Profinite Groups, Arithmetic, and Geometry. (AM-67), Volume 67 9781400881857

Table of contents :
CONTENTS
PREFACE
CHAPTER I – PROFINITE GROUPS
§0. Preliminaries on Projective and Inductive Limits
§1. Profinite Groups
§2. Supernatural Numbers and the Sylow Theory
CHAPTER II – COHOMOLOGY OF PROFINITE GROUPS
§1. δ-functors and the Definition of the Cohomology of Profinite Groups
§2. Behavior of the Cohomology as a Functor, Induced Modules
§3. Restriction, Transfer (co-restriction), and Cup-products
§4. A Utilitarian View of Spectral Sequences
CHAPTER III – COHOMOLOGICAL DIMENSION
§1. Definition and Elementary Properties
§2. The Case of Cohomological Dimension One
§3. The Structure of Profinite p-groups; Šafarevič-Golod’s Theorem
§4. Šafarevič’s Solution of the Class Tower Problem
CHAPTER IV – GALOIS COHOMOLOGY AND FIELD THEORY
§1. Resume of Krull's Galois Theory and First Results
§2. The Cohomological Dimension of Fields
§3. Algebraic Dimension, Quasi-algebraic Closure, and the Property Cr
§4. Existence of Fields of Arbitrary Cohomological Dimension
§5. Counter-examples in the Theory of Cohomological Dimension and Forms
CHAPTER V – LOCAL CLASS FIELD THEORY
§1. Local Fields and Their Extensions – A Resumé
§2. Cohomological Triviality, Tate and Nakayama Theorems, and Herbrand Quotients
§3. The Brauer Group of a Local Field
§4. The Existence Theorem (Part I: Formal Existence Theorem, and beginning of the proof.)
§5. The Existence Theorem (Part II: The Hilbert Symbols and completion of the proof.)
CHAPTER VI – DUALITY
§1. Dualizing Modules and Poincaré Groups
§2. Grothendieck Topologies and Cohomology
§3. Galois, Étale, and Flat Cohomology
§4. Finite Group Schemes and Their Cohomology Over a Local Field
§5. Tate-Nakayama Duality
§6. Duality Theorem for Finite Group Schemes
BIBLIOGRAPHY

Citation preview

Annals of Mathematics Studies Number 67

PROFINITE GROUPS, ARITHMETIC, AND GEOMETRY

BY

STEPHEN S. SHATZ

PRINCETON UNIVERSITY PRESS AND UNIVERSITY OF TOKYO PRESS

1972

Copyright ©

1972, by Princeton University Press A L L RIGHTS RESERVED

L C Card: 77-126832 IS B N : 0-691-08017-8 A M S 1968: Primary-—2050, 2051, 1067, 1068 Secondary— 1015, 1077, 1078, 1435, 1440, 1810, 1815, 1820, 5532

Published in Japan exclusively by University o f T ok yo Press; in other parts o f the world by Princeton University Press

Printed in the United States o f Am erica

For Marilyn, Geoffrey, and Adria

PREFACE This short volume reproduces in its first five chapters the essential content of a one semester course I gave at the University of Pennsylvania in the Spring of 1968. The last chapter covers material I had hoped to in­ clude, but could not for lack of time. My aim in giving such a course was to give students a body of material upon which some of the modern research in Diophantine geometry and higher arithmetic is based, and to do this in a way which emphasized the many interesting roads out of these elementary foundations. I wanted to enable them to start reading the literature in those areas of mathematics I personally find very interesting. The enthusiastic reception of my able c la ss convinced me that this material w as deserving of a larger audience, and I resolved to write up the notes for the course. Anyone conversant with this subject w ill note how indebted I am to many authors. Foremost among them is J. P . Serre from whose works I have taken a great d e a l,* and there is John Tate whose unpublished work and unmistakeable stamp appear here in many places. R esponsibility for any errors is, of course, my own. The highly personal nature of these notes has not blinded me to their many faults, notably the lack of good examples and exercises, and the uneven tempo of exposition. In attempting to retain the flavor of the original lectures, I have allowed these and other faults to stand; I hope they do not vitiate the aim of the project. I have also included a large but not ex­ haustive bibliography; it covers the main references, and the bibliographies in these works and w ill serve as a good guide to the literature.

Especially from [S C L , SG] where the smooth treatment is impossible to improve upon.

v ii

C O N TEN TS P R E F A C E .................................................................................................... CHAPTER I -

vii

P R O F IN IT E G R O U PS

§0.

Prelim inaries on Projective and Inductive Limits

................

3

§1.

Profinite Groups ...........................................................................

7

§2.

Supernatural Numbers and the Sylow Theory ..........................

12

C H A P T E R I I - CO H O M O LO G Y O F P R O F IN IT E G R O U PS §1.

5-functors and the Definition of the Cohomology of Profinite Groups ............................................................................

§2.

16

Behavior of the Cohomology as a Functor, Induced Modules ..........................................................................................

23

§3.

Restriction, Transfer (co-restriction), and Cup-products ......

33

§4.

A Utilitarian V iew of Spectral Sequences

..............................

39

§1.

Definition and Elementary Properties ......................................

53

§2.

The C a se of Cohomological Dimension One

63

§3.

The Structure of Profinite p-groups; Safarevic-G olod’s

C H A P T E R I I I - C O H O M O LO G IC A L DIMENSION

§4.

..........................

T h eorem ..........................................................................................

69

Safarevic’s Solution of the C la s s Tower Problem ..................

86

C H A P T E R I V - G A LO IS CO H O M O LO G Y A N D F IE L D T H E O R Y §1.

Resume of K rulFs G alois Theory and F irst Results

............

93

§2.

The Cohomological Dimension of F ie ld s .................................

97

§3.

A lgebraic Dimension, Q u asi-algebraic Closure, and the Property

§4.

C f ............................................................................ 101

Existence of F ie ld s of Arbitrary Cohomological Dimension ........................................................

ix

119

CONTENTS

X

§5.

Counter-examples in the Theory of Cohomological Dimension and Forms

.................................................................. 121

C H A P T E R V - L O C A L C LA SS F IE L D T H E O R Y §1.

L o c a l F ie ld s and Their Extensions — A Resume ....................129

§2.

Cohomological Triviality, Tate and Nakayama Theorems, and Herbrand Quotients ................................................................ 136

§3.

The Brauer Group of a L o c a l F i e l d .......................................... 152

§4.

The Existence Theorem (Part I: Formal Existence Theorem, and beginning of the proof.)

§5.

..................................... 166

The Existence Theorem (Part II: The Hilbert Symbols and completion of the p r o o f.).............................................................. 173

C H A P T E R V I - D U A L IT Y ..................................183

§1.

D ualizing Modules and Poincare Groups

§2.

Grothendieck T opologies and C ohom ology.............................. 192

§3.

G alois, Etale, and F lat Cohomology ........................................ 204

§4.

Finite Group Schemes and Their Cohomology Over a L o c a l F ield

.................................................................................. 217

§5.

Tate-Nakayam a Duality ................................................................ 230

§6 .

Duality Theorem for Finite Group Schemes ............................ 236

B IB L IO G R A P H Y ............................................................................................248

PROFINITE GROUPS, ARITHMETIC, AND GEOMETRY

CHAPTER I P R O F IN IT E G R O U PS §0. Prelim inaries on projective and inductive lim its* L et A be a set (the index set) and suppose S^, for a e A is a family of sets (groups, rings, etc.) indexed by A. We assume A is partially ordered (by a < ft) and sa tisfie s the Moore-Smith Property: (V a ,/ 3 )(3 y )(a < y

and j8 < y)

.

Now we assume given the follow ing data: (1 )

(V «,/ 3 )(a < £ = = »

3 0 / :

Sa -» S jQ)

,

and the maps (group-homomorphisms, ring homomorphisms, etc.) are con­ sistent in the sense that: given a,/3,y with a < /3 < y, then the induced diagram

commutes. Furthermore, we insist that cf>£ =* id, all a. Any system, as above, w ill be called a direct mapping family of sets (groups, rings), and denoted iS

From now on, we w ill concentrate

on the set-theoretic formulation, leaving to the reader the task of formu­ lating all definitions and results for groups, rings, or objects of other categories._________________________ *

Only the concept of lim based on a directed index set, not on the more general

notion of “ index category/’ is introduced here.

3

PROFINITE GROUPS

4

If instead of the equation (1 ), we write (V a ,/ 8 )fo < j8 = - > 3 ^ a :

(2 )

- Sfl)

and demand the obvious consistency, we get a projective (o r inverse) mapping family of sets (e tc .) —this time with the notation

Examples. ( I D )

SQ *

Z, A *

natural numbers: 1 ,2 ,..., and the order­

ing on A is by cardinality (i.e ., the usual ordering).

1 : Z-> Z

is

‘ ‘multiplication by p ,” where p is a fixed prime number. ( I P ) Same as (I D ), only in this case (2 D ) A *

is multiplication by

p.

natural numbers, partially ordered by division. Sn ■ Z/nZ,

and if n ™ is multiplication by m/n. (2 P ) Same Sn and A as (2D ). T h is time

is the natural projection

Z/m Z -► Z/nZ. (3 D ) G is a group; A is the family of all finite subgroups of G, par­ tially ordered by inclusion. SQ is element of A. The map

a considered as subgroup of G, not as

is the natural inclusion Sa -* S^g .*

(3 P ) G is a group, A is the family of all normal subgroups of G of finite index. If a is in A, SQ is G/a, a finite group. Partially order A v i a : a S^g

/3 C a as subgroups; let cj>^ be the natural projection

Sa . (O bserve (3 P ) generalizes (2 P ).)

If T is a set, and if \jja \ Sa -► T are given maps, one for each a t A , which are consistent with the cf>QP

in the sense that the diagram

commutes for all a < j8 ; then we may inquire as to how “ faithfully }T , ifsa \ p o ssesse s all the information contained in }Sa ,

We assume

G is abelian.

” ?

More exactly,

§0. PROJECTIVE AND INDUCTIVE LIMITS

5

we may ask: D oes there exist a “ universal” object {T , xlJa) a e pS L ^rom which the others may be recaptured? (F o r projective systems the if/Q take T to Sa for all a, and the diagram is changed mutatis mutandis.) A s is usual in such circumstances, what we are asking for is the representability o f a cer­ tain functor —or, in older terminology, the solution o f a universal mapping problem.

The functor we w ish to represent is F where F (T ) = {fam ilies { ^ ^ ^ 1

: Sa -» T

and

diagram (3 ) commutes for all a < ^3 i . The representing object w ill then be a set S and a “ universal” family {0 a ia e A» where a : SQ -* S and the rf>a are consistent. In terms of univer­ sa l mapping properties, the object we seek w ill have the Direct universal mapping property (D U M P ): (a )

£ A *s a set anc* a consistent family indexed by A ; ^ ol : Sa

(b )

S and

Given any other pair {T , 4fa ^CL€f^ having (a ), there exists a unique map cf> : S - T so that the diagram

commutes for every a e A. (T h e reader may formulate the analogous universal mapping problem for inverse fam ilies by turning arrows around and so arrive at a formulation of the P ro je c tiv e universal mapping property (P U M P ).) Again, as is usual, if the objects we seek exist then they w ill be unique up to unique isomorphism. Any object satisfying the DUM P is called the

PROFINITE GROUPS

6

direct limit of the family j SQ,

} —by abuse of notation it is written S = dir lim Sa . a

Similarly, any object satisfying the PU M P is the projective limit of the family {SQ, cf>^\ — written S = proj lim

.

For our purposes, the follow ing theorem is sufficient. TH EOREM 1. In the categories of sets, groups, rings, and topological groups, direct and inverse limits always exist. Proof: We content ourselves with a sketch—and that only for the case of sets. The other categories are entirely similar and obvious modifications of our proof yield the desired results. Q Let iSa , ! be a direct mapping family of sets. We may and do a s­ sume that the Sa are disjoint, and we form their union S = U a Sa . In S introduce the relation, ~ , by: If x e S, (3 y )(a

y e S, (sa y x f Sfl, y e S^g) then x ~ < y, /3 < y ) such that

One sees trivially that ~

0 a ^ (x )

y

if andonly if

= 0 ^ (y ) .

is an equivalence relation, and we may form

S * § / ~ . Let cj>a : Sa -> S be the composition: Sfl -* S -> S / ~ . It is easily checked that (S, cf>a ^a e \

has the DUMP.

Now let {Sa ,

yP(Xy) = X0 > .

Let cf>a be the a*h projection restricted to S. One checks that iS ,0 a ia £ y^ has the PU M P. (Note: It may happen that S = 0.)

Q .E .D .

§1.

PROFINITE GROUPS

7

Examples. T h ese examples refer to the earlier ones on page (I D ) dir lim lS fl,

4.

1 = i-^ | r, s e Z !

n

p

( I P ) proj lim {Sn,

i = (0 )

(2 D ) d ir lim {S n, 0 nm ! = Q /Z

(2 P ) proj lim |Sn, > (2 ). Since G is profinite, there are finite groups

and maps p : G^g

Gfl

GQ such that G * proj lim GQ. Hence, there are maps

cf>a '. G -» Ga , for all a. Let UQ » ker (3 ). Let N be the connected component of identity in G.

If {U a f denotes the family of all open, normal subgroups of G, let

=*

Ua fl N for all a. The groups NQ are open, normal subgroups of N; as N is connected, we must have Na = N for all a. But then N =

nN a

a

= f l N n u = N f l f l U = N n | l! a a a a

.

(The last equality by hypothesis (2 ).) Thus N = { l l , and (3 ) is proved. (3 ) = >

(1). It follow s from our hypothesis, and fromTheorem 17,

Chapter III of [ P ] , that given any open neighborhood U of 1 in G, there exists an open subgroup H of G with H C U . Since G is compact and H is open, the index (G : H ) is finite, and therefore ( G : N q (H )) < oo. From this, it follow s that the distinct conjugates x H x "

1

of H in G are finite

in number, so their intersection K forms an open, normal subgroup of G contained in H; hence, in U . A s U was an arbitrary open neighborhood of 1, we see that the intersection of a ll open, normal subgroups of G re­ duces to the identity. Let {U Q ! denote the family of all open, normal sub­ groups of G (so that we have shown f l a Ua =* {l i ).

The groups G/Ua =

Gfl form in a natural way a projective mapping family of finite groups. Our theorem now follow s from the more general L E M M A 1. L e t G be a compact, Hausdorff group; let GQ, Sa

be two

families of closed subgroups of G, indexed by the directed set A. Assum e that Sa a, Ga 3 G^g and Sa D S^g. Then the groups Ha =

*orm a ProJe c ^ v e mapping family in a natural way,

and

proj lim H = proj lim G / S„ = fl G /fl S„ a u a a a

.

§1.

PROFINITE GROUPS

9

Proof: If /3 > a, then S^g C SQ; so one has the maps

H|8

= G;S / Sj3 -

Ga / S „ -

^

given by Ha , for each a and they are con­ sistent with the

Hence, by the PU M P , we obtain a map cf>: H =

n c a / n s a -> r

.

Here, map means continuous group homomorphism, so when 0 is proven bijective the compact Hausdorff property of the groups in question w ill show that cf> is automatically a homeomorphism. L et f

e H, and assume 0 ( £ ) = 1 f L

1 ---------- ► r

h

shows that *Aa ( £ ) ** 1 for a ll a. Thus f 1 in H and cf> is

any a, the set of a ll ay with £a

Sa for all a ; hence, £ represents

*

£.

But £ a

€ Ga

3^ .

because a ll the £a

be the limit of this subnet. Now given

C Ga if av > a , and as Ga is closed in e fla Ga . Let f be the coset of £

£ e H. A ll we need now show is a is final in the av ; hence,

G, we deduce that £ e Ga . Thus mod fla Sa , i.e .,

= a (x ) € Ha for each a.

e Ga such that £a mod Sa

p o s s e s s e s a converging subnet

lie in G and G is compact. Let f

lim

6

injective. Given x e T , let

A s Ha is Ga / Sa , there exists a ^ The net

The diagram

f ° r every a.

The xa are consistent; hence, for all

mod Sa . Given any a , for any /3 > a, we can find

/8

> a,

PROFINITE GROUPS

10

s aj3€ Sa

s a/8 * Tlie ^scq8®/3 form a net in Sa ’ so there is

with €/3 “

a converging subnet. Look at the subnet of this newly chosen subnet formed by those /3 beyond some number ay of our previous indexing set. We have .lim p m new family

s p . ^

s

e S

and hence,

| = lim

= lim ^ s ajg = £a s a .

The last equation when unscrambled in terms of £ and if,a( h

= xf l .

says precisely

Q .E .D .

Here are some facts concerning profinite groups whose proofs are easy exercises. (F act 1)

Every closed subgroup and every quotient group of a profinite

group is profinite. (F act 2)

The product or projective limit of a family of profinite groups

is profinite. (F act 3)

If S is any subgroup of a profinite group G, then its closure

S is given by S * (F act 4)

n 1U|U is an open normal subgroup of G and S C U I. If G is an abelian group, then G is profinite if and only if

its Pontrjagin dual is a torsion group.

In this way, the category of pro-

finite abelian groups is the dual of the category of torsion abelian groups. An important theorem for the cohomology theory of profinite groups is: THEOREM 3. (C ross-section Theorem) L e t G be a profinite group and let N be a closed normal subgroup. Then there exists a continuous crosssection for the map n : G -► G/N. That is, there is a continuous map (f> : G/N -* G (not a homomorphism in general) such that where A : G/N

G/S

is a cross-

section (continuous of course) of the projection G/S -* G/N. C a ll the set of such pairs ^P and partially order $P in the usual fashion, viz: < S , A>




(1 ) T C S

and (2 ) the diagram

G/N

G / T -------------------- ► G/S commutes (i.e ., A (x ) = S)x(x)). Now $P 4 0

as < N , id> e ^P.

family of $P, let S = fl

If < S a ,Aa > is a totally ordered sub­

S . By Lemma 1, G/S = proj lim G/S • so, as a

A^g(x) -► Aa (x ) for /3 > a under the map G/S^g -* G/Sfl , we see that for each x e G/N, the family {Aa (x )i represents a point A (x ) e G /S. Clearly, < S ,A > belongs to ^P, and we have proved ^P is inductive. By Zorn’s lemma, there exists a maximal pair < S ,A > necessary to prove that S *

i l l Were

in ^P. It is only

S 4- 111, there would exist an open

normal subgroup U of G with S ^ U . Therefore, S H U < S. We shall con­ struct the section G/S -> G / S flU , and this w ill yield the desired contra­ diction. L et G ■* G/S fl U , S = S/S fl U , then G/S is G /S is to construct the section G / S -» G

and our problem

for the profinite group G . But

S is isomorphic to SU/U which is finite (being contained in G /U ). Hence, our problem is reduced to the case of a finite normal subgroup. A s S is finite, there is an open normal subgroup V The projection

G_ -> G/S^ when restricted to V

a ll groups are compact Hausdorff, image in G /S .

of G such that S fl V = {1}.

V

is a monomorphism.

is topologically isomorphic to its

The inverse map of this isomorphism is a continuous map

from an open subgroup of

G/S

onto an open subgroup of G.

lation of this mapping to the finitely many cosets in G/S tion G/S -> G.

Since

Q .E .D .

Now a trans­

yields our s e c ­

PROFINITE GROUPS

12

C O R O L L A R Y . If M, N are closed, normal subgroups of the profinite group G, with N D M, then there exists a section G/N

G/M.

Remark. One of the features of the theory of profinite groups which makes it important for various problems in number theory and geometry is that, in large measure, many mapping properties and number-theoretic prop­ erties of finite groups are (when attention is paid to their proper formula­ tion) in reality valid for the larger c la ss of profinite groups. An example is the cross-section theorem above; others follow in the next section.

§2. Supernatural numbers and the Sylow Theory. D E F IN IT IO N 2. A supernatural number is a formal product

n pp p where the product is taken over all prime numbers and np is an integer

0

n^ * 0

for all q 4- P (np = * 0 ). G

is a p-group < = >

|G| is a p-power.

The first indication that things are all right is PR O P O S IT IO N 1. (L a g ra n g e ’s Theorem). L e t S 2 K be closed sub­ groups of a profinite group G. Then (G : K ) = (G : S ) ( S : K )

Proof: ( G : K ) = l.c.m. (G /U a : K/K H Ua ), where the Ua range over all open, normal subgroups of G.

Now K/K fl Ufl = KUa / U

and

SUa 3 KUa . Hence,

(G /U a : K/K n Ua )

= (G/Ua : SUa /Ua )(S U fl/Ua : KUa /Ufl) =

However, S fl Ua -

(G : SUa )(S / S n Ua : K/K n Ua )

.

Sa is open, normal in S, and the family of such is

final in the set of a ll open, normal subgroups of S. Therefore, (G /U a : K/K n Ua ) » (G : SUa )(S / S a : K/K fl Sa )

.

It is easy to see (but requires a small separate argument which is left to the reader) that (G : K ) -

hence, (G : K ) «

l.c.m .(G : SUQ) l.c.m, (S/Sa : K/K fl Sa ) a a

(G : S )(S : K ).

;

Q .E .D .

A p-S ylow group S of a profinite group G is a closed sub-p-group such that (G : S) is prime—to—p. THEOREM 4. (Sylow Theorem for Profinite Groups). L e t G be a profinite group, and let p be a prime number. Then (1 )

G p o sse sse s p-S ylow subgroups;

(2 )

If T is any p-subgroup of G then T is contained in some p-Sylow subgroup of G.

PROFINITE GROUPS

14

(3 ) Any two p-Sylow subgroups of G are conjugate in G. Proof: We shall give two proofs of part (1 ) of this theorem. The first is due to John Tate, the second to J .-P . Serre. For the first, we let S be the set of all subgroups H C G such that (G : H ) is prime-to-p. Since G e S, § is non-empty and we partially order it by inclusion. If (Ha ! is a chain in S, let H = f la Ha - Let U be open, normal in G, then HU D H and is open in G. B ecau se the Ha are compact, there exists an index a for which Ha C H U. Thus (G : H U ) divides (G : H a ); hence, (G : H U ) is prime-to-p. It follow s that H e S ; whence, S is inductive. Choose a mini­ mal element, P , of S. I claim |P| is a p-power. If not, there would exist an open, normal subgroup U of G with (P : P fl U ) not a p-power. A s P / P D U is finite, it p o sse sse s a p-Sylow subgroup Q/P fl U, with P ^ Q. But then, (P : Q) is prime-to-p, and Q is smaller than P in S, a contra­ diction. The second (Serre’s ) proof of (1 ) uses the following well-known fact: The projective limit of a non-empty family of finite sets is non-empty. This being said, let {Ua i be the family of all open, normal subgroups of G, and let P ( U Q) be the set of all p-Sylow subgroups of G/Ua . By the ordinary Sylow Theorem, the sets P ( U a ) are non-empty and finite. More­ over, if /3 > a, then U^g C Ufl, so the surjection G/U^g -♦ G/Ua yields a map P(U^g) -> P (U fl). By our remark, there exists an element $ in proj lim P (U ). What is such a a

It is a family P

of subgroups of a

G/Ua such that each Pa is a p-Sylow subgroup of G/Ua and such that for /3 > a, P^g -» P a exists and is consistent for y ;> /3 ^ a,

In other words,

is a projective mapping family of p-Sylow sugroups from G/Ua , for each a. Let P = proj lim P , then P is a p-group. I claim (G : P ) is prime-to-p. But, a (G :P ) * «

l.c.m .(G /U a : P / P H Ufl) = l.c.m. (G /Ua : Pa ) prime-to-p .

Fin ally, to prove (2 ) and (3 ) it su ffices to prove: If T is a p-subgroup

§2. SUPERNATURAL NUMBERS AND THE SYLOW THEORY

15

of G, there is some a e G with T ° C P, where P is a given p-Sylow sub­ group of G. Here, the notation T a means o ’" 1 To.

Choose any open nor­

mal subgroup U of G, and consider T U and P U . We know that P U / U is a p-Sylow subgroup of G/U, so by finite Sylow Theory, there exists a Ojj in G/U with (* )

(T U / U )CTU C (P U / U )

Let R|j be the set of all o^j in G such that (1 ) R jj 4 0 for any U,

. sa tisfie s ( * ) above. Then

(2 ) R y is a union of cosets of U in G so is

closed in G, and (3 ) the sets R ^ have the finite intersection property. (The last assertion follow s from

Rv - Ru

if V C U .) By the compactness

of G, the intersection f l y R jj is non-empty; and it is clear that o e fl^j R ^ is the required element of G.

Q .E .D .

Examples. (4 ) If G is a group generated by a set \xa \ of elements, let S be the set of all normal subgroups U of G such that (a ) G/U is finite, (b ) almost all x„ lie in U. L e t G = proj lim G/U, then G is a U called the profinite completion of G. F or example if G is Z we get Z as in example (2 P ). (5 ) Let G = F (X ) be the free-group on X. Its profinite completion is called the free-profinite group on X. If we restrict attention to those sub­ groups U for which in addition (c ) (G : U ) is a p-power, then we get F p (X ) (or in example (4 ) G p) which is called the free-profinite p-group on X (or the p-profinite completion of G ). If X is the set with one element, we have F (X ) = Z , F p(X ) = Z p = p-adic integers (s e e example (2 P )) = p-Sylow subgroup of Z.

C H A P T E R II CO H O M O LO G Y OF P R O F IN IT E GROUPS § 1 . 8 -functors and the definition of the cohomology of profinite groups. Let (2 be an abelian category (think of modules over a ring, or abelian groups on which some group acts via automorphisms), then we have the notion of an exact sequence of objects of A '

This means ( 1)

0

►A

0

(2 ► A"

ker 0 = Im 0 ; and to say that 0 -------

A ' -------►A -------► A " ------ - 0

0

0

is exact means that 0 is injective, 0

is surjective, and ker 0 = Im 0.

An exact sequence as in (1 ) is called a short exact sequence. Suppose we have a family of functors T ° , T 1, T 2, ..., from (2 to the category of abelian groups. T h is family of functors is called a 8-functor (or exact, connected sequence of functors) iff (1 )

For all short exact sequences as above and for all n > 0, there

exists a homomorphism Sn : T n( A " ) -+ T n+ 1 ( A 0

so that the long sequence

obtained by using the 8n , viz: 0 - » T ° ( A ' ) -* T ° ( A ) - T ° ( A " ) ----- T k A ' ) - * T ^ A ) - T 1 ( A " ) -> T 2( A ' ) -* T 2 (A ) -> ••• is exact (in particular, we assume T ° is left-exact!), and

16

§1. S-FUNCTORS AND COHOMOLOGY

17

(2 ) The big diagram 0 -. T ° ( A ') - > T ° ( A ) - » ••• -> T 'X a )-* T n( A " ) -* T ^ ^ C A ') -> •••

0 -» T ° ( B ' ) - T ° ( B ) -» ••• -> T n(B ) -♦ T n( B " ) - T ^ C B ' ) - * ••• (induced from the little commutative diagram 0 - A '- * A - A " - * 0 I I I 0 -> B '- » B -> B " - > 0

)

is everywhere commutative. We sh all now construct the

main 5-functor to be used in this course.

Let G be a profinite group, let A be an abelian group on which G acts as a group of automorphisms. We thus have a map G x A -> A given by ->

A is continuous. E x e rc ise : Given a £ A , let U a = \a e G |a a = a! = the stabilizer of a in G. Then A is a G-module if and only if U a is an open subgroup of G. If U is a given subgroup of G, let A u = {a e A |(Vcr £ U ) (era = a)}. Then, A is a G-module if and only if A = U {A u |U is open in G}. L et (?(G) be the category of G-modules and G-homomorphisms—it is an abelian category. Given an integer n ;> 0, given A £ Ob S (G ), set (2)

C n( G , A ) = { f | f : Gn -> A and f is continuous].

Here, G n means the cartesian product of G with itse lf n times and by G ° we understand a set with one element {0!. Consequently, C ° ( G , A ) :^ ^ A .

COHOMOLOGY OF PROFINITE GROUPS

18

Under pointwise operations C n(G , A ) forms an abelian group, and, obviously, C n(G , — ) is a functor from fl(G ) to abelian groups. Proposition 2. The functor C n(G , — ) is exact for all n. Proof: If 0 - > A ' - > A - > A " - > 0

is exact, one se e s trivially that

0 - C n(G , A ' ) -* C n(G , A ) is exact. Given f e C n(G , A " ) , A

C n(G , A " )

consider any section of the homomorphism

A " , i.e., any map 6 : A"-> A which when composed with A -> A*' yields

id. A s A, A " are discrete, 6 is continuous; consequently, 6 o f is con­ tinuous, hence, lie s in C n(G , A ). Clearly

f is the image of 6 o f under cf> .

Q .E.D . If f £ C n(G ,A ), we define S f e C n + 1 ( G , A ) by the formula

n

i= 1

E xercises: (a ) 8 : C n -> C n +

1

is a homomorphism for each A , in fact,

it is a morphism of functors; (b )

(4)

8 8= 0

i.e., C ° (G , A ) 5 C ^ G , A ) I C 2 (G , A ) I . ..

is a complex (composition of two su ccessive maps is zero). The complex (4 ) is called the standard complex. U sin g the standard complex, we shall define our 8 -functor. We set

§1. S-FUNCTORS AND COHOMOLOGY

19

Z n(G , A ) = ker ( C n (G , A ) — » C n + X(G , A )) = group of n cocycles of G with coefficients in A B n(G , A ) =

Im (C n - 1 ( G , A )

C n(G , A ))

(5 ) = group of n coboundaries of G with coeffs. in A Hn(G , A ) = Z n(G , A )/ B n(G , A ) = n**1 cohomology group of G with coefficients in A.

P R O P O S IT IO N 3. The sequence of functors iH n( G , —) }

is a 8-functor

on (3(G) to the category of abelian groups. Proof: It is completely trivial that a short exact sequence 0 — > A ' — > A — > A " — >0 induces the exact sequences

0 — >H °(G , A ' ) — >H °(G , A ) — -»H °(G , A " ) ,

H n(G , A ' ) — >Hn(G , A ) — >Hn(G , A " )

,

n > 0,

and that a small diagram involving two short exact sequences yields com­ mutative diagrams for a ll n > 0.

T o define the connecting homomorphisms,

5 , we use the snake lemma and Proposition 2 as follow s:

0 ------► C n(G , A ' ) ---------- C n(G , A ) ---------- C n( G , A " ) ------ - 0

0------►Cn + H g , A ' ) ------- C n + [f

1 (G ,

A

Cn+

1 (G ,

A " ) ------ - 0

e H n( G , A " ) , choose