Divisor theory in module categories

DIVISOR THEORY I N MODULE CATEGORIES

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NORTH-HOLLAND MATHEMATICS STUDIES

14

Notas de Matematica (53) Editor: Leopoldo Nachbin

Universidade Federaldo Rio de Jarmiro and University of Rochester

Divisor Theory in Module Categories

W. V. VASCONCELOS Rutgers University

1974

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM OXFORD AMERICAN ELSEVIER PUBLISHING COMPANY, INC. - NEW YORK

@ North-Holland Publishing Company - 1974 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form o r by any means, electronic, mechanical, photocopying, recording OF otherwise, without the prior permission of the copyright owner.

Lib1 auy of Congress Catalog Card Number: 74-84871 North-Holland ISBN for this Series: 0 7204 2700 2 North-Holland ISBN .for this Volume: 0 7204 2715 0 American Elsevier ISBN: 0 444 I 0 737 I

PUBLISHERS:

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NORTH-HOLLAND PUBLISHING COMPANY, LTD. - OXFORD SOLE DISTRIBUTORS FOR THE U.S.A. A N D CANADA:

AMERICAN ELSEVIER PUBLISHING COMPANY, INC. 52 VANDERBILT AVENUE, NEW YORK, N.Y. 10017

PRINTED IN THE NETHERLANDS

Preface Heuristically the divisor d ( E ) o f an A-module E i s t h e i d e a l o f A n a c k i n y t h e most i n f o r m a t i o n on E .

A nrime c a n d i d a t e

f o r t h i s r o l e , t h e a n n i h i l a t o r o f E , l a c k s decent f u n c t o r i a l p r o p e r t i e s . I n s t e a d , a g e n e r a l i z a t i o n of a n o t h e r o f t h e c l a s s i cal d i v i s o r s

n l a y s a more v i s i b l e r o l e i f one works i n t h e

f o l l o w i n g s e t t i n g . D e f i n e a d i v i s o r on a s u h c a t e y o r y C o f mod(R) a s an a d d i t i v e - w i t h r e s n e c t t o s h o r t e x a c t s e q u e n c e s mapping from C i n t o some s e m i - g r o u n S o f i d e a l s . An o u t s t a n d i n p example i s t h a t f o u n d i n t h e c a t e g o r y T o f f i n i t e l y g e n e r a t e d t o r s i o n modules o f f i n i t e p r o j e c t i v e d i m e n s i o n o v e r a N o c t h e r i a n r i n g A . I n t h i s c a s e one may d e f i n e a d i v i s o r f u n c t i o n from T i n t o t h e s e m i - g r o u p Inv(A) o f i n v e r t i b l e i d e a l s and o b t a i n an

e x a c t sequence

-

A

Ko(A)-

This d

d -

Ko(T)-

Inv(A)

i s d e f i n e d i n t h e u s u a l manner : Am

If

- - 4

E

A"

1.

0

i s a p r e s e n t a t i o n o f t h e module E , t h e d e t e r m i n a n t a l i d e a l F(E) g e n e r a t e d by t h e m i n o r s of o r d e r

n o f t h e m a t r i x 0 i s changed

into

For t h e l a r g e r category of a l l f i n i t e l y generated t o r s i o n modules t h i s f u n c t i o n i s a d d i t i v e o n l y i f t h e r i n g i s i n t e R r a l l y closed. In f a c t , i n t h i s c a s e , it i s t h e only a d d i t i v e d(A/xA)

mapping s a t i s f y i n g

=

xA.

D i v i s o r s a r i s e a l s o i n c a t e g o r i e s o f modules g e n e r a t e d by s p h e r i c a l modules

-

modules s h a r i n g c e r t a i n h o m o t o p i c p r o p e r V

Preface

VI

t i e s o f A o r t h e c a n o n i c a l modules o f bfacaulay r i n g s . S p e c i f i c a l l y we t a k e a module G s a t i s f y i n g : i) ii)

IIomA(G,G) = A , and i ExtA(G,G) = 0 f o r i > 0

and c o n s i d e r modules s u s c e n t i b l c o f a r e n r e s e n t a t i o n GmAs

r$

4l

c;"

-E -

0.

may be viewed as a m a t r i x , an a n p r o p r i a t e S c h a n u e l ' s lemma

w i l l make t h e c o r r e s n o n d i n q d e t e r m i n a n t a l i d e a l w e l l d e f i n e d . As e a r l i e r t h e d i v i s o r w i l l t a k e v a l u e s i n Inv(A) whenever

E

admits a f i n i t e G - r e s o l u t i o n . A consequence i s t h a t t h i s d i v i s o r does n o t depend on t h e s n h e r i c a l module u s e d i n t h e f i n i t e resolution. In t h e l a s t chapter a d i v i s o r i s defined i n the catepory

o f modules o f f i n i t e i n j e c t i v e d i m e n s i o n b u t u s i n g t h e symme t r i c a l notion of co-presentation. These n o t e s a r e a t r a n s c r i p t o f some l e c t u r e s on d i v i s o r t h e o r y g i v e n a t R u t g e r s I J n i v e r s i t y i n t h e S p r i n g o f 1974. I t was f e l t n e c e s s a r y t o i n c l u d e an e x p o s i t i o n o f t h e homoloyy of N o e t h e r i a n - which was t a k e n a s synonymous w i t h t h e t h e o r y o f blacaulay r i n g s - t o make t h e whole s u f f i c i e n t l v s e l f - c o n t a i n e d . O f c o u r s e t h e t i m e l i m i t s n r e c l u d e d any d i s c u s s i o n o f t h e i n t e -

r e s t i n g p o r t i o n o f t h a t t h e o r y d c a l i n c w i t h how ' l n c a u l a v r i n e s arise. ' l a j o r y o r t i o n s o f t ' i e s e n o t e s were j o i n t l y worked o u t w i t h

. J e f f r e y lfawson : t h i s i s a t l e a s t t h e c a s e f o r t h e w h o l e o f Chapter 5

and ( 3 . 1 5 ) , t h e main r e s u l t o f C h a n t e r 3 . These and

h i s ongoincr, r e s e a r c h oq h i q l i e r c l i v i s o r i a l i d e a l s will a l s o b e p a r t o f h i s t h e s i s . To J u d i t h S a l l v we a r e i n d e b t e d f o r

Preface v a l u a b l e comments on an e a r l i e r v e r s i o n o f t h e n o t e s and

VII

COT

-

r e c t i o n s o f some o f t h e most o f f e n s i v e e r r o r s . F i n a l l y , t h e f i n a n c i a l sunport of t h e National Science Foundation i s p r a t e f u l l y acknowledged.

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Table o f Contents

Preface

V

Chapter 1 : 1.1 1.2 1.3 1.4

. .. .

Chapter 2 : 2.1 2.2 2.3 2.4 2.5 2.6 Appendix

.. . .. ..

Chapter 3 : 3.1 3.2 3.3 Appendix

.. ..

Chapter 4 :

4.1 4.2 4.3 4.4 4.5

.. .. .

Chapter 5 : 5.1 5.2 5.3

.

..

Bibliography Index

Local Algebra

...... 1 ............... 4 ................ 8 . . . . . . . . . . 13

Noetherian and Coherent Rings Local Rings Flatness F i t t i n g ' s Invariants Homology of Local Rings

. . . . . . . . . . . . 16 . . . . . . . . . . . . . . . . . 19 . . . . . . . . . . . . . 28

Koszul Complexes Depth Macaulay Rings P r o j e c t i v e and I n j e c t i v e Dimensions E u l e r C h a r a c t e r i s t i c s o f Modules Gorenstein Rings Rings o f Type One

.. .. .. 36 33 . . .. .. .. .. .. .. .. .. .. .. .. 4524

Divisorial Ideals

. . . . . . .. .. ... ... ... ....... ... ... ... 55 63 72 . . . . . . . . 80

Composition i n Id(A) Divisors Modules o f Dimension One Higher D i v i s o r i a l I d e a l s

S p h e r i c a l Modules and D i v i s o r s

.... . . .. .. .. ..

A Theorem o f Gruson Change of Rings and Dimensions S p h e r i c a l Modules Elementary P r o p e r t i e s R e s o l u t i o n s and D i v i s o r s

. . . . . . 82 .. ... ... ... ... ... 998404 . . . . . . 98

I-divisors

. . . . . . . . . . . . . .1 0 4 . . . . .1 0 7 . . . . . . . . . . .1 0 9 . . . . . . . . . . . . . . . . . . . . . 117 . . . . . . . . . . . . . . . . . . . . . 120

Construction E u l e r C h a r a c t e r i s t i c s o f Inj(A) D i v i s o r s on I n j (A)

IX

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Chapter 1 Local A l g e b r a I n t h i s c h a p t e r - o f a s k e t c h y n a t u r e - a r e grouped some o f t h e b a s i c f a c t s o f commutative a l g e b r a t h a t s h a l l be u s e d throughout - Noetherian r i n g s , Krull dimension, f l a t n e s s , e t c . [6, 2 0 , -341 w i l l be s o u r c e s f o r t h e p r o o f s , which w i l l n o t , a s

a r u l e , be s u p p l i e d h e r e . 51.1

Noetherian and coherent r i n g s . Although some o f t h e n o t i o n s d i s c u s s e d i n t h e s e n o t e s do

not require t h e commutativity of the r i n g s involved, t h i s w i l l b e n e c e s s a r y f o r some c o n s t r u c t i o n s .

Thus we assume t h e b l a n k e t

c o n d i t i o n t h a t a l l r i n g s a r e commutative, w i t h 1, and t h a t modules are u n i t a l . Given a r i n g

A

-

and an A-module E w e s a y t h a t E i s

f i n i t e l y generated i f there is a s u r j e c t i o n $A-E. n summands

A n = A O

...

a r e t h e images o f t h e ' b a s i s ' e l e m e n t s Of A n , en v = ( x l , ...,xn) e An w i l l be c a l l e d a r e l a t i o n on t h e e i ' s i f

If

el,

9

The s e t o f s u c h e l e m e n t s i s a submodule K o f t o be f i n i t e l y p r e s e n t e d i f , b e s i d e s , K That t h e f i n i t e n e s s o f

K

Lemma.

E

is said

i s f i n i t e l y generated.

does n o t depend on how w e p r e s e n t

E f o l l o w s from:

(1.1)

An.

(Schanuel ' s lemma) 1

Let

2

Local A l e e b r a

be exact sequences with

P

and

P' A - p r o j e c t i v e s .

Then

K ' 8 P.

K 8 P'

Proof.

Well-known.

Definitions. submodule o f

An A-module

E

i s Noetherian i f every

i s f i n i t e l y generated, o r equivalently, i f

E

e v e r y a s c e n d i n g c h a i n o f submodules o f

is stationary.

E

As

s e v e r a l c o n s t r u c t i o n s on modules show a f i n i t a r y c h a r a c t e r , t h e f o l l o w i n g w i l l be u s e d a t times.

E

i s coherent i f every

f i n i t e l y g e n e r a t e d submodule i s f i n i t e l y p r e s e n t e d . A

The r i n g

i t s e l f i s N o e t h e r i a n i f i t is N o e t h e r i a n a s an A-module,

t h a t i s every i d e a l is f i n i t e l y generated.

Similarly f o r the

notion of coherent r i n g . (1.2)

Lemma.

If

E

i s a f i n i t e l y generated (resp.

f i n i t e l y p r e s e n t e d ) module o v e r a N o e t h e r i a n ( r e s p . c o h e r e n t ) ring

A

then

Proof. Remark.

E

i s Noetherian (resp. coherent).

[?I. Although N o e t h e r i a n r i n g s a r e c o h e r e n t t h i s i s

no l o n g e r t h e c a s e f o r modules a s c o h e r e n c e i s ' m o r e ' o f a relative notion. A c h a i n o f prime i d e a l s ( o r p r i m e s f o r s h o r t ) i n a f i n i t e sequence

A

is

Local Alnehra

o f d i s t i n c t prime i d e a l s o f

r

A.

3

is the length of the chain.

A , K-dim(A), i s t h e supremum o f t h e

The K r u l l d i m e n s i o n o f

The h e i g h t o f a prime

l e n g t h s o f a l l c h a i n s o f prime i d e a l s .

p,

h t ( p ) , i s t h e supremum o f t h e l e n g t h s o f c h a i n s w i t h e l e -

p.

ments c o n t a i n e d i n

If

I

i s an i d e a l , h t ( 1 ) i s d e f i n e d

a s t h e infimum o f t h e h e i g h t s o f p r i m e s c o n t a i n i n g (1.3)

Theorem.

p

The h e i g h t o f a prime

i n a Noetherian

r i n g i s f i n i t e and e q u a l t o t h e minimal number

xl,

...,xn

e p

Proof.

such t h a t

A

p i s a minimal prime i d e a l o v e r

zero-divisor for

E

E

p

if

The s e t o f z e r o - d i v i s o r s o f

e c E.

A s s ( E ) w i l l denote

z(E) =

Proposition.

If

i s Noetherian, each E.

0

# e c E

x c A with

w i l l be d e n o t e d

E

(1.4)

generated then Ass(E)

i s minimal o v e r an

E # 0 , an e l e m e n t

If

i f there is

h i l a t o r o f an e l e m e n t of

as a

A and a n A-module E, w e s a y t h a t

is associated t o

the s e t o f such primes.

...,xn

p.

i d e a l o f t h e form I = a n n i h i l a t o r o f

A

xl,

We s h a l l r e f e r t o t h e e l e m e n t s

Given a commutative r i n g

p

of elements

h a s a u n i q u e maximal i d e a l t h e n K-dim(A)

system of parameters f o r

a prime

n

[E, 341.

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

is finite.

I.

u

xe = 0 .

z(E).

P .

pcAss (E)-

pcAss(E) i s a c t u a l l y t h e a n n i -

I f , besides,

is f i n i t e

is a

[z, 341.

E

is f i n i t e l y

L o c a l Algehra

4

(1.5)

Proposition.

If

such t h a t

Mi/Mi-l

- A/pi,

... s M n

C_

E~

=

= M,

= prime i d e a l .

[=I.

Proof.

11.2

sM1

there is a f i l t r a t i o n

A

u l e over t h e Noetherian r i n g 0

i s a f i n i t e l y g e n e r a t e d mod-

E

Local Rinvs. The J a c o h s o n r a d i c a l o f a r i n g

.J(A) o f t h e maximal i d e a l s o f h a s a u n i q u e maximal i d e a l

is the intersection

The r i n g

A.

m_

IJ;

A

equals

l y t h e s e t of n o n - i n v e r t i b l e elements of

A J(A)

A.

k- = A/m- w i l l be c a l l e d t h e r e s i d u e f i e l d o f Let module.

A

be a l o c a l r i n g and

i s a s e t of elements o f b a s i s , t h e submodule +

m_E.

(1.6)

That

F

of

E = F

E / m-E .

E

In t h i s c a s e A.

E,

v(E), is the

Indeed, i f

whose images i n

E

and i s p r e c i s e -

a f i n i t e l y generated A-

The minimum number o f g e n e r a t o r s o f

dimension of t h e k - v e c t o r space

E = F

E

i s local i f it

el,.

. . ,en

form a

E/gE

they generate i s such t h a t

i s a consequence of t h e u b i q u i t o u s

P r o p o s i t i o n . (Nakayama's lemma)

f i n i t e l y g e n e r a t e d A-module and

F

Let

E

he a

a submodule s a t i s f y i n g

E = F + J(A) E .

Then we have Pro0 f

.

F = E.

I f we l e t

G = E/F,

we have

G = JG

and w e

Local Algebra

h a v e t o show of

G.

G = 0.

. . ,gn

gl,.

Let

5

be a s y s t e m o f g e n e r a t o r s

There e x i s t elements a

1< i , j ~n

e J

ij

such t h a t

c

Pi =

aijgj

*

j

We have

det(6ij-a..)G 13

a e J , is invertible,

= 1 + a , with

det(sij-aij)

since

= 0;

G = 0.

R e p e a t e d u s e o f ( 1 . 6 ) w i l l b e made i n t h e f o l l o w i n g form: Let

-

b e a f i n i t e l y g e n e r a t e d A-module and

E

$ : Em

En

a homomorphism r e a l i z e d by m u l t i p l i c a t i o n by a m a t r i x 1 < i < m , 1:

ive then

j!

S

2 n , with e n t r i e s i n

S

of

x,y e S

and

i f f there is

x - y e S.

=-->

S-'E

z e S or

with

If

s t r u c t u r e and

i t s e l f i t endows

A

ES

€or

p.

As

p

S = A\p

E

x

(e,x) S/-

-

(f,y)

is

AS

with a ring

A

E

If

is c a l l e d t h e l o c a l i -

itself is not a multiplicative

s e t t h e r e w i l l b e no c o n f u s i o n when Viewed as a r i n g

by

S

i s an As-module i n t h e w e l l known way.

ES

at

x

z(ye-xf) = 0.

p i s a prime i d e a l , E

i s an A-module an

E E

set i f

and h a s a n a t u r a l g r o u n s t r u c t u r e .

ES

Applied t o t h e r i n g

zation of

4

is called a multiplicative

A

e q u i v a l e n c e r e l a t i o n i s d e f i n e d on

written

If

J.

E = 0.

A subset 0

j

(aij) is surject-

ES

is written

E

P'

h a s a s i m p l i f i e d prime i d e a l s t r u c t u r e :

Local Algebra

6

i t s primes a r e i n one-one correspondence w i t h t h e primes of contained i n

p.

ma1 i d e a l

.

PAP

In p a r t i c u l a r

A

A

i s a l o c a l r i n g w i t h maxi-

E

I t s r e s i d u e f i e l d w i l l be d e n o t e d by

k(p).

The u s e f u l n e s s o f t h i s c o n s t r u c t i o n , t h a t can be e x t e n d e d t o module homomorphisms r e s t s p r i m a r i l y on (1.7) i)

([c]):

Proposition. If

S

i s a m u l t i p l i c a t i v e s e t and f

E-+F-&G

i s an e x a c t s e q u e n c e o f m o d u l e s , t h e n

- fS

ES

FS

%

GS

is also exact. ii)

E = 0 iff E

E

= 0

f o r e v e r y prime i d e a l .

A p r o j e c t i v e r e s o l u t i o n o f an A-module E i s an e x a c t sequence P.

w i t h Pi n

:

Pn

.. . p1

A-projective.

(pdA E = n )

that

...

Pn+l = 0 .

E

fl

~

Po-E-O

i s s a i d t o have p r o j e c t i v e d i m e n s i o n

i f there is a resolution with

n

l e a s t such

I n t h i s c a s e i t f o l l o w s i m m e d i a t e l y from ( 1 . 1 )

t h a t whenever o n e s t a r t s b u i l d i n g a r e s o l u t i o n f o r E , t h e k e r n e l of Pn-l

f n - l , Pn-2

is a l w a y s p r o j e c t i v e .

If

E

is a

f i n i t e l y p r e s e n t e d module o v e r a c o h e r e n t r i n g t h e n from ( 1 . 2 ) one c o n c l u d e s t h a t E a d m i t s a p r o j e c t i v e r e s o l u t i o n w i t h f i n i t e l y g e n e r a t e d terms.

In t h e s e c a s e s a consequence o f (1.7) i s

t h a t pdA E = s u p (pdA A).

E

P E’

where

p

runs o v e r t h e primes of

7

Local Algebra

of A i s defined as

The g l o b a l dimension

s u p 1 pdAE,

f o r a l l A-modules 1. Theorem. ( H i l b e r t ' s s y z y g i e s theorem) I f A i s

(1.8) any r i n g

g l dim(A[t])

=

g l dim(A)

+

1,

where t i s a n i n d e t e r m i n a t e . I n p a r t i c u l a r t h e p o l y n o m i a l r i n g in

n

i n d e t e r m i n a t e s over a f i e l d h a s g l o b a l dimension An

sequence I'

with Ii

o f a module E i s an

injective resolution

: 0

-

0

E -1

...

-I1-

n.

exact

- 1" -

...

an i n j e c t i v e A-module. The n o t i o n o f i n j e c t i v e dimen -

s i o n i s s i m i l a r l y defined while t h e

g l o b a l dimension

'

of A

c o i n c i 4 e s w i t h t h a t d e f i n e d above. L e t A b e a N o e t h e r i a n l o c a l r i n g o f maximal i d e a l

m_

and

k . Among t h e numbers residue f i e l d -

K r u l l dim(A)

,

dim,(m_/m_')

and

g l o b a l dim(A) we have t h e f o l l o w i n g r e l a t i o n s : g l dim(A) K r u l l dim(A)

( [El1

.

(1.9)

Theorem.

i)

g l dim(A) <

ii)

L

dimk(m_//m2)

-

The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t : m

.

dirnk(m_/m_*) = K r u l l dim(A).

Moreover i n e i t h e r case one h a s e q u a l i t y o f a l l t h r e e dimensions

.

Local Algebra

8

A r i n g s a t i s f y i n g t h e s e c o n d i t i o n s will b e c a l l e d r e g u l a r .

34]): The f o l l o w i n g g i v e s a p i c t u r e o f r e g u l a r r i n g s ( [ -

(1.10)

Theorem.

Let

be a l o c a l r i n g .

A

r e g u l a r i f f t h e graded r i n g a s s o c i a t e d t o

m_

Then

A

is

i s a polynomial

ring.

xl,

Concretely, i f generating s e t f o r

m

...,xn

(n =

a r e t h e e l e m e n t s i n a minimal ) we h a v e a homomorphism

-

of graded r i n g s

with

k(Xi)

=

class o f

A is regular i f f

x 1.

g/g2. The a s s e r t i o n i s t h a t

in

i s an i s o m o r p h i s m .

This i s t h e c a s e , f o r

i n s t a n c e , w i t h t h e r i n g o f f o r m a l power s e r i e s i n

n

variables

over a f i e l d . The f o l l o w i n g change o f r i n g s r e s u l t w i l l be u s e d r e p e a t e d l y ([%]). (1.11) and

x

an e l e m e n t o f A

t o both for

i

Proposition.

'>

and

E.

Let

A

be a r i n g , E

t h a t i s a nonzero d i v i s o r with respect

A

We h a v e t h e n an isomorphism o f f u n c t o r s

0:

i+l Ext A (-,E)

E x t iA/ ( x ) ( - , E / x E )

i n t h e category o f R/(x)-modules.

§1.3

Flatness. Let

A

an A-module,

be a r i n g ,

E

an A-module and

Local Algebra

... M i + l

( E.1)

- - P.1

?Iiel

i

9

...

b e an a r b i t r a r y e x a c t s e q u e n c e o f A-modules.

. ..

( i\lQAE)

b.I Q E i A

Mi+l@AE

i s a l s o e x a c t , we s h a l l s a y t h a t

the right

exactness

n e s s o f E amounts t o : MQAE # 0

whenever

of

.. .

bli-l@AE

i s A -f l a t , o r simply f l a t

E

when t h e r i n g i s w e l l u n d e r s t o o d .

If

A s (-)eAE

already preserves the f l a t -

short exact seouences,

(-)BAE

preserves injections.

)I # 0 , t h e n

If also

w i l l be c a l l e d f a i t h f u l l y

E

flat.

... , a n ]

a = {al,

If

-

i s a sequence of elements of

A, t h e

module o f r e l a t i o n s on t h e a i l s i s t h e k e r n e l o f

0 : A" R

,.(a)

= ker(4t31E)

...,x n ) = c x i. a i ' { ( e l , ...,e n ) e E w i t h

A,

+ (x,,

is then

C aiei

= 0)

and w i l l be c a l l e d t h e module o f r e l a t i o n s on t h e a i l s w i t h coefficients in (1.12) A-module

E.

i)

E

ii)

E.

is flat. g

For e v e r y s e q u e n c e i ) =->

t h e elements i n

5. F

F

ii):

Let

e x a c t sequence

$81

a s above

(a)

-

RA(a)E =

RE(a).

be t h e i d e a l g e n e r a t e d by

C o n s i d e r t h e exact s e q u e n c e A

An

i s a f r e e module,

F Q E

RA(g)E C R E ( @ .

The f o l l o w i n g a r e e q u i v a l e n t f o r an

Proposition.

Proof.

where

Notice t h a t

An 8 E

A/(s)-

0

Tensoring with

- 0Q1

A Q E

E

we obtain the

A / ( a-) Q E

-

0.

Local Algebra

10

im($ 8 1 ) =

Since

R ~ ( & ) E , we have ( i i ) .

Actually, i n the

l a n g u a g e o f d e r i v e d f u n c t o r s , t h e argument shows t h a t Tor:(A/(z),E) i i ) =>

i):

where

@I

jective.

RE(%)/

=

RA(&)E

.

Consider t h e diagram

is i n j e c t i v e , With

P

L = k e r JI

i s a f r e e module and and

J,

is sur-

K = J,-’(cb(F)), w e o b t a i n t h e

diagram L @ E - P @ E - G @ E - O

T

T

w i t h o b v i o u s maps.

From i t f o l l o w s t h a t t h e v e r t i c a l map on

t h e r i g h t w i l l b e i n j e c t i v e i f t h e two maps e n d i n g i n are injective.

We may t h e n assume t h a t

I t i s a l s o clear t h a t

G

P 8 E

i s a f r e e module.

may b e assumed o f f i n i t e r a n k .

G

An

e a s y i n d u c t i o n on r a n k t a k e s c a r e o f t h e r e m a i n d e r o f t h e p r o o f , t h e s t a r t i n g c a s e b e i n g s u p p l i e d by t h e h y p o t h e s i s . I t i s c l e a r t h a t f r e e modules a r e f l a t a s a r e d i r e c t A b i t s u r p r i s i n g i s t h e converse

limits of f r e e modules. f l a t modules ([z]):

f r e e modules. amounts to

are d i r e c t l i m i t s of f i n i t e l y generated

The l o c a l i z a t i o n p r o c e d u r e o f 2 1 . 2 r e a l l y

ES = E 8 AS; c o n s e q u e n t l y

A criterion for flatness:

(1.13)

Theorem.

Let

2

: A

-

AS

B

i s a f l a t A-module.

be a l o c a l

Local A l g e b r a

11

homomorphism o f t h e l o c a l N o e t h e r i a n r i n g

B-module

E

(k(m_)c r ) .

(B,n_)

l o c a l coherent r i n g

is f l a t over

(A,m_)

A f i n i t e l y presented

iff

A

A Torl(A/m_,E) = 0

.

We b e g i n by r e m a r k i n g t h a t i f

Proof.

A g e n e r a t e d A-module t h e n Torl(M,E)

is a finitely

M

i s a f i n i t e l y generated B-

- - ... - - -

module.

Indeed, i f

( F)

.,.

F1

Fn

Fo

i s a p r o j e c t i v e r e s o l u t i o n of

M

EI

0

by f i n i t e l y g e n e r a t e d f r e e o f B-modules i s made up o f

F@ B

A-modules, t h e complex

into the

f i n i t e l y g e n e r a t e d c o h e r e n t modules and t h u s i t s homology groups a r e f i n i t e l y generated.

A c c o r d i n g t o ( 1 . 5 and 1 . 1 2 ) i t

A

i s enough t o show t h a t

Torl(A/p,E)

p.

then

If

K-dim(A/p) = 0

p

=

m_

Torl(A/p,E) Suppose t h a t

K-dim(A/E) > 0

A/p

0-

Tensoring with

E

and by h y p o t h e s i s

= 0

Let

a e m_

- .a

A/p

yields

T o r l (A/p,E)

5 T o r l (A/p,E)

But K-dim ( A / ( p , a ) ) < K-dim(A/E)

then

p; by i n d u c t i o n

Torl ( A / p , E )

= a.Torl

p

A/(p,a)

-

and c o n s i d e r t h e

-

0

.

T o r l (A&,a) , B )

.

and t h u s by ( 1 . 5 ) t h e f i r s t

module a d m i t s a f i l t r a t i o n w i t h f a c t o r s l y containing

.

and t h a t t h e s t a t e m e n t h o l d s f o r

a l l primes o f lower dimension. sequence

f o r each prime i d e a l

= 0

A/p',

Torl(A/(p,a),E)

(A/p,E)

,

with = 0.

p' p r o p e r But

which by Nakayama's lemma

Jmcal A l e e h r a

12

( 1 . 6 ) and t h e i n i t i a l remarks f o r c e s T o r l ( A / p , E ) Remark.

= 0.

Without r e l a t i v e f i n i t e n e s s c o n d i t i o n s t h e does n o t ensure t h e f l a t n e s s of

v a n i s h i n g o f Torl(A/m_,E) = 0 E as e a s y examples show.

Let

be a l o c a l N o e t h e r i a n r i n g and

fA,m_)

a finite-

E

l y g e n e r a t e d A-module. A t o p o l o g y i s i n d u c e d on E by d e c l a r r i n g t h e submodules m - E t o be a s y s t e m o f n e i g h b o r h o o d s f o r 'This w i l l be r e f e r r e d t o a s t h e m - -adic topology o f

0.

Theorem.

(1.14)

module o f

r

1

( A r t i n - R e e s ' lemma)

s > 0

T h e r e e x i s t s an i n t e g e r

E.

Let

E.

be a s u b -

F

such t h a t f o r

s , w e have

r-s.

Fn m -r * E = mProof.

See [ 34]

S

(Fn m_ * E ) .

f o r a s l i g h t l y more g e n e r a l s t a t e m e n t .

A s c o n s e q u e n c e s we c o n c l u d e t h a t t h e m_-adic t o p o l o g y o f

F

i s i n d u c e d from t h a t o f

derive

F = m_*F which by ( 1 . 6 ) i m p l i e s E

m-adic topology of *

Denote by A

-

*

A 8 E

E

=

n m-r * E r

F = 0 --i.e. the

E

with respect t o the

A

i s an A-module.

-

Actually the canonical

h

E

I:

i s Hausdorff.

t h e completion of

E

m_-adic t o p o l o g y ; map

E , a n d , by p u t t i n g

i s an isomorphism and

A

*

A

is a faith-

4]). f u l l y f l a t homomorphism ( [ 3The main f a c t a b o u t (1.15)

Theorem.

2

t h a t we s h a l l u s e i s

(Cohen's theorem)

image o f a power s e r i e s r i n g

R = D[[xl,

2

([k]):

i s a homomorphic

...,x n ] ]

where

i s e i t h e r a f i e l d o r a complete d i s c r e t e v a l u a t i o n r i n g .

D

13

Local A l g e b r a

§

Fitting's invariants.

1.4

Given a f i n i t e l y g e n e r a t e d A-module

E

we a t t a c h t o it

a s e q u e n c e o f i d e a l s which g e n e r a l i z e t h e c l a s s i c a l e l e m e n t a r y divisors.

Let A ( a ) L An --+

be a p r e s e n t a t i o n o f

0

-+

A(a)

with

E

E

a f r e e A-module o f u n d e t -

ermined rank. Definition. ideal

F (E)

r 4

matrix if

r

2

For an i n t e g e r

0 < r < n

we c a l l t h e

g e n e r a t e d by t h e m i n o r s o f o r d e r

the r-th Fitting ideal of

E.

n - r of t h e

We p u t

Fr(E) = A

n.

That

Fr(E)

d o e s n o t depend on t h e p r e s e n t a t i o n f o l l o w s

r a t h e r e a s i l y from ( 1 . 1 ) . i s t h e following:

+

If

-

Another manner o f d e f i n i n g : F

A-modules, d e f i n e t h e ' o r d e r ' o f

?(I$) = C f ( + ( F ) ) where

f

If

k

-

: A

4

o("-X 4 )

d e n o t e s t h e n - r e x t e r i o r power o f Remark.

i s a homomorphism o f

G

runs over

t h e n p u t i n t h e n o t i o n above

B

Fr(E)

t o be t h e i d e a l HomA(G,A). = Fr(E)

We c o u l d

where

'Ar

4

4.

i s a r i n g homomorphism and

E

i s a f i n i t e l y g e n e r a t e d A-module i t f o l l o w s i m m e d i a t e l y from the f i r s t definition that

h(Fr(E))-B. Fr(E gAB) = -

t h e f i r s t i n v a r i a n t i s viewed a s an i d e a l o f The p r o p e r t y t h a t

E

(Here

B obviously.)

be a p r o j e c t i v e module i s e a s i l y

expressed using F i t t i n g ' s i n v a r i a n t s .

Thus i f

E

i s a mod-

u l e of f i n i t e p r e s e n t a t i o n since i t s p r o j e c t i v i t y i s decided

L o c a l AlEebra

14

a t each l o c a l i z a t i o n

E

E

([c]):

w e may s t a t e

(1.16) P r o p o s i t i o n .

is projective i f f its Fitting's

E

i n v a r i a n t s have t h e p r o p e r t y t h a t a t each l o c a l i z a t i o n t h e y are e i t h e r 0

or

A

A

P

P'

Another p r o p e r t y o f t h e

Fr(E)'s

defined t o be t h e a n n i h i l a t o r of

c -

I1(E)"

E

F p )

is:

I1(E)

If

is

then

c_ I p )

*

More g e n e r a l l y w e c o u l d d e f i n e t h e i n v a r i a n t f a c t o r s o f E : Ir(E)

--

attached t o E

t h e no name i n v a r i a n t s o f

d e n o t e s t h e s e t o f submodules o f elements, then

r

and d e r i v e s i m i l a r

F i n a l l y we d e f i n e s t i l l a n o t h e r -set o f i d e a l s

relations.

by

E

being t h e a n n i h i l a t o r of

E

t h a t can be g e n e r a t e d

Kr(E) = C a n n i h i l a t o r

runs over t h e elements of

Ir

If

E.

E / F , where

F

r.

The r e l a t i o n s h i p between t h e p r i m e i d e a l s c o n t a i n i n g these F ' s ,

p at

1 ' s and K's i s n o t d i f f i c u l t t o d e t e r m i n e .

be a prime i d e a l and l e t

p.

Denote by v ( p ; E )

o f t h e A -module

E

P

(" ('IE) )

i) ii)

x(E/N)

. We

p

If

Conversely, i f r

E

t h e minimal number o f g e n e r a t o r s

I t f o l l o w s e a s i l y t h a t V(E:

P'

E) =

v(P;E) > r.

iff

,&

Kr,

where

Localizing a t

by

be t h e l o c a l i z a t i o n o f

P

have t h e n

p 2 Ir = 0,

E

Let

there is

i s a submodule g e n e r a t e d by

N

E w e conclude v(2;E)

e l e m e n t s and

x e Kr \ p such t h a t

5 r

E

P

= N

P

and

Thus

elements.

v(p:E) < r.

t h e r e i s a submodule

y # p, yE C N .

r

N

generated

Local Algebra

p 2 Kr iii)

iff

v(p;E) > r

Finally, let

and

IS

rad(Kr) = r a d ( I r + l ) .

p

p z F r ; by l o c a l i z i n g a t

and u s i n g

a p r e v i o u s remark w e may t a k e a minimal r e s o l u t i o n o f (i.e. the entries of

ring).

0 l i e i n t h e maximal i d e a l o f t h e l o c a l

The hypothesis then implies

verse is a l s o clear. TO sum up :

E

p 2 Fr

Thus

rad(Fr)

=

v(p;E)

iff

5 r.

v(p;E)

rad(Kr) = r a d ( I r + l ) .

The con-

'> r + l .

Chapter 2 Ilomology o f L o c a l Rings The p o i n t o f view t a k e n i n t h i s c h a p t e r i s t h a t t h e t h e o r y o f Macaulay - o r Cohcn-’lacaulay

- r i n g s i s almost

synonymous w i t h s t u d y i n g t h c homology o f l o c a l N o e t h e r i a n U n f o r t u n a t e l y l i t t l e s p a c e i s d e v o t e d t o g i v i n g ways

rings.

i n which s u c h r i n g s a r i s e i n a s y s t e m a t i c manner.

For some

o f t h e s c a s p e c t s we r e f e r t o [ lS, 301 and t h e b i b l i o g r a p h i e s there.

Koszul c o m p l e x e s .

52.1

Throughout

w i l l be a commutative r i n g .

A

W e begin

w i t h a d i s c u s s i o n o f what i s p e r h a p s t h e most i n t e r e s t i n g complex i n commutative a l g e b r a . Let of

b e an .\-module and

E

ential

d+

on

6

1

sends

h(E)

...

d (e A d+

+

For an e l e m e n t

E.

n AE

,

h(E)

e HomA(E,A)

i n degree

into

n-1 A E

one d e f i n e s a d i f f e r -

n , by t h e f o r m u l a

i he ) = c ( - l ) + ( e i ) e l A

n

the e x t e r i o r algebra

... h e i ~... h e n h

:

and, a s e a s i l y checked,

(db)’ = 0 . When

E

i s a f i n i t e l y g e n e r a t e d module we t a k e , h o w e v e r ,

an a l t e r n a t e a p p r o a c h . First w e r e c a l l t h e n o t i o n o f t e n s o r p r o d u c t o f c h a i n complexes o f A-modules.

Let

16

Homology o f L o c a l Rings

17

...

(X' ,dl):

be two c h a i n complexes

(Xi = 0 , i < 0 a l w a y s ) .

(X Q X ' ,

a)

i s defined as

Let

a

b e an e l e m e n t o f t h e r i n g

A

and l e t

Aa

be t h e

complex d e f i n e d a s (Aa)i

= 0

for

i # 0,l

= A

for

i = 0,l

= m u l t i p l i c a t i o n by

dl

a.

The Koszul complexes w e s h a l l be i n t e r e s t e d i n a r e b u i l t up o f s u c h p i e c e s and o f modules viewed a s comnlexes i n t h e u s u a l way.

Thus i f

i s an A-module we w r i t e

E

E a = (Aa) Q E

,

which h a s a s m e a n i n g f u l homology g r o u p s : HO(Ea) = E/aE of

a

in

A.

5

=

E = An

$(rl,

=

(0 : a ) = a n n i h i l a t o r E

Ix,,

. . . ,xn1

i s a sequence o f elements

The Koszul complex a s s o c i a t e d t o

K. K.(&;A)

H1(Ea)

E.

Suppose now in

and

(x;A) = Ax -

1

-

Q

..

8 A xn

i s defined a s

.

i s t h e n t h e e x t e r i o r a l g e b r a comp l e x as so c i a t e d t o and t h e map

...,r n )

=

1 rixi

@ : An

.

A

defined a s

Homology o f L o c a l R i n e s

18

F i n a l l y we s h a l l w r i t e

K.(&;E)

K.(x:A) -

for

e x p r e s s i o n s f o r two o f t h e homology g r o u n s o f easily written:

(5)

particular i f the ideal zero d i v i s o r s o f

E,

and

K.(x:E)

lln = (0 :

are

5).

E does n o t c o n s i s t e n t i r e l y o f

lIn = 0 , an o b s e r v a t i o n we s h a l l u s e

K. ( 5 ; - )

notice that (2.1)

i s an e x a c t f u n c t o r on A-modules.

Proposition.

Let

be a c h a i n complex a n d l e t

C.

b e a c h a i n complex o f f r e e modules w i t h

Fi = 0

for

There i s t h e n an e x a c t sequence 0

-

-

€1 (I1 ( C . ) Q F , ) o

q

Proof.

(1

- f

h

(FO).

-&

F.

h

h

g

H (C.@F.)

-+

H

1

(H

q-1

i > 1.

-

(C.)@F.)

0

C o n s t r u c t t h e s e a u e n c e o f complexes 0

with

In

B e f o r e we l i s t some t e c h n i c a l f a c t s on s u c h comnlexes

often.

F.

... , x n ) E

fiO = E / ( x l ,

The

Q E.

( F O ) O = F",

(F1).

-

0

h

(Fo)l = 0 ,

(Fl)o

= 0,

(F1)l

= F1,

f

and

t h e c o r r e s p o n d i n g i n j e c t i o n and p r o j e c t i o n mappings.

Tensoring with

C-

a n d w r i t i n g t h e homology s e q u e n c e we g e t

k[q(C.@(F1).)

-t.

Ha-l(C.@(FO).I

A

Note t h a t

H

(C.QP(F1).) = Hq(C.)@ F1,

q+l H (C.) Q F o and t h a t 9

cntial of

F..

a

Hq(C.@(FO).) =

i s , up t o a s i g n , j u s t t h e d i f f e r -

Taking a l l t h i s i n t o account w e o b t a i n t h e

d e s i r e d sequence. W r i t e f o r any complex

C,,

Cx = C , @ A x .

.

Homology o f L o c a l Rings

(2.2)

Proposition.

19

For any c h a i n complex

C.,

x H(Cx) = 0 .

Proof.

Consider

Ax

and

A

X ,x

= Ax Q A x .

Define chain

maps

f ( a ) = (a,O)

where

and

g ( a , b ) = a+b

i n dimension

1.

This

i m p l i e s a monomorphism

T e n s o r now

by

Cx

and t a k e t h e homology t o g e t t h e f o l l o w i n g s e q u e n c e

(2 . l ) :

f

Ilq(Cx)

As

f,

*

Hq(cx,x)

-

(-1 p - I x

H (I -1 (CX)

Hq-l(Cx)

i s a monomornhism, m u l t i n l i c a t i o n by

*

is the n u l l

x

map.

Depth.

52.2.

I n t h i s s e c t i o n we s h a l l d e f i n e a n u m e r i c a l i n v a r i a n t f o r i d e a l s which p l a y s a r o l e comparable t o t h a t o f i t s h e i r r h t . S p e c i f i c a l l y t h e s i t u a t i o n i s as follows: and

E

an A-module.

respect t o

E

If

x1 e I

w e can a s k w h e t h e r

zero d i v i s o r s o f

E/xlE.

Let

I

be an i d e a l

i s n o t a zero d i v i s o r with I

c o n s i s t s e n t i r e l y of

I n t h i s manner a s e q u e n c e

Homology o f L o c a l Rings

20

x1,x2,

xi

...

o f elements of

a r i s e s w i t h the property t h a t

I

is n o t a zero d i v i s o r f o r

E/(xl

,...,x

~ - ~ ) E . Given

. .. C ( x , , . .. , x i )

n o e t h e r i a n c o n d i t i o n s , t h e sequence

(x,) C

c...

w i l l consist e n t i r e l y of

s t a b i l i z e s and e v e n t u a l l y

x1 , x 2 , .

..

.., x i , . . . ) E .

E/(xl,.

zero d i v i s o r s of

I

We s h a l l r e f e r t o

as a r e g u l a r E - s e q u e n c e , o r s i m p l y E - s e q u e n c e .

A f i r s t q u e s t i o n h e r e i s w h e t h e r t h e maximal number one o b t a i n s i s i n d e p e n d e n t o f t h e c h o s e n s e q u e n c e . question is:

What i s t h i s n o t i o n good f o r ?

Another

We s h a l l answer

t h e f i r s t i n t h e a f f i r m a t i v e and g i v e some i n s t a n c e s where i t c a n be u s e d i n a r a t h e r n a t u r a l way. ( 2 . 3 ) Theorem.

in

A

Let

xl,.

..,xn

I.

Let

g e n e r a t i n g an i d e a l E # IE

t e d A-module w i t h

Let

i n g Koszul complex. Hq(K.) # 0 .

and l e t q

be a sequence o f elements

E

K.(x;E)

If

Proof.

be t h e l a r g e s t i n t e g e r f o r which

a

q

Since

HO(K.)

= E/IE,

If

q # n

p i c k an

which i s n o t a z e r o d i v i s o r o f

t h e e x a c t sequence 0

+

E.

are zero d i v i s o r s f o r

s a t i s f y i n g o u r assumption.

I

have

# 0 , we h a v e by a p r e v i o u s remark

H,(K.)

in

I

We c a l l t h i s number t h e I - d e p t h o f

(0 : I ) # 0 and s o a l l e l e m e n t s o f I E E. We u s e d e c r e a s i n g i n d u c t i o n on q .

element

be t h e c o r r e s p o n d -

Then a l l maximal r e g u l a r E - s e q u e n c e s i n

t h e same l e n g t h n - q .

there is a

be a f i n i t e l y g e n e r a -

E

-.a

E

E/aE + 0 ;

by t h e f u n c t o r i a l i t y o f t h e Koszul complex w e g e t

E.

Form

Homology o f L o c a l Rinqs

From t h i s s e q u e n c e we have a-I1 (K. ( 5 ; " ) = 0

By ( 2 . 2 ) Ilq

lli(K.

I n t h e s n e c i a l c a s e where A = grade o f

A If

and l c t

a +

Let

(K,(x:E/aE))

Theorem.

E

Let

tie an i d e a l o f a N o e t h e r i a n r i n g

.J

,JE f E .

b e a f i n i t e l y g e n e r a t e d .A-module w i t h A

then

(.T,a)-depth

.J-depth I:. F i r s t observe t h a t

C/(.J,a)E # 0

f o r otherwise

which would c o n t r a d i c t Nakayama's lemma.

= E/JE,

K. b e t h e K o s z u l complex o f

aenerators of

,J

and

Fa

=

K.

L

A Aa,

r e l a t i v e t o a system o f that is,

complex r e l a t i v e t o a s y s t e m o f g e n e r a t o r s o f

F.

i s a Koszul

(J,a).

Bv

we have

(2.1)

Let

=

(1+1

I.

Proof. a(E/.JE)

i > q+l.

we c a l l I - d e p t h

1: = A

i s i n t h e .Jacobson r a d i c a l o f

L < 1

I{

if

0

=

Induction ends i t .

(K. ( 5 : E ) ) .

(2.4)

(x;E/aE))

and t h u s

(1

'1

q

be t h e i n t e g e r which d e t e r m i n e s t h e . J - d e n t h of

t h e n have lemma.

Ho(Hq(K.)8Aa) = Hq(K.)/aHq(K.)

#

0

by Nakayama's

The c o n c l u s i o n now f o l l o w s from ( 2 . 3 ) as

' l o n g e r ' by one t h a n

E.

is

F.

K..

Assume f o r t h e r e m a i n d e r o f t h i s s e c t i o n t h a t N o e t h e r i a n and modules a r e f i n i t e l y g e n e r a t e d .

A

is

We

Homology o f L o c a l Rings

22

Corollary.

(2.5)

t h e Jacobson r a d i c a l of

. . ,xn)

I = (xl,.

Let

A

and l e t

E

be c o n t a i n e d i n

be an A-module.

If

I - d e p t h E = n , t h e n t h e x ' s , i n any o r d e r , form an E - s e q u e n c e . Let

Proof. J-depth E

. . , x , - ~ ) ; by

.J = ( x , , .

n - 1 and by i n d u c t i o n

=

E - s e q u e n c e i n any o r d e r .

Let

xl,

Iil(K.B.Ax

(2.1)

xn

n

. . .'Xn- 1

form a r e g u l a r

be t h e Koszul complex c o r -

K.

responding t o f i r s t n - 1 generators. that

t h e Drevious theorem

Our a s s u m p t i o n i n c l u d e s

) = 0 ( a c t u a l l y i t i s e q u i v a l e n t ) and s o , by

i s n o t a zero d i v i s o r of

E/JE.

Clearly t h i s

suffices. Remarks. If

i)

K.

i s t h e Koszul complex a s s o c i a t e d t o a sequence

x- and t h e A-module

E , f o r any A - a l g e b r a

B

we have

where t h e y ' s a r e t h e images i n K.(y;EBAB) = K.(x;E)BAB,

of the x ' s .

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

H(K.(y;EBB))

= H(K.(x;E))B -

R

B

i s A - f l a t we h a v e

I f a p p l i e d t o t h e case o f

B.

l o c a l i z a t i o n s we h a v e I-depth E where

p

i n f 11 - d e p t h ED)

=

P

runs o v e r t h e primes such t h a t (E/IE)

P

#

0.

P r a c t i c a l l y t h i s s a y s t h a t i n c o m p u t i n g t h e d e p t h w e m i g h t as well r e s t r i c t ourselves t o local rings. ii)

From t h e p r e c e d i n g we see t h a t t h e g r a d e o f an i d e a l

does n o t decrease under l o c a l i z a t i o n s .

w e s h a l l refer t o Pp-depth

A

P

For a prime i d e a l

a s t h e l o c a l grade of

p.

p

Homology o f L o c a l Rings

23

S i m i l a r i t y with indeterminates : The s i m i l a r i t y between r e g u l a r ( i . e . A - )

s e q u e n c e s and

i n d e t e r m i n a t e s i s i l l u s t r a t e d by t h e f o l l o w i n g c o n s i d e r a t i o n s . A

Let

b e a l o c a l N o e t h e r i a n r i n g (more g e n e r a l l y a c o h e r e n t

ring) containing a f i e l d

xl,

and l e t

k-

. . . ,xn m_

A-sequence c o n t a i n e d i n t h e maximal i d e a l make

A

R = k -[ t l , .

an a l g e b r a o v e r k-, by l e t t i n g

ring over

ti-x.

t

t h e p r e m i e r example o f a r e g u l a r s e q u e n c e , k-.

R-projective resolution of K.(

t

;R) 8 A = K.(

o t h e r than

wise, the subring A

:R)

i s an

is A-regular,

k_[xl,

...,x n ]

A

of

Stated other-

i s isomorphic t o

R

i s f l a t over i t . Let

be an i d e a l o f

I

g e n e r a t e d by ' p o l y n o m i a l s ' i n t h e x i ' s w i t h c o e f f i c i e n t s i n

k ; then -

of

... , x n

a f l a t R-algebra.

A consequence is t h e following: A

(t

K.

being

R Tor 1( k- , A ) = 0 , which a c c o r d i n g

In p a r t i c u l a r

0.

xl,

... ,t n

x- ; A ) h a s t r i v i a l homology a t d i m e n s i o n s

t o ( 1 . 1 2 ) s u f f i c e s t o make A

and

As

We can

t h e polynomial

tl,

=

A.

of

..,tn 1 , -

1'

be a r e g u l a r

R.

pdAI I n .

Indeed,

- - ... - - -

By H i l b e r t ' s s y z y g i e s theorem t h e r e i s a r e s o l u t i o n 0

where t h e

Fn-l

Fils

A

algebra over

I.

Fo

are R-free.

jective resolution of If

i s an i d e a l

I.

I = IOA where

I.

Tensoring with

A

We c a n s e e t h a t

0

yields a pro-

pdRIO = pdAI.

d o e s n o t c o n t a i n a f i e l d w e can s t i l l make S = 22 [ t l ,

...,t n ] ,A)

TOT;(Z

If characteristic

A = p

n

, we

=

and s i m i l a r l y c o n c l u d e

o

for

i >

o

.

can s t i l l conclude

A

an

24

Homology o f L o c a l R i n g s

S

Tori(L,A) module. Tor:(iZ

Indeed

is a

A

/ p Z ,A) = 0

Z!

for

yields 0 = T o rS Z(L',A)

... , t n ]

[t,,

i > 1, i . e .

L-

0-

P

i s a Z - t o r s i o n f r e e S-

L

= 0 , i > 0 , whenever

n *P

-

L-

module and

f l a t dimSA < 1. L'-

T o r lS( L , A )

Then

0

n

*P

T o r lS( L , A )

and t h e d e s i r e d c o n c l u s i o n . Again a c o n s e q u e n c e i n t h i s c a s e i s t h a t i f ideal of

g s n e r a t e d by p o l y n o m i a l s i n t h e x i ' s w i t h i n t e g -

A

r a l c o e f f i c i e n t s , and such t h a t

S/Io

P

. . ,xn)A

i s ZZ - t o r s i o n f r e e , t h e n A = 0

then

could be # 0

Z

P

pdAI = p d S I o .

C A

but

T h e r e i s a c a s e , however,

Torl(L,A).

where we might e v e n d r o p t h e h y p o t h e s i s t h a t xl,.

. . ,xn

i n a commutative r i n g monomials i n t h e

xi's.

is

and t h u s i n f l u e n c e t h e

S

ZZ - t o r s i o n f r e e n e s s o f

Assume

S

I o , the corresponding i d e a l of

If characteristic

H n (x,,.

i s an

I

A

be c o h e r e n t .

form a r e g u l a r s e q u e n c e i n any o r d e r A

and l e t

I

b e an i d e a l g e n e r a t e d by

Assume a l s o t h a t

A

i2 - t o r s i o n

is

free. Theorem.

(2.6)

Let

I.

be t h e i d e a l o f

r e s n o n d i n g monomials i n t h e t ' s : t h e n over, i f

(xl

,...,xn)

S

pdAI < pdSIO.

in the corMore-

i s c o n t a i n e d i n r a d ( I ) , t h e n pdAI

=

pdSIO = n - 1 . We s h a l l g i v e a p r o o f t h a t i n c l u d e s t h e s t a t e m e n t s on t h e e q u a l i t i e s of dimensions i n t h e preceding d i s c u s s i o n .

Homology o f L o c a l Rings

Proof.

We show t h a t

S/Io

f a c t o r s t h a t a r e isomorphic t o S

Tori(S/IO,A) PdAI

for

= 0

2s

admits a f i l t r a t i o n w i t h I t w i l l follow then t h a t

ZZ.

and whence t h e f i n i t e n e s s o f

i > 0

The key t o t h e p r o o f i s t h e f o l l o w i n g (2.7)

Lemma.

Let

xl,.

an A - s e q u e n c e i n any o r d e r . monomials i n

x2,

... , x n .

Then

o f t h e monomials g e n e r a t i n g

i s g e n e r a t e d by monomials i n monomials i n

*

ing

x3,

...,xn.

Pass t o t h e r i n g

images o f

Let

x2,.

. . ,xn

be e l e m e n t s c o n s t i t u t i n g b e an i d e a l g e n e r a t e d by

J

txl e J

W e may assume t h a t

Proof.

v e L.

. .,xn

x2

x2,

,

c o n s t i t u t e an

u = wxl e (K,L)

say

and

.

v, u e K,

+

A/(xl)-sequence.

Next we s u b s t i t u t e f o r

txl

v

u*x$ e L*, whence

u

J

and deduce

in the equation

(t-wx2)x1 e L .

Since

x1,x3

i s a l s o an A - s e q u e n c e we have a g a i n by i n d u c t i o n on

t-wx2 e L , hence

Writ-

W e make i n d u c t i o n on t h e

w e (K,L).

and f i n d

K

j u s t bv

L

sum o f t h e d e g r e e s o f t h e monomials g e n e r a t i n g

= ux2 +

where

n o t i n g t h e homomorphic

f o r homomorphic image we have

u e (xl , L )

(J.

E

J = (x2K,L)

t x l = ux2

,

t

a c t u a l l y o c c u r s i n one

... ,xn

We have A/(xl)

implies

Write

.J.

([GI):

,...,xn

n,

t e (x2K,L) = .T.

Now u s e t h e t ' s f o r t h e x ' s i n t h i s lemma and assume a l l

t ' s e n t e r i n t h e composition o f where

L

IO.

i s g e n e r a t e d by monomials i n

I1 = ( t l , L ) .

Then

I1/Io

S/L.

proof of t h e f i r s t p a r t of ( 2 . 6 ) .

Say

I.

=

(tlK,L),

. . , tn .

t2,.

Let

Induction ends e a s i l y t h e

Homology o f L o c a l Rings

26

- ... - - -

Let u s now p r o v e t h e e q u a l i t y o f d i m e n s i o n s . 0

-+

Pr

be a S - p r o j e c t i v e r e s o l u t i o n o f ated.

Since

I = I.

I.

Po

Io, w i t h

Let

0

f i n i t e l y gener-

Pi

has a f i n i t e projective resolution

Q A

by f i n i t e l y g e n e r a t e d A - p r o j e c t i v e m o d u l e s , we have t h a t pdAI = Sup { p d

1

I

We may assume

ZZ [ t l , .

maximal i d e a l o f

., , t n ]

l o n g e r S - s e q u e n c e f o r any p r i m e g r a d e I.

a).

p e

...,t n

p,tl, As

n-3

the

pdSIO = n - 1 .

-...-

is a

pdS(S/Io)

and c o n s i d e r a minimal S - r e s o l u t i o n f o r 0-L-F

(n > 2

(as

( [ E l o r ( 2 . 2 4 ) ) we c o n c l u d e

pdAI < n - 1

S

a t t h e i n v e r s e image o f t h e

pdSIO f n - 1

A.

A.

running o v e r t h e prime i d e a l s ‘of

l o c a l t h e n and s t i l l w r i t e f o r

A

localization of

p

2

Assume I.

F o’Io’o

as l o w e r cases are e a s i l y c o n s i d e r e d ) ; t e n s o r i n g w i t h

A we g e t

S

Torl(L,A) 0-F

and

= 0

- 0F

n-1

L 8 A n-2

((1.16)). ideal of

L. L

Since a l l t h e e n t r i e s o f S however, t h i s makes

Fn-l

We g e t from t h e above

are all @

0

or

A

l i e i n t h e maximal Q A = Torl(L,A)

= 0,

which i s a c o n t r a d i c t i o n . (2.8)

Corollary.

If

s e q u e n c e i n any o r d e r t h e n

,...,xn) i s pdA(xl,. . . ,xn)’ (xl

Let

-L-0

be a minimal S - f r e e r e s o l u t i o n f o r t h a t the F i t t i n g ’ s invariants of

i s a f r e e A-module.

a r e g u l a r A= n-1.

27

Homology o f L o c a l Rings

Kelation w i t h Ext: I f A i s a commutative r i n g and I i s an i d e a l o f A , w e write V ( I ) , t h e v a r i e t y o f I , f o r t h e s e t o f primes of A c o n t a i n i n g I . I f M i s a module, Sunp(bl), t h e s u p p o r t o f bl, w i l l d e n o t e t h e

primes

hln # 0 . I f M i s f i n i t e l y g e n e r a t e d c l e a r l y

p f o r which

-

.

Suvp ( Y ) = V ( a n n (M) )

The f o l l o w i n g i s b a s i c : Lemma. I f M i s a f i n i t e l y g e n e r a t e d module o v e r t h e

(2.9)

Noetherian r i n g A , then

Ass (HomA(M,N)) = Supp(M)n A s s (N) Proof. I f

-

p E

Ass(HomA(M,N)), t h e n a s HomA(M,N)

HornA (M , N ) # 0 , p

E

Sunp(M). I f p- = a n n ( f ( M ) ) ,

submodule o f N

p

E

E E E

and

E

u n i q u e maximal i d e a l o f A . Thus i f

=

f(M) is a

Supp(M) n A s s ( N ) n

E

N

is the

i s such t h a t

p,

ann(n) =

0 E

P

Ass(N).

C o n v e r s e l y , w e may assume p

and E

.

# Hom(M/pM,An)&Hom(M,An)C

Hom(M,N)

Ass (HomA(M,N)).

(2.10)

C o r o l l a r y . Let A be a N o e t h e r i a n r i n g and l e t I be

an i d e a l o f A and l e t M be a f i n i t e l y g e n e r a t e d A-module. The following a r e equivalent: i)

Hom(N,M)

= 0

f o r a l l f i n i t e l y g e n e r a t e d A-modules

N with Supp(N)SV(I).

ii)

Hom(N,M) = 0 f o r some f i n i t e l y g e n e r a t e d A-module N

w i t h Supp(N) = V ( 1 ) . iii)

Ass(M)nV(I) = 0

.

flomology o f Local Rings

28

iv)

There e x i s t s t

E

such t h a t t i s M-regular.

I

An e a s y i n d u c t i o n y i e l d s , C o r o l l a r y . Let A be a N o e t h e r i a n r i n p , l e t 1 be an

(2.11)

i d e a l o f A, l e t b l be a f i n i t e l y g e n e r a t e d A-module and l e t n be an i n t e g e r . The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t : Exti(N,bl) = 0

i) N

with

Supp(N)C V(1)

ii) N

with

Exti(N,M) = 0 Supp(N) = V ( 1 )

iii)

f o r a l l f i n i t e l y g e n e r a t e d A-modules and a l l i n t e p e r s i < n . f o r some f i n i t e l y g e n e r a t e d A-module and a l l i n t e g e r s i < n .

There e x i s t e l e m e n t s t l , .

..,tn

E

I

f o r m i n e an

? > I - r e g u l a rs e q u e n c e . 52.3

Macaulay r i n g s . Assume A t o be a N o e t h e r i a n r i n g t h r o u g h o u t t h i s s e c t i o n .

Let I be an i d e a l o f A and l e t .J be an i d e a l g e n e r a t e d by a maximal A-sequence i n I :

then

I c z (A/J)

.

I n p a r t i c u l a r some

prime i d e a l c o n t a i n i n g I w i l l a l s o be c o n t a i n e d i n z ( A / J )

and

s o will have t h e same g r a d e as I . In o t h e r w o r d s , cgrade I

=

i n f { g r a d e p; p minimal o v e r I } ht(p)

2 g r a d e p, f o r i f

xl,

.

I n g e n e r a l , f o r a p r i m e E,

...,xn

i s a r e g u l a r sequence i n

p, p c o n t a i n s n r o p e r l y some prime i d e a l minimal o v e r t h e i d e a l g e n e r a t e d by n-l

x ~ , . . . , x ~ -b~u t: s u c h p r i m e h a s h e i g h t a t l e a s t

by i n d u c t i o n . D e f i n i t i o n . A Macaulay ( o r Cohen-Macaulay) r i n g A i s one

f o r which

h e i g h t = g r a d e f o r e a c h i d e a l . S i m i l a r l y an A-module

E i s s a i d t o be a Macaulay module i f

I-depth E = height(I/J)

Homology o f L o c a l R i n g s f o r every i d e a l I?ann(E)

= J;

29

h e r e I / J i s viewed as an i d e a l

of A / J . By t h e r e m a r k s above and t h e u s u a l d e f i n i t i o n o f h e i g h t o f an i d e a l we may r e s t r i c t c o n s i d e r a t i o n t o p r i m e i d e a l s . P r o p o s i t i o n . I f h t ( 2 ) = g r a d e 2 f o r e a c h maximal

(2.12)

i d e a l , t h e n A i s a Macaulay r i n g . P r o o f . I f 2 i s maximum among t h e p r i m e s w i t h g r a d e E h t ( p ) , l e t pcm_ = maximal i d e a l ; we may assume a l s o t h a t A i s a l o c a l r i n g . Let (2.4)

while

g r a d e (p,a)

a

E

ht(2,a)

m_\p. Then g r a d e ( 2 , a ) 5 l + g r a d e p l+ht(,)

thus contradicting

by

ht(p,a) =

.

(2.13)

Corollary. I f

pg9

a r e i m m e d i a t e p r i m e s ( i . e . no

p r i m e s i n - b e t w e e n ) i n a Macaulay r i n g A , t h e n h t ( q ) = h t ( p ) + l . I n p a r t i c u l a r two s a t u r e d c h a i n s o f p r i m e s b e t w e e n two f i x e d p r i m e s h a v e t h e same l e n g t h . Remarks. i ) L e t A b e a Macaulay r i n g . Then AS and A / ( x ) a r e a l s o Macaulay r i n g s i f S i s a m u l t i p l i c a t i v e s e t and x i s a n o n z e r o d i v i s o r . The power s e r i e s r i n g A [ [ t ] ]

i s Macaulay as t

l i e s i n t h e J a c o b s o n r a d i c a l . A s f o r A [ t ] t h e s i t u a t i o n i s more i n t e r e s t i n g : L e t m_ b e a maximal i d e a l o f B = A [ t ] and l o c a l i z e at

FfIA

=

2. Then

h t ( m ) = l + h t ( p ) and m_ = ( p , f ) , where f may

b e t a k e n t o b e monic. But t h e n i t i s c l e a r t h a t 2 - d e p t h A = 2 - d e p t h B/(f), ii)

s i n c e t h i s l a s t module i s A - f r e e .

The p r e m i e r example o f a Macaulay r i n g i s a r e g u l a r

r i n g A : I n t h i s c a s e f o r e a c h maximal i d e a l IJ, m_Am i s a c t u a l l y

-

Homology o f L o c a l Rings

30

g e n e r a t e d by a r e p l a r A m - s e q u e n c e . iii)

-

I f A i s a f i n i t e l v e e n e r a t e d domain o v e r t h e f i e l d k-,

by h ' o e t h e r ' s n o r m a l i z a t i o n t h e o r e m ( [ ZS]) R

k

isomorphic t o a polynomial r i n g over

f i n i t e l y g e n e r a t e d a s an R-module.

t h e r e is a subring such t h a t A i s

I t i s well-known

(e.g. (2.23))

t h a t A l l a c a u l a y amounts t o s a y i n g t h a t 4 i s R - p r o j e c t i v e .

Now we c o n s i d e r t h e n o t i o n o f type

o f a l o c a l Macaulay

r i n g . Let A be a N o e t h e r i a n r i n g and l e t E be an A-module. We q u o t e from [3 ] t h e f o l l o w i n g f a c t s on a minimal i n j e c t i v e resolution of E : E-

0-

Each

Ii = 0 I(A/p)epi(p)

envelope o f A / p

I0

- I 1 ... where

I(A/p)

is the injective

and p i ( p ) = dim,(n)Exti(A/p,E) L

P'

We s h a l l

w r i t e p i (p;E) when m e n t i o n o f t h e module becomes n e c e s s a r y . Observe t h a t i f E i s f i n i t e l y g e n e r a t e d t h e n a l l p i ( p ) f i n i t e . Also, i n t h i s c a s e , pi(@ = 0

are

f o r i < p-depth E

according t o (2.11). Assume now E = A , A

For a p r i m e

a d - d i m e n s i o n a l l o c a l Macaulay r i n g .

p t h e f i r s t nonzero pi(p)

W e shall c a l l the integer pr(p)

occurs a t i = h t ( p ) = r.

t h e z-type o f A o r t h e t y p e

o f A a t p. pd(m_) w i l l be s i m p l y c a l l e d t h e t y p e o f A . An i n t e r p r e t a t i o n o f t h e t y p e o f A i s o b t a i n e d i n t h e f o l l o w i n g manner: Let x l , . . . ,xd be a maximal A-sequence i n m; a s d Ext (A/m,A) = Hom (A/E,A/ (5) 3 P d ( E t A ) = 'Jo ( 5 ) ;A/ ( 5 ) ); A/

(53

b u t t h i s l a s t q u a n t i t y can be i n t e r p r e t e d a s t h e number o f i r r e d u c i b l e components o f t h e m-primary i d e a l ( 5 ) . (2.14)

P r o p o s i t i o n . Let A be a N o e t h e r i a n r i n g and l e t M

Homology o f L o c a l Rings

p be an e l e m e n t o f

be a f i n i t e l y g e n e r a t e d A-module. Let Ass(M),

31

l e t x be a n o n z e r o d i v i s o r r e l a t i v e t o M w i t h M/xM # 0 ,

and l e t m be a p r i m e minimal o v e r ( p , ~ ) .Then m - is associated

2l.1~ (m;M/xM).

and v0 (p;M)

t o M/xM

P r o o f . We may assume t h a t A i s l o c a l and m_ i s i t s maximal

O:p be t h e submodule o f M a n n i h i l a t e d by p. and

i d e a l . Let d e n o t e by

p*

the ideal

( p , x ) / ( x ) o f A/(x) = A*. S i m i l a r l y l e t

O:p* be t h e submodule o f

a n n i h i l a t e d by p. I n t h e

M/xM = M*

s u r j e c t i on M*-

M-

t h e elements of a = x a ' ; since

a

E

0:p map

x ( 0 : p ) . Thus we have (0

O:p*. I f

into

pa = xpa' = 0

0

and

a E O:p

maps i n t o 0 ,

x i s a nonzero d i v i s o r ,

an i n j e c t i o n

:El / x ( 0 :p)L(0 :p") *

View now t h i s embedding as a s e q u e n c e of A/p-modules. Moreover,

since

m_

i s minimal o v e r ( p , x ) , A / p

has

~ / pa s i t s o n l y n o n -

t r i v i a l p r i m e i d e a l and t h e modules a r e A r t i n i a n . We now compute approximately t h e i r l e n g t h s . (2.15)

Lemma. Let A b e a l o c a l r i n g w i t h maximal i d e a l

m

a n d l e t I be a n m-primary i d e a l . Then f o r any f i n i t e l y g e n e r a t e d module M , L ( 0 : I ) z L ( A / I ) * l ( O :m) M

M-

.

P r o o f . We u s e i n d u c t i o n on L ( A / I ) ; i f L(A/I) = 1, "_ = I and t h e r e i s n o t h i n g t o show. I n any e v e n t , A/I c o n t a i n s a sub-. module i s o m o r p h i c t o A/I 0

-

with&(A/I) = L ( A / J ) get

A/;

+ 1.

g i v i n g r i s e t o an e x a c t s e q u e n c e

- A/1

A/J

p p l y i n g Horn(- ,M)

-0 t o t h i s s e q u e n c e we

Homology o f L o c a l Rings

32

0

-

- ( 0 : I ) - (0:m)-

(0:J)

w i t h t h e a n n i h i l a t o r modules t a k e n i n h l and l ( 0 : I )

'<

l(0:J)

+

l(0:m -) . We a p p l y t h i s t o t h e c a s e o f V/xN p l a y i n g M and I = ( p , x ) - ) = .e(A/(n,x))uO(m:bZ/xFI). t o get : X(O:p)_i l ( A / ( p , x ) ) l ( O : m

(2.16)

Lemma. Let A be a o n e - d i m e n s i o n a l l o c a l domain and

l e t M be a f i n i t e l y g e n e r a t e d A-module. Assumc t h a t x i s n o t a z e r o d i v i s o r r e l a t i v e t o Pi. Then e(al/x?l) = L ( A / ( x ) ) d ( b l ) , where d(M) i s t h e d i m e n s i o n o f t h e v e c t o r s p a c e

B1

(0)

over K , t h e

f i e l d of quotients of A. P r o o f . Given E.l t h e r e i s a c h a i n ?I = M

= ... ? M y

'41

0-.1-

o f submodules s u c h t h a t

= 0

i s e i t h e r A o r A / m-. I t follows

b!i/bfi+l

e a s i l y t h a t t h e number o f f a c t o r s i s o m o r p h i c t o A i s p r e c i s e l y d(M). C o n s i d e r t h e d i a g r a m 0 0-

PI1-

.XJ.

Fil

-U/E.x - -A/p M0

.XJ.

?lo

where p = 0 , ~ .Assume f i r s t p =

m:

and

X(Ml/xFll)

A/=

-Ml/xFll -

= L ( M g / ~ ? 4 0 ) ,d(PI1)

0

s i n c e x i s a nonzero d i v i s o r

r e l a t i v e t o h i , we h a v e t h e s e o u e n c e 0 -

0

t

blO/~?lg

= d(?4,,).

-

A/m_--

0

The a n a l y s i s o f t h e

o t h e r c a s e i s s i m i l a r and an e a s y i n d u c t i o n c o m p l e t e s t h e p r o o f I f w e a p p l y t h i s t o O:p

i n t h e n o t a t i o n o f ( 2 . 1 4 ) w e can

c o m p l e t e t h e p r o o f o f t h e n r o p o s i t i o n a s l ( ( O : p-) / x ( O : p ) ) = l ( A / ( p , x ) ) . d ( O : n ) = L ( A / ( p , x ) ) .uO(p;W.

Homology o f L o c a l Rings (2.17)

33

C o r o l l a r y . The t y p e o f a Macaulay r i n g d o e s n o t

i n c r e a s e s under l o c a l i z a t i o n . Projective

92.4

&

i n j e c t i v e dimensions.

The e m p h a s i s i n t h i s s e c t i o n i s on h o m o l o g i c a l d i m e n s i o n s o f f i n i t e l y g e n e r a t e d modules o v e r a l o c a l N o e t h e r i a n r i n g A . m_ w i l l d e n o t e t h e maximal i d e a l o f A and k - i t s residue f i e l d .

We b e g i n w i t h some remarks on c o n n e c t e d s e q u e n c e s o f l i n e a r f u n c t o r s from mod(A)

,

A-

t h e category of f i n i t e l y generated

A-modules, i n t o mod(B), where B i s a l o c a l N o e t h e r i a n r i n g and B i s a l o c a l homomorphism. The f u n c t o r s we have m o s t l y

h-: A ->

i n mind a r e Ext ' s and T o r ' s . P r o p o s i t i o n . Let { T i ,

(2.18)

i > 0 1 be a c o n n e c t e d

s e q u e n c e o f ( c o ) c o n t r a v a r i a n t f u n c t o r s o f mod(A). F o r e a c h h1

E

mod(A)

s u p {d(M)l

write

d(M) = s u p { i

.

We s h a l l r e f e r t o a l s o write

d(M)

d(M) = T-dim(M) )

I

Ti(FI) # 0 )

.

Then

d(k) =

a s t h e T - d i m e n s i o n of M ( a n d w i l l and t o

d(k)

as t h e cohomologi-

c a l dimension o f T. P r o o f . We g i v e a p r o o f i n t h e c o n t r a v a r i a n t c a s e . I f d ( k -)

i s i n f i n i t e t h e r e i s n o t h i n g t o p r o v e . Assume d ( k ) f i n i t e a n d l e t n = s + l . Our c l a i m i s t h a t

= s is

Ti = 0 f o r i

2 n.

C l e a r l y by t h e h a l f - e x a c t n e s s o f t h e T ' s , Ti(M) = 0 f o r a module M o f f i n i t e l e n g t h and i

L n . Assume t h i s t r u e f o r modules o f

d i m e n s i o n l e s s t h a n r and l e t p be a p r i m e w i t h dim(A/p) = r . Pick

x

E

m_\p and form t h e e x a c t s e q u e n c e

Homology o f L o c a l Rings

34

Alp-

0 -

.x -

A/p

- A/(p,x)

.

- 0

By t h e c o n n e c t e d n e s s i t y i e l d s

-Tn+1( A / ( p , x ) )

T " ( A / ~ )3 T ~ w ~ )

= 0.

The l i n e a r i t y o f t h e T ' s and Nakayama's lemma f o r c e s Tn(A/p)=O. (2.19)

i > 0)

P r o p o s i t i o n . Let I T i ,

with

some T i # 0

be a c o n n e c t e d s e q u e n c e o f A - l i n e a r f u n c t o r s . I f T-dim(k) - < t h e n f o r any bl

E

mod(A), T-dim(F1)

m,

FI = T - d i m (k).

+ m-depth -

Proof. First observe t h a t i f m - i s a s s o c i a t e d t o t h e module M t h e r e is a sequence 0

and

T r ( k-) # 0

-k -F1-

C

-

for

r

-

0

( 2 . 1 8 ) , Tr(M) # 0 .

= T - d i m (k ) f o r c e s , by

I f , on t h e o t h e r h a n d , x i s a n o n z e r o d i v i s o r r e l a t i v e t o M , by Nakayama's lemma

we g e t

TS(Fl) # 0

s + l = T-dim(M/xM).

if

A f t e r t h e s e remarks i t f o l l o w s t h a t

-

T-dim(k) -

is

T-dim(b1)

p r e c i s e l y t h e l e n g t h o f a maximal M-sequence i n m. i Let E be a f i n i t e l y g c n e r a t e d module and p u t T i = ExtA(-,E].

Thus T - d i m (k) i s f i n i t e i f € E has f i n i t e i n j e c t i v e dimension. Since

T-dim(A) = 0 , we have i n t h i s c a s e (2.20)

C o r o l l a r y . idAE = m-depth A

=

T-dim(M)

+

m-depth P1

f o r any f i n i t e l y g e n e r a t e d module M . Now w e examine t h e p r o j e c t i v e d i m e n s i o n o f a f i n i t e l y g e n c r a t e d module r.1

u s i n g T o r ' s . Let

r

= pd >I

and l e t x l , . .

.

PXS

be a maximal ? I - s c q u e n c e i n 1 ' 1. I t f o l l o w s e a s i l y t h a t pd M/(x)M = r

(2.21)

+ s.

Proposition. I f

pd F I <

and

m_

E

Ass(PI),

then

Homology o f L o c a l Rings

35

pd M = m_-depth A . Proof. Let K. (&:A)

xl,

... , x n

he a maximal A-seouence i n

t h e c o r r e s p o n d i n F Koszul comnlex. But

= O:(x) - # 0 Hn(K.(x;A)PM) M

and t h u s

pd Y

m_

and

Tor, ( A / ( 5 ) ,‘I) =

n.

B e f o r e p r o v i n g t h e r e v e r s e i n e q u a l i t y we r e c a l l a b a s i c r e s u l t o f t h e l i n c a r a l g e b r a o v e r commutativc r i n g s

([El) :

P r o p o s i t i o n (McCoy’s t h e o r e m ) . L e t A b e a commutan be a homomorphism o f A-modules t i v e r i n g and l e t Q, : Mm --> M (2.22)

g i v e n by an

n x m matrix

(aij).

g e n e r a t e d by t h e m i n o r s o f o r d e r

Q, i s i n j e c t i v e i f f t h e i d e a l m

d o e s n o t a n n i h i l a t e a non-

z e r o e l e m e n t o f M. Back t o t h e p r o o f o f ( 2 . 2 1 ) = n

: Let M be a module w i t h pd M

and l e t

-

-

...

on Fo M0 Fn-l Fn be a minimal p r o j e c t i v e r e s o l u t i o n of M . Assume t h a t Fnhas rank 0-

t ; l e t I b e t h e i d e a l g e n e r a t e d by t h e m i n o r s o f $n

of order t .

We c l a i m t h a t I c o n t a i n s a r e g u l a r A-sequence o f n e l e m e n t s a t l e a s t . Let

x1

b e an e l e m e n t o f I which i s n o t a z e r o d i v i s o r

o f A ( a c h i e v e d t h r o u g h McCoy’s t h e o r e m ) . T e n s o r 0-

by

A/(xl)

Fn

- Fn-l @Il

...

F1

Im(+l)-

0

t o g e t a new e x a c t s e q u e n c e which r e p r e s e n t s

a

minimal p r o j e c t i v e r e s o l u t i o n o f t h e A/(xl) -module im(Q,l)BdA/(x,) I n d u c t i o n now e n d s t h e p r o o f . (2.23)

C o r o l l a r y ( E q u a l i t y of Auslander-Buchsbaum)

.

Let A

be a l o c a l N o e t h e r i a n r i n g and l e t M be a f i n i t e l y g e n e r a t e d

.

Homology o f L o c a l Rings

36

A-module w i t h pd hi < (2.24)

A. Y = m-depth -

Then pd bf + !-depth

m.

C o r o l l a r y . Let M b e a f i n i t e l y g e n e r a t e d module

o v e r t h e N o e t h e r i a n r i n g A and l e t

p

p

g r a d e p; m o r e o v e r , i f pd 1\1 i s f i n i t e , g r a d e P r o o f . pdAbi t h e proof t h a t

2 pdA M p P

depth A

A-sequence i n p. As 2

i s a prime

g?p, 1

P

= depth A =

P

=

=

l o c a l g r a d e E.

l o c a l grade

g r a d e p: Let

xl,.

pd M >

Then

Ass(M).

E

..,xr

p. Now

for

be a maximal

c o n s i s t s o f zero d i v i s o r s of A / ( & ) ,

E

Ass(A/(x)).

there

L o c a l i z i n g a t 9 wc c o n c l u d e

> pd M = d e p t h A pd E! = r 9 P P'

The argument a l s o shows t h a t p i s a c t u a l l y an e l e m e n t A s s ( A / ( x-) ) .

of

A d i f f i c u l t question i n t h i s a r e a i s t h e following

( A u s l a n d e r ) : D e f i n e t h e g r a d e o f a module t o be t h e g r a d e o f i t s a n n i h i l a t o r ; i s i t t h e c a s e t h a t g r a d e bl

+

K r u l l dim(M)

=

K r u l l dim(A)? S e e [ _ 1 7 ,30] f o r a deep d i s c u s s i o n o f t h i s nroblem.

§2.5

of

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

modules.

Here a c l o s e r l o o k a t t h e n a t u r e o f f i n i t e p r o j e c t i v e r e s o l u t i o n s i s t a k e n . Let A b e a commutative r i n g and l e t 0-

Fn

...

F1

-

M-

Fo-

b e a f i n i t e f r e e r e s o l u t i o n o f t h e module M.

characteristic of M,

x(M), x(M)

=

C(-l)irank(Fi).

c'cpend o f t h e c h o s e n r e s o l u t i o n ( s e e

x that

Define t h e Euler

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

I t i s e a s i l y s e e n from S c h a n u e l ' s lemma

properties of

0

that

[El

x(M)

does n o t

f o r t h e elementary

w e s h a l l u s e ) . W i t h o u t f u r t h e r ado w e u s e

Homology o f L o c a l Rings t h a t x (M)

s t a y s t h e same f o r any f l a t change o f r i n g s .

Definition. I f $ : E

~

(

$

= 1sup{

$

is a

: E-F

uPlhl

I I$ +

r

E = M@)m

If

37

where

and

-F

i s a homomorphism o f A-modules,

03. F = M@)"

M-matricial

a r e A-modules, w e s a y t h a t homomorphism i f $ i s g i v e n

as

u : Am-An.

I n t h i s t e r m i n o l o g y t h e s t a t e m e n t o f McCoy's theorem i s : a m a t r i c i a l homomorphism $ : PIernrank($) = m

is injective i f f

Men

and t h e i d e a l g e n e r a t e d by t h e

m x m

minors of

u d o e s n o t a n n i h i l a t e a n o n z e r o e l e m e n t o f Y. F i n a l l y , u : Am

- An

minors of u , (2.25)

define

.

P r o p o s i t i o n . Let

0-

Fn

r x r

I ( u ) = i d e a l g e n e r a t e d by a l l

r = rank(u)

if

'n Fn-l -

...

Fo

- FI -

be a f i n i t e f r e e r e s o l u t i o n o f t h e module ? I .

If all

0

I(Oi) = A

t h e n M i s a p r o j e c t i v e module. The p r o o f p r o c e e d s by showing t h a t

Fn s p l i t s o f f

Fn-l

and i n d u c t i o n . Note t h a t t h e c o n v e r s e a l s o h o l d s . (2.26)

P r o p o s i t i o n . Let M b e a module a d m i t t i n g a f i n i t e

f r e e r e s o l u t i o n . Then

i) ii)

x(W 2

x(M)

0.

> 0

i i i ) x(M) = 0

(2.27)

i f f M is faithful. iff

ann(Y)

is faithful.

Lemma. Let 0-F-G-H

U

V

Homology o f L o c a l Rings

38

b e an e x a c t s e q u e n c e o f f r e e A-modules. Then r a n k ( u )

+

rank(v)

= rank((;).

P r o o f . T e n s o r t h e s e q u e n c e ( i . e . c h a n p t h e r i n g ) by A [ t ] =

p o l y n o m i a l r i n g i n t . Then I ( u ) b e i n g a f i n i t e l y g e n e r a t e d

f a i t h f u l i d e a l a c q u i r e s a nonzero d i v i s o r i n A [ t ] . Localize a t t h e powers o f a n o n z e r o d i v i s o r o f I ( u ) A [ t ] ; F h a A [ t l S t h e n s p l i t s

o f f G@A[tIS and r a n k ( v ) = r a n k ( ( ; ) - r a n k ( I m ( u ) ) . S i n c e t h e c h a n g e s o f r i n g s l e a v e t h e r a n k s u n c h a n g e d , we have t h e d e s i r e d equality. Proof o f ( 2 . 2 6 ) t h a t a l l I($i)

: P a s s t o A [ t ] ; t h e e a r l i e r argument shows

are f a i t h f u l i d e a l s - the case of I($n) being

t h e s t a t e m e n t o f McCoy's t h e o r e m . I f we now l o c a l i z e a t t h e s e t o f r e g u l a r e l e m e n t s o f A [ t ] , t h e s e q u e n c e s p l i t s D i e c e w i s e and i ) , i i ) and i i i ) f o l l o w . (2.28)

C o r o l l a r y . I f A i s a l o c a l N o e t h e r i a n r i n g and Eil

i s a f i n i t e l y g e n e r a t e d module o f f i n i t e p r o j e c t i v e d i m e n s i o n , t h e n M i s e i t h e r f a i t h f u l o r i t s a n n i h i l a t o r c o n t a i n s a nonzero divisor. We now c o n s i d e r some o f t h e same q u e s t i o n s f o r modules o f f i n i t e i n j e c t i v e dimension o v e r a l o c a l Noetherian r i n g A. (2.29)

P r o p o s i t i o n . L e t M b e a f i n i t e l y g e n e r a t e d module

o v e r A o f f i n i t e i n j e c t i v e d i m e n s i o n . Then M i s e i t h e r f a i t h f u l or its annihilator In Chapter 5

is faithful. w e s h a l l d i s c u s s an E u l e r c h a r a c t e r i s t i c f o r

M and a d i f f e r e n t p r o o f o f ( 2 . 2 9 ) . Now w e n e e d a few lemmas.

Homology o f L o c a l Rings

39

Let M be a n o n z e r o , f i n i t e l y g e n e r a t e d module o f f i n i t e i n j e c t i v e d i m e n s i o n o v e r t h e l o c a l r i n g A . We saw i d N = depth A which i s a l s o t h e maximum v a l u e f o r t h e d e p t h o f any f i n i t e l y g e n e r a t e d A-module a c c o r d i n g t o ( 2 . 2 0 )

(This f a c t has provided

H o c h s t e r w i t h an o p e n i n g f o r h i s s o l u t i o n o f v a r i o u s q u e s t i o n s 18]). In p a r t i c u l a r , i f on t h e homology o f N o e t h e r i a n r i n g s [i d M = 0 , A has a l s o Krull dimension 0 such a s A/p,

p

a s o t h e r w i s e a module

= minimal p r i m e , would p r o v i d e an example w i t h

depth > 0. (2.30)

i d bZ = 1, t h e n A i s a Macaulay r i n g .

Lemma. I f

P r o o f . From ( 2 . 2 0 ) i t f o l l o w s t h a t A may be assumed t o be c o m p l e t e ; i f p i s a minimal p r i m e w i t h

K-dim A = K-dim(A/p)

I

and B i s t h e i n t e g r a l c l o s u r e o f A/P- (which i s f i n i t e l y g e n e r a ([2 8 ] ) we have :

t e d a s an A/p-module

m-depth B > 2

if

> 2 by [ 3 4 ] . Thus A i s o n e - d i m e n s i o n a l . K-dim A -

(2.31)

Lemma [Abhyankar-liartshorne's lemma). Let I and J

be n o n z e r o i d e a l s i n a commutative r i n g A s u c h t h a t 1 - J = 0 . Then

grade ( I + J )

i s a t most o n e .

P r o o f . We can assume t h a t x E I

n.J,

r i n g . Let

then

i = r(i+j)

n,J

= 0

f o r otherwise, i f 0 #

x ( I + J ) = 0 . We can e v e n assume A t o be a l o c a l

x = i+j, i

is clear that

I

E

I and j

E

i # 0, j # 0 . Also,

yields

.J, b e a nonzero d i v i s o r : it i $

A ( i + j ) for an equation

( 1 - r ) i = r j , which i s a c o n t r a d i c t i o n i f

r i s 'a u n i t o r n o t . F i n a l l y , ( I + J ) i C _A ( i + j ) , t h a t is grade(I+J)

= 1.

I n o r d e r t o a p p l y t h i s t o o u r q u e s t i o n , l e t I be t h e a n n i h i l a t o r o f M , l e t J be t h e a n n i h i l a t o r of I and l e t J ' b e t h e

Homology o f L o c a l R i n g s

40

a n n i h i l a t o r o f J . By t h e lemma, g r a d e ( J + J ' ) < 1 , i n f a c t = 1. -

F o r o t h e r w i s e I and J would be c o n t a i n e d i n t h e same minimal nrime E o f A : Y

would t h e n be a n o n z e r o , n o n f a i t h f u 1 , i n j e c t i v e

P

! Let p t h e n be a e;rade

module o v e r t h e l o c a l A r t i n i a n r i n g A

?!. J + J ' ; it i s e a s i l y seen t h a t

one prime i d e a l c o n t a i n i n g h a s d e p t h o n e . We c l a i m t h a t

.J

n_

A

# 0 - t h u s implying t h a t I

which i s n o t t r i v i a l , c o n s i s t s o f z e r o d i v i s o r s . But t h i s indeed t h e case

,J' # A

as

P

E

i s c l e a r l y impossible. This

P

P' is

is

t h e r e q u i r e d r e d u c t i o n . We c a n make a f r e s h s t a r t and assume t h a t Fl h a s i n j e c t i v e d i m e n s i o n one o v e r t h e l o c a l r i n g .4 o f d i mension o n e . T h e r e a r e two c a s e s t o e x a m i n e .

m

i)

is not associated t o M.

Ilere t h e o n l y p r i m e s a s s o c i a t e d t o M a r e t h e minimal p r i m e s of A containing I . L e t

n

0-34-1

- I

1

be a minimal i n j e c t i v e r e s o l u t i o n o f h4.

- 0

I"

i s a d i r e c t sum

of

I ( A / 2 ) = i n j e c t i v e e n v e l o p e o f A/p, f o r t h e v a r i o u s 1 p r i m e s o f h e i g h t 0 c o n t a i n i n g I ; I i s a d i r e c t sum o f c o p i e s copies of

o f I (A/m). -

Let

pl,.

. . ,yr

be t h e above minimal p r i m e s . We can

t h e n p i c k x i n some o t h e r minimal prime (one c o n t a i n i n g J ) b u t n o t i n any o f t h e pi's. biap t h e s e q u e n c e i n t o i t s e l f v i a m u l t i p l i c a t i o n by x and u s e t h e s n a k e lemma t o g e t 0-

X

-

31

X

In

-

X

I1

-

M/xM

-In

/XI

o

-I

1 /XI1-&

S i n c e x i s n o t i n any o f t h e p i ' s , i t a c t s a s a u n i t i n I ( A / p i ) and t h u s

([?I)

= ?I/xPI. But

i s an i n j e c t i v e A/(x)-module

# 3l/xM

i s a f i n i t e l y generated i n j e c t i v e

and t h u s

0

Homology o f L o c a l Rings module o v e r

By an e a r l i e r remark

A/(x).

41

R/(x) i s then

A r t i n i a n r i n g , which i s a c o n t r a d i c t i o n a s x was t a k e n

an in

a

minimal p r i m e . ii)

m- i s a s s o c i a t e d t o Y .

We u s e t h e n o t a t i o n o f i ) . A minimal i n j e c t i v e r e s o l u t i o n o f M now l o o k s l i k e 0-

I 0 @ Et-

bl-

ES

-0

and t h e i n t e g e r s t a n d s a r e d e t e r m i n e d b y 1 t = dimk(Ilom(A/m,M)) and s = d i m k ( E x t (A/m,M)). Since is

E = I(A/m)

where

-

-

a s s o c i a t e d t o b l , t > 0 . Let p

be a prime n o t r e p r e s e n t e d i n

I o , t h a t i s a m i n i m a l prime o f J . A p n l y i n g

-

0-

Hom(A/p,M)

exact

1 a s Ext (A/p,M)

Hom(A/p,I* d E t ) for

= 0

fiom(A/n,-) we g e t

- Hom(A/p,ES) -

d e n t h A/E

0

= 1. A n o t h e r way

to

write t h i s l a s t sequence i s 0-

But 1'.

E

Io = 0 as

IcfP

p

P

I

0

d

Et-

P

ES-O.

P

i s n o t c o n t a i n e d i n any o f t h e p r i m e s o f

Thus we g e t

o r , i n o t h e r words, t h a t

bl

P

i s a n o n z e r o module o f f i n i t e

l e n g t h a n d i n j e c t i v e d i m e n s i o n o n e o v e r A l p . From t h e d u a l i t y theory of

[27] i t

Define g(C) g(C)

follows t h a t

t = s.

f o r any A-module C o f f i n i t e l e n g t h t o be = l e n g t h Hom(C,M)

I t is clear that g(-)

- length Ext1 (C,?I).

i s additive with respect t o short exact

sequences. Since g(k) = t - s = 0

and any module o f

finite

Homology o f L o c a l Rings

42

L's,

l e n g t h i s an e x t e n s i o n o f

we have g ( C )

= 0

f o r any s u c h

module. Let

be t h e l a r g e s t submodule o f f i n i t e l e n g t h o f PI.

)lo

Also l e t

be a n o n z e r o d i v i s o r o f A

x

i s not associated t o

M*

= ?l/b10

a g a i n we g e t 0

-?'lo -M-

But

X

xb!*

and

0

=

X

M*

xYO = 0 . m_

such t h a t

# 0 . UsinR t h e s n a k e lemma

- bIO

t h u s length(MO)

+

*

?I*/xM -0.

M/xbl

length(M*/xM*)

=

length(M/xbl). On t h e o t h e r h a n d t h e e a c t s e q u e n c e 'X

0-A-A-

yields that A/(x)

has

Hom(A/(x) , M )

A/(x) = \I0

and

-

0

E x t 1 ( A / ( x ) ,hi)

= M/xM.

As

f i n i t e l e n g t h , g ( A / ( x ) ) = 0 , which i m n l i e s

length(M*/xM*)

= 0 , c o n t r a d i c t i n g Nakayama's

lemma.

We now r e t u r n t o t h e e x a m i n a t i o n o f t h e i d e a l s

I(@i)

in

a f i n i t e f r e e r e s o l u t i o n . The b r o a d e s t s t a t e m e n t on t h e s e 7 , -3 0 ] ) : i d e a l s is contained i n the following ( [_ (2.32)

Theorem. Let A be a N o e t h e r i a n r i n g and l e t bf

be

a f i n i t e l y g e n e r a t e d A-module. L e t M.

...

:

VTk

@ k LI'k-1. . . -

be an M - m a t r i c i a l complex w i t h i s e x a c t ( i . e . Hi(M.) i)

ii)

= 0

€ o r l a r g e i . Then

>.I.

for i > 0) i f f

I ( @ k ) - d e p t h Fl > k; rank($k)

Proof. I f the ' l a s t '

= 0

ri

I(+)

M.

+

rank(@k-l) = rk-l.

i s e x a c t , p i c k i n g an

M-regular element i n

- it e x i s t s by McCoy's t h e o r e m - and l o c a l i -

z i n g a t t h e m u l t i p l i c a t i v e s e t i t g e n e r a t e s w e may r e d u c e t h e

Homology o f L o c a l Rings

43

l e n g t h o f t h e complex : i i ) would t h e n f o l l o w much a s f o r v e c t o r s p a c e s ; i ) i s a g a i n a d i r e c t c o n s e q u e n c e o f McCoy's theorem.

For t h e c o n v e r s e , t h e f i r s t t h i n g t o n o t e i s s i n c e t h e ideals

contain

I(+)

M-regular elements

it i s s t a b l e under

l o c a l i z a t i o n s . Let n b e t h e l a r g e s t i n t e g e r statement of

ii)

r a n k ( O n ) = rn

makes

rn # 0 ; t h e

with

and t h u s $n i s

injective BY P c k i n g an e l e m e n t t h a t i s M - r e g u l a r

statement ted

ii)

i m p l i e s t h a t t h e homology o f

in

I ( + k ) , the

M.

is annihila-

by an M - r e g u l a r e l e m e n t . We may assume A l o c a l and a l s o

t h a t t h e homology g r o u p s a l l have f i n i t e l e n g t h . We a r g u e by i n d u c t i o n on n : 0

Mrn

-

. . . M r 2 - L -0 ,

M'n-1

L = image ( + 2 ) , i s t h e n an e x a c t s e q u e n c e . I t a l s o f o l l o w s t h a t

as

> n , depth L > 2 . Let depth M -

c o n s i d e r t h e sequence 0 -L

If

K = kernel($1)

-

K-

II-

and

0.

x i s an e l e m e n t i n t h e a n n i h i l a t o r o f M t h a t i s P I - r e g u l a r ,

wc get 0

-tl -L / x L - K / x K -1-1 -0 ,

and t h e maximal i d e a l o f A w i l l b e a s s o c i a t e d t o L/xL, contradicting (2.33)

> 1, u n l e s s d e p t h L/xL -

11 = 0 .

C o r o l l a r y . Let

...

-

Fo-E-Q F1 be a f i n i t e f r e e r e s o l u t i o n o f t h e module E . Let M b e O-Fn

thus

a

Homology o f L o c a l Rings

44

f i n i t e l y g e n e r a t e d A-module w i t h t h e f o l l o w i n g p r o p e r t y : I f I

rc

i s an i d e a l o f g r a d e

n , t h e n I - d e p t h bl = r . Then t h e

sequence 0

-FnPF1

. ..

FIPP.l

-

I:O@Pl

-EPbl -

0

is also exact. Gorenstein r i n g s .

52.6

W e s t a r t b y l o o k i n g a t t h e s u n a o r t of a

f i n i t e l y genera-

t e d A-module o f f i n i t e i n j e c t i v e d i m e n s i o n ( A i s a N o e t h e r i a n local ring). (2.34)

P r o p o s i t i o n . i ) Sunn(b1)

s a t i s f i e s the equal chain

c o n d i t i o n - i . e . a l l maximal c h a i n s o f n r i m e s between two f i x e d p r i m e s have t h e same l e n g t h . ii)

K-dim(M) = d e p t h A - i n f { d e p t h A p , n - minimal o v e r

-

t h e a n n i h i l a t o r of Fl). P r o o f . Let = id M

P

p be an e l e m e n t o f

) = 1

1 s x c g\c, w e have Extr(A/p,M)

-

we claim t h a t

depth A

Extr+'(A/

= depth A

- with

( p , ~ ,&I)% )

P

= r+l. Indeed, i f

I I

- E x t r +1( A / ( ~ , X,)M )

5Extr(A/p,M)

Extr(A/p,Ft)p # 0 , we a l s o g e t

since

r

I f 1 i s a p r i m e immediate o v e r

by ( 2 . 2 0 ) .

K-dim(A /PA

Supp(b1) : t h e n

Extr(A/p,M)

1

#

;

0 and t h u s

# 0 by Nakayama's lemma. T h i s shows t h a t

> r+l. 9For t h e r e v e r s e i n e q u a l i t y w e b e g i n by showing t h a t

depth A

grade p = depth A

P

f o r any

p

E

Supp(bt). Let p be maximum w i t h

Homology o f L o c a l R i n g s with the property t h a t

depth A

P

> grade

be a maximal A-sequence i n p. Then t h e a s s o c i a t e d primes of A / ( x ) , Exti(A/(x),E.l)

-

grade p

9

= 0

E

p and l e t x l , . - .

xS

i s c o n t a i n e d i n one of

s a y 9. W e t h e n have

i > s . By ( 2 . 2 0 )

for

45

and t h u s by t h e p r e c e d i n g

s = depth A

9

=

1 = p.

I n p a r t i c u l a r i f 1\1 i s a f a i t h f u l module K-dim(b1) = dim(A) and A i s a Macaulay r i n g . A t t h i s p o i n t we i n s e r t B a s s ' s c o n j e c t u r e : I f a l o c a l r i n g A admits a nonzero f i n i t e l y g e n e r a t e d module M o f f i n i t e i n j e c t i v e d i m e n s i o n , t h e n A i s a Macaulay r i n g . T h i s q u e s t i o n h a s been s e t t l e d ( [ 1 7 , 3 0 ] )

for a l l local

r i n g s o f e q u a l c h a r a c t e r i s t i c and f o r s p e c i a l m o d u l e s . D e f i n i t i o n . A l o c a l Noetherian r i n g A i s s a i d t o be Gorenstein r i n g i f

idAA <

I t follows t h a t prime

A

2

a

a.

i s a l s o a Gorenstein r i n g f o r each

p. G l o b a l G o r e n s t e i n r i n g s c o u l d t h e n b e d e f i n e d i n t h e

same manner a s above o r a s b e i n g l o c a l l y G o r e n s t e i n ; c l e a r l y t h e s e d e f i n i t i o n s would a g r e e f o r r i n g s o f f i n i t e K r u l l dimension.

I f A is an

n-dimensional

(local) Gorenstein r i n g , then

Extn(A/m,A) -

i)

# 0

i s t h e o n l y n o n z e r o o f s u c h Ext m o d u l e s . To examine t h i s module l e t xl,

...,xn

be a r e g u l a r A-sequence; by changing t h e

r i n g ( i . e . u s i n g ( 1 . 1 1 ) ) we g e t Extl(A/m,A)

= HornA/

-

(A/m_,A/ (XI I and E x t i /

-

(A/m_,A/

(XI 1

= 0

Homology o f L o c a l Rings

46

f o r i > 0 . Thus A / ( x ) indecomposable

i s a s e l f - i n j e c t i v e r i n g . Since

i t must be t h e i n j e c t i v e e n v e l o p e o f e a c h o f

i t s nonzero submodules, i n p a r t i c u l a r o f iIomA/

-

(A/F,A/

We r e p l a c e

it is

( 5 )1

*

A;

= A / m-.

i)

(1

m

c g

.

Hence

by t h e c o n d i t i o n s

A i s a Macaulay r i n g ;

ii)

Extn(A/m,A) = A/!.

iii)

The meaning o f

iii)

becomes c l e a r i n a minimal i n j e c t i v e

r e s o l u t i o n o f A : t h a t f o r each prime i d e a l p of h e i g h t r , then I(A/p) summand o f

appears only once i n such r e s o l u t i o n - a s I r . Thus

a

A h a s c o n s t a n t t y p e = 1. S t i l l a n o t h e r

way o f i n t e r p r e t i n g i i i )

is :

the ideal

(x)

generated

by

any s y s t e m o f p a r a m e t e r s o f a G o r e n s t e i n r i n g i s i r r e d u c i b l e . Wc may e v e n p r o v e a c o n v c r s e : (2.35)

Theorem. Let A be a l o c a l N o e t h e r i a n r i n g s u c h

t h a t e v e r y s y s t e m of p a r a m e t e r s g e n e r a t e s an i r r e d u c i b l e i d e a l . Then A i s a G o r e n s t e i n r i n g . P r o o f . To make an i n d u c t i o n on t h e d i m e n s i o n d o f A

we

..

c o n t a i n s some n o n z e r o d i v i s o r . Let x l , . , n n x d be a s y s t e m o f p a r a m e t e r s and l e t I n = ( x l , . . , x d ) . Then Ind - 1 , that contradicts Inf I n - 1 f o r o t ~ i e r w i s e l y d = n n i s a l s o a system o f Nakayama's lemma. Now e a c h s e t x l , ,xd f i r s t show t h a t

m

.

...

p a r a m e t e r s and s o t h e i r r e d u c i b i l i t y of In

I n ' s a r e a l l i r r e d u c i b l e i d e a l s . Rut t h e implies t h a t

i d e a l s pronerly containing In. Since fI(In:!)

= O:;,

i.e.

In:m

is contained i n a l l

In = 0 , we have

0 =

m_ c o n t a i n s some n o n z e r o d i v i s o r , s a y x .

Homology o f L o c a l Rings

41

P a s s t o A / ( x ) ; t h e same h y p o t h e s i s f o r s y s t e m s of p a r a m e t e r s i s i n h e r i t e d by A / ( x ) . Remark. I t can be shown t h a t a l t h o u g h t h e number o f i r r e d u c i b l e components o f t h e i d e a l g e n e r a t e d by a s y s t e m o f p a r a m e t e r s i n a l o c a l Macaulay r i n g i s c o n s t a n t t h e c o n v e r s e d o e s n o t always h o l d . The s t a t e m e n t above s a y s t h a t i f t h i s constant is

one

t h e n t h e r i n g i s i n d e e d blacaulay.

I n t h e same s p i r i t o f ( 2 . 3 5 ) i s (2.36)

Theorem. Let A be a l o c a l N o e t h e r i a n r i n g

such

t h a t f o r e v e r y i d e a l I g e n e r a t e d by a s y s t e m o f p a r a m e t e r s

i s A / I - f r e e . Then A i s a blacaulay r i n g .

I/Iz

S i n c e i d e a l s g e n e r a t e d by A - s e q u e n c e s have t h i s ' i n d e p e n d e n c e ' p r o p e r t y , ( 2 . 3 6 ) i s a c h a r a c t e r i z a t i o n o f Macaulay rings. P r o o f . dim .4 = 1 : Let

x be a s y s t e m o f p a r a m e t e r s ( s . o .

p . f o r s h o r t ) ; t h e n xn f o r a l l i n t e g e r s n i s a l s o a s . 0 . p . .

assumption i s t h a t (x"). then

(xn)/(xn)'

= A/(xn),

that is

(x2")

The

: xn =

We c l a i m t h a t x i s n o t a z e r o d i v i s o r : i n f a c t , i f r x = 0 n r x = 0 and so r E (x") f o r a l l n . By t h e i n t e r s e c t i o n

theorem

r = 0.

dim A > 1 : Assume t h e s t a t e m e n t t r u e f o r r i n g s o f l o w e r d i m e n s i o n . Let

x l , ...,xd

be a s . 0 . p . .

Pass t o A ' = A/(xl);

it

i s c l e a r t h a t A ' i n h e r i t s t h e independence p r o p e r t y f o r i d e a l s g e n e r a t e d by s . 0 . p .

I s .

By i n d u c t i o n t h e n x j , .

.., x i

form an

A ' - s e q u e n c e . I n terms o f A t h i s means t h a t (x1):Xz = ( x i ) ,

...,

( X 1 , * * . 9 X d - l ) : X d= ( X 1 9 " ' 9 X d - 1 ) *

Ilomology o f L o c a l R i n e s

48

Since

. . 7 ~n d - 1 , x d i s

n xl,.

..

we have

theorem

, t h e r e l a t i o n s above

S . O . ~ .

we s u b s t i t u t e xn1' f o r i < d . F i n a l l y , i f r x d n n r E (x,,. , x ~ - ~ and ) again by the i n t e r s e c t i o n

h o l d i f f o r xi = 0

also a

r = 0

and

xd

i s a n o n z e r o d i v i s o r . The same c a n be

s a i d o f x1 and s o i f we add t o t h e r e l a t i o n s above

we c o n c l u d e t h a t t h e x i ' s

0:x

1

= 0,

form an A-sequence.

Canonical modules: (2.37)

Theorem. L e t A be a l o c a l G o r e n s t e i n r i n e . Then

any module o f f i n i t e i n j e c t i v e d i m e n s i o n a l s o h a s f i n i t e pmject i v e dimension. We g i v e a p r o o f f o r f i n i t e l y g e n e r a t e d modules o n l y ([2 4 ] ) . A c t u a l l y G o r e n s t e i n r i n g s can b e c h a r a c t e r i z e d by t h i s

19]. p r o p e r t y a s i n [-

Pro0 f

.

Let E be s u c h a module,map a f r e e module o v e r i t 0-L-F-E-0,

CS)

and n o t e t h a t I. a l s o h a s f i n i t e i n j e c t i v e d i m e n s i o n . I f

i s less than

depth of E

L = 1 m-depth

+

dim A , by ( 2 . 1 1 ) w e have t h a t

m-depth E . We can t h e n assume t h a t E h a s maxi-

mum d e p t h a l r e a d y . The c l a i m i s t h a t Hom(E,-) o f ( S ) w e g e t 0

-

Hom(E,L)

e x a c t . Since Ext'(E,L)

the

-

Hom(E,F)

-

(S) s p l i t s . T a k i n g

Hom(E,E)

m-depth E = dim A , from ( 2 . 2 0 )

-

1 E x t (E,L)

it follows t h a t

= 0.

Thus w e c o n c l u d e t h a t

H(A), t h e c a t e g o r y o f f i n i t e l y g e -

n e r a t e d module o f f i n i t e p r o j e c t i v e d i m e n s i o n , c o i n c i d e s w i t h

49

Homology o f L o c a l Rings I (A), s i m i l a r l y defined f o r t h e i n j e c t i v e dimension. H(A) n I ( A ) #

Conjecture : I f

for a local ring A , then

A i s a Gorenstein r i n g .

Given a l o c a l r i n g A , how t o f i n d modules o f f i n i t e i n j e c t i v e d i m e n s i o n ? In view o f B a s s ' s c o n j e c t u r e i t i s s a f e r

to

r e s t r i c t o u r s e l v e s t o Macaulay r i n g s . Let A h e a l o c a l Slacaulay r i n g ; an e a s y examnle i s o b t a i n e d i n t h e f o l l o w i n y manner. Let

5

h e a maximal A-sequence i n m_

i n j e c t i v e e n v e l o p e o f A/p1

and l e t E h c t h e

a s an .A/(&)-module: t h e n E i s

a

f i n i t e l y g e n e r a t e d A-module and t h e u s u a l chanEe o f r i n g s - v i z . ( 1 . 1 1 ) - shows t h a t i t has f i n i t e i n j e c t i v e d i m e n s i o n . What i s much h a r d e r i s t o f i n d examnles o f h i g h e r K r u l l d i m e n s i o n . There i s a ca se

however where t h i s can be done : Assume t h a t

A c a n he w r i t t e n

a s B/I, with R a l o c a l Gorenstein r i n g (e.g.

when A i s a l o c a l i z a t i o n o f an a f f i n e a l g e b r a ) (2.38)

n-d ExtB ( A , B )

Theorem. I f dim B = n

([El).

and dim A = d , t h e n R =

i s an A-module o f f i n i t e i n j e c t i v e d i m e n s i o n .

P r o o f . We p r o v e i n f a c t t h a t

d ExtA(-,Q) provides a d u a l i -

z i n g f u n c t o r f o r t h e c a t e g o r y o f A-modules o f f i n i t e l e n g t h . This f a c t , with

A = R , was i m p l i c i t i n o u r e a r l i e r d i s c u s s i o n

of Gorenstein r i n g s . S i n c e A i s a Flacaulay r i n g o f d i m e n s i o n d , we c a n f i n d a x

r e g u l a r R-sequence R

of length

n-d

i n I . Using ( l . l l ) ,

= €iomB/(x)( B / I , B / ( 5 ) ); -

dim A a s B/(&) i s a l s o a d G o r e n s t e i n r i n g . Then l e t T ( - ) = E x t A ( - , Q ) ' I f

we may t h u s assume t h a t dim B

=

HomoloFy o f L o c a l Rings

50 0

hq t

- - V" $!

0

is a s e q u e n c e o f f i n i t e l y g e n e r a t e d A-modules w i t h s u n n o r t i n

h), - we

may t a k e a s y s t e m o f p a r a m e t e r s = 0 . Notice t h a t (t)bI

such t h a t

t

t

= tl,.

..,td

in

m

i s a l s o R - r e g u l a r . Apply

t h e change o f r i n g s o f (1.11) t o have

T(M) = ExtA(M,fi) d As

fi/(E_>fi

HomB/ ( t I)

lifting

(t), B / ( t ' ),)w e

= fIomB,(tt) ( A /

( M , B / ( C t ) ) , where

t.

Since

B / (t')

= HomA/(t)

(?f,fi/(t)fi) *

f i n a l l y get

T(M) =

t t i s a system o f parameters f o r B

i s s e l f - i n j e c t i v e , w e a r e done.

fi w i l l be c a l l e d t h e c a n o n i c a l

module o f A .

In Chapter 4

w e d i s c u s s t h e b r o a d e r c l a s s o f s p h e r i c a l modules t o which R b e 1 o n g s . A more e x t e n s i v e t h e o r y o f ' c a n o n i c a l ' modules for r i n g s which a r e n o t n e c e s s a r i l y Macaulay i s f o u n d i n

[El.

C o r o l l a r y . I f A i s a ?facaulay r i n g t h a t i s a lJFD

(2.39)

and a homomorphic image o f a G o r e n s t e i n r i n g , t h e n A i s Gorenstein r i n g

a

([El).

P r o o f . Let R be t h e module c o n s t r u c t e d above - n o t needed h e r e t o assume A l o c a l . . S i n c e A i s a domain i t i s e a s i l y s e e n t h a t R may be i d e n t i f i e d t o an i d e a l o f A . As A i s a U F D $ dI

where

d

is the g.c.d.

o f n . Changing fi by I

=

we may

> 2 . But i n t h e e x a c t s e q u e n c e assume t h a t g r a d e R 0-n-A-

depth

= depth- A

A/Q-

0

( f o r any l o c a l i z a t i o n ) f o r c e s d e p t h A/fi

2

dim A - 1 , which i s a c o n t r a d i c t i o n . Thus R i s D r i n c i n a l and A

is a Gorenstein r i n g . Remark. The module R o f ( 2 . 3 8 ) i s as o b s e r v e d a hlacaulay

Homology o f L o c a l Rings

51

module o f t h e same d i m e n s i o n a s A . A n o t h e r P r o n e r t y i t e n j o y s is that

i > 0

Ilom(R,R) = A : t h i s f o l l o w s from and from

over A / ( t ) ,

Exti(R,n)

for

Q / (t ) Q b e i n g t h e i n j e c t i v e e n v e l o p e o f A / m-

whenever t i s a system o f p a r a m e t e r s .

([GI)t o

T h i s l a s t f a c t h a s been u s e d

show t h e c o n v e r s e ,

t h a t i s , i f R i s a Macaulay module s a t i s f y i n g

(t)( A / m-)

= 0

f o r some

s.0.p.

t,

n/(t)n

=

t h e n A i s a homomornhic image

o f a G o r e n s t e i n r i n g . P r o o f : Let B = A @ R be t h e t r i v i a l e x t e n s i o n o f A by R. The maximal i d e a l o f B

is

t h a t B i s G o r e n s t e i n , f i r s t o b s e r v e t h a t €3 i s a

m_

@ 9 . 'To show

Xacaulay A -

module; n e x t , by r e d u c i n g modulo a maximal A - s e q u e n c e w e may assume t h a t A i s an A r t i n i a n r i n g - R i s t h e n t h e i n j e c t i v e To c o m p l e t e t h e p r o o f i t i s enough t o d e t e r -

e n v e l o p e o f A/!.

mine t h e s o c l e o f B : a s 9 i s a f a i t h f u l A-module t h i s i s evidently

0:m) = B/(m_ 0 Q).

(0,

R-

How do G o r e n s t e i n r i n g s a r i s e ? : -P r a c t i c a l l y t h e o n l y way o f c o n s t r u c t i n g G o r e n s t e i n r i n g s

w e h a v e d i s c u s s e d i s by t a k i n g a r e g u l a r l o c a l r i n g B d i v i d i n g o u t by an i d e a l

I = (x,,

... , x r )

g e n e r a t e d by a r e q u l a r

B-se-

q u e n c e . I n p a r t t h i s d i f f i c u l t y i s germane t o t h e n a t u r e o f G o r e n s t e i n r i n g s as B/I w i t h B r e g u l a r which i s G o r e n s t e i n , must b e a c o m p l e t e i n t e r s e c t i o n i f t h e i d e a l I i s n o t l a r g e . Let A be a t w o - d i m e n s i o n a l Macaulay r i n g o f t y p e

Denote by v (M) (2.40)

u

(A).

t h e minimal number o f g e n e r a t o r s o f a module M .

P r o p o s i t i o n . I f I i s an i d e a l o f g r a d e 2 and

p r o j e c t i v e dimension 1,

u (A/I)

=

.JJ

(A) (v ( I ) - 1 ) .

Homology o f L o c a l Rings

52

I n p a r t i c u l a r , i f I i s i r r e d u c i b l e i t i s g e n e r a t e d by two e l e m e n t s and A i s a l s o a G o r e n s t e i n r i n g .

P r o o f . Let

0

- F1 a F g - I - O

be a minimal f r e e r e s o l u t i o n o f I . T a k i n g

exact with and a l s o

I$2

Ext 1 (&,I)

tiom(&,-) w e g e t

t h e t r i v i a l maps. Thus Ext 1 ( k , I ) =Ext 2 (k_,F1) = Hom(k,A/I) -

= (I:m -) / I

= k’(A’l). -

Finally

we g e t p ( A / I ) = p ( A ) - r a n k ( F 1 ) = p ( A ) ( v ( I ) - l ) . The s t a t e m e n t i s c l e a r l y s t i l l t r u e f o r Macaulay r i n R s o f h i g h e r d i m e n s i o n - b u t s t i l l w i t h g r a d e I = 2 , p d I = 1. The f o l l o w i n g i s o f a more d e l i c a t e n a t u r e : (2.41)

Theorem. Let B be a r e g u l a r l o c a l r i n g and l e t I

be an i d e a l o f h e i g h t t h r e e

such t h a t A = B / I i s a Gorenstein

r i n g . Then I i s m i n i m a l l y g e n e r a t e d by an odd ( n o t always t h r e e ) number o f g e n e r a t o r s . 8 ] o r [40]. P r o o f . See [ -

Appendix : Rings o f t y p e o n e . We may e x t e n d t h e d e f i n i t i o n o f t y p e o f a l o c a l i l a c a u l a y r i n g g i v e n e a r l i e r t o an a r b i t r a r y N o e t h e r i a n r i n g A , by ing that

t h e t y p e o f A a t a arime p

I t c a n be shown t h a t t h i s number

say-

is

1.

1

always

([GI).On

t h e o t h e r h a n d , we saw t h a t f o r G o r e n s t e i n r i n g s t h e t y p e i s

53

llomology o f L o c a l Rings 1 a t a l l primes. C o n j e c t u r e : Rings o f t y p e one a r e C o r e n s t e i n r i n g s . O f course t h e d i f f i c u l t y l i e s i n nroving t h a t such r i n g s

a r e Nacaulay. This i s v e r i f i e d h e r e i n i t s simplest case : be such a r i n g of t y p e o n e . I f

dim A = 1. Let (A,;)

0

0 - A - I

-1

1

i s a minimal i n j e c t i v e r e s o l u t i o n o f A , t h e h y p o t h e s i s means

lies

where t h e p i t s a r e t h e minimal p r i m e s o f A . The problem

i n proving t h a t 2 i s not a s s o c i a t e d t o A , i . e . t h a t Step 1 :

Write

I(A/m_)

= E ; a p p l y Horn(-,E) t o t h i s s e q u e n c e t o g e t

Hom(E,E) 0

a d i c completion of A

8 Hom(Q,E)

A

Call + ( i ) n i r = L; then

E-

Pi

,...,x r , f ) ,

embeds i n E . S i n c e

A

N o e t h e r i a n A-module

As Hom(E,E) = A = m-

O,$(a) = a(xl

ir/L

bm(A,E)-o A

's.

(fc]), we w r i t e

-

-

0 IlOm(Q,E)

where w e lumped i n Q t h e v a r i o u s

1'

0.

=

r 2 1.

f{orn(E,E)

A

r

ir/L

h

xic A .

is a

A

and E i s an A r t i n i a n A-module ( [ 2 7 1 ) , L

A

Ar/L

r

5

i s a module o f f i n i t e l e n g t h . T h i s c l e a r l y i m p l i e s t h a t 1

and, i f

r = 1 that

x = x1

i s n o t c o n t a i n e d i n any

h

minimal p r i m e o f A a s Step 2 :

LC

(x).

r = 0.

Let y be an e l e m e n t o f A

such t h a t

*

A

yA = xA ( p o s s i b l e a s

Homology o f L o c a l Rings

54 h

xA

A

i s an

EA-primary i d e a l ) . To t h e p r e s e n t a t i o n a p p l y

Hom(A/ ( y ) ,-1 t o g e t

Since

YA

E = x E , w e have

Y Hom(A/ ( y 2 ) ,- )

Y

A =

YE

Y

YE.

E . Now a p p l y t o t h e p r e s e n t a t i o n

to get 2A

0Y

and t h e n

- 'X

0-

-

Y

2E

- YZE 'X

2 A . T h i s i m p l i e s t h a t 0 :yr- A/(y) ; a s YA = Y i s an i n j e c t i v e A / ( y ) - m o d u l e , t h e i n c l u s i o n above s p l i t s .

E Y But a s

yA =

A/(y) i s indecomposable

possible.

(0:y)

+

(y) = A is

not

Chapter 3 nivisorial Ideals I n t h i s c h a p t e r w e s t u d y some s e m i - g r o u p s t r u c t u r e s on s u b s e t s o f i d e a l s o f a commutative r i n g A . The c l a s s i c a l examp l e i s t h a t of t h e d i v i s o r i a l i d e a l s of a completely i n t e g r a l l y c l o s e d domain s o s u c c e s s f u l l y u s e d i n t h e s t u d y o f t h e f a c t o r i a l i t y . For o t h e r r i n g s t h i s composition has r e c e i v e d s c a n t a t t e n t i o n and a p u r p o s e h e r e i s t o remedy somewhat t h i s s i t u a t i o n . Higher grade i d e a l s a l s o admit a composition b u t l a c k t h u s f a r any o f t h e p r o p e r t i e s o f g r a d e one d i v i s o r i a l i d e a l s i n the sense prescribed i n the Preface. I n t h e s e c o n d s e c t i o n t h e e m p h a s i s i s on t h e d i v i s o r o f

Auslander-MacRae-Mumford d e f i n e d on t h e c a t e g o r y o f f i n i t e l y g e n e r a t e d t o r s i o n modules o f f i n i t e p r o j e c ’ i v e

dimension.

An

e x a c t sequence r e l a t i n g t h e Grothendieck group o f t h i s c a t e g o r y to

Inv(A) = i n v e r t i b l e i d e a l s o f A

and

KO(A)

is useful

in

t h i s s t u d y and s e r v e s a s a model f o r t h e more g e n e r a l t r e a t m e n t of Chapter 4 . The l a s t s e c t i o n i s d e v o t e d t o c o m p u t a t i o n s o f t h e l e n g t h

of t h e t o r s i o n p a r t o f a module o f d i m e n s i o n o n e , t o c o n d i t i o n s e n s u r i n g t h e s p l i t t i n g o f s e q u e n c e s o f s u c h m o d u l e s , a n d t o an e x p o s i t i o n o f a r e s u l t o f Burch c h a r a c t e r i z i n g t h e

torsion-

f r e e modules o f r a n k one and p r o j e c t i v e d i m e n s i o n o n e . 53.1

Composition i n Id(A). L e t A be a commutative r i n g which w i l l b e a l t e r n a t i v e l y

N o e t h e r i a n o r c o h e r e n t and d e n o t e by I d ( A ) t h e s e t o f f i n i t e l y 55

Divisorial Ideals

56

g e n e r a t e d i d e a l s o f A t h a t c o n t a i n a r e g u l a r e l e m e n t . We w i l l be c o n c e r n e d w i t h v a r i o u s p a r t i a l c o m p o s i t i o n s i n I d ( A ) , t h a t

i s w i t h group s t r u c t u r e s - o r s e m i - g r o u p o n e s f a i l i n g t h a t t h a t may e x i s t f o r s u b s e t s o f I d ( A ) . A key remark w i l l show t h a t t h e r e a r e q u i t e a few o f t h o s e . I f I i s an i d e a l c o n t a i n i n g a r e p l a r e l e m e n t , HomA(I,A)

may be i d e n t i f i e d

with

1 - l = Ix

E

KI x I G A } , where K s t a n d s

( I - I ) - l may, i n t u r n , be I S ( I -1) -1

f o r t h e t o t a l r i n g of q u o t i e n t s o f A . and

i d e n t i f i e d t o an i d e a l of A

.

D e f i n i t i o n . An i d e a l I o f A i s s a i d t o be r e f l e x i v e d i v i s o r i a l i f I = ( I -1) -1

or

.

The r e a s o n f o r t h e f i r s t t e r m i n o l o g y i s c l e a r w h i l e t h e o t h e r w i l l soon become s o . N o t i c e t h a t a s A i s c o h e r e n t , 1 - l w i l l be a f i n i t e l y g e n e r a t e d f r a c t i o n a l i d e a l o f A . To p l a c e I

and I - '

on an e q u a l f o o t i n g we may t h e n e x t e n d t h e d e f i n i t i o n

o f Id(A) t o i n c l u d e a l l r e g u l a r f i n i t e l y g e n e r a t e d f r a c t i o n a l i d e a l s o f A . The d e f i n i t i o n above may t h e n be e x t e n d e d t o a l l such i d e a l s . For any i d e a l I , 1 - l

w i l l be r e f l e x i v e

and ( I

-1 -1 is )

the smallest reflexive ideal containing I . In o r d e r t o c o n s i d e r a c o m p o s i t i o n on D i v ( A ) , t h e s u b s e t of a l l d i v i s o r i a l i d e a l s i n Id(A), w e c o n s i d e r f i r s t a c l a s s o f prime i d e a l s o f A p l a y i n g a r o l e i n q u e s t i o n s

of

divisibility. Define ( a ) :b

P(A) =

ip

E

Spec(A)

I p

i s minimal o v e r an i d e a l

f o r some r e g u l a r e l e m e n t a ) . (3.1)

Lemma. Let I be a f i n i t e l y g e n e r a t e d i d e a l o f A

Divisorial Ideals I

c o n t a i n i n g a r e g u l a r element: i f

Q

51

f~o r any 2

P(A), 1-l =

E

A. The converse a l s o h o l d s .

Proof. I f

x

E

x = b/a, w i t h a regular, l e t

1 - l ' say

a prime i d e a l minimal over ( a ) : b . Then Conversely, l e t

1s p =

XISA i m p l i e s

minimal o v e r (a) :b

p. be

ICp.

and assume-

1 - l = A. There i s t h e n s $ p and an i n t e g e r n > 0 n n-1. s I E ( a ) : b . Pick n l e a s t ; t h e n ( s b / a ) I I C A and

such t h a t

&I-'& A , a contradiction.

(sbja) (3.2)

C o r o l l a r y . Let A be an i n t e g r a l domain o f q u o t i e n t

and l e t

field K

i s a prime i d e a l

fA[t]

iff

i s n o t c o n t a i n e d i n any E i n A[t]

if

(3.3)

a,b

... +

f = antn +

E

a.

be a polynomial i n t .

i s prime

fK[t]

P(A). In p a r t i c u l a r

...,a,)

and (a,, a+bt

i s prime

form a r e g u l a r A-sequence.

P r o p o s i t i o n . Two r e f l e x i v e i d e a l s i n Div(A) a r e

e q u a l i f f t h e y a g r e e i n each

A

E'

E

E

P(A).

P r o o f . Consider t h e map a s s o c i a t e d t o a r e f l e x i v e i d e a l I

Let

x

E

ker(0)

and l e t

L = {y

E

A Iyx

E

11. Notice t h a t A

b e i n g c o h e r e n t L i s a f i n i t e l y g e n e r a t e d i d e a l . Pass t h e equation

L = I:x

o v e r t o t h e polynomial r i n g

add t o x an element

b t with

A[t]. W e may now

b r e g u l a r i n I w i t h o u t changing

t h e e q u a t i o n , t h a t i s , w e may assume t h a t x i s r e g u l a r (No need t o assume a t t h i s p o i n t t h a t x-lL-'?

I - l ; b u t as L - l = A

A [ t ] is c o h e r e n t ! ) . x L C I by (3.1) w e conclude x

E

I . By f a i t h f u l l y f l a t d e s c e n t t h e c o n c l u s i o n f o l l o w s .

yields -1 -1 (I )

Divisorial Ideals

58

I f I i s an e l e m e n t o f Id(A) w e w r i t e D ( 1 )

.

f o r ( I -1) -1

By

g o i n g o v e r t o A [ t ] a s above we may assume f o r t h e p u r p o s e s h e r e t h a t e v e r y i d e a l i n Id(A) h a s a g e n e r a t i n g s e t c o n s i s t i n g o f r e g u l a r e l e m e n t s . Thus i f I - 1 = ( d l d,), di = r e g u l a r i n K ,

,...,

D(1) = A : I - ' t h a t says : D ( I )

=nAdil

is the intersection of a l l princinal ideals

c o n t a i n i n g I . S t r i c t l y s p e a k i n g we c a n assume t h e above o n l y a f t e r a f a i t h f u l l y f l a t chanpe o f r i n g s . For two e l e m e n t s I,.J E I ) i v ( A ) , d e f i n e I o J = D ( 1 J ) . (3.4)

Lemma. The c o m p o s i t i o n

'0'

is associative.

P r o o f . Let I J G A d ; t h e n I J d - l G A o r Jd-'C (I-')-';

also

D(ID(.J))

Ad _ > J J - ' d > J ( I - l ) - ' .

-1

and J - ' d

_>

T h i s s a y s t h a t D(1J) =

f o r any two i d e a l s . F i n a l l y a p p l y

t o get (D(I)oD(J))oD(K) = D ( 1 J K )

I

t h i s t o D(D(1J)K)

= D(I)o(D(J)oD(K)).

We s h a l l s t i l l n o t e by Div(A) t h i s s e m i - g r o u p s t r u c t u r e on t h e s e t o f d i v i s o r i a l i d e a l s . Let u s f i r s t d e s c r i b e i t s i n v e r t i b l e elements - A p l a y s t h e r o l e of t h e i d e n t i t y . IoJ

=

A implies ((IJ)-')-'

or that

= .4

( I J ) - l = A and

from ( 3 . 1 ) I J i s n o t c o n t a i n e d i n any prime p - E P(A). A

P

s a y s t h a t In i s an i n v e r t i b l e i d e a l o f A -

inverse; also (3.5) i f f ID

.J

=

I-'.

and J,, i s -

-

its

W e may summarize t h i s i n

Proposition. I

E

Div(A) i s i n v e r t i b l e ( i n D i v ( A ) )

i s an i n v e r t i b l e i d e a l o f A

L

P

(IJ)D =

n

f o r each p

E

P(A).

I n t h e n e x t s t a t e m e n t , i f A i s n o t a domain assume t h a t E i s an i d e a l .

I l i v i s o r i a1 I de a 1s

59

P r o p o s i t i o n . J,et A bc a c o h e r e n t l o c a l r i n g where

(3.6)

m

t h e maximal i d e a l

P(A).

E

Let I: lic a f i n i t e l y p r e s e n t e d

t o r s i o n - l e s s module ( i . e . 1, i s a submodule o f a n r o d u c t o f i f E * = Horn A (E,A) i s 4 - f r e e t h e n E i s A - f r c e .

copies of A ) :

P r o o f . Let .J = t r a c e ( E ) : .J i s a r e c u l a r i d e a l o f .A.

I f ,J

=

A , C will admit a summand i s o m o r n l i i c t o A and wc n r o c e e d hy

i n d u c t i o n on t h e rank o f I:.

-1 ,J f 1 8

of E ” ; but then

...

I f .T # .\, l c t -1 A .J fn

=

f 1 7 . . . 7 fn h e a b a s i s and , l - ’ = I.

F.”

C o r o l l a r y . I.ct :1 h c a X o e t h e r i a n r i n c r and l e t li he

(3.7)

a f i n i t e l y g e n e r a t e d t o r s i o n - l e s s m o d u l e . Then E i s A - n r o j e c tive i f f

I:*

i s A - p r o j e c t i v e a n d e v e r y r e g u l a r A-scauencc o f

two e l e m e n t s i s a l s o E - r e q u l a r . Proof. In t h e sequence

C

P

=

0

11

€ o r each

E

P ( A ) . Thus I = a n n ( C )

Apply Hom(A/I,-) t o g e t

lfom(A/I,C)

=

has q r a d e > 2.

1

Ext ( A / I , C ) = 0 by ( 2 . 1 1 )

t h a t i s c l e a r l v impossible i f I # A .

(3.8)

‘Theorem. ijiv(A) i s a c r o u p i f f

An

is a valuation

&

p

domain f o r e a c h

E

P(i\).

P r o o f . Write V f o r A

clear

that if all

II

and

“_ f o r

pA . . From above i t i s E

V ‘ s a r e v a l u a t i o n domains D i v ( 4 ) i s a

g r o u p . Let u s p r o v e t h e c o n v e r s e by f i r s t showing t h a t V i s a domain. L e t

x

E

V

and l e t

t h e i d e a l ( J , a ) i s by ( 3 . 6 )

.J = 0 : x ; i f a i s r e R u l a r i n

m_,

i n v e r t i b l e , s a y , (.J,a) = V c . J i s

t h e n = Lc and a s c i s r e p u l a r , ,T = L

and by Nakayama’s lemma

Divisorial Ideals

60

J = 0 - s i n c e A i s coherent

i s f i n i t e l y generated. m - i s , by

<J

t h e same argument a d i r e c t e d u n i o n o f p r i n c i n a l i d e a l s , t h a t i s , m- i s a f l a t i d e a l .

Lemma. L e t ( V , m-)

(3.9)

be a l o c a l c o h e r e n t domain w i t h p

a f l a t i d e a l . Then V i s a v a l u a t i o n domain. x , y be n o n z e r o e l e m e n t s o f m_: w i t h I = ( x , y )

P r o o f . Let

consider a presentation K-VL-I-

0-

S i n c e t h e mapping

IPm -

sequence K / mK -

0-

Im -

-

0.

is injective, we get the exact

(V/m)

2

- I/mI- - 0

showing t h a t a s K i s f i n i t e l y g e n e r a t e d I must be p r i n c i p a l . The n e x t q u e s t i o n t o c o n s i d e r i s t h a t when d i v i s o r i a l composition i s j u s t o r d i n a r y m u l t i p l i c a t i o n o f i d e a l s . This w i l l n o t be h a n d l e d h e r e e x c e p t i n t h e c a s e where Id(A) =

Div(A). The a d j u s t m e n t t o t h e more g e n e r a l c a s e i s n o t t o o d i f f i c u l t t o perceive. For an i d e a l I o f A we have t h e e x a c t s e q u e n c e 0

- - A

1-l

1 Ext (A/I,A)

which s a y s t h a t I i s r e f l e x i v e i f f

-

0

I i s the annihilator of the

module E x t l ( A / I ,A). I f x i s a n o n z e r o d i v i s o r i n I , by t h e 1 u s u a l change o f r i n g s Ext (A/I,A) = HomA/(x) ( M I ,A/ (XI 1 = a n n i h i l a t o r of

I / ( x ) i n A/ ( x ) . Thus i f e v e r y i d e a l i n Id(A) i s

r e f l e x i v e , A/ ( x ) h a s t h e f o l l o w i n g d o u b l e a n n i h i l a t o r p r o p e r t y : For e v e r y f i n i t e l y g e n e r a t e d i d e a l J o f A / ( x ) , O : ( O : J )

= J. In

t h e p a r t i c u l a r c a s e o f a N o e t h e r i a n r i n g , t h i s would f o r c e

D i v i s o r i a 1 I d e a1s

61

A/(x) t o b e an A r t i n i a n r i n g and a c t u a l l y a F r o b e n i u s r i n g . We s h a l l c o n s i d e r more g e n e r a l l y c o h e r e n t r i n g s . Assume, b y c h a n g i n g n o t a t i o n , t h a t A i s t h e r i n g A / ( x ) a b o v e . F o r an i d e a l I o f A w e h a v e t h e s e q u e n c e ann(1) = J

0-

f

If

E

-

Hom(I,A), J f = 0

I = (L,a)

f ( 1 ) C O : J = I . Let u s p r o v e

means

This i s c l e a r i f I i s a principal i d e a l ;

t h a t Hom(1,I) = A / J . if

- E x t 1(A/I,A) - 0 .

A -Hom(I,A)

assume t h a t t h e s t a t e m e n t h o l d s f o r L .

4(L)

an e l e m e n t o f lfom(1 , I ) , by t h e p r e c e d i n g b e realized by X

r

@(x) = rx

i.e.

x

for

t r i c t i o n o f $I t o ( a ) i s r e a l i z e d by A s . Xs(x)

and t h u s

r-s

( O : ( O : a ) ) = K , Thus q

E

0 : a . But t h e n X (3.10)

E

0:K. But O:(O:L

r-s

E

O : L + O:a, s a y

agrees with

r-p

Q

and may t h u s

L . Assume t h e r e s -

E

K = L n ( a ) , hr(x) =

On +

cL

I f 4 is

0 : a ) = (O:(O:L))

r-s = p+q, p

E

3 i'

0:L,

on b o t h L and ( a ) .

Theorem. L e t A be a l o c a l c o h e r e n t r i n g o f maximal

i d e a l F c o n t a i n i n g a n o n z e r o d i v i s o r . I f e v e r y i d e a l o f Id(A)

i s reflexive then A-module.

Exti(E,A) = 0 f o r every f i n i t e l y presented

I n p a r t i c u l a r e v e r y f i n i t e l y p r e s e n t e d A-module i s

reflexive. P r o o f . Let E be a f i n i t e l y p r e s e n t e d module. A p r e s e n t a tion 0-

with

L-

F-E-0

F f i n i t e l y g e n e r a t e d and f r e e l e a d s t o

Ext'(E,A).

Ext 1(L,A) =

Let x be a n o n z e r o d i v i s o r i n y; c o n s i d e r t h e map

i n d u c e d by m u l t i p l i c a t i o n by x on L : by Nakayama's lemma i t i s enough t o show t h a t

E x t L ( L / x L , h ) = 0 . L/xL a d m i t s a f i l t r a t i o n

.

L/xL = L O 1 L1 1 , . 2 L n = 0

Divisorial Ideals

62

w i t h f a c t o r s o f t h e form A / I , I a f i n i t e l y g e n e r a t e d i d e a l - a s A i s c o h e r e n t . As

x E I, I

I d ( A ) . The p r e v i o u s d i s c u s s i o n 2 t h e n i m p l i e s - v i a c h a n z e o f r i n g s - t h a t Ext (A/I,A) = 0 . E

l h e s t a t e m e n t on t h e r e f l e x i v i t y f o l l o w s : i f A'" i S

A

- -

An

E

a p r e s e n t a t i o n o f E and we w r i t e

0

D(E) f o r c o k e r ( H o m ( $ , l ) )

we g e t t h e well-known s e q u e n c e

1 Ext (D(E),A)-

0-

E

- E** - E x t 2 (D(E),A) -

0,

and t h e c o n c l u s i o n f o l l o w s . ( 3 . 1 0 ) c a n a l s o b e s t a t e d hy s a y i n g t h a t A h a s F P ( f o r finitely

p r e s e n t e d ) i n j e c t i v e d i m e n s i o n 1.

The f o l l o w i n g i s due t o Gruson i n a n o t h e r c o n t e x t :

sion

(3.11)

Theorem, Let A be a c o h e r e n t domain o f FP-dirnen-

1. I f

B = i n t e g r a l closure of A i s f i n i t e l y generated,

t h e n B i s a P r C f e r domain. P r o o f . Let C = HomA(B,A); C may be i d e n t i f i e d t o an i d e a l o f B and as HornB(-,C) = HomA(-PBB,A) = HomA ( - , A ) ,

HornB(-,C) i s

a d u a l i z i n g f u n c t o r on t h e f i n i t e l y g e n e r a t e d t o r s i o n - f r e e modules o v e r B (Note t h a t B i s a c o h e r e n t d o m a i n ) . Let I b e a f i n i t e l y g e n e r a t e d i d e a l o f B t h a t we want t o

prove i n v e r t i b l e . Let B

.J = IIomB(I,B)

and n o t e t h a t HomB(J,J) =

s i n c e B i s i n t e g r a l l y c l o s e d (Only u s e o f t h e i n t e g r a l l y

closed condition in proof). HomB(I,B) = HomB(IPC,C) = HomB(IC,C). Let T = I C HomB(IC,C)& C ; a s HomB(T,C) = HomB(ICBIIomB(IC,C) ,C) =

Hom(Hom(IC,C) ,Hom(IC,C)) = Iiom(Hom(1,B) ,Hom(I ,B)) = B , w e

Divisorial Ideals

63

conclude T = C. Let L = I * H o m ( I , R ) , t h e t r a c e i d e a l of I . L C = IC Hom(IC,C) = C

i m p l i e s I, = A and I i s i n v e r t i b l e .

I n t h e l a n g u a g e o f c a n o n i c a l modules: (3.12)

C o r o l l a r y . I f I i s a r e f l e x i v e i d e a l of a o n e - d i -

mension Macaulay r i n g and Ilom(1,I) = A , t h e n I i s i n v e r t i b l e . P r o o f . L e f t a s an e x e r c i s e . Remark. Let C be an e l e m e n t i n Id(A) s u c h t h a t Hom(C,C) = A.

I f we c o n s i d e r t h e s u b s e t

J = Nom(Hom(J,C) , C ) ,

DivC(A)

of a l l i d e a l s s a t i s f y i n e

t h e n v e r y much t h e same t h e o r y c o u l d bc i n s t e a d . T h i s phenomenon w i l l be p r e s e n t

d e v e l o p e d f o r DivC(A) i n t h e next chapter. 53.2

Divisors. L e t C be a f u l l s u b c a t e g o r y o f mod(A); we r e c a l l t h a t an

E u l e r c h a r a c t e r i s t i c on C

i s a d d i t i v e mapping

x:C-M

M an a b e l i a n g r o u p . E q u i v a l e n t l y ,

x

: Ko(C)

i s a g r o u o homomorphism

- M,

Ko(C ) t h e G r o t h e n d i e c k g r o u p o f C

To e n s u r e

x

.

more l a t i t u d e , w e might a l s o c o n s i d e r a d d i t i v e

mappings from C i n t o s e m i - g r o u p s . Definition. A divisor

on C i s an a d d i t i v e mapping

w i t h v a l u e s i n a semi-group of i d e a l s o f A.

x

on C

Divisorial Ideals

64

These mappings are v e r y r e l u c t a n t t o show up and i n t h e s e n o t e s w e s h a l l c o n s i d e r a few o f t h e more i n t e r e s t i n g o n e s t h a t a r e known. We b e g i n w i t h a d i s c u s s i o n o f a v e r y f r u i t f u l c a s e . i f it is annihila-

W e c a l l a module E o v e r A t o b e t o r s i o n

t e d by a r e g u l a r e l e m e n t o f A . Let T b e t h e c a t e g o r y o f f i n i t e l y g e n e r a t e d t o r s i o n modules o v e r A t h a t admit a r e s o l u t i o n 0-

- Pn-l

'n

...

Po

- E

by f i n i t e l y g e n e r a t e d p r o j e c t i v e modules. Let

0

KO(T) d e n o t e i t s

G r o t h e n d i e c k g r o u p . Among t h e e l e m e n t s o f T a r e t h o s e o f t h e f u l l s u b c a t e g o r y T1

E E T w e may ma? a f r e e

Notice t h a t i f E

-

o f modules o f p r o j e c t i v e dimension o n e . A/(x)-module Eo o n t o

x h e r e i s a r e g u l a r element i n t h e a n n i h i l a t o r of E

thus derive a resolution 0-E

w i t h E i E T1.

n

-

En_1

...

Eo

- E

We q u o t e i n f u l l t h e f o l l o w i n g

-

and

0

[z]:

Theorem. Let 4 be an a b e l i a n c a t e g o r y and l e t

(3.13)

P C M be f u l l s u b c a t e g o r i e s o f 4

s a t i s f y i n g t h e following :

i ) P and M a r e c l o s e d u n d e r f i n i t e d i r e c t sums. i i ) If M

t h e n M'

i s e x a c t i n 4 and M , M "

0-MI-M-M"-0 E

b4.

i i i ) I f M i s an o b j e c t o f M , t h e r e i s an e x a c t s e q u e n c e 0--Pd

with a l l

Pi

E

...

Po-M-O

P.

Then t h e i n c l u s i o n P G M i n d u c e s an isomorphism : K p )

KO(W

*

From t h e remarks above we t h e n have

Ko(T1)

Ko(T)

.

E

Div i s o r i a 1 I de a 1 s

65

Now a l o o k a t t h e t o r s i o n modules o f n r o j e c t i v e d i m e n s i o n o n e . Let O - K - A " - E - OU

be a p r e s e n t a t i o n o f E . As t h e a n n i h i l a t o r o f E c o n t a i n s a r e g u l a r e l e m e n t K h a s c o n s t a n t r a n k and i s t h u s a f i n i t e l y Ren e r a t e d module. Let

& ( E ) be t h e i d e a l q e n e r a t e d by t h e m i n o r s

of order n of u , that i s , f r e e of rank n , i.e.

&(E)

d(E)

d(E)

= FO(E).

Because K i s l o c a l l y

i s l o c a l l y zenerated by a r e g u l a r element,

i s an i n v e r t i b l e i d e a l of A.

(3.14)

Lemma.

: TI

-

A is additive.

P r o o f . Let

E'

0-

be an e x a c t s e q u e n c e i n T1.

Inv(A) = i n v e r t i b l e i d e a l s o f

-E -

To show

0

El'

& ( E ' ) *d(E") = d(E)

we

may assume t h a t A i s a l o c a l r i n g and c o n s t r u c t r e s o l u t i o n s

0-E'-E

10

with the F ' s

As

u ' and u"

det(u"),

10

- E"-

f r e e modules.The m a t r i x

0

10

u i s g i v e n by

a r e s q u a r e m a t r i c e s we have d e t ( u ) = d e t ( u ' ) .

as d e s i r e d .

d e x t e n d s t h e n t o an a d d i t i v e mapping a c t u a l l y d e f i n e d i n t h e manner :

:T

-

Inv(A)

Divisorial Ideals

66

If

...

En

0-

is a resolution of E

E

- -

E0

E

T by modules Ei

An i m m e d i a t e c o n s e q u e n c e i s t h a t homomorphism

T1 t h e n

E

.

d(E) = d ( E o ) - d ( E 1 ) - l . .

-

0

d(En) ( - 1 P

-

-

A

if

R

i s a Ting

w i t h B t o r s i o n - f r e e a s an A-module, t h e n we have = KO(T1(~))

KO(T(A))

where @ [ E l = [EfdB]

a -A

Inv(A)

and $ ' ( I ) = I B .

'This a p p l i e s e s p e c i a l l y t o l o c a l i z a t i o n s and n o l y n o m i a l extensions. (3.15)

Lemma. For p

T(A,)

P(A)

E

= T i

(A ) .

1

2

P r o o f . I t i s enough t o show t h a t i f 0-F-G-bl-0

U

i s an e x a c t s e q u e n c e i n mod(A

free

!?

3

with

F,G

f i n i t e l y generated

and I?l t o r s i o n - f r e e , t h e n i t s p l i t s . W e may assume t h a t

a l l t h e e n t r i e s o f u l i e i n t h e maximal i d e a l o f A

p

a r e g u l a r element such t h a t A/(x)

s o r i n g t h e sequence b y

i s minimal o v e r

we g e t

ufW/(x)

P'

Let x be

( x ) : b . Tenstill injective

t h u s c o n t r a d i c t i n g McCoy's t h e o r e m . I t follows t h a t

i d e a l of A

P

for

f o r each p

E

E

E

P(A).

T(A)

p

E

E

P(A),

A lrIc ( X I }

.

If p

a contradiction.

d(E)

",

This forces

i n t e g r a l i d e a l of A ( [ 3 2 ] ) : Write J = Cr

,

i s an i n t e g r a l

d(E)

d(E) = Ix-1

t o be a n

and l e t

i s a minimal p r i m e o v e r J we g e t

Divisorial Ideals

67

Remark. An a l t e r n a t e a p p r o a c h t o d e f i n i n g d on T when A i s

a c o h e r e n t r i n g and a v o i d i n g t h e d i r e c t u s e o f ( 3 . 1 3 ) i s t h e following : Define

d(E)

t o he t h e d i v i s o r i a l i d e a l o f A a s s o -

ciated t o the 0-th Fitting ideal of E , FO(E). the various A

P

it f o l l o w s t h a t

By g o i n g o v e r t o

d p r o v i d e s an a d d i t i v e mapping

from T i n t o t h e s e m i - g r o u p D i v ( A ) . U s i n g now i n d u c t i o n on t h e p r o j e c t i v e d i m e n s i o n o f t h e module i t w i l l f o l l o w t h a t t h e c o m p o s i t i o n i s j u s t p l a i n m u l t i p l i c a t i o n . The o f modules i n T t h e n show t h a t

'0'

T -resolutions

1 d ( E ) i s i n v e r t i b l e . The d i f f e -

r e n c e i n a p p r o a c h i s t h e n by d e f i n i n g

d(E)

= (F,(E)-')-'

one

s t a r t s o f f w i t h an i d e a l which w e know t o be an i n v a r i a n t o f E . (3.16)

C o r o l l a r y . L e t I be an i d e a l o f f i n i t e p r o j e c t i v e

d i m e n s i o n . Then t h a t is

I = d ( h / I ) - J , where J i s an o v e r d e n s e i d e a l ,

J - l = A.

P r o o f . I i s t h e 0 - t h F i t t i n g i d e a l o f t h e module A/I and d, by t h e d e f i n i t i o n o f -

a product

I

d(A/I).

c ( A / I ) J . T a k i n g i n v e r s e s twice

( I - l ) - l = d ( E ) * ( J -1) -1

-

(3.17)

We may t h e n e x p r e s s I a s

or

(J-l)-'

= A.

Theorem. L e t A b e a l o c a l c o h e r e n t r i n g i n which

every f i n i t e l y g e n e r a t e d i d e a l has f i n i t e p r o j e c t i v e dimension. Then A i s a G . C . D .

domain.

P r o o f . T h a t A i s a domain f o l l o w s from ( 2 . 2 6 ) . two n o n z e r o e l e m e n t s of t h e maximal i d e a l

m

Let a , b b e

o f A and w r i t e I

=

( a , b ) . A p p l y i n g t h e d e c o m p o s i t i o n above t o I we g e t -1 ( a , b ) = G(cr,B) where (a,f3) = A.6 i s t h e d e s i r e d gcd. T h i s i s e s s e n t i a l l y MacRae's t h e o r e m

([El)t h a t

says t h a t

Divisorial Ideals

68

i n a Noetherian r i n g every two-generated i d e a l has n r o j e c t i v e

-.

dimension 0 , l o r

I n p a r t i c u l a r it p r o v i d e s a g e n e r a l i z a t i o n

of t h e f a c t o r i a l i t y o f r e g u l a r l o c a l r i n g s .

We s h a l l d i s c u s s t h e e x t e n t t o which d - is universa1,that i s , an i s o m o r p h i s m . For t e c h n i c a l r e a s o n s we s h a l l assume A t o be N o e t h e r i a n a l t h o u g h w i t h a b i t more of c a r e t h e p r o o f s c o u l d be f i n e s s e d t o i n c l u d e c o h e r e n t r i n g s . We may t h e n t a k e A t o be connected - 0 , l a r e t h e only idempotents

of A.

. 1

Denote by

KO(A) t h e reduced Grothendicck g r o w of t h e

c a t e g o r y of f i n i t e l y g e n e r a t e d D r o j e c t i v e modules o v e r A. For [ P ] - [ A n ] (n = r a n k ( P ) ) , a g e n e r a t o r i n 'jTo(A), A

: KO(A)

-

where I$ : P-

An

A i s a group

by A ( [ P I - f A n l )

KO ( T )

define

[bf] - [A/Fo (M) 1

i s some embedding and

hl = c o k e r ( I $ )

.

That

homomorphism w i l l s o o n b e shown. Theorem. The f o l l o w i n g i s a n e x a c t s e q u e n c e

(3.18) N

KO(A)

h

KO(T)

6 -

Inv(A)

-

0.

Proof. W e b e g i n b y v e r i f y i n g t h a t A i s a group mapping: Let

0-

$(P)

0-

O(P)

- - -

-

be two embeddings o f P i n t o An. sor x

such t h a t

x$(P)s o(P)

An

M

0

A"-

N-

o

Since t h e r e i s a nonzero d i v i -

, we

have o n l y t o c o n s i d e r t h e

case $ ( P ) c a ( P ) . The s e q u e n c e s above l e a d t o a s e q u e n c e 0-

Notice t h a t

a(P)/$(P)

a(P)/$(P)

= P/$(P)

- - M

N

0.

f o r some $ : P C P .

We s h a l l u s e t h e f o l l o w i n g t h a t i s b a s e d on an i d e a

Divisorial Ideals

69

o f Kramer ( [ 22]): (3.19)

Lemma. I f

fl

i s exact with F f r e e , then

- - F -M 4

F

0

[MI = [A/detQ.A] i n K O ( T ) .

P r o o f . Let ( a . .) b e a n n x n m a t r i x r e p r e s e n t i n g Q i n t h e 17

c a n o n i c a l b a s i s of F . The i d e a l g e n e r a t e d by t h e column (all,

...,a n l )

i s r e g u l a r by McCoy's theorem. We r e c a l l P r o p o s i t i o n . Let ( a , J ) b e a r e g u l a r i d e a l o f t h e

(3.20)

N o e t h e r i a n r i n g A . T h e r e is t h e n a n o n z e r o d i v i s o r of t h e form a+j, j

J.

-E

[%I.

Proof.

Apply t h i s t o

J = (azl,.

a = all,

.., a n l ) .

We may t h e n

change t h e b a s i s o f F t o make a n o n z e r o d i v i s o r t o a p p e a r i n t h e upper l e f t c o r n e r o f t h e m a t r i x f o r 0. Assume t h e n a . ,

II

to

b e a n o n z e r o d i v i s o r . Let

ri 0

o

...

all...

0

0

........... 0 0 ...a l l and write 0

and

CI

= a-$

-

. We

colter(@)

have t h e e x a c t s e q u e n c e

- coker(0) - coker(a) - 0

det(u> = det(a).det(+).

c l e a r while

Rut [ c o k e r ( a ) ] = [A/deta.A]

is

[ c o k e r ( a ) ] = [A/detu.A] by i n d u c t i o n o n t h e r a n k

s i n c e t h e m a t r i x of a is e q u i v a l e n t t o o n e

having

a l l e l e m e n t s i n t h e f i r s t column z,ero b u t f o r t h e f i r s t o n e .

Divisorial Ideals

70

The completion of the proof that A is well-defined and a homomorphism is now clear. Since

{ o A = 0 , let u s show the

exactness of (3.18) by constructing a map 4

:

Inv(A)

- Kg ( T ) / imA

in the following manner: If I is an invertible ideal of A , pick x I C A . Put @(I)

x a regular element of A with in Ko(T)/imA.

[A/xI] -[A/xA]

=

That A is well-defined will again follow from the

proof that it is additive. Let I,J be integral invertible ideals of

A.

I/JI is a

rank one projective module over A/J. Let pl,.,.,& be the non-embedded prime ideals of J. Pick b a nonzero divisor in I such that

(b)pi

by letting

f(l*)

prime of Ass(A/J) we have

[A/J] 0

we have

I for all i t s . Define pi f I/JI A/J

=

-

=

b*. f is injective as it is s o at each

by construction. Let L [L]

+

=

=

coker(f) ; in KO(T)

[I/JI]. On the other hand, from

- I/JI - A/JI - A/I - 0

[A/JI]

=

[I/JI]

[A/I], and the additivity of

+

will follow from showing [L] Notice that

ann(L)c

J

E

4

imA.

and as an A/J-module it vanishes Since L is a

when we localize at the elements of Ass(A/J). module o f projective dimension finite over A/J

this means that

its annihilator contains a nonzero divisor and thus, as an Amodule, grade L 2 2 , and consequently d(L) If x is a nonzero divisor in ann(L), resolution 0

- M - Fn/xFn ..

,

FO/xFO

=

A.

we can find a

-L -

0

Divisorial Ideals with

Fi A-free

- that d

additivity of [A/&(M)]

=

A we get from the

d(M) is a principal ideal. This makes

rank(Fn) [A/xA]

-

d(L)

pd M < 1. Since

and

71

+

... =

0. Using this relation with

that derived from the sequence above we get [Ll = Z (-lli([Fi/xFi]-rank(Fi) [A/xA]) and [L]

E

(-l)n+l([M]-[A/&(M)]),

+

imA.

(3.21)

Corollary. If A is one-dimensional or a local

&

Noetherian ring, then

: KO(T)

- Inv(A)

Proof. The local case comes from

is an isomorphism.

-

KO(A)

=

0

and the one-

dimensional from the proof above. Remark. The following considerations were pointed out by H.Bass to show that

6

is not in general an isomorphism. Suppose

A is a commutative Noetherian ring of finite global dimension. Then we have a commutative exact sequence KO(T) I

Inv(A)

J

- KO@) -Pic(A) I

J

0

whence an epimorphism

0

0

0

ker(&)

- ker(det) -

0 . Now

ker(det) # 0 in general. For instance, if A = R[x,y,z], x 2 + y2 + z2 = 1, we have Pic(A) = 0 and KO(A) = 2/22. Let

now A be a coherent domain and C denote the category

of finitely presented torsion modules over A and let us search for the additive (3.22)

maps from C into Div(A).

Theorem. Let

: C

- Div(A)

be additive and

Divisorial Ideals

72

& ( A / ( x ) ) = ( x ) . Then A i s i n t e g r a l l y c l o s e d

such t h a t

d(M)

f o r each M

(F0(M)-')-'

=

Proof. I f M

E

C.

E

C there is a filtration

M = M 2M3.., ?Mn 0 - 1-

= 0

I i a f i n i t e l y g e n e r a t e d i d e a l . We t h e n

with

Mi/Mi+l

have

&(MI = & ( A / I O ) o . . . O ~ _ ( A / I ~ _ ~ ) .

A/Ii,

Lemma. I f I i s a n i n t e g r a l i d e a l i n D i v ( A ) , t h e n

(3.23)

d(A/I)

and

= I.

Proof. I = n A x i ; t h e r e i s XI =nAxxi

and a s

x

such t h a t

I

E

&(A/xI) = x d(A/I)

X X ~ EA .

Thus

w e may assume t h a t I i s

an i n t e r s e c t i o n of p r i n c i p a l i n t e g r a l i d e a l s . If

y

E

IGAx, t h e s u r j e c t i o n

t o (y) = d ( A / ( y ) ) = d ( A / I ) o d ( K ) h a n d , IS Ax forcing

=

The c o n c l u s i o n o f ( 3 . 2 2 ) argument above shows '0')

Since

&(A/I)

y A/I

( x ) . Thus

I

and

(FO(-)

-1 -1 )

Modules

of

dimension

leads

d ( A / I ) . On t h e o t h e r

-

-0

A/(x)

d(A/I)

= I.

now f o l l o w s from ( 3 . 8 ) : The

t o be i n v e r t i b l e i n Div(A) (under

agree

coincide globally. 53.3

E

- A/I - 0

AD i s a v a l u a t i o n domain f o r e a c h

and t h u s

A(-)

and

leads t o a surjection

d(A/I)cd(A/(x))

A/(y)

a t each A

E

p

E

P(A).

they

one.

These modules a r e s t u d i e d h e r e p a r t i c u l a r l y w i t h r e g a r d t o t h e i r t o r s i o n submodules. (a)

Length o f t h e t o r s i o n submodule:

Divisorial Ideals

73

Let L(M) d e n o t e t h e l e n g t h o f a module M o v e r t h e N o e t h e r i a n r i n g A . We b e g i n w i t h a consequence o f ( 3 . 1 8 ) (3.24)

[El:

C o r o l l a r y . Let A be a o n e - d i m e n s i o n a l N o e t h e r i a n

r i n g and l e t E b e a f i n i t e l y g e n e r a t e d t o r s i o n module o f p r o j e c t i v e d i m e n s i o n one. Then

L(E)

=

Proof. Since t h e elements of T(A)

L(A/d-(E)).

have f i n i t e l e n g t h ,

i s a n a d d i t i v e f u n c t i o n ; a s E i s e q u i v a l e n t t o A/d(E) in

L(-)

e(E) = L ( A / & ( E ) ) .

KO(T),

U n f o r t u n a t e l y t h e l e n g t h o f t h e t o r s i o n submodule o f a f i n i t e l y g e n e r a t e d module o f p r o j e c t i v e dimension one c a n n o t always be e x p r e s s e d i n t e r m s of F i t t i n g ' s i n v a r i a n t s . There a r e however a few c a s e s where t h i s i s p o s s i b l e ( [ 2 2 , 3 9 ] ) . i)

,ith

Let E be a module p r e s e n t e d a s

Q(1)

=

v = (al,.

.. , a n ) . $(u)

determined e a s i l y a s ment

x

with

by t h e a i l s ) .

xu = yv

E

and

The t o r s i o n submodule T(E)

T(E) c a n be

i f f there i s a regular ele-

y / x I S A ( I i s the i d e a l generated

I t follows t h a t

T(E)

I-'/A.

Note t h a t

I = F (E). n-1

ii)

A complementary c a s e i s t h a t a t o r s i o n module E w i t h

a presentation An+'

0 An - E -

0.

S i n c e E i s a t o r s i o n module, we may assume - c h a n g i n g t h e b a s i s of

An+'

i f necessary -

generated by

the first

that

$

r e s t r i c t e d t o t h e submodule

n b a s i s v e c t o r s of

An+l i s injective.

Divisorial Ideals

14

I f we c a l l L t h e image i n An o f t h i s submodule, we have

- A ~ / L- E -

$(A~+~)/L

0 -

If

all

...

a

anl

...

a nn

where

d

we have L(An/L) = &(A/dA) t h e d e t e r m i n a n t of t h e l e f t

0.

an , n + l i s a s i n t h e c a s e above

n x n b l o c k of 4 . + ( A n + l ) / L on t h e

o t h e r hand i s a c y c l i c module A / J w i t h J =

r

I

A

E

By C r a m e r ' s r u l e t h e n

r(al,n+l,.

. . , an , n + l)

J = (d):FO(E)

1.

E L

and f i n a l l y

l(E) =

C(A/dA) - l ( A / ( d ) : F O ( E ) ) = l ( ( d ) : F O ( E ) / ( d ) ) = l ( F 0 ( E ) - l / A ) . C o r o l l a r y . L e t A be a n o n e - d i m e n s i o n a l a f f i n e

(3.25)

domain o v e r a f i e l d o f c h a r a c t e r i s t i c 0 i n t e r s e c t i o n . If Qk(A)

which i s a c o m p l e t e

d e n o t e s i t s module of k - d i f f e r e n t i a l s ,

t h e n l ( T ( R k ( A ) ) ) = l ( A / J ) , where J i s i t s J a c o b i a n i d e a l . P r o o f . We know t h a t Rk(A) i s a module of p r o j e c t i v e dimension one o v e r A w i t h a r e s o l u t i o n ( l o c a l l y ) : 0

Note

-

F1(Qk(A))

=

An

0 An+'

J . Applying

-

nk(A)

-

0.

Horn(-,A) t o t h i s s e q u e n c e

+ Hom(An,A) - Ext 1( Q i ( A ) t

Hom(A"+l,A)

and

l(Ext'(Rk(A),A))

,A)

=

L(A/J).

Since

0,

A i s a complete i n t e r -

s e c t i o n - whence a G o r e n s t e i n r i n g - Ext'(-,A)

'reads' the

t o r s i o n submodule o f a module and t h e c o n c l u s i o n f o l l o w s .

Divisorial Ideals (b)

75

S p l i t t i n g t h e t o r s i o n submodule: Now we t a k e up t h e q u e s t i o n o f when t h e t o r s i o n p a r t of a

module o f p r o j e c t i v e d i m e n s i o n one s p l i t s . M a i n l y w e f o l l o w

[25].

Let

Am

4 An - E - 0

be a p r e s e n t a t i o n of E

...

and l e t

I o ~ I , ~ be t h e s e q u e n c e

o f F i t t i n g ' s i d e a l s o f E . L e t 1, b e t h e f i r s t n o n z e r o i d e a l and assume i t c o n t a i n s a n o n z e r o d i v i s o r o f A ; t h i s i s e q u i v a l e n t t o saying t h a t

i s a p r o j e c t i v e K-module, K t h e t o t a l r i n g

EBK

of q u o t i e n t s of A according t o (1.16). (3.26)

Proposition. If Ir is A-projective

then

pd E = 1

and i t s t o r s i o n submodule s p l i t s . P r o o f . We may assume t h a t A i s a l o c a l r i n g and i f

_.___

...

all

al m

f4= i s t h e m a t r i x of

i n the canonical bases,

r$

t e d by t h e minor

d

of t h e

upper l e f t c o r n e r . Let

el,

then Ir i s genera-

n - r x n-r submatrix s i t t i n g i n the

..., en

be t h e corresponding gene-

r a t o r s o f E ; from t h e r e l a t i o n s n C a . .e = 0 j = l 11 j

(j=1,

...,n-r)

w e g e t by m u l t i p l y i n g by t h e c o f a c t o r s of t h e h - t h column and adding de Since

dj

h

+

n C d.e

J j j-n-r+l

i s d i v i s i b l e by d

=

0

.

we get t h a t

Divisorial Ideals

76

eh

n +

c (dj/d)e j =n-r+l

j

i s a n n i h i l a t e d by d and hence l i e s i n T ( E ) . I t f o l l o w s t h a t t h e

...,

e n* o f t h e l a s t r e l e m e n t s g e n e r a t e E/T(E). e *n - r + l ' I f K i s t h e t o t a l r i n g of q u o t i e n t s o f A , a s o b s e r v e d ,

images

i s a f r e e module o f r a n k e x a c t l y r . Thus

(E/T(E))PK = EPK E/T(E)

i s A-free

and T(E) s p l i t s o f f E .

Remark. A c t u a l l y t h e h y p o t h e s i s t h a t I r b e a f i n i t e l y generated f a i t h f u l p r o j e c t i v e i d e a l s u f f i c e s t o ensure t h a t E h a s a f i n i t e p r e s e n t a t i o n . We would n e e d t o modify - f o r t h e c a s e o f a r b i t r a r y commutative r i n g - t h e d e f i n i t i o n o f t o r s i o n submodule : e

E

E is torsion i f f

ann(e) i s a f a i t h f u l i d e a l .

F u r t h e r d e c o m p o s i t i o n o f T(E) c o u l d p e r h a p s b e o b t a i n e d b y r e q u i r i n g t h a t some

Fi(E)

be a l s o p r o j e c t i v e f o r

i > r.

U n f o r t u n a t e l y o n l y i n t h e l o c a l c a s e o r when A i s N o e t h e r i a n o f dimension one t h i s c a n e a s i l y be c a r r i e d o u t . (c)

A_ c r i t e r i o n f o r s p l i t t i n g : Let A b e a o n e - d i m e n s i o n a l Macaulay r i n g and

be an e x a c t s e q u e n c e o f modules o f p r o j e c t i v e dimension 5 1. I f t h i s s e q u e n c e s p l i t s s o does t h e c o r r e s p o n d i n g s e q u e n c e (beware h e r e ) o f t o r s i o n submodules. (3.27)

T(E')

8 T(E")

Theorem. I f A i s a G o r e n s t e i n r i n g and T(E) t h e n t h e s e q u e n c e above s p l i t s .

A t t h i s t i m e w e c o u l d assume t h e s p l i t t i n g o f

Divisorial Ideals

or

T(E)

= T(E')

- T(E) - T(E")

T(E')

0-

0 T(E")

I7

a r i s e s i n some o t h e r way. O f c o u r s e

i f s u c h a d e c o m p o s i t i o n h o l d s t h e s e q u e n c e o f t o r s i o n submodu-

l e s i s e x a c t a t t h e r i g h t by l e n g t h c o n s i d e r a t i o n s . We assume t h e second case t o h o l d i n o r d e r t o p r e s e n t a p r e t t y r e s u l t of E i s enbud- Hamsher :

(3.28)

Theorem. Let

0-

E

-F - - 0 G

b e an e x a c t s e q u e n c e o f f i n i t e l y g e n e r a t e d modules o v e r t h e Noetherian r i n g A. I f

F

5

E 8 G , then t h e sequence s p l i t s .

P r o o f . According t o [ l , P r o p . 6 . 5 ] i t i s enough t o show t h a t MP(-)

m a i n t a i n s t h e s e q u e n c e e x a c t f o r any f i n i t e l y g e n e r a t e d

module M . We may assume t h a t A i s a l o c a l r i n g . I f M h a s f i n i t e

* (MPE) 8 (MPG)

length

MPF

C(MBG)

and t h u s MPE

s a y s t h a t L(MPF) = [(MPE) +

- - MPG MPF

0

must b e e x a c t on t h e l e f t . L e t now M be an a r b i t r a r y ( f . g . ) module and c o n s i d e r t h e s e q u e n c e s 0-L-MMPE-C-0 0-

C

-MPF - MPG - 0 .

T e n s o r b o t h s e q u e n c e s by A/mn -

by t h e f i n i t e l e n g t h c a s e . Thus i n t e r s e c t i o n theorem

Proof o f ( 3 . 2 7 )

(IJ

=

maximal i d e a l o f A) t o g e t

Lsm_".(MPE)

f o r a l l n . By t h e

L = 0. : The s e q u e n c e ( S ) s p l i t s i f t h e s e q u e n c e

Divisorial Ideals

78

- Ext 1( E , E ' )

1 Ext (E",E')

is exact

-

1 Ext ( - , A )

a t the l e f t . As

on t o r s i o n modules, t h e s e q u e n c e 0-

1 Ext (E",A)

-

1 Ext (E,A)

1 Ext ( E ' , E ' )

-0

is a d u a l i z i n g f u n c t o r

-

1 Ext (E',A)

-0

i s ' d u a l ' t o t h a t o f t h e t o r s i o n submodules and t h u s s p l i t s and t a k i n g i n t o a c c o u n t t h e n a t u r a l

a l s o . Tensoring with E ' equivalence

1

Ext (M,A)P( - )

- Ext 1(M,-)

f o r a module M

( f . g . ) o f p r o j e c t i v e d i m e n s i o n one we h a v e t h e d e s i r e d e n d . Q u e s t i o n : Does t h e same s t a t e m e n t i n Theorem ( 3 . 2 7 ) h o l d s for arbitrary (d)

Torsion

1 - d i m e n s i o n a l Macaulay r i n g s ?

free

modules

of p r o j e c t i v e

dimension one and r a n k

one : -

Here w e g i v e a n e x p o s i t i o n o f a theorem o f Burch d e s c r i b i n g s u c h modules. L e t A b e a commutative r i n g and l e t

0

-

P

9 An - E -0

b e a r e s o l u t i o n o f E; assume t h a t 0 i s t h e o n l y e l e m e n t k i l l e d by a f i n i t e l y g e n e r a t e d f a i t h f u l i d e a l o f A

and t h a t P is a

p r o j e c t i v e module o f rank n - 1 ( T h i s w i l l be t h e c a s e o f a f a i t h f u l i d e a l o f p r o j e c t i v e dimension o n e ) . By c o n s i d e r i n g a p a r t i t i o n o f t h e u n i t y ( [ 6-] )

tJ=

assume t h a t P = An-'

... ...

...

and w r i t e

a 1, n - 1 a

f o r t h e m a t r i x o f 6. We want t o r e l a t e E t o t h e i d e a l g e n e r a t e d by t h e minors o f o r d e r

n - 1 of 4 . Write

ei f o r t h e image i n E

Divisorial Ideals

19

of the

i - t h c a n o n i c a l b a s i s v e c t o r o f An. The r e l a t i o n s n E aijei = 0 ( j = l , n-1) i=l y i e l d by C r a m e r ' s r u l e

...,

d.e = d.e J i l j where di i s t h e minor o b t a i n e d by d e l e t i n g t h e i - t h row. L e t D = (dl,.

. . ,dn)

and t e n t a t i v e l y d e f i n e $

by + ( E r i d i ) zridi

=

D-E

F i r s t note t h a t

Eriei.

=

1

0 y i e l d s (cridi)ek

=

+

d k ( E r1. e 1. )

is well-defined as and t h u s Eriei

is

a n n i h i l a t e d by D , a f a i t h f u l i d e a l by McCoy's theorem. Thus

J,

is a n isomorphism and D h a s a l s o p r o j e c t i v e d i m e n s i o n o n e . (3.29)

Lemma. I n c a s e E i s an i d e a l o f A J, c a n be r e a l i -

zed by m u l t i p l i c a t i o n by an e l e m e n t o f A . Proof. A s

J,

E

Hom(D,A), t h e c l a i m w i l l b e e s t a b l i s h e d i f

we p r o v e t h a t t h e n a t u r a l

homomorphism

A

-

Hom(D,A) i s a n

isomorphism. To p r o v e e q u a l i t y change t h e r i n g t o A [ t ] . A s D h a s f i n i t e p r e s e n t a t i o n - b e i n g i s o m o r p h i c t o E - w e have ( D [ t ] , A [ t ] ) . I n A [ t ] however D [ t ] i s A[tI n an i d e a l c o n t a i n i n g a n o n z e r o d i v i s o r , s a y d l t + + dnt

HomA(D,A)PA[t] = Hom

...

We c a n now p r o c e e d a s i n [ E l

and assume t h a t D c o n t a i n s a

n o n z e r o d i v i s o r a l r e a d y . Hom(D,A)

i s t h e n D-'.

t e n s o r i n g t h e r e s o l u t i o n o f E by A/bA w e g e t $BA/bA 0 (A/bA)n-l (A/bA)n

-

-

-

If

E/bE

e x a c t . Using McCoy's t h e o r e m a g a i n , t h e F i t t i n g o r d e r one o f E / b E (3.30)

is s t i l l f a i t h f u l , forcing

Theorem. I f

0

.

-A n - l L

An-

a/b

E

D-l,

-0

i n v a r i a n t of

a

E

bA. 1-

0

Divisorial Ideals

80

i s e x a c t and I i s a n i d e a l o f A , t h e n I = dD

where D i s t h e

n - 1 x n - 1 minors of 4 .

i d e a l g e n e r a t e d by t h e

Appendix : Higher d i v i s o r i a l i d e a l s . For an i d e a l I o f g r a d e = 1 a t e d w i t h I can b e r e a l i z e d (1-l)

-'

=

the divisorial ideal associ-

as

ann(ExtA 1 ( A / I ,A))

.

This su g g e s t s t h e following c o n s t r u c t i o n f o r i d e a l s o f h i g h e r grade. D e f i n i t i o n . D(1) = a n n ( E x t i ( A / I , A ) )

-a

( 3 . 3 1 ) Lemma. I f then

D(I/(a))

=

where

grade I = r .

is a A-sequence o f l e n g t h r-1 i n I ,

D(I)/(a).

P r o o f . T h i s f o l l o w s i m m e d i a t e l y from t h e f a c t t h a t (A/ 1 , A / (a) 1

E x t i ( A / I ,A) = when

a

i s an A-sequence o f l e n g t h n c o n t a i n e d i n I .

Notice t h a t i f a i s an A-sequence of l e n g t h r-1 i n I , where g r a d e I = r , t h e n

D(I/(a>)

=

((I/(a))-')-'

which i s

an i n t e r s e c t i o n o f p r i n c i p a l f r a c t i o n a l i d e a l s o f A/(a)

-indeed

f i n i t e l y many o f them. With t h i s i n mind w e c o n s i d e r an

A-sub-

module o f t h e q u o t i e n t r i n g o f A L =

where

ai

i s a n A-sequence o f l e n g t h r

would l i k e t o show t h a t D(1)

since

9 ti-1(Zi) L = D(1)

D(1) = D ( D ( 1 ) ) .

and

t i l ( a i ) 31. We

and we may assume t h a t I =

Divisor i a l Ideal s

a

Suppose

81

i s an A-sequence i n I o f l e n g t h

( c / E ) (A/ (a)) i s a p r i n c i p a l f r a c t i o n a l i d e a l of ing

I/@),

then

i n v e r s e images o f

t

u,

and

aJt

A/ (a) - contain-

chosen

w i t h t chosen t o be a nonzero

d i v i s o r by prime avoidance. Moreover and e i t h e r

r-1. I f

/ I w i t h t and u s u i t a b l y

t-'(a,u)

-

=

5 , u i s an

A-sequence

i s an A-sequence o r t i s a u n i t . I f we

let

T be t h e i n t e r s e c t i o n o f a l l such f r a c t i o n a l i d e a l s o f A , t h e n T c o n t a i n s L. Furthermore, T n A / ( a ) = I / @ )

(5) i s c o n t a i n e d i n b o t h

r e f l e x i v e . But

since I/(a) i s

T n A and

I

so

T n A = I . Hence we would be through i f we could show t h a t L i s

contained i n A. I f any o f t h e components o f t h e i n t e r s e c t i o n f o r L

is

c o n t a i n e d i n A we a r e through, s o we assume t h i s i s n o t t h e a , u ) as above. Now i s an A-sequence c a s e . Consider t -1( -

a,t

and a g a i n by prime avoidance we can f i n d a nonzero d i v i s o r such t h a t v

=

a , v i s an A-sequenceJ V , E

t mod@).

v

i s an A-sequence, and

This l a s t c o n d i t i o n i n s u r e s t h a t

v

-1

( 5 , ~2 ) I . We

can a l s o f i n d a nonzero d i v i s o r o f t h e form s = v+b where bE (a) and v , s i s an A-sequence. Since t - l ( a , u ) 2 s - 1( a , u ) _>L, we f i n d t h a t ( v , s ) CL-'. (v,s)-'?

(L-')-'Z?L.

Taking i n v e r s e s a g a i n we o b t a i n

But ( V , S ) - ' = A

since grade(v,s) = 2 .

T h e r e f o r e we have proved t h a t L i s c o n t a i n e d i n A a s d e s i r e d , that is : (3.32)

Theorem. L

=

D(1).

Chanter 4 Snhc r i c a l ' f o d u l c s and

ivisors

One o f t h e aims o f t h i s c ! i a n t e r i s t o s t u d v t h e modules t h a t can be o h t a i r l e d a s ?iomomorn!iic imaTes o f d i r e c t sums o f c o n i e s of a f i x e d modulr C .

4 r e s u l t o f Cruson s a y s t h a t i f C

i s f i n i t e l v g c n e r a t c d and f a i t h f u l t h e c a t e e o r y

mod(A) comes

c l o s e t o h e i n n s o obtained, a s e a c h module '1 a d m i t s a f i n i t e f i l t r a t i o n w i t h f q c t o r s t h a t a r e imapes o f sums o f c o n i e s o f G . A p p l i c a t i o n s o f t h i s a r e made t o chanqe o f rinrzs and homoloeic a l dimensions. Calling

C(G)

t h e catecrory p e n e r a t e d by

I;,

we t u r n t o t h e

e x a m i n a t i o n o f t h e modules i n C ( G ) t h a t admit r e s o l u t i o n s by d i r e c t sums o f G I s , o r more e s n e c i a l l v f i n i t e r e s o l u t i o n s . The s i t u a t i o n becomes t h e n r i p e f o r a n p l y i n g t h e p r o c e d u r e o f Chapter 3

and a s s i g n i n E a d i v i s o r t o s u c h m o d u l e s . E x n e c t e d l y some

r e s t r i c t i o n s must be p l a c e d - t o o b t a i n a d e p r e e o f i n v a r i a n c e on t h e module G . A c l a s s o f modules - s n h e r i c a l o n e s - f o r which t h e p r o c e d u r e works o n t i m a l l y i s d i s c u s s e d . 54.1

A f Gruson. - t h e o r e m o-

(4.1)

Theorem. L e t E be a f i n i t e l y g e n e r a t e d f a i t h f u l

module o v e r t h e commutative r i n g A . Then e v e r y A-module a d m i t s a f i n i t e f i l t r a t i o n o f submodules whose f a c t o r s a r e q u o t i e n t s

of d i r e c t sums o f c o p i e s o f E . I t i s a s i f e v e r y f i n i t e l y g e n e r a t e d f a i t h f u l module 82

is

Snhe r i c a 1 ?fodul e s

83

‘ p i e c e w i s e ’ a g e n e r a t o r f o r t h e c a t e g o r y mod(A). P r o o f . I t i s enough t o show t h i s f o r A i t s e l f : Let I0 C I c - 1-

... G A

be a s e o u e n c e o f i d e a l s o f A s u c h t h a t f o r each

i there is a surjection

Let

Q

:

A(L)

?1

-”

-

0

be a p r e s e n t a t i o n o f t h e module

PI. Then t h e s e r i e s o f submodules o f ‘I q i v e n by

~ ( 1 ; ~ )h)a s

thc desired properties. For t h e p r o o f o f t h e r i n q A i t s e l f i n v a r i a n t s of E.

Let U

An

we u s e t h e F i t t i n p

-E -

0

be a f r e e p r e s e n t a t i o n o f E

and d e n o t e bv ( a - . ) , 1 5 j 5 n ,

i c L , t h e m a t r i x o f u. L e t

Fr(E) be t h e i d e a l g e n e r a t e d by

11

t h e minors of o r d e r n - r of t h i s m a t r i x . The s m a l l e s t i n t e g e r r s u c h t h a t

Fr f 0

nonzero; t h e s m a l l e s t i n t e g e r m such t h a t

m 2 r . We n o t i c e t h a t module ( ( 1 . 1 6 ) )

Fm = A

and i s

e x i s t s and

m = r means t h a t E i s a p r o j e c t i v e

and i n t h i s c a s e we a r e done. d = m-r.

We s h a l l r e a s o n by i n d u c t i o n on

Suppose

exists

d > 0. L e t

I = ann(E/FrE).

i n d u c t i o n h v p o t h e s i s t o t h e A/I-module is f a i t h f u l (xECIECFrE

implies

F i t t i n g i d e a l i s t r i v i a l and i t s

We have a l s o t h a t

InsF r .

x

E

We c a n a p p l y t h e

E/IE

: (a)

I ) , and (b)

t h i s module

its

r-th

m-th i d e a l i s A / I . This gives rise t o a f i l t r a t i o n

where e a c h f a c t o r i s a n A/I-module. Thus t o c o m p l e t e t h e p r o o f

S p h e r i c a l Modules

84

i t i s enough t o show t h a t

c o p i e s o f E . Let

Fr i s a q u o t i e n t o f a d i r e c t sum o f

ll be t h e d e t e r m i n a n t o f t h e

n - r = s minor

j , 1 5 k , t 5 s , and l e t u s l o o k f o r a l i n e a r k’jl form on E whose image c o n t a i n s D. One c h o o s e s an i n t e g e r t

d e f i n e d by

(ai

between 1 and n and d i s t i n c t from t h e i k ’ s : t h i s i s p o s s i b l e since

r > 0 . We c o n s i d e r t h e f o l l o w i n g l i n e a r form on An c

ail,jl det

a a L

As t h e m i n o r s o f o r d e r

s+l

.

s”1

.

it’ll

*..

a +is -

... . ... a i , J., ...

a. lt,js

x1

X

-

S

Xt -L

a r e z e r o , t h i s form i s t r i v i a l on

t h e image o f u : i t d e f i n e s t h e n , bv p a s s a g e t o t h e q u o t i e n t , a l i n e a r form on E . On t h e o t h e r h a n d , i t t r a n s f o r m s t h e

t-th

v e c t o r o f t h e c a n o n i c a l b a s i s o f An i n t o D. W e p o i n t o u t some immediate c o n s e q u e n c e s . (4.2)

C o r o l l a r y . Let F be a r i g h t e x a c t f u n c t o r which

commutes w i t h a r b i t r a r y d i r e c t sums and i s t r i v i a l on E : t h e n

F is t r i v i a l . (4.3)

C o r o l l a r y . Let

E be a f i n i t e l y g e n e r a t e d f a i t h f u l

module o v e r t h e commutative r i n g A If 54.2

and l e t kl be an A-module.

EPM = 0 , t h e n bl = 0 .

Change o f r i n g s and d i m e n s i o n s . The change o f r i n R s p r o b l e m i n h o m o l o g i c a l d i m e n s i o n

theory is the following:

Sphe r i c a l Module s

Let

f : A-

85

be a homomorphism o f r i n g s and l e t E be

B

a module o v e r B . How a r e

pdAE

and

pdBE ( o r some o t h e r d i -

mension) r e l a t e d ? I n g e n e r a l one h a s t h a t pdAE < pdAB

(1)

pdgE.

+

T h i s f o l l o w s , f o r i n s t a n c e , from t h e s p e c t r a l s e q u e n c e

Ep 7 q = Ext[(E ,Ext:(B 2

(2) where C

,C)

-

( [ lo])

Extl(E,C)

P

may be any A-module. I f

d = pdAB

b o t h f i n i t e , t h e r e are s e v e r a l i n s t a n c e s that

s e q u e n c e a l l o w s one t o c o n c l u d e

e = pdBE

and

where

Et'e

are

this spectral

# 0 , and h e n c e

e q u a l i t y i n (1) above. Here w e w i l l s t u d y t h e c a s e where €3 i s n o t o n l y f i n i t e l y s e n e r a t e d a s an A-module b u t a d m i t s a r e s o l u t i o n (3)

...

0 -Pd

PI -P

o-

B-0

where t h e P i ' s

a r e f i n i t e l y g e n e r a t e d p r o j e c t i v e modules. d I n t h i s c a s e t h e module ExtA(B,A) p l a y s a s i p n i f i c a n t

role

and we b e g i n w i t h an e l e m e n t a r y d i s c u s s i o n o f i t . A f i r s t

property t o notice

i s t h a t i f C i s any A-module

we have a

c a n o n i c a 1 i s omo r p h i s m ExtA(B,C) d

-5

ExtA(B,A)MAC. d

T h i s a r i s e s e i t h e r from a s p e c t r a l s e q u e n c e argument o r more s i m p l y from t h e f o l l o w i n g c o n s i d e r a t i o n s . As o n l v t h e module s t r u c t u r e o f B i s i n v o l v e d we may assume t h a t

pd B = 1. W r i t e

then a r e s o l u t i o n 0-

w i t h P o ,P1

-

p1

Po-

B-0

f i n i t e l y g e n e r a t e d p r o j e c t i v e modules. A m l y i n g

HomA(-,C) w e g e t

Sphe r i c a 1 "odul e s

86

HomA(Po,C)

1 -HomA(P1,C) - ExtA(B,C) -0 .

The r e s u l t now f o l l o w s from o h s e r v i n , e t h e n a t u r a l e q u i v a l e n c e as

HomA(Pi,-) = Horn A ( P i , A ) @ , , ( - ) , l y generated.

Pi

i s n r o j e c t i v e and f i n i t e -

We a r e g o i n g t o c o n s i d e r two t y n e s o f c o n d i t i o n s on B : f o r d n l a v s an e s s e n t i a l r o l e , and b o t h t h e module T = ExtA(B,A) which a r e , u n d e r g e n e r a l c i r c u m s t a n c e s , n e c e s s a r y f o r e q u a l i t y i n ( 1 ) . T h i s s i t u a t i o n we s h a l l , a t t i m e s , a b b r e v i a t e bv s a v i n g t h a t t h e r e i s chanpe of r i n g s .

(a)

bfacaulay e x t e n s i o n s : We assume h e r e t h a t

(Condition

h.0

i ExtA(B,A) = 0

for

i < d.

This r e s t r i c t i o n i s reminiscent of t h e condition a f a c t o r r i n g o f a r e g u l a r l o c a l r i n g must s a t i s f v t o be a Yacaulay r i n g .

In t h i s c a s e , i f we annly

IlomA(-,A) t o t h e s e q u e n c e ( 3 )

t w i c e , t a k i n g i n t o account the r e f l e x i v i t y of t h e p r o j e c t i v e m o d u l e s , and t h e maps between them, we o b t a i n t h a t d Ext,,(T,A) 2 B . I n p a r t i c u l a r , we c o n c l u d e t h a t T i s a f a i t h f u l module. (4.4)

Theorem. Let

B

be an e x t e n s i o n o f A s a t i s f y i n g

c o n d i t i o n s ( 3 ) and ( h r ) a b o v e . Then f o r any R-module E pdgE = e

<

-,we

have

pdRE = pdAB

+

with

ndBE.

P r o o f . According t o ( 2 ) i t w i l l b e enough t o show t h a t f o r some

C

= A (L)

E x t i ( E , E x t i ( B , C ) ) = Ext;(TaAC)

= ExtE(E,T(L))

#

0.

S p h e r i c a 1 \lo d u l e s

87

Remark. F o r t h e i n j e c t i v e and f l a t a n a l o g u e s o f ( 4 . 4 )

one

c a n p r o c e e d i n a s i m i l a r way t o show t h e e q u a l i t y i n t h e d i mension f o r m u l a

u s i n g t h e following s p e c t r a l sequences = Ext!(Tor

E;7q

A (B,C) , E ) 0

and = T o r B (E,Tor"(B,C))

E!7q

P

respectively

4

-Extt(C,E) n

-Tor, (C , E l A

D

.

L e t u s i n d i c a t e how t o p r o c e e d

i n t h e i n j e c t i v e c a s e (one

u s e s t h e same t y p e o f argument - and module - i n t h e f l a t c a s e . i Let !.I be an i n j e c t i v e A-rnodulc. Because ExtA(B,A) = 0 , i < d , much i n t h e same way as b e f o r e

it follows

HomA(T7M). Let ExtE(T,E) # 0 Let

0

ExtE(C,E) # 0

that

A

Tord(B,hi) =

f o r some B-module C : t h e n

also.

-T - M

Extpe(M,E) = 0

be an A - i n j e c t i v e e n v e l o p e o f T .

implies t h a t

ExtE(HomA(T,?I),E) = 0 . S i n c e

T i s f i n i t e l y g e n e r a t e d and f a i t h f u l as

a B-module, t h e r e i s

an e x a c t s e q u e n c e 0-B-

o b t a i n e d by mapping

1

Tn

- ( x l ,.. . , x n )

E

T",

where t h e

x 's i

form a g e n e r a t i n g s e t f o r T . But i n t h i s c a s e ExtB(HomA(B,M),E) e = 0 a l s o and t h e n Exti(HomA(B,T),E) = 0 .

As

there is a surjection HomA(B,T) g i v e n by $(f) = f ( 1 ) (b)

-T $

we c o n t r a d i c t

0

ExtE(T,E) # 0 .

F i n i t e l y g e n e r a t e d modules :

If E i t s e l f , i n the previous considerations, is f i n i t e l y

Sphe r i c a l Modules

88

g e n e r a t e d t h e r e i s n o t a n e e d t o impose s u c h a s t r o n g r e s t r i c t i o n on B . Theorem. Let A and B be r i n g s , w i t h B l o c a l and f i -

(4.5)

n i t e l y g e n e r a t e d a s an A-module. Assume t h a t B a d m i t s a f i n i t e r e s o l u t i o n by f i n i t e l y g e n e r a t e d p r o j e c t i v e A-modules.

Let E b e

a B-module a d m i t t i n g a f i n i t e f r e e r e s o l u t i o n o v e r B . Then pdAE

=

pdAB

+

pdRE.

T h i s r e s u l t , c o n t r a r y t o ( 4 . 4 ) h a s an e l e m e n t a r y v i a of a t t a c h t h r o u g h t h e d e v i c e o f minimal p r o j e c t i v e r e s o l u t i o n s . P r o o f . IJsing t h e n r e v i o u s n o t a t i o n n o t i c e t h a t

A( R , A ) ) = Ext; ( E ,R ) PBExtA d ( B ,A) Ext; ( E , E x t d a s E h a s a f i n i t e p r e s e n t a t i o n . S i n c e b o t h modules i n t h e p r o duct a r e nonzero

and t h e r i n g R i s l o c a l , t h e s t a t e m e n t

follows. (c)

Some

equalities of dimensions:

B e f o r e we d e r i v e o t h e r c o n s e q u e n c e s of ( 4 . 4 ) and ( 4 . 5 ) r e c a l l t h e d e f i n i t i o n of t h e f i n i t i s t i c p r o j e c t i v e d i m e n s i o n o f a r i n z A - FPD(A) f o r s h o r t - : FPn(A)

i s o b t a i n e d by t a k i n p

i n t h e d e f i n i t i o n of t h e g l o b a l d i m e n s i o n o f A o n l y t h o s e mod u l e s o f f i n i t e d i m e n s i o n . b l o r e o v e r , i f we o n l y c o n s i d e r t h e f i n i t e l y g e n e r a t e d modules we o b t a i n fPD(A). (4.6)

Theorem. Let A b e a c o h e r e n t l o c a l r i n g and l e t I

be a f i n i t e l y g e n e r a t e d i d e a l w i t h p d A I fPD(A) = pd,(A/I)

+

<w.

Then

fPD(A/I).

S n h e r i c a l hlodules

89

When A i s N o e t h e r i a n t h e r e i s a s i m n l e way - v i a t h e complementary n o t i o n of denth - t o v e r i f y t h i s e q u a l i t y . For t h e r e s t o f t h i s s e c t i o n we s h a l l assume a l l u n s D e c i f i e d A-modules t o be o f f i n i t e n r e s e n t a t i o n . (4.7)

Lemma. L e t (A,m) -

be a c o h e r e n t l o c a l r i n g and l e t

E be a f i n i t e l y p r e s e n t e d A-module. Then

A - # 0 Tor (E,A/m) n

pdAE = n i f f

A and Torn+l(E,A/m_) = 0 .

P r o o f . Observe t h a t t h e r e i s a r e s o l u t i o n

... F 7- where e a c h Fi $.

1

$1 F1-Fq-E-r) i s a f r e e module o f f i n i t e r a n k and t h e m a t r i x $2

has a l l of i t s e n t r i e s i n

m.

'The r e s u l t f o l l o w s i n j u s t t h c

same way a s i n t h e N o e t h e r i a n c a s e : s a y , d i m e n s i o n s h i f t i n c l and Nakayama's lemma. Proof of ( 4 . 6 ) f o l l o w s from A / I , Write

fPD(A) 2 ndA(A/I)

: That

+

fPD(A/I)

a s we o b s e r v e t h a t A/I i s a l s o c o h e r e n t .

d = pdA(A/I)

and l e t 1: b e an A-module o f f i n i t e

p r o j e c t i v e d i m e n s i o n and c o n s i d e r a minimal r e s o l u t i o n 0-

N -

Fd-l

...

Fg

- E

0,

where a pfii0fi.i s e v e r a l o f t h e t e r m s ahove c o u l d h e t r i v i a l . A s T o r 4h + i ( E , A / I ) = T o rAi ( N , A / I ) , T o r 4i ( N , A / I ) = 0 f o r i > 0 . Consider t h e s p e c t r a l senuence E2 P9

= 9

T o r A / ' (A/;,Tor:(N,A/I)) I,

Tor:"

(A/!,N/I?J)

n

Torn(A/m_,N) A

= Tor,(A/m,N) A -

.

S p h e r i c a l ?lodules

90

But t h i s isomorphism s a y s t h a t ndAN = pd pdAE > d

we c o n c l u d e

inequality (4.8)

fPn(A) 2

pdAE = d d

+

+

pd

A/ 1

(N/IN) and i f A/ 1 (N/IN) t o Eet h e o t h e r

fPD(A/I).

C o r o l l a r y . Let (A,m) - b e a c o h e r e n t l o c a l r i n g and

l e t I be a f i n i t e l y g e n e r a t e d i d e a l s u c h t h a t : ( a ) p d A I and ( b )

the radical of I is m -. Then

P r o o f . According t o ( 4 . 6 )

fPD(A) <

<

-,

m.

( by p a s s i n g t o t h e r i n g A / I )

we may assume t h a t A i s p r i m a r y r i n g . Suppose s a y t h a t @ O-F1-Fo-E-O

i s a minimal r e s o l u t i o n o f t h e module E . By McCoy's theorem t h e minors of rank t o r , impossible §4.3

rank(F1) o f @ have a

unless

nontrivial annihila-

F1 = 0 .

S p h e r i c a l modules. I n t h i s s e c t i o n we t a k e up t h e q u e s t i o n s t a t e d a t t h e

o u t s e t on when i s a module E a homomorphic image o f a d i r e c t

sum o f c o p i e s o f a f i x e d module G . We assume t h r o u g h o u t N o e t h e r i a n c o n d i t i o n s . Thus t h e s i t u a t i o n t r a n s l a t e s on w h e t h e r t h e n a t u r a l map

i s s u r j e c t i v e . A c t u a l l y o u r i n t e r e s t w i l l f o c u s on t h o s e mod u l e s E a d m i t t i n g r e s o l u t i o n s by sums o f c o p i e s o f G . I f s u c h r e s o l u t i o n s a r e g o i n g t o show any form o f i n v a r i a n c e t h e cond i t i o n i)

i ExtA(G,G) = 0

for

i

>

0,

s h o u l d be p r e s e n t . A companion r e s t r i c t i o n , i f t h e n o t i o n o f

Sphe r i c a l Nodule s

91

G-dimension i s t o be d e f i n e d , i s t h a t t h e module E b e s u c h t h a t i ExtA(G,E) = 0

ii)

for i > 0.

T h i s e v i d e n t l y a l l o w s f o r a form o f S c h a n u e l ' s lemma: i t w i l l appear q u i t e n a t u r a l l y . F i n a l l y , and in o r d e r t o a t t a c h a d i v i s o r t o modules t h a t admit f i n i t e G - r e s o l u t i o n s , we s h a l l n e e d iii)

HomA(G,G) = A .

In o r d e r t o m o t i v a t e t h e u s e f u l n e s s of t h i s c o n t e x t we d i s c u s s t h e following r e s u l t ( a l s o nroved by T.Culliksen)

P r o p o s i t i o n . Let (A,m) be a local Artinian r i n e

(4.9)

with

L

1

.

= 0 . Let G be a f i n i t e l y z e n e r a t e d A-module

that

s a t i s f i e s t h e con d i t i ons HomA(G,G) = A , ExtA(G,C) 1 = 0.

(a) (b) Then G

A

or G

I ( = i n j e c t i v e envelone o f

P r o o f . C o n s i d e r t h e man anplying t h e f u n c t o r Hom(1,I) 2

A

-$I : GPHom(G,I)

k

= A/!).

- I . By

€ { o m ( - , I ) we q e t

- Hom(GPHom(G,I) , I )

=

Hom(G,fiom(G,I) , I ) )

€?om(G,G). But t h e two e n d s o f t h i s s e r i e s

a r e isomorphic t o

and t h e combined map i s c l e a r l y t h e i d e n t i t y a f t e r t h e

a p p r o p r i a t e i d e n t i f i c a t i o n s ; we c o n c l u d e t h a t $ I

i s an

isomorphism. Denote now by

r(E)

(resp.v(E))

t h e dimension of t h e s o -

c l e o f t h e module E ( r e s p . minimal number o f g e n e r a t o r s o f E ) . Because

Horn(-,I)

isomorphism above

i s a s e l f - d u a l i z i n g f u n c t o r we g e t from t h e that

r(G).v(G) = v ( 1 )

= r(A) = v ( A ) ,

the

S p h e r i c a l Modules

92

t y p e o f t h e l o c a l r i n g A.

In p a r t i c u l a r t h i s e q u a l i t y says t h a t

i f A i s a G o r e n s t e i n r i n g o r more R e n e r a l l v t h e dimension o f t h e s o c l e o f A i s a ? r i m e number

then C

3

A or I.

Let us now g o back t o t h e c o n d i t i o n s o f t h e n r o n o s i t i o n . I’Jrite and

s(E)

€ o r t h e s o c l e o f t h e module F.. In o u r c a s e s ( A )

s ( G ) = EC

p

([El).

a s c a n h e e a s i l y s!iorm

be a minimal n r e s e n t a t i o n o f G. Innlyincr Ilom(-,C;) (1))

=

and u s i n n

we z c t t h c e x a c t s e q u e n c e

9 Since

-4 -

-I l o m ( L , G ) -

Gv(c)

LCmAu(G) -we g e t t h a t

Ilom(L/m_L,s(G)).

IIom(l,,G) = fIom(I,,m_G) iz

Noting t h a t

J , / m-L = Torl(l;,k_)

0. =

and t a l i i n n

lcngtlis o f t h e sequence w e g e t A

L(Torl(G,k)).r(G)

+

l ( 4 ) = r(G)-b(C).

A n o t h e r :*lay o f cornnutin? t h e l e n g t h o f t h e T o r i s v i a fl

-

A Torl(C,l;)

- GBm - - G/zG -

G

-0:

we g e t now A

l ( T o r l ( c ; , k-) ) = u ( C )

+

u(C)*r(%) - L(C).

Taking i n t o account t h e r e l a t i o n s :

we f i n d a f t e r

b(A) = r(A)

+ 1,

b(G)

+ r(C),

= v(G)

and

an e l e m e n t a r y s u b s t i t u t i o n t h a t 2 2 (r(G) - l ) ( u ( G ) - 1 ) = 0 .

Example. T o show t h e e x i s t e n c e o f a module s u c h a s G o v e r an A r t i n i a n r i n g t h a t i s n e i t h e r A o r I A = K[x,y],

K a f i e l d , (x,y)

L

= 0,

consider the case:

and I t h e i n j e c t i v e e n v e l o p e

S n h e r i c a l Plodules o f A/;;

I f A.

note

G = IR,I3.

93 3

Let I3 = . ' l [ u , v ] , ( u , v ) - = 0 , and l e t

G i s c e r t a i n l y n o t isomornhic t o t h e i n j e c t i v e

envelope of t h e r e s i d u e f i e l d o f B - a s t h e dimension o f t?ie socle of

C;

i s 2 . I t i s a l s o not isomornhic t o R .

D e f i n i t i o n . A f i n i t e l v g e n e r a t e d module i n n r i n g A w i l l be c a l l e d conditions

snherical

C;

over a Noether-

i f it s a t i s f i e s

i ) and i i i ) a b o v e . \\re write Snh(A)

the

for the

s o h c r i c a l modules o v e r A . Remarks. The f o l l o w i n g o b s e r v a t i o n s w i l l be u s e d i n

the

se quel without f u r t h e r ado. (i)

If G is a

p r o j e c t i v e A-module

t h e n IPG

it i s l o c a l l y isomorphic t o

l o c a l l y isomorphic

C;.

i s again a

s p h e r i c a l module a s

C o n v e r s e l y , i f G and G '

s p h e r i c a l modules

r a n k one p r o j e c t i v e module I

then

and c l e a r l y

( i i ) I f E i s a d i r e c t summand o f cellation E is

and I i s a r a n k one

s p h e r i c a l A-module

C;",

Hom(G,G')

IPG

are is a

= G'.

G s p h e r i c a l , by l o c a l c a n -

i s o m o r p h i c a t e a c h l o c a l i z a t i o n t o a Gm (m

p o s s i b l y v a r y i n g with t h e l o c a l i z a t i o n ) . Conversely, suppose E

i s l o c a l l y i s o m o r p h i c t o a Gm; t h e r e i s a p r e s e n t a t i o n @ Gn-E-0

w h i c h , we c l a i m , s p l i t s . Assume A l o c a l

and view 4 a s a m a t r i x

w i t h e n t r i e s i n A . As a l l o f i t s e n t r i e s c a n n o t l i e i n 2

by

Nakayama's lemma, w e e a s i l y g e t a s p l i t t i n g g o i n g . Problem. The d e t e r m i n a t i o n o f Sph(A)

Sph(A[t])

i n terms o f

looms a c h a l l e n g i n g q u e s t i o n i n view a l r e a d y o f t h e

difficulties

met i n t h e s i m p l e r

Pic(Alt1).

S p h e r i c a l Modules

94

Elementary p r o p e r t i e s .

54.4 (a)

If G is a

s p h e r i c a l A-module and

r i n g homomorphism

then

GPAR

is a

h: A -

-

is a flat

B

s p h e r i c a l R-module.(The

converse i s t r u e i f B i s f a i t h f u l l y f l a t . )

P r o o f . Immediate. (b)

A s Hom(C,,G) = A f o r a

s p h e r i c a l module G , A and

C,

have

Thus i f x i s a r e g u l a r

t h e same a s s o c i a t e d p r i m e s by ( 2 . 9 ) .

e l e m e n t o f A i t i s a l s o G - r e a u l a r . AnnlyinE Hom(G,-) t o .x O - G G G/xG 0

-

-

G / x C t o be s p h e r i c a l . C o n v e r s e l y , i f x i s an e l e m e n t

we g e t

i n t h e J a c o b s o n r a d i c a l o f A t h a t i s b o t h A - r e g u l a r and G-regular

then t h e converse h o l d s .

We can a l s o c o n c l u d e t h a t i f G i s S - s p h e r i c a l for a l o c a l ring A

then

m-depth G = d e p t h A .

Gorenstein l o c a l r i n g Given a

then A

I t follows t h a t i f A i s a

C,.

s p h e r i c a l module C; t h e modules s a t i s f y i n g i Ext (C,,E) = 0 € o r i > 0 ,

w i l l p l a y an i n t r i g u i n g r o l e l a t e r . F o r s u c h modules we define a f i l t r a t i o n

C II (G)

C(G) C as follows C(G)

= 1 E \ t h e n a t u r a l man $E : GPHom(G,E)-E

C(C)o

=

i s onto ?,

E [ w i t h O E an isomorphism I .

N o t i c e t h a t E b e l o n g s i n C(G) 0

-K -

Gn -E

i f i t c a n be w r i t t e n a s

-

0.

Sp h e r i c a 1 Modules I f we a p p l y Hom(G,-) diagram (?)

-

1

0-K

and f o l l o w w i t h GP(-) w e o b t a i n t h e

-

GPIIom(G,K)

95

GWIlom(G,Gn)

I

- GWHom(G,E) - GBExt(G,K) I

E-fl

Gn

where t h e u p p e r secluence b e g i n s w i t h an i n j e c t i o n and e n d s w i t h a surjection. If Thus

K

E

C(G)

Thus

E

E

C(C)o

E

E

C(G)o

and w e a l s o r e c o g n i z e ( ? )

E

G"-

S c h a n u e l ' s lemma : Suppose

(i)

Ext(G,K) = 0 .

a s Tor(C,,Ifom(I;,E)).

implies the existence of a nresentation G~---

(c)

we c o n c l u d e t h a t

0-

E

-n. cb

K-Gm-

and

C(G)o

E

- - c;" -

E-0

J,

(ii)

0

L

L

a r e two G - p r e s e n t a t i o n s o f E . Then

E-

F)

Gm

0

=

K 8 G".

P r o o f . As i n t h e p r o o f o f t h e S c h a n u e l ' s lemma n r o p e r

we define

(x,y)

X =

E

t h e e x a c t sequences 0

0-

- KL

Gm x G n

1

$ ( x ) = $ ( y ) l and have

- x -cmX

Gn-

0 0

t h a t s p l i t by t h e p r e c e d i n g r e m a r k s . C(G),

may now b e d e f i n e d a s c o n s i s t i n g o f t h e modules

a d m i t t i n g G - r e s o l u t i o n s o f a r b i t r a r y l e n g t h . I f we now e x t e n d t h e n o t i o n o f G - r e s o l u t i o n by a l l o w i n g d i r e c t summands o f GmTs t o p a r t i c i p a t e , we define

H(G)

a s c o n s i s t i n g o f modules w i t h

a f i n i t e G - r e s o l u t i o n . For t h e modules i n H ( G ) we s h a l l s e e i t h a t t h e c o n d i t i o n ExtA(G,-) = 0 , i > 0 , i s a u t o m a t i c a l l y satisfied.

S n h e r i c a l hlodules

96

Let G,G' b e s p h e r i c a l A-modules and x a n o n z e r o d i v i s o r i n

(d)

t h e <Jacobson r a d i c a l o f A . P r o o f . Apply

-

*

G1/xG1 then G

G'.

.x - c;' - T,'/xc;' -Hom(C,G') - Hom(G,G') - Hom(G,G'/xG')

Ext 1(G,G')

Since

*

Hom(G,-) t o

0

0

I f C/xG

0 :

C,'

X

EXt

Ext 1( G , G ' / x G ' ) A

1( G , G

= Ext 1 A/

)

-

XI

Ext 1 ( G , G ' / x G ' ) .

(G/xG,G'/xG')

and G/xG

2

GlxG'

1 s p h e r i c a l A / ( x ) - m o d u l e , Ext A ( C , G ' ) = 0 by Nakayama's lemma. Let now 6 EHom(G,G') b e s u c h t h a t i t i n d u c e s t h e

is a

isomorphism

C/xG

9

G'/xG';

surjective. Similarly

by Nakavama's

lemma a g a i n

@

is

i s a homomorphic imaqe o f GI. Thus

C,

G and G ' a r e i s o m o r n h i c by an s t a n d a r d a r g u m e n t .

(c)

On t h e number o f isomornhism c l a s s e s o f s n h e r i c a l m o d u l e s :

Let A be a l o c a l blacaulay r i n g : a c c o r d i n g t o ( d ) , t h e man Sph(A)

- Snh(A/x\)

o b t a i n e d by r e d u c i n p modulo a n o n z e r o d i v i s o r i s i n j e c t i v e . I n o r d e r t o e s t i m a t e ( a t l e a s t whether i s f i n i t e ) t h e c a r d i n a l i t y

o f Sph(A)/

-

t h e n t h a t A i s an A r t i n i a n r i n g .

we assume

Let I d e n o t e t h e i n j e c t i v e e n v e l o p e o f A/E. leads t o

G*

= IIom(G,I)

llom(G*,G*)

Sph(A) a l s o . I n d e e d

E

*

Then G;cSnh(A)

Ilom(G*PG,I)

s i n c e we saw i n (4.9) t h a t

G@G*

*

flom(1,I)

=

A.

I . A s A i s Noetherian

we

r e c a l l from [ E , C h a p . V I ] t h e isomornhism liomA(Exti(X,Y) , Z )

Tori(X,HomA(Y A ,Z))

when Z i s A - i n j e c t i v e and X f i n i t e l y g e n e r a t e d . FlakinR G

and Z = I

we get

X = Y =

S D h e r i c a l Modules i A ExtA(G*,G*) = Tori(G*,G)

97

A i = (ExtA(G,G)* for i * Tor.(C;,G*) 1

>

0.

I n t h i s f a s h i o n w e o b t a i n an i n v o l u t i o n a c t i n g on S?h(A). Q u e s t i o n . By an abuse o f n o t a t i o n s t i l l w r i t e Sph(A) f o r t h e isomorphism c l a s s e s o f s p h e r i c a l modules.

( a ) I s t h e number

o f e l e m e n t s i n S p h ( A ) , f o r a l o c a l Macaulay r i n g A , f i n i t e ? ( b ) I f Sph(A)

#

is

{A},

Sph(A)

an e v e n number? ( I t i s e a s y

t o s e e t h a t i f t h e r e s i d u e f i e l d o f A i s f i n i t e t h e n Sph(A) i s

.

finite ) The p r e c e d i n g o b s e r v a t i o n s a l s o s u g g e s t : (f)

I f A i s a l o c a l Flacaulay r i n g

module t h e n

Sph(A)

-

admittint! a canonical

is surjective, that i s ,

Snh(A/xA)

e v e r y e l e m e n t o f Sph(A/xA) i s l i f t a b l e . Althouqh we c a n n o t p r o v e t h i s y e t , p o r t i o n s o f ( e ) can b e e x t e n d e d t o h i g h e r d i m e n s i o n s . We u s e an argument o f [ 1 3 , 3 6 ] . (4.10)

P r o p o s i t i o n . Let G b e an s p h e r i c a l module o v e r

t h e N o e t h e r i a n r i n g A and l e t E be a module o f f i n i t e i n j e c t i v e d i m e n s i o n . Then $ E : GPHom(G,E)

-E

i s an isomorphism.

P r o o f . We assume E f i n i t e l y g e n e r a t e d . Again w e u s e t h e s p e c t r a l sequences of

[El Torn(G ,E.xtq (G,E)) and

E x t D ( E x t q ( G , G ) , E ) t h a t c o n v e r g e t o t h e same l i m i t . By ( 2 . 2 0 ) i Ext ( G , E ) = 0 f o r i > 0 s i n c e d e n t h C; = d e p t h A with r e s p e c t t o any i d e a l . T h e s e s e q u e n c e s t h e n y i e l d GPHom(G,E) (4.11)

Corollary. I f

module, t h e n Ifom(G ,n)

E

G

Sph (A).

E

2

E.

Sph(A)

and R i s a c a n o n i c a l

S p h e r i c a l Zlodules

98

P r o o f . F i r s t l y , Ilom(llom(G,R) , l f o m ( C ; , R ) )

=

Ilom(GPIIom(C,fi) , Q ) = llom(9,R) = A . \ i c x t , t o s ~ i o w E x t i ( l l o m ( ~ , ~,11om(C;,n)) ) = 0 for i > 0 : i\pply l i x t J ( - ,R) t o t h i s module and r e t s u c c c s s i v c l v LxtJ (Exti ( C k , G * ) ,G)

i

= T o r . (C,:$ ,T-.xt [C;*

1

,Q)

T o r . ( ( ; * , , T o r i ( f ; , I l o m ( n , R ) ) = 'Tor. ( C * , C ) J

(p,)

=

1

=

0,

C;*

=

Hom(G,R).

R e f l e x i v i t y o f s p h e r i c a l modules : Let C E Snln(;t):

i f C;

i s r e f l e x i v c and 2 i s a l o c a l 4 r t i n i a n r i n n , C = Ilom(llom(G, 2) , A )

yiclds the relation r(C) = ~(1lom(G,~'~))-r(A) in the n o t a t i o n o f ( 4 . 0 ) .

S i n c e w e have a l s o

we c o n c l u d e v(Iiom(C,A)) = 1

r(A) = r ( G ) - v ( G ) ,

and C i s A - f r e e .

4ssume now t h a t 4 i s a S , - r i n E .

I f G i s r e f l e x i v e we may

1

by t h e p r e c e d i n g i d e n t i f y G t o an i d e a l o f A . As A = Ilom(C,,C;) = End(Hom(Hom(G,A) , A ) ) ,

f o r e a c h l o c a l i z a t i o n a t a prime o f h e i e h t 1 w e c o n c l u d e IIom(G,A)

i s a f r e e A-module by ( 3 . 1 2 )

that

and h e n c e G i s A-free.

Thus we c o n c l u d e t h a t i f ,4 i s a S 2 - r i n g t h e n G i s r e f l e x i v e i f f it i s isomorphic t o a d i v i s o r i a l i d e a l t h a t i s p r i n c i p a l a t t h e nrimes o f P ( A ) . (4.12)

C o r o l l a r y . I f A i s an UFD

t h e n Sph(A) = { A ] .

T h i s c o u l d be viewed as a g e n e r a l i z a t i o n o f ( 2 . 3 9 ) . 54.5

Resolutions

and

Given a s p h e r i c a l

Divisors. module G , an E u l e r c h a r a c t e r i s t i c i s

Sph e r i c a 1 ?4odules

a t t a c h e d t o t h e modules i n l i ( G )

99

and a d i v i s o r t o t h e modules i n

H ( C ) w i t h t o r s i o n ; t h i s s u b s e t w i l l b e d e n o t e d by H(G)O and t h e

a s s o c i a t e d d i v i s o r runs p a r a l l e l t o t h a t o f Chapter 3. Assume A t o b e a c o n n e c t e d r i n g . F o r any module E which i s a d i r e c t summand o f a Gn w e may a t t a c h a u n i q u e r a n k i n any o f t h e two ways: i ) A s Hom(Gn,G) i s f r e e o f r a n k n t h e r a n k o f E i s r a n k o f t h e p r o j e c t i v e module i i ) E @ I: = G"

the

Hom(E,G).

implies t h a t a t each l o c a l i z a t i o n E i s

i s o m o r p h i c t o a d i r e c t sum o f r c o p i e s o f G - l o c a l c a n c e l l a t i o n ; t h a t r i s u n i q u e and a g r e e s w i t h t h e r a n k i n i ) . (a)

Euler c h a r a c t e r i s t i c : Let E b e a module i n H ( C ) . 0-G""

r c P

Let

...-I;

'1 - $1 G

0 - E - 0

be a f i n i t e G - r e s o l u t i o n . i

D e f i n i t i o n . x G ( E ) = C(-1) ri. T h i s i n t e g e r i s n o t d e p e n d e n t on t h e r e s o l u t i o n and i t s n u l l i t y i s r e l a t e d t o t h e e x i s t e n c e - j u s t as i n ( 2 . 2 6 ) -

of

n o n z e r o d i v i s o r s of A i n t h e a n n i h i l a t o r o f E . We show t h i s f a c t i n two d i f f e r e n t ways. Because

Hom(G,G)

= A, we

may view t h e r e s o l u t i o n a s a G -

m a t r i c i a l complex and a p p l y ( 2 . 3 2 )

i) ii)

I(+k)-depth G rank($k)

+

2

:

k;

rank(+k-l) = rk.

As e v e r y r e g u l a r A-sequence i s a l s o a r e g u l a r G - s e q u e n c e ,

Sn he r i c a l P4odule s

100

i f we a p p l y Hom(G,-) t o t h i s s e q u e n c e w e o h t a i n a f r e e complex t h a t i s a l s o e x a c t . I f we now t e n s o r t h i s l a s t comnlex hv G we o b t a i n t h e o r i g i n a l comnlex

9,

and

-E

: GBl[lom(C;,E)

i s an isomorphism. In p a r t i c u l a r t h e h y p o t h e s i s t h a t t h e mo-

dules i n H(C)

s a t i s f y the condition i n C(G)

T h i s shows t h a t t h e i n t e q e r x G ( E ) c h a r a c t c r i s t i c of Hom(G,E) t i v e d i m e n s i o n . Note t h a t

eouals

i s n o t needed.

0

t h e usual Euler

which i s a module o f f i n i t e p r o j e c Tori(G,flom(G,E))

= 0

for i > 0, a

remark t o he u s e d l a t e r . (4.13)

Hom(G,-) and G @ ( - ) a r e

C o r o l l a r y - . The f u n c t o r s

inverse equivalences o f the categories (4.14)

Corollary-. x G ( E )

2

ll(G)

and H ( A ) .

0 ; x G ( E ) = 0 i f f ann(E)

con-

t a i n s a n o n z e r o d i v i s o r o f A. S i m i l a r l y we c o u l d d e f i n e a G-dimension f o r t h e modules i n Ii(G)

: assume A t o be a l o c a l r i n g

l a s t i n t e g e r f o r which n o t h i n g new i s o b t a i n e d

and d e f i n e G-dim(E) a s t h e

E x t r ( E , C ) # 0 . I n t h i s c a s e however a s t h i s number i s j u s t

m-depth A -

m-depth E. Another a p p r o a c h i s t o use t h e S c h a n u e l ' s lemma o f 5 4 . 3 ~ t o g e t h e r w i t h t h e t e c h n i q u e s o f l o c a l c a n c e l l a t i o n and d e v e l o p a t h e o r y o f d i m e n s i o n and minimal r e s o l u t i o n s . Remark. Without much d i f f i c u l t y one can s e e t h a t i f i n a s h o r t e x a c t s e q u e n c e two modules a r e i n H ( G )

then

SO

is the

t h i r d one. (4.15)

Remark. Let G , G '

be two s p h e r i c a l modules. The

S n h e r i c a l llodules e v i d e n c e seems t o p o i n t t h a t

101

is rather sparse i f

Il(G)nli(G')

C; and G' a r e n o t l o c a l l y i s o m o r n h i c . Supnose 4 i s a l o c a l rincl

and l e t E b e a module w i t h G-dim(E) = 1: i f E

E

we must

lf(C')

GI-dim(E) = 1 , b y an e a r l i e r r e m a r k . Let

a l s o have t h a t

dl

0 -G1-

be a minimal C - r e s o l u t i o n o f E

E-!I

Go-

- i.e. the e n t r i e s of

C$

are in

t h e maximal i d e a l o f A . We o b t a i n 0

-

Hom(G' , G I )

-FIom(G' ,Go)-

Hom(C;' , E )

-

1 By Nakayama's lemma w e t h e n have Kxt ( G ' , C ) = 0 . T e n s o r now

t h e s e q u e n c e above w i t h C;' t o q e t t h e d i a e r a m

-p

0

-

yo

G'PHom(G' ,G1)

I

0 -

-

- G'PHom(G'

1

,Go)

- G'PMom(G' , E ) 11

0

n

E -

G1

-

where t h e m i d d l e h o r i z o n t a l s e q u e n c e i s e x a c t as t h e module Torl(G',Hom(G',E))

= 0.

we a l s o c o n c l u d e t h a t

From t h e

minimality o f t h e r e s o l u t i o n

r$G i s s u r j e c t i v e . K1

hand a r e d i r e c t sums of t h e k e r n e l o f them i s r e a l l y t h e ' m a t r i x ' 0

w e g e t t h a t r$G

.

$G

and K O on t h e o t h e r and t h e man b e t w e e n

lemma again

By Nakayama's

i s an i s o m o r n h i s m . C o u n t i n g m i n i m a l numbers o f

g e n e r a t o r s we h a v e

v

( G I )

* v (Hom(C; ' , G ) )

account t h e symmetrical r e l a t i o n

= v(G)

.

Takinp i n t o

we c o n c l u d e v(Hom(G' ,G)) = 1.

T h i s l a s t module b e i n g f a i t h f u l i t i s i s o m o r p h i c t o A and G

G'. (4.16)

Remark. A s t a t e m e n t more g e n e r a l t h a n 5 4 . 4 b i s t h e

S p h e r i c a l Modules

102

f o l l o w i n g : L e t G be a s p h e r i c a l A-module and l e t I be

a

G o r e n s t e i n i d e a l , t h a t i s , an i d e a l s a t i s f y i n g g r a d e I = pd A/I = r , and

i)

E x t i ( A / I , A ) = A/I.

ii) Then

G/IG

E

Sph(A/I).

P r o o f . We may assume t h a t A i s a l o c a l r i n g . C o n s i d e r t h e following convergent s p e c t r a l sequence E x t I / I ( G / I G , E x t 2 ( A / I ,G))-Exti(G/IG,G). S i n c e e v e r y r e g u l a r A-sequence i s G - r e g u l a r , Ext!(A/I,G) for

q < r

and

Exti(A/I,G)

Ext;(A/I,A)WC

2

= 0

G/IG. Thus we

g e t t h e isomorphism ExtP,/I (C/IG,G/IG) Let now

Fr- 4

0-

Fr-l

Ext!+r(G/IG,C).

. . . Fo

-

A/I-

0

be a minimal p r o j e c t i v e r e s o l u t i o n o f A / I . The c o n d i t i o n i i ) makes Fr

of rank 1

and $(1) = (al,

..., a n )

E

Fr-l

with t h e

a ' s a minimal g e n e r a t i n g s e t f o r t h e i d e a l I . T e n s o r t h i s i sequence w i t h G t o g e t a G - r e s o l u t i o n o f l e n g t h r f o r G/IG and c o n s e q u e n t l y Ext!+r(G/IG,G) Exti(G/IG,G) (b)

2

A/I

= 0

for

t o complete t h e proof.

Divisors : Now w e a t t a c h an i n v e r t i b l e i d e a l

module E i) tion

D > 0 . We a l s o g e t

E

H(G)

.

d(E)

t o any t o r s i o n

Let E be a t o r s i o n module i n C(G)(>; c o n s i d e r a p r e s e n t a -

Sphe r i c a 1 Mo du 1e s

G Define

d(E)

-E -

9 G"

= (F($)-')-',

where

I03

0.

d e n o t e s t h e i d e a l gene-

F(+)

r a t e d by t h e m i n o r s o f o r d e r n o f t h e m a t r i x . By t h e S c h a n u e l ' s

d(E)

lemma o f 1 4 . 4 ~ we g e t t h a t

does n o t depend on t h e

-

C;

presentation. ii)

Suppose now

E

obtain

:

H(G)

Gn

0-

(where G i

E

...

Go-

E-

0

i s a d i r e c t summand o f a Gm)

t h e module

Hom(G,E)

E

. Applying d(E)

and

H(A)

Hom(G,-) w e

is the divisor

d(E) i s an i n t r o d u c e d i n C h a p t e r 3 o f t h e module Hom(G,E). Thus invertible ideal. i i i ) Suppose now

0

# E

E

H(G)nH(G')

for d i s t i n c t s p h e r i c a l

modules. As w e r e m a r k e d , i t i s p l a u s i b l e t h a t H(G) = H ( G ' )

at

l e a s t i n t h e l o c a l case. Nevertheless t o o b t a i n t h e e q u a l i t y of t h e two r e f l e x i v e i d e a l s

i t i s enough t o

and $ , ( E )

$,(E)

compare them a t t h e l o c a l i z a t i o n s

c a s e we u s e ( 4 . 1 5 ) t h a t says E = 0 or G

E

p

Ap,

E

P

P(A).

But i n t h i s

G'

P'

To sum up : (4.17)

Theorem. Let Sph(A)

denote t h e s e t of s p h e r i c a l

modules o v e r A and l e t E be a module i n

H(G)O.

Then & ( E )

i s an i n v e r t i b l e i d e a l and d o e s n o t depend on G . R e s t r i c t e d t o a fixed

H(G) t h i s d i v i s o r i s a d d i t i v e .

Chapter 5 I -divisors A d i v i s o r i s a t t a c h e d t o f i n i t e l y g e n e r a t e d modules o v e r a

N o e t h e r i a n r i n g t h a t does n o t q u i t e a r i s e i n t h e manner o f C h a p t e r 4. I t i s o b t a i n e d by mimicking t h e c o n s t r u c t i o n o f t h e standard F i t t i n g ' s divisors but using i n j e c t i v e resolutions.

Thus i f A is a N o e t h e r i a n r i n g and E i s a f i n i t e l y gener a t e d module w e c o n s i d e r a r e s o l u t i o n 0-E-I w i t h 'I

and I1

o - I@

1

i n j e c t i v e modules. We assume f u r t h e r t h a t , f o r

e a c h prime p, I(A/p) a p p e a r s f i n i t e l y o f t e n o n l y i n t h e i n d e composable r e p r e s e n t a t i o n s o f I o and I1 - and t h i s can always b e a r r a n g e d by

[?I.

o

may t h e n b e viewed a s a m a t r i x and t h e

d e t e r m i n a n t a l i d e a l s formed much a s done e a r l i e r . To overcome s e v e r a l t e c h n i c a l d i f f i c u l t i e s - as these i d e a l s l i v e i n a p r o d u c t o f p - a d i c c o m p l e t i o n s - s h a l l b e assumed t h a t p r i m e s o f h e i g h t one and g r a d e one a r e t h e same. The g l u i n g c a n t h e n b e c a r r i e d o u t and d i v i s o r i a l i d e a l s i n A o b t a i n e d . A f t e r a summary e x a m i n a t i o n o f i t s p r o p e r t i e s t h i s d i v i s o r

i s a p p l i e d t o t h e c a t e g o r y o f modules o f f i n i t e i n j e c t i v e d i mension where i t t u r n s o u t t o be i n v e r t i b l e . 85.1

Construction. Let A be a N o e t h e r i a n r i n g o f t y p e S 2 - t h e g r a d e r e s t r i c -

t i o n above. T h i s i s a minor c o n s t r a i n t a s t h e modules o f f i n i t e injective

dimension a r e e f f e c t i v e l y d e f i n e d o v e r t h e s e r i n g s . 104

I -divisors

105

Let E b e a f i n i t e l y g e n e r a t e d t o r s i o n module and c o n s i d e r an i n j e c t i v e p r e s e n t a t i o n as above. In t h e s e t a t e d primes o f E c o n s i d e r t h o s e primes

pl,.

o f associ-

Ass(E)

. .,pn o f

height

o n e . I f t h e r e i s no s u c h prime i n Ass(E) w e p u t I - d i v i s o r o f E =

d i v ( E ) = A . Assume t h i s i s n o t t h e c a s e and l e t S be t h e mul-

t i p l i c a t i v e s e t n (A\P).

Localize the p r e s e n t a t i o n a t S t o g e t

over t h e one-dimensional semi0 1 The i n j e c t i v e modules ( I ) s and ( I ) s a r e d i r e c t

a n i n j e c t i v e p r e s e n t a t i o n o f ES l o c a l r i n g AS.

sums o f t h e I ( A / p i ) ' s t h e moment A i s AS.

o n l y . Change n o t a t i o n and assume f o r

The r e s o l u t i o n t a k e s a more e x p l i c i t form :

?

O-E-

I(A/pi)

I'.

1

We r e c a l l t h a t i)

@

(Ex]) :

HomA(I(A/p) , I ( A / p ) ) = Ap

5. 1

.

8 I ( A / ~ ~ ) i

(where

'

o v e r a module

denotes completion with respect t o the a d i c topology of t h e c o r r e s p o n d i n g maximal i d e a l ) . ii)

HomA(I(A/p),I(A/q))

f o r .incomparable p r i m e s p,g.

= 0

A l t o g e t h e r t h e s e remarks l e a d t o a view o f

where Bi Di(E)

of

ri x si block with e n t r i e s i n g e n e r a t e d by t h e minors o f o r d e r

i s ,an A

pi e a s i l y seen t o be a p . A 1

Pi

-primary; Localize

becomes a module o f f i n i t e l e n g t h functor sequence

( - ) * = Homi

pi

(-,I(A/pi))

-

as a matrix

@

A

Ei

ri

E a t pi

The i d e a l

,

o f Bi i s

-

and E

and a p p l y t h e d u a l i z i n g t o obtain the exact

pi

I-divisors

106

t ~ i^r Ai-Ai-E*-

(t=transpose)

0

pi

pi

and t h e i d e a l Di(E) j u s t d e f i n e d i s t h e 0 - t h F i t t i n g i d e a l o f E X . In p a r t i c u l a r it i s independent of t h e chosen i n j e c t i v e

presentation. There i s a unique primary i d e a l

of A such t h a t

gi

A

giApi = Di(E).

I t t h e n f o l l o w s t h a t we may f i n d an i d e a l q

i n A such t h a t

(%A

A

pi

)

=

Di(E)

f o r each of t h e pi's.

We now go b a c k t o t h e o r i g i n a l s i t u a t i o n b e f o r e t h e l o c a l i z a t i o n a t S . We p i c k an i d e a l Q i n A o n l y a s s o c i a t e d p r i m e s and w i t h (5.1)

Let

with

pl,.

. . ,E,

-

for its

QAS = 9.

D e f i n i t i o n . d i v ( E ) = (Q

-1 -1 )

=

D(Q).

Example. Let k b e a f i e l d and l e t t b e a n i n d e t e r m i n a t e . 3 4 3 4 5 B = k [ [ t , t ] ] L A = k [ [ t , t , t ] ] 6 k [ [ t ] ] . Notice t h a t

B i s a G o r e n s t e i n r i n g and t h a t

=

Homg(B,A)

is a canonical

module f o r A . I t i s c l e a r t h a t R = ( t 3 , t 4 ) . Let E b e t h e module

n/t5Q.

E has a presentation 2 W E 0-L-A

0 3 where L h a s f o r g e n e r a t o r s t h e e l e m e n t s ( t ,O), ( 0 , t 3 ) , ( t 4 , t5 ), 5 4 ( t , t ) and ( t 6 , t 5 ) . Thus t h e F i t t i n g d i v i s o r o f E , d(E) = 3 3 4 5 t ( t ,t ,t ). As for its I-divisor : 0 -

n

t 3L i - 2 - E - 0

leads t o 0-A-

F i n a l l y , as

.t

1 ExtA(E,n)

A-

- 0.

E x t i ( E , Q ) = HomA(E,I(A/m), - we c o n c l u d e t h a t 1 d(ExtA (E,R))

=

3 t A

=

div(E).

I-divisors 55.2

107

Euler c h a r a c t e r i s t i c s of I n j (.h). For a S o e t h e r i a n r i n c A we d e v c l o n a t h e o r y o f E u l e r c h a t h e c a t e p o r y o f f i n i t e l y i r e n e r a t e d mo-

racteristics for Inj (A),

d u l e s o f f i n i t e i n j e c t i v e d i m e n s i o n a s c o n s i d e r e d i n [_ 3 6 ,_ 35]. In t h e n e x t s e c t i o n a d i v i s o r t h c o r v f o r t h e s u b c a t e c o r y l n j c o n s i s t i n R o f t h e modules i n I n j ( A ) w i t h t o r s i o n w i l l

Inj(A)O

be d i s c u s s e d . Both depend c r u c i a l l y on i n some n a r t i a l d u a l i t y 30] e s t a b l i s h e d i n [-

between Ini(.4) and t h e modules o f f i n i t e

p r o j e c t i v e dimension. Without l o s s o f q e n e r a l i t y l e t 2 be a c o n n e c t e d r i n g . Let C b e a f i n i t e l y g e n e r a t e d module o f f i n i t e i n j e c t i v e d i m e n s i o n ,

and w r i t e 0

- E - I o - I 1 ...

f o r a minimal i n j e c t i v e r e s o l u t i o n o f E . w i t h d e p t h An = r

In

-0

F o r a p r i m e i d e a l n-

write

L

x(p:E)

=

Theorem. x(n_:E)

(5.2)

(-l)r-ipi(~:E).

C

i =

x(E)

t h a t does n o t denend on p. x ( E ) = 0

i s a non-neqative i n t e p e r

i f f t h e a n n i h i l a t o r of E

contains a r e p u l a r element. 'That

x

i s t h e n an E u l e r c h a r a c t e r i s t i c on I n j ( A ) f o l l o w s

from t h e meaning o f t h e p i t s . P r o o f . Because o f t h e c o n n e c t e d n e s s of A w e may assume t h a t A i s a l o c a l r i n g o f maximal i d e a l m_.

Let

r

= depth A:

h

let A

denote t h e m - - a d i c c o m n l c t i o n o f 4 . As I A ( A / ~ ) = I i ( A / m _ ) ,

in calculating

x(m_;E) we may a s ' w e l l assume t h a t .4 i s a com-

plete local ring.

I-divisors

108

The s e t t i n g i s now r e a d y f o r an a p p l i c a t i o n o f

([El):

Theorem. Let A be a c o m n l e t e l o c a l r i n R o f d e n t h r

(5.3)

and E b e a f i n i t e l y g e n e r a t e d module o f f i n i t e i n j e c t i v e dimens i o n . 'Then t h e module

b! = E x t i ( I ( A / m -) , E ) h a s t h e followincr

p r o p e r t i e s : i ) PI i s a f i n i t e l y p e n e r a t e d module o f f i n i t e i p r o j e c t i v e d i m e n s i o n ; i i ) Supp(fiI) = S u p p ( E ) : i i i ) ExtA(M,A) = Hom(W:-l(E) -

,I(,\/!)),

where

s t a n d f o r t h e l o c a l cohomo-

I{;(-)

-

logy groups. Proof of ( 5 . 2 ) Br-i(Fl)

where

: ( a ) I t w i l l b e f i r s t shown t h a t u i ( E )

B.(bf)

I

A

= dimk(Tor.(k,M)) j -

i n a minimal f r e e r e s o l u t i o n o f '1.

= rank o f t h e j - t h

=

term

In t h i s c a s e x ( n-: E ) would

equal t h e Euler c h a r a c t e r i s t i c defined i n 92.5: t h e statement

on t h e p o s i t i v i t y and r e l a t i o n t o t h e r e g u l a r i t y o f t h e a n n i h i l a t o r would f o l l o w from

i i ) a b o v e . To show t h e e q u a l i t y o f t h e

and c o n s i d e r t h e convergent s p e c t r a l sequences with i t h e same l i m i t 1-1

Tor (k ,Extq ( I ,E) ) P -

and

E x t P (Extq (k,I ) ,E) (I = I (A/m_) ) .

I n t h e p r o o f o f ( 5 . 3 ) i t emerges t h a t E x t i ( E , I ) i C o n s e q u e n t l y w e h a v e Torr-i(&,bf) = E x t ( k-, E )

= O for i <

r.

.

( b ) T h i s s t e p c o n s i s t s i n showing t h a t i f

i d e a l t h e n x(p:E) = x ( m _ ; E ) .

2 i s any nrime

In t h i s we may assume t h a t x(m_;E)

> 0 , t h a t i s , E i s a f a i t h f u l A-module. From ( 2 . 3 4 ) t h i s makes A a Eiacaulay r i n g .

I f A admjtted a canonical

- as is t h e case

h

f o r A - we would have QPHom(Q,E) A E Exti(Q,E) = 0 , i > 0

(4.10) (2.20)

and

I-divisors

109

and c o u l d t h e n b e g i n a r e s o l u t i o n

O-Lwith fiO

clo-E-fl

a d i r e c t sum o f c o p i e s o f 9

and I, a n o t h e r module o f

m-denth L

f i n i t e i n j e c t i v e dimension: a l s o ,

i n f f m-depth -

a,

m-depth E + 1 1 . We c o u l d t h e n f a s h i o n a r e s o l u t i o n

with

Ri

nr-l

K -

0-

= d i r e c t sum o f R ' s

b e i s o m o r o h i c t o some Q

n

...

R"

-E -

0

m-denth K = r . K would t h e n

and

and t h e c o n s t a n c y o f x ( 2 : E )

would

f o l l o w much a s i n 9 4 . 5 .

To a v o i d t h e u s e o f t h e c a n o n i c a l module we r e a s o n a s f o l l o w s : Let

p. Let

r,

also

(5.4)

be a minimal p r i m e i d e a l o f c\ c o n t a i n e d i n

be a minimal p r i m e o f A l y i n q ahovc nd. L e m m a . ( [ g ] ) J,et (A,?)

- (R,;)

homomorphism o f l o c a l N o e t h e r i a n r i n g s and l e t Vacaulay r i n g o f type A-module o f t y p e and

rg(B/m_B)

=

be a f l a t l o c a l B / m-B be a

r . Let F. be a ( f . ~ . )Vacaulav

r A ( E ) = s . Then EMB i s a ' l a c a u l a v R-module

rg(EBB) = r * s . I f w e u s e t h e 0 - d i m e n s i o n a l c a s e o f t h i s lemma t i e p e t x(m_;E) = x ( m _ ; E ) = x ( h : E )

= x(n+;E)

and f i n a l l y l e t

p p l a y t h e r o l e o f m_. 85.3

Divisors

on

I n j (A)'.

The c o n s t r u c t i o n o f d i v ( - ) i n 1 5 . 1

applied t o the torsion

modules i n I n j ( A ) y i e l d s a d i v i s o r i a l i d e a l which w i l l be shown t o b e i n v e r t i b l e . N o t i c e t h a t t h e - h y p o t h e s i s t h a t A he o f t y p e S2

i s f u l f i l l e d a s Bass's c o n j e c t u r e i s r e s o l v e d f o r r i n g s

I -divisors

110

o f l o c a l depth one. lie d i s c u s s f i r s t t h e change o f t h e i n j e c t i v e dimension o f

k

a module a f t e r a f l a t c h a n s e o f r i n g s

: A

-

B.

Next, w e

show t h a t u n d e r m i l d c o n d i t i o n s on t h e f i b e r s o f t h e morphism h t h e d i v i s o r introduced i n 5 5 . 1 behaves p r o p e r l y . Additional

f a c t s may be found i n [ _ 12,_ 37]. Let A h e a N o e t h e r i a n r i n p and l e t E be an A-module o f

h_ : A

f i n i t e i n j e c t i v e d i m e n s i o n . Now l e t

-B

be a f l a t

homomorphism o f r i n g s . W e are interested in the conditions that make

BBAE

a

B-module of f i n i t e i n j e c t i v e d i m e n s i o n . In t h i s

g e n e r a l i t y s i m p l e examFles show t h e n a t u r a l i t y of t h e f o l l o w i n g c o n d i t i o n s : ( a ) T h e K r u l l d i m e n s i o n s o f A and B a r e f i n i t e ; ( b ) The f i b e r s o f h-, i . e . t h e r i n g s Bt?,k(n_)

are

Gorenstein r i n g s . (5.5)

Theorem. IdB(BeAE) <

m

.

P r o o f . Let 0 - E - I

0

- 1

1

...

1n-o

be an i n j e c t i v e r e s o l u t i o n o f E . S i n c e t h e K r u l l d i m e n s i o n o f a r i n g bounds t h e ( f i n i t e ) i n j e c t i v e d i m e n s i o n o f modules ( [ 4-] ) and t h e I i

a r e d i r e c t sums o f indecomposable i n j e c t i v e s , we

may assume t h a t that

BRAE = B t?

E = I(A/p).

P AP

In c o n s i d e r i n g R a A E

we notice

E . We may t h e r e f o r e assume t h a t t h e r i n g s i n

q u e s t i o n a r e l o c a l r i n g s : (A,;),

(B,P4). I f

BPAI(A/m_), from t h e s t r u c t u r e o f

T(A/m). The p r o o f w i l l now

@ = B

we get 0 =

f o l l o w by i n d u c t i o n on t h e d i m e n s i o n o f t h e f i b e r B / m-B . (i)

e claim t h a t i n t h i s case dim B / m-B = 0 : W

BPE i s

I-divisors t h e i n j e c t i v e e n v e l o p e o f B/P!.

111

F i r s t n o t i c e t h a t BBE i s an

e s s e n t i a l e x t e n s i o n o f B/mB, a s fIomB(B/mB,BPE) = BPliom(A/m,E). -

AS B/mB - i s a G o r e n s t e i n rinp. o f d i m e n s i o n 0 , i t i s an e s s e n t i a l

e x t e n s i o n o f P,/P,l.

1Ve may t h e n w r i t e

0

- RPE

IB(B/?l).

To show e q u a l i t y above i t i s enough t o v e r i f y t h a t llomB(H/mnB,) -

t a k e s t h e same v a l u e on t h e two modules f o r e a c h n . T h i s r e d u ces t h e question t o t h e consideration of Artinian r i n p s , i . e . 1 1 t h a t ExtB(B/rnB,BOE) = 0 i m p l i e s ExtB(B/”I,BWE) = 0 . S u n n o s e wc want t o c o m p l e t e t h e diaernm

t h e r e s t r i c t i o n o f Q t o mB To p r o v e t h a t

can be l i f t e d t o a map I):B -BBE.

o may b e l i f t e d i t s u f f i c e s t o show t h a t

i s l i f t a b l e . A s i m p l e i n d u c t i o n on

!In

+

mB

0

-

u

leads t o the

desired result. dim B / m-B >

(ii)

I)

: Sunpose t h a t

idB(BhdE) =

m.

Among

a l l prime i d e a l s o f B l y i n g above m_ n i c k a minimal one s u c h that

i d B (BPE)Q =

Q

a.

Change t h e n o t a t i o n and assume Q = 31. Let

x b e a n o n z e r o d i v i s o r i n t h e r a d i c a l o f B/m_B; t h e n x i s a nonzero d i v i s o r i n B (S)

is exact

0

and B/xB i s A - f l a t . Thus t h e s e q u e n c e

-

BhdE

BBE

-

(B/xB)BE-O

and by t h e i n d u c t i o n h y p o t h e s i s idBlxB(B/xBBE) <

By t h e change o f d i m e n s i o n t h e o r e m for large n ( say

1

+

dim B )

E x t nB (-,EMF)

.x

idB(B/xB@E) <

we have n

m

m.

a l s o . Thus

I-divisors

112

and i n p a r t i c u l a r

n Extg(I%/M,I?@E) = 0. Suppose f o r some prime

i d e a l P o f C , Ext:(R/P,RWE) a

Thus ;\

'3:

n i c k P maximal and l e t

The sequence

M\P.

E

#

Exti(R/P,T?WE) i s d i v i s i b l e

(PnA)

u n i q u e l y bv e v e r y c l e m e n t i n

and s h o u l d be l e f t unchanged i f we l o c a l i z e a t t h e

multiplicative set

S = A\

( P n . 4 ) . But

BWT.6d.4S = 0 .

'The s i m p l e s t cxnmplc where ( 5 . 3 ) a p p l i e s i s

R = .A[t].

More p c n e r a l l y , i f

T? = A [ t ] / I

t i v e ideal of A [ t ]

and t h e c o n d i t i o n o n t h e f i b e r s i s automa-

t i c a l l y s a t i s f i e d . Suppose Gorenstein f i b e r s

I3

i s / \ - f l a t then I i s a p r o j c c -

= A[x,y]/T

i s A-flat with

and l e t us examine n i i i c k l y what t h i s

portends f o r I . IVe f i r s t remark t h a t ,J t h e i d e a l p c n e r a t e d 1)y t h e c o e f f i -

c i e n t s o f t h e v o l y n o m i a l s i n I i s e e n e r a t e d by onc i d e m p o t e n t ( [ 40]).

Without s a c r i f i c e o f g e n e r a l i t v we may assume .J = 4.

Suppose A t o be l o c a l w i t h maximal i d e a l 2. 'I'ensorinc! t h e sequence A[x,v]

O-I-

w i t h A/m

we g e t 0

I f R / m-R

- I/mI - A/;[x,y] -

-

-

€3

R / m-R

-

i s t o be a Gorenstcin rinn then e i t h e r

i s p r i n c i p a l , o r (11)

I/mT -

0.

( a ) I/mI -

i s q e n e r a t e d by two e l e m e n t s

w i t h o u t a common f a c t o r . We can e a s i l y c o n c l u d e

I -divisors

C o r o l l a r y . I i s an i d e a l o f p r o j e c t i v e d i m e n s i o n

(5.6) < 1 and -

113

B = K1 x R 2

i d e a l and B2

where

B1

a arojective 1 ( i . e . with f i n i t e f i b e r s ) .

quasi-finite

= A[x,y]/I1,

I

To have an i d e a o f more g e n e r a l f l a t e x t e n s i o n s o f f i n i t e t y p e o f a r i n g A we l o o k a t t h e f o l l o w i n g e x t e n s i o n o f t h e I i i l b e r t ' s syzygies theorem. Let A be a commutative r i n g - n o t n e c e s s a r i l y N o e t h e r i a n -

t] = and l e t E h e a module o f f i n i t e p r e s e n t a t i o n o v e r B = A [A[tl,

...,t n ]

- o r more g e n e r a l l y a f l a t A - a l p e b r a o f f i n i t e

presentation, with regular fibers. P r o p o s i t i o n . I f E i s a f l a t A-module

(5.7)

pdBE

5

n.

P r o o f . The key e l e m e n t i s t h e f a c t t h a t E - u n d e r t h e s e c i r c u m s t a n c e s - a d m i t s an i n f i n i t e f i n i t e p r e s e n t a t i o n a c c o r d a n c e w i t h [14,Prop.11.3.9.1] Pn-l

O-L-

with t h e P i ' s

:

...

PO

- E

in

O

f i n i t e l y g e n e r a t e d p r o j e c t i v e B-modules and L

f i n i t e l y p r e s e n t e d . L i s a l s o a f l a t A-module. Now we show t k a t L is a projective

B; l e t

p

= PnA.

a projective

B-module. We may l o c a l i z e a t a p r i m e P o f By I l i l b e r t ' s s y z y g i e s t h e o r e m ( L / P L ) ~

Bp/pBp-module. 0

-

K-

To c o n c l u d e F-

a minimal f r e e p r e s e n t a t i o n o f L p .

s t i l l exact

is

write

LpTensor w i t h

0

k(p)

s i n c e L i s A - f l a t . By Nakayama's lemma

t o get

K = 0.

Now we r e t u r n t o t h e c o n s i d e r a t i o n o f t h e f i n i t e l y g e n e -

I-divisors

114

r a t e d t o r s i o n modules o v e r a r i n g A and t h e i r d i v i s o r s . Suppose

-

h - : A

and l e t

E l i e s i n Inj(A)'

B

be a f a i t h f u l l y f l a t

homomorphism o f r i n g s w i t h f i n i t e K r u l l d i m e n s i o n and G o r e n s t e i n f i b e r s a t t h e maximal i d e a l s (The c a s e t o k e e p i n - and i t s c o m p l e t i o n ) . mind i s t h a t o f a l o c a l r i n g (A,m)

h_

P r o o f . We may assume t h a t l o c a l r i n g s . That as

Ext:(A/m,E) -

E

Inj(A)

F.

i s a l o c a l homomorphism o f

can be i n t e r p r e t e d f o r A l o c a l

f o r l a r g e n. The p r o o f o f ( 5 . 5 )

= 0.

Nakayama's lemma, t o t h e c o n c l u s i o n

RPE

E

leads, via

Inj(B).

Let Q be a p r i m e i d e a l of B , o f h e i g h t 1

and m i n i m a l o v e r -1 t h e a n n i h i l a t o r J O B o f BOE, where J = annAE. L e t P = h (Q)

.

Then P i s a l s o a h e i g h t 1 p r i m e minimal o v e r , J . Complete Ap and

B

Q

w i t h r e s p e c t t o t h e t o p o l o g i e s d e f i n e d by t h e maximal i d e a l s A

i n e a c h r i n g . As a resolution

0

Now i t f o l l o w s

-

a d m i t s a c a n o n i c a l module R t h e r e i s

Ap

-n

" L

R"-

i

-

by Nakayama's lemma - t h a t RRBO

of i n j e c t i v e d i m e n s i o n one and a l s o o f r a n k one

i s e x a c t , EBB

Q

E

maximal i d e a l o f h

f o r Bg.

Inj(l3 )

Q

h

B

Q'

0.

h

i s a module as

and t h e e n t r i e s o f +@l l i e i n t h e h

Thus

RPBO

i s a l s o a c a n o n i c a l module h

From t h i s i t f o l l o w s

that

div(EBB)

h

= d e t ( @ B 1 ) B Q=

A

det$*R

(2'

(5.9)

ideal.

Theorem. I f E

E

Inj(.\)

0

, d i v ( E ) i s an i n v e r t i b l e

I-divisors Since the d e f i n i t i o n of

115

d i v ( E ) 'commutes' w i t h l o c a l i z a -

t i o n , A may be assumed l o c a l : u s i n g now ( 5 . 8 ) w e may t a k e A t o be complete w i t h r e s p e c t t o t h e ;-adic

tonologv.

P r o o f . Let

M b e t h e module o f

(5.3).

1Ve show t h a t t h e

Fitting divisor

&(?I) = d i v ( E ) . Let

pl,... ,TI* h e t h e n r i m e s

o f h e i g h t one i n Supp(E) = S u p p ( V ) . To show t h e e q u a l i t y above s u f f i c e s t o check t h e l o c a l i z a t i o n s a t t h e s e n r i m e s .

Let R be a d - d i m e n s i o n a l r e g u l a r l o c a l r i n e mapping o n t o A . As t h e K r u l l d i m e n s i o n o f E i s

d e p t h A - 1 by ( 2 . 3 4 ) t h e r e

i s an i s o m o r p h i s m g i v e n by l o c a l d u a l i t y ExtA(bl,A) 1 with

ExtR d - r + l( E , R )

r = d e p t h A . Let P he one o f t h e

ti's

and 0 a n r i m e o f l?

l y i n g above P . L o c a l i z i n g t h i s i s o m o r p h i s m a t

Notice t h a t

d - r + l = K r u l l d i m e n s i o n RCf = t

0

we o b t a i n

and t h a t E p i s an

I1 -module o f f i n i t e l e n g t h . The s e c o n d module can a l s o be

Q

w r i t t e n , from t h e s n e c t r a l s e q u e n c e o f [ g , X V I . 5 ]

as

E x 2Ap ( E P , E x tRo t-l(Ap,Po)). Ext t - 1 ( A p , R Q )

Since

clude t h a t

':q

div(E)p

i s t h e c a n o n i c a l module f o r A p , we c o n -

i s t h e f l - t h F j t t i n q i d e a l o f t h e module

E x t l ( ? I , A ) . Rut i t i s c l c a r t h a t

?I

h a v e t h e same i n v a r i a n t s a s

= 1

Q

Ap

p

module.

ndATp

Q

and

=

and

?Ip

Ext'

AP

('In,Ap)

is a torsion

We may now s t a t e an i n t e r e s t i n g a p p l i c a t i o n . L e t E h e a

f i n i t e l y g e n e r a t e d module of f i n i t c i n j e c t i v e d i m e n s i o n o v e r t h e l o c a l r i n g A . Let

%J = a n n ( E ) .

I -divisors

116

C o r o l l a r y . I f g r a d e .J = 1 , t h e n A i s a 3lacaulay

(5.10)

ring, P r o o f . Write

Ad = d i v ( E )

and l e t p1

,...,qn

be t h e

p r i m e i d e a l s o f h e i 2 h t onc above ,J. By t h e p r o n e r t i e s o f d i v ( - ) t h e i d e a l Ad

s h a r e s a l l o f t h e s e p r i m e s and has no o t h e r

a s s o c i a t e d n r i m e s . Thus K r u l l dim (A/dA) = sup{ K r u l l dim ( A / p . ) -1 But i f

r

dimension

= depth A ,

a l l o f t h o s e modules

\/pi have K r u l l

r-1. S i n c e t h e K r u l l d i m e n s i o n o f A/dA

d i m e n s i o n o f A l e s s one

we g e t

r

1.

= dim A .

is t h e

B ih 1i o g r anhy

f11

F!.Auslander,

C o h e r e n t F u n c t o r s , i n Proc.Conf . C a t e g o r i c a l

Algebra, S p r i n g e r - V e r l a q , 1965, 189-231

-

- and D.Xuchsbaum, l l o m o l o g i c a l d i m e n s i o n i n l o c a l

r i n g s , Trans.Amer.~lath.Soc. 85 (1957), 390-405.

f 31

H.Bass, On t h e u b i q u i t y of G o r e n s t e i n r i n g s , Math. Z e i t s c h r . 82 ( 1 9 6 3 ) , 8 - 2 8 .

f41

-

-

,

I n j e c t i v e dimension i n Noetherian r i n g s , T r a n s .

Amer.Math.Soc. 151

f61

-

1 0 2 (1962), 18-29.

- , A . H e l l e r and P..G.Swan, T h e W h i t e h e a d Rroup o f a

p o l y n o m i a l e x t e n s i o n , Pub1 . r l a t h . I€IES 2 2 , P a r i s , 1964. N.Bourbaki,

\

Algebre Commutative, Chaps. T

- V I I , Hermann,

P a r i s , 1960-65. [71 81 [91

D.Buchsbaum and D . E i s e n b u d , What makes a complex e x a c t ? , J.Algebra 25 ( 1 9 7 3 ) , 2 5 9 - 2 6 8 .

-

-

- , Gorenstein i d e a l s of height t h r e e , P r e p r i n t .

I,.Burch, On i d e a l s o f f i n i t e h o m o l o g i c a l d i m e n s i o n i n

64 ( 1 9 6 8 ) ,

l o c a l r i n g s , Proc.Camh.Phi1.Soc.

941-348.

H.Cartan and S . E i l e n b e r g , HomoloEical AlRebra, P r i n c e t o n U n i v e r s i t y P r e s s , P r i n c e t o n , 1956. i H.-B.Foxbv, On t h e p i n a m i n i m a l i n j e c t i v e r e s o l u t i o n , Math.Scand.

20 ( 1 9 7 1 ) , 1 7 5 - 1 8 6 .

-

- , I n j e c t i v e modules under f l a t b a s e c h a n e e , P r e n r i n t .

-

- , On G o r e n s t e i n m o d u l e s a n d r e l a t e d m o d u l e s , V a t h .

Scand. 3 1 ( 1 9 7 2 ) , 267-281. A.Grothendieck, Eliments Flath.IHES

32, P a r i s ,

de

I

I

Geometrie A l g i b r i q u e , Publ.

1965.

T . H . G u l l i k s e n , On t h e l e n g t h o f f a i t h f u l m o d u l e s o v e r A r t i n i a n l o c a l r i n g s , Vath.Scand.

117

31 (1972), 78-82.

B i b 1i o e r anhy

118

. J . H e r z o g a n d C.Kunz,

&

k a n o n i s c h e '_ l o d_ u l -ei n e s Cohen

Placaulay R i n q s , L e c t u r e s N o t e s i n V a t h e m a t i c s S p r i n g e r - V e r l a c , B e r l i n , 1971.

-

238,

' l . i l o c l i s t e r , I)een l o c a l r i n v s , P r e n r i n t .

-

J . I , . R o b e r t s , A c t i o n s o f r e d u c t i v e q r o u p s on r e

- and

-

g u l a r r i n p s and Cohen - ' l a c a u l a y r i n o s , B u l l . h e r . ? l a t h . SOC. J 8

(1974), 181-284. On t h e v a n i s h i n c l o f l i m ' " ) ,

C.IJ..Jcnscn,

.J.AlFchra 15

(197O), 1 5 1 - 1 6 6 . I . K a n l a n s k v , Commutative A l g e b r a , A l l v n and B a c o n , B o s t o n , 1371. -

-

, R - s e o u e n c c s and h o m o l o g i c a l d i m e n s i o n , Yaqoya

ilat11.J.

g

( 1 9 h L ) , 135-200.

f l . Krgmer, L i n c g c Anwendunpen d c r G - F u n c t i o n von VacRae,

Arch.Yath. 2 2 ( 1 9 7 2 ) , 47Q-490. D.l,azar, Autour de l a n l a t i t u d e , Rull.Soc.Pfath.

France

37 ( 1 9 6 9 ) , 8 1 - 1 2 8 .

C,.Levin a n d W.V.\.asconcelos,

H o m o l o v i c a l d i m e n s i o n s and

25 ( 1 9 6 8 ) , 3 1 5 - 3 2 3 . 'lacaulay r i n E s , P a c i f i c 7.rlnth. .J.I,inman,

On t h e . J n c o b i a n i d e a l o f t h e module o f d i f f e -

r c n t i a l s , Proc.Amer.hlath.Soc. 2 1 (1960), 422-426. R.\facKae, On an a n n l i c a t i o n o f t h e F i t t i n e i n v a r i a n t s ,

2

,J . A l ~ e b r a

(1965)

,

153-163.

E . M a t l i s , I n j e c t i v e modules o v e r N o e t h e r i a n r i n e s , 8 (1')58), 511-528. P a c i f i c J.b!ath. I I . \ f a t s u m u r a , Commutative A l g e b r a , Renjamin,New Y o r k , l 0 7 0 . :r.P.%irthv,

A n o t e on f a c t o r i a l r i q e s , A r c h . " a t h .

15

(1964) , 4 1 S - 4 2 O . C . P e s k i n e and I , . S z n i r o ,

1)imension n r o j e c t i v e f i n i e e t

c o h o m o l o e i e l o c a l e , Puh1.Flath.IIlES 4 2 , P a r i s , 1973.

D.Rees, The g r a d e o f a n i d e a l o r m o d u l e , P r o c . C a m b . P h i 1 . SOC. 53 ( 1 9 5 7 ) , 2 8 - 4 2 .

R i h 1i o p r a p h y

D.E.Rush,

1 I9

The G - f u n c t i o n o f PlacRae, P r e n r i n t .

P.Samue1, On u n i q u e f a c t o r i z a t i o n d o m a i n s , T a t a I n s t i t u t e o f F u n d a m e n t a l I k s e a r c h , Bomhav, 1 9 6 4 . [34 1

.J. - P . S e r r e , A l g t b r e L o c a l e - ' l u l t i p l i c i t g s , L e c t u r e s

Notes i n 'lathematics

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11,S n r i n e e r - V e r l a e ,

B e r l i n , 1965.

- , Sur lcs modules p r o j c c t i f s , Seminaire I h b r c i l ,

P a r i s , 1960-61. R . Y . S h a r p , F i n i t e l y g e n e r a t e d modulcs o f f i n i t e i n j e c t i v e d i m e n s i o n o v e r c e r t a i n Cohen - % l a c a u l a y r i n r s , P r o c . London f l a t h . S o c . 25 ( 1 9 7 2 ) , 3 0 3 - 3 3 8 .

-

- , The C o u s i n c o m p l e x a n d i n t e E r a l e x t e n s i o n s o f

Noetherian domains, Proc.Camb.Phil.Soc.

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407 - 4 1 5 . - , Gorenstein modules, : l a t h , Z e i t s c h r . 115 (197n), 117-139. ~

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34

(1973), 39-43. W.V.Vasconcelos, 105-107.

(1970),

-

-

3

Simnle f l a t e x t e n s i o n s , .J.Alpehra 16

-

The c o m m u t a t i v e r i n p s o f q l o h a l d i m e n s i o n

Seminar n o t e s , RutEers I l n i v e r s i t y , 1973.

e,

Index

Associated prime, 3 C a n o n i c a l module, 48 Coherent r i n g , 2 Depth, 1 9 D i v i s o r , 63 D i v i s o r i a l i d e a l , 56 F i n i t i s t i c d i m e n s i o n , 88 F l a t module, 9 G o r e n s t e i n r i n g , 44 Grothendieck group, 6 3 H i g h e r d i v i s o r i a l i d e a l , 80 I n j e c t i v e dimension, 7 Invariant factors, 14 Koszul complex, 1 6 K r u l l dimension, 3 Macaulay r i n g , 28 Macaulay e x t e n s i o n , 86 M a t r i c i a l complex, 37 No-name i n v a r i a n t , 1 4 P r o j e c t i v e dimension, 6 Regular r i n g , 8 S p h e r i c a l module, 9 3 S u p p o r t o f a module, 2 7 Type o f a r i n g , 30 V a r i e t y o f a module, 2 7 Zero d i v i s o r , 3

120