Chain Conjectures in Ring Theory

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann 647 Louis J. Ratliff, Jr. Chain Conjectures in Ring Theo...

Author: L.J. Jr. Ratliff

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Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

647 Louis J. Ratliff, Jr.

Chain Conjectures in Ring Theory An Exposition of Conjectures on Catenary Chains

Springer-Verlag Berlin Heidelberg New York 1978

Author Louis J. Rattiff, Jr. Department of Mathematics University of California Riverside, CA 92521 U.S.A.

AMS Subject Classifications (1970): 13A15, 13 B20, 13C15, 13H99 Secondary: 13B25, 13J15 ISBN 3-540-08758-3 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-08758-3 Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1978 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210

To my Mother and Step-Father Ruth and Earl McCracken

ACKNOWLEDGMENTS

I want to thank Steve McAdam and David E. Rush for many helpful and stimulating conversations concerning the chain conjectures.

Their

help shows up in many (frequently indiscernable) ways in these notes. I want to express my sincere gratitude to the National Science Foundation for their support of my research on these problems

(Grants

28939, 28939-1, 28939-2, MPS71-O2929-A03, MCS76-06009, and MCS77-00951, and to the University of California, Riverside for my Sabbatical year in residence (1976-77) in which I essentially completed writing these notes. Finally, I am very grateful to Mrs. Jane Scully for her very excellent typing and her constant good humor and patience.

PREFAC E These notes are concerned with a number of open problems, catenamy chain conjectures,

the

some of which are of quite long-standing.

In the hope of attracting some new people to do research on these problems, an attempt has been made to make these notes understandable to non-experts in this area.

This shows up partly in the history given

for the terminology (Chapter I), the discussion of the recently solved problems

(Chapter 2), the explanation of where the conjectures come

from and their history (Chapters 4 - 13), the examples (Chapter 14), and the discussion of the related open problems (Chapter 15).

Also,

numerous references have been included in the proofs, and many of the referenced results are summarized in Appendix A to help decrease the amount of time the reader must spend searching for a result in research journals.

For those more familiar with this area, there are a number

of new conjectures given, many implications between the conjectures are shown to hold, numerous equivalences of the more important conjectures are proved, and an extensive bibliography is included.

The chain con-

jectures are interesting and important problems, and it is my hope that the material in these notes will help in making these conjectures more widely known and in determining which of the conjectures are true.

TABLE

CHAPTER

0

INTRODUCTION

CHAPTER

I

DEFINITIONS

CHAPTER

2

SOME RECENTLY

OF CONTENTS

........................................ AND

BASIC

SOLVED

I

RESULTS

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

5

PROBLEMS

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

14

CHAPTER

3

SOME

CHAPTER

4

THE CHAIN

CONJECTURE

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

C HA P T E R

5

THE DEPTH

CONJECTURE

AND

CHAPTER

6

THE

CHAPTER

7

THE DESCENDED

CHAPTER

8

THE STRONG AVOIDANCE CONJECTURE AND THE AVOIDANCE CONJECTURE ........................................

67

CHAPTER

9

THE UPPER

70

CHAPTER

i0

THE TAUT-LEVEL

CHAPTER

Ii

THE CATENARY

CHAPTER

12

THE NORMAL

CHAPTER

13

COMMENTS

CHAPTER

14

SOME EXAMPLES

CHAPTER

15

SOME RELATED

APPENDIX

A

APPENDIX

B

(CATENARY)

CHAIN

H-CONJECTURE

THE WEAK

GB-CONJECTURE

DEPTH

CONJECTURE---

AND THE

GB-CONJECTURE

CONJECTURE

O N M. N A G A T A ' S

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

85

AND CONJECTURE

(K)

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

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

CHAIN

75 79

A SUMMARY OF KNOWN EQUIVALENCES FOR CERTAIN CHAIN CONDITIONS ........................................ NOTES

53 57

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

....................................... QUESTIONS

20 45

--- 63

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

CONJECTURE

(3.3.1)

.......

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

CONJECTURE

CHAIN

CHAIN

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

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

CONJECTURE

ON

CONJECTURES

PROBLEM

EXAMPLE(S)

......

88 90 93 99 114

BIBLIOGRAPHY

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

125

SUPPLEMENTAL

BIBLIOGRAPHY

128

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

TABLE

OF N O T A T I O N

INDEX

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

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

131 132

CHAPTER 0 INTRODUCTION All rings in these notes are assumed to be commutative with an identity.

The definitions concerning

the various chain conditions on

a ring are given in (i.I), and the undefined the same as that in M. Nagata's Local Rings

terminology [N-6].

is, in genera~

(We also use

c

as in [N-6] to mean proper subset.) Many conjectures concerning saturated chains of prime ideals in Noetherian domains and in integral extension domains of such a ring h~ve appeared in the literature. tures, are collected

together

These, and some similar new conjec-

in these notes in order to see what impli-

cations exist between them and to give a number of equivalences of the more important ones.

Besides

this, there is given:

a brief history

of the terminology and a brief list of some of the most important results that have recently been proved in this area; a summary of where the conjectures have previously appeared with the known results concerning

them;

in the literature some examples

together

to show that the

conjectures hold for certain classes of local domains and, on the other hand,

to show that there are quasi-local domains for which the conjec-

tures do not hold; a brief list of some related open problems; extensive bibliography of papers The chain conjectures

and, an

in this area.

say that one or another of the various chain

conditions holds for certain very large classes of local domains. these same chain conditions are hypothesized problems

Now

to hold in many research

in algebraic geometry and commutative algebra,

so it is impor-

tant to determine whether or not these conjectures hold, and any results to this effect should be of quite general interest. As already noted,

the main purpose of these notes is to establish

implications between these conjectures and to prove some equivalences of the more important of the conjectures.

There are a number of reasons

why consideration of these implications and equivalences are of interest and importance, but only two will be mentioned. if (and when) one or another of these conjectures

The first is that

is shown to hold,

then the conjectures which are implied by it, together with the statements equivalent to it, will also be known to hold and will be readily available for use.

The second (and more important)

reason is that the

known conjectures have, to date, withstood the many efforts to settle them, so it is hoped that possibly one of the new conjectures of the new equivalences) will be more readily decidable. real possibility,

since some of the equivalences

(or one

(This is a

to the conjectures

sound so reasonable that they clearly should hold, and since the equivalences vary over a fairly wide range of concepts, such as:

integral

extension domains; valuation rings; Rees rings; associated graded rings, polynomial rings; Henselizations;

completions;

elements; and, ideals of the principal class.)

analytically independent And if this turns out

to be the case, then some new insight into the other conjectures should result and be of help in future work in this area. Chapter I contains the definitions of the various chain conditions for a ring, some comments on the history of the terminology is given, and some of the basic facts concerning these concepts are listed. In Chapter 2, a brief summary of some of the recently proved important results in this area is given. Chapter 3 contains the conjectures

(both old and new), and in this

chapter a number of implications between these conjectures are established.

It turns out that almost all the conjectures lie between (im-

plicationwise) jecture

the Chain Conjecture

(3.3.8).

Conjecture

(3.3.2) and the Catenary Chain Con-

(They all lie between (3.3.1) and the Normal Chain

(3.3.9).)

Then, to indicate how things currently stand, two

additional statements are also considered.

One is somewhat stronger

than (3.3.1) and is false, and the other is somewhat weaker than the

Normal Chain Conjecture Chapters

and is true.

4 - 12 have the same general

form.

ters one of the more important conjectures

In each of these chap-

from Chapter 3 is singled

out for attention,

and there is given some indication of where, when,

and why it arose.

Then a summary of where

erature and of the results it is also given,

it has appeared

that have previously

and, finally,

in the lit-

been proved concerning

a number of new equivalences

of it are

proved. Chapter

13 is concerned with

(3.3.1)

- the one conjecture

in Chap-

ter 3 that I think may not be true - and a brief explanation of why I have doubts about this conjecture Chapter

14 contains

some examples

hold for certain classes for all quasi-local

is given. to show that the conjectures

of local domains,

domains,

but that they do not hold

and in Chapter 15 some related open prob-

lems are briefly discussed. In Appendix A quite a few characterizations conditions

(Hi-local

references

were proved are also given.

The first is that throughout

to make reference

to many known results

chain conditions,

in this appendix

paper

etc.) are listed and

is a very important part of these notes for at least

three reasons.

reference

first chain condition,

to where the characterizations

This appendix

various

ring,

of certain of the chain

in the literature

reference),

in this appendix

in which the result is proved.

result.)

in Chapters

stated

this will diminish research papers

(and more important)

reason is

of one or more of the various chain

4 - 13 are in terms of these chain conditions,

and so each of these characterizations

the

rather than to the research

(Hopefully,

A second

that a number of the equivalences conjectures

concerning

we usually give a

the amount of time the reader must spend in searching to find a referenced

it is necessary

and since many of these are explicitly

(with an appropriate

to the result

these notes

gives rise to a different

(in

appearance) Finally,

equivalence

of the chain conjecture

under consideration.

these lists can also be used to give a number of equivalences

of the other conjectures in detail

in Chapters

Appendix the answer

mentioned

in Chapter 3 that are not considered

4 - 13.

B is concerned with M. Nagata's

to the chain problem of prime

to Chapter 2) is no.

ideals

A brief description

some of their properties (that have previously

are established,

appeared

example(s)

to show that

(see the introduction

of the examples

is given,

and some additional uses

in the literature)

of these examples

is

also given. The Bibliography

is in two parts.

The first part lists the sev-

enty books and research papers which are referred and then follows a Supplemental portant papers these notes.

Bibliography

to in these notes,

that lists additional

in this area which were not specifically (The papers

mental Bibliography

[1,2,3,4,5,7,19,20,22,36,43]

are definitely

non-Noetherian

referenced

However,

a number of the methods

are of use for the chain conjectures, should be included

in flavo~ and are

notes understandable

and results

theory of

in these papers

so it was felt that these papers

in the Supplemental

As noted in the Preface,

in

in the Supple-

mainly concerned with a different aspect of the dimension rings.

im-

Bibliography.)

an attempt has been made to make these

to non-experts

in this area, and I hope the mater-

ial in them will help attract new people

to do research

in this area.

CHAPTER DEFINITIONS In this chapter,

the various

fined and some comments Following quently

chain conditions

on the history

this, a few basic

used

results

the definitions

At least a few of these definitions some of the concepts (I.I)

DEFINITION.

(I • i.i) saturated height maximal

case

= I .

ideals

ideals

such that

in

in

Po

(f.c.c.)

Po

A

in what

follows.

be new to the reader,

c

and let

a = altitude A .

PI

c

A

c

in case,

in case

is minimal

Po c

satisfies

(1.5).

will probably

ideals

--- c Pk

-..

Pk

in

for each

is

k .

A

Pk

The chain

chain

is maximal.

ideals

chain of prime

in

In this

the first chain c0ndition f o r prime

of !ength

mcpil k

,

is a

(that

in case each maximal

is a

a

i = l,...,k

it is a saturated

and

is

A

is, a maximal ~hain of Rrime ideals A

are given. that are fre-

that will be needed

be a ring,

ideals

it will be said that

(1.1.2)

in (1.2)

The length of the chain

chain of prime

of prime

A

A chain of prime

Pi/Pi_l

of the terminology

are quite new.

Let

chain of prime

on a ring are de-

on these definitions

in this paper are listed

(I.i) contains

since

i

AND BASIC RESULTS

k)

ideals

in

A

has

length

=a°

(1.1.3) P c Q

in

A

(1.1.5)

fies

satisfies

for each pair of prime

A/z

prime

satisfies

ideal

ideals

z

P c Q

for prime ideals

in

the f.c.c,

the chain condition

fox each pair of prime

ideals

the f•c.c.

the sec0n d chain condition

for each minimal

domain of A

in case,

satisfies

satisfies

in case,

gral extension

in case,

is qatenary

A , (A/P)Q/p

(1.1.4) (s.c.c.)

A

A , each inte-

and

for prime in

A

depth z = a . ideals

(c.c.)

, (A/P)Q/p

satis-

in

the

the s.c.c. (1.1.6)

A

is level

in case all maximal

ideals

A

have

same height and all minimal A

is taut

in

A

(respectively,

, height (1.1.7)

(o.h.c.c.) in

A

in case

ideal

taut-level)

satisfies A

, the integral

prime

A

in

closure

A

H-ring,

of saying

an

A

Hi+l-ring , and,

in the integral -

of

A/z

A

Ci-ring

for each

closure

of

for each prime

satisfies

(i ~ 0)

that

height

ideal

for prime prime

ideals

ideal

z

the c.c. for each

(that is,

height

height

i

P +

be said that

A

is an

Hl-ring. ) in case

A

is an

Hi-ring ,

i P E Spec A , all maximal

have

P

= a)

in case,

is an

(i ~ O)

A/P

the same depth.

for each minimal

i = 1 , it will usually

is a

have

(respectively,

, depth P = a - i

(For

A

1½ chain condition

Hi-ring

depth P = a) instead

the

in

in case,

is taut and,

is an

P

(1.1.9)

ideals

P + depth P E [l,a} A

(1.1.8)

prime

the same height

ideals

(= depth P = a

i)

(i.I.I0) integral (P c Q

A

is a

extension

ring of

are adjacent

(I.I.II)

GB-ring

If

A

in case A

contract

is an integral

altitude

formula

mula)

in case,

for all finitely

PNA

Ap~ A + trd B/A quotient

If

A

(respectively,

trd D/C

then

integral

for all

prime

A

ideals.

satisfies

altitude

domains

B

P E Spec B

domain

D

the transcendence over

forover

A

such that

Bp + trd (B/P)/(A/(PnA))

denotes

If

extension

is a semi-local Nuasi-unmixed)

divisors

(1.1.13) special

to adjacent

in each

= altitude degree of the

the quotient

field of its

C

(1.1.12)

mal) prime

generated

altitude

field of the integral

subdomain

mixed

, where

ideals

the dominating

(respectively,

ideal),

A

domain,

(respectively,

P £ Spec B

is a maximal

in

prime

height Q/P = I .)

the

and for all

in case adjacent

of zero A

(Noetherian)

in case all

in the completion

is a local

(A s ; M 1 , M2)

A

then

(~espectively,

of

ring with maximal of

ring,

A

have ideal

is a principal

A

is un-

all mini-

the same depth. M

, then a

(or, simple)

integral extension ring ideals

M I = (M,E)A s

(1.1.14)

A'

A s = A[x]

and

~

of

A

that has exactly two maximal

= (M,x-I)A s , and

MINM 2 = M .

denotes the integral closure of

quotient ring, and if

A

is semi-local,

Henselization and completion of

then

A

AH

in its total

and

A*

denote the

A , respectively.

The "mcpil k" terminology was introduced by S. McAdam and myself in 1976, in [RMc], since the phrase for which it is an abbreviation is quite long and occurred very frequently in

[~Mc].

(Of course, maximal

chains of prime ideals had been considered by many authors in many papers prior to [RMc] .) a

mcpil k

Some known equivalences of the existence of

in a local ring are listed in (A.5), and some additional

related results are given in (1.5). The f.c.c., s.c.c., c.c., quasi-unmixed,

unmixed, and altitude

formula are conditions that are well known to hold for local domains of classical algebraic geometry.

The first papers I know of in which

this terminology is used and in which these concepts are studied in their own right are M. Nagata's 1956 paper [N-3], for:

f.c.c., s.c.c.,

quasi-unmixed, and unmixed; and his 1962 book [N-6], for: altitude formula.

(He used "dimension formula" for "altitude formula"

in 1959, in [N-5], and this is also used in [ZS-2]. ralizes the classical result that if F[XI,...,X n] , where

F

This formula gene-

is a prime ideal in

F

n

=

height P + depth P = n

and

The French school terminology for these con-

"bi-equidimensional"

mally equidimensional"

P

is a field, then

depth P = trd (Fn/P)/F .) cepts is:

c.c. and

[G-I, (16.1.4)], for "f.c.c.;" "for-

[G-2, (7.1.9) ] , for "quasi-unmixed; " "strictly

formally equidimensional"

[G-2, (7.1.9) and (7.2.1)], for "unmixed;"

and, "universally catenary"

[G-2, (5.6.1) and (5.6.2)], for "altitude

formula" (but universally catenary is defined for Noetherian rings while altitude formula is defined for integral domains, so the matching of the terminology is nor perfect in this case).

(See (2.6).)

There

does not seem to be very good corresponding

terminology

for "s.c.c."

Or for "c.c." The "catenary" example,

see [G-l,

(16.1.4)].

for this condition called

terminology

is also due to the French school (Nagata did not introduce

in [N-3] or in [N-6].)

the "chain condition

for prime

I used "saturated chain condition (The French word cat~naire

so this is why I used the parentheses

ideals" in [ZS-2, p. 326], and

I introduced 1972,

chainlike,

or chain,

in the title of these notes.) s.c.c.,

catenary,

and c.c. con-

and (1.3), and quite a few known characteri-

zations of a local domain that satisfies listed in (A.9) and (A.II),

ideals" in [R-2] and [R-3].

to catenary,

A few basic results on the f.c.c., ditions are given in (1.2)

terminology

This condition on a ring is

for prime

translates

- ~or

the f.c.c,

or the s.c.c,

are

respectively. formula"

terminology

in

in [R-5], in order to show that two different

definitions

in the

literature

the "dominating

for a Noetherian

altitude

domain to satisfy

the altitude

formula are

equivalent. The word ture.

"level" has been used to mean various

For example,

things

D. Rees used it to mean quasi-unmixed,

in the literain

[Re],

and it has elsewhere been used to mean that all minimal prime ideals in a ring have the same depth, and, on the other hand,

that all maximal

ideals in a ring have the same height. The "taut" and "taut-level" and myself in [McR-2].

However,

be t~aced back to I. S. Cohen's see (2.2).)

conditions were introduced by S. McAdam the concept of such a ring can clearly 1954 paper

The reason for considering

(Concerning

these two conditions

an integral extension domain of a catenary [McR-2, Proposition

[C-2].

is that

local domain is taut, by

12], and it was hoped that by studying

this condi-

tion it could be shown that a taut (respectively,

taut-level)

local domain is catenary

the f.c.c.).

(respectively,

this,

satisfies

semi(Con-

cerning

this, see Chapter

i0.)

A number of known characterizations

a taut local ring and of a taut semi-local

of

domain are listed in (A-7)

and (A.8), respectively. The "o.h.c.c." was

introduced by M. E. Pettit,

Jr. and myself

in

[RP], and therein it is shown that this condition on a local ring stands in relation to the s.c.c, much as the taut condition stands to the f.c.c. terminology

A deeper study of this condition

for this condition

tion does fall intermediate main, by [~-18,(2.3)].

is made in [R-18].

is not particularly

to the f.c.c,

the o.h.c.c,

(The

good, but the condi-

and the s.c.c,

for a local do-

A number of known characterizations

local domain that satisfies

in relation

of a semi-

are listed in (A.10), and two

basic results on this condition are given in (1.4). I introduced

the "H" condition

tion 4], and it was noted in JR-6] was hoped that every Henselian

for a ring in 1971, that the reason

local domain is an

is true, then the Chain Conjecture

(3.3.2)

H

in [R-4, Secwas used is it

H-domain.

(If this

holds - see (4.1.1)

(4.1.4) .) M. E. Pettit,

Jr. generalized

the "C." conditions i

in 1973,

the "H" condition

in [P]

These conditions

tions on a local domain than the s.c.c., study of these finer

to the "H i

and

are finer condi-

and so it was hoped that a

conditions would yield some new information

on

chains of prime ideals that would help in the study of the chain conjectures. satisfies

A number of known characterizations one of these conditions

I introduced

are listed in (A.3) and (A.4).

the "GB" terminology

in [R-16], but the concept of a

GB-ring can be traced back to W. K~ull's this, see (2.1).)

of a local ring that

1937 paper

(Concerning

It turns out that a local domain satisfies

if, and only if, it is catenary

and is a

GB-ring,

Now the catenary condition has been deeply studied search papers,

[Kr].

but only widely

scattered

by

JR-16,

the s.c.c. (3.10)].

in a number of re-

results on the

GB

condition

10 were known prior to [R-16] tween" - it readily GB-ring

and

follows

[R-19].

B

of

A

P c Q

in

A

such that

PNA c p

P

in

B

such that

P c P

c Q .

GD

and the

concerning

a

to the

Finally,

c QUA

of GB-rings

I introduced

in [R-6, Section 4].

(4.7)].

conjectures

GU

conditions.)

considerably

will be given.

extensions"

examples

minology

this, however,

is new understand

of these definitions,

IN-6, Example

i ~ 0 , that for all

203-205]

and I used this concept (3.3.8),

to

in

that a number of the chain and a

is given in [D].

in (i.i)

to help those for w h o m the ter-

some of the relationships

between several

on them.

s.c.c. = f.c.co = taut-level ~ H i

C i = H i , and that a ring A

is a

for all

Ci-ring and an

Hi-ring

i ~ altitude A .

A taut-level taut-level

ideal in

ring

or fails

one m a x i m a l

is an

in 1973,

2, pp.

facts on the concepts

we make a few comments

It is clear that

terminology

in terms of special extensions,

some needed basic

Before

ideal

is given in (A.6).

deeper study of such extension rings

(1.5)

ideal

in (15.3), and a list of

It will be seen in these notes

In (1.2)

in an inte-

An open question

of the Catenary Chain Conjecture,

can be characterized

is a

This c o n d i t i o n on a ring is some-

(see A p p e n d i x B) involve special extensions,

[R-6,

ideals

A

, then there exists a prime

the "special

M. Nagata's

give a c h a r a c t e r i z a t i o n

for "going be-

that a ring

are prime

G B - l o c a l domain is m e n t i o n e d

known characterizations

stands

such that there exists a prime

p

w h a t comparable

GB

from the definition

if, and only if, w h e n e v e r

gral extension ring

(The

ideal in

A .

Ho-ring

Also,

is a taut ring, and a taut ring is either

to be taut-level A

and/or

only because

a taut local ring is an

Hi-ring , for follows

from

there exists a height

there exists a depth one m i n i m a l prime Hi-ring , for

if, and only if, it is taut-level),

local ring is an It readily

A

i ~ 1

(and

and a taut semi-

i ~ 2 .

[R-5, Theorem 2.2]

(see (A.9.1) =

(A.9.5))

11 that the following R

satisfies

taut; and,

statements

the f.c.c.; R

is an

are equivalent

R

is catenary;

Hi-ring , for

for a local domain

R

is taut-level;

i = l,...,a-2

, where

R :

R

is

a = alti-

tude R . Also, by

[RP,

(3.11)],

for a local domain R

satisfies

for

R :

R

the o.h.c.c,

i = 0,1,...,a-2

the following satisfies

and

, where

R

(See (B.3.2).)

Also,

Hi-rings

(respectively,

but are not

Hj-rings,

j = 0,i ..... m .)

203-205]

~See

statements

satisfies

and,

in the case

the o.h.c.c.,

Ci-rings ) for for

R

is a

and

(respectively,

C~-rings,

for

(B.4.2).)

that contain basic

facts on the concepts

in the remainder of these notes.

2.22 - 2.24].)

that are

= altitude R ,

(A sketch of the proofs of these statements

in [R-4, Remarks

Ci-ring,

but not the s.c.c.

in the first two of these follow quite readily

definitions.

the c.c.;

m = 0 , there exists

i = m+l,...,a

j = i ..... m

(B.4.1)

that are needed

R

m ~ 0 , there are local domains

We now give four remarks in (I.I)

is level;

that satisfies for

the s.c.c.;

are equivalent

a = altitude R .

By [N-6, Example 2, ppo a local domain

R'

statements

(1.2)

All

the

from the can be found

is concerned w i t h catenary

rings

and the f.c.c. (1.2) main

A

REMARK.

ideals

I

S)

A .

enary

If

in

(1.2.2) maximal

A

A

hold for an integral do-

(1.2.3)

is catenary M

if,

If A

then

As/IA s

and for all m u l t i p l i c a t i v e l y

A

ideals

is catenary,

in

if, and only if,

and only

statements

:

(1.2.1)

in

The following

A . AM

if, and only (Note,

if

satisfies

AM

closed

AM

height M < =

for all

subsets

S

is catenary, , then

AM

(0

for all is cat-

the f.c.c.)

altitude A < = , then is level and

if,

is catenary,

A

satisfies

is catenary,

the f.c.c,

for each maximal

if, ideal

12 M

in

A , and this holds if, and only if, (1.2.4)

satisfies

If

B

is an integral

the f.c.c.,

(1.3)

then

A

A

is level and catena~y.

extension domain of

satisfies

in relation

and

B

the f.c.c.

is concerned with the c.c. and the s.c.c.

c.c. stands

A

to s.c.c, much as catenary

It shows that

stands

in relation

to f.c.c. (1.3) main

REMARK.

The following

statements

hold for an integral do-

A : (1.3.1)

c.c.,

If

A

satisfies

for all ideals

subsets

S

I

(0 ~ S)

(1.3.2)

A

in

in

satisfies

for all maximal

then

satisfies

(1.3.3)

If

A

the c.c. ideals

the c.c.

A , and this holds

satisfies

is level and

If

B

If

A

satisfies

then

the f.c.c.

(respectively,

(respectively, in [N-6,

(even when follows

AM

A .

A

(Note, AM

AM if

closed

satisfies height M < = ,

satisfies

satisfies

satisfies

A

the s.c.c.)

the s.c.c,

the c.c.,

if, and only if,

if,

for each maximal is level and

A

satisfies (34.2)].

extension domain of

satisfies

(respectively,

(respectively,

the c.c.). However,

A

B

the c.c.),

is catenary).

B

is integral over

the c.c.),

then

For the c.c., neither

is a local domain and

and

the reader might expect that if

such that satisfies

A

the s.c.c.

the s.c.c.

and (1.3.4),

are integral domains

is catenary

stated

A

satisfies

Because of (1.2.4) A ~ B

in

is an integral

the s.c.c.,

(1.3.5) A

the

the c.c.

(1.3.4)

then

M

altitude A < = , then

ideal

satisfies

satisfies

if, and only if,

if, and only if,

A

in

As/IA S

and for all multiplicatively

and only if, M

then

A .

the c.c., AM

the c.c.,

B

and

B

is catenary

this is, in fact,

of these statements

is a finite

from [N-6, Example 2, pp. 203-205].

A

A

is true

A-algebra),

(See (B.3.4)

as

and (B.5.1).)

13 On the other hand,

if

B

and if, for each maximal then

A

is catenary

from the statements (1.4) gives (B.3.12)

R/P

Let

R

(2.3)].

(1.4.2)

[R-18,

(2,9)

the o.h.c.c.

to semi-local

rings

(It is shown in is necessary

statements

in (1.4).)

ring that sat-

hold:

is catenary.

and (2.10)].

For each

P E Spec R, Rp

and

the o.h.c.c.

REMARK.

[RMc,

for a local domain

(1.5.1)

of a

mcpil

n

in an integral

There

exists

a

(2.14)]. (R,M) mcpil

The following

statements

are

: n

in some

integral

extension

do-

R .

(1.5.2) R

There exists

such that

(1.5.3)

There

a minimal

prime

ideal

z

in the completion

depth z = n . exists

a

mcpil n + k - depth Q

R k = R[X 1 ..... Xk]

, Q E Spec R k , and

(1.5.4)

a

There exists

(There are two additional

here. )

as follows

domain.

equivalent

in (1.5)

the c.c.), (1.3)~

is concerned with the existence

(1.5)

of

satisfies

(respectively,

R

the c.c.,

B , height MnA = height M < ~ ,

Then the following

[R-18,

main of

in

satisfies

be a s e m i - l o c a l ~ o e t h e r i a n )

(1.4.1)

extension

(respectively,

two facts concerning

REMARK.

satisfy

M

(respectively, in (1.2)

the o.h.c.c.

(1.5)

R

ideal

that the restriction

(1.4) isfies

is catenary

that are given

in

RkQ

, where

MR k c Q .

mcpil n+l statements [RMc,

in

in

R[X](M,X)

equivalent

(2.14)],

to the statements

but they will not be needed

CHAPTER 2 SOME RECENTLY

SOLVED PROBLEMS

In any discussion of the various chain conditions in a Noetherian such results.

The first is due to I. S. Cohen in 1946,

local ring is catenary

ideals,

is the second result just alluded

cal domains

(for example,

rings), but,

regular

is no, he constructed

to.

He showed,

is yes for quasi-unmixed

the answer

is no.

a family of local domains

R

and

in

and are due to a maximal

all have length

the integral closure examples

included

m + i R'

of

the case

R

m ~ I)

such that

examples problem.

are (essentially) (The examples

Example, pp. 327-329].

o f these examples

Since 1956 (and because

ideals

R'

.)

in Noetherian

(5.6.11)],

Appendix B contains

the answer

and conjectures

R

cesses will now be mentioned,

in (His

is catenary

To date, his to the chain

[M, (14.E)],

concerning

and

a brief description them.)

to the chain problem

domains have appeared

of these have recently been settled.

M'

in [N-6, Example 2, pp. 203-205],

together with some facts concerning

number of other problems

ideal

the only known counter-examples

and similar examples are given in [G-2, [ZS-2,

such that

height M' = m + i .

ideal in

are reproduced

images of

such that the short chains

m = 0 , and for this case

and there exists a height one maximal

lo-

To show that the

altitude R = r + m + I (r ~ I R

in [N-3,

local rings or homomorphic

in general,

This

and its solution by M. Nagata,

Theorem i and Section 3], that the answer

answer

(1.1.3).

to ask is if every local domain is catenary.

is the chain problem of p q ~ e

Macaulay

in [C-l, Theo-

since a local domain is a dense subspace of its completion,

a natural question

in 1956,

ideals

ring, there are two results which stand above all other

rem 19], and it shows that a complete Therefore,

for prime

is no), a

chains of prime

in the literature,

and some

A few of the more important

suc-

and we begin by noting a non-Noetherian

solution of a problem of considerably

longer standing.

15 (2.1) question:

In 1937, in [Kr, p. 755], W. Krull asked the following if

B

is an integral domain that is integral over its inte-

grally closed subdomain in

A , and if

B , then does it follow that

P c Q

P~A c QnA

are adjacent prime ideals are adjacent prime ideals?

(That is, is every integrally closed domain a GB-ring (i.I.i0)?)

In

1972, I. Kaplansky gave a negative answer to this question in [K]. [K], the ring

A

In

is not the integral closure of a Noetherian domain

(nor is it a K~ull domain), and the problem is still open for this case. (If the Chain Conjecture

(3.3.2) holds, then the answer, for this case,

is yes, by (3.6.1) = (3.6.4).) (2.2)

In 1956, in [Y], M. Yoshida asked if a taut-level (1.1.6)

local domain

R

must be catenary.

(This question was clearly suggested

by I. S. Cohen's discussion of the chain problem in 1954, in [C-2, p. 655], and by M. Nishi's 1955 paper

[Ni].)

2.2], I gave an affirmative answer to this.

In 1972, in JR-5, Theorem And, in 1973, in [McR-2,

Proposition 7] and in [R-14, (2.15)], S. McAdam and I showed that if R

is a taut-level local ring, then

R

satisfies the f.c.c.

(1.1.2).

(Related to this, in 1972, in answer to a question asked in [R-5, Remark 2.6(iv)], W. Heinzer gave an example of a taut-level quasi-local domain that is not catenary.

(See (14.6).)

Some additional results related

to this are given in (2.8).) (2.3)

In 1956, in [N-3, Problem i, p. 62], M. Nagata asked (*):

can (0) in the completion prime divisors? M-transform of and of

p # M}) R

R

of a local domain

(R,M)

have imbedded

And, in 1959, in IN-5, Section 4], he asked if the R

(= U M -n = [B, Corollary 1.6] R (w) = ~[Rp ; p E Spec R

must be a finite

R-algebra when the integral closure

R'

is quasi-local, and he then commented that if the answer was yes,

then he could prove that the answer to (*) was no and that the Chain Conjecture holds, but if the answer was no, then it was almost certain

16

that an example of such a ring could be used to show the answer to (*) was yes.

(In 1970, in [F-M, p. 120], M. Flexor-Mangeney showed that

(*) and the finiteness of

R (w)

are, in fact, equivalent problems;

that is, the answer to (*) is no if, and only if, finite

R-module, when

R'

is quasi-local.)

R (w)

is always a

In 1970, in [FR, Proposi-

tion 3.3], D. Ferrand and M. Raynaud gave an example of a local domain R

such that

is local and R (w) is not a finite , the zero ideal in R has an imbedded prime divisor.

this

R'

R , R # R'

(since

R (w)

is not finite over

R-module, and (Of course, for

R) , and it is

still an open problem if the answer to (*) is yes, when grally closed.)

R

is inte-

Related to (*), in 1948, in [Z], O. Zariski asked if

every normal local domain is analytically irreducible.

In 1955 and

1958, in [N-2] and [N-4] (whose titles are self-explanatory), M. Nagata gave a negative answer.

Also related to (*), in 1953, in [N-l, Conjec-

ture i], M. Nagata asked if there exists a Henselian local domain R @ such that (0) in R is not primary, and he gave an affirmative answer to this in 1958, in [N-4] - and [FR, Proposition 3.3] showed that (0)

in such

R

can even have an imbedded prime divisor.

these questions are related to the chain conjectures,

since it is known

JR-2, Theorem 3.1] that a local domain is quasi-unmixed and only if, it satisfies the s.c.c.

(1.1.4).

(All of

(1.1.12)

if,

Concerning this, see

(2.5) .) (2.4)

In [N-3, Problem I, p. 62], it was asked if

(0)

in

R

can have an imbedded prime divisor when all minimal prime ideals in R*

have the same depth; that is, does there exist a quasi-unmixed local

domain that is not unmixed (1.1.12)?

[FR, Proposition 3.3]

(together

with [R-2, Proposition 3.5]) gives an affirmative answer to this. (Prior to this answer, I proved in 1970, in JR-3, Proposition 5.13], that the answer is yes if, and only if, there exists a Henselian local domain

R

such that

R (I) = n [ ~

; p E Spec R

and

height p = 13

is

17 not a finite

R-algebra.

Compare this with the result in IF-M, p. 120]

mentioned in (2.3).)

There is a related problem [N-3, Problem 2, p. 62]

which is still open:

is a factor domain of an unmixed local domain

unmixed?

R

A partial answer to this was given by M. Brodmann in 1974, in

[Bro, (5.9)], where it was shown that most finitely many height one [R-7, Remark 3.2(2)] (2.5)

R/p

p E Spec R .

This also follows from

together with [N-6, (18.11)].

In 1969, in JR-2, Theorem 3.1], I proved the following con-

ditions are equivalent for a local domain tude formula (i.I.ii); unmixed.

is unmixed, for all but at

R

R :

R

satisfies the alti-

satisfies the s.c.c.; and,

R

is quasi-

And, in 1972, in [R-5, Theorem 3.3], I showed these are also

equivalent to:

R

satisfies the dominating altitude formula (I.i. Ii);

thus showing that two different definitions of the altitude formula which had appeared in the literature are equivalent for Noetherian domains.

The question of the equivalence of the first two of these condi-

tions arose in 1959, in [N-5], when M. Nagata gave a proof showing that if

R

satisfies the s.c.c., then every locality over

s.c.c, and the altitude formula.

Unfortunately,

satisfies the s.c.c, if the integral closure of and this is still an open problem - see (3.3.9). lence of the second and third conditions, [N-6, (34.6)] it was proved that quasi-unmixed.

R

R

satisfies the

the proof used: R

R

satisfies the f.c.c.; Concerning the equiva-

in [N-3, Theorem i] and in

satisfies the s.c.c., if

R

is

But the proof in [N-3] was based on a result that is

closely related to the just noted open problem, and the proof in IN-6] was not completely clear, so some added details for this implication were given in [R-l, Corollaries 2.6 and 2.7] and in [R-l, pp. 283-284]. (2.6)

In 1965, in [G-2, (7.1.12)(i)], A. Grothendieck asked if

a local ring generated

R

which is universally catenary (that is, every finitely

R-algebra is catenary)

is f o ~ a l l y

catenary (that is, for

18

each minimal prime ideal in [G-2, (7.2.10)(ii)],

z

in

R ,

R/z

R

is universally catenary and

for all

R

is catenary.

catena ~

(R/p) (I)

p E Spec R), the condition:

be replaced by:

And,

he asked if, in parts (c), (d), and (e) of the

defining theorem of a strictly f o ~ a l l y is,

is quasi-unmixed).

R

local ring is a finite

R

(that

R/p-algebra,

is universally catenary;

could

An affirmative answer was given to

each question in 1970, in Theorems 2.6 and 5.4 in [R-3]. (2.7) satisfies

Circa 1970, I. Kaplansky asked if each local domain which the s.c.c,

is a homomorphic

image of a Macaulay ring.

A

negative answer is given by [FR, Proposition 3.3] together with IN-6, (34.10)].

That is, [FR, Proposition 3.3] gives an example of a quasi-

unmixed local domain

R

that is not unmixed

(see (2.4) above), and

[N-6, (34.10)] says that a factor domain of a Macaulay local ring is unmixed.

Finally, a quasi-unmixed

local domain satisfies

the s.c.c.,

by (A.II.I) = (A.II.4). (2.8)

In 1975, in [Fu-l, Lemma 6] (together with [R-2, Theorem

3.6]), K. Fujita showed that there exists a Noetherian Hilbert domain that satisfies the f.c.c, but not the s.c.c.

(M. Nagata's examples

[N-3] showed this could happen for a local domain, and Fujita's example was based on Nagata's. in 1973, in [R-II, [R-II,

See (B.3.11).)

(2.19) and (2.20)].

This answers a question I asked Prior to this I showed, in

(2.22)], that there exists a Noetherian Hilbert domain that does

not satisfy the c.c.

(See (B.5.7).)

Fnjita also gave, in [Fu-l, Propo-

sition, p. 478], a negative answer to a question I asked in 1971, in [R-4, Remark 2.25] and in [R-14, (2.14)], with an example of a Noetherian Hilbert domain that is taut-level but does not satisfy the f.c.c. (Compare this with the fact that a taut-level local domain satisfies the f.c.c., as noted in (2.2).) ~M~2,

Theorem 4] that

R<X)

Following this, S. McAdam showed in

satiafies the f.c.c,

if it is taut-level,

19 where U[N

R

is a semi-local

domain and

; N and NNR are maximal

to the Taut-Level

R<X> = R[X] S,

ideals}.

Conjecture

(3.9.5).)

Fujita also showed in [Fu-l,

pp. 482 and 484]

main

that is not taut and there exists an

main that is not catenary. the question

(2.9)

to the

It is known

local domains

exists a

mcpil n

in

R

R .

I showed that there exist chain of prime ideals in

R .

ideals

(3.3.2)

(3.1)]

that if

(i.i.I)

in

S

S

(3.15)]. such

R

In [R-20,

R c S

S

and

are R ,

is if a maxi-

to a maximal (2.10)] S

chain of

(see (B.4.8)), has a maximal

to a maximal chain of prime (4), p. 87]

that if this

then the Chain Conjecture

[R-20] also contains

examples

chains of prime

that were shown by Nagata's

be shown in local (rather than semi-local)

(S,N)

is integral over

such that

Henselian,

number of other things concerning extension domains

(R,M)

must contract

(It is pointed out in [R-10,

does not hold.)

do-

if, and only if, there

that does not contract

can be shown to hold with

H-quasi-local

From this, a natural question in

[R-IO,

do-

(3.3.6).

R ~ S , NNR = M , and

mal chain of prime ideals prime ideals

H-Conjecture

mcpil n in

H-Noetherian

in [R-4, Remark 2.25], and both of these

[R-IO,

such that

then there exists a

that there exists an

The first of these results answers part of

I asked in 1971,

results are related

S' = R[X] -

(The last two results are related

Propositions, (1.1.8)

with

showing that a

ideals in integral

examples

integral

can,

in fact,

extension

domains.

CHAPTER SOME In this chapter, rated

chains

a number

of prime

(CATENARY) many

gram showing

the major

given on p. 44.) but my feeling

between

is all of them, 13.)

except

satu-

domain are considered, are proved.

that are proved

in this chapter

of the statements

(3.3.1),

should hold.

stronger

and

(A dia-

(if any)

Two additional

is somewhat

and the other

concerning

these statements

It is unknown which

one of which

and is false,

(conjectures)

in a Noetherian

implications

ing (3.3.1), see Chapter in (3.5),

CHAIN CONJECTURES

statements

ideals

of relationships

3

statements

is

hold,

(Concern-

are also given,

than any of the conjectures

is somewhat weaker

than any of the conjec-

tures and is true. The following lows.

two lemmas will be needed quite often

For the applications

take

S = R' (3.1)

of (3.1)

but occasionally LEMMA.

Let

R

and let

S

exists

a height

one maximal

maximal

ideals

i__n_n S .

Let

statements

be a ring such that

S

B = Rib,i/b]

B

B

(3.1.3)

A prime

ideal.

and

S = R such that

R c_ S c_ R'

Assume

S

and let

b

altitude there

in all height

is in all other maximal

C = S[i/b]

(3.1.4)

B

domain ,

one

ideals

Then the following

is an

ideal in

B c__ C c_ B' = R'[1/b]

, and

C

ideals.

is a local domain,

(respectively,

verse holds

in

we will usually

domain

1 - b

is a semi-local

(3.1.2)

mal

ideal

such that

has no height one maximal

B

take

fol-

hold:

(3.1.1)

in

we will

be a semi-local

R > 1

in

in these notes,

in what

C)

P

in

if

R

R[b]

is a local domain. (respectively,

if, and only if,

H-domain , if

R

is an

if there are no height one maximal

P

in

S)

is lost

is a height one maxi-

H-domain, ideals

in

and the conR .

21 (3.1.5)

B

(respectively, (3.1.6)

(respectively, C) S)

B

if, and only if,

R

is taut. (respectively, C)

(respectively, S) (3.1.7)

is taut-level,

is catenary if, and only if,

R

is catenary.

B'

satisfies the c.c. if, and only if,

R'

satisfies

the c.c. This is p~oved in detail in [D], so we simply note here that it is proved in much the same way as the local domain case was proved in JR-10, Section 4]. The following lemma, which will be needed in a number of results below, generalizes if

R

is an

is an in

[R-21, (9.2.1)].

That is, [R-21, (9.2.1)] says that

H-semi-local domain such that

altitude R > 1 , then

H-domain if, and only if, there are no height one maximal ideals

R' , and this is true if, and only if,

R'

is level.

need the more general result that is given in (3.2). that the hypothesis an

R'

depth p E [0,a-l}

However, we

It should be noted

in (3.2) is satisfied if

H-domain, is taut (or taut-level), or satisfies the f.c.c.

S.C.C.)

is

(or the

.

(3.2)

LEMMA.

R = a > 1 . [O,a-l]

R

Let

R

be a semi-local domain such that

Assume that, for each height one

Then, for each height one

and for each maximal ideal

M'

altitude

p E Spec R , depth p E

p' E Spec R' , depth p' E ~O,a-l]

i__n_n R' , height M' ~ [l,a}

Moreover,

the following statements are equivalent: (3.2.1)

R

(3.2.2)

R'

(3.2.3)

There does not exist a height one maximal ideal in

(3.2.4)

There does not exist an integral extension domain of

and

R'

are

H-domains

(1. i. s).

is level (1.1.6).

that has a height one maximal ideal.

R' R

;

22 (3.2.5)

R

Proof. in

R'

and

Assume

first

so

is level and is an Therefore

Co-domains

(3.2.3)

is an

H-domain,

implies

one maximal

H-domain

(3.2.2)

and

together w i t h

one maximal

ideals

in

R

(by hypothesis),

by assumption

concerning

either of (3.2.1)

(3.2.1)

(1.1.9).

that there are no height

R

the statements

Now, and

are

Then there are no height

gral dependence),

and

R'

and

[R-21,

height M'

and

ideals

(by inte-

and thus

R'

(9.2.1)].

depth p'

hold,

(3.2.1).

or (~.2.2) (3.2.2)

implies

imply

(3.2.3),

(3.2.5),

since

since

R

a > 1 ,

and

R'

and

H -domains. o (3.2.5) domain of

R

an integral a height dence,

=

(3.2.4),

extension

there exists (3.2.5)

Finally,

domain ideal,

S by

a height

(since

of

ideal,

one maximal

2.9],

ideal

that

(3.2.4)

a height

fies the conditions

on

[Mc-l, Theorem

(3.1.1)

that

B' = C = R'[i/b~ has already statements

nine statements are listed,

R , by

been proved,

We now begin

B = R[b,i/b]

R'[I/b]

depth p'

to consider

concerning

R'

S

has depen-

, and this con-

and

chains

implications

ideals.

R' satis-

Therefore, H-domain,

hold, by

ideals

from

domain and by what so the

~3.1.3),

chain conjectures.

implies

hold.

Rib]

H-semi-local

height M'

of prime

in

8], so it follows

is an

the various

ideal

Then

is level and is an

and it is shown that each

if any of the reverse

in

be as in (3.1).

has no height one maximal

concerning

and

so, by integral

one maximal

C

(3.1.3)

S c R'

= (3.2.3).

, B , and

and

extension

then there exists

such that

[R-4, Lemma

that there exists

b , S = R'

R

an integral

a > I)

it is clear

Now assume and let

if there exists

that has a height one maximal

one maximal

tradicts

since

q.e.d.

In (3.3),

in a local domain

the next.

I do not know

There are four additional

23 conjectures

that could be given in (3.3), but we prefer to give them

later in these notes. separated

from (3.3) because

and (3.3.4) (3.8.2),

The first two are given in (3.4), and they are

(see(3.4)).

they are very closely

(3.3.2) = (3.8.2) = (3.3.3).

to consider

local case later in this chapter

the chain conjectures the conjectures

(from (3.6.4)

One reason for not giving

is that all the other conjectures and (3.3.9).

Another

on).

that (3.8.2)

in

concerning

for the non-

The fourth

in this chapter) given statement

con-

till then

in these notes are intermediate

(3.3)

(15.1).

easier to prove that a

implies one of the equivalences statement

preceding

to

(and a number of other results

it is sometimes considerably

4 - 13) of the succeeding ment itself.

this conjecture

reason is mentioned

It turns out that to prove

lences,

(and (3.13))

that is not given in (3.3) is (15.1), and it is clear that

(3.3.9) = (15.1).

(3.3.1)

Conjecture,

The reason for not including

a local domain first, and then consider

jecture

to (3.3.3)

The third is the Strong Avoidance

and it is shown in (3.8) and (3.14.1)

(3.3) is that we prefer

related

(proved in Chapters

rather than the succeeding

state-

When this is the case, we will make use of these equiva-

and then use care in Chapters

4 - 13 to keep away from circu-

lar proofs. Six of the statements appeared previously named conjecture,

in (3.3) are named,

in the literature

(3.3.4),

and five of these have

under these names.

and the remaining

The other

three conjectures

are new

in these notes. (3.3) for

THEOREM.

For the following

statements,

(3.3.i) = (3.3.i+I),

i = i .... ,8 : (3.3.1)

If

P E Spec R , R/P (3.3.2)

R

is a local domain such that, for all nonzero

satisfies

the s.c.c.,

The Chain Conjecture

sure of a local domain satisfies

then

holds;

the c.c.

R

is catenary.

that is, the integral clo-

24 (3.3.3)

The Depth Conjecture

domain and

p 6 Spec R

p E Spec R

such that

(3.3.4)

holds;

is such that p c P

and

height

P £ Spec R

P = 1 , then there exists

R

is a local

P > 1 , then there

exists

depth p = depth P + 1 .

The W e a k Depth Conjecture

local domain and

that i_~s, i f

holds;

is such that

p E Spec R

that i_~s, i f

height

P = h

such that

R

is a

and

depth

height p = 1

and

depth p ~ h . (3.3.5) ture holds

If

in

(3.3.6)

(R,M)

is a local domain,

then the W e a k Depth Conjec-

R[X](M,X ) The

H-Qonjecture

holds;

that is, an

H-local

domain

is

catenary. (3.3.7) sion of

R

If

R

is an

(1.1.13),

then

(3.3.8)

T h e Catenary

gral closure

of a catenary

(3.3.9)

RSM. i

domain and

is an

Chain Conjecture local

domain

domain

R

Rs

H-domain, holds;

satisfies

The Normal Chain Conjecture

gral closure o f ! local fies

H-local

holds;

satisfies

is a special for

exten-

i = 1,2

that is, the intethe c.c.

that is, if the inte-

the f.c.c.,

then

it suffices,

by

R

satis-

the s.c.c. Proof.

(4.1.2),

To prove

to prove

satisfies

that

(3.3.1) = 0.3.2),

that if (3.3.1)

the s.c.c.

For this,

Then it may clearly be assumed P ~ Spec R , so

R/P

let

R

that

then a Henselian be a Henselian

a = altitude

R > 1 .

and so, by induction

Therefore,

(3.3.1),

the s.c.c.

since

is Henselian,

R

by

satisfies

R

the s.c.c.,

local domain

local domain.

is Henselian,

satisfies R

holds,

(4.1.1)

Let

on

a , R/P

is catenary. by

(0) #

(A.II.14)

Hence, =

(A. ii. I). Assume Spec R

(3.3.2)

such that

such that

P'nR = P

holds, height and

let

R

P > 1 . height

be a local domain,

and let

Then

P' £ Spec R'

there exists

P' = height

P .

Let

c E P'

P E

such

25 that P'

c

is not in any maximal

, and let

p'

height P'/p' so

R'/p'

maximal

= I .

Now

is catenary,

ideals

depth P'/p' fore,

be a prime

in

ideal in

R' by

R'/p'

R' R'

(1.2.1).

c E p' c P'

by hypothesis

Also,

(by the choice

and

that does not contain such that

is catenary,

= altitude R'/p'

p = p'AR ~ P

ideal in

P'/p'

of

c)

- height P'/p'

and

(1.3.5),

is contained

, so

and

in all

depth P' =

= depth p' - I .

There-

depth p = depth p' = depth P' + I = depth

P + i , so (3.3.2) = (3.3.3). Assume a

(3.3.3)

height h

(3.3.4) prime

ideal

so assume

PI c P

It is clear Assume R

(3.1.6)

assumed

by

since

(A. II.19), height

Therefore

holds,

let

(R,M)

S = R')

be an

Then, by

, to prove

(A.3.1) = (A.3.17), R

is not catenary,

so there exists

R

D

is not an

is catenary,

Assume

(3.3.6)

that

D = R[X](M,X ) D

(3.1.4),

holds,

it may be

(A.II.I)

depth P = I

and

Thus, by There-

so (3.3.5) = (3.3.6).

p

ideal in Now,

by

is a contradiction.

let

R

be an

H-local domain,

be a height one prime

height p' = I

R'

H-domain.

(A.9.1) = (A.9.4).

ideal in

be a prime

ideal in

is an

such that

and this

and sup-

(3.1.2),

is not catenary,

P E Spec D

H-domain,

holds,

Let

Then,

from a finite

H-local domain,

(3.3.6)

R , and let

p'

there exists a

follows

(3.1.1),

a special e x t e n s i o n of Rs .

be

h -<- i , then clearly

(3.3.3),

(3.3.4)

P

(3.3.4) = (3.3.5).

P + depth P < altitude D , by

(3.3.5), fore

Then, by

If

that there does not exist a height one maximal

Therefore, Also,

R .

depth PI = depth P + i = 2 .

is not catenary. (with

ideal in

and let

of this.

that

(3.3.5)

be a local domain,

h > i .

height PI -<- h - I .

pose that

R

such that

number of repetitions

and

let

and depth one prime

holds,

clearly,

holds,

L = RSN

L' = (R')(RS_N)

to prove

that

L

, where ideal in

such that is an

N

let

Rs

be

is a maximal L , and let

p'NL = p , so

H-domain,

it may clearly

26

be assumed

that

altitude

= altitude

L - 1 , as will now be shown.

(and hypothesis), (1.3.5).

R'

Also,

[i , altitude

if

R}

in

L'

it follows

M'

, then

that

=

(11.i.1)

(1.2.4),

R

holds.

in

R'

Rs

is integral

and,

(6.1.4)

is catenary, , then

if

N'

L'

by

height M' E over

R ,

is a maximal

L = altitude , so

R}

.

Hence

depth p =

Therefore,

holds and let

R'

Then

hence

L

(11.1.8),

appeared

This

(3.3.4)

4 of [R-12]

Also,

is also

is an

and

(11.1.8)

two different

Chain Conjecture

(4.5)].

Finally,

the f.c.c.,

Therefore,

satisfies by

(and of

R'

q.e.d.

on where

these

and some equiva-

(3.3.1))

will be given

theorems

it was noted (below) that the

were given

in this chapter.

in the last paragraph

implies

the

H-Conjecture

H-Conjecture implies

in [R-5, Remark 3.5(ii)] proof

Chain Conjecture

and

in the introduction

implies

the Normal

the [R-6, of

Chain Con-

jecture. The reason

foe including

by

the s.c.c.,

(1.3.4),

some comments

it was noted without

that the Catenary

satisfies

in the literature

(3.4.1) proofs

be a local domain

the case for the other named conjec-

= (3.3.6),

Catenary

R'

in the remaining

that

R

the c.c.

the s.c.c.,

in the introduction,

4 - 13.

R

satisfies

satisfies

have previously

Concerning

[R-4]

(3.3.8)

the f.c.c.

that are considered

of Section

R'

(3.3.7) =

of each of the named conjectures

in Chapters tures

that

and the f.c.c.,

and so

As mentioned

lences

so

L - 1 .

since clearly

by (3.3.8),

the c.c.

conjectures

(by (1.2.1))

(6.1.1) =

= (3.3.7).

satisfies

hence,

by (1.3.3),

since

L' - 1 = altitude

assume

R'

satisfies

Thus,

ideal

by

depth p'

(3.3.8).

Finally, such that

the c.c.,

height p' + depth p' = altitude

= (3.3.8),

=

Namely,

height N' E [i , altitude

so (3.3.6)

(3.3.7)

depth p' ~ 1 , and so

is a maximal

is catenary

depth p' = altitude H-domain,

satisfies

, by (3.2).

L' = (R')(RS_N) ideal

L > 1 , so

(3.3.5)

is explained

following

(3.4).

27 Also,

concerning (3.4)

main

REMARK.

(R,M)

see the comment preceding

Consider

the following

If there exists

and

Then

(3.4.1)

(3.3.3)

Proof. (3.4.2)

holds

It is clear

= (3.3.5),

that

P , so

(5.1.1)

= (5.1.3)

If

height

height Qh-I = h-i (5.1.3)

by

Ph-3

height Qh-2 = h-2)

by

(*): that

(3.3.3)

holds

and

(3.3.3)

Ph

the

and let

(R,M)

(3.4.1)

Qh-I

the

and

P

holds,

so

mcpil h+l

Then,

by

(3.3.3)

and

P

of

such that

I

=

=

(5.1.3)),

depth Qh-i = 2

and

such that

P

Again by of (5.1.3)),

depth Qh-2 = 3

of this give a chain of prime such that height

depth Qh-i =

QI = i

and

q.e.d.

(3.3.5)

(3.3.3) = (*), and

(3.4.1)

that

I

In particular,

holds,

the Depth Conjecture

and that

Ph-2 c Qh-i c Ph = P) " and

= (3.3.5).

i ~ h .

Repetitions

.

= (3.4.2)

be a

.-. c Qh-2 c Qh-I c Ph = P

depth QI = h , so (3.4.1)

clear

height

... c Ph = P c M

Qh-2 E Spec R

height Qh-i = h-i

Along w i t h

(3.3.3)

Qh-I ~ Spec R

(with

(and

ment

such that

(3.4.1)

= (3.3.4)

(0) c PI c

, since

can be replaced

and

that

Ph-2

Ph-2

(0) c QI c

(3.4.1)

and

Pi = i , for

(with

(and

i+l

that

Let

can be replaced

ideals

height P = h

D = R[X](M,X )

h ~ i , then it is clear

Ph-i

(5.1.1) =

for a local do-

such that

p E Spec R

= (3.3.4)

so assume

h > I .

through

for

= (3.4.1)

be as in (3.4.1). assume

statements

depth p = h .

(3.4.2)

and

holds

(3.4), for

if we consider

the state-

D = R[X](M,X ) ; then it is

(*) = (3.4.2)

as in the proof

that

= (3.4.1).

The reason for including chain conjectures (see

P E Spec R

depth P = I , then there exists

p = I

(3.10).

:

(3.4.1) and

(3.3.7),

(14.1))

hold

(3.3.5)

in (3.3)

for local domains

is that certain of the

of the form

and if this can be shown to hold

D = R[X](M,X )

for the W e a k Depth

28 Conjecture

(or for (3.4.1)

lows from

(3.3)

jecture,

or for the Depth Conjecture),

(or (3.4) or the preceding

the Catenary

Chain Conjecture,

paragraph)

then it fol-

that the

H-Con-

and the Normal Chain Conjecture

also hold. (3.5)

REMARK.

(3.5.1)

(3.3.1),

and is false:

nonzero

P ~ Spec D , D/P

(3.5.2) true:

if

f.c.c.,

is either

with

then

Proof. 2, pp.

R

ideals

R > i)

, hence exists

is catenary,

then

statement

is weaker

(R,M) m = 0 R'

Then

satisfies

a mcpil n

the o.h.c.c.

in an integral

D

Therefore

one prime

ideal

is not catenary, D , by

(3.5.2) R

If

satisfies

(A.II.19), There

(3.3.9)

and

the

in [N-6, Example

domain with

and

that,

exactly

D/P

has a

domain of

if

R , then

Hence,

n £ [2,a+l}

is catenary. (by (1.2.1),

if there

, by (1.5.1)

1 , so

ideal

Hence,

for each

since

there is a

D/Q = (D/P)/(Q/P))

mcpil

two

= altitude

for each height one prime

and then

R'

R'

Therefore,

= (A. IO.II).

is catenary

P ~ Q

since

and is

satisfies

is catenary

extension

(A.10.1)

it follows

Q E Spec D , D/Q

< altitude

so

in

, by

D , depth P E If,a]

height

R'

(1.1.7).

D = R[XJ(M,X ) , then

nonzero

R

is a regular

exists a mcpil n

in

than

- one of height one and the other of height R

for all

is catenary.

be the local domain

R]

P

D

and if

n C [i , a = altitude

(1.5.4).

such that,

than

the s.c.c.

Let

(since

is stronger

local domain or is of the form

a local domain,

in the case

the c.c.

statement

is a local domain

a Henselian

satisfies

(3.5.1)

maximal

there

(L,M)

203-205]

satisfies

D

The following

R

L[X](M,X )

if

The following

D

.

has a

However, mcpil

2

(1.5.1) = (1.5.4). R'

satisfies

the s.c.c.,

by

the f.c.c., (A.II.I)

then

R

= (A.II.14)

does, and

by (1.2.4),

(A.II.I)

q.e.d. is another

result

that is closely

related

to (3.3.8)

and

29 that is true, namely: satisfies 3.13].

the s.c.c.,

if

R

for all non-maximal

(This is closely related

(see the next paragraph), (3.3.8), R/P

is a catenary

namely

satisfies

[R-6,

(4.3)]:

the s.c.c.,

if

maximal

ideals

M

in

satisfies

other hand,

z

satisfies

z

in

satisfies

the Catenary Chain Conjecture mentioned

six theorems

of the reverse

is analytiunramified, ring M

A

becomes

for

satis-

in

and

A , "a Noe-

depth z = alti-

in place of f.c.c. such as: 5);

On the (I) the

(2) the H-Conjec-

(3) the equivalence

just above.

of

The concept of

that the reader consider us-

it is applicable.)

to the Chain Conjecture

In each of the theorems,

implications

for all

in this chapter contain some additional

that are intermediate

mal Chain Conjecture.

R

ideals

to Chapter

is a useful one, and I recommend

conjectures

ring

the f.c.c,

to Chapter 6); and,

ing it after proving any theorem where

leads

is analytically

unramified,

leads to an open problem,

ture (see the introduction

then

if, and only if, for each minimal

(see the introduction

The remaining

R

is analytically

(3) same as (2) with s.c.c,

Depth Conjecture

inverting

ring

height M = altitude A"

A , A/z

it sometimes

local domain,

R" ; (2) "a Noetherian

the f.c.c,

to

This sometimes

"a semi-local R/z

it

"height" and "depth" and

is analytically

in

and

[R-4, Corollary

P E Spec R .

if, and only if, for all maximal

A

tude A" ; and,

RM

becomes

ideals

the f.c.c,

therian ring prime ideal

is a catenary

interchange

if, and only if,

all minimal prime fies the f.c.c,

R"

Rp

that is equivalent

(I) "a semi-local

if, and only if,

cally unramified

AM

R

for all nonzero

such as:

then

to (3.3.8), because on "inverting"

"quotient ring" and "factor ring."

to a true result, unramified

P E Spec R

we get a statement

(By "inverting" we mean: interchange

local domain,

and the Nor-

I do not know if any

hold.

The three new statements

in (3.6) are concerned with

and in (3.6.4) we come to the first non-local

GB-domains,

domain conjecture.

In

30 (3.6.4)

attention could be restricted

to the local domain case, by

(7.4.1) ~ (7.4.2), but for historical (3.6.4)

for arbitrary

Noetherian

reasons we prefer

domains.

(Note that (3.6.4)

restricted version of W. Krull's question (3.6) for

R

THEOREM.

i = 1,2,3,4

For the following

The Chain Conjecture

(3.6.2)

If

R

GB-domain

(3.6.3)

The

(3.6.5)

is a

GB-Conjecture

R'

is a

If (3.6.1) R

GB-domain,

R'

holds;

is level,

then

that is, if

the n

R

is a

R

i__ss

GB-domain.

that is, the integral closure

GB-domain.

holds and

satisfies

holds;

i__ss~uasi-local,

GB-Conjectu~e

domain

then

(3.6.i) = (3.6.i+I),

(3.3.2) holds.

The Normal Chain Conjecture

Proof. is level,

statements,

(I.I.I0).

a local domain such that

of a Noetherian

in [Kr, p. 755] - see (2.1).)

is a local domain such that

The Descended

(3.6.4)

is a

:

(3.6.1)

is a

to state

R

(3.3.9) holds.

is a local domain such that

the s.c.c., by (4.1.I) = (4.1.2),

by (A.II.I) = (A.II.13),

and so (3.6.2)

R'

so

R

holds.

It is clear that (3.6.2) = (3.6.3). Assume is a in

(3.6.3)

GB-domain A'

holds and let

A

if, and only if,

, by (A.6.1) = (A.6.5).

not in any other maximal

be a Noetherian

A' M, Now,

ideal in

is, for each maximal

if

A'

c E M'

is the integral closure of the local domain

fore,

to prove that (3.6.4) holds, A'

Therefore

is quasi-local, A'

is a

and then

GB-domeain,

is a

ideal

M'nA

c

that

GB-domain,

A' M'

is

, then

A[C]M,nA[c ]

it may be assumed A

Then

is such that

that lies over

A' M,

and

domain.

A

Thereis local

by (3.6.3).

by (A.6.1) = (A.6.2), hence

(3.6.4)

holds. Finally, satisfies

if (3.6.4) holds and

the f.c.c.,

then

R'

R

is a local domain such that

satisfies

the f.c.c,

and is a

R'

GB-domain,

31 by (3.6.4), R

so

satisfies (3.7)

R'

satisfies

the s.c.c., PROPOSITION.

Conjecture

Assume

main such that and satisfies (A.II.13), (11.1.4),

R'

holds and let

is level.

the f.c.c.,

Then

so

R

R

R

be a catenary

is a

satisfies

GB-domain

local do-

(by (3.6.2))

the s.c.c., by (A.II.I) holds, by (ii.i.i)

three more named conjectures,

appeared

in the literature

two of which have

under these names, and the third,

is new in these notes. THEOREM.

i = i,...,5

For the following

statements,

The Chain Conjecture

(3.3.2)

(3.8.2)

The Strong Avoidance

Conjecture

is a saturated NI,...,N h

there exists

chain of prime

are prim e ideals

q E Spec R

height P + i , and (3.8.3)

such that

q E Spec A

(3.8.4) semi-local

in

If

~in$

z c p c N

that is, if

R

such that

A

P c

tins

R

N ~ UN i , then

(so

holds;

P c q c N that is, if

ideals in a Noetherian

in

such that

holds;

P c q c N , q ~ UN i , height q =

The Avoidance Conjecture

are prime ideals

holds.

ideals in a semi-local

depth q = depth N + i

is a saturated chain of prime NI,...,N h

(3.8.i) = (3.8.i+i),

:

(3.8.1)

exists

holds, q.e.d.

then the Catena~x Chain

hence the Catenary Chain Conjecture

(3.8)

and if

Hence

q.e.d.

previously

Q c N

(3.10)].

and so (3.6.5)

If (3.6.2) holds,

(3.6.2)

(3.8) contains

for

by (1.3.4),

[R-16,

(3.3.8) holds.

Proof.

(3.8.2),

the s.c.c., by

such that

P c q c N

~in$

is saturated). P c Q c N A

and if

N ~ UN i , then there

is saturated and

q ~ DN i .

is a maximal chain of prime ideals

R , then there exists

q C Spec R

such that

in a height q

= i = depth q . (3.8.5)

The Upper Conjecture

holds;

that is,

if

(R,M)

is a local

32

domain and if there exists a either

there exists a

(3.8.6)

Nh

mcpil n

The Catenary

Proof.

Assume

P c q c N

in

R

in

D = R [ X ] ( M , X ) , then

o__r_r n = 1 .

Chain Conjecture

(3.8.1)

be as in (3.8.2).

that

mcpil n + 1

holds and let

(3.3.8)

R , P c Q ~ N , and

Then to show the existence

is saturated,

it may clearly be assumed

of

that

P = (0)

q c N

For this case,

is known ring and

A

, depth q = depth N + 1 , and

[Mc-l, T h e o r e m

i] that if

P

, then at most finitely many

height q'/P = 1

lows that

(3.8.2)

holds.

q ~ UN i .

is a prime

such

q ~ DN i ,

it will be

q E Spec R Then,

such

since

it

ideal in a N o e t h e r i a n

q' ~ Spec A

are such that

NI,... ,

q E Spec R

depth q = depth N + 1 , and

shown that there exist infinitely many height one that

holds.

such that

P c q'

height q' > height P + 1 , it fol-

Therefore we assume

to b e g i n w i t h that

P =

(0) Let S ~ R'

S

be a finite

integral

extension

and there exists a one-to-one

mal ideals

in

mal ideal

M

S

and the m a x i m a l

in

S , SM

(SM)' = R ' ( S _ M ) Therefore

S

satisfies

satisfies

correspondence

ideals

satisfies

in

R'

the s.c.c.,

the s.c.c.

the c.c., by

domain of

such that

between

.

Then,

by

(1.3.4),

(by hypothesis (1.3.2),

R

so

and

S

the maxi-

for each maxisince

(1.3.2)).

is catenary,

by

(1.3.5). Let

(0) c Q' ~ N'

that lies over nary).

Let

(0) c Q ~ N , so

q' c N' and

ideal in ideals

in

S

chain of prime ideals

height N' = 2

(since

I = {N" E Spec S ; N"~R E {NI,...,Nh} }

known that there exist that

be a saturated

infinitely many height one

q' ~ U = (U{N"

and

N' ~ M])

{q'MR

; q' ~ N'

of them have height one, by

; N" E I})U(U{M

S

in

is cate-

Assume

it is

q' E Spec S ; M

S

such

is a maximal

Then there are infinitely many pri~ne and

q' ~ U}

, and all but finitely many

[Mc-l, T h e o r e m 7].

depth q'NR = depth q' = depth N' + 1 , since

Also, q'

q'NR c N

is contained

and

in a

33

maximal

ideal

Therefore

M

existence and

S

only if

q' ~_ U[N"

and since

; N" E I}

Therefore

of infinitely many height one

S

is catenary. Moreover,

it remains

q' E Spec S

since

N' ~_ U , let

a minimal prime divisor , ~ UUU 1 , w h e r e

ql'

of

a I E N' alS

, ~ U .

such that

to show the

such that

ql' c N'

Let

q2'

of

a2S

.

a 3 E N'

Repetitions q' c N'

, ~ UUUIUU 2 , w h e r e

of this show the existence

such that

Assume

such that

q2' c N'.

there exists

holds and let

Let

q ~ UNiA S , and this follows

in by

R/z

domain and

= (8.3.1) = (8.3.2). such that

+ 1 = 1 , by Assume

and

R

(3.8.5)

there exists a

holds and let

mcpil n+l > 2 let

D .

Then, by

Q1 c

... c Qn+l = Pn+l

i = I .... ,n~

Qi ~

in

(0) c P1 c

[HMc, Corollary

(so

(R,M)

= (9.1.1) = (9.1.6),

For this,

is a

in

1.5],

D

ideal in

(Qi NR)D)

•

, by

,

(5.2).

height q > height z

and

D

be as in (3.8.5). to prove

D , then there exists a .-- c Pn+l

be a

there exists a

Let

R/z

holds.

it suffices

such that

mcpil 2

there exist infinitely many

a~e such that

(3.8.4)

that

be as in (3.8.4).

z c q , height q/z = 1 = depth q/z

[Mc-l, Theorem i], (3.8.4)

to prove

is saturated and

(0) c p/z c N/z

Therefore

q

be

from (3.8.2).

z c p c N

Since only finitely many of these

R .

NI,...,N h

, so there exists a height one depth one prime

q E Spec R

in

PA S c q c NA S

immediately

holds and let

is a semi-local

(3.8.3)

Then, by

, P c Q c N , and

such that

~ N']

holds.

S = A - N D N I U ' ' ' U N h , so it suffices

q E Spec A S

Ass~une (3.8.3)

A

!

E Spec S ; a 2 E q2

of infinitely many height one

q' ~ U , so (3.8.2)

(3.8.2)

as in (3.8.3).

R/z

U 2 = U[q 2

a2

Then

!

Then

q' c

Then there exists

U 1 = U~q I' E Spec S ; a I E ql' c N'}

there exists a minimal prime divisor Let

q'nR

q' ~ U .

For this,

E N'

N' c M

depth q'AR = depth N' + i = depth N + I .

~_ UN i , since

N'

in

that if mcpil n

mcpil n+l > 2 mcpil n+l

(0) c

height QiNR = i-I , fo~ q = Qn_inR

, so

height q =

in

34

= n-2

and

mcpil

3

in

extension 1.8].

D/qD ~

domain

Therefore,

ideal R/q

qD c Qn-i

in

Then

(0) c Qn_i/qD c Qn/qD c Qn+i/qD

(R/q)[XI(M/q,X)

S

of

R/q

by

(3.8.4),

, by (A.5.5) = (A.5.4), since

so (3.8.5)

holds.

Finally, domain.

(3.8.5)

Then to prove

a = altitude

, there exists

and let

(3.8.6)

holds

Let

be a height

P

R[X](M,X ) , and let

d = depth P , so

fore there exists

mcpil d+l

hypothesis, so (3.8.6)

a

and so holds,

(3.8.6)

The proof of (3.8.1) (3.8.2)

can be replaced

in (8.2) exists"

in (3.8.2)

D - 2

= (3.8.2)

can be replaced

Proposition

3.8]

mcpil n

R/q

in

R , and

be a catenary

two prime

(since

Hence

D

ideal

a > i) so

local

in

There-

~-domain,

q.e.d.

that the "there exists"

Conjecture

D =

d+l = a , by

is an

= (II.I.!),

shows

many." holds,

in

It is shown then the "there

by "there are infinitely

that the Chain Conjecture = (3.8.5)

a

in

in

it may clearly be assumed

d > 0

= (ii.I.ii)

ideal

mcpil 2

(R,M)

by "there are infinitely

The proof of (3.8.4)

one depth one prime

R , by (3.8.5),

that if the Strong Avoidance

out assuming

[HMc,

in

d = altitude by

a

holds

that

R > 1 .

a height

there exists

integral

2 , by [HMc, Theorem

a height one depth one prime

height q = n-2

assume

mcpil

there exists

hence

is a

, so there exists a finite

that has a

S , so there exists

Therefore,

that

"

many"

(with-

holds).

is quite

that the Avoidance

similar

Conjecture

to the proof implies

in

the Upper

Conjecture. In [HMc, implies

Proposition

3.7],

the Upper Conjecture

it was noted without Taut-Level

proof

Conjecture

domains.

(see (3.14.3)

(see

appeared

below),

(3.14.4)

and in [Mc-l, p. 728]

Conjecture

in the literature

implies

the

below).

in (3.9) are concerned

Only one of these new statements

has previously

that the Depth Conjecture

that the Avoidance

(3.9.5)

The four new statements

it was shown

is named, under

with semi-local (3.9.5),

this name.

and it

It should

35 be noted

that

(3.9.2)

(The semi-local

versions

for local comains and

is the semi-local

version of the

of the other previously

are considered

in (3.13),

H-Conjecture.

considered

(4.4),

(7.3),

conjectures (9.3),

(11.2),

(12.2) .) (3.9)

for

THEOREM.

i = i,...,5

For the following

statements,

(3.9.i)

= (3.9.i+i),

:

(3.9.1)

The Strong Avoidance

Conjecture

(3.8.2)

holds.

(3.9.2)

If

R

is an

H-semi-local

domain,

then

R

If

R

is an

H-semi-local

domain,

then,

for each maximal

satisfies

the

f.c .c. (3.9.3) ideal

M

in

R , RM

(3.9.4) mal

ideal

M

(3.9.5) taut-level

is an

If

R

in

R , RM

H-domain.

is a taut semi-local is an

The Taut-Level

semi-local

(3.9.6)

The Catenary

Proof.

then

(3.13.1)

together

that the semi-local

(3.9.2),

level in

semi-local

R .

prove

that

Qn

n = altitude by

(3.8.2)

hence

applied

Also,

n = height

(3.9.1)

(3.3.8) (3.8.2)

and

H-Conjecture

R

satisfies

Conjecture

For this,

(0) c Q1 c

the semi-

it is shown

(3.9.2)

... c Qn

the f.c.c.,

to

Qn-2 c Qn-i c Qn

ideal

in

R

let

R

be a

height Qn-I = n-I

, by

that

(3.9.5)

holds,

so(3.9.1)

be a tautmcpil n

it suffices that

to n > 1 .

' it may be assumed

that contains [McR-2,

Qn-i + depth Qn-i = altitude

= (3.9.2).

implies

in (3.14.2)

= (3.9.5).

is a

holds.

R , and it may clearly be assumed

is the only maximal

depth Qn-I = 1 . Therefore

that

that

R

the f.c.c.

and the Taut-Level

domain and let

Then to prove

Therefore, that

once we know (3.8.2)

for each maxi-

that is, if

satisfies

holds,

Depth Conjecture

imply

holds;

Chain Conjecture

the semi-local

=

R

It is shown in (3.14.1)

local Depth Conjecture

then,

H-domain.

Conjecture

domain,

domain,

Qn-I

, so

Corollary

R , so

(3.9.5)

8]. holds,

36

It is clear that Assume either in

R

(3.9.3).

(3.9.3) holds and let

a = altitude R = I , then

(3.9.3).

R

is an

Therefore,

mal ideal in

R .

is a taut-level an

(3.9.2) =

H-domain.

R

be a taut semi-local domain.

or there are no height one m a x i m a l

H-domain,

so each

assume

a > i

Then, w i t h

b

is an

H-domain, by

and

S = R

as in (3.1),

(3.1.1)

for each m a x i m a l

and

B = R[I/b]

(3.1.5),

ideal

M

in

so

R

is an

H-domain, by

(and

is an

H-domain,

M = i .

Therefore,

Assume and let

and it is clear

M

be a m a x i m a l

let

R

be a taut-level

ideal in

R .

a , to show that

to p r o v e that

RM

show that

is an

RM

RM

is catenary,

R by

H.-domain,

Then,

RM

is an

for some

i

(1.2.3).

height p = i . RM/PRM ~

i < a)

Then

(R/P)M/p

RM/PRM = a-

i

'

for

i = i

for

R/p

since

(3.9.4), Let

.,a

p E Spec R

H-domain, by RM

is an

the f.c.c.,

height

it suffices

it suffices

by (A.9 i) =

"

'

so assume

is taut, by

is an

(A.3.5).

RM

is an

such that

i = l,...,a

hence

(A.9 5). "

H.-domain,l

p c M

and

[McR-2, P r o p o s i t i o n 5], so (3.9.4).

H.-domain,

Therefore,

to

"

Also,

so

RM

altitude is an

l

main, b y (A.3.1) = H.-domain,

if

height M = alti-

For this, '"

H-domain, by

(i ~

(3.9.3)

semi-local domain,

since

satisfies

i

Now

is

(3.9.4) holds.

(3.9.4) holds,

tude R = (say)

that

B

such that

h e i g h t M > i , R M = R [ I / b ] M R [ I / b ] = BMB (3.1.3));

ideals

and there exists a h e i g h t one maxi-

semi-local domain, by Therefore,

RM

If

it follows

that

RM

Hi+l -dois an

(3.9.5) holds.

l

Finally,

assume

(3.9.5) holds and let

gral e x t e n s i o n domain of a local domain (I0.i.i) ~

(10.1.2),

(11.1.6) =

(ii.i.i),

In [R-14,

A

is catenary.

A

R .

be a taut finite inteThen, by h y p o t h e s i s and

Therefore

(3.9.6) holds, by

q.e.d.

(2.14)],

it was noted w i t h o u t proof

that the C a t e n a r y

C h a i n C o n j e c t u r e holds,

if the following c o n d i t i o n

(formally stronger

than

(3.9.4))

holds:

if

R

is a taut semi-local domain,

then,

for each

37

maximal

ideal

M

in

R

,

RM

is taut.

Also,

it was noted w i t h o u t

p r o o f in the r e m a r k p r e c e d i n g P r o p o s i t i o n 9 in [McR-2]

that (3.9.5) =

(3.9.6). By (3.9.2) = whenever holds.

R

(3.9.5),

is an

it follows

that if

H - s e m i - l o c a l domain,

RM

is an

H-domain

then the T a u t - L e v e l C o n j e c t u r e

This was a quite u n e x p e c t e d result to me.

It c e r t a i n l y seems

like the c u r r e n t state of k n o w l e d g e about semi-local domains should be s u f f i c i e n t to show that either However,

even w h e n

a local s u b d o m a i n (see (3.10.2)) e x t e n s i o n of

L

R

=

is an

to hold even w h e n

(by (3.3.7),

(3.3.7) ~

l_~f R

a local sub-domain, If

since

R

H-domain is a special

(3.10.2)).

in (3.10) are very closely related to those in (3.9).

For the following statements,

(3.10.2) = (3.3.7) =

(3.10.2)

RM

and this c o n t i n u e s

THEOREM.

(3.10.1)

H - s e m i - l o c a l d o m a i n that is integral over

L , it is not k n o w n if each

The statements

(3.10)

is an

(3.9.2) holds or that it does not hold.

(3.10.3) ~ is an

then R

a local sub-domain,

R

is an

(3.9.2) =

(3.10.1)

(3.3.8).

H - s e m i - l o c a l d o m a i n that is integral over satisfies

the f.c.c.

H - s e m i - l o c a l d o m a i n that is integral over

then, for each m a x i m a l

ideal

M

in

R , RM

is an

H-domain. (3.10.3)

If

R

is a taut semi-local d o m a i n that is a finite in-

tegral e x t e n s i o n o f a loca___~lsub-domain,

Proof. so assume

R

R . , let

that

Let

(3.3.7) holds,

A

c C M

A = Lo[C]

radical of

~i

is an

It is clear that (3.9.2) = (3.10.1) = let

is integral over a local s u b d o m a i n in

then

A

.

such that , and let

Then

L

l-c

R

be an

H-domain.

(3.10.2) =

(3.3.7),

H - s e m i - l o c a l d o m a i n that

L o , and let

M

be a m a x i m a l

is in all other m a x i m a l

L = Lo + J , where

J

ideal

ideals in

is the J a c o b s o n

is a local d o m a i n and it is readily seen that

is a special e x t e n s i o n of

L .

Also,

L

is an

H-domain,

since

R

38 is integral AM~ A

over

is an

H-domain,

by (A.3.1)

that if

R

L

and

Lo-algebra,

fices

by

to prove

..., height M

M

(3.10.2)

and

Lo

in

AMn A , so

R

and let

A .

AM

Therefore,

that each of the rings

R

L

such

dependent

on

L[b,

(3.1.2)).

to prove

= (3.3.8),

that

such that

is as in the proof of

be a taut finite

Then,

(Ii.i.I)

(1.2.2),

by

(note

Lo-module

is integrally

A

sub-domain. =

a finite b

is an

holds.

is a local domain

, so if

B = R[i/b]

(11.1.6)

over

by (3.3.7),

as in the proof of (3.9.3) = (3.9.4)

holds

that is, by ideals

hence

L c R c L'

sion domain of a local

Therefore,

is integral

is a local domain,

(3.10.3)

it suffices,

H-domain.

then there exists

then

L[b,I/b]

Assume

maximal

RM

= (A.3.3),

= (3.10.3),

(3.9.4),

catenary;

is an

Now

is local and

(3.9.3) =

R

is as in (3.10.3)

is a finite

i/b]

and

H-domain.

(3.10.2)

that

L

that

to prove

satisfies

integral (3.3.8) that

extenholds,

A

the f.c.c.,

is for all

by

(A.9.1)

= (A.9.5),

it suf-

AM

is an

H.-domain,l

for

, and the proof of this is similar

i = i,

to the proof of (3.9.4)

= (3.9.5). Finally, (3.10.3)

(3.3.8) = (3.10.3),

(by (1.2.2)),

The next

= (3.11.2)

for all

H-local

If

R

P £ Spec R H

= (11.1.6)

statements,

(3.3.2) =

(3.11.1)

= (3.3.8):

is a local domain (1.1.14),

and these are concerned

domains.

For .the following

(3.11.3) = (3.10.2)

(3.11.1)

= (ii.i.i)

q.e.d.

and w i t h

THEOREM. =

(3.3.8)

theorem has three new statements,

with Henselizations (3.11)

since

such that

R'

is level,

then,

depth P = depth PNR .

(3.11.2)

If

R

is an

H-local

domain,

then

RH

(3.11.3)

If

R

is an

H-local

domain,

then,

for all maximal

ideals

M'

in

R'

, R' M,

is an

H-d0main.

is an

H-ring.

39 Proof. that

R'

Assume

(3.3.2)

is level.

(4.1.2),

Then

so (3.11.1)

Assume

holds and let R

satisfies

holds, by

(3.11.1)

one prime

(3.2),

ideals

Therefore, let

the s.c.c., by

holds and let

R

be an

that

RH

If there are no height one maximal

is level, by

in

assume

b , S = R'

so

RH

RH

be a local domain such

is an

H-ring,

H-local domain. is an

H-ring,

ideals

in

by

lie over height one prime

B

be as in (3.1),

so

(3.1.4),

proved,

BH

= R'[i/b]

and

is an

, by

(3.1.1).

H-ring.

(3.1.1),

depth z = a}

, b y IN-6,

2, p. 188].

Therefore,

(since RH

RH/I

is an

and

that

(3.11.2)

be a m a x i m a l

ideal in

(3.2). R'

Let

, let

c E M'

R'

that

=

R)

ideal in

R'

b

and

is an

in

B'

H-

, by

and since

H-ring,

[N-6, Ex.

by (A.3.2) = (A.3.1)

the same total quotient I

B'

I = ~[z E Spec R H ;

and since

if

; M'

Ex. 2, p. 188] and

since

(A.3.5),

RH so

(43.18)] is an CH

.

let

c

ring). z

Thus

is a mini-

is a maximal

(3.2)), hence

H-local domain,

to show that

ideal

(3.11.3)

and let

holds,

C = R[C]p z

in

RH

, so

(A.3.1) = (A.3.3).

C' = R' M,

such that

together w i t h

(by hypothesis),

it

height M' = altitude R,

is not in any other maximal

, and let

H-ring

be an

height M' > 1 , so

and its proof

is, by

R

Then,

such that

P = M'nR[c]

(by [N-6,

Now,

in

ideals

depth z E [height M'

holds,

there exists a minimal prime CH

have

(by [N-6,

(since height

and its proof together w i t h

is an

R'

holds.

clearly may be assumed

in

RH/I

R H , then

R'} = [l,a]

Assume

by

(43.18)]

a =

by what has already been

, where

H-ring, by the d e f i n i t i o n of

ideal in

M'

B H = (RH/I)[b]

, then

ideals

Also, by the choice of

(RH/I)[b]

mal prime ideal in

(3.11.2)

Therefore,

If

so assume

B = R[b,I/b]

local domain and there are no height one maximal (3.1.2),

R'

(3.11.1)

there exists a height one maximal

, and

(4.1.1)

(A.II.I) = (A.II.15).

altitude R = 1 , then it is clear a > 1 .

R

ideal Then

(RH/z)[c]

=

[N-6, Ex. 2, p. 188]).

RH/z

is, by

Therefore,

(A.3.1)

since

C

is

40

a dense R' M,

subspace

is a n

(3.11.3)

integrally ideal

clearly Let so

in

.

be a s s u m e d , and

(A.3.!)

over

H-domain,

mains

(by i n t e g r a l

each maximal

M'

ideal

that

such

H-domain

and

A' = L'

s u c h that

=

AN

a > i)

Finally, it r e m a i n s H-local

d o m a i n a n d let

shown

that

that

h e i g h t M' > 1 . ideal

R' M,

Now,

in

M'

is a n

R'

, let

by

AN

is, by

(3.10.2),

and so

and let

b

A = R[c]

Therefore , S = R'

(3.2),

B

that

RM

is

that

L1

is

are

H-dofor

Also, one p r i m e

A

is an

p' c N'

in

h e i g h t N' = a .

(since

L' N,

that For

ideal this,

is an

Hence

H-

that

L1 =

is, b y

R'

this, R'

=

let

(3.3.8), R

be a n

, so it m u s t b e

it m a y c l e a r l y b e a s s u m e d c

is n o t N = M'NA

one maximal A

(3.10.2)

in

, and let

R' M, = (AN)'

assume

, and

so

, and

H-domain.

ideals

such that

if there are no h e i g h t

H-domain,

A

(in (3.10)),

holds.

For

c E M'

it m a y

be a h e i g h t

, so

- 1

be a maximal

Let

is an

(A.3.3).

p

(3.11.3).

H-domain.

R'

=

let

it has a l r e a d y b e e n s h o w n =

and

L - 1 , so it f o l l o w s

(3.10.2)

(3.10.2)

A

(height N > 1 , s i n c e

= h e i g h t N'

be a m a x i -

(3.10.2)

to p r o v e

is an

N'NA = N

that is

so by h y p o t h e s i s ,

there e x i s t p r i m e and

M

and

L

is),

, L' M,

p c N = M~A

a n d so

to s h o w that

maximal

R

L' - 1 = a l t i t u d e

since

R' M,

since

L

this,

domain

a = altitude R > 1 .

(3.3.7) =

it s u f f i c e s

Hence

holds.

H-domain,

that

over

Therefore,

Then

H-domain,

is a n

of

For

A' = L'

p'nA = p

= altitude

is a n

(A.3.3).

(3.2).

h e i g h t N / p ~ h e i g h t N'/p' domain)

Therefore

in

, by

A

RM

is i n t e g r a l

dependence,

h e i g h t M' E [l,a] in

that

is not l o c a l and

.

(3.1!.3)

L ° , and let

be as in the p r o o f

ideal

and so

(3.11)] .

subdomain

R

R

[R-9,

H-semi-local

that

(A.3.1)

by

be a n

to p r o v e

L 1 = AMn A by

R

Then,

L

H-domain, = A.3.3),

let

on a l o c a l

is a local domain,

integral an

R

by

is a n

holds,

dependent

c , A L

CH , C

H-domain,

Assume

mal

of

ideals

integral

is an

in a n y o t h e r , so

(AN)'

R'

, then

in

dependence,

H-domain,

by

has a h e i g h t one m a x i m a l

be as in (3.1).

Then

=

hence

(A.3.1) ideal

B = R[b,i/b]

is

41

an

H-local

domain and there are no height

R'[i/b]

, by

A[b,I/b]

~ B' = R'[I/b]

over

(3.1.2),

A[b,I/b] N,

has already holds,

(3.1.4), and

, where

been proved

and

one maximal

(3.1.1).

Therefore,

R' M,

is an

B' =

B c

is integral

, it follows

H-domain,

in

since

R' M, = R'[I/b]M,R,[I/b ]

N' = M'R'M,NA[b,I/b] that

ideals

from what

and so (3.11.3)

q.e.d.

(3.12)

R~RK.

By (6.1.1)

= (6.1.9),

the

H-Conjecture

implies

(3. II. 2). In (3.9.2)

the semi-local

In (3.13) we make this conjecture, ture.

some comments

of the

concerning

of the Depth Conjecture,

(The semi-local

jectures

version

versions

the semi-local

of the other previously

(7.3),

conjectures

the Strong Avoidance

as follows

(3.13)

from (3.9),

REMARK.

(11.2),

(3.13),

Together with

version

of

considered

to the local versions,

will be shown in (4.4),

jecture,

(9.3),

was given.

and of the W e a k Depth Conjec-

for a local domain are equivalent

lie between

H-Conjecture

and

(12.2).)

Conjecture

(3.14.1),

(3.9.2),

and

consider

These

and the

conas three

H-Con-

(3.3). the following

statements: (3.13.1) that and

that

If

R

is a semi-local

height P > 1 , then there exists

P E Spec R

p E Spec R

is such

such that

p c P

depth p = depth P + 1 . (3.13.2)

If

height

P = h

such that Then (3.13.1)

R

is a semi-local and

height p = 1 it is clear ~ (3.3.3),

and

(3.13.2)

that

to "(3.3.6)

(3.9.3)."

and

for each maximal

P E Spec R

=

= (3.3.6), (3.3.4);

(3.3.3) = (3.3.4).

ideal

exists

is such

p E Spec R

depth p ~ h .

that (3.9.2)

and

domain and

depth P = 1 , then there

as in the proof

then,

domain and

(For, M

if in

R

(3.9.2)

and

Also, is an

R , RM

= (3.9.3),

(3.13.i) (3.9.2)

= (3.13.2),

is equivalent

H-semi-local is an

H-domain,

domain, by

(3.9.3),

42 so R

RM

satisfies

is level

isfies

(since

R

the f.c.c.,

by

(3.3.3)

= (3.3.4)

even been able

between

Moreover,

(3.13.1)

The following

(3.13.1)

H-domain),

it was

seen that so

R

sat-

shown in (3.3) case

that

I have not

(3.9.2). gives

four more

implications

that have already been considered. statements

hold,

hold:

Conjecture

(3.13.1)

If the Taut-Level

(3.8.2)

holds,

then

holds.

Conjecture

(3.9.5)

and the semi-local

then the semi-local

H-Conjecture

holds,

(3.14.3)

[HMc, Proposition

3.7].

then the Upper Conjecture

(3.14.4)

[Mc-l, p. 728].

then the Taut-Level

Proof.

(3.14.1)

in a semi-local

such that

follows

from

R

(3.8.2)

(3.8.2)

and

(3.9.5)

Conjecture

height height

(3.8.3)

holds.

holds and let

that there exists

(3.3.3)

holds.

If the Avoidance

such that

q c q' c P

If the Depth Conjegture

(3.8.5)

Conjecture

Assume

domain

Spec R

P

P > 1 .

Let

P/q = 2 .

p ~ Spec R

be a prime

ideal

q , q' E

Then it readily such that

p c P

depth p = depth P + 1 , so (3.8.2) = (3.13.1). (3.14.2)

semi-local

Assume

domain.

(3.9.5)

Suppose

P = h , depth P = d , and H-domain, (so

so by

height

there

R

such that

exists

hold and let P E Spec R

Then

there exists

and

height

(3.13.1)

h+d < a

(3.13.1),

P1 -<- h-l)

and

ideals

Pi+l -~ height

R

.

R

Hheight

is an

such that

Repetitions

Pk-I c

be an

such that

h > 1 , since

P! ~ Spec R

depth P1 = d+l

that there exists a chain of prime in

=

in this chapter

Depth Conjecture

Depth Conjecture (3.9.2)

and is an

If the Strong Avoidance

(3.14.2)

and it is readily

but for the semi-local

that

THEOREM.

the semi-local

and

= (3.3.6),

theorem

(3.14.1)

(3.3.6);

(1.2.3).)

the named conjectures

(3.14)

holds,

by

is semi-local

to prove

The final

holds,

the f.c.c.,

P1 c P

of this show

''' c P1 c P (k--< h)

Pi - 1 , depth

Pi+l

= depth Pi

43

+ i = d+i+l d+k-i

, and

height Pk-i = I .

Therefore

(3.9.5),

assume PI c

Assume

there exists a

R

, so

is taut-level,

holds,

let

mcpil n+l > 2

in

that

.

Then, by

height p E [0,I}

in

Proposition ideal

q

mcpil

in

R .

.

If

in

R

says

such that

(3.14.4) was essentially

shows

(3.9.2),

since

Qn-i

the f.c.c.,

be a local domain,

D = R [ X ] ( M , X ) , say

and

(0) c

(see (5.4)),

depth PI = n .

i = n-i if

3.6]

and

Let

p =

PI = pD

, so

, and so there exists a

height p = 0 , then

[Mc-l,

that there exists a height one prime

holds, by proved

(3.8.3)

that it may be assumed

that contains

(R,M)

depth q = n-!

R , and so (3.8.5)

(3.9.1) =

satisfies

height p = i , then

On the other hand,

i (iii) = (i)]

R

[HMc, Proposition

height PI = i

depth p = depth pD - i = depth PI mcpil n

so

, and this

holds.

(3.3.3)

''' c Pn+l = (M,X)D

it may be assumed PINR

Thus

and so (3.9.2)

(3.14.3)

depth Pk-i =

a-I = depth Pk-I = d+k-I ~ d+h-i < a-i

is a contradiction. by

In particular,

(9.1.1) ~ (9.1.6).

in the last half of the proof of

applied

that

Qn

, so again there exists a

to

Qn-2 c Qn-i c Qn

is the only maximal

also

ideal in

R

' q.e.d.

The proof of (3.14.3)

is the same as that in [HMc, Proposition

3.7]° A diagram of the implications have been proved gram,

in this chapter

between

is given on the next page.

the numbers under a conjecture

the numbers

on the lines b e t w e e n

implication

is proved.

the named conjectures

indicate w h e r e

the conjectures

that

In the dia-

it is stated,

indicate where

and

the

b-t

O

to,3 • r-~-I C'3 ~ v

E

o0

E

~

M •

•

Mm

~

M rn

~,,ol

CHAPTER 4 THE CHAIN CONJECTURE The main result in this chapter,

(4.1), gives nine statements that

are equivalent to the Chain Conjecture.

But, before giving these

equivalences, a brief history of this conjecture will be given. Since M. Nagata's example showing that the answer to the chain problem of prime ideals is no was not integrally closed (see the introduction to Chapter 2 and (B.I)), a natural follow-up question is if every integrally closed local domain is catenary, and the Chain Conjecture is a generalization of this question (since, by (4.5), the Chain Conjecture is equivalent to:

the integral closure of a local domain

is catenary). This conjecture was (essentially) given by M. Nagata in 1956, in [N-3, Problem 3", p. 62].

In IN-3, Problems 3 and 3', p. 62], two

equivalences of the conjecture were given (see (4.1.1) = (4.1.2) (4.1.3)), and, as mentioned in (2.3), it was indicated in [N-5] this conjecture holds, if the answer to (2.3)(*) is yes, as was noted in (2.3).)

that

u

and

H - 1 . is an

u

I/u

(But the answer

Also, in [N-3, Problem 4, p. 62], the

following result was mentioned without proof: local domain and

is no.

if

(H,M)

is a Henselian

is an element in the quotient field of are not in

that

H' , then

H

such

altitude H[U]MH[u ] = altitude

This is equivalent to saying that every Henselian local domain

H-domain, by (A.3.1) ~ (A.3.7) (and (A.2)), and this, in turn,

is equivalent to saying that the Chain Conjecture holds, by (4.1.1) (4.1.4). Next, A. Grothendieck asked the following question in 1965, in [G-2, p. 103]: domain

if in every finite integral extension domain of a local

R , all maximal ideals have the same height,

isfy the s.c.c.?

then does

R

sat-

This is equivalent to the Chain Conjecture, as follows

from (4.1.1) = (4.1.2).

Also, in 1967, in [G-3, (18.9.6)(ii)], he asked

46 if every Henselian local domain is quasi-unmixed;

and this is equiva-

lent to the Chain Conjecture, by (4.1.1) = (4.1.2) and (A.II.I) (A.II.4). Then, at the end of his 1972 paper weaker question mentioned above:

[K], I. Kaplansky asked the

is an integrally closed Noetherian

domain necessarily catenary? In 1973, in JR-If, (2.20)], I showed that the Chain Conjecture holds for level Noetherian Hilbert domains, is satisfied:

if the following condition

if

D

is a level Noetherian Hilbert domain,

is level, fo~ all

b

in the quotient field of

D .

then

D[b]

But this condition

does not hold, as follows from K. Fujita's result mentioned in (2.8). Finall~ the Chain Conjecture does hold for a large class of local domains

(the local domains that satisfy the s.c.c.,

(for example, com-

plete local domains, regular local rings, and homomorphic

images of

Macaulay rings IN-6, (34.8)]), by (A.II.I) = (A.II.7)), but it is known that it does not hold for at least some quasi-local domains.

Namely,

J. Sally showed in 1970, in [S], that there exists an integrally closed quasi-local domain that is not catenary, and I. Kaplansky's example mentioned in (2.1) also showed this. In (4.1), a number of equivalences of the Chain Conjecture are given, and among these are that certain local domains are either domains or have a

mcpil n

or satisfy the f.c.c, or the s.c.c.

HNow

there are a great many ways of saying that a local domain satisfies one of these conditions

(see (A.3), (A.5),

(A.9), and (A.II)), and

each of these ways gives rise to a formally different equivalence of the Chain Conjecture.

Since some of the characterizations of a number

of the other conjectures considered in Chapter 3 also involve these three conditions, as well as other conditions

(such as

C i , GB ,

o.h.c.c., etc.), it was decided to list in Appendix A some of the equivalences of these four conditions

(and also of:

an

Hi-local ring (A.3);

47 a

Ci-local

ring

(A.4);

a

taut semi-local

domain

a corresponding

reduction

in Chapters Thus, w i t h

(A.8);

equivalences

(A.5),

(A.9),

for the reader's

THEOREM.

(4.1.1)

The following

(4.1.2)

If

m a i n such that

satisfies R

R'

in mind, we now given nine

(We restate

the Chain Conjecture

This will also be done for 5 - 13.)

statements (3.3.2)

are equivalent:

holds:

the integral

is either

a Henselian

is level,

then

R

local domain or a local do-

satisfies

the s.c.c.

is as in (4.1.2),

then

R

satisfies

(4.1.4)

If

R

is as in (4.1.2),

then

R

is an

(4.1.5)

If

R

is as in (4.1.2),

then

R

(4.1.6)

If

R

is a local domain and there exists a

in

ideal

in

(4.1.8)

If

R

If

R

is such that

for all maximal (4.1.9) H-domain (4.1.10) in in

, then there exists

H-domain.

(1.1.14)

a

the f.c.c.

is an

H-ring.

depth n

height n

maximal

R' is a local

R , then there exists

u E F

R

closure

the c.c.

R

(4.1.7)

ideal

that are given

If

prime

a

and then make

(4.1.3)

minimal

(A.7);

and the other conjectures.

(A.II)

in Chapters

ring

(A.10)),

of equivalences

convenience.

The Chain Conjecture

of a local domain

n+l

and

of the Chain Conjecture.

(4.1)

an

a taut local

the o.h.c.c.

in the number

the other named conjectures

ideal

and,

(A.6);

4 - 13 of the Chain Conjecture (A.3),

in (4.1.1)

GB-ring

a

R

or an If

height n

maximal

altitude M'

R[u] < altitude

in

R'

such that

is as in (4.1.2),

then

in

a

mcpil n

i__nn

R' field

R , then

F

and

i/u E M'R'M,

height M' = altitude D = R[X](M,X )

, R .

is either

H2-domain. (R,M)

is a local domain and there exists

D = R[X](M,X ) , then there exists D

ideal

is a local domain with quotient

ideals

If

domain and there exists

that contains

a height

a monic polynomial.

one depth

a

mcpil n

prim e

48

Proof. that

R'

hence R' by

Assume

(4.1.1)

is level.

R'

(If

is level.)

satisfies (1.3.4),

holds and let

R

is Henselian,

Then

the s.c.c.,

R

R'

be a local domain such

then

R'

is quasi-local,

is level and satisfies

by (1.3.3),

hence

R

the c.c.,

satisfies

so

the s.c.c.,

and so (4.1.1) = (4.1.2).

(4.1.2) = (4.1.3),

by

(1.3.5),

and it is clear

that

~4.1.3) =

(4.1.4). Assume

that (4.1.4)

is a Henselian plies

local domain,

that (4.1.I) (4.1.2) =

Namely,

that

, then

let

depth z > 1

holds,

exists a

(4.1.1)

depth n

a

(1.5.1),

so

and

that this im-

(4.1.5) = (4.1.2),

R > 1 .

If

z

ideals

in

R'

ideal

p

R

(4.1.5).

3.5],

since

, by hypothesis).

such that

in

Then it may

is a minimal prime

(by [R-2, Proposition

holds,

let

R

z c p

and

Thus,

height

[Mc-l, T h e o r e m i], there

such that Therefore

z c p , hence

R

minimal prime

A[R']

A

be a local domain, ideal in

of

R

is an integral

mcpil n , by the Going Up Theorem.

the c.c.,

R

is quasi-unmixed,

by (A.II.4) = (A.II.I).

integral extension domain =

(2.4)]

be as in (4.1.2).

p E Spec R*

depth z = depth p + 1 = a , by

Assume

H-domain w h e n e v e r

[R-6,

height > height z + 1 = 1 , by

exists a height one prime

so (4.1.2)

R

a = altitude

since only finitely many have

is an

and it is known

there are no height one maximal

p/z = 1

R

(4.1.5), by (A. II.I) = (A.II.16),

clearly be assumed R

Then

holds.

as will now be shown.

ideal in

holds.

there exists a

height

n

R*

and assume

Then there exists an

that has a

mcpil n , by

extension domain of Therefore, maximal

there

since

ideal in

R' R'

R'

(1.5.2) that has satisfies

, so (4.1.6)

holds. (4.1.6) (4.1.6) =

implies

that a Henselian

(4.1.2), by

Assume

(4.1.6)

(11.1.4) =

holds,

let

local domain

is quasi-unmixed,

so

(Ii.i.i). R

be a local domain,

and assume

there

49

exists a

mcpil n

ideal in

R

n

maximal

in

, by

Then there exists a

(1.5.1) = (1.5.2).

ideal in

(4.1.7)

R .

R'

, by

Therefore

(4.1.6),

implies all H e n s e l i a n

depth n

minimal prime

there exists a

height

and so (4.1.7) holds.

local domains

satisfy

the f.c.c.,

so (4.1.7) = (4.1.3). Assume suppose that

(4.1.1)

that

holds,

let

I/u ~ M'R' M,

R , F , and

, for some m a x i m a l

height M' = altitude R .

so there exists a maximal Now,

there is a finite

Then

ideal

N

integral

in

ideals

c.c.

(by hypothesis),

altitude

formula.

R'

.

Thus

height N + trd

+ trd R'[u]/R'

That is, since

(= altitude R)

Therefore

N

Assume

(4.1.8)

does not hold. u E F

u

and

i/u

contradicts

(4.1.8).

It is known

(4.1.2) = Assume

since

R'

that

R'

= M'

such that between

the

satisfies satisfies

the the

= height M'

height N = height M' ~ altitude

holds.

(for

local domain

R'

and

i = i

(R,M)

and

altitude R[u] <

and see (A.2)),

and this

(4.1.8) = (4.1.4). that

Ci_l-domain.

D

in (4.1.9)

Therefore,

is an

Hi-domain

(4.1.9) = (4.1.4)

(4.1.9). holds and let

mcpil n+l

minimal prime

there exists a c E M'

is a

(4.1.6)

there exists a depth n

R

such

that the Henselian part of (4.1.4)

are not in

(A.4.1) = (A.4.7)

if, and only if, and

Therefore

R

= altitude R'[u]

so (4.1.8)

altitude R , by (A.3.1) ~ (A.3.9)

of

(R'[u]/N)/(R'/M')

Then there exists a H e n s e l i a n

such that

A

is maximal,

holds and suppose

R'

NNR'

correspondence

says

altitude R[u]

R , and this is a contradiction,

in

such that

Therefore,

[R-2, T h e o r e m 3.10]

M'

and

, by [ZS-2, p. 325],

extension domain

maximal

and in

ideal

R'[u]

and such that there is a one-to-one A

be as in (4.1.8),

i ~ M'R'[u]

A ~ R'

in

u

c

be a local domain such that

D = R[X](M,X )

ideal i n R *

height n

such that

in

(R,M)

maximal

Then there exists a

by (i 5.4) = (1.5.2) ideal

M'

in

is not in any other m a x i m a l

R'

.

so, by

(4.1.6)

Therefore,

with

ideal in

R'

B =

50

R[C]M,AR[c ]

has

altitude

= n .

there exists a height one depth K

contains

a monic polynomial

Finally, assume

(4.1.10)

the~e exists a

exists a

R

exists a

in

height n holds,

(13.7)].

tially

Also,

thequotient

(V/N)/(R'/M') (4.2.2)

ideal in

R

, by

and

Then there

so it follows

extension domain

from A =

Therefore

there

JR-4, Lemma 2.9], hence

the proof of (4.1.1) = (4.1.6) =

Two additional

(2.4)]

(4.1.2)

to prove

that

(13.6) is essen-

(4.1.6)

is

R

over

R

M'

of the Chain Con-

in the integral closure

(discrete v a l u a t i o n

such that:

= height M'

R' ~ V ; NNR'

i)

, where

ring) = M'

R ; is of the first kind , IN-13,

(2.19.1)];

is a local domain such that

field of

JR-21,

ideal

R , every DVR

field of

If

characterizations

namely:

; (V,N) C ~]

the quotient

~ = {(V,N)

; V

R'

(V,N)

; and,

in V

is

(that is, trd

and,

R'

is level,

is a v a l u a t i o n

R , R ~ V , NNR = M , and

then ring in

altitude V = alti-

(13.6)].

(It is shown in [R-21]

.)

(4.1.6) = (4.1.10).

height = n .

R'

such that

be a local domain,

integral

ideal of

D

(4.1.1) = (4.1.4) was given in [R-21,

integral over a locality

R'

Thus

minimal prime

ideal in

For each maximal

of a local domain

(4.2.2))

in

D , by (1.5.2) = (1.5.4),

maximal

REMARK.

(4.2.1)

tude R}

R

K

,

q.e.d.

jecture are known,

R' = N[V

ideal

= (M,c)R[c]

to the Chain Conjecture.

(4.2)

of

let

the same as that given in [RMc,

equivalent

M'NR[c]

B = D/K .

holds,

that has a m a x i m a l

A n o t h e r proof that and

and

since

prime

that there exists a principal

of

(4.1.6)

n

depth n

mcpil n + 1

(4.1.10) R[c]

assume

Hence,

has many properties

that

A[V ; (V,N) E ~}

that are analogous

(with

~

as in

to the known properties

51

(4.3) contains (4.3) exists a

REMARK. depth n

depth n

some comments (4.3.1)

ideal in holds

(4.3.3)

(3.4.2)

together w i t h

that

RH

(4.3.1)

is a dense is given

(4.3.3)

It is known

(4.3.3)

follows

(4.3.4) exists a

Let

(R,M) in

height P = n

be a m i n i m a l prime Then,

(z,X)D

since

R

depth zNR + i) The final

and

(A.5.4)

that if

(R,M)

ideal in

if, and only

D .

Therefore

= (4.1.1). local ring and assume Then there exists

D

is complete, ideal

such that

(A.5.1) = z c P

depth z = n+l (since

(since

and .

p/XD = znR

D/p = R/(znR)

there P E Spec D

(A.5.4).

Let

height P/z

Therefore

p =

is a minimal and

depth z =

, q.e.d. theorem in this chapter

shows

that this conjecture

just as well have b e e n stated for an arbitrary N o e t h e r i a n

(4.4)

is a local

D = R[X](M,X )

depth P = 1 , by

depth p = n

local rings.

3.5].

in

D = R[X](M,X ) .

is a height one prime ideal),

for complete

depth one prime

in

to (3.3.2).

R*

be a complete

ideal

is equivalent

holds

mcpil n+l

and

that some such height

[N-6, Ex. 2, p. 188] and the fact

(A.5.1) =

from (4.1.10)

mcpil n+l

such that

of

height n

if there

, then there exists a

the c o n d i t i o n

in [R-2, P r o p o s i t i o n

then there exists a

if, there exists a

= n .

from

subspace

to:

n = i .

in (4.1.10)

follows

(4.3.2)

domain,

for

a monic p o l y n o m i a l

The c o n d i t i o n

Proof.

R

in (4.1).

RH

(4.1.6)

ideal contains

is equivalent

ideal in

(4.3.2)

(4.3.4)

prime

(4.1.6)

m i n i m a l prime

minimal prime

one prime

z

on the statements

THEOREM.

The following

statements

could

domain.

are equivalent:

(4.4.1)

The Chain Conjecture

(4.1.1)

(4.4.2)

The integral closure

of a N o e t h e r i a n

holds. domain is catenary.

52 (4.4.3)

The integral closure of a Noetherian domain satisfies the

c.c. Proof. Assume is level. so =

R

(4.4.3) = (4.4.2), by (1.3.5). (4.4.2) holds and let Then

R'

R

be a local domain such that

R'

satisfies the f.c.c., by hypothesis and (1.2.3),

satisfies the f.c.c., by (1.2.4), hence (4.4.1) holds, by (4.1.3)

(4.1.1).

Finally, assume (4.4.1) holds and let Then to prove that to prove that if the c.c.

A' M'

For this,

A

be a Noetherian domain.

satisfies the c.c., it suffices, by (1.3.2), is a maximal ideal in

A' M,

A' , then

A' M,

satisfies

is the integral closure of a local domain

(as in the proof of (3.6.3) ~ (3.6.4)), so

A' M,

satisfies the c.c.,

by hypothesis, and so (4.4.3) holds, q.e.d. (4.5) to (*):

REMARK.

The Chain Conjecture (4.1.1) is also equivalent

the integral closure of a local domain is catenary.

Proof.

(4.1.1) = (*), by (1.3.5), and (*) = (4.1.1), by the proof

of (4.4.2) = (4.4.1), q.e.d.

CHAPTER 5 THE DEPTH CONJECTURE AND THE W E A K DEPTH CONJECTURE The Depth Conjecture cussion between I]:

if

R

arose from trying to "invert"

(3.5) and (3.6))

the following

is a local domain and

P £ Spec R

result

(see the dis-

[Mc-l, Theorem

is such that

P > 1 , then all but finitely many

Q E Spec R

and

height Q = height P + 1 .

height Q/P = 1

are such that

(B.5.8) we give an example the Depth Conjecture

arose from a natural

mented on in [P, p. 72].)

in 1972,

in a preliminary version

(This fact was briefly com-

S. McAdam mentioned

in [HMc, Proposition

the Depth Conjecture

in

implies

(3.4.1)

Related 2.4(2)],

implies

it was noted in [R-12] the

and

A

is a

j = i , i + l,...,a

.

Di-domain

Also,

Houston showed that a local domain is a

domain

A

in JR-4, Corollary

is such that

(that is, for all

in 1976, (R,M)

A

is a

alti-

P E Spec A D.-domain, J

in [Hou-l, Theorem 12], E. is a

D.-domain

if

and only

Di+l-domain.

the Depth Conjecture

for all catenary

that the formally

in 1971,

depth P = i , height P = a - i) , then

Finally,

in

H-Conjecture.

I showed that if a Noetherian

R[X](M,X )

Then,

is new in these notes, but as mentioned

to these two depth conjectures,

tude A = a < ~ such that

the one in (B.5.7).

the Upper Conjecture.

after the proof of (3.3), stronger

than)

3.7], he and E. Houston showed that this

The Weak Depth Conjecture

if,

so

in [Mc-l, Remark, p. 720], and he then gave an example similar

conjecture

for

In

follow-up question.

H-Conjecture.

to (but somewhat more complicated 1975,

and

and therein it was shown that the Chain Conjecture =

the Depth Conjecture = the

1974,

P c Q

to show that the result does not invert,

I stated the Depth Conjecture of these notes,

such that

depth

local domains,

and the W e a k Depth Conjecture

hold

but they do not hold for all Noetherian

54 domains

or quasi-local

domains.

For example,

in [Fu-l,

p. 482], K. Fujita gave an example of a N o e t h e r i a n nitely many maximal And,

in (14.7),

ideals

that does not satisfy

Proposition,

domain w i t h

the Depth Conjecture.

it is shown that b o t h the Depth Conjecture

Depth Conjecture

infi-

fail to hold for some n o n - N o e t h e r i a n

and the W e a k

quasi-local

do-

mains. (5.1) gives (5.1)

two equivalences

THEOREM.

(5.1.1)

The following

The Depth Conjecture

domain and

P ~ Spec R

p E Spec R

such that

(5.1.2)

If

P

then there exists P+I

of the Depth Conjecture. statements (3.3.3)

is such that p c P

and

is a height p E Spec R

are equivalent:

holds:

if

R

is a local

height P > 1 , then there exists

depth p = depth P + 1 . two p~ime

such that

ideal in a local domain p c P

and

R ,

depth p = depth

. (5.1.3)

i_~n R

If

such that

nitely many

R

is a local ring,

I c P

and

p E Spec R

P E Spec R , and

I

is an ideal

height P/I > 1 , then there exist

such that

I c p c P

and

infi-

depth p = depth P

+I

Proof. (5.1.1) = I

(5.1.3) = (5.1.1)

(5.1.2),

so assume

be as in (5.1.3).

P/q = 2 p c P

Then, by and

Let

(with

that

(5.1.2)

q E Spec R

(5.1.2),

p , so (5.1.3)

, and it is clear

holds and let

such that

there exists

depth p = depth P + I .

infinitely many such

I = (0))

Therefore, holds,

R , P , and

I ~ q c P

p ~ Spec R by

and

such that

(5.2),

enough to be explicitly

stated.

q c

there exist

(3.8.4)

of (5.1), and it will be used in the proof of (5.3) and in Chapter quite readily from two results

height

q.e.d.

(5.2) has already been used in the proof of (3.8.3) =

It follows

that

and 8.

in [Mc-l], but is important

55

(5.2) ideals

PROPOSITION.

i__qn! semi-local

there exist

Let

tins

s

infinitely many

= height P + I , and

P c Q c N

be a saturated chain of prime

such that

q £ Spec S

depth Q = depth N + i . such that

depth q = depth N + I

(so

Then

P c q c N , height q P c q c N

is satu-

rated). Proof. S/P N/P

P/P c Q/p c N/P

, so there exist in

S/P

tion 2]. P c q c N follows

and shows

in

q/P c

[Mc-l,

q E Spec S

Proposi-

such that

depth q = depth N + i , so the conclusion

since only finitely many of these

> height P + i , by

stronger

and

ideals

ideals

depth q/P = depth N/P + i , by

there exist infinitely many

is saturated

The final

chain of prime

infinitely many height one prime

such that

Therefore

is a saturated

[Mc-l, T h e o r e m

q

are such that

height q

i], q.e.d.

theorem in this chapter

sharpens

that the W e a k Depth C o n j e c t u r e

[HMc, P r o p o s i t i o n

is equivalent

3.6]

to a considerably

looking result.

(5.3)

THEOREM.

(5.3.1)

The fol!owing

statements

The W e a k Depth C o n j e c t u r e

cal domain and

P ~ Spec R

= i , then there exists

(3.3.4)

is such that

p ~ Spec R

are equivalent: holds:

if

height P = h

such that

R

and

is a lodepth P

height p = I

and

depth p -~ h . (5.3.2)

If

height P = h

and

mcpil n , say and

(R,M)

is a local domain and

P E Spec R

depth P = d , then there exist

(0) c PI c

-.. c Pn = M

depth Pi = n - i , for

, i__nn R

...,n - i , there are i n f i n i t e l y many choices

n =< h + d

such that

i = l,...,n - i .

In fact, for

is such that

Pi

and a

height Pi = i for each

with

i = i,

the remain-

der of the chain unchanged. Proof. and let

It is clear

(R,M)

and

P

that

(5.3.2) = (5.3.1),

be as in (5.3.2).

so assume

(5.3.1)

Then there exists

holds

Q C Spec R

58

such that

height Q = h ÷ d - 1

d = 0

d = 1 , and

or

so there e x i s t and

Pn = M

i = l,...,n i - 1 . Pi c

it f o l l o w s

n ~ h + d

d e p t h P1 = n - 1 , by

Pn-i c

.

Assume

Pi+l

(5.4) jecture was

and

for

(5.3.1). n

[HMc,

and

Pi = i

Depth Conjecture

(0) c

height

holds.

P1 c

P1 c

depth

(5.2)

result

domain ... c

R

... c

Pi = n - i , for height to

Pi-i = Pi-I c Pi-i

f r o m this, q.e.d.

, there

P / P i = n - i , for

P1 = 1

s u c h that

for the D e p t h Con-

given a

Pn = P

,

d e p t h P i = d e p t h Pi+l

to (5.3)

3.6]:

if

d > i)

P2 c

applied

P i E Spec R

follows

Proposition

in a N o e t h e r i a n

height

by

if

height

(I ~ i ~ n - i)

Then,

readily

given

ideals

' so

is c l e a r

(A.5.4),

(0) = Po c

P1 i

=

(this

s u c h that

h e i g h t P i = i , and

The analogous

c h a i n of p r i m e

Let

infinitely many

The conclusion

P

(A.5.1)

through

REMARK.

ideal

from

P1 C Spec R

i = 1 .)

is s a t u r a t e d ,

in

depth Q = 1

that for some

' there exist

+ 1 = n - i .

a prime

mcpil

(This h o l d s

Pi+l

c Pi c

be a

and

in

mcpil n

exists R

up to

a maximal

s u c h that

i = l,...,n

- 1 , if the

CHAPTER 6 THE This conjecture,

H-CONJECTURE

like the Depth Conjecture, arose from trying to

"inve~t" a known result, namely [R-5, Remark 2.6(i)]: cal domain and all depth one altitude R - I , then

R

P ~ Spec R

if

are such that

R

is a lo-

height P =

is catenary.

The conjecture was first stated in 1971, in [R-4, p. 1096] also in [R-4, Remark 2.5(b)]

(and

in a different form), and it was pointed

out in 1972, in [R-6, p. 225], that the reason

"H"

it is hoped that all Henselian local domains are

was used is that

H-domains (that is,

by (4.1.1) = (4.1.4), that the Chain Conjecture holds). As mentioned after the proof of (3.3), two different proofs that the

H-Conjecture implies the Catenary Chain Conjecture were given in

[R-5, Remark 3.5(ii)] and JR-6, (4.5)], and in [R-12, p. 130] it was shown that (3.4.1) implies the

H-Conjecture.

Also, in [R-21, (13.8)

and (13.10)], it was shown that the

H-Conjecture holds, if, for all

local domains

is level,

(R,M)

such that

R'

~[V ; (V,N) E ~}

is as in (4.2.2)) either satisfies the f.c.c, or is taut.

(where

Moreover,

it was noted without proof in the comment following (5.7) in [R-21] that (6.1.5) is equivalent to this conjecture. Finally,

it is known that this conjecture does not hold for some

nonlocal domains. there exist

For example, W. Heinzer's example,

(14.6), shows that

H-quasi-local domains that are not carenary.

Also, K.

Fujita in [Fu-l, Proposition, p. 484] gave a different example of an H-quasi-local domain that is not catenary, and in [Fu-l, Proposition, p. 482] he gave an example of an

H-Noetherian domain that is not taut.

And, another example of a noncatenary

H-quasi-local domain is given in

(14.7). The main result in this chapter,

(6.1), gives eleven equivalences

58 of the

H-Conjecture.

Among these are that certain local rings are

Hi-local rings, or are taut, or satisfy the f.c.c., the o.h.c.c., or the s.c.c., so we remind the reader that a number of equivalences of these conditions are listed in (A.3) and (A.7) - (A.II). The list of equivalences of the

H-Conjecture given in (6.1) is

very similar to the list of equivalences of the Catenary Chain Conjecture given in (ii.i).

Since the lists are given in approximately the

same order, the reader should have no difficulty in matching the corresponding characterizations.

However, two comments should be made.

First, it follows immediately from (A.10.1) ~ (A.10.12) that (6.1.4) (6.1.9).

The reason for including this equivalence is that in compar-

ing the conclusions of the corresponding statements in (6.1) and (II.i), it seems like it should be possible to replace the conclusion of (6.1.9) with:

RH

is an

H-ring; but I have been unable to prove this.

(Con-

cerning this, see (3.11) and (3.12).) The second comment is that in one case, the corresponding statements have quite different numbers.

Name-

ly, (6.1.7) corresponds to (11.1.12), since, by (A.4.1) = (A.4.7), the conclusion of (11.1.12) all height one

is equivalent to:

R/p

is a

Co-domain , for

p ¢ Spec R ; and this has been replaced by the weaker

conclusion given in (6.1.7).

(If it were not for this weakening of the

conclusion of the corresponding

(11.1.12),

then (6.1.7) would not have

been included, since it follows from (A.3.1) = (A.3.5) that (6.1.6) (6.1.7) .) (6.1) (6.1.1)

THEOREM. The

The following statements are equivalent:

H-Conjecture

(3.3.6)holds:

an

H-local domain is

catenary. (6.1.2)

If

R

is an

H-local domain,

(6.1.3)

If

R

is an

H-local domain, then there exists a taut

integral extension domain of (6.1.4)

If

R

is an

then

R'

is taut.

R . H-local domain , then

R

satisfies the

59

o.h.c.c. (6.1.5) then

R

If

R

satisfies

is an

(6.1.6)

If

R

is an

H-local domain,

then

R

(6.1.7)

If

R

is an

H-local domain,

then

R/p

(6.1.8) sion of

If

R

is an R sMi

If

is level,

is an

~-domain.

is an

H-domain,

p E Spec R .

R , then

(6.1.9)

R

H-local domain and

is c atenary

is an

Rs

(i = 1,2)

H-local domain,

is a special exten-

.

then

RH

is taut.

(6.1.10)

If

R

is an

H-local domain , then

R*

(6.1.11)

If

R

is an

H-local domain,

D = R[X](M,X )

If

(R,M)

then

is an

H-ring. is

~-domain. (6.1.12)

in

M

such that Proof.

height

It is known

is an

H-local domain,

is catenary,

then, for all

(b,c)R = 2 , B = R[c/b](M,c/b ) [McR-2, Propositions

gral extension domain of a local domain R

R'

the s.c.c.

for all height one

an

H-local domain such that

R

b,c

is an

H-domain.

12 and 7] that an inte-

is taut if, and only if,

so (6.1.i) = (6.1.2) and (6.1.3) = (6.1.1), and it is

clear that (6.1.2) = (6.1.3). Since

(6.1.1)

by [R-6, (4.5)],

implies that the Catenary Chain Conjecture holds,

(6.1.1) = (6.1.4).

rather than to (3.3.6) = (3.3.8)

(The reason for referring

is that in proving

to [R-6]

(3.3.6) = (3.3.7)

= (3.3.8), use was made of (6.1.1) = (6.1.4).) (6.1.4) implies that if then

R'

satisfies

R

is an

H-local domain and

the s.s.c., by (1.3.3),

R'

is level,

so (6.1.4) = (6.1.5), by

(1.3.4). Assume level,

(6.1.5) holds and let

R

be an

then (6.1.5) = (6.1.1), by (1.3.5),

level. ideal in

Then R'

altitude R > i

H-local domain. so assume that

R'

If

R' is

is not

and there exists a height one maximal

, by (3.2), so, with

b , S = R' , and

B

as in (3.1),

60 B = R[b,i/b] if

R

is an

is catena~y,

R'[I/b]

by

by (3.1.2),

(1.3.5),

It is clear (A.3.1)

that

(6.1.i)

(6.1.6)

~-domain.

altitude

holds

Then,

R - i

and

Repetitions

catenary,

by (A.9.5)

(6.1.4)

=

then

and

(A.9.1), by

so

= (6.1.9),

(6.1.4)

= (6.1.10),

(A.10.15)

(6.1.12),

= (A.10.1)

if

and only

Also,

B' =

implies

that

and so (6.1.1)

and (6.1.6)

B

holds.

= (6.1.7),

by

assume

be a height

by

A/P* ~- (R/P)IX]

b y (6.1.12),

(A.10.1)

(since

R*

so

R

R

is taut-level,

(6.1.i)

R/p =

R/p

is an

so

R

is

holds.

= (A.10.6) and

so

is

H3-domain , b y (A.3.1)

and

(1.2.1),

and if

height M i E [i , altitude

is taut, and so (6.1.8)

(A.10.1)

by

by (6.1.7),

is an

hence

domain,

R],

= (6.1.3).

= (A.10.12). = (A.10.15), is an

and

(6.1.10)

H-ring only

if

R*

= (6.1.4), is taut

by

, by

(A.10.1)

(6.1.12)

holds,

two prime

ideal

A = R[c/b]

P

= PA

and

, so

let in

R R .

, and let

be an

(6.1.Ii)

Let

H-local b,c

in

N = (M,c/b)A

is a height one prime

=

so

depth P = altitude

domain, P .

Then, P*c

Also,

height N/P* = altitude R - 2 , so (6.1.12)

and

such that

ideal,

height N/P* = depth P + I .

JR-9, Lemma 2.7],

hence

= (A.I0.19),

= (A.3.5).

JR-9, Lemma 2.7],

R

R

H-local

p £ Spec R , altitude

H-domain,

(A.10.1)

Rs

(b,c)R = 2 , let

= altitude

be an

local rings a~e catenary)).

by (A.3.1)

Finally,

R

is catenary

(6.1.4) = (6.1.II),

and

= (6.1.6),

is an

Rs

(3.2),

(since complete

by

so (6.1.5)

of this show that

(6.1.4)

height

(3.1.6).

is catena~y,

Therefore

= (6.1.8),

holds,

by (1.2.2)

P

R

and let

R/p

(A.3.5).

let

(3.2),

is catenary and

for all height one

H2-domain , by (6.1.6).

by

(3.1.4),

and

hence

B

= (A.3.5).

Assume

(6.1.8)

domain and

is level, by (3.1.1)

is catenary,

an

H-local

N

,

height N R - 1 ,

= (6.1.6),

q.e.d.

A different

proof

that

(6.1.1)

= (6.1.6)

was given

in [P,

(8.7)].

61

For (6.2), we briefly recall one definition. an ideal in a Noetherian ring with respect to where

t

I

A , then the Rees ring

is the graded subring

is an indeterminate and

(6.2)

REMARK.

Namely,

if

~(A,I)

~(A,I) = A[tl,u]

I

is

of

A

of

A[t,u],

u = I/t .

Two additional equivalences of the

H-Conjecture

should be mentioned, namely: (6.2.1) If a

Ci+l-domain;

(R,M)

R

is a

is catenary,where

respect to Proof.

Ci-local domain, then

D = R[X](M,X )

is

and,

(6.2.2) If ~N

is a

(b,c)R

Co-local domain,

~ = ~(R,(b,c)R)

and

~

then, for all

is the

b,c

Rees ring of

in R

is the maximal homogeneous ideal in

The equivalence of the

M ,

with ~ .

H-Conjecture and (6.2.1) was given

in JR-9, (3.23)]. The proof of the equivalence of the

H-Conjecture and (6.2.2) will

be omitted, since it involves a number of the special properties of Rees rings. It should be noted from (4.2.1) and (e) in the introduction to Chapter II that the Chain Conjecture and the Catenary Chain Conjecture can be characterized in terms of certain DVR's that dominate A similar characterization of the

R' M, .

H-Conjecture holds, and, in fact,

follows from (6.1.1) = (6.1.4) and (A.10.1) = (A.10.21). It was noted in (3.13) that the semi-local is equivalent to "(3.9.3) and (3.3.6)." tant to know under what circumstances

H-Conjecture,

(3.9.2),

Because of this, it is impor-

(3.9.3) holds.

Now it is shown

in (14.1) and (14.3) that certain of the chain conjectures hold for local domains of the form

R[X](M,X)

(where

and for semi-local domains of the foL-m R[X] S local domain and

S = R[X] - U[(M,X)

; M

(R,M) (where

is a local domain) R

is a semi-

is a maximal ideal in

R}).

Thus it might be thought that (3.9.3) should be easier to verify for

62 R[X]s

- but we show in (6.3)

it is for the general (6.3)

that it is no easier

case.

REMARK.

(3.9.3)

is equivalent

local domain such that

D = R[X] S

local domain,

is an

in

then

DN

and let maximal is an

It is clear

R

be an

ideals

H-domain,

(3.9.3) assume

R'

H-semi-local

local domain,

that

R'

Let

by

(3.1.1)

Bp

is an

as above)

is a semi-

is an

H-semi-

ideal

N

RM

in

is an

Also,

b

M

if

[Hou-2,

Theorem 1.9].

H-domain,

(*) holds

(3.2), and so Therefore,

by

(A.3.17)

(3.9.3)

B

be as in (3.1)

(with

S = R')

(3.1.4).

H-domain,

Therefore,

holds,

, so

ideal in

R[b]

maximal

ideal in

B

L , so

B ~ RM[b,I/b] ~ L , and so

that lies over

is local, by the choice of L = B(M,I_b)B

is an

b)

P

H-domain,

= (A.3.1),

q.e.d.

M

M .

Also,

ideal B

is an

ideals in

Now

B .

(M,I-b)B RM~

(M,I-b)B

in

L'

(3.2), and so

Let

M

b , (M,l-b)R[b]

B'

,

(since

is a

= (say)

RM[b,i/b]

is a maximal

(since RM

be a

is the only

B(M,I_b)B

as already noted.

ideals by

and

in

L = RM[b,I/b ]

H-domain,

there are no height one maximal is an

ideal

Then, by the choice of

that lies over

so

by what has already been shown,

for each maximal

maximal

H-

and so

and that there exists a height one maximal and

D

by (*),

is an

= (A.3.1),

a = altitude R = I , then

R .

(A.3.13)

by

domain and there are no height one m a x i m a l

and

, L'

H-domain,

R , D(M,X)D = R M [ X ] ( M R M , X )

ideal in

B')

that

If there are no height one

is an

maximal

by

R

for each maximal

domain.

R'

domain, by

a > i

H-semi-local

in

S

if

(3.9.3) = (*), so assume

, then

ideal

so

holds.

that

H-semi-local

in

for each maximal

so

(with

to (*):

D .

Proof.

in

for this case than

ideal,

Therefore,

since

there are none is an

H-domain,

CHAPTER 7 THE DESCENDED GB-CONJECTURE AND THE GB-CONJECTURE The Descended

GB-Conjecture

is new in these notes, but questions

closely related to it were asked in 1976, in [R-IO, (3.15) and (3.16)], [R-12,

(4.9)], and JR-19, (7.1.3)].

In JR-10, (3.16)(4)],

that the Chain Conjecture implies this conjecture.

it was noted

Also, the charac-

terization of this conjecture given in (7.2) was proved in [R-16, (4.3)]. In the comment following that with is not a

R

(7.1.3) in [R-19], it was pointed out

a local domain as in [N-6, Example 2, pp. 203-205],

GB-domain, but

was mentioned in (2.9).

R'

is.

(See (B.4.5)).

In both of these cases,

R

Another such example R'

is not quasi-local

(and is not even level), so some condition (such as in ((3.6.2) or (3.6.3)) must be placed on

R'

when considering a conjecture of this

type. The history of the "non-Noetherian" (2.1).

The '~oetherian"

GB-Conjecture

GB-Conjecture was given in (given in (3.6.4)) was essen-

tially asked in 1972, by I. Kaplansky in the introduction

to [K], and

I somewhat more specifically asked it in 1976, in the comment between (2.1) and (2.2) in [R-16].

In (4.2) of this latter paper this conjec-

ture was essentially characterized

as in (7.4.3).

Finally, both of these conjectures hold for local domains that satisfy the s.c.c., by [R-16, (3.10) and (3.1)], but I. Kaplansky's example in [K] shows that there exist integrally closed quasi-local domains for which neither of these conjectures hold. (7.1) gives two characterizations ture.

of the Descended Chain Conjec-

For this result and for (7.4), the reader should refer to (A.6)

for some equivalences (7.1) (7.1.1)

THEOREM.

of the

GB

condition.

The following statements are equivalent:

The Descended GB-Conjecture

(3.6.3) holds:

if

R

is a

64

local domain such that (7.1.2) and if that

If

R

S = R[c]

R'

is ! principal

c 2 + rc ~ R , for some

(7.1.3)

With

ideals

in

Proof.

R

R'

in

R'

that

so it will be omitted. then

R'

lows from the Going Up T h e o r e m

GB-Conjecture

is a

is quasi-local,

that

In [R-16,

is equivalent then,

ideals

to (and easier

if

for all prime

ideals

such in

S

R .

the proof

by

that

that

(A~6.1) = (A.6.2),

GB-domain.

And

it readily fol-

q.e.d.

it was shown that the Descended R

is a local domain such that

ideals

height Q/P > i , the Upper Conjecture and there are no height one maximal

in

(7.1.3) = (7.1.1),

(4.3)]

to:

R

and the p~oof

than)

GB-domain,

holds by the d e f i n i t i o n of a

REMARK.

is ~uasi-local,

GB-domain and adjacent

(7.1.1) = (7.1.2),

(7.4.4) = (7.4.1),

(7.2)

is a

lie over adjacent prime

is similar

(7.1.3)

R'

GB-domain.

R .

as in (7.1.2),

holds,

is a

integral e x t e n s i o n domain of

(7.1.2) = (7.1.1)

so

R

r E R , then a d j a c e n t prime

ideals

It is clear

If (7.1.1)

then

is a local domain such that

lie over adjacent prime

prime

is quasi-local,

P c Q

(3.8.5)

ideals

We next show that this conjecture

in

in

holds

R

for

R'

such that L = RQ/PRQ

L'

could have been stated for a

more general case. (7.3) alent

to (*):

one-to-one ~hen

PROPOSITION.

A

if

A

is a N o e t h e r i a n

correspondence

is a

Proof. holds and let

that

between

GB-Con~ecture

(7.1.1)

is equiv-

domain such that there exists a

the maximal

ideals

in

A

an___d_d A'

,

GB-domain. It is clear A

A

that

(*) = (7.1.1),

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

to-one c o r r e s p o n d e n c e to prove

The Descended

is a

between

that

(7.1.1)

domain such that there exists a one-

the maximal

GB-domain,

so assume

ideals

it suffices

in

A

to prove

and that,

A'

Then,

for each

65 maximal

ideal

M

in

and this follows

A

by

(A.6.1) = (A.6.5),

from the hypothesis,

in this chapter

THEOREM°

(7.4.1)

The

of a Noetherian (7.4.2) R'

is a

and i f

The following

GB-Conjecture

domain

If

gives

q.e.d.

three characterizations

of

R

is a

statements

(3.6.4)

If

R

S = R'[c]

~

prime (7.4.4)

holds when

so assume

c

closure

such that

R'

i_~s level,

is a local

such that

R'

is quasi-local ,

R

S

and

It is clear

(7.4.4)

is a

GB-domain,

(A.6.5),

in

is quadratic

that

R'

domain

holds

is a

extension

lie over adjacent S

as in (7.4.3),

over

then

that

adjacent

prime

R

GB-domain

such that

Then, w i t h

ideal R

[R-4, Lemma 2.9]

M'

its integral

closure

some integral lows from

and since

extension

s E L Let

such that K°

L

R'

of (7.4.3)

suppose

a height

b 2 - sb E L

and

3.2]

L[b]

of the natural

domain of

p = Po~R '

one maximal

ideal

in

domain and

one maximal

For these

R' M,

there exist

a Noetherian

is a height

to

is quasi-local

extension and

Then,

(A.6.1)

Therefore,

q = QoNR '

L)

, by R'

3.5 and Remark

be the kernel

in

lies between

there

domain of

[R-2, Proposition

R'

that

is local and

, there exists

(since

in

domain.

to prove

in some integral

height q/p > I , where

L = (R'/p)q/p

ideals

, then

(7.4.3) = (7.4.4),

be a Noetherian

that

Po c Qo

R'

the conclusion

it is sufficient

for each maximal

ideals

prime

(7.4.1) = (7.4.2) =

and let

so it may be assumed

domain of

R'

in the proof of (3.6.3) = (3.6.4)).

ideal.

the integral

is a local domain

ideals

prove

and

holds:

GB-domain.

is a free integral

With

Proof.

L', by

are equivalent:

GB-domain.

(7.4.3)

R'

GB-domain,

GB-Conj ecture. (7.4)

~s

is a

immediately

The lasttheor~m the

, AM

ideal in

same reasons,

it fol-

that there exist

b E L'

has a height one maximal homomorphism

of

L[X]

66

onto

Lib]

and let

K

be the pre-image

pR' [X]H c K , KNR'q = pR'q R'q[X]

such that

monic

quadratic

ideal

in

K/p*

, and

is maximal

division

and

A

prime

.

p

A

so

since

f

algorithm,

is a free

ratic

integral

that contract dicts

in

(7.4.4). (7°5)

following

extension R'

Let

and,

this

readily

A

A

prime

in

A

the rings

give some additional GB-Conjecture.

= (f)

and

p

contains

a free quadprime

ideals

and this contra-

ideals

domain

ideals

to a maximal in (7.1) and

that the A

:

A

in each integral

to a saturated

equivalences

(by the

Therefore

from the definition

chain of prime

contracts

p

are

q.e.d.

for an integral

A

(since

pR'q c qR'q

closed)).

ideals,

, P =

= qR'q

that has adjacent

follows

a

domain of

that there exists

= (7.4.1),

in

QnR'q

and monic

chain of prime

contracts

together w i t h

ture and the

R'

are equivalent

each maximal

sion domain of Using

(7.4.4)

each saturated

tension domain of ideals;

domain of

It readily

statements

GB-domain;

so it follows

in

contains

extension

Now

is integrally

q

K

N

A = R' q[X] /p*

Therefore

is quadratic R'

Then

ideal

Also,

, and

R'q)

c A .

q

to non-adjacent

Therefore

REMARK.

over

R'

(since

R' q -algebra,

a maximal

is an integral

is integral

R'q[X]

be a height one prime

f E p* c N .

Then

in

height N/K = i .

f , so let

ideals,

no linear polynomials

a

and

such that

Q = N/p*

not adjacent

A

, and there exists

polynomial

R'q[X]

K°

, P c Q , height Q/P = i , PnR'q = pR'q

R'q Q

K c N

of

ex-

chain of prime

in each integral chain of prime (7.4),

is

extenideals.

the reader can

of the Descended

GB-Conjec-

CHAPTER 8 THE STRONG AVOIDANCE CONJECTURE AND THE AVOIDANCE CONJECTURE The Strong Avoidance Conjecture is new in these notes, and it results from combining the Avoidance Conjecture and the semi-local Depth Conjecture.

(See (3.8.2) = (3.8°3) and (3.14.1).)

The Avoidance Conjecture arose by imposing an additional condition on the following familiarresult in commutative algebra p. 240]:

if

P1 c P2 c

therian ring

A

and

--. c Pn

NI,...,N h

is a chain of prime ideals in a Noeare prime ideals in

UN i , then there exists a chain of prime ideals Pn' = Pn

such that

[ZS-I, Lemma,

P.'j ~_ UN i , for

A

such that

Pn

PI c P2' c --- c

j = 2,...,n .

This conjecture was first stated by S. McAdam in 1974, in [Mc-l, Question i, p. 728], and in the more general form of (3.8.3) in 1975, in [HMc, p. 752]. in [ ~ c ,

As mentioned after the proof of (3.8), it was shown

Proposition 3.8] that this conjecture implies the Upper Conjec-

ture, and it was noted in [Mc-l, p. 728] that this conjecture implies the Taut-Level Conjecture.

Also, in [Mc-4, Theorem 3], it was recently

shown that this conjecture holds for non-extended prime ideals little-heisht two in (QnR)R[X] # Q R[X]

R[X]

(that is,

Q ~ Spec R[X]

Q

of

is such that

and there exists a saturated chain of prime ideals in

of the form

(0) c p c Q)

(Some results that are closely re-

lated to this last result are given in [Mc-2].) Finally, by [ZS-I, Lemma , p. 240], the Strong Avoidance Conjecture holds for catenary local domains and the Avoidance Conjecture holds for catenary Noetherian rings.

On the other hand, in (14.7) it is shown

that there exists a quasi-local domain for which neither of these conjectures hold. (8.1)

g~ves one equigalence of the Strong ~voidance Conjecture.

(Two additional equivalences follow from (8.2).)

68

(8.1)

THEOREM.

(8.1.1) Q c N

The following

statements

The Strong A v o i d a n c e Conjecture

is a saturated chain of prime

and if

NI,...,N h

there exists

ar___e~prime

q E Spec R

height P + 1 , and (8.1.2) semi-local that

If

ideals

ideals

in

such that

R

domain

R

Proof.

many

(5.2)

q E Spec R

holds,

N ~ UN i , then

(so

P c q c N

is saturated).

NI,...,N h

are prime

ideals

ideals in a

in R

q E Spec R

such

such that

that

(8.1.1) = (8.1.2),

(on passing such that

to

R/P)

that

Then it follows

(8.1.2)

f~om

that there exist infinitely

P c q c N

Therefore,

so assume

is saturated,

q ~ UN i , and

since only finitely many of these

height q > height P + 1 , by

[Mc-l, T h e o r e m I],

(8.1.1)

q.e.d.

(8.2) (8.1.2)

REMARK.

By (5.2),

the "there exists"

in (8.1.1)

and in

can be replaced by "there are infinitely m a n y . "

(8.3) gives one equivalence provides

two additional

(8.3)

THEOREM.

(8.3.1)

NI,...,N h

q E Spec A

(8.3.2) a semi-local

The following

If

ideals

in

such that (0) c p c N

domain

statements

Conjecture

chain of prime

are prime

of the A v o i d a n c e

Conjecture,

and

(8.4)

equivalences.

The A v o i d a n c e

is a saturated

exists

R

depth q = depth N + 1 .

depth q = depth N + 1 . are such that

P c

ring

is a saturated chain of prime

and if

It is clear

and

if

P c q c N, q ~ DN i , height q =

holds and let the n o t a t i o n be as in (8.1.1). (8.1.2)

holds:

in a semi-local

N ~ UN i , then there exists a height one

q c N, q ~ UN i , and

q

(3.8.2)

such that

depth q = depth N + 1 (0) c Q c N

are equivalent:

ideals A

(3.8.3)

are equivalent: holds:

in a N o e t h e r i a n

such that

P c q c N

if ring

A

and if

N ~ UN i , then there

is saturated and

is a m a x i m a l chain of prime

R , then there exists

P c Q c N

q E Spec R

q ~_ UN i . ideals

such that

in

89

(0) c q c N contains

is saturated

and

It is clear

that

N

is_s the only maximal

ideal in

R

that

q .

Proof. and let

A

, P c Q c N , and

As/PA S , w h e r e

NI,...,N h

S = A - (NUNIU'''UNh)

that there exists

q E Spec A

q ~ UN i , so (8.3.1) holds, (8.4)

(8.3.1) = (8.3.2),

REMARK.

The

so assume

be as in (8.3.1).

.

Then it follows

such that

P c q c N

(8 3.2) holds Let

R =

from (8.3.2)

is saturated

and

and in (8.3.2)

can

q.e.d.

"there exists"

in (8.3.1)

be replaced by "there are infinitely m a n y . " Proof.

The proof of [Mc-l, P r o p o s i t i o n

N, NI,...,N h , and R = A s / P A S ; and, such that in

R

A

(0) ~ q c M

follows

domain

S = A - (NUNID'''UNh);

is a

mcpil 2

and

M

q E Spec R

is the only m a x i m a l

q , if there is one such

q .

ideal

The c o n c l u s i o n

from this, q.e.d.

Concerning that if

and with:

that w i t h P ~ 0 c

M = NAs/PA S ; there are infinmtely many

that contains

readily

as in ~8.3.1)

2] shows

(8.1.2)

(0) c P c Q R , then

infinite

set.

properly

contain

and

(8.3.2),

it is known

is a saturated chain of prime

I = {p E Spec R ; (0) c p c Q

However, U~p

[R-4, P r o p o s i t i o n

; p E I} .

(See (B.3.9).)

ideals in a local

is saturated}

it is shown in [HMc, Example

2.2]

3.2]

that

is an Q

may

CHAPTER 9 THE UPPER CONJECTURE (A nonextended prime ideal to

Q

in

R[X]

QnR , and the name of this conjecture derives from this.)

Nagata's examples

when there does not exist a R > i) , but for a R

M.

[N-6, Example 2, pp. 203-205] show that if

is a local domain, then there may exist a

in

is said to be an upper

mcpil I

mcpil n + I > 2

(in his examples).

mcpil 2

in in

R

in

(R,M)

D = R[X](M,X )

(that is, when

D , there exists a

altitude mcpil n

Since this also holds for all other local

domains for which this has been tested, the Upper Conjecture arose from asking the natural question. This conjecture was first stated by S. McAdam in 1974, in [Mc-l, Question 3, p. 728], and it was stated in [HMc, p. 750] as: is a local domain and

Q

exists a

mcpil m

R[X]Q} ~ [2] U [n+l ; there exists a

in

R]

Now it is known [R-10, (5.5.6)]

in

R[X]Q

in

is an upper to

if, and only if, there exists a

M

in

R[X]

if

, then

[m ; there mcpil n

that there exists a mcpil m

in

(R,M)

mcpil m

R[X](M,X )

so (3.8.5) is equivalent to the statement of this conjecture as given in [ ~ c ,

p. 750].

This conjecture was stated (essentially) as in (3.8.5)

in [RMc, (2.21)] and in [R-12, (4.10.3)], and (9.2) contains a number of equivalences of this conjecture that were given in these two papers. Finally,

in (14.1) it is shown that this conjecture holds for Hen-

selian local domains and local domains of the fo~m

L[X](M,X ) , while

in (14.5) it is shown that there exists a quasi-local domain for which this conjecture does not hold. The characterizations of the Upper Conjecture in (9.1) and (9.2) involve the existence of a

mcpil n

in a local domain.

The reader is

referred to (A.5) for some equivalences of this condition. (9.1) (9.1.1)

THEOREM.

The following statements are equivalent:

The Upper Conjecture

(3.8.5) holds:

if

(R,M)

is a io-

71

cal d o m a i n and if there exists a either there exists a (9.1.2) in

R'

If

If

in a p r i n c i p a l

R

R

R

If

MR k c Q

in

mcpil n > I

R

in

R[c] R

of

R

of

R

mcpil n > i

such that

, then there exists a

mcpil n > i mcpil n

is a local d o m a i n and there exists a

R k = R[X I .... ,Xk]

R[c]

.

(k > 0)

, then either there exists a

and

in

R .

mcpil n

Q E Spec R k

is such

mcpil n - k + depth Q

in

o_rr n - k + d e p t h Q = I (9.1.6)

n+l > 2

If

in

(R,M)

is a local domain and there exists a

R[X](M,X)

Proof.

, then there exists a

(9.1.1) = (9.1.2), by

by the Going Up Theorem, (9.1.4) = R

By (9.2.3) =

(9.1.2) = (9.1.3),

(9.1.3) =

(9.2.1),

R

domain

A = R[c]

there exists a is not local. d £ Pn

R

that has a

mcpil n Let

C = Rid] in

(9.1.4).

it suffices

to prove if

in some integral

mcpil n

in

R .

For this,

says there exists a p r i n c i p a l integral e x t e n s i o n

of

in

(0) c

such that

mcpil n

, then there exists a

mcpil

i_~n R .

(9.2.1) = (9.2.3),

is a local d o m a i n and there exists a m c p i l n > I

[P~Ic, T h e o r e m I.i0]

let

mcpi! n

and it is clear that

(9.1.1):

e x t e n s i o n d o m a i n of

a

mcpil n

mcpil n

Rs

(R,M)

(Rk) Q , w h e r e

that

n = I

is a local domain and there exists a

in a special e x t e n s i o n

in

or

integral e x t e n s i o n d o m a i n

If

(9.1.5)

R

D = R [ X ] ( M , X ) , then

is a local domain and there exists a

, then there exists a (9.1.4)

in

in

is a local d o m a i n and there exists a

then there exists a

(9.1.3)

R'

R

mcpil n

m c p i l n+l

I - d Then

APn

Cp nC

' by

R , by

PI c

mcpil n .

If

(A.5.2) = (A.5.1),

... c Pn

be a

mcpil n

is in all other m a x i m a l is integral over (A.5.2) =

A

(A.5.1),

is local, so assume in

ideals in

CPnOC

A

then A

, let A

, and

, so there exists

hence there exists a

n

mcpil n

in

of

Then

C .

e x t e n s i o n of

C . L

Let

L = R + J , where

J

is the J a c o b s o n radical

is local and it is readily seen that

L , so there exists a

mcpil n

in

C

is a special

L , by hypothesis,

72

and so t h e r e e x i s t s With

a

mcpil

the n o t a t i o n

if and o n l y

if there

(9.2.1)

=

exists

, by

that

exists

(9.1.6)

a

a

prime

(1.5.2) =

mcpil

n = 0

assume

that

n = 0 .

exists

a

mcpil

holds,

n

n+l

in

or

let in

R

n = i . Then

n = 0

REMARK.

in

R

and)

The

prime

R

that

If

in

R

following

R

R

g = ~ a

Proof.

(7.5)

(Rk) Q s u c h that

(9.1.5) =

implies

If

(9.1.1),

(9.1.1)

mcpil

(9.1.6), and assume

n + l > 2 , then

(9.1.6).

If

(9.1.1)

holds,

n+l ~ 2 ,

holds,

so

so there

q.e.d.

of this c o n j e c t u r e .

are equivalent:

holds. there exists

, then there exists

, where

mcpil

n

in

R'}

(9.2.1) = by

JR-12,

shows that

g = [R ; R

in some

if t h e r e e x i s t s

ideals

(9.2.4),

R

is a field,

statements

t h a t has a

local domain and either maximal

in

a

a

depth n > 1

mcpil n

there e x i s t s

in

R .

an i n t e g r a l

n > 1 , then t h e r e e x i s t s

a

R .

exists only

in

, so

equivalences

is a l o c a l d o m a i n and

d o m a i n of in

R

is a l o c a l d o m a i n and

ideal

by

, and so (9.1.1)

If

(9.2.4) there

n

be a local domain,

n = I , then

it f o l l o w s

(9.2.2)

n

z

D = R[X](M,X )

If

The Upper Conjecture

(9.2.3)

mcpil

(A.5.1).

mcpil

ideal

as d e s i r e d ,

(9.2.1)

extension

a

(1.5.3)

(R,M)

In (9.2) w e list four o t h e r k n o w n

minimal

=

that the U p p e r C o n j e c t u r e

mcpil

then either

(9.2)

(A.5.2)

there e x i s t s

a minimal

, by

it is c l e a r

that t h e r e e x i s t s there

R

(9.2.2).

Finally, so a s s u m e

in

of (9.1.5),

depth z = n - k + depth Q by

n

a

integral

mcpil

altitude

n

in

R = 1

is a local d o m a i n s u c h extension R}

and

d o m a i n of ~ = [R ; R

that R

(if

is a

or t h e r e a r e no h e i g h t one

.

(9.2.2)

=

(4.10.3)],

if the

(9.2.3),

by

[P~Mc, (2.22)],

and

(9.2.1)

q.e.d.

GB-Conjecture

holds

and

R

is an inte-

73

grally closed

local domain,

so the

GB-Conjecture

(9.2.1)

~

then

implies

R

R

satisfies

satisfies

the c o n d i t i o n in (9.2.3),

the Upper Conjecture,

(9.2,3).

The set

g

of (9.2.4)

d o m a i n is in it. (15.4.5),

is of some interest,

Some q u e s t i o n s c o n c e r n i n g

and in (B.3.5)

it is shown that

g g

does not imply that

R/P

or

Rp

are in

g

since every

GB-local

are m e n t i o n e d

in

is not closed under fac-

t o r i z a t i o n nor under l o c a l i z a t i o n - that is,

R E g

and

P E Spec R

.

The final result in this chapter shows that this c o n j e c t u r e e q u i v a l e n t to its semi-local

(9.3)

n+l R}

THEOREM.

The Upper C o n j e c t u r e

(9.3.2)

If

n+l > 2

in

N~R

U[(M,X)

, where

Assume

ideal

N

R<X)

in

ideals}

= R[X] S,

is a m a x i m a l R

or

ideal in

n = 1 .

in

let

m c p i l n+l > 2 R<X>

with

S' = R[X]

, then there exists a

(9.3.1) holds,

there exists a

maximal

mcpil n

; M

mcpil

is a s e m i - l o c a l d o m a i n and there exists a

are m a x i m a l

Proof. assume

R

R<X>

(9.1.1) holds.

S = R[X]

, then either there exists a If

a~e equivalent:

is a semi-local d o m a i n and there exists a

R[X] S , w h e r e

(9.3.3)

and

The following statements

R

is

version.

(9.3.1)

in

by

R in

mcpil

- U[N

mcpil n

; N

in

R .

be a semi-local domain, and R<X)

T h e n there exists a

such that there exists a

m c p i l n+l

in

Jr

R<X) N .

Let

ideal in

M = NAR

R , by

a

in

Assume

of

R<X)

in

mcpil n

and

Then N

M

is an upper

R M [ X ] ( M R M , X ) , by

in

RM

is a m a x i m a l

, by (9.3.1),

to

JR-10,

MRM

,

(5.5.6)].

so there exists

R , and so (9.3.3) holds.

(9.3.3) holds,

there exists a

= NR<X)NnRM[X]

m c p i l n+l > 2

there exists a

mcpil n

N

[Hou-2, P r e l i m i n a r i e s ] ,

so there exists a Therefore

and

m c p i l n+l

and the m a x i m a l

let in

D = R[X] S D .

ideals

in

Then, D

be as in (9.3.2), and assume since

D

is a q u o t i e n t ring

lie over m a x i m a l

ideals

in

74

R(X)

, there

then

there

exists

clearly

n E

n = 0

Then

n -- 0

. in

exists

R

Finally,

[0,i]

a

a

mcpil mcpil

.

If

it f o l l o w s , and

so

n+l

n

n = 1 that

(9.3.2)

it is c l e a r

in

that

in R

R(X) , by

, then R

Therefore, (9.3.3).

(9.3.2)

is a field,

If

holds, so

there

if

n+l ~- 2

, then

so a s s u m e

that

exists

holds. (9.3.2)

=

(9.3.1),

n+l > 2

q.e.d.

a

,

mcpil

CHAPTER I0 THE TAUT-LEVEL CONJECTURE It is known [McR-2, Proposition main of a catenary local domain

R

12] that an integral extension dois taut, and is taut-level,

is level (by [McR-2, Proposition 12] and (3.2)).

Therefore,

if

R'

since the

Catenary Chain Conjecture holds if every integral extension domain of R

is catenary, by [R-6, (4.3)], the Taut-Level Conjecture arose from

generalizing

the natural question to ask about integral extension do-

mains of a catenary local domain. I stated this conjecture

in 1971, in JR-4, Remark 3.14(iii)], and

it was more explicitly stated in [Mc-l, Question 2, p. 728], in Remark (i) preceding Proposition 9 in [McR-2], and in [R-14, last two references, (in [McR-2]),

(2.14)].

In these

it was noted without proof that (I0.i.i) = (10.1.7)

that (10.1.2) = (10.1.6)

equivalent conditions

(in [R-14]), and that these

imply the Catenary Chain Conjecture.

was noted in [R-4, Remark 3.14(iii)]

Also,

it

(together with [McR-2, Propositions

12 and 13]) that this conjecture implies the Normal Chain Conjecture. Moreover,

a remark relating this conjecture

(13.1)) was given in [McR-I,

to Conjecture

(2.10)], and in [R-21,

(K) (see

(3.9)] it was shown

that this conjecture holds, if, for each taut semi-local domain for all maximal ideals (4.2.2)

(generalized

Finally, for

Henselian

N

in

I = n[V ; (V,N) E ~}

to semi-local domains)),

IN

(with is an

~

R

and

as in

H-domain.

in (14.3) it is shown that the Taut-Level Conjecture holds semi-local

rings and certain localizations

of

R[X]

But, on the other hand, an example of a taut-level semi-local ring (not a domain) which is not catenary was given preceding Corollary 8 in [McR-2], and the existence of additional examples showing the conjecture must be restricted

to semi-local domains has already been noted in (2.2)

and (2.8). Seven characterizations

of this conjecture are given in (i0.i).

76 The reader various

should

refer

conditions

(I0.I)

mentioned

THEOREM.

(i0.I.I) taut-level

to (A.7)

The followin$

semi-local If

(10.1.3)

R

for some equivalences

of the

in (I0.i).

The Taut-Level

(10.1.2)

- (A.II)

domain,

statements

Conjecture then

R

are equivalent:

(3.9.5)

holds:

satisfies

is a taut semi-local

if

R

is a

the f.c.c.

domain,

then

R

is catenary.

I_~f R

is a taut semi-loca! ' domain,

then

R'

satisfies

If

R

is a taut semi-local

domain,

then

R

If

R

is a taut-semi-local

domain and

the c.c. (10.1.4)

satisfies

the o.h.c.c. (10.1.5) then

R

satisfies

(10.1.6)

(10.1.7) taut-leve~

level,

R

R

for all

then

is a taut semi-local

ideals If

(10.1.8)

M

in

If

R

(i0.I.i),

Assume

If

R'

is taut , where

by

(3.2).

Let

b

is a taut-level B

satisfies

(10.1.2) semi-local

(i0.i.i)

holds

then

R

holds.

, S = R'

semi-local by

, and domain,

S

RM

is taut,

domain,

then

R/P

is

domain such that

R'

is

is the complement

in

R[X]

with

M

and let

R

Therefore,

maximal

so

R

assume

B = R[b,I/b]

(I0.i.I),

(3.1.1) so

R

R .

is catenary, R'

a height one maximal

by

in

be a taut semi-local

is level,

and there exists

the f.c.c.,

and so (i0.i.2)

(M,X)R[X]

is level,

R > 1

semi-local

is a taut semi-local

and so (10.1.2)

so altitude

then

P E Spec R .

of the union of the ideals

domain.

@omain,

R .

is a taut-level

D = R[XJ S

Proof.

is level,

the s.c.c.

If

for all maximal

R'

is not level, ideal

be as in (3.1), and

(3.1.5).

is catenary,

by

in so

R'

,

B

Therefore by

(3.1.6),

and let

R

be a taut

is level,

so

holds.

= (10.1.4):

domain.

If

R'

Assume

(10.1.2)

is level,

then

holds R

R

is cate-

77 nary and level, by (10.1.2), Also,

R

satisfies

each finite integral extension domain

(A.8.1) = (A.8.3)) level),

so

A

the f.c.c. so

and so

and

A'

is level

is catenary

Therefore

R

(by [N-6,

and level, by satisfies

A

the f.c.c., by (1.2.3).

of

R

(10.14)],

(10.1.2),

the s.c.c., by (A.II.I) = (A.II.7),

holds.

Therefore

assume

S = R'

, and

B

is not level,

be as in (3.1),

local domain such that and (3.1.5).

Therefore

been shown, by (3.1.7),

B'

B'

satisfies

so (10.1.4)

is a taut semi-local

If

R'

(10.1.5)

is level,

fore, assume the proof

R'

so

R'

B

R

is catenary,

R

satisfies

ideal

M

in

(10.1.6) local domain

, by (3.2).

and

Let

b ,

is a taut-level

semi-

ideals, by (3.1.1)

hence

R'

satisfies

the c.c.,

holds.

domain such that

holds and let

then (i0.I.I)

R

Therefore

R

B

R'

satis-

by (1.3.4). semi-local

, and

B

is taut-level

by (10.1.5) R

then

that if

and (1.3.5).

b , S = R'

Then

implies

is level,

holds, by (10.1.5)

the f.c.c.,

by (3.1.6).

holds and

R'

be a taut-level

is not level and let

satisfies

and (10.1.3)

so (10.1.3) = (10.1.5),

the f.c.c., by (1.2.3),

If (10.1.2)

altitude R > 1

B = R[b,i/b]

the s.c.c.,

that (10.1.2) = (10.1.4).

level, hence

satisfies

is level, by (3.2), so by what has already

fies the s.c.c., by (1.3.3), Assume

so

ideal in

It is clear that (10.1.4) = (10.1.3), R

is

and so (10.1.4)

has no height one maximal

B'

A

R'

the s.c.c., by (A.II.I) = (A.II.8),

satisfies

there exists a height one maximal

since

and so

R'

R'

is taut (by

There-

be as in and

and (1.3.5),

B'

is as in (10.1.6),

R , RM

satisfies

the f.c.c.,

implies

that, for each maximal

R , ~

satisfies

is

and so

is level and catenary,

and so (i0.I.i)

domain.

hence

holds.

then, for each maximal

by (1.2.2), ideal

M

so (10.1.6)

holds.

in a taut semi-

the f.c.c., by (A.9.5) = (A.9.1),

so

(i0~1.6) = (lO 1.2), By (1.2.2). Assume and let

(i0.i.I)

holds,

P E Spec R .

Then

let R/P

R

be a taut-level satisfies

semi-local

the f.c.c.

domain,

(by (1.2.1),

78 since

R

does),

Assume and let

so

R/P

(10.1.7)

holds,

(0) c P1 c

on

hence

(i0.I.I)

and

altitude

holds

then, RM

by (A.II.I)

altitude

[RMc,

R by

satisfies

assume ideal

and

holds,

in

in JR-14,

ideals

.

so

R .

Also,

height

satisfies

M

in

domain

the f.c.c.,

such that

R , height M = alti-

by hypothesis

R + 1 .

holds,

Then

and

(1.3.3).

is catenary

Therefore,

let

it follows

So,

and

that

R

R

be as in (10ol.5),

R[X] S

and let M

height M = height MD = altitude

(= RM[X]MRM[X])

(A.8.5).

hence

= DMD

Therefore

satisfies

RM

, so

RM(X )

satisfies

the s.c.c.,

satisfies

the s.c.c.,

by

(1.3.3),

that,

in fact,

D = 1 = the by

and so

q.e.d.

domain

The proof given

Pk/PI

in

holds.

R .

RM(X )

so it follows

semi-local

R

R[XI(M,X ) = ~I[X](MRM,X )

The proof of (10.1.5) level,

R , so

domain,

ideals

the f.c.e.,

is a taut semi-local

the s.c.c.,

(10.1.8)

(A.8.1) =

(2.15)],

(10.1.5)

chain of prime

satisfies

for all maximal

is taut, and so (10.1.8)

s.e.c.,

R

R[X](M,X ) = altitude

be a maximal

semi-local

R - 1 = depth P1 = height

k = altitude

and

~ (A.II.19),

Finally,

holds.

holds.

is level,

tude R

(10.1.7)

be a taut-level

R , R/P 1

Therefore

If (10.1.5) R'

R

hence

be a maximal

altitude

altitude

Pk/PI = k - 1 .

let

... c Pk

Then, by hypothesis, by induction

is taut-level,

that

= (10.1.8) (10.1.8)

such that

that

(10.1.8)

(4.29.3)].

R'

shows

is equivalent

is level,

= (10.1.5)

then

to: D

given above

if

D R

is tautis a taut

is taut-level. is the same as that

CHAPTER ii THE CATENARY CHAIN CONJECTURE I stated this conjecture in 1971 in the introduction of [R-4]. The original reason for considering this conjecture was that if a catenary local domain, then, for all nonmaximal prime ideals R' , height P' + depth P' = altitude R' and

R'p,

R

is

P'

in

satisfies the s.c.c.,

by (A.8.1) = (A.8.3) and JR-4, Corollary 3.12] - so it seems that it should be possible to prove that the fact that

R'

R'

satisfies the c.c. from this and

is a Krull domain.

One equivalence of this conjecture has already been given in (3.10.3) and a number of equivalences of this conjecture have previously appeared in the literature. (a)

For example:

every special extension of a catenary local domain is catenary

JR-6, (4.7)]; (b)

if

local domain ideals

N

(c)

S

R , then

in if

is a finite integral extension domain of a catenary

S

satisfies the s.c.c., for all maximal

and for all nonzero

(R,M)

for all nonzero

SN[I/bJ

b

b

in

M ,

~

[R-II, (2.12.2)]; R'

is catenary, where

R

with respect to

geneous ideal in

~

JR-9, Question 3.22];

if

NS N

is a catenary local domain and

the Rees ring of

(d)

in

bR

and

~

is level, then,

~ = ~(R,bR)

is

is the maximal homo-

R

is a catenary local domain, then

R

is taut [R-14,

R

is a catenary local domain, then, for each maximal

(3.18)] ; and, (e) ideal that

M'

if in

R' , every DVR

R c_ V , NnR' = M'

, and

(V,N) V

in the quotient field of

R

is integral over a locality over

such R

is of the first kind [R-13, (2.19.2)]. (See also [R-6, (4.3)J and [P,(8.8)].)

Of these,

(a) and (d) are

sharpened in (Ii.i.$) and (ii.i. I0), respectively, and (e) follows from (ii.I.I) = (11.1.3) and (A.10.1) = (A.10.21).

80 Also, this conjecture holds, if, for all H-domain, whenever

R

is an

P E Spec R , Rp

H-local domain, by [R-20, last paragraph],

and a remark relating this conjecture to Conjecture was given in [McR-I, (2.9)].

is an

(K) (see (13.1))

And, as noted after the proof of (3.3),

it was proved in [R-5] and in [R-6] that the

H-Conjecture implies this

conjecture, and it was noted without proof in the introduction of [R-4] that this conjecture implies the Normal Chain Conjecture. It is known [R-3, Theorem 2.21] that if

L

is a Henselian local

ring, a complete local ring, or a local ~ing of the form with

(R,M)

if,

L

a local ring), then

satisfies the s.c.c.

L

R[X](M,X )

satisfies the f.c.c, if, and only

(so this conjecture holds for such

L)

It is interesting to note that it is shown in (ii.i.I) = (11.1.4) that one of the characterizations

of the Catena~y Chain Conjecture is that

the f.c.c, and the s.c.c, are equivalent conditions for all local domains

R

such that

R'

is level.

this conjecture are given in (Ii.I).

Eleven other characterizations of (It should be noted that the state-

ments are arranged to correspond to the order of the statements in the characterizations of the

H-Conjecture,

(6.1), and this is quite differ-

ent from the order in which they are proved to be equivalent. the reader is referred to (A.3),

(A.4), and (A.9)

Also,

(A.II) for some

equivalences of the various conditions mentioned in (ii.I).) (Ii.i) (II.i.I)

THEOREM.

The following statements are equivalent:

The Catenary Chain Conjecture

(3.3.8) holds:

the inte-

gral closure of a catenary local domain satisfies the c.c. (11.1.2)

If

R

is a catenary local domain, then

R'

(11.1.3)

If

R

is a catenary local domain, then

R

If

R

is a catenary local domain such that

is catenary. satisfies

the o.h.c.c. (11.1.4) then

R

R'

is level,

satisfies the s.c.c.

(11.1.5)

If

R

is a catenary local domain, then

R

is a

C I-

81

domain. (11.1.6) finite catenary

I f ~ rin$

A

is taut

(~espectively,

taut-level)

integral e x t e n s i o n d o m a i n of a local domain (respectively,

(11.1.7) H-domain,

If

R

satisfies

(11.1.8)

If

e x t e n s i o n of

R

(11.1.9)

If

ideals

R

Rs Mi

A

R' M,

is an

local domain , then

M'

is ~ catenary

R , then

R , then

is

the f.c.c.).

is a catenary

for all maximal

and is

in

local domain and

is an

is ! catenary

R'

H-domain

Rs

is a special

(i = 1,2)

local domain,

then

RH

is an

H-

ring. (II.i.i0)

If

R

is a catenary

(ii.I.ii)

If

(R,M)

local domain,

then

R

is an

H-

ring.

R[X] (M,X)

is an

(11.1.12) prime

ideal

p

(11.1.13) in

M

by

(1.3.5),

With

R

i_~n R

, D/pD

If

Assume

R

RSN by

is an (3.2).

such that R'

, and

that

D =

local domain,

(II.I.i) =

for all is an

b,c

H-domain.

(11.1.3) = (11.1.2),

(1.2.1). be a special exten-

Fix

i , let

N = M i , and let

height p = 1

and

p c N .

Then to prove

Then

that

p' c M'

is an

hence

that

height N = altitude R > I,

, so there exists a height one

(11.1.7), RSN

then,

R .

height M' = altitude R , by

R - 1 , so

(11.1.3),

(R s ; MI, M2)

it may be assumed

p'nR s = p

for each height one

H-domain.

(11.1.7), by

R c R s c R'

= altitude R - 1 , by altitude

is an

local domain

H-domain,

as in (ii.i.ii),

holds and let

such that

Now

then

(b,c)R = 2, B = R [ c / b ] ( M , c / b )

(11.1.2) =

(11.1.7)

D

is ! qatenary

height

sion of a catenary p E Spec R s

and

It is clear and

local domain,

q-domain.

such that Proof.

is a catenary

p' E Spec R'

, for some maximal

ideal

(3.2).

height M'/p'

Therefore

it follows

H-domain,

that

M'

height N/p =

and so (11.1.8)

holds.

in

82 Assume suffices special

(ii.i.8)

to prove

RSM

R

that

Then to prove

is a catenary

R , then

is catenary,

N = M i , so

to prove

holds.

that if

extension of

is, that let

that

Rs

for

is catenary

and

ideal RS/p

altitude either by

in

Rs

such that

is catenary, R - 1 , by

since

(4.8)(2)].

R/(pnR)

If

is, by

ial e x t e n s i o n of is catenary.

or

to prove

(since RS/p

Fix

p

i

and

that

height

is a height one

height N/p = altitude R - 1 For this,

, then

RS/p

, then, by induction on Rs

that

Therefore

height N/p =

height N = altitude R)

that

it

is a

(4.7)];

(3.2).

On the other hand,

it follows

holds, Rs

.

is a special extension of

(1.2.1).

Therefore

, by

that if

p c N , then

RS/p = R/(pnR)

R/(pnR)

[R-6,

, by (1.2.2).

(A.9.1) = (A.9.6).

(11.1.8)

RS/p = R/(pNR)

[R-6,

by

by

it may clearly be assumed

N = altitude R , and it suffices prime

is catenary,

i = 1,2

(Ii.I.I)

local domain and

height N E ~i , altitude R}

RSN

that

Also,

R/(pNR)

,

is catenary, if

RS/p

is a spec-

altitude R , RS/p

is catenary,

and so (ii.I.i)

holds. (11.1.3) = (11.1.4) = (ii.I.i) (6.1.5)

= (6.1.i)

(1.3.5)

and

(3.1.4),

(11.1.3) by

(but use

(A.4.1) = (A.4.7),

(A.4.1),

hence

(II.I.5)

holds.

to prove for

prime

i

(A.10.1)

R

= (11.1.3),

C l - d o m a i n , by

is a

(3.1.7)

in place of

R - 2

Then

(11.1.5),

R/p

(11.1.5) =

by

it suffices,

by

p

is a

be a

is catenary, by (1.2.1), R

is a

and so

by [RP,

R

then,

(A.4.7) =

local domain is a

For this, assume

and so

(A.3.5).

local domain,

Co-domain,

and let

(II.i. II),

(A.3.1) =

(A.4.5) = (A.4.1),

implies a catenary

(i --< i < altitude R - 2) R .

= (A.IO.!0),

is a catenary

C l - d o m a i n , by

(11.1.5)

(11.1.5)

ideal in

and

(ii.i. Ii) = (11.1.12),

that if

is a

i = l,...,altitude

for some

a

that

= (A.II.7)

p E Spec R , R/p

R

To show that

by

and

implies

for all height one

(A.II.I)

respectively).

= (11.1.5),

(11.1.12)

much as in the proof of (6.1.4) =

(3.13)], Ci-domain, Ci-domain ,

height so

C i + l - d o m a i n , by

R/p

i is

(A.4.5) =

83 (A.4.1).

Therefore,

Assume

it follows

(11.1.3)

that

(11.1.5)

holds and let

A

be a taut finite

sion domain of a local domain

R .

Propositions

is catenary,

= (A.10.5)

12 and 7], so

(since a semi-local

catenary,

by (1.4.1)),

Assume R

(11.1.6)

is taut.

(A.8.1)

(11.1.8)

holds,

(ii,i.3) Assume p

holds

(11.1.6)

finite

= (11.1.9),

by

holds,

pR H , so

(11.1.9). R - 1

(ii.I.5)

holds.

(11.1.3)

let

assume

holds

it may be assumed ideal

and

I - (e/d)

and

c

in

by

is

local domain,

domain

by

A

of

(11.1.6),

so

R

is

and so

be a catenary

local domain,

in

P

R .

that

(R/p)'

=

(where

p

R

R .

(R/p)'

d

is proper).

height

(M,c/b)

= altitude

R - i , by

R , by

(11.1.13).

RH/pR H=- (R/p) H)

that

(6.1.4)

p

, so

= (6.1.10).

be a height

to prove

in

M/p

ideals

in

and

p

one prime

that (11.1.5) Let

such that (R/p)'

c

so

it follows

so that

be

e/d C N b

(R/p)[e/d]

(M,c/b)

(4.7), pp. 49-50]. 2.7],

N

Let

e , respectively,

[R-9, Lemma Thus,

R - i ,

height =

(since

Now it may be assumed [P,

prime

(A.3.5).

d,e

and

(b,c)R = 2 , by the proof of

has

is not quasi-local.

, and let

of

(R/p)'

since

Then

= pR[I/b]nR[c/b])

= N@(R/p)[e/d]

in

and

be a minimal

depth P = altitude

holds and let

is in all other maximal in

Let

ideal

(A.3.1)

height

= altitude

(A.10.1)

(A.10.12).

P = 1 , hence

local domain

be p~eimages

(M/p,e/d)

ideal

(11.1.13)

in a catenary

R[c/b]/p*

R

=

Ex. 2, p. 188],

= (11.1.13),

ideal

a maximal

extension

= (ii.i. I0), as in the proof

Finally,

and

the o.h.c.c,

be a catenary

each maximal

(by [N-6,

(ii.I.II)

by [McR-2,

by hypothesis

is catenary,

(A.10.1)

height

Therefore,

altitude

is catenary,

exten-

(1.2.1).

(11.1.9)

of

R

A

integral

holds.

integral

so

R

that satisfies

and let

= (A.8.3), by

Then

ring

be a height one prime

divisor by

hence

Then every

taut, by

let

A

= (11.1.3).

that Therefore

height

(M,c/b)/p*

height N =

84 altitude R - i = depth p , so

R

is a

Cl-domain , hence

(11.1.5) holds,

q.e.d. In [P, (8.8)], (Ii.I.I) H-domain"

it was proved that (ii.i.I) = (Ii.i. I0) and that

is equivalent

to (11.1.12)

replaced by "catenary."

equivalence

to (ii.I)

and to (11.1.13), both with "an Also, an additional

(quite technical)

is given in this same result in [P].

This chapter will be closed by showing

that this conjecture

could

have been stated for an arbitrary catenary Noetherian domain. (11.2)

THEOREM.

The following statements

are equivalent:

(11.2.1)

The Catenary Chain Conjecture

(11.2.2)

If

A

is a catenary Noetherian domain,

then

A'

is

If

A

is a catenary Noetherian domain,

then

A'

satis-

(ii.i.i) holds.

catenary. (11.2.3) fies the c.c. Proof.

(11.2.3) = (11.2.2), by

by (11.1.2) = (ii.i.i).

Therefore assume

be a catenary Noetherian domain, and let (AM)'

M = M'nA satisfies

per and Therefore q.e.d.

.

Then

AM

let

M'

satisfies

that (11.2.1) holds, be a maximal

is catenary,

the c.c., by hypothesis.

A' M, = (A'(A_M)) N , so A'

(1.3.5), and (11.2.2) = (11.2.1),

A' M,

by (1.2.1), Now

satisfies

ideal in so

let A'

A ,

A'(A_M ) =

N = M'A'(A_M )

is pro-

the c.c., by (1.3.1).

the c.c., by (1.3.2), and so (11.2.3) holds,

85 (12.1.3)

If

R

is a catenary

quasi-local , then

R

satisfies

(12.1.4)

R

is a local domain such that

f.c.c.,

Assume

is level,

(12.1.1)

then

R'

s.c.c., b y (12.1.1), R'

hence

ideal in

R'

be as in (3.1), so

R'[I/b]

R'

satisfies

integral extension domain of

is taut-level

Therefore

it follows

fires the s.c.c.,

holds and let

satisfies

is not level,

one maximal C

is

the R

the f.c.c.

Proof.

assume

R'

the s.c.c.

then every free quadratic

satisfies

R'

If

local domain such that

R' so

R

be as in (12.1.2).

the f.c.c.,

so

R

satisfies

does by (A.II.I) = (A.II.7). altitude R > 1

, by hypothesis.

the Therefore,

and there exists a height

Let

b , S = R'

, B , and

B = Rib,i/b]

is a local domain and

B' = C =

and catenary,

by (3.1.2),

(3.1.5),

from what has already been proved

so

If

R'

satisfies

and

that

the c.c., by (3.1.7),

(3.1.6). B'

saris-

and so (12.1.2)

holds. Assume

(12.1.2)

is catenary,

holds and let

by (A.9.1) = (A.9.3),

nary and taut.

Therefore

f.c.c.,

satisfies

so

R'

the s.c.c.,

by (1.3.4),

Assume satisfies R'

and

(12.1.3)

A

and

ideal

the f.c.c., hence

R'(A_N )

satisfies

from (1.3.3)

AN

that

satisfies

is level,

the c.c.

the s.c.c., by (1.3.3).

A

R

so

in

Hence

satisfies

so

R

R'

is cateand the

satisfies

be a local domain such that R-algebra

A , (AN)' = R'(A_N)

by (1.2.1),

R'

(by (12.1.2))

such that

have the same number of maximal N

Then

holds.

be a finite

AN

the s.c.c.,

R c A

ideals.

Then,

is quasi-local

is catenary, by (12.1.3).

R'

and

by (A.9.2) = Therefore

the s.c.c., by (A.II.I) = (A.II.7),

so it follows

R'

satisfies

satisfies

the s.c.c.

s.c.c., b y (1.3.4), and so (12.1.1) Finally,

R'

and so (12.1.3)

Let

R'

(A.9.1),

R'

be as in (12.1.3).

and

holds and let

the f.c.c.

for each maximal satisfies

R

it is clear that

Hence

R

the

holds.

(12.1.1) = (12.1.4),

so assume

that

CHAPTER 12 THE NORMAL CHAIN CONJECTURE This conjecture arose from M. Nagata's incomplete proof of Proposition la in [N-3], in 1956.

A related Proposition ib also has an in-

complete proof, and both results were repeated in 1959, in [N-5, pp. 85-86], and a corrected version of these results was given in 1962, in IN-6, (34.3)].

(Since the reference to [N-6, (33.10)] in the proof of

[N-6, (34.3)] was not explained,

this result was reproved and sharpened

in [R-2, Theorem 3.11].) One equivalence of this conjecture follows easily from [R-2, Corollary 3.12]

(see (12.1.1) = (12.1.4)), and it was noted in the comment

following

(2.3.4) in [R-10] that this conjecture holds if the following

condition holds: domain

B

if there exists a

of a Noetherian domain

mcpil n

in an integral extension

A , then there exists a

mcpil n

in

BNA' It follows from comparing

(12.1.1) = (12.1.3) with (II.I.i)

(11.1.4) that this conjecture is very closely related to the Catenary Chain Conjecture.

Also, it is shown in (14.1) that this conjecture

holds for Henselian local domains and local domains of the fo~n L[X] (M,X) (12.1) gives three equivalences of this conjecture and the reader should refer to (A.8), (A.9), and (A.II) to obtain some additional equivalences. (12.1)

THEOREM.

(12.1.1)

The following statements are equivalent:

The Normal Chain Conjecture

sral closure of a local domain

R

(3.3.9) holds:

if the inte-

satisfies the f.c.c., then

R

satis-

fies the s.c.c. (12.1.2) nary, then

R'

If

R

is a local domain such that

satisfies the c.c.

R'

is taut and cate-

87 (12.1.4) holds, let f.c.c., and let If

R

R[c]

c E R' , then

be a local domain such that

R'

satisfies the

be a quadratic integral extension domain of

R[c]

satisfies the f.c.c., by (1.2.4).

If

then there are no linear polynomials in

K = Ker (R[X] ~ R[c~)

X2+rX+s E K , for some

c

r,s C R

(since

Therefore, by the Division Algorithm, so

R[c]

is a free

hypothesis.

R-algebra, hence

Therefore

R

is quadratic over

it follows that R[c]

R .

c ~ R', and R) .

K = (X2+rX+s)R[X],

satisfies the f.c.c., by

satisfies the s.c.c., by (A.II.9) = (A.II.I),

and so (12.1.1) holds, q.e.d. (The proof that (12.1.4) = (12.1.1) is essentially the same as that given in [R-2, Corollary 3.12].) It follows from (12.1.4) that if "catenary" is inherited by flat finite integral extension domains,

then this conjecture holds.

The final result in this chapter shows that this conjecture could have been stated for a more general case. (12.2)

PROPOSITION.

equivalent to (*):

if

fies the f.c.c., then Proof.

Then

A

A A

A'

satis-

satisfies the s.c.c. (*) = (12.1.i), so assume that (12.1.I)

be a Noetherian domain such that

A'

satisfies the

satisfies the f.c.c., by (1.2.4), so to prove that

satisfies the s.c.c,

it suffices to prove that

for all maximal ideals A'(A_M )

(12.1.1) is

is a Noetherian domain such that

It is clear that

holds and let f.c.c.

A

The Normal Chain Conjecture

is level (since

M

in A'

AM

A , by (1.3.3).

satisfies the c.c.,

For this,

(41)' =

is) and catenary (by (1.2.1)), so

satisfies the f.c.c., by (1.2.3).

Therefore

by (12.1.1), and so (*) holds, q.e.d.

AM

A

(AM)'

satisfies the s.c.c.,

CHAPTER 13 COMMENTS ON (3.3.1) AND CONJECTURE I think that all the conjectures exception of (3.3.1), The reasons Kaplansky

of [McR-I],

(13.1)

Conjecture

in a Noetherian

domain

implies

(see [H],

R

(K).

(say

M'

P

and

Q

(K) holds,

p £ Spec A

R. Heitmann's

then the Chain Con-

However,

that Conjecture

construction

height M = n = height N in

~N

.

such that and

N')

R'

and

(0)

of Heitmann

is a regular domain with two maximal 1 < height M' < height N'

would be a completely

R

failure of the chain problem of prime

Jacobson radical of

ideals

M'nN')

R

If this is

(since M. Nagata's

is false.

examples in the At

can be constructed. that this conjecture

L[X](M,X)

(13.2) gives one equivalence

ideals

new example of the

R'), and would show that (3.3.1)

it is shown in (14.2)

domains of the form

in

local

(and such that

have infinitely many prime ideals contained

I do not know if such an

Finally,

(R;M,N)

to obtain a noncatenary

the case,

R

domain

is the only prime ideal

ideals contained

for noncatenary

(K) does not

It seems to me that it should be possible

such that

then such an

it is known

in [HI shows that given

n > 1 , there exists a regular semi-local

there are no nonzero prime

present,

in the

are height two prime ideals

so (3.3.1) = (3.3.2).

to adjust this construction R

If

A , then there exists a height one

(3.3.1),

contained

domain

Stated

the question becomes:

[Mc-3], or [Mc-5, Theorem 7])

an integer

in

in [Hoc, p. 67].

that if Conjecture

In particular,

such that

by M. Hochster

p c PnQ .

It can be proved

hold.

(3.3.1).

for these doubts have to do with a question asked by I.

terminology

jecture

in Chapter 3, with the

hold, but I have strong doubts about

and mentioned

such that

mentioned

(K)

of (3.3.1).

holds for local

89

(13.2)

THEOREM.

(13.2.1) all nonzero

(3.3.1)

The following holds:

if

P ~ Spec R , R/P

statements

R

are equivalent:

is a local domain such that,

satisfies

the s.c.c.,

then

R

for

is cate-

na ry. (13.2.2)

If

E Spec R , R/P Proof.

R

is a local domain such that,

satisfies

It is clear

holds and let

R

for all nonzero

the s.c.c., that

(13.2.1)

then =

R

p E Spec R .

so there exists a height one

Let

P

is an

(13.2.2),

be a local domain such that

for all nonzero H-domain.

so assume

R/P

(13.2.2)

satisfies

be a nonzero prime

p 6 Spec R

P

such that

the s.c.c.,

ideal in

p c P .

R ,

Then

height P/p + depth P/p = altitude R/p = altitude R - 1 , by hypothesis. Therefore

height P + depth P = altitude R , hence

(A.9.5) = (A.9.1), (13.3) pp.

203-205]

and so (13.2.1)

REMARK.

It follows

in the case

be replaced by:

R

m = 0

satisfies

holds,

R

by

q.e.d.

quite readily

from

[N-6, Example

that the c o n c l u s i o n

the s.c.c.

is catenary,

(Concerning

in (3.3.1) this,

2,

cannot

see (B.3.3).)

CHAPTER 14 SOME EXAMPLES A number of examples are given in this chapter to show that some of the named conjectures in Chapter 3 hold for certain classes of local domains, and, on the other hand, that there are quasi-local domains for which certain of these conjectures do not hold.

We begin by showing

that a number of the conjectures hold for a Henselian local domain and for a local domain of the form (14.1)

EXAMPLE.

R[X](M,X)

The Upper Conjecture,

the Catenary Chain Conjec-

ture, and the Normal Chain Conjecture hold for Henselian local domains and local domains of the form

D = R[X](M,X ) , where

(R,M)

is a local

domain. Proof. D[Y](M,X,y )

By (1.5.3) = (1.5.4), there exists a if, and only if, there exists a

Upper Conjecture holds for exists a mcpil n+l

in

D

D .

mcpil n+2

mcpil n+l

in

in D , so the

Also, by (1.5.1) = (1.5.4), there

if, and only if, there exists a

some integral extension domain of

R .

Therefore,

mcpil n

in

since every integral

extension domain of a Henselian local domain is quasi-local,

the Upper

Conjecture holds for Henselian local domains, by (A.5.2) = (A.5.1). [R-3, Theorem 2.21] shows that

D

and Henselian local domains

are catenary if, and only if, they satisfy the s.c.c., so these rings satisfy the Catenary Chain Conjecture, by (A.II.I) = (A.II.7). Finally, for the Normal Chain Conjecture,

this was shown in

(3.5.2), q.e.d. (14.2)

REMARK.

Proof.

If

(3.3.1) also holds for

D/XD = R

D

as in (14.1).

satisfies the s.c.c., then

f.c.c., by (A. II.I) = (A.II.19), so (3.3.1) holds for

D

satisfies the

D , q.e.d.

91

If (3.3.1) holds for Henselian local domains, then the Chain Conjecture holds, by the proof of (3.3.1) = (3.3.2). (14.3)

EXAMPLE.

mains of the fo~m D[(M,X)R[X]

; M

Henselian semi-local rings and semi-local do-

D = R[XI S , where

R

is a maximal ideal in

is semi-local and

S = R[X] -

R} , satisfy the Taut-Level

Conjecture. Proof.

If

R

is a taut-level semi-local Henselian ring, then,

for each minimal prime ideal Proposition 5], and

R/z

z

in

R , R/z

is taut, by [McR-2,

is a Henselian local domain.

satisfies the f.c.c., by (A.9.5) = (A.9.1), and tude R , hence Also, if

R D

R , ~[X]M~M[X] Therefore

RM

RM[X](MRM,X )

Thus

R/z

altitude R/z = alti-

satisfies the f.c.c. is taut-level,

= DMD

then, for each maximal ideal

M

in

satisfies the s.c.c., by [McR-2, Proposition 9].

satisfies the s.c.c., by [RMc, (2.15)], so

D(M,X ) =

satisfies the f.c.c., by [R-3, Theorem 2.21].

Also,

height (M,X)D = altitude D , so

D

satisfies the f.c.c., by (1.2.3),

q.e.d. An alternate proof that the Taut-Level Conjecture holds for

D

is given in Remark (b) preceding Corollary 5 in [Mc-2]. (14.4)

REMARK.

(14.4.1) tude R ~ i

If

or

R

The following statements are readily verified: is a quasi-local domain such that either

altitude R = 2

all the named conjectures hold for

R .

and

R'

is level, then (3.3.1) and

in Chapter 3, except the Upper Conjecture,

(Some hold vacuously.)

And, if the Upper Conjecture is

restated using (9.2.3), then the Upper Conjecture also holds for (For the (3.8. ~ (14.4.2)

If

version of the Upper Conjecture, R

the named conjectures

alti-

R .

see (14.5).)

is a complete local domain, then (3.3.1) and all in Chapter 3 hold for

R .

92

We now show that there are q u a s i - l o c a l the named c o n j e c t u r e s noted the

in C h a p t e r 3 do not hold.

that this is the case for: H-Conjecture,

Descended

by

[Fu-l] or W.

(14.5)

by W.

EXAMPLE.

for w h i c h some of

It has already b e e n

the C h a i n Conjecture,

G B - C o n j e c t u r e and the

Level Conjecture,

domains

Heinzer's example GB-Conjecture,

by

[S] or

[K];

(see (14.6));

by

[F]; and,

the

the Taut-

Heinzer's example.

The Upper C o n j e c t u r e does not hold for c e r t a i n

q u a s i - l o c a l domains.

Proof.

Let

R

be the q u a s i - l o c a l domain m e n t i o n e d

p a r a g r a p h of Section 2 in [Sei], so = 3 , hence

altitude R = 1

the Upper C o n j e c t u r e does not hold for

In (14.6) and

(14.7), we use the fact

in the last

and

altitude R[X]

R , q.e.d.

[L, (3.1)]

that each finite

p a r t i a l l y o r d e r e d set w i t h a m i n i m u m element and a m a x i m u m element is, in fact,

isomorphic

to

Spec R , for some q u a s i - l o c a l Bezout d o m a i n

R .

B o t h examples are readily verified.

(14.6) b

EXAMPLE.

(Wo Heinzer

, c , d , e , f , g , h]

and

a c c c

f c h

with

[R-14,

(4.29.1)].)

Let

S = [a ,

a c b c d c f c h , a c c c e c g c h ,

(~ resular h e x a g o n split down a d i a g o n a l and then

one p a r t of the cut slid on the other part to the former center). with

R

~ N u a s i - l o c a l Bezout domain such that

not satisfy either the

(14.7) c c e c

f

EXAMPLE. an_d

d o m a i n such that

Spec R ~ S , R

does

H - C o n j e c t u r e or the T a u t - L e v e l Conjecture.

Let

a c b c d c

S = [a f .

, b

, c

, d , e , f}

Then, w i t h

R

with

Avoidance Conjecture,

the D e p t h Conjecture,

and the

fail to hold for

I do not k n o w any examples or the N o r m a l C h a i n C o n j e c t u r e

a c b c

~ q u a s i - l o c a l Bezout

Spec R m S , the Strong A v o i d a n c e Conjecture,

H-Conjecture

Then,

the

the W e a k D e p t h Conjecture,

R .

for w h i c h the C a t e n a r y Chain C o n j e c t u r e fail to hold.

CHAPTER 15 SOME RELATED QUESTIONS In this chapter, chain conjectures

are briefly discussed.

chain conjecture. jecture, ever,

a number of questions

that are related to the

The first is actually

It comes from a weakening

another

of the Normal Chain Con-

and it could have been given in Chapter 3 as (3.3.10).

it was decided

main difficulty

to delay giving it until this point,

in settling

culty in showing

this problem

is essentially

that the Normal Chain Conjecture

is this same difficulty

that arises

that (15.1a) has an affirmative may well provide

the needed

in (15.1a)

answer,

information

How-

since the

the same diffi-

holds.

In fact,

it

- and if it can be shown

then the same method of proof to show that (15.1) and (3.3.9)

hold. (15.1) does

R

If

satisfy

(15.1a) nary,

R

is an integrally

closed catenary

then

the s.c.c.?

is a very special case of (15.1).

since

local domain,

altitude R = 3

(R

H-domain

in (15.1a)

is cate-

and

R

is an

(since height one

If

R

is a local UFD (unique factoriza-

prime ideals are principal).) (15.1a)

JR-12,

tion domain)

(4.4)].

such that

altitude R = 3 , does

It was noted in JR-19, such a ring R

R

a

(6.1)]

that (15.1a)

is catenary

[R-3, Theorem 2.21] shows

in (15.1a),

so, if

Finally,

R

R

the s.c.c.?

is equivalent

is also Henselian,

11.2.4],

for a more general

that the answer

sion domains of

satisfy

to:

is

GB-domain?

noted by H. Seydi in [Sey, Corollary

3.12]

R

then, as

the answer is yes.

result.)

Also,

(See

[R-2, Corollary

is yes if all free quadratic

integral exten-

are catenary.

in [Fu-2, Proposition],

K. Fujita constructed

a quasi-

94

local I~D

R

of altitude

does not satisfy (15.2)

three

that is not catenary,

and so this

R

the s.c.c.

Three questions

on semi-local

UFD's.

(15.2. i)

Are all local Henselian

(15.2.2)

Are all local UFD's of altitude

(15.2.3)

Are all semi-local

UFD's of altitude

taut-level

four catenary?

four catenary?

UFD's

of altitude

four

catenary ? For all three questions, are no

mcpil

and if

P

(since

Rp

(15.3)

that if

(since height prime

(15.2.3) JR-16,

is a special

a

is the ring, ideals

R , the

are principal)

Rp

is catenary to

P = 3 .

are special

cases

of the

case of the Taut-Level

If

then there

, so it is sufficient

height

(15.2.2)

(6.1.3)].

= R[X]MR[X ]

It is known

and

in

Rp <= 3)

altitude

(15.2.1)

R

one prime

ideal

height M/P = i , then

that

and

R(X)

R

is a UFD and

(Note

is

in

is a non-maximal

show that if

jecture,

2

note

H-Con-

Conjecture.)

(R,M)

is a local

GB-domain,

then

(4.1)]

that

is equivalent

GB-domain?

[R-19,

(2.8) and

(15.3)

to

each of the following: (a)

Is such a ring

(b)

For such a ring

R , is

R[X]

(c)

For such a ring

R , is

R[X] N

mal ideal

N

(15.4) extension

R[X]

Questions

domains

(15.4.1) in some

in

such that

a

chains

of a local domain

(R,M)

(2.3)].

extension

(a)

Is

(b)

Is there a finite

GB-domain? a

GB-domain,

for some maxi-

NnR = M ?

on saturated

JR-10,

integral

R catenary?

If

domain

of prime

(0) c QI c B

of

ideals

... c Qn

in integral

is a

mcpil n

R , then:

height Qi = height QiN(B~R ') ? integral

extension

domain

A

of

R

that

95 has a

mcpil n , say

height Qi

and

(c)

be chosen as a subring

of

A

be chosen as a subring

(e)

Can

A

in (b) be chosen

as a principal

in [R-10,

that the answer

B

height

i = l,...,n

Can

if

Pi =

?

B

and such that

of

BNR'

? integral

extension

R ?

R = R'

for each

a Noetherian

i = l,...,n

R[bi] c R'

height Qi

and

(15.4.2) of

R

p. 79]

, and that the answers

lies between

R[c]

' for

(d)

It was noted

main

A

' such that

?

domain of

if

... c Pn

depth Pi = depth Qi

If so, can

Pi = Qi NA

yes,

(0) c PI c

, there exists

Concerning mcpil

a principal

ideal

depth Pi = depth Qi

R

(if and)

(15.4.2),

exists

a

n

exists

a principal

Pi

' by

only

a

(c) are yes, Also,

extension

do-

height Pi =

domain

extension

domain

in some integral a

[HMc, Theorem

extension

extension

integral

integral

if there exists

in some integral

closure.

such that

mcpil n

it is known

integral

and

is

[R-4, Lemma 2.9].

there exist a principal

such that there exists

sion domain of

to (15.4.1)(b)

domain and its integral

that has a prime

Does

to (15.4.1)(a)

mcpil n i. I0]

domain of Rib]

of

in

extenR[c]

?

that if there R , then there

R

that has a

mcpil n .

(15.4.3) infinitely

many

that

height

...,n

?

For in

If

(0) c PI c mcpil n

in

PI'' = height Pi

(15.4.3),

R , by [McR-2,

it is known Lemma

--. c Pn R , say and

is a

mcpil n

(0) c PI' c

in

R , are there

... c P n ' , such

depth P.'I = depth Pi , for

that there exist

I] and its proof.

there may be only finitely many choices

infinitely

many

It is also known

for a given

Pi

i = I,

mcpil n

[R-22]

that

For example,

98

if

n = 3

and

height P2 = 3

(and little height P2 = 2)

may be only finitely many prime Q = 3

and

ideals

Q

in

R

, then there

such that

height

such that

R

little height Q = 2

(15.4.4)

If

R % S

therian and

S

in

(0) c P1 c

R , say

are q u a s i - l o c a l

is integral

P.1 = d.1 , then does

over

domains

R , and if there exists a

... c Pn ' such that

mcpil n

height Pi = hi

there exist such a chain in

S ?

is Noe-

Does

and

depth

the converse

hold?

Concerning a

mcpil n

(15.4.4),

in

R

(15.4.5) a

mcpil n

mcpil n where

in

S

If

mains and

R[X](M,X )

the property m e n t i o n e d that if

R

as above

inherits

(4.1.2)

(where

is catenary

Does

R

and

by

R

inherit

domain

N

R'

only if SN

inherit

is a maximal

ideal

this property? local do-

local domain)

It is also known

[R-12,

is quasi-local,

from

have

(4.8.2)]

then each

SN

R .

in (15.4.5) (1.5.1) =

ideal

and

S .

that there exists

(if and)

is an arbitrary

is closely

N

related

to the

(1.5.4).)

there exist a N o e t h e r i a n

and has a m a x i m a l

R

in

(4.1.3) I that Henselian

(R,M)

this property

(3.8.5),

mcpil n

the property

over

in (15.4.5).

(Note that the property

(15.5)

has

that there exists

R , does every integral

has this property and

Upper C o n j e c t u r e

R

is integral

JR-12,

(3.1)]

extension domain of

Does every locality over

It is known

[R-10,

if there exists a

(4.6)].

in some integral

this property, S ?

if and only

[R-12,

there exists a

in

it is known

Hilbert D l - d o m a i n

such that

A

that

I < height N < alti-

tude A ?

If the answer holds.

(For,

if

to (15.5) (R,M)

is no,

then the Catenary Chain Conjecture

is a catenary

local domain,

then

D = R[X](M,X )

97 is a local zero

D2-domain , by

f E (M,X)D

D[i/f]

, D[I/f]

is a Noetherian

Corollary

is a

Hilbert

3.11 and Theorem

catenary.

Now,

E [I , altitude Lemma

(A.9.1)

2.1]

(15.6)

(A.9.16).

Therefore,

Dl-domain , by domain.

4.11],

Also,

by

it can be proved

that

D[i/f]

, then it follows

f

is arbitrary)

Chain Conjecture [R-3, Remark

holds,

5.12].

in

and

D[i/f]

R}

N

that

by

D

Questions

D[i/f]

and

[R-4, is

, height N

from

is an

(II.i.ii)

2.1],

(1.2.2)

ideal

= altitude

for each non-

JR-4, Lemma

if, for each maximal

(and since

the Catenary

=

[R-4,

~-domain,

so

= (II.I.i).)

on unmixedness

of a local

domain. (15.6. i) algebra,

is

If R

(15.6.2) is

R

satisfies

R

is quasi-unmixed

R (I)

is a finite

R-

(15.6.3)

If

R

is integrally

(15.6.4)

See the questions

If

is a finite

and

R (I)

is a finite

R-algebra,

unmixed?

R (I)

does not exist a height 5.7(i)]. answer

Therefore,

to (15.6.1)

(A.II.4).

Also,

mentioned

R-algebra

one maximal

is yes, by note

it is known

is a finite

closed,

(3.2),

that if

and

unmixed?

in (2.3) and

in

R'

(ii.i.I)

that

=

, then R (I)

[R-3, Lemma 5.11(1)]

R-algebra,

R

(2.4).

R > 1 , then there

, by

Chain Conjecture

R = R'

to assume

is

altitude

ideal

if the Catenary

it is not necessary

Finally, R (I)

and

quasi-unmixed? If

R

case

the f.c.c,

[R-3, Corollary holds,

(11.1.4),

then the

and

(A.II.I)

R (I) = R , so in this is a finite

that if

R

R-algebra.

is unmixed,

and it is shown in (B.5.2)

then

that the con-

verse does not hold. (15.7) separated

(D. Rush)

maximal

Concerning

Does every

Cohen-Macaulay (15.7),

Henselian

module

M

local domain such that

in [Hoc, p. 69], M. Hochster

R

have a

Supp M = Spec R ? asked

if maximal

98 Cohen-Macaulay modules with good properties can be constructed, and some affirmative results have recently been obtained. [Ru, Corollary 2]:

if

R

One is given in

is a complete equicharacteristic

that has a maximal Cohen-Macaulay module as in (15.7), then fies the f.c.c.

local ring R

satis-

If this continues to hold for Henselian local domains,

then by (4.1.1) = (4.1.3), the Chain Conjecture is translated into a problem in homological algebra by (15.7).

Since many results are cur-

rently being discovered in the area of Big Cohen-Macaulay modules, and since powerful homological methods are readily available, an affirmative answer to this question may go a long way toward settling the Chain Conj ec ture.

APPENDIX A A SUMMARY OF KNOWN EQUIW~LENCES FOR CERTAIN CHAIN CONDITIONS In this Appendix we list a number of characterizations the chain conditions considered

in these notes.

of nine of

These lists can be

used to give formally different characterizations

of the various charac-

terizations of the chain conjectures given in Chapters 4 - 13, and they can be used in like manner for the other chain conjectures mentioned in Chapter 3. Some of the lists are quite long, but in no case is the list exhaustive - that is, quite a few other such characterizations conditions have appeared in the literature.

However,

the lists do in-

clude at least most of the more important characterizations, indicate the variety of the characterizations has appeared in the literatu~e.

of these

and they

of each condition that

(A few of the characterizations

are,

in fact, new, but these follow quite readily from previously known characterizations.) statements

In the lists an attempt has been made to group the

that are concerned with similar concepts near to each other.

Also, an effort has been made to keep approximately appearance of similar concepts characterizations

are weak,

verify than the definition. show some unexpected

the same order of

in the different lists.

Some of the

in the sense that they should be easier to Others are strong,

things hold,

in the sense that they

if the characterized condition holds.

Before giving the lists, we need the following notation and lemma. (A°I)

NOTATION.

be analytically Let

Let

(R,M)

independent elements

Z = (0):biR , for all large

modulo a depth cal ring

Z , and let k

be a local ring and let in

i , let

R , so o

A = (R/Z)[Cl°/b ° . . . . c k o /b°]

by

R(Cl/b .... ,Ck/b )

is not nilpotent.

denote residue class

prime ideal, by [R-4, Remark 4.4(i)], A(M/Z)A

b

b,Cl,...,c k

Then

(M/Z)A

is

so we denote the lo-

I O0

The following lemma will help clarify the applications of (A.3.1) (A.3.7) and (A.3.1) = (A.3.9) in Chapter 4. (A.2) then

LEMMA.

b,c

If

(R,M)

are analytically independent in

exists a maximal ideal in

M'

in

R'

R

b,c

are in

M ,

if and only if there

such that

c/b

and

b/c

are not

R' M, Proof.

in

is a local domain and

R

By [R-3, Lemma 4.3],

if and only if

holds if and only if maximal ideal lent to

c/b

MR[c/b]

in

and

b/c

are analytically independent

is a depth one prime ideal°

M'R'[c/b~

M'

b,c

Now this

is a depth one prime ideal, for some

R' , by integral dependence, and this is equivaare not in

R' M,

, by [ZS-2, Corollary, p. 20],

q.e.d. Before each of the next nine remarks a summary of the results in Chapters 3 - 13 in which the condition has appeared is given. cases a secondary list is included in brackets

(In some

[ ] listing results for

which the characterized condition is only partly applicable - such as R'

satisfies the c.c., for (A.II). For (A.3), see:

(3.11.2),

(3.11.3),

(6.3), (11.1.8), and (13.2.2).

(3.3.6), (3.12),

(11.1.9),

[See also:

(3.3.7),

(4.1.4),

(4.1.5),

(II.I.I0), (3.1.4),

(3.9.4),

(4.1.9), all of (6.1),

(Ii.I.ii),

(3.2.1),

(3.10.2), (3.10.3),

(11.1.12),

(3.9.2),

(11.1.13),

(3.9.3),

(3.10.1),

(3.14.2), and (11.1.7).] (A.3) cal ring

where

REMARK. (R,M)

The following statements are equivalent for a lo-

, where is an

a = altitude R :

(A.3.1)

R

(A.3.2)

There exists

g = [S ; S

Hi-ring. S

in

depth z ~ i] .

such that

S

is an

Hi-ring ,

is a quasi-local integral extension ring of

that, for each minimal prime ideal or

g

z

in

S , either

z@R

R

such

is minimal

101

(A.3.3)

Every

S

in

g

is an

Hi-ring , w h e r e

~

is as in

(A.3.2). (A.3.4)

R/(Rad R)

(A.3.5)

For each fixed

such that a-j

or

j

(0 ~ j ~ i) is an

and for each

Hi_j-ring

p E Spec R

and either

depth p =

depth p ~- i-j Every

can be extended

in

H.-ring. l

height p = j , R/p

(A.3.6) R

is an

set of

i+l

to a set of

analytically a

independent

analytically

elements

independent

in

elements

R . (A.3.7)

independent is an

For each fixed elements

k (0 ~ k-~

b,c I ..... c k

in

Hi_k-ring and its altitude (A.3.8) If

R(X)

i = 1

= R[X]MR[X ]

and

R

are also equivalent (A.3.9)

and for all analytically

R , R(Cl/b .... ,Ck/b )

is either

is an

= a-k

to the following

or is

(see (A.I))

~ i-k .

Hi-ring.

is an integral domain,

For all a n a l y t i c a l l y

altitude R[c/b]

i)

then these statements

statements:

independent

elements

b,c

in

R ,

= a . .u

(A.3.10) quotient ideals

For all elements

field of N

in

R[u]

(A.3.11) then

height

in

such that such that

For all (M,u)R[u]

If, moreover, ideals

R

R'

u

in an algebraic

MR[u]

# R[u]

closure

F

of the

and for all m a x i m a l

MR[u] ~_ N , height N £ [l,a}

u E F

as in (A.3.10),

if

(M,u)R[u]

# R[u]

,

£ [l,a}

either

a = 1

or there are no height one maximal

, then these statements

are also equivalent

to the follow-

ing statements:

R

(A.3.12)

R'

(A.3.13)

There exists a finite algebraic

such that

is an

altitude

H-domain.

S = a

(A.3.14)

With

u , F

(A.3.15)

With

u

and

and , and F

S' N

is an

extension domain

S

of

H-domain.

as in (A.3.10),

as in (A.3.11),

height

height N = a . (M,u)R[u]

= a .

102

(A.3.16) of

R

Altitude R[u] = a , for all

such that (A.3.17)

D = R[X] (M,X)

(A.3.18)

If

is an

p E Spec R[X](M,X) then

R'

H-domain. has height one and contains a

depth p = a .

(A.3.19)

R' = N[V ; (V,N) E ~}

(A.3.20)

J' = n[N ; (V,N) ~ L'} , where

is the Jacobson radical of Proof.

in the quotient field

i/u ~ J' , the Jacobson radical of

linear polynomial,

J'

u

, where

is as in (4.2.2). is as in (4.2.2) and

R'

(A.3.1) - (A.3.8) are all equivalent,

by [RP, Section 2

and (3.15) ]. (A.3.1), JR-21,

(A.3.9)

(9.12),

- (A.3.11),

and (A.3.13)

are all equivalent,

by

(9.13), and (9.5)].

(A.3.1) = (A.3.12) = (A.3.17), by Finally,

(A.3.1),

all equivalent,

(A.3.14)

by JR-17,

For (A.4), see:

-(A.3.16),

(2.11),

(6.2.1),

[R-6,

(2.8),

(3.2)].

and (A.3.18)

(2.9),

(A.3.20) are

(2.6), and (2.7)], q.e.d.

(6.2.2), and (11.1.5).

[See also

(3.2.5). ]

(A. 4) cal ring

REMARK. (R,M)

The following statements are equivalent for a Io-

, where is a

a = altitude R :

(A.4.1)

R

Ci-ring.

(A.4.2)

There exists

S C g

such that

S

is a

Ci-ring , where

is as in (A.3.2). (A.4.3)

Every

(A.4.4)

R/(Rad R)

(A.4.5)

For each fixed

such that a-j

or

S E g

is a

is a

height p = j , R/p

Ci-ring , where

C.-ring. i j (0 --< j :< i) is a

is as in (A.3.2).

and for each

Ci_j-ring and either

depth p =< i-j

(A.4.6)

R(X) = R[X]MR[X]

is a

(A.4.7)

R[X] (M,X)

Hi+l-ring.

is an

g

Ci-ring.

p E Spec R depth p =

103

(A.4.8)

For all

an

Hi+l-ring, where

of

R

= ~(R,bR) ~

bR

For all

b

and

R

~ = ~(R,bR)

is

the Rees ring

• = (M,u,tb)~

as in (A.4.8),

~

ideal in

is an R

H.-ring,~ where

with respect to

bR

~ .

then these statements are also

to: There exists

cally independent elements is a

k (i ~ k < i) b,c I ..... c k

such that, for all analyti-

in

R , R(Cl/b ..... Ck/b)

(see

Ci_k-ring.

(A.4.11)

For all

pendent elements

k (I ~ k ~ i)

b,c I ..... c k

in

and for all analytically

R , R(Cl/b .... ,Ck/b)

inde-

(see (A.I))

Ci_k-ring. (A.4.12)

For each

(M,y)R[y] # R[y] a

with

is an integral domain,

(A.4.10)

=

height bR = 1 , £(R,bR)

is the associated graded ring of

equivalent

is a

= ~

is the maximal homogeneous

If

(A.I))

such that

£(R,bR)

with respect to (A.4.9)

and

b E M

or is

~

Proof.

i

y

in the quotient field of

, R[Y](M,y )

is an

R

such that

H.-ringl and its altitude is either

.

(A.4.1)

(A.4.5),

(A.4.7), and (A.4.12) are all equiva-

lent, by [RP, Section 3]. (A.4.1) = (A.4.6) = (A.4.10) = (A.4.11), by JR-15, and,

(A.4.1) = (A.4.8) = (A.4.9), by JR-9, (3.1.2),

(4.1) and (4.2)];

(2.14.3), and (2.15)],

q.e.d~ For (A.5), see: (4.1.10),

(5.3.2),

[See also: (5.2),

(3.4.1),

(5.4),

(3.8.2),

(3.4.2),

(3.8.5),

(3.14.3),

(4.1.7),

(7.2), all of (9.1), of (9.2), and of (9.3).

(3.8.3),

(3.8.4),

(3.14.1),

(3.14.4),

(5.1.2),

(8.1.2), and (18.3.2).]

(A.5) local domain (A.5.1)

REMARK. (R,M)

The following statements are equivalent for a :

There exists a

mcpil n

in

R .

104

(A.5.2) of

R

There exists

a quasi-local

such that there exists (A. 5.3)

Every

integral

(A. 5.4)

There

exists

There

exists

integral

a mcpil n extension

in

domain

S

S .

domain

P C Spec R

extension

S

of

such that

R

has a mcpil n .

height P = n-I

and

depth P = 1 (A.5.5)

a depth one prime (A.5.6) P

Every

such that

integral

domain of

height P = n-i

extension

domain of

(A.5.7)

There

exists

(A.5.8)

There

exists a quasi-local

of

R

a mcpil

Every quasi-local

is such that there exists Proof. [R-10

(A.5.1),

(3.1)

(A.5.1) (A.5.3)

=

n

such that there exists

(A.5.9)

by

P

extension

R

R

that has

. has a prime

ideal

as in (A.5.5).

(S,N)

R

ideal

an integral

and

in

integral

integral

and

= R[X]MR[X ]

a mcpil n

a mcpil n

(A.5.2),

R(X)

extension

in

extension in

S(X)

(A.5.7)

domain

S(X) = S[X]Ns[X ] domain

(S,N)

of

= S[X]Ns[X ]

(A.5.9)

are all equivalent,

(3.10)].

=(A.5.3),

by the Going Up Theorem,

and it is clear

that

(A.5.1).

(A.5.1)

=

(A.5.4),

by

[Mc-l,

Theorem

5], and clearly

(A.5.4)

=

(A. 5. l). Finally, = (A.5.6), For (7.2),

by

it is clear [R-10,

(A.6),

(7.3),

(A. 6)

that

(2.8)],

see:

REMARK.

=

(A.5.4)

= (A.5.5),

and

and

(3.6.3),

(3.6.4),

(3.7),

all of (7.1),

(7.5).

The following

statements

are equivalent

for a ring

A : (A. 6. I)

A

(A.6.2)

Every

is a

GB-ring.

integral

(A.5.5)

q.e.d.

(3.6.2),

all of (7.4),

(A.5.6)

extension

ring of

A

is a

GB-ring.

105

(A.6.3)

A/z

is a

GB-ring,

for each minimal

prime

ideal

(A.6.4)

A/I

is a

GB-ring,

for each ideal

(A.6.5)

AM

is a

GB-ring,

for each maximal

(A.6.6)

AS

is a

GB-ring,

for each multiplicatively

z

in

A .

set

S

(0 ~ S)

If

A =R

in

lent to the following

Every principal

to adjacent

prime

(A.6.8)

R/p

A

ideal

M

. in

A .

closed

sub-

then these

statements

are also equiva-

statements:

is such that all pairs R

in

A .

is a local domain,

(A.6.7)

I

integral

of adjacent

extension

prime

domain

ideals

in

R[c]

R[c]

of

R

contract

in

ideals.

is a

GB-ring,

for each height one

p ~ Spec R , and

R (I) c R' (A.6.9)

For all

in some integral a mcpi! n

in

p c q

extension

in

Spec R , if there

domain of

L = (R/p)q/p

For all

p c q

in

Spec R

there does not exist a height one maximal

(3.3),

(A.6.1)

and

For

, then there exists

(A.6.6)

such that ideal

in

are all equivalent,

height q/p > i , ((R/p)q/p)'

by

[R-16,

(3.1),

(3.5)].

(A.6.1) = (A.6.1) ~

a mcpil n

L .

(A.6.10)

Proof.

exists

(A.6.7) = (A.6.9),

(A.6.8) ~ (A.7),

(A.7) local ring

see:

REMARK° (R,M)

, where

R

(A.7.2)

There

and such that

(6.1.9)

by

JR-16,

JR-19,

and

(4.2)

(5.3) and

and

(4.3)];

(5.2)],

and

q.e.d.

(10.1.6).

The following

(A.7.1)

that minimal

(A.6.10),

by

statements

a = altitude

are equivalent

for a

R :

is taut. exists

prime

ideals

S

is taut.

an integral in

S

extension

lie over minimal

ring prime

S

of ideals

R

such in

R

106

(A.7.3) mal prime

Every

ideals

in

extension

(A.7.5)

For all

P E Spec R

(A.7.7)

Every maximal

is an

satisfies

or

a

and has altitude

in

R

is taut.

P +

i = 1 .... ,a-2

= a-k

R(X)

(A.7.11)

For all

the f.c.c,

p

elements

and

elements

altitude

independent (see

b,c

R(c/b)

elements

(A.I))

in = a-i

R , .

b,Cl,...,c k

satisfies

the f.c.c.

.

= R[X]MR[X ]

ideal

independent

elements.

, R(Cl/b ..... Ck/b )

(A.7.10)

prime

such that mini-

depth P ~ 1 , height

independent

For all analytically

(I ~ k < a)

geneous

R

ideals

set of analytically

For all analytically

(see A.I))

R

prime

such that

Hi-ring , for

has either one element

(A.7.9)

of

.

R

R(c/b)

S

is taut.

(A.7.6)

(A.7.8)

ring

lie over minimal

R/(Rad R)

R

in

S

(A.7.4)

depth P E [l,a]

in

integral

is taut.

b 6 M

such that

height bR = 1 , every homo-

in

such that

height p > 1

~

height ~/p = a + 1 - height p , where

~ = ~(R,bR)

is such that

and

~

are as in

(A. 4.8). Proof.

(A.7.1)

(A.7.3)

are equivalent,

by

[McR-2,

Proposition

121. (A.7.1), [RP,

by

and

(3.15),

(2.3.4),

(A.7.1)

= (A.7.5),

JR-9, For

(3.18.1)], (A.8),

(3.14.4), and

(A.7.4),

(2.16.2), by

- (A.7.10)

(2.14.2),

[R-14,

are all equivalent,

(2.14.3),

(2.13.1)];

and~

and

by

(2.8.2)].

(A.7.1)

= (A.7.11),

(3.10.3),

(3.14.2),

q.e.d.

see:

(6.1.3),

(A.7.6)

(3.1.5),

all of

(3.9.4),

(i0.I),

and

(3.9.5), (11.1.6).

[See also:

(6.1.2)

(12.1.2) . ] (A.8)

local

domain

REMARK.

The following

R , where

statements

a = altitude

R :

are equivalent

for a semi-

107

a

(A.8.1)

R

(A.8.2)

There exists a taut integral

(A.8.3)

Every

(A.8.4)

For each depth one

(A.8.5)

For each non-maximal

and

Rp

domain of

R

R .

is taut. .

P E Spec R , height P + depth P =

the s.c.c. b

in the Jacobson radical of

R , Rb

the s.c.c. There exists a non-zero

such that

Rb

(A.8.8) (that is, ideal

extension domain of

P E Spec R , height P = a-i

For each non-zero

(A.8.7) R

integral e x t e n s i o n

satisfies

(A.8.6) satisfies

is taut.

in

are a n a l y t i c a l l y R)

R[c/b]

and

S

maximal

ideal in

, AS

independent

independent

is taut and

is the complement R

in the Jacobson radical of

the f.c.c.

For all a n a l y t i c a l l y

b,c

M

satisfies

b

such that

elements

in

RM

b,c

A

in

R

, for some maximal

altitude A S = a-i in

b,c

, where

of the ideals

are analytically

MA

A

=

with

M

independent

a

in

RM

(A.8.9)

For all

dent elements wh~re

k = l,...,z-2

b,Cl,...,c k

in

A = R[Cl/b,...,Ck/b]

ideals

MA

with

analytically

R(X)

~deal

Proof.

in

and

(2.13.2),

(3.10),

(3.21),

(13.2.1). (3.10.1), (12.2).]

see:

(6.1.8),

RM

altitude A S = a-k

is the complement R

such that

in

A

,

of the

b,Cl,...,c k

are

.

is taut, where

(A.8.4)

S = R[X]

are equivalent by - (A.8.10)

(4.17),

(3.3.1),

(10.1.6),

[See also: (4.4.2),

S

(A.8.8)),

ideal in

- (A.8.3)

(A.8.1)

(4.1.3),

(see

indepen-

- U[MR[X]

; M

is

R}

(A.8.1)

(A.9),

in

= R[X] S

12]; and,

For

and

a maximal

independent

(A.8.10) a maximal

M

R

and for all analytically

(i0.I.i),

and

(3.3.9),

(10.1.2),

(4.10.1)],

(3.3.8),

all of (II.i),

(3.1.6),

Proposition

are all equivalent,

(4.20),

(3.3.6),

[McR-2,

(3.5.2),

(3.7),

(12.1.4),

(3.9.2),

all of (11.2),

[R-14,

q.e.d.

(3.5.1),

(12.1.3),

by

(3.12),

and

(3.9.5),

(12.1.2),

and

108 REMARK.

(A.9) local

R

domain

(R,M)

(A. 9. i)

R

(A. 9.2)

There

that

The f o l l o w i n g , where

a = altitude

is c a t e n a r y exists

statements

are e q u i v a l e n t

for a

R :

(equivalently,

a quasi-local

satisfies

integral

the f.c.c.).

extension

domain

of

is catenary.

(A.9.3)

All q u a s i - l o c a l

integral

extension

domains

of

R

are

catenary. (A.9.4) + depth

and

For all

P C Spec R

(A.9.5)

For all

(A.9.6)

With

height

P E Spec R , h e i g h t

at m o s t

p = I , then

and

R/p

(A.9.8)

Rb

satisfies

the f.c.c.,

for some

(A.9.9)

Rb

is level,

for some

(see

For all

a

P

(see

p E Spec R

is catenary.

0 # b E M

.

0 # b E M

0 # b E M

of p a r a m e t e r s

if

.

.

Xl,...,x a

in

R , height

i = i ..... a . set of a n a l y t i c a l l y

independent

elements

elements.

For all a n a l y t i c a l l y (A.I))

is c a t e n a r y

For each

and

For e a c h

and

indepen]e~7 altitude

k = l,...,a-i

elements

is c a t e n a r y

(A.9.14)

systems

Each m a x i m a l

independent

b,Cl,...,c k

altitude

elements

R(e/b)

in

R

R(X)

(A.9.16)

Every

= R[X]MR[X ] depth

= a - I .

in

R , R(Cl/b,...,Ck/b )

, altitude

= a-k

(see

.

R ( C l / b ..... Ck/b ) =

is catenary.

two p r i m e

ideal

in

D = R[X](M,X)

has

= a - I .

Proof.

(A.9.1)

- (A.9.3),

(A.9.7),

,

and for each set of a n a l y t i -

R ( C l / b .... ,Ck/b)

k = I .... ,a-2

b,c

(A.I)).

(A.9.15)

height

depth p = a - I

for all

(A.9.13)

a-k

exceptions,

the s.c.c.,

(A.9.12)

(A.]))

many

satisfies

contains

cally

finitely

Rb

(A.9. ii)

R(c/b)

P = I , height

P + depth P = a

(A.9.7)

(Xl,... ,xi)R = i , for

R

depth

P = a .

(A°9.10)

in

such that

(A.9.8),

(A.9.11),

(A.9.12),

109

and

(A.9.15), (A.9.1)

2.6(i)]; Remark

are all equivalent = (A.9.4) =

(A.9.1)

(A.9.1)

by

For

(A.IO) semi-local

(4.15)];

and,

(11.1.3).

The following

a = altitude

(A.IO.2)

R

is taut and all finitely R

R'

(A.IO.4)

There

All

(A.IO.6)

R

R

(A.9.1)

=

(10.1.4).]

are equivalent

for a

R :

the o.h.c.c. generated

algebraic

exten-

satisfies

the o.h.c.c.

exists an integral

extension

domain of

R

that

integral

extension

domains

is taut and all principal

of

R

the o.h.c.c.

integral

satisfy

extension

domains

are catenary.

(A.10.7) such that (A.10.8) and

(A.10.9)

R

is taut and all integral

2

c -rc E R , for some For each maximal altitude R

extension

(A.IO.IO)

R[c]

of

ideal

M

in

R , RM

satisfies

the

RM ~ [l,a}

is taut and,

R

domains

r E R , are catenary.

for all nonzero

depth P > 1 , there does not exist a height

zero

(2.21)];

the o.h.c.c.

(A.10.5)

o.h.c.c,

[R-D,

are catenary.

(A.IO.3)

of

[R-II,

[See also

statements

satisfies

of

and

13], q.e.d. and

R , where

[R-4, eemma 3.16] by

JR-4,

in [R-4].

2.2 and Remark

=~ (A.9.9),

R

satisfies

R

by

(A.10.1)

sion domains

R[c]

(6.1.4)

REMARK. domain

by

Corollary

(A.IO), see:

[R-5, Theorem

(A.9.1)

= (A.9.14),

[Hou-l,

by

~ (A.9.10),

respectively;

= (A.9.13)

(A.9.16),

(A.9.5),

= (A.9.6)

2.6(ii)],

by the main theorem

is taut and

R/P

P C Spec R

one maximal

satisfies

such that

ideal

the s.c.c.,

in

(R/P)'

for all non-

P C Spec R . (A.10.11)

m a i n of

If there exists a mcpil

R , then

(A.10.12)

RH

n £ [l,a] is taut.

n

in an integral

extension

do-

11o

(A.10.13)

RH

(A.10.14)

R

satisfies

the o.h.c.c.

is taut and

depth P = depth PNR

, for all nonminimal

P E Spec R H (A.IO.15)

R

is taut.

(A.10.16)

R

satisfies

(A.10.17)

R(X)

U[MR[X]

; M

satisfies

is a maximal

(A.10.18)

the o.h.c.c. the o.h.c.c.,

ideal in

For each maximal

and there exists

i

is an

for

H.-domain, J

ideal

(A.10.20)

For all

M

in

such that

j = i+l,...,a-i

(A.IO.18)

R(X)

= R[X]

-

R}

(i ~ i ~ a-2)

(A.10.19)

where

holds w i t h

R , height M E [l,a] RM[X I .... ,Xi](M,X I .... ,Xi)

.

"there exists"

replaced by "for

all."

maximal

ideals

M

is catenary and (A.IO.21) R ~ V, N~R

in

y R

in the quotient such that

Every DVR

(V,N)

is a maximal

M'

in

(A.I0.22)

to

I

ideal,

and

ideals

have

ideals

I

relevant

ideal in case

q.e.d.

, A = R[y] (M,y)

field of

R

such that

is integral over a locality over

V = R' M,

, for some height one maxi-

I

in

R

such that graded

depth I = 0 , all

ring of

R

with

depth E [l,a]

spect to

to not containing

V

in the associated

For all ideals

relevant

Proof.

and for all

R'

have

(A.I0.23) maximal

# R[y]

in the quotient

For all ideals

the minimal prime respect

(M,y)R[y]

R

altitude A E [l,a}

R, is either of the first kind or mal ideal

field of

I

in

R

in the Rees ring

height E [l,a} H

such that ~ = ~(R,I)

(An ideal

is homogeneous

H

in

depth I = 0 , all of ~

R

w i t h re-

is a maximal

and is m a x i m a l w i t h respect

tl . )

These are all shown to be equivalent

in [R-18,

Section 4],

11t

For

(A.II),

(11.1.4), (3.3.2), and

see:

(12.1.3), (3.3.8),

(3.3.1),

(3.3.9),

all of (13.2), (3.7),

and

(4.4.3),

(3.5.2),

(13.3).

(10.1.3),

(4.1.2), [See also;

(10.1.5),

(6.1.5), (3.1.7)

(11.2.3),

(12.1.2),

(12.2).] (A.II)

REMARK.

local domain

The following

(R,M)

, where

statements

a = altitude

are equivalent

for a

R :

(A.II.I)

R

satisfies

the s.c.c.

(A.II.2)

R

satisfies

the altitude

(A.II.3)

R

satisfies

the dominating

(A.II.4)

R

is quasi-unmixed.

(A.II.5)

All finitely

generated

There

an integral

formula. altitude

integral

formula.

domains

over

R

satisfy

domain of

R

that

R

the s.c.c.

the c.c. (A.II.6) satisfies

extension

the s.c.c.

(A. II.7)

All

(A.II.8)

All principal

satisfy

integral

extension integral

domains

of

extension

satisfy

domains

R[c]

of

R

the f.c.c.

(A.II.9) some

exists

All

R[c]

r E R , satisfy (A.II.10)

R

as in (A.I!.8)

such that

c 2 - rc E R , for

the f.c.c.

is catenary

and,

for all

p C Spec R

such that

depth p > 1 , there does not exist a height one maximal (A.II.II)

R

(A.II.12)

R'

for all

is a

Ci-ring , for

satisfies

i = O,l,...,a-2

the f.c.c,

and

R/p

ideal

in

(R/p)'

.

satisfies

the s.c.c.,

(0) # p E Spec R .

(A.II.13)

R

is catenary

and is a

(A.II.14)

RH

satisfies

the f.c.c.

(A. 11.15)

R

is catenary

(A. Ii. 16)

R

satisfies

(A. ii. 17)

R(X)

and

GB-domain.

depth P = depth P~R

Spec R H . the f.c.c.

= R[X]MR[X ]

satisfies

the s.c.c.

, for all

P E

112

R[X]

(A.II.18)

R[X]

is catenary°

(A.II.19)

D = R[X](M,X)

(A.II.20)

R<X>

is taut-level,

- U[N E Spee R[X] (A.II.21)

is an

(A.II.22) (M,y)R[y]

for

i

is proper,

y

V

with

S =

.

.

in the quotient

A = R[Y](M,y )

, and

= R[X] S

(i ~ i ~ a-l), R[X I ..... Xi](M,X 1 .... ,Xi)

Every DVR (V,N)

R ~ V, NNR = M

R<X)

j = i, i + l,...,a-i

For all

(A.II.23)

where

; MR[X] c N]

For some

Hj-domain,

is catenary.

field of

R

is catenary and

in the quotient

such that altitude A = a .

field of

R

such that

is integral over a locality over

R

is of the

first kind. (A.II.24)

For each ideal

has a basis of the integral

closure

B

= e(B)

(with

R

w i t h respect

to

(A.II.28)

to

I

I

ideals

, for all large

have

I

relevant

C ~ B

in C

ideals

R

(In)a ,

such that

is a reduction of

M-primary

in the associated

ideals

graded

I)

ring of

the same depth. I

(or, for all

ideals

M-primary

in the Rees

ring

ideals

~(R,I)

of

I) R

have the same height.

There exists an

M-primary

p'

of

ideal and

I

in

R

such that

height p'N~(R,I)

= 1 ,

u~(R,I)'

There exists an

M-primary

is generated by a system of parameters

ideal and

I

in

R

such that

~[i/u]n~ (I) c ~'

, where

= ~ (R,I)

(A.II.30)

I

n)

(or, for all

is generated by a system of parameters

(A.II.29)

(that is,

height = height I .

denoting multiplicity),

For all ideals

for all prime divisors

I

M-primary

class

R , every prime divisor of

I n , has

For all ideals

R , all maximal

w i t h respect

I

e

of

R , all the minimal prime

(A.II.27) in

R

CB n = B n+l

(A. II.26) in

in

For all

(that is,

of the principal

height I e l e m e n t ~ in

(A.II.25) e(C)

I

For all ideals

I

in

R, ~ [ i / u ] ~

(I) c ~'

, where

113

= ~ (R,I)

(A. ii. 31)

For all

b E M, £(R,bR)

is taut, where

£(R,bR)

is as

in (A.4.8). (A. ii. 32) Proof. and,

L (I) c L'

, for all localities

(A.II.I) = (A.II.2) = (A.II.4),

(A.II.3) = (A.II.4), (A.II.I)

= (A.II.5)

orem 3.6]; and,

R .

by [R-2, Theorem 3.1];

= (A.II.8)

by [R-2, Corollary

= (A.II.9),

It is clear that (A.II.I) = (A.II.6), [N-6,

over

by JR-5, Theorem 3.3]. = (A.II.18),

(A.II.I)

L

3.7 and The-

by JR-2, Theorem 3.11].

and (A.II.6) = (A.II.I), by

(34.2)]. (A.II.I)

= (A.II.12)

= (A.II.7),

= (A.II.IO)

by the definition

= (A.II.I),

of s.c.c. ; and,

by [R-6,

(A.II.7)

(4.3)] and [R-3, Theorem

2.19]. (A.II.I), and

JR-16,

(A.II.II),

(3.10)],

(A.II.19),

and

respectively,

by [R-3, Theorem

(A.II.14)

(A.II.13)

implies

R

are equivalent,

and (A.II.I) = (A.II.14)

(A.II.14),

by [McR-2,

(A.II.I),

is catenary

(A.II.4) Theorem

since

(since

[RMc,

JR-15,

(2.15)],

(2.6)],

P E Spec R H , so

- (A.II.32)

and Theorem 4.16],

(A.II.27)

and

and

(A.II.14)

(A. II.15)

im-

all equivalent,

5], [RP, (3.11)]

JR-13,

3.1 and 3.8]; and, finally,

(3.18.6)],

to (A.II.24)

to-

(2.11)].

are all equivalent,

are all equivalent,

and JR-9,

related

- (A.II.23)are

[Mc-2, Corollary

[RP, (3.16)],

and (A.II.24)

An example

= (A.II.I))

7].

and (A.II.20)

2.29] and JR-3, Theorems

and (A.II.28)

(A.II.14)

height P = height PNR , so (A.II.15) =

Proposition

(A.II.17),

by, respectively, gether with

= (A.II.16)

height P = height PAR). And, conversely,

plies R H is taut-level,

(3.11)]

2.21].

height P + depth P = altitude R H , for all = (A.II.15)(sin=e

by [RP,

by JR-8, (A.II.4)

by [Saw, Corollary

q.e.d.

is given in (B.5.6).

4.17

APPENDIX NOTES ON M. NAGATA'S

B

CHAIN PROBLEM EXAMPLE(S)

In this a p p e n d i x we first summarize M. Nagata's [N-3, Section 31

(and repeated

in IN-6, Example

family of local domains

to show that the answer

of prime

Then,

useful

ideals is no.

ideals, we prove a few properties

list a number of the ways in (B.3) - (B.5)• (B.I)

over ~<j

K+J

be elements

K(X)

, let

Let

Let

, where

J

with maximal

in

K[[XI]

A A _ ( M U N ) , R' and

in [N-3]

j > i , let

not regular•

Finally,

radical

= m+l

- (B.2.5) were

sion domains, (Bt2)

in [R-6,

REMARK.

i = I,

independent

, W = A N , R' = VNW of

R

e(J)

that

R

R =

is a local domain R , R' =

two maximal

of

J

facts concerning

is one

shown to hold

R

ideals

and

is catenary w h e n and only w h e n

essentially

, and

,

R'

, height NR' = r+m+l

the m u l t i p l i c i t y

We now list a few a d d i t i o n a l ((B.2.2)

and

zij = z i

is the integral closure of

, height MR'

Also,

hold:

for

(and in [N-6])

J , R'

K = R'/NR'

X,YI,...,Y m

that are a l g e b r a i c a l l y

is a regular domain w i t h exactly

NR')

of the rings.

A = K [ X , [ z i j ] , Y I , . . . , Y m ] , M = (X,YI,...,Ym)A

is the jacobson

ideal

been used in the litera-

aij.Xj (aij E K

zi = ~ i

to be very

in (B.2), and then

be a field and let

, N = (X-l,Zl,...,zr,Yl,...,Ym)A

It is proved

(MR'

K

Zil = z i , and,

aik~/XJ-i

V = AM

Let

of a

about saturated chains

We b e g i n w i t h the c o n s t r u c t i o n

CONSTRUCTION•

...,r > 0)

203-205])

in

to the chain p r o b l e m

of them,

they have previously

be indete~ninates •

(m ~ 0)

2, pp.

these rings have proved

in answering a number of other questions

of prime

ture,

since

construction

and

R'/MR'

and

R

=

is

m = 0 • R'

for all special exten-

(4.8)].)

W i t h the above notation,

the following statements

115

(B.2.1)

R'

(B.2.2)

R' = R[a]

(B.2.3)

For all

leJ)

or

R'/I

is a s p e c i a l

There

ideals

p # J

g i v e n by

(B.2.5) prime.

If

ideals

exists in

Rp = R ' p

and either:

(a)

; or,

fore,

in cases

Co) and

divisor

of

(b,c)R

and

For

is

(c)

if

in

k+l Since

prime

and

J c_ R .

Therefore

and so

b ~ R'

, so

such that

X-I ~ NR',

, then

X = (1/k)(a-r)

(J,X)R'

q = qR' (if

q There-

i m 2

bR'

m = 0

and

if

prime

= (bR)a =

al,.•.,a k

= I

and

= MR'

R'/MR' and

and

are

(al' ... '

= K = R'/NR',

b - k 2 E NR'

so

and

J =

(B.2.2). there

Thus,

since

+ (b-k2)X E

R' = R + XK = R[X] r E R

that

(J,X-I)R'

to p r o v e

= (b-kl)(1-X)

, for some , and

, it follows

it suffices

since

, so

E R + aK

then

if

= (MR')(NR')

b - k I + (kl-k2)X

a = r + kX

R'

is semi-

imbedded

, then

(a I , ... ,a k )R'

b - k I C MR'

b ~ R + X~

for

is the only

(1.1.13),

Then,

in

ideal•

MR'NNR'

by d e f i n i t i o n

Let

X E MR'

height

'

Hence,

if

0 # k C K ,

R' = R + XK c R + aK c R[a]

R' (B.2.3)

follows

easily

from

.

the

, ~ NR ')

p-primary,

~ bR

I c J)

= q~NMR'

q . c_ M R ,

m > 0 • R (I) = W

S , then

(B.2.1)

kl,k 2 E K

qR'

S = {XZl,...,Xzr,XYI,...,XYm}

is a h e i g h t

, ~ R

is not

= J .

Therefore,

a C R'

(if

pR'

= qR p NR'

; (b)

bR:cR

= (X2-X,Zl ..... zr,Y I ..... Y L~L )R'

exist

q

c E bR:J,

elements

(B.2.2)

and

Moreover,

If

NR'

, then

.

(B.2.8)

Proof•

in

~ J)

(if

P ~ {MR' ,NR'}

ideal

R

(if

between

domain.

,

if

R/(I~R)

ideals

b ~ J , J

bR

= R/(InR)

local

= q~nNR' qi

a E R', ¢ R .

correspondence

each n o n z e r o

R (I) = R'

ak,X-l)R'

q

.

R'/I

ring of

is a r e g u l a r

qR'

(c),

(B.2 7)

distinct

, either

the p r i m e

(if

R

, for all

extension

p-primary

= q*

, ~_ MR')

R'

is a p r i m e

q

qR'

in

and

Rp

p # J

NR'

(B.2.6)

I

a one-to-one

R

, so

In fact ' if

ring of

= R + a R = R + aK

is a special

(B. 2.4) prime

extension

(B.2.2)

and

the fact

that

J

is the

116

maximal

ideal

in

R , and

the conductor

of

R

If

q * c NR'

(B.2.5)

in

(B.2.4)

q = qR' = q*

(if

from the fact that

J

is

R' , then

= q ~ qR' ~ q*N J , so either or

follows

"~ = q*N (NR' NMR' ) = q"NJ = q^AR

q*NMR'

q = qR' m q*n MR'

q * c_ MR' ~NR'

= J)

(if

q * ~ NR, ~ MR')

(c) is proved

in a similar

way. (B.2.6) 2.3], and

Let

0 # b E J .

bX £ bR'nR, ~ b R

prime divisor of (see (B.2.4)),

bR

and

then

a primary decomposition of

bR

and

(since

~

bR c

it follows

that

of

R , so

Hence

If

R

and If

such that

Rp = R'

R'

(B.2.8)

,

follows

domain w h o s e maximal is a unit in

W

is a

ideal in

R'

, so from

is a prime divisor

J .

, by

, so

bRp

is the only imbedThen

cJ c bR

IN-6,

(12.3)],

, then

Therefore,

, so so

d/b E R'

for each

, and so

,

r' E R',

br' E (b,d)R

bR' = (bR) a = (b,d)R

, and

.

height MR' > i , so, by between

(B.2.4),

the height one prime

Rp = R' p,

the height one prime p'

and so

in

R'

, and so

there ideals

R (I) = R'(1)

W = R'NR,

ideal is generated by NR'

ideals

p

in

that are contained

R(1) = (R,NR,)(1)

from the facts:

; and,

p # J

height MR' = i , so there exists a one-to-

ideals "

J

so

r' = r + s(d/b)

(bR)a

if

prime

(bR)a , ~ hR

(B.2.2).

such that

between

p

c/b E R'

= (bR) a = (b,c)R

m = 0 , then

the height one prime

(B.2.4),

d E

correspondence in

one c o r r e s p o n d e n c e

X

, by

JR-8, Lemma

is a prime divisor of

c E bR:J, ~ bR

if

m > 0 , then

p'

Now,

that

pRp

Also,

such that

bR'

exists a one-to-one

= R'

.

(bR) a ~ bR' c_ (b,d)R ~

(B.2.7)

in

Also,

Finally,

that

it follows

Let

J = bR:cR

r,s E R

so it follows

p

bR

.

(bR)a

, by

(BR')R'p = BR'p = bRp

local ring, by

R' = R + (d/b)R

there exist

bR c

(bR)aR p ~

of

bR

c E b R ' N R = (bR)a

(bR)a = b R ' N R

is the c o r r e s p o n d i n g

(bR)a)

is a regular

ded prime divisor

, so P

bRp ~

Then

= R'

NR'

= W

R

in

and NR'

"

is a regular local

X-l,z I .... 'Zr'Yl'''''Ym

is the only maximal

ideal in

R'

;

that

117

contains

X-I

, q.e.d.

Quite a few additional and

R'

are proved

interesting

used in a number of papers

erties; and,

that show:

in the literature

are listed

the existence

that c e r t a i n hypotheses

that the converse

strict a t t e n t i o n

each remark. applications

in a N o e t h e r i a n

of rings

that have c e r t a i n prop-

We re-

m = 0 , in (B.3); m ~ 0 , in (B.4); and,

Also,

in (B.3)

- (B.5) are new,

a brief e x p l a n a t i o n

so a

as to why each

importance will be given following

as in A p p e n d i x A,

of these rings

ring.

for a k n o w n result cannot be weakened;

is of interest and/or Finally,

ideals

in (B.3) - (B.5), and among these

A few of the results

proof of these will be given. of the results

to answer c e r t a i n ques-

of c e r t a i n k n o w n results does not hold.

to the case:

m > 0 , in (B.5).

R

this family of local rings has been

saturated chains of prime

Some of these applications are results

facts concerning

in [D].

As has already b e e n mentioned,

tions c o n c e r n i n g

and useful

there are quite a few other

in the literature

that are not included

(B.3) - (B.5), but the lists do show the variety of uses

in

that have been

made of these rings. (B.3) and let

REMARK.

m = 0

Let the n o t a t i o n be as in (B.I),

Then the following

(B.3.1)

[McR-2, p. 74].

R

statements

is taut-level,

let

D = R[T](j,T),

hold: but

R'

is only taut,

not taut-level. (B.3.2)

R

satisfies

(B.3.3)

For all nonzero

and for all n o n - m a x i m a l does not satisfy (B.3.4) not satisfy (B.3.5)

the o.h.c.c.,

but not the s.c.c.

p ~ Spec R , R/p

P E Spec R , Rp

satisfies

satisfies

the s.c.c.,

the s.c.c., but

R

the s.c.c.

[R-2, R e m a r k 3.9].

R'

satisfies

the c.c., and

R

does

the c.c. [R-12,

(4.1.7)].

With

g

as in (9.2.4), D E S

, but

I18

D/TD

and

DjD

are not in

(B.3.6) fires

[R-II,

the c.c., (B.3.7)

is an

D[I/T]

H2-domain,

is not catenary}

height

in

[R-9,

J , £ = £(R,bR)

in

J , D[i/b]

satis-

the c.c.

(3.2)].

If

is not closed

Q = i = depth Q}

b

If

r = 2 , then

D

Hi-domain.

3.1].

[HMc, Example

(B.3.10)

preceding

but is not an

[HMc, Example

(B.3.9)

For all nonzero does not satisfy

[R-15, comment

(B.3.8) Dp

but

(2.16)].

r = I , then

in the Zariski

3.2].

If

topology.

r = i , then

is an infinite

(3.18.5)].

H = [P E Spec D ;

If

set, but

(J,T)D # U[Q

r = i , then,

is homogeneously

I = [Q E Spec D ;

taut-level,

; Q E I}.

for all nonzero but

£

b

is not an

H-domain. (B.3.11) F

is zero,

[Fu-l,

then

that satisfies

Lemma

r = i

[R-18, B[T]

(3.2)

then

o.h.c.c.,

but the following

is taut;

the o.h.c.c.,

for all

the o.h.c.c.,

but

is a Noetherian

and

(3.3)].

is a Noetherian

BIT]'

and the characteristic Hilbert

of

domain

but not the s.c.c.

(B.3.11),

B[T]'

If

B = PJIK[X,zI,I/X]

the f.c.c,

(B.3.12)

6].

and

domain

the o.h.c.c.;

does not satisfy

is as in

B[T]

and,

Moreover,

B

that satisfies

do not hold:

p C Spec B[T]

B[TI,T2]/P

r = i

Hilhert

statements

satisfies

If

the

is catenary;

B[T]p

satisfies

B[TI,T 2]

satisfies

the o.h.c.c.,

for some

P E Spec B[TI,T 2] Proof. and

R'

(B.3.2)

satisfies

satisfies

R

is taut

the c.c.,

the o.h.c.c.

R

(since

since

R

R'

is catenary

is a regular

does not satisfy

(since

domain,

the s.c.c.,

m = 0))

so

since

R R'

is

not level. (B.3.3) then R'/p' Also,

p' ~ J

If

(since

satisfies if

(0) # p E Spec R m = 0)

the c.c., by

, so

and R'/p'

(1.3.1),

J # P E Spec R , then

p' C Spec R'

so

= R/p R/p

Rp = R'(R.p )

, by

lies over

(B.2.3).

satisfies

p ,

Now

the s.c.c.

is a regular

local ring,

,

119

by

(B.2.4),

so

the s.c.c.,

by

(B.3.1)

Rp

(B.3.2),

shows

tension domains, (B.3.2) "o.h.c.c." If

LQ

that "taut-level"

By

shows

(A.6.1)

L[I/b]

satisfies

is, by

[McR-2,

together w i t h

Proposition

(A.II.I)

(0) # 0 # P , then (B.3.3)

shows

(1.3.1)

ex-

12].

= (A.II.7))

implies

(A.6.4)

and

the s.c.c, shows

shows

(A.6.6),

(34.2)]

the class

to factor rings and quotient

(A.9.7)

shows

satisfies

that

for all

L/O

0 # b E p .

of

GB-rings (B.3.5)

(L,P)

shows

is

shows g .

is such

0 # b E P , then (B.3.6)

and

is false.

related class of rings

for some

O E

does not hold.

rings.

that if a local domain

the s.c.c.,

and

that

that the converse

that the c.c. part of IN-6,

the s.c.c.,

does not satisfy

is not inherited by integral

that this does not hold for the closely (A.9.8) =

R

condition on a local domain than "s.c.c."

the s.c.c.

closed under passage

that

Finally,

is a local domain that satisfies

is such that

(B.3.4)

"taut"

(1.1.7)

is a w e a k e r

satisfy

the s.c.c.

q.e.d.

whereas

(and

(L,P)

Spec L

satisfies

L[I/b]

that this

does not hold for the c.c. In [R-15, and is an

k

is a positive

it is shown that if integer

Hi-domain , for some

j = 0,1,...,i that

(3.1)]

.

(B.3.7)

i -< k , then

shows

k < i < altitude D k

such that

(L,P)

D k = L[TI,...,T k] (P,Tl,...,Tk) Dk

Hj-domain,

that this does not hold for

to know w h e t h e r

prime spectrum of a ring is or is not closed,

i

for such

a given subset of the and

(B.3.8)

answers

this

H . (B.3.9)

shows

that a certain g e n e r a l i z a t i o n

T h e o r e m does not hold. ideal

is an

i

It is always of interest

for

is a local domain

P

of the Principal

That is, if it were always

of little height

two in a N o e t h e r i a n

P = U{p E Spec A ; (0) c p c P

is saturated}

Ideal

true that a prime

domain

A

was

such that

, then an interesting

120

generalization For

(B.3.10),

of the form it were

of the Principal a homogeneous

H~ , where

H

were always

ideal

in

£

is defined

is a homogeneous

true that homogeneous

£(L,bL)

Ideal Theorem could be shown to hold.

taut,

taut-level

(where

ideal

in

~(R.bR)

local domains

(L,P)

to be an ideal If

of the form

is a local domain and

0 #

b E P)

, then it could be shown that the Catenary

Chain Conjecture

holds,

by

(3.22)].

[R-9,

(B.3.11)

(2.10.1),

answers

It is known that satisfies satisfies

Noetherian

REMARK.

[P, p. 8].

or

.

i ~ m+l

is an

R

satisfies

JR-19,

and

S/P

facts cannot be extended

be as in (B.I), statements

Hi-domain

if

if, and only

is a

Di-domain

the s.c.c.,

S' satisfy to all

let

D = R[T](M,T ),

hold: if, and only

if, either

if,

.

i ~ m+l

if, and only if, either

for all non-maximal

P £ Spec R ,

R'

but

m > 0 .

final paragraph].

JR-12, p. 124].

integral

extension

exist a

mcpil n

(B.4.7)

g

is taut;

is a

GB-domain,

R

GB-domain.

(B.4.6)

where

S'

ring

the o.h.c.c.

Ci-domain

is not catenary

L = D+J'

is catenary;

that these

R

[P. p. 68].

is not a

is a semi-local

.

is a

Rp

S

(2.20)].

P E Spec S , Sp

Let the notation

(B.4.3)

(B.4.5)

S

and

in [R-!I,

2] that if

for all

shows

R

R

I asked

Then the following

i ~ m+l

(B.4.4) but

and,

(B.4.2)

i = 0

(3.18.6),

then:

that satisfy

m ~ 0

or

Section

the o.h.c.c.,

rings

(B.4.1) i = 0

[R-18,

(Bo3.12)

(B.4) and let

a question

the o.h.c.c.;

the o.h.c.c.

(3.18.1),

domain of in

[R-20,

, where

If there exists

J'

R(T)

a

mcpil n

in some

= R[T]jR[T ] , then there may not

R(T) (2.8.1)].

Let

is the Jacobson

is as in (9.2.4),

L

S = R'[T](R[T]_ _ radical

is a finite

of

local

-(J,T))

S .

Then

integral

and let D E g , extension

121

domain of

D , and there exists

height PAD

.

(B.4.8)

JR-20,

there exists a

mcpil m÷l

chain of prime

Proof. C.-domain let

ideals

for

(since

of (B.3.3).

Clearly by

D , S , and

as in (B.4.7),

that does not contract

R

is a

to a maximal

C r + m + l - d o m a i n , and Finally,

let

height p = i , and let

Then

height P = height p

, so

R/p = R'/P

Thus

L

height P <

L

(B.4.1)

such that

m+l ~ i)

such that

D .

i < m+l

PAR = p .

With

in

in

(B.4.2)

p C Spec R

that

(2.10)].

P E Spec L

(R/p)'

so

is not a

i E (m+l,

. r~m+l}

p E Spec R'

(by (B.2.4))

satisfies

is level,

R

R

and

such

P ~ J

the s.c.c., by the proof is a

C.-domain,

by

(B.4.1).

i

(B.4.4) After

follows

from (B.2.4),

defining an

is of interest that is an

q.e.d.

H.-ringl (respectively,

C.-ring,l Di-ring)'

to know if there exists a n o n - c a t e n a r y

H.-ringl (respectively,

(0 < i < altitude

L).

(B.4.1)

local domain

C.-ringl , Di-ring )

- (B.4.3)

it

for some

L

i

show that such rings do in-

deed exist. By

JR-4, Corollary

3.13],

a local domain

and only if, for all nonmaximal and

p E Spec L , Lp

height p + depth p = altitude L .

that the c o n d i t i o n

is a

(B.4.4)

GB-domain.

an integral

(B.4.5)

shows

is catena~y

satisfies (for

"height p + depth p = altitude

By (A.6.1) = (A.6.2), domain

(L,P)

the s.c.c.

m > 0)

L"

if,

shows

is necessary.

extension domain of a that the converse

GB-local

of this does

not hold. By [RMc, a

mcpil n

(2.14)], if

(L,P)

in some integral

and only if, there exists a does not continue (B.4.7)

and

to hold (B.4.8)

is a local domain,

extension domain of mcpil n

in

C

for local domains answer questions

then there exists

C = L[T](p,T )

(B.4.5)

shows

of the form

asked

in [R-IO,

L(T)

if,

that this .

(3.15)]

and

122 they show that two bad things w h i c h were shown to be possible by Nagata's

examples

can also be shown to be possible

than a semi-local) (B.5) and let

but

m > 0 .

R

R

Let the notation be as in (B.I),

Then the following

[R-2, Remark 3.9].

statements

R'

JR-3, Remark 5.12].

let

D = R[T](j,T),

hold:

is catenary and satisfies

is neither catenary nor satisfies

(B.5.2)

(rather

integral extension domain.

REMARK.

(B.5.1)

in a local

the c.c.,

the c.c.

R (I)

is a finite

R-algebra and

is not quasi-unmixed. (B.5.3)

elements and

JR-3, p. 127].

b,c

R

height JR[c/b] (B.5.4)

lytically

If

z

JR[c/b]

r = m = 1 , then

is a depth one prime

elements

R(c/b)

b,c

in

is catenary,

R , but With

ideal

local ring that has a regular (z,c)L*

R

for all ana-

is not catenary.

q = (X,Y I .... ,Ym_I)R'NR ,

, Q = R[TI,T2](J,TI,T2 ) , and

such that

independent

< height J - 1 .

[R-8, Example 2.28(a)].

p = (X-I)R'~R complete

such that

independent

(B.5.5)

ideal

in

There exist a n a l y t i c a l l y

L = 0/(pNq)O,

element

is a non-maximal

c

L*

is a

and a minimal prime

prime

ideal of height

> 1 (B.5.6)

[R-8, Example

(B.5.5), and if is a height (Ii)a

is the m a x i m a l

i _m 1 ,

[R-II,

and

(2.22)].

L

m = r = I , if L , then

class

in

L

L

is as in

I = (YI,Zl)L

and

(Ii)a:P

is not quasi-unmixed.

D[I/T]

such that there exists a maximal

If

ideal in

two ideal of the principal

, for all (B.5.7)

P

2.28(b)].

ideal

is a N o e t h e r i a n P

in

D[I/T]'

Hilbert domain such that

height P < height PnD[I/T] (B.5.8)

If

m = 1 , then there exists a prime

ideal

P

in

D

such that height P = 2 , depth P = 1 , and there exist infinitely * , p* many p E Spec D such that p c P and depth > depth P + 1 . (B.5.9)

[R-20,

(2.6)].

q =

((X-I,y I .... ,Ym_I)R'~R,T-Ym)D

E Spec D

123

is

such

= m+2

that

Q =

(q,T)D

~

Spec

D and

m+r+2

ideal in prime

(B.5.8)

R'

height

q +

1

p

> i .

in

in

ideal in

R'

m = i , there exists a height two m a x i m a l

, and so there exists a height one depth one prime

R , by

R

Let

Since

, so there exist infinitely many height one depth one

ideals

ideal

[Me-l, T h e o r e m 7].

such that

and

Q = (q,T)D

depth Q = r+l

.

, so

(B.5.1) it shows

depth

q

p*

, so

be a height one prime r+l = depth q

height P = 2 = height Q ,

Then, by

exist infinitely many height one prime c PNQ

Let

depth q = altitude R - 1 , so

P = (p,T)D

depth P = 1 , and

p

Q >

.

Proof.

*

= height

[Mc-5, Theorem 3], there

ideals

p

in

D

such that

= depth Q + 1 > 2 = depth P + 1 , q.e.d.

shows that the c.c. part of [N-6,

that the analogous

statement

(34.2)]

for "catenary"

is false, and

in place of "c.c."

is also false. It is k n o w n main,

then

[R-3, Lermna 5.11]

R (I)

is a finite

R-algebra.

verse of this does not hold. problem R

, R (I)

if integrally = R

R

is an unmixed

(B.5o2)

(As mentioned local domains

in (4.1.I) = (4.1.4),

exists a H e n s e l i a n

local d o m a i n

(A.3.1) = (A.3.7),

there are a n a l y t i c a l l y height PL[c/b] non-Henselian

shows

that the con-

in (15.6.3),

it is an open

are unmixed

that is not an

does there exist such

independent

elements

(B.5.3)

b,c

shows

[R-4, T h e o r e m 4.12]

H-domain (L,P) L

;

such that

such that

that this can happen for

that a local domain

if, and only if, for all a n a l y t i c a l l y

b,c

L , L(c/b) shows

in

if there

local domains.

catenary

(B.5.4)

- and for such

it is an open p r o b l e m

(L,P)

< height P - i

It is known

in

local do-

.)

As noted

that is, by

closed

that if

is catenary and

that the hypothesis

independent

altitude L(c/b)

"altitude L(c/b)

(L,P)

is

elements

= altitude L - i .

= altitude L - I"

124

is necessary. It is known B'

, where

ideal

in

(B.5.5)

B B'

[R-8, Lemma 2.22]

is a Noetherian such that

shows

that if

ring,

(z,b)B'

and if

# B'

that the hypothesis

b

is a regular z

, then

is a minimal height

about being

element

(z,b)B'

integrally

in

prime = 1 .

closed

is

necessary. It is known quasi-unmixed in

L

[R-8, Corollary

if

and only

such that

large

i .

(B.5.6)

placed by "there (B.5.7) not satisfy result

height

shows

if for all ideals

I = altitude

shows

Finally,

if

to hold

ring

(L,P)

is

of the principal

L - 1 , (Ii)a:P = (Ii)a

Fujita's

Noetherian

example

that there are such domains

continue

I

that the hypothesis

that t h e r e a r e

the c.c.

(P,T)B[T]

that a local

class

, for all

"for all" cannot be re-

exists."

(B.5.8)

height

shows

2.31]

that

B

[Mc-l,

PB[T]

Theorem

prime

shows

domains

that do

the stronger

the f.c.c.

I] cannot be "inverted."

ring and

+ 1

for non-extended

(B.3.11)

that satisfy

is a Noetherian

= height

Hilbert

(B.5.9) ideals

P E Spec B , then shows in

that this does not

BIT]

BIBLIOGRAPHY [B] [Bro ]

[C-l] [c-2] [D] [FR] [F-M]

J. Brewer, The ideal transform and overrings of an integral domain, Math. Z. 107(1968), 301-306. M. Brodmann, Uber die minimale dimension der assoziierten primideal der komplettion eines lokalen integ~itaetsbereiches," Ph.D. Dissertation, University of Basel, Switzerland, 1974. I. S. Cohen, On the structure and ideal theory of complete local rings, Trans. Amer. Math. Soc. 59(1946), 54-106. , Lengths of prime ideal chains, Amer. J. Math. 76(1954), 654-668. L. D~chene,'%djacent integral extension domains,"Ph.D. Dissertation, University of California, Riverside, in preparation. D. Ferrand and M. Raynaud, Fibres formelles d'un anneau local Noeth~rien, Ann. Sci. Ecole Norm. Sup. 3(1970), 295-311. M. Flexor-Mangeney~ Etude de l'assassin du compl~t~ d'un anneau local Noetherien, Bull. Soc. Math. France 98(1970), 117-125.

[Fu-l]

K. Fujita, Some counterexamples related to prime chains of integral domains, Hiroshima Math. J. 5(1975), 473-485.

[Fu-2]

, Three dimensional unique factorization domain which is not catenarian, 7 page preprint.

[G-l]

A. Grothendieck, "El~ments de G~om~trie Algebrique," IV (Premiere Partie), Inst. Hautes Etudes Sci. Publ. Math. Presses Universitaires de France, Paris, "France, 1964. "El~ments de G~om~trie Algebrique," IV (Seconde Pattie), Inst. Hautes Etudes Sci. Publ. Math. Presses Universitaires de France," Paris, France, %96 .5~--

[G-2]

[G-3]

"El6ments de G~om~trie Algebrique," IV (Quartrieme Pattie), Inst. Hautes Etudes Sci. Publ. Math. Presses Universitaires de France, Paris, France, 1967.

[H]

R. C. Heitmann, preprint.

[Hoe]

M. Hochster, "Topics in the Homological Theory of Modules over Commutative Rings," Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 24, Amer. Math. Soc., Providence, RI, 1975. E. G. Houston and S. McAdam, Chains of primes in Noetherian rings, Indiana Univ. Math. J. 24(1975), 741-753.

[HMc] [Hou-l] [Hou-2]

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TABLE A'

OF N O T A T I O N

,

(1.1.14),

p. 7.

AH ,

(I.i.14),

p. 7.

A

(i.I.14),

p. 7.

,

Ia ,

(A.II.24),

p.

p. 112.

e(I)

,

(A.II.25),

R (I)

,

(2.4),

p.

16.

R (w)

,

(2.3),

p.

15.

R<X)

,

(2.8),

p.

19.

R(X)

,

(15.3),

112.

p. 94.

R ( C l / b .... ,Ck/b ) , (A.I), ~(A,I)

,

p. 99.

p. 61.

£(R,bR)

,

(A.4.8),

p.

103.

~(R,bR)

,

(A.4.9),

p.

103.

(4.2.2),

p.

50.

,

INDEX A adjacent prime ideals, (I.I.I0), p. 6. altitude formula, (I.i.ii), p. 6. analytically independent elements in semi-local ring, (A.8.8), p. 107.

bi-equidimensional

ring, p. 7.

H-Conjecture, (3.3.6), p. 24. H-ring, (1.1.8), p. 6. H.-ring, (1.1.8), p. 6. 1

length (of a chain of prime ideals), (i.I.i), p. 5. level ring, (1.1.6), p. 5. little height two, p. 67.

Catenary Chain Conjecture, (3.3.8), p. 24. catenary ring, (1.1.3), p. 5. c.c., (1.1.5), p. 5. chain condition for prime ideals, (1.1.5), p. 5. Chain Conjecture, (3.3.2), p. 23. chain problem of prime ideals, p. 14. Ci-ring , (1.1.9), p. 6. Conjecture (K), (13.1), p. 88.

maximal chain of prime ideals, (I.i.i), p. 5. maximal relevant ideal, (A.I0.23), p. Ii0. mcpil, (i.I.i), p. 5. M-transform, (2.3), p. 15.

Depth Conjecture, (3.3.3), p. 24. Descended GB-Conj ec ture, (3.6.3), p. 30. D.-ring, p. 53. 1 dominating altitude formula, (I.I.ii), p. 6. DVR, (4.2.1), p. 50.

o.h.c.c., (1.1.7), p. 6. one and a half chain condition for prime ideals, (1.1.7), p.6.

f.c.c., (1.1.2), p. 5. first chain condition for prime ideals, (I.i.2), p. 5. first kind (of valuation ring), (4.2.1), p. 50. formally catenary ring, (2.6), p. 17. formally equidimensional ring, p. 7.

GB-Conjecture, (3.6.4), p. 30. GB-ring, (i.I.I0), p. 6.

Normal Chain Conjecture, p. 24.

principal class, p. 112.

quasi-unmixed

(3.3.9),

(A.II.24),

ring,

(1.1.12), p.6.

R

reduction (of an ideal), (A.II.25), p. 112. Rees ring, p. 61.

saturated chain of prime ideals, (I.i.i), p. 5. s.c.c., (1.1.4), p. 5. second chain condition for prime ideals (1.1.4), p. 5. semi-local Depth Conjecture, (3.13.1), p. 41.

semi-local H-Conjecture, (3.9.2), p. 35. semi-local Weak Depth Conjecture, (3.13.2), p. 41. special extension ring, (1.1.13), p. 6. strictly formally catenary ring, p. 18. strictly formally equidimensional ring, p. 7. Strong Avoidance Conjecture, (3.8.2), p. 31.

UFD, p. 93. universally catenary ring, (2.6), p. 17. Upper Conjecture, (3.8.5), p. 31. upper ideal, p. 70.

valuation ring of first kind, (4.2.1), p. 50.

T taut ring, (1.1.6), p. 6. Taut-Level Conjecture, (3.9.5), p. 35. taut-level ring, (1.1.6), p. 6. trd D/C, (i.I.ii), p. 6.

Weak Depth Conjecture, (3.3.4) p. 24.