Cohomology of Number Fields - Heidelberg Universityschmidt/NSW2e/NSW2.1.pdf · Cohomology of Number Fields by ... cohomology associated to a Grothendieck topos is sufﬁciently covered

Cohomologyof

Number Fields

by

Jürgen NeukirchAlexander Schmidt

Kay Wingberg

Second Editioncorrected version 2.1, September 2013

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Vorwort

Als unser Freund und Lehrer Jürgen Neukirch Anfang 1997 starb, hinter-ließ er den Entwurf zu einem Buch über die Kohomologie der Zahlkörper,welches als zweiter Band zu seiner Monographie Algebraische Zahlentheoriegedacht war. Für die Kohomologie proendlicher Gruppen, sowie für Teile derKohomologie lokaler und globaler Körper lag bereits eine Rohfassung vor,die schon zu einer regen Korrespondenz zwischen Jürgen Neukirch und unsgeführt hatte.

In den letzten zwei Jahren ist, ausgehend von seinem Entwurf, das hiervorliegende Buch entstanden. Allerdings wussten wir nur teilweise, was JürgenNeukirch geplant hatte. So mag es sein, dass wir Themen ausgelassen haben,welche er berücksichtigen wollte, und anderes, nicht Geplantes, aufgenommenhaben.

Jürgen Neukirchs inspirierte und pointierte Art, Mathematik auf hohemsprachlichen Niveau darzustellen, ist für uns stets Vorbild gewesen. Leidererreichen wir nicht seine Meisterschaft, aber wir haben uns alle Mühe gegebenund hoffen, ein Buch in seinem Sinne und nicht zuletzt auch zum Nutzen seinerLeser fertig gestellt zu haben.

Heidelberg, im September 1999 Alexander SchmidtKay Wingberg



Introduction

Number theory, one of the most beautiful and fascinating areas of mathe-matics, has made major progress over the last decades, and is still developingrapidly. In the beginning of the foreword to his book Algebraic Number Theory,J. Neukirch wrote

" Die Zahlentheorie nimmt unter den mathematischen Disziplineneine ähnlich idealisierte Stellung ein wie die Mathematik selbstunter den anderen Wissenschaften." ∗)

Although the joint authors of the present book wish to reiterate this statement,we wish to stress also that number theory owes much of its current strongdevelopment to its interaction with almost all other mathematical fields. Inparticular, the geometric (and consequent functorial) point of view of arithmeticgeometry uses techniques from, and is inspired by, analysis, geometry, grouptheory and algebraic topology. This interaction had already started in the1950s with the introduction of group cohomology to local and global classfield theory, which led to a substantial simplification and unification of thisarea.

The aim of the present volume is to provide a textbook for students, as wellas a reference book for the working mathematician on cohomological topicsin number theory. Its main subject is Galois modules over local and globalfields, objects which are typically associated to arithmetic schemes. In viewof the enormous quantity of material, we were forced to restrict the subjectmatter in some way. In order to keep the book at a reasonable length, wehave therefore decided to restrict attention to the case of dimension less thanor equal to one, i.e. to the global fields themselves, and the various subringscontained in them. Central and frequently used theorems such as the globalduality theorem ofG.POITOU and J.TATE, as well as results such as the theoremof I. R. ŠAFAREVIČ on the realization of solvable groups as Galois groups overglobal fields, had been part of algebraic number theory for a long time. But theproofs of statements like these were spread over many original articles, someof which contained serious mistakes, and some even remained unpublished. Itwas the initial motivation of the authors to fill these gaps and we hope that theresult of our efforts will be useful for the reader.

In the course of the years since the 1950s, the point of view of class fieldtheory has slightly changed. The classical approach describes the Galois groups

∗)“Number theory, among the mathematical disciplines, occupies a similar idealized positionto that held by mathematics itself among the sciences.”



viii Introduction

of finite extensions using arithmetic invariants of the local or global groundfield. An essential feature of the modern point of view is to consider infiniteGalois groups instead, i.e. one investigates the set of all finite extensions of thefield k at once, via the absolute Galois group Gk. These groups intrinsicallycome equipped with a topology, the Krull topology, under which they areHausdorff, compact and totally disconnected topological groups. It proves tobe useful to ignore, for the moment, their number theoretical motivation and toinvestigate topological groups of this type, the profinite groups, as objects ofinterest in their own right. For this reason, an extensive “algebra of profinitegroups” has been developed by number theorists, not as an end in itself, butalways with concrete number theoretical applications in mind. Nevertheless,many results can be formulated solely in terms of profinite groups and theirmodules, without reference to the number theoretical background.

The first part of this book deals with this “profinite algebra”, while thearithmetic applications are contained in the second part. This division shouldnot be seen as strict; sometimes, however, it is useful to get an idea of howmuch algebra and how much number theory is contained in a given result.

A significant feature of the arithmetic applications is that classical reciprocitylaws are reflected in duality properties of the associated infinite Galois groups.For example, the reciprocity law for local fields corresponds to Tate’s dualitytheorem for local cohomology. This duality property is in fact so strong that itbecomes possible to describe, for an arbitrary prime p, the Galois groups of themaximal p-extensions of local fields. These are either free groups or groupswith a very special structure, which are now known as Demuškin groups. Thisresult then became the basis for the description of the full absolute Galoisgroup of a p-adic local field by U. JANNSEN and the third author.

The global case is rather different. As was already noticed by J. TATE,the absolute Galois group of a global field is not a duality group. It is thegeometric point of view, which offers an explanation of this phenomenon: theduality comes from the curve rather than from its generic point. It is thereforenatural to consider the étale fundamental groups πet1 (Spec(Ok,S)), where S isa finite set of places of k. Translated to the language of Galois groups, thefundamental group of Spec(Ok,S) is a quotient of the full group Gk, namely,the Galois group Gk,S of the maximal extension of k which is unramifiedoutside S. If S contains all places that divide the order of the torsion of amoduleM , the central Poitou-Tate duality theorem provides a duality betweenthe localization kernels in dimensions one and two. In conjunction with Tatelocal duality, this can also be expressed in the form of a long 9-term sequence.The duality theorem of Poitou-Tate remains true for infinite sets of places Sand, using topologically restricted products of local cohomology groups, thelong exact sequence can be generalized to this case. The question of whether



Introduction ix

the group Gk,S is a duality group when S is finite was positively answered bythe second author.

As might already be clear from the above considerations, the basic techniqueused in this book is Galois cohomology, which is essential for class field the-ory. For a more geometric point of view, it would have been desirable to havealso formulated the results throughout in the language of étale cohomology.However, we decided to leave this to the reader. Firstly, the technique of sheafcohomology associated to a Grothendieck topos is sufficiently covered in theliterature (see [5], [139], [228]) and, in any case, it is an easy exercise (at leastin dimension ≤ 1) to translate between the Galois and the étale languages. Afurther reason is that results which involve infinite sets of places (necessarywhen using Dirichlet density arguments) or infinite extension fields, can bemuch better expressed in terms of Galois cohomology than of étale cohomo-logy of pro-schemes. When the geometric point of view seemed to bring abetter insight or intuition, however, we have added corresponding remarks orfootnotes. A more serious gap, due to the absence of Grothendieck topologies,is that we cannot use flat cohomology and the global flat duality theorem ofArtin-Mazur. In chapter VIII, we therefore use an ad hoc construction, thegroup BS , which measures the size of the localization kernel for the first flatcohomology group with the roots of unity as coefficients.

Let us now examine the contents of the individual chapters more closely.The first part covers the algebraic background for the number theoretical ap-plications. Chapter I contains well-known basic definitions and results, whichmay be found in several monographs. This is only partly true for chapter II: theexplicit description of the edge morphisms of the Hochschild-Serre spectralsequence in §2 is certainly well-known to specialists, but is not to be found inthe literature. In addition, the material of §3 is well-known, but contained onlyin original articles.

Chapter III considers abstract duality properties of profinite groups. Amongthe existing monographs which also cover large parts of the material, we shouldmention the famous Cohomologie Galoisienne by J.-P. SERRE andH.KOCH’sbook Galoissche Theorie der p-Erweiterungen. Many details, however, havebeen available until now only in the original articles.

In chapter IV, free products of profinite groups are considered. These areimportant for a possible non-abelian decomposition of global Galois groupsinto local ones. This happens only in rather rare, degenerate situations forGalois groups of global fields, but it is quite a frequent phenomenon forsubgroups of infinite index. In order to formulate such statements (like thearithmetic form of Riemann’s existence theorem in chapter X), we develop theconcept of the free product of a bundle of profinite groups in §3.



x Introduction

Chapter V deals with the algebraic foundations of Iwasawa theory. We willnot prove the structure theorem for Iwasawa modules in the usual way by usingmatrix calculations (even though it may be more acceptable to some mathe-maticians, as it is more concrete), but we will follow mostly the presentationfound in Bourbaki, Commutative Algebra, with a view to more general situa-tions. Moreover, we present results concerning the structure of these modulesup to isomorphism, which are obtained using the homotopy theory of modulesover group rings, as presented by U. JANNSEN.

The central technical result of the arithmetic part is the famous global dualitytheorem of Poitou-Tate. We start, in chapter VI, with general facts about Galoiscohomology. Chapter VII deals with local fields. Its first three sections largelyfollow the presentation of J.-P. SERRE in Cohomologie Galoisienne. Thenext two sections are devoted to the explicit determination of the structureof local Galois groups. In chapter VIII, the central chapter of this book, wegive a complete proof of the Poitou-Tate theorem, including its generalizationto finitely generated modules. We begin by collecting basic results on thetopological structure, universal norms and the cohomology of the S-idèle classgroup, before moving on to the proof itself, given in sections 4 and 6. Inthe proof, we apply the group theoretical theorems of Nakayama-Tate and ofPoitou, proven already in chapter III.

In chapter IX, we reap the rewards of our efforts in the previous chapters. Weprove several classical number theoretical results, such as the Hasse principleand the Grunwald-Wang theorem. In §4, we consider embedding problemsand we present the theorem of K. IWASAWA to the effect that the maximalprosolvable factor of the absolute Galois group of Qab is free. In §5, we givea complete proof of Šafarevič’s theorem on the realization of finite solvablegroups as Galois groups over global fields.

The main concern of chapter X is to consider restricted ramification. Ge-ometrically speaking, we are considering the curves Spec(Ok,S), in contrastto chapter IX, where our main interest was in the point Spec(k). Needless tosay, things now become much harder. Invariants like the S-ideal class groupor the p-adic regulator enter the game and establish new arithmetic obstruc-tions. Our investigations are guided by the analogy between number fieldsand function fields. We know a lot about the latter from algebraic geometry,and we try to establish analogous results for number fields. For example,using the group theoretical techniques of chapter IV, we can prove the numbertheoretical analogue of Riemann’s existence theorem. The fundamental groupof Spec(Ok), i.e. the Galois group of the maximal unramified extension of thenumber field k, was the subject of the long-standing class field tower problemin number theory, which was finally answered negatively by E. S. GOLOD and



Introduction xi

I. R. ŠAFAREVIČ. We present their proof, which demonstrates the power of thegroup theoretical and cohomological methods, in §8.

Chapter XI deals with Iwasawa theory, which is the consequent conceptualcontinuation of the analogy between number fields and function fields. Weconcentrate on the algebraic aspects of Iwasawa theory of p-adic local fieldsand of number fields, first presenting the classical statements which one canusually find in the standard literature. Then we prove more far-reaching resultson the structure of certain Iwasawa modules attached to p-adic local fields andto number fields, using the homotopy theory of Iwasawa modules. The analyticaspects of Iwasawa theory will merely be described, since this topic is cov-ered by several monographs, for example, the book [246] of L. WASHINGTON.Finally, the Main Conjecture of Iwasawa theory will be formulated and dis-cussed; for a proof, we refer the reader to the original work of B. MAZUR andA. WILES ([134], [249]).

In the last chapter, we give a survey of so-called anabelian geometry, aprogram initiated by A. GROTHENDIECK. Perhaps the first result of this theory,obtained even before this program existed, is a theorem of J. NEUKIRCH andK. UCHIDA which asserts that the absolute Galois group of a global field, as aprofinite group, characterizes the field up to isomorphism. We give a proof ofthis theorem for number fields in the first two sections. The final section givesan overview of the conjectures and their current status.

The reader will recognize very quickly that this book is not a basic textbookin the sense that it is completely self-contained. We use freely basic algebraic,topological and arithmetic facts which are commonly known and contained inthe standard textbooks. In particular, the reader should be familiar with basicnumber theory. While assuming a certain minimal level of knowledge, wehave tried to be as complete and as self-contained as possible at the next stage.We give full proofs of almost all of the main results, and we have tried notto use references which are only available in original papers. This makes itpossible for the interested student to use this book as a textbook and to findlarge parts of the theory coherently ordered and gently accessible in one place.On the other hand, this book is intended for the working mathematician as areference on cohomology of local and global fields.

Finally, a remark on the exercises at the end of the sections. A few of themare not so much exercises as additional remarks which did not fit well into themain text. Most of them, however, are intended to be solved by the interestedreader. However, there might be occasional mistakes in the way they are posed.If such a case arises, it is an additional task for the reader to give the correctformulation.



xii Preface to the Second Edition

We would like to thank many friends and colleagues for their mathemat-ical examination of parts of this book, and particularly, ANTON DEITMAR,TORSTEN FIMMEL, DAN HARAN, UWE JANNSEN, HIROAKI NAKAMURA andOTMAR VENJAKOB. We are indebted to Mrs. INGE MEIER who TEXed a largepart of the manuscript, andEVA-MARIA STROBEL receives our special gratitudefor her careful proofreading. Hearty thanks go to FRAZER JARVIS for goingthrough the entire manuscript, correcting our English.

Heidelberg, September 1999 Alexander SchmidtKay Wingberg

Preface to the Second Edition

The present second edition is a corrected and extended version of the first.We have tried to improve the exposition and reorganize the content to someextent; furthermore, we have included some new material. As an unfortunateresult, the numbering of the first edition is not compatible with the second.

In the algebraic part you will find new sections on filtered cochain com-plexes, on the degeneration of spectral sequences and on Tate cohomology ofprofinite groups. Amongst other topics, the arithmetic part contains a newsection on duality theorems for unramified and tamely ramified extensions, acareful analysis of 2-extensions of real number fields and a complete proof ofNeukirch’s theorem on solvable Galois groups with given local conditions.

Since the publication of the first edition, many people have sent us listsof corrections and suggestions or have contributed in other ways to this edi-tion. We would like to thank them all. In particular, we would like to thankJAKOB STIX and DENIS VOGEL for their comments on the new parts of thissecond edition and FRAZER JARVIS, who again did a great job correcting ourEnglish.

Regensburg and Heidelberg, November 2007 Alexander SchmidtKay Wingberg



Contents

Algebraic Theory 1

Chapter I: Cohomology of Profinite Groups 3§1. Profinite Spaces and Profinite Groups . . . . . . . . . . . . . . 3§2. Definition of the Cohomology Groups . . . . . . . . . . . . . . 12§3. The Exact Cohomology Sequence . . . . . . . . . . . . . . . . 25§4. The Cup-Product . . . . . . . . . . . . . . . . . . . . . . . . . 36§5. Change of the Group G . . . . . . . . . . . . . . . . . . . . . . 45§6. Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . 60§7. Cohomology of Cyclic Groups . . . . . . . . . . . . . . . . . . 74§8. Cohomological Triviality . . . . . . . . . . . . . . . . . . . . . 80§9. Tate Cohomology of Profinite Groups . . . . . . . . . . . . . . 83

Chapter II: Some Homological Algebra 97§1. Spectral Sequences . . . . . . . . . . . . . . . . . . . . . . . . 97§2. Filtered Cochain Complexes . . . . . . . . . . . . . . . . . . . 101§3. Degeneration of Spectral Sequences . . . . . . . . . . . . . . . 107§4. The Hochschild-Serre Spectral Sequence . . . . . . . . . . . . . 111§5. The Tate Spectral Sequence . . . . . . . . . . . . . . . . . . . . 120§6. Derived Functors . . . . . . . . . . . . . . . . . . . . . . . . . 127§7. Continuous Cochain Cohomology . . . . . . . . . . . . . . . . 136

Chapter III: Duality Properties of Profinite Groups 147§1. Duality for Class Formations . . . . . . . . . . . . . . . . . . . 147§2. An Alternative Description of the Reciprocity Homomorphism . 164§3. Cohomological Dimension . . . . . . . . . . . . . . . . . . . . 171§4. Dualizing Modules . . . . . . . . . . . . . . . . . . . . . . . . 181§5. Projective pro-c-groups . . . . . . . . . . . . . . . . . . . . . . 189§6. Profinite Groups of scdG = 2 . . . . . . . . . . . . . . . . . . 202§7. Poincaré Groups . . . . . . . . . . . . . . . . . . . . . . . . . 210§8. Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220§9. Generators and Relations . . . . . . . . . . . . . . . . . . . . . 224



xiv Contents

Chapter IV: Free Products of Profinite Groups 245§1. Free Products . . . . . . . . . . . . . . . . . . . . . . . . . . . 245§2. Subgroups of Free Products . . . . . . . . . . . . . . . . . . . . 252§3. Generalized Free Products . . . . . . . . . . . . . . . . . . . . 256

Chapter V: Iwasawa Modules 267§1. Modules up to Pseudo-Isomorphism . . . . . . . . . . . . . . . 268§2. Complete Group Rings . . . . . . . . . . . . . . . . . . . . . . 273§3. Iwasawa Modules . . . . . . . . . . . . . . . . . . . . . . . . . 289§4. Homotopy of Modules . . . . . . . . . . . . . . . . . . . . . . 301§5. Homotopy Invariants of Iwasawa Modules . . . . . . . . . . . . 312§6. Differential Modules and Presentations . . . . . . . . . . . . . . 321

Arithmetic Theory 335

Chapter VI: Galois Cohomology 337§1. Cohomology of the Additive Group . . . . . . . . . . . . . . . 337§2. Hilbert’s Satz 90 . . . . . . . . . . . . . . . . . . . . . . . . . 343§3. The Brauer Group . . . . . . . . . . . . . . . . . . . . . . . . 349§4. The Milnor K-Groups . . . . . . . . . . . . . . . . . . . . . . 356§5. Dimension of Fields . . . . . . . . . . . . . . . . . . . . . . . 360

Chapter VII: Cohomology of Local Fields 371§1. Cohomology of the Multiplicative Group . . . . . . . . . . . . . 371§2. The Local Duality Theorem . . . . . . . . . . . . . . . . . . . 378§3. The Local Euler-Poincaré Characteristic . . . . . . . . . . . . . 391§4. Galois Module Structure of the Multiplicative Group . . . . . . . 401§5. Explicit Determination of Local Galois Groups . . . . . . . . . . 409

Chapter VIII: Cohomology of Global Fields 425§1. Cohomology of the Idèle Class Group . . . . . . . . . . . . . . 425§2. The Connected Component of Ck . . . . . . . . . . . . . . . . . 443§3. Restricted Ramification . . . . . . . . . . . . . . . . . . . . . . 452§4. The Global Duality Theorem . . . . . . . . . . . . . . . . . . . 466§5. Local Cohomology of Global Galois Modules . . . . . . . . . . 472§6. Poitou-Tate Duality . . . . . . . . . . . . . . . . . . . . . . . . 480§7. The Global Euler-Poincaré Characteristic . . . . . . . . . . . . 503§8. Duality for Unramified and Tamely Ramified Extensions . . . . . 513



Contents xv

Chapter IX: The Absolute Galois Group of a Global Field 521§1. The Hasse Principle . . . . . . . . . . . . . . . . . . . . . . . . 522§2. The Theorem of Grunwald-Wang . . . . . . . . . . . . . . . . . 536§3. Construction of Cohomology Classes . . . . . . . . . . . . . . . 543§4. Local Galois Groups in a Global Group . . . . . . . . . . . . . 553§5. Solvable Groups as Galois Groups . . . . . . . . . . . . . . . . 557§6. Šafarevič’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 574

Chapter X: Restricted Ramification 599§1. The Function Field Case . . . . . . . . . . . . . . . . . . . . . 602§2. First Observations on the Number Field Case . . . . . . . . . . . 618§3. Leopoldt’s Conjecture . . . . . . . . . . . . . . . . . . . . . . 624§4. Cohomology of Large Number Fields . . . . . . . . . . . . . . 642§5. Riemann’s Existence Theorem . . . . . . . . . . . . . . . . . . 647§6. The Relation between 2 and∞ . . . . . . . . . . . . . . . . . . 656§7. Dimension of H i(GTS ,ZZ/pZZ) . . . . . . . . . . . . . . . . . . 666§8. The Theorem of Kuz’min . . . . . . . . . . . . . . . . . . . . . 678§9. Free Product Decomposition of GS(p) . . . . . . . . . . . . . . 686§10. Class Field Towers . . . . . . . . . . . . . . . . . . . . . . . . 697§11. The Profinite Group GS . . . . . . . . . . . . . . . . . . . . . . 706

Chapter XI: Iwasawa Theory of Number Fields 721§1. The Maximal Abelian Unramified p-Extension of k∞ . . . . . . 722§2. Iwasawa Theory for p-adic Local Fields . . . . . . . . . . . . . 731§3. The Maximal Abelian p-Extension of k∞Unramified OutsideS . . 735§4. Iwasawa Theory for Totally Real Fields and CM-Fields . . . . . 751§5. Positively Ramified Extensions . . . . . . . . . . . . . . . . . . 763§6. The Main Conjecture . . . . . . . . . . . . . . . . . . . . . . . 771

Chapter XII: Anabelian Geometry 785§1. Subgroups of Gk . . . . . . . . . . . . . . . . . . . . . . . . . 785§2. The Neukirch-Uchida Theorem . . . . . . . . . . . . . . . . . . 791§3. Anabelian Conjectures . . . . . . . . . . . . . . . . . . . . . . 798

Literature 805

Index 821



Algebraic Theory



Chapter I

Cohomology of Profinite Groups

Profinite groups are topological groups which naturally occur in algebraicnumber theory as Galois groups of infinite field extensions or more generally asétale fundamental groups of schemes. Their cohomology groups often containimportant arithmetic information.

In the first chapter we will study profinite groups as objects of interest inthemselves, independently of arithmetic applications, which will be treated inthe second part of this book.

§1. Profinite Spaces and Profinite Groups

The underlying topological spaces of profinite groups are of a very specifictype, which will be described now. To do this, we make use of the conceptof inverse (or projective) limits. We refer the reader to the standard literature(e.g. [160], [79], [139]) for the definition and basic properties of limits. Allindex sets will be assumed to be filtered.

(1.1.1) Lemma. For a Hausdorff topological space T the following conditionsare equivalent.

(i) T is the (topological) inverse limit of finite discrete spaces.

(ii) T is compact and every point of T has a basis of neighbourhoods con-sisting of subsets which are both closed and open.

(iii) T is compact and totally disconnected.

Proof: In order to show the implication (i) ⇒ (ii), we first recall that theinverse limit of compact spaces is compact (see [15] chap.I, §9, no.6, prop.8).Therefore T is compact. By the definition of the inverse limit topology andby (i), every point of T has a basis of neighbourhoods consisting of sets of the



4 Chapter I. Cohomology of Profinite Groups

form f−1(W ), where W is a subset of a finite discrete space V and f : T → Vis a continuous map. These sets are both open and closed.

For the implication (ii)⇒ (iii) we have to show that the connected componentCt of every point t ∈ T equals {t}. Since T is compact, Ct is the intersectionof all closed and open subsets containing t (see [15] chap.II, §4, no.4, prop.6).Since T is Hausdorff, we obtain Ct = {t}.

It remains to show the implication (iii)⇒ (i). Let I be the set of equivalencerelations R ⊆ T × T on T , such that the quotient space T/R is finite anddiscrete in the quotient topology. The set I is partially ordered by inclusionand is directed, because R1 ∩ R2 is in I if R1 and R2 are. We claim that thecanonical map φ : T → lim

←− R∈IT/R is a homeomorphism.

First we see that the map φ is surjective, because for an element {tR}R∈I ∈lim←− R∈I

T/R, the sets (pR ◦ φ)−1(tR) are nonempty and compact. Since I isdirected, finite intersections of these sets are also nonempty and compactnessthen implies that φ−1({tR}R∈I) =

⋂R∈I(pR ◦ φ)−1(tR) is nonempty.

For the injectivity it suffices to show that for t, s ∈ T , t /= s, there exists anR ∈ I such that (t, s) ∈/ R. But since s is not in the connected component of t,there exists a closed and open subsetU ⊆ T with t ∈ U , s ∈/ U (see [15] chap.II,§4, no.4, prop.6). Then the equivalence relation R defined by "(x, y) ∈ R ifx and y are both in U or both not in U" is of the required type. The proof iscompleted by the remark that a continuous bijection between compact spacesis a homeomorphism. 2

In fact one immediately verifies that we could have chosen the inverse systemin (i) in such a way that all transition maps are surjective.

(1.1.2) Definition. A space T is called a profinite space if it satisfies theequivalent conditions of lemma (1.1.1).

A compactness argument shows that a subset V ⊆ lim←−

Xi of a profinite spaceis both closed and open if and only if V is the pre-image under the canonicalprojection pi : X → Xi of a (necessarily closed and open) subset in Xi forsome i. Every continuous map between profinite spaces can be realized asa projective limit of maps between finite discrete spaces. Without giving anexact definition, we want to note that the category of profinite spaces withcontinuous maps is the pro-category of the category of finite discrete spaces.

Recall that a topological group is a group G endowed with the structure ofa topological space, such that the group operations G → G, g 7→ g−1, and



§1. Profinite Spaces and Profinite Groups 5

G × G → G, (g, h) 7→ gh, are continuous. The reader will immediatelyverify that the inverse limit of an inverse system of topological groups is justthe inverse limit of the groups together with the inverse limit topology on theunderlying topological space.

(1.1.3) Proposition. For a Hausdorff topological group G the following con-ditions are equivalent.

(i) G is the (topological) inverse limit of finite discrete groups.

(ii) G is compact and the unit element has a basis of neighbourhoods consist-ing of open and closed normal subgroups.

(iii) G is compact and totally disconnected.

Proof: (i) ⇒ (iii): The inverse limit of compact and totally disconnectedspaces is compact and totally disconnected.(ii)⇒ (i): Assume that U runs through a system of neighbourhoods of the unitelement e ∈ G, which consists of open normal subgroups. Then the canonicalhomomorphism φ : G→ lim

←− UG/U is an isomorphism:

To begin with, φ is injective, because G is Hausdorff. In order to show thesurjectivity, let x = {xU}U ∈ lim←− U G/U . Denoting the canonical projectionby φU : G→ G/U , we have the equality

φ−1(x) =⋂U

φ−1U (xU ).

The intersection on the right side is taken over nonempty compact spaces andfinite intersections of these are nonempty. Hence φ−1(x) is nonempty, andtherefore φ is surjective. Furthermore, φ is open, hence a homeomorphism.Finally, for every such U , the groupG/U is discrete and compact, hence finite.(iii) ⇒ (ii): By (1.1.1), the underlying topological space of G is profinite,hence every point has a basis of neighbourhoods consisting of open and closedsubsets. Note that an open subgroup is automatically closed, because it is thecomplement of the union of its (open) nontrivial cosets. Let U be an arbitrarychosen, closed and open neighbourhood of the unit element e ∈ G. Set

V := {v ∈ U |Uv ⊆ U}, H := {h ∈ V |h−1 ∈ V }.We claim that H ⊆ U is an open (and closed) subgroup in G. We first showthat V is open. Fix a point v ∈ V . Then uv ∈ U for every u ∈ U and thereforethere exist neighbourhoods Uu of u and Vu of v, such that UuVu ⊆ U . The opensets Uu cover the compact space U and therefore there exists a finite subcover,Uu1 , . . . , Uun , say. Then

Vv := Vu1 ∩ · · · ∩ Vun




is an open neighbourhood of v contained in V . Hence V is open and alsoH := V ∩ V −1, since the inversion map is a homeomorphism. It remains toshow that H is a subgroup. Trivially e ∈ H and H−1 = H by construction.We now check that xy ∈ H if x, y ∈ H . First we have Uxy ⊆ Uy ⊆ U , and soxy ∈ V . In the same way we obtain y−1x−1 ∈ V , hence xy ∈ H . This provesthat H is an open subgroup of G contained in U . In particular, H has finiteindex in G and there are only finitely many different conjugates of H . Theintersection of these finitely many conjugates is an open, closed and normalsubgroup of G contained in U . 2

(1.1.4) Definition. A Hausdorff topological group G satisfying the equivalentconditions of (1.1.3) is called a profinite group.

Without further mention, homomorphisms between profinite groups are al-ways assumed to be continuous and subgroups are assumed to be closed. Sincea subgroup is the complement of its nontrivial cosets and by the compactnessof G, we see that open subgroups are closed and a closed subgroup is open ifand only if it has finite index. IfH is a (closed) subgroup of the profinite groupG, then the set G/H of coset classes with the quotient topology is a profinitespace. If H is normal, then the quotient G/H is a profinite group in a naturalway.

In principle, all objects and statements of the theory of finite groups havetheir topological analogue in the theory of profinite groups. For example, theprofinite analogues of the Sylow theorems are true (see §6). We make thefollowing

(1.1.5) Definition. A supernatural number is a formal product∏p

pnp ,

where p runs through all prime numbers and, for each p, the exponent np is anon-negative integer or the symbol∞.

Using the unique decomposition into prime powers, we can view any naturalnumber as a supernatural number. We multiply supernatural numbers (eveninfinitely many of them) by adding the exponents. By convention, the sum ofthe exponents is∞ if infinitely many summands are non-zero or if one of thesummands is∞. We also have the notions of l.c.m. and g.c.d. of an arbitraryfamily of supernatural numbers. In particular, any family of natural numbershas an l.c.m., which is, in general, a supernatural number.




(1.1.6) Definition. Let G be a profinite group and let A be an abelian torsiongroup.

(i) The index of a closed subgroup H in G is the supernatural number(G : H) = l.c.m.(G/U : H/H ∩ U ),

where U ranges over all open normal subgroups of G.

(ii) The order of G is defined by#G = (G : 1) = l.c.m.

U#(G/U ).

(iii) The order of A is defined by#A = l.c.m. #B ,

where B ranges over all finite subgroups of A.

Given closed subgroups N ⊆ H ⊆ G, we have(G : N ) = (G : H)(H : N ).

Furthermore, the order #A of an abelian torsion groupA is just the order of theprofinite group Hom(A,Q/ZZ).

(1.1.7) Definition. Let G be a profinite group. An abstract G-module M isan abelian group M together with an action

G×M →M, (g,m) 7→ g(m)such that 1(m) = m, (gh)(m) = g(h(m)) and g(m + n) = g(m) + g(n) for allg, h ∈ G, m,n ∈M .

A topological G-module M is an abelian Hausdorff topological group Mwhich is endowed with the structure of an abstract G-module such that theaction G×M →M is continuous.

For a closed subgroup H ⊆ G we denote the subgroup of H-invariantelements in M by MH , i.e.

MH = {m ∈M | h(m) = m for all h ∈ H}.

(1.1.8) Proposition. Let G be a profinite group and let M be an abstractG-module. Then the following conditions are equivalent:

(i) M is a discrete G-module, i.e. the action G×M →M is continuous forthe discrete topology on M .

(ii) For every m ∈M the subgroup Gm := {g ∈ G | g(m) = m} is open.(iii) M =

⋃MU , where U runs through the open subgroups of G.




Proof: If we restrict the map G ×M → M to G × {m}, then m ∈ M haspre-image Gm × {m}. This shows (i)⇒(ii). The assertion (ii)⇒(iii) is trivialbecausem ∈MGm . Finally, assume that (iii) holds. Let (g,m) ∈ G×M . Thereexists an open subgroup U such thatm ∈MU . Therefore gU ×{m} is an openneighbourhood of (g,m) ∈ G×M mapping to g(m). This shows (i). 2

In this book we are mainly concerned with discrete modules and so the termG-module, without the word “topological” or “abstract”, will always mean adiscrete module.

If (Ai)i∈I is a family of discrete G-modules, then their direct sum⊕

i∈I Ai,endowed with the componentwise G-action g((ai)i∈I) = (g(ai))i∈I , is again adiscrete G-module, but this is not necessarily true for the product. The tensorproduct

A⊗B = A⊗ ZZ B

of two discrete modules endowed with the diagonal action g(a⊗b) = g(a)⊗g(b)is a discrete module. The set Hom(A,B) = Hom ZZ (A,B) becomes an abstractG-module by setting g(φ)(a) = g(φ(g−1(a))). Its subgroup of invariants

HomG(A,B) = Hom(A,B)G

is the set of G-homomorphisms from A to B. If A = AU for some opensubgroup U ⊆ G, then Hom(A,B) is a discrete G-module. This is the case,for example, if G is finite or if A is finitely generated as a ZZ-module.

The groupsZZ,Q,ZZ/nZZ, IFq are always viewed as trivial discreteG-modules,i.e. G-modules with trivial action of G.

So far we have considered totally disconnected compact groups. If A is anytopological group, then the connected component A0 (of the identity) of A isa closed subgroup. We have the following general facts for which we refer to[170] sec. 22, and [15] chap. III, §4.6.

(1.1.9) Proposition. Let A be a locally compact group. Then

(i) A0 is the intersection of all open normal subgroups of A, and A/A0 is thelargest totally disconnected quotient.

(ii) A0 is generated by every open neighbourhood of 1 in A0.

(iii) If A→B is a continuous surjective homomorphism onto the locallycompact group B, then the closure of the image of A0 is the connectedcomponent of 1 in B.




An essential tool for working with locally compact abelian groups is a dualitytheorem due toL. S.PONTRYAGIN. We consider the group IR/ZZ as a topologicalgroup with the quotient topology inherited from IR.

(1.1.10) Definition. Let A be a Hausdorff, abelian and locally compact topo-logical group. We call the group

A∨ := Homcts(A, IR/ZZ)

the Pontryagin dual of A.

Given locally compact topological spaces X, Y , the set of continuous mapsMapcts(X, Y ) carries a natural topology, the compact-open topology. A sub-basis of this topology is given by the sets

UK,U = {f ∈ Mapcts(X, Y ) | f (K) ⊆ U},

where K runs through the compact subsets of X and U runs through the opensubsets of Y . For the proof of the following theorem we refer to [170], th. 5.3or [146], th. 23, [186], th. 1.7.2.

(1.1.11) Theorem (Pontryagin Duality). If A is a Hausdorff abelian locallycompact topological group, then the same is true for A∨ endowed with thecompact-open topology. The canonical homomorphism

A −→ (A∨)∨,

given by a 7−→ τa : A∨ → IR/ZZ, φ 7→ φ(a), is an isomorphism of topologicalgroups. Thus ∨ defines an involutory contravariant autofunctor on the cate-gory of Hausdorff abelian locally compact topological groups which moreovercommutes with limits. Furthermore, ∨ induces equivalences of categories

(abelian compact groups) ∨⇐⇒ (discrete abelian groups)(abelian profinite groups) ∨⇐⇒ (discrete abelian torsion groups).

For an (abstract) abelian group A we use the notation

A∗ = Hom(A,Q/ZZ).

Clearly, if A is a discrete torsion group, then A∨ ∼= A∗ and we will frequentlyalso write A∗ instead of A∨, at least if we are not interested in the topologyof the dual. If A is abelian and profinite, then it is an easy exercise to see




that every continuous homomorphism φ : A → IR/ZZ has finite image. If,moreover, A is topologically finitely generated, then every subgroup of finiteindex is open in A, and hence also in this case A∨ ∼= A∗.

Now assume that we are given a family (Xi)i∈I of Hausdorff, abelian topo-logical groups and let an open subgroup Yi ⊆ Xi be given for almost all i ∈ I(i.e. for all but finitely many indices). For consistency of notation, we putYi = Xi for the remaining indices.

(1.1.12) Definition. The restricted product∏i∈I

(Xi, Yi)

is the subgroup of∏Xi consisting of all (xi)i∈I such thatxi ∈ Yi for almost all i.

The restricted product is a topological group, and a basis of neighbourhoodsof the identity is given by the products∏

j∈J

Uj ×∏i∈I\J

Yi,

where J runs over the finite subsets of I and Uj runs over a basis of neigh-bourhoods of the identity of Xj .

Basic examples of restricted products are the product of groups (Yi = Xi forall i) and the direct sum of discrete groups (Yi = 0 for all i). We will write∏

i∈I

Xi

for short if it is clear from the context what the Yi are. The restricted productis again a Hausdorff, abelian topological group.

(1.1.13) Proposition. If all Xi are locally compact and almost all Yi arecompact, then the restricted product is again an abelian locally compact group.

For the Pontryagin dual of the restricted product, there is a canonical iso-morphism

(∏i∈I

(Xi, Yi) )∨ ∼=∏i∈I

(X∨i , (Xi/Yi)∨).

Proof: The product of compact topological spaces is compact, therefore therestricted product is locally compact under the given conditions. Furthermore,since Yi is compact and open inXi, the same is true for (Xi/Yi)∨ inX∨i . Hencealso

∏i∈I

(X∨i , (Xi/Yi)∨) exists and is locally compact.




A sufficiently small open neighbourhood of 0 ∈ IR/ZZ contains no nontrivialsubgroups. Therefore a continuous homomorphism φ :

∏(Xi, Yi)→ IR/ZZ

annihilates Yi for almost all i. In other words, the restriction of φ to Xi lies inthe subgroup (Xi/Yi)∨ ⊆ X∨i for almost all i. This yields a bijection between(∏

i∈I(Xi, Yi) )∨ and

∏i∈I

(X∨i , (Xi/Yi)∨), which easily can be seen to be a

homeomorphism. 2

Exercise 1. Show that an injective (resp. surjective) continuous map between profinite spacesmay be represented as an inverse limit over a system of injective (resp. surjective) mapsbetween finite discrete spaces.

Exercise 2. Let X be a profinite space and let X0 ⊆ X be a closed subspace. Show thatevery continuous map f : X0 → Y from X0 to a finite discrete space Y has a continuousextension F : X → Y (i.e. F |X0 = f ) and that any two such extensions coincide on an openneighbourhood of X0 in X .

Exercise 3. Let G,H be profinite groups. Show that

Hom(G,H) = lim←−V ⊆H

lim−→U⊆G

Hom(G/U,H/V ),

where the limits are taken over all open normal subgroups V of H and U of G.

Exercise 4. If K ⊆ H are closed subgroups of the profinite group G, then the projectionπ : G/K → G/H has a continuous section s : G/H → G/K.Hint: LetX be the set of pairs (S, s), where S is a closed subgroup such thatK ⊆ S ⊆ H ands is a continuous section s : G/H → G/S. Write (S, s) ≤ (S′, s′) if S′ ⊆ S and if s is thecomposite of s′ and the projection G/S′ → G/S. Then X is inductively ordered. By Zorn’slemma, there exists a maximal element (S, s) of X . Show that S = K.

Exercise 5. A morphism φ : X → Y in a category C is called a monomorphism if for everyobject Z of C and for every pair of morphisms f, g : Z → X the implication "φ ◦ f = φ ◦ g⇒ f = g" is true. The morphism φ is called an epimorphism if it is a monomorphism in theopposite category Cop (the category obtained from C by reversing all arrows).(i) Show that the monomorphisms in the category of profinite groups are the injective homo-morphisms.(ii) Show that the epimorphisms in the category of profinite groups are the surjective homo-morphisms.Hint for (ii): First reduce the problem to the case of finite groups. Assume that there is anepimorphism φ : G→ H of finite groups which is not surjective. Assume that (H : φ(G)) ≥ 3(otherwise φ(G) is normal inH) and choose two elements a, b ∈ H having different nontrivialresidue classes modulo φ(G). Let S be the (finite) group of set theoretic automorphisms ofH .Let s ∈ S be the map H → H which interchanges the cosets aφ(G) and bφ(G) and which isthe identity on the other left cosets modulo φ(G). Then consider the maps f and g defined byf (h1)(h2) = h2h−11 and by g(h) = s

−1f (h)s.




§2. Definition of the Cohomology Groups

The cohomology of a profinite group G arises from the diagram

· · · −→−→−→−→ G×G×G−→−→−→ G×G −→−→ G,

the arrows being the projections

di : Gn+1−→Gn, i = 0, 1, . . . , n,given by

di(σ0, . . . , σn) = (σ0, . . . , σ̂i, . . . , σn),

where by σ̂i we indicate that we have omitted σi from the (n + 1)-tuple(σ0, . . . , σn). G acts on Gn by left multiplication.

From now on, we assume allG-modules to be discrete. For everyG-moduleA we form the abelian group

Xn = Xn(G,A) = Map (Gn+1, A)

of all continuous maps x : Gn+1−→A, i.e. of all continuous functionsx(σ0, . . . , σn) with values in A. Xn is in a natural way a G-module by

(σx)(σ0, . . . , σn) = σx(σ−1σ0, . . . , σ−1σn).

The maps di : Gn+1−→Gn induce G-homomorphisms d∗i : Xn−1−→Xn andwe form the alternating sum

∂n =n∑i=0

(−1)id∗i : Xn−1−→Xn.

We usually write ∂ in place of ∂n. Thus for x ∈ Xn−1, ∂x is the function

(∗) (∂x)(σ0, . . . , σn) =n∑i=0

(−1)i x(σ0, . . . , σ̂i, . . . , σn).

Moreover, we have the G-homomorphism ∂0 : A → X0, which associates toa ∈ A the constant function x(σ0) = a.

(1.2.1) Proposition. The sequence

0−→A ∂0

−→X0 ∂1

−→X1 ∂2

−→X2−→ . . .is exact.

Proof: We first show that it is a complex, i.e. ∂∂ = 0. ∂1 ◦ ∂0 = 0 isclear. Let x ∈ Xn−1. Applying ∂ to (∗), we obtain summands of the formx(σ0, . . . , σ̂i, . . . , σ̂j, . . . , σn) with certain signs. Each of these summandsarises twice, once where first σj and then σi is omitted, and again where first



§2. Definition of the Cohomology Groups 13

σi and then σj is omitted. The first time the sign is (−1)i(−1)j and the secondtime (−1)i(−1)j−1. Hence the summands cancel to give zero.

For the exactness, we consider the map D−1 : X0 → A, D−1x = x(1), andfor n ≥ 0 the maps

Dn : Xn+1−→Xn, (Dnx)(σ0, . . . , σn) = x(1, σ0, . . . , σn).These are homomorphisms of ZZ-modules, not of G-modules. An easy calcu-lation shows that for n ≥ 0

(∗) Dn ◦ ∂n+1 + ∂n ◦Dn−1 = id.If x ∈ ker(∂n+1) then x = ∂n(Dn−1x), i.e. ker(∂n+1) ⊆ im(∂n) and thusker(∂n+1) = im(∂n) because ∂n+1 ◦ ∂n = 0. 2

An exact sequence of G-modules 0 → A → X0 → X1 → X2 → . . . iscalled a resolution ofA and a family (Dn)n≥−1 as in the proof with the property(∗) is called a contracting homotopy of it. The above resolution is called thestandard resolution.

We now apply the functor “fixed module”. We set for n ≥ 0

Cn(G,A) = Xn(G,A)G.

Cn(G,A) consists of the continuous functions x : Gn+1 → A such thatx(σσ0, . . . , σσn) = σx(σ0, . . . , σn)

for all σ ∈ G. These functions are called the (homogeneous) n-cochains ofG with coefficients in A. From the standard resolution (1.2.1) we obtain asequence

C0(G,A) ∂1

−→ C1(G,A) ∂2

−→ C2(G,A)−→ . . . ,

which in general is no longer exact. But it is still a complex, i.e. ∂∂ = 0, andis called the homogeneous cochain complex of G with coefficients in A.

We now setZn(G,A) = ker (Cn(G,A) ∂

n+1

−→ Cn+1(G,A)),Bn(G,A) = im (Cn−1(G,A) ∂

n

−→ Cn(G,A))

and B0(G,A) = 0. The elements of Zn(G,A) and Bn(G,A) are called the(homogeneous) n-cocycles and n-coboundaries respectively. As ∂∂ = 0, wehave Bn(G,A) ⊆ Zn(G,A).

(1.2.2) Definition. For n ≥ 0 the factor group

Hn(G,A) = Zn(G,A)/Bn(G,A)

is called the n-dimensional cohomology group of G with coefficients in A.




For computational purposes, and for many applications, it is convenient topass to a modified definition of the cohomology groups, which reduces thenumber of variables in the homogeneous cochains x(σ0, . . . , σn) by one. LetC 0(G,A) = A and C n(G,A), n ≥ 1, be the abelian group of all continuousfunctions y : Gn−→A. We then have the isomorphism

C0(G,A)−→C 0(G,A), x(σ) 7−→ x(1),

and for n ≥ 1 the isomorphism

Cn(G,A)−→C n(G,A),x(σ0, . . . , σn) 7→ y(σ1, . . . , σn) = x(1, σ1, σ1σ2, . . . , σ1 · · ·σn),

whose inverse is given by

y(σ1, . . . , σn) 7→ x(σ0, . . . , σn) = σ0y(σ−10 σ1, σ−11 σ2, . . . , σ−1n−1σn).

With these isomorphisms the coboundary operators ∂n+1 : Cn(G,A)−→Cn+1(G,A) are transformed into the homomorphisms

∂n+1 : C n(G,A)−→C n+1(G,A)

given by

(∂1a)(σ) = σa− a for a ∈ A = C 0(G,A),(∂2y)(σ, τ ) = σy(τ )− y(στ ) + y(σ) for y ∈ C 1(G,A),(∂n+1y)(σ1, . . . , σn+1) = σ1y(σ2, . . . , σn+1)

+n∑i=1

(−1)iy(σ1, . . . , σi−1, σiσi+1, σi+2, . . . , σn+1)

+(−1)n+1y(σ1, . . . , σn) for y ∈ C n(G,A).

SettingZ n(G,A) = ker (C n(G,A) ∂

n+1

−→ C n+1(G,A))Bn(G,A) = im (C n−1(G,A) ∂

n

−→ C n(G,A)) ,

the isomorphisms Cn(G,A) −→∼ C n(G,A) induce isomorphisms

Hn(G,A) ∼= Z n(G,A)/Bn(G,A).

The functions in C n(G,A),Z n(G,A),Bn(G,A) are called the inhomo-geneous n-cochains, n-cocycles and n-coboundaries. The inhomogeneouscoboundary operators ∂n+1 are more complicated than the homogeneous ones,but they have the advantage of dealing with only n variables instead of n + 1.

For n = 0, 1, 2 the groups Hn(G,A) admit the following interpretations.




The group H0(G,A): We have a natural isomorphism C0(G,A)→A,x 7→ x(1), by which we identify C0(G,A) with A. Then, for a ∈ A,(∂1a)(σ0, σ1) = σ1a−σ0a, or (∂1a)(σ) = σa− a in the inhomogeneous setting,so that

H0(G,A) = AG.

The group H1(G,A): The inhomogeneous 1-cocycles are the continuousfunctions x : G −→ A such that

x(στ ) = x(σ) + σx(τ ) for all σ, τ ∈ G.

They are also called crossed homomorphisms. The inhomogeneous1-coboundaries are the functions

x(σ) = σa− awith a fixed a ∈ A. If G acts trivially on A, then

H1(G,A) = Homcts(G,A). ∗)

The groupH1(G,A) occurs in a natural way if we pass from an exact sequence

0 −→ A i−→ B j−→ C −→ 0of G-modules to the sequence of fixed modules. Then we lose the exactnessand are left only with the exactness of the sequence

0 −→ AG −→ BG −→ CG.The groupH1(G,A) now gives information about the deviation from exactness.In fact we have a canonical homomorphism

δ : CG −→ H1(G,A)extending the above exact sequence to a longer one. Namely, for c ∈ CG wemay choose an element b ∈ B such that jb = c. For each σ ∈ G there is anaσ ∈ A such that iaσ = σb − b. The function σ 7→ aσ is a 1-cocycle andwe define δc to be the cohomology class of this 1-cocycle in H1(G,A). Thedefinition is easily seen to be independent of the choice of the element b. Ifδc = 0, then aσ = i−1(σb− b) = σa− a, a ∈ A, so that b′ = b− ia is an elementof BG with jb′ = c. This shows the exactness of the sequence

0 −→ AG −→ BG −→ CG δ−→ H1(G,A).We shall meet this again in a larger frame in §3.

The group H1(G,A) admits a concrete interpretation using the concept oftorsors. Since this concept may be more fully exploited in the framework ofnon-abelian groups A, we generalize H1(G,A) as follows.

∗)Since G is automatically understood as a topological group, we usually write Hom(G,A)instead of Homcts(G,A).




A G-group A is a not necessarily abelian group with the discrete topologyon which G acts continuously. We denote the action of σ ∈ G on a ∈ A byσa, so that σ(ab) = σa σb. A cocycle of G with coefficients in A is a continuousfunction σ 7→ aσ on G with values in the group A such that

aστ = aσ σaτ .

The set of cocycles is denoted by Z 1(G,A). Two cocycles a, a′ are said tobe cohomologous if there exists a b ∈ A such that a′σ = b

−1aσσb. This is an

equivalence relation in Z 1(G,A) and the quotient set is denoted by H1(G,A).It has a distinguished element given by the cocycle aσ = 1.

AG-set is a discrete topological spaceX with a continuous action ofG. LetA be aG-group. AnA-torsor is aG-setX with a simply transitive right actionX × A −→ X, (x, a) 7→ xa, of A which is compatible with the G-action onX . This means that for every pair x, y ∈ X there is a unique a ∈ A such thaty = xa, and σ(xa) = σxσa. For example, if

1 −→ A −→ B j−→ C −→ 1

is an exact sequence of G-groups, then the cosets j−1(c) for c ∈ CG are typicalA-torsors. It is clear what we mean by an isomorphism of A-torsors. Letnow TORS (A) denote the set of isomorphism classes of A-torsors. It has adistinguished element given by the A-torsor A, and is thus a pointed set.

(1.2.3) Proposition. We have a canonical bijection of pointed sets

H1(G,A) ∼= TORS (A).

Proof: We define a map

λ : TORS (A) −→ H1(G,A)

as follows. Let X be an A-torsor and let x ∈ X . For every σ ∈ G there is aunique aσ ∈ A such that σx = xaσ. One verifies at once that aσ is a cocycle.Changing x to xb changes this cocycle to b−1aσσb, which is cohomologous.We define λ(X) to be the class of aσ.

We define an inverse µ : H1(G,A) −→ TORS (A) as follows. Let the set Xbe the group A. We let G act on X in the twisted form

σ ′x = aσ · σx.

The action of A on X is given by right multiplication. In this way, X becomesan A-torsor and this defines the map µ. Replacing aσ by b−1aσσb, we havean isomorphism x 7→ b−1x of A-torsors. One now checks that λ ◦ µ = 1 andµ ◦ λ = 1. 2




Remark: If 0 −→ A i−→ B j−→ C −→ 0 is an exact sequence ofG-modules and if we identify in the exact sequence

0 −→ AG −→ BG −→ CG δ−→ H1(G,A)

H1(G,A) with TORS (A), then the map δ is given by δc = j−1(c).

The group H2(G,A): We return to the case that A is abelian. The inhomo-geneous 2-cocycles are the continuous functions x : G × G −→ A such that∂x = 0, i.e.

x(στ, ρ) + x(σ, τ ) = x(σ, τρ) + σx(τ, ρ).

Among these we find the inhomogeneous 2-coboundaries as the functions

x(σ, τ ) = y(σ)− y(στ ) + σy(τ )

with an arbitrary 1-cochain y : G −→ A.The 2-cocycles had been known before the development of group cohomo-

logy as factor systems and occurred in connection with group extensions. Toexplain this, we assume that eitherA orG is finite, in order to avoid topologicalproblems (but see (2.7.7)).

The question is: how many groups Ĝ are there, which have theG-module Aas a normal subgroup and G as the factor group (we write A multiplicatively).To be more precise, we consider all exact sequences

1 −→ A −→ Ĝ −→ G −→ 1

of topological groups (i.e. of profinite groups if A is finite, and of discretegroups if G is finite), such that the action of G on A is given by

σa = σ̂aσ̂−1,

where σ̂ ∈ Ĝ is a pre-image of σ ∈ G. If��f

1GĜA1

1GĜ′A1

is a commutative diagram of such sequences with a topological isomorphism f ,then we call these sequences equivalent, and we denote the set of equivalenceclasses [Ĝ] by EXT(G,A). This set has a distinguished element given by thesemi-direct product Ĝ = A o G (see ex.1 below).

(1.2.4) Theorem (SCHREIER). We have a canonical bijection of pointed sets

H2(G,A) ∼= EXT(G,A).




Proof: We define a mapλ : EXT(G,A) −→ H2(G,A)

as follows. Let the class [Ĝ] ∈ EXT(G,A) be represented by the exact sequence

1 −→ A −→ Ĝ −→ G −→ 1.We choose a continuous section s : G −→ Ĝ of Ĝ −→ G, and we set σ̂ = s(σ).Such a section exists (see §1, ex.4). Regarding A as a subgroup of Ĝ, everyγ̂ ∈ Ĝ has a unique representation

γ̂ = aσ̂, a ∈ A, σ ∈ G,

and we haveσ̂a = σ̂aσ̂−1σ̂ = σaσ̂.

The elements σ̂τ̂ and σ̂τ are both mapped onto στ , i.e.

σ̂τ̂ = x(σ, τ )σ̂τ ,

with an element x(σ, τ ) ∈ A such that x(σ, 1) = x(1, σ) = 1. Since σ̂ is acontinuous function of σ and A is closed in Ĝ, x(σ, τ ) is a continuous mapx : G × G −→ A. The associativity (σ̂τ̂ )ρ̂ = σ̂(τ̂ ρ̂) yields that x(σ, τ ) is a2-cocycle:

(σ̂τ̂ )ρ̂ = x(σ, τ )σ̂τ ρ̂ = x(σ, τ )x(στ, ρ)(στρ)̂ ,

σ̂(τ̂ ρ̂) = σ̂x(τ, ρ)τ̂ ρ = σx(τ, ρ)σ̂τ̂ρ = σx(τ, ρ)x(σ, τρ)(στρ)̂ ,

i.e.x(σ, τ )x(στ, ρ) = σx(τ, ρ)x(σ, τρ).

We thus get a cohomology class c = [x(σ, τ )] ∈ H2(G,A). This class does notdepend on the choice of the continuous section s : G −→ Ĝ. If s′ : G −→ Ĝis another one, and if we set σ̃ = s′(σ), then σ̃ = y(σ)σ̂, y(σ) ∈ A, andσ̃τ̃ = x̃(σ, τ )σ̃τ . For the 2-cocycle x̃(σ, τ ) we obtain

σ̃τ̃ = x̃(σ, τ )y(στ )σ̂τ = x̃(σ, τ )y(στ )x(σ, τ )−1σ̂τ̂

= x̃(σ, τ )x(σ, τ )−1y(στ )y(σ)−1 σ̃y(τ )−1τ̃

= x̃(σ, τ )x(σ, τ )−1y(στ )y(σ)−1 σy(τ )−1σ̃τ̃ ,

i.e. x̃(σ, τ ) = x(σ, τ )y(σ, τ ) with the 2-coboundary

y(σ, τ ) = y(σ)y(στ )−1 σy(τ ).

The cohomology class c = [x(σ, τ )] also does not depend on the choice of therepresentative 1 −→ A −→ Ĝ −→ G −→ 1 in the class [Ĝ]. Namely, if��

f

��1GĜ′A1

1GĜA1




is a commutative diagram and σ̂′ = f (σ̂), then

σ̂′τ̂ ′ = f (σ̂)f (τ̂ ) = f (σ̂τ̂ ) = f (x(σ, τ )σ̂τ ) = x(σ, τ )(σ̂τ )′,

i.e. the group extensions Ĝ′ and Ĝ yield the same 2-cocycle x(σ, τ ). We thusget a well-defined map

λ : EXT(G,A) −→ H2(G,A).In order to prove the bijectivity, we construct an inverse µ : H2(G,A) −→EXT(G,A). Every cohomology class c ∈ H2(G,A) contains a normalized2-cocycle x(σ, τ ), i.e. a cocycle such that

x(σ, 1) = x(1, σ) = 1.

Namely, if x(σ, τ ) is any 2-cocycle in c, then we obtain from the equalityx(στ, ρ)x(σ, τ ) = x(σ, τρ) σx(τ, ρ) that

x(σ, 1) = σx(1, 1), x(1, ρ) = x(1, 1).

Setting y(σ) = x(1, 1) for all σ ∈ G, we obtain a 2-coboundary

y(σ, τ ) = y(σ)y(στ )−1 σy(τ ) ,

and the 2-cocycle x′(σ, τ ) = x(σ, τ )y(σ, τ )−1 has the property that

x′(σ, 1) = x(σ, 1)(σx(1, 1))−1 = 1, x′(1, τ ) = x(1, τ )x(1, 1)−1 = 1.

Let now x(σ, τ ) be a normalized 2-cocycle in c. On the set Ĝ = A × G withthe product topology we define the continuous multiplication

(a, σ)(b, τ ) = (x(σ, τ )a σb, στ ).

This product is associative because of the cocycle property:((a, σ)(b, τ ))(c, ρ) = (x(σ, τ )a σb, στ )(c, ρ)

= (x(στ, ρ)x(σ, τ )a σb στc, στρ) = (x(σ, τρ) σx(τ, ρ)a σb στc, στρ)

= (a, σ)(x(τ, ρ)b τc, τρ) = (a, σ)((b, τ ), (c, ρ)).

(1,1) is an identity element:

(a, σ)(1, 1) = (x(σ, 1)a, σ) = (a, σ) = (x(1, σ)a, σ) = (1, 1)(a, σ)

and ([σ−1x(σ, σ−1) σ−1a]−1, σ−1) is an inverse of (a, σ) since

(a, σ)([σ−1x(σ, σ−1) σ

−1a]−1, σ−1) = (a σ(σ

−1a)−1, σσ−1) = (1, 1).

In this way Ĝ = A × G becomes a group with the product topology, and themaps a 7→ (a, 1) and (a, σ) 7→ σ yield an exact sequence

1 −→ A −→ Ĝ −→ G −→ 1.Setting σ̂ = (1, σ), we have σ̂−1 = (σ−1x(σ, σ−1)−1, σ−1) and

σ̂(a, 1)σ̂−1 = (x(σ, 1) σa, σ)(σ−1x(σ, σ−1)−1, σ−1) = (σa, 1).

We thus obtain an element [Ĝ] in EXT(G,A). This element does not dependon the choice of the normalized 2-cocycle x(σ, τ ) in c. For, if x′(σ, τ ) =x(σ, τ )y(σ, τ )−1 is another one, y(σ, τ ) = y(σ)y(στ )−1 σy(τ ) is a 2-coboundary,




and if Ĝ′ is the group given by the multiplication onA×G via x′(σ, τ ), then themap f : (a, σ) 7→ (y(σ)a, σ) is an isomorphism from Ĝ to Ĝ′ and the diagram��

f

� 1GĜ′A1

1GĜA1

is commutative, noting that y(1) = 1 because 1 = x′(1, σ) = x(1, σ)y(1)−1

= y(1)−1. Therefore [Ĝ] = [Ĝ′], and we get a well-defined map

µ : H2(G,A) −→ EXT(G,A).This map is inverse to the map λ constructed before. For, if x(σ, τ ) is the2-cocycle produced by a section G −→ Ĝ, σ 7→ σ̂, of a group extension

1 −→ A −→ Ĝ −→ G −→ 1,then the map f : (a, σ) 7→ aσ̂ is an isomorphism of the group A×G, endowedwith the multiplication given by x(σ, τ ), onto Ĝ. This proves the theorem.

2

It is a significant feature of cohomology theory that we don’t have concreteinterpretations of the groups Hn(G,A) for dimensions n ≥ 3 in general. Thisdoes, however, not at all mean that they are uninteresting. Besides their naturalappearance, the importance of the higher dimensional cohomology groups isseen in the fact that the theory endows them with an abundance of homomorphicconnections, with which one obtains important isomorphism theorems. Thesetheorems give concrete results for the interesting lower dimensional groups,whose proofs, however, have to take the cohomology groups of all dimensionsinto account.

Next we show that the cohomology groups Hn(G,A) of a profinite group Gwith coefficients in a G-module A are built up in a simple way from those ofthe finite factor groups of G. Let U, V run through the open normal subgroupsof G. If V ⊆ U , then the projections

Gn+1 −→ (G/V )n+1 −→ (G/U )n+1

induce homomorphisms

Cn(G/U,AU ) −→ Cn(G/V,AV ) −→ Cn(G,A),which commute with the operators ∂n+1. We therefore obtain homomorphisms

Hn(G/U,AU ) −→ Hn(G/V,AV ) −→ Hn(G,A).The groups Hn(G/U,AU ) thus form a direct system and we have a canonicalhomomorphism lim

−→ UHn(G/U,AU ) −→ Hn(G,A).




(1.2.5) Proposition. The above homomorphism is an isomorphism:

lim−→U

Hn(G/U,AU ) −→∼ Hn(G,A) .

Proof: Already the homomorphismlim−→U

C.(G/U,AU ) −→ C.(G,A)

is an isomorphism of complexes. The injectivity is clear, since the maps

C.(G/U,AU )→C.(G,A)are injective.

Let conversely x : Gn+1 −→ A be an n-cochain of G. Since A is discrete, xis locally constant. We conclude that there exists an open normal subgroup U0of G such that x is constant on the cosets of Un+10 in G

n+1. It takes values inAU0 , since for all σ ∈ U0 we have

x(σ0, . . . , σn) = x(σσ0, . . . , σσn) = σx(σ0, . . . , σn).

Hence x is the composite ofGn+1 −→ (G/U0)n+1

xU0−→ AU0

with an n-cochain xU0 of G/U0, and is therefore the image of the element inlim−→ U

Cn(G/U,AU ) defined by xU0 . This shows the surjectivity. Since thefunctor lim

−→is exact, we obtain the isomorphisms

lim−→U

Hn(G/U,AU ) ∼= Hn(lim−→U

C.(G/U,AU ))∼= Hn(C.(G,A))

= Hn(G,A). 2

Finally we introduce Tate cohomology. We do this for finite groups here,and will extend the theory to profinite groups in §8. Let G be a finite group forthe remainder of this section.

We consider the norm residue group

Ĥ0(G,A) = AG/NGA,

where NGA is the image of the norm map∗)

NG : A−→A, NG a =∑σ∈G

σa.

∗)The name “norm” is chosen instead of “trace”, because in Galois cohomology this mapwill often be written multiplicatively, i.e. NG a =

∏σ∈G

σa.




We call the groups

Ĥn(G,A) ={AG/NGA for n = 0,

Hn(G,A) for n ≥ 1

the modified cohomology groups. We also obtain these groups from a com-plex. Namely, we extend the standard complex (Cn(G,A))n≥0 to

Ĉ.(G,A) : C−1(G,A) ∂0−→ C0(G,A) ∂1−→ C1(G,A) ∂2−→ . . . ,where C−1(G,A) = C0(G,A) and ∂0x is the constant function with value∑σ∈G x(σ). We then obtain the modified cohomology groups for all n ≥ 0 as

the cohomology groups of this complex,

Ĥn(G,A) = Hn(Ĉ.(G,A)).Besides the fixed module AG, we have also a “cofixed module” AG = A/IGA,where IGA is the subgroup of A generated by all elements of the formσa − a, a ∈ A, σ ∈ G. AG is the largest quotient of A on which G actstrivially. We set

H0(G,A) = AG.

If G is a finite group, then IGA is contained in the group

NGA = {a ∈ A | NGa = 0},

and we setĤ0(G,A) = NGA/IGA.

The norm NG : A−→A induces a map NG : H0(G,A) −→ H0(G,A), andthe proof of the following proposition is obvious.

(1.2.6) Proposition. We have an exact sequence

0 −→ Ĥ0(G,A) −→ H0(G,A)NG−→ H0(G,A) −→ Ĥ0(G,A) −→ 0.

The group Ĥ0(G,A) is very often denoted by Ĥ−1(G,A) for the followingreason. For a finite group G one can define cohomology groups Ĥn(G,A) forarbitrary integral dimensions n ∈ ZZ as follows:

For n ≥ 0, let ZZ[Gn+1] be the abelian group of all formal ZZ-linear combi-nations

∑a(σ0,...,σn)(σ0, . . . , σn), σ0, . . . , σn ∈ G, with its obvious G-module

structure. We consider the (homological) complete standard resolution ofZZ, i.e. the sequence of G-modules X. = X.(G,ZZ)

. . . −→ X2∂2−→ X1

∂1−→ X0∂0−→ X−1

∂−1−→ X−2 −→ . . .




where Xn = X−1−n = ZZ[Gn+1] for n ≥ 0, and the differentials are defined forn > 0 by

∂n(σ0, . . . , σn) =n∑i=0

(−1)i(σ0, . . . , σi−1, σi+1, . . . , σn)

∂−n(σ0, . . . , σn−1) =∑τ∈G

n∑i=0

(−1)i(σ0, . . . , σi−1, τ, σi, . . . , σn−1) ,

while ∂0 : X0→X−1 is given by

∂0(σ0) =∑τ∈G

τ .

The (cohomological) complete standard resolution of A is defined as thesequence of G-modules X. = X.(G,A) = Hom(X., A)

. . . −→ X−2 ∂−1−→ X−1 ∂

0

−→ X0 ∂1

−→ X1 ∂2

−→ X2 −→ . . .

where X−1−n = Xn = Hom(Xn, A) = Map(Gn+1, A) for n ≥ 0 and ∂n =Hom(∂n, A) for n ∈ ZZ. X. is a complex. Using the maps

D−n : X−n+1 −→ X−n

given by

(Dnx)(σ0, . . . , σn) = x(1, σ0, . . . , σn) for n ≥ 0,

(D−1x)(σ0) = δσ0,1x(1)∗) for n = 1,

(D−nx)(σ0, . . . , σn−1) = δσ0,1x(σ1, . . . , σn−1) for n ≥ 2,

we get

Dn ◦ ∂n+1 + ∂n ◦Dn−1 = id

for all n ∈ ZZ. From this we conclude that the above complex X. is exact.For every n ∈ ZZ, we now define then-th Tate cohomology group Ĥn(G,A)

as the cohomology group of the complex

Ĉ.(G,A) = ((Xn)G)n∈ ZZat the place n:

Ĥn(G,A) = Hn(Ĉ.(G,A)).Clearly, for n ≥ 0 we get the previous (modified) cohomology groups, and itis immediate to see that Ĥ−1(G,A) is our group Ĥ0(G,A) = NGA/IGA. Moregenerally, the Tate cohomology in negative dimensions can be identified withhomology (see §8).

∗) i.e. δσ,τ = 0 if σ /= τ and δσ,σ = 1.




Exercise 1. Let G be a profinite group and A a G-group. Assume that either G or A is finite.The semi-direct product is a group

Ĝ = Ao GcontainingA andG such that every element of Ĝ has a unique presentation aσ, a ∈ A, σ ∈ G,and (aσ)(a′σ′) = a σa′σσ′. We then have a group extension

1→ A→ Ĝ π→ G→ 1and the inclusion G ↪→ Ĝ is a homomorphic section of π. Two homomorphic sectionss, s′ : G→ Ĝ of π are conjugate if there is an a ∈ A such that s′(σ) = as(σ)a−1 for all σ ∈ G.Let SEC (Ĝ→ G) be the set of conjugacy classes of homomorphic sections of Ĝ π→ G. Thenthere is a canonical bijection of pointed sets

H1(G,A) ∼= SEC (Ĝ→ G).

Exercise 2. There is the following interpretation of H3(G,A). Consider all possible exactsequences

1 −→ A i−→ N α−→ Ĝ π−→ G −→ 1,whereN is a group with an action σ̂ : ν 7→ σ̂ν of Ĝ satisfying α(ν)ν′ = νν′ν−1, ν, ν′ ∈ N , andα(σ̂ν) = σ̂α(ν)σ̂−1, ν ∈ N, σ̂ ∈ Ĝ. Impose on the set of all such exact sequences the smallestequivalence relation such that

1 −→ A −→ N −→ Ĝ −→ G −→ 1is equivalent to

1 −→ A −→ N ′ −→ Ĝ′ −→ G −→ 1,whenever there is a commutative diagram!"#$%&'()*

Ĝ′N ′

1GA1

ĜN

in which the vertical arrows are compatible with the actions of Ĝ and Ĝ′ on N and N ′ (butneed not be bijective). If EXT 2(G,A) denotes the set of equivalence classes, then we have acanonical bijection

EXT 2(G,A) ∼= H3(G,A)(see [18], chap. IV, th. 5.4).

Exercise 3. Let G be finite and let (Ai)i∈I be a family of G-modules. Show that

Hr(G,∏i∈I

Ai) =∏i∈I

Hr(G,Ai)

for all r ≥ 0.

Exercise 4. An inhomogeneous cochain x ∈ Cn(G,A), n ≥ 1, is called normalized ifx(σ1, . . . , σn) = 0 whenever one of the σi is equal to 1. Show that every class in Hn(G,A) isrepresented by a normalized cocycle.

Hint: Construct inductively cochains x0, x1, . . . , xn ∈ Cn(G,A) and y1, . . . , yn ∈Cn−1(G,A) such that

x0 = x, xi = xi−1 − ∂yi, i = 1, . . . , n,

yi(σ1, . . . , σn−1) = (−1)i−1xi−1(σ1, . . . , σi−1, 1, σi, . . . , σn−1).Then xn is normalized and x− xn is a coboundary.



§3. The Exact Cohomology Sequence 25

§3. The Exact Cohomology Sequence

Having introduced the cohomology groups Hn(G,A), we now turn to thequestion of how they behave if we change the G-module A. If

f : A −→ Bis a homomorphism of G-modules, i.e. a homomorphism such that f (σa)= σf (a) for a ∈ A, σ ∈ G, then we have the induced homomorphism

f : Cn(G,A)→ Cn(G,B), x(σ0, . . . , σn) 7→ fx(σ0, . . . , σn),and the commutative diagram+, ∂n+1-./

f

0∂n+1

1f

2· · ·Cn+1(G,B)Cn(G,B)· · ·

· · ·Cn+1(G,A)Cn(G,A)· · ·

.

In other words, f : A −→ B induces a homomorphismf : C.(G,A) −→ C.(G,B)

of complexes. Taking homology groups of these complexes, we obtain homo-morphisms

f : Hn(G,A) −→ Hn(G,B).Besides these homomorphisms there is another homomorphism, the “connect-ing homomorphism”, which is less obvious, but is of central importance incohomology theory. For its definition we make use of the following generallemma, which should be seen as the crucial point of homological algebra.

(1.3.1) Snake lemma. Let3 i4 j567i′

8j′

9α

:β

;γ

C ′B′A′0

0CBA

be a commutative diagram of abelian groups with exact rows. We then have acanonical exact sequence j?@A δBCD i′E j′F coker(j′).coker(γ)coker(β)coker(α) ker(γ)ker(β)ker(α)ker(i)




Proof: The existence and exactness of the upper and the lower sequence isevident. The crucial cohomological phenomenon is the natural, but slightlyhidden, appearance of the homomorphism

δ : ker(γ) −→ coker(α).

It is obtained as follows. Let c ∈ ker(γ). Let b ∈ B and a′ ∈ A′ be elementssuch that

jb = c and i′a′ = βb.

The element b exists since j is surjective and a′ exists (and is uniquely deter-mined by b) since j′βb = γjb = γc = 0. We define

δc := a′ mod α(A).

This definition does not depend on the choice of b, since if b̃ ∈ B is anotherelement such that jb̃ = c and i′ã′ = βb̃, ã′ ∈ A′, then j(b̃ − b) = 0, i.e.b̃ − b = ia, a ∈ A, so that i′(ã′ − a′) = β(b̃ − b) = βia = i′αa, and thusã′ − a′ = αa, i.e. ã′ ≡ a′ mod α(A).

Exactness at ker(γ): δc = 0 means a′ = αa, a ∈ A, which implies β(b− ia) =i′a′ − i′αa = 0, i.e. b− ia ∈ ker(β) and j(b− ia) = c.

Exactness at coker(α): Let a′ ∈ A′ such that i′a′ ≡ 0 mod β(B), i.e. i′a′ =βb, b ∈ B. Setting c = jb, we have by definition δc = a′ mod α(A). 2

We now show that every exact sequence of G-modules

0 −→ A i−→ B j−→ C −→ 0

gives rise to a canonical homomorphism

δ : Hn(G,C) −→ Hn+1(G,A)

for every n ≥ 0. We consider the commutative diagramGHIJK∂A

LMNOP∂B

Q∂C

0.Cn+1(G,C)Cn+1(G,B)Cn+1(G,A)0

0Cn(G,C)Cn(G,B)Cn(G,A)0

It is exact, which is seen by passing to the inhomogeneous cochains (see alsoex.1). By the snake lemma, we obtain a homomorphism

δ : ker(∂C) −→ coker(∂A).




(1.3.2) Theorem. For every exact sequence 0 −→ A → B → C → 0 ofG-modules, the above homomorphism δ induces a homomorphism

δ : Hn(G,C) −→ Hn+1(G,A)and we obtain an exact sequence

0 −→ AG −→ BG −→ CG δ−→ H1(G,A)−→· · ·

· · · −→ Hn(G,A) −→ Hn(G,B) −→ Hn(G,C) δ−→ Hn+1(G,A) −→ · · · .

Proof: Setting Cn(G,A) = Cn(G,A)/Bn(G,A) and similarly for B and Cin place of A, we obtain from the above diagram the commutative diagramRSTU

∂A

VWXY∂B

Z∂C

Zn+1(G,C),Zn+1(G,B)Zn+1(G,A)0

0Cn(G,C)Cn(G,B)Cn(G,A)

which is obviously exact. Noting that

ker(∂A) = Hn(G,A) and coker(∂A) = Hn+1(G,A),

the snake lemma yields an exact sequence[\]^_ δ`abc Hn+1(G,C).Hn+1(G,B)Hn+1(G,A) Hn(G,C)Hn(G,B)Hn(G,A)This proves the theorem. 2

The homomorphism δ : Hn(G,C) −→ Hn+1(G,A) is called the connectinghomomorphism, or simply the δ-homomorphism, and the exact sequence inthe theorem is called the long exact cohomology sequence.

Remark: If the group G is finite, then, using the unrestricted complex

. . . −→ Xn−1(G,A) −→ Xn(G,A) −→ Xn+1(G,A) −→ . . . (n ∈ ZZ)mentioned in §2, we get by the same argument an unrestricted long exactcohomology sequence

· · · δ−→ Ĥn(G,A) −→ Ĥn(G,B) −→ Ĥn(G,C) δ−→ Ĥn+1(G,A) −→ · · · .

The connecting homomorphism δ has the following compatibility properties.




(1.3.3) Proposition. Ifde if jghf

iji′

kj′

lmh

ng

0C ′B′A′0

0CBA0

is an exact commutative diagram of G-modules, then the diagramso δpg

qδ

rf

Hn+1(G,A′)Hn(G,C ′)

Hn+1(G,A)Hn(G,C)

are commutative.

Proof: This follows immediately from the definition of δ. If cn ∈ Hn(G,C)and if bn ∈ Cn(G,B) and an+1 ∈ Cn+1(G,A) are such that jbn = cn and ian+1 =∂n+1bn, then δcn = an+1, and fδcn = fan+1 = fan+1. On the other hand, settingc′n = gcn, b′n = hbn, a′n+1 = fan+1, we have j′b′n = c′n, i′a′n+1 = ∂n+1b′n, sothat

δgcn = δc′n = a′n+1 = fan+1 = fδcn. 2

In order to avoid repeated explanations it is convenient to introduce thenotion of δ-functor. Let A and B be abelian categories. An exact δ-functorfrom A to B is a family H = {Hn}n∈ ZZ of functors Hn : A −→ B togetherwith homomorphisms

δ : Hn(C) −→ Hn+1(A)defined for each short exact sequence 0 −→ A −→ B −→ C −→ 0 inA withthe following properties:(i) δ is functorial, i.e. ifstuvwxyz{|}

0C ′B′A′0

0CBA0

is a commutative diagram of short exact sequences in A, then~ δδ

Hn+1(A′)Hn(C ′)

Hn+1(A)Hn(C)

is a commutative diagram in B.




(ii) The sequence

· · · −→ Hn(A) −→ Hn(B) −→ Hn(C) δ−→ Hn+1(A) −→ · · ·

is exact for every exact sequence 0→A→B→C→ 0 in A.

If a family of functors Hn is given only for an interval −∞≤ r ≤n≤ s≤∞,then one completes it tacitly by setting Hn = 0 for n ∈/ [r, s].

In this sense the family of functors Hn(G,−) (completed by Hn(G,−) = 0for n < 0) is a δ-functor from the category of G-modules into the category ofabelian groups. A curious property of δ-functors is their “anticommutativity”.

(1.3.4) Proposition. Let {Hn} be an exact δ-functor from A to B. If000

0C ′′CC ′0

0B′′BB′0

0A′′AA′0

000

is a commutative diagram in A with exact rows and columns, then δδ

δ

−δ

Hn+1(A′)Hn(A′′)

Hn(C ′)Hn−1(C ′′)

is a commutative diagram in B.

Proof: It simplifies the proof if we assume thatA is a category whose objectsare abelian groups (together with some extra structure), as then we may provestatements by “diagram chases” with elements. We may do this, since it can beshown that every small abelian category may be fully embedded into a categoryof modules over an appropriate ring in such a way that exactness relations arepreserved, and in any case we shall apply the proposition only to the categoryof G-modules.




Let D be the kernel of the composite map B −→ C ′′, so that the sequence

0 −→ D −→ B −→ C ′′ −→ 0

is exact. Leti : A′ −→ A⊕B′

be the direct sum of the maps A′ −→ A and A′ −→ B′ and let

j : A⊕B′ −→ B

be the difference d1 − d2 of the maps d1 : A −→ B and d2 : B′ −→ B. Onechecks at once that we get an exact sequence

0 −→ A′ i−→ A⊕B′ j−→ D −→ 0

and that the diagram ¡¢id

£pr1

¤¥¦§¨©ªid

«−id

¬−pr2

®¯id

°±²³C ′′CC ′B′A′

C ′′BDA⊕B′A′

C ′′B′′A′′AA′

of solid arrows is commutative. This diagram can be commutatively com-pleted by homomorphisms D → A′′ and D → C ′ , since im(D → B′′) ⊆im(A′′ → B′′) andA′′ → B′′ is injective, and since im(D → C) ⊆ im(C ′ → C)and C ′ → C is injective. From this we obtain the commutative diagram´ δµ δ¶

id

·¸id

¹δ

ºδ

»id

¼½−id

¾δ

¿δ Hn+1(A′)Hn(C ′)Hn−1(C ′′)

Hn+1(A′)Hn(D)Hn−1(C ′′)

Hn+1(A′)Hn(A′′)Hn−1(C ′′)

and the proposition follows. 2

From the exact cohomology sequence, we often get important isomorphismtheorems. For example, if

0 −→ A −→ B −→ C −→ 0




is an exact sequence of G-modules and if Hn(G,B) = Hn+1(G,B) = 0, then

δ : Hn(G,C) −→ Hn+1(G,A)

is an isomorphism. For this reason it is very important to know which G-modules are cohomologically trivial in the following sense.

(1.3.5) Definition. A G-module A is called acyclic if Hn(G,A) = 0 for alln > 0. A is called cohomologically trivial (welk in German, flasque inFrench) if

Hn(H,A) = 0

for all closed subgroups H of G and all n > 0.

Important examples of cohomologically trivial G-modules are the inducedG-modules given by

IndG(A) = Map (G,A),

where A is any G-module. The elements of IndG(A) are the continuousfunctions x : G −→ A (with the discrete topology on A) and the action ofσ ∈ G on x is given by (σx)(τ ) = σx(σ−1τ ).

If G is a finite group, then we have an isomorphism

IndG(A) ∼= A⊗ ZZ[G]

given by x 7→∑σ∈G

x(σ) ⊗ σ, where ZZ[G] = {∑σ∈G

nσσ |nσ ∈ ZZ} is the group

ring of G.

(1.3.6) Proposition. (i) The functor A 7→ IndG(A) is exact.

(ii) An induced G-module A is also an induced H-module for every closedsubgroup H of G, and if H is normal, then AH is an induced G/H-module.

(iii) If one of the G-modules A and B is induced, then so is A ⊗ B. If G isfinite, the same holds for Hom(A,B).

(iv) If U runs through the open normal subgroups of G, then

IndG(A) = lim−→U

IndG/U (AU ).

We leave the simple proof to the reader (for (ii) use ex.4 of §1 to find ahomeomorphism G ∼= H ×G/H). As mentioned above, the very importanceof the induced G-modules lies in the following fact.


c©

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