Arithmetic Duality Theorems

Arithmetic Duality TheoremsSecond Edition

J.S. Milne

August 7, 2004

2

This work is available at http://www.jmilne.org/math/

First edition published by Academic Press 1986.

c 2004 J.S. Milne.

This work is licensed under a Creative Commons License:

http://creativecommons.org/licenses/by-nc-nd/2.0/

Briefly, you are free to copy the work for noncommercial purposes under certain conditions(see the link for a precise statement).

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3

Preface to the first edition.

In the late fifties and early sixties, Tate (and Poitou) found some important duality theoremsconcerning the Galois cohomology of finite modules and abelian varieties over local andglobal fields.

About 1964, Artin and Verdier extended some of the results toetale cohomology groupsover rings of integers in local and global fields.

Since then many people (Artin, Bester, Begueri, Mazur, McCallum, the author, Roberts,Shatz, Vvedens’kii) have generalized these results to flat cohomology groups.

Much of the best of this work has not been fully published. My initial purpose inpreparing these notes was simply to write down a complete set of proofs before they wereforgotten, but I have also tried to give an organized account of the whole subject. Only afew of the theorems in these notes are new, but many results have been sharpened, and asignificant proportion of the proofs have not been published before.

The first chapter proves the theorems on Galois cohomology announced by Tate in histalk at the International Congress at Stockholm in 1962, and describes later work in thesame area. The second chapter proves the theorem of Artin and Verdier onetale coho-mology and also various generalizations of it. In the final chapter improvements using flatcohomology are described.

As far as possible, theorems are proved in the context in which they are stated: thustheorems on Galois cohomology are proved using only Galois cohomology, and theoremson etale cohomology are proved using onlyetale cohomology.

Each chapter begins with a summary of its contents; each section ends with a list of itssources.

It is a pleasure to thank all those with whom I have discussed these questions over theyears, but especially M. Artin, P. Berthelot, L. Breen, S. Bloch, K. Kato, S. Lichtenbaum,W. McCallum, B. Mazur, W. Messing, L. Roberts, and J. Tate.

Parts of the author’s research contained in this volume have been supported by theNational Science Foundation.

Finally, I mention that, thanks to the computer, it has been possible to produce thisvolume without recourse to typist, copy editor, or type-setter.

4

Preface to the second edition.

A perfect new edition would fix all the errors, improve the exposition, update the text, and,of course, being perfect, it would also exist. Unfortunately, these conditions are contradic-tory. For this version, I have translated the original word-processor file into TEX, fixed allthe errors that I am aware of, made a few minor improvements to the exposition, and addeda few footnotes.

Significant changes to the text have been noted in the footnotes. The numbering isunchanged from the original (except for II 3.18). All footnotes have been added for thisedition except for those on p30 and p244.

There are a few minor changes in notation: canonical isomorphisms are often denoted�D rather than�, and, lacking a Cyrillic font, I use III as a substitute for the Russian lettershah.

I thank the following for providing corrections and comments on earlier versions: Ching-Li Chai, Matthias Fohl, Cristian Gonzalez-Aviles, David Harari, Eugene Kushnirsky, BillMcCallum, Bjorn Poonen, Joel Riou, and others.

Since most of the translation was done by computer, I hope that not many new misprintshave been introduced. Please send further corrections to me at [email protected].

20.02.2004. First version on web.07.08.2004. Proofread against original again; fixed many misprints and minor errors; im-proved index; improved TEX, including replaced III with the correct CyrillicX.

Contents

I Galois Cohomology 90 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Duality relative to a class formation . . . . . . . . . . . . . . . . . . . . . 222 Local fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 Abelian varieties over local fields . . . . . . . . . . . . . . . . . . . . . . . 414 Global fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 Global Euler-Poincare characteristics . . . . . . . . . . . . . . . . . . . . . 636 Abelian varieties over global fields . . . . . . . . . . . . . . . . . . . . . . 687 Application to the conjecture B-S/D . . . . . . . . . . . . . . . . . . . . . 868 Abelian class field theory, in the sense of Langlands . . . . . . . . . . . . . 929 Other applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Appendix A: Class field theory for function fields . . . . . . . . . . . . . . . . . 113

II Etale Cohomology 1230 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1231 Local results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1312 Global results: preliminary calculations . . . . . . . . . . . . . . . . . . . 1433 Global results: the main theorem . . . . . . . . . . . . . . . . . . . . . . . 1544 Global results: complements . . . . . . . . . . . . . . . . . . . . . . . . . 1655 Global results: abelian schemes . . . . . . . . . . . . . . . . . . . . . . . . 1726 Global results: singular schemes . . . . . . . . . . . . . . . . . . . . . . . 1797 Global results: higher dimensions . . . . . . . . . . . . . . . . . . . . . . 181

III Flat Cohomology 1890 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1891 Local results: mixed characteristic, finite group schemes . . . . . . . . . . 2012 Local results: mixed characteristic, abelian varieties . . . . . . . . . . . . . 2123 Global results: number field case . . . . . . . . . . . . . . . . . . . . . . . 2184 Local results: mixed characteristic, perfect residue field . . . . . . . . . . . 2225 Two exact sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2296 Local fields of characteristicp . . . . . . . . . . . . . . . . . . . . . . . . 2347 Local results: equicharacteristic, finite residue field . . . . . . . . . . . . . 2418 Global results: curves over finite fields, finite sheaves . . . . . . . . . . . . 2489 Global results: curves over finite fields, Neron models . . . . . . . . . . . . 25310 Local results: equicharacteristic, perfect residue field . . . . . . . . . . . . 258

5

6 CONTENTS

11 Global results: curves over perfect fields . . . . . . . . . . . . . . . . . . . 261Appendix A: Embedding finite group schemes . . . . . . . . . . . . . . . . . . . 264Appendix B: Extending finite group schemes . . . . . . . . . . . . . . . . . . . 268Appendix C: Biextensions and Neron models . . . . . . . . . . . . . . . . . . . 271

Index 292

CONTENTS 7

Notations and Conventions

We list our usual notations and conventions. When they are not used in a particular section,this is noted at the start of the section.

A global field is a finite extension ofQ or is finitely generated and of finite transcen-dence degree one over a finite field. Alocal field is R, C, or a field that is locally compactrelative to a discrete valuation. Thus it is a finite extension ofQp, Fp((T )), or R. If v is aprime of a global field, thenj jv denotes the valuation atv normalized in the usual way sothat the product formula holds, andOv D fa 2 K j jajv � 1g. The completions ofK andOv relative toj jv are denoted byKv andbOv.

For a fieldK, Ka andKs denote the algebraic and separable algebraic closures ofK,and Kab denotes the maximal abelian extension ofK. For a local fieldK, Kun is themaximal unramified extension of ofK. We sometimes writeGK for the absolute Galoisgroup Gal(Ks=K) of K andGF=K for Gal(F=K). By char(K) we mean the characteristicexponent ofK, that is, char(K) is p if K has characteristicp 6D 0 and is1 otherwise. Fora Hausdorff topological groupG, Gab is the quotient ofG by the closure of its commutatorsubgroup. Thus,Gab is the maximal abelian Hausdorff quotient group ofG, andGab

K D

Gal(Kab=K).If M is an abelian group (or, more generally, an object in an abelian category) andm is

an integer, thenMm andM (m) are the kernel and cokernel of multiplication bym on M .Moreover,M(m) is the m-primary component

Sm Mmn and Mm�div is the m-divisible

subgroupT

n Im(mnWM ! M). Thedivisible subgroup1 Mdiv of M is

Tm Mm-div. We

write TmM for lim �

Mmn and cM for the completion ofM with respect to the topologydefined by the subgroups of finite index (sometimes the subgroups are restricted to thoseof finite index a power of a fixed integerm, and sometimes to those that are open withrespect to some topology onM ). WhenM is finite, [M] denotes its order. A groupM isof cofinite-typeif it is torsion andMm is finite for all integersm.

As befits a work with the title of this one, we shall need to consider a great manydifferent types of duals. In general,M � will denote Homcts(M, Q=Z), the group of con-tinuous characters of finite order ofM . Thus, ifM is discrete torsion abelian group, thenM � is its compact Pontryagin dual, and ifM is a profinite abelian group, thenM � is itsdiscrete torsion Pontryagin dual. IfM is a module overGK for some fieldK, thenM D

denotes the dual Hom(M, Ks�); whenM is a finite group scheme,M D is the Cartier dualHom(M, Gm). The dual (Picard variety) of an abelian variety is denoted byAt . For avector spaceM , M _ denotes the linear dual ofM .

All algebraic groups and group schemes will be commutative (unless stated otherwise).If T is a torus over a fieldk, thenX �(T ) is the group Homks(Gm, Tks) of cocharacters(also called the multiplicative one-parameter subgroups).

There seems to be no general agreement on what signs should be used in homologicalalgebra. Fortunately, the signs of the maps in these notes will not be important, but thereader should be aware that when a diagram is said to commute, it may only commute upto sign. I have generally followed the sign conventions in Berthelot, Breen, and Messing1982, Chapter 0.

1This should be called the subgroup of divisible elements — it contains the largest divisible subgroup ofM but it need not be divisible itself. A similar remark applies to them-divisible subgroup.

8 CONTENTS

We sometimes useD to denote a canonical isomorphism,2 and the symbolsXdfD Y and

X Ddf Y mean thatX is defined to beY , or thatX equalsY by definition.In Chapters II and III, we shall need to consider several different topologies on a scheme

X (always assumed to be locally Noetherian or the perfection of a locally Noetherianscheme). These are denoted as follows:

Xet (small etale site) is the category of schemesetale overX endowed with theetaletopology;

XEt (big etale site) is the category of schemes locally of finite-type overX endowedwith theetale topology;

Xsm (smooth site) is the category of schemes smooth overX endowed with the smoothtopology (covering families are surjective families of smooth maps);

Xqf (small fpqf site) is the category of schemes flat and quasi-finite overX endowedwith the flat topology;

Xfl (big flat site) is the category of schemes locally of finite-type overX endowed withthe flat topology;

Xpf (perfect site) see (III 0).The category of sheaves of abelian groups on a siteX� is denoted byS(X�).

2And sometimes, in this edition,�D.

Chapter I

Galois Cohomology

In ~1 we prove a very general duality theorem that applies whenever one has a class for-mation. The theorem is used in~2 to prove a duality theorem for modules over the Galoisgroup of a local field. This section also contains an expression for the Euler-Poincare char-acteristic of such a module. In~3, these results are used to prove Tate’s duality theorem forabelian varieties over a local field.

The next four sections concern global fields. Tate’s duality theorem on modules overthe Galois group of a global field is obtained in~4 by applying the general result in~1 tothe class formation of the global field and combining the resulting theorem with the localresults in~2. Section 5 derives a formula for the Euler-Poincare characteristic of such amodule. Tate’s duality theorems for abelian varieties over global fields are proved in~6,and in the following section it is shown that the validity of the conjecture of Birch andSwinnerton-Dyer for an abelian variety over a number field depends only on the isogenyclass of the variety.

The final three sections treat rather diverse topics. In~8 a duality theorem is provedfor tori that implies the abelian case of Langlands’s conjectures for a nonabelian class fieldtheory. The next section briefly describes some of the applications that have been made ofthe duality theorems: to the Hasse principle for finite modules and algebraic groups, to theexistence of forms of algebraic groups, to Tamagawa numbers of algebraic tori over globalfields, and to the central embedding problem for Galois groups. In the appendix, a classfield theory is developed for Henselian local fields whose residue fields are quasi-finite andfor function fields in one variable over quasi-finite fields.

In this chapter, the reader is assumed to be familiar with basic Galois cohomology (thefirst two chapters of Serre 1964 or the first four chapters of Shatz 1972), class field theory(Serre 1967a and Tate 1967a), and, in a few sections, abelian varieties (Milne 1986b).

Throughout the chapter, whenG is a profinite group, “G-module” will mean “dis-creteG-module”, and the cohomology groupH r (G, M) will be defined using continuouscochains. The category of discreteG-modules is denoted byModG .

9

10 CHAPTER I. GALOIS COHOMOLOGY

0 Preliminaries

Throughout this section,G will be a profinite group. By a torsion-freeG-module, we meanaG-module that is torsion-free as an abelian group.

Tate (modified) cohomology groups

(Serre 1962, VIII; Weiss (1969).)When G is finite, there areTate cohomology groupsH r

T (G, M), r 2 Z, M a G-module, such that

H rT (G, M) D H r (G, M), r > 0,

H 0T (G, M) DM G=NGM, whereNG D

P�2G�,

H �1T (G, M) D Ker(NG)=IGM , whereIG D f

Pn�� j

Pn� D 0g ,

H �rT (G, M) D Hr�1(G, M), �r < �1.

A short exact sequence ofG-modules gives rise to a long exact sequence of Tate cohomol-ogy groups (infinite in both directions).

A complete resolutionfor G is an exact sequence

L� D � � � ! L2

d2

! L1

d1

! L0

d0

! L�1

d�1

! L�2! � � �

of finitely generated freeZ[G]-modules, together with an elemente 2 LG�1 that generates

the image ofd0. For any complete resolution ofG, H rT (G, M) is ther th cohomology group

of the complex HomG(L�, M). The mapd0 factors as

L0

�! Z

�! L�1

with �(x)e D d0(x) and�(m) D me. If we let

LC� D � � � ! L2

d2

! L1

d1

! L0

L�� D L�1

d�1

! L�2! L�3! � � � ,

then

H r (G, M) D H r (HomG(LC� , M)), r � 0,

Hr (G, M) D H �r�1(HomG(L�� , M)), r � 0.

By thestandard resolutionLC� for G we mean the complex withLC

r D Z[Gr ] and theusual boundary map, so that Hom(LC

� , M) is the complex of nonhomogeneous cochainsof M (see Serre 1962, VII 3). By thestandard complete resolutionfor G, we mean thecomplete resolution obtained by splicing togetherLC

� with its dual (see Weiss 1969, I-4-1).Except for Tate cohomology groups, we always setH r (G, M) D 0 for r < 0.

For any bilinearG-equivariant pairing ofG-modules

M �N ! P

0. PRELIMINARIES 11

there is a family of cup-product pairings

(x, y) 7! x [ yWH rT (G, M)�H s

T (G, N)! H rCsT (G, P)

with the following properties:(0.1.1)dx [ y D d(x [ y)I

(0.1.2)x [ dy D (�1)deg(x)d(x [ y)I

(0.1.3)x [ (y [ z) D (x [ y) [ zI

(0.1.4)x [ y D (�1)deg(x) deg(y)y [ xI

(0.1.5)Res(x [ y) D Res(x) [ Res(y)I

(0.1.6)Inf(x [ y) D Inf(x) [ Inf(y)I

(d Dboundary map, ResDrestriction map; InfDinflation map).

THEOREM 0.2 (TATE-NAKAYAMA ). Let G be a finite group,C a G-module, andu anelement ofH 2(G, C ). Suppose that for all subgroupsH of G

(a) H 1(H, C ) D 0, and(b) H 2(H, C ) has order equal to that ofH and is generated byRes(u).

Then, for anyG-moduleM such thatTorZ1(M, C ) D 0, cup-product withu defines anisomorphism

x 7! x [ uWH rT (G, M)! H rC2

T (G, M ˝ C )

for all integersr .

PROOF. Serre 1962, IX 8.

Extensions ofG-modules

For G-modulesM andN , define ExtrG(M, N) to be the set of homotopy classes of mor-phismsM �

! N � of degreer , whereM � is any resolution ofM by G-modules andN � is any resolution ofN by injectiveG-modules. One sees readily that different resolu-tions ofM andN give rise to canonically isomorphic groups Extr

G(M, N). On takingM �

to be M itself, we see that ExtrG(M, N) D H r (HomG(M, N �)), and so ExtrG(M,�)

is the r th right derived functor ofN 7! HomG(M, N)WModG ! Ab. In particular,ExtrG(Z, N) D H r (G, N).

There is a canonical product

(f, g) 7! f � gWExtrG(N, P)� ExtsG(M, N)! ExtrCsG (M, P)

such thatf � g is obtained fromf WN �! P � andgWM ! N � by composition (hereN �

andP � are injective resolutions ofN andP). Forr D s D 0, the product can be identifiedwith composition

(f, g) 7! f ı gWHomG(N, P)� HomG(M, N)! HomG(M, P).

When we takeM D Z, and replaceN andP with M andN , the pairing becomes

ExtrG(M, N)�H s(G, M)! H rCs(G, N).


An r -fold extension ofM by N defines in a natural way a class in ExtrG(M, N) (see

Bourbaki Alg. X 7.3 for one correct choice of signs). Two such extensions define thesame class if and only if they are equivalent in the usual sense, and forr � 1, every ele-ment of ExtrG(M, N) arises from such an extension (ibid. X 7.5). Therefore Extr

G(M, N)

can be identified with the set of equivalence classes ofr -fold extensions ofM by N .With this identification, products are obtained by splicing extensions (ibid. X 7.6). Letf 2 ExtrG(N, P); then the mapg 7! f � gWExtrG(M, N) ! ExtrCs

G (M, P) is ther -foldboundary map defined by anyr -fold extension ofN by P representingf.

A spectral sequence for Exts

Let M andN beG-modules, and write Hom(M, N) for the set of homomorphisms fromM to N as abelian groups. Forf 2 Hom(M, N) and� 2 G, define�f to bem 7!

�(f (��1m)). Then Hom(M, N) is a G-module, but it is not in general a discreteG-module. For a closed normal subgroupH of G, set

Hom(M, N) D[U

Hom(M, N)U (union over the open subgroupsU, H � U � G)

D ff 2 Hom(M, N) j �f D f for all � in someU g.

ThenHomH (M, N) is a discreteG=H -module, and we defineExt rH (M, N) to be ther th

right derived functor of the left exact functor

N 7! HomH (M, N)WModG ! ModG=H .

In the case thatH D f1g, we drop it from the notation; in particular,

Hom(M, N) D[U

Hom(M, N)U

with U running over all the open subgroups ofG. If M is finitely generated, thenHomH (M, N) D

HomH (M, N), and soExt r

H (M, N) D ExtrH (M, N)I

in particular,Hom(M, N) D Hom(M, N)

(homomorphisms as abelian groups).

THEOREM 0.3. LetH be a closed normal subgroup ofG, and letN andP beG-modules.Then, for anyG=H -moduleM such thatTorZ1(M, N) D 0, there is a spectral sequence

ExtrG=H (M, ExtsH (N, P)) H) ExtrCs

G (M ˝Z N, P).

This will be shown to be the spectral sequence of a composite of functors, but first weneed some lemmas.

0. PRELIMINARIES 13

LEMMA 0.4. For any G-modulesN and P and G=H -moduleM , there is a canonicalisomorphism

HomG=H (M,HomH (N, P))�D�! HomG(M ˝Z N, P).

PROOF. There is a standard isomorphism

HomG=H (M,Hom(N, P))�D! Hom(M ˝Z N, P).

TakeG-invariants. On the left we get HomG(M, Hom(N, P)), which equals

HomG(M, HomH (N, P))

becauseM is aG=H -module, and equals HomG(M,HomH (N, P)) becauseM is adis-creteG=H -module. On the right we get HomG(M ˝Z N, P).

LEMMA 0.5. If I is an injectiveG-module andN is a torsion-freeG-module, thenHomH (N, I)

is an injectiveG=H -module.

PROOF. We have to check that

HomG=H (�,HomH (N, I))WModG=H ! Ab

is an exact functor, but (0.4) expresses it as the composite of the two exact functors�˝Z N

and HomG(�, I).

LEMMA 0.6. Let N andI beG-modules withI injective, and letM be aG=H -module.Then there is a canonical isomorphism

ExtrG=H (M,HomH (N, I))�D! HomG(TorZr (M, N), I).

PROOF. We use a resolution ofN

0! N1! N0! N ! 0

by torsion-freeG-modules to compute TorZr (M, N). Thus TorZ1(M, N) and TorZ0(M, N) D

M ˝Z N fit into an exact sequence

0! TorZ1(M, N)!M ˝Z N1!M ˝Z N0! TorZ0(M, N)! 0

and TorZr (M, N) D 0 for r � 2. For each open subgroupU of G containingH , there is ashort exact sequence

0 > HomG(Z[G=U ]˝Z N, I) > HomG(Z[G=U ]˝Z N0, 1) > HomG(Z[G=U ]˝Z N1, 1) > 0

jj jj jj

HomU (N, I) HomU (N0, I) HomU (N1, I)


The direct limit of these sequences is an injective resolution

0! HomH (N, I)! HomH (N0, I)! HomH (N1, I)! 0

of HomH (N, I), which we use to compute ExtrG=H (M,HomH (N, I)). In the diagram

HomG=H (M,HomH (N0, I))˛��! HomG=H (M,HomH (N1, I))??y�D

??y�D

HomG(M ˝Z N0, I)ˇ��! HomG(M ˝Z N1, I).

we have

Ker(˛) D HomG=H (M,HomH (N, I)), Coker(˛) D Ext1G=H (M,HomH (N, I))

Ker(ˇ) D HomG(TorZ0(M, N), I), Coker(ˇ) D HomG(TorZ1(M, N), I).

Thus the required isomorphisms are induced by the vertical maps in the diagram.

We now prove the theorem. Lemma 0.4 shows that HomG(M˝ZN,�) is the compositeof the functorsHomH (N,�) and HomG=H (M,�), and Lemma 0.6 shows that the first ofthese maps injective objectsI to objects that are acyclic for the second functor. Thusthe spectral sequence arises in the standard way from a composite of functors (Hilton andStammbach 1970)1.

EXAMPLE 0.7. LetM D N D Z, and replaceP with M . The spectral sequence thenbecomes the Hochschild-Serre spectral sequence

H r (G=H, H s(H, M)) H) H rCs(G, M).

EXAMPLE 0.8. LetM D Z andH D f1g, and replaceN andP with M andN . Thespectral sequence then becomes

H r (G, Exts(M, N)) H) ExtrG(M, N).

WhenM is finitely generated, this is simply a long exact sequence

0! H 1(G, Hom(M, N))! Ext1G(M, N)! H 0(G, Ext1(M, N))! H 2(G, Hom(M, N))! .

In particular, when we also have thatN is divisible by all primes occurring as the order ofan element ofM , then Ext1(M, N) D 0, and so

H r (G, Hom(M, N)) D ExtrG(M, N).

EXAMPLE 0.9. In the case thatN D Z, the spectral sequence becomes

ExtrG=H (M, H s(H, P)) H) ExtrCsG (M, P).

The map ExtrG=H (M, P H ) ! ExtrG(M, P) is obviously an isomorphism forr D 0; thespectral sequence shows that it is an isomorphism forr D 1 if H 1(H, P) D 0, and that itis an isomorphism for allr if H r (H, P) D 0 for all r > 0.

1Better Shatz 1972, p50.

0. PRELIMINARIES 15

REMARK 0.10. Assume thatM is finitely generated. It follows from the long exact se-quence in (0.8) that Extr

G(M, N) is torsion forr � 1. Moreover, ifG andN are writtencompatibly asG D lim

�Gi andN D lim

�!Ni (Ni is aGi-module) and the action ofG onM

factors through eachGi, then

ExtrG(M, N) D lim�!

ExtrGi(M, N).

REMARK 0.11. LetH be a closed subgroup ofG, and letM be anH -module. ThecorrespondinginducedG-moduleM� is the set of continuous mapsaW G ! M such thata(hx) D h�a(x) all h 2 H , x 2 G. The groupG acts onM� by the rule:(ga)(x) D a(xg).The functorM 7! M�WModH ! ModG is right adjoint to the functorModG ! ModH

“regard aG-module as anH -module”; in other words,

HomG(N, M�)�D! HomH (H, N), N aG-module, M anH -module.

Both functors are exact, and thereforeM 7!M� preserves injectives and the isomorphism

extends to isomorphisms ExtrG(N, M�)

�D! ExtrH (N, M) all r . In particular, there are

canonical isomorphismsH r (G, M�)�D! H r (H, M) for all r . (Cf. Serre 1964, I 2.5.)

Augmented cup-products

Certain pairs of pairings give rise to cup-products with a dimension shift.

PROPOSITION0.12. Let

0!M 0!M !M 00

! 0

0! N 0!N ! N 00

! 0

be exact sequences ofG-modules. Then a pair of pairings

M 0�N ! P

M �N 0! P

coinciding onM 0 �N 0 defines a canonical family of (augmented cup-product) pairings

H r (G, M 00)�H s(G, N 00)! H rCsC1(G, ).

PROOF. See Lang 1966, Chapter V.

REMARK 0.13. (a) The augmented cup-products have properties similar to those listed in(0.1) for the usual cup-product.

(b) Augmented cup-products have a very natural definition in terms of hypercohomol-ogy. The tensor product

(M 0 dM

! M 1)˝ (N 0 dN

! N 1)

of two complexes is defined to be the complex with

M 0˝N 0 d0

!M 1˝N 0

˚M 0˝N 1 d1

!M 1˝N 1


with

d0(x ˝ y) D dM (x)˝ y C x ˝ dN (y),

d1(x ˝ y C x0˝ y 0) D x ˝ dN (y)� dM (x0)˝ y 0.

With the notations in the proposition, letM �D (M 0

! M) andN �D (N 0

! N). Alsowrite P[�1] for the complex withP in the degree one and zero elsewhere. Then the hy-percohomology groupsHr (G, M �), Hr (G, N �), andHr (G, P[�1]) equalH r�1(G, M 00),H r�1(G, N 00), andH r�1(G, P) respectively, and to give a pair of pairings as in the propo-sition is the same as to give a map of complexes

M �˝N �

! P[�1].

Such a pair therefore defines a cup-product pairing

Hr (G, M �)�Hs(G, N �)! HrCs(G, P[�1]),

and this is the augmented cup-product.

Compatibility of pairings

We shall need to know how the Ext and cup-product pairings compare.

PROPOSITION0.14. (a) LetM � N ! P be a pairing ofG-modules, and consider themapsM ! Hom(N, P) and

H r (G, M)! H r (G,Hom(N, P))! ExtrG(N, P)

induced by the pairing and the spectral sequence in (0.3). Then the diagram

H r (G, M) � H s(G, N) ! H rCs(G, P) (cup-product)# jj jj

ExtrG(N, P) � H s(G, N) ! H rCs(G, P) (Ext pairing)

commutes (up to sign).(b) Consider a pair of exact sequences

0!M 0!M !M 00

! 0

0! N 0! N ! N 00

! 0

and a pair of pairings

M 0�N ! P

M �N 0! P

coinciding onM 0�N 0. These data give rise to canonical mapsH r (G, M 00)! ExtrC1

G (N 00, P),and the diagram

H r (G, M 00) � H s(G, N 00) ! H rCsC1(G, P) (augmented cup-product)# jj jj

ExtrC1G (N 00, P) � H s(G, N 00) ! H rCsC1(G, P) (Ext pairing)

commutes (up to sign).

0. PRELIMINARIES 17

PROOF. (a) This is standard, at least in the sense that everyone assumes it to be true.There is a proof in a slightly more general context in Milne 1980, V 1.20, and Gamst andHoechsmann 1970, contains a very full discussion of such things. (See also the discussionof pairings in the derived category in III 0.)

(b) The statement in (a) holds also ifM , N , andP are complexes. If we regard the pairof pairings in (b) as a pairing of complexesM �

�N �! P[�1] (notations as (0.13b)) and

replaceM , N , andP in (a) with M �, N �, andP[�1], then the diagram in (a) becomesthat in (b). (Explicity, the mapH r (G, M 00)! ExtrC1

G (N 00, P) is obtained as follows: thepair of pairings defines a map of complexesM � ! Hom(N �, P [�1]), and hence a mapHr (G, M �)! Hr (G,Hom(N �, P [�1]); butHr (G, M �) D Hr�1(G, M), and there is anedge morphismHr (G,Hom(N �, P [�1])!ExtrG(N �, P [�1]) D ExtrG(N 00, P).)

Conjugation of cohomology groups

Consider two profinite groupsG andG 0, a G-moduleM , and aG 0-moduleM 0. A ho-momorphismf WG 0 ! G and an additive maphWM ! M 0 are said to becompatibleifh(f (g0) �m) D g0

� h(m) for g02 G 0 andm 2 M . Such a pair induces homomorphisms

(f, h)r�WH

r (G, M)! H r (G 0, M 0) for all r.

PROPOSITION0.15. Let M be aG-module, and let� 2 G. The mapsad(�) D (g 7!

�g��1)WG ! G and��1D (m 7! ��1m)WM !M are compatible, and

(ad(�), ��1)r�WH

r (G, M)! H r (G, M)

is the identity map for allr .

PROOF. The first assertion is obvious, and the second needs only to be checked forr D 0,where it is also obvious (see Serre 1962, VII 5).

The proposition is useful in the following situation. LetK be a global field andv aprime of K. The choice of an embeddingKs

! Ksv over K amounts to choosing an

extensionw of v to Ks, and the embedding identifiesGKvwith the decomposition group

Dw of w in GK . A second embedding is the composite of the first with ad(�) for some� 2 G (becauseGK acts transitively on the extensions ofv to Ks). Let M be aGK -module. An embeddingKs! Ks

v defines a mapH r (GK , M) ! H r (GKv, M), and the

proposition shows that the map is independent of the choice of the embedding.

Extensions of algebraic groups

Let k be a field, and letG D Gal(ks=k). The category of algebraic group schemes overk

is an abelian categoryGpk (recall that all group schemes are assumed to be commutative),and therefore it is possible to define Extr

k(A, B) for objectsA andB of Gpk to be the set ofequivalence classes ofr -fold extensions ofA by B (see Mitchell 1965, VII). Alternatively,one can chose a projective resolutionA� of A in the pro-categoryPro-Gpk , and defineExtrk(A, B) to be the set of homotopy classes of mapsA�

! B of degreer (see Oort 1966,I 4, or Demazure and Gabriel 1970, V 2). For any objectA of Gpk , A(ks) is a discreteG-module, and we often writeH r (k, A) for H r (G, A(ks)).


PROPOSITION0.16. Assume thatk is perfect.(a) The functorA 7! A(ks)WGpk ! ModG is exact.(b) For all objectsA andB in Gpk , there exists a canonical pairing

Extrk(A, B)�H s(k, A)! H rCs(k, B).

PROOF. (a) This is obvious sinceks is algebraically closed.(b) The functor in (a) sends anr -fold exact sequence inGpk to anr -fold exact sequence

in ModG, and it therefore defines a canonical map Extrk(A, B)! ExtrG(A(ks), B(ks)). We

define the pairing to be that making

ExtrG(A, B) � H s(k, A) ! H rCs(k, A)

# jj jj

ExtrG(A(ks), B(ks)) � H s(G, A(ks)) ! H rCs(G, B(ks))

commute.

PROPOSITION0.17. Assume thatk is perfect, and letA andB be algebraic group schemesoverk. Then there is a spectral sequence

H r (G, Extsks(A, B)) H) ExtrCsk

(A, B).

PROOF. See Milne 1970a.

COROLLARY 0.18. If k is perfect andN is a finite group scheme overk of order prime tochar(k), thenExtrk(N, Gm) �D ExtrG(N(ks), ks�) all r.

PROOF. Clearly Homks(N, Gm) D HomG(N(ks), ks�), and the table Oort 1966, p II14-2, shows that Exts

ks(N, Gm) D 0 for s > 0. Therefore the proposition implies thatExtrk(N, Gm) D H r (G, HomG(N(ks), ks�), which equals Extr

G(N(ks), ks�) by (0.8).

Topological abelian groups

Let M be an abelian group. In the next proposition we writeM ^ for them-adic completionlim �

n M=mnM of M , and we letZm DQ`jmZ` D Z^ andQm D

Q`jmQ` D Zm ˝Z Q.

PROPOSITION0.19. (a) For any abelian groupM , M ^D (M=Mm�div)

^; if M is finite,thenM ^

DM(m), and ifM is finitely generated, thenM ^DM ˝Z Zm.

(b) For any abelian groupM , lim�!

M (mn) D (M ˝Z Q=Z)(m), which is zero ifM istorsion and is isomorphic to(Qm=Zm)r if M is finitely generated of rankr .

(c) For any abelian group,TmM D Hom(Qm=Zm, M) D Tm(Mm�div); it is torsion-free.

(d) Write M � D Homcts(M, Qm=Zm); then for any finitely generated abelian groupM , M �

D (M ^)� andM ��DM ^.

(e) LetM be a discrete torsion abelian group andN a totally disconnected compactabelian group, and let

M �N ! Q=Zbe a continuous pairing that identifies each group with the Pontryagin dual of the other.Then the exact annihilator ofNtors is Mdiv, and so there is a nondegenerate pairing

M=Mdiv �Ntors! Q=Z.

0. PRELIMINARIES 19

PROOF. Easy.

Note that the proposition continues to hold if we takem D “Q

p”, that is, we takeM ^

be the profinite completion ofM , Mm�div to beMdiv, M(m) to beMtor, and so on.We shall be concerned with the exactness of completions and duals of exact sequences.

Note that the completion of the exact sequence

0! Z! Q! Q=Z! 0

for the profinite topology is

0! bZ! 0! 0! 0,

which is far from being exact. To be able to state a good result, we need the notion of a strictmorphism. Recall (Bourbaki Tpgy, III 2.8) that a continuous homomorphismf WG ! H oftopological groups is said to be astrict morphismif the induced mapG=Ker(f )! f (G)

is an isomorphism of topological groups. Equivalently,f is strict if the image of everyopen subset ofG is open inf (G) for the subspace topology onf (G). Every continuoushomomorphism of a compact group to a Hausdorff group is strict, and obviously everycontinuous homomorphism from a topological group to a discrete group is strict. TheBaire category theorem implies that a continuous homomorphism from a locally compact� -compact group onto2 a locally compact group is a strict morphism (Hewitt and Ross1963, 5.29; a space is� -compactif it is a countable union of compact subspaces).

Recall also that it is possible to define the completionG^

of a topological group whenthe group has a basis of neighbourhoods(Gi) for the identity element consisting of normalsubgroups; in fact,G D lim

�iG=Gi. In the next proposition, we writeG� for the full

Pontryagin dual of a topological groupG.


G 0f! G

g! G 00

be an exact sequence of abelian topological groups and strict morphisms.(a) Assume that the topologies onG 0, G, andG 00 are defined by neighbourhood bases

consisting of subgroups; then the sequence of completions is also exact.(b) Assume that the groups are locally compact and Hausdorff and that the image ofG

is closed inG 00; then the dual sequence3

G 00�! G�

! G 0�

is also exact.

PROOF. We shall use that a short exact sequence

0! A! B ! C ! 0

2The original had “to” for “onto”, but the inclusion of the discrete groupZ into Zp is continuous withoutbeing strict.

3Here� denotes the full Pontryagin dual, which coincides with Hom(�, Q=Z) on abelian profinite groups.


of topological groups and continuous homomorphisms remains exact after completion pro-vided the topology onB is defined by a neighbourhood basis consisting of subgroups andA andC have the induced topologies (Atiyah and MacDonald 1969, 10.3).

By assumption, we have a diagram

G= Im(f )�D��! Im(g)x?? ??yb

G 0f��! G

g��! G 00??ya

x??G 0=Ker(f )

�D��! Im(f ).

When we complete, the mapa remains surjective, the middle column remains a short exactsequence, andb remains injective because in each case a subgroup has the subspace topol-ogy and a quotient group the quotient topology. Since the isomorphisms obviously remainisomorphisms, (a) is now clear.

The proof of (b) is similar, except that it makes use of the fact that for any closedsubgroupK of a locally compact abelian groupG, the exact sequence

0! K! G ! G=K! 0

gives rise to an exact dual sequence

0! (G=K)�! G�

! K�! 0.

Note that in (b) of the theorem, the image ofG in G 00 will be closed if it is the kernelof a homomorphism fromG 00 into a Hausdorff group.

The right derived functors of the inverse limit functor

The category of abelian groups satisfies the condition Ab5: the direct limit of an exactsequence of abelian groups is again exact. Unfortunately, the corresponding statementfor inverse limits is false, although the formation of inverse limits is always a left exactoperation (and the product of a family of exact sequences is exact).4

4For an inverse system(An) indexed byN,

� � � ! An

un! An�1 ! � � � ,

lim �

An and lim �

1 An are the kernel and cokernel respectively of

QnAn

1�u��!

QnAn, ((1� u)(ai))n D an � unC1anC1,

and lim �

i An D 0 for i > 1. Using the snake lemma, we find that a short exact sequence of abelian groups

0! (An)(fn)! (Bn)

(gn)! (Cn)! 0

0. PRELIMINARIES 21

PROPOSITION0.21. LetA be an abelian category satisfying the condition Ab5 and havingenough injectives, and letI be a filtered ordered set. Then for any objectB of A and anydirect system(Ai) of objects ofA indexed byI , there is a spectral sequence

lim �

(r) ExtsA(Ai , B) H) ExtrCsA (lim�!

Ai , B)

wherelim �

(r) denotes ther th right derived functor oflim �

.

PROOF. Roos 1961.

PROPOSITION0.22. Let (Ai) be an inverse system of abelian groups indexed byN with itsnatural order.

(a) For r � 2, lim �

(r)Ai D 0.

(b) If eachAi is finitely generated, thenlim �

(1)Ai is divisible, and it is uncountablewhen nonzero.

(c) If eachAi is finite, thenlim �

(1)Ai D 0.

PROOF. (a) See Roos 1961.(b) See Jensen 1972, 2.5.(c) See Jensen 1972, 2.3.

COROLLARY 0.23. Let A be an abelian category satisfying Ab5 and having enough in-jectives, and let(Ai) be a direct system of objects ofA indexed byN. If B is such thatExtsA(Ai , B) is finite for all s andi, then

lim �

ExtsA(Ai , B) D ExtsA(lim�!

Ai , B).

The kernel-cokernel exact sequence of a pair of maps

The following simple result will find great application in these notes.

PROPOSITION0.24. For any pair of maps

Af! B

g! C

gives rise to a six-term exact sequence

0! lim �

An ! lim �

Bn ! � � � ! lim �

1Cn ! 0.

It is known (and easy to prove) that if an inverse system of abelian groups(An)n2N satisfies the Mittag-Lœffler condition, then lim

�

1 An D 0, however, the “well-known” generalization of this to abelian categoriessatisfying Ab4� (see, for example, Jannsen, Uwe, Continuousetale cohomology. Math. Ann. 280 (1988), no.2, 207–245, Lemma 1.15, p. 213) is false: Neeman and Deligne (A counterexample to a 1961 ”theorem” inhomological algebra. With an appendix by P. Deligne. Invent. Math. 148 (2002), no. 2, 397–420) constructan abelian categoryA in which small products and direct sums exist and are exact, i.e., which satisfies Ab4and Ab4�; the opposite category has the same properties, and inside it there is a inverse system(An)n2N withsurjective transition maps (hence(An) satisfies Mittag-Lœffler) such that lim

�

1 An 6D 0.Since Roos 1961 contains no proofs and some false statements, it would be better to avoid referring to it.

Thus, this subsection should be rewritten.


of abelian groups, there is an exact sequence

0! Ker(f )! Ker(g ıf )! Ker(g)! Coker(f )! Coker(g ıf )! Coker(g)! 0.

PROOF. An easy exercise.

NOTES. The subsection “A spectral sequence for Exts” is based on Tate 1966. The rest ofthe material is fairly standard.

1 Duality relative to a class formation

Class formations

Consider a profinite groupG, aG-moduleC , and a family of isomorphisms

invU WH2(U, C )

�! Q=Z

indexed by the open subgroupsU of G. Such a system is said to be aclass formationif(1.1a)for all open subgroupsU � G, H 1(U, C ) D 0, and(1.1b) for all pairs of open subgroupsV � U � G, the diagram

H 2(U, C )ResV ,U

��! H 2(V , C )??yinvU

??yinvV

Q=Zn��! Q=Z

commutes withn D (U WV ). The map invU is called theinvariant map relative toU .WhenV is a normal subgroup ofU of index n, the conditions imply that there is an

exact commutative diagram

0 ��! H 2(U=V , C V ) ��! H 2(U, C )ResV ,U

��! H 2(V , C ) ��! 0

�

??yinvU=V �

??yinvU �

??yinvV

0 ��! 1nZ=Z ��! Q=Z

n��! Q=Z ��! 0

in which invU=V is defined to be the restriction of invU . In particular, for a normal opensubgroupU of G of indexn, there is an isomorphism

invG=U WH2(G=U, C U )

�!

1nZ=Z,

and we writeuG=U for the element ofH 2(G=U, C U ) mapping to1=n . ThusuG=U is theunique element ofH 2(G=U, C U ) such that invG(Inf(uG=U )) D 1=n.

LEMMA 1.2. LetM be aG-module such thatTorZ1(M, C ) D 0. Then the map

a 7! a [ uG=U WHrT (G=U, M)! H rC2

T (G=U, M ˝Z C U )

is an isomorphism for all open normal subgroupsU of G and integersr.

1. DUALITY RELATIVE TO A CLASS FORMATION 23

PROOF. Apply (0.2) toG=U , C U , anduG=U .

THEOREM1.3. Let(G, C ) be a class formation; then there is a canonical maprecG WCG!

Gab whose image inGab is dense and whose kernel is the groupT

NG=U C U of universalnorms.

PROOF. TakeM D Z and r D �2 in the lemma. AsH �2T (G=U, Z) D (G=U )ab and

H 0T (G=U, C U ) D C G=NG=U C U , the lemma gives an isomorphism

(G=U )ab �! C G=NG=U C U .

On passing to the projective limit over the inverses of these maps, we obtain an injectivemapC G=

TNG=U U ! Gab. The map recG is the composite of this with the projection of

C G ontoC G=T

NG=U U . It has dense image because, for all open normal subgroupsU ofG, its composite withGab! (G=U )ab is surjective.

The map recG is called thereciprocity map.

QUESTION 1.4. Is there a derivation of (1.3), no more difficult than the above one, thatavoids the use of homology groups?

REMARK 1.5. (a) The following description of recG will be useful. The cup-product pair-ing

H 0(G, C )�H 2(G, Z)! H 2(G, C )

can be identified with a pairing

h , iWC G� Homcts(G, Q=Z)! Q=Z

and the reciprocity map is uniquely determined by the equation

hc,�i D �(recG(c)) all c 2 C G, � 2 Homcts(Gab, Q=Z).

See Serre 1962, XI 3, Pptn 2.(b) The definition of a class formation that we have adopted is slightly stronger than the

usual definition (see Artin and Tate 1961, XIV) in that we require invU to be an isomor-phism rather than an injection inducing isomorphisms

H 2(U=V , C V )�! (U WV )�1Z=Z

for all open subgroupsV � U with V normal inU . It is equivalent to the usual definitionplus the condition that the order ofG (as a profinite group) is divisible by all integersn.

EXAMPLE 1.6. (a) LetG be a profinite group isomorphic tobZ (completion ofZ for thetopology of subgroups of finite index), and letC D Z with G acting trivially. Choose atopological generator� of G. For eachm, G has a unique open subgroupU of indexm,and�m generatesU . The boundary map in the cohomology sequence of

0! Z! Q! Q=Z! 0


is an isomorphismH 1(U, Q=Z)! H 2(U, Z), and we define invU to be the composite ofthe inverse of this isomorphism with

H 1(U, Q=Z) D Homcts(U, Q=Z)f 7 !f (�m)��! Q=Z.

Note that invU depends on the choice of� . Clearly (G, Z) with these maps is a classformation. The reciprocity map is injective but not surjective.

(b) Let G be the Galois group Gal(Ks=K) of a nonarchimedean local fieldK, and letC D Ks�. If I D Gal(Ks=Kun), then the inflation mapH 2(G=I, Kun�)! H 2(G, Ks�)

is an isomorphism, and we define invG to be the composite of its inverse with the isomor-phisms

H 2(G=I, Kun�)ord! H 2(G=I, Z)

invG=I

! Q=Zwhere invG=I is the map in defined in (a) (with the choice of the Frobenius automorphismfor � ). Define invU analogously. Then(G, Ks�) is a class formation (see Serre 1967a,~1,or the appendix to this chapter). The reciprocity map is injective but not surjective.

(c) Let G be the Galois group Gal(Ks=K) of a global fieldK, and letC D lim�!

CL

whereL runs through the finite extensions ofK in Ks andCL is the idele class group ofL. For each primev of K, choose an embedding ofKs into Ks

v overK. Then there is aunique isomorphism invG WH 2(G, C )! Q=Z making the diagram

H 2(G, C )invG

��! Q=Z??y k

H 2(Gv, Ks�v )

invv

��! Q=Z

commute for allv (including the real primes) with invv the map defined in (b) unlessv isreal, in which case it is the unique injection. Define invU analogously. Then(G, C ) is aclass formation (see Tate 1967a,~11). In the number field case, the reciprocity map is sur-jective with divisible kernel, and in the function field case it is injective but not surjective.

(d) LetK be a field complete with respect to a discrete valuation having an algebraicallyclosed residue fieldk, and letG D Gal(Ks=K) . For a finite separable extensionL of K,let RL be the ring of integers inL. There is a pro-algebraic groupUL over k such thatUL(k) D R�

L. Let�1(UL) be the pro-algebraicetale fundamental group ofUL, and let

�1(U ) D lim�!

�1(UL), K � L � Ks, [LWK] <1.

Then�1(U ) is a discreteG-module and(G,�1(U )) is a class formation. In this case thereciprocity map is an isomorphism. See Serre 1961, 2.5 Pptn 11, 4.1 Thm 1.

(e) LetK be an algebraic function field in one variable over an algebraically closed fieldk of characteristic zero. For each finite extensionL of K, let CL D Hom(Pic(XL),�(k)),whereXL is the smooth complete algebraic curve overk with function fieldL and�(k) isthe group of roots on unity ink. Then the duals of the norm maps Pic(XL0) ! Pic(XL),L0� L, make the family(CL) into a direct system, and we letC be the limit of the

system. The pair(G, C ) is a class formation for which the reciprocity map is surjective butnot injective. See Kawada and Tate 1955 and Kawada 1960.

(f) For numerous other examples of class formations, see Kawada 1971.


The main theorem

For eachG-moduleM , the pairings of~0

ExtrG(M, C )�H 2�r (G, M)! H 2(G, C )inv! Q=Z

induce maps˛r (G, M)WExtrG(M, C )! H 2�r (G, M)�

In particular, forr D 0 andM D Z, we obtain a map

˛0(G, Z)WC G! H 2(G, Z)�

D Homcts(G, Q=Z)�D Gab.

LEMMA 1.7. In the case thatM D Z, the maps r (G, M) have the following description:

˛0(G, Z)WC G! Gab is equal torecG ;

˛1(G, Z)W0! 0I

˛2(G, Z)WH 2(G, C )�D! Q=Z is equal to invG .

In the case thatM D Z=mZ, the maps r (G, M) have the following description: thecomposite of

˛0(G, Z=mZ)W (C G)m! H 2(G, Z=mZ)�

with H 2(G, Z=mZ)� � (Gab)m is induced byrecG;

˛1(G, Z=mZ)W (C G)(m)! (Gab)(m) is induced byrecG ;

˛2(G, Z=mZ)WH 2(G, C )m!1

mZ=Z is the isomorphism induced byinvG .

PROOF. Only the assertion about0(G, Z) requires proof. As we observed in (1.5a),recG WH

0(G, C )! H 2(G, Z)� is the map induced by the cup-product pairing

H 0(G, C )�H 2(G, C )! H 2(G, C ) �D Q=Z

and we know (0.14) that this agrees with the Ext pairing.

THEOREM 1.8. Let (G, C ) be a class formation, and letM be a finitely generatedG-module.

(a) The map˛r (G, M) is bijective for all r � 2, and ˛1(G, M) is bijective for alltorsion-freeM . In particular,ExtrG(M, C ) D 0 for r � 3.

(b) The map 1(G, M) is bijective for allM if ˛1(U, Z=mZ) is bijective for all opensubgroupsU of G and allm.

(c) The map 0(G, M) is surjective (respectively bijective) for all finiteM if in addition˛0(U, Z=mZ) is surjective (respectively bijective) for allU andm.

The first step in the proof is to show that the domain and target of˛r (G, M) are bothzero for larger .


LEMMA 1.9. For r � 4, ExtrG(M, C ) D 0; whenM is torsion-free,Ext3G(M, C ) is alsozero.

PROOF. Every finitely generatedG-moduleM can be resolved

0!M1!M0!M ! 0

by finitely generated torsion-freeG-modulesMi. It therefore suffices to prove that for anytorsion-free moduleM , ExtrG(M, C ) D 0 for r � 3. Let N D Hom(M, Z). ThenN ˝ZC �D Hom(M, C ) asG-modules, and so (0.8) provides an isomorphism Extr

G(M, C ) �DH r (G, N ˝Z C ). Note that this last group is equal to lim

�!H r (G=U, N ˝Z C U ) where the

limit is over the open normal subgroups ofG for which N UD N . The theorem of Tate

and Nakayama (0.2) shows that

a 7! a [ uG=U WHr�2(G=U, N)! H r (G=U, N ˝Z C U )

is an isomorphism for allr � 3. The diagram

H r�2(G=U, N)��! H r (G=U, N ˝Z C U )??y(U WV )Inf

??yInf

H r�2(G=V , N)��! H r (G=V , N ˝Z C V )

commutes because Inf(uG=U ) D (U WV )uG=V and Inf(a [ b) D Inf(a) [ Inf(b). AsH r�2(G=U, N) is torsion forr � 2 � 1, and the order ofU is divisible by all integersn,the limit lim

�!H r�2(G=U, N) (taken relative to the maps(U WV )Inf) is zero forr � 2 � 1,

and this shows thatH r (G, N ˝Z C ) D 0 for r � 3.

PROOF OFTHEOREM 1.8. Lemma 1.9 shows that the statements of the theorem are truefor r � 4, and (1.7) shows that they are true forr � 2 whenever the action ofG on M istrivial. Moreover, (1.9) shows that Ext3

G(Z, C ) D 0, and it follows that Ext3G(Z=mZ, C ) D

0 because Ext2G(Z, C) is divisible. Thus the theorem is true whenever the action ofG on

M is trivial. We embed a generalM into an exact sequence

0!M !M�!M1! 0

with U an open normal subgroup ofG such thatM UDM and

M� D Hom(Z[G=U ], M) D Z[G=U ]˝Z M .

As H r (G, M�) D H r (U, M) and ExtrG(M�, C ) D ExtrU (M, C ) (apply (0.3) toZ[G=U ],M , andC ), there is an exact commutative diagram (1.9.1)

�! ExtrG(M1, C ) �! ExtrU (M, C ) �! ExtrG(M, C ) �! ExtrC1G (M1, C ) �! � � �??y˛r (G,M1)

??y˛r (U,M)

??y˛r (G,M)

??y˛rC1(G,M1)

�! H 2�r (G, M1)��! H 2�r (U, M)�

�! H 2�r (G, M)��! H 1�r (G, M1)�

�! � � �


The maps 3(U, M), ˛4(G, M1), and˛4(U, M) are all isomorphisms, and so the five-lemma shows that 3(G, M) is surjective. Since this holds for allM , ˛3(G, M1) is alsosurjective, and now the five-lemma shows that˛3(G, M) is an isomorphism. The sameargument shows that2(G, M) is an isomorphism. IfM is torsion-free, so also areM�

andM1, and so the same argument shows that˛1(G, M) is an isomorphism whenM istorsion-free. The rest of the proof proceeds similarly.

EXAMPLE 1.10. Let(G, Z) be the class formation defined by a groupG � bZ and a gener-ator� of G. The reciprocity map is the inclusionn 7! �n

WZ ! G. AsbZ=Z is uniquelydivisible, we see that both0(U, Z=mZ) and˛1(U, Z=mZ) are isomorphisms for allm,and so the theorem implies that˛r (G, M) is an isomorphism for all finitely generatedM ,r � 1, and˛0(G, M) is an isomorphism for all finiteM .

In fact,˛0(G, M) defines an isomorphism HomG(M, Z)^!H 2(G, M)� for all finitely

generatedM . To see this, note that HomG(M, Z) is finitely generated and Ext1(M, Z) isfinite (becauseH 1(G, M) is) for all finitely generatedM . Therefore, on tensoring the firstfour terms of the long exact sequence of Exts withbZ, we obtain an exact sequence

0! HomG(M1, Z)^! HomU (M, Z)^

! HomG(M, Z)^! Ext1G(M1, Z)! � � � .

When we replace the top row of (1.9.1) with this sequence, the argument proving the theo-rem descends all the way tor D 0.

WhenM is finite, Extr (M, Z) D 0 for r 6D 1 and

Ext1(M, Z) D Hom(M, Q=Z) DM �.

Therefore ExtrG(M, Z) D H r�1(G, M �) (by (0.3)), and so we have a non-degenerate cup-product pairing

H r (G, M)�H 1�r (G, M �)! H 1(G, Q=Z) �D Q=Z.

When M is torsion-free, Extr (M, Z) D 0 for r 6D 0 and Hom(M, Z) is the lineardualM _ of M . Therefore Extr (M, Z) D H r (G, M _), and so the mapH r (G, M _) !

H 2�r (G, M)� defined by cup-product is bijective forr � 1, and induces a bijectionH 0(G, M _)^

! H 2(G, M)� in the caser D 0.

EXAMPLE 1.11. LetK be a field for which there exists a class formation(G, C ) withG D Gal(Ks=K), and letT be a torus overK. The character groupX �(T ) of T is afinitely generated torsion-freeG-module withZ-linear dual the cocharacter groupX�(T ),and so the pairing

ExtrG(X �(T ), C )�H 2�r (G, X �(T ))! H 2(G, C ) �D Q=Z

defines an isomorphism

ExtrG(X �(T ), C )! H 2�r (G, X �(T ))�

for r � 1. According to (0.8),

ExtrG(X �(T ), C ) D H r (G, Hom(X �(T ), C )), and

Hom(X �(T ), C ) D X�(T )˝ C.


Therefore the cup-product pairing

H r (G, X�(T )˝ C )�H 2�r (G, X �(T ))! H 2(G, C ) �D Q=Z

induced by the natural pairing betweenX�(T ) andX �(T ) defines an isomorphism

H r (G, X�(T )˝ C )! H 2�r (G, X �(T ))�, r � 1.

REMARK 1.12. Let(G, C ) be a class formation. In Brumer 1966 there is a very usefulcriterion for G to have strict cohomological dimension2. Let V � U � G be opensubgroups withV normal inU . We get an exact sequence

0! Ker(recV )! C V recV! V ab

! Coker(recV )! 0

of U=V -modules which induces a double connecting homomorphism

d WH r�2T (U=V , Coker(recV ))! H r

T (U=V , Ker(recU )).

The theorem states that scdp(G) D 2 if and only if, for all such pairsV � U , d inducesan isomorphism on thep-primary components for allr . In each of the examples (1.6a,b,d)and in the function field case of (c), the kernel of recV is zero and the cokernel is uniquelydivisible and hence has trivial cohomology. In the number field case of (c), the cohomologygroups of the kernel are elementary2-groups, which are zero if and only if the field is totallyimaginary (Artin and Tate 1961, IX 2). Consequently scdp(G) D 2 in examples (1.6a,b,c,d)except whenp D 2 andK is a number field having a real prime.

On the other hand, letK be a number field and letGS be the Galois group overK of themaximal extension ofK unramified outside a set of primesS . The statement in Tate 1962,p292 that scdp(GS) D 2 for all primesp that are units at allv in S (except forp D 2 whenK is not totally complex) is still unproven in general. As was pointed out by A. Brumer, itis equivalent to the nonvanishing of certainp-adic regulators.

A generalization

We shall need a generalization of Theorem 1.8. For any setP of rational prime numbers,we define aP -class formationto be a system(G, C, (invU )U ) as at the start of this sectionexcept that, instead of requiring the maps invU to be isomorphisms, we require them to beinjections satisfying the following two conditions:

(a) for all open subgroupsV andU of G with V a normal subgroup ofU , the map

invU=V WH2(U=V , C V )! (U WV )�1Z=Z

is an isomorphism, and(b) for all open subgroupsU of G and all primes in P , the map on -primary compo-

nentsH 2(U, C )(`)! (Q=Z)(`) induced by invU is an isomorphism.Thus whenP contains all prime numbers, aP -class formation is a class formation in thesense of the first paragraph of this section, and whenP is the empty set, aP -class formationis a class formation in the sense of Artin and Tate 1961. Note that, in the presence of the

2. LOCAL FIELDS 29

other conditions, (b) is equivalent to the order ofG being divisible by 1 for all ` in P . If(G, C ) is a class formation andH is a normal closed subgroup ofG, then(G=H, C H ) is aP -class formation withP equal to the set primessuch that 1 divides(GWH).

If (G, C ) is aP -class formation, then everything said above continues to hold providedthat, at certain points, one restricts attention to the` -primary components for in P .(Recall (0.10) that Extr

G(M, N) is torsion forr � 1.) In particular, the following theoremholds.

THEOREM 1.13. Let (G, C ) be aP -class formation , let be a prime inP , and letM bea finitely generatedG-module.

(a) The map r (G, M)(`)WExtrG(M, C )(`)! H 2�r (G, M)�(`) is bijective for allr �2, and˛1(G, M)(`) is bijective for all torsion- freeM .

(b) The map 1(G, M)(`) is bijective for allM if ˛1(U, Z=`mZ) is bijective for all opensubgroupsU of G and allm.

(c) The map 0(G, M) is surjective (respectively bijective) for all finite-primary M ifin addition˛0(U, Z=`mZ) is surjective (respectively bijective) for allU andm.

EXERCISE 1.14. LetK D Q(p

d) whered is chosen so that the 2-class field tower ofK

is infinite. LetKun be the maximal unramified extension ofK, and letH D Gal(Ks=Kun).Then(GK=H, C H ) is aP -class formation withP D f2g. Investigate the mapsr (GK=H, M)

in this case.

Notes: Theorem 1.8 and its proof are taken from Tate 1966.

2 Local fields

Unless stated otherwise,K will be a nonarchimedean local field, complete with respect tothe discrete valuation ordWK� ! Z, and with finite residue fieldk. Let R be the ring ofintegers inK, and letKun be a largest unramified extension ofK. Write G D Gal(Ks=K)

andI D Gal(Ks=Kun). As we noted in (1.6b),(G, Ks�) has a natural structure of a classformation. The reciprocity map recG WK

�! Gab is known to be injective with dense

image. More precisely, there is an exact commutative diagram

0 ��! R� ��! K�ord��! Z ��! 0??y�

??y ??y0 ��! Iab

��! Gab��! bZ ��! 0

in which all the vertical arrows are injective andIab is the inertia subgroup ofGab. Thenorm groups inK� are the open subgroups of finite index. See Serre 1962, XIII 4, XIV 6.

In this sectionN ^ will denote the completion of a groupN relative to the topologydefined by the subgroups ofN of finite index unlessN has a topology induced in a naturalway from that onK, in which case we allow only subgroups of finite index that are open rel-ative to the topology. With this definition,(R�)^

D R�, and the reciprocity map defines anisomorphism(K�)^

! GabK . WhenM is a discreteG-module, the group HomG(M, Ks�)


inherits a topology from that onKs, and in the next theorem HomG(M, Ks�)^ denotesits completion for the topology defined by the open subgroups of finite index5. AsbZ=Zis uniquely divisible, 0(G, Z=mZ) and˛1(G, Z=mZ) are isomorphisms for allm. Thusmost of the following theorem is an immediate consequence of Theorem 1.8.

THEOREM 2.1. LetM be a finitely generatedG-module, and consider

˛r (G, M)WExtrG(M, Ks�)! H 2�r (G, M)�.

Then˛r (G, M) is an isomorphism for allr � 1, and˛0(G, M) defines an isomorphism(of profinite groups)

HomG(M, Ks�)^! H 2(G, M)�.

The^ can be omitted ifM is finite. The groupsExtrG(M, Ks�) andH r (G, M) are finitefor all r if M is of finite order prime to char(K), and the groupsExt1G(M, Ks�) andH 1(G, M) are finite for all finitely generatedM whose torsion subgroup is of order primeto char(K).

PROOF. We begin with the finiteness statements. Forn prime to char(K), the cohomologysequence of the Kummer sequence

0! �n(Ks)! Ks� n! Ks�

! 0

shows that the cohomology groups are

H r (G,�n(Ks)) D �n(K) K�=K�n 1nZ=Z 0

r D 0 1 2 � 3.

In particular, they are all finite.Let M be a finiteG-module of order prime to char(K), and choose a finite Galois

extensionL of K containing allmth roots of1 for m dividing the order ofM and such thatGal(Ks=L) acts trivially onM . ThenM is isomorphic as a Gal(Ks=L)-module to a directsum of copies of modules of the form�m, and so the groupsH s(Gal(Ks=L), M) are finitefor all s, and zero fors � 3. The Hochschild-Serre spectral sequence

H r (Gal(L=K), H s(Gal(Ks=L, M)) H) H rCs(G, M)

now shows that the groupsH r (G, M) are all finite because the cohomology groups of afinite group with values in a finite (even finitely generated forr � 1) module are finite.This proves thatH r (G, M) is finite for all r and allM of finite order prime to char(K),and Theorem 1.8 shows that all the˛r (G, M) are isomorphisms for finiteM , and so thegroups ExtrG(M, Ks�) are also finite.

5(In original.) If n is prime to the characteristic ofK, thenK�n is an open subgroup of finite index inK�. It follows that every subgroup ofK� (hence of HomG(M, Ks�)) of finite index prime to char(K) isopen. In contrast, when the characteristic ofK is p 6D 0, there are many subgroups of finite index inK� thatare not closed. In fact (see Weil 1967, II 3, Pptn 10),1 C m �

QZp (product of countably many copies of

Zp), and a proper subgroup ofQ

Zp containing Zp can not be closed.

2. LOCAL FIELDS 31

Let M be a finitely generatedG-module whose torsion subgroup has order prime tochar(K). In proving thatH 1(G, M) is finite, we may assume thatM is torsion-free. LetL be a finite Galois extension ofK such that Gal(Ks=L) acts trivially onM . The exactsequence

0! H 1(Gal(L=K), M)! H 1(Gal(Ks=K), M)! H 1(Gal(Ks�=L), M)

shows thatH 1(G, M) is finite because the last group in the sequence is zero and the firstis finite. Theorem 1.8 implies thatr (G, M) is an isomorphism forr � 1 and all finitelygeneratedM , and so Ext1G(M, Ks�) is also finite.

It remains to prove the assertion about˛0(G, M). Note that 0(G, Z) defines an iso-morphism(K�)^

! Gab, and so the statement is true ifG acts trivially onM . Let L be afinite Galois extension ofK such Gal(Ks=L) acts trivially onM . Then HomG(M, Ks�) D

HomG(M, L�), and HomG(M, L�) contains an open compact group HomG(M,O�w),

whereOw is the ring of integers inL. Using this, it is easy to prove that the maps

0! HomG(M1, Ks�)! HomG(M�, Ks�)! HomG(M, Ks�)!

in the top row of (1.9.1) are strict morphisms. Therefore the sequence remains exact whenwe complete the first three terms (see 0.20), and so the same argument as in (1.8) completesthe proof.

COROLLARY 2.2. If M is a countableG-module whose torsion is prime tochar(K), then

˛1(G, M)WExt1G(M, Ks�)! H 1(G, M)�

is an isomorphism.

PROOF. Write M as a countable union of finitely generatedG-modulesMi and note thatExt1G(M, Ks�) D lim

�Ext1G(Mi , Ks�) by (0.23).

For any finitely generatedG-moduleM , write M DD Hom(M, Ks�). It is again a

discreteG-module, and it acquires a topology from that onKs.

COROLLARY 2.3. Let M be a finitely generatedG-module whose torsion subgroup hasorder prime to char(K). Then cup-product defines an isomorphism

H r (G, M D)! H 2�r (G, M)�

for all r � 1, and an isomorphism (of compact groups)

H 0(G, M D)^! H 2(G, M)�.

The groupsH 1(G, M) andH 1(G, M D) are finite.

PROOF. As Ks� is divisible by all primes other than char(K), Extr (M, Ks�) D 0 for allr > 0, and so ExtrG(M, Ks�) D H r (G, M D) for all r (see 0.8).


COROLLARY 2.4. LetT be a commutative algebraic group overK whose identity compo-nentT ı is a torus6. Assume that the order ofT =T ı is not divisible by the characteristic ofK, and letX �(T ) be the group of characters ofT . Then cup-product defines a dualitiesbetween:ı the compact groupH 0(K, T )^ (completion relative to the topology of open sub-

groups of finite index) and the discrete groupH 2(G, X �(T ));ı the finite groupsH 1(K, T ) andH 1(G, X �(T ));ı the discrete groupH 2(K, T ) and the compact groupH 0(G, X �(T ))^ (completion

relative to the topology of subgroups of finite index).In particular, H 2(K, T ) D 0 if and only ifX �(T )G D 0 (whenT is connected, this lastcondition is equivalent toT (K) being compact).

PROOF. TheG-moduleX �(T ) is finitely generated without char(K)-torsion, andX �(T )DD

T (Ks), and so this follows from the preceding corollary (except for the parenthetical state-ment, which we leave as an exercise — cf. Serre 1964, pII-26).

REMARK 2.5. (a) If the characteristic ofK is p 6D 0 andM has elements of orderp, thenExt1G(M, Ks�) and H 1(G, M) are usually infinite. For example Ext1

G(Z=pZ, Ks�) D

K�=K�p andH 1(G, Z=pZ) D K=}K, }(x) D xp� x, which are both infinite.

(b) If n is prime to the characteristic ofK andK contains a primitiventh root of unity,thenZ=nZ � �n noncanonically and(Z=nZ)D �D �n canonically. The pairing in (2.3)gives rise to a canonical pairing

H 1(K,�n)�H 1(K,�n)! H 2(G,�n ˝ �n) �D �n.

The groupH 1(K,�n) D K�=K�n, and the pairing can be identified with

(f, g) 7! (�1)v(f )v(g)f v(g)=gv(f )WK�=K�n

�K�=K�n! �n

(see Serre 1962, XIV 3).If K has characteristicp 6D 0, then the pairing

Ext1G(Z=pZ, Ks�)�H 1(G, Z=pZ)! H 2(G, Ks�) �D Q=Z

can be identified with

(f, g) 7! p�1 Trk=Fp(Res(f

dg

g))WK�=K�p

�K=}K! Q=Z

(see Serre 1962, XIV 5 or III 6 below).

Unramified cohomology

A G-moduleM is said to beunramified if M ID M . For a finitely generatedG-module,

we writeM d for the submodule Hom(M, Run�) of M D D Hom(M, Ks�). Note that ifMis unramified, thenH 1(G=I, M) makes sense and is a subgroup ofH 1(G, M). Moreover,whenM is finite,H 1(G=I, M) is dual to Ext1G=I (M, Z) (see 1.10).

6Such an algebraic group is called a group of multiplicative type.

2. LOCAL FIELDS 33

THEOREM 2.6. If M is a finitely generated unramifiedG-module whose torsion is primeto char(k), then the groupsH 1(G=I, M) andH 1(G=I, M d) are the exact annihilators ofeach other in the cup-product pairing

H 1(G, M)�H 1(G, M D)! H 2(G, Ks�) �D Q=Z.

PROOF. From the spectral sequence (0.3)

ExtrG=I (M, ExtsI (Z, Ks�)) H) ExtrCsG (M, Ks�)

and the vanishing of Ext1I (Z, Ks�) �D H 1(I, Ks�), we find that

Ext1G=I (M, Kun�)�! Ext1G(M, Ks�).

From the split-exact sequence ofG-modules

0! Run�! Kun�

! Z! 0

we obtain an exact sequence

0! Ext1G=I (M, Run�)! Ext1G=I (M, Kun�)! Ext1G=I (M, Z)! 0,

and so the kernel of Ext1G(M, Ks�) ! Ext1G=I (M, Z) is Ext1G=I (M, Run�). It is easy to

see from the various definitions (especially the definition of invG in 1.6b) that

Ext1G(M, Ks�)˛1(G,M)��!

�H 1(G, M)�??y ??yInf�

Ext1G=I (M, Z)˛1(G=I,M)��!

�H 1(G=I, M)�

commutes. Therefore the kernel of

Ext1G(M, Ks�)! H 1(G=I, M)�

is Ext1G=I (M, Run�). Example (0.8) allows us to identify Ext1G(M, Ks�) with H 1(G, M D)

and Ext1G=I (M, Run�) with H 1(G=I, M d), and so the last statement says that the kernel ofH 1(G, M D) ! H 1(G=I, M)� is H 1(G=I, M d). (WhenM is finite, this result can alsobe proved by a counting argument; see Serre 1964, II 5.5.)

REMARK 2.7. A finiteG-moduleM is unramified if and only if it extends to a finiteetalegroup scheme over Spec(R). In Chapter III below, we shall see that flat cohomology allowsus to prove a similar result to (2.6) under the much weaker hypothesis thatM extends to afinite flat group scheme over Spec(R) (see III 1 and III 7).


Euler-Poincare characteristics

If M is a finiteG-module, then the groupsH r (G, M) are finite for allr and zero forr � 2.We define

�(G, M) D[H 0(G, M)][H 2(G, M)]

[H 1(G, M)].

THEOREM 2.8. LetM be a finiteG-module of orderm relatively prime to char(K). Then

�(G, M) D (RWmR)�1.

We first dispose of a simple case.

LEMMA 2.9. If the order ofM is prime to char(k), then�(G, M) D 1.

PROOF. Let p D char(k). The Sylowp-subgroupIp of I is normal inI , and the quotientI=Ip is isomorphic tobZ=Zp (see Serre 1962, IV 2, Ex 2). AsH r (Ip, M) D 0 for r > 0, theHochschild-Serre spectral sequence forI � Ip shows thatH r (I, M) D H r (I=Ip, M Ip),and this is finite for allr and zero forr > 1 (cf. Serre 1962, XIII 1). The Hochschild-Serrespectral sequence forG � I now shows thatH 0(G, M) D H 0(G=I, M I ), thatH 1(G, M)

fits into an exact sequence

0! H 1(G=I, M I )! H 1(G, M)! H 0(G=I, H 1(I, M))! 0,

and thatH 2(G, M) D H 1(G=I, H 1(I, M)). But G=I �D bZ, and the exact sequence

0! H 0(bZ, N)! N��1! N ! H 1(bZ, N)! 0

(with � a generator ofbZ; see Serre 1962, XIII 1) shows that

[H 0(bZ, N)] D [H 1(bZ, N)]

for any finitebZ-module. Therefore,

�(G, M) D[H 0(G=I, M I )][H 0(G=I, H 1(I, M))]

[H 1(G=I, M I )][H 1(G=I, H 1(I, M))]D 1.

Since both sides of equation in (2.8) are additive inM , the lemma allows us to assumethatM is killed by p D char(k) and thatK is of characteristic zero. We shall prove thetheorem for allG-modulesM such thatM D M GL , whereL is some fixed finite Galoisextension ofK contained inKs. Let G D Gal(L=K). Our modules can be regarded asFp[G]-modules, and we letRFp

(G) , or simplyR(G), be the Grothendieck group of thecategory of such modules. Then the left and right hand sides of the equation in (2.8) definehomomorphisms�`,�r WR(G)! Q>0. As Q>0 is a torsion-free group, it suffices to showthat�` and�r agree on a set of generators forRFp

(G) ˝Z Q. The next lemma describesone such set.

LEMMA 2.10. Let G be a finite group and, for any subgroupH of G, let IndGH be the

homomorphismRFp(H)˝Q! RFp

(G)˝Q taking the class of anH -module to the classof the corresponding inducedG-module. ThenRFp

(G)˝Q is generated by the images oftheIndG

H asH runs over the set of cyclic subgroups ofG of order prime top.

2. LOCAL FIELDS 35

PROOF. Write RF (G) for the Grothendieck group of finitely generatedF[G]-modules,Fany field. Then Serre 1967b, 12.5, Thm 26, shows that, in the case thatF has characteristiczero,RF (G) ˝ Q is generated by the images of the maps IndG

H with H cyclic. It followsfrom Serre 1967b, 16.1, Thm 33, that the same statement is then true for any fieldF .Finally Serre 1967b, 8.3, Pptn 26, shows that, in the case thatF has characteristicp 6D 0,the cyclic groups ofp-power order make no contribution.

It suffices therefore to prove the theorem for a moduleM of the form IndGH N . LetK0D LH , let R0 be the ring of integers inK0, and letn be the order ofN . Then

�(G, M) D �(Gal(Ks=K0), N)

(RWmR) D (RW nR)[K 0WK]D (R0

W nR0),

and so it suffices to prove the theorem forN . This means thatwe can assume thatGis a cyclic group of order prime top. ThereforeH r (G, M) D 0 for r > 0, and soH r (G, M) D H r (Gal(Ks=L), M)G .

Let�0 be the homomorphismR(G)! R(G) sendingM to (�1)i [H i(Gal(Ks=L), M)],where [*] now denotes the class of * inR(G).

LEMMA 2.11. The following formula holds:

�0(M) D �dim(M) � [KWQp] � [Fp[G]].

Before proving the lemma, we show that it implies the theorem. Let� WRFp(G)! Q>0

be the homomorphism sending the class of a moduleN to the order ofN G. Then� ı�0 D �

and�([Fp[G]]) D p, and so (2.11) shows that

�(M) D ��0(M) D p�[K WQp]�dim(M)D 1=(RWmR).

It therefore remains to prove (2.11). On tensoringM with a resolution ofZ=pZ byinjectiveZ=pZ[G]-modules, we see that cup-product defines isomorphisms ofG-modules

H r (Gal(Ks=L), Z=pZ)˝M ! H r (Gal(Ks=L), M),

and so�0(M) D �0(Z=pZ) � [M].

Let M0 be theG-module with the same underlying abelian group asM but with the trivialG-action. The map

� ˝m 7! � ˝ �m

extends to an isomorphism

Fp[G]˝M0

�! Fp[G]˝M,

and sodim(M) � [Fp[G]] D [Fp[G]] � [M].


The two displayed equalities show that the general case of (2.11) is a consequence of thespecial caseM D Z=pZ.

Note that

H 0(Gal(Ks=L), Z=pZ)D Z=pZ,

H 1(Gal(Ks=L), Z=pZ) D H 1(Gal(Ks=L),�p(Ks))�D (L�=L�p)�,

H 2(Gal(Ks=L), Z=pZ) D (�p(L))�,

whereN � denotes Hom(N, Fp) (still regarded as aG-module; as Hom(�, Fp) is exact, itis defined for objects inR(G)). Therefore

�0(Z=pZ)�D [Z=pZ]� [L�=L�p]C [�p(L)].

Let U be the group of unitsR�L in RL. From the exact sequence

0! U=U p! L�=L�p

! Z=pZ! 0,

we find that[Z=pZ]� [L�=L�p] D [U (p)],

and so

�0(Z=pZ)�D �[U (p)]C [�p(L)],

D �[U (p)]C [Up].

We need one last lemma.

LEMMA 2.12. Let W andW 0 be finitely generatedZp[H ]-modules for some finite groupH . If W ˝Qp � W 0

˝Qp asQp[H ]-modules, then

[W (p)]� [Wp] D [W 0(p)]� [W 0p]

in Fp[H ].

PROOF. One reduces the question easily to the case thatW � W 0 � pW , and for such amodule the lemma follow immediately from the exact sequence

0! W 0p ! Wp ! W=W 0

! W 0(p)! W (p)

! W=W 0! 0

given by the snake lemma.

The exponential map sends an open subgroup ofU onto an open subgroup of the ringof integersRL of L, and so (2.12) shows that

[U (p)]� [Up] D [R(p)L ]� [(RL)p] D [R

(p)L ].

The normal basis theorem shows thatL � Qp[G][K WQp] (as G-modules), and so (2.12)implies that

[R(p)L ] D [K W Qp] � [Fp[G]].

As [Fp[G]]� D [Fp[G]], this completes the proof of (2.11).

2. LOCAL FIELDS 37

Archimedean local fields

Corollaries 2.3, 2.4 and Theorem 2.8 all have analogues forR andC.

THEOREM 2.13. (a) Let G D Gal(C=R). For any finitely generatedG-moduleM withdualM D

D Hom(M, C�), cup-product defines a nondegenerate pairing

H rT (G, M D)�H 2�r

T (G, M)! H 2(G, C�)�D!

12Z=Z

of finite groups for allr.(b) Let G D Gal(C=R). For any commutative algebraic groupT over R whose

identity component is a torus, cup-product defines dualities betweenH rT (G, X �(T )) and

H 2�rT (G, T (C)) for all r .

(c) LetK D R or C, and letG D Gal(C=K). For any finiteG-moduleM

[H 0(G, M)][H 0(G, M D)]

[H 1(G, M)]D jmjv.

PROOF. (a) Suppose first thatM is finite. AsG has order2, the`-primary componentsfor ` odd do not contribute to the cohomology groups. We can therefore assume thatM is2-primary, and furthermore that it is simple. ThenM D Z=2Z with the trivial action ofG,and the theorem can be proved in this case by direct calculation.

WhenM D Z the result can again be proved by direct calculation, and whenM D Z[G]

all groups are zero. Since every torsion-freeG-module contains a submodule of finite indexthat is a direct sum of copies ofZ or Z[G], this proves the result for such modules, and thegeneral case follows by combining the two cases.

(b) TakeM D X �(T ) in (a).(c) The complex case is obvious becauseH 0(G, M) D M andH 0(G, M D) D M D

both have orderm, H 1(G, M) D 0, andjmjv D m2. In the real case, let� generateG, andnote that form 2M andf 2M D

((1� �)f )(m) D f (m)=�(f (�m))

D f (m) � (f (�m)) (because� D ��1)

D f ((1C �)m).

Therefore1 � � WM D ! M D is adjoint to1 C � WM ! M , and so, in the pairingM D�M ! C�, (M D)G andNC=RM are exact annihilators. Consequently

[M] D [(M D)G ][NC=RM] D[H 0(G, M D)][H 0(G, M)]

[H 0T (G, M)]

,

and the periodicity of the cohomology of cyclic groups shows that[H 0T (G, M)] D [H 1(G, M)].

As [M] D m D jmjv, this proves the formula.


Henselian local fields

Let K be the field of fractions of an excellent Henselian discrete valuation ringR with finiteresidue fieldk. (See Appendix A for definitions.) It is shown in the Appendix that the pair(GK , Ks�) is a class formation, and that the norm groups are precisely the open subgroupsof finite index. The following theorem generalizes some of the preceding results.

THEOREM2.14. LetM be a finitely generatedG-module whose torsion subgroup is primeto char(K).

(a) The map r (G, M)WExtrG(M, Ks�) ! H 2�r (G, M)� is an isomorphism for allr � 1, and˛0(G, M) defines an isomorphism (of compact groups)HomG(M, Ks�)^

!

H 2(G, M)�. The^ can be omitted ifM is finite. The groupsExtrG(M, Ks�) andH r (G, M)

are finite for all r if M is finite, and the groupsExt1G(M, Ks�) andH 1(G, M) are finitefor all finitely generatedM.

(b) If K is countable, then for any algebraic groupA overK,

˛1(G, A(Ks))WExt1G(A(Ks), Ks�)! H 1(G, A(Ks))�

is an isomorphism, except possibly on thep-primary component whenchar(K) D p 6D 1.(c) Cup-product defines isomorphismsH r (G, M D)! H r (G, M)� for all r � 1, and

an isomorphismH 0(G, M D)^! H 2(G, M)� of compact groups. The groupsH 1(G, M D)

andH 1(G, M) are both finite.

PROOF. (a) LetbR be the completion ofR. There is a commutative diagram

0 ��! R��! K�

��! Z ��! 0??y ??yrec

??y0 ��! bR�

��! G ��! bZ ��! 0.

All the vertical maps are injective, and the two outside vertical maps have cokernels thatare uniquely divisible by all primes 6D char(K). Therefore the reciprocity mapK�

! G

is injective and has a cokernel that is uniquely divisible prime to char(K). The first twoassertions now follow easily from (1.8). The finiteness statements follow from the fact thatGal(Ks=K) D Gal(bKs=bK)

(b) The groupA(Ks) is countable, and therefore it is a countable union of finitelygenerated submodules. The statement can therefore be proved the same way as (2.2).

(c) The proof is the same as that of (2.3).

REMARK 2.15. (a) Part (a) of the theorem also holds for modulesM with p-torsion, exceptthat it is necessary to complete Ext1

G(M, Ks�). For example, whenM D Z=pZ, the map˛1 is

K�=K�p! Hom(GK , Z=pZ).

BecauseK is excellent, the mapK�=K�p ! bK�=bK�p is injective and induces an iso-

morphism(K�=K�p)^�! bK�=bK�p. We know Gal(Ks=K) D Gal(bKs=bK), and so in this

case the assertion follows from the corresponding statement forbK.

2. LOCAL FIELDS 39

(b) As was pointed out to the author by M. Hochster, it is easy to construct nonexcellentHenselian discrete valuation rings. Letk be a field of characteristicp, and choose anelementu 2 k[[t ]] that is transcendental overk(t). Let R be the discrete valuation ringk(t, up) \ k[[t ]], and consider the HenselizationRh of R. Then the elements ofRh areseparable overR (Rh is a union ofetaleR-subalgebras), and sou =2 Rh, butu 2 (Rh)^

D

k[[t ]].

Complete fields with quasi-finite residue fields

EXERCISE2.16. LetK be complete with respect to a discrete valuation, but assume that itsresidue field is quasi-finite rather than finite. (See Appendix A for definitions.) Investigateto what extent the results of this section continue to hold forK. References: Serre 1962,XIII, and Appendix A for the basic class field theory of such fields; Serre 1964, pII-24,pII-29 for statements of what is true; Vvedens’kii and Krupjak 1976 and Litvak 1980 for aproof of (2.3) for a finite module in the case the field has characteristic zero.)

d -local fields

A 0-local field is a finite field, and ad -local field for d � 1 is a field that is completewith respect to a discrete valuation and has a(d � 1)-local field as residue field. IfKis d -local, we shall writeKi, 0 � i � d , for the i-local field in the inductive definitionof K. We write�`n for the GK -modulef� 2 Ksj �`

n

D 1g, �`1(r) for lim�!

�˝r`n , and

Z`(r) for lim �

�˝r`n . If M is an`-primary GK -module, we setM(r) D M ˝ Z`(r) and

M �(r) D Hom(M,�`1(r)).

THEOREM 2.17. LetK be ad -local field withd � 1, and let` be a prime6D char(K1).(a) There is a canonical trace map

H dC1(GK ,�`1(d))�! Q`=Z`.

(b) For all GK -modulesM of finite order a power of , the cup-product pairing

H r (GK , M �(d))�H dC1�r (GK , M)! H dC1(GK , Q`=Z`(d)) �D Q`=Z`

is a nondegenerate pairing of finite groups for allr.

PROOF. For d D 1, this is a special case of (2.3). Ford > 1, it follows by an easyinduction argument from the next lemma.

LEMMA 2.18. Let K be any field complete with respect to a discrete valuation, and letk

be the residue field ofK. For any finiteGK -module of order prime to char(k), there is along exact sequence

� � � ! H r (Gk , M I )! H r (GK , M)! H r�1(Gk , M(�1)I )! H rC1(Gk , M I )! � � �

whereI is the inertia group ofGK .


PROOF. Let char(k) D p, and letIp be ap-Sylow subgroup ofI (soIp D 1 if p D 1).ThenI 0

Ddf I=Ip is canonically isomorphic toQ` 6DpZ`(1) (see Serre 1962, IV 2). The

same argument that shows thatH r (G, M) D M G, MG, 0 for r D 0, 1, > 2 whenG D bZandM is torsion (Serre 1962, XIII 1), shows in our case that

H r (I, M) D H r (I 0, M) D

8<:

M I , for r D 0

M(�1)I , for r D 1

0, for r > 1.

(I.1)

The lemma therefore follows immediately from the Hochschild-Serre spectral sequence forG � I.

Write KrR for ther th QuillenK-group of a ringR.

COROLLARY 2.19. LetK be a2-local field, and letm be an integer prime tochar(K1) andsuch thatK contains themth roots of1. Then there is a canonical injective homomorphism(K2K)(m)

! Gal(Kab=K)(m) with dense image.

PROOF. On takingM D Z=mZ in the theorem, we obtain an isomorphism

H 2(G,�m ˝ �m)! H 1(G, Z=mZ)�.

But H 1(G, Z=mZ) D Homcts(G, Z=mZ), and so this gives us an injectionH 2(G,�m ˝

�m) ! (Gab)(m) with dense image. Now the theorem of Merkur’ev and Suslin (1982)

provides us with an isomorphism(K2K)(m) �! H 2(G,�m ˝ �m).

Theorem 2.17 is a satisfactory generalization of Theorem 2.3 in the case that the char-acteristic drops fromp to zero at the first step. The general case is not yet understood.

Some exercises

EXERCISE 2.20. (a) LetG be a profinite group, and letM be a finitely generatedG-module. WriteT D Hom(M, C�), and regard it as an algebraic torus overC. Let G act onT through its action onM . Show that

Ext0G(M, Z)DX�(T )G ,

Ext1G(M, Z) D �0(T G),

ExtrG(M, Z) D H r�1(G, T ) for r � 2.

If M is torsion-free, show that ExtrG(M, Z) D H r (G, X�(T )).

(b) Let K be a local field (archimedean or nonarchimedean), and letT be a torus overK. Let T _ be the torus such thatX �(T _) D X�(T ). Show that the finite groupH 1(K, T )

is dual to�0(T _G) and thatH 1(K, T _) is canonically isomorphic to the groupT (K)� ofcontinuous characters of finite order ofT (K). (In ~8 we shall obtain a similar descriptionof the group of generalized characters ofT (K).) [Hint: To prove the first part of (a), usethe spectral sequence (0.8)

H r (G, Exts(M, C�)) H) ExtrCsG (M, C�)

3. ABELIAN VARIETIES OVER LOCAL FIELDS 41

and the exponential sequence0! Z! C! C�! 0. ]

Reference: Kottwitz 1984.

EXERCISE2.21. LetK be a2-local field of characteristic zero such thatK1 has character-istic p 6D 0. Assume

(a) K hasp-cohomological dimension� 3 and there is a canonical isomorphism

H 3(G,�pn ˝ �pn)! Z=pnZ

(Kato 1979,~5, Thm 1);(b) if K contains a primitivepth root of 1, then the cup-product pairing is a nondegener-

ate pairing of finite groups (ibid.~6).Prove then that (2.17) holds forK with ` D p.

EXERCISE2.22. LetK D k((t1, ..., td)) with k a finite field, and letp D char(k). Define�(r) D Ker(˝r

K=k,dD0! ˝r

K=k), whereC is the Cartier operator (see Milne 1976). Show

that there is a canonical trace mapH 1(GK , �(d))�D! Z=pZ, and show that the cup-product

pairings

H r (GK , �(r))�H 1�r (GK , �(d � r))! H 1(GK , �(d)) �D Z=pZ

are nondegenerate in the sense that their left and right kernels are zero. Letd D 2, andassume that there is an exact sequence

0! K2Kp! K2K! �(2)! 0

with the second map being dlogdlogW K2K ! �(2) . (In fact such a sequence exists:the exactness at the first term is due to Suslin 1983; the exactness at the middle term is atheorem of Bloch (Bloch and Kato 1986); and the exactness at the last term has been provedby several people.) Deduce that there is a canonical injective homomorphism(K2K)(p)

!

(GabK )(p). (These results can be extended to groups killed by powers ofp rather thatp itself

by using the sheaves�n(r) of Milne 1986a.)

NOTES. The main theorems concerning local fields in the classical sense are due to Tate.The proofs are those of Tate except for that of (2.8), which is due to Serre (see Serre 1964,II 5). Theorem 2.17 is taken from Deninger and Wingberg 1986.

3 Abelian varieties over local fields

We continue with the notations at the start of the last section. In particular,K is a local field,complete with respect to a discrete valuation ord, and with finite residue fieldk. WhenG

andH are algebraic groups over a fieldf , we write ExtrF (G, H) for the group formed inthe categoryGpF (see~0).

Let A be an abelian variety overK, and letAt be the dual abelian variety. The Barsotti-Weil formula (Serre 1959, VII,~3) states thatAt(Ks) D Ext1K s(A, Gm).


LEMMA 3.1. For any abelian varietyA over a perfect fieldF , there is a canonical isomor-phism

H r (F, At)! ExtrC1F (A, Gm),

all r � 0.

PROOF. The group ExtrF s(A, Gm) is shown to be zero forr � 2 in (Oort 1966, Pptn 12.3),and HomF s(A, Gm) D 0 because all maps from a projective variety to an affine variety areconstant. This together with the Barsotti-Weil formula shows that the spectral sequence(0.17)

H r (Gal(F s=F), ExtsF s(A, Gm)) H) ExtrCsF (A, Gm)

degenerates to a family of isomorphismsH r (F, At)�! ExtrC1

F (A, Gm).

In particular Ext1K (A, Gm) D At(K) whenK has characteristic zero, and the Ext grouptherefore acquires a topology from that onK. Recall that there is a canonical pairing (0.16)

ExtrK (A, Gm)�H 2�r (K, A)! H 2(K, Gm),

and an isomorphism invG WH2(K, Gm)

�D! Q=Z (1.6b). Therefore there is a canonical map

˛r (K, A)WExtrK (A, Gm)! H 2�r (K, A)�.

THEOREM 3.2. If K has characteristic zero, then1(K, A) is an isomorphism of compactgroups

Ext1K (A, Gm)��! H 1(K, A)�

and˛2(K, A) is an isomorphism of torsion groups of cofinite type

Ext2K (A, Gm)��! A(K)�.

For r 6D 1, 2, ExtrK (A, Gm) andH 2�r (K, A) are both zero.

We first need a lemma.

LEMMA 3.3. In the situation of the theorem,A(K) contains an open subgroup of finiteindex isomorphic toRdim(A); thereforeA(K) D A(K)^ (completion for the profinite topol-ogy), and

[A(K)(n)]=[A(K)n] D (R W nR)dim(A).

PROOF. The existence of the subgroup follows from the theory of the logarithm (see Mat-tuck 1955 or Tate 1967b, p168–169), and the remaining statements are obvious.

PROOF OF3.2. From0! An! A

n! A! 0

we get the rows of the following diagram

0 ��! ExtrK (A, Gm)(n)��! ExtrK (An, Gm) ��! ExtrC1

K (A, Gm)n ��! 0??y˛r (K,A)(n)

??y˛r (K,An)

??y˛rC1(K,A)n

0 ��! (H 2�r (K, A)n)��! H 2�r (K, An)�

��! H 1�r (K, A)(n)��! 0


As is explained in (0.18), ExtrK (An, Gm)

�! ExtrG(An(Ks), Ks�) for all r , and if we take

˛r (G, An) to be the map r (G, An(Ks)) of ~2, then it is clear that the diagram commutes.As ˛r (G, An) is an isomorphism of finite groups for allr , we see that

˛r (K, A)(n)WExtrK (A, Gm)(n)

! (H 2�r (K, A)n)�

is an injective map of finite groups for allr , and, in the limit,

lim �

˛r (K, A)(n)W lim �

ExtrK (A, Gm)(n)! (H 2�r (K, A)tors)

�

is injective. As Ext1K (A, Gm) D At(K), the lemma shows that

Ext1K (A, Gm) D lim �

Ext1K (A, Gm)(n).

Thus we have shown that˛1(K, A) is injective.We next show thatH r (K, A) D 0 for r � 2. For r > 2, this follows from the fact that

G has cohomological dimension2 (see 2.1). On takingr D 0 in the above diagram, we getan exact commutative diagram

0 ��! 0 ��! HomK (An, Gm) ��! Ext1K (A, Gm)??y ??y�

??y0 ��! (H 2(K, A)n)�

��! H 2(K, An)��! H 1(K, A)�.

As the right hand vertical arrow is injective, the snake lemma shows thatH 2(K, A)n D 0,and therefore thatH 2(K, A) D 0. Because

H r (G, At) �D ExtrC1K (A, Gm),

this also shows that ExtrK (A, Gm) D 0 for r 6D 1, 2.

We now prove that 1(K, A) is an isomorphism. We have already seen that it is aninjective mapAt(K) ! H 1(K, A)�, and it remains to show that the mapsAt(K)(n)

!

(H 1(K, A)n)� are surjective for all integersn. As these maps are injective, this can beaccomplished by showing that the groups have the same order. LetM D An(Ks) andM D D At

n(Ks). Then (2.8) shows that

�(G, M) D (R W nR)�2dD �(G, M D),

whered is the dimension ofA, and (3.3) shows that

[A(K)(n)]=[A(K)n] D (R W nR)dD [At(K)(n)]=[At(K)n].

From the cohomology sequence of

0!M ! A(Ks)n! A(Ks)! 0

we find that

�(G, M) D[A(K)n][H 2(G, M)]

[A(K)(n)][H 1(K, A)n]


or1

(RW nR)2dD

1

(R W nR)d

[H 0(G, M D)]

[H 1(G, A)n].

As H 0(G, M D) � At(K)n, this can be rewritten as

[H 1(G, A)n] D (R W nR)d [At(K)n] D [At(K)(n)],

which completes the proof that˛1(K, A) is an isomorphism.It remains to show that 2(K, A) is an isomorphism. The diagram at the start of the

proof shows that 2(K, A) is surjective, and we know that it can be identified with a mapH 1(K, At) ! A(K)�. The above calculation withA and At interchanged shows that[H 1(K, At)n] D [A(K)(n)] for all n, which implies that 2(K, A) is an isomorphism.

COROLLARY 3.4. If K has characteristic zero, then there is a canonical pairing

H r (K, At)�H 1�r (K, A)! Q=Z,

which induces an isomorphism of compact groupsAt(K)�! H 1(K, A)� (caser D 0) and

an isomorphism of discrete groups of cofinite-typeH 1(K, At)�! A(K)� (caser D 1).

For r 6D 0, 1, the groupsH r (K, A) andH r (K, At) are zero.

PROOF. Lemma 3.1 allows us to replace ExtrK (A, Gm) in the statement of the theorem with

H r�1(K, At).

REMARK 3.5. There is an alternative approach to defining the pairings

H r (K, At)�H 1�r (K, A)! H 2(K, Gm)

of (3.4). For any abelian varietyA overK, write Z(A) for the group of zero cycles onAK s

of degree zero (that is, the set of formal sumsniPi with Pi 2 A(Ks) andni D 0). There isa surjective mapS WZ(A)�A(Ks) sending a formal sum to the corresponding actual sumonA, and we writeY(A) for its kernel. There are exact sequences

0! Y(At)! Z(At)! At(Ks)! 0,

0! Y(A)! Z(A)! A(Ks)! 0.

Let D be a divisor onAt� A, and leta andb be elements ofY(At) andZ(A) such that

the support ofD does not meet the support ofa � b. The projectionD(a) of D � (a � b)

ontoA is then defined and, becausea is in Y(A), it is principal, sayD(a) D div(f ). It isnow possible to define

D(a, b) D f (b)dfDQ

b2supp(b)f (b)ordb(b)2 Ks�.

Now letD be a Poincare divisor onAt�A, and letDt be its transpose. A reciprocity law

(Lang 1959, VI 4, Thm 10), shows that the pairings

(a, b) 7! D(a, b)WY(At)�Z(A)! Ks�

(b, a) 7! Dt(b, a)WY(A)�Z(At)! Ks�


satisfy the equalityD(a, b) D Dt(b, a) if a 2 Y(At) andb 2 Y(A). They therefore giverise to augmented cup-product pairings (0.12)

H r (K, At)�H 1�r (K, A)! H 2(K, Gm).

It is possible to show that these pairings agree with those in (3.4) (up to sign) by check-ing that each is compatible with the pairings

H r (K, Atn)�H 2�r (K, An)! H 2(K, Gm)

defined by theen-pairingAtn � An ! Gm. Alternatively, one can show directly that the

mapsH r (K, A)! ExtrC1G (At(Ks), Ks�) defined by this pair of pairings (see 0.14b) equal

those defined by the Barsotti-Weil formula. (In fact the best way of handling these pairingsis to make use of biextensions and derived categories, see Chapter III, especially AppendixC.)

REMARK 3.6. WhenK has characteristicp 6D 0, (3.4) can still be proved by similarlyelementary methods provided one omits thep-parts of the groups. More precisely, writeA(K)(non-p) for lim

�A(K)(n) wheren runs over all integers not divisible byp, and let

H r (K, A)(non-p) DL

` 6DpH r (K, A)(`) for r > 0.

Then the pair of pairings in (3.5) defines augmented cup-products

H r (K, At)�H 1�r (K, A)! Q=Z,

which induce an isomorphism of compact groups

At(K)(non-p)�! H 1(K, A)(non-p)� (caser D 0)

and an isomorphism of discrete groups of cofinite type

H 1(K, At)(non-p)�! A(K)(non-p)� (caser D 1).

For r 6D 0, 1, the groupsH r (K, At)(non-p) andH r (K, A)(non-p) are zero.Probably this can be proved by the same method as above, but I have not checked this.

Instead I give a direct proof.There is a commutative diagram

0 ��! At(K)(n)��! H 1(K, At

n) ��! H 1(K, At)n ��! 0??y ??y ??y0 ��! H 1(K, A)�

n ��! H 1(K, An)��! A(K)(n)�

��! 0

for all n prime to p. As the middle vertical arrow is an isomorphism (2.3), we see onpassing to the limit that

H 1(K, At)(non-p)! A(K)(non-p)�


is surjective. To show that it is injective, it suffices to show that for anyn prime top,H 1(K, At)n ! A(K)(n)� is injective, and this we can do by showing that the two groupshave the same order. There is a subgroup of finite index inA(K) that is uniquely divisibleby all integers prime top (namely, the kernel of the specialization mapA(R) ! A0(k),whereA is the Neron model ofA; it follows from Hensel’s lemma that this is uniquelydivisible prime to the characteristic ofk). Consequently[A(K)n] D [A(K)(n)]. Now thesame argument as in the proof of (3.2) shows that the groups in question have the sameorder. The rest of the proof is exactly as in (3.2).

In ~7 of Chapter III, we shall use flat cohomology to prove that (3.4) is valid even forthep-components of the groups.

REMARK 3.7. The duality in (3.4) extends in a rather trivial fashion to archimedean localfields. LetG D Gal(C=R) and letA be an abelian variety overR. Then the pair of pairingin (3.5) defines a pairing of finite groups

H rT (G, At(C))�H 1�r

T (G, A(C))! H 2T (G, C�) �D 1

2Z=Z

for all integersr . The pairing can be seen to be nondegenerate from the following diagram

0 ��! H r�1T (G, At) ��! H r

T (G, At2) ��! H r

T (G, At) ��! 0??y ??y ??y0 ��! H r�1

T (G, A)��! H 2�r

T (G, A2)��! H 1�r

T (G, A)��! 0.

Part (a) of (2.13) shows that the middle arrow is an isomorphism, and the two ends of thediagram show respectively that

H rT (G, At)! H 1�r

T (G, A)�

is injective for allr and surjective for allr .The groupA(R)ı is a connected, commutative, compact real Lie group of dimension

dim(A), and therefore it is isomorphic to(R=Z)dim(A). The norm mapA(C) ! A(R) iscontinuous andA(C) is compact and connected, and so its image is a closed connectedsubgroup ofA(R). Since it contains the subgroup2A(R) of A(R), which has finite indexin A(R), it must also be open, and therefore it equalsA(R)ı. ConsequentlyH 0

T (G, A) D

�0(A(R)). The exact sequence

0! A(R)ı2! A(R)2! �0(A)! 0

shows that[�0(A)] � 2dim(A)

D [A(R)2],

and soH 1(R, At) 6D 0 if and only if [A(R)2] > 2dim(A). For example, whenA is anelliptic curve,H 1(R, A) 6D 0 if and only if (in the standard form) the graph ofA in R� Rintersects thex-axis in three points.

WhenA is an algebraic group over a fieldk, we now write�0(A) for the set of con-nected components (for the Zariski topology) ofA over ks; that is,�0(A) D Aks=Aı

ks

regarded as aG-module.


PROPOSITION3.8. LetA be an abelian variety overK, and letA be its Neron model overR. Then

H 1(G=I, A(Kun)) D H 1(G=I,�0(A0))

whereA0 is the closed fibre ofA=R. In particular, if A has good reduction, then

H 1(G=I, A(Kun)) D 0.

PROOF. Let Aı be the open subgroup scheme ofA whose generic fibre isA and whosespecial fibre is the identity component ofA0. BecauseA is smooth overR, Hensel’slemma implies that the reduction mapA(Run) ! A0(ks) is surjective (see for exampleMilne 1980, I 4.13), and it follows that there is an exact sequence

0! Aı(Run)! A(Run)! �0(A0)! 0.

MoreoverA(Run) D A(Kun) (becauseA ˝R Run is the Neron model ofA ˝K Kun), andso it remains to show thatH r (G=I,Ao(Run)) D 0 for r D 1, 2.

An element of H 1(G=I,Aı(Run)) can be represented by anAı-torsorP . AsAı0 is a

connected algebraic group over a finite field, Lang’s lemma (Serre 1959, VI.4) shows thattheAı

0-torsorP ˝R k is trivial, and soP(k) is nonempty. Hensel’s lemma now impliesthatP(R) is nonempty, and so D 0.

Finally, for eachn, H 2(G=I,Aı(Run=mn)) D 0 becauseG=I has cohomological di-mension1, and this implies thatH 2(G=I,Aı(Run)) D 0 (Serre 1967a, I 2, Lemma 3).

REMARK 3.9. The perceptive reader will already have observed that the proof of the propo-sition becomes much simpler if one assumes thatA has good reduction.

REMARK 3.10. (a) LetR be an excellent Henselian discrete valuation ring with finiteresidue field, and letK be the field of fractions ofR. For any abelian varietyA overK, letA(K)^ be the completion ofA(K) for the topology defined byK. Let bK be the completionof K. Then

(i) the mapA(K)^! A(bK) is an isomorphism;

(ii) the mapH 1(K, A)! H 1(bK, A) is an isomorphism.Therefore the augmented cup-product pairings

H r (K, At)�H 1�r (K, A)! H 2(K, Gm)

induce isomorphismsAt(K)^! H 1(K, A)� andH 1(K, At)! A(K)�.

To prove (i) we have to show that every element ofA(bK) can be approximated arbi-trarily closely by an element ofA(K), but Greenberg’s approximation theorem (Greenberg1966) says that every element ofA(bR) can be approximated arbitrarily closely by an ele-ment ofA(R), andA(bR) D A(bK).

The injectivity of H 1(K, A) ! H 1(bK, A) also follows from Greenberg’s theorem,because an element ofH 1(K, A) is represented by a torsorP over K, which extends toa flat projective schemeP overR; if P(bK) is nonempty, thenP(R=mi) is nonempty forall i, which (by Greenberg’s theorem) implies thatP(R) is nonempty. For the surjectivity,one endowsH 1(bK, A) with its natural topology, and observes thatH 1(K, A) is dense in it(because, for any finite Galois extensionL of K, Z1(L=K, A) has a a natural structure asan algebraic group (Milne 1980, p115), and so Greenberg’s theorem can be applied again).Proposition 3.8 then shows that the topology onH 1(bK, A) is discrete.


EXERCISE3.11. Investigate to what extent the results of this section continue to hold whenK is replaced by a complete local field with quasi-finite residue field.

NOTES. The duality betweenH 1(K, At) andH 1(K, A) in (3.4) was the first major theo-rem of the subject (see Tate 1957/58); it was proved before (2.3), and so can be regardedas the forerunner of the rest of the results in this chapter. The proof of Theorem 3.2 ismodelled on a proof of Tate’s of (3.4) (cf. Milne 1970/72, p276). The description of thepairing in (3.4) given in (3.5) is that of Tate’s original paper. Proposition 3.8 can be foundin Tate 1962 in the case of good reduction; the stronger form given here is well known.

4 Global fields

Throughout this section,K will be a global field, andS will be a nonempty set of primes ofK, containing the archimedean primes in the case theK is a number field. IfF � K, thenthe set of primes ofF lying over primes inS will also be denoted byS (or, occasionally,by SF ). We writeKS for the maximal subfield ofKs that is ramified overK only at primesin S , andGS for Gal(KS=K). Also

RK,S DTv =2SOv D fa 2 K j ordv(a) � 0 for all v =2 Sg

denotes the ring ofS -integers inK. For each primev we choose an embedding (overK) of Ks into Ks

v, and consequently an extensionw of v to Ks and an identification ofGv Ddf Gal(Ks

v=Kv) with the decomposition group ofw in GK .Let P denote the set of prime numbers` such that 1 divides the degree ofKS over

K. If K is a function field, thenP contains all prime numbers becauseKS containsKks

whereks is the separable closure of the field of constants ofK. If K is a number field, thenP contains at least all the primes` such that RK,S D RK,S (that is, such thatS containsall primes dividing`) because for such primes,KS contains the mth roots of 1 for allm.(It seems not to be known7 how largeP is in the number field case; for example, ifK D QandS D f`,1g, is P the set of all prime numbers?8)

For a finite extensionF of K contained inKS , we use the following notations:JF D the group of ideles ofF IJF,S D f(aw) 2 JF j aw D 1 for w =2 Sg �D

Q0

w2SF �w (restricted topological product

relative to the subgroupsbO�w);

RF,S DTw=2S Ow D ring of SF -integers (D integral closure ofRK,S in F );

EF,S D R�F,S D group ofSF -units;

CF,S D JF,S=EF,S D group ofSF -idele classes;

UF,S D f(aw) 2 JF j aw 2 bO�

w for w =2 S , aw D 1 otherwiseg �DQw=2S

bO�w .

DefineJS D lim

�!JF,S , RS D lim

�!RF,S , ES D lim

�!EF,S ,

CS D lim�!

CF,S , US D lim�!

UF,S ,

7Haberland (1978) simply assumes thatP is always the set of all prime numbers. Thus, many of histheorems are false, or at least, not proven. This is unfortunate, since his book has been widely used as areference.

8As far as I know, this question is still unanswered.

4. GLOBAL FIELDS 49

where the limit in each case is over all finite extensionsF of K contained inKS .

WhenS contains all primes ofK, we usually drop it from the notation. In this caseKS D Ks, GS D GK , andP contains all prime numbers. MoreoverJF,S D JF , RF,S D

F , EF,S D F �, andCF,S D CF is the idele class group ofF . Since everything becomesmuch simpler in this case, the reader is invited to assumeS contains all primes on a firstreading.

A duality theorem for the P -class formation(GS , CS)

LetCS(F) D CF=UF,S ; we shall show that(GS , CS) is aP -class formation withC Gal(KS=F)

S D

CS(F). Note that whenS contains all primes,(GS , CS) is the class formation(G, C ) con-sidered in (1.6c) andCS(F) D CF .

LEMMA 4.1. There is an exact sequence

0! CF,S ! CF=UF,S ! IdF,S ! 0,

whereIdF,S is the ideal class group ofRF,S . In particular, if S omits only finitely many

primes, thenIdF,S D 1 andCF,S

�D! CF=UF,S .

PROOF. Note thatF �\UF,S D f1g andJF,S \ (F �

�UF,S) D EF,S (intersections insideJF ). ThereforeUF,S can be regarded as a subgroup ofCF and the injectionJF,S ,! JF

induces an injectionCF,S ,! CF=UF,S . The cokernel of this last map is

JF=JF,S � UF,S � F� �D (

Lv =2SZ)=Im(F �),

which can be identified with the ideal class group ofRF,S . If S omits only finitely manyprimes, thenRK,S is a Dedekind domain with only finitely many prime ideals, and anysuch ring is principal.

PROPOSITION4.2. The pair(GS , CS) is aP -class formation andC GS

S D CK=UK,S .

PROOF. As we observed in (1.6c),(G, C ) is a class formation. Therefore(GS , C HS ),whereHS D Gal(Ks=KS), is aP -class formation (see the discussion preceding 1.13). Thenext two lemmas show that there is a canonical isomorphismH r (GS , C HS )! H r (GS , CS)

for all r � 1, and since the same is true for any open subgroup ofGS , it follows that(GS , CS) is also aP -class formation.

PROPOSITION4.3. There is a canonical exact sequence

0! US ! C HS ! CS ! 0.

PROOF. WhenS is finite, on passing to the direct limit over the isomorphismsCF,S

�!

CF=UF,S we obtain an isomorphismCS

�! C HS=US , which gives the exact sequence. In

the general case, we have to show that lim�!

IdF,S D 0. Let L be the maximal unramifiedextension ofF (in Ks) in which all primes ofS split, and letF 0 be the maximal abeliansubextension ofL=F . ThusF 0 is the maximal abelian unramified extension ofF in which


all primes ofS split (that is, such that all primes inS are mapped to 1 by the reciprocitymap). Class field theory (Tate 1967a, 11.3) gives us a commutative diagram

IdF,S

�D��! Gal(L=F)ab Gal(F 0=F)??y ??yV

idF 0,S

�D��! Gal(L=F 0)ab

with V the transfer (that is, Verlagerung) map. The principal ideal theorem (Artin and Tate1961, XIII 4) shows thatV is zero. Since similar remarks hold for all finite extensionsF of K contained inKS , we see that lim

�!IdF,S D 0 (direct limit over suchF), and this

completes the proof.

LEMMA 4.4. With the above notations,H r (GS , US) D 0 for r � 1. Therefore the co-homology sequence of the sequence in (4.3) gives isomorphismsCS(K) ! C

GS

S andH r (GS , C HS )! H r (GS , CS), r � 1.

PROOF. By definition,

H r (GS , US) D lim�!F

H r (Gal(F=K), UF,S) D lim�!F

H r (Gal(F=K),Qw=2SF

bO�w).

The cohomology of finite groups commutes with products, and so

H r (Gal(F=K),Q

w=2SF

bO�w) D

Qv =2SK

H r (Gal(F=K),Qwjv

bO�w),

DQv =2SK

H r (Gal(Fw=Kv),Qwjv

bO�w),

where in the last productw denotes the chosen primew lying overv. Now

H r (Gal(Fw=Kv), bO�w) D 0, r � 1,

becausev is unramified inF (cf. Serre 1967a, Pptn 1), and this completes the proof because(C HS )GS D CK (Tate 1967a, 8.1).

We writeDS(F) andDF for the identity components ofCS(F) andCF . WhenK is afunction field, the idele groups are totally disconnected, and so their identity componentsreduce to the identity element.

LEMMA 4.5. Assume thatK is a number field. ThenDS(K) D DK UK,S=UK,S . It isdivisible, and there is an exact sequence

0! DS(K)! CS(K)rec! Gab

S ! 0.

PROOF. WhenS contains all primes ofK, this is a standard part of class field theory; infactDK is the group of divisible elements inCK (Artin and Tate 1961, VII, IX). The identitycomponent ofCS(K) is the closure of the image of the identity component ofCK . AsUK,S

is compact,CK ! CS(K) is a proper map, and so the image of the identity component

4. GLOBAL FIELDS 51

is already closed. This proves the first statement, andDS(K) is divisible because it is aquotient of a divisible group. The image ofUK,S in Gab is the subgroup fixingKS \Kab,which is also the kernel ofGab ! Gab

S , and the existence of the exact sequence followsfrom applying the snake lemma to the diagram

UK,S ��! Gal(Kab=KS \Kab) ��! 0??y ??y0 ��! DK ��! CK ��! Gal(Kab=K) ��! 0.

THEOREM 4.6. LetM be a finitely generatedGS -module, and let 2 P .(a) The map

˛r (GS , M)(`)WExtrGS(M, CS)(`)! H 2�r (GS , M)�(`)

is an isomorphism for allr � 1.(b)9 Let K be a number field, and choose a finite totally imaginary Galois extensionL

of K contained inKS and such thatGal(KS=L) fixesM ; if P contains all prime numbersor if M is a finite module such that[M]RS D RS , then there is an exact sequence

Hom(M, DS(L))NL=K

! HomGS(M, CS)

˛0

! H 2(GS , M)�! 0.

9For (b) to be true, it is necessary to takeL sufficiently large. What follows is an email from Bill McCal-lum.

Here is the point: suppose you have a totally imaginary fieldK which does not satisfy Leopoldt’s conjec-ture for some primep, i.e., there is a non-trivial kernel to the map

E0˝ Zp ! completion atp.

Let S be the primes abovep and infinity. LetCS (K) andDS (K) be as [in the book]. ThenDS (K) willhave an infinitelyp-divisible part coming from the primes at infinity, and an extrap-divisible part comingfrom the Leopoldt kernel (given an element of the Leopoldt kernel, construct a sequence of ideles by takingits pn-th roots at primes abovep and infinity,1 everywhere else). Now, I believe that the kernel of the map

˛0(GS , Z=pZ)WHom(Z=pZ, CS )! H 2(GS , Z=pZ)�

should be just the part ofDS (K) coming from the infinite primes, no more. I think the statement would becorrect if you replaced “choose a finite totally imaginary Galois extension L of K” with “choose a sufficientlylarge finite ...”

Specifically, sufficiently large would be to adjoin enoughp-power roots of unity, because the inverse limitof the Leopoldt kernels under norm is zero as you go up the cyclotomic tower. So the norms would capturejust the infinite part, as required.

I noticed that Tate says “sufficiently large” in his Congress announcement of 1962, although as you pointout in the book he later makes the mistake of saying thatGS has strict cohomological dimension2, whichmeans that Leopoldt’s conjecture is satisfied and the hedge isn’t necessary!

If I am right, you would have to correct the statement [in the proof of (b)]: “It follows easily that thesequence is exact wheneverGS acts trivially onM andL D K.” I tried to verify this and ran into the problemthat I had to assumeH 3(GS , Z) D 0, which is the point of Tate’s mistake. You would have to replace thissentence with a more detailed argument, and the hypothesis would be “wheneverGS acts trivially onM andL is sufficiently large.”


(c) LetK be a function field; for any finitely generatedGS -module, there is an isomor-phism

HomGS(M, CS)^

! H 2(GS , M)�

where^ denotes the completion relative to the topology of open subgroups of finite index.

PROOF. Assume first thatK is a number field. Lemma 4.5 shows that, for all` 2 P

and allm, ˛1(GS , Z=`mZ) is bijective and 0(GS , Z=`mZ) is surjective. Thus it followsfrom (1.13) that part (a) of the theorem is true for number fields and that˛0(GS , M)(`) issurjective for finiteM .

For (b), note first that whenM D Z andL D K, the sequence becomes that in thelemma. It follows easily that the sequence is exact wheneverGS acts trivially onM andL D K. Let M andL be as in (b), and consider the diagram (1.9.1) in the proof of (1.8):

HomGS(M1, CS) ��! HomU (M, CS) ��! HomGS

(M, CS) ��! � ��! � � �??y ??y ??y ??y�

H 2(GS , M1)��! H 2(U, M)�

��! H 2(GS , M)��! � ��! � � � .

HereU D Gal(KS=L). All vertical maps in the diagram are surjective, and so we get anexact sequence of kernels:

Ker(˛0(GS , M1))! Ker(˛0(U, M))! Ker(˛0(GS , M))! 0.

We have already observed that the kernel of˛0(U, M) is Hom(M, DS(L)), and thereforethe kernel of 0(GS , M) is the imageNL=K (Hom(M, DS(L)) of this in HomG(M, CS).

When K is a function field, recG WCK ! Gab is injective with dense image. Moreprecisely, there is an exact sequence

0! CK ! Gab! bZ=Z! 0

and the first arrow induces a topological isomorphism offa 2 CK j jaj D 1g onto the opensubgroup Gal(Ks=Kks) of Gab (Artin and Tate 1961, 8.3). From this we again get an exactsequence

0! CGS

S ! GabS !

bZ=Z! 0

As bZ=Z is uniquely divisible, part (a) of the theorem follows in this case directly from(1.8). Part (c) can be proved by a similar argument to that which completes the proof of(2.1).

We next reinterpret (4.6) as a statement about the cohomology of an algebraic torusT

over K. Let AF,S DQ0

w2SFw be the ring ofS -adeles ofF , and letAS D lim�!

AF,S ,where the limit is again over finite extensions ofK contained inKS . As for any algebraicgroup overK, it is possible to define the setT (AF,S) of points ofT with values inAF,S ,and we letT (AS) D lim

�!T (AF,S). If T is split by KS , thenT (AS) D X�(T ) ˝Z JS .

This suggests the definitionT (RS) D X�(T ) ˝Z ES . A cocharacter� 2 X �(T ) definescompatible maps

T (RS)! ES , T (AS)! JS ,

4. GLOBAL FIELDS 53

and hence a mapT (AS)=T (RS)! JS=E�S D CS . We have therefore a pairing

X �(T )� T (AS)=T (RS)! CS ,

which induces cup-product pairings

H r (GS , X �(T ))�H 2�r (GS , T (AS)=T (RS))! H 2(GS , CS)! Q=Z.

COROLLARY 4.7. Let T be a torus overK split by KS , and let` 2 P . Then the cup-product pairings defined above induce dualities between:

the compact groupH 0(GS , X �(T ))^ (`-adic completion) and the discrete group

H 2(GS , T (AS)=T (RS))(`)I

the finite groupsH 1(GS , X �(T ))(`) andH 1(GS , T (AS)=T (RS))(`); and,whenP contains all the prime numbers, the discrete groupH 2(GS , X �(T )) and the

compact groupH 0(GS , T (AS)=T (RS))^ (completion of the topology of open subgroupsof finite index).

PROOF. As we saw in (1.11), ExtrGS

(X �(T ), CS) D H r (GS , X�(T )˝ CS). On tensoringthe exact sequence

0! ES ! JS ! CS ! 0

with X�(T ), we find thatX�(T )˝ CS D T (AS)=T (RS). Therefore

ExtrGS(X �(T ), CS) D H r (GS , T (AS)=T (RS)),

and part (a) of the theorem gives us isomorphisms

H r (GS , T (AS)=T (RS))(`)�! H 2�r (GS , X �(T ))�(`), r � 1.

As H s(GS , X �(T )) is obviously finite fors D 1 and is finitely generated fors D 0, thisproves the first two assertions. In the function field case, we also have an isomorphism

H s(GS , T (AS)=T (RS))^! H 2(GS , X �(T ))�.

In the number field case, completing the exact sequence

Hom(X�(T ), DS(L))! H 0(GS , T (AS)=T (RS))! H 2(GS , X �(T ))�! 0

given by (4.6b) yields the required isomorphism because the first group is divisible.

Statement of the main theorem

The rest of this section is devoted to stating and proving Tate’s theorem (Tate 1962, Thm3.1), which combines the dualities so far obtained for local and global fields. From now on,M is a finitely generatedGS -module the order of whose torsion subgroup is a unit inRS .

For v a prime ofK, let Gv D Gal(Ksv=Kv). In the nonarchimedean case, we write

k(v) for the residue field atv, andgv D Gal(k(v)s=k(v)) D Gv=Iv. The choice of the


embeddingKs ,! Ksv determines mapsGv ! GK ! GS , and using these maps we

obtain localization mapsH rS(GS , M) ! H r (Gv, M) for eachGS -moduleM . We write

H r (Kv, M) D H r (Gv, M) except in the case thatv is archimedean, in which case we setH r (Kv, M) D H r

T (Gv, M). ThusH 0(R, M) D M Gal(C=R)=NC=RM andH 0(C, M) D 0.When v is archimedean andM is unramified atv, we write H r

un(Kv, M) for the im-age ofH r (gv, M) in H r (Gv, M). Thus,H 0

un(Kv, M) D H 0(Kv, M), H 1un(Kv, M) �D

H 1(gv, M), and, unlessM has elements of infinite order,H 2un(Kv, M) D 0. A finitely

generatedGS -moduleM is unramified for all but finitely manyv in S , and we defineP r

S(K, M) to be the restricted topological product of theH r (Kv, M) relative to the sub-groupsH r

un(Kv, M). Thus

P 0S(K, M) D

Yv2S

H 0(Kv, M)

with the product topology (it is compact ifM is finite);

P 1S(K, M) D

Y0

v2SH 1(Kv, M)

with the restricted product topology (it is always locally compact because eachH 1(Kv, M)

is finite by (2.1)). IfM is finite, then

P rS(K, M) D

Lv2SH r (Kv, M)

(discrete topology) forr 6D 0, 1.

LEMMA 4.8. For any finitely generatedGS -moduleM , the image of

H r (GS , M)!Yv2S

H r (Kv, M)

is contained inP rS(K, M).

PROOF. Let 2 H r (GS , M). Then arises from an element 0 of H r (Gal(L=K), M)

for some finite Galois extensionL of K contained inKS , and for allv that are unramifiedin L, the image of in H r (Kv, M) lies inH r

un(Kv, M).

The lemma provides us with mapsr WH r (GS , M) ! P rS(K, M) for all r . When

necessary, we writerS(K, M) for ˇr .

LEMMA 4.9. Assume thatM is finite. Then the inverse image of every compact subset ofP 1

S(K, M) under the map 1S(K, M) is finite (in other words, the map is proper relative to

the discrete topology onH 1(GS , M)).

PROOF. After replacingK with a finite extension contained inKS , we can assume thatGS acts trivially onM . Let T be a subset ofS omitting only finitely many elements, andlet

P(T ) DY

v2SrT

H 1(Kv, M)�Yv2T

H 1un(Kv, M).

4. GLOBAL FIELDS 55

ThenP(T ) is compact by Tikhonov’s theorem, and every compact neighbourhood of1 inP 1

S(K, M) is contained in such a set. It suffices therefore to show that the inverse image of

P(T ) is finite. An element of this set is a homomorphismf WGS !M such thatKKer(f )

S is

unramified at all primesv in T . ThereforeKKer(f )

S is an extension ofK of degree dividingthe fixed integer[M] and unramified outside the finite setS r T . It is a well knownconsequence of Hermite’s theorem (see, for example, Serre 1964, pII-48) that there areonly finitely many such extension fields, and therefore there are only finitely many mapsf.

DefineXr

S(K, M) D Ker(ˇrWH r (GS , M)! P r

S(K, M)).

For a finiteGS -moduleM , we write

M DD Hom(M, K�

S ) D Hom(M, ES).

It is again a finiteGS -module, and if the order ofM is a unit in RK,S , then M D D

Hom(M, Ks�) andM DD is canonically isomorphic toM .The results (2.3), (2.6), and (2.13) combine to show that for allr 2 Z, P r

S(K, M) is thealgebraic and topological dual ofP 2�r

S (K, M D). Therefore there are continuous maps

rD r

S(K, M D)WP rS(K, M D)! H 2�r (GS , M)�

with r the dual of 2�r .

THEOREM 4.10. LetM be a finiteGS -module whose order is a unit inRK,S .(a) The groupsX1

S(K, M) andX2S(K, M D) are finite and there is a canonical nonde-

generate pairingX1

S(K, M)�X2S(K, M D)! Q=Z.

(b) The map 0S(K, M) is injective and 2

S(K, M D) is surjective; forr D 0, 1, 2,

Im(ˇrS(K, M)) D Ker( r

S(K, M D)).

(c) For r � 3, ˇr is a bijection

H r (GS , M)!Yv real

H r (Kv, M).

Consequently, there is an exact sequence of locally compact groups and continuous homo-morphisms

0 ��! H 0(GS , M)ˇ0

��! P 0S(K, M)

0

��! H 2(GS , M D)�??yH 1(GS , M D)�

1

�� P 1S(K, M)

ˇ1

�� H 1(GS , M)??yH 2(GS , M)

ˇ2

��! P 2S(K, M)

2

��! H 0(GS , M D)��! 0.


The groups in this sequence have the following topological properties:

finite compact compactcompact locally compact discretediscrete discrete finite

The finiteness ofX1S(K, M) is contained in (4.9); that ofX2

S(K, M D) will followfrom the existence of the nondegenerate pairing in (a). The vertical arrows in the abovediagram will be defined below; alternatively they can be deduced from the nondegeneratepairings in (a) because the cokernels of 0 and 1 areX2

S(K, M D)� andX1S(K, M D)�

respectively.

EXAMPLE 4.11. (i) For any integerm > 1 and any set of primesS of density greaterthan1=2, X1

S(K, Z=mZ) D 0; consequently,X2S(K,�m) D 0 under the same condition

providedm is a unit inRK,S .(ii) If S omits only finitely many primes ofK andm is a unit inRK,S , thenX1

S(K,�m) D

0 or Z=2Z; consequently,X2S(K, Z=mZ) D 0 or Z=2Z under the same conditions.

To see (i), note that

H 1(GS , Z=mZ)DHom(GS , Z=mZ)

H 1(Kv, Z=mZ) D Hom(Gv, Z=mZ).

Therefore an element ofX1S(K, Z=mZ) corresponds to a cyclic extension ofK in which

all primes ofS split. The Chebotarev density theorem shows that such an extension mustbe trivial whenS has density greater that1=2.

To see (ii), note thatX1S(K,�m) is the kernel ofK�=K�m

!L

v2SK�v =K

�mv , that

is, it is the set of elements ofK� that are localmth powers modulo those that are globalmth powers. This set is described in Artin and Tate 1961, X.1, where the “special case” inwhichX1

S(K,�m) 6D 0 is also determined.

Proof of the main theorem

The proof of the theorem will consist of identifying the exact sequence in the statement ofthe theorem with the ExtGS

(M D ,�)-sequence of

0! ES ! JS ! CS ! 0,

except that, in the number field case, HomGS(M D , JS) and HomGS

(M D , CS) must bereplaced by their quotients byNL=K Hom(M D ,

Qv archL

�w) andNL=K Hom(M D , DL,S)

for any fieldL as in (4.6b).In fact we shall consider more generally a finitely generatedGS -moduleM . In this

case we writeM d for the dual ofM loosely regarded as a group scheme over Spec(RK,S).More precisely, whenM is being regarded as aGS -module, we letM d

D Hom(M, ES).For v =2 S , M is agv-module, and we writeM d

D Hom(M, bOun�v ); for v 2 S , M is a

Gv-module, and we writeM dD Hom(M, Ks�

v ). It will always be clear from the context,which of these three we mean.

4. GLOBAL FIELDS 57

LEMMA 4.12. Let M be a finitely generatedGS -module such that the order ofMtors is aunit in RK,S .

(a) The groupExtrGS(M, ES) D H r (GS , M d), all r � 0.

(b) For v =2 S , H r (gv, M d) D Extrgv(M, bOun�

v ); for r � 2, both groups are0.

PROOF. (a) AsES is divisible by all integers that are units inRK,S , this is a special caseof (0.8).

(b) As bOun�v is divisible by all integers dividing the order ofMtors, this is again a special

case of (0.8). Thegv-modulebOun�v is cohomologically trivial (Serre 1967a, 1.2), and so

an easy generalization to profinite groups of (Serre 1962, IX 6, Thm 11) shows that thereexists a short exact sequence ofgv-modules withI0 andI1 injective. It is obvious fromthis that Extrgv

(M, bOun�v ) D 0 for r � 2.

LEMMA 4.13. In addition to the hypotheses of (4.12), assume either thatM is finite or thatS omits only finitely many primes. Then

HomGS(M, JS) D

Q0

v2SH 0(Gv, M d)

(D P 0S(K, M d) if K is a function field) and

ExtrGS(M, JS) D P r

S(K, M d), r � 1.

PROOF. We consider finite subsetsT of S satisfying the same hypotheses asS relativeto M , namely,T contains all archimedean primes plus those nonarchimedean primes atwhichM is ramified, and the order ofMtors is a unit inRK,T . Let

JF,S�T DQw2T F �

w �Qw2SrT

bO�v .

ThenJS D lim�!F,T

JF,S�T (limit over F andT with F � KT and splittingM ), and so

(0.10) shows that

ExtrGS(M, JS) D lim

�!F,T

ExtrGal(F=K)(M, JF,S�T ).

Since Exts commute with products in the second place (to see this, compute them by takinga projective resolution of the term in the first place), on applying (0.11) we find that

ExtrGF=K(M, JF,S�T ) D

�Qv2T

ExtrGFw=Kv(M, F �

w )

��

Qv2SrT

ExtrGFw=Kv(M, bO�

w)

!.

As bOun�v is cohomologically trivial, (0.9) shows that forv 2 S � T,

ExtrGFw=Kv(M, bO�

Fw) D Extrgv

(M, bOun�v ),

and we have already seen that

Extrgv(M, bOun�

v D H r (gv, M d).


On combining these statements, we find that

ExtrGS(M, JS) D lim

�!F,T

Qv2T

ExtrGFw=Kv(M, F �

w )�

Qv2SrT

H r (gv, M d)

!!.

For r � 1, (0.9) shows that we can replace ExtrGFw=Kv

(M, F �w ) with ExtrGv

(M, K�v,s),

which equalsH r (Gv, M d) by (0.8). Hence

ExtrGS(M, JS) D lim

�!T

Qv2T

H r (Gv, M d)�Q

v2SrT

H r (gv, M d)

!,

which equalsQv2SH 0(Gv, M d) in the case thatr D 0, and equals

Q0

v2SH 1(Gv, M)dfD

P 1S(K, M) for m D 1.

For r � 2, H r (gv, M d) D 0 by (4.12), and so

ExtrGS(M, JS) D lim

�!F

�Lv2S

ExtrGFw=Kv(M, F �

w )

�(limit over all F � Ks, F � K)

DLv2S

lim�!F

ExtrGFw=Kv(M, F �

w )

!.

In the case thatS contains almost all primes, lim�!F

Fw D Ksv and so

lim�!F

ExtrGFw=Kv(M, F �

w ) D ExtrGv(M, Ks�

v ).

In the case thatM is finite, we know that if divides the order ofM , thenS containsall primes lying over . Therefore lim

�!FH 2(Gal(Ks

v=Fw), Ks�v )(`) D lim

�!Br(Fw)(`) D 0,

and the spectral sequence (0.9)

ExtrGal(Fw=Kv)(M, H s(Gal(Ksv=Fw), Ks�

v )) H) ExtrGv(M, Ks�

v )

shows that again lim�!

ExtrGFw=K(M, F �

w ) D ExtrGv(M, Ks�

v ). From (0.8) we know that

ExtrGv(M, Ks�

v ) D H r (Gv, M d) (D H r (Kv, M d)), and so this completes the proof of thelemma.

REMARK 4.14. Without the additional hypotheses, (4.13) is false. For example, letK D Q,S D f1g, and letM D Z. ThenGS D f1g, and so ExtrGS

(Z, JS) D 0 for r > 0, but

P 2S(K, M d) D H 2(Gal(C=R), C�) D 1

2Z=Z.

Now assume thatM is finite. On using (4.12), (4.13), and (4.6) to replace the terms inthe sequence

� � � ! ExtrGS(M D , ES)! ExtrGS

(M D , JS)! ExtrGS(M D , CS)! � � � ,

4. GLOBAL FIELDS 59


0! H 0(GS , M)!Qv2S

H 0(Kv, M)! HomGS(M D , CS)

! H 1(GS , M)! P 1S(K, M)! H 1(GS , M D)�

! H 2(GS , M)! P 2S(K, M)! H 0(GS , M D)�

! H 3(GS , M)!Lv real

H 3(Gv, M)! 0

and isomorphisms

H r (GS , M)�D!

Lv real

H r (Kv, M), r � 4.

This is the required exact sequence except for the first three terms in the number fieldcase and the surjectivity ofP 2

S(K, M D) ! H 0(GS , M)�. But this last map is dual toH 0(GS , M) ! P 0

S(K, M), which is injective. (Note that ifM 6D 0 in the number fieldcase, thenS must contain at least one nonarchimedean prime.) For the first three terms ofthe sequence in the number field case, consider the exact commutative diagram:

Hom(M D ,Qv archL

�v ) �! Hom(M, CS(L))??yNL=K

??yNL=K

0 �!H 0(GS , M) �!Qv2SH 0(Gv, M) �! HomGS

(M D , CS) �! Ker(ˇ1) �! 0??y ??yP 0

S(K, M) �! H 2(GS , M)�??y ??y0 0 .

The map Hom(M D ,Qv archL

�v )! Hom(M D , DS(L)) is always an isomorphism on tor-

sion, and therefore it is an isomorphism in our case. The snake lemma now gives us anexact sequence

0! H 0(GS , M)! P 0S(K, M)! H 2(GS , M D)�

! . . . ,

which completes the proof of the theorem. (An alternative approach is to note that the firsthalf of the sequence can be obtained as the algebraic and topological dual of the secondhalf.)

Consequences

COROLLARY 4.15. If S is finite andM is a finiteGS -module whose order is a unit inRK,S , then the groupsH r (GS , M) are finite for allr.

PROOF. In this case the groupsP rS(K, M) are finite, and so the finiteness ofH 0(GS , M) is

obvious and that ofH 1(GS , M) andH 2(GS , M) follows from the finiteness ofX1S(K, M)

andX2S(K, M).


COROLLARY 4.16. LetM be a finiteGS -module whose order is a unit inRK,S . Then, forany finite subsetT of S omitting at least one finite prime ofS , the map

H 2(GS , M)!Lv2T

H 2(Gv, M)

is surjective. In particular, in the number field case the map

H 2(K, M)!Lv2T

H 2(Kv, M)

is surjective.

PROOF. Let v0 be a finite prime ofK not in T . In order to prove the corollary, it sufficesto show that for any elementa D (av) of P 2

S(K, M), it is possible to modifyav0so as

to get an element in the image ofˇ2. Theorem 4.10 shows that, in the duality betweenP 0

S(K, M D) andP 2S(K, M), the image of 2 is the orthogonal complement of the image

of ˇ0. Let � be the character ofP 0S(K, M D) defined by a, and let�0 be its restriction to

H 0(GS , M D). The mapH 0(GS , M)! H 0(Kv0

, M)

is injective, and every character ofH 0(GS , M D) extends to one ofH 0(Kv0, M D). Choose

such an extension of�0 and leta0v0

be the element ofH 2(Kv0, M) corresponding to it by

duality. When the componentav0of a is replaced byav0

� a0v0

, thena becomes orthogonalto Imˇ0 and is therefore in the image ofˇ2.

COROLLARY 4.17. For any number fieldK,

H 0(GK , Z) D Z,

H 2(GK , Z) D Hom(CK=DK , Q=Z),

H 2r (GK , Z) D (Z=2Z)t for 2r � 4, wheret is the number of real primes ofK, and

H r (GK , Z) D 0 for r odd.

PROOF. The assertions forr � 2 are obvious. According to (1.12),GdfD GK con-

tains an open subgroupU of index 2 having strict cohomological dimension 2. ThereforeH r (G, Z[G=U ]) D H r (U, Z) D 0 for r � 3. Let � generateG=U . The exact sequence

0! Z1C�! Z[G=U ]

1��! Z[G=U ]

� 7 !1! Z! 0

gives rise to isomorphismsH r (G, Z)! H rC2(G, Z) for r � 3. Forr � 4,

H r (G, Z) D H r�1(G, Q=Z)

D lim�!

H r�1(G, 1nZ=Z)

4.10D lim�!

Qv realH

r�1(Kv,1nZ=Z)

DQv realH

r (Kv, Z).

(We applied (4.10) withS the set of all primes ofK.) If r is odd,H r (R, Z) D 0, and if ris even,H r (R, Z) �D Z=2Z, and so this completes the proof.

4. GLOBAL FIELDS 61

COROLLARY 4.18. For any prime that is a unit inRS ,

H r (GS , ES)(`)!Lv real

H r (Gv, K�v,s)(`)

is an isomorphism, allr � 3. In particular, whenr � 3, H r (GS , ES)(`) D 0 if ` or r isodd.

PROOF. From the sequence

0! ES ! K�S !

Lv =2S

Z! 0

(hereS denotes the set of primes ofKS lying over a prime ofS), we get an exact sequence

H 2(GS , ES)! Br(K)!Lv =2S

Br(Kv)

(cf. A.7). Therefore, the mapH 2(GS , ES) !Lv real

Br(Kv) is surjective. The Kummer

sequence

0! �`n ! ES

`n

! ES ! 0

gives us the first row of the next diagram

H 2(GS , ES) ��! H 3(GK ,�`n) ��! H 3(GK , ES)`n ��! 0??ysurj.

??y�

??yLv real

Br(Kv) ��!Lv real

H 3(Gv,�`n) ��!Lv real

H 3(Gv, Ks�v )`n ��! 0,

and the five-lemma proves our assertion forr D 3. One now proceeds by induction, usingthe continuation of the diagram.

REMARK 4.19. LetF be a finite extension ofK contained inKS , and letHS D Gal(KS=F).Then the following diagram is commutative:

0 ��! H 0(GS , M) ��! P 0S(K, M) ��! H 2(GS , M D)�

��! H 1(GS , M) ��! . . .??yRes

??yRes

??yCor�??yRes

0 ��! H 0(HS , M) ��! P 0S(F, M) ��! H 2(HS , M D)�

��! H 1(HS , M) ��! . . . .

This follows from the commutativity of the following diagrams:

� � � ��! ExtrGS(M, ES) ��! ExtrGS

(M, JS) ��! ExtrGS(M, CS) ��! � � �??y ??y ??y

� � � ��! ExtrHS(M, ES) ��! ExtrHS

(M, JS) ��! ExtrHS(M, CS) ��! � � �

andExtrGS

(M, CS) ��! H 2�r (GS , M D)�??y ??yCor�

ExtrHS(M, CS) ��! H 2�r (HS , M D)�.


An explicit description of the pairing betweenX1 and X2

Finally we shall give an explicit description of the pairing

X1S(K, M)�X2

S(K, M D)! Q=Z.

Representa 2 X1S(K, M) and a0

2 X2S(K, M D) by cocycles 2 Z1(GS , M) and

˛02 Z2(GS , M D). Write ˛v and˛0

v for the restrictions of and˛0 to Gv. Then for eachv 2 S , we have a0-cochainˇv and1-cochainˇ0

v such thatdˇv D ˛v anddˇ0v D ˛0

v.The cup-product [ ˛0 2 Z3(GS , ES), and asH 3(GS , ES) has no nonzero elements oforder dividing the[M], there is a2-cochain� (for GS) with coefficients inES such that˛ [ ˛0

D d�. Thend(ˇv [ ˛0v) D d�v D d(˛v [ ˇ

0v) andd(ˇv [ ˇ

0v) D ˛v [ ˇ

0v � ˇv [ ˛

0v,

and so, for eachv, (˛v [ ˇ0v) � �v and(˛0

v [ ˇv) � �v are cocycles representing the sameclass, saycv, in H 2(Gv, Ks�

v ). Setha, a0i D

Pinvv(cv). It is easy to see that this element

is independent of the choices made, and one can show that it is equal to the image of(a, a0)

under the pairing constructed in the proof of the theorem.

Generalization to finitely generated modules

We note that in the course of the proof of (4.10) we have shown the following result.

THEOREM 4.20. Assume thatS omits only finitely many primes ofK, and letM be afinitely generated10 module overGS such that the order ofMtors is a unit inRK,S .

(a) The groupX2S(K, M d) is finite and is dual toX1

S(K, M).(b) There is an exact sequence of continuous homomorphisms

H 1(GS , M)� ��

Q0H 1(Kv, M d) �� H 1(GS , M d)??y

H 2(GS , M d) ��! ˚H 2(Gv, M d) ��! H 0(GS , M)��! 0,

and forr � 3 there are isomorphisms

H r (GS , M d)�D!Qv realH

r (Kv, M d).

(c) In the function field case, the sequence in (b) can be extended by

0! H 0(GS , M d)^!Q0

H 0(Gv, M d)^! H 2(GS , M)�

! . . .

where^ denotes completion with respect to the topology of open subgroups of finiteindex.

PROOF. To obtain (b) and (c), write down the Ext(M,�)-sequence of

0! ES ! JS ! CS ! 0

and use (4.6), (4.12), and (4.13) to replace various of the terms. Part (a) is a restatement ofthe fact that the sequence in (b) is exact atH 2(GS , M d).

10Meaning finitely generated as an abelian group.

5. GLOBAL EULER-POINCARE CHARACTERISTICS 63

COROLLARY 4.21. Let T be a torus overK. If S omits only finitely many primes, thenthere are isomorphisms

H r (GS , T )�D!˚v realH

r (Kv, T )

for all r � 3. In particular H r (GS , Gm) D 0 for all odd r � 3.

PROOF. TakeM D X �(T ) in the theorem.

I do not know to what extent Theorem 4.20 holds withM andM d interchanged, butR. Kottwitz has shown that for any torusT over a number fieldK, andr D 1, 2, there is acanonical nondegenerate pairing of finite groups

Xr (K, T )�X3�r (K, X �(T ))! Q=Z.

For r 6D 1, 2, Xr (K, T ) andXr (K, X �(T )) are zero. HereXr (K, T ) is defined to bethe kernel ofH r (K, T )!

Qall vH

r (Kv, T ). See also (II 4) below.

NOTES. Theorem 4.10 is due to Tate (see Tate 1962 for an announcement with a briefindication of proof). Parts of the theorem were found independently by Poitou (1966,1967). The above proof of (4.10) generalizes that in Tate 1966, which treats only thecase thatS contains all primes ofK. There is also a proof in Haberland 1978 similarlygeneralizing Poitou 1967. Corollaries 4.16 and 4.17 are also due to Tate (cf. Borel andHarder 1978, 1.6, and Serre 1977, 6.4).

Proofs of parts of the results in this section can also be found in Takahashi 1969, Uchida1969, Bashmakov 1972, and Langlands 1983, VII 2.

5 Global Euler-Poincare characteristics

LetK be a global field, and letS be a finite nonempty set of primes including all archimedeanprimes. As in~4, we writeKS for the largest subfield ofKs that is ramified overK onlyat primes inS , GS for Gal(KS=K), andRK,S for the ring ofS -integers

Tv =2SOv. Let M

be a finiteGS -module whose order is a unit inRS . We know from (4.15) that the groupsH r (GS , M) are finite for allr , and we would like to define�(GS , M) to be the alternatingproduct of their orders. However, whenK is a real number field, the cohomology groupswill in general be nonzero for an infinite number of values ofr (see 4.10c), and so this isnot possible. Instead, we abuse notation, and set

�(GS , M) D[H 0(GS , M)][H 2(GS , M)]

[H 1(GS , M)].


THEOREM 5.1. With the above definition,11

�(GS , M) DYv arch

[H 0(Gv, M)]

j[M]jv.

REMARK 5.2. (a) In the function field case, the theorem says simply that�(GS , M) D 1.In the number field case, (2.13c) shows that

[H 0(Gv, M)]

j[M]jvD

[H 1(Gv, M)]

[H 0(Gv, M D)],

and (2.13a) shows that[H 1(Gv, M)] D [H 1(Gv, M D)], which equals[H 0T (Gv, M D)] be-

cause the Herbrand quotient of a finite module is1. Therefore the formula can also bewritten as

�(GS , M) DYv arch

[H 0T (Gv, M D)]

[H 0(Gv, M)].

(b) BecauseS is finite, all groups in the complex in Theorem 4.10 are finite, and so theexactness of the complex implies that

�(GS , M) � �(GS , M D) DQv2S

�(Kv, M) (5.2.1)

where�(Kv, M) D [H 0(Kv, M)][H 1(Kv, M)]�1[H 2(Kv, M)] (notations as in~4). Ac-cording to (2.8),�(Kv, M) D j[M]jv if v is nonarchimedean, and obviously

�(Kv, M) D [H 0(Kv, M)] D [H 0T (Gv, M)]

if v is archimedean. By assumptionj[M]jv D 1 if v =2 S , and so the product formula showsthat Y

v2S

�(Kv, M) DYv arch

[H 0(Kv, M)]

j[M]jv.

Now (2.13c) allows us to rewrite this asYv arch

[H 0T (Gv, M)][H 0

T (Gv, M D)]

[H 0(Gv, M)][H 0(Gv, M D)].

Therefore (5.2.1) is also implied by (5.1), and conversely, in the case thatM � M D ,(5.2.1) implies the theorem.

11Since the notation is confusing, I should give an example. LetM D Z=2Z and letRS D OK [ 12], so that

S consists of the infinite primes and those dividing2. The groupH 0(Gv, M) DM Gv D Z=2Z (it is not theTate cohomology). On the other handj � jv is the normalized valuation (the one that goes into the productformula). Thusj[M]jv D 2 if v is real, andj[M]jv D 4 if v is complex. Thus, the formula says that

�(GS , M) D1

2s

wheres is the number of complex primes.


(c) The theorem can sometimes be useful in computing the order ofH 1(GS , M). Itsays that

[H 1(GS , M)] D [H 0(GS , M)] � [H 2(GS , M)] �Yv arch

j[M]jv=[H0(Gv, M)],

and we know by (4.10) thatH 2(GS , M) fits into an exact sequence

0!X2S(K, M)! H 2(GS , M)!

Lv2S

H 2(Kv, M)! H 0(GS , M D)�! 0.

By duality, [X2S(K, M)] D [X1

S(K, M D)] and[H 2(Kv, M)] D [H 0(Kv, M D)], and sothe theorem is equivalent to the statement

[H 1(GS , M)] D [X1S(K, M D)]

[H 0(GS , M)]

[H 0(GS , M D)]�

Yv2S

[H 0(Kv, M D)] �Yv arch

j[M]jv

[H 0(Gv, M)].

The method of the proof of Theorem 5.1 is similar to that of (2.8). Let'(M) bethe quotient of�(GS , M) by the right hand side of the equation. We have to show that'(M) D 1. The argument in (5.2b) shows that (4.10) implies that'(M)'(M D) D 1, andso in order to prove the theorem for a moduleM , it suffices to show that'(M) D '(M D)

LEMMA 5.3. The map' from the category of finiteGS -modules toQ>0 is additive.

PROOF. Let0!M 0

!M !M 00! 0

be a short exact sequence, and consider the truncated cohomology sequence

0! H 0(GS , M 0)! � � � ! H 4(GS , M 00)! H 5(GS , M 0)0! 0,

whereH 5(GS , M 0)0 is the kernel of the boundary mapH 5(GS , M 0) ! H 5(GS , M).According to (4.10), forr � 3, we can replaceH r (GS ,�) with

P rS(K,�) D

Lv archimedean

H r (Gv,�).

Now [P 3S(K, M)] D [P 4

S(K, M)] because the Herbrand quotient of a finite module is one,and so the sequence leads to the equality

�(M 0) � �(M 00) D �(M) � [P 5S(K, M 0)0],

whereP 5S(K, M 0)0 denotes the kernel of the mapP 5

S(K, M 0) ! P 5S(K, M). Because of

the periodicity of the cohomology of a finite cyclic group,[P 5S(K, M 0)0] D [C ], where

C D Ker(Lv real

H 1(Gv, M 0)!Lv real

H 1(Gv, M)).

From the exact sequence

0!Lv arch

H 0(Gv, M 0)!Lv arch

H 0(Gv, M)!Lv arch

H 0(Gv, M 00)! C ! 0


we see that

[C ] DYv arch

[H 0(Gv, M 0)] � [H 0(Gv, M 00)]

[H 0(Gv, M)].

As [M 0][M 00] D [M], it is now clear that'(M 0)'(M 00) D '(M).

The lemma shows that it suffices to prove the theorem for a moduleM killed by someprimep, and the assumptions onM require thatp be a unit inRS . Choose a finite GaloisextensionL of K, L � KS , that splitsM and contains a primitivepth root of 1 (primitive4th root in the case thatp D 2). Let G be Gal(L=K). We need only consider modulesM

split by L. Note that' defines a homomorphism from the Grothendieck groupRFp(G) to

Q>0. An argument as in the proof of Theorem 2.8 (using 2.10) allows us to replaceK bya larger field, and consequently assume thatG is a cyclic group of order prime top. NotethatL is totally imaginary, and soH r (Gal(KS=L), M) D 0 for r � 3 (by 4.10). It followsthat there is a well-defined homomorphism�0

WRFp(G) ! RFp

(G) sending the class[M]

of M in RFp(G) to

[H 0(Gal(KS=L), M)]� [H 1(Gal(KS=L), M)]C [H 2(Gal(KS=L), M)].

As Hom(�, Fp) is exact, it also defines a functor�W RFp(G)! RFp

(G).

LEMMA 5.4. For a finiteFp[G]-moduleM , there are the following formulas:(a) �0(M D) D [M]� � �0(�p).

(b) [M] � [Fp[G]] D dimFp(M) � [Fp[G]].

PROOF. (a) On tensoring a resolution of�p by Hom(M, Fp), we see that the cup-productpairing arising from

(�,f ) 7! (x 7! �f (x))W�p � Hom(M, Fp)!M D

defines isomorphisms

H r (Gal(KS=L),�p)˝ Hom(M, Fp)! H r (Gal(KS=L), M D)

for all r (recall that Gal(KS=L) acts trivially onM and�p). This gives the formula.(b) Let M0 denoteM regarded as aG-module with the trivial action. As we observed

in ~2,� ˝m 7! � ˝ �m extends to an isomorphismFp[G]˝M0! Fp[G]˝M , and thisgives (b).

On applying both parts of the lemma, we see that

�0(M D) � [Fp[G]]� D [M]� � [Fp[G]]� � �0(�p) D dim(M) � [Fp[G]]� � �0(�p).

Similarly�0(M) � [Fp[G]]� D dim(M) � [Fp[G]]� � �0(�p).

Let � be the homomorphismRFp(G) ! Q>0 sending the class of a moduleN to the

order ofN G. Then� ı �0D �, and so on applying� to the above equalities, we find that

�(M) D �(M D).


Let v be a real prime ofK. If Lw 6D Kv, thenp must be odd, and so[M Gv ] D [MGv],

which equals[(M D)Gv ]. This shows that the factors of'(M) and'(M D) correspondingto v are equal. It is now clear that'(M) D '(M D), and we have already noted that this isimplies that'(M) D 1.

REMARK 5.5. In the function field case there is a completely different approach to thetheorem. LetK D Kks (composite insideKS ), and letH D Gal(KS=K). Let g(K) bethe genus ofK, and lets be the number of primes ofK lying over primes inS . ThenH isan extension of a groupH 0 having2g(K)C s generators and a single well-known relation(the tame fundamental group of the curve overks obtained by omitting the points ofS) bya pro-p group,p D char(K). Using this, or a littleetale cohomology, it is possible to showthatH r (H, M) is finite for all finiteH -modulesM of order prime top (cf. Milne 1980,V 2). Also, it follows from (4.10) thatH r (H, M) D 0 for r > 2. The Hochschild-Serrespectral sequence forH � G reduces to short exact sequences

0! H r�1(H, M)g ! H r (GS , M)! H r (H, M)g! 0

in which g D G=H D Gal(ks=k) D h�i and the two end groups are defined by theexactness of

0! N g! N

��1! N ! Ng ! 0.

It follows from the first set of exact sequences that

�(GS , M) D[H 0(H, M)g] � [H 1(H, M)g] � [H 2(H, M)g]

[H 0(H, M)g] � [H 1(H, M)g] � [H 2(H, M)g]

and from the second that this product is equal to1.

An extension to infiniteS

As we observed above, in the case thatS is finite, all groups in the complex in (4.10) arefinite, and therefore the alternating product of their orders is one. Oesterle (1982/83) showsthat, whenS is infinite, it is possible to define natural Haar measures on the groups in thecomplex, and prove that (in an appropriate sense) the alternating product of the measures isagain one. For example, the measure to take onP 1

S(K, M) is the Haar measure for whichthe compact subgroupH 1(gv, M Iv) (product over all nonarchimedeanv) in has measure1 (note thatH 1(gv, M Iv) D H 1

un(Kv, M) if M is unramified atv). The main result ofOesterle 1982/83 can be stated as follows.

THEOREM 5.6. Let K be a global field, letS be a (possibly infinite) set of primes ofK,and letM be a finiteGS -module. Assume thatS contains all archimedean primes and allprimes for which[M] is not a unit. Relative to the Haar measure onP 1

S(K, M) definedabove, a fundamental domain forP 1

S(K, M) modulo the action of the discrete subgroupH 1(GS , M)=X1

S(K, M) has finite measure

[X1S(K, M)][H 0(GS , M D)]

[X1S(K, M D)][H 0(GS , M)]

Yv archimedean

[H 0(Gv, M)].


PROOF. Suppose first thatS is finite. Then the groups are all finite, and the measure of thefundamental domain in question isQ

[H 1(Kv, M)]Q[H 1(gv, M I )]

[X1S(K, M)]

[H 1(GS , M)].

From (5.2c) we know that this is equal to

[X1S(K, M)][H 0(GS , M D)]

[X1S(K, M D)][H 0(GS , M)]

Yv2S

[H 1(Kv, M)]

[H 1(gv, M I )] � [H 0(Kv, M D)]

Yv archimedean

[H 0(Gv, M)]

j[M]jv.

As[H 1(gv, M I )] D [H 0(gv, M I )] D [H 0(Gv, M)]

for v nonarchimedean (we set it to zero forv archimedean) and[H 0(Kv, M D)] D [H 2(Kv, M)],we see that the middle term isY

v2S

�(Kv, M)�1�

Yv archimedean

[H 0T (Gv, M)].

In (5.2b) we showed thatYv2S

�(Kv, M)�1D

Yv arch

j[M]jv=[H0T (Gv, M)].

This verifies the theorem in this case. For an infinite setS , one chooses a suitably largefinite subsetS 0 of S and shows that the theorem forS is equivalent to the theorem forS 0

(see Oesterle 1982/83,~7).

NOTES. Theorem 5.1 is due to Tate (see Tate 1965/66, 2.2, for the statement together withhints for a proof). Detailed proofs are given in Kazarnovskii 1972 and Haberland 1978,~3. The above proof differs from previous proofs in that it avoids any calculation of thecohomology of�n.

In his original approach to Theorem 4.10, Tate proved it first in the case thatS isfinite by making use of a counting argument involving (presumably) the formula (5.2.1) for�(GS , M)�(GS , M D) in order to show thatX1

S(K, M) andX2S(K, M D) have the same

order. He deduced it for an infiniteS by passing to the limit. (See Tate 1962, p192.)Theorem 5.6 is taken from Oesterle 1982/83.

6 Abelian varieties over global fields

Throughout this sectionK will be a global field, andA will be an abelian variety overK.The letterS will always denote a nonempty set of primes ofK containing all archimedeanprimes and all primes at whichA has bad reduction. We continue to writeKS for the largestsubfield ofKs containingK that is ramified only at primes inS , GS for Gal(KS=K), andRK,S for the subring

Tv =2SOv of K. The letterm is reserved for an integer that is a unit in

RK,S ; thusjmjv D 1 for all v =2 S . For example, it is always permitted to takeS to be the

6. ABELIAN VARIETIES OVER GLOBAL FIELDS 69

set of all primes ofK, and in that casem can be any integer prime to char(K). As usual,we fix an embedding ofKs into Ks

v for each primev of K.

For an abelian groupM , M ^ denotes them-adic completion lim �n

M=mnM . If X isan algebraic group overK, then we often writeH r (GS , X ) for H r (GS , X(KS)) (equalto H r (K, X) Ddf H r (GK , X(Ks)) in the case thatS contains all primes ofK). WhenX is an algebraic group overKv, we setH r (Kv, X ) D H r (Gv, X(Ks

v)) except whenv isarchimedean, in which case we set it equal toH r

T (Gv, X(Ksv)). By H r (�, X(m)) we mean

lim�!n

H r (�, Xmn) and byH r (�, TmX) we mean lim �n

H r (�, Xmn).

The weak Mordell-Weil theorem

The Mordell-Weil theorem says thatA(K) is finitely generated. The first step in its proofis the weak Mordell-Weil theorem: for some integern > 1, A(K)=nA(K) is finite. Weprove a stronger result in (6.2) below.

LEMMA 6.1. Let A andB be abelian varieties overK having good reduction outsideS ,and letf WA ! B be an isogeny whose degree is a unit inRK,S . Write Af for Ker(f ).Then all points inAf (Ks) have their coordinates inKS , and there is an exact sequence

0! Af (KS)! A(KS)f! B(KS)! 0.

In particular, there is an exact sequence

0! Am(KS)! A(KS)m! A(KS)! 0.

PROOF. Let P 2 B(K); its inverse imagef �1(P) in A is a finite subscheme ofA. Weshall show that this finite subscheme splits overKS , which implies thatP lies in the imageof A(KS)! B(KS). WhenP is taken to be zero,f �1(P) is Af , and so this shows thatAf is split overKS , i.e., thatAf (KS) D Af (Ks).

By assumption,A andB extend to abelian schemesA andB over Spec(RK,S). Themapf extends to a finite flat mapf WA ! B which, because its degree is prime to theresidue characteristics ofRK,S , is alsoetale. Our pointP extends to a sectionP of B overSpecRS , andf �1(P) is a finiteetale subscheme ofA over Spec(RS). Any such schemesplits overRS , which implies thatf �1(P) splits, and proves the lemma. (For more detailson such things, see Milne 1986b,~20.)

The lemma yields exact sequences

� � � ! H r (GS , Af )! H r (GS , A)f! H r (GS , B)! � � �

� � � ! H r (GS , Am)! H r (GS , A)m! H r (GS , A)! � � �

PROPOSITION 6.2. (Weak Mordell-Weil theorem) For any integern prime to char(K),A(K)=nA(K) is a finite group.


PROOF. Givenn, we can choose a finite setS of primes ofK satisfying the conditions inthe first paragraph and such thatn is a unit inRK,S . Then (6.1) provides us with an exactsequence

0! An(KS)! A(KS)n! A(KS)! 0.

The cohomology sequence of this gives an injectionA(KS)(n) ,! H 1(GS , An), and wehave seen in (4.15) that this last group is finite.

To deduce the full Mordell-Weil theorem from (6.2), one uses heights (see Lang 1983,V).

The Selmer and Tate-Shafarevich groups

The Tate-Shafarevich groupA classifies the forms ofA for which the Hasse principle fails.The Selmer group gives a computable upper bound for the rank ofA(K). The differencebetween the upper bound and the actual rank is measured by Tate-Shafarevich group.

LEMMA 6.3. Let a be an element ofH 1(K, A). Then for all but finitely many primesv ofK, the image of a inH 1(Kv, A) is zero.

PROOF. As H 1(K, A) is torsion,na D 0 for somen, and asH 1(K, An) ! H 1(K, A)n

is surjective, there is ab 2 H 1(K, An) mapping toa. For almost allv, An(Ks) is anunramifiedGv-module andb maps intoH 1

un(Kv, An) (see 4.8). Thereforea maps intoH 1(gv, A(Kun

v )) for almost allv, but (3.8) shows that this last group is zero unlessv is oneof the finitely many primes at whichA has bad reduction.

TheTate-Shafarevich groupXS(K, A) is defined to be the kernel of

H 1(GS , A)!L

v2SH 1(Kv, A).

TheSelmer groupsSS(K, A)m andSS(K, A, m) are defined by the exact sequences

0! SS(K, A)m! H 1(GS , Am)!L

v2SH 1(Kv, A)

0! SS(K, A, m)! H 1(GS , A(m))!L

v2SH 1(Kv, A)

The second sequence can be obtained by replacingm with mn in the first sequence andpassing to the direct limit. Therefore

SS(K, A, m)! lim�!

n

SS(K, A)mn .

WhenS contains all primes ofK, we drop it from the notation. Thus,

X(K, A) D Ker(H 1(K, A)!Q

all vH1(Kv, A))

S(K, A, m) D Ker(H 1(K, A(m))!Q

all vH1(Kv, A)).

PROPOSITION6.4. There is an exact sequence

0! A(K)(m)! SS(K, A)m!XS(K, A)m! 0.


PROOF. Apply the snake lemma to the diagram

0 0??y ??ySS(K, A)m ��! XS(K, A)m??y ??y

0 ��! A(K)(m) ��! H 1(GS , Am) ��! H 1(GS , A)m ��! 0??y ??y ??y0 ��!

Lv H 1(Kv, A)

D��!

Lv H 1(Kv, A) ��! 0

PROPOSITION6.5. There are exact sequences

0! H 1(GS , A(m))! H 1(K, A(m))!L

v =2SH 1(Kv, A)

0! H 1(GS , A)(m)! H 1(K, A)(m)!L

v =2SH 1(Kv, A)

PROOF. Forv =2 S , there is a commutative diagram

H 1(GS , A(m)) ��! H 1(gv, A(Kunv ))??y ??y

H 1(K, A(m)) ��! H 1(Gv, A(Ksv)).

According to (3.8),H 1(gv, A(Kunv )) D 0, and so the diagram shows that the image of

H 1(GS , A(m)) in H 1(K, A(m)) is contained in the kernel of

H 1(K, A(m))!L

v =2SH 1(Kv, A).

Conversely, leta lie in this kernel. We may assume thata is the image of an elementb of H 1(K, Am) (after possibly replacingm by a power). To prove that the first sequenceis exact, it suffices to show thatb (hence a) is split by a finite extension ofK unramifiedoutsideS . After replacingK by such an extension, we can assume thatAm(K) D Am(Ks)

(because of 6.1). Thenb corresponds to a homomorphismf WGal(Ks=K) ! Am(K),and it remains to show that the subfieldKf of Ks fixed by the kernel off is unramifiedoutsideS . This can be checked locally. Ifv =2 S , then, by assumption, the image ofb inH 1(Kv, Am) maps to zero inH 1(Kv, A). It therefore arises from an elementcv of A(Kv).The closure ofKf in Ks

v is Kv(m�1cv), which is unramified by (6.1).


The exactness of the second exact sequence can be derived from the first. In the diagram

lim�!

A(K)(m) lim�!

A(K)(m)??y ??y0 ��! H 1(GS , A(m)) ��! H 1(K, A(m)) ��! ˚v =2SH 1(Kv, A)??y ??y ??y0 ��! H 1(GS , A)(m) ��! H 1(K, A)(m) ��! ˚v =2SH 1(Kv, A)??y ??y

0 0

the exactness of the bottom row follows from the exactness of the rest of the rest of thediagram (use the snake lemma, for example).

COROLLARY 6.6. For all S andm (as in the first paragraph),

XS(K, A)(m) DX(K, A)(m)

SS(K, A, m) D S(K, A, m).

PROOF. The kernel-cokernel sequence (see 0.24) of the pair of maps

H 1(K, A)(m)ˇ!L

all vH1(Kv, A)(m)

pr!L

v =2SH 1(Kv, A)(m)

is0!X(K, A)(m)! H 1(GS , A)(m)!

Lv2SH 1(Kv, A)(m)! � � � ,

because (6.5) allows us to replace Ker(prıˇ) with H 1(GS , A)(m). This sequence identifiesX(K, A)(m) with XS(K, A)(m). The second equality is proved by replacingH 1(K, A)(m)

in the proof withH 1(K, A(m)).

REMARK 6.7. Recall (4.15) thatH 1(GS , Am(KS)) is finite whenS is finite. Thereforeits subgroupSS(K, A)m is finite whenS is finite, and (6.4) then shows thatXS(K, A)m

is finite. It follows now from (6.6) thatX(K, A)m is finite, and (6.4) in turn shows thatS(K, A)m is finite. Consequently,S(K, A)(m) andX(K, A)(m) are extensions of finitegroups by divisible groups isomorphic to direct sums of copies ofQ`=Z`, ` dividing m. Itis widely conjectured thatX(K, A) is in fact finite.

Definition of the pairings

The main results in this section will concern the continuous homomorphisms

ˇ0WA(K)^

!Qv2SH 0(Kv, A)^ (compact groups)

ˇrWH r (GS , A)(m)!

Lv2SH r (Kv, A)(m), r 6D 0, (discrete groups).


Write XrS(K, A, m) D Ker(ˇr ). ThusX1

S(K, A, m) D XS(K, A)(m), which we haveshown to be independent ofS . We also write

XrS(K, A(m)) D lim

�!n

XrS(K, Amn) D Ker(H r (GS , A(m))!

Qv2SH r (Kv, A(m)))

XrS(K, TmA) D lim

�nXr

S(K, Amn) D lim �

n Ker(H r (GS , Amn)!Qv2SH r (Kv, Amn)).

LEMMA 6.8. For anyr � 2, there is a canonical isomorphism

XrS(K, A, m)

�D!Xr

S(K, A(m)).

PROOF. For eachr � 2 there is an exact commutative diagram:

0 �! H r�1(GS , A)˝Qm=Zm �! H r (GS , A(m)) �! H r (GS , A)(m) �! 0??yˇr�1(A)˝1

??yˇr (A(m))

??yˇr (A)(m)

0 �!Lv2S

H r�1(Kv, A)˝Qm=Zm �!Lv2S

H r (Kv, A(m)) �!L

v2S H r (Kv, A)(m) �! 0.

As r � 1 � 1, the groupsH r�1(GS , A) andH r�1(Kv, A) are both torsion, and so theirtensor products withQm=Zm are both zero. The diagram therefore becomes

H r (GS , A(m))��! H r (GS , A)(m)??yˇr (A(m))

??yˇr (A)(m)Lv2S

H r (Kv, A(m))��!

Lv2S

H r (Kv, A)(m),

from which the result is obvious.

PROPOSITION6.9. For r D 0, 1, 2, there are canonical pairings

h , iWXrS(K, A, m)�X2�r

S (K, At , m)! Q=Z.

PROOF. There is a unique pairing making the diagram

X0S(K, A, m)�X2

S(K, At , m) > Q=Z

jj

X1S(K, TmA)

_

�X2S(K, At(m))

^

�

> Q=Z

commute. Here the bottom pairing is induced by theem-pairing and the pairings in~4,the first vertical arrow is induced by the mapH r (GS , A) ! lim

�H rC1(GS , Am), and the

second vertical map is the isomorphism in (6.8). This defines the pairing in the caser D 0,and the caser D 2 can be treated similarly.

The definition of the pairing in the caser D 1 is more difficult. We will in fact define apairing

h , iWXS(K, A)m �XS(K, At)m! Q=Z.


Since the Tate-Shafarevich groups are independent ofS , we takeS to be the set of allprimes ofK. If � is a global cohomology class, cocycle, or cochain, we write�v for thecorresponding local object.

Let a 2X(K, A)m anda02X(K, At)m. Choose elementsb andb0 of H 1(GK , Am)

andH 1(GK , Atm) mapping to a anda0 respectively. For eachv, a maps to zero inH 1(Kv, A),

and so it is obvious from the diagram

A(Kv) ��! H 1(Kv, Am) ��! H 1(Kv, A) x??A(Kv) ��! H 1(Kv, Am2)

that we can liftbv to an elementbv,1 2 H 1(Gv, Am2) that is in the image ofA(Kv).

Suppose first thata is divisible bym in H 1(GK , A), saya D ma1, and choose an ele-mentb1 2 H 1(GK , Am2) mapping toa1. Thenbv,1�b1,v maps to zero underH 1(Kv, Am2)!

H 1(Kv, Am), and so it is the image of an elementcv in H 1(Kv, Am). We define

ha, a0i D

Pinvv(cv [ b0

v) 2 Q=Z

where the cup-product is induced by theem-pairing Am � Atm ! Gm, and invv is the

canonical mapH 2(Kv, Gm)�D! Q=Z.

In the general case, let be a cocycle representingb, and lift it to a cochain 1 2

C 1(GK , Am2). Choose a cocyclev,1 2 Z1(Gv, Am2) representingbv,1, and a cocycleˇ0 2 Z1(GK , At

m) representingb0. The coboundarydˇ1 of ˇ1 takes values inAm, anddˇ1 [ ˇ

0 represents an element ofH 3(GK , Ks�). But this last group is zero (by 4.18 or4.21), and sodˇ1 [ ˇ

0D d� for some2-cochain�. Now12 (ˇv,1 � ˇ1,v) [ ˇ

0v � �v is a

2-cocycle, and we can define

ha, a0i D

Pinvv((ˇ1,v � ˇ1,v) [ ˇ

0v � "v) 2 Q=Z.

It is not too difficult to check that the pairing is independent of the choices made.

REMARK 6.10. (a) IfB is a second abelian variety overK having good reduction outsideS andf WA! B is an isogeny, then

hf (a), bi D ha,f t(b)i, a 2XrS(K, A, m), b 2X2�r

S (K, Bt , m).

This follows from the fact that the local pairings are functorial.(b) LetD be a divisor onA rational overK, and let'D WA! At be the corresponding

homomorphism sendinga 2 A(Ks) to the class ofDa�D, whereDa is the translateDCa

of D. Thenhc,'D(c)i D 0 for all c 2X1S(K, A, m). See Tate 1962, Thm 3.3. This can be

proved by identifying the pairing defined in (6.9) with that defined in (6.11) below, whichwe check has this property. See also (II 5)

12Poonen suggestsv,1 andˇ1,v should be interchanged.


REMARK 6.11. There is a more geometric description of a pairing on the Tate-Shafarevichgroups, which in the case of elliptic curves reduces to the original definition of Cassels1962, ~3. An elementa of X(K, A) can be represented by a locally trivial principalhomogeneous spaceX over K. Let Ks(X) be the function field ofX ˝K Ks. Thenthe exact sequence

0! Ks�! Ks(X)�

! Q! 0

leads to a commutative diagram

Br(K) ��! H 2(GK , Ks(X)�) ��! H 2(GK , Q) ��! 0??y ??y ??y0 ��!

Lall v

Br(Kv) ��!Lall v

H 2(Gv, Ksv(X)�) ��!

Lall v

H 2(Gv, Q).

The zero at top right comes from the fact thatH 3(GK , Ks�) D 0 (see 4.21). The zero atlower left is a consequence of the local triviality ofX . Indeed, consider an arbitrary smoothvariety Y over a fieldk. The map Br(Y ) ! Br(k(Y )) is injective (Milne 1980, II 2.6).The structure mapY ! Spec(k) induces a map Br(k)! Br(Y ), and any element ofY(k)

defines a section to this map, which is then injective. In our situation, we have a diagram

0 ��! Br(Kv) ��! Br(XKv) ��! Br(XK s

v) ??yinj

??yinj

Br(Kv) ��! Br(Kv(X)) ��! Br(Ksv(X))

from which the claimed injectivity is obvious.The exact sequence

0! Q! Div0(X ˝Ks)! Pic0(X ˝Ks)! 0

yields a cohomology sequence

H 1(GK , Div0(X ˝Ks))! H 1(GK , Pic0(X ˝Ks))! H 2(GK , Q)! � � � .

A trivialization A˝Ks �! X˝Ks determines an isomorphism Pic0(X˝Ks)

�! Pic0(A˝

Ks). Because the trivialization is uniquely determined up to translation by an element ofA(Ks) and translations by elements inA(Ks) act trivially on Pic0(A˝Ks) (Milne 1986b,9.2), the isomorphism is independent of the choice of the trivialization. A similar argumentshows that it is aGK -isomorphism. Therefore the sequence gives a mapH 1(GK , At) !

H 2(GK , Q). Let a02 X(K, At), and letb0 be its image inH 2(GK , Q). Thenb0 lifts to

an element ofH 1(GK , Ks(X)�), and the image of this in H 1(Kv, Ksv(X)�)) lifts to an

element(cv) 2 ˚Br(Kv). Defineha, a0i D

Pinvv(cv) 2 Q=Z. Note that the cokernel of

Br(K)! ˚Br(Kv) is Q=Z, and soha, a0i can also be described as the image ofb0 under

the map defined be the snake lemma. As the principal homogeneous spaceX is uniquelydetermined up to isomorphism bya, this shows thatha, a0i is well-defined.

It is easy to prove that ifX(K, A) is mapped toX(K, At) by means of a polarizationdefined by aK-rational divisor13, then the pairing onX(K, A) is alternating. LetP 2

13This condition was omitted in the original.


X(Ks); then�P D P C˛(�) where(˛(�)) is a cocycle representinga. The map'D sendsQ 2 A(Ks) to the class ofDQ � D in Pic0(A), and soa0 is represented by the cocycle(˛0(�)) 2 Z1(GK , At), where˛0(�) is represented by the divisorE� D D˛(�) �D. Now

use the trivializationQ 7! P CQWA˝Ks �! X ˝Ks to identify Pic0(A) with Pic0(X).

Then one sees immediately that(˛0�), regarded as a crossed homomorphism into Pic0(XK s),

lifts to a crossed homomorphism into Div0(XK s). Therefore the image ofa0 in H 2(GK , Q)

is zero, and soha, a0i D 0.

We leave it to the reader to check that this pairing agrees with that defined in (6.9).

REMARK 6.12. WhenA is the Jacobian of a curveX over K, there is yet another de-scription of a pairing on the Tate-Shafarevich groups. WriteS for the canonical mapDiv0(X ˝Ks)! A(Ks).

Let a 2 X(K, A) be represented by 2 Z1(GK , A(Ks)), and let˛v D dˇv withˇv 2 Z0(Gv, A(Ks

v)). Write

˛ D S(a), a 2 C 1(GK , Div0(XK s))

ˇv D S(bv), bv 2 C 0(Gv, Div0(XK sv)).

Thenav D dbv C (fv) in C 1(Gv, Div0(XK sv)), wherefv 2 C 1(Gv, Ks

v(X)�). Moreoverda D (f ), f 2 Z2(GK , Ks(X)�). Let a0 be a second element ofX(K, A) and definea0,b0v, f

0v , andf 0 as fora. Define

� D f 0[ a� f [ a0

2 C 3(GK , Ks�).

Thend� D 0, and so, asH 3(GK , Ks�) D 0, � D d" for some" 2 C 2(GK , Ks�). Set14

v D f0v [ av � b0

v [ Resv(f )� "v 2 C 2(GK , Ks�v )

where[ denotes the cup-product pairing induced by(h, c) 7! h(c). Then v is a2-cocyclerepresenting acv 2 Br(Kv), and we let

ha, a0i D

Pinvv(cv) 2 Q=Z.

One shows without serious difficulty that the choices in the construction can be made sothatha, a0

i is defined and that it is independent of the choices.15

14In the original, the" was omitted (cf. the proof of 6.9). As Bjorn Poonen pointed out to me, without it, v need not be a cocycle. For more on the pairing, see:

Gonzalez-Aviles, Cristian D. Brauer groups and Tate-Shafarevich groups. J. Math. Sci. Univ. Tokyo 10(2003), no. 2, 391–419.

Poonen, Bjorn; Stoll, Michael. The Cassels-Tate pairing on polarized abelian varieties. Ann. of Math. (2)150 (1999), no. 3, 1109–1149.

The first reference includes a proof that it coincides with the pairing in (6.9).15The original claimed without proof that the pairing is always alternating, but this is not true — see the

paper of Poonen and Stoll mentioned in an earlier footnote.


The main theorem

We shall need to consider the duals of the mapsˇr . Recall (3.4, 3.6, and 3.7) thatH r (Kv, A)

is dual toH 1�r (Kv, At), except possibly for thep-components in characteristicp. There-fore there exist maps,

1WL

v2SH 1(Kv, A)(m)! At(K)�(m) (discrete groups)

0WQv2SH 0(Kv, A)^

! H 1(GS , At)(m)� (compact groups)

such that r (A) D ˇ1�r (At)�.

THEOREM 6.13. (a) The left and right kernels of the canonical pairing

X1(K, A)(m)�X1(K, At)(m)! Q=Z

are the divisible subgroups ofX1(K, A)(m)) andX1(K, At)(m).(b) The following statements are equivalent:

(i) X1(K, A)(m) is finite;(ii) Im(ˇ0) D Ker( 0) and the pairing betweenX0

S(K, A, m) andX2S(K, At , m)

is nondegenerate.(c) The map 2 is surjective with kernel the divisible subgroup ofH 2(GS , A)(m), and

for r > 2, ˇr is an isomorphism

H r (GS , A)(m)�!

Lv real

H r (Kv, A)(m).

REMARK 6.14. (a) Much of the above theorem is summarized by the following statement:if X(K, A)(m) andX(K, At)(m) are finite, then there is an exact sequence with contin-uous mapsMv real

H 2(Kv, At)(m)� �> H 2(GS , At)(m)� > H 0(GS , A)^

H 1(GS , A)(m) < H 1(GS , At)(m)� < 0 Y

v2S

H 0(Kv, A)^

_ˇ0

Mv2S

H 1(Kv, A)(m)

_ˇ1

1

> H 0(GS , At)^� > H 2(GS , A)(m)ˇ2

>>Mv real

H 2(Kv, A)(m)

The unnamed arrows exist because of the nondegeneracy of the pairings defined in (6.9).(b) We shall see in (6.23) and (6.24) below that ifS contains almost all primes ofK,

thenˇ0 andˇ2 are both injective. In this case, the above sequence can be shortened to afour-term sequence:

0!X(K, A)(m)! H 1(GS , A)(m)!L

v2SH 1(Kv, A)(m)! H 0(GS , At)^�! 0.

In particular, whenS contains all primes ofK and the Tate-Shafarevich groups are finite,then the dual of the exact sequence

0!X(K, A)! H 1(GS , A)!L

v2SH 1(Kv, A)! B! 0.


is an exact sequence except possibly for thep-components in characteristicp 6D 0. HereBis defined to be the cokernel of the preceding map. In the second sequence,H 0(Kv, At) D

At(Kv) unlessv is archimedean, in which case it equals the quotient ofA(Kv) by itsidentity component (see 3.7). The termAt(K)^ is the profinite completion ofAt(K),which is equal to its closure in

QH 0(Kv, At) (see 6.23b).

(c) If X(K, A) is finite, then so also isX(K, At). To see this note that there is anintegerm and mapsf WA! At andgWAt

! A such thatfg D m D gf . Therefore thereare mapsX(f )WX(K, A) ! X(K, At) andX(g)WX(K, At) ! X(K, A) whosecomposites are both multiplication bym. It follows that the kernel ofX(g) is contained inX(K, At)m. Whenm is prime to the characteristic, we observed in (6.7) thatX(K, At)m

is finite, and an elementary proof of the same statement form a power of char(K) can befound in Milne 1970b (see also Chapter III). Hence the kernel ofX(g) is finite, and thisshows thatX(At) is finite.

We begin the proof of (6.13) with part (c). As we saw in the proof of (6.8), whenr � 2,there is a commutative diagram

H r (GS , A(m))��! H r (GS , A)(m)??yˇr (A(m))

??yˇr (A)(m)Lv2S

H r (Kv, A(m))��!

Lv2S

H r (Kv, A)(m),

AsH r (Kv, A) is zero whenr � 2 andv is nonarchimedean (see 3.2), the sum at lower rightneeds to be taken only over the real primes. Whenr > 2, ˇr (A(m)) is an isomorphism(see 4.10c), and sor (A) is an isomorphism. Whenr D 2, (4.10) shows that the cokernelof ˇ2(A(m)) is

lim�!

Atmn(K)�

D

�lim �

Atmn(K)

��

D�TmAt(K)

��,

which is zero becauseAt(K) is finitely generated (by the Mordell-Weil theorem). Considerthe diagram

H 2(GS , Am) �! H 2(GS , A)(m)m�! H 2(GS , A)(m) �! H 3(GS , Am)??yˇ2(Am)

??yˇ2(A)

??yˇ2(A)

??yˇ3(Am)Lv real

H 2(Kv, Am) �!Lv real

H 2(Kv, A)(m) �!Lv real

H 2(Kv, A)(m) �!Lv real

H 3(Kv, Am)

The first vertical arrow is surjective by (4.16). We have just shown thatˇ2(A) is sur-jective, and we know that 3(Am) is an isomorphism by (4.10c). Therefore we have asurjective map of complexes, and so the sequence of kernels is exact, from which it fol-lows that Ker(ˇ2(A)) is divisible bym. On repeating this argument withm replaced bymn we find that Ker(ˇ2(A)) is divisible by all powers ofm. Since it obviously containsH 2(GS , A)(m)div, this shows that it equalsH 2(GS , A)(m)div. This completes the proof ofpart (c).


We next prove part (b). Letv 2 S , and consider the diagram

0 ��! A(K)(m) ��! H 1(GS , Am) ��! H 1(GS , A)m ��! 0??y ??y ??y0 ��! H 0(Kv, A)(m)

��! H 1(Kv, Am) ��! H 1(Kv, A)m ��! 0.

On replacingm with mn and passing to the inverse limit, and then replacing the bottomrow by the restricted product over allv in S , we obtain an exact commutative diagram

0 ��! A(K)^ ��! H 1(GS , TmA) ��! TmH 1(GS , A) ��! 0??yˇ0

??y ??yˇ1

0 ��!Qv2S

H 0(Kv, A)^��!

Q0

v2S H 1(Kv, TmA) ��!Q0

v2S TmH 1(Kv, A) ��! 0.

The snake lemma now gives an exact sequence

0!X0S(K, A)!X1

S(K, TmA)! TmX(K, A)!

QH 0(Kv, A)^

Im(ˇ0)

�! (H 1(GS , At(m))�.

Here we have used (4.10) to identify the cokernel of the middle vertical map with a sub-group of

lim �

H 1(GS , Atmn)�D (lim�!

H 1(GS , Atmn))�

D H 1(GS , At(m))�.

Consider the maps

Qv2S

H 0(Kv, A)^ 0

�! (H 1(GS , At)(m)�)�0

�! H 1(GS , At(m))�

the second of which is the dual ofH 1(GS , At(m)) � H 1(GS , At)(m) and is thereforeinjective; consequently, Ker(�0

ı 0) D Ker( 0). The composite�0ı 0 is the composite

of the projection Qv2SH 0(Kv, A)^

!H 0(Kv, A)^

Im(ˇ0)

with �. Since Im(ˇ0) goes to zero under�0ı 0, we see that it must also be mapped to zero

by 0, that is, 0ıˇ0D 0 (without any assumptions). We also see that Ker( 0) D Im(ˇ0)

if and only if � is injective, which is equivalent toX1S(K, TmA) ! TmX(K, A) being

surjective.Consider on the other hand the first part

0!X0S(K, A)!X1

S(K, TmA)! TmX(K, A)

of the above exact sequence and the isomorphism

X2S(K, A)(m)

� X2

S(K, A(m))


in (6.8). Clearly the duality betweenX1S(K, TmA) andX2

S(K, A(m)) arising from (4.10)induces a duality betweenX0(K, A) andX2(K, A)(m) if and only if the mapX1

S(K, TmA)!

TmX(K, A) is zero.On combining the conclusions of the last two paragraphs, we find that the following

two statements are equivalent:(*) X0

S(K, A) andX2S(K, At) are dual and Im(ˇ0) D Ker( 0)

(**) X1S(K, TmA)! TmX(K, A) is both surjective and zero.

Clearly (**) is equivalent toTmX(K, A) being zero, butTmX(K, A) D 0 if and only ifthem-divisible subgroup ofX(K, A)(m) is zero, in which case the group is finite. Thisproves the equivalence of statements (i) and (ii) in (b).

In preparing for the proof of (a), we shall need a series of lemmas. Since the statementof (a) does not involveS , we can choose it to be any set we wish provided it satisfies theconditions in the first paragraph of this section. We always take it to be finite.

LEMMA 6.15. Leta 2L

v2S H 1(Kv, Am), and consider the pairingPh , ivW

QH 1(Kv, Am)�

LH 1(Kv, At

m)! Q=Z, hav, a0viv D invv(av [ a0

v).

Thenha, a0i D 0 for all a0 in the image ofSS(K, At)m !

Lv2S H 1(Kv, At

m) if andonly if a can be writtena D a1 C a2 with a1 and a2 in the images of

QH 0(Kv, A) !Q

H 1(Kv, Am) andH 1(GS , Am)!Q

H 1(Kv, Am) respectively.

PROOF. The dual of the diagram

˚H 1(Kv, Atm)

I@@

@ˇ1

˚H 1(Kv, At)

_

< H 1(GS , Atm) < SS(K, At)m < 0

is QH 1(Kv, Am)

@@

@ 1

RQH 0(Kv, A)

^

> H 1(GS , Atm)� > (SS(K, At)m)� > 0.

Let a 2Q

H 1(Kv, Am). If a maps to zero in(SS(K, At)m)�, then 1(a) is the image ofan elementb in H 0(Kv, A). Let a1 denote the image ofb in H 1(Kv, Am); thena � a1 isin the kernel of 1. But according to (4.10), the kernel of 1 is the image ofH 1(GS , Am),and soa� a1 D a2 for somea2 2 H 1(GS , Am).

LEMMA 6.16. LetX0(A) be the subgroup ofX(K, A) of elements that become divisiblebym in H 1(GS , A). Then there is an exact sequence

0!X0(K, A)!X(K, A)!X2S(K, Am).


PROOF. Consider

X(K, A)m��! X(K, A) ��! X2

S(K, Am)??y ??y ??yH 1(GS , A)

m��! H 1(GS , A) ��! H 2(GS , Am)??y ??y ??yL

v2S

H 1(Kv, A)m��!

Lv2S

H 1(Kv, A) ��!Lv2S

H 1(Kv, Am).

An elementa in X(K, A) maps to zero inX2S(K, A) if and only if it maps to zero in

H 2(GS , Am), and this occurs if and only if its image inH 1(GS , A) is divisible bym.

LEMMA 6.17. Let a 2X0(K, A). Thena 2 mX(K, A) if and only ifha, a0i D 0 for all

a0 2X(K, At)m.

PROOF. If a D ma0 with a0 2X(K, A), then

ha, a0i D hma0, a0

i D ha0, ma0i D 0

for all a02 X(K, At)m. Conversely, assume thata satisfies the second condition, and

let a1 2 H 1(GS , A) be such thatma1 D a; we have to show thata1 can be modified tolie in X(K, A). Choose a finite setS satisfying the conditions at the start of this sectionand containing allv for which a1,v 6D 0. If a1 is replaced by its sum with an element ofH r (GS , A), then it is still zero outsideS (see the proof of 6.5). Defineb1, bv,1, andcv asin (6.9); thusb1 2 H 1(GS , Am2) and maps toa1, bv,1 2 H 1(Kv, Am2) and maps tobv,andcv 2 H 1(Kv, Am) and maps tobv,1 � b1,v. We shall show that there is an elementb0 2 H 1(GS , Am) such that

b0,v � cv � �b1,v mod A(Kv)(m)

for all v. This will complete the proof, because thena1 C a0, with a0 the image ofb0 inH 1(GS , A), lies inX(K, A) and is such thatm(a1 � a0) D ma1 D a.

According to (6.14), an elementb0 will exist if and only ifPvhcv, b0

vi D 0 for allb0 in SS(K, A)m. But, by definition of the pairing on the Tate-Shafarevich groups (6.9),Pvhcv, b0

vi D ha, a0i wherea0 is the image ofb0 in X(K, At)m, and our assumption on a

is that this last term is zero.

We now complete the proof of part (a) of the theorem. Note that because the groups aretorsion, the pairing must kill the divisible subgroups. Consider the diagram

0 ��! X0(K, A)=mX(K, A) ��! X(K, A)=mX(K, A) ��! X2S(K, Am)??y ??y ??y�

0 ��! (X(K, At)m= Im X1(K, Atm))�

��! (X(K, At)m)�

��! X1S(K, At

m)�.


The top row comes from (6.16) and the bottom row is the dual of an obvious sequence

X1S(K, At

m)!X(K, At)m! Coker! 0.

The first vertical map is the injection given by Lemma 6.17, and the third vertical arrowis the isomorphism of (4.10). A diagram chase now shows that the middle vertical ar-row is also injective. On passing to the limit over powers ofm, we obtain an injectionX(K, A)^

! X(K, At)(m)�. But X(K, A)^D X(K, A)=X(K, A)m�div, and so the

left kernel in the pairing

X(K, A)(m)�X(K, At)(m)! Q=Z

is X(K, A)m�div. Therefore

[X(K, A)=X(K, A)m�div] � [X(K, At)=X(K, At)m�div].

Since this holds for allA, we also have

[X(K, At)=X(K, At)m�div] � [X(K, At t)=X(K, At t)m�div]

D [X(K, A)]=X(K, A)m�div].

It follows that all these orders are equal, and therefore that the right kernel isX(K, At)m�div.

REMARK 6.18. If A has dimension one andm is prime, thenX2S(K, Am) D 0 (see 9.6),

and soX0(K, A)m DX(K, A)m (see 6.17). Therefore in this case it is significantly easierboth to define the pairing on the Tate-Shafarevich groups and to prove its nondegeneracy.

Complements in the case thatS contains almost all primes

We shall now show how a theorem of Serre (1964/71) can be used to improve some of theseresults whenS omits only finitely many primes.

PROPOSITION6.19. Let m be an integer prime to char(K), and letG be the image ofGal(Ks=K) in TmA. Then the groupH 1(G, TmA) is finite.

PROOF. Whenm is prime, this is proved in Serre 1964/71, II 2, and the result for a com-positem follows immediately.

We give a second proof of (6.19) based on a theorem of Bogomolov and a lemma ofSah. Note that (6.1) shows that the action ofGK on TmA factors throughGS . Also that,becauseTmA is a Zm-module,Zm is a subring of End(TmA) andZ�

m is a subgroup ofAut(TmA).

LEMMA 6.20. For any prime 6D char(K), the image ofGS in Aut(TÀ) contains an opensubgroup ofZ�

`.

PROOF. Theorem 3 of Bogomolov 1981 shows that (at least whenK is a number field),for any prime , the Lie algebra of the image ofGK in Aut(TÀ) contains the scalars. Thisimplies that the image ofGK is open inZ�

`.


LEMMA 6.21. Let G be a profinite group andM a G-module. For any element� of thecentre ofG, H r (G, M) is annihilated byx 7! �x � x.

PROOF. We first allow� to be any element ofG, not necessarily a central element. Themaps

g 7! �g��1WG ! G, m 7! ��1mWM !M,

are compatible, and so define automorphisms˛rWH r (G, M)! H r (G, M). According to

(0.15),˛r is the identity map. In the case that� central, one sees by looking on cochainsthat˛r is the map induced by theG-homomorphism��1WM !M . Consequently,� actsas the identity map onH r (G, M), as claimed by the lemma.

We now (re-)prove 6.19. It follows from (6.20) that there is an integeri and an element� 2 G such that�x D (ì

� 1)x, all x 2 TÀ. Now (6.21) shows thatiH 1(G, TÀ) D 0.Corollary 4.15 implies thatH 1(G, A`n) is finite for all n, and so the inverse limit of theexact sequences

H 1(G, A`n)ì

! H 1(G, A`nCi )! H 1(G, Aì )

is an exact sequence

H 1(G, TÀ)0! H 1(G, TÀ)! H 1(G, Aì ),

and soH 1(G, TÀ) is a subgroup of the finite groupH 1(G, Aì ).

PROPOSITION6.22. If S omits only finitely many primes ofK, then the map

H 1(GS , TmA)!Qv2SH 1(Kv, TmA)

is injective.

PROOF. Write X1S(K, TmA) for the kernel of the map in the statement of the proposition.

Then there is an exact commutative diagram

0 ��! H 1(G, TmA) ��! H 1(GS , TmA) ��! H 1(G 0, TmA)??y ??y ??y0 ��!

Lv2S

H 1(Gv, TmA) ��!Lv2S

H 1(Gv, TmA) ��!Lv2S

H 1(G 0v, TmA)

in which G and Gv are the images ofG and Gv in Aut(TmA) and G 0 and G 0v are the

kernels ofG � G and Gv � Gv. The two right hand groups consist of continuoushomomorphisms, and so the Chebotarev density theorem shows that the right hand verticalmap is injective. It follows that the subgroupX1

S(K, TmA) of H 1(GS , TmA) is containedin H 1(G, TmA), and is therefore torsion. SinceTmH 1(GS , A) is torsion free, the sequence

0! A(K)^! H 1(GS , TmA)! TmH 1(GS , A)

now shows that any elementc of X1S(K, TmA) is in A(K)^. But for any nonarchimedean

primev, the mapA(K)^! A(Kv)^ is injective on torsion points, and soc D 0.


COROLLARY 6.23. AssumeS omits only finitely many primes.(a) There is an injection

TmX1(K, A)!Qv2SH 0(Kv, A)^=A(K)^.

(b) There is a sequence of injective maps

A(K)^! lim �

SS(K, A)m!Qv2SH 0(Kv, A)^.

In particular, X0S(K, A) D 0. The kernel ofA(K) ! A(K)^ is the subgroup of

elements with finite order prime tom.

PROOF. Consider the diagram

0 ��! A(K)^��! H 1(GS , TmA)

a��! TmH 1(GS , A) ��! 0??y ??yc

??yb

0 ��!Qv2SH 0(Kv, A)^

��!Qv2SH 1(Kv, TmA)

d��!

Qv2STmH 1(Kv, A) ��! 0.

The vertical arrow markedc is injective, and that markedb has kernelTmX(K, A). There-fore part (a) follows from the snake lemma. The first map in part (b) is the inclusionKer(a) ,! Ker(b ı a). The second is the injection Ker(d ı c) ,! Ker(d).

COROLLARY 6.24. LetS be as in the proposition. The map

H 2(GS , A)!Lv real

H 2(Kv, A)

is an isomorphism; in particular,X2S(K, A) D 0.

PROOF. We have a commutative diagram

0 ��! H 1(GS , A)˝Qm=Zm ��! H 2(GS , A(m)) ��! H 2(GS , A)(m) ��! 0??y ??y ??y0 ��! ˚H 1(Kv, A)˝Qm=Zm ��! ˚H 2(Kv, A(m)) ��! ˚H 2(Kv, A)(m) ��! 0.

BecauseH 1(Kv, A) is torsion, its tensor product withQm=Zm is zero. Therefore a nonzeroelement ofX2

S(K, A)(m) would give rise to a nonzero element ofX2S(K, A(m)), but this

group is dual toX1S(K, TmAt), which the proposition shows to be zero. Therefore the

map is injective, and it was shown to be surjective in (6.13c).

REMARK 6.25. Note that (6.23b) solves the congruence subgroup problem for subgroupsof A(K) of index prime to the characteristic ofK: any such subgroup contains a subgroupdefined by congruence conditions. (In fact, that was Serre’s purpose in proving (6.19).)

On combining the above results with Theorem 6.13, we obtain the following theorem.

THEOREM 6.26. Assume thatS omits only finitely many primes ofK.


(a) The left and right kernels of the canonical pairing

X(K, A)(m)�X(K, At)(m)! Q=Z

are the divisible subgroups ofX(K, A)(m) andX(K, At)(m).

(b) The Tate Shafarevich groupX(K, A)(m) is finite if and only ifIm(ˇ0) D Ker( 0),in which case there is an exact sequence

0!X(K, A)(m)! H 1(GS , A)(m)!Mv2S

H 1(Kv, A)(m)! At(K)^�! 0.

(c) The groupsXrS(K, A, m) are zero forr 6D 1, and forr � 2, ˇr is an isomorphism

H r (GS , A)�D�!

Mv real

H r (Kv, A)(m).

REMARK 6.27. The Tate-Shafarevich group is not known16 to be finite for a single abelianvariety over a number field. However, there are numerous examples where it has beenshown that some componentX(K, A)(m) is finite. The first examples of abelian varietiesover global fields known to have finite Tate-Shafarevich groups are to be found in Milne1967 and Milne 1968. There it is shown that, for constant abelian varieties over a functionfield K, the Tate-Shafarevich group is finite17 and has the order predicted by the conjectureof Birch and Swinnerton-Dyer (see the next section for a statement of the conjecture; anabelian variety over a function fieldK is constantif it is obtained by base change from anabelian variety over the field of constants ofK). See also Milne 1975, where (among otherthings) it is shown that the same conjecture is true for the elliptic curve

Y 2D X(X � 1)(X � T )

overk(T ), k finite.

NOTES. Theorem 6.13 was proved by Cassels in the case of elliptic curves (Cassels 1962,1964) and by Tate in the general case (announcement Tate 1962). So far as I know, nocomplete proof of it has been published before.18 The survey article Bashmakov 1972contains proofs of parts of it, and Wake 1986 shows how to deduce (6.22), (6.23), and (6.24)from (6.19); both works have been helpful in the writing of this section in the absence ofTate’s original proofs.19 20

16Only a few months after the book was sent to the publisher, Rubin proved that the Tate-Shafarevichgroups of some elliptic curves overQ with complex multiplication are finite, and not long after that Kolyvaginproved similar results for some modular elliptic curves.

17Of course, this implies the finiteness of the Tate-Shafarevich group of an abelian variety that becomesconstant over a finite extension of the ground field.

18In fact, before the publication of the original version of this book, no proof of Theorem 6.13 was avail-able.

19David Harari and Tamas Szamuely have shown that the global duality theorems for tori and abelian va-rieties can be combined to give a duality theorem for one-motives (arXiv:math.NT/0304480, April 30,2003).

20This section should be rewritten in terms of generalized Selmer groups.


7 An application to the conjecture of Birch and Swinnerton-Dyer

The results of the preceding two sections will be applied to show that the conjecture ofBirch and Swinnerton-Dyer, as generalized to abelian varieties by Tate, is compatiblewith isogenies (except possibly for isogenies whose degree is divisible by the character-istic of K). We begin by reviewing the statement of the conjecture in Tate 1965/66,~1.Throughout,A andB will be abelian varieties of dimensiond over a global fieldK, andG D Gal(Ks=K).

L-series.

Let v be a nonarchimedean prime ofK, and letk(v) be the corresponding residue field.If A has good reduction atv, then it gives rise to an abelian varietyA(v) overk(v). Thecharacteristic polynomial of the Frobenius endomorphism ofA(v) is a polynomialPv(T )

of degree2d with coefficients inZ such that, when we factor it asPv(T ) DQ

i(1� aiT ),then

Qi(1 � am

i ) is the number of points onA(v) with coordinates in the finite field ofdegreem overk(v) (see, for example, Milne 1986b,~19). It can be described also in termsof VÀ Ddf Q` ˝ TÀ. Let Dv � Iv be the decomposition and inertia groups atv, and letFrv be the Frobenius element ofDv=Iv. Then (6.1) shows thatIv acts trivially onTÀ,and it is known (ibid.) that

Pv(A, T ) D det(1� (Frv)T jVÀ), ` 6D char(k).

For any finite setS of primes ofK including the archimedean primes and those whereA has bad reduction, we define theL-seriesLS(s, A) by the formula

LS(s, A) DQv =2SPv(A, Nv�s)�1

whereNv D [k(v)]. Because the inverse rootsai of Pv(T ) have absolute valueq1=2, theproduct is dominated by�K (s � 1=2)2d , and it therefore converges for<(s) > 3=2. It iswidely conjectured thatLS(s, A) can be analytically continued to a meromorphic functionon the whole complex plane. This is known in the function field case, but in the numberfield case it has been verified only for modular elliptic curves,21 abelian varieties withpotential complex multiplication, and some other abelian varieties.

Let! be a nonzero global differentiald -form onA. As� (A,˝dA) has dimension1,! is

uniquely determined up to multiplication by an element ofK�. For each nonarchimedeanprimev of K, let�v be the Haar measure onKv for whichOv has measure1, and for eacharchimedean prime, take�v to be the usual Lebesgue measure onKv. With these choices,we have�(cU ) D jcjv�v(U ) for anyc 2 K� and compactU � Kv. Just as a differentialon a manifold and a measure onR define a measure on the manifold, so do! and�v definea measure onA(Kv), and we set

�v(A,!) D

ZA(Kv)

j!jv�dv

21Hence all elliptic curves defined overQ!

7. APPLICATION TO THE CONJECTURE B-S/D 87

(see Weil 1961). Let� be the measureQ�v on the adele ringAK of K, and setj�j DR

AK=K�. For any finite setS of primes ofK including all archimedean primes and those

nonarchimedean primes for whichA has bad reduction or such that! does not reduce to anonzero differentiald -form onA(v), we define

L�S(s, A) D LS(s, A)

j�jdQv2S�v(A,!)

.

This function is independent of the choice of!; if !0D c! is a second differentiald -form

on A having good reduction outsideS , thenc must be a unit at all primes outsideS , andso the product formula shows thatQ

v2S�v(A,!0) DQv2S�v(A,!).

The functionL�S(s, A) depends on the choice ofS , but its asymptotic behaviour ass ap-

proaches1 does not, because ifv is a prime at whichA and! have good reduction atv,then it is known that�v(A,!) D [A(k(v))]=(Nv)d (ibid., 2.2.5), and it is easy to see thatthis equalsPv(A, Nv�1).

Heights

The logarithmic height of a pointx D (x0 W ... W xm) in Pm(K) is defined by

h(x) D log

�Qall v

max0�i�m

fjxijvg

�.

The product formula shows that this is independent of the representation ofx. Let D be avery ample divisor onA. After replacingD with D C (�1)�D, we may assume thatD islinearly equivalent to(�1)�D. Let f WA ! Pn be the embedding defined byD, and fora 2 A(K), let'D(a) be the point inAt(K) represented by the divisor(D C a)�D. Thenthere is a unique bi-additive pairing

h , iWAt(K)�A(K)! R

such thath'D(a), ai C 2h(f (a)) is bounded onA(K). The discriminant of the pairing isknown to be nonzero. The pairing is functorial in the sense that iff WA! B is an isogeny,then the diagram

At(K)� A(K) > R

jj

Bt(K)

^

f t

� B(K)_f

> Rcommutes. (See Lang 1983, Chapter V.)

Statement.

In order to state the conjecture of Birch and Swinnerton-Dyer we need to assume that thefollowing two conjectures hold forA:


(a) the functionLS(s, A) has an analytic continuation to a neighbourhood of 1;(b) the Tate-Shafarevich groupX(K, A) of A is finite.

The conjecture then asserts:

lims!1

L�S(s, A)

(s � 1)rD

[X(K, A)] � j detha0i , aj ij�

At(K)WP

Za0i

�� (A(K)W

PZai)

(B-S/D)

wherer is the common rank ofA(K) andAt(K), and(a0i)1�i�r and(ai)1�i�r are families

of elements ofAt(K) andA(K) that are linearly independent overZ.

LEMMA 7.1. Let A andB be isogenous abelian varieties over a global fieldK, and letSbe a finite set of primes including all archimedean primes and all primes at whichA or B

has bad reduction.(a) The functionsLS(s, A) andLS(s, B) are equal. In particular, if one function can be

continued to a neighbourhood ofs D 1, then so also can the other.(b) Assume that the isogeny has degree prime to the char(K). If one ofX(A) of X(B)

is finite, then so also is the other.

PROOF. (a) An isogenyA! B defines an isomorphismVÀ�! V`B, and so the polyno-

mialsPv(T ) are the same forA and forB.

(b) Let f WA ! B be the isogeny, and letAf be the kernel off . EnlargeS so thatdeg(f ) is a unit inRK,S . Then (6.1) gives us an exact sequence

� � � ! H 1(GS , Af )! H 1(GS , A)f! H 1(GS , B)! � � � .

According to (4.15),H 1(GS , Af ) is finite, and so the kernels off WH 1(GS , A)! H 1(GS , B)

anda fortiori X(f )WX(K, A)!X(K, B) are finite. Therefore ifX(K, B) is finite, soalso isX(K, A), and the reverse implication follows by the same argument from the factthere exists an isogenygWB ! A such thatg ı f D deg(f ).

Before stating the main theorem of this section, it is convenient to make another defini-tion. If f WX ! Y is a homomorphism of abelian groups with finite kernel and cokernel,we define

z(f ) D[Ker(f )]

[Coker(f )].

LEMMA 7.2. (a) If X andY are finite, thenz(f ) D [X ]=[Y ].

(b) Consider maps of abelian groupsXf! Y

g! Z; if any two ofz(f ), z(g), and

z(g ı f ) are defined, then so also is the third, andz(g ı f ) D z(g)z(f ).

(c) If X � D (0! X 0! � � � ! X n! 0) is a complex of finite groups, thenQ[X r ](�1)r

DQ

[H r (X �)](�1)r

.

(d) If f �WX � ! Y � is a map of exact sequences of finite length, andz(f r ) is definedfor all r , thenz(f r )(�1)r

D 1.

PROOF. Part (b) is obvious from the kernel-cokernel sequence of the two maps. Part (d) isobvious from the snake lemma whenX � andY � are short exact sequences, and the generalcase reduces to that case. The remaining statements are even easier.


THEOREM 7.3. Assume that the abelian varietiesA andB are isogenous by an isogeny ofdegree prime to the char(K). If the conjecture of Birch and Swinnerton-Dyer is true forone ofA or B, then it is true for both.

PROOF. We assume that the conjecture is true forB and prove that it is then true forA. Letf WA! B be an isogeny of degree prime to the characteristic ofK, and letf t WBt ! At

be the dual isogeny. Choose an element!B 2 � (B,˝dB=K ), and let!A be its inverse

imagef �!B on A. Fix a finite setS of primes ofK including all archimedean primes, allprimes at whichA or B has bad reduction, all primes whose residue characteristic dividesthe degree off , and all primes at which!B or !A does not reduce to a nonzero globaldifferential form. Finally choose linearly independent families of elements(ai)1�i�r ofA(K) and(b0

i)1�i�r of Bt(K), wherer is the common rank of the groups ofK-rationalpoints on the four abelian varieties, and letbi D f (ai) anda0

i D ft(b0

i). Then(a0i)1�i�r

and(bi)1�i�r are linearly independent families of elements ofAt(K) andB(K). The proofwill proceed by comparing the corresponding terms in the conjectured formulas forA andfor B.

The functoriality of the height pairings shows that

hf t(b0j ), aii D hb

0j ,f (ai)i,

and this can be rewritten asha0

j , aii D hb0j , bii.

Therefore,detha0

j , aii D dethb0j , bii.

The diagram

0 ��!P

Zai ��! A(K) ��! A(K)=P

Zai ��! 0??y�

??yf (K)

??y0 ��!

PZbi ��! B(K) ��! B(K)=˙Zbi ��! 0

and its analogue forf t , we see that

z(f (K)) D

�A(K)W

PZai

��B(K)W

PZbi

� , z(f t(K)) D

�Bt (K)W

PZb0

i

��At (K)W

PZa0

i

� .We have seen in (7.1) and (6.14c) that the finiteness ofX(B) implies that ofX(A),

X(At), andX (Bt), and so (6.13a) shows that the two pairings in the following diagramare nondegenerate,

X(A)�X(At) > Q=Z

jj

X(B)_X(f )

�X(Bt)

^

X(f t )

> Q=ZTherefore,[CokerX(f )] D [KerX(f t)], and so we have equalities

[X(A)]

[X(B)]D z(X (f )) D [KerX(f )]

[KerX(f t )].


Finally, consider the mapf (Kv)WA(Kv) ! B(Kv). By definition!A D f�!B, and

so�v(U,!A) D �v(f U,!B) for any subsetU of A(Kv) that is mapped injectively intoB(Kv). Therefore,

�v(A(Kv),!A) D [Kerf (Kv)] � �v(f (A(Kv)),!B).

Since�v(f (A(Kv)),!B) D [Coker(f (Kv))]

�1� �v(B(Kv)),!B),

we see thatz(f (Kv)) D �v(A,!A)=�v(B,!B),

and soL�(s,A)

L�(s,B)D

Q�v(B,!B)Q�v(A,!A)

DQv2S

z(f (Kv))�1.

On combining all the boxed formulas, we find that to prove the theorem it suffices to showthat Q

v2S

z(f (Kv)) D[KerX(f t)]

[KerX(f )]

z(f (K))

z(f t(K))(7.3.1)

Consider the commutative diagram

0 ��! H 0(GS , M) ��!Lv2S

H 0(Kv, M)??yH 2(GS , M D)�??y

0 ��! Coker(f (K)) ��! H 1(GS , M) ��! H 1(GS , A)f ��! 0??y'0

??y' ??y'00

0 ��!Lv2S

Coker(f (Kv)) ��!Lv2S

H 1(Kv, M) ��!Lv2S

H 1(Kv, A)f ��! 0??y 0

??y ??y 00

0 ��! H 1(GS , Bt)�f t ��! H 1(GS , M D)�

��! (Coker(f t(K))��! 0

in which the rows are extracted from the cohomology sequences of

0!M ! A(KS)f! B(KS)! 0

0!M ! A(Ksv)

f! B(Ks

v)! 0

0!M D! Bt(KS)

f t

! At(KS)! 0

respectively, and middle column is part of the exact sequence in Theorem 4.10. Therows are exact. The duality betweenB(Kv) andH 1(Kv, Bt) induces a duality betweenB(Kv)=fA(Kv) andH 1(Kv, Bt)f t , and the map 0 is the dual of the composite

H 1(GS , Bt)f t !L

v2SH 1(Kv, Bt)f t

�!L

v2S (B(Kv)=fA(Kv))� .


The map 00 is the dual of the composite

At(K)=f tBt(K)!L

v2SAt(Kv)=ftBt(Kv)

�!L

v2S(H 1(Kv, A)f )�.

The two outside columns need not be exact, but it is clear from the diagram that they arecomplexes.

The serpent lemma and a small diagram chase give us an exact sequence

0! Ker(' 0)! Ker(')! Ker(' 00)! Ker( 0)= Im(' 0)! 0.

As Ker(' 00) D Ker(X(f )), we obtain the formula

[Ker' 0]

[Ker']

[KerX(f )]

[Ker 0= Im' 0]D 1.

From the first column, we get (using (7.2c) and that Coker( 0) D (KerX(f t))�)

[Coker(f (K))]Qv2S [Coker(f (Kv))]

[H 1(GS , Bt)f t ] D[Ker' 0]

[Ker 0=Im' 0][KerX(f t)].

From the third row, we get

1 D[H 1(GS , Bt)f t ]

[H 1(GS , M D)][Coker

�f t(K)

�].

From the middle column we get (using thatH 0(GS , M) D Kerf (K), . . . )

1 D[Kerf (K)]Q[Kerf (Kv)]

Yv arch

[H 0(Kv, M)]

[H 0T (Kv, M)]

[H 2(GS , M D)]

[Ker'].

Finally, we have the obvious equality

[Kerf t(K)] D [H 0(GS , M D)].

On multiplying these five equalities together, we find thatYv2S

z(f (Kv)) D[KerX(f t)]

[KerX(f )]

z(f (K))

z(f t(K))� �(GS , M D) �

Yv arch

[H 0(Kv, M)]

[H 0T (Kv, M)]

.

Theorem 5.1 (in the form (5.2a)) shows that the product of the last two terms on the rightof the equation is1, and so this completes the proof of the theorem.

REMARK 7.4. Since in the number field case the conjecture of Birch and Swinnerton-Dyeris not known22 for a single abelian variety, it is worth pointing out that the above argumentsapply to them-primary components of the groups involved: ifX(K, A)(m) is finite andhas the order predicted by the conjecture, then the same is true of any abelian varietyisogenous toA.

22As (foot)noted earlier, this is no longer the case.


REMARK 7.5. We mention two results of a similar (but simpler) nature to (7.3).Let A be an abelian variety over a finite separable extensionF of the global fieldK.

ThenA gives rise to an abelian varietyA� overK by restriction of scalars. The conjectureof Birch and Swinnerton-Dyer holds forA overF if and only if it holds forA� overK (seeMilne 1972, Thm 1).

Let A be an abelian variety over a number fieldK, and assume that it acquires complexmultiplication overF , and thatF is the smallest extension ofK for which this is true.Under certain hypotheses onA, it is known that the conjecture of Birch and Swinnerton-Dyer holds forA over K if and only if it holds forAF over F (ibid. Corollary to Thm3).

NOTES. For elliptic curves, Theorem 7.3 was proved by Cassels (1965). The general casewas proved by Tate (announcement Tate 1965/66, Theorem 2.1). The above proof wasexplained to me by Tate in 1967.

8 Abelian class field theory, in the sense of Langlands

Abelian class field theory for a global fieldK defines a reciprocity map recK WCK !

Gal(Ks=K)ab that classifies the finite abelian extensions ofK. Dually, one can regardit as associating a character� ı recK of CK with each (abelian) character� of Gal(Ks=K)

of finite order; the correspondence is such that theL-series of� and� ı recK are equal. Itis this second interpretation that generalizes to the nonabelian situation. For any reductivegroupG over a local or global fieldK, Langlands has conjectured that it is possible toassociate an automorphic representation ofG with each “admissible” homomorphism ofthe Weil groupWK of K (Weil-Deligne group in the case of a local field) into a certaincomplex groupLG; theL-series of the automorphic representation is to equal that of theWeil-group representation. In the case thatG D Gm, the correspondence is simply thatnoted above. For a general reductive group, the conjecture is difficult even to state since itrequires a knowledge of representation theory over adele groups (see Borel 1979). For atorus however the statement of the conjecture is simple, and we shall prove it in this case.First we prove a duality theorem (8.6), and then we explain the relation of the theorem toLanglands’s conjectural class field theory. In contrast to the rest of these notes, in this sec-tion we shall consider cohomology groupsH r (G, M) in which G is not a profinite group.The symbolH r (G, M) will denote the group constructed without regard for topologies,andH r

cts(G, M) will denote the group defined using continuous cochains. As usual, whenG is finite,H r

T (G, M), r 2 Z, denotes the Tate group. For a topological groupM ,M �D Homcts(M, Q=Z) D group of characters ofM of finite order;

M uD Homcts(M, R=Z) D group of characters ofM (the Pontryagin dual ofM)I

M 0 D Homcts(M, C=Z) D Homcts(M, C�) D group of generalized characters ofM I

M |D Hom(M, C�) D group of generalized (not necessarily continuous) characters

of M . WhenM is discrete,M 0DM |.

As usual, whenK is a global field, we writeCK for the idele class group ofK. In orderto be able to give uniform statements, we sometimes writeCK for K� whenK is a localfield.

8. ABELIAN CLASS FIELD THEORY, IN THE SENSE OF LANGLANDS 93

Weil groups

First we need to define theWeil group of a local or global fieldK. This is a triple(WK ,', (rF )) comprising a topological groupWK , a continuous homomorphism'WWK !

Gal(Ks=K) with dense image, and a family of isomorphismsrF WCF

�! W ab

F , one for eachfinite extensionF � Ks of K, whereWF D '�1(GF ). (Here, as always,W ab

K is thequotient ofWK by theclosureW c

K of its commutator subgroup.)For any finite extensionF of K, defineWF=K D WK=W c

F ; then, ifF is Galois overK,there is an exact sequence

0! CF ! WF=K ! GF=K ! 0

whose class inH 2(GF=K , CF ) is the canonical class (that is, the element denoted byuGF=K

in the second paragraph of~1). The topology onWF=K is such thatCF receives its usualtopology and is an open subgroup ofWF=K . The full Weil groupWK is equal to the inverselimit lim �

WF=K (as a topological group).

EXAMPLE 8.1. (a) LetK be a nonarchimedean local field. The Weil groupWK is the densesubgroup ofGK consisting of elements that act as an integral multiple of the Frobeniusautomorphism on the residue field. It therefore contains the inertia subgroupIK of GK ,and the quotientWK=IK is Z. The topology onWK is that for whichIK receives theprofinite topology and is an open subgroup ofWK . The map' is the inclusion map, andrF

is the unique isomorphismF �! W ab

F such thatrF followed by' is the reciprocity map.(b) Let K be an archimedean local field. IfK D C, thenWK is C�, ' is the trivial

mapC�!Gal(C=C), andrK is the identity map. IfK is real, thenWK D Ks�

t jKs�

(disjoint union) with the rulesj2D �1 andjzj�1

D z (complex conjugate). The map'sendsKs� to 1 andj to the nontrivial element ofGK . The maprK s is the identity map,andrK is characterized by

rK (�1) D jW cK

rK (x) D x12 W c

K for x 2 K, x > 0.

(c) LetK be a function field in one variable over a finite field. The Weil groupWK is thedense subgroup of Gal(Ks=K) of elements that act as an integral multiple of the Frobeniusautomorphism on the algebraic closure of the field of constants. It therefore contains thegeometric Galois groupGKks D Gal(Ks=Kks) � GK , and the quotient ofW by GKks isZ. The topology onWK is that for whichGKks receives the profinite topology and is anopen subgroup ofWK . The map' is the inclusion map, andrF is the unique isomorphismCF ! W ab

F such thatrF followed by' is the reciprocity map.(d) Let K be an algebraic number field. Only in this case, which of course is the most

important, is there no explicit description of the Weil group. It is constructed as the inverselimit of the extensions corresponding to the canonical classesuGF=K

(see Artin and Tate1961, XV, where the Weil group is constructed for any class formation, or Tate 1979).

Let K be a global field. For each primev of K, it is possible to construct a commutative


diagram

WKv

'v

��! GKv??y ??yWK

'��! GK ,

(see Tate 1979, 1.6.1). We shall assume in the following that one such diagram has beenselected for eachv.

Some cohomology

We regard the cohomology and homology groups as being constructed using the standardcomplexes. For example,H r (G, M) D H r (C �(G, M)) whereC �(G, M) consists ofmaps[g1, . . . , gr ] 7! ˛(g1, . . . , gr )WGr

!M . WhenG is finite, the groupsH �1T (G, M)

andH 0T (G, M) are determined by the exact sequence

0! H �1T (G, M)!MG

NG

�!M G! H 0

T (G, M)! 0.

LEMMA 8.2. Let G be a finite group, and letQ be an abelian group regarded as aG-module with trivial action. IfQ is divisible, then for allG-modulesM , the cup-productpairing

H r�1T (G, Hom(M, Q))�H �r

T (G, M)! H �1T (G, Q) � Q

induces an isomorphism

H r�1T (G, Hom(M, Q))! Hom(H �r

T (G, M), Q)

for all r .

PROOF. This is proved in Cartan and Eilenberg 1956, XII 6.4.

Let (G, C ) be a class formation, and letG be the quotient ofG by an open normalsubgroupH . The pairing

(f, c) 7! f (c ˝�)W (C H˝M)|

� C H!M |

and the canonical classu 2 H 2(G, C H ) define maps

a 7! a [ uWH rT (G, (C H

˝M)|)! H rC2T (G, M |).

LEMMA 8.3. For all finitely generated torsion-freeG-modulesM and all r , the map

� [ uWH rT (G, (C H

˝M)|)! H rC2T (G, M |)

is an isomorphism.


PROOF. The diagram

H rT (G, (C H

˝M)|)�H �r�1T (G, C H

˝M)) > H �1T (G, C�)� C�

jj

H rC2T (G, M |)

_�[u

� H �r�3T (G, M)

^

u[�

! H �1T (G, C�)� C�

commutes because of the associativity of cup products:

(a [ u) [ b D a [ (u [ b), a 2 H rT (G, (C H

˝M)|), b 2 H �r�3T (G, M).

The two pairings are nondegenerate by (8.2), and the second vertical map is an isomorphismby virtue of the Tate-Nakayama theorem (0.2). It follows that the first vertical map is anisomorphism.

Note that

(C H˝M)| df

D Hom(C H˝M, C�) D Hom(C H , Hom(M, C�)) D Hom(C H , M |).

Therefore, the isomorphism in the above lemma can also be written

� [ uWH rT (G, Hom(C H , M |))

��! H rC2

T (G, M |).

Let0! C H

! W ! G ! 1

be an exact sequence of groups corresponding to the canonical classu 2 H 2(G, C H ). ForanyW -moduleM , the Hochschild-Serre spectral sequence gives an exact sequence

0! H 1(G, M C )Inf�! H 1(W , M)

Res�! H 1(C H , M)G �

�! H 2(G, M C ).

The map� (the transgression) has the following explicit description: leta 2 H 1(C H , M)G,and choose a1-cocycle˛ representing it; extend to a 1-cochainˇ on W ; thendˇ is a2-cocycle onG, and the class it represents is�(a).

LEMMA 8.4. If C H acts trivially onM , then the transgression

� WH 0(G, Hom(C H , M))! H 2(G, M)

is the negative of the map� [ u induced by the pairing

Hom(C H , M)� C H!M.

PROOF. WriteW DS

C Hwg (disjoint union of right cosets), and letwgwg0 D (g, g0)wgg0 .Then( (g, g0)) is a2-cocycle representingu. Let ˛ 2 HomG(C H , M), and define byˇ(cwg) D ˛(c), c 2 CH . Then

dˇ(g, g0)dfD dˇ(wg,wg0)

D gˇ(wg0)� ˇ(wgwg0)C ˇ(wg)

D 0� ˛( (g, g0))C 0

D �˛( (g, g0)),

which equals�(˛ [ )(g, g0). Therefore�(˛) D �˛ [ u.


The duality theorem

Let K be a local or global field (we could in fact work abstractly with any class formation),and letF be a finite Galois extension ofK. Let M be a finitely generated torsion-freeGF=K -module. Then

M 0 dfD Homcts(M, C�) D Hom(M, C�) DM |

are againGF=K -modules. We shall use the notationM 0 when we wish to emphasize thatM 0 has a topology. We frequently regard these groups asWF=K -modules.

Write WF=K DFwgCF (disjoint union of left cosets). For any homomorphism

˛WCF !M , the map Cor(˛)WWF=K !M | such that

(Cor(˛))(w) DP

g2G

wg˛(w�1g wwg0), wwg0 � wg modCF 0

is a cocycle, and so we have a map

CorWH 1(CF , M |)! H 1(WF=K , M |),

called thecorestrictionmap. It is independent of the choice of coset representatives (seeSerre 1962, VII 7, or Weiss 1969, p81). It is clearly continuous, and so maps continuoushomomorphisms to continuous cocycles.

LEMMA 8.5. The corestriction mapCorWH 1(CF , M |)! H 1(WF=K , M |) factors throughH 1(CF , M |)G, G D GF=K .

PROOF. Let ˛ 2 Hom(CF , M |) andh 2 G. Then(h˛)(w) D wh˛(w�1hwwh) (this is the

definition), and so

Cor(h˛)(w) DP

gwgwh˛(w�1h w�1

g wwg0wh)

whereg0 is such thatwwg0 � wg modCF . The family(wgwh)g2G is also a set of cosetrepresentatives forCF in WF=K , andw(wg0wh) � (wgwh) modCF . Therefore the class ofCor(h˛) is the same as that of Cor(˛), and so Cor((h� 1)˛) D 0 in H 1(WF=K , M |).

THEOREM 8.6. For any finitely generated torsion-freeGF=K -moduleM , the corestrictionmap defines an isomorphism

Homcts(CF , M 0)GF=K

��! H 1

cts(WF=K , M 0).

PROOF. Throughout the proof, we writeG for GF=K . We shall first prove that the core-striction map defines an isomorphism

Hom(CF , M |)G ! H 1(WF=K , M |)

and then show (in Lemma 8.9) that it makes continuous homomorphisms correspond tocontinuous cocycles.


Consider the diagram in which we have writtenC for CF (8.6.1)

0 �!H �1T (G, Hom(C, M |)) �! Hom(C, M |)G

NG

��! Hom(C, M |)G�!H 0

T (G, Hom(C, M)|)??y�

??yCor

??yid

??y�

0 �! H 1(G, M |) �!H 1(WF=K , M |) �! H 1(CF , M |)G�! H 2(G, M |).

The top row is the sequence defining the Tate cohomology groups of Hom(CF , M |). Thebottom row can be deduced from the Hochschild-Serre spectral sequence or else can beconstructed in an elementary fashion. The two isomorphisms are those in Lemma 8.3. Thethird square (anti-) commutes because of (8.4). We shall prove in the next two lemmasthat the first two squares in the diagram commute. The five-lemma will then show thatCorWHom(CF , M |)G ! H 1(WF=K , M |) is an isomorphism. Finally Lemma 8.9 willcomplete the proof.

LEMMA 8.7. The first square in (8.6.1) commutes.

PROOF. We first show that Cor maps an element ofH �1T (G, Hom(CF , M |)) into the sub-

groupH 1(G, M |) of H 1(WF=K , M |). Let ˛ be a homomorphismCF ! M |, and letc 2 CF andw 2 W . Then

(Cor(˛))(cw) DX

g

wg˛(w�1g cwgw

�1g wwg0)

D

Xg

(wg˛)(c)C (Cor˛)(w)

D (N˛)(c)C (Cor˛)(w).

Therefore, ifN˛ D 0 (that is,˛ 2 H �1T (G, Hom(CF , M |))), then (Cor )(w) depends

only on the class ofw in G, and so Cor(˛) arises by inflation from an element ofH 1(G, M |).It remains to show that the restriction of Cor toH �1

T (G, Hom(CF , M |)) is �u. Notethat

(Cor(˛))(h) DX

g

g(˛(w�1g whwh�1g))

D

Xg

(g˛)(whwh�1gw�1g )

D

Xg

(g˛)(u(h, h�1g)).

To obtain the middle equality, we have used that

w�1g whwh�1g D c H) whwh�1g D wgc D (gc)wg H) whwh�1gw

�1g D gc

and thatg(˛(c)) D (g˛)(gc).

It is difficult to give explicit descriptions of cup-products when both negative and posi-tive indices are involved. We shall use the exact sequence

0! IG ! Z[G]! Z! 0


to shift the problem. This sequence remains exact when tensored withM | and Hom(CF , M |),and the boundary maps in the resulting cohomology sequences give the horizontal maps inthe following diagram:

H �1T (G, Hom(CF , M |))

d�1

��! H 0T (G, Hom(CF , M |)˝ IG)??y�[u

??y�[u˝1

H 1(G, M |)d1

��! H 2(G, M |˝ IG).

Both boundary maps are isomorphisms, and� [ u is the unique map making the diagramcommute. If we can show that the diagram still commutes when this map is replaced withCor, we will have proved the lemma. This we do by an ugly cocycle calculation.

Note first thatd�1 andd1 have the following descriptions:

d�1(˛) D N(˛˝1) D N(˛˝1)�(N˛)˝1 DX

g

g˛˝(g�1),˛ 2 Hom(CF , M |), N˛ D 0

d1(ˇ)(g1, g2) D g1ˇ(g2)˝ (g1 � 1), ˇ 2 Z1(G, M |), g1, g2 2 G.

If ˛ 2 Hom(CF , M |) hasN˛ D 0, then

(d1ı Cor˛)(g1, g2) D

Xg2G

g1 � (g˛)(u(g2, g�12 g))˝ (g1 � 1)

and(d�1˛ [ (u˝ 1))(g1, g2) D

Xg

(g˛)(u(g1, g2))˝ (g � 1).

An element ofM |˝ IG can be written uniquely in the formmg ˝ (g � 1). Therefore

a general element ofC 1(G, M |˝ IG) is of the form

PFg ˝ (g � 1) with Fg a map

G !M |, and a coboundary inB2(G, M |˝ IG) can be written

d(X

g

Fg˝(g � 1))(g1, g2) DXg

(g1 � Fg�11 g(g2)� Fg(g1g2)C Fg(g1))˝ (g � 1)�

Xg

g1.Fg(g2)˝ (g1 � 1).

In obtaining the second expression, we have used that

g1(P

Fg(g2)˝ (g � 1)) DP

g1 � Fg(g2)˝ (g1g � g1)

DP

g1 � Fg(g2)˝ (g1g � 1)�P

g1 � Fg(g2)˝ (g1 � 1)

DP

g1 � Fg�11 g(g2)˝ (g � 1)�

Pg1 � Fg(g2)˝ (g1 � 1).

PutFg(g2) D (g˛)(u(g2, g�1

2 g)I

then

(dP

Fg ˝ (g � 1)� (d�1˛) [ (u˝ 1)C (d1ı CorD ˛)(g1, g2)P

(g˛)(g1u(g2, g�12 g�1

1 g) � u(g1g2, g�12 g�1

1 g)�1� u(g1, g�1

1 g) � u(g1, g2)�1)˝ (g � 1).


When we puth D g�12 g�1

1 g, this becomes

(g˛)(g1u(g2, h) � u(g1g2, h)�1� u(g1, g2h) � u(g1, g2)�1)˝ (g � 1),

and each term in the sum is zero becauseu is a2-cocycle. Therefore

dP

(Fg ˝ (g � 1)) D (d�1˛) [ (u˝ 1)� (d1ı Cor(˛)),

which completes the proof of the lemma.

LEMMA 8.8. The composite

H 1(CF , M |)Cor�! H 1(WF=K , M)| Res

�! H 1(CF , M |)

is equal to the normNG. Hence the second square in (8.6.1) commutes.

PROOF. For˛ 2 Z1(CF , M |) andw 2 WF=K , Cor(˛)(w) DP

g wg˛(w�1g wwg0). When

w 2 CF , this becomes Cor(˛)(w) DP

g2G g˛(g�1wg) D (NG˛)(w).

LEMMA 8.9. Let ˛ 2 Hom(CF , M 0); then˛ 2 Homcts(CF , M 0) if and only ifCor(˛) 2

Z1cts(WF=K , M 0).

PROOF. Clearly˛ 2 Z1(WF=K , M 0) is continuous if and only if its restriction toCF iscontinuous. Therefore (8.8) shows that it suffices to prove that a homomorphismf WCF !

M 0 is continuous if and only ifNGf is continuous. SinceNG is continuous, there is acommutative diagram

Homcts(CF , M 0)N��! Homcts(CF , M 0)G

��! H 0T (G, Homcts(CF , M 0)) ��! 0??y ??y ??y

Hom(CF , M 0)N��! Hom(CF , M 0)G

��! H 0T (G, Hom(CF , M 0)) ��! 0,

from which it follows that it suffices to show that

H 0T (G, Homcts(CF 0 , M 0))! H 0

T (G, Hom(CF , M 0))

is injective. In fact, following Labesse 1984, we shall prove much more.

LEMMA 8.10. For all r , the map

H rT (G, Homcts(CF , M 0))! H r

T (G, Hom(CF , M 0))

is an isomorphism.

PROOF. We consider the cases separately.(a) K local archimedean.The only nontrivial case hasK D R andF D C. Here

CK D C�, and we shall use the exponential sequence

0! Z! C! C�! 0.


From it we get exact sequences

0! Hom(C�, M 0)! Hom(C, M 0)! Hom(Z, M 0)! 0

(becauseM 0 is divisible) and

0! Homcts(C�, M 0)! Homcts(C, M 0)! Homcts(Z, M 0)! 0

(becauseM 0 is a connected commutative Lie group). The groups Hom(C, M 0) and Homcts(C, M 0)

are uniquely divisible, and so are cohomologically trivial. Therefore, we can replaceCF inthe statement of the lemma withZ, but then it becomes obvious becauseZ is discrete.

(b) K local nonarchimedean.Here CF D F �. From Serre 1967a, 1.4, we knowthatF � contains a cohomologically trivial open subgroupV ; moreoverV contains a fun-damental system(Vn) of neighbourhoods of zero with eachVn an open subgroup, suchthat V =Vn is cohomologically trivial. (For example, whenF is unramified overK, itis possible to takeV D O�

K .) Now, becauseM 0 is divisible, Serre 1962, IX 6, Thm9, shows that Hom(V , M 0) and Hom(V =Vn, M 0) are also cohomologically trivial. AsHomcts(V , M) D lim

�!Hom(V =Vn, M 0), we see that it also is cohomologically trivial.

A similar argument to the above, using the sequence

0! V ! F �! F �=V ! 0

shows that it suffices to prove the lemma withCF replaced withF �=V , but this group isdiscrete.

(c) K global. HereCF is the idele class group. DefineV � CK to beQ

Vv whereVv D bO�

v for v a nonarchimedean prime that is unramified inF andVv is a subgroupas considered in (b) for the remaining nonarchimedean primes. This group has similarproperties to the groupV in (b). It therefore suffices to prove the lemma withCF replacedwith CF=V . In the function field case this is discrete, and in the number field case it is anextension of a finite group byR� (with trivial action). In the first case the lemma is obvious,and in the second the exponential again shows thatR� is the quotient of a uniquely divisiblegroup by a discrete group.

This completes the proof of the theorem.

COROLLARY 8.11. Let K be a global or local field, and letM be a finitely generatedtorsion-freeGK -module. There is a canonical isomorphism

((C ˝M)GK )0�D�! H 1

cts(WK , M 0)

whereC DS

CF .

PROOF. Let F be a finite Galois extension ofK splitting M . Any continuous crossedhomomorphismf WWK ! M 0 restricts to a continuous homomorphism onWF . BecauseM is commutative,f must be trivial onW c

F and so factors throughWK=W cF Ddf WF=K .

Consequently, the inflation mapH 1cts(WF=K , M 0)! H 1

cts(WK , M 0) is bijective.Next note that Hom(CF , M 0) D (CF ˝M)0. I claim that the canonical map(CF ˝

M)0G ! ((CF˝M)G)0 is an isomorphism. Note that this is obviously so when0 is replaced


with |, because(CF ˝M)G is the maximal subgroup ofCF ˝M on whichG acts trivially,and so((CF ˝M)G)| is the maximal quotient group on whichG acts trivially, that is, it is((CF ˝M)|)G. The diagram

0 ��! H �1(G, (CF ˝M)0) ��! (CF ˝M)0G ��! ((CF ˝M)0)G??y�

??y ??yinjective

0 ��! H �1(G, (CF ˝M)|) ��! (CF ˝M)|G ��! ((CF ˝M)|)G

shows that the middle vertical arrow is injective. Now the diagram

(CF ˝M)0G ��! ((CF ˝M)G)0??yinjective

??y�

(CF ˝M)|G ��! ((CF ˝M)G)|

shows that(CF ˝M)0G ! ((CF ˝M)G)0 is injective, which proves the claim since the

map is obviously surjective. To complete the proof of the corollary, note that

Hom(CF , M 0)GF=KD (CF ˝M)0

GF=KD ((CF ˝M)GF=K )0

D ((C ˝M)GK )0,

and so the corollary simply restates the theorem.

REMARK 8.12. (a) After making the obvious changes, the above arguments show that thereis a canonical isomorphism

((C ˝M)GK )u! H 1

cts(WK , M u).

(b) ReplaceM in (8.11) with its linear dual. ThenH 1cts(WK , M 0) becomesH 1

cts(WK , M˝

C=Z) and((C ˝M)GK )0 becomes HomGK(M, C )0. On the other hand, (4.10) gives us

an isomorphismH 2cts(GK , M) ! HomGK

(M, C )�, andH 2cts(GK , M) D H 1

cts(GK , M ˝

Q=Z) D H 1cts(WK , M ˝Q=Z). These results and their relations can be summarized as fol-

lows: for any finitely generated torsion-freeG-moduleM , there is a commutative diagram

H 1cts(WK , M ˝ (Q=Z)) �> H 1

cts(WK , M ˝ (R=Z)) �> H 1cts(WK , M ˝ (C=Z))

HomWK(M, C )�_�

� > HomWK(M, C )u_�

� > HomWK(M, C )0._�

in which the horizontal maps are defined by the inclusions

Q=Z ,! R=Z ,! C=Z.

Application to tori

Let T be a torus over a fieldK. Thedual torus T _ to T is the torus such thatX�(T _)

is the linear dualX �(T ) of X�(T ). WhenK is a global field, we say that an element ofH 1

cts(WK , M 0) is locally trivial if it restricts to zero inH 1cts(WKv

, M 0) for all primesv.


THEOREM 8.13. LetK be a local or global field, and letT be a torus overK.(a) WhenK is local, H 1

cts(WK , T _(C)) is canonically isomorphic to the group of con-tinuous generalized characters ofT (K).

(b) WhenK is a global, there is a canonical homomorphism fromH 1cts(WK , T _(C))

onto the group of continuous generalized characters ofT (AK )=T (K). The kernel isfinite and consists of the locally trivial classes.

PROOF. Take M D X�(T ) in the statement of (8.11). Then Hom(M, R�) D T _(R)

for any ring R containing a splitting field forT . In particular,M 0 D T _(C), and soH 1

cts(WK , M 0) D H 1cts(WK , T _(C)).

WhenK is local,((X�(T ) ˝ CF )G)0D (T (F)G)0

D T (K)0, which proves (a). In theglobal case, on tensoring the exact sequence

0! F �! JF ! CF ! 0

with X�(T ), we obtain an exact sequence

0! T (F)! T (AF )! X�(T )˝ CF ! 0,

and hence an exact sequence

0! T (K)! T (AK )! (X�(T )˝ CF )G! H 1(G, T (F)).

The last group in this sequence is finite, and so we have a surjection with finite kernel((X�(T )˝ CF )G)0 � (T (AK )=T (K))0. This, composed with the isomorphism

H 1cts(WF=K , T _(C))

��! ((X�(T )˝ CF )G)0

of the theorem, gives the map.There is a commutative diagram:

H 1cts(WK , T _(C)) ��! (T (AK )=T (K))0??y ??yQ

H 1cts(WKv

, T _(C)) ��!Q

T (Kv)0.

We have just seen that the lower horizontal map is an isomorphism, and the second verticalmap is injective because it is the dual of a surjective map. Therefore, the kernels of the tworemaining maps are equal, as claimed by the theorem.

Re-interpretation as class field theory

Let T be a torus overK, let M D X�(T ), and letT _ be the torus such thatX �(T _) DM .Let GK act onT _(C) D Hom(M, C�) through its action onM , and defineLT to bethe semi-direct productT _(C) o GK . It is complex Lie group with identity componentLT ı D T _(C). A continuous homomorphism'WWK !

LT is said to beadmissibleifit is compatible with the projections ontoGK . Two such homomorphisms' and' 0 aresaid to beequivalent if there exists at 2 LT ı such that' 0(w) D t'(w)t�1 for all w.Write˚K (T ) for the set of equivalence classes of admissible homomorphisms, and define˘K (T ) to beT (K)0 whenK is local and(T (K )=T (K))0 whenK is global.


THEOREM 8.14. There is a canonical mapK (T )! ˘K (T ); whenK is local, the mapis an isomorphism, and whenK is global, it is surjective with finite kernel.

PROOF. Any continuous homomorphism'WWK !LT can be written' D f � � with

f and� maps fromWK into LT ı andGK respectively. One checks immediately that' isan admissible homomorphism if and only iff is a1-cocycle and� is the mapWK ! GK

given as part of the structure ofWK . Moreover, every1-cocycle arises in this way, and two'’s are equivalent if and only if the corresponding1-cocycles are cohomologous. Thus thetheorem follows immediately from (8.13).

L-series

Let K be a nonarchimedean local field. For any representation� of WK on a finite-dimensional complex vector spaceV , theL-series

L(s, �) D (det(1� �(Fr)N(�)�sjV I )�1

whereFr is an element ofWK mapping to1 under the canonical mapWK ! Z,� is a localuniformizing parameter, andI is the inertia group. For a global fieldK and representation� of LT , the Artin-HeckeL-seriesL(s, �) is defined to be the product of the localL-series at the nonarchimedean primes. (It is possible also to define factors correspondingto the archimedean factors, but we shall ignore them.) ForS a finite set of primes, we letLS(s, �) be the product of the local factors over all primes not inS.

Assume now thatT splits over an unramified Galois extensionF of K. On tensoring

0! O�F ! F �

! Z! 0

with X�(T ), we obtain an exact sequence

0! T (OF )! T (F)! X�(T )! 0

with T (OF ) a maximal compact subgroup ofT (F). The usual argument (Serre 1967, 1.2)shows thatH 1(GF=K , T (OF )) D 0, and so there is an exact sequence

0! T (OK )! T (K)! X�(T )GF=K ! 0.

Let � be a generalized character ofT (K), and assume that it is trivial onT (OK ) (we thensay that� is unramified). Such a� gives rise to a generalized character ofX�(T )GF=K ,which we can extend to a generalized characterQ� of X�(T ). Because

Hom(X�(T ), C�) D X �(T )˝ C�D T _(C) D LT 0 ,

we can viewQ� as an element of this last group. Letr be a representation ofLT (as a pro-algebraic group) on a finite-dimensional complex vector spaceV . We define theL-series

L(s,�, r ) D det(1� r( Q�, �)N(!)�sjV )

where� is an element of Gal(Ks=K) restricting to the Frobenius automorphism onF .


Now let K be a global field, and let� be a generalized character ofT (AK )=T (K). Byrestriction, we get generalized characters�v of K�

v for eachv. Let F be a finite Galois ex-tension ofK splittingT , and choose a finite set of primesS of K including all archimedeanprimes, all primes that ramify inF , and all primesv for which�v is ramified. Define theautomorphicL-series

LS(s,�, r ) DYv =2S

L(s,�v, rv)

whererv is the restriction ofr to the localL-group.

THEOREM 8.15. (a) LetK be a local field, and letT be a torus overK splitting over anunramified extension ofK. For all ' 2 ˚(T ) and all representationsr of LT ,

L(x, r ı ') D L(s,�, r )

where� is the character ofT (K) corresponding to' in (8.14).(b) Let K be a global field, and letT be a torus overK. Let ' 2 ˚(T ), and let�

be the corresponding element of(T ). Choose a setS of primes ofK containing allarchimedean primes, all primes that ramify in a splitting field forT , and all primesv suchthat�v is ramified. Then, for all representationsr of LT ,

LS(s, r ı ') D LS(s,�, r ).

PROOF. Only (a) has to be proved, and we leave this as an exercise to the reader.

The general conjecture

Let K be a global field, and letG be a reductive group overK. ThenG is determined bycertain linear data (a root datum), and the groupLG0 is defined by the dual data. The fullL-groupLG is defined to be a semi-direct productLG0 o GK . The set (G) of equiva-lence classes of admissible homomorphismsWK !

LG is defined analogously to the caseof a torus, but the analogue of a generalized character ofT (AK )=T (K) is more difficultto define. SinceG(AK ) is neither commutative or compact, its interesting representationsare infinite dimensional. The correct notion is that of an irreducible automorphic represen-tation ofG. Langlands conjectures23 that it is possible to associate with each' 2 ˚(G)

a (nonempty) set of irreducible automorphic representations ofG. If � is associated with', then theL-series of' and� are related as in (8.15b): letr be a complex representationof LG; corresponding to almost all primesv of K, it is possible to define a localL-seriesfor � and r ; for each of these primesv, the localL-series for� and r is equal to thecorresponding factor of the Artin-HeckeL-series ofr'. See Borel 1979.

NOTES. The results in this section were proved in Langlands 1968 and again in Labesse1984. While the above proof of (8.6) borrows from the proofs in both papers, it is somewhatsimpler than each. For applications of the theorems, see Kottwitz 1984, Labesse 1984, andShelstad 1986.

23For G D GLn, Langlands’s conjecture has been proved for function fields (Drinfeld, Lafforgue, et al.)and for local fields (Harris, Taylor, Henniart, et al.).

9. OTHER APPLICATIONS 105

9 Other applications

We explain a few of the other applications that have been made of the duality theorems in~2 and~4.

The Hasse principle for finite modules

Let K be a global field, and letM be a finite module overGK . We say that theHasseprinciple holds forM if the map

ˇr (K, M)WH 1(K, M)!Q

all vH1(Kv, M)

is injective.

EXAMPLE 9.1. (a) LetF=K be a finite Galois extension of degreen such that the greatestcommon divisorr of local degrees[Fw W Kv] is strictly less thann. (For example, letK D Q andF D Q(

p13,p

17); thenn D 4 and the local degrees are all1 or 2.) Considerthe exact sequence

0!M ! (Z=nZ)[G]�! Z=nZ! 0

in which G D Gal(F=K) and � is the augmentation mapP

n�� 7!P

n� . From itscohomology sequence, we obtain an isomorphismZ=nZ ! H 1(G, M). Let c generateH 1(G, M); thenrc is a nonzero element ofH 1(K, M) mapping to zero in all the localcohomology groups. ThereforeX1(K, M) 6D 0, and the Hasse principle does not hold forM.

(b) From the duality theorem (4.10), we see that, forM as in (a),X2(K, M) 6D 0. Fora more explicit example (based on the failure of the original form of the Grunwald theorem)see Serre 1964, III 4.7.

In view of these examples, the theorem below is of some interest. For a moduleM ,we writeK(M) for the subfield ofKs fixed by Ker(GK ! Aut(M)). ThusK(M) is thesmallest splitting field ofM . A finite groupG is said to be -solvableif it has a compositionseries whose factors of order divisible by` are cyclic.

THEOREM 9.2. Let M be a finite simpleGK -module such thatM D 0 for some prime ,and assume thatGal(K(M)=K) is an`-solvable group.

(a) If S is a set of primes ofK with Dirichlet density one, then the mapping

ˇ1S(K, M)WH 1(K, M)!

Yv2S

H 1(Kv, M)

is injective.(b) If ` 6D char(K), then the mapping

ˇ2(K, M)WH 2(K, M)!Yall v

H 2(Kv, M)

is injective.


Note thatˇrS is not quite the same as the map in~4. However the next lemma shows

that Ker 1S(K, M) D X1

S(K, M). For any profinite groupG andG-moduleM , defineH 1

� (K, M) to be the kernel of

H 1(G, M)!YZ

H 1(Z, M),

where the product is over all closed cyclic subgroupsZ of G. WhenG D GK , we alsowrite H 1

� (K, M) for H 1� (G, M). The next result explains the significance of this notion

for the theorem. As always, for each primev of K, we choose an extensionw of v to Ks.

LEMMA 9.3. LetM be a finiteGK -module, and letF � Ks be a finite Galois extension ofK containingK(M). LetS be a set of primes ofK with Dirichlet density one, and let

ˇ1S(F=K, M)WH 1(GF=K , M)!

Yv2S

H 1(GFw=Kv, M)

be the map induced by the restriction maps. Then there is a commutative diagram

Ker(ˇ1S(F=K, M))

�D> Ker(ˇ1

S(K, M))

\ \

H 1� (Gal(F=K), M)

�D> H 1

� (GK , M)

The inclusions become equalities when all the decomposition groupsGal(Fw=Kv) arecyclic.

PROOF. There is an exact commutative diagram

0 ��! H 1(Gal(F=K), M)Inf��! H 1(K, M) ��! H 1(F, M)??yˇ1

S(F=K,M)

??yˇ1S

(K,M)

??yˇ1S

(F,M)

0 ��!Qv2S

H 1(Gal(Fw=Kv), M)Inf��!

Qv2S

H 1(Kv, M) ��!Qv2S

H 1(Fw, M).

The Chebotarev density theorem shows thatˇ1S(F, M) is injective. The inflation map

therefore defines an isomorphism of the kernels of the first two vertical maps, which givesus the isomorphism on the top row. The isomorphism on the bottom row can be provedby a similar argument. The Chebotarev density theorem shows that all cyclic subgroupsof Gal(F=K) are of the form Gal(Fw=Kv) for some primeswjv with v 2 S , and soclearly H 1

� (Gal(F=K), M) � Ker(ˇ1(F=K, M)). The reverse inclusion holds if all thedecomposition groups are cyclic.

We say that theHasse principle holds for a finite groupG (and the prime`) ifH 1

� (G, M) D 0 for all finite simpleG-modulesM (with `M D 0). Note that the Hasseprinciple obviously holds forG if all of its Sylow subgroups are cyclic.


LEMMA 9.4. Let1! G 0

! G ! G 00! 1

be an exact sequence of finite groups. If the Hasse principle holds forG 0 andG 00 relativeto the prime , then it holds forG and`; conversely, if the Hasse principal holds forG and`, then it holds forG 00 and`.

PROOF. Let M be a simpleG-module such thatM D 0. As G 0 is normal inG, M G0

is stable underG, and so eitherM G0

D 0 or M G0

D M . In the first case, there is acommutative diagram

H 1(G, M)Res��! H 1(G 0, M)??y ??yQ

H 1(Z, M)Res��!

QH 1(Z \G 0, M)

in which the upper restriction map has kernelH 1(G 00, M G0

) D 0. When regarded as aG 0-module,M is semisimple because, for any nonzero simpleG 0-moduleN of M , M isa sum of simple modulesgN , g 2 G. Therefore, if the Hasse principle holds forG 0 and`,then the right hand vertical arrow is an injection. Consequently, the first vertical arrow isalso an injection, and this shows that the Hasse principle holds forG and`.

In the case thatM G0

DM , we consider the diagram

0 ��! H 1(G 00, M) ��! H 1(G, M) ��! H 1(G 0, M)??y ??y ??y0 ��!

QH 1(ZG 0=G 0, M) ��!

QH 1(Z, M) ��!

QH 1(Z \G 0, M).

The right hand vertical arrow is an injection becauseG 0 acts trivially onM and the groupsZ\G 0 generateG 0. The left hand vertical arrow has kernelH 1

� (G 00, M) because the groupsZG 0=G 0 run through all cyclic subgroups ofG 00, and so we see that if the Hasse principleholds forG 00 and` then it holds also forG and`.

We use the same diagram to prove the converse part of the lemma. A simpleG 00-modulecan be regarded as a simpleG-module such thatM G0

DM . Therefore, the diagram showsthatH 1

� (G 00, M) D 0 if H 1� (G, M) D 0.

PROPOSITION9.5. (a) The Hasse principle holds for a finite group (and the prime`) whenit holds for all the composition factors of the group (and`).

(b) If G is `-solvable, then the Hasse principle holds forG and`.(c) A solvable group satisfies the Hasse principle.

PROOF. Part (a) follows by induction from the lemma. Part (c) follows from (a) and theobvious fact that the Hasse principle holds for a cyclic group. Part (b) follows from (a)and (c) and the additional fact that the higher cohomology groups of a module killed by`

relative to a group of order prime toare all zero.


We now prove Theorem 9.2. Lemma 9.3 shows that

Kerˇ1S(K, M) D Kerˇ1

S(K(M)=K, M) � H 1� (GK(M)=K , M),

and (9.5b) shows that this last group is zero, which proves part (a) of the theorem. From(4.10) we know that Ker(ˇ2(K, M)) is dual to Ker(ˇ1(K, M D)). ClearlyM D is simpleif M is, and the extensionK(M D) is `-solvable ifK(M) is because it is contained inK(M)(�`). Therefore part (b) of the theorem follows from part (a).

COROLLARY 9.6. If M � Z=`Z�Z=`Z (as an abelian group) for some prime` not equalto the characteristic ofK, thenX2(K, M) D 0.

PROOF. If M is simple (or semisimple) as aGK -module, this follows directly from thetheorem. The remaining case can be proved directly.

NOTES. The groupsH r� (G, M) were introduced by Tate (see Serre 1964/71). Theorem 9.2

and its proof are taken from Jannsen 1982. For an elementary proof of (9.6), see Cassels1962,~5.

The Hasse principle for algebraic groups

In this subsection,G will be a connected (not necessarily commutative) linear algebraicgroup over a number fieldK. We say thatG satisfies theHasse principle

H 1(K, G)!Yall v

H 1(Kv, G)

if is injective. It is known (Kneser 1966, 1969; Harder 1965/66) that ifG is semisimpleand simply-connected without24 factors of typeE8, thenH 1(Kv, G) D 0 for all nonar-

chimedeanv, andH 1(K, G)�!Qv realH

1(Kv, G).

THEOREM 9.7. Let G be a simply connected semisimple group, and let'WG ! G 0 bea separable isogeny. LetM be the kernel of'(Ks)WG(Ks) ! G 0(Ks), and assume thatX2(K, M) D 0. If the Hasse principle holds forG, then it also holds forG 0.

We first prove a lemma.

LEMMA 9.8. Let M be a finite moduleGK whose order is not divisible by char(K), andassumeS omits only finitely many primes ofK.

(a) The cokernel ofH 1(K, M) !Qv =2SH 1(Kv, M) is canonically isomorphic to the

dual of(Kerˇ1S(K, M D))=(Ker ˇ1(K, M D)).

(b) If eachv =2 S has a cyclic decomposition group inK(M), thenH 1(GK , M) !Lv =2S H 1(Kv, M) is surjective. In particularH 1(K, M)!

Lv real H

1(Kv, M) issurjective.

24In 1989, Chernousov removed this condition.


PROOF. (a) From (4.10) we know there is an exact sequence

H 1(K, M D)ˇ1

�! P 1S(K, M D)�

Yv =2S

H 1(Kv, M D) 1

�! H 1(K, M)�.

Therefore the kernel-cokernel sequence (0.24) of the pair of maps

H 1(K, M D)! P 1S(K, M D)�

Yv =2S

H 1(Kv, M D)! P 1S(K, M D)


0! Kerˇ1(K, M D)! Kerˇ1S(K, M D)!

Yv =2S

H 1(Kv, M D) �! H 1(K, M)�.

The exactness at the third term says that Ker(ˇ1S)=Ker(ˇ1) D Ker( ), but this last group

is the dual of the cokernel ofH 1(K, M)!Qv =2SH 1(Kv, M).

(b) Let F be a finite Galois extension ofK containingK(M D). According to theChebotarev density theorem, for each primev =2 S having a cyclic decomposition groupin Gal(F=K), there is a primev0 2 S having the same decomposition group. Thereforeif an elementc of H 1(GF=K , M D) maps to zero inH 1(GFw=Kv

, M D) for all v in S , thenit maps to zero for allv. Hence Ker 1

S(F=K, M) D Ker ˇ1(F=K, M), and Lemma 9.3shows that this implies that Ker1S(K, M) D Ker ˇ1(K, M). Now (a) implies (b).

PROOF OFTHEOREM 9.7. Consider the diagram of pointed sets:

H 1(K, M) ��! H 1(K, G) ��! H 1(K, G 0) ��! H 2(K, M) ��!??y ??yinjective

??y ??yinjectiveQall v

H 1(Kv, M) ��!Qall v

H 1(Kv, G) ��!Qall v

H 1(Kv, G 0) ��!Qall v

H 2(Kv, M) ��! .

(See Serre 1961, VII, Annexe.) Ifc 2 H 1(K, G 0) maps to zero inH 1(Kv, G 0) for all v,then it lifts to an elementb 2 H 1(K, G). As we observed above,H 1(Kv, G) D 0 for allnonarchimedeanv. For each archimedean primev, the imagebv of b in H 1(Kv, G) liftsto an elementav of H 1(Kv, M). According to (9.8), there is an elementa 2 H 1(K, M)

mapping toav for all archimedeanv. Now b� a0, wherea0 is the image of a inH 1(K, G),maps toc in H 1(K, G 0) and to 0 inH 1(Kv, G) for all v. The last condition shows thatb � a0 (hencec) is zero. This shows that the kernel ofH 1(K, G 0) !

QH 1(Kv, G) is

zero, and a standard twisting argument (cf. Kneser 1969, I 1.4) now allows one to showthat the map is injective.

COROLLARY 9.9. Let G be a semisimple algebraic group overK without factors of typeE8. Then the Hasse principle holds forG under each of the following hypotheses:

(a) G has trivial centre;(b) G is almost absolutely simple;(c) G is split by a finite Galois extensionF of K such that all Sylow subgroups of

Gal(F=K) are cyclic;


(d) G is an inner form of a group satisfying (a), (b), or (c).

PROOF. A group with trivial centre is a product of groups of the formRF=K G with G anabsolutely simple group overF , and so (a) follows from (b). An absolutely almost simplegroup is an inner form of a quasi-split almost simple group, and such a group is split by aextension whose Galois group is a subgroup of the group of automorphisms of its Dynkindiagram. But this automorphism group is either trivial or isZ=2Z or S3. Therefore (b)follows from (c) and (d). LetG be split by an extensionF as in (c), and letM be thekernel of QG(Ks) ! G(Ks) where QG is the universal covering group ofG. ThenM isa sum of Gal(Fs=F) modules of the form�m, and so Gal(Fs=F) acts trivially onM D .Therefore

X1(K, M D) D Kerˇ1(F=K, M D) � H 1� (Gal(F=K), M D) D 0,

and soX2(K, M) D 0. Finally (d) is obvious from the fact that the Gal(Ks=K)-moduleM is unchanged whenG is replaced by an inner form.

NOTES. Theorem 9.7 is proved in Harder 1967/68, Theorem 4.3.2, and in Kneser 1969,pp77-78. Part (a) of Corollary 9.9 is proved in Langlands 1983, VII 6.25 All of the resultsin this subsection are contained in Sansuc 1981.

Forms of an algebraic group

The next result shows that (under certain conditions) a family of local forms of an algebraicgroup arises from a global form.

THEOREM 9.10. Let K be an algebraic number field,S a finite set of primes ofK, andG

an absolutely almost simple algebraic group overK that is either simply connected or hastrivial centre. Then the canonical map

H 1(K, Aut(G))!Qv2SH 1(Kv, Aut(G))

is surjective.

PROOF (SKETCH): Let QG be the universal covering group ofG, and letM D Ker( QG(Ks)!

G(Ks)). Consider the diagram

� � � ��! H 1(K, QG) ��! H 1(K, G) ��! H 2(K, M) ��! � � �??y ??y ??y� � � ��!

Lv2S

H 1(Kv, QG) ��!Lv2S

H 1(Kv, G) ��!Lv2S

H 2(Kv, M) ��! � � �

Corollary 4.16 shows that the final vertical map is surjective. We have already noted that thefirst vertical map is surjective whenG has no factors of typeE8, but in fact this conditionis unnecessary. Next one shows that the mapH 1(K, G) ! H 2(K, M) is surjective,

25See also Satz 4.3.2 of: Harder, Gunter, Berichtuber neuere Resultate der Galoiskohomologie halbein-facher Gruppen. Jber. Deutsch. Math.-Verein. 70 1967/1968 Heft 4, Abt. 1, 182–216.


and a diagram chase then shows thatH 1(K, G) ! ˚v2SH 1(Kv, G) is surjective. Oneshows that it suffices to prove the theorem for a splitG, in which case Aut(G) is the semi-direct productG o Aut(D) of G with the automorphism group of the Dynkin diagramof G. The proof of the theorem then is completed by showing thatH 1(K, Aut(D)) !

˚H 1(Kv, Aut(D)) is surjective.

For the details, see Borel and Harder 1978, where the theorem is used to prove theexistence of discrete cocompact subgroups in the groups of rational points of reductivegroups over nonarchimedean local fields of characteristic zero.

The Tamagawa numbers of tori

We refer the reader to Weil 1961 for the definition of the Tamagawa number�(G) of alinear algebraic groupG over a global field.

THEOREM 9.11. For any torusT over a global fieldK

�(G) D[H 1(K, T _)]

[X1(K, T )]

whereX1(K, T ) is the kernel ofH 1(K, T )!Q

all vH1(Kv, T ) andT _ is the dual torus

defined by the relationX �(T ) D X�(T _).

PROOF. Let

'(T ) D �(T )[X1(K, T )]

[H 1(K, T _)].

The proof has three main steps:(i) ' is an additive function on the category of tori overKI

(ii) '(Gm) D 1;(iii) for any finite separable extensionF of K, '(ResF=K T ) D '(T ).

Once these fact have been established the proof is completed as follows. The functorT 7! X�(T ) defines an equivalence between the category of tori overK and the categoryRepZ(GK ) of continuous representations ofGK on freeZ-modules of finite rank, and sowe can regard' as being defined on the latter category. Then (i) says that' induces ahomomorphismK0(RepZ(GK ))! Q>0. A theorem (Swan 1960) shows that[X ] � [X 0]

is a torsion element ofK0(RepZ(GK )) if X ˝ Q D X 0˝ Q, and, asQ>0 is torsion-

free, ' is zero on torsion elements ofK0(RepZ(GK )). Therefore' takes equal valueson isogenous tori. Artin’s theorem on characters (Serre 1967b, 9.2) implies that for anytorusT , there exists and integerm and finite separable extensionsFi andEj of K suchthat T m

�Q

ResFi=K Gm is isogenous toQ

ResEj=K Gm. Now (ii) and (iii) show that'(T )m

D 1, and therefore'(T ) D 1.Statements (ii) and (iii) are easily proved (they follow almost directly from the defini-

tions), and so the main point of the proof is (i). This is proved by an argument, not dis-similar to that used to prove Theorem 7.3, involving the duality theorems. See Ono 1961,1963, and also Oesterle 1984, which corrects errors in Ono’s treatment of the function fieldcase.


The central embedding problem

Let S be a finite set of primes of a global fieldK, and letGS be the Galois group overKof the maximal extension ofK unramified outsideS . Let

1!M ! E ! G ! 1

be an extension of finite groups withM in the centre ofE, and let'WGS � G be asurjective homomorphism. Theembedding problemfor E and' is the problem of finding asurjective homomorphism WGS � E lifting '. Concretely this means the following: thehomomorphism' realizesG as the Galois group of an extensionF of K that is unramifiedoutsideS , and the embedding problem asks for a fieldF 0 that is Galois overK with GaloisgroupE, is also unramified outsideS , containsF , and is such that the mapE � G inducedby the inclusion ofF into F 0 is that in the sequence. For eachv in S , letGv be the image ofGal(Ks

v=K) in G, and letEv be the inverse image ofGv in E. Then the (local) embeddingproblem asks for a homomorphism Gal(Ks

v=Kv) � Ev lifting Gal(Ksv=Kv) � Gv.

Let � be the class of the extension inH 2(G, M). If the embedding problem has asolution, then� clearly is sent to zero by the mapH 2(G, M)! H 2(GS , M) defined by'. The converse is also true if� 6D 0 andM is a simpleG-module. Thus Theorem 9.2 hasthe following consequence.


1!M ! E ! G ! 1

be a nonsplit central extension of finite groups, and let'WGK � G be a surjective homo-morphism. IfM is a simpleG-module andG is solvable, then the embedding problem forE and' has a solution if and only if the corresponding local problem has a solution forall v.

PROOF. The necessity of the condition is obvious. For the sufficiency, note that when thelocal problem has a solution, the image of� in H 2(Kv, M) is zero for allv. According to(9.2), this implies that� is zero.

Unfortunately, the proposition does not lead to a proof of Shafarevich’s theorem (Sha-farevich 1954): for any number fieldK and finite solvable groupG, there exists an exten-sionF of K with Galois groupG.

For other applications of the duality theorems to the embedding problem, see for exam-ple Haberland 1978, Neumann 1977, and Klingen 1983.

Abelian varieties defined over their fields of moduli

Let A be a polarized abelian variety defined overQ. The obstruction toA having a modelover its field of moduli is a class� in H 2(G, Aut(A)). In the case that Aut(A) is abelian,the duality theorems can sometimes be helpful in studying this element.

APPENDIX A: CLASS FIELD THEORY FOR FUNCTION FIELDS 113

Abelian varieties andZp-extensions

The duality theorems (and their generalizations to flat cohomology) have been used in thestudy of the behaviour of the points on an abelian variety as one progresses up aZp-towerof number fields. See for example Manin 1971, Mazur 1972, Harris 1979, and Rubin 1985.

Appendix A: Class field theory for function fields

Most of the accounts of class field theory either omit the case of a function field or makeit appear harder than the number field case. In fact it is easier (at least for those knowing alittle algebraic geometry). In this appendix we derive the main results of class field theoryexcept for the existence theorem for a function field over a finite field. As a preliminary,we derive class field theory for a Henselian local field with quasi-finite residue field. Wealso investigate to what extent the global results hold for a function field over a quasi-finitefield.

Local class field theory

A field k is quasi-finiteif it is perfect and if the Galois groupG(ks=k) is isomorphic to theprofinite completionbZ of Z. The main examples of quasi-finite fields are the finite fieldsand the power series fieldsk0((t)) with k0 an algebraically closed field of characteristiczero, but there are others. For example, any algebraic extensionk 0 of a quasi-finite fieldkwhose degree[k 0

W k] is divisible by only a finite power of each prime number is quasifinite.Also, given an algebraically closed fieldK, one can always find a quasi-finite fieldk havingK as its algebraic closure (Serre 1962, XIII 2, Ex 3).

Whenever a quasi-finite fieldk is given, we shall always assume that there is also givenas part of its structure a generator�k of Gal(ks=k), or equivalently, a fixed isomorphism'k W�(Gk) ! Q=Z where�(Gk) is the character group Homcts(Gk , Q=Z) of Gk . Therelation between� and'k is that'k(�) D �(�) for all � 2 �(Gk). A finite extension ofa quasi-finite fieldk is again quasi-finite with generator�` D �

[`Wk]

k. Whenk is finite, we

always take�k to be the Frobenius automorphisma 7! aq, q D [k].

Note that the Brauer group of a quasi-finite field is zero, becauseGk has cohomologicaldimension one andks� is divisible. LetR be a discrete valuation ring with residue fieldk D R=m. Write f for the reduction of an element ofR or R[X ] modulom. We saythat R is Henselian if it satisfies the conclusion of Hensel’s lemma: wheneverf is amonic polynomial with coefficients inR such thatf factors asf D g0h0 with g0 andh0 monic and relatively prime, thenf itself factors asf D gh with g andh monic andsuch thatg D g0 andh D h0. Hensel’s lemma says that complete discrete valuation ringsare Henselian, but not all Henselian rings are complete. For example, letv be a prime ina global fieldK, and letOv be the ring of elements ofK that are integral atv. Choosean extensionw of v to Ks, let Kdec be the decomposition field ofw in Ks, and letOh

v

be the ring elements ofKdec that are integral with respect tow. Alternatively, choose anembedding ofKs into Ks

v, and letOhv D Ks

\ bOv. ThenOhv is a Henselian local ring, called

theHenselizationof Ov. See, for example, Milne 1980, I 4.


Now let R be a Henselian discrete valuation ring with quasi-finite residue fieldk, andwrite K for its field of fractions. Many results usually stated only for complete discretevaluation rings hold in fact for Henselian discrete valuation rings (often the proof uses onlythat the ring satisfies Hensel’s lemma). For example, the valuationv on K has a uniqueextension to a valuation (which we shall also writev) on Ks. As usual we writeKun forthe maximal unramified subextension ofKs overK, andRun for the integral closure ofRin Kun.

PROPOSITIONA.1. There is a canonical isomorphisminvK WBr(K)! Q=Z.

We first show that Br(Kun) D 0. Let D be a skew field of degreen2 overKun. BecauseRun is also a Henselian discrete valuation ring, the valuation onKun has a unique extensionto each commutative subfield ofD, and therefore it has a unique extension toD. The usualargument in the commutative case shows that, for this extension,n2 D ef . Let˛ in D havevalue1=e; thenKun[˛], being a commutative subfield ofD, has degree at mostn, and soe � n. On the other handf D 1 because the residue field ofR is algebraically closed, andit follows thatn D 1.

The exact sequence

0! H 2(Gal(Kun=K), Kun�)! Br(K)! Br(Kun)

now shows that Br(K) D H 2(Gal(Kun=K), Kun�).

LEMMA A.2. The mapH 2(Gal(Kun=K), Kun�)ord! H 2(Gal(Kun=K), Z) is an isomor-

phism.

PROOF. As0! Run�

! Kun� ord! Z! 0

is split as a sequence of Gal(Kun=K)-modules, the map in question is surjective. Letc liein its kernel, and let be a cocycle representingc. Associated withc there is a centralsimple algebraB overK (Herstein 1968, 4.4), and if is chosen to take values inR�, thenthe same construction that givesB gives an Azumaya algebraB0 overR that is an orderin B. The reductionB0 ˝R k of B0 is a central simple algebra overk, and therefore isisomorphic to a matrix algebra. An elementary argument (Milne 1980, IV 1.6) shows thenthatB0 is also isomorphic to a matrix algebra, and this implies thatc D 0.

We define invK to be the unique map making

H 2(Gal(Kun=K), Z) D Br(K)invK

��! Q=Z�D

x??d �D

x??'k

H 1(Gal(Kun=K), Q=Z) �(Gk)

commute. It is an isomorphism. IfF is a finite separable extension ofK, then the integralclosureRF of R in F is again a Henselian discrete valuation ring with quasi-finite residuefield, and one checks easily from the definitions that

invF (Res(a)) D [F W K] invK (a), a 2 Br(K).


Therefore(GK , Ks�) is a class formation in the sense of~1.We identify the cup-product pairing

H 0(GK , Ks�)�H 2(GK , Z)! H 2(GK , Ks�)

with a pairingh , iWK�

��(GK )! Br(K).

THEOREM A.3. (Local reciprocity law). There is a continuous homomorphism

(�, K)WK�! Gal(Kab=K)

such that(a) for each finite abelian extensionF � Ks of K, (�, K) induces an isomorphism

(�, F=K)WK�=NF=K F �! Gal(F=K)I

(b) for any� 2 �(G) anda 2 K�, �(a, K) D invK ha,�i.

PROOF. As is explained in~1, this theorem is a formal consequence of the fact that(GK , Ks�)

is a class formation.

COROLLARY A.4. Let F1 and F2 be extensions ofK such thatF1 \ F2 D K, and letF D F1F2. If all three fields are finite abelian extensions ofK, thenNF �

D NF �1 \NF �

2

and(NF �1 )(NF �

2 ) D K�.

PROOF. According the theorem,a 2 NF � if and only if (a, K) acts trivially onF ; thisis equivalent to(a, K) acting trivially onF1 andF2, or to it lying in NF �

1 \ NF �2 . The

second equality can be proved similarly.

REMARK A.5. WhenF=K is unramified,� lifts to a unique (Frobenius) elementQ� inGal(F=K), and(�, K)WK�

! Gal(F=K) sends an elementa of K� to Q�ord(a). In par-ticular, (a, F=K) D 1 if a 2 R�. WhenF=K is ramified, the description of(a, F=K) ismuch more difficult (see Serre 1967a, 3.4).

We say that a subgroupN of K� is a norm group if there exists a finite abelian ex-tensionF of K such thatN D NF=K F �. (The name is justified by the following result:if F=K is any finite separable extension ofK, thenNF=K F �

D NL=K L� whereL is themaximal abelian subextension ofF ; see Serre 1962, XI 4.)

REMARK A.6. The reciprocity map defines an isomorphism lim �

K�=N ! Gab (inverse

limit over the norm groups inK�), and so to understandGab fully it is necessary to deter-mine the norm groups. This is what the existence theorem does.

Case 1: K is complete andk is finite. This is the classical case. Here the normgroups ofK� are precisely the open subgroups of finite index. Every subgroup of finiteindex prime to char(K) is open. The image of the reciprocity map is the subgroup ofGab of elements that act as an integral power of the Frobenius automorphism onks. Thereciprocity map is injective, and it defines an isomorphism of the topological groupR�

onto the inertia subgroup ofGab. See Serre 1962, XIV 6.


Case 2:K is Henselian with finite residue field. We assume thatR is excellent. Thisis equivalent to the completionbK of K being separable overK. From the description of itgiven above, it is clear that the Henselization of the local ring at a prime in a global field isexcellent. Under this assumption:

(i) every finite separable extension ofbK is of the formbF for a finite separable extensionF of K with [F W K] D [bF W bK]I

(ii) K is algebraically closed inbK, and hencebK is linearly disjoint fromKa overK.

To prove (i), writebF D bK[˛], and letF D K[ˇ] with ˇ a root of a polynomial inK[X ]

that is close to the minimal polynomial of˛ overK (cf. Lang 1970, II 2). To prove (ii), notethat if K is not algebraically closed inbK, then there is an element˛ of bK that is integralover R but which does not lie inR. Let f (X) be the minimal polynomial of over K.As ˛ is integral overbR, it lies in bR, and sof has a root inbR. An approximation theorem(Greenberg 1966) now says thatf has a root inR, butf was chosen to be irreducible overK. ThusK is algebraically closed inbK, and combined with the separability ofbK overK,this implies thatbK is linearly disjoint fromKa (Lang 1958, III 1, Thm 2).

On combining these two assertions, we find thatF 7! bF defines a degree-preservingbijection from the set of finite separable extensionsF of K to the set of similar extensionsof bK. Moreover,NF �

D N bF �\ K� for eachF becauseNF � is dense inN bF � and

Greenberg’s theorem implies thatNF � is open inK�. It follows that the norm groups ofK� are again precisely the open subgroups of finite index.

Case 3: K is complete with quasi-finite residue field. In this case every subgroup ofK of finite index prime to char(k) is a norm group, but not every open subgroup of indexa power of the characteristic ofk is. In Whaples 1952–54, various characterizations of thenorm groups are given using the pro-algebraic structure onK�.

Case 4:K is Henselian with quasi-finite residue field. We leave this case to the readerto investigate.

Global class field theory

Let X be a complete smooth curve over a quasi-finite field(k, �), and letK D k(X). Theset of closed points ofX will be denoted byX 0 (thusX 0 omits only the generic point ofX ). To each pointv of X 0, there corresponds a valuation (also writtenv) of K, and wewrite Kv for the completion ofK with respect tov andRv for the ring of integers inKv.The residue fieldk(v) is a quasi-finite field of degree deg(v) over k with �deg(v) as thechosen generator of Gal(k(v)s=k(v)). Write av for the image of an element a of Br(K)

in Br(Kv), and define invK WBr(K) ! Q=Z to bea 7! invv(av) where invv is invKv. Let

X Ddf X ˝k ks beX regarded as curve overks, and letK D ks(X) be its function field.We write JacX be the Jacobian variety ofX.

THEOREM A.7. There is an exact sequence

0! H 1(Gk , JacX (ks))! Br(K)!Mv2X 0

Br(Kv)invK

! Q=Z! 0.

PROOF. We use the exact sequence ofGk-modules

0! ks�! K

�! Div(X)! Pic(X)! 0


where Div(X) DL

v2X 0 Z is the group of (Weil) divisors onX . From the cohomologysequence of its truncation

0! ks�! K

�! Q! 0

we obtain an isomorphismH 2(Gk , K�

)�D! H 2(Gk , Q). Note that

H 2(Gk , K�

) D Ker(Br(K)! Br(K)),

and recall that Tsen’s theorem (Shatz 1972, Theorem 24) states that Br(K) D 0, and soH 2(Gk , Q) D H 2(Gk , K

�) D Br(K).

The cohomology sequence of the remaining segment

0! Q! Div(X)! Pic(X)! 0

of the sequence is

H 1(Gk , Div(X))! H 1(Gk , Pic(X))! H 2(Gk , Q)

! H 2(Gk , Div(X))! H 2(Gk , Pic(X))! 0.

But H r (Gk , Div(X)) DL

v2X 0 H r (Gk , Dv), whereDv DdfL

w 7 !vZ is theGk-moduleinduced by the trivialGk(v)-moduleZ, and so

H r (Gk , Div(X)) DL

v2X 0H r (Gal(ks=k(v)), Z).

In particular,H 1(Gk , Div(X)) D 0 andH 2(Gk , Div(X)) DL

v2X 0 �(Gk(v)).

Almost by definition of JacX , there is an exact sequence

0! JacX (ks)! Pic(X)! Z! 0.

As JacX (ks) is divisible (Milne 1986b, 8.2) andk has cohomological dimension one,H 2(Gk , JacX (ks)) D 0, and soH 2(Gk , Pic(X)) D H 2(Gk , Z) D �(Gk). These resultsallow us to identify the next sequence with the required one:

� � � �!H 1(Gk , Pic(X)) �!H 2(Gk , Q) �!H 2(Gk , Div(X)) �! H 2(Gk , Pic(X)) �! 0 0 �! H 1(Gk , JacX ) �! Br(K) �!

L�(Gk(v))

P�! �(Gk) �! 0.

The map can be identified with invK WL

v Br(Kv)! Q=Z.

We define thegroup of idelesJK of K to be the subgroup ofQv2X 0K�

v comprisingthose elementsa D (av) such thatav 2 R�

v for all but finitely manyv. The quotient ofJK

by K� (embedded diagonally) is theidele class groupof CK of K. We setJ D lim�!

JF

andC D lim�!

CF (limit over all finite extensionF of K, F � Ks).

COROLLARY A.8. If H 1(Gk0 , JacX (ks)) D 0 for all finite extensionsk 0 of k, then it ispossible to define on(GK , C ) a natural structure of a class formation.


PROOF. An argument similar to that in (4.13) shows that

H r (GK , J ) DL

v2X 0H r (Gv, Ks�v ), r � 1,

where for eachv in X 0 a choicew of an extension ofv to Ks has been made in orderto identify K with a subfield ofKs

v andGv Ddf GKvwith a decomposition group inG.

Consider the diagram

0 �! Br(K) �!L

v2X 0 Br(Kv)invK

��! Q=Z �! 0 0 �!H 1(GK , C ) �!H 2(GK , Ks�) �!

Lv2X 0 H 2(Gv, Ks�

v ) �! H 2(GK , C ) �! 0

whose top row is the sequence in (A.7) and whose bottom row is the cohomology sequenceof

0! Ks�! J ! C ! 0.

The zero at lower left comes from Hilbert’s theorem 90, and the zero at lower right comesfrom the fact thatGK has cohomological dimension� cd(k)C1 D 2. This diagram shows

that H 1(GK , C ) D 0 and that there is a unique isomorphism invK WH2(GK , C )

�! Q=Z

making

H 2(GK , C )invK

��! Q=Zx?? H 2(Gv, Ks�

v )invv

��! Q=Zcommute for allv. The same assertions are true for any finite separable extensionF of K,and it obvious that the maps invF satisfy the conditions (1.1).

COROLLARY A.9. If k is algebraic over a finite field, then(GK , C ) is a class formation.

PROOF. Lang’s lemma shows thatH 1(Gk , A) D 0 for any connected algebraic groupA ifk is finite. If k is algebraic over a finite field, then any element ofH 1(Gk , A) is representedby a principal homogeneous space, which is defined over a finite field and is consequentlytrivial by what we have just observed.

In (A.14) below, we shall see examples of fieldsK=k for which the conditions of (A.8)fail. We now investigate how much of class field theory continues to hold in such cases.

Fix an extension of eachv to Ks, and hence embeddingsivWGv ! GK . Define(�, K)WJK ! Gal(Kab=K) by

(a, K) DQv2X 0

iv(av, Kv), a D (av).

For any finite abelian extensionF of K, this induces a mapping(�, F=K)WJK=NF=K JF !

Gal(F=K) such that(a, F=K) DQ

iv(av, Fw=Kv) whereFw denotes the completion ofF at the chosen prime lying overv. It follows from (A.5) above, and the fact that onlyfinitely many primes ofK ramify in F , that this last product is finite (and that the previousproduct converges).


LEMMA A.10. For all a in K�, (a, K) D 0.

PROOF. Consider the diagram:

JK � �(G) >M

Br(Kv)invK> Q=Z

jj jj

H 0(GK , J ) �H 2(GK , Z) > H 2(GK , J )

^

jj

H 0(GK , Ks�)

^

�H 2(GK , Z) > H 2(GK , Ks�).

^

The two lower pairings are defined by cup-product, and the top pairing sends(a,�) toinvK (

Phav,�jGvi) D �((a, K)), a D (av) (hereh , i is as in A.3). It is obvious from

the various definitions that the maps are compatible with the pairings. Ifa 2 K�, then thediagram shows that�(a, K) lies in the image of Br(K) in Q=Z, but Br(K) is the kernel ofinvK , and so�(a, K) D 0 for all �. This implies that(a, K) D 0.

The lemma shows that there exist maps

(�, K)WCK ! Gal(Kab=K)

(�, F=K)WCK=NCF ! Gal(F=K), F=K finite abelian.

We shall say that thereciprocity law holdsfor K=k if, for all finite abelian extensionsF=K, this last map is an isomorphism.

Unfortunately the reciprocity law does not always hold because there can exist abelianextensionsF=K in which all primes ofK split, that is, such thatFw D Kv for all primesv. This suggests the following definition: letF be a finite abelian extension ofK, and letK0 be the maximal subfield ofF containingK and such that all primes ofK split in K0;thereduced Galois groupGF=K of F overK is the subgroup Gal(F=K0) of Gal(F=K).

PROPOSITIONA.11. For any finite abelian extensionF of K, the map(�, F=K) induces

an isomorphismCK=NCF

�D! GF=K .

PROOF. For each primev, the image of Gal(Fw=Kv) is contained inGF=K , and so theimage ofJ in Gal(F=K) is also contained inGF=K .

It clearly suffices to prove the surjectivity of(�, F=K) in the case thatF=K0 is cyclic ofprime order. Then there exists a primev such thatFw 6D Kv, andK�

v ! Gal(Fw=Kv) �

GF=K is surjective by local class field theory.To prove the injectivity, we count. IfF1 andF2 are finite abelian extensions ofF such

thatF1 \ F2 D K andF1F2 D F , then it follows from (A.4) thatNCF1\NCF2

D NCF

and(NCF1)(NCF2

) D NCF . As GF1=K \ GF2=K D 1 andGF1=K � GF2=K D GF=K , itsuffices to prove thatCK=NCF andGF=K have the same order forF=K cyclic of primepower order.


Let F=K be cyclic of prime power, and consider the diagram

Br(F=K) ��!Lv2X 0

Br(Fw=Kv)invK

��! Q=Zx??�

x??�

K�=NF � ��! JK=NJF ��! CK=NCF ��! 0.

The top row is part of the sequence in (A.7), and the bottom row is part of the Tate coho-mology sequence of

0! F �! JF ! CF ! 0.

The first two vertical arrows are the isomorphisms given by the periodicity of the cohomol-ogy of cyclic groups. The order of the image of invK is the maximum of the orders of theBr(Fw=Kv), and the order of Br(Fw=Kv) is [Fw W Kv]. Thus the order of the image equalsthe order ofG(L=K). From the diagram, we see that it is also the order ofCK=NCF .

For any curveY over a quasi-finite field, we define theBrauer group Br(Y ) of Y

to be the kernel of Br(F) !L

v2Y 0 Br(Fv), whereF is the function field ofY . Thisdefinition will be justified in A.15 below. Note that Theorem A.7 shows that Br(Y ) D

H 1(Gk , JacY (ks)).

PROPOSITIONA.12. The following statements are equivalent:(a) the reciprocity law holds forK=kI

(b) for all finite cyclic extensionsF=K, the sequence

Br(F=K)!L

v2X 0 Br(Fw=Kv)! [F WK]�1Z=Z! 0

is exact;(c) for all finite cyclic extensionsF=K, H 1(Gal(F=K), Br(Y )) D 0, whereBr(Y ) is the

Brauer group of the projective smooth curve with function fieldF.

PROOF. It follows from (A.11) that the reciprocity law holds forK=k if and only ifGF=K D GF=K for all finite abelian extensionsF=K, and it suffices to check this forcyclic extensions. But, as we saw in the above proof, for such an extension the order ofG(F=K) is the order of the cokernel of Br(F=K)!

LBr(Fw=Kv). The equivalence of

(a) and (b) is now clear.Consider the exact commutative diagram

Br(X) ��! Br(K) ��!L

Br(Kw) ��! Q=Z ��! 0??y ??y˛ ??y ??yn

0 ��! (Br(F)=Br(Y ))Gal(F=K) ��! (L

Br(Fw))Gal(Fw=Kv) ��! Q=Z

wheren D [F W K]. From the Hochschild-Serre spectral sequences, we get exact sequences

0! Br(F=K)! Br(K)! Br(F)Gal(F=K)! H 3(Gal(F=K), F �)

0! Br(Fw=Kv)! Br(Kv)! Br(Fw)Gal(Fw=Kv)! H 3(Gal(Fw=Kv), F �

w ),


and from the periodicity of the cohomology of cyclic groups, we see that

H 3(Gal(F=K), F �) D H 1(Gal(F=K), F �) D 0,

H 3(Gal(Fw=Kv), F �v ) D H 1(Gal(Fw=Kv), F �

w ) D 0.

Thus the preceding diagram gives an exact sequence of kernels and cokernels,

Br(F=K)!M

Br(Fw=Kv)! n�1Z=Z! Coker(˛)! 0.

But, as Br(K)! Br(F)Gal(F=K) is surjective,

Coker(˛) D Coker�Br(F)Gal(F=K)

! (Br(F)=Br(Y ))Gal(F=K)�

,

which equalsH 1(Gal(F=K), Br(Y )) becauseH 1(Gal(F=K), Br(F)) D 0 (look at theHochschild-Serre spectral sequence). Thus (b) is equivalent to (c).

We say that theHasse principleholds forK=k if the mapK�=NF �!L

K�v =NF �

w

is injective for all finite cyclic field extensionsF of K.

PROPOSITIONA.13. The following are equivalent:(a) the Hasse principle holds forKI(b) H 1(Gk , JacX (ks)) D 0I

(c) Br(X) D 0.

In particular, the Hasse principle holds forK=k if k is algebraic over a finite field.

PROOF. As K�=NF �� Br(F=K) for F=K finite and cyclic, we see that that the Hasse

principle holds forK=k if and only if Br(F=K)!L

Br(Fw=Kv) is injective for allF=Kfinite and cyclic. As Br(K) D 0, Br(K) D [Br(F=K) where the union runs over all finitecyclic extensions. Thus the Hasse principle holds forK if and only if Br(K)!

LBr(Kv)

is injective, but the kernel of this map is Br(X) D H 1(Gk , JacX (ks)).

REMARK A.14. Let k0 be an algebraically closed field of characteristic zero, and letk

be the quasi-finite fieldk0((t)). In this case there exist elliptic curvesE over k withH 1(k, E) 6D 0, and there exist function fieldsK overk with finite extensionsF linearlydisjoint from ks such that every prime ofK splits completely inF (see Rim and Whap-les 1966). In lectures in 1966, Rim asked (rather pessimistically) whether the followingconditions on a quasi-finite fieldk are equivalent:

(a) k is algebraic over a finite field;(b) H 1(Gk , A) D 0 for all connected commutative algebraic group varieties overk;(c) the reciprocity law holds for allK=kI

(d) the Hasse principle holds for allK=k.

We have seen that (a)H) (b) H) (c),(d), but (b) does not imply (a). In fact Jar-dan has shown (1972, 1974) that ifk is finitely generated overQ, then for almost all� 2 Gal(ks=k), the fixed fieldk(�) of � is quasi-finite and has the property that everyabsolutely irreducible variety over it has a rational point; thus (b) holds fork(�).


REMARK A.15. We useetale cohomology to show that the group Ker(Br(K)!˚Br(Kv))

is indeed the Brauer group ofX . Let� WX ! Spec(k) be the structure morphism, and con-sider the exact sequence of sheaves

0! Gm! g�Gm! DivX ! 0

onXet (see Milne 1980, II 3.9; hereg is the inclusion of the generic point intoX and DivXis the sheaf of Weil divisors). On applying the right derived functors of��, we get a longexact sequence of sheaves on Spec(k)et which we can regard asGk-modules. The sequenceis

0! ks�! K

�! Div(X)! Pic(X)! 0,

which is exactly the sequence considered in the proof of Theorem A.7. Here it tells us that��Gm D ks�, R1��Gm D Pic(X), andRr��Gm D 0 for r � 2. Therefore the Lerayspectral sequence for� reduces to a long exact sequence

� � � ! H r (Gk , ks�)! H r (Xet, Gm)! H r�1(Gk , Pic(X))! � � � .

From this we can read off thatH 2(Xet, Gm) D H 1(Gk , Pic(X)), which proves what wewant becauseH 2(Xet, Gm) is equal to the Brauer group ofX.

EXERCISE A.16. Investigate to what extent the results in the second section remain truewhen the fieldsKv are replaced by the Henselizations ofK at its primes.

NOTES. Class field theory for complete fields with quasi-local residue fields was first de-veloped in Whaples 1952/54 (see also Serre 1962). The same theory for function fieldsover quasi-finite fields was investigated in Rim and Whaples 1966.

Chapter II

Etale Cohomology

In ~1 we prove a duality theorem forZ-constructible sheaves on the spectrum of a Henseliandiscrete valuation ring with finite residue field. The result is obtained by combining theduality theorems for modules over the Galois groups of the finite residue field and thefield of fractions. After making some preliminary calculations in~2, we prove in~3 ageneralization of the duality theorem of Artin and Verdier toZ-constructible sheaves onthe spectrum of the ring of integers in a number field or on curves over finite fields. Inthe following section, the theorems are extended to certain nonconstructible sheaves andto tori; also the relation between the duality theorems in this and the preceding chapteris examined. Section 5 treats duality theorems for abelian schemes,~6 considers singularschemes, and in~7 the duality theorems are extended to schemes of dimension greater thanone.

In this chapter, the reader is assumed to be familiar with the more elementary parts ofetale cohomology, for example, with Chapters II and III of Milne 1980. All schemes areendowed with theetale topology.

0 Preliminaries

We begin by reviewing parts of Milne 1980. Recall thatS(Xet) denotes the category ofsheaves of abelian groups onXet (smalletale site).

Cohomology with support on a closed subscheme

(Milne 1980, p73-78, p91-95)

Consider a diagram

Zi,! X

j - U

in which i andj are closed and open immersions respectively, andX is the disjoint union

123

124 CHAPTER II. ETALE COHOMOLOGY

of i(Z) andj(U ). There are the following functors between the categories of sheaves:

S(Zet)

i�

i�

!

i!

S(Xet)

j!

j�

!

j�

S(Uet).

Each functor is left adjoint to the one listed below it; for example, HomZ (i�F, F 0) �DHomX (F, i�F 0). The functorsi�, i�, j!, andj� are exact, andi! andj� are left exact. Thefunctorsi�, i!, j�, andj� map injective sheaves to injective sheaves. For any sheafF onX , there is a canonical exact sequence

0! j!j�F ! F ! i�i�F ! 0. (II.1)

PROPOSITION0.1. (a) For any sheavesF onU andF 0 onX , the restriction map

ExtrX (j!F, F 0)! ExtrU (F, j�F 0)

is an isomorphism forr � 0; in particular,

ExtrX (j!Z, F 0) �D H r (U, F 0jU ), r � 0.

(b) For any sheavesF onX andF 0 onU , there is a spectral sequence

ExtrX (F, Rsj�F 0) H) ExtrCsU (F jU, F 0).

(c) For any sheavesF onX andF 0 onZ, there is a canonical isomorphism

ExtrX (F, i�F 0)�D�! ExtrZ (i�F, F 0), r � 0.

(d) For any sheafF onX , there is a canonical isomorphism

H rZ (X, F) �D ExtrX (i�Z, F ), r � 0I

consequently, for any sheafF onZ,

H rZ (X, i�F) �D H r (Z, F), r � 0.

(e) For any sheavesF onZ andF 0 onX , there is a spectral sequence

ExtrZ (F, Rsi!F 0) H) ExtrCsX (i�F, F 0).

(f) For any sheafF onX , there is a long exact sequence

� � � ! H rZ (X, F)! H r (X, F)! H r (U, F)! � � � .

0. PRELIMINARIES 125

PROOF. (a) If F 0! I � is an injective resolution ofF 0, thenj�F 0

! j�I � is an injec-tive resolution ofj�F 0, becausej� is exact and preserves injectives. On passing to thecohomology groups in

HomX (j!F, I �) �D HomU (F, j�I �),

we obtain isomorphisms ExtrX (j!F, F 0) �D ExtrU (F, j�F 0).

(b) Asj� is left exact and preserves injectives, and HomX (F,�)ıj��D HomU (j!F,�),

this is the spectral sequence of a composite of functors (Milne 1980, Appendix B, Theorem1).

(c) The proof is the same as that of (a): asi� is exact and preserves injectives, onforming the derived functors of HomX (F, i�(�)) �D HomZ (i�F,�), we obtain canonicalisomorphisms Extr

X (F, i�(�)) �D ExtrZ (i�F,�).(d) From the exact sequence (see (II.1))

0! HomX (i�Z, F )! HomX (Z, F )! HomX (j!Z, F )

we see that

HomX (i�Z, F ) �D Ker(� (X, F)! � (U, F))dfD �Z (X, F).

Hence HomX (i�Z,�) �D �Z (X,�), and on passing to the derived functors we obtain therequired canonical isomorphism. The second statement can be obtained by combining thefirst with (c).

(e) Asi! is left exact and preserves injectives, and HomZ (F,�) ı i! �D HomU (i�F,�),this is the spectral sequence of a composite of functors.

(f) Form the ExtX (�, F )-sequence of

0! j!Z! Z! i�Z! 0

(see (II.1)) and apply (a) and (d).

The exact sequence in (0.1f) is referred to as thecohomology sequence of the pairX � U.

Extensions of sheaves

We generalize Theorem 0.3 of Chapter I to theetale topology.If Y is a Galois covering of a schemeX with Galois groupG, then for anyG-module

M , there is a unique locally constant sheafFM on X such that� (Y, FM ) D M (as aG-module) (Milne 1980, III 1.2). In the next theorem we use the same letter forM andFM .

For anyetale mapf WU ! X , letZU D f!Z. Then every sheaf onXet is a quotient of adirect sum of sheaves of the formZU (cf. the second proof of III 1.1, Milne 1980), and sucha sheaf is flat. Therefore, flat resolutions exist inS(Xet). The conditionT orZ

r (M, N) D 0

for r > 0 in the next theorem means that, for any flat resolutionF �! N of N , M ˝Z

F �!M ˝Z N is a resolution ofM ˝Z N .


THEOREM0.2. LetY be a finite Galois covering ofX with Galois groupG, and letN andP be sheaves onXet. Then, for anyG-moduleM such thatT orZ

r (M, N) D 0 for r > 0,there is a spectral sequence

ExtrG(M, ExtsY (N, P)) H) ExtrCsX (M ˝Z N, P).

In particular, there is a spectral sequence

H r (G, ExtsY (N, P)) H) ExtrCsX (N, P).

The second spectral sequence is obtained from the first by takingM D Z. After a fewpreliminaries, the first will be shown to be the spectral sequence of a composite of functors.

LEMMA 0.3. For any sheavesN and P on X and G-moduleM , there is a canonicalisomorphism

HomG(M, HomY (N, P)) �D HomX (M ˝Z N, P).

PROOF. Almost by definition of tensor products, there is a canonical isomorphism

HomY (M,HomY (N, P)) �D HomY (M ˝Z N, P).

BecauseM becomes the constant sheaf onY ,

HomY (M,HomY (N, P)) �D Hom(M, HomY (N, P))

(homomorphisms of abelian groups). On takingG-invariants, we get the required isomor-phism.

LEMMA 0.4. If I is an injective sheaf onX andF is a flat sheaf onX , thenHomY (F, I)

is an injectiveG-module.

PROOF. We have to check that the functor HomG(�, HomY (F, I))WS(Xet) ! Ab is ex-act, but Lemma 0.3 expresses it as the composite of two exact functors� ˝Z F andHomX (�, I).

LEMMA 0.5. Let N andI be sheaves onX with I injective, and letM be aG-module. IfT orZ

r (M, N) D 0 for r > 0, thenExtrG(M, HomY (N, I)) D 0 for r > 0.

PROOF. By assumption, a flat resolutionF � ! N of N gives a resolutionM ˝Z F � !

M ˝Z N of M ˝Z N , and it follows from the injectivity ofI that HomX (M ˝Z N, I)!

HomX (M ˝Z F �, I) is then a resolution of HomX (M ˝ N, I). In particular, on takingM D Z[G], we get a resolution

HomX (Z[G]˝Z N, I)! HomX (Z[G]˝Z F �, I) (II.2)

of HomX (Z[G]˝Z N, I). But

HomX (Z[G]˝Z F, I) �D HomG(Z[G], HomY (F, I)) �D HomY (F, I)


for any sheafF , and so (II.2) can be regarded as a resolution

HomY (N, I)! HomY (F �, I)

of theG-module HomY (N, I). In fact (0.4), this is an injective resolution of HomY (N, I),which we use to compute Extr

G(M, HomY (N, I)). From (0.3) we know that

HomG(M, HomY (F �, I)) �D HomX (M ˝Z F �, I),

and we have already noted that this last complex is exact except at the first step. Conse-quently ExtrG(M, HomY (N, I)) D 0 for r > 0.

We now prove Theorem 0.2. Lemma 0.3 shows that

HomX (M ˝Z N,�)

is the composite of the functors HomY (N,�) and HomG(M,�), and Lemma 0.5 showsthat the first of these sends injective objectsI to objects that are acyclic for the secondfunctor. We therefore obtain the spectral sequence as that associated with a composite offunctors (Milne 1980, Appendix B, Theorem 1).

COROLLARY 0.6. LetM andN be sheaves onX , and letY be a finite Galois covering ofX . Then

ExtrY (M, N) �D ExtrX (��M, N).

PROOF. On applying Theorem 0.2 withM D Z[G], we find that

HomG(Z[G], ExtrY (M, N)) �D ExtrX (Z[G]˝Z M, N),

but the first group is ExtrY (M, N), and, in the second,Z[G]˝M D ��

�M.

Pairings

For any sheavesM , N , andP onX , there are canonical pairings

ExtrX (N, P)� ExtsX (M, N)! ExtrCsX (M, P),

which can be defined in the same way as the pairings in (I~0). Also, if X is quasi-projectiveover an affine scheme (as all our schemes will be), then we can identify the cohomologygroups with theCech groups (Milne 1980, III 2.17) and use the standard formulas (ibid. V1.19) to define cup-product pairings

H r (X, M)�H s(X, N)! H rCs(X, M ˝N).

Recall also (ibid. III 1.22) that there is a spectral sequence

H r (X, Ext sX (M, N)) H) ExtrCs

X (M, N)

whose edge morphisms are mapsH r (X,HomX (M, N))! ExtrX (M, N). As we noted inthe proof of Lemma 0.3, a pairingM�N ! P corresponds to a mapM ! HomX (N, P).


PROPOSITION0.7. Let M � N ! P be a pairing of sheaves onX , and consider thecomposed map

H r (X, M)! H r (X,HomX (N, P))! ExtrX (M, N).

Then the diagram

H r (X, M)�H s(X, N) > H rCs(X, P) cup-product pairing

jj jj

ExtrX (N, P)_

�H s(X, N) > H rCs(X, P) Ext pairing

commutes.

PROOF. See Milne 1980, V 1.20.

The Cech complex

Let F be a sheaf onXet. For anyU etale overX , let C �(V =U, F) be the Cech complexcorresponding to a covering(V ! U ), and defineC�(F)(U ) to be lim

�!C �(V =U, F) (direct

limit over theetale coverings(V ! U )). ThenC�(F) is a complex of presheaves onX ,and we letC �(U, F) D � (U, C�(F)) be the complex of its sections overU . In the nextproposition, we writeHr (F) for the presheafU 7! H r (U, F).

PROPOSITION0.8. Assume thatX is quasi-projective over an affine scheme.(a) For any sheafF onX , H r (C�(F)) �D Hr (F) andH r (C �(X, F)) �D H r (X, F).

(b) For any morphismf WY ! X , there is a canonical mapf �C�(F) ! C�(f �F),which is a quasi-isomorphism iff is etale.

(c) For any pair of sheavesF andF 0, there is a canonical pairingC�(F) � C�(F 0)!

C�(F ˝ F 0) inducing the cup-product on cohomology.(d) Let X be the spectrum of a fieldK, and letF be the sheaf onX corresponding to

theGK -moduleM . ThenC �(X, F) is the standard resolution ofM (defined usinginhomogeneous cochains).

PROOF. (a) By definition,H r (C �(V =U, F)) D LH r (V =U, F), and therefore

H r (C�(F)(U )) D lim�!

H r (C �(V =U, F)) D lim�!LH r (V =U, F) D LH r (U, F).

Under our assumptions, theCech groups agree with derived-functor groups, and so thissays that

H r (� (U, C�(F))) �D H r (U, F)dfD � (U,Hr (F)),

which proves both the equalities.(b) For anyetale mapV ! X , there is a canonical map� (V , F)! � (V(Y ),f

�(F)).In particular, whenU is etale overX and(V ! U ) is anetale covering ofU , then thereis a canonical map� (V r , F )! � (V r

(Y ),f �(F)) (hereV r denotesV �U V �U ...), all r .

On passing to the limit overV , we obtain a map

� (U, Cr (F))! � (U(Y ), Cr (f �F))


for all r , and these maps give a map of complexesC�(F)! f�C�(f �F). By adjointness,we get a mapf �C�(F) ! C�(f �F). The last part of the statement is obvious because,whenf is etale,

H r (f �C�(F)) �D Hr (F)jU �D Hr (F jU ) �D H r (C�(f �F)).

(c) For eachU ! X , the standard formulas define a pairing of complexes

� (U, C�(F))� � (U, C�(F 0))! � (U, C�(F ˝ F 0)),

and these pairings are compatible with the restriction maps.(d) If U is a finite Galois covering ofX with Galois groupG, then it is shown in Milne

1980, III 2.6, thatC �(U=X, F) is the standard complex for theG-moduleF(U ). Theresult follows by passing to the limit.

Constructible sheaves

Let X be a scheme of Krull dimension one. A sheafF on such a scheme isconstructibleif there is a dense open subsetU of X such that

(a) for some finiteetale coveringU 0 ! U , the restriction ofF to U 0 is the constantsheaf defined by a finite group;

(b) for all x =2 U , the stalkFx of F is finite.It is said to beZ-constructibleif its restriction to some suchU 0 is the constant sheaf de-fined by a finitely generated group and the stalksFx are finitely generated. Note that aconstructible sheaf isZ-constructible and aZ-constructible sheaf is constructible if andonly if it is torsion.

The constructible sheaves form an abelian subcategory ofS(Xet) and if

0! F 0! F ! F 00

! 0

is exact, thenF is constructible if and only ifF 0 andF “ are constructible. For a morphism� locally of finite type,�� carries constructible sheaves to constructible sheaves, and when� is finite,�� has the same property. Similar statements hold forZ-constructible sheaves.

PROPOSITION0.9. If X is quasi-compact, then every sheaf on X is a filtered direct limitof Z-constructible sheaves. Moreover, every torsion sheaf is a filtered direct limit of con-structible sheaves.

PROOF. Let F be a sheaf onX , and consider all pairs(gWU ! X, s) with g etale,Uaffine, ands a section ofF overU . For each such pair, we have a mapZ! F jU sending1 to s. This induces a mapg�Z! F andg�Z is Z-constructible (becauseg is finite overa dense open subset ofX ). Therefore the image ofg�Z in F is Z-constructible, and it isclear that the union of all subsheaves of this form isF . WhenF is torsion, then each of thesubsheaves, being torsion andZ-constructible, is constructible.


Mapping cones

For a complexA�, A�[1] denotes the complex with(A�[1])rD ArC1 and the differential

d rD �d rC1

A . Let uWA�! B� be a map of complexes. Themapping coneC �(u)

corresponding tou is the complexA�[1]˚B� with the differentiald r D �d rC1A CurC1Cd r

B.ThusC r (u) D ArC1

˚ Br , and the differential is(a, b) 7! (�da, ua C db). There is anobvious injectioniWB� ,! C �(u) and an obvious projectionpWC �(u)[�1] � A�, and thedistinguished triangle corresponding tou is

C �(u)[�1]�p! A� u

! B� i! C �(u).

By definition, every distinguished triangle is isomorphic to one of this form.A short exact sequence

0! A� u! B� v

! C �! 0

gives rise to a distinguished triangle

C �[�1]w! A� u

! B� v! C �

in whichw is defined as follows: letqWC �(u) ! C � bev on B� and zero onA�[1]; thenq is a quasi-isomorphism, and so we can definew to be(�p) ı q�1[�1]. A distinguishedtriangle of complexes of sheaves onXet

C �[�1]p! A� u

! B� i! C �,

gives rise to a long exact sequence of hypercohomology groups

� � � ! Hr (X, A�)! Hr (X, B�)! Hr (X, C �)! HrC1(X, A�)! � � �

PROPOSITION0.10. (a) A morphism of exact sequences of complexes

0 ��! A��! B�

��! C ��! 0??ya

??yb

??yc

0 ��! D� ��! E� ��! F � ��! 0

defines a distinguished triangle

C �(c)[�1]! C �(a)! C �(b)! C �(c).

(b) Assume that the rows of the diagram

C �[�1] ��! A��! B�

��! C �??y ??yF �[�1] ��! D�

��! E��! F �

1. LOCAL RESULTS 131

are distinguished triangles and that the diagram commutes; then the diagram can be com-pleted to a morphism of distinguished triangles.

(c) For any maps

A�u��! B�

v��! C �

of complexes, there is a distinguished triangle

C �(v)[�1]! C �(u)! C �(v ı u)! C �(v).

PROOF. The statements are all easy to verify. (Note that (b) and (c) are special cases of theaxioms (TR2) and (TR3) (Hartshorne 1966, I 1) for a triangulated category; also that thedistinguished triangle in (c) is the analogue for complexes of the kernel-cokernel sequence(I 0.24) of a pair of maps.)

1 Local results

Except when stated otherwise,X will be the spectrum of an excellent Henselian discretevaluation ringR with field of fractionsK and residue fieldk. For example,R could be acomplete discrete valuation ring or the Henselization of the local ring at a prime in a globalfield. We shall use the following notations:

Ks

j I D Gal(Ks=Kun)

ks� Run

� Kun G D Gal(Ks=K)

j j j g D Gal(ks=k) D G=I

k � R � K

Speck D xi! X

j u D SpecK

Preliminary calculations

We compute the cohomology groups ofZ andGm.

PROPOSITION1.1. (a) Let F be a sheaf onu; then H r (X, j!F) D 0 for all r � 0, andconsequently there is a canonical isomorphism

H r (u, F)�D! H rC1

x (X, j!F), all r � 0.

(b) For any sheafF on X , the mapH r (X, F) ! H r (x, i�F) is an isomorphism allr � 0.

PROOF. (a) The cohomology sequence of the pairX � u (see 0.1f)

� � � ! H rx (X, j!F)! H r (X, j!F)! H r (u, j!F ju)! � � �

shows that the first part of the statement implies the second.


Let M be the stalkFu of F atu regarded as aG-module. The functorF 7! i�j�F canbe identified with

M 7!M IWModG ! Modg.

The equality Homg(N, M I ) D HomG(N, M) for N a g-module shows thati�j� has anexact left adjoint, namely, “regard theg-module as aG-module”, and soi�j� preservesinjectives. Consider the exact sequence (II.1)

0! j!F ! j�F ! i�i�j�F ! 0.

If F is injective, this is an injective resolution ofj!F becausej� andi� preserve injectives.

As H 0(X, j�F)! H 0(X, i�i�j�F) is the isomorphismM G�D! (M I )g, the cohomology

sequence of the sequence shows thatH 0(X, j!F) D 0 for all F and thatH r (X, F) D 0

for all r � 0 if F is injective. In particular, we see that ifF is injective, thenj!F is acyclicfor � (X,�).

Let F ! I � be an injective resolution ofF . Thenj!F ! j!I� is an acyclic resolution

of j!F , and soH r (X, j!F) �D H r (� (X, j!I�)). But H r (� (X, j!I

�)) �D (Rrf )(F) wheref is the functorF 7! � (X, j!F) D 0, and soH r (X, j!F) D 0 for all r.

(b) The cohomology sequence of

0! j!j�F ! F ! i�i�F ! 0

yields the required isomorphisms.

COROLLARY 1.2. For all r , H r (X, Z) D H r (g, Z); in particular, whenk is finite,

H r (X, Z) D Z 0 Q=Z 0

r D 0 1 2 � 3.

PROOF. This follows immediately from part (b) of the proposition.

LEMMA 1.3. If k is algebraically closed, thenH r (K, Gm) D 0 for all r � 1.

PROOF. The assumption thatR is excellent entails thatbK is separable1 over K, and wehave seen in (I A.6) thatK is algebraically closed inbK. ThereforebK is a regular extensionof K, and so we can apply Shatz 1972, Theorem 27, p116, to obtain thatK is aC1 field. Itfollows thatK has cohomological dimension at most 1, and soH r (K, Gm) D 0 for r � 2

(Serre 1964, II.3).

LEMMA 1.4. If k is perfect, thenRrj�Gm D 0 for all r > 0; therefore

ExtrX (F, j�Gm) �D Extru(F ju, Gm)

for all sheavesF onX and all r.

1A field K is separableover a subfieldk if either the characteristic is0 or the characteristic isp 6D 0 and

K is linearly disjoint fromk1p overk (Jacobson 1964, p166). It is aregular extensionof k if it is separable

overk andk is algebraically closed inK (Zariski and Samuel 1960, p229).


PROOF. The stalks ofRrj�Gm are (see Milne 1980, III 1.15)

(Rrj�Gm)x�D H r (Kun, Gm) D 0, r > 0 (by 1.3)

(Rrj�Gm)x�D H r (Ks, Gm) D 0, r > 0.

This proves the first assertion, and the second follows from (0.1b).

PROPOSITION1.5. Assume thatk is finite.(a) For all r > 0, H r (X, Gm) D 0.

(b) We haveH r

x (X, Gm) D 0 Z 0 Q=Z 0

r D 0 1 2 3 > 3.

PROOF. (a) As Rrj�Gm D 0 for r � 0, H r (X, j�Gm) �D H r (K, Gm) all r . Moreover,H r (K, Gm) D 0 for r 6D 0, 2 and H 2(K, Gm) �D H 2(k, Z) �D Q=Z (cf. Lemma 1.3or I A.1). On the other hand,H r (X, i�Z) �D H r (x, Z) for all r (0.3c), and so the exactsequence

0! Gm! j�Gm

ord! i�Z! 0

gives rise to an exact sequence

0! H 0(X, Gm)!K� ord! Z! H 1(X, Gm)! 0! 0!

H 2(X, Gm)! H 2(K, Gm)�D! H 2(k, Z)! 0! H 3(X, Gm)! 0! � � � ,

which yields the result.(b) Consider the cohomology sequence of the pairX � u (see 0.1f),

0 �!H 0x (X, Gm) �!H 0(X, Gm) �! H 0(K, Gm) �!H 1

x (X, Gm) �!H 1(X, Gm) �! � � � R�

ord�! K� 0.

From the part we have displayed, it is clear thatH 0x (X, Gm) D 0 andH 1

x (X, Gm) �D Z. Theremainder of the sequence gives isomorphismsH r (K, Gm) �D H rC1

x (X, Gm) for r � 1,from which the values ofH r

x (X, Gm), r � 2, can be read off.

COROLLARY 1.6. Assume thatk is finite. If n is prime to char(k), thenH rx (X,�n) �D

Z=nZ for r D 2, 3, andH rx (X,�n) D 0 otherwise.

PROOF. As n is prime to char(k), the sequence

0! �n! Gm

n! Gm! 0

is exact, and the result follows immediately from (1.5).


REMARK 1.7. (a) Part (a) of (1.5) can also be obtained as a consequence of the followingmore general result (Milne 1980, III 3.11(a)): ifG is a smooth commutative group schemeover the spectrum of a Henselian local ring, thenH r (X, G) �D H r (x, G0) for r � 1, wherex is the closed point ofX , andG0 is the closed fibre ofG=X .

Alternatively, (1.1b) shows thatH r (X, Gm) �D H r (x, i�Gm). Obviously i�Gm cor-responds to theg-moduleRun�, and it is not difficult to show thatH r (g, Run�) D 0 forr > 0 (cf. I A.2).

(b) There is an alternative way of computing the groupsH rx (X, Gm). Wheneverk is

perfect, (1.4) shows thatH r (X, j�Gm) ! H r (u, Gm) is an isomorphism for allr , and itfollows immediately thatH r

x (X, j�Gm) D 0 for all r . Therefore the exact sequence

0! Gm! j�Gm

ord! i�Z! 0

leads to isomorphismsH rx (X, i�Z)

�! H rC1

x (X, Gm), and we have seen in (0.1d) thatH r

x (X, i�Z) �D H r (x, Z). ThereforeH rx (X, Gm) �D H r�1(x, Z) for all r.

This argument works wheneverk is perfect. For example, whenk is algebraicallyclosed, it shows thatH r

x (X, Gm) D 0, Z, 0, ... for r D 0, 1, 2, ... from which it followsthat, ifn is prime to char(k), thenH r

x (X,�n) D 0, 0, Z=nZ, 0, ... forr D 0, 1, 2, ... . Thesevalues ofH r (X,�n) are those predicted by the purity conjecture (Artin, Grothendieck, andVerdier 1972/73, XIX).

SinceRr i!Gm is the sheaf onz associated with the presheafz07! H r

z0(X0, Gm) (here

X 0! X is the etale covering ofX corresponding toz0

! z), we see that ifk is anyperfect field, thenR1i!Gm

�D Z andRr i!Gm D 0 for r 6D 1, and consequently that if(n,char(k)) D 1, thenR2i!�n

�D Z=nZ andRr i!�n D 0 for r 6D 2. For this last statementconcerning�n it is not even necessary to assume thatk is perfect.

The duality theorem

We now assume that the residue fieldk is finite. There are two natural candidates for a tracemapH 3

x (X, Gm)! Q=Z. The first is that in (1.5), namely, the composite of the inverse of

H 2(u, Gm)�D! H 3

x (X, Gm) with H 2(u, Gm) D H 2(GK , K�s )

invK

! Q=Z. The second is that

in (1.7b), namely, the composite of the inverse ofH 2(x, Z) D H 2x (X, Z)

�D! H 3

x (X, Gm)

with H 2(x, Z) D H 2(Gk , Z)invk

! Q=Z. From the definition of invK (see I 1.6) it is clearthat the two methods lead to the same map.

Recall (0.1d) that for any sheafF on X , H rx (X, F) �D ExtrX (i�Z, F ), and so there is a

canonical pairing

ExtrX (F, Gm)�H 3�rx (X, F)! H 3

x (X, Gm).

On combining the pairing with the trace map, we obtain a map

˛r (X, F)WExtrX (F, Gm)! H 3�rx (X, F)�.

Before we can state the theorem, we need to endow HomX (F, Gm) with a topology.We shall see below that the restriction map

HomX (F, Gm)! Homu(F ju, Gm) �D HomG(Fu, Ks�)


is injective. The last group inherits a topology from that onK�s , and we give HomX (F, Gm)

the subspace topology. WhenF isZ-constructible, all subgroups of HomX (F, Gm) of finiteindex prime to char(K) are open.

THEOREM 1.8. (a) Let F be aZ-constructible sheaf onX without p-torsion if K hascharacteristicp 6D 0.

i) The map 0(X, F) defines an isomorphism

HomX (F, Gm)^! H 3(X, F)�

(completion for the topology of open subgroups of finite index).ii) The groupExt1X (F, Gm) is finitely generated, and1(X, F) defines an isomor-

phismExt1X (F, Gm)^

! H 3(X, F)�

(completion for the topology of subgroups of finite index).iii) For r � 2, the groupsExtrX (F, Gm) are torsion of cofinite-type, andr (X, F)

is an isomorphism.(b) LetF be a constructible sheaf onX , and assume thatK is complete or thatpF D F

for p D charK. Then the pairing


x (X, Gm) �D Q=Z.

is nondegenerate; ifpF D F , then all the groups are finite.

We first consider a sheaf of the formi�F , F aZ-constructible sheaf onx. Recall (0.1d),that for such a sheafH r

x (X, i�F) �D H r (x, F), all r.

LEMMA 1.9. For any sheafF on x, there is a canonical isomorphismExtr�1x (F, Z)

�D!

ExtrX (i�F, Gm), all r � 1.

PROOF. From the exact sequence

0! Gm! j�Gm! i�Z! 0


� � � ! ExtrX (i�F, Gm)! ExtrX (i�F, j�Gm)! ExtrX (i�F, i�Z)! � � � .

But ExtrX (i�F, j�Gm) �D Extru(i�F ju, Gm) in view of (0.1b) and (1.4), and the secondgroup is zero becausei�F ju D 0. Also ExtrX (i�F, i�Z) �D Extrx(F, Z) by (0.1c), and so thesequence gives the required isomorphisms.

It is obvious from the various definitions that the diagram

ExtrX (i�F, Gm)�H 3�rx (X, i�F) > H 3

x (X, Gm)�D Q=Z

jj

Extr�1x (F, Z)

^�D

� H 3�r (x, F) > H 2(x, Z)

^�D

�D Q=Z


commutes. The lower pairing can be identified with the pairing in (I 1.10). We deduce:Ext1(i�F, Gm) is finitely generated and1(X, i�F) defines an isomorphism Ext1(i�F, Gm)^

!

H 2x (X, i�F)� (completion for the profinite topology);2(X, i�F) is an isomorphism of fi-

nite groups; 3(X, i�F) is an isomorphism of torsion groups of cofinite-type; for all othervalues ofr , the groups are zero. WhenF is constructible, it corresponds to a finiteg-module, and so all the groups are finite (and discrete). This completes the proof of thetheorem for a sheaf of the formi�F.

We next consider a sheaf of the formj!F , with F aZ-constructible sheaf onu withoutp-torsion. Consider the diagram

ExtrX (j!F, Gm)�H 3�rx (X, j!F) > H 3

x (X, Gm)�D Q=Z

Extru(F, Gm)_

�D

� H 2�r (u, F)

^�D

> H 2(u, Gm)

^�D

�D Q=Z

in which the first isomorphism is restriction fromX to u (see 0.1a), and the two remainingisomorphisms are boundary maps in the cohomology sequence of the pairX � u (see (1.1)and (1.5)). It is again clear from the various definitions that the diagram commutes. Thelower pairing can be identified with that in (I 2.1). We deduce:ı HomX (j!F, Gm) is finitely generated and0(X, j!F) defines an isomorphism

HomX (j!F, Gm)^! H 3

x (X, j!F)�

(completion for the topology of open subgroups of finite index);ı ˛1(X, j!F) is an isomorphism of finite groups;ı ˛2(X, i�F) is an isomorphism of torsion groups of cofinite type;ı for all other values ofr , the groups are zero.

WhenF is constructible, it corresponds to a finiteG-module, and all the groups are finite(and discrete).

This completes the proof of the theorem whenF � i�i�F or F � j!j�F andF is

without p-torsion. For a generalZ-constructible sheafF without p-torsion, we use theexact sequence

0! j!(F jU )! F ! i�i�F ! 0,

and apply the five-lemma to the diagram

� ��! ExtrX (i�i�F, Gm) ��! ExtrX (F, Gm) ��! ExtrX (j!(F jU ), Gm) ��! �??y�

??y�

??y ??y�

??y�

� ��! H 3�rx (X, i�i�F)� ��! H 3�r

x (X, F)� ��! H 3�rx (X, j!(F jU ))� ��! �

This leads immediately to a proof of (a) of the theorem forr � 2. For r < 2, one only hasto replace the first four terms in the top row of the diagram with their completions:

0 ��! Hom(F, Gm)^��! Hom(j!(F jU ), Gm)^

��! Ext1X (i�i�F, Gm)^��! � � �??y ??y�

??y�

0 ��! H 3x (X, F)�

��! H 3x (X, j!(F jU ))�

��! H 2x (X, i�i�F)�

��! � � �


Note that the top row is exact by virtue of (I 0.20a).The remaining case, whereK is complete andF is constructible withp torsion, can be

treated similarly.

COROLLARY 1.10. Letp D chark.

(a) Let F be a locally constant constructible sheaf onX such thatpF D F , and letF D D Hom(F, Gm). Then there is a canonical nondegenerate pairing of finitegroups

H r (X, F D)�H 3�rx (X, F)! H 3

x (X, Gm) �D Q=Z.

(b) LetF be a constructible sheaf onu such thatpF D F , and letF D D Homu(F, Gm).The pairingF D

� F ! Gm extends to a pairingj�F D� j�F ! Gm, and the re-

sulting pairing

H r (X, j�F D)�H 3�rx (X, j�F)! H 3

x (X, Gm) �D Q=Z

is nondegenerate.

PROOF. (a) We shall use the spectral sequence (Milne 1980, III.1.22)

H r (X, Ext sX (F, Gm)) H) ExtrCs

X (F, Gm)

to show that the term ExtrX (F, Gm) in the theorem can be replaced withH r (X, F D). Ac-

cording to Milne 1980, III.1.31, the stalk ofExtsX (F, Gm) at x is Exts(Fx, R�

un) (Ext asabelian groups). This group is zero fors > 0 becauseR�

un is divisible by all primes di-viding the order ofFx. The stalk ofExt s

X (F, Gm) at u is Exts(Fx, K�s ), which is zero for

s > 0 by the same argument. ThereforeExt sX (F, Gm) is zero fors > 0, and the spectral

sequence collapses to give isomorphismsH r (X, F D) �D ExtrX (F, Gm).

(b) On applyingj� to the isomorphismF D�D! Homu(F, Gm), we obtain an isomor-

phismj�F D�D! j�Homu(F, Gm). But

j�Homu(F, Gm) �D j�Hom(j�j�F, Gm) �D HomX (j�F, j�Gm)

(see Milne 1980, II 3.21), and from the Ext sequence of

0! Gm! j�Gm! i�Z! 0

and the vanishing of HomX (j�F, i�Z) �D Homx(i�j�F, Z) we find thatHomX (j�F, j�Gm) �DHomX (j�F, Gm). Thusj�F D �D HomX (j�F, Gm) and the existence of the pairingj�F D

�

j�F ! Gm is obvious.Next we shall show thatExt r

X (j�F, Gm) D 0 for r > 0. Let M D Fu. If M ID M ,

thenj�F is locally constant, and we showed in the proof of part (a) of the corollary that thehigherExt ’s vanish for such sheaves. IfM I D 0, thenj�F D j!F , andExt r

X (j!F, Gm) �Dj�Ext r

u (F, Gm). The stalk ofj�Ext ru (F, Gm) atx is H r (Kun, M D). BecauseM has order

prime top, H r (I, M D) �D H r (I=Ip, M D), whereIp is thep-Sylow subgroup ofI . ButH r (I=Ip, M D) is zero forr > 1, andH 1(I=Ip, M D) is dual toH 0(I=Ip, M), whichequalsM I (cf. the proof of I 2.18). By assumption, this is zero. BecauseI is normal


in GK , everyGK -module has a composition series whose quotientsQ are such that eitherQID Q or QI

D 0. Our arguments therefore show thatExt rX (j�F, Gm) D 0 for r > 0.

The spectral sequence

H r (X, Ext sX (j�F, Gm)) H) ExtrCs

X (j�F, Gm)

therefore reduces to a family of isomorphismsH r (X, j�F D)�D! ExtrX (j�F, Gm), and the

corollary follows from the theorem.

REMARK 1.11. (a) Part (a) of the theorem is true without the condition thatF has nop

torsion,p D charK, provided one endows Ext1X (F, Gm) with a topology deduced from

that onK and defines Ext1X (F, Gm)^ to be the completion with respect to the topology

of open subgroups of finite index. Note that˛1(X, i�Z) is the natural inclusionZ ,! bZ,and that 1(X, j!Z=pZ) for p D charK is an isomorphism of infinite compact groupsK�=K�p

! (K=}K)� whenK is complete. The first example shows that it is necessaryto complete Ext1X (F, Gm) in order to obtain an isomorphism, and the second shows thatit is necessary to endow Ext1

X (F, Gm) with a topology coming fromK because not allsubgroups of finite index inK�=K�p are open.

(b) By using derived categories, it is possible to restate (1.8) in the form of (1.10)for any constructible sheafF such thatpF D F . Simply setF D D RHom(F, Gm)

(an object in the derived category of the category of constructible sheaves onX), andnote thatHr (X, F D) D ExtrX (F, Gm). (The point of the proof of (1.10) is to show thatH r (RHom(F, Gm)) D 0 for r > 0 whenF is locally constant.)

Singular schemes

We now letX D SpecR with R the Henselization of an excellent integral local ring ofdimension1 with finite residue fieldk. Then R is again excellent, but it need not bereduced. Letu D fu1, ..., umg be the set of points ofX of dimension0. ThenOX,ui

is a fieldKi, and the normalizationQR of R is a product of excellent Henselian discretevaluation ringsRi such thatRi has field of fractionsKi (see Raynaud 1970, IX). We havea diagram

fx1, . . . , xmgQi> QX <

Qju

��

�j

x_ i

> X_�

with QX D Spec QR andx andxi the closed points ofX and SpecRi respectively. Forh inthe total ring of fractions ofR, define

ord(h) DP

[k(xi)W k(x)] � ordi(h)

where ordi is the valuation onKi. One can define a similar map for anyU etale overX ,and so obtain a homomorphism ordW j�Gm ! i�Z. DefineG be the complex of sheavesj�Gm! i�Z onX.


LEMMA 1.12. (a) For all r > 0, Rrj�Gm D 0; therefore

ExtrX (F, j�Gm) �D Extru(F ju, Gm)

for all sheavesF onX and all r .(b) For all r , there is a canonical isomorphismH r�1(x, Z)! H r

x (X, G).

PROOF. (a) The map� is finite, and therefore�� is exact. Asj� D ��Qj�, this shows that

Rrj�Gm�D ��Rr Qj�Gm, which is zero by (1.4).

(b) From (a) and (0.1) we see that

H rx (X, j�Gm) �D ExtrX (i�Z, j�Gm) �D Extru(i�Zju, Gm) D 0,

all r . The exact sequence

� � � ! H rx (X, G)! H r

x (X, j�Gm)! H rx (X, i�Z)! � � �

now leads immediately to the isomorphism.

We define the trace mapH 3x (X, Gm)

�D! Q=Z to be the composite of the inverse of

H 2(x, Z)�D! H 3

x (X, Gm) and invk WH 2(g, Z)�D! Q=Z.

THEOREM 1.13. For any constructible sheafF onX ,


x (X, Gm) �D Q=Z,

is a nondegenerate pairing of finite groups.

As in the case thatX is regular, it suffices to prove this for sheaves of the formi�F andj!F.

LEMMA 1.14.For all r , ExtrX (i�F, j�Gm) D 0; therefore the boundary mapsExtr�1x (F, Z)!

ExtrX (i�F, Gm) are isomorphisms.

PROOF. The proof is the same as that of (1.9).

The theorem fori�F now follows from the diagram:

ExtrX (i�F, G)�H 3�rx (X, i�F) > H 3

x (X, G)�D Q=Z

Extr�1x (F, Z)

^�D

� H 3�r (x, F)

^�D

> H 2(x, Z)

^�D

�D Q=Z.

We next consider a sheaf of the formj!F.

LEMMA 1.15. For all r , H r (X, j!F) D 0; therefore the maps

H r�1(u, F)! H rx (X, j!F)

are isomorphisms.



0 ��! ��Qj!F ��! ��

Qj�F 0 ��! j!F ��! j�F ��! i�i�j�F ��! 0.

Because(��Qj!F)x D 0, the map of��

Qj!F into i�i�j�F is zero, and therefore the image of��Qj!F is contained inj!F . The resulting map��

Qj!F ! j!F induces isomorphisms on thestalks and therefore is itself an isomorphism. The first assertion now follows from (1.1),and the second is an immediate consequence of the first.

The theorem forj!F now follows from the diagram:

ExtX (j!F, G) � H 3�rx (X, j!F) > H 3

x (X, G) �D Q=Z

Li Extrui

(F, Gm)_

�D

�L

iH2�r (ui , Fi)

^�D

>L

iH2(ui , Gm)

^

�D(Q=Z)m.

^P

Higher dimensional schemes

We obtain a partial generalization of (1.8) tod -local fields. Recall from (I 2) that a0-localfield is a finite field, and that ad -local field is a field that is complete with respect to adiscrete valuation and has a(d � 1)-local field as residue field. Ifp is either1 or a primeandM is an abelian group or sheaf, we writeM(non�p) for lim

�!Mm where the limit is

over all integers prime top. We also write�1(r) for the sheaf lim�!

�˝rm on Xet (limit over

all integersm).

THEOREM 1.16. Let K be ad -local field withd � 2, and letp D char(K1) whereK1

is the1-local field in the inductive definition ofK. Let X be SpecR with R the discretevaluation ring inK, and letx andu be the closed and open pointsSpeck andSpecK ofX .

(a) There is a canonical isomorphism

H dC2x (X,�1(d))(non�p)

�D! Q=Z(non�p).

(b) For any constructible sheafF onX such thatpF D F,

ExtrX (F,�1(d))�H dC2�rx (X, F)! H dC2

x (X,�1(d))! Q=Z

is a nondegenerate pairing of finite groups, allr.

PROOF. (a) Let i andj be the inclusions ofx andu respectively intoX . As we observedin (1.7),Rr i!�m

�D Z=mZ for r D 2 and is zero otherwise. On tensoring both sides with�˝d�1

m , we find thatRr i!�˝dm�D �˝d�1

m for r D 2 and is zero otherwise. Next, on passing


to the direct limit, we find thatRr i!�1(d)(non�p) �D �1(d � 1)(non�p) for r D 2 andis zero otherwise. Now the spectral sequence

H r (x, Rsi!�1(d)) H) H rCsx (X,�1(d))

shows thatH dC2x (X,�1(d))(non�p) �D H d(x,�1(d�1))(non�p), which equals(Q=Z)(non-

p) by (I 2.17).

We give a second derivation of this trace map. Note that (1.1) implies thatH dC1(u,�1(d))�D!

H dC2x (X, j!�1(d)). MoreoverH dC2

x (X, j!�1(d)) ! H dC2x (X,�1(d)) is an isomor-

phism becausek has cohomological dimensiond (this is implied by (I 2.17) applied tok).Therefore we have a trace map

H dC2x (X,�1(d))(non-p) � H dC1(u,�1(d))(non-p) � (Q=Z)(non-p).

The inductive approach we adopted to define the trace map in (I 2.17) shows that the twodefinitions give the same trace map are equal.

(b) As in the previous cases, it suffices to prove this for sheaves of the formi�F andj!F.

LEMMA 1.17. For all r , there are canonical isomorphisms

Extr�2x (F,�1(d � 1))! ExtrX (i�F,�1(d)).

PROOF. BecauseRr i!�1(d) D 0 for r 6D 2 andR2i!�1(d) �D �1(d � 1), the spectralsequence,

Extrx(F, Rsi!�1(d)) H) ExtrCsX (i�F,�1(d)),

collapses to give the required isomorphisms.

The theorem fori�F now follows from the diagram,

ExtrX (i�F,�1(d)) �H dC2�rx (X, i�F) > H dC2

x (X,�1(d)) > Q=Z

jj

Extr�2x (F,�1(d � 1))

^

�

� H dC2�r (x, F) > H d(x,�1(d � 1))

^

�

> Q=Z

and (I 2.17) applied tok.

Let F be a sheaf onu. The theorem forj!F follows from the diagram,

ExtrX (j!F,�1(d))�H dC2�rx (X, j!F) ! H dC2

x (X,�1(d))�D Q=Z

Extru(F,�1(d))_�

� H dC1�r (u, F)

^

�

> H dC1(u,�1(d))

^

�

�D Q=Z

and (I.2.17) applied toK. This completes the proof of Theorem 1.16.For any ringA we writeKrA for ther th QuillenK-group ofA, and for any schemeX ,

we writeKr for the sheaf onXet associated with the presheafU 7! Kr� (U,OU ).


THEOREM 1.18. LetK andp be as in (1.16).(a) There is a canonical isomorphism

H dC2x (X,K2d�1)(non�p)

�D! (Q=Z)(non�p).

(b) For any constructible sheafF onX such thatpF D F ,

ExtrX (F,K2d�1)�H dC2�rx (X, F)! H dC2

x (X,K2d�1)! Q=Z

is nondegenerate pairing of finite groups.

The main part of the proof is contained in the next lemma. Recall (Browder 1977)that for any ringA, there areK-groups with coefficientsKr (A, Z=mZ) fitting into exactsequences

0! Kr (A)(m)! Kr (A, Z=mZ)! Kr�1(A)m! 0.

Also that for any ringA and integerm that is invertible inA, there is a canonical map�m! K2(A, Z=mZ). Using the product structure on the groupsKr (A, Z=mZ), we obtaina canonical map�m(r)! K2r (A, Z=mZ)! K2r�1(A).

LEMMA 1.19. LetX be any scheme. Ifm is invertible onX , then there is an exact sequence

0! �m(d)! K2d�1

m! K2d�1! 0

of sheaves onXet.

PROOF. This is fairly direct consequence of the following two theorems.(a) Letk be an algebraically closed field; thenK2rk is uniquely divisible for allr , and

K2r�1k is divisible with torsion subgroup equal to�1(r) (Suslin 1983b, 1984).(b) If R is a Henselian local ring with residue fieldk and m is invertible in R, then

Kr (R, Z=mZ)�D! Kr (k, Z=mZ) for all r (Gabber 1983; see also Suslin 1984).

For any field extensionL=k of degreepn, there are maps

f�WKr (k)! Kr (L), f �WKr (L)! Kr (k)

such thatf � ı f� D pn D f� ı f�. ThereforeKr (k)(non�p) ! Kr (L)(non�p) is an

isomorphism. This remark, together with (a) and a direct limit argument, implies that fora separably closed fieldk with char(k) D p, (K2rk)(non�p) is uniquely divisible for allr and(K2r�1k)(non�p) is divisible with torsion subgroup equal to�1(r)(non�p). Interms ofK-groups with coefficients, this says thatK2r (k, Z=mZ) �D K2r�1(k)m

�D �m(r)

andK2r�1(k, Z=mZ) D 0 for all m prime top.

Let R be a strictly Henselian local ring. From the diagram

0 ��! K2r (R)(m) ��! K2r (R, Z=mZ) ��! K2r�1(R)m ��! 0??y ??y�D

??y0 ��! K2r (k)(m) ��! K2r (k, Z=mZ)

��! K2r�1(k)m ��! 0

2. GLOBAL RESULTS: PRELIMINARY CALCULATIONS 143

we see thatK2r�1(R)m

�D! K2r�1(k)m, and therefore that the map�m(r)! K2r�1(R)m

is an isomorphism. AsK2r�1(R, Z=mZ) �D K2r�1(k, Z=mZ) D 0, we know thatK2r�1(R)(m)D

0, and therefore that the sequence

0! �m(r)! K2r�1(R)m! K2r�1(R)! 0.

is exact. This implies the lemma because the exactness of a sequence of sheaves can bechecked on the stalks.

LEMMA 1.20. Let X be the spectrum of a Henselian discrete valuation ring with closedpointx; then for any sheafF onX , H r

x (X, F) is torsion forr � 2.

PROOF. Let u be the open point ofX , and consider the exact sequence

� � � ! H 1(u, F)! H 2x (X, F)! H 2(X, F)! � � � .

From (1.1) we know thatH r (X, F)�D! H r (x, i�F) (excellence is not used in the proof of

(1.1)), andH r (x, i�F) andH r (u, F ju) are both torsion forr > 0 because they are Galoiscohomology groups. The lemma follows.

We now complete the proof of Theorem 1.18. The exact sequence in the lemma leadsto an exact sequence (ignoringp-torsion)

0! H dC1x (X,K2d�1)˝Q=Z! H dC2

x (X,�1(d))! H dC2x (X,K2d�1)! 0,

which shows thatH dC2x (X,�1)(non�p) ! H dC2

x (X,K2d�1)(non�p) is an isomor-

phism because the first term in the sequence is zero. Similarly, ExtrX (F,�1)

�D! ExtrX (F,K2d�1)

for all r , and so (1.16) implies (1.18).

REMARK 1.21. The corollary is a satisfactory generalization of (1.8) in the case thatK1

has characteristic zero. The general case, where the characteristic jumps fromp to zero atsome later stage, is not yet understood. For a discussion of what the best result should be,see~7 below.

NOTES. Part (b) of Theorem 1.8 is usually referred to as the local form of the duality the-orem of Artin and Verdier, although Artin and Verdier 1964 only discusses global results.In Deninger 1986c it is shown that the result extends to singular schemes whenGm is re-placed byG. The extension to higher dimensional schemes in Theorems 1.16 and 1.18 istaken from Deninger and Wingberg 1986. The key lemma 1.19 has probably been provedby several people.

2 Global results: preliminary calculations

Throughout this section,K will be a global field. WhenK is a number field,X denotesthe spectrum of the ring of integers inK, and whenK is a function field,k denotes thefield of constants ofK andX denotes the unique connected smooth complete curve over


k havingK as its function field. The inclusion of the generic point intoX is denoted bygWSpecK ! X , and we sometimes write� for SpecK. For any open subsetU of X ,U 0 is the set of closed points ofU , often regarded as the set of primes ofK correspondingto points ofU . The residue field at a nonarchimedean primev is denoted byk(v). Thefield Kv is the completion ofK at v if v is archimedean, and it is the field of fractionsof the HenselizationOh

v of Ov otherwise. For a sheafF on X or an open subset ofX ,we sometimes writeFv for the sheaf on SpecKv obtained by pulling back relative to theobvious map

fvWSpecKv ! SpecK! X.

Note that whenv is nonarchimedean, there is a commutative diagram

SpecKv ��! SpecK??y ??ySpecOh

v ��! X.

For an archimedean primev of K and a sheafF on Spec(Kv), we setH r (Kv, F ) �DH r

T (Gv, M) (notation as in I~0) whereM is the Gv-module corresponding toF andGv D Gal(Kv,s=Kv). ThereforeH r (K, F) is zero for allr 2 Z when v is complex,andH r (Kv, F ) is isomorphic toH 0

T (Gv, M) or H 1(Gv, M) according asr is even or oddwhenv is real. We letgv D Gal(k(v)s=k(v)).

The cohomology ofGm

PROPOSITION2.1. LetU be an open subset ofX , and letS denote the set of all primes ofK (including the archimedean primes) not corresponding to a point ofU . Then

H 0(U, Gm) D � (U, OU )�,

H 1(U, Gm) D Pic(U ),

there is an exact sequence

0! H 2(U, Gm)!Lv2S

Br(Kv)

Pinvv

��! Q=Z! H 3(U, Gm)! 0.

andH r (U, Gm) D

Lv real

H r (Kv, Gm), r � 4.

PROOF. Let gW � ,! U be the inclusion of the generic point ofU . There is an exactsequence (Milne 1980, II 3.9)

0! Gm! g�Gm,� ! DivU ! 0 (II.3)

with DivU DL

v2U 0iv�Z the sheaf of Weil divisors onU . The same argument as in (1.4)shows thatRsg�Gm D 0 for s > 0, and so the Leray spectral sequence forg degeneratesto a family of isomorphisms

H r (U, g�Gm,�)�D! H r (K, Gm), r � 0.


Clearly,H r (U, DivU ) D

Lv2U 0H r (v, i�Z) D

Lv2U 0H r (k(v), Z),

and we know from (I A.2) that

H r (k(v), Z) �D Z 0 Br(Kv) 0

r D 0 1 2 � 3.

The cohomology sequence of (II.3) therefore gives exact sequences

0! H 0(U, Gm)! K�!

Lv2U 0

Z! H 1(U, Gm)! 0,

0! H 2(U, Gm)! Br(K)!Lv2U 0

Br(Kv)! H 3(U, Gm)! H 3(K, Gm)! 0,

H r (U, Gm)�D! H r (K, Gm), r � 4.

The first sequence shows thatH 0(U, Gm) andH 1(U, Gm) have the values claimed in thestatement of the proposition. Global class field theory (Tate 1967a; I A.7) provides an exactsequence

0! Br(K)!L

all v Br(Kv)! Q=Z! 0.

From this and the second of the above sequences, we see that the kernel-cokernel sequenceof the pair of maps

Br(K)!L

all v Br(Kv)!L

v2U 0 Br(Kv)

is the required exact sequence

0! H 2(U, Gm)!L

v2S Br(Kv)

Pinvv

��! Q=Z! H 3(U, Gm)! H 3(K, Gm)! 0,

To complete the proof in the number field case, we use thatH r (K, Gm) �DL

v realHr (Kv, Gm)

for r � 3 (see I 4.21), and thatH 3(R, Gm) D 0. In the function field case, we know thatK has cohomological dimension� 2, which implies thatH r (K, Gm) D 0 for r � 3.

REMARK 2.2. (a) IfS contains at least one nonarchimedean prime, then the mapXinvvW

Lv2S Br(Kv)! Q=Z

is surjective, and so in this case there is an exact sequence

0! H 2(U, Gm)!L

v2S Br(Kv)! Q=Z! 0,

and

H r (U, Gm) �D

� Lv realH

0T (Kv, Gm), r even, r � 4

0, r odd, r � 3K a number field,

H r (U, Gm) D 0, r � 3, K a function field.

(b) If K has no real primes, thenH 2(X, Gm) D 0, H 3(X, Gm) D Q=Z, andH r (X, Gm) D

0 for r � 4.


Cohomology with compact support

We shall definecohomology groups with compact supportH rc (U, F), r 2 Z, that take

into account the real primes. They will fit into an exact sequence

� � � ! H rc (U, F)! H r (U, F)!

Lv =2U H r (Kv, Fv)! H rC1

c (U, F)! � � � (II.4)

(sum over all primes ofK, including any archimedean primes, not corresponding to a point

of U ). In particular,H rc (U, F)

�D!

Lv realH

r�1(Kv, Fv) for r < 0. The groupH rc (U, F)

differs fromH r (X, j!F) by a group killed by2, and equals it except in the case thatK is anumber field with a real embedding. The reader who is prepared to ignore the prime2 canskip this subsection.

Let F be a sheaf onU , and writeC�(F) for the canonicalCech complex ofF de-fined in ~0. Thus� (U, C�(F)) D C �(U, F), andH r (C �(U, F)) D H r (U, F). Foreach primev, there is a canonical mapf �

v C�(F) ! C�(Fv) and therefore also a mapC �(U, F) ! C �(Kv, Fv). As we noted in (0.8),C �(Kv, Fv) can be identified with thestandard (inhomogeneous) resolutionC �(Mv) of the Gv-moduleMv associated withFv.Whenv is real, we writeS�(Mv) (or S�(Kv, Fv)) for a standard complete resolution ofMv, and otherwise we setS�(Mv) �D C �(Mv). In either case there is a canonical mapC �(Mv) ! S�(Mv). On combining this with the previous maps, we obtain a canonicalmorphism of complexes

uWC �(U, F)!L

v =2U S�(Mv)

(sum over all primes ofK not in U ). We define2 Hc(U, F) to be the translateC �(u)[�1]

of the mapping cone ofu, and we setH rc (U, F) D H r (Hc(U, F)).

PROPOSITION2.3. (a) For any sheafF on an open subschemeU � X , there is an exactsequence

� � � ! H rc (U, F)! H r (U, F)!

Lv =2U H r (Kv, Fv)! H rC1

c (U, F)! � � � .

(b) A short exact sequence

0! F 0! F ! F 00

! 0

of sheaves onU , gives rise to a long exact sequence of cohomology groups

� � � ! H rc (U, F 0)! H r

c (U, F)! H rc (U, F 00)! � � � .

(c) For any closed immersioniWZ ,! U and sheafF onZ,

H rc (U, i�F) �D H r (Z, F).

(d) For any open immersionj WV ,! U and sheafF onV ,

H rc (U, j!F) �D H r

c (V , F).

2In the original, this was writtenHc .


Therefore, for any sheafF onU , there is an exact sequence

� � � ! H rc (V , F jV )! H r

c (U, F)!L

v2V rU H r (v, i�vF)! � � �

(e) For any finite map� WU 0! U and sheafF onU 0, there is a canonical isomorphism

H rc (U,��F)

�D! H r

c (U 0, F ).

PROOF. (a) This is obvious from the definition ofHc(U, F) and the properties of mappingcones (see~0).

(b) From the morphism

0 ��! C �(U, F 0) ��! C �(U, F) ��! C �(U, F 00) ��! 0??yu0

??yu

??yu00

0 ��!L

S�(Kv, F 0v) ��!

LS�(Kv, Fv) ��!

LS�(Kv, F 00

v ) ��! 0

of short exact sequences of complexes, we obtain a distinguished triangle

Hc(U, F 00)[�1]! Hc(U, F 0)! Hc(U, F)! Hc(U, F 00).

(see 0.10a). This yields the long exact sequence.(c) Since the stalk ofi�F at the generic point is zero,H r (Kv, (i�F)v) D 0 for all v.

ThereforeH rc (U, i�F)

�D! H r (U, i�F), and the second group is isomorphic toH r (Z, F).

Before proving (d), we need a lemma.

LEMMA 2.4. Under the hypotheses of (d), there is a long exact sequence

� � � ! H r (U, j!F)! H r (V , F)!L

v2U rV H r (Kv, F )! � � � .

PROOF. The cohomology sequence of the pairU � V is

� � � ! H rU �V (U, j!F)! H r (U, j!F)! H r (V , F)! � � � .

By excision (Milne 1980, III.1.28),H rU �V (U, j!F) �D

Lv2U rV H r

v (Uv, j!F) whereUv D

SpecOhv, and according to (1.1),H r

v (Uv, j!F) �D H r�1(Kv, F ). The lemma is now obvi-ous.

PROOF OF(2.3) CONTINUED. On carrying out the proof of the lemma on the level ofcomplexes, we find that the mapping cone of

C �(U, j!F)! C �(V , j!F jV ) D C �(V , F)

is quasi-isomorphic toL

v2U rV C �(Kv, Fv). The cokernel ofLv =2U S�(Kv, Fv)!

Lv =2V S�(Kv, Fv)

is alsoL

v2U rV C �(Kv, F ), and so the mapping cone ofHc(U, j!F) ! Hc(V , F) isquasi-isomorphic to the mapping cone of a map

Lv2U rV C �(Kv, F )!

Lv2U rV C �(Kv, F ).

The map is the identity, and thereforeHc(U, j!F) ! Hc(V , F) is a quasi-isomorphism.


This completes the proof of the first part of (d), and to deduce the second part one only hasto replaceH r

c (U, j!F) with H rc (V , F) in the cohomology sequence of (see (II.1))

0! j!j�F ! F ! i�i�F ! 0.

Finally, (e) of the proposition follows easily from the existence of isomorphismsH r (U,��F)�D!

H r (U 0, F ) andH r (Kv,��F)�D! H r (K0

v, F ).

PROPOSITION2.5. (a) For any sheavesF andF 0 onU � X , there is a canonical pairing

h , iWExtrU (F, F 0)�H sc(U, F)! H rCs

c (U, F 0).

(b) For any pairingF � F 0! F 00 of sheaves onU � X , there is a natural cup-product

pairingH r

c (U, F)�H sc(U, F 0)! H rCs

c (U, F 00).

(c) The following diagram commutes:

H r (U,Hom(F, F 0))�H sc(U, F)!H rCs

c (U, F 0) (cup-product)

# jj jj

ExtrU (F, F 0) �H sc(U, F)!H rCs

c (U, F 0) (Ext pairing).

PROOF. (a) For example, represent an element of ExtrU (F, F 0) by anr -fold extension, and

takeH rc (U, F)! H rCs

c (U, F 0) to be the correspondingr -fold boundary map.(b) The cup-product pairing on theCech complexes (Milne 1980, V.1.19)

C �(U, F)� C �(U, F 0)! C �(U, F 00)

combined with the cup-product pairing on the standard complexes (Cartan and Eilenberg1956, XII)

S�(Kv, Fv)� S�(Kv, F 0v)! S�(Kv, F 00

v )

gives a natural pairing

Hc(U, F)�Hc(U, F 0)! Hc(U, F 00).

(c) Combine (I 0.14) with (0.7).

The cohomology ofGm with compact support

PROPOSITION2.6. LetU be an open subscheme ofX . ThenH 2c (U, Gm) D 0, H 3

c (U, Gm) D

Q=Z, andH rc (U, Gm) D 0, r > 3.

PROOF. Part (a) of (2.3) gives exact sequences

0! H 2c (U, Gm)! H 2(U, Gm)!

Lv =2U

Br(Kv)! H 3c (U, Gm)! H 3(U, Gm)! 0

0! H 2rc (U, Gm)! H 2r (U, Gm)!

Lv real

H 2r (Kv, Gm)! H 2rC1c (U, Gm)! H 2rC1(U, Gm)! 0,

(2r � 4). WhenU 6D X , these sequences and (2.2a) immediately give the proposition, but(2.3d) and the next lemma show thatH r

c (U, Gm) does not depend onU if r � 2.


LEMMA 2.7. For any closed immersioniWZ ,! U with i(Z) 6D U , H r (Z, i�Gm) D 0 allr � 1.

PROOF. It suffices to prove this withZ equal to a single pointv of U . Theni�Gm corre-sponds to thegv-moduleOun�

v , and soH r (Z, i�Gm) �D H r (gv,Oun�v ). The sequence

0! Oun�v ! Kun�

v

ord! Z! 0

is split as a sequence ofgv-modules, and soH r (gv,Oun�v ) is a direct summand ofH r (gv, Kun�

v ).Therefore Hilbert’s theorem 90 shows thatH 1(gv,Oun�

v ) D 0, and we know from (I A.2)that H 2(gv,Oun�

v ) D 0. As gv has strict cohomological dimension 2, this completes theproof.

REMARK 2.8. (a) LetK be a number field, and letR be the ring of integers inK. Thenthere is an exact sequence

0! H 0c (X, Gm)! R�

!L

v realK�v =K

�2v ! H 1

c (X, Gm)! Pic(R)! 0,

where Pic(R) is the ideal class group ofR. In particular,

H 0c (X, Gm) �D fa 2 R�

jsign(av) > 0 all realvg

D groups of totally positive units inK.

Let Id(R) be the group of ideals inR. Then

H 1c (X, Gm) �D Id(R)=f(a)ja 2 K�, sign(av) > 0 all realvg

�D group of ideal classes ofK in the narrow sense

(see Narkiewicz 1974, III,~2, ~3).The cohomology sequence with compact support of

0! Gm! g�Gm!L

v2X 0iv�Z! 0

isH 0

c (X, g�Gm)!L

v nonarch.Z! H 1c (X, Gm)! H 1

c (X, g�Gm).

The exact sequence given by (2.3a)

0! H 0c (X, g�Gm)! K�

!L

v realK�v =K

�2v ! H 1

c (X, g�Gm)! 0

shows thatH 0c (X, g�Gm) is the group of totally positive elements ofK� andH 1

c (X, g�Gm) D

0. Let Id(R) be the group of ideals inR. Then

H 1c (X, Gm) �D Id(R)=f(a)ja 2 K�, sign(av) > 0 all realvg

�D group of ideal classes ofK in the narrow sense

(see Narkiewicz 1974, III,~2, ~3).(b) UnfortunatelyH 1

c (X, Gm) is not equal to the group of isometry classes of Hermitianinvertible sheaves onX (the “compactified Picard group ofR” in the sense of Arakelovtheory; see Szpiro 1985,~1). I do not know if there is a reasonable definition of theetalecohomology groups of an Arakelov variety. Our definition of the cohomology groups withcompact support has been chosen so as to lead to good duality theorems.


Cohomology of locally constant sheaves

Let U be an affine open subset ofX , and letS be the set of primes ofK not correspondingto a point ofU . Since to be affine in the function field case simply means thatU 6D X ,S will satisfy the conditions in the first paragraph of I 4. With the notations of (I 4),GS D �1(U, �), and the functorF 7! F� defines an equivalence between the categoryof locally constantZ-constructible sheaves onU and the category of finitely generateddiscreteGS -modules. We writeQU for the normalization ofU in KS ; thus QU D SpecRS

where, as in (I 4),RS is the integral closure ofRK,S in KS .

PROPOSITION2.9. Let F be a locally constantZ-constructible sheaf on an open affinesubschemeU of X , and letM D F�. ThenH r (U, F) is a torsion group for allr � 1, andH r (U, F)(`) D H r (GS , M)(`) for all r if ` is invertible onU or ` D char(K).

PROOF. The Hochschild-Serre spectral sequence forQU =U is

H r (GS , H s( QU , F j QU )) H) H rCs(U, F).

As H 0( QU , F) D M , we have to show thatH s( QU , F j QU ) is torsion fors > 0 and thatH s( QU , F j QU )(`) D 0 if ` is invertible onU or equals char(K). By assumptionF j QU isconstant, and so there are three cases to considerW F j QU D Z=`Z with ` invertible onU ,F j QU D Z=pZ with p D char(K), andF j QU D Z.

The first cohomology group can be disposed off immediately, because

H 1( QU , F) �D Hom(�1( QU , �), F( QU )),

and�1( QU , �) is zero.Now let F j QU D Z=`Z with ` a prime that is invertible inRS . ThenZ=`Z � �` on QU ,

and the remark just made shows that the cohomology sequence of

0! �` ! Gm

`! Gm! 0

is

0! Pic( QU )`! Pic( QU )! H 2( QU , Z=`Z)! Br( QU )` ! 0.

The Picard group ofQU is the direct limit of the Picard groups of the finiteetale coveringsU 0 of U and so is torsion. The sequence shows that Pic( QU )(`) D 0, and soH 2( QU , Z=`Z)

injects into Br( QU ). Let L � KS be a finite extension ofK containing the th roots of 1,and consider the exact sequence (see 2.2a)

0! Br(RL,S)!L

w2SLBr(Lw)! Q=Z! 0.

Let L0 be a finite extension ofL; it is clear from the sequence and local class field theorythat an element a of Br(RL,S)` maps to zero in Br(RL0,S) if ` divides the local degreeof L0=L at all w in SL. Let H be the Hilbert class field ofL. Then the prime idealcorresponding tow becomes principal inH with generatorcw say. The fieldL0 generatedoverH by the elementsc1=`

w , w 2 SL, splitsa. As L0 is contained inKS , this argumentshows that lim

�!Br(L)` D 0, and therefore that Br( QU )(`) D 0. HenceH 2( QU , Z=`Z) D 0.


Finally, (2.2) shows thatH r (UL, Gm) D 0 for r > 2, whereUL D SpecRL,S , becauseLhas no real primes, and soH r ( QU , Gm)(`) D 0 for all r > 2.

In the caseF D Z=pZ, p D char(K), we replace the Kummer sequence with theArtin-Schreier sequenceW

0! Z=pZ! O QU

}! O QU ! 0, }(a) D ap

� a.

As H r ( QUet,O) �D H r ( QUZar,O), which is zero forr � 1, we see thatH r ( QU , Z=pZ) D 0

for r � 2.

Finally considerZ. The next lemma shows thatH r ( QU , Z) is torsion forr > 0, and sofrom the cohomology sequence of

0! Z`! Z! Z=`Z! 0

and the results in the preceding three paragraphs, we can deduce thatH r ( QU , Z)(`) D 0 forr > 0 if ` D char(K) or ` is invertible onU .

LEMMA 2.10. Let Y be a normal Noetherian schemeY , and letF be a constant sheafon Y . For any r > 0, the cohomology groupH r (Y, F) is torsion, and it is zero ifF isuniquely divisible.

PROOF. We may assume thatY is connected. LetgW � ,! Y be its generic point. Theng�g�F �D F and the stalks ofRrg�(g�F), being Galois cohomology groups, are tor-sion (see Milne 1980, II 3.7, III 1.15). IfF is constant and uniquely divisible, thenRrg�(g�F) D 0 for r > 0, and the Leray spectral sequence shows thatH r (Y, F) �DH r (�, g�F) D 0 for r > 0. Now let F be constant. In provingH r (Y, F) is torsion, wemay assumeF to be torsion free. For such a sheaf, the cohomology sequence of

0! F ! F ˝Q! (F ˝Q)=F ! 0

shows thatH r�1(Y, (F˝Q)=F) maps ontoH r (Y, F) for r � 1 becauseF˝Q is uniquelydivisible. This completes the proof as(F ˝Q)=F is torsion.

COROLLARY 2.11. Let U be an open subscheme ofX , and letS denote the set of primesof K not corresponding to a point ofU .

(a) For all r < 0, H rc (U, Z) �

Lv realH

r�1(Kv, Z); in particular, H rc (U, Z) D 0 if r is

even and< 0.

(b) There is an exact sequence

0 ! H 0c (U, Z)! Z!

Lv2SH 0(Kv, Z)! H 1

c (X, Z)! 0.

If S contains at least one nonarchimedean prime, thenH 0c (U, Z) D 0.

(c) There is an exact sequence

0! H 2c (U, Z)! H 2(U, Z)!

Lv2SH 2(Kv, Z)! H 3

c (U, Z)! H 3(U, Z)! 0.

For all primes` that are invertible onU or equal the characteristic ofK, there is anexact sequence

0! H 2c (U, Z)(`)! H 2(GS , Z)(`)!

Lv2SH 2(Kv, Z)(`)!

H 3c (U, Z)(`)!H 3(GS , Z)(`)! 0.


(d) For all r � 4 , H rc (U, Z) D 0.

PROOF. All the statements follow from the exact sequence

� � � ! H rc (U, Z)! H r (U, Z)!

Lv2SH r (Kv, Z)! � � � .

For r < 0, H r (U, Z) D 0, and so the sequence gives isomorphismsLv2SH r�1(Kv, Z)

�D! H r

c (U, Z).

For r 6D 0, 1, 2,L

v2SH r (Kv, Z) DL

v realHr (Kv, Z). SinceH r (R, Z) D 0 for odd r ,

these calculations prove (a).As H 0(U, Z) D Z and H 1(U, Z) D Homcts(�1(U, �), Z) D 0, we have an exact

sequence

0! H 0c (U, Z)! Z!

Lv2SH 0(Kv, Z)! H 1

c (U, Z)! 0.

WhenS contains a nonarchimedean prime, the middle map is injective, and so this proves(b).

The first part of (c) is obvious from the fact that

H 1(Kv, Z) D 0 D H 3(Kv, Z)

for all v, and the proposition allows us to replaceH r (U, Z)(`) with H r (GS , Z)(`) for theparticular`.

For (d), we begin by showing thatH rc (U, Z)(`) D 0 for r � 4 when` is a prime that is

invertible onU . Consider the diagram

H r�1(GS , Q=Z) ��!L

v realHr�1(Kv, Q=Z)??y ??y

H r (GS , Z) !L

v realHr (Kv, Z).

The vertical arrows (boundary maps) are isomorphisms forr � 2 becauseQ is uniquelydivisible, and Theorem I 4.10c shows that the top arrow is an isomorphism on the`-primarycomponents forr � 4 if ` is invertible onU . Therefore the maps

H r (GS , Z)(`)!L

v realHr (Kv, Z)(`)

are isomorphisms forr � 4 and all` that are invertible onU . As H 3(R, Z) D 0, thisproves thatH r

c (U, Z)(`) D 0 for r � 4 and such . SinceH rc (U, F) D H r

c (U [1=`], F ) forany r � 4, this completes the proof except for thep-primary component in characteristicp. We may assume thatU is affine. The cohomology sequence of

0! Z=pZ! OU

}! OU ! 0

shows thatH r (U, Z=pZ) D 0 for r � 2. ThereforeH r (U, Z)(p) D 0 for r � 3, and thisimplies thatH r

c (U, Z)(p) D 0 for r � 4.

REMARK 2.12. It has been conjectured that scd`(GS) D 2 for all primes` that are in-vertible inRK,S . This would imply that the map

Lv2SH 2(Kv, Z)(`)! H 3

c (U, Z)(`) in(2.11c) is surjective on the-primary components for such.



Let F be a constructible sheaf onU such thatmF D 0 for somem that is invertible onU .We shall see that the groupsH r (U, F) andH r

c (U, F) are all finite, and so it makes senseto define

�(U, F) D[H 0(U, F)][H 2(U, F)]

[H 1(U, F)][H 3(U, F)], �c(U, F) D

[H 0c (U, F)][H 2

c (U, F)]

[H 1c (U, F)][H 3

c (U, F)].

THEOREM 2.13. Let F be a constructible sheaf onU such thatmF D 0 for somem thatis invertible onU.

(a) The groupsH r (U, F) are finite, and

�(U, F) DYv arch

[F(Kv)]

[H 0(Kv, F )]j[F(Ks)]jv.

(b) The groupsH rc (U, F) are finite, and

�c(U, F) DYv arch

[F(Kv)].

PROOF. (a) Choose an open affine subschemeV of U such thatF jV is locally constant.Theorem 2.9 shows thatH r (V , F) �D H r (GS , M) for all r , whereS is the set of primesof K not inV andM is theGS module corresponding toF jV . Therefore Theorem (I 5.1)shows that

�(V , F jV )[H 3(V , F jV )] D �(GS , M) DYv arch

[H 0(Gv, M)]=j[M]jv.

As H 3(V , F jV )�D!

Qv archH

3(Kv, M) (by I 4.10c), and the groupsH r (Kv, M) for afixed archimedean primev all have the same order (recall that they are Tate cohomologygroups), this proves the result forF jV , and it remains to show that�(U, F) D �(V , F jV ).The sequence

� � � !L

v2U rV H rv (Oh

v , F )! H r (U, F)! H r (V , F)! � � �

shows that�(U, F) D �(V , F jV )�Q�v(Oh

v , F ), and the sequence

� � � ! H rv (Oh

v , F )! H r (Ohv , F )! H r (Kv, F )! � � �

shows that�v(Ohv , F ) D �(Oh

v , F )�(Kv, F )�1. But F(Kv,s) has order prime to theresidue characteristic ofK, and so (I 2.8) shows that�(Kv, F ) D 1. Moreover (see 1.1)H r (Oh

v , F ) D H r (gv, F(Ohv)) andF(Oh

v) is finite, and so it is obvious that�(Ohv , F ) D 1

(see Serre 1962, XIII1).(b) The sequence (2.3d)

� � � ! H rc (V , F jV )! H r

c (U, F)!L

v2U rV H r (v, i�F)! ...


shows that�c(U, F) D �c(V , F jV ) for any open subschemeV of U , and so we canassume thatU 6D X and thatF is locally constant. There is an exact sequence

0!Qv archH

�1(Kv, F )! H 0c (U, F)! H 0(U, F)!

Qv =2U H 0(Kv, F )!

� � � ! H 3c (U, F)! H 3(U, F)!

Qv archH

3(Kv, F )! 0

because (see I 4.10c)H 3(U, F) !Qv archH

3(Kv, F ) is surjective (in fact, an isomor-phism). As the groupsH r (Kv, F ) for v archimedean all have the same order,

�c(U, F) D �(U, F)�Qv2X rU�(Kv, F )�1

�Qv arch[H

0(Kv, F )].

According to (I 2.8),�(Kv, F ) D j[F(Ks)]jv, and so

�c(U, F) DQv arch[F(Kv)]v �

Qv =2U j[F(Ks)]j

�1v .

But j[F(Ks)]jv D 1 for v 2 U , and soQv =2U j[F(Ks)]jv D 1 in virtue of the product

formula, and so we obtain the formula.

REMARK 2.14. (a) LetF be a locally constant sheaf onU with mF D 0 for somem that isinvertible onU . In the next section, we shall show thatH r (U, F) is dual toH 3�r

c (U, F D)

for all r . This implies that�(U, F)�c(U, F D) D 1. If we letM be theGS -module modulecorresponding toF , then (2.13) shows that

�(U, F)�c(U, F D) DQv arch

[H 0(Gv, M)][H 0(Gv, M D)]

j[M]jv[H 0(Kv, M)],

which (I 2.13c) shows to be one. Thus our results are consistent.(b) Assume thatU 6D X and thatF is locally constant. ThenH 0(U, F) ,! H 0(Kv, F )

and, of course,H �1(U, F) D 0. ThereforeQ

H �1(Kv, F )�D! H 0

c (U, F), and so (2.13b)becomes in this case

[H 2c (U, F)]

[H 1c (U, F)][H 3

c (U, F)]DQv arch

[F(Kv)]

[H 0(Kv, F )].

NOTES. So far as I know, K. Kato was the first to suggest defining cohomology groups“with compact support” fitting into an exact sequence

� � � ! H rc (X, F)! H r (X, F)!

Qv realH

r (Kv, Fv)! � � �

(letter to Tate, about 1973). Our definition differs from his, but it gives the same groups.

3 Global results: the main theorem

Statements

We continue with the notations of the last section. From (2.6) (and its proof) we know that

there are trace mapsH 3c (U, Gm)

�D! Q=Z such that

3. GLOBAL RESULTS: THE MAIN THEOREM 155

(a) for anyV � U ,

H 3c (V , Gm)

�D��! Q=Z??y

H 3c (U, Gm)

�D��! Q=Z

commutes;(b) for anyv =2 U ,

Br(Kv)inv��! Q=Z??y

H 3c (U, Gm)

�D��! Q=Z

commutes.On combining the pairings

ExtrU (F, Gm)�H 3�rc (U, F)! H 3

c (U, Gm)

with this trace map, we obtain maps

˛r (U, F)WExtrU (F, Gm)! H 3�rc (U, F)�.

THEOREM 3.1. LetF be aZ-constructible sheaf on an open subschemeU of X.

(a) For r D 0, 1, ExtrU (F, Gm) is finitely generated andr (U, F) defines isomorphisms

ExtrU (F, Gm)^! H 3�r

c (U, F)�

where^ denotes the completion for the topology of subgroups of finite index.For r � 2, ExtrU (F, Gm) is a torsion group of cofinite-type, andr (U, F) is anisomorphism.

(b) If F is constructible, then


c (U, Gm)

is a nondegenerate pairing of finite groups for allr 2 Z.

Note that (b) implies thatH r (U, F) is finite if F is a constructible sheaf onU with-out char(K)-torsion, because (2.3a) and (I 2.1) show that thenH r (U, F) differs fromH r

c (U, F) by a finite group.Before beginning the proof of the theorem, we list some consequences.

COROLLARY 3.2. For any constructible sheafF on an open subschemej WU ! X of X

and prime number, there is a canonical nondegenerate pairing of finite groups

ExtrU (F, Gm)(`)�H 3�r (X, j!F)(`)! H 3(X, j!Gm)(`) D (Q=Z)(`), r 2 Z,

except when D 2 andK is number field with a real prime.


PROOF. According to (2.3d),H rc (U, F) �D H r

c (X, j!F) for any sheafF on U (not neces-sarily constructible), and it is clear from (2.3a) thatH r

c (X, j!F) differs fromH r (X, j!F)

by at most a group killed by2, and that it differs not at all ifK has no real embedding.Thus, the statement follows immediately from (b) of the theorem.

COROLLARY 3.3. LetF be a constructible sheaf onU such thatmF D 0 for somem thatis invertible onU , and letF D D RHom(F, Gm) (an object of the derived category ofS(Uet)).

(a) If F is locally constant, thenH r (F D) D 0 for r > 0; thus in this caseF D can beidentified with the sheafHom(F, Gm).

(b) There is a canonical nondegenerate pairing of finite groups

H r (U, F D)�H 3�rc (U, F)! H 3(U, Gm) �D Q=Z, r 2 Z.

PROOF. Part (a) can be proved by the argument in the proof of Corollary 1.10(a). Part (b)is obvious from the theorem, becauseHr (U, F D) �D ExtrU (F, Gm).

Define3

Dr (U, F) D Im(H rc (U, F)! H r (U, F)).

COROLLARY 3.4. LetF be a locally constant constructible sheaf onU such thatmF D F

for somem invertible onU . Then there is a nondegenerate pairing of finite groups

Dr (U, F)�D3�r (U, F D)! Q=Z

for all r 2 Z.

PROOF. From (2.3a), there is an exact sequence

0! Dr (U, F)! H r (U, F)!L

v =2U H r (Kv, F ),

and (3.3) and (I 2.3) show that the dual of this is an exact sequenceLv =2U H 2�r (Kv, F D)! H 3�r

c (U, F D)! Dr (U, F)�! 0.

But this second sequence identifiesDr (U, F)� with D3�r (U, F D) (apply (2.3a) again).

The proof of the theorem is rather long and intricate. In (3.5 — 3.8) we show thatit suffices to prove the theorem withU replaced by an open subset. Proposition 3.9 andCorollary 3.10 relate the theorem onU to the theorem onU 0 for some finite covering ofU .In Lemma 3.12 it is shown that the groups vanish for larger whenK has no real primes,and hence proves the theorem for suchK andr . Lemma 3.13 allows us to assume thatK

has no real primes. In (3.14 — 3.17) the theorem is proved by an induction argument forconstructible sheaves, and we then deduce it for allZ-constructible sheaves.

3The groupsDr (U, F) are the intersection cohomology groups ofF for the middle perversity.


The proof

Throughout the proof,U will be an open subscheme of the schemeX . We setbr (U, F)

equal to the map ExtrU (F, Gm)^ ! H 3�r (U, F)� induced by r (U, F) whenr D 0, 1,

and equal to r (U, F) otherwise.

LEMMA 3.5. Theorem 3.1 is true ifF has support on a proper closed subset ofU.

PROOF. We can assume that our sheaf is of the formi�F wherei is the inclusion of asingle closed pointv into U . According to (2.3c),

H rc (U, i�F) �D H r (v, F ).


0! Gm! g�Gm!L

u2U 0iu�Z! 0


� � � ! ExtrU (i�F, Gm)! ExtrU (i�F, g�Gm)!L

u2U 0 ExtrU (i�F, iu�Z)! � � � .

As we observed in the proof of (2.1),Rsg�Gm D 0 for s � 1, and so4

ExtrU (i�F, g�Gm) �D Extr�(i�F j�, Gm),

which is zero for allr becausei�F j� D 0. Moreover (0.1c) shows that ExtrU (i�F, iu�Z) �D

Extru(i�uiv�F, Z), which equals 0 unlessu D v, in which case it equals Extr

v(F, Z). There-fore the sequence gives isomorphisms

Extr�1v (F, Z)

�D! ExtrU (i�F, Gm) (II.5)

for all r . Let M be thegv-module corresponding toF . Then we have a commutativediagram

ExtrU (i�F, Gm)�H 3�rc (U, i�F) > H 3

c (U, Gm)�D Q=Z

jj jj

Extr�1gv

(M, Z)

^�D

�H 3�r (gv, M) ! H 2(gv, Z) �D Q=Z

and so the theorem follows in this case from (I 1.10).[There is an alternative proof of (II.5). One can show (as in 1.7b) thatRr i!Gm

�D Z forr D 1 and is zero otherwise. The spectral sequence (0.1e)

Extrv(F, Rsi!Gm) H) ExtrCsU (i�F, Gm)

now yields isomorphisms Extr�1v (F, Z)

�D! ExtrU (i�F, Gm).]

4The functorg� has an exact left adjointg�, and so it is left exact and preserves injectives (cf. (0.1b)).


LEMMA 3.6. For any Z-constructible sheaf onU , the groupsExtrU (F, Gm) are finitelygenerated forr D 0, 1, torsion of cofinite-type forr D 2, 3, and finite forr > 3. If F isconstructible, all the groups are finite.

PROOF. Note that ExtrU (Z, Gm) �D H r (U, Gm), and so forF D Z the values of ExtrU (Z, Gm)

can be read off from (2.1). In particular, the lemma is true forZ, hence forZ=nZ, and sofor all constantZ-constructible sheavesF .

Next suppose thatF is locally constant andZ-constructible. It then becomes constanton some finite Galois covering� WU 0

! U with Galois groupG, say, and the spectralsequence (see 0.2)

H r (G, ExtsU 0(F jU0, Gm)) H) ExtrCs

U (F, Gm),

shows that ExtrU (F, Gm) differs fromH 0(G, ExtrU 0(F jU 0, Gm)) by a finite group. There-fore the lemma forF jU 0 implies it forF .

Finally, let F be an arbitraryZ-constructible sheaf, and letV be an open subset ofU

on whichF is locally constant. Writej andi for the inclusions ofV and its complementinto U . The Ext sequence of

0! j!j�F ! F ! i�i�F ! 0


� � � ! Extr�1U rV (i�F, Z)! ExtrU (F, Gm)! ExtrV (F, Gm)! � � �

(use (II.5) and (0.1a)). As Extr�1U rV (i�F, Z) is finitely generated forr D 1, finite for r D 2,

torsion of cofinite-type forr D 3, and0 for all other values ofr (see I 1.10), the lemmafollows.

LEMMA 3.7. Let0! F 0

! F ! F 00! 0

be an exact sequence ofZ-constructible sheaves onU . If (3.1) holds for two out of thethree sheavesF 0, F , andF 00, then it also holds for the third.

PROOF. Because Ext1U (F 0, Gm) is finitely generated, its image in the torsion group Ext2

U (F 00, Gm)

is finite. Therefore the sequence

� � � ! Extr (F 00, Gm)! Extr (F, Gm)! Extr (F 0, Gm)! � � �

remains exact after the first six terms have been replaced by their completions. On the otherhand, Hom(�, Q=Z) is exact becauseQ=Z is injective, and so the lemma follows from thefive-lemma.

LEMMA 3.8. Let V be a nonempty open subscheme ofU , and letF be aZ-constructiblesheaf onU ; the theorem is true forF on U if and only if it is true for the restriction ofFto V .


PROOF. Write j for the open immersionV ,! U and i for the complementary closedimmersionU r V ,! U . Then (see (0.1a))

ExtrU (j!F jV , Gm) �D ExtrV (F jV , Gm),

and (see (2.3d))H r

c (U, j!F jV ) �D H rc (V , F jV ).

It follows thatbr (U, j!F jV ) can be identified withbr (V , F jV ). Therefore the theorem istrue forj!(F jV ) on U if and only if it is true forF jV on V . Now (3.7, 3.5) and the exactsequence

0! j!j�F ! F ! i�i�F ! 0

show that the theorem is true forF onU if and only if it is true forj!(F jV ) onU.

The lemma shows that it suffices to prove Theorem 3.1 for locally constant sheaves and“small” U .

LEMMA 3.9. Let� WU 0! U be the normalization ofU in a finite Galois extensionK0 of

K.(a) There is a canonical norm mapNmW��Gm,U 0 ! Gm,U .

(b) For everyZ-constructible sheafF onU 0, the composite

N WExtrU 0(F, Gm)! ExtrU (��F,��Gm)Nm! ExtrU (��F, Gm)

is an isomorphism.

PROOF. (a) Let V ! U be etale. ThenV is an open subset of the normalization ofU

in some finite separableK-algebraL. By definition� (V ,��Gm) �D � (V 0,O�V 0) where

V 0Ddf U 0

�U V . As V 0 is etale overU 0, it is normal, and as it is finite overV , it must bethe normalization ofV in the finite GaloisL-algebraK0

˝K L:

U 0 � V 0 U 0

�U V

�

??yfinite � 0

??yU

etale �� V

K0�! K0

˝K Lx?? x??K �! L.

Consequently, the norm mapK0 ˝K L ! L induces a map� (V ,��Gm) ! � (V , Gm),and for varyingV these maps define a map of sheaves NmW��Gm,U 0 ! Gm,U .

(b) Let j WV ! U 0 be an open immersion such that�V Ddf �jV is etale. Then Milne1980, V 1.13, shows that the map

ExtrV (j�F, Gm)! ExtrU (��j!j�F, Gm)

defined by the adjunction map�V !��V Gm! Gm is an isomorphism for allr . The compos-

ite of this withExtrU (j!j

�F, Gm) �D ExtrV (j�F, Gm)

(see 0.1a) is the map in (b). Therefore, (b) is true for the sheafj!j�F .


For a sheaf of the formiv�F , v 2 U , (b) again follows from (ibid., V 1.13) because thesequence of maps can be identified with

Extr�1��1(v)

(F, Z)! Extr�1v (��F,��Z)

Tr! Extr�1

v (��F, Z)

(for the trace map��Z! Z, see ibid. V 1.12; for the identification, see (II.5)).

For the general case, apply the five-lemma to the diagram obtained from the exactsequence (II.1)

0! j!j�F ! F ! i�i�F ! 0.

LEMMA 3.10. Let� WU 0! U be as in Lemma 3.9. For anyZ-constructible sheafF 0 on

U 0, br (U 0, F ) is an isomorphism if and only ifbr (U,��F) is an isomorphism.

PROOF. From the norm map (3.8)��GmU 0 ! Gm, we obtain a map Nm,

H 3c (U 0, Gm)

(2.3e) ��

�D

H 3c (U,��Gm)! H 3

c (U, Gm).

For anyw 7! v =2 U , the diagram

Br(K0w) ��! H 3

c (U 0, Gm)??yNm

??yNm

Br(Kv) ��! H 3c (U, Gm)

commutes. The left hand arrow commutes with the invariant maps, and so the right handarrow commutes with the trace maps. The diagram

ExtrU 0(F, Gm) � H 3�rc (U 0, F ) > H 3

c (U 0, Gm)

ExtrU (��F, Gm)

�D

_N

�H 3�rc (U,��F)

�D

^

(2.3e)

! H 3c (U, Gm)

_Nm

commutes, and the lemma follows.

REMARK 3.11. In Lemmas 3.9 and 3.10, it is only necessary to assume thatK0 is separableoverK.

LEMMA 3.12. (a) If F is constructible, thenH rc (U, F) is zero forr > 3, and if F is

Z-constructible, then it is zero forr > 4.(b) If F is constructible andK has no real primes, thenExtrU (F, Gm) D 0 for r > 4.

PROOF. (a) Let F be constructible. According to (2.3d), we can replaceU by an opensubset, and hence assume thatF is locally constant and, in the number field case, thatmF D 0 for some integerm that is invertible onU . We have to show (see 2.3a) thatH r (U, F) !

Lv realH

r (Kv, F ) is an isomorphism forr � 3. But (2.9) identifies thiswith the map

H r (GS , M)!L

v realHr (Kv, M), M D F�, (II.6)


and (I 4.10c) shows that (II.6) is an isomorphism forr � 3 except possibly whenK is afunction field and the order ofM is divisible byp. In the last case we can assume thatF

is killed by some power ofp and have to show thatH r (U, F) D 0 for r > 3. From thecohomology sequence of

0! Z=pZ! OU

}! OU ! 0

we see thatH r (U, Z=pZ) D 0 for r � 2 (becauseH r (Uet,OU ) D H r (UZar,OU ) D 0 forr > 1). In general there will be a finiteetale covering� WU 0 ! U of degreed prime top such thatF jU 0 has a composition series whose quotients are isomorphic to the constantsheafZ=pZ (apply Serre 1962, IX, Thm 3, Thm 2). ThenH r (U 0, F jU 0) D 0 for r � 2,and as the composite

H r (U, F)! H r (U 0, F jU 0)trace! H r (U, F)

is multiplication byd (see Milne 1980, V.1.12), this proves thatH r (U, F) D 0 for r � 2.Now letF beZ-constructible. ThenFtors is constructible, and so it suffices to prove the

result forF=Ftors: we can assume thatF is torsion-free. Then

H r�1c (U, F=mF) � H r

c (U, F)m, H r�1c (U, F=mF) D 0 for r > 4,

and so it remains to show thatH rc (U, F) is torsion forr > 4. Again, (2.3d) allows us to as-

sumeF is locally constant. Proposition 2.3a shows thatH rc (U, F) differs fromH r (U, F)

by a torsion group forr > 1, and we saw in (2.9) thatH r (U, F) is torsion whenr > 0.(b) BecauseK has no real primes,H r (U, F) D H r

c (U, F) D 0 for r > 3. If F hassupport on a closed subschemeZ, the lemma is obvious from the isomorphism

Extr�1Z (F, Z)

�D! Extr (i�F, Gm)

of (3.5). As usual, this allows us to assume thatF is locally constant. ThenExt rU (F, Gm) D

0 for r > 1 (see the proof of 1.10a), and forr D 0, 1, it is torsion, and is therefore a directlimit of constructible sheaves (0.9). HenceH r (U, Exts

U (F, Gm)) D 0 for r > 3, and soExtr (F, Gm) D 0 for r > 4.

LEMMA 3.13. Assume thatbr (X, Z) is an isomorphism for allr wheneverK has no realprimes. Then Theorem 3.1 is true.

PROOF. WhenK has no real primes, the assumption implies that Theorem 3.1 is true forconstant sheaves onX , and (3.8) then implies that it is true for constant sheaves on anyopenU � X .

According to Lemma 3.8, it suffices to prove Theorem 3.1 for pairs(U, F) with F lo-cally constant and with2 invertible onU in the number field case. We prove thatbr (U, F)

is an isomorphism in this case by induction onr . Note that Lemma 3.12a implies thatbr (U, F) is an isomorphism whenr < �1 — assume it to be an isomorphism whenr < r0. For a pair(U, F) as above, there exists a finiteetale covering� WU 0

! U such thatU 0 is the normalization ofU in field K0 with no real primes andF becomes constant onU 0. LetF� D ��

�F . The trace map (Milne 1980, V.1.12)F�! F is surjective (on stalks


it is just summation,PWF d

v ! Fv), and we writeF 0 for its kernel. From the commutativediagram

Extr0�1U (F�, Gm) ��! Extr0�1

U (F 0, Gm) ��! Extr0

U (F, Gm) ��! Extr0

U (F�, Gm)??y�

??y�

??y 3.10

??y�

H 4�r0(U, F�)� ��! H 4�r0(U, F 0)� ��! H 3�r0(U, F)� ��! H 3�r0(U, F�)�

we see thatbr0(U, F)WExtr0(F, Gm) ! H 3�r0(U, F)� is injective. (Forr D 0, 1, it isnecessary to replace the groups on the top row with their completions; see the proof of(3.7).) SinceF 0 is also locally constant,br0(U, F 0) is also injective, and the five-lemmaimplies thatbr0(U, F) is an isomorphism.

For a constructible sheafF , we define r (U, F)WH rc (U, F)! Ext3�r

U (F, Gm)� to bethe dual of r (U, F).

LEMMA 3.14. For any Z-constructible sheafF on U , there is a finite surjective map�1WU1! U , a finite map�2WU2! U with finite image, constantZ-constructible sheavesFi onUi, and an injective mapF !˚i�i�Fi .

PROOF. Let V be an open subset ofU such thatF jV is locally constant. Then there is afinite extensionK0 of K such that the normalization� WV 0

! V of V in K0 is etale overV andF jV 0 is constant. Let�1WU1 ! U be the normalization ofU in K0, and letF1 bethe constant sheaf onU1 corresponding to the group� (V 0, F jV 0). Then the canonical mapF jV ! ��F jV 0 extends to a mapWF ! �1�F1 whose kernel has support onU � V .Now takeU2 to be anetale covering ofU �V on which the inverse image ofF onV �U

becomes a constant sheaf, and takeF2 to be the direct image of this constant sheaf.

Note that Lemma 3.13 shows that it suffices to prove Theorem 3.1 under the assumptionthat K has no real primes. From now until the end of the proof of the theorem we shallmake this assumption.

LEMMA 3.15. (a) Letr0 be an integer� 1. If for all K, all constructible sheavesF onX ,and all r < r0, ˇr (X, F) is an isomorphism, thenr0(X, F) is injective.

(b) Assume that for allK, all constructible sheavesF onX , and allr < r0,ˇr (X, F) isan isomorphism; further assume thatˇr0(X, Z=mZ) is an isomorphism whenever�m(K) D

�m(Ks). Then˛r0(X, F) is an isomorphism for allX and all constructible sheavesF.

PROOF. (a) Let F be a constructible sheaf on someX , and letc 2 H r0(X, F). Thereexists an embeddingF ! I of F into a torsion flabby sheafI on X . According to (0.9),I is a direct limit of constructible sheaves. AsH r0(X, I) D 0, and cohomology commuteswith direct limits, this implies that there is a constructible sheafF� onX and an embeddingF ,! F� such thatc maps to zero inH r0(X, F�). LetQ be the cokernel ofF ! F�. ThenQ is constructible, and a chase in the diagram

H r0�1(X, F�) �! H r0�1(X, Q) �! H r0(X, F) �! H r0(X, F�)??y�

??y�

??y ??y�

Ext4�r0

X (F�, Gm)��! Ext4�r0

X (Q, Gm)��! Ext3�r0

X (F, Gm)��! Ext3�r0

X (F�, Gm)�


shows that r0(c) 6D 0. Since the argument works for allc, this shows that r0(X, F) isinjective.

(b) LetF be a constructible sheaf onX . For a suitably small open subsetU of X , therewill exist a finite Galois extensionK0 of K such that the normalizationU 0 of U in K0 isetale overU , F jU 0 is constant, and�m(K) D �m(Ks) for somem with mF D 0. In theconstruction of the preceding lemma, we can takeU1 to be the normalization ofX in K0.Let F� D �1�F1 ˚ �2�F2; then (3.10) and (3.5) show respectively thatˇr0(X,�1�F) andˇr0(X,�2�F) are isomorphisms. The mapF ! F� is injective, and we can construct adiagram similar to the above, except that now we know thatˇr0(X, F�) is an isomorphism.Therefore r0(X, F) is injective, and asQ is constructible we have also thatˇr0(X, Q) isinjective. The five-lemma now shows thatˇr0(U, F) is an isomorphism.

LEMMA 3.16. Theorem 3.1 is true for all constructible sheavesF onX.

PROOF. We proveˇr (X, F) is an isomorphism by induction onr . For r < 0 it is anisomorphism by (3.12).

To compute the group ExtrX (Z=mZ, Gm), we use the exact sequence

� � � ! ExtrX (Z=mZ, Gm)! H r (X, Gm)m! H r (X, Gm)! � � � .

By definitionH 0(X, Z=mZ) D Z=mZ, and it follows from (2.2b) that Ext3(Z=mZ, Gm) �Dm�1Z=Z. The pairing is the obvious one, and soˇ0(X, Z=mZ) is an isomorphism. Now(3.15b) implies that 0(X, F) is an isomorphism for allF.

Lemma 3.15a shows that1(X, F) is always injective. The order ofH 1(X, Z=mZ)

is equal to the degree of the maximal unramified abelian extension ofK of exponentm.By class field theory, this is also the order of Pic(X)(m) �D [Ext2X (Z=mZ, Gm)], and soˇ1(X, Z=mZ) is an isomorphism for allX . It follows thatˇ1(X, F) is an isomorphism forall X andF .

Lemma 3.15a again shows thatˇ2(X, F) is always injective. To complete the proof,5

it remains to show (by 3.15) thatr (X, Z=mZ) is an isomorphism for allr � 2 when�m(K) D �m(Ks).

Fix a K (hence anX ) such that�m(K) D �m(Ks) (and, of course,K has no realprimes). Initially assumem is prime to the characteristic ofK. Choose a dense openU � X on whichm is invertible, and letiWX r U ,! X be its closed complement. Fromthe exact sequence (II.1), p124, we obtain a diagram

� � � ��! H rc (U, Z=mZ) ��! H r (X, Z=mZ) ��! H r (X, i�(Z=mZ)) ��! � � �??yˇr (U,Z=mZ)

??yˇr (X,Z=mZ)

??yˇr (X,i�(Z=mZ))

� � � ��! � ��! � ��! � ��! � � �

From it, we see that r (U, Z=mZ) is an isomorphism forr � 1 and an injection forr D 2.For r D 3, ˇr (U, Z=mZ) arises from a pairing

HomU (Z=mZ, Gm)�H 3c (U, Z=mZ)! H 3

c (U, Gm).

5At this point in the original, I effectively assumed thatˇ2(X, F) is an isomorphism (see the diagramp288). I thank Joel Riou for pointing this out to me.


Any isomorphismZ=mZ�! �m � Gm induces an isomorphism

H rc (U, Z=mZ)

��! H r

c (U,�m)�H rc (U, Gm)

1m

Z=Z � Q=Z

from which it follows that 3(U, Z=mZ) is an isomorphism. As

H rc (U, F) D 0 D Ext3�r

U (F, Gm) for r > 3,

this shows that r (U, Z=mZ) is an isomorphism for allr except possibly forr D 2, whenit is injective. Now (2.13) (cf. 2.14a) shows thatH 2

c (U, Z=mZ) and Ext1U (Z=mZ, Gm) �DH 1(U,�m) have the same order, and soˇ2(U, Z=mZ) is an isomorphism for allr .

It remains to treat the sheafZ=pZ with p 6D 0 the characteristic ofK. From thecohomology sequence of

0 ��! Z=pZ ��! OU

}��! OU ��! 0

and the Ext sequence of

0 ��! Zp��! Z ��! Z=pZ ��! 0

we see thatH r (X, Z=pZ) D 0 D Ext3�r

X (Z=pZ, Gm) for r > 2.

Thus,ˇr (X, Z=pZ) is an isomorphism for allr except possiblyr D 2, when it is injective.Using the sequences just mentioned, one can show thatH 2(X, Z=pZ) and Ext1X (Z=pZ, Gm)

have the same order,6 which completes the proof.

We now complete the proof of Theorem 3.1 by proving thatbr (X, Z) is an isomorphismfor all r (recall that we are assumingK has no real primes). We are concerned with themaps

˛r (X, Z)WH r (X, Gm)! H 3�r (X, Z)�, r 6D 0, 1,br (X, Z)WH r (X, Gm)! H 3�r (X, Z)�, r D 0, 1.


0 �! H r (X, Gm)^�! lim �n

ExtrC1(Z=nZ, Gm) �! lim �n

H rC1(X, Gm)n �! 0??y ??y�

??yH 3�r (X, Z)�

�! H 2�r (X, Q=Z)��! H 2�r (X, Q)�

For 2 � r � 1 (that is, forr � 1), H 2�r (X, Q) D 0 D H 3�r (X, Q) (see 2.10), and soH 2�r (X, Q=Z) ! H 3�r (X, Z) is an isomorphism. Forr � 1, H rC1(X, Gm) is finitelygenerated, and so lim

�H rC1(X, Gm)n D 0 (see I 0.19). Therefore it is obvious from the

diagram thatbr (X, Z) is an isomorphism forr � 1. For r D 3, the map is the obviousisomorphismQ=Z! Z�, and for all other values ofr , both groups are zero.

6Alternatively, one can avoid counting by using the original proof, which actually works in this case.

4. GLOBAL RESULTS: COMPLEMENTS 165

REMARK 3.17. 7For a locally Noetherian schemeY , let Ysm denote the category of smoothschemes overY endowed with theetale topology. Letf WYsm! Yet be the morphism ofsites defined by the identity map. Thenf� is exact and preserves injectives (Milne 1980,III.3.1), and it follows that ExtrYsm

(f �F, F 0) �D ExtrYet(F,f�F) for all sheavesF on Yet, all

sheavesF 0 on Ysm, and allr . ThereforeUet can be replaced byUsm in the above resultsprovided one defines aZ-constructible sheaf onYsm to be the inverse image byf of aZ-constructible sheaf onYet.

NOTES. Corollary 3.2 in the number field case is the original theorem of Artin and Verdier(announcement in Artin and Verdier 1964). As far as I know, no complete proof of thetheorem has been published before, but Mazur 1973 contains most of the ingredients. Itand the notes of a 1964 seminar by Mazur were sources for this section. We note thatTheorem 3.1 improves the original theorem in three respects: by taking into account thearchimedean primes, it is able to handle the2-torsion; it includes the function field case;and it allows the sheaves to beZ-constructible instead of constructible. To my knowledge,several people have extended the original theorem to the function field case, but the onlypublished account is in Deninger 1984. Deninger (1986) showed how to deduce theZ-constructible case from the constructible case. In Zink 1978 there is an alternative methodof obviating the problem with 2-primary components in the original theorem.

4 Global results: complements

This section is concerned with various improvements of Theorem 3.1. We also discuss itsrelation to the theorems in Chapter I. The notations are the same as in the preceding twosections.

Sheaves without sections with finite support

Let F be a sheaf on an open subschemeU of X . For anyV etale overU , a sections 2 � (V , F) is said to havefinite supportif sv D 0 for all but finitely manyv 2 V .

PROPOSITION4.1. LetF be aZ-constructible sheaf on an open affine subschemeU of X .If F has no sections with finite support, thenExt1U (F, Gm) andH 2

c (U, F) are finite, and˛1(U, F) is an isomorphism.

PROOF. Note that HomU (Z, Gm) D O�U , which is finitely generated, and that Ext1

U (Z, Gm) �DPic(U ), which is finite (because, in the function field case, it is a quotient of Pic0(X)). Itfollows immediately that Ext1

U (F, Gm) is finite if F is constant. As we observed in (3.14),there is a finite surjective map� WU 0

! U , a constantZ-constructible sheafF 0 onU 0, anda morphismF ! ��F 0 whose kernel has support on a proper closed subset ofU . As F

has no sections with finite support, we see that the map must be injective. LetF 00 be itscokernel. In the exact sequence

Ext1U (��F 0, Gm)! Ext1U (F, Gm)! Ext2(F 00, Gm),

7This was Remark 3.18 in the original.


Ext1U (��F 0, Gm) �D Ext1U 0(F 0, Gm) (see 3.9) and so is finite, and Ext2(F 00, Gm) is tor-sion, and so Ext1

U (F, Gm) (being finitely generated) has finite image in it. This provesthat Ext1U (F, Gm) is finite, and Theorem 3.1 implies that˛1(X, F) is an isomorphism. Itfollows thatH 2

c (U, F) is also finite.

Nonconstructible sheaves

We say that a sheafF on U � X is countableif F(V ) is countable for allV etale overU . For example, any sheaf defined by a group scheme of finite type overU is countable.Fix a separable closureKs of K. If F is countable, then there are only countably manypairs(s, V ) with V an open subset of the normalization ofU in a finite subextension ofKs ands 2 F(V ). Therefore the construction in (0.9) expressesF as a countable union ofZ-constructible sheaves (of constructible sheaves ifF is torsion).

PROPOSITION4.2. Let F be a countable sheaf on an open subschemeU of X , and con-sider the map r (U, F)WExtrU (F, Gm)! H 3�r (U, F)�.

(a) For r � 2, the kernel of r (U, F) is divisible, and it is uncountable when nonzero;for r D 0 or r > 4, ˛r (U, F) is injective.

(b) For r � 2, ˛r (U, F) is surjective.(c) If F is torsion, then r (U, F) is an isomorphism for allr .(d) If U is affine andF has no sections with finite support, then˛2(U, F) is an isomor-

phism and 1(U, F) is surjective.

PROOF. Write F as a countable union ofZ-constructible subsheaves,F DS

Fi. Then(see I 0.21 and I 0.22) there is an exact sequence

0! lim �

(1) Extr�1U (Fi , Gm)! ExtrU (F, Gm)! lim

�ExtrU (Fi , Gm)! 0,

and lim �

(1) Extr�1U (Fi , Gm) is divisible (and uncountable when nonzero) if each group Extr�1(Fi , Gm)

is finitely generated, and it is zero if each group Extr�1(Fi , Gm) is finite. Theorem 3.1 pro-vides us with a map Extr (Fi , Gm)! H r (U, Fi)

� which is injective for allr and is surjec-tive for r � 2; it is an isomorphism for anyr for which the groups Extr (Fi , Gm) are finite.On passing to the inverse limit, we obtain a map lim

�ExtrU (Fi , Gm)! H 3�r

c (U, F)� withthe similar properties. The proposition is now obvious from (3.1) and (4.1).

COROLLARY 4.3. Let G be a separated group scheme of finite type over an open affinesubschemeU of X . Then˛2(U, G)WExt2U (G, Gm)! H 1

c (U, G)� is an isomorphism. IfGdefines a torsion sheaf, thenr (U, G) is an isomorphism for allr.

PROOF. If a sections of G overV agrees with the zero section on an open subset ofV ,then it agrees on the whole ofV (becauseG is separated overV ). ThusG (when regardedas a sheaf) has no sections with support on a finite subscheme, and the corollary resultsimmediately from part (d) of the proposition.


EXAMPLE 4.4. In particular,

Ext2U (A, Gm)�D�! H 1

c (U, A)�, A a semi-abelian scheme overU,

Ext2U (Gm, Gm)�D�! H 1

c (U, Gm)�, and

ExtrU (Ga, Gm)�D�! H 3�r (U, Ga)� for all r when char(K) D p 6D 0.

(In fact the groups ExtrU (G, Gm), computed for the smalletale site, seem to be rather patho-

logical. For example, ifk is a finite field, then Homk(Gm, Gm) � Gal(ks=k) D bZ.)

EXERCISE 4.5. (a) Show that there are only countably manyZ-constructible sheaves onU . (HintW Use Hermite’s theorem.)

(b) Show that there are uncountably many countable sheaves on SpecZ. (HintW Con-sider sheaves of the form

Lp prime ip�Fp.)

Tori

We investigate the duality theorem whenF is replaced by a torus. By atorus over aschemeY , we mean a group scheme that becomes isomorphic to a product of copies ofGm on a finiteetale covering ofY . The sheaf of charactersX �(T ) of T is the sheafV 7! HomV (T , Gm) (homomorphisms as group schemes). It is a locally constantZ-constructible sheaf. In the next theorem,^ denotes completion relative to the topology ofsubgroups of finite index.

THEOREM 4.6. LetT be a torus on an open subschemeU of X .(a) The cup-product pairing

H r (U, T )�H 3�rc (U, X �(T ))! H 3

c (U, Gm) �D Q=Z

induces isomorphisms

H r (U, T )^! H 3�r

c (U, X �(T ))� for r D 0, 1,

H r (U, T )! H 3�rc (U, X �(T ))� for r � 2.

If U is affine, thenH 1(U, T ) is finite.(b) Assume thatK is a number field. The cup-product pairing

H r (U, X �(T ))�H 3�rc (U, T )! H 3

c (U, Gm) �D Q=Z


H r (U, X �(T ))^! H 3�r

c (U, T )� for r D 0, 1,

H r (U, X �(T ))! H 3�rc (U, T )� for r � 2.


PROOF. (a) The sheafX �(T ) is locally isomorphic toZdim(T ), and soExt rU (X �(T ), Gm)

is locally isomorphic to the sheaf associated with the presheafV 7! H r (V , Gm)dim(T ). It istherefore zero forr > 0. As Ext0

U (X �(T ), Gm) D HomU (X �(T ), Gm) D T , the spectralsequence

H r (U, ExtsU (X �(T ), Gm) H) ExtrCs

U (X �(T ), Gm)

gives isomorphismsH r (U, T )��! ExtrU (X �(T ), Gm) for all r . Thus (a) follows from

(3.1) and (4.1).(b) Consider the diagramQ

v =2U

H r�1(Kv, X �(T )) �! H rc (U, X �(T )) �! H r (U, X �(T )) �!

Qv =2U

H r (Kv, X �(T ))??y ??y ??y ??yQv =2U

H 3�r (Kv, T )��! H 3�r (U, T )�

�! H 3�rc (U, T ) �!

Qv =2U

H 2�r (Kv, T ))�.

Replace the groupsH 0 andH 1c (U, X �(T )) with their completions. Then (I 2.4) shows that

the mapsQ

H r (Kv, X �(T ))!Q

H 2�r (Kv, T )� are isomorphisms (provided one com-pletesH 0(Kv, X �(T ))), and (a) shows that the mapsH r

c (U, X �(T ))! H 3�r (U, T )� areisomorphisms (provided one completesH r

c (U, X �(T )) for r D 0, 1). Now the five-lemmashows thatH r (U, X �(T )) ! H 3�r

c (U, T )� is an isomorphism for allr (provided onecompletesH 0(U, X �(T ))).

COROLLARY 4.7. AssumeK is a number field. There are canonical isomorphisms

Dr (U, X �(T ))! D3�r (U, T )�

where

Dr (U, X �(T )) D Im(H rc (U, X �(T ))! H r (U, X �(T )), r 6D 0,

D0(U, X �(T )) D Im(H 0c (U, X �(T ))^

! H 0(U, X �(T ))^), r D 0,

Dr (U, T ) D Im(H rc (U, T )! H r (U, T )), r 6D 0, 1,

Dr (U, T ) D Im(H rc (U, T )^

! H r (U, T )^), r D 0, 1.

PROOF. Part (a) of the theorem and (I 2.4) show that the dual of the sequence

0! Dr (U, X �(T ))! H r (U, X �(T ))!L

H r (Kv, X �(T ))

(complete the groups forr D 0) is an exact sequenceLH 2�r (Kv, T )! H 3�r

c (U, T )! Dr (U, X �(T ))�! 0,

(complete the groups for3� r D 0, 1) which identifiesDr (U, X �(T ))� with D3�r (U, T ).


Duality for Exts of tori

We wish to interpret (4.6b) in terms of Exts, but for this we shall need to use the bigetale siteXEt on X and the flat siteXfl. Recall that for any locally Noetherian schemeY , YEt is the category of schemes locally of finite type overY endowed with theetaletopology, andYfl is the same category of schemes endowed with the flat topology. Alsof denotes the morphismYfl ! YEt which is the identity map on the underlying cate-gories. For the rest of this section,Ext r

Yfl(F, F 0) denotes the sheaf onYEt associated with

the presheafV 7! ExtrVfl(F, F 0). Note thatV 7! HomVfl (F, F 0) is already a sheaf, and so

� (V ,HomVfl (F, F 0)) D HomVfl (F, F 0).

PROPOSITION4.8. For any sheafF onYEt and smooth group schemeG of finite type overY , there is a spectral sequence

H r (YEt, ExtsYfl

(f �F, G)) H) ExtrCsYEt

(F, G).

PROOF. If F 0 is an injective sheaf onYf l , thenf�F 0 is also injective (Milne 1980, III 1.20),and soHomYEt(F,f�F 0) is flabby (ibid. III 1.23). HenceH r (YEt,HomYEt(F,f�F 0)) D 0

for r > 0. But for anyV locally of finite type overY ,

� (V ,HomYEt(F,f�F 0)) D HomVEt(F,f�F 0)

D HomVfl (f�F, F 0)

D � (V ,HomYfl (f�F, F 0)).

ThereforeH r (YEt,HomYfl (f�F, F 0)) D 0 for r > 0, which means thatHomYfl (f

�F, F 0)

is acyclic for� (YEt,�). Next note that

� (YEt,HomYfl (f�F, F 0)) D HomYfl (f

�F, F 0) D HomYEt(F,f�F 0),

and so there is a spectral sequence

H r (YEt, Ext sYfl

(f �F, F 0)) H) RrCs˛(F 0)

where˛ D HomYEt(F,�) ı f�. There is an obvious spectral sequence

ExtrYEt(F, Rsf�F 0) H) RrCs˛(F 0).

On replacingF 0 with G in this spectral sequence and using thatRsf�G D 0 for s > 0

(ibid. III 3.9), we find thatRrCs˛(G) D ExtrCsYEt

(F, G). The result follows.

COROLLARY 4.9. For any sheafF on Yet and smooth group schemeG on Y , there is aspectral sequence

H r (Yet, Ext sYfl

(f �F, G)) H) ExtrCsYet

(F, G)

wheref now denotes the obvious morphismYfl ! Yet and ExtsYfl

(f �F, G) denotes thesheaf onYet associated withV 7! ExtsVfl

(f �F, F 0).


PROOF. Let f 0WYEt ! Yet be the obvious morphism. For any sheavesF on Yet and

F 0 on YEt, HomYEt(f0�F, F 0) D HomYet(F,f 0

�F 0). As f 0� is exact and preserves injec-

tives, this shows that ExtrYEt

(f 0�F, F 0) D ExtrYet(F,f�F 0) for all r . Moreover the sheaf

ExtsYfl

(f �F, Gm) of the corollary is the restriction toYet of the corresponding sheaf in(4.8), and so the result follows from (4.8) becauseYEt andYet yield the same cohomologygroups (see Milne 1980, III 3.1).

PROPOSITION 4.10. Let Y be a regular scheme, and letp be a prime such thatp! isinvertible onY . ThenExt r

Yfl(T , Gm) D 0 for 0 < r < 2p � 1.

PROOF. Since this is a local question, we can assume thatT D Gm. From Breen 1969,~7, we know thatExt r

Ufl(Gm, Gm) is torsion forr � 1. Let ` be prime, and consider the

sequence

� � � ! Ext r�1Yfl

(�`, Gm)! Ext rYfl

(Gm, Gm)`�! Ext r

Yfl(Gm, Gm)! � � � .

If Y is connected, then this sequence starts as

0! Z`�! Z! Z=`Z

0�! Ext1

Yfl(Gm, Gm)

`�! � � � .

Therefore,Ext1Yfl

(Gm, Gm) D 0. We shall complete the proof by showing thatExt rYfl

(�`, Gm) D

0 for all ` if 0 < r < 2p � 2.If ` is invertible onY , then�` is locally isomorphic toZ=`Z, and soExt r

Yfl(�`, Gm) is

locally isomorphic toExt rYfl

(Z=`Z, Gm). There is an exact sequence

� � � ! Ext rYfl

(Z=`Z, Gm)! Ext rYfl

(Z, Gm)! Ext rYfl

(Z, Gm)! � � � .

But Ext rYfl

(Z, Gm) is the sheaf (for theetale topology) associated with the presheafV 7!

H r (Vet, Gm); it is therefore zero forr > 0 and equal toGm for r D 0. The sequencetherefore shows thatExt r

Yfl(Z=`Z, Gm) D 0 for r > 0.

Next assume that is not invertible onY . Our assumption implies that for all primesq < p, q�` D �`. Therefore the main theorem of Breen 1975 shows thatExt r

Yfl(�`, Gm) D

0 for 1 < r < 2p � 2, and forr D 1 the sheaf is well-known to be zero (see Milne 1980,III 4.17).

THEOREM 4.11. Assume thatK is a number field. LetT be a torus on an open subschemeU of X , and assume that6 is invertible onU .

(a) The groupExtrUEt(T , Gm) is finitely generated forr D 0, finite forr D 1, and torsion

of cofinite type forr D 2, 3.(b) The map r (U, F)WExtrUEt

(T , Gm)! H 3�rc (Uet, T )� is an isomorphism for0 < r �

4, and˛0(U, T ) defines an isomorphismHomU (T , Gm)^ ! H 3c (U, T )� (as usual,

^ denotes completion for the topology of finite subgroups).

PROOF. The lemma shows that the spectral sequence in (4.9) gives isomorphisms

ExtrUEt(T , Gm)

��! H r (U, X �(T ))

for r � 4. Therefore the theorem follows from (4.6).


REMARK 4.12. (a) The only reason we did not allowK to be a function field in (4.6b) and(4.11) is that this case involves additional complications with the topologies.

(b) It is likely that (4.11) holds withXEt replaced by the smooth siteXsm. If one knewthat the direct image functorf�, wheref is the obvious morphismXEt! Xsm, preservedinjectives, then this would be obvious.

(c) It is not clear to the author whether or not pathologies of the type noted in Breen1969b should prevent Extr

UEt(T , Gm) being dual toH 3�r

c (U, T ) for all r � 0 (and withoutrestriction on the residue characteristics).

Relations to the theorems in Galois cohomology

In this subsection,U is an open affine subscheme ofX andS is the set of primes ofK notcorresponding to a point ofU . We also make use of the notations in I 4; for example,GS D

�1(U, �). For a sheafF on U , we writeF DD Hom(F, Gm). WhenM is aGS -module

such thatmM D 0 for some integerm that is invertible onU , M D D Hom(M, Ks�).

PROPOSITION4.13. LetF be a locally constant constructible sheaf onU such thatmF D

0 for somem that is invertible onU , and letM D F� andN D M D be theGS -modulescorresponding toF andF D .

(a) The groupDr (U, F) D XrS(K, M) and Dr (U, F D) D Xr

S(U, M d); conse-quently, the pairing

Dr (U, F)�D3�r (U, F D)! Q=Zof (3.4) can be identified with a pairing

XrS(K, M)�X3�r

S (K, M D)! Q=Z.

(b) The groupExtrU (F, Gm) D H r (GS , N) andH rc (U, F) D Extr�1

GS(N, CS); conse-

quently the pairingExtrU (F, Gm)�H 3�r

c (U, F)! Q=Z,

of (3.1) can be identified with a pairing

H r (GS , N)� Ext2�rGS

(N, CS)! Q=Z.

(c) The long exact sequence

� � � ! H rc (U, F)! H r (U, F)!

Mv2S

H r (Kv, Fv)! � � �

can be identified with a long exact sequence

� � � ! H 3�r (GS , N)�! H r (GS , M)!

Mv2S

H r (Kv, M)! � � � .

PROOF. (a) Compare the sequences

0 ��! XrS(K, M) ��! H r (GS , M) ��!

Lv2S H r (Kv, M)??y ??y�

??y�

0 ��! Dr (U, F) ��! H r (U, F) ��!L

v2S H r (Kv, F )


the second of which arises from the sequence in (2.3a) and the definition ofDr (U, F).(b) As Ext r

U (F, Gm) D 0 for r > 0, ExtrU (F, Gm) D H r (U, F D), and (2.9) showsthatH r (U, F D) D H r (GS , N). The second isomorphism can be read off from the longexact Extr (F,�)-sequence corresponding to the sequence of sheaves defined by the exactsequence ofGS -modules

0! R�S !

LK�v ! CS ! 0.

(c) It follows from (2.9) thatH r (U, F) D H r (GS , M), and it is obvious thatH r (Kv, F ) D

H r (Kv, M). According to (3.3),H rc (U, F) D H 3�r (U, F D)�, and (2.9) again shows that

H 3�r (U, F D) D H 3�r (U, N).

For aGS -moduleM , write M d D Hom(M, R�S ).

PROPOSITION4.14. Let T be a torus onU , and letX �(T ) be its sheaf of characters. IfM D X �(T )� is theGS -module corresponding toX �(T ), thenM d D T�. For all ` thatare invertible onU and allr � 1, Dr (U, T )(`) DXr

S(K, M)(`) andDr (U, X �(T ))(`) D

XrS(U, M d)(`); consequently, the pairing

Dr (U, T )(`)�D3�r (U, X �(T ))(`)! (Q=Z)(`)

of (4.8) can be identified forr � 1 with a pairing

XrS(K, M d)(`)�X3�r

S (K, M)(`)! (Q=Z)(`).

PROOF. In the course of proving (2.9), we showed thatH r ( QU , Gm)(`) D 0 for all r > 0.ThereforeH r (U, T )(`) D H r (GS , T )(`).

REMARK 4.15. Presumably, the maps are the same as those in Chapter I. Once this hasbeen checked, some of the results of each chapter can be deduced from the other. It is notsurprising that there is an overlap between the two chapters: to give a constructible sheafonX is the same as to give aGS -moduleM for some finite set of nonarchimedean primesS together with Gal(Ks

v=Kv)-modulesMv for eachv 2 S and equivariant mapsM !Mv

(see Milne 1980, II 3.16).

Galois cohomology has the advantage of being more elementary thanetale cohomology,and one is not led to impose unnecessary restrictions (for example, thatS is finite) as issometimes required for theetale topology. Etale cohomology has the advantage that moremachinery is available and the results are closer to those that algebraic topology wouldsuggest.

NOTES. Propositions 4.1 and 4.2 are taken from Deninger 1986.

5 Global results: abelian schemes

The notations are the same as those listed at the start of~2. In particular,U is always anopen subscheme orX . As in (I 6), we fix an integerm that is invertible onU and writeM ^Ddf lim �

M=mnM for them-adic completion ofM .

5. GLOBAL RESULTS: ABELIAN SCHEMES 173

Let A be an abelian scheme overU , and letA be its generic fibre. AsA is properover U , the valuative criterion of properness (Hartshorne 1977, II 4.7) shows that everymorphism Spec(K) ! A extends to a morphismU ! A, that is,A(K) D A(U ). Asimilar statement holds for anyV etale overU , which shows thatA representsg�A onUet.In fact (see Artin 1986, 1.4)A representsg�A onUsm.

PROPOSITION5.1. (a) The groupH 0(U,A) is finitely generated; forr > 0, H r (U,A) istorsion andH r (U,A)(m) is of cofinite-type; the map

H r (U,A)(m)!Yv arch

H r (Kv, A)(m)

is surjective forr D 2 and an isomorphism forr > 2.(b) For r < 0, Y

v arch

H r�1(Kv,A)! H rc (U,A)

is an isomorphism;H 0c (U,A) is finitely generated;H 1

c (U,A) is an extension of a torsiongroup by a subgroup which has a natural compactification;H 2

c (U,A) is torsion; andH 2

c (U,A)(m) is of cofinite-type; forr � 3, H rc (U,A)(m) D 0.

PROOF. (a) The groupH 0(U,A) D A(U ) D A(K), which the Mordell-Weil theoremstates is finitely generated. As Galois cohomology groups are torsion in degree> 1,the Leray spectral sequenceH r (U, Rsg�A) H) H rCs(K, A) shows that the groupsH r (U, g�A) are torsion forr > 0 becauseRsg�A is torsion fors > 0 andH r (K, A)

is torsion forr > 0. Finally, the finiteness ofH r (U,A)m follows from the cohomologysequence of

0! Am! A m�! A! 0

becauseH r (U,Am) is finite for all r (by 2.13). On replacingm with mn in this cohomol-ogy sequence and passing to the direct limit overn, we obtain an exact sequence

0! H r�1(U,A)˝Qm=Zm! H r (U,A(m))! H r (U,A)(m)! 0. (5.1.1)

The first term in this sequence is zero forr > 1 because thenH r�1(U,A) is torsion. Hence

H r (U,A(m))��! H r (U,A)(m) for r � 2. AsAmn is locally constant,H r (U,Amn) D

H r (GS ,Amn(KS)) (by 2.9). ThereforeH r (U,Amn)!Qv archH r (Kv, Amn) is surjective

for r D 2 (by I 4.16) and an isomorphism forr � 3 (by I 4.10c), and it follows thatH r (U,A)(m)!

Qv archH r (Kv, A)(m) has the same properties.

(b) All statements follow immediately from (a) and the exact sequence

� � � ! H rc (U,A)! H r (U,A)!

Mv =2U

H r (Kv, A)! � � � .

The dual abelian schemeAt to A is characterised by the fact that it represents thefunctorV 7! Ext1V (A, Gm) on Usm (generalized Barsotti-Weil formula, see Oort 1966, III18). AsHom(A, Gm) D 0, the local-global spectral sequence for Exts gives rise to a mapH r (U,At)! ExtrC1

U (A, Gm) all r . On combining this with the pairing

ExtrU (A, Gm)�H 3�rc (U,A)! H 3

c (U, Gm) �D Q=Z


we get a pairing

H r (U,At)�H 2�rc (U,A)! H 3

c (U, Gm) �D Q=Z.

(For a symmetric definition of this pairing in terms of biextensions, see Chapter III.) Wedefine

D1(U,A) D Im(H 1c (U,A)! H 1(U,A)) D Ker(H 1(U,A)!

Yv =2U

H 1(Kv, A)).

It is a torsion group, andD1(U,A)(m) is of cofinite-type.

THEOREM 5.2. (a) The groupH 0(U,At)(m) is finite; the pairing

H 0(U,At)(m)�H 2c (U,A)! Q=Z

is nondegenerate on the left and its right kernel is them-divisible subgroup ofH 2c (U,A).

(b) The groupsH 1(U,At)(m) andH 1c (U,A)(m) are of cofinite-type, and the pairing

H 1(U,At)(m)�H 1c (U,A)(m)! Q=Z

annihilates exactly the divisible subgroups.(c) If D1(U, At)(m) is finite, then the compact groupH 0(U, At)^ is dual to the discrete

torsion groupH 2c (U, A)(m).

PROOF. BecauseH rc (U,Amn) is finite for all n andr , passage to the inverse limit in the

sequences

0! H r�1c (U,A)(mn)

! H rc (U,Amn)! H r

c (U,A)mn ! 0

yields an exact sequence

0! H r�1c (U,A)^

! H rc (U, TmA)! TmH r

c (U,A)! 0, (5.2.1)

where we have writtenH rc (U, TmA) for lim

�H r

c (U,Amn). Note thatTmH rc (U,A) is

torsion-free and is nonzero only if the divisible subgroup ofH rc (U,A)(m) is nonzero.

Corollary 3.3 provides us with nondegenerate pairings of finite groups

H r (U,Atmn)�H 3�r

c (U,Amn)! Q=Z,

and hence a nondegenerate pairing (of a discrete torsion group with a compact group)

H r (U,At(m))�H 3�rc (U, TmA)! Q=Z.

For r D 0, this shows that the finite group

H 0(U,At)(m) D H 0(U,At(m)) D A(K)(m)

is dual toH 3c (U, TmA), and (5.2.1) shows that this last group equals

H 2c (U,A)^

D H 2c (U,A)=H 2

c (U,A)m-div


becauseH 3c (U,A) D 0. This completes the proof of (a).

From (5.1.1) we obtain an isomorphism

H 1(U,At(m))=H 1(U,At(m))div��! H 1(U,At)(m)=H 1(U,At)(m)div,

and the left hand group is dual toH 2c (U, TmA)tor (see I 0.20e). The sequence (5.2.1) gives

an isomorphismH 1c (U,A)^

tor

�D�! H 2

c (U, TmA)tor, and

H 1c (U,A)^

tor D H 1c (U,A)(m)=H 1

c (U,A)(m)div.

This completes the proof of (b).

For (c), consider the diagram

0 ��! H 0(U,At)^ ��! H 1(U, TmAt) ��! TmH 1(U,At) ��! 0??y ??y�

??y0 ��! H 2

c (U,A)(m)��! H 2

c (U,A(m))��! (H 1

c (U,A)˝Qm=Zm)��! 0.

It shows that the mapH 0(U,At)^! H 2

c (U, A)(m)� is injective, and that it is an isomor-phism if and only ifTmH 1(U,At) ! (H 1

c (U,A) ˝ Qm=Zm)� is injective. On applyingHom(Qm=Zm,�) to the exact sequence

0! D1(U,At)! H 1(U,At)!Yv =2U

H 1(Kv, At),

we obtain the top row of the following diagram:

0 ��! TmD1(U,At) ��! TmH 1(U,At) ��!Q

TmH 1(Kv, At)??y ??y�

(H 1c (U,A)˝Qm=Zm))�

��!Q

(H 0(Kv, A)˝Qm=Zm)�

Our assumption onD1(U,At) implies thatTmD1(U,At) D 0, and so the diagram showsthat TmH 1(U,At) ! (H 1

c (U,A) ˝ Qm=Zm)� is injective. This completes the proof of(c).

COROLLARY 5.3. The groupD1(U,A) is torsion, andD1(U,At)(m) is of cofinite-type;8

there is a canonical pairing

D1(U,At)(m)�D1(U,A)(m)! Q=Z

whose left and right kernels are the divisible subgroups of the two groups.

PROOF. The first statement follows directly from the definition ofD1(U,A). For thesecond statement, we use the commutative diagram

0 ��! D1(U,At)(m) ��! H 1(U,At)(m) ��!Qv =2U H 1(Kv,At)??y ??y�

0 ��! D1(U,A)(m)��! H 1

c (U,A)(m)��!

Qv =2U H 0(Kv,A)�.

8Recall that throughout this section, we are assuming thatm is invertible onU .


It demonstrates that there is a mapD1(U,At)(m) ! D1(U,A)(m)� whose kernel obvi-ously contains the divisible subgroup ofD1(U,At)(m). The kernel of the second verticalmap is zero, and that of the first is divisible. A diagram chase now shows that the kernelD1(U,At)(m) ! D1(U,A)� is divisible. Because of the symmetry of the situation, thisimplies that the right kernel of the pairing is also divisible.

EXERCISE5.4. Let'D WA! At be the map defined by a divisor onA. Show that for alla 2 D1(U,A), h'D(a), ai D 0.

We now show how the above results can be applied to the Tate-Shafarevich group.Write S for the set of primes ofK not corresponding to a point ofU.

LEMMA 5.5. The mapH 1(U,A) ! H 1(K, A) induces isomorphismsH 1(U,A)��!

H 1(GS , A) andD1(U,A)��!X1(K, A).

PROOF. BecauseA D g�A, the Leray spectral sequence forg gives an exact sequence

0! H 1(U,A)! H 1(K, A)! � (U, R1g�A).

This sequence identifiesH 1(U,A) with the set of principal homogeneous spaces forA

overK that are split by the inverse image of someetale cover ofU , or equivalently, thathave a point inKun

v for eachv (recall thatKv is the field of fractions ofOhv). From the

Hochschild-Serre spectral sequence forKunv overKv and (I 3.8), we see that the restriction

mapH 1(Kv, A)! H 1(Kunv , A) is injective. Therefore we have an exact sequence

0! H 1(U,A)! H 1(K, A)!Mv2U

H 1(Kv, A). (5.5.1)

We have seen (I 3.10) that the mapsH 1(Kv, A) ! H 1(bKv, A) are injective, and so oncomparing this sequence with that in (I 6.5), we see immediately thatH 1(U,A)(m) �DH 1(GS , A)(m). On combining (5.5.1) with the exact sequence

0! D1(U,A)! H 1(U, A)!Mv =2U

H 1(Kv, A),


0! D1(U,A)! H 1(K, A)!Mall v

H 1(Kv, A),

and this shows thatD1(U, A) DX1(K, A).

THEOREM 5.6. LetA be an abelian variety overK.

(a) There is a canonical pairing

X1(K, At)�X1(K, A)! Q=Z

whose kernels are the divisible subgroups of each group.


(b) AssumeX1(K, A)(m) is finite. Then the dual of the exact sequence

0!X1(K, A)(m)! H 1(K, A)(m)!Mall v

H 1(Kv, A)(m)! B! 0


0 X1(K, At)(m) H 1(K, A)(m)�

Yall v

At(Kv) At(K)^ 0.

PROOF. (a) Fix a prime` and chooseU so that` is invertible onU and A has goodreduction at all primes ofU . ThenA andAt extend to abelian schemes onU , and thelemma shows that the pairing

D1(U,At)(`)�D1(U,A)(`)! Q=Z

of (5.2) can be identified with a pairing

X1(K,At)(`)�X1(K,A)(`)! Q=Z.

(b) ChooseU so small thatm is invertible on it andA has good reduction at all primesof U . The assumption implies thatX1(K, At) is also finite. Therefore, (5.2) shows thatthe dual of the sequence

0!X1(K, A)(m)! H 1(U,A)(m)!Mv =2U

H 1(Kv, A)(m)! H 2c (U,A)(m)! � � �


0 X1(K,At)(m) H 1(U,A)(m)�

Yv =2U

At(Kv)^ At(K)^

� � � .

Now pass to the direct limit in the first sequence and to the inverse limit in the sec-ond over smaller and smaller open setsU . According to (I 6.25), the mapAt(K)^

!Qall v At(Kv)

^ is injective, and so the result is obvious.

REMARK 5.7. (a) We have now defined three pairings

X1(K, At)�X1(K, A)! Q=Z.

For the sake of definiteness, we shall refer to the pairing in (5.6) as theCassels-Tate pairing.(b) For any abelian schemeA on a regular schemeY , Breen’s theorems (Breen 1969a,

1975) imply thatExt rY (A, Gm) D 0 for r D 0 or 1 < r < 2p � 1, wherep is a prime

such thatp! is invertible onY . SinceExt1Yfl

(A, Gm) D At , we see that ExtrUEt

(A, Gm) D

H r�1(U,At) for r � 4, provided 6 is invertible onU . In particular Ext2UEt(A, Gm) D

H 1(U,At), which is countable. By way of contrast, (4.4) implies that Ext2Uet

(A, Gm) iscountable if and only if the divisible subgroup ofH 1

c (U,A) is zero. Thus if Ext2UEt(A, Gm) D

Ext2Uet(A, Gm) then the divisible subgroup ofX1(K, A) is zero (and there is an effective

procedure for finding the rank ofA(K)!).


Finally we show that, by usingetale cohomology, it is possible to simplify the lastpart of the proof of the compatibility of the conjecture of Birch and Swinnerton-Dyer withisogenies.

Let f WA! B be an isogeny of degree prime to charK. The initial easy calculationsin (I 7) showed that to prove the equivalence of the conjecture forA andB, one must showthat

z(f (K)) DYv =2U

z(f (Kv)) � z(f t(K)) � z(X1(f )).

In this formula, for a nonarchimedean primev, Kv denotes the completion ofK rather thanthe Henselization, but it is easy to see that this does not change the value ofz(f (Kv)).

Let m be the degree off , and choose an open schemeU * X on whichm is invertibleand which is such thatf extends to an isogenyf WA! B of abelian schemes overU . Theexact commutative diagram

0 ��! A(U ) ��!Qv =2U

H 0(Kv, A) ��! H 1c (U,A) ��! X1(K, A) ��! 0??y ??y ??y ??y

0 ��! B(U ) ��!Qv =2U

H 0(Kv, B) ��! H 1c (U,B) ��! X1(K, B) ��! 0

shows thatz(f (K)) � z(H 1

c (f )) DYv =2U

z(H 0(Kv,f )) � z(X1(f )).

To prove the compatibility, it therefore remains to show that

z(H 1c (f )) � z(f t(K)) D

Yv arch

z(H 0(Kv,f )) � z(f (Kv))�1.

Let F be the kernel off WA! B. The exact sequence

0! Coker(H 0c (f ))! H 1

c (U, F)! H 1c (U,A)! H 1

c (U,B)! H 2c (U, F)

! H 2c (U,A)! H 2

c (U,B)! H 3c (U, F)! 0

shows that

�c(U, F) � [H 0c (U, F)]�1

D [Coker(H 0c (f ))]�1

� z(H 1c (f ))�1

� z(H 2c (f )).

But�c(U, F)[H 0

c (U, F)]�1D

Yv arch

[F(Kv)][H0(Kv, F )]�1

by 2.14b,z(H 2

c (f )) D z(f t(K))�1

by duality, andCokerH 0

c (f ) DYv arch

CokerH �1(Kv,f )

6. GLOBAL RESULTS: SINGULAR SCHEMES 179

becauseQ

H �1(Kv, A)��! H 0

c (U,A) andQ

H �1(Kv, B)��! H 0

c (U,B). Therefore

z(H 1c (f )) � z(f t(K)) D

Yv arch

[F(Kv)]�1[H 0(Kv, F )][Coker(H �1(Kv,f ))],

and it remains to show that for all archimedean primesv

[F(Kv)]�1� z(f (Kv)) D [H 0(Kv, F )]�1

� z(H 0(Kv,f )) � [Coker(H �1(Kv,f ))].


0! CokerH �1(Kv,f )! H 0(Kv, F )! KerH 0(Kv,f )! 0,

we see that this comes down to the obvious fact that Cokerf (Kv)! CokerH 0(Kv,f ) isan isomorphism (cf. I 3.7).

NOTES. This section interprets Tate’s theorems on the Galois cohomology of abelian vari-eties over number fields (Tate 1962) in terms ofetale cohomology.

6 Global results: singular schemes

We now letX be an integral scheme whose normalization is the spectrum of the ring ofintegers inK (number field case) or the unique complete smooth curve withK as its func-tion field (function field case). The definition in~2 of cohomology groups with compactsupport also applies to singularX : for any sheafF on an open subschemeU of K, thereis an exact sequence

� � � ! H rc (U, F)! H r (U, F)!

Mv2S

H r (Kv, F )! � � �

whereS is the set of primes ofK not corresponding to point of the normalization ofU .Using (1.12) – (1.15), it is possible to prove an analogue of (2.3).

Let u 2 U , and leth 2 K�. Thenh can be writtenh D f=g with f, g 2 Ou, and wedefine

ordu(h) D length(Ou=(f )) � length(Ou=(g)).

This determines a homomorphismK�! Z (see Fulton 1984, 1.2). Alternatively, we

could defineordu(h) D [k(v)W k(u)]ordv(h)

where the sum is over the points of the normalization ofU lying overu (ibid. 1.2.3). Onecan define similar maps for each closed point lying overu on schemeV etale overU andso obtain a homomorphism orduWg�Gm! iu�Z. DefineG to be the complex of sheaves

g�Gm

Pordu

��!

Mu2U 0

iu�Z

on Uet. Note that ifU is smooth, then we can identifyG with Gm. We shall frequentlymake use of the fact thatRsg�Gm D 0 for s > 0. This follows from the similar statementfor the normalizationQU of U , and the fact thatQU ! U is finite.


LEMMA 6.1. For all open subschemesU of X , there is a canonical trace map

TrWH 3c (U, G)

�D�! Q=Z

such that(a) wheneverU is smooth,Tr is the map defined at the start of~3;(b) wheneverV � U , the diagram

H 3c (V , G)

Tr��! Q=Z??y

H 3c (U, G)

Tr��! Q=Z

commutes.

PROOF. The proof is similar to the smooth case. LetS be the set of primes ofK notcorresponding to a point in the normalization ofU , and assume first thatU 6D X . Fromthe definition ofG, we obtain a cohomology sequence

0! H 2(U, G)! H 2(K, Gm)!Mu2U 0

Q=Z! H 3(U, Gm)! 0.

The middle map sends an elementa of Br(K) toP

invu(a) where

invu(a) DPv 7 !u

[k(v)W k(u)] invv(a).

The kernel-cokernel exact sequence of the pair of maps

Br(K)!Mall v

Br(Kv)!Mv2U 0

Q=Z

provides us with the top row of the following diagram:

H 2(U, G) ��! B ��! Q=Z ��! 0 ??y ??y0 ��! H 2(U, G) ��!

Lv2S Br(Kv) ��! H 3

c (U, G) ��! 0.

HereB D Ker(L

all v Br(Kv) !L

v2U 0 Q=Z). Write B D B 0˚�L

v2S Br(Kv)�. Then

B 0 maps to zero inQ=Z, and as it is the kernel of the middle vertical map, this shows thatthe mapQ=Z! H 3

c (U, G) is an isomorphism.For U D X , one can remove a smooth point and prove as in (2.6) thatH 3

c (X, G) D

H 3c (X r fxg, G).

As in ~3, the trace map allows us to define maps

˛r (U, F)WExtrU (F, G)! H 3�rc (U, F)�

for any sheafF onU.

7. GLOBAL RESULTS: HIGHER DIMENSIONS 181

THEOREM 6.2. Let F be aZ-constructible sheaf on an open subsetU of X . For r � 2,the groupsExtrU (F, G) are torsion of cofinite-type, andr (U, F) is an isomorphism. Forr D 0, 1, the groupsExtrU (F, G) are of finite-type, and r (U, F) defines isomorphisms

ExtrU (F, Gm)^! H 3�r

c (U, F)�

where^ denotes completion relative to the topology of subgroups of finite index. IfF isconstructible, then r (U, F) is an isomorphism of finite groups for allr 2 Z.

We begin by proving the theorem whenF has support on a finite subset.

LEMMA 6.3. Theorem 6.2 is true ifF has support on a proper closed subset ofU.

PROOF. We can assume that our sheaf isi�F wherei is the inclusion of a single closedpointv into U . From the analogue of (2.3) for singular schemes, we see thatH r

c (U, i�F) D

H r (v, F ). As in (1.14), ExtrU (i�F, g�G) D 0, and so Extr�1x (F, Z)! ExtrU (F, Gm) is an

isomorphism. We have a commutative diagram

Ext1U (F, G) � H 3�rc (U, F) > H 3(U, G)

Extr�1gx

(M, Z)

^

�

�H 3�r (gx, M)

_�

> H 2(gx, Z)

�

_

and so the theorem follows in this case from (I 1.10).

Now let F be a sheaf onU , and letj WV ,! U be a smooth open subscheme ofU . For F jV , the theorem becomes (3.1). Since Extr

U (j!F jV , G) D ExtrV (F, G) andH r

c (U, j!F jV ) D H rc (V , F), this implies that the theorem is true forj!F jV . The lemma

shows that the theorem is true fori�i�F , and the two cases can be combined as in (3.7) toprove the general case.

NOTES. Theorem 6.2 is proved in Deninger 1986 in the case thatU D X andF is con-structible.

7 Global results: higher dimensions

The notations are the same as those in~2. Throughout,� WY ! U will be a morphism offinite-type, and we defineH r

c (Y, F) D H rc (U, R�!F).

PROPOSITION7.1. If F is constructible andmF D 0 for some integerm that is invertibleonU , then the groupsH r (Y, F) are finite.

PROOF. For any constructible sheafF on Y , the sheavesRr��F are constructible (seeDeligne 1977, 1.1). Therefore it suffices to prove the proposition forU itself, but we havealready noted that (3.1) implies the proposition in this case.


REMARK 7.2. In particular, the proposition shows thatH 1(Y, Z=mZ) is finite for all m

that are invertible onY , and this implies that�ab1 (Y )(m) is finite. Under some additional

hypotheses, most notably that� is smooth, one knows (Katz and Lang 1981) that the fullgroup�1(Y )ab is finite except for the part provided by constant field extensions in thefunction field case.

Letv be an archimedean prime ofK. In the next proposition, we writeYv for Y�U SpecKv

andYv for Y�U SpecKsv.

PROPOSITION7.3. Let� WY ! U be proper and smooth, and letF be a locally constantconstructible sheaf onY such thatmF D 0 for somem that is invertible onU . Then

�(Y, F) DYv arch

�(Yv, F jYv)Gv

j�(Yv, F jYv)jv,

where�(Yv, F jYv)

GvdfD

Yr

[H r (Yv, F jYv)Gv ](�1)r

.

PROOF. The proper-smooth base change theorem (Milne 1980, VI 4.2) shows that thesheavesRs��F are locally constant and constructible for allr , and moreover that(Rs��F)v D

H s(Yv, F jYv) for all archimedean primesv. Therefore (2.13) shows that

�(GS , Rs��F) DYv arch

[H s(Yv, F jYv)Gv ]

j[H s(Yv, F jYv)]jv.

On taking the alternating product of these equalities, we obtain the result.

REMARK 7.4. (a) A similar result is true for�c(Y, F).

(b) The last result can be regarded as a formula expressing the trace of the identity mapon Y in terms of the schemesYv, v archimedean. For a similar result for other maps, seeDeninger 1986b.

Before stating a duality theorem for sheaves on such aY , we note a slight improvementof (3.1). Just as in the case of a single sheaf, there is a canonical pairing of (hyper-) Extand (hyper-) cohomology groups

ExtrU (F �, Gm)�H 3�rc (U, F �)! H 3

c (U, Gm),

for any complex of sheavesF � onU . Consequently, there are also maps

˛r (U, F �)WExtrU (F �, Gm)! H 3�rc (U, F �)�.

LEMMA 7.5. Let F � be a complex of sheaves onU that is bounded below and such thatH r (F �) is constructible for allr and zero forr >> 0. Then

˛r (U, F �)WExtrU (F �, Gm)! H 3�rc (U, F �)�

is an isomorphism of finite groups.


PROOF. If F consists of a single sheaf, this is (3.1b). The general case follows from thiscase by a standard argument (see, for example, Milne 1980, p280).

We write ExtrY,m(F, F 0) for the Ext group computed in the category of sheaves ofZ=mZ-modules onYet. Let F be a sheaf killed bym; if F 0 is anm-divisible andF 0

! I �

is an injective resolution ofF 0, thenF 0m! I � is an injective resolution ofF 0

m, and so

ExtrY,m(F, F 0m)

dfD H r (HomY (F, I �

m)) D H r (HomY (F, I �))dfD ExtrY (F, F 0).

In particular, ifF is killed bym andm is invertible onY , then ExtrY,m(F,�m) D ExtrY (F, Gm).

Let � WY ! U be smooth and separated with fibres pure of dimensiond , and letm

be an integer that is invertible onU . Then there is a canonical trace mapR2d�!�˝dm

�D�!

Z=mZ (Milne 1980, p285). OnU , �m is locally isomorphic toZ=mZ, and so when the

trace map is tensored with�m, it becomes an isomorphismR2d�!�˝dC1n

�D�! �m. As

Rr��˝dC1n D 0 for r > 2d , there is a canonical trace map

H 2dC3c (Y,�˝dC1

m )�D�! H 3

c (Y,�m)�D�! Z=mZ,

and hence a pairing

ExtrY,m(F,�˝dC1m )�H 2dC3�r

c (Y, F)! H 2dC3c (Y,�˝dC1

m ) �D Z=mZ.

THEOREM7.6. LetY ! U be a smooth separated morphism with fibres pure of dimensiond , and letF be a constructible sheaf onY such thatmF D 0 for somem that is invertibleonU . Then

ExtrY,m(F,�˝dC1m )�H 2dC3�r

c (Y, F)! H 2dC3c (Y,�˝dC1

m ) �D Z=mZ


PROOF. The duality theorem in Artin, Grothendieck, and Verdier 1972/73, XVIII, showsthat there is a canonical isomorphism

R�!(RHomY,m(F,�˝dC1m [2d]))

�D�! RHomU,m(R�!F,�m).

(See also Milne 1980, p285.) On applyingR� (U,�) to the left hand side, we get a com-plex of abelian groups whoser th cohomology group is ExtrC2d

Y,m (F,�˝dC1m ). On applying

the same functor to the right hand side, we get a complex of abelian groups whoser th co-homology group is Extr

U,m(R�!F,�m). This is equal to ExtrU (R�!F, Gm), and Lemma 7.5

shows thatExtrU (R�!F, Gm)

��! Hom(H 3�r

c (U, R�!F), Z=mZ).

By definition,H 3�rc (U, R�!F) D H 3�r

c (Y, F), and so this proves the theorem.

For any sheafF on Y such thatmF D 0, write F(i) D F ˝ �˝im and F D(i) D

Hom(F,�˝im ). Note thatF D in the old terminology is equal toF D(1) in the new.


COROLLARY 7.7. Let � WY ! U be a smooth separated morphism with fibres pure ofdimensiond , and letF be a locally constant constructible sheaf onY such thatmF D 0

for somem that is invertible onU . Then cup-product defines a nondegenerate pairing offinite groups

H r (Y, F D(d C 1))�H 2dC3�rc (Y, F)! H 2dC3

c (Y,�˝dC1m ) D Z=mZ,

for all r.

PROOF. In this case ExtrY,m(F,�˝dC1m ) D H r (Y, F D(d C 1)).

As usual, we letKi be the sheaf onY defined by thei th QuillenK-functor.

COROLLARY 7.8. Let F be a constructible sheaf onY such that nF D 0 for some prime` invertible onU . Assume thatH 2dC2

c (Y,K2dC1) is torsion. Then there is a trace map

H 2dC3c (Y,K2dC1)(`)

��! Q`=Z`, and the canonical pairing

ExtrY (F,K2dC1)�H 2dC3�rc (Y, F)! H 2dC3

c (Y,K2dC1)(`) � Q`=Z`

is a duality of finite groups.

PROOF. Recall (1.19), that for anym that is invertible onU , there is an exact sequence ofsheaves

0! �˝iC1m ! K2iC1

m�! K2iC1! 0.

Therefore ExtrY,`n(F,�˝dC1`n ) D ExtrY (F,K2dC1). Also, there is an exact sequence

0! H 2dC2c (Y,K2dC1)˝Q`=Z` ! H 2dC3

c (Y,�˝dC1`1 )! H 2dC3

c (Y,K2dC1)(`)! 0.

Since we have assumedH 2dC2c (Y,K2dC1) to be torsion, the first term of this sequence is

zero, and so we obtain isomorphisms

H 2dC3c (Y,K2dC1)(`) � H 2dC3

c (Y,�˝dC1`1 ) � Q`=Z`.

The corollary now follows directly from (7.6).

REMARK 7.9. For any regular schemeY of finite type over a field, it is known Milne 1986,7.1, thatH r (Y,Ki) is torsion forr > i. (The condition thatY be of finite type over a fieldis only required so that Gersten’s conjecture can be assumed.)

ASIDE 7.10. One would like to weaken the condition thatY is smooth overU in the aboveresults to the condition thatY is regular. The purity conjecture inetale cohomology will berelevant for this. It states the following: LetiWZ ! Y be a closed immersion of regularlocal Noetherian schemes such that for eachz in Z, the codimension ofZ in Y at z is c,and letn be prime to the residue characteristics; then(Rr i!)Z=mZ D 0 for r 6D 2c, and(R2ci!)Z=mZ D �˝�c

n (Artin, Grothendieck, and Verdier 1972/73, XIX).The author is uncertain as to the exact conditions under which the proof of the conjec-

ture is complete. See ibid., XIX 2.1, and Thomason 1984.The strategy for passing from the smooth case to the regular case is as followsW replace

U by its normalization inY , and note that the theorem will hold on an open subsetV of Y ;now examine the mapY r V ! U . (Compare the proof of the Poincare duality theoremVI 11.1 in Milne 1980, especially Step 3.)


In the case thatY D U , Corollary 7.8 is much weaker than Theorem 3.1 becauseit requires thatF be killed by an integer that is invertible onU . We investigate someconjectures that lead to results that are true generalizations of (3.1). We first consider theproblem of duality forp-torsion sheaves in characteristicp.

For a smooth varietyY over a field of characteristicp 6D 0, we letWn˝iY =k

be the sheafof Witt differential i-forms of lengthn onY (Illusie 1979). Define�n(i) to be the subsheafof Wn˝

iY =k

of locally logarithmic differentials (see Milne 1986a,~1). The pairing

(! ,!0) 7! !^!0WWn˝

i�Wn˝

j! Wn˝

iCj

induces a pairing�n(i)� �n(j)! �n(i C j).

THEOREM 7.11. Let Y be a smooth complete variety of dimensiond over a finite fieldk.

Then there is a canonical trace mapH dC1(Y, �n(d))�D�! Z=pnZ, and the cup-product

pairing

H r (Y, �n(i))�H dC1�r (Y, �n(d � i))! H dC1(Y, �n(d)) �D Z=pnZ


PROOF. When dim Y � 2 or n D 1, this is proved in Milne 1976. The extension to thegeneral case can be found in Milne 1986a.

COROLLARY 7.12. The canonical pairing

ExtrY,pn(Z=pnZ, �n(d))�H dC1�r (Y, Z=pnZ)! H dC1(Y, �n(d)) �D Z=pnZ


PROOF. One sees easily that ExtrY,pn(Z=pnZ, �n(d)) �D H r (Y, �n(d)), and so this follows

immediately from the theorem.

COROLLARY 7.13. Let Y be a smooth complete variety of dimensiond � 2 over a finite

field k. Then there is a canonical trace mapH dC2(Y,Kd)(p)��! Qp=Zp, and for anyn

there are nondegenerate pairings of finite groups

ExtrY (Z=pnZ,Kd))�H dC2�r (Y, Z=pnZ)! H dC2(Y,Kd)(p) �D Qp=Zp.

PROOF. The key point is that there is an exact sequence

0! Ki

pn

�! Ki ! �n(i)! 0

for i � 2. When i D 0, 1, the exactness of the sequence is obvious; wheni D 2, itsexactness at the first and second terms follows from theorems of Suslin (1983a) and Bloch(Bloch and Kato 1986) respectively. Now the corollary can be derived in the same man-

ner as (7.8); in particular the trace map is obtained from the mapsH dC1(Y, �1(d))��!

H dC2(Y,Kd)(p) andH dC1(Y, �1(d))��! (Q=Z)(p), where�1(i) D lim

�!�n(i).


REMARK 7.14. It should be possible to extend the last three results to noncomplete vari-etiesY by using (in the proof of 7.11) cohomology with compact support for quasicoherentsheaves (see Deligne 1966 or Hartshorne 1972). However, one problem in extending themto all constructible sheaves is that the purity theorem for the sheaves�n(i) is weaker thanits analogue for the sheaves�n(i) (see Milne 1986,~2). Nevertheless, I conjecture that forany constructible sheafF of Z=pnZ-modules on a smooth varietyY of dimensiond overa finite field,

ExtrY,pn(F, �n(d))�H dC1�rc (Y, F)! H dC1

c (Y, �n(d)) �D Z=pnZ

is a nondegenerate pairing of finite groups.I do not conjecture that the (7.13) holds for varieties of all dimensions.

Let Y be a smooth complete surface over a finite field. Then we have dualitiesW

ExtrY (Z=pnZ,K2)�H 4�r (Y, Z=pnZ)! H 4(Y,K2)(p) �D Qp=Zp, p D chark,

ExtrY (Z=`nZ,K3)�H 5�r (Y, Z=`nZ)! H 5(Y,K3)(`) �D Q`=Z`, ` 6D chark.

These are similar, but the numbers do not agree! It appears that in order to obtain a uni-form statement, the sheavesKi will have to be replaced by the objectsZ(i) conjecturedin Lichtenbaum 1984 to exist in the derived category ofS(Yet) for any regular schemeY . (Beilinson has independently conjectured the existence of similar objects inS(YZar).)These are to have the following properties:

(a)Z(0) D Z, Z(1) D Gm[�1].

(b`) For` 6D p, there is a distinguished triangle

�˝i`n [�1]! Z(i)

`n

�! Z(i)! �˝i`n .

This implies that there is an exact sequence

� � � ! H r (Y, Z(i))`n

�! H r (Y, Z(i))! H r (Y,�˝i`n )! � � � .

(c) There are canonical pairingsZ(i)� Z(j)! Z(i C j).

(d) H 2r�j (Z(i)) D Grr Kj up to small torsion, andH r (Z(i)) D 0 for r > i or r < 0

(alsoH 0(Z(i)) D 0 except wheni D 0).(e) If Y is a smooth complete variety over a finite field, thenH r (Y, Z(i)) is torsion for

all r 6D 2i, andH 2r (Y, Z(r)) is finitely generated.In the present context, it is natural to ask that the complex have the following additional

properties whenY is a variety over a field of characteristicpW

(bp) There is a distinguished triangle

�n(i)[�i � 1]! Z(i)pn

�! Z(i)! �n(i)[�i].

This implies that there is an exact sequence

� � � ! H r (Y, Z(i))pn

�! H r (Y, Z(i))! H r�i(Y, �n(i))! � � � .

(f) (Purity) If iWZ ,! Y is the inclusion of a smooth closed subscheme of codimensionc into a smooth scheme andj > c, thenRi!Z(j) D Z(j � c)[�2c].


THEOREM7.15. Let� WY ! U be smooth and proper with fibres pure of dimensiond . Let` be a prime, and assume that either` is invertible onU or ` DcharK andY is complete.Assume that there exist complexesZ(i) satisfying(b`) and thatH 2dC3

c (Y, Z(d C 1)) istorsion. Then there is a canonical isomorphism

H 2dC4c (Y, Z(d C 1))(`)

�D�! (Q=Z)(`),

and the cup-product pairing

H r (Y, Z(i))(`)�H 2dC4�rc (Y, Z(d C 1� i))(`)! H 2dC4

c (Y, Z(d C 1))(`) �D Q=Z(`)

annihilates only the divisible subgroups.

PROOF. Assume first that 6D char(K). The same argument as in the proof of (7.8) showsthe existence of an isomorphism

H 2dC3c (Y,�˝dC1

`1 )��! H 2dC4

c (Y, Z(d C 1)).

This proves that a trace map exists. Now the exact sequence

0! H r�1(Y, Z(i))˝ (Q=Z)(`)! H r�1(Y,�˝i`1)! H r (Y, Z(i))(`)! 0

shows thatH r�1(Y,�˝i`1) modulo its divisible subgroup is isomorphic toH r (Y, Z(i))(`)

modulo its divisible subgroup. Similarly

0! H rc (Y, Z(i))^

! lim �

H rc (Y,�˝i

`m)! T`HrC1c (Y, Z(i))! 0

shows thatH rc (Y, Z(i))(`) modulo its divisible subgroup is isomorphic to the torsion sub-

group of lim �

H rc (Y,�˝i

`m). Now the theorem follows from (7.7) using (I 0.20e).The proof when D p is similar.

Consider the following statement:

(*) for any smooth varietyY of dimensiond over a finite field and any con-structible sheafF onY , there is a duality of finite groups

ExtrY (F, Z(d))�H 2dC2c (Y, F)! H 2dC2

c (Y, Z(d)) �D Q=Z.

WhenF is killed by somem prime to the characteristic ofk and we assume (b) and thatH 2dC1

c (Y, Z(d)) is torsion, this can be derived from (7.6) in the same way as (7.8). WhenF is ap-primary sheaf, it is necessary to assume the conjectured statement in (7.14).

THEOREM 7.16. Let� WY ! U be a smooth proper morphism with fibres of dimensiond .Assume there exist complexesZ(i) satisfying the conditions (a), (b), and (f); also assume(*) above, and thatH 2dC3

c (Y, Z(d C 1)) is torsion, so that there exists a canonical trace

mapH 2dC4c (Y, Z(d C 1))

�D�! Q=Z. Then for any locally constant constructible sheafF

onY , there is a nondegenerate pairing of finite groups

ExtrY (F, Z(d C 1))�H 2dC4�ic (Y, F)! H 2dC4

c (Y, Z(d C 1)) �D Q=Z.


PROOF. First assumeF has support onYZ for some closed subschemeZ of U , and writei for the closed immersionYZ ,! Y . The spectral sequence

R HomYZ(F, Ri!Z(d C 1)) D R HomY (i�F, Z(d C 1))

shows thatExtr�2

YZ(F, Z(d � 1)) D ExtrY (i�F, Z(i)).

Thus this case of the theorem follows from the induction assumption.Next suppose thatmF D 0 for somem that is invertible onU . In this case then the

theorem can be deduced from (7.6) in the same way as (7.8).Next suppose thatpnF D 0, wherep DcharK. In this case the statement reduces to

(*).The last two paragraphs show that the theorem holds for the restriction ofF to YV for

some open subschemeV of U , and this can be combined with the statement proved in thefirst paragraph to obtain the full theorem.

In (4.11), we have shown that there is a nondegenerate pairing

ExtrU (F ˝ Z(1), Z(1))�H 4�rc (U, F ˝ Z(1))! H 4

c (Z(1)) �D Q=Z.

for any torsion-freeZ-constructible sheafF on U . For a finite fieldk, we also know thatthere is a nondegenerate pairing

Extrk(F ˝ Z(0), Z(0))�H 2(k, F ˝ Z(0))! H 2(k, Z(0)) �D Q=Z.

This suggests the following conjecture.

CONJECTURE7.17. 9For any regular schemeY of finite-type overSpecZ and of (absolute)

dimensiond , there is a canonical trace mapH 2dC2c (Y, Z(d))

�D�! Q=Z. For any locally

constantZ-constructible sheafF onY , the canonical pairing

ExtrY (F ˝L Z(i), Z(d))�H 2dC2�rc (Y, F ˝L Z(i))! H 2dC2

c (Y, Z(d)) �D Q=Z


ExtrY (F ˝L Z(i), Z(d))��! H 2dC2�r

c (Y, F ˝L Z(i))�, r 6D 2(d � i),

Ext2d�2iY (F ˝L Z(i), Z(d))

��! H 2C2i

c (Y, F ˝L Z(i))�.

The conjecture has obvious implications for higher class field theory.Finally, we mention that Lichtenbaum (1986) has suggested a candidate forZ(2) and

Bloch (1986) has suggested candidates forZ(r), all r . Also Kato (1985/6) has generalized(7.11) to a relative theorem, and in the case of a surface Etesse (1986a,b) has generalized itto other sheaves.

NOTES. This section owes much to conversations with Lichtenbaum and to his criticismsof an earlier version.

9In the original, this was miss-labelled 7.16.

Chapter III

Flat Cohomology

This chapter is concerned with duality theorems for the flat cohomology groups of finiteflat group schemes or Neron models of abelian varieties. In~1 - ~4, the base scheme isthe spectrum of the ring of integers in a number field or a local field of characteristic zero(with perfect residue field of nonzero characteristic). In the remaining sections, the basescheme is the spectrum of the rings of integers in a local field of nonzero characteristicor a curve over a finite field (or, more generally, a perfect field of nonzero characteristic).The appendices discuss various aspects of the theory of finite group schemes and Neronmodels.

The prerequisites for this chapter are the same as for the last: a basic knowledge of thetheory of sites, as may be obtained from reading Chapters II and III of Milne, 1980. Allschemes are endowed with the flat topology.

The results of the chapter are more tentative than those in the first two chapters. Oneproblem is that we do not yet know what is the correct analogue for the flat site of the notionof a constructible sheaf. The examples of Shatz 1966 show that for any nonperfect fieldK, there exist torsion sheavesF over K such thatH r (Kfl, F ) is nonzero for arbitrarilyhigh values ofr . In particular, no duality theorem can hold for all finite sheaves over such afield. We are thus forced to restrict our attention to sheaves that are represented by finite flatgroup schemes or are slight generalizations of such sheaves. Another problem is that fora finite flat group schemeN over an algebraically closed fieldk, the groups Extr

k(N, Gm)

computed in the category of flat sheaves overk need not vanish forr > 0 (see Breen1969b); they therefore do not agree with the same groups computed in the category ofcommutative algebraic groups overk, which vanish forr > 0.

0 Preliminaries

We begin by showing that some of the familiar constructions for theetale site can also bemade for the flat site.

Cohomology with support on closed subscheme


189

190 CHAPTER III. FLAT COHOMOLOGY

Zi�! X

j � U

in which i andj are closed and open immersions respectively, andX is the disjoint unionof i(Z) andj(U ).

LEMMA 0.1. The functorj�WS(Xfl)! S(Ufl) has an exact left adjointj!.

PROOF. For any presheafP on U , we can define a presheafj!P on X as follows: for anymorphismV ! X of finite type, set

� (V , j!P) D ˚P(Vf )

where the sum is over all mapsf 2 HomX (V , U ) andVf denotesV regarded as a schemeover U by means off . One checks easily thatj! is left adjoint to the restriction functorjpWP(Xfl)! P(Ufl) and that it is exact. Leta be the functor sending a presheaf onXfl to

its associated sheaf. Then the functor

S(Ufl) ,! P(Ufl)j!

�! P(Xfl)a�! S(Xfl),

is easily seen to be left adjoint toj�. It is therefore right exact. But it is also a compositeof left exact functors, which shows that it is exact.

LEMMA 0.2. There is a canonical exact sequence

0! j!j�Z! Z! i�i�Z! 0.

PROOF. The maps are the adjunction maps. We explicitly compute the two end terms. Let'WV ! X be a scheme of finite type overU . WhenV is connected,' factors throughU in at most one way. Therefore, in this case, the presheafj!Z takes the valueZ on V if'(V ) � j(U ) and takes the value 0 otherwise. It follows thatj!Z is the sheaf

V 7! � (V , j!Z) D Hom(� 00(V ), Z)

where� 00(V ) is the subset of�0(V ) of connected components ofV whose structure mor-

phisms factor throughU . Since� (V , Z) D Hom(�0(V ), Z), it is obvious thatj!j�Z! Z

is injective.For anyV , � (V , i�Z) D Hom(�0('�1Z), Z), which is zero if and only if'(V )\Z D

;. It clear from this that the sequence is exact at its middle term, and that an element of� (V , i�Z) lifts to � (Vi , Z) for eachVi in an appropriate Zariski open covering ofV . Thiscompletes the proof.

The mapF 7! Ker(� (X, F)! � (U, F)) defines a left exact functorS(Xfl)! Ab,and we writeH r

Z (X,�) for its r th right derived functor.

PROPOSITION0.3. LetF be a sheaf onXfl.(a) For all r , H r

Z (X, F) D ExtrX (i�Z, F).(b) For all r , ExtrX (j!Z, F) D ExtrU (Z, j�F).


(c) There is a long exact sequence

� � � ! H rZ (X, F)! H r (X, F)! H r (U, F)! � � � .

PROOF. (a) On applying Hom(�, F ) to the exact sequence in (0.2), we get an exact se-quence

0! HomX (i�Z, F )! HomX (Z, F )! HomU (Z, j�F)

or,0! HomX (i�Z, F )! � (X, F)! � (U, F).

Therefore HomX (i�Z, F )��! H 0

Z (X, F), and on taking the right derived functors weobtain the result.

(b) Note that, because it has an exact left adjoint,j� preserves injectives. It is also exact(Milne 1980, p68). Therefore we may derive the equality HomX (j!Z, F ) D HomU (Z, j�F)

and obtain an isomorphism ExtrX (j!Z, F ) � ExtrU (Z, j�F).

(c) The ExtX (�, F )-sequence arising from the exact sequence in (0.2) is the requiredsequence.

Cohomology with compact support

Let X be the spectrum of the ring of integers in a number field or else a complete smoothcurve over a perfect field, and letK be the field of rational functions onX . For any opensubschemeU of X and sheafF on Ufl, we shall definecohomology groups with compactsupportH r

c (Xfl, F ) having properties similar to their namesakes for theetale topology. Inparticular, they will be related to the usual cohomology groups by an exact sequence

� � � ! H rc (U, F)! H r (U, F)!

Lv2X rU H r (Kv, F )! � � �

whereKv is the field of fractions of the HenselizationOhv of Ov.

Let Z be the complement ofU in X , and letZ0DSv2X rU SpecKv (disjoint union).

ThenZ0 D lim �

V �X U , where the limit is over theetale neighbourhoodsV of Z in X W

V � V �X U

��

��

Z - X?etale

� U.?

Let i 0 be the canonical mapi 0WZ0! U , and letF ! I �(F) be an injective resolution

of F on Ufl. Theni 0� is exact and preserves injectives, and soF jZ0! I �(F)jZ0 is an

injective resolution ofF jZ0. There is an obvious restriction map

uW� (U, I �(F))! � (Z0, I �(F)jZ0),

and we define1 Hc(U, F) to be the translateC �(u)[�1] of its mapping cone. Finally, wesetH r

c (U, F) D H r (Hc(U, F)).

1In the original, this was denotedHc .


PROPOSITION0.4. (a) For any sheafF on an open subschemeU � X , there is an exactsequence

� � � ! H rc (U, F)! H r (U, F)!

Lv2X rU

H r (Kv, Fv)! � � � .

(b) For any short exact sequence

0! F 0! F ! F 00

! 0

of sheaves onU , there is a long exact sequence of cohomology groups

� � � ! H rc (U, F 0)! H r

c (U, F)! H rc (U, F 00)! � � � .

(c) For any sheafF onU and open subschemeV of U , there is an exact sequence

� � � ! H rc (V , F jV )! H r

c (U, F)!L

v2U rV

H r (Ohv , F )! � � � .

(d) If F is the inverse image of a sheafF0 on Uet , or if F is represented by a smoothalgebraic space, thenH r

c (Ufl, F ) D H r (Xet, j!F).(e) For any sheavesF andG onU , there are canonical pairings

ExtrU (F, G)�H sc(U, F)! H rCs

c (U, G).

PROOF. (a) Directly from the definition ofHc(U, F), we see that there is a distinguishedtriangle

Hc(U, F)! � (U, I �(F))! � (Z0, I �(F)jZ0)! Hc(U, F)[1].

AsH r (� (U, I �(F))) D H r (U, F)

andH r (� (Z0, I �(F)jZ0)) D H r (Z0, F jZ0) D

Lv2X rU

H r (Kv, F ),

we see that the required sequence is simply the cohomology sequence of this triangle.(b) From the morphism

0 ��! � (U, I �(F 0)) ��! � (U, I �(F)) ��! � (U, I �(F 00)) ��! 0??yu0

??yu

??yu00

0 ��! � (Z0, I �(F 0)jZ0) ��! � (Z0, I �(F)jZ0) ��! � (Z0, I �(F 0)jZ0) ��! 0

of short exact sequences of complexes, we may deduce (II 0.10a) the existence of a distin-guished triangle

Hc(U, F 00)[�1]! Hc(U, F 0)! Hc(U, F)! Hc(U, F 00).

This yields the required exact sequence.


(c) LetF ! I �(F) be an injective resolution ofF onU , and consider the maps

� (V , I �(F))a�!

Lv =2V

� (Kv, I �(F))b�!

Lv =2U

� (Kv, I �(F))˚L

v2U rV

�v(Ohv , I �(F))[1].

The mapb is such thatH r (b) is the sum of the identity maps

H r (Kv, F )! H r (Kv, F ) (v 2 X r U )

and the maps in the complex

H r (Ohv , F )! H r (Kv, F )! H rC1

v (Ohv , F ), v 2 U r V .

From these maps, we get a distinguished triangle (II 0.10c)

C �(b)[�1]! C �(a)! C �(bıa)! C �(b).

ClearlyC �(a) D Hc(V , F)[1]. We shall show that there exist isomorphismsC �(bıa) �

Hc(U, F)[1] and C �(b) �L

v2U rV

� (Ohv , I �(F))[1] (in the derived category). Thus the

cohomology sequence of this triangle is the required sequence.From

� (V , I �(F))bıa�!

Lv =2U

� (Kv, I �(F))˚L

v2U rV

�v(Ohv , I �(F))[1]

c�!

Lv2U rV

�v(Ohv , I �(F))[1]

we get a distinguished triangle

C �(c)[�1]! C �(b ı a)! C �(c ı b ı a)! C �(c).

But C �(c ı b ı a) � � (U, I �(F))[1] andC �(c) �Lv =2U

� (Kv, I �(F))[1], which shows that

C �(b ı a) � Hc(U, F)[1].Finally, the statement aboutC �(b) is obvious from the distinguished triangles (forv 2

U r V )

� (Ohv , I �(F))! � (Kv, I �(F))! �v(Oh

v , I �(F))[1]! � (Ohv , I �(F))[1].

(d) SinceH r (Uet, F )��! H r (Ufl, F ) and

H r ((SpecKv)et, F )��! H r ((SpecKv)fl, F )

(Milne 1980, III 3), this follows from a comparison of the sequence in (0.4a) and with thecorresponding sequence for theetale topology.

(e) Letc0 2 H rc (U, F), and regard it as a homotopy class of maps of degreer

c0WZ! Hc(U, F)[r ].

Let c 2 ExtrU (F, F 0), and regard it as a homotopy class of maps of degrees,

cW I �(F)! I �(F 0)[s].

On restricting the maps in this class, we get a similar class of maps

cjZ0W I �(F)jZ0

! (I �(F 0)jZ0)[s].

The last two maps combine to give a morphismHc(U, F)! Hc(U, F 0)[s], and we definehc, c0i to be the composite of this morphism withc.


REMARK 0.5. (a) LetiWZ,!Y be a closed immersion, and letF be a sheaf onZ. Theproof in Milne 1980, II 3.6, of the exactness ofi� for theetale topology (hence the equalityH r (Uet, i�F) D H r (Zet, F )), fails for the flat topology.

(b) Note that the sequence in (c) has the same form as (II 2.3d) except that in the lattersequence it has been possible to replaceH r (Oh

v , F ) with H r (v, i�F). In the case of theflat topology, this is also possible ifF is represented by a smooth algebraic space (Milne1980, III 3.11).

REMARK 0.6. (a) In the case thatX is the spectrum of the ring of integers in a number field,it is natural to replaceHc(U, F) with the mapping cone of

Lv arch

S�(Kv, Fv)! Hc(U, F).

Then the cohomology groups with compact support fit into an exact sequence

� � � ! H rc (U, F)! H r (U, F)!

LvH

r (Kv, F )! � � �

where the sum is now over all primes ofK, including the archimedean primes, not inU ,and for archimedeanv,

H r (Kv, F ) D H rT (Gal(Ks

v=Kv), F(Ksv)).

(b) In the definition ofH rc (U, F) it is possible to replaceKv with its completion. Then

the sequence in (0.4a) will be exact withKv the completion ofK atv, and the sequence in(0.4c) will be exact withOh

v replaced withbOv. This approach has the disadvantage that thegroups no longer agree with theetale groups (that is, (0.4d) will no longer hold in general).

The definition given here of cohomology groups with compact support is simple andleads quickly to the results we want. I do not know whether there is a more natural defini-tion nor in what generality it is possible to define such groups.

Topological duality for vector spaces

Let k be a finite field withk elements, and letV be a locally compact topological vectorspace overk. Write V _ for the topological linear dual ofV , V _

D Homk,cts(V , k).

THEOREM 0.7. The pairing

V _� V ! C�, (f, v) 7! exp(2� i

pTrk=Fp

f (v))

identifiesV _ with the Pontryagin dual ofV .

PROOF. Let V � be the Pontryagin dual ofV . The pairing identifiesV _ with a subspace ofV �. Clearly the elements ofV _ separate points inV , and soV _ is dense inV �. But V _ islocally compact, and so it is an open subset of its closure inV �; hence it is open inV �. Asit is a subgroup, this implies that it is also closed inV � and so equalsV �.

Let R be a complete discrete valuation ring of characteristicp 6D 0 having a finiteresidue fieldk, and letK be the field of fractions ofR. The choice of a uniformizing

parametert for R determines an isomorphismK��! k((t)) carryingR ontok[[t ]]. Define

a residue map resWK! k by setting res(P

aiti) D a�1.


COROLLARY 0.8. LetV be a freeR-module of finite rank, and letV _ be itsR-linear dual.Then the pairing

V _� (V ˝K)=V ! C�, (f, v) 7! exp(2� i

pTrk=Fp

(res(f (v)))

identifiesV _ with the Pontryagin dual of(V ˝K)=V .

PROOF. Consider first the case thatV D R. ThenR is isomorphic (as a topologicalk-vector space) to the direct product of countably many copies ofk, andK=R is isomorphicto the direct sum of countably many copies ofk. The pairing

(a, b) 7! res(ab)WR�K=R! k

identifiesR with the k-linear topological dual ofK=R, and so (0.7) shows thatR��!

(K=R)�. In the general case, the pairing

V _� (V ˝K)=V ! k, (f, v) 7! res(f (v))

similarly identifiesV _ with k-linear topological dual of(V ˝ K)=V , and so again theresult follows from the proposition.

The Frobenius morphism

For any schemeS of characteristicp 6D 0, theabsolute Frobenius mapFabsWS ! S isdefined to be the identity map on the underlying topological space anda 7! ap onOS . Itis functorial in the sense that for any morphism� WX ! S , the diagram

XFabs �� X??y� ??y�

SFabs �� S

commutes, but it does not commute with base change. Therelative Frobenius mapFX=S

is defined by the diagram

<Fabs

X < X (p) <FX=S

X, X (p) dfD X �S,Fabs S.

��

�

S_�

<Fabs

S_�(p)

For any morphismT ! S , FX=S �T id D FX �S T=T .A schemeS is said to beperfectif FabsWS ! S is an isomorphism. For example, an

affine scheme SpecR is perfect if thepth power mapa 7! apWR! R is an isomorphism.

In the case thatS is perfect, it is possible to identify� (p)WX (p)

! S with F �1absı� WX ! S

andFX=S with Fabs.A finite group schemeN over a schemeS is said to haveheighth if F h

N =S WN ! N (ph)

is zero butF h�1N =S is not zero. For any flat group schemeN , there is a canonical morphism

V D VN WN(p)! N , called theVerschiebung(see Demazure and Gabriel 1970, IV, 3, 4).


The Oort-Tate classification of group schemes of orderp

Let� D Z[�, (p(p�1))�1]\Zp, where� is a primitivepth root of1, and the intersection istaken insideQp. We consider only schemesX such that the unique morphismX ! SpecZfactors through Spec�. For example,X can be any scheme of characteristicp because�hasFp as a residue field. The following statements classify the finite flat group schemes oforderp overX (see Oort and Tate 1970, or Shatz 1986,~4).

(0.9a)It is possible to associate a finite flat group schemeNLa,b

of orderp overX witheach triple(L, a, b) comprising an invertible sheafL overX , an elementa 2 � (X,L˝p�1),and an elementb 2 � (X,L˝1�p) such thata˝b D wp for a certain universal elementwp.

(0.9b) Every finite flat group scheme of orderp overX isomorphic toNLa,b

for sometriple (L, a, b).

(0.9c)There exists an isomorphismNLa,b

��! NL0

a0,b0 if and only if there is an isomor-

phismL ��! L0 carryinga to a0 andb to b0.

(0.9d)For allX -schemesY ,

NLa,b(Y ) D fy 2 � (Y,L˝OY )j y˝p

D a˝ yg.

(0.9e)The Cartier dual ofNLa,b

is NL_

b,a.

(0.9f) WhenX has characteristicp, wp D 0. If a D 0 in this case, thenN has heightone, and thep-Lie algebra ofNL

a,bisL with thep-power mapf 7! f (p)

D b ˝ f .

Duality for unipotent perfect group schemes

When the ground field is finite, our duality theorems will be for the cohomology groupsendowed with the structure of a topological group. When the ground field is not finite, itwill be necessary to endow the cohomology groups with a stronger structure, namely thestructure of a perfect pro-algebraic group, and replace Pontryagin duality with Breen-Serreduality. We now describe this last duality.

Let S be a perfect scheme of characteristicp 6D 0. TheperfectionX pf of anS -schemeX is the projective limit of the system

XredF � X

(p�1)red

F � � � �

F � X

(p�n)red

F � � � �

It is a perfect scheme, and has the universal property that HomS(X, Y ) D HomS(X pf, Y )

for any perfectS -schemeY .Let S be the spectrum a perfect fieldk. A perfectS -schemeX is said to bealgebraic

if it is the perfection of a scheme of finite type overS . From the corresponding fact for thealgebraic group schemes overS , one sees easily that the perfect algebraic group schemesoverX form an abelian category. Define theperfect siteSpf to be that whose underlyingcategory consists of all algebraic perfectS -schemes and whose covering families are thesurjective families ofetale morphisms. For any algebraic group schemeG overS , the sheafon Spf defined byG is represented byGpf. One checks easily, that any sheaf onSpf that isan extension of perfect algebraic group schemes is itself represented by a perfect algebraicgroup scheme.

We writeS(pn) for the category of sheaves onSpf killed by pn.


THEOREM 0.10. For all r > 0, ExtrS(p)(Gpfa , Gpf

a ) D 0.

PROOF. See Breen 1981, where the result is proved with the base scheme the spectrum ofany perfect ring of characteristicp 6D 0.

LEMMA 0.11. Letf WSfl ! Spf be the morphism of sites defined by the identity map. Forany affine commutative algebraic group schemeG on S , f�G is represented byGpf andRrf�G D 0 for r > 0.

PROOF. We have already observed thatf�G is represented byGpf. If G is smooth, thenRrf�G D 0 for r > 0 because of the coincidence of flat andetale cohomology groups ofsmooth group schemes (Milne 1980, III 3.9). We calculateRrf�G for G D p and�p byusing the exact sequences

0! �p ! Gm

F�! Gm! 0

0! p ! Ga

F�! Ga! 0.

SinceF is an automorphism ofGpfm andGpf

a , we haveRrf��p D 0 D Rrf� p for all r .The general case follows from these case because, locally for theetale topology, anyG hasa composition series whose quotients areGm, Ga, �p, p, or anetale group scheme.

By ap-primary group scheme, we mean a group scheme killed by a power ofp.

LEMMA 0.12. LetG be a perfectp-primary affine algebraic group scheme onS ; let U beits identity component, and letD D G=U . ThenD is etale andU has a composition serieswhose quotients are all isomorphic toGpf

a .

PROOF. In view of the exactness off�, this is a consequence of the structure theorem foraffine commutative algebraic group schemes onS .

Let G(pn) be the category of perfect affine algebraic group schemes onS killed by pn,and letG(p1) D

SG(pn). Note that (0.12) shows thatG(p1) can also be described as

the category of perfect unipotent group schemes onS .

LEMMA 0.13. Let G be a perfect unipotent group scheme onS ; let U be its identitycomponent, and letD D G=U .

(a) If G is killed bypn, then there exists a canonical isomorphism

RHomS(pn)(G, Z=pnZ)�D�! RHomS(G, Qp=Zp).

(b) HomS(G, Qp=Zp)�D�! HomS(D, Qp=Zp), which equalsD�, the Pontryagin dual

of D.

(c) Ext1S(G, Qp=Zp)

�D�! Ext1

S(U, Qp=Zp), which is represented by a connected unipo-tent perfect group scheme; ifU has the structure of aWn(k)-module, then there is acanonical isomorphism ofWn(k)-modules

Ext1S(G, Qp=Zp)

�D�! HomWn(k)(U, Wn(OS)).


(d) Ext rS(G, Qp=Zp) D 0 for r > 1.

PROOF. (a) Choose an injective resolutionI � of Qp=Zp. The usual argument in the caseof abelian groups shows that an injective sheaf is divisible. Therefore the kernelI �

pn ofpnW I �! I � is a resolution ofZ=pnZ, and it is obvious that it is an injective resolution in

S(pn). Consequently

RHomS(pn)(G, Z=pnZ) D Hom�S(pn)(G, I �

pn) D Hom�S(G, I �) D RHomS(G, Qp=Zp).

(b,c,d) WhenD D Z=pZ, the sequence

0! HomS(Z=pZ, Q=Z)! HomS(Z, Q=Z)p�! HomS(Z, Q=Z)! Ext1

S(Z=pZ, Q=Z)! 0

shows thatRHomS(D, Q=Z) D HomS(D, Q=Z) D D�.

A generaletale groupD is locally (for theetale topology) an extension of copies ofZ=pZ,and so the same equalities holds for it.

If U D Gpfa , then (0.10) shows that the exact sequence

0! HomS(p)(Gpfa , Gpf

a )1�F�! HomS(p)(Gpf

a , Gpfa )! Ext1

S(p)(Gpfa , Z=pZ)! � � �

yields an isomorphismRHomS(p)(Gpf

a , Z=pZ) � G[�1]

where G is the cokernel of1 � F . It is well-known (see, for example, Serre 1960)that G D Gpf

a . ThereforeExt1S(Gpf

a , Z=pZ) D Ext1S(p)

(Gpfa , Q=Z) is connected, and

Ext rS(Gpf

a , Q=Z) D 0 for r 6D 1. A general connected unipotent groupU is an extension ofcopies ofGpf

a , and soExt rS(U, Q=Z) is connected forr D 1 and zero forr 6D 1.

Suppose thatU has the structure of aWn(OS)-module. The Artin-Schreier sequence

0! Z=pnZ! Wn(OS)F�1�! Wn(OS)! 0

gives a morphismWn(OS) ! (Z=pnZ)[1] in the derived category ofS(pn), and hence acanonical homomorphism

HomWn(OS )(U, Wn(OS))! RHomS(pn)(U, Z=pnZ),

which isWn(OS)-linear for the given structure onU . One checks easily that

Ext1Wn(OS )(U, Wn(OS)) D 0:

asU has a filtration by subWn(OS)-modules such that the quotients are linearly isomorphicto Gpf

a , it suffices to show that this homomorphism is an isomorphism forU D Gpfa , which

is assured by (0.10). It follows that the homomorphism is an isomorphism.The assertions (b), (c), and (d) now result in the general case from making use of the

exact sequence0! U ! G ! D! 0.


For any perfect connected unipotent groupU , we writeU _ for Ext1S(U, Qp=Zp). Let

Db(G(p1)) be the full subcategory of the derived category ofS(Spf) consisting of thosebounded complexes whose cohomology lies inG(p1). For anyG� in Db(G(p1)), defineG�t D RHomS(G�, Qp=Zp).

THEOREM 0.14. For any G� in Db(G(p1)), G�t also liesDb(G(p1)), and there is a

canonical isomorphismG��D�! G�t t . There exist canonical exact sequences

0! U rC1(G�)_! H �r (G�t)! Dr (G�)�

! 0,

whereU r (G�) is the identity component ofH r (G�) andDr (G�) D H r (G�)=U r (G�).

PROOF. The cohomology sheaves ofRHomS(G�, Qp=Zp) are the abutment of the spec-tral sequence

Er,s2 D Ext r

S(H �s(G�), Qp=Zp) H) Ext rCsS (G�, Qp=Zp).

After (0.13),Er,s2 D 0 for r 6D 0, 1, so that the spectral sequence reduces to a family of

short exact sequences

0! Ext1S(H rC1(G�), Qp=Zp)! Ext�r

S (G�, Qp=Zp)! HomS(H r (G�), Qp=Zp)! 0,

which are the required exact sequences. They imply moreover thatG�t is in Db(G(p1))

becauseG(p1) is stable under extension. Finally one shows that the homomorphism ofbidualityG�

! G�t t is an isomorphism by reducing the question to the cases ofZ=pZ andGa, which both follow directly from (0.13).

REMARK 0.15. Denote by Extrk(G, H) the Ext group computed in the category of affine

perfect group schemes overk. Each such Ext group can be given a canonical structure asa perfect group scheme, and (0.10) implies that, whenG andH are unipotent, Extr

k(G, H)

agrees withExt rS(G, H).

Pairings in the derived category

We review some of the basic definitions concerning pairings in the derived category. Formore details, see Gamst and Hoechsmann 1970 or Hartshorne 1966.

Fix a schemeX , endow it with a Grothendieck topology, and writeS(X) for the re-sulting category of sheaves. WriteC(X) for the category of complexes inS(X), andK(X) for category with the same objects but whose morphisms are homotopy classes ofmaps inC(X). The derived categoryD(X) is obtained fromK(X) by formally invert-ing quasi-isomorphisms. Thus, for example, a mapA�

! B� and a quasi-isomorphism

B�� C � define a morphismA� ! C � in D(X). As usual,C C(X), C �(X), and

C b(X) denote respectively the categories of complexes bounded below, bounded above,and bounded in both directions. We use similar notations for the homotopy and derivedcategories. SinceS(X) has enough injectives, for everyA� in C C(X), there is a quasi-

isomorphismA��! I(A�) with I(A�) a complex of injectives, and there is a canonical


equivalence of categoriesIC(X) ! DC(X), whereIC(X) is the full subcategory ofKC(X) whose objects are complexes of injective objects.

Recall that a sheafP is flat if �˝P WS(X)! S(X) is exact. For any bounded-above

complexA�, there is a quasi-isomorphismP(A�)��! A� with P(A�) a complex of flat

sheaves. IfB� is a second bounded-above complex, thenP(A�) ˝ B� is a well-definedobject ofD(X), which is denoted byA�

˝L B�. Despite appearances, there is a canonical

isomorphismA�˝

L B� �D B�˝

L A�.Let M andN be flat sheaves onX , and letA� andB� be objects ofC b(X). There is a

canonical pairing

ExtrX (M, A�)� ExtsX (N, B�)! ExtrCsX (M ˝N, A�

˝L B�)

that can be defined as follows: represent elementsf 2 ExtrX (M, A�) andg 2 ExtsX (N, B�)

as homotopy classes of mapsf WM ! I(A�)[r ] andgWN ! I(B�)[s]; thenf ˝ g isrepresented by

M ˝N ! I(A�)[r ]˝ I(B�)[s]� � P(I(A�))˝ I(B�)[r C s].

The pairing is natural, bi-additive, associative, and symmetric (up to the usual signs). It alsobehaves well with respect to boundary maps (Gamst and Hoechsmann, ibid.). A similardiscussion applies whenN is not flat — replace it with a flat resolution.

Consider a mapA� ˝L B�! G�. The above discussion gives a pairing

H r (X, A�)� ExtsX (M, B�)! ExtrCsX (M, G�),

There is also the usual (obvious) pairing

ExtrX (B�, G�)� ExtsX (M, B�)! ExtrCsX (M, G�).

The mapA�˝

L B�! G defines a mapA�

! Hom(B�, C �) and hence edge morphismsH r (X, A�)! ExtrX (B�, G�).

THEOREM 0.16. The following diagram commutes:

H r (X, A�) �ExtsX (M, B�)!ExtrCsX (M, G�)

jj

ExtrX (B�, G�)_

�ExtsX (M, B�)!ExtrCsX (M, G�).

_

PROOF. See Gamst and Hoechsmann 1970.

NOTES. The definition of cohomology groups with compact support for the flat topologyis new, and will play an important role in this chapter.

The duality for unipotent perfect group schemes has its origins in a remark of Serre(1960, p55) that Ext’s in the category of unipotent perfect group schemes over an alge-braically closed field can be used to define an autoduality of the category. For a detailedexposition in this context, see Begueri 1980,~1; Serre in fact worked with the equivalentcategory of quasi-algebraic groups. The replacement of Ext’s in the category of perfectgroup schemes with Ext’s in the category of sheaves, which is essential for the applicationswe have in mind, is easy once one has Breen’s vanishing theorem (0.10). Our expositionof the autoduality is based on Berthelot 1981, II, which, in turn, is based on Milne 1976.

Most of the rest of the material is standard.

1. LOCAL RESULTS: MIXED CHARACTERISTIC, FINITE GROUP SCHEMES 201

1 Local results: mixed characteristic, finite group schemes

Throughout this section,R will be a Henselian discrete valuation ring with finite residuefield k and field of fractionsK of characteristic zero. In particular,R is excellent. We usethe same notations as in (II 1): for example,X D SpecR andi andj are the inclusionsof the closed pointx and the open pointu of X into X . The characteristic ofk will bedenoted byp and the maximal ideal ofR by m.

LEMMA 1.1. LetN be a finite flat group scheme overR.(a) The mapN(R)! N(K) is a bijective, andH 1(X, N)! H 1(K, NK ) is injective;

for r � 2, H r (X, N) D 0.(b) The boundary mapH r (K, N)! H rC1

x (X, N) defines isomorphisms

H 1(K, N)=H 1(R, N)�D�! H 2

x (X, N)

H 2(K, N)�D�! H 3

x (X, N)I

for r 6D 2, 3,H r

x (X, N) D 0.

PROOF. (a) AsN is finite, it is the spectrum of a finiteR-algebraA. The image of anyR-homomorphismA! K is finite overR and is therefore contained inR. This shows thatN(R) D N(K). An elementc of H 1(X, N) is represented by a principal homogeneousspaceP overX (Milne 1980, III 4.3), andc D 0 if and only if P(R) is nonempty. AgainP is the spectrum of a finiteR-algebra, and so ifP has a point inK, then it already has apoint inR.

From Appendix A, we know that there is an exact sequence

0! N ! G ! G 0! 0

in which G andG 0 are smooth group schemes of finite type overX . According to Milne1980, III 3.11,H r (X, G) D H r (k, G0) andH r (X, G 0) D H r (k, G 0

1) for r > 0, whereG0 andG 0

1 are the closed fibres ofG andG 0 overX . The five lemma therefore shows that

H r (X, N)��! H r (k, N0) for r > 1. We now use that there is an exact sequence

0! N0! G1! G 01! 0

with G1 and G 01 smooth connected group schemes overk (for example, abelian vari-

eties). By Lang’s lemma,H r (k, G1) D 0 D H r (k, G 01) for r > 0, and it follows that

H r (k, N0) D 0 for r > 1.(b) This follows from the first statement, because of the exact sequence

� � � ! H rx (X, N)! H r (X, N)! H r (K, N)! � � � .

and the fact thatH r (K, N) D 0 for r > 2 (K has cohomological dimension2).

REMARK 1.2. The proof of the lemma does not use thatK has characteristic zero. Thesame argument as in the proof of (a) shows thatH r (X, N) D 0 for r � 2 if N is a finiteflat group scheme over any Noetherian Henselian local ring with finite residue field.


Let F be a sheaf onX . The pairing

ExtrX (F, Gm)� ExtsX (i�Z, F )! ExtrCsX (i�Z, Gm)

can be identified with a pairing

ExtrX (F, Gm)�H sx(X, F)! H rCs

x (X, Gm)I

see (0.3a). SinceGm is a smooth group scheme, the natural mapH rx (Xet, Gm)! H r

x (Xfl, Gm)

is an isomorphism for allr , and so (see II 1) there is a canonical trace map

H 3x (Xfl, Gm)

�D�! Q=Z.

Let N be a finite group scheme overX . The sheaf defined by the Cartier dualN D of N

can be identified withHom(N, Gm), and the pairingN D�N ! Gm defines a pairing

H r (X, N D)�H 3�rx (X, N)! H 3

x (X, Gm) �D Q=Z.

This can also be defined using the edge morphismsH s(X, N D)! ExtsX (N, Gm) and theExt-pairing (0.16).

THEOREM 1.3. For any finite flat group schemeN onX ,

H r (X, N D)�H 3�rx (X, N)! H 3

x (X, Gm) �D Q=Z

is a nondegenerate pairing of finite groups, allr .

We shall give two proofs, but first we list some corollaries.

COROLLARY 1.4. For any finite flat group schemeN over X , H 1(X, N D) is the exactannihilator ofH 1(X, N) in the pairing

H 1(K, N D)�H 1(K, N)! H 2(K, Gm) �D Q=Z

of (I 2.3).

PROOF. The diagram

H 1(X, N D)�H 1(X, N)!H 2(X, Gm) D 0

H 1(K, N D)

_

�H 1(K, N)

_

!H 2(K, Gm)

_

�DQ=Z

shows thatH 1(X, N D) andH 1(X, N) annihilate each other in the pairing. Forr D 1,(1.1) allows us to identify the pairing in the theorem with

H 1(X, N D)�H 1(K, N)=H 1(X, N)! H 2(K, Gm).

Thus we see that the nondegeneracy of the pairing in this case is equivalent to the statementof the corollary.


COROLLARY 1.5. LetN be a finite flat group scheme onX . For all r < 2p � 2,

ExtrX (N, Gm)�H 3�rx (X, N)! H 3

x (X, Gm) �D Q=Z


PROOF. Let N(p) be thep-primary component ofN . According to Breen 1975,

Ext rX (N(p), Gm) D 0 for 1 � r < 2p � 2,

and as we explained in the proof of (II 4.10), this implies thatExt rX (N, Gm) D 0 for

1 < r < 2p � 2 (Ext1X (N, Gm) D 0 by Milne 1980, III 4.17, andExt r

X (N(`), Gm) D 0

for r > 1 and` 6D p becauseN(`) is locally constant for theetale topology). HenceH r (X, N D) D Extr (N, Gm) for r < 2p � 2.

Write f WXfl ! Xet for the morphism defined by the identity map.

COROLLARY 1.6. Let N be a quasi-finite flat group scheme overX whosep-primarycomponentN(p) is finite overX . LetN D be the complex of sheaves such that

N D(`) D

(HomXfl(N(`), Gm) if ` D p

f �RHomXet(N(`), Gm) if ` 6D p.

ThenH r (X, N D)�H 3�r

x (X, N)! H 3x (X, Gm) �D Q=Z


PROOF. For each 6D p, H sx(X, N(`)) D H s

x(Xet, N(`)) and

H r (X, N D(`)) D H r (Xet, RHomXet(N(`), Gm)) D ExtrXet(N(`), Gm).

Therefore, for the prime-to-p components of the groups, the corollary follows from (II 1.8).For thep component it follows immediately from the theorem.

QUESTION1.7. Does there exist a single statement that fully generalizes both (1.3) and (II1.8b)?

The first proof of Theorem 1.3

The first proof is very short, but makes use of (A.6). We begin by proving a duality resultfor abelian schemes.

PROPOSITION1.8. LetA be an abelian scheme overX , and letAt be its dual. Then thepairing

H r (X,At)�H 2�rx (X,A)! H 3

x (X, Gm) D Q=Z

defined by the canonical biextensionAt˝

LA! Gm (see Appendix C) induces an isomor-phismH 0(X,At)^

! H 2x (X,A)� (^ denotes the completion for the profinite topology);

for r 6D 0, both groups are zero.


PROOF. LetA andA0 be the open and closed fibres respectively ofA=X . ThenH r (X,A) D

H r (x,A0) for r > 0 (see Milne 1980, I 3.11), andH r (X,A0) D 0 for r > 0 by Lang’slemma. Moreover,A(X) D A(K) becauseA is proper overX . ThereforeH r

x (X,A) iszero forr � 1 and equalsH r�1(K, A) for r > 1. HenceH 2

x (X,A) D H 1(K, A), andH r

x (X,A) D 0 for all other values ofr . Consequently, whenr D 0, the pairing becomes

H 0(K, At)�H 2(K, A)! H 2(K, Gm) �D Q=Z,

and both groups are zero for all other values ofr . The proposition now follows from (I3.4).

PROOF OFTHEOREM 1.3. We now prove (1.3). Note that the Lemma 1.1 implies thatH 0(X, N) andH 1(X, N) are finite(NK is a finite etale group scheme). According to(A.6) and (A.7),N can be embedded into an exact sequence

0! N ! A! B! 0

with A andB abelian schemes overX . This leads to an exact cohomology sequence

0! H 2x (X, N)! H 2

x (X,A)! H 2x (X,B)! H 3

x (X, N)! 0,

which we regard as a sequence of discrete groups.There is a dual exact sequence

0! N D! Bt

! At! 0

(see Appendix C), which leads to a cohomology sequence

0! H 0(X, N D)! H 0(X,Bt)! H 0(X,At)! H 1(X, N D)! 0

The two middle terms of the sequence have natural topologies, and the two end termsinherit the discrete topology. Therefore the sequence remains exact after the middle twoterms have been completed. The theorem now follows from the diagram

0 ��! H 0(X, N D) ��! H 0(X,Bt)^��! H 0(X,At)^

��! H 1(X, N D) ��! 0??y ??y�

??y�

??y0 ��! H 3

x (X, N)��! H 2

x (X,B)��! H 2

x (X,A)��! H 2

x (X, N D)��! 0

The second proof of Theorem 1.3

The second proof will usep-divisible groups, for whose basic theory we refer the reader toTate 1967b or Shatz 1986. In order to simplify the argument, we shall assume throughoutthatR is complete.

Let H D (H�, i�)��1 be ap-divisible group overX . Let L be a finite extension ofK,and letRL be the integral closure ofR in L. Then the group of pointsH(RL) of H withvalues inRL is defined to be lim

�iH(RL=m

iL), whereH(RL=m

iL) Ddf lim

�!vH�(RL=m

iL).

Let MH DS

H(RL) whereL runs over the finite extensions ofK contained inKs. ThenMH becomes a discrete module under the obvious action of Gal(Ks=K).


LEMMA 1.9. (a) The groupH(R) is compact if and only if its torsion subgroup is finite.(b) The group of elements ofMH fixed byGK is H(R).(c) The sequence ofGal(Ks=K)-modules

0! H�(Ks)!MH

p�

�!MH ! 0

is exact.

PROOF. (a) LetH ı be the identity component ofH . ThenH ı(R) is an open subgroup ofH(R), andH(R)=H ı(R) is torsion. SinceH ı(R) is compact and its torsion subgroup isfinite (H(R) is isomorphic toRdim(R)), the assertion is obvious.

(b) It suffices to show thatH(RL)GD H(R) for L a finite Galois extension ofK with

Galois groupG. When we writeH D SpfA, H(RL) is the set of continuous homomor-phismsA! RL, and so the assertion is obvious.

(c) The sequences

0! H�(RL=miRL)! H(RL=m

iRL)p�

�!H(RL=miRL)

are exact, and so on passing to the inverse limit, we obtain an exact sequence

0! H�(RL)! H(RL)p�

�! H(RL).

The termH�(RL) has its usual meaning, and we have observed in (1.1) thatH�(RL) D

H�(L). Therefore on passing to the direct limit we obtain an exact sequence

0! H�(Ks)!MH

p�

�!MH .

It remains to show thatpWMH !MH is surjective. IfH is etale, thenMH D H(ks),which is obviously divisible byp. If H is connected, sayH D SpfA, then the mappWH ! H turnsA into a freeA-module of finite rank, and so the divisibility is againobvious. The general case now follows from the fact that

0! H ı(RL)! H(RL)! H et(RL)! 0

is exact for allL (see Tate 1967b, p168).

Let H t be thep-divisible group dual toH (ibid. 2.3), so that(H t)� D H D� .

PROPOSITION 1.10. Assume that the torsion subgroups ofH(R) and H t(R) are bothfinite. Then there is a canonical pairing

H 1(K, MH t )�H(R)! Q=Z

which identifies the discrete groupH 1(K, MH t ) with the dual of the compact groupH(R).


PROOF. From the cohomology sequence of the sequence in (1.9c), we get an exact se-quence

0! H(R)(p�)! H 1(K, H�)! H 1(K, MH )p� ! 0.

I claim that the first map in the sequence factors throughH 1(X, H�),!H 1(K, H�). Notefirst that it is possible to defineH(R0) for any finite flatR-algebraR0. Let P 2 H(R);then the inverse imageP underp�WH ! H (regarded as map of functors of finite flatR-algebras) is a principal homogeneous space forH� overX whose generic fibre representsthe image ofP in H 1(K, H�). This proves the claim.

As we observed in the proof of (1.4), the images ofH 1(X, H D� ) andH 1(X, H�) anni-

hilate each other in the nondegenerate pairing

H 1(K, H D� )�H 1(K, H�)! H 2(K, Gm) �D Q=Z.

Therefore, the images ofH t(R)(p�) andH(R)(p�) annihilate each other under the samepairing, and so the diagram

0 ��! H(R)(p�)��! H 1(K, H�)??y�D

0 ��! (H 1(K, MH t )p� )� ��! H 1(K, H t�)

� ��! H t(R)(p�)�

shows that the pairing induces an injectionH(R)(p�),!(H 1(K, MH t )p� )�. In the limitthis becomes an injection

lim �

H(R)(p�) ,! (lim�!

(H i(K, MH t )p� )�D H 1(K, MH t )�,

and because of our assumption onH(R), lim �

H(R)(p�)D H(R).

We therefore have a injectionH(R)! H 1(K, MH t )�, and to prove that it is surjective,it suffices to show that[H(R)(p)] D [H 1(K, MH t )p]. This we do using an argument similarto that in the proof of (I 3.2). From (I 2.8), we know that

�(K, H1) D (RWpR)�hD �(K, H t

1)

whereh is the common height ofH andH t . The logarithm map (Tate 1967b, 2.3), and ourassumptions onH(R) andH t(R) show thatH(R) andH t(R) contain subgroups of finiteindex isomorphic toRd andRd 0

respectively whered andd 0 are the dimensions ofH andH t . Therefore

[H(R)(p)]=[H(R)p] D (RWpR)d ,

[H t(R)(p)]=[H t(R)p] D (RWpR)d 0

.

From the cohomology sequence of

0! H t1(Ks)!MH t

p�!MH t ! 0

we see that

�(K, H t1) D

[H t(R)p][H 2(K, H t1)]

[H t(R)(p)][H 1(K, MH t )p]


or1

(RWpR)hD

1

(RWpR)d 0D

[H 0(K, H1)]

[H 1(K, MH t )p].

But d C d 0D h (Tate 1967b, Pptn 3), and so this shows that

[H 1(K, MH t )p] D (RWpR)d [H(R)p] D [H(R)(p)],

as required.

REMARK 1.11. The proposition is false without the condition that the torsion subgroupsof H(R) andH t(R) are finite. For example, ifH D (Z=p�Z)��1, thenH(R) (D Qp=Zp)andH 1(K, MH t ) are both infinite and discrete, and so can not be dual. IfH D (�p� )��1,then

H(R) D fa 2 R j a � 1 mod mg

andH 1(K, MH t ) D Hom(Gal(Ks=K), Qp=Zp), which are not (quite) dual.

We now complete the second proof of (1.3). Forr D 0, the pairing can be identifiedwith the pairing

H 0(K, N D)�H 2(K, N)! H 2(K, Gm) �D Q=Z

of (I 2.3), and forr 6D 0, 1 both groups are zero. This leaves the caser D 1, and wesaw in the proof of (1.4) that this case is equivalent to the statement thatH 1(X, N D) andH 1(X, N) are exact annihilators in the duality betweenH 1(K, N D) andH 1(K, N). Weknow that the groups in question do annihilate each other, and so

[H 1(K, N)] � [H 1(X, N)][H 1(X, N D)],

and to show that they are exact annihilators it suffices to prove that equality holds.According to (A.4) and (A.7), there is an exact sequence

0! N ! H'�! H 0

! 0

with H andH 0 bothp-divisible groups. Moreover from the construction of the sequence,it is clear thatH , H 0, and their duals satisfy the hypotheses of (1.10). WriteH r (X, H) forlim�!

H r (X, H�), and let

H 0(R)(')D Coker('WH(R)! H 0(R)),

H 1(X, H)' D Ker('WH 1(X, H)! H 1(X, H 0)).

The cohomology sequence of the above sequence and its dual

0! N D! H 0t 't

�! H t! 0

show that

[H 1(X, N)] D [H 0(R)(')][H 1(X, H)' ] � [H 0(R)(')] and

[H 1(X, N D)] D [H t(R)('t )][H 1(X, H 0t)'t ] � [H t(R)('t )].


On combining the three inequalities, we find that

[H 1(K, N)] � [H 1(X, N)][H 1(X, N D)] � [H 0(R)(')][H t(R)('t )].

It follows from (1.10) thatH 0t(R)'t

�! H t(R) is dual toH 1(K, MH )'�! H 1(K, MH 0),

and so[H t(R)('t )] D [H 1(K, MH )' ]. But [H 1(K, N)] D [H 0(R)(')][H 1(K, MH )' ],from which it follows that all of the above inequalities are equalities. This completes thesecond proof of (1.3).

REMARK 1.12. The above argument shows that for any isogeny'WH ! H 0 of p-divisible

groups overX , H 0(R)(')�D�! H 1(X, Ker(')) andH 1(X, H)' D 0, provided the torsion

subgroups ofH(R) andH t(R) are finite.

A duality theorem for p-divisible groups

If A is an abelian variety overK, then (I 3.4) shows that there is an exact sequence

0! A(K)˝Qp=Zp ! H 1(K, A(p))! At(K)�(p)! 0.

Our next result is the analogue of this forp-divisible groups. Recall thatH r (K, H) Ddf

lim�!

H r (K, H�).

PROPOSITION1.13. Assume thatR is complete, and letH be ap-divisible group overXsuch that the torsion subgroups ofH(R) andH t(R) are finite. ThenH 1(X, H) D 0, andthere is an exact sequence

0! H(R)˝Q=Z! H 1(K, H)! H t(R)�! 0.

PROOF. On applying Remark 1.12 to the isogenyp�WH ! H , we find thatH(R)(p�) ��!

H 1(X, H�) andH 1(X, H)p� D 0. As H 1(X, H) is p-primary, the equality shows that itis zero. From (1.4) we know there is an exact sequence

0! H 1(X, H�)! H 1(K, H�)! H 1(X, H t�)

�! 0.

On using the isomorphism to replace the first and third terms in this sequence, we obtainan exact sequence

0! H(R)(p�)! H 1(K, H�)! H t(R)(p�)�

! 0.

Now one has only to pass to the direct limit to obtain the result.


If N is a finite flat group scheme overX , then the groupsH r (X, N) are finite for allr andzero forr > 1. We define�(X, N) D [H 0(X, N)]=[H 1(X, N)]. Let N D SpecB. Recallthat theorder of N is defined to be the rank ofB overR, and thediscriminant idealof N

is the discriminant ideal ofB overR.


THEOREM 1.14. Let N be a finite flat group scheme overX , and letn be its order anddits discriminant ideal. Then(RW d) is annth power and

�(X, N) D (RW d)�1=n.

WhenN is etale,H r (X, N) D H r (g, N(Run)), and so both sides of the equation are1. This allows us to assume thatN is local. We can also assume thatR is complete becausepassing to the completion does not change either side. Consider an exact sequence

0! N ! H'�! H 0

! 0

with H andH 0 connectedp-divisible groups. AsH 1(X, H) D 0,

�(X, N) D z('(R))dfD [Ker'(R)]=[Coker'(R)].

Write H D SpfA andH 0 D SpfA0. ThenA andA0 are power series rings ind variablesover R, whered is the common dimension ofH andH 0. The map' corresponds to ahomomorphism'a

WA0! A makingA into a freeA0-module of rankn. It also defines a

mapd'aW˝1

A0=R ! ˝1A=R.

LEMMA 1.15. Choose bases for 1A0=R and˝1

A=R, and let�0 and� be the corresponding

basis elements forVd

˝A0=R andVd

˝A=R overA0 andA respectively. IfVdd'aWVd

˝A0=R !Vd

˝A=R

maps�0 to a�, a 2 A, thenNA=A0a generates the discriminant ideal ofA overA0.

PROOF. This follows from the existence of a trace map TrWVd

˝A=R !Vd

˝A0=R; seeTate 1967b, p165.

Let !A=R and!A0=R be theR-modules of invariant differentials onH andH 0 respec-

tively. The inclusion!A=R ! ˝1A=R induces isomorphisms!A=R ˝R A

��! ˝1

A=R and

!A=R

��! ˝1

A=R ˝A R. Let � and� 0 be basis elements forVd

!A=R andVd

!A0=R. Ontaking� and�0 to be� ˝ 1 and� 0

˝ 1 in the lemma, we find thatd'a(� 0) D a� , a 2 R,and that the discriminant ideal ofA over A0 is generated byan. SinceA ˝A0 R D B,whereB D � (N,ON ), this shows that the discriminant ideald of N is generated byan. Itremains to show that�(X, N) D (RW aR).

Let T (H) andT (H 0) be the tangent spaces toH andH 0 at zero. They are dual to!A=R and!A0=R, and soaR is equal to the determinant ideal of the map ofR-modulesd'WT (H)! T (H 0). Recall Tate 1967b, 2.4, that there exists a logarithm map logWH(R)!

T (H) ˝R K, and that if we choose an isomorphismA � R[[X1, ..., Xd ]], then for anycwith cp�1 < jpj, log gives an isomorphism between

H(R)c D fx 2 H(R)j jxij � c all ig

andT (H)c D f� 2 T (H)j j�(Xi)j � c all ig D pcT (H).


From the commutative diagram

H(R)c

'c

��! H 0(R)c

�

??ylog �

??ylog

T (H)c

(d')c

��! T (H 0)c

we see that(RWdet(d')) D (RWdet(d')c) D [Coker('c)]. But

[Coker('c)] D z(')(H(R)WH(R)c)(H 0(R)WH 0(R)c)�1

and(H(R)WH(R)c) D pd(c�1)D (H 0(R)WH 0(R)c). Therefore

(RW aR) D (RWdet(d')) D z(') D �(X, N).

This completes the proof of Theorem 1.14.For a finite flat group schemeN overX , define2

�x(X, N) D[H 2

x (X, N)]

[H 3x (X, N)]

.

COROLLARY 1.16. Let N be a finite flat group scheme overX , and letn be its order andd its discriminant ideal. Then

�x(X, N) D (RW nR)(RW d)�1=n.

PROOF. From the cohomology sequence ofX � u, we find that

�x(X, N) D �(X, N)�(K, N)�1D (RW dR)�1=n(RW nR)

(see (1.14) and (I 2.8).)

As H r (X, N) is dual toH 3�rx (X, N D), �(X, N)�x(X, N D) D 1. Therefore (1.14)

and (1.16) imply thatd(N)d(N D) D (nn). This formula can also be directly deduced fromthe formulas

NmB=R D(N) D d(N), D(N)D(N D) D (n),

whereD(N) andD(N D) are the differents ofN andN D andB D � (N,ON ) (see Ray-naud 1974, Pptn 9, and Mazur and Roberts 1970, A.2).

Let N be a quasi-finite, flat, separated group scheme overX . BecauseX is Henselian,there is a finite flat group schemeN f

� N having the same closed fibre asN ; moreover,N=N f is etale and it is the extension by zero of generic fibre (cf. Milne 1980, I 4.2c). Inthis case we writenf for the order ofN f anddf for its discriminant ideal.

COROLLARY 1.17. LetN be as above.(a) �(X, N) D (RW df )�1=nf

;(b) �x(X, N) D (RW nR)(RW df )�1=nf

.

2The original hadH 1x in the denominator, which is clearly wrong (1.1).


PROOF. (a) From the cohomology sequence of

0! N f! N ! N=N f

! 0

we find that�(X, N) D �(X, N f )�(X, N=N f ). Therefore it suffices to prove the formulain the cases thatN D N f or N f

D 0. In the first case, it becomes the formula in (1.14),and in the secondN D j!NK , and so both sides are1.

(b) Again, it suffices to prove the formula in the casesN D N f or N f D 0. In the first

case the formula becomes that in (1.16). In the second,H r (K, NK )��! H rC1

x (U, N),and so

�x(X, N) D �(K, NK )�1D (RW nR)

by (I 2.8).

COROLLARY 1.18. Let H D (H�)��1 be ap-divisible group overX ; then�(X, H�) D

(RWpd�R), whered is the dimension ofH .

PROOF. According to Tate 1967b, Pptn 2, the discriminant ideal ofH� is generated bypd�ph�

. Therefore�(X, H�) D ((RWpd�R)ph�

)p�h�

D (RWpd�R).

Extensions of morphisms

For each finite group schemeN overX , defineh(N) to be the pair(NK , H 1(X, N)). Amorphismh(N)! h(N 0) is aK-morphism'K WNK ! N 0

K such thatH 1('K )WH 1(K, N 0K )!

H 1(K, N 0K ) mapsH 1(X, N) into H 1(X, N 0).

THEOREM 1.19. The functorN 7! h(N) of finite group schemes overX is fully faithful;that is, a homomorphism'K WNK ! N 0

K extends to a homomorphism'WN ! N 0 ifand only ifH 1('K ) mapsH 1(X, N) into H 1(X, N 0), and the extension is unique when itexists.

PROOF. This is the main theorem of Mazur 1970b.

Examples

Assume thatK contains thepth roots of1. For eacha, b 2 R with ab D p, there is awell-defined finite flat group schemeNa,b overR given by the classification of Oort andTate (cf. 0.9). It is a finite group scheme of orderp and discriminantap. Therefore�(X, Na,b) D (RW aR).

BecauseK contains apth root of 1, m Ddf ord(p)=(p � 1) is an integer. We say thatNa,b splits genericallyif its generic fibre is isomorphic toZ=pZ. This is equivalent toabeing a nonzero(p � 1)st power inR. Choose a uniformizing parameter� in R. Thenthe generically split group schemes of orderp overR correspond to the pairs(a, b) witha D � (p�1)i, 0 � i � m, andb D p=a. For example, ifa D 1, thenN D Z=pZ, and ifa D � (p�1)m

D (unit)p, thenN D �p. LetU D R� andU (i)D fa 2 U j ord(1�a) � ig.


PROPOSITION 1.20. Let N be a finite flat generically split group scheme of orderp

over X . Then there is a nonzero map'WN ! �p, and for any choice of', the mapH(')WH 1(X, N) ! H 1(X,�p) identifiesH 1(X, N) with the subgroupU (i)U p=U p ofH 1(X,�p) D U=U p, wherei D pm� ord(discN)=(p � 1).

PROOF. Let N D Na,b, and supposea D � (p�1)i. Then, for anyX -schemeY , N(Y ) D

fy 2 � (Y,OY )j ypD ayg, and soy 7! y� (m�i) defines a morphism of functorsN(Y )!

�p(Y ) and hence a nonzero mapN ! �p. For the proof of the proposition, one first showsthat the image ofH 1(X, N) is contained inU (i)U p=U p and then uses (1.14) to show thatit equals this group. See Roberts 1973.

More explicitly, if N D Na,b with a D � (p�1)i, thenH 1(X, N) D U (j)U p=U p,wherej D p � ord(p)=(p � 1) � pi. For example, ifa D 1 so thatN D Z=pZ, thenH 1(X, N) D U (pm)U p=U p, and if N D � (p�1)m so thatN D �p, thenH 1(X, N) D

U (0)U p=U p.

REMARK 1.21. In the examples in (1.20), the mapH 1(X, N)! H 1(Xi , N) is an isomor-phism fori >> 0. This is true for any finite group schemeN , as can be easily deducedfrom the exact sequence

H(R)! H 0(R)! H 1(X, N)! 0

arising from a resolution ofN by p-divisible groups.

NOTES. Theorems 1.3 and 1.14 are due to Mazur and Roberts (Mazur and Roberts 1970;Mazur 1970a). The second proof of (1.3) and the proof of (1.14) are taken from Milne1973. The first proof of (1.3) is new. Theorem 1.19 is due to Mazur (1970b).

2 Local results: mixed characteristic, abelian varieties

The notations are the same as in~1. Except in the last two results,X will be endowed withits smooth topology.

Let A be an abelian variety overK, and letA be its Neron model overX . As inAppendix C, we writeAı for the open subgroup scheme ofA whose closed fibreAı

x isconnected. There is an exact sequence of sheaves onXsm

0! Aı! A! i�˚ ! 0.

We often regard as a Gal(ks=k)-module. Recall that for any submodule� of ˚ , A�denotes the inverse image of� in A. There is an exact sequence

0! Aı! A� ! i�� ! 0 (2.0.1)

PROPOSITION 2.1. The mapA� (X) ! � (x) arising from (2.0.1) is surjective, andH r (X, A� ) ! H r (x,� ) is an isomorphism forr � 1; therefore,H r (X,A� ) D 0

for r � 2.

2. LOCAL RESULTS: MIXED CHARACTERISTIC, ABELIAN VARIETIES 213

PROOF. According to Milne 1980, III 3.11,H r (X,A� ) D H r (x,A�x ) for r > 0, andLang’s lemma implies thatH r (x,Aı

x) D 0 for r > 0. Therefore the cohomology sequenceof (2.0.1) leads immediately to the result.

LEMMA 2.2. For any� , there is an exact sequence

˚(x)! (˚=� )(x)! H 1(X,A� )! H 1(K, A),

in which the last map is the restriction map; in particular, ifGal(ks=k) acts trivially on˚ ,thenH 1(X,A� )! H 1(K, A) is injective.

PROOF. We first consider the case that� D ˚ . ThenA� D A, and asA D j�A, theLeray spectral sequence forj shows immediately that the mapH 1(X,A)! H 1(K, A) isinjective. In the general case, the lemma can be deduced from the diagram

H 1(X,A� ) ��! H 1(X,A)??y�D

??y�D

˚(x) ��! (˚=� )(x) ��! H 1(x,� ) ��! H 1(x,˚).

LEMMA 2.3. We have

H rx (X,A� ) D

�0, r 6D 1, 2

(˚=� )(x), r D 1

and there is an exact sequence

0! � (x)! ˚(x)! (˚=� )(x)! H 1(X,A� )! H 1(K, A)! H 2x (X,A� )! 0.

PROOF. Consider the exact sequence

0! H 0x (X,A� )! H 0(X,A� )! H 0(K, A)! H 1

x (X,A� )! H 1(X,A� )! � � � .

ObviouslyA� (X)! A(K) is injective, which shows thatH 0x (X,A� ) D 0. AsH r (X,A� )

andH r (K,A� ) are both zero forr > 1, the sequence shows thatH rx (X,A� ) D 0 for

r > 2.Take� D ˚ , so thatA� D A; thenA(X)! A(K) is an isomorphism and (2.2) shows

thatH 1(X,A)! H 1(K, A) is injective. Therefore the sequence shows thatH 1x (X,A) D

0. In the general case the exact sequence

0! H 0x (X,˚=� )! H 1

x (X,A� )! H 1x (X,A)

gives an isomorphism(˚=� )(x)��! H 1

x (X,A� ). The existence of the required exactsequence follows from

H 1x (X,A� ) �!H 1(X,A� ) �! H 1(K, A) �!H 2

x (X,A� ) �!H 2(X,A� ) �! 0??y�D

˚(x) �! (˚=� )(x) �!H 1(X,A� ) �!H 1(K, A).


We now consider an abelian varietyA overK, its dual abelian varietyB, and a PoincarebiextensionW of (B, A) byGm. Recall (C.12) thatW extends to a biextension of(B� 0

,A� )

by Gm if and only if � 0 and� annihilate each other in the canonical pairing˚ 0 � ˚ !

Q=Z.

LEMMA 2.4. If � 0 and� are subgroups of 0 and˚ that annihilate each other, then thefollowing diagrams commuteW

H 1(K, B) � H 0(K, A) > H 2(K, Gm)

H 1(X,B� 0

)

^

�H 1x (X,A� )

_

> H 3x (X, Gm),

_�D

H 1(K, B) � H 0(K, A) > H 2(K, Gm)

H 2x (X,B� 0

)

_

�H 0(X,A� )

^

> H 3x (X, Gm).

_�D

PROOF. In the first diagram, the first vertical arrow is the restriction map, and the secondand third arrows are boundary mapsH r (u,�) ! H rC1

x (X,�). Since the top pairing isdefined by the restriction tou of the biextension defining the bottom pairing, the commu-tativity is obvious. The proof that the second diagram commutes is similar.

THEOREM 2.5. The canonical pairing 0� ˚ ! Q=Z is nondegenerate; that is, Conjec-

ture C.13 holds in this case.

PROOF. The groups and the pairing are unchanged when we replaceK with its completion.After making an unramified extension ofR, we can assume that Gal(ks=k) acts triviallyon ˚ and˚ 0. By symmetry, it suffices to show that the pairing0 � ˚ ! Gm is leftnondegenerate, and for this, it suffices to show that the pairing

H 1(x,˚ 0)�H 0(x,˚)! H 1(x, Q=Z) �D Q=Z

is left nondegenerate.The canonical pairing of 0 with ˚ is so defined that

B ��! Ext1X (A, j�Gm)??y ??y

i�˚0��! HomX (i�˚, i�Q=Z)

commutes (see C.11). Alternatively, we can regard it as being the unique homomorphism˚ 0! Ext1

x(˚, Z) making

B ��! Ext1X (A, j�Gm)??y ??y

i�˚0��! Ext1

X (i�˚, i�Z)


commute. From this we get a commutative diagram

H 1(X,B) ��! Ext2X (A, j�Gm)??y ??yH 1(X, i�˚

0) ��! Ext2X (i�˚, i�Z).

These maps are used to define the two lower pairings in the following diagram

H 1(K, B) �H 0(K, A) > H 2(K, Gm) �D Q=Z

H 1(X,B)

[

^

inj

�H 0(X,A)

^�D

> H 2(X, j�Gm)

^�D

H 1(X, i�˚0)

_�D

�H 0(X,˚)

__

surj

> H 2(X, i�Z),

_�D

and so the the diagram commutes (the upper arrows are all restriction maps). The toppairing is nondegenerate (I 3.4), and so the lower two pairings are left nondegenerate. Thisproves the theorem.

COROLLARY 2.6. Suppose that� 0 and� are exact annihilators under the canonical pair-ing of˚ 0 and˚ . Then the map

B� 0

! Ext1Xsm

(A� , Gm)

defined by the extension ofW is an isomorphism (of sheaves onXsm).

PROOF. See (C.14).

THEOREM 2.7. Assume that� 0 and� are exact annihilators. Then the pairing

H r (X,B� 0

)�H 2�rx (X,A� )! H 3

x (X, Gm) �D Q=Z

defined by the canonical biextension of(B� 0

, A� ) byGm induces an isomorphism

H 2x (X, A� )

�D�! B� 0

(X)�

of discrete groups forr D 0 and an isomorphism of finite groups

H 1(X, B� 0

)�D�! A� (X)�

for r D 1. For r 6D 0, 1, both groups are zero.


˚ 0 (x) �! (˚ 0=� )0(x) �! H 1(X,B� 0

) �! H 1(K, B) �! H 2x (X,B� 0

) �! 0??y�D

??y�D

??ya

??y�D

??yb

H 1(x,˚)��! H 1(x,� )�

�! H 1x (x,A� )�

�! H 0(K, A)��! H 0(X,A� )�

�! 0.


The top row is the exact sequence in (2.3), and the bottom row is the dual of the cohomologysequence of the pairX � u. That the last two squares commute is proved in (2.4). Thefirst two vertical maps are the isomorphisms induced by the canonical pairing between˚ 0 and˚ . Thus the first square obviously commutes, and second was essentially shownto commute in the course of the proof of (2.5). It follows from the diagram thata isinjective andb is surjective. But the two groupsH 1(X,B� 0

) andH 1x (X,A� ) have the

same order (see (2.1) and (2.3)), and soa is an isomorphism. This in turn shows thatb isan isomorphism.

REMARK 2.8. (a) Once (2.7) is acquired, it is easy to return and prove (2.5): the mapH 1(x,˚ 0)! H 0(x,˚)� can be identified with the isomorphismH 1(X,B0)! H 1

x (X,Aı)�

given by the (2.6).(b) Let bX D SpecbR. Then it follows from (I 3.10) that the mapsH r

x (X,A� ) !

H rx (bX,A� ) are isomorphisms for allr , and thatH r (X,A� ) ! H r (bX,A� ) is an iso-

morphism for allr > 0. The mapA(X) ! A(bX) is injective and its image includes thetorsion subgroup ofA(bX); A(bX) is the completion ofA(X) for the topology of subgroupsof finite index.

(c) WhenR is complete,B� 0

(X) is compact. Therefore in this case the pairing inducesdualities between:

the compact groupB� 0

(X) and the discrete groupH 2x (X,A� )I

the finite groupH 1(X,B� 0

) and the finite groupA� (X).

Write Bfng for the complexB n�! Bn˚ 0

andAfng for the complexA˚nn�! Aı. The

pairingsB˝LAı! Gm[1] andBn˚ 0

˝LA˚n ! Gm[1] defined by a Poincare biextension

induce a pairingBfng ˝L Afng ! Gm in the derived category of sheaves onXsm (seeGrothendieck 1972, VIII 2).

PROPOSITION2.9. The mapBfng ˝L Afng ! Gm defines nondegenerate pairings

H r (X,Bfng)�H 3�rx (X,Afng)! H 3

x (x, Gm) �D Q=Z

of finite groups for allr .

PROOF. From the exact sequences of complexes

0! Bn˚ 0

[�1]! Bfng ! B! 0

0! Aı[�1]! Afng ! A˚n ! 0

we get the rows of the diagram

� � � ��! H r�1(X,Bn˚ 0

) ��! H r (X,B(n)) ��! H r (X,B) ��! � � �??y�D

??y ??y�D

� � � ��! H 3�rx (X,A˚n)� ��! H 3�r

x (X,A(n))� ��! H 2�rx (X,Aı)� ��! � � � .

Since the diagram obviously commutes, the theorem follows from (2.7).


COROLLARY 2.10.Assume thatn is prime to the characteristic ofk or thatA has semistablereduction. Then for allr , there is a canonical nondegenerate pairing of finite groups

H rx (Xfl,Bn)�H 3�r (Xfl,An)! H 3

x (Xfl, Gm) �D Q=Z.

PROOF. The hypothesis implies thatB n�! Bn˚ 0

andA˚nn�! Aı are surjective when

regarded as a maps of sheaves for the flat topology (see C.9). HenceBn � Bfng andAn � Afng, and so

H r (Xfl,Bn) � H r (Xfl,Bfng) � H r (Xsm,Bfng)

andH r

x (Xfl,An) � H rx (Xfl,Afng) � H r

x (Xsm,Afng).

Curves overX

By exploiting the autoduality of the Jacobian, it is possible to use (2.9) to prove a dualitytheorem for a curve overX .

THEOREM 2.11. Let� WY ! X be a proper flat map whose fibres are pure of dimensionone. Assume that the generic fibreYK is smooth and connected, that the special fibreYx isconnected, and that there is a section to� . Assume further thatPic�Y =X D J , whereJ isthe Neron model of the Jacobian ofYK . Then there is a canonical duality of finite groups

H r (Y,�n)�H 5�rYx

(Y,�n)! H 3x (X, Gm) �D Q=Z.

PROOF. We use the Leray spectral sequence of� . Under the hypotheses,

R0��n�D �n,

R1��n�D Ker(J n

�! J ),

R2��n�D Z=nZ,

Rr��n D 0, for r > 2.

MoreoverJ D J ı. On takingA D J D B in the (2.9), we find thatH r (X, R1��n) isdual toH 3�r

x (X, R1��n) for all r . The result can be obtained by combining this dualitywith the duality ofH r (X, R0��n) andH 3�r

x (X, R2��n).

For conditions onY=X ensuring that the hypotheses of the theorem hold, see the lastfew paragraphs of Appendix C. Our hypotheses are surely too stringent. Because of this,we make the following definition. LetX be the spectrum of an excellent Henselian dis-crete valuation ring (not necessarily of characteristic zero) with finite residue field, and let� WY ! X be a proper flat morphism whose generic fibre is a smooth curve. If there is acanonical pairing

R��n �R��n! Gm[2].

extending that on the generic fibre and such that the resulting pairing

H r (Y,�n)�H 5�rYx

(Y,�n)! H 3(X, Gm) �D Q=Z

is nondegenerate, then we shall say that thelocal duality theorem holds forY=X andn.

NOTES. This section is based on McCallum 1986.


3 Global results: number field case

Throughout this section,X will be the spectrum of the ring of integersOK in a number fieldK. For an open subschemeU of X , U [1=n] denotes Spec� (U,OX )[1=n], andH r

c (U,�)

denotes the flat cohomology group with compact support as defined in (0.6a) (thus, it takesaccount of the infinite primes).

Finite sheaves

Let U be an open subscheme ofX . As Gm is smooth,H rc (Ufl, Gm) D H r

c (Uet, Gm), and

so (see II 3) there is a canonical trace mapH 3c (U, Gm)

�D�! Q=Z. Therefore, for any sheaf

F onU , there is a canonical pairing


c (U, Gm) �D Q=Z

(see 0.4e).Let f WUfl ! Uet be the morphism of sites defined by the identity map. Recall Milne

1980, V 1, that the constructible sheaves onUet are precisely those sheaves that are repre-sentable byetale algebraic spaces of finite-type overU ; moreover, if QF representsF onUet, then it representsf �F onUfl.

THEOREM 3.1. Let U be an open subscheme ofX , and letF be a sheaf onUfl such thatnF D 0 for some integern. Assume

(i) the restriction ofF to U [1=n]fl is represented by anetale algebraic space of finite-type overU [1=n];

(ii) for each v 2 U r U [1=n], the restriction ofF to (SpecOv)fl is represented by afinite flat group scheme.LetF D be the sheaf onU such thatF D

jU [1=n] D f �R HomU [1=n]et(F, Gm) andF DjV D

HomVfl(F, Gm) for any open subschemeV of U whereF jV is represented by a finite flatgroup scheme.

Then there are canonical mapsF D! RHomU (F, Gm), henceH r (U, F D)! ExtrU (F, Gm),

and the resulting pairing

H r (U, F D)�H 3�rc (U, F)! H 3�r

c (U, Gm) �D Q=Z


PROOF. We first note that onV [1=n] D V \ U [1=n], F is represented by a finite flatetalegroup scheme whose order is prime to the residue characteristics. Therefore

RHomV [1=n]et(F, Gm) D HomV [1=n]et(F, Gm),

and so the requirements onF D coincide onV [1=n], which shows thatF D exists.For all r ,

H rc (U [1=n]et, F ) D H r

c (U [1=n]fl, F ),

H r (U [1=n]fl, F D) D H r (U [1=n]et, F D).

3. GLOBAL RESULTS: NUMBER FIELD CASE 219

Therefore, for the restriction ofF to U [1=n], the theorem becomes (II 3.3). To pass fromU [1=n] to the whole ofU , one uses the diagram

� � � ��! H r (U, F D) ��! H r (U [1=n], F D) ��!L

v2U rU [1=n]

H rC1v (Oh

v , F D) ��! � � �??y ??y�D

??y�D

� � � ��! H 3�rc (U, F)�

��! H 3�rc (U [1=n], F )�

��!L

v2U rU [1=n]

H 2�r (Ohv , F )�

��! � � �

and (1.3).

COROLLARY 3.2. Let N be a finite flat group scheme overU , and letN D be its Cartierdual. Then

H r (U, N D)�H 3�rc (U, N)! H 3

c (U, Gm) �D Q=Zis a nondegenerate pairing of finite groups for allr .

PROOF. When the sheafF in (3.1) is taken to be that defined byN , thenF D is the sheafdefined byN D .

COROLLARY 3.3. Let N be a quasi-finite flat separated group scheme overU , and letnN D 0. Assume that there exists an open subschemeV of U such that

(i) V contains all pointsv of U whose residue characteristic dividesnI

(ii) N jV is finite;(iii) if j denotes the inclusion ofV [1=n] intoU [1=n], then the canonical mapN jU [1=n]et!

j�j�(N jV [1=n]et) is an isomorphism.LetN D D HomUfl(N, Gm). Then the canonical pairing

H r (U, N D)�H 3�rc (U, N)! H 3

c (U, Gm) �D Q=Z


PROOF. BecauseN jV is finite,N DjV is the Cartier dual ofN jV . Therefore the theorem

shows thatH r (V , N D) is finite and dual toH 3�rc (V , N). The corollary therefore follows

from

� � � ��! H r (U, F D) ��! H r (V , F D) ��!L

v2U rV

H rC1v (Oh

v , F D) ��! � � �??y ??y ??y�D

� � � ��! H 3�rc (U, F)�

��! H 3�rc (V , F)�

��!L

v2U rV

H 2�r (Ohv , F )�

��! � � � .

and (II 1.10b).

Let A be an abelian variety overK, and letA andB be the Neron minimal models overU of A and its dualB. Let n be an integer such thatA has semistable reduction at allvdividing n. There are exact sequences

0! Bn! B! Bn˚ 0

! 0

0! An! A˚n ! Aı! 0.


The Poincare biextension of(B, A) by Gm extends uniquely to biextensions of(B,Aı) byGm and of(Bn˚ 0

,A˚n) by Gm. Therefore (cf.~1), we get a canonical pairing

Bn �An! Gm.

COROLLARY 3.4. LetBn andAn be as above. Then

H r (U,Bn)�H 3�rc (U,An)! H 3

c (U, Gm) �D Q=Z

is a nondegenerate pairing of finite groups for allr .

PROOF. Over the open subsetV whereA has good reduction,An is a finite flat groupscheme with Cartier dualBn, and so overV , the corollary is a special case of (3.2). To passfrom V to U , use (2.10).


We extend (II 2.13) to the flat site. LetN be a quasi-finite flat separated group scheme overU . For each closed pointv 2 U , letnfv be the order of the maximal finite subgroup schemeNfv of N �U Spec(Oh

v), anddfv be the discriminant ofN f

v overOhv . Also, we set

�(U, N) D[H 0(U, N)][H 2(U, N)]

[H 1(U, N)][H 3(U, N)], �c(U, N) D

[H 0c (U, N)][H 2

c (U, N)]

[H 1c (U, N)][H 3

c (U, N)].

THEOREM 3.5. LetN be quasi-finite, flat, and separated overU . Then

�(U, N) DQ

v2X rU

j[N(Ks)]jv �Qv2U

(Ohv W d

fv )�1=n

fv �

Qv arch

[N(Kv)]

[H 0(Kv, N)]

and�c(U, N) D

Qv2U

(RW dfv )�1=nf

�Qv arch

[N(Kv)].

PROOF. Let V be an open subset ofU such thatN jV is finite and has order prime to theresidue characteristics ofV , so that, in particular,N jV is etale. The exact sequence

� � � !Q

v2U rV

H rv (U, N)! H r (U, N)! H r (V , N)! � � �

shows that�(U, N) D �(V , N) �Qv2U rV �v(Oh

v , N), and (II 2.13) and (1.17b) showrespectively that

�(V , N) DYv arch

[N(Kv)]

[H 0(Kv, N)]j[N(Ks)]jv

and�v(Ohv , N) D j[N(Ks)]j

�1v (RW d

fv )�1=n

fv . The formula in (a) follows immediately.

The exact sequence

� � � ! H rc (V , N)! H r

c (U, N)!L

v2U rV H r (Ohv , N)! � � �

shows that�c(U, N) D �c(V , N) �Q�v(Oh

v , N) and (II 2.13) and (1.17a) show respec-tively that�c(V , N) D

Qv arch[N(Kv)] and�(Oh

v , N) D (RW dfv )�1=nf

.

3. GLOBAL RESULTS: NUMBER FIELD CASE 221

Neron models

Let A be an abelian variety overK, and letA be its Neron model. ThenA=AıDdf˚ DL

v iv�˚v (finite sum) where v D i�v (Av=A0

v).

PROPOSITION3.6. Let� be a subgroup of , and letA� be the corresponding subschemeofA.

(a) The groupH 0(U,A� ) is finitely generated; forr > 0, H r (U,A� ) is torsion andof cofinite-type; the mapH r (U,A� ) !

Lv arch H r (Kv, A) is surjective forr D 2

and an isomorphism forr > 2.(b) For r < 0,

Lv arch H r (Kv,A� ) ! H r

c (U,A� ) is an isomorphism;H 0c (U,A� )

is finitely generated;H 1c (U,A� ) is an extension of a torsion group by a subgroup

which has a natural compactification;H 2c (U,A� ) is torsion and of cofinite-type; for

r � 3, H rc (U,A� ) D 0.

PROOF. Fix an integerm, and letV be an open subscheme ofU such thatm is invertibleon U andA is an abelian scheme overV . Then all statements are proved in (II 5.1) forAjV andm. The general case follows by writing down the usual exact sequences.

Let B be the dual abelian variety toA, and letB be its Neron model. LetB=B0 Ddf

˚ 0 DL

v iv�˚0v. For any subgroups� D

Lv iv��v and� 0 D

Lv iv��

0v of ˚ and˚ 0,

the Poincare biextension overK extends to a biextension overU if and only if each�vannihilates each� 0

v in the canonical pairing. In this case we get a map

B� 0

˝L A� ! Gm[1].

THEOREM 3.7. Suppose that�v and� 0v are exact annihilators at each closed pointv.

(a) The groupH 0(U,B� 0

)tors is finite; the pairing

H 0(U,B� 0

)�H 2c (U,A� )! Q=Z

is nondegenerate on the left and its right kernel is the divisible subgroup ofH 2c (U,A� ).

(b) The groupsH 1(U,B� 0

) andH 1c (U,A� )tors are of cofinite-type, and the pairing

H 1(U,B� 0

)�H 1c (U,A� )tors ! Q=Z

annihilates exactly the divisible groups.(c) If the divisible subgroup ofX1(K, A) is zero, then the compact groupH 0(U,B� 0

)^

(completion for the topology of subgroups of finite index) is dual to the discrete tor-sion groupH 2

c (U,A� ).

PROOF. Fix an integerm, and choose an open subschemeV of U on whichm is invertibleandA andB have good reduction. Theorem II 5.2 proves the result overV for the m-components of the groups. To pass from there to them-components of the groups overU ,use (2.7). Asm is arbitrary, this completes the proof.


Curves overU

For a proper map� WY ! U and sheafF onYfl, we defineH rc (Y, F) to beH r

c (U, R��F).

THEOREM 3.8. Let � WY ! U be a proper flat map whose fibres are pure of dimensionone and whose generic fibre is a smooth geometrically connected curve. Assume that forall v 2 U , Y �U SpecOh

v ! SpecOhv satisfies the local duality theorem forn (see~2).

Then there is a canonical nondegenerate pairing of finite groups

H r (Y,�n)�H 5�rc (Y,�n)! H 3

c (U, Gm) �D Q=Z.

PROOF. Choose an open subschemeV of U such thatn is invertible onV and�j��1(V )

is smooth. For�j��1(V ) the statement becomes that proved in Theorem II 7.7. LetZ D

Y r YV . To pass fromV to U , use the exact sequences

� � � ! H rZ (Y,�n)! H r (Y,�n)! H r (YV ,�n)! � � �

and

� � � ! H rc (YV ,�n)! H r

c (Y,�n)!L

v2U rV

H r (Y �U Spec(Ohv),�n)! � � �

and note thatH rZ (Y,�n) D

Lv2U rV H r

v (Y �U Spec(Ohv),�n).

NOTES. Theorem 3.1 was proved by the author in 1978. Earlier Artin and Mazur hadannounced the proof of a flat duality theorem overX (neither the statement of the theoremnor its proof have been published, but two corollaries are stated3 in Mazur 1972, 7.2, 7.3;I believe that their original theorem is the special case of (3.3) in whichU D X andn isodd).

4 Local results: mixed characteristic, perfect residue field

In this section we summarize the results of Begeuri 1980. Throughout,X will be the spec-trum of a complete discrete valuation ringR whose field of fractionsK is of characteristiczero, and whose residue fieldk is perfect of characteristicp 6D 0. (Essentially the sameresults should hold ifR is only Henselian.) We letm be the maximal ideal ofR, and we letXi D SpecR=miC1.

Some cohomological properties ofK

PROPOSITION4.1. If k is algebraically closed, then for any torusT overK, H r (K, T ) D

0 all r � 0.

PROOF. Let L be a finite Galois extension ofK with Galois groupG. ThenH 1(G, L�) D

0 by Hilbert’s theorem 90, andH 2(G, L�) D 0 because the Brauer group ofK is trivial(K is quasi-algebraically closed, Shatz 1972, p116). These two facts show thatL� isa cohomologically trivialG-module (Serre 1962, IX 5, Thm 8). ChooseL to split T .Then Hom(X �(T ), L�) D T (L), and (ibid. Thm 9) shows that Hom(X �(T ), L�) is alsocohomologically trivial because Ext1(X �(T ), L�) D 0.

3And made essential use of.

4. LOCAL RESULTS: MIXED CHARACTERISTIC, PERFECT RESIDUE FIELD 223

COROLLARY 4.2. Assume thatk is algebraically closed, and letN be a finite group schemeoverK.

(a) For all r � 2, H r (K, N) D 0.(b) LetK0 be a finite Galois extension ofK, and letG D Gal(K0=K). ThenH r

T (G, H 1(K0, N))

is finite for all r 2 Z, and is isomorphic toH rC2T (G, N(K0)). The canonical homo-

morphismH0(G, H 1(K0, N))! H 1(K, N) deduced from the corestriction map isan isomorphism.

PROOF. (a) ResolveN by tori,

0! N ! T0! T1! 0,

and apply the proposition.(b) From the above resolution, we get an exact sequence

0! N(K0)! T0(K0)! T1(K0)! H 1(K0, N)! 0.

Since the middle twoG-modules are cohomologically trivial, the iterated coboundary mapis an isomorphismH r

T (G, H 1(K0, N))! H rC2T (G, N(K0)). The last statement is proved

similarly (see Begeuri 1980, p34).

The algebraic structure onH r(X, N)

For anyk-algebra�, letWi(�) be the ring of Witt vectors over� of lengthi, and letW (�)

be the full Witt ring. For any schemeY overW (k) and anyi, theGreenberg realizationof leveli, Greeni(Y ), of Y is the scheme over overk such that

Greeni(Y )(�) D Y(Wi(�))

for all k-algebras� (see Greenberg 1961). Note thatR has a canonical structure as aW (k)-algebra, and so for any schemeY overX , we can defineGi(Y ) to be the Greenbergrealization of leveli of the restriction of scalars ofY , ResX=Spec(W (k)) Y . ThenGi(Y ) ischaracterized by the following condition: for anyk-algebra�,

Gi(Y )(�) D Y(R˝W (k) Wi(�)).

In particular,Gi(Y )(k) D Y(R=piR) D Y(Xi�1). Note thatG1(Y ) D Y ˝R k D Yk . Forvaryingi, theGi(Y ) form a projective systemG(Y ) D (Gi(Y )). The perfect group schemeassociated withGi(Y ) will be denoted by4 Gi(Y ). Thus

Gi(Y )(�) D Y(R˝W (k) Wi(�))

for any perfectk-algebra� andGi(Y )(k) D Y(R=piR). We let G(Y ) be the perfectpro-group scheme(Gi(Y )).

WhenG is a smooth group scheme overX , we let5 V (!G) be the vector group associ-ated with theR-module!G of invariant differentials onG.

4In the original, this is denotedGi(Y ).5In the original, this is denotedV(!G).


PROPOSITION4.3. Let G be a smooth group scheme overR. For all i � 1, Gi(G) is asmooth group scheme overk, and for all i 0

� i, there is an exact sequence ofk-groups

0! Gi(V (!G))! GiCi0(G)! Gi0(G)! 0.

In particular,GiC1(G)! Gi(G) is surjective with kernel!G ˝R k, andGi(G) is an exten-sion ofGk by a smooth connected unipotent group. The group schemeGi(G) is connectedif and only if its special fibre is connected. The dimension ofGi(G) is ei � dim(Gk) wheree is the absolute ramification index ofR.

PROOF. We may assume thatk is algebraically closed and apply Begueri 1980, 4.1.1.

LEMMA 4.4. Let0! N ! G0

'�! G1! 0

be an exact sequence ofR-groups withG0 and G1 smooth and connected andN finite.For all i � 1, thek-groupCoker(Gi(')) is smooth, and whenk is algebraically closed itsgroup ofk-points isH 1(Xi�1, N).

PROOF. The first statement follows from the fact thatGi(') is a homomorphism of smoothgroup schemes overk. For the second, note thatH r (Xi�1, G) D H r (k, Gk) D 0 forr > 0, and so we have a diagram

Gi(G0)(k) ��! Gi(G1)(k) ��! Coker(Gi(')(k)) ��! 0 ??y�

G0(Xi�1) ��! Gi(Xi�1) ��! H 1(Xi�1, N) ��! 0

Define QH 1(Xi , N) to be the sheaf on Spec(k)qf associated with the presheaf� 7!H 1(Xi ˝W (k) W (�), N). Then the lemma realizesQH 1(Xi , N) as an algebraic group, andthe next lemma shows that this realization is essentially independent of the choice of theresolution.

LEMMA 4.5. Let

0! N ! G 00

'0

�! G 01! 0

be a second resolution ofN by smooth algebraic groups. Then there is a canonical iso-

morphismCoker(Gi('))�D�! Coker(Gi('

0)).

PROOF. It is easy to construct a diagram

0 ��! G 00 ��! G 0

0 ��! 0??y ??y ??y0 ��! N ��! G0 ˚G 0

0

'00

��! G ��! 0 ??y ??y0 ��! N ��! G0

'��! G1 ��! 0


with G a smooth algebraic group. When we applyGi, the resulting diagram gives anisomorphism

Coker(Gi('00))

��! Coker(Gi(')),

and a similar construction gives an isomorphism

Coker(Gi('00))

��! Coker(Gi('

0)).

We now regard QH 1(Xi , N) as an algebraic group, and we writeQH 1(X, N) for the pro-algebraic group( QH 1(Xi , N))i�0.

For the definition of theabsolute differentD of a finite groupN scheme overR, werefer the reader to Raynaud 1974, Appendice. It is an ideal inR.

THEOREM 4.6. Let N be a finite flat group scheme of order a power ofp overX . For alli � 0, the smooth algebraick-group QH 1(Xi , N) is affine, connected, and unipotent. Thereexists an integeri0 such that QH 1(X, N) ! QH 1(Xi , N) is an isomorphism for alli � i0.The group schemeQH 1(X, N) has dimensionord(D) whereD is the different ofN .

PROOF. We may assume that the residue field is algebraically closed and apply Begeuri1980, 4.2.2.

PROPOSITION4.7. A short exact sequence

0! N 0! N ! N 00

! 0

of finite flatp-primary group schemes gives rise to an exact sequence of algebraic groups

0! G(N 0)! G(N)! G(N 00)! QH 1(X, N 0)! QH 1(X, N)! QH 1(X, N 00)! 0.


We write H1(Xi , N) andH1(X, N) for the perfect algebraic groups associated withQH 1(Xi , N) and QH 1(X, N). Suppose thatk is algebraically closed. If

0! N ! G0! G1! 0

is a smooth resolution ofN andi is so large thatN(R) \ piG0(R) D 0, then the kerneland cokernel of the map

G(G0)(R)(pi)! G(G1)(R)(pi)

areN(R) andH1(X, N)(k) respectively (ibid. p44–45).


The algebraic structure onH 1(K, N)

Let T be a torus overK. According to Raynaud 1966,T admits a Neron model overX W this is a smooth group schemeT over X (not necessarily of finite type) such thatT (Y ) D T (YK ) for all smoothX -schemesY . Write G(T ) for G(T ). It is a perfectpro-algebraic group overk whose set of connected components�0(G(T )) is a finitelygenerated abelian group, equal to the set of connected components of the special fibre ofT .

LEMMA 4.8. LetN be a finite group scheme overK, and let

0! N ! T0

'�! T1! 0

be a resolution ofN by tori. The cokernel ofG(')WG(T0) ! G(T1) is a pro-algebraicperfect group scheme, and whenk is algebraically closed it hasH 1(X, N) as its group ofk-points.

PROOF. Begeuri 1980, 4.3.1.

We writeH1(K, N) for the cokernel ofG('). It is a perfect pro-algebraic group schemewith group of pointsH 1(K, N) whenk is algebraically closed. An argument as in the proofof (4.5) shows thatH1(K, N) is independent of the resolution. The identity component ofH1(K, N) is unipotent.

PROPOSITION4.9. Let N be a finite flatp-primary group scheme overX . Then the stan-dard resolution defines a closed immersion

H1(X, N)! H1(K, N).


THEOREM 4.10. For any finiteK-groupN , the perfect group schemeH1(K, N) is affineand algebraic. Its dimension isord([N ]), where[N ] is the order ofN .

PROOF. The basic strategy of the proof is the same as that of the proof of (I 2.8); seeBegeuri 1980, 4.3.3.

The reciprocity isomorphism

Assume first thatk is algebraically closed. For any finite extensionK0=K, let UK 0 D

G(Gm,R0) whereR0 is the ring of integers inK0. Then the norm mapNR0=RWResR0=R Gm,R0 !

Gm,R induces a surjective mapUK 0 ! UK of affine k-groups. LetVK 0 be the kernel ofthis map, and letV o

K 0 be the identity component ofVK 0. Then we have an exact sequence

0! �0(VK 0)! UK 0=V ıK 0 ! UK ! 0.


Assume thatK0 is Galois overK, and let t 0 be a uniformizing parameter inK0. Thehomomorphism

Gal(K0=K)abD H �2(Gal(K0=K), Z)! (UK 0=V o

K 0)(k),

sending� 2 Gal(K0=K) to the class of�(t 0)=t 0 in UK 0 allows us to identify the precedingexact sequence with an exact sequence

0! H �2(Gal(K0=K), Z)! UK 0=V ıK 0

N�! UK ! 0.

On passing to the inverse limit over the fieldsK0, we get an exact sequence

0! Gal(Kab=K)! lim �

UK 0=V ıK 0 ! UK ! 0.

As UK is connected, this sequence defines a continuous homomorphism

recK W�1(UK )! Gal(Kab=K)

and the main result of Serre 1961 is that this map is an isomorphism.It is also possible to show that�1(U ) Ddf lim

�!�1(UK 0) is a class formation, and so

define recK as in (I 1).Recall (Serre 1960, 5.4) that for any perfect algebraic groupG and finite perfect group

N , there is an exact sequence

0! Ext1k(�0(G), N)! Ext1k(G, N)! Homk(�1(G), N)! 0.

In particular, whenG is connected Ext1k(G, N)

�D�! Hom(�1(G), N). (We are still assum-

ing thatk is algebraically closed.) Therefore, recK gives rise to an isomorphism

Hom(Gal(Kab=K), Z=pnZ) D H 1(K, Z=pnZ)�D�! Hom(�1(UK ), Z=pnZ) D Ext1K (UK , Z=pnZ).

If we assume thatK contains thepnth roots of1, and we replaceZ=pnZ with �pn(K),then the isomorphism becomes

˚nWH1(K,�pn)

�D�! Ext1K (UK ,�pn).

Both groups have canonical structures of perfect algebraic groups.

PROPOSITION4.11. The map n is a morphism of perfect algebraic groups.

PROOF. See Begeuri 1980, 5.3.2.

When we drop the assumption thatk is algebraically closed, we obtain an isomorphism

recK W �(UK )! Gal(Kab=K)

where�(UK ) is the maximal constant quotient of�1(UK ). See Hazewinkel 1969.


Duality for finite group schemes overK

THEOREM 4.12. LetN be a finite group scheme overK; then there is a canonical isomor-phism of connected perfect unipotent groupsH1(K, N)ı

! (H1(K, N D)ı)_.

PROOF. We can assume thatk is algebraically closed and apply Begeuri 1980, 6.1.6.

This result can be improved by making use of derived categories (ibid. 6.2). Assumethatk is algebraically closed, and let

Mn D category of finite group schemes overK killed by pn,

Qn D category of perfect pro-algebraic groups overk killed by pn,

Sn D category of sheaves on(Speck)pf killed by pn.Let C WDb(Mn) ! Db(Mn), S WDb(Qn) ! Db(Qn), andBWDb(Sn) ! Db(Sn) be thefunctors defined respectively by Cartier duality, Serre duality, and Breen-Serre duality (see~0; hereDb(�) denotes the derived category obtained from the categoryKb(�) of boundedcomplexes and homotopy classes of maps). ThenH1

WMn ! Qn admits a left derivedfunctor, and we have a commutative diagram (up to an isomorphism of functors):

Db(Mn)LH1

��! Db(Mn)can��! Db(Mn)??yC

??yS

??yB

Db(Mn)LH1

��! Db(Mn)can��! Db(Mn)I

(4.12.1)

moreover,(canıLH1)(N)��! RH0(N)[1]. See Begeuri 1980, 6.2.4.

Duality for finite group schemes overR

THEOREM4.13. For any finite flatp-primary group schemeN overX , there is a canonicalisomorphism ofk-groups

H1(X, N)�D�! (H1(K, N D)ı=H1(X, N D))t .

PROOF. Ibid. 6.3.2..

Duality for tori

Let T be a torus overK, and letT be its Neron model overX .

THEOREM 4.14. The pairingH 0(K, X �(T ))� T (K)! Z defines isomorphisms

H 0(K, X �(T ))�D�! Hom(�0(Tk), Z)

H 1(K, X �(T ))�D�! Ext1k(�0(Tk), Z) (finite groups)

H 2(K, X �(T ))�D�! Homcts(�1(T (K)), Q=Z).

PROOF. Ibid. 7.2.

5. TWO EXACT SEQUENCES 229

Duality for abelian varieties

Let A be an abelian variety overK, and letA be its Neron model overX . We writeG(A)

for G(A) and�i(A) for �i(G(A)).

THEOREM 4.15. LetA be an abelian variety overK.(a) The pairing�0(Ak)� �0(At

k)! Q=Z defined in (C.11) is nondegenerate.

(b) There is a canonical isomorphism

H 1(K, At)��! Ext1k(G(A), Q=Z).

PROOF. (a) We can assume thatk is algebraically closed, and in this case the result isproved in Begeuri 1980, 8.3.3.

(b) From (ibid. 8.3.6) we know that the result holds ifk is algebraically closed; todeduce the result in the general case, apply the Hochschild-Serre spectral sequence to theleft hand side and the spectral sequence (I 0.17) to the right hand side.

COROLLARY 4.16. Assume thatAk is connected. Then there is a nondegenerate pairingof Gal(Kun=K)-modules

H 1(Kun, At)� �1(A)! Q=Z.

PROOF. Again we can assume thatk is algebraically closed. As we noted above, for anyconnected perfect group schemeG overk and finite perfect group schemeN , Ext1k(G, N) D

Homk(�1(G), N). This shows that Ext1k(G(A), Q=Z) D Homk(�1(G(A), Q=Z), and so

the result follows from the theorem.

NOTES. The results in this section are due to Begeuri 1980. Partial results in the samedirection were obtained earlier by Vvedens’kii (see Vvedens’kii 1973, 1976 and earlierpapers).

5 Two exact sequences

We write down two canonical short resolutions that are of great value in the proof of dualitytheorems in characteristicp. Throughout,X will be a scheme of characteristicp 6D 0.

The first exact sequence

The first sequence generalizes the sequences

0! Z=pZ! Ga

1�F�! Ga! 0

0! p ! Ga

F�! Ga! 0

to any group schemeN D that is the Cartier dual of a finite group scheme of height one.Note that in each sequence,Ga is the cotangent space toN .


Let N be a finite flat group scheme overX of height1, and leteWX ! N be the zerosection. LetI � ON be the ideal defining the closed immersione (so that(ON =I)je(X) D

OX ), and letInf 1X (N) Ddf Spec(ON =I2) be the first order infinitesimal neighbourhood

of the zero section. ThenI=I2 is the cotangent space!N of N over X . Locally onX ,there is an isomorphism of pointed schemes

N � Spec(OX [T1, ..., Tm]=(Tp1 , ..., T p

m )),

and thereforeI=I2

� (T1, ..., Tm)=(T 21 , ..., T 2

m)

(see Messing 1972, II 2.1.2). In particular,!N is a locally freeOX -module of finite rank,and hence it defines a vector groupV (!N ) over X . We shall almost always write!N

for V (!N ). This vector group representsMorX -ptd(Inf 1X (N), Gm) viewed as a functor of

schemes overX . (The notationX -ptd means that the morphisms are required to respectthe canonicalX -valued points of the two schemes.) The Cartier dualN D of N representsHomX (N, Gm), and so the inclusionInf 1

X (N),!N defines a canonical homomorphism�(N)WN D

! !N .Recall from~0 that the Verschiebung is a mapV WN (p)

! N . It induces a map!N !

!N (p), and on combining this with the canonical isomorphism!N (p)�D !

(p)N , we obtain a

homomorphism'0W!N ! !(p)N . The relative Frobenius morphism for the vector group!A

overX is also a homomorphism'1W!N ! !(p)N , and we define' D '0 � '1.

THEOREM 5.1. For any finite flat group schemeN of height one overX , the sequence

0! N D ��! !N

'�! !

(p)N ! 0

is exact.

For N D �p andN D p the sequence becomes one of those listed above. In the casethatX is an algebraically closed field, everyN has a composition series whose quotientsare isomorphic to�p or to p, and the theorem can be proved in this case by induction onthe length ofN (see Artin and Milne 1976, p115).

LEMMA 5.2. The sequence is a complex, that is,' ı � D 0.

PROOF. This can be proved by direct calculation (ibid., p114).

The next lemma shows that, whenX in Noetherian, the theorem follows from the casethatX is an algebraically closed field.

LEMMA 5.3. Let0! G 0 �

�! G'�! G 00

! 0

be a complex of flat group schemes of finite type over a Noetherian schemeX . Assume thatfor all geometric pointsx of X , the sequence of fibres

0! G 0x ! Gx ! G 00

x ! 0

is exact. Then the original sequence is exact.


PROOF. The faithful flatness of the'x, combined with the local criterion for flatnessGrothendieck 1971, IV 5.9, implies that' is faithfully flat. Thus Ker(') is flat and offinite type, and by assumption� factors through it. Now the same argument shows that�WN ! Ker(') is faithfully flat. Finally, the kernel of� is a group scheme overX whosegeometric fibres are all zero, and hence is itself zero.

We now complete the proof of (5.1). It suffices to check the exactness of the sequencelocally on X , and so we can assume thatX is quasi-compact. Then there will exist aNoetherian schemeX0, a finite flat group schemeN0 overX0, and a mapX ! X0 suchthatN D N0�X0

X . We know that the sequence forN0 is exact, but since the constructionof the sequence commutes with base change, this proves that the sequence forN is exact.

EXAMPLE 5.4. SupposeN has orderp. Then it can be writtenN D NL0,a in the Oort-Tate

classification (0.9) withL an invertible sheaf onX anda 2 L˝1�p. Its dualN D D NL_

a,0 .The cotangent sheaf!N is equal toL_, and when we identifyL_(p) with L_˝p, the map'in the sequence in the theorem becomes

z 7! z˝p� a˝ zWL! L_˝p.

In this case the exactness of the sequence is obvious from the description Oort and Tategive of the points ofN D (see 0.9d).

REMARK 5.5. The theorem shows that every finite flat group schemeN over X whosedual has height one gives rise to a locally freeOX -moduleV of finite rank and to a linearmap'0WV ! V(p). To recoverN from the pair(V ,'0), simply form the kernel of'0 �

'1 where'1 is the relative Frobenius ofV (regarded as a vector group). These remarkslead to a classification of finite group schemes of this type that is similar, but dual, tothe classification of finite flat group schemes of height one by theirp-Lie algebras (seeDemazure and Gabriel 1970, II,~7).

The second exact sequence

We now let� WX ! S be a smooth map of schemes of characteristicp, and we assumethat S is perfect. WriteX 0 for X regarded as anS -scheme by means of� 0 D Fabsı � .BecauseS is perfect, we can identify(X 0,� 0) with (X (1=p),� (1=p)) (see~0), and whenwe do this, the relative Frobenius mapFX (1=p)=S WX

(1=p)! X becomes identified with

the absolute Frobenius mapF D Fabs. For example, ifS D SpecR andX D SpecAfor someR-algebraiWR ! A, thenX 0

D SpecA, with A regarded as anR-algebra bymeans ofa 7! i(a)p, and the Frobenius mapX X 0 corresponds toa 7! ap

WA ! A.We write˝1

X=S,clfor the sheaf of closed differential forms onX relative toS , that is,

˝1X=S,cl

D Ker(d W˝1X=S ! ˝1

X=S). We regard 1X=S and˝1

X=S,clas sheaves onXet.

Again N is a finite flat group scheme of height one onX , and we letn be the Liealgebra ofN (equal to the tangent sheaf ofN overX).

THEOREM 5.6. Let f WXfl ! Xet be the morphism of sites defined by the identity map.ThenRrf�N D 0 for r 6D 1, and there is an exact sequence

0! R1f�N ! n˝OX˝1

X 0=S,cl

�! n˝OX

˝1X=S ! 0.


We first show thatRrf�N D 0 for r 6D 1. According to (A.5), there is an exactsequence

0! N ! G0! G1! 0

with G0 andG1 smooth. AsRrf�G D 0 for r > 0 if G is a smooth group scheme (seeMilne 1980, III 3.9), it is clear thatRrf�N D 0 for r > 1 (this part of the argument worksfor a finite flat group scheme over any scheme). The sheaff�N is the sheaf defined byNon Xet, which is zero becauseN is infinitesimal and all connected schemesetale overXare integral.

As far as the sequence is concerned, we confine ourselves to defining the maps; for theproof of the exactness, see Artin and Milne 1976,~2. First we need a lemma.

LEMMA 5.7. The mapH 1(X, N)! H 1(X 0, N) is zero.

PROOF. SinceN has height one,FN =X WN ! N (p) factors throughe(p)(X) whereeWX !

N is the zero section. Therefore,FabsWN ! N factors throughe(X), which meansthat the image ofa 7! ap

WON ! ON is contained inOX � ON . By descent theory,this last statement holds for any principal homogeneous spaceP of N overX W there is amap�WOP ! OX whose composite with the inclusionOX ,!OP is thepth power map.

The compositeOP

��! OX

id�! OX 0 is anOX -morphism, and therefore defines anX -

morphismX 0! P . This shows thatP becomes trivial overX 0. Since all elements of

H 1(X, N) are represented by principal homogeneous spaces, this proves the lemma.

Let P be a principal homogeneous space forN over X . Then the lemma shows thatthere is a trivialization' 0

WX 0! P , and the fact thatN is purely infinitesimal implies

that ' 0 is unique. It gives rise to two mapsX 0�X X 0

D X 00! P , namely,' 0

ıp1

and' 0ıp2. Their difference is an element00 of N(X 00) that is zero if and only if' 0 is

arises fromN(X). Now if N denotes the nilradical ofOX 00, thenOX 00=N � OX 0 andN =N 2

� ˝1X 0=S . Since˛00 is trivial on X 0, the restriction of 00 to Spec(OX 00=N 2)

defines a map!N ! ˝1X 0=S , whose image can be shown to lie in1

X 0=S,cl. This map can

be identified with an element ofn˝˝1X 0=S .

The same construction works for anyU etale overX ; for such aU , we get a mapH 1(Ufl, N)! n˝OU 0˝

1U 0=S,cl

. AsR1f�N is the sheaf associated withU 7! H 1(Ufl, N),this defines a map

R1f�N ! n˝OU 0 ˝1U 0=S,cl ,

which we take to be the first map in the sequence.

LEMMA 5.8. There exists a unique mapC W˝1X=S,cl

! ˝1X=S with the properties:

(i) C(f p!) D f C(!), f 2 OX , ! 2 ˝1X=S,cl

I

(ii) C(!) D 0 if and only if! is exact;(iii) C(f p�1df ) D df .

PROOF. Every closed differential 1-form is locally a sum of exact differentials and differ-entials of the formf p�1df , and so (ii) and (iii) completely describeC . For the proof thatC exists, see Milne 1976, 1.1, and Katz 1970, 7.2.


Note that (ii) says thatC is p�1-linear. According to our conventions,1X 0=S D ˝

1X=S

as sheaves of abelian groups onX 0D X , butf 2 OX acts asf p on˝1

X 0=S . Therefore,when regarded as a map1

X 0=S,cl! ˝1

X=S , C isOX -linear. Define 0W n ˝ ˝1X 0=S,cl

!

n˝˝1X=S to be1˝ C .

Recall (Demazure and Gabriel 1970, II,~7), thatn has the structure of ap-Lie algebra,that is, there is a mapn 7! n(p)

W n ! n such that(f x)(p)D f px(p). Also we have a

canonical inclusion 1X 0=S,cl

! ˝1X=S (becauseX 0

D X). Define 1W n ˝ ˝1X 0=S,cl

!

n˝˝1X=S to ben˝! 7! n(p)

˝!, and to be 0� 1; thus (n˝!) D n˝C!�n(p)˝!.

EXAMPLE 5.9. LetN D NL0,b

, b 2 � (X,L˝1�p) (in the Oort-Tate classification (0.9)).Then we can describe explicitly. It is the map

L˝˝1X 0=S,cl ! L˝˝1

X=S , x ˝ ! 7! x ˝ C! � (b ˝ x)˝ ! .

If L D OX , then theOX -structure on 1X 0=S,cl

is irrelevant, and so we can identify it with˝1

X=S,cl. The map then becomes

(! 7! C! � b!)W˝1X=S,cl ! ˝1

X=S .

For example, ifb D 1, thenN D �p andR1f��p D O�X =O

�pX ; the sequence is

0! O�X =O

�pX

dlog�! ˝1

X=S,cl

C �1�! ˝1

X=S ! 0, dlog(f ) Ddf

f.

If b D 0, thenN D p andR1f� p D OX =OpX ; the sequence is

0! OX =OpX

d�! ˝1

X=S,cl

C�! ˝1

X=S ! 0.

The canonical pairing of the complexes

We continue with the notations of the last subsection. SetWrite for the complex

U �(N) D (n˝˝1X 0=S,cl

�! n˝˝1

X=S),

V �(N D) D (!N

'�! !

(p)N ).

The terms on the right are complexes supported in degrees zero and one.

PROPOSITION5.10. There is a canonical pairing of complexes

V �(N D)� U �(N)! U �(�p).

PROOF. In order to define a pairing of complexes, we have to define pairings

( , )0,0WV0(N D)� U 0(N)! U 0(�p)

( , )1,0WV1(N D)� U 0(N)! U 1(�p)

( , )0,1WV0(N D)� U 1(N)! U 1(�p)


such that (v, u)0,0 D ('v, u)1,0 C (v, u)0,1

for all (v, u) 2 V 0(N D)� U 0(N). If we set

(˛, n˝ !0)0,0 D ˛(n)!0

(ˇ, n˝ !0)1,0 D ˇ(n)!0

(˛, n˝ !)0,1 D ˛(n)! ,

then it is routine matter to verify that these pairings satisfy the conditions (ibid.,~3).

Note that Theorems 5.1 and 5.6 give us quasi-isomorphismsRf�N D ��! V �(N D)

and Rf�N��! U �(N)[�1]. Also, that the pairingN D � N ! �p gives a pairing

Rf�N D˝

L Rf�N ! Rf��p.

PROPOSITION5.11. The following diagram commutes (in the derived category)

Rf�N D˝

L Rf�N ��! Rf��p??y�

??y�

V �(N D)˝L U �(N)[�1] ��! U �(�p)[�1].

PROOF. This is a restatement of Artin and Milne 1976, 4.6.

NOTES. This section summarizes Artin and Milne 1976.

6 Local fields of characteristicp

Throughout this section,K will be a local field of characteristicp 6D 0 with finite residuefield k. Let R be the ring of integers inK. A choice of a uniformizing parametert for R

determines isomorphismsR��! k[[t ]] andK

��! k((t)).

We shall frequently use that any group schemeG of finite-type overK has a compo-sition series with quotients of the following types: a smooth connected group scheme; afinite etale group scheme; a finite group scheme that is local withetale Cartier dual; a finitegroup scheme that is local with local Cartier dual. We shall refer to the last two groupschemes as being local-etale and local-local respectively. A finite local group scheme hasa composition series whose quotients are all of height one, and a finite local-local groupscheme has a composition series whose quotients are all isomorphic top (Demazure andGabriel 1970, IV,~3,5).

Cech cohomology

Fix an algebraic closure ofKa of K. For any sheafF on (SpecK)fl and finite extensionL of K, we write LH r (L=K, F) for the r th cohomology group of the complex of abeliangroups

6. LOCAL FIELDS OF CHARACTERISTIC P 235

0! F(L)! F(L˝K L)! � � � ! F(˝rK L)! F(˝rC1

K L)! � � � (6.0.1)

In the case thatL is Galois overK with Galois groupG, this complex can be identifiedwith the complex of inhomogeneous cochains of theG-moduleF(L) (see Shatz 1972,p207, or Milne 1980, III 2.6). We defineLH r (K, F) to be lim

�!LH r (L=K, F) whereL runs

over the finite field extensions ofK contained inKa.

PROPOSITION6.1. For any group schemeG of finite-type overK and anyr � 0, thecanonical map LH r (K, G)! H r (K, G) is an isomorphism.

The proof uses only thatK is a field of characteristicp. The first step is to show that ashort exact sequence of group schemes leads to a long exact sequence ofCech cohomologygroups. This is an immediate consequence of the following lemma.

LEMMA 6.2. For any short exact sequence

0! G 0! G ! G 00

! 0

of group schemes of finite type overK and anyr , the sequence

0! G 0(˝rK Ka)! G(˝r

K Ka)! G 00(˝rK Ka)! 0

is exact.

PROOF. We show thatH 1(˝rK Ka, G)(D lim

�!H 1(˝r

K L, G)) is zero. For any finite exten-sion L of K, ˝r

K L is an Artin ring. It is therefore a finite product of local rings whoseresidue fieldsLi are finite extensions ofL. If G is smooth,H 1(˝r

K L, G) DQLH 1(Li , G)

(see Milne 1980, III 3.11), and so obviously lim�!

H 1(˝rK L, G) D 0. It remains to treat

the case of a finite group schemeN of height one. Denote SpecrK L by X . Then (5.7)

shows that the restriction mapH 1(X, N) ! H 1(X (p�1), N) is zero. SinceX (p�1)D

Lp�1

˝K � � � ˝K Lp�1

, we again see that lim�!

H 1(˝rK L, G) D 0.

We next need to know thatLH r (K, G) andH r (K, G) are effaceable in the category ofgroup schemes of finite-type overK.

LEMMA 6.3. LetG be a group scheme of finite-type overK.(a) For any c 2 LH r (K, G), there exists an embeddingG ,! G 0 of G into a group

schemeG 0 of finite-type overK such thatc maps to zero inLH r (K, G 0).(b) Same statement withLH r (K, G) replaced byH r (K, G).

PROOF. In both cases, there exists a finite extensionL of K, L � Ka, such thatc maps tozero in LH r (L, G) (or H r (L, G)). TakeG 0 to be ResL=K G and the map to be the canonicalinclusion G,!ResL=K G. In the case ofCech cohomology, a simple direct calculationshows thatc maps to zero in LH r (K, G 0), and in the case of derived-functor cohomology,H r (K, G 0) D H r (L, G).


We now prove the proposition by induction onr . Forr D 0 it is obvious, and so assumethat it holds for allr less than somer0. For any embeddingG,!G 0, G 0=G is again a groupscheme of finite type overK, and we have a commutative diagram

� � � ��! LH r0�1(K, G 0) ��! LH r0�1(K, G 0=G) ��! LH r0(K, G) ��! LH r0(K, G 0)??y�D

??y�D

??y ??y� � � ��! H r0�1(K, G 0) ��! H r0�1(K, G 0=G) ��! H r0(K, G) ��! H r0(K, G 0).

Let c be a nonzero element ofLH r0(K, G), and chooseG,!G 0 to be the embedding givenby (6.3a); then a diagram chase shows that the image ofc in H r0(K, G) is nonzero. Letc02 H r0(K, G), and chooseG,!G 0 to be the embedding given by (6.3b); then a diagram

chase shows thatc0 is in the image of LH r0(K, G)! H r0(K, G). As c andc0 are arbitraryelements, this shows thatLH r0(X, G) ! H r0(X, G) is an isomorphism and so completesthe proof.

First calculations

Note that for any finite group schemeN overK, H 0(K, N) is finite.

PROPOSITION6.4. LetN be a finite group scheme overK.(a) If N is etale-local, thenH r (K, N) D 0 for r 6D 0, 1.(b) If N is local-etale, thenH r (K, N) D 0 for r 6D 1, 2.(c) If N is local-local, thenH r (K, N) D 0 for r 6D 1.

PROOF. (a) SinceN is etale,H 1(K, N) D H 1(Gal(Ks=K), N(Ks)), and because itsCartier dual is local,N must havep-power order. Therefore the assertion follows fromthe fact thatK has Galoisp-cohomological dimension1.

(b) We can assume thatN has height one, and then (5.6) shows thatRrf�N D 0 forr 6D 1. ThereforeH r (K, N) D H r�1(Ket, R1f�N), andR1f�N is ap-torsion sheaf.

(c) We can assume thatN D p. In this case the statement follows directly from thecohomology sequence of

0! p ! Ga

F�! Ga! 0.

The topology on the cohomology groups

The ring˝rK L has a natural topology. IfG is an affine group scheme of finite type overK,

thenG(˝rK L) also has a natural topology: immerseG into some affine spaceAn and give

G(˝rK L) � An(˝r

K L) the subspace topology. One checks easily that the topology is inde-pendent of the immersion chosen. Therefore, for any group schemeG of finite type overK,G(˝r

K L) has a natural topology, and the boundary maps in (6.0.1) are continuous becausethey are given by polynomials. EndowZr (L=K, G) � C r (L=K, G)(D G(˝rC1

K L)) withthe subspace topology, andLH r (L=K, G) D Zr (L=K, G)=Br (L=K, G) with the quotienttopology. We can then giveH r (K, G) the direct limit topology: a mapH r (K, G)! T iscontinuous if and only if it defines continuous maps onLH r (L=K, G) for all L.


LEMMA 6.5. Let G be a group scheme of finite type overK, and letL � Ka be a finiteextension ofK.

(a) The groupH r (X, G) is Hausdorff, locally compact, and� -compact (that is, a count-able union of compact subspaces).

(b) The maps in the cohomology sequence arising from a short exact sequence of groupschemes are continuous.

(c) The restriction mapsH r (K, G)! H r (L, G) are continuous.(d) WhenG is finite, the inflation mapInfW LH 1(L=K, G)! H 1(K, G) has closed image

and defines a homeomorphism ofLH 1(L=K, G) onto its image.(e) Cup-product is continuous.

PROOF. (a) The groupsC r (L=K, G) are Hausdorff,� -compact, and locally compact. AsZr (L=K, G) is a closed subspace ofC r (L=K, G), it has the same properties. Also theimageBr (L=K, G) of C r�1(L=K, G) is a locally compact subgroup of a Hausdorff group,and so is closed. HenceH r (L=K, G) is the quotient of a Hausdorff,� -compact, locallycompact space by a closed subspace, and it therefore inherits the same properties.

(b) Obvious.(c) It suffices to note that, for anyL0

� L, the mapsC r (L0=K, G) ! C r (L0=L, G)

are continuous.(d) It suffices to show that for anyL0 � L, the inflation mapLH 1(L=K, G)! LH 1(L0=K, G)

is closed. The mapG(L˝K L)! G(L0˝K L0) is closed, and therefore its restriction to

Z1(L=K, G)! Z1(L0=K, G) is also closed. SinceB1(L0=K, G) is compact (it is finite),the mapZ1(L0=K, G)! LH 1(L0=K, G) is closed, and the assertion follows.

(e) Obvious.

REMARK 6.6. (a) Let0! N ! G ! G 0

! 0

be an exact sequence, and suppose thatH 1(K, G) D 0. ThenH 0(K, N) has the subspacetopology induced fromH 0(K, N) ,! G(K) and H 1(K, N) has the quotient topologyinduced fromG 0(K) � H 1(K, N).

(b) Our definition of the topology onH r (K, G) differs from, but is equivalent to, thatof Shatz 1972, VI.

EXAMPLE 6.7. (a) The cohomology sequence of

0! Z=pZ! Ga

}�! Ga! 0

shows thatH 1(K, Z=pZ) D K=}K. This group is infinite, and it has the discrete topologybecauseR=}R is a finite open subgroup ofK=}K.

(b) From the Kummer sequence we find thatH 1(K,�p) D K�=K�p. This group iscompact becauseR�=R�p is a compact subgroup of finite index.

(c) The groupH 1(K, p) D K=Kp. The subgroupR=Rp of K=Kp is compact andopen, and the quotientK=RKp is an infinite discrete group.

(d) The groupH 0(K, Gm) equalsK� with its locally compact topology. The groupH 2(K, Gm) equalsQ=Z with the discrete topology becauseH 2(K, Gm) D

SH 2(L=K, Gm),

andH 2(L=K, Gm) is finite and Hausdorff.


Table 6.8.

G H 0(K, G) H 1(K, G) H 2(K, G)

etale-local finite, discrete torsion, discrete 0

local-etale 0 compact finite, discrete

local-local 0 locally compact 0

torus locally compact finite, discrete discrete

To verify the statements in the table, first note that, because the topologies onLH r (L=K, G)

andH r (K, G) are Hausdorff, they are discrete when the groups are finite. Next note thatLH r (L=K, G) contains LH r (RL=RK , G) as an open subgroup. Moreover, ifG is Z=pZ,�p, p, or Gm, each assertion follows from (6.7). It is not difficult now to deduce that theyare true in the general case.

Duality for tori

Let M be a finitely generated torsion-free module for Gal(Ks=K). As M becomes a mod-ule with trivial action over some finite separable extensionL of K, it is represented byan etale group scheme locally of finite-type overK. The method used above for groupschemes of finite-type defines the discrete topology on the groupsH r (K, M).

THEOREM 6.9. Let T be a torus overK and letX �(T ) be its group of characters. Thenthe cup-product pairing

H r (K, T )�H 2�r (K, X �(T ))! H 2(K, Gm) �D Q=Z

defines dualities between:the compact groupH 0(K, T )^ (completion relative to the topology of open subgroups

of finite index) and the discrete groupH 2(K, X �(T ))I

the finite groupsH 1(K, T ) andH 1(K, X �(T ))I

the discrete groupH 2(K, T ) and the compact groupH 0(K, X �(T ))^ (completion rel-ative to the topology of subgroups of finite index).

PROOF. AsT is smooth,H r (K, T ) D H r (Gal(Ks=K), T (Ks)), and the groupH r (K, X �(T )) D

H r (Gal(Ks=K), X �(T )). All the groups have the discrete topology exceptH 0(K, T ) D

T (K), which has the topology induced by that onK. The theorem therefore simply restates(I 2.4).

Finite group schemes

We now letN be a finite group scheme overK.

THEOREM 6.10. For any finite group schemeN overK, the cup-product pairing

H r (K, N D)�H 2�r (K, N)! H 2(K, Gm) �D Q=Z

identifies each group with the Pontryagin dual of the other.


PROOF. After (I 2.3) we may assume thatN is ap-primary group scheme. Suppose firstthatN is etale. Then there is a short exact sequence of discrete Gal(Ks=K)-modules

0!M1!M0! N(Ks)! 0

with M0 andM1 finitely generated and torsion-free (as abelian groups). Dually there is anexact sequence

0! N D! T 0

! T 1! 0

with T 0 andT 1 tori. Because the cohomology groups of the modules in the first sequenceare all discrete, the dual of its cohomology sequence is exact. Therefore we get an exactcommutative diagram

� � � ��! H r (K, N D) ��! H r (K, T 0) ��! H r (K, T 1) ��! � � �??y ??y ??y� � � ��! H r (K, N)�

��! H r (K, M0)��! H r (K, M1)�

��! � � � .

For r � 1, the second two vertical arrows in the diagram are isomorphisms, and this shows

thatH r (K, N D)�D�! H r (K, N)� for r � 2. The diagram also shows that the image of

H 1(K, N D) in H 1(K, N)� is dense. AsH 1(K, N D) is compact, its image is closed andso equalsH 1(K, N)�. If H 1(K, N D)! H 1(K, N) were not injective, then there wouldexist an elementb 2 T 1(K) that is in the image ofT 0(K)^ ! T 1(K)^, but which is notin the image ofT 0(K) ! T 1(K). I claim that the image ofT 0(K) is closed inT 1(K).Let L be a splitting field forT 0 andT 1, and consider the diagram

0 ��! Hom(X �(T 0), R�L) ��! T 0(K) ��! Hom(X �(T 0), Z) ��! � � �??y ??y ??y

0 ��! Hom(X �(T 1), R�L) ��! T 1(K) ��! Hom(X �(T 1), Z) ��! � � �

(Hom’s asG-modules). Letb 2 T 1(K). If the image ofb in Hom(X �(T 1), Z) is notin the image fromT 0, thenb � Hom(X �(T ), R�

L) is an open neighbourhood ofb that isdisjoint from the image ofT 0(K). Therefore we may assumeb 2 Hom(X �(T 1), R�

L); butHom(X �(T 0), R�

L) is compact and so its complement in Hom(X �(T 1), R�L) is an open

neighbourhood ofb. This proves the claim and completes the proof of the theorem in thecase thatN or its dual isetale.

Next assume thatN D p. Here one shows that the pairing

H 1(K, p)�H 1(K, p)! Q=Z

can be identified with the pairing

K=Kp�K=Kp

! p�1Z=Z � Q=Z, (f, g) 7! p�1 Trk=Fp(res(fdg)),

(see Shatz 1972, p240-243); it also follows immediately from the elementary case of (5.11)in whichN D p), and this last pairing is a duality.

The next lemma now completes the proof.


LEMMA 6.11. Let0! N 0

! N ! N 00! 0

be an exact sequence of finite group schemes overK. If the theorem is true forN 0 andN 00,then it is true forN .

PROOF. Proposition I 0.22 and the discussion preceding it show that the bottom row of thefollowing diagram is exact, and so this follows from the five-lemma:

� � � ��! H r (K, N 00D) ��! H r (K, N D) ��! H r (K, N 0D) ��! � � �??y�D

??y ??y�D

� � � ��! H 2�r (K, N 00)��! H 2�r (K, N)�

��! H 2�r (K, N 0)��! � � � .

REMARK 6.12. (a) It is possible to give an alternative proof of Theorem 6.10 using thesequences in~5. After (6.11) (and by symmetry), it suffices to prove the theorem for agroupN of height one. Then the exact sequences in~5 yield cohomology sequences

0! H 0(K, N D)! V 0(N D)! V 1(N D)! H 1(K, N D)! 0

0! H 1(K, N)! U 0(N)! U 1(N)! H 2(K, N)! 0.

HereV 0(N D) is theK-vector space!N andU 1(N) is theK-vector spacen˝˝1K=k

. Thepairing(˛, n˝!) 7! res(˛(n)!) identifies!N with thek-linear dual ofn˝˝1

K=k, and so

the pairing(˛, n˝ !) 7! p�1 Trk=Fp

res(˛(n)!)

identifies!N with the Pontryagin dual ofn ˝ ˝1K=k

(see 0.7). SimilarlyV 1(N D) is thePontryagin dual ofU 0(N). Therefore the pairing of complexes in (5.10) shows that thedual of the first of the above sequences can be identified with an exact sequence

0! H 1(K, N D)�! U 0(N)! U 1(N)! H 0(K, N D)�

! 0.

Thus there are canonical isomorphismsH 1(K, N)�D�! H 1(K, N D)� andH 2(K, N)

�D�!

H 0(K, N D)�, and (5.11) shows that these are the maps given by cup-product.(b) It is also possible to deduce a major part of (6.10) from (I 2.1). LetN be etale

overK. ThenH r (Kfl, N) D H r (Ket, N), and so we have to show that ExtrKet

(N, Gm) D

H r (Kfl, N D). Note (Milne 1980, II 3.1d) thatf �(N jKet) D N , wheref W (SpecK)fl !

(SpecK)et is defined by the identity map. From the spectral sequence ExtrKet

(N, Rsf�Gm) H)

ExtrCsKfl

(f �N, Gm) and the vanishing of the higher direct images ofGm, we see that ExtrKet

(N, Gm) D

ExtrKfl(N, Gm) for all r . But N is locally constant on(SpecK)fl, and the exact sequence

� � � ! H r (X, Gm)n�! H r (X, Gm)! ExtrKfl

(Z=nZ, Gm)! � � �

and the divisibility ofGm on the flat site show thatExt rKfl

(Z=nZ, Gm) D 0 for r > 0.ThereforeExt r

Kfl(N, Gm) D 0 for r > 0, and the local-global spectral sequence for Exts

shows that ExtrKet(N, Gm) D H r (Kfl, N D).

7. LOCAL RESULTS: EQUICHARACTERISTIC, FINITE RESIDUE FIELD 241

REMARK 6.13. Much of the above discussion continues to hold ifK is the field of fractionsof an excellent Henselian discrete valuation ring with finite residue field. For example, ifN is etale, thenH r (K, N) D H r (bK, N) for all r becauseK and bK have the same abso-lute Galois group, andN(Ks) D N(bKs); if N is local-etale, thenH 1(K, N) is dense inH 1(bK, N) andH 2(K, N) D H 2(bK, N); if N is local-local, thenH 1(K, N) is dense inH 1(bK, N). The mapH r (K, N D) ! H 2�r (K, N)� given by cup-product is an isomor-phism forr 6D 1, and is injective with dense image forr D 1.

NOTES. The main results in this section are taken from Shatz 1964; see also Shatz 1972.Theorem 6.10 was the first duality theorem to be proved for the flat topology and so can beregarded as the forerunner of the rest of the results in this chapter.

7 Local results: equicharacteristic, finite residue field

Throughout this section,R will be a complete discrete valuation ring of characteristicp 6D

0 with finite residue fieldk. As usual, we use the notationsX D SpecR and

Speck D xi�! X

j � u D SpecK.


Let N be a finite flat group scheme overX . As in (1.1), we find that

H 0(X, N) D N(X) D N(K) D H 0(K, NK ),

H 1(X, N),!H 1(K, N),

H r (X, N) D 0, r > 1,

and

H 2x (X, N) D H 1(K, N)=H 1(X, N),

H 3x (X, N) D H 2(K, N),

H rx (X, N) D 0, r 6D 2, 3.

In the last section we defined topologies on the groupsH r (K, N). We endowH r (X, N)

with its topology as a subspace ofH r (K, N), and we endowH rx (X, N) with its topology as

a quotient ofH r (K, N). With respect to these topologiesH 0(X, N) is discrete (and finite),andH 1(X, N) is compact and Hausdorff;H 2

x (X, N) andH 3x (X, N) are both discrete (and

H 3x (X, N) is finite).

THEOREM 7.1. For any finite flat group schemeN overX , the canonical pairings

H r (X, N D)�H 3�rx (X, N)! H 3

x (X, Gm) �D Q=Z

define dualities between:the finite groupsH 0(K, N D) andH 3

x (X, N)I

the compact groupH 1(X, N D) and the discrete torsion groupH 2x (X, N).


Before giving the proof, we list some corollaries.

COROLLARY 7.2. For any finite flat group schemeN over X , H 1(X, N D) is the exactannihilator ofH 1(X, N) in the pairing

H 1(K, N D)�H 1(K, N)! H 2(K, Gm) �D Q=Z

of (6.10).

PROOF. As in the proof of (1.4), one sees easily that the corollary is equivalent to the caser D 1 of the theorem.

COROLLARY 7.3. LetN be a finite flat group scheme overX . For all r < 2p � 2,

ExtrX (N, Gm)�H 3�rx (X, N)! H 3

x (X, Gm) �D Q=Z


PROOF. The proof is the same as that of (1.5).

Write f WXfl ! Xet for the morphism of sites defined by the identity map.

COROLLARY 7.4. Let N be a quasi-finite flat group scheme overX whosep-primarycomponentN(p) is finite overX . LetN D be the complex of sheaves such that

N D(`) D

(HomXfl(N(`), Gm) if ` D p

f �RHomXet(N(`), Gm) if ` 6D p.

ThenH r (X, N D)�H 3�r

x (X, N)! H 3x (X, Gm) �D Q=Z


PROOF. For the prime-to-p components of the groups, the corollary follows from (II 1.8);for thep-primary component, it follows immediately from the theorem.

PROOF OF7.1. Assume first thatN has height one. Then the first exact sequence in~5leads to a cohomology sequence

0! H 0(X, N D)! H 0(X,!N )! H 0(X,!(p)N )! H 1(X, N D)! 0,

and the second leads to a cohomology sequence

0! H 2x (X, N)! H 1

x (X, n˝˝1X 0)! H 1

x (X, n˝˝1X )! H 3

x (X, N)! 0.

The pairing(˛, n˝ w) 7! ˛(n)wW!N � n˝˝1

X ! ˝1X (� OX )

realizesH 0(X,!N ) as theR-linear dual ofH 0(X, n˝˝1X ); therefore (0.8) shows that the

compact groupH 0(X,!N ) is the Pontryagin dual of the discrete groupH 1x (X, n˝˝1

X ) D

H 0(X, n ˝ ˝1X ) ˝K=H 0(X, n ˝ ˝1

X ). Similarly the compact groupH 0(X,!(p)N ) is the


Pontryagin dual ofH 1x (X, n ˝ ˝1

X 0), and so the pairing in (5.10) gives a commutativediagram

0 �! H 0(X, N D) �! H 0(X,!N ) �! H 0(X,!(p)N ) �! H 1(X, N D) �! 0??y�

??y�

0 �! H 3x (X, N)� �! H 1

x (X, n˝˝1X )� �! H 1

x (X, n˝˝1X 0)

� �! H 2x (X, N)� �! 0.

The diagram provides isomorphismsH 0(X, N D)��! H 3

x (X, N) andH 1(X, N D)��!

H 2(X, N), which we must show are those given by the pairing in the theorem. For this weretreat to the derived category. In (5.11) we saw that there is a commutative diagram:

Rf�N D˝

L Rf�N ��! Rf��p??y�

??y�

V �(N D)˝L U �(N)[�1] ! U �(�p)[�1].

From this we get a commutative diagram

H r (X, N D) � H 3�rx (X, N) > H 3

x (X,�p)

H r (X, V �(N D))

_

�H 2�rx (X, U �(N))

_

> H 2x (X, U �(�p)),

_�

which exactly says that the two pairs of maps agree. This completes the proof of thetheorem whenN has height one.

Next we note that forr D 0 the theorem follows from (6.10), and that forr D 1 it isequivalent to (7.2). Since this last statement is symmetric betweenN andN D , we see thatthe theorem is also true ifN is the dual of a finite group scheme of height one. Every finitegroup scheme overX has a composition series each of whose quotients is of height one oris the dual of a height one group (this is obvious overK, and one can apply (B.1) to obtainit overX), and so the theorem follows from the obvious fact that it is true for any extensionof groups for which it is true.

REMARK 7.5. The original proof of the Theorem 7.1 (Milne 1970/72, 1973) was moreexplicit. We include a sketch. It clearly suffices to prove (7.2). WritedK (� K=Kp) anddR (� R=Rp) for the images of the mapsd WK! ˝1

K=kandd WR! ˝1

R=k.

Assume first thatN D p. Then the diagram

H 1(X, N D)�H 1(X, N)

H 1(K, N D)

_

�H 1(K, N)

_

> H 2(K, Gm) �D Q=Z

(7.5.1)


R=Rp� dR

K=Kp_

� dK_

> p�1Z=Z � Q=Z,


where the bottom pairing is(f,!) 7! p�1Trk=Fpres(f!). It is obvious that the upper

groups are exact annihilators in the lower pairing.Let N D Na,0 in the Oort-Tate classification (0.9) witha D t (p�1)c. Then the diagram

(7.5.1) can be identified with

t�cpR=((t�cpR) \ }R)�dlog(K�) \ t cpRdt

K=}K_

� dlog(K�)_

> p�1Z=pZ � Q=Z

where the lower pairing is(f,!) 7! Trk=Fp(res(f!)). It is easy to check that the upper

groups are exact annihilators in the lower pairing.These calculations prove the theorem wheneverN D N0,0 (that is,N D p), N D Na,0

wherea D t c(p�1), or N D N0,b whereb D t c(p�1). The general case can be reduced tothese special cases by means of the following statements (Milne 1973, p84-85):

(i) for any finite group scheme ofp-power orderN overR, there exists a finite exten-sionK0 of K of degree prime top such that overR0, N has a composition series whosequotients have the above form;

(ii) the theorem is true overK if it is true over some finite extensionK0 of K of degreeprime top.

REMARK 7.6. LetN be a finite flat group scheme overX . We define

�(X, N) D[H 0(X, N)]

[H 1(X, N)]

when both groups are finite. Then the same proof as in (1.14) shows that whenNK is etale,

�(X, N) D (RW d)�1=n,

wheren is the order ofN andd is the discriminant ideal ofN overR. Note that

H 1(X, N) is infinite () NK is notetale() d D 0,

and so, if we interpret1=1 as 0, then the assertion continues to hold whenNK is notetale.

It is possible to prove a weak form of (1.19).

EXERCISE 7.7. (a) LetN andN 0 be finite flat group schemes overR; then a homomor-phism'WNK ! N 0

K extends to a homomorphismN ! N 0 if and only if, for all finitefield extensionsL of K, H 1('L)WH 1(L, NL) ! H 1(L, NL) mapsH 1(RL, NL) intoH 1(RL, N 0

L).(b)6 7 Use (a) to prove the characteristicp analogue of the main theorem of Tate 1967b:

if G and G 0 are p-divisible groups overR, then every homomorphism'WGK ! G 0K

extends to a unique homomorphismG ! G 0.

6(In original.) The author does not pretend to be able to do part (b) of the problem; the question is stillopen in general.

7Tate’s theorem was extended to characteristicp by de Jong (Homomorphisms of Barsotti-Tate groupsand crystals in positive characteristic. Invent. Math. 134 (1998), no. 2, 301–333; erratum, ibid. 138 (1999),no. 1, 225).


Abelian varieties and Neron models

We extend (I 3.4) and the results of~2 to characteristicp. Note that in the main results, thecoefficient groups are smooth, and so the cohomology groups can be computed using theetale topology (or even using Galois cohomology). The proofs however necessarily involvethe flat site.

THEOREM 7.8. LetA be an abelian variety overK, and letB be its dual. The pairings

H r (K, B)�H 1�r (K, A)! H 2(K, Gm) �D Q=Z

induced by the Poincare biextensionW of (B, A) byGm define dualities between:the compact groupB(K) and the discrete groupH 1(K, A)I

the discrete groupH 1(K, B) and the compact groupA(K).For r 6D 0, 1, all the groups are zero.

We note first that the pairing is symmetric in the sense that the pairing defined byW

and by its transposeW t are the same (up to sign) (see C.4). Therefore it suffices to showthat the map 1(K, A)WH 1(K, B) ! A(K)� is an isomorphism and thatH r (K, B) D 0

for r � 2. Consider the diagram

0 ��! B(K)(n) ��! H 1(K, Bn) ��! H 1(K, B)n ��! 0??y ??y�

??y0 ��! H 1(K, A)�

n ��! H 1(K, An)��! A(K)(n)�

��! 0

Theorem 6.10 shows that the middle vertical arrow is an isomorphism, and it follows that˛1(K, A)n is surjective for alln. As A(K) is a profinite group, its dualA(K)� is torsion,and so this proves that1(K, A) is surjective. To show that it is injective, it suffices toprove thatH 1(K, B)` ! A(K)(`)� is injective for all primes. For 6D p, we saw in (I 3)that this can be done by a counting argument. Unfortunately, for` D p the groups involvedare not finite (nor even compact), and so we must work more directly with the cohomologygroups of finite group schemes.

We dispose of the statement thatH 2(K, B) D 0 (note thatH r (K, B) D 0 for r > 2

becauseK has strict (Galois) cohomological dimension2). Consider the diagram

H 1(K, B) ��! H 2(K, Bn) ��! H 2(K, B)n ��! 0??y ??y ??yA(K)�

��! An(K)��! 0

which is just a continuation to the right of the previous diagram. We have seen that the firstvertical arrow is surjective, and the second is an isomorphism by (6.10). A diagram chasenow shows thatH 2(K, B)n D 0 for all n, and soH 2(K, B) is zero.

The next lemma will allow us to replaceK by a larger field.

LEMMA 7.9. If for some finite Galois extensionL of K, ˛1(L, AL) is injective, then˛1(K, A) is injective.


PROOF. SinceK is local, the Galois groupG of L over K is solvable, and so we mayassume it to be cyclic. There is an exact commutative diagram

0 �! H 1(G, B(L)) �! H 1(K, B) �! H 1(L, B)G �! H 0T (G, B(L)) �! H 2(K, B) D 0??y ??ysurj

??y�

??y0 �! H 0

T (G, A(L))��! A(K)�

�! A(L)�G�! H 1(G, A(L))�

�! 0

in which the top row is part of the sequence coming from the Hochschild-Serre spectralsequence (except that we have replacedH 2

T (G, B(L)) with H 0T (G, B(L))), and the bottom

row is the dual of the sequence that explicitly describesH 0T andH 1

T for a cyclic group. Thesecond and third vertical arrows are˛1(K, A) and˛1(L, A), and the first and fourth areinduced by 1(K, A) and by the dual of 1(K, B) respectively. From the right hand end ofthe diagram we see that

H 0T (G, B(L))! H 1(G, A(L))�

is an isomorphism, and by interchangingA andB we see that

H 0T (G, A(L))! H 1(G, B(L))�

is an isomorphism. Thus all vertical maps but the second are isomorphisms, and the five-lemma shows that it also is an isomorphism.

To proceed further, we need to consider the Neron modelsA andB of A andB. Leti�˚ D A=Aı, and writeA� for the subscheme ofA corresponding to a subgroup� of ˚ .

LEMMA 7.10. (a) The mapA� (X) ! � (x) is surjective, and the mapH r (X, A� ) !

H r (x,� ) is an isomorphism for allr � 1; thereforeH r (X, A� ) D 0 for r � 2.(b) There is an exact sequence

˚(x)! (˚=� )(x)! H 1(X,A� )! H 1(K, A).

(c) We have

H rx (X,A� ) D

(0 if r 6D 1, 2

(˚=� )(x) if r D 1,

and there is an exact sequence

0! � (x)! ˚(x)! (˚=� )(x)! H 1(X,A� )! H 1(K, A)! H 2x (X,A� )! 0.

PROOF. The proofs of (2.1–2.3) apply also in characteristicp.

After we make a finite separable field extension,A (andB) will have semistable reduc-tion andAp andBp will extend to finite group schemes overX . The group (k)p then hasorderp� where� is the dimension of the toroidal part of the reductionAı

0 of Aı (equal tothe dimension of the toroidal part of the reduction ofBı). The extension of the Poincarebiextension to(Bı,Aı) defines a pairing

H 2x (X,Bı)�H 0(X,Aı)! H 3

x (X, Gm) �D Q=Z,


which the proposition allows us to identify with a mapH 1(K, B) ! Aı(X)�. Clearlythis map is the composite of1(K, A)WH 1(K, B) � A(K)� with A(K)� � Aı(X)�. Inorder to complete the proof of the theorem, it suffices therefore to show that the kernel ofH 1(K, B)p ! Aı(X)(p)� has order[A(K)(p)=Aı(X)(p)] D [˚(k)(p)].

From the diagram

0 ��! ˚ 0(k)(p) ��! H 2x (X,Bı

p) ��! H 2x (X,Bı)p ��! 0??y ??y ??y

0 ��! H 0(X,Aıp)��! Aı

p(X)(p)��! 0,

we see that it suffices to show that the kernel ofH 2x (X,Bı

p) ! H 0(X,Aı

p)� has order[˚ 0(k)(p)][˚(k)(p)].

The mapH 2x (X,Bı

p)! H 0(X,Aıp)� is the composite of the maps

H 2x (X,Bı

p)! H 2x (X,Bp)! H 0(X,Ap)�

! H 0(X,Aıp)�.

The middle map is an isomorphism (6.1) and the remaining two maps are surjective withkernels respectivelyH 1

x (X,˚ 0p) D H 1(x,˚ 0

p) andH 0(x, p)�. The shows that the kernelof H 2

x (X,Bıp)! H 0(X,Aı

p)� has the required order. (See also Milne 1970/72.)

Once (7.8) is acquired, the proofs of (2.5) to (2.10) apply when the base ring has char-acteristicp. We merely list the results.

THEOREM 7.11. The canonical pairing 0� ˚ ! Q=Z is nondegenerate; that is, Con-

jecture C.13 holds in this case.

COROLLARY 7.12. Suppose that� 0 and � are exact annihilators under the canonicalpairing on˚ 0 and˚ . Then the map

B� 0

! Ext1Xsm

(A� , Gm)

defined by the extension of the Poincare biextension is an isomorphism (of sheaves onXsm).

THEOREM 7.13. Assume that� 0 and� are exact annihilators. Then the pairing

H r (X,B� 0

)�H 2�rx (X,A� )! H 3

x (X, Gm) �D Q=Z

defined by the canonical biextension of(B� 0

,A� ) byGm induces an isomorphism

H 2x (X,A� )

�D�! A� 0

(X)�

of discrete groups forr D 0 and an isomorphism of finite groups

H 1(X,B� 0

)�D�! A� (X)�

for r D 1. For r 6D 0, 1, both groups are zero.


REMARK 7.14. Assume thatR is an excellent Henselian discrete valuation ring, and letbX D SpecbR. Then it follows from (I 3.10) that the mapsH rx (X,A� )! H r

x (bX,A� ) areisomorphisms for allr , andH r (X,A� )! H r (bX,A� ) is an isomorphism for allr > 0.The mapA(X)! A(bX) is injective and maps onto the torsion subgroup ofA(bX); A(bX)

is the completion ofA(X) for the topology of open subgroups of finite index.

Write Bfng for the complexB n�! Bn˚ 0

andAfng for the complexA˚nn�! Aı.

The pairingsB ˝L Aı! Gm[1] andBn˚ 0

˝L A˚n ! Gm[1] defined by the Poincare

biextension induce a pairingBfng ˝L Afng ! Gm in the derived category of sheaves onXsm.

THEOREM 7.15. The mapBfng ˝L Afng ! Gm defines nondegenerate pairings

H r (X,Bfng)�H 3�rx (X,Afng)! H 3

x (x, Gm) �D Q=Z

for all r .

THEOREM 7.16. Assume thatn is prime top or thatA has semistable reduction. Then forall r , there is a canonical nondegenerate pairing

H rx (Xfl,Bn)�H 3�r (Xfl,An)! H 3

x (Xfl, Gm) �D Q=Z.

Curves overX

It is possible to prove an analogue of Theorem 2.11. Note that the methods in Artin andMilne 1976,~5, can be used to prove a more general result.

NOTES. Theorem 7.1 was proved independently by the author (Milne 1970/72, 1973) andby Artin and Mazur (unpublished). The above proof is new. Theorem 7.8 was also provedby the author (Milne 1970/72) (Shatz 1967 contains a proof for elliptic curves with Tateparametrizations). The stronger forms of it are due to McCallum (1986).

8 Global results: curves over finite fields, finite sheaves

Throughout this section,X will be a complete smooth curve over a finite fieldk. Thefunction field ofX is denoted byK, andp is the characteristic ofk. For a sheafF on anopen subschemeU of X , H r

c (U, F) denotes the cohomology group with compact supportas defined in (0.6b). Thus, there exist exact sequences

� � � ! H rc (U, F)! H r (U, F)!

Mv2X rU

H r (Kv, F )! � � �

� � � ! H rc (V , F)! H r

c (U, F)!M

v2U rV

H r (bOv, F )! � � �

with Kv and bOv the completionsof K andOv at v. With this definition, a short exactsequence of sheaves gives rise to a long exact cohomology sequence, and there is a pairing

8. GLOBAL RESULTS: CURVES OVER FINITE FIELDS, FINITE SHEAVES 249

between Ext groups and cohomology groups with compact support (see 0.4b and 0.4e), butin general the flat cohomology groups with compact support will not agree with theetalegroups even for a sheaf arising from anetale sheaf or a smooth group scheme (contrast0.4d).

The duality theorem

WhenN is a quasi-finite flat group scheme onU , we endowH r (U, N) with the discretetopology.

LEMMA 8.1. For any quasi-finiteetale group schemeN on an open subschemeU of X ,H r

c (U, N) D H rc (Uet, N) all r .

PROOF. Let QKv be the field of fractions ofOhv . Then, as we observed in (6.13),H r ( QKv, N)

��!

H r (Kv, N) for all r , and asH r ( QKv,et, N)��! H r ( QKv, N), the lemma follows from com-

paring the two sequences

� � � ��! H rc (Uet, N) ��! H r (Uet, N) ��!

Lv2X rU

H r ( QKv,et, N) ��! � � �??y �

??y �

??y� � � ��! H r

c (U, N) ��! H r (U, N) ��!L

v2X rU

H r (Kv, N) ��! � � � .

THEOREM 8.2. Let N be a finite flat group scheme over an open subschemeU of X . Forall r , the canonical pairing

H r (U, N D)�H 3�rc (U, N)! H 3

c (U, Gm) �D Q=Z

defines isomorphismsH 3�rc (U, N)

��! H r (U, N D)�.

After Lemma 8.1 and Theorem II 3.1, it suffices to prove the theorem for a groupscheme killed by a power ofp. We first need some lemmas.


! N ! N 00! 0

be an exact sequence of finite flat group schemes onU . If the theorem is true forN 0 andN 00, then it is true forN .

PROOF. Because the groups are discrete, the Pontryagin dual of

� � � ! H r (U, N 00D)! H r (U, N D)! H r (U, N 0D)! � � �

is exact. Therefore one can apply the five-lemma to the obvious diagram.

LEMMA 8.4. Let V be an open subscheme ofU . The theorem is true forN on U if andonly if it is true forN jV onV .


PROOF. This follows from (7.1) and the diagram

� � � ��! H 3�rc (V , N) ��! H 3�r

c (U, N) ��!L

v2U rV

H 3�r (bOv, N) ��! � � �??y ??y ??y�

� � � ��! H r (V , N D)��! H r (U, N D)�

��!L

v2U rV

H rv (bOv, N D)�

��! � � � .

LEMMA 8.5. The theorem is true ifU D X andN or its dual have height one; moreover,the groups involved are finite.

PROOF. Assume first thatN has height one. The first exact sequence in~5 yields a coho-mology sequence

� � � ! H r (X, N D)! H r (X,!N )! H r (X,!(p)N )! � � �

and the second a sequence

� � � ! H rC1(X, N)! H r (X, n˝˝1X 0)! H r (X, n˝˝1

X )! � � � .

The canonical pairing of complexes (5.10) together with the usual duality theorem for co-herent sheaves on a curve show that the finite-dimensionalk-vector spacesH r (X,!N ) andH r (X,!

(p)N ) are thek-linear (hence Pontryagin) duals of thek-vector spacesH 1�r (X, n˝

˝1X ) andH 1�r (X, n˝˝1

X ), and moreover that there is a commutative diagram

� � � ��! H 3�r (X, N) ��! H 2�r (X, n˝˝1X 0) ��! H 2�r (X, n˝˝1

X ) ��! � � �??y ??y�

??y�

� � � ��! H r (X, N D)��! H r�1(X,!

(p)N )�

��! H r�1(X,!N )��! � � � .

The diagram gives an isomorphismH 3�r (X, N)��! H r (X, N D)� for all r , and (5.11)

shows that this is the map in the statement of the theorem.In this case the groupsH r (X, N) andH r (X, N D) are finite, and the statement of the

theorem is symmetric betweenN andN D . Therefore, the theorem is proved also if thedual ofN has height one.

We now prove the theorem. LetN be a finite group scheme overU . Lemma 8.4 allowsus to replaceU by a smaller open subset, and so we can assume thatN has a compositionseries all of whose quotients have height1 or are the Cartier duals of groups of height1. Now Lemma 8.3 allows us to assume thatN (or its dual) has height1. According toProposition B.4,NK extends to a finite flat group schemeN on X which is of height one(or has a dual of height one). After again replacingU by a smaller open set, we can assumethatN jU D N . According to (8.5), the theorem is true forN on X , and (8.4) shows thatthis implies the same result forN jU D N .

8. GLOBAL RESULTS: CURVES OVER FINITE FIELDS, FINITE SHEAVES 251

COROLLARY 8.6. LetN be a finite flat group scheme onU . For all r < 2p�2, the pairing

ExtrU (N, Gm)�H 3�rc (U, N)! H 3

c (U, Gm) �D Q=Z

defines isomorphismsH 3�rc (U, N)! ExtrU (N, Gm)�.

PROOF. Under the hypotheses,H r (U, N D) D ExtrU (N, Gm) (see the proof of (1.5)), andso this follows immediately from the theorem.

COROLLARY 8.7. Let N be a quasi-finite flat group scheme overU whosep-primarycomponentN(p) is finite overU . LetN D be the complex of sheaves such that

N D(`) D

�HomUfl(N(`), Gm) ` D p

f �RHomUet(N(`), Gm) ` 6D p.

Then the pairing

H r (U, N D)�H 3�rc (U, N)! H 3

c (U, Gm) �D Q=Z

defines isomorphismsH 3�rc (U, N)

��! H r (U, N D)� for all r .

PROOF. For thep-primary component ofN , this follows directly from the theorem; forthe`-primary component, 6D p, Lemma 8.1 shows that it follows from (II 1.11b).

PROBLEM 8.8. The groupH r (U, N D) is torsion, and soH r (U, N D)� is a compact topo-logical group. The isomorphism in the theorem therefore givesH r

c (U, N) a natural topol-ogy as a compact group. Find a direct description of this topology.


WhenU 6D X , the groupsH r (U, N) will usually be infinite, even whenN is a finiteetalegroup scheme overU (for example,H 1(A1, Z=pZ) D k[T ]=}k[T ], which is infinite).This restricts us to considering the caseU D X .

LEMMA 8.9. For any finite flat group schemeN overX , the groupsH r (X, N) are finite.

PROOF. WhenN or its dual have height one, we saw that the groups are finite in (8.5). Inthe general case,NK will have a filtration all of whose quotients are of height one or haveduals that are of height one, and by taking the closures of the groups in the filtration, weget a similar filtration forN (cf. B.1). The lemma now follows by induction on the lengthof the filtration.

WhenN is a finite flat group scheme onX , we define

�(X, N) D[H 0(X, N)][H 2(X, N)]

[H 1(X, N)][H 3(X, N)].

PROBLEM 8.10. Find a formula for�(X, N).


Let X D X ˝k ks, and let� D Gal(ks=k). If the groupsH r (X , N) are finite, then itfollows immediately from the exact sequences

0! H r�1(X , N)� ! H r (X, N)! H r (X , N)� ! 0

given by the Hochschild-Serre spectral sequence forX overX , that�(X, N) D 1. WhenN is etale, the finiteness theorem inetale cohomology (Milne 1980, VI 2.1) shows thatH r (X , N) is finite, and a duality theorem (see~11) shows that the same is true whenN isthe dual of anetale group. Otherwise the groups are often infinite. For example,

H 1(X , p) D Ker(F WH 1(X ,OX )! H 1(X ,OX )),

which is finite if and only if the curveX has an invertible Hasse-Witt matrix. Nevertheless,�(X, p) D 1.

Let N D NLa,0 in the Oort-Tate classification. Then (cf. 5.4), we have an exact sequence

0! N ! L'�! L˝p

! 0

with '(z) D z˝p� a ˝ z. Therefore�(X, N) D q�(L)��(L˝p) whereq is the order ofk.

But the Riemann-Roch theorem shows that

�(L) D deg(L)C 1� g

�(L˝p) D p deg(L)C 1� g,

and so�(X, N) D p(p�1) deg(L).

It is easy to constructN for which deg(L) 6D 0: takeL0 to be any invertible sheaf of degree> 0; then for somer > 0, � (X,L˝r(p�1)

0 ) 6D 0, and so we can takeN D NLa,0 with

L D L˝r0 anda any element of� (X,L˝(p�1)).

PROBLEM 8.11. As we mentioned above, the groupsH rc (U, N) have canonical compact

topologies. Is it possible extend the above discussion to�c(U, N) by using Haar measures?

REMARK 8.12. We show that, for any schemeY proper and smooth over a finite fieldk ofcharacteristicp, the groupsH r (Y,�p) are finite for allr and

�(Y,�p)dfD [H r (Y,�p)](�1)r

D 1.

From the exact sequence (special case of (5.6))

0! R1f��p ! ˝1Y =k,cl

C �1�! ˝1

Y =k ! 0

we see that it suffices to show that the groupsH r (Y,˝1Y =k,cl

) andH r (Y,˝1Y =k

) are allfinite and that�(Y,˝1

Y =k,cl) D �(Y,˝1

Y =k). Consider the exact sequences of sheaves on

Yet,

0! OY

F�! OY ! dOY ! 0

9. GLOBAL RESULTS: CURVES OVER FINITE FIELDS, NERON MODELS 253

and0! dOY ! ˝1

Y =k,cl

C�! ˝1

Y =k ! 0.

From the cohomology sequence of the first sequence, we find thatH r (Yet, dOY ) is finitefor all r and is zero forr > dim(Y ); moreover�(Y, dOY ) D 1. From the cohomologysequence of the second sequence, we find thatH r (Yet,˝

1Y =k,cl

) is also finite for allr andzero for r > dim(Y ), and that�(Y,˝1

Y =k,cl) D �(Y,˝1

Y =k), which is what we had to

prove.

REMARK 8.13. It has been conjectured that for any schemeY proper over SpecZ, thecohomological Brauer groupH 2(Y, Gm) is finite. The last remark shows that when theimage of the structure map ofY is a single point(p) in SpecZ, thenH r (Y, Gm)p is finitefor all r .

NOTES. In the very special case thatU D X andN is constant without local-local factors,Theorem 8.2 can be found in Milne 1977, Thm A 2, with a similar proof. The caseU D X

and a generalN is implicitly contained in Artin and Milne 1975.

9 Global results: curves over finite fields, Neron models

The notations are the same as those in the last section. In particular,X is a complete smoothcurve over a finite field,U is an open subscheme ofX , andH r

c (U, F) is defined so thatthe sequence

� � � ! H rc (U, F)! H r (U, F)!

Mv =2U

H r (Kv, F )! � � �

is exact withKv thecompletionof K.Let A be an abelian variety overK, and letA andB be the Neron models overU of

A and its dualB. If either A has semistable reduction orn is prime top, there are exactsequences

0! Bn! B! Bn˚ 0

! 0

0! An! A˚n ! Aı! 0.

The Poincare biextension of(B, A) by Gm extends uniquely to biextensions of(B,Aı) byGm and of(Bn˚ 0

,A˚n) by Gm. Therefore (cf.~1), we get a canonical pairing

Bn �An! Gm

in this case.

PROPOSITION9.1. Assume thatn is prime top or thatA has semistable reduction at allprimes ofK. Then the pairing

H r (U,Bn)�H 3�rc (U,An)! H 3

c (U, Gm) �D Q=Z

induces an isomorphismH 3�rc (U,An)! H r (U,Bn)� for all r .


PROOF. Let V be an open subset whereA (hence alsoB) has good reduction. OverV ,An

is a finite flat group scheme with Cartier dualBn, and so the proposition is a special case of(8.2). To pass fromV to U , we use the diagram

� � � ��! H 3�rc (V ,An) ��! H 3�r

c (U,An) ��!L

v2U rV

H 3�r (bOv,An) ! � � �??y�

??y ??y� � � ��! H r (V ,Bn)� ��! H r (U,Bn)� ��!

Lv2U rV

H rv (Oh

v ,Bn)� ��! � � � .

ObviouslyH rv (Oh

v ,Bn)��! H r

v (bOv,Bn), and Corollary 7.16 shows thatH 3�r (bOv,An)!

H rv (bOv,Bn)� is an isomorphism. Therefore the proposition follows from the five-lemma.

As usual, we writeA=Aı

D ˚ D ˚iv�˚v (finite sum).

PROPOSITION9.2. There are exact sequences

0! Aı(X)! A(K)!˚˚v(k(v))! H 1(X,Aı)!X(K, A)! 0

and0!X(K, A)! H 1(X,A)!˚H 1(v,˚v).

PROOF. Let U be an open subscheme ofX such thatAjU is an abelian scheme; in partic-ular,AjU D Aı

jU . As in (II 5.5), we have an exact sequence

0! H 1(U,A)! H 1(K, A)!Yv2U

H 1(Kv, A).

There is an exact sequence

0 �!Aı(X) �!A(U ) �!L

v2X rU

H 1v (Oh

v ,Aı) �!H 1(X,Aı) �!H 1(U,A) �!L

v2X rU

H 2v (Oh

v ,Aı) A(K)

Lv2X rU

˚v(k(v))L

v2X rU

H 1( QKv, A)I

(for the second two equalities, see (7.10)). According to (I 3.10), the field of fractionsQKv

of Ohv can be replaced byKv in H 1( QKv, A). The kernel-cokernel exact sequence of

H 1(K, A)!Yv2X

H 1(Kv, A)!Yv2U

H 1(Kv, A)


0!X(K, A)! H 1(U,A)!Y

v2X rU

H 1(Kv, A),


and it follows from this and the six-term sequence thatX(K, A) is the image ofH 1(X,Aı)

in H 1(U,A). The first exact sequence can now be obtained by truncating the six-term exactsequence, and the second sequence can be obtained by comparing the last sequence abovewith

0! H 1(X, A)! H 1(U,A)!Y

v2X rU

H 2v (Oh

v ,A).

COROLLARY 9.3. For any� � ˚ , H 1(X, A� ) is torsion and of cofinite type.

PROOF. It suffices to prove this with� D ;. Then the group equalsX(K, A), whichis obviously torsion. It remains to show thatX(K, A)p is finite. There is an elementaryproof of this in Milne 1970b. It can also be proved by using (8.12) in the case of a surfaceto show thatX(K, A)p is finite whenA is a Jacobian variety, and then embedding anarbitrary abelian variety into a Jacobian to deduce the general case.

Let B=B0 Ddf ˚0D ˚iv�˚

0v. For any subgroups� D ˚iv��v and� 0

D ˚iv��0v of

˚ and˚ 0, the Poincare biextension overK extends to a biextension overU if and only ifeach�v annihilates each� 0

v . In this case we get a map

B� 0

˝L A� ! Gm[1].

THEOREM 9.4. Suppose that�v and� 0v are exact annihilators at each closed pointv.

(a) The the kernels of the pairing

H 1(U,B� 0

)�H 1c (U,A� )tors! Q=Z

are exactly the divisible groups.(b) If X1(K, A) is finite, thenH 0(U,A� )^ is dual toH 2

c (U,B� 0

).

PROOF. If A has good reduction onU , this can be proved by the same argument as in (II5.2) (using 8.2). This remark shows that the theorem is true for someV � U , and to passfrom V to U one uses (7.13).

COROLLARY 9.5. The Cassels-Tate pairing (II 5.7a)

X1(K, B)�X1(K, A)! Q=Z

annihilates only the divisible subgroups.

PROOF. This follows from (9.4) and the diagramL˚ 0v(k(v)) ��! H 1(X,Bı) ��! X1(K, B) ��! 0??y ??y ??yL

H 1(v,˚v)� ��! H 1(X,A)� ��! X1(K, A)� ��! 0

because (7.11) shows that the first vertical map is an isomorphism.


Application to the conjecture (B-S/D) for Jacobians

Recall that the index of a curveC over a fieldF is the greatest common divisor of thedegrees of the fieldsF 0 overF such thatC has a rational point inF 0. Equivalently, it isthe least positive degree of a divisor onC .

In this subsection, we letY be a regular connected surface overk, and we let� WY ! X

be a proper morphism such that(i) the generic fibre� is a smooth geometrically connected curve overKI

(ii) for all v 2 X , the curveYKvhas index one.

We writeA for the Jacobian of the generic fibre of� .

PROPOSITION9.6. (a) The orders of the Brauer group ofY and the Tate-Shafarevich groupof A are related by

ı2[Br(Y )] D [X(K, A)]

whereı is the index ofYK .(b) The conjecture of Artin and Tate (Tate 1965/66, Conjecture C) holds forY if and

only if the conjecture of Birch and Swinnerton-Dyer (I 7, B-S/D) holds forA.

PROOF. (a) Once (9.5) is acquired, the proof in Milne 1981 applies.(b) It is proved in Gordon 1979 that (a) implies (b).

COROLLARY 9.7. Let A be a Jacobian variety overK arising as above. The followingstatements are equivalent:

(i) for some prime (` D p is allowed), the -primary component ofX(K, A) is finite;(ii) the L-seriesL(s, A) of A has a zero ats D 1 of order equal to the rank ofA(K);(iii) the Tate-Shafarevich groupX(K, A) is finite, and the conjecture of Birch and

Swinnerton-Dyer is true for A.

PROOF. After part (b) of the theorem, the equivalence of the three statements (i), (ii),and (iii) for A follows from the equivalence of the corresponding statements forY (Milne1975).8

The behaviour of conjecture (B-S/D) with respect top-isogenies.

We partially extend Theorem I 7.3 to the case ofp-isogenies.

THEOREM 9.8. Let f WA ! B be an isogeny of abelian varieties overK, and letN bethe kernel of its extensionf WA ! B to the Neron models ofA and B over X . Assumethat either the degree off is prime top or that A and B have semistable reduction atall points ofX and H r (X ˝ ks, N) is finite for all r . Then the conjecture of Birch andSwinnerton-Dyer is true forA if and only if it is true forB.

8In Milne 1975, it is assumed thatp is odd, but this condition is used only in the proof of Theorem 2.1 ofthe paper. The proof of that theorem uses my flat duality theorem for a surface, which in turn uses Bloch’spaper (listed as a preprint), which assumes p odd. Illusie (Ann. Sci.Ecole Norm. Sup. (4) 12 (1979), no. 4,501–661) does not require that p be odd, so if you replace the reference to Bloch by a reference to Illusie youcan drop the condition from my duality paper (Ann. Sci. Ecole Norm. Sup. 9 (1976), 171-202), and hencefrom my 1975 paper.


PROOF. After (I 7.3), we may assume that the degree off is a power ofp. The initialcalculations in (I 7) show that in order to prove the equivalence, one must show that

z(f (K)) D

Yv2X

�v(A,!A)

�v(B,!B)

!� z(f t(K)) � z(X1(f )).

(Note that we can not replace the local terms withz(f (Kv)) because the cokernel off (Kv)

need not be finite.) LetN ıD Ker(f ı

WAı! Bı). From the exact sequence

0! N ı! Aı

! Bı! 0

we get an exact sequence

0! N ı(X)! Aı(X)! Bı(X)! H 1(X, N ı)! H 1(X,Aı)! H 1(X,Bı)

! H 2(X, N ı)! H 2(X,Aı)! H 2(X,Bı)! H 3(X, N ı)! 0.

The sequence shows that

z(H 0(f ı)) D z(H 1(f ı)) � z(H 2(f ı))�1� �(X, N ı).

But �(X, N 0) D 1 (because we have assumed that the groupsH r (X ˝ ks, N) are finite;cf. the discussion following (8.10)), and (9.3) shows thatz(H 2(f ı))�1 D z(H 0(f t)).Therefore it remains to show that

z(f (K))

z(H 0(f ı))D

z(X(f ))

z(H 1(f ı))

Yv2X

�v(A,!A)

�v(B,!B)

!.

From (9.2), we get a diagram

0 ��! Aı(X) ��! A(K) ��!L˚v(k(v)) ��! H 1(X,Aı) ��! X(K, A) ��! 0??y ??y ??y ??y ??y

0 ��! Bı(X) ��! B(K) ��!L˚ 0v(k(v)) ��! H 1(X,Bı) ��! X(K, B) ��! 0

which shows thatz(f (K))

z(H 0(f ı))D

z(X(f ))

z(H 1(f ı))�

Yv

[˚v(k(v))]

[˚ 0v(k(v))]

.

It remains to show that�v(A,!A)

�v(B,!B)D

[˚v(k(v))]

[˚ 0v(k(v))]

,

but this follows from the formula

�v(A,!v) D [˚v(k(v))]=Lv(1, A),

!v a Neron differential onA˝Kv, for which we have no reference to offer the reader.


REMARK 9.9. It was first pointed out in Milne 1970, p296, that, because the groupH 1(X˝

ks, N) may be infinite,X1(K, A)p may be infinite whenK is a function field with alge-braically closed field of constants. This phenomenon has been studied in the papers Vve-dens’kii 1979a, 1979b, 1980/81, which give criteria for the finitenessXp (and hence ofthe groupsH r (X ˝ ks, N) in the above theorem).

PROBLEM 9.10. Prove the above result for every isogenyf WA! B.

There seems to be some hope that the method used in (II 5) may be effective in thegeneral case; the groups are no longer finite, but they are compact.

Duality for surfaces

It is possible to prove a similar result to (3.8) (see also Artin and Milne 1976).

10 Local results: equicharacteristic, perfect residue field

Throughout this section,X D SpecR whereR is a complete discrete valuation ring withalgebraically closed residue fieldk. We letXi D SpecRi, whereRi D R=miC1.


In the equicharacteristic case, the Greenberg construction becomes a special case of Weilrestriction of scalars: for eachi � 0, thek-algebra structure onRi defines a map i WXi !

Speck, and for any group schemeG overX , Gi(G) D ResXi=k G. We writeG(G) for thepro-algebraic group(Gi(G))i�0 onX .

LEMMA 10.1. (a) The functorG 7! G(G) from smooth group schemes onX to pro-algebraic groups onSpec(k) is exact.

(b) WhenG is smooth and has connected fibres,Gi(G) is smooth and connected for alli.

(c) WhenN is a finite flat group scheme of height one,Gi(N) is connected for alli.

PROOF. See Bester 1978, 1.1, 1.2.

Let�0 be the functor sending a pro-algebraic group scheme overk to its maximaletalequotient, and let�r be ther th left derived functor of�0. Write�r (G) for �r (G(G)). WhenN is a finite flat group schemeX , we choose a resolution ofN by smooth connected formalgroups

0! Ni ! Gi ! Hi ! 0, i � 0,

and defineFi(N) D Coker(�1(Gi)! �1(Hi)). It is an pro-etale group scheme onX , andwe letF(N) be the pro-etale group scheme(Fi(N))i�0.

Write˛i for the morphismXi ! Speck, andRrb� for the functorF 7! lim �

Rr˛i,�(F jXi).If G is a group scheme of finite-type overX , thenb�G D G(G), and if N is a finite flatgroup scheme overX , thenR1b�N is representable by a pro-algebraic group scheme overk.

10. LOCAL RESULTS: EQUICHARACTERISTIC, PERFECT RESIDUE FIELD 259

LEMMA 10.2. LetN be a finite flat group scheme overX .(a) The group schemeF(N) is independent of the choice of the resolution.(b) A short exact sequence

0! N 0! N ! N 00

! 0

of finite group schemes defines an exact sequence

0! �1(N 0)! �1(N)! �1(N 00)! F(N 0)! F(N)! F(N 00)! 0.

(c) There is an exact sequence

0! �0(N)! F(N)! �1(R1b�N)! 0

of pro-sheaves onX .

PROOF. See Bester 1978, 3.2, 3.7, 3.9.

Write�1 for the direct system of finite group schemes(�pn).

LEMMA 10.3. There is a canonical isomorphismH 2x (X,F(�1))

�D�! Qp=Zp.

PROOF. From the Kummer sequence

0! �pn ! Gm

pn

�! Gm! 0


0! �1(Gm,X )pn

�! �1(Gm,X )! F(�pn)! 0.

This yields a cohomology sequence

0! H 2Z (X,�1(Gm,X ))! H 2

Z (X,�1(Gm,X ))! H 2Z (F(�pn))! 0.

But the higher cohomology groups of the universal covering group ofG(Gm,X ) are zero,and so

H 2x (X,�1(Gm,X )) D H 1

x (X, Gm,X ) D K�=R� �D ZI

thusH 2Z (F(�pn)) �D Z=pnZ.

Assume thatN is killed bypn. Then the pairing

N D�N ! �pn

induces a pairingN D� F(N)! F(�pn),

and hence a pairing

H 2x (X, N D)� F(N)! H 2

x (X,F(�pn)) �D Z=pnZ.


THEOREM 10.4. The above pairing defines an isomorphism

H 2x (X, N D)! Homk(F(N), Z=pnZ).

PROOF. If N (or its dual) has height one, this can be proved using the exact sequence in~5. The general case follows by induction on the length ofNK . See Bester 1978, 2.6.

As in ~4, we can endowH 1(X, N) andH 1(K, N) with the structures of perfect pro-algebraic group schemes overk. We write H1(X, N) and H1(K, N) for these groupschemes. Note thatH1(X, N) is the perfect group scheme associated withR1b�N . For anyfinite group schemeN overX , the mapH1(X, N) ,! H1(K, N) is a closed immersion,and we writeH2

x(X, N) for the quotient group.

THEOREM 10.5. For any finite flatp-primary group scheme overX , there is a canonicalisomorphism

H1(X, N)�D�! (H2

x(X, N D)ı)t .

PROOF. This follows from the commutative diagram (see 10.2c)

0 �! Homk(�1(H 1(X, N)), Q=Z) �! Homk(F(N), Q=Z) �! Homk(�0(N), Q=Z) �! 0??y ??y�

??y0 �! H2

x(X, N D)ı�! H2

x(X, N D) �! �0(H2x(X, N D)) �! 0,

and the isomorphism

Homk(�1(H1(X, N)), Qp=Zp))�D�! Ext1k(H1(X, N), Qp=Zp).

REMARK 10.6. The above proof shows that the dual of the continuous partH2x(X, N)ı of

H2x(X, N) is H1(X, N) and the dual of its finite part�0(H2

x(X, N)) is the finite part�0(N)

of H0(X, N). Write Hx(X, N) andH(X, N) for the canonical objects in the derived cat-egory such thatH r (Hx(X, N)) D Hr

x(X, N) andH r (H(X, N)) D Hr (X, N). Then thecorrect way to state the above results is that there is a canonical isomorphism

Hx(X, N D)�D�! H(X, N)t [2]

where thet denotes Breen-Serre dual (0.14).

Abelian varieties

Let A be an abelian variety overK, and letA be its Neron model overX . We write�r (A)

for �r (A) andG(A) for G(A).

CONJECTURE10.7. There is a canonical isomorphism

H 1(K, At)! Ext1k(G(A), Q=Z).

In particular, ifAk is connected, thenH 1(K, At)�D�! Homk(�1(A), Q=Z).

11. GLOBAL RESULTS: CURVES OVER PERFECT FIELDS 261

The second part of the statement follows from the first, as in (4.16). For the componentsof the groups prime top, the conjecture is proved in Ogg 1962 and Shafarevich 1962.

In the case thatA has good reduction,F(Apn) D �1(A)(pn) and H 2x (X, At)pn �

H 2x (X, At

pn) and so the conjecture can be obtained by passing to the limit in (10.4). SeeBester 1978, 7.1.

One can also show by a similar argument to that in (7.9) that it suffices to prove theresult after passing to a finite separable extension ofK.

Finally, one can show that the result is true ifA is an elliptic curve with a Tate parametriza-tion (cf. Shatz 1967). In this case there is an exact sequence

0! Zn7 !qn

��! L�! A(L)! 0

for all fieldsL finite overK. Therefore

H 1(K, A)��! H 2(K, Z)

��! Homcts(Gal(Ks=K), Q=Z).

On the other hand,A(K) D R� � (Z=ord(q)), and so

Ext1k(G(A), Qp=Zp)��! Ext1k(G(Gm,R), Qp=Zp).

As Ext1k(G(Gm,R), Qp=Zp) D Homk(�1(Gm,R), Qp=Zp) (see Serre 1960, 5.4), the dualityin this case follows from the class field theory of Serre 1961.

It is to be hoped that the general case can be proved by the methods of~7.

REMARK 10.8. The discussion in this section holds with only minor changes when theresidue fieldk is an arbitrary perfect field.

NOTES. This section is based on Bester 1978. Some partial results in the same directionwere obtained earlier by Vvedens’kii (1973, 1976, and earlier papers).

11 Global results: curves over perfect fields

Throughout this section,S D Speck with k a perfect field of characteristicp 6D 0, and� WX ! S is a complete smooth curve overS . Again we defineH r

c (U, F) so that thesequence

� � � ! H rc (U, F)! H r (U, F)!

Lv =2U H r (Kv, F )! � � �

is exact withKv the completion ofK atv.Let N be a finite flat group scheme overU � X , and writeRr��N andRr�!N for the

sheaves on the perfect siteSpf associated withS 07! H r (US 0 , N) andS 0

7! H rc (US 0 , N).

THEOREM 11.1. (a) The sheavesRr��N andRr�!N are representable by perfect groupschemes onS .

(b) The canonical pairing

R��N D�R�!N ! R�!Gm

�D Q=Z[�2]

induces an isomorphism

R��N D! RHomS(R�!N, Q=Z[�2]).


We begin the proof with the case thatX D U .

LEMMA 11.2. The theorem is true ifU D X andN or its dual have height one.

PROOF. Assume first thatN has height one. The first exact sequence in~5 yields an exactsequence

� � � ! Rr��N D! Rr��!N ! Rr��!

(p)N ! � � �

and the second a sequence

� � � ! RrC1��N ! Rr��(n˝˝1X 0)! Rr��(n˝˝1

X )! � � � .

Since two out of three terms in these sequences are vector groups, it is clear thatRr��N

andRr��N D are represented by perfect algebraic groups. The usual duality theorem forcoherent sheaves on a curve show that thek-vector spacesH r (X,!N ) andH r (X,!

(p)N )

are thek-linear duals (hence Breen-Serre duals) of thek-vector spacesH 1�r (X, n˝˝1X )

andH 1�r (X, n ˝ ˝1X 0). The pairing (5.10) induces an isomorphismR��V �(N D)

��!

R��U �(N)t [1]. SinceR��N D ��! R��V �(N D) andR��N

��! R��U �(N)[�1], this

(together with (5.11)) shows thatR��N D ��! (R��N)t [2], as required. (For more details,

see Artin and Milne 1976.)


! N ! N 00! 0

be an exact sequence of finite flat group schemes onU . If the theorem is true forN 0 andN 00, then it is true forN .

PROOF. This is obvious from (0.14).

LEMMA 11.4. Let V be an open subscheme ofU . The theorem is true forN on U if andonly if it is true forN jV onV .

PROOF. This follows from the distinguished trianglesMv2U rV

Hv(bOv, N D)! R��(N D)! R��(N jV )D!

Mv2U rV

Hv(bOv, N D)[1]

R�!(N jV )! R�!N !M

v2U rV

H(bOv, N)! R�!(N jV )[1],

and (10.6).

We now prove the theorem. LetN be a finite group scheme overU . After replacingU by a smaller open subset we can assume thatN has a composition series all of whosequotients have height1 or are the Cartier duals of groups of height1. Now Lemma 11.3shows that we can assume thatN (or its dual) has height1. According to Appendix B,NK extends to a finite flat group schemeN on X which is of height one (or has a dual ofheight one). After again replacingU by a smaller open set, we can assume thatN jU D N .According to (11.2), the theorem is true forN onX , and (11.4) shows that this implies thesame result forN jU D N .

11. GLOBAL RESULTS: CURVES OVER PERFECT FIELDS 263

Neron models

We now assume the ground fieldk to be algebraically closed. LetA be an abelian varietyoverK, and letA be its Neron model overX .

LEMMA 11.5. The restriction mapH 1(X,A) ! H 1(K, A) identifiesH 1(X,A) withX1(K, A).

PROOF. The argument in the proof of Proposition (9.2) shows again that there is an exactsequence

0!X1(K, A)! H 1(X,A)!M

H 1(v,˚v),

but in the present case, the final term is zero.

In the proof of the next theorem,we shall use without proof thatR2��A has no con-nected part.The argument that the tangent space toR2��A should equalR2��(tangentsheaf toA), which is zero becauseX is a curve, makes this plausible. This assumption isnot needed ifA has good reduction everywhere.

THEOREM 11.6. There is an exact sequence

0! H 1(X,A)! H 1(K, A)!Mall v

H 1(Kv, A)! (TpAt(K))�

with At the dual abelian variety toA; in particular, if A has no constant part (that is, theK=k-trace ofA is zero), thenH 1(K, A)!

Lall v H 1(Kv, A) is surjective.

PROOF. Let U be an open subscheme ofX . The cohomology sequence of the pairX � U

0! H 1(X,A)! H 1(U,A)!Qv =2U

H 2v (Ov,A)! H 2(X,A)! � � �

can be rewritten as

0!X(K, A)! H 1(U,A)!Qv =2U

H 1(Kv, A)! H 2(X,A)! � � � .

Because the residue fields at closed points are algebraically closed, for any openV � X ,

H 2c (V ,A)

��! H 2(X,A). Choose aV such thatAjV is an abelian scheme. There is an

exact sequence

0! H 1c (V ,A)˝Qp=Zp ! H 2

c (V ,A(p))! H 2c (V ,A)(p)! 0.

InterpreteH rc (V ,�) asRr (�jV )!. Then Theorem 11.1 shows that�0(H 2

c (U,A(p))) isdual toH 0(U, TpAt) D Tp(At(K)) (andH 2

c (U,A(p))ı is dual toH 1(U, TpA)ı). Fromour assumption, the mapH 2

c (V ,A(p))! H 2(X,A)(p) factors through�0(H 2c (U,A(p))),

and so we can replaceH 2(X,A)(p) in the sequence withTp(At(K))�. Now pass to thedirect limit over smaller open setsU .

If A has no constant part, thenAt(K) is finitely generated by the generalized Mordell-Weil theorem Lang 1983, Chapter 6, and soTp(At(K)) D 0.


REMARK 11.7. The last theorem is useful in the classification of elliptic surfaces withgiven generic fibre. See Cossec and Dolgachev 1986, Chapter 5.

PROBLEM 11.8. Extend as many as possible of the results in Raynaud 1964/5, II, to thep-part.

NOTES. In the caseU D X , Theorem 11.1 is in Artin and Milne 1976.

Appendix A: Embedding finite group schemes

An embeddingof one group scheme into a second is a map that is both a homomorphismand a closed immersion.

THEOREMA.1. LetR be a local Noetherian ring with perfect residue fieldk, and letN bea finite flat group scheme overSpecR. Writem for the maximal ideal ofR, Ri for R=miC1,and Ni for N ˝R Ri. Then there exists a family of embeddings'i WNi ,! Ai such thatAi is an abelian scheme overRi and'iC1 ˝ Ri D 'i for all i. Consequently there is anembedding of the formal completionbN of N into a formal abelian schemeA overSpf(bR).

PROOF. To deduce the last sentence from the preceding statement, note that the family(Ai) defines a formal schemeA over Spf(bR) and that the'i define an embedding ofbNintoA (see Grothendieck and Dieudonne 1971,~10.6).

We first prove the theorem in the case thatR D k is an algebraically closed field ofcharacteristic exponentp. The only simple finite group schemes overk areZ=`Z (` 6D p),Z=pZ, �p, and p. The first three of these can be embedded in any nonsupersingularelliptic curve overk, and the last can be embedded in any supersingular elliptic curve. Weproceed by induction on the order ofN . Consider an exact sequence

0! N 0! N ! N 00

! 0

in which N 0 and N 00 can be embedded into abelian varietiesA0 and A00. Let e be theclass of this extension in Ext1

k(N 00, N 0), and lete0 be the image ofe in Ext1k(N 00, A0). AsExt2k(A00=N 00, A0) D 0 (see Milne 1970a, Thm 2),e lifts to an elementQe of Ext1k(A00, A0),andN embeds into the middle term of any representative ofQe:

0 ��! N 0��! N ��! N 00

��! 0 (D e)??y ??y 0 ��! A0

��! X ��! N 00��! 0 (D e0) ??y ??y

0 ��! A0 ��! A ��! A00 ��! 0 (D Qe).

We next consider the case thatR D k is a perfect field. The first step implies thatN

embeds into an abelian variety over a finite extensionk 0 of k, Nk0 ,! A. Now we can formthe restriction of scalars (Demazure and Gabriel 1970, I,~1, 6.6), of this map and obtain an

APPENDIX A: EMBEDDING FINITE GROUP SCHEMES 265

embeddingN ,!Resk0=k Nk0 ,! Resk0=k A. The fact thatk 0=k is separable implies thatResk0=k A is again an abelian variety, because

(Resk0=k A)˝k ksD Resk0˝kks=ks A D A

[k0Wk]

ks .

To complete the proof, we prove the following statement: letR be a local Artin ringwith perfect residue fieldk D R=m; let I be an ideal inR such thatmI D 0, and letR D R=I ; let N be a finite flat group scheme overR, and let'WN ˝R R ,! A be anembedding ofN ˝R R into an abelian scheme overR; then' lifts to a similar embedding' overR.

LEMMA A.2. Let X be a smooth scheme overR, and letL(X) be the set of isomorphismclasses of pairs(Y, ) whereY is a smooth scheme overR and is an isomorphism

Y ˝R��! X . LetT X0

be the tangent sheaf onX0 Ddf X ˝R k.(a) The obstruction to liftingX to R is an element 2 H 2(X0, T X0

)˝k I .(b) When nonempty,L(X) is a principal homogeneous space forH 1(X0, T X0

)˝k I .(c) If X is an abelian scheme overR, then˛ D 0.

PROOF. See Grothendieck 1971, III 6.3, and Oort 1971.

The lemma shows that there is an(A, ) 2 L(A). AsA is smooth, the zero section ofA overR lifts to a section ofA overR. Now the rigidity of abelian schemes (Mumford1965, 6.15) implies that the group structure onA lifts to a group structure onA, and itfollows thatA is an abelian scheme. (See also Messing 1972, IV 2.8.1.) It remains to showthat' lifts to an embedding' of N intoA (after possibly changing the choice ofA).

LEMMA A.3. There is an exact sequence

HomR(N,A)! HomR(N ,A)! Ext1k(N0, T0(A0)˝k I)

whereT0(A0) is the tangent space at zero toA0 and we have used the same notation forthe vector spaceT0(A0)˝k I and the vector group it defines.

PROOF. In disagreement with the rest of the book, we shall writeYFl for the big flat site onY (category of all schemes locally of finite-type overY with the flat topology) andYfl forthe small flat site (category of all schemes locally of finite-type andflat overY with the flattopology). There is a well known short exact sequence

0! T0(A0)˝k I ! A(R)! A(R)! 0.

A similar sequence exists withR replaced by any flatR-algebra, and so, if we writei and{

for the closed immersions Speck ,! SpecR and SpecR ,! SpecR, then there is an exactsequence

0! i�(T0(A0)˝k I)! A! {�A! 0

of sheaves on(SpecR)fl. This yields an exact sequence

HomR,fl(N,A)! HomR,fl(N, {�A)! Ext1R,fl(N, i�(T0(A0)˝k I))


where the groups are computed in(SpecR)fl. Note that HomR,fl(N, {�A) D HomR,fl({�N,A)

and that (becauseN , N ,A andA are all in the underlying categories of the sites)

HomR,fl(N,A) D HomR(N,A), HomR ,fl({�N,A) D HomR(N ,A)

(the right hand groups are the groups of homomorphisms in the category of group schemes).Let f W (SpecR)Fl ! (SpecR)fl be the morphism of sites defined by the identity map.Thenf� is exact and preserves injectives, and so Extr

R,Fl(f�F, F 0) D ExtrR,fl(F,f�F 0) for

any sheavesF on (SpecR)fl andF 0 on (SpecR)Fl . In our casef �N D N (see Milne1980, II 3.1d), and so we can replace Ext1

R,fl(N, i�(T0(A0) ˝k I)) in the above sequencewith the same group computed in the big flat site onR. Next Rr i�(T0(A0) ˝k I) D

0 for r > 0, becauseT0(A0) ˝k I is the sheaf defined by a coherent module and soH r (VFl, T0(A0) ˝k I) D H r (VZar, T0(A0) ˝k I) D 0 for r > 0 whenV is an affinek-scheme. Therefore

ExtrR,Fl(N, i�(T0(A0)˝k I)) D Extrk,Fl(i�N, T0(A0)˝k I)

for all r . Finally Ext1k,Fl(N0, T0(A0)˝k I) D Ext1k(N0, T0(A0)˝k I).We know thatTA0

is the free sheafT0(A0) ˝k OA0. On tensoring the isomorphism

Ext1k(A0, Ga)��! H 1(A0,OA0

) (Serre 1959, VII 17) withT0(A0) ˝k I we get an iso-morphism,

Ext1k(A0, T0(A0)˝k I)��! H 1(A0, TA0

)˝k I

The inclusionN0 ,! A0 defines a map

Ext1k(A0, T0(A0)˝k I)! Ext1k(N0, T0(A0)˝k I),

which is surjective because Ext2k(�, Ga) D 0; see Oort 1966, p.II 14-2, and (I 0.17). Con-

siderH 1(A0, T A0

)˝k I??ysurj

HomR(N,A) ��! HomR(N ,A) ��! Ext1k(N0, T0(A0)˝k I).

It is clear from this diagram and Lemma A.2b that if'WN ! A does not lift to a map'from N toA, then a different choice ofA can be made so that' does lift. The lifted map' is automatically an embedding.

COROLLARY A.4. In addition to the hypotheses of the theorem, assume thatR is completeand thatN has order a power ofp. ThenN can be embedded in ap-divisible groupschemeH overR.

PROOF. Take thep-divisible group scheme associated with the formal abelian schemeA.

We next consider the problem of resolving a finite flat group scheme by smooth groupschemes. LetN be a finite flat group scheme over a Noetherian schemeS . Then thefunctorMorS(N, Gm) is representable by ResN =S Gm, which is a smooth affine group

APPENDIX A: EMBEDDING FINITE GROUP SCHEMES 267

scheme of finite type overS . Note thatN D is (in an obvious way) a closed subgroup ofMorS(N, Gm). Write N i for N �S ...�S N , and letMorS(N 2, Gm)sym be the kernel ofthe map

MorS(N 2, Gm)!MorS(N 2, Gm)

sendingf to the functionfsym, wherefsym(x, y) D f (y, x)f (x, y)�1. Finally, letZ2(N, Gm)sym

be the kernel of the boundary map

d WMorS(N 2, Gm)sym!MorS(N 3, Gm),

df (x, y, z) D f (y, z)f (xy, z)�1f (x, yz)f (x, y)�1. The image of the boundary map

d WMorS(N, Gm)!MorS(N 2, Gm), df (x, y) D f (xy)f (x)�1f (y)�1,

is contained inZ2(N, Gm)sym.

THEOREM A.5. The sequence

0! N D!MorS(N, Gm)

d�! Z2

S(N, Gm)sym! 0

is an exact sequence of affine group schemes onS . The final two terms are smooth overS .

PROOF. See Begeuri 1980, 2.2.1.

The exact sequence in the theorem is called thecanonical smooth resolutionof N D .

THEOREM A.6. Let N be a finite flat group scheme over a Noetherian schemeS . Lo-cally for the Zariski topology onS , there is a projective abelian schemeA overS and anembeddingN ,! A.

PROOF. The idea of the proof is to construct (locally) a smooth curve� WX ! S overS and a principal homogeneous spaceY for N D over X . Such aY defines an elementof R1��N D , and cup-product with this element defines a mapN D HomS(N D , Gm)!

R1��Gm D PicX=S whose image is in the abelian scheme Pic0X=S � PicX=S . One shows

that the map is a closed immersion. For the details, see Raynaud 1979 and Berthelot, Breen,and Messing 1982, 3.1.1.

REMARK A.7. It is possible to construct quotients by finite flat group schemes (see Dieudonne1965, p114). Therefore from (A.1), we get exact sequences

0! Ni ! Ai ! Bi ! 0

with Bi an abelian scheme, and from (A.4), we get an exact sequence

0! N ! H ! H 0! 0

with H 0 ap-divisible group overR. Finally, (A.6) shows that (locally)N fits into an exactsequence

0! N ! A! B! 0

with A andB projective abelian schemes.


REMARK A.8. LetS be the spectrum of a discrete valuation ringR with field of fractionsK, and letN be a quasi-finite flat separated group scheme overS . If the normalizationQN of N in NK is flat overR, thenN is a subgroup of an abelian schemeA (because it

is an open subgroup ofQN by Zariski’s main theorem, andQN is a closed subgroup of anabelian scheme). Conversely, ifN is a subgroup of an abelian schemeA overS , then itsnormalization is flat (becauseNK � (AK )n for somen, and the closure ofNK in An isflat, see Appendix B).

The quotientA=N is represented by an algebraic space (Artin 1969, 7.3), but it is notan abelian scheme unlessN is finite because it is not separated (the closure of the zerosection is QN=N).

NOTES. Theorem A.1 is due to Oort (1967), and Theorem A.6 is due to Raynaud. LemmaA.3 is an unpublished result of Tate; the above proof of it was suggested to me by Messing.

Appendix B: Extending finite group schemes

Let R be a discrete valuation ring with field of fractionsK. In Raynaud 1974, p271, itis asserted that, whenK has characteristicp, every finite group scheme overK killed bya power ofp extends to a finite group scheme overR (the statement is credited to Artinand Mazur). Our first proposition provides a counterexample to this assertion. Then weinvestigate some cases where the group does extend. First we recall a well known lemma.

LEMMA B.1. LetR be a discrete valuation ring with field of fractionsK, and let

0! N 0! N ! N 00

! 0

be an exact sequence of finite group schemes overR. Assume thatN extends to a finite flatgroup schemeN overR. Then there exists a unique exact sequence

0! N 0! N ! N 00

! 0

of finite flat group schemes overR having the original sequence as its generic fibre.

PROOF. The groupN 0 is the closure ofN 0 in N , andN 00 is the quotient ofN by N 0.(Alternatively,N 0 is such that� (N 0,ON 0) is the image of� (N ,ON ) in � (N 0,ON 0), andN 00 is such that� (N 00,ON 00) D � (N ,ON ) \ � (N 00,ON 00).)

Now let R be a discrete valuation ring of characteristicp 6D 0. Consider an extensionof Z=pZ by�p overKW

0! �p ! N ! Z=pZ! 0.

Such extensions are classified by Ext1K (Z=pZ,�p), and the following diagram shows that

APPENDIX B: EXTENDING FINITE GROUP SCHEMES 269

this group is isomorphic toK�=K�p:

HomK (Z, Gm) K�??yp

HomK (Z, Gm) K�??yHomK (Z=pZ, Gm) ��! Ext1K (Z=pZ,�p)

��! Ext1K (Z=pZ, Gm) ��! 0 ??y

0 Ext1K (Z, Gm) 0

Let ˛(N) be the class of the extension inK�=K�p, and leta(N) be ord(˛(N)) regardedas an element ofZ=pZ.

PROPOSITIONB.2. The finite group schemeN extends to a finite flat group scheme overR if and only ifa(N) D 0. Therefore, there exists a finite group scheme overK killed byp2 that does not extend to a finite flat group scheme overR.

PROOF. Assume thatN extends to a finite flat group schemeN overR, and let

0! N 0! N ! N =N 0

! 0,

be the extension of the sequence given by Lemma B.1. The isomorphismZ=pZ��!

(N =N 0)K extends to a mapZ=pZ! N =N 0 overRW simply map each section ofZ=pZover R to the closure of its image under the first map. Similarly, the Cartier dual of the

isomorphismN 0K

��! �p extends to a mapZ=pZ! (N 0

K )D , and the dual of this map is amapN 0

! �p whose generic fibre is the original isomorphism. Now after pulling back byZ=pZ ! N =N 0 and pushing out byN 0

! �p, we obtain an extension ofZ=pZ by �p

overR whose generic fibre is exactly the original extension. Thus we see thatN extendsoverR if and only if the original extension ofZ=pZ by �p extends overR. The result isnow obvious from the commutative diagram,

Ext1K (Z=pZ,�p)�D��! Ext1K (Z=pZ, Gm)

�D��! H 1(K,�p)

�D��! K�=K�px?? x?? x?? x??

Ext1R(Z=pZ,�p)�D��! Ext1R(Z=pZ, Gm)

�D��! H 1(R,�p)

�D��! R�=R�p

becauseR�=R�p is the kernel ofK�=K�p ord�! Z=pZ.

REMARK B.3. The same argument shows that an extensionN of Z=pZ by Z=pZ need notextend to an extension ofZ=pZ by Z=pZ overR, because

Ext1K (Z=pZ, Z=pZ) D H 1(K, Z=pZ) �D K=}K,

Ext1R(Z=pZ, Z=pZ) D H 1(R, Z=pZ) �D R=}R.


However,N does extend to an extension ofZ=pZ by some finite flat group scheme overR. Indeed, in (7.5) we note that ifN 0

� Na,0 in the Oort-Tate classification (0.9) witha Dt c(p�1), thenH 1(R,N 0) is the image oft�cpR in K=}K, and so ifc is chosen sufficientlylarge, the class of the extension in Ext1

K (Z=pZ, Z=pZ) will lie in Ext1R(N 0, Z=pZ).

PROPOSITIONB.4. Let X be a regular quasi-projective scheme of characteristicp 6D 0

over a ringR, and letK be the field of rational functions onX . Any finite flat group schemeN of height one overK extends to a finite flat group scheme of height one overX .

PROOF. We first prove this in the case thatN has orderp. ThenN D N0,b for someb 2 K. We have to show that there exists an invertible sheafL on X , a trivialization

LK

��! K, and a global section of L˝1�p corresponding tob under the trivialization.

Let D be a Weil divisor such thatD � 0 and(b) � D, and letL D O(D). Then under theusual identification ofO(D)K with K,

� (X,L˝1�p) D � (X,O((1� p)D) D fg 2 K j(g) � (p � 1)Dg.

Clearlyb 2 � (X,L˝1�p).

We now consider the general case. Recall (Demazure and Gabriel (1970), II,~7) that a(commutative)p-Lie algebraV on a schemeY of characteristicp 6D 0 is a coherent sheafof OY -modules together with a map'WV ! V such that'(x C y) D '(x) C '(y) and'(ax) D ap'(x). With each locally freep-Lie algebraV there is a canonically associatedfinite flat group schemeN D G(V) of height� 1. Moreover, whenY is the spectrum ofa field, every finite group schemeN is of the formG(V) for somep-Lie algebraV. Notethat to give' is the same as to give anOY -linear mapV ! V(p).

PROOF. Thus let(V ,') be thep-Lie algebra associated withN overK. We have to showthat (V ,') extends to ap-Lie algebra overX . ExtendV in some trivial way to a locallyfree sheafV on X , and regard' as a linear mapV ! V (p). Then' is an element ofHomK (V , V (p)) and we would like to extend it to a section ofHomOX

(V ,V(p)). This willnot be possible in general, unless we first twist by an ample invertible sheaf. LetL be sucha sheaf onX , and writeV(r) for V ˝ L˝r . Then

HomOX(V(r),V(r)(p)) D HomOX

(V(r),V(p)(pr)) D HomOX(V ,V(p))(pr � p),

and so for a sufficiently highr , HomOX(V(r),V(r)(p)) will be generated by its global

sections (Hartshorne 1977, II 7). Therefore, we can write' DP˛i'i with ˛i 2 K and

'i 2 HomOX(V(r),V(r)(p)). Now choose a divisorD such that(˛i) � D for all i. Then

' is a global section of

HomOX(V(r)˝OX (D),V(r)˝OX (D)(p)) D HomOX

(V(r),V(r)(p))˝OX ((p�1)D).

COROLLARY B.5. Let X be a regular quasi-projective scheme of characteristicp 6D 0

over a ring R, and letK be the field of rational functions onX . Any finite flat groupschemeN whose Cartier dual is of height one overK extends to a finite flat group schemeoverX .

APPENDIX C: BIEXTENSIONS AND NERON MODELS 271

PROOF. Apply the proposition toN D , and take the Cartier dual of the resulting finite flatgroup scheme.

REMARK B.6. It is also possible to prove (B.5) directly from (5.5).

NOTES. The counterexample (B.2) was found by the author in 1977.

Appendix C: Biextensions and Neron models

Throughout,X will be a locally Noetherian scheme endowed with either the smooth or theflat topology.

Biextensions

Let A, B, andG be group schemes overX (commutative and of finite type as always). Abiextensionof (B, A) by G is a schemeW with a surjective morphism� WW ! B �X A

endowed with the following structure:(a) an actionW �B�A GB�A! W of GB�A onW makingW into aGB�A-torsor;(b) aB-morphismmBWW �B W ! W and a sectioneB of W overB makingW into a

commutative group scheme overB; and(c) anA-morphismmAWW �A W ! W and a sectioneA of W overA makingW into

a commutative group scheme overA.These structures are to satisfy the following conditions:

(i) if GB ! W is the mapg 7! eB � g, then

0! GB ! W��! AB ! 0

is an exact sequence of group schemes overBI

(ii) if GA! W is the mapg 7! eA � g, then

0! GA! W��! BA! 0

is an exact sequence of group schemes overA;(iii) the following diagram commutes

(W �A W )�B�B (W �A W )mA�mA

> W �B W

@@

@mB

R

jj W

��

��mA

(W �B W )�A�A (W �B W )mB�mB

> W �A W

(The “equality” at left is(w1,w2Iw3,w4) $ (w1,w3Iw2,w4)). See Grothendieck 1972,VII.


EXAMPLE C.1. LetA andB be abelian varieties of the same dimension over a field. Wecall an invertible sheafP on B � A a Poincare sheafif its restrictions tof0g � A andB�f0g are both trivial and if�(B�A,P) D ˙1. It is known (Mumford 1970,~13, p131),that then the map of functorsb 7! (b� 1)�P WB(T )! Pic(AT ) identifiesB with the dualabelian varietyAt of A. Moreover, for any abelian varietyA over a field, there exists anessentially unique pair(B,P) with P a Poincare sheaf onB �A (ibid. ~8, ~10-12).

Let P be a Poincare sheaf onB � A. With P, we can associate aGm-torsorW D

IsomB�A(OB�A,P) (less formally,W is the line bundle associated withP with the zerosection removed). For each pointb 2 B, Pb is a line bundle onA, andWb has a canonicalstructure of a group scheme overA such that

0! Gm! Wb ! A! 0

is an exact sequence of algebraic groups (Serre 1959, VII,~3). This construction can becarried out universally (onB), and gives a group structure toW regarded as anA-schemewhich is such that

0! GmB ! W ! AB ! 0

is an exact sequence of group schemes overB. By symmetry, we get a group structureon W regarded as a group scheme overA, and these two structures form a biextension of(B, A) by Gm. Any biextension arising in this way from a Poincare sheaf will be called aPoincare biextension.

WhenA, B, andG are sheaves onXfl (or Xsm), it is possible to modify the above defini-tion in an obvious fashion to obtain the notion of a biextension of(B, A) by G: it is a sheafof setsW with a surjective morphismW ! B �A having the structure of aGB�A-torsorand partial group structures satisfying the conditions (i), (ii), and (iii). WhenA, B, andG

are group schemes, we write BiextX (B, AIG) for the set of biextensions of(B, A) by G,and whenA, B, andG are sheaves, we write BiextXfl (B, AIG) (or BiextXsm(B, AIG)) forthe similar set of sheaves. Clearly, there is a map

BiextX (B, AIG)! BiextXfl (B, AIG)

and also BiextX (B, AIG)! BiextXsm(B, AIG) whenG is smooth overX).

PROPOSITIONC.2. LetA, B, andG be group schemes overX . If G is flat and affine overX , then the map

BiextX (B, AIG)! BiextXflB, AIG)

is bijective, and ifG is smooth and affine overX , then

BiextX (B, AIG)! BiextXsm(B, AIG)

is bijective.

PROOF. The essential point is that, in each case, torsors in the category of sheaves arerepresentable, and torsors in the category of schemes are locally trivial for the respectivetopologies (see Milne 1980, III 4.2 and 4.3).


Consider a biextension (of schemes)W of (B, A) by G. Given anX -schemeT and aT -valued pointt of B, we can pull-back the sequence in (i) and so obtain an extension

0! GT ! W (t)! AT ! 0.

This gives us a mapB(T )! Ext1T (AT , GT ), which can be shown to be a group homomor-phism. In this manner, a biextension of(B, A) by G defines homomorphisms of sheavesB ! Ext1

Xfl(A, G) andA ! Ext1

Xfl(B, G). A biextension of sheaves determines similar

maps. This has a pleasant restatement in terms of derived categories.

PROPOSITIONC.3. There is a canonical isomorphism

BiextXfl(B, AIG)��! HomXfl(B ˝

L A, G[1])

(and similarly for the smooth topology).

PROOF. See Grothendieck 1972, VII 3.6.5.

Given a biextension of(B, A) by G, we can define pairings

H r (X, B)�H s(X, A)! H rCsC1(X, G)

in three different ways: directly from the mapB ˝L A ! G[1], by using the mapB !Ext1(A, G), or by using the mapA! Ext1(B, G).

PROPOSITIONC.4. The three pairings are equal (up to sign).

BiextX (B, AIG)! BiextXsm(B, AIG)

PROOF. See Theorem 0.15.

Let W be a biextension of(B, A) by G. For any integern, the mapB ˝L A ! G[1]

defines in a canonical way a mapBn ˝An ! G (Grothendieck 1972, VIII 2). Up to sign,the following diagram commutes

0 ��! Bn ��! Bn��! B??y ??y ??y

Hom(An, G) ��! Ext1(A, G)n��! Ext1(A, G)

(and similarly withA andB interchanged). The pairingBn �An! G defines pairings ofcohomology groups.

COROLLARY C.5. The diagram

H r (X, B) �H s(X, A) > H rCsC1(X, G)

jj

H rC1(X, Bn)

_

�H s(X, An)

^

> H rCsC1(X, G)

commutes.

PROOF. This is obvious from the definitions.

WhenW is a Poincare biextension onB � A, the pairingBn � An ! Gm identifieseach group with the Cartier dual of the other. Forn prime to the characteristic, it agreeswith Weil’s en-pairing.


Neron models

From now onX is a Noetherian normal integral scheme of dimension one with perfectresidue fields. The fundamental theorem of Neron (1964) on the existence of canonicalmodels can be stated as follows.

THEOREM C.6. Let gW �! X be the inclusion of the generic point ofX into X . For anyabelian varietyA over�, g�A is represented onXsm by a smooth group schemeA.

PROOF. For a modern account of the proof, see Artin 1986.

The group schemeA is called theNeron modelof A. It is separated and of finitetype, andA� D A. It is obviously uniquely determined up to a unique isomorphism. Itsformation commutes withetale mapsX 0

! X and with Henselization and completion. LetAı be the open subscheme ofA having connected fibres. Then there is an exact sequenceof sheaves onXsm

0! Aı! A! ˚ ! 0

in which˚ is a finite sumL

iv�˚v with a˚v finite sheaf onv.Assume now thatX is the spectrum of a discrete valuation ringR, and writeK andk

for the field of fractions ofR and its (perfect) residue field. As usual,i andj denote theinclusions of the closed and open points ofX into X .

If the identity componentAı0 of the closed fibreA0 of A is an extension of an abelian

variety by a torus, thenA is said to havesemistable reduction. In this case, the formationof Aı commutes with all finite field extensionsK! L.

THEOREMC.7. There exists a finite separable extensionL ofK such thatAL has semistablereduction.

PROOF. There are several proofs; see for example Grothendieck 1972, IX 3, or Artin andWinters 1971.

Let 'WA! B be an isogeny of abelian varieties overK. From the definition ofB, wesee that' extends uniquely to an homomorphism'WA ! B. Write 'ı for the restrictionof ' toAı, and letN D Ker('0); it is group scheme overX .

PROPOSITIONC.8. Let'WA! B be the map defined by an isogenyA! B. The follow-ing conditions are equivalent:

(a) 'WA! A is flat;(b) N is flat overX ;(c) N is quasi-finite overX ;(d) ('0)˝R k is surjective.

When these conditions are realized, the following sequence is exact onXflW

0! N ! Aı'ı

�! Bı! 0.

PROOF. The same arguments as those in Grothendieck 1972, IX 2.2.1, suffice to prove thisresult.


COROLLARY C.9. LetA be an abelian variety overK, and letn be an integer. The follow-ing conditions are equivalent:

(a) nWA! A is flat;(b) nWAı

! Aı is surjective;(c) Aı

n is quasi-finite;(d) n is prime to the characteristic ofk or A has semistable reduction.

PROOF. It is easy to see from the structure ofAı ˝R k that condition (d) of the corollaryis equivalent to condition (d) of the proposition. The corollary therefore follows directlyfrom the proposition.

Biextensions of Neron models

From now on, we endowX with the smooth topology. Also we continue to assume thatX

is the spectrum of a discrete valuation ring, and we writex for its closed point. LetW be aPoincare biextension onB � A, and writei�˚

0 andi�˚ for B=Bı andA=Aı respectively.According to Grothendieck 1972, VIII, there is a canonical pairing ofGal(ks=k)-modules

˚ 0� ˚ ! Q=Z

which represents the obstruction to extendingW to a biextension of(B,A) by Gm. Wereview this theory, but first we need a lemma.

If � is a submodule of , then we writeA� for the inverse image ofi�� in A. ThusA has the same generic fibre asA andAx=Aı

x D � .

LEMMA C.10. For any submodule� � ˚ , there is a canonical isomorphism

j�Ext1Ksm

(A, Gm) �D Ext1Xsm

(A� , j�GmK )I

thereforeB �D Ext1Xsm

(A� , j�GmK ).

PROOF. We first show thatR1j�Gm (computed for the smooth topology) is zero. ForeachY smooth overX , R1j�GmjYet is the sheaf (for theetale topology) associated withthe presheafU 7! Pic(UK ). We shall show that in fact the sheaf associated withU 7!

Pic(UK ) for the Zariski topology is zero.Let y 2 Y , and let� be a uniformizing parameter forR. We have to show that

Pic(OY,y [��1]) D 0. Let Y 0D SpecOY,y and writei 0 andj 0 for the inclusionsZ0,!Y 0

and U 0,!Y 0 corresponding to the mapsOY,y � OY,y=(�) andOY,y,!OY,y [��1]. IfZ0D ;, thenOY,y [��1] D OY,y, and the assertion is obvious. In the contrary case,Z0 is

a prime divisor on the regular schemeY 0 (becauseY is smooth overX), and so there is anexact sequence

0! Gm! j 0�Gm! i 0

�Z! 0.

The cohomology sequence of this is

� � � ! Pic(Y 0)! Pic(U 0)! H 1(Y 0, i 0�Z)! � � � .

But Pic(Y 0) D 0 becauseY 0 is the spectrum of a local ring, andH 1(Y 0, i 0�Z) D H 1(Z0, Z),

which is zero becauseZ0 is normal. Therefore Pic(U 0) D 0.


As j�A� D A, there is a canonical isomorphism of functors

j�HomK (A,�) �D HomX (A� , j��)

(see Milne 1980, II 3.22). We form the first right derived functor of each side and evaluateit at Gm. BecauseHomK (A, Gm) D 0, on the left we getj�Ext1

K (A, Gm), and becauseR1j�Gm D 0, on the right we getExt1

X (A� , j�Gm), which proves the lemma.

Let B��! Ext1

K (A, Gm) be the map defined byW . On applyingj�, we get a map

B ��! j�Ext1

K (A, Gm)��! Ext1

X (A, j�GmK ). From the exact sequence

0! Gm! j�Gm! i�Z! 0

we get an exact sequence

HomX (A, i�Z)! Ext1X (A, Gm)! Ext1

X (A, j�Gm)! Ext1X (A, i�Z).

But Ext rX (A, i�Z) D i�Ext r

x(i�A, Z), andi�A D Ak becauseA is in the underlying cate-gory of Xsm (see Milne 1980, II 3.1d). ThereforeHomX (A, i�Z) D i�Homx(Ax, Z) D 0

andExt1X (A, i�Z) D i�Ext1

x(Ax, Z) D i�Homx(Ax, Q=Z) D i�Homx(˚, Q=Z). Thisgives us the lower row of the diagram below.

LEMMA C.11. There is a unique map 0! Homx(˚, Q=Z) making

0 ��! Bı ��! B ��! i�˚0 ��! 0??y ??y�

??y0 ��! Ext1

X (A, Gm) ��! Ext1X (A, j�Gm) ��! i�Homx(˚, Q=Z) ��! 0

commute.

PROOF. Obviously the composite of the maps

B! Ext1X (A, j�Gm)! i�Homx(˚, Q=Z)

factors throughB=Bı D i�˚0.

To give a map of sheaves 0 ! Homx(˚, Q=Z) is the same as to give a map ofGal(ks=k)-modules˚ 0

! Hom(˚, Q=Z), or to give an equivariant pairing 0� ˚ !

Q=Z. We shall refer to the pairing defined by the map in the lemma as thecanonicalpairing.

PROPOSITION C.12. The biextensionW of (B, A) by Gm extends to a biextension of(B� 0

, A� ) by Gm if and only if � 0 and � annihilate each other in the canonical pair-ing between 0 and˚ . The extension, if it exists, is unique.


PROOF. For a detailed proof, see Grothendieck 1972, VIII 7.1b. We merely note that it isobvious from the following diagram (extracted from C.11)

0 ��! Bı��! B� 0

��! i��0

��! 0??y ??yinj

??y0 ��! Ext1

X (A� , Gm) ��! Ext1X (A� , j�Gm) ��! i�Homx(� , Q=Z) ��! 0

thatB� 0

! Ext1X (A� , j�Gm) factors throughExt1

X (A� , Gm) if and only if � 0 maps tozero in Hom(� , Q=Z).

CONJECTUREC.13. The canonical pairing 0� ˚ ! Q=Z is nondegenerate.

The conjecture is due to Grothendieck (ibid., IX 1.3). We shall see in the main body ofthe chapter that it is a consequence of various duality theorems for abelian varieties.

PROPOSITIONC.14. If � 0 annihilates� , then there is a commutative diagram

0 ��! B� 0

��! B ��! i�(˚ 0=� 0) ��! 0??y ??y�

??y0 ��! Ext1

X (A� , Gm) ��! Ext1X (A� , j�Gm) ��! i�Homx(� , Q=Z) ��! 0

If conjecture (C.13) holds and� 0 is the exact annihilator of� , or if � D 0 and� 0D ˚ 0,

then all the vertical maps are isomorphisms.

PROOF. The diagram can be constructed the same way as the diagram in (C.11). The finalstatement is obvious.

The Raynaud group

Assume now thatA has semistable reduction, and write˚0 for Ax=Aıx. Overk there is an

exact sequence

0! T ! Aıx ! B ! 0

with B an abelian variety overk. The next theorem shows that this sequence has a canoni-cal lifting to R.

For any group schemeG overX , we writebG for the formal completion ofG along theclosed pointx of X (Hartshorne 1977, II 9).

THEOREM C.15. There is a smooth group schemeA# overR and canonical isomorphismsbA ��! (A#)^ and bAı

��! (A#ı)^.

(a) There is an exact sequence overR

0! T ! A#ı! B! 0

with T a torus andB an abelian scheme; the reduction of this sequence modulo themaximal ideal ofR is the above sequence.


(b) Let˚ D A#=A#ı; then˚ is a finiteetale group scheme overR whose special fibreis˚0.

(c) LetN D (A#ı)p; thenN is the maximal finite flat subgroup scheme of the quasi-finiteflat group schemeAı

p, and there is a filtration

Ap D (Aıp)K � N � Tp � 0

with N D NK , Tp D (Tp)K , andN=Tp D (Bp)K .(d) Let At be the dual abelian variety toA, and denote the objects corresponding to it

with a prime. The nondegenerate pairing of finite group schemes overK

A0p �Ap ! Gm

induces nondegenerate pairings

N 0=T 0p �N=Tp ! Gm

A0p=T 0

p �N ! Gm.

(e) AssumeR is Henselian. IfA(K)p D A(Ka)p, then p has orderp�, where� is thedimension of the maximal subtorus ofAx.

PROOF. For (a), (b), and (c) see Grothendieck 1972, IX.7.(d) The restriction of the pairing onA0

p�Ap to N 0�N extends to a pairingN 0�N !Gm induced by the biextension of(At0 ,Aı) by Gm. This pairing is trivial onT 0

p andTp, and the quotient pairing onB0

p � Bp is that defined by the canonical extension of aPoincare biextension of(B0

K ,BK ) by Gm. This shows thatT 0p andTp are the left and right

kernels in the pairingN 0�N ! Gm. The pairingA0

p=T 0p �N ! Gm is obviously right

nondegenerate. ButA0p=T 0

p has orderp2d�� whered is the common dimension ofA andAt and� is the common dimension ofT andT 0, andN has orderp�C2˛ where˛ is thecommon dimensions ofB andB0. As d D � C ˛, this proves that the pairing is also leftnondegenerate.

(e) From the diagram

0 ��! Aı(R) ��! A(K) ��! ˚(k) ��! 0??yp

??yp

??yp

0 ��! Aı(R) ��! A(K) ��! ˚(k) ��! 0


0! Aı(R)p ! A(K)p ! ˚(k)p ! Aı(R)(p).

Let a 2 ˚(k)p. There will exist a finite flat local extensionR0 of R such thata maps tozero inAı(R0)(p) (becausepWAı

! Aı is a finite flat map), and so the image ofa in

˚(k 0) lifts to A(K0)p. By assumption,A(K)p��! A(K0)p, and soa lifts to A(K)p. This

shows that0! Aı(R)p ! A(K)p ! ˚(k)p ! 0

is exact, and the result follows by counting.

The groupA# is called theRaynaud group scheme.


Neron models and Jacobians

Let X again be any Noetherian normal integral scheme of dimension one, and let� WY !

X be a flat proper morphism of finite-type. Recall that PicY =S is defined to be the sheafon XEt associated with the presheafX 0

7! Pic(Y �X X 0). Write P D PicY =X . WhenP is representable by an algebraic space, thenP �

D Pic�Y =X is defined to be the subsheafof P such that, for allX -schemesX 0, P(X 0) consists of the sections� whose image in(Px=P

ıx)(X 0) is torsion for allx 2 X .

Assume now thatY is regular, that the fibres ofY over X are all pure of dimension

1, and thatY� is a smooth and geometrically connected. Moreover, assume thatOX

��!

��OY universally (that is, as sheaves onXfl). For each closed pointx of X , definedx to bethe greatest common divisor of the multiplicities of the irreducible components ofYx (themultiplicity of Yi � Yx is the length ofOYi,yi

whereyi is the generic point ofYi).

THEOREM C.16. (a) The functorP is representable by an algebraic space locally of finitetype overX .

(b) If dx D 1 for all x, thenP � is representable by a separated group scheme overX .(c) Assume that the residue fields ofX are perfect. Under the hypothesis of (b),P � is

the Neron model of the Jacobian ofY�.

PROOF. (a) Our assumption thatOS

��! ��OX universally says that� is cohomologically

flat in dimension zero. Therefore the statement is a special case of a theorem of Artin(1969b).

(b) This is a special case of Raynaud 1970, 6.4.5.(c) This is a special case of Raynaud 1970, 8.1.4.

REMARK C.17. The hypotheses in the theorem are probably too stringent.

The autoduality of the Jacobian

Let C be a smooth complete curve over a fieldk. Then there is a canonical biextension of(J, J ) by Gm, and the two mapsJ ! Ext1

k(J, Gm) are isomorphisms (and differ only by

a minus sign) (see Milne 1986c,~6, or Moret-Bailly 1985). It is this biextension which wewish to extend to certain families of curves.

Let � WY ! X be a flat projective morphism with fibres pure of dimension one andwith smooth generic fibre; assume thatY is regular and thatY has a sections over X .Endow bothY and X with the smooth topology. ThenR1��Gm is representable by asmooth group scheme PicY =X overX . Write P 0 for the kernel of the degree map on thegeneric fibre; thusP 0(X) is the set of isomorphism classes of invertible sheaves onY

whose restriction tos(X) is trivial and whose restriction toY� has degree zero. Note thatPicY =X D P 0

˚ Z.

THEOREM C.18. There exists a biextension of(P 0, P 0) by Gm whose restriction to thegeneric fibre is the canonical biextension.

280

PROOF. In the case thatP has connected fibres, this follows from the result Grothendieck1972, VIII 7.1b that for any two group schemesB andA over X with connected fibres,and any nonempty open subsetU of X , the restriction functor

BiextX (A, BIGm)! BiextU (A, BIGm)

is a bijection. See also Moret-Bailly 1985, 2.8.2. For the general case, we refer the readerto Artin 1967.

CONJECTUREC.19.Assume thatY is the minimal model of its generic fibre. Then the mapsP 0 ! Ext1

X (P 0, Gm) (of sheaves for the smooth topology) induced by the biextension in(C.18) are isomorphisms.

A proof of this conjecture has been announced by Artin and Mazur (Artin 1967), atleast in some cases. We shall refer to this as theautoduality hypothesis.

NOTES. The concept of a biextension was introduced by Mumford (1969), and was devel-oped by Grothendieck (1972). Apart from Neron’s Theorem C.6 and the theorems of Artinand Raynaud (C.16), most of the results are due to Grothendieck. The exposition is partlybased on McCallum 1986.

... and so there ain’t nothing more to write about, and I am rotten glad of it, because ifI’d knowed what a trouble it was to make a book I wouldn’t a tackled it and ain’t agoingto no more. But I reckon I got to light out for the Territory ahead of the rest, because AuntSally she’s going to sivilize me and I can’t stand it. I been there before.

H. Finn

281

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Index

absolute different, 225admissible homomorphism, 102augmented cup-product, 15autoduality hypothesis, 280

Barsotti-Weil formula, 41, 173biextension, 271Brauer group, 120Breen-Serre duality, 196, 228

canonical pairing, 276canonical smooth resolution, 267Cartier dual, 7Cassels-Tate pairing, 177, 255Cech complex, 128class formation, 22

P, 28cofinite-type, 7cohomology group

Tate, 10cohomology groups

with compact support, 146, 191cohomology sequence of a pair, 125complete resolution, 10conjecture

duality, 188Neron components, 277of Birch and Swinnerton-Dyer, 88

constructible, 129Z, 129

countable sheaf, 166cup-product

augmented, 15

different of a finite group scheme, 210discriminant ideal, 208divisible subgroup, 7dual

Cartier, 7Pontryagin, 7

dual torus, 101duality

Breen-Serre, 196

embedding, 264embedding problem, 112excellent, 116extensions

of algebraic groups, 17of modules, 11of sheaves, 125

fieldd-local, 39global, 7local, 7quasi-finite, 113

finite support, 165Frobenius map, 195

global field, 7Greenberg realization, 223group

Brauer, 120Selmer, 70Tate-Shafarevich, 70Weil, 93

group of multiplicative type, 32group scheme

p-primary, 197

Hasse principle, 105, 106, 108, 121height, 87

of a finite group scheme, 195Henselian, 113Henselization, 113homomorphism

compatible of modules, 17

ideles, 117induced module, 15invariant map, 22inverse limits, 20

kernel-cokernel exact sequence, 21

local duality theorem holds, 217local field, 7

292

INDEX 293

d, 39

mapcorestriction, 96Frobenius, 195invariant, 22reciprocity, 23transfer, 50Verlagerung, 50

mapping cone, 130module

induced, 15morphism

strict, 19

Neron model, 274norm group, 115

orderof a finite group scheme, 208

pairingCassels-Tate, 177

pairingscompatibility of, 16in the derived category, 199

perfect scheme, 195algebraic, 196

perfectionof a scheme, 196

Poincare biextension, 272Poincare sheaf, 272Pontryagin dual, 7

quasi-finite field, 113

Raynaud group scheme, 278reciprocity law, 119reciprocity map, 23reduced Galois group, 119resolution

complete, 10standard, 10

semistable reduction, 274sheaf

constructible, 129flat, 200

� -compact, 19site

big etale, 8big flat, 8etale, 8perfect, 196small fpqf, 8

smooth, 8solvable, 105spectral sequence

for Exts, 12Hochschild-Serre, 14

splits generically, 211standard complete resolution, 10standard resolution, 10strict morphism, 19subgroup

divisible, 7

Tamagawa number, 111Tate cohomology group, 10theorem

abelian class field theory, 96, 103, 104algebraic structure of cohomology group, 226an exact sequence, 230, 231behaviour of B-S/D for isogenies, 256cohomology of finite flat group schemes, 225compatibility of B-S/D, 89compatibility of pairings, 200duality

for $d$-local fields, 140for a class formation, 29for a d-local field, 39for a global field, 51, 55for a local field, 30for a torus, 167, 170for abelian schemes, 247, 248, 255, 263for abelian varieties, 77, 229, 245for abelian variety, 42for an abelian scheme, 174for archimedean local field, 37for class formation, 25for constructible sheaves, 135, 139, 155for finite components, 247for finite flat group schemes, 202for finite group schemes, 228, 238, 241, 249,

260, 261for flat sheaves, 218for Henselian local field, 38for higher dimensional schemes, 142, 183, 185,

187for Neron components, 214for Neron models, 215, 217for perfect group schemes, 199for Tate-Shafarevich groups, 176for tori, 102, 228, 238

embedding finite group schemes, 264, 267Euler-Poincare characteristic, 209Euler-Poincare characteristics, 153, 220existence of Neron models, 274global class field theory, 116

294 INDEX

global Euler-Poincare characteristic, 63, 67Hasse principle, 105, 108, 110local Euler-Poincare characteristic, 34local reciprocity law, 115semistable reduction, 274spectral sequence for Exts, 12spectral sequence of exts, 126Tamagawa number, 111Tate-Nakayama, 11topological duality for vector spaces, 194unramified duality, 33weak Mordell-Weil, 69

torus, 7, 27dual, 101over a scheme, 167

unramified character, 103unramified module

module, 32

Verlagerung map, 50Verschiebung, 195

Weil group, 93

Arithmetic Duality Theorems

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