Capitulation and transfer kernels

K. W. G

RUENBERG

A. W

EISS

Capitulation and transfer kernels

Journal de Théorie des Nombres de Bordeaux, tome

12, n

o

_{1 (2000),}

p. 219-226

<http://www.numdam.org/item?id=JTNB_2000__12_1_219_0>

© Université Bordeaux 1, 2000, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Capitulation

and

Transfer Kernels

par K. W. GRUENBERG et A. WEISS

RÉSUMÉ. On sait _{que pour} une extension

_galoisienne

finie

_K/k

d’un _corps de

_nombres,

_{le noyau du}

_morphisme

d’extension _Clk ~

ClK s’identifie au _noyau

_{X (H)}

du transfert

_{H/H’ ~}

_A,

où H =

Gal(K/k),

A =

Gal(K/K)

_et

K

est le corps de classes de Hilbert

de K.

Lorsque

le groupe G =

Gal(K/k)

_est

abélien,

H. Suzuki _a

montré

_que

_{|G| divise}

_|X(H)|.

Nous

_appelons

_{noyau de} transfert pour G tout groupe abélien

fini X _qui s’écrit

_X(H)

_pour un certain groupe H tel _que A ~

H ~ G.

_Après

avoir caractérisé les _noyaux de transfert en termes

de

_{représentations}

entières de

_G,

nous _{montrons que} X est un

noyau de transfert pour le groupe abélien G si et seulement si on

a

|G| X =

0

_et

_{|G| divise |X|,}

ce _qui fournit une nouvelle

démonstration du résultat de Suzuki.

ABSTRACT. If

_K/k

is a finite Galois extension of number fields

with Galois _group

_G,

then the kernel of the

_capitulation

_map

Clk

~ _ClK of _{ideal class groups is}

_isomorphic

to the kernel

_X(H)

of the transfer _map

_{H/H’ ~}

_A,

where H =

Gal(K/k),

A =

Gal(K/K)

and K

is the Hilbert class field of K. H. Suzuki

_proved

that when G is

_abelian,

_|G|

divides

_|X(H)|.

We call a finite

abelian group X a transfer kernel for G if X ~

X(H)

for some

group extension A ~ H ~ G.

After

_{characterizing}

transfer kernels in terms of

_integral

rep-resentations of

_G,

we show that X is a transfer kernel for the

abelian _group G if and

_only

_{if |G|X}

= 0 and

|G| divides |X|.

Our

arguments

give

a new

_proof

of Suzuki’s result.

Let

_K/k

be a finite unramified Galois extension of number fields with

Galois _group G. The

_capitulation

kernel for is the kernel of the

nat-ural

_homomorphism

of ideal class _groups

_Clk

e

_CIK.

Suzuki

_[S]

_proved

that when G is

_abelian,

its order

_G~

divides the order of the

_capitulation

kernel. This remarkable result

_encapsulates

much of the information

pre-viously

available about

_{capitulation.}

We refer to _{the surveys}

_[J]

and

_[M]

Manuscrit requ le 23 _septembre 1999.

for relevant

_background.

Our aim here is to

_explain

a new

approach

to

Suzuki’s theorem.

The transition to _group

_theory

_(reviewed

_{in §}

₁₎

allows one to

_interpret

capitulation

kernels as transfer

_kernels,

_by

which we mean the

following:

given

a finite _group

G,

then a finite abelian _{group X} is a

transfer

kernel

for

G if there exists a _group extension A - H ~ G with A finite abelian so that

X is

_isomorphic

to the kernel of the transfer

_homomorphism

_{H] -4}

A. We shall prove the

_following

result.

Theorem 1.

_If

G is a

finite

abelian _group, then the

finite

additive _group

X is a

_transfer

kernel

_for

G

_if,

and

_{only if,}

= 0 and

I G I

divides

We outline what follows. In

_§1

we translate the

problem

into an

equiv-alent one on G-module extensions over

AG,

the

_augmentation

ideal of the

integral

group

ring

ZG. Then

§2,

the core of the _paper, is an

analysis

of the

common structural

_properties

of transfer kernels for G. This makes

possi-ble the

_proof

of Theorem 1 in

_§3.

In our final

§4

we collect some comments

and

_questions.

1. TRANSLATIONS

We

_begin

with the classical result of E. Artin.

Proposition

1. The

_capitulation

kernel

_for

_{Klk is}

a

transfer

kernel

for

G.

Here is a sketch of the

_proof.

Let K be the Hilbert class field of K and

A =

Gal( K / K).

If H =

Gal(K/k)

then there is _a commutative square

from which the

_proposition

follows

_by

_taking

kernels.

Proposition

2. The

_finite

additive _{group X} is a

transfer

kernel

for

G

if,

and

_{only if,}

there exists a G-module extension A ~ B --~ AG with A

finite

and X -

_H-1(G,

_B).

Proof.

This result is clear from the functorial

_relationship

between _group extensions over G and G-module extensions over OG

(cf.

[G]

§10.5).

As

this is not the usual

_approach

in the literature we sketch it here. A _{group extension}

yields

the G-module extension

where B = _T is induced from _7r: AH _--t A(3 and

j(a)

is the

appropriate

coset of

_{i(a) -}

1.

Conversely, given

(2),

let

Then H is a _{group with}

_{multiplication}

_{x.y -}

T(x)y

+ x + y, its

identity

element is 0 and the inverse of x is

x-1

=

-g-lx,

where

T(x) - g -

1.

The module

_homomorphism

T

gives

the group

homomorphism

H -+ G via

z e

_r(x)

+ 1 with kernel A.

If B arises from the _{group extension}

_(1),

then the hidden group H in B

gives

an extension

equivalent

to H:

where

_{u(h) = (h - 1)}

+ AA.AH. The G-coinvariants on

_B,

namely

BG

=

B/(dG)B,

are

_naturally

_isomorphic

to

AH/(AH)2,

whence to

_{H/(H, H~}

and so to

!Il

Notice that

H/~H, H~ ~

BG

is

_just

_{~(H, H~ H x}

+

(AG)B.

We claim there is a commutative _square

where

G

is the norm

_endomorphism

and the lower

isomorphism

is

induced

_{by j .}

-In view of

₍₃₎

we _may

replace

H

by

H and

view j

as inclusion. Take a

transversal

_G,

for A in H. If x E H with

_T(x)

= J~ -

_1,

then the

image

of x under transfer is

_11~

which is the same as

_1:g

and so the transfer

_image

is

Gx

as

required.

Proposition

2 follows

_{by taking}

kernels. D 2. TRANSFER KERNELS

Let G be a finite _group, not

_necessarily

abelian. Put A =

ZG/(G)

and

identify

AG

with If M is a

_ZG-module,

then

dG(M)

denotes the

minimum number of module

_generators

of M.

Theorem 2. The

_following

are

_equivalent:

(a)

X is a

_transfer

kernel

_for

G;

(b)

X is

_isomorphic

to the cokernel

_of

a

homomorphism

_{cp :}

Am

where m &#x3E;

_dG(AG)

and U is a

finitely generated

G-submodule

of

Q

(c)

X is

_isomorphic

to

MG

for

sorrce

finitely generated

G-module

M,

where

GM

= 0 and

QM

contains _a

QG-copy

of

QA;

(d) IGIX

= 0 and there exists _a

surjective homomorphism

X -

MG

with

M as in

(c).

Proof.

(a)

~

_(b).

_Using

_Proposition

2 we _may, and

shall,

assume X ri

H-1 (G,

B),

where A - B - AG. Take a free resolution of

_B,

so

deter-mining

m and S in the

_{following diagram:}

Now

_{H-1(G, B) =}

_(we

use Tate

cohomology

throughout)

and

the exact sequence

SG --+ S -4* U gives

where J is the

_{connecting homomorphism.}

Since A is

_finite,

_QS

=

QR ri

whence

SG -

and Note that U is Z-torsion-free and so U is contained in

QU.

Also

GU

= 0

gives

H-1 (G,

U)

=

UG

and

H~(G,

U)

= 0. Thus

UG 6 )Am --*

X is exact.

(b)

~

_(c).

The exact _sequence A -

_AG

of G-modules

_stays

exact when we

apply

Hom(U, -)

because U is Z-free. So we obtain the

exact _sequence

where the

_right

hand term is 0 because it is

_isomorphic

to

which is 0 because

_HomG(U,7G)

= 0 _as

It follows that the

_{given homomorphism cp :}

_{UG --~}

_AG

lifts to a

G-homomorphism Q :

U e A"’t

_{with QG}

_giving

the commutative

_diagram

-

with M =

_Coker,B.

This induces an

_epimorphism

M - X and

_hence,

by

taking

G-coinvariants,

an

isomorphism

X.

Finally, QM (1) QU

contains

_(a

_G-copy

_of)

_{QA"~ ,}

whence

_QM

contains

_QA.

(c)

~ (d)

is clear since

_{IGI I}

annihilates

_MG

=

H-1 (G, M).

(d)

~ (b) .

Choose m &#x3E;

_{dG (AG),}

_{d(X ) ~}

and take a G-free

presentation

L -4 M of

M.,

Since

GM =0,

LG

=

thereby giving

the exact sequence where U =

L/LG,

and

hence the exact _sequence

_UG

Am--*

_G

_MG.

Choose a :

_AJ-X

_G and let

{3: X - MG

be the

_given

_{homomorphism.}

Thus

_{Coker iG ^_r}

_Imoa,

which

_implies,

_by

the Lemma

_below,

that

_{ImiG -}

_{Ker,8a. Consequently}

Kera is

_isomorphic

to a

_subgroup

D of

ImiG.

There exists a _map of

UG

onto D

_(remember

we are

dealing

with finite abelian

groups),

giving

the

composite

p :

UG -~

D "

Am,

with Kera.

Again

using

the

Lemma

_below,

_{Cokerp ri}

X and so

UG

Am

-X is exact.

_Finally,

QU

e

_{QM ~ QA"}

shows that

_QA

_implies

_QU

C

Lemma.

_(i)

Given

_{epimorpltisms}

_{fi , f 2 :}

_{Az -}

_{X ,}

then

_{Ker f 2 .}

(ii)

Given

_{epimorphisms gi :}

_{Xi ,}

i =

l, 2,

with

Kergl ^-r

Kerg2,

then

_X2.

Proof.

In

_(i)

the

_{homomorphisms}

_{f1, f2}

are free

_{presentations}

of X as

So Schanuel’s Lemma and the Krull-Schmidt

_property

give

the result. For

_(ii),

dualise with

_respect

to

_Q/Z

and obtain

_XZ

by

(i).

(b)

~

_(a).

Our aim is to _prove that the X of

_(b)

is a transfer kernel in

the module-theoretic sense of

Proposition

2. We use the

_isomorphism

given by

integral duality: ~

~

_(x

H

~.x).

_{Hence p}

_corresponds

to a

uniquely

determined extension 7~’"~ ~--&#x3E; S -H U whose associated

_connecting

homomorphism

H-1(G,

U)

e is _cp

_(e.g.

11.1 in

_[GW]).

Thus

UG ~

H°(G,

S)

is exact and so X -

H°(G,

S).

Take a free

_{presentation R "}

ZGm --* AG of AG and embed S in R with cokernel A. This can be done because

Taking

the

_pushout

_along

R - A

_gives

a

_{diagram exactly}

like

(4)

_except

that A

_might

not be finite. _{In any} case

H-1(G, B).

It remains to find a submodule L of A so that

A/L is

finite and

H-1 (G,

B)

=

H-1(G, B/L).

First note that = 0

by

the middle column of

(4)

and

_{QSG --}

Hence

_BG

is finite and so

A/A

n AG.B is finite. Pick

a torsion-free G-submodule L of finite index in A n Then

A/L

is

finite;

also

GL

=

0,

whence

LG

= 0. The _exact

sequence L Y B -

B/L

then

_gives

the exact _sequence

which finishes the

_proof.

0

3. PROOF OF THEOREM 1

To _prove Theorem 1 it

_suffices,

in view of

_Proposition

2 and Theorem

_2,

to establish the

_equivalence

of

(i)

X is a

_finite

additive _group such that

IGIX

= 0 and

IGI

divides

with

(ii)

X is

_isomorphic

to

MG

for

some

finitely generated

G-module

M,

where

GM

= 0 and

QM

contains a

QG-copy of QA.

(i)

#

(ii).

_{By (d)}

of Theorem 2 it suffices to _prove X has a transfer

kernel for G as a

_homomorphic

_image.

We shall show that _any

_image

of X

of order

_~G~

is a transfer kernel.

Change

notation and call this

image

X.

So we have and shall use induction on when X =

0,

then G = 1 and so we can take M = 0.

Now let X =

Xl

Since

~X~,

_so G has _an

image

G =

GIG,

of order

_p’

and then

_{IG11 =}

_{By induction,}

for an

appropriate Ml.

Define M =

A,

where A =

ZG/(G)

is a

G-module

_by

inflation. Then

OM

= 0 since

G1M1

=

_0,

whence

(ii)

~

_(i).

We

_repeat

the classical

_argument.

Take a free

ZG-presentation

F - M of

_M,

with F = Since

MG

is

finite,

the kernel of

FG - MG

is

_isomorphic

to

_FG

and so

MG

is the cokernel of an

_{endomorphism f}

of

FG.

It follows that

_{det f}

=

Since F -~

FG

maps

Ker(F

e

M)

onto the

_image

of

_{f ,}

there is a

ZG-endomorphism

f

of F such that

fc

=

f

and

Coker f

_{maps onto} M.

(det

/)M

= 0

implies

det

/

= nG for _a suitable

integer

_n

(since

QA

₉

QM) .

Finally,

with e

_denoting

the

_augmentation

on

ZG,

e

det f

= det

f

and

thus

_nlGI

=

E(nd) =

4. REMARKS

(1)

Is the converse of

_Proposition

1 true? This is a fundamental

_problem.

An even

_stronger

form of this is the

following: given

a _group extension A --t

H e G with A

_abelian,

does there exist an unramified Galois extension L with Galois _{group H} so that L is the Hilbert class field K of the fixed field

K of A?

It should be noticed that _any

_{group H}

can be realised as the Galois

group of an unramified extension

L/k ([L],

_p.

121).

Then L C K and the

difficulty

lies in

_ensuring

that L = K.

(2)

Suppose

X is a finite additive _group such that

IGIX

= 0. Then

(a)

if X is a transfer kernel for

G,

IG/[G, G]I

I

divides

I X 1;

_;

(b)

if

_~G~

divides

_IXI,

then X is a transfer kernel for G.

Both these facts are variations of

_§3;

for

(b)

one must first show that

_if,

for each

_prime

_p, the

_p-primary

part of X is a transfer kernel for a

Sylow

p-subgroup

of

_G,

then X is one for G.

However,

neither

_(a)

nor

(b)

has a converse if G is a non-abelian _p-group.

This is obvious for

_{(b) (take A = 1).}

For

_(a),

if X is

_Z-cyclic

and of order , then X cannot be a transfer kernel for G: for if X -

MG

with M as in

(c)

of Theorem

_2,

then

_lifting

a

_generator

of

_MG

to M

_gives

a

G-homomorphism

A - M which becomes an

_isomorphism

on coinvariants

(by

Nakayama’s

Lemma and

_QA);

then

_{IXI =}

_IGI

_forcing

G to be abelian.

REFERENCES

[G] K.W. _Gruenberg, Relation Modules of Finite Groups. CBMS Monograph 25, Amer. Math.

Soc., Providence, R.I., 1976.

[GW] K.W. _Gruenberg, A. Weiss, Galois invariants for units. Proc. London Math. Soc. 70

(1995), 264-284.

[J] J.-F. _Jaulent, L’état actuel du _problème de la _{capitulation.} Sem. de Théorie des Nombres de _{Bordeaux, 1987-1988, exposé} no. 17.

[L] S. _{Lang, Algebraic} Number _{Theory. Springer,} New York, 1994.

[M] K. _{Miyake, Algebraic investigations of} Hilbert’s Theorem 94, the principal ideal theorem and the _{capitulation problem. Expo.} Math. 7 _(1989), 289-346.

[S] H. _{Suzuki, A generalization of Hilbert’s Theorem 94. Nagoya Math. J. 121 (1991),} 161-169.

K. W. GRUENBERG

School of Mathematical Sciences

Queen Mary and Westfield College

Mile End Road

London El 4NS, England

E-mail : _{K. W. Gruenbergtqmv. ac. uk} A. WEISS

Department of Mathematics

University of Alberta Edmonton

Canada, TG6 2G1