On a conductor discriminant formula of McCulloh

ILLINOISJOURNAL OFMATHEMATICS Volume40, Number 2, Summer1996

ON

A CONDUCTOR DISCRIMINANT FORMULA

OF

MCCULLOH

BART

DE

SMIT

I.

Introduction

For

certain finiterings

E,

McCullohhas indicated a canonical constructionofan orderT(E)in a Galoisalgebra

TQ(E)

over

Q,

whose Galois

group

isthe unit

group

E* of E.

In

the casethat E

Z/nZ

for somenon-negative integern, the Galois

algebraisthe nthcyclotomicfieldand

T (E)

is itsring of integers. McCullohhasused theseorderstogeneralize Stickelbergerrelations

[3], [4].

The construction ofT(E),

which isexplainedin Section2,works for all self-dual orquasi-Frobenius rings

E.

The conductor discriminantformulaforcyclotomicfields

[5,

Theorem

3.11

ex-pressesthe discriminant of acyclotomic ring of integersas a

product

ofconductors.

A

generalization of this formula to certain orders

T(E)

was usedbyMcCullohto

prove Stickelberger typeformulas for theminus-part of theclass

group

of

T

(E);see theremarkafter Theorem3in

[3]. In

atalk in Durham in 1994 McCulloh

posed

the

questionof whether thefollowing generalizationholds for all commutative self-dual finiterings E:

(1.1)

AT(E)/Z

H

"]f(x)"

ZHom(E*,*)

The conductor

_f

isthelargestE-ideal afor which )f factors

through

(E/a)*,

and

the norm

.M(a)

of an E-ideal a is its index as an additivesubgroup of

E

(or,more

precisely,the Z-idealgenerated by this index). The main resultof this note is the

following.

THEOREM 1.2. Let E be a

_self-dual

_finite

commutative ring. The conductor

product

I-Ix

_Af(x)

with_X ranging overthe homomorphisms E*

C*,

is adivisor

of

Are)/,.

We

have

_Are)/z

I-Ix

_Af(fx

_if

andonly

_if

E

is aprincipalideal ring.

Theproofisgivenin Section3, togetherwithanexplicitformula for

A

re)/z. The

easiestexamplewhere

(1.1)

fails is

E

_F2

V4],the

group

ringoverthe fieldoftwo elementsof the abeliangroup of type (2,

2). In

thiscase, the left handside is

224,

and therighthand side is

222

In

Section4weshowthat one canoften changetheringstructure of

E

tothat of aprincipalidealringwithoutchangingtheorder

T (E).

ReceivedDecember 6, 1994.

1991 MathematicsSubjectClassification.Primary 11R33;Secondary11S45,11T99.

(C)1996bythe Board ofTrusteesof the University of Illinois Manufactured in the United States of America

For

non-commutativeself-dual rings

E,

McCullohhas

suggested

comparingthe discriminant

_{ArF_,)/,}

withthe conductorproduct

I-Ix

A/’(fx)

x). Here

the

product

is taken over the irreducible_{complex characters X} ofE*. Theconductorof X isthe largesttwo-sidedE-ideal a for whichthe representation

E*

-->

GLxI

(C)

associated toX factors through

(E/a)*.

This notionofconductor canbefoundin

Lamprecht [2,

3.2]. At

presentit is notevenknown if oneinequality holdsin thisgenerality.

2. Terminology

2.1. Self-dual rings. The dual

D(A)

of a finite abelian

group A

is defined tobe

the

group Hom(A,/zoo), where/zo

isthe

group

ofroots ofunityin a fixedalgebraic

closure

Q

of

Q.

Let E

be a finiteringwith

(not

necessarily commutative). The dual

D(E+)

ofthe additive

group

_E+

of

E

has aright-E-modulestructuregivenby (tpe)(x)

99(ex)

for all e,x

E

and 99

D(E+). We

say that

E

is self-dual if

D(E+)

isfreeof rank as aright-E-module. This isequivalentto sayingthat

E

is

injectiveas amodule over itself,andthat

E

isaquasi-Frobenius ring 1,

57-58]. A

finite commutativeringisself-dualif and

only

if it is Gorenstein.

2.2. Galoisalgebras. There is a(contravariant) equivalence of categories

be-tweenfinite

separable algebras

over

Q

andfinitef2-sets, where f2

Gal(Q/Q).

Here

f2 isa profinite

group,

and an f2-set isunderstoodtobe adiscrete set on whichf2

actscontinuously. Under thisequivalence,an

algebra

A/Q

correspondstothef2-set ofring homomorphisms Horn(A,

Q),

and a f2-set

X corresponds

to the

Q-algebra

Mapf

(X,

Q)

consistingoff2-equivariant

maps

x

-->

Q.

Givinga

separable algebra A

the structureofa Galois

algebra

with Galois

group

Gisthesameasgivingafight-G-actionon the f2-set

X

thatit

corresponds

to, insuch

awaythat thefollowingtwoconditionsare satisfied:

(i) for allcr f2, x

X

and g

G

wehave(trx)g cr(xg); (ii) forall x,y

X

thereisauniqueg

G

withxg y.

Thefirst condition

says

that

X

isa(f2,

G)-space,

and thesecondcondition

says

that

X

isaprincipal

homogeneous G-space.

2.3. Definitionof the order

T(E).

Suppose

E

isself-dual finite ring. The

group

ring

Q[E+]

of the additive

group

of

E

is afinite

separable algebra

over

Q.

A Q-algebra

homomorphism

Q[E+]

--

isjusta

group

homomorphism form

_E+

to

*,

sothe f2-set associated to

Q[E+]

isthe set

D(E+)

Hom(E+,/z),

with 2-action induced fromtheactionon/z C

Q.

Since

E

isself-dual,

D(E+)

isafree fight-E-moduleofrank 1.

Let

Sbe thesubsetof

D(E+)

consistingofthe generators

340 BART DESMIT

now definethealgebra

TQ(E)

tobe

Map(S, Q). We

havecanonicalsurjective ring homomorphisms

Q[E+]

Map(D(E+),

)

J

Mapn(S,

)

TQ(E).

SinceSis the setof generators ofafree right-E-moduleof rank1,ithasaright-action

ofthe group E*, makingit into aprincipal

homogeneous

E*-space.

This actionalso respects the left action off2 on

S,

so that

TQ(E)

is a Galois

algebra

over

Q

with Galoisgroup E*.

The order

T (E)

is defined to betheprojectionin

_TQ(E)

of

Z[

_E+

], or, equivalently,

asthe

Z-algebra

generated

by

theimageof

_E+

in

TQ(E).

It

isanorderinaproduct

of a number ofcopies of

Q((n),

where n isthe characteristic ofE. TheZ-rankof

T(E)

is #E*.

3. Proof of the theorem

In

this section we

prove

Theorem

(1.2)

and wegiveanexplicit formulaforthe

discriminantof

T (E)

in termsof thestructure of

E.

Let E

beaself-dual finite commutativering. Since

E

is Artinian, it is aproduct oflocalrings.

We

firstshowthatwe canreduce tothe case that

E

islocal.

Suppose

that

E

isaproductof two finite commutative rings:

E

El

E2.

Then

El

and

E2

areself-dual.

Moreover,

wehave

T(E)

T(EI)

(R)z

T(Ez),

sothat

Are)/z

mr2 Ar

_#E;’.

_Writing

_C(E)

_{fortheconductor product of}

_E,

T(E)/Z,_T(E2)/Z,whereri

onechecks

easily

that

C(E)

C(EI)r2C(E2)

rt _Also,

_E

_{isaprincipalidealringif} andonlyifboth

_El

and

_E2

are. Thus,thetheorem follows for

E

if we know itfor

_El

and

_E2.

We

may now assume that

E

islocal. Fix aJordan-H01derfiltration of

E

as an E-module:

(.)

0=EkCEk_l

_{C...cEI CE0=E.}

Thismeansthat each

Ei

isan ideal in

E

andthat thequotients

Ei/Ei+l

are simple

E-modules.

But

the

only

simple E-module

(up

toisomorphism)istheresidue field

k(E)

of

E,

so wehave

#Ei

qk-i,

where q #k(E). Thediscriminant of

T(E)

isgiven bythefollowinglemma. Again,thecyclotomic caseiswellknown[5,

Prop.

2.1

].

LEMMA

3.1.

_If

E

is a

_finite

localcommutative

_self-dual

ring with residue

field

of

cardinalityq,then#E

qk

withk

Z,

and

AT(E)/Z

q(kq-k-l)q

k-Proof.

Since

E

isself-dual,

E

has auniqueminimal

non-zero

ideal

H,

andthe

0 isexactlythe setof those

p

6

D(E+)

that vanish on thelargestE-ideal contained in the kernel oftp. Therefore, the charactersof

_E+

whichare notE-modulegenerators

of

D(E+)

areexactlythe charactersof

E+/H,

and itfollowsthat the canonical

map

Q[E+]----+Q[E+/H]

x

TQ(E)

is anisomorphism of

Q-algebras.

Under thisisomorphism,

Z[

_E+

is

mapped

to asubalgebra of

Z[

_E+

/

H]

x

T

(E),

whose index wedenotebyi.

We

wanttocomputethis index. The

group

ring

Z[E+

surjectsto

T

(E),and the kernel is the set of H-invariants

Z[E+]

H,

wherewelet

H

act on

_E+

bytranslation. Thus,wehave a commutativediagramwith exactrows:

0

Z[E+]

H

Z[E+]

T(E)

0

0

Z[E+/H]

Z[E+/H]xT(E)

---+

T(E)

O.

Note

that

Z[E+]

n isgenerated by formal

H-coset

sumsof

E.

Sincesucha coset-sum ismappedto q times the cosetelementin

Z[E+/H],

and

Z[E+/H]

hasZ-rank

q-,

itfollows that the cokernel of the leftmost vertical

map

hascardinality

qqk-.

By

the snakelemma it follows that

qqk-.

The discriminant of the

group

ring

Z[A

ofan abelian

group A

ofordern is n

n,

so one finishesthe

proof

by notingthat

AZ[E+I/Z

qkq’

AT(E)/Z

i2AZ[E+/HI/Z

q2qt,_q(k_l)q,_

q

(kq-k-l)qk- _["]

We

return totheproofof the theorem. Thering

E

is stilllocal.

For

each with < < k thequotient ring

E/Ei

isa local ringof order

qi.

The unitsof

E/Ei

areexactly the elements not contained in its maximal ideal, so

(E/Ei)*

has order

si

qi

qi-1.

Putting

so

this alsoholds for 0.

For

each character

X E*

--

/zolet

J:x

be thelargestE-ideal

Ei

in our filtration

(.)

for which

X

factors

over

_(E/Ei)*.

Thisdepends onthe choice of theJordan-H61derfiltration

(.). For

each with0 < <kitisclearthatexactlysicharacters of

E*

factorover

(E/Ei)*.

Thisimpliesthat the numberofcharacters_Xof

E*

with

_Ei

is_si si-1 if

-7/:

0.

It

follows that

k

H

N(*Z)"-H

(Ei)si-si-’"

zED(E*) i=1 Since

N’(Ei)

qN’(Ei-1)

for

#

0 this isequalto

k-I

q-So(qk)s,

H

q-Si

q-l+k(qk--q’-’)-(q’---l)

q(kq-k-1)q

t’-

AT(E)/Z.

i=l

Thismeansthatthe conductor discriminantformula holds forthe conductors

*

rather

thanfor

.

The first statement ofthetheoremnowfollowsfromthe observationthat

342 BART DE SMIT

Ifthe idealsof

E

arelinearly ordered

by

inclusion then

every

ideal of

E

occurs in (,),and wehave

_[

x.

Conversely,if

x

for all characters

_:

of

E*,

then

the idealsof

E

arelinearly ordered.

To

see this, let

I

be an idealof

E

and choose

maximal under the condition that

Ei

3

I.

We may

assumethat

I

E

sothat > 1.

For every

character

:

of

E*

thatvanishes on

+

I,

the assumption that

implies thatitalsovanishes on /

_{Ei. By}

duality offiniteabelian

groups

itfollows

that

+

Ei

+

I

andtherefore

I

E.

It

remains toshow thata finitelocal ring

E

isaprincipalidealringifandonlyif its ideals areordered linearlybyinclusion.

To

see

"only

if" note that

every

ideal is

oftheform x

E

for >0ifthemaximal idealof

E

is

generated

byx.

To

prove

"if"

suppose

that x,y

E.

If theidealsare orderedlinearly,thenx

E

C y

E

ory

E

C x

E,

sothe ideal(x, y)is

equal

to

(x)

orto(y).

But

then

any

non-emptysetof generators

of an E-ideal can be thinned out to a set of element;i.e.,

E

isaprincipalidealring.

Thiscompletesthe

proof

of

(1.2).

4. Changingthering structure

Ifone isonlyinterested in the structure ofthe order

T (E),

then onecan sometimes

change

the ring structureof

E

tothat ofa principal ideal ring, withoutchanging

theisomorphism classof

T(E).

In

our

example E

Y2[V4],

wherethe conductor

discriminant formula fails tohold, one

may say

thatwejust pickedthe

wrong

ring

structure on

E+,

becausethe

group

ring

E’

Y2[C4]

of the cyclic

group

of order4,

is aprincipalidealringfor whichT(E)and

T(E’)

areisomorphic.

A

more general

construction isgivenin the nextproposition.

PROPOSITION 4.1.

Suppose

that

E

is a

_finite

_self-dual

commutative ringand

_E+

ishomogeneous, i.e.,

_free

over

Z/nZ

where n char

E.

Then there exists a

_finite

commutativeprincipalideal ring

E’,

anisomorphism

of

abeliangroups

_E+

-

E’+,

and anisomorphism

of

Z-algebras

T

E)

-

T E’)

suchthat thediagram

E+

E’+

T(E)

T(E’).

iscommutative.

Proof

By

writing

E

asa

product

oflocal rings,we

may

assumethat

E

islocal.

Let

p and q

pf

be the characteristic and cardinality ofits residuefield. The characteristic n of

E

is alsoa

power

of p.

We

let r betherankof

_E+

over

Z/nZ.

The p-torsion

subgroup

of

E

has size

pr

and since it isanE-ideal,

pr

isa

power

of

q. Thisimplies thatrisdivisibleby

f,

and wepute

r/f.

of

K

by

Or,,

andlet

E’

be thering

Or,/nOr.

Thering

E’

isclearlyaprincipalideal

ring,whichalso impliesthat it isself-dual. Both

_E+

and

E.

arefree over

Z/nZ

of rank r, and the minimal non-zero ideals

H

and

H’

of

E

and

E’

arebothelementary

abeliansubgroupsoforder q.

It

is nothardtosee thatthereexists anisomorphism ofabelian

groups

_E+

--

_{E_}

that

maps H

to

H’.

Thisisomorphisminducesanisomorphism

Z[E+]

_---Z[E.]

of

Z-algebras. We

claimthat this induces anisomorphism of quotients

T (E)

T

(E’).

To

see this, werecall fromthe

proof

of

Lemma

3.1thatthekernel ofthe

map Z[

E/

--T

(E)isgenerated by formalsums of

H-cosets

of

E.

Thesesumsclearly

map

to

H’-cosetsums in

E’.

El

Onecan do the same construction forproducts of

homogeneous

rings.

For

non-homogeneous local ringsthe statement in the proposition

may

fail tohold.

To

see this, consider thering

Z[X]/(2X, X

2

₊

_{4),whichis theonly self-dualcommutative}

ring

E

withadditive

group

oftype (8,2) forwhichtheZ-rankof

T (E)

is8.

REFERENCES

1. C.W.Curtis andI.Reiner, Representationtheory_offinitegroupsand associativealgebras,Interscience,

NewYork,1962.

2. E.Lamprecht, Strukturund Relationen allgemeiner GaussscherSummeninendlichen Ringen, LJ.

ReineAngew.Math. 197(1957),1-26.

3. L. R. McCulloh, "Stickelbergerrelationsin classgroupsand Galois modulestructure"inJournes

Arithmtiques1980,Cambridge University Press, Cambridge1982,pp.194-201.

4. Galoismodulestructure_ofabelianextensions,J.ReineAngew.Math.3751376 (1987), 259-306.

5. L.C.Washington,Introductiontocyclotornicfields,GraduateTextsin Math. $3,Springer-Verlag,New

York,1982.

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