International Mathematical Conference 1982: Proceedings (Mathematics Studies)

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PROCEEDINGS OF THE INTERNATIONAL MATHEMATICAL CONFERENCE, SINGAPORE 198 1

This Page Intentionally Left Blank

NORTH-HOLLAND MATHEMATICS STUDIES

74

Proceedings of the International Mathematical Conference, Singapore 1981

Edited by

LOUIS H. Y. CHEN TZE BENG NG and

M. J. WICKS MathematicsDepartment National University of Singapore

1982

NORTH-HOLLAND PUBLISHING COMPANY

- AMSTERDAM

NEW YORK

. OXFORD

North-Holland Publishing Company, 1982 AN rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwbe, without the prior permission of the copyright owner.

ISBN: 0 444 86510 I

Publishers: NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM. NEW YORK OXFORD Sole distributors for the U.S.A.and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52 VANDERBILT AVENUE, NEW YORK, N.Y. 10017

PRINTED IN THE NETHERLANDS

V

PREFACE

The present volume contains t h e Proceedings of a conference and workshop held a t t h e National University of Singapore i n June 1981.

The following

contributions t o t h e conference a r e not included i n t h e s e proceedings. D. L. Burkholder, Topics i n martingale theory and Brownian motion.

D. L . Burkholder, Martingale transforms and t h e geometry o f Banach

spaces. M . Fukushima, On absolute c o n t i n u i t y of r e v e r s i b l e d i f f u s i o n processes.

B. H. Neumann, Not q u i t e i n n e r automorphisms of groups. We would l i k e t o thank t h e many people, e s p e c i a l l y Drs. Arjen Sevenster, Mathematics e d i t o r of North-Holland Publishing company who cooperated t o make t h i s volume p o s s i b l e .

L.H.Y.

CHEN

T.B. NG M.J. WICKS


Vii

CONTENTS

V

ix xi xii xv Conference Lectures M. Nagata, Applications of the theory of valuation rings ....................... J.

3

W. S. Cassels,

Rational quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

N. Iwahori, Some topics on Coxeter groups and Weyl groups

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

9 27

H. Nagao, Some correspondences in the representation theory of finite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

T. Hida, White noise analysis and its applications

43

6.

Hartley, Powers of the augmentation ideal

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

____-_____________-----------------49

P. Erdss, Some of my favourite problems which recently have been solved H. Matsumura, Recent developments in the theory of excellent rings

-------

59

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

81

E. C. Milner, On the decomposition of partially ordered sets into directed sets

---

85

Workshop Lectures H. Matsumura, Homological methods in commutative algebra

M. Nagata, Commutative algebra and algebraic geometry

T. Hida, Calculus of Brownian functionals M.

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

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

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

Fukushima, Markov processes and functional analysis

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

93

125

155

187


ix

INTRODUCTION

The International Mathematical Conference was held at the National University of Singapore from 1 to 13 June, 1981. It was jointly organized by the Singapore Mathematical Society and the Department of Mathematics of the National University of Singapore. The sponsors of the conference were UNESCO, the International Mathematical Union (IMU), the Mathematical Society of Japan (MSJ) and the Southeast Asian Mathematical Society (SEAMS). UNESCO made a grant for the financial support of participants from within the Southeast Asian region, while IMU contributed to the travel expenses of some of the invited speakers. The contribution of MSJ is witnessed by the presence of six invited speakers from Japan. The initiative for the conference came from SEAMS which adopted it as one of its major activities for 1981. There were 132 participants from 16 countries. A conference of this size would not have been possible without adequate financing. Among the contributions the largest came from the Lee Foundation. The conference also received financial support from Shell Eastern Petroleum (Pte) Ltd and National Kap (Pte) Ltd. The facilities for the conference were provided by the National University of Singapore. In the planning of the conference the Organizing Committee received valuable advice and assistance from J. W. S. Cassels, S . S . Chern, B. H.Neumann (IMU), K. It;, Y . Kawada (MSJ) and W. S . Tan (SEAMS). The Organizing Committee would like to express its deep gratitude to the above organizations and individuals and to the participants for making the conference one of the most stimulating and fruitEul mathematical meetings in Southeast Asia.

Peng Tsu Ann Chairman Organizing Committee


xi

ORGANIZING COMMITTEE

Chairman : PENG Tsu Co-Chairman : TEH Hoon Secretary : Louis H. Assistant S e c r e t a r i e s :

Ann Heng Y . CHEN

LA1 Choy Heng LEONG Yu Kiang

Treasurer Members

:

KOH Khee Meng

: CHEW K i m Lin CHONG Chi Tat LEE Peng Yee Malcolm WICKS

SUBCOMMITTEES

Accommodation

: CHENG Kai Nah

CHAN Gin Hor

LEE Peng Yee NG Peng Nung

TEH Hoon Heng

Finance

:

Hospital it y

: Malcolm WICKS

PENG Tsu Ann Malcolm WICKS CHOW Yoong Ming LAM Lay Yong

WAN Fook Sun

Programnes

:

Stephen T . L . CHOY

CHEN Chum Chong

LEONG Yu Kiang

R . SUBRAMANIAN WOO Kai Yuen

Pub1 i c a t i o n s

:

LEE Peng Yee

CHONG Chi Tat LING Kiong Doong NG Tze Beng

Public Relations

:

Louis H.Y. CHEN

KOH Khee Meng

TAN S i e Keng

CHANG Boon Chong

Leonard Y .H. YAP

Registration

:

LA1 Choy Heng

LIM Chee Seng

Transport

:

SHEE Sze Chin

CHANG Kai Meng

P e t e r C.T. R I N G S . J. WILSON

YAP K i m Yew

xii

LIST OF WORKSHOP LECTURES

1 June

-

5 June

M . Nagata

Connnutative algebra and algebraic geometry

H . Matsumura

HomotogicaZ methods i n c o m t c t i v e algebra

D. L . Burkholder

Topics i n martingale theory and Bromian motion

T. Hida

Calculus of Brownian functionals

M . Fukushima

Markov processes and functional analysis

LIST OF INVITED LECTURES

8 June

9 June

10 June

11 June

12 June

M. Nagata

Applications of theory of valuation rings

J . W . S . Cassels

Rational quadratic f o m s

B . H . Neumann

Not quite inner automorphisms of groups

D. L . Burkholder

Martingale transforms and the geometry of Banach spaces

N . Iwahori

Some topics on Weyl groups and Coxeter groups

H . Nagao

SevemZ correspondences i n the representation theory of f i n i t e groups

T. Hida

White noise analysis and i t s applicatiovrs

B. Hartley

Powers of the augmentation ideal

P . Erd&

Some of my f a v o w i t e problems which recently have been solved

H. Matsumura

Recent developments in the theory of excellent rings

E . C . Milner

Some recent r e s u l t s i n the theory of partially ordered s e t s

M. Fukushima

On absolute continuity of reversible diffusion processes

LIST OF SHORT TALKS

9 June

Z . T. Hou

Closure of renmaZ sequences f o r circle-operation (read by 2 . S . Liang)

H. H . Teh

A generalization of S. EiZenberg's t h e o q

foms i n the studg of format Zangwrges

on n o m l

List of Lectures and Short Talks C. T. Chong

Global and local a d d s s i b i l i t y : II. Major subsets and automorphisms

S. Darwis

Nuees dynamiques : a method f o r classification

G. D. Hu and Y. C. Chang

10 June

the optimum approximation solution i n the order f o r principal oomponent analysis (read by 2. Wang)

On

Z. Soejoeti

Joint asymptotic normality of the least squares estimators and the mean square error i n the usual linear regression model

M. F. Chen

On construction problem f o r weakly symnetrizable &-processes (read by S . J. Yan)

S . Nualtaranee

On the maximum principle for a subhmonic function i n a cone and i n a cylinder

K. F. Tien

Relationships between analytical regularization and regularization by surgery

S. Yodintra

The cardinal number of the s e t of &percomplex nwnbers

G. H. Chan

A conjecture involving generalized matrix functions

K. M. Chong

A stu& of nmrtingales using spectral Orders

S. J . Yan

Potentiality and reversibitity for general speed fmctions

M. 'F. Chen and W. D. Ding

Xiii

L. H. Y. Chen

Martingale convergence d a the square +ction

S. M. Nababan

Oscillation c r i t e r i a for second order nonlinear inequalities

A . Buraczewski

A generalization of detsnninant theory of Fredholm

operators i n Banach spaces

11 June

S . T. L. Choy

Integml representation of multiplicative, involution p r e s e d n g opemtors i n L(C,(S,A) ,B)

C. C. Chen and K. M. Koh

On the strong p&i$ of the l a t t i c e o f sublattices of a f i n i t e distributive l a t t i c e

C. C. Chen and K. M. Koh

On the l m e r length of the Zattice of s u b k t t i c e s of a f i n i t e d i s t r i b u t i v ~planar l a t t i c e

C. C. Chen K. M. Koh and K. L. Teo

On the length of the l a t t i c e s of subalgebras of a Stone algebm

Z. S. Liang

On Delphic-semigroups i n stochastic point processes

xiv

List of Lectures and Short Talks M. F. Chen

and X. G.

Zhen

M. 2. S h a h a r i r

Uniqueness criterion f o r &-processes ( r e a d by S . J. Yan) A derivation of the Schroedinger equation via the

Fresne 1 stochastic i n tegra 1 C. R. Hwang

Limit problem for some linear global constraint

2. Wang

Last e x i t distributions and m a x i m excursion for Brownian motion

Z . T. Hou and M. F. Chen

Markov processes and f i e t d t h e o q ( r e a d by S. J. Yan)

D. Avis

Cornbinatorial aspects of computational geometry

G . L . Chia

and C. K. L i m

Some results on superconpact graphs

R. B . Eggleton and R . J . Armstrong

Covering l a t t i c e points with discs

C . H. Kan

Norrri-attaining vectors of operators on L spaces

P

xv

LIST OF PARTICIPANTS

ABDUL RAZAK BIN SALLEH, Universiti Kebangsaan Malaysia, Malaysia. ABU OSMAN BIN MD TAP, Universiti Kebangsaan Malaysia, Malaysia. ARIFIN, ACHMAD, Institut Teknologi Bandung, Indonesia. AVIS, DAVID, McGill University, Canada. BURKHOLDER, D.L., University of Illinois at Urbana-Champaign, U.S.A. BURACZEWSKI, ADAM, University o f Papua New Guinea, Papua New Guinea. CARRILLO, LUIS D., Iligan Institute of Technology, Philippines. CASSELS, J.W.S., University o f Cambridge, U.K. CASTILLO, ZANITA C., Philippines. CHAN, GIN HOR, National University o f Singapore, Singapore. CHAN, KAI MENG, National University of Singapore, Singapore. CHANG, KOK WAH, University of Calgary, Canada. CHEN, CHUAN CHONG, National University o f Singapore, Singapore. CHEN, LOUIS H.Y., National University o f Singapore, Singapore. CHENG, KAI NAH, National University o f Singapore, Singapore. CHIA, GEK LING, University of Malaya, Malaysia. CHONG, CHI TAT, National University of Singapore, Singapore. CHONG, KONG MING, University o f Malaya, Malaysia. CHOW, YOONG MING, National University of Singapore, Singapore. CHOY, STEPHEN T.L., National University of Singapore, Singapore. CHUA, SENG KEE, National University o f Singapore, Singapore. DARWIS, SUTAWANIR, Institut Teknologi Bandung, Indonesia. DEFENSOR, CORAZON G., University o f the Philippines, Philippines. DEL ROSARIO, ESTELITA B., Polytechnic University of the Philippines, Philippines. EGGLETON, R.B., University of Newcastle, Australia. ELEAZAR, MARIA N., Philippines. ERDOS, PAUL, Hungarian Academy o f Sciences, Hungary. FUKUSHIMA, MASATOSHI, Osaka University, Japan. HADIWIDJOJO, MOEHARTI, IKIP, Yogyakarta, Indonesia. HANG, KIM HOO, National University o f Singapore, Singapore. HANTINGAN, ARMAND, Universitas Palangka Raya, Indonesia. HARAHAP, ACHMAD SAID, Universitas Sumatera Utara, Indonesia. HARAHAP, MARWAN, Universitas Sumatera Utara, Indonesia. HARTLEY, B., University o f Manchester, U.K. HEE, JUAY GUAN, National University of Singapore, Singapore. HENG, SIEW JOO, National University of Singapore, Singapore. HIDA, TAKEYUKI, Nagoya University, Japan. HUANG, HUE1 CHUEN, National University of Singapore, Singapore. HWANG, CHII-RUEY, Academia Sinica, Taipei.

xvi

List ofParticipants

IWAHORI, NAGAYOSHI, University o f Tokyo, Japan. KAN, CHARN HUEN, National University of Singapore, Singapore. KAROT, SOMAK, Silpakorn University, Thailand. KAROT, WAREE, Silpakorn University, Thailand. KIU, SUN-WAH, Hong Kong Polytechnic, Hong Kong. KOH, KHEE MENG, National University o f Singapore, Singapore. KWAK, HYO-CHUL, Jeonbug National University, Korea. LAID CHOY HENG, National University of Singapore, Singapore. LAM, CHUN CHOON, University of Malaya, Malaysia. LAM, LAY YONG, National University o f Singapore, Singapore. LAMBINICIO, ARACELI, Polytechnic University of the Philippines, Philippines. LEE, CHEE QWUN, National University of Singapore, Singapore. LEE, PENG YEE, National University of Singapore, Singapore. LEUAR, BOON CHAR, National University of Singapore, Singapore. LIANG, ZHI-SHUN, Zhongshan University, Guangzhou. LIEW, MA1 HENG, National University of Singapore, Singapore. LIEW, NYUK CHOY, Singapore. LIM, CHONG KEANG, University o f Malaya, Malaysia. LIM, KIM LEONG, National University of Singapore, Singapore. LING, KIONG DOONG, National University o f Singapore, Singapore. MAK, KOK SUN, University of Malaya, Malaysia. MANALASTAS, PABLO R. JR., Ateneo de Manila University, Philippines. MANGARA, NABABAN, Universitas Sumatera Utara, Indonesia. MARASIGAN, JOSE A,, Ateneo de Manila University, Philippines. MATSUMURA, HIDEYUKI, Nagoya University, Japan. MILNER, E.C., University of Calgary, Canada. MITCHELL, SIDNEY S., Chulalongkorn University, Thailand. NAAK-IN, WATTAYA, Ministry of Education, Thailand. NABABAN, SAHALA, Institut Teknologi Bandung, Indonesia. NACHROWI, University of Indonesia, Indonesia. NAGAO, HIROSI , Osaka University, Japan. NAGATA, MASAYOSHI, Kyoto University, Japan. NAIR, BHAVANI, St. Anthony's Convent, Singapore. NAIR, SARADAMMA, Singapore. NAJADILAGA, I. JUNADI, Indonesia. NEBRES, B.F., Ateneo de Manila University, Philippines. NEUMA", B.H., Australian National University & C.S.I.R.O., Australia. NG, PENG NUNG, National University of Singapore, Singapore. NG, SOK BENG, University of Malaya, Malaysia. NG, TZE BENG, National University of Singapore, Singapore.

List of Participants

xvii

NUALTARANEE, SAWAI, Chulalongkorn University, Thailand. NUQUI, HONEST0 G., University of the Philippines, Philippines. O'CONNOR, WILLIAM JAMES, Capricornia Institute o f Advanced Education, Australia. PEMIMPIN, SAIGIAN, Universitas Swnatera Utara, Indonesia. PENG, TSU ANN, National University of Singapore, Singapore. PUNIM, NARONG, Srinakharinwirot University, Thailand PURBRICK, KEITH JAMES, Institute of Education, Singapore. QUEK, TONG SENG, National University of Singapore, Singapore. RAMOS, ROLAND0 E., University of the Philippines, Philippines. RATANAPRASERT, CHAWEWAN, Thailand. RATANAPRASERT, PAIBOOL, Thailand. RUIZ, MARI-JO P., Ateneo de Manila University, Philippines. SAW, SUTAIP LENG CHOOI, National University o f Singapore, Singapore. SHAHARIR, M. ZAIN, Universiti Kebangsaan Malaysia, Malaysia. SIM, CHIAW HOCK, University of Malaya, Malaysia. SIM, MU1 KIANG, National University o f Singapore, Singapore. SISWANTO, MASNAH, Indonesia. SOEJOETI, ZANZAWI, Gadjah Mada University, Indonesia. SOEMARTOJO, NOENIEK, University of Indonesia, Indonesia. SOETOMO, Indonesia. SOON, YO TIONG, Singapore. SUBRAMANIAN, RAMANAWAN, National University of Singapore, Singapore. SY, POLLY WEE, University of the Philippines, Philippines. TAN, CHOON PENG, University o f Malaya, Malaysia. TAN, GEOK CHYE, National University o f Singapore, Singapore. TAN, KENG HIAN, National University of Singapore, Singapore. TAN, SIE KENG, National University of Singapore, Singapore. TAN, WANG SENG, Universiti Sains Malaysia, Malaysia. TANG, IRVING CHE-HONG, Hong Kong Polytechnic, Hong Kong. TEH, HOON HENG, National University of Singapore, Singapore. TEO, KEE LEONG, Massey University, New Zealand. TEO, SIONG KHOW, Nat?onal University o f Singapore, Singapore. TIEN, KUANG-FU, National Taiwan University, Taipei. TUNG, PETER C. T., National University of Singapore, Singapore. VENEGAS, AURORA C . , Polytechnic University of the Philippines, Philippines. VILLA, NORMITA A., Polytechnic University of the Philippines, Philippines. WAN, FOOK SUN, National University of Singapore, Singapore. WANG, ANN LEE, University of Malaya, Malaysia. WANG, ZIKUN, Nankai University, Tienjin.

xviii

List of Participants

WICKS, M.J., National University of Singapore, Singapore. WILSON, S.J., National University of Singapore, Singapore. WIRJOSUDIRDJO, SUNARDI, Institut Teknologi Bandung, Indonesia. WONG, KOK YONG, University o f Malaya, Malaysia. WONG, PENG CHOON, University o f Malaya, Malaysia. WOO, KAI YUEN, National University of Singapore, Singapore. YAN, SHI-JIAN, Beijing Normal University, Beij ing. YANG, HWAI NIEN, National Chengchi University, Taipei. YAP, KIM YEW, National University of Singapore, Singapore. YAP, LEONARD Y.H., National University of Singapore, Singapore. YODINTRA, SMAI, Chiangmai University, Thailand. YOUNG, AH LIM, National University of Singapore, Singapore. ZAHARIN BIN YUSOFF, Universiti Sains Malaysia, Malaysia. ZAINODIN HAJI JUBOK, Universiti Kebangsaan Malaysia, Malaysia.

CONFERENCE LECTURES


Roceedinm of the International Mathematical Conference L.H. Y. Chen, T.B. Ng,M J . wlcks (eds.) @ North-Hdlandhblishing Compmy. 1982

APPLICATIONS OF THE THEORY OF VALUATION RINGS Masayoshi Nagata Department of Mathematics Kyoto University Kyoto Japan

The theory of valuation rings was founded in connection with number theory. The valuations concerned are multiplicative valuations having values in non-negative real numbers. Then W. Krull [ l ]defined the notion of an additive valuation (called an "Allgemeine Bewertung") having values in an ordered additive group. This is a generalization of a p-adic valuation which Krull employed to study commutative rings. We shall review first some of the well known and important applications of the theory of valuations (81). Then we prove a generalization of a well known existence theorem o f valuation rings ( 8 2 ) . In the last section, we give a theorem which gives a sufficient condition for a field to be finitely generated, as an application of the theory of valuation rings. Furthermore we add some modification of the theorem, In this article, by a ring, we mean a commutative ring with identity. 1. REVIEWS ON WELL KNOWN AND IMPORTANT APPLICATIONS, Krull [2 ]applied the theory of valuation rings especially to study normal rings, and obtained the theory of Krull rings (called "Diskrete Hauptordnung"). The notion of Krull ring is a generalization of that of noetherian normal ring and plays an important r$le in studying noetherian rings. Furthermore, Krull made use of the generalized valuations in many other situations. One basic fact is that a normal ring is the intersection of valuation rings containing it. This was given by Krull [ l 1. 0. Zariski [l] employed valuation rings in order to study correspondences between algebraic varieties. A basic fact for this may be expressed that a valu-

tion ring of the function field, k(V), of a variety V over a field k characterizes a f u l l set of simultaneous specializations of corresponding points (in the family of birationally equivalent varieties) to a given generic point of V. In the case of multiplicative valuations, their completions play an important

3

M. Nagata

4

r6le in several situations. An additive valuation does not have a completion, except for the rank one case and the field case; these cases are practically contained in the case of multiplicative valuations. But it has the henselization, introduced by Nagata [l ] , and maximal completions, introduced by Kaplansky [ l ] (cf. Schilling [l] , cf. also Krull [l] ) . The following theorem was proved by using the notion of maximal completions (see Nagata [ 4 1 ) : THEOREM 1.1. (1) Asswne that K i s an ordered f i e l d . Then V = { x e K 1 - r < x < r for some r e Q 1 (Q is the field of rational numbers) i s a valuation ring with There i s a natural homomaximal ideal M = { x e K 10 < r e Q implies -1< x < r } morphism $ of V i n t o the real nwnber f i e l d R so that (i) the kernel of $ i s M and (ii) $a = r (a e V, r e R) i f and only i f t > a > s f o r any t,s 6 Q such that t >r>s.

.

Conversely, ( 2 ) assume t h a t K i s a f i e l d , V i s a valuation ring o f K with m&mal idea2 M, and t h a t $ i s a homomorphism of V i n t o the real nwnber f i e l d R with kernel M, then K can be made i n t o an ordered f i e l d so that V, M, $ i n (1) above coincide with the given ones. 2. EXISTENCE THEOREM

It is well known that if R is an integral domain contained in a field K and > P is a descending chain of prime ideals in R, then there is a P13P2> valuation ring V of K with prime ideals Q 3 Q 3....>% such that Qin R = Pi 1 2 (i = 1,. ,n).

...

..

This existence theorem can be generalized as follows : THEOREM 2.1. If R i s an integral domin which i s contained i n a f i e l d K and i f { PA 1 Xe A 1 i s a linearly ordered (under containment r e l a t i o n ) s e t of prime ideaZs i n R, then there i s a valuation ring V of K which has prime ideal Q, f o r each X e A such that Q, I\ R = PA . Before proving this theorem, we prove a lemma as follows : LEMMA 2.2. Let K, R and P be as above. If an element a of K is integral over R, then R" = R [a] has prime ideals Pi (for all X e A ) such that {Py I X e A 1 is linearly ordered and P i n R = P A '

PROOF. Take 0 # d e R such that a' = ad is a root of a monic polynomial f (X) over R which is irreducible over the field, k, of fractions of R. Then, as for R' = R La1] , prime ideals lying over Pa correspond to irreducible factors of (f(X) modulo Pa). Therefore, R' has a linearly ordered set I P; I X e A } of prime ideals such that Pin R = Pa , Thus we may assume that a 8 k. Set

The theory of valuation rings

5

2

I = I b e R I ba e R I . If PA I, then there is a unique prime ideal Q, of R" which lies over PA . Therefore, considering + = (the natural homomorphism modulo the union of all P, not containing I), we have only to show the existence of similar set o f primes in +(R") with respect to $(R). Repeating these two steps, we see the existence of the required P y . PROOF OF THEOREM 2.1. We consider the set F of pairs of subrings S of K containing R and a linearly ordered set of prime ideals (P' I h e A }of S such that x P; n R = PA . (We consider only subrings S which have such linearly ordered set of prime ideals.) Then F is obviously an inductive set, and has a maximal member, say (V, { Q, I X e A I ) . We want to show that V is the required valuation ring of K. For this purpose, it suffices to show that x e K, x 4 V, x-l $ V give us a contradiction. By virtue of Lemma 2.2, we see that V is integrally closed in K. From the normality of V, we see that V [x] 2 V LX] / I with I generated by {cX - d I c,d e V, cx = d 1. Let I and I1 be the set consisting of these c and d respectively. If Q, does not contain .I then there is a unique prime ideal of V [x] If Q, contains both .I and 11, then lying over Q, Therefore, the V [x] /Q, V [x] is isomorphic to a polynomial ring over V/Q, maximality of V implies that there is a heA such that

.

b = qX(q Similarly, considering V [x-l]

6

1 Q,, b B Q,

, we

.

**.

(1).

see the existence of

,

e A , c #!

Q, (c

e V)

and P e Q, such that c = px -1

...

(2).

Multiplying (1) and (Z), we have bc = pq.

.

\

,'con-

Then b,c are not in hence bc is not in Q We may assume that Q , Z Q !J tradicting bc = pq e Q , This completes the proof of Theorem 2.1.

u

3 . FINITENESS OF A FIELD.

THEOREM 3.1. Assume that an integral domain R contains a f i e l d k and has a prime ideal P such that height P + trans .degk R/P

trans .degkR

.. .

(1).

Then the f i e l d of fractions of R/P i s finitell! generated over k, provided that the f i e l d K of fractions of R i s f i n i t e l y generated over k. The equality actually holds i n (1).

PROOF. We may assume that R is a valuation ring of K and that P is the maximal ideal. Letting h be the height of P, we take the chain of prime ideals of

M. Nagata

6 R : P = P

0

3

PI

. . . 3 Ph

3

I0 1 .

.

For each i = 1,. . , h , we choose an element

...,

s o t h a t xi 4 Pi.

Take elements x ~ + ~ , x of R ( n = h t t r a n s . degk n R / P ) so t h a t t h e i r r e s i d u e c l a s s e s modulo P a r e a l g e b r a i c a l l y independent over k . Then x l , , . . , x a r e a l g e b r a i c a l l y independent over k . Therefore K i s a f i n i t e

xi of Pi-l

a l g e b r a i c extension of k ( x l , . s i o n h and V / ( P n V )

.., xn) .

V = R n k(xl,

i s isomorphic t o k ( x h + l , . . . ,

.., xn) ] . k(xh+l

) is o f Krull dimen-

Since R i s an extension

xn).

o f V t o K, by t h e well known i n e q u a l i t y , we have k(xl,. V/(POV)

. . ., x

[R/P : V / ( P n V ) ]

5 [K :

This shows t h a t R / P i s a f i n i t e a l g e b r a i c extension of

,. . . , xn) .

This completes t h e proof of Theorem 3 . 1 .

In o r d e r t o make s i m i l a r o b s e w a t i o n i n t h e c a s e o f unequal c h a r a c t e r i s t i c , we g i v e a modification a s follows : THEOREM 3 . 2 . Assume that an integral domain R contains a discrete valuation ring

k with a prime element p ( f 0 ) .

Assume that P i s a prime ideal of R such that

p a p and

height P + t r a n s . d e g

k/pk

R/P ,trans.degk

R + 1

...

(2)

If the f i e l d K of fractions of R i s f i n i t e l y generated over k, then the f i e l d of fractions of R / P i s f i n i t e l y generated over k/pk. The equality holds i n ( 2 ) . The proof i s r a t h e r s i m i l a r t o t h a t o f Theorem 3 . 1 , and we omit i t .

Furthermore,

as i s e a s i l y seen by t h e proof above, Theorems 3 . 1 and 3 . 2 a r e modified a s Theorems 3 . 3 and 3 . 4 below. THEOREM 3 . 3 . Let P be a prime ideal of an integral domain R, containing a f i e l d

k and contained i n a f i n i t e l y generated f i e l d K over k. ring (V,M) of K dominating R such that

If there i s a valuation

P ( r a t i o n a l rank o f V) + trans.degk V/M L t r a n s . d e g k K,

then the f i e l d of fractions of R / P i s f i n i t e l y generated over k . THEOREM 3 . 4 . Let P be a prime ideal of an integral domin R , containing a d i s -

crete valuation ring, k , with a prime element p ( # 0) contained i n P. Assume that R i s contained i n a f i n i t e l y generated f i e l d K over k. If there i s a valuation ring (V,M) of K dominating R

P

such that

( r a t i o n a l rank of V) + t r a n s . d e g

k/pk

VIM

trans.degk K + 1

then the f i e l d of fractions of R / P i s f i n i t e l y generated over k/pk. We add h e r e another remark.

,

7

The theory of valuation rings

REMARK 3.6. In Theorem 2.1 (or 2.2). the assumption (1) (or (2)) is important. For instance, the rational function field Q(x,y) of two variables over the rational number field Q has a discrete valuation ring V whose residue class field is an infinite algebraic extension of Q. Such an example is constructed as follows*

.

Take infinitely many prime numbers p1 < p < . . and set bi = Jpi , K = Q(bl,b2:. . .). 2 K [x,y] is imbedded in the formal power series ring ]x[K so that y = Zbixl. This imbedding gives u s a discrete valuation ring V ’ of K(x,y) such that V’/xV1 = K. Then V = Q(x,y)nVl has residue class field isomorphic to K. The proof follows by applying the following lemma to the completion of V. LEMMA 3.7. Let k be a field and K a separable algebraic extension field of k. Assume that V is a complete discrete valuation ring containing k and dominated by K[x]

. ..).

,

If V contains x and Cc.xi (ci e K), then V contains ko = k(c 1’ c2’

PROOF. We proceed with an induction on the suffix i of ci. Assume that c. are 3 in V for all j less than i. Then zi = C. c xn-i is in V. Let fi(X) be the nLi n irreducible polynomial over ki = k(c c.1-1) having ci as a root. Then fi(X) modulo the maximal ideal of V has a root ci. Therefore, by virtue o f Hensel Lemma, we see that ci is in V.

REFERENCES Kaplansky, I. Maximal fields with valuations, Duke J. 9(1942), [l]

303-321.

Krull, W. Allgemeine Bewerungstheorie, J. Reine Angew. Math. 167(1931), 160-196. [l] L2J ijber die Zerlegung der Hauptideale in algemeinen Ringen, Math. Ann. 105 (1931) , 1-14. Nagata, M. On the theory of Henselian rings, Nagoya Math. J. 5(1953), 45-57. [l] -, 11, Mem. Coll. Sci. Univ. Kyoto 28(1953-54), 109-120.

PI

c31 -, 111, Mem. C o l l . Sci. Univ. Kyoto 32(1959-60), 93-101. [4] Field theory, Marcel Dekker, New York and Basel, 1977. Schilling, O.F.G. [l] Valuation theory, A.M.S. Math. Surveys 4(1950).

M. Nagata

8

Zariski, 0. [l]

Foundations of a general theory of birational correspondences, Trans. Am. Math. S O C . 53(1943), 490-542.

Boceedings of the International Mathematical Conference L A Y . chen, T.B.Ng, MJ. Wicks (edr.) 0North-HollandPublishing Company,1982

9

RATIONAL QUADRATIC FORMS J .W.S

. Cassels

Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, U.K.

1. INTRODUCTION

The theory of quadratic forms over the rational field Q or over the ring of rational integers is far too extensive to deal with in a single lecture. Our subject here is the so-called "localglobal" theorems and their limitations, that is, the information that can be obtained from p-adic considerations. Some of the results extend to algebraic number-fields or in other ways, but we shall not discuss these extensions here. (cf. O'Meara (1963)). Much of the material is necessarily contained in my recent book (Cassels (1978)), where the reader will find proofs etc. This review is, however, intended to be self-contained and there is a little extra, particularly about ternary forms. It is a mildly updated version, in English, of a lecture which I delivered to the Jahrestagung of the Deutscher Mathemmatiker-Vereinigung at Hamburg in 1979 and which is published in their Jahresbericht (E (19801, 81-93). We are grateful to the Deutscher Mathematiker-Vereinigung and to B.G. Teubner GMBH, Stuttgart, the owners of the copyright, for permission to publish it here. For any prime p we denote by Q 2 respectively the field of -PI -P p-adic numbers and the ring of p-adic integers, with the usual convention that & = -2 = is the field of real numbers. The number of variables is n and we write x = (xl,...,xn) etc. 2.

BEHAVIOUR OVER FIELDS

There is a very close relation between the behaviour of rational quadratic forms over Q and over the Q It is given by the ori-P' ginal local-global principle of Hasse (1923, 1923a):

.

THEOREM 2 .l. ( "Weak Hasse Principle") TWO q u a d k a t i c dokrnd f(x), g(x) dehined o v e h Q ake e q u i v a e e n t owe4 Q p k e c i ~ e l ywhen t h e y ahe e q u i v a l e n t owe& a l l Qp ( i n c l u d i n g p = m ) .

J. W.S. Cassels

10

THEOREM 2.2.

("Strong Hasse Principle").

dedined oueh

non t h i u i a l l y o u e h

Q hephebentb 0 when it doed 6 0 oueh e u e h y Q

f

(x)

d o owe& e u e h y

Q -P

COROLLARY.

0

phecideey

p = -1.

(inc.

-P

Applying Theorem 2.2 to

A q u a d h a t i c 6 o h m f (If)

2 f(x) - b x ~ + we ~have the

hephedentb

b

L

Q

ouc h

Q

phecideey when

id doed

( i f l c . p = -1.

The adjectives "weak" and "strong" are conventional because Theorem 2.1 can be deduced from Theorem 2.2 by quite general arguments (Witt (1936)). Further, there are fields (e.g. R(t)) for which there is an analogue of Theorem 2.1 but not of Theorem 2.2. In essence, Theorem 2.1 was given by Minkowski (1890). He could not formulate it so succinctly because Hensel had yet to invent the p-adic numbers, though in a sense Minkowski already thought p-adically. Theorem 2.2 also shows the advantage of the p-adic language both for enunciations and proofs. The case n = 3 is essentially a classical theorem of Legendre (1798) and Gauss (1801). The case n > 3 was treated by Meyer (1884) but his statements are lengthy and his proofs complicated and probably not entirely correct (cf. Dickson (1930)). Incidentally, the book of Dickson (1930) shows how long it took for the p-adic viewpoint to get accepted. In the book (and also in Mordell's review of it in the Mathematical G a z e t t e ) there is no mention of Hasse's papers. 3.

INTEGRAL EQUIVALENCE

Two integral quadratic forms which are equivalent over all (inc.

m)

need not be equivalent over

fl = 2x2

+ 41y2; f2

=

x2

+ 82y2

z.

Z

-P

For example, (3.1)

are equivalent over every Z (since 2,41 is a square in -P 2 respectively): but they cannot be equivalent over be241' -2 cause fl does not represent 1 integrally. A further example is

z

g1 = 2x2

-

41y2; g2 = x2

-

82y2.

(3.2)

Here the inequivalence over is not so immediate. We have encountered a logical peculiarity of the definition of equivalence, namely that it does not furnish an algorithm which decides infallibly whether two forms are equivalent or not. Indeed it was f o r a long time not known whether

Rational quadratic forms x

2

-

3y2

-

2yz

-

23z2

;

x2

-

7y2

-

6yz

are equivalent (Dickson (1930), p. 147). problem of effectivity.

11

-

llz

2

(3.3)

We shall come back to this

%

The forms which are equivalent to one another over every (inc. form a genus. They have the same determinant. A genus contains only finitely many - equivalence classes. If f,g are in the same genus and M is any natural number, then there is a form g* which is 2 - equivalent to g and such that

for every integral vector equivalence classes of f

m)

5. It is thus impossible to separate the and

q

by local criteria.

The power of the local-global paradigm i s , however, far from being exhausted. Two forms f, g in the same genus are by definition certainly Q - equivalent for every p and so they are Q -P equivalent by the weak Hasse Principle (Theorem 2.1). In other words, there is a Q-vector space V of dimension n endowed with a quadratic form $I and there are elements bl,...,bn, cl,...,c-n of V such that $4xp1 + $(xlcl

+

... + xn2n1

... + xn-nc

=

) =

f (xl,.

..,Xn)

;

g(xl I . . . , xn)

.

(3.5)

We can now renew the local attack in the hope of obtaining further information about the equivalence or inequivalence of f and g. LATTICES

4.

We are thus led to the following set-up. Let V be an n-dimensional Q vector space and $I a (regular) quadratic form on V. The orthogonal and the proper orthogonal group are denoted respectively by O ( V ) and O+(V). A lattice A in V is a free 2 - module + xnkn, where kl, b of rank n (i.e. the set of xlbl + -n ,xn run through 2 indepenare linearly independent and xl, dently). Two lattices A , r are called equivalent if r = ah for some u E O+(V). In the way which was sketched at the end of the last section, there is a one-one correspondence between equivalenceclasses of lattices and proper equivalence-classes of forms (i.e. where the determinant of the integral transformation is required

-

...

...

...,

J. W.S. Cassels

12

to be

+ 1).

For every prime p the 8-vector space V has a "localization", namely the Q - vector space V = V @Q Qp. We denote the exten-P P also by 4 . A Z - lattice is, by definition, sion of 4 to V P -P In particular, A = A Q Z a free Z - module of rank n in V -P P P -P is the "localization" of A . Two lattices A,r are in the same genus if for every p ( # a) there is a u E O+(Vp) such that P A = a r There is a one-one correspondence between the genera of P P P: lattices in V and the genera of those forms which are rationally equivalent to 4 .

.

z

Lattices behave particularly well under localization: LEMMA 4.1. d m o b t aLl

(a) Let

A,r

be g l o b a l l a t t i c e d .

Then

A

P

=

r

P

doh

p.

(b) Let A be a g l o b a l L a t t i c e and d o h evehy p ( # m ) l e t = Ap doh a t m o d t a l l be a - l a t t i c e . Suppode t h a t r p. Then t h e h e i n a u n i q u e l y detehmined g l o b a l l a t t i c e r buch t h a t r P = r(P) doh a l l p. Here "almost all" means "with at most finitely many exceptions". the localization of a Here and in what follows, we denote by r P are just given locally. global lattice r , whereas the I'(p)

r (p)

zp

adele is a set 5 = 15 1 , where p runs through all the primes P + and m and where Ep E 0 (Vp) for all p and 5, E 0 (Ap) for almost all p. Here A is a global lattice and O + ( A p ) denotes + the subgroup of 0 (Vp) which leaves Ap invariant. By the Lemma, the definition of an adele is independent of the choice of A . The + adeles form a group OA(V) which operates on the lattices of a genus according to the prescription: r = [ A means

An

(4.1)

By the Lemma r is uniquely given by E , A . The determination of the equivalence classes in a genus may thus be reduced to a question + about the group OA(V). We shall introduce a new classification of lattices, namely into spinor genera, but must first say something about the spin group.

Rational quadratic forms

5.

13

THE SPIN GROUP

The orthogonal group O+(V) has a double covering Spin = Spin(V) I which is simply connected for n 2 3. Over the ground field 5 this goes back to Hamilton, who described the proper orthogonal group of ordinary euclidean 3-dimensional space by means of his quaternions. The quaternions are the algebra u = u0 -

+ u1i + u2j + u3kl

for multiplication.

u'

=

of the

-

uo

There is an involution uli

-

u2j

- u3k

for which (5.4)

N(g) =

= g'g =

U J '

u2

+

u1 2

+

u2 2 -+ u3 2

(5.5)

gives a homomorphism of the multiplicative group of the non-zero quaternions into the positive reals. We identify 3-dimensional euclidean space with the "pure" quaternions x = x1 i + x2j + x3kl which are characterized by XI -

The norm

N(x) =

-x2 x

= -x -

.

is the square of the length of

y = ~ - lwhere , E V and u # 0 is a quaternion. check that y' = -y, that is y. E V . Further, y2 that is

=

= x2uu-1 = x2 - 1

(5.7)

5.

Now put

It is easy to

(5.8)

J. W.S.Cassels

14

u : -x

T

-1

uxu -

->

( 5 . 8 bis)

is an orthogonal transformation of V. By continuity, det TU = +1, so TU E O+(V). For g E V one checks that - T z 5 is the reflecin the plane perpendicular to g ; aEd hence it follows tion zf easily that every u E O+(V) is of the form u = T for some u_. 2 Finally, T = Tv precisely when y = Xg for some X E g . We may 11 thus normalize to N ( g ) = 1. Let Spin be the group of the 2 with + N ( g ) = 1. Then we have constructed an epimorphism Spin -> 0 (V) with kernel 2 1.

x

If we take an arbitrary ground field K instead of R, most of the above argument goes through. The exception is that the normalization to N(u) = 1 in u = TU is in general no longer possible. Instead, there is a homomorphismo+(v) -> (where K* given by

(5.8 ter)

K*/(K*)~

is the multiplicative group of non-zero elements of

u = T

-

N(u) - (K*12.

->

(5.8

K)

quater)

This is the so-called spinor-norm. For general K we still have a homomorphism of the group Spin defined by N(u) = 1 onto O + ( V ) The kernel is still 2 1 but the image is now the subgroup 0 of O+(V) consisting of the a with spinor-norm 1.

.

For general quadratic forms the story is similar, if one uses the so-called Clifford algebra instead of the quaternions. There is an algebraic group Spin, which is simply-connected for n 2 3 , and there are homomorphisms 8:

e : o+(v) The image 0 (V) of ( 5 . 9 ) (5.9) consists of 2 1. 6.

SPINOR

+

Spin ->

0 (V) -->

(5.9) ( 5 .lo)

K*/(K*)*.

is the kernel of ( 5 . 1 0 )

,

and the kernel of

GENERA

We define two lattices h , r to belong to the same spinor genus if there is a global 5 E O+(V) and there are local rl E O(Vp) (that 4P is, np E 0 (Vp) with spinor-norm 1) such that


15

The subdivision of lattices into spinor genera is obviously coarser than that into equivalence classes, but finer than that into genera. We now return to the action of the adele group on the set of lattices. that is r Suppose that r = 5 A for an adele 5 = {cpl, P - SPAPThen (6.1) is the same as

5, for some

pP

E

=

srl P1-1 P

(6.2)

+

0 (Ap). This, again, is equivalent to

+

Here 8 ( . ) is the spinor-norm on 0 (Vp). We are now in a commuP tative situation and have an effective procedure to determine whether two lattices are in the same spinor genus or not. The number of spinor genera in a genus is clearly a power of 2 : in the simplest cases it is equal to 1. The name "spinor genus" was introduced by Eichler (1952a), but with a somewhat different meaning. The above definition is due to Kneser (1959) and is now standard (cf. O'Meara (1963)). Other definitions of spinor genus from another standpoint have been given by Jones and Watson (19561, Watson (1956, 1960). (In Cassels (1978) it is asserted that the definition in Watson (1960) is equivalent to that of Kneser, but that is rubbish. The latter is presumably equivalent to the second definition given in Watson (1956)). There is another way to treat spinor genera. Let A be a latt ce in the Q-quadratic space V,$, and suppose that $ induces a classically-integral quadratic form of determinant d on A Let r be another lattice of the same genus, and define I =I(A,r) to be the index of A n r in A (and so in r ) . Under the hypothesis that I is prime to 2d, the spinor genus of r depends only on that of A and on the image of I in the group

n Q;/(Q;I2.

As every lattice A

of the genus is equivalent to

PI 2d

a lattice I: of this kind, we obtain all the spinor genera of the genus in this way. We can translate what has just been said into the language of quadratic forms. Let f(x) be a classically-integral form of determinant d. Suppose that the form

J. W.S. Cassels

16

is also integral, where I is an integer prime to 2d. Then g is in the same genus. The spinor genus of g depends only on that of f and on the image of I in Q*/(Q;)2. p I 2d-P

n

If the integral forms f,g are in the same genus, we can get from one to the other by a chain of transformations each of which is either an equivalence or of the kind just described. If €,g are in the same spinor genus, then a stronger statement is true. Let q be any prime not dividing 2d. Then there is a chain of transformations leading from f to g for which always I = q. This gives a convenient way to determine all the equivalence classes in a spinor genus (more general in Kneser (1957)). 7.

CLASS NUMBERS OF SPINOR GENERA

For indefinite forms with n 2 3 the class number of a spinor genus is always 1. This follows from a "strong approximation theorem" for the spin group, which is a special case of a theorem which holds generally for simply-connected groups (Kneser (1966), Platonov (1969)). Thus we have an effective procedure to determine whether two indefinite forms with n 2 3 are equivalent or not. (For n = 2 we can do this by the classical reduction theory. For definite forms the question is clearly trivial). Eisenstein (18511, on the basis of his tables of the classes of indefinite ternary forms, had already conjectured that a genus of such forms contains only a single class. In fact the genera which he considered contained each only a single spinor genus. The first theoretical results about the class numbers of indefinite ternary forms were obtained by A . Meyer (1891, 1893, 1894-96) in a difficult series of papers (cf. also Dickson (1930)). The situation is quite different for definite forms. Magnus (1937) showed that there are only finitely many genera with n 2 3 whose class number is less than any given bound (cf. also Watson (1963, 1963b) and Gersten (1972)). The genera with class number 1 were determined by Watson (1963a, 1972, 1974, 1975). 8. WEIGHT OF GENERA The weight (mass) of a definite form f (or of the equivalence class + to which it belongs) is defined to be l/O (f)r where O+(f) is the


17

order of its group of proper integral automorphisms. The weight of a genus is the sum of the weights of the classes in it. Eisenstein observed that the weight is the sensible way to generalize the notion of class number for binary forms. He himself and H.J.S. Smith gave formulae for the weight when n = 3 . Minkowski gave a formula for n 2 3 , but containing an incorrect power of 2. For indefinite forms, the weight is defined in terms of the measure of a fundamental region of the real orthogonal group with respect to the group of integral automorphisms. Siege1 gave formulae for the weight both in the definite and the indefinite cases. Recently they have been put in the succinct shape: the Tamagawa number of an orthogonal group is 2. (cf. Weil (1961): Kneser (1974) has given a proof which works with orthogonal groups only).

9.

REPRESENTATION BY GENERA

integer b is said to be globally representable by the form f if there is an g E En with f(g) = b. The representation is primitive if al,...,an have no common factor. Similarly, one defines (for p = m representability and primitive representability over Z -P every representation with 2 # 0 is primitive). We say that b is everywhere locally (primitively) representable if this is the case over every Z -P' THEOREM 9.1. SUppO6e t h a t b i b evehy~Uhehe .toca.t.ty p h i m i t i v e L y h e p n e s e n t a b l e b y f. Then it .i6 g l o b a l l y p h i m i t i v e . t y h e p h e b e n t a b l e b y home 6anm f* 0 6 t h e 6 U m e genU6 a6 f.

An

The example f = 2x 2 -+ 41y2, b = 1 arily equivalent to f.

all

shows that

f*

is not necess-

a2 of b by f are in the same orbit if Two representations g2 - Tgl for some proper (det = + 1) automorphism T. All the representations of b by f lie in a finite number of orbits. The weight of an orbit is defined to be the weight of f. There are formulae for the total weight of the representations of an integer by a genus, which are intimately connected with the formulae for the weight of a genus (cf. Weil (1962)). In particular, as was noted in essence already by Minkowski, the total weight is given by a product over all primes p, and this can be regarded as a quantitative localglobal principle. (There is a similar interpretation for the "singular series" produced by the Hardy-Littlewood circle method).

18

J. W.S. Cassels

We thus have a satisfactory theory of representation of integers by genera. The theory of representations by individual forms remains unsatisfactory, especially for n = 3. In the remainder of this is lecture we discuss what has been achieved so far. If b E everywhere locally primitively representable by f but is not globally primitively representable, we shall say that b is an exception for f.

z

10. REPRESENTATION BY FORMS, n

2

4

In general there are exceptions for definite quadratic forms, for example

does not represent b = 1. (For definite quaternary forms which represent all natural numbers, see Ramanujan (19171, Willerding (1948)). The number of exceptions is finite, however, and there are explicit bounds for them. For diagonal forms this was done by Kloosterman using the Hardy-Littlewood circle method, and his proof was extended to all definite forms with n 2 4 by Tartakowsky (1929). There are more precise results which describe the distribution of the representations in space (Ross and Pall (19461, Pommerenke (1959), Watson (1960a), Malygev (1962, Chapter 3) , Peters (1979)). It is possible to argue without analytic tools, but then the proofs become more complicated (Hsia, Kitaoka and Kneser (1978), Cassels (1978, Chap. 11, 59)). For in'definite forms (n 2 4) Siegel (1951) showed by his deep analytic methods that there are no exceptions. Independently, Watson (1955) gave a simple proof. Siegel's results went further, however, and showed that the weight of the representations of an integer b is the same for all forms of a genus. Using the group of adeles Kneser (1961) showed comparatively simply that for n 2 4 the total weight of the representations by a spinor genus is the same for all spinor genera of a genus. (The result was enunciated only for indefinite forms, but the proof is valid also f o r definite ones, cf. Hsia (1976)). For indefinite forms, where a spinor genus is the same as an equivalence class (57), Kneser's result only confirms that of Siegel, but for definite forms it gives new information.


19

11. REPRESENTATIONS, n = 2

It is easy to see that a prime q can be represented by at most two classes of forms of given determinant d. As the class number of a genus can be arbitrarily large, we conclude that exceptions are the rule. 12.

REPRESENTATIONS, n

=

3

This is the most difficult case. The number of variables is too small for analytic estimations to work, but there is not the good algebraic structure which exists for n = 2 . It turns out that exceptions actually occur, but are rare. Consider the form (Siege1 (1951)) f1 = x2 - 2y2 + 6422

.

(12.1)

It represents any b 1 (mod 8 ) everywhere locally primitively. We show that it does not primitively represent the b = m2, m 3 (mod 8). Otherwise, we should have x2

-

2y2 = (m

+

82)

(m

-

82).

(12.2)

There must be a prime p 5 3 (mod 8 ) which divides m an odd power: and this easily leads to a contradiction.

+

82

to

In fact the genus of fl consists of two spinor genera, each of one class. A representative of the other class is

-

f2 = (2x + z ) ~ 2y2 An odd square b 5 1 (mod as m timately related is equivalent to 5 1, but is I

+ 16z2 .

(12.3)

m 2 is represented by fl or by f2 according 8) or m 5 3 (mod 8 ) . This phenomenon is into the arguments at the end of 96. If f in (6.4) fl or f2, then g is equivalent to f if in the other class if I 5 3 (mod 8 ) .

=

For indefinite ternary forms, Jones and Watson showed that the modulo exceptions lie in a finite number of classes bc2 (c E squares(2), The following result holds both for definite and indefinite forms. If b is representable by some genus but not by 2 every spinor genus in the genus, then the same is true for bq , where q is any odd prime not dividing the determinant of the genus, and conversely (Watson (1960, Theorem 7 5 ) = Cassels (1978, Chap. 11, Lemma 7.1)). The proof is quite simple. In a paper which has

z)

J. W.S.Cassels

20

already been mentioned, Kneser (1961) considered the weights of the representations of an integer b by the spinor genera of a genus and showed that only two cases are possible: (i) the weights are all equal, or (ii) the genus splits into two "half-genera", in each of which the weights are equal. The case (ii) can occur only for b in a finite number of classes modulo squares (see also Hsia (1976), Estes and Hsia (1979) and Schulze-Pillot (1980)). Jones and Watson (1956) had already obtained similar results when b is represented by some but not all the spinor genera of a genus. For definite ternary forms the situation is more obscure. Watson (1976) has determined the classes of such forms with square-free determinant which have no exceptions (cf. also Jones and Pall 2 (1939)). The form (3) x2 + y2 + 102 , which was considered by Ramanujan (19171, appears to have precisely 18 exceptions, of which the greatest is 2719 (Gupta (1941, 1973)). The number of exceptions can be arbitrarily large, as the following simple example shows: 2 2 if fo = qx + ry2 + s z , where q,rrs are primes with 2 < q < r < s, then every b q (mod 8 ) in 0 < b < q is clearly an exception. In fact the number of exceptions is usually large (Watson (1954)). We can make the following CONJECTURE 12.1.

Let

f

b e a d e d i n i t c ? t e h n a h y q u a d h a t i c dohm.

T h e n t h e h e ake o n l y d i n i t e l y many i n t e g e h b by t h e o p i n a h genuo

06

f, b u t not b y

f

b

w h i c h ahe a e p h e n e n t e d

i,tbe.td.

The first results of this nature were obtained by Linnik (1939, 1940). His investigations were continued by his pupil Malygev (cf. Malygev (1962)). They consider only forms f of a special kind: in particular its genus contains only a single spinor genus. Their arguments require an auxiliary prime g , which may be chosen arbitrarily subject to mild restrictions, and they consider only the b which satisfy (-db/q) = 1, Here d is the discriminant of Legendre symbol. They show that tions which satisfy (12.4). The plicated, but the result has the

(12.4)

f and the left hand side is a there are only finitely many excepproof is highly ingenious and comfollowing defects:

(i) one cannot get rid of (12.4).


21

(ii) even in principle, there is no bound for the exceptions (i.e. the result is ineffective). In fact they use Siegel's estimate for the class number of binary quadratic forms, which is well known to be ineffective. (iii) they consider only forms of a special kind. If we assume the Generalized Riemann Conjecture (GRH), the first two defects disappear. For given b one can then find a q satisfying (12.4) which is so small that one can estimate uniformly. Further, on the GRH the Siege1 estimate becomes effective, and indeed what is needed was proved already by Hecke (cf. Landau (1918)). To be honest, I have not studied the Linnik-Malygev proof closely enough to be sure that it does not contain further hidden ineffectivities. Recently, Peters (1978) has combined the arguments of Linnik and Malygev with those of Kneser (1961). He obtains a weak form of Conjecture 12.1 subject to a condition (12.4). (The precise formulation is on p. 70, lines 7-10 of his paper). On the GRH the condition (12.4) again disappears, and presumably the result becomes effective. The methods of Linnik and Malygev also give information about the distribution in space of the representations of b (MalyLev (1962)1 . Similar results have been obtained by Ahrensdorf and Johnson (1979) with quite different methods, but their results still contain a condition of the type (12.4). NOTES

a

i

and

1 aixi2

+ 2 1 aijxixj i 1<j aij are integral,

(1) A form

is classically-integral if the

(2) It is not asserted that all of the integers in the finitely many classes modulo squares are exceptions. For example, all odd squares are exceptions for the genus given by (12.1) , (12.3) , but 64 = fl(O,O,l) = f2(- 1,0,2) is not an exception. The result mentioned in the next part of the same paragraph shows that the set of exceptions is precisely the union of classes {bit21 , where c runs through the odd integers prime to the discriminant. Herr SchulzePillot has shown me a case in which an infinite number of bi is required, This answers a question which was left open in the German version of this paper.

J. W.S. Cassels

22

(3) The genus of but two classes. 2x2

+

2y2

+

2yz

x2

+

y2

+

10z2 contains only one spinor genus

A representative of the other class is

+

322.

REFERENCES For further references, see the book Cassels (1978) and the reports O'Meara (1976), Hsia (1979). Ahrenstorf, R.F.; Johnson, D. (1979): Uniform distribution of integral points on 3-dimensional spheres via modular forms. J. Number theory 11, 218-238. Cassels, J.W.S. London.

(1978): Rational quadratic forms. Academic Press,

Dickson, L.E. (1930): Studies in the theory of numbers. Univ. of Chicago Press. Eichler, M. (1952): Die &-mlichkeitsklassen indefiniter Gitter. Math.2. 55, 216-252. Eichler, M. (1952a): Quadratische Formen und orthogonale Gruppen. Grundlehren d. Math. Wiss 63, Springer, Berlin-Gdttingen-Heidelberg. Eisenstein, G. (1851): Tabelle der reducirten positiven quadratischen Formen, nebst den Resultaten neuerer Forschungen. J. reine angew. Math.%, 140-190. Anhang: 227-242 (=Math.Werke 2, 637-702)

.

Estes, D.R.; Hsia, J.S. (1979): Exceptional integers of some ternary quadratic forms. (Preprint)

.

Gauss, C.F. (1801): Disquisitiones arithmeticae. Fleischer, Lipsiae, (=Leipzig)

.

Gersten, L.J. (1972): The growth of class numbers of quadratic forms, Amer.J.Math. 94, 221-236. Gupta, H. (1941): Some idiosyncratic numbers of Ramanujan. Proc. India Acad.Sci. A 12, 519-520. Gupta, H. (1977): Ramanujan's ternary quadratic form x 2+y 2+1Oz 2

.

Res.Bull.Punjab Univ. (N.S.124 (1973), No 1-2,57 (1977). Hasse, H. (1923): fiber die Darstellbarkeit von Zahlen durch quadratische Formen im Kdrper der rationalen Zahlen. J.reine angew. Math.=, 129-148 (=Math.Abh. bd. 1, 3-22).


23

Hasse, H. (1923a): cber die iquivalenz quadratischer Formen im Kdrper der rationalen Zahlen.J.reine angew.152, 205-224 (=Math.Abh.Bd.l, 23-42). Hsia, J.S. (1976): Representation by spinor genera. Pacific J.Math.

63, 147-152. Hsia, J.S. (1979): Arithmetic theory of integral quadratic forms. To appear in the proceedings of the conference "Recent developments in number theory", Queen's University, Kingston, Ontario. Hsia, J.S.; Kitaoka, Y.; Kneser, M. (1978): Representations of positive definite forms. J.reine angew.Math.301, 132-141. Jones, B.W.; Watson, G.L. (1956): On indefinite ternary quadratic forms. Canadian J.Math.8, 592-608. Kloosterman, H.D. (1926): On the representation of numbers in the form ax2+by2+cz2+dt2. Acta Math.%, 407-464. Kneser, M. (1956): Klassenzahlen indefiniter quadratischer Formen. Arch.Math. (Basel)l, 323-332. Kneser, M. (1957): Klassenzahlen definiter quadratische Formen. Arch.Math. (Basel)8, 241-250. Kneser, M. (1961): Darstellungsmasse indefiniter quadratischer Formen. Math.Z.11, 188-194. Kneser, M. (1965): Starke Approximation in algebraischen Gruppen 1. J.reine angew.Math.218, 190-203. Kneser, M. (1966): Strong Approximation. Algebraic groups and discontinuous sub-groups. (Proc.Sympos.Pure Math., Boulder, Colo. 1965)pp.187-196. Amer.Math.Soc. Kneser, M. (1974): Quadratische Formen. Mathematisches Institut, Gdttingen. Lecture notes. Landau, E. (1918): ifber die Klassenzahl imagintlr-quadratischer Zahlkdrper.Nachr.k.Gesell.Wiss.Gdttingen.Math.-Phys.Kl. 1918, 285-295. Legendre, A.M.

(1798): Essai sur la thgorie des nombres. Paris.

Linnik, Ju.V. (1939): Odna obgzaja teorema o predstavlenii zisel otdel'nymi ternarnymi formami.Izv.Akad.Nauk SSSR (ser.mat.)?, 87-108. V

Linnik, Ju.V. (1940): 0 predstavlenii bol 'gih ;isel po1ozite:l 'nymi ternarnymi formami. 1zv.Akad.Nauk SSSR (ser.mat.)Q, 363-402.

24

J. W.S. Cassels

Magnus, W. (1937): ifber die Anzahl der in einem Geschlecht enthaltenen Klassen von positivdefiniten quadratischen Formen. Math.Ann. 114, 465-475 -+ I&, 643-644. Malygev, A.V. (1962): 0 predstavlenii celyh sisel pologitel 'nymi formami.Trudy Mat.Inst.Steklov 65, (212 p p . ) . Meyer, A. (1884): Mathematische Mitteilungen 1. uber die Kriterien ftlr die Aufldsbarkeit der Gleichung ax2+by2+cz2+du2=O in ganzen Zahlen. 2.ffber die Aufldsung der Gleichung ax2+by2+cz2+du2+ev2=0 in ganzen Zahlen. Vierteljahrschrift Naturf.Gesel1.ZUrich 29, 209-222. Meyer, A. (1893): Note zu der Abhandlung tlber terndre Eormenim 98. Band dieses Journals. J.reine angew.Math.E, 87-88. Meyer, A. (1894-1896): Cfber indefinite terndre quadratische Formen. J.reine angew.Math.z, 186-206; 114, 233-254; 115, 150-182; 116, 307-325. Minkowski, H. (1890): Cfber die Bedingungen, unter welchen zwei quadratischen Formen mit rationalen Koeffizienten ineinander rational transformiert werden kdnnen. J.reine angew.Math.E, 5-26 (=Werke 1, 219-239). O'Meara, O.T. (1963): Introduction to quadratic forms. Grundlehren der Math.Wiss.No.117. Springer, Berlin, Gdttingen and Heidelberg. O'Meara, O.T. (1976): Hilbert's eleventh problem. Proc.Symposia Pure Math. ( A M S ) z , 379-400. Peters, M. (1978): Dasstellungen durch definite terntlre quadratische Formen. Acta.Arith.2, 57-80. Peters, M. (1979): Exceptions of integral quadratic forms. J.reine angew.Math.314, 196-199. Platonov, V.P. (1969): Problema sil 'nor approksimacii i gipoteza Knezera-Titsa dlja algebraiseskih grupp. Izvestija Akad .Nauk SSSR (ser.mat.112, 1211-1219 + g(1970) , 775-777. Pommerenke, C. (1959): ober die Gleichverteilung von Gitterpunkten auf rn-dimensionalen Ellipsoiden, Acta Arith.5, 227-257. Ramanujan, S . (1917): On the expression of a number in the form 2 2 ax2+by +cz +du2. Proc.Cambridge Philos .Soc.GI 11-21 (=Collected papers (Cambridge University Press 1927), 169-178. See also pp.341343).


25

ROSS, A.C.; Pall, G. (1946): An extension of a problem of Kloosterman. Amer.J.Math.68, 59-65. Schulze-Pillot, R. (1980): Darstellung durch Spinorgeschlechter terndrer quadratischer Formen. J.Number Theory 12, 529-540. Siegel, C.L.(1951): Indefinite quadratische Formen und Funktionen105-142). theorie I.Math.Ann.124,17-54(=Ges.Abh.Bd.l, Tartakovskii, V. (=Tartakowsky)(1929): Die Gesamtheit der Zahlen die xS) ( s 2 4) dartellbar sind. durch eine quadratische Form F(x1,x2, 1zv.Akad.Nauk SSSR 1929, 111-122, 165-196(=Bull.Acad.Sci.de 1'URSS).

...,

Watson, G.L. (1954): The representation of integers by positive ternary quadratic forms, Mathematika 1, 104-110. Watson, G.L.(1955): Representation of integers by indefinite quadratic forms. Mathematika 2, 32-38. Watson, G.L.(1956): The equivalence of quadratic forms. Canad.J. Math.?, 526-548. Watson, G.L.(1957): Bounded representations of integers by quadratic forms. Mathematika 2, 32-38. Watson, G.L.(1960): Integral quadratic forms. Cambridge tracts in mathematics and mathematical physics. No.51. Cambridge University Press. Watson, G.L. (1960a): Quadratic diophantine equations. Philos.Trans. Roy.Soc.London 253, 227-254. Watson, G.L. (1961): Indefinite quadratic diophantine equations. Mathematika 32-38.

s,

Watson, G.L. (1963): The class-number of a positive form. Proc. London Math.Soc. (3113, 549-576. Watson, G.L. (1963a): One-class genera of positive quadratic forms. J.London Math.Soc.38, 387-392. Watson, G.L. (1963b): Positive quadratic forms with small classnumbers. Proc.London Math.Soc.(3)Gr 577-592. Watson, G.L. (1972): One-class genera of positive ternary quadratic forms I. Mathematika 19, 96-104. Watson, G.L. (1974): One-class genera of positive quaternary quadratic forms. Acta Arithmetica 24, 461-475. Watson, G.L. (1975): One-class genera of positive quadratic forms in at least 5 variables. Acta Arith.26, 309-327.

26

J. W.S. Camels

Watson, G.L. (1975a): One-class genera of positive ternary quadratic forms 11. Mathematika 22, 1-11. Watson, G.L. (1976): Regular positive ternary quadratic forms. J. London Math.Soc. (2113, 97-102. Weil, A. (1961): Adeles and algebraic groups. Lecture notes. Institute for Advanced Study, Princeton. Weil, A. (1962); Sur la theorie des formes quadratiques. Colloque sur la theorie des groupes algebriques, Bruxelles, 1962, pp.9-20. CBRM Librairie Universitaire, Louvain et Gauthier-Villars, Paris (=Oeuvres Scientifiques 11, 471-484). Witt, E. (1936): Theorie der quadratischen Formen in be1 ebigen Kdrpen. J.reine angew.Math.z, 31-44. Willerding, M.F. (1948): Determination of all classes of positive quaternary quadratic forms which represent all (positive integers. Bull.Arner.Math.Soc.5, 334-337.

Proceedings of the International Mathematical Conference LN.Y. Chen, T.B. Ng,M.J. Wicks (edr.) 0North-HollandPublishing Company, 1982

27

SOME TOPICS ON COXETER GROUPS AND WEYL GROUPS Nagayoshi Iwahori Department of Mathematics Faculty of Science University of Tokyo Hongo, Bunkyo-ku, Tokyo JAPAN

Starting from a brief introduction to Coxeter groups and Weyl groups, we shall describe several new results concerning these groups which are recently found in Japan. They are as follows: (i) finite Coxeter groups W of rank > 2 is characterized by the lattice L(W) consisting of the subgroups of W, (ii) the finiteness of the order of a group given in a certain presentation is shown to be equivalent to the finiteness of the order of a Coxeter group associated with the given presentation in an explicit manner, (iii) the generating function of B. Kostant's partition function (relative to positive roots) is characterized by several functional properties. Also a remarkable property of this generating function is given.

5 1. PRELIMINARIES We begin with recalling the notion of Coxeter groups. This notion appeared in H. S . M. Coxeter's paper [ 3 ] which classified all discrete subgroups generated by reflections of the full group of isometries of a finite dimensional Euclidean space. Coxeter groups are now widely known and used in several fields (e.g. in algebraic groups) due to the basic works on Coxeter complexes and buildings of J. Tits [ 7 1 and the Chapters 4-6 by N. Bourbaki's "Groupes et algGbres de Lie" [ 2 1 together with several works of G. Lusztig et al. (see e.g. [ 4 I.) Definition: Let S be a set and let m be a map from S x S into { 1, 2, 3, * . } U {-I. The pair (S, m) is called a Coxeter matrix if (i) s E S m ( s , s ) = 1, and (ii) s E S, t E S , s # t z> m(s, t) = m(t, s)>, 2.

-

For example, if S = { sl, s 2 , s3, s4 1 ,by putting m = m(si, s j ) , ij following matrix M = (m ) gives an example of a Coxeter matrix: ij

the

Every Coxeter matrix (S, m) defines a non-oriented, loopless graph r provided with edge-multiplicities as follows: the vertex set of r is S; two vertices s E S, t E S(s # t) are connected by an edge if and only if m(s, t) > 2 ; furthermore if m(s, t) > 2 , then the "multiplicity" m(s, t) is attached t o the edge connecting the vertices s and t. (For the sake of convenience we use the

N. Iwahori

28

following conventions: if m(s, t) = 3 ,

then instead of

we denote

; if m(s, t) = 4 , then instead of

it by

.) The graph

we denote it by thus obtained is called the Coxeter diagram of the given

r

Coxeter matrix (s, m). For example, the Coxeter diagram of the Coxeter matrix given in (1) is as follows.

It is clear that the Coxeter diagram of

m) characterizes ( S , m) uniquely.

(S,

Definition: A Coxeter group (or more precisely, a Coxeter system) associated with a given Coxeter matrix ( S , m) is the group W = W(S, m) given by the following presentation. Let F be the subset of S x S consisting of all ( s , t) such that m ( s , t)< a . Then W is generated by the set S together with the defining relations (st)m(s' Example 1. If Thus W

(S,

t, = 1

for all

, then W

m) is

( s , t)

=

E

<s, t

F.

1

s2 = t2 = (st)" = 1>

.

is a dihedral group of order 2n.

Example 2.

If ( S , m) is

A

, then W

2 3 = CS,~,rl s2 = t = r2 = (st)

. W is an infinite group. One can realize group of isometries of a plane generated by 3 reflections of the

= (tr)3 = ( r ~ ) = ~1 >

as the lines obtained by extending the 3 sides of an equi-lateral triangle in a plane. Example 3.

If

(S,

m)

is

with the symmetric group @

2-9-

--

*

W 3

, then W(S, m) is isomorphic

1 2 n-1 on n letters.

A fundamental theorem about Coxeter groups is a criterion for the finiteness of

W(S,m). Since the natural map S + W(S, m) is known to be injective, the finiteness of W(S, m) implies that S is finite. So henceforth we assume S to be finite. We call the cardinality of S the rank of W(S, m). Now define a real symmetric matrix Q = (q(s,t)) (s, t E S ) by q(s,t) = -cos(n /m(s,t)) ( s , t E S ) . Then it is known([ 2 1) that W(S, m) is finite if and only if Q is positive definite. Now decompose the Coxeter diagram r of ( S , m) into the "connected-components" rl, rr. Then it isknown that W(S, m) is isomor-

...

w1 x w 2 x x w r where Wi is the Coxeter group associated to ri (ldid r). Thus the finiteness-criterion of W(S, m) is reduced to the case where the Coxeter diagram r of ( S , m) is connected. Then the fundamental criterion is as follows ( [ 2 3):

phic to the direct product

Theorem. Let W(S, m) be a Coxeter group with finite S and connected Coxeter diagram. Then W(S, m) is a finite group if and only if its Coxeter diagram is one of the following diagrams. (The suffix in the following means the number of vertices.)

Coxeter groups and Weyl groups

§

2.

29

CHARACTERIZATION OF FINITE COXETER GROUPS BY ITS SUBGROUP-LATTICE

Let G1, G2 be finite groups. Denote by L(G ) the lattice consisting of all i subgroups of G. as the elements of L(G ) together with the partial ordering H Z K defined b$ the inclusion relation 8 3 K . The natural question "does L(G1) L(G2) imply G1 G2 ? " has the answer "No!" in general. As for the relationship between G and L(G) see M. Suzuki[ 6 ] Thus it may be an for which the answer to the interesting question to find finite groups G1 above question is "Yes!".

.

Recently T. Uzawa (University of Tokyo) established the following interesting case. Theorem (T. Uzawa[8 3). Let W = W(S, m) a finite Coxeter group of rank>2. Let H be a finite group such that L(W) Z L(H). Then W ; H. Let us describe briefly the essential ideas in the proof. Let S = Is,,

*.*,snj

and a: L(W) + L(H) be the lattice isomorphism. Consider the image K = $(<si>) under of the subgroup <si> of W generated by the element si. Then

L(K) Q L ( < s i>) implies that K is a cyclic group of a prime order p i' Now the condition on the rank of W implies rather easily that there exists an element U S W - { s } of order 2 such that siu = usi. (One has to pay a little bit of attention only for the cases (A3),(B3),(D4),(13)

; the other cases are

easy.) Then Q ( < u > ) is a cyclic group of a prime order R. Now (<Si, u>) = ) Z L(). However the,Hasse-diagramof L(<si, d) is easily seen to be of the form <Si,

@ 1

Hence L() has the same Hasse-diagram. However then one can prove that

N. Iwahori

30

is isomorphic with the Klein's four group C2 X C2 (Cn means the cyclic group of order n). Thus the order pi of ti is 2. Thus, if one can show that (t t )m(si'sj) = 1 (for every i,j with i t j), then there exists a surjective i j Then JI must be injective, because if the kernel N of homomorphism ~JJ : W-H. J, is not {l] , we get L(W/N) f L(H) and L(W) Z L(H) which is impossible.

The proof of (t.t )m(siysj)zl is easy if m(si,sj) = 2 because of L(<si,s.>) Ij J L( ) . For the case m ( s s . ) = 3, the Hasse-diagram L(<si,s.>) j i' 3 J (i.e. of L( ) is i' j

'1-

From this picture Uzawa concludes that is either isomorphic to <Si,Sj> @3 or to Cj X C3. However Z C3 X C3 since ti is of order 2. Thus one concludes (t t )3 = 2 . i j A similar (but a slightly more technical) proof works for the remaining cases m(si,s.) = 4 or 5 and the proof can be accomplished. (See[8 1 for the detail.) 3

5 3.

SOME ANALOGUES OF COXETER GROUPS We give here some analogue of the finiteness criterion of Coxeter groups stated in 51. Recently J. Murakami (University of Tokyo) established the following modified version.

1

(14 i 9 j< r), nij (14 i # jg r), ij k .(li i< r, 14 j< s ) , m..(16 i, j6 s) be the positive integers such that iJ 1J ni3 2 (16 i< r), nij = nji (16 i =) jg r), mii = 1 (14 (i s ) . Let G be a group with the following presentation: generators : A1, A 2 , - * * ,Ar; B1, B2, B defining relations : 2n 2 A i = 1 (16 ig r), B. = 1 (16 j < s ) , i J = 1 (16 i P k i r), (A.A )"ij = 1 (16 i j< r), (AiG1)";k lj (A.B.)kij = 1 (16 i\< r, 14 j\<s ) , (B.B.lmij = 1 (I< i, ji s ) . Theorem (J. Murakami[ 5

).

Let ni(l< i4 r), n

...,

+

1 J

1 3

Coxeter groups and Weyl groups m(fi, e.) = m(e 3

j'

= k

f

i

31

(IdiGr, 16j<s),

ij

m(f-i, e.) = m(ej, f-i) = kij 3

(1( id r, lg jg s ) ,

m(ei, e.) = m(ej, ei) = m

(I< i , j < s ) . ij Then G is finite i f and only if W i s finite. Furthermore, when G i s finite, then the order of G coincides with the order of W. (Remark. G is in general not isomorphic to W even if G i s finite.) 3

Example 1. Let G = < A ~ ,

I

a= 4= (A~

= (A

A-1) 3 = i> 1 2

Then the corresponding W has the following Coxeter diagram

-Po 2

i.e. that of

(F4).

1

-1

f-2,

Thus G i s finite and of order 1152.

Example 2.

Let G = 2 2 1 2 Then the corresponding W has the following Coxeter diagram

.

P i.e. that of

(E6).

Thus G

e2

i s finite and of order 51840.

5 4. ON A TOPIC RELATED WITH WEYL GROUPS One may define Weyl groups as the direct product of finitely many Coxeter groups W X W2 X . . . X Wr such that (i) each W has a connected Coxeter diagram ri and 1

ri

i

(12(k)) with k f (2,3,4,6}. However the significance of the Weyl groups exists in the relationship with the theory of algebraic group. But since we have no space here speaking about it, let us take another route to describe our results. (ii) each

is not

(H3) or

(H5) or

Let E = RR be an n-dimensional Euclidean space with positive definite inner product ( , ). A finite subset II = (a --*,a,)of E is called a system of 1' simple roots if the following conditions are satisfied : (i)II forms a base of the real vector space E and (ii) 2(ai, a j ) / ( a j , a j ) is a non-positive integer for every i , j (1 6 i # j

a). R

It is known (e.g.[2])that if Il i s a system of simple roots in E = R group W = W(lT) generated by the reflections wi(l 0.

Let

r)

i

and

a r e a l l of

rl

(g) I),

V(ICC

i

(9)

I)

' 0.

Therefore Z (x(h) hEL

(4.2)

- r 5 ( h ) )m

+l(x(g)

- r c ( g ) ) W r (mod

From ( 4 . 1 )

(

~

~

?&

1.

and ( 4 . 2 ) w e have t h a t

( x l nG 1 and hence

o

5

(xc

(rc,rl) = r 6 ,r)) (mod 50 i i s a unique element of I r r ( C i I o =

1

1, such t h a t

+5 0 ~ (mod ) PI.

Lastly l e t

5 E

Irr(C.1'

3

j # i

with

and l e t

g E C

j'

V

and hence

(mod"$

0

If

v ( l C G ( g )1 ) = 0 , t h e n

D(K) =

p

pr

v(lCC (g)

).

1)

= 0

and f o r

K = gr E C a ( r )

j

Pi,

and hence

X B ( K ) = 0.

r j K = gG E z ( R * r ) n Z(R*G) , w e have O = A ( K ) E w ( K ) =

B

X

Since

x1

(mod% 1

41

The representation theory offinite groups and hence

x ( g ) E 0 (mod

7)

and w e have ( 4 . 3 ) .

J

3

where

L

ranges over

0

CE(C.1 3

(mod

and

Thus

1, g E L.

This completes t h e proof.

REFERENCES

[l] J . L. A l p e r i n : Proc. of Conference of f i n i t e g r o u p s I Academic P r e s s (1975) I 341-356. [2] W. F e i t : R e p r e s e n t a t i o n s of f i n i t e g r o u p s , Yale U n i v e r s i t y (1969). [3] G . Glauberman: Canad. J. Math., 2 0 ( 1 9 6 8 ) , 1465-1488. [4] I . M. I s a a c s : Amer. J. Math., 9 5 ( 1 9 7 3 ) , 594-637. : P a c i f i c J. Math., 5 3 ( 1 9 7 4 ) , 171-188. 151 : J. A l g e b r a , 5 1 ( 1 9 7 8 ) , 476-490. [6 1 [ 7 ] T. R. Wolf: I l l i n o i s J . Math., 2 2 ( 1 9 7 8 ) , 327-340.


Roceedings of the International Mathematical Confererrce L.H.Y. Chen. T.B. Ng, M.J. Wicks (edp.) 0North-Holland AtMishing Company, 1982

43

WHITE NOISE ANALYSIS AND ITS APPLICATIONS Takeyuki Hida Department o f Mathematics Faculty o f Science Nagoya U n i v e r s i t y Chi kusa-ku, Nagoya , 464 JAPAN

White noise may be r e a l i z e d as the time d e r i v a t i v e d B ( t ) / d t = 1 b ( t ) of a Brownian motion { B ( t ) , t E R 1 . The c o l l e c t i o n { b ( t ) l can be thought o f as a continuous analogue o f a sequence o f independent , i d e n t i c a l l y d i s t r i b u t e d Gaussian random variables. Then { f l ( t ) l can therefore be taken t o be the system o f variables o f Brownian f u n c t i o n a l s which are nonlinear f u n c t i o n a l s o f b ( t ) ' s o r e q u i v a l e n t l y o f B ( t ) ' s . With t h i s choice o f variables, we d e f i n e t h e a r t i a l derivat i v e s l i k e a/ab(t) w i t h respect t o the O(t! and then proceed t o introduce stochastic p a r t i a l d i f f e r e n t i a l equations i n v o l v i n g ( a / a b ( t ) ) ' s and m u l t i p l i c a t i o n s by the B ( t ) ' s . By the use o f these operators, we s h a l l be able t o discuss the causal calculus, where the time e v o l u t i o n i s e x p l i c i t l y taken i n t o account. I n t e r e s t i n g a p p l i c a t i o n s can be found i n t h e f i e l d s o f Quantum Mechanics and E l e c t r i a l Engineering. Cont e n t s

51.

Introduction Generalized Brownian f u n c t i o n a l s

52.

Causal Calculus

SO.

53. Concluding remarks References

T. Hi&

44

50. I n t r o d u c t i o n . We s h a l l discuss the analysis o f f u n c t i o n a l s , n o n l i n e a r i n general, o f Brownian motion { B ( t ) l . They are c a l l e d Brownian f u n c t i o n a l s and are o f t e n expressed as d f u n c t i o n a l s o f white noise { b ( t ) l w i t h b ( t ) = E B ( t ) , since the l a t t e r may be viewed as a continuous analogue o f the independent i d e n t i c a l l y d i s t r i b u t e d Gaussian random v a r i a b l e s and i s t h e r e f o r e e a s i e r t o be d e a l t with, So f a r , two basic t o o l s have been used and i n f a c t they are p l a y i n g very important r o l e s i n our analysis. These t o o l s are

1 ) I n t e g r a l representation o f Brownian f u n c t i o n a l s by the use o f 7 - t r a n s f o r m , and 2)

I n f i n i t e dimensional r o t a t i o n group.

While, as i s seen i n t h e Workshop Lecture Notes [5], a new t o o l has been proposed and was revealed t o be a powerful t o o l i n our analysis, i t i s i n f a c t t h e method o f the causal calculus, where the passage o r e v o l u t i o n o f time should be taken i n t o account e x p l i c i t l y . We are then l e d n a t u r a l l y t o the idea o f t a k i n g white noise { b ( t ) l t o be a system o f variables of Brownian functionals, and t h e r e f o r e we may t h i n k o f t h e a as w e l l as t h e i r a d j o i n t p a r t i a l d i f f e r e n t i a l operators o f the form a * aUt) operators (-1 We may a l s o t h i n k o f the m u l t i p l i c a t i o n by b ( t ) and the a&t) Gauss kernel so as t o propose the so-called i n f i n i t e dimensional harmonic analysis.

-

.

With t h i s sketch of the theory i n mind some recent developments w i l l be discussed i n t h i s paper. Of course, no attempt i s made a t e i t h e r g e n e r a l i t y o r completeness a t present , however we have hopes t h a t t h i s approach would be a good c o n t r i b u t i o n t o the f u t u r e development o f the white noise analysis. 1.

Generalized Brownian f u n c t i o n a l s .

Since ( b ( t ) l i s taken t o be the basic system o f variables, i t i s q u i t e reasonable t o take polynomials i n b ( t ) ' s and t h e i r exponential f u n c t i o n s as basic f u n c t i o n a l s o f white noise, although they are defined o n l y as generalized Brownian functionals. Indeed, our class ( L 2 ) - o f generalized Brownian f u n c t i o n a l s were introduced so as t o i n v o l v e the polynomial: having been renormalized i n an a d d i t i v e manner. Exponential functions o f the B ( t ) ' s , as a r e s u l t , accept m u l t i p l i c a t i v e renormalization t o become generalized Brownian f u n c t i o n a l s . Recalling such idea, we s h a l l have a s h o r t review o f the d e f i n i t i o n as w e l l as some basic properties o f (L2 ) - f u n c t i o n a l s . We s t a r t w i t h the measure space o f p ) , where A* i s the dual space o f the white noise, denote i t by Schwartz space d , 28 i s the Bore1 f i e l d generated by c y l i n d e r subsets o f 8* and )-I i s the p r o b a b i l i t y measure given by the c h a r a c t e r i s t i c f u n c t i o n a l C ( 6 ) = 1 be the space o f m u l t i p l e Wiener e x p [ - ~ l l ~ 1 1 ~ ] .Set (L2) = L 2 ( # ' , p ) , and l e t i n t e g r a l s o f degree n. Then (L2) i s the Wiener-It6 decomposition. n The 7 - t r a n s f o r m defined by

@,a,

;hf, =z@q

White noise analysis gives an isomnrDhism between w i t h kernel C ( s - n), (5, ,)

45

and the reproducing kernel H i l b e r t space IF If J ' i s r e s t r i c t e d t o &n, then we have

.

E

(The i n t e g r a l representation. )

4'n

where L (R ) holds t h a t

i s the space o f a l l s y n e t r i c

L2(Rn)-functions.

I n addition, i t

and therefore we e s t a b l i s h

where

3

= IU(c); U

i s given by ( 2 ) f o r cp c g n 1 and i t i s topologized so t h a t

isomorphisms ( 3 ) hold. There i s a simple way t o compute the f u n c t i o n a l U(c) Namely, the 3 - t r a n s f o r m due t o Kubo and Takenaka:

gives us the associated

i n the expression (2).

U-functional i n such a way t h a t

where cp i s n o t necessarily i n

gnb u t can be a r b i t r a r y .

The isomorphisms ( 3 ) extend t o

(4) O n i n a natural manner, where H (R ) - {symmetric generalized func i o n s i n the Sobolev space H-m(Rn) o f order - m l . The space (L2)- o f generalized Brownian functionals i s t o be the sum o f the

(5)

:

T.Hida

46

This d e f i n i t i o n has some freedom f o r choosing a topology t o sum up x h - n ) ' s (see 2 [3] and [ a ] ) , however we understand, i n t h i s note, ( L ) - t o be the dual space o f m

@;bC,(,",

the t e s t f u n c t i o n a l s (L2)+ =

where

NAn)i s

defined t o be the

------.

n=O

subspace isomorphic t o the s y m e t r i c Sobolev space H(nt1)'2(Rn) t i o n o f the isomorphism ( 3 ) .

A t y p i c a l example o f an '#,!,-n)-functional

under the r e s t r i c -

is

: i ( t ) n : = n!H ( b ( t ) ; 1 n

2 ), which i s obtained by a d d i t i v e renormalization from i ( t ) n , f o r which t h e r e i s no way t o give a d e f i n i t e value w i t h o u t m o d i f i c a t i o n . (Hn(x;

0')

i s the Hermite polynomial o f degree

Another example i s the exponential f u n c t i o n o f

n w i t h parameter

IJ

i ( t ) : e.g.

exp[ib(t) + 1/2dtl i s a member o f

2 ( L ).

We may define polynomials i n B ( t ) , the second d e r i v a t i v e o f Brownian motion, s t i l l by using the a d d i t i v e renormalization. I n t h i s case, we must take h i g h e r order Sobolev space; f o r instance, i n order t o have polynomial i n

B ( t ) , i t i s enough f o r us t o have

:B(t)n:,

-1)

hT H

a renormalized n (R )

i n place o f

m*)'2(Rn) i n (3). We can even i n v i t e a s u i t a b l y renormalized exponential function of B ( t ) , c a l l i t :exp[iB(t)]:, t o a c l a s s o f generalized f u n c t i o n a l s e.g. i n a manner t h a t we understand i t as a generalized Brownian f u n c t i o n a l whose 7 - t r a n s f o r m i s given by C(c)exp[c'(t)]. But, a t present, these polynomials and exponential functions are o f l e s s i n t e r e s t t o us. Thus, s p e c i f i c a t i o n o f NA-") i n ( 4 ) i s a matter o f choice and o f the a p p l i c a t i o n s t h a t one has i n mind. §2.

Causal calculus.

a is

The p a r t i a l d i f f e r e n t i a l operator

defined i n the f o l l o w i n g manner (see aNt) [5] Chapter 5 f o r d e t a i l s ) : L e t U ( c ) be associated w i t h cp E ( L 2 ) If U has t h e Fr6chet d e r i v a t i v e U ' ( t ) , then cp i s said t o be d i f f e r e n t i a b i e a t t

5

and we define

a

c

p =-y'(U;(t)).

ab(t) The operator

a a&t)

i s o f t e n denoted simply by

at.

The a d j o i n t operator

a;

Whlte noise analysis i s defined i n the usual way.

Higher order operators such as

a l s o defined, and m u l t i p l i c a t i o n by

i ( t ) i s expressed as

47

atas,

2 atat, etc.

b(t)- =

a; + at.

are

By using these operators we c a r r y on the causal calculus, some f i e l d s o f which are 1isted be1ow.

a)

Characterizations o f Brownian f u n c t i o n a l s .

By using stochastic p a r t i a l d i f f e r e n t i a l equations i n v o l v i n g at, a,; fl(t). etc. we can characterize some elementary generalized Brownian f u n c t i o n a l s (see [5]). As a r e s u l t , we can see the properties o f the generalized f u n c t i o n a l o f the form

,

: e x p [ i c ( t ( ~ ) ~ d u ] : , e.g. i t might be viewed as a continuous analogue o f the Gauss 1 Rn i f i c i s taken t o be -7 Also, such a f u n c t i o n a l has an

kernel on

.

i n t e r e s t i n g a p p l i c a t i o n i n quantum mechanics (see [B]). b)

Applications t o p r e d i c t i o n theory.

As was proposed i n [4] we can apply operators at, a; etc. t o s t o c h a s t i c 2 processes l i v i n g i n (L ) t o discuss p r e d i c t i o n problems f o r them. D e t a i l s are omitted here. c)

The Wiener expansion.

I t i s an important and even serious problem t o obtain the kernels o f the i n t e g r a l

representation for the given Brownian f u n c t i o n a l s o r f o r s t o c h a s t i c processes 2 l i v i n g i n (L ). L e t cp be a Brownian f u n c t i o n a l which i s i n the domain o f products o f the {Fn} be the system o f the associated kernels. Then we have

at's,

and l e t

was discussed by N. Wiener by Fn using the so-called Lee-Wiener e l e c t r i c a l c i r c u i t s (see e.g. [2] Appendix 4 ) . Although the former i s a t h e o r e t i c a l approach, one can s t i l l t h i n k o f ways o f p r a c t i c a l use and make some comparisons w i t h Wiener's method. An a l t e r n a t i v e way o f o b t a i n i n g the kernel

Now consider the case where cp i s a generalized Brownian f u n c t i o n a l and i s i n t h e domain o f products o f the at's. The kernel Fn, which i s a generalized function, i s uniquely determined i n such a way t h a t

T. Hiah

48

where the bracket denotes the canonical b i l i n e a r form t h a t l i n k s

2 (L2)+ and (L ) - .

53. Concluding remarks. A few remarks are now i n order. i)As i s i n s p i r e d by the exponential f u n c t i o n discussed i n 62 a), we have hopes t o develop the Fourier Analysis over t h e space A* w i t h measure U. An i n t e r e s t i n g approach has already been s t a r t e d by H.-H.,Kuo [7] and others. I t would be f i n e i f one can f i n d a close connection w i t h Levy's Laplacian operator (see [ l ] ) . i i ) There are l o t s o f examples t h a t can be characterised by t h e method t h a t was i l l u s t r a t e d i n 92 a). I t seems an i n t e r e s t i n g problem t o f i n d a method t h a t could deal w i t h the c h a r a c t e r i z a t i o n o f generalized Brownian f u n c t i o n a l s i n a systematic way. i i i ) P. L&y [l] has suggested t o us a geometrical consideration o f i n f i n i t e dimensional manifolds i n $* w i t h p ( i n our understanding). I n v e s t i g a t i o n along t h i s l i n e , t a k i n g i n t o consideration t h e i n f i n i t e dimensional r o t a t i o n group, would be i n t e r e s t i n g .

[References] P. Levy, Probl&nes concrets d'analyse f o n c t i o n n e l l e , G a u t h i e r - V i l l a r s , 1951 T. Hida, Brownian motion. Iwanami 1975 ( i n Japanese); English e d i t i o n , Springer-Verlag, Applications o f Math. 11, 1980.

[31

[41 [51

------

, Analysis o f Brownian f u n c t i o n a l s . Carleton Mathematical Lecture Notes no. 13, 2nd e d i t i o n 1978, Carleton Univ. Ottawa.

_------, Causal calculus and an a p p l i c a t i o n . Professor P. Masani volume, North-Holl and, t o appear. , Calculus o f Brownian functional s . Lecture Notes , Workshop a t National U n i v e r s i t y o f Singapore, June 1 5, 1981 ( t h i s Proceedings).

_------

-

c r 71

Japau Academy 56 ser. A (1980), 376-380, 411 -416.

[81

L. S t r e i t and T. Hida, Generalized Brownian f u n c t i o n a l s and the Feynman

61 I . Kubo and S. Takenaka, Calculus o f Gaussian w h i t e noise. I , 11. Proc. H. -H. KUO, Fourier-Wiener transform on Brownian f u n c t i o n a l s . P r e p r i n t .

integral

. Prepri n t .

Proceedings of the International Mathematical Conference LJ&Y. Chen, T.B. Ng,M.J. Wicks (eds.) 0North-Holland fiblishing Company, 1982

49

POWERS OF THE AUGMENTATION IDEAL B. H a r t l e y

Department of Mathematics University of Manchester Manchester M13 9PL England

Some s e l e c t e d r e s u l t s and problems are discussed r e l a t i n g t o t h e powers of t h e augmentation i d e a l of an i n t e g r a l group r i n g . The main t o p i c s d e a l t w i t h a r e dimension subgroups and the i n t e r s e c t i o n of the powers. INTRODUCTION This is not intended a s an exhaustive survey of t h i s r a t h e r wide a r e a , b u t simply a s an i n t r o d u c t i o n t o some of the problems and r e s u l t s i n it. F u l l e r treatments of some of t h e s e topics can be found i n [20], [21].

1. FREE GROUPS AND FORMAL POWER SERIES RINGS We begin by d i s c u s s i n g dimension subgroups. These seem t o have been considered f i r s t by Magnus [ I s ] , who d e a l t only with f r e e groups, and introduced dimension subgroups a s follows. L e t A be t h e r i n g of polynomials over t h e i n t e g e r s Z i n the non commuting indeterminates XA ( A E A ) . Thus, each elementof A i s a f i n i t e formal l i n e a r combination of monomials

xal x x 2

*** X ~ t

( t b 0, Ai

E

A).

Monomials a r e multiplied by j u x t a p o s i t i o n , and t h e m u l t i p l i c a t i o n is extended by l i n e a r i t y t o give a d i s t r i b u t i v e m u l t i p l i c a t i o n on t h e whole of A. Note t h a t A has an i d e n t i t y , t h e "empty monomial". The monomial (1) is s a i d t o have degree t , and more generally, any non-zero l i n e a r combination of monomials of degree t i s c a l l e d a homogeneous element of degree t . Thus each element a E A i s uniquely of the form a = a + a ( a 5 0) tl

where

atj

t2

i s homogeneous of degree

tj

A = A. @ A 1

where

At

@

+ .'. + and

t l < tp
L2 >

... >

L LC

It can then be seen t h a t

L

is

adapted t o t h e s e r i e s >

LC+1

3

0.

Thus t h e r e a r e i n t e g e r s

such t h a t

,

bi

..., b s

t

.

i s a b a s i s of L t I f U i s t h e u n i v e r s a l enveloping a l g e b r a of L, then by the Poincarg-Birkhoff-Witt Theorem, U has a Q-basis c o n s i s t i n g of t h e ordered products

We give to

bi

a weight

p(bi)

defined by

and give the b a s i s element (1) the weight

E t denote the Q-subspace of U spanned by t h e elements (1) of weight 2 t . It i s an easy matter t o show by induction on s-i, and f o r f i x e d i, by induct i o n on 2, t h a t i f (1) has weight t , then

Let

I f i = s t h i s i s c l e a r since b, c o m u t e s with t h e o t h e r f a c t o r s ; i t i s a l s o c l e a r i f i c j l , Otherwise, we s u b s t i t u t e = b . bi + [bi, b j l ] . j, ~1 K i s t h e i d e a l of U spanned by t h e elements (1) w i t h

b.b 1

It follows t h a t i f

R > 0,

then ICEt 6 Et+l f o r a l l t 2 0. Hence since c l e a r l y L n E c + l = 0, we deduce t h a t Kc+l

Kt

ii >

*.. >

and

K C + l = 0.

K/KC+’ of the by r i g h t

U

(3)

Because of t h e way L was constructed, {exp u : u E L) (formed i n i> i s a group isomorphic t o G*. Now (expu) - 1 a l s o a n n i h i l a t e s t h e f a c t o r s of ( 3 ) . Thus we o b t a i n a homomorphism of QG* i n t o U which i s monomorphic on G* b u t zero on Aq(G*Ic+l. Hence G* n (1

+ A (G*)‘+’)

Q

=

1.


53

The c a l c u l a t i o n used t o e s t a b l i s h (2) is of course similar t o t h a t given i n [7], but it i s much e a s i e r i n t h e l i n e a r context. We t u r n our a t t e n t i o n t o t h e i n t e g r a l case. G and n 5 1,

It i s immediate t h a t f o r a l l groups

where f o r b r e v i t y we w r i t e

In p a r t i c u l a r , i f G is f i n i t e l y generated t o r s i o n f r e e n i l p o t e n t , then by Theorem 2.2, Dc+l(G) = 1, and i t i s n o t hard t o see as i n Section 1 t h a t G operates f a i t h f u l l y a s a group of a d d i t i v e automorphisms of ZG/JA(G)C+l, where JA(G)C+l/A(G)C+l i s t h e t o r s i o n subgroup of ZG/A(G)C+l. In t h i s way we can obtain a f a i t h f u l r e p r e s e n t a t i o n of G by unipotent m a t r i c e s over Z. Although t h e e x i s t e n c e of such a r e p r e s e n t a t i o n i s q u i t e easy t o e s t a b l i s h d i r e c t l y (see [30], p.231, the importance of t h i s one i s t h a t i t i s canonical. Grunewald and Segal e x p l o i t t h i s i n one of t h e f i r s t s t e p s of t h e i r r e c e n t s o l u t i o n of the isomorphism problem f o r f i n i t e l y generated n i l p o t e n t groups [ 6 1. It enables them t o transform t h i s problem t o a c e r t a i n conjugacy problem f o r n i l p o t e n t subgroups of GL(n, Z). Not a g r e a t deal i s known about the i n t e g r a l dimension subgroups i n g e n e r a l , however. It was suspected f o r a long time t h a t they coincided with t h e terms of the lower c e n t r a l s e r i e s (see [ 5 I ) , and t h e r e was some evidence t o support t h i s . However it was eventually r e f u t e d by Rips [ 2 6 ] , who proved THEOREM 2.3. There e x i s t s a f i n i t e 2-group L D ~ ( G ) =I 2.

G

of nilpotency c l a s s 3 such t h a t

The b e s t general r e s u l t t o d a t e seems t o be contained i n t h e b e a u t i f u l but somewhat d i f f i c u l t paper of Sjogren [27], of which t h e a u t h o r has w r i t t e n a s i m p l i f i e d version [ 81. To s e t it i n context, n o t e t h a t by (4) and Theorem 2.2, we know t h a t , f o r a l l groups G and n b 1,

-

D (G)/yn(G) i s periodic. -n Sjogren's r e s u l t improves on t h i s observation a s follows.

...

THEOREM 2.4.

There e x i s t i n t e g e r s cl! c2, Dn(G)/yn(G) has f i n i t e exponent d i v i d i n g cn.

such t h a t f o r a l l groups

The i n t e g e r s cn a r e e x p l i c i t l y given a s follows. multiple of { l , 2, m] and define

...,

Then

Thus

PIC,

Let

G,

bm b e t h e l e a s t common

n- 1 c = n a (k) k . n k=1 c1 = c 2 = c 3 = 1, c,, = 2, i f and only i f n 2 p + 1.

COROLLARY 2.5. ( i )

Dn(G) = yn(G) ( i i ) Dn(G) = yn(G)

and c5 = 24. Hence w e have

Also i f

p

i s a prime, then

f o r a l l G and n = 1, 2, 3. if n p + 1 and G i s a p-group.

The f i r s t p a r t of t h i s i s w e l l known; t h e second p a r t was proved f o r

n 6 p

by

B. Hartley

54

Moran [19].

Tahara [29] has improved Sjogren's r e s u l t f o r

THEOREM 2.6. For a l l

D5(G)/y5(G)

G,

n = 5:

has exponent d i v i d i n g 6. of odd o r d e r

G

A t p r e s e n t t h e r e seems t o be no known example of a f i n i t e group such t h a t Dn(G) > yn(G) f o r some n.

It is a l s o not known whether m

n

m

D,(G) =

n= 1 f o r a l l groups

G.

n

Y,(G)

(5)

n=l

Write m

Then (5) i s equivalent t o PROBLEM 1.

Is it t r u e t h a t

Du(G) = 1 f o r a l l n i l p o t e n t groups

G7

In i n c r e a s i n g o r d e r of optimism, we may s t a t e PROBLEM 2.

If

G

i s n i l p o t e n t , does it follow t h a t

PROBLEM 3. Does t h e r e e x i s t a f u n c t i o n n i l p o t e n t groups G of c l a s s c ?

f(c)

Dn(G) = 1 f o r some

such t h a t

n?

Df(c)(G) = 1 f o r a l l

Not a g r e a t d e a l i s known about these matters. I f G i s n i l p o t e n t , then D (G) l i e s i n the c e n t r e of t h e t o r s i o n subgroup of G [9]. An o s t e n s i b l y s t r o n g e r W statement follows from

-

l ) c ( c + 1) + 1, THEOREM 2.7. [ 9 1 Let $(b, c ) = (b group of c l a s s $ c with t o r s i o n subgroup T. Then

and l e t

G

be a n i l p o t e n t

(G) < Db(T) i f b 2 1. D$(b,c) Work of KusYkule; [13] l e a d s t o a s i m i l a r r e s u l t , b u t w i t h a f u n c t i o n whose o r d e r of magnitude i s somewhat l a r g e r . Evidently t h i s r e s u l t "reduces" Problems 1 and 2 t o t h e case of a t o r s i o n group. Thus we can o b t a i n f o r i n s t a n c e COROLLARY 2.8. exponent, then

I f G i s n i l p o t e n t and each Sylow subgroup of Dn(G) = 1 f o r some n.

G

has f i n i t e

Elementary arguments reduce Problem 3 t o t h e f i n i t e case. 3. THE INTERSECTION OF THE POWERS OF A(G) There i s a l a r g e l i t e r a t u r e on t h e s t r u c t u r e of t h e a d d i t i v e groups A(G)"/A(G)"+' and ZG/A(G)n. We s h a l l bypass t h i s and concentrate our a t t e n t i o n on t h e intersection m

n A(G)". n=l I f we wish t o study t h i s i d e a l , a n a t u r a l f i r s t aim i s t o determine when it i s zero. There i s a n a t u r a l connection between t h i s and c e r t a i n r e s i d u a l n i l p o t e n c e questions i n group theory. For example we have A ( G Y=

THEOREM 3.1. (Baumslag [ 2 1 , P a s s i [22]). L e t F b e f r e e , RQ, and Then F/R' i s r e s i d u a l l y n i l p o t e n t i f and only i f A(G)W = 0.

Necessary and s u f f i c i e n t c o n d i t i o n s f o r

A(G)'" t o be zero, when

G

G = F/R.

i s nilpotent,


55

were given by the author i n [12]. The general c a s e was handled by Lichtman [ 141. To s t a t e h i s r e s u l t , we d e f i n e a group H t o be discriminated by n i l p o t e n t groups of f i n i t e prime power exponent i f and only i f , f o r each f i n i t e set h,, hn of non i d e n t i t y elements of H, t h e r e e x i s t s a n i l p o t e n t group K of f i n i t e prime power exponent and a homomorphism 6 : H -+ K such t h a t $ < h i ) 1 (1 6 i f n ) .

...,

THEOREM 3.2. nilpotent or exponent.

A(GIW= 0 i f and only i f e i t h e r G i s r e s i d u a l l y t o r s i o n f r e e is discriminated by n i l p o t e n t groups of f i n i t e prime power

G

An important s t e p i n the proof of t h i s i s t o show that A(G)w = 0 when G i s t o r s i o n f r e e n i l p o t e n t . This was roved by t h e a u t h o r i n [12], b u t a mare e l e g a n t proof has s i n c e been i n d i c a t e d by [mel'kin [28], using the correspondence between d i v i s i b l e t o r s i o n f r e e n i l p o t e n t groups and Lie a l g e b r a s over t h e r a t i o n a l s mentioned i n Section 2. In t h e general case, t h a t i s , when A(G)W i s n o t n e c e s s a r i l y zero, i t i s reasona b l e t o t r y to generate i t i n some simple way. There a r e A c e r t a i n elements which f a i r l y obviously l i e i n i t , and which p l a y an important r o l e i n determining necessary conditions f o r A(G)w t o vanish. For example, given a prime p, we may define a n element x E G t o be of i n f i n i t e p-height, i f t h e equation ypn = x is soluble i n

for a l l

G

n 2 0. (1

-

Then x)(i

- y)

E

A(G)~

i f x has i n f i n i t e p-height i n G and y i s a p-element of G. In t h e n i l p o t e n t case, t h e elements of t h i s form come f a i r l y c l o s e t o generating A(GIW. More p r e c i s e l y , l e t G be n i l p o t e n t , T be i t s t o r s i o n subgroup, and Tp be t h e Sylow p-subgroup of T. The s e t of elements of i n f i n i t e p-height i n G i s a subgroup G(p) of G ( a s G i s n i l p o t e n t ) . Let G*(p)/T be t h e subgroup of elements of i n f i n i t e p-height i n G/T, and Tp(p) = Tp n G(p) denote t h e subgroup of elements of i n f i n i t e p-height i n Tp. THEOREM 3.3.

[9

3.

If

G

i s n i l p o t e n t , then

In favourable cases, a s f o r instance when t h i s reduces t o A ( G )=~

Tp

h a s f i n i t e exponent f o r a l l

p,

A(G(P))A(T~)zG.

P I n studying questions about A(G)W, one i s led q u i t e quickly t o consider t h e r e l a t i o n between the powers of the augmentation i d e a l of G and those of c e r t a i n subgroups of G. For example, t h e statement t h a t A(G)W= 0 i f G i s t o r s i o n f r e e n i l p o t e n t follows q u i t e e a s i l y from THEOREM 3.4. [ 9 ] [12] I f G i s a f i n i t e l y generated t o r s i o n f r e e n i l p o t e n t grouE of c l a s s f c and H 6 G, then a ( c l c a n ZH 5 A ( H ) ~ f o r a l l

a 2 1.

Two r e s u l t s w i t h a s i m i l a r flavour a r e the following, from which resuLts l i k e Theorem 2.7 can be deduced. A s p e c i a l c a s e of t h e f i r s t one i s i n Kuskulez [13].

B. Hartley

56

group of class f c and K is a normal subgroup of G such that G/K is torsion free, we have

THEOREM 3.6. [lo] Let p be a given prime and c, N be given natural numbers. Then there exists a function $(a) (depending on p, c, N) such that if G = KH is a nilpotent group of class f c, where K d G and H has exponent dividing pN, we have

A(G)@(~)

n ZK s A(K)~

for all a 3 1.

In addition to A ( G ) ' , some attention has been paid to the higher transfinite powers, their intersection, and the corresponding "dimension subgroups". See for instance [ 31 [Ill. Questions about powers of A(G) and dimension subgroups are sometimes studied from the standpoint of stable representations; several Russian authors take this For example, one may ask, for which groups G point of view (see [24] [25]). does there exist a faithful G-module containing a descending series of length a of submodules, on the factors of which G operates trivially? The answer is that such a mdule exists if and only if G n (1 + A(GIa) = 1, and direct sums of copies of ZG/A(GIa are free in the category of these modules. REFERENCES Amayo, R. and Stewart, I.N., Infinite Dimensional Lie Algebras (Noordhoff, Leyden, 1974). Baumslag, G., Wreath products and extensions, Math. 2. 81 (1963) 286-299. Gruenberg, K.W. and Roseblade, J.E., The augmentation terminals of certain locally finite groups, Canad. J. Math. 24 (1972) 221-238. Grin, O., b e r die Faktorgruppen freier Gruppen I, Deutsche Math. , Jahrgang 1. Heft 6 (1936) 772-782. GrGn, 0., Zusammenhang zwischen Potenzbildung und Kommutatorbildung, J. reine angew. Math. 182 (1940) 158-177. Grunewald, F.J. and Segal, D., Some general algorithms I: Arithmetic groups, Ann. Math. 112 (1980) 531-583; 11: Nilpotent groups, ibid.112 (1980) 585-617. Hall, P., The Edmonton Notes on Nilpotent Groups (Queen Mary College, London, 1969). Hartley, B., Dimension and lower central subgroups, Preprint, University of Manchester, 1978. Hartley, B., Powers of the augmentation ideal in group rings of infinite nilpotent groups, J. London Math. SOC., to appear. Hartley, B., An intersection theorem for powers of the augmentation ideal in groups rings of certain nilpotent p-groups, to appear. Hartley, B., Augmentation powers of locally finite groups, Proc. London Math. SOC. (3) 23 (1976) 1-24. Hartley, B., The residual nilpotence of wreath products, Proc. London Math. SOC. (3) 20 (1970) 365-392.

51

Powers of the augmentation ideal Y

Y

[13] Kuskulei, A.Kh., On f i n i t e l y s t a b l e r e p r e s e n t a t i o n s of n i l p o t e n t groups, L a t v i a n Math. Yearbook 16 (1973) 39-45 (Russian). [14] Lichtman, A . I . , The r e s i d u a l n i l p o t e n c e of t h e augmentation i d e a l and t h e r e s i d u a l n i l p o t e n c e of some c l a s s e s of groups, I s r a e l J. Math. 26 (1977) 276-293. [15] Magnus, W., Beziehungen zwischen Gruppen und I d e a l e n i n einem s p e z i e l l e n Ring, Math. Annalen 111 (1935) 259-280. [16] Magnus, W., h e r beziehungen zwischen h'dheren Komutatoren, J . r e i n e angew. Math. 177 (1937) 105-115.

[17] Magnus, W., Karrass, A., and S o l i t a r , D.,

Combinatorial Group Theory

( I n t e r s c i e n c e , New York, 1966). [18] Mal'cev, A . I . , Generalized n i l p o t e n t a l g e b r a s and t h e i r a d j o i n t groups, Mat. Sb. N.S. 25 (67) (1949) 347-366 (Russian); h e r . Math. SOC. T r a n s l . (2) 69 (1968) 1-21. [19] Moran, S., Dimension subgroups modulo n, Proc. Cambridge P h i l o s . SOC. 68 (1970) 578-582. [20] P a s s i , I.B.S., Group Rings and t h e i r Augmentation I d e a l s , Lecture Notes i n Mathematics, Vo1.715 (Springer-Verlag, B e r l i n , 1979). [21] P a s s i , I.B.S., The a s s o c i a t e d graded r i n g of a group r i n g , B u l l . London Math. SOC. 10 (1978) 241-255. [22] P a s s i , I.B.S., A n n i h i l a t o r s of r e l a t i o n modules 11, J. Pure Appd. Algebra 6 (1975) 235-237. [23] Passman, D.S., York, 1977).

The Algebraic S t r u c t u r e of Group Rings ( I n t e r s c i e n c e , New

[24] P l o t k i n , B . I . , Remarks on s t a b l e r e p r e s e n t a t i o n s of groups, Trudy Mosk. Mat. Obi:. 29 (1973) 191-206 (Russian); Trans, Moscow Math. SOC. 29 (1973) 185-200. [25] P l o t k i n , B . I . , V a r i e t i e s i n r e p r e s e n t a t i o n s of f i n i t e groups. L o c a l l y s t a b l e v a r i e t i e s . Matrix groups and v a r i e t i e s of r e p r e s e n t a t i o n s , Uspekhi Mat. Nauk 34 (1979) 65-94. [26] Rips, E., 342-346.

On t h e f o u r t h i n t e g e r dimension subgroup, I s r a e l J. Math. 12 (1972)

[27] Sjogren, J.A.,

Dimension and lower c e n t r a l subgroups, J. Pure Appd. Algebra

14 (1979) 175-194.

[28] Smel'kin, A.L., On t h e connection between Lie a l g e b r a s and groups, Uspekhi Mat. Nauk 33 (1978) 193-194. [29] Tahara, K., The augmentation q u o t i e n t s of group r i n g s and t h e f i f t h dimension subgroups, J. Algebra, t o appear. [30] Wehrfritz, B.A.F.,

I n f i n i t e Linear Groups (Springer-Verlag,

B e r l i n , 1973).

[31] W i t t , E . , Treue D a r s t e l l u n g Liescher Ringen, J. r e i n e angew. Math. 1 7 7 (1937) 152-160.


Rmeedings of the International Mathematical Conference L.H. Y. Chen, T.B. Ng,MJ. Wicks (eds.) @North-HollandPublishing Company, 1982

59

SOME OF MY FAVOURITE PROBLEMS WHICH RECENTLY HAVE BEEN SOLVED

Paul Erdos Hungarian Academy o f Science Budapest Hungary

During my very long mathematical l i f e , which has now extended f o r 50 y e a r s , I made conjectures i n d i f f e r e n t s u b j e c t s .

My conjectures i n s e t theory most of

which a r e j o i n t work with Hajnal, Rado, Milner and o t h e r s I s h a l l p r a c t i c a l l y ignore since several survey papers appeared on t h i s s u b j e c t

-

a new one would

c l e a r l y be needed but I am c e r t a i n l y not competent t o w r i t e it a l o n e .

Thus I w i l l

concentrate mainly on number theory, geometry and various branches o f a n a l y s i s 1

and a l s o j u s t f o r completeness I mention some of my few c o n j e c t u r e s i n topology.

I w i l l almost e n t i r e l y d i s c u s s only solved o r a t l e a s t p a r t i a l l y solved conject u r e s and i n o r d e r not t o make t h e paper too long I do not attempt completeness, a l s o s i n c e I have t o f i n i s h t h e paper i n somewhat o f a hurry, I s h a l l have t o r e l y g r e a t l y on my memory which d e s p i t e my enormous age, i s s t i l l q u i t e good but i s not ( a n d i n - f a c t never was) i n f a l l i b l e and so I apologise f o r t h e omissions.

First of a l l I s t a t e some o f my o l d e s t c o n j e c t u r e s a l l from t h e e a r l y t h i r t i e s (1930's not 1830's). 1,

Let 1 5 al
c Ln

I o f f e r e d 10,000 (U.S.)

consecutive

some c o n s t a n t c > O ,

d o l l a r s f o r a proof o f

I proved t h a t for i n f i n i t e l y many n (3) and c o n j e c t u r e d t h a t f o r every k

(4) Very r e c e n t l y Maier proved (4) i n a v e r y ingenious way.

Nevertheless I am cer-

t a i n t h a t , i f we put

then l i m fk+l(x) / f k ( x ) = 0. X-h=

I can not even prove that t h e l i m i n f i n (5) i s 0 .

I t would be of i n t e r e s t t o

t r y t o prove (5) f o r o t h e r sequences d e f i n e d by number t h e o r e t i c p r o p e r t i e s , e . g . f o r t h e s q u a r e f r e e numbers, b u t I have not succeeded t o g e t any i n t e r e s t i n g r e s u l t

Some of my favourite problems

65

so f a r . I prove t h a t

(6)

l i m inf

Pnt1 - Pn log n

< 1.

No doubt t h e value o f t h e l i m i n f i s 0.

I never could prove

(7) but of course t h e r e i s no doubt t h a t t h e value is 0 h e r e too. R. A. Rankin, The d i f f e r e n c e between consecutive prime numbers, J . London Math.

SOC. 13 (1938), 242-247. H. Maier, Chains of l a r g e gaps between consecutive primes, Advances i n Math. 39

(1981), 257-269.

P. Erdos, Problems and r e s u l t s on t h e d i f f e r e n c e s of consecutive primes, Publicat i o n e s Math. Delrecen 1 (1949), 33-37. P. Erdos, The d i f f e r e n c e o f consecutive primes, Duke Math. J. 6 (1940), 438-441.

54.

Sidon asked me n e a r l y 50 years ago t o f i n d a sequence a < a2 < . . . f o r which 1 t h e sums a . + a . a r e a l l d i s t i n c t and f o r which a tends t o i n f i n i t y a s slowly a s 1 3 k p o s s i b l e . Sidon c a l l e d t h e s e sequences B2 sequences and proved t h a t t h e r e i s a

B2 sequence with ak < c k 4 and he asked f o r an improvement. He expected t h a t t h e r e 2+E should be a B2 sequence s a t i s f y i n g ak< k f o r a l l k > ko(E). I observed t h a t t h e greedy algorithm gives

%< ck3

( t h i s was a l s o observed by Chowla and Mian who

f u r t h e r conjectured t h a t t h e greedy algorithm gives ak < k2+' I conjectured t h a t t h e r e i s a B2 sequence s a t i s f y i n g

%

f o r some 0 < c < 1 ) .

= o(k 3) .

This modest

looking conjecture remained open u n t i l very r e c e n t l y A j t a i , Komlos and Szemergdi proved i t by a new and ingenious method. a B2 sequence f o r which ak < k

3-E

Unfortunately t h e i r method does not give

holds f o r some p o s i t i v e

>O.

B

M. A j t a i , J . Koml6s and E . Szemergdi, On f i n i t e Sidon sequences, European J . Comb.

2 (1980), 1-11.

For f u r t h e r problems and r e s u l t s on t h e s e and r e l a t e d problems s e e t h e e x c e l l e n t book o f H . Halberstam and K . F . Roth, Sequences,

i,Oxford

Univ Press 1966 and A.

S t a k r , Geloste und ungeloste Fragen uber Busen d e r Naturlichen Zahlenreihe, I and 11, J . r e i n e u. angew. Math. 194 (19551, 40-65 and 111-140. 85.

Let f ( n ) be an a d d i t i v e function f o r which f ( n + l )

then f ( n ) = c log n

.

-

I f u r t h e r conjectured t h a t i f f ( n + l )

f ( n ) -+O.

-

I proved t h a t

f ( n ) < c then f (n) =

cllogn + g(n) where g(n) i s an a d d i t i v e function s a t i s f y i n g Ig(n) I -4 . Then every G(n;m) contains a triangle (xl, x2, x3) for which

where v(x) is the valency or degree of x. In fact we formulated a more general conjecture (for k(r) instead of k(3)). Edwards proved (1) and he in fact proved our conjecture nearly in its full generality. B. Bollobis, Cycles modulo k, Bull. London Math. Sco. 9(1977), 97-98. C. S . Edwards, Complete subgraphs with largest sum of vertex degrees, Coll. Math. SOC. J. Bolyai 18, Combinatorics, Edited by A. Hajnal V. T. S ~ S ,North Holland 1978, 293. For many further problems and results see the excellent book of B. Bollobis, Extremal graph theory, London Math. SOC. Monographs No.11, Acad. Press 1978.

P. Erdos

12

56.

V . T. S6s and I observed t h a t i f I S \ = n and AiCS

lAil

= 3, 1

5 i 5 t and

i f we f u r t h e r assume t h a t f o r 1 5 i < j 5 t , IAi fl A . I # 1 then max t = n e q u a l i t y 1 i f and only i f n I 0 (mod 4 ) . We f u r t h e r conjectured t h a t i f IS1 = n, AiCS,

1 5 i 5 t k , \ A i ( = k and IAinA.I # 1 f o r every 1 5 i J.

< j

5

k then f o r n > no(k)

n -2 Max t k = ( k - 2 ) '

(1)

(1) was proved f o r k = 4 by Katona and by P. Frankl i n t h e general case.

I f u r t h e r conjectured t h a t i f I S 1 = n, A k c S , 1 that

5

i

5

t n , and i f we f u r t h e r assume

+

I A ~ I I A . ~r f o r every 1 5 i < j 5 tn 1 1 ( t h e s i z e of t h e A ' s i s not r e s t r i c t e d here) and t h a t En < r < ( T - E ) n , then (2)

tn < ( 2 - E )

n

.

A s f a r as I know (2) i s s t i l l open though P. Frankl has many i n t e r e s t i n g r e s u l t s

which seem t o make (2) more p l a u s i b l e .

P. Frankl, Families of f i n i t e s e t s containing no two i n t e r s e c t i n g i n a s i n g l e t o n , Bull. Australian, Math. SOC. 17(1977), 125-134.

P. Frankl and R. M . Wilson, I n t e r s e c t i o n theorems with geometric consequences, Combinatorica 1 no. 4(1981). IV

Here I d i s c u s s some of my problems i n a n a l y s i s .

Since t h i s paper i s already longer

than I (and probably t h e e d i t o r s ) planned I w i l l be very b r i e f .

§I. I f i r s t mention some problems i n function theory.

In a paper w r i t t e n i n

Hungarian (Some remarks on a paper of K8v6riY Mat Lapok 7 (1956), 214-217) t h e following two problems were r a i s e d : ( i ) Is t h e r e an e n t i r e function f ( z ) f o r which f o r every i n f i n i t e sequence o f integers n < n2 < (ni) t h e s e t Sn i s everywhere dense, where S is the i=l i ni (nil s e t of roots of f (z) ?

. ..

( i i ) Let H1,

H2,...,

a f i n i t e l i m i t point. quence n

O and is module-finite over Ap, then it is quasiexcellent. In [2] he proved that such A is even excellent. b) Myself[Q] in 1972 proved that if A is finitely generated over a field k of characteristic zero, then the completion problem is affirmative. This result was generalized by Valabrega[91 in 1974 to the case of arbitrary field k. Cf. also [lo]. c) Marot[3] in 1973 proved the lifting problem for (P) = Nagata. His result was used in the work of Valabrega cited above. d ) Rotthaus proved the lifting problem for (P) = quasi-excellent in the following cases: 1) A is semi-local “ 6 1 , 19781, ii) A is universally catenary, dim A < and A 3 Q (173, 1980).

.

84

H.Matsumura

e) Nishimura proved in 1980 that the lifting problem for (P) = Gring was false in general. f) Seydi[ll] proved that formal power series rings over a universally catenary ring are universally catenary. But Greco [13] showed that the lifting problem for (P) = universally catenary was negative. In fact, he constructed a three-dimensional semi-local domain A and an ideal I such that A is I-adically complete and separated and A/I is excellent, but such that A itself is not catenary. Thus the result of Rotthaus [6] cannot be generalized to the case (P) = excellent. On the other hand, Greco proved that if A is I-adically complete and separated, A/I is universally catenary and A is catenary, then A is universally catenary. Rotthaus' proof of d)ii) above has two remarkable features. One is that it uses Hironaka's resolution of singularities, and according to her the result of [7] will remain valid if one replaces the assumption A Z Q by the assumption that A contains a field, provided that the resolution of singularities in characteristic p is proved. The other is that she views A as an inverse limit of semi-local rings and applies her previous result. Viewing a ring as an inverse limit of semi-local rings is a bold new idea, and overcoming enormous difficulties she succeeded in making this idea work. It will be interesting to see if this idea can be applied to other problems in the theory of noetherian rings. REFERENCES

Kunz, E., Characterizations of regular local rings of characteristic p, Amer. J. Math. 91(1969), 772-784. Kunz, E., On noetherian rings of charcteristic p, ibid. 98(1976), 999-1013. Marot, J., Sur les anneaux universellement japonais, C.R.Paris, 277(1973), 1029-1031. Matsumura, H., Formal power series rings over polynomial rings I, in: Kusunoki, Y. et a1 (eds), Number Theory, Algebraic Geometry and Commutative Algebra in Honor of Yasuo Akizuki, (Kinokuniya, Tokyo 1973). Rotthaus, C., Nicht ausgezeichnete, universe11 japanische Ringe, Math. Z. 152(1977)I 107-125. Rotthaus, C., Komplettierung semilokaler quasiausgezeichneter Ringe, Nagoya Math. J. 76(1979), 173-180. Rotthaus, C., Zur Komplettierung ausgezeichneter Ringe, Math. Ann. 253(1980), 213-226. Grothendieck, A., EGA ( = Elgments de G6om6trie Alg6brique)IV-2, Publ. IHES No.24, 1965. Valabrega, P., On the excellent property €or power series rings over polynomial rings, J. Math. Kyoto Univ., 15(1975), 378-395. Matsumura, H., Noetherian rings with many derivations, in: Bass. H. et a1 (eds.), Contributions to Alqebra (a collection of papeks dedicated -to E. .Kolchin), Academic PGess, New York 1977 [ll] Seydi, H., Anneaux henseliens et conditions de chaines, Bull. SOC. Math. France 99(1970), 9-31. [12] Nishimura, J., On ideal-adic completion of noetherian r rigs J. Math. Kyoto Univ.,21(1981), 153-169. [13] Greco, S., A note on universally catenary rings, Nagoya Math.J (to appear).

.

Proceedings of the International Mathematical Conference L.H.Y. Chen, T.B. Ng, M.J. Wicks (eds.) 0North-HollandEtrblishing Company, 1982

85

ON THE DECOMPOSITION OF PARTIALLY ORDERED SETS INTO DIRECTED SETS

E. C . Milner

Ma thema ti cs Department U n i v e r s i t y o f Calgary A1 berta Canada

and

Mathematics Department National U n i v e r s i t y o f Singapore Kent Ridge Singapore

51. INTRODUCTION Let (P, 5 ) be a p a r t i a l l y ordered s e t . { x

e P : x 5 a f o r some a e A 1 , and by

a e A 1

.

Also we w r i t e

a(

A(

) for

t h e set

For A S P denote by

I x e P : x 2 a f o r some 1 . mo elements a , b e P

t h e set

U

A ) when A = { a

a r e comparable i f a 5 b o r b 5 a , and they a r e compatible i f t h e r e i s c e P such t h a t a 5 c and b 5 c .

a t o t a l l y ordered s e t , i . e . , a p a r t i a l l y

A *is

ordered s e t i n which each p a i r o f elements i s comparable. t o t a l l y unordered s e t , no two elements being comparable.

An a n t i c h a i n i s a A p a r t i a l l y ordered s e t

(P, 5 ) i s d i r e c t e d i f every p a i r of elements i n P i s compatible, and a subset A C_ P i s undirected i f no two elements o f A a r e compatible

& P.

Note t h a t , i f

A

A i s a d i r e c t e d s u b s e t of P, then so a l s o is

If

v

i s a c a r d i n a l number, we w r i t e (P,

an a n t i c h a i n of P, and (P, 5 ) e =) O ' ( P ) .

and (P, 5 ) e

5)

if

I

e

U

CE)(v) i f

I
2K and l e t P b e t h e f u n c t i o n s f w it h r a n g e ( f ) s { 0 , l 1 and domain(f) a f i n i t e s u b s e t o f

and l e t P be o r d e r e d by e x t e n s i o n .

In t h i s l e c t u r e we w i l l prove t h e f o ll o w in g theorem o f Milner and P r i k r y

[3 ]

which g i v e s t h e i n f i n i t e analogue t o t h e above theorems o f Dilworth and Erdgs-

Tarski.

where A(p) is t h e smallest c a r d i n a l g r e a t e r t h a n

1 u" . V x i t follows t h a t xm, x ' a r e a l s o compatible and t h i s is a contran- m m diction.

93. PROOF OF THEOREM. Suppose (P,

51

e r)'(v).

I f Q C P is not d i r e c t e d , then t h e r e is some two-element subset I(Q) 5 Q whose members a r e not compatible @ Q . Now l e t Q be any subset of P . o r d i n a l number y(Q)
0 be an integer. Consider the following conditions: (1*) there exists an M-sequence of length n in I; (1) there exist an integer r & 0 and an M[xl,...,x r ]-sequence of length n in IR[xl,..., Xrl : (2) for every R-module N such that I& we have i ExtR(N, M) = 0 (i < n) i (3) Extft(R/I, M) = 0 (i < n); (4) there exists a finite R-module N such that i M) = 0 (i < n). Supp(N) = V ( 1 ) and ExtR(N, Then we have i) ii) iii)

+

(1") 3 (1) +(2) (3) +(4): if R is noetherian then (1) through ( 4 ) are equivalent to each other: if R is noetherian and M is meta-finite (cf.1.2.21, then all the conditions (1*) through ( 4 ) are equivalent to each other.

Proof. i) The implications (1*)3 (1) and (2) *(3) 3 ( 4 ) are trivial. Let us prove (1) 3(2). By (1.4.4) we may replace R by R[xl, ...,xrl. Therefore it suffices to prove (2) under the assumption (1). Since alVN = 0 for v large and since O:M al" = 0, we have HomR(N,M)=O.

99

Homological methodr in commutative algebra If n

>

1, w e g e t by i n d u c t i o n on n E x t i ( N , M/alM)

.

( i < n-1)

= 0

From t h e s h o r t e x a c t s e q u e n c e

a O'M&M*M/aM+O

w e g e t t h e l o n g e x a c t sequence a

. . . + Extk(N,M)

ExtA(N,M)

--f

1

Extk(N,M/alM)

--+

i+l ExtR (N,M)

a

1 ,

i on E x t ( N , M ) f o r i < n. R V al N = 0 , t h e r e f o r e t h e m u l t i p l i c a t i o n by al i n d u c e s a n i l p o t e n t

which shows t h a t a l i s a n o n - z e r o - d i v i s o r But

T h e r e f o r e Exti(N,M) = 0

endomorphism on N , hence a l s o on Extk(N,M). f o r i < n.

=+

ii) W e prove ( 4 )

F o r n = 1, (1) i s e q u i -

( 1 ) by i n d u c t i o n on n.

v a l e n t t o s a y i n g t h a t I c o n t a i n s a l a t e n t non-zero-divisor

on M ( c f .

1 . 3 . 4 and 1 . 3 . 6 ) , and s i n c e I i s f i n i t e l y g e n e r a t e d ( R i s n o e t h e r i a n !)

0

it i s a l s o equivalent to

# u E

I u = 0.

M,

(1.2.2).

Since

I

HomR(R/I, M ) = 0

by ( 1 . 3 . 3 ) .

Suppose

E A s s ( M ) s u c h t h a t a n n ( u ) _C P by Np # 0 by a s s u m p t i o n . Then Np/PNp

There e x i s t s P

C

P

w e have

# 0 by Nakayama, and s i n c e t h i s l a s t i s a module o v e r t h e f i e l d k:= Rp/PRp

t h e r e e x i s t s a s u r j e c t i v e l i n e a r mapping

On t h e o t h e r hand t h e r e e x i s t s a n i n j e c t i o n Ass(M)

,

and l o c a l i z i n g w e g e t a n i n j e c t i o n

R/P

Np/PNp + k .

+M

k 4 Mp.

because

P

E

Composing

-+

Np + N p / P N p 3 k Mp w e o b t a i n HomR ( N p , M p ) # 0 , hence HomR(N,M) # 0 by ( 1 . 4 . 2 1 , b u t t h i s i s a c o n t r g d i c t i o n . T h e r e f o r e 0 : I = 0. M If n 1, w e know a l r e a d y t h a t I c o n t a i n s a l a t e n t n o n - z e r o - d i v i s o r P u t R* = R[x], I * = IR*, M* = M[x], a n d l e t f b e a non-zeroon M . d i v i s o r on M* i n I * . W e have

>

ExtR*(N*, i M*) = Ext;(N, by ( 1 . 4 . 3 ) .

M ) @ R*

On t h e o t h e r hand t h e c o n d i t i o n Supp(N) = V ( 1 ) i s e q u i -

valent t o (ann N)R*, hence

=

fir and a s R* i s f l a t o v e r R w e have

e, i.e.

-

Supp(N*) = V ( I * ) .

ann N* = Thus R*, I * ,

M* and N* s a t i s f y ( 4 ) , and t h e exact s e q u e n c e f 0 4 M* M* --c M*/fM* --+ 0 shows

Exti,(N*,

M*/fM*) = 0

( i< n - 1 ) .

Our a s s e r t i o n ( 1 ) i s now

immediate from t h e i n d u c t i o n h y p o t h e s i s .

iii) When R i s

n o e t h e r i a n and M i s m e t a - f i n i t e I c o n t a i n s a non-

z e r o - d i v i s o r i f f it c o n t a i n s a l a t e n t n o n - z e r o - d i v i s o r ,

by ( 1 . 2 . 4 ) .

T h e r e f o r e t h e argument used i n t h e p r o o f o f ii) shows ( 4 ) * ( 1 * ) .

H. Matsumuru

100 2.2

Depth

(2.2.1)

DEFINITION.

L e t R b e a r i n g , I b e a n i d e a l o f R and M be a n

The supremum o f t h e l e n g t h s o f M-sequences i n I w i l l b e

R-module. d e n o t e d by

..

If

d(1,M).

xl,

are i n d e t e r m i n a t e s t h e n

x2,...

, x ] M[xl,.. .,xr1) i s a n i n c r e a s i n g f u n c t i o n i n r , and r i t s l i m i t w i l l b e d e n o t e d by d e p t h ( 1 , M I . [Remark: T h i s d e f i n i t i o n i s due t o H o c h s t e r a n d N o r t h c o t t and i s t a k e n from [ 8 ] p . 1 4 8 , 1 4 9 . Note t h a t N o r t h c o t t c a l l s d ( 1 , M ) t h e c l a s s i c a l g r a d e and d e p t h ( 1 , M ) d(IR[xl,.

t h e t r u e ( o r p o l y n o m i a l ) g r a d e , and d e n o t e them by GrR(I:M)

respectively.]

(2.2.2)

When R i s n o e t h e r i a n w e have d e p t h ( 1 , M) = sup{ i

I

grR(I;M) and

Extft(R/I, M) # 0 1 .

When R i s n o e t h e r i a n and M i s m e t a - f i n i t e w e a l s o h a v e d e p t h ( 1 , M) = d ( 1 , M ) . These a r e o b v i o u s c o n s e q u e n c e s o f ( 2 . 1 . 4 ) . (2.2.3)

I n t h e g e n e r a l case w e have

d e p t h ( 1 , M) = sup{ d e p t h ( I ' , M ) 1 , I' where I ' r a n g e s o v e r a l l f i n i t e l y g e n e r a t e d i d e a l s c o n t a i n e d i n I . T h i s i s e a s y from t h e d e f i n i t i o n ; c f . (2.2.4)

[ 8 ] p.149 Th.11.

According t o o u r d e f i n i t i o n o f M-sequence w e h a v e d e p t h ( 1 , 0 ) = d ( 1 , 0) =

depth(R,M) = d(R,M) =

m,

m.

On t h e o t h e r hand, i f t h e i d e a l I i s g e n e r a t e d by n e l e m e n t s a n d i f M

# I M , then

For t h e proof c f . (2.2.5) Proof.

r ) s u c h t h a t

l a t e n t M-sequence i n I , t h e n w e w i l l s a y t h a t ma1 l a t e n t M-sequence i n I .

fl,...,fn,g fl,...,f

is a i s a maxi-

n When w e d o n o t c o n s i d e r p o l y n o m i a l ex-

t e n s i o n s w e o m i t t h e word ' l a t e n t ' .

The key p o i n t o f t h e t h e o r y o f

d e p t h i s t o show t h a t e v e r y maximal l a t e n t M-sequence i n I h a s t h e

101

Homological methods in commu tative algebra

I n t h e c l a s s i c a l c a s e where R i s n o e t h e r i a n and M i s a

same l e n g t h .

t h i s c a n b e p r o v e d by means o f E x t , or Koszul com-

f i n i t e R-module,

When R i s n o e t h e r i a n ( a n d M i s g e n e r a l )

p l e x , o r l o c a l cohomology.

I n t h e most g e n e r a l

t h i s i s an immediate c o n s e q u e n c e o f T h . ( 2 . 1 . 4 ) .

c a s e N o r t h c o t t gave a c o m p l e t e l y e l e m e n t a r y a n d e l e g a n t p r o o f i n [ 8 ] pp.136-151.

I n t h e n e x t theorem w e w i l l d e r i v e t h e same r e s u l t from

(2.1.4).

(2.2.7)

L e t R b e a r i n g , I a n i d e a l and M a n R-module.

THEOREM.

flr...,f

Let

b e a maximal l a t e n t M-sequence i n I .

n

Then

d e p t h ( 1 , M ) = n. Proof.

Suppose

depth(1,M)

>

n.

Then t h e r e i s a l a t e n t M-sequence

g l r . - . r gn+l i n I. L e t r b e a n i n t e g e r s u c h t h a t a l l f i a n d a l l g j a r e i n IR[xl,...,xrI, and p u t R* = R[xl, xrl, M* = M [ x ~ , . . . , x ~ I ,

...,

W e c a n f i n d a f i n i t e l y g e n e r a t e d i d e a l J o f R* c o n t a i n i n g i s maximal and c o n t a i n e d i n I * . S i n c e f l , . . . , f j n w e have HomR*(R*/J, M*/(f l , . . . , f n ) M * ) # 0 b y ( 1 . 3 . 3 ) . On t h e o t h e r i s an M*-sequence and so w e h a v e t h e s h o r t e x a c t hand, f l , . . . , f n I * = IR*.

a l l f i and a l l g

sequences 0 Mo = M*

where

and

4

Mi

Mi >- ,

i+l

Mi

+Mi+l

.,fi)M*.

= M*/(fl,..

3

0

(OSiCn)

The a s s o c i a t e d l o n g

e x a c t sequences y i e l d e x a c t sequences 0 as

+

Extk;l(R*/J,Mi)

fi+l E J .

+

ExtR* k- 1( R * / J ,

Then, i f n ) O ,

Mi+l)

+

ExtR*(R*/J, k Mi)

+

0

f o r i = n-1 and k=O w e o b t a i n

0 ExtR*(R*/J, Mn-l)

= 0,

and f o r i = n - 1 and k = l w e o b t a i n

1

ExtR* (R*/J,

0 'v ExtR* (R*/JI

Mn-l)

Mn)

# 0.

I n t h e same way w e s u c c e s s i v e l y o b t a i n n n- 1 E x t R *(R*/J, M*) 2' ExtR* (R*/J,

M1)

>

.. .

0 ExtR* (R*/J,

91 * * * g n + l i s a n M*-sequence i n J , and ( 2 . 1 . 4 ) n E x t R * ( R * / J , M*) = 0. C o n t r a d i c t i o n .

But

(2.2.8)

COROLLARY.

L e t R , I , M be as above and l e t

a l a t e n t M-sequence i n I .

Put

R* = R [ x l l . . . , x

and assume t h a t a l l f i a r e i n IR*.

r 1 , M*

THEOREM.

Let

4: R * R '

# 0.

shows

fll...,f

P

= M[xll...,x

be

r1

Then )M*) + p . P b e a homomorphism o f r i n g s , I b e

d e p t h ( 1 , M ) = d e p t h ( I R * , M*/(f l , . . . , f (2.2.9)

Mn)

H.Matsumura

102

an ideal of R and M be an R'-module.

Then we have

depth(1, M) = depth(IR', MI, where in the left hand side M is viewed as R-module via $. Proof. If alr...ra n is an M-sequence in I, then $(al),...,$(an) is an M-sequence in IR'. This is obvious from the definitions. Moreover, $ induces a homomorphism $ : R[x1,...,xr1 +. R'[xl,...,xr]. r This already proves the Therefore depth(1, M) ,< depth(IR', M) assertion of the theorem if depth(1,M) = m . Suppose depth(1,M) = n < m. Then there exist an integer r >/ 0 and an M[xlr ,xr 1-sequence flr...,fn in IR[xlr...,x,]. Putting R* = R[x1 r . - * r ~ ~ M* l r = M[xlr xr] we have depth(IR*, M*/(f l,...,fn)M*) = 0. If we can show ,xr], M*/(fl, ,fn)M*) = 0 then we will have that depth(1R' [x,, depth(IR', M) = n by (2.2.8). Replacing R by R* and M by M*/(flr.. ,fn)M* we have thus reduced to proving that depth(1, M) = 0 means depth(IR', M) = 0. Let J' be a finitely generated ideal of R' contained in IR'. Then there exists a finitely generated ideal J of R contained in I such that J' JR'. We have depth(J,M) 5 depth(1,M) = 0 and depth(J',M) 5 depth(JR',M). Therefore if we can show depth(JR',M) = 0 then depth(J',M) = 0 and hence depth(IR',M) = suy depth(J',M) = 0. Hence we may suppose that I is finitely geneJ rated. Then depth(1,M) = 0 means O:MI = 0 by (1.3.3). But 0:MI = 0:M IR', therefore depth(IR',M) = 0 by ( 1 . 3 . 3 ) . Q.E.D.

.

...

...,

...

.. .

.

(2.2.10) The preceding theorem enables us to reduce most problems about depth to the case where R is noetherian. In fact, if I is finitely generated: I = Ra + + Ram , then consider a subring S of R such that S is generated by a finite number of elements as 2algebra and contains all ai ' s . Of course such S exists and is + Sam, then JR = I. noetherian. Moreover, if we put J = Sal + Therefore it holds that depth(1,M) = depth(J,M).

...

...

(2.2.11) LEMMA. Let R-modules. Put d' = Then: i) if d' ii) if d' iii) if d'

0 -+ M' + M +. M" -+ 0 be an exact sequence of depth(1,M') , d = depth(1,M) , d"= depth(1,M"). > d, then =

d, then

< d, then

d" = d. d" >/ d-1. d" = d'-1.

Proof. When R is noetherian these assertions are easy to prove (and easy to remember) from the long exact sequence of Ext in view of (2. 2.2).

Homological methodr in commutative algebra

103

In

I n t h e g e n e r a l c a s e assume f i r s t t h a t I i s f i n i t e l y g e n e r a t e d . t h i s c a s e t h e t e c h n i q u e of ( 2 . 2 . 1 0 ) n o e t h e r i a n case.

r e d u c e s t h e problem t o t h e

are a f i n i t e l y generated i d e a l J

When I i s n o t f i n i t e l y g e n e r a t e d b u t d , d ' , d "

a l l f i n i t e , w e c a n c h o o s e by ( 2 . 2 . 3 ) contained i n I such t h a t and w e a r e done.

(JIM"),

d = d e p t h ( J , M ) , d ' = d e p t h ( J , M ' ) , d"= depth Lastly, i f

l y generated i d e a l J C I such t h a t

d'=

> d , t h e n f o r any f i n i t e depth(J,M) = d w e

m

>

depth(J,M')

have d e p t h ( J , M " ) = d . T h e r e f o r e , t a k i n g supremum w i t h r e s p e c t t o J w e o b t a i n i ) . The case d ' < d = i s similar. I f d ' = d = m, f o r e a c h i n t e g e r n > 0 w e can f i n d a f i n i t e l y g e n e r a t e d i d e a l J C I s u c h t h a t d e p t h ( J , M ' ) b n and d e p t h ( J , M ) b n , and t h e n w e h a v e d e p t h ( J , M " ) 3 n-1. L e t t i n g n t e n d t o i n f i n i t y w e o b t a i n d" = m . I f S i s a m u l t i p l i c a t i v e s u b s e t o f R t h e n it i s c l e a r t h a t

(2.2.12)

d e p t h ( S - l I , S-lM)

depth(1,M).

I f R i s a l o c a l r i n g w i t h maximal i d e a l m t h e n depth(m,M) i s u s u a l l y d e n o t e d by depth(M) o r d e p t h R ( M ) . n o e t h e r i a n and M i s f i n i t e , t h a t

depth(P,M) = d e p t h ( M p ) .

was s t a t e d t h a t

c o r r e c t e d i n t h e 2nd e d i t i o n . ) (2.2.13)

I t c a n happen, even when R i s

THEOREM.

depth(P,M)

T h i s shows dim

Mp

dim R

d i m M = dim R,

and

shows for i

9

and

-

There-

/ d i m

-

d i m R/P.

R/P.

R/P

+

On t h e o t h e r dim R P ( t h e

dim R/P

> d i m Rp and

= depth(P,M) = dim Mp

= dim Mp

Therefore

9

d e p t h Mp.

+

dim R = dim R/P

Rp.

(2.3.8)

L e t R be a Cohen-Macaulay

LEMMA.

( = system of parameters)

s.0.p. Proof.

an s . 0 . p .

4

depth

>/

.

= P/ann(M)

l a s t i n e q u a l i t y f o l l o w s from t h e d e f i n i t i o n o f d i m e n s i o n ) . depth M

C.-

P 2 ann M I a n d w e c a n r e p l a c e R by

Theorem ( 2 . 3 . 2 )

t h i s means

Mp i s a

d e p t h Mp = depth(P,M) a n d

= R/ann(M) a n d

Exti(R/P, M) = 0 By ( 2 . 2 . 2 )

R then

depth(P,M) , dim M e t c . ( c f . 2 . 2 . 9 ) .

ann M = 0 .

dim Mp = dim Rp.

B

where

R/ann(M) w i t h o u t c h a n g i n g

hand

f o r every

local r i n g .

Then e v e r y

f o r R i s an R-sequence.

L e t m b e t h e maximal i d e a l o f R a n d a l , . . . , a

f o r R.

( n = dim R) b e n I f P E A s s ( R ) t h e n dim R/P = n by ( 2 . 3 . 6 1 , h e n c e

...,an

i n R/P would g e n e r a t e a T h e r e f o r e al i s m/P-primary i d e a l and w e would have dim R/P< n - 1 ) . a n o n - z e r o - d i v i s o r on R , and R/alR i s a Cohen-Macaulay l o c a l r i n g of dimension n-1. By i n d u c t i o n on n w e see t h a t a 2 , ,an i s a n R/alRs e q u e n c e . Hence a l , a 2 , . . . , a n i s a n R-sequence i n m. al

P ( o t h e r w i s e t h e images o f a 2 ,

...

(2.3.9)

A r i n g R i s c a l l e d a Cohen-Macaulay r i n g i f i t i s n o e t h e r i a n

and Rp i s a Cohen-Macaulay l o c a l r i n g f o r e v e r y P E Spec R. t u e of (2.3.7) (2.3.10)

By v i r -

f o r e v e r y maximal i d e a l m.

it s u f f i c e s t h a t Rm i s C.-M.

The h e i g h t of a prime i d e a l P i n a r i n g R i s t h e supremum

o f t h e l e n g t h s o f t h e d e s c e n d i n g c h a i n s of p r i m e i d e a l s s t a r t i n g from P I and it i s d e n o t e d by h t ( P ) . Thus h t ( P ) = d i m Rp. The h e i g h t o f

an i d e a l I ( f R) i s d e f i n e d by ht(1) = inf { ht(P)

1

P € V(1) 1 .

L e t R b e a n o e t h e r i a n r i n g and I b e a n i d e a l f R; l e t A s s R ( R / I )

=

H.Matsumura

106

(P1,...,PS~. If ht(Pi) = ht(1) for all i, then I is said to be unmixed. We say that "the unmixedness theorem holds in R" if, for every r 3 0, every ideal I (fR) of height r generated by r elements is unmixed. Recall that, by the "principal ideal theorem" of Krull, every minimal prime over-ideal of such I as above must be of height r. Therefore such I is unmixed iff it has no "embedded primes". (A non-minimal element of AssR(R/I) is called, by abuse of language, an embedded prime of I.) The condition for r = 0 impies that (0) is unmixed, i.e. that Ass(R) has no non-minimal elements. (2.3.11) THEOREM. A noetherian ring R is Cohen-Macaulay iff the unmixedness theorem holds in it. Proof. Suppose the unmixedness theorem holds in R. Then it holds in Rp for every prime ideal P, hence we may assume R is a local ring with maximal ideal rn. Put dim R = n. By assumption every P E Ass(R) is of height 0 , hence if n 0 we can find a l E rn which is a nonzero-divisor on R. Since the principal ideal (a,) is unmixed of height 1, if n 1 we can choose a2 E m which is not in any prime in AssR(R/(al)). Then al,a2 is an R-sequence in m. Continuing in this way we can find an R-sequence al,a2, an in m, hence depth R = dim R and R is Cohen-Macaulay. Conversely, suppose R is Cohen-Macaulay. In order to prove unmixedness theorem we may localize, s o we again assume that R is local. Let al,...,ar E rn be such that ht(a l,...,a ) = r. Then all minir ma1 prime over-ideals of (al,...,ar) are of height r, and so if dim R > r we can choose ar+l E m which is not in any of them. Then ht(al,:..,ar+l) = r+l. In this way we can choose ar+ll...lan (n = dim R) such that all an is an s.0.p. for R. By lemma (2.3.8) al,...,a n is an R-sequence, hence ar+l,...,an is an R/(al, ,ar) -sequence in rn. Since the images of ar+l,...,an in fi = R/(a 1'"" is an s.0.p. for we see that is a Cohen-Macaulay local ring, ar) or what is the same thing, R is a C.-M. R-module. Hence AssR(%) has no non-minimal elements, i.e. (al,...,a r ) is unmixed.

>

>

...,

...,

...

(2.3.12) Cohen-Macaulay rings are very important. One can prove that if R is C.-M. then the polynomial rings R[xl,...,xm] are C.-M., and the regular local rings are C.-M. (cf.[7]). Historically, Macaulay proved the unmixedness theorem in polynomial rings over a field in 1916, and 1.S.Cohen proved the same for regular local rings in 1946. One can characterize Cohen-Macaulay local rings in terms of multiplicity: a noetherian local ring R is C.-M. iff for some (resp.

Homological methodr f n commutative algebra

107

every) ideal q generated by an s.o.P., e(q) = e(R/q) holds, where e( 1 denotes the multiplicity and l!( denotes the length of R-module. In early Fifties M. Nagata and P. Samuel studied Cohen-Macauly rings from this standpoint, while J.-P. Serre and M. Auslander- D. Buchsbaum found the notion of depth (called homological codimension), introducing homological methods in commutative algebra. In the Sixties A . Grothendieck established the theory of local cohomology, which has become one of the most popular tools in commutative algebra. A noetherian local ring (R,m) is C.-M. iff the local cohomology modules Hi(R) is 0 for all i # dim R. In the Seventies some of the most important achievements in commutative algebra consisted in showing that certain concrete rings were C.-M. Also, Hochster [ 5 ] showed that every noetherian local ring R containing a field has a "big Cohen-Macaulay module", i.e. a (not necessarily finite) R-module M such that if al,...,a n is an s.0.p. for R then it is an M-sequence with M # (al,...,an)M, and he derived many useful consequences from this. It is a big open problem to extend the existence theorem of Hochster to the unequal characteristic case. Another important progress in the Seventies was the discovery of the concept of Buchsbaum rings. This class of rings is defined by a generalization of the definition of C.-M. rings in terms of multiplicity and is being intensively studied in East Germany and in Japan. We now have the following hierarchy of noetherian local rings: {regular} C (complete intersection) C {Gorenstein} C {Cohen-Macaulay} {Buchsbaum).

c

53.

3.1

EXACTNESS CRITERION OF A FINITE COMPLEX Determinantal Ideals

(3.1.1) Let R be a ring. We consider matirces with entries in R. A t x t minor of C is the deterLet C = (cij) be a p x q matrix minant of a t x t submatrix of C. We will denote the ideal of R generated by all the t x t minors of C by It(C), and call it the determinantal ideal of C of order t. We put It(C) = 0 if t min(p,q) and Io(C) = R. Thus we have I0 (C) = R zI1(C) 2 12(C) 2 If A is a p x r matrix and B is an r x q matrix, then it is well known that we have It(A)It(B) C It(A) A It(B). It(AB)

.

...

c

For a proof, cf. 181 p.5-7.

H. Matsumura

108

I n p a r t i c u l a r , i f U i s an i n v e r t i b l e p x p m a t r i x and V i s an i n v e r t i b l e q x q matrix, then

It(UAV)

= It(A)

for all t

0.

(3.1.2)

L e t F, G b e f r e e R-modules o f f i n i t e r a n k p and q , r e s p e c -

tively.

L e t @:

F

G be a l i n e a r mapping.

Choosing a b a s i s e l , . . . , o f G, w e o b t a i n a matrix C = (c ) e i j P q l d e f i n e d by $ ( e1. ) = C j = l c i j e j (1,C i ,C p ) A change o f b a s e s t a n s f o r m s C t o UCV w i t h U and V i n v e r t i b l e . Therefore t h e determi-+

o f F and a b a s i s e i , . . . , e '

.

nantal ideals I t ( C )

depend o n l y o n @,

and so w e c a n p u t

I t ( @:= ) It(C)

3 0.

for t

(3.1.3)

The d e t e r m i n a n t a l i d e a l s have many i n t e r e s t i n g p r o p e r t i e s . ( C ) i s g e n e r a t e d by P- 1 I n g e n e r a l , i f C is a p x q

F o r i n s t a n c e i f C i s a p ~ ( p - 1 )m a t r i x t h e n I p m i n o r s , b u t i t s h e i g h t i s a t most 2 . m a t r i x and 0

0 , and i f P E Supp(M), t h e n depth(Ip,Mp) >/ depth(1,M) and

Ips # 0.

Therefore

rank($klM), k = 1,2,...,n,

under l o c a l i z a t i o n and t h e c o n d i t i o n s

are i n v a r i a n t

i),ii),iii) c o n t i n u e t o h o l d

H.Matsumura

114

%

as l o n g as And i f

# 0 , and i f

%

i s o b v i o u s l y exact.

= 0 t h e n F.@$

( F . @ M ) p i s exact f o r a l l P t h e n F.@M i s e x a c t .

Thus w e assume t h a t R i s a n o e t h e r i a n l o c a l r i n g , and w e p r o c e e d by d o u b l e i n d u c t i o n on n and dim R. i n McCoy's theorem ( 3 . 1 . 2 ) .

The case n = 1 w a s a l r e a d y p r o v e d

On t h e o t h e r h a n d , i f dim R = 0 , t h e n

t h e maximal i d e a l m i s n i l p o t e n t , so t h a t t h e hypothesis i) i m p l i e s

depth(m,M) = 0 .

for k = l,...,n.

I($k,M) = R

c a s e t h e e x a c t n e s s o f F.@M follows from ( 3 . 4 . 3 ) . n > 1, dim R > 0

assume t h a t then

i s exact.

0 0

C@M

-+

Then b o t h -+

0

Fn

-f

-+

$A-l:

rank(@n)

-+

-+

Fn

.

r a n k Fn

,

by t h e i n d u c t i o n h y p o t h e s i s .

0

-f

Fn@M

-f

-+

-+

:

i s exact.

0

+

Mn

Since each

-

Fn-18M

3 Mn-l

\

and ii) shows

-+

... -$0F

n

4

C@M

-+

1 and and

Mn-l

+

T h e r e f o r e t h e Lemma i s t r u e f o r n = 1.

0

c

[1] lemma 3 ,

Mn-l

and

c e e d by i n d u c t i o n on n .

W e have

+

H (C) = 0 o r depth(1, Hk(C)) k Then C . i s exact.

f o r 1 6 k $ n. Proof.

k

Mn

+

31.

o f R-modules, where n

(1) d e p t h ( 1 , \ )

0

C.:

+

Mn-3

*

The complex

... -+ Mo

s a t i s f i e s t h e c o n d i t i o n o f t h e l e m m a , h e n c e i t i s e x a c t by t h e induction hypothesis. 3.5

The lemma is proved.

Proof o f N e c e s s i t y

(3.5.1)

LEMMA.

( [ 8 ] p.100)

L e t R b e a r i n g , A be a p x q m a t r i x

over R and B b e q x t m a t r i x over R s a t i s f y i n g non-negative i n t e g e r s such t h a t

p

+ v

> q.

AB = 0 .

L e t l ~ , vb e

Then some power o f I ( A ) lJ

annihilates I V ( B ) . Proof.

L e t D be a

l ~ xl~ minor o f

of D a n n i h i l a t e s I v ( B ) .

A.

We w i l l show t h a t some power

Suppose t h e c o n t r a r y .

a prime i d e a l P which c o n t a i n s

Then t h e r e e x i s t s

a n n ( I V ( B ) ) b u t n o t D.

Replacing

w e o b t a i n a s i t u a t i o n where R is l o c a l and I ( A ) = R , I v ( B ) lJ But t h e n w e can f i n d i n v e r t i b l e matrices U a n d V s u c h t h a t

R by Rp

# 0.

VAU

=if-. :] r

where E i s t h e i d e n t i t y m a t r i x of o r d e r u . Then t h e f i r s t l~ r o w s -1 lJ = Iv(B) # 0 o f U B must be zero, and t h i s c o n t r a d i c t s I v ( U - l B )

116

H. Ma tsumura

+ v

s i n c e 1~ (3.5.2)

> q. L e t R b e a r i n g and

LEMMA.

$ F + $G - + H

S:

be a complex o f f i n i t e f r e e R-modules. such t h a t S@M i s e x a c t .

Suppose

M # 0

Let

b e a n R-module

depth(I($,M),M) > 0.

Then w e have

(1) d e p t h ( I ( J , , M ) , M ) > 0 , and rank(J,,M) = r a n k G .

F i r s t o f a l l w e r e p l a c e R by R/ann(M)

Proof. =

+

rank($,M)

(2)

rank(-,M),

.

Then w e h a v e r a n k ( - )

I(-) = I(-,M).

If $ = $ = 0

G @ M = 0 , b u t G @ M i s a d i r e c t sum of q:=

then

c o p i e s o f M and M # 0 .

rank G

r a n k G = 0 , and t h e lemma i s t r i -

Therefore

v i a l i n t h i s case. If $ = 0

and

$

# 0 , t h e n t h e lemma f o l l o w s from McCoy's theorem ( 3 .

and

JI

= 0,

2.2). If $

# 0

= R , depth(R,M) =

u E M

then

t h e n w e have

m .Therefore

I ( $ ) C$

such t h a t I($) $ P ,

%

# 0.

and

= I($)

I(J,,M)

O:M I ( $ ) = 0 , i f 0 #

Since

t h e r e e x i s t s a prime i d e a l P

Then, r e p l a c i n g R by Rp w e may assume

w i t h o u t changing rank ( $ ,M)

I($) = R

rank($) = 0

Thus (1) h o l d s .

CQ.

G = Im($)

.

Then w e have by ( 3 . 4 . 1 )

0 GI

and F@M -+ G @ M i s s u r j e c t i v e by a s s u m p t i o n . and w e have r a n k ( $ ) = r a n k G a s wanted.

Therefore

G ' = 0,

Suppose now $ # 0 , 3, # 0 , and p u t r a n k ( $ ) = p , r a n k ( $ ) = v . First w e c l a i m t h a t p + v 6 r a n k G . I n f a c t , s i n c e v = rank($,M) t h e r e I f u + v > rank G then we e x i s t s u E M such t h a t I v ( J , ) u # 0. would have I p ( $ ) r I v ( $ ) u = 0 f o r some r > 0 by t h e p r e c e d i n g

lemma, b u t t h i s i s a b s u r d s i n c e 0 : I ( $ 1 = 0. M u I n o r d e r t o p r o v e u + v = r a n k G , t h e r e f o r e , w e have o n l y t o p r o v e t h a t t h e assumption p

+

v < rank G

leads t o a contradiction.

t h i s p u r p o s e w e may l o c a l i z e a s l o n g a s M r e m a i n s non-zero

For

(because

v d o e s n ' t i n c r e a s e and p d o e s n ' t c h a n g e ) . So w e may assume t h a t R i s l o c a l and I ( $ ) = R [ j u s t a s w e d i d i n t h e case $ # 0 and J, = 01. Then G = Im($) @ G ' w i t h r a n k G I = r a n k G p, and J, i s zero on

-

Im($). I(J,,M)

T h e r e f o r e J, i n d u c e s =

I(J,I,M).

w e must have

-

p.

Thus

Since

$I:

GI

-+

H

and

0 + G1@M + H@M

rank(Jl,M) = r a n k G' by ( 3 . 2 . 2 ) . p

+ v

< rank G

rank(JI,M) = r a n k ( $ ' , # ) ,

i s e x a c t by a s s u m p t i o n , But

r a n k GI = r a n k G

c a n n o t h o l d and ( 2 ) is p r o v e d .

Homological methods in commutative algebra To p r o v e (1), s u p p o s e of M such t h a t Spec(R)

u # 0 have

I($)# 0

O:M

I ( $ ) u = 0.

Since

I ( @$ ) P

such t h a t

and t a k e a non-zero e l e m e n t u

I(@)

4

/ann(u)

ann(u) 5 P.

and

, we

Then

P€

can find I($,)

=

%

and

S i n c e (2) h o l d s o v e r Rp a s w e l l a s o v e r R w e must

i n Mp.

rank($,M) = r a n k ( $ p , M p ) ,

hence

0 Im($)

and

= I ( I $ ' , M ) ~where

G'@M

$ I :

G'

(3.2.2) w e must have

-+

+

H

Thus w e

I($)% = I($pfMp).

may assume t h a t R i s l o c a l , I ( @ = ) R and G = G'

117

O:M

I($) # 0 .

But t h e n

i s i n j e c t i v e , with I ( $ ) = I($,M)

H(QM

i s t h e i n d u c e d mapping.

T h e r e f o r e by

O:M I ( $ ) = 0 , c o n t r a d i c t i o n .

End of t h e p r o o f o f t h e n e c e s s i t y p a r t o f Theorem (3.3.2).

(3.5.3)

R i s a r i n g , M i s a n R-module,

W e assume:

i s a complex o f f i n i t e f r e e modules s u c h t h a t

C.@M

is e x a c t .

We

wish t o p r o v e

( i ) d e p t h ( 1 $krM)r M I 3 k ( l S k , 0.

shows ( i i ) a n d

successively t o

Fk Fk-l + Fk-2f d e p t h ( I ( $ k , M ) , M) > 0 .

t o c o n c l u d e ( i i i )a n d

...,2,

(l$k 1. L e t R* = R [ X be a polynomial

lf...,sl

r i n g o v e r R and p u t C? :

-% 4J*

O+Fi

b e t h e complex I ( $ k r M)R*,

M* = M@R*.

depth(I($,

w e may r e p l a c e R by R*. ment

ak

i n I(4JkfM)

... 4Ji

FA-l+ Then

C.@R*.

Let 3

i s e x a c t , and

Ct@M*

I($$, M*) =

M*), M*) = d e p t h ( I ( @ k , M ) , M ) .

Therefore

T h i s e n a b l e s u s t o f i n d a n M - r e g u l a r ele-

f o r e a c h k.

a n M-regular e l e m e n t and

F;

a = ala 2...an.

Put

a E I($kfM)

f o r a l l k.

Then

a

is

W e have a s h o r t

e x a c t s e q u e n c e o f complexes 0 ---* C.@M

%

CSM

+

C.@(M/aM)

+

0

and by a f u n d a m e n t a l t h e o r e m of h o m o l o g i c a l a l g e b r a w e o b t a i n a l o n g e x a c t s e q u e n c e o f homology:

... 4 H k ( C . @ M ) Hk-l(C.QM)

3 Hk(C.@M)

+

-+ Hk-l(C.@M)

--$

Hk(C.@(M/aM)) Hk-l(C.@(M/aM))

+...

118

H . Matsumura

Since Hk(C.@M) = 0 for k ) O , Therefore, if we put

we have

C'. :

0

3

Fn 3 Fn-l

--f

Hk(C.@(M/aM))

...+ F2

= 0

for k > l .

F1 then C'.@M is exact. If M/aM = 0 then depth(I($k,M) ,MI = m for 1 5 k 6 n and we are done. Suppose M/aM # 0. Since C'. is shorter than C. the theorem holds for C', hence depth(I($k, M/aM)

>/

+

(2

k-1

Moreover, rank($n, M/aM) = rank Fn = rank(Qn, M) rank($k+lr M/aM) + rank($k, M/aM)

=

6 k $ n). and

rank Fk

(2 ,< k

/ 1, this completes the proof of the theorem (3.3.2).

>

(3.5.4) C.:

COROLLARY. 0

3

Let R be a ring and n 'i, Fn-l 4 Fn _ -+Fl

...

$1 --+

F0

0

be a complex of finite free R-modules. Let M be an R-module ( # 0 ) . Then, in order that C.@M is exact, a) it is necessary that we have (1 6 k n) , i) depth(I(@k,M),MI = m ii) rank($n, M) = rank F n' and ( 1 s k < n); iii) rank($k+lrM) + rank($k,M) = rank Fk

M)

.

01

V

TI(

is a left-exact functor.

d e n o t e d by

i

HI(M)

I t s r i g h t d e r i v e d f u n c t o r s are a n d are c a l l e d t h e l o c a l cohomology

( i = 0 , 1 , 2 , . ..)

of M with respect to the ideal I (or with supports i n V ( 1 ) ) .

W e have

= s m Extk(R/IV,M)

€I;( M )

.

V

F o r t h e g e n e r a l t h e o r y o f l o c a l cohomology, c f . G r o t h e n d i e c k , L o c a l Cohomology, S p r i n g e r L e c t u r e Notes 4 1 , 1967 and Herzog-Kunz,

D e r k a n o n i s c h e Modul e i n e s Cohen-Macaulay R i n g s , S p r i n g e r L e c t u r e Notes 238,1971.

Cf. a l s o R.Y.Sharp,

Local Cohomology Theory i n Commutative A l g e b r a , Q u a r t . J . Math. Oxford 2 1 (19701, 425-434.

(4.3.2)

W e have

(4.3.3)

THEOREM.

H:(M)

= 0

if

i > dim M.

L e t ( R , m ) b e a n o e t h e r i a n l o c a l r i n g a n d M be a

f i n i t e -R-module o f d i m e n s i o n n . H:(M)

z

Then

0.

C o n s e q u e n t l y , w e have

d i m M = sup { i ( H i ( M ) # 0 (4.3.4)

L e t R be a noetherian r i n g ,

module ( f

0)

.

1.

I a n i d e a l a n d M a f i n i t e R-

Then depth(1,M) = i n f

{ i ( H i ( M 1 # 03.

T h e r e f o r e , u n d e r t h e h y p o t h e s i s o f ( 4 . 3 . 3 ) , M i s Cohen-Macaulay i f f H ~ ( M )#

o

f o r o n l y one v a l u e of i .

Homological methods in commutative algebra 4.4

123

Syzygies

(4.4.1) Let R be a ring. The notion of k-th syzygy module is defined inductively as follows: every R-module is a 0-th syzygy; if M is a (k-1)-th syzygy and if there exists an exact sequence

0 3 N + F +M'O

with F free, then N is a k-th syzygy. (4.4.2)

An R-module M is said to be S{ if

depth M >/ min(k, dim R p ) RP for all prime ideals P of R. Note that this condition is slightly different from the condition (Sk) of EGA, which is defined by depth M >/ min(k, dim RP It is easy to check that a k-th syzygy module over a Cohen-Macaulay ring is S.; The converse is true under some additional finiteness assumption, as we see in the following theorem. (4.4.3) THEOREM. Let R be a Cohen-Macauly ring and M # 0 be a finite R-module of finite projective dimension. Then M is S{ iff it is a k-th syzygy.

Proof. Cf. Auslander-Bridger, 1oc.cit. p.127. (4.4.4) Last year G.Evans and P.Griffith jointly proved the following theorem, which has many applications commutative algebra and algebraic geometry and had been known as Syzygy Conjecture for some years.

SYZYGY THEOREM. Let R be a local Cohen-Macauly domain which contains a field. Let M be a finite R-module of finite projective dimension. rank M, then M is free. If M is a k-th syzygy and k For the proof they use the preceding therem of Auslander-Bridger as well as the big C.-M. module of Hochster (cf.(2.3.12)). The following theorem is one of the applications, given by Evans-Griffith, of the Syzygy Theorem. (4.4.5)

THEOREM. Let R be a regular local ring containing a field. If I is an unmixed ideal of height two generated by three elements, then R/I is Cohen-Macaulay. Proof. Let dim R = n.

Then

dim R/I

,

prove depth R/I >/ n-2. We may assume n 3. fices to show pdR(R/I) 5 2. (It is well known ring R is C.-M. and every R-module is of finite By assumption there is an exact sequence of the

By (4.1.4) it sufthat a regular local projective dimension.) form

O-+M+R 34 1 4 0 . We claim that M is S.; In fact, if P is a prime ideal and ht(P) & 3 then P is not an associated prime of I (since I is unmixed), hence HomR R/P, R/I) = 0. It follows from this and from depth(P,R) = ht(P) i (cf. 2.3.7) we obtain ExtR(R/P,M) = 0 (i=O,1,2). Hence depth I+$, 0 = 0, Extk(R/P,R) = 0 b3. If ht(P)= 2 it holds that ExtR(R/P,I) (i=O 1), hence Extk(R/P,M) = 0 (i=O,l). Therefore depth Mp 3 2. The case ht (P) 5 1 is easy. but on the other hand rank M = 2. (Here Thus we have that M is S,; rank M means the vector space dimension of M@K, K being the quotient field of R.) Therefore M is free by Syzygy Theorem, and O 4 R 2 + R 43R + R / I + O is a free resolution of R/I.

Thus we have

pdRR/I 6 2, as wanted.

REFERENCES [l] Buchsbaum,D. and Eisenbud,D., What makes a complex exact ? J.of Algebra 25(1973), 259-268. [2] Cartan,H. and Eilenberg,S., Homological Algebra (Princeton University Press, 1956). [ 3 ] Eagon,J.A. and Northcott, D.G., On the Buchsbaum-Eisenbud theory of finite free resolutions, J. fur reine u. angew. Mathematik, 262/263 (19731, 205-219. [4] Eagon,J.A. and Northcott, D.G., Ideals defined by matrices and a certain complex associated with them, Proc. Roy. SOC. London 269 (1962), 188-204. [5] Hochster, M., Topics in the homological theory of modules (American Math. SOC., Regional Conference Series 24, 1975). [6] Hochster, M., Grade-sensitive modules and perfect modules, Proc. London Math. SOC. 29(1974), 55-76. [7] Matsumura, H., Commutative Algebra (Benjamin, 1st ed.1969, 2nd ed. 1980). [8] Northcott, D.G. , Finite Free Resolutions (Cambridge University Press, Cambridge Tracts in Math. 71, 1976). [9] Northcott, D.G., Introduction to Homological Algebra (Cambridge University Press., 1960). [lo] Peskine,C. and Szpiro, L . , Dimension projective finie et cohomologie locale, Publ. IHES 42(1973), 47-119.

Roceedin@ of the International Mathematical Conference L.H. Y. Chen, T.B. Ng, MJ. Wicks (eds.) 0North-Hdlandhblishing C o m p y , 1982

125

COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY Masayoshi Nagata Department o f Mathematics Kyoto U n i v e r s i t y Kyoto Japan

0. INTRODUCTION.

In recent years algebraic geometry has been studied in a very abstract manner and this has brought it closer to commutative algebra. On the other hand, in order to learn commutative algebra it is often useful to look at the geometry which underlies its problems. For this purpose, the classical approach to algebraic geometry is helpful. Here we shall discuss some basic notions of algebraic geometry and their connection with commutative algebra, especially the connection with polynomial rings. A ring will always be a commutative ring with 1. We begin with the following remark:

...,

Let k, K be the rings with k a subring of K, and let XI, Xn be variables (i.e. . .,XJ commuting indeterminates) over k. A polynomial f (Xl, ,Xn) from k [X,. defines a mapping of Kn into K such that each (bl,. ,bn) of Kn is mapped to Addition in the polynomial ring k [X1 f(bl ,..., b,). Xn1 is defined by the operations of k so that the sum of two polynomials will map (bl, b,) to the sum of the images of (bl, b,) under these polynomials. There is a similar result for multiplication and products.

. ..

.. ,...,

...,

.

...,

One sees immediately from the definition of a polynomial ring that properties of k will have a lot to do with properties of k [X1, Xn] ; some representative examples of this will be discussed in the first half of 51.

...,

In the second half of 8 1 we review results which are related to Noether's normalisation theorem. Although it is important to study polynomial rings without any restriction on the coefficient ring k, it is also true that the most basic case occurs when k is a field. Here we feel that the ring k [X1, X], has rather simple structure. This does not mean that questions are easy if they are related to polynomial rings

...,

M.Naga ta

126

over a field : there are difficult problems which are due to the simplicity of the structure o f polynomial rings. Thus we shall give most attention to the case where k isa field. This corresponds geometrically to affine n-space (i.e. ndimensional affine space) over k. Again, we may feel that affine spaces have simple structure and are more easily understood than projective spaces. But the geometry of affine space is more understandable via the geometry of projective space. Some topics related to this fact will be discussed in 5 2 . As an example of the complexity of phenomena related to polynomial rings, or equivalently to rational varieties, we shall discuss in 13 the birational correspondences of ruled surfaces. The discussion is further related to the automorphism groups of polynomial rings and Cremona groups. These are considered in 14 and 1 5 . Finally, in 16, we discuss group actions on an affine ring. A list of principal references is given at the end. 1. ELEMENTARY PROPERTIES OF POLYNOMIAL RINGS

.

Consider the polynomial ring F = klXl,. ., over a ring k. Then properties of k are carried over to or reflected in F. We observe some typical examples of such properties of k.

3

The first example concerns zero-divisors. Since k may be taken as a subring of F, it is obvious that zero-divisors of k are zero-divisors of F. In the converse direction there is THEOREM 1.1. If f is a zero-divisor in F, then there is a non-zero b in k such that bf = 0 (i.e. all coefficients of f are annihilated by b). Moreover, if 0 g e F is such that fg = 0, then b may be chosen in the ideal o f k generated by the coefficients of g.

+

PROOF. We order the monomials X;l.. . Xen by the lexicographical order of the n series of exponents; thus, dl ... xnd, x;1..* x,"" > x1 if either e > dl, or there is j > 1 such that e =di for i < j and e > d i j j' Suppose f, g are as stated. If f = O , there is nothing to prove, so we suppose f # 0. We then have s,t > 0 such that s

f=

1

i=l

t

aimiy g =

1 b.n., j=1 J J

where ai, b. are non-zero elements of k and mi, n. are monomials such that 1

1

127

Commutative algebra and algebraic geometry ml

>

... >

mS

and n1 >

... > nt '

The proof i s by induction on s + t . I f s t t = 2, we have s = t = 1 and 0 = fg = a b m n 1111'

Hence albl = 0 and we may t a k e b = b l . Now suppose s + t > 2 and assume t h e r e s u l t holds f o r products with "fewer terms". The leading term of f g i s alblmlnl

and we must again have albl = 0 .

The cases

( i ) a g # 0, and ( i i ) alg = 0 a r e taken s e p a r a t e l y . 1 (i)

Since albl = 0, a g has fewer terms than g . 1

Further

f a 1g = a 1f g = 0, so t h e induction hypothesis a p p l i e s .

Hence t h e r e i s b i n k with bf = 0, b # 0,

and b is i n t h e i d e a l of k generated by t h e c o e f f i c i e n t s of alg.

The i d e a l

generated by alb *,..., albt i s contained i n t h e i d e a l generated by t h e c o e f f i c i e n t s o f g, s o b i s i n t h e l a t t e r .

-

( i i ) We now have a g = 0. Taking f = f a m we g e t 1 1 11 0 = f g = (fl+alml)g = f l g + mlalg = f l g . The induction hypothesis now a p p l i e s t o f l , g t o give b, 0 bf

1

= 0 and b i s i n t h e i d e a l generated by bl,...,

a b . = 0, 1 1 1

5

5 t , and hence a 1b

j

= 0.

bt.

# b

e k , such t h a t

Since alg = 0, we have

I t follows t h a t

bf = b(fl+alml) = bfl = 0 .

Q.E.D.

The following i s an easy consequence. I f k i s an i n t e g r a l domain, then F = k[X

COROLLARY 1 . 2 .

lJ...JXdi s a l s o an

i n t e g r a l domain. This i s equivalent t o COROLLARY 1.3.

I f P i s a prime i d e a l i n k, then PF i s a prime i d e a l i n F, where

PF i s t h e i d e a l of F generated by P.

.,Yn]

PROOF.

Since k / P is an i n t e g r a l domain, we have (k/P) [Y1,..

domain.

I t i s easy t o see t h a t t h e l a t t e r and F/PF a r e isomorphic.

i s an i n t e g r a l Thus F/PF i s

an i n t e g r a l domain and PF is a prime i d e a l . We s t a t e without proof t h e following t h r e e r e s u l t s which a r e very familiar; they appear i n most o f t h e standard t e x t books.

THEOREM 1 . 4 . k[X]

I f k i s a unique f a c t o r i s a t i o n domain, then t h e polynomial r i n g

i s a l s o a unique f a c t o r i s a t i o n domain.

Consequently, F = k[Xl,.,.,X,,]

is a

128

M. Nagata

unique f a c t o r i s a t i o n domain THEOREM 1.5.

I f k i s a f i e l d , k[X]

is a unique f a c t o r i s a t i o n domain.

THEOREM 1.6.

I f k is a noetherian r i n g , then F i s a l s o noetherian

We r e c a l l t h a t a r i n g i s noetherian i f it s a t i s f i e s t h e ascending chain c o n d i t i o n . DEFINITION.

A normal domain is an i n t e g r a l domain which i s i n t e g r a l l y closed i n

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

We n o t e t h a t i n t h e c a s e k i s a f i e l d t h e f a c t t h a t k i s i n t e g r a l l y closed i s t h e same a s k being a l g e b r a i c a l l y c l o s e d . A basic r e s u l t i s

I f k i s a noetherian normal domain, t h e n P also i s a n o e t h e r i a n

THEOREM 1 . 7 .

normal domain. In o r d e r t o prove t h e theorem, besides Theorems 1.5 and 1.6, we a l s o need I f k i s a noetherian normal domain, then

THEOREM 1 . 8 .

(i)

f o r every prime i d e a l P o f height 1, t h e r i n g k

P

= Ia/b

I a,b

e k

6 b 6 P 1

i s a d i s c r e t e v a l u a t i o n r i n g ; and

( i i ) k i s t h e i n t e r s e c t i o n of a l l such kp. (Here, P i s of height 1 ( i n an i n t e g r a l domain) means t h a t i f Q i s a prime i d e a l such t h a t 0 C Q

LP, then

either Q = 0 or

4

= P.)

The proof of t h i s theorem, which helps t o c h a r a c t e r i s e t h e normality of a noetherian domain, w i l l be found i n any standard t e x t book and i s omitted. For t h e proof o f Theorem 1.7 we begin with Gauss’ Lemma : LEMMA 1.9. PROOF.

If k i s a unique f a c t o r i s a t i o n domain, then k i s a normal domain.

Assume t h a t f / g ( f , g i n k and g # 0) i s i n t e g r a l over k .

a r e d > 0 and c l , . . . ,

(f/dd d- 1 Hence f d + c l f g + ,

+

..

cl(f/g)

+ cdg

a f a c t o r of f d , and then of f .

d-1

= 0.

+

... +

Cd

= 0.

I t follows t h a t any prime f a c t o r o f g i s

But we may assume t h a t f , g have no common prime

f a c t o r , so it follows t h a t g has no prime f a c t o r s , i . e . g i s a u n i t . We can proceed now t o

Then t h e r e

cd i n k such t h a t

Q.E.D.

Commutative algebra and algebraic geometry PROOF OF THEOREM 1 . 7 .

Let P be a prime i d e a l o f k o f height 1 and consider This i s a unique f a c t o r i s a t i o n domain by Theorem 1.5; hence

.,XA.

Ff’ = kp[X1,..

a normal domain by Lemma 1 . 9 .

. ., X A

F = k[X1,. F;.

Let f be i n t h e f i e l d o f f r a c t i o n s o f Since F* i s normal , f i s i n P Q.E.D.

and suppose f i s i n t e g r a l over F .

Thus f e n F ; ,

129

and by Theorem 1.8, t h e l a t t e r i s F .

Now we d i s c u s s some r e s u l t s t h a t a r e c l o s e l y r e l a t e d t o t h e famous theorem known as t h e normalisation theorem of Noether. This r e s u l t i s s t a t e d a s THEOREM 1.10.

a field k.

(i)

zl,.

Let t h e r i n g R be generated by a f i n i t e number of elements over

Then t h e r e a r e elements z l , . . . , z t

.., z t

of R such t h a t

a r e a l g e b r a i c a l l y independent over k , i . e . k[zl

n a t u r a l l y isomorphic t o k[X1,.

. . , X t ] ; and

. . ,zt] .

( i i ) R i s i n t e g r a l over k[zl,.

Moreover, i f R = k[X1

,.. . , z t]

,.. .,xnj

is

and P i s a

given non-zero i d e a l i n R, then we may choose z1 from P. The proof r e q u i r e s some p r e l i m i n a r i e s . LEMMA 1.11.

f b k. (i)

Let f be a polynomial i n F = klX1,.

Then t h e r e a r e elements Y 1 , . . . , Y

Y1 = f ; and

( i i ) F i s i n t e g r a l over R = k[Y1,

n

. ., X A

, where

k i s a f i e l d and

i n F such t h a t

...,Yn].

Furthermore, (1) f o r any given p o s i t i v e i n t e g e r q we can choose mi,

m u l t i p l e s of

q, i n such a way t h a t mi Y. = X . + X1 1

1

,i

(2) I f k i s i n f i n i t e , we can choose c

Y. = X . + ciX1, 1

PROOF.

1

= 2,...,

n.

i n k so t h a t i i = Z,..,, n.

Let q be given and t a k e t a m u l t i p l e of q such t h a t t > degf.

mi = For each monomial M = X

dl

ti-l

,

mi

Yi = X i +X1

,

i = 2,...,

.. . Xndn d e f i n e t h e weight

Now s e t

n.

of M t o be w(M) =Cdimi. Note t h a t

t h e ordering o f monomials by t h e i r weights coincides with t h e lexicographical order of t h e exponents i n t h e i r r e v e r s e o r d e r , i . e . as dn,

..., d l ’

We now l e t M be t h e monomial of g r e a t e s t weight t o occur i n f , so t h a t f = cM + monomials o f l e s s e r weights, c e k .

I f g i s t h e polynomial i n X1,

Yn such t h a t

Y2,...,

~~

g = f(X1, Y2 then we see t h a t

-xl , ..., Yn m2

X’I”

1,

M.Naga ta

130

g = ( - l ) m cXycM) + terms of lower degree i n XI where m = d

2

.. .+

t

Introducing Y1 = f , d i v i d i n g through by ( - l ) m c and

dn.

rearranging, shows t h a t X1 i s i n t e g r a l over R = k[Y1,..., Further, s i n c e X. = Y i

i

Y.I = X.1

c.X

t

+

c

11'

e k.

i

..., Yn;

it i s of t h e form

terms o f lower degree i n XI,

where a i s ( t h e value o f ) a polynomial i n c 2 , . . . , c n . may choose t h e ci s o t h a t a # 0.

Set h = height I .

The e a r l i e r argument can now be r e p e a t e d .

...,Xn]

,.. .,Yn]

;

(3) Yh+i = \+i + gi with g. e PIXl, k # 0, gi e P X],':

[ X p , ...,

t h e r e a r e Y;,

R' = k[Y;,

Let I be an i d e a l

. . .,%la

,.. .,Yh;

1

s a t i s f y i n g corresponding c o n d i t i o n s f o r 1'.

..., Y; ] .

Then I '

n R'

and

In case p = t h e c h a r a c t e r i s t i c of

Take an i d e a l I ' such t h a t I ' G I , h e i g h t I' = h - 1 .

.. ., YA

= IiChY;R'.

Then h e i g h t ( I

n R')

1

Then

Set

,

= h -1

Therefore, t h e r e i s an element f e I n R ' such t h a t f # I ' n R ' .

height ( I n R I ) = h . Considering f ( 0 , .

o f F such t h a t (1)

(2) I fRli s generated by Y1

If h = 0, then I = 0, and we may s e t X. = Y . .

We employ induction on h.

5 1.

Q.E.D.

over a f i e l d k, and l e t P be t h e prime

Then t h e r e a r e elements Y 1 , . . . , Y n

F i s i n t e g r a l over R = k[Y1

PROOF.

Since k i s i n f i n i t e , we

(Normalization theorem for polynomial r i n g s )

of t h e polynomial r i n g F = k[X,,

Assume t h a t h

.

Here we s e t

Now express f as a polynomial i n X 1, Y2,

field.

ynl

mi XI , we s e e t h a t F i s i n t e g r a l over R .

-

For (2), we now assume k i s i n f i n i t e .

THEOREM 1 . 1 2 .

,

. . , 0, Y i ,.. ., Y i ) , we

may assume t h a t f e F' = k ( Y i , ,

. .,YA] .

Apply Lema 1.11 t o t h i s f and F ' , and we s e e t h e e x i s t e n c e of Yh = f , Y h + l J . . . ,

.,Yn] and Yh+i + Y; ( i 21 1. Set .., YnJ we s e e (1) and (3) immediately ( i n

Yn such t h a t F ' i s i n t e g r a l over k[YhJ.. Yi = Y;

for i < h .

p # 0, we choose m

Then, with Y1,. i

Iich YiR which i s a I i-h YiR. Q.E.D.

t o be m u l t i p l e s o f p ) .

Set zi = $Y

1.12.

Q.E.D.

Let I be t h e kernel o f t h e n a t u r a l s u r j e c t i o n J, of t h e

,.. .,Xn]

. ( i =l,..., h+i

J,(k[Y lJ...,Yn]) i f g(z l J . . . J z

A s f o r (2), s i n c e I n R c o n t a i n s

prime i d e a l of height a t least h, we s e e t h a t I n R =

PROOF OF THEOREM 1.10.

polynomial r i n g k[X1

case

Then we have Y1

t o R. n -h).

= k[z l J . . . J z n-h

,... ,Yn

Then R = $(k[X l J . . . , X n l )

1.

I f g e k[ZIJ...JZn-h]

n-h ) = 0, then g(Yh+lJ...J Y ) e I , and

as i n Theorem 1 . 1 2 . i s i n t e g r a l over (polynomial r i n g ) and

g = 0 by (2) i n Tneorem

131

Commutative algebra and algebraic geometry THEOREM 1.13.

Assume t h a t R i s an i n t e g r a l domain generated by a f i n i t e number

of elements over a f i e l d k . Po = 0, ( i i ) P

t

I f Po, P 1 , . . . ,

Pt a r e prime i d e a l s such t h a t ( i )

.

is maximal, ( i i i ) PoC P1 E.. G Pt and ( i v ) t h e r e i s no prime i d e a l

Q such t h a t P i - l c Q C P i for any i , then t must be t h e transcendence degree o f R over k .

PROOF.

We proceed by induction on t .

independent elements z l , . . . , z

.

...,

By Theorem 1.10, t h e r e a r e a l g e b r a i c a l l y

over k such t h a t R i s i n t e g r a l over

I f t = 0, then R i s f i e l d .

k[zl, zu] over F implies t h a t F is a f i e l d , and then u = 0.

Assume now t h a t t > 0.

l a s t statement of Theorem 1.10, we may assume t h a t z1 e P1. 1

which is i n t e g r a l over F/zlF

prime i d e a l s Pi/P1. COROLLARY 1.14.

By t h e

Since R i s i n t e g r a l

n F = z1F . Then we apply 1 2 k[z2, z , and t o t h e U

over F, height (P n F ) = l J 1 and t h i s implies t h a t P our induction t o R/P1,

F =

Then t h e i n t e g r a l dependence of R

Thus we have t - 1 = u - 1 and t = u .

..., 1 Q.E.D.

I f an i n t e g r a l domain R i s f i n i t e l y generated over a f i e l d k ,

then f o r any prime i d e a l P of R, i t holds t h a t height P + Krull dim R/P = Krull dim R = t r a n s . degkR. I f a r i n g R i s f i n i t e l y generated over a f i e l d k and i f M i s a

COROLLARY 1.15.

maximal i d e a l of R, then R/M i s a l g e b r a i c over k . I f a r i n g R i s f i n i t e l y generated over a f i e l d k and i f I i s an

WEOREM 1.16.

i d e a l of R, then t h e i n t e r s e c t i o n o f a l l maximal i d e a l s containing I coincides with t h e r a d i c a l

0 of

I) = { f e R l f m

I f o r some m))

PROOF.

B

I (

.

( t h e i n t e r s e c t i o n o f a l l prime i d e a l s c o n t a i n i n g ( H i l b e r t zero-point theorem.)

I t s u f f i c e s t o show t h e case where I i s prime.

i t s maximal i d e a l

MI.

Then considering R/I, we

Let f be a nonzero element o f R and consider R [ l / f ]

may assume t h a t I = 0. over k .

fi =

R[l/f]/M'

Thus M ' n R i s a maximal i d e a l o f R not containing f .

and

i s algebraic

i s a l g e b r a i c over k , and R / ( M ' n R )

Q.E.D.

We add here some o t h e r important a p p l i c a t i o n s o f t h e normalization theorem.

For

t h e purpose, we f i r s t prove some f a c t s on derived normal r i n g s . LEMMA 1 . 1 7 .

Let R be a noetherian r i n g with t o t a l q u o t i e n t r i n g Q.

An

R-submodule M of Q i s f i n i t e l y generated i f and only i f t h e r e i s a nonzero-divisor a of F such t h a t aMG R .

This follows from a r e s u l t which a s s e r t s t h a t i f an i n t e g r a l domain R i s i n t e g r a l over a normal i n t e g r a l domain F, then f o r any i d e a l I o f R , it holds t h a t height I = height ( I n F ) . This r e s u l t i s a c o r o l l a r y t o t h e Going-down Theorem.

M.Nagata

132

I f p a r t : M is contained i n a-1R which i s f i n i t e R-module, and hence M

PROOF.

i s noetherian.

The only i f p a r t i s easy.

Q.E.D.

THEOREM 1.18.

Let R be a normal r i n g , f(X) a monic polynomial over R, a a r o o t of f(X) i n an extension f i e l d , and R* t h e derived normal r i n g of R[a] Let f v ( X )

.

be t h e d e r i v a t i v e of f(X), and l e t D be t h e d i s c r i m i n a n t o f f ( X ) .

Then we have

D R * C f ' ( a ) R * S R[a]. PROOF.

Let t h e r o o t s o f f(X) be u1 = a , U 2 , . . . , ~ r

( r =deg f ) , and s e t gi(X) =

li,l

Therefore Then f'(X) = lgi(X) and f ' ( a ) = gl(a) = (ul - u i ) . I t s u f f i c e s t o prove t h e i n c l u s i o n f ' ( a ) R * C R[a] assuming t h a t

f(X)/(X - u i ) . DR*E f ' ( a ) R * .

f(X) i s i r r e d u c i b l e . a i s separable.

I f a i s i n s e p a r a b l e , then f v ( a ) = 0.

Hence we assume t h a t

Let T be t h e l e a s t Galois extension o f R c o n t a i n i n g R*, l e t G be Let u1 = 1,

t h e Galois group, and l e t H be t h e subgroup corresponding t o R*.

..., u r

u2,

be elements of G such t h a t

Let b be an a r b i t r a r y element o f R*.

1'

= ui.

Then G i s t h e d i s j o i n t union o f

Write g,(X) = c r - l ~ r - l +

k i . gi(x) = gl(x)'i.

liDjbUicjuiaj =

Zi

l j ( libui

... +

c

( c ~ =- ~1; ci e R[al).

Then b f ' ( a ) = bgl(a) =

I:=,

U'

b 'g. (a) =

Since b, c . a r e H-invariants, we s e e t h a t 3 i s G-invariant and hence i s i n R. Thus b f ' ( a ) e R(a1. Q.E.D.

buicjui

COROLLARY 1.19.

cgi)aj.

Let R be a noetherian normal r i n g , K i t s f i e l d of f r a c t i o n s ,

and l e t L be a f i n i t e s e p a r a b l e a l g e b r a i c extension f i e l d of K.

If a ring R ' ,

such t h a t R C. R'L L, i s i n t e g r a l over R , then R ' is f i n i t e as an R-module. Let a e L be such t h a t L = K(a) and l e t c o , . .

PROOF.

coan +...+

= 0 (co f 0 ) .

c

a i s i n t e g r a l over R.

Then, t a k i n g coa i n s t e a d of a , we may assume t h a t

normal r i n g R* of R[a]

i s a f i n i t e R[a]-module, Q.E.D.

['l

...,zt D

'* *

Thus R ' is a

hence over R , t o o .

In Theorem 1.10, i f R i s an i n t e g r a l domain which i s s e p a r a b l e over

t h e f i e l d of f r a c t i o n s o f R i s separably generated over k ) , then t h e

k (i.e.,

zl,

i n R be such t h a t

Then Lemma 1.17 and Theorem 1.18 show t h a t t h e derived

noetherian R-module. LEMMA 1.20.

., c

D

can be chosen so t h a t R i s separably a l g e b r a i c (and i n t e g r a l ) over 't]

'

For t h e proof, t h e r e a d e r i s assumed t o know some b a s i c f a c t s on d e r i v a t i o n s PROOF, al,...,an

I t s u f f i c e s t o prove t h e c a s e where t h e c h a r a c t e r i s t i c p # 0 . be a s e t o f g e n e r a t o r s f o r R over k such t h a t a,,+l,...,

s e p a r a t i n g transcendence base.

Then we apply t h e proof o f Theorem 1.10.

c o n s t r u c t i o n of Yi i n Theorem 1 . 1 2 , we see t h a t ar+l P being t h e prime f i e l d .

Let

an form a

-

P zi e P(al

,

. ..,

P ah]

By our

,

Therefore, f o r every d e r i v a t i o n D o f k ( a l,..., a,,),

we

Commutative algebra and algebraic geometry

133

.

..

we have Dah+i = Dz. ( i = 1, . ,n -h; n-h = t .) This shows that if D is a derivation over k(zl, zt), then D is derivation over k(ah+l,,..,an). Since form a separating tanscendence base, it follows that D = 0, and ah+l,..., a zl, ..., zt are as required. Q.E.D.

...,

Now we come to another application of the normalization theorems. THEOREM 1.21. Let R be a finitely generated integral domain over a field k, and let L be the field of fractions of R. Then the integral closure R* of R in a finite algebraic extension field L' of L is a finite R-module, and hence R* is finitely generated over k. If L ' is separably generated over k, then we choose zl, ..., zt as in Lemma 1.20, and we prove the assertion by Corollary 1.19. In the general case, we choose z1, . , zt as in Theorem 1.10. There is a finite purely inseparable extension k' of k such that L'v K' is separable over k'. Then the integral PROOF.

..

,..

.,zt] is a finite T-module. T is a finite module over closure T* of T = k' [zl F = k[z l,...rzt , and T* is finite over F. Since F C R C_R* C T * , we see the result. Q.E.D.

3

COROLLARY 1.22. Let R be a finitely generated integral domain over a field k, and let R* be the derived normal ring of R. Let P* be a prime ideal of R*, and set P = P * n R . Then height P* = height P. PROOF. Since R* is finitely generated, we have height P* = trans.degkR* trans.degkR*/P* = trans.degkR - trans.degkR/P = height P (Corollary 1.14). Q.E.D. Although the following theorem is known for general noetherian rings, we prove this restricted form as an application of the normalization theorems. THEOREM 1.23.

If a ring R is finitely generated over a field k and if I is an

ideal of R generated by r elements ( I # R ), then for every minimal prime divisor P of I, it holds that height P 'r. PROOF. (1) The case where r = 1 : It suffices to show a contradiction assuming the existence of a chain of prime ideals P 3 P' 3 PI'. Considering R/P", we may assume that R is an integral domain. By Corollary 1.22, we may assume that R is normal. Let Q1,...,Qm be minimal prime divisors of I different from P, and let g be an element of Q, n (I Q, outside of P. Considering RIg-l], we may assume that P is the unique minimal prime divisor of I. Let zl,...,zt be as in Theorem 1.10. Set F = k[zl,...,z ] Let L, M be fields of fractions of R, F, t respectively, and consider a finite normal extension L* of M containing L, the (f) of a generator f of I, and integral closure R* of F in L*, the norm f* = N L/M

...

.

M.Nagata

134

P* i s a minimal prime d i v i s o r o f fR*.

a prime i d e a l P* o f R* l y i n g over P .

Since F i s a UFD, t h e r e i s a prime i d e a l Q o f F c o n t a i n i n g f * , contained i n P f l F By t h e going-down theorem, t h e r e i s a prime i d e a l Q* of R*

and o f height 1.

Then f * e Q*, and some conjugate o f f i s i n

lying over Q and contained i n P*. Then f e Q*'

with a a i n G(R*/F)

Q*. height Q = 1, and t h e r e f o r e Q*'n

implies t h a t Q*a

nR

.

R) = h e i g h t Q*'

We have h e i g h t (Q*'n

R i s a minimal prime d i v i s o r of fR.

=

This

= P and h e i g h t P = 1.

(2) m e general c a s e : We prove t h e a s s e r t i o n by induction on r .

be a s e t o f generators f o r I , and s e t I ' = d i v i s o r of I ' contained i n P .

li -n

+ m = (B, C ) .

Therefore, if (C,C) < 0, then (B, C) < 0 and B, C must have a common component. Q. E . D. 2

These r e s u l t s have a l o t t o do with Cremona transformations o f P

, because

of t h e

following f a c t . (1) I f p e P 2 , then d i l P 2 i s b i r e g u l a r t o F1 ; and (2) i f F i s a P nonsingular r a t i o n a l p r o j e c t i v e surface, then e i t h e r F i s b i r e g u l a r t o P 2 o r t h e r e 'MEOREM 3 . 6 .

is one Fn and a r e g u l a r r a t i o n a l mapping o f F t o Fn.

Here, F1, Fn a r e those

given i n Theorem 3.4. For t h e proof, s e e f o r i n s t a n c e Nagata [13

1.

As f o r Cremona transformations of P2, we s h a l l have some d i s c u s s i o n o f them i n 85.

M. Naga ta

144

This theorem o f t e n helps our understanding of r a t i o n a l s u r f a c e s , because nons i n g u l a r r a t i o n a l p r o j e c t i v e s u r f a c e s a r e e i t h e r P2 o r Fn o r a s u r f a c e blown up from some Fn.

It

One simple example i n t h i s d i r e c t i o n may be cubic s u r f a c e s .

i s well known t h a t a cubic nonsingular s u r f a c e i n P 3 h a s 27 l i n e s on i t .

But

it i s a l s o i n t e r e s t i n g t o c h a r a c t e r i z e t h e s e s u r f a c e s i n t h e following way : THEOREM 3.7.

Let p l , .

. . ,p6 be

ordinary p o i n t s on p2 such t h a t ( i ) no t h r e e o f

them a r e c o l l i n e a r ; and ( i i ) t h e r e i s no conic going through a l l o f them.

Then

t h e r a t i o n a l s u r f a c e obtained by q u a d r a t i c d i l a t a t i o n s with c e n t e r s a t a l l o f pi i s a nonsingular cubic s u r f a c e i n P3.

Conversely, any nonsingular c u b i c s u r f a c e

i n P3 i s b i r e g u l a r t o some one obtained a s above. (See, f o r

We omit t h e proof, because we need some knowledge of cubic s u r f a c e s . i n s t a n c e , Nagata [14]

and van d e r Waerden [22]

.)

By t h e way, we n o t e t h a t t h e 27 l i n e s a r e ( i ) t o t a l transforms of pl, . . . , p 6 ; ( i i ) proper transforms o f conics going through f i v e o f pl, . . . , p 6 ;

and ( i i i ) proper

transforms of l i n e s going through two o f p1,...,p6.

4 . AUTOMORPHISM GROUPS OF POLYNOMIAL RINGS

. ., Xn

Let k be a r i n g , X1,.

indeterminates and s e t F = k[XID., ,,Xn]

.

Then t h e

o b j e c t we s h a l l observe i n t h i s s e c t i o n i s t h e automorphism group AutkF, i . e . t h e group of automorphisms of F which f i x every element o f k. I f n = 1, then t h e s t r u c t u r e of AutkF i s e a s i l y seen, and we omit t h e case.

If

n = 2 and i f k i s a f i e l d , then we have a good r e s u l t which w i l l be s t a t e d l a t e r (Theorem 4.1).

Even when n = 2 , i f k i s not a f i e l d , we do not have any s a t i s -

factory r e s u l t yet.

If n

3, then even i f k i s a f i e l d , we do not have any good

result yet. In t h i s s e c t i o n , we review some f a c t s on AutkF. THEOREM 4.1.

F i r s t we s t a t e :

Assume t h a t k i s a f i e l d and n = 2 .

We write x,y i n s t e a d o f X1,

Then Aut F i s generated as an amalgamated product by two subgroups A and J :

X2. k A i s t h e s e t of

(I

i n AutkF such t h a t

ox = ax + by + c

(a, b, c e k)

oy = ex + f y + g

(e, f , g e k)

af

-

be # 0.

J i s t h e s e t of T i n AutkF such t h a t TX

= ax + f ( y )

TY

= by + c

# a e k ; f ( y ) e kry]) (b, c e k, b # 0 ) . (0

145


There a r e several known proofs of t h i s r e s u l t ; one proof i s found i n Nagata [19]. One immediate adaption of t h i s r e s u l t t o t h e case n = 3 would be t h e following question : Assume t h a t k i s a f i e l d and n = 3.

QUESTION 4.2.

Let A be t h e subgroup o f

AutkF whose elements a r e a f f i n e ( l i n e a r ) transformations.

Let J be t h e group con-

s i s t i n g of elements u o f AutkF such t h a t

i s AutkF generated by A and J ?

Our c o n j e c t u r e on t h i s question i s negative, because t h e following element

'I

of

AutkF seems not t o be i n t h e group generated by A and J .

In order t o s e e t h a t t h i s d e f i n e s an element o f AutkF, we n o t e f i r s t t h a t T(X3X1

t

2

x2) =

x3x1

t

x2.

x1

-+

x1

Then we f i n d t h a t t h e following gives t h e i n v e r s e of

T : +

2

2X2(X3X1

X2)

+

-

X3(X3X1

+

2 X2),

Several people, including David Wright, have asked t h e following questions :

...,

F for i = 1, n and l e t A be k[X1,. .,Xi] t h e subgroup o f AutkF c o n s i s t i n g o f a f f i n e transformations. Do A, H1, Hn-l

In Aut F, s e t Hi = Aut

QUESTION 4.3.

.

k

...,

generate AutkF ? QUESTION 4.4.

I u

e AutkF

Set Vi = k

I uVi

QUESTION 4 . 5 . s e t Fn = k[X1,

4 Vi 1

.

t

kX1

Do K1,.

t

... + kXi

. ., Kn

for i = l,...,n.

Consider i n f i n i t e l y many indeterminates X1,

.. ., Xn]

f o r n = 0,1,2,

Let Ki be

generate AutkF ?

.. . .

X2,

..., Xn, ... and

For each n > 1, l e t Jn be t h e subgroup

of Aut F c o n s i s t i n g of elements u such t h a t uXi = aiXi t f i ( 0 # ai e k; f i eFi-l) k n f o r i = n , n-1, , 1. Let An be Kn above ( t h e same as A) and l e t Gn be t h e subgroup generated by An and Jn. For m > n we d e f i n e t h a t uXm = Xm i f u eAutkFn

. .,

M.Nagata

146

and we make AutkFn a subgroup o f AutkFm. Now we ask i f every AutkFn i s a subgroup o f t h e group generated by a l l t h e G i . There i s a completely d i f f e r e n t problem, on which one may f e e l t h a t t h e r e may be something t o do with t h e s e groups :

QUESTION 4 . 6 . (Jacobian problem).

. ..

Assume t h a t k i s a f i e l d o f c h a r a c t e r i s t i c

,..

, f i n F = k[Xl , ,Xn] , consider t h e polynomial zero. With polynomials f l , n Assume t h a t t h e jacobian matrix (afi/aX.) mapping (X1,...,Xn) -+ ( f l , . . . , f n ) . J has a nonzero constant determinant. Does it follow t h a t t h e mapping i s b i r e g u l a r ? One s e e s e a s i l y t h a t i f b i r a t i o n a l i t y o f t h e mapping i s known, then t h e bireguThere a r e s e v e r a l people who t r i e d t h e problem, b u t , a t t h e

l a r i t y follows.

moment, even i f n = 2 ( t h e c a s e n = 1 i s t r i v i a l l y e a s y ) , t h e r e a r e known only some p a r t i a l r e s u l t s , most o f which a r e due t o S .

Abhyankar and T. T . Moh.

5. CREMONA GROUPS The Cremona group of Pn i s t h e group o f b i r a t i o n a l mappings o f Pn t o i t s e l f , o r , equivalently, t h e group of K-automorphisms o f t h e r a t i o n a l f u n c t i o n f i e l d K(x l , . , . , ~ n ) of Pn.

For n = 1, every b i r a t i o n a l mapping o f P1 i s a l i n e a r ( p r o j e c t i v e ) transformation, hence, i n t h e form of AutKK(x), it i s obtained a s {x

+.

(ax + B ) / ( c x + d )

I a,b,c,d

e K , ad

For n = 2 , t h e r e is a good r e s u l t , f i r s t claimed by M .

by Castelnuovo [3 Jung [ 7

1 , etc.) .

THEOREM 5.1.

]

.

# bc 1 . Noether and then proved

Several o t h e r proofs were given l a t e r (Alexander [l]

,

One Eormulation of t h e r e s u l t can be s t a t e d as follows :

The Cremona group of P2 is generated by l i n e a r p r o j e c t i v e t r a n s -

formations and an a r b i t r a r y q u a d r a t i c transformation. Here, a q u a d r a t i c transformation i s defined a s follows. p o i n t s which a r e not c o l l i n e a r .

Let p , q, r be t h r e e

Take l i n e a r forms f , g, h, i n t h e homogeneous

coordinate r i n g H = K[x, y, z] which d e f i n e t h e t h r e e l i n e s C, C ' , C" going through two of p , q, r .

Then t h e correspondence (x, y, z) -+ (gh, h f , f g ) =

( f - l , g-l, h - l ) defines a b i r a t i o n a l mapping of P2 t o i t s e l f such t h a t ( i ) it i s b i r e g u l a r a t any point o u t s i d e o f C U C ' U C" ; ( i i ) t h e proper transform o f each o f C , C ' , C" i s a point ; and ( i i i ) each of p, q, r corresponds t o a l i n e . p r e c i s e l y , t h i s can be s t a t e d as follows : c e n t e r s p, q, r .

More

Consider q u a d r a t i c d i l a t a t i o n s with

Then t h e proper transforms o f C, C ' ,

C t t a r e nonsingular r a t i o n a l

curves with s e l f - i n t e r s e c t i o n number -1, hence they can be c o n t r a c t e d t o p o i n t s

141


(noting that they do not intersect each other),

Then the new surface is a P

2

.

We should note here that in the statement of Theorem 5.1, since all linear trans-

formations are allowed, to allow one special quadratic transformation amounts the same as to allow all quadratic transformations. Therefore, in terms of AutkK(x,y), Theorem 5.1 can be stated as : THEOREM 5 . 2 . If x,y are algebraically independent elements over the algebraically closed field k, then Aut K(x,y) is generated by a quadratic transformation k defined by (x,y) -+ (x-’, y-l) and the group of projective linear transformations

{

(x,t)

-[

,

m]

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

eK,

i:,”fI1. d e f # 0

These formulations are made from the coordinate-wise viewpoint. In order to analyze the geometric feature of a Cremona transformation, it is worthwhile to observe it in the birational class of rational surfaces with identification of points as in 13. At this point, Theorem 5.1 is equivalent to : THEOREM 5.3. An arbitrary birational mapping of a P 2 to another P2 is the product of quadratic transformations. (This means that there is a sequence of P2 and quadratic transformations ii :

so that the given birational transformation is the product of these T ~ . )

But, as a matter of fact, Theorem 5 . 3 is too inefficient in the sense that, in order to practice the transformation, blowing up of biregularly corresponding points is often required. In order to explain such a situation and to have a more efficient formulation, we review two kinds of Cremona transformations : generalized quadratic transformations and Jonquieres transformations. The term “quadratic” transformation derived from the fact that the transformation is defined by forms of degree two (gh, hf, fg in the previous notation). For this fact, the three points need not be ordinary points. So let us observe linear systems of conics of P‘ having three base points p, q, r where p is assumed to be an ordinary point. If q, r are also ordinary points, then (i) p, q, r are not collinear, then we have the usual quadratic transformation; (ii) if they are collinear, then the system has a fixed component, and this case is omitted. So we assume that q is an infinitely near point to p of order one. CASE 1.

Assume that r is an ordinary point. Let L and L ’ be the lines going through p + q and p + r respectively. If L = L’, then the linear system has L as a fixed component, and we omit the case. So we assume that L # L’. Then the

148

M.Nagata

l i n e a r system d e f i n e s a Cremona transformation which t u r n s t o t h e following (modulo l i n e a r transformation) : (x,y,z)

+

(yz, z2, xy).

This i s obtained by

successive q u a d r a t i c d i l i t a t i o n s with c e n t e r s p, q, r and c o n t r a c t i o n s o f proper transforms of L, L ' .

Then t h e proper transform o f t h e curve on t h e f i r s t

d i l a t a t i o n came from p. CASE 2 .

Assume t h a t r i s i n f i n i t e l y near t o p .

If r i s not i n f i n i t e l y near

t o q, then t o go through p + q + r implies t h a t p must be a double p o i n t and every member of t h e l i n e a r system i s t h e sum o f two l i n e s going through p . a system does not d e f i n e a b i r a t i o n a l mapping. t o q.

Such

Therefore r i s i n f i n i t e l y n e a r Then r cannot be on t h e proper

Let L be t h e l i n e going through p + q.

transform of L a f t e r t h e d i l a t a t i o n s with c e n t e r s p, q (otherwise L becomes a fixed component o f t h e l i n e a r system).

If C i s an i r r e d u c i b l e member o f t h e

l i n e a r system, p = (0, 0, 1) and i f L = V(x), then t h e equation f o r C is of t h e form xz + ax2 + bxy + cy2 ( c # 0 ) .

On t h e o t h e r hand, L + (any l i n e going

through p) i s i n t h e l i n e a r system, a s we can compute by our r u l e i n 13.

We s e e

t h a t t h e transformation i s defined (modulo l i n e a r transformations) by :

where c depends on t h e p o s i t i o n of r and hence may be made 1 by a l i n e a r t r a n s f o r mation.

In terms of d i l a t a t i o n , we s e e t h a t t h e transformation i s made by :

First apply q u a d r a t i c d i l a t a t i o n s with c e n t e r s p, q, r ( r can be assumed t o be

of o r d e r two), then c o n t r a c t t h e proper transform of L, then t h e proper transform of t h e r a t i o n a l curve t h a t came from q , and then t h e one t h a t came from p . These two types of Cremona transformations a r e generalized q u a d r a t i c transformat i o n s ; as we have seen, they and t h e usual q u a d r a t i c transformations a r e t h e only Cremona transformations defined properly by l i n e a r systems o f c o n i c s . Let u s observe h e r e Theorem 5.3 i n t h e s e c a s e s .

As f o r Case 1, what we do is t o

t a k e a p o i n t s which i s not on e i t h e r o f t h e l i n e s L, L ' . q u a d r a t i c transformation with c e n t e r s p, r , s .

p o i n t s from t h e l i n e s going through p + s , r + s . q u a d r a t i c transformation with c e n t e r s q , q u a d r a t i c transformation.

rl,

Then f i r s t apply

Let r ' , p ' be t h e c o n t r a c t e d

p'.

Then t h e second s t e p i s t h e Thus we o b t a i n t h e generalized

In Case 1, we need t o blow up one e x t r a p o i n t .

Case 2

can be handled s i m i l a r l y , but we have t o blow up two e x t r a p o i n t s . Let us go t o t h e notion of a Jonquieres transformation.

I t i s defined t o be a

Cremona plane transformation defined by an i r r e d u c i b l e l i n e a r system L o f curves o f degree d having a (d-1)-ple p o i n t , d

2.

We have seen above t h a t t h e c a s e

d = 2 i s nothing but a usual o r a generalized q u a d r a t i c transformation, and we s h a l l mainly observe t h e c a s e d > 2 .

By t h e genus formula, we s e e t h a t an irre-

d u c i b l e curve o f degree d having a (d-1)-ple is a r a t i o n a l curve.

Computing t h e


149

dimension of L, o r variable intersections of members of L, we see that L must be a linear system having the (d-1)-ple point and 2d - 2 other base points. These base points may be infinitely near to some points, If one observes carefully, one sees that Jonquieres transformations are interpreted in terms of elementary transformations as follows : Take a point p1 of P2 and apply quadratic dilatation. Then we obtain a ruled surface F1 (under the notation of Theorem 3 . 4 ) . Let E be the total transform of pl. Apply successive elementary transformations with centers p2, ...,pd all on either E or its proper transforms and we obtain Fd. Then apply successive elementary transformations with centers P ~ + ~ , . , .p2d-l , all outside of proper transforms of E and we obtain another F1. Then F1 is contracted to P 2 . p1 is the (d-1)-ple point of L and p2, ..., are the other base points. (Only in P2d-l case d = 2 , we may take p2 outside of E and we obtain a usual quadratic transformation .) It terms of the linear system L, it is hard to describe the positions of p2, ..., p2d-l (because there is some restriction, as we saw even in the case d =2). But, in terms of elementary transformations, we can state the restriction to be that the elementary transformations can be done successively as described above. Furthermore, that p1 is a (d-1)-ple point of members of L implies that each line going through p1 meets each member of L at only one point (unless the member contains the line). This means that when we blow up pl, then on F1, every member of L becomes a section ( o r , a section t some generators). Therefore Jonquieres transformations are rather understandable Cremona transformations. The following version of Theorem 5.1 would be better than Theorem 5.3, though Theorem 5.3 looks more beautiful : Every birational mapping of a P2 to another P2 is factored into the product of Jonquieres transformation. THEOREM 5.4.

For the detail of these topics, see for instance, Nagata [13]

.

we pointed out in 12, our knowledge on higher dimensional cases is very little, and how to control the cases would be an important subject to be clarified in the future.

As

6 . GROUP ACTIONS ON AFFINE RINGS

When we say that a group G acts on a ring R, we understand that there is given a homomorphism $ of G into Aut R ( =the automorphism group of R) and the action of G on R is defined by oa = ($o)a for u e G, a e R .

M.Nagata

150

We d e f i n e a r a t i o n a l a c t i o n a l i t t l e d i f f e r e n t l y .

Namely, i f G i s an a l g e b r a i c

group over a f i e l d k, R a f i n i t e l y generated r i n g over k and i f K i s an a l g e b r a i c a l l y closed f i e l d i n which we consider coordinates o f p o i n t s (k C-K), then by saying t h a t a r a t i o n a l a c t i o n o f G on R i s defined over k , we understand t h a t ( i ) G a c t s on RO k K i n t h e sense s t a t e d above and ( i i ) f o r each a e R and f o r bn of R = R f 1 each connected component Gi of G , t h e r e a r e elements b l , and r a t i o n a l functions f l ,

..., f n e k(Gi)

oa = l f i ( o ) b i f o r a l l o e G i .

...,

on t h e connected component Gi,

so t h a t

Note t h a t ( i i ) above i s n e a r l y equivalent t o

assuming t h e same f o r only one component, but n o t e x a c t l y . In our observations i n t h i s s e c t i o n , t h e r o l e o f k i s not important, and we consider t h e c a s e k = K f o r s i m p l i c i t y . The condition on t h e e x i s t e n c e o f f i and bi looks t o be s l i g h t , but i s r e a l l y very strong, because o f t h e following f a c t : THEOREM 6 . 1 .

I f an a l g e b r a i c group G a c t s r a t i o n a l l y on an a f f i n e r i n g R , then

f o r each a e R , t h e module M a PROOF.

loeG oaK i s a f i n i t e

With bi a s above, we have

1oeCO oaK E. 1b.K;

K-module. s i n c e t h e right-hand s i d e

i s a f i n i t e K-module, t h e left-hand s i d e i s a l s o a f i n i t e K-module.

applied t o any Ta( T B G), o f t h e index of Go.

The same i s

and t h e f i n i t e n e s s o f Ma follows from t h e f i n i t e n e s s

Q.E.D.

This r e s u l t means t h a t i f we t a k e a s e t o f g e n e r a t o r s a l , consider t h e sum M of t h e modules Mai,

...,

a f o r R and i f we m then M i s a r e p r e s e n t a t i o n module f o r G

and we o b t a i n a r a t i o n a l r e p r e s e n t a t i o n of G, i . e . a r e p r e s e n t a t i o n p of G i n CL(n,K) ( f o r some n) such t h a t every e n t r y p . . ( o ) of pa i s a r a t i o n a l f u n c t i o n on 11

Go;

i f N i s t h e kernel of t h e r e p r e s e n t a t i o n , then every element o f N a c t s

t r i v i a l l y on R. THEOREM 6 . 2 .

Thus : I f an a l g e b r a i c group G a c t s r a t i o n a l l y on a f i n i t e l y generated

r i n g R , then t h e a c t i o n i s given v i a an a f f i n e a l g e b r a i c group.

Namely, t h e r e is

a r a t i o n a l r e p r e s e n t a t i o n p of G i n GL(n; K) (with n a n a t u r a l number) and a s e t o f elements a l ,

..., an o f

R which generate R and such t h a t laiK i s a r e p r e s e n t a -

t i o n module f o r G and t h e a c t i o n o f each element

(I

o f G is given by

Note t h a t t h i s i s a very d i f f e r e n t phenomenon from t h e p r o j e c t i v e c a s e We add h e r e two remarks whose proofs a r e easy.

?he f i r s t i s

151


Assume t h a t a r i n g R i s generated by al,...,an over t h e f i e l d K ;

THEOREM 6 . 3 .

that al,

...,a

a r e l i n e a r l y independent over K and t h a t a subgroup G of GL(n,K)

i s given so t h a t f o r each p B G

(a,,

. ..,an)

(aly

. . . ,an)v

Let G* be t h e c l o s u r e o f G i n GL(n,K) (under t h e

defines a K-automorphism o f R . Zariski topology).

-

Then G* is an a l g e b r a i c group and t h e a c t i o n o f G i s n a t u r a l l y

extended t o t h a t o f G*. The o t h e r is r e l a t e d t o t h e f i e l d s o f r e f e r e n c e s and t h e r e f o r e we must consider t h e case k # K . Assume t h a t a r a t i o n a l a c t i o n o f an a l g e b r a i c group G on a f i n i t e l y

THEOREM 6 . 4 .

generated r i n g R = k[a,,

. . .,an)

i s defined over t h e f i e l d k .

If k ' is a field

k ' c K , then t h e r a t i o n a l a c t i o n is n a t u r a l l y extended t o t h a t o f

such t h a t k

G on R O k k ' , and i n t h i s c a s e i f u e G i s a k ' - r a t i o n a l p o i n t , then u ' d e f i n e s

a k'-automorphism of R e k ' . k

This i s s i m i l a r l y a p p l i e d t o an extension o f K .

A s f o r group a c t i o n s , t h e r i n g s of i n v a r i a n t s play an important r o l e on s e v e r a l

occasions. In general, i f a group G a c t s on a r i n g R, i . e . t h e r e is given a homomorphism J1 of G i n t o Aut R and t h e a c t i o n i s defined so t h a t ua = $ua f o r any u e G, then t h e G s e t of G-invariants i n R forms a subring of R . The subring i s denoted by R

.

Under t h e given circumstances, i f N i s a normal subgroup of G, then N G a c t s n a t u r a l l y on R and t h e a c t i o n i s p r a c t i c a l l y an a c t i o n o f G / N . LEMMA 6 . 5 .

I f R i s a f i n i t e l y generated r i n g over a noetherian r i n g k and i f R

LEMMA 6.6.

i s i n t e g r a l over i t s subring S containing k , then S i s f i n i t e l y generated over k.

In p a r t i c u l a r , i f a f i n i t e group G a c t s on a f i n i t e l y generated r i n g R over a G noetherian r i n g k so t h a t t h e a c t i o n of G on k i s t r i v i a l , then R i s f i n i t e l y generated over k . Proofs a r e easy, and we omit them.

Our main i n t e r e s t l i e s i n t h e case where an a l g e b r a i c group G a c t s r a t i o n a l l y on a f i n i t e l y generated r i n g R over a ground f i e l d k .

As we can s e e e a s i l y by our

observation above, t h e c a s e i s p r a c t i c a l l y equivalent t o t h e following case : (1)

G i s a subgroup of GL(n,K) f o r some n ;

(2)

al, an generate t h e r i n g R over K (with t h e same n a s above), and a r e l i n e a r l y independent over K ;

(3)

For any u

...,

(al,.

6

G, t h e automorphism o f R defined by o maps ( a l ,

. .,aJu .

...,an)

to

M.Nagata

152

Here K denotes as before an algebraically closed field in which we consider coordinates of points. Under the circumstances, the closure G* of G in GL(n,K) under the Zariski topology is an algebraic group; and G* acts rationally on R. Furthermore, RG = RG* . We recalled this formulation for two reasons. One is that we also have to discuss the case where k # K. The other is to explain one classical proof of the following fact. Here, we define as follows : linear algebraic group G is said to be linearly reductive if every rational representation of G is completely reducible.

A

A linear algebraic group G is said to be geometrically reductive (or simply

reductive) if the radical of G is a t o r u s group. It is known that a linear algebraic group G is linearly reductive if and only if either (i) the ground field is of characteristic zero and the radical of G is a torus group; or (ii) the ground field is of non-zero characteristic, say p, and the connected component Go of 1 of G is a torus group and furthermore [G : Go] is not a multiple of p. As for geometrically reductive groups, the validity of the Mumford conjecture below, proved by Haboush [4 ] , is important.

Mumford conjecture. Let G be a linear algebraic group, G E GL(n,K), such that its radical is a torus group and assume that G acts rationally on an affine ring R over K. If M is a G-stable finite K-module and if f is an element of R which is G-invariant modulo M, i.e. f - af e M for any u e G, then there is a G-invariant g of the form fm + clfm-’ + , , + c with coefficients ci in the K-module Mi = m the K-module generated by { nl...n. I n e M 1 , 1 j

.

THEOREM 6.7. If G is a linearly reductive algebraic group and if G acts rationally on a finitely generated ring R over the field K, then RG is finitely

generated over K. A proof of Theorem 6 . 7 in the classical case (see Weyl [24]

, cf.

1

Nagata [12 ) proceeds along the following lines : On one hand, we prove the theorem in the case that G is a compact Lie group (instead of an algebraic group) where the case can be proved using inte grations; another basis for the proof is that any linearly reductive algebraic group G with K = (the complex number field) contains a compact Lie subgroup which is dense in G under the Zariski topology.


153

Although this itself cannot be applied to the positive characteristic case, the idea can be adapted in many cases. A proof o f Theorem 6.7 good for an arbitrary characteristic case was given by Nagata [18] by proving the following : THEOREM 6.8. Under the circumstances of Theorem 6.7, (i) if I is an ideal of RG, then I R n R G = I ; and (ii) if $ is a G-admissible K-homomorphism of the ring R onto another ring R', then it holds that $(RG) = RIG, generalization of these results to the geometric reductive case was given by Nagata [17] ; Theorem 6.7 with the same conclusion, namely, Theorem 6.10 below; and Theorem 6.8 with modified conclusion, namely Theorem 6.9 below. A

THEOREM 6.9. Assume that a geometrically reductive group G acts rationally on a finitely generated ring R over the field K. (i) If hl,...,hr are elements of (ii) If $ is a G-admissible RG , then ( lhiR)nRG is nilpotent modulo lhiRG K-homorphism of the ring R onto another ring R' and if h e RIG, then there is a G power hS of h so that hS e $(RG). Consequently, R" is integral over @(R ) .

.

THEOREM 6.10. K.

Under the circumstances as above, RG is finitely generated over

REFERENCES [l]

J. W. Alexander On the factorization of Cremona plane transformations, Trans. her. Math. SOC 17(1916), 295-300.

[2]

G . Castelnuovo, Recerche generali sopra i systemi lineari di curve piani, Mem. Accad. Sci. Torino, C1. Sci. Fis. Mat. Natur. ser. 2, 42(1892), 3-43.

~31

, La transformazioni generatrici dei grouppo Cremoniano nel piano, Atti Accad. Sci. Torino, C1. Sci. Fis. Mat. Natur. 36(1901) 861-874.

[4]

W. Haboush, Reductive groups are geometrically reductive, Ann. of Math. (2) 102 (1975), 67-83.

[s]

H. P. Hudson, Cremona transformations, Cambridge Univ. Press, 1927.

(61

H. W. E. Jung, Zusammensetzung von Cremonatransformationen der Ebene aus quadratischen Transformationen, Crelle J. 180(1939),

97-109.

[71

, b e r ganze birationale Transformationen der Ebene, Crelle J. 184 (1942), 161-174.

[8]

W. van der Kulk, On polynomial rings in two variables, Nieuw Arch. Wisk. (3) l(1953) , 33-41,

M.Naga ta

154

[91

D. Mumford, Geometric invariant theory, Springer, Berlin, 1965.

[lo] M. Nagata, A treatise on the 14th problem of Hilbert, Mem. Coll. Sci. Univ. Kyoto, A-Math. 30(1956-57), 57-70; addition and correction, ibid., 197-200.

, On , On

I111

PI

the 14th problem of Hilbert, h e r . J. Math. 81(1959), the 14th problem of Hilbert, Shgaku 12(1960),

766-772.

203-209.

(Japanese)

, On

[13]

rational surfaces. I, Mem. Coll. Sci. Univ. Kyoto, A-Math.

32(1959-60), 351-370.

, On

C141

33(1960-61), [151

271-293.

, Complete reducibility of rational representations of a matric group, J. Math. Kyoto Univ. l(1961-62), 87-99.

PI C171

rational surfaces. 11, Mem. Coll. Sci. Univ. Kyoto A-Math.

, Note on

orbit spaces, Osaka Math. J. 14(1962),

21-31.

, Invariants of a group in an affine ring, J. Math. Kyoto Univ. 3(1963-64), 369-377. , Lectures on the fourteenth problem of Hilbert, Tata Inst. F.

1181

R.9

1965.

, On automorphisms group o f k[x,y], ~191 Kinokuniya, Tokyo, 1972, 12 01

, Field

Lectures Math. Kyoto Univ.,

theory, Marcel Dekker, New York and Basel, 1977.

1211 D. Rees, On a problem of Zariski, Illinois J. Math. 2(1958),

145-149.

[22] B. L. van der Waerden, Einfurung in die algebraische Geometrie, Springer, Berlin, 1939. [231 R. WeitzenbEck, h e r die Invarianten von linear Cruppen, Acta Math. 58(1932), 231-293. [241 H. Weyl, The classical groups, Princeton Univ.

[25]

0. Zariski, Foundations of general theory of birational correspondences, Trans. her. Math. SOC. 53(1943),

PI

Press, 1939.

490-542.

, Intgrpretations alggbraico-g6om6triquesdu de Hilbert, Bull. Sci. Math. 78(1954),

, Introduction to

quatorzisme problzme

155-168.

the problem of minimal models in the theory of algebraic surfaces, Publ. Math. SOC. Japan 4(1958).

Proceedngs of the International Mathematical Conference L.H.Y. Chen, T.B. Ng, MJ. Wicks leak.) 0 North-HollandAtblishing Compny, I982

155

CALCULUS OF BROWNIAN FUNCTIONALS

Takeyuki Hida Department o f Mathematics F a c u l t y o f Science Nagoya U n i v e r s i t y Chikusa-ku, Nagoya, 464 Japan L e t t B ( t ) } be a one-dimensional s tandard Brownian mot ion. N on-lin e a r f u n c t i o n a l s o f I B ( t ) } o f t h e f o r m f ( B ( t ) ; t E T ), T an i n t e r v a l , w i l l be c a l l e d Brownian f u n c t i o n a l s . The a n a l y s i s o f such Brownian f u n c t i o n a l s p l a y s very i m p o r t a n t r o l e s n o t o n l y i n t h e t h e o r y o f s t o c h a s t i c processes b u t a l s o i n v a r i o u s f i e l d s o f a p p l i c a t i o n s such as e l e c t r i c a l enginee r i n g , quantum mechanics and o t h e r s , where n o i s e s or f l u c t u a t i o n s are involved. I n these notes we s h a l l d i s c u s s 1. Basic p r o p e r t i e s o f a Brownian motion.

2 o f Brownian f u n c t i o n a l s . 2 I n t e g r a l representation o f (L )-functionals. (L )

2.

White n o i s e and t h e c l a s s

3. 4. 5.

G e n e r a l i z e d Brownian f u n c t i o n a l s . Causal c a l c u l u s .

156

T.Hiah

Not a t io ns . Basic p r o b a b i l i t y space : (n, B, P) Brownian mo t io n : B(t, u), t E T, d White n o i s e : B(t, U) = $(t, Brownian f u n c t i o n a l

: Q(b(t), t

E

w c fi w)

T)

IFLOW DIAGRAM I

p

(Prob. d i s t r . o f

(b(t)})

7-transform r

IF

R.K.H.S.

n

1

Generalized Functionals

Appl ic a t i ons , Causal Cal c u l us

The LBvy A n a l y s i s

Calculus of Brownian functionals CHAPTER 1.

157

BROWNIAN MOTION

T h i s c h a p t e r i s devoted t o a background f o r o u r a n a l y s i s , i n p a r t i c u l a r , we s h a l l d i s c u s s some b a s i c p r o p e r t i e s o f a Brownian motion. 1.1. D e f i n i t i o n s . We s t a r t w i t h d e f i n i t i o n s . d e f i n e d on a p r o b a b i l i t y space (n, B, P ) . D e f i n i t i o n 1.1. The c o l l e c t i o n i f any f i n i t e l i n e a r combination

2=

Let

I X (w); A

X,(W)'S

be random v a r i a b l e s

A I i s c a l l e d a Gaussian system

E

z

a X i s a Gaussian random v a r i a b l e . k kAk

,...,

,X X, ) i s Gaussian i n A1 n d i s t r i b u t i o n . I f i n p a r t i c u l a r i t s covariance m a t r i x V i s non-singular, the d e n s i t y o f t h e d i s t r i b u t i o n i s expressed i n t h e f o r m I f J- i s a Gaussian system, then any

m = (m1, m2,...,

where

mn)

The c o l l e c t i o n I B ( t , w ) ; t Brownian motion, i f a.

B(0,

b.

IB(t), t

O)

TI

Note.

e.g. we

c

TI, T = R

I

[ 0 , -)

or

x'

i s the transpose o f

or

[a, b],

x.

i s a (standard)

i s a Gaussian system such t h a t

-

B(t+h) where t h e symbol 2 and v a r i a n c e 0

i s t h e mean v e c t o r and

and

= 0 E

(X

B(t)

G(m,

02)

i s subject t o

G(0, [ h i ) ,

t, t+h

E

T,

stands f o r t h e Gaussian d i s t r i b u t i o n w i t h mean 0

.

I n what f o l l o w s t h e v a r i a b l e w o f random v a r i a b l e s i s o f t e n o m i t t e d , denote s i m p l y b y X i n s t e a d o f X(W).

T r i v i a l it i e s . i ) E ( B ( t ) ) = 0; Covariance E ( B ( t ) B ( s ) ) = min ( t , s ) = t h s . i i ) S e t AB = B ( b ) - B ( a ) f o r A = [a, b]. I f h , A ~ , . . .A,, overlapping, then

A B ~ S a r e m u t u a l l y independent.

i s an a d d i t i v e process. i i i ) The p r o b a b i l i t y d i s t r i b u t i o n o f g i v e n by

where iv)

tl < t2

.--

If I B ( t ) , t

The same f o r {x-'B(x2t),

x#

[O, - ) I ,

t

0.

(The space-time t r a n s f orma t ion) 1.2. Sample f u n c t i o n p r o p e r t i e s . We a r e i n t e r e s t e d i n t h e r e g u l a r i t y o f sample f u n c t i o n s o f a Brownian motion.

A) U n i f o r m c o n t i n u i t y . The parameter s e t T i s now t a k e n t o be ( P . L&y)

Theorem 1.1. such t h a t (1 1

(B(t,

whenever

t, t '

Let

c

-

B(t',

W)

[0, 1 1 and

c

1,

z

Fo r a l m o st a l l

w)I 5

-

It

[0, 11.

c h ( t

-

there exists

W,

t'(log(t

-

6 = 6(~)>0

t'1-l

t ' I < 6.

6) The second v a r i a t i o n . Let

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

TI

[O, 11 ' i n t o sub-intervals

n = { A ~ } , u

=

[O, 11,

A;(S

i.e.

A ~ S non-overlapping.

k Set

6 ( r r ) = max I t k t 1

The second v a r i a t i o n

-

t k l , where 2 TI f due t o

Take a sample f u n c t i o n

= [tk, TI

o f a function

f i n e r t h an

Given a sequence T .,I

f

i s d e f i n e d t o be

B ( t , W ) o f a Brownian motion, and s e t 2 = nnB(., 2 V,,(W)

Theorem 1.2.

tktl1.

inn,

n

2

0)

w).

o f p a r t i t i o n s such t h a t

nntl

is

Then

i) For almost a l l

w

the l i m i t 2 2 l i m Vn(w) E V ( w )

nexists. ii) If,i n a d d i t i o n , t h e as

n

-+

m,

t hen we have

{nn}

satisfies the further condition that

TI,,)

+

0

159

Calculus of Brownfanfunctionah V2(w) = 1,

(2)

For p r o o f , see e.g.

a.e.

t h e book [Z] Chapter 2.

1.3. C o n s t r u c t i o n o f a Brownian motion. I n t h i s s u b s e c t i o n we p r e s e n t two i m p o r t a n t methods o f c o n s t r u c t i n g a Brownian motion. I t s h o u l d be n o t e d t h a t t h e c o n s t r u c t i o n s themselves i l l u s t r a t e b a s i c p r o p e r t i e s o f a Brownian m o t i o n . A)

L 6 v y ' s method.

The i d e a comes f r o m t h e f o l l o w i n g

P r o p o s i t i o n . For any t decomposition o f B ( t ) :

E

(a, b )

c

[O, 13 we a r e g i v e n t h e f o l l o w i n g

where

X ( t ) = G(0, l ) , independent o f

-

u ( t ) = J(t

a)(b

-

t)/(b

-

{B(s), s

E

[0, a] u [b,

111

a)

I t deserves mention t h a t t h e c o v a r i a n c e f u n c t i o n o f t h e Gaussian process { X ( t ) , t E ( a , b ) ] i s t h e square r o o t o f t h e anharmonic r a t i o ( a , b ; s , t).

We a r e now r e a d y t o show t h e L e v y ' s method o f c o n s t r u c t i n g a Brownian motion. Construction. i ) Prepare a system o f independent i d e n t i c a l l y d i s t r i b u t e d ( i . i . d . ) v a r i a b l e s {Yn; n 2 11, each o f which i s s u b j e c t t o G(0, 1 ) . ii) Form { X l ( t ) , t c [0, 111 as i s shown b y F i g . 1

random

1

0

Fig. 1

iii)

L e t Tn

n z 1, and s e t n, by

denote t h e s e t of a l l b i n a r y numbers

To =

u

nzl

Tn.

Now d e f i n e t h e sequence

k/2"', Xn(t,

k = 0, l,...,p - 1 w), inductively i n

,

160

T.Hiah

I

n t l. ..

$Xn(t+2-n,

w) +

t

E

u)] + 2 - 5 k ( w ) ,

xn(t-2-n,

Tn+l

-

Tn, k = k ( t ) = 2"'

+ ~1 ( n t2- 1 ) ,

(See Fig. 2 )

t

E

k k+l [--, -1.

zn zn

Fig. 2

iv)

We can then prove t h a t

lim Xn(t,

W)

n+-

E ( X ( t ) ) = 0,

= X(t,

Paley-Wiener's method.

e x i s t s , and

E(X(t)X(s)) = t

We have thus obtained a Brownian motion B)

W)

A

s.

{ X ( t , w ) ; t . c [O, 111.

T

The parameter s e t

[O, Zn], and we f o l l o w the f o l l o w i n g steps. Fourier s e r i e s expansion.

i s taken t o be the i n t e r v a l

The main idea i s the use o f the

i ) Prepare a system o f i . i . d . random v a r i a b l e s each o f which i s subject t o G(0, 1 ) . i i ) Form complex Gaussian random variables

IXk(w), Y k ( w ) ;

k = 0,+1,+2,-..),


161

They a r e m u t u a l l y independent, i d e n t i c a l l y d i s t r i b u t e d and n o r m a l i z e d . i i i ) Then, f o r m a random F o u r i e r s e r i e s

iv)

Having n o r m a l i z e d t h e

Zl(t,

W)

:

--1

we a r e g i v e n a complex Brownian m o t i o n and E ( Z ( t ) m ) = t A s. v)

{Z(t, w); t

E

[0, 2 ~ 1 1: E ( Z ( t ) ) = 0

Set

--1 Z(t, W) = 2

2{ X ( t , W ) + i Y ( t ,

w)IY

X(t, w ) ,

Y(t,

real.

w)

Then we have t h e f o l l o w i n g e x p r e s s i o n s :

s i n n t t Yn(w)(cos n t

-

1) Y

G

n

(4)

r(t,

-

w) =

1) t Yn(w)sin n t -Xn(w)(cos n t yo(w) t x ' I G n G n t

'0'

where

f:

stands for t h e sum

The two processes

q

n=

t

2.

nz-1

[O, 2r1,

T. Hida

162

are standard Brownian motions and they a r e m u t u a l l y independent. I n c i d e n t a l l y we g i v e a d e f i n i t i o n o f a complex Gaussian system. L e t Z ( W ) complex-valued random v a r i a b l e on ( a , B, P ) and be expressed i n t h e form

where m i s t h e e x p e c t a t i o n o f Val ued.

Z(U),

and where

X(U)

and

Y(u)

be a

are r e a l -

D e f i n i t i o n 1.2. i ) I f X ( W ) and Y(u) i n t h e decomposition ( 5 ) a r e independent and have t h e same Gaussian d i s t r i b u t i o n w i t h z e r o mean, then Z(U) i s c a l l e d a complex Gaussian random v a r i a b l e .

-

i i ) L e t Z = { Z A ( u ) ; A t A } be a system o f complex-valued random v a r i a b l e s . I f any f i n i t e l i n e a r combination Z c j z A . , c . ' s b e i n g complex numbers, i s a J J J complex Gaussian random v a r i a b l e , then 1, i s c a l l e d a complex Gaussian system. I t i s easy t o see t h a t t h e complex Brownian motion a complex Gaussian system.

{Z(t, w);

t

[0, 2a]}

forms

A complex Gaussian system enjoys s i m i l a r p r o p e r t i e s t o a ( r e a l ) Gaussian system.

Exercise 1.

Let

{Zl(w),

Z 2 ( ~ ) } be a complex Gaussian system.

I f they are

u n c o r r e l ated, then they are independent. Exercise 2.

Let

I ={ Z A ( u ) ;

A ) be a complex Gaussian system, and l e t

A E

be t h e decomposition as i n ( 5 ) . Z A ( u ) = mA + X A ( u ) + iY,(u) and {YA(w); A t A} a r e b o t h Gaussian systems.

Then

{XA(w); A E A }

Before c l o s i n g t h i s chapter, we make a s h o r t o b s e r v a t i o n on sample f u n c t i o n c o n t i n u i t y o f a Brownian motion. For a continuous f u n c t i o n

f

we s e t

I f w(6) < c.6', then f i s s a i d t o be i n L i p a. Zygmund says t h e f o l l o w i n g : Suppose t h a t a f u n c t i o n f has t h e F o u r i e r s e r i e s expansion

1 f = -a 2 0 Then

m

m

+ q a n c o s nx + q b n s i n nx. n: n=

163


Take a t o be ( 1 / 2 ) - ~ and compare t h e above r e s u l t t o t h e u n i f o r m c o n t i n u i t y o f a sample f u n c t i o n e s t a b l i s h e d i n §1.2. A ) . One w i l l , then, be a b l e t o see some i d e a c o n c e r n i n g t h e r e g u l a r i t y o f sample f u n c t i o n s o f a Brownian motion. CHAPTER 2.

White Noise and t h e H i 1b e r t space

( L2) .

F u n c t i o n a l s o f a Brownian m o t i o n { B ( t ) } , c a l l them Brownian f u n c t i o n a l s , can be d e a l t w i t h more e a s i l y i f t h e y a r e expressed as f u n c t i o n a l s o f t h e w h i t e n o i s e d {b(t)}, b(t) 5 B ( t ) . F o r one t h i n g , { b ( t ) } i s viewed as a c o n t i n u o u s analogue o f Gaussian p r i c e f o r t h e use o f it process -indeed, sample f u n c t i o n s a r e

i . i . d . random v a r i a b l e s . We s h a l l however have t o pay a i s no more an o r d i n a r y s t o c h a s t i c i t , because b(t) i s a g e n e r a l i z e d s t o c h a s t i c process i n t h e sense t h a t t h e a l l generalized functions.

T h i s c h a p t e r i s t h e r e f o r e devoted t o a q u i c k r e v i e w o f t h e t h e o r y o f p r o b a b i l i t y d i s t r i b u t i o n s o f g e n e r a l i z e d s t o c h a s t i c processes, and f u r t h e r p r o v i d e s a 2 d i s c u s s i o n on t h e H i l b e r t space ( L ) c o n s i s t i n g o f Brownian f u n c t i o n a l s w i t h 2 f i n i t e v a r i a n c e , i n p a r t i c u l a r t h e s o - c a l l e d W i e n e r - I t 6 decomposition o f ( L ) . These serve a l s o as background t o what w i l l be c o n s i d e r e d i n t h e f o l l o w i n g chapters. 2.1.

White n o i s e . 1 { B ( t ) ; t c R 1 be a s t a n d a r d Brownian motion. The w h i t e n o i s e 1 { b ( t ) ; t c R 1 i s o b t a i n e d by t a k i n g t h e t i m e d e r i v a t i v e o f Brownian m o t i o n : d B ( t ) , where t h e d e r i v a t i v e i s d e f i n e d i n t h e sense o f g e n e r a l i z e d b(t) : Let

f u n c t i o n ; namely t h e v a l u e

B(c)

evaluated a t a t e s t f u n c t i o n

~ ( s ) ( w )E b ( 5 ) = -[B(t)f,'(t)dt,

6

C J

Proposition.

i) E

iii)

(b(5); 5

b(5)

J}

i s continuous and l i n e a r i n

W)

5 f o r almost a l l

b(6)

w

.

i s a Gaussian system.

The c h a r a c t e r i s t i c f u n c t i o n o f t h e random v a r i a b l e

Remark.

i s g i v e n by

( t h e Schwartz space).

(As was n o t e d b e f o r e we o m i t t h e p r o b a b i 1it y paramer

ii)

5

8(c)

is

i s o f t e n w r i t t e n as

and i t i s a Wiener i n t e g r a l o f

5

w i t h r e s p e c t t o t h e random measure

dB(t).

The f i r s t t h i n g f o r us t o do i s t o e s t a b l i s h t h e p r o b a b i l i t y d i s t r i b u t i o n o f white noise an i d e a .

{ b ( c ) ; 5 c J I .The f o l l o w i n g f i n i t e dimensional analogue g i v e s us

164

T.Hfda Random v e c t o r 3

( X1,

X2

,..., Xn)

C h a r a c t e r i s t i c f u n c t i o n q ( z l y z 2 y . - - ,zn) = E ( e rzkXk)

F : ( P ( z ~ , z ~ , - - - , z ~= )

Distribution function

6

e iezkxkdF(x)

(The Bochner Theorem) For t h e case o f w h i t e n o i s e we must have System

{B(c); c

E

dl C(6) = E(eib(e))

Characteristic functional Distribution

p

: C(6) =

= e

-; l c112

fii<x’e>d~(x).

Now we s h o u l d n o t e t h a t t h e i n t e r g r a n d o f t h e l a s t i n t e g r a l i n v o l v e s t h e ! and $*. T h i s suggests t o us t h a t canonical b i l i n e a r form < , > t h a t l i n k s C t h e p r o b a b i l i t y d i s t r i b u t i o n p i s t o be i n t r o d u c e d t o t h e d u a l space $* o f t h e Schwartz space $, t h r o u g h which r u n s t h e v a r i a b l e 5 o f t h e c h a r a c t e r i s t i c f u n c t i o n a l . We s h o u l d t h e r e f o r e f i x t h e measurable space o f t h e f o r m ( J * ) B ) on which t h e measure p i s i n t r o d u c e d . I t i s q u i t e r e a s o n a b l e t o t a k e 8 t o b e t h e s i g m a - f i e l d generated by t h e c y l i n d e r subsets o f d*, s i n c e a l l t h e v a r i a b l e s B(6) a r e i n t e r e s t e d f o r us. With t h i s measurable space ( $ * , B ) we a r e now ready t o s t a t e t h e Bochner-Minlos theorem which guarantees t h e e x i s t e n c e o f t h e p r o b a b i l i t y d i s t r i b u t i o n on o u r space.

2.2.

The Bochner-Minlos Theorem.

The theorem i n q u e s t i o n can be a p p l i e d n o t o n l y t o t h e case o f w h i t e n o i s e b u t a l s o t o more general c h a r a c t e r i s t i c f u n c t i o n a l s . Even t h e b a s i c space can be g e n e r a l i z e d t o a n u c l e a r space, however, t o f i x t h e i d e a we t a k e t h e Schwartz space $. T h i s space i s t o p o l o g i z e d b y an i n c r e a s i n g system o f H i l b e r t i a n norms 2 1 { n t 01, b e i n g t h e L (R )-norm. The c o m p l e t i o n o f d w i t h r e s p e c t t o t h e norm w i l l be denoted by Jn. We f o r m t h e d u a l space o f dn,

(1 (In;

keeping

Theorem 2.1. i) ii) iii)

11 [lo

11 lln 11 [lo t o be t h e b a s i c norm,

If a functional

I( Ilp-continuous

f o r some

C(c) p

>

and denote by J;

on

or

$,.

satisfies

0,

positive definite, C ( 0 ) = 1,

t h e n t h e r e e x i s t s a p r o b a b i l i t y measure

p

on

(J*, 63)

such t h a t

Thus we have

165

Ci~lculusof Brownian functionals

Moreover if t h e i n j e c t i o n :T type, then

:

J',

i s supported by $;

1-1

-P

Jp

E

J-n.

w i t h some

n > p

i s o f Hilbert-Schmidt

For t h e p r o o f o f t h i s theorem we r e f e r t o t h e book I . M. G e l ' f a n d and N. Ya. V i l e n k i n , G e n e r a l i z e d f u n c t i o n s , v o l . 4 (1964), o r t o t h e book [2] Chapter 3. Examples. 2

-?- 115tI' = e

: white noise w i t h variance

u

2

.

(1)

C(c)

(2) (3)

gives the p r o b a b i l i t y d i s t r i b u t i o n o f C'(c) = e More general case :

-i1I~'1l2

d2 B ( t ) = +(t) dt

where m(c) i s t h e mean f u n c t i o n a l and K ( c , 11) i s t h e c o v a r i a n c e f u n c t i o n a l . O f course m and K a r e assumed t o be continuous on and J x $, r e s p e c t i v e ly. 2 1 E x e r c i s e 3. Suppose t h a t m and K i n Example 3) be c o n t i n u o u s on L (R ) 2 1 2 1 and L ( R ) 8 L ( R ) . Then, can one say t h a t t h e a s s o c i a t e d p r o b a b i l i t y d i s t r i b u t i o n i s supported by t h e space $-,? Coming back t o t h e w h i t e noise, we b r i e f l y mention some o f i t s b a s i c p r o p e r t i e s . a ) White n o i s e has independent values a t e v e r y moment, which means t h a t i f c l ( t ) c 2 ( t ) 5 0, then

i n o t h e r words,

<x,

el>

and

<x,

c2>

a r e m u t u a l l y independent.

b ) White n o i s e i s a s t a t i o n a r y g e n e r a l i z e d s t o c h a s t i c process. T h i s mean t h a t t h e p r o b a b i l i t y d i s t r i b u t i o n i s i n v a r i a n t under t h e s h i f t o f t i m e . We can paraphrase t h i s p r o p e r t y i n terms o f c h a r a c t e r i s t i c f u n c t i o n a l : I f St denotes t h e o p e r a t o r on J g i v e n by

then we have

C(Stc)

C(c)

f o r any

t.

c ) White n o i s e has a f l a t spectrum. We a r e i n t e r e s t e d i n t h e s p e c t r a l decompos i t i o n o f b ( t ) o r o f a sample f u n c t i o n x, s i n c e w h i t e n o i s e i s s t a t i o n a r y . Let

be t h e F o u r i e r t r a n s f o r m o f

5.

Then t h e F o u r i e r t r a n s f o r m

of

x

E

d*

166

T. Hida

i s defined v i a

The variance o f

< i ,F,>

2 11511 , which i s expressed i n the form

is

2 L

l i ( h ) I2dh,

which t e l l s us t h a t the spectrum i s f l a t . n

2.3.

The H i l b e r t space

(LL).

F i t h t h e measure p each member x o f J* i s v i p e d as a sample f u n c t i o n o f B, s i n c e u i s the p r o b a b i l i t y d i s t r i b u t i o n o f { B ( t ) l . Hence, a f u n c t i o n ~ ( x ) o f x i s thought o f as a f u n c t i o n a l o f Brownian motion (or o f w h i t e n o i s e ) . We a r e thus given a r e a l i z a t i o n of Brownian f u n c t i o n a l s on t h e measure space ( J * , B , P). Set

; complex-valued, B-measurable and square

integrable w i t h respect t o For any f i x e d <x, f> f o r

5 , <x, p 2 1 f E L (R ) .

i s an example o f

u

( L2 ) - f u n c t i o n a l .

I t extends t o

The c o l l e c t i o n

forms a Gaussian system. The c l o s e d l i n e a r span ( w i t h complex c o e f f i c i e n t s ) o f 2 i s a subspace o f ( L ), c a l l i t g1,

9

The f o l l o w i n g a s s e r t i o n s a r e s t r a i g h t f o r w a r d . Proposition.

i ) Polynomials i n

<x, p ' s

are a l l l i v e i n

c o l l e c t i o n o f a l l such polynomials forms a dense subset o f i i ) Exponential f u n c t i o n s o f t h e form

forms an algebra, which i s dense i n 2.4. <x,

(7)

cj >Is

o f t h e form c

TT Hn.(<x, J

Moreover t h e

(L'). 2 ( L ).

The Wiener-It6 decomposition o f

Take a complete orthonormal system

2

(L ), 2 (L ).

2

1

1 6 . 1 i n L ( R ) , and form polynomials i n J

cj>/n),c

a constant.

Calculus of Brownian fwnctionals

167

Such a polynomial i s c a l l e d t h e Fo u r i e r - H e r m i t e polynomial based on { 5 . 1 , and J z n i s t h e degree o f t h e p o l y n o m i a l . Two Fourier-Hermit e polynomials a r e j J 2 orth ogonal i n ( L ) i f a s s o c i a t e d f i n i t e sequences ( n . 1 ’ ~ a r e d i f f e r e n t , i n J p a r t i c u l a r degrees a r e d i f f e r e n t . L e t An be t h e subspace o f

2 (L )

spanned b y t h e F ourier-Hermit e polynomials o f

I t i s easy t o see t h a t t h e subspace A,,

degree n. choic e o f

{c,.).

i s independent o f t h e

J 2 We have t h us o bt a i n e d t h e Wi e n e r - I t 6 decomposition o f ( L ) : 2 Theorem 2.2. The H i l b e r t space ( L ) a d m i t s a d i r e c t sum decomposition :

Definition.

A member o f

4 i s c a l l e d a m u l t i p l e Wiener i n t e g r a l

o f degree

n.

E xe rc is e 4.

Use t h e P r o p o s i t i o n i ) i n 52.3 t o prove t h a t t h e .c o l l e c t i o n o f a l l We can even t h e F o urie r-H ermit e p o l y n o m i a l s spans a dense subspace o f (L’). choose t h e F ourie r - H e r m i t e p o l y n o m i a l s based on a f i x e d 1 6 . 1 t o have t h e same J conc lus ion. CHAPTER 3.

The 3 - t r a n s f o r m .

I n t e g r a l R e p r e sent at ion.

To develop o u r a n a l y s i s we s h a l l use two b a s i c t o o l s ; one i s t h e 7 - t r a n s f o r m and t h e o t h e r i s t h e i n f i n i t e dimensional r o t a t i o n group. I n t hese l e c t u r e s t h e former w i l l p l a y a dominant r o l e , w h i l e t h e l a t t e r w i l l be touched upon o n l y briefly. 3.1.

I n t e g r a l representation.

7:

Introduce t h e transformation

and s e t

Obvious ly lF

i s a v e c t o r space.

Proposition.

Vector spaces

IF

and

( L2 )

a r e isomorphic under

7.

F u r t h e r t h e space lF i s made t o be t h e RKHS (Reproducing k e r n e l H i l b e r t space) w i t h k e r n e l C(6 - n), ( 5 , n) E Rxd. I n f a c t , we can prove t h e f o l l o w i n g Theorem 3.1. matio n

Let

IF

be t h e RKHS w i t h k e r n e l

g i v e s an isomorphism

C(s

-

0).

Then, t h e t r a n s f o r -

T.Hi&

168

Example 1.

Take a u n i t vector

q,

l r l j = 1, from L 2 (R1 ) .

And s e t

Then we have

and hence

under

7. Take a Fourier-Hermite polynomial based on

Example 2.

In.} J

:

Then, we have

where

n = z k . j

Note.

and F

i s the symmetrization o f

J

By the symmetrization o f

(n!)-lX(un(l), n

u ~ ( ~ ) , - * *un(,,)I, ,

F(ul.

u 2 , - - - , un)

we mean t h e f u n c t i o n

where the sum extends over a l l permutations o f

169

Calculus of Brownian functfonals

(1, 2,..., n). A function symmetri z a t i on.

i s s a i d t o be s y m n e t r i c ifi t agrees w i t h i t s

F

We i n t r o d u c e a n o t a t i o n .

--

e n L (R

L ~ ( R " ) ; F i s symmetric)

{F

Based on t h e above two examples we can prove o u r main theorem : Theorem

i) For cp

3.2.

E

4

there e x i s t s a function

F

E

5'n L (R )

such t h a t

is one- to-one. ii)

Definition.

The r e p r e s e n t a t i o n o f cp

t h e i n t e g r a l r e p r e s e n t a t i o n , and

e s t a b l i s h e d i n t h e above theorem i s c a l l e d

F i s c a l l e d the kernel o f the representation.

I t i s not e d t h a t , so f a r as cp i s r e s t r i c t e d t o An, we a r e g i v e n a represen2 t a t i o n o f cp i n terms o f L - f u n c t i o n o v e r f i n i t e dimensional, i n f a c t n - d i mensional, E uc lidea n space.

For a general

cp

in

(L 2 ) we f i r s t have t h e expansion

cp = C c p n y

n

and t h en have a sequence o f k e r n e l s

I n view o f t h i s we may w r i t e

cpn

Ednd,,

T. Hida

170

(L')

@

=

mL / 2 \ (nR ) .

n=

An a l t e r n a t i v e approach t o t h e i n t e g r a l r e p r e s e n t a t i o n i s as f o l l o w s : L e t t h e T a y l o r s e r i e s expansion o f

Since each

Cn(&, n)

C(s

-

11)

be

i s p o s i t i v e d e f i n i t e , t h e r e i s a RKHS IFn w i t h k e r n e l

Noting t h a t

(Cn(*.

rll),

n2)&

c,,(-,

= 0,

m # n,

(

being the s c a l a r product i n

we have t h e d i r e c t sum decomposition o f

F

:

Apply 7-l t o o b t a i n

(L2)

=x@bi, w i t h

7-'(IFn)

n

Addendum. For cp

3 -transform

(due t o Kubo and Takenaka)

€ A n we have (Tcp)(s)= C ( C ) i n U ( s ) .

For a general

cp =z(pn,

n

we d e f i n e

Set

=An,

IF,

Cn.

Cnlculus of Brownian functionals Proposition.

171

We have

(5cp) ( 5

(11)

= C(6)

s

e<Xy5’cp(x)du(x).

Use t h e f o r m u l a ( 1 ) t o o b t a i n t h e

E x e r c i s e 1.

4- t r a n s f o r m

o f exponential

f u n c t i o n discussed i n Example 1 i n 53.1. 3.2.

S t o c h a s t i c processes.

We a r e i n t e r e s t e d i n s t a t i o n a r y s t o c h a s t i c processes w h i c h a r e always i n and share t h e t i m e - s h i f t w i t h w h i t e n o i s e . Let

{St}

-

be t h e s h i f t on

g i v e n i n 52.2 : ( S t c ) ( u ) = ~ ( u t ) .

The

2

(L ) St

f o r m an a b e l i a n group :

S t Ss The

ITt; t

E

R 1 1 w i t h Tt

= S;

syt

= St+sy

i s a f l o w on

E

R1

.

@*,@,

u ) , namely

parameter group o f p-measure p r e s e r v i n g t r a n s f o r m a t i o n on c a l l e d t h e f l o w o f Brownian m o t i o n . Introduce

Ut,

A*.

i t i s a one-

It i s often

a u n i t a r y o p e r a t o r , by

(Utcp)(x) = cp(Ttx),

x

EL?*,

We are, then, g i v e n a one-parameter u n i t a r y group Set

cp

E

{Ut;

(L2).

t

E

1 R 1 a c t i n g on

2 (L ).

1 2 Then { X ( t ) , t E R 1 i s a s t a t i o n a r y s t o c h a s t i c process l i y i n g i n t h e (L ) . It should be n o t e d t h a t I X ( t ) l shares t h e t i m e - s h i f t w i t h {B(t)} (or x € A * ) . T h i s means t h a t { X ( t ) } i s viewed as a mathematical model d e s c r i b i n g a n o n l i n e a r d e v i c e h a v i n g w h i t e n o i s e i n p u t , as i s i l l u s t r a t e d b y F i g . 3.

Fig. 3

I n o r d e r t o analyse such a s t o c h a s t i c process we s h a l l make use o f t h e i n t e g r a l representation established i n the l a s t section together w i t h the properties o f t h e f l o w o f Brownian motion.

T.Hida

172

Let

Pn

be t h e o r t h o g o n a l p r o j e c t i o n down t o t h e subspace

P

u

Then we have

UtP,.

n t

We t h e r e f o r e have a decomposition o f

4 ' ,.

X(t) :

Thus, { X n ( t ) , , n t 01, i s a f a m i l y o f m u t u a l l y o r t h o g o n a l s t a t i o n a r y s t o c h a s t i c {Xn(t); n

processes

2

0). Each process

{Xn(t))

t i e s t h a t can be expresses i n terms o f t h e Flow

enjoys t h e important properITt}.

Set

Then we have i) ii)

Bt

3

BS,

V Bt

for

t > s,

By

t

iii) T t q = Proposition.

We have

I n view o f t h i s p r o p o s i t i o n t o g e t h e r w i t h t h e above p r o p e r t i e s i )- iii)t h e f l o w I T t ) i s s a i d t o be a Kolmogorov f l o w , and B0 i s t a k e n t o b e a g e n e r a t o r o f t h e

at.

The general t h e o r y o f Kolmogorov f l o w t e l l s us t h a t { T t } has a c o u n t a b l y 2 ( L ) e {ll;i n p a r t i c u l a r , i t i s e r g o d i c .

Lebesgue spectrum on t h e space C o r o l l a r y 1. i)

L:

ii)

V

Lt2 -- L 2 @, dt, P )

I f we s e t

i s monotone i n c r e a s i n g i n L t = (L'),

t

(L/:

, then

we have

t,

t h e l a t t i c sum)

t

i i i ) n L z = 111. t

Note.

The same p r o p e r t i e s h o l d f o r 14n,t

= 1%

2

n Lt.

Calculus of Brownian functionals Corollary 2. istic.

I f cp

2 La f o r some a, then

E

Namely, i f S t ( X )

173

X(t) = U p

i s p u r e l y non-determin-

denotes the sigma-field generated by

{X(s); s s t},

then we have

L i n gn,then t h e kernel

Proposition. I f cp i s a member o f representation s a t i s f i e s

un) = 0, when some

F(u1,.--,

Proposition.

Let F(ul,**-,

the kernel f o r

Proposition.

I f cp i s i n

i i ) L e t F and Ft respectively. Then

Theorem 3.3. PnX(t, x ) . i) ii)

Let

>

0.

be the kernel associated w i t h

(0

in

'.InThen .

is

Utcp

i)

un)

u j

F o f the i n t e g r a l

16, so

i s the c o n d i t i o n a l expectation

be the kernels associated w i t h cp and

{X(t, x ) , a t r

t

E

R}

be an

E(cp/\Lt).

E(cp/JSt),

2 (L )-martingale.

Set

Xn(t,

x) =

Then

{Xn(t, x ) ,

at,

t

E

1 R 1

There e x i s t s a f u n c t i o n

i s a martingale and i t i s purely non-deterministic. Fn

such t h a t the kernel associated w i t h

Xn(t,

x)

i s given by

The proofs o f the above pro o s i t i o n s and the theorem are n o t so d i f f i c u l t ; however, i f necessary, see !2] Chapter 4.

3.3.

Stochastic i n t e g r a l s .

This section aims a t g i v i n g the d e f i n i t i o n o f the I t 6 type s t o c h a s t i c i n t e g r a l i n a general case and discussing a p p l i c a t i o n s t o the case o f white noise. Therefore, the p r o b a b i l i t y space i s , a t f i r s t , taken t o be a general (a, B, P). Let a Brownian motion { B ( t , w ) , t

E

[0, 111 be given on s s.

Bs the o - f i e l d generated by the B(u), u

(n, B, P).

Denote by

T. Hi&

174

Let At' A.l) A.2)

At At

t 3

E

Bt and

[0, 11, be an increasing f a m i l y o f

o - f i e l d s such t h a t

f o r every t, {B(t+s) - B ( t ) , s t 01 are mutually independent.

Integrand f ( t , W ) i s assumed t o s a t i s f y f.1) f ( t , W ) i s B(R) xB-measurable, f.2) f ( t , .) i s At-measurable f o r any t, 1 f.3) f(t, w)2dt < a.e. 0

-

1

Under these assumptions the s t o c h a s t i c i n t e g r a l

i s defined. Proposition.

For constants

x1

and

The i n t e g r a l

i s continuous i n

t

f o r almost a l l

W.

We can a l s o t h i n k o f the s t o c h a s t i c i n t e g r a l o f the form

m

Ifi n p a r t i c u l a r

f2du]
01 of Markovian kernels is said to be a Wtion function if (2.1.1) PtPSU = Pt+sU’ t, s > 0 , lJ 6 % (2.1.2) pO(x,B) = tix(B) &-measure (2.1.3) lim ptu(x) = u(x) x 6 X, u Co(X). t+O The equation (2.1.1) is called the Chapman-Kolmogorov equation. A transition function pt is called conservative if pt(x,X) = 1. t > 0 , x € X , namely, pt(x,.) is a probability measure. A transition function can always be made to be a conservative transition function on an enlarged space XA = X U A by setting x 4 X, Pt(X,X) = P~(X, B - (A}) (1 - ptl(x))~IA~(B), Pt(A,B) = &{A(B) Usually A is added as a point at infinity if X is non-compact and as an extra point if X is compact. We may think of a transition function pt(x,B) as the probability of finding a moving particle in the set B at time t under the condition that it starts from x at time 0. Then A may well be said to be a death point.

-

+

A transition function can be viewed as a collection of linear operators on the space o f bounded measurable functions. The property (2.1.1) is nothing but the semigroup property of this collection. invariant, then If each p t makes the space Cw(X) the transition function Ep,, t > 01 is said to be a Feller transition function o r Feller semigroup. In general, a collection of linear operators I Tt, t > 01 on a Banach space V is called a strongly continuous contraction semigroup if TtTs = Tt+s, t,s > 0, llTtlI 2 1, IITtu - ull -+ 0 , t f 0,II 1 I being the Banach norm. Then the infinitesimal generator A of {Tt, t > 0 ) is defined by Ttu - u (2.1.4) Au = lim t+O with domain D(A) = {u 6 V : the limit in (2.1.4) exists in the sense of II 1 I 1. The semigroup property and (2.1.4) formally lead us to the equation a (Tp) (2.1.5) - A(Tt4 at which may be solved symbolically as Ttu = exp(tA)u. The HilleYosida theorem concerns necessary and sufficient conditions for a linear (but unbounded generally) operator A on V to be a generator of a unique strongly continuous contraction semigroup on

V. In particular, a Feller transition function {p , t > 01 is a strongly continuous contraction semigroup on th& space Cm(X) endowed with the supremum norm Ibll, = sup Iu(x)I . In fact, X€X


191

[btullm llulJ is trivial. (2.1.3) implies the weak continuity lim c v , ptu - u> = 0 for u 6 C m ( X ) and any bounded signed measure v , which in turn means the strong continuity of p t’ In this case, the Hille-Yosida theorem mentioned above can be stated as follows : a linear operator A on the space Cw(X) is the generator of a Feller transition function if and only if (A. 1) D(A) is dense in C m ( X ) , (A. 2) if u E Cm(X) takes a maximum at x = x0’ then Au(xo) 6 0, (A. 3) for any X > 0 and v 6 C m ( X ) , the equation ( A - A)u = v has a solution u e D(A). Example 1. Let X = Rd the d-dimensional Euclidean space. Suppose a transition function pt is conservative and spatially d homogeneous in the sense that p t (x,B) = pt(x+y, B+y), y e R . Then pt is Feller because of the expression ptu(x) = For instance {Rd Pt(O,dy)u(x+y)

-

@

(2.1.6) pt(x,dy) = (27~t)d/ 2 exp( )dy is such a transition function. Let A be the generator on Cm(Rd) of the transition function (2.1.6) and let L be a linear operator defined bv 2 ‘ d Lu = 1 Au ( = a D(L) = C O2 ( R d ) , 2 i=l 1 2 where Co is the space of twice continuously differentiable functions. Using the Taylor expansion of u(x+y) around x, it is then easy to see (2.1.7) L C A. In this sense, the equation (2.1.5) can be interpreted as the heat equation. A related transition function on ( 0 , ~ )defined by 2 2 - (2~t)-~/’exp(- *)I dy pt(x,dy) = {(Z~t)-~/’exp(- k$-is still Feller but non-conservative : pt(x, (0,m)) < 1, x > 0. In this case also Co((O,m)) 2 C D(A) and Au = 1 u ” , u E. C 2o ( ( O , m ) ) .

7 ’ 9

It is in general hard to check directly the Feller property for non-spatially homogeneous transition functions. Moreover, as is illustrated in the above example, an explicit expression of the generator A is available only on a subset of the domain D(A). What we practically know is that A is a closed extension of some concretely given linear operator L with domain D(L) consisting of smooth functions. Kolmogorov [9] first presented a typical expression o f the operator L as an elliptic differential operator: d 2 d (2.1.8) Lu = a. .(x) a ‘I + xbi(x) + c(x)u i,j=l 1J axiaxj i=l axl ~

(Caij(x)titj 0 , c(x) o 1. He actually proved that, underdsome regularity conditions for the transition function pt on R , the parabolic differential equation

M. Fukushima

192

must hold with

L

of ( 2 . 1 . 8 ) .

A fundamental question arises : under what conditions on the coefficients aij, bi, c of the operator L, does the equation (2.1.9) admits a transition function as a solution ? This question has been answered directly or indirectly in several ways. One is a probabilistic way of solving the associated stochastic differential equations ( [ 8 ] - , [ 1 2 ] ) which has been quite successiful especially in the multidimensional cases. But we do not go into this direction now. The second way is to rely upon the theory of The third is the way of partial differential equations ( [ 4 ] ) . functional analysis yo construct, by taking a closed extension of L, an operator A = L satisfying the Hille-Yosida conditions(A.l), This method was successful in one-dimensional (A.2) and ( A . 3 ) . case but rather hard to be used in the multidimensional situations. From now, we like to present another way of a functional analytical character, namely, an L2-theoretic method rather than the C-theory. Since the main concern of probabilists is not a transition function but a stochastic process behind it called a Markov process, the method of SDE is in a sense more favorable than the analytical methods. But LZ-theoretic methods provides us with many Markov processes which can not be covered by the SDE method. Furthermore, as the title of the historical paper of Kolmogorov suggested, the theory of Markov processes can not get rid of the influence of the analysis on the state space X. Instead we make full use of a relevent functional analysis, that is , the theory of Dirichlet form. An L'-theoretic approach toward transition functions is based on the following considrations. Denote by m a positive Radon measure on X such that m is positive on each non-empty open set. L 2 (X;m) stands for the L 2 -space of m-square integrable functions on X with inner product ( u , v) = {Xu(x)v(x)m(dx). Let pt be a transition function. m is said to be an excessive measure with respect to pt if

lx

m(dx) pt(x,B) 2 m(B),

B C @(XI.

Lemma 1 If m is excessive with respect to a transition function p,, then {pt, t > 0 ) determines uniquely a strongly continuous Tt is Markovian in the semigroup {Tt, t > 01 on L 2 (X;m). sense that 0 2 Ttu 6 1 m-a.e. follows from 0 6 u 1, u G L 2 (X;m). Proof By Schwarz inequality, we have for u 6 C o ( X ) , (ptu(x)) 2 2 2 I ptl(x) ptu (XI 5 ptu' (XI; and consequently (ptu, ptu) 6 2 Hence pt extends from Co(X) to ptu (x)m(dx) 6 ( u , u ) . L (X;m) as a unique linear operator Tt with llTtllLZ 2 1.

In

Semigroup property

Tt are clear.

TtTS = Tt+s and the Markovian property of each The strong continuity of Tt follows from


(ptU - u,PtU - u) 2.2 -

2 {(u,u) - (

u , ptu))

-+

t+O

0, u B C,(X).

193

q.e.d.

Dirichlet forms and symmetric Markov processes

We say that a transition function

jX

pt

is m-symmetric if

jx

(2.2.1) P,U(X) v(x)m(dx) = u(x) ptv(x)m(dx), u , v E @+. Evidently m is then an excessive measure of pt and , according to the preceding lemma, an m-symmetric transition function gives rise to a unique strongly continuous contraction semigroup {Tt, t > 0 ) on L 2 (X;m). In this case, Tt is not only Markovian but also a symmetric operator on L 2 (X;m). Such a semigroup is completely characterized by the notion of the Dirichlet form. In this subsection, we quickly mention relevent basic notions and relations by refering the readers to [S] for further details. densely defined non-negative definite symmetric bilinear form E on L2(X;m) is simply said to be a symmetric form. Its domain is We let denoted by D [ E ] . E a ( u , v) = E(u, v) + a ( u , v), U, v 6 D[E]. If D[E] is comlete with respect to the El-metric, then a symmetric We say that a symmetric form E on form E is called closed. L2(X;m) is Markovian if, for any E > 0 , there exists a real func t ion $,It), t 6 R 1 , satifying

A

& $,(t) 0 2 $,(t) such that u 6 D[E] (2.2.2)

-E

& 1

$,(t) = t

+E,

& t

- $,(t')

implies v

- t' =

$,(u)

for t e [0,11, for t' < t, E D[E] and E(v,v) , I E(u,u).

We call a Markovian closed symmetric form on L 2 (x;m) a Dirichlet form. A Dirichlet form E on L2(X;m) is in one-to-one correspondence to a strongly continuous contraction semigroup {Tt,t > 01 of Markovian symmetric operators on Lz(X,m). The correspondence is given by D[E]

=

(2.2.3) E(u,v)

=

1

( u , u - T u) < m ) t t+O 1 u , v 6 D[E]. lim -t- ( u , v - Ttv),

{u C L'

: lim

t +o

We have seen that any m-symmetric transition function gives rise to such a semigroup. We call the associated Dirichlet form the A natural question Dirichlet form of the transition function. arises : conversely, given any Dirichlet form, is it the Dirichlet form of some m-symmetric transition function ? Exam le 2 The transition function (2.1.6) on Rd is symmetric tce*w to the Lebesgue measure. Its Dirichlet form E on L2(Rd) is given by D[E] = H 1 (Rd ) (2.2.4) Vu Vv dx E(u, v) =

IRd

M. Fukushima

194

where H 1 (Rd) is the Sobolev space of order 1 : H 1 (Rd) = {u 6 L 2 : au Indeed, by taking the Fourier transform axi C L 2 , 1 g i d l . 2 of u 6 L and using the formula ( 2 . 2 . 3 ) , we easily get ( 2 . 2 . 4 ) . The Markovian property of the form ( 2 . 2 . 4 ) can directly be seen as We may assume that u is absofollows. Take u G H1(Rd). lutely continuous on almost all lines parallel to the axes. Take for any E > 0 a C1-function $,(t) satisfying ( 2 . 2 . 2 ) . Then

jRd \ v Q E ( U > l dx

=

'I,

7 d

\VU\

2

which means the Markovian property of

\@;(U)\2

dx & E ( u ,

U),

E.

Given a transition function p on X, one can construct a Markov f rocess governed by pt, namel?, there exists a collection o xA with state space M = { Q , Q, Xt, PX} b t i c processes X,

such that for each

x h X

PX -a.s. Px(Xt+s E A l a t ) = P,(X~,A) (2.2.6) Px(Xo = x) = 1 where A is any element of @(X) and Dt denotes a sub a-field of @ making Xs, s 6 t, measurable. Suppose that the sample path X.(W) is right continuous and has left limits PX -a.s. for each x X. We then say that M has the strong Markov property if, for some right continuous o-fields {C$)t>o, equation ( 2 . 2 . 5 ) holds with time t being replaced by any @}-stopping time. If in addition almost every sample path is left continuous along any increasing sequence of stoppint times, then M is said to be a h ; ; process: In particular a strong Markov process whose sample is continuous up to the death time ~ ( w ) = inf {t > 0 : X,(W) Px-a.s., x f X, is called a diffusion (process). A) (2.2.5)

In regard to the question raised before Example 2 , we introduce two notions about Dirichlet forms. A Dirichlet form E on L 2 ( X ; m ) is called regular if the space D[E]nC,(X) is dense both in D[E] and in Co(X) E is said to be local if E(u, v ) vanishes whenever u, V C D[E] have disjoint compact support. Now the answer ty the above question is as follows : if a Dirichlet form E on L (X;m) is regular, then E admits a Hunt process, namely, E is the Dirichlet form of the m-symmetric transition function of some Hunt process M. The associated Hunt process M is unique in a certain sense. In this case, M can be taken to be a diffusion if and only if E is local.

.

Those strong statements would become void if there were not practical means of constructing regular local Dirichlet forms from some concrete data. Fortunately we possess the following useful assertions : if a symmetric form E with D[E]_dense in Co is Markovian and closable (that is, E admits a closed extension), then the closure E is a regular Dirichlet form. If D[E] satisfies some additional mild condition and if E is local, then E as also local.


Consider a non-negative function p t LioC(Rd). 1 d) , E(u, v) = lRd VUVV p dx, D[E] = CO(R

Example 3 (2.2.7)

195

Then

$

defines a Markovian local symmetric form on L2(Rd;pdx). If either (2.2.8) inf p(x) > 0 for any compact set K C Rd, x K or 2 d) , (2.2.9) the distribution derivatives Dx P are in Lloc(R i then E is closable on L2 (Rd ;pdx). Then the closure E' becomes a local regular Diri hlet form and consequently admits a diffusion process Mp on R .

s

A recent paper [ 7 ] concerns the conditions for the function p that the associated diffusion Mp is mutually absolutely continuous with respect to the d-dimensional Brownian motion. Brownian motion is by definition a diffusion process with transition function (2.1.6). In other words,Brownian motion is associated with the Dirichlet form (2.2.4) . Therefore it is a version o f the diffusion MD for p = 1. so

53

CAPACITY

3.1 Capacitary potentials and hitting probabilities -

Given a regular Dirichlet form E on L 2 (X;m), we define the associated capacity Cap(A) for an open set A C X by 1 m-a.e.on A}. Cap(A) = inf E1(u,u), where LA = { u E D[E]: u u6 LA We then extend Cap(A) for any set A C X as the outer capacity yielding a Choquet capacity (see subsection 1.2). The notions !!q.e.'! and !'quasi-continuous''will be used with respect to this When E is transient (or equivalently the as ciated capacity. Hunt process is transient), then the 0-order capacity Cap( 69 (A) can be introduced in the same way as above but by replacing El and D[E] with E and the extended Dirichlet space respectively. This replacement does not change the notions I'q.el' and "quasicontinuous". Cap(O)(A) related to the Dirichlet form (2 2 . 4 ) for d = 3 coincides with classical Newtonian capacity of A C Rs up to a multiplicative constant. Hence there The set L is closed and convex for open A C X. exists a uhique element eA in LA minimizing E1(u,u), which is said to be the e uilibrium potential or the capacitary potential of It then h:lds that 0 6 eA & 1 and A. (3.1.1)

Cap(A)

=

El(eA, eA). be a Hunt process associated with the

regular Dirichlet form E. probability pA of A by

We defige the (1-order) hitting

-U.

(3.1.2) where Ex ure Px.

pA(x)

=

Ex( e A ; uA
0 : Xt(u)EA}

is the expectation with respect to the probability meas-

M. Fukushima

196

Theorem 1

For an open set A

with

LA 9

4 , pA

is a version of

eA' Proof Let ITt, t > 01 be the L 2 -semigroup corresponding to E and M. Then both eA and pA are 1-supermedian : e-tTteA 6 -t It suffices to show the inequality eA3 TtPA & PA' (3*1.3)

PA

6

eA,

because we can then apply formula (2.2.3) to get pA c L and (here we use the fact that eA 6 AL 2 ) , which E1(pA,pA) 9 El(eA,eA) implies PA

=

eA'

To see (3.1.3), take a Borel version eA of eA and let h be any h(x)m(dx) = 1. non-negative function with Let Yt(w) = e-teA(Xt ( w ) ) , then (yt, a t , Ph.m) tBo is easily seen to be a supermartingale : E~.,,(Y~~@) 6 y S phmm-a.s., s < t. Choose any finite set D C ( 0 , m ) with min D = a, max D = b and let a(D ;A) = min {t 6 D : Xt 6 A). The optional sampling theorem for the supermartingale yields -u(D;A) : a ( D : A ) < b, & Eh.m(Ya(D:A)) 6 Eh.m(Ya) 6 ( h , eA)* Eh .m(e By making D increase to a dense subset of (0, b) and b + m, we get (3.1.3). q.e.d.

Ix

When E i transient, we may consider the 0-order capacitary (Opotf, ntial e(if o f an open set A along with Cap(O)(A). e A 2 in the extended Dirichlet space but may not belong to L ( X ; m ) . But we still have the identity (3.1.4) pio) is a version of eA( 0 ) , In fact we get where p p ) = Px( aA < m) . same way as above and hence the inequality

pA('1

eA( O )

in the

p p ) I\ aRaf 6 eiO)fi aRaf , a > 0 , f continuous 2 0, where Ra is the resolvent of pt. Since both hand sides are a-supermedian, we obtain the equality by the same argument as in the m. above proof. It now suffices to let a -f

Theorem 1 enables us to identify diverse potential theoretic notions relevent to the Dirichlet form E with those relevent to the Hunt process M. Here are some of the ikentifications : (3.1.A) Cap(N) = 0, N C X for some Borel N 2 N, PhSm(ur <m) = O e f o r some Borel N > N, m($ = 0 and for any x E x - N, pX( xt or xt- (i 5 for some t 2 0 ) = 0 . (3.1.B) An open J. Cap(An) J. 0 Px(lim aA < 5 ) = 0 q.e. x. n n L.


197

(3.1.C) A function u locally in D[E] is quasi-continuous if and only if u(Xt) is right continuous in t 2 0 and lim u(Xt,) t'+t = u(Xt-), t > 0, Px-a.s. for q.e. x 6 X. "Only if" part of (3.1.C) is a direct consequence of (3.1.B). "If" part can be shown by using a fact that any function u locally in D[E] (i.e., f o r any relatively compact open set G there exists v 6 D[E] sucJh tkat u = v m-a.e. on G ) admits a quasicontinuous version u : u is quasi-continuous and F = u m-a.e. 3.2 -

Tests of irreducibility and attainability

Let E and M be as in the preceding subsection. We further assume that E is local and consequently M is a diffusion. A Borel set B is called T t -invariant if TtIB = IBTt, namely, Tt(IBu) = IB - Ttu for any u 6 Lz, and t > 0. B is said to be M-invariant if Px( ux-B < m) = 0 for any x B. We say that an increasing sequence {FnI of closed sets is a nest if Cap(X-Fn) -+ 0. Following Lejan[ll],we call a Borel set B uasi-o en (resp. quasi-closed) if there exists a nest {F,) s u c W n F n is open (resp. closed) in Fn for each n. F o r Borel sets B , u and B, B is said to be a modification of B if m ( E e B) = 0. Theorem 2 Following conditions are equivalent for a Borel set B. B is Tt -invariant. (i) (ii) u E D[E] IB.u E D[E]. (iii) IB is locally in D[E]. (iv) IE is quasi-continuous f o r some modification B of B. (v) B is quasi-open and quasi-closed f o r some modification of B. (vi) X can be decomposed as X = B1 + BZ + N where Bl(resp. B2) is a modification o f B (resp. X-B), both B1 and B2 are M-invariant and m(N) = 0.

-

h.

Remark 1 Condition (v) is the same as saying that there exists a nest {Fn} such that B n F n and Fn - B are closed for each n. This simplifies corresponding statements in [l]. The set N in condition (vi) is o f zero capacity according to ( 3 . 1 . A ) . Proof

(i)=)(ii)

:

o f identity { E A , X > 01

D[E]

= {U

L2 : 2

jmA 0

Tt and D[E] on L2(X;m)

d(EXu, u)

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