Conference on Commutative Algebra

Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B Eckmann, Z0rich

311 Conference on Commutative Algebra Lawrence, Kansas 1972

Edited by James W. Brewer and Edgar A. Rutter The University of Kansas, Lawrence, KS/USA

Springer-Verlag Berlin.Heidelberg • New York 1 973

A M S Subject Classifications (1970): 13-02, 1 3 A 0 5 , 13B20, 13Cxx, 1 3 D 0 5 , 13E05, 13E99, 13F05, 13F20, 1 3 G 0 5 , 1 3 H 1 0 , 1 3 H 1 5 , 1 3 H 9 9 , 13J05

I S B N 3-540-06140-1 Springer-Verlag Berlin - H e i d e l b e r g - N e w Y o r k I S B N 0-387-06140-1 Springer-Verlag N e w Y o r k . H e i d e l b e r g • B e r l i n This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the pubiisher, the amount of the fee to bc determined by agreement with the publisher. © by Springer-Verlag Berlin • Heidelberg 1973. Library of Congress Catalog Card Number 72-96859. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach]Bergstr.

PREFACE

This volume tributed

contains

to a Conference

The University 1972.

The Conference Foundation

like

on Commutative

of Kansas

Science

to express

a collection

during

Algebra

the week of May

received

under Grant

of articles

funding

held at 8 - May 12

from the National

No. G.P.

our appreciation

con-

33192

and we would

to the Foundation

for its

support. To make possible Proceedings, volume

the early

and thereby

to insure

has been prepared

manuscripts

supplied

the correction

delay

of minor

in the appearance

to include

the paper

appearance

by returning

therefore

their

timeliness,

In instances

would

have

of the Proceedings, rather

led to a

the editors

than delay

it to the author

take the responsibility

the

from the typewritten

errors

as received

of these

chose

its

for correction.

for all misprints

We

which may

the reader. We should

participants thanks

directly

by the contributors.

where

bedevil

appearance

must

Buchsbaum, who were

like to thank

for their

go to Professors Robert

involved

without

whose

Gilmer,

and cooperation.

Shreeram

Irving

Special

thanks

Phil Montgomery

also must

help.

Finally

it would

Abhyankar,

Kaplansky

in the Conference

assistance

Conrad,

efficiency

each of the Conference

enthusiasm

from

David

and Pierre

its outset

go to our Kansas and Paul McCarthy to the countless

colleagues,

and

Phelps details

involved

Edgar A. Rutter August,

and

for her

W. Brewer

Lawrence,

Paul

for their advice

such an undertaking. James

Samuel

not have been possible.

we wish to thank Deborah

in attending

Special

1972

in

TABLE SHREERAM

ABHYANKAR

OF CONTENTS

~Notes by A. SATHAYE)

On Macaulay's

Examples

1

JIMMY T. ARNOLD Prime

Ideals

J. W. BREWER, W. J. HEINZER Krull

in Power

P. R. MONTGOMERY,

Dimension

J. W. BREWER

Series

On a Problem D. DAVIS

PAUL EAKIN

Rings

26

46

in Linear Algebra

and ANTHONY

Ideals

Rings

57

HEINZER

Problem

for Rings

and E. GRAHAM EVANS,

Three Conjectures nomial Rings

50

V. GERAMITA

in Polynomial

and WILLIAM

DAVID EISENBUD

of the Functor

and DAVID EISENBUD

A Cancellation

ROBERT

and

and E. A. RUTTER

DAVID A. BUCHSBAUM

Maximal

17

E. A. RUTTER

of Polynomial

A Note on the Faithfulness S 8 R (-)

EDWARD

Rings

About Modules

61

JR. Over Poly78

GILMER

Prefer-like Conditions on the Set of Overrings of an Integral Domain ~DWARD

90

D. DAVIS

Integrally

Closed

Pairs

103

WILLIAM HEINZER Noetherian

Intersections

of Integral

Domains

II 107

MELVIN HOCHSTER Cohen~Macaulay

Modules

120

IRVING KAPLANSKY Commutative

Rings

153

Vl R. E. MACRAE On Euclidean

Rings

R. E. MACRA E and PIERRE Subfields Fields

of Index

of Algebraic

Functions

167

SAMUEL 2 of Elliptic

Function 171

JOE L. MOTT The Group tions

of Divisibility

and Its Applica194

JACK OHM Homological LOUIS

Dimension

J. RATLIFF,

JR.

Chain Conjectures JUDITH

209

and Euler Maps

222

and H-Domains

SALLY

On the Number Dimension O

of Generators

of Ideals

of 239

WOLMER V. VASCONCELOS Rings

of Global

Dimension

Two

243

PARTICIPANTS

Gover, E.

Orbach, K.

Albis, V.

Grams, A.

Orzech, G.

Arnold, J. T.

Greenberg,

Barger, S. F.

Hays, J.

Paeholke, K.

Bennett, B.

Hedstrom, J.

Parker, T.

Bertholf, D.

Heinzer, W.

Prekowitz,

Bittman, R.

Heitman, R.

Ratliff, L.

Blass, J.

Hinkle, G.

Razar, M.

Boisen, M.

Hoehster, M.

Robson, J. C.

Boorman, E.

Huckaba, J.

Sally, J.

Brown, D.

Kaplansky,

Buchsbaum, D.

Kelly, P.

Sathaye, A.

Butts, H. S.

Kohn, P.

Shannon, D.

Davis, E. D.

Kuan, Wei-Eihn

Sheets, R.

Eagon, J.

Larsen, M.

Sheldon, P.

Eakin, P.

Larson, L.

Shores, T.

Eggert, N.

Levin, G.

Smith, W.

Eisenbud, D.

Lewis, J.

Snider, R.

Enoehs, E.

Lu, Chin-Pi

Stromquist, W.

Entyne, S.

MacAdam,

Thaler, A.

Epp, S.

McDonald,

Evans, G.

MaeRae, R.

Fields, D.

Magarian,

Fischer, K.

Mott, J.

Wiegand,

Gabel, M.

Murthy, M.

Woodruff, D.

Geramita, A.

Nichols, J.

Van der Put, M.

Gilbert, J.

Ohm, J.

Vasconeelos, W.

Gilmer, R.

O'Malley, M.

Vekovius, A.

Abhyankar,

S.

B.

I.

S. B.

Orzeeh, M.

B.

Samuel, P.

Wadsworth,

A.

Wiehman, M. E.

Wiegand, R. S.

ON M A C A U L A Y ' S

EXAMPLES

Lectures by Professor Abhyankar N o t e s b y A- S a t h a y e

§l

Introduction. ideals m

Pm

INTRODUCTION

in the p o l y n o m i a l

ible c u r v e s

in a f f i n e extended

space will

3-space with to the

ous

time, b u t

After

individuals

The n o t e was

service

1 -- a r e s i d u a t i o n

In §2 w e c o n s t r u c t

(m - 2)

not h a v e any

f r o m the

f) so t h a t it does n o t

and the c u r v e s

of d e g r e e

fixed p o i n t o u t s i d e .

the d e t a i l e d

Let

Y, Z3,

k

each

m a 1

needs

at l e a s t

Note:

We note

irreducible

be

there exists m

a prime

of the set, raises

in

following

proof.

closed 3

ideal

theorem along

field and

variables p

m

c k[X,

over

A = k.

Y, Z3

Then so t h a t

for P m

generators.

that for

surface

original

ring

it do

in §4.

an a l g e b r a i c a l l y

the p o l y n o m i a l

through

of the e x a m p l e s ,

p r o o f of the

lines as in M a c a u l a y ' s

lie on any curve

(m - i)

This c o n s t r u c t i o n

are c o l l e c t e d

Theorem:

and p r o v e

t h e o r e m to be u s e d later on. 1 ~ m ( m - i) p o i n t s in the p l a n e

some q u e s t i o n s w h i c h

k[X,

Purdue S e m i n a r b y

P. Russell o f f e r e d

fix u p t e r m i n o l o g y

for the c o n s t r u c t i o n

the same

Also

it w o u l d

note.

of the note.

besides being useful

§3 p r o v i d e s

thought

a set of

(in fact on a c e r t a i n c u r v e of degree

in

t h e m to v a r i -

an e x p l a n a t o r y

Professor Abhyankar.

in the w r i t i n g

is w r i t t e n

'well known'

t o d a y and s e e m in n e e d of

Professor Abhyankar to p u b l i s h

In the r e m a i n i n g p a r t of §i w e Proposition

thus p r o v i d i n g e x a m -

and r e e x p l a i n i n g

p r e p a r e d b y A. S a t h a y e

and d i s c u s s i o n s w i t h

The 3-

3.

are r a t h e r m y s t e r i o u s

public

in a f f i n e

a l t h o u g h precise,

the p r o o f s

irreduc-

at the o r i g i n .

p r o b a b l y p a r t of the

o v e r the years,

many suggestions

generators,

that the proof,

rediscovering

be a w o r t h w h i l e

singularities

ring of the o r i g i n

m

assertions which were

Macaulay's proof.

local

n e e d at least

are ideals d e f i n i n g

l o c a l r i n g of d i m e n s i o n

It seems h o w e v e r

of p r i m e

ring in 3 v a r i a b l e s w h i c h

s t i l l n e e d at least

in a r e g u l a r

with many

given examples

In fact all the e x a m p l e s

same

ples

PRELIMINARIES

In [i] p. 36 M a c a u l a y has

generators.

ideals

AND

m = i, 2

one can take

and an i r r e d u c i b l e

curve

the ideal of an

respectively

and h e n c e w e

assume

i.I

m ~ 3.

After

fixing

m

we shall denote

General

Terminoloq Z

Let

be an a l g e b r a i c a l l y

k

k[X 0 . . . . . over

Xn]

k.

degree

Then i}

Let ible.

closed

field.

be the g ~ a d e d p o l y n o m i a l R (n) =

where

0

ring

(--). H

R (n)

{~I u, v E R (n), function Let r i n g of

i u, v

field of

coordinate

in p r o j e c t i v e homogeneous

c R (n)

H

at

homogeneous

is d e f i n e d b y

i.e.

those

Q(H)

r e s p e c t to

is i r r e d u c -

Q(H)

~(H)

H.

interchange

R (n) .

A divisor so that

divisors

clearly

We d e n o t e if

so t h a t

Let

minimal

prime

for w h i c h

H

for all

Thus

there

D(m)

denote v ~ 0],

the

D

m

on

D(m)

prime

ideals

in

m

The

local

are n o n s i n g u l a r

as a b o v e

R (n)

r e s p e c t to

and

Om

Om

in

is a

H

H = 0

is a m a p

D

for all b ~ t

for all

is d e n o t e d b y

m.

and p r i m e d i v i s o r s

given by

m O m

d e p e n d i n g on the c o n t e x t .

: ~(H)

prime -->

ideals

Z,

in

m £ ~(H).

(D + E) (m) = D(m)

~(H).

Define

The

+ E(m).

D ~ E

if and

T h e n the s e m i g r o u p of d i v i s o r s

~(H)

and d i v i s o r s

in

.

Let

the g r o u p of

finitely many

form a group by defining

a E(m)

ideal.

is a one to one c o r r e s p o n d e n c e

the g r o u p of all d i v i s o r s by

D ~ 0

~(H)

which

the set of all m i n i m a l h o m o g e n e o u s

integers

by

O m = [~ E Q(H) I v ~ m } .

of the f i r s t k i n d w i t h

denote

H

and then in fact a p r i m e d i v i s o r of the f i r s t

between minimal homogeneous

Thus w e m a y

of

modulo

of the s a m e degree,

be a h o m o g e n e o u s

m

p r i m e d i v i s o r of kind with

variables

r i n g of the h y p e r s u r -

n-space.

We s h a l l c o n s i d e r only h y p e r s u r f a c e s i,

P.

H.

m

codimension

by

m

R (n) =

~ R! n) where R! n) = [u I u i=0 i 1 is h o m o g e n e o u s of all d e g r e e s .

is the h o m o g e n e o u s

of d e g r e e

P

(n + i)

H E R! n) for some i so t h a t H ~ 0 and H 1 the images of v a r i o u s q u a n t i t i e s in R (n)

face

only

Let in

Denote

a bar

of

the ideal

~(H)

D

are c a l l e d

effective. For with

O m.

m £ ~(H)

we

shall d e n o t e b y

F o r a set of h o m o g e n e o u s ~v E Om]

and

vm(S)

0 ~ u E R (n),

homogeneous,

define

If

u

< U > = . < u > - .

For

ideal g e n e r a t e d b y

define

{U E R(n) I U

S c R (n)

associated

define

= rain [vm(a) [ a E SOm].

U E R (n), U homogeneous u For 0 # ~ E Q(H) define D E ~(H)

the v a l u a t i o n

elements

S O n = {~I u E S,

vm({u]).

vm

< u > E 9(H) and

by

(m)

~ = u ~ 0,

(~) £ ~ ( H )

I(D) C ~ R (n) homogeneous,

~ = 0

=

define

so that

to b e

For

(~) =

the h o m o g e n e o u s or



z D].

For

m E ~(H)

we have

the d i v i s o r

Dm(m' )

Clearly

divisors

elements

of

a divisor We = C.

~(H)

: A --> degree We

defined

fix,

for

f = O(F)

It f o l l o w s

defines

an

~(U(F))

1.2

Pointsets

correspondence

a prime

Z] = B Y,

and

Y,

B

that

Y,

without

identity

shall

be denoted Y Z = W r h ( ~ , ~, ~),

Z))

Note

the

divisors.

R (3) = k[X,

of the

with

divisor.

of p r i m e

Z]

We have which

u(h(X,

are defined

on

Z, W]

the g r a d e d

map

from

by

~.

A Let

where

r =

= the

defines

F

an i r r e d u c i b l e

is h o m o g e n e o u s

and w e

which

shall

assume

is n o n s i n g u l a r

an i r r e d u c i b l e ,

F E A

of degree that

~(F)

in c o d i m e n s i o n nonsingular

i.

curve.

= K.

f on

on

f

being f

shall be

be

in the p r o j e c t i v e

a vector ~ = 0

define

Fi(L,

called

one d i m e n s i o n a l

s e t of p o i n t s

[L, ~] = [h I h E L, space again.

also

= F E C

divisors

by points

L c Bn

the notes,

f

since

~(f)

f

of

(i.e.

surface

~(f)

The e f f e c t i v e

We

A. B,

u(F)

then

= Q,

multiplicities)

Let

by

Dm

k[X,

, m ~ 3

m

that

that

Put

by

m = m,.

Y,

the r e m a i n d e r

E B

irreducible

It is c l e a r

~(f)

by

if

combination

ring

of)

i

by

h.

now

so t h a t m).

C

of

ring

m # m'

call

R (2) = k[X,

shall be denoted

(the u n d e r l y i n g

if

in o n e - o n e

linear

the u n d e r l y i n g

0

defined

[

are

also

a formal

shall write

Also,

type

and w e

is j u s t

structure to

of this

=

D m 6 ~(H)

on

For

< h > ~ 29]

29) = the

(with

divisors

pl~ne.

We

on

shall

f

denote

f.

space.

or

pointsets

prime

fixed

a pointset which

part

29

we

is c l e a r l y

of

L

on

define a vector

~

as

follows. If

h E [L, 29]

Fi (L, 29)(M)

= min

A pointset

Remark

I:

implies

~

then

[ < h > ( M ) I h E {L, 29], 29

is c a l l e d

It is e a s y

h E [B n, 29},

~ = O,

~ 0

to s e e then

29

n-free

that is

if

~ if

29

Fi (L, ~}) = 29. ~ 0]

for

Otherwise

M E @(f).

F i [ B n, 29] = ~).

is

n-free

(n + l ) - f r e e .

and

there

exists

With each pointset where

~

w e d e f i n e d e g r e e of

the sum is e x t e n d e d

pointset

if

~(M)

A point defines

~ 1

we have

into ~ ( ~ ( F ) ) D = @*

for

consists

for all

M c ~,

the m i n i m a l

E ~(f)

o v e r all

i.e.

M;

otherwise

~

homogeneous

prime

ideal This

M ~ c ~. induces

a * . Any

(0, 0, 0,

G E A

with

l)

~(G)

a simple

is s i n g u l a r . prime

ideal Thus

D E ~(~(F))

B,

from ~(f) so that

divisor,

in the p r o j e c t i v e = g E B

in

for e a c h

a map

shall be c a l l e d a h o m o g e n e o u s

through

It is c l e a r that if

is c a l l e d

a minimal homogeneous

w h i c h we d e n o t e b y

of lines

~

~* = M ~ E 9(u (F)).

@ E ~(f)

~ = deg ~ = ~ ~(M)

M E ~(f).

since

it

3-space.

for some

n

then

n

< ~ G > : *. A pointset d i m k [B n, ~] n-gg

for

1.3

~

is c a l l e d

= max

n = 0,

[0,

!:

+ @ = , g'

6 Bs_ t

Proof:

there

the f o l l o w i n g

Let

~,

that

exists

u

~

If

n

min

g,

in

vm(h)

to p r o v e

nB(m ~

Note

~ : , and there

0 # ~ E Bt' exists

of

and h e n c e

gB(m ) This

u E [; u / 0

all the a s s o c i a t e d

the u n m i x e d n e s s

(We a p p l y

t h e o r e m all of

the t h e o r e m to the

of

g

if

h B(~)

for all the a s s o c i a t e d being

if

Now

a valuation

= n N

~'

are c o n t a i n e d

in

and we

that w e h a v e

c

"B(m)"

prime

ideals

s i m p l y get from proved



where m

~

above

Thus m

of

a

none of the a s s o c i -

so that E~

a

prime

ring,

m i n [Vm(U) I u E ~ B ( m ) ] gB

ideal

f.)

for an a s s o c i a t e d

if and o n l y

B(m)

follows

for then so that

< g ' > : - = @.

and t h e n take r e s i d u e s m o d u l o

that

n'

s - t,

being homogeneous

~ E n

But

for some

E g B . g

if and o n l y

ideals

~ vm(g).

R e m a r k 2:

~

B

[v~(u) I u E ~ B ( m ) ] .

Hence_

~ = u~ of d e g r e e

is the p r i m a r y c o m p o n e n t

c nB(m )

ated p r i m e

that

[3] p. 203. f

h B(m ) c n B ( m ) .

~B(m)

s ~ t

that

m , then c l e a r l y

it is e n o u g h ~,

so that

g' : u

to s h o w

t h e m are m i n i m a l .

ideal

be p o i n t s e t s

are homogeneous, and by

generated by

theorem.

Then clearly

to p r o v e

It is w e l l - k n o w n of

is c a l l e d gg if ~ is 1 d i m k B n = ~ (n + i) (n + 2).

Note:

residuation

is h o m o g e n e o u s

g' E Bs_t,

Thus we w a n t

~

if

< g ' > : @.

It is e n o u g h

primes

@

0 @ h E V s.

so that

it follows

m - i.

generic)

f

We s h a l l p r o v e

proposition

(n-geometrically

dim k B n - deg ~].

i, 2 . . . . .

R e s i d u a t i o n on

n-gg

n'B (m) = B ( m ) " if and o n l y if

and h e n c e

I(-~) = g ~

~ E gB

.

whenever

the

only associated prime hold

in the g e n e r a l

1.4

The t a n q e n t s Let

G E A,

for all b u t Then

d

of

case w h e n

g B B

and this

is r e p l a c e d b y

G ~ 0.

Then

o(G)

finitely many values

~ (G) = the l e a d i n g S c A

=

of

the o r d e r of

For any set I c A, ~(I)B

are m i n i m a l

R (n)

result will

as in i.i.

at the o r i g i n

is c a l l e d

define

ideals

G

is a h o m o g e n e o u s

n.

Let

G = gd"

where

Ord G.

We d e f i n e

F E I.

Then

Z(I)B

I,

~ say,

t a n g e n t set of

: [~(H) I H 6 S]. -i ideal in B. ~ (£(I)B)

u

homogeneous].

1.5

Divisors

also defines

(Assume

on

~(F)

so t h a t

if

and the B e r t i n i

£ - E =

of

For any

l i n e a r s y s t e m if, w i t h

over

k.

If

or

defines

f

the

I.

Sup-

called

the

= m i n [vm(~) I u E ~(I)B,

fixed

on

40 E L,

~ @ 0,

~(F)

for some effective

(h) = £ - £0

~ c Cn,

ideal

Theorem

£, E

(g)

l e c t i o n of l i n e a r l y e q u i v a l e n t

[h E ~I h = 0

on

We

Z ( I ) B ~ 0.)

R e c a l l that two d i v i s o r s arly e q u i v a l e n t

a pointset

~(m)

0

Z (0) = O.

~(S)

t a n g e n t c o n e at the o r i g i n of the a f f i n e v a r i e t y d e f i n e d b y pose

gn

d = min In I gn ~ 0].

and is d e n o t e d by

form of

define

gn " gn E B n

are said to be

g E ~.

Let

divisors.

the set of

for some

L

space

be a c o l -

is c a l l e d

functions

£ EL]

is a v e c t o r

L

k,

a

!'

is a v e c t o r over

line-

= space

we obtain

a linear system putting

(If w e

fix

£0 = '

In general, L'

by

h0~'

denoted by

replacing

for s o m e h1 [~21

generated by

go E ~, 40

then by

£~

0 ~ h 0 E ~.

0 ~ h i E i']

i'

: [~

has

I g E i}.) 0 the e f f e c t of r e p l a c i n g

In p a r t i c u l a r ,

depends

on

L

the s u b f i e l d

of

o n l y and w i l l be

k(L).

Given a linear system the fixed c o m p o n e n t of

L,

L,

we define

the d i v i s o r

Fi(L),

called

by

Fi(L) (m) = rain [£(m) I A E L]

for any

mE

ar system,

~(~(F)). Fi(L')

Bertini's

= 0

Then and

L' = [4 - Fi(L) k(L)

t h e o r e m now states

A E L]

is a g a i n

a line-

= k(L'). the

following:

If

L'

is a l i n e a r

system without transcendence exists for

fixed c o m p o n e n t d e g r e e of

E 6 L'

In fact,

k(L')

such that

"most" m e m b e r s

of

of

the t r a n s c e n d e n c e

Prolx~stition 2: 1 deg ~ = 7 m ( m -

L'

Proof:

i:

set.

Also

of

k(L')

§2

THE

is at least

2,

then there

(This is true e v e n

4.2

[2] p. 61 w e can d e d u c e

i.e. not a p r i m e d i v i s o r ,

over

k

is at m o s t

that then

i.

POINTSET

shall prove

There exists ~

is

so t h a t

g E [Bm_l,

a simple pointset

(m - 2) gg ~ ~ 0,

p},

and

> ~

~ # 0

~

(m-

on

f

and for any

with

so that

i) free,

(M)

i.e.

there

M E @(f)

= ~(M).

In a d d i t i o n

m-free.

First we

Lemma

k

and s u c h that the

is a p r i m e d i v i s o r .

is r e d u c i b l e ,

degree

i),

g E Bm_ 2

is t h e n

over

= 0)

L'.)

In this s e c t i o n w e

is no

E

Fi(L')

f r o m the p r o o f of Thm.

if e v e r y m e m b e r

there exists

(i.e.

Let

shall p r o v e

~ c Bn,

let

~

the f o l l o w i n g

n < m - 1

be any point.

lemmas.

be a v e c t o r Then

space

d i m [~, ~]

and

~

a point-

- 1

d i m [~, ~ + 9~] < d i m [~, ~9].

Corollary

i:

d i m [i, ~}

Corollary

2:

If

d i m [~5, ~ + ~}

Lemma

2:

Lemma

3:

exists

~

~2

g EIBm_I,

and

a simple pointset

is

d i m [~, 29} # 0,

then

- i.

~

so that

deg ~ =

gg.

be a p o i n t s e t ~ ~ 0

~ m ( m - I).

with Also

of d e g r e e

1 ~ m ( m - i).

= ~2 + ~' ~

is

where

(m - 2) gg

~'

T h e n there is a p o i n t s e t

if and o n l y

if

F'

(m - 2) gg.

Lemma is

and

Let

of d e g r e e is

Fi [~, ~} (M) = ~(M)

= d i m [~, ~}

There exists

1 m ( m - i) 2

> d i m ~ - deg ~.

4:

(m-

Let 2) gg.

~

be Then

a s i m p l e p o i n t s e t of d e g r e e ~

is

(m-

i) free.

1 ~ m ( m - i)

which

Corollary

3:

£

is

m-free.

The p r o o f of P r o p o s i t i o n and C o r o l l a r y

We w i l l

d < m

then

there and

use w i t h o u t

~ ~ 0

i:

further

since d e g r e e

If

is n o t h i n g

exists

follow

from L e m m a s

3, 4

d i m {~, ~}

to prove.

h E [~, ~},

comment

of

then

0 ~ h ~ Bd

and

dim {~, ~ + ~]

d i m [i, ~]

h / 0

if

f = m.

= 0

If

that

Since

> 0

~ c B

then c l e a r l y with

n

= 0

n < m,

and there

~ ~ 0

> Jg. Clearly

To prove

[~, ~ + ~] c [~, ~}

the s e c o n d

inequality

d i m {~, ~ + ~] < d i m [~, ~]

clearly

3.

Note:

Proof L e m m a

2 will

~ ~,

i.e.

~ ~ + 9~.

(~) Let

then

g ~ [~, ~},

= ~(~)

(~) Thus

so that there

Then

~(~)

g / 0.

< d i m {~, ~].

to c o n s i d e r is

< (~)

Then

Otherwise

= (~).

> ~(~)

{~, ~}

This p r o v e s

d i m {~, ~ + ~]

the case w h e n

h E {~, ~9},


~(~)

+ 1

we h a v e

so that

g + ih E {£, ~ + ~}.

space by

h

and

[~, ~ + ~].

the result.

Proof C o r o l l a r y

i:

Follows

by

induction

on

deg ~

using

the i n e q u a l -

ity

d i m {~, ~}

Proof C o r o l l a r y there

exists

2:

Since

h 6 [!, ~],

[~, ~ + 9~] ~ {~, ~].

Thus

- 1 ~ dim {~, ~ + ~ .

Fi {i, ~] (~) = ~9(~) ~-~ 0

from L e m m a

d i m {~, ~ + ~] < d i m {~, ~},

Proof 1 Lemma

There so that

Put

2:

exists

~t

~0

is

= 0

so that

hence

We shall

prove

the

a simple

pointset

and

(~)

d i m [~, ~}

= ~(~).

1 it follows

that

~ 0

Then d i m {~, ~]

the result.

following

~t

more

of d e g r e e

general

t,

result.

t = 0,

1 ....

gg.

which

clearly

satisfies

the c o n d i t i o n s .

Assume

- 1

~O'

~i .....

~t-i

are

ns, d = d i m Vs, d /~t = ~ t - i such

that

that

gd

+ ~

already

O ~ s ~ t - i. is s i m p l e

and

V ( t _ l ) , d ~ [0}

Put

We shall

gg.

Vs, d = [Bd,

find

For each

choose

gd

a point

V(t_l), d

E V(t_l), d

/gs] ~

and

so t h a t

, 1 < d < m

so t h a t

so

gd # 0

~ 0.

Let

Nt

g £ Nt

be

the

implies

~t = ~t-I then

+ 9~"

s e t of all

(7~) We claim

= 0

If

such

and

that

nt, d = m a x

implies

defined.

induction

such

that

~ ( t - l ) (~) = 0.

~

Put

Vt, d : [Bd, ~ t ] ,

[0,

dim B d - deg ~t].

nt, d = 0.

nt, d = d i m Vt, d

Also

n(t_l), d = 0

hypothesis

O = max

so t h a t

Choose

that

if

n ( t _ l ) , d = 0, t h e n c l e a r l y by

gd"

such

[O,

dim B d - deg ~(t-l)

dim Bd - deg ~(t_l)]

~ O.

dim

Hence

Bd - deg ~t

~ O

and nt, d = m a x If

n ( t _ l ) , d a i, t h e n b y

n(t-l),d Also by Corollary there

[0,

exists

2,

induction

nt, d = n ( t _ l ) , d - 1

gd E V ( t _ l ) , d , gd @ 0

nt, d : d i m

2 Lemma

2:

curve

f

choose

the p i e c e

denoted k[~',

~' (f),

~']

An element defined

It is e n o u g h

contains

for

a pointset

outside has

where h E Bd

X'

hypothesis

= dim B d - deg ~(t-l)

F i ( V ( t _ l ),d ) (~) : 0 = ~ ( t - l ) (~)"

Proof

dim B d - deg ~t].

the

affine

induces by

the c h o i c e

of

= 0

so t h a t

affine

piece

B d - d e g 29t a O.

to p r o v e having

line

a

since by (~)

Hence

- ~Y

that

some

the d e s i r e d

Z = 0.

coordinate

= ~X , ¥'

~ E ~' (f)

and

> i.

and

k-valued

ring

properties.

(Note

Z ~ f.)

k[X',

f' (X' , Y') function

h' (~) = ~ - r e s i d u e

This

Y']/f'(X',

piece,

Y')

: f(X' , Y' , i). h'

of

o f the We

on h(X',

£' (f) Y',

i).

=

9 Now

deg

f

= deg

{h' I h E Bd~,

f = m,

a vector

and hence space

dimension

s(d)

= 71

(d + i ) ( d

of degree

< d

in

X'

h 6 Bd h'(~)

and = 0.

h 6 Bd ,

and

~ 6 @(f'). Thus

h' (~i)

It

follows

Y'.

that

{Bd,

They

~]

Bd

Bd

functions

Let

~ = ~i

if a n d o n l y

d < m,

k-valued

+ 2).

w I .....

on

a ~

+

@' (f)

for

B~.

if a n d o n l y

"'" + ~ t "

~i

of

be monomials

Ws(d)

form a basis

Then clearly

for s i m p l e

a ~

of

for

Let if

~ @'(f)"

and

if

= 0,

i = i,

2 .....

is i s o m o r p h i c

t.

to t h e s p a c e

of solutions

of

s(d) i~=~lliwi(7~j)

To

find

~

~i .....

with

the r e q u i r e d 1 (t = ~ m ( m - I),

~t

(wi(~j))

has maximal

d = 0,

...,

i,

= 0,

properties all

possible

m - i.

For

j = 1 .....

~

is

gg.

now

appeal

set and If

V

a vector

d i m V ~ t,

E V

such

that

Applying so t h a t ible.

to the

- rank

= s(d)

- min

then

this

s(d),

so that

s(d)]

the m a t r i x

for

there

exist

(w i(~j)) {t,

s(d)

s(d)]

- t]

elementary

k-valued

lemma:

functions

~i ....

' ~t

Let

on

6 S

S,

and

S k

be

a

a field.

~i'

....

~t

= 6ij. S = @' (f)

(wi(~j)),

Then clearly

i = 1 .....

of

to

the m a t r i x

{0,

following

space

Ui(~j)

{t,

= s(d)

= max

We

distinct)

min

to d e t e r m i n e

then

d i m {B d, ~]

and

it is e n o u g h

~i

rank

t.

f o r any

and

i,

V = B m _ 2,

j = 1 .....

d ~ m - i,

j = 1 .....

t

has

we

find

the m a t r i x

maximal

~i .....

t = s ( m - 2)

(wi(~j)),

possible

rank,

as

required.

Proof Lemma exists theorem d e g J'

3:

By C o r o l l a r y

0 ~ g 6 { B m _ I, J} deg = m(m 1 = ~ m ( m - I).

It is e n o u g h

-

1

with I)

to p r o v e

d i m {Bm_l, a

we have

that

if

j

/~

~ m

and since by

< g > = J + J'

J

so t h a t

is n o t

there

Bezout's

with

(m - 2) g g

~t

is i n v e r t -

then

~'

10 is n o t

(m - 2) gg,

Thus 3.

let

Then by

= dim

Bezout's

(Bm_3)

there

- 1

exists Then

1 there

exists dim

Proof

Lemma

g'

4:

6 Bm_ 2

We have

g 6 { B m _ I, ~]

~ =

Hence

~ {Bm_2, (~) h~h~

£]

= 0.

NOW

~

R' (~) = 0

some and

h~ 7,

then

(~)





for any

R(~)

~ =

= J'

~ E @(f)

# 0

6 B1

R(~)

R(~)

E 9(f).

= i. Then

= 0

by

so t h a t

for s o m e

(~)

= 0.

of

Fix

g

= 0 = R(~)

R' (~) m i.

~

= i,

with

we have

as a b o v e

0 ~ g E I B m _ I, ~]

follows

and we have

For each so t h a t

we may choose h ~

exists

= 1 : R(~).

(Existence

Then

+ J

so t h a t

a point with

= 0,

+ J'.

there

and h e n c e

E { B m _ 2, ~ - ~} (£ - ~)

(~)

(~)

= £ + £'.

Proposition

= R(~).

exists

~

Hence

(m - 2) gg.

(~ - ~)

that

= I.

and

I.

and by

d i m I B m _ 2, ~}

E { B m _ 2, F - ~] (~)

and

+ J'

1 and

with

assume

@ 0

where

there

be

Corollary

+ ~

< g > = £ + £'.

Thus we may choose

(~ - ~)

~ 0

pointset m ( m - 3) - i) = 2

by Corollary

Choosing

let

is n o t

(~)

and hence

E I B m _ I, @}

and write If

m 1

['

g'

that

by

+ J'

and

the c a s e w h e n

Then we may write d i m I B m _ 2, ~ - ~]

=

J'

J, J'. for some

1 - ~ m(m

- 2) m 1

so t h a t

to s h o w

so t h a t

consider

hypothesis.

~ 0

£}

so t h a t

+ £'

in

= J + £

deg J = m(m

£ { B m _ 3, J]

J']

symmetric

and

d i m {Bm_3,

= J + J + ~'

{Bm_2,

First

theorem

0 ~ h'

being

h @ 0

so t h a t

(~)

Proof C o r o l l a r y that by R e m a r k

3: 1

that

$(G)

exists

3:

d i m {Bm_2,

is

so that

ideal

and

have

Proof:

Consider

other hand

E B m) .

(m)

= 0.

and is

GW,

the v e c t o r

space

+ ~I

Fi(l~l)

GX,

~ 0,

determined

for all

= i,

that

is

a contradiction.

0 # g E {Bm_ I, ~],

F(~)

o(H)

Obviously and

if

~

> 0

D P

where G

implies

E IBm,

Fi(l~l)

so

~]

so that

] c C--re.

> ~*.

On the

= * is not h o m o g e n e o u s

is h o m o g e n e o u s

and since

G Z

E ~

F

is

so that

degree

of

k ( l ~ I)

Theorem

there

exists

a prime divisor

+ H E

= ~ *

II

+ m.

m

P

F' (~) = 0.

/~* ~ Fi(ILI)-

so that t r a n s c e n d e n c e

XGW

so

Then there

(F, G + H ) A = ~

with

0 ~ ~ (H) E ~

G Y ,

0 ~ G E A

F u r t h e r we can c h o o s e

= ~*.

3, c l e a r l y

Let

= ~ + ~'.

i.e.

H 6 A

Fi(l!l)

2.

over

k

2.

for some

by

~.

THEOREM

and

Put d i f f e r e n t l y ,

By B e r t i n i ' s i.e.

E B

points,

Hence

k ( l ~ I) : ~

and w r i t e

~(~)

m-free by Corollary Now

~]

there exists

(h = ~(H)

F(~)

3 there e x i s t s

no c o m m o n

that

with

PROOF OF THE

G(H)

= {X~W

We c l a i m

g~ = g'

- ~}

m free.

m = I(~).

and

~

F'

is u n i q u e l y

be as in P r o p o s i t i o n

= g E [Bm_l,

is a prime F'

~

d i m {Bm_2,

take

~] ~ i,

As in L e m m a ~

Let

H 6 A

i, and g~

for all

so that

+¢~. Since

So we may

z 1

§3

Proposition

= 0.

by L e m m a

in

and we get

F']

- ~]

E I£I ',

12 S

With each divisor

~ =

~ n i m i on ~ (F) w e can a s s o c i a t e i=l d e g m i = d e g r e e of the i r r e d u c i b l e c u r v e

deg ~ = ~

n i deg m i

where

on

defined by

the ideal

~(F)

faces y i e l d s

for

U 6 Cn,

m i.

Then Bezout's

theorem

for s u r -

U ~ 0

deg = nm.

Thus

deg <XGW+

H> = m

2

.

Also

for any

~ £ @(f),

deg

(~*) = i,

i.e.

the d e g r e e of a h o m o g e n e o u s p r i m e d i v i s o r is i. Hence clearly 1 1 d e g ~* = d e g ~ = : m ( m - i) so t h a t d e g (m) = : m ( m + i) > i. In particular

m

clude

k ~ 0

that

is not a h o m o g e n e o u s

since otherwise Now

k~W

to the p r i m e As

m

since otherwise = ~'* + < W >

+ H ~ W~

m

would

since

Hence we can con-

is h o m o g e n e o u s .

Also

H ~ 0

n o t be prime.

H 6 A

and h e n c e

W

does not b e l o n g

ideal of m .

in R e m a r k 2 we can c o n c l u d e

and s i n c e

prime divisor.

W

does n o t d i v i d e

XG W

that

(IGW

+ :

+ H)C = I(~*)

we have

n I(m)

in a f f i n e

3-space

(kG + H,

where

~(P)

that

~(~) Since

= I(m),

~(~)

F)A = ~ n P

= I(~*).

It is s t r a i g h t f o r w a r d

to c h e c k

= I(~) . k ~ 0

w e may

take

k = 1

and w e h a v e

the r e q u i r e d

G + H. F o r the last r e m a r k we with

R(Y~) = 1

there

is

simply have

0

to o b s e r v e

~ g~ 6 { B m _ I, ~]

t h a t for e a c h

so that

(~)

= i.

We m a y t h e n take a l i n e a r c o m b i n a t i o n and

(~)

= (Y~) = i.

~ C~g~ = g so t h a t C~ £ k R(~) =l It is c l e a r t h a t G = -l(g) is the

required element.

P r o p o s i t i o n 4: satisfies

Proof:

Let

~'.

~ 0

For

3 the t a n g e n t set

~

of

~'.

to p r o v e

so t h a t

= 0

~(F")

for all

7~

t h r o u g h any p o i n t of

F i { B m _ I, ~} (~)

= ~, = 0

for e a c h and

o(F~)

12

t h a t if

6 {Bin_ I, ~

with ~'.

Y~ w i t h

~(F')

~ 0 with

~' (Y~) > 0, This

i.e.

is p o s s i b l e

~' (Y~) > 0

6 {Bm_ I, ~}.

then

we

find

A suitable

P

15 (general)

linear combination

We c l a i m that Now of

F" ~ P

is h o m o g e n e o u s

F'F"

6 P n ~ =

(F, G + H ) A

is

> ~' (~).

Since =

~ + ~'

(f, g)B

if

where

~ 0.

=

and n e i t h e r

E

's

m

serve

(7~)

= 0

E Bd

~ 0

> ~'.

with

and e i t h e r

In the

d > m - 1

~(F')

l a t t e r case, so t h a t

= 0

~'

Ord(F')

>

i).

(m-

Proposition fact if

5:

U I,

For

~, ~',

.... U s

P

as b e f o r e ,

generate

P,

[ Bm_l,

~'~

are in

Bm_ 1

Proof:

F o r the f i r s t s t a t e m e n t

generate

gl E IBm_ I, ~'}, -i

=

((X, Y, Z)A) m-l.

F' E P,

(m-

G + H)A)B

(gl + hl) Now

Also

if

pointset exists

gl ~ 0

E P,

for t h e n

0 ~ gl E Bm_ 1 h = ~(H), 3'.

Now

0 ~ h I E Bm

there

IBm_l,

it is e n o u g h to p r o v e t h a t exists

h I E Bm

which

= ~' + 3.

< h g l > = < g > + J + 3' < h l > = I + 3'. m

h g I = gh 1 so that HG 1 = GH 1 + UF

13

E ~(P).

Also

= ~ + ~'.

= ~ + 3'

and b y Then

for

so that

+ hl))

we have by construction

with

and in

Z(Ui)

.

gl = ~ ( ~ - l ( g l

and

~'} c ~(p)

then t h o s e o f the

for some

Proposition

1 there

14 where -i G1 = o

-i (gl),

H1 = c

(G).

Hence (G + H ) G 1 = G(G 1 + H I) + UF. Now

G + H, F E P

the same r e a s o n as G1 + H1 E P

so that

F" ~ P

G(G 1 + HI)

E P.

Also

G ~ P

in the p r o o f of P r o p o s i t i o n 4).

(for Hence

as r e q u i r e d .

For the s e c o n d s t a t e m e n t U 1 .....

Us

of

P,

any

£(alU 1 + Now

"'" + asU s), £ (U i) E B r,

observe

that g i v e n a set of g e n e r a t o r s

0 ~ gl E {Bin_I, ~'}

a I, r >

is of the f o r m

.... a s E A. (m - i) b y C o r o l l a r y

4.

Since

0 ~ gl E

Bin_ 1 ,

~ (a i) Z (U i) • (U i ) EBm_ 1

gl =

(ai) Ek This p r o v e s

the s e c o n d s t a t e m e n t .

P r o o f of the Theorem: t h a t the p r i m e

By C o r o l l a r y

ideal

P

needs

§4

R e m a r k 3:

If the p o i n t s e t

1

at l e a s t

R e m a r k 4: ~'

One w a y

P

~'

~' (~) ~ 1 i n s t e a d of

(m - 2) gg.) Bertini's pointsets question

is

to p r o v e

can be a r r a n g e d

so that to

set of

m

It follows

generators.

~'

in §3 c o u l d be p r o v e d

to be

4 c o u l d be s h a r p e n e d

(m - i)

to p r o v e

that

~'.

that

~'

is

(m - i) free is to s h o w t h a t

to b e a s i m p l e p o i n t s e t d i s j o i n t implies ~.

~ m.

SOME Q U E S T I O N S

free t h e n the p r o o f of P r o p o s i t i o n the t a n g e n t

d i m {Bm_ I, ~'}

~(~)

(By L e m m a

This c o u l d be done

t h e o r e m on v a r i a b l e

= 0). 3,

~'

Then one m a y is

to the

g ~ 0].

0

~

(i.e.

apply L e m m a 4

(m - 2) gg

in c h a r a c t e r i s t i c

singularities

{ < g > - ~I g E { B m _ I, ~},

from

since

~

by a p p l y i n g

l i n e a r s y s t e m of

W e ask the f o l l o w i n g

in g e n e r a l :

Q i:

Does t h e r e e x i s t g 6 Bm_ 1 so that 1 d e g ~' = ~ m ( m - I) and b o t h ~, ~' are disjoint?

14

= ~ + ~', (m - 2) gg,

deg ~ = simple

and

is

15 Q 2:

g E Bm[ 1

Let

d e g @ = d e g @'

Another Lemma

2.

be

= ~ m(m

member

Does

it

i)

~

is

(m - 2) gg

~

is

(m-

way would

i)

be

and w r i t e

follow

ii)

possible

We

a generic - i).

to h a v e

Let

g E Bm_ 1

with



= ~ + ~'.

does

it f o l l o w

that

~'

is

Clearly we may

Remark

above

5:

a lemma

of the

type

of

ask

free,

in the

= F + F',

free?

Q 3:

gg



that

As

assume,

Given

(m - i)

if n e c e s s a r y ,

that

~

is

(m - i)

free?

that

~

is

(m - 2)

gg

question.

in R e m a r k

3,

if

~'

is

(m - i)

free,

then

~'

is the

t a n g e n t set of P. Thus if w e c o u l d a r r a n g e ~' to be of the 1 ~' = ~ m ( m - i)~ the r e s u l t i n g c u r v e t h e n w o u l d h a v e o n l y one (set t h e o r e t i c a l l y )

at the o r i g i n .

Q 4:

can we arrange 1 ~' = ~ m ( m - i ) ~ ?

Remark

6:

Just

pointsets distinct such

in the

have

fixed

no

through

set

~

on

simple

~

we

so that

~

and

point

outside

~.

then

the

f.

Thus

in g e n e r a l

image

For

a set

~

as above,

of degree

n

through

In c h a r a c t e r i s t i c one

can

a~o

and

of the

on a c u r v e as

f

prove

not

equal

a more

start with

there

curve

One has

may

f

form

we may

formal

clearly

0, that

of d e g r e e

If w e of

~

we

define

sums

of

corresponds

to

would

m - 2 through

a nonsingular be

the

curve

required

is

min

{n I t h e r e

is a n o n s i n g u l a r

?

using

Lemma

indeed

to z e r o

pass

f

1 ~ m ( m - i)

of

of d e g r e e (m - i)

could on

~

ask:

what ~]

a set

is no c u r v e

the c u r v e s

Q 5:

induction

free

on

tangent

ask

multiplicities) pointset

~,

characteristic

we

type

in the plane.

plane

through

(m - i)

pointsets

(without

Every

Hence

to b e

defined

to c o n s t r u c t

passing

f

as w e

points.

Thus points

~'

in the p l a n e

a pointset

or

4 and

a type

the m i n i m u m

the q u e s t i o n

generalized

version

15

is

of d e s c e n d i n g (m - i).

remains

open.

of Q 5, viz.

In

16 Q 6: For a pointset nonsingular

~

in the plane h o w to find

curve of degree

n

through

~}

min {n I there

is a

?

BIBLIOGRAPHY

[1]

Macaulay, Hafner

[2]

"The Algebraic

Service Agency

Matsusaka,

Memoirs

Ser A, Vol. XXVI,

[3]

Zariski,

Samuel,

Theory of Modular

Systems",

Stechert

(Reprint) ~, 1964.

of the College 1950,

of Science,

Univ.

of Kyoto,

p. 51-52.

"Commutative

Algebra"

Company.

16

Vol.

II, Van Nostrand

PRIME IDEALS IN POWER SERIES RINGS

Jimmy T. Arnold

Virginia Polytechnic Institute and State University

ABSTRACT. In this paper we wish to briefly review some known results concerning the ideal structure of the formal power series ring m[[x]]. As the title indicates, primary consideration will be given to prime ideals in R[[X]]. We begin by discussing some of the basic difficulties which arise in relating the ideal structure of R[[X]] with that of R. We then consider the Krull dimension of R[[X]] and, finally, we review some results on valuation overrings of D[[X]], where D is an integral domain.

i.

Introduction.

Our notation and terminology are essentially that of [8].

Throughout, R denotes a conm~utative ring with identity and D is an integral domain with quotient field K.

If R has total quotient ring T, then by an overring S of R

we mean a ring S such that R c S c T. w and w

The set of natural numbers will be denoted by

is the set of nonnegative integers.

If A is an ideal of R, then we let

O

A[[X]] = If(X) = ~'~l=o a'xil I a.1 E A for each i E Wo} and we denote by AR[[X]] ideal of R[[X]] which is generated by A. q.f.

the

We denote the quotient field of D[[X]] by

(D[[X]]).

2.

Quotient overrinss and extended ideals.

Let S denote a multiplicative

system in

R.

In studying the ideal structure of the polynomial ring R[X], one important and

widely used technique is to pass to a quotient ring R S of R and utilize the fact that Rs[X ] is a quotient ring of R[X] - namely, Rs[X ] = (R[X])s. 35.11 of [8] and Theorems 36 and 171 of [I0].)

(cf. Theorem

One particularly important aspect of

this technique is that in studying the integral domain D[X], one is able to make considerable use of the ideal structure of the Euclidean domain K[X].

For example,

one can easily describe the essential valuation overrimgs of D[X] in terms of the essential valuation overrings of D and K[X] [4, Lemma I].

17

Unfortunately,

one canno~

in general, employ such techniques in studying power series rings.

In fact, it is

clear from the following result, proved by Gilmer in [7], that D[[X]] and K[[X]] seldom share the same quotient field.

THEOREM I.

The following conditions are equivalent.

(I)

~.!.(D[[X]]) = ~.!.(K[[X]]).

(2)

K[[X]] = (D[[X]]) D _ (0)"

(3)

If [ai}i= 1 is a collection of nonzero elements of D, then Ni~ 1 aid # (0).

In [13, Theorem 3.8], Sheldon restates Theorem 1 in the following more general form.

THEOREM 2.

If S is a multiplicative system of D, then the following statements

are equivalent. (I)

~.i.(Ds[[X]] ) = ~.~ (D[[X]]).

(2) Ds[[X]] ~ (D[[X]])D- (0)" (3)

Fo___rrevery sequence [si}i= 1 of elements of S, Ni= 1 aiD # (0).

Even when the conditions of Theorem 2 hold, it need not be the case that Ds[[X]] = (D[[X]]) S.

In fact, Ds[[X]] may not be a quotient overring of D[[X]].

For example, if we let V denote a rank two discrete valuation ring with prime ideals (0) c p c M, then Vp[[X]] ~ (V[[X]]) V - (0)' however, Vp[[X]] is not a quotient overring of V[[X]].

In [13], Sheldon further investigates the relationship between

q.f.(D[[X]]) and q.f. (~[[X]]), where D 1 is an overring of D.

He shows, for example,

that D is completely integrally closed if and only if q.f.(D[[X]]) # q.f.(Dl[[X]] ) for each overring D 1 properly containing D [13, Theorem 3.4].

He also shows that if

= a iD = (0), then q.f (D[[X /a]]) has infinite transcendence a E D - (0) and n i=l degree over q.f (D[[X]]). Another useful technique in considering ideals in R[X] is the following: (*)

Let B be an ideal of R[X] and set A = B N R.

ideal in the polynomial ring (R/A)[X] ~ R[X]/AR[X].

Then B = B/AR[X] is an

Moreover, B A (R/A) = (0).

(cf. Theorems 28, 31 and 39 of [I0].) Even though we do have the isomorphism (R/A)[[X]] ~ R[[X]]/A[[X]] for power 18

series rings, it is not generally true that A[[X]] = AR[[X]]. of R[[X]] with A = B n R, it may happen that B ~ A[[X]].

Thus, if B is an ideal

The following result is

given by Gilmer and Heinzer in [9, Proposition I].

THEOREM 3.

Let A be an ideal of R.

Then A[[X]] = AR[[X]]

if and only if for

each countably generated ideal C c A, there exists a finitely generated ideal B such that C c B c A.

Gilmer and Heinzer also note that the equality A[[X]] = AR[[X]] need not imply that A is finitely generated.

For example, if V is a valuation ring with maximal

ideal M and whose set of prime ideals is of ordinal type ~, the first uncountable ordinal,

then MV[[X]]

= M[[X]], but M is not finitely generated

[9, p. 386].

Howeve~

in order that we be able to employ globally the method described in (*), we need that A[[X]] = AR[[X]]

for each ideal A of R.

The following result is an easy consequence

of Theorem 3.

THEOREM 4.

The equality A[[X]] = AR[[X]] holds for each ideal A of R if and

only if R is Noetherian.

If Q is a prime ideal of R[[X]] and P = Q n R, then Q ~ P 4 ~ X ] ] .

Thus, if we

wish to apply the reduction technique of (*) only to prime ideals of R[[X]], is sufficient to have P[[X]] = ~PR[[X]]

for each prime ideal P of R.

then it

In considering

radical ideals in R[[X]], some results concerning nilpotent elements in R[[X]] will be useful. Let f(X) = E. ~

i=0

a. X i E R[[X]]. 1

If R has characteristic n > O, then Fields

has shown in Theorem I of [5] that f(X) is nilpotent if and only if there is a k positive integer k such that a. = 0 for each i E w . l

Further, Fields gives an

O

example to show that one cannot drop the assumption that R has finite characteristic, k that is, in an arbitrary ring R it may happen that a i = 0 for each i E ~o' yet, f(X) is not nilpotent. f(X).

Let Af denote the ideal of R generated by the coefficients of

If we impose the condition that ak = 0 for each a E Af, then there exists

m E w such that mA~ = (0). characteristic"

This satisfactorily imitates the assumption of "finite

and, in fact, yields that g(X) is nilpotent for each g(X) E Af[[X]].

19

More generally, we have the following result [i, Lemma 4].

LEMMA.

Let A be an ideal of R and suppose there is a positive inteser k such

that ak = 0 for each a E A.

Then f(X) is nilpotent

for each f(X) E A[[X]].

The author does not know if f(X) nilpotent implies the existence of k E m such that a k = 0 for each a E Af.

Thus, in the following theorem, which is irmnediate

from the above lemma, we are able only to give sufficient conditions on an ideal A in order that A[[X]] ~ A R [ [ X ] ] .

THEOREM 5.

Let A be an ideal of R and suppose that for each countably senerated

ideal C c A there exists a positive integer k and a finitely generated ideal B c A such that ck E B for each c E C.

Then A[[X]] ~

JA-R[[X]].

We shall say that an ideal A is an ideal of strong finite type (or, an SFTideal) provided there exists k E w and a finitely generated ideal B c A such that a

k

E B for each a E A.

We say that the ring R satisfies the SFT-property if each

ideal of R is an SFT-ideal [2]. property,

then R satisfies

It is easy to see that if R satisfies

the SFT-

the ascending chain condition for radical ideals,

R has Noetherian prime spectrum.

The converse does not hold.

that i%

For example, a rank

one nondiscrete valuation ring does not satisfy the SFT-property. Following Theorem 3, we referred to an example of a valuation ring V with maximal ideal M such that M is not finitely generated, yet My[x]] then, My[x]] = ~ [ [ X ] ] .

But M is not an SFT-ideal since

of a finitely generated ideal. A[[X]] ~ A R [ [ X ] ] ,

THEOREM 6.

= MV[[X]].

Clearly

it is not even the radical

However, if we impose globally the condition that

then we obtain the following result [I, Theorem i].

Th__£efollowin$ conditions are equivalent.

(i)

R satisfies the SFT-property.

(2)

A[[X]] ~

(3)

P[[X]] = ~

~

for each ideal A of R. for each prime ideal P of R.

In view of the somewhat analogous nature of Theorems 4 and 6, it seems reasonable to ask whether the condition that P[[X]] = PRy[X]]

20

for each prime ideal

P of R implies that R is Noetherian.

3.

Krull dimension in R[[X]].

The author does not know the answer.

We say that R has dimension n, and write dim R = n,

provided there exists a chain Po c PI c •

... c pn of n + I prime ideals of R, where

Pn # R, but no such chain of n + 2 prime ideals. if dim R = n, then n + 1 ~ dim R[X] ~ 2n + i. n + 1 ~ dim R[[X]], however,

Seidenberg has shown in [II] that

It is easy to see that one also h a ~

the following result shows that dim R[[X]] has no upper

bound [I, Theorem i].

THEOREM 6.

If R does not satisfy the SFT-property,

then dim R[[X]] = =.

In order that a PrHfer domain D satisfy the SFT-property,

it is necessary and

sufficient that for each nonzero prime ideal P of D, there exists a finitely generated ideal A such that p2 c A c p [2, Proposition 3.1].

In particular,

a valuation

ring V satisfies the SFT-property if and only if it contains no idempotent prime ideals.

A rank one nondiscrete valuation ring,

therefore, provides a simple example

of a Prefer domain which does not satisfy the SFT-property.

An integral domain D is

called an almost Dedekind domain provided DM is a rank one discrete valuation ring for each maximal ideal M of D.

An almost Dedekind domain which is not Dedekind

[8, p. 586] provides another example of a Prufer domain which does not satisfy the SFT-property. If D is a Prefer domain with dim D = n, then Seidenberg has shown in [12, Theorem 4] that dim D[X] = n + I.

The analogous statement for power series rings

does not hold, for, as we have just seen, D need not satisfy the SFT-property. we may have dim D[[X]] = ~. satisfy the SFT-property, rings.

However,

Thus,

if we restrict our attention to rings which

then we can extend Seidenberg's result to power series

Namely, we get the following result [2, Theorem 3.8].

THEOREM 7.

Let D be a Prefer domain with dim D = n.

are equivalent. (i)

D satisfies the SFT-property

(2)

Dim D[Ex]]

(3)

Dim D[[X]] < ~.

= n + 1.

21

The followin$ statements

It can be easily shown that if R is any ring with dim R = 0, then statements (i) - (3) of Theorem 7 are equivalent which

(i) - (3) are not equivalent.

for R.

The author knows of no example for

In an attempt to find such an example, one

might be lead to consider rings R which satisfy the SFT-property but for which dim R[X] > 1 + dim R.

To construct one such ring,

simple transcendental extension of k.

let k be a field and K = k(t) a

Suppose that V = K + M is a rank one discrete

valuation ring with maximal ideal M (e.g., take V = K[[Y]]).

If we set D = k + M,

then D satisfies the SFT-property and dim D = I, yet, dim D[X] = 3 [8, Appendix 2]. In this case, however, we get that dim D[[X]] = 2 [9]. From the statement of Theorem 6, one can conclude only that there is no bound on the lengths of chains of prime ideals of R[[X]].

The proof given in [I, Theorem

i] shows, more strongly, that R[[X]] contains an infinite ascending chain of prime ideals.

It is easy to see, then, that when statements

(I) - (3) of Theorem 7 are

equivalent

for a rin~ R, then they are equivalent to the statement "R[[X]] satisfies

the a.c.c,

for prime ideals".

The proof given in [I] for Theorem 6 shows, in fact,

that if R does not satisfy the SFT-property, P of R and an infinite chain Q1 c Q2 c Qi n R = P for each i E w.

then there exists a nonzero prime ideal

... of prime ideals of R[[X]]

such that

Foll~qing a proof given by Fields in [6, Len~na 2.4],

Arnold and Brewer show in [3, Proposition i] that if Q is a prime ideal of R[[X]] such that Q N R = M is a maximal ideal of R, then either Q = M + (X) or Q ~ My[x]]. Thus, if D is a one dimensional integral domain which does not satisfy the SFTproperty,

then there exists a maximal ideal M of D and an infinite chain

Q1 c Q2 c

... of prime ideals of D[[X]] such that MD[[X]] c Q1 c Q2 c

... c M[[X]].

It follows from Theorem 16.10 of [8] that if P is a prime ideal of the Prefer domain D, then each prime ideal of D[X] contained in P[X] is the extension of a prime ideal of D.

Our remarks in the preceding paragraph show that this need not

occur in power series rings.

THEOREM 8. SFT-property,

We do have the following somewhat analogous result.

Let D be a finite dimensional Prefer domain which satisfies the

and let P be a prime ideal of D.

contained in P[[X]] have the form plY[X]]

The only prime ideals of D[[X]]

for some prime ideal PI o f D.

22

4.

Valuation overrinss of D[[XI].

The domain (D[X])p[x]

Let P be a prime ideal of the integral domain D.

is a valuation ring if and only if Dp is a valuation ring.

For

Krull domains, the analogous statement for power series rings holds [9, Theorem 2]. In [3] Arnold and Brewer have considered the following question:

If P is a prime

ideal of the domain D, when is (D[[X]])p[[x] ] a valuation ring?

Necessary conditions

are given in the following theorem [3, Theorem I].

THEOREM 9.

Let P be a prime ideal of the intesral domain D.

is ~ valuation rin$, then Dp is a rank one discrete valuation rin$. (D[[X]])p[[x]]

If (D[[X]])p[[X]] Moreover,

is rank one discrete.

The proof given in [3] for Theorem 9 shows the following:

Let (0) c Q c p

= I anDp be prime ideals of D and assume that Dp is a valuation ring with QDp = N n= for some a E P - Q.

Then (D[[X]])Q[[X] ] is not a valuation ring.

Thus, if P is a

prime ideal of any valuation ring V, with dim V = n > i, then (V[[X]])p[[x]] a valuation ring.

is not

This shows that the condition that Dp be a rank one discrete

valuation ring is not sufficient to insure that (D[[X]])p[[x] ] is a valuation ring. Our observations in the previous section yield another simple example. almost Dedekind domain which is not Dedekind, of D and an infinite chain QI c Q2 c MD[[X]] c QI c Q2 c ... c M[[X]].

If D is an

then there exists a maximal ideal M

... of prime ideals of D[[X]] such that

In view of Theorem 9, (D[[X]])M[[X]]

is not a

valuation ring. The following theorem gives sufficient conditions

for (D[[X]])p[[x] ] to be a

valuation ring [3, Theorem 2].

THEOREM I0.

Suppose that P is a prime ideal of the intesral domain D and that

Dp is a rank one discrete valuation Tins. then (D[[X]])pD[[X]]

If PD[[X]] is ~ prime ideal of D[[X]],

is a rank one discrete valuation Tins.

In particular,

if

PD[[X]] = P[[X]], then (D[[X]])p[[x] ] is a valuation ring.

While the assumption that PD[[X]] is assumption that PD[[X]] = P[[X]],

prime

is ostensibly more general than the

the authors in [3] are unable to provide an

example of a prime ideal P which satisfies the hypotheses of Theorem 10 and which has

23

the property that DP[[X]] is prime but PD[[X]] # P[[X]].

If V is a rank one non-

discrete valuation ring with maximal ideal M, it is the case that MV[[X]] is prime, yet, MV[[X]] # M[[X]] (see eerana I of [3] for a proof due to M. Van der Put). The conditions given in Theorem I0 are not necessary, for one can construct a Krull domain D with a minimal prime ideal P such that PD[[X]] is not prime (in particular, PD[[X]] # P[[X]]), yet, (D[[X]])p[[x]] is a valuation ring [3]. Although the following theorem gives necessary and sufficient conditions in order that (D[[X]])p[[x] ] be a valuation ring, these conditions are not entirely satisfactory since they are not given in terms of the ideal structure of D[3, Theorem 3].

THEOREM ii.

Let P be a prime ideal of the intesral domain D and suppose that

Dp is a rank one discrete valuation rin$.

The followin~ conditions are equivalent.

(i)

(D[[X]])p[[X] ] is a valuation rin$.

(2)

P(D[[X]])p[[x]] = P[[X]](D[[X]])p[[x] ].

(3)

Dp[[X]] n ~.~.(D[[X]]) ~ (D[[X]])p[[x] ].

REFERENCES

i.

Arnold J., Krull dimension in power series rings, Trans. Amer. Math. Soc. (to appear).

2.

, Power series rings over Prefer domains, Pacific J. Math. (to appear).

3.

Arnold, J. and Brewer, J., On when (D[[X]])p[[x] ] is a valuation ring (submitted).

4.

.. . . . . . . Proc. Amer. Math. Soe

5.

, Kronecker function rings and flat D[X] - modules, 27 (1971), 483-485.

Fields, D., Zero divisors and nilpotent elements in power series rings, Proc. Amer. Math. Soc., 27 (1971), 427-433.

6.

, Dimension theory in power series rings, Pacific ~. Math 35 (1970), 601-611.

24

7.

Gilmer, R., A note on the quotient field of the domain D[[X]], Proc. Amer. Math. Soc. 18 (1967), 1138-1140.

8.

, '~ultiplicative Ideal Theory", Queen's Papers on Pure and Applied Mathematics, No. 12, Kingston, Ontario, 1968.

9.

Gilmer, R. and Heinzer, W., Rings of formal power series over a Krull domain, Math. Zeit. 106 (1968), 379-387.

i0.

Kaplansky, I., "Commutative Rings", Allyn and Bacon, Boston, 1970.

Ii.

Seidenberg, A., A note on the dimension theory of rings, Pacific J. Math. 3 (1953), 505-512.

12.

,

A note on the dimension theory of rings II, Pacific J. Math.

4 (1954), 603-614.

13.

Sheldon, P., How changing D[[X]] changes its quotient field, Trans. Amer. Math. Soc. 159 (1971), 223-244.

25

KRULL DIMENSION OF POLYNOMIAL RINGS

J.W° Brewer t P.R. Montgomery~ University of Kansas,

E.A. Rutter

Lawrence,

Kansas

and W.J. Heinzer Purdue University,

Lafayette,

Indiana

ABSTRACT. Let Q be a prime ideal of R[X], ..., X ] and let P = Q ~ R. This paper investigates~the relationship between the ranks of Q and P[X], ..., Xn]. The results are used to recover some weIl-known results concerning the Krull dimension of R[XI, ..., X ]. The paper also contains a number of examples r~lated to questions w h i c h arose in connection with the above investigation.

Let R be a commutative unitary ring with XI, minates over R. Prufer domain,

..., X n indeter-

It is well-known that if R is a Noetherian ring or a then dim (R[X 1 . . . . .

Xn]) = n + dim (R), where dim (S)

denotes the Krull dimension of the ring S.

It is customary [3] and [9]

to treat these two classes of rings in a different Kaplansky [7], for the case of a single variable,

fashion.

However,

treats the two cases

in a unified manner by introducing the notion of a strong S-ring. strong S-rings respected polynomial

extensions,

If

then this approach

could be extended to several variables by means of an induction argument.

But strong S-rings do not respect polynomial

extension as we

shall see in Section 2 and in fact, we shall show by examples that it is not possible to give an n-variable definition of strong S-ring in 26

such a way that the approach can be extended. ment is to be given for several variables, found.

So, if a unified treat-

another approach must be

The properties of strong S-rings which are crucial for proving

that dim (R[XI]) = 1 + dim (R) are:

(i) rank (P) = rank (PtXI]) for

each prime ideal P of R and (ii) if Q is a prime ideal of ~ X I] with Q D (Q n R)[XI] , then rank (Q) = 1 + rank (Q n R)[XI]. extend these ideas to n variables, rank (P[X 1 .....

In trying to

one sees readily that rank (P) =

Xn] ) in the classical cases mentioned above and there-

fore the problem is to relate the rank of a prime Q of R[X I, ..., X n] with that of (Q n R)[X 1 .....

Xn].

The principal positive result of

this paper is Theorem i, which states that for an arbitrary commutative ring R and an arbitrary prime ideal Q of R[X 1 . . . . .

Xn] , rank (Q) =

rank ((Q n R)[X I . . . . .

X n]) + rank (Q/(Q N R)[X I .....

rank ((Q n R)[X 1 .....

Xn] ) + n.

X n])

An appealing feature of Theorem i is

that its proof is not only brief but also elementary and therefore it seems to make available

from scratch, more readily than ever before,

information about the ranks of prime ideals of R[X 1 ..... Moreover,

Xn].

not only can Theorem 1 be used to prove the classical dimen-

sion theorems, but it also yields immediately that dim (K[XI,

.... Xn]) = n, when K is a field.

Some other applications

include a greatly simplified proof of the "Special Chain Theorem" of Jaffard and the fact that if all maximal ideals of R[XI,

..., X n] have

the same rank, then R is a Hilbert ring. Perhaps the most interesting part of the paper is Section 2 which is given over to the construction of counterexamples which arise in Section i.

In particular,

to questions

it is shown that strong

S-rings do not respect polynomial extension and that if all maximal

27

ideals of R[X I] have the same rank, it need not be true that all maximal ideals of R have the same rank. Our notation is essentially that of [7].

However, we shall

write "r(P)" in place of "rank (P)" for the rank of a prime ideal P of the ring R.

I.

The Main Theorem and Some Applications

We proceed to the proof of our main theorem pausing only to prove a preliminary result.

LEMM

i.

Let Q be a prime ideal of R[X I] and let P = Q A R.

If Q D P[Xl] , then r(Q) = r(P[Xl] ) + i, and for each integer n > i, r(Q[X2,

.... Xn]) = r(P[X I . . . . . Proof~

Xn]) + i.

Both assertions are obvious if r(Q) = =

In the finite

case to prove that r(Q) = r(P[ X I] ) +I, we induct on r(P). then r(P[ X I] ) = 0.

If r(P) = O,

Since three distinct primes of R[ X I] cannot have

the same contraction to R, P[ X I] is the unique prime ideal properly contained in Q and r(Q) = i = i + r(P[ Xl] ).

Assume the result to be

true for all k < m, where m > 0 and r(P) = m.

To prove that r(Q) =

r(P[X I]) + i it suffices to prove that r(Ql) ~< r(P[X I] ) for each prime ideal QI c Q.

Thus, we let PI = QI n R.

If PI -- P' then

P[XI] C QI c Q, QI = e[xl] and r(Ql) ~< r(P[X I]).

If el c p, the

induction hypothesis implies that r(Ql) ~< r(P IX I] ) + i ~< r(P[ X I] ). i As for the second assertion considering R[X I, ..., X n] as R[X I][X 2 . . . . , X n] we have that NIX 2 . . . . . (R N Q)[X2, as R[X2,

..., X n]) = P[X 2 . . . . .

x n] .

Xn] N Q[X2,

..., Xn] =

Then regarding R[XI,

..., X n]

..., X n][X I] , it follows from the first part of the lemma that

28

4

r(Q[X 2, .... Xn]) = r(P[Xl,

THEOREM i.

..., Xn] ) + i.

If Q is a prime ideal of R[XI,

Q n R = P, then r(Q) = r(e[Xl, r(P[Xl,

.... X n] with

.... x n]) + r(Q/P[Xl,

..., x n]) ~
2.

dim (D[XI, YI' "''' Yn]/M[XI' YI . . . . .

r(M[XI' YI' "''' Yn ]) = 2.

But

Yn ]) = dim ((D/M)[X I .... ,Yn ]) =

n + i and dim (D[XI, YI' "''' Yn ]) = n + 3.

r(e) = i = r(P[YI,

To show

then r(P) = 2 since R is integrally closed but

not a valuation domain.

r(P) = 3.

Thus, by Arnold's

Put R = D[X I] and let P be a prime ideal of R.

that r(P) = r(P[X I . . . . . arise.

integrally closed

Hence, we must have that

If P n D = (0), then Rp is a DVR and

.... Yn]).

By Lemma i, r(P[YI,

If e n D # (0) and if P # M[XI] , then .... Yn ]) = r(M[XI' YI . . . . .

Yn ]) + i = 3.

REFERENCES

[i]

Arnold, J.T., On the dimension theory of overrings of an integral domain, Trans. Amer. Math. Soc. 138 (1969), 313-326.

[2]

Endo, S., On semi-hereditary rings, J. Math. Soc. Jap. 13 (1961), 109-119.

[3]

Gilmer, R.W., Multiplicative Ideal Theory, Queen's Papers on Pure and Applied Mathematics,

Kingston, Ontario,

44

1968.

20

[4]

Heinzer, W. J., J-Noetherian integral domains with i in the stable range, Proc. Amer. Math. Soc. 19 (1968), 1369-1372.

[S]

Jaffard, P., Theorie de la Dimension dans les Anneaux de Polynomes, Gauthier-Villars,

[6]

Kaplansky,

Paris, 1960.

I., Commutative Rings, Queen Mary College Mathematics

Notes, London, 1966. [7] [8]

, Commutative Rings, Allyn and Bacon, Boston, 1971. Seidenberg, A., A note on the dimension theory of rings, Pac. J. Math. 3 (1953), 505-512.

[9]

, On the dimension theory of rings, Pac. J. Math. 4 (1954), 603-614.

45

A NOTE ON THE FAITHFULNESS OF THE FUNCTOR S ~R(-)

J.W. Brewer and E.A. Rutter University of Kansas, Lawrence,

Kansas

ABSTRACT. If R and S are rings with S a unitary extension of R, the faithfulness of the funetor S ~ ( - ) is studied.

Let R and S be unitary rings with S an extension of R. the functor S ~R (-) is additive,

Since

it is faithful if and only if it

reflects zero maps [ 2, pp. 52 and 56].

It is natural to ask whether

or not this is equivalent to the apparently weaker condition that S ~R (-) reflects zero objects.

Of course,

flat extension of R [2, Prop. 7.2, p. 57]. is to show that in general, however,

this is the case if S is a The purpose of this note

this is not so.

In fact, we prove

that S ~R (-) is faithful if and only if S is a pure extension of R in the sense of Cohn [I] and we give an example of integral domains R and S having the following two properties:

Let 0

(I)

S is not a pure extension of R and

(2)

S ~R(- ) reflects zero objects.

>E

> F be an exact sequence of right R-modules.

The

sequence is said to be pure if and only if the sequence 0

> E ~R M

> F ~R M is exact for each R-module M.

46

When R is a subring

of S and the sequence O--->R--->S is pure, we say S is a pure extension of R.

Theorem.

Let R C S be rings, R a subring of S.

the functor S RR(-) be faithful,

In order that

it is necessary and sufficient

that S

be a pure extension of R.

Proof.

Since the functor S RR(-)

and only if it reflects

is additive,

the zero mapping;

each nonzero R-homomorphism

Assuming

the mapping

that S is a pure exten-

sion of R, let f : M'--->M be a nonzero R-homomorphism following commutative

Since the vertical

if

that is, if and only if for

f : M'-->M of R-modules,

1 S R f : S RRM'--->S RR M is nonzero.

it is faithful

and consider the

diagram.

S i R M~

ls~f>s iRM

R ~R M'

iR~f> R ~R M

arrows are monomorphisms and since 1 R ~ f is nonzero,

i S N f is nonzero and it follows that S RR(-)

is faithful.

Suppose now that S is not a pure extension of R and let M be an R-module

such that 0-->R ~RM-->S ~R M is not exact.

with M, we see that this is equivalent

and denote by f the injection of M' into M.

f

commutes

>

M

M

and by the definition of f, eMo f = O.

47

Let M' = ker eM

The diagram

is f>s

M'

R ~R M

to the condition that the

canonical map eM : M-->S ~R M is not a monomorphism.

s RM'

Identifying

But eM,(M' ) generates

S ~R M' as an S-module and therefore i S M f = 0.

We conclude that

S MR(-) is not faithful.

Before providing the example promised in the introduction, we make some preliminary observations. R a subring of S.

Let R ! S be unitary rings with

Theorem 2.4 of [i]

shows that S is a pure extension

of R if and only if each system of linear equations with coefficients in R which is solvable in S is solvable

in R.

Thus,

if S is a pure

extension of R, then each linear equation which is solvable in S is solvable in R.

We observe that this latter condition is equivalent to

the condition that AS N R = A for each right ideal A of R. the desired example, of K.

To construct

let K and F be fields with F a proper subfield

If X is an indeterminate over K denote by S the ring of formal

power series in X over K.

Then S = K + M, where M is the maximal

of S, and if We set R = F + M, then M is a common maximal and S.

AS N R ~ A.

that aS A R

Let s E S ,

PaR.

By the

it suffices to find an ideal A of R such that s ~ R and let a e M ,

aS n R = aR, then as = at for some t E R

a ~ 0.

Then a s e R and if

which is absurd.

It follows

It remains to show that S ~R (-) reflects

Suppose N is an R-module such that S ~R N = 0. S-module

ideal of R

We verify first that S is not a pure extension of R.

above observations,

ideal

Then E ~R N = 0, for any

E since E ~R N = (E ~S S) ~R N = E ~S (S ~R N) = 0.

lar, S/M ~R N = 0 and M ~R N = 0. MN via the mapping m ~ n-->mn.

zero objects.

In particu-

Thus, MN = 0 since M ~R N maps onto

Since M c R, there is a natural

R-monomorphism 0-->R/M-->S/M.

As M is a common ideal of R and S, it

annihilates both R/M and S/M.

Hence,

the usual way as R/M-modules.

If this is done, then the R-homomorphisms

and the R/M-homomorphisms

these modules may be regarded in

between them are identical. 48

Thus, the above

monomorphism splits since R/M is a field.

Therefore,

the sequence

0-->R/M ~RN--->S/M HRN is exact and hence R/M ~R N = 0.

But R/M ~R N =

N/MN and since MN = 0, we conclude that N = 0.

REFERENCES

i.

P. Cohn, On the free product of associative rinss, I.

Math. Zeit. 71 (1959), 380-398. 2.

B. Mitchell, Theory of Categories, New York:

Press, 1965.

49

Academic

On A PROBLEM IN LINEAR ALGEBRA David A. Buchsbaum and David Eisenbud

In its vaguest and most tantalizing

form~ the problem referred to in the

title of this paper is to say something about the solution of systems of linear equations over a polynomial ring

R = K[XI,...,Xn]

, where

K

is a field.

difficulty is~ of course, that a system of linear equations over

R

The

may not

possess a set of linearly independent solutions from which all solutions may be obtained by forming linear combinations with coefficients

in

R .

Hilbert's Syzygy Theorem suggests a promising approach to this problem: ~i

Let

be a system of linear equations~ which we will regard as a map of free

R-modules~

say

~i: F l - - >

Fo

If we form a free resolution of the cokernel of

~i ' that is an exact sequence of free R-module of the form

°n- ~

~n (i)

Fn

>Fn_ I

~2

~Fn_ 2

>: ..

>F 2

>F I

~>Fo

'

then the Syzygy Theorem tells us that the system of linear equations represented by

~n-1

cisely,

does have a full set of linearly

independent solutions.

More pre-

it tells us this in the graded or local cases; in general~ Ker ~n-i

merely stably free. may replace

Thus, replacing

Fn

by

Ker ~n-i

is

in the sequence (i), we

(i) by a finite free (or stably free) resolution of the form

(2)

o

>F n

~n >Fn-i

>...

>FI

>F o

In order to use the approach to linear equations that Hilbert's Theorem suggests~

it seems to be necessary to study the relations that must hold between

the maps

~i

in a finite free resolution of the form of

establishing a relationship between the maps [6].

Letting

Theorem i.

(3)

nAX

(2).

The first result,

~i ' was also obtained by Hilbert

denote the n th exterior power of a module X, we state it as;

Suppose that

R

0

is a noetherian ring.

>R n

~2

>R n+l

~i

If

>R

is an exact sequence of free R-modules, then there exists a non-zerodivisor

50

a c R

n

such that~ after making the canonical identification ~Rn+l*

Rn+l

, we have

ARn* ~ R

and

n ~i = aA ~2" "

(Hilbert actually proved this only when

R

is the graded polynomial ring in

two variables over a field; he did it to illustrate the application of the Hilbert function.

The first proof that works in the generality in which we have stated

the theorem is due to Butch [5].

The most elementary proof is contained in [7].

We shortly give a new proof of this result.) Theorem i can be applied to a special case of Grothendieck's Theorem in a way that we will now describe. polynomial rings to regular

To do so, we w i l l shift our point of view from

local rings; just as in the polynomial case, every

modnle over a regular local ring has a finite free resolution. Grothendieck's ring, and let

x ¢ S

erated R-module

lifting problem is the following: be such %hat

R = S/(x)

let

is regular.

S

be a regular local

Given a finitely gen-

M , does there exist a finitely generated S-module

M~

such

that

(2)

x

If a module

is a non-zerodivisor M # satisfying

on

M# ?

(i) and (2) exists, it is called a lifting of

M

That the lifting problem is closely related to the structure of free resolutions is shown by the next lemma. Lemma 2.

Let

S

be any ring, and let

x e S

be a non-zerodivisor.

Let

R = S/(x) , and let

~2

F: F 2

~l > FI

be an exact sequence of free R-modules.

> F°

Suppose that

# # is a sequence of free S-module which reduces modules then Coker ( 4 ) Proof:

is a lifting of

(x) to

If

~01 ~02 =

0 ,

Coker(~l).

The hypothesis allows us to construct a short exact sequence of complexes

o

>~F #

x

>~#

>~

> 0

The associated exact sequence in homology contains

(4)

Hl(F)

> H o ( ~ #)

x

> Ho ( ~ # )

51

>Ho(IF)

> 0

On the other hand, HI(IF) = 0

Ho( IF#) = Coker(~#l) ~ Ho(IF) = Coker(~l) and, by hypothesis

Thus

(4) becomes

0--.>

Coker(~l#)

x

> Coker(~)

> Coker({01)

> 0 ,

and the lemma is proved. Lemma 2 implies the liftability of various classes of modules. in the situation of the lifting problem, any module one can be lifted:

If

0

> FI

>F

For example,

M of homological dimension

is a free presentation of

M ~ then

O

there exists a homomorphism ~#i

of free

(for example, to construct

one can choose bases of

~I

each element of the matrix of

~i).

S-modules which reduces to FI

By lemma 2, Coker (~#i)

and

~i modulo

F2

x

and lift

is a lifting of

M

Theorem i can be applied to the lifting problem through the use of lemma 2. Together they imply the liftability of cyclic modules of homological dimension 2.

For, if (3) is the free resolution of such a module, we may choose a map

# ~2

of free S-modules and an element

a # e S which reduce, modulo (x), to

# and a respectively.

Letting

# #

q01

~02

# =

, it is easy to see that

a

#

~i q02 = 0

By theorem i,

~i

reduces to

and shows that Coker(~l) is a lifting of

~i

modulo

Coker(~l)

(x) , so lemma 2 applies

.

We now return to the problem with which we began: what can one say about the relations between the maps

q0i in (2) ? If the F. were vector spaces, the l exactness of (2) would be equivalent to a condition on the ranks of the maps q0i . In the case of, say, a local ring

R , what additional condition or condi-

tions must be imposed in order to ensure exactness of a sequence of free R-modules? Let

In order to answer this, we introduce some terminology and notation. R

be a commutative ring and

free R-modules. r

that

r

~

> G

a map of finitely generated

to be the largest integer

r

such

r

A~: AF

ideal of

~: F

We define the rank of

R

>

AG

is not zero.

If

generated by the minors of

The definition of

I(~)

of order

~ .

I(~)

F

and

It can also be shown that the map

k , a map kAG* ® kAF

~

52

the

r .

makes sense if one chooses bases of

writes the matrix associated to induces, for every

r = rank (~)~ we denote by ~

>R (where G* = HomR(G,R))

G

and

~:F-->G and

With this notation, we can state Theorem 3 [2]. Let

R

be a noetherian commutative ring, and ~n

~: 0

~i

-> Fn

>Fn_ I

a complex of free R-modules.

i)

ii)

Then

> ---

~

is an exact sequence if and only if

rank(~k+l) + rank(~ k) = rank(Fk) l(~k)

>F o

>F I

for

k = i, .... n

contains an R-sequence of length

k

for

and

k = l,...,n

We make the convention that condition ii) is automatically satisfied if l(~n) = R. As a result, when

R

is a field, condition i) becomes the only (and the usual)

condition for the exactness of a complex of vector spaces. Theorem 3 provides one very quick proof of Theorem i. • : 0

>R n-I (aij)>R n (Yi) >R

~2 = (aij) and

>R/I

is exact.

~i = (Yi) ' we know by Theorem 3 that

I(~2) contains an R-sequence of length two. the minors of order 0

>0

For suppose

>R (Ai) >B n

n-i ~

But

of the matrix (aij) . >R n-I

Letting

rank(~2) = n-i

I(~ ) = (Ai) where 2 Thus, the sequence

and [Ai}

are

is exact since the composition is clearly zero

and Theorem 3 applies. - 0

Because the sequence

>R

(Yi) >R n

~

>R n-I

is of order two,

we get a map of complexes

0

>R (Yi)

0-->R and thus

Yi = a~.l for

>R n

q0~

>Rn_ 1

(~i) >Rn i = l,...,n

Although Theorem 3 provides us with a nice proof of Theorem i, it is not clear how that gets us any further into the problem of determining relations among the maps of an arbitrary finite free resolution.

In [4], we have exploited

Theorem 3 to obtain the following two results: Theorem 4 [4~ Th. 3-1]. Let

R

be a noetherian ring, let (2) be an exact se-

quence of free R-modules, and let r k

= rank (~k).

53

Then for each

k, 1 < k < n,

there exists an unique homomorphism

i)

an =

ii)

rn rn A ~n: R = AFn

for each

rk > A Fk_l~

ak: R

rk_l A F{_ I

such that

rn > AFn-1

k < n , the diagram rk A ~k

rk A Fk

rk > A Fk_ I

R/ a commutes. Using the maps

a kl :

a k , we may define maps

rki~ A FR_ I

to be

> F~_ I

the composite:

rk-1

-i

rk+l

A Fk_l

r n: ~ Fk_ I @

where

a k @1 rk

= R e A F~_I - - ~ r k +i A F~_ I

rk+l

AF~_le

>F[_ I

AF~_I

n

>F~_I

is the usual action of

AFk_ I

on

AF{_ I

(see Ill). Theorem

5.[4, Th. 6.1].

Let notation and hypothesis be as in Theorem 4. rk-I k ~ 2 , there exist maps bk: F{ > A Fk_ I making the diagram

rk-i A Fk

rk-i A ~k >

Then for

r -I k A Fk_l

ak- I

bk F* k

commute. Theorems minors

4 and 5 give us fairly strong information about the relations

of orders

rk

and

rk-i

of the maps

If we have the exact sequence

54

~k

of the

in a finite free resolution

(2).

F;

then

O

.> Rm-2

O3

> Rm

~2

> R3

O1

> R

,

r I = 2 , r I -i = i , and it is possible to use Theorems 4 and 5 to express

the maps

~i

and

O2

completely

in terms of the minors of

O3

of order

m-2

(see [4]). Consequently just as Theorem i was applied to the lifting problem for cyclic modules of homological dimension 2, Theorems 4 and 5 may be applied to show that cyclic modules

R/I

of homological dimension 3 may be lifted, provided

I

is generated by 3 elements. In the general case of a cyclic module results applied to a resolution of

0

> R p - -O3 ~

R/I

of homological dimension 3, our

R/I:

R m - -O2 ~

Rn

~i

> R

only give us information about the minors of order

> R/I

n-I

and

> 0

n-2

of

~i

One

might therefore hope that further information about the lower order minors of and

O2

might be obtained from information about the lower order minors of

This idea, of course,

is completely demolished when one finds resolutions

~i O3 .

of the

form (5)

0

particularly

> R

if

I

O3

> Rn

On

> R n - -O1>

R

contains an R-sequence of length three.

local ring, this turns out to be the case precisely when ring.

It is

> R/I

If

R/I

R

> O,

is a regular

is a Gorenstein

possible to show, however, that the lifting problem for 5) in the

Gorenstein case reduces to a problem of lifting

0

where

> Rp

J

¢3

> R m - ~2 - ~

~i

> R

>R/J

> 0

is an ideal generated by four elements and contains an R-sequence of

length three.

Since

p = m-4+l = m-3 , the map

than those of order p,

provided m -3 > i

it possible to have an ideal i)

R4

R/I

ii)

h~

iii)

I

i.e.

¢3

will have lower order minors

m > 4 .

I in a regular local ring

is a Gorenstein ring R/I = n

and

is minimally generated by

n + i elements?

55

The question arises: is R

such that

In [3], we show that this situation cannot arise.

In fact, the question arises

as to what restrictions there are on the number of generators a regular local ring

R

when

R/I

is

Gorenstein.

dim R/I = 0 , we have found such ideals

If

of an ideal

dim R = 3

I

in

and

I generated by five, seven, and nine

elements, but none that are generated by an even number of elements. This appears to be R e d up with questions about the possible skew-symmetry of the map

0

> R

~3

~2

> Rn

(or probable)

in the resolution of such an ideal:

~2

> Rn

~i

> R

R / I ~

0

A computer program worked out by R. Zibman for the Brandeis PDP-10 computer has provided numerous examples of such ideals information on the problem.

I

and may yield same helpful

In any event, it seems evident that our problem in

linear algebra has many ramifications,

the nature of which we are just beginning

to discover.

Bibliosraphy

i.

N.

Bourbaki, Elements de Mathematiques, Algebre, Ch. III, Hermann,

2.

D.A. Buchsbaum and D. Eisenbud, What makes a complex exact, J. Alg. to appea~

3.

, Remarks on ideals and resolutions,

1970.

Symposia Math.

to appear. , Some structure theorems for finite free resolutions,

4. To appear. 5.

L. Butch, On ideals of finite homolo~ical dimension in local rings, Proc. Cam. Phil. Soc. 64 (1968) 941-946.

6.

D. Hilbert, Uber die Theorie der al~ebraischen Formen, Math. Ann. 473-534.

7.

I. Kaplansky, Commutative Rin~s, Allyn and Bacon, 1970.

56

(1890),

MAXIMAL

IDEALS

IN P O L Y N O M I A L

RINGS

E d w a r d D. D a v i s and A n t h o n y V. G e r a m i t a Dept. of M a t h . r SUNYA, A l b a n y , N. Y. 12222, U S A Dept. of M a t h . , Q u e e n ' s U n i v . , K i n g s t o n , Ont., C a n a d a

T h i s n o t e is c o n c e r n e d w i t h e s t i m a t e s of the m i n i m a l n u m b e r of g e n e r a t o r s for m a x i m a l ideals in p o l y n o m i a l r i n g s -- e s t i m a t e s e x t e n d i n g the w e l l k n o w n facts of the case w h e r e i n the c o e f f i c i e n t r i n g is a field.

As ever,

all r i n g s

are c o m m u t a t i v e w i t h

the m i n i m a l n u m b e r of g e n e r a t o r s c o u r s e be i n f i n i t e . ) b u t in m a n y

cases

the p o l y n o m i a l Hilbert's

shall obtain

sharp estimates

a generalization result

m e n t of the N u l l s t e l l e n s a t z This m e a n s

that there exists

reader who

cares

ideals

only

in some e a s i l y o b t a i n e d ,

consequence

= n if R is a f i e l d

Noetherian

of

[8].

R and as c o r o l l a r y

We the

H i l b e r t ring,

for

ideals.

as above, and P = MnR.

We n e e d one

rendering

of the K r u l l - G o l d m a n

[4, 6]: R/P is a G - d o m a i n

[5, Thm.

t in R s u c h t h a t P R rings

is m a x i m a l . t can m a n a g e w i t h o u t

We n e e d a l s o the u p p e r s e m i c o n t i n u i t ~ I finitely

ists t in A - Q s u c h t h a t ~(IAt)

i d e a l of

infinitely many maximal

for N o e t h e r i a n

of a r i n g A, w i t h

denote

(This n u m b e r m a y of

A well known

for R a r e g u l a r

polished

I.

L e t v(I)

for M a m a x i m a l

for a r b i t r a r y

a Dedekind domain with

from Kaplansky's

fact.)

of ~(M)

is that ~(M)

F r o m n o w on R, S and M are fact

interested

ring S = R[Xl,...,Xn].

sharpest possible example,

of the i d e a l

We are h e r e

Nullstellensatz

1 ~ 0.

generated

= ~(IAQ).

57

of ~:

treat23]. (The this

For I and Q

and Q p r i m e ,

t h e r e ex-

THEOREM. (I)

For P maximal,

(2)

In any case,

(3)

For P maximal

Proof. R/P;

W i t h R, S, P, M as above, ~(M)

~(M)

semicontinuity

ring

Then:

__< ~ ( P R p ) + n + l . ~(M)

< u(PRp)+n.

S/PS is a p o l y n o m i a l

(I) is t h e n an i m m e d i a t e

the c o e f f i c i e n t

finite.

< ~(P)+n.

and n p o s i t i v e ,

For P maximal

a s s u m e ~(P)

consequence

is a field.

As

for

ring over

of the k n o w n (2), o b s e r v e

case w h e r e i n t h a t the u p p e r

of ~ and the fact t h a t R/P is a G - d o m a i n

existence

of t in R such t h a t P R t is m a x i m a l

(i) then,

~(MSt)

~ ~(P~p)+n.

the f i e l d

guarantee

the

and 9 ( P R t) = ~(PRp).

By

T a k e m in M c o m a x i m a l w i t h

t and let I

be the i d e a l of S g e n e r a t e d by m and ~ ( P R g e n e r a t e MS t. of S.

Then I = M because

As for

(3), n o t e

n o m i a l r i n g S/PS shows s h o w t h a t it s u f f i c e s say

t h a t the N u l l s t e l l e n s a t z that M ~ R [ X I ]

~ ~(PRp)

tf-p, w h e r e Hence

tr+tf*,

where

ated maximal

(reR, pcP). it s u f f i c e s

f* d e n o t e s

REMARKS.

of

apply

in the p r o o f of

(I) to c o n c l u d e

shows

domain

of

image

is p r i n c i p a l ; (2) we h a v e

2. one

M = gS+PS.

1 = tr+p =

gener-

(Observe t h a t this alone.

a n d A/I

And somewhat

semilocal,

~(I)

can a v o i d use of the f a c t a b o u t

(2) by a s s u m i n g ~(M)

t h a t ~(MSM)

Let g =

For Q a f i n i t e l y

~ ~(QAQ)+I.

generated

(i)

of f in A.

"upper semicontinuity"

~ ~(P~)+n; than

and M a n o n p r i n c i p a l

that one cannot

t h a t tA = A:

(2) gives:

~(Q)

3. In g e n e r a l no b e t t e r e s t i m a t e Dedekind

t as in the p r o o f

the c a n o n i c a l

For I finitely

l + m a x { ~ ( I A Q ) IQ m a x i m a l } . ) G-domains

f a c t and

In this case M / P S

to c h e c k

i d e a l of a r i n g A,

generally:

This

to the p o l y -

T h e n s i n c e rg = f - p f - p r ,

i. T a k i n g n = 0 in

f a c t is a c o n s e q u e n c e more

Taking

applied

for any R - a l g e b r a A s u c h t h a t tA = A.

1 = tr+p

for A = S/gS,

is m a x i m a l .

to take n = i.

f g e n e r a t e s M m o d PS.

t h a t ~(PA)

)+n e l e m e n t s of M w h i c h P ISQ = M S Q for e v e r y m a x i m a l i d e a l Q

then

maximal

ideal.

Localize

at P;

apply Remark

(2) is p o s s i b l e :

in g e n e r a l w e a k e n

58

finite:

Take R = S a

4. R e m a r k

the h y p o t h e s i s

i.

on n in

3 (3).

-

Nor

can

one

for

n = 1 wherein

principal

weaken

regular is

Rp

ideal

generated

nonmaximal

is), by

ring,

COROLLARY. tive is

number

generated

Question:

dim(R)

by

Yes =

2

Dedekind

in

domain

proof

Suppose

S with

show

that

~(Q)

the

product

I = HJ,

ponents From

of

of

this

I and

with

J.

~(I) ~ ideals. for

It 2 if,

to an follows for

I a radical

ideal that

ideal.

~(M)

M is

a non-

If S M is

= dim(SM) ; whence

necessarily

maximality known

for

so

by

for

taking

P

R a

R a field:

a polynomial

a regular

ring

Noetherian

the

cases:

in a p o s i -

Hilbert

ring

one

deduces

Q = JK,

and

routinely

--

where

For of

the

"divide"

of For

is

this

the has

R a PID

these

59

n =

--

of

i.

remarks

R a

discovered

For the

Q any proof

of

if Q n R

is

proper

are

ideal

1 primary

com-

component

is

that

of

I.

I is

comaximal

J / ( J ~ R ) S is.

by

and

treatment.

e.g.,

distinct

observed

n = i,

Heinzer

b y H --

(K6R) S a n d

product been

by W.

height

if

--

-- w a s

0-dimensional

K =

[i]);

corollary

I a nonzero

Q / ( Q n R ) S is p r i n c i p a l J~R

(Endo

adapts

is p r i n c i p a l

"the"

proviso?

Heinzer-Davis

domain

ideals.

J is

and

ring"

1

ideals

the

one

= the

Geramita

intersection

--

of

maximal

a Dedekind

the

"Hilbert

dim(R)

case

by

nonzero,

(I:H)

case

of

many

maximal

examplep

A special

P's

well

2 if Q / ( Q n R ) S

decomposition

S-isomorphic

-- n o t

without

Gilmer,

Q~R

H is

J =

[2]

is e s s e n t i a l l y

~

and

examples

= dim(Rp)+n.

then

A special

R is

distinct where

over

valid

by

that

of

to

S,

now

domain

> 0,

insure

semilocal

(3)

ideal

(3)

of

of

n

ideal

infinitely

Swan,

creates

sequence.

[3]).

with

by

The

proper

certain

d i m ( S M)

a theorem

maximal

result

(Geramita

independently Davis.

results

easily

P = 0.

and

if w e

one local

sequence

So

a regular

this

Pz

that

indeterminates

Is

Answer:

so

P maximal

Every

of

and

4).

there

on

for which

a regular

(Remark

-

a 1-dimensional

R Notherian

(iff

Hilbert

hypothesis

R is

maximal

Assume

M

the

3

Hence

maximal

Gilmer: special

~(I)

~

cases

2

-

of

a t h e o r e m of Serre

Question:

How much of Serre's

domains? setting

[7]: v(I)

Observe

S-module

-

~ 2 if ~(IS N) ~

2 for every m a x i m a l N.

theorem survives

for arbitrary

Dedekind

that the validity of the theorem in the more general

is an immediate

sharpening

4

corollary

of the F o r s t e r - S w a n

of the Eisenbud-Evans

conjectured

bound on the number of generators

of an

[0].

REFERENCES

[0] Eisenbud,

D. and Evans,

over p o l y n o m i a l [i] Endo,

S.

15

E. D. Pac.

[3] Geramita,

[4] Goldman,

(1963)

A. V.

O.

20

(1967)

(1951)

I.

[6] Krull, W.

Jacobsonscher

No.

[8] Zariski,

J. Math.

class,

R-sequences

correspondences,

...,

rings,

to appear.

and the Hilbert Nullstellensatz,

Rin~s,

Ringe, Math.

Sur les modules

Foundation

and

197-209.

Allyn and Bacon,

Hilbertscher

Z. 54

(1951)

projectifs,

Boston

354-387.

Seminaire

Dubreil-Pisot

of a general theory of b i r a t i o n a l Trans.

A.M.S.

60

53

(1970).

Nullstellensatz,

2 (1960/61).

O.

Soc.

136-140.

Commutative

Dimensiontheorie,

J-P.

rings,

ideals in p o l y n o m i a l

Hilbert rings

[5] Kaplansky,

[7] Serre,

over p o l y n o m i a l

339-352.

Maximal

Z. 54

efficiently

these proceedings.

Ideals of the p r i n c i p a l

J. Math.

Math.

G e n e r a t i n g modules

rings,

Projective modules

Japan

[2] Davis,

E. G.

(1943)

490-542.

A CANCELLATION PROBLEM FOR RINGS PAUL EAKIN and WILLIAM HEINZER 0.

Introduction. Suppose A and B are rings. Let us say that A and B are stably equivalent if there is an

integer n such that the polynomial rings A [ X 1. . . . . X n] and B[Y 1. . . . . Yn ] are isomorphic. A number of recent investigations [CE], [AEH], [EK], [BR], [H] are concerned with the study of this equivalence. The most obvious question one might ask is: (0.1)Cancellation Problem. Suppose A is a ring. What conditions on A guarantee that for any ring B, A is stably equivalent to B only if A is isomorphic to B?

This cancellation problem was first investigated in [CE] and later in [AEH], [EK], [BR] and [H]. In [H], Hochster provides an example of two stably equivalent rings which are not isomorphic. In addition to providing an example, [HI also draws our attention to the connection between the cancellation problem for rings and the well-studied cancellation problem for modules. In the study of the cancellation problem for rings the following concepts have proved useful. (0.2) A ring A is said to be invariant provided that whenever A is stably equivalent to some ring B, then A is isomorphic to B. (0.3) A ring A is said to be strongly invariant provided that whenever o: A [ X 1 . . . . . X n] ~ B[Y 1 . . . . ,Yn ] is an isomorphism o f polynomial rings, then o(A) = B.

Obviously, strongly invariant rings are invariant. The most obvious example of a strongly invariant ring is the integers Z. On the other hand there are invariant rings which are not strongly invariant. The following isa result from [AEH]. (0.4) Suppose A is a domain o f transcendence degree one over a field, then A is invariant. Moreover A is strongly invariant unless there is a field k such that A = k [ X ] . This result tells us that if k is a field then A = k [X] is an invariant ring. However A is n o t strongly invariant, for if A = k [X] and B = k[Y] then the identity is an isomorphism of A [Y] o n t o B [X] which does not take A o n t o B. A fair a m o u n t can be said about rings of Krull dimension one which are not strongly invariant [AEH], but this case is not completely settled. In particular there is a gap in our knowledge o f the case when A is a Dedekind domain. In section five we have a discussion of the cancellation problem for Dedekind domains and present some questions whose affirmative answer would affirmatively settle the cancellation problem for all one dimensional noetherian domains. One way t o approach t h e cancellation problem for a particular ring A is to write

A [ X 1. . . . . X n] = B[Y 1. . . . . Yn ] for,some ring B and look at D = ANB. In the case of domains, if the transcendence degree of A over D is at

61

most one and S denotes the non zero elements of D, t h e n e i t h e r A = B o r A

s -~ B s ~ k [ X ]

where k is the

quotient field of D. This follows from the result on domains of transcendence one over a field mentioned above, with information like A s = k [ t ] , a bit of additional hypothesis will often yield results like A ~ B-,~ D[X].

When the transcendence of A over D is greater than one, the problem becomes considerably more

complex. In particular, the cancellation problem for C [ X , Y ] , the polynomials in t w o variables over the complex numbers is not settled. C.P. Ramanujam has taken a particularly sophisticated topological approach to this question JR]. He characterizes the complex 2-space as a non singular, contractible algebraic surface which is simply connected at infinity. To our knowledge, this has not yielded a solution to the cancellation problem for C [ X , Y ] ,

C

the complex numbers.

We begin this article by examining Hochster's example from [ H ] . We observe that this provides a nice introduction to an interesting class of rings, called locally polynomial rings in [ES]. (Given a ground ring R and an algebra A over R, we say that A is locally a polynomial ring over R if Ap = A®RR p is a polynomial ring over Rp for every prime p of R). Over a fairly substantial collection of rings we give a number of characterizations of locally polynomial rings "in one variable." In particular we show that if D is a domain, the affine (ie. finitely generatedring extension) locally polynomial rings in one variable over D are precisely the symmetric algebras of the invertible ideals of D. We then use this result to indicate an approach to the cancellation problem for k [ X , Y ] ,

k an algebraically closed field.

Before proceeding let us make the usual stipulations: all rings are commutative with identity and all modules are unital.

1.

Symmetric Algebras and Non Invariant Rings. In this section we give Hochster's example of a non

invariant ring. This example, in addition to showing that stable equivalence of rings is not a trivial relationship (i.e. isomorphism) also serves to remind "commutative algebraists" that its nice to know some topologyespecially the theory of vector bundles. Let us recall the definition and basic properties of the symmetric algebra of a module.

(1.1) Definition. Let R be a ring and M an R module. A symmetric algebra of M over R is a pair (~0,S(M)) consisting of an R isomorphism ~ and an R algebra S(M) such that ~: M ~ S(M) and such that if a is any Rhomomorphism of M into an R algebra A, then there is a unique R-homomorphism h such that the following diagram commutes M

~ S(M)

Every module has a symmetric algebra which is unique up to an R-isomorphism [C]. In fact, if M is an R module, S(M) is R isomorphic

to the "'abelianization'" of the tensor algebra of M over R [C]. Another

realization of the symmetric algebra of a module which is often quite computable is the following.

62

Let M be a module and write 0

-~

K

~

F

~

M

-*

0

exact

where F is a free module, say F = ~ RZi. Let {Xi)ie [ be indeterminates over R and consider J(K) = ie[ {~; riX i e R [ { X i ) I ~ riZi ¢ K). Let 6L denote the ideal in R [ { X i } i d ] generated by J. Then R[M] = R [ {X i) ie]]/6"(

is the symmetric algebra of M over R. To see this one observes that R [M] is a graded,

homogeneous ring over R whose module of 1-forms is isomorphic to M. Thus we have an R-homomorphism h such that M

i

a [M]

S(M)

commutes.

Let z i = rt(Zi). Then we have a mapping R [ { X i } i ¢ l]

~

S(M)

given by

xi~0(zi).

Since J C Ker £ we have a well defined mapping R [M] ~ S(M) which is surjective. Since ho£ = id we have R [M] is R-isomorphic to S(M). Using the above, or referring to [C] one can easily see the following well known facts. (1.2) Lemma. Let R be a ring, then (i)

If F is a free R module on a set of cardinality ], S(F) is R isomorphic to the polynomials over R in

a set of indeterminates having cardinality I. (ii)

If A and B are R modules then S(A~B) is R isomorphic to S(A) ORS(B).

Recall that t w o R-modules M and N are stably equivalent if there is a free module F such that M(DF ~ N@F. In this case, if F is free on n generators then, taking symmetric algebras we have S(M)®RS(F) ~ S(MI~F) ~ S(N(~F) ~ S(N)®RS(F ). S(F) ~ R[X 1 . . . . . X n] . S ( M ) [ X 1 . . . . . X n]

But Hence we have

~ S(M)®R R[X1 . . . . . Xn] ~ S(N)®R R[X1 . . . . . Xn] ~ S(N)[X1 . . . . . Xn]"

Thus stably equivalent modules give rise to stably equivalent rings. This provides the basic idea of Hochster's example: find two stably equivalent modules which are not isomorphic, then prove that their respective symmetric algebras are not isomorphic. (1.3) Lemma. Let R be a ring and M and N two finitely generated R modules. If S(M) and S(N) denote the respective symmetric algebras then S(M) is R isomorphic to S(N) if and only if M is isomorphic to N.

63

4

Proof. Clearly M ~- N : ~ : S(M) is R-isomorphic to S(N). Conversely, suppose S(M) is R-isomorphic to S(N), say by an R-isomorphism ~o. Now S(M) is a homogenous graded ring whose module of 1-forms is isomorphic to M. Identify M with these 1-forms and let {r/i)iei be a basis for M. Then write ¢(rt i) = j~OCXij e S(N). Here aij is a homogenous element of S(N) of degree j. Since ¢ is an R-isomorphism, we have ~°(rti-a°i) = j~=l aij • Denote this element by 3,i. Then since S(M) = R [ (r/i} i e [ ] = R [ {r/i-C~oi} ] we have S(N) = R [ ~3,i}iei ] . By grading it follows that (c~1 i}ie[ generates the 1-forms of S(N), which is a module isomorphic to N. There is then a well defined surjective R homomorphism a: M -* N

given by

o(r/i) = a i l

which takes M onto N. By symmetry there is a surjection o': N ~ M. But o'oa is a surjective mapping of M onto itself. Thus o'oa is an isomorphism [V] and so is o. Thus, by this lemma we see that if M and N are finitely generated, stably equivalent, non isomorphic modules, then S(M) and S(N) cannot be R-isomorphic. The idea now is to find two such modules and prove that any supposed isomorphism of the symmetric algebras would (essentially) have to be an R isomorphism. To this end we recall the following from [S]. n Let k d e n o t e t h e real n u m b e r s and R = k [ X o , . . •, X n ] / i ~=0 X i 2 - 1 = k [ x 0 , x 1 . . . . ,Xn]. Let F denote the free m o d u l e on generators e o , . . . , e n and define o: F ~ F b y

o(ei) = xi(E xie i) . T h e n one has

o 0 ~ Hence

PI9 R~

P

~

F

~

R n+l = Rn~

(Zxiei)R ~ R.

0

exact.

Moreover P ~ R n (ie

P is free) if and o n l y if n=1,3,7.

The notation of this statement holds for the next two lemmas.

(1.4) Lemma. The ring R above is strongly invariant. Proof. Suppose 7': R IX 1. . . . ,X n] -~ B [Y1 . . . . . Yn ] . We may assume R [X I . . . . . X n] = B [Y1 . . . . . Yn ] and show R = t3. Suppose xi=boi+bliYn

+'" ' + b k i Y k

w h e r e b i j e B [ Y 1. . . . . Y n _ l ].

Then ~ x i 2 = 1 implies ~b2,i = 0 if k is greater than zero. But R[X 1 . . . . . X n] is formally real (ie. 0 cannot be written non trivially as a sum of squares there. Thus x i = bol for each i and x i e B [ Y 1 . . . . . Yn--1 ] • By further reduction, x i e B for each i and hence R C B. But it now follows easily that R = B. (1.5) Lemma. If ¢: S(P) ~ R [X o . . . . . X n _ 1 ] then up to an automorphism of R, ~0 is an R isomorphism. Proof. By adding an indeterminate and extending trivially we have

64

~0:S(P)[Y] ~ R[X o ..... Xn_ I ] [ Z ] . Since S(P)[Y] = R [Z o ..... Zn], one concludes that, up to an automorphism a of R, ~ois an R isomorphism. However one then has S(P) ~

a [ x o ..... X n - 1 ]

~1

R[Xo ..... X n - l ]

and a-lo~0 is an R isomorphism. Thus, with the notation of the previous lemmas, if R IX 1..... X n] is isomorphic to S(P), then P ~ Rn by (1.3). So for n # 1,3,7 we see that the rings S(P) and R [X 1..... X n] are non isomorphic, stably equivalent rings. (1.6) Remark. In case n = 2 in the above, the integral domain R has the following properties (i)

R is strongly invariant

(ii)

R [X] is invariant, but not strongly invariant

(iii) R[X,Y] is not invariant.

Proof. We have already seen (i) and (iii). To see (ii) we observe that the proof of (1.4) actually shows somewhat more. We have seen that if R IX 1..... X n] = B [Y]. Then R C B. We can thus appeal to the following result from [AEH]. Let D be a unique factorization domain (UFD) and let (X i) be indeterminates over A. If R is a UFD such that D C R C D[ {Xi} ] and such that R has transcendence degree one over D, then R is a polynomial ring over D.

Remark ( 1.6) has to be considered when one gets overly optimistic about the prospects of proving k[X,Y] invariant, k a field.

2.

Locally Polynomial Rings. Let D be a ring and A an algebra over D. If for every prime p C D, Ap =

Dp ~D A is a polynomial ring over D, we say that A is locally a polynomial ring over D. These rings are investigated in [ES]. Since the formation of symmetric algebras respects change of ring, we find that the symmetric algebras of locally free modules provide plenty of examples of locally polynomial rings. (2.1) Lemma. If M 1 and M 2 are modules over a ring R such that S(M1)[X 1..... X n] ~ S(M2)[Y 1..... Yn ] R (ie. the algebras are R-isomorphic]. Then for every prime p C R, S(M 1) O Rp Rp S(M2) ® Rp.

Proof. By lemma (1.3) we have that M 1 and M 2 are stably equivalent modules. Thus MI(~RR p and M2®RR p are stably equivalent Rp modules. Thus MI®RR p is Rp isomorphic to M2~RR p by a theorem of Bass which assures us that stably equivalent modules over a local ring are isomorphic [B]. Thus S(MI®RR p) = S(M1)®RR p is Rp isomorphic to S(M2)®RR p = S(M2®R Rp).

65

With reference to the preceeding lemma we see that if S(M 1) is a domain and S(M 1 ) is stably equivalent to S(M2), then even though S(M 1) may not be isomorphic to S(M2), they still have isomorphic quotient fields. Referring to Hochster's example then, we see that this is not a counterexample to the following "bi-rational cancellation problem". Suppose A and B are domains and A[X] ~ B[Y] is an isomorphism o f polynomial rings. Are the quotient fields of A and B isomorphic?

This question is considered in [ A E H ] .

It is certainly related to the following, well studied question of

Zariski: Zariski Problem, Let K and K' be finitely generated fields over a field k. Assume that simple transcendental extensions of K and K' are k-isomorphic to each other. Does it follow that K and K' are kisomorphic to each other? 1

3.

Some Interesting Types of Ring Extensions. In studying the cancellation problem we have been led to

consider the following possible relationships between a ring R and its extension A. (l)

A is the symmetric algebra of a projective R module

(2)

A is projective in the category of R algebras. That is, given a diagram of R-algebras and R

homomorphisms. A

B

~

C

~

0

with the bottom row exact, there is a R homomorphism 0 : A -+ B which makes the diagram

A

B~C~O

(3)

commute.

A is a retract of a polynomial ring over R. That is, there is an idempotent endomorphism of some

polynomial ring R[ (Xi}ie I] which has A as its range. (4)

There exists an R-algebra A* such that A®RA* is a polynomial ring over R.

(5)

There are indeterminates (Xc~) and {Yi3} such that R[ {Xc~}] = A[ (YI3}].

(6)

There exist indeterminates {Xc~} such that R C A C D[ {Xc~} ] and A is an inert subring o f

R[ {Xa) ]. That is, if a,b are non zero elements of R[ {Xc~}] and ab e A then a e A and b e A.

1

This question is studied by Nagata in [ T V R ] .

He notes there that it was raised by Zariski at a 1949

Paris Colloquium on algebra and the theory of numbers.

66

7 (7)

A is locally a polynomial ring over R.

A t present the relationships among these properties is not clear. However, some connections do f o l l o w readily. (3.1) Lemma. With regard to the above conditions we have the following implications: (7)

~

(1)

=

(2)

4

~

5

¢=~ (3)

f

In case R is a domain we have 15) ~ (6), and if A is an affine ring of transcendence degree one over the domain R then (7) ~ (1). Proof. (1) ~ (7)

Let P be a projective R module and Q a prime of R. Then SR(P)®RR Q = SRQ[P®RRQ].

But P®RRQ is free, hence SRQ[P®RRQ] is a polynomial ring. (2) ~(Rp,

of e l e m e n t s

that we w i l l use

the e x i s t e n c e

and p

set of p r i m e

is a p r i m e

For the r e a d e r ' s [E-E,1]

m cM,

ideal

generated

of g e n e r a t o r s

M at p for every p r i m e

is the p

number

~(~,~)

of a p r i m e @t

be a f i n i t e l y

i d e a l of R , then dim(p)

If M is an R - m o d u l e ,

m is b a s i c ment

is the m i n i m a l

M

•

If

that

: Let

R

be f i n i t e l y ideal

p

generated

of

R ,

m l , . . . , m u ~ M'

(r,ml)

c R @ M

of the f o r m ml+rm~, The

other

result

be a n o e t h e r i a n R-modules,

and

w i t h dim R = d < ~ . suppose

M'

is

(dim(p)+l)-fold

generate

M'

, and if

is basic, where

ring,

then there u m' ¢ i~2 Rmi"

is that b a s i c

basic

r c R

is a b a s i c

elements

that for in

M

is g i v e n element

in p r o j e c t i v e

at

such

of

modules

M

are

unimodular:

Lemma

i [E-E,I].

generated and only

Section The Let ring.

S

If

projective

R

is a c o m m u t a t i v e R-module,

if it g e n e r a t e s

2.

an

a free direct

we w i s h to c o n s i d e r

be a n o e t h e r i a n

ring,

and

let

d = dim R .

79

and P

element

summand

The Conjectures, and a S u r v e y

conjectures

Set

then

ring,

of

of K n o w n

is a f i n i t e l y

m ~ P

is b a s i c

P .

Special

Cases.

are the following: R = S[x]

be the p o l y n o m i a l

if

i)

If

M

is a finitely

for every prime

ideal

In particular, P 2)

generated

if

p

P

such that

R , then

M

is a projective

~(Rp,~)

has a basic

R-module

~ d

element.

of rank

d , then

has a free summand.

If

P

is a finitely

and if

Q

generated

is a finitely

Q • P m Q e P' , then 3)

of

R-module

Let

M

be a finitely

of primes

of

M

R-module

generated projective

of rank > d

R-module

such that

P m P,. generated

R-module,

R such that dim(p)

n = Then

projective

max(~(R pe@ v'Mp)

can be generated

< d .

and let

@

be the set

Set

+ dim(p))

by

n elements.

All three statements become true [E-E,I] if d is replaced b y d+l. It is easy to see that given their generality, these conjectures are the strongest suppose

that

J

possible.

To see that

is a maximal

M = J ~...~ J (d-i times). for if

(rl,...,rd_l)

radical

of the ideal

ideal of height

Then

M

d

and

element,

contradicting

then

J

Krull's

i ,

let

cannot have a basic

were a basic ZRr i

this is so for conjecture

element

,

would be the

principal

ideal

theorem. To see that conjecture 2 fails be the coordinate

S3 .

it is known that ~

2 x 3 - I), and let

Then

rank P = 2 = dim R-I

then

~ m P/XP ~ S 2

which

As for conjecture

where

ideal K

elements

I

~

free. and

2

d-l,

3 , Murthy

, and

~ ~ S

R = S[x]

R ~ P m R 3.

However,

[Mur,2]

let

S

of

is free, but and if

P = ~®S R P -~ R 2 ,

has given an example

2, in a ring of the form

such that

implies

be the cokernel

If we set

but r e q u i r ~ 3 generators

Conjecture

by

is a contradiction.

of height

is a field,

~

is projective

is not

then

unmixed

d

ring of the real 2-sphere,

2 2 S ~IR[Xl,X2,X3]/(Xl+X2+ sJxI'x2'x3)-~

if we replace

I

can be generated

R =

of an K[X,Y,Z]

locally by 2

globally.

that if

R = K[XI,...,X d]

80

with

K

a field,

then every projective

of rank > d

is free.

This is a weak form of

Serre's problem. The following Corollary illustrates

the application

of conjecture

3: Corollary to Conjecture ring, and let

I

rated locally by

3 : Let

R = S[X]

be an ideal of g

elements.

R . Then

be a noetherian polynomial

Suppose that I

I

can be gene-

can be generated b y

max( d , g+dim R/I) elements. The Forster-Swan Theorem [E-E,I] that

I

implies in the above situation

can be generated by max(d+l,

g + dim R/I)

elements. It would be tempting to formulate a stronger version of conjecture 2, to parallel

[E-E,I,Thm.Aiib].But

this stronger form is false, as is

any form strong enough to imply the Stable Range Theorem for the ring K[XI,...,~], [Vas]

where

shows that

K

d+l

is the field of real numbers.

is best possible value in this case).

We do not know whether, of the conjectures,

(An example in

E(d,R)

for a ring R

satisfying this hypothesis

is transitive

on unimodular

elements

(See [ B a s s ~ ] ) . We will now ennumerate the special cases of our conjectures that we have found in the literature. The best known of these special cases is Seshadri's if

S

is a principal ideal ring, then every projective

is free.

This has been generalized by

and Murthy [Mur,I]

Serre,

to the case in which

S

a free module and an ideal.

S[X]-module

[Set], Bass [Bass,l]

is any 1-dimensional

ring with only finitely many non-regular maximal ideals; the theorem says that ar d .

is free.

theorem

, where

Geramita

rows

is i m m e d i a t e

3

S

dimension,

P

of

conjecture

for

on u n i m o d u l a r

conjecture

the p r o b l e m

in w h i c h

R = S[X]

2 in case

So[XI,...,Xn]

S[X]

1 ,

is a o n e - d l m e n -

our c o n j e c t u r e

from the structure

[Endo]

S

ring of p o s i t i v e

conjecture

In [Ger], of

if

is t r a n s i t i v e

for i n s t a n c e

has a b a s i c

ring over a s e m i - l o c a l

h i m to p r o v e

As for c o n j e c t u r e

euclidean

[B-S]

ring over a s e m i - l o c a l

d = dim R , then E(n,R)

M

is free,

results

2] p r o v e s

for a

.]

2 . Endo

His

in [E-E,2]

if , in c o n j e c t u r e

in [E-E,2]

S[X,Y]

domain.

in [Bass,

7

that

R , then

the result

[E-E,1,

every projective

implies

of

We now turn to c o n j e c t u r e when

the p r o o f we gave

in case

dim S=O;

for m o d u l e s for m a x i m a l

over a ideals

is a s e m i l o c a l conjecture

is a D e d e k i n d

domain

3 .

prinfor

Davis and Geramita form

[D-G] prove it for m a x i m a l

R = S o [ X I , . . . , X n]

positive

, where

interesting

implied b y a theorem of Murthy holds if

field,

is an a r b i t r a r y

over rings of the semilocal ring of

dimension.

A particularly

3

So

ideals

R

and

is

M

Section 3 • In this

is any ideal of

2 Special

Cases

section we will R

cludes a number were m e n t i o n e d

form

3 always holds

Theorem

i .

S

conjectures

Remarks.

The hypothesis

only finitely many prime tion in the proof

section.

S

maximal height.

A version

S

is

We will also prove that con-

modules

of rank 1 .

ring with a noetherian

R = S[XI,..,Xn].

S

Set

d = dim R .

to be that

may be p r o v e d

of the primes

I using j-primes,

version

has

The only m o d i f i c a -

of Theorem

of the

of

S

etc.,

just as we will prove Theorem

some n o n - c o m m u t a t i v e

Then

S

is the replacement

by the i n t e r s e c t i o n of Theorem

spectrum

R •

can be weakened

that would be n e c e s s a r y of

, where

in

This result in-

ideals of m a x i m a l height.

Jacob saa radical

[E-E,I]

dimension.

of section 2 hold for

on

over a

special cases of the conjectures w h i c h

be a semi-local

the three

in 3 variables

R = S [ X I, .... X n]

for p r o j e c t i v e

of dimension > 0 , and let

shows that conjecture

all three of our conjectures

ring of positive

in the previous

Jecture

is

R .

establish

of the known

Let

, which

3

of the Conjectures

has the

a semi-local n o e t h e r i a n

Presumably,

[Mur,2]

the ring of p o l y n o m i a l s

the case in which

style of

special case of conjecture

of in the

I .

i , as in [E-E,I],

is also true. All three parts

Lemma 2: Let Let K ~ M

R

of Theorem

I depend on the following

be a n o e t h e r i a n

be finitely g e n e r a t e d

ring,

and let

R-modules, 83

I

lemma.

be an ideal of

and let

t

R

.

be an integer

such that for every p r i m e at

p .

Suppose

a)

(r,k)

b)

The

Then

image

in

M

Proof

of T h e o r e m

of

k

p

The

element

M/IM

k'

Let

J on

R

in

M

such that

such that

statement

S

follows

R-module

k + rk'

is

"

and

radical

shows from

that

of

S , and

dim(R/I)

the first b e c a u s e

generates

a free

summand

set

< d . a basic

[E-E,1,

1]. the f i r s t

statement,

R

contains

I , we h a v e

which

,

(M/IM)~)

w h e r e we h a v e w r i t t e n (R/I)

ing the

~

, with

M

We w i l l is a p r i m e i d e a l of

basic

of

Thus

ideal

m + m' R

of

m' = m ( m o d

M/IM

m'

c IM

This

p

by

Since

[E-E,

element.

I].

Apply-

element

such that

at all p r i m e s

shows that

I •

m + m'

"

q N S

On the other hand,

I).

modulo

to the b a s i c

w i t h ht q = d , then

q m I .

p in

P e @d-i

is b a s i c

every p r i m e

to this b a s i c

t = d-l, exists

for

~ d ,

element

reduces

there

at every p r i m e

ideal

lo I since ~ +

a

K = M and

show that

S .

p(Rp,Mp)

which

c R e M , we see that in

we n o t e that,

for the r e d u c t i o n

b e an e l e m e n t

Lemma

is b a s i c

=

< d , tl.ere exists

m c M

(l,m)

c IK

be the J a c o b s o n

To p r o v e

Let

are e l e m e n t s

basic

is basic.

an e l e m e n t

in a p r o j e c t i v e

~((R/I)~

dim

(dimt(P)+l)-fold

k ¢ K

for a l l p c @t

l:

second

and

in

The h y p o t h e s i s

l)

of

at

' K is

is b a s i c , and

exists

basic

Lemma

r ¢ R

~ R ~ M

there

I = JR.

that

P ~ @t

of

R .

If

q

is a m a x i m a l

m + m' is b a s i c

m + m'

is b a s i c

moduat

q,

as required. 2.) there

We b e g i n b y m a k i n g

exists

is e n o u g h at a time.

a projective

to be able

the f a m i l i a r

module

to c a n c e l

Thus we may

suppose

Q'

reduction

such that

the r a n k

1 free

R S P m R~P'.

84

to the

QeQ'

is free,

summands Let

case

of

Q = R: so it

Q~Q'

one

f: ReP' + R e P

be

8 the isomorphism.

(l,O)eR@~

is clearly basic,

= (r,m o) e R ~ P , then

(r,mo)

is basic.

morphism

carries

(r,mo)

a

of

a commutative

This

R @ P

diagram with exact R

~

R @ P'

~ P' ~ 0

0

~

R

-~

R ~ P

~ P

a , we again

use

[E-E,

to show that there

such that

E o + r~l

is basic

to denote

reduction

modulo

We can now apply the (r, mo+ rml)

in part basic

map

, it follows

If

~

is basic

in

P

, we have written

-

and

t = d-I

there exists

to the ele-

an element

at all primes

that the element

a map

follows

denotes P ~ R

Lemma 1 ,

m

R ~ P

taking

the map

carrying

generates 1

a =

the pattern

p e @d-I

m = mo +

m to

a free

in [Bass,

R ~ P carrying

1 to

l-r

(such maps

summand

of P), and

" As

rml+ rm 2 is

2

~ and [E-E, ml+m 2 ,

exist because, y

denotes

the

to -m , then we may take

1 + ~ ?+~+?~

3)

mI

P .

l,Cor.4]:

by

K = M

By the Lemma,

mo+rml+rm 2

I , above

and the fact

exists an element

P/IP , where

Lemma with

The rest of the proof

denotes

in

I Theorem A]

I .

e R ¢ P.

such that

in

form:

P ~ P.'

(R/I) < d

m 2 ¢ IP

We will thus obtain

~ 0

dim

ment

(1,O).

rows of the following

~

To construct that

f((l,O))

We will show that some auto-

to

0

shows that

so if

Unless n < max(d + ~(Rp,M) ) , ~ dimp=d

~

1

i +?O

/

this is the conclusion

of the

%4o

usual Forster-Swan p of dimension primes

p

theorem

d such that

of dimension

[E-E,I]. %4

< d -i

Therefore

there must

0 , and so, a f o r t i o r i , such that 85

~

@ 0 .

exist primes

there

Thus n

exist

~ d .

Now suppose exact s e q u e n c e (* )

~(R,M)

It f o l l o w s

the

pattern

is b a s i c

which

Rt =

, Lemma

i]

Rm @ P , by

[E-E,I

We w i l l t h e n h a v e

rank

t-l.

contradicts

P -~ M , so

and

K

)-fold b a s i c is

in

It is easy to see that

Rt .

Thus

contains which

of

[E-E,

a basic

r e d u c e s to

to the b a s i c k'

c IK

P

"

Since

~ @d-i

the p r o o f

tern

K

i, Thm. element [

contains

such that

k + k'

k+k' = m

is b a s i c

The main of

Rt

Instead,

at w h i c h

difference between with

p

k ¢ K

in

dim p < d ,

the i d e a l

I ,

as is

K

in

K

Rt/IR t

in

be an e l e m e n t

e x i s t s an ideals

I , it f o l l o w s as in

R t , as r e q u i r e d .

l]

Lemma follows

the p a t -

so c l o s e l y that we

we w i l l r e m a r k on the p o i n t s

before

We a s s u m e

in that

him.

L e m m a 2 of this paper,

A = R

t

dim P = d .

at a l l p r i m e

of this

in [E-E,

R

L e m m a w i t h t = d-I

c h a n g e s m u s t b e made.

a c o p y of [E-E,1]

[E-E, 1]

the

modulo

The p r o o f

of T h e o r e m A g i v e n

of T h e o r e m A

Let

is b a s i c

of i~ that

it in detail.

at

the i m a g e of

Applying

in

the r e a d e r has

A ii) b)

p

e R ~ R t , we see that t h e r e

of P r o o f of L e m m a 2 :

the p r o o f

ideal

is Just as b a s i c

IR t .

indu-

is

s u c h that

is b a s i c

w i l l not g i v e

p

K

p

R t / IR t .

is

elements.

of

primes

k

of the p r o o f

element

such that

of

(l,k)

a basic

that

P .

= t , p r o v i n g the theorem.

at e v e r y p r i m e

implies

modulo

t-I

that

at

P

Rt ~ M

t > n , it f o l l o w s

if a p r i m e

~

epimorphism

~(R,M)

contains

Rt

A]

the

Rt

module

above,

can b e g e n e r a t e d b y

that

in R t / I R t

element

element

Sketch

K

in R t

d-fold basic

then the i m a g e

M

our a s s u m p t i o n

a n d the a s s u m p t i o n

(dim p + l

R t , so that

P -- t - I > d , so b y p a r t 2),

It r e m a i n s to s h o w t h a t (*)

in

, for some p r o j e c t i v e

On the o t h e r hand,

ces an e p i m o r p h i s m

From

show that

Cor. 5

e K

This

], we w i l l

of [E-E,I,

m

of r a n k

is a short

~ M ~ 0

there e x i s t s an e l e m e n t

free

that t h e r e

of the f o r m

0 ~ K ~ Rt

Following

= t > n.

and Theorem

, is that in L e m m a 2, m I is a s s u m e d

86

i0

basic mod I , and we w i s h to produce T h e o r e m A~) w h i c h essentZally

lie in

elements

a i (in the notation

I . (The appearance

trivial change).

b y a number of a p p l i c a t i o n s

The elements of Lemma 3

of the sets

a. 1

of

are actually

[E-E,1].

of Lemma 3 , it suffices to prove that we may take elements

aj

in Lemma 3

don't matter).

Lemma 3 , in section 5 of [E-E,I] in the notation

of

@t

is an obtained

In the notation ale I

(the other

Turning to the proof

, we see that

of

aI

of

is chosen

so that,

[E-E,I], m I + aalm I

is basic

in

M

Suppose that of this

at a certain finite mI

is basic mod I, and that

list that contain

ml+aalml basic mod Thus it suffices contain

I .

list of primes

I .

pl,...,pv

Pl '''''Pv

Then any choice of

.

are the primes

ale I

makes

I, and therefore basic at each of the primes ~Y"Pv''

to deal with the primes

To do this, we first

pv,+l,...,pv

choose

aI

, w h i c h do not

in the way that the

proof

of Lemma 3 instructs u s - not n e c e s s a r i l y in I. Then we p i ~ k s¢l v such thac s ~ u . P i ; this is possible, since otherwise I w o u l d b e conl=V ' + 1 tained in one of t~e primes in the union. We can now replace a I by

sale I

and continue with the proof as given in [E-E,1].

The next theorem proof

covers a special case of conjecture

is similar to that in

T h e o r e m 2.

Let

polynomial

ring,

S

P

ring,

then

We p r o c e e d b y induction S

is artinian.

lemma,

in this case

N

on

87

of rank i .

If

If

dim S = 0 ,

radical.

By Nakayama's

P/NP).

is a direct p r o d u c t

is a direct product

be the

elements.

be its nilpotent

N = O, so that S R

d

R-module

d-1 = dim S .

~(R,P) = ~(R/NR,

Thus we may assume But

Let

and let R = S[X]

be a projective

dim R = d , then P can b e g e n e r a t e d b y

Proof.

The

[E-E,2].

be a n o e t h e r i a n

and let

3).

of p r i n c i p a l

of fields.

II

ideal domains,

so

Now suppose subset which S

P

is cyclic.

that dim S > 0 .

is the complement

Let

~ g S

of the union

be the multiplicative of the minimal primes

of

.

Then rated

SM

over

exists

has dimension S [X] = R

an element

by one element

u ¢ M

(3)

0 , so by induction, Pl¢

P~

can he gener-

P " It follows

that there

such that

uP g Rpl Since

u

dim S/(u)

is not in any of the minimal primes

< dim S •

R/(u)-module

P/uP

Thus,

by induction,

can be generated

by

of

S ,

the rank 1 projective d-I element

~2,...,~d

,

that is , d

(4) We claim that if modulo

(h) to

P2'''''Pd

are any elements

of

P

which reduce

~ 2 , . . . , ~ d , then d

p

V =

2

RPi

i=l It is enough to prove ]prime ideal

q

of

that this holds after R .

If

u ¢ q

localizing

at an arbitrary

then

d i=l follows

immediately

(5) follows

from

q

(3).

from Nakayama's

If, on the other hand,

lemma and

$8

(4).

This

u c q , then

concludes

the proof.

12

REFERENCES Bass, H., [Bass, I] "Torsion Free and Projective Modules," Trans. A.M.S. 102 (1962), 319-327. [Bass, 2] "K-Theory and Stable Algebra " Publ. Math. IHES no. 22 (196~), 5-60. [Bass, 3] "Modules Whic h Support Nonsingular Forms," J. Alg. 13 (1969), 246-252. Bass, H., and Schanuel, S. [B-S] "The Homotop[ Theory of Projective Modules," Bull. A.M.S. 68 (1962), 425-428. Davis, E., and Geramita, A., [D-G] "Maximal Ideals in Polynomial Rings." these Proceedings. Eisenbud, D., and Evans, E. [E-E,1] "Generating Modules Efficiently: Theorems from Algebraic K-Theory." j. Alg., To appear. [E-E, 2] "Every Algebraic Set in n-space is the Intersection of n Hypersurfaces." To appear. Endo, S. [Endo] "Projective Modules Over Polynomial Rings." Soc. Japan 15 (1963), 339- 352.

J. Math

Geramita, A. [Ger], "Maximal Ideals in Polynomial Rings ," Queen's University Mathematical Preprint no. 1971-56. KapLansky, I., [Kap] Infinite Abelian Groups, Revised edition, The University of Michigan, Ann Arbor, (19~9). Kronecker, L., [Kro] "Grundz~ge eine arithmetischen Theorie der algebraischen Grossen," J. Reine. Angew. Math. 92(1882), 1-123. Murthy, P. [Mur,1] "Projective Modules Over a Class of Polynomial ;Rings," Math. Z. 88 (1965), 184-189. [Mur, 2] "Generators for Certain Ideals in Regular Rings of Dimension three" To appear. Serre, J.-P. [Ser] "Sur les Modules ProJectifs," Sem. Dubriel-Pisot. t. 14 , 1960-1961, no. 2 . Seshadri, C.~[Ses] "Triviality of Vector Bundles over the Affine Space K~ '' Proc. Nat. Acad. Sci. 44(1958), 456-458. Vaserstein, L. [Vas] "Stable Rank of Rings and Dimensionality of Topological Spaces~ Functional Analysis and its Applications 5 , (1971), I02-110.

Brandeis University Waltham, Massachusetts M.I.T. Cambridge, Massachusetts

89

PRUFER-LIKE CONDITIONS ON THE SET OF OVERRINGS OF AN INTEGRAL DOMAIN

Robert Gilmer, Department of Mathematics Florida State University, Tallahassee, Florida, 32306

This paper considers ten conditions on the set of overrings of an integral domain D with identity. Each of these conditions is satisfied if D is a Prufer domain. Relations among the conditions are discussed, and several related questions are mentioned.

Let

D

be an integral domain with identity with quotient field

--Overring of

D,

we mean a subring of

of overrings of (0)

D

D,

K

containing

D.

K.

We denote by

By an ~

the set

and we consider the following eleven conditions.

is a Prufer domain; that is, finitely generated nonzero ideals of

are invertible. (i)

Each valuation overring of

D

valuation ring) is a quotient ring of

(that is, an overring of

D

D.

(2)

Each overring of

D

is a Prufer domain.

(3)

Each overring of

D

is integrally closed.

(4)

Each overring of

D

is flat as a D-module.

(5)

Each overring of

D

is an intersection of quotient rings of

(6)

If

J • Z,

If

J • Z

and if

prime ideal of

J

lying over

(8)

J • Z

and if

of

that is a

then the prime ideals of

J

D.

are extensions of prime ideals

D. (7)

ideals of (9)

If J

lying over

The set

Z

P

is a prime ideal of

dim J _< dim D

D,

then there is at most one

D,

then a chain of prime

P. P

is a prime ideal of

has at most one member.

is closed under addition--that

Jl + J2 = {Xl + x2 I x i • Ji } (i0)

P

is in

for each

Z. J

in

~.

90

is, if

Jl' J2 • Z'

then

In the sequel we discuss relations among the preceding conditions. lar, conditions D

(0) - (4)

are equivalent and they imply conditions

is integrally closed, then conditions

Conditions

(i) - (3)

(0) - (9)

and Some Related Questions

(0)

is a Prufer domain I if and only if each valuation overring of

ring of

D.

J, V • E

It then follows easily that

with

quotient ring of

J ~ V,

(5) - (i0); if

are equivalent.

W. Krull in [42, p. 554] established the equivalence of D

In particu-

and if

V

(0)

and

(2)

is a quotient ring of

and D

(1)--that is,

is a quotient

are equivalent, for if D,

then

V

is also a

J.

Since a Prefer domain is integrally closed, the implication from the equivalence of

(0)

and

(2).

(0) ÷ (3)

follows

E. D. Davis proved the reverse implication

in [12, p. 198]; he also extended his results to commutative rings

R

with identity

with few zero divisors, the definition of such a ring being that the set of zero divisors of

R

is a finite union of prime ideals of

Several questions related to the equivalence of sidered in the literature.

R. (0)

and

(3)

have been con-

A dual question that is easy to answer is:

conditions is each subring of

D

with identity integrally closed?

Under what

The answer to

this question is contained in [20, Theorem i].

THEOREM A.

field

The following conditions are equivalent.

(i)

Each subring of

(2)

D

Q,

D

with identity is integrally closed.

is an algebraic extension field of a finite field or

D

has quotient

the field of rational numbers.

iKrull uses the term "Multiplikation ring" in [42] instead of "Prufer domain". As Krull noted, his use of this term was in conflict with its meaning in older literature. Current usage of the term multiplication ring has reverted to that of the older literature--a commutative ring R in which the containment relation A ! B for ideals implies the existence of an ideal C such that A = BC (that is, "every divisor is a factor"); see [27] for more on multiplication rings. The first use of the term "Prefer domain" (or rather, "Prefer ring") that I have found in the literature appears in H. Cartan and S. Eilenberg [ii, p. 133]; I. Kaplansky repeats the terminology in [40]. The term "Prefer ring", as it is currently used, includes commutative rings with zero divisors; see [32] and [46, Chapter i0]. 9~

It follows from Theorem A that if each subring integrally closed, then each such J/A

J

J

of

D

with identity is

is a Euclidean domain with the property that

is finite for each nonzero ideal

A

of

J.

In [19, Theorems 3, 4], R. Gilmer gives necessary and sufficient conditions, reminiscent of those in

(2)

of Theorem A, in order that each subring of

D

should

be Noetherian; W. Borho [8] has investigated this question for rings (not necessarily commutative) with zero divisors, and the "Noetherian pairs" of A. Wadsworth [54] are another variant of the same question 2. In [27], Gilmer and J. Mott considered several questions of the following type: Determine necessary and sufficient conditions in order that each integrally closed subring of

D

should have property

P.

To give an indication of the results of [27],

we cite a portion of its Theorem 3.1.

THEOREM B.

The following conditions are equivalent.

(i)

Each integrally closed subring of

(2)

Either

characteristic (3)

K p ~ 0

has characteristic and

0

D and

is completely integrally closed. K/Q

is algebraic, or

K

has

tr.d. K/GF(p) S i.

Each integrally closed subring of

K

is a PrSfer domain.

In analogy with Noetherian pairs, we could, of course, define an integrally closed pair to be a pair

(R, S)

that

S

R

is a subring of

of commutative rings with a common identity such

and each subring of

S

containing

R

is integrally

closed 3 .

Flat Overrings

In [50], F. Richman proved that ring of

D

is flat as a D-module.

the structure of

D

D

is a PrHfer domain if and only if each over-

This particular equivalence is useful in relating

to that of an overring

J

of

D,

primarily because of Theorem C,

2For more on questions of the form "characterize rings R such that each subring of R has property P", see [19, Theorem 5], [25], and [30]. 3Several persons, including E. D. Davis, A. Grams, I. Kaplansky, T. Parker, and A. Wadsworth, have shared with me some observations about such pairs after the conclusion of my talk; see, for example, the next paper (by Davis) in these Proceedings. 92

which we cite later. and

(4)

At this point, we remark that the equivalence of

(0), (3),

generalizes to the case of rings with zero divisors; see [32] and [46,

Theorem 10.18]. There are several papers in the literature that touch on the subject of flat overrings.

The papers [i, 2, 3] of T. Akiba dwell on the topic, while [50], [45],

[35], [32], and [5] also contain some of the general theory of flat overrings.

As a

nice summary statement, we cite a result that appeared in a preprint of [5].

THEOREM C. T,

and let

Let

R'

R

be a commutative ring with identity with total quotient ring

be a subring of

T

containing

R

such that

R'

is flat as an

R-module. (a) and

There is a generalized multiplicative

AR' = R'

PR'

for each

Let

A

and

(b)

As : A R ' .

(c)

AR' = R'

(d)

Let

Q

is prime in

R,

and let

be a P-primary ideal of R',

QR'

(f)

If

S

in

R

such that

R' : RS

S. A'

if and only if there exists

A' = (A' n R)R'.

m

in

B be ideals of

(e)

is

R

PR'-primary,

A' = (a~, ..., a') m

bl, ..., b n • R A'

A

system 4

B ~ S

R'.

such that

such that

A ~_ B.

QR' c R'.

P = PR' n R,

and

Then

PR' c R',

Q = QR' n R.

is finitely generated, then there exist i -< i -< m,

such that for

be an ideal of

i -< j < n,

a'b. • R i ]

and

n

:

i=l j=l

a'b.R'. m ]

4By a generalized multiplicative system in a commutative ring T, we mean a nonempty family S = {I } of nonempty subsets of T (in forming generalized quotient rings, there is no loss of generality in assuming that each such that

I I 8 = {El xiYi I x i • I , Yi • 18} • S

alized quotient ring

TS

of

in the total quotient ring of

T

with respect t o T

such that

xl

for all S, c T

I ~

is an ideal of and

B.

T)

By the gener-

we mean the set of elements for some

I

in

S.

x

The no-

tion of a generalized quotient ring, which obviously generalizes the concept of a quotient ring with respect to a regular multiplicative system, seems to have originated with W. Krull [44, Section 2]. More recently, D. Kirby [41], L. Budach [9], M. Griffin [32], W. Heinzer, J. Ohm, and R. Pendleton [35], J. Arnold and J. Brewer [5], and H. S. Butts and C. Spaht [i0] have used the concept. It is interesting to note that the set of generalized quotient rings of D i n c l u d e s the set of overrings J of D that can be written as an intersection of looalizations of D [35, Prop. 4.3 and Cot. 4.4]. 93

(g)

(A n B)R'

(h)

](AR') = (/A)R'.

(i)

If

B

: AR'

n BR'.

[AR':BR'] = [A:B]R'.

is finitely generated, then

Of several homological characterizations terms of overrings, here.

H. Storrer [53] has proved that

and only if the injection map of commutative rings for each

J

D

in

into ~.

domain if and only if the D-module

Conditions

(5) - (8)

of Prefer domains, we mention one, in

J

Storrer also proved that

D[s] @D D[s]

implies

D.

(7)

are satisfied if

Moreover, part

and that D

D

is a Prefer

is torsion-free for each

s

in

K 5.

and Some Related Results

satisfied for each flat overring (6)

is a Prefer domain if

is an epimorphism in the category of

As we have previously observed, a flat overring of localizations of

D

J

(7)

(e) of

D

is an intersection of

of Theorem C shows that condition D.

implies

(5)

is

It is clear that (8).

Consequently,

conditions

(5) - (8)

is a Prefer domain.

E. D. Davis asked in [12, p. 200] if a domain intersection of localizations of alized quotient ring of

D)

D

( and

hence

D

for which each overring is an

each overring of

is, in fact, a Prefer domain.

to this question is affirmative if each prime ideal of

D

D

is a gener-

He proved that the answer has finite height.

In

[21], Gilmer and Heinzer conducted a thorough investigation into the theory of a domain

D

for which each overring is an intersection of localizations;

such a domain a QQR-domain or a domain with the QQR-property.

they called

Gilmer and Heinzer

proved that the integral closure of a QQR-domain is a Prefer domain; moreover, valuation overring of a VQR-domain) D

D

is an intersection of localizations

of

D

(that is,

and if the ascending chain condition for prime ideals holds in

is a Prefer domain.

Finally,

if each D D,

is then

[21] contains examples that show that a VQR-domain

need not have the QQR-property, and that a QQR-domain need not be a Pr[fer domain.

5A. Hattori [33] proved that D is a Prefer domain if and only if for all torsion-free D-modules M and N, M 8 D N is torsion-free; some of Hattori's results were generalized by S. Endo in [13]. 94

It seems appropriate

at this point to mention

and Heinzer

in [21] to obtain their examples.

imal ideal

M,

D

if

¢

is a subring of

structure

of

DI

ring extension such domains to provide field of

D,

then

the domains

one frequently

is realized

this reason,

D I : ~-I(D)

assumes

as a subring

of

consider

over

ring.

V;

such that Krull's

the closely related ring

({Xl}),

S

have property

example

is in a paper

and

B

amounts

arithmetisch

The

A

as a

construc-

f(X, Y)/g(X,

to taking

Y) e k.

J = k + M, k[X, Y](X)

where

R

Let Y)

k

J be

in two

In terms of a where

M

= k(Y) + M.

is a subring S[{X~}]

[A

Krull

closed domain

is the following:

f(0, Y)/g(0,

closed),

D + M

D + M

integrally

functions

ring

and for

[43, p. 670] by Krull.

this is an example

is integrally

is not endlich

use of the

of a domain with property

Krull uses the same domain as an example

M.

is, the residue

can be found in the classical

D + (X, Y)K[X,

D

and if

can be realized by

D I = D + M,

is an ideal of the polynomial ring

integrally

v-operation

J

R + B,

(in modern terminology,

where

of

valuation

domain that is not completely Y],

of

2 of [16] as the

quasi-local

and

construction

To obtain an example BA

containing

V = £ + M--that

the earliest

g(0, Y) ~ 0

then such a construction

of H. Prefer 6.

A = V/M,

and the structure

in Appendix

be the set of all rational

k

V

in this case,

His description

ideal of the rank one discrete

commutative in

J

construction,

maximal

that

of a one-dimensional

that is not a valuation a field and let

V

onto

ring with max-

have usually been used in the literature

In the form just described,

gives there an example

D + M

of

DI

tion that I have found in the literature

variables

is a subring

I refer to such a construction

construction.

is a valuation V

of

used by Gilmer

and hence a rather wide range of properties

Because

examples,

V

of

reflects both the structure

D I.

V

If

is the canonical homomorphism

A,

of

the construction

is the If we

of the contained

paper [49, p. 19] that does not

of an integrally

closed

Prufer gave the example

closed and

D ~ K.

of an integrally

In [42, p. 569],

closed domain on which the

brauchbar.

6prufer's paper [49] represents the first study of Prufer domains, per se, in the literature. With proper modesty, Pr~fer's term for an integral domain with identity in which nonzero finitely generated ideals are invertible was a domain with property I_BB. 95

Another important use of the

D + M

construction is in the paper [52, p. 604]

of A. Seidenberg; Seidenberg used such domains to prove that if tive integers such that domain

J

such that

n + i ~ k ~ 2n + i,

dim J = n

and

n

and

k

are posi-

then there is an integrally closed

dim J[X] = k.

In recent years, the

D + M

construction and related construction for polynomial rings have appeared frequently in the literature 7, and some recent papers have contained theorems concerning the structure of such constructions; see [29], [22], [16, Appendix 2], [7]. Gilmer has generalized some of these theorems to finite intersections We return to our discussion of relations between conditions

(0), (5) - (8).

One of the examples of Gilmer and Heinzer in [21] shows that conditions and

(9)

do not imply

(0).

On the other hand, conditions

(7)

In [18], n hi= I (D.I + M.).

and

(5), (6), (8)

are equiv-

alent and are, in fact, equivalent to the condition that the integral closure of is Prefer [16, Th. 16.10 and Th. 22.2].

Consequently, conditions

(0) - (8)

D

are

equivalent for an integrally closed domain. If each overring of equivalence of

(0)

and

D (2)

is a quotient ring of that

D

D,

then it follows from the

is a Prefer domain.

Such domains were con-

sidered independently by Davis [12] and by Gilmer and Ohm [28]. say that

D

has the QR-property if each overring of

A Prefer domain need not have the QR-property.

D

Following [28], we

is a quotient ring of

D.

In fact, the following theorem was

proved independently by Davis, Gilmer and Ohm, and by O. Goldman [31].

THEOREM D.

If

D

is Noetherian, then

D

has the QR-property if and only if

is a Dedekind domain with torsion class group.

In [28], Gilmer and Ohm prove that a Prefer domain power of each finitely generated ideal of strengthened this result to:

D.

A

of

has the QR-property if a

is principal; in [48], Pendeleton

The Prufer domain

if for each finitely generated ideal ideal of

D

D

D,

D /A

has the QR-property if and only is the radical of a principal

Must a domain with the QR-property have torsion class group?

This

7in fact, I have heard the statement made in jest that if an example exists, it can be realized in the form D + M. Since each domain with identity is trivially of the form, the statement is literally true, but in making this observation, I avoid the spirit of the speaker. 96

D

question remained open for several years; it was answered in the negative by Heinzer in [34].

The example that Heinzer gave (not a

D + M

construction)

is related to a

construction of Goldman in [31]. The search for a characterization quotient rings is not lost, however.

of Prufer domains in terms of overrings being Both Arnold and Brewer [5] and Butts and Spaht

[i0] have obtained such a characterization. that

D

Specifically, Theorem 1.5 of [5] states

is a Prufer domain if and only if each overring of

for some generalized multiplicative

system

S = {I }

Arithmetic Relations on the Set of Overrings of

In the terminology of [24], Jl + J2 = {al + a2 I ai ~ Ji }

D

D

is of the form

of invertible ideals of

D.

D

is a A-domain if for all

is a subring of

DS

K.

Jl" J2 e Z,

A Prefer domain is a A-domain;

this follows since a valuation ring is obviously a A-domain and since a domain

D

a A-domain if and only if

More

generally,

DM

is a A-domain for each maximal ideal

M

of

D.

is

a domain with the QQR-property is a A-domain [24, Theorem 5], and hence a

A-domain need not be a Prefer domain.

It is true, however, that the integral closure

of a A-domaln is a PrHfer domain. Condition

(9)

is distinguished

it is primarily arithmetic in nature. and only if

xy e D[x] + D[y]

for all

from the other conditions on our list because In fact, it is true that x, y e K.

D

is a A-domain if

Several characterizations

of

Prufer domains, in terms of arithmetic relations on the set of ideals, are known. As a fair summary of these results, we state Theorem E (see [38] and [16, Section 21]).

THEOREM E.

The following conditions are equivalent.

(a)

D

is a Prefer domain.

(b)

A n (B + C) = (A n B) + (A n C)

(c)

(A + B)(A n B) : AB

(d)

A(B n C) = AB n AC

(e)

A:B + B:A = D

(f)

(A + B):C : A:C + B:C

for all ideals

A, B, C

of

D

with

C

finitely

C:(A n B) : C:A + C:B

for all ideals

A, B, C

of

D

with

A

and

for all ideals

for all ideals for all ideals

for all

A, B

A, B A, B, C

of

A, B, C

of" D.

D. of

D.

finitely generated ideals of

D.

generated. (g)

97

B

finitely generated. (h)

AB = AC,

zero, implies that

for ideals

A, B, C

of

D

with

A

finitely generated and non-

B = C.

Several of the conditions of Theorem E have been considered for commutative rings

R

with identity [46, pp. 150,151].

This is especially true of condition

--the condition that the lattice of ideals of tributive.

R,

under

+

and

n,

(b)

should be dis-

L. Fuchs in [14] refers to such rings as arithmetischer Ringe, and C. U.

Jensen has investigated these rings (arithmetical rings, in Jensen's terminology) in a series of papers (see, in particular, [39]). Besides [24], a few other papers that have touched on the question of characterizing Prefer domains of

D

D

in terms of arithmetic relations on the set of overrings

are [47], [17], and [23].

Dimension Theory of Overrings

It follows from part overring of

D.

(e)

of Theorem C that

Thus condition

generally, condition

(i0)

(i0)

dim J N dim D

is satisfied if

follows from condition

D

(8).

D',

J

is a flat

is a Prufer domain; more Since dimension is pre(i0)

served by integral extensions, it is clear that condition if and only if it is satisfied for

if

the integral closure of

is satisfied for D.

But unlike the

other conditions we have considered, an integrally closed domain satisfying need not be a Prefer domain.

If

D

(i0)

For finite dimensional domains, the following theorem

gives a characterization of domains satisfying

THEOREM F.

D

has finite dimension

(i0).

n,

then the following conditions are

equivalent. (a)

dim J -< n

for each

J

in

E.

(b)

The valuative dimension of

D,

(c)

dim D[XI, ...

dim

V

D,

is

n°

X ] = 2n. n

A proof of Theorem F is given in [16, pp. 346-9]; Arnold proves a generalization of Theorem F in Theorem 6 of [4].

The notion of valuative dimension was introduced

by P. Jaffard in [36] (see also [37, Chapitre IV]);

98

dim

V

D

is defined to be

i0

sup {dim V I V

is a valuation

always holds,

and it follows

domain

D

or if

integrally

domain if and only if

where D

(i0)

dim D I

is a positive

than or equal to dim D k = k struction

and

k,

F that

condition

closed Noetherian is at most

i.

dim

domain There

closure D = i

v

proved by Seidenberg integer

and if

dim D = dim

m

dim v D k = m.

[16, p. 572].

In general,

dim v D = r < ~,

then for

but

if

D

D

is a Prefer

is satisfied DI

for each

is a Prefer case in which

is a Prufer domain--the that the integral

case

closure of

in [51, p. 511]; see also [15]. ~

or is a positive

Dk

closed domain

integer greater Dk

can be realized by a

such that D + M

con-

{dim D, dim D[XI] ,

for example,

s a r,

DI,

of

the sequence

can be wild indeed;

D

(i0)

implies

is

Such a domain

v

dim D ~ dim v D

is one nontrivial

then there exists an integrally

dim D[XI, X2] , ...} But if

The inequality

In particular,

This result--that

is a Prufer domain--was k

D}.

implies that the integral

dim D = i.

If

of

from Theorem

is Noetherian.

finite-dimensional

condition

overring

dim D[XI,

see [7, Section

5] and [6].

..., X s] = r + s

[36, Th~oreme

Resume

In summary, conditions ditions

conditions

(5) - (i0).

(5) - (9)

Finally,

integrally D

n

The integral

are equivalent, closure

and these conditions

of a domain satisfying

is a Pr{ifer domain, but the domain of Example

that the combination closed.

(0) - (4)

of conditions

for each positive

closed domain

Dn

(5) - (i0) integer

such that condition

one of the con-

4.1 of [21] shows

does not imply that

n > i,

imply

D

is integrally

there is an n-dimensional (i0)

is satisfied

for

Dn,

but

is not a Pr[fer domain.

.REFERENCES

[i]

Akiba,

T. Remarks

on generalized

quotient rings,

Prec.

Japan Acad.

40, 801-806

(1964). [2]

Remarks

on generalized

rings

of quotients

II, J. Math. Kyoto Univ.

5,

39-44 (1965). [3]

Remarks 205-212

on generalized

rings of quotients

(1969). 99

III, J. Math.

Kyoto Univ.

9,

3].

ii

[4]

Arnold, J. T. On the dimension theory of overrings ef an integral domain, Trans. Amer. Math. Soc. 138, 313-326 (1969).

[5]

Arnold, J. T., and Brewer, J. W. On flat overrings, ideal transforms, and generalized transforms of a commutative ring, J. Algebra 18, 254-263 (1971).

[6]

Arnold, J. T., and Gilmer, R. The dimension sequence of a commutative ring, in preparation.

[7]

Bastida, E., and Gilmer, R. Overrings and divisorial ideals of rings of the form D + M,

preprint.

[8]

Borho, W. Die torslonsfreien Ringe mit lauter Noethersehen Unterringen, preprint.

[9]

Budach, L. Quotientenfunktoren und Erweiterungstheorie, Math. Forschungsberichte

22, VEB Deutscher Verlag der Wiss. Berlin, 1967. [i0]

Butts, H. S., and Spaht, C. G. Generalized quotient rings, to appear in Math. Nach.

[ii]

Cartan, H., and Eilenberg, S. Homological Algebra, Princeton Univ. Press, Princeton, N. J. (1956).

[12]

Davis, E. D. Overrings of commutative rings. II. Integrally closed overrings, Trans. Amer. Math. Soe. ii0, 196-212 (1964).

[13]

Endo, S. On semi-hereditary rings, J. Math. Soc. Japan 13, 109-119 (1961).

[14]

Fuchs, L. 0ber die Ideale arithmetischer Ringe, Comment. Math. Helv. 23, 334-341

(1949). [15]

Gilmer, R. Domains in which valuation ideals are prime powers, Arch. Math. 17, 210-215 (1966).

[16]

Multiplicative

Ideal Theory, Queen's University, Kingston, Ontario

(1968). [17]

On a condition of J. Ohm for integral domains, Canad. J. Math. 20, 970-983 (1968).

[18]

Two constructions of Prufer domains, J. Reine Angew. Math. 239/240, 153-162 (1969).

[19]

Integral domains with Noetherian subrings, Comment. Math. Helv. 45, 129-134 (1970).

[20]

Domains with integrally closed subrings, Math. Jap. 16, 9-11 (1971).

100

12

[21]

Gilmer, R., and Heinzer, W. Intersections of quotient rings of an integral domain, J. Math. Kyoto Univ. 7, 133-150 (1967).

[22]

On the number of generators of an invertible ideal, J. Algebra 14, 139-151 (1970).

[23]

Gilmer, R., and Huckaba, J. A. The transform formula for ideals, J. Algebra 21, 191-215 (1972).

[24] [25]

A-rings, preprint. Gilmer, R., Lea, R., and O'Malley, M. Rings whose proper subrings have property P,

[26]

to appear in Acta Sci. Math. (Szeged).

Gilmer, R., and Mott, J. L. Multiplication rings as rings in which ideals with prime radical are primary, Trans. Amer. Math. Soc. 114, 40-52 (1965).

[27]

Integrally closed subrings of an integral domain, Trans. Amer. Math. Soc. 154, 239-250 (1971).

[28]

Gilmer, R., and Ohm, J. Integral domains with quotient overrings, Math. Ann. 53, 97-103 (1964).

[29]

Primary ideals and valuation ideals, Trans. Amer. Math. Soc. 117, 237-250 (1965).

[30]

Gilmer, R., and O'Malley, M. Non-Noetherian rings for which each proper subring is Noetherian, to appear in Math. Scand.

[31]

Goldman, O. On a special class of Dedekind domains, Topology 3, 113-118 (1964).

[32]

Griffin, M. Prefer rings with zero divisors, J. Reine Angew. Math. 239/240,

55-67 (1969). [33]

Hattori, A. On Prufer rings, J. Math. Soc. Japan 9, 381-385 (1957).

[34]

Heinzer, W. Quotient overrings of an integral domain, Mathematika 17, 139-148 (1970).

[35]

Heinzer, W., Ohm, J., and Pendleton, R. On integral domains of the form P

[36]

minimal, J. Reine Angew. Math. 241, 147-159 (1970).

Jaffard, P. Dimension des anneaux de polynomes.

La notion de dimension

valuative, C. R. Acad. Sci. Paris Ser. A-B 246, 3305-3307 (1958). [37]

Theorie de la Dimension dans les Anneaux de Polynomes, GauthierVillars, Paris, 1960.

101

n Dp,

13

[38] [39] [40]

Jensen, C. U. On characterizations of Prefer rings, Math. Scand. 13, 90-98 (1963). Arithmetical rings, Acta Math. Acad. Sci. Hungar. 17, 115-123 (1966). Kaplansky, I. A characterization of Prufer rings, J. Indian Math. Soc. (N.S.) 24, 279-281 (1960).

[41]

Kirby, D. Components of ideals in a commutative ring, Ann. Mat. Pura Appl. (4) 71, 109-125 (1966).

[42]

Krull, W. Beitrage zur Arithmetik kommutativer Integrit~tsbereiche, Math. Z. 41, 545-577 (1936).

[43]

Beitr~ge zur Arithmetik kommutariver Integritatsbereiche. II. v-ldeale und vollstandig ganz abegeschlossene Integrit~tsbereiche, Math. Z. 41, 665-679 (1936).

[44]

Beitr~ge zur Arithmetik kommutativer Integrit~tsbereiche. VIII. Multiplikativ abgeschlossene Systeme yon endlichen Idealen, Math. Z. 48, 533-552 (1943).

[45]

Larsen, M. D. Equivalent conditions for a ring to be a P-ring and a note on flat overrings, Duke Math. J. 34, 273-280 (1967).

[46]

Larsen, M. D., and McCarthy, P. J. Multiplicative Theory of Ideals, Aaademic Press, New York, 1971.

[47]

Ohm, J. Integral closure and

(x, y)n = (xn, yn), Monatsh. Math. 71, 32-39 (1967).

[48]

Pendleton, R. L. A characterization of Q-domains, Bull. Amer. Math. Soc. 72, 499-500 (1966).

[49]

Prefer, H. Untersuchungen ~ber die Teilbarkeitseigenschaften in Korpern,

J.

Reine Angew. Math. 168, 1-36 (1932). [50]

Richman, F. Generalized quotient rings, Proc. Amer. Math. Soc. 16, 794-799 (1965).

[51]

Seidenberg, A. A note on the dimension theory of rings, Pacific J. Math. 3, 505-512 (1953).

[52] [53]

On the dimension theory of rings II. Pacific J. Math. 4, 603-614 (1954). Storrer, H. H. A characterization of Prefer rings, Canad. Math. Bull. 12, 809812 ( 1 9 6 9 ) .

[54]

Wadsworth, A. Noetherian pairs and form.,

Thesis,

Chicago,

the 1972.

102

function

field

of a q u a d r a t i c

INTEGRALLY

CLOSED PAIRS

E d w a r d D. Davis D e p a r t m e n t of M a t h e m a t i c s S t a t e U n i v e r s i t y of New Y o r k at A l b a n y A l b a n y , N. Y. 12222 U.S.A.

In the d i s c u s s i o n f o l l o w i n g G i l m e r ' s a d d r e s s K a p l a n s k y i n t r o d u c e d the n o t i o n " i n t e g r a l l y c l o s e d pair", r a i s e d the q u e s t i o n of c h a r a c t e r i z i n g t h e s e p a i r s and c o n j e c t u r e d an a p p r o x i m a t e answer. Mott wisely conjectured t h a t the m e t h o d s of Davis' t h e s i s [i] m i g h t w e l l p r o v i d e an answer. This n o t e e s t a b l i s h e s the v a l i d i t y of b o t h c o n j e c t u r e s , at l e a s t in the N o e t h e r i a n case, and s o m e w h a t m o r e g e n e r a l l y , for K r u l l d o m a i n s .

INTEGRALLY

C L O S E D P A I R S OF K R U L L D O M A I N S .

cellent discussion

of " K r u l l d o m a i n s "

The r e a d e r w i l l

in G i l m e r ' s

r e g a r d the n o t i o n

as so m u c h e s o t e r i c a ,

closed Noetherian

domains"

book

(A,B)

closed

if A is a s u b r i n g of B and all i n t e r m e d i a t e

and B)

are i n t e g r a l l y

closed.

ideals

of A, Ap m e a n s

the l o c a l i z a t i o n

the i n t e r s e c t i o n parenthetical theorem

c l o s e d if, B algebraic height

observation

is this w r i t e r ' s

THEOREM

i.

A b i t of n o t a t i o n :

of the l o c a l i z a t i o n s concluding

rings

of A w i t h

respect

(2) is unique, (Consequently:

consisting

the p a i r

is i n t e g r a l l y

(A,B)

of A w i t h such.

closed,

of m a x i m a l

ideals w h i c h

t h e n B is a l o c a l i z a t i o n

of i n v e r t i b l e m a x i m a l

is c o m p l e m e n t a r y

ideals.)

10.3

(i) A a f i e l d w i t h respect

to a set of

The e x c l u d e d set in

t h o s e m a x i m a l M for w h i c h MB = B.

For A N o e t h e r i a n b u t not a field,

integrally

The

conjecture.

only maximal

of e x a c t l y

of P.

of K a p l a n s k y ' s

(2) B a l o c a l i z a t i o n

that excludes

to P -- i.e.,

of the f o l l o w i n g

and o n l y if, one of the f o l l o w i n g holds:

1 primes

(including A

for P a set of p r i m e

the s t a t e m e n t

understanding

"integrally

is i n t e g r a l l y

of A at the p r i m e s

For A a K r u l l domain,

o v e r A;

[2]; s h o u l d he

he may s u b s t i t u t e

The p a i r of d o m a i n s

f i n d an ex-

if the p a i r

of A w i t h

to a u n i q u e l y

respect

(A,B)

is

to a set

determined

set

-

proof

The

progresses

via

blanket

assumption:

algebra

over

closed,

B is n e c e s s a r i l y

case

(i) ; h e r e a f t e r

system

a sequence

(A,B)

a domain

-

of

lemmas

is i n t e g r a l l y

abounds

we

2

closed.

in s u b a l g e b r a s

algebraic

assume

over

A not

for w h i c h Since

which

A.

This

a field.

ring between

A S and B S is of the

ring between

A and

B;

therefore

the p a i r

this

freely

We

LEMMA

shall

i.

We

A,

element

o v e r A; we

lose n o t h i n g

A[x]

is for

a ring

we

following

nonunit height

whence

A[x]

If A is

assume points:

of A lies 1 primes;

if,

and o n l y

The

existence

from these

if, and

points

a consequence

the

local

closed,

local

N for w h i c h

proof

every

A = Ap,

and

N contains unicity

from

of

lemma

domain

for M; Ap

A[x]

M and

over

observe

Consequently

Lemma.

i/x is in A. and

and

[i]

in any

for

Thus:

case

104

further

B

B = AN

1 prime;

N subsets

M of

P such

P is of

details. comment

domain;

the

in

set

each

of A's

P, A M c o n t a i n s

ring that

only

x lies

explicit

of A is a K r u l l

M excludes

of:

that

use w i t h o u t

is a v a l u a t i o n

that

of A,

= A[zx].

then

case,

1 of

height

a subset 2;

B is a l g e b r a i c

the o b s e r v a t i o n

localization one

of A.

NB ~ B.

of t h e o r e m

A a Krull

comment.

y in A and x i n t e g r a l

Nakayama's

of A in the

integrally

explicit

z ~ 0 a nonunit

and x in B-A,

in at l e a s t

and

For

= A by

i/x is in A f o l l o w s

consult

Henceforth

in A.

of

f o r m C S for C a

field

Since

f o r m x/y w i t h

is i n t e g r a l l y

s e t of p r i m e s

That

A [ x 2] --

the

A[zx]

of q u o t i e n t s

N the

Proof.

2.

the

A local.

disposes

is also

and w i t h o u t

in the q u o t i e n t

taking

t h a t x lies

= A+zA[x];

LEMMA

by

of B is of

show

that because

fact

B is c o n t a i n e d

Proof. each

use

integrally

S a multiplicative

every

(As,B S)

a

the p o l y n o m i a l

remark

For

now make

are n o t

in A,

closed.

we

for e a c h

AN

P in

P.

B = A M follow

maximal

ideals

is

-

3.

LEMMA

Proof. of

-

If A is l o c a l and B ~ A,

S i n c e B ~ A, we

P excluding

of A i m p l i e s

3

t h e n A is of d i m e n s i o n

i.

lose n o t h i n g by a s s u m i n g B = A M for M a s u b s e t

a s i n g l e p r i m e P.

Then

that B is a r i n g of q u o t i e n t s

that B = A x for x in P but in no m e m b e r of M.

Were

there

a n o n u n i t y n o t in P, t h e n y / x w o u l d be in B-A, b u t x/y n o t in A -- a contradiction

LEMMA pair

4.

(A,AN)

Proof.

of l e m m a 2.

For

Finally

the s u f f i c i e n c y

N the set of n o n m a x i m a l

is i n t e g r a l l y

ideals

Careful examination

the p r o o f of l e m m a

s e l e c t a here,

elsewhere.

Call

pair

Except (A,B)

grally

is m e r e l y

closed.

c l o s e d if, valuation and o n l y

2.

if,

ring,

For A local,

and o n l y ring;

general

setting;

closed pairs

A a valuation

THEOREM

relatively

integrally

if, B = Ap

if

rings

-- m o s t

results;

we

for p u b l i c a t i o n

integrally integrally

closed closed

an i n t e g r a l l y

closed

c l o s e d p a i r w i t h B inte-

The c e n t r a l p o i n t of the g e n e r a l i z a t i o n

integrally

examples:

more general

case of A a field,

a relatively

2 is v a l i d in the m o r e atively

(A,B)

all i n t e r m e d i a t e

for the t r i v i a l

in M~A;

of the a b o v e a r g u m e n t s

2 -- r e v e a l s

a p a i r of d o m a i n s

If M ~ A is

In e i t h e r case C M = A M ~ A.

l e a v i n g the d e t a i l e d e x p o s i t i o n

if A is a s u b r i n g of B w i t h in B.

to P, the

r i n g C.

of N c o n t a i n e d

M ~ A is in P, then AM~ A is a v a l u a t i o n ring.

present

belonging

i d e a l of an i n t e r m e d i a t e

n o t in P, then A M ~ A = A M for M the m e m b e r s

importantly

(2):

closed.

Let M be a m a x i m a l

GENERALIZATIONS.

of

whence

in the

is t h a t l e m m a

local case,

rel-

are c o m p o u n d e d o u t of the two o b v i o u s

a n d the t r i v i a l e x a m p l e

the p a i r

(A,B)

of A = B.

is r e l a t i v e l y

for P a p r i m e w i t h P = PAp

for A not a field,

the p a i r is i n t e g r a l l y

in a d d i t i o n A is i n t e g r a l l y

105

closed.

integrally and A/P

a

c l o s e d if,

-

We

now p l a y a b i t l o o s e w i t h

fast-Noethersche composition of h e i g h t primes

and a l s o

associated

theorem

3.

in the s e n s e

For

closed

s i o n i).

i.

Since

(A,B)

if, and only if,

decomposition

domain

de-

to p r i n c i -

is e i n b e t t u n g s f r e i ,

A,

the p a i r

(A,B)

of p r i n c i p a l

ideals

is u n i q u e

if,

and o n l y

to a set of m a x i m a l

determined

ring

for A e i n b e t t u n g s f r e i .

closed

re-

that excludes (of d i m e n -

ideals M for w h i c h MB = B can serve

integrally

respect

is r e l a t i v e l y

B is a l o c a l i z a t i o n of A w i t h

as

(Therefore:

if, B is a l o c a l i -

ideals

ideals M for w h i c h A M is a v a l u a t i o n

set is u n i q u e l y

a primary

if its only

ideals M for w h i c h A M is a v a l u a t i o n

is r e l a t i v e l y

set of m a x i m a l

admits

a domain

and at m o s t a f i n i t e n u m b e r

a Krull

fast-Noethersche

set, w h i c h

z a t i o n of A w i t h

Call

case Of:

The set of m a x i m a l

the e x c l u d e d

[3]:

nonunits

of the p r i m a r y

s p e c t to a set of a s s o c i a t e d p r i m e s only such maximal

of K r u l l

call s u c h a d o m a i n e i n b e t t u n g s f r e i

1 is a s p e c i a l

integrally

two n o t i o n s

lies in at l e a s t one

are of h e i g h t

THEOREM

-

if e a c h of its n o n z e r o

1 primes;

pal i d e a l s

4

complementary ring;

this

to a

latter

for A e i n b e t t u n g s f r e i . )

REFERENCES

[1] Davis,

E. D.

Overrings

closed overrings,

[2] Gilmer,

R. W.

of c o m m u t a t i v e Trans.

Multiplicative

[3] Krull,

W.

Ontario

Math.

-- No.

II.

(1964)

Integrally 196-212.

Queen's

12, Q u e e n ' s

Papers

on P u r e

University,

(1968).

Einbettungsfreie,

Oberringe,

ii0

I d e a l Theory,

and A p p l i e d M a t h e m a t i c s Kingston,

A.M.S.

rings.

fast-Noethersche

Nachr.

21

106

(1960)

Ringe

319-338.

und ihre

NOETHERIAN

INTERSECTIONS

OF INTEGRAL

DOMAINS

II

William Heinzer* Purdue

If

R

and

would

like

that

D = R~V

that

R

are integral

to c o n s i d e r

or

V

A sample

If

when

R

R

ring and

if

K?

(ii) when

that

V

that

are

R

V

of

as in (i),

quasi-local,

and was

extended

to the p r e s e n t

and myself.

Access

Section in S e c t i o n

of w h e t h e r

i.

Some

Throughout quotient called *This

our results

remarks

this s e c t i o n K.

R

The

R

radical

research

supported

was

of

and

is a rank

the case w h e n form

V

one

then

of

if

denote

out

(i)

and

to us in our

Ohm.

(ii);

while

(but do not resolve)

noetherian

domain,

R} again be n o e t h e r i a n . of integral

integral

domains ideals

R.

107

by Jack Ohm

jointly with

of the m a x i m a l

by NSF Grant

is

to Jack Ohm in that many

on i n t e r s e c t i o n s V

R

jointly

was u s e f u l

and also discuss

intersection

the J a c o b s o n

R

is n o e t h e r i a n

on q u e s t i o n s

one prime

and

if

D.

a 2-dimensional

is a h e i g h t

general

field

(iii),

for

conclude

is the following:

V

D

manuscript

given here were w o r k e d

1 contains

R* = N { R p I P

for

I am also i n d e b t e d

2 we deal with

the q u e s t i o n must

to P r e k o w i t z ' s

results

say

are n o e t h e r i a n .

Prekowitz

of the other

we

can one

is i r r e d u n d a n t ,

to Bruce

of the result.

K,

is also n o e t h e r i a n ?

and

D = R~V

1 is due

extension

can one

questions

radical

field when

(iii)

D = R~V

localizations

and

(i)

D = RNV?

Jacobson

such

quotient

questions:

of

the same h y p o t h e s i s

and only

with

can say on these

has n o n z e r o

both

Result

can one say

of w h a t we

With

field

is a l o c a l i z a t i o n

valuation

2)

domains

the f o l l o w i n g

has q u o t i e n t

are n o e t h e r i a n ,

i)

V

University

GP-29326AI

domains. with of

R

is

2 i.i Lemma. Jacobson Proof:

If

V

is a v a l u a t i o n

radical It will

field of

D.

a ~ V,

then

I/(l+a)

e

suffice

If

1.2 Remark.

ring

V

k

clear

{V }

as

{a e All + ra

that

any e l e m e n t

implies in

radical

radical

is not

radical

zero,

that

D = RNV

X

field

K.

in the q u o t i e n t

If

a e J

in

V,

of

T

and

so

and

that n o n z e r o

radical. ideals

in

V

a rank

has q u o t i e n t

of subrings

then

T =

~ V

one

field K.

R = k[X],

in

A

for all

is a unit

again has For R

D or

T.

r e A}, V

and

This

is

of a ring

A

and the fact is a unit

in

T.

R and V

b o t h have n o n z e r o

quotient

field

having V

of

ring

the p r o p e r t y

radical

in each

if

of some has

radical

of the J a c o b s o n

from 1.3 that

D = RNV

for

= k.

in the J a c o b s o n

which

even

an i n d e t e r m i n a t e ,

J ,

is a unit

Jacobson

quotient

contract

K, then

field

D

K

to n o n z e r o

ideals

D. This yields

the f o l l o w i n g

1.5 P r o p o s i t i o n . K

which

is a c o l l e c t i o n

It follows

has n o n z e r o

has n o n z e r o

is c o n t a i n e d

in

from the c h a r a c t e r i z a t i o n

Jacobson

J

R

R

quotient

a e D.

D = RAV

is c o n t a i n e d

1.4 Remark.

and

then

is a field, then

If

~ J

K

a e V,

has J a c o b s o n

has J a c o b s o n

that

has

that

it need not follow

V = k[I/X](I/X),

if

is a unit

R

if

1.3 Remark.

and

on

= D.

If

For example,

D = RAV

to show

a e J

1 + a RAV

valuation

J, then

ring

and if

has

R

quotient

If

n 1 {Vi}i=

has n o n z e r o field

extension

of I.i.

is a finite

Jacobson

radical,

set of v a l u a t i o n then

D = RN(

rings

on

A ni=l Vi)

K.

In c o n n e c t i o n w i t h

i.i and 1.5, we w o u l d

like

to raise

the

following: 1.6 question. with

quotient

If field

R

and

V

K, must

are

1-dimensional

D = RNV

108

quasi-local

have q u o t i e n t

field

domains K?

3 The following l-dimensionality 1.7 Example. Let

simple

example

assumption

Let

k

shows

the necessity

in 1.6.

be a field and let

R = k[X, X2y]

and

2-dimensional

and are subrings element

of

R

the property Y-degree,

of the formal power

Given that conditions natural

on

ideals

R V

of

quotient

V

of

V

is a G-domain,

follow

to be almost V-module

that an element

element

of K

Proof.

V* if

to

D*

D

D*

radical

V

field k(X, Y) = K,

k[[X,

Any

Y]]

is greater

has

than

V

generated

has smaller

if

V

V[e]

field

quotient

[7, p. 1210

radical

field.

may be assumed

of

V

and

It would

to be an

in

K

and let

of the nonzero N

V[~]

is said

is nonzero.

with

every

has quotient

has quotient N

is a nonzero

field K.

be the con-

primes

K, it will suffice

109

V

in a finite

D = RAV

D* = R A V *

field

so

field of

VCV*CK

V, then

of the intersection

has quotient

Jacobson

prime

if the

ring extension

is contained

and

only if)

V * is a G-domain,

a

is a G-domain

if the nonzero

of the quotient

over

has quotient

K,

rings.

is a G-domain integral

V

for

field

or equivalently

has nonzero

~

and looking

has quotient

if the conductor

(and obviously

VCV*CK,

To see that

If almost

Suppose

traction

over

or equivalently

1.8 Proposition.

Since

R

D = RNV

integral

are

is greater than the

to be that

intersection,

of rank one valuation

We recall

field

seems

from 1.8 that in such an example

intersection

in

Y-degree

D = RAV

is a finitely

but

V

k[[X, Y]].

series

the

Jacobson

V

I am aware of no example where V

ring

is said to be a G-domain

have nonzero

field of

series

V

that

to put on V

R and

= k.

in order

domain

.

(xyZ,y)

of each monomial

has nonzero

hypothesis

(an integral

R nv

be indeterminates.

Y]

as a formal power

and for any element Thus

Y

local rings having quotient

that the X-degree

the X-degrse.

and

'

regular

regarded

X

V = k[XY 2

(x,xZy) both

of the

of

ideal

V*. of

D*.

to show that

N

4

is contained

in the quotient

of

n,

n

~,

say

and

a

field of

n+l

field of

is contained are in

RAV

precisely

If

V

= D, so

G-domains

R

valuation

R

V

constructed

V

in

V[~].

Thus

is in the quotient

Let

obvious

V~z

has quotient VI,...,V n

Vi

is contained

valuation

of

K.

R

has quotient

has quotient

K.

Apply

if there

in

[14].

extension

almost field

field

of rank one quasi-local

rings have been Perhaps

these

of 1.8. with quotient

integral K

field

over

field

V..z

if (and obviously

K

Then only if)

K. with quotient

in only a finite number

field

exist

Jacobson

K.

field

K

of rank one

radical,

In particular,

then

V 1 A...NV

n

1.5 and i.i0.

1.12 Remark. ideals

containing

of 1.8,

is

1.6.

has nonzero

D = RAV In... AV n field

rings

of 1-dimensional

be G-domains

such that each

If

V

are intersections

be G-domains

has quotient

rings

over

does not have quotient

and Ribenboim

VI,.. , V n

Let

then it is easily

of rank one valuation

nv n

I.ii Corollary.

integral

examples

with

nV*n

almost

RAV

V iCV~CK

D* = R A V ~ A . . .

G-domain,

in answering

the following

D = RAV In...

A 1-dimensional is a G-domain,

VI,...,V n

quotient

then some power

V

closed

application

exist which

in [i0]

would be useful

and suppose

K

by repeated

are intersections

i.i0 Corollary.

if

of

n+i/n

of the rank one valuation

Interesting

by Nagata

We note

of

such that

and

rings.

which

examples

Hence,

and

K, then such

prime

a = ~

is an integrally

the intersection

[4, p. 359].

Proof.

~ e N,

in the conductor

seen that the set of elements

domains

If

D.

1.9 Remark.

V

D.

field

domain with

so for example,

are 1-dimensional K, then

only a finite number

semi-local

Vl n . . . N V n

110

it follows

from i.ii

(noetherian)

has quotient

of

field

domains K

that with

and in

5 fact

RAVIN

Jacobson

... N V n

to the q u e s t i o n

field

K

whenever

of w h e n

R

or

R

has n o n z e r o

V

is a l o c a l i z a t i o n

D = R ~V.

1.13 P r o p o s i t i o n . a valuation D = RNV

Let

1 + x

V

R

P

denote

then

y / ( l + xy)

of

R,

are in

D

of

e P + yD P

R

rank one v a l u a t i o n x e RkV

and

and

R

ideal Proof.

of

ideal of

ring such

If

R

empty.

Let

b ~ 0.

Since

sufficiently Moreover,

xn

of

then

I/(I + xy)

for

and

x e J\V,

V

then

e (DNJ)\P.

is a u n i t

R

D.

If

and x/(l+x)

Jacobson

in

P

If

and

R.

Hence

R

is m a x i m a l .

is

If

x ~ J\V,

then

and h e n c e

a unit

radical

R ~ V,

then

xny ~ V

has n o n z e r o D = RNV

J

and

J ~ V.

in

D.

for some p o s i t i v e

Jacobson

D, then

V

V

r a d i c a l J,

is c e n t e r e d

is a

For if

is an i r r e d u n d a n t

integer

V

n.

is

intersection, on a m a x i m a l

for some m u l t i p l i c a t i v e

of the i n t e r s e c t i o n If

y e V,

large p o s i t i v e in

Hence

~ = a/(x n + b)

Since

y = a/B, y e Dp,

R

and so

and

b E J

B = b / ( x n + b) Dp = V.

111

xnv 0 .

(E) ,

we

R-free.

We devote the next section to our remaining suggested approach to the higherdimensional case. purposes,

We conclude this section with the remark that, for certain

adjoining round brackets indeterminates is harmless in looking for

maximal

C-M

modules over a complete local domain.

In other words, if several indeterminates, S .

R .

homomorphism

Then

is a complete local domain and

then if

To see this, represent

local ring

R[t]

(S,P)

S(t)

S

S(t)

is module-finite over for some

n .

R(t) ,

(R',P')

of

R

We can choose an

such that

h

R' .

homomorphism m ,

.

But since

in

R'

denote generators for

P .

R[%l/h]

R'

(R,P)

If

is

0 th

integer

such that

h

extension domain R'/P'

and then we

specializes to a

R-free we can view

S ~ T~m(R)

,

which induces a R' ~ ~m(R)

for

.

deficiency

is a finitely generated

M

definition depends on the choice of The

~ R'

in

such that

denotes a local ring and

M / 0

to associate certain integers with

notation).

such that

h

DEFICIENCIES AND SYMBOLIC POWERS

Throughout this section,

d

t

and then we have a homomorphism

5.

R

R-free module-finite

This gives a homomorphism S -~(R')

and we have a

does not vanish identically on

can choose values for the variables unit of

module, so does

We can choose a polynomial

with some coefficient outside the maximal ideal of 1/hi).

C-M

denotes

as a module-finite domain extension of a complete

f: S(t) ~ n ( R ( t ) )

f(S) ~ n ( R [ t ,

some

possesses a maximal

(t)

PI' "''' Pw R-module, we want

which we call the deficiencies Pl' " ' "

Pw

defo~ = def0M

L(M) A p d M = 0 ,

where

142

of

M

(the

but we shall ignore this in our we define to be the smallest

L(M) = H~(M) = [m E M: ptm = 0

for

24 some integer

t] = U t AnnMPt .

indeterminates,

To define the

w s i = Zj= 1 tijP~~ ,

let

defkM : def~(t)M(t)/(s! , ..., Sk)M(t)

k th

deficiency let

I < i < k ,

M

M

occurs when

k = D(M) ,

kw

and let

.

It is trivial to see that the first nonvanishing module

tij. be

deficiency of a nonzero

and we regard it as a measure of how far away

is from having depth one more than it actually does. The significance

Theorem 5.1. D(M) ~ k ,

R(t)

.

Le__~t M

N be a nonzero submodule of defkM P M . Then D(N) ~ k+l .

Nc

The question of whether

Thus, we may adjoin

deficiency,

Sl, ..., sk

(M/N)(t) ~ M(t)/N(t) '

kw

denote

,

and hence on

not an associated prime of

N(t)

N(t)'

have an element

N(t)

(t) in

M(t)

u 6 (s I, ..., Sk)M(t)

P(t)

Working modulo

(Sl, ..., Sk)M(t )

defoM (t)

is - 0

Suppose we

We must show

so it suffices to show

u' E L(M(t)') , since P ( t ) u ~ (s I, . ., . Sk)N(t) . .~ (Sl, . defkM u' 6 P(t) M(t)' , since u E N c pdefkMM . Hence,

we see that

, Sk)M(t)

,

and

,

M(t)' = 0 ,

as required.

We can now complete the argument in the fourth proof of given in

D(N(t)) 2 k .

(Sl, ..., Sk)N(t ) .

that

u 6 (Sl, ..., Sk)M(t ) ,

It

0 ~ N(t)' ~ M(t)' - (M(t)/N(t))'

P(t)u~

k th

and

Then it remains to show that

N P(t)

of

"general position".

u E (Sl, ..., Sk)N(t ) = N(t) ~ (Sl, ..., Sk)M(t ) ,

i.e.

D(M/N) ~ k .

as in the definition

that

u E L(M(t)')

with

..., Sk)M(t ) = (Sl, ..., Sk)N(t ) .

such that

.

(R,P)

such that

as well, so that

Since

N(t) N (sl,

of the following:

is unaffected by tensoring with

Sl, ..., sk

...., Sk).

is exact, we have that of

M

is a regular sequence on both

®R(t) R(t)/(Sl'

u

D(N) ~ k+l

indeterminates

and we obtain a sequence

follows that

Let

be a finitely generated module over

and let

Suppose that Proof.

of this notion for us here is a consequence

§3 •

143

(E)

in dimension

2

25 Corollary 5.2.

Let

(R,P)

be ~ Complete local domain ~n~

Drime of coheight at least one. Proof. all m

m ,

Let

d = deflR .

Clearly,

since any element of

is so large that

intersection

P - Q

Q(m) ~ pd ,

D(R)

Q(m)

and

we are done.

D(R/Q (m))

Q

is

0 ,

R

of

nopzero

m ,

D(Q (m)) ~ 2 .

R/Q (m)

Hence,

if

is a domain the

and since pt

~

are at least one for

on

But since

is contained in a given power

R

P

is complete it

for all sufficiently

m .

In fact, suppose that whenever define

Q(m)[M]

Lemma 5.3.

for some

Le___tt (R,P)

b__eeprimes o__ff R . integer.

Let

M

a

localizing,

I) implies since

R

in

R-P ,

i) ~

M

au E pmM]

F = Rk .

By the Artin-Rees

large integer

m ,

and

~(t)[M]

R

we can embed

Q(m) c p t *

Q(m) c p t *

we only need to show that

Q~

Q(m) c Q.(m)

is any prime of ,

3)

(37.8)

M

lying over

irreducible

t*

and

R .

m ,

since

To show that

we have

~

Since

144

S

Q~(m) = O ,

and we are done.

At this point we want to propose that a systematic

By

such that

~m Q(m)S = O . Q ,

of

in a free module

for large

be the completion of

S

by Corollary

is analytically

Let

if

be a given

Q(m)[M] c i l

Q(m)[M] c Q(m)[F] c pt*F .

is a domain,

t

Q ~ QI

over a complete regular local ring.

Then it suffices to show that

m ,

Q c QI '

irreducible.

lemma we can choose an integer

S

so that

closed, o__rr

2) implies

(R,P)

over

we

o__rr

is analytically --

RQI

,

R

Then we have:

3)

is torsion-free

for large

•

is integrally

is module-finite

Since

a prime of

M ~ (M/QmM) ® R Q

2) RQI

3) obviously,

QI = P "

Q

R-module and let

= P ,

we can assume instead that

ptM c pt*F .

R-module and

be a torsion-free

Then for every sufficiently

Nagata [13] ,

is an

be a complete local domain an__~dlet

Suppose either that:

Proof.

M

to be the kernel of the natural map

Q(m)[M] = [u E M:

while

large

is a nonzerodivisor

of the symbolic powers of

follows that large

Then for all sufficiently

Q

study of deficiencies

26 (defiQ(m)[M])

and residual deficiencies

(defiM/Q(m)[M])

of symbolic powers of

primes on modules be made, because understanding the behavior of these functions as m - ~

could well lead to a proof of Conjecture

(E) .

The general idea is to look for filtrations where the inclusions are proper, and complete local domain I)

(R,P)

D(M/Mi) ~ I ,

M

0 = M 0 c ~ - ~ ... c M n : M ,

is a module of dimension

n

over a

such that the filtration satisfies the conditions:

0 < i < n-I (so that

D(Mi+I/Mi) ~ I

for each

i

as well),

and defn_i_l(Mi+2/M i )

2) ~i+i/Mi ~ P

Mi+l/Mi , O < i

2 ,

then eventually the

N.

--

Theorem 5.1. D(M/Ni) ~ I and

D(Ni)

~ 2 ,

we have that

D(M) ~ 2 ,

D(M) ~ 2 .

D(M/N i) ~ 2 ,

is a height one prime of

instead of requiring require that that

RQ

and

We can then apply

i ~ def2M

taken together

D(N i) ~ 3 •

However, if we are taking the sequence Q

by

In fact, by a first application of Theorem 5.1, the facts that deflM M and N i c P imply that D(Ni) ~ 2 , and then since D(M/N i)

Theorem 5.1 again:

where

3 ,

i

are both

imply that

have depth

R ,

D(M/Ni) ~ 2 ,

defl(M/Ni)

N. l

is a discrete valuation ring.

Q(m)[M] ,

we do not need to know quite as much:

so that

be bounded as

to be a subsequence of

defl(M/Ni)

i -- =

: 0 ,

we only need to

However, we shall need to assume

This is true for all but finitely many

height one primes in a complete local domain since the singular locus is closed.

Proposition 5.4.

Le___tt (R,P)

be a complete local domain of dimension 3 •

Let

Q

be a height on___eeDrime of

Let

M

be a finitely generated torsion-free module over

defl(M/Q(i)[M] )

is bounded as

R

such that

i ~ ~ .

RQ

is a discrete valuation ring. R .

Suppose that

Then for all sufficiently large

145

i ,

27

D(Q(i)[M]) = 3 . Proof.

Since

height two prime bound

QI

is complete and the singular locus is closed we can choose a of

defl(M/Q(i)[M])

Lemma 5.3)

lemma).

and choose

and

M I = Q~J)[M] c PJ'M .

RQI

is regular.

Let

Choose

Let

j

d = def2M I .

k

(by Choose

i

(which is possible from Lemma 5.3 and the Artin-Rees

D(MI/Q(i)[M]) ~ 2 .

Remark 5.5.

Q c QI

J' 2 max [k, deflM ] .

defl(M/Q(i)[M]) ! k ,

D(Q(i)[M]) 2 3 ,

M

such that

Q(i)[M] c pdM I

Since

follows that

R

so large that

so large that

Let

R

as required.

and

Since

MI/Q(i)[M ] c Pk(M/Q(i)[M]) Q(i)[M] c pdM 1 ,

,

it

we have that

(We have found a good filtration.)

It is possible to bound deficiencies in a more "functorial" way.

be a finitely generated

R-module,

T = R + Pz + p2z2 + ... = R[Pz] c R[z] , Then there is a functor

F

F(M) = M + PMz + p2Mz2 +

...

from c

(R,P) local. where

z

is an indeterminate over

R-modules to graded

M[z] = M ® R R [ Z ]

,

Let R .

T-modules: F(M) = Im(M®RT ~ M®RR[Z])-

i.e.

Then

defo(M) S since

~T(F(M))

~(H~T(F(M))

= Z i (H~(M) N piM)zi

and

,

H~(M) 0 piM

contribution to the described length for each

i ,

makes a nonzero

0 < i < defoM .

It follows at

once that

0 def i(M) _< g(HpT(t ) (F(M(t) ®R(t)(R(t)/(Sl,

We next want to make estimates of the behavior of

..., si))) ) .

defl(M/Q(i)[M])

under good

conditions.

Proposition 5.6. M

Let

(R,P)

be ! complete local domain of dimension 3, let

be a finitely generated torsion-free

prime of

R

has depth 2.

such that

RQ

R-module, and let

Q

be a height one

is discrete valuation ring and such that

Then there is a constant

c

146

such that

M/Q(1)[M]

defl(M/Q(i)[M]) J ci

fo__Er

28

all i . Proof. that in

Choose

yQ(1)[M] R(t),

for

x E Q

= xM .

such that

Let

s

y

of

def I .

Then

y , s

We shall begin by proving by induction on

w E Hi = ~ ( t ) ( M ( t ) / ( Q ( t ) ( i ) [ M ( t ) ] + s M ( t ) ) ) y i-I w E E i = Q(t) (i) [M(t) ]+sM(t)

Ei ,

and

in

P - Q

such

.

P

.

that

i f and o n l y i f y i-1 w

On the one hand, if

Rad ( y i - l , Q ( i )

On the other hand, assume

i

of

is a system of parameters

(M/Q(1)[M]) ®R(t) = M ( t ) / Q ( t ) ( 1 ) [ M ( t ) ]

and

( y i - 1 Q(i), s ) w c

and choose

be a general linear combination of generators

as in the definition

R(t)/QR(t)

Rq = XRQ

w E H.

is in

s) D Rad (Q, y, s) = P ( t ) If

i = 1 ,

Ei

then

.

we have that

1

y~w + sv E E 1 •

But since

and since

is a system of parameters

y, s

• + sv 6 Q(t)(1)[ M(t)] y~w Now suppose

M(t)/Q(t)(1)[M(t)]

and

Q(t)(i)[M(t)] c Q(t)(1)[M(t)] Suppose

w = w' + sv' ,

We then obtain

yw' = xw* ,

that

yi-2w*

is in

is in

It follows that

Ei_ I .

w = y l-1 w

Then

w

,

,

where

•

is in

Q(t I l)[M(t)]+sM(t),

Since

is in

Q(t)(1)[M(t)]+sM(t) and

v"

v'

show that

d

such that

I

w

is in

yQ(t)(i)[M(t)]

By the induction hypothesis,

say

(Q(t), s, y) •

Suppose that

w

w' = yiw" .

yi_lw,, E E i

Let

w' 6 yiM(t) Choose

But then

j

and

we then have

is in

as required.

c = 2d .

We need only

H. N P(t)2diM(t)

i

where

.

and

y(yJw' + sv") = x(yJw * + sv*)

.

Then

i

(Q, s, y)2iM(t) c QiM(t) + sM(t) + yiM(t)

w = w' + h

M(t)

yi-lw = yi-l(w' + sv') =

P(t) d c

H. A p(t)ciM(t) c E.

is in

.

then must be in

yi-lw' + s(yi-lv ') = yi-2xw* + s(yi-lv ') E xEi_ I + sM(t) c E i , Now choose

R(t)/Q(t)

from the fact that

is in

Q(t)(1)[M(t)]

yJ(yw') + s(yv")

Q(t)(i-l>[M(t) ] .

is in

of depth 2 over

R(t)/Q(t)

it follows that w'

Then

yv" = xv* .

" y3w* + sv*

so

,

where

as well.

C-M

yJw + sv E Q(t)(i)[M(t)]

yJw' + sv" E Q(t)(i)[M(t)]

Q(t)(1)[M(t)]

for

we can deduce that

i > I ,

is

and

h E E i c H.m '

such that

yJw' E E i •

w' = ylw" E E.I '

so that

.

Since

so that Then

QiM(t) + sM(t) = EicHi, w' E H i N yiM(t)

yj+iw"

E Ei ,

so that

w = w' + h E E i + E i = E i ,

as required.

This result falls short of what we would like to have:

147

,

some additional

29

condition is n e e d e d to guarantee boundedness of the residual deficiency.

But the

conditions of the hypothesis are very easy to satisfy, so that it might well be possible to meet extra conditions. The following lemma shows how to construct situations which satisfy the hypothesis of Proposition 5.6.

D. Eisenbud suggested trying this type of

construction to me.

Lemma 5.7. such that

Let

(R,P)

be a local domain, and let

RQ is a discrete valuation ring.

torsion-free module over R-module

M

Proof.

such that

R/Q .

Le___~t E

E

Then there is a finitely generated torsion-free

M/Q(1)[M] m E .

The localization of

in a free

0 ~ E ~ F

E

(R/Q)-module

at the prime

0

F = (R/Q) t

of K

R/Q of

is a finiteR/Q .

way that

(R/Q) p

E

as

Im f ,

where

maps onto the image of g: R p ~ R t .

R-modules:

Let

E

We can

in such a way that the inclusion

becomes an isomorphism when we localize at the prime

can therefore represent

of free

be a height one prime

be a finitely generated

dimensional vector space over the field of fractions embed

Q

0

of

R/Q .

f: (R/Q) p ~ (R/Q) t = F in

F .

M = g(R p)

We can lift M c Rt

f

We

in such a to a map

g

is torsion-free.

Since the diagram

Rp _onto > M ~-__> R t

o I

n t

O

commutes onto

E .

E = f(u(RP))

u

¢~r

\o

, ~

(R/Q)p onto,>~, ~

= v(g(RP)) : v(M)

,

v

n t O + (R/Q)o

and there is an induced map

It remains to see that the kernel of this map is precisely

¢

of

Q(1)[M]

and for this purpose it suffices to compute the kernel after localizing at may therefore assume that so that the inclusion of free and consequently f r e % It is clear that to

E

induced by

(R,Q) E

in

is a discrete valuation ring and that (R/Q) t

is an isomorphism.

Then

M

Q .

is torsion-

and we are trying to show that the kernel of

is surjective.

It suffices to show

148

We

R/Q : K ,

¢

Q~i is contained in the kernel, and we know that the map of ¢

M

is

QM. M/QM

3O that this map is an isomorphism, or, equivalently, that Since

¢

is surjective, dimK(M/QM ) ~ t ,

dimK(M/QM ) = rank M ~ t ,

a maximal

C-M

module over

Q

such that

module

R

E

such that

M~

Rt ,

and we are done.

Now, given a complete local domain height one prime

and since

dimK(M/~ ) = dimKE = t .

RQ

over

(R,P)

of dimension 3 we can choose a

is a discrete valuation ring.

R/Q ,

We can construct

and we can then construct a torsion-free

M/Q(1)[M] e E .

This verifies the statement that the

hypothesis of Proposition 5.6 is easily satisfied. To see that some extra condition really ~ needed to guarantee that defl(M/Q(i)[M] )

is bounded, not merely

Example 5.8. ideal

Let

(x,yz) c R ,

and let

Q = zR .

, where Then

Q(i)[M ] = (xz i ,yz i ) ~ (x,y)

depth 2, but that

R = k[[x,y,z]]

O(i) ,

defl(M/Q(i)[M])

we consider the following:

k

is a field.

Let

M

M/Q(1)[M] ~ x(R/zR) ~ R/Q

has depth 2 for all

i ,

be the has

which shows

is not bounded.

Despite this, there are interesting cases where the idea of Proposition 5-4 succeeds.

We conclude this paper with one class of such examples.

Example 5.9. Let

f

Let

k

be an algebraically closed field of characteristic

be an irreducible homogeneous polynomial in

XI, X2, f

k[Xl, X2, X3]

is a homogeneous system of parameters but

that the partial derivatives of

f

simultaneously only at

(One example is

(0,0,0).

conditions guarantee that

such that

X 2 ~ (XI, X2, f) 3

with respect to

X1, X2, X3

f = ~

and such

vanish

+ X~ + X~ .)

R 1 = k[Xl, X2, X3]/(f ) = k[Xl, x2, x3]

0 .

These

is integrally

closed but not proper in the sense of Chow [4], P. 817, and it follows from the Theorem on p. 818 of [4] that the Segre product of R 2 = k[Yl, y2 ] R = k[xiYj:

is not Cohen-Macaulay.

i = i, 2, 3

and

R1

with the polynomial ring

This Segre product

j = I, 2] c Rl[Yl, y2]

closed ring of dimension 3 which is not Cohen-Macaulay. depth 3 module using the ideas of this section.

is then an integrally We shall construct a

(We could localize at the

homogeneous maximal ideal and complete without essentially changing the situation,

149

31 but it will be simpler to continue in the graded case.) Let

Q = yl~[yl,Y2] N R .

R/Q ~ k[XlY2,x2Y2,x3Y2] ~ R I R .

We shall show that

Then

Q(i)

is a

note that

Q(i) = Yli R I[YI'Y2] A S .

that proof.

xlY2~ x2Y 2

for all

C-M

D(R/Q (i)) = 2

where

y~

i ,

Then

Then

f0x2 = g0xl

in

~

,

f0y]'Pyq

and since

i .

Xl, x 2

150

modulo

In fact, we shall To see this,

Yi' Y2 g

=

(yl) .

i = i, 2, 3] We shall show

which will complete the

fx 2 = gx I ,

and

follows easily.

Yl

R/Q (i) ,

bihomogeneous pieces of the same bidegree in =

is a height one prime of

which certainly suffices.

is the image of

f

in fact,

Hence, R/Q (i) = k[xiY7', xiY2:

fx2Y 2 = gxlY 2 .

we might as well assume that

Q

module for large

is a regular sequence in

Suppose that

is a domain;

is two-dimensional, and

show that

= Sl[Yl, y2]/(yl) ,

R/Q c ~ [ y 2 ]

and

f, g

break into

for which this is true, and g0ylPy q ,

is an

where

f0' go E R I .

Rl-sequence , the result

32

BIBLIOGRAPHY

I.

H. Bass,

2.

D. Buchsbaum and D. Eisenbud,

On the ubiquity of Gorenstein rings,

Math. Z. 82 (1963), 8-28.

What makes a complex exact?,

to appear in

J. of Alg. 3.

,

Remarks on ideals and resolutions, preprint.

4.

W. L. Chow,

5.

D. Ferrand and M. Raynaud,

On unmixedness theorem,

Amer. J. Math.

86 (1964), 799-822.

Fibres formelles d'un anneau local

Noeth6rien, Annales Sci. d e l'Ecole Normale Superieure 3 (1970), 295-312. 6. IV.

A. Grothendieck (with J. Dieudonn@),

(Seconde partie),

El@ments de g$om@trie alg@brique,

Publications math@matiques de l'I. H. E. S. n ° 24, Paris,

1965. 7.

A. Grothendieck (notes by R. Hartshorne),

Verlag Lecture Notes in Mathematics 8.

M. Hochster,

9.

,

I0.

,

Springer-

No. 41, 1967.

Grade-sensitive modules and perfect modules, Contracted ideals from integral extensions,

I. Kaplansky,

II.

Local cohomology,

Commutative rings,

Allyn and Bacon,

preprint. preprint.

Boston, 1971.

Topics in commutative ring theory, I. and IIl.,

duplicated

notes. 12.

G. Levin and W. Vasconcelos,

Pacific J. Math.

Homological dimensions and Macaulay rings,

25 (1968), 315-323.

13.

M. Nagata,

Local rings,

14.

C. Peskine and L. Szpiro,

Interscience Tracts 13,

Dimension projective finie et cohomologie locale,

Thesis (Orsay, Serie A, N ° d'Ordre 781), 15.

D. Rees,

16.

J. P. Serre,

in Mathematics 17.

to appear in Publ. I. H. E. S.

The grade of an ideal or module,

Philosophical Society

New York, 1962.

Proc. of the Cambridge

53 (1957), 28-42. Alg~bre locale.

Multiplicit@s.

Springer-Verlag Lecture Notes

No. II, 1965.

R. Y. Sharp,

Application of dualizing complexes to finitely generated

modules of finite injective dimension,

preprint.

151

33

Comments I.

There

is a serious

theorem

in [9]

Question

4.1).

if

R

the result

a field,

and the general

of a weak

It can be shown classes

get a version

where ,

Sl,S 2

of

case

is

(E) [9].

that

the

is an Rn-sequence ,

so that we need only divide

of all g-perfect

of Conjecture

152

case

form of Conjecture

in the 2-dimensional

lim G(Rn) -+ n out by these instead

after

still holds

in a revised version

[Rn/(Sl,S2)],

generate

of the main

However,

This will be shown

2.

gap in the proof

(see the first paragraph

contains

a consequence

and Corrections

(F)

modules

to hold.

to

C O M M U T A T I V E

RINGS

Irvin~ Kaplansky

i. I N T R O D U C T I O N I have

c h o s e n to s p e a k on the subject

a t o p i c w h i c h has maturity

opportunity

to o b s e r v e that

a historical touch time

f a s c i n a t e d me for years,

as one of the b a s i c

to take the hasten

on the recent to r e c o r d

The

to s u r v e y

subject

lightly

field begins,

number

The

each

defined

by

therefore

pen has

eras,

signalled

quadratic

of the i n t e g e r s

functions

or r a t h e r

by the a p p e a r -

established,

to

painstaking

an i n s i g h t

w i t h the

field.

to a

At the

of any

and so did the p a r a l l e l This

is the

"one-dimensional"

case we have to look to the

d u r i n g the

in the i n t r o d u c t i o n

they g a i n e d

of a q u a d r a t i c

of one v a r i a b l e .

geometry

did,

forms a m o u n t e d

the r i n g of i n t e g e r s

For the n - d i m e n s i o n a l

" r e s t e d on a most

3- THE F I R S T

Let me

and spend m o r e

century mathematics

binary

field became well

has a p t l y w r i t t e n ,

of the a x i o m a t i c

of

and K r o n e c k e r ,

of w o r k on a l g e b r a i c

from which

of 19th

theory

account

of a l g e b r a i c

geometers

like

paper.

Dedekind,

case of our subject. stream

I will

authoritative

into five well

four p e r i o d s ,

as so m u c h

complete

of Kummer,

algebraic

perspective.

background,

N. B o u r b a k i ' s

g r o w n to

I would

[6] are a c c o m p a n i e d

standard.

classical

rings,

ERA

Disquisitiones. G a u s s ' s

theory

which

era and then

2. THE P R E H I S T O R I C

virtually

chapters

erudite

seems to me to d i v i d e

ance of a v e r y i m p o r t a n t

hands

seven

on the

history,

and one w h i c h has of m a t h e m a t i c s .

it in h i s t o r i c a l

Bourbaki's

Noetherian

in detail.

a "prehistoric"

The

"structures"

note up to his u s u a l

comparatively

yet

of c o m m u t a t i v e

~OJ,

study

not always

19th century. that

A. W e i l

the a l g e b r a i c

of n u m e r o u s

special

found a m o n g m o d e r n

examples,

exponents

creed".

PERIOD

19th c e n t u r y b e g a n w i t h a m o n u m e n t a l

piece

of work;

it also ended

w i t h one. Hilbert's memoir (i) He p r o v e d nomials

[14] m a d e

that

three

extremely

if K is a f i e l d and

in n v a r i a b l e s ,

then

every

important

contributions.

R=K[Xl,...,Xn] the

ideal of R is f i n i t e l y

153

r i n g of p o l y generated.

-

L a t e r this was recast the Hilbert finitely theorem

and s l i g h t l y

basis theorem:

generated,

u n i t e d at one stroke

expressible

are a g a i n p o l y n o m i a l s . The

combination This

"syzygies"

investigated

generated

is to say,

observations

a finite

any p o l y n o m i a l

not,

set of in I is

coefficients

however,

that

ordinarily

be

by r e l a t i o n s

geometers.

For d e e p e r i n s i g h t

he next

ul,...,u r. He r e c o g n i z e d that he was now d e a l i n g

rather than

an ideal,

and p r o c e e d e d to the

This

is

decisive

... + Urf r = 0

but s h o w e d

"second

you reach O, or to state it otherwise, reached.

call

ideal

Hilbert's

of fl,...,f r w i t h

is m e a s u r e d

by the a l g e b r a i c

the r-ples

with a module

ring R every

t h e r e is t h e r e f o r e

expression will

lack of u n i q u e n e s s

ulf1+ called

form we now

case by case.

in K[xl,...,Xn]

as a l i n e a r

to the

in R[x].

is true

for I, say fl,...,f r. That

generators

unique.

generalized

a large n u m b e r of f i n i t e n e s s

been m a d e

(2) If I is an i d e a l

-

if in a c o m m u t a t i v e

t h e n the same

that had p r e v i o u s l y

2

tour de force was

it was

syzygies".

again

finitely

after n steps

Theorem:

after n-1 steps a free module

a farsighted

anticipation

is

of h o m o l o g i c a l

algebra. I in K[x1,...,Xn]

(3) G i v e n an i d e a l homogeneous

ideals,

s t u d i e d the

asymptotic number

degree

contained

Subsequently treatment Three

in I, and was

this t u r n e d

later,

Nullsteliensatz. of f u n c t i o n a l

of l i n e a r l y

I would

another

forms

at some point.

ideal

fundamental

Analogously,

in n v a r i a b l e s

t h i n k of R as c o n s i s t i n g

o v e r the

consists

of f u n c t i o n s

generation

complex

be the r i n g of

c l o s e d f i e l d K, and

space of n - t u p l e s

over K.

ideal in R is the set of all polynomials

The p o l y n o m i a l

is a good deal h a r d e r

theorem.

theorem

But here

i~ u n c o u n t a b l e .

is a curious remark:

(I say " c u r i o u s "

154

t h a n the

continuous

the N u l l s t e l l e n s a t z

because,

functions

functions

every maximal at some point.

Theorem:

the

if X is a

of all the

let R=K[x~ .... , x ~

on the

rigorous

theorem:

the y o u n g e r

algebraically

polynomial.

varieties.

the r i n g of c o n t i n u o u s

in C(X)

Hilbert

of a g i v e n

for the m o d e r n

of a l g e b r a i c

like to try to i n t e r e s t

space and C(X)

only w i t h

characteristic

so let me b e g i n by r e c a l l i n g that

every maximal

polynomials

independent

led to a c e r t a i n

proved

dealt

can be g e n e r a t e d by forms)

out to be a key t o o l

Hilbert[15~

analysts,

Hausdorff

on X, t h e n vanishing

(actually Hilbert

ideals w h i c h

of i n t e r s e c t i o n m u l t i p l i c i t i e s

years

compact

i.e.,

vanishing function

is easy if K

a m o n g o t h e r things,

in H i l b e r t ' s

-

day one did not

specify

of c o m p l e x n u m b e r s . ) me b r i e f l y recast

repeat

K,

it.

Given

element

that the e l e m e n t s This

The N u l l s t e l l e n s a t z ideals

and points.

between prime course

not just

the p r i m e

he p r o v e d primary

that

This

additions

were m a d e

up to his

Cambridge

tract

of his time, recently. himself

~7,

cannot p.68~

linearly

inde-

between

require

algebra

[21] w h e n

as an i n t e r s e c t i o n

powers.

in a n u m b e r of p a p e r s Macaulay

of his w o r k won full

was

leading

far a h e a d

appreciation

w i t h the

of

as we can expect

of prime

respects

fail to be i m p r e s s e d

is of

and geometry.

us to study all ideals,

as a p r o d u c t

In m a n y

found M a c a u l a y ' s

correspondence This

step was t a k e n by L a s k e r

by M a c a u l a y

[23].

between maximal

varieties.

is as good an a n a l o g u e

and some a s p e c t s

One

in l i n e a r

for R is a v e c t o r

to a o n e - t o - o n e

an e x p r e s s i o n

of an i n t e g e r

Important

over K, are

algebraic

the b r i d g e

The next

admits

theorem

of the r e p r e s e n t a t i o n

can be

contain a

exercise

correspondence

can pass

and g e o m e t r y

ideals.

field let

over K.

in b u i l d i n g

any ideal

ideals.

a ranging

and i r r e d u c i b l e

of b o t h a l g e b r a

closed)

(and a nice

if K is u n c o u n t a b l e ,

F r o m this one

step

[20], but

i d e a l M in R, our p r o b l e m

sets up a o n e - t o - o n e

ideals

a vital

The needs

i/(u-a),

is a c o n t r a d i c t i o n dimension

K was the

[19] and

R/M c o i n c i d e s w i t h K. Now if the f i e l d R/M

It is a fact

algebra)

of c o u n t a b l e

in

(since K is a l g e b r a i c a l l y

u.

pendent. space

a s s u m e d that

is c o n t a i n e d

a maximal

that

is l a r g e r t h a n K it m u s t transcendental

-

it b e i n g t a c i t l y

The p r o o f

to the s t a t e m e n t

3

fact that

w o r k to be " t e i l w e i s e

only

Krull

recht m ~ h s a m " .

4. THE S E C O N D P E R I O D The

s e c o n d era was

tance

inaugurated

of this p a p e r

exaggeration

a x i o m the

ideals.

conclusion

generated, A very

intersection expressible

of H i l b e r t ' s

simple,

general

of a finite

number

as an i n t e r s e c t i o n that u s h e r e d

irreducible

is p r i m a r y .

ideal

It is e n t i r e l y a p p r o p r i a t e later

called

paper

[28]. The i m p o r -

it is s u r e l y not m u c h of m o d e r n

basis

she takes

theorem:

of two

larger

that rings

155

ideal

is

condition

on

any ideal

ideals

ones).

in the new era:

"Noetherian".

chain

shows that

of i r r e d u c i b l e

as a sole

that every

the ascending

argument

of an

algebra.

of the b a c k g r o u n d

or equivalently

charming argument

were

that

to call her the m o t h e r

After a leisurely description

finitely

by Emmy N o e t h e r ' s

is so great

(i.e.

And t h e n

is the ideals not comes the

a short p r o o f that

w i t h the a s c e n d i n g

chain

any

condition

-

Let me take the normalization ring

in w h i c h

is n i l p o t e n t Let

4

-

space to r e c o r d the proof.

that

the i r r e d u c i b l e

0 is i r r e d u c i b l e (for this

is what

ideal

I do it w i t h the h a r m l e s s

is O.

So: we have

and have to p r o v e that it m e a n s

to say that

a Noetherian

every

zero-divisor

0 is a p r i m a r y

ideal).

ab=O w i t h b~O. Let I

ascend,

be the a n n i h i l a t o r of a n . The ideals ( I ) n u l t i m a t e l y b e c o m e stable, say at I k. We claim

and t h e r e f o r e

Ik~Rak:o.

x=ya k lies in the i n t e r s e c t i o n .

For s u p p o s e

O:xak:ya 2k,

Then

Y c I2k. But I2k :Ik, so yak:o, x:O. By the i r r e d u c i b i l i t y of O, e i t h e r I k or Ra k is O. But I k c o n t a i n s b. Hence Rak:o and a is n i l p o t e n t as required. In a s u b s e q u e n t treatment

of rings

(subsequently book

D~

[29], E m m y N o e t h e r

w i t h the p r o p e r t i e s

called Dedekind

taught

I insert Artin

paper

all these

at this point

descending

things

as the n a t u r a l algebras

it.

Natural

dimensional rings w h i c h and r e m a i n rings

are not

the next n a t u r a l

rings w h i c h rings.

are f i n i t e l y

Among

PERIOD

In the e a r l y

1920's

commutative

at the t i m e

[17] gave

and m a k e s

The t h i r d era was the

day K r u l l has

rings w i t h p e n e t r a t i o n

of 1935

principal

inaugurated

ideal theorem.

37 of

worse

~7].

Krull

The

one on p a g e

common.

Infinite-

Noetherian

substantially

under

(possibly non-commutative)

over

commutative class

Noetherian

of rings!

to carry on the t r a d i t i o n .

investigated

every

and d i s c e r n m e n t .

aspect

of

His E r g e b n i s s e

subject

as he saw it

r e a d i n g today.

by the p a p e r

gives this

of

of the A r t i n i a n

commutative

for this

arose

rings

I am b e g i n n i n g

~167

in w h i c h

Krull proved

( W a r n i n g to the g r o u p - t h e o r i s t s :

n o t h i n g to do w i t h the p r i n c i p a l matters

of

those

Artinian

are hard to w o r k w i t h

with

over fields class

In ~927

subject But

are not

a b r o a d view of the

stimulating

note.

of rings;

the

tangible

algebras,

a name

a s t r o n g new voice

up to the p r e s e n t

monograph

is the

we n e e d

rings

that,

generated modules

o t h e r things,

5. THE T H I R D

I argue

algebras

step

number theory

of a l g e b r a i s t s .

class

of r i n g theory.

the most

finite-dlmensional

largely mysterious.

generation

and c o n t r o v e r s i a l

of A r t i n i a n

axiomatic

van der W a e r d e n ' s

way to g e n e r a l i z e

perhaps

and f i n i t e - d i m e n s i o n a l

control,

Right

rings,

long,

For forty years t h e s e

to a p i e c e

examples

division

in a l g e b r a i c

call the dual

chain c o n d i t i o n .

finite-dimensional to doubt

to a whole

one m i g h t

have been r e g a r d e d

found Before

a parenthetical

[i] i n t r o d u c e d what

w i t h the

rings).

gave her c e l e b r a t e d

ideal t h e o r e m

of group theory.

name to two t h e o r e m s :

25 is a c o m p a r a t i v e 156

this has

see p a g e s

triviality.)

To m a k e 25 and

-

As N o r t h c o t t theorem rings.

justly

arrived,

remarks

to B o u r b a k i ' s

ideals

one p a s s e s

theorem;

is minimal

relatively

a prime

that

of the

his s e v e n c h a p t e r s

rank of a p r i m e

chain.

over an i d e a l

saying

in a Noetherian

Theorem:

over a principal

ideal,

. . . 3 Pn of prime

ring,

ideal has rank ~ i. F r o m

e a s i l y to the k i n g - s i z e p r i n c i p a l

ideal m i n i m a l

subject.

ideal theorem.

a c h a i n P = P o ~ PI

of l e n g t h n, but no l o n g e r

a prime ideal which this

exists

ideal

s h a d o w of p o l y n o m i a l

asinorum

thoroughness

I i n t r o d u c e the

that P has r a n k n if t h e r e

u n t i l the p r i n c i p a l

a pale

a kind of pons

have not yet r e a c h e d the p r i n c i p a l To state the t h e o r e m ,

i05-i06],

rings were but

it r e m a i n s

It is a t e s t i m o n i a l

-

E30, PP.

Noetherian

To this day

5

ideal

g e n e r a t e d by n e l e m e n t s

has rank < n. K r u l l was

of course

inspired

and t h e i r h o m o m o r p h i c is in that

context

I note that Rees

Rees

in

variety

about p o l y n o m i a l

rings

the p r i n c i p a l

ideal t h e o r e m

of c l a s s i c a l p r o p e r t i e s

of the d i m e n -

and of the d i m e n s i o n

gave a s p a r k l i n g new p r o o f

[31~, a p a p e r that

of an i n t e r s e c t i o n .

of the p r i n c i p a l

also m a r k e d the a p p e a r a n c e

ideal

of the A r t i n -

lemma.

The p r i n c i p a l

ideal theorem yields

in a N o e t h e r i a n strengthened

ideals

r i n g s a t i s f y the

form,

a fixed prime.

Of course,

and a f o r t i o r i

I see the next technique

paper

[3~

was

n o t e d that

"local"

nation

can be f o u n d

"quasi-local"

be a r e a s o n a b l e The two m a j o r

have

Krull

on local rings. although

need

The

it was not

in c o m p l e t e

has stuck.

generality. to

It is to be

Noetherian.

concept

It is r e a s o n -

s h o u l d h a v e the

convention

I suggest

short

is not u n i v e r s a l l y

"not n e c e s s a r i l y

literature.

for a c o m m u t a t i v e

choice

the

circumlocution

in the

c h a i n on all

enough there

c a l l e d such a r i n g a " S t e l l e n ring"

important

arisen:

from

ideal r e d u c e s m a n y p r o b l e m s

to i n c o r p o r a t e

that the most

and the d r e a d f u l

local ring"

~8~

it a r r i v e d

at a m a x i m a l ideal.

in a

from a fixed prime.

~938 p a p e r

r e n a m i n g to "local

But two d i f f i c u l t i e s

accepted,

but p e r v e r s e l y

getting established,

is here m e a n t

able in m a t h e m a t i c s name.

chain condition

ideals

b o u n d on chain d e s c e n d i n g

ascending

in 1946 that

localization

but C h e v a l l e y ' s

ideals,

as K r u l l ' s

case of a single m a x i m a l

ring",

descending

on chains

landmark

In p a r t i c u l a r ,

i n f o r m a t i o n that the p r i m e

by a x i o m we have the a s c e n d i n g

on p r i m e

of l o c a l i z a t i o n

until U z k o v ' s

the

to wit w i t h a u n i f o r m

not be a u n i f o r m b o u n d

the

facts

B r i e f l y put,

a consequence

sion of an a l g e b r a i c

theorem

by the k n o w n

images.

Noetherian

that the

r i n g w i t h one m a x i m a l

desig-

ideal w o u l d

on w h i c h to agree.

accomplishments

in

~8]

157

are the i n t r o d u c t i o n

of the t e c h -

-

nique of completing special

a local ring,

class of local rings.

local ring R can be generated

if this m a x i m u m

is attained.

[7] "regular"

not the happiest meanings

choice

and the recognition

time a definition

of regularity

REMARK.

This account

alternative restricted

at the latest

for regularity,

corresponded

of a nonsingular

(i) If R is regular,

(2) Is a regular

sequence rings,

raised

local rings that

effort,

Cohen's

for geometric

thesis

stubbornly

complete

regular

local rings.

a kind of climax theorems

local rings,

now had a m e t h o d localize

easily.

in answering But the

then

local

which came close to

for launching

(hopefully)

a systematic ideal,

work your way

The method works with gratifying should

for this

for complete

at a general maximal

theory,

frequency,

learn how to use it. But it does

e.g. the two problems m e n t i o n e d

above

continued

to hold out.

A pat.tern began to appear repeatedly. for a special geometric

rather

also succeeded

the geo-

remained open.

[9] provided

structure ring.

29

He gave structure

and every young ringtheorist not always work,

geometry,

[41, p.

regular

back to the original

domain?

(1) in the affirmative

The subject

use Cohen's

factorization

Zariski

attack on a proposed problem: complete,

ideal in R, is Rp (the localiza-

settles

of developments.

especially

original paper.

arise in algebraic

for both problems,

being decisive.

in Krull's

local ring a unique

the second affirmatively

The late I.S.

concept

This was fully set

to P) regular?

interpretation

case,

to a very classical

on a variety.

and P is a prime

tion of R relative

general

used

Chevalley

[4~.

were promptly

With additional

2,947

somewhat.

point

metrical

has

different

Krull and Chevalley and furthermore

It turned out that regularity

For the regular

"regular"

[39] of an entirely

indeed,

Two questions

count,

in order and It was perhaps

of all, there was already at that

is over-simplified:

his local rings

forth by Zariski

accepted.

for rings.

definitions

that

called R a "p-Reihenring"

Another name was doubtless

is now universally

and, worst

ideal M of a

then n is the maximal

in R. Krull

due to von Neumann

concept

of a certain

if the maximal

by n elements,

since,

in mathematics;

-

Recall that

length of a chain of prime ideals

Chevalley's

6

class of Noetherian

case. Workers

A certain theorem would get proved

rings,

usually

in the field relentlessly

to remove the restriction.

This was partly

158

something accepted

like the the challenge

for the usual reason:

the

-

elegance

and c l a r i t y

a n s w e r to current diophantine integers

and a n t i c i p a t e d

geometry

calls

since

in this

local

no n i l p o t e n t

elements.

Chevalley This

the exact

its q u o t i e n t that

the

course

integral

is p e r h a p s did not

is f i n i t e l y

closure

the

in

subject

of

o v e r the

the

inter-

Let R be a

c o m p l e t i o n R ~ has local domain,

in a s p l e n d i d p a p e r

not [3~,

for e v e r y r i n g f b e t w e e n R and

generated

of T is a f i n i t e l y

that this

a little m o r e

for a g e n e r a l

[26]. But Rees,

c o n d i t i o n needed:

it is s t a n d a r d

it was

rings

generalize.

[8] p r o v e d that

is not true

closed

field which

for i n s t a n c e ,

of p o l y n o m i a l

which

case the t h e o r e m

domain.

But p a r t l y

images.

instance

e v e n if R is i n t e g r a l l y pinpointed

needs;

for a study

and t h e i r h o m o m o r p h i c

geometric

-

of a good g e n e r a l i z a t i o n .

I w i l l give an e x p l i c i t esting

7

condition

o v e r R, we m u s t

generated

is f u l f i l l e d

have

T-module.

Of

in the g e o m e t r i c

case. Incidentally,

there

an integrally

closed geometric

again

is a c o m p a n i o n t h e o r e m

local

a domain.

Nagatats

example

in

generalization.

It w o u l d

be n i c e

to have

showing exactly

what

6. THE F O U R T H

is n e e d e d

by the a r r i v a l Let us r e c a l l (i.e.

k e r n e l KI, If the gical

a direct

summand

finished

N o w it turns behaves

Krull's

theorem,

was h e r a l d e d

was a n n o u n c e d

the job

in

E34].)

that

a

a k e r n e l K2,

etc.

is K

we say that the h o m o l o n =n; the value of n does not

out.

If no K

Buchsbaum in

is p r o j e c t i v e n chains of syzygies.

and Serre.

[2], w i t h

(The

full d e t a i l s

Let R be local w i t h m a x i m a l

if and only if d(M) O.

If

x e D

containing

x

is such that such that

system containing

containing

M . g

S.

Hence,

Furthermore,

is the group of divisibility of the g

valuation ring

Dp

[23].

Finally, I apply Theorem 2.1 to obtain a ring theoretic proof of a well known result about realizations of an Z-group by subgroups of reals.

First, observe the

following proposition.

PROPOSITION 4,7.

If

D

.

valuation overrings of in the center of some

is a Bezout domain and if

.

D

.

{V }

.

such that each non-zero non-unit

V ,

then

is a collection of

C~

xD = na (xV~ n D)

and

x

of

D

is contained

D = n~ Ve.

The proof is similar to that of [13, p. 42] needing only the additional observation that

[zD:xD]

is finitely generated [13, p. 301].

An archimedean totally ordered group can be embedded as a subgroup of the reals. This fact led Clifford [5] to conjecture that an archimedean Z-group has a realization by subgroups of

R.

Similarly, Krull [19] conjectured that a completely integrally

closed domain is an intersection of rank one valuation rings. terexample to both conjectures in [25].

Nakayama gave a coun-

Realizations by subgroups of

R

are obtained

by applying the following corollary to Proposition 4.7.

COROLLARY 4.8.

If

D

is a Bezout domain such that each non-zero non-unit of

D

is contained in a prime ideal of height one, then the group of divisibility can be embedded as an Z-subgroup of a cardinal product o f subgroups of the reals.

Proof. P

By Proposition 4.7,

D

runs through all prime ideals of

is of height one,

Dp

is an intersection of valuation rings

Dp

D

Since

of height one.

is a rank one valuation ring, and 2O5

Apply Theorem 2.2. G(Dp)

where

is a subgroup of

P R.

13

The next theorem in its present form appeared in a paper by Conrad, Harvey, and Holland [8, p. 164], but it was originally proved for vector lattices by Nakayama [26].

I add part 3 for completeness.

THEOREM 4.9.

(Nakayama, Conrad, Harvey, Holland)

ordered abelian group. i)

G

Suppose

G

is a lattice-

The following are e~uivalent.

is 0-isomorphic to a sublattice of the cardinal product of copies of the

additive group of real numbers. 2)

For each non-zero

g • G,

there is a value

Mg

of

g

such that

G/M

i_~s g

~-isomorphic to a subgroup of the real numbers. 3)

G

is the group of divisibility of a Bezout domain

zero non-unit

Proof.

x

That

of

D

3)

such that each non-

is contained in a prime ideal of height one.

implies

Finally, assume

i) 2)

is the content of Corollary 4.8.

implies

2).

D

with

G

holds.

If

v

is the canonical semi-valuation onto

of

g

such that

g

saturated multiplicative

Suppose G,

S

of

D.

Since

x

let

is contained in the reals. system

Clearly

i)

By Theorem 2.1, there is a Bezout domain

as its group of divisibility.

G/M

D

is a non-zero non-unit of g = v(x).

By [23]

'

M

Let g

M

g

D.

be a value

corresponds to a

G/M

is totally ordered, S is g the complement of a prime ideal P of D. Further G(D S) = G/M and G/M a subg g group of the reals imply that D S is a rank one valuation ring and that P is a prime ideal of height one.

REFERENCES i.

Arnold, J. On the ideal theory of the Kronecker function ring and the domain

D(X),

Canad. J. 21, 558-563 (1969). 2.

Birkhoff, G. Lattice-ordered groups, Annals of Math. 43, 298-331 (1942).

3.

Bourbaki, N. Algebra Commutative, Chapitre 7, Herman, Paris, 1965.

4.

Brewer, J° and Mott, J. L. On integral domain of finite character, J. Reine Angew. Math. 241, 34-41 (1970).

5.

Clifford, A. H. Partially ordered abelian groups, Ann. of Math. 41, 465-473 (1940).

206

14

6.

Cohen, I. S. and Kaplansky, I. Rings with a finite number of primes, I. Trans. Amer. Math. Soc. 60 (1946), 468-477.

7.

Conrad, P. F. and McAlister, D. The completion of a lattice-ordered group, J. Australian Hath. Soc. 9, 189-208 (1969).

8.

Conrad, P., Harvey, J., and Holland, C. The Hahn embedding theorem for abelian lattice-ordered groups, Trans. Amer. Math. Soc. 108, 143-169 (1963).

9.

Dedekind, R. Ueber Zerlegungen von Zahlen durch ihre grossten gemeinsamen Teiler, Gesammelte Math. Werke, Vol. II, Chelsea, New York, 103-147 (1969).

i0.

Dieudonn~, J. Sum la Th~orie de la DivisibilitY, Bull. Soc. Math. Prance 49, 1-12 (1941).

ii. 12. 13.

Fuchs, L. Partially ordered algebraic systems, Pergamon Press, New York, 1963. The generalization of the valuation theory, Duke Math. J. 18, 19-26 (1951). Gilmer, R. W. Multiplicative

ideal theory, Queens' Papers, Lecture Notes No. 12,

Queen's University, Kingston, Ontario, 1968. 14.

Heinzer, W. Some remarks on complete integral closure, J. Australian Math. 9, 310-314 (1969).

15.

J-Noetherian integral domains with i in the stable range, Proc. Amer. Math. Soc. 19, 1369-1372 (1968).

16.

Jaffard, P. Contribution ~ la th4orie des groupes ordonn4s, J. Math. Pures Appl. 32, 203-280 (1953).

17.

Extension des groupes r@ticul6s et applications, Publ. Sci. Univ. Alger. Ser. AFI, 197-222 (1954).

18.

Un eontre-exemple

concernant les groupes de divisibilite, C. R. Acad.

Sci. Paris 243, 1264-1268 (1956). 19. 20.

Krull, W. Allgemeine Bewertungetheorie,

J. Reine Angew. Math. 167, 160-196 (1931).

Be itr[ge zur arithmetik kommutativer Integrit[tsbereiche;

I, II, Math.

Z. 41, 545-577; 665-679 (1936). 21.

Lorenzen, P. Abstrakte Begrundung der multiplicativen

Idealtheorie, Math. Z. 45,

533-553 (1939). 22.

MacNeille, H. Partially ordered sets, Trans. Amer. Math. Soc. 42, 416-460 (1937).

23.

Mott, J. L. The convex directed subgroups of a group of divisibility, submitted for publication. 2O7

15

24.

Nagata, M° Local rings, Interscience New York, 1962.

25.

Nakayama, T. On Krull's conjecture concerning completely integrally closed integrity domains, I, II, Proc. Imp. Acad. Tokyo 18, 185-187; 233-236 (1942); III, Proc. Japan Acad. 22, 249-250 (1946).

26.

_

_

Note on lattice-ordered groups, Proc. Imp. Acad. Tokyo 18, 1-4 (1942).

27.

Ohm, J. Some counterexamples related to integral closure in

D[[X]], Trans. Amer.

Math. Soc. 122, 321-333 (1966). 28.

Semi-valuation

and groups of divisibility, Canad. J. Math. 21, 576-591

( 1969 ). 29.

Prefer, H. Untersuchungen ~ber die Teilbarkeitseigenschaften

in K~rpern, J. Reine

Angew. Math. 168, 1-36 (1932). 30.

Ribenboim, P. Conjonction d'ordres dans les groupes abelian ordonn~s, An. Acad. Brasil. Ci. 29, 201-224 (1957).

31.

Un th~or~me de realisation de groupes r~ticul@s, Pacific J. Math. i0, 305-308 (1960).

32.

Sheldon, P. A counter example to a conjecture of Heinzer, submitted for publication.

33.

Ward, W. Residuated lattices, Duke Math. J. 6, 641-651 (1940).

34.

van der Waerden, B. L. Modern Algebra, Vol. II, 2nd English edition, Ungar, New York, 1950.

208

HOMOLOGICAL

DIMENSION AND EULER MAPS*

Jack Ohm, Purdue U n i v e r s i t y and Louisiana

I.

Introduction.

for

~¢

If

is a map

multiplicative

f

~¢¢

is a collection

of ~

on short

f(M)

exact sequences M

for

dimension

fractional

of

and Kaplansky's

direct certain

~

sum.

[SE]).

approach

otherwise

treatment

Roughly,

and a subset

the set of elements

O

in

of

G

is the

is that of MacRae

[M],

group of

to these what

examples.

I have followed [K].

of

~

generated

having

R-modules

free R-modules

(Compare

the paper

[M]

also Bass

[B]

are closed under

extended

finite

~,

under

to an Euler map on

~-resolution.

The

~

and

can be taken to be the set of

and the set of all finitely

respectively, are more

while

the

complicated

_~ and

in §3.

* This research was

My

I feel is a proper

~¢, both of which

it can be uniquely

give the Euler map of MacRae discussed

sum of the ranks of

the idea is to begin with a set of

for the first of the above examples

generated

free resolution,

If then an Euler map can be defined on

conditions

all finitely

if

R.

is in providing

for the exposition;

R-modules

example

is

For example,

into the m u l t i p l i c a t i v e

ideals

only claim to o r i g i n a l i t y

and Swan-Evans

which

M; here the group

Another

I shall give here a unified

of MacRae

~¢.

an Euler map

an Euler map from the set of torsion R-modules of

finite projective invertible

G

which have a finite

in a resolution

group of integers.

who has defined

setting

from

can be taken to be the alternating

the free modules additive

of R-modules,

into an abelian group

-~ is the set of R-modules then

State U n i v e r s i t y

supported by NSF Grant GP-29104A.

209

and

G

which

will be

2.

Dimension.

operation

+.

possibility avoid

Let

required

never

enter

sequences

a collection

~

n m 3, M i e ~ ,

(M n .... ,MI)

~2.

For any

(M n, • .. ,MI) (i.e.

M

the picture;

and any

can be added

For every

M ~,

~4.

(0,M',M,0)

e ~=

~5.

(Factoring

through

(M n .... ,Mi+ I, Ki, ~6. such ~7.

(0,A,B,C,0) that

(0,A',B',C,0) and

An element (Mn,...,MI) (Mi+l,Mi)

of certain

there

is

given

MI) ,

axioms:

for every

1 ~ i < n,

. . . + M, M I + M . Mi_l,

terms

,MI)

of any element

~

of ~ ) .

c ~.

= M. images exists

0) ~ ~

and

e ~

that

successive

there

and

).

i

such

and

e ~=

(0,A,B,C,0)

exists

M e ~

e such

1 ~ i < n,

e ~.

there

(0,K,M',A,0)

that

that

(0, Ki, Mi,...,MI)

(0,K,M,B,0)

e ~

there

For every K i e ~¢i such

of pullbacks).

e ~=

(0,A',M,B,0)

such

.,Mi+2, . .Mi+ .1

M'

an exact

2 ~ j ~ n-i.

(0,0,M,M,0,0)

(0,M',M,C,0)

(Existence

i

to two

~3.

that

down

In p a r t i c u l a r ,

seven

~

to

to write

to define

(Mn,...,M2,

the following

such

= (M n .

e ~

of the form

and one

instead

suppose

to an

However,

the existence

(Mi+j, .... Mi+l,Mi) j

category.

properties.

only

respect

sequence,

and then

we shall

satisfies

and

M e~ e~

(Mn,...,MI)

Thus,

e ~

i = 1 .... ,n - 2

these

with

chosen

theorems

having

of sequences

which

of an exact

we have

for our

matters.

semigroup

in an abelian

considerations,

to be something

morphisms

~i.

the notion

w o u l d be to work

the properties

exact

be an abelian

We need

extraneous

sequence

~

exists

M'

e ~¢

e 5~. 5~ and that

(0,A,M,B',0)

e ~

e ~.

of the set

is exact,

~

we shall

and shall write

Ki=

will

be called

call a

Ki

given by

im (Mi+l,Mi).

210

an exact ~5

sequence.

If

an image

for

2.1 ~i,

Consequences ~3,~4, ii)

of the axioms,

i)

(0,M,0)

¢ ~

~ M = 0,

by

and ~ 5 . For any

M',M

¢ _~f, (0,M',M'

+ M,M,0)

e ~,

by ~ i ,

~2,

and ~ 3 .

Note

that

R-modules

~¢'

of i s o m o r p h i s m

ring R) u n d e r

sum,

by the exact

~:M ÷ B, take

~':B'

is the s e m i g r o u p

(for a fixed

are s a t i s f i e d if

if

~ C,

take

appropriate

M'

sequences

= ~-I(A);

M = {(b,b')

diagrams

9

o

K

K

for ~ 6

direct

if

~ B ~ B'[~(b) and

d~7

the above

in the usual

and for ~ 7 ,

classes

sense.

of

axioms For

~

6,

~:B ÷ C,

= ~'(b')}.

The

are the following:

eT.

?

o

A'

A'

I

0- - ÷M'-_+M__÷C --+0 + + 0 + Aa + B + C + 0

Given P ¢~ any

K,K',L' there

(0,M,L'

+

0

0

an exact

will

exact,

+

sequence

be called c~

+ P,B,0)

(0,A,B,C,0),

projective

such that

exists

0-+A-÷M+ 0 +A+B i

M e Jl

relative

are exact.

that

One

+

+

0

0

A,B,C to

(0,K',L',A,0) such

_÷B'__+0 + + C + 0

c J l,

(0,A,B~C,0) and

have

if for

(0,K,P,C,0)

(0,K',M,K,0)

should

an element

are

and

in m i n d

the f o l l o w i n g

diagram:

o

9

+

0---÷

K'

....

L' 0

+

o

+

M

+

....

L'+P t

A

÷

0

B

that p r o j e c t i v e s s p l i t ,

in the sense

and

P

to

relative

+

C

÷

0

0

that

(0,A,B,P,0)

211

+ 0

P

0

Note

projective

K--

(0,A,B,P,0) implies

exact

B = A + P.

One

sees this by taking M = 0

by 2.1,

relative

K = K' = 0

and hence

to every short

then we shall

Let

O

relation

on

simply

and

L' = A

B = A + P exact

call

P

by

in the above.

~4.

sequence

If

P

(0,A,B,C,0),

Then

is projective A,B,C

e ~,

projective.

be a subsemigroup

of ~d~.

We define

an equivalence

.~" as follows:

For any

MI,M 2 e ~ ,

MI--M 2

if there exist

PI,P2

~ O

such that

M1 + P1 = M2 + P2" The equivalence P ~ O

~*

such that

M.

of

O

M + P ~ 0},

M e ~,

0,

n ~ 0,

We use

finite of

0*

an exact

{M ~ ~

Ithere exists

and one easily verifies

Res(O~)

M, do(M),

sequence

will be called

length}.

of length

If

an

to denote

n;

we define

Proposition.

and

n > 0.

Let

that

exist

P0*,PI*

~O*,

and apply

If

we define

O

M

n

for

G-resolution the

of

G-dimension

has an

G-resolution

an induction

such that

exists

are in

is exact. n > i,

be exact with

P.*I ~ 0 *

PI.* -- Pi,i

= 0,...,n

is exact.

there

P1 = PI* + Q

(K,Pn*,...,P2*,PI,P0,M,0) sequence.

of length

has an

such that

Pie

Proof.

required

(0,Pn,...,P0,M,0),

dG(0 ) = -I.

(K,Pn,...,P0,M,0)

and

{M e ~ I M

n

and such that

P0 = P0* + Q

G-resolution

(K,Pn*,... 'P0*'M'0)

Then there

Since

of the form

M # O e Res(O~),

to be the least

2.2

~5

is then

= 0*.

If Pi¢

class

factor

hypothesis.

212

Q ~ O

0.

If

such that

By J~2,

n = i,

through

this

is the

K 0 = im(P],P 0)

via

2.3

Corollary.

d~(M)

2.4

Res(~,~)

= do,(M)

if

d o , ( M ) # 0,

Proposition.

Suppose

(0'K'Pn - 1 ' "'''P0 'M'0) Pi,Pi ' g G *

Proof.

and

Proceed

(0,K',M',0) n = i:

M + P = M' suffices assume

+ P'.

P0,P0'

¢

and the s p l i t t i n g K + P0'

there

Hence

step,

factor

K 0 ~ K 0'

be an element

(0,Pn*,...,P0*,M,0).

~g

Let

having

The above

shows

via

rather For this

§2 use

only

us to a b b r e v i a t e

~2,

it

Similarly,

we may

of p u l l b a c k s that

K 0 = im(Pi,P 0)

and

n = 1

case,

K ~ K'

of

are p r o j e c t i v e ,

an

O*

than

check

reason, G*

a given all

we shall

to c o m p u t e

our n o t a t i o n

213

to

and then

and let

resolution and let

Then

that w h e n

one need only e x a m i n e

of

allows

corollary

that

lemma)

K 0 = im(P0*,M),

1 m i ~ n

remainder

M = M'.

through

by the

K i = im(Pi*,Pi_l*) ,

one.

such

on a s u m m a n d

the elements

of

and

K -- K'

Suppose

a minimal

g O

Schanuel's

2.5

do,(M),

P,P'

when

and

~a4.

(as w i t h

hypothesis.

compute

by

of p r o j e c t i v e s

by the i n d u c t i o n

projective,

= K'

by adding

= 0.

are exact with

n = 0:(0,K,M,0)

exist

~,(M)

K ~ K'

existence

K 0' = i m ( P l ' , P 0 ' ) .

M ~ 0

n.

then

are p r o j e c t i v e

from the

induction

Corollary.

~

if

It follows

= K' + P0"

For the

on

the p r o p o s i t i o n ~.

of

then

K = M -- M'

Therefore

to p r o v e

0 ~ dG(M ) ~ 1

M ~ M'

by i n d u c t i o n

M ~ M',

and

M g Res(O~/~,

(0,K',P'n-l'" . "'P0 ''M''0) If

imply

If

the elements

and

n ~ 0.

exact

Since

= Res~Y*,~.

d~,(M) =

min{ilK i ~O*}.

the e l e m e n t s

of

~*-resolution ~*-resolutions henceforth

O

are

for

M

and select

in the

dimension,

which

in turn

d(M)

~,(M)

and

for

to

Res

for

R e s ~

remainder

2.6

of

§2 that

Corollary.

then

d(M)

Proof.

Apply

be d e l e t e d

then

Proof.

If

adding

Suppose > 0,

that

(ii)

then

is in

d(M)

n > 0

implies

from

2.6 that

a summand

d(K)

are p r o j e c t i v e .

~ 0

are in Res

the h y p o t h e s i s

c Res, d(M)

Res,

and

that

P ~ O ~,

M'

and

= 1 + d(K);

then there

(0,K,P,M,0)

(0,Pn,...,PI,KI,0)

M -- M',

~ Res

can

(0,K,P,M,0)

while

if

d(M)

is = 0,

to

is exact

= n > 0, for

P

and

and

and let

M.

Factoring

is a m i n i m a l

K ~ 0, K 1 ~ 0. d(K)

= n - i.

P ~ O ~

d(M)

Suppose we may

P = K + M;

and

= 1 + d(K).

(0,Pn,...,P0,M,0) through

K 1 =im(Pi,P0),

O~-resolution

Since

M via ~ 2 ,

of p r o j e c t i v e s ,

exist

K -- K 1 now

for

by 2.4,

d(M)

assume

= 0. P,M ~ O

and hence

K I. it By •

K ~O ~

~ 0.

Let

and factor

(0,Pn,...,P0,M,0) through

is the r e q u i r e d

2.8

M,K

G~-resolution

T h e n by s p l i t t i n g and

for the

2.6.

Suppose

be a m i n i m a l

follows

M'

G

assume

2.5.

in 2.8 that

M ~ 0

such

we have

and

o~

also

~ 0.

(i)

Also,

show

d(M)

d(K)

K ~ Res

M ~ 0

2.4 and

i)

If

ii)

the elements

If

from

Lemma.

exact.

We shall

= d(M').

We shall

2.7

(= R e s ( G ~ ) .

Lemma.

If

be a m i n i m a l

K =im(Pi,P0).

Then

O~-resolution K ~ Res

and

sequence.

M c Res

and

M' ~ M,

214

then

M'

c Res.

for

M,

(0,K,P0,M,0)

Proof.

It s u f f i c e s

M + P g Res.

That

consequence

+ P,M~0)

M = 0

by 2.1,

By

and

~6,

M c Res

of ~ 2 .

(0,P,M

P' e O *

to see that

there

is exact by

such that

projectives,

P

M e Res.

2.9

Theorem

such

that

(dimension

(0,A,B,C,0)

Res,

then the third

d(B)

~ max{d(A),d(C)};

d(C)

=

Proof.

d(A)

+

Case

induction

then

the t h e o r e m

is exact.

If any two of

Assume

d(C).

If

A

is immediate.

so we may

d(C)

follows; e 6"

= 1 + d(K).

element

of

and hence

and

O

to

an

in

B,C,

hypothesis,

and

of

K + P ~ Res and

L e Res

be elements A,B,C

holds,

L e Res,

in Res.

then

of ~

are in

then

and

d(C)

that

= n > -I.

By 2.7(i~,

= 1 + d(K')

O-summand

does

not

Then

P

there

+ P,B,0)

and hence

alter

2, we may

exists

215

and

dimension

assume

L ~ ~

are exact.

B ~ Res,

there

and

2.6 and 2.8, by adding

via ~

Then

(0,K',P',A,0)

d(A)

and

and

and the

that

by

by

A = B

B = C

A # 0.

such

We p r o c e e d

C = 0

A = 0, then

e Res

Res,

(0,L,P'

are

assume

and such

is p r o j e c t i v e .

(0,K',L,K,0)

is exact.

occurs,

then that

If

also

K,K'

Since

of b e i n g

= -i,

B ~ 0.

are exact

the p r o p e r t y

this

C

Assume

theorem

(0,K,P,C,0)

A,B,C

inequality

and

d(C)

also

P,P'

is exact

Let

and if the

exist

Then by s p l i t t i n g

theorem).

C # 0, and hence

exist

+ P,0)

L = K + P; and

When

there

i.

(i):

on

By 2.7,

implies

that

(0,L,P',M,0)

is also.

is an i m m e d i a t e

M + P = 0

are exact.

implies

if and only

M + P e Res.

(0,K,P',M such

(0,L,P',M,0)

But

M + P e Res

M + P # 0.

e ~

projective

M ~ Res

Moreover,

K ~ Res

and

imply

2.1.

assume

L

P e 6,

suppose

so we may

(0,K,L,P,0)

K ~ Res.

implies

Conversely,

exists

since

for

or

on an P e 6

such that

By the i n d u c t i o n

and also

if

d(L) If

~ max{d(K'),d(K)} d(B)

> 0,

dimensions

then by

of

K',L,K

respectively, A,B,C.

If

2.7(i)

d(B)

Then exact such M

and

to

d(B)

that

and

A

there

implies

N'

By 2.4, implies

f

of

for any short f(A)f(C) sequence

J

e Res

N ~ N';

such

and hence

A e Res

by case

be a subset

into

a (multiplicative)

A simple

of

~.

as

and

B # 0.

exists Case

N e~

(i), and then

(iii):

C e Res

implies

(0,N',Q',C,0) by

2.8.

Then

is M

M i e ~,n

for

group

G

with

A,B,C

~ ~,

shows ~ 3,

An Euler map

that

such

Let a)

O i c Oi+ 1 If

be subsets

(0,A,B,C,0)

of

is exact

one has

~

such

that

and

A,C

are

in

Gi,

then

B c ~. l

b)

For any

M ~ ~i+l'

there

and

exist

216

~

is that

then for any exact

f(M1) = f ( M 2 ) f ( M 3 ) - l f ( M 4 ) . . .

3.1

is

(ii):

abelian

(0,A,B,C,0)

induction

(0,Mn,...,MI,0),

We can

(ii).

~

sequence

implies

(0,M,Q,B,0)

there

that

N ~ Res

Let

exact

= f(B).

Q' ~ G *

to

from

e Res

B e Res

by case Case

down

(i).

are exact.

N e Res

the

follows

A,B

such that

is exact.

and

carries

A e Res.

so assume

+ i.

A,B,C

case

(ii))

implies

(0,M,N,A,0)

(0,N,Q,C,0)

Euler maps.

a map

Res

(C~e

and hence by ~ 6

are

of

thus

concludes

M e Res

and

and then

than those

that

B = 0, and

= d(K')

and the t h e o r e m

B,C e Res

Q e O*

d(K)

+ i;

K',L,K

This

to e s t a b l i s h

(iii))

in

less

B e ~*

(0,A,B,C,0).

implies ~ d(L)

for

(0,N,Q,C,0)

exist

N e Res

then

= 1 + d(M);

since

exact.

d(B)

are all one

of the case

exist

and

C e Res

3.

(C~e

dispose

there

holds

2.7(i),

= 0,

It only remains

before


h .

N = M'R'M,OA h

in

Assume

If there exists

(2)

>

x E R' M,

A = R[x], h = height MA + i < height

P = [M' ; M' (I)

N

as in (b),

(2.6), but (2.7) and (2.8) together lend

PROPOSITION.

such that, w i t h

and

R'

support to the chain conjecture.

(2.8)

then

in

(c) is accounted for by the fact that

if the chain conjecture holds,

indicates how to prove

M"

N

such that

there are always only finitely many maximal

let

ideal

height M " = height M A + i , and, for each

there exists a maximal

fore,

then, for all such

Then

is a maximal

M' E ~

x

such that

o__Er I/x

ideal in

is in

A

R' M,

R' M,

such that

is an H-domain

(say, x) MA ~ N

and

and

height

.

(3) Then,

Assume every

for each maximal

N > h , there exists Moreover,

M' E P ideal M' E P

is such that N

in

A

such that

there exists a maximal

ideal

R' M,

is an H-domain.

such that

MAc

N

x E R' M,

and

N = M'R'M,NA

M"

in

R'

and height

such that

i

height

M" = h . Before proving

(2.8),

it should be noted that, since

an H-domain s there exist such is a quotient ring of

R[x]

x E F and of

(2.7). R[i/x]

Further,

N ~ (M,x)A

R[i/x]

such that

Proof.

and the maximal MR[l/x] c N*

ideals and

(I) follows from (2.5).

228

since

isn't R[x,i/x]

, it is clear that there

exists a one-to-one correspondence between the ideals that

R

N

in (3) such

N* ~ (M,i/x)R[I/x]

in

height N* > h . For (2), if neither

x

nor

i/x

is in

R' M,

, then

Corollary,

p.

thesis

(2.5).

and

diction

R'[x] , and For

and

M'

MA

A

height

height

M'B

height

M'

maximal

ideal

PnA = MA height

N > height

ideal

M"

height

, and

height

is in

integral A

R' M,

,

dependence

, NnR = M

or

(so

MA

that

MAc

let

B

~ BN,

which

.

lies over

and

by

height

N'

i/x

But

is in

R'M,.

i/x 6 N'B N,

Finally,

depth

so there

is a

P = M"R'[x]

Hence,

, for,

,

R' ~ B ~ R' M,

= N .

ideal,

with

or

so

, so

, and so

, and so

, and

such

R' M,

height

x

N

M' = N'NR' is in

in

Hence

MAc

B = R'[x]

I/x

ideal

one p r i m e

P = height

in

N , and

M'R'M,nA

such that,

such

ideal

x E R' M,

Therefore

A

inequality)

~ R'M,

Hence

R'

R ' [ x ] / M '~=

I/x

M'B ~ height MA

MA + 1 .

is a depth in

by in

the contra-

, since

or

in

x

one prime

R'[i/x]

= N' MA

, depth P = I,

(2.5),

height

M" =

MA + 1 = h , q.e.d.

3. Because interest

SOME CHARACTERIZATIONS OF H - D O M A I N S

of the e q u i v a l e n c e

to study

indicated

The fixed

H-domains.

in (4.1)

characterizations

this

ideal

Then

- 1 = (2.5)

, then

x

[i0,

~ h , by hypo-

implies

= MA

Then,

M' ~ (by the a l t i t u d e

M'R'M,~B implies

.

N' = h e i g h t R'

is a c o n t r a d i c t i o n .

MA > 0

(M'*NA

be a m a x i m a l

is a depth

N' = h e i g h t

implies

are

in

M'R'M,[X]

ideal

M' > h .

N'

height

one prime

dependence,

Therefore

be a m a x i m a l

ideal

i/x 6 M'R' M,

which

N

and

MA + 1 ~ h e i g h t

If

N ~ height

is a m a x i m a l

, so

integral

is a m a x i m a l

N > h , let

then

M'* = height

N = M'R'M,~A

, N

let

N'NA = N

if not,

by

is a d e p t h

M,* ~ height MA

Let

(3),

height

A/(M'*NA))

over

height

that

and This,

over

x 6 R' M,

of N)

20],

h ~ height

is integral say

M ' * = M'R'[x]

and

of (i) and Some

(4.5)

of an H - d o m a i n

notation

of

further

below.

(2)

in (2.4),

reasons

For

these

to study

H-domains

reasons,

some

will

be g i v e n

in this

(2.3) w i l l

continue

to be used

section.

229

it is of some

section. throughout

The m a i n it lends R

reason

support

is c a t e n a r y ,

(3.1)

for

the f o l l o w i n g

to the H - c o n j e c t u r e then

R

THEOREM.

a[xl~[x]

including

Then

is c a t e n a r y

Let

R

(4.1),

X

theorem

since

it is k n o w n

[7, T h e o r e m

that

if

4.111.

be an i n d e t e r m i n a t e ,

is an H - d o m a i n

is b e c a u s e

if and only

and let if

R

R

=

is an

H- doma i n . Proof.

Assume

height

one p r i m e

p -- 1

and

first

ideal

x = X modulo

S = R[r2x]

NS[x]

r E M

such

If

(2.5).

is the q u o t i e n t Let

are a n a l y t i c a l l y NS[x]

depth

that

p * = altitude If

rx

= a-i

over

is i n t e g r a l

over

R

is i n t e g r a l

over

R

, so

elements Thus

, then

Thus

R[x]

in

depth p

a > 1

is a non-

and w h o s e S

so

is an HAlso,

r 2 )r 2 X

[6, L e m m a = a-i

and

, hence

N = a .

ideal,

S

be a height

there

Hence

height

p

, then

R .

is a d e p t h one p r i m e

(2.5)°

let

R* /pR * = a l t i t u d e

p nR = (0)

field of

N = (MR*/p*)NS

independent

and

p = p nR # (0)

domain which

= MR[X]/(p"NR[X])

height

R

is an H - d o m a i n

is a l g e b r a i c

is a local

field

R

, as desired.

(p*nR[X])

zero e l e m e n t

domain

in

p * = p R * , hence

R/p = d e p t h p = a-i

quotient

that

4.3],

, and so

hence

R

is an

H- domain. Conversely, prime so

ideal

in

let R °

R

be an H - d o m a i n

Then

pR

is a h e i g h t

d e p t h p = (as in the first p a r t

hence

R The

is an H-domain, following

(3.2) terminate. (i)

Then R

is a h e i g h t (2)

R'

(3)

P

proof.

p

be a h e i g h t

one p r i m e

ideal

in

depth pR*

one R

in S e c t i o n

= a-I

4.

Le____tt P = R [ X ] ( M , X ) R [ X ] , w h e r e

is an H - d o m a i n ideal

and in

statements R (I) c_ R' R]

X

is an inde-

are e q u i v a l e n t : , where

R (I) = N[Rp

; p

R (I) c R'

if

.

is an H-domain. is an H-domain. It is k n o w n

,

q.e.d.

the f o l l o w i n g

one p r i m e

let

of this proof)

t h e o r e m w i l l be used

THEOREM.

and

[6, C o r o l l a r y

230

5.7(1)]

that

,

I0 and

only

p'NR

if,

= i

for each height Therefore,

dependence),

(i)

Assume If p

(2) holds

= depth p+l

x = X modulo

L = a .

domain

the q u o t i e n t (3.1)).

Let

S'

with

and

c

b

as in the p r o o f integral cS':bS' since ideal

altitude

R[x]

= S[x]

R[x]

p'NR

that = i

is k n o w n mal

R'

and

Now

[4,

Hence,

R

is a h e i g h t

of

S'

(M,x)R[x]

that

ideal

a height

x E p'

, altitude

such

that

one m a x i m a l

l-x

R[X](M,x)R[x]

isn't an H-domain.

Hence

is in all = i < a .

height

in

= a-i

,

(3) ~olds. as in the last

then

is a h e i g h t p

in

in

this

p ' N R = I , and

Then

R

such R

is an

Therefore,

implies so (3)

ideals

that

in

P

implies

(i),

q.e.d. (3.3)

COROLLARY.

Let

R

be an H-domain

23I

such

it

one maxi-

since

R'

to height

p'NR > i .

the other m a x i m a l But

(by

= depth

xR[x]

R'

Hence,

ideal

of

(11.13)],

, it suffices

ideal

d e p t h p = depth p' = depth p ' n R < a-I is a height

[4,

xS'[x]

follows

p'

one prime

and

x = c/b

Hence

height

that either

R

is the only m a x i m a l

R (I) ~ R'

Suppose

that

is an H - d o m a i n

L = a .

one prime

, where

over

(by

depth

R

is a local

Since

depth

P .

implies

(as in the p r o o f

S .

hence

in depth

to show

is integral

is an H - d o m a i n

To p r o v e

(2)

R , there

the same

x , altitude

ideal

, hence

Theorem,

Therefore,

since

then that

= pP

it remains

implies

(10.14)]),

2.11]

p'

, height

(by integral

m S'/(cS':bS')

(2)

S' - I = a-I

exists

p

S

closure

S, S'[X]/ES'[x]

[7, P r o p o s i t i o n or there

R'

one prime

over

that are

the integral

(3.1). p'

L

5.7(1)].

H-domain, with

if

such

[6, C o r o l l a r y

ideal

that

of

in

L = R[X](M,x)R[x]

and

is a l g e b r a i c

containing

(3) holds,

of the p r o o f

x

S

and

and

then let

L = P/p*

of (2.7)).

=

If

prove

in

dependence

in

, so

of

be

p = i

by the L y i n g - O v e r

~ R[x] ~ L

fields

be a h e i g h t

p NR = ~ ) ,

Since

S = R[r2x]

p

height

(since,

(p*NR[X])

p'

depth p' = depth p ' N R

and let

If

ideal

(2).

, then

= a

is an H-domain).

altitude

since

implies

p = p AR # (0)

one prime

that

R (I) c R'

II and let

R n = R[XI,...,X n]

pendent

over

R .

PNR = M, Rnp Proof. (3.1)

Then,

AQ

The proof

that

remainder

Rip

(2) If

such that prime

H-domain,

over

n .

R1

R)

Hence,

(i) If

R

in

Rn

If

such that

and

Using

n = 1 , then by

P = (M,f)R 1 , for

is integral

since

over

A = R[f]

Q = PnA = (M,f)A

, so

QR 1 = P , it follows the case for

from

n = 1 , the

q.e.d.

is an H-domain,

one maximal

is an H-domain has height

p' c M'

ideal

in

a .

and

R (I) _c R'

For,

if

height p ' N R > 1 . R

then either R'

R (I) c R'

(as in the proof

is a maximal

there exists

depth p' < a-i

But this and

ideal

a height

Since

R

[7, Proposition

AND THE CATENARY

(3.2) will be used

two conjectures

The notation

, then every maximal in

R'

one

is an

2.11]

imply

isn't an H-domain.

ON THE H-CONJECTURE

a new formulation

M'

(by (i))

such that

In this section, between

P

inde-

(i) in (3.2)).

the contradiction

4.

Then

height M' ~ a , then

ideal

on

is straightforward,

a height

R

R'

ideal

MR 1 c P , hence

is an H-domain.

(3) implies

in

(3.2).

REMARK.

or there exists

ideal

that

f 6 P .

of the proof

(3.4)

that

is by induction

is transcendental

is an H-domain

(2.5)

are alsebraically

is an H-domain.

some monic polynomial f

XI,...,X n

for each prime

it may be assumed

(hence

, where

CHAIN CONJECTURE

to show an implication

w h i c h have appeared

in the literature.

of one of these conjectures of (2.3) will continue

Then,

will be given.

to be used

throu~hout

this

section. (4.1)

H-CONJECTURE:

The converse

(4.2) satisfies

If

R

is an H-domain,

of the conjecture

CATENARY

clearly

then

R

is catenary.

holds.

CHAIN CONJECTURE:

If

R

is catenary,

then

R'

is equivalent

If

R

is catenary,

then

R'

the c.c.

This conjecture

to:

232

12 is catenary.

Some other equivalences

are given

in the following

theorem. (4.3) ments

THEOREM.

If

R

is catenary,

then the following

state-

are equivalent:

(i)

R'

(2)

Every

satisfies

the c.c.

integral

domain w h i c h

is a finite R-algebra

is

catenary. (3)

For each height one prime

ideal

p

in

R, R/p

satisfies

the s.c.c. Proof. assumed

If

that

Assume R-algebra~

a = I , then the theorem

(i) holds, and let

let

N

it is known

that

D

is integral

height N = i

in

prime

ideal

Then

depth P' = i

P' c Q , then

D

R = R/p

R-algebra. prove

~

Hence

(2) holds, , and let To prove

satisfies

an integral

domain

, for some height is catenary, and

R

A

B = AN

.

To prove

, and let

is catenary, P'nB = P

Q

and

, hence

by

P

so

be a depth

D = C(A~N ) (i).

ideal

Let

Then

P'

be a

P' = height P . in

D

such that

height Q = height QnC = a

height P = height P' = a-i

A

For this,

3.1(1)], Let

height

is a maximal

that

is catenary.

height N = a > I .

D

if

in

is a finite

[7, Propo-

, and so

B

is

(2.2).

Assume let

and

and,

that

C = R'[A]

such that

QnR = M

sition 3.1(1)]. catenary

B

ideal

domain w h i c h

or a [7, Proposition

that

B , let

over

in

be an integral

to prove

it may be assumed ideal

A

be a maximal

it suffices

one prime

so it will be

a > i .

is catenary

clearly

is easy,

and

let ~

that

p

be a height

be an integral R

satisfies

the f.c.c. A

which

it follows

are catenary)

ideal

from

p

the s.c.c,

A .

233

ideal

R ,

is a finite

For this,

R-algebra in

in

it suffices

3.11].

[7, Proposition

that every maximal

ideal

domain which

[5, Theorem

is a finite

one prime

one prime

to

there is

such that

A/p ~ =

Then

(2) implies

3.1(1)]

(since

in

~

R, A,

has height a-l.

13 Hence

(3) holds. Assume

one maximal

(3) holds, and assume, ideals.

at first,

Then it is known that

[5, Proposition 3.3], and so

R'

R

satisfies

remains to consider the case w h e n

R'

that

R'

satisfies

the s.c.c.

is a catena~y

the s.c.c. Thus it

has a height one maximal

In this case, Lemma 4.4 below shows that, for some R[x,i/x]

has no height

x E R', L =

local domain whose integral closure

R'[i/x]

has no height one maximal

ideals and the only prime ideals in

w h i c h blow up in

are the height one maximal

clearly

R'

R'[i/x]

satisfies

the c.c.,

and, as already noted, for

L

p"nR[x] 3.1(1)],

So, let , and let so

R/p

dependence,

R'[I/x]

p"

R'[i/x]

satisfies

satisfies

.

Then

satisfies

the s.c.c.,

height p = 1

the s.c.c.

satisfies

, since, by the proof of (4.4),

ideals

N 1 = (M,x)

and

N 2 = (M,l-x)

Another equivalence

Thus,

the s.c.c.; if (3) holds L , let

p' =

[7, Proposition

(by (3)), hence, by integral

the s.c.c.

R[x]/p'

R'

ideals.

be a height one prime ideal in

p = p'NR

R[x]/p'

if

ideal.

Finally,

R[x] and

L/p" =

has only two maximal

p' c N2, ~ N 1 , q.e.d.

of the catenary chain conjecture will be

given in (4.7) below. (4.4)

LEMMA.

Assume

R

i__sscatenary and

b e an integral domain w h i c h is integral over maximal

ideal

an element

N

i__nn A

x E R' R[x,I/x]

(2)

Every maximal

(3)

The prime ideals in

Proof.

ideal in

R'

height N < a , then there exists

ideal in

A[x,I/x]

R'

has height a ; and,

w h i c h blow up in

R'[I/x]

are the

height N = 1

[7,

ideals.

If such

N Let

exist,

then necessarily

B = R'[A]

Q'nA = N , and let and

If there is a

is a catenary local domain;

Proposition 3.1(1)]. such that

R

A

such that:

(I)

height one maxim~l

B

such that

a > 1 , and let

height Q = 1

, let

Q'

Q = Q'nR' [4, (10.14)].

234

be a maximal Then

Q

So, let

ideal in

is a maximal x

be an

14

element w h i c h is in a prime ideal in a height one maximal maximal

ideals in

ideal, and let

R'

such that

it may be assumed that maximal

and

local domain.

Also,

holds), and Thus,

y

Then

B[i/x]

is a unit in

and

height N 2 = a , so

R'

is integral over

N 2 = (M,I-x)R 1 , and

L = R[x,i/x]

has no height one maximal

R'[i/x] x

Clearly

has exactly two

= RIN 2

is the integral closure of

it follows from the choice of

A[x,i/x]

be an element in all other

R 1 = R[x]

N 1 = (M,x)R 1

R'[I/x]

if and only if the ideal is

x+y = u

u = 1 .

ideals, namely,

height N 1 = 1

R'

and

ideals.

L

and over

[4, (10.14)] Therefore,

is a (so (3)

A[x,I/x] that if (i) holds,

then (2) follows from [7, Proposition 3.1(1)]. To prove that

L

is catenary,

a local domain w i t h maximal R[x]

Let

and let

P'

ideal

Then

Further,

C

in

RI)

, hence

Hence,

since

C

THEOREM.

is catenary L

is

R ~ C c R1 =

L , let

(since

C

J

P = P'nR 1 ,

is the conductor

height P' = height P =

depth P' = depth P (= height N2/P ) , and

P' = height p = a-i , and so (4.5)

J = NINN 2 , and

Lp, = Rip = Cp

of the integral closure of

P = depth p .

C = R + (NInN 2) , so

be a depth one prime ideal in

p = PnC .

height p .

let

depth

[7, Theorem 3.2], height

is catenary

(2.2), q.e.d.

If the H-conjecture holds,

then the catenary

chain conjecture holds. Proof. satisfies M'

in

Let

the c.c.

R', R' M,

this holds if every in

M'

R'

R

be a catena~y it suffices

satisfies

height

and

R

height

Let

p'

p'nR > 1 , then

[7, Proposition 2.11], tion, so

the s.c.c.

a > 1

R

isn't an

p'nR = 1 , hence

is an H-domain.

Therefore

by (3.2), hence, by hypothesis,

To prove that

to prove that, for each maximal [7, Remark 2.23(iv)].

height M' = 1 , so by (4.4)

has height If

local domain.

P

ideal Clearly

it may be assumed that

be a height one prime ideal depth p' < a-i , hence, by

H-domain. R (I) ~

R'

This is a contradic-

[6,

Corollary

P = R[X](M,X)R[X ] is catenary.

235

R'

5.7(1)]

is an H-domain,

But it is known

15 [6, Theorem

2.21]

that

the s.c.c.

Hence

R'

(4.6) grally

P

satisfies

COROLLARY.

the s.c.c.,

local domain

This follows

the s.c.c,

(if and)

if and only

from

holds,

satisfies

(4.5),

only

if

R

satisfies

q.e.d.

If the H-conjecture

closed catena~y

Proof.

is catenary

then every

the s.c.c.

since a local domain

if it satisfies

inte-

the c.c.

satisfies

[7, Remark

2.23(v)],

q.e.d. (4.7)

indicates

conjecture

another

approach

to proving

the catenary

chain

w h i c h might prove more accessible.

(4.7)

THEOREM.

The catenary

if the following

condition

is such that

A

has exactly

N 2 = (M,l-x)

and

chain conjecture

holds:

If

R

if and only

is catena~x and

two maximal

NInN 2 = M

holds

ideals

is the maximal

A = R[x]

N I = (M,x)

ideal

in

and

R , then

A

is catenary. Proof. dition,

so assume

altitude assumed

the catenary

that

R

that

a > i .

R

the s.c.c.,

it suffices

is a finite

implies

and the condition

satisfies

By (4.3),

domain w h i c h

chain conjecture

is catenary

R = a = i , then

an integral B

By (4.3)

integral

holds.

hence

to prove

the con-

it may be

that if

extension

If

of

B

is

R , then

is catenary. For this,

if

B

so it may be assumed height (4.4)

one maximal

Then Let

that

ideal

it may be assumed Let

l-x

is local,

N

and

R I = R + (NInN2) Therefore,

B(ANNI)

= BN

B

is catena~y

isn't local. B , then

BN

that every maximal ideal

in

the other maximal

N I = (M,x)A

is catenary,

in

be a maximal

is in all

3.2].

B

then

ideals

RI

is integral

A

to prove over

ideal

B .

in

B

x E N Let

B(ANN2)

by a .

such that A = R[x] ideals

in

A

.

[7, Theorem

so to prove is catenary

AN1 , h e n c e i s c a t e n a r y

236

is a

Hence,

local domain

is catena~y,

that

N

3.2],

has height

are the maximal

is a catenary

by hypothesis,

it suffices

in

if

is catenary.

B , and let

N 2 = (M,I-x)A , so

Clearly,

[7, Theorem

that (since

[7, Theorem

B

16 3.2]).

Since

AN2

and

B(ANN2)

satisfy

the conditions

B , it follows by finitely many repetitions

that

B

on

R

and

is catenary,

q.e.d. Call the type of extension tion of special extensions

could prove rewarding,

might give the needed results R

is.

In the following

properties

REMARK. and let

the following

statements

is catenary whenever

If

I

be a special extension of

A/I

A , then either

Rp = Ap

R/(InR)

(if (but

an integral domain).

in

.

Then

A/I = R/(InR)

is a special extension of

There is a one-to-one p # M

R .

y E A, ~ R, A = R[y]

is an ideal in , or

catenary)

hold:

(2)

not necessarily

R

correspondence

and the prime

Moreover,

ideals

for all prime

between the prime

P ~ [NI,N2]

ideals

Q

in

in

A

given

A, QAQ = qAQ ,

q = QNR . (4)

over

in particular,

be a (not necessarily

A = R[x]

For each element

where

A

and,

a proof of some of the more evident

(R,M)

(i)

I ~ NInN 2 = M)

by

remark,

Let

local domain,

ideals

to show that

An investiga-

of special extensions will be given.

(4.8)

(3)

ring in (4.7) special.

R (5)

It follows from (2) and (3) that

is locally unramified

(see [4, pp. 144-145]). If

p # M

In fact,

if

either:

(a) qA = q~

N I) ; or, (b) that

A

q

is a prime ideal in

is a p-primary

(c) qA = q*nN 2

qANI = NIANI

similar comment holds Proof.

(if

q

ideal in c NInN2)

(if

, hence

qi

; (b) qA = q &N I

is semi-prime. q

= qRpnA

(if

q

It follows

isn't p-primary,

for

,

c N2,

in case

i ~ 2

A

in case (c).

(i) follows from

•

q

pA

R , then, with

q * c NI, ~ N2)

A = R + xR

from (i) and an easy computation, For (5), if

R , then

M = R:A , (2) follows

and (3) follows

*

~ N I , then

and

*

*

from

M = R:A . *

q nN 2 = q nM = q NR = q ~ qA ~ q NR .

237

17

It follows that either q = qA = q

(if

q

q = qA = q NN 2

~ N1NN2)

(if

q

~ NI, ~_ N 2)

or

, q.e.d.

BIBLIOGRAPHY

[z]

I. Kaplansky, Adjacent prime ideals, J. Algebra 20(197~, 94-97.

[2]

W. Kmull, Beitr~ge zur Arithmetik kommutativer Integrit~tsbereiche. III Zum Dimensionsbegriff der Idealtheorie, Math. Zeit. 42(1937), 745-766.

[3]

M. Nagata, On the chain problem of prime ideals, Nagoya Math. ~. 10(1956), 51-64.

[4]

M. Nagata, Local Rings, Interscience Tracts 13, Interscience, New York, 1962.

[5]

L. J. Ratliff, Jr., On quasi-unmixed local domains, the altitude formula, and the chain condition for prime ideals, (I), Amer. J. Math. 91(1969), 508-528.

[63

L. J. Ratliff, Jr., On quasi-unmixed local domains, the altitude formula, and the chain condition for prime ideals,

(II),

Amer. J. Math. 92(1970), 99-144o

[7]

L. J. Ratliff, Jr., Characterizations of catenary rings, Amer. J. Math. 93(1971), 1070-1108.

[8]

L. J. Ratliff, Jr., Catenary rings and the altitude formula, .Amer. J. Math., (forthcoming).

[9]

J. Sally, Failure of the saturated chain condition in an integrally closed domain, Abstract 70T-A72, Notices Amer. Math. Soc. 17(1970), 560.

[103

O. Zariski and P. Samuel, Commutative Algebra, Vol. II, Van Nostrand, New York, 1960

238

ON THE NT~]3~ OF GENERATORS OF IDEALS OF DIF~NSION ZERO Judith Sally Rutgers University

ABSTRACT. This note contains generalizations of two results of Abhyankar [!] which give a bound for the embedding dimension of certain local rings in terms of the mu!tiolicity and the dimension of the ring.

Let R be a Noetherian ring. mension of R/I is O.

An ideal I has dimension 0 if the (Krull) di-

The multiplicity of an ideal I of dimension 0 is denoted /~(I);

the length of an R-module A is denoted A(A).

If (R, M) is a local ring and I is any

ideal, v(I) denotes the number of elements in a minimal basis of I, i.e., v(I) is the 4imension of I/IM as a vector space over R/M.

When I = M, /~(M) = / ~ R )

and v(M)

is the embedding dimension of R. Theorem i. ideal.

Let (R, M) be a d-dimensional local Macaulay ring.

Then, v(i) .< d + ~ ( I )

Proof. = ~(R). d ,0.

Let I be an M-primary

We may assume I / (O).

-

~(R/I).

The proof is by induction on d.

Since ~(R)~/ X(R/I) + k(I/IM), we have that

If d = O, then/%(I)

v(I).< /~(I) - ~(R/I).

By passing to R(X), .~q(X) and IR(X) as in Nagata [3, p.18, p.71] , we may

assume that R/M is an infinite field.

By Theorem 22.3 in Nagata [3], there is a non-

zero divisor x in I but not in IM such that x is superficial for I. rin~ R/xR we have, hy induction,

v(I) = v(I/xP) + 1.

In the Macaulay

v(I/xR) ~< d - 1 +/~I/xR) - A((R/xR)/(I/x_R)).

Since x is superficial and a non-zero divisor,/~(I/xR) =/~(I).

Remark.

Assume

Since x is not in IM

Thus, v(I).~ d +/~(I) - ~(R/I).

Abhyankar's proof in [i] for the case I = M can be modified to give a proof

of Theorem I.

However, since his proof uses a fact about systems of parameters in a

239

2 local ~ c a u l a y ring which can be derived from Theorem i, a different proof is given here.

The bound obtained in Theorem i is best possible. Dles of local Macaulay rings (R, M) with v(M) = d + ~ ( M )

The following are exam-

- i, i.e., emdim(R) = d +

R(R) - I, and of an M-primary ideal I of R with v(1) = d +/K(1) - k(R/I). a field.

L e t / ~ b e any integer ~ I. 0 For d = O, take R a = K[[Xl, .... x _l]]/(x I . . . .

x~_ 1

Let K be

,X~_l)2 , where Xl, . . . .

are indeterminates, and take I to be any ideal. For d = i, take R aI = K[[x~,x~+l,

,x2~-l]], where x is an indeterminate,

snd take I = (x~,xa+l, . . ,x2~-2). For d • l ,

d 1 take R~ = R~[[tl, ... ,td_l]], where t I ....

,td_ 1

are indeter-

minates, and take I = (x~,x~÷I, ... ,x2a-2,tl , ... ,td_l).

In order to give a global version of Theorem I we need the following definition.

An ideal I of a ring R is rank unmixed if all primes minimal over I have

the same rank (i.e., height). Corollary.

Let R be a d-dimensional Macaulay ring.

Let I be a non-zero ideal of

dimension 0 and rank r.

If I is rank unmixed, then I can be generated by max(d + I,

r + ~ ( I ) - I) elements.

If R is semi-local, max(l, r +/~(I) - i) elements are enough.

Proof.

We will show that I can be !ocallv generated by r +/~(I) - i elements.

desired conclusion will then follow from the Forster - Swan Theorem [h]. Pk he the primes minimal over I P / Pl' "'" 'Pk"

The

Let PI' "'''

Since i has dimension O, IRp = Rp for all primes

By Theorem I, v(IRp.) ~ r +~(IRp.) - i, for i, ... ,k. k l l

is rank unmixed,/~(I) = i~=l~(IRp.=.). 1

Thus we have that v(IRp. )~ l

i = I, ... ,k.

240

Since I

r +/~(I) - I, for

3 Abhyankar also proves in [i] that if (R, M) is a d-dimensional local ring such that R/M is algebraically closed and A, the associated graded ring of R, is a domain, then emdim(R) & d +~(R)

- 1.

Theorem 2 below extends this result to M-pri-

mary ideals Theorem 2. closed.

Let (R, M) be a d-dimensional local ring such that R/M is algebraically

Let I be an M-primary ideal.

If the graded ring G = R/M + I/IM + I2/I2M +...

is a domain, then v(I) 4

Proof.

Assume I ~ (0).

~In/InM). p.23~]).

d +21(I)

- 1.

Let H(G, n) denote the polynomial which, for large n, gives

H(G, n) is of degree t - i, where t = dim G

(cf. Zariski and Samuel [5,

Let a/(t - I)I be the leading coefficient of H(G, n).

Since R/M is alge-

braically closed, we have as in Abhyankar [I], that a + t - 1 ~dim[I/IM

(also cf. Abhyankar [2, (12.3.5)]). t = d and that2~(I) ~ a.

: R/M] = v(I),

To prove the theorem it is enough to show that

Let B - R/I + I/I 2 + ....

B has dimension d.

B/(M/I)B, so that (~'~I)B is a prime homogeneous ideal of B.

But, since ~

some positive integer j, we have that ( ~ I ) B is the nilradical of B. dim G = dim B/(M/I)B = dim B = d. n, gives A(In/In+l).

Now G = I for

Thus, t =

Let H(B, n) denote the polynomial which, for large

H(B, n) has degree d - 1 and leading coefficient~(I)/(d

Now H(B, n) ~ H(G, n) because

- 1)I

l(In/I n+l) = l(In/MI n) + ~(MIn/I n+l) and, since both

polynomials have the same degree, we have that~t(I) ~ a .

It is interesting to note that although the bound obtained in Theorem 2 is, in general, larger than the one in Theorem i, it is best possible.

Let (R, M) be the

local, non-Macaulay domain of Nagata's Example 2 [3, 0.203-205] for the case m = O, r = 1.

(We may take K algebraically closed.)

241

Let xR' and N be the maximal ideals of

R', the integral closure of R.

Then M = xR'~ N = xN.

isfy the hypotheses of Theorem 2. v(M) =

d +~(M)

- 1 and v(I) =

Let I = x2N.

R, M and I sat-

v(M) = v(I) = 2; d = 2;/~(M) =/~(I) = I. d +~(I)

Hence,

- i.

REFERENCES 1

Abhyankar, S. S., "Local rings of high embedding dimension", Amer. J. Math 89

(1967), lO73-1077. 2

Abhyankar, S. S., Resolutions of Singularities of Embedded Al~ebraic Surfaces, Academic Press, New York, 1966.

3

Nagata, M., Local Rings, Interscience, New York, 1962.

h

Swan, N. G., "The number of generators of a module", Math. Zeit. 102 (1967), 318-322.

5

Zariski, O. and Samuel, P., Commutative Algebra, vol. II, D. Van Nostrand, New York, 1960

242

RINGS OF GLOBAL DIMENSION

TWO

by Wolmer V. Vasconcelos Department

of Mathematics,

New Brunswick,

Rutgers University

New Jersey

08903

This paper gives a change of rings theorem for homological dimensions which seems especially suited for the study of the commutative rings of global dimension two. Elsewhere we gave a description of the local rings of global dimension two that when applied to the global case yields an idea of what the spectrum of such rings look like "down" from a maximal ideal. Here an attempt is made to describe the spectrum "up" from a minimal prime ideal.

I. CHANGE OF RINGS As a general tive.

Bourbaki

elementary

proviso,

[3]

IN DIMENSION

all rings considered

will be the basic

source

TWO here will be commuta-

for terminology

and

notions.

I.Kaplansky but effective

in [12] proved

the following

result which

is a simple

dimensions

of rings and

tool in the study of homologieal

modules. THEOREM x

i.i.

a central

Let

(Change o f rings element

B = A/(x)

tive dimension

in dimension

of A which

is neither

(proj dim

for short)

With A commutative

supported

over B. If the

of E over B is finite,

= 1 + proj

by National

where

Science

243

divisor.

projecthen

dim B(E).

this result was generalized

[7,10] to the case of B = A/I

Partially

Let A be any ring and

a unit nor a nonzero

and let E be a nonzero module

proj dim A(E)

Jensen

one)

by Cohen and

I is a faithfully

Foundation

projee-

Grant GP-19995.

2 rive ideal of A. Observe Vasconoelos proved

[14]

recently

that

finitely

generated.

that projectivity

local property

for the Zariski

remark

that,

implies

such ideals

change of rings In general,

to

On the other hand~

(and thus projective

topology

in dimension

with

are according

one~

of Spec(A).

Gruson

[8]

dimension)is

In particular

it is enough to consider

a

this

the

I = (x).

if E is a module over the ring

B = A/I one always

has (2)

proj dim A ( E ) ! proj dim B(E)

This follows

for instance

from the spectral

C may be any A-module Now we will consider

be a finitely not always

generated

(cf.

[5, XVI.5]).

the dimension

where x,y

any module.

in (2) holds.

local ring of dimension

form a regular

E is a finitely

generated

In fact~

A-sequence~ B-module

p.82])

and thus we cannot have always situation

from sitting skirted

dimension

morphism.

In fact~

that its grade

ideals

overdens~.

THEOREM

1.2.

summand

(Change o_ff rings

one and let B = A/I.

holds

in (2) if

but not for

n-l,

its fini-

to Gruson

(cf.

[8.

in (2). Thus we must

>HOmA(I~A)

reasons,

I

are

be an iso-

and if A is noetherian

this

For brevity we call such

two)

generated

Let A be a commutative ideal of projective

If E is a nonzero

244

It is

I = x(x~y)

more elementary

A

in dimension

finitely

one.

of A. Both conditions

map

is at least two.

let I be an overdense

dimension

equality

such ideal is faithful

implies

ring~

according

and also prevent,for

by asking that the canonical

and

of B is

Let I

for instance,

[i, Cot 2.12])

is also n-i

in a pmoper direct

Thus,

n > 1

as the Krull dimension

projective

of (i.i).

dimension

then equality

(cf.

tistic

avoid this

two analogue

ideal of A of projective

the case that equality

if A is a regular

sequence

Ext E,C)

E ,q = Ext CE,E tX B, where

+ proj dim A(B).

B-module

with finite

3 projective

dimension

over B, then

pro~ dim A(E) Examples

of such idealstend

sion two as we shall see. manner tries

in A. Let

proj

to abound

Other examples

: Let A be a domain

of ¢. If

= 2 + proj dim B(E).

can be found

and let ~ be an

I = D(¢) be the ideal generated

= 1

is particularly

and I is overdense

easy to check

z

in the following

of A, x I C A }

domains,

zy + 1

dimen-

by the n x n-minors

from Butch's

in G.C.D.

of global

n x (n+l) matrix with en-

I -I : {x e K : field of quotients

dim A(I)

PROOF.

in domains

= A, then

theorem

[4]. This

e.g. with A = k[x,y,z]

.

We begin with some remarks.

(i) I is of finite presentation: Let

0

be a projective

resolution

is also finitely each prime Actually

>K

module

we have only to consider

Kp

the above

sequence

in P and thus

Ip = Ap

the unity" with

(cf.

for which

Kf = Afn-I

multiplicative

, where

have that for some

i.e.

denotes

E ~ (0)

fi = f

same as proj dim B(E), projectivity.

Kf

system of powers

for any B = A/I-module

is free of rank

of

At the same time

primes;

let P it

n-1 Kp = Ap .

and thus

pro~

dim (E) = -~. By a there

there exists localization

is a "partition (fl,...,fm)

= A

f = fi" While this is happening,

and

proj dim

dimension,

proj dim Bf(Ef)

to the local

245

of

of K at the

of finite projective

Ef # (0)

according

n-l.

at P. As I is faithful,

[3, Chap. II, p.138])

K is free,

to show that for

to An-l:

First we pose that if E = (0) then of Bourbaki

and we claim it

this for the minimal

(2) K may be taken to be isomorphic

theorem

>0

For that it is enough

ideal P the localized

be contained

>I

of I. Thus K is projective

generated.

be one such and localize cannot

>A n

is the

characterization

(If) = 1

since If

will

of

Af-projec-

4 tive would make it equal to Af. Assume then that we have an exact sequence (***) (3)

0

>A n-I

¢ >A n

>I

>0.

I is generated by the (n-l) x (n-l)-minors of ¢: Let D = (dl,...,d n) be the ideal generated by the minors of ~.

Using the notation in Kaplansky's dja i. Define the following map note that

is well-defined

(Zr.d.)a. = 0 i i ]

[ll,p.148,Ex.8],

~ : I

>D

as Zr.a.l i = 0

we have d.a. = l]

~(Zria i) = Zrid i. First yields

d~(Zria i ) ]

and thus Zr.d. = 0 as I is faithful. i l

=

On the other hand,

since I is overdense @ is realized by multiplication by an element d in A

and

D = dI.

Since I is torsion-free

theorem that d is a unit in A (4)

If C is a nonzero Tensor O

(***)

with

>Tor~(I,C)

By (3) above

D : I

and hence

B-module C ~ 0

0

>cn-i

~C >C n

implies that each

>0.

(n-l)-minor of ¢C Tor~(B,C)

Ext~(B,C)

>C n

is annihi-

: ker¢ C ~ 0.

~ 0 :

t¢c > c n - l - - > E x t ~ ( I , C )

it readily follows that

Now we claim that A/I-module.

>I ~ C

,C) to (***) to get

>HOmA(I,C)

from which

~ 0

to get

For any nonzero B-module C Apply HOmA(

I = D.

Tor~(B,C)

lated by D and thus by McCoy's theorem (5)

it follows by McCoy's

T : Ext~(I,A)

Ext~(I,C)

>0

: Ext~(I,A) ~ C.

is a finitely generated faithful

But this follows from the exact sequence 0

>HOmA(I,A)

>A n

>A n-I

>Ext~(I,A)

>0

and the well-known companion exact sequence 0

1 1 >EXtA(EXtA(I,A),A)

>I

.->HOmA(I,A),A)

The conclusion now follows from Gruson's result such a module T, T ~ C = 0

implies

246

C = 0.

2 1 >EXtA(EXtA(I,A),A).

[8,p.58]

that for

We are now ready to prove the theorem. Let E be a nonzero module with The first

B-

proj dim B(E) = n. Notice that by (4) pro~ dim A(E) [ 2. case to examine is then

Proj dim B(E) : 1 : Let

0

>F

>F' - - > E

.-

be a projective resolution of E as a B-module. and remark

Set

C : Ext~(E,F)

C # 0. Consider the sequences 0

>HOmA(I,F)

0

>L

obtained by applying HomB(E,

>0

>F n

>F n-I

HomA(

>L

>0

>Extl(l,F)

>0

,F) to (***) and then splicing. Applying

) to both sequences and observing

proj dim B(E) = i, we get

the diagram Ext~(E,HOmA(l,F))

>Ext~(E,F n)

>Ext~(E,L)

....>0

E~t~(E,F n-l)

with would

exact mean

row

and

column.

Notice

that

= 0

Ext~(E,Ext~(I,F))

that t

[Ext~(E,F)]n

~Ext(E,F) >[Ext~(E,r)]n-i

>0

is exact. We know however that the cokernel of this map is Extl(I,A) Q Ext~(E,F)

which by (5) is distinct from 0.

Passing now to the spectral sequence

P

we see that Ext~(E,F) ~ 0

E P2'q: 0 for and

p > 2, g > 1

but

~ 0. Thus E 2,1 2

proj dim A(E) : 3.

Proj dim B(E) : n > 1 : Write exact with F

0

>L

> F

B-projective. Then

proj dim A(L) = n+l.

>E - - ~ proj dim A(F) : 2

and by induction

Thus, by dimension shifting, proj dim A(E) =

247

6 n+2,

as desired.

REMARK.

It is clear how the procedure

and injective

analogues

of the change

2. FINITELY Let A be, unless bal dimension

two.

to be the presence results

ideals

coherence

is missing

sitting

two.

IDEALS a commutative

ring of glo-

in the study of these rings

[ 16, Prop.

provide

a clear picture ideals.

first been discovered

at least of

Examples by Jensen

where

Portions is duplicated pro~ective

and J~ndrup; to

ideals,

if and only i f Min(A),

is compact

in coherent

flat").

of local rings of global

rings

of

(This is also equivalen ~ to "the

o f A i_ss absolutely

of the structure

the subspace

of finite

global

dimension

dimension

two

with

= free.

PROPOSITION finitely

the

3.4 ] :

2.1. A is coherent

prime

seems

as in the former case the

above overdense

have

total ring o f quotients

rives

in dimension

specified,

section

the flat

we will steer away from this case and limit ourselves

PROPOSITION minimal

of rings

or lack of coherence,

of the preceding

normally

here yields

GENERATED

The great divider

the prime

quoting

otherwise

followed

2.2.

Let A be a commutative

generated

are free

coherent

ideal has finite projective

( i__nnparticular

i f A is local)

ring such that every

dimension.

If projec-

then A is a G.C.D.

domain. PROOF

(Sketch).

a domain.

[15] it follows

Let us now apply the method

each finitely "common

From Vasconcelos

generated

divisor

ideal",

and J overdense. we claim that

With

proj dim

of MaRae

that such a ring is

[13] of associating

ideal I of finite projective i.e.

an ideal

I = (a,b),

(d)

J = (e,f)

(e,f) < I. Indeed,

248

dimension

such that and

its

I = dJ

a = de, b = df

consider

the sequence

to

7 0 with

¢(x,y)

and hence

= xe-yf.

>A 2

domain

>0

xe-yf = O, x / f e

in the

for which rank one projectives

if and only if the projective

by two elements

If projectives

generated

I = P.J, with P projective

are free.

dimension

Then

of each

is at most one.

are not assumed

has that every finitely

(e,f) -I = A

by (f,e).

the rest of the argument

ideal ~enerated REMARK,

->(e,f)

K is generated

Let A be a domain

A is a G.C.D.

~

If (x,y)c K, i.e.

x eAf. Thus

We isolate LEMMA.

>K

to be free in (2.2) one still

ideal I admits

a decomposition

and J overdense.

The main use of (1.2)

for coherent

rings

of global dimension

two is the following COROLLARY

2.3.

Let I be a finitely

then A/I is an artinian I are finitely PROOF.

ring.

I__nnparticular

overdense the prime

ideal of A; ideals

above

generated.

By (1.2) the finitistic

and thus by Bass'theorem coherent

generated

projective

dimension

[2] A/I is perfect.

being a coherent

A-module,

of A/I is zero

Since A/I is also

by Chase's

result

[6] A/I is

artinian. COROLLARY

2.4.(Noetherianization)

two with Spee(A)

noetherian

and

Let A be a ring o f global Ap noetherian

dimension

for each Drime

ideal P.

Then A is noetherian. First we consider PROPOSITION

2.5.

the subspace of domains; LEMMA.

Let A be a ring of global

of maximal

ideals,

in particular

I of A (i.e.

two with Max(A),

Then A is a finite product

A is coherent.

Let A be a commutative

pure ideal

noetherian.

dimension

A/I

ring with Max(A) is A-flat)

249

noetherian.

is generated

Then each

by one idempotent.

8

PROOF.

Recall that purity for the ideal I means

ideal P the localization Let I $ 0

be pure.

subset of Max(A) is minimal. with

As

= 1

yields

For each

and

a e I

V(a)

= V(b)

if

e = ra. Notice that V(e)

(a). Pick a such that V(a)

# 0 )

then

(a,l-b)

= A. Write

A(l-e).

that

If there is an element c

ideal of A(l-e)

is properly

ra +

we claim

I = (e) @ (l-e)I observe

in every maximal V(e+c)

we may write a = ab

e is an idempotent;

= V(a); with

not contained

(l-e)I

let V(a) be the closed

and hence

(l-e)I is a pure ideal of the ring (l-e)I

or (I).

(a) = aI (checked by localization)

(e) = I. Clearly

of

0

defined by the ideal

b e I. Thus

s(l-b)

Ip

that for each prime

contained

(c exists

in V(e), which is

impossible. The proof of (2.5) now follows of A is finite. A is a domain

Assume then A indecomposable;

(cf.[17])

PROOF of (2.4): Heinzer-Ohm's generated finite.

and thus the minimal

at each l o c a l i z a t i o n primes are pure ideals.

By the preceding we may assume A to be a domain.

theorem

I = P.J

(weakly)

associated

with P projective

primes of A/I is

and J overdense.

is enough to prove this locally in the Zariski topology, Thus yields

C

>(d)/dJ

Ass(A/I)c

while a prime minimal

over

>A/I

>A/(d)

A s s ( A / ( d ) ) u Ass(A/J).

ideal is associated (d)

By

[9] it is enough to show that for each finitely

ideal I the set of

Write

: Clearly the number of idempotents

assume

P = (d).

>0

By (2.3)

to A/(d)

Since it

Ass(A/J)

is finite

if and only if it is

since A is locally noetherian. REFERENCES

[i]

M.Auslander Annals Math.

[2]

H.Bass,

and D.Buchsbaum, 68 (1958),

Finitistic

semi-primary

Codimension

and Multiplicity,

625-657.

dimension

and a h o m o l o g i c a l

rings, Trans.Amer. Math. Soc.

250

generalization

9 5 (1960),

466-488.

of

9

[3]

N.Bourbaki,

[4]

L. Burch, On ideals of finite homological dimension in local

Algebre Commutative,

rings, Proc. Camb. Phil. Soc.

Is]

H.Cartan and S.Eilenberg,

Hermann,Paris,

64 (1968),

1961-65.

941-948.

Homological Algebra,

Princeton

University Press, 1956.

[6]

S.U.Chase,

Direct product of modules, Trans.

Amer.Math. Soc. 9 7

(1960), 457-473. [7]

J.M. Cohen, A note on homological (1969),

[8]

J.Algebra

ii

483-487.

L.Gruson

and M.Raynaud,

Inventines Math.

[9]

dimension,

•

%

Crzteres

de platitude

et

°

•

•

de pro]ectzvzte,

•

13 (1971), 1-89.

W. Heinzer and J.Ohm, Locally noetherian commutative rings,

Trans.

Amer. Math. Soc. 158 (1971), 273-284.

[10]

C.U.Jensen,

Zeit.

[ll]

106 (1968),

I.Kaplansky,

[12] [13]

Some remarks on a change of rings theorem, Math.

395-401.

Commutative Rings, Allyn and Bacon,

Boston, 1970.

, Fields and Rings, University of Chicago Press, R.E.MacRae,

On an application of the Fitting invariants,

1969.

J.

Algebra 2 (1965), 153-169.

[14]

W.V.Vasconcelos, Amer. Math. Soe.

[15]

On projective modules of finite rank, Proc.

22 (1969),

, Annihilators tion, Proc. Amer. Math. Soc.

[16]

, Finiteness

430-433. of modules with a finite free resolu29 (1971), 440-442.

in projective ideals, J~Algebra

(to appear). [17]

, The local rings of global dimension two, Proc. Amer.Math. Soc.

(to appear).

251