On the cokernel of the Witt decomposition map

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

G

ABRIELE

N

EBE

On the cokernel of the Witt decomposition map

Journal de Théorie des Nombres de Bordeaux, tome

12, n

o

_{2 (2000),}

p. 489-501

<http://www.numdam.org/item?id=JTNB_2000__12_2_489_0>

© Université Bordeaux 1, 2000, tous droits réservés.

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

On the cokernel

of

the

Witt

_{decomposition}

_map

par GABRIELE NEBE

Dedicated to _Jacques Martinet

RÉSUMÉ. Soit R un anneau de Dedekind et K son _corps de

frac-tions. Soit G un _groupe fini. Si R est un anneau d’entiers

p-adiques,

alors

_{l’application 03B4}

de

_{décomposition}

de Witt entre le

groupe de Grothendieck-Witt des KG-modules bilinéaires et celui

des RG-modules bilinéaires de torsion est _surjective. Pour les

corps de nombres K, on demontre

_{que 03B4}

est _surjective si G est un

groupe

nilpotent

d’ordre impair, et on donne des

_{contre-exemples}

pour des groupes d’ordre pair.

ABSTRACT. Let R be a Dedekind domain with field of fractions

K and G a finite _group. We show that, if R is a

_ring

of

p-adic

_integers,

then the Witt

_{decomposition map 03B4}

between the

Grothendieck-Witt _group of bilinear KG-modules and the one of

finite bilinear RG-modules is _surjective. For number fields

is also

_surjective,

if G is a

_nilpotent

_group of odd _order, but there are

_{counterexamples}

for _groups of even order.

1. Introduction

In

_{[Dre 75],}

A. Dress defines Grothendieck-Witt _groups

_{GW (R, G)}

for finite _groups G and Dedekind domains R. If K is the field of fractions of

R,

then there is an _{exact sequence}

where p

runs

_through

all maximal ideals of R. The

_{map 6}

is called the

Witt

_{decom position}

_{map. In} the first section of this _paper, the _necessary

terminology

is

_introduced,

to define these and more

general

Witt _groups

for orders with involution. In our

_terminology

the _groups

GW(R, G)

are

denoted

_by

_W(RG,

_{° ),}

where ° is the R-linear involution on the _group

ring

RG defined

_{by g°}

=

g-1

for all

g E G.

Dress asked to calculate the cokernel of 6. This _paper is intended to

answer this

_question

in some cases. Section 4 shows that 6 is

_surjective

for

can be used to show that in the case of number fields

K,

the

composition

6.

of 6 with the

_projection

onto

_°)

is

_surjective

for all

_prime

ideals p

of R

_(Theorem

_4.6).

The

_example

R =

Z,

G =

C4

and p

= 5 shows

that this is not true for the classical

_{decomposition}

_map of Brauer.

J. Morales

_{([Mor 90])}

_investigates

the sequence

(*)

for p-groups

G,

where

p is an odd

_prime,

and number fields K. He shows that in this situation the

cokernel of ð is

_isomorphic

to the

_{exponent-2-subgroup}

of the ideal class

group of K as in the classical case G = 1. This theorem can be

_easily

generalized

to

_nilpotent

groups G of odd order

(Theorem

5.2).

Using

an

induction theorem

_[Dre

_75,

Theorem

_2]

_(cf.

Theorem

_4.1),

one

immediately

gets

that 6 is

_surjective

for _groups of odd

_order,

if the class number of K is odd

_(see

Theorem

_5.3),

which is shown in

_{[Miy 90]}

for K =

Q.

But in

general 6

is not

_surjective

for K

_{= Q}

as one sees

_{by looking}

at dihedral

groups of order

2p

(see

Section

5.2).

The methods to

investigate

the

se-quence

(*)

in Section 5.1

heavily depend

on Morita

theory

for hermitian

forms. Therefore this

_theory

is revisited in Section 3.

2. Hermitian and covariant forms

Throughout

the _paper let R be a Dedekind domain with field of fractions K. Let A be a

_K-algebra

with K-linear involution ° and A = A° _an R-order

in A that is invariant under the involution °. Let V be a

_right

A-module

and L C V be a A-lattice in

V,

i.e. a

finitely generated

R-module that

spans V as a vector space over K such that LA = L.

Definition 2.1.

_(h)

An R-bilinear form h : L x L -~ A is called

_hermitian,

if =

h(l2, 11)’

and

h(ll, l2~) - h(ll, 12)A

for all

ll, 12

_E

L,

A e A.

(c)

An R-bilinear form b : L x L -~ R is called

_covariant,

if

_b(ll,

₁₂₎

=

b(l2,

11)

and

_{b(lIÀ, 12)}

_{= b(ll, 12A-)}

for

_{all ll, 12}

E

_L,

A E A.

For a

right

A-module M let M* :=

HomA(M,

A).

Then M* is

_naturally

a

left A-module and becomes a

right

A-module

by letting

=

A° f (m)

for all m E

_M,

A E A and

_f

E M*. The hermitian forms

_correspond

bijectively

to the

_symmetric

_{A-homomorphisms h}

- h E

_HomA(M,

_M*)

defined

_by

_h(m)(m’)

:=

h(m, m’).

Similarly

covariant forms

correspond

to

_symmetric

elements of

_M#),

where

M#

:=

R)

is a

right

A-module

_{by letting}

_{( f ~ a)(m) := f (ma°),}

for all m E

_{M, A}

E A and

f

In

_particular

if there is a functorial

_isomorphism

M* for all

A-lattices

_M,

then the

_categories

of hermitian and covariant forms are

equiv-alent. One can show that this is true if A ^-_’

_R)

as a bimodule

(see

[ARS

97,

Proposition

IV.3.8]).

Since the

_isomorphism

M# E£

M* is

is that A = RG is a _group

ring

of some finite _group G. Then the

_concepts

of

hermitian and covariant forms are

equivalent

and are used

simultaneously,

according

to which notion is more convenient to work with.

In

_{[Dre 75]}

a _sequence of

_equivariant

Witt _{groups is}

investigated.

To

in-troduce this sequence

naturally,

one also needs hermitian and covariant A-torsion modules. If M is a A-torsion

module,

then define

M* :=

A /A)

and

M#

:=

HomR (M, K/R) .

Also hermitian _resp.

covariant forms on a torsion module take values in

A/l1

_resp.

K/R.

Definition 2.2. Let M be either a A-lattice or a A-torsion module and let

h _resp. b be a hermitian _{resp. covariant} form on M. Then h _resp. b are

called

_regular,

if h :

M 2013~ M* _resp.

b :

M 2013~

M#

are

isomorphisms.

A

regular

hermitian or covariant module is called

_{meta bol ic,}

if it contains a

A-submodule U with U =

U 1

.

The set of

_{isometry-classes}

of

_regular

hermitian _resp. covariant modules forms a semi group with

_respect

to

_orthogonal

sums.

Introducing

the rela-tions

_{[M, h) -}

0 for all metabolic hermitian _resp. covariant

_modules,

one

obtains a _group, called the

equivariant

Witt _group of hermitian _resp.

covari-ant

_(torsion)

_modules,

denoted

_by

_{Wh(A, °)}

_resp.

_{W (A, °) (WTh(A, °)}

resp.

WT(A,

°)

for torsion

modules).

Let

_{(Y, h)}

be a hermitian A-module. For _any A-lattice L in

_V,

the

hermitian dual lattice

_Lh

:=

Iv

E V

_{I h(v, 1)}

E A for all 1 E

L}

is a

A-module

_isomorphic

to L* . Note that

_{(L, h)}

is

_regular,

if and

_only

if

_Lh

= L

(i.e.

L is a unimodular

R-lattice).

The lattice

(L, h)

is called

_{i ntegra I,}

if

L C

_Lh.

For _any

_integral

A-lattice

_{(L, h),}

the hermitian form h induces a

hermitian form h on the A-torsion module

by

Analogous

notations are used for covariant A-modules

(V, b).

In

par-ticular

Lt

:=

f v

E

_V

_{I b(v, l)}

E R for all I E

_L}

is the dual lattice of L

with

_respect

to b and the covariant form b induced on the A-torsion

mod-ule for _any

_integral

lattice L is

_b(v

+

_{L, w}

+

_L)

:=

b(v, w)

+ R E

K

Lemma 2.3

_{([Tho 84]jDre 75], [Mor 88]).}

Let

_{(V, h)}

_resp. a

her-mitian _resp. covariant A-module and L an

integral

A-lattice in V . Then

[~/L~] =

E

_WTh(A,

_°)

_resp. = _E

WT(A, °)

for

all

_integral

M in V.

Since the

_mapping

in the lemma _maps metabolic modules to metabolic torsion

_modules,

one obtains a well defined _map

Definition 2.4. 6 is called the Witt

_{decomposition}

map

Putting

t([L, b])

=

[L 0

K,

b],

for _any

_regular

A-lattice

_{(L, b),}

it is clear that 6 o t = 0. One

gets

an exact _sequence

(cf. [Dre

75,

Theorem

_5]).

_Working

with hermitian

_modules,

one obtains an

analogous

exact _sequence

(*h).

3. Morita

_theory

for hermitian modules

Let A = A° be _an R-order in the

separable K-algebra

A with involution

o. Let

_{(L, h)}

be a

regular

hermitian A-lattice with

endomorphism

_ring

O :=

Endn(L).

Then L is a left O-module and h induces an involution

-on 0 by

Lemma 3.1

_(cf.

_[Mor

_90,

_{§3], [Miy}

_90,

_{§3], [Knu}

_91,

_{§I.9j). Let (V, h)}

be a

regular

hermitian A-rrtodule and L C V be a

_projective

A-lattice such that

(L, h) is

regular.

Let D :=

EndA(V)

and O :=

EndA (L)

and assume that

L* 00 L ~ A and L OA L* = 0 as bimodules. Then there are

isomorphisms

ol

0’, 0"

such that

commutes,

where - is the involution on D

(and

on

0)

induced

_by

h.

Proof.

Let

_0,

_{~’, 0"}

be the

_mappings

defined on

[Mor

_90,

_p.

214,215]:

For

any hermitian A-lattice let :=

(O(M), 0(o))

where

O(M) =

Hom A (L , M)

is an

O-right

module via

( f ~o)(l)

:=

f (ol)

for all

f

E

_{0 (M),}

1 E

L, o

E O. To define the hermitian form let

_{EndA(L) =}

O for

_{f, 9}

E

_Homn(L,

_M)

be the

_composition

where ~ :

M ~

_M*,

m ~

’ljJ( m, .)

is the

_isomorphism

induced

_{by 0.}

One

easily

checks that

_{Ø( 1/J)}

is an O-hermitian form.

The inverse

_{of 0}

is defined

_by

_{~-1 (N, -y)}

=

(~-1 (N), ~-1 (-y) ),

where

because L* 00 L ~ A

_by

_assumption.

_{If ,}

I

then

Similarly

one shows that

₀

04&#x3E;-1

= id.

The

_{mapping 0’}

is defined

_analogously

and also

_0"

is

_similarly

defined

_by:

is the

_composition

One

_easily

sees that these _{maps map} metabolic modules to metabolic ones and that the

diagram

is commutative. 0

The next

_(trivial)

rule _says that the sequence

(*)

(or (*~))

is a direct _sum,

if the R-order A

_decomposes

as a direct sum of two involution invariant

orders:

Lemma 3.2. Let E =,E’ E A be a central

idempotent.

Then _any

(hermitian

or

covariant~

A-module

(L, h)

_decomposes

as the

_orthogonal

sum

(LE, h)

1

(L(1 - E), h)

yielding

a direct sum

decomposition

such that

commutes.

Recall that a

_regular

hermitian or covariant A-module M is called

anisotropic,

if the

_only

A-submodule

_U

M with U C is U =

fol.

If U

TJ1

M is a submodule of

_M,

then one

_easily

sees that M is

equiv-alent to

_(with

the induced

_regular

hermitian or covariant

_form)

in

the

_{corresponding}

Witt _group

_(see

_e.g.

_[Scha

_85,

Lemma

_5.1.3]).

Therefore each element of the Witt _group has an

_{anisotropic representative.}

Since the different

_primary

_components

of hermitian or covariant A-torsion

mod-ules are

_orthogonal

to each other and _any

_p-primary

_component

of an

anisotropic

A-torsion module is annihilated

_by

the

_prime

ideal _p

_R,

the

anisotropic

A-torsion modules are

_orthogonal

sums of _{0R A-modules}

with a hermitian or bilinear form that takes values in

respectively

group of torsion A-modules to the one of covariant or hermitian modules over Artinian

_algebras.

Remark 3.3. There is a

(non

canonical)

_isomorphism

where p

runs

_through

the maximal ideals of the Dedekind domain R.

For

_algebras

_(A

or

R/p

_0R

A)

over

_fields,

the

_anisotropic

modules are

semisimple,

because for _every submodule U of an

_anisotropic

module V one

has U n = 0 and hence V = U Q3 This shows the

following

lemma.

Lemma 3.4

_(see

_e.g.

_[Dre

_75,

Lemma

_4.2]).

Let

_~4R

be a maximale ideal.

Then _any

_anisotropic

_RIP

_0R A-module is an

_orthogonal

sum

of simple

regular

hermitian or covariant

R/ p

_0~ A-modules.

4. The

_surjectivity

of 6 for local fields.

A. Dress _proves in

_{[Dre 75]}

an

_analogon

to a theorem of Brauer on

in-duced characters: Let G be a finite _group and let

and

1t2

:=

1t2

:=

{U

G

U

has a

_cyclic

normal

_subgroup

of

_2-power

index}.

Theorem 4.1

_([Dre

_75,

Theorem

_2]).

For _any Dedekind domain R the

in-duction

_yields

a

_{surjective mapping}

- , - , __1- - - ,

The theorem of Brauer can be used to show that for a finite extension K of

_Qp

with valuation

_{ring R}

and residue class field k := where

p = 7rR is the maximal ideal of

R,

the

decomposition

map from the

ring

of

generalized

characters of G over K to that over k is

_surjective

_{(see [Ser}

_77,

Chapter

17~).

The same

_{method, using}

Theorem 4.1 also shows that the _sequence

is exact. Most of the exactness is

_already

shown in

_[Dre

_75,

Theorem

5].

The

_only

_missing

_ingredient

is the

_surjectivity

of the

_composition

ð7r :

W(KG, °)

-j

W(kG, °)

of 6 with the

isomorphism

WT(RG, °) =

W(kG,

°) (Remark 3.3)

given by

multiplication

with 7r, i.e.

b~~(V, B)~ _

If the _group order is invertible in

_R,

then this

_surjectivity

follows

_by

the

_general

Morita

_theory

in the last section. But _{it is easy} to establish a

slightly

stronger

result:

Lemma 4.2. Assume that

_IGI

is invertibl e in R. Let M be a

_simple

kG-module and

_{(bl, ... , bn)}

a k-basis

_of

the _space

_of

covariant

_forms

on

M. Then there is a

simple

KG-module

V,

a lattice

L C V,

and an

R-basis

_{(Bl, ... , Bn)}

_of

the _space

_{of integral}

covariant

_forms

on

_L,

such that

(M, bZ)

for i

=

1, ... ,

_n.

Proof.

[Ser

77,

15.5]

shows that there is a KG-module V such that M E£

L/7rL

for _any RG-lattice L in V. Let

_B~

be _any

_symmetric

bilinear form

on L such that

_bi

(mod 7r)

and define

_Bi

:=

1

(i

=

1, ... ,

n) .

Since

= bi

_(mod

7r)

for

_{all g}

E

_G,

the forms

_BZ

are G-invariant forms

_{lifting bi}

_(i

=

l, ... ,

n).

They

form an R-basis of the lattice

of all

_integral

covariant forms on L since their reductions modulo 7r form

such a k-basis for

L/7rL.

Moreover

LB. -

L for all the forms

_Bi.

Therefore

2

and

_(M,

_bi)

for i =

1, ... ,

n. 0

Z Z I

For

_{elementary subgroups}

the

_surjectivity

of 6 in

_(*)

can be seen

_by

number theoretical

_arguments:

Theorem 4.3

_([Neb

_99,

Satz

_4.3.6]).

G : = C : P be the semidirect

_product

of

a

cyclic

_group C

of

order not divisible

_by

the

_prime

l and an

1-group

P.

Then is

_surjective.

More

_{precisely, for}

_every

_simple

_regular

kG-module

_{(M, b)}

there is a

regular

RG-lattice

_{(L, B)}

such that

_{(M, b).}

Proof.

The first

_part

of the

_proof

follows

_closely

the one of

[Ser

_77,

Theorem

41].

Let _p :=

char(k) .

By

Remark 3.4 it suffices to show that all

_simple

kG-modules M that have a

_regular

G-invariant

_symmetric

bilinear form b are in the

_image

of 6.

So let

_{(M, b)}

be such a

simple

orthogonal

kG-module. . _{Assume first that}

₁₀

p. Then the

Sylow-p-subgroup

S of G is normal in

G and therefore acts

_trivially

on

M,

so M can be viewed as a

kG/S-module.

Since

_lllG/SI

the theorem follows from Lemma 4.2. 0 _{We now assume that 1} =

p.

By

induction we assume that M is a faithful

kg-module. Since the centralizer of C in P is a normal

_p-subgroup

of G and hence acts

_trivially

on

M,

we assume that

Cp (C)

= 1 _so P _acts

faithfully

on C. Now

_implies

that M is a

semisimple

kC-module. Let

M =

Ma

be _a

decomposition

of M into

_kC-isotypic

_components.

Since M is an irreducible

_kG-module,

G

_permutes

the

_Ma

_{transitively.}

Let

Ga

=

C :

Pa

be the stabilizer of

Ma .

Then M =

Ind8o:

and

Ma

is _an

irreducible

_Ga-module.

Since

_Ma

is a

_homogeneous

_kc-module,

the

_image

of the

_{representation}

_End(Ma)

is a field

1~~~),

where (

is a

primitive

root of

_unity.

Since

_{char(k) =}

_p and

Pa

is a _p-group,

there is

_{0 =I vEMa}

such that _vg = v for

all g

E

_P,.

Then

_Ma

is

a

Ga-invariant

_subspace

of

_Ma,

because C is normal in

_Ga,

and therefore

Ma

=

kv.

Identifying v

with 1

E k,

_we

identify

Ma

with

l.

Then P,

acts as

Galois

_{automorphisms}

on

_Ma.

Let k

=

K(

be the unramified extension

N N N

of K with residue class field

k ^-_’

_{fil/ j5}

where R

=

R[(]

is the

ring

of

integers

in

K and j5

= the maximal ideal of

R.

The

homomorphism

C -~ k*

lifts

_uniquely

to a

homomorphism

C

R*,

which makes

R

into a

RC-module. Since the Galois groups

Gal(K/K)

and

Gal(k/k)

are

canonically

isomorphic,

the group

P,

acts

_naturally

on R as Galois

automorphisms.

This make R into an

RG,-Iattice,

with

M~.

We now consider the invariant form b on M. To this _purpose let

M:

=

be the dual kC-module.

_Distinguish

two cases:

a)

M:

as kc-modules.

b)

M:

as kC-modules.

If one also identifies

M:

with

k,

then - :

~-4

C-1

is a k-linear

Galois

_automorphism

of k in case

a)

but not in case

b).

In case

a)

the module

Ma

has a

kGa-invariant

regular

_symmetric

bilinear

form b’ :

_{Ma x Ma -~ k,}

_{b’(x, y)}

:= Since the diff erent

isotypic

components

are

orthogonal

in this _case, the module

(M, b)

is induced from

(Ma,

By

induction on

~G~

we _may assume that M =

Ma

and G =

G~.

Let 1~+ :=

Fix(-)

be the fixed field of - in k. Then the

symmetric

C-invariant bilinear forms are the

forms bz :

(x,y)

H

tracek/k(xzy)

with

z E

k+.

_Clearly

_bz

is

_P-invariant,

if and

_only

if z E

k+

lies in the fixed field

k+

of P in

1~+.

In

_particular

the form b

_{= bz’}

for some z’ E

1~+.

Since - is a Galois

automorphism

of k

fixing

1~,

the

_{map - :}

K -~

K,

(

H

(-1

defines _a Galois

_automorphism

of

K/K.

Let

K+ K

_{be the}

fixed field of

_{(P, -~}

_Gal(K/K)

with

_ring

of

_integers

R+ and maximal

ideal

_pR+

=:

~+.

Then the G-invariant

_symmetric

bilinear forms

on k

are

the forms

_{Bz :}

_{(x, g)}

H

tracek/K(xzy)

with z E

K+. Let

Z’ E

R+

be a

preimage

of z’ E

R+/~+ _

~+,

Then Z’ E

(R+)*

is a unit and

(R, Bz~)

is a

_regular

covariant RG-module with

_(M,

_b).

Now consider the case

b),

that

Mf .

Since M is self

dual,

the

module

Ma

is

_isomorphic

to some other

_isotypic

_component

_Mo.’

of M. As

above we _may assume

by

induction that M =

Ma

₊ Then m k ₊ k

where the action of a

_{generator g}

of C is

(x

+

_g)g

:=

_x(

₊

yç-l.

Now

the C-invariant

_symmetric

bilinear forms on M are of the form

_{bz :}

_(xl

+

gl, x2 +

_y2)

H

trace(xlZY2

₊

YlZX2)

with z _E

1.

One also checks that

_bz

a),

these invariant forms can be lifted to invariant forms on R +

R

and one

finds a

preimage

of

(M, b).

0

With Theorem 4.1 this allows to conclude the

_surjectivity

of 6 for

arbi-trary

groups G.

Corollary

4.4. Let R be the vaduation

_ring

in a

_finite

extension K

_of

_Qp

with maximal ideal and residue class

_field

k =

R/ 7r R.

Let G be a

finite

group. Then there is an _{exact sequence}

Proof.

The

_proof

is based on the

following

commutative

diagram:

The vertical arrows are

surjective

by

Theorem

4.1,

so it is

enough

to

show the claim for the

_{elementary subgroups}

U E 9

U ? C2.

In

_particular

it suffices _{to prove} the

_corollary

for such _groups G that contain a

_cyclic

normal

_subgroup

of

_1-power

index for some

_prime

i. But such a _{group is}

isomorphic

to C : P for an

_1-group

P. Therefore the

_corollary

follows from

Theorem 4.3. 0

Here it is essential that A is a _group

ring.

For

arbitrary

symmetric

orders one

easily

constructs

_{counterexamples}

to the

_surjectivity

of

_ð7r:

where R :=

Z2.

Let A be the sublattice of Q3

R2 x 2

with R-basis

Then A is

_symmetric

with

respect

to where

_{T r2}

is the reduced

trace of the i-th

_component

R2 x 2 .

_Taking

the

_transpose

in each

_component

defines an involution ° on A. If

(V, B)

is a

_{simple regular}

covariant

_{Q2 0 R}

A-module,

then

_{S2(Y, B)}

= 0 _or

62(( B) = (8,1)

1

_(S,1),

where S is the

simple

A-module. Therefore

₆₂

is not

_surjective.

The

_surjectivity

of 6 for

_p-adic

fields has the

_important

_consequence, that for number fields

_K,

the

_composition

of 8 with the

_projection

on one

component

W(KG, °)

°)

is

_surjective.

This is in

_general

not true for the classical

_{decomposition}

map: Let G ^--’

C4.

Then G has

only

3 irreducible

_{representations}

over

Q,

but 4 irreducible

_{representations}

over

F5.

Therefore the 5-modular

_{decomposition}

map over the rationals is not

Theorem 4.6. Let G be an

_arbitrary

_group and R the

_ring

of

_integers

in a number

field

K. Let p a R be a

_prime

ideal

of

R. Then the

composition

7rp o 6

=: 6p : W(KG,

°)

°)

is

surjective.

Proof.

As above it suffices _{to prove} the theorem for the groups G = C : P as in Theorem 4.3. Let k :=

_R/p

and G be such a _group and

(M, b)

a

_simple

regular

kG-module. Let

_Rp

be the

_completion

of R

_{at p}

with maximal ideal

_7rR.

=

Rp

_{OR p.} Then Theorem 4.3

yields

_a

regular

RpG-lattice

(L" B,),

such that

_{(M, b) =}

_{((L,) B/L" rB,).}

Let V be the irreducible

p p p

irB.

p

KG-module such that

_Wp

_L

is a constituent of

_Vp

= V.

Then there is a

_regular

G-invariant form

_Vp

x

_Kp

and an

lattice

_{Lp 9 V.}

such that

_{(M, b) ^--’}

Let L’ :=

Lp

n V. _

Then L’ is a lattice for the localization

_R

of R at _p

_(cf.

_[Rei

_75,

Theorem

(5.2)(ii)~).

Let L be any RG-lattice in V such that

_R~~,~

0R L = L’. Then

Rp

0R L

= Let

IGI

=

paq

with

pXq

and r E Z with _{rq - 1}

_{(mod p).}

Choose _any

_symmetric

bilinear form B’ : L x L ~

_R,

such that B’ -

_Bp

(mod pa)

and let

_{B(v, w) :=}

for all v, w E V. Then

B is G-invariant and

_integral

on L and B -

_{Bp (mod p).}

Using

the same

--construction for R with _any element 7ro E R

with,7ro

== 1r

_(mod

p 2),

one

finds

_(k

_0p

_{B) =}

_{(M, b).}

D

5. The cokernel of 6 for number fields.

In this section let K be a number field and R its

_ring

of

_integers.

If G is a finite _group then one has a

forgetful

_map:

W(RG, °)

~

W(R),

_mapping

an

orthogonal

RG-lattice

(L, b)

onto the

_underlying

R-lattice

_{(L, b).}

Let

Wo(RG, °)

be the kernel of this _map.

_Analogously

one defines

Wo(KG, °)

and

_{° ) .}

Then one has an exact

diagram

where

_Co

and C are the

respective cokernels,

W(R), W(K),

and

WT(R)

are the classical Witt _groups of

regular

_symmetric

bilinear forms. The

exactness of the last row is shown in

[Scha

85,

Theorem

6.6.11]

(cf.

also

Remark 5.1. For all finite _groups G and number fields

_K,

the cokernel of 6 has an

epimorphic

_image

C(K)IC(K)2.

5.1.

_Groups

of odd order. This section

_mainly

intends to

_give

a _survey on known

results,

though

most of them are stated more

generally

as the ones in the literature.

Theorem 5.2

_(cf.

_[Mor

_90,

_Corollary

_3.10]

_{for p-groups}

_G).

Let G be a

nilpotent

group

of

odd order and let K be a number

_field

with

_ring

_of

inte-gers R. Then the cokernel

of

6 is

isorrzorphic

to the

exponent-2-factor

group

C(K)IC(K)2

of

the class _group

_C(K)

_of

K.

Proof.

Let G

_P,

x ... x

P,"L

where Pi

is the

largest

normal

pi-subgroup

of G and _{pl, - P,,,t} are distinct

_primes.

We

_proceed

_by

induction on m to

show that the restriction

₆₀

of 6 to

_{Wo(KG, °) in}

_diagram

_(**)

is

_surjective.

If K is not

_totally

_real,

then

_{Wo(KG, °)}

= 0

_(cf.

_[Mor

_{90, Proposition}

3.3~)

and we are done. So assume that K is a

totally

real number field.

If m = 0 then G = 1 and the statement is trivial. For _m =

1,

the

theorem is

_[Mor

_90,

Theorem

_3.9].

Assume that m &#x3E; 0. Let S :=

{p

4 R ~ E _{p, p}

_prime

ideal

_}.

First we show that

°)

is in the

_image

of

60.

To this

purpose let p E S. Then _pi E _p for some 1 i m.

Since Pi

is normal in

G,

it acts

_trivially

on all

_simple

So the

_simple

_orthogonal

R/pG-modules

are modules for

G/Pi.

By

the induction

hypotheses

these

modules are in the

_image

of

60.

Hence the cokernel of

60

is an

_epimorphic

image

of the factor _group

Therefore the cokernel of

₆₀

is

_isomorphic

to the

_{corresponding}

cokernel

_Cs

in the localized situation defined

_by

the _{exact sequence}

l12

is a maximal order in KG

where

_{Al, ... ,}

are maximal orders in the

simple

constituent of

_KG,

that

do not

_correspond

to the trivial

_{representation}

of KG. Moreover

_Ai

=

l12

since K is

_totally

real.

_By

Lemma 3.2

_°)

_{° ) .}

TG-l 1=

As in the

_proof

of

_[Mor

_90,

Theorem

_3.4

one finds for _every

_simple

self dual KG-module V an G-covariant form on V and an

R[£]G-lattice

that is self

II

dual with

_respect

to this form. The

_{endomorphism ring}

of this lattice is

the Morita

_theory

of Section

_3,

one _proves

Cs

= 0 _as in

[Mor

90,

Theorem

3.9].

D

To deduce the

_surjectivity

of

₆₀

for

_arbitrary

_groups of odd

_order,

one

has to show the

_surjectivity

of the induction _map

_{(Theorem 4.1)}

also for

Wo

and

_WTO,

which I could not establish. So we have to restrict to the case that the class number of K is odd to show:

Theorem 5.3

_(cf.

_[Miy

_90,

Theorem

_C]

for K =

Q).

Let G be a _group

of

odd order and assume that odd. Then 6 is

_surjective.

Proof.

This follows

_immediately

from the

_surjectivity

of 6 for the

_elementary

subgroups

of G shown in Theorem 5.2 and Theorem 4.1. D

5.2. A

_{counterexample}

to the

_surjectivity

of 6 for K =

Q:

Dihedral _groups.

Proposition

5.4. Let _p &#x3E; 2 be a

_prime,

G :=

(.r,?/ ~ I

~p =

y2 =

1) ~ D2p

the dihedral _group

_of

order

_2p

and K :=

Q[(p

+

_{~P 1~}

the maximal real

_{subfield of}

the

_{p-th cyclotomic field.}

Then

is exact.

Proof.

It remains to show that

Using

the

_{argumentation}

in

_[Scha

_85,

_p

_176/177]

that shows the

surjec-tivity

of the

_{Witt-decomposition}

_map for G = 1 and K =

Q,

one sees that

GW(Fp,

G)

GW(IFr, G)

is in the

_image

of 6

_(this

follows also from the

_surjectivity

of 6 for

C2

_[Mor

90,

Theorem

_2.3]).

Let S :=

Z[~].

Then

(*)

is exact if and

_only

if

is exact. The _group

_ring

SG is

_isomorphic

to

SC2

s3

R(p~2x2,

where R :=

7G~~p -E- ~p 1~

is the

_ring

of

_integers

in K. Hence

_by

Lemma

_3.2,

the _sequence

(*)p

is a direct sum of two sequences. One

easily

sees that the _sequence

is exact. So we

_only

have to deal with the other direct summand of

_SG,

which is Morita

_equivalent

to

_R(p~.

To

_apply

Lemma 3.1 one has to

con-struct a unimodular hermitian SG-lattice in the irreducible

QG-module

V

of dimension

_p-1.

But V can be identified with

Q [(p],

where x acts as

mul-tiplication

by

the

_{primitive p-th}

root of

_unity

_(p

and _y as the Galois

auto-morphism

(p

-

(~- 1.

Then h : V x V - G defined

_by

h( (;, (p3) :

:= E G

is an G-hermitian form on V. Let L :=

S[(p].

Then

(L, h)

is a

_regular

SG-lattice.

_By

Lemma 3.1 one can

replace

by

R[]

=

p p _p

if one considers Witt groups of hermitian forms. But the involution on

]

is

_trivial,

so

by

a classical result

(see

_e.g.

[Scha

_85,

Theorem

6.6.11~,

[MiH

73,

Example

IV.3.4~),

the

_following

_{sequence is} exact:

where 7

runs

through

the

prime

ideals of the Dedekind domain

R[-!].

The

prime

ideal of R over _p is

_{generated by}

_{(~p -}

_(;1)2

and hence

_principal.

Therefore the class _group

_C(K)

of fractional R-ideals in K is

_isomorphic

to the class _group

_{C(R[)] )}

of fractional

_R[)]-ideals

and the cokernel of 6 is

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Gabriele NEBE

Abteilung Reine Mathematik Universitat Ulm

89069 Ulm

Germany