Topics in Cohomology of Groups_Serge Lang.pdf

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg E Takens, Groningen

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Serge Lang

Topics in Cohomology of Groups

~ Springer

Author

Serge Lang Mathematics Derpartment Yale University, Box 208 283 10 Hillhouse Avenue New Haven, CT 06520-8283, USA

L i b r a r y of Congress C a t a l o g i n g - i n - P u b l i c a t i o n Data

Lang, S e r g e , 1927- [ R a p p o r t su r ]a c o h o m o ] o g i e des g r o u p e s . E n g l i s h ] Top i cs in cohomo]ogy oF g roups / Serge Lang.

p, cm. - - ( L e c t u r e n o t e s in ma thema t i c s ; 1625) I n c ] u d e s b i b ] i o g r a p h i c a ] r e ~ e r e n c e s (p . - ) and i n d e x . ISBN 3 - 5 4 0 - 6 1 1 8 1 - 9 ( a ] k , p a p e r ) 1. C ]ass F l e ] d t h e o r W. 2. Group t h e o r y . 3. Homo;ogy t h e o r y .

I , T i t l e , I I . S e r i e s : L e c t u r e no tes in ma thema t i cs ( S p r i n g e r - V e r ] a g ) ; 1625. QA247.L3513 1996 5 1 2 ' . 7 4 - - d c 2 0 9 6 - 2 6 6 0 7

The first part of this book was originally published in French with the title "Rapport sur la cohomologie des groupes" by Benjamin Inc., New York, 1996. It was translated into English by the author for this edition. The last part (pp. 188-215) is new to this edition.

Mathematics Subject Classification (1991): 11S25, 11S31, 20J06, 12G05, 12G10

ISBN 3-540-61181-9 Springer-Verlag Berlin Heidelberg New York

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C o n t e n t s Chapter I. Existence and Uniqueness

w The abs t rac t uniqueness theorem . . . . . . . . . . . . . . . . . . . . . . . 3 w Notat ions , and the uniqueness theorem in Mod(G) . . . . . . 9 w Existence of the cohomological functor on Mod(G) . . . . . 20 w Explicit computa t ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 w Cyclic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Chapter II. Relations with Subgroups

w Various morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 w Sylow subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 w Induced representa t ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 w Double cosets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Chapter III. Cohomological Triviality

w The twins theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 w The tr iplets theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 w Spli t t ing module and Tate 's theorem . . . . . . . . . . . . . . . . . . . 70

Chapter IV. Cup Products

w Erasabi l i ty and uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 w Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 w Relat ions with subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 w The tr iplets theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 w The cohomology ring and dual i ty . . . . . . . . . . . . . . . . . . . . . . 89 w Per iodic i ty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 w The theorem of Ta te -Nakayama . . . . . . . . . . . . . . . . . . . . . . . . 98 w Explicit Nakayama maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

VI

Chapter V. Augmented Products

w Def in i t ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 w Ex i s t ence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 w Some p rope r t i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Chapter VI. Spectral Sequences

w Def in i t ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 w T h e Hoehsch i ld -Ser re spec t ra l sequence . . . . . . . . . . . . . . . 118 w Spec t r a l sequences a n d cup p r o d u c t s . . . . . . . . . . . . . . . . . . 121

Chapter VII. Groups of Galois T y p e (Unpublished article of Tate)

w Def in i t ions a n d e l e m e n t a r y p rope r t i e s . . . . . . . . . . . . . . . . . 123 w C o h o m o l o g y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 w Cohomolog ica l d imens ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 w Cohomolog ica l d imens ion _< 1 . . . . . . . . . . . . . . . . . . . . . . . . . 143 w T h e tower t h e o r e m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 w Galois g roups over a field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Chapter VIII. Group Extensions

w M o r p h i s m s of ex tens ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 w C o m m u t a t o r s a n d t r ans fe r in an ex tens ion . . . . . . . . . . . . 160 w T h e def la t ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Chapter IX. Class formations

w Def in i t ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 w T h e rec ip roc i ty h o m o m o r p h i s m . . . . . . . . . . . . . . . . . . . . . . 171 w Weil g roups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Chapter X. Applications of Galois Cohomology in Algebraic Geometry ( f rom le t te rs of Ta te )

w Tors ion- f ree m o d u l e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 w F in i t e m o d u l e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 w T h e Ta te pa i r ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 w (0, 1 ) -dua l i ty for abe l i an variet ies . . . . . . . . . . . . . . . . . . . . . 199 w T h e full dua l i t y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 w B r a u e r g roup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 w Ideles a n d idele classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 w Idele class c o h o m o l o g y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

P r e f a c e The Benjamin notes which I published (in French) in 1966 on the

cohomology of groups provided missing chapters to the Artin-Tate notes on class field theory, developed by cohomological methods. Both items were out of print for many years, but recently Addison- Wesley has again made available the Artin-Tate notes (which were in English). It seemed therefore appropriate to make my notes on cohomology again available, and I thank Springer-Verlag for pub- lishing them (translated into English) in the Lecture Notes series.

The most basic necessary background on homological algebra is contained in the chapter devoted to this topic in my Algebra (derived functors and other material at this basic level). This material is partly based on what have now become routine constructions (Eilenberg-Cartan), and on Grothendieck's influential paper [Gr 59], which appropriately defined and emphasized 5-functors as such.

The main source for the present notes are Tate's private papers, and the unpublished first part of the Artin-Tate notes. The most significant exceptions are: Rim's proof of the Nakayama-Tate theorem, and the treatment of cup products, for which we have used the general notion of multilinear category due to Cartier.

The cohomological approach to class field theory was carried out in the late forties and early fifties, in Hochschild's papers [Ho 50a], [Ho 50b], [HoN 52], Nakayama [Na 41], [Na 52], Shafarevich [Sh 46], Well's paper [We 51], giving rise to the Weft groups, and seminars of Artin-Tate in 1949-1951, published only years later [ArT 67].

As I stated in the preface to my Algebraic Number Theory, there

are several approaches to class field theory. None of them makes any other obsolete, and each gives a different insight from the others.

The original Benjamin notes consisted of Chapters I through IX. Subsequently I wrote up Chapter X, which deals with applications to algebraic geometry. It is essentially a transcription of weekly installment letters which I received from Tate during 1958-1959. I take of course full responsibility for any errors which might have crept in, but I have made no effort to make the exposition anything more than a rough sketch of the material. Also the reader should not be surprised if some of the diagrams which have been qualified as being commutative actually have character -1.

The first nine chapters are basically elementary, depending only on standard homological algebra. The Artin-Tate axiomatization of class formations allows for an exposition of the basic properties of class field theory at this elementary level. Proofs that the axioms are satisfied are in the Artin-Tate notes, following Tate's article [Ta 52]. The material of Chapter X is of course at a different level, assuming some knowledge of algebraic geometry, especially some properties of abelian varieties.

I thank Springer Verlag for keeping all this material ill print. I also thank Donna Belli and Mel Del Vecchio for setting the manu- script in AMSTeX, in a victory of person over machine.

Serge Lang New Haven, 1995

C H A P T E R I Existence and Uniqueness

w T h e a b s t r a c t u n i q u e n e s s t h e o r e m

We suppose the reader is familiar with the terminology of abelian categories. However, we shall deal only with abelian categories which are categories of modules over some ring, or which are obtained from such in some standard ways, such as categories of complexes of modules. We also suppose that the reader is acquainted with the standard procedures constructing cohomological functors by means of resolutions with complexes, as done for instance in my Algebra (third edition, Chapter XX). In some cases, we shall summarize such constructions for the convenience of the reader.

Unless otherwise specified, all functors on abelian categories will be assumed additive. What we call a 6-functor (following Grothendieck) is sometimes called a c o n n e c t e d s e q u e n c e of functors . Such a functor is defined for a consecutive sequence of integers, and transforms an exact sequence

O ~ A ~ B ~ C ~ O

-into an exact sequence

�9 ..---~ HP(A) ~ HP(B) -~ HP(C) ~ HP+I(A) -- - . . . .

functorially. If the functor is defined for all integers p with - z c < p < co, then we say that this functor is cohomologica l .

Let H be a 5-functor on an abelian category 9.1. We say tha t H is e r a s a b l e by a subset 9Yt of objects in A if for every A in 9.[ there exists l~A E 92t and a monomorph i sm CA : A ~ !~A such tha t H(MA) = 0. This definition is slightly more restr ict ive than the usual general definition (Algebra, Chapte r XX, w but its condit ions are those which are used in the for thcoming applications. An e r a s i n g f u n c t o r for H consists of a functor

M ' A ---,M(A)

of 9.1 into itself, and a monomorphism r of the ident i ty in M , i.e. for each object A we are given a monomorph i sm

~A : A - - + M A

such that , if u �9 A ~ B is a morphism in 92, then there exists a morph i sm M(u) and a commuta t ive d iagram

0

> A ~*> M(A)

"I I M(,) , B > M ( B )

SB

such tha t M(uv) = M(u)M(v) for the composite of two morph i sms u, v. In addit ion, one requires H(MA) = 0 for all A E 9.1.

Let X(A) = XA be the cokernel of eA. morph i sm

X(u) : XA X s

For each u there is a

such tha t the following d iagram is commutat ive :

0 , A ) M A ) X A ) 0

.I I I 0 , B ' MB , X e ~ 0,

and for the composite of two morphisms u, v we have X(uv) = X(u)X(v). We then call X the c o f u n c t o r of M.

Let P0 be an integer, and H = (H p) a 8-functor defined for some values of p. We say tha t M is an e r a s i n g f u n c t o r fo r H in dimension > p0 if HP(MA) = 0 for all A E 9.1 and all p > p0.

I . l 5

We have similar notions on the left. Let H be an exact 6-functor on 9/. We say that H is c o e r a s a b l e by a subset gYr if for each object A there exists an epimorphism

r lA" M A --* A

with i~/IA E g~, such that H(i~VfA) -- 0. A c o e r a s i n g f u n c t o r M for H consists of an epimorphism of M with the identity. If r / is such a functor, and u : A ---* B is a morphism, then we have a commutative diagram with exact horizontal sequences:

0 ' YA ~ MA 'a'-+ A ) 0

IU(")

0 ) Y s ' M s ~ B ~0 rib

and YA is functorial in A, i.e. Y(uv) = Z (u )Y (v ) .

R e m a r k . In what follows, erasing fur.Lctors will have the additional property that the exact sequence associated with each object A will split over Z, and therefore remain,; exact under tensor products or horn. An erasing functor into an abelian category of abetian groups having this property will be said to be sp l i t t i ng .

T h e o r e m 1.1. F i r s t u n i q u e n e s s t h e o r e m . Let 9.1 be an abelian category. Let H , F be two 6-fvnctors defined in degrees 0, 1 (resp. 0 , - 1 ) with values in the same abelian category. Let (~20, ~1) and (~0, ~ ) be 6-raorphisms of H into F, coinciding in dimension 0 (resp. (~-1 ,~0) and (~-1 ,~o)) . Suppose that H 1 is erasable (resp. H -1 is coerasable). Then we have r = ~1

(resp. ~-1 = ~-1) .

Proof. The proof being self dual, we give it only for the case of indices (0, 1). For each object A E 9/we :have an exact sequence

0 --+A ~ MA ---+ XA ----* 0

and H 1 (MA) = 0. There is a commutative diagram

H~ , H ~ ~" , H i ( A ) , 0

Fo(MA) , F o ( x a ) 6F -:. F t (A) , 0

with horizontal exact sequences, from which it follows that 6H is surjective. It follows at once that )91 = ~1.

In the preceding theorem, )91 and ~1 are given. One can also prove a result which implies their existence.

T h e o r e m 1.2. Second u n i q u e n e s s t h e o r e m . Let 92 be an abelian category. Let H, F be 5-functors defined in degrees (0, 1) (resp. 0 , - 1 ) with values in the same abelian category. Let )90 : H ~ -~ F ~ be a morphism. Suppose that H 1 is erasable by injectives (rasp. H -1 is coerasable by projectives). Then there exists a unique morphism

) 9 1 : H 1 ---* F 1 (resp.)9_1 : H -1 --+ F -1)

such that (~o,)91) (resp. ()9o,~-1)) is also a &morphism. The association )90 ~ )91 is functorial in a sense made explicit below.

Proof. Again the proof is self dual and we give it only in the cases when the indices are (0, 1). For each object A E 92 we have the exact sequence

O-"* A - ' + l~/f A --+ X A ---*0

and H I ( M A ) = 0. We have to define a morphism

c21(A) : Hi(A) ~ FI(A)

which commutes with the induced morphisms and with 5. We have a commutative diagram

H~ , H~

F~ , F~

~ ~ H i ( A ) , 0

FI(A) 5F

with exact horizontal sequences. The right surjectivity is just the erasing hypothesis. The left square commutativity shows that Ker 6 H is contained in the kernel of 6F~90(XA). Hence there exists a unique morphism

c21(A) : Hi(A) ---, F I (A)

1.1 7

which makes the right square commutative. We shall prove that ~a(A) satisfies the desired conditions.

First, let u : A -+ B be a morphism. From the hypotheses, there exists a commutative diagram

0 ~ A ~ MA ~ X A ~ 0

0 , B , M B , X B ~ 0

the morphism M(u) being defined because MA is injective. The morphism X(u) is then defined by making the right square commutative. To simplify notation, we shall write u instead of M(u) and X(u).

We consider the cube: 614

H~ .H I(A) ,

oo.). , o ( x , ) ~ Vl(B)

F~ .Hi(B)

We have to show that the right face is commutative. We have:

~I(B)HI(u)SH = !pl(B)SHH~

_- (~F~P0H0(u)

= ,~FF~

= F ' ( u ) S r ~ o

= F I ( u ) ~ , ( A ) S , .

We have used the fact (implied by the hypotheses) that all the faces of the cube are commutative except possibly the right face. Since 3H is surjective, one gets what we want, namely

~pl(B)Hl(u) = Fa(u)~pa(A).

The above argument may be expressed in the form of a useful general lemma.

If, in a cube, all the ]aces are c o m m u t a t i v e except possibly one, and one o f the arrows as above is surject ive, then this face is also c o m m u t a t i v e .

Next we have to show that ~1 commutes with 6, that is (~P0,cPl) is a 6-morphism. Let

0 ~ A' --* A ~ A" ~ 0

be an exact sequence in 9/. Then there exist morphisms

v : A ~ MA, and w : A" ---* X A ,

making the following diagram commutative:

0 ) A' ~ A , A" ) 0

0 ) A ' ' ]VIA, ) X A ' ) 0

because MA, is injective. There results the following commutative diagram:

H~ '')

,, Ho(XA ,) �9 H ' (A ' )

~o / ~ ) ~ ~(A')

FO(XA ,) �9 r I(A') 6F

1.2 9

We have to show that the right square is commutative. Note that the top and bottom triangles are commutative by definition of a 6-functor. The left square is commutative by the hypothesis that c20 is a morphism of functors. The front square is commutative by definition of ~I(A') . We thus find

(r = I(A')6HH~ (top triangle)

= 6FqOoH~ (front square)

= ~ r F ~ (left square)

- - 6F~0 (bottom triangle),

which concludes the proof.

Finally, let us make explicit what we mean by saying that ~1 depends functorially on ~0- Suppose we have three functors H, F, E defined in degrees 0,1; and suppose given ~0 : H ~ ~ F ~ and ~0 : F ~ ~ E ~ Suppose in addition that the erasing functor erases both H 1 and F 1 . We can then construct ~1 and r by applying the theorem. On the other hand, the composite

r = ~0 " H ~ ---+ E ~

is also a morphism, and the theorem implies the existence of a morphism

{?1 " H1 --+ E 1

such that ({?0, (71) is a 6-morphism. By uniqueness, we obtain

{71 = r o ~I .

This is what we mean by the assertion that c21 depends functorially on ~o.

w N o t a t i o n , a n d t h e u n i q u e n e s s t h e o r e m in Mod(G)

We now come to the cohomology of groups. Let G be a group. As usual, we let Q and Z denote the rational numbers and the integers respectively. Let Z[G] be the group ring over Z. Then

10

ZIG] is a free module over Z, the group elements forming a basis over Z. Multiplicatively, we have

ct 6 G cr , r

the sums being taken over all e lements of G, but only a finite number of a~ and b,- being r 0. Similarly, one defines the group algebra k[G] over an arbi t rary commuta t ive ring k.

The group ring is often denoted by P = Fa. It contains the ideal I 6 which is the kernel of the a u g m e n t a t i o n h o m o m o r p h i s m

e : Z[G] ~ Z

defined by 6 (~--] n~a) = )-~ n~. One sees at once tha t I a is Z-free, with a basis consisting of all e lements a - e, with ~ ranging over the e lements of G not equal to the unit element. Indeed, if ~ n~ = O, then we may write

) - ' n , , a = , , ~ - ~ n ~ , ( o - e ) .

Thus we obtain an exact sequence

0 ~ z a ~ z [ a ] --+ Z ~ 0,

used constant ly in the sequel. The sequence splits, because ZIG] is a direct sum of I a and Z �9 ea (identified with Z).

Abel ian groups form an abelian category, equal to the category of Z-modules , denoted by Mod(Z). Similarly, the category of modules over a r ing R will be denoted by Mod(R).

An abelian group A is said to be a G - m o d u l e if one is given an opera t ion (or action) of G on A; in other words, one is given a map

G x A - - - ~ A

satisfying

(oT)a = c~(Ta) e . a = a o(a -5 b) = aa + o'b

1.2 11

for all a, r E G and a, b E A. We let e = ea be the unit element of G. One extends this operation by linearity to the group ring Z[G]. Similarly, if k is a commutative ring and A is a k-module, one extends the operation of G on A to k[G] whenever the operation of G commutes with the operation of k on A. Then the category of k[G]-modules is denoted by Modk(G) or Mod(k, G).

The G-modules form an abelian category, the morphisms being the G-homomorphisms. More precisely, if f : A --+ B is a morphism in Mod(Z), and if A, B are also G-modules, then G operates on Horn(A, B) by the formula

(~r f ) (a )=cr ( f (a - la ) ) for s E A and crEG.

If there is any danger of confusion one may write [a]f to denote this operation. If [cr]f - f , one says that f is a G - h o m o m o r p h i s m , or a G - m o r p h i s m . The set of G-morphisms from A into B is an abelian group denoted by Horns(A, B). The category consisting of G-modules and G-morphisms is denoted by Mod(G). It is the same as Mod(Pa ) .

Let A E Mod(G). We let A a denote the submodule of A consisting of all elements a E A such that aa = a for all a E G. In other words, it is the submodule of fixed elements by G. Then A G is an abelian group, and the association

H~ �9 A ~ A a

is a functor from Mod(G) into the category of abelian groups, also denoted by Grab . This functor is left exact.

We let x a denote the canonical map (in the present case the identity) of an element a E A G into H~(A) .

T h e o r e m 2.1. Let Ha be a cohomological functor on Mod(G) with values in Mod(Z), and such that H~ is defined as above. Assume that H ~ ( M ) = 0 if M is injective and r > 1. Assume also that H~(A) = 0 for A E Mod(G) and r < O. Then two such cohomological functors are isomorphic, by a unique morphism which is the identity on H~(A) .

This theorem is just a special case of the general uniqueness theorem.

12

C o r o l l a r y 2.2. If G = {e} then H$(A) = 0 /or all r > O.

Proof. Define HG by letting H~(A) = A e and H~(A) = 0 for r # 0. Then it is immediately verified that He is a cohomologica/ functor, to which we can apply the uniqueness theorem.

C o r o l l a r y 2.3. Let n E Z and let nA " A - -* A be the morphism a ~-+ na for a E A. Then H~(nA) = n H (where H stands for Hb(A)) .

Proof. Since the coboundary 8 is additive, it commutes with multiplication by n, and aga/n we can apply the uniqueness theorem.

The existence of the functor HG will be proved in the next section.

We say that G o p e r a t e s t r iv ia l ly on A if A = A a, that is cra -- a for all a E A and ~ C G. We always assume that G operates trivially on Z, Q, and Q/Z.

We define the abehan group

AG = A / I A a .

This is the factor group of A by the subgroup of elements of the form (or - e)a with cr E G and a C A. The association

A ~-+ AG

is a functor from Mod(G) into Grab.

Let U be a subgroup of finite index in G. We may then define the t r a c e

S ~ : A U --* A G by the formula SU(a) = E 6a, C

where {c} is the set of left cosets of U in G, and 6 is a representative of c, so that

G= [.3 ~U. C

If U = {e}, then G is finite, and in that case the t r ace is written SG, so

SG(a) = E (;a. ~6G

For the record, we state the following useful lemma.

1.2 13

L e m m a 2.4. Let A ,B , C be G-modules. Let U be a subgroup of finite indez in G. Let

A - L B--L C-% D

be morphisms in Mod(G), had suppose that u, w are G-morphisms while v is a U-morphism. Then

S (wvu) = ws (v)u

Proof. Immediate.

We shall now describe some embedding functors in Mod(G). These will turn out to erase some cohomological functors to be defined later. Indeed, injective or projective modules will not suffice to erase cohomology, for several reasons. First, when we change the group G, an injective does not necessarily remain injective. Second, an exact sequence

0 ---* A ~ J ---* A" --~ 0

with an injective module J does not necessarily remain exact when we take its tensor product with an arbitrary module B. Hence we shall consider another class of modules which behave better in both respects.

Let G be a group and let B be an abelian group, i.e. a Z- module. We denote by MG(B) or M(G, B) the set of functions from G into B, these forming an abelian group in the usual way (adding the values). We make Ms(B) into a G-module by defining an operation of G by the formulas

= for a .

We have trivially (ar)( f ) = a(r f ) . Furthermore:

P r o p o s i t i o n 2.6. Ze~ G ~ be a subgroup of G and let G = U x~G' be a coset decomposition. For f E M(G,B) let

Ot

f~ be the f~nction in M(G' ,B) such tha~ f~(y) = f (x~y) for y E G'. Then the map

Ot

14

is an isomorphism

M ( G , B ) ~-~ H M ( G " B ) ot

in the category of G'-moduIes.

The proof is immediate, and Proposition 2.5 is a special case with G' equal to the trivial subgroup.

Let A 6 Mod(G), and define

~A : A ---* Ma(A)

by the condition that eA(a) is the function fa such that fa(a) = aa for all a 6 A and cr E G. We then obtain an exact sequence

(1) 0 ---* A ~A MG(A) ---+ XA ---* 0

in Mod(G). Furthermore, this sequence splits over Z, because the map

M y ( A ) ~ A given by f ~ f(e)

splits the left arrow in this sequence, i.e. composed with 6A it yields the identity on A. Consequently tensoring this sequence with an arbitrary G-module B preserves exactness.

We already know that Ma is an exact functor. In addition, if f : A ---* B is a morphism in Mod(G), then in the following diagram

0 , A ~A ' MG(A) ' X A , 0

fl ~MG(f) ~X( f )

0 ~ B ) M a ( B ) ) X B ~ 0

(2)

~B

the left square is commutative, and hence the right square is commutative. Therefore, we find:

T h e o r e m 2.5. Let G be a group. Notations as above, the pair (MG,e) is an embedding functor in Mod(G). The associated exact sequence (1) splits over Z for each A 6 Mod(C).

In the next section, we shall define a cohomological functor HG on Mod(G) for which (Ma, ~) is an erasing functor. By Proposition 2.6, we shall then find:

1.2 15

C o r o l l a r y 2.6. Let G' be a subgroup of G, and consider Mod(G) as a subcategory of Mod(G') . Then Ha, is a cohomological func- tot on Mod(G), and (Ma , r is an erasing functor for Ha, .

Thus we shall have achieved our objective of finding a serviceable erasing functor simultaneously for a group and its subgroups, be- having properly under tensor products. The erasing functor as above will be called the o r d i n a r y e r a s ing func to r .

R e m a r k . Let U be a subgroup of finite index in G. Let A, B E Mod(G). Let f �9 A ---* B be a U-morphism. We may take the trace

s u ( f ) �9 X ---+ B

which is a G-morphism. Furthermore, considering Mod(G) as a subcategory of Mod(U), we see that (IV/a, r is an embedding functor relative to U, that is, there exist U-morpkisms IV/a(f) and X ( f ) such that the diagram (2) is commutative, but with vertical U-morphisms.

Applying the trace to these vertical morphisms, and using Lemma 2.4, we obtain a commutative diagram:

(3)

0 ~ A ~A , M a ( A ) ' X A ) 0

0 , B , M a ( B ) , X B ~ 0 ~B

T h e c a s e o f f i n i t e g r o u p s

For the rest of this section, we assume that G is finite.

We define two functors from Mod(G) into Mod(Z) by

H~ " A ~-+ AC / S a A

H a 1 : A ~ A s o / I a A .

We denote by AsG the kernel of Sa in A. This is a special case of the notation whereby if f : A --, B is a homomorphism, we let A / be its kernel.

16

We let

~c : A a ---+ H ~ ( A ) = AG/SGA.

n~V: Asc --~ H a l ( A ) = Asa / IGA.

be the canonical maps. The proof of the following result is easy and straightforward, and will be left to the reader.

T h e o r e m 2.7. The functors H~ l and H~ form a 5-functor if one defines the coboundary as follows. Let

0 --~ A' -L A -~ A" ~ 0

be an exact sequence in Mod(G). For a" E A"sa "we define

5mG(a") = x G ( u - l S G v - l a'').

The inverse images in this last formula have the usual meaning. One chooses any element a such that va = a", then one takes the trace SG. One shows that this is an element in the image of u, so we can take u -1 of this element to be an element of A 'c, whose class modulo ScA ' is well defined, i.e. is independent of the choices of a such that va = a". The verification of these assertions is trivial, and left to the reader. (Cf. Algebra, Chapter III, w

T h e o r e m 2.8. Let H a be a cohomologicaI functor on Mod(G) (with finite group G), with value in Mod(Z), and such that H~ is as above. Suppose that H ~ ( M ) = 0 if M is injective and r <= O. I f FG is another cohomologicaI functor having the same properties, then there exists a unique isomorphism of Ha with FG which is the identity on H~

Proof. This is a particular case of the uniqueness theorem.

C o r o l l a r y 1.9. I f G is trivial then H~(A) = 0 for all r E Z.

C o r o l l a r y 2.10. Let n E Z and suppose G finite. For A in Mod(G) we have Hb(nA ) = n H (abbreviating H 5 by H).

Both corollaries are direct consequences of the uniqueness theorem, like their counterpart for the other functor, as in Corollaries 2.2 and 2.3.

1.2 17

Let K1, K2, K be commutative rings, and let T be a biadditive bifunctor

T : Mod(K1) x Mod(K2) ---+ Mod(K).

Suppose we are given an action of G on A1 E Mod(K1) and on A2 E Mod(K2), so A1 E Mod(KI[G]) and A2 E Mod(K2[G]). Then T(AI,A2) is a K[G]-module, under the operation T(cr, a) if T is covariant in both variables, and T(cr -1 , a) if T is contravariant in the first variable and covariant in the second. This remark will be applied to the case when T is the tensor product or T = Horn.

A K[G]-module A is called K[G]- regula r if the identity 1A is a trace, that is there exists a K-morphism v : A ~ A such that

1A = s a ( v ) .

When K = Z, a K[G]-regular module is simply called G-regular .

P r o p o s i t i o n 2.11. Let K1, K2, K be commutative rings as above, and let T be as above. Let Ai C Mod(Ki) (i = 1,2) and suppose A~ is K,[G]-regular for i = 1,2. Then T(A1,A2) is K[G]-regular.

Proof. Left to the reader.

P r o p o s i t i o n 2.12. Let G' be a subgroup of the finite group G. Let A C Mod(K,G). If A is K[G]-regular then d is also K[G']- regular. If G' is normal in G, then A G' is K[G/G']-regular.

Proof. Write G = U G'xi expressing G as a right coset decomposition of G'. Then by assumption, we can write 1A in the form

r6G' i

with some K-morphism v. The two assertions of the proposition are then clear, according as we take the double sum in the given order, or reverse the order of the summation.

P r o p o s i t i o n 2.13. Let A E Mod(K,G). Then A is K[G]- projective if and only if A is K-projective and K[G]-regular.

Proof. We recall that a projective module is characterized by being a direct summand of a free module. Suppose that A is K[G]- projective. We may then write A as a direct summand of a free

18

K[G]-module F = A | B, with the natural injection i and projection 7r in the sequence

A ~ - - ~ A •

with rri = 1A, and both i, 7r are K[G]-homomorphisms. Since F is K[G]-free, it follows that 1F = Sa(v) for some K-homomorphism v. Then by Lemma 2.4,

1A = 7clFi = 7rSe(v)i = Sa(Trvi),

whence A is K[G]-regular. Conversely, let A be K-projective and K[G]-regular. Let

71"

F---+ A ---~ O

be an exact sequence in Mod(K, G), with F being K[G]-free. By hypothesis, there exists a K-morphism i f : A --~ F such that 7riK = 1A, and there exists a K-morphism v : A --+ A such that 1A = SG(v). We then find

~ s ~ ( i ~ v ) = s ~ ( ~ i ~ v ) = Sc(v) = 1~,

which shows that SG(iKv) splits % whence A is a direct summand of a free module, and is therefore K[G]-projective. This proves the proposition.

Wi th the same type of proof, taking the trace of a projection, one also obtains the following result.

Proposition 2.14. In Mod(G), a direct summand of a G- regular module is also G-regular. In particular, every projective module in Mod(G) is also G-regular.

Proof. The second assertion is obvious for free modules, whence it follows from the first assertion for projectives.

For finite groups we have a modification of the embedding rune- for defined previously for arbitrary groups, and this modification will enjoy stronger properties. We consider the following two exact sequences:

(3)

(4)

o ~ Ia ~ Z[G] & Z ~ 0

o - - , Z - - ~ Z[G] ~ Ya ~ O.

1.2 19

The first one is just the one already considered, with the augmentation homomorphism c. The second is defined as follows. We embed Z in Z[G] on the diagonal, that is

C I " ?'/ e---~ n ~ - ~ 0".

a E G

Since G acts trivially on Z, it follows that s t is a G-homomorphism. We denote its cokernel by Jc .

Proposition 2.15. The exact sequences (3) and (4) split in Mod(Z).

Pro@ We already know this for (3). For (4), given ~ = ~2 n~a in ZIG] we have a decomposition

aEG ~#e

~EG ~#e

which shows that Z[G] is a direct sum of c'(Z) and another module, as was to be shown.

Given any A C Mod(G), taking the tensor product (over Z) of the split exact sequences (3) and (4) with A yields split exact sequences by a basic elementary property of the tensor product, with G-morphisms eA = e | 1A and el4 = e' | 1A, as shown below:

(4A)

(5A)

0 ~ I v | ~A Z |

0 - - - , A = Z N A ~Z[G] | !

CA

As usual, we identify Z | A with A. Let M e be the functor given by

M~(A) = Z[G] | A.

We observe that Ma(A) is G-regular. In the next section, we shall define a cohomological functor on Mod(G) for which M a will be an erasing functor.

20

Let f �9 A ---+ B be a G-morphism, or more generally, suppose G ~ is a subgroup of G and A, B �9 Mod(G), while f is a G~-morphism. Then

M a ( f ) = 1 | f

is a GI-morphism.

w E x i s t e n c e of t h e cohomolog i ca l f u n c t o r s on Mod(G)

Although we reproduced the proofs of uniqueness, because they were short, we now assume that the reader is acquainted with standard facts of general homology theory. These are treated in Alge- bra, Chapter XX, of which we now use w especially Proposition 8.2 giving the existence of the derived functors. We apply this proposition to the bifunctor

T ( A , B ) = H o m a ( A , B ) for A , B �9 Mod(G),

with an arbitrary group G. We have

Homa(Z, A) = A a.

We then find:

T h e o r e m 3.1. Let X be a projective resolution of Z in Mod(G). Let H(A) be the homology of the complex Homa(X,A) . Then H = {H ~} is a cohomology functor on Mod(G), such that

Hr(A) = O if r < O.

H~ = A a.

Hr(A) = 0 if A is injective in Mod(G) and r >= 1.

This cohomology functor is determined up to a unique isomorphism.

For the convenience of the reader, we write the first few terms of the sequences implicit in Theorem 3.1. From the resolution

�9 "---* X1 ---* X0 ---+ Z --* 0

we obtain the sequence

0 --~ Horns(Z, A) --~ Horns(X0, A ) - ~ Homo(X1, A ) - ~

1.3 21

so the cohomology sequence arisihg from an exact sequence

0 ---* A' --~ A ---. A" ---. 0

starts with an exact part

0 ---* A 'c ~ A a ~ A ''G ~ HI (A ' ) .

Dually, we work with the tensor product. We let / a as before be the augmentation ideal. For A 6 Mod(G), I a A is the G-module generated by all elements aa - a with a 6 A, and even consists of such elements. We consider the functor A ~ A a = A / f a A from Mod(G) into Grab. For A, B E Mod(G) we define

T G ( A , B ) : A | B = (A | B)G.

Then Ta is a bifunctor

T a " Mod(G) x Mod(G) --* Grab,

covariant in both variables. From Algebra, Chapter XX, Proposi- tion 3.2', we find:

T h e o r e m 3.2. Let X be a projective resolution o~ Z in Mod(G). Let Ta be as above, and let H = {Hr} be the homology of the complex TG(X , A) . Then H is a homological functor such that:

Hr(d ) : 0 i / r >0 .

Ho(A) = AG.

H~(A) : 0 i~ A is projective in Mod(G) and r >= 1.

The explicit determination of Ho(A) = A v comes from the fact that

X1 | A --* Xo | A --* Z | A ---* 0

is exact, and that Z | A is functorially isomorphic to Ao.

Given a short exact sequence 0 ---* A' --+ A ~ A" ---* 0, the long homology exact sequence starts

�9 "---* H I ( A " ) - * A b --~ A a ~ A" G ---~ O.

The previous two theorems fit a standard pattern of the derived functor. In some instances, we have to go back to the way these functors are constructed by means of complexes, say as in Algebra, Chapter XX, Theorem 2.1. We summarize this construction as follows for abelian categories.

22

T h e o r e m 3.3. Let 92, ~8 be abeIian categories. Let

Y : 9 2 --~ C(f8)

be an exact functor to the category of complexes in ~3. Then there exists a cohomologicaI functor H on 92 with values in ~ , such that Hr(A) = homology of the complex Y(A) in dimension r. Given a short exact sequence in 92:

0 ~ A' ~ A" ---* A -5 ~ 0

and therefore the exact sequence

0---+ Y(A ' ) - -~ Y ( A ) ---, Y(A")---+ O,

the coboundary is given by the usual formula Y(u) - l d Y ( v ) -1 .

For the applications, readers may take ~ to be the category of abelian groups, and 92 is Mod(G) most of tke time.

C o r o l l a r y 3.4. Let 921,92 be abelian categories and F a bif~nc- tor on 921 • 92 with values in ~ , contravariant (resp. covariant) in the first variable and covariant in the second. Let X be a complex in C(921) such that the functor A F(X ,A) on 92 is exact. Then there exists a cohomological functor (resp. homolog- icaI functor) H on 92 with values in ~ , obtained as in Theorem 3.3, with F ( X , A ) = Y(A) .

Next we deal with finite groups, for which we obtain a non-trivial cohomological functor in all dimensions, using constructions with complexes as in the above two theorems.

F i n i t e g roups . Suppose now that G is finite, so we have the trace homomorphism

S = S a : A - - * A

for every A E Mod(G). We omit the index G for simplicity, so the kernel of the trace in A is denoted by As. We also write [ instead of Ia as long as G is the only group under consideration. It is clear t-hat I A is contained in As and the association

A ~ A s / I A

is a functor from Mod(G) into Grab. We then have T a t e ' s t heo - rem.

1.3 23

T h e o r e m 3.5. Let G be a finite group. There is a cohomological functor H on rood(G) with values in Grab such that:

H ~ is the functor A ~-~ A G / S a A .

H (A) = 0 i f A is in j ec t i ve and r > 1

H~(A) = 0 if A is projective and r is arbitrary.

H is erased by G-regular modules, and thus is erased by M o .

Proof. Fix the projective resolution X of Z and apply the two bifunctors | and Homo to obtain a diagram:

""* X 1 Q G A ""* X o | H o m G ( X o , A )

Z | HoraG(Z,A)

0 0

"* H o m G ( X 1 , A )

We have AG = Z | and Homo(Z,A) = A ~ The right side with Homo comes from Theorem 3.1 and the left side comes from Theorem 3.2. We shall splice these two sides together. The trace maps Ao --* A ~ and yields a morphism of the functor A ~ Ao to the functor A ~ A G. Hence there exists a unique homomorphism 5 which makes the following diagram commutative.

X I ~ G A "-'+ X o @ A ~ H o m a ( X o , A ) ---* H o m G ( X 1 , A ) --*

T T

A G ----* A G SG

0 0

The upper horizontal line is then a complex. Each Xr | A may be considered as a functor in A, and similarly for Homo(Xr, A). These functors are exact since X~ is projective. Furthermore, 5 is a morphism of the functor XO| into the functor Homo(X0, .). We let

Y,-(A) = { Homo(X~,A) for r => 0

X - ~ - I | A for r < 0.

Then Y ( A ) is a complex, and A ~ Y ( A ) is exact, meaning that if 0 --* A' --~ A --* A" ~ 0 is a short exact sequence, then

0 ~ Y(A')--~ Y ( A ) ~ Y ( A " ) ~ 0

24

is exact. We are thus in the standard situation of constructing a homology functor, say as in Algebra, Chapter XX, Theorem 2.1, whereby Hr(A) is the homology in dimension r of the complex Y(A). In dimensions 0 and -1, we find the functors of Theorem 3.1 and 3.2, thus proving all but the last statement of the theorem, concerning the erasability.

To show that G-regular modules erase the cohomology, we do it first in dimension r > 0. There exists a homotopy (Cf. Algebra, Chapter X_X, w i.e. a family of Z-morphisms

Dr " X r - ' * Xr+l

such that idr = idx~ = 0 r + l D r + Dr-lOt.

(Cf. the Remark at the end of w loc. cir.) Let f : X,- ---* A be a cocycle. By definition, fOr+l = 0 and hence

f = f o id~ = fDr-1 Or.

On the other hand, by hypothesis there exists a Z-morphism v �9 A ---* A such that 1A = SG(V). Thus we find

f = 1Af = S c ( v ) f = Sv(vf ) = Sc(vfD~-lOr)

=SG(vfDr-1)O~,

which shows that f is a coboundary, in other words, the cohomology group is trivial.

For r = 0 we obviously have H~ = 0 if A is G-regular. For r = - 1 , the reader will check it directly. For r < -1 , one repeats the above argument for r => 1 with the tensor product, essentially dualizing the argument (reversing the arrows). This concludes the proof of Theorem 3.3.

Alternatively, the splicing used to prove Theorem 3.3 could also be done as follows, using a complete resolution of Z .

Let X = (Xr)r> 0 be a G-free resolution of Z, with Xr finitely generated for all r, acyclic, with augmentation ~. Define

X - r - 1 = Hom(Xr, Z) for r >__ 0.

1.3 25

Thus we have defined G-modules Xs for negative dimensions s. One sees immediately that these modules are G-free. If (ei) is a basis of X~ over Z[G] for r >= 0, then we have the dual basis (e v) for the dual module. By duality, we thus obtain G-free modules in negative dimensions. We may sphce these two complexes around 0. For simplicity, say X0 has dimension 1 over Z[G] (which is the case in the complexes we select for the apphcations). Thus let

x 0 = w i t h = 1.

Let ~v be the dual basis of X - i = Horn(X0, Z). We define 00 by

gEG

We can illustrate the relevant maps by the diagram:

o~=8-1 8o X - 2 ~ X - l " X0 �9

/ z

/ o

81 X1 ~

The boundaries ~9-~-i for r >= 0 are defined by duality. One verifies easily that the complex we have just obtained is acyclic (for instance by using a homotopy in positive dimension, and by dualis- ing). We can then consider Homa(X, A) for A variable in Mod(G), and we obtain an exact functor

A ~ Homa(X, A)

from Mod(G) into the category of complexes of abelian groups. In dimension 0, the homology of this complex is obviously

H~ = AC/SaA,

26

and the uniqueness theorem applies.

The cohomological functor of Theorem 3.5 for finite groups will be called the specia l cohomology, to distinguish it from the cohomology defined for arbitrary groups, differing in dimension 0 and the negative dimensions. We write it as HG if we need to specify G in the notation, especially when we shall deal with different groups and subgroups. It is uniquely determined, up to a unique isomorphism. When G is fixed throughout a discussion, we continue to denote it by H. Thus depending .on the context, we may write

H(A) = H G ( A ) = H(G,A) for A e Mod(G).

The s t a n d a r d c o m p l e x

The complex which we now describe allows for explicit computations of the cohomology groups. Let X~ be the free Z[G]-module having for basis r-tuples ( a l , . . . , err) of elements of G. For r = 0 we take X0 to be free over Z[G], of dimension 1, with basis element denoted by (.). We define the boundary maps by

r

§ ~-~(--1)J(crl , . . . ,cr jo ' j+l , . . . ,vrr) j--1

+

We leave it to the reader to verify that d d = 0, i.e. we have a complex called the s t a n d a r d complex.

The above complex is the non-homogeneous form of another s tandard complex having nothing to do with groups. Indeed, let S be a set. For r = 0, 1, 2 , . . . let Er be the free Z-module generated by (r + 1)-tuples ( x 0 , . . . , xr) with x 0 , . . . , x~ 6 S. Thus such (r + 1)- tuples form a basis of Er over Z. There is a unique homomorphism

dr+ 1 : Er+l -+ Er

such that

r+ l

d r + l ( x o , . . . , X r ) = ~ f i -~( - -1)J(xo , . . . ,X j , . . . ,Xr+i) , j=O

1.3 27

where the symbol ~j means that this term is to be omitted. For r = 0 we define r : E0 ~ Z to be the unique homomorphism such that r = 1. The map s is also called the a u g m e n t a t i o n . Then we obtain a resolution of Z by the complex

---* E~+I ---* E~ ---* . . . ---* E0 ---* Z --~ 0.

The above standard complex for an arbitrary set is called the ho- m o g e n e o u s s t a n d a r d complex . It is exact, as one sees by using a homotopy as follows. Let z E S and define

h" E,, ---, E,.+I by h(:co,. . . ,:r,,.)= (z, x o , . . . , z , . ) .

Then it is routinely verified that

dh + hd = id

Exactness follows at once.

and dd = O. r

Suppose now that the set S is the group G. Then we may define an action of G on the homogeneous complex E by letting

~( ,0 , . . . , ~ ) = ( ~ 0 , . . . , ~ r ) .

It is then routinely verified that each E~ is Z[G]-free. We take z = e. Thus the homogeneous complex gives a Z[G]-free resolution of Z.

In addition, we have a Z[G]-isomorphism X ~ E between the non-homogeneous and the homogeneous complex uniquely determined by the value on basis elements such that

( ~ 1 , . . . , ~ ) ~ (e , ,1 , ~ 1 ~ 2 , . . . , ~ , 2 . . . ~ ) .

The reader will immediately verify that the boundary operator c9 given for X corresponds to the boundary operator as given on E under this isomorphism.

If G is finite, then each Xr is finitely generated. We may then proceed as was done in general to define the standard modules X8 in negative dimensions. The dual basis of {(Crl,..., cry)} will be denoted by {[~1,- . . ,~r]} for ~ >__ 1. The dual basis of (.) in dimension 0 will be denoted by [-]. For finite groups, we thus obtain:

28

T h e o r e m 3.6. Let a be a finite group. Let X = {Xr}(r E Z) be the standard complex. Then X is Z[G]-free, acyclic, and such that the association

A ~ Horns(X, A)

is an exact functor of Mod(G) into the category of complexes of abeIian groups. The corresponding cohomological functor H is such that H~ = Aa/SGA.

E x a m p l e s . In the standard complex, the group of 1-cocycles consists of maps f : G ---* A such that

f (a) + crf(v) = f (ar) for all o, r E G.

The 1-coboundaries consist of maps f of the form f(~) = aa - a for some a E A. Observe that if G has trivial action on A, then by the above formulas,

HI(A)=Hom(G,A) .

In particular, H 1 (Q/Z) = G is the character group of G.

The 2-cocycles have also been known as fac tor sets, and are maps f (a , r) of two variables in G satisfying

f(o', T) + f(o'r,p) = o'f(r,p) + f(cr, vp).

In Theorem 3.5, we showed that for finite groups, H is erased by Ms. The analogous statement in Theorem 3.1 has been left open. We can now settle it by using the standard complex.

T h e o r e m 3.7. Let G be any group. Let B E Mod(Z). Then for all subgroups G' of G we have

Hr(G' ,Ma(B)) = 0 for r > O.

Proof. By Proposition 2.6 it suffices to prove the theorem when G' = G. Define a map h on the chains of the standard complex by

h: Cr(G, MG(B)) ---. Cr- ' (G, Ma(B))

1.4

by letting

One verifies at once that

f = hdf + dhf,

whence the theorem follows. (Cf. Algebra, Chapter XX, w

29

w E x p l i c i t c o m p u t a t i o n s

In this section we compute some low dimensional cohomology groups with some special coefficients.

We recall the exact sequences

o-+ la z - - , o

0-- , Z---, q - ~ Q / Z - - ~ 0 .

We suppose G finite of order n. We write I = 1c and H = HG for simplicity. We find:

H - 3 ( Q / Z ) ~ H-2 (Z) ~ H - I ( I ) = I / I 2 H - 2 ( Q / Z ) ~ H - I ( Z ) ~ H~ = 0 H - I ( Q / Z ) ~ H~ ~ H~(I) = Z / n Z H ~ ~ H~(Z) ~ H2(I) = 0 H~(Q/Z) ~ H2(Z) ~ H3(I) = G.

The proof of these formulas arises as follows. Each middle term in the above exact sequences annuls the cohomological functor because ZIG] is G-regular in the first case, and Q is uniquely divisible in the second case. The stated isomorphisms other than those furthest to the right axe then those induced by the coboundary in the cohomology exact sequence.

As for the values furthest to the right, they are proved as follows.

For the first one with 1/12, we note that every element of I has trace 0, and hence H - I ( I ) = 1/12 directly from its definition as in Theorem 2.7.

For the second line, we immediately have H - I ( Z ) = 0 since an element of Z with trace 0 can only be equal to 0.

30

For the third line, we have obviously H~ = Z / n Z from the definition.

For the fourth line, Hi (Z) = 0 because from the standard complex, this group consists of homomorphism from G into Z, and G is finite.

For the fifth line, we find G because H i ( Q / Z ) is the dual group Hom(G, Q/Z) .

R e m a r k . Let U be a subgroup of G. Then

Hr(U,Z[G]) = Hr(U, Q) = 0 for r E Z.

Hence the above table applies also to a subgroup U, if we replace H a by H u and replace na = # (G) by nu = # (U) in the third line, as well as G by U in the last line.

As far as I / I 2 is concerned, we have another characterization.

P r o p o s i t i o n 4.1. Let G be a group (possible infinite). Let G c be its commutator group. Let [a be the ideal of ZIG] generated by all elements of the form cr - e. Then there is a functoriaI isomorphism (covariant on the category of groups)

a / a c . za / by c - e) +

Proof. We can define a map G --+ 1/12 by cr ~-+ (~ - e) + 12 . One verifies at once that this map is a homomorphism. Since I / I 2 is commutative, G c is contained in the kernel of the homomorphism, whence we obtain a homomorphism G/G c ---+ I / I 2. Conversely, I is Z-free, and the elements (or - e) with cr E G, ~ # e form a basis over Z. Hence there exists a homomorphism I ~ G / G c defined by the formula (a - e) ~-+ crG c, cr # e. In addition, this homomorphism is trivial on 12, as one verifies at once. Thus we obtain a homomorphism of 1 / I 2 into G/G c, which is visibly inverse of the previous homomorphism of G / G c into [ / I 2. This proves the proposition.

- In particular, from the first line of the table, we find the isomorphism

H - 2 ( Z ) ..~ G /G ~,

obtained from the coboundary and the isomorphism of Proposition 4.1. This isomorphism is important in class field theory.

1.4 31

We end our explici t c o m p u t a t i o n s wi th one m o r e result on H 1 .

P r o p o s i t i o n 4 .2 . Let G be a group, A E Mod(G) , and let a E H I (G ,A ) . Let {a(cr)} be a 1-cocycle representing a. There exists a G-morphism

f :Ia---~ A

such that f ( ~ r - e) = a(~), i.e. one had f E (Hom(IG, A)) a. Then the sequence

0 ---* A = Horn(Z, A) --* Hom(Z[G] , A) --* Horn(Is ,A) ---* 0

is exact, and taking the coboundary with respect to this short exact sequence, one has

5(~Gf) = -o~.

Proof. Since the e lements (a -. e) fo rm a basis of Ia over Z, one can define a Z - m o r p h i s m f sat isfying f(o" - e) = a(~) for ~ :f ie . T h e fo rmu la is even valid for cr = e, because p u t t i n g ~ = r = e in the fo r mu la for t he c o b o u n d a r y

a(~r) = a(cr) + era(r),

we f ind a(e) = O. We claim tha t f is a G-morph i sm . Indeed, for a, r E G we find:

f ( c r ( r - e ) ) = f ( a r - or) = f ( ( c r r - e ) - (or - e ) )

= f ( a r - e ) - f ( ~ r - e )

= a ( a r ) - - a ( o - )

---- ~ a ( ~ )

= ~ r f ( r - e ) .

To c o m p u t e x a f we first have to find a s t a n d a r d c o c h i n of Hom(Z[G] , A) in d imens ion 0, m a p p i n g on f , t h a t is an e lement

-f ' E H o m ( Z [ G ] , A ) whose res t r ic t ion to IG is f . Since

Z[G] = rc + Z~

is a direct sum, we can define f ' by prescr ib ing t ha t f ' (e) = 0 and f ' is equal to f on I s . One t hen sees t ha t go = ~ f ' - f ' is a cocycle

32

of dimension 1, in Horn(Z, A), representing x c f by definition. I claim that under the identification of Horn(Z, A) with A, the map g~ corresponds to -a(cr). In other words, we have to verify that gz(e) = --a(cr). Here goes:

g a ( e ) = ( o f ' ) ( e ) - f ' ( e )

= o-f ' (o --1 )

= o f ( o "-1 - e )

= f ( e - o )

= - a ( ~ ) ,

thus proving our assertion and concluding the proof of Proposition 4.2.

w Cycl ic g roups

Throughout thin section we let G be a finite cyclic group, and we let ~ be a generator of G

The main feature of the cohomology of such a cyclic group is that the cohomology is periodic of period 2, as we shall now prove.

We start with the &functor in two dimensions

H51 and H~.

Recall that g : A v -+ H~ A) = A G / S G A and

ax: As~ ---+ H - I ( G , A ) = A s a / I a A

are the canonical homomorphisms. We are going to define a cohomological functor directly from these maps. For r E Z we let:

f H - I ( G , A ) i f r is odd H~(G,A) / H~ A) if r is even.

We then have to define the coboundary. Let

0 ---+ A' --~ A --L A" ---+ 0

1.5 33

be a short exact sequence in Mod(G). For each r E Z we define x~ and ~x~ in the natural way given the above definition, and for a" E A " a we pick any element a E A such that va = a" and define

~ , x r ( a " ) - ~r(cra - a).

Similarly, in odd dimensions, for a" E Asc we pick a E A such that va = a" and we define

= x r ( S G a ) .

It is immediately verified that ~, is well-defined (depending on the choice of generator or), that is independent of the choice of a such that va = a". It is then also routinely and easily verified that the sequence {H r} (r E Z) with the coboundary ** is a cohomological functor. Since it vanishes on G-regular modules, and is given as before in dimension < 0, it follows that it is isomorphic to the special functor defined previously, by the uniqueness theorem. Di- rectly from this new definition, we now see that for all r E Z and A E Mod(G) we have the periodicity

H " + 2 ( G , A ) = H " ( G , A ) .

Of course, by truncating on the left we can define a similar functor in the ordinary cue. We put H ~ = A c as before, and for r _> 1 we let:

H - I ( G , A ) i f r i s o d d

H " ( A ) = H O ( G , A ) if r is even.

We define the coboundary as before, so we find a cohomological functor which is periodic for r >__ O, and 0 in negative dimensions. Again the uniqueness theorem shows that it coincides with the functor defined in the previous sections. The beginning of the cohomology sequence reads:

0 ---+ A '~ ~ A a --~ A ' 'a ----+ H - I ( G , A )

and it continues as for the special functor.

The cohomology sequence for the special functor can be conve- niently written as in the next theorem.

34

T h e o r e m 5.1. Let. G be f ini te cyclic and let a be a generator . Le t

0 ---* A I ---* A --~ A" ---, 0

be a shor t exact sequence in Mod(G). is exact:

Then the fo l lowing hexagon

H-1 .H-I(A '')

HI(A ') = H-I(A ') H~ ')

H~ H~

Suppose given an exact hexagon of finite abelian groups as shown:

H2

H6 Ha

\ / Hs �9 H4

and let hi be the order of Hi, that is hi = (Hi : 0). Let

f i : Hi ---* Hi+l with i mod 6

be the corresponding homomorphism in the diagram. Then

hi = (Hi : f i - l H i - 1 ) ( f i - l H i - 1 : O) = m i m i - 1 .

Hence m o m l m 2 r n s m 4 m 5 h l h a h s

1 - - ---- r n l m 2 r n s r n 4 m s r n s h2h4h6 "

We may apply this formula to the exact sequence of Theorem 5.1. Assume that each group Hi(A) is finite, and let h i ( A ) = order of Hi(G, A). Then

h l ( A ' ) h t ( A " ) h 2 ( A ) 1 =

h l ( d ) h 2 ( A ' ) h 2 ( A " ) "

1.5 35

Now let A E Mod(G) be arbitrary. If hi(A) and h2(A) are finite, we define the H e r b r a n d q u o t i e n t h2D(A) to be

h2(A) h 2 / l ( A ) - h i (A)

(A._, :SoA) Asc : (a - e)A)"

If h~(A) or h2(A) is not finite, we say that the Herbrand quotient is no t def ined. This Herbrand quotient is in fact a Euler characteristic, cf. for instance Algebra, Chapter XX, w If A is a finite abelian group in Mod(G), then the Herbrand quotient is defined. The main properties of the Herbrand quotient are contained in the next theorems.

T h e o r e m 5.2. H e r b r a n d ' s L e m m a . Let G be a finite cyclic group, and let

0 ~ A' ---* A ---* A" ---* 0

be a short exact sequence in 1VIod(G). /] two out of three Her- brand quotients h2 / l (A ' ) , h2 / l (A) ,h2 / l (A" ) are defined so is the third, and one has

h 2 / l ( A ) = h 2 / l ( A ' ) h 2 / l (AU) .

Proof. This follows at once from the discussion preceding the theorem. It is also a special case of Algebra, Chapter XX, Theorem 3.3.

T h e o r e m 5.3. Let G be finite cyclic and suppose A E Mod(G) is finite. Then h2/l(A ) = O.

Proof. We have a lattice of subgroups:

A J

ASa A~,_~

I I (~ - e)A SeA

0

The factor groups A / A s c and SaA are isomorphic, and so are A / A , _ , and (a - e)A. Computing the order (A: 0) going around

36

bo th sides of the hexagon, we find that the factor groups of the two vertical sides have the same order, that is

(Asc : (a - e)A) = (A~_~ : SGA).

Tha t h2/l(A) = 1 now follows from the definitions.

Finally we have a result concerning the Herrbrand quotient for trivial action.

T h e o r e m 5.4. Let G be a finite cyclic group of prime order p. Let A E Mod(G). Let t(A) be the Herbrand quotient relative to the trivial action of G on the abelian group A, so that

t(A) - (Ap 0 ) ( A / p A ' O )

Suppose this quotient is defined. Then t(AG), t (Av) , and h2/l(A) are defined, and one has

h2/l(A) p-1 = t(AC)P/t(A) = t(AG)P/t(A).

Proof. We leave the proof as an exercise (not completely trivial).

C H A P T E R II Relat ions with Subgroups

This chapter tabulates systematically a number of relations between the cohomology of a group and that of its subgroups and factor groups.

w Var ious m o r p h i s m s

(a) C h a n g i n g t h e g r o u p G. Let ), : G' ~ G be a group h o m o m o r p h i s m . Then • gives rise to an exact functor

~ : M o d ( a ) - ~ Mod(C')

because every G-module can be considered as a G'-module if we define the action of an element cr ~ E G ~ on an element a E A by

~'a = ~(~')a.

We may therefore consider the cohomoiogical functor Ho, o ~;~ (or the special functor He, o ~:~ if G' is finite) on Mod(G).

In dimension 0, we have a morphism of functors

g ~ --+ H~, o ~

given by the inclusion A ~ ~ A G' = (~x(A)) c ' . If in addition G and G' are finite, then we have a morpkism of functors

H a ---. H a, o ~

38

given by the homomorphism Aa/SGA --~ AG'/SG, A, with G' acting on A in the manner prescribed above, via A.

By the uniqueness theorem, there exists a unique morphism of cohomological functor (5-morphism)

A* : HG "-+ HG' o ~x or HG--~HG, o ~ x ,

the second possibility arising when G and G' are finite. We shall now make this map )~* explicit in various special cases.

Suppose A is surjective. Then we call A* the l if t ing morphism, and we denote it by lif(gG,. In this case, G may be viewed as a factor group of G ~ and the lifting goes from the factor group to the group. On the other hand, when G ~ is a subgroup of G, then A* will be called the r e s t r i c t ion , and will be studied in detail below.

Let A E Mod(G) and B E Mod(G'). We may consider A as a G'-module as above (via the given A). Let v : A --* B be a G'-morphism. Then we say that the pair (A,v) is a m o r p h i s m of (G, A) to (G', B). One can define formally a category whose objects axe pairs (G, A) for which the morphisms are precisely the pairs (k, v). Every morphism (k, v) induces a homomorphism

(A,v). : H~(G,A) ~ H~(G',B),

and similarly replacing H by the special H if G and G' are finite, by taking the composite

H" (G ,A) x*, H r ( G ' , A ) H~ , ( , )H~(G , B)"

Of course, we should write more correctly Hr (G ', ~5~(A)), but usu- ally we delete the explicit reference to r when the reference is clear from the context.

P r o p o s i t i o n 1.1. Let (A,v) be a morphism of (G,A) to G',B),

is a morphism of (G,A) to (G",C), and one has

II .1 39

Proof. Since ~* is a morphism of functors, the following diagram is commutative:

H~(G,,~x(A)) Ha,(~) H~(G,,B)

<~ ~" H~(G '' , C;~@x(A)) Hr(G '', q~(B)) .

HG,,(v)

Consequently we find

( ~ , ~ , _ , ) . = H c , , ( ~ ) o x~)*

= Hg,,(w) o Ho,,(v) o ~* o A*

= HG,,(~) o < H ~ , ( . ) o ;,*

= ( ~ , ~ ) . ( ~ , ~) . ,

thus proving the proposition.

(b) R e s t r i c t i o n . This is the case when A is an injection, so that we may consider G' as a subgroup of G. We therefore have for A E Mod(G):

resv G, " H~(G,A)---+ H~(G',A),

and similarly for the special functor H ~ when G is finite. One verifies at once that for r > 0 the restriction homomorphism is obtained from the standard complex by restricting a cochain { f (~r l , . . . , cry)} as a function of r-tuples of elements of G to r-tuples of elements in G', because the restriction is a morphism of cohomological functors to which we can apply the uniqueness theorem. In dimension O, the restriction is induced by the inclusion mapping.

We have transitivity:

Proposition 1.2. Let G' D G" be ~ubgroup8 of G. Then on Ha, or HG if G is finite, we have

rest;, o re s$ = res$ , .

Proof. Immediate from Proposition 1.1.

40

(c) I n f l a t i on . Let A : G ---, G/G' be a surjective homomorphism. Let A E Mod(G). Then A G' is a G/G'-module for the obvious action induced by the action of G, trivia[ on G', and of course A G' is also a G-module for this operation. We have a morphism of inclusion

u : A a' ~ A

in Mod(G), which induces a homomorphism

H~(u) = u~ " H~(G,A a') ~ H~(G,A) for r => 0.

We define inflation

int~a/c'- H ~ ( G / G ' , A a') ---, H~(G,A)

to be the composite of the functorial morphism

H " ( G / G ' , A a' ) .-_, H " ( G , A a' )

followed by the induced homomorphism ur for r => 0. Note that inflation is NOT defined for the special cohomology functor when G is finite.

In dimension 0, the inflation therefore gives the identity map

(A G' )G/G' --+ A G.

In dimension r > 0, it is induced by the cochain homomorphism in the standard complex, which to each cochain {f (~h , . . . ,~r)} with ~i E G/G' associates the cochain { f ( a l , . . . ,~rr)} whose values are constant on cosets of G'.

We have already observed that if G acts trivially on A, then H i ( G , A) is simply Horn(G, A). Therefore we obtain:

P r o p o s i t i o n 1.3. Let G' be a normal subgroup o / G and suppose G acts trivially on A. Then the inflatwn

ini~a/c'- H I ( G / G ' , A a') ~ H I ( G , A )

induces the inflation of a homomorphism ~ : G/G' --* A to a homomorphism X " G --~ A.

Let G' be a normal subgroup of G. We may consider the association

F G : A ~ A G'

II.1 41

as a functor, not exact, from Mod(G) to Mod(G/G') . Inflation is then a morphism of functors (but not a cohomological morphism)

Ha~a, o FG, --+ Ha

on the category Mod(G). Even though we are not dealing with a cohomological morphism, we can still use the uniqueness theorem to prove certain commutativity formulas, by decomposing the inflation into two pieces.

As another special case of Proposition 1.1, we have:

P r o p o s i t i o n 1.4. Let G', N be subgroups of G with N normal in G, and N contained in G'. Then on Hr (G/N , A N) we have

' G/N in~G/N. in~GG ,IN o resG,/N = resG G, o

We also have transitivity, also as a special case of Proposition 1.1.

P r o p o s i t i o n 1.4. Let G ---* G1 ---* G2 be aurjective group ho- momorphisma. Then

i n g ' o ing~ = ini~a =.

(d) C o n j u g a t i o n . Let U be a subgroup of G. For ~ E G we have the conjugate subgroup

v = = =

The notation is such that U (~) = (U~) ~'. On Mod(G) we have two cohomological functors, Hu and Hu- . In dimension 0, we have an isomorphism of functors

A U- - ~ A U~ given by a ~ r - l a .

We may therefore extend this isomorphism uniquely to an isomorphism of/-/u with Hu. which we denote by a , and which we call c o n j u g a t i o n .

Similarly if U is finite, we have conjugation ~, on the special functor Ha --~ Hu~.

42

P r o p o s i t i o n 1.5. If or E U then a, is the identity on Hu (reap. H u if U is finite).

Proof. The assertion is true in dimension O, whence in all dimensions.

Let f : A --* B be a U-morphism with A, B E Mod(G). Then

f~ = [cr -1I f = f[~r] : A ---* B

is a U~-morphism. The fact that a, is a morphism of functors shows that

o = o rv(f)

as morphisms on H(U, A) (and similarly for H replaced by H if U is finite).

If U is a normal subgroup of G, then or, is an automorphism of Hu (resp. H u if U is finite). In other words, G acts on Hu (or Hu) . Since we have seen that or, is trivial if ~ E U it follows that actually G/U acts on Hu (resp. Hu).

P r o p o s i t i o n 1.7. Let V C U be subgroups of G, and let ~r E G. Then

a, ores = r e s . oct.

on Hu (resp. H u if U is finite).

P r o p o s i t i o n 1.8. Let V C U be subgroups of G o f finite index, and let a E G. Suppose V normal in U. Then

in ~ o ~ r , = ( r . o i n IV

on H(U/V, AV), with A E Mod(G).

Both the above propositions are special cases of Proposition 1.1.

(e) T h e t r a n s f e r . Let U be a subgroup of G, of finite index. The trace gives a morphism of functors H~] ~ U~ by the formula

S U �9 A U ---, A G,

and similarly in the special case when G is finite, H~r --~ H ~ by

S U . A U / S u A -., A a / S a A .

II.1 43

The unique extension to the cohomological functors will be denoted by tr U, and will be called the t ransfer . The following proposition is proved by verifying the asserted commutativity in dimension 0, and then applying the uniqueness theorem. In the case of inflation, we decompose this map in its two components.

P ropos i t i on 1.9. Let V C U C G be subgroups o/finite index in G. Then on H v (resp. H v ) we have

(I) t ra g o tr y :tra v.

(2) ~ . o t r V = t r vocr. for cr C a.

(3) If V is normal in a , then on H~(U/V, AV) with r >= 0 we have

. U/V = t r Uoin~U/v. inf~a/v o r, rG/V

The next result is particularly important.

P ropos i t i on 1.10. Let U be a subgroup of finite index in G. Then on Ha (resp. H a ) we have

tr U ores a = (G" U),

where (G :U) on the right abbreviates (G" U)H, i.e. multiplication by the index on the cohomology functor.

Proof Again the formula is immediate in dimension O, since restriction is just inclusion, and so the trace simply multiplies elements by (G : U). Then the proposition follows in general by applying the uniqueness theorem.

Coro l l a ry 1.11. Suppose G finite of order n. Then for all r E Z and A E Mod(C) we have nH~(G,A) = 0.

Proofi Take U = {e} in the proposition and use the fact that g ~ ( e , A ) = 0 .

Coro l l a ry 1.12. Suppose G finite, and A E Mod(G) finitely generated over Z. Then H"(G, A) is a finite group for all r E Z.

Proof. First Hr(G, A) is finitely generated, because in the standard complex, the cochains are determined by their values on the

44

finite number of generators of the complex in each idmension. Since Hr(G, A) is a torsion group by the preceding corollary, it follows that it is finite.

Corollary 1.13. Suppose G finite and A E Mod(G) is uniquely divisible by every integer m 6 Z, m # O. Then Hr(G, A) = 0 for all r E Z .

Proposition 1.14. Let U C G be a subgroup of finite index. Let A, B 6 Mod(G) and let f : A ---* B be a U-morphism. Then

H a ( S V ( f ) ) = t rv o H v ( f ) o res t ,

and similarly with H instead of H when G is finite.

Proof. We use the fact that the assertion is immediate in dimension 0, together with the technique of dimension shifting. We also use Chapter I, Lemma 2.4, that we can take a G-morp~sm in and out of a trace, so we find a commutative diagram

0 , A , MG(A) , XA , 0

! 1 l 0 , B , MG(B) , XB , 0

the three vertical maps being s U ( f ) , S U ( M ( f ) ) and s U ( x ( f ) ) respectively. In the hypothesis of the proposition, we replace f by X ( f ) : X A ----+ X B , and we suppose the proposition proved for X ( f ) . We then have two squares which form the faces of a cube as shown:

r e s HG(XA)

HG(XB) ~ res

~ ( A { H c ( S ~ ( x ( I ) ) )

HG(B) " tr

: Hu(XA)

-

\ Hu(A)

Hu(B)

The maps going forward are the coboundary homomorphisms, and are surjective since MG erases cohomology. Thus the diagram allows an induction on the dimension to conclude the proof. In the case of the special functor H, we use the dual diagram going to the left for the induction.

II . 1 45

C o r o l l a r y 1.15. Suppose G finite and A,B E Mod(G). Let f �9 A ---, B be a Z-morphism. Then SG(f) " A --~ B induces 0 on all the cohomology groups.

Proof. We can take U = {e} in the preceding proposition.

E x p l i c i t f o rmu la s

We shall use systematically the followihg notation. We let {c} be the set of right cosets of a subgroup U of G (not necessarily finite). We choose a set of coset representatives denoted by ~. If cr E G, we denote by ~ the representative of Uc. We may then write

G = U U ~ : U ~ - I U c c

since {~--1} is a system of representatives for the left cosets of U in G. By definition, we have U~ = Uc, whence for all a E G, we have

caca 1 E U.

We now give the explicit formula for the transfer on standard cochalns. It is induced by the cochain map f ~ trU(f) given by

trU(f)(cr0,. . . ,or,) = E c - l f ( c a ~ " " ' ccrr-~-~l )" c

For a non-homogeneous cochaln, we have the formula

t rGU(f)(~r l , . . . , a t ) = - -

, . . . , C0.10.2 C0.10. 1 - 1

c

, . . . , r r 1).

(f) T r a n s l a t i o n . Let G be a group, U a subgroup and N a normal subgroup of G. Let A E Mod(G). Then we have a lattice

46

of submodules of A: A

I AUnN

J A U A N

/ AUN

I A a

We have UN/N ,.m U/(U O N), and U acts on A N since G acts on A N. Furthermore U O N leaves A N fixed, and so we have a homomorphism called t r a n s l a t i o n

t s l , " H~(UN/N,A N) ~ H~(U/(U O N),A UnN)

for r __> 0. The isomorphism UN/N .~ U/(U N N) is compatible with the inclusion of A N in A vnN. Similarly, if G is finite, we get the translat ion for the special cohomology H instead of H, with r_>O.

Taking G arbitrary and r _>_ O, we have a commutative diagram:

H,.(G/N, AN) res Hr(UN/N, AN ) inf H"(UN, A)

l t s l

inf I u r ( v / ( u n N) ,A vnN) 1 ~

l i n f

H"(G,A) , H~(U,A) , H"(U,A) r e s

The composition, which one can achieve in three ways,

tsl , " H"(G/N,A N) --~ H~(U,A)

will also be called t r a n s l a t i o n , and is denoted tsl, .

In dimension - 1 , we have the following explicit determination of cohomology.

II.1

Proposition 1.16. Let A E Mod(G).

(1) For a E dsa we have SU(a) E As~ and

res~n~a(a) = n<(SU(a)).

(2) For a E As~ we have a E Asa and

trU>ru(a) = ~G(a).

(3) Let a E Asv and cr E G. Then a - l a E A s w

(7 ,~u( a) = >I<UO'(O "- la) .

47

Let G be a finite group and U a subgroup.

and

Proof. In each case, one verifies explicitly that the morphism of the functor H -1 given by the expression on the right of the formulas is a 8-morphism, for the pair of functors (H -1, H~ We can then apply the uniqueness theorem. The verification is done routinely, and is left to the reader, who will use the explicit determination of 8 in Chapter I, Theorem 2.7.

Roughly speaking, Proposition 1.16 asserts that the restriction and transfer correspond respectively to the trace and the inclusion (so the order is reversed with respect to dimension 0). Conjugation just consists in applying a-1 to a cochain representing a cohomology class.

In Chapter I, w we gave an explicit determination of H -2 (G, Z). We shall now indicate how the transfer, restriction and conjugation behave with respect to this determination.

We recall the isomorphisms:

H-2(G,Z) ~--* H - I ( G , I a ) = I a / I ~ "~, G/G c.

If T E G, we denote by ~,- the element of H-2(G, Z) which corresponds to ~-G c under the above isomorphism.

Directly in terms of groups, we have some natural homomorphisms as follows. If ~ : G -* G is a homomorphism, then we have an induced homomorphism

: u / c

48

In particular, if U is a subgroup of G, we have the canonical homomorphism

inc. �9 U/U ~ ---. GIG ~

induced by the inclusion.

If U is of finite index in G then we have the transfer from group theory

Tr~ = =a/C~177 c - - . U l y ~

defined by the product

Tr ( V c) = UC)

Cf. for instance Artin-Tate, Chapter XlII, w

We shall now see that the transfer and restriction on H -2 correspond to the inclusion and transfer on the groups (so the order is reversed).

T h e o r e m 1.17. Let G be a finite group and U a subgroup. Then:

1. The transfer t rU: H-2(U,Z) ~ H - 2 ( G , Z ) corresponds to the natural map U/U c ----. G/G c induced by the inclusion of U in G. Thus we may write

tr(C,-) = C,-.

2. The restriction res G" H-2(G, Z) ---. H-2(U, Z) corresponds to the transfer of group theory. Thus we may write

res (C ) =

. Conjugation o ' . : H-2(U,Z) --. I4.-2(U~,Z) corresponds to the map of U/U c into U~/(U~) c induced by conjugation with a E G, so we may write

I I . 1 4 9

Proof. Since Z[U] is natural ly contained in Z[G] we obtain a commuta t ive diagram

0 , zv z [ u ] , z , 0

0 > Iv Z[G] , Z , 0

the vertical maps being inclusions, and the map on Z being the identity. The horizontal sequences are exact. Consequently we obtain a commuta t ive diagram

H - 2 ( U , Z)

H - 2 ( U , Z)

, H-I(U, Iu) ~ Z~IX~:

~ i n c .

, H - I ( U , I c ) ~(IG)s~ / Iu Ia .

The coboundaries are isomorphisms, and hence inc. is also an isomorphism. Thus we may write, as we have done, (fG)sr:/ fufc ins tead of Iu/I~]. In dimension - 1 we may then use the explicit de te rmina t ion of H -2 from the preceding proposition, which we do case by case.

Let ~- E U. Then v - e is in (IG)su, and we have

trU~<u(T - - e) = ~G(T - - e),

which proves the first formula.

Next let cr E G. We have

where c ranges over the right cosets of U, and ~ is a representative of c. Fur thermore

C C

50

because c ~ ca permutes the cosets. Since caca 1 is in U, we may rewrite this equality in the form

~ ( ~ a ~ -~ _ ~ ) ( ~ - ~) + ~ ( ~ a ~ -~ _ ~) C C

= ~ ( ~ a ~ -~ - ~) mod •177 C

If we apply ~x to both sides, one sees that the second formula is proved, taking into account the formula for the transfer in group theory, which ~ves

Tr~(aa c) = [ I ( ~ a ~ - i uo).

As to the third formula, it is proved similarly, using the equalities

= ( , _ ~)(~-1 _ ~ ) + ( ~ _ ~)

- - - - ( r -e ) m o d I u f a .

This concludes the proof of Theorem 1.17.

w Sylow s u b g r o u p s

Let G be a finite group of order N. For each prime p I _IV there exists a Sylow subgroup Gp, i.e. a subgroup of order a power of p such that the index (G : Gp) is prime to p. Furthermore, two Sylow subgroups are conjugate.

In particular, if A E Mod(G) then Hr(Gp, A) is well defined, up to a conjugation isomorphism.

By Corollary 1.11 we know that Hr(G,A) is a torsion group. Therefore

H~(G, A ) = G H " ( G , A , p ) , pIN

where H~(a, A, p) is the p-primary subgroup of Hr(a, A), i.e. consists of those elements whose period is a power of p. In particular, if G = Gp is a p-group, then

H~(G,A) = H~(G,A,p).

II.2 51

T h e o r e m 2.1. Let Gp be a p-Sylow subgroup of G. Then for all r E Z, the restriction

resg, "H~(G,A ,p) - -~ H r ( G , A )

is injective, and the transfer

tr~" . H~(Gp, A) --~ H~(G,A ,p )

is surjective. We have a direct sum decomposition

H~(Gp,A) = Im r e s t , + Ker t r~ p.

Proof. Let q = (Gp : e) be the order o f G p a n d m = ( G : Gp). These integers are relatively prime, and so there exists an integer m ~ such tha t rn'm = 1 rood q. For all a E H~(Gp, A) we have

a = rn'rnc~ -- rn' �9 t r ~ p r e s t , (c~) = ~ra' G, rn' �9 resgp (c~),

whence the inject ivi ty and surjectivity foUow as asserted. For the third, we have for/3 E Hr(Gp, A),

= r e s . ~ ' t r ( Z ) + ( # - r e s . ~ ' t r ( # ) ) ,

the restr ict ion and transfer being taken as above. One sees immediately tha t the first term on the right is the image of the restrict ion, and the second term is the kernel of the transfer. The sum is direct, because if/3 = res(a) , tr(fl) = 0, then

t r ( r e s ( ~ ) ) = . ~ = 0 ,

whence m'rno~ = ~ = 0 and so t3 = 0. This concludes the proof.

C o r o l l a r y 2.2. Given r E Z , and A E Mod(G), the map

pin

gives an injective homomorphism

W ( V , a ) -~ 1-I H~(GP'A) �9 pIN

52

C o r o l l a r y 2.3. /f H~(Gp, A) is of finite order for all p ] N, then so is H~(G, A), and the order of this latter group divides the product of the orders of H~(Gp,A) for all p i N .

C o r o l l a r y 2.4. /f H~(Gp,A) = 0 for alI p, then H~(G,A) = O.

w Induced representations

Let G be a group and U a subgroup. We are going to define a functor

M U : Mod(U) --* Mod(G).

Let B E Mod(U). We let MU(B) be the set of mappings from G into B satisfying

a f ( x ) = f ( a x ) for ~ rEU and x E G .

One can also write

MU(B) = (MG(B)) U.

The sum of two mappings is taken as usual summing their values, so M u ( B ) is an abelian group. We can define an action of G on MU(B) by the formula

(af)(x) = f (xa) for cr, x E G.

We then have transitivity.

Proposition 3.1. Let V C U be subgroups of G. functors

M U o M v and M v

are isomorphic in a natural way.

Then the

We leave the proof to the reader.

We use the same notation as in w with a right coset decomposition {c} of U in G, and chosen representatives ~. We continue to use B E Mod(U).

II.3 53

P r o p o s i t i o n 3.2. Let G be a group and U a subgroup with right cosets {c}. Then the map f ~ resf, which to an element f �9 M g ( B ) associates its restriction to the coset representatives {~}, is a Z-isomorphism

M U(B) - -* M ( G / U , B )

where M(G/U , B) is the additive group of maps from the coset space G /U into B.

Pro@ The formula f(o'~') = af(~-) for ~r �9 U and r �9 G shows that the values f(~) of f on coset representatives determine f , so the restriction map above is injective. Furthermore, given a map fo : G / U ~ B, if we define f0(e) = fo(c), then we may extend f0 to a map f : G ~ B by the same formula, so the proposition is clear.

P r o p o s i t i o n 3.3. Let G be a group and U a subgroup. Then M U is an additive, covariant, exact functor of Mod(U) to Mod(G).

Proof. Let h : B ---* B' be a surjective morpkism in Mod(U), and suppose f ' : G/U ---* B' is a given map. For each value f ' (5) there exists an element b E B such that h(b) = f'(5). We may then define a map f : G/U ~ B such that f o b = ft. From this one sees that M ~ ( B ) ---* M U ( B ') is surjective. The rest of the proposition is even more routine.

T h e o r e m 3.4. Let G be a group and U a subgroup. The bifunctors

Home(A, Mg (B)) and Som (d, 8) from Mod(G) x MoG(U) to Mod(Z) are isomorphic under the following associations. Given f e Homc(d, MU(B)), we let f~ be the map A ~ B such that f l (a) = f(a)(e). Then f l is in Homu(A, B). Conversely, given h E Homu(A, B) and a e A, letg~ be defined byg~(cr) = h(cra). Thena~-+ga is in Homa(A, MU(B)) . The maps f ~-* f l , a ~-* (a ~ g~) are inverse to each other.

Proof. Routine verification left to the reader.

The above theorem is fundamental, and is one version of the basic formalism of induced representations. Cf. Algebra, Chapter XVIII, w

54

C o r o l l a r y 3.5. We have (MU(B)) a = B v.

Proof. Take A = Z in the theorem.

C o r o l l a r y 3.6. If B is injective in Mod(U) then MU(B) is injective in Mod(G).

Proof. Immediate from the definition of injectivity.

T h e o r e m 3.7. Let G be a group and U a subgroup. The map

Hr(G, MU(B))----, Hr(U,B)

obtained by composing the restriction res~ followed by the U- , orph,s, g g(e), is an iso orph,s, for r >= 0

Proof. We have two cohomological functors Ha o MG U and Hu on Mod(U), because M g is exact. By the two above corollaries, they are both equM to 0 on injective modules, and are isomorphic in dimension 0. By the uniqueness theorem, they are isomorphic in all dimensions. This isomorphism is the one given in the statement of the theorem, because if we denote by Tr : M g ( B ) ~ B the U-morphism such that 7rg = g(e), then

Hu(~r) o r e s t " Hu o M U G ""*Hu

is clearly a 6-morphism which, in dimension 0, induces the prescribed isomorphism (MU(B)) a on B U. This proves the theorem.

Suppose now that U is of finite index in G. Let A = Mg(B) . For each. coset c, we may define a U-endomorphism 1re : A ~ A of A into itself by the formula:

0. ifcr C U 7r~(f)(cr) = f(cra) if cr E U.

Indeed, zrc is additive, and if ~- E U, then

r(zrcf)(cr) = (wcf)(Tcr) for all ~ E G.

Indeed, if cr ~ U then both sides are equal to O; and if cr E U, then we use the fact that f E MU(B) to conclude that they axe equal.

Let us denote by A1 the set of elements f E MU(B) such that f(~r) = 0 if a ~ U. Then AI is a U-module, as one verifies at once.

II.3 55

T h e o r e m 3.8. Let U be of finite index in G. Then:

(i) Let A1 be the U-submodule of elements f �9 l~U(B) such that f(cr) = 0 if cr ~ U. Then

M U ( B ) = O ~ - I A , , r

and every such f can be written uniquely in the form

s =

(ii) The map f ~-~ f(e) gives a U-isomorphism A1 ---+ B.

Proof. For the first assertion, let a = re0 with r �9 U. Then

E c-l(rrcf)((r)

=

If c # Co then the corresponding term is O. Hence in the above sum, there will be only one term # O, with c = co. In tlMs case, we find the value f(r~0) = f(~) . This shows that f can be written as asserted, and it is clear that the sum is direct. Finally A1 is U-isomorphic to B because each f I A1 is uniquely determined by its value f(e), taking into account that r f (e) = f ( r ) for r e U. This same fact shows that we can define f I A1 by prescribing f ( e ) = b �9 B and f ( ~ ) = ~b. This proves the theorem.

We continue to consider the case when U is of finite index in G. Let A �9 Mod(G). We say that A is s emi loca l for U, or r e l a t i v e to U, if there exists a U-submodule A1 of A such that A is equal to the direct sum

A = O c - lAl" r

We then say that the U-module A1 is the local c o m p o n e n t . It is clear that A is uniquely determined by its local component, up to an isomorphism. More precisely:

56

P r o p o s i t i o n 3.9. Let A1,A t E Mod(U).

(i)

(ii)

Let f l : A1 "~ A~ be a U-isomorphism, and let A ,A ' be G-modules, which are semilocal for U, with local components A1 and A~ respectively. Then there exists a unique G-isomorphism f : A ---* A' which extends f l .

Let A E Mod(G) and let A1 be a Z-submodule of A. Sup- pose that A is direct sum of a finite number of ~rA1 (for cr E G}. Then A is semilocal for the subgroup of elements v E G such that ~'A1 = A1.

Proof. Immediate.

Theorem 3.8 and Proposition 3.9 express the fact that to each U-module B there exists a unique G-module semilocal for U, with local component (U, B).

P r o p o s i t i o n 3.10. Let A E Mod(G), U a subgroup of finite index in G, and A semilocal for U with local component A1. Let rl : A ~ A1 be the projection, and ~r its composition with the inclusion of Ax in A. Then

1A =

in other words, the identity on A is the trace of the projection.

Proof. Every element a E A can be written uniquely

a - ~ c , lac E "

with ac E A1. By definition,

= Z

The proposition is then clear from the definitions, taking into account the fact that if c, c t are two distinct cosets, then ~r~ac, = O.

K G is finite, one can make the trace more explicit.

II.3 57

Proposition 3.11. Let G be a group, and U a subgroup of finite index. Let A E Mod(G) be semilocal for U with local component A1. Then an element a E A is in A G if and only if

a = ~ ~-1al with some al E A U. C

I f G is finite, then a E SGA if and only ira1 E SuA1 in the above formula. The functors

H~(MU(B)) and H~(B)

(with variable B E Mod(U)) are isomorphic.

Proof. One verifies at once that for the first assertion, if an element a is expressed as the indicated sum with al E A1 and a E A G then the projection maps A a into A U. Since we already know that the projection gives an isomorphism between A c and A U it follows that all elements of A c are expressed as stated, with al E A U. If G is finite, then for b E A we have

\ r E U /

and the second assertion follows directly from this formula.

For finite groups, MG U maps U-regular modules to G-regular modules. This is important because such modules erase cohomology.

P r o p o s i t i o n 3.12. Let G be finite with subgroup U. I f A is semilocal for U with local component (U, A1) and if A1 is U- regular, then A is G-regular.

Proof. If one can write 1A1 = Su ( f ) with some Z-morphism f, then

1A = ScU(~rSu(f)) = SGU(Su(~f)) = Sa(zrf),

which proves that A is G-regular.

From the present view point, we recover a result already found previously.

58

C o r o l l a r y 3.13. Let G be a -finite group, with subgroup U and B in Mod(U). If B is U-regular then M g ( B ) is G-regular.

T h e o r e m 3.14. Let U be a subgroup of finite index in a group G. Suppose A 6 Mod(G) is semilocal for U, with local component A1. Let Trl : A ~ A1 be the projection and inc : Al ~ A the inclusion. Then the maps

Hu(~rl ) ores G

are inverse isomorphisms

and tr u o Hu(inc)

H " ( G , A ) , = ) Hr(U, A1).

I f G is finite, the same holds for the special functor H ~ instead of H".

Proof. The composite

A ,~1 A1 inc A

is a U-morphism of A into itself, which we denoted by ~-. We know that the identity 1A is the trace of this morphism. We can then apply Proposition 1.15 to prove the theorem for H. When G is finite, and we deal with the special functor H, we use Corollary 3.13, the uniqueness theorem on cohomological functors vanishing on U-regular modules, the two functors being

HG o MG U and Hu.

We thus obtain inverse isomorphisms of Hr(G, A) and Hr(U, A1). This concludes the proof.

R e m a r k . Theorem 3.14 is one of the most fundamental of the theory, and is used constantly i/1 algebraic number theory when considering objects associated to a finite Galois extension of a number field.

w D o u b l e cosets

Let G be a group and U a subgroup of finite index. Let S be an arbitrary subgroup of G. Then there is a disjoint decomposition of G into double cosets

G = UUTS = UST-Iu, 7 7

I I .4 59

with {7} in some finite subset of G (because U is assumed of finite index), representing the double cosets. For each 7 there is a decomposition into simple cosets:

s = U(s n ub ] )~ = U ~41( s n ub]), r~ r~

where 7- t ranges over a finite subset of S, depending on 7. Then we claim that the elements {')'%} form a family of right coset representatives for U in G, so that

1", T-y "y,T.y

is a decomposition of G into cosets of U. The proof is easy. First, by hypothesis, we have

a = U U u7(s n u[71)~, r~ -y

and every element of G can be written in the form

u177-1u27% = u7% , with ul ,u2 E U.

Second, one sees that the elements {')'%} represent different cosets, for if

U7T-I = U7%~,,

then 7 = 7' since the 7 represent distinct double cosets, whence r- / and T 7' represent the same coset of 7-1U7, and are consequently equal.

For the rest of this section, we preserve the above notation.

P r o p o s i t i o n 4.1. On the cohomological functor Hu (resp. H u if G is finite) on Mod(G), the following morphisms are equal:

res~ o tr~ ~ sneer1 u[.,] ---- ~r S o r e s s n u [ ~ , ] o 7 * .

y

Proof. As usual, it suffices to verify this formula in dimension 0. Thus let a E A v. The operation on the left consists in first taking

60

the trace $U(a), and then applying the restriction which is just the inclusion. The operation on the right consists in taking first 7-1a, and then applying the restriction which is the inclusion, followed by the trace

trsSnU['r](7-1a) = ~ 7"~17 - l a

according to the coset decomposition which has been worked out above. Finally taking the sam over 7, one finds tr U, which proves the proposition.

C o r o l l a r y 4.2. I f U is normal, then for A E Mod(G) and a E H"(U, A) we have

res tr ( ) =

and similarly for H if G is finite.

Proof. Clear.

Let again A E Mod(G) and a E H(U,A). s t ab l e if for every ~r E G we have

We say that A is

reSunu[,]a,[c~) = resUnu[,](a).

If U is normal in G, then a is stable if and only if a ,a = a for all a E G .

P r o p o s i t i o n 4.3. Let a E H"(U, A) for A E Mod(G). I f o~ = rest(/3) for some ~ E H~(G,A) then a is stable.

Proof. By Proposition 1.5 we know that ~r,/3 = /3. Hence we find

reSunu[a] resg(Z) o a,(o~) = reSunu[a] o ~r, o

[] [1

= resanut l(Z)

If we unwind this formula via the intermediate subgroup U instead of U[a], we find what we want to prove the proposition.

II.4

Proposition 4.4. If c~ E H~(U, A) is stable, then

resg o trU(~) - - - - (G" U)o~.

61

Proof. We apply the general formula to the case when U = `9. The verification is immediate.

Suppose finally that A is semilocal for U, with local component A1, projection ~rl " A ---* A1 as before. Then elements of ,9 possibly do not permute the submodules Ac transitively. However, we have:

Proposition 4.5. Let ~ E G. Then cr E "97-1U " i f and only if ~A1 is in the same orbit of "9 as 7-1A1. For each 7, the sum

~'~-1~-IA1

r~

is an ,9-module, semilocaI for "9 N U[7], with local component v~-17-1A1 ,

Proof. The assertion is immediate from the decomposition of g into cosets r~-lT-1U.

If ,9 and U are both of finite index in G, then the reader will observe a symmetry in the above formulas, in particular in the double coset decomposition. In particular, we can rewrite the formula of Proposition 4.3 in the form

. u [ - ~ ] n s s (*) resg o tr~ -- E 7~-x o ~ru[~] o resu[,]ns.

-y

We just take into account the commutativity of 7. and the other maps, replacing U by 9̀, ,9 by U and 7 by 7 -1 .

C H A P T E R III Cohomological Triviality

In this chapter we consider only finite groups, and the special functor HG, such that HG(A) = A a / S c A . The main result i3 Theorem 1.7.

w T h e tw ins t h e o r e m

We begin by auxi].iaxy results. We let Fp = Z / p Z for a prime p. We always assume G has trivial action on Fp.

P r o p o s i t i o n 1.1. Let G be a p-group, and A E Mod(G) finite of order equal to a p-power. Then A c = 0 implies A = O.

Proof. We express A as a disjoint ,mion of orbits of G. For each z E A we let Gz be the isotropy group, i.e. the subgroup of elements a E G such that az = x. Then the number of elements in the orbit Gx is the index (G : G=). Since G leaves 0 fixed, and

(AO) - E

where rni is the number of orbits having pi elements, it follows that either A - 0 or there is an element z # 0 whose orbit also has only one element, i.e. z is fixed by G, as was to be shown.

III .1 63

C o r o l l a r y 1.2. Let G be a p-group, f f A is a simple p-torsion G-module then A ~ Fp.

Proof. Immediate.

C o r o l l a r y 1.3. The radical of Fp[G] is equal to the ideal fp generated by all elements (a - e) over Fp.

Proof. A simple G-module over Fp[G] is finite, of order a power of p, so isomorphic to Fp, annihilated by Ip which is therefore contained in the radical. The reverse inclusion is immediate since F,[G]/r, ~ F,.

P r o p o s i t i o n 1.4. Let G be a p-group and A an Fp[G]-module. The following conditions are equivalent.

1. Hi(G, A) = 0 for some i with - c ~ < i < oo.

2. A is G-regular.

3. A is G-free.

Proof. Since Fp is a field every Fp-module is free in Mod(Fp). If A is G-free then A is obviously G-regular. Conversely, if A is G-regular, with local component A1 for the unit element of G, then A1 is a direct sum of Fp a certain number of times, and the G-orbit of a factor Fp is G-isomorphic to Fp[G], so A itself is a G-direct sum of such G-modules isomorphic to Fp [G], thus proving the equivalence of the last two conditions.

It suffices now to prove the equivalence between the first and third conditions. Abbreviate f = Ip for the augmentation ideal of Fp[G]. Then A / I A is a vector space over Fp. Let {aj} be representatives in A for a basis over Fp, so that A is generated over Fp [G] by these elements aj and IA . Let E be the Fp [G]-free module with free generators ~j. There is a G-morphism E ---* A such that ~zj ~ aj. Let B be the image of E in A, so that A = B + IA . Since I is nilpotent, I '~ = 0 for some positive integer n, and we find

A = B + I A + B + I B + I2A = .-. = B + I B + . . . + I " A = B

by iteration. Hence the map E --~ A is surjective. kernel. We have

E / I E = A / I A

Let A' be its

64

and so A' C IE. Hence SaA' = 0. Since the following sequence is e x a c t

O--+ A ' --+ E--~ A- '~ O,

and 13 is G-regular, one finds Hr(G, A) = H~+I(G, A') for all r E Z.

Suppose that the index i in the hypothesis is -2 . Then H - I ( G , A') = 0, so A'sa = IA' . Since A' = A'sa, we find A' = IA ' and hence A' = 0 so E ~ A.

On the other hand, if i # -2 , we know by dimension shifting that there exists a G-module C such that H~(U, C) ~ H~+I(U, A) for some integer d, all r E Z and all subgroups U of G, and also H-2(G, C) = 0. Hence C is Fp[G]-free, so cohomologically trivial, and therefore similarly for A, so in particular H-2(G, A) = 0. This proves the proposition.

P r o p o s i t i o n 1.5. Let G be a p-group, A E Mod(G). Suppose there exists an in~egeri such ~hat I-Ii(G,A) = Hi+I(G, A) = 0. Suppose in addition that A is Z-free. Then A is G-regular.

Proof. We have an exact sequence

0 --+ A ~ A ~ A / p A --~ 0

and hence 0 = H i ( A ) - - . Hi(A/pA) - -~ H i + I ( A ) = 0,

the functor H being H a . Therefore H ' ( A / p A ) = 0. Since A / p A is an Fp[G]-module, we conclude from Proposition 1.3 that A / p A is Fp[G]-free, and therefore G-regular.

Since we supposed A is Z-free, we see immediately that the sequence

0 --* HomA(A,A) ~ HomA(A,A) --* H o m z ( A , A / p A ) ---+ 0

is exact. But Homz(A, A/pA) is G-regular, so

p = p . : Hi(Homz(A, A)) --~ Hi (Homz(A,A))

is an automorphism. Hence so is its iteration, and hence so is multiplication by the order (G : e) since G is assumed to be a p-group. But (G : e). = 0, whence all the cohomology groups Hr (Homz(A,A) ) = 0 for r E Z. In particular, H~ = 0. From the definitions, we conclude that the identity 1A is a trace, and so A is G-regular, thus proving Proposition 1.5.

I I I . 1 65

C o r o l l a r y 1.6. Hypotheses being as in the proposition, then A is projective in Mod(G).

Proof. Immediate consequence of Chapter I, Proposition 2.13.

Let G be a finite group and A E Mod(G). We define A to be cohomolog i ca l l y t r i v i a l if H~(U, A) = 0 for all subgroups U of G and all r E Z.

T h e o r e m 1.7. T w i n t h e o r e m . Let G be a finite group and A C Mod(G). Then A is cohomologically trivial if and only if for each p [ (G " e) there exists an integer i v such that

Hip(Gp,A) : Hi~+I(Gp,A) : 0

for a p-Sylow subgroup G v of G.

Proof. Let E E ZIG] be G-free such that

O-~ A '-~ E-~ A-~ O

is exact. Then

Hi~+I(Gv,A) = Hi~+2(Gp,A') = 0.

Since A' is Z-free, it is also Gp-regular by Proposition 1.5. Hence ! for all subgroups Gp of Gv we have H~(G~, A) = 0 for all r C Z.

Since there is an injection

0 ~ H~(G',A)-~ 1-IH~(G'p,A) P

for all subgroups G' of G by Chapter II, Corollary 2.2. It follows that A is cohomologically trivial. The converse is obvious.

C o r o l l a r y 1.8. Let G be finite and A E Mod(G). The following conditions are equivalent:

1. A is cohomologicaIly trivial.

2. The projective dimension of A is <= 2.

3. The projective dimension of A is finite.

6 6

Proof. Recall from Algebra that A has projective dimension _ _< s < oc if one can find an exact sequence

O---, P1---, P2---~...---, Ps---~ A- - ,O

with projectives Pi" One can complete this sequence by introducing the kernels and cokernels as shown, the arches being exact.

0

Therefore

/ \x3 / / \ / \

0 0 0 0

H r ( G ' , A ) = Hr+I (G ' ,X~_I ) . . . . = H~+~-I(G' , P 1 ) = 0 .

It is clear that a G-module of finite projective dimension is cohomologically trivial.

Conversely, let us write an exact sequence

O--* A '--* P---~ A---* O

where P is Z[G]-free. Then A' is Z-free, and by Proposition 1.5 it is also Gp-regular for all p. We now need a lemma.

L e m m a 1.9. Suppose M C Mod(G) is Z-free and Gp-regular for all primes p. Then M is G-regular, and so projective in Mod(G).

Proof. We view H~ Homz(M,M)) as being injected in the product

H H ~ p

and we apply the definition. We conclude that M is G-regular. Since M is Z-free it is Z[G]-projective by Corollary 1.6.

We apply the lemma to M = A' to conclude that the projective dimension of A is __< 2. This proves Corollary 1.8.

III.1 67

C o r o l l a r y 1.10. Let A E Mod(G) be cohomoIogically trivial, and let M E Mod(G) be without torsion. Then A | M is cohomologically trivial.

Proof. There is an exact sequence

O--~ PI ---~ P2---+ A--+ O

with P1,P2 projective in Mod(G). Since M has no torsion, the sequence

O ---* PI @ M --~ P2 @ M --* A | M ---* O

is exact. But P1, P2 are G-regular (direct summands in free modules, so regular), whence Pi | M is cohomologically trivial for i = 1, 2, whence A | M is cohomologically trivial.

More generally, we have one more result, which we won't use in the sequel.

Suppose A is cohomologically trivial, and that we have an exact sequence

O---~ PI ---~ P2-- , A----~O

with projectives P1, P2. Then we have an exact sequence

--. Torl(P2, M) --~ Tor l (A ,M) ---* P~| --* P~NM ~ A N M ~ 0

for an arbitrary M E Mod(G). Furthermore Torl(P2, M) = 0 since P2 has no torsion (because ziG] is Z-free). By dimension shifting, and similar reasoning for Horn, we find:

T h e o r e m 1.11. Let G be a finite group, A , B E Mod(G). and suppose A or B is cohomologicaIly trivial. Then for all r E Z and all subgroups G' of G, we have

H~(G', A @ B) ~ H~+2(G ', TorZ(A, B))

H ~ ( G ' , H o m ( A , B ) ) ~ H ~ - 2 ( G ' , E x t ~ ( A , B ) ) .

C o r o l l a r y 1.12. Let G , A , B be as in the theorem. Then A | is cohomoIogicaIly trivial if and only if TorZ(A,B) is cohomologically trivial; and Horn(A, B) is cohomologicalIy trivial if and only if Ext~(d , B) is cohomoIogicaUy trivial.

68

C o r o l l a r y 1.13. Let G , A , B be as in the theorem. Then A | is cohomologically trivial if A or B is without p-torsion for each prime p dividing (G �9 e).

w T h e t r i p l e t s t h e o r e m

Let f : A --~ B be a morphism in Mod(G). Let U be a subgroup of G, and let

f~ : H ' (U, A)---+ Hr(U, B)

be the homomorph.isms induced on cohomology. Actually we should write f~,u but we omit the index U for simplicity. We say that f is a c o h o m o l o g y i s o m o r p h i s m if f~ is an isomorphism for all r and all subgroups U. We say that A and B axe c o h o m o l o g i c a l l y e q u i v a l e n t if there exists a cohomology isomorphism f as above.

T h e o r e m 2.1. Let f : A ---. B be a morphism in Mod(G), and suppose there ezists some i E Z such that f i-1 is surjective, fi is an isomorphism, and fi+l is injective, for all subgroups U of G. Then f is a cohomology isomorphism.

Proof. Suppose first that f is injective. We shall reduce the general case to this special case later. We therefore have an exact sequence

O - - . A f--~a 9-.C---~0,

with C = B / f A , and the corresponding cohomology sequence

----* Hi_I(U,A) f ~ HI_I(U,B) gl-_~1 HI_t(U,C)

6--.LQ Hi(U,A) ~ HI(U,B) ~ Hi(U,C) - - -

6~+._.._~1 HI+X(U,A ) 1.~ HI(U,B )

We shall see that Hi - I (U, C) = Hi(U, C) = 0. As to Hi - I (U, C) = 0, it comes from the fact that fi-1 surjective implies gi-1 = O, and fi being an isomorphism implies 6i-1 = 0. As to Hi(U, C) = 0, it comes from the fact that fi surjective implies gi = 0, and fi+l injective implies 6i+1 = 0. By the twin theorem, we conclude that Hr(U, C) = 0 for all r E Z, whence f~ is an isomorphism for all r.

We now reduce the theorem to the preceding case, by the method of the mapping cylinder. Let us put MG(A) = A and aA = a. We

III.2 69

have an injection c : A---* A.

We map A into the direct sum B | A by

f : A - - - , B | such that f(a)=f(a)+e(a).

One sees at once that f is a morphism in Mod(G). We have an exact sequence

O--~ A Z B| h--~C--~O

where C is the cokernel of f .

We also have the projection morphism

p : B | defined by p(b+~)=b.

Its kernel is .4, and we have f = p f, whence the commutative diagram:

We then obtain the diagram

Hi(i)=O

l , Hi(a) s_~ HI(BOA) hi HI(C) ~ H~+I(A)

HI+I(A)=0

Hi-I(C)

0

~ A i ~ B | ~ C ~0

B

70

the cohomology groups H being H u for any subgroup U of G. The triangles are commutative.

The extreme vertical maps in the middle are 0 because .4 = MG(A), and consequently pi is an isomorphism, which has an inverse p~-i We have put

gi = hiP-i 1

From the formula f = p f we obtain fi = Pill. We can therefore replace Hi (B | A) by H ' ( B ) in the horizontal sequence, and we obtain an exact sequence which is the same as the one obtained in the first part of the proof. Thus the theorem is reduced to this first part, thus concluding the proof.

w T h e s p l i t t i n g m o d u l e a n d Ta te ' s t h e o r e m

The second cohomology group in many cases, especially class field theory, plays a particularly important role. We shall describe here a method to kill a cocycle in dimension 2.

Let G have order n and let a E H2(G,A). Recall the exact sequence

Ia z[a] z 0,

which is Z-split, and induces an isomorphism

5 H r ( G , Z ) --, H~+I(G,s for all r.

T h e o r e m 3.1. Let A E Mod(a) and (~ E H2(G,A) . There exists A' E Mod(G) and an exact sequence

O---~ A--~ A '--~ Ic---~ O,

splitting over Z, such that a = 66~, where ~ is the generator of H~ corresponding to the class of 1 in H~ = Z / n Z , and

~t,OL ~ O,

in other words, c~ splits in A'.

Proof. We define A' to be the direct sum of A and a free abelian group on elements x , ( a E G, a # e). We define an action of G on

III.3 71

A' by means of a cocycle {aa,~-} representing a. We put z, = a,,, for convenience, and let

(TX~- --~ T a r - - X a -~- a a , r .

One verifies by brute force that this definition is consistent by using the coboundary relation satisfied by the cocycle {a~,,~}, namely

~ a z , r - - a A # , r + a ) ~ , # r - - a # , r - - O.

One sees trivially that a splits in A'. Indeed, (aa,,-) is the coboundary of the cochain (x#).

We define a morphism v : A' ---+ Ia by letting

v ( a ) - - O f o r a e A a n d v ( x a ) - a - e f o r r

Here we identify A as a direct summand of A'. The map v is a G-morphism in light of the definition of the action of G on A'. It is obviously surjective, and the kernel of v is equal to A.

There remains to verify that a = 66~. The coboundary 6r is represented by the 1-cocycle b~ = ~ - e in IG, representing an element /3 of Hi (G, Ic) . We find 6/3 by selecting a cochain of G in A', for instance (z~,), such that v(z~) = ba. The coboundaxy of (Za) represents 6/3, and one then sees that this gives a, thus proving the theorem.

We define an element A E Mod(G) to be a class m o d u l e if for every subgroup U of G, we have H~(U,A) = 0, and if H2(G,A) is cyclic of order (G : e), generated by an element a such that r e s t ( a ) generates H2(U,A) and is of order (U" e). An element a as in this definition will be called f u n d a m e n t a l . The terminology comes from class field theory, where one meets such modules. See also Chapter IX.

T h e o r e m 3.2. Let G , A , a , u : A --+ A' be as in Theorem 3.1. Then A is a class module and a is fundamental if and only if A' is cohomologically ~rivial.

Proof. Suppose A is a class module and a fundamental. We have an exact sequence for all subgroups U of G:

0---, Hi(A)---+ HI (A ') ---+ H~(I)---+ H2(A)--+ H2(A ') ---+ O.

72

The 0 furthest to the right is due to the fact t&at

H2( Ia ) = H i (Z) = 0.

Since a = 6fl and fl = 6~,

and H i ( U , / ) is cyclic of order (U : e) generated by fl, it follows that

H i ( I ) ~ H2(A)

is an isomorphism. We conclude that Hi(U, A' ) = 0 for all U.

Since we also have the exact sequence

H2(A)---+ H2(A ') ---* H 2 ( / ) = 0,

and a splits in A' , we conclude that H2(U, A') = 0 for aLl U. Hence A ~ is cohomologicaUy trivial by the twin theorem.

Conversely, suppose A' cohomologically trivial. Then we have isomorphisms

H i ( I ) & H2(A) and H~ ~ Hi(z),

for all subgroups U of G. This shows that H2(U, A) is indeed cyclic of order (U : e), generated by 66~. This concludes the proof.

C H A P T E R IV Cup Products

w Erasability and uniqueness

To treat cup products, we have to start with the general notion of multilineaz categories, due to Cartier.

Let 92 be an abelian category. A structure of m u l t i l i n e a r cat- e g o r y on 92 consists in being given, for each (n + 1)-tuple A1 , . . . ,A ,~ ,B of objects in 92, an abelian group L ( A 1 , . . . , A n , B ) satisfying the following conditions�9

M U L 1. For n = 1 ,L(A ,B) = Hom(A,B).

M U L 2. Let

f l : A n x . . . • Azm ---* B1

f~: A ~ I • 2 1 5 --* B~ g : B1 x . . . x B ~ --* C

be multilinear. Then we may compose g ( f l , . . . , f,-) in L ( A n , . . . , A,.,~,, C), and this composition is multilinear in g, f l , . . . , f r .

M U L 3. With the same notation, we have g ( id , . . . , i d ) = g.

74

M U L 4. The composition is associative, in the sense that (with obvious notation)

g(k(h . . . ) , k ( h . . . ) , . . . , fr(h. . . )) = g ( k , . . . , / r ) ( h . . . ).

As usual the reader may think in terms of ordinary multilinear maps on abelian groups. These define a multilinear category, from which others can be defined by placing suitable conditions.

E x a m p l e . Let G be a group. Then Mod(G) is a multilinear category if we define L (A1 , . . . , A, , B) to consist of those Z-multilinear maps O satisfying

O(aal,...,aa,~) =aO(al , . . . ,an)

for all a E G and a i E Ai.

We can extend in the obvious way the notion of functor to multilinear categories. Explicitly, a functor F : 92 --0 ~ of such a category into another is given by a map T : f ~ T( f ) = f . , which to each multilineax map f in 92 associates a multilinear map in B, satisfying the following condition. Let

f l : A1 x . . x AN1

fp: Anp_~ x . . x A n p

�9 B1

.B ,

g : B 1 x . . x Bp " C

be multilinear in 92. Then we can compose g( f l , . . . , fp) and T g ( T f l , . . . , Tfp). The condition is that

T ( g ( f l , . . . , L ) ) = T g ( T f l , . . . , T L ) and T ( i d ) = id.

We could also define the notion of tensor product in a multilinear abelian category. It is a bifunctor, bilinear, on 92 x 92, satisfying the universal mapping property just as for the ordinary tensor product. In the applications, it will be made explicit in each case how such a tensor product arises from the usual one.

Furthermore, in the specific cases of multilinear abelian categories to be considered, the category will be closed under taking tensor products, i.e. if A, B are objects of 9/, then the linear factorization of a multilinear map is also in 92.

IV. 1 75

Let now E1 = (EC~) , . . . ,E= = (E~") and H = (H r) be 6- functors on the abelian category 92, which we suppose multi l inear, and the functors have values in a mult i l inear abelian category ~ . We assume tha t for each value of p l , . . �9 pn taken on by E l , . . . , E,~ respectively, the sum pl + ' " + p,~ is among the values taken on by r. By a c u p p r o d u c t , or c u p p i n g , of E 1 , . . . , E , ~ into H we mean tha t to each mult i l inear map 8 E L ( A 1 , . . . , A , , B ) we have associated a mult i l inear map

8(p) = 8, = 8p,,...,p,~ �9 E ~ ( A 1 ) x . . . x E ~ " ( A n ) --~ H P ( B )

where p = Pl + " �9 �9 + P,~, satisfying the following conditions.

C u p 1. The association 8 ~-+ 8(p) is a functor from the multil inear category 92 into ~ for each (p) = ( p l , - - - , p , ) -

C u p 2. Given exact sequences in 92,

0 ---+ A~ ~ Ai ---+ A~' --+ 0

0 ~ B' ---* B --+ B" --* 0

and mult i l inear maps f ' , f ' , f " in 9 / m a k i n g the following diagram commutat ive:

A l x . . . xA~ x . . . x A , ---+ A t x. .- xA~ x . . . x A ,

B ' ~ B

---+ A 1 X - . . • .-. x A ,

.[ f"

--+ B"

then the d iagram

f,, E ~ I ( A 1 ) x . . . x E ~ I ( A ' / ) x . . . x E ~ " ( A . ) ) H P ( B '')

E ~ ( & ) x...• Ef'+~(A'J x-..• E~"(A,,) ~', H,+~(B')

has character ( - 1)Pl +.--+p,-1, which m e a n s

s , id) = ( - 1 ) P ~ + + P ' - ~ 6 o s

The accumulat ion of indices is inevitable if one wants to take all possibilities into account. In practice, we most ly have to deal with the following cases.

76

First , let H be a cohomological functor. For each n _> 1 suppose given a cupping

H • 2 1 5

( the p roduc t on the left occurr ing n t imes) such tha t for n = 1 the cupping is the identity. Then we say tha t H is a c o h o m o l o g i c a l c u p f u n c t o r .

Next, suppose we have only two factors, i.e. a cupping

E x F - - ~ H

from two &functors into another . Most of the t ime, ins tead of indexing the induced maps by their degrees, we simply index t hem b y a , .

If 92 is closed under the tensor product , then by C u p 1 a cohomological cup functor is uniquely de te rmined by its values on the canonical bil inear maps

A x B - - - ~ A |

Indeed, if f �9 A x B --+ C is bilinear, one can factorize f t h rough A |

A x B ~174 ~~ C

where 8 is bi l inear and T is a morphism in 92. Thus certain theorems will be reduced to the s tudy of the cupping on tensor products .

Let E = ( E p) and F = (Fq) be 5-functors wi th a cupping into the 5-functor H = (Hr ) . Given a bil inear map

AxB--- -~C

and two exact sequences

0 > A ' ~ A > A " ~ 0

0 ~ C ' , C , C " ~ 0

such tha t the bi l inear map A x B ----* C induces bil inear maps

A I x B ----* C I and A" x B ~ C ' ,

IV. 1 77

then we obtain a commuta t ive diagram

EP(A '') x Fq(B) , HP+q (C")

6

Ep+I(A ') x Fq(B) = HP+q+I(C')

and therefore we get the formula

6 ( a " ~ ) = ( 6 a " ) f l with c~" E EP(A '') and ~ E F q ( B ) .

P r o p o s i t i o n 1.1. Let H be a cohomological cup functor on a multilinear category 9.1. Then the product

(a, fl) ~-~ (-1)Pq/3~ for a r HP(A) ,~ E Hq(B)

also defines a cupping H • H ---* H, making H into another cohomoIogical cup functor, equal to the first one in dimension O.

Proof. Clear.

R e m a r k . Since we shall prove a uniqueness theorem below, the preceding proposi t ion will show that we have

=

Now we come to the quest ion of uniqueness for cup products . It will be applied to the uniqueness of a cupping on a cohomological functor H given in one dimension. More precisely, let H be a cohomological cup functor on a mult i l inear category 9/. W h e n we speak of the cup functor in dimension 0, we mean the cupping

H ~ • ~ • x H ~ ~

which to each mult i l inear map 0 : A1 • " - • An ~ B associates the mult i l inear map

00: H~ • "" • H~ ---* H~

We note tha t to prove a uniqueness theorem, we only need to deal with two factors (i.e. bil inear morphisms) , because a cupping of several functors can be expressed in terms of cuppings of two functors, by associativity.

78

T h e o r e m 1.2. Let 92 be an abelian multiIinear category. Let E = ( E ~ 1) and H = ( H ~ ~) be two exact 6-f~nctors, and let F be a functor of 9~ into a muItiIinear category lB. Suppose E 1 is erasable by (M, ~), that 9.1, f8 are closed under tensor products, and that for all A, B E 91 the sequence

O --~ A | B ~A| MA | B ---+ XA | B ---+ O

is exact. Suppose given a cupping E • F ---+ H. Then we have a commutative diagram

E~ • F ( B) ---+ H~ | B)

i L id El(A) x F ( B) ---, H i ( A | B)

and the coboundary on the left is surjective.

Proof. Clear.

Coro l la ry 1.3. Hypotheses being as in the theorem, if two cuppings E x F ~ H coincide in dimension O, then they coincide in dimension i.

Proof. For all a E E l ( A ) there exists ~ E Z~ such that a = 6~, and by hypothesis we have for/3 E F(B),

= (6 )Z =

whence the corollary follows.

T h e o r e m 1.4. Let G be a group and let H be the cohomologicaI functor Ha from Chapter I, such that H~ = A G for A in Mod(G). Then a cupping

H x H - - - + H

such that in dimension O, the cupping is induced by the bilinear map

(a,b) ~ O(a,b) for O : A x ---* C and a E AG, b E B c,

IV. 1 79

is uniquely determined by this condition.

Proof. This is just a special case of Theorem 1.2, since in Chap- ter I we proved the existence of the erasing functor (M, ~) necessary to prove uniqueness.

We shall always consider the functor Ha as having its structure of cup functor as we have just defined it in Theorem 1.4. Its existence will be proved in the next section.

If G is finite, then in the category Mod(G) we have an erasing functor M e for the special cohomology functor H e . So we now formulate the general situation.

Let E = ( E ' ) , F = ( F ' ) , and H = (H r) be three exact ~- functors on a multilinear category 9.1. We suppose given a cupping E x F ~ H. As in Chapter I, suppose given an erasing functor M for E in dimensions > p0. We say that M is spec ia l if for each bilinear map

O : A x B - + C

in 9.1, there exists bilinear maps

M ( 0 ) : M A x B ~ M c and X ( O ) : X A x B ~ X c

such that the following diagram is commutative:

A x B ~ MA X B ~ XA X B

o~ M(O)~ M(O)~

C ~ M e ~ X c

T h e o r e m 1.5 ( r igh t ) . Let E = ( E P) ,F = (Fq) and H = (H r) be three exact ~-functors on a multilinear category PJ. Suppose given a cupping E x F ---+ H. Let M be a special erasing functor for E and for H in all dimensions. Then there is a commutative diagram associated with each bilinear map 0: A • B--+ C:

EP(XA) X Fq(B) �9 HP+q(c)

id b

EP+I(A) x Fq(B) " HP+I"~I(C)

80

and the vertical maps are isomorphisms.

Proof. Clear.

Of course, we have the dual situation when M is a coerasing functor for E in dimensions < p0. In this case, we say that M is spec ia l if for each 0 there exist bilinear maps M(0) and Y(O) making the following diagram commutative:

YA X B ) MA X B ) A x B

Y(O) 1 M(O) 1 O[ Ire ~ M c ~ C

One then has:

T h e o r e m 1.5 ( lef t ) . Let E, F, H be three ezact &functors on a multilinear category 91. Suppose given a cupping E x F ---* H. Let M be a special coerasing functor for E and H in all dimensions. Then for each bilinear map 0 : A x B ~ C we have a commutative diagram

EP(A) x Fq(B) , HP+q(C)

id id

EP+q+I(YA) x Fq(B) * HP+q+I(C)

and the vertical maps are isomorphisms.

C o r o l l a r y 1.6. Let E , F , H be as in the preceding theorems, with values p = O, 1 (resp. O, -1 ) ; q = O; r = O, 1 (resp. O, -1 ) . Let M be an erasing functor (resp. coerasing) for E and H. Sup- pose given two cuppings E x F ~ H which coincide in dimension O. Then they coincide in dimension 1 (resp. -1 ) .

We observe that the choice of indices O, 1 , - 1 is arbitrary, and the corollary applies mutatis mutandis to p, p + 1, or p, p - 1 for E, q arbitrary for F, a n d p + q , p + q + l (resp. p + q , p + q - 1 ) for H.

C o r o l l a r y 1.7. Let H be a cohomological functor on a multilinear category P.l. Suppose there exists a special erasing and coeras- lug functor on H. Suppose there are two cup functor structures on H, coinciding in dimension O. Then these cuppings coincide in all dimensions.

IV.1 81

P r o p o s i t i o n 1.8. Let G be a finite group and G' a subgroup. Let H be the cohomological functor on Mod(G) such that H(A) = H(G' ,A) . Suppose given in addition an additional structure of cup functor on H. Then the erasing functor A ~ M s ( A ) (and the similar coerasing functor) are special.

Proof. Let 0 : A • B ~ C be bilinear. Viewing MG as coerasing, we have the commutative diagram

_ f G | 1 7 4 , Z [ G ] | 1 7 4 , Z | 1 7 4

l l 1 z o | , z [ a ] | , z |

where the vertical maps are defined by ~ | a ~ b ~-, ~ | ab. We have a similar diagram to the right for the erasing functor.

From the general theorems, we then obtain:

T h e o r e m 1.9. Let G be a group and H = Ha the ordinary cohomologicaI functor on Mod(G). Then for each multiIinear map A1 x . . . • Am ---+ B in Mod(G), if we define

x (a l ) . . .>c(a ,~)=~(a l . . . a ,~ ) for a, E A~,

then we obtain a muItiIinear functor, and a cupping

H ~ • • ~ ~ H ~

A cup functor structure on H which is the above in dimension 0 is uniquely determined. The similar assertion holds when G is finite and H is replaced by the special functor H = H c .

As mentioned previously, existence will be proved in the next section. For the rest of this section, we let H denote the ordinary or special cohomology functor on Mod(G), depending on whether G is arbi trary or finite.

C o r o l l a r y 1.10. Let G be a group and H the ordinary or special functor if G is finite. Let A1, A2, A3, A12, A123 be G- modules. Suppose given multilinear maps in Mod(G):

A1 x A2 --* A12 AI~ • A3 ---+ A123

82

whose composite gives rise to a multilinear map

A1 x A2 x A3 --* A123.

Let ai E HP~(Ai). Then we have associativity,

=

these cup products being taken relative to the multilinear maps as above.

Proof. One first reduces the theorem to the case when A12 = A1 | and A123 = A1 @A2| The products on the right and on the left of the equation satisfy the axioms of a cup product, so we can apply the uniqueness theorem.

R e m a r k . More generally, to define a cup functor structure on a cohomological functor H on an abelian category of abelian groups 92, closed under tensor products, it suffices to give a cupping for two factors, i.e. H x H ---* H. Once this is done, let

al e H ~ l ( A 1 ) , . . . , a n e H ~ ( A n ) .

We may then define

~ l . . . ~ n = (. �9 (~1~)~3) . �9 �9 ~ ) ,

and one sees that this gives a structure of cup functor on H, using the universal property of the tensor product.

Co ro l l a ry 1.11. Let G be a group and H = HG the ordinary cup functor, or the special one if G is finite. Let 8 : A • B --+ C be bilinear in Mod(G). Let a E A G, and let

8~ : B ---* C be defined by 8~(b) = ab.

Then Oa is a morphism in Mod(G). /f

Hq(Oa) = 0~o : H a ( B ) ---* n q ( c )

is the induced homomorphism, then

x(a ) f l=0~ . ( f l ) for / 3 E H q ( B ) .

Proof. The first assertion is clear from the fact that a E A c implies r = aab. If q = 0 the second assertion amounts to the definition of the induced mapping. For the other values of q, we apply the uniqueness theorem to the cuppings H ~ x H ---* H given either by the cup product or by the induced homomorphism, to conclude the proof.

IV.2 83

C o r o l l a r y 1 .12. Suppose G finite and 0 : A x B --+ C bilinear in Mod(G). Then for a �9 A ~ and b �9 Bsa we have

J4(a) U n<(b) = n~(ab).

Pro@ A direct verification shows tha t the above formula defines a cupping of H ~ and ( H ~ H -1) into ( H ~ H - l ) . Under the s ta ted hypotheses , we clearly have ab �9 Csa, so the formula makes sense, and is valid.

w Exis tence

We shall see tha t on Mod(G) the cup p roduc t is induced by a p roduc t of cochains, and we shall give the explicit formula for cochains in the s t anda rd complex. First we give an axiomatic version so the reader sees the general s i tuat ion in the f ramework of abel ian categories. But as usual, the reader may th ink of Mod(G) and abel ian groups for the categories men t ioned in the next theorem.

T h e o r e m 2.1. Let 91 be a multilinear category of abelian groups. Let

A ~ Y ( A )

be an exact functor of 91 into the category of complexes in 91. Suppose that for each bilinear map 0 : A • B --+ C in 91 we are given a bilinear map

Y r ( d ) • Y~(B) ~ Yr+' (C)

which is functoriaI in A, B, C (eovariant) and such that if f C Y~(A) and g �9 Y ' ( B ) , then

5( f g) = (6 f )g + (--1)r f(Sg).

Let H be the cohomological functor associated to the functor Y. Then there exists on H a structure of cup functor, induced by the above bilinear map. This structure satisfies the property of the three exact sequences, as it is described below.

The p r o p e r t y o f the three exact sequences is the following. Consider three exact sequences in 91:

0 > A I ~ A ~ A" ~ 0

0 > B I > B > B" > 0

0 > C' ~ C > C" > 0

84

Suppose given a bilinear map A x B ---+ C in 92 such that

A'B = O, AB' C C', A'B C C', A"B" C C".

Then for a" E H~(A '') and ~" E H~(B '') we have

6(a"UlS")=(6a")Uf' + ( -1) 'a " U (6,8").

It is a pain to write down in complete detail what amounts to a routine proof. One has to combine the additive construction with multiplicative considerations on the product of two complexes, cf. for instance Exercise 29 of Algebra, Chapter XX. More generally, we observe the following general fact. Let 9 /be a multilinear abelian category and let Corn(92) be the abelian category of complexes in 92. Then we can make Corn(92) into a multilinear category as follows. Let K, L, M be three complexzes in 92. We def ine a b i l inea r m a p

O : K x L - - - ~ M

to be a family of bilinear maps

Sr,~ : K ~ x L ~ ---. M ~+~

satisfying the condition

for x E K ~ and y E L ~. We have indexed the coboundaries 6M,6L, 6K according to the complexes to which they belong. If we omit all indices to simplify the notation, then the above condi- t ion reads

6(x �9 y) = 6x �9 y + (-1)~x �9 6y.

Exercise 29 ioc. cit. reproduces this formula for the universal bilinear map given by the tensor product. Then the cup product is induced by a product on representing cochains in the cochain complex. We leave the details to the reader. As we shall see, in practice, one can give explicit concrete formulas which allow direct verification.

We shall now make the theorem explicit for the standard complex and Mod(G).

IV.2 85

L e m m a 2.2. Let G be a group and Y (G ,A) the homogeneous standard complex for A E Mod(G). Let A • B ~ C be bilinear in Mod(G). For f C Y~(G,A) and g e Ys (G,A) define the product f g by the formula

(fg)(cr0, . . . , at+l)---- f(ao,...,a~)g(cr~,...,cr~+s).

Then this product satisfies the relation

6(f g) : (S f)g + (-1)~f(Sg).

Proof. Straightforward.

In terms of non-homogeneous cochains f , g in the non-homogeneous standard complex, the formula for the product is given by

( fg ) (a l , . . . , a r+s) = f(~rl, . . . ,O'r)(~l,...,(Trg(O'r+l,...,O'r+s)).

T h e o r e m 2.3. Let G be a group and let HG be the ordinary cohomological functor on Mod(G). Then the product defined in Lemma 2.2 induces on Ha a structure of cup functor which satisfies the property of the three exact sequences. Furthermore in dimension O, we have

~(ab) = x(a)x(b) for a E A G and b e B c.

Proof. This is simply a special case of Theorem 2.1, taking the lemma into account, giving the explicit expression for the cupping in terms of cochains in the standard complex. The property of the three exact sequences is immediate from the definition of the coboundary. Indeed, if we are given cochains f " and g" representing a" and/~ ' , their coboundaries are defined by taking cochains f , g in A , B respectively mapping on f " , g ' , so that fg maps on f ' g " . The formula of the lemma implies the formula in the property of the three exact sequences.

We have the analogous result for finite groups and the special functor.

86

L e m m a 2.4. Let G be a finite group. Let X be a complete, Z[G]-free, acyclic resolution of Z, with augmentation ~. Let d be the boundary operation in X and let

d ' = d | and d " = i d x |

be those induced in X | Then there is a family of G-morphisms

hr,s : Xr+s ~ X~ | Xs for - oo < r, s < oo,

satisfying the following conditions:

(i) h~,~d = d'h,-+l,~ + d"h,-,s+l

( i i ) (~ | ~)h0,0 = ~.

Proof. Readers will find a proof in Caxtan-Eilenberg [CaE].

T h e o r e m 2.5. Let G be a finite group and HG the special cohomology functor. Then there exists a unique structure of cup functor on Ha such that if 8 : A x B --+ C is bilinear in Mod(G) then

x(a)x(b) = x(ab) for a e AG, b e B G.

Furthermore, this cupping satisfies the three exact sequences property.

Proof. Let A E Mod(G), and let X be the standard complex. Let Y(A) be the cochain complex Homa(X, A). For

f E Y ' ( A ) = i o m c ( X ~ , A ) and g E YS(A),

we can define the product fg E Y~+~(A) by the composition of canonical maps

Z r w s hr s f@g O' ~ X ~ | , A | ----* C,

where 8' is the morphism in Mod(G) induced by 8, that is

fg = 8 ' ( f Q g)hr,~.

One then verifies without difficulty the formula

6(fg) = (Sf)g + ( -1)~f(Sg) ,

IV.3 87

and the rest of the proof is as in Theorem 2.3.

w Relations with subgroups

We shall tabulate a list of commutativity relations for the cup product with a group and its subgroups.

First we note that every multilinear map 8 in Mod(G) induces in a natural way a multilinear map 8' in Mod(G') for every subgroup G' of G. Set theoretically, it is just 8.

T h e o r e m 3.1. Let G be a group and 8 : A • B ---+ C bilinear in Mod(G). Let G' be a subgroup of G. Let H denote the ordinary cup functor, or the special cup functor if G is finite, except when we deal with inflation in which case H denotes only the ordinary functor. Let the restriction res be from G to G'. Then:

(1) res(a/~) = (res a)(res/3) for ~ e Hr (G ,A) and ~ e H~(G,B) .

(2) tr((res a)/Y) = a( t r /3 ' ) for a E Hr (G ,A) and/3' E HS(G' ,B) ,

the transfer being taken from G' to G. Similarly,

t r (a ' ( res f l ) ) = (tr a')fl for a' C Hr (G ' ,A ) and fl E H*(G,B) .

(3) Let G l be normal in G. Then 8 induces a bilinear map

A G' • B G' ____+ C G'

and for ~ C H r ( G / G ' , A a ' ) , f l e H S ( G / G ' , B a') we have

inf(afl) = (inf c~)(inf fl).

Proof. The formulas are immediate in dimension O, i.e. for r = s = O. In each case, the expressions on the left and on the right of the stated equality define separately a cupping of a cohomological functor into another, coinciding in dimension O, and satisfying the conditions of the uniqueness theorem. The equalities are therefore valid in all dimensions. For example, in (1) we have two cuppings of HG • HG -+ Ha, given by

(a, fl) ~ res(afl) and (a, fl) ~ (res a)(res fl).

88

In (3), we find first a cupping

Ha~a, x Ha~a, ~ H a

to which we apply the uniqueness theorem on the right. We let the reader write out the details. For the inflation, we may write explicitly one of these cuppings:

H ( G / G ' , A a') x H ( G / G ' , B a ' ) cup H(a/a,,Ca,)--. H(a, Ca).

w T h e t r i p l e t s t h e o r e m

We shall formulate for cup products the analogue of the triplets theorem. We shall reduce the proof to the preceding situation.

T h e o r e m 4.1. Let G be a finite group and 8 : A x B ---* C bilinear in Mod(G). Fix c~ C HP(G,A) for some index p. For each subgroup G' of G let c~' = res~,(o~) be the restriction in HP(G I, A). For each integer s denote by

c~: H~(G', B)--* H~(G ', C)

the homomorphism /Y ~ a~/~ ~. Suppose there exists an index r is an isomorphism, and ~ is such that C~r+ 1 1 is surjective, ~r C~r+l

injective, for all subgroups G ~ of G. Then a~s is an isomorphism for all s.

Proof. Suppose first that r = 0. We then know by Corollary is the homomorphism (8a). induced by an element 1.11 that o~8

a C A a, where Oa : B ---* C is defined by b ~ 0(a,b) = ab. The theorem is therefore true if p = 0 by the ordinary triplets theorem. We note that the induced homomorphism is compatible with the restriction from G to G t.

We then prove the theorem ill general by ascending and descending induction on p. For example, let us give the details in the case of descending induction to the left. We have E = F = H. There exists ~ C H r ( G , X A ) such that a = 54, where XA is the cokernel in the dimension shifting exact sequence as in Theorem 1.14 of Chapter II. The restriction being a morphism of functors, we have ~'8 = 54'8 for all s. It is clear that cd8 is an isomorphism (resp. is

IV.5 89

injective, resp. surjective) if and only if ~'8 is an isomorphism (resp. is injective, resp. surjective). Thus we have an inductive procedure to prove our assertion.

w The cohomology ring and duality

Let A be a ring and suppose that the group G acts on the additive group of A, i.e. that this additive group is in Mod(G). We say that A is a G-r ing if in addition we have

o(ab) = (o'a)(ab) for all o" e a,a,b r A.

Suppose A is a G-ring. Then multiplication of n elements of A is a multilinear map in the multilinear category Mod(G).

Let us denote by H(A) the direct sum

H(A) = ( ~ HP(A), - - O o

where H is the ordinary functor on Mod(G), or special functor in case G is finite. Then H(A) is a graded ring, multiplication being first defined for homogeneous elements c~ C HP(A) and/3 E Hq(A) by the cup product, and then on direct sums by linearity, that is

We then say that H(A) is the c o h o m o l o g y r ing of A.

One verifies at once that if A is a commutative ring, then H(A) is anti-commutative, that is if a r HP(A) and/3 E Hq(A) then

o~# = ( - l )Pq#oz.

Since by definition a ring has a unit element, we have 1 E A a and x(1) is the unit element of H(A). Indeed, for fl E Ha(A) we have

x ( 1 ) # = 0i , /3 -- #,

90

because 01 : a ~ la = a is the identity.

Let A be a G-ring and B �9 Mod(G). Suppose that B is a left A-module, compatible with the action of G, that is the map

A x B - - * B

defined by the action of A on B is bilinear in the multilinear category Mod(G). We then obtain a product

HP(A) x Hq(B) ---+ H'+q(B)

which we can extend by linearity so as to make the direct sum H(B) = @ Hq(B) into a graded H(A)-module. The unit element of H(A) acts as the identity on H(B) according to the previous remarks.

Let B, C �9 Mod(G). There is a natural map

Hom(B,C) x B ~ C

defined by (f , b) ~ f(b). This map is bilinear in the multilinear category Mod(G), because we have

([a]f)(ab) = a f a - l a b = a(fb)

for f �9 Horn(B, C) and b �9 B. Thus we obtain a product

(%/3) ~ ~/3, for ~ �9 HP(Hom(B,C)) and /3 �9 Hq(B).

T h e o r e m 5.1. Let

0 ---* B' ~ B ---* B" -* 0

be a short exact sequence in Mod(G), let C �9 Mod(G), suppose the Horn sequence

0 ~ Hom(B", C)--~ H o m ( B , C ) ~ Hom(B', C ' ) ~ 0

is exact. loe have

and

Then for/3" C Hq-I(B ") and ~' �9 HP(Hom(B',C)),

(~o')/3" + (-1)P~'(5r = 0.

IV.5 91

Proof. We consider the three exact sequences:

0 ---+ H o m ( B " , C) --~ Horn(B, C) --+ Hom(B ' , C) ---+ 0

0 ---+ B' ~ B --+ B" ~ O.

0 ---* C --+ C -+ 0 --+ 0,

and a bi l inear map from M<~d(G), in the middle, inducing mappings as in the existence theorem for the cup product . Since qo'fl" = 0, we find the present result as a special case.

We may rewri te the result of Theorem 5.1 in the form of a dia- g ram

H ~ ( H o m ( B ' , C ) ) ~ H~(B ') , H~+~(C)

q < H~+I(Hom(B",C)) , H ' - I ( B ''1 , H~+'(C)

which has charac ter ( - 1 ) r+l .

In d imension 0, we find:

P r o p o s i t i o n 5.2. Let f �9 H o m G ( B , C ) = ( H o m ( B , C ) ) c, and �9 H~(B). Then

g(f)13 = f,~. If G is finite and H is the special functor, then

x(f ) u . ( b ) = . ( f (b)) for b �9 Bs,,.

Proof. In dimension 0, this is an old result. The assertion concerning dimensions -1 and 0 is a special case of the uniqueness theorem, Corollary 1.12.

There is a homomorph i sm

h~,s: H " ( H o m ( B , C)) --+ Hom(H"(B),H"+s(C)),

obta ined f rom the bil inear map

Hr(Hom(B,C)) • H~(B) ---+ gr+s(C).

In impor t an t cases, we shall see tha t h~,s is an isomorphism. We shall now give a cri terion for this in the case of the special functor .

92

T h e o r e m 5.3. Let G be finite, and H = HG the special func- tot. Suppose that for C fixed and B variable in Mod(G), and two fixed integers p0, qo the map hpo,q o is an isomorphism. Then hp,q is an isomorphism for all p, q such that p + q = Po + qo.

Proof. We are going to use dimension shifting. We consider the exact sequence

0 -~ I | ~ Z[G] @B ---, Z | = B ~ O,

which we horn into C. Since the sequence splits, we obtain an exact sequence

0 ~ Horn(B, C) ~ Hom(Z[G] | B, C) --~ Hom(! | B, C) ---+ 0.

Applying the diagram following theorem 5.2, we find:

H P ( H o m ( I | h~,~ ,

,l H ' + I (Horn(B, C))

hp+l,q+t

Hom(Hq (_r | B), HP+q(C))

Hom(Hq-1 (B), HP+q(C) ).

The vertical coboundaries axe isomorphisms because the middle object in the exact sequence is G-regular, and so annuls the cohomology. This concludes the proof going from p to p + 1. Going the other way, we use the other exact sequence

B Z[a] O B - , : o B

and we let the reader finish this side of the proof.

As an apphcation, we shall prove a duality theorem. Let B be an abehan group. As before, we define its dual group by /3 = Horn(B, Q/Z) . It is the group of characters of finite order, which we consider as a discrete group. Its elements will be called simply c h a r a c t e r s . Let B E Mod(G). We consider B as an abelian group to ge t /~ .

We have an isomorphism

~ - ~ �9 H - ~ ( Q / Z ) --~ (Q/Z)n

IV.5 93

between H - I ( Q / Z ) and the elements of order n = (G: e) in Q/Z.

In addition, we have a bilinear map in Mod(G):

/} • B ~ Q/Z ,

and consequently a corresponding bilinear map of abelian groups

H-q(/}) x Hq-I(B) ~ H-I(Q/Z).

T h e o r e m 5.4. D u a l i t y T h e o r e m . The h o m o m o r p h i s m

H-q(/})--+ Hq-I(B) ^

which to each 9o E U-q(B) associates the character fl ~ :~<-l(~fl), is an i somorphism, so we have

H-q(/~) = Hq- I (B) ^.

Proof. From the definitions, we see that the theorem amounts to proving that h-q,q-1 is an isomorphism. According to the preceding theorem, it suffices to show that h0,-1 is an isomorphism. Since ho,q is an induced homomorphism, we can make h0,-1 explicit in the present case, as follows. We have a homomoprhism

/}alSGB ~ (BsG/ IB) ^,

obtained by associating to an element f E /} the character b ~-+ f (b ) for b E Bsa. We have to prove that this map is an isomorphism.

For the surjectivity, let f0 : Bs --+ ( Q / Z ) , be a homomorphism vanishing on I B . We can extend f0 to a homomorphism f of B into Q / Z because Q / Z is injective. Furthermore f is i n / } o because

f(crb) - a f ( b ) = f ( a b ) - f (b ) = f ( a b - b) = 0

by hypothesis. This proves surjectivity.

For the injectivity, let f E /}a and suppose f (Bs ) = 0. Since B / B s is isomorphic to SB, there exists g E (SB) ̂ such that

f (b) = g ( sb ) for b e B.

94

We extend g to a homomorphism of B into Q/Z , denoted by the same letter. Then f = Sg, because

(Sg)(b)-- Eo'go'-lb= Eg(T-lb--g (E o'-lb) ---gSb: f(b).

This concludes the proof of hte duality theorem.

Consider the special case when B = Z. Then

/} = 2 = Hom(Z, Q / Z )

and we find:

C o r o l l a r y 5 . 5 . n - q ( Q / Z ) ~ H q - I ( Z ) ^

Applying the coboundary homomorphism arising from the exact sequence

0 - , z - , q - ~ q / z - ~ 0,

we obtain:

C o r o l l a r y 5.6.

H-p-I(Q/Z)

The following diagram is commutative:

x HP(Z) , H-I(Q/Z)

6 ] id 6

H-P(Z) x HP(Z) �9 H~

The vertical maps are isomorphisms, and thus

H-P(Z) ~ HP(Z) ̂ .

Proof. Since Q is uniquely divisible by n, its cohomology groups are trivial, and hence the coboundaries are isomorphisms, so the corollary is clear.

C o r o l l a r y 5.7. Let M C Mod(G) be Z-free. commutative diagram:

Then one has a

HP-I(Hom(M,Q/Z)) x H-P(M) �9 H-I(Q/Z)

id

HP(Hom(M, Z)) • H-P(M) , H~

IV.6 95

where the vertical maps are isomorphisms. Thus we obtain a canonical isomorphism

HP(Hom(M, Z)) ~ H-P(M) ^.

Proof. This is an immediate consequence of the fact that Q is G-regular (the identity is the trace of l /n ) , and so Horn(M, A) is also G-regular, and so annuls cohomology. The sequence

0 --+ Hom(M, Z) ---+ Horn(M, Q) --+ Horn(M, Q/Z) ---+ 0

is exact. We can then apply the definition of the cup functor to conclude the proof.

It will be convenient to use the following terminology. Suppose given a bilinear map of finite abelian groups

F ' • F --+ q / z .

This induces a homomorphism F' ---+ F ^. If F t ---+ F ^ is an isomorphism, then we say that F r and F are in pe r fec t dua l i t y under the bilinear map. Each of Theorem 5.4, Corollary 5.5 arid Corol- lary 5.6 establish a perfect duality of cohomology groups in their conclusions, when B is, say, finitely generated.

w P e r i o d i c i t y

In Chapter I, we saw that the cohomology of a finite cyclic group is periodic. We shall now give a general criterion for periodicity for an arbitrary finite group G. This section will not be used in what follows.

Let r E Z be fixed. An element C E Hr(G, Z) will be said to be a m a x i m a l g e n e r a t o r if ~" generates H~(G, Z) and if r is of finite order (G: e).

T h e o r e m 6.1. Let G be a finite group and ~ E Hr(G, Z). The following properties are equivalent.

M A X 1. ~ is a maximal generator.

M A X 2. ( is of order (G : e).

M A X 3. There exists an element ~--1 E H-r(G, Z)

96

such that C-1C = 1.

M A X 4. For all A E Mod(G), the map

a ~ Ca of Hi(G,A) --* Hi+~(G,A)

is an isomorphism for all i.

Proof. That M A X 1 implies M A X 2 is trivial.

Assume M A X 2. Suppose r has order (G: e). Since H-~(Z) is dual to H"(Z) by Corollary 5.6, the existence of r follows from the definition of the dual group, so M A X 3 is satisfied.

Assume M A X 3. The maps

a ~ Ca for a �9 Hi(A) and/3 ~ ~-1/3 for/3 �9 Hi+~(A)

are inverse to each other, up to a power ( -1 ) ~, and are therefore isomorphisms, thus proving M A X 4.

Assume M A X 4. We take A = Z a n d i = 0 in the preceding assertion, and we use the fact that H~ is cyclic of order (G : e). Then M A X 1 follows at once.

The uniqueness of ~-1 in Theorem 6. I, satisfying condition M A X 3, is clear, taking into account that H-~(Z) is the dual group of H~(Z).

P r o p o s i t i o n 6.2. Let C C H~(G,Z) be a maximal generator. Then so is ~-1. I f C1 is a maximal generator of Hs(G, Z) for some s, then CC1 i~ a maximal generator.

Proof. The first assertion follows from M A X 3; the second from M A X 4.

An integer rn will be said to be a c o h o m o l o g y pe r i od of G if Hm(G, Z) contains a maximal generator, or in other words, Hm(G,Z) is cyclic of order (G : e). The anticommutativity of the cup product shows that a period is even.

P r o p o s i t i o n 6.3. Suppose m is a cohomoIogy period of G. Let U be a subgroup of G and let ( C Hm(G,Z) be of order Ca: e).

IV.6 97

Then resg(0 has order (U e) and a ohomoIogy period 4 U.

Proof. Since trgresg(0 = ( a U ) r

it follows that the order of the restriction of C to U is at least (U : e). Since it is at most equal to (U : e), it is a period.

P r o p o s i t i o n 6.4. Let Gp be a p-SyIow subgroup of G and let E H~(Gp, Z) be a maximal generator. Let n be a positive

integer such that

k ~ = 1 m o d ( G p : e )

for all integers k prime to p. Then ['~ E Hnr(Gp, Z) is stable, i.e. ~r.((~) = ~'~ for all cr e G, and

G'(cl has order (Gp :e).

Proof. Since (r, is an isomorphism, and since the restriction of a maximal generator is a maximal generator, one concludes that the elements

G [~] fl = resG~nGp[cq(~ n) and resGpnGp[~] o 0",(~ n) = A

are both maximal generators in Hr(Gp a Gp[(r],Z). Hence there exists an integer k prime to p such that one is equal to k times the other, i.e. kfl = ,\. Taking the n-th power we get

k~fl~ = A ~"

From the definition of n, with the fact that (Gp : e) kills/3 and A, and with the commutativity of the cup product and the indicated operations, we find that [~ is stable. This being the case, we know from Proposition i.I0 of Chapter II that

Since (G : Gp) is prime to p, it follows that the transfer followed by the restriction is injective on the subgroup generated by ~n Thus the transfer is injective on this subgroup. From this one sees that the period of this transfer is the same as that of ~n, whence the same as that of [, thus concluding the proof.

98

C o r o l l a r y 6.5. Let G be a finite group. Then G admits a cohomological period > 0 if and only if each Sylow subgroup Gp has a period > O.

Proof. If G has a period > 0, the proposition shows that Gp also has one. Conversely, suppose that ~'~ E Hr(Gp, Z) is a maximal generator. Let

c , =

Then the order of <p is the same as that of ~'p by Proposition 6.4. Let

Then ~ is an element of order (G : e), so G has a cohomological period > 0, as was to be shown.

The preceding corollary reduces the study of periodicity to p- ~oups . One can show easily:

P r o p o s i t i o n 6.6. Let G = Gp be a p-group. Then G admits a cohomological period > 0 if and only if G is cyclic or G is a generalized quaternion group.

We omit the proof.

w T h e t h e o r e m s of T a t e - N a k a y a m a

We shall now go back to the theorem concerning the splitting module for a class module as in Chapter III, w We recall that if A' E Mod(G) is cohomologically trivial and M is a G-module without torsion, then A' @ M is cohomologically trivial.

T h e o r e m 7.1. (Ta te ) . Let G be a finite group, M E Mod(G) without torsion, A E Mod(G) a class module, and a E H2(G, A) fundamental. Let

a t : Hr(G, M) ---* Hr+2(G,A | M)

be the cup product relative to the bilinear map A x M --+ A | M, i.e. such that

a ~ ( A ) = a U A for A E H ~ ( G , M ) .

IV.7 99

T h e n a~ is an i s o m o r p h i s m fo r all r E Z .

Proof . As in the main theorem on cohomological triviality (The- orem 3.1 of Chapter III, we have exact sequences

0 ~ A ~ A I ) I , 0

0 , A | ~ A ' | ~ I | ) O.

The exactness of the second sequence is due to the fact that the first one splits.

In addit ion, A' | is cohomologically trivial. Let us put fl = 5~. Then

ar (~) = a~ = (6~)~ = ~ ( ~ ) .

If we now use the exact sequences

0 , i , z [ a ] , z , 0

0 ) I | , Z [ G ] | ) Z | , 0,

we find

The coboundaries 6 are isomorphisms, in one case because Z[G] | is G-regular, in the other case by the main theorem on cohomological triviality. To show that a~ is an isomorphism, it will suffice to show that ~ : A ~-~ ~A is an isomorphism. But this is clear because it is the identity, as one sees by making explicit the canonical i somorphism Z | M ~ M. This concludes the proof of Theo rem 7.1.

We can rewrite the commutat ive diagram arising from the theorem in the following manner.

H ~ • H " ( M ) ~ H " ( Z |

H i ( I ) x H r ( M ) , H r + I ( I |

H2(A) x H r ( M ) , H~+2(A| M).

100

The vertical maps 5 are isomorphisms, and the cup product on top corresponds to the bilinear map Z • M ~ Z | M = M, so the isomorphism induced by ~r is the identity.

If we take M = Z and r = -2 , we find

H2(A) • H - 2 ( Z ) ~ H~

We know that H-2(Z) = G / G c and so we find an isomorphism

c / a c .~ H~ = AG/SGA.

We shall make this isomorphism more explicit below.

We also obtain an analogous theorem by taking Horn instead of the tensor product, and by using the duality theorem.

T h e o r e m 7.2. Let G be a finite group, M E Mod(G) and Z- free, A E Mod(G) a class module. Then for all r C Z, the biIinear map of the cup product

Hr(G, Hom(M,A)) • H2- (G,M) H (G,A)

induces an isomorphism

H~(G, Hom(M,A) ) ~ H2-~(G,M) A

Proof. We shift dimensions on A twice. Since A' and ZIG] are Z-free, it follows that the sequences

0 , H o m ( M , Z ) ----* H o m ( M , A ' ) , H o m ( M , _ r ) , 0

0 , H o m ( U , I ) - - - H o m ( M , Z [ G ] ) , H o m ( M , Z ) ----* 0

are exact. By the definition of the cup product one finds commutative diagrams as follows, where the vertical maps are isomorphisms.

Hr -2 (Hom(M,Z) ) x H2- r (M) , H~

H ~ - l ( H o m ( m , / ) ) x H2-~(M) , H i ( I )

lid W(Hom(M,A)) • H2- (M) ,

IV.8 101

The bilinear map on top is that of Corollary 5.7, and the theorem follows.

Selecting M = Z we get for r = 0:

H~ x H2(Z)--~ H2(A),

this being compatible with the bilinear map

A @ Z ~ A such that (a ,n)~-+na.

We know that 5 : H i ( Q / Z ) ~ H2(Z) is an isomorphism, so we find the pairing

H~ x H i ( Q / Z ) = (~ --, H2(G,A)

which is a perfect duality since G is finite. One can give an explicit determination of this duality in terms of standard cocycles as follows.

T h e o r e m 7.3. Let A be a class module in Mod(G). Then the

perfect duality between H ~ and G is induced by the following pairing. For a C A G and a character X : G ~ Q/Z , we get a 2-cocycle

a~,~- = [x'(a) + x ' ( v ) - x ' (av)]a,

where X' is a lifting of X in Q. The expression m brackets is a 2-cocycIe of G in Z. The cocycle (a~,~-) represents the class

u @.

Thus the perfect duality arises from a bilinear map

A a x G ~ H2(G,A)

which we may write (a, x) ~ a U Sx,

whose kernel on the left is SGA, and the kernel on the right is 0.

w Explicit Nakayama maps

Throughout this section we let G be a finite group.

102

In Chapter I, w we had an isomorphism

H - I ( G , Z) - ~ G/G c

by means of a sequence of isomorphisms

H - 2 ( Z ) ,,~ H - I ( I ) ~ I / I 2 ,~ C / C r

If ~- C G, we denote by ~r the element of H-2 (Z) corresponding to the coset ~'G r in G/G c. So by definition

r =

where ~ is the coboundary associated to the exact sequence

0 IG z[a] z 4 0 .

On the other hand, we now have a cup product

H~(A) • H-2 (Z) ---, H~-2(A)

for A E Mod(G), associated to the natural bilinear map Z • ---+ A. We are going to make this cup product explicit for r ~ 1, in terms of cochains from the standard complex, and the description of H-2 (Z) given above.

To start, we give a special case of the cup product under dimension shifting. We consider as usual the exact sequence

0 ~ 1 ~ ZIG]-~ Z ~ 0

and its dual

0 ~ Hom(Z,A) ---, Hom(Z[G],A) ~ Horn(I, A) ~ 0.

We have a bilinear map in Mod(G)

Hom(Z[G],A) • Z[G]-+ A given by (f,A) ~ f(A).

IV.8 103

There results a pairing of these two exact sequences into the exact sequence

0 - . A ---. A ----~ 0 --~ 0,

to which one can apply the commutative diagram following Theo- rem 5.1 to get:

P r o p o s i t i o n 8.1. The following diagram has character -1:

H~ x H - I . ( I ) , H - I ( A )

HI(Hom(Z,A)) • H-2(Z) , H - ' ( A )

On the other hand, we know that in dimension 0, the cup product is given by the induced morphisms. By Corollary 1.12, we see that the cup product in the top line is given by the maps

>c(f) U m ( a - e ) = m ( f ( c r - e ) ) for f � 9

We now pass to the general case.

T h e o r e m 8.2. Let a = a(cr l , . . . ,a~) be a standard cochain in c ~ ( a , A) for r >= 1 For each ~ ~ a , d e , he a map

a~+a*~"

by the formulas:

of C"(G,A) ---+ C"-2(G,A)

Then for r >= 1 we have the relation

( ~ a ) , ~ = ~(~,T) .

If a is a cocycle representing an element a of Hr(G,A) , then (a * ~') represents aU ~ �9 Hr-2(G,A) .

(u, ~-)(.) = c(~-) if r = I

(a �9 T)(.) = E a(p, T) if r = 2 pEG

(a,~-)(o-~,. . . ,~-, ._~)= E a (o ' , , . . . ,~ , ._~ ,p , r ) if ,~ > 2. pGO

104

Proof. Let us first give the proof for r = 1 and 2. We let a = a(cr) be a 1-cochain. We find

((6a) �9 7-)(.) -= E ( 6 a ) ( p , 7") P

--= E ( P a ( 7 - ) -- a(pT-) + a(p)) P

= ~ ;a(7-) P

= SG(a(7-))) - S a ( ( a * 7-)(.)) = ( ~ ( a , 7-))(.),

which proves the commuta t iv i ty for r = 1.

Next let r = 2 and let a(~, 7-) be a 2-cochain. Then:

( ( ~ a ) , 7 - ) ( ~ ) = ~ ( 6 a ) ( ~ , p, 7-.) P

= ~ ( ~ a ( p , 7 - ) - a(~p, 7-)+ a(~, pT-) - a(~, p)) O

= ~ ~a(p, 7-) - a(p, 7-)

P

= (o - ~ ) ( ( a , 7-)(.)) = ( ~ ( a , 7-))(~),

which proves the commuta t iv i ty relat ion for r = 2. For r > 2, the proof is entirely similar and is left to the reader.

From the commuta t iv i ty relation, one obtains an induced homomorph i sm on the cohomology groups, namely

~ - " H ' ( A ) ~ H ' - 2 ( A ) for r >__ 2.

For r = 2, we have to note tha t if the cochain a(~z) is a coboundary , tha t is

a ( ~ ) = ( ~ - e ) b for some b E A ,

IV.8 105

then ( a , r ) ( . ) = ( r - e)b is in IDA. Thus ~,- is a morphism of functors. It is also a &morphism, i.e. ~ commutes with the coboundary associated with a short exact sequence. Since

is also a 5-morphism of H ~ to H r-2, to show that they are equal, it suffices to show that they coincide for r = 1, because of the uniqueness theorem.

Explicitly, we have to show that if a E H 1 (A) is represented by the cocycle a(a), then of tO r is represented by (a �9 r)(-), that is

of U (~- = n<(a(r)).

This is now clear, because of the diagram:

~ ( f ) • (~ - ~) , , ( f ( ~ - ~ ) )

[ ( S X ( / ) = --Of] X (5--1(T -- e ) I ) of U Cr-

The coboundary on the left comes from Proposition 8.1. This concludes the proof.

C o r o l l a r y 8.3. If of 6 Hi (A) is represented by a standard cocycle a(~), then/or each ~ e a we have a(~) e As~ and

of U ~,- = >K(a(v)) 6 H - I ( A ) .

If of 6 H2(A) is represented by a standard cocycle a(a, 7), then /or each ~ C C we have Ep a@, ~) C A% and

C o r o l l a r y 8.4. The duality between Hi(G , Q / Z ) andH-2(G,Z) in the duality theorem is consistent with the identification of H I ( G , Q / Z ) with G and of H-2(G,Z) with G/G c.

The above corollary pursues the considerations of Theorem 1.17, Chapter II, in the context of the cup product. We also obtain further commutativity relations in the next theorem.

106

Propos i t ion 8.5. Let U be a subgroup of G. (i) For T e U,A �9 Mod(G) ,a �9 Hr (G ,A) we have

trV(r U r e s t ( a ) ) = r_ U (~.

(ii) zf u i~ normal, m = (U : e), and o~ �9 H r ( a / U , A v) ~,ith r >= 2, then

m . i n ~ / ~ (r U ~) = r U i n ~ / ~ (~).

If r = 2, m. in~v/v is induced by the maps

B a ---+ mBa and mSuB ---+ SaB

for B �9 Mod(G/U).

Proof. As to the first formula, since the transfer corresponds to the map induced by inclusion, we can apply directly the cup product formula from Theorem 3.1, that is

t r (a U res(/3)) = t r (a) U/3.

One can also use the Nakayama maps, as follows. For T fixed, we have two maps

a~-+~rUa and a~-+ t rg (~rUresG(a ) )

which are immediately verified to be 5-morphisms of the cohomological functor H a into H a with a shift of 2 dimensions. To show that they are equal, it will suffice to do so in dimension 2. We apply Nakayama's formula. We use a coset decomposition G = U ~u as usual. If f is a cocycle representing a, the first map corresponds to

f ~ E f(p' 7). pea

The second one is

sG ( E f (P 'T)) = E c,oeU

IV.8 107

Using the cocycle relation

f(e,p) + f(ep, r) - f(e, pr) = ef(p, r),

the desired equality falls out.

This method with the Nakayama map can also be used to prove

the second part of the proposition, with the lifting morphism lif~G/U replacing the inflation ing/U. One sees that m . lif~J v is a a- morphism of Ha/u to Ha on the category Mod(G/U), and it will suffice to show that the a-morphism.

a ~ ( ~ U li~/u(a) and a ~ m . l i ~ / u ( ( , U a )

coincide in dimension 2. This follows by using the Nakayama maps as in the first case.

If G is cyclic, then H-2(G, Z) has a maximal generator, of order (G : e), and - 2 is a cohomological period. If a is a generator of G, then for all r E Z, the map

H~(G,A) -+ H~-2(G,A) given by a ~-+ ~ U a

is aaa isomorphism. Hence to compute the restriction, inflation, transfer, conjugation, we can use the commutativity formulas and the explicit formulas of Chapter II, w

C o r o l l a r y 8.6. Let G be cyclic and suppose (U : e) divides the order of G/U. Then the inflation

i n ~ / U - HS(G/U,A U) ---+ HS(G,A)

is 0 fo r s > 3.

Proof. Write s = 2r or s = 2 r + l with r > 1. Let cr be a generator of G. By Proposition 8.5, we find

(~, U inf(a) = in:f((~ U a).

But (~ U a has dimension s - 2. By induction, its inflation is killed by (U : e) "-1 , from which the corollary follows.

The last theorem of this section summarizes some cornmutativ- ities in the context of the cup product, extending the table from Chapter I, w

108

T h e o r e m 8.7. Let G be finite of order n. Let A E Mod(G) and a E H2(A) = H2(G,A). The following diagram is commutative.

H0(A) x H2(XZ) ' H2(A)

lu~ Io~ • H (Z) . H~ = z / n z

l i d 16 16

H-2(Z) x Hi(Q/Z) �9 H-I(Z/Z)

a / a o x d . ( z / z ) .

The vertical maps are isomorphisms in the two lower levels. I f A is a class module and a a fundamental element, i.e. a generator of H2(A), then the cups with a on the first level are also isomorphisms.

Proof. The commutative on top comes from the fact that all elements have even dimension, and that one has commutativity of the cup product for even dimension. The lower commutativities are an old story. If A is a class module, we know that cupping with

gives an isomorphism, this being Tate's Theorem 7.1.

R e m a r k . Theorem 8.7 gives, in an abstract context, the reciprocity isomorphism of class field theory. If G is abelian, then G c = e and H~ = A a / S G A is both isomorphic to G and dual to G. On one hand, it is isomorphic to G by cupping with a, and identifying H-2(Z) with G. On the other hand, if X is a character of G, i.e. a cocycle of dimension 1 in Q /Z , then the cupping

~(a) X ~X ~-~ x(a) U ~X

gives the duality between A G / S G A and H 1 (Q/Z) , the values being taken in H2(A). The diagram expressed the fact that the identification of H~ A) with G made in these two ways is consistent.

C H A P T E R V Augmented Products

w Definitions

In Tate's work a new cohomological operation was defined, satisfying properties similar to those of the cup product, but especially adjusted to the applications to class field theory and to the duality of cohomology on connection with abelian varieties. As usual here, we give the general setting which requires no knowledge beyond the basic elementary theory we are carrying out.

Let 92 be an abelian bilinear category, and let H, E, F be three 5-functors on 92 with values in the same abelian category ~3. For each integers r, s such that H r, E s are defined, we suppose that F r+s+l is also defined. By a Ta t e p r o d u c t , we mean the data of two exact sequences

O _ . _ . , A ' A + A J A " _ . . _ , O

0 ---, B' -L B j B" ~ 0

and two bilinear maps

A' • B ---, C and A • B' o C

coinciding on A I • B'. Such data, denoted by (A,B, C), form a category in the obvious sense. An a u g m e n t e d c u p p i n g

H•

110

associates to each Tare product a bilinear map

U~u, : H " ( A " ) x E S ( B '') ~ Fr+~+l(c)

satisfying the following conditions.

A C u p 1. The association is functorial, in other words, if u : (A, B, C) ---* (A, B, C) is a morphism of a Tate product to another, then the diagram

Hr(A '') x E'(B") �9 F r+'+l

~(u) lE(u) IF(u)

H~(fit '') x e ' ( / ~ " ) �9 F r + ' + l ( C )

is commutative.

A C u p 2. The augmented cupping satisfies the property of dimension shifting namely: Suppose given an exact and commutative diagram:

0 0 0

! 1 l 0 ~ A ' ~ A ~ A"

0 ~ M ' ~ M , M ~'

1 1 l 0 , X ' , X ~ X "

1 ! l 0 0 0

and two exact sequences

O ~ B ' ~ B- - - . B " ~ O

0 ----* C ' ---* M e ---* X c ~ 0,

~ 0

~ 0

~ 0

V.1 111

as well as bilinear maps

A' x B - - - ~ C A x B' --~C

X ' x B--* X c X x B' ---* X c

B' x B--~ M c M x B'---+ M c

which are compatible in the obvious sense, left to the reader, and coincide on A t x B' resp. M ' x B ~, resp. X' x B', then

Hr(x,,) • E'(B") -

l,a i, H~+I(A '') x E'(B")- " F~+'+2(C)

is commutative. Similarly, if we shift dimensions on E, then the similar diagram will have character -1 .

If H is erasable by an erasing functor M which is exact, and whose cofunctor X is also exact, then we get a uniqueness theorem as in the previous situations.

Thus the agreed cupping behaves like the cup product, but in a little more complicated way. All the relations concerning restriction, transfer, etc. can be formulated for the augmented cupping, and are valid with similar proofs, based as before on the uniqueness theorem. For example, we have:

P r o p o s i t i o n 1.1. Let G be a group. Suppose given on Ha an augmented product. Let U be a subgroup of finite index. Given a Tate product (A, B, C), let

and

Then

t ! a 6 H~(G,A")

/3" 6 H~(U,B") .

= trU/m'~ U.og Z") G,- ,

Proof. Both sides of the above equality define an augmented cupping Ha • Hu --* Ha, these cohomological functors being taken on the multilinear category Mod(G). They coincide in dimension (0, O) and 1, as one determines by an explicit computation, so the general uniqueness theorem applies.

112

P r o p o s i t i o n 1.2. For a" E H~(U, A) and/3" E Hs(U, B " ) and ~r E G, we have

~.(c~" u~.g ~") = ~.c~" u~.g ~.~".

Similarly, if U is normal in G and o/' E H " ( G / U , A"u) , 8" ~ Hs(a/V,B"U), we have for the inZa*ion

inf(~" U ~~ Y') = inf(~") U aug in f (~")

Of course, the above ~tatements hold for the special functor H . when G is finite, except when we deal with inflation.

We make the augmented product more explicit in dimensions ( -1 , 0) and 0, as well as (0, 0) and 1, for the special functor H e .

D i m e n s i o n s ( -1 , 0) a n d 0. We are given two exact sequences

0 - - ~ A ' ~ A j A ' ' ~ 0

0 - - , B' 2* B ~ B " ~ 0

as well as a Tare product, that is bilinear maps in Mod(G):

A I x B ~ C and A x B I ~ C

coinciding on A I x B I. We then define the augmented product by

x<(a 'l) U~ug >c(b 'l) = ;4(alb -- abl),

b I where a l, are determined as follows. We choose a E A such that ~ . b I j a a II and b E B such that jb b" Then a I, are uniquely

determined by the conditions

ia' = SG(a) and ib' = Sc(b).

D i m e n s i o n s (0, O) a n d 1. We define

~(a") U~ug x(b") = cohomology class of the cocycle a~b + abe,

is determined by the formulas where the cocycle az

j a = a" and ia~ = ~ra - a,

and similarly for b'.

w Existence

The existence is given in a way similar to that of the cup product. We shall be very brief. First an abstract statement:

V.3 113

T h e o r e m 2.1. Let 91 be a multilinear abelian category, and suppose given an exact bilinear functor A ~-+ Y(A) from 91 into the bilinear category of complexes in an abelian category ~ . Then the corresponding eohomological functor H on P.J has a structure of augmented cup functor, in the manner described below.

Recall from Chapter IV, w that we already described how the category of complexes forms a bilinear category. For the application to the augmented cup functor, suppose that (A,B, C) is a Tate product. We want to define a bilinear map

Hr(A '') x H~(B '') ---+ Hr+S+l(c).

We do so by a bilinear map defined on the cochains as follows. Let c~" and r be cohomology classes in H~(A '') and H*(B '') respectively, and let f " , g " be representative cochains in Y~(A), Y*(B) respectively, so that j f = f " and jg = g". We view i as an inclusion, and we let

h = S f .g + (-1)~ f .Sg ,

the products on the right being the Tate product. Then we define o/' U~ug fl" to be the cohomology class of h.

One verifies tediously that this class is independent of the choices made in its construction, and one also proves the dimension shifting property, which is actually a pain, which we do not carry out.

w S o m e p r o p e r t i e s

T h e o r e m 3.1. Let the notation be as in Theorem 2.1 with a Tare product (A ,B ,C) . Then the squares in the following diagram from left to right are commutative, resp. of character ( - 1 ) r, resp. commutative.

H~(A ,) ---. H~(A) --~ H~(A,,) 6_~ H~+I(A ,)

X X X X

H ' ( B ) *-- H '+I (B ') ~- H ' ( B " ) 0-- H ' ( B )

u]. u~ iU~ug ~u

H~+~+~(C) -~ H~+~+~(C) -~ H~+,+~(C) -~ H~+~+~(C)

114

The morphisms on the bottom line are all the identity.

Proof. The result follows immediately from the definition of the cup and augmented cup in terms of cochain representatives, both for the ordinary cup and the augmented cup.

The next property arose in Tate's application of cohomology theory to abelian varieties. See Chapter X.

T h e o r e m 3.2. Let the multilinear categories be those of abelian groups. Let m be an integer > 1, and suppose that the following sequences are exact:

0 ~ A"~ --, A" m A" --~ 0

O ~ B ~ ~ B" "~) B" ~ O.

Given a Tate product (A, B, C), one can define a bilinear map

A2 • B: c

as follows. Let a" C A"~ and b" C B"~. Choose a E A and b C B such that j a = a" and jb = b". We define

(a", b") = ma.b - a.rnb.

Then the map ( a ' , b " ) H (an,b H) is bilinear.

Proof. Immediate from the definitions and the hypothesis on a Tate product.

T h e o r e m 3.3. Let (A, B, C) be a Tate product in a multilinear abeIian category of abelian groups. Notation as in Theorem 3.2, we have a diagram of character (--1)r-l :

H r ( A ~ ) , H r ( A '')

X X

H~+I(B~) , ~ H~(B ")

gr+~+l(C) , Hr+S+a(C) id

V.8 115

Note that the coboundary map in the middle is the one associated with the exact sequence involving B~ and B" in Theorem 3.2. The cup product on the left is the one obtained from the bilinear map as in Theorem 3.2.

CHAPTER VI

Spectral Sequences

We recall some definitions, but we assume that the reader knows the mater ia l of Algebra, Chapter XX, w on spectral sequences, their basic constructions and more elementary properties.

w Definitions

Let 9 /be an abelian category and A an object in 9/. A f i l t r a t i o n of A consists in a sequence

F = F ~ D F 1 D F 2 D . . . D F n D F n+l = 0 .

If F is given with a differential (i.e. endomorphism) d such that d 2 = 0, we also assume that dF p C F p for all p = 0 , . . . ,n, and one then calls F a f i l t e r e d d i f f e r e n t i a l o b j e c t . We define the graded object

G r ( F ) = O G r P ( F ) where G r P ( F ) = F P / F p+I.

p>o

We may view Gr(F) as a complex, with a differential of degree 0 induced by d itself, and we have the homology H(GrPF) .

Fi l tered objects form an additive category, which is not necessarily abelian. The family Gr(A) defines a covariant functor on the category of filtered objects.

VI.1 117

of: A s p e c t r a l s equence in A is a family E = (EPr 'q, E n) consisting

(1) Objects EPr ,q defined for integers p, q, r with r > 2.

(2) Morphisms d~'q �9 E~,q ---* E~ +' 'q-~+l such that

d p + r , q - r + l o c~ p 'q = O. r - - T

(3) Isomorphisms

_~,~"q �9 Ker(d~ 'q ) / Im(d~ -~'q+~-I ) ~ ~ + 1

(4) Filtered objects E '~ in A defined for each integer n.

We suppose that for each pair (p, q) we have d~'q = 0 and d~ -~'q+~-I = 0 for r sufficiently large. It follows that E~ 'q is independent of r for r sufficiently large, and one denotes this object by E~q . We assume in addition that for n fixed, F P ( E ~) = E ~ for p sufficiently small, and is equal to 0 for p sufficiently large.

Finally, we suppose given:

(5) Isomorphisms ~P,q �9 E ~ q -+ GrP(EP+q).

The family {E'~}, with filtration, is called the a b u t m e n t of the spectral sequence E, and we also say that E a b u t s to {E n} or converges to {E'~}.

By general principles concerning structures defined by arrows, we know that spectral sequences in 92 form a category. Thus a morphism u : E --+ E' of a spectral sequence into another consists in a system of morphisms.

U P , q �9 ~i~,P,q __+ ]:i~P,q a n d u '~ " E = ---+ E ' 'n . ~ r m •

compatible with the filtrations, and commuting with the morphisms d~ ,q, ~ , q and ~ ,q . Spectral sequences in P.l then form an additive category, but not an abelian category.

A s p e c t r a l f u n c t o r is an additive functor on an abelian category, with values in a category of spectral sequences.

We refer to Algebra, Chapter XX, w for constructions of spectral sequences by means of double complexes.

118

A spectral sequence is called p o s i t i v e if EP~ 'q = 0 for p < 0 and q < O. This being the case, we get:

E~ 'q ~ EP~; q for r > sup(p, q + 1)

E n = 0 for n < 0

Fm(E ~) = 0 if m > n

F ro (E" ) = E n if m __< O.

In what follows, we assume that all spectral sequences are positive.

We have inclusions

E n : F~ '~) D F I ( E '~) D . . . D Fn(E '~) D F ~ + ~ ( E n ) : O.

The isomorphisms

~0,~: E0,~ --~ Gr0(E n) = FO(En) /FI (E TM) = E,~/FI(E,~)

Z-,0. E- ; 0 ___, G r " ( E '~) = F'~(E n)

will be called the edge , or extreme, isomorphisms of the spectral sequence.

Theorem 9.6 of Algebra, Chapter XX, shows how to obtain a spectral sequence from a composite of functors under certain conditions, the Grothendieck spectral sequence. We do not repeat this result here, but we shall use it in the next section.

w The Hochschild-Serre spectral sequence

We now apply spectral sequence to the cohomology of groups. Let G be a group and H c the cohomological functor on Mod(G). Let N be a normal subgroup of G. Then we have two functors:

A ~-+ A N of M o d ( G ) i n t o Mod(G/N)

B ~ B G/N of Mod(G/N) into Grab (abelian groups).

Composing these functors yields A ~-+ A c. Therefore, we obtain the Grothendieck spectral sequence associated to a composi te of functors, such that for A E Mod(G):

E~'q(A) = HP(G/N, Hq(N,A) ) ,

vI.2 119

with G/N acting on Hq(N, A) by conjugation as we have seen in Chapter II, w Furthermore, this spectral functor converges to

E~(A)=H~(G,A).

One now has to make explicit the edge homomorphisms. First we have an isomorphism

~o,~. E~(A)---+ H~(G,A)/FI(H~(G,A)),

where F 1 denotes the first term of the filtration. Furthermore

E~ = H~(N,A) G/N

and E~g~(A) is a subgroup of E2'~(A), taking into account that the spectral sequence is positive. Hence the inverse of fl0,,~ yields a monomorphism of H~(G,A)/FI(H~(G,A)) into H'~(N, d), and induces a homomorphism

H'~(G,A)-+H~(N,A).

P r o p o s i t i o n 2.1. This homomorphism induced by the inverse of ~o,n is the restriction.

Proof. This is a routine tedious verification of the edge homomorphism in dimension 0, left to the reader.

In addition, we have an isomorphism

9 ,0 A)),

whose image is a subgroup of Hn(G, A). Dually to what we had

previously, E~g~ is a factor group of E~ '~ = H~(G/N, AN). Composing the canonical homomorphism coming from the d~ '~ and fin,0 we find a homomorphism

Hn(G/N,A N) --+ Hn(G,A).

120

P r o p o s i t i o n 2.2. This homomorphism is the inflation.

Proof. Again omitted.

Besides the above edge homomorphisms, we can also make the spectral sequence more explicit, both in the lowest dimension and under other circumstances, as follows.

T h e o r e m 2.3. Let G be a group and N a normal subgroup. Then for A E Mod(G) we have an exact sequence:

0---+ HI (G/N ,A N) inf HI(G,A) res HI(N,A)C/N d2

d2 H2(G/N, AN) inf H2(G,A).

The homomorphism d2 in the above sequence is called the t r a n s - g ress ion , and is denoted by tg, so

tg" HI(N,A) a /x -+ H2(G/N, AN).

This map tg can be defined in higher dimensions under the following hypothesis.

T h e o r e m 2.4. If Hr(N,A) = 0 for 1 <= r < s, then we have an exact sequence:

0 ~ H*(G/N,A N) inf HS(G,A) re L H,(N,A)C/N tg

tg H,+I(G/N, AN) in~ Hs+I(G,A).

For computations, it is useful to describe tg in dimension 1 in terms of cochains, so we consider tg as ill Theorem 2.3, in dimension 1, and we have:

An element a E H2(U/N, A N) can be written as

a = tg(/~) with/3 E HI(N,A) G/N

if and only if there exists a cochain f E CI(G, A) such that:

1. The restriction of f to N is a 1-cocycle representing/3.

2. We have 6f = inflation of a 2-cocycle representing a.

In case many groups H~(N, A) are trivial, the spectral sequence gives isomorphisms and exact sequences as in the next two theorems.

VI.3 121

T h e o r e m 2.5. Suppose Hr(N, A) = 0 for r > O. Then

p,o . H P ( G / N , A N) --~ HP(G,A ) ~2

is an isomorphism for all p >= O.

The hypothesis in Theorem 2.5 means that all points of the spectral sequence are 0 except those of the bottom line. Furthermore:

T h e o r e m 2.6. Suppose that Hr(N, A) = 0 for r > 1. Then we have an infinite exact sequence:

0 --* H I ( G / N , H ~ --~ H I ( G , A ) --* H ~ --*

d2 --. H 2 ( G / N , H ~ --~ H 2 ( G , A ) --. H I ( G / N , H I ( N , A ) ) --.

d2 --~ H S ( G / N , H ~ --. H 3 ( G , A ) --. H 2 ( G / N , H I ( N , A ) ) --.

The hypothesis in Theorem 2.6 means that all the points of the spectral sequence are 0 except those of the two bottom lines.

w S p e c t r a l s e q u e n c e s and cup products

In this section we state two theorems where cup products occur within spectral sequences. We deal with the multilinear category Mod(G), a normal subgroup N of G, and the Hochschild-Serre spectral sequence.

T h e o r e m 3.1. The spectral sequence is a cup functor (in two dimensions) in the following sense. To each bilinear map

A x B - - - * C

there is a cupping determined functorially

E~'q(A) x E~"q'(B) ~ E,'+P"q+q' (C)

such that for a C E~'q(A) and/3 e E~ ''q'(B) we have

d,-(o~ �9 #/) = (d,~o~). ~ + ( -1)P+qo~ �9 (d, .~).

122

If we denote by U the usual cup product, then for r = 2

The cupping is induced by the bilinear map

Hq(N,A) • Hq' (N,B) ---* g q+q' (N,C).

Finally, suppose G is a finite group, and B 6 Mod(G). We have an exact sequence with arrows pointing to the left:

c~ ,n N N i n c O+---Ba/SG/NBN~ BGr B /IG/NB ( BNG/N/IG/NBN+---O,

or in other words

Oe---H~ G/N,BN)(----H~ G,B)+--Ho( G /N,BN)+--H-I( G/N,BN)+---O

This exact sequence is dual to the inflation-restriction sequence, in the following sense.

T h e o r e m 3.2. Let G be a group, U a normal subgroup of finite index, and ( A , B , C ) a Tate product in Mod(G). Suppose that (A U,B U, C U) is also a Tare product. Then the following diagram is commutative:

HI(G/U,A,,U) i M H' (G ,A" ) ' : ~ , HI(U,A '')

X X X

0 < HO(GIU, B,,U) < r HO(G,B,) ( s~ Ho(U,B,, )

H2(G/U,C U) , H2(C,C) , H2(U,C). inf t r

The two horizontal sequences on top are exact.

C H A P T E R V I I

Groups of Galois Type

(Unpublished article of Tate)

w Def in i t i o n s a n d e l e m e n t a r y p r o p e r t i e s

We consider here a new category of groups and a cohomological functor, obtained as limits from finite groups.

A topological group G will be said to be of Galois t y p e if it is compact, and if the normal open subgroups form a fund~.mental system of neighborhoods of the identity e. Since such a group is compact, it follows that every open subgroup is of finite index in G, and is therefore closed.

Let S be a closed subgroup of G (no other subgroups will ever be considered). Then S is the intersection of the open subgroups U containing S. Indeed, if r E G and a ~ S, we can find an open normal subgroup U of G such that Ucr does not intersect S, and so US = SU does not contain c~. But US is open and contains S, whence the assertion.

We observe that every closed subgroup of finite index is also open.Warning: There may exist subgroups of finite index which are not open or closed, for instance if we take for G the invertible power series over a finite field with p elements, with the usual topology of formal power series. The factor group G/G p is a vector space over

124

Fp, one can choose an intermediate subgroup of index p which is not open.

Examples of groups of Galois type come from Galois groups of infinite extensions in field theory, p-adic integers, etc.

Groups of Galois type form a category, the morphisms being the continuous homomorphisms. This category is stable under the following operations:

1. Taking factor groups by closed normal subgroups. 2. Products. 3. Taking closed subgroups. 4. Inverse limits (which follows from conditions 2 and 3).

Finite groups are of Galois type, and consequently every inverse limit of finite groups is of Galois type. Conversely, every group of Galois type is the inverse limit of its factor groups G / U taken over all open normal subgroups. Thus one often says that a Group of Galois type is prof in i te .

The following result will allow us to choose coset representatives as in the theory of discrete groups, which is needed to make the cohomology of finite groups go over formally to the groups of Galois type.

Proposition 1.1. Let G be of Galoia type, and let S be a closed subgroup of G. Then there exists a continuous section

G/S --, G,

i.e. one can choose representatives of left coseta of S in G in a continuous way.

Proof. Consider pairs (T, f ) formed by a closed subgroup T and a continuous map f : G / S ---, G / T such that for all x E G the coset f ( x S ) = yT is contained in xS. We define a partial order by putt ing ( T , F ) <= (T1, f l ) if T C T1 and f l ( zS ) C f ( x S ) . We claim that these pairs are then inductively ordered. Indeed, let { (Ti, fi) } be a totally ordered subset. Let T = A Ti. Then T is a closed

i

subgroup of G. For each x E G, the intersection

is a coset of Ti, and is closed in S. Indeed, the finite intersection of such cosets f i (zS) is not empty because of the hypothesis on

VII. 1 12~

the maps fi. The intersection Af i ( zS ) taken over all indices i is therefore not empty. Let y be an element of this intersection. Then by definition, yTi = fi(zS) for all i, and hence yT C fi(zS) for all i. We define f ( zS) = yT. Then f ( zS) C zS.

The projective limit of the homogeneous spaces G/Ti is then canonically isomorphic to G/T, as one verifies immediately by the compactness of the objects involved. Hence the continuous sections G/S ~ G/Ti which are compatible can be lifted to a continuous section G/S ~ G/T. By Zorn's lemma, we may suppose G/T is maximal, in other words, T is minimal. We have to show that r ~ e .

In other words, with the subgroup S given as at the beginning, if S # e it will sumce to find T :~ S and T open in S, closed in G, such that we can find a section G/S ~ S/T. Let U be a normal open in G, U V I S # S, and put U V I S = T. I f G = [ . J z i U S i s a coset decomposition, then the map

x i u S ~ x i u T for u E U

gives the desired section. This concludes the proof of Proposition 1.1.

We shall now extend to closed subgroups of groups of Galois type the notion of index. By a s u p e r n a t u r a l n u m b e r , we mean a formal product

I I pnp

taken over all primes p, the exponents np being integers _>_ 0 or oe. One multiplies such products by adding the exponents, and they are ordered by divisibility in the obvious manner. The sup and inf of an arbitrary family of such products exist in the obvious way. If S is a closed subgroup of G, then we define the i n d e x (G : S) to be equal to the supernatural number

(G �9 S ) = 1.c.m. (G : V) , V

the least common multiple 1.c.m. being taken over open subgroups V containing S. Then one sees that (G : S) is a natural number if and only if S is open. One also has:

126

P r o p o s i t i o n 1.2. Let T C S C G be closed subgroups of G. Then

(a : s)(s : T)= (a : T).

If (Si) is a decreasing family of closed subgroups of G, then

(a:ns i )=lcm.(a:s , ) i

Pro@ Let us prove the first assertion. Let m, n be integers => 1 such that m divides (G : S) and n divides (S : T). We can find two open subgroups U, V of G such that U D S, V D T, m divides ( a : U) and n divides (S : V N S). We have

(a : s n V) = (a : U)(U: U n Y).

But there is an injection S / ( V N S) --~ U/(U N V) of homogeneous spaces. By definition, one sees that rnn divides (G : T), and it follows that

( G : S ) ( S : T ) divides ( G : T ) .

One shows the converse divisibility by observing that if U D T is open, then

(a : u) = (a : us)(us: u) and (US :U) = (S: s o u),

whence (G : T) divides the product. This proves the first assertion of Proposition 1.2. The second assertion is proved by applying the first.

Let p be a fixed prime number. We say that G is a p -g roup if (G : e) is a power of p, which is equivalent to saying that G is the inverse limit of finite p-groups. We say that S is a Sylow p - s u b g r o u p of G if S is a p-group and (G : S) is prime to p.

Proposition 1.3. Let G be a group of Galois type and p a prime number. Then G has a p-Sylow subgroup, and any two such subgroups are conjugate. Every closed p-subgroup S of G is contained in a p-Sylow subgroup.

Proof. Consider the family of closed subgroups T of G containing S and such that (G : T) is prime to p. It is partially ordered by descending inclusion, and it is actually inductively ordered since

VII. 1 127

the intersection of a totally ordered family of such subgroups contains 5' and has index prime to p by Proposition 1.2. Hence the family contains a minimal element, say T. Then T is a p-group. Otherwise, there would exist an open normal subgroup U of G such that (T : T n U) is not a p-power. Taking a Sylow subgroup of the finite group T/ (T n U) = TU/U, for a prime number ~ p, once can find an open subgroup Vof GH such that (T : T N V) is prime to p, and hence (5' : 5"NV) is also prime t o p . Since 5' is a p - group, one must have 5' = 5' N V, in other words V D S, and hence also T O V D 5'. This contradicts the minimality of T, and shows that T is a p-group of index prime to p, in other words, a p-Sylow subgroup.

Next let 5'1,5'2 be two p-Sylow subgroups of G. Let 5'l(U) be the image of 5' under the canonical homomorphism G --+ G/U for U open normal in G. Then

(G/U: 5"iN~U) divides (G: 5'lU),

and is therefore prime to p. Hence 5'l(U) is a Sylow subgroup of G/U. Hence there exists an element a C G such that S2(U) is conjugate to SI(U) by a(U). Let Fu be the set of such a. It is a closed subset, and the intersection of a finite number of Fu is not empty, again because the conjugacy theorem is known for finite groups. Let cr be in the intersection of all Fu. Then 5"~ and 5'2 have the same image by all homomorphisms G ~ G/U for U open normal in G, whence they are equal, thus proving the theorem.

Next, we consider a new category of modules, to take into account the topology on a group of Galois type G. Let A C Mod(G) be an ordinary G-module. Let

Ao = U AU'

the union being taken over all normal open subgroups U. Then A0 is a G-submodule of A and (A0)0 = A0. We denote by Galm(G) the category of G-modules A such that A = A0, and call it the category of Ga lo i s modu le s . Note that if we give A the discrete topology, then Galm(G) is the subcategory of G-modules such that G operates continuously, the orbit of each element being finite, and the isotropy group being open. The morphisms in Galm(G) are the ordinary G-homomorphisms, and we still write HomG(A,B) for A, B C Galm(G). Note that Galm(G) is an abelian category

128

(the kernel and cokernel of a homomorphism of Galois modules are again Galois modules).

Let A e Galm(G) and B �9 Mod(G). Then

H o m a ( A , B ) : Homa(A, Bo )

because the image of A by a G-homomorphism is automatically contained in B0. From this we get the existence of enough injectives in Galm(G), as follows:

P r o p o s i t i o n 1.4. Let G be of Galois type. I f B C Mod(G) is injective in Mod(G), then Bo is injective in Galm(a) . / f A C Galm(G), then there ezists an injective M C Galm(a) and a monomorphism u : A ---+ M.

Thus we can define the derived functor of A ~-* A a in Galm(G), and we denote this functor again by Ha, so

H~ : H ~ ( A ) = A a

as before.

Proposition 1.5. Let G be of Galois type and N a closed normal subgroup of G. Let A C Galm(G). If A is injective in Galm(G), then A N is injective in Galm(G/N).

Proof. If B C Galm(G/N) , we may consider B as an object of Galm(G), and we obviously have

Homc(B ,A) = H o m a / N ( B , A N)

because the image of B by a G-homomorphism is automatically contained in A N. Considering these Horn as functors of objects B in Galm(G/N) , we see at once that the functor on the right of the equality is exact if and only if the functor on the left is exact.

w C o h o m o l o g y

(a) Existence and uniqueness. One can define the cohomology by means of the standard complex. For A C Galm(G), let us

VII.2 129

p ut:

C"(G,A) = 0 if r = 0

C ~ = A

C~(G,A) = groups of maps f ' G " ~ A ( fo r t >0 ) ,

continuous for the discrete topology on A.

We define the coboundary

8~.C~(G,A)----~C~+I(G,A)

by the usual formula as in Chapter I, and one sees that C(G, A) is a complex. Furthermore:

P r o p o s i t i o n 2.1. The functor A ~ C(G, A) is an exact func- tot of Galm(G) into the category of complexes of abelian groups.

Proof. Let 0 --+ A' --, A --+ A" ~ 0

be an exact sequence in Galm(G). Then the corresponding sequence of standard complexes is also exact, the surjectivity on the right being due to the fact that modules have the discrete topology, and that every continuous map f : G ~ - . A" can therefore be lifted to a continuous map of G" into A.

By Proposition 1.5, we therefore obtain a &functor defined in all degrees r E Z and 0 for r < 0, such that in dimension 0 this functor is A ~-~ A a. We are going to see that this functor vanishes on injectives for r > 0, and hence by the uniqueness theory, that this 8-functor is isomorphic to the derived functor of A ~ A a, which we denoted by He.

T h e o r e m 2.2. Let G be a group of Galois type. Then the co- homoIogical functor Ha on Galm(G) is such that:

H r ( G , A ) = O for r > O.

H ~ = A c.

H~(G,A) = 0 if A is injective, r > O .

Proof. Let f(c~l, . . . ,cr~) be a standard cocycle with r _>_ 1. There exists a normal open subgroup U such that f depends only

130

on cosets of U. Let A be injective in Galm(G). There exists an open normal subgroup V of G such that all the values of f are in A v because f takes on only a finite number of values. Let W = U N V. Then f is the inflation of a cocycle f of G / W in A W. By Proposition 1.5, we know that A W is injective in Mod(G/W). Hence fi = 6~ with a cochain ~ of G / W in A W, and so f = 5g if g is the inflation of G to G. Moreover, g is a continuous cochMn, and so we have shown that f is a coboundary, hence that H~(G, A) = O.

In addition, the above argument also shows:

T h e o r e m 2.3. Let G be a group of Galois type and A E Galm(G). Then

Hr(G,A) ,~ dir l imH~(G/U, AU),

the direct limit dir lim being taken over all open normal subgroups U of G, with respect to inflation. Furthermore, Hr(G, A) zs a torsion group for r > O.

Thus we see that we can consider our cohomological functor -FIG in three ways: the derived functor, the limit of cohomology groups of finite groups, and the homology of the standard complex.

For the general terminology of direct and inverse limits, cf. Al- gebra, Chapter III, w and also Exercises 16 - 26. We return to such limits in (c) below.

R e m a r k . Let G be a group of Galois type, and let C Galm(G). If G acts trivially on A, then similar to a previous remark, we have

HI(G,A) = cont hom(G,A),

i.e. HI(G, A) consists of the continuous homomorphism of G into A. One sees this immediately from the standard cocycles, which are characterized by the condition

f ( a ) + f ( r ) = f ( a r )

in the case of trivial action. In particular, take A = Fp. Then as in the discrete case, we have:

Let G be a p-group of Galois type. If HI(G, Fp) = 0 then G = e, i.e. G is trivial.

VII.2 131

Indeed, if G # e, then one can find an open subgroup U such that G/U is a finite p-group -# e, and then one can find a non-trivial homomorpkism A : G/U ---+ Fp which, composed with the canonical homomorphism G ~ G/U would give rise to a non-trivial element of H 1 (G, Fp).

(b) C h a n g i n g t h e group. The theory concerning changes of groups is done as in the discrete case. Let A : G ~ ~ G be a continuous homomorphism of a group of Galois type into another. Then A gives rise to an exact functor

r Galm(G) ~ Calm(G'),

meaning that every object A C Galm(G) may be viewed as a Galois modute of G t. If

T: A--.. A I

is a morphism in Galm(G'), with A e Galm(G),A' e Galm(G'), with the abuse of notation writing A instead of O:,(A), the pair (A, ~) determines a homomorphism

Hr(A, ~2) = (A,~). : H"(G,A) ~ H"(G',A'),

functorially, exactly as for discrete groups.

One can also see this homomorphism explicitly on the standard complex, because we obtain a morphism of complexes

C(A,r : C(G,A) ~ C(G',A')

which maps a continuous cochain f on the cochain c 2 o f o A r.

In particular, we have the inflation, lifting, restriction and conjugation:

inf: H"(G/N,A N) ~ H"(G,A)

lif: H"(G/N,B) --* H"(G,B)

res: H"(G,A) ~ H"(S,A)

cr.: H"(S,A)---+ H"(S~,o'-IA),

for N closed normal in G, S closed in G and ~ E G.

132

All the commutativity relations of Chapter II are valid in the present case, and we shall always refer to the corresponding result in Chapter II when we want to apply the result to groups of Galois type.

For U open but not necessarily normal in G, we also have the transfer

t r : H~(U,A) ---+ Hr(G,A)

with A C Galm(G). All the results of Chapter II, w for the transfer also apply in the present case, because the proofs rely only on the uniqueness theorem, the determination of the morphism in dimension 0, and the fact that injectives erase the cohomology functor in dimension > 0.

(c) L imi t s . We have already seen in a naive way that our cohomology functor on Galm(G) is a limit. We can state a more general result as follows.

T h e o r e m 2.4. Let (Gi, s and (Ai, r be an inverse directed family of groups of Galois type, and a directed system of abelian groups respectively, on the same set of indices. Suppose that for each i, we have Ai E Galm(Gi) and that for i <= j , the homomorphisms

"~ij " Gj ~ Gi and ~Pij " Ai --+ Aj

are compatible. ]Let G = inv lim Gi and A = dir l imAi. Then A has a canonically determined structure as an element of Galm( G), such that for each i, the maps

~i " G ~ Gi and ~i " Ai --~ A

are compatible. Furthermore, we have an isomorphism of complexes

O: C(G,A) ~, dir lim c ( a i , A i ) ,

and consequently isomorphisms

O, "Hr (G,A) - -* dir lim Hr(Gi ,Ai) .

Proof. This is a generalization of the argument given for The- orem 2.3. It suffices to observe that each cochain f : G r ---+ A is uniformly continuous, and consequently that there exists an open normal subgroup U of G such that f depends only on cosets of U,

VII.2 133

and takes on only a finite number of values. These values are all represented in some A~. Hence there exists an open normal subgroup Ui of Gi such that /\ll(Ui) C U, and we can construct a cochain fi " G[ --~ Ai whose image in Cr(G,A) is f. Similarly, we find that if the image of fi in C~(G, A) is O, then its image in C"(Gj,Aj) is also 0 for some j > i , j sufficiently large. So the theorem follows.

We apply the preceding theorem in various cases, of which the most important axe:

(a) When the Gi are all factor groups G/U with U open normal in G, the homomorphisms ~ij then being surjective.

(b) When the G~ range over all open subgroups containing a closed subgroup S, the homomorphisms s then being inclusions.

Both cases are covered by the next lemma.

L e m m a 2.5. Let G be of Galois type, and let (Gi) be a family of closed subgroups, Ni a closed normal subgroup of Gi, indexed by a directed set {i}, and such that Nj C Ni and Gj C Gi when i <=j. Then one has

inv lim

Proof. Clear.

Note that Theorem 2.3 is a special case of Theorem 2.4 (taking into account Lemma 2.5). In addition, we get more corollaries.

C o r o l l a r y 2.6. Let G be of Galois type and A E Galm(G). Let S be a closed subgroup of G. Then

H r ( S , A ) = i n v lira Hr(V,A), V

the inverse limit being taken over all open subgroups V of G containing S.

C o r o l l a r y 2.7. Let G, (Gi), (Ni) be as in Lemma 2.5. Let A E galm(G) and let N = N Ni. Then

H r ( ( N G i ) / ( N N ~ ) , A N ) ~ dir lira Hr(Gi/Ni ,AN') ,

134

the limit being taken with respect to the canonical homomorphisms.

Proof. Immediate, because

AN = U A Ni = dir lira A Ni

because by hypothesis A E Galm(G).

C o r o l l a r y 2.8. Let G be of Galois type and A E Galm(G). Then

H~(G,A) = d i r lira H~(G,E)

where the limit is taken with respect to the inclusion morphisms E C A, for all submoduIes E of A finitely generated over Z.

Proof. By the definition of the continuous operation of G on A, we know that A is the union of G-submodules finitely generated over Z, so we can apply the theorem.

Thus we see that the cohomology group H~(G, A) are limits of cohomology groups of finite groups, acting on finitely generated modules over Z. We have already seen that these are torsion modules for r > 0.

C o r o l l a r y 2.9. Let rn be an integer > O, and A E Galm(G). Suppose

rn d : A--+ A

is an automorphism, in other words that A is uniquely divisible by m. Then the period of an element of H~(G, A) for r > 0 is an integer prime to m. I f mA is an automorphism for all positive integers m, then H~(G, A) = 0 for all r > O.

(d) The erasing f u n c t o r , and induced representations. We are going to define an erasing functor Mc on Galm(G) similar to the one we defined on Mod(G) when G is discrete.

Let S' be a closed subgroup of G, which we suppose of Galois type. Let B E Galm(S) and let M g ( B ) be the set of all continuous maps g : G ~ B (B discrete) satisfying the relation

= for s, a.

VII.2 135

Addition is defined in IvIS(B) as usual, i.e. by adding values in B. We define an action of G by the formula

(rg)(z) = g(x~) for r , z e G.

Because of the uniform continuity, one verifies at once that MaS(B) E Galm(a) .

Taking into account the existence of a continuous section of G/S in G in Proposition 1.1, one sees that:

MS(B) is isomorphic to the G~,Iois module of all continuous maps G/S --+ G.

Thus we find results similar to those of Chapter II, which we summarize in a proposition.

P r o p o s i t i o n 2.10. Notations as above, M~ is a covariant, additive exact functor from Galm(S) into Galm(G). The bifunctors

Homa(d, MSa(B)) and Homs(d ,B)

on Galm(G) x Galm(S) are isomorphic. If B is injective in Galm(S), then MSG(B) is iniective with Ga.lm(G).

The proof is the same as in Chapter II, in hght of the condition of uniform continuity and the lemma on the existence of a cross section.

T h e o r e m 2.11. Let G be of Galois type, and S a closed subgroup. Then the inclusion S C G is compatible with the homomorphism

g ~ g(e) of Mg(B) --+ B,

giving rise to an isomorphism of functors

Ha o M~ ,~ Hs.

rn part icular, if S = e, then H r (G, M s ( B ) ) = 0 for r > O.

Proof. Identical to the proof when G is discrete. For the last assertion, when S = e, we put Ma = M~.

In particular, we obtain an erasing functor MG =/VI~ as in the discrete case. For A E Galm(G), we have an exact sequence

0---+ A ~A Ma(A)---+ X(A)---+ O,

136

where CA is defined by the formula g A ( a ) : ga and g~(c~) = o 'a for ~rEG.

As in the discrete case, the above exact sequence splits.

C o r o l l a r y 2.12. Let G be of Galois type, S a closed subgroup, and B 6 Galm(S). Then Hr(S, Mg(B)) = 0 for r > O.

Proof. When S = G, this is a special case of the theorem, taking S = e. If V ranges over the family of open subgroups containing S, then we use the fact of Corollary 2.6 that

H ~ ( S , A ) = d i r lim Hr(V,A).

It will therefore suffice to prove the result when S = V is open. But in this case, Ms(B) is isomorphic in Galm(V) to a finite product of M~(B) , and one can apply the preceding result.

C o r o l l a r y 2.13. Let A E Galm(G) be injective. Then H~(S, A) = 0 for all closed subgroups S of G and r > O.

Proof. In the erasing sequence with gA, we see that A is a direct factor of MG(A), so we can apply Corollary 2.12.

(e) C u p p r o d u c t s . The theory of cup products can be developed exactly as in the case when G is discrete. Since existence was proved previously with the standard complex, using general theorems on abelian categories, we can do the same thing in the present case. In addition, we observe that

Galm(G) is closed under taking the tensor product,

as one sees immediately, so that tensor products can be used to factorize multitinear maps. Thus Galm(G) can be defined to be a multilinear category. If A1, . . . ,An, B are in Galm(G), then we define f : A1 x .. . x AN --* B to be in L(A1, . . . ,An ,B) if f is multilinear in Mod(Z), and

f (c ra l , . . . , a a n ) = c r f ( a l , . . . , a n ) for all a e G ,

exactly as in the case where G is discrete.

We thus obtain the existence and uniqueness of the cup product, which satisfies the property of the three exact sequences as in the

VII.2 137

discrete case. Again, we have the same relations of commutativity concerning the transfer, restriction, inflation and conjugation.

(f) S p e c t r a l s equence . The results concerning spectral sequences apply without change, taking into account the uniform continuity of cochains. We have a functor F : Galm(G) --+ Galm(G/N) for a closed normal subgroup N, defined by A ~ A N. The group of Galois type G / N acts on H~(N, A) by conjugation, and one has:

P r o p o s i t i o n 2.14. If N is closed normal in G, then H~(N, A) is in Galm(G/N) for d E Galm(G).

Proof. If ~ E N, from the definition of c%, we know that ~r, = id. We have to show that for all cr E Hr(N, A) there exists an open subgroup U such that ~ ,a = a for all ~r E U. But by shifting dimensions, there exist exact sequences and coboundaries f l , . . . 6~ such that

a = f l , . . . ,f,-a0 with s0 E H ~ for some B E Galm(G).

One merely uses the erasing functor r times. We have

and we apply the result in dimension 0, which is clear in this case since a, denotes the continuous operation of cr E S'.

Since the functor A ~ A N transforms an injective module to an injective module, one obtains the spectral sequence of the composite of derived functors. The explicit computations for the restriction, inflation and the edge homomorphisms remain valid in the present case.

(g) Sylow s u b g r o u p s . As a further application of the fact that the cohomology of Galois type groups is a limit of cohomology of finite groups, we find:

P r o p o s i t i o n 2.15. Let G be of Galois type, and A E Galm(G). Let S be a closed subgroup of G. If (G: S) is prime to a prime number p, then the restriction

res : Hr(G,A) H (S,A)

induces an injection on H~ ( G, A, p).

Proof. If S is an open subgroup V in G, then we have the transfer and restriction formula

t ro res(a) = (G: V)a,

138

which proves our assertion. The general case follows, taking into account that

H~(S,A)=dir lira H~(V,A)

for V open containing S.

w C o h o m o l o g i c a l d i m e n s i o n .

Let G be a group of Galois type. We denote by Galmtor(G) the abelian category whose objects are the objects A of Galm(G) which are torsion modules, i.e. for each a �9 A there is an integer n 7 ~ 0 such that na = O. Given A �9 Galm(G), we denote by Ator the submodule of torsion elements. Similarly for a prime p, we let Ap, denote the kernel of p~ in A, and Ap~ is the union of all Ap- for all positive integers n. We call Ap~ the submodule of p - p r i m a r y e l e m e n t s . As usual for an integer m, we let Am be the kernel of mA, SO

Ator = U Am and Ap~ = U Ap,

the first union being taken for m �9 Z, rn > 0 and the second for n>O.

The subcategory of elements A �9 Galm(G) such that A = Ap~ (i.e. A is p-primary) will be denoted by Galmp(G).

Let n be an integer > 0. We define the notion of cohomologica l d i m e n s i o n , abbreviated cd, and s t r i c t cohomolog ica l d i m e n s i o n , abbreviated scd, as follows.

c d ( a ) = < n if and and A

cdp(G) __< n if and and A

sod(G) =< n if and and A

scdp(G) __< n

only if H~(G, A) = 0 for all r > n E Galmtor(G)

only if H~(G,A,p) = 0 for all r > n �9 Galmtor (G)

only if H~(G, A) = 0 for all r > n �9 Galm(a)

if and only if H"(G,A,p) = 0 f o r r > n

and A E Galm(G).

We note that cohomological dimension is defined via torsion modules, and the strict cohomological dimension is defined by means of arbitrary modules (in Galm(G), of course).

VII .3 139

Since H ~ ( G , A ) = G H ~ ( G , A , p )

p

one sees that

cd(G) = sup cdp(G) and scd(G) = sup scdp(G). p p

For all A E Gaimtor(G) we have A = UAp~ , the direct sum being taken over all primes p. Hence

Hr(a,a)

To determine cdp(G), it will suffice to consider H"(G, Afo ), because if we let A' be the p-complementary module (p)

' U A(p) = Am with m prime to p,

then Alp ) is uniquely determined by pn for all integers n > 0,

so pn induces an automorphism of H~(G,A[p)) for r > 0, and

H ~ A' (G, (p)) is a torsion group. Hence Hr(G, A(p)) does not contain

any element whose torsion is a power of p, and we find:

P r o p o s i t i o n 3.1. Let A E Galmtor(G). Then the homomorphism

Hr(G,A, ~ ) ~ Hr(G,A,p)

induced by the inclusion Ap~ C A is an isomorphism for all r.

Corollary 3.2. In the definition of cdp( G), one can replace the condition A E Gaimtor(G) by A E Gaimp(G).

We are going to see that the strict dimension can differ only by I from the other dimension.

P r o p o s i t i o n 3.3. Let G be of GaIois type, and p prime. Then

cdp(G) _-< scdp(G) _-< cdp(G) + 1,

140

and the same inequalities hold omitting the indez p.

Proof. The first inequality is trivial. For the second, consider the exact sequence

0 ---* pA 2_~ A --, A /pA --* 0

O ---~ A p ---~ A J p A -* O

and the corresponding cohomology exact sequences

Hr+l(pA) i. Hr+I(A ) --+ H~+I(A/pA)

H~+l(dp)--- , H~+I(A) J'~ H~+l(d /pd) .

We assume that c@(G) < n and r > n. Since ij = p, we find i , j , = p,. We have g~+l(Ap) = 0 by definition, and also H~+I(A/pA) = 0. One then sees that j , is bijective and i, is surjective. Hence p, is surjective, i.e. H~(A) is divisible by p, and hence by an arbitrary power of p. The elements of H~(G, A, p) being p-primary, it follows that H ~+l (G, A, p) = 0. This proves the proposition.

For the next result, we need a lemma on the erasing functor M s .

L e m m a 3.4. Let G be of Galois type, and S a closed subgroup. Let B �9 Galmto,(S) (resp. Galmp(S)). Then M S ( B ) is in Galmto,(S) (resp. Galmp(a)). If, in addition B is finitely generated over Z, and S is open, then M S ( B ) is finitely generated over Z.

Proof. Immediate from the definitions.

P r o p o s i t i o n 3.5. Let S be a closed subgroup of H.

cd, __< cd,(G) and scdp(S) =< scdp(G),

and equality holds if ( G : S) is prime to p.

Then

Proof. By Theorem 2.11, we know that Hr(G, MS(B)) ~ H~(S,B) for all B C Galm(S). The assertions are then immediate consequences of the definitions, together with the fact that (G : S) prime to p implies that the restriction is an injection on the p-primary part of cohomology (Proposition 2.15).

As a special case, we find:

VII .3 141

C o r o l l a r y 3.6. Let Gp be a p-Sylow subgroup of G. Then

cdp(G) = cdp(Gp) = cd(Gp),

and similarly with scd instead of cd. Furthermore,

cd(G) = sup cd(G,) and scd(G) = sup scd(Gp). p p

We now study the cohomological dimension, and leave aside the strict dimension. First, we have a criterion in terms of a category of submodules, easily described.

P r o p o s i t i o n 3.7. We have cdp(G) <= n if and only if Hn+I(G, E) = 0 for all elements E E Galm(G) such that E is finite of p-power order, and simple as a G-module.

Pro@ Implication in one direction is trivial, taking into account that E is uniquely divisible by every integer rn prime to p, and therefore that

H'~+I(G,E) = H"+I(G,E,p).

Conversely, suppose H~+I(G,E) = 0 for all E as prescribed. Let A C Galmtor(G) have finite p-power order. If A # 0, then there is an exact sequence

0 ---+ A' ---+ A --+ A" --~ 0

with A' simple. The order of A" is strictly smaller then the order of A, and the exact cohomology sequence shows by induction that H"+I(G,A) = 0. Then let A E Galmp(G). Then A is a direct limit of finite submodules, and we can apply Corollary 2.8. It follows that H'~+I(G, A) = 0 for A E Galmp(G). Using the erasing functor MG, one can then proceed by induction, taldng into account the fact that MG maps Galmp(G) into Galmp(G), and one finds Hr(G,A) = 0 for r > n and A ~. Galm,(a). We can conclude the proof by applying Corollary 3.2.

L e m m a 3.8. Let G be a p-group of Galois type, and let A E Galmp(G). If A G = 0 then A = O. The only simple module A E Galmp(G) is Fp.

Proof. We already proved this lemma when G is finite, and the general case is an immediate consequence, because G acts continuously on A.

142

T h e o r e m 3 .9 . L e t G = Gp be a p-group of GaIois type. Then cd(G) <= n if and only if H'~+I(G,F, ) = 0.

Proof. This is immediate from Proposition 3.7 and the lemma.

T h e o r e m 3.10. Let G be a group of Galois type. The following condition~ are equivalent:

cd(G) = 0; scd(G) = 0; G - - e .

If G is a p-group, then cdp(G) = 0 implies G = e.

Proof. It will clearly suffice to prove that if cd(G) = 0 then G is trivial, so suppose cd(G) = 0. For every p-Sylow subgroup Gp of G, we have cd(Gp) = 0 (as one sees from the induced representation), and

cd (a , ) = cdp(ap).

Hence Ht(Gp, Fp) = 0. But Gp acts trivially on Fp so Ht(Gp, Fp) is just the group of continuous homomorpkism cont hom(Gp, Fp). If G r e, then there exists an open normal subgroup U such that G/U is a finite p-group, is equal to e. One could then construct a non-trivia/ homomorphism of G/U into Fp, contradicting the hypothesis, and concluding the proof.

To show that certain cohomology groups are not 0 in certain dimensions greater than some integer, we have the following criterion.

L e m m a 3.11. Let G be a p-group of Galois type and cd(a ) = n < co. / f E E Ga/mp(G) has finite order and E # O, then H'~(G, E) # O.

Proof. By Lemma 3.8, there is an exact sequence

0 ~ E' --* E ~ E I' --~ 0

with a maxima/ submodute E ' of E. Since Hn(G, Fp) • 0 by hypothesis, one has, again by hypothesis, the exact sequence

Hn(G,E) ~ H~'(G, F p ) ~ H'~+I(G,E ') = O,

which shows that H'~(G, E) cannot be trivia/.

As an application, we give a refinement of Proposition 3.5.

VII.4 143

Proposition 3.12. Let G be of Galois type, and let S be a cloned subgroup of G. If ordT(G. S) is finite and cdT(G) < 0% then cdT(S) = c@(G).

Proof. Let S T be a p-Sylow subgroup of S, and similarly G T a p-Sylow subgroup of G containing S T. Then

OrdT(Gp : ST)+ OrdT(G : GT)= ordT(G �9 Sp)

= ordp(G" S) + OrdT(S" S T)

Hence ordp(G T �9 ST) = ordT(G �9 S). This reduces the proof to the case when G is a p-group, and S is open in G. Suppose

n = cd(G) < co.

Then by Lemma 3.11,

H " ( S , F , ) = Hn(G, Mg(Fp)) # O,

because MS(Fp) has p(a:s) elements. This concludes the proof.

Corollary 3.13. If 0 < ordp(a �9 e) < 0% then cdp(a) = co. In fact, if G is a finite p-group, then Hr(G, Fp) # 0 for all r > 0 .

From this corollary, one sees the cohomological dimension is in- teresting only for infinite groups. We shall give below examples of Galois groups with finite cohomological dimension.

w Cohomological dimension __< 1.

Let us first remark that if G is a group of Galois type with scdp(G) =< 1, then scdp(G) = 0 and hence every p-Sylow subgroup Gp of G is trivial. Indeed, we have by hypothesis

0 = H2(Gp,Z) ~ HI(Gp,Q/Z) = cont hom(Gp, Q/Z)

from the exact sequence with Z, Q and Q/Z. That Q is uniquely divisible by every integer # 0 implies that its cohomology is 0 in dimensions > O. At the end of the preceding section, we saw that if Gp # e then we can find a non-trivial continuous homomorphism

1 4 4

of Gp into Fp, which can be naturally imbedded in Q /Z , and one sees therefore that Gp = e, thus proving our assertion.

We then consider the condition cdp(G) = 1. We shall see that this condition characterizes certain topologically free groups.

We define a group of Galois type to be p -ex t ens ive if and only if for every finite group F and each abelian p-subgroup E normal in F, and every continuous homomorphism f : G ~ F /E , there exists a continuous homomorphism f : G ---* F which makes the following diagram commutative:

G

F

�9 E / E f

Proposition 4.1. We have cdp(G) __< 1 if and only if G is p-extensive.

Proof. Suppose first that cdp(G) __< 1. We are given F, E, f as above. As usual, we may consider E as an F/E-module, the operation being that of conjugation. Consequently, E is in Galm(G) via f , namely for a C G and x C E we define

ax = f (a )x .

For each a C F/E , let u~ be a representative in F. Put

- - 1 C a , r = ? . t a U ~ - ~ o . r .

Then (ea,~-)is a 2-cocycle in C2(F/E,E) , and consequently (cf(a),i(,-)) is a 2-cocycle in C2(G, E). By hypothesis, there exists a continuous map a ~-~ a~ of G in E such that

e f ( a ) , f ( r ) = a~,r/aaaar.

We define ](a) by

f (a ) = a~u1(~ ).

V I I . 4 145

From the definit ion of the action of G on E, we have

- 1 o'a : uf(~)auf(~).

Thus we find

f ( o ' ) f ( a ) = aau f ( z )aru fo . ) = a~a~uf(z )u i (r ) r

= a~,a~ ef(~),f(~-)ui(~. )

: ao.ruf(o.r)

= f(~77"),

which shows tha t fi is a homomorphism. It is continuous because ( a~) i s a continuous cochain, and cr ~ f ( a ) is continuous. Further- more, it is clear that f is a lifting of f , i.e. that the diagram as in the definit ion of p-extensive is commutat ive.

Conversely, let E C Galmtor(G) be of finite order, equal to a p- power, and let a E H2(G, E). We have to prove that a = 0. Since E is finite, there exists an open normal subgroup U such that U leaves E fixed, i.e. E = E v, and E is therefore a G/U-module . Taking a smaller open subgroup of U if necessary, we can suppose wi thout loss of generality that a comes from the inflation of an e lement in H 2 ( G / U , E ) , i.e. there exists a0 E H 2 ( G / U , E ) such

�9 ~G/UI tha t a = m I c <a0). Let F be the group extension of G / U by E corresponding to the class of a0, so that we have G / U = F / E , and let

f: G-~F/E

be the corresponding homomorphism. We are then in the same si tuat ion as in the first part of the proof, and (ef(~),f(~)) is a 2- cocycle represent ing a. Since f now exists by hypothesis, we define a~ : f ( ~ ) u ~ ) . The same computa t ion as before shows that

o" a~,~- = a~a~.ef(~),f(~.),

and since (aa) is clearly a continuous cochain, one sees that (ef( ,) , f0-)) is a coboundary, in other words a = 0. This concludes the proof�9

R e m a r k . In the definition of p-extensive, wi thout loss of generality, we may assume that f is surjective (it suffices to replace F

146

by the inverse image of f (G) in F/E) . However, we cannot require that f is surjective. For instance, let G be the Galois group of the separable closure of a field k. Then F / E is the Galois group of a finite extension K / k , and the problem of finding f surjective amounts to finding a finite Galois extension L D K D k such that F is its Galois group, a problem considered for example by Iwasawa, Annls of Math. 1953.

We shall now extend the extension property to the situation when we can take F, E to be of Galois type.

P r o p o s i t i o n 4.2. Let G be of Galois type and p-extensive. Then the p-extension property concerning (G, f , F, F / E) is valid when F is of Galois type (rather than finite), and E is a closed normal p-subgroup.

Proof. We suppose first that E is finite abelian normal in F. There exists an open normal subgroup U such that U O E = e. Let f l : G ~ F / E U be the composite of f : G --+ F / E with the canonical homomorphism F / E ~ F /EU. We can lift f l to a continuous homomorphism f l : G ~ F /U by p-extensivity for F1 = F/U and El = E U / ( U N E), and f~. We have a homomorphism

(f, f l ) : G --~ (F /E) x (FLU),

and the canonical map i : U A E = e. The image of of i because f and f l lift extension problem in the

F ~ (F /E) x ( F / U ) is an injection since G under (f, f l ) is contained in the image 5 . Hence f = (f, f l ) : G ---* F solves the present case.

We can now deal with the general case. We want to lift f : G --~ F /E . We consider all pairs (E ' , f ' ) where E ' is a closed subgroup of E normal in F, and fr : G ~ F I E ~ lifts f . By Zorn's lemma, there is a maximal pair, which we denote also by (E, f ) . We have to show that E = e. I r E # e, then there exists a non trivial element 8 E HI(E , Fp). This character vanishes on an open subgroup V, and has therefore only a finite number of conjugates by elements of F, i.e. it is a Galois module of F / E . Let E1 be the intersection of the kernel of 8 and all its conjugates. Then E1 is a closed subgroup of E, normal in F, and by the first part of the proof we can lift f to fl : G ~ F/E1, which contradicts the hypothesis that (E, f ) is maximal, and concludes the proof of Proposition 4.2.

VII.4 147

Next, we connect cohomological dimension with free groups. We fix a prime p.

Let X be a set and Fo(X) the free group generated by X in the ordinary meaning of the word (d. Algebra, Chapter I, w We consider the family of normal subgroups U C X such that:

(i) U contains all but a finite number of elements of X.

(ii) U has index a power of p in Fo(X).

We let Fp(X) be the inverse limit

Fp(X) = inv lira Fo/U

taken over all such subgroups U. We call Fp(X) the p rof in i t e free p - g r o u p g e n e r a t e d by X. Thus Fp(X) is a group of Galois type.

Let G be a group of Galois type and G o the intersection of all the kernels of continuous homomorphisms ~ : G ---. Fp, i.e.

E H](G, Fp). Then HI(G, Fv) is the character group of G/G ~ The converse is also true by Pontrjagin duality.

By definition, if P is a finite p-group, then the continuous homomorphisms f : Fp(X) --~ P are in bijection with the maps f0 : X ---* P Such that fo(x) = e for all but a finite number of x e X. Hence gl(Fp(X),Fp) is a vector space over Fp, of finite dimension equal to the cardinality of X, and having a basis which can be identified with the elements of X.

Furthermore, we see that Fp(X) is p-extensive, and that cd Fp(Z) =<_ 1. Indeed, we can take f surjective in the definition of p-extensive, and F is then a finite p-group. One uses the freeness of F0 to see immediately that Fp(X) is p-extensive. We shM1 prove the converse.

L e m m a 4.3. Let G be a p-group of Galois type, S a closed subgroup. Then SG ~ = G implie~ S = G.

Proof. This is essentially an analogue of Nakayama's lemma in commutative algebra. Actually, one can prove the lemma first for finite groups, and then extend it to the infinite case, to be left to the reader.

148

T h e o r e m 4.4. Let G be a p-group of Galois type. Then there exists a proj~nite #ee p-group, Fp(X) and a continuous homomorphism

O : Fp(X) --+ G

such that the induced homomorphism

Hi(G, Fp)---, HI(Fp(X) ,F , )

is an isomorphism. The map 0 is then surjective. If cd(G) __< 1, then ~ is an isomorphism.

Proof. From the preceding discussion, to obtain an isomorphism

HI(G, Fp) --~ HI(Fp(X),Fp)

it suffices to take for X a basis of HI(G, Fp) and to form Fp(X). By duality, we obtain an isomorphism

F (X)/Fp(X) ~ a / a ~

whence a homomorphism

g: Fp(x) a / a ~

Since Fp(X) is p-extensive, tative diagram

we can lift g to G, to get the commu-

G

G ( x ) _ -~ g

G/G ~

and 9 is surjective by the lemma. Suppose finally that cd(G) __< 1, and let N be the kernel of g. Then we obtain an exact sequence

O---~HI(G,Fp) inf)Hl(Fp(X),Fp) ~')HI(N,Fp)G--+H2(G,Fp)=O.

We have H2(G, Fp) = 0 by the assumption cd(G) __< 1. The inflation is an isomorphism, and hence Hi(N, Fp) a = 0. By Lemma 3.8, we find HI(N, Fp) = 0, i.e. N has only the trivial character, whence N = e, thus proving the theorem.

VII.5 149

Corollary 4.5. Let G be a p-group of Galois type. Then the following conditions are equivalent:

G is profinite free;

G is p-extensive;

cd(G) __< 1.

We end this section with a discussion of the condition cd(G) =< 1 for factor groups. Let G be of Galois type. Let T be the intersection of all subgroups of G which are the kernels of continuous homomorphisms of G into p-groups of Galois type. Then G/T is a p-group which we denote by G(p), and which we call the m a x i m a l p - q u o t i e n t of G. One can also characterize T by the condition that it is a closed normal subgroup satisfying:

(a) (G" T ) i s a p-power.

(b) H I ( T , F ; ) = 0.

The characterization is immediate.

Propos i t ion 4.6. Let G be a group of Galois type. Then cdp(a) < 1 implies cdpG(p) < 1.

Proof. Consider the exact sequence

O--'+H 1 (G/T , Fv) --"+H 1 (G,Fv) ---+H 1 (T, Fv)G/T ___~H2(G/T,Fv) --"+0,

with a 0 on the right because of the assumption cdp(G) =< 1. By the characterization of T we have HI(T, Fp) = 0, whence H2(G/T, Fp) = 0 which suffices to prove the proposition by Theo- rem 3.9.

In the Galois theory, G(p) is the Galois group of the maximal p-extension of the ground field, and G is the Galois group of the algebraic closure. Cf. w below for applications to this context.

w The tower theorem

In many cases, one gets information on a group G be considering a normal subgroup N and the factor group G/N. We do this for eohomological dimension, and we shall find

ca(c) __< ca(N) + ca(a/x),

150

and similar with cdp instead of cd. We use the spectral sequence with

E2'~=H~(G/N, HS(N,A)) converging to H(G,A)

for A E Galm(G). There is a filtration of H n ( a , A ) such that the successive quotients are isomorphic to F, ~'' for r + s = n, and

~ ' ~ Hence H'~(G, A) = 0 _~F, ~'s is a subgroup of a factor group of ~2 �9 whenever H"(G/N,H~(N,A)) = 0, which occurs in the following cases:

r > cd(G/N) and

r > sca(G/N) and

s > cd(N) and

s > sea(N) and

s > 0 or A E Galmtor(G);

s arbitrary;

A E GaAmtor(G);

A E Galm(G).

From this we find the theorem:

T h e o r e m 5 .1 . Let G be of Galois type and N a closed normal subgroup. Then for all primes p,

cdp(G) =< cdp(G/N) + c@(N),

and similarly with cd instead of cdp.

As an application, suppose that G/N is topologically cyclic, and c@(N) =< 1. Then cdp(G) __< 2. This happens in the following cases: G is the Galois group of the algebraic closure of a totally imaginary number field, or a p-adic field. Indeed, in each case, one can construct a cyclic extension (maximM unramified in the local case, cyctotomic in the global case), which decomposes G into a subgroup N and factor G/N as above. In the next sections, we shall give a criterion with the Brauer group to show that cd(N) =< 1 for suitable N.

w G a l o i s g r o u p s o v e r a f i e ld

Let k be a field and k~ its separable closure. Let

ak = Gal(zc / )

VII.6 151

be the Galois group. If K is a Galois extension of k, we let GK/k be its Galois group. Then GK is normal in Gk and the factor group G k / G K is GK/k. All these groups are of Galois type, with the Krull topology.

We shall use constantly Hilbert's Theorem 90, that for the multiplicative group K*, we have

H I ( G h - / k , K *) = O.

Note that K* C Galm(GK/k). In the additive case, with the additive group K +,

H " ( G K / k , K + ) = 0 for all r > 0 .

One sees this reduction to the case when K is finite Galois over k, so there is a normal basis showing that K + is semilocal with local group reduced to e, whence the cohomology is trivial in dimension > 0 .

Next, we give a result in characteristic p.

T h e o r e m 6.1. Let k have characteristic p > 0 and let k(p) be the maximal p-extension with Galois group G(p) over k. Then cd G(p) <= 1, and so G(p) is profinite free. The number of generators is equal to d imF,(k+/pk+) , where px = x p - x.

Proof. We recall the Kummer theory exact sequence

0 --, G --* C A k~+ --, 0,

whence the cohomology sequence

0 ---+ Fp ---* k + ~ k + ~ H I ( G k , F p ) --+ H l ( G k , k +) = O.

Consequently,

k+/fak + ~ H 1 (Gk, Fp) = cont hom(Gk, Fp).

Since k(p) is the maximal Galois p-extension of k, it has no Galois extension of p-power degree, and hence we have an exact sequence

0 -- , G --* k(p) + --* k(p) + --' 0.

152

By the remarks made at the beginning of this section, we get from the exact cohomology sequence that H2(a(p), Fp) = o, and hence by the criterion of Theorem 3.9 that cd G(p) _<_ 1. Moreover, the beginning of this same exact sequence yields

k+ 2+ k+ ~ HI(G(p),F,)___+ ff l(G(p),]c(p)+)= O,

whence an isomorphism

k+/pk + ,~ HI(G(p),Fp)),

which gives us the desired number of generators.

We now go to characteristic r p using the multiplicative Kum- met sequence instead of the additive one. Cohomological dimension will be studied via Galois cohomology in k~*.

T h e o r e m 6.2. Let k be a field of characteristic r p and containing a p-th root of unity. Let k(p) be the mazimal p-eztension, with Galois group G(p). Then cd(G(p)) <= n if and only if:

(i) Hn(G(p),k(p) *) is divisible by p.

(ii) H'~+l(G(p),k(p) *) = O.

Pro@ We consider the exact sequence

0 ~ Fp --* k(p)* ~ k(p)* ~ 0,

where Fp gets embedded on the group of p-th roots of unity, and the map on the right is taking p-th powers. We obtain the cohomology exact sequence

H'~( k(p)')---+ H"( k(p)*)--+ H'~+t(Fp)--+ H'~+l( k(p)*)--+ H'~+t( k(p) *)

with acting group G(p). The theorem follows at once from this exact sequence.

C o r o l l a r y 6.3. ( K a w a d a ) The GaIois group G(p) = Gk(p)/k is a profinite p-group if p = char k. If p 7~ char k and the p-th roots of unity are in k, then it is profinite free if and only if H2(G(p),k(p)) = O.

Proof. One always has Hl(k(p) *) = 0, and the rest follows from the preceding two theorems.

Theorem 6.1 can be translated in terms of cohomology with values in ]c*

8 "

VII.6 153

T h e o r e m 6.4. Let k be a field and p prime ~ char k. Let n be an integer > O. Then cdv(Gk ) <__ n if and only if:

(i) Hn(CE,~; )= 0 is divisible by p

(ii) H'~+I(GE, k*,p) = 0

for all finite separable extensions E of k of degree prime to p.

Proof. The Kummer sequence

0 ---+ Fp --+ k2 v, k: ---, 0

yields the cohomology sequence with groups Gk:

H'~(k:)--~H'~(k:)---..+H"+t(Fp)---+H'*+I(k:)-~H'~+I(k~)---+H'*+2(Fv).

Suppose cdv(Gk ) _< n. Then Hn+l(Fp) = H'~+2(Fv) = 0 by Proposition 3.7. Conditions (i) and (ii) are then clear, taking Proposition 3.5 into account. The converse can be proved in a similar way, from the fact that if G v is a p-Sylow subgroup of Gk, then

* r Hr(Gv,k : ) = inv h m H (GE, k~),

the limit being taken over all finite separable extensions E of k of degree prime to p. The Galois groups GE constitute precisely the set of open subgroups of Gk containing Gp, or its conjugates, which amounts to the same thing.

C o r o l l a r y 6.5. If H2(GE, k*) = 0 for all finite separable ez- te,~sions E of ~, then cd(ak) __< 1.

Proof. We have HI(Ge, k2) = 0 automatically, and we apply the theorem for the p-component when p ~: char k. If p = char k, then we saw in Theorem 6.1 that the cohomological dimension is __< 1.

The above corollary provides the announced criterion in terms of the Brauer group, because that is what H 2 k* ( ~ ) amounts to.

T h e o r e m 6.6. Let K be an extension of k. Then

cdp(a~-) <__ tr deg~'/~ +cdp(ak)

154

(By definition, tr deg is the transcendence degree.)

Proof. If in a tower K D K1 D k the assertion is true for K/K1 and for K1/k, then it is true for K/k . We are therefore reduced to the cases when ei ther K / k is algebraic, in which case GK is a closed subgroup of Gk, and the assertion is trivial; or when K is a pure t ranscendenta l extension K = k(x), in which case we have a field d iagram as follows.

T T k , k ~

G~

By Tsen 's theorem and the corollary of Theorem 6.3, we know that cd(Gk,(z)) ~ 1. The tower theorem shows that

cd(Gk(.)) =< cd(ak) + 1,

thus proving the theorem.

T h e o r e m 6.7. In the preceding theorem, there is equality if cdp(Gk) < oo (p r char k) and K is finitely generated over k.

Proof. The assertion is aga in transit ive in towers, and we are reduced ei ther to the case of a finite algebraic extension, when we can apply Proposi t ion 3.12, or to K = k(x) purely transcendental . For this last case, we need a lemma.

L e m m a 6.8. Let G be of Galois type, T a closed normal subgroup such that cdp(T) =< 1. If cdp(G/T) <= n, then there is an isomorphism

H'~+I(G,A) ~ Hn(G/T, HI(T ,A))

for all A E Galmtor(G).

Proof. We have Hr(T, A) = 0 for r > 1 and the spectral sequence becomes an exact sequence

0 ~ Hn+I(G/T ,A T) ---* Hn+I(G,A) ~ H'~(G/T, HI(T ,A))

--~ Hn+2(G/T, A T) --+ O,

v i i . 6 155

whence the lemma follows.

Coming back to the theorem, put

G =Gk( , ) and T = Gk(,),lk,(,).

We refer to the diagram for Theorem 6.6. We may replace k be its extension corresponding to a Sylow subgroup of Gk, that is we may suppose that Gk is a p-group. We have Gk = G/T. Let us now take A = Fp in the lemma. Suppose

n = cdp(G/T) < oc.

We must show that Hn+i(G, A) # 0. By the lemma, this amounts to showing that Hn(G/T, HI(T, Fp)) # 0. Since the p-th roots of unity are in k (p # char k), Kummer theory shows that

HI(T, Fp) = cont holn(T, Fp)

is G/T-isomorphic to ks(z)*/ks(x) *p. The unique factorization in ks(x) shows that this group contains a subgroup G-isomorphic to Fp. On the other hand, this group is a direct sum of its orbits under G/T, and one of these orbits is Fp. Hence H n+l (G, Fp) # 0 as was to be shown.

The theorem we have just proved, and which occurs here at the end of the theory, historically arose at its beginning. Its conjecture and the sketch of its proof are due to Grothendieck.

C H A P T E R V I I I Group Extensions

w Morphisms of extensions

Let G be a group and A an abelian group, bo th wr i t ten multiplicatively. An extension of A by G is an exact sequence of groups

We can then define an action of G on A. If we identify A as a subgroup of U, then U acts on A by conjugation. Since A is assumed commutat ive , it follows that elements of A act trivially, so U / A = G acts on A.

For each cr E G and a E A we select a.,1 element ua E U such tha t j u , , = or, and we put

Each e lement of U can be wri t ten uniquely in the form

u = au,7 with a E G and a E A.

T h e n there exist elements a~,,- E A such tha t

UO.U r ~ a t r , T U o - r ,

and (a.,,-) is a 2-cocycle of G in A. A different choice of u,, would give rise to another cocycle, differing from the first one by

VIII. 1 157

a coboundary. Hence the cohomology class a of these cocycles is a well defined element of H2(G, A), determined by the extension, i.e. by the exact sequence.

Conversely, suppose given an element a 6 H2(G,A) with G given, and A abelian in :VIod(G). Let (a~,~) be a cocycle representing a. We can then define an extension of A by G as follows. We let U be the set of pairs (a, or) with a 6 A and cr E G. ~vVe define multiplication in U by

(a, cr)( b, r) = (aaba~,,-, or).

One verifies that U is a group, whose unit element is -1 (a,,~, e). The existence of the inverse of (a, #) is determined at once from the definition of multiplication. Defining j (a, ~) = ~r gives a homomorphism of U onto G, whose kernel is isomorphic to A, under the correspondence

a ~---* (aa;~,e).

Thus we get a group extension of A by G.

Extensions of groups form a category, the morphisms being triplets of homomorphisms (f, F, ~) which make the following diagram commutative:

0 > A ) U > G ) 0

0 ) B , V , H , 0

We have the general notion of isomorphism in this category, but we look at the restricted notion of extensions U, U' of A by G (so the same A and G). Two such extensions will be said to be i s o m o r p h i c if there exists an isomorphism F : U ~ U' making the following diagram commutative:

A - ~ U , G

d I A > U ' ) G

Isomorphism classes of extensions thus form a category, the morphisms being given by isomorphisms F as above.

158

Let (G, A) be a pair consisting of a group G and a G-module A. We denote by E(G, A) the isomorphism classes of extensions of A by G. For G fixed, A ~ E(G, A) is a functor on Mod(a). We may summarize the discussion

T h e o r e m 1.1. On the category Mod(G), the functors H2(G,A) and E ( G , A) are isomorphic, by the bijection estab- lished at the beginning of the section.

Next, we state a general result providing the existence of the homomorphism F when pairs of homomorphisms (r f ) are given, from a pair (G, A) to a pair (G', A').

T h e o r e m 1.2. Let G ~ U/A and G' ..~ U' /A' be two extensions. Suppose given two homomorphisms

~2 : G----~ G' and f " A ---+ A'.

There exists a homomorphism F" U ~ U' making the diagram commutative:

A i j , U , G

A' , U' ) G' i' j '

if and only if:

(1) f is a G-homomorphism, with G acting on A ~ via T.

(2) f . a = ~*a, where a , a ~ are the cohomology classes of the two extensons respectively, and f . , ~* are the morphisms induced by the morphisms of pairs

(id, f ) : (G,A) --+ (G,A') and (c2,id)" (G' ,A') ~ (G,A') .

Proof. We begin by showing that the conditions are necessary. Without loss of generality, we let i be an inclusion. Let (u~) and (u~,) be representatives of a and e~ respectively in G and G'. For u = au~ in U we find

F(u) = F ( a u ~ ) = F ( a ) F ( u ~ ) = f (a)F(u~) .

VIII . 1 159

One sees that F is uniquely determined by the data F(u~). h a v e

I j F ( u ~ ) = r = ~o" = j'u~,,.

Hence there exist elements c~ E A' such that

We

: Ccr t~tocr.

It follows that F is uniquely determined by the data (c~), which is a cochain of G in A'. Since F is a homomorphism, we must have

F(u( au- 1) = F ( u o . ) f ( a ) f ( u o . ) -1

These conditions imply:

(I) f ( o a ) = ~ f l a ( f a ) for a E A and o ' E G .

( i i ) fa., =

for the cocycles (a~,~-) and (b~,,~,) associated to the representatives (u~) and (u ' ) . By definition, these two conditions express precisely the conditions (1) and (2) of the theorem. Conversely, one verifies that these conditions are sufficient by defining

F(au~) = f ( a ) c ~ u ~ .

This concludes the proof.

We also want to describe more precisely the possible F in an isomorphism class of extensions of A by G. We work more generally with the situation of Theorem 1.2. Let f, ~ be fixed and let

F1, F2 " U ~ U'

be homomorphisms which make the diagram of Theorem 1.2 commutative. We say that F1 is equ iva len t to F2 if they differ by an inner automorphism of U' coming from an element of A', that is there exists a' E A' such that

Fl(u) = a'F2(u)a '-1 for al l u E U.

This equivalence is the weakest one can hope for.

160

T h e o r e m 1.3. Let f , ~ be given as in Theorem 1.2. Then the equivalence classes of homomorphisms F as in this theorem form a principal homogeneous space of H I ( G , A ' ) . The action of H I ( G , A ') on this space is defined as follows. Let (u~) be representatives of G in U, and (z~) a 1-cocycIe of G in A'. Then

(zF)(aua) = f (a)zaF(ua) .

Proof. The straightforward proof is left to the reader.

Corol lary 1.4. If H I ( G , A r) = O, then two homomorphisms F1, F2 : U ---* U I which make the diagram of Theorem 1.2 commutative are equivalent.

w C o m m u t a t o r s and transfer in an extens ion .

Let G be a finite group and A E Mod(G). We shall write A multiplicatively, and so we replace the trace by the norm N = NG. We consider an extension of A by G,

O - - - , A J - ~ E J G - - ~ O ,

and we suppose without loss of generality that i is an inclusion. We fix a family of representatives (u , ) of G in E, giving rise to the cocycle (a~,,-) as in the preceding section. Its class is denoted by a. We let E r be the commutator subgroup of E. The notations will remain fixed throughout this section.

Propos i t i on 2.1. The image of the transfer

Tr : E / E ~ ~ A

is contained in A G, and one has: (1) Tr (aE c) = 11 uaau-~ 1 = N a ( a ) for a E A.

erEG

(2) Wr(u~ Ec) = 11 u~u~u~-I = YI a~,~ (the Nakayama map). aEG aEG

Proof. These formulas are immediate consequences of the definition of the transfer.

VIII.2 161

P r o p o s i t i o n 2.2. One ha~ I c A C E ~ @ A C AN. For the cup product relative to the pairing Z x A ---* A, we have

a U H-a(G, Z) = ma((E ~ @ A)/IGA).

Proof. We have at once aa/a = uaau j l a -1 6 E ~ @ A. The other stated inclusion can be seen from the fact that tr is trivial on E r and applying Proposition 2.1. Now for the statement about the cup product, recall that a subgroup of an abelian group is determined by the group of characters f : A ---+ Q / Z vanishing on the subgroup. A character f : A ---+ Q / Z vanishes on E ~ @ A if and only if we can extend f to a character of E/E% because

A/(EC@A) c E / E c.

The extension of a character can be formulated in terms of a commutative diagram such as those we considered previously, and of the existence of a map F, namely:

A ) E ) G

Q / z , Q / z , 0

The existence of F is equivalent to the conditions:

(a) f is a G-homomorphism. (b) = 0.

From the definition of the cup product, we have a commutative diagram: H-3(Z) x H2(Q/Z) ---+ H-I(Q/Z)=(Q/Z)N

idl ft ~f* H-3(Z) x I-I2(a) --~ H-I(A)=AN/1GA.

The duality theorem asserts that H-a (Z) is dual to H2(Q/Z) . In addition, the effect of f , on H - I ( A ) is induced by f on A N / I a A .

Suppose that f is a character of A vanishing on AN. Then

f , ( a U H-a (A) ) = 0, and so f , ( a ) U H-a (Z) = 0.

162

Since H - 3 ( Z ) is the character group of H2(Q/Z) , we conclude that f.(ol) = 0. The converse is proved in a similar way. This concludes the proof of Proposition 2.2.

In addition, Proposition 1.1 also gives:

T h e o r e m 2.3. Let

O ~ A ~ E ~ G ~ O

be an eztension, and a 6 H2(G,A) its cohomology class. Then the following diagram is commutative:

0 :A/E ~nA i E/EC

0 ' NA " A G

= G/G c = H - 2 ( G , Z ) �9 0

" H~ A) ' 0

where N , i , j are the homomorphisms induced by the norm, the inclusion, and j respectively; and a-2 denotes the cup product with a on H-2(G, Z).

Proof. The left square is commutative because of the formula for the norm in Proposition 2.1(1). The transfer maps E / E c into A a by Proposition 2.1. The right square is commutative because the Nakayama map is an explicit determination of the cup product, and we can apply Proposition 2.1(2).

The next two corollaries are especially important in the application to class modules and class formations as in Chapter IX. They give conditions under which the transfer is an isomorphism.

C o r o l l a r y 2.4. Let E / A = G be an extension with corresponding cohomology class a 6 H2(G,A). With the three homomorphisms

Tr : E / E ~ ~ A a a - 3 : H - 3 ( G , Z ) ~ H - I ( G , A ) a - 2 : H-2(G, Z) ---* H~ A)

we get an exact sequence

0 --* H - I ( G , A ) / I m a-3 ~ Ker Tr ~ Ker a -2 --~ 0

viii.3 163

and an isomorphism

0 ~ A a / I m Tr --~ H ~ ~-2 ~ O.

Proof. Chasing around diagrams.

C o r o l l a r y 2.5. If ~-2 and ~-3 are isomorphism,, then the transfer on AG / N A is an isomorphism in Theorem 2.3.

The situation of Corollary 2.5 is realized for class modules or class formations in Chapter IX.

w T h e d e f l a t i o n

Let G be a group and A E Mod(G), written multiplicatively. Let EG be an extension of A by G. Let N be a normal subgroup of G and EN = j - I ( N ) , so we have two exact sequences:

O -.+ A --., E G J G ---, O

O---+ A--+ E N---+ N---+ O.

Then EN is an extension of A by N, and if ~ E H2(G, A) is the cohomology class of E a then res~(~) is the cohomology class of EN.

One sees that EN is normal in Ec , and in fact EG/EN ,,~ G/N. We obtain an exact sequence

0 --+ EN --+ Ea ~ G / N --* O.

Since EN is not necessaily commutative, we factor by E~v to get the exact sequence

0 ~ EN/E~N --* EG/E~N --* G / N ---* 0

giving an extension of E N / E ~ by G/N, called the f ac to r ex t en - s ion corresponding to the normal subgroup N of G. The group

164

lattice is as follows.

A

EG /

EN. / \ \ /

A N \

e

This factor extension corresponds to a cohomology class/3 in H2(G/N, EN/E~) . We can take the transfer

Tr: EN/ECN ~ A N ,

which is a G/N-homomorphism, the operation of G/ N on E N / E ~ being compatible with that of E~/E~ . Consequently, there is a induced homomorphism

Tr," H~(G/N ,E N /E~N) ---+ H~(G/N, AN).

The image of Tr,(/~) depends only on a. Hence we get a map

def" H2(G,A) --* H2(G/N ,A N) such that a ~ Tr,(/~).

We call this map the def la t ion . It may not be a homomorphism, but we shall see that for G finite, it is. First:

T h e o r e m 3.1. Let S be a subgroup of a finite group G. Fix right coset representatives of S in G, and for a 6 G let ~ be the representative of Sa. Let A 6 Mod(G) and let (aa,T) be a 2-cocycle of G in A. Let EG be the extension of A by S obtained from the restriction of this cocycle to S. Let (ua) be representatives of G in EG. Let 7(a, T) = G T a T 1. Then

Es -I TrA (ueueu~'v) = H aP,7" pES

Proof. This comes directly from the formulas of Theorem 2.1.

VIII.3 165

Corollary 3.2. Let G be a finite group, N normal in G. A E Mod(G). Then on H2(G,A) , we have

in f~c/Nodeg/N = ( N " e).

Let

Proof. One computes wi th the explicit formulas on cocycles. Note tha t the group A being wri t ten multiplicatively, the expression

(N �9 e) on the right is really the map a ~ a (N:*) for a E H2(G, A).

T h e o r e m 3.3. Let G be a finite group and N a normal subgroup. Then the deflation is a homomorphism. If a r H2(G, A) is represented by the cocycle (a~,~), then def(a) is represented by the cocycle

pEN pEN

H --1 a~ p j -ap ,aa p,-5-- ~.

pEN

As in Theorem 3.1, ~ denotes a fixed coset representative of the coset N a, and "), = ~a--Y -1 = 7(a, T).

Proof. The proof is done by an explicit computat ion, using the explicit formula for the t ransfer in Theorem 3.1. The fact tha t the deflation is a homomorph i sm is then apparent from the expression on the r ight side of the equality. One also sees from this r ight expression tha t the expression on the left is well defined. The details are left to the reader.

C H A P T E R IX Class F o r m a t i o n s

w Definitions

Let G be a group of Gedois type, with a fundamental system of open neighborhoods of e consisting of open subgroups of finite index U, V, . . . . Let A E Gedm(G) be a Galois module. We then say that the pair (G, A) is a class f o r m a t i o n if it satisfies the following two axioms:

C F 1. For each open subgroup V of G one has HI(V,A) = O.

Because of the inflation-restriction exact sequence in dimension 1, this axiom is equivalent to the condition that for all open subgroups U, V with U normal in V, we have

HI(V/U, AU)=o.

E x a m p l e . If k is a field and K is a Galois extension of k with Galois group G, then (G, K*) satisfies the axiom C F 1.

By C F 1, it follows that the inflation-restriction sequence is exact in dimension 2, and hence that the inflations

inf : H2(V/U,A U) ---+ H2(V,A)

are monomorphisms for V open, U open and normal in V. We may therefore consider H2(V,A) as the union of the subgroups

IX.1 167

H2(V/U, Au). It is by definition the B r a u e r g r o u p in the preceding example. The second axiom reads:

CF 2. For each open subgroup V of G we are given an embedding

i n v y : H2(V, A) ---+ Q / Z denoted a ~ invv(a) ,

called the i nva r i an t , satisfying two conditions:

(a) If U C V are open and U is normal in V, of index n in V, then invv maps H2(V/U,A U) onto the subgroup ( Q / Z ) , consisting of the elements of order n in Q/Z.

(b) If U C V are open subgroups with U of index n in V, then

invy o res V = n.invv.

We note that if (G : e) is divisible by every positive integer m, then inva maps H2(G,A) onto Q/Z , i.e.

invG: H2(G,A) ---+ Q / Z

is an isomorphism. This is the case in both local and global class field theory over number fields:

In the local case, A is the multiplicative group of the algebraic closure of a p-adic field, and G is the Galois group.

In the global case, A is the direct limit of the groups of idele classes. On the other hand, if G is finite, then of course inva maps H2(G,A) only on ( Q / Z ) , , with n = (G: e).

Let G be finite, and (G, A) a class formation. Then A is a class module. But for a class formation, we are given an additional structure, namely the specific fundamental elements a C H 2 (G, A) whose invariant is 1/n (rood Z).

Let (G, A) be a class formation and U C V open subgroups with U normal in Y. The element a e H2(V/U,A U) C H2(V,A), whose V-invariant invy((~) is 1/(V : U), will be called t h e f u n d a m e n t a l class of H2(V/U, Au), or by abuse of language, of V/U.

168

P r o p o s i t i o n 1.1. Let U C V C W be three open subgroups of G, with U normal in W. If a is the fundamental class of W/U then resW(a) is the fundamental class of V/U.

Proof. This is immedia te from C F 2(b) .

C o r o l l a r y 1.2. Let (G,A) be a class formation.

(i) Let V be an open subgroup of G. Then (If, A) is a class formation, and the restriction

res : H2(G,A) --+ H2(V,A) is surjective.

(ii) Let N be a closed normal subgroup. Then (G /N ,A N) is a class formation, if we define the invariant of an element in H 2 ( V N / N , A N) to be the invariant of its inflation in H2(VN, A).

Proof. Immediate .

P r o p o s i t i o n 1.3. Let (G, A) be a class formation and let V be an open subgroup of G. Then:

(i) The transfer preserves invariants, that is for a E H2(V, A) we have

invG t rY(a ) = invv(a ) .

(ii) Conjugation preserves invariants, that is for a 6 H2(V, A) w e have

invv[a] (a , a ) = i n v v ( a )

Proof. Since the restr ict ion is surjective, the first assert ion follows at once from C F 2 and the formula

tr o r e s = ( G ' V ) .

As for the second, we recall tha t or. is the ident i ty on H2(G, A). Hence

invv[#] o #, o res G = invv[#lresG[# l o ~,

= (G" V[a]) inva o ~r,

= ( G : V) inva

= invyres~y .

IX.1 169

Since the restriction is surjective, the proposition follows.

T h e o r e m 1.4. Let G be a finite group and (G,A) a class formation. Let a be the fundamental element of H2(G,A) . Then the cup product

a~: H~(G, Z) ---. H~+2(G, A)

is an isomorphism for all r 6 Z.

Proof. For each subgroup G' of G let a ' be the retriction to ' the cup product taken on the G'-cohomology. By G', and let a r

' satisfies the the triplets theorem, it will suffice to prove that a r hypotheses of this theorem in three successive dimensions, which we choose to be dimensions - 1 , 0, and +1.

For r = -1 , we have Hi(G, A) = 0 so a ' l is surjective.

For r = 0, we note that H~ has order (G' : e) which is the same order as H2(G ', A). We have trivially

=

which shows that a~ is an isomorphism.

For r = 1, we simply note that Hi (G, Z) = 0 since G is finite and the action on Z is trivial. This concludes the proof of the theorem.

Next we make explicit some commutativity relations for restriction, transfer, inflation and conjugation relative to the natural isomorphism of Hr(G, A) with Hr-2(G, Z), cupping with a.

P r o p o s i t i o n 1.5. Let G be a finite group and (G,A) a class formation, let a 6 H2(G,A) be a fundamental element and a' its restriction to G' for a subgroup G' of G. Then for each pair of vertical arrows pointing in the same direction, the following diagram is commutative.

w ( a , z)

res I tr

H~(a ', Z)

Ot r

H~+2(G, A)

res I tr

'~ Hr+2(G', A) o'r

170

Proof. This is just a special case of the general commutativity relations.

P r o p o s i t i o n 1.6. Let G be a finite group and (G,A) a class formation. Let U be normal in G. Let a E H2(G,A) be the fundamental element, and 6~ the fundamental element for G/U. Then the following diagram is commutative for r >= O.

H (G/U,Z ) e,, H +2(G/U, AU)

l l;~ H"(C, Z) ) Hr+2(G,A)

Proof. This is just a special case of the rule

inf(a U ~3 ) = inf(a) U inf(~).

We have to observe that we deal with the ordinary functor H in dimension r >= 0, differing from the special one only in dimension 0, because the inflation is defined only in this case. The left homomorphism is (U : e)inf for the inflation, given by the inclusion. Indeed,we have

( a : e) = ( a : U)(U: e),

so inf(~) = (U" e)a, and we can apply the above rule.

Finally, we consider some isomorphisms of class formations. Let (G,A) and (G',A') be class formations. An i s o m o r p h i s m

(A , f ) : (G ' ,A ' ) - -~ (G ,A)

consists of a pair isomorphism A : G --* G' and f �9 A' --~ A such that

inva(A, f ) . ( a ' ) = invv,(a ' ) for a' e H2(G',A').

From such an isomorphism, we obtain a commutative diagram for U normal in V (subgroups of G):

Hr (V/U,Z) ~" , Hr+2(V/U,A U)

H"(AV/AU, Z) ) H"+2(AV/AU, A '>'v)

IX.2 171

where a, a ' denote the fundamental elements in their respective H 2 .

Conjugation is a special case, made explicit in the next proposition.

P r o p o s i t i o n 1.7. Let (G, A) be a class formation, and U C V two open subgroups with U normal in V. Let r E G and the fundamental element in H2(V/U, AU). Then the following diagram is commutative.

H~(V/U,Z)

nr(V[,]/U[,l,Z)

Olr , Hr+~(V/U, A u)

l ~. , H~+2(V[~'I/U[r],AU[d).

w T h e r e c i p r o c i t y h o m o m o r p h i s m

We return to Theorem 8.7 of Chapter IV, but with the additional structure of the class formation. From that theorem, we know that if (G, A) is a class formation and G is finite, then G/G e is isomorphic to Aa/SGA = H~ The isomorphism can be realized in two ways. First, directly, and second by duMity. Here we start with the duality. We have a bilinear map

given by

A a x G. ~ H2(G,A)

(a, x) ~ ~(a) u 6x.

Following this with the invariant, we obtain a bilinear map

(a,x) ~ invv(~(a) U6x) of A a • G ~ Q /Z ,

whose kernel on the left is SaA and whose kernel on the right is trivial. Hence AG/SaA ~ G/G c, both groups being dual to G. We

172

recall the commuta t ive diagram:

H~ x H2(Z)

Uc~

H-2(Z)

1 H-2(Z)

1 G/G ~

• H2(Z)

6

x HI(Q/Z)

. H~(A)

1~ ~ , H~

~ - ~ ( q / z )

x 4 . ( q / z ) .

which we apply to the fundamenta l cocycle ~ E H2(G,A) , with n = ( G : e) and invc (~) = 1/n. We have

~(1) U a = c~ and inva (x (1 ) U c~) = invG(c~) = 1In.

Thus using the invariant from a class formation, at a finite level, we obta in the following fundamenta l result.

T h e o r e m 2.1. Let G be a finite group and (G, A) a class formation. For a E A c let ~ be the element of GIG c corresponding to a under the above isomorphism. Then for all characters X of G we have

x( o) = invc(, (a) u

An element a E G is equal to a~ if and only if for all characters X,

X(a) = inva(>c(a) U 8X).

The map a ~-* aa induces an isomorphism A G / S G A ,~ G / G ~.

The e lement aa in the theorem will also be denoted by (a, G).

Let now G be of Galois type and let (G, A) be a class formation. Then we have a bilinear map

H~ x H i ( G , Q / Z ) --~ H2(G,A) , i . e .A a x G ---, H2 (G ,A)

IX.2 173

with the ordinary functor H ~ by the formula

(a, X) ~ a U 6X,

where we identify a character X with the corresponding element of Hi(G, Q/Z) , and we identify H~ with A a. Since inflation commutes with the cup product, we see that if U is normal open in G, then the following diagram is commutative:

H~ A) • Hi(G, Q/Z) �9 H2(G, A)

inf inf inf

HO(G/U,A U) x HX(G/U,Q/Z) " H2(G/U, AU)

The inflation on the far left is simple the inclusion A U C A, and the inflation in the middle is that of characters.

In particular, to each element a C A c" we obtain a character of Hi(G, Q / Z ) given by

X ~ inva(a U 5X).

We consider Hi(G, Q / Z ) as a discrete group. Its character group is G/Gr according to Pontrjagin duality between discrete and compact groups, but G c now denotes the closure of the commutator group. Thus we obtain a homomorphism

reca : A a ---, G/G c

which we call the r e c i p r o c i t y h o m o m o r p h i s m , characterized by the property that for U open normal in G, and a E A c we have

recG/u(a) = (a, G/U) = (a, G/Gcu).

Similarly, we may replace U be any normal closed subgroup of G. This is called the c o n s i s t e n c y of the reciprocity mapping. As when G is finite, we denote

reca(a) = (a, C).

The next theorem is merely a formal summary of what precedes for finite factor groups, and the consistency.

174

T h e o r e m 2.2 . Let G be a group of Galois type and (G,A) a class formation. Then there exists a unique homomorphism

recG : A G --+ GIG c denoted a ~-* (a,G)

satisfying the property

inva(a U 5X) = x(a, G)

for all characters X of G.

Recall t ha t if ~ : G1 ---+ G2 is a g roup h o m o m o r p h i s m , t h e n induces a h o m o m o r p h i s m

This also holds for a con t inuous h o m o m o r p h i s m of groups of Galois type , where G c denotes the closure of the c o m m u t a t o r group.

T h e nex t t h e o r e m summar izes the fo rmal i sm of class f o r m a t i o n t heo ry and the rec iproc i ty mapp ing .

T h e o r e m 2 .3 . Let G be a group of Galois type and ( G , A ) a class formation.

(i) I f a C A a and S is a closed normal subgroup with factor group ~ : G ---* G/S , then recG/S = Ac o recG, that is

(a, G /T) = At(a, G).

(ii) Let V be an open subgroup of G. Then r ecv = Tray o r eco , that is for a E A Q,

(a, V) = Tray(a, G).

(iii) Again let V be an open subgroup of G and let ~ : V --* G be the inclusion. Then recG o S y = s o recy , that is for a C AV~

( S V ( a ) , G ) = ~C(a,V).

(iv) Let V be an open subgroup of G and a C A v. L e t 7 C G. Then

(Ta, V r ) = ( a , V ) T.

IX.2 175

These properties are called respectively cons i s t ency , t r a n s f e r , t r a n s l a t i o n , and c o n j u g a t i o n for the reciprocity mapping.

Proof. The consistency property is just the commutativity of inflation and cup product. We already used it when we defined the symbol (a, G) for G of Galois type. The other properties are proved by reducing them to the case when G is finite. For instance, let us consider (ii). To show that two elements of V / V c are equal, it suffices to prove that for every character X : V ~ Q / Z the values of X on these two elements are equal. To do this, there exists an open normal subgroup U of G with U C V such that x(U) = 0. Let G = G/U. Then the following diagram is commutative:

G/GC T, , V / V C x , Q / Z

1 1 l a l a ~ > V l V c , Q/Z

Tr X

the vertical maps being canonical. Furthermore, by consistency, w e h a v e

( a , G ) U = ( a , G / U ) = ( a , G ) .

This reduces the property to the finite case G.

But when G is finite, then we can also write

(a, C ) = o- < , ~<c(a) = ~,, U o~,

where ~ is the element of H-2(G, Z) ~ GIG c corresponding to a, and a is the fundamental class. The restriction r e s t ( a ) = a ' is the fundamental class of H2(V, A), and we know that

r u ~' = res~((~, U ~) .

Since res~(mG(a)) = ~v(a),

one sees that Tr(a, G) = (a, Y).

For (iii), note that the diagram is commutative,

X v / w , a / a ~ - �9 Q / z

1 l 1 X

V l W . a l a ~ , QZ

176

where as previously U C V is normal in G and G = G/U, V = V/U. This reduces the property to the case when G is finite. In this case, let

:,~ : v / v ~ __, a / a ~

be the homomorphism induced by inclusion. Let a be the fundamental class of (G,A) . Then r e s t ( a ) = a ' is the fundamental class of (V, A). The transfer and cup product are related by the formula

tr(~ u o~') = r u o,.

But the transfer amounts to the trace on H~ = A v, so the assertion is proved.

The fourth property is just a transport of structure for algebraically defined notions and relations.

We state one more property somewhat different from the others.

T h e o r e m 2.4. Limitat ion Theorem. Let G be of Galois type, V an open subgroup, and (G,A) a class formation. Then the image of SV(A V) by the reciprocity mapping reca is contained in VGr ~, and we have an isomorphism induced by recG, namely

r e c a : A a / S V A v ~-~ G / V G ~.

Proof. The first assertion is Property (iii) of Theorem 2.3. Con- versely, since V is open, we may assume without loss of generality that G is finite. In this case, there exists b E A u such that he(b,V) = (a,G). By this same Property (iii), this is equal to (S~(b), G). But we know that the kernel of reca is equal to SaA. Hence a and SV(b) are congruent mod SG(A). Since

Sa(A) C sV(A),

we have proved the theorem.

C o r o l l a r y 2.5. Let G be finite and (G,A) a class formation. Let a ' = a / a c and A' = A ~~ Then S~(A) = S~,(A') and reca,reca, are equal, their kernels being Sa(A).

Note that G t = G / G c can be written G ~b, and can be viewed as the maximal abelian quotient of G in Corollary 2.5. The corollary shows that the information in the reciprocity mapping is entirely concerned with this maximal abelian quotient.

IX.2 177

T h e o r e m 2.6. Let G be abelian of Galois type. Let (G,A) be a class formation. Then the open subgroups V of G are in bijection with the subgroups of A of the form SV(AV), called the t r a c e g roup . If we denote this subgroup by B v , and U is an open subgroup of G, then U C V if and only if B v C Bu, and B u v = B u N B v . I f in addition B is a subgroup of A a such that B D B v for some open subgroup V of G, then there exists U open subgroup of G such that B = Bu.

Pro@ All the assertions are special cases of what has previously been proved, except possibly for the last one. But for this one, one may suppose G finite and consider ( G / V , A v) instead of (G,A) . We let U = reca(B), and we find an isomorphism B / S G ( A ) ..~ U, to which we apply Theorem 2.4 to conclude the proof.

A subgroup B of A a will be called admis s ib l e if there exists V open subgroup of G such that B = SBV(Aa). We then write B = By. The next reuslt is an immediate consequence of Theorem 2.6 and the basic properties of the reciprocity map.

C o r o l l a r y 2.7. Let G be a group of Galois type and (G,A) a class formation. Let B C A c be admissible, B = Bu, and suppose U normal, G /U abelian. Let V be an open subgroup of G and put

C = ( sV) - I (B) ,

so C is a subgroup of A u. Then C is admissible for the class formation (V,A) , and C corresponds to the subgroup U n V of V.

In the next section, we discuss in greater detail the relations between class formations and group extensions. However, we can already formulate the theorem of Shafarevich-Weil. Note that if G is of Galois type and U is open normal in G, then U/U ~ is a Galois module for G, or in other words, U ~ is normal in G. Consequently, G/U acts on U/U c, and we obtain a group extension

(1) O - - , U / U C ~ G / U c ~ G / U --,0.

If in addition (G, A) is a class formation, then the reciprocity mapping

recg : A g ~ U/U c

is a G/U-homomorphism.

178

T h e o r e m 2.8 ( S h a f a r e v i c h - W e i l ) . Let G be of Galois type, U open normal in G, and (G, A) a class formation. Then

recu. : H2(G/U,A U) ~ H2(G/U,U/U ~)

maps the fundamental class on the class of the group extension (1). There exists a family of coset representatives (~ )~a of U in G such that if aa,e is a cocycle representing a, then

(a~,e, U) = o'rcrT- 1UC.

Proof. Let V C U be open normal in G. Ultimately, we let V tend to e. By the deflation operation of Chapter VIII, Theorem 3.2, there exists a cocycle ba,e representing the fundamental class of H2(G/V, A V) and representatives a of U/V such that

(2) ae,e:Su/y(b*,e/bT(a,~),~--e) I I bp,7(a,,-), peV/V

where 7(a, T) : aT-a'r -1. Therefore, we find

(aa,eU/V) = avery 1VUC.

We take a limit over V as follows, let C1,... , Cm be the cosets of U in G. They are closed and compact. The product space (C1, . . . ,Cm) is compact. If a l , . . . ,am are representatives of U/V satisfying (2), then any representatives of the cosets #1 V, . . . , #m V will also satisfy (2). The subset (# iV, . . . ~,,~ V) is closed in (C1, . . . , Cm). From the consistency of the reciprocity map, these subsets have the finite intersection property. Hence their intersection taken over all V is not empty, and there exists representatives

of the cosets of U in G which all give the same a,,e. The theorem is now clear.

w Well groups

Let G be a group of Galois type and (G, A) a class formation. At the end of the preceding section, we saw the exact sequence

0 c / u c a / u c a / u

IX.3 179

for every open subgroup U normal in G. Furthermore, U/U ~ is isomorphic to the factor group AU/sU(A) . We now seek an extension X of A v by G / U and a commutative diagram

0 ~ A ~' , X , G / U , 0

o , u i u c , a l U , a l U , o

satisfying various properties made explicit below. The problem will be solved in the following discussion.

We start first with the finite case, so let G be finite. By a Wel l g r o u p for (G,A) we mean a triple (E,g , {fv}), consisting of a group E and a surjective homomorphism

E 9-~G-.-~O

(so a group extension) such tha t , if we put Eu = g - l ( U ) for U an open subgroup of G (and so E = EG), then f u is an isomorphism

f u " A v --~ E u / E ~ .

These data are assumed to satisfy four axioms:

W 1. For each pair of open subgroups U C V of G, the following diagram is commutative:

A v "f~ ~ E u / E ~

inc T T Tr

A v , Ev /E~ , Iv

W 2. For x E EG and every open subgroup U of G, the diagram is commutative:

nU fv ~ E u / E ~

1 1 AuM , Eu[~]/E~M

fV[~l

The verticM isomorphisms are the natural ones arising from x.

180

Let U C V be open subgroups of G, and U normal in V. Then we have a canonical isomorphism

E v / E u ~ V /U

and an exact sequence

(3) 0 ~ Eu/E~] ~ Ev/E~] ---+ V/U ---, O.

Then V/U acts on Eu/E~r and W 2 guarantees that fu is a G/U- isomorphism. This being the case we can formulate the third axiom.

W 3. Let fu . : H 2 ( V / U , A U) ~ H2(V/U, Eu/E~]) be the induced homomorphism. Then the image of the fundamental class of (V/U, A U) is the class corresponding to the group extension defined by the exact sequence (3).

Finally we have a separation condition.

W 4. One has E c = e, in other words the map f~ �9 A ~ E~ is an isomorphism.

T h e o r e m 3.1. Let a be a finite group and (G, A) a class formation. Then there exists a Well group for (G,A). Its uniqueness will be described in the subsequent theorem.

Pro@ Let Ea be an extension of A by G,

O ---~ A----~ E c ~ G---* O

corresponding to the fundamental class in H2(G, A). This extension is uniquely determined up to inner automorphisms by elements of A, because H 1 (G, A) is trivial (Corollary 1.4 of Chapter VIII), and we have an isomorphism

f~:A---+E~,

so W 4 is satisfied.

For each U C G, we let Eu = g - l ( U ) , the extreme cases being given by A and EG. Thus we have an exact sequence

O -* A --+ E u --~ U --* O

IX.3 181

of subextension, and its class in H2(U, A) is the restriction of the fundamental class, i.e. it is a fundamental class for (U, A).

Consequently, if U is normal in G, we may form the factor extension

0 ---+ Eu/E~ ~ Ea/E~ ~ G/U ~ O.

By Corollary 2.5 of Chapter VIII, we know that the transfer

Tr" Eu/E~ ~ A U

is an isomorphism, and one sees at once that it is a G/U-isomorphism. Its inverse gives us the desired map

fu " A U ~ Eu/E~.

It is now easy to verify that the objects (Ea,g, {fu}) as defined above form a Weil group. The Axioms W 1~ W 2, W 4 are immediate, taking into account the transitivity of the transfer and its functoriality. For W 3, we have to consider the deflation. In light of the "functorial" definition of Ec , one may suppose that V = G in axiom W 3. If a is the fundamental class in H2(G, A), then (U : e)a is the inflation of the fundamental class in (G/U, Au). By Corollary 3.2 of Chapter VIII, one sees that the deflation of the fundamental class of (G,A) to (G/U,A U) is the fundamental class of (G/U, AU). Since fg is the inverse of the transfer, one sees from the definition of the deflation that axiom W 3 is satisfied. This concludes the proof of existence.

We now consider the uniqueness of a Weil group. Suppose G finite, and let (G, A) be a class formation, let (E, g, { f~: }) be two Weil groups. An i s o m o r p h i s m ~ of the first on the second is a group isomorphism

~ :E----~ E'

satisfying the following conditions:

I S O W 1. The diagram is commutative:

E g ~ G

I id E ! ~ G .

182

From I S O W 1 we see that ~ ( E u ) = E~u for all open subgroups U, whence an isomorphism

�9 ~ E u / E t g. E u / E b '

The second condition then reads:

I S O W 2. The diagram

A U fv , E u / E b

id I 2~~ ]opt / i~Tlc A U ) J..~u/~_,U

Ib

is commutative for all open subgroups U of G.

T h e o r e m 3.2. Let G be a finite group. Two Well groups associated to a class formation (G ,A) are isomorphic. Such an isomorphism is uniquely determined up to an inner automorphism of E ~ by elements of E'~.

Proof. Let ~ be an isomorphism�9 The following diagram is commutative by definition.

0 > A A ) E r ~ E , G

0 , A ~ ' E ' , E e , , G f' e

0

~0.

Conversely, we claim that any homomorphism ~ which makes this diagram commutative is an isomorphism of Well groups. Indeed, the exactness of the sequences shows that ~ is a group isomorphism of E on E r, and ~ ( E u ) = E b for all subgroups U of G. Hence qp induces an isomorphism

! C : E u / E b --, E v / E , v .

I X . 3 183

We consider the cube: inc A u -...<

id Eu / E~]

A v 1 b ~ u l inc

i tr Eu/Ev

, A

* Ee

--.... . E"

Tr' The top and bottom squares are commutative by I S O W 1.

The back square is clearly commutative. The front face is commutative because the transfer is functorial. The square on the right is commutative because of the commutative diagram in Theorem 3.2. Hence the left square is commutative because the horizontal morphisms are injective.

Thus the study of a Weil isomorphism is reduced to the study of in the diagram. Such ~ always exists since the group extensions

have the same cohomology class. Uniqueness follows from the fact that Hi(G, A) = O, using Theorem 1.3 of Chapter VIII, which was put there for the present purpose.

We already know that a class formation gives rise to others by restriction or deflation with respect to a normal subgroup. Similarly, a Weil group for (G, A) gives rise to Weil groups at intermediate levels as follows.

T h e o r e m 3.3. Let (G, A) be a class formation, and suppose G finite. Let ( EG, gG, F~) be the corresponding Well group, where ~e is the family of isomorphisms {fu} for subgroups U of G. Let V be a subgroup of G. Let

Ev = g~l(V) and gv = restriction of gG to V; ~v = subfamily of ~c consisting of those fv such that

U c V .

Then:

(i) (Ev, gv, ~v) is a Well group associated to (V, A). (ii) I f V is normal in G, then (EG/E~,gG,~G) is a Well group

associated with (G/V, AV), the family ~G consisting of the isomorphism

f v " A v ---* E u / E 5 "~ (Eu /E~) / (ES /E~) ,

where U ranges over the subgroups of G containing V.

184

Proof. Clear from the definitions.

The possibility of having Weil groups associated with factor groups in a consistent way will allows us to take an inverse limit. Before doing so, we first show that the reciprocity maps are induced by the isomorphisms f u of the Well group.

T h e o r e m 3.4. Let G be a finite group and (G, A) a class formation. Let (E~ ,g ,~ ) be an associated Weil group. Let V be a subgroup of G. Then the following diagram i8 commutative.

A v fv , Ev/E~

l li.c A G ) EG/E b

la

Proof. Since we have not assumed that V is normal in G, we have to reduce the oroof to this special case by means of a cube:

inc A v ' A

r

E a / E b ' Ee/E

The vertical arrow Scv on the back face is defined by means of representatives of cosets of V in G. The front vertical arrow S' is defined to make the right face commutative. In other words, we lift these representatives in EG be means of 9 -1. Thus if G = U aiV we choose ui 6 E c such that

and we define

g ( ~ r i ) = u i

g'(x) = H x" ' (mod E~). i

We note that E a = U u iEv , in other words that the ui represent the cosets of E v in Ea. Then the front face is commutative, that is

S'(Tr'(u)) = Tr(inc,(u)) for u 6 E v / E ~ ,

IX.3 185

immediately from the definition of the transfer. It then follows that the left face is commutative, thus finishing the proof.

Corollary 3.5. Let G be finite and (G,A) a class formation. Let (EG,g,~) be an associated Well group. I f U C V are subgroups of G, then fu and f v induce isomorphism~:

A V / S U ( A U) ,~ E v / E u E ~ / and ( sU) - l ( e ) ~ (Eu N E~,)E~.

If U is normal in V, then the first isomorphism i8 the reciprocity mapping, taking into account the isomorphism E w / E u .~ V/U.

Note that Corollary 3.5 is essentially the same result as Theorem 2.8. The proof of Corollary 3.5 is done by expliciting the transfer in terms Of the Nakayama map, and the details are left to the reader.

In practice, in the context of class field ~heory, the group A has a topology (idele classes globally or multiplicative group of a local field locally). We shall now sketch the procedure which axiomatizes this topology, and allows us to take an inverse limit of Weft groups.

Let G be a group of Galois type and A E Galm(G). We say that A is a topological Galois module if the following conditions are satisfied:

T O P 1. Each A U (for U open subgoup of G) is a topological group, and if U C V, the topology of A v is induced by the topology of A U.

T O P 2. The group G acts continuously on A and for each o- E G, the natural map A v ---. A U[~'] is a topological isomorphism.

Note that if U C V, it follows that the trace S U �9 A U --+ A v is continuous.

Let G be of GMois type and A E Galm(G) topological. If (G, A) is a class formation, we then say it is a topological class formation. By a Well group associated to such a topological class formation, we mean a triplet (Ec , g, 5) consisting of a topological group Ea , a morphism g : E c --+ G in the category of topological groups (i.e. a continuous homomorphism) whose image is dense in G (so that for each open subgroups V D U with U normal in G we have an isomorphism E w / E u ~ V/U), and a family of topological isomorphisms

f v " Au ~ E u / E b

186

(where E L is the closure of the commutator group), satisfying the following four axioms.

W T 1. For each pair of open subgroups U C V of G, the following diagram is commutative:

A U . : v > E u / E ~

inc A v > E v / E ~

Iv

Note that the transfer on the right makes sense, because it extends continuously to the closure of the commutator subgroups.

W T 2. Let x C E a be such that a = g(x). Then for all open subgroups U of G the following diagram is commutative:

A U Iv , E u / E ~

E ~ A u['] > Eur~l/ U[z] fu[~] L J - -

W T 3. If U C V are open subgroups of G with U normal in V, then the class of the extension

0 --+ A v ,~ E u / E ~ -~ E v / E ~ ---', E v / E u ~ V / U --', 0

is the fundamental class of H 2 ( V / U , AU).

W T 4. The intersection N E~ taken over all open subgroups U of G is the unit element e of G.

To prove the existence of a topological Weft group, we shall need two sufficient conditions as follows.

W T 5. The trace S U �9 A U --~ A V is an open morphism for each pair of open subgroups U C V of G.

IX.3 187

W T 6. The factor group AU/A V is compact.

Then there exists a topological Weil group associated to the formation.

T h e o r e m 3.6. Let G be a group of Galois type, A C Galm(G), and (G,A) a topological class formation satisfying W T 5 and W T 6.

Proof. The proof is essentialy routine, except for the following remarks. In the uniqueness theorem for Weil groups when G is finite, we know that the isomorphism ~ is determined only up to an inner automorphism by an element of A = E,. When we want to take an inverse limit, we need to find a compatible system of Well groups for each open U, so the topological A v intervene at this point. The compactness hypothesis is sufficient to allow us to find a coherent system of Weil groups for pairs (G/U, A U) when U ranges over the family of open normal subgroups of G. The details are now left to the reader.

C H A P T E R X

Applications of Galois Cohomology in Algebraic Geometry

by J o h n Tate

Notes by Serge Lang

1959

Let k be a field and Gk the Galois group of its algebraic closure (or separable closure). It is compact, totally disconnected, and inverse limit of its factor groups by normal open subgroups which axe of finite index, and axe the Galois groups of finite extension.

We use the category of Galois modules (discrete topology on A, continuous operation by G) and a cohomological functor Ha such that H(G, A) is the limit of H(G/U, A U) where U is open normal in G.

The Galois modules Galm(G) contains the subcategory of the torsion (for Z) modules GMmtor(G). RecM1 that G has cohomological dimension _<_ n if Hr(G,A) = 0 for all r > n and all A E GMmtor(G), mad that G has strict cohomological dimen-

X.1 189

sion < n if HT(G,A) = 0 for r > n and all A C Galm(G).

We shall use the tower t h e o r e m that if N is a closed normal subgroup of G, then ca (a ) _< cd(a/N)+cd(N), as in Chapter VII, Theorem 5.1. If a field k has trivial Brauer group, i.e.

H2(GE,fF) = 0 (all E/k finite)

where ft = k8 (separable closure) then cd(Gk) _< 1. By definition, a p-adic field is a finite extension of Qp. The maximal unramified extension of a p-adic field is cyclic and thus of cd _< 1. Hence:

If k is a p-adic field, then cd(Gk) _< 2.

This will be strengthened later to scd(Gk) _< 2 (Theorem 2.3).

w Torsion-free modules

We use principally the dual of Nakayama, namely: Let G be a finite group, (G, A) a class formation, and M finitely generated torsion free (over Z, and so Z-free). Then

Hr(G, Hom(M,A)) x H2-T(G,M)---* H2(G,A)

is a dual pairing, (i.e. puts the two groups in exact duality).

We suppose k is p-adic, and let ft be its algebraic closure. We know Gk has ed(Gk) _< 2 by the tower theorem. We shall eventually show scd(ak) _< 2.

Let X = Horn(M, ft*). Then X is isomorphic to a product of f~* as Z-modules, their number being the rank of M, and we can define the operation of Gb on X in the natural manner, so that X C Galm(Gk).

By the existence theorem of local class field theory for L ranging over the finite extensions of k, the groups

NL/kXL

are cofinal with the groups nXk, if we write XL = X GL and similarly ML = M aL �9 This is clear since there exists a finite extension K of k which we may take Galois, such that M aK = M. Then for L D K, our statement is merely local class field theory's existence

190

theorem, and then we use the n o r m NL/K to conclude the proof (transitivity of the norm).

We shall keep K fixed with the property that MK = M. We wish to analyze the cohomology of X and M with respect to Gk.

P r o p o s i t i o n 1.1. H r ( G k , X ) = 0 for r > 2.

Proof. X is divisible and we use cd _< 2, with the exact sequence

0 ~ X to r ---+ X ~ X / X t o r ---+ O,

where Xtor is the torsion part of X.

T h e o r e m 1.2. Induced by the pairing

X x M---~ ft*

we have the pairings

P0. H2(Gk,X) • H~ --, g2(ak , f~ *) = Q/Z P1. H I ( G k , X ) x H I ( G k , M ) --~ Q/Z

P2. H~ • H2(Gk,M) --~ Q/Z

In P0, H 2 ( a k , X ) = M~ = by definition Hom(Mk, Q/Z). In P1, the two groups are finite, and the pairing is dual. In P2, H 2 ( G k , M ) is the torsion submodule of Hom(Xk, Q/Z), i.e.

H 2 ( G k , M ) X ^ . =( k)tor

Proof. The pairing in each case is induced by inflation in a finite layer L D K D k. In P0, the right hand kernel wil be the intersection of NL/kML taken for all L D K, and this is merely [L : K ] N K / k M K which shrinks to 0. The kernel on the left is obviously 0.

In P1, the inflation-restriction sequence together with Hilbert's Theorem 90 shows that

inf: H I ( G K / k , X K ) ---, H I ( G L / k , X L )

is an isomorphism, and the trivial action of GL/K o n M shows similarly that

inf : H 1 (aKIk, M) --, H l(Gglk, M)

X.2 191

is an ismorphism. It follows that both groups are finite, and the duality in the limit is merely the duality in any finite layer L D K.

In P3 , we dualize the argument of P0, and note that He(Gk, M) will produce characters on Xk which are of finite period, i.e., which are trivial on some nXk for some integer n. Oth- erwise, nothing is changed from the formalism of P0.

w Finite m o d u l e s

The field k is again p-adic and we let A be a finite abelian group in Galm(Gk). Let B = Hom(A, f~*). Then A and B have the same order, and B E Galm(Gk). Since f~* contains all roots of unity, B = A, and A = / ) once an identification between these roots of unity and Q / Z has been made.

Let M be finitely generated torsion free and in Gaim(Gk) such that we have an exact sequence in Gaim(Gk):

O----, N----~ M--~ A---. O.

Since f~* is divisible, i.e. injective, we have an exact sequence

O , - Y ~ - - X ~ - - B . - - O

where X = Horn(M, f~* ) and Y = Horn(N, f~*).

By the theory of cup products, we shall have two dual sequences

(B) H i ( B ) ~ H i ( x ) ~ Hi(Y)---* H2(B)---* H2(X)-+ H2(Y)

(A) H i (A) ~ H i ( M ) --- Hi(N) ~- H~ ~ H~ ~ H~

with H : Hck. If one applies Hom(., Q /Z) to sequence (A) we

obtain a morphism of sequence (B) in%o Hom((A), Q/Z) . The 5- lemma gives:

T h e o r e m 2.1. The cup product induced by A x B ---* f~* gives an exact duality

H2(Gk,B) x H~ --~ H2(G fF).

192

T h e o r e m 2.2. With A again finite in Galm(Gk) and B = Horn(A, ~*) the cup product

HI(Gk,A) x HI(Gk,B) --+ H2(Gk,~ *)

give8 an exact duality between the H 1, both of which are finite groups.

Proof. Let us first show they are finite groups. By the inflation- restriction sequence, it suffices to show that for L finite over k and suitably large, HI(GL,A) is finite. Take L such that GL operates trivially on A. Then HI(GL,A) = c o n t Hom(GL,A) and it is known from local class field theory or otherwise, that GL/G~ G~L is finite.

Now we have two sequences

(s) (A)

H~ --+ H~ ~ H~ Hi(B) - -+ H i ( x ) - - + Hi(Y)

H2(A) +- H2(M)+- H 2 ( N ) + - Hi (A) +- H i ( M ) + - Hi(N).

We get a morphism from sequence (A) into the torsion part of Hom((B), Q/Z)) , and the 5-1emma gives the desired result.

Next let n be a large integer, and let us look at the sequences above with A = M / n M and N = M. We have the left part of our sequences

0--+ H~ H~

H3(M) +- H3(M)+-- H2(A)+- H2(M)

I contend that H2(M) ---+ H2(A) is surjective, because every character of H~ is the restriction of some character of H~ since Bk N nXk = 0 for n large. Hence the map

n : H3(M) --+ H3(M)

is injective for all n large, and since we deal with torsion groups, they must be 0. This is true for every M torsion free finitely generated, in Galm(Gk). Looking at the exact sequence factoring out the torsion part, and using cd _< 2, we see that in fact:

X.2 193

T h e o r e m 2.3. We have scd(Gk) < 2, i.e. H r = 0 . f o r t > 3 - - G k - -

and any object in Galm(Gk).

We let X be the (multiplicative) Euler characteristic, cf. Algebra, Chapter XX, w

T h e o r e m 2.4. Let k be p-adic, and A finite Gk Galois module. Let B = Hom(d, fl*). Then x ( G k , A ) = IIAIIk.

Pro@ The Euler characteristic X is multiplicative, so can assume A simple, and thus a vector space over Z/gZ for some prime g. We let AK = A OK. For each Galois K/k , either Ak = 0 or Ak = A by simplicity.

Case 1. Ak = A, so Gk operates trivially, so order of A is prime ~. Then

h ~ h l = ( k * ' k * t ) , h 2 ( A ) = h ~

So the formula checks.

Case 2. Bk = B. The situation is dual, and checks also.

C a s e 3. Ak = 0 and Bk = 0. Then

X(Gk,A) = 1/hl(Gk,A).

Let K be maximal tamely ramified over k. Then GK is a p- group. If AK = 0 = A aK then f # p (otherwise GK must operate trivially). Hence HI(GK,A) = 0. The inflation restriction sequence of Gk, GK and GI~'/k shows H 1 (Gk, A) = 0, so we are done.

Assume now AK # O, so AK = A. Let L0 be the smallest field containing k such that ALo = A. Then L0 is normal over k, and cannot contain a subgroup of ~-power order, otherwise stuff left fixed would be a submodule # 0, so all of A, contradicting L0 smallest. In particular, the ramification index of Lo/k is prime to

Adjoin ~-th roots of unity to L0 to get L. Then L has the same properties, and in particular, the ramification index of L/k is prime to *.

194

Let T be the inertia field. Then the order of GL/T is prime to ~.

L

I GL/T

T

I GT/k

GL/T

Hence Hr(GL/T,A) = 0 for all r > O. By spectral sequence, we conclude Hr(GL/k,AL) = Hr(GT/k,AT) all r > O. But GT/k is cyclic, AT is finite, hence HI(GL/k,AL) and H2(GL/k,AL) have the same number of elements. In the exact sequence

O------+HI(GL/k ,AL)------+HI(Gk ,A)-'-+HI(GL ,A)GL/k --+H2(GL/k ,A)"-+O

we get 0 on the right, because H2(Gk, A) is dual to H~ B) = Bk = 0. We can replace H2(GL/k, A) by HI(GL/k, A) as far as the number of elements is concerned, and then the hexagon theorem of the Herbrand quotient shows

hl(Gk, A) = order HI(GL, A) aL/k .

Since GL operates trivially on A, the H 1 is simply the horns of GL into A, and thus we have to compute the order of

HOmaL/k (GL, A).

Such horns have to vanish on G t and on the commutator group, so if we let G2 be the abelianized group, then * *t GL/G n . But this is Gn/k-isomorphic to L*/L *t, by local class field theory. So we have to compute the order of

HOmaL/k (L*/L *t, A).

If ~ # p, then L*/L *t is GL/k-isomorphic to Z/gZ • #t where #t is the group of t - th roots of unity. Also, GL/k has trivial action

X.3 195

on Z/eZ. HOmGL/k

So no horn can come from that since Ak = 0. of #~, if f is such, then for all ~ ff GL/k,

As for

f ( a ( ) = ~f ( ( ) .

But a ( = (" for some y, so a = f ( ( ) generates a submodule of order g, which must be all of A, so its inverse gives an element of Bk, contradicting Ba = 0. Hence all Go/k-horns are 0, so Q.E.D.

If g = p, we must show the number of such homs is 1/HAIIk. But according to Iwasawa,

L*/L *p = Z/pZ x #p x Zp(GL/k) m.

Using some standard facts of modular representations, we are done.

w The 'rate pairing

Let V be a complete normal variety defined over a field k such that any finite set of points can be represented on an afs k- open subset of V. We denote by A = A(V) its Albanese variety, defined over k, and by B = B(V) its Picard variety also defined over k. Let D~(V) and De(V) be the groups of divisors algebraically equivalent to 0, resp. linearly equivalent to 0. We have the Picard group D~(V)/D~(V) and an isomorphism between this group and B, induced by a Poincar6 divisor D on the product V x B, and rational over k.

For each finite set of simple points S on V we denote by Pics(V)

n /n(1) where Da,s consists of divisors alge- the factor group ~,a,s/L,~, s

braically equivalent to 0 whose support does not meet S, and r)(1) ~'t,S is the subgroup of Da,s consisting of the divisors of functions f such that f (P) = 1 for all points P in S. Then there are canonical surjective homomorphisms

Pics,(V) ~ Pics(V)--+ Pic(V)

whenever S' D S.

Actually we may work rationally over a finite extension K of k which in the applications will be Galois, and with obvious definitions, we form

n in( I ) Pics, K(V) = L'~,S,t~'/L'~,S, Is-

196

the index K indicat ing ra t ional i ty over K .

We may form the inverse limit inv liras Pics, K(V) . For our purposes we assume merely tha t we have a group Ca,K together wi th a coherent set of surjective homomorph isms

cps " Ca,K ~ Pics, K(V)

thus defining a homomorph i sm ~ (their l imit) whose kernel is de- no ted by UK. We have therefore the exact sequence

(1) o --, u/*. co ,K B ( K ) -+ o.

We assume th roughou t tha t a divisor class (for all our equiva- lences) which is fixed under all e lements of G/,- contains a divisor ra t ional over K . Similarly, we shall assume th roughou t tha t the sequence

(2) 0 --+ Z~,K --~ Zo,K --~ A ( K ) -+ 0

is exact , (where Z0 are the 0-cycles of degree 0, and Z~ is the kernel of Albanese) , for each finite extension K of k.

Relat ive to our exact sequences, we shall now define a Tate pairing.

Since Ca,K is essentially a project ive limit, we shall use the exact sequence

r l l / n ( 1 ) 0 --* ~_,~,s,/*./~s,/*. ---* Pies,/,- j B ( k ) --* 0

because if o e~ D S, then we have a commuta t ive and exact d iagram

I n ( 1 ) 0 ' DI,S,,K/~.,e,S,,/*- , Pies,,/*" ~ B ( k ) , 0

$ ; l i d D /n(1) 0 ) ~,S,K/L,~,S,I~ ~ Pics, K ~ B ( k ) ~ 0

0

Now we wish to define a pair ing

Zo,K x UK ~ K*.

X.3 197

Let u E UK, and a E ZO,K. Write

a = E nQQ

where the Q are distinct algebraic points. Let S be a finite set of points containing all those of a, and rational over K. Then u has a representative in Pics, K and a further representative function f s defined over K, and defined at all points of a. We define

<.,u> -1 : fs(~) = [I fs(Q) "q"

It is easily seen that (a, u> -1 does not depend on the choice of S and f s subject to the above conditions. It is then clear that this is a bilineax pairing.

We define a further pairing

Z.,K x C~, K -+ K*

D "D (1) Let a~ follows. Let 7 E C~,K, so 7 : 5mTs,Ts E ~,S,K/ <S,K" a E Z~,K and let S contain supp(a). Let b be a 0-cycle on B, rational over K, corresponding to the point b = iV. Let X s be a divisor on V rational over K, representing 7s. Let D be a Poincax@ divisor whose support does not meet that of S • b. We define:

<a, 7} : [~D(b) - Xs](a) m(~,b)

observing that t D ( b ) - X s is the divisor of a function whose support does not meet S and is thus defined at a.

Using the reciprocity law of [La 57], see Mso [La 59], Chapter VI, w Theorem 10, one verifies that this is independent of the successive choice of S, b, Xs, and D subject to the above conditions.

One verifies finally that our pairings agree on Za x U.

Thus to summarize: we have exact sequences

o ~ z~,K(v)~ Zo,K(v)-~ A(K)-, o

o ~ c%: --~ C~,K(V) --~ B(K) -~ o

198

and we have a Tate pairing:

(a, 7} [tD(b) -Xs](a) K* = D(a,b) E

where b E Zo(B) maps on the point b E B, the same point as 7 E Ca(V), S is a finite set of points containing supp(a), Xs represents 7s, and

(ll,@ = fs(ll) -1

where fs is a function representing u. We take S' so large that everything is defined.

Proposition 3.1. The induced bilinear map on (A,,~,Bm) coincides with em(a,b), i.e. with tn(mb, a)/n(ma, b).

Proof. Clear. We are using [La 57] and [La 59], Chapter VI.

The above statements refer to the Tate augmented product of Chapter V. The augmented product exists whenever one is given two exact sequences

0 --~ A' ~ A :-~ A" --+ 0

0 --+B' --+B J-~ B" - - ,0

an object C, two pairings A x B' ~ C and A' x B --~ C which agree on A' x B' . Such an abstract situation induces an augmented product

H~(A '') x HS(B '') 2& Hr+s+l(c)

which may be defined in terms of cocycles as follows. If f " and g" are cocycles in A" and B" respectively, their augmented cup is represented by the cocycle

6 f U g + ( - 1 ) dim I fU6g

where j f = f" and jg = g", i.e. f and g are cochains of A and B respectively pulled back from f " and g".

In dimensions (0, 1), the most important for what follows, we may make the Tate pairing explicit in the following manner. Let (b~) be a 1-cocycle representing an element/3 E HI(Gt(/k,B(K))

X.4 199

and let a E A(K) represent c~ C H~ Let a C Zo,k(V) belong to a, and let b~ be in ZO,K(B) and such that S(b~) = b~. Let D be a Poincar~ divisor on V • B whose support does not meet a • b~ for any a. Then it is easily verified that putt ing b = b~ + crbT -- b~-, the cocycle

tD(b, a)

represents a U~ug j3.

w The (0, 1) duality for abelian varieties

We assume for the rest of this section that k is p-adic.

T h e o r e m 4.1. The augmented product of the Tats pairing described in Section 3 induces a duality between H~ and HI(Gk,B), with values in g2 (Gk ,~ )= Q/Z .

Proof. According to the general theory of the augmented cupping, we have for each integer m > 0,

0 ---+ A ( k ) / m A ( k ) ~ H I ( G k , A m ) ---. H I ( G k , A ) m ---+ 0

0 ---+ ( H I ( G k , B ) m ) ^ --+ H I ( G k , B m ) ^ ---. ( B ( k ) / r n B ( k ) ) ^ ---* 0

a morphism of the first sequence into the second. Since the pairing between Am and Bm is an exact duality, so is the pairing between their H 1 by Theorem 2.1. We wish to prove the end vertical arrows are isomorphisms, and for this we count. We have:

(A(k) : mA(k)) <_ hl(B)m (S(k) : roB(k)) <_ hl(A)m

hl(A)m(A(k) : mA(k))= hi(Am)

hl(B)m(B(k) : roB(k))= hl(Bm)

(A(k) : rnA(k)) (B(k) :roB(k)) (A(k)m : O) = II nllkr = (B(k)m "0)

(by cyclic cohomology, trivial action)

x(Am) = IIm2rllk = x(Bm) and h~ = h2(Am) by duality. Put t ing everything together, we get equality in the first inequalities; this proves the desired isomorphism.

200

T h e o r e m 4.2. We have H2(Gk,B) = O. abelian varieties, better than scd < 2.)

Proof. We have an exact sequence

(This is special for

0---+ H I ( B ) / m H I ( B ) - - , H2(Bm)---+ H 2 ( B m ) ~ H2(B)m --* O.

But H2(Bm) is dual to H~ and in part icular has the same number of elements. Also, H i ( B ) being dual to H~ we see that H I ( B ) / m H I ( B ) is dual to H~ = A(k)m, which is also H~ Hence the two terms on the left have the same number of elements, since H2(B)m = 0 for all m, so 0 since it is torsion group.

Now we have the duali ty for H 1 , H ~ in finite layers.

T h e o r e m 4.3. Let K / k be finite Galois with group G = GK/k. Then the pairing

H~ x HI (G ,B(K) ) ~ H2(G,K *)

is a duality.

Proof. This follows from the abstract fact that restriction is dual to the transfer valid for any Tate pairing and the induced augmented cupping.

If one uses the inflation-restriction sequence, together with the commuta t iv i ty derived abstractly for d2, and Theorem 4.2, we get the following -4- commuta t ive diagram, put t ing U = GK, and G = GK/k,

U~.ug tr HX(U,A) a /u X (BU)G/u * H2(U,~*)G/Lr * H2(G,fl *)

H~(a/V,A u) X HI(a/U,B u) , g~(a/u,a "v) �9 H2(a,a ") U~g inf

Identifying H2(G, fl*) with q / z we get the duali ty between H 2 and H - 1 :

X.5 201

Identifying H2(G, ~2") with Q / Z we get the duality between H 2 and H -1"

T h e o r e m 4.4. I f K / k is a finite Galois extension with group G, then the augmented cupping

H 2 ( G , A ( K ) ) x H - I ( G , B ( K ) ) ---* H 2 ( G , K *)

is a perfect duality.

w T h e full d u a l i t y

We wish to show how the following theorem essentially follows from the (0, 1) duality without any further use of arithmetic, only from abstract commutative diagrams.

T h e o r e m 5.1. Let k be a p-adic field, A and B an abelian variety and its Picard variety defined over k, and consider the Tate pairing described in w Then the augmented cupping

H I - " ( G K / k , A ( K ) ) x H " ( G K / k , B ( K ) ) ---* H 2 ( G K / k , K *)

puts the two groups (which are finite) in ezact duality.

(Of course, the right hand H 2 is (Q/Z)n , where n = (GK/k : e).)

As usual, the A means Horn into Q/Z .

Put GK = U and G = Gk. We have a compact discrete duality

A U • !-/I(u, B) --+ H2(U,~ *) = Q/Z

and we know from this that A U is isomorphic to Hi(U, B) A G/U-module. Hence the commutative diagram say for r ~ 3:

as a

H~-'(G/U,H~(U,B) ̂ ) • H'-2(a/U,H~(U,B)) ~_. H-~(G/U,Q/Z)

H~-'(G/U,A ~) x H~-2(G/U,H~(U,S)) --- H-t(a/U,Q/Z). U

The top line comes from the cup product duality theorem, and the arrow on the left is am ismorphism, as described above.

202

Of course, we have H-I(G/U, Q/Z) = (Q/Z),, if n is the order of G/U. Note a/so that the Q /Z in the lower right stands for H2(U, Q*), because of the invariant isomorphism.

Now from the spectral sequence and Theorem 4.2 to the effect that H 2 of an abelian variety is trivial over a p-adic field, we get an isomorphism

d2 " H"-2(G/U, HI(U,B))---* H"(G/U,H~

and we use another abstract diagram:

HI-~(G/U,A ~r) • H"-2(G/U, HI(U,B)) ~ H-t(G/U, H2(U,~2*))

id~ d21 Hz-"(G/U,A u) • H"(G/U,H~ , FI2(G/U,n "u)

U~ug

In order to complete it to a commutative one, we complete the top line and the bot tom one respectively as follows:

I'I-~(G/U,H=(U, ~*) --" H2(U,n*)G/U ~ H2(G, ~2.) l n c

�9 ,t d"

H2(G/U,12 "u) [nf )'H2(G,~ *)

and since the transfer and inflation perserve invariants, we see that our duality has been reduced as advertised.

We observe that we have the ordinary cup on the top line and the augmented cup on the bottom. The top one is relative to the A U, H I (U, B) duality, derived previously.

w The Brauer g r o u p

We continue to work with a variety V defined over a p-adic field k. We assume V complete, non-singular in codimension 1, and such that any finite set of points can be represented on an ~.fl~ne k-open subset of V.

We let G = Gk and all cohomology groups in this section will be taken relative to G. We observe that the function field ~(V) has group G over k(V), and we wish to look at its cohomology.

X.6 203

By Hilbert 's Theorem 90 it is trivial in dimension 1, and hence we look at it in dimension 2: It is nothing but that part of the Brauer group over k(V) which is split by a constant field extension. We make the following assumptions.

A s s u m p t i o n 1. There exists a O-cycle on V rational over k and of degree 1.

A s s u m p t i o n 2. Let N S denote the N6ron-Severi group of V (it is finitely generated). Then the natural map

Div(V) a~ ---* N S a~ = NSk

of divisors rational over k into that part of N6ron-Severi which is fixed under Gk is surjective, i.e. every class rational over k has a representative divisor rational over k.

These assumptions can be translated into cohomology, and it is actually in this latter form that we shall use them. This is done as follows.

To.begin with, note that Assumption 1 guarantees that there is a canonical map of V into its Albanese variety defined over k (use the cycle to get an origin on the principal homogeneous space of Albanese). Hence by pull-back from Albanese, given a rational point b on the Picard variety, there is a divisor X E D~(V) rat ional over k such that CI(X) = b. In other words, the map

D,(V) C. B(k) is surjective. Now consider the exact sequence

H~ H ~ ~ g l ( D ~ ) --, HI(Div(V)) .

Then Div(V) Ck is a direct sum over Z of groups generated by the irreducible divisors, and putting together a divisor and its conjugates, we get

Div(V/G = O G z z xE~

where ~ ranges over the prime rational divisors of V over k and X ranges over its algebraic components. Now the inside sum is semilocal, and by semilocal theory we get HI(GK, Z) where K = k x is the smallest field of definition of X. This is 0 because GK is of Galois type and the cohomology comes from finite things. Thus:

204

P r o p o s i t i o n 6 .1 . H I ( D i v ( V ) ) = O.

F rom this, one sees tha t our Assumpt ion 1 is equivalent wi th the condi t ion Hi(D=) = O.

Now looking at the other sequence

H~ H~ Hi(Dr)--+ HI(D,)

we see tha t Assumpt ion 2 is equivalent to H 1 (De) = O. Thus:

Assumptions 1 and 2 are equivalent with

HI(D~) = 0 and HI(Dt) = O.

Now we have two exact sequences

0

1

0 . Hi(B) . H2(Dl) , H2(Da) �9 0

l

and f rom them we get a surjective map

~ " H2( f l (V) *) ---, H2(D=)--+ O.

We define H~(~(V)*) to be its kernel, and call it the u n r a m i f i e d par t of the S r a u e r group H2(~(V)*). In view of the exact cross we get a map

H2(~(V) *) ~ Hi(B) .

X . 6 2 0 5

Thus

H2(f~(V)*) /H~(a(V) *) ,~ H2(Da)

H 2 ( a ( V ) * ) / H 2 ( a ) ~ H i ( B ) ~-, Char(A(k))

H2(fl *) ~ O / Z = Char(Z)

where Char means continuous character (or here equivalently character of finite order, or torsion part of .4(k) = Hom(A(k), Q/Z) ) . This gives us a good description of our Brauer group, relative to the filtration

2 �9 H 2 g2(f~(V) *) D H,~(a(Y) ) D (a*) D O.

We wish to give a more concrete description of Hu 2 above, making explicit its connection with the Tate pairing. Relative to the sequence

0 ~ A ( k ) ~ Zk/Zo,,k ~ Z ~ 0

and taking characters Char, we shall get a commutatiove exact diagram as follows:

0 * C h a r ( Z ) , Char(Zk/Z~, .k) , C h a r ( A ( k ) ) , 0

0 . H~( f l ") . H2 ( f I (V ) ") ,, H I ( B ) . 0

The two end arrows are as we have just described them, and are isomorphisms. We must now define the middle arrow and prove commutativity.

Let v C H2~(Q(V)*). For each prime rational 0-cycle p of V over k we shall define its reduction mod p,v o C H2(f~ *) and then a character Zk/Z~,,k by the formula

Ov(a) = E up inv(vp) P

whenever a is a rational O-cycle,

o=Evp.p, v p E Z .

We shall prove that 8v vanished on the kernel of Albanese, whence the character, and then we shall prove commutativity.

206

D e f i n i t i o n o f vp. Let (f~,T) be a representat ive cocycle. By definit ion it splits in Da, so that there is a divisor X~ C Da such that

(f~,~) = X~ + ~X~ - X ~ .

For each a, choose a function g~ such that X~ = (g~) at p. Put

! then (f~,r) = 0 at p, i.e. f'~,~, is a unit at p. We now put

aa,T : H f~,~(P)' PCp

where {P} ranges over the algebraic points into which p splits. We contend tha t (aa,~-) is a cocycle, and that its class does not depend on the choices made during its construction.

Let (f*,~.) be another representat ive cocycle which is a unit at p, and obta ined by the same process. Then

f* = f ' . 5g

with (5g)r = g~,g]/g~- a unit at p. Hence taking divisors,

+ (gT) - = o

at all points in supp(p). Let this support be S, and let Div S be the group of divisors passing through some point of p. Then H l ( D i v S) = 0 by the same argument as in Proposi t ion 6.1 (semilocal and H i ( Z ) = 0) and hence taking the image of (gr in Div s we conclude tha t there exists a divisor X E Div s such that (gr = a X - X. Let h be a function such that X = (h) at p. Replace gr by g~,h 1-~. Then still f* = f ' �9 5g and now gr is a unit at p, for all ~. From this we get

H (f*/f')~,~(P) = H (hg).,T(P) PEP PEp

from which we see tha t it is the boundary of the 1-cochain

H g~(P)" PEP

X.6 207

Thus we have proved our reduction mapping

73 e--+ Up

well defined.

Now the image of v in H ~ (B) is by definition reprsented by the cocycle CI(X~) : b~ (notation as in the above paragraph) and if a r Zo,k then

0v(.) = i s ( a ) Uaug/3

where/3 is represented by the cocycle (b~).

Thus 0v vanishes on the kernel of Albanese, and the right side of our diagam is commutative.

As for the left side, given w E H2(~*), represented by a cocycle (c~,~), then wp is represented by

I I m C ~ T ~ C~rT

P E p

where m = deg(p). Then

O~(a) = Z ( d e g p)upinv(co) P

= deg(a), inv(w)

whence commutativity. This concludes the proof of the following theorem.

T h e o r e m 6.2. There is an isomorphism

2 * Char(Z/Z~)k H (a(V) )

under the mapping Ov and the diagram

0 ------+ C h a r ( Z ) ~ C h a r ( Z / Z . ) k -----~ C h a r ( A ( k ) ) ~ 0

T t T 0 ~ H2( f~ *) ~ H 2 ( a ( V ) *) , H ~ ( B ) ----* 0

which is exact and commutative.

We conclude this section with a description of H2(Da) also in terms of characters.

208

We have the exact sequence

0 ---* D~ ---. Div --~ NS ~ 0

and hence

0--~ HI(NS)--+ H 2 ( D ~ ) ~ H 2 ( D i v ) ~ H 2 ( N S ) ~ 0

the last 0 by scd _< 2.

Now H2(Div) is easy to describe since Div is essentially a direct sum. In fact,

Divk = @ @ Z ' X xE~

where the sum is taken over all prime rational divisors ~ over k and all algebraic components X in ~. Using the semilocal theory, we get

H2(Div) = @ H2(Gk, 'Z) = @ H I ( G k , 'Q/Z)

= G Char(k?)

where k~ is the smallest field of definition of an algebraic point X in ~ and Gk~ is the Galois group over k~. Thus we see that our H 2 is a direct sum of character groups.

T h e Case of Curves . If we assume that V has dimension 1, i.e. is a non singular curve, then this result simplifies considerably since NS = Z is infinite cyclic, and we have also

N S = N & = ( N S ) G~ .

We have

Hx(~vS) = O, H~(NS) = H~(Z) = Char(k*)

and we get the commutative diagram

0 , H2(D~,) , H2(Div) , H2(Z) , 0

o , @~ Char(k?) , @ Char(kj) , Char(k*) , 0

X.6 209

where the @0 on the left means those elements whose sum gives 0. The morphism on the lower right is given by the restriction of a character from k~ to k* and the sum mapping. Thus an element

of H2(Div) is given by a vector of characters (Xv,p) where p ranges over the prime rational cycles of V, i.e. the ~ since cycles and divisors coincide.

We observe also that by Tsen's theorem, H2(f~(V)*) is the full Brauer group over k(V) since ft(Y) does not admit any division algebras of finite rank over itself.

Finally, we have slightly better information on H 2"

P r o p o s i t i o n 6.3. If V is a curve, then

2 * 2 Hu(ft(V) ) = N H , ( f t ( V ) * ) P

where H~ consists of those cohomology classes having a cocycIe representative (f~,~) in the units at p.

We leave the proof as an exercise to the reader.

We shall discuss ideles for arbitrary varieties in the next section. Here, for curves, we take the usual definition, and we then have the same theorem as in class field theory.

P r o p o s i t i o n 6.4. Let V be a curve, and for each p let k(V)p be the completion at the prime rational cycle p. Let Br(k(V)) be the Brauer group over k(Y), i.e. g2(Gk, k(Y)8) where k(Y)8 is the separable (= algebraic) closure of k(Y), and similarly for Br(k(V),) . Then the map

Br(k(V)) -~ H Br(k(V)p) P

is injective.

One can give a proof based on the preceding discussion or by proving that Hi (Ca) = O, just as in class field theory. We leave the details to the reader.

210

w Ide les a n d idele c lasses

Let k be a field and V a complete normal variety defined over k and such that any finite set of points can be represented on an affine k-open subset. By a cycle, we shall always mean a 0-cycle.

For each prime rational cycle p over k on V we have the integers 0p, the units Up and maximal ideal nap in k(V).

There are several candidates to play the role of ideles, and we shall describe here what would be a factor group of the classical ideles in the case of curves. We let

Fp = k(V)*/(1 + nap).

Then we let Ik be the subgroup of the Cartesian product of all the Fp consisting of the vectors

f = ( . . . , f p , . . . ) fpcFp

such that there exists a divisor X, rational over k, such that

X = (fp) at p for all p.

(In the case of curves, this means unit almost everywhere.) We call this divisor X (obviously unique) the d iv i so r a s s o c i a t e d w i t h t h e idele f, and write X = (f).

We have two subgroups Ia,k and Ie,k consisting of the ideles whose divisor is algebraically equivalent to 0 and linearly equivalent to 0 respectively.

Since every divisor is linearly equivalent to 0 at a simple point, we have an exact sequence

0 - ~ 5,k ~ I~,k ~ B ( k ) ~ 0

where B is the Picard variety of V, defined over k.

As usual, we have an imbedding

K(V)* c Ik

on the diagonal: if f C k(V)*, then f maps on ( . . . , f , f , f , . . . ) (of course in the vector, it is the class of f rood 1 + nap).

X.7 211

We recall our Picard groups Pies(V) associated with a finite set of points of V and here we assume that S is a finite set of prime ra-

tional cycles. We have Pics, k(V) = Da,s,k(V)/D~l),k where D~,s,k consists of the divisors on V algebraically equivalent to 0, not pass-

ing through any point of S, and rational over k, and n(1) consists ~'t,S,k of those which are linarly equivalent to O, belonging to a function which takes the value 1 at all points of S, and is defined over k.

We contend that we have a surjective map

Ws : Ia,k ~ Pics, k

for each S as follows. Given f in I~,k there exists f C k(V)* such that we can write

f = f fs with f ' = 1 E Fp

for all p E S. This is easily proved by moving the divisor of f by a linear equivalence, and then using the Chinese remainder theorem in an affine ring of an affine open subset of V. We then put

~ s ( f ) = Cls( ( f ' ) )

where Cls is the equivalence class mod r}(1) Our collection of "'t,S,k" maps qPs is obviously consistent, and thus we can define a mapping

�9 I a , k ~ l i m Pics, k(V).

For our purposes here, we denote by Ca,k the image of qp in the limit and call it the group of idele classes. This is all right:

Content ion . The kernel of ~ is k(V)*.

Pro@ If f is in the kernel, then for all S there exists a function f s such that

f = f s f s

where fs is 1 in S, and ( fs) = O. All f s have the same divisor, namely (f). Looking at one prime p in S, we see that all f s are equal to the same function f , and we see that f is simply the function f .

212

We have the un i t ideles I~,k consisting of those ideles whose divisor is 0, the idele classes Ck = I k / K ( V ) * , and also the obvious subgroups of idele classes:

ca,k = h ,k /k(V)*

C..~ = k ( V ) * I . . ~ / k ( V ) * = Z~., /k*.

We keep working under Assumptions 1 and 2, of course. In that case, if K is a finite Galois extension of k, we have the two fundamental exact sequences

(i) (2)

0 ---* Z~,K --~ ZO,K ---* A ( K ) ~ 0

o ---, C~,l,- ~ c o , K ---, B ( K ) -~ o

in the category of GK/k-mpodules . For the limit, with respect to f~ one will of course take the injectve limit over all K.

From the definition, we see that

= I I k(p), p/k

where k(p) is the residue class field of the prime rational cycle p over k, i.e. k(p) = op/r%.

If K / k is finite Galois, then we write

z ,K = I I k(v)* WK

where gl ranges over the prime rational cycles over K.

w Idele class cohomology

Aside from the fundamental sequences (1) and (2), we have three sequences.

0 ---* Z O , K ~ Z K ~ Z ~ 0

0 ~ K* ~ I,,,~: ~ Cu,K ~ 0

0 - - - ~ 0 ----~ K * --* K * ----~ 0

X.8 213

and pairings giving rise to cup products:

ZK • I~,,K ~ K*

defined in the obvious manner: Given f E I~,,g and a cycle

a = E v p .p,

the pairing is

It induces pairings

(a , f ) = H f~P.

Zo,K • C.,K --+ K*

Z • K* --* K*

ZO,K • K* ~ 0

and we get an exact commutat ive diagram from the cup product

H ~ ( K *) . Hr(/.) �9 H~(c~) ,.

H 2 - r ( Z ) ^ . H ~ - ~ ( Z ) ^ . H ~ - ~ ( z o ) ^ .

taking into account that

H2(Gtc/k,K*) = (Q/Z)n

H ' + I ( K *)

H I - ~ ( Z ) ^

From the exact sequence in the last section, we get

Hr-I(B) . Hr(c.) . Hr(c.) ~ H'(B) ~ H~+I(c.)

H 2 - ~ ( A ) ̂ ,. H 2 - ~ ( Z ) o ) ̂ ~.. H 2 - ' ( z ~ ) ^ ~ H ~ - ' ( A ) ^ , H ~ - ' ( Z o ) ^

and ~4 is induced by the augmented cup, the others by the cup.

where n = (G �9 e) the cup products taking their values in this H 2. Here, as in the next diagram, H is taken with respect to GK/k, we omit the index K on the modules, and r C Z so H = HCK/k is the special functor.

214

T h e o r e m 8.1. All ~ are isomorphisms.

Proof. We proceed stepwise.

991 is an isomorphism by Tate's theorem. 9~ by a semilocal analysis and again by Tate's theorem. 9~3 by the 5-1emma and the result for ~1 and 9~2. 9~4 by the augmented cup duality already done. ~ by the 5-1emma and the result for 9~3 and 9~4.

So that 's it.

C o r o l l a r y 8.2. HI(GI~-/k, Ca,K) = O.

Proof. It is dual to H I ( z ~ ) which is 0 since we assumed the existence of a rational cycle of degree 1.

In the case of a curve, if we had worked with the true ideles JK instead of our truncated ones [K, we would also have obtained (essentially in the same way) the above corollary. Thus from the sequence

0 ~ K(V)* ---* Ja,K ~ Ca,~: ~ 0

we would get exactly

0--+ H2(K(V) *) ---. H2(Ja,K)

thus recovering the fact that an element of the Brauer group which splits locally everywhere splits globally (H 2 is taken with GK/k).

Furthermore, the curves exhibit one more duality, a self duality, of our group Fp. This is a local question. We take k a p-adic field, K a finite extension, Galois with group GK/k, and consider the power series k((t)) and K((t)) . We let F be our local group

F = K(( t ) )* / (1 + m)

where m is the maximal ideal. Then 1 + m is uniquely divisible, and so its cohomology is trivial. Hence

Hr(GK/k ,K( ( t ) ) *) : Hr(GK/k ,F) .

We have the exact sequence

0 ---* K* ---~ F ---* Z ---* 0

X.8

and a pairing

defined by

K((t))* • K((t))* K*

( f , g ) --~ ( - - 1 ) ~ ~176176 '

which induces a pairing

F x F - - - * K*.

Now we get the commutative diagram

0 ~ H r ( K *) ~ H r ( F ) ~ H * ( Z ) ----* 0

; i l

0 ~ H 2 - * ( Z A) ---* H 2 - ~ ( F ) ^ ~ H 2 - ~ ( t ( A ) ~ 0

and by the five lemma, together with Tate's theorem, the middle arrow is an isomorphism. Hence

H~(F) is dual to H2-T(F)

by the cup product.

215

we see that

Bibliography [ArT 67]

[CaE 56]

[Gr 59]

[Ho 50a]

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[HoN 52]

[HoS 53]

[Ka 55a]

lEa 555]

[Ka 63]

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H. CARTAN and S. EILENBERG, Homological Algebra, Princeton Univ. Press 1956

A. GROTHENDIECK, Sur quelques points d'alg~bre ho- mologique, Tohoku Math. Y. 9 (1957) pp. 119-221

G. HOCHSCHILD, Local class field thoery, Ann.Math. 51 No. 2 (1950) pp. 331-347

G. HOCHSCHILD, Note on Artin's reciprocity law, Ann. Math. 52 No. 3 (1950) pp. 694-701

G. HOCHSCHILD and T. NAKAYAMA, Cohomology in class field theory, Ann.Math. 55 No. 2 (1952) pp. 348-366

G. HOCHSCHILD and J.-P. SERRE, Cohomology of group extensions, Trans. AMS 74 (1953) pp. 110-134

Y. KAWADA, Class formations, Duke Math. J. 22 (1955) pp. 165-178

Y. KAWADA, Class formations III, Y. Math. Soc. Japan 7 (1955) pp. 453-490

Y. KAWADA, Cohomology of group extensions, J. Fac. Sci. Univ. Tokyo 9 (1963) pp. 417-431

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S. LANG, Divisors and endomorphisms on abelian varieties, Amer. Y. Math. 80 No. 3 (1958) pp. 761-777

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J. MILNE, Arithmetic Duality Theorems, Academic Press, Boston, 1986

T. NAKAYAMA, Uber die Beziehungen zwischen den Fak- torensystemen und der Normklassengruppe eines galoiss- chen Erweiterungskhrpers, Math. Ann. 112 (1936) pp. 85-91

T. NAKAYAMA, A theorem on the norm group of a finite extension field, Yap. J. Math. 18 (1943) pp. 877-885

T. NAKAYAMA, Factor system approach ot the isomorphism and reciprocity theorems, Y. Math. Soc. Japan 3 No. 1 (1941) pp. 52-58

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Complementary References

A. ADEM and R.J. MILGRAM, Cohomology of Finite Group~, Springer-Verlag 1994

K. BROWN, Cohomology of Groups, Springer-Verlag 1982

S. LANG, Algebraic Number Theory, Addison-Wesley 1970; Springer- Verlag 1986, 2nd edn. 1994

S. MAC LANE, Homology, Springer-Verlag 1963, 4th printing 1995

Table of N o t a t i o n

Ap~ : Elements of A annihilated by a power of p

A~, : If ~ is a homomorphism, kernel of ~0 in A

.A: Ham(A, Q / Z )

A a : Elements of A fixed by G

Am : Kernel of the homomorphism m A :: A --* A such that a ~ rna

cd : Cohomological dimension

F v : Z/pZ

G : Character group, Ham(G, Q / Z )

x a : Natural homomorphism of A G onto H~ A) or H~ A) n<a : Natural homomorphism of As into H - I ( G , A)

Galm(G) : Galois modules

Galmv(G ) : Galois modules whose elements are annihilated by a p-power

Galmtor(G) : Torsion Galois modules

G c : Commuta to r group, or c losureof commuta tor if G is topological

G v : p-Sylow subgroup of G

Grab : Category of abelian groups

hl/2 : Herbrand quotient, order of H 2 divided by order of H 1

Ha : Functor such that Ha(A) = A a

H a : Functor such that H a ( A ) = A a / S a A

I a : Augmenta t ion ideal, generated by the elements a - e, a E G

222

MG(A) : Funct ions (sometimes continuous) from G into A

M a : Z[G] | A

M s : Induced funct ions

Mod(G) : Abelian category of G-modules

Mod(Z) : Abelian category of abelian groups

s c d : Strict cohomological d imension

SG : The relative trace, f rom a subgroup U of finite index, to G

S a : The trace, for a finite group G

Tr : Transfer of group theory

tr : Transfer of cohomology

Z[G] : Group ring

I n d e x

Abutment of spectral sequence 117

Admissible subgroup 177

Augmentation 10, 27

Augmented cupping 109, 198

Bilinear map of complexes 84

Brauer group 167, 202, 209

Category of modules 10

Characters 28

Class formation 166

Class module 71

Coerasing functor 5

Cofunctor 4

Cohomological cup funetor 76

Cohomological dimension 138

Cohomological period 96

Cohomology ring 89

Complete resolution 23

Conjugation 41,174

Consistency 173

Cup functor 76

224

Cup product 75

Cyclic groups 32

Deflation, def 164

Delta-functor 3

Double cosets 58

Duality theorems 93, 190, 192, 199

Edge isomorphisms 118

Equivalent extensions 159

Erasable 4

Erasing functor 4, 15, 134

Extension of groups 156

Extreme isomorphisms 118

Factor extension 163

Factor sets 28

Filtered object 116

Filtration 116

Fundamental class 71,167

G-module 10

G-morphism 11

G-regular 17

Galm(G) 127

Galmp(G) 138

Galmtor(G) 138

Galois group 151,195

Galois module 127

Galois type 123

Grab 11

Herbrand lemma 35

Herbrand quotient 35

Hochschild-Serre spectral sequence 118

225

HomG(A, B) 11

Homogeneous standard complex 27

Idele 210

Idele classes 211

Induced representation 52, 134

Inflation inf~/c' 40

Invariant invc 167

Lifting morphism 38

Limitation theoreem 176

Local component 55

Maximal generator 95

Maximal p-quotient 149

Ma(A) 13, 19

Mod(G) I0

Mod(R) lo

MS(B) 52

Morphism of pairs 38

Multilinear category 73

Nakayama maps 101

Periodicity 95

p-extensive group 144

p-group 50, 126

Positive spectral sequence 118

Profinite group 124, 147

Projective 17

Reciprocity law 198

Reciprocity mapping 173

Regular 17

Restriction res~ 39

Semilocal 71

226

Shafarevich-Weil theorem 177

Spectral functor 117

Splitting functor 5

Splitting module 70

Standard complex 26, 27

Strict cohomological dimension 138, 193

Supernatural number 125

Sylow group 50, 126, 137

Tate pairing 195

Tare product 109

Tate theorems 23, 70, 98

Tensor product 21

Topological class formation 185

Topological Galois module 185

Trace 12, 15

Transfer of cohomology tra 43

Transfer of group theory Tra 48, 160, 174

Transgression tg 120

Translation 46, 174

Triplet theorem 68, 88

Triplet theorem for cup products 88

Twin theorem 65

Uniqueness theorems 5, 6

Unramified Brauer group 204

Weil group 179, 185

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