The rings of dimension two 0824764471, 9780824764470

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the rings of dimension two

W o lm e r V V a sco n ce lo s

THE RINGS OF DIMENSION TWO

PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes Executive Editors

—

Monographs, Textbooks, and Lecture Notes

Earl J. Taft R utgers University New Brunswick, New Jersey Edwin Hewitt University of W ashington Seattle, W ashington

Chairman of the Editorial Board S. Kobayashi U niversity of California, Berkeley Berkeley, California

Editorial Board M asanao Aoki University of California, Los Angeles Glen E. Bredon Rutgers University Sigurdur Helgason M assachusetts In stitu te of Technology G. Leitm an U niversity of California, Berkeley W.S. M assey Yale University Irving Reiner University of Illinois at Urbana-Champaign Paul J. Sally, Jr. University of Chicago Jane Cronin Scanlon R utgers U niversity M artin Schechter Yeshiva U niversity Julius L. Shaneson R utgers U niversity

LECTURE NOTES IN PURE AND APPLIED MATHEMATICS 1. 2.

N. Jacobson, Exceptional Lie Algebras L.-Â. Lindahl and F. Poulsen, Thin Sets in Harmonic Analysis

3.

/. Satake, Classification Theory of Semi-Simple Algebraic Groups

4.

F. Hirzebruch, W. D. Newmann, and S. S. Koh, Differentiable Manifolds and Quadratic Forms

5.

I. Chavel, Riemannian Symmetric Spaces of Rank One

6.

R. B. Burckel, Characterization of C(X) Among Its Subalgebras

7.

B. R. McDonald, A. R. Magid, and К. C. Smith, Ring Theory: Proceedings of the Oklahoma Conference

8.

Yum-Tong Siu, Techniques of Extension of Analytic Objects

9.

S. R. Caradus, W. E. Pfaffenberger, and Bertram Yood, Calkin Algebras and Algebras of Operators on Banach Spaces

10.

Emilio O. Roxin, Pan-Tai Liu, and Robert L. Sternberg, Differential Games and Control Theory

11.

Morris Orzech and Charles Small, The Brauer Group of Commutative Rings

12.

S. Thomeier, Topology and Its Applications

13.

Jorge M. Lopez and Kenneth A. Ross, Sidon Sets

14.

W. W. Comfort and S. Negrepontis, Continuous Pseudometrics

15.

Kelly McKennon and Jack M. Robertson, Locally Convex Spaces

16.

M. Carmeli and S. Malin, Representations of the Rotation and Lorentz Groups: An Introduction

17.

George B. Seligman, Rational Methods in Lie Algebras

18.

Djairo Guedes de Figueiredo, Functional Analysis: Proceedings of the Brazilian Mathematical Society Symposium

19.

Lamberto Cesari, Rangachary Kannan, and Jerry D. Schuur, Nonlinear Functional Analysis of Differential Equations: Proceedings of the Michigan State University Conference

20.

Juan Jorge Schaffer, Geometry of Spheres in Normed Spaces

21.

Kentaro Yano and Masahiro Kon, Anti-Invariant Submanifolds

22.

Wolmer V Vasconcelos, The Rings of Dimension Two Other volumes in preparation

THE RINGS OF DIMENSION TWO Wolmer V. Vasconcelos Department o f Mathematics Rutgers University The State University of New Jersey New Brunswick. New Jersey

MARCEL DEKKER, INC.

NEW YORK AND BASEL

COPYRIGHT ©

1976 by MARCEL DEKKER, INC.

ALL RIGHTS RESERVED.

Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. MARCEL DEKKER, INC. 270 Madison Avenue, New York, New York LIBRARY OF CONGRESS CATALOG CARD NUMBER: ISBN:

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Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 PRINTED IN THE UNITED STATES OF AMERICA

10016 76-592

To Aurea

-

INTRODUCTION

A motivating element in these notes on homological dimensions of commuta­ tive rings is the following problem:

Let A be a ring, and let E be a fi­

nitely generated A-module with generators

f^,

(or syzygies) of the f^'s are the n-tuples

f^.

The relations

(gj,...,g ) e An

such that

g.f. + ••• + g f = 0 61 1 bn n They are made into an A-module К which, if finitely generated, allows one to "find" the relations among a finite set of generators.

If this proce­

dure can be carried out a certain number of times, we get a resolution:

The problem is whether the integer r is sufficiently large to ensure that from the rth stage the resolution can be indefinitely continued using fi­ nitely generated modules.

In the language of Ref. 11, Chap. 1, when do

modules with a finite r-presentation have a finite infinite presentation? To formalize this question, we define the A-dimension of A [A-dim(A) for short] as the infimum of such integers— if any exists.

Thus

Noetherian

rings are those with A-dimension 0, while the rings of A-dimension at most 1, named coherent, are those for which finitely generated ideals are fi­ nitely presented.

Note that, according to Weyl [65], before succeeding in

proving his 'basis theorem' Hilbert in fact proved the coherence of the polynomial ring in several indeterminates over a field. The question of determining the A-dimension of a ring becomes simpler if we were to know that projective resolutions stop after a certain number of steps;

such rings will be called "regular," and its (grandiosely named)

global dimension [gl dim(A) for short] is the least number of steps that works for all modules.

Although the A-dimensions of a ring A and a poly­

nomial ring В = A[x^,...,xn ] may differ, a measure of stability is expressed in the inequality

A-dim(B) _< 1 + gl dim(A).

v

One of the main results in

INTRODUCTION

vi

these notes is the proof that if A is coherent and of global dimension two, then В above is also coherent.

A feature of the proof is, once sufficiently

many structural details of A are revealed, to throw in all indeterminates simultaneously— in marked contrast to the Hilbert's basis theorem--to keep better track of the coefficients. The bulk of the notes is dedicated to a leisurely discussion and assemblage of the pieces of homological dimension theory needed for a description of the rings of global dimension two. Very early it became clear that the classical examples of such rings, Noetherian rings and Prüfer rings of appropriate size, were being used with singular frequency.

Thus, the running theme became the examination of the

extent with which such rings make up all rings of global dimension two. most pleasing aspect of this program concerned the local rings;

The

they were

shown to be either Noetherian rings, valuation domains, or a hybrid obtained by attaching one of the former class on the top of a valuation ring.

Guided

by this view, the next step was the examination of the domains of global dimension two;

here the results were far from complete, perhaps because of

the diverse ways such rings arise, as illustrated by the examples scattered throughout.

Noteworthy is the result that a domain which is not a Prüfer

ring must contain a finitely generated maximal ideal.

This was, however,

as far as the analysis proceeded except for the examination of its more immediate consequences.

Finally, for arbitrary rings

not necessarily

domains) it is clear that at least for "half" of them, the coherent rings, the difficulties are just the same as for domains.

In this case a promis­

ing sheaf representation over a compact space exists with fibers which are domains of dimension at most two.

The noncoherent rings were just identi­

fied with the inexcusable purpose of steering away from them. Every chapter contains a description of its contents. summary is as follows:-

An overall

Chapter 0 is a brief survey of homological dimension

theory dealing mostly with the relationship among the following dimensions associated to a ring:

Krull, global, weak, and finitistic.

Chapter 1

gives descriptions of rings in which the last three dimensions assume the value one;

this is needed later since such rings appear as rings of resi­

dues or fractions of rings of dimension two. results on local rings.

Chapters 2 and 4 contain the

Chapters 3 and 5 are of less specialized interest

since the first is concerned with descent of projectivity, and the other with change of rings and dimensions.

A consequence of Chapter 3 is that

it allows, in a rather painless way, construction of local rings of finite

vu

INTRODUCTION global dimension but with many zero divisors.

Chapter 6 exploits the local

theorem and change of rings to treat the nature of the ring modulo a finitely generated ideal.

Chapter 7 deals mostly with domains in terms of

infinitely many local and lower

dimensional residual conditions.

Chapter

8 uses the accumulated information to show the coherence of the polynomial ring over a coherent ring of global dimension two.

It concludes with a

discussion of the ^.-dimension of rings. The author is grateful to the lively audience of a seminar held at Rutgers University in the spring of 1973, where the subject was discussed, and especially to Brian Greenberg and Hu Sheng, both of whom spoke of their own work (Chapters 4 and 6, respectively).

Greenberg has also played a

role in the current form of Chapter 8. We are equally indebted to Laurent Gruson and Christian Jensen for a spirited correspondence that we, at times (rather shamelessly!), transcribe freely into the text. Finally, we thank Irving Kaplansky who, so many years ago, made this subject attractive to us and continued to show interest in its development. A work about the bibliography:

Although plentiful, it is certainly

not exhaustive and no exaggerated care was taken in providing direct attrib­ ution of all the results contained here. Wolmer V. Vasconcelos

CONTENTS

Introduction

v

Chapter 0

Preliminaries

1

Global dimensions Noetherian rings Dimension zero

1 6 7

Dimension One

9

1. 2. 3. Chapter 1 1. 2. 3. 4. 5. Chapter 2 1. Chapter 3 1. 2. Chapter 4 1. 2. 3. 4. Chapter 5 1. 2. 3. 4. 5. 6. Chapter 6 1. 2. 3. 4. 5.

Tor-dimension one Global dimension one Finitistic dimension one The minimal spectrum Generating ideals efficiently

9 10 13 14 16

Local Rings (I)

19

Classification

19

Conductors, Cartesian Squares, and Projectivity

25

Conductors and the descent of projectivity Cartesian squares (I)

25 28

Local Rings (II)

31

F-rings Cardinality conditions Fiber products Global dimension of F-rings

31 32 35 36

Change of Rings and Dimensions

41

Macaulay extensions Finitely generated modules Some equalities of dimensions Divisibility Local rings of Tor-dimension two Difficulties in higher dimensions

42 46 46 48 54 56

Finitely Generated Ideals

59

Coherence Burch's theorem revisited The Д -theorem Generating prime ideals efficiently Noetherianization

ix

59 61 •63 66 67

CONTENTS

X Chapter 7 1. 2. 3.

Domains

71

Xß-Noetherian domains Symmetric algebras Sheaf representation

71 75 82

Chapter 8

Coherence of Polynomial Rings

85

1. 2. 3. 4.

A remark of Gruson's Rings of global dimension two The X-dimension of a ring Stability

85 86 89 93

References Index

97 101

THE RINGS OF DIMENSION TWO

Chapter О PRELIMINARIES

All rings to be considered are commutative— and this will be emphasized occasionally--and have an identity;

modules are imitai.

References 11 and

36 should be used for terminology and basic notions of commutative algebra. In this chapter we start assemblying various results in the theory of homological dimensions to be repeatedly referred to.

To set the stage, we

begin with the definitions of our main objects of interest. 1 . Global dimensions 0.1

Definition.

Let E be a module over the ring A;

the projective

dimension of E (рОдЕ for short or simply pd E when no danger of confusion exists) is finite and equal to n if there is an exact sequence 0 -- ► P --- -------- ► P, ---- Pn ---- E ---- 0 n 1 0 with P^ A-projective and n least.

Otherwise the projective dimension of E

is said to be infinite. 0.2

definition.

Global dimension of A = gl dim(A) = sup{pd E}, over

all A-modules. The following result conveniently reduces the task of determining the global dimension of a ring A [2]. 0,3

Lemma

[Global dimension lemma),

gl dim(A) = sup pd{A/l}, where

I runs over the ideals of A. Proof: we must show

Assume sup pd A/I

= n < 00, and let E be an A-module for which

pd E < n + 1, that is

Ex^

(E,F) = 0, for any A-module F.

Consider an injective presentation of F, 0 —

F—

I0 —

I, —

------ - V i —

where the I^'s are injective modules.

Since

C —

°

Ext^(E,C) = Ext^+*(E,F), it

suffices to show that C is injective. This will be done, by Baer's lemma [14], if Ext^(A/I,C) = 0 for any ideal I. But this last module is iso-

1

PRELIMINARIES

2 morphic to 0.4

ExtA

(A/I,F) = 0.

Remark.

If A is a Noetherian ring, it is enough to consider the

ideals generated by three elements [12,27,38], or more efficiently maximal ideals, which reduces the task in the case of a local ring to the examina­ tion of a single ideal [3,56]. 0.5

Definition.

Let E be an A-module;

E is said to be А -flat (or

simply flat) if for an arbitrary exact sequence of A-modules,

CM)

M. ,--- - M . --- «-М. . 1+1

1

1-1

the sequence (E ® A

M)

E ® д M. .--- - E ®. M.--- * E ®. M. , A l+l A l A l-l

is also exact. Since the functor

E ® A (-)

already preserves right exactness, the

flatness of E amounts to the fact that If

a_ = (a^, ..., an >

E ® A (-)

preserves injections.

is a sequence of elements of A, the module of

relations on the a.'s is the kernel of l ф: АП ----А, Rg(a) = ker(lE ® ф)

Ф(х., ..., x ) = Ex.a.

is then

{(e^

..., e ) e En

with

E a ^

= 0};

it

will be called the -------------------module of relations on the a.'s with coefficients in E. l Кд(а)Е c J?E (a).

Notice that 0.6

Proposition.

(1)

E is flat

(2)

For every sequence a as above, Кд(а)Е = Rg(a_).

Proof: a_.

The following are equivalent for an A-module E.

(1) => (2): Let (a) be the ideal generated by the elements in

Consider the exact sequence F J L . An —

where F is a free module.

A -- - А/ (a)--- - 0

Tensoring with E, we obtain the exact sequence

E ® F —® t E ® АП 1- ^ i > E ® A ----- - E ® A/ ( a ) ------- 0 (where the A's were dropped harmlessly in ® д ) . we have (2).

Since

im(l ® ф) = RA (a)E,

Actually, in the language of derived functors, the argument

shows that Tor^(A/(a),E) = RE (a)/RA (a)E (2) => (1): Consider the diagram

GLOBAL DIMENSIONS

3 P

l* f

-4-

G

where ф is injective, P is a free module, and ф is surjective. ker ф and К = ф 1 (ф(Р)), we obtain the diagram

With

L =

E 0 L ---* E 0 P -- ► E 0 G ---* 0

I

I

t

E 0 L --- - E 0 К --- -E ® F -- * 0 with obvious maps.

From this it follows that the vertical map on the right

will be injective if the two maps ending in then assume that G is a free module. sumed to be of finite rank.

E 0 P

are injective.

We may-

It is also clear that G may be as­

An easy induction on rank takes care of the

remainder of the proof, the starting case being supplied by the hypothesis. Let E

be a flat A-module, and consider a presentation

where F is free with basis {f }. a If к = la.f is an element of K, tensoring this sequence by А / (a), 1 £ = {aj, ..., a^}, yields by (0.6) an exact sequence and к e (a)F. Thus к = £a^g^,

e K.

If we define

9: F — *-К

by 0(fa

C* , + К n-1 n

Let u denote the endomorphism obtained when the above construction is n applied to the module C . The following relations are easily checked:

5

GLOBAL DIMENSIONS (1)

фип = ф

(2)

Let

u (с) = 0 for с е С n п (3) u u = u for n > m n m п F be the direct sum of a countable set of copies of P.

We claim that

the sequence 0 ---- F -X— F A * E ---►O is exact, with ф(х1 ,х2,...) = фСЕх.^ and

X(x 1, x2, . . . ) = (xr

x 2 - u1(x1) , . . . , xn - un_1(xn l ) , . . . )

It is clear that ф is surjective and x is injective. ker ф.

For the converse, let

is, Ex. plies

belongs to un (Ex^) = 0.

(x^,x2 ,...)

ker ф = К, x. e C By (3)

From (1), im x

be an element of ker ф, that

for a suitable C .

we may assume

x^ = 0

for

Then (2) im-

i > n.

By repeat­

ed use of (3), we check that XfXj, u 1 (x1) + x 2 , u 2 (xx + x2) + x3 , ...) = Cx1,x2,...) to conclude the proof. The following definition is another

side of the global dimension of

a ring. 0.12

Definition♦

The finitistic projective dimension of A is defined

as in (0.2) by taking only the modules with finite projective dimension. We abbreviate it as

FPD(A).

This dimension together with the Tor-dimension of A are indeed the two faces of the global dimension [32]. 0.13

Theorem.

Let A be

a ring for which FPD(A) is finite.

Then any

flat module has finite projective dimension. In particular, the finiteness of

gl dim(A)

simultaneous finiteness of FPD(A) and Tor-dim(A).

is a consequence of

In general, these three

numbers are distinct, with the only relationship being contained in formula

the

the

gl dim(A) = sup{Tor-dim(A), FPD(A)}.

With

d(A) denoting any of the three dimensions above [35], a way to

produce rings with finite dimensions is provided by the following classi­ cal result.

PRELIMINARIES

6 0.14

Theorem

(Hilbert syzygies' theorem).

ring of polynomials in one indeterminate over A. Proof:

Let

A[x] denote

the

Then d(A[x]) = d(A) + 1.

(This proof uses a device introduced by Hochschild, together

with some changes of rings and homological dimensions results [35].) Let E be an A[x]-module;

construct the sequence

0 -- ► A[x] ® A E - A - A[x] ® A E —

E ---- 0

where ф(хп ® e) = xn

xe - x

n+1 л* ® e

cf>(f(x) ® e) = f(x)e It is easy to see that this sequence of A[x]-modules and A[x]-homomorphisms is exact.

The proof proceeds now as follows:

be viewed as an A[x]-module via of Ref. 35 then says that

f(x) = f(0)e.

If E is an A-module, it can The change of rings result

p d ^ ^ E = pdAE + 1.

If now E is an A[x]-module, the sequence above says that рйдЕ >_ PdA |-xjA[x] ® E. Thus

pdA ^ j E

The first inequality follows because A[x] is A-free.

PdA [-x ]E 1 PdAE + 1. By taking sup over either all modules or only those of finite projec­

tive dimension, we prove the syzygies' theorem for gl dim(A) or FPD(A). The case of Tor-dim(A) is similar. 2.

Noetherian rings We recall the definition of a regular (Noetherian) local ring.

Let

A be a local ring of maximal ideal m. the dimension of the

If the Krull dimension of A equals 2 A/m-vector space m/m , A is said to be regular. This

remarkable class of rings is characterized homologically in the following theorem [3,56]. 0.15

Theorem

therian ring.

(Auslander-Buchsbaum-Serre theorem).

Let A be a Noe­

Then gl dim(A) is finite if and only if A has finite Krull

dimension and the localization

A is regular for each maximal ideal m. H —

For a commutative Noetherian ring the global and the weak dimensions coincide according to (0.3) and (0.10).

Of much more recent vintage is the

determination of the finitistic dimension of a Noetherian ring A [26]. 0.16

Theorem.

FPD(A) = Krull dimension of A.

The surprising fact is that this equality had not been proved even for low-dimensional cases other than for Artinian rings, for which it is tri­ vial, or for regular rings as in (0.15).

1

DIMENSION ZERO 3.

Dimension zero It is not particularly difficulty to "identify" the rings for which

the dimensions defined above are zero. 0.17 ----

gl dim(A) = 0

Thus we have the following.

A = K. x ••• x к

means

1

n

К. = field 1

Indeed, gl dim(A) = 0 simply means that all modules are projective, that is, A is a semisimple (commutative) ring. 0.18

Tor-dim(A) = 0 means that A is a von Neumann ring, that is,

every ideal

Aa

is generated by one idempotent.

To see this, let I be a finitely generated ideal and consider the sequence 0 — -I

—

A — ►A/I

— ►0

Since A/I is flat, by (0.7) there is a homomorphism generating set for I.

u: A — »-I

fixing a

Thus the sequence splits and I is generated by one

idempotent. 0.19

[6].

FPD(A) = 0 means that

a ring with a unique prime ideal a sequence

a^, a^, ...

A = B^ x ... x B^, where each B^ is

which is T-nilpotent (f.e., if we pick

of elements in P, then for some m, aj • a^

a^

is zero). Indeed assume FPD(A) = 0 and let a^, a^, ... be a sequence of elements in A. Construct the A-homomorphism ф: A[x] -«-A[x] where ф(хП ) = xn - an+^xn+^.

ф is clearly injective and

mension at most one.

E = coker ф has projective di­

The assume E to be projective;

note that

E = J-E,

where J is the ideal generated by the a^'s. Write

E Ф G - A[x], and let ф be the projective of A[x] onto E.

If

e e E, write e = b„ + b,x + ••• + b xn 0 1 n and denote gets

c(e) = (bQ , ..., bn ) .

e = ф(е), and since

ama' s lemma we can find

Applying ф to the equation above, one

E = J-E, we conclude j £ J

with

c(e) = J-c(e).

We apply these remarks to the class of 1 in E. we conclude

(1 - j)e = 0 also. ,

1-

.

j

^ ,,

1 - j .

Since

(1 - j)c(e) = 0

is then in the image of ф, n

n+l.

* V 1 - aix) + • • • + V х - V i x }

an easy calculation now shows

By Nakay­

(1 - j)c(e) = 0.

PRELIMINARIES

8

(1 - j)ax • a2 ••• an+1 = 0 We are now in a position to prove (0.19).

Let us begin by showing

that A is zero-dimensional, that is, every prime ideal is maximal. pick

a e m

\ £, £ c w

above we get for some

distinct prime ideals.

With

a =

If not,

= a^ = - • •

r e A, (1 - ra)an+^ = 0

and

1 - га e g_C m_, a contradiction. Next we show that A is a semilocal ring.

1 = A/(nil radical)

quence of orthogonal idempotents cheaply; that

Assume not.

Since the ring

is a von Neumann ring, we can find an infinite se­ e*, e*, ... in B.

there is no need to assume

e^, e^, __

I = (e^, e^, ... ) is a flat ideal of A.

Lift them to A (but

are orthogonal).

Notice

By (0.11), A/I has projec­

tive dimension at most one, and I will be generated by a single element. Thus we can write unique prime of B^.

A = B. x — X B, B . = local. Let P. be the 1 n 1 1 FPD(B^) = 0 also, the argument above shows that

Since

P^ is T-nilpotent. The converse is immediate. Remark.

The last result is already an indication on how much simpler

the determination of these dimensions for commutative rings is compared with the noncommutative cases.

For another instance, in Chapter 1 we shall

see how painless it is to give a reasonable description of the commutative rings of global dimension one, whereas for noncommutative rings the similar situation is not totally satisfactory.

In higher dimensions the noncommu­

tative case is chaotic, to put it mildly.

Chapter 1 DIMENSION ONE

Here we discuss briefly what it means for a commutative ring to have dimension, global dimension or finitistic projective dimension one.

TorThe

need for such a sketch will become apparent later as those properties will show up in rings of fractions or residues of rings of dimension two.

This

chapter concludes with a discussion on generating efficiently finitely ge­ nerated ideals in rings of dimension one. 1 . Tor-dimension one 1.1

Lemma.

Proof: {A,m}.

A finitely generated flat module over a local ring is free.

Let E be a finitely generated flat module over the local ring

Among all the minimal generating sets for E pick one

e^, —

, e^

with a relation r. f 0, s < n l —

+ r e =0 s s

r lel + of shortest length.

From the flatness of E we may write e. = Ea..f. i il 1

E r .a.. = 0 i l]

With all a . . e m we would have il E = m.E + (e — 1 s+1 and thus s = 0, by .Nakayama's lemma. а

1,1

(£ m .

e ) n

Assume then

s > 0

and let, say,

But in this case we have a relation

r2 es+l’ 1.2 Proposition.

en }*

A has Tor-dimension £ 1 if and only if

Ap

is a

valuation domain for each prime ideal P. Proof:

Lemma 1.1 says that every finitely generated ideal of a local

ring V of Tor-dim £ 1 is free and thus V is a valuation domain.

9

As the

DIMENSION ONE

10

flatness is decided at the localizations, the statement follows. 1.3

Examples.

(a)

Let A be the ring of all algebraic integers; then

since every finitely generated ideal of A is principal dimension one.

[36], A has Tor-

As A is a countable ring, it follows from Jensen's lemma

(0.11) that every ideal has projective dimension at most one and thus glo­ bal dim(A) _< 2. (b)

(It is clearly not one, as will be seen shortly.)

Let E be a countable direct sum of copies of

tion and multiplication defined component-wise. define a multiplication according to the rule

Let

Z/2Z A = Z®E,

with addi­ and

(m,e) • (n,f) = (me, mf +

ne + ef).

It follows easily that for each prime ideal P of A, Ap is either

7LI2TL

ZZ^-py

or

Thus A is still another ring of Tor-dimension one and,

as we shall see, of global dimension two. 2.

Global dimension one Our approach to a description of such rings, г.е. of hereditary rings,

will be through the examination of the conditions that make a projective ideal finitely generated. First we recall the notion of trace of a projective module E over the commutative ring A. over

НоШд(Е,А).

It is simply the ideal

J = J(E) = Ef(E), where f runs

Equivalently, J is the ideal generated by the coordinates

of all elements of E whenever a decomposition

E

G = F (free) is given.

From this second interpretation it follows that if momorphism, then

h: A —

В is a ring ho­

J (E ® B) = 7z(J(E))B.

At this point we insert, for quick reference, the following result which will be indispensable throughout [34]. 1.4

Theorem.

A projective module E over a local ring is free.

1.5

Proposition.

Let J be the trace of a projective module.

finitely generated, then Proof :

J = Ae,

If J is

e an idempotent.

Let J be the trace of E and let P be a prime ideal of A.

If

Ep = 0, Jp = 0 by the remarks above, whereas if Ep ^ 0, (1.4) says that Ep is free and thus Jp = Ap . tions).

In particular,

J =

(checked at the localiza­

If J is finitely generated, the so-called determinantal trick then

yields the desired conclusion. Note that in all cases referred to as pure ideals.

A/J

is a flat A-module. Such ideals will be

11

GLOBAL DIMENSION ONE 1.6 A.

Proposition.

Let I be a projective ideal of the commutative ring

If I is not contained in any minimal prime ideal, it is finitely gene­

rated. Proof:

Let J be the trace ideal of I.

If J = A, it would mean that

we have an equation n 1 = Ef.(x.) j l l

f. e Horn.(I,A), x. e I l A k ’ iy l

Thus for x e I we have n n x = Zxf.(x.) = Zf.(x)x. 1 l i j l l and the x^'s generated I. I C J;

We show this to be the case for J.

if J is not A, let M be a maximal ideal containing it.

Notice that By hypothe­

sis M is not a minimal prime ideal and will thus properly contain one such, say, P. above, M/P.

Consider the canonical homomorphism I

A/P = I/PI

is a projective

A — A/P;

by the remarks

A/P-module of trace

(J + P)/P

But an integral domain (viz. A/P) cannot admit a pure ideal [viz.

(J + P)/P] other than the trivial ones, and this is ruled out here.

This

contradiction proves the statement. The first use to be made of (1.6) is under the most favorable condi­ tions: study of the reduced rings (i.e., rings without nilpotent elements) in which every ideal not contained in any minimal prime ideal is projective. (Notice that if A has global dimension one, every localization of it is a domain;

thus it is reduced).

For brevity we call such rings almost here­

ditary. 1.7

Theorem.

Let A be almost hereditary.

(1)

A is semihereditary

Then:

every finitely generated ideal is

projective) (2)

A/P is a Dedekind domain for each prime ideal P

(3)

A/I is Artinian for every ideal I not contained in any minimal

prime, and I is a product of maximal ideals. Proof: (a) domain. are

The statements will follow from a list of observations on A.

For every prime ideal M, the localization A^ is an integral Since A is reduced, it is enough to show that the minimal primes

two by two comaximal.

suppose

Consider

P + Q С M, a maximal ideal.

P, Q

distinct minimal primes, and

By hypothesis

ideal; in A^^ both P and Q survive

and

ment

x = z(x + y)

x + y, x e P , у e QM .

unit, y e Pw

'

M

and

Now

PM + QM

P + Q

is a projective

is generated by one ele­ for some z e A ^

if z is a

Q C P ; if z is not a unit, (1 - z)x = zy gives M M

DIMENSION ONE

12

x e Qw and Pw C Qw . Either way we get a contradiction. М М М (b) The annihilator of each element of A is generated by one idempotent. M,

Let x e A

and write

1^ = 0 or A^

a pure ideal,

I = annihilator of x.

and in particular

I П

(x) = lx = 0

and

J is contained in a minimal prime P,

2

1 = 1 .

Let

J = (x) © I .

By (a), for each prime J = (x,I);

since I is

On the other hand, if

Ap is a field and the image of x in

Ap is trivial; consequently, Ip = Ap, which is impossible.

Thus J is a

projective ideal, and according to (1.6) it is finitely generated. Then I 2 is also finitely generated and since 1 = 1 , I = Ae for some idempotent element e. Now we prove (1):

Let I = (x^, ..., x ) be a finitely generated ideal

which must be proved projective.

If

Ae^ = annihilator of x^, e^ = idempo­

tent, then Let

Ае,П П Ae = Ae (e = e,•••e ) is the annihilator of I. 1 n v 1 n' J = (e,I); as before we conclude that J = Ae © I and that J is not

contained in a minimal prime.

Hence it is projective, and I is also pro­

jective. As for (2), (a) says that each minimal prime P is a pure ideal and thus

A/P

is a flat epimorphic image of A.

It obviously inherits the de­

fining property of A and being a domain, it is a Dedekind domain. Finally for (3), if I is as stated, A/I is a Noetherian ring by (1.6), and since A has Krull dimension one by (2), there are only finitely many maximal ideals containing I;

since each localization

A^ is a discrete

valuation ring, it follows that I is a product of maximal ideals. In gneral, the remaining properties of A are hidden in the nature of some of its associated rings, e.g.3 total ring of quotients, Boolean ring of idempotents.

Finally, we discuss a phenomenon— studied in Ref.43 by

valuation theory and in Ref.8 by sheaf theory--precisely, that it does not take much to make an almost hereditary ring hereditary. 1.8

Theorem.

(1)

A is hereditary

(2a)

A is almost hereditary

(2b)

The total ring of quotients К of A is hereditary.

Proof : of A,

The following are equivalent for a commutative ring A.

That (1) implies (2) is clear.

IK can be written

IK = © L K e ^

As in Ref.43, if I is an ideal

ei = idempotent.

at each localization A^ (M = prime ideal), the

Since A is a domain

e^'s lie in A and thus

I П Ke. = I П Ae. = Ie.. On the other hand, Ie. © A(1 - e.) is an ideal 1 i l i l of A clearly contained in no minimal prime ideal and is hence projective.

13

FINITISTIC DIMENSION ONE Since

I = Ф Е 1 е д we are through.

Remark.

The closeness of hereditary rings to a finite direct product

of Dedekind domains (i.e.} hereditary domains) reposes then on the behavior of its minimal primes.

Thus, it will be Noetherian if such ideals are fi­

nite in number (since they are comaximal, the Chinese Remainder theorem takes over) or finitely generated [by Cohen's theorem and (1.6)].

There

should exist, however, more natural conditions with the same consequence especially in the case ideal).

For although

Minimax(A) = 0 (г.е.} no minimal prime is a maximal "hereditary" plus "Minimax(A) = 0"

does not neces­

sarily imply that A is Noetherian (see examples in Ref.8), it comes very close since the nature of the minimal primes is largely determined by the maximal ideals in this case. 3.

Finitistlc dimension one Although the rings with

FPD(A) = 1

general theory is rather sparse.

occur in various contexts, their

After treating the decomposition of pro­

jective ideals into a finite product of projective primary ideals, we con­ sider a condition (coherence) which often precipitates its Noetherianess. 1.9

Theorem.

dimension one.

Let A be a commutative ring of finitistic projective

Then every projective ideal not contained in any minimal

prime ideal is a product of projective primary ideals, in which the factors are unique up to order. Proof:

Let I be such ideal;

faithful ideal.

by (1.6) it is a finitely generated

Thus according to the change of rings theorem in dimension

one [35] for any A/I-module, E of finite projective dimension one has pdAE = 1 + р й д д Е

and thus

FPD(A/I) = 0.

Since A/I is commutative and

perfect, we may write

A/I =

x ••• x B^, where

is a local ring of

Krull dimension zero.

This decomposition yields in A a representation

I -Q r-Л where Q. is the primary ideal corresponding to the ith component of A/I. Note that each Q. is finitely generated, by I plus 1 - e^, where e.^ is some element of A mapping into the identity of B^.

As for the projectivity of

Q^, since it is a faithful ideal, it is enough [60] to show that it is principal at each localization A^, which is clear.

The uniqueness of the

Q.'s is also immediate. 4i We recall the notion of a coherent ring.

Let E be a finitely gene­

DIMENSION ONE

14 rated module over a ring A.

E is said to be finitely presented if there is

an exact sequence

Such modules have extremely good properties with respect to localization and tensorization with infinite products.

The rings for which every fi­

nitely generated ideal is finitely presented are called coherent. 1.10

Proposition.

A ring is coherent if and only if the direct pro­

duct of every family of flat modules is a flat module. Proof:

See Ref.15 for the proof and several other ways of expressing

coherence. 1.11

Proposition.

FPD(A) = 0.

Let A be a commutative ring which -is coherent and

Then A is Artinian.

Proof:

According to (0.19) we may assume that A is a local ring with

a T-nilpotent maximal ideal P. show P finitely generated.

To prove the statement, it is enough to

From the coherence condition it suffices to

show that P is the annihilator of a single element of A, which follows from the T-nilpotency of P. The coherent rings of finitistic projective dimension one are not so lightly disposed of.

In cases of domains however, we have the following

proposition. 1.12

Proposition.

Let A be a coherent domain with FPD(A) = 1.

Then

A is a Noetherian ring of Krull dimension one. Proof: ring

A/aA

Let P be a nonzero prime ideal of A;

pick

0 / a e P. The

has, by the change of rings theorem of Ref.35, finitistic pro­

jective dimension zero.

Since A/aA is also coherent, it follows from

(1.11) that A/aA is Artinian.

Thus P is finitely generated and A is

Noetherian by Cohen's theorem. 4.

The minimal spectrum The rings of Tor-dimension one have the property of being domains at

each localization, i . e they are rings in which principal ideals are flat (PF-rings for brevity).

They will be PP-rings (for "principal ideals are

projective" as in the case of rings of global dimension one) when some additional conditions are imposed on the topology of Min(A), the subspace of Spec(A) consisting of the minimal primes.

15

THE MINIMAL SPECTRUM 1.13 Proposition.

The following conditions are equivalent for a com­

mutative ring A: (1)

A is a PP-ring

(,2a)

A is a PF-ring

(2b)

Min(A) is compact.

Proof:

It will be enough to show that. (2) implies (1), the reverse

implication being clear.

Let f e A, and write

I = annihilator of f.

Since A is locally a domain, I is a pure ideal of A. the ideal

J = (f,I)

On the other hand,

is not contained in any minimal prime.

Thus we can

write Min (A) =

U D geJ g

= {P e Min(A), g t P}

with D g

By the compactness of Min(A) we can write Mi n (A3 as the union of finitely many D^'s, an, we have В

pdA E = pdAB + pdBE Proof: ule

C =

According to (2) it will be enough to show that for some mod­

e IA

=

a

(i\

Extg(E,Extd (B,C)) = Extg(E, T ® д C) = Ext®(E,TCl)) ф 0

This will be accomplished by an argument due to Gruson [25]. 5.2

Proposition.

the commutative ring A.

Let E be a finitely generated faithful module over Then every A-module has a finite composition se­

ries whose factors are quotients of direct sums of copies of E. Thus every such module is piecewise a generator for the category of A-modules. Proof:

It is enough to show this for A itself:

be a sequence of ideals of A such that for each 4k : E (ri} -- - Ii/Ii l — Let

m ф: A v

--- ► M

i

Let

there is a surjection

► 0

be a presentation for a module M.

submodules of M given by ф

(1^

J)

IQ c I c ... c A

Then the series of

has the desired properties.

For the proof in the ring case, we recall the definition of the

43

MACAULAY EXTENSIONS Fitting invariants of the module E [36]:

Let

A(I) — ï. A" —

E -- 0

be a free presentation of E and denote by of u.

Let

= F^[E)

of this matrix. (1) (2)

u . ^ . „ T the matrix xi 1 1 J 1 n, 1 e I be the ideal generated by the minors of order n - r

We note the following properties of these ideals [36]: F1 C

Foc

c F c

n n+1 If J is the annihilator of E, J c

The smallest integer r, such that smallest integer m, such that m = r

(

FQ 0.

Let

d = m - r.

I = Ann^CE/F^E),

hypothesis to the A/I-module

E/IE:

(a)

We can apply the induction

this module is faithful [xE c I E c

F^E => x e I], and (b) its rth Fitting ideal is trivial and its mth Fitting ideal is A/I. On the other hand, by property (2) above, ln c F . This gives rise to the filtration

F c F + in_1 c r

r

where each factor is an A/I-module.

0.

We consider the following linear

form on АП :

h ’h

(Xj,

Since the minors of order age of u;

det V

Ji

V

Ji

a. 11 ’^s

• a. 1s ’^ s V

Js

s + 1 are zero, this form is trivial on the im­

it defines then, by passage to the quotient, a linear form on E.

On the other hand, it transforms the tth vector of the canonical basis of

An into D .

CHANGE OF RINGS AND DIMENSIONS

44

We point out some immediate consequences. 5.3

Corollary.

Let

F

be a right exact functor which commutes with

arbitrary direct sums and is trivial on E;

then F is trivial.

In the situation of (5.1), consider the right exact functor on the category of B-modules,

F ( - ) = Extg(E, T ® B ( - ) ) and use the result above to complete the proof. 5.4

Remark.

For the injective and flat analogs of (5.1) one can pro­

ceed in a similar way to show the equality in the dimension formula using the following spectral sequences EV2 ’4 = ExtP(To/(B,C),E) = > q p

ExtJJ(C,E)

and E^’4 = TorB (E,TorA (B,C)) ■— => TorA (C,E) 2 p q n

p

respectively. Let us indicate how to proceed in the injective case (one uses the same type of argument, and module, in the flat case). tive A-module.

Because

way as before that В-module C;

Ext^(B,A) = 0 ,

TorA (B,M) = Нотд (Т,М).

by (5.3),

Let M be an injec­

i < d, it follows in much the same

Ext„(T,E) + 0 also.

Let

Extg(C,E) Ф 0

for some

Let

D

0 be an A-injective envelope of T. 6 Extg(HomA (T,M),E) = 0. Since T

► T — *- M Ext^+ e (M,E) = 0 implies that the module ^ is finitely generated and faithful as a

В -module, there is an exact sequence 0 -- * В *— ► ТП obtained by mapping for T.

e

1 into

(x^,...,*^)

But in this case, Ext^fHom (B,M),E) = 0 also and then the module

в

Extg(HomA (B,T),E) = 0.

A

Since there is a surjection Нотд (В,Т) —

given by ф Cf) = f(l), we contradict 5.5 Remark. (1)

where the x^®s form a generating

T — ► 0 Extg(T,E) / 0.

The theorem above applies in the following cases:

В = А / (x),

x a nonzero divisor, to give Ref. 35, Theorem 3.

MACAULAY EXTENSIONS (2)

В = A/I,

45 where I is a faithfully projective ideal.

According to

(1.6) such ideals are finitely generated and we obtain the result of Ref. 3L Theorem 0.16 puts very strict conditions on

В • A/I

change of rings to apply for this pair of rings.

in order for

In particular, if A is a

regular local ring, then change of rings applies to A/I if and only if it is a Macaulay ring, that is, if the condition (M) holds. The particular case we shall need [64] concerns ideals of projective dimension one in rings of global dimension two. 5.6

Definition.

We say that a finitely generated ideal I of a com­

mutative ring A is overdense if the canonical map

A — -Ноп1д(1,А)

is an

isomorphism. Such ideals are faithful and if the ring A is Noetherian, this is equivalent to asking that a regular sequence of length two be inside I [5]. These ideals tend to abound in rings of global dimension two which also have Tor-dimension two. 5.7

Theorem.

ideal with

Let A be a commutative ring, and let I be an overdense

Р^д1 = 1.

If E is an A/I-module with

p d д д E < °°, then

pdAH - 2 ♦ pdA/IE Proof:

Let 0 -- - К -- ► АП -- - I -- - 0

be a projective resolution of I. generated and (b)

We have to show that (a)

Ext*(A/I,A) = 0 ,

К is finitely

i = 0, 1.

That (a) holds will follow from the fact that it is a projective mo­ dule of constant rank

n - 1, I being faithful.

As for (b), consider the sequence 0 -- - I — ► A -- - A / I ---* 0 that yields 0

Нот (А/1 ,A) ---* Horn (A, A)

* Hom(I,A) — ► Ext|(A/I,A) -- - 0

As an application of the change of rings for Tor-dimension, consider the following example. 5.8

Example.

Let P be a finitely generated prime ideal of

where A is a Prüfer domain and x is an indeterminate over A. an invertible ideal, then P is maximal.

В = A[x],

If P is not

First notice that as В is coherent

46

CHANGE OF RINGS AND DIMENSIONS

[26] and Tor-dim(B) = 2 (from Theorem 0.14),

pdgP = 1.

check, or will follow from (6,5), that P is overdense. of rings, the finitistic flat dimension of В/P is 0.

It is easy to Thus by the change

However, since this

ring is a domain, P must be maximal. 2,

Finitely generated modules

If E itself, in the previous considerations, is finitely generated, there is no need to impose such a strong restriction on B. 5.9 Theorem.

Let A and В be rings, with В local and finitely gene­

rated as an A-module.

Assume that В admits a finite resolution by finitely

generated projective A-modules. free resolution over B.

Let E be a В-module admitting a finite

Then pdAE = pdAB + pdßE

This result [38], contrary to (5.1), has an elementary via of attack through the device of minimal projective resolutions. Proof:

Using our previous notation, notice that Ext®(E,ExtA (B,A)) = Extg(E,B) ® g ExtjJ(B,A)

since E has a resolution by finitely generated projective modules.

Since

both modules in the product are nonzero and the ring В is local, the state­ ment follows. 3,

Some equalities of dimensions

A consequence of

section 2 is that it provides estimates for the fi­

nitistic projective dimension of the ring B, to wit FPD(B) £ FPD(A) - pdAB. Here, by restricting to the case of the fPD (that is, where one takes in the definition of FPD only those modules which have a resolution by fi­ nitely generated projective modules), we get a case of equality [52]. the same time we shall initiate a discussion of the local rings of

At

Tor-

dimension two. In Chapter 2, of considerable help was the fact that local rings of global dimension two were coherent domains.

In Chapter 3 we saw an exam­

ple of a ring of Tor-dimension two but not coherent.

We shall see that

coherence in local rings of Tor-dimension two implies not only its inte­ grality but also the steps

1 to 4

of the proof of (2.2).

FINITELY GENERATED MODELES 5.10

Theorem.

47

Let A be a coherent local ring, and let I be a finitely

generated ideal with

P ^ I < 00.

Then

fPD(A) = pd (A/I) + fPD(A/I) When A is Noetherian, there is a simple way (via the notion of depth [5]) to verify the equality.

We need the following easy lemma.

For the

rest of this section we shall unspecified A-modules to be of finite presen­ tation. 5.11

Lemma.

Let

{A,M}

be a coherent local ring and let E be a fiд Then Р^дЕ = n if and only if Torn (E,A/M) / 0,

nitely presented A-module. and Tor^+ 1 (E,A/M) = 0. Proof:

Observe that there is a resolution

where each F. is a free module of finite rank and the matrix f. has all of l l its entries in M. The result follows in just the same way as in the Noe­ therian case, say, dimension shifting and Nakayama's lemma. Proof of (5.10): pdA (A/I) + fPD(A/I) Write

Since

A/I

is also a coherent ring,

fPD(A)

£

follows from (5.9).

d = pdA (A/I), let E be an A-module of finite projective dimen­

sion, and consider the minimal resolution 0 —

N —

Fd-i—

------ * Fo —

* E —

0 Д

where several of the terms above could be trivial. TorA (N,A/I), TorA (N,A/I) = 0 for i > 0.

Since

Tor^+(j(E,A/I) =

Consider the spectral sequence E2

= TorA /I(A/M,TorA (N,A/I)) = >

p,q Since

E2 = 0, P>4

p

q

TorA (A/M,N)

p

q > 0, TorA/I(A/M,N/IN) - TorA (A/M,N)

By (5.11) this isomorphism says that we conclude that

P ^ N = pdA^j(N/IN), and if Р d,

pdAE = d + PdA/j(N/IN)

to get the inequality

fPD(A) £

d + fPD(A/I). 5.12

Corollary.

Let

{A,M}

be a coherent local ring, and let I be

a finitely generated ideal such that of I is M.

Then

fPD(A) < 00.

(1)

pdA I < °°> and (2)

the radical

CHANGE OF RINGS AND DIMENSIONS

48

Proof: According to (5.10) (by passing to the ring A/I), we may as­ sume that A is a primary ring.

Suppose, say, that

is a minimal resolution of a module E. of order

rank(Fj) of f

in our situation unless

By McCoy's theorem [36] the minors

have a nontrivial annihilator, which is impossible F., = 0.

Observe that (5.12) is valid under the condition that every proper finitely generated ideal of A/I has a nontrivial annihilator, again by McCoy's theorem.

Those are the local rings in which the maximal ideal lies

in the assassinator of A [39].

We shall use this remark in one case that

involves a maximal ideal that is minimal over an ideal nonzero divisor. 5.13 Remark.

Then with

(x):b, with x a

I = (x), it will follow that

Let A be a coherent ring.

fPD(A) = 1.

The determination of the

Tor-dimension of A can be changed into the determination of that dimension of its localizations, Tor-dim(A) = sup{Tor-dim(Ap)} One can, in fact, discard some of these primes.

In

X = Spec(A), consider

Y = {P e X I Spec(Ap) N {P} is quasi-compact in Spec(Ap)}

Claim: ф: A — - В =

p§y^P

*s f^^hfully flat.

Granted this, we

obviously have Tor-dim(A) = sup{Tor-dim(Ap), P e Y)

Proof of plain: В is flat by the coherence of A.

Suppose

PB = В

for some prime P of A.

This means that there are x,,...,x in P so that 1 n (Xj,...,x^) is not contained in any prime of Y, clearly a contradiction. A consequence of this remark, later to be improved upon, follows:

Let

A be a coherent ring with gl dim(A) = Tor-dim(A) - 2 Then there is a maximal ideal

P e Y, because otherwise every Ap , P e Y

would be a valuation domain [by (2.2)].

4. Divisibility In order to study divisibility in a coherent local ring of finite Tordimension, we examine first the nature of the annihilator of a module with

DIVISIBILITY

49

a finite resolution by finitely generated free modules. Let E be such a module over the commutative ring A: 0 — ► F

n

--- -------- -

F, — 1

— ► E— ► 0

0

Define the Euler characteristic of E [62] as X(E) = EC-l^rankfF.) 5.14

Proposition.

The following statements hold:

(a)

x(E) 1 0.

(b)

X(E) = 0

if and only if

(c)

x CE) > 0

implies that Ann(Ann(E)) = 0.

Proof:

Ann(E) î 0.

If S is a multiplicative set of A, clearly X(E 0

forces

1=0.

Ap , a 0-ring. Since

As remarked earlier, Ep

0 / x e Ap, we get

Ep = 0.

is

Thus

Conversely, by localizing at a minimal prime

ideal we conclude the converse part of (b) Suppose now x CE) = 0, that is, 1 ^ 0 , of Ann(I).

In this case, let P be a prime ideal minimal over the annihi­

lator L of x; Ip t Ap 5.15

and let x be a nonzero element

notice that

I c L.

Again, Ep is a free module and since

we get X(Ep) > 0, a contradiction. Corollary. 0

* P n

Let E be an A-module with a resolution ► •• • --- - P, — ► Pn 1 0

by finitely generated projective modules.

Then

E — ► 0 Ann(Ann(E)) is generated

by one idempotent. Proof:

We may define a function

crete topology

by

r(P) = xCEp).

r: Spec(A)— ” 7Z,

1L with the dis­

Since x(-) is made up of a sum of con­

CHANGE OF RINGS AND DIMENSIONS

50

tinuous functions (the rank function of a finitely generated projective mo­ dule

is continuous [11]), the support of r is both open and closed. On the other hand, the relations above between Ann(E) and the Oth

Fitting invariant J of E imply that zation.

Ann(Ann(E)) = Ann(J) in each locali­

Thus Ann(J) has an open-closed support and is locally 0 or Ap;

then Ann(J) is generated by one idempotent [16]. 5.16 Corollary,

Let A be a coherent local ring such that every prin­

cipal ideal has finite projective dimension. Let A be a coherent domain.

Then A is a domain.

(In the following we could avoid the

blanket hypothesis of domain by assuming the existence of nonzero divisors in the appropriate places.)

In questions of divisibility in A the follow­

ing set of prime ideals plays a role sufficiently rich on which to base a theory of divisors [55]. type

P = all prime ideals minimal over ideals of the

(a):b = (r e A | rb e (a)}. We remark on some of the properties of P.

If I is a fractionary ideal of A, denote the set {x e К | xl c A}, К the field of quotients of A, by I \ An ideal I is called reflexive if (I

) 1 “ I, that is, if it is reflexive as an A-module.

is always reflexive for any ideal and that xive ideal containing I . (a) If I is not contained in any (b)

Notice that I-'*'

(I ^) ^ is the smallest refle­

P e P, I-'*' = A.

If I is a reflexive ideal, then 1 =

pQ p 1?

Since (a) is immediate, let us prove (b).

Let x be an element in the

right-hand side of the expression above. Then L = I:x = {r e A j rx e 1} is not contained in any P e P. Thus Lx с Iyields L-1 зxl-1. By (a), L

*= A; and by a second inversion and the

reflexivity of I, weget x

e L

R be the set of finitely generated fractionary ideals I of A having the property that Ip is invertible for each P e P. On R we define Let

a composition I 0 J = CCI . J)-1)"1 That

" о "

is a composition in R is clear by localization.

associativity, just notice that

(I о J) о L

and

I

As for the

о (j о l )

are both

reflexive, and thus by (b) above it is enough to verify equality at each Ap, P e P, which is clear.

DIVISIBILITY

51

Note that if one of the ideals above I or J is invertible, then I 0 J = I • J,

that is, just the ordinary multiplication of ideals.

We now define, essentially in the manner of Ref. 42 (extended to the non-Noetherian case, e.g.3 in Ref. 52), a function from torsion modules (finitely presented) of finite projective dimension into R. Let

be exact, with F^,F

finitely generated projective modules.

Let I be the

Oth Fitting invariant of E, and define d(E) = (I-1)'1 the divisor of E.

That

d(E) is really in R results from the next lemma

and the comments following (5.13). 5.17

Lemma.

Proof:

If

Р^дЕ = 1, then