-
Cohomologyof
Number Fields
by
Jürgen NeukirchAlexander Schmidt
Kay Wingberg
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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
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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.”
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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
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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.
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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
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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.
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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
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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
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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
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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
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Algebraic Theory
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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
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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
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§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
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6 Chapter I. Cohomology of Profinite Groups
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.
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§1. Profinite Spaces and Profinite Groups 7
(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.
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8 Chapter I. Cohomology of Profinite Groups
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.
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§1. Profinite Spaces and Profinite Groups 9
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
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10 Chapter I. Cohomology of Profinite Groups
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.
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§1. Profinite Spaces and Profinite Groups 11
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.
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12 Chapter I. Cohomology of Profinite Groups
§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
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§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.
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14 Chapter I. Cohomology of Profinite Groups
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.
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§2. Definition of the Cohomology Groups 15
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).
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16 Chapter I. Cohomology of Profinite Groups
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
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§2. Definition of the Cohomology Groups 17
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).
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18 Chapter I. Cohomology of Profinite Groups
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
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§2. Definition of the Cohomology Groups 19
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,
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20 Chapter I. Cohomology of Profinite Groups
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).
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§2. Definition of the Cohomology Groups 21
(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.
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22 Chapter I. Cohomology of Profinite Groups
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 −→ . . .
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§2. Definition of the Cohomology Groups 23
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.
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24 Chapter I. Cohomology of Profinite Groups
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.
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§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)
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26 Chapter I. Cohomology of Profinite Groups
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).
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§3. The Exact Cohomology Sequence 27
(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.
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28 Chapter I. Cohomology of Profinite Groups
(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.
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§3. The Exact Cohomology Sequence 29
(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.
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30 Chapter I. Cohomology of Profinite Groups
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
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§3. The Exact Cohomology Sequence 31
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.
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