowastn) (seesect14 (d))

Kahn also raises the issue whether πr is surjective or not (loc cit) We canprove it to be non-surjective for biquaternion division algebras a la Platonov

Proposition 313Let k = Qp((t1))((t2)) for a prime p Suppose A = (a t1) otimes (b t2)is a biquaternion division k-algebra for a b isin Qtimesp Then π1 is notsurjective

Proof In Example I10 we saw that SK1(A) sim= Z2 Using (14) cd(Qp) = 2and Br(Qp) = QZ [Ser2 Ch II sect51 amp Prop 15] we find that H4

4 (k) sim=Z4 We can also add a fourth primitive root of unity to k as this does notchange the Brauer group In this case we have the Bloch-Kato isomorphismH4

4 (k) sim= K4(k)4

We now prove H44A(k) sim= Z2 Under the Bloch-Kato isomorphism

K2(k)2 sim= 2Br(k) the class of A corresponds to a t1+ b t2 isin K2(k)2(sect11 (b)) so that H2(k microotimes2

4 ) cup [A] is isomorphic to (K2(k)4) middot (2a t1 +2b t2) As the isomorphism H4

4 (k) sim= Z4 is retrieved by taking tworesidues part3

t1 and part4t2 it suffices to determine the group (cfr (110))

part3t1 part

4t2

((K2(k)4) middot (2a t1+ 2b t2)

)

By the definition of residues on Milnor K-groups [Mil5 sect2] it is clear thatthis equals (K1(Qp)4) middot 2a + (K1(Qp)4) middot 2b As we assumed thatSK1(A) is not trivial a cannot be a square by Wangrsquos theorem This meansthat (K1(Qp)4) middot 2a+(K1(Qp)4) middot 2b is not trivial On the other handit has 2-torsion inside K2(Qp)4 sim= Z4 so that indeed H4

4A(k) sim= Z2

Then π1 Z4 rarr Z2 is the ldquomodulo 2rdquo map and m1 Z2 rarr Z4 is thecanonical injection Suslin proves ρS06Ak SK1(A)(k) rarr H4

4A(k) is nottrivial (113) so it is the identity map on Z2 It is then clear that this cannever factor through H4

4 (k) so that π1 is clearly not surjective

(b) Determining factors ndash We prove that for the product of two symbolalgebras of degree n the factor dA appearing in Proposition 312 onlydepends on the invariant ρ and the characteristic of k

78


Proposition 314Let ρ be a moderate invariant of SK1 with values in H4

otimesr Letfurthermore p be equal to zero or to any prime and let m be an integernot divisible by p Then there exist an integer i(pm) isin Zm2 suchthat for any field k of char(k) = p containing a primitive m-th rootof unity ξm and for any product A = (a b)m otimes (c d)m of two symbolk-algebras

mr(ρA) = i(pm) ρKahnA isin Inv4(SK1(A)Hlowastm2) sub Zm2

Remark 315 ndash Although i(pm) is in general not uniquely determinedwe can take a canonical representant as we know Inv4(SK1(A)Hlowastm2) iscyclic This comes down to taking the class in Zm2 satisfying therequired relation and such that the representant in 0 m2 minus 1 isas low as possible It also of course depends on the invariant We addan index if necessary to stress which invariant is compared to Kahnrsquosinvariant Moreover it also depends on the exact definition of the injectionInv4(SK1(A)Hlowastm2) sub Zm2 but this can be chosen in a canonical waydue to the results of Kahn [Kah3 Def 113]

Proof Take k the prime field of characteristic p and set kprime = k(ξm) for anm-primitive root of unity ξm isin ks Denote by T = (t1 t2)m otimes (t3 t4)m theproduct of two Azumaya symbol algebras over R = kprime[tplusmn1

1 tplusmn12 tplusmn1

3 tplusmn14 ] where

t1 t2 t3 t4 are variables and where Azumaya symbol algebras are definedusing the same relations as used for symbol algebras over a field TakeK = kprime(t1 t2 t3 t4) and T = TK = (t1 t2)m otimes (t3 t4)m the product of therespective symbol algebras over K By Proposition 312 we find a uniquedT isin Zm2 such that

mr(ρT ) = dT ρKahnT (33)

We prove dT only depends on m and p

So suppose F is a field of characteristic p containing anm-th primitive root ofunity so that kprime sub F Take any product A = (a b)motimes (c d)m of two symbolalgebras of degree m over F Now A can be obtained from TF = T otimesR F byspecialising t1 t2 t3 t4 to a b c d respectively

Moreover (a b c d) defines a k-rational point x of Spec(F [tplusmn11 tplusmn1


4 ])Take Ox to be the local ring of Spec(F [tplusmn1


3 tplusmn14 ]) in x with maximal

79


ideal M It is clear that the completion Ox of Ox with respect to the M -adictopology is F -isomorphic to Rprime = F [[u1 u2 u3 u4]] where u1 = t1 minus a u2 =t2minus b u3 = t3minus c and u4 = t4minusd (see also [Gro1 Thm 1964]) Under theisomorphism Br(Rprime) sim= Br(F ) from sect212 (a) it is clear that ARprime = Aotimes Rprimeis an Azumaya Rprime-algebra mapping to A Furthermore the F -isomorphismof Ox with Rprime gives an isomorphism Br(Ox) sim= Br(Rprime) In its turn this givesan isomorphism Br(Ox) rarr Br(F ) with inverse given by taking the tensorproduct over F with Ox It sends the class of TOx to the class of A

Let K prime = F ((u1))((u2))((u3))((u4)) then A otimesF K prime is Brauer-equivalent toTOx otimesOx K

prime sim= TKprime By Corollary 23 SK1(A) sim= SK1(TKprime) Furthermore(14) gives an injection H4

m2(F )rarr H4m2(K prime) The diagram

SK1(A)

sim=

ρ H4

m2(F )

SK1(TKprime)ρ

H4m2(K prime)

commutes for both mr(ρ) and ρKahn (by definition of an invariant) Then by(33) and functoriality of the arguments we get mr(ρA) = dTρKahnA

(c) Non-triviality of Kahnrsquos invariants ndash As mentioned in Remark 111ρKahn is not-trivial for biquaternion algebras (of index 4) We generalisethis to the product of two cyclic algebras a la Platonov (Ex I10) Forthat purpose we compare ρKahn to ρS91 as this invariant is non-trivial forPlatonovrsquos examples (sect14 (a)) This means that we have to work withHlowastnAotimes2 for suitable n and A (In the same way as in Proposition 313 thesegive also examples of non-trivial π2)

Theorem 316Let k be p-adic field containing a n3-th primitive root unity and letF = k((t1))((t2)) Suppose A = (a t1)n otimes (b t2)n is a division F -algebra then ρKahnA is not trivial If n = q1 middot middot qr for differentprimes qi then

Inv4(SK1(A)Hlowastn2) sim= Zn

Moreover if n is odd the integer iS91(0 n) isin Zn2 defined inProposition 314 for ρS91 is not trivial

80


Proof We know SK1(A) sim= Zn by Example I10 Furthermore H4n2(F ) =

Zn2 (arguments as in the proof of Proposition 313)

To calculate H4n2Aotimes2(F ) we use an analogous argument as in the proof

of Proposition 313 If n is odd we also find H4n2Aotimes2(F ) sim= Zn as in

this case perk(Aotimes2) = perk(A) If n is even perk(Aotimes2) = n2 so thatH4n2Aotimes2(F ) sim= Z(2n) In either case m2 H4

n2Aotimes2(F ) rarr H4n2(F ) is the

canonical injection (m2 is the multiplication by m for m = n if n odd andm = n2 if n even)

Suslin proves ρS91A is not trivial (on the field F ) [Pla Thm 48] If n isodd ρKahnA is not trivial (on F ) by Proposition 312 and hence by definitioniS91(0 n2) 6= 0 isin Zn2 If n is even a similar argument as in the proof ofProposition 312 gives the non-triviality of ρKahnA (mutatis mutandis m byn2)

By the bound on the invariant group (sect14 (d)) and a Brauer decompositionof A with a related decomposition of invariants in primary parts theisomorphism statement follows

322 Wild case

Now we continue the comparison in the wild case Using a lift we cangeneralise the statement to any central simple algebra with some loss ofinformation This does let us prove a relation between the several i(p n)rsquos

Let A be a central simple k-algebra of indk(A) = n and perk(A) = m Wedefine the functors of graded groups for r an integer

Hlowastn k-fieldsrarr Groups F 7rarr (H in(F ))igt0 and

HlowastnAotimesr k-fieldsrarr Groups F 7rarr (H in(F )(Kiminus2(F ) middot r[AF ])ige2

They are in general no cycle module as to obtain a cycle module we have toadd in an extra field L (see Definitions 223 amp 228)

We again have a morphism

mr Inv4(SK1(A)HlowastnAotimesr)rarr Inv4(SK1(A)Hlowastn)

81


induced by the multiplication for any field extension F of kmr H4

nAotimesr(F ) middotmrarr H4nm(F )rarr H4

n(F )Note that we can also define a map πr as in sect321 (a)


otimesr Suppose kis a field of char(k) = p gt 0 and let A = [a b)potimes [c d)p be the productof two p-algebras over k then

mr(ρA) = i(0 p) ρKahnA

Proof Let (KR k) be a p-ring The lifted Azumaya R-algebra B of A is(after base extension to K) a product of two symbol algebras of degree pThis follows from the injection H2

p2(k) rarr H2p2(K) (see Remark 222) and

from the description of the image of A and BK in the second cohomologygroups as described in Remarks 12 and 220

The result follows immediately from the injectionsInv4(SK1(BK)Hlowastp2) rarr Inv4(SK1(A)Hlowastp2) and

Inv4(SK1(BK)Hlowastp2BotimesrK

) rarr Inv4(SK1(A)Hlowastp2Aotimesr)

defined by lifting invariants (Theorem 243) and the relations for ρBK andρKahnBK (Proposition 314)

Remark 318 ndash In the view of Remark 219 we could even refine thestatement in the moderate case Let (KR k) be a p-triple and A =(a b)n otimes (c d)n a product of two symbol k-algebras for n isin ktimes thena similar statement holds as A lifts to the central simple K-algebra(a b)n otimes (c d)n where a b c d isin R are lifts from a b c d (see Remark12 and sect11 (c))If ρA = ρA then i(p n) is a multiple of i(0 n) in Zn Indeed ρKahnA isa generator of Inv4(SK1(A)Hlowastn) sub Zn and for some integer λ

i(p n)ρKahnA = mr(ρA) = i(0 n) ρKahnA = i(0 n)λ ρKahnA

In particular i(p n) = i(0 n) if ρKahnA = ρKahnA so that the integersi(p n) would not depend on the characteristic of the base field

82


323 Formula on the centre

We can now generalise the formula of Merkurjev on the centre of abiquaternion algebra ([Mer2 Ex p70] ndash see also [KMRT Ex 1723] andsect314 (c)) to the tensor product of two symbol algebras We first prove ageneral formula and later we prove a finer result using Theorem 316

(a) General result ndash We again use cohomological invariants however notinvariants of algebraic groups as in sect13 but rather invariants as introducedin [GMS Ch I] These are also natural transformations of functors butrather a natural transformation of a functor B k-fields rarr Sets into afunctorH k-fieldsrarr Ab For the natural transformation cause we considerH to be a functor k-fieldsrarr Sets

Proposition 319Let p be equal to 0 or to any prime and let n gt 0 be an integer notdivisible by p There exists an integer j(p n) such that the followingformula holds for any field k of char(k) = p containing a primitiven2-th root of unity ζ and for A = (a b)n otimes (c d)n any product of twosymbol k-algebras (for a b c d isin ktimes)

ρKahnAk([ζ]) = ϕ[j(p n)h4

mk(a b c d)]isin H4

n2(k)

Here ϕ is the canonical map H4m(k)rarr H4

n2(k) (for m = n2)

Remark 320 ndash Remark that microotimesin2sim= Zn2 as Γk-modules for any i gt

0 as k contains an n2-th primitive root of unity Note also thatϕ[h4mk(a b c d)

]= mprime h4

n2k(a b c d) for mprime = n2m and that that ϕis injective The former follows from the definitions and the latter followsfrom the long exact sequence in Galois cohomology associated with

0rarr Zmrarr Zn2 rarr Zmprime rarr 0

which by the Bloch-Kato isomorphism comes down to

rarr K3(k)n2 rarr K3(k)mprime rarr K4(k)mϕrarr K4(k)n2

Now K3(k)n2 rarr K3(k)mprime is clearly surjective so that ϕ is indeedinjective

83


Remark 321 ndash This expression is indeed compatible with the biquaternioncase keeping in mind diagrams (113) and (31) Also the integer j(p n)in the theorem is not uniquely determined but can be picked canonicallyby taking the smallest positive integer satisfying the relation Moreoverj(p n) depends on the n-th primitive root of unity used in the definitionof the symbol algebra and of the choice of n2-th primitive root of unityζ We are interested in the invertibility of j(p n) modulo m and thereforethe exact choices do not matter so we do not incorporate them in thenotation

Proof As ρKahn has m-torsion (Lemma 110) we can assume ρKahnAk([ζ])to have values in H4

m(k)

Let k be the prime field of characteristic p and set kprime = k(ζ) for ζ isin ka primitive n2-th root of unity Take T = (t1 t2)n otimes (t3 t4)n over F =kprime(t1 t2 t3 t4) We prove the formula for T The proof ends by specialisingto A as in the proof of Proposition 314

Let B k-fields rarr Sets be the functor attaching to a field extension F ofk the Galois cohomology group H1(F microm)4 and H associating H4(F microotimes4

m )with F Then ρKahn induces a cohomological invariant of B into H Indeedusing the isomorphism H1(F microm) sim= Ftimes(Ftimes)m we associate with any fourrepresentants a b c d isin Ftimes of classes inH1(F microm) the value ρKahnAF ([ζ]) isinH4m(F ) sim= H4(F microotimes4

m ) sim= K4(F )m (for A = (a b)n otimes (c d)n)

Using a full description of all possible invariants of B into H of [Gar Prop21 amp sect31] and [GMS Ex 165] we find that rn(ρKahnTF ([ζ])) can bewritten in K4(F )m as sum of pure symbols of the form λz1 z2 z3 z4where λ is an integer and each zi is either a tj or an element of k Weprove that only t1 t2 t3 t4 occurs By specialising t1 to 1 we obtainT1 = (1 t2)notimes(t3 t4)n from T But then SK1(T1) = 0 by Wangrsquos theorem sothat ρKahnT1F ([ζ]) = 0 This induces that for all (non-trivial) pure symbolsz1 z2 z3 z4 appearing in ρKahnTF ([ζ]) one of the zi has to equal t1 (as theother ones are zero by the specialisation above) Three other specialisationsgive the result

Remark 322 ndash In the same way as in Remark 318 there is a compatibilitybetween the j(p n)rsquos Let k be a field of char(k) = p gt 0 containing ann2-th primitive root of unity ζ and take A = (a b)n otimes (c d)n a tensorproduct of two symbol k-algebras of degree n isin ktimes Take (KR k) a p-

84


triple associated with k then A lifts again to BK = (a b)l otimes (c d)l wherea b c d isin R are lifts from a b c dUnder the injection H4

m(k) rarr H4m(K) (for m = n2) induced by (14)

ϕ[h4mk(a b c d)

]is sent to ϕ

[h4mK(a b c d)

](with an abuse of

notation for ϕ from Proposition 319) This follows from a splitting forMilnorrsquos K-Theory (17)Now ζ lifts to a primitive n2-th root of unity ζ isin R Then by definition ofρKahnA and Proposition 319 it follows that

ρKahnA([ζ]) = ϕ[j(0 n)h4

mk(a b c d)] (34)

On the other hand by the definition of ρKahnA as a generator

ρKahnA([ζ]) = λ ρKahnA([ζ]) = λϕ[j(p n)h4

mk(a b c d)]

for an integer λ If ρKahnA = ρKahnA we can again take j(p n) = j(0 n)so that the integers j(p n) would not depend on the characteristic

Remark 323 ndash In wild characteristics (ie when p |n) a formula as abovedoes not make sense as there are no non-trivial p2-th roots of unity Sosimilar as in sect314 (c) we cannot generalise this formula to wild invariantsby means of a lift

(b) Non-triviality of factor ndash We prove the non-triviality of the factorappearing in Proposition 319 This uses the non-triviality of ρKahn forPlatonovrsquos examples (Theorem 316) First we recall some notions relatedto tori See [CTS1] as a reference for more details

Denote for a finite separable field extension K of k by RKk(Gm) the torusobtained by Weil restriction of scalars from K to k (see eg DefinitionB1) Denote furthermore the kernel of the multiplication map RKk(Gm)rarrGmk by R1

Kk(Gm) and the cokernel of the injection Gmk rarr RKk(Gm) byRKk(Gm)Gm Furthermore for any k-torus T we denote by T (k)R theR-equivalence classes of T (k) The dual T of a k-torus T is the charactergroup Hom(TGm) The dual of RKk(Gm) is clearly the free abelian groupZ[Γ] for Γ = Gal(Kk) The dual of R1

Kk(Gm) is then JΓ the cokernel ofthe norm

Zrarr Z[Γ] a 7rarrsumγiisinΓ

aγi

85


The dual of RKk(Gm)Gm is the kernel IΓ of the augmentation map

Z[Γ]rarr Z sumγiisinΓ

niγi 7rarrsumγiisinΓ

ni

Recall that a k-torus F is called flabby (flasque) if F is a flabby Γk-module ieExt1(F P ) = 0 for any permutation Γk-module P (for equivalent definitionssee ibid Lem 1) A flasque resolution of a k-torus T is an exact sequenceof k-tori

0rarr S rarr E rarr T rarr 0with E quasi-trivial (ie E is a permutation module) and S flabby Thisalways exists and if T is split by a field K then E and S can also be chosento be split by K

Theorem 324Let k be a p-adic field containing a n3-th primitive root of unity andlet F = k((t1))((t2)) If A = (a t1)n otimes (c t2)n is a division F -algebrathen

ρKahnAF ([ζ]) = ϕ[λh4

mF (a t1 c t2)]isin H4

n2(F )

for ζ an n2-th primitive root of unity m = n2 and an integer λ 6equiv 0mod m (and ϕ as in Proposition 319) A fortiori j(0 n) 6equiv 0 mod mfor any n

Proof We know by Theorem 316 that ρKahnA SK1(A)(F ) rarr H4n2(F ) is

not trivial and moreover SK1(A)(F ) sim= Zn and H4n2(F ) sim= Zn2 We prove

that the image of micron2(F ) sim= Zn2 inside SK1(A)(F ) is all of SK1(A)(F ) Inthat case ρKahnA([ζ]) is not trivial in H4

n2(F ) (and in H4m(F ) sim= Zm) so

that j(0 n) 6equiv 0 mod m

To prove the statement let K = k( nradica nradicb) and Γ = Gal(Kk) sim= Zn times

Zn Then by taking residues on F with respect to t1 and t2 Platonovproves SK1(A)(F ) sim= Hminus1(Γ Ktimes) where the cohomology group is a Tatecohomology group (see eg [Wei1 Def 624]) - also use [Pla Thms 417amp 57] and [Wad (615)]) On the other hand Hminus1(Γ Ktimes) = T (k)R forT = R1

Kk(Gm) [CTS1 Prop 15] The resulting isomorphism SK1(A)(F ) sim=T (k)R is a specialisation morphism (in t1 and t2) [Wad (69) amp (610)]so that the composite micron2(F ) rarr SK1(A)(F ) sim= T (k)R is the canonical

86


morphism micron2(k)rarr T (k)R It suffices to prove that the surjectivity of thelatter

First take a flabby resolution 1 rarr S rarr E rarr T rarr 1 of K-split tori thenH1(k S) = T (k)R (loc cit Thm 2) The evaluation morphism S times S rarrGm induces a perfect pairing [Nak Tat]

H1(k S)timesH1(k S)rarr H2(kGm) sim= QZ

Moreover H1(k S) sim= H1(Γ S(K)) This follows from the inflation-restriction exact sequence [GS 3314] and H1(KS) = 0 The pairing abovecan be modified to a pairing

H1(Γ S(K))timesH1(Γ S(K))rarr Br(Kk) sim= Zn2Z

Now note that micron2 sub T so that we get a dual map T rarr Zn2Z Using theflabby resolution and the pairing T (k)times T (K)rarr Ktimes we get the followingcommutative diagram of pairings

H1(k S) times H1(k S)

sim=

H2(kGm) sim= QZ

H1(Γ S(K))

sim=OO

times H1(Γ S(K))

Br(Kk)

OO

T (k)

OO

times H2(Γ T (K))

Br(Kk)

micron2(k)

OO

times H2(ΓZn2) Br(Kk)

The bottom pairing is perfect as micron2(k) sim= Zn2 note that the bottom squarecomes from the compatibility of the pairings

T (k) times T (K)

Ktimes

micron2(k)

OO

times Zn2 Ktimes

87


As H1(k S) = T (k)R sim= Zn to prove the surjectivity of micron2(k) rarrT (k)R it suffices to prove the injectivity of H1(k S) rarr H2(ΓZn2)Since H1(Γ E(K)) = 0 this comes down to proving the injectivity ofH2(Γ T )rarr H2(ΓZn2) This morphism fits into an exact sequence

H2(Γ IΓ)rarr H2(Γ T )rarr H2(ΓZn2)

because of the exact sequence of group functors

0rarr micron2 rarr T rarr RKk(Gm)Gm rarr 0

Clearly T rarr RKk(Gm)Gm factors through RKk(Gm) so thatH2(Γ IΓ)rarrH2(Γ T ) factors through H2(ΓZ[Γ]) which is trivial by Shapirorsquos LemmaThis proves the desired injectivity

Remark 325 ndash Note that the proof also defines an invariant of the torusT with values inside H4

n2

88

Conclusion

ldquoChi tace e chi piega la testa muore ognivolta che lo fa chi parla e chi cammina

a testa alta muore una volta solardquomdash Giovanni Falcone

Overall in this text we studied invariants of SK1 On the one hand wedefined wild invariants starting from existing moderate invariants using liftsand appropriate cycle modules On the other hand we compared invariantsand proved ρKahn is not trivial for Platonovrsquos examples of non-trivial SK1This gives a different way of looking at Suslinrsquos conjecture (Conjecture I12)

Conjecture C1Let k a field and A a central simple k-algebra of indk(A) containing asquare factor then Suslinrsquos invariant is not trivial for SK1(A)

Remark C2 ndash By Suslinrsquos invariant we mean either ρS06A or ρS06Adepending on char(k) and indk(A) Clearly a positive answer to thisconjecture would imply Suslinrsquos conjecture Therefore one could callthis conjecture a strong version of Suslinrsquos conjecture For biquaternionalgebras this conjecture is true by Theorem I20 and Remark 240 Wecan also rephrase this question for other invariants and obtain a modifiedconjecture

Again by the index reduction formula (Proposition I14) it suffices to answerthe question for central simple k-algebras A of indk(A) = p2 (p prime) UsingTheorems I16 and I17 we can also reduce the question to verifying it forcyclic division algebras of the form [(a b)p otimes [(c d)p as in Proposition I19

We now try to attack this problem with the techniques from Chapters 2 and3

89

CONCLUSION

(a) Lifting and specialising invariants ndash By lifting central simple algebrasfrom positive characteristic to characteristic zero as in sect212 (a) we obtainthe following result

Proposition C3Let (KR k) be a p-triple A a central simple k-algebra and B thelifted R-Azumaya algebra If Suslinrsquos (strong) conjecture holds for Athen it also holds for BK

Proof Recall that indk(A) = indK(BK) The statement on Suslinrsquosconjecture follows from Corollary 23 The one on Suslinrsquos strong conjectureholds as by definition ρS06BK maps to ρS06A under a morphism (see Theorem243)

Inv4(SK1(BK)HlowastrLBK

)rarr Inv4

(SK1(A)HlowastrLA

)

Remark C4 ndash Whether the inverse of Proposition C3 holds is an openquestion and does not follow formally from the definition Indeed supposeSK1(A) = 0 ie SK1(A otimesk kprime) = 0 for any field extension kprime of k ThenSK1(BK otimesK K prime) = SK1(A otimesk kprime) = 0 for any p-extension (K prime Rprime kprime) of(KR k) But it is not sure that SK1(BK otimesK F ) = 0 for any extensionF of K If we reformulate this in the setting of sect242 then the inversetranslates into a possible injectivity of the morphism (213)

To the author the constructions introduced in this thesis do not seem to giveimmediate ways of making strong reductions of characteristics It would behowever interesting to do so and to be able to define one of the dotted arrowsin the ldquodiagramrdquo beneath where we abbreviate Suslinrsquos conjecture to SC andSuslinrsquos strong conjecture to SSC

SC positive characteristic SC characteristic 0

SSC positive characteristic SSC characteristic 0

90

CONCLUSION

(b) Comparing invariants ndash Using Theorem 324 and the Bloch-Katoisomorphism we find the following result in moderate characteristic

Corollary C5Let k be a field containing an l2-th root of unity (for l 6= char(k)any prime) and let A = (a b)l otimes (c d)l be any product of two symbolk-algebras If a b c d 6= 0 isin KM

4 (k)l then SK1(A) 6= 0

Proof In characteristic 0 this follows immediately from the injectivity of ϕ(Remark 320) and j(0 l) 6equiv 0 mod l (Theorem 324) In characteristic pthis follows analogously from (34)

By a result of Rost-Serre-Tignol there is little hope that this gives a generalway to approach Suslinrsquos conjecture (in moderate characteristics) Theyprove that given k contains a primitive 4-th root of unity the biquaternionk-algebra (a b)otimes (c d) is cyclic if and only if 〈〈a b c d〉〉 = 0 isin Wq(k) [RSTThm 3] By Milnorrsquos conjecture for quadratic forms (sect312) the latterinduces a b c d = 0 isin KM

4 (k)2 However Theorems I16 and I20 givecyclic biquaternion algebras A with SK1(A) 6= 0

(c) Overall viewpoint ndash Apart from the questions posed above it would alsobe interesting to find more examples of non-triviality of any of the existinginvariants It seems a very hard task to do so but a small improvementcould turn out to be a large step towards proving Suslinrsquos conjecture

91

Verification of cycle modulerules

Appendix A

ldquoMathematics is no more computationthan typing is literaturerdquo

mdash John Allen Paulos

In this appendix we verify that HlowastpnL of Definition 223 verifies the rules ofcycle modules as in sect12 (a) Recall that the data D1-D4 are given in sect221(d) (e) and (f)

Proposition A1Let (KR k) be a p-triple with (L S L) a finite Galois p-extensionThen HlowastpnL of Definition 223 respects the rules R1a-R3e FD and Cof cycle modules

Proof Rules R1a-R3e follow immediately from the definition of both HlowastpnLand its data D1-D4 Only rules R1c and R3b are maybe not straightforwardobtainable R1c relies on the universal property of tensor products R3b isproved by passing to completions and using [Ser1 Ch 2 Thm 1] (see eg[GS Cor 7311 amp Prop 741]) The proof of rule FD follows as in theclassical case of finite support of divisors [Har Ch II Lem 61]

We deduce now rule C from the fact that it holds for Milnor K-groups [Kat4]The residues partK for Milnor K-groups are explained in sect12 (d) To avoida K-cacophony we replace (KR k) by (FR F ) Let X be an integral R-scheme local of dimension 2 We suppose first that the structure morphismX rarr Spec(R) is surjective Then X = X timesR F is an F -scheme and Y =X timesRF is an F -scheme both of dimension 1 Furthermore char(F (X)) = 0and char(F (Y )) = p So we have to verify that the composition of residues

93

VERIFICATION OF CYCLE MODULE RULES

gives a complex (where y0 is the closed point of X and q ge 2)

Hq+1pnL(F (X))rarr

oplusxisinX(1)

HqpnL(F (x))oplus

oplusyisinY (0)

HqpnL(F (y))rarr Hqminus1

pnL(F (y0))

(A1)

We describe both the appearing groups and residues with K-groups as thisallows us to use rule C for MilnorrsquosK-groups We start describing the groupsby K-theory

bull The group Hq+1pnL(F (X))

AsΓ = Gal(Fnr(X)F (X)) sim= Gal(FnrF ) sim= Gal(F sF )

we know that cdp(Γ) le 1 [Ser2 Ch II Prop 3] The spectral sequenceof Hochschild-Serre

Est2 = Hs

(Γ Ht(Fnr(X) microotimesqpn )

)=rArr Hs+t(F (X) microotimesqpn )

induces an isomorphism

H1(Γ Hq(Fnr(X) microotimesqpn )) sim= ker

[Hq+1pn (F (X))rarr Hq+1

pn (Fnr(X))]

Furthermore the Bloch-Kato isomorphism gives usHq(Fnr(X) microotimesqpn ) sim=Kq(Fnr(X))pn So we get an isomorphism

H1(Γ Kq(Fnr(X))pn) sim= ker


pn (Fnr(X))]

(A2)

and hence an inclusion

Hq+1pnL(F (X)) sub H1(Γ Kq(Fnr(X))pn) (A3)

bull The group HqpnL(F (x)) for x isin X(1)

In the same way as above we get an inclusion

HqpnL(F (x)) sub H1(Γ Kqminus1(Fnr(x))pn) (A4)

bull The group HqpnL(F (y)) for y isin Y (0)

Let y isin Y (0) thenHqpn(F (y)) sim= H1

(F (y) νn(qminus1)F (y)s

)by (25) The

isomorphism of Bloch-Kato-Gabber νn(q minus 1)F (y)ssim= Kqminus1(F (y)s)pn


H1(F (y) Kqminus1(F (y)s)pn) sim= Hq+1

pn (F (y))

94


which also induces an inclusion

HqpnL(F (y))

sim= ker[H1(F (y) Kqminus1(F (y)s)pn

)rarr H1(L(y) Kqminus1(F (y)s)pn

)]sub ker

[H1(F (y) Kqminus1(F (y)s)pn

)rarr H1(F s(y) Kqminus1(F (y)s)pn

)]

(A5)

This last term is isomorphic to H1(Γ (Kqminus1(F (y)s)pn)ΓFs(y))

by theinflation-restriction sequence [GS Prop 3314]

bull The group Hqminus1pnL(F (y0)) for y0 the closed point of X

As above

Hqminus1pnL(F (y0)) sub H1

(Γ (Kqminus2(F (y0)s)pn)ΓFs(y0)

) (A6)

Let us now explain the residues by means of K-theory

bull The residue partx Hq+1pnL(F (X))rarr Hq

pnL(F (x)) for x isin X(1)The valuation attached to x induces a residue partx but also a Γ-equivariant residue partKx Kq(Fnr(X))pn rarr Kqminus1(Fnr(x))pn (asGal(Fnr(x)F (x)) sim= Γ) Hence this induces a morphism (which wegive the same name by a slight abuse of notation)

partKx H1(Γ Kq(Fnr(X))pn)rarr H1(Γ Kqminus1(Fnr(x))pn)

Lemma A2 (infra) induces that partKx is compatible with partx under theinclusions (A3) and (A4) in a commutative diagram

Hq+1pnL (F (X))

partx

H1(Γ Kq(Fnr(X))pn)

partKx

HqpnL (F (x)) H1(Γ Kqminus1(Fnr(x))pn

)

(A7)

bull The residue party Hq+1pnL(F (X))rarr Hq

pnL(F (y)) for y isin Y (0)Lemma A2 shows that under the injection (A5) im(party) ends up in

95


H1(Γ Kqminus1(F s(y))pn) On the other hand the valuation attached to

y induces a Γ-equivariant residue partKy Kq(Fnr(X)) rarr Kqminus1(F s(y)

)and hence a morphism

partKy H1(Γ Kq(Fnr(X))pn)rarr H1(Γ Kqminus1(F s(y))pn

)

Lemma A2 shows that we have a commutative diagram which explainsthe compatibility of party and partKy under the inclusions (A3) and (A5)

Hq+1pnL (F (X))

party

H1(Γ Kq(Fnr(X))pn)

partKy

HqpnL

(F (y)

) H1(Γ Kqminus1(F s(y))pn

)

(A8)

bull The residue partxy0 HqpnL(F (x))rarr Hqminus1

pnL(F (y0)) for x isin X(1)Lemma A2 shows that under the inclusion (A6) im(partxy0) is mappedinto H1(Γ Kqminus2(F s(y0))pn

) On the other hand we have a Γ-

equivariant residue partxKy0 Kqminus1(Fnr(x)) rarr Kqminus2(F s(y0)) giving onthe cohomological level a morphism

partxKy0 H1(Γ Kqminus1(Fnr(x))pn)rarr H1(Γ Kqminus2(F s(y0))pn)

Again Lemma A2 guarantees that partxKy0 is compatible with partxy0 underthe inclusions (A4) and (A6) so that we get a commutative diagram

HqpnL(F (x))

partxy0

H1(Γ Kqminus1(Fnr(x))pn)

partxKy0

Hqminus1pnL(F (y0)) H1(Γ Kqminus2(F s(y0))pn

)

(A9)

bull The residue partyy0 HqpnL(F (y))rarr Hqminus1

pnL(F (y0)) for y isin Y (0)In this situation we also have a residue partyy0 on the cohomology groupsand a Γ-equivariant residue in K-theory partyKy0 Kqminus1(F s(y)) rarr

96


Kqminus2(F s(y0)) (for y isin Y (0)) Then partyKy0 induces a morphism on thecohomological level

partyKy0 H1(Γ Kqminus1(F s(y))pn)rarr H1(Γ Kqminus2(F s(y0))pn)

Lemma A2 shows once more a compatibility of partyKy0 with partyy0 underthe inclusions (A5) and (A6)

HqpnL(F (y))

partyy0

H1(Γ Kqminus1(F s(y))pn)

partyKy0


)

(A10)

In total we have a collection of residues

H1(Γ Kq(Fnr(X))pn)minusrarroplus

xisinX(1)

H1(Γ Kqminus1(Fnr(x))pn)oplus

oplusyisinY (0)

H1(Γ Kqminus1(F s(y))pn

)minusrarr H1(Γ Kqminus2(F s(y0))pn

)

We know this is a complex as Milnorrsquos K-groups respect rule C [Kat3] Thecommutative diagrams (A7A8A9A10) then show that (A1) is a complexas well

If the structure morphism is not surjective X is either an F -scheme or anF -scheme If X is an F -scheme the cycle module consists of kernels of usual(moderate) Galois cohomology groups Rule C then follows immediatelyfrom rule C in the moderate case If X is an F -scheme we can rewrite (A1)using (25) and the isomorphism of Bloch-Kato-Gabber as

H1(Γ Kq(F s(X ))pn)rarr

oplusxisinX (1)

H1(Γ Kqminus1(F s(x))pn)

rarr H1(Γ Kqminus2(F s(x0))pn)

where x0 is the closed point of X This is again a complex as the residues areagain compatible with the residues from Milnorrsquos K-theory (see Lemma A2in the case ldquoy and y0rdquo) and as rule C holds for MilnorrsquosK-theory [Kat3]

97


Lemma A2Let X be an integral R-scheme local of dimension 2 with surjectivestructure morphism then the diagrams (A7A8A9A10) arecommutative

Proof We have to prove four situations let us treat them case by case

bull Diagram (A7) is commutative for x isin X(1)The Bloch-Kato isomorphism Kq(Fnr(X))pn sim= Hq(Fnr(X) microotimesqpn ) isdefined by the Galois symbol and hence commutes with the usualresidue on Hq(Fnr(X) microotimesqpn ) (with section given by the cup productwith a class of an uniformiser πx of the valuation associated with x)[GS Prop 751] One deduces the result from this as the isomorphism(A2) is an inflation and as partx also has a section given by the cupproduct with the class of πx

bull Diagram (A8) is commutative for y isin Y (0)Recall that we also have to verify that im(party) is contained inH1(Γ Kqminus1(F s(y))pn) As the residue party is defined by a section wecan take w otimes x2 otimes otimes xq isin Hq

pnL(F (y)) with w isin Wn(F (y)) andx2 xq isin Otimesy (Oy being the valuation ring corresponding to thevaluation associated with y) If πy is an uniformiser of Oy it is theresidue of

i(w) cup hqpnF (X)(πy x2 xq) isin Hq+1pnL (F (X))

Hence it corresponds to((σ(a)minus a)πy x2 xq

)σisin H1 (Γ Kq(Fnr(X))pn)

where a(p)minus a = w with a isin Wn(F (y)) and where we consider (σ(a)minusa) as an element of ZpnZ On the other hand w otimes x2 otimes otimes xqcorresponds to(

(σ(a)minus a)x2 xq)σisin H1(Γ Kqminus1(F (y)s)pn)

This implies the commutativity and that ((σ(a)minus a)x2 xq)σ isindeed an element of H1(Γ Kqminus1(F s(y))pn) as partKy has its images inthis group

98


bull Diagram (A9) is commutative for x isin X(1)The verification follows in an analogous way as the previous case

bull Diagram (A10) is commutative for y isin Y (0)The isomorphisms

νn(qminus1)F (y)ssim= Kqminus1(F (y)s)pn νn(qminus2)F (y0)s

sim= Kqminus2(F (y0)s)pn

and the residue Kqminus1(F (y)s)rarr Kqminus2(F (y0)s) induce a residue

νn(q minus 1)F (y)s rarr νn(q minus 2)F (y0)s defined by

aotimes π0 otimes x2 otimes otimes xqminus1 7rarr aotimes x2 otimes xqminus1

Here a isin Wn(Ov) and xi isin Otimesv where Ov is the valuation ringassociated with the valuation v induced by y0 with uniformiser π0By the definition of the residue partyy0 (see Remarks 222 and 227) it isclear that these residues are compatible

99

Elementary obstruction andWeil restriction

Appendix B

ldquoThe dream begins with a teacher who believesin you who tugs and pushes and leads youto the next plateau sometimes poking you

with a sharp stick called lsquotruthrsquordquomdash Dan Rather

ndash Dedicated to the memory of Joost van Hamel ndash

In this appendix we treat the subject of a first paper of the author [Wou1]It is not related to questions about SK1 but rather concerns the existenceof rational points on varieties The methods used though are similar to theones used in the main core of this article Galois cohomology homology It is this setting that made the author familiar with these techniques Theauthors owes a lot to Joost van Hamel for introducing him to this subjectThis appendix is dedicated to his memory

B1 Introduction

For a field k and a variety X over k (ie a separated k-scheme of finitetype) questions concerning k-rational points of X have been studied sinceages Different aspects arise in this area of research In this appendix wefocus on a certain obstruction to the existence of a rational point namely theelementary obstruction introduced by Colliot-Thelene and Sansuc [CTS2Sec 22]

In this appendix we denote by k a separable closure1 of k and Γk by Γ If Xis a smooth geometrically integral variety over k the elementary obstructionob(X) of X is defined as the class of the exact sequence of left Γ-modules

OB(X) = 1rarr ktimes rarr k(X)times rarr k(X)timesktimes rarr 1

1This conflicts with the conventions posed for the rest of this thesis This notationhowever keeps up with most of the publications on this subject

101

ELEMENTARY OBSTRUCTION AND WEIL RESTRICTION

as Yoneda extension in Ext1Γ(k(X)timesktimes ktimes) Note that we use the commonnotation k(X) for the function field of X = X timesk k Analogously we denotek[X] to be ring of regular functions on X If X contains a k-rational pointthen ob(X) = 0 [CTS2 Prop 222] Furthermore if k[X]times = k

times the classof

E(X) = 1rarr ktimes rarr k(X)times rarr Div(X)rarr Pic(X)rarr 1

in Ext2Γ(Pic(X) ktimes) is denoted by e(X) Colliot-Thelene and Sansuc showthat the morphism

δ Ext1Γ(k(X)timesktimes ktimes)rarr Ext2Γ(Pic(X) ktimes)

which arises in the long exact sequence induced by

1rarr k(X)timesktimes rarr Div(X)rarr Pic(X)rarr 1

is injective and that δ(ob(X)) = e(X) [CTS2 Prop 224] This is aconsequence of Shapirorsquos Lemma and Hilbert 90 Therefore it is also justifiedto say e(X) is the elementary obstruction of X In this paper we mainly usethis definition for the elementary obstruction

Several authors have been wondering whether the elementary obstructionbehaves well under classical geometric constructions A first observation isthat the elementary obstruction is a birational invariant since birationallyequivalent varieties have isomorphic function fields Wittenberg proves beingzero behaves well under rational maps [Wit2 Lem 312] Borovoi Colliot-Thelene and Skorobogatov wonder whether being zero behaves well underbase extension (ie whether ob(X) = 0 implies ob(X timesk K) = 0 for K afield extension of k and X a smooth geometrically integral variety over k)[BCTS Sec 2] They give several (partial) positive answers to this questionWittenberg gives a positive answer to this question for arbitrary (smoothproper geometrically integral) X when K is a p-adic or real closed field[Wit2 Cor 323] or when k is a number field and the Tate-Shafarevichgroup of the Picard variety of X is finite [Wit2 Cor 332] He also gives anegative answer to this question by producing a counterexample over C((t))(unpublished)

In this appendix we focus on the question whether being zero behaveswell under the Weil restriction of varieties To describe the problem moreexplicitly we first recall the definition of the Weil restriction

102

INTRODUCTION

Definition B1Let k be a field and kprime a finite field extension of k Let X be a varietydefined over kprime We say a variety RkprimekX over k is the Weil restriction(of scalars) of X to k if there is a kprime-morphism ϕ RkprimekX timesk kprime rarr Xsuch that for any k-variety Y and kprime-morphism f Y timesk kprime rarr X aunique k-morphism g Y rarr RkprimekX exists such that ϕ gprime = f Heregprime Y timesk kprime rarr RkprimekX timesk kprime is the kprime-morphism induced by g If theWeil restriction exists it is unique up to k-isomorphism

The following proposition guarantees the existence of the Weil restriction

Proposition B2Let k be a field k a separable closure and kprime a finite subextensionof k in k Denote Γ = Gal(kk) H = Gal(kkprime) and let X bea quasiprojective variety over kprime The Weil restriction RkprimekX of Xexists and

RkprimekX timeskprime k =prod

[σ]isinHΓσX

Here σX is the k-variety obtained by base extension from X timesk k byσ k rarr k and HΓ are the right cosets of H in Γ The kprime-morphismϕ RkprimekX timesk kprime rarr X is obtained by descent theory from its baseextension ϕ RkprimekX rarr X the projection onto the factor (id)X

For the proof see [Mil2 Prop 1626] Remark that if [σ] = [τ ] isin HΓ theuniversal property of fibre products guarantees σX and τX to be isomorphicas k-varieties The universal property of the Weil restriction gives also abĳection betweenRkprimekX(k) andX(kprime) as rational points are equivalent withsections of the structure morphism It is then natural to ask the followingquestion

Question B3Let k be a field and kprime a finite field extension Suppose X is a smoothgeometrically integral variety over kprime such that the Weil restrictionRkprimekX exists Does e(X) = 0 implies e(RkprimekX) = 0 and vice versa

103


We answer this question partially positively First we give a result on productvarieties as the Weil restriction is closely related to product varieties byProposition B2

B2 Product varieties

Let X and Y be two smooth geometrically integral varieties over a field kthen the following theorem is a merely homological result

Theorem B4The multiplication π k(X)timesktimes oplus k(Y )timesktimes rarr k(X timesk Y )timesktimesinduces a morphism by pullback

πlowastprime Ext1Γ(k(X timesk Y )timesktimes ktimes)rarr

Ext1Γ(k(X)timesktimes ktimes)oplus Ext1Γ(k(Y )timesktimes ktimes)

such that πlowastprime(ob(Xtimesk Y )) = (ob(X) ob(Y )) If k[X]times = ktimes = k[Y ]timesthen the Γ-morphism ψ Pic(X)oplusPic(Y )rarr Pic(X timesk Y ) defined bypullback of linebundles induces a morphism

ψlowastprime Ext2Γ(Pic(X timesk Y ) ktimes)rarr Ext2Γ(Pic(X) ktimes)oplus Ext2Γ(Pic(Y ) ktimes)

such that ψlowastprime(e(X timesk Y )) = (e(X) e(Y )) Even more πlowastprime and ψlowastprime

commute with the natural inclusions

Ext1Γ(k(Y )timesktimes ktimes)oplus Ext1Γ(k(Y )timesktimes ktimes)

δ

Ext2Γ(Pic(X) ktimes)oplus Ext2Γ(Pic(Y ) ktimes)

Ext1Γ(k(X timesk Y )timesktimes ktimes)

δ

πlowastprime

++VVVVVV

Ext2Γ(Pic(X timesk Y ) ktimes)ψlowastprime

++VVVVV

If π or ψ is an isomorphism then e(XtimeskY ) = 0 (resp ob(XtimeskY ) = 0)if and only if e(X) = 0 and e(Y ) = 0 (resp ob(X) = 0 and ob(Y ) = 0)

104

PRODUCT VARIETIES

Remark B5 ndash If X and Y are smooth geometrically integral varietiessatisfying k[X]times = ktimes = k[Y ]times then Xtimesk Y is also smooth geometricallyintegral and by a result of Rosenlicht [Ros1 Thm 2] it satisfies k[X timeskY ]times = k

times So speaking about e(X timesk Y ) in the second case does makesense

Proof If we denote the canonical isomorphism

Ext1Γ(k(X)timesktimes oplus k(Y )timesktimes ktimes)rarr


by ϕ then πlowastprime = ϕ πlowast is the required morphism where

πlowast Ext1Γ(k(X timesk Y )timesktimes ktimes)rarr Ext1Γ(k(X)timesktimes oplus k(Y )timesktimes ktimes)

is the pullback of 1-extensions by π We now prove the assertion on theelementary obstruction

We surely have a morphism of short exact sequences which consists ofproduct morphisms

1

1

ktimes oplus ktimes

π1 ktimes

k(X)times oplus k(Y )times

π2 k(X timesk Y )times

k(X)timesktimes oplus k(Y )timesktimes

π3=π k(X timesk Y )timesktimes

1 1105


Denote the left short exact sequence by E(X)oplusE(Y ) The right short exactsequence is E(X timesk Y ) By the general theory of Yoneda extensions [MLCh III] we get

ϕminus1(e(X) e(Y )) = [π1(E(X)oplus E(Y ))] = [E(X timesk Y )π3] = πlowast(e(X timesk Y ))

where π1(E(X) oplus E(Y )) denotes the pushforward of the Yoneda extensionE(X)oplus E(Y ) by π1 and E(X timesk Y )π3 denotes the pullback of the Yonedaextension E(X timesk Y ) by π3 This proves the first part

The second part is proved analogously using Γ-morphisms π4 Div(X) oplusDiv(Y ) rarr Div(X timesk Y ) and ψ Pic(X) oplus Pic(Y ) rarr Pic(X timesk Y ) Thecommutativity assertion follows from the following morphism of short exactsequences

1

1


π3 k(X timesk Y )timesktimes

Div(X)oplusDiv(Y )

π4 Div(X timesk Y )

Pic(X)oplus Pic(Y )

π5=ψ Pic(X timesk Y )

1 1

This induces a morphism of long exact sequences by Shapirorsquos lemma andHilbert 90 containing the required diagram

So we see that in any case e(X) = 0 and e(Y ) = 0 (resp ob(X) = 0 andob(Y ) = 0) if e(X times Y ) = 0 (resp ob(X times Y ) = 0) If ψ (resp π) is anisomorphism ψlowastprime (resp πlowastprime) is so too so in one of these cases the inverseimplication holds as well (recall that e(minus) = 0 if and only if ob(minus) = 0)

106

PRODUCT VARIETIES

Remark B6 ndash A known result says that if X and Y are varieties overseparable closed field k then as groups the morphism ψ Pic(X) oplusPic(Y )rarr Pic(X timesk Y ) defined by pull-backs has a section This sectionrestricts a line bundle on X timesk Y to x0 times Y and X times y0 where x0 and y0are base points on X and Y So as groups Pic(X) oplus Pic(Y ) is a directsummand of Pic(X timesk Y ) This looks interesting to get more informationon the structure of Ext2Γ(Pic(X timesk Y ) ktimes)In our case however X and Y are defined over a not necessarily separablyclosed field k and ψ Pic(X) oplus Pic(Y ) rarr Pic(X timesk Y ) is a Γ-morphismThe section however is not necessarily a Γ-morphism since the base pointsdo not have to behave well (if we do not know anything about the existenceof k-rational points on X and Y ) So we cannot use this result toextend the previous theorem in a direct way However we do retrievethe injectivity of the Γ-morphism ψ

Of course ψ Pic(X) oplus Pic(Y ) rarr Pic(X timesk Y ) does not need to bean isomorphism the product of an elliptic curve with itself delivering acounterexample [Har Ch IV Ex 410] We can however give sufficientconditions for ψ to be an isomorphism This involves the notion ofthe relative Picard functor and the Picard variety If X is a smoothgeometrically integral projective variety over a field k we denote the relativePicard functor by P icXk (see definition in the proof of Proposition B7)which is representable by a group variety Pic(X) the Picard variety Denoteby Pic0(X) the zerocomponent of Pic(X) (See [BLR Ch 8] for moreinformation)

Proposition B7If X is projective and Pic0(X) = 0 then ψ Pic(X) oplus Pic(Y ) rarrPic(X timesk Y ) is a Γ-isomorphism

Proof By Remark B6 we know that ψ is injective so it is sufficient to provecoker ψ = 0 By definition

P icXk(Y ) = Pic(X timesk Y )Pic(Y ) sim= Homk(Y Pic(X))

Any f isin Homk(Y Pic(X)) has a connected image but since Pic0(X) = 0the connected components of Pic(X) are its points So Homk(Y Pic(X))

107


consists of the constant maps onto a point of Pic(X) This does not dependon Y so

Homk(Y Pic(X)) sim= Homk(kPic(X)) sim= Pic(X)Because these isomorphisms are induced by the representability of the Picardfunctor

coker ψ = Pic(X timesk Y )Pic(Y )Pic(X)

sim=Pic(X)Pic(X)

= 0

Proposition B8If X is quasiprojective char(k) = 0 and Pic(X) is finitely generatedthen Pic(X)oplus Pic(Y ) sim= Pic(X timesk Y )

Proof Say X sub X1 for a projective variety X1 Since char(k) = 0 thereexists a (smooth projective) Hironaka desingularisation X prime of X1 As X issmooth X is isomorphic to an open of X prime So without loss of generality weassume X to be an open part of X prime The exact sequence

DivXprimeX(X prime)rarr Pic(X prime)rarr Pic(X)rarr 0

induces Pic(X prime) to be finitely generated as Pic(X) and DivXprimeX(X) arefinitely generated (DivXprimeX(X) are the divisors on X prime with support outsideX)

It suffices to prove Pic(X prime timesk Y ) sim= Pic(X prime) oplus Pic(Y ) as this also inducesPic(X timesk Y ) sim= Pic(X)oplus Pic(Y ) Indeed there is a commutative diagram

0 Pic(X prime)oplus Pic(Y )

Pic(X prime timesk Y )

0 Pic(X)oplus Pic(Y )

Pic(X timesk Y )

0 0

108

WEIL RESTRICTION

where the vertical arrows are the surjective restriction morphisms If theinjection of the first row turns out to be an isomorphism then the injectionof the bottom row is also surjective hence it is an isomorphism

Because Pic(X prime) is finitely generated we have Pic0(X prime) = 0 Indeed ifPic0(X prime) 6= 0 then Pic0(X prime) is an abelian variety of dimension m gt 0whose group of k-points is finitely generated as Pic(X prime) = Homk(kPic(X prime))is finitely generated On the other hand the group of k-points of an abelianvariety is divisible [Fre Thm 2] But a divisible non-trivial finitelygenerated group does not exist In this way we get a contradiction andso the proposition follows by Proposition B7

Consequently we obtain the following result

Corollary B9Let X and Y be smooth geometrically integral varieties over a fieldk with k[X]times = ktimes = k[Y ]times Let k be a separable closure of k andΓ = Gal(kk) If one of the following conditions holds

(i) X is projective and Pic0(X) = 0 or

(ii) X is quasiprojective char(k) = 0 and Pic(X) is finitelygenerated

then


is an isomorphism such that ψlowastprime(e(X timesk Y )) = (e(X) e(Y ))

So if one of the conditions is true e(X timesk Y ) = 0 if and only if e(X) = 0and e(Y ) = 0

B3 Weil restriction

Knowing more on the case of product varieties we proceed to theWeil restriction Throughout this section we assume that kprime is a finite

109


subextension of a field k in k Denote H = Gal(kkprime) and let X be a smoothgeometrically integral quasiprojective variety over kprime The Weil restrictionof X from kprime to k exists by Proposition B2 and we abbreviate it as R

Proposition B10The natural H-morphism k(X)times rarr k(R)times induces a pullback of 1-extensions

Πlowast Ext1Γ(k(R)timesktimes ktimes)rarr Ext1H(k(X)timesktimes ktimes)

with Πlowast(ob(R)) = ob(X) If furthermore k[X]times = ktimes then the

natural H-morphism Pic(X) rarr Pic(R) induces a pullback of 2-extensions

Φlowast Ext2Γ(Pic(R) ktimes)rarr Ext2H(Pic(X) ktimes)

with Φlowast(e(R)) = e(X) As in Proposition B4 these morphismscommute with the natural inclusions sending ob(minus) to e(minus)

Remark B11 ndash The natural H-morphisms mentioned in the propositionare induced by Proposition B2 This proposition gives a kprime-morphismϕ Rtimesk kprime rarr X retrieved by descent from the k-projection ϕ R rarr XThis morphism ϕ gives by pullback of principle divisors and line bundlesthe required H-morphisms

Remark B12 ndash As in Remark B5 it is true that k[R]times = ktimes provided

k[X]times = ktimes So it makes sense to speak about e(R) if at first glance we

only require k[X]times = ktimes

Proof We give the proof of the assertion on 2-extensions The assertion on1-extensions follows in the same way The commutative part follows as inProposition B4

Denote the H-morphism Pic(X)rarr Pic(R) by ϕprime This induces a pullback

ϕprimelowast Ext2H(Pic(R) ktimes)rarr Ext2H(Pic(X) ktimes)

If we use the forgetful map

π Ext2Γ(Pic(R) ktimes)rarr Ext2H(Pic(R) ktimes)

110

WEIL RESTRICTION

we get the required morphism Φlowast = ϕprimelowast π To prove Φlowast(e(R)) = e(X) weuse the morphism E(X)rarr E(R) of H-extensions

1 ktimes

id

k(X)times

Div(X)

Pic(X)

ϕprime

1

1 ktimes k(R)times Div(R) Pic(R) 1

As it is clear that the H-equivalence class of E(R) equals π([e(R)]) we getfrom elementary homological reasons

Φlowast(e(R)) = ϕprimelowast(π([e(R)])) = [E(X)] = e(X)

So e(R) = 0 implies e(X) = 0 We proceed figuring out when the converse istrue This holds in the very same situation as the converse holds for productvarieties To prove this we use the notion of induced group module withsome corresponding notation Let G be a profinite group H a subgroupof G and A a left H-module then the induced G-module is IndGH(A) =Z[G]otimesZ[H]A where Z[G] is considered as a right Z[H]-module This is a leftG-module the G-action is defined by γprime(γ otimes a) = γprimeγ otimes a for any a isin A andγ γprime isin G If A and B are left H-modules and f Ararr B is an H-morphismthen we get an induced G-morphism

IndGH(f) IndGH(A) 7rarr IndGH(B) defined by γ otimes a 7rarr γ otimes f(a)

for a isin A and γ isin G If B is also a left G-module we write IndGH(f)prime for theG-morphism π IndGH(f) with

π IndGH(B)rarr B defined by γ otimes b 7rarr γb

If E is an exact sequence

A1f1

A2f2

A3

then we get an induced exact sequence IndGH(E)

IndGH(A1)f1

IndGH(A2)f2

IndGH(A3)

111


where we have denoted fi = IndGH(fi) for sake of simplicity

Theorem B13If k[X]times = k

times and if one of the two following conditions is true



then Φlowast of Proposition B10 is an isomorphism

Proof We prove this result by giving another description of Φlowast

If ϕprime is the H-morphism Pic(X) rarr Pic(R) as defined in the proof ofProposition B10 the induced Γ-morphism IndΓ

H(ϕprime)prime IndΓH(Pic(X)) rarr

Pic(R) gives a pullback of 2-extensions

IndΓH(ϕprime)primelowast Ext2Γ(Pic(R) ktimes)rarr Ext2Γ(IndΓ

HPic(X) ktimes)

Furthermore say πprime is the forgetful map

πprime Ext2Γ(IndΓH(Pic(X)) ktimes)rarr Ext2H(IndΓ

H(Pic(X)) ktimes)

and letilowast Ext2H(IndΓ

H(Pic(X)) ktimes)rarr Ext2H(Pic(X) ktimes)

be the pullback by i Pic(X) rarr IndΓH(Pic(X)) L 7rarr id otimes L We have the

following situation

Ext2Γ(Pic(R) ktimes)π Ext2H(Pic(R) ktimes)

ϕprimelowast Ext2H(Pic(X) ktimes)

Ext2Γ(IndΓH(Pic(X)) ktimes)

πprime

IndΓH(ϕprime)primelowast

Ext2H(IndΓH(Pic(X)) ktimes)

ilowastltltzzz

We prove Φlowast = ϕprimelowast π is an isomorphism by proving that ilowast πprime IndΓH(ϕprime)primelowast

is an isomorphism and that the diagram above commutes The latter followsdirectly from elementary homological reasons

112

WEIL RESTRICTION

To prove the former first observe that ilowast πprime is an isomorphism by ShapirorsquosLemma as it has an inverse IndΓ

H(id)primelowast IndΓH with

IndΓH Ext2H(Pic(X) ktimes)rarr Ext2Γ(IndΓ

H(Pic(X)) IndΓH(ktimes))

[E] 7rarr [IndΓH(E)]

and IndΓH(id)primelowast the pushforward

Ext2Γ(IndΓH(Pic(X)) IndΓ

H(ktimes))rarr Ext2Γ(IndΓH(Pic(X)) ktimes)

by IndΓH(id)prime IndΓ

H(ktimes) rarr ktimes This is indeed an inverse by elementary

homological reasons

So it remains to prove IndΓH(ϕprime)primelowast is an isomorphism We first choose a set

of representatives σ1 σn of the classes of HΓ with σ1 = id

If Condition (i) or (ii) is true then pullback along all components

ψ noplusi=1

Pic(σiX)rarr Pic(R)

is an isomorphism of H-modules by Proposition B7 and B8 We provethere is a 1-1 correspondence τ IndΓ

H(Pic(X)) rarroplusn

i=1 Pic(σiX) and thatψ τ = IndΓ

H(ϕ)prime This induces IndΓH(ϕ)prime to be an isomorphism

First remark that for all i = 1 n base extension by σi induces a bĳectionBi Pic(X) rarr Pic(σiX) which does not need to be a H-morphism as Hdoes not necessarily commute with σi There are also H-morphisms ψi Pic(σiX)rarr Pic(R) induced by projection on the i-th factor so ψ =

sumni=1 ψi

and ψ1 = ϕprime It is easy to see that the Bi and ψi relate as σminus1i ψi(Bi(L)) =

ψ1(L) for any L isin Pic(X)

To define τ it satisfies defining τ(γ otimes L) for any L isin Pic(X) and γ isin ΓSuppose γ = σih for h isin H and 1 le i le n then we set τ(γ otimes L) with 0 as[σj ]-components for j 6= i and Bi(hL) as [σi]-component This is well definedand as all the Bi are bĳections τ is indeed a 1-1 correspondence Even more

ψ τ(γ otimes L) = ψi(Bi(hL)) = σiψ1(hL) = γψ1(L) = IndΓH(ϕ)prime(γ otimes L)

So if one of the two conditions holds e(X) = 0 if and only if e(R) = 0

113

Bibliography

[AG] Maurice Auslander and Oscar Goldman The Brauer group of acommutative ring Trans Amer Math Soc 97367ndash409 1960

[Alb1] Adrian Albert Normal division algebras of degree four over an algebraicfield Trans Amer Math Soc 34(2)363ndash372 1932

[Alb2] Adrian Albert Simple algebras of degree pe over a centrum ofcharacteristic p Trans Amer Math Soc 40(1)112ndash126 1936

[Bae] Ricardo Baeza Quadratic forms over semilocal rings Lecture Notes inMathematics Vol 655 Springer-Verlag Berlin 1978

[BCTS] Mikhail Borovoi Jean-Louis Colliot-Thelene and Alexei SkorobogatovThe elementary obstruction and homogeneous spaces Duke Math J141(2)321ndash364 2008

[BK] Spencer Bloch and Kazuya Kato p-adic etale cohomology Publ MathInst Hautes Etudes Sci (63)107ndash152 1986

[Bla] Altha Blanchet Function fields of generalized Brauer-Severi varietiesComm Algebra 19(1)97ndash118 1991

[BLR] Siegfried Bosch Werner Lutkebohmert and Michel Raynaud NeronModels volume 21 of Ergebnisse der Mathematik und ihrer Grenzgebiete3 Folge Springer Verlag Berlin 1990

[Bou] Nicolas Bourbaki Elements de mathematique volume 864 of LectureNotes in Mathematics Masson Paris 1981 Algebre Chapitres 4 a 7

[BT] Hyman Bass and John Tate The Milnor ring of a global field InAlgebraic K-theory II ldquoClassicalrdquo algebraic K-theory and connectionswith arithmetic (Proc Conf Seattle Wash Battelle Memorial Inst1972) pages 349ndash446 Lecture Notes in Math Vol 342 Springer Berlin1973

[Car] Pierre Cartier Questions de rationalite des diviseurs en geometriealgebrique Bull Soc Math France 86177ndash251 1958

[Coh] Irvin Cohen On the structure and ideal theory of complete local ringsTrans Amer Math Soc 5954ndash106 1946

[CT] Jean-Louis Colliot-Thelene Birational invariants purity and the Gerstenconjecture In K-theory and algebraic geometry connections withquadratic forms and division algebras (Santa Barbara CA 1992)volume 58 of Proc Sympos Pure Math pages 1ndash64 Amer Math SocProvidence RI 1995

[CTS1] Jean-Louis Colliot-Thelene and Jean-Jacques Sansuc La R-equivalencesur les tores Ann Sci Ecole Norm Sup (4) 10(2)175ndash229 1977

115

BIBLIOGRAPHY

[CTS2] Jean-Louis Colliot-Thelene and Jean-Jacques Sansuc La descente sur lesvarietes rationnelles II Duke Math J 54375ndash492 1987

[Dra] Peter Draxl Skew Fields volume 81 of London Mathematical SocietyLecture Note Series Cambridge University Press Cambridge 1983

[EKLV] Helene Esnault Bruno Kahn Marc Levine and Eckart Viehweg TheArason invariant and mod 2 algebraic cycles J Amer Math Soc11(1)73ndash118 1998

[Fre] Gerhard Frey On the structure of the class group of a function fieldArch Math 3833ndash40 1979

[Gar] Skip Garibaldi Cohomological invariants exceptional groups and spingroups Mem Amer Math Soc 200(937)xii+81 2009 With an appendixby Detlev W Hoffmann

[Gil1] Philippe Gille Invariants cohomologiques de Rost en caracteristiquepositive K-Theory 2157ndash100 2000

[Gil2] Philippe Gille Le probleme de Kneser-Tits Asterisque (326) 2009Seminaire Bourbaki no 983

[GMS] Skip Garibaldi Alexander Merkurjev and Jean-Pierre Serre Cohomo-logical invariants in Galois cohomology volume 28 of University LectureSeries Amer Math Soc 2003

[Gro1] Alexander Grothendieck Elements de Geometrie Algebrique IV Etudelocale des schemas et des morphismes de schemas Premiere Partievolume 20 of Publ Math Inst Hautes Etudes Sci Bures-sur-Yvette1964

[Gro2] Alexander Grothendieck Le groupe de Brauer I Algebres drsquoAzumayaet interpretations diverses Seminaire Bourbaki 9199ndash219 1964-1966Expose No 290

[GS] Philippe Gille and Tamas Szamuely Central Simple Algebras and GaloisCohomology volume 101 of Cambridge studies in advanced mathematicsCambridge University Press Cambridge 2006

[Har] Robin Hartshorne Algebraic Geometry volume 52 of Graduate Texts inMathematics Springer Science+Business Media Inc New York 1977

[Izh] Oleg Izhboldin On the cohomology groups of the field of rational functionsIn Mathematics in St Petersburg volume 174 of Amer Math Soc TranslSer 2 pages 21ndash44 Amer Math Soc Providence RI 1996

[Kah1] Bruno Kahn Applications of weight-two motivic cohomology Doc MathJ DMV 1395ndash416 1996

[Kah2] Bruno Kahn Formes quadratiques sur un corps volume 15 of CoursSpecialises Societe Mathematique de France 2008

[Kah3] Bruno Kahn Cohomological approaches to SK1 and SK2 of central simplealgebras Preprint 2009

[Kat1] Kazuya Kato Galois cohomology of complete discrete valuation fields InAlgebraic K-Theory volume 967 of Lecture notes in mathematics pages215ndash238 Berlin 1982

116

BIBLIOGRAPHY

[Kat2] Kazuya Kato Symmetric bilinear forms quadratic forms and MilnorK-theory in characteristic two Invent Math 66(3)493ndash510 1982

[Kat3] Kazuya Kato A Hasse principle for two-dimensional global fields JReine Angew Math 366142ndash183 1986

[Kat4] Kazuya Kato Milnor K-theory and the Chow group of zero cycles InApplications of algebraic K-theory to algebraic geometry and numbertheory Part I II (Boulder Colo 1983) volume 55 of Contemp Mathpages 241ndash253 Amer Math Soc Providence RI 1986

[KMRT] Max-Albert Knus Alexander Merkurjev Markus Rost and Jean-PierreTignol The book of involutions volume 44 of Amer Math Soc ColloqPubl 1998

[Knu] Max-Albert Knus Quadratic and Hermitian forms over rings volume294 of Grundlehren der Mathematischen Wissenschaften Springer-VerlagBerlin 1991

[KO] Max-Albert Knus and Manuel Ojanguren Theorie de la Descenteet Algebres drsquoAzumaya volume 389 of Lecture Notes in MathematicsSpringer-Verlag Berlin 1974

[Lan] Serge Lang Algebra volume 211 of Graduate Texts in MathematicsSpringer-Verlag New York third edition 2002

[Lic] Stephen Lichtenbaum The construction of weight-two arithmeticcohomology Invent math 88183ndash215 1987

[Mat] Hideyuki Matsumura Commutative ring theory volume 8 of CambridgeStudies in Advanced Mathematics Cambridge University PressCambridge 1986 Translated from the Japanese by M Reid

[Mer1] Alexander Merkurjev Generic element in SK1 for simple algebras K-Theory 7(1)1ndash3 1993

[Mer2] Alexander Merkurjev K-theory of simple algebras In K-theory andalgebraic geometry connections with quadratic forms and division algebras(Santa Barbara CA 1992) volume 58 of Proc Sympos Pure Math pages65ndash83 Amer Math Soc Providence RI 1995

[Mer3] Alexander Merkurjev Invariants of algebraic groups J reine angewMath 508127ndash156 1999

[Mer4] Alexander Merkurjev The group SK1 for simple algebras K-Theory37(3)311ndash319 2006

[Mil1] James Milne Etale cohomology volume 33 of Princeton MathematicalSeries Princeton University Press Princeton NJ 1980

[Mil2] James Milne Algebraic Geometry Taiaroa Publishing Erehwon 5thedition Februari 2005 httpwwwjmilneorg

[Mil3] James Milne Algebraic groups and arithmetic groups 2006 httpwwwjmilneorgmath

[Mil4] James Milne Algebraic number theory 2009 httpwwwjmilneorgmath

117

BIBLIOGRAPHY

[Mil5] John Milnor Algebraic K-theory and quadratic forms Invent Math9318ndash344 19691970

[ML] Saunders Mac Lane Homology volume 114 of Die Grundlehren derMathematischen Wissenschaften Springer Verlag Berlin 1967

[Nak] Tadasi Nakayama Cohomology of class field theory and tensor productmodules I Ann of Math (2) 65255ndash267 1957

[NM] Tadasi Nakayama and Yozo Matsushima Uber die multiplikative Gruppeeiner p-adischen Divisionsalgebra Proc Imp Acad Tokyo 19622ndash6281943

[OVV] Dmitri Orlov Alexander Vishik and Vladimir Voevodsky An exactsequence for KMlowast 2 with applications to quadratic forms Ann of Math165(1)1ndash13 2007

[Pan] Ivan Panin Splitting principle and K-theory of simply connectedsemisimple algebraic groups Algebra i Analiz 10(1)88ndash131 1998

[Pie] Richard Pierce Associative algebras volume 88 of Graduate Texts inMathematics Springer-Verlag New York 1982 Studies in the History ofModern Science 9

[Pla] Vladimir Platonov The Tannaka-Artin problem and reduced K-theoryMath USSR Izv 10(2)211ndash243 1976 English translation

[PTY] A V Prokopchuk S V Tikhonov and V I Yanchevskiı Ob obxih

lementah v gruppah SK1 dl central~nyh prostyh algebr (Genericelements in the groups SK1 for central simple algebras) Vestsı NatsAkad Navuk Belarusı Ser Fız-Mat Navuk (3)35ndash42 126 2008

[Ras] Wayne Raskind Abelian class field theory of arithmetic schemes InK-theory and algebraic geometry connections with quadratic forms anddivision algebras (Santa Barbara CA 1992) volume 58 of Proc SymposPure Math pages 85ndash187 Amer Math Soc Providence RI 1995

[Ros1] Maxwell Rosenlicht Toroidal algebraic groups Proc Amer Math Soc12984ndash988 1961

[Ros2] Markus Rost Chow Groups with Coefficients Doc Math J DMV1319ndash393 1996

[Ros3] Markus Rost The basic correspondence of a splitting variety 1998 Notesdownloadable from his website

[RST] Markus Rost Jean-Pierre Serre and Jean-Pierre Tignol La forme tracedrsquoune algebre simple centrale de degre 4 C R Math Acad Sci Paris342(2)83ndash87 2006

[RTW] J-F Renard Jean-Pierre Tignol and Adrian Wadsworth GradedHermitian forms and Springerrsquos theorem Indag Math (NS) 18(1)97ndash134 2007

[RTY] Ulf Rehmann Sergey Tikhonov and Vyacheslav Yanchevskiı Symbolsand cyclicity of algebras after a scalar extension Fundam Prikl Mat14(6)193ndash209 2008

118

BIBLIOGRAPHY

[San] Jean-Jacques Sansuc Groupe de Brauer et arithmetique des groupesalgebriques lineaires J reine angew Math 32712ndash80 1981

[Sch] Colette Schoeller Groupes affines commutatifs unipotents sur un corpsparfait Bulletin de la SMF 100241ndash300 1972

[Ser1] Jean-Pierre Serre Corps Locaux Publications de lrsquoInstitut deMathematique de lrsquoUniversite de Nancago Hermann Paris 1968

[Ser2] Jean-Pierre Serre Galois Cohomology Springer Monographs inMathematics Springer-Verlag Berlin 2002

[SGA] Schemas en groupes III Structure des schemas en groupes reductifsSeminaire de Geometrie Algebrique du Bois Marie 196264 (SGA3) Dirige par M Demazure et A Grothendieck Lecture Notes inMathematics Vol 153 Springer-Verlag Berlin 19621964

[Sus] Andrei Suslin SK1 of division algebras and Galois cohomology InAlgebraic K-theory volume 4 of Adv Soviet Math pages 75ndash99 AmerMath Soc Providence RI 1991

[SVdB] Aidan Schofield and Michel Van den Bergh The index of a Brauer classon a Brauer-Severi variety Trans Amer Math Soc 333(2)729ndash7391992

[Tat] John Tate The cohomology groups of tori in finite Galois extensions ofnumber fields Nagoya Math J 27709ndash719 1966

[Voe] Vladimir Voevodsky On Motivic Cohomology with Zl coefficientsPreprint 2009

[Wad] Adrian Wadsworth Valuation theory on finite dimensional divisionalgebras In Valuation theory and its applications Vol I (Saskatoon SK1999) volume 32 of Fields Inst Commun pages 385ndash449 Amer MathSoc Providence RI 2002

[Wan] Shianghaw Wang On the commutator group of a simple algebra AmerJ Math 72323ndash334 1950

[Wed] Joseph Wedderburn On hypercomplex numbers London M S Proc2(6)77ndash118 1908

[Wei1] Charles Weibel An introduction to homological algebra volume 38 ofCambridge Studies in Advanced Mathematics Cambridge UniversityPress Cambridge 1997

[Wei2] Charles Weibel The norm residue isomorphism theorem J Topol2(2)346ndash372 2009

[Wit1] Ernst Witt Zyklische Korper und Algebren der Charakteristic p vomGrad pn J reine angew Math 176126ndash140 1937

[Wit2] Olivier Wittenberg On albanese torsors and the elementary obstructionto the existence of 0-cycles of degree 1 Math Ann 340(4)805ndash838 2008

[Wou1] Tim Wouters The elementary obstruction and the Weil restrictionManuscripta Math 128(2)137ndash146 2009

[Wou2] Tim Wouters Comparing invariants of SK1 Preprint 2010[Wou3] Tim Wouters Lrsquoinvariant de Suslin en caracteristique positive To appear

in Journal of K-Theory 2010

119

Glossary

In the glossary k represents a field F a field extension of it A a centralsimple k-algebra and X a k-scheme For some definitions we need furtherassumptions on the objects used See the exact definition for the rightassumptions

〈a1 an〉 quadratic n-form 66〈〈a1 an〉〉 n-fold Pfister form 66Ab the category of commutative groups ix[(a b) either (a b) or [a b) 4(a b) biquaternion k-algebra with char(k) 6= 2 4[a b) biquaternion k-algebra with char(k) = 2 4[a b)p p-algebra 4(a b)p symbol algebra 4[(a b)p either (a b)p or [a b)p 4Ai(XMj) i-th homology group of weight j of the Gersten

complex associated with X and M21

Ai(XMj)mult multiplicative subgroup of A0(XMj) 23AF base extension of A to F ixA0(XMj) reduced subgroup of A0(XMj) 23

simBr Brauer-equivalent 2Br(Fk) ker[Br(k)rarr Br(F )] 2Br(k) Brauer group of k 2nBr(k) part of n-torsion of Br(k) 14

cd(k) cohomological dimension of k ixcdp(k) p-cohomological dimension of k (for a prime p) ixchar(k) characteristic of kCqpn(k) logarithmic differentials of k (char(k) = p) 42

deg(A) degree of A 3Dqpn(k) Wn(k)otimes (ktimes)otimesq (char(k) = p) 41

121

GLOSSARY

Fnr maximal unramified extension of a discretevalued field F

x

(FOv κ(v)) valuation triple associated with a discretevaluation v on F

36

Gal(Fk) Galois group of F over kΓK absolute Galois group of k ixGm Spec(Z[T Tminus1]) ixGroups the category of groups ix

H i+1m (F ) H i+1

pl (F ) oplus H i+1r (F ) if char(F ) = p and m =

plr with p - r14 41

H i+1nAotimesr(F ) relatif H i+1

n (F ) with respect to Aotimesr 16HlowastmL cycle module associated with H i+1

m (F ) 21 45HlowastnLAotimesr relatif cycle module associated with H i+1

nAotimesr(F ) 21 4857

H i+1pnnr(F ) unramified cohomology 43

hnpF differential symbol of F of degree n (char(F ) =p)

46

hnmF Galois symbol of F of degree n isin Ftimes andweight m

15

I(k) fundamental ideal of W (k) 67indk(A) index of A 3Invj(GM) invariants of degree j of a group functor G in

a cycle module M23

InWq(k) In(k) middotWq(k) 67InWq(k) InWq(k)In+1Wq(k) 67InW primeq(k) In(k) middotW primeq(k) 67InW primeq(k) InW primeq(k)In+1W primeq(k) 67

Jq(k) certain subgroup of Dpn(k) (char(k) = p) 42

κ(v) residue field of a discrete valuation v xk algebraic closure of k ixk-fields the category of field extensions of k ix(Kk σ a) cyclic algebra 3Kn(F ) n-th Milnor K-group of F 15ks separable closure of k ix

122

GLOSSARY

k((t1)) ((tn)) n-fold iterated Laurent series field over k ix

Mn(k) matrix algebra of ntimes n matrices over kmicrom the Γk-module of m-th roots of unity in ks ixmicrom(k) m-th roots of unity in k x

n integer defined using a prime decomposition ofn

27

NFk norm of a finite field extension F of k 5NrdAk reduced norm of A 5Nrpσk Pfaffian norm of A 65νn(q) kernel of the Cartier morphism 42

Ωqk q-differentials on k 41Ov valuation ring of a discrete valuation v x

perk(A) period of A 3PGLinfin projective linear group scheme 32PGSp(A σ) certain group scheme associated with A with

symplectic involution σ69

Pic(X) Picard variety of X 107P icXk Picard functor of X 107Prdak(X) reduced characteristic polynompial of a isin A 5Prpσak(X) Pfaffian characteristic polynompial of a isin A 65

R-fields the category of R-algebras which fields 18ρBIA KMRTrsquos invariant of SK1(A) with A a

biquaternion k-algebra67

ρKahnA Kahnrsquos 2006 invariant of SK1(A) 27ρKahnA Kahnrsquos 2006 generalised invariant of SK1(A) 58ρrA Kahnrsquos r-th invariant of SK1(A) 27ρrA Kahnrsquos r-th generalised invariant of SK1(A) 58ρRostA Rostrsquos invariant of SK1(A) with A a biquater-

nion k-algebra25

ρS06A Suslinrsquos 2006 invariant of SK1(A) 26ρS06A Suslinrsquos 2006 generalised invariant of SK1(A) 58ρS91A Suslinrsquos 1991 invariant of SK1(A) 25ρS91A Suslinrsquos 1991 generalised invariant of SK1(A) 58R1kprimek(Gm) ker(Rkprimek(Gm)rarr Gm) 85

Rkprimek(Gm)Gm coker(Gm rarr Rkprimek(Gm)) 85

123

GLOSSARY

Rkprimek(Y ) Weil restriction of scalars to k 103

SB(A) Severi-Brauer variety of A 26Sets the category of sets ixSK1(A) reduced Whitehead group of A 5SK1(A) reduced Whitehead group functor of A 8SL1(A) k-points of the special linear group of A 9SL1(A) special linear group of A 9Symd(A σ) symmetrised elements in A under involution σ 65

T dual of a torus T 85TrFk trace of a finite field extension F of k 5TrdAk reduced trace of A 5Trpσk Pfaffian trace of A 65

W (k) Witt ring of k 65Wn(k) Witt p-vectors of length n on k (char(k) = p) 41Wq(k) Witt group of k 65W primeq(k) subgroup of Wq(k) consisting of even-

dimensional non-singular quadratic spaces67

X(i) set of points of codimension i of X xX(F ) F -rational points of X ixXF base extension of X to F ix

124

Index

Azumaya algebra 31

biquaternion algebra 11Bloch-Kato isomorphism 15Bloch-Kato-Gabber isomorphism 46Brauer class 2Brauer group 2Brauerrsquos decomposition theorem 9Brauer-equivalence 2

cohomological invariant 13cyclic algebra 3

differential symbol 46discrete valuation xdivision algebras 1dual torus 85

elementary obstruction 101equicharacteristic x

flasque resolution 86flasque torus 86fundamental ideal 67

Galois symbol 15Gersten complex 21group functor 13

Henselrsquos lemma a la Grothendieck 32hyperbolic involution 67

index reduction formula 9invariant 13involution 64

KMRTrsquos invariant 63Kneser-Tits problem 9

lifted Azumaya algebra 31logarithmic differentials 41

Milnor K-groups 15Milnorrsquos conjectures 68mixed characteristic xMorita invariance of SK1 6Morita isomorphism 5multiplicative subgroup 23

orthogonal involution 64

p-algebra 4p-extension 36p-ring 33p-triple 36Pfaffian characteristic polynomial 65Pfaffian norm 65Pfaffian trace 65Pfister form 66Picard functor 107Picard variety 107Platonovrsquos examples 7

R-field 18R-valuation triple 36reduced Whitehead group 6residue morphism 16rigidity 32

Severi-Brauer variety 26special linear group 6splitting field 2Suslinrsquos conjecture 8Suslinrsquos strong conjecture 89symbol algebra 4symmetrised elements 65

125

INDEX

symplectic involution 64

Tannaka-Artin problem 7Tate twist 14

unramified cohomology 43unramified division algebra 32

valuation triple 36

Wangrsquos theorem 7Wedderburnrsquos theorem 1Weil restriction 102Weil restriction of scalars 103Whitehead group 5Witt group 65Witt ring 65Witt vectors 34Witt-equivalence 65

126

And now the end is nearAnd so I face the final curtainMy friends Irsquoll say it clearIrsquoll state my case of which Irsquom certain

Irsquove lived a life thatrsquos fullIrsquove travelled each and every highwayAnd more much more than thisI did it my way

Frank Sinatra

Arenberg Doctoraatsschool Wetenschappen amp TechnologieFaculteit Wetenschappen

Departement WiskundeAfdeling Algebra

Celestĳnenlaan 200B - bus 2400 3001 Leuven

Dankwoord

Abstract

Samenvatting

Contents


Introduction

SK1 of a central simple algebra

Wangs theorem and Suslins conjecture

Reductions of the problem

Overview of the thesis


Cohomology groups

Cycle modules

Invariants agrave la Merkurjev

Invariants of SK1

Lifting and specialising invariants

Moderate case

Wild case

General case

Some remarks


Invariants for biquaternion algebras

Kahns invariant

Conclusion

Verification of cycle module rules

Elementary obstruction and Weil restriction

Introduction

Product varieties

Weil restriction

Bibliography

Glossary

Index


Tim Wouters

PromotorenProf Philippe GilleProf Joost van HamelProf Willem Veys

ExamencommissieProf Jan Denef Proefschrift voorgedragenProf Paul Igodt tot het behalen van deProf Johannes Nicaise graad van Doctor in deProf Jean-Pierre Tignol Wetenschappen (Wiskunde)Prof Jan Van Geel

Mei 2010









soobwestvu

Dankwoord







iii

DANKWOORD






Tim WoutersMei 2010

iv

Abstract





v

SAMENVATTING

Samenvatting





vi

Contents

Dankwoord iii

Abstract v

Samenvatting vi

Contents vii


Introduction 1







12 Cycle modules 18




21 Moderate case 29

vii

CONTENTS

22 Wild case 40

23 General case 56

24 Some remarks 58




Conclusion 89



B1 Introduction 101



Bibliography 115

Glossary 121

Index 125

viii










ix








x

Introduction










1

INTRODUCTION









2

INTRODUCTION



dimk(A)








i=0 Kxi with


3

INTRODUCTION









4











A 00 1

)isin GLn+1(R)

Then define

GLinfin(R) =⋃ngt0




)

5

INTRODUCTION















)

6




SK1(D)

sim=











7

INTRODUCTION











8




I)



k









9

INTRODUCTION










10



sumpminus1i=0 X










11

INTRODUCTION



12


Chapter 1






A(K)

ϕK M(K)

A(K prime)ϕKprime

M(K prime)




13















14

COHOMOLOGY GROUPS



n times

J





mF (xn) (13)


m (F ))ige0




m (F )







15










m(κ(v))rarr 0 (14)


parti+1v H i+1


m(κ(v))





m (F )





n ) cup r[A])



16

COHOMOLOGY GROUPS











cup r[A]


cup r[BK ]


cup r[A]

0




nBotimesrK(F )





17


12 Cycle modules





R-fieldsrarr Ab







18

CYCLE MODULES












ϕlowastw partw


partv ϕlowast = 0



19





partxy =sumz|y

ϕlowastz partz




0 =sum

xisinX (1)




20

CYCLE MODULES


rarroplus

xisinX(iminus1)


oplusxisinX(i)


oplusxisinX(i+1)





)ige0 and


)ige2





nBotimesrK)


21







x1 x2 xn 7rarr 0









22














23












24

INVARIANTS OF SK1


SK1(A)sim=

ρAF


H4n2A(F )

part3t1part4t2

H2n2(k)part3


n2 ) cup [A])

(110)







ax21 + bx2


4 minus dx25 + cdx2

6 (112)




25



SK1(A)(k)

H4nA(k)




SK1(A)(k)

id

ρS06 ker[H4


]rA

SK1(A)(k)ρRost

ker[(H4

2 (k)rarr H42 (k(Y ))

]

(113)





]




26

INVARIANTS OF SK1














27




H4n(k) sim= H4

pe11


(k)


28


Chapter 2


mdash Isaac Newton



21 Moderate case


211 Strategy


29




of two steps








Lift and specialise

30

MODERATE CASE


SK1(A)(kprime)A

ED




nAotimesr(kprime) 0




212 Lifting objects






31












32

MODERATE CASE










33










sumiisinZ ait

i with

34

MODERATE CASE







0 H i+1nAotimesr(k)

ulowast

H i+1nBotimesrK

(K) parti

vlowast

H inAotimesr(k)

ulowast

0


nBotimesrK(K prime)

parti H i

nAotimesr(kprime) 0



nBotimesrK(K prime)


35







SK1(A)(k)sim=

SK1(BK)(K)



36

MODERATE CASE


213 The lift






nAotimesr(kprime)

SK1(BK)(K prime)

sim=

OO

ρprimeKprime

H4nBotimesrK

(K prime)

(21)


= (Hj







37









H1(k micron)

cup r[A]


cup r[Ak(G)]



H3n(k)

H3n(k(G))

part3

oplusxisinG(1) H2

n(k(x))

H3nAotimesr(k) H3


oplusxisinG(1) H2

nAotimesr(k(x))



xisinG(1)


38

MODERATE CASE














nAotimesr(kprime)











39




22 Wild case





40

WILD CASE





andΩ1kZ and






xdy1

y1and and dyq


y1and and dyq

yqmod dΩqminus1

k






q times


xdy1

y1and and dyq





41


Then Cqpn(k) = Dq




defined by















1 ktimesspn

ktimessdlog

νn(1)ks 1


0 νn(q)ks Cqpn(ks)

Fminus1 Cq


(24)

42

WILD CASE









p (k) [GS Prop 925]











rarr H1(FZpnZ)

43




Wn(κ(v)s) 0





















44

WILD CASE






Hj+1pnL(F ) =

ker[Hj+1








45





pn(F ) sim= Cqpn(E)








dx1

x1and and dxm

xm


a + dbb







46

WILD CASE



pn (F ) defined by










0 Hq+1pn (κ(v))

Hq+1pnnr(F )

Hqpn(κ(v))

0




47


















48

WILD CASE








49



0 Hq+1pnL(κ(v))

ilowast Hq+1pnL(F )

part HqpnL(κ(v)) 0


OO


OO


OO

0


H2pn(κ(v))

sim=

ilowast H2pnnr(F )

sim=

pnBr(κ(v))i

pnBrnr(F )

(210)




)στisin H2

pn(F )



pn(κ(v)) inpnH


)στ

So


)στisin H2

pn(F )


50

WILD CASE


H2pnL(κ(v))

sim=

ilowast H2pnL(F )

sim=


pnBr(LotimesK FF )



)= ilowastK

(Kqminus1(κ(v))




)= partK

(Kqminus1(F )

)middot r[Aκ(v)]



)= ψK

(Kqminus2(κ(v))





51




ulowast

H i+1pnLBotimesrK

(K) parti

vlowast

H ipnLAotimesr(k)

ulowast

0







222 The lift


H0Zar(GH3

pn) sim= H0Zar(GH3

pn)H3pn(k)




H0Zar(GH3

pn) rarr H0Zar(GH3

pn)

52

WILD CASE



H0Zar(GH3




2 (U) (where KQ2






pn)



53








Zar(GH3pnL) sub H0




[H3pn(k(G))rarr H3

pn(ks(G))]


[H0

Zar(GH3pn) rarr H0

Zar(GH3pn)]







pnLAotimesr(kprime)

SK1(BK)(K prime)

sim=

OO

ρprimeKprime

H4pnLBotimesrK

(K prime)


= (HjpnLBotimesrK

)jge2


54

WILD CASE


ktimes

middotr[A]

k(G)timespart1

middotr[Ak(G)]

oplusxisinG(1) Z


H3pn(k)

H3pn(k(G)) part3

oplusxisinG(1) H2

pn(k(x))

H3pnAotimesr(k) H3


oplusxisinG(1) H2

pnAotimesr(k(x))






55



SK1(BK)ρK

H4pnLBotimesrK

(K)

SK1(BL)ρL

H4pnLBotimesrK

(L)




23 General case






56

GENERAL CASE





pn



(SK1(BK)Hlowast

eLBotimesrK






eLAotimesr(kprime)

SK1(BK)(K prime)

sim=

OO

ρprimeKprime

H4eLBotimesrK

(K prime)


57



24 Some remarks








58

SOME REMARKS


A(F ((t))

) ρrarrMj

(F ((t))










(kprime((t))

)rarr Hjminus1



gives us0 = ρ(α(0)) = ρ(α(1)) = ρ([a b])

59








eLAotimesr(F ((t))

)rarr H3

eLAotimesr(F )







60

SOME REMARKS


0 A0(RH4)

A0(KH4)

A0(kH3)

0











Inv4(GGGK Hlowast)

ϕ


61






62


Chapter 3










63









64












65










21+ +anx2

n




66
















67













0 SK1(A)(F )

=

ρRostAF H4

2 (F )

ψ

H42 (F (Y ))

sim=

0 SK1(A)(F )ρBIAF

I4(F ) I4(F (Y ))



68















69





Symd([a b) σ1

)= 〈v〉 and Symd

([c d) σ2

)= 〈1〉



Symd ((4a+ 1 b) τ1) = 〈1 i j〉 and


)= 〈1 i ij〉


70



Symd ((4c+ 1 d) τ2) = 〈1〉 and


)= 〈1 j ij〉





Symd(BK τ)





Symd(BK τ prime)



71














H44A(kprime)

ilowast


ψkprime

ilowast

I3Wq(kprime)

j

H44BK (K prime)

rB H4

2 (K prime)sim=

ψKprime

I3Wq(K prime)

(32)


72












34(x1 x2 x3)


32(x1 x2 x3)


Hn2 (kprime)

sim=

ϕ

I3Wq(kprime)

j


sim= I3Wq(K prime)



bda1

a1and da2

a2and da3


73









12 (K prime)








74






Then








75













76


321 Moderate case




nm(F )rarr H4n(F )


πr H4n(F )rarr H4

nAotimesr(F )










77







4 (k) sim= K4(k)4






part3t1 part

4t2

((K2(k)4) middot (2a t1+ 2b t2)

)


4A(k) sim= Z2





78








3 tplusmn14 ] where










79






SK1(A)

sim=

ρ H4

m2(F )

SK1(TKprime)ρ

H4m2(K prime)






80











322 Wild case








81


















82







mk(a b c d)]isin H4

n2(k)


n2(k) (for m = n2)



]= mprime h4






83




m(k)







84




ϕ[h4mk(a b c d)

]is sent to ϕ

[h4mK(a b c d)

](with an abuse of



mk(a b c d)] (34)



mk(a b c d)]








aγi

85





ni






n2(F )










86









sim=

H2(kGm) sim= QZ

H1(Γ S(K))

sim=OO

times H1(Γ S(K))

Br(Kk)

OO

T (k)

OO

times H2(Γ T (K))

Br(Kk)

micron2(k)

OO



T (k) times T (K)

Ktimes

micron2(k)

OO

times Zn2 Ktimes

87








n2

88

Conclusion








89

CONCLUSION





)rarr Inv4

(SK1(A)HlowastrLA

)





90

CONCLUSION



4 (k)l then SK1(A) 6= 0





91


Appendix A







93



Hq+1pnL(F (X))rarr

oplusxisinX(1)

HqpnL(F (x))oplus

oplusyisinY (0)


pnL(F (y0))

(A1)





Est2 = Hs






pn (Fnr(X))]




pn (Fnr(X))]

(A2)









)by (25) The




pn (F (y))

94



HqpnL(F (y))



)]sub ker



)]

(A5)




As above



) (A6)






Hq+1pnL (F (X))

partx

H1(Γ Kq(Fnr(X))pn)

partKx


)

(A7)



95






)


Hq+1pnL (F (X))

party

H1(Γ Kq(Fnr(X))pn)

partKy

HqpnL

(F (y)


)

(A8)







HqpnL(F (x))

partxy0


partxKy0


)

(A9)



96





HqpnL(F (y))

partyy0


partyKy0


)

(A10)



xisinX(1)


oplusyisinY (0)



)




oplusxisinX (1)




97













98










99


Appendix B





B1 Introduction





101



times the classof









102

INTRODUCTION





[σ]isinHΓσX




103













δ



δ

πlowastprime

++VVVVVV


++VVVVV


104

PRODUCT VARIETIES










1

1

ktimes oplus ktimes

π1 ktimes





1 1105






1

1



Div(X)oplusDiv(Y )


Pic(X)oplus Pic(Y )


1 1



106

PRODUCT VARIETIES







107





sim=Pic(X)Pic(X)

= 0









Pic(X timesk Y )

0 0

108

WEIL RESTRICTION







then




B3 Weil restriction


109


















110

WEIL RESTRICTION


1 ktimes

id

k(X)times

Div(X)

Pic(X)

ϕprime

1









A1f1

A2f2

A3


IndGH(A1)f1

IndGH(A2)f2

IndGH(A3)

111













HPic(X) ktimes)



H(Pic(X)) ktimes)




following situation




πprime



ilowastltltzzz



112

WEIL RESTRICTION











homological reasons




ψ noplusi=1







sumni=1 ψi






113

Bibliography















115

BIBLIOGRAPHY


















116

BIBLIOGRAPHY


















117

BIBLIOGRAPHY


















118

BIBLIOGRAPHY




















119

Glossary








121

GLOSSARY


x


36


H i+1m (F ) H i+1


plr with p - r14 41







46


15


a cycle module M23




122

GLOSSARY




27









nion k-algebra25



123

GLOSSARY







124

Index

Azumaya algebra 31
















125

INDEX




valuation triple 36


126



Frank Sinatra




Dankwoord

Abstract

Samenvatting

Contents


Introduction






Cohomology groups

Cycle modules


Invariants of SK1


Moderate case

Wild case

General case

Some remarks



Kahns invariant

Conclusion



Introduction

Product varieties

Weil restriction

Bibliography

Glossary

Index









soobwestvu

Dankwoord







iii

DANKWOORD






Tim WoutersMei 2010

iv

Abstract





v

SAMENVATTING

Samenvatting





vi

Contents

Dankwoord iii

Abstract v

Samenvatting vi

Contents vii


Introduction 1







12 Cycle modules 18




21 Moderate case 29

vii

CONTENTS

22 Wild case 40

23 General case 56

24 Some remarks 58




Conclusion 89



B1 Introduction 101



Bibliography 115

Glossary 121

Index 125

viii










ix








x

Introduction










1

INTRODUCTION









2

INTRODUCTION



dimk(A)








i=0 Kxi with


3

INTRODUCTION









4











A 00 1

)isin GLn+1(R)

Then define

GLinfin(R) =⋃ngt0




)

5

INTRODUCTION















)

6




SK1(D)

sim=











7

INTRODUCTION











8




I)



k









9

INTRODUCTION










10



sumpminus1i=0 X










11

INTRODUCTION



12


Chapter 1






A(K)

ϕK M(K)

A(K prime)ϕKprime

M(K prime)




13















14

COHOMOLOGY GROUPS



n times

J





mF (xn) (13)


m (F ))ige0




m (F )







15










m(κ(v))rarr 0 (14)


parti+1v H i+1


m(κ(v))





m (F )





n ) cup r[A])



16

COHOMOLOGY GROUPS











cup r[A]


cup r[BK ]


cup r[A]

0




nBotimesrK(F )





17


12 Cycle modules





R-fieldsrarr Ab







18

CYCLE MODULES












ϕlowastw partw


partv ϕlowast = 0



19





partxy =sumz|y

ϕlowastz partz




0 =sum

xisinX (1)




20

CYCLE MODULES


rarroplus

xisinX(iminus1)


oplusxisinX(i)


oplusxisinX(i+1)





)ige0 and


)ige2





nBotimesrK)


21







x1 x2 xn 7rarr 0









22














23












24

INVARIANTS OF SK1


SK1(A)sim=

ρAF


H4n2A(F )

part3t1part4t2

H2n2(k)part3


n2 ) cup [A])

(110)







ax21 + bx2


4 minus dx25 + cdx2

6 (112)




25



SK1(A)(k)

H4nA(k)




SK1(A)(k)

id

ρS06 ker[H4


]rA

SK1(A)(k)ρRost

ker[(H4

2 (k)rarr H42 (k(Y ))

]

(113)





]




26

INVARIANTS OF SK1














27




H4n(k) sim= H4

pe11


(k)


28


Chapter 2


mdash Isaac Newton



21 Moderate case


211 Strategy


29




of two steps








Lift and specialise

30

MODERATE CASE


SK1(A)(kprime)A

ED




nAotimesr(kprime) 0




212 Lifting objects






31












32

MODERATE CASE










33










sumiisinZ ait

i with

34

MODERATE CASE







0 H i+1nAotimesr(k)

ulowast

H i+1nBotimesrK

(K) parti

vlowast

H inAotimesr(k)

ulowast

0


nBotimesrK(K prime)

parti H i

nAotimesr(kprime) 0



nBotimesrK(K prime)


35







SK1(A)(k)sim=

SK1(BK)(K)



36

MODERATE CASE


213 The lift






nAotimesr(kprime)

SK1(BK)(K prime)

sim=

OO

ρprimeKprime

H4nBotimesrK

(K prime)

(21)


= (Hj







37









H1(k micron)

cup r[A]


cup r[Ak(G)]



H3n(k)

H3n(k(G))

part3

oplusxisinG(1) H2

n(k(x))

H3nAotimesr(k) H3


oplusxisinG(1) H2

nAotimesr(k(x))



xisinG(1)


38

MODERATE CASE














nAotimesr(kprime)











39




22 Wild case





40

WILD CASE





andΩ1kZ and






xdy1

y1and and dyq


y1and and dyq

yqmod dΩqminus1

k






q times


xdy1

y1and and dyq





41


Then Cqpn(k) = Dq




defined by















1 ktimesspn

ktimessdlog

νn(1)ks 1


0 νn(q)ks Cqpn(ks)

Fminus1 Cq


(24)

42

WILD CASE









p (k) [GS Prop 925]











rarr H1(FZpnZ)

43




Wn(κ(v)s) 0





















44

WILD CASE






Hj+1pnL(F ) =

ker[Hj+1








45





pn(F ) sim= Cqpn(E)








dx1

x1and and dxm

xm


a + dbb







46

WILD CASE



pn (F ) defined by










0 Hq+1pn (κ(v))

Hq+1pnnr(F )

Hqpn(κ(v))

0




47


















48

WILD CASE








49



0 Hq+1pnL(κ(v))

ilowast Hq+1pnL(F )

part HqpnL(κ(v)) 0


OO


OO


OO

0


H2pn(κ(v))

sim=

ilowast H2pnnr(F )

sim=

pnBr(κ(v))i

pnBrnr(F )

(210)




)στisin H2

pn(F )



pn(κ(v)) inpnH


)στ

So


)στisin H2

pn(F )


50

WILD CASE


H2pnL(κ(v))

sim=

ilowast H2pnL(F )

sim=


pnBr(LotimesK FF )



)= ilowastK

(Kqminus1(κ(v))




)= partK

(Kqminus1(F )

)middot r[Aκ(v)]



)= ψK

(Kqminus2(κ(v))





51




ulowast

H i+1pnLBotimesrK

(K) parti

vlowast

H ipnLAotimesr(k)

ulowast

0







222 The lift


H0Zar(GH3

pn) sim= H0Zar(GH3

pn)H3pn(k)




H0Zar(GH3

pn) rarr H0Zar(GH3

pn)

52

WILD CASE



H0Zar(GH3




2 (U) (where KQ2






pn)



53








Zar(GH3pnL) sub H0




[H3pn(k(G))rarr H3

pn(ks(G))]


[H0

Zar(GH3pn) rarr H0

Zar(GH3pn)]







pnLAotimesr(kprime)

SK1(BK)(K prime)

sim=

OO

ρprimeKprime

H4pnLBotimesrK

(K prime)


= (HjpnLBotimesrK

)jge2


54

WILD CASE


ktimes

middotr[A]

k(G)timespart1

middotr[Ak(G)]

oplusxisinG(1) Z


H3pn(k)

H3pn(k(G)) part3

oplusxisinG(1) H2

pn(k(x))

H3pnAotimesr(k) H3


oplusxisinG(1) H2

pnAotimesr(k(x))






55



SK1(BK)ρK

H4pnLBotimesrK

(K)

SK1(BL)ρL

H4pnLBotimesrK

(L)




23 General case






56

GENERAL CASE





pn



(SK1(BK)Hlowast

eLBotimesrK






eLAotimesr(kprime)

SK1(BK)(K prime)

sim=

OO

ρprimeKprime

H4eLBotimesrK

(K prime)


57



24 Some remarks








58

SOME REMARKS


A(F ((t))

) ρrarrMj

(F ((t))










(kprime((t))

)rarr Hjminus1



gives us0 = ρ(α(0)) = ρ(α(1)) = ρ([a b])

59








eLAotimesr(F ((t))

)rarr H3

eLAotimesr(F )







60

SOME REMARKS


0 A0(RH4)

A0(KH4)

A0(kH3)

0











Inv4(GGGK Hlowast)

ϕ


61






62


Chapter 3










63









64












65










21+ +anx2

n




66
















67













0 SK1(A)(F )

=

ρRostAF H4

2 (F )

ψ

H42 (F (Y ))

sim=

0 SK1(A)(F )ρBIAF

I4(F ) I4(F (Y ))



68















69





Symd([a b) σ1

)= 〈v〉 and Symd

([c d) σ2

)= 〈1〉



Symd ((4a+ 1 b) τ1) = 〈1 i j〉 and


)= 〈1 i ij〉


70



Symd ((4c+ 1 d) τ2) = 〈1〉 and


)= 〈1 j ij〉





Symd(BK τ)





Symd(BK τ prime)



71














H44A(kprime)

ilowast


ψkprime

ilowast

I3Wq(kprime)

j

H44BK (K prime)

rB H4

2 (K prime)sim=

ψKprime

I3Wq(K prime)

(32)


72












34(x1 x2 x3)


32(x1 x2 x3)


Hn2 (kprime)

sim=

ϕ

I3Wq(kprime)

j


sim= I3Wq(K prime)



bda1

a1and da2

a2and da3


73









12 (K prime)








74






Then








75













76


321 Moderate case




nm(F )rarr H4n(F )


πr H4n(F )rarr H4

nAotimesr(F )










77







4 (k) sim= K4(k)4






part3t1 part

4t2

((K2(k)4) middot (2a t1+ 2b t2)

)


4A(k) sim= Z2





78








3 tplusmn14 ] where










79






SK1(A)

sim=

ρ H4

m2(F )

SK1(TKprime)ρ

H4m2(K prime)






80











322 Wild case








81


















82







mk(a b c d)]isin H4

n2(k)


n2(k) (for m = n2)



]= mprime h4






83




m(k)







84




ϕ[h4mk(a b c d)

]is sent to ϕ

[h4mK(a b c d)

](with an abuse of



mk(a b c d)] (34)



mk(a b c d)]








aγi

85





ni






n2(F )










86









sim=

H2(kGm) sim= QZ

H1(Γ S(K))

sim=OO

times H1(Γ S(K))

Br(Kk)

OO

T (k)

OO

times H2(Γ T (K))

Br(Kk)

micron2(k)

OO



T (k) times T (K)

Ktimes

micron2(k)

OO

times Zn2 Ktimes

87








n2

88

Conclusion








89

CONCLUSION





)rarr Inv4

(SK1(A)HlowastrLA

)





90

CONCLUSION



4 (k)l then SK1(A) 6= 0





91


Appendix A







93



Hq+1pnL(F (X))rarr

oplusxisinX(1)

HqpnL(F (x))oplus

oplusyisinY (0)


pnL(F (y0))

(A1)





Est2 = Hs






pn (Fnr(X))]




pn (Fnr(X))]

(A2)









)by (25) The




pn (F (y))

94



HqpnL(F (y))



)]sub ker



)]

(A5)




As above



) (A6)






Hq+1pnL (F (X))

partx

H1(Γ Kq(Fnr(X))pn)

partKx


)

(A7)



95






)


Hq+1pnL (F (X))

party

H1(Γ Kq(Fnr(X))pn)

partKy

HqpnL

(F (y)


)

(A8)







HqpnL(F (x))

partxy0


partxKy0


)

(A9)



96





HqpnL(F (y))

partyy0


partyKy0


)

(A10)



xisinX(1)


oplusyisinY (0)



)




oplusxisinX (1)




97













98










99


Appendix B





B1 Introduction





101



times the classof









102

INTRODUCTION





[σ]isinHΓσX




103













δ



δ

πlowastprime

++VVVVVV


++VVVVV


104

PRODUCT VARIETIES










1

1

ktimes oplus ktimes

π1 ktimes





1 1105






1

1



Div(X)oplusDiv(Y )


Pic(X)oplus Pic(Y )


1 1



106

PRODUCT VARIETIES







107





sim=Pic(X)Pic(X)

= 0









Pic(X timesk Y )

0 0

108

WEIL RESTRICTION







then




B3 Weil restriction


109


















110

WEIL RESTRICTION


1 ktimes

id

k(X)times

Div(X)

Pic(X)

ϕprime

1









A1f1

A2f2

A3


IndGH(A1)f1

IndGH(A2)f2

IndGH(A3)

111













HPic(X) ktimes)



H(Pic(X)) ktimes)




following situation




πprime



ilowastltltzzz



112

WEIL RESTRICTION











homological reasons




ψ noplusi=1







sumni=1 ψi






113

Bibliography















115

BIBLIOGRAPHY


















116

BIBLIOGRAPHY


















117

BIBLIOGRAPHY


















118

BIBLIOGRAPHY




















119

Glossary








121

GLOSSARY


x


36


H i+1m (F ) H i+1


plr with p - r14 41







46


15


a cycle module M23




122

GLOSSARY




27









nion k-algebra25



123

GLOSSARY







124

Index

Azumaya algebra 31
















125

INDEX




valuation triple 36


126



Frank Sinatra




Dankwoord

Abstract

Samenvatting

Contents


Introduction






Cohomology groups

Cycle modules


Invariants of SK1


Moderate case

Wild case

General case

Some remarks



Kahns invariant

Conclusion



Introduction

Product varieties

Weil restriction

Bibliography

Glossary

Index

Dankwoord







iii

DANKWOORD






Tim WoutersMei 2010

iv

Abstract





v

SAMENVATTING

Samenvatting





vi

Contents

Dankwoord iii

Abstract v

Samenvatting vi

Contents vii


Introduction 1







12 Cycle modules 18




21 Moderate case 29

vii

CONTENTS

22 Wild case 40

23 General case 56

24 Some remarks 58




Conclusion 89



B1 Introduction 101



Bibliography 115

Glossary 121

Index 125

viii










ix








x

Introduction










1

INTRODUCTION









2

INTRODUCTION



dimk(A)








i=0 Kxi with


3

INTRODUCTION









4











A 00 1

)isin GLn+1(R)

Then define

GLinfin(R) =⋃ngt0




)

5

INTRODUCTION















)

6




SK1(D)

sim=











7

INTRODUCTION











8




I)



k









9

INTRODUCTION










10



sumpminus1i=0 X










11

INTRODUCTION



12


Chapter 1






A(K)

ϕK M(K)

A(K prime)ϕKprime

M(K prime)




13















14

COHOMOLOGY GROUPS



n times

J





mF (xn) (13)


m (F ))ige0




m (F )







15










m(κ(v))rarr 0 (14)


parti+1v H i+1


m(κ(v))





m (F )





n ) cup r[A])



16

COHOMOLOGY GROUPS











cup r[A]


cup r[BK ]


cup r[A]

0




nBotimesrK(F )





17


12 Cycle modules





R-fieldsrarr Ab







18

CYCLE MODULES












ϕlowastw partw


partv ϕlowast = 0



19





partxy =sumz|y

ϕlowastz partz




0 =sum

xisinX (1)




20

CYCLE MODULES


rarroplus

xisinX(iminus1)


oplusxisinX(i)


oplusxisinX(i+1)





)ige0 and


)ige2





nBotimesrK)


21







x1 x2 xn 7rarr 0









22














23












24

INVARIANTS OF SK1


SK1(A)sim=

ρAF


H4n2A(F )

part3t1part4t2

H2n2(k)part3


n2 ) cup [A])

(110)







ax21 + bx2


4 minus dx25 + cdx2

6 (112)




25



SK1(A)(k)

H4nA(k)




SK1(A)(k)

id

ρS06 ker[H4


]rA

SK1(A)(k)ρRost

ker[(H4

2 (k)rarr H42 (k(Y ))

]

(113)





]




26

INVARIANTS OF SK1














27




H4n(k) sim= H4

pe11


(k)


28


Chapter 2


mdash Isaac Newton



21 Moderate case


211 Strategy


29




of two steps








Lift and specialise

30

MODERATE CASE


SK1(A)(kprime)A

ED




nAotimesr(kprime) 0




212 Lifting objects






31












32

MODERATE CASE










33










sumiisinZ ait

i with

34

MODERATE CASE







0 H i+1nAotimesr(k)

ulowast

H i+1nBotimesrK

(K) parti

vlowast

H inAotimesr(k)

ulowast

0


nBotimesrK(K prime)

parti H i

nAotimesr(kprime) 0



nBotimesrK(K prime)


35







SK1(A)(k)sim=

SK1(BK)(K)



36

MODERATE CASE


213 The lift






nAotimesr(kprime)

SK1(BK)(K prime)

sim=

OO

ρprimeKprime

H4nBotimesrK

(K prime)

(21)


= (Hj







37









H1(k micron)

cup r[A]


cup r[Ak(G)]



H3n(k)

H3n(k(G))

part3

oplusxisinG(1) H2

n(k(x))

H3nAotimesr(k) H3


oplusxisinG(1) H2

nAotimesr(k(x))



xisinG(1)


38

MODERATE CASE














nAotimesr(kprime)











39




22 Wild case





40

WILD CASE





andΩ1kZ and






xdy1

y1and and dyq


y1and and dyq

yqmod dΩqminus1

k






q times


xdy1

y1and and dyq





41


Then Cqpn(k) = Dq




defined by















1 ktimesspn

ktimessdlog

νn(1)ks 1


0 νn(q)ks Cqpn(ks)

Fminus1 Cq


(24)

42

WILD CASE









p (k) [GS Prop 925]











rarr H1(FZpnZ)

43




Wn(κ(v)s) 0





















44

WILD CASE






Hj+1pnL(F ) =

ker[Hj+1








45





pn(F ) sim= Cqpn(E)








dx1

x1and and dxm

xm


a + dbb







46

WILD CASE



pn (F ) defined by










0 Hq+1pn (κ(v))

Hq+1pnnr(F )

Hqpn(κ(v))

0




47


















48

WILD CASE








49



0 Hq+1pnL(κ(v))

ilowast Hq+1pnL(F )

part HqpnL(κ(v)) 0


OO


OO


OO

0


H2pn(κ(v))

sim=

ilowast H2pnnr(F )

sim=

pnBr(κ(v))i

pnBrnr(F )

(210)




)στisin H2

pn(F )



pn(κ(v)) inpnH


)στ

So


)στisin H2

pn(F )


50

WILD CASE


H2pnL(κ(v))

sim=

ilowast H2pnL(F )

sim=


pnBr(LotimesK FF )



)= ilowastK

(Kqminus1(κ(v))




)= partK

(Kqminus1(F )

)middot r[Aκ(v)]



)= ψK

(Kqminus2(κ(v))





51




ulowast

H i+1pnLBotimesrK

(K) parti

vlowast

H ipnLAotimesr(k)

ulowast

0







222 The lift


H0Zar(GH3

pn) sim= H0Zar(GH3

pn)H3pn(k)




H0Zar(GH3

pn) rarr H0Zar(GH3

pn)

52

WILD CASE



H0Zar(GH3




2 (U) (where KQ2






pn)



53








Zar(GH3pnL) sub H0




[H3pn(k(G))rarr H3

pn(ks(G))]


[H0

Zar(GH3pn) rarr H0

Zar(GH3pn)]







pnLAotimesr(kprime)

SK1(BK)(K prime)

sim=

OO

ρprimeKprime

H4pnLBotimesrK

(K prime)


= (HjpnLBotimesrK

)jge2


54

WILD CASE


ktimes

middotr[A]

k(G)timespart1

middotr[Ak(G)]

oplusxisinG(1) Z


H3pn(k)

H3pn(k(G)) part3

oplusxisinG(1) H2

pn(k(x))

H3pnAotimesr(k) H3


oplusxisinG(1) H2

pnAotimesr(k(x))






55



SK1(BK)ρK

H4pnLBotimesrK

(K)

SK1(BL)ρL

H4pnLBotimesrK

(L)




23 General case






56

GENERAL CASE





pn



(SK1(BK)Hlowast

eLBotimesrK






eLAotimesr(kprime)

SK1(BK)(K prime)

sim=

OO

ρprimeKprime

H4eLBotimesrK

(K prime)


57



24 Some remarks








58

SOME REMARKS


A(F ((t))

) ρrarrMj

(F ((t))










(kprime((t))

)rarr Hjminus1



gives us0 = ρ(α(0)) = ρ(α(1)) = ρ([a b])

59








eLAotimesr(F ((t))

)rarr H3

eLAotimesr(F )







60

SOME REMARKS


0 A0(RH4)

A0(KH4)

A0(kH3)

0











Inv4(GGGK Hlowast)

ϕ


61






62


Chapter 3










63









64












65










21+ +anx2

n




66
















67













0 SK1(A)(F )

=

ρRostAF H4

2 (F )

ψ

H42 (F (Y ))

sim=

0 SK1(A)(F )ρBIAF

I4(F ) I4(F (Y ))



68















69





Symd([a b) σ1

)= 〈v〉 and Symd

([c d) σ2

)= 〈1〉



Symd ((4a+ 1 b) τ1) = 〈1 i j〉 and


)= 〈1 i ij〉


70



Symd ((4c+ 1 d) τ2) = 〈1〉 and


)= 〈1 j ij〉





Symd(BK τ)





Symd(BK τ prime)



71














H44A(kprime)

ilowast


ψkprime

ilowast

I3Wq(kprime)

j

H44BK (K prime)

rB H4

2 (K prime)sim=

ψKprime

I3Wq(K prime)

(32)


72












34(x1 x2 x3)


32(x1 x2 x3)


Hn2 (kprime)

sim=

ϕ

I3Wq(kprime)

j


sim= I3Wq(K prime)



bda1

a1and da2

a2and da3


73









12 (K prime)








74






Then








75













76


321 Moderate case




nm(F )rarr H4n(F )


πr H4n(F )rarr H4

nAotimesr(F )










77







4 (k) sim= K4(k)4






part3t1 part

4t2

((K2(k)4) middot (2a t1+ 2b t2)

)


4A(k) sim= Z2





78








3 tplusmn14 ] where










79






SK1(A)

sim=

ρ H4

m2(F )

SK1(TKprime)ρ

H4m2(K prime)






80











322 Wild case








81


















82







mk(a b c d)]isin H4

n2(k)


n2(k) (for m = n2)



]= mprime h4






83




m(k)







84




ϕ[h4mk(a b c d)

]is sent to ϕ

[h4mK(a b c d)

](with an abuse of



mk(a b c d)] (34)



mk(a b c d)]








aγi

85





ni






n2(F )










86









sim=

H2(kGm) sim= QZ

H1(Γ S(K))

sim=OO

times H1(Γ S(K))

Br(Kk)

OO

T (k)

OO

times H2(Γ T (K))

Br(Kk)

micron2(k)

OO



T (k) times T (K)

Ktimes

micron2(k)

OO

times Zn2 Ktimes

87








n2

88

Conclusion








89

CONCLUSION





)rarr Inv4

(SK1(A)HlowastrLA

)





90

CONCLUSION



4 (k)l then SK1(A) 6= 0





91


Appendix A







93



Hq+1pnL(F (X))rarr

oplusxisinX(1)

HqpnL(F (x))oplus

oplusyisinY (0)


pnL(F (y0))

(A1)





Est2 = Hs






pn (Fnr(X))]




pn (Fnr(X))]

(A2)









)by (25) The




pn (F (y))

94



HqpnL(F (y))



)]sub ker



)]

(A5)




As above



) (A6)






Hq+1pnL (F (X))

partx

H1(Γ Kq(Fnr(X))pn)

partKx


)

(A7)



95






)


Hq+1pnL (F (X))

party

H1(Γ Kq(Fnr(X))pn)

partKy

HqpnL

(F (y)


)

(A8)







HqpnL(F (x))

partxy0


partxKy0


)

(A9)



96





HqpnL(F (y))

partyy0


partyKy0


)

(A10)



xisinX(1)


oplusyisinY (0)



)




oplusxisinX (1)




97













98










99


Appendix B





B1 Introduction





101



times the classof









102

INTRODUCTION





[σ]isinHΓσX




103













δ



δ

πlowastprime

++VVVVVV


++VVVVV


104

PRODUCT VARIETIES










1

1

ktimes oplus ktimes

π1 ktimes





1 1105






1

1



Div(X)oplusDiv(Y )


Pic(X)oplus Pic(Y )


1 1



106

PRODUCT VARIETIES







107





sim=Pic(X)Pic(X)

= 0









Pic(X timesk Y )

0 0

108

WEIL RESTRICTION







then




B3 Weil restriction


109


















110

WEIL RESTRICTION


1 ktimes

id

k(X)times

Div(X)

Pic(X)

ϕprime

1









A1f1

A2f2

A3


IndGH(A1)f1

IndGH(A2)f2

IndGH(A3)

111













HPic(X) ktimes)



H(Pic(X)) ktimes)




following situation




πprime



ilowastltltzzz



112

WEIL RESTRICTION











homological reasons




ψ noplusi=1







sumni=1 ψi






113

Bibliography















115

BIBLIOGRAPHY


















116

BIBLIOGRAPHY


















117

BIBLIOGRAPHY


















118

BIBLIOGRAPHY




















119

Glossary








121

GLOSSARY


x


36


H i+1m (F ) H i+1


plr with p - r14 41







46


15


a cycle module M23




122

GLOSSARY




27









nion k-algebra25



123

GLOSSARY







124

Index

Azumaya algebra 31
















125

INDEX




valuation triple 36


126



Frank Sinatra




Dankwoord

Abstract

Samenvatting

Contents


Introduction






Cohomology groups

Cycle modules


Invariants of SK1


Moderate case

Wild case

General case

Some remarks



Kahns invariant

Conclusion



Introduction

Product varieties

Weil restriction

Bibliography

Glossary

Index

DANKWOORD






Tim WoutersMei 2010

iv

Abstract





v

SAMENVATTING

Samenvatting





vi

Contents

Dankwoord iii

Abstract v

Samenvatting vi

Contents vii


Introduction 1







12 Cycle modules 18




21 Moderate case 29

vii

CONTENTS

22 Wild case 40

23 General case 56

24 Some remarks 58




Conclusion 89



B1 Introduction 101



Bibliography 115

Glossary 121

Index 125

viii










ix








x

Introduction










1

INTRODUCTION









2

INTRODUCTION



dimk(A)








i=0 Kxi with


3

INTRODUCTION









4











A 00 1

)isin GLn+1(R)

Then define

GLinfin(R) =⋃ngt0




)

5

INTRODUCTION















)

6




SK1(D)

sim=











7

INTRODUCTION











8




I)



k









9

INTRODUCTION










10



sumpminus1i=0 X










11

INTRODUCTION



12


Chapter 1






A(K)

ϕK M(K)

A(K prime)ϕKprime

M(K prime)




13















14

COHOMOLOGY GROUPS



n times

J





mF (xn) (13)


m (F ))ige0




m (F )







15










m(κ(v))rarr 0 (14)


parti+1v H i+1


m(κ(v))





m (F )





n ) cup r[A])



16

COHOMOLOGY GROUPS











cup r[A]


cup r[BK ]


cup r[A]

0




nBotimesrK(F )





17


12 Cycle modules





R-fieldsrarr Ab







18

CYCLE MODULES












ϕlowastw partw


partv ϕlowast = 0



19





partxy =sumz|y

ϕlowastz partz




0 =sum

xisinX (1)




20

CYCLE MODULES


rarroplus

xisinX(iminus1)


oplusxisinX(i)


oplusxisinX(i+1)





)ige0 and


)ige2





nBotimesrK)


21







x1 x2 xn 7rarr 0









22














23












24

INVARIANTS OF SK1


SK1(A)sim=

ρAF


H4n2A(F )

part3t1part4t2

H2n2(k)part3


n2 ) cup [A])

(110)







ax21 + bx2


4 minus dx25 + cdx2

6 (112)




25



SK1(A)(k)

H4nA(k)




SK1(A)(k)

id

ρS06 ker[H4


]rA

SK1(A)(k)ρRost

ker[(H4

2 (k)rarr H42 (k(Y ))

]

(113)





]




26

INVARIANTS OF SK1














27




H4n(k) sim= H4

pe11


(k)


28


Chapter 2


mdash Isaac Newton



21 Moderate case


211 Strategy


29




of two steps








Lift and specialise

30

MODERATE CASE


SK1(A)(kprime)A

ED




nAotimesr(kprime) 0




212 Lifting objects






31












32

MODERATE CASE










33










sumiisinZ ait

i with

34

MODERATE CASE







0 H i+1nAotimesr(k)

ulowast

H i+1nBotimesrK

(K) parti

vlowast

H inAotimesr(k)

ulowast

0


nBotimesrK(K prime)

parti H i

nAotimesr(kprime) 0



nBotimesrK(K prime)


35







SK1(A)(k)sim=

SK1(BK)(K)



36

MODERATE CASE


213 The lift






nAotimesr(kprime)

SK1(BK)(K prime)

sim=

OO

ρprimeKprime

H4nBotimesrK

(K prime)

(21)


= (Hj







37









H1(k micron)

cup r[A]


cup r[Ak(G)]



H3n(k)

H3n(k(G))

part3

oplusxisinG(1) H2

n(k(x))

H3nAotimesr(k) H3


oplusxisinG(1) H2

nAotimesr(k(x))



xisinG(1)


38

MODERATE CASE














nAotimesr(kprime)











39




22 Wild case





40

WILD CASE





andΩ1kZ and






xdy1

y1and and dyq


y1and and dyq

yqmod dΩqminus1

k






q times


xdy1

y1and and dyq





41


Then Cqpn(k) = Dq




defined by















1 ktimesspn

ktimessdlog

νn(1)ks 1


0 νn(q)ks Cqpn(ks)

Fminus1 Cq


(24)

42

WILD CASE









p (k) [GS Prop 925]











rarr H1(FZpnZ)

43




Wn(κ(v)s) 0





















44

WILD CASE






Hj+1pnL(F ) =

ker[Hj+1








45





pn(F ) sim= Cqpn(E)








dx1

x1and and dxm

xm


a + dbb







46

WILD CASE



pn (F ) defined by










0 Hq+1pn (κ(v))

Hq+1pnnr(F )

Hqpn(κ(v))

0




47


















48

WILD CASE








49



0 Hq+1pnL(κ(v))

ilowast Hq+1pnL(F )

part HqpnL(κ(v)) 0


OO


OO


OO

0


H2pn(κ(v))

sim=

ilowast H2pnnr(F )

sim=

pnBr(κ(v))i

pnBrnr(F )

(210)




)στisin H2

pn(F )



pn(κ(v)) inpnH


)στ

So


)στisin H2

pn(F )


50

WILD CASE


H2pnL(κ(v))

sim=

ilowast H2pnL(F )

sim=


pnBr(LotimesK FF )



)= ilowastK

(Kqminus1(κ(v))




)= partK

(Kqminus1(F )

)middot r[Aκ(v)]



)= ψK

(Kqminus2(κ(v))





51




ulowast

H i+1pnLBotimesrK

(K) parti

vlowast

H ipnLAotimesr(k)

ulowast

0







222 The lift


H0Zar(GH3

pn) sim= H0Zar(GH3

pn)H3pn(k)




H0Zar(GH3

pn) rarr H0Zar(GH3

pn)

52

WILD CASE



H0Zar(GH3




2 (U) (where KQ2






pn)



53








Zar(GH3pnL) sub H0




[H3pn(k(G))rarr H3

pn(ks(G))]


[H0

Zar(GH3pn) rarr H0

Zar(GH3pn)]







pnLAotimesr(kprime)

SK1(BK)(K prime)

sim=

OO

ρprimeKprime

H4pnLBotimesrK

(K prime)


= (HjpnLBotimesrK

)jge2


54

WILD CASE


ktimes

middotr[A]

k(G)timespart1

middotr[Ak(G)]

oplusxisinG(1) Z


H3pn(k)

H3pn(k(G)) part3

oplusxisinG(1) H2

pn(k(x))

H3pnAotimesr(k) H3


oplusxisinG(1) H2

pnAotimesr(k(x))






55



SK1(BK)ρK

H4pnLBotimesrK

(K)

SK1(BL)ρL

H4pnLBotimesrK

(L)




23 General case






56

GENERAL CASE





pn



(SK1(BK)Hlowast

eLBotimesrK






eLAotimesr(kprime)

SK1(BK)(K prime)

sim=

OO

ρprimeKprime

H4eLBotimesrK

(K prime)


57



24 Some remarks








58

SOME REMARKS


A(F ((t))

) ρrarrMj

(F ((t))










(kprime((t))

)rarr Hjminus1



gives us0 = ρ(α(0)) = ρ(α(1)) = ρ([a b])

59








eLAotimesr(F ((t))

)rarr H3

eLAotimesr(F )







60

SOME REMARKS


0 A0(RH4)

A0(KH4)

A0(kH3)

0











Inv4(GGGK Hlowast)

ϕ


61






62


Chapter 3










63









64












65










21+ +anx2

n




66
















67













0 SK1(A)(F )

=

ρRostAF H4

2 (F )

ψ

H42 (F (Y ))

sim=

0 SK1(A)(F )ρBIAF

I4(F ) I4(F (Y ))



68















69





Symd([a b) σ1

)= 〈v〉 and Symd

([c d) σ2

)= 〈1〉



Symd ((4a+ 1 b) τ1) = 〈1 i j〉 and


)= 〈1 i ij〉


70



Symd ((4c+ 1 d) τ2) = 〈1〉 and


)= 〈1 j ij〉





Symd(BK τ)





Symd(BK τ prime)



71














H44A(kprime)

ilowast


ψkprime

ilowast

I3Wq(kprime)

j

H44BK (K prime)

rB H4

2 (K prime)sim=

ψKprime

I3Wq(K prime)

(32)


72












34(x1 x2 x3)


32(x1 x2 x3)


Hn2 (kprime)

sim=

ϕ

I3Wq(kprime)

j


sim= I3Wq(K prime)



bda1

a1and da2

a2and da3


73









12 (K prime)








74






Then








75













76


321 Moderate case




nm(F )rarr H4n(F )


πr H4n(F )rarr H4

nAotimesr(F )










77







4 (k) sim= K4(k)4






part3t1 part

4t2

((K2(k)4) middot (2a t1+ 2b t2)

)


4A(k) sim= Z2





78








3 tplusmn14 ] where










79






SK1(A)

sim=

ρ H4

m2(F )

SK1(TKprime)ρ

H4m2(K prime)






80











322 Wild case








81


















82







mk(a b c d)]isin H4

n2(k)


n2(k) (for m = n2)



]= mprime h4






83




m(k)







84




ϕ[h4mk(a b c d)

]is sent to ϕ

[h4mK(a b c d)

](with an abuse of



mk(a b c d)] (34)



mk(a b c d)]








aγi

85





ni






n2(F )










86









sim=

H2(kGm) sim= QZ

H1(Γ S(K))

sim=OO

times H1(Γ S(K))

Br(Kk)

OO

T (k)

OO

times H2(Γ T (K))

Br(Kk)

micron2(k)

OO



T (k) times T (K)

Ktimes

micron2(k)

OO

times Zn2 Ktimes

87








n2

88

Conclusion








89

CONCLUSION





)rarr Inv4

(SK1(A)HlowastrLA

)





90

CONCLUSION



4 (k)l then SK1(A) 6= 0





91


Appendix A







93



Hq+1pnL(F (X))rarr

oplusxisinX(1)

HqpnL(F (x))oplus

oplusyisinY (0)


pnL(F (y0))

(A1)





Est2 = Hs






pn (Fnr(X))]




pn (Fnr(X))]

(A2)









)by (25) The




pn (F (y))

94



HqpnL(F (y))



)]sub ker



)]

(A5)




As above



) (A6)






Hq+1pnL (F (X))

partx

H1(Γ Kq(Fnr(X))pn)

partKx


)

(A7)



95






)


Hq+1pnL (F (X))

party

H1(Γ Kq(Fnr(X))pn)

partKy

HqpnL

(F (y)


)

(A8)







HqpnL(F (x))

partxy0


partxKy0


)

(A9)



96





HqpnL(F (y))

partyy0


partyKy0


)

(A10)



xisinX(1)


oplusyisinY (0)



)




oplusxisinX (1)




97













98










99


Appendix B





B1 Introduction





101



times the classof









102

INTRODUCTION





[σ]isinHΓσX




103













δ



δ

πlowastprime

++VVVVVV


++VVVVV


104

PRODUCT VARIETIES










1

1

ktimes oplus ktimes

π1 ktimes





1 1105






1

1



Div(X)oplusDiv(Y )


Pic(X)oplus Pic(Y )


1 1



106

PRODUCT VARIETIES







107





sim=Pic(X)Pic(X)

= 0









Pic(X timesk Y )

0 0

108

WEIL RESTRICTION







then




B3 Weil restriction


109


















110

WEIL RESTRICTION


1 ktimes

id

k(X)times

Div(X)

Pic(X)

ϕprime

1









A1f1

A2f2

A3


IndGH(A1)f1

IndGH(A2)f2

IndGH(A3)

111













HPic(X) ktimes)



H(Pic(X)) ktimes)




following situation




πprime



ilowastltltzzz



112

WEIL RESTRICTION











homological reasons




ψ noplusi=1







sumni=1 ψi






113

Bibliography















115

BIBLIOGRAPHY


















116

BIBLIOGRAPHY


















117

BIBLIOGRAPHY


















118

BIBLIOGRAPHY




















119

Glossary








121

GLOSSARY


x


36


H i+1m (F ) H i+1


plr with p - r14 41







46


15


a cycle module M23




122

GLOSSARY




27









nion k-algebra25



123

GLOSSARY







124

Index

Azumaya algebra 31
















125

INDEX




valuation triple 36


126



Frank Sinatra




Dankwoord

Abstract

Samenvatting

Contents


Introduction






Cohomology groups

Cycle modules


Invariants of SK1


Moderate case

Wild case

General case

Some remarks



Kahns invariant

Conclusion



Introduction

Product varieties

Weil restriction

Bibliography

Glossary

Index

Abstract





v

SAMENVATTING

Samenvatting





vi

Contents

Dankwoord iii

Abstract v

Samenvatting vi

Contents vii


Introduction 1







12 Cycle modules 18




21 Moderate case 29

vii

CONTENTS

22 Wild case 40

23 General case 56

24 Some remarks 58




Conclusion 89



B1 Introduction 101



Bibliography 115

Glossary 121

Index 125

viii










ix








x

Introduction










1

INTRODUCTION









2

INTRODUCTION



dimk(A)








i=0 Kxi with


3

INTRODUCTION









4











A 00 1

)isin GLn+1(R)

Then define

GLinfin(R) =⋃ngt0




)

5

INTRODUCTION















)

6




SK1(D)

sim=











7

INTRODUCTION











8




I)



k









9

INTRODUCTION










10



sumpminus1i=0 X










11

INTRODUCTION



12


Chapter 1






A(K)

ϕK M(K)

A(K prime)ϕKprime

M(K prime)




13















14

COHOMOLOGY GROUPS



n times

J





mF (xn) (13)


m (F ))ige0




m (F )







15










m(κ(v))rarr 0 (14)


parti+1v H i+1


m(κ(v))





m (F )





n ) cup r[A])



16

COHOMOLOGY GROUPS











cup r[A]


cup r[BK ]


cup r[A]

0




nBotimesrK(F )





17


12 Cycle modules





R-fieldsrarr Ab







18

CYCLE MODULES












ϕlowastw partw


partv ϕlowast = 0



19





partxy =sumz|y

ϕlowastz partz




0 =sum

xisinX (1)




20

CYCLE MODULES


rarroplus

xisinX(iminus1)


oplusxisinX(i)


oplusxisinX(i+1)





)ige0 and


)ige2





nBotimesrK)


21







x1 x2 xn 7rarr 0









22














23












24

INVARIANTS OF SK1


SK1(A)sim=

ρAF


H4n2A(F )

part3t1part4t2

H2n2(k)part3


n2 ) cup [A])

(110)







ax21 + bx2


4 minus dx25 + cdx2

6 (112)




25



SK1(A)(k)

H4nA(k)




SK1(A)(k)

id

ρS06 ker[H4


]rA

SK1(A)(k)ρRost

ker[(H4

2 (k)rarr H42 (k(Y ))

]

(113)





]




26

INVARIANTS OF SK1














27




H4n(k) sim= H4

pe11


(k)


28


Chapter 2


mdash Isaac Newton



21 Moderate case


211 Strategy


29




of two steps








Lift and specialise

30

MODERATE CASE


SK1(A)(kprime)A

ED




nAotimesr(kprime) 0




212 Lifting objects






31












32

MODERATE CASE










33










sumiisinZ ait

i with

34

MODERATE CASE







0 H i+1nAotimesr(k)

ulowast

H i+1nBotimesrK

(K) parti

vlowast

H inAotimesr(k)

ulowast

0


nBotimesrK(K prime)

parti H i

nAotimesr(kprime) 0



nBotimesrK(K prime)


35







SK1(A)(k)sim=

SK1(BK)(K)



36

MODERATE CASE


213 The lift






nAotimesr(kprime)

SK1(BK)(K prime)

sim=

OO

ρprimeKprime

H4nBotimesrK

(K prime)

(21)


= (Hj







37









H1(k micron)

cup r[A]


cup r[Ak(G)]



H3n(k)

H3n(k(G))

part3

oplusxisinG(1) H2

n(k(x))

H3nAotimesr(k) H3


oplusxisinG(1) H2

nAotimesr(k(x))



xisinG(1)


38

MODERATE CASE














nAotimesr(kprime)











39




22 Wild case





40

WILD CASE





andΩ1kZ and






xdy1

y1and and dyq


y1and and dyq

yqmod dΩqminus1

k






q times


xdy1

y1and and dyq





41


Then Cqpn(k) = Dq




defined by















1 ktimesspn

ktimessdlog

νn(1)ks 1


0 νn(q)ks Cqpn(ks)

Fminus1 Cq


(24)

42

WILD CASE









p (k) [GS Prop 925]











rarr H1(FZpnZ)

43




Wn(κ(v)s) 0





















44

WILD CASE






Hj+1pnL(F ) =

ker[Hj+1








45





pn(F ) sim= Cqpn(E)








dx1

x1and and dxm

xm


a + dbb







46

WILD CASE



pn (F ) defined by










0 Hq+1pn (κ(v))

Hq+1pnnr(F )

Hqpn(κ(v))

0




47


















48

WILD CASE








49



0 Hq+1pnL(κ(v))

ilowast Hq+1pnL(F )

part HqpnL(κ(v)) 0


OO


OO


OO

0


H2pn(κ(v))

sim=

ilowast H2pnnr(F )

sim=

pnBr(κ(v))i

pnBrnr(F )

(210)




)στisin H2

pn(F )



pn(κ(v)) inpnH


)στ

So


)στisin H2

pn(F )


50

WILD CASE


H2pnL(κ(v))

sim=

ilowast H2pnL(F )

sim=


pnBr(LotimesK FF )



)= ilowastK

(Kqminus1(κ(v))




)= partK

(Kqminus1(F )

)middot r[Aκ(v)]



)= ψK

(Kqminus2(κ(v))





51




ulowast

H i+1pnLBotimesrK

(K) parti

vlowast

H ipnLAotimesr(k)

ulowast

0







222 The lift


H0Zar(GH3

pn) sim= H0Zar(GH3

pn)H3pn(k)




H0Zar(GH3

pn) rarr H0Zar(GH3

pn)

52

WILD CASE



H0Zar(GH3




2 (U) (where KQ2






pn)



53








Zar(GH3pnL) sub H0




[H3pn(k(G))rarr H3

pn(ks(G))]


[H0

Zar(GH3pn) rarr H0

Zar(GH3pn)]







pnLAotimesr(kprime)

SK1(BK)(K prime)

sim=

OO

ρprimeKprime

H4pnLBotimesrK

(K prime)


= (HjpnLBotimesrK

)jge2


54

WILD CASE


ktimes

middotr[A]

k(G)timespart1

middotr[Ak(G)]

oplusxisinG(1) Z


H3pn(k)

H3pn(k(G)) part3

oplusxisinG(1) H2

pn(k(x))

H3pnAotimesr(k) H3


oplusxisinG(1) H2

pnAotimesr(k(x))






55



SK1(BK)ρK

H4pnLBotimesrK

(K)

SK1(BL)ρL

H4pnLBotimesrK

(L)




23 General case






56

GENERAL CASE





pn



(SK1(BK)Hlowast

eLBotimesrK






eLAotimesr(kprime)

SK1(BK)(K prime)

sim=

OO

ρprimeKprime

H4eLBotimesrK

(K prime)


57



24 Some remarks








58

SOME REMARKS


A(F ((t))

) ρrarrMj

(F ((t))










(kprime((t))

)rarr Hjminus1



gives us0 = ρ(α(0)) = ρ(α(1)) = ρ([a b])

59








eLAotimesr(F ((t))

)rarr H3

eLAotimesr(F )







60

SOME REMARKS


0 A0(RH4)

A0(KH4)

A0(kH3)

0











Inv4(GGGK Hlowast)

ϕ


61






62


Chapter 3










63









64












65










21+ +anx2

n




66
















67













0 SK1(A)(F )

=

ρRostAF H4

2 (F )

ψ

H42 (F (Y ))

sim=

0 SK1(A)(F )ρBIAF

I4(F ) I4(F (Y ))



68















69





Symd([a b) σ1

)= 〈v〉 and Symd

([c d) σ2

)= 〈1〉



Symd ((4a+ 1 b) τ1) = 〈1 i j〉 and


)= 〈1 i ij〉


70



Symd ((4c+ 1 d) τ2) = 〈1〉 and


)= 〈1 j ij〉





Symd(BK τ)





Symd(BK τ prime)



71














H44A(kprime)

ilowast


ψkprime

ilowast

I3Wq(kprime)

j

H44BK (K prime)

rB H4

2 (K prime)sim=

ψKprime

I3Wq(K prime)

(32)


72












34(x1 x2 x3)


32(x1 x2 x3)


Hn2 (kprime)

sim=

ϕ

I3Wq(kprime)

j


sim= I3Wq(K prime)



bda1

a1and da2

a2and da3


73









12 (K prime)








74






Then








75













76


321 Moderate case




nm(F )rarr H4n(F )


πr H4n(F )rarr H4

nAotimesr(F )










77







4 (k) sim= K4(k)4






part3t1 part

4t2

((K2(k)4) middot (2a t1+ 2b t2)

)


4A(k) sim= Z2





78








3 tplusmn14 ] where










79






SK1(A)

sim=

ρ H4

m2(F )

SK1(TKprime)ρ

H4m2(K prime)






80











322 Wild case








81


















82







mk(a b c d)]isin H4

n2(k)


n2(k) (for m = n2)



]= mprime h4






83




m(k)







84




ϕ[h4mk(a b c d)

]is sent to ϕ

[h4mK(a b c d)

](with an abuse of



mk(a b c d)] (34)



mk(a b c d)]








aγi

85





ni






n2(F )










86









sim=

H2(kGm) sim= QZ

H1(Γ S(K))

sim=OO

times H1(Γ S(K))

Br(Kk)

OO

T (k)

OO

times H2(Γ T (K))

Br(Kk)

micron2(k)

OO



T (k) times T (K)

Ktimes

micron2(k)

OO

times Zn2 Ktimes

87








n2

88

Conclusion








89

CONCLUSION





)rarr Inv4

(SK1(A)HlowastrLA

)





90

CONCLUSION



4 (k)l then SK1(A) 6= 0





91


Appendix A







93



Hq+1pnL(F (X))rarr

oplusxisinX(1)

HqpnL(F (x))oplus

oplusyisinY (0)


pnL(F (y0))

(A1)





Est2 = Hs






pn (Fnr(X))]




pn (Fnr(X))]

(A2)









)by (25) The




pn (F (y))

94



HqpnL(F (y))



)]sub ker



)]

(A5)




As above



) (A6)






Hq+1pnL (F (X))

partx

H1(Γ Kq(Fnr(X))pn)

partKx


)

(A7)



95






)


Hq+1pnL (F (X))

party

H1(Γ Kq(Fnr(X))pn)

partKy

HqpnL

(F (y)


)

(A8)







HqpnL(F (x))

partxy0


partxKy0


)

(A9)



96





HqpnL(F (y))

partyy0


partyKy0


)

(A10)



xisinX(1)


oplusyisinY (0)



)




oplusxisinX (1)




97













98










99


Appendix B





B1 Introduction





101



times the classof









102

INTRODUCTION





[σ]isinHΓσX




103













δ



δ

πlowastprime

++VVVVVV


++VVVVV


104

PRODUCT VARIETIES










1

1

ktimes oplus ktimes

π1 ktimes





1 1105






1

1



Div(X)oplusDiv(Y )


Pic(X)oplus Pic(Y )


1 1



106

PRODUCT VARIETIES







107





sim=Pic(X)Pic(X)

= 0









Pic(X timesk Y )

0 0

108

WEIL RESTRICTION







then




B3 Weil restriction


109


















110

WEIL RESTRICTION


1 ktimes

id

k(X)times

Div(X)

Pic(X)

ϕprime

1









A1f1

A2f2

A3


IndGH(A1)f1

IndGH(A2)f2

IndGH(A3)

111













HPic(X) ktimes)



H(Pic(X)) ktimes)




following situation




πprime



ilowastltltzzz



112

WEIL RESTRICTION











homological reasons




ψ noplusi=1







sumni=1 ψi






113

Bibliography















115

BIBLIOGRAPHY


















116

BIBLIOGRAPHY


















117

BIBLIOGRAPHY


















118

BIBLIOGRAPHY




















119

Glossary








121

GLOSSARY


x


36


H i+1m (F ) H i+1


plr with p - r14 41







46


15


a cycle module M23




122

GLOSSARY




27









nion k-algebra25



123

GLOSSARY







124

Index

Azumaya algebra 31
















125

INDEX




valuation triple 36


126



Frank Sinatra




Dankwoord

Abstract

Samenvatting

Contents


Introduction






Cohomology groups

Cycle modules


Invariants of SK1


Moderate case

Wild case

General case

Some remarks



Kahns invariant

Conclusion



Introduction

Product varieties

Weil restriction

Bibliography

Glossary

Index

SAMENVATTING

Samenvatting





vi

Contents

Dankwoord iii

Abstract v

Samenvatting vi

Contents vii


Introduction 1







12 Cycle modules 18




21 Moderate case 29

vii

CONTENTS

22 Wild case 40

23 General case 56

24 Some remarks 58




Conclusion 89



B1 Introduction 101



Bibliography 115

Glossary 121

Index 125

viii










ix








x

Introduction










1

INTRODUCTION









2

INTRODUCTION



dimk(A)








i=0 Kxi with


3

INTRODUCTION









4











A 00 1

)isin GLn+1(R)

Then define

GLinfin(R) =⋃ngt0




)

5

INTRODUCTION















)

6




SK1(D)

sim=











7

INTRODUCTION











8




I)



k









9

INTRODUCTION










10



sumpminus1i=0 X










11

INTRODUCTION



12


Chapter 1






A(K)

ϕK M(K)

A(K prime)ϕKprime

M(K prime)




13















14

COHOMOLOGY GROUPS



n times

J





mF (xn) (13)


m (F ))ige0




m (F )







15










m(κ(v))rarr 0 (14)


parti+1v H i+1


m(κ(v))





m (F )





n ) cup r[A])



16

COHOMOLOGY GROUPS











cup r[A]


cup r[BK ]


cup r[A]

0




nBotimesrK(F )





17


12 Cycle modules





R-fieldsrarr Ab







18

CYCLE MODULES












ϕlowastw partw


partv ϕlowast = 0



19





partxy =sumz|y

ϕlowastz partz




0 =sum

xisinX (1)




20

CYCLE MODULES


rarroplus

xisinX(iminus1)


oplusxisinX(i)


oplusxisinX(i+1)





)ige0 and


)ige2





nBotimesrK)


21







x1 x2 xn 7rarr 0









22














23












24

INVARIANTS OF SK1


SK1(A)sim=

ρAF


H4n2A(F )

part3t1part4t2

H2n2(k)part3


n2 ) cup [A])

(110)







ax21 + bx2


4 minus dx25 + cdx2

6 (112)




25



SK1(A)(k)

H4nA(k)




SK1(A)(k)

id

ρS06 ker[H4


]rA

SK1(A)(k)ρRost

ker[(H4

2 (k)rarr H42 (k(Y ))

]

(113)





]




26

INVARIANTS OF SK1














27




H4n(k) sim= H4

pe11


(k)


28


Chapter 2


mdash Isaac Newton



21 Moderate case


211 Strategy


29




of two steps








Lift and specialise

30

MODERATE CASE


SK1(A)(kprime)A

ED




nAotimesr(kprime) 0




212 Lifting objects






31












32

MODERATE CASE










33










sumiisinZ ait

i with

34

MODERATE CASE







0 H i+1nAotimesr(k)

ulowast

H i+1nBotimesrK

(K) parti

vlowast

H inAotimesr(k)

ulowast

0


nBotimesrK(K prime)

parti H i

nAotimesr(kprime) 0



nBotimesrK(K prime)


35







SK1(A)(k)sim=

SK1(BK)(K)



36

MODERATE CASE


213 The lift






nAotimesr(kprime)

SK1(BK)(K prime)

sim=

OO

ρprimeKprime

H4nBotimesrK

(K prime)

(21)


= (Hj







37









H1(k micron)

cup r[A]


cup r[Ak(G)]



H3n(k)

H3n(k(G))

part3

oplusxisinG(1) H2

n(k(x))

H3nAotimesr(k) H3


oplusxisinG(1) H2

nAotimesr(k(x))



xisinG(1)


38

MODERATE CASE














nAotimesr(kprime)











39




22 Wild case





40

WILD CASE





andΩ1kZ and






xdy1

y1and and dyq


y1and and dyq

yqmod dΩqminus1

k






q times


xdy1

y1and and dyq





41


Then Cqpn(k) = Dq




defined by















1 ktimesspn

ktimessdlog

νn(1)ks 1


0 νn(q)ks Cqpn(ks)

Fminus1 Cq


(24)

42

WILD CASE









p (k) [GS Prop 925]











rarr H1(FZpnZ)

43




Wn(κ(v)s) 0





















44

WILD CASE






Hj+1pnL(F ) =

ker[Hj+1








45





pn(F ) sim= Cqpn(E)








dx1

x1and and dxm

xm


a + dbb







46

WILD CASE



pn (F ) defined by










0 Hq+1pn (κ(v))

Hq+1pnnr(F )

Hqpn(κ(v))

0




47


















48

WILD CASE








49



0 Hq+1pnL(κ(v))

ilowast Hq+1pnL(F )

part HqpnL(κ(v)) 0


OO


OO


OO

0


H2pn(κ(v))

sim=

ilowast H2pnnr(F )

sim=

pnBr(κ(v))i

pnBrnr(F )

(210)




)στisin H2

pn(F )



pn(κ(v)) inpnH


)στ

So


)στisin H2

pn(F )


50

WILD CASE


H2pnL(κ(v))

sim=

ilowast H2pnL(F )

sim=


pnBr(LotimesK FF )



)= ilowastK

(Kqminus1(κ(v))




)= partK

(Kqminus1(F )

)middot r[Aκ(v)]



)= ψK

(Kqminus2(κ(v))





51




ulowast

H i+1pnLBotimesrK

(K) parti

vlowast

H ipnLAotimesr(k)

ulowast

0







222 The lift


H0Zar(GH3

pn) sim= H0Zar(GH3

pn)H3pn(k)




H0Zar(GH3

pn) rarr H0Zar(GH3

pn)

52

WILD CASE



H0Zar(GH3




2 (U) (where KQ2






pn)



53








Zar(GH3pnL) sub H0




[H3pn(k(G))rarr H3

pn(ks(G))]


[H0

Zar(GH3pn) rarr H0

Zar(GH3pn)]







pnLAotimesr(kprime)

SK1(BK)(K prime)

sim=

OO

ρprimeKprime

H4pnLBotimesrK

(K prime)


= (HjpnLBotimesrK

)jge2


54

WILD CASE


ktimes

middotr[A]

k(G)timespart1

middotr[Ak(G)]

oplusxisinG(1) Z


H3pn(k)

H3pn(k(G)) part3

oplusxisinG(1) H2

pn(k(x))

H3pnAotimesr(k) H3


oplusxisinG(1) H2

pnAotimesr(k(x))






55



SK1(BK)ρK

H4pnLBotimesrK

(K)

SK1(BL)ρL

H4pnLBotimesrK

(L)




23 General case






56

GENERAL CASE





pn



(SK1(BK)Hlowast

eLBotimesrK






eLAotimesr(kprime)

SK1(BK)(K prime)

sim=

OO

ρprimeKprime

H4eLBotimesrK

(K prime)


57



24 Some remarks








58

SOME REMARKS


A(F ((t))

) ρrarrMj

(F ((t))










(kprime((t))

)rarr Hjminus1



gives us0 = ρ(α(0)) = ρ(α(1)) = ρ([a b])

59








eLAotimesr(F ((t))

)rarr H3

eLAotimesr(F )







60

SOME REMARKS


0 A0(RH4)

A0(KH4)

A0(kH3)

0











Inv4(GGGK Hlowast)

ϕ


61






62


Chapter 3










63









64












65










21+ +anx2

n




66
















67













0 SK1(A)(F )

=

ρRostAF H4

2 (F )

ψ

H42 (F (Y ))

sim=

0 SK1(A)(F )ρBIAF

I4(F ) I4(F (Y ))



68















69





Symd([a b) σ1

)= 〈v〉 and Symd

([c d) σ2

)= 〈1〉



Symd ((4a+ 1 b) τ1) = 〈1 i j〉 and


)= 〈1 i ij〉


70



Symd ((4c+ 1 d) τ2) = 〈1〉 and


)= 〈1 j ij〉





Symd(BK τ)





Symd(BK τ prime)



71














H44A(kprime)

ilowast


ψkprime

ilowast

I3Wq(kprime)

j

H44BK (K prime)

rB H4

2 (K prime)sim=

ψKprime

I3Wq(K prime)

(32)


72












34(x1 x2 x3)


32(x1 x2 x3)


Hn2 (kprime)

sim=

ϕ

I3Wq(kprime)

j


sim= I3Wq(K prime)



bda1

a1and da2

a2and da3


73









12 (K prime)








74






Then








75













76


321 Moderate case




nm(F )rarr H4n(F )


πr H4n(F )rarr H4

nAotimesr(F )










77







4 (k) sim= K4(k)4






part3t1 part

4t2

((K2(k)4) middot (2a t1+ 2b t2)

)


4A(k) sim= Z2





78








3 tplusmn14 ] where










79






SK1(A)

sim=

ρ H4

m2(F )

SK1(TKprime)ρ

H4m2(K prime)






80











322 Wild case








81


















82







mk(a b c d)]isin H4

n2(k)


n2(k) (for m = n2)



]= mprime h4






83




m(k)







84




ϕ[h4mk(a b c d)

]is sent to ϕ

[h4mK(a b c d)

](with an abuse of



mk(a b c d)] (34)



mk(a b c d)]








aγi

85





ni






n2(F )










86









sim=

H2(kGm) sim= QZ

H1(Γ S(K))

sim=OO

times H1(Γ S(K))

Br(Kk)

OO

T (k)

OO

times H2(Γ T (K))

Br(Kk)

micron2(k)

OO



T (k) times T (K)

Ktimes

micron2(k)

OO

times Zn2 Ktimes

87








n2

88

Conclusion








89

CONCLUSION





)rarr Inv4

(SK1(A)HlowastrLA

)





90

CONCLUSION



4 (k)l then SK1(A) 6= 0





91


Appendix A







93



Hq+1pnL(F (X))rarr

oplusxisinX(1)

HqpnL(F (x))oplus

oplusyisinY (0)


pnL(F (y0))

(A1)





Est2 = Hs






pn (Fnr(X))]




pn (Fnr(X))]

(A2)









)by (25) The




pn (F (y))

94



HqpnL(F (y))



)]sub ker



)]

(A5)




As above



) (A6)






Hq+1pnL (F (X))

partx

H1(Γ Kq(Fnr(X))pn)

partKx


)

(A7)



95






)


Hq+1pnL (F (X))

party

H1(Γ Kq(Fnr(X))pn)

partKy

HqpnL

(F (y)


)

(A8)







HqpnL(F (x))

partxy0


partxKy0


)

(A9)



96





HqpnL(F (y))

partyy0


partyKy0


)

(A10)



xisinX(1)


oplusyisinY (0)



)




oplusxisinX (1)




97













98










99


Appendix B





B1 Introduction





101



times the classof









102

INTRODUCTION





[σ]isinHΓσX




103













δ



δ

πlowastprime

++VVVVVV


++VVVVV


104

PRODUCT VARIETIES










1

1

ktimes oplus ktimes

π1 ktimes





1 1105






1

1



Div(X)oplusDiv(Y )


Pic(X)oplus Pic(Y )


1 1



106

PRODUCT VARIETIES







107





sim=Pic(X)Pic(X)

= 0









Pic(X timesk Y )

0 0

108

WEIL RESTRICTION







then




B3 Weil restriction


109


















110

WEIL RESTRICTION


1 ktimes

id

k(X)times

Div(X)

Pic(X)

ϕprime

1









A1f1

A2f2

A3


IndGH(A1)f1

IndGH(A2)f2

IndGH(A3)

111













HPic(X) ktimes)



H(Pic(X)) ktimes)




following situation




πprime



ilowastltltzzz



112

WEIL RESTRICTION











homological reasons




ψ noplusi=1







sumni=1 ψi






113

Bibliography















115

BIBLIOGRAPHY


















116

BIBLIOGRAPHY


















117

BIBLIOGRAPHY


















118

BIBLIOGRAPHY




















119

Glossary








121

GLOSSARY


x


36


H i+1m (F ) H i+1


plr with p - r14 41







46


15


a cycle module M23




122

GLOSSARY




27









nion k-algebra25



123

GLOSSARY







124

Index

Azumaya algebra 31
















125

INDEX




valuation triple 36


126



Frank Sinatra




Dankwoord

Abstract

Samenvatting

Contents


Introduction






Cohomology groups

Cycle modules


Invariants of SK1


Moderate case

Wild case

General case

Some remarks



Kahns invariant

Conclusion



Introduction

Product varieties

Weil restriction

Bibliography

Glossary

Index

Contents

Dankwoord iii

Abstract v

Samenvatting vi

Contents vii


Introduction 1







12 Cycle modules 18




21 Moderate case 29

vii

CONTENTS

22 Wild case 40

23 General case 56

24 Some remarks 58




Conclusion 89



B1 Introduction 101



Bibliography 115

Glossary 121

Index 125

viii










ix








x

Introduction










1

INTRODUCTION









2

INTRODUCTION



dimk(A)








i=0 Kxi with


3

INTRODUCTION









4











A 00 1

)isin GLn+1(R)

Then define

GLinfin(R) =⋃ngt0




)

5

INTRODUCTION















)

6




SK1(D)

sim=











7

INTRODUCTION











8




I)



k









9

INTRODUCTION










10



sumpminus1i=0 X










11

INTRODUCTION



12


Chapter 1






A(K)

ϕK M(K)

A(K prime)ϕKprime

M(K prime)




13















14

COHOMOLOGY GROUPS



n times

J





mF (xn) (13)


m (F ))ige0




m (F )







15










m(κ(v))rarr 0 (14)


parti+1v H i+1


m(κ(v))





m (F )





n ) cup r[A])



16

COHOMOLOGY GROUPS











cup r[A]


cup r[BK ]


cup r[A]

0




nBotimesrK(F )





17


12 Cycle modules





R-fieldsrarr Ab







18

CYCLE MODULES












ϕlowastw partw


partv ϕlowast = 0



19





partxy =sumz|y

ϕlowastz partz




0 =sum

xisinX (1)




20

CYCLE MODULES


rarroplus

xisinX(iminus1)


oplusxisinX(i)


oplusxisinX(i+1)





)ige0 and


)ige2





nBotimesrK)


21







x1 x2 xn 7rarr 0









22














23












24

INVARIANTS OF SK1


SK1(A)sim=

ρAF


H4n2A(F )

part3t1part4t2

H2n2(k)part3


n2 ) cup [A])

(110)







ax21 + bx2


4 minus dx25 + cdx2

6 (112)




25



SK1(A)(k)

H4nA(k)




SK1(A)(k)

id

ρS06 ker[H4


]rA

SK1(A)(k)ρRost

ker[(H4

2 (k)rarr H42 (k(Y ))

]

(113)





]




26

INVARIANTS OF SK1














27




H4n(k) sim= H4

pe11


(k)


28


Chapter 2


mdash Isaac Newton



21 Moderate case


211 Strategy


29




of two steps








Lift and specialise

30

MODERATE CASE


SK1(A)(kprime)A

ED




nAotimesr(kprime) 0




212 Lifting objects






31












32

MODERATE CASE










33










sumiisinZ ait

i with

34

MODERATE CASE







0 H i+1nAotimesr(k)

ulowast

H i+1nBotimesrK

(K) parti

vlowast

H inAotimesr(k)

ulowast

0


nBotimesrK(K prime)

parti H i

nAotimesr(kprime) 0



nBotimesrK(K prime)


35







SK1(A)(k)sim=

SK1(BK)(K)



36

MODERATE CASE


213 The lift






nAotimesr(kprime)

SK1(BK)(K prime)

sim=

OO

ρprimeKprime

H4nBotimesrK

(K prime)

(21)


= (Hj







37









H1(k micron)

cup r[A]


cup r[Ak(G)]



H3n(k)

H3n(k(G))

part3

oplusxisinG(1) H2

n(k(x))

H3nAotimesr(k) H3


oplusxisinG(1) H2

nAotimesr(k(x))



xisinG(1)


38

MODERATE CASE














nAotimesr(kprime)











39




22 Wild case





40

WILD CASE





andΩ1kZ and






xdy1

y1and and dyq


y1and and dyq

yqmod dΩqminus1

k






q times


xdy1

y1and and dyq





41


Then Cqpn(k) = Dq




defined by















1 ktimesspn

ktimessdlog

νn(1)ks 1


0 νn(q)ks Cqpn(ks)

Fminus1 Cq


(24)

42

WILD CASE









p (k) [GS Prop 925]











rarr H1(FZpnZ)

43




Wn(κ(v)s) 0





















44

WILD CASE






Hj+1pnL(F ) =

ker[Hj+1








45





pn(F ) sim= Cqpn(E)








dx1

x1and and dxm

xm


a + dbb







46

WILD CASE



pn (F ) defined by










0 Hq+1pn (κ(v))

Hq+1pnnr(F )

Hqpn(κ(v))

0




47


















48

WILD CASE








49



0 Hq+1pnL(κ(v))

ilowast Hq+1pnL(F )

part HqpnL(κ(v)) 0


OO


OO


OO

0


H2pn(κ(v))

sim=

ilowast H2pnnr(F )

sim=

pnBr(κ(v))i

pnBrnr(F )

(210)




)στisin H2

pn(F )



pn(κ(v)) inpnH


)στ

So


)στisin H2

pn(F )


50

WILD CASE


H2pnL(κ(v))

sim=

ilowast H2pnL(F )

sim=


pnBr(LotimesK FF )



)= ilowastK

(Kqminus1(κ(v))




)= partK

(Kqminus1(F )

)middot r[Aκ(v)]



)= ψK

(Kqminus2(κ(v))





51




ulowast

H i+1pnLBotimesrK

(K) parti

vlowast

H ipnLAotimesr(k)

ulowast

0







222 The lift


H0Zar(GH3

pn) sim= H0Zar(GH3

pn)H3pn(k)




H0Zar(GH3

pn) rarr H0Zar(GH3

pn)

52

WILD CASE



H0Zar(GH3




2 (U) (where KQ2






pn)



53








Zar(GH3pnL) sub H0




[H3pn(k(G))rarr H3

pn(ks(G))]


[H0

Zar(GH3pn) rarr H0

Zar(GH3pn)]







pnLAotimesr(kprime)

SK1(BK)(K prime)

sim=

OO

ρprimeKprime

H4pnLBotimesrK

(K prime)


= (HjpnLBotimesrK

)jge2


54

WILD CASE


ktimes

middotr[A]

k(G)timespart1

middotr[Ak(G)]

oplusxisinG(1) Z


H3pn(k)

H3pn(k(G)) part3

oplusxisinG(1) H2

pn(k(x))

H3pnAotimesr(k) H3


oplusxisinG(1) H2

pnAotimesr(k(x))






55



SK1(BK)ρK

H4pnLBotimesrK

(K)

SK1(BL)ρL

H4pnLBotimesrK

(L)




23 General case






56

GENERAL CASE





pn



(SK1(BK)Hlowast

eLBotimesrK






eLAotimesr(kprime)

SK1(BK)(K prime)

sim=

OO

ρprimeKprime

H4eLBotimesrK

(K prime)


57



24 Some remarks








58

SOME REMARKS


A(F ((t))

) ρrarrMj

(F ((t))










(kprime((t))

)rarr Hjminus1



gives us0 = ρ(α(0)) = ρ(α(1)) = ρ([a b])

59








eLAotimesr(F ((t))

)rarr H3

eLAotimesr(F )







60

SOME REMARKS


0 A0(RH4)

A0(KH4)

A0(kH3)

0











Inv4(GGGK Hlowast)

ϕ


61






62


Chapter 3










63









64












65










21+ +anx2

n




66
















67













0 SK1(A)(F )

=

ρRostAF H4

2 (F )

ψ

H42 (F (Y ))

sim=

0 SK1(A)(F )ρBIAF

I4(F ) I4(F (Y ))



68















69





Symd([a b) σ1

)= 〈v〉 and Symd

([c d) σ2

)= 〈1〉



Symd ((4a+ 1 b) τ1) = 〈1 i j〉 and


)= 〈1 i ij〉


70



Symd ((4c+ 1 d) τ2) = 〈1〉 and


)= 〈1 j ij〉





Symd(BK τ)





Symd(BK τ prime)



71














H44A(kprime)

ilowast


ψkprime

ilowast

I3Wq(kprime)

j

H44BK (K prime)

rB H4

2 (K prime)sim=

ψKprime

I3Wq(K prime)

(32)


72












34(x1 x2 x3)


32(x1 x2 x3)


Hn2 (kprime)

sim=

ϕ

I3Wq(kprime)

j


sim= I3Wq(K prime)



bda1

a1and da2

a2and da3


73









12 (K prime)








74






Then








75













76


321 Moderate case




nm(F )rarr H4n(F )


πr H4n(F )rarr H4

nAotimesr(F )










77







4 (k) sim= K4(k)4






part3t1 part

4t2

((K2(k)4) middot (2a t1+ 2b t2)

)


4A(k) sim= Z2





78








3 tplusmn14 ] where










79






SK1(A)

sim=

ρ H4

m2(F )

SK1(TKprime)ρ

H4m2(K prime)






80











322 Wild case








81


















82







mk(a b c d)]isin H4

n2(k)


n2(k) (for m = n2)



]= mprime h4






83




m(k)







84




ϕ[h4mk(a b c d)

]is sent to ϕ

[h4mK(a b c d)

](with an abuse of



mk(a b c d)] (34)



mk(a b c d)]








aγi

85





ni






n2(F )










86









sim=

H2(kGm) sim= QZ

H1(Γ S(K))

sim=OO

times H1(Γ S(K))

Br(Kk)

OO

T (k)

OO

times H2(Γ T (K))

Br(Kk)

micron2(k)

OO



T (k) times T (K)

Ktimes

micron2(k)

OO

times Zn2 Ktimes

87








n2

88

Conclusion








89

CONCLUSION





)rarr Inv4

(SK1(A)HlowastrLA

)





90

CONCLUSION



4 (k)l then SK1(A) 6= 0





91


Appendix A







93



Hq+1pnL(F (X))rarr

oplusxisinX(1)

HqpnL(F (x))oplus

oplusyisinY (0)


pnL(F (y0))

(A1)





Est2 = Hs






pn (Fnr(X))]




pn (Fnr(X))]

(A2)









)by (25) The




pn (F (y))

94



HqpnL(F (y))



)]sub ker



)]

(A5)




As above



) (A6)






Hq+1pnL (F (X))

partx

H1(Γ Kq(Fnr(X))pn)

partKx


)

(A7)



95






)


Hq+1pnL (F (X))

party

H1(Γ Kq(Fnr(X))pn)

partKy

HqpnL

(F (y)


)

(A8)







HqpnL(F (x))

partxy0


partxKy0


)

(A9)



96





HqpnL(F (y))

partyy0


partyKy0


)

(A10)



xisinX(1)


oplusyisinY (0)



)




oplusxisinX (1)




97













98










99


Appendix B





B1 Introduction





101



times the classof









102

INTRODUCTION





[σ]isinHΓσX




103













δ



δ

πlowastprime

++VVVVVV


++VVVVV


104

PRODUCT VARIETIES










1

1

ktimes oplus ktimes

π1 ktimes





1 1105






1

1



Div(X)oplusDiv(Y )


Pic(X)oplus Pic(Y )


1 1



106

PRODUCT VARIETIES







107





sim=Pic(X)Pic(X)

= 0









Pic(X timesk Y )

0 0

108

WEIL RESTRICTION







then




B3 Weil restriction


109


















110

WEIL RESTRICTION


1 ktimes

id

k(X)times

Div(X)

Pic(X)

ϕprime

1









A1f1

A2f2

A3


IndGH(A1)f1

IndGH(A2)f2

IndGH(A3)

111













HPic(X) ktimes)



H(Pic(X)) ktimes)




following situation




πprime



ilowastltltzzz



112

WEIL RESTRICTION











homological reasons




ψ noplusi=1







sumni=1 ψi






113

Bibliography















115

BIBLIOGRAPHY


















116

BIBLIOGRAPHY


















117

BIBLIOGRAPHY


















118

BIBLIOGRAPHY




















119

Glossary








121

GLOSSARY


x


36


H i+1m (F ) H i+1


plr with p - r14 41







46


15


a cycle module M23




122

GLOSSARY




27









nion k-algebra25



123

GLOSSARY







124

Index

Azumaya algebra 31
















125

INDEX




valuation triple 36


126



Frank Sinatra




Dankwoord

Abstract

Samenvatting

Contents


Introduction






Cohomology groups

Cycle modules


Invariants of SK1


Moderate case

Wild case

General case

Some remarks



Kahns invariant

Conclusion



Introduction

Product varieties

Weil restriction

Bibliography

Glossary

Index

Cohomological invariants of SK - COnnecting REpositoriestu as sugg´er´e, trouvait sa place. Merci...

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Transcript of Cohomological invariants of SK - COnnecting REpositoriestu as sugg´er´e, trouvait sa place. Merci...