Topics in Cohomology of Groups

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. ( T a t e ) . 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 : H r ( G , M) ---* H r + 2 ( G , A | M) be the cup product relative to the bilinear map A x M --+ A | M , i.e. such that

a~(A)=aUA

for

AEH~(G,M).

IV.7

99

T h e n a~ is a n i s o m o r p h i s m f o r all r E Z . P r o o f . As in the main theorem on cohomological triviality (Theo r e m 3.1 of Chapter III, we have exact sequences 0

~

A

0

, A|

~

AI

)

~ A'|

I

, 0

~ I|

) O.

T h e exactness of the second sequence is due to the fact that the first one splits. In addition, A' | Then

is cohomologically trivial. Let us put fl = 5~. a r ( ~ ) = a~ = (6~)~ = ~ ( ~ ) .

If we now use the exact sequences 0

0

,

i

,

z[a]

,

, Z[G]|

) I|

z

,

) Z|

0

, 0,

we find T h e 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 isomorphism Z | M ~ M. This concludes the proof of T h e o r e m 7.1. We can rewrite the commutative diagram arising from the theorem in the following manner. H~

•

H"(M)

~ H"(Z|

Hi(I)

x

Hr(M)

,

Hr+I(I|

H2(A)

x

Hr(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 Zfree, 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, H o m ( M , A ) ) ~ H 2 - ~ ( G , M ) A Proof. We shift dimensions on A twice. Since A' and ZIG] are Z-free, it follows that the sequences 0

, Hom(M,Z)

----*

0

, Hom(U,I)

---

Hom(M,A')

, Hom(M,_r)

Hom(M,Z[G])

, Hom(M,Z)

, 0 ----* 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(Hom(m,/))

x

H2-~(M)

,

Hi(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

We know that 5 : H i ( Q / Z ) ~ find the pairing H~

(a,n)~-+na.

H2(Z) is an isomorphism, so we

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 40.

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 Theorem 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

f9

We now pass to the general case. T h e o r e m 8.2. Let a = a ( c r l , . . . , a ~ ) be a standard cochain in c ~ ( a , A) for r >= 1 For each ~ ~ a , d e , he a map

a~+a*~"

of

C " ( G , A ) ---+C " - 2 ( G , A )

by the formulas: (u, ~-)(.) = c(~-) (a 9 T)(.) = E a(p, T)

if r = I if r = 2

pEG

(a,~-)(o-~,...,~-,._~)= E a(o',,...,~,._~,p,r)

if ,~ > 2.

pGO

Then for r >=1 we have the relation

( ~ a ) , ~ = ~(~,T). If a is a cocycle representing an element a of H r ( G , A ) , then (a * ~') represents a U ~ 9 H r - 2 ( G , A ) .

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 c o m m u t a t i v i t y 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 c o m m u t a t i v i t y relation for r = 2. For r > 2, the proof is entirely similar and is left to the reader. F r o m the c o m m u t a t i v i t y relation, one obtains an i n d u c e d homom o r p h i s m on the cohomology groups, n a m e l y ~-"H'(A)

~ H'-2(A)

for

r >__2.

For r = 2, we have to note t h a t if the cochain a(~z) is a coboundary, t h a 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)

[ ( S X ( / ) = --Of]

•

(~ - ~)

,

,

X

(5--1(T -- e)

I

)

(f(~

- ~))

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 H i ( A ) is represented by a standard cothen/or each ~ e a we have a(~) e As~ and

cycle a(~),

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 E p a@, ~) C A% and

C o r o l l a r y 8.4. The duality between H i ( 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 P r o p o s i t i o n 8.5. Let U be a subgroup of G. (i) For T e U,A 9 M o d ( G ) , a 9 H r ( G , A ) we have trV(r

(ii)

U rest(a))

= r_ U (~.

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 ---+m B a

and m S u B ---+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)) = tr(a) U/3. One can also use the Nakayama maps, as follows. For T fixed, we have two maps

a~-+~rUa

and

a~-+trg(~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)) = c,oeU E

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 amorphism 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 f o 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 cornmutativities 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)

lu~

x

•

lid

H2(XZ)

H (Z)

'

.

16

H2(A)

Io~

H~

= z/nz

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. If 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.

CHAPTER

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 T a t e p r o d u c t , we mean the data of two exact sequences O_._.,A'A+AJA"_.._,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") ~(u) lE(u)

9 Fr+'+l IF(u)

H~(fit'')

9 Fr+'+l(C)

x e'(/~")

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

0

~ M'

~ M

, M ~'

~0

1

1

0

l

, 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

V.1

111

as well as bilinear maps

A' x B - - - ~ C A x B' --~C

B' x B--~ M c M x B'---+ M c X' x B--* X c X x B' ---* X 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 H~+I(A'')

x

E'(B")-

i, " 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 H a an augmented product. Let U be a subgroup of finite index. Given a Tate product (A, B, C), let

a

t!

6 H~(G,A")

and /3" 6 H~(U,B"). Then

U.og Z") =

trU/m'~ G,- ,

Proof. Both sides of the above equality define an augmented cupping H a • H u --* H a , 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 inf(~") 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 0 --,

jA''~0

B' 2* B ~

B" ~

0

as well as a Tare product, that is bilinear maps in Mod(G): AI x B ~ C

and

AxB 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), where a l, bI are determined as follows. We choose a E A such that ja a II and b E B such that jb ~ b" . Then a I, b 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,

where the cocycle az is determined by the formulas 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

H r ( 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 = Sf .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

Some properties 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 ,) X

H'(B) u]. H~+~+~(C)

---.

H~(A)

--~

X

*-- H ' + I ( B ') u~ -~ H~+~+~(C)

H~(A,,)

6_~ H~+I(A ,)

X

~-

H'(B")

iU~ug -~ H~+,+~(C)

X

0--

H'(B) ~u

-~ 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: Hr(A~)

,

X

H~+I(B~)

H r ( A '') X

,

~

H ~ ( B ")

, Hr+S+a(C)

gr+~+l(C) id

V.8

115

Note that the coboundary m a p 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 m a p as in Theorem 3.2.

CHAPTER

VI

Spectral Sequences We recall some definitions, but we assume that the reader knows the material of Algebra, Chapter XX, w on spectral sequences, their basic constructions and more elementary properties.

w

Definitions

Let 9 / b e 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

~ DF 1 DF 2 D...DF

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 t h e n 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 Gr(F)=OGrP(F p>o

)

where

G r P ( F ) = F P / F p+I.

We m a y view G r ( F ) as a complex, with a differential of degree 0 i n d u c e d by d itself, and we have the homology H ( G r P F ) . Filtered 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

A s p e c t r a l s e q u e n c e in A is a family E

=

(EPr'q, E n) consisting

of: (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 -o rc~p'q + l= r

--T

O.

(3) Isomorphisms _~,~"q 9 Ker(d~ 'q ) / I m ( 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 ~~i~,P,q q r , 9

]:i~P,q

m•

and

u '~ " E = ---+ E ''n.

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)

En = 0

for n < 0

F m ( E ~) = 0

if m > n

Fro(E") = E n

if m __
In what follows, we assume that all spectral sequences are positive. We have inclusions

E n : F~

'~) D F I ( E '~) D . . . D F n ( E '~) D F ~ + ~ ( E n ) : O.

T h e isomorphisms ~0,~: E0,~ --~ Gr0(E n) = F O ( E n ) / F I ( E

TM) =

E,~/FI(E,~)

Z-,0. E - ; 0 ___, G r " ( E '~) = F'~(E n) will be called the e d g e , or extreme, isomorphisms of the spectral sequence. T h e o r e m 9.6 of Algebra, Chapter XX, shows how to obtain a spectral sequence from a composite of functors u n d e r 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. T h e n we have two functors:

A ~-+ A N of M o d ( G ) i n t o Mod(G/N) B ~ B G/N of M o d ( G / N ) into Grab (abelian groups). Composing these functors yields A ~-+ A c. Therefore, we obtain the Grothendieck spectral sequence associated to a composite of functors, such that for A E Mod(G):

E~'q(A) = HP(G/N, H q ( 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. 2.3. Let G be a group and N a normal subgroup. Then for A E Mod(G) we have an exact sequence: Theorem

0---+ H I ( 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 r e s s i o n , and is denoted by tg, so tg" H I ( 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) reL 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 H I ( 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 H r ( N , A) = 0 for r > O. Then

~2p,o . H P ( G / N , A N) --~ H P ( G , A ) 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 H r ( 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

AxB---*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) • H q ' ( 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:

O+---Ba/SG/NBN~ c ~ , n BGr

B N /IG/NB N (i n c 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)

iM

X

0 <

H'(G,A")

' : ~ , H I ( U , A '')

X

HO(GIU, B,,U) < r

H2(G/U,C U)

X

HO(G,B,) ( s~

, H2(C,C) inf

The two horizontal sequences on top are exact.

Ho(U,B,, )

, H2(U,C). tr

CHAPTER

VII

Groups of Galois Type (Unpublished article of Tate) w

Definitions and elementary properties

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 G a l o i s 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 p r o f i n i t e . 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 ) = y T is contained in xS. We define a partial order by putting ( T , F ) <=(T1, f l ) if T C T1 and f l ( z S ) 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 ( z S ) is not empty because of the hypothesis on

VII. 1

12~

the maps fi. The intersection A f i ( z S ) 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 ( z S ) = yT. Then f ( z S ) 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

xiuS~xiuT

for

uE 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 II

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 : n s i ) = l c mi . ( a : s , ) 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 r o u p 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 " N V ) 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 G a l o i s m o d u l e 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. If B C Mod(G) is injective in Mod(G), then Bo is injective in G a l m ( a ) . / f A C Galm(G), then there ezists an injective M C G a l m ( 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 H o m c ( 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 G a l m ( 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

Cohomology

(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 C~

if

r=0

= A

C~(G,A) = groups of maps

f'G"~A

(fort >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 functot 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 H e . T h e o r e m 2.2. Let G be a group of Galois type. Then the cohomoIogical functor Ha on Galm(G) is such that:

H r ( G , A ) = O for H~

r > O.

= 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

H r ( G , A ) ,~ dir limH~(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. Aland also Exercises 16 - 26. We return to such limits in (c) below.

gebra, Chapter III, w

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

H I ( 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(ar) 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 g r o u p . 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 c2 o f o Ar. 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) ---+H r ( 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 i m i 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 limAi. 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, " H r ( G , A ) - - * dir lim H r ( G i , A i ) . Proof. This is a generalization of the argument orem 2.3. It suffices to observe that each cochain uniformly continuous, and consequently that there normal subgroup U of G such that f depends only

given for Thef : G r ---+ A is exists an open 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 V

Hr(V,A),

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. If 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 , a n d induced representations. We are going to define an erasing functor M c 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:

M S ( 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 bifunc-

tors 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 particular, 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 M a = 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, M g ( 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, M s ( 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 ( A 1 , . . . , A n , B ) if f is multilinear in Mod(Z), and f(cral,... ,aan)=crf(al,...,an)

for all

aeG,

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 e q u e n c e . The results concerning spectral sequences apply without change, taking into account the uniform continuity of cochains. We have a functor F : Galm(G) --+ G a l m ( 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 We have to show that for all cr E Hr(N, A) there exists subgroup U such that ~ , a = a for all ~r E U. But by dimensions, there exist exact sequences and coboundaries such that a = f l , . . . ,f,-a0

with s0 E H ~

~r, = id. an open shifting f l , . . . 6~

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) S y l o w 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 number p, then the restriction res : H r ( G , A )

(G: S) H

is prime to a prime

(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 tro 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

Cohomological dimension.

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, S O

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 c o h o m o l o g i c a l d i m e n s i o n , abbreviated cd, and s t r i c t c o h o m o l o g i c a l d i m e n s i o n , abbreviated scd, as follows. cd(a)=
if and only if H~(G, A) = 0 for all r > n and A E Galmtor(G)

cdp(G) __ n and A 9 Galmtor (G) sod(G) =< n if and only if H~(G, A) = 0 for all r > n and A 9 Galm(a) scdp(G) __
0 for r > n

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)=GH~(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 = U A p ~ , 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'(p) be the p-complementary module

A(p) ' = U 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 ~(G, A'(p)) is a torsion group. Hence H r ( 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 homomor-

phism 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 / p A --* 0 O ---~ A p ---~ A J p A -* O and the corresponding cohomology exact sequences

Hr+l(pA) i. H r + I ( A ) --+ H~+I(A/pA) H~+l(dp)---, H~+I(A) J'~ H ~ + l ( d / p d ) . 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. Then cd, __
and

scdp(S) =< scdp(G),

and equality holds if ( G : S) is prime to p.

Proof. By Theorem 2.11, we know that Hr(G, for all B C Galm(S). The assertions are then quences of the definitions, together with the fact to p implies that the restriction is an injection part of cohomology (Proposition 2.15). As a special case, we find:

MS(B)) ~ H~(S,B) immediate consethat (G : S) prime on the p-primary

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 c d ( 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 c d ( 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

H n ( 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 interesting 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) ~ H I ( G p , 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

144

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 - e x t e n s i v e 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:

F

G

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 Ca, r

=

--1 ?.taU~-~o. r .

Then (ea,~-)is a 2-cocycle in C 2 ( 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 ef(a),f(r) =

a~,r/aaaar.

We define ](a) by

f ( a ) = a~u1(~ ).

VII.4

145

From the definition of the action of G on E, we have -1

o'a : uf(~)auf(~). Thus we find f ( o ' ) f ( a ) = a a u f ( z ) a r u f o . ) = a~a~uf(z)ui(r) r

= a~,a~ ef(~),f(~-)ui(~. ) ao.ruf(o.r) = f(~77"), :

which shows t h a t fi is a h o m o m o r p h i s m . It is continuous because ( a ~ ) i s a continuous cochain, and cr ~ f ( a ) is continuous. Furthermore, it is clear that f is a lifting of f, i.e. that the diagram as in the definition of p-extensive is commutative. Conversely, let E C Galmtor(G) be of finite order, equal to a ppower, 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 - m o d u l e . Taking a smaller open subgroup of U if necessary, we can suppose w i t h o u t loss of generality that a comes from the inflation of an element in H 2 ( G / U , E ) , i.e. there exists a0 E H 2 ( G / U , E ) such ~G/UI that a = mIc
f: G-~F/E be the corresponding h o m o m o r p h i s m . We are t h e n in the same situation as in the first part of the proof, and (ef(~),f(~)) is a 2cocycle representing a. Since f now exists by hypothesis, we define a~ : f ( ~ ) u ~ ) . T h e same c o m p u t a t i o n 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, without loss of generality, we m a y 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 fl : G ~ F / E U be the composite of f : G --+ F / E with the canonical homomorphism F / E ~ F / E U . We can lift fl to a continuous homomorphism fl : 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 : F ~ ( F / E ) x ( F / U ) is an injection since U A E = e. The image of G under (f, fl ) is contained in the image of i because f and fl lift 5 . Hence f = (f, fl ) : G ---* F solves the extension problem in 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 H I ( 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 r o f i n i t e free p - g r o u p g e n e r a t e d b y 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 H I ( 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. Lemma

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 H i ( G , Fp)---, H I ( F p ( 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, we can lift g to G, to get the commutative diagram 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 H i ( N , Fp) a = 0. By Lemma 3.8, we find H I ( 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.

P r o p o s i t i o n 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 H I ( T , Fp) = 0, whence H2(G/T, Fp) = 0 which suffices to prove the proposition by Theorem 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

T h e tower t h e o r e m

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) __
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 ~ ' ~ 9 Hence H'~(G, A) = 0 _~F, ~'s is a subgroup of a factor group of ~2 whenever H"(G/N,H~(N,A)) = 0, which occurs in the following cases:

r > cd(G/N)

and

s > 0 or A E Galmtor(G);

r > sca(G/N)

and

s arbitrary;

s > cd(N)

and

A E GaAmtor(G);

s > sea(N)

and

A E Galm(G).

From this we find the theorem: Theorem

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

Galois groups over a field

Let k be a field and k~ its separable closure. Let a k = 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"(GK/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 The number of generators is equal to d i m F , ( k + / p k + ) , where px = x p - x. cd G(p) <= 1, and so G(p) is profinite free.

Proof. We recall the Kummer theory exact sequence 0 --, G --* C

A k~+ --, 0,

whence the cohomology sequence 0 ---+ Fp ---* k + ~

k+ ~

HI(Gk,Fp)

--+ 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,)___+ ffl(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 Kummet sequence instead of the additive one. Cohomological dimension will be studied via Galois cohomology in k~*. Theorem

6.2. Let k be a field of characteristic r p and con-

taining 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). exact sequence.

The theorem follows at once from this

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 H n + l ( F p ) = 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 H r ( G v , k : ) = inv h m H (GE, k~), *

r

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 ezte,~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 ~ ) amounts to. of the Brauer group, because that is what H 2 ( k* 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, t h e n it is true for K/k. We are therefore reduced to the cases when either 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 transcendental extension K = k(x), in which case we have a field diagram as follows.

T

T

k

,

k~

G~

By Tsen's t h e o r e m and the corollary of T h e o r e m 6.3, we know that cd(Gk,(z)) ~ 1. T h e 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. T h e assertion is a g a i n transitive in towers, and we are reduced either to the case of a finite algebraic extension, when we can apply Proposition 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 ~ H n + I ( G / T , A T) ---* Hn+I(G,A) ~ H'~(G/T, HI(T,A)) --~ Hn+2(G/T, A T) --+ O,

vii.6

155

whence the lemma follows. Coming back to the theorem, put G = G k ( , ) 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.

CHAPTER

VIII

Group Extensions w

Morphisms of extensions

Let G be a group and A an abelian group, b o t h w r i t t e n multiplicatively. An extension of A by G is an exact sequence of groups We can t h e n define an action of G on A. If we identify A as a s u b g r o u p of U, then U acts on A by conjugation. Since A is a s s u m e d commutative, 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 t h a t j u , , = or, and we put

E a c h element of U can be written 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 t h a t UO.U r ~

atr,TUo-r,

a n d (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. ~vVedefine multiplication in U by

(a, cr)( b, r) = (aaba~,,-, or). One verifies that U is a group, whose unit element is (a,,~, -1 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

d A

>U'

,G

I )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 H 2 ( G , A ) and E ( G , A ) are isomorphic, by the bijection established 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 ,U j ,G

A'

i'

, U'

j'

) G'

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~). We have

I

jF(u~) = r

= ~o" = j'u~,,.

Hence there exist elements c~ E A' such that :

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(uo.)f(a)f(uo.) -1

These conditions imply: (I) f ( o a ) = ~ f l a ( f a ) (ii)

fa.,

for

aEA

and

o'EG.

=

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 e q u i v a l e n t to F2 if they differ by an inner automorphism of U' coming from an element o f A', that is there exists a' E A' such that Fl(u) = a'F2(u)a '-1

for all

u E

This equivalence is the weakest one can hope for.

U.

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. Corollary 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 e x t e n s i o n .

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---,AJ-~EJG--~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.

P r o p o s i t i o n 2.1. The image of the transfer Tr : E / E ~ ~ A is contained in A G, and one has: (1) T r ( a E 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) = m a ( ( E ~ @ A)/IGA). Proof. We have at once aa/a = u a a u 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) cE/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

)

Q/z

E

) G

, 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 H-3(Z) x

ft

~f*

I-I2(a) --~ H-I(A)=AN/1GA.

The duality theorem asserts that H - a ( Z ) is dual to H 2 ( 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 H 2 ( 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: 2.3. Let

Theorem

O~A~E~G~O be an eztension, and a 6 H 2 ( G , A ) its cohomology class. Then the following diagram is commutative: 0

:A/E ~nA

0

'

NA

i

E/EC

"

=

G/G c = H-2(G,Z)

"

A G

H~

A)

9

0

' 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

The deflation

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 E c , and in fact E G / E N ,,~ G / N . We obtain an exact sequence 0 --+ EN --+ E a ~ 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 a c t o r e x t e n s i o n corresponding to the normal subgroup N of G. The group

164 lattice is as follows.

/

EG

EN.

/\ A

\ / AN

\ e

This factor extension corresponds to a cohomology class/3 in H2(G/N, E N / 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) --* H 2 ( G / N , A N) such that a ~ Tr,(/~). We call this map the d e f l a t i o n . 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. Let Then on H2(G,A), we have

A E Mod(G).

inf~c/Nodeg/N = ( N "

e).

Proof. One computes with the explicit formulas on cocycles. Note t h a t the group A being written 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

H pEN

pEN

--1

a~ p j - a p , a a 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 computation, using the explicit formula for the transfer in Theorem 3.1. The fact t h a t the deflation is a h o m o m o r p h i s m is then apparent from the expression on the right side of the equality. One also sees from this right expression t h a t the expression on the left is well defined. The details are left to the reader.

CHAPTER

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. W e m a y 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: C F 2. For each open subgroup V of G we are given an embedding invy:

H2(V, A) ---+Q / Z

denoted a ~ invv(a),

called the i n v a r i a n 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 immediate from C F 2 ( b ) . Corollary

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 r e s : 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 H 2 ( V N , 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 r Y ( a ) = i n v v ( a ) . (ii) Conjugation preserves invariants, that is for

a 6

H2(V,A)

we have

invv[a] (a, a) = i n v v ( a )

Proof. Since the restriction is surjective, the first assertion follows at once from C F 2 and the formula tr o r e s = ( G ' V ) . As for the second, we recall that or. is the identity 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 G', and let a r' the cup product taken on the G'-cohomology. By the triplets theorem, it will suffice to prove that a r' satisfies the hypotheses of this theorem in three successive dimensions, which we choose to be dimensions - 1 , 0, and +1. For r = - 1 , we have H i ( 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 H i ( 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.

Proposition

1.5. Let G be a finite group and (G,A) a class formation, let a 6 H 2 ( 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

Ot r

H~+2(G, A)

tr

H~(a ', Z)

res I o'r

tr

'~ Hr+2(G', A)

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,,

l

H +2(G/U, AU) l;~ ) Hr+2(G,A)

H"(C, Z)

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)

H"(AV/AU, Z)

~" ,

Hr+2(V/U,A U)

) 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. Olr

H~(V/U,Z)

,

Hr+~(V/U, A u)

l ~. nr(V[,]/U[,l,Z)

w

,

H~+2(V[~'I/U[r],AU[d).

The reciprocity homomorphism

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 A a / S G A = 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

Aa

x

G. ~

H2(G,A)

given by

(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 S a A and whose kernel on the right is trivial. Hence A G / S a A ~ G/G c, both groups being dual to G. We

172 recall the c o m m u t a t i v e diagram:

H~

x

H2(Z)

. H~(A)

Uc~ H-2(Z)

1 1

H-2(Z)

G/G ~

1~~ •

H2(Z)

, H~

6 x HI(Q/Z)

~-~(q/z)

x

. (q/z).

4

which we apply to the f u n d a m e n t a l cocycle ~ E H 2 ( G , A ) , with n = ( G : e) and i n v c ( ~ ) = 1/n. We have ~(1) U a = c~ and i n v a ( x ( 1 ) U c~) = invG(c~) = 1In. Thus using the invariant from a class formation, at a finite level, we obtain the following f u n d a m e n t a 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 G I G 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 ~. T h e element aa in the t h e o r e m will also be d e n o t e d by (a, G). Let now G be of Galois type and let (G, A) be a class formation. T h e n we have a bilinear m a p

H~

x H i ( G , Q / Z ) --~ H 2 ( G , A ) , i . e . A a x G ---, H 2 ( 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

HO(G/U,AU)

x HX(G/U,Q/Z)

inf

" 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 h a t if ~ : G1 ---+ G2 is a g r o u p h o m o m o r p h i s m , t h e n induces a homomorphism

T h i s also holds for a c o n t i n u o u s h o m o m o r p h i s m of g r o u p s of Galois t y p e , w h e r e G c d e n o t e s t h e closure of t h e c o m m u t a t o r g r o u p . T h e n e x t t h e o r e m s u m m a r i z e s t h e f o r m a l i s m of class f o r m a t i o n theory and the reciprocity mapping. 2 . 3 . Let G be a group of Galois type and ( G , A ) a

Theorem

class formation. (i) If 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 e c v = Tray o r e c o ,

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 r e c y , 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 c o n s i s t e n c y , 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/VC

1

x

Q/Z

,

1

ala ~

> VlV

l Q/Z

,

c

Tr

X

the vertical maps being canonical. Furthermore, by consistency, we have

(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- <, ~
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

,

1 VlW

a/a

~ -

l .

ala

9

1

X ~

Q/z

,

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 ~ assertion is proved.

= A v, so the

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. L i m i t a t i o n T h e o r e m . 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 V G r ~, 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. Conversely, 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 r o u p . 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 . If 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 a d m i s s i b 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/UC~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 ( # i V , . . . ~,,~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/uc

a/u

IX.3

179

for every open subgroup U normal in G. Furthermore, U/U ~ is isomorphic to the factor group A U / s U ( A ) . We now seek an extension X of A v by G / U and a commutative diagram 0

o

~

A ~'

,

, uiu c

X

, G/U

, 0

, alU

, alU

, 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 W e l 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 t h a t , if we put E u = 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:

Av

"f~ ~ E u / E ~

inc T

T Tr

Av

, Ev/E~, Iv

W 2. For x E EG and every open subgroup U of G, the diagram is commutative:

nU

fv ~

1

Eu/E~

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

Ev/Eu ~ 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 f u is a G/Uisomorphism. 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 E a 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 E u = 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 ---+E u / E ~ ~ Ea/E~ ~ G/U ~ O. By Corollary 2.5 of Chapter VIII, we know that the transfer

Tr" E u / 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 E c , 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

9E u / E b

~ E u' / E t g.

The second condition then reads: I S O W 2. The diagram

AU

fv

, Eu/E b

id I A U

)

2~~ ]opt / i~Tlc 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 )Er

~ E

,G

0

0

, A

~ f'

, E'

,G

~0.

, E e'

e

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

: Eu/Eb

--, E v! / E , v C.

IX.3

183

We consider the cube:

Au id

inc

-...<

,

Eu / E~]

Av 1 b ~ u l

A

*

Ee

inc

--.... i tr Eu/Ev

Tr'

. E"

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 f v such that UcV.

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 "~ ( E u / E ~ ) / ( E S / 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. 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. Theorem

Av

fv , E v / E ~

l

li.c

AG

la

) EG/E b

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

Ea/Eb

' 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 g(~ri)

= ui

and we define

g'(x) = H x " ' ( m o d E~). i

We note that E a = U u i E v , in other words that the ui represent the cosets of E v in E a . 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. If U C V are subgroups of G, then f u and f v induce isomorphism~: A V / S U ( A U) ,~ E v / E u E ~ / and ( s U ) - 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 m o d u l e 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 g r o u p associated to such a topological class formation, we mean a triplet ( E c , g, 5) consisting of a topological group E a , 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: AU

.:v

>

Eu/E~

inc Av

> Ev/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: AU

A u[']

Iv

,

>

fu[~]

Eu/E~

~ Eur~l/ E U[z] 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 ~

-~ Ev/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.

CHAPTER X Applications of Galois Cohomology in Algebraic Geometry by John 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 H a 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 c a ( 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 m o d u l e s

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 (transitivity of the norm).

norm

NL/K to conclude the proof

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 t o r ---+ X

~

X/Xto

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.

H2(Gk,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 similarly that

GL/K o n

inf : H 1(aKIk, M ) --, H l(Gglk, M )

M shows

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 P 3 , we dualize the argument of P 0 , 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. Otherwise, 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 ) ~

Hi(x)~

Hi(Y)---* H2(B)---* H2(X)-+ H2(Y)

(A) H i ( A ) ~ H i ( M ) --- H i ( N ) ~- H~

~ H~

~ H~

with H : Hck. If one applies Hom(., Q / Z ) to sequence (A) w e obtain a morphism of sequence (B) in%o Hom((A), Q / Z ) . The 5lemma 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 inflationrestriction 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~ H i ( B ) - - + H i ( x ) - - + Hi(Y) H2(A) +- H2(M)+- H 2 ( N ) + - H i ( 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~ H 3 ( M ) +- H3(M)+-- H2(A)+- H2(M) I contend that H 2 ( 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 : H 3 ( M ) --+ H 3 ( 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 and any object in Galm(Gk). --

Gk

=0.fort

>3

--

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. C a s e 1. Ak = A, so Gk operates trivially, so order of A is prime ~. Then h~

hl=(k*'k*t),

h2(A)=h~

So the formula checks. C a s e 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 pgroup. If AK = 0 = A aK then f # p (otherwise GK must operate trivially). Hence H I ( G K , 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 GL/T

T I GT/k

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 into A, and thus we have to compute the order of

GL

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 GL/G * *t 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. So no horn can come from that since Ak = 0. As for HOmGL/k of #~, if f is such, then for all ~ ff GL/k, 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 kopen 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 P i c s ( V ) n /n(1)s where Da,s consists of divisors algethe factor group ~,a,s/L,~, braically equivalent to 0 whose support does not meet S, and ~'t,S r)(1) 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 indicating rationality over K . We m a y form the inverse limit inv liras Pics, K ( V ) . For our purposes we assume merely t h a t we have a group Ca,K t o g e t h e r w i t h a coherent set of surjective h o m o m o r p h i s m s cps " Ca,K ~ Pics, K ( V ) thus defining a h o m o m o r p h i s m ~ (their limit) whose kernel is den o t e d by UK. We have therefore the exact sequence

(1)

o --, u/*.

co,K

B ( K ) -+ o.

We assume t h r o u g h o u t t h a t a divisor class (for all our equivalences) which is fixed u n d e r all elements of G/,- contains a divisor r a t i o n a l over K . Similarly, we shall assume t h r o u g h o u t t h a 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. Relative to our exact sequences, we shall now define a Tate pairing. Since Ca,Kis essentially a projective limit, we shall use the exact sequence rll

/n(1)

0 --* ~_,~,s,/*./~s,/*. ---* Pies,/,- j

B ( k ) --* 0

because if oe~ D S, t h e n we have a c o m m u t a t i v e a n d exact d i a g r a m 0

'

0

)

I n ( 1)

DI,S,,K/~.,e,S,,/*$ D ~,S,K/L,~,S,I~ /n(1)

,

Pies,,/*" ;

~ B(k) lid

,

~

Pics, K

~ B(k)

~ 0

0 Now we wish to define a pairing Zo,K x UK ~

K*.

0

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* (1) Let a~ follows. Let 7 E C~,K, so 7 : 5 m T s , T s E D ~,S,K/"D <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:
z~,K(v)~ Zo,K(v)-~ A(K)-,

o ~

c%: --~

C~,K(V) --~ B(K) -~

o

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), X s 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 f U 6 g 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 putting 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 g 2 ( G k , ~ ) = Q / Z . Proof. According to the general theory of the augmented cupping, we have for each integer m > 0, 0 ---+

A(k)/mA(k)

0 ---+ ( H I ( G k , B ) m )

~

HI(Gk,Am)

---.

^ --+ H I ( G k , B m )

HI(Gk,A)m

---+ 0

^ ---. ( 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. Putting 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. (This is special for abelian varieties, better than scd < 2.)

Proof. We have an exact sequence

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 particular has the same n u m b e r 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 n u m b e r of elements, since H2(B)m = 0 for all m, so 0 since it is torsion group. Now we have the duality for H 1 , H ~ in finite layers. Theorem

4.3. Let K / k be finite Galois with group G = GK/k.

Then the pairing H~

x H I ( G , B ( K ) ) ~ H 2 ( 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 a u g m e n t e d cupping. If one uses the inflation-restriction sequence, together with the c o m m u t a t i v i t y derived abstractly for d2, and T h e o r e m 4.2, we get the following -4- c o m m u t a t i v e diagram, p u t t i n g U = GK, and

G = GK/k, U~.ug

HX(U,A) a/u X

(BU)G/u

tr

*

H~(a/V,Au) X HI(a/U,Bu)

H2(U,~*)G/Lr

, g~(a/u,a"v) U~g

Identifying H2(G, fl*) with q / z and H - 1 :

*

H2(G,fl *)

9

H2(a,a ")

inf

we get the duality between H 2

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. If K / k is a finite Galois extension with group G, then the augmented cupping

H2(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) --+ H 2 ( 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: H~-'(G/U,H~(U,B) ^) • H'-2(a/U,H~(U,B))

H~-'(G/U,A ~)

as a

~_. H-~(G/U,Q/Z)

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 G/U. Note a/so that the Q / Z in the lower right stands for H2(U, Q*), because of the invariant isomorphism.

of

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)) ~

id~

H-t(G/U, H2(U,~2*))

d21

Hz-"(G/U,A u) • H"(G/U,H~

,

U~ug

FI2(G/U,n "u)

In order to complete it to a commutative one, we complete the top line and the bottom one respectively as follows: I'I-~(G/U,H=(U, ~*) --" H2(U,n*)G/U ~

H2(G, ~2.)

lnc

H2(G/U,12 "u)

[nf

9,t d" )'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 group

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~ = N S k 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) rational over k such that C I ( 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 H I ( G K , 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 Proposition

6 . 1 . H I ( D i v ( V ) ) = O.

F r o m this, one sees t h a t our Assumption 1 is equivalent with the condition Hi(D=) = O. Now looking at the other sequence

H~

H~

Hi(Dr)--+ HI(D,)

we see t h a t A s s u m p t i o n 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 r o m t h e m we get a surjective m a p ~ " H 2 ( 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 p a r t of the S r a u e r group H2(~(V)*). In view of the exact cross we get a m a p

H2(~(V) *) ~ Hi(B).

X.6

205

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 g2(f~(V) *) D H,~(a(Y) ) D H 2 (a*) D O.

We wish to give a more concrete description of Hu2 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

*

Char(Z)

,

Char(Zk/Z~,.k)

,

Char(A(k))

,

0

0

.

H ~ ( f l ")

.

H 2 ( f I ( V ) ")

,,

HI(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 representative cocycle. By definition 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. P u t

!

t h e n (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 t h a t (aa,~-) is a cocycle, and that its class does not d e p e n d on the choices m a d e during its construction. Let (f*,~.) be another representative cocycle which is a unit at p, and o b t a i n e d by the same process. T h e n

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 t h r o u g h some point of p. T h e n H l ( D i v S) = 0 by the same argument as in Proposition 6.1 (semilocal a n d H i ( Z ) = 0) and hence taking the image of (gr in Div s we conclude t h a 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~,h1-~. T h e n still f* = f ' 95g 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 t h a t it is the b o u n d a r y 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 C~ T ~

II

C~rT

m

PEp

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

H 2 (a(V) *)

Char(Z/Z~)k

under the mapping Ov and the diagram 0 ------+C h a r ( Z ) ~

T 0 ~

H 2 ( f ~ *) ~

Char(Z/Z.)k

-----~ C h a r ( A ( k ) ) ~

t H2(a(V)

0

T *)

,

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 --~ N S ~ 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 C a s e o f C u r v e s . If we assume that V has dimension 1, i.e. is a non singular curve, then this result simplifies considerably since N S = Z is infinite cyclic, and we have also NS

= N&

= ( N S ) G~ .

We have

Hx(~vS) = O, H~(NS) = H~(Z) = Char(k*) and we get the commutative diagram 0

,

o

,

H2(D~,)

@~

Char(k?)

,

H2(Div)

,

@ Char(kj)

H2(Z)

,

,

Char(k*)

,0

,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 H 2 ( D i v ) i s 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 * Hu(ft(V) ) = N H , (2f 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 H i ( C a ) = O, just as in class field theory. We leave the details to the reader.

210 w

I d e l e s a n d idele classes

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=(...

,fp,...)

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 i v i s o 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 P i e s ( V ) associated with a finite set of points of V and here we assume that S is a finite set of prime rational 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 passing through any point of S, and rational over k, and n(1) ~'t,S,k consists 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 "'t,S,k" r}(1) Our collection of maps qPs is obviously consistent, and thus we can define a mapping 9 Ia,k

~

lim

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: Contention.

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 = fsfs 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 u n i t i d e l e s I~,k consisting of those ideles whose divisor is 0, the i d e l e 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 G K / k - m p o d u l e s . 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 ~

ZK

~

0 ~ K * ~ I,,,~: ~ 0---~0

Z ~

0

Cu,K ~ 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= Evp

.p,

the pairing is ( a , f ) = H f~P. It induces pairings Zo,K • C.,K --+ K* Z • K* --* K*

ZO,K • K* ~ 0

and we get an exact c o m m u t a t i v e diagram from the cup product H ~ ( K *)

H2-r(Z)

Hr(/.)

.

^

.

H~-~(Z)

^

9 H~(c~)

,.

H ' + I ( K *)

.

.

HI-~(Z)

H~-~(zo)

^

^

taking into account that

H2(Gtc/k,K*) = (Q/Z)n 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. From the exact sequence in the last section, we get Hr-I(B)

.

Hr(c.)

H2-~(A) ^

,. H 2 - ~ ( Z ) o ) ^

.

Hr(c.)

~.. H 2 - ' ( z ~ ) ^

~ H'(B)

~ H~+I(c.)

~H~-'(A) ^

,

H~-'(Zo) ^

and ~4 is induced by the a u g m e n t e d cup, the others by the cup.

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--+ H 2 ( 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

H r ( G K / k , K ( ( t ) ) *) : Hr(GK/k,F). We have the exact sequence 0 ---* K* ---~ F ---* Z ---* 0

X.8

215

and a pairing

K((t))* • K((t))*

K*

defined by (f,g)

--~ ( - - 1 ) ~

~176176

'

which induces a pairing

FxF---*

K*.

Now we get the commutative diagram 0 ~

H r ( K *)

; 0 ~

~

Hr(F)

~

i

H 2 - * ( Z A) ---* H 2 - ~ ( F ) ^ ~

H*(Z)

----* 0

l H2-~(t(A)

~

0

and by the five lemma, together with Tate's theorem, we see that the middle arrow is an isomorphism. Hence H~(F) by the cup product.

is dual to H2-T(F)

Bibliography [ArT 67] E. ARTIN and J. TATE, Class Field Theory, Benjamin 1967; Addison Wesley, 1991 [CaE 56] H. CARTAN and S. EILENBERG, Homological Algebra, Princeton Univ. Press 1956 [Gr 59] A. GROTHENDIECK, Sur quelques points d'alg~bre homologique, Tohoku Math. Y. 9 (1957) pp. 119-221

[Ho 50a] G. HOCHSCHILD, Local class field thoery, Ann.Math. 51 No. 2 (1950) pp. 331-347 [Ho 50b] G. HOCHSCHILD, Note on Artin's reciprocity law, Ann. Math. 52 No. 3 (1950) pp. 694-701 [HoN 52] G. HOCHSCHILD and T. NAKAYAMA, Cohomology in class field theory, Ann.Math. 55 No. 2 (1952) pp. 348-366 [HoS 53] G. HOCHSCHILD and J.-P. SERRE, Cohomology of group extensions, Trans. A M S 74 (1953) pp. 110-134

[Ka 55a] Y. KAWADA, Class formations, Duke

Math. J. 22 (1955)

pp. 165-178 l E a 555] Y. KAWADA, Class formations III, Y. Math. Soc. Japan 7 (1955) pp. 453-490

[Ka 63] Y. KAWADA, Cohomology of group extensions, Sci. Univ. Tokyo 9 (1963) pp. 417-431

J. Fac.

218 [Ka 69] Y. KAWADA, Class formations, Proc. Syrup. Pure Math. 2O AMS, 1969 [KaS 56] Y. KAWADA and I. SATAKE, Class formations II, Y. Fac. Sci. Univ. Tokyo 7 (1956) pp. 353-389

[KaT 55]

Y. KAWADA and J. TATE, On the Galois cohomology of unramified extensions of function fields in one variable, Am. J. Math. 77 No. 2 (1955)pp. 197-217

[La 57] S. LANG, Divisors and endomorphisms on abelian varieties, Amer. Y. Math. 80 No. 3 (1958) pp. 761-777

59]

S. LANG, AbeIian Varieties, Interscience, 1959; Springer Verlag, 1983

[La 66]

S. LANG, Rapport sur la cohomologie des groupes, Benjamin 1966

[La

[La 71/93] S. LANG, Algebra, Addison-Wesley 1971, 3rd edn. 1993 [Mi S6] J. MILNE, Arithmetic Duality Theorems, Academic Press, Boston, 1986 [Na 36] T. NAKAYAMA, Uber die Beziehungen zwischen den Faktorensystemen und der Normklassengruppe eines galoisschen Erweiterungskhrpers, Math. Ann. 112 (1936) pp. 85-91

[Na 43]

T. NAKAYAMA, A theorem on the norm group of a finite extension field, Yap. J. Math. 18 (1943) pp. 877-885

[Na 41] T. NAKAYAMA, Factor system approach ot the isomorphism and reciprocity theorems, Y. Math. Soc. Japan 3 No. 1 (1941) pp. 52-58

[Na 52]

T. NAKAYAMA, Idele class factor sets and class field theory, Ann. Math. 5 5 No. 1 (1952) pp. 73-84

[Na 53]

T. NAKAYAMA, Note on 3-factor sets, Kodai Math. Rep. 3 (1949) pp. 11-14

[Se 73/94] J.-P. SERRE, Cohomologie Galoisienne, Benjamin 1973, Fifth edition, Lecture Notes in Mathematics No. 5, Springer Verlag 1994 [Sh 46] I. SHAFAREVICH, On Galois groups of p-adic fields, Dokl. Akad. Nauk SSSR 53 No. 1 (1946) pp. 15-16 (see also Collected Papers, Springer Verlag 1989, p. 5)

219

[Ta 52] J. TATE, The higher dimensional cohomology groups of class field theory, Ann. Math. 56 No. 2 (1952) pp. 294-27 [Ta 62] J. TATE, Duality theorems in Galois cohomology over number fields, Proc. Int. Congress Math. Stockholm (1962) pp. 288-295 [Ta 66] J. TATE, The cohomology groups of tori in finite Galois extensions of number fields, Nagoya Math. Y. 27 (1966) pp. 709-719 [We 51] A. WEIL, Sur la th6orie du corps de classe, Y. Math. Soc. Japan 3 (1951) pp. 1-35

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; SpringerVerlag 1986, 2nd edn. 1994 S. MAC LANE, Homology, Springer-Verlag 1963, 4th printing 1995

Table of Notation Ap~ : Elements of A annihilated by a power of p A~, : If ~ is a homomorphism, kernel of ~0 in A .A: H a m ( 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 Fv: Z/pZ G : Character group, Ham(G, Q / Z ) x a : Natural h o m o m o r p h i s m of A G onto H~

A) or

H~

A)

n
G v : p-Sylow subgroup of G G r a b : Category of abelian groups

hl/2 : H e r b r a n d quotient, order of H 2 divided by order of H 1 Ha : Functor such that H a ( A ) = A a H a : Functor such that H a ( A ) = A a / S a A I a : A u g m e n t a t i o n ideal, generated by the elements a - e, a E G

222

MG(A) :

Functions (sometimes continuous) from G into A

M a : Z[G] | A M s : Induced functions Mod(G) : Abelian category of G-modules M o d ( Z ) : Abelian category of abelian groups s c d : Strict cohomological dimension SG : T h e relative trace, from a subgroup U of finite index, to G S a : T h e trace, for a finite group G Tr : Transfer of group theory tr : Transfer of cohomology Z[G] : G r o u p ring

Index 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