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Title

The structure of quasi-parameter ideals and

their associated graded rings

Author(s)

松岡,直之

Citation

URL

http://hdl.handle.net/10291/11036

Rights

Issue Date

2008

Text version

ETD

Type

Thesis or Dissertation

DOI

生ψ／一・9・・2＞

The structure of quasi−parameter ideals

  and七heir associated graded rings

 （擬i巴系イデアルとその随伴次数環の構造解析）

指導教員 後藤 四郎教授

学位請求者 基礎理工学専攻数学系

      松岡 直之

Iwould like to express my profbund respect and gratitude to my supervisor Professor Shiro Goto of Meiji University fbr his warm guidance and support during the preparation of my thesis．

Ialso thank my co−workers， Ryo Takahashi of Shinshu University， Satoru Kimura of

Meiji University， and Tran Thi Phuong of Ton Duc Thang University， fbr their discus− sions and suggestions． Finally， I wish to express my thank to my family for their support and trust．

CONTENTS

Acknowledgement

Preface

10n m−full powers of parameter ideals

  1．1  1ntroduction ．．．．．．．．．．．．．．．．．   1．2 Preliminaries ．．．．．．．．．．．．．．．．   1．3 Proofs of Theorem 1，1．1 and Corollary 1．1．2

2 Quasi−socle ideals in a Gorenstein local ring

  2．1  1ntroduction ．．．．．．．．．．．．．．．．．．   2．2 Preliminaries ．．．．．．．．．．．．．．．．．   2．3 Proof of Theorem 2．1．1．．．．．．．．．．．．   2．4 Examples ．．．．．．．．．．．．．．．．．．．

3 Quasi−socle ideals in Gorenstein numerical semigroup rings

3．1  1ntroduction ．．．．．．．．．．．．．．． 3．2 The main result and the proof．．．．． 3．3 The case where H＝〈α，α十1＞．．．．．

3．4 Examples ．．．＿．．．．．．．．．．．

・       g       o       ・       o       ●       ●       ●       ・       ・       ●       ●       o o       ●       o       ■       ●       9       ■       ，       ，       ●       ●       ・       ● ●       ●       o       o       ●       ●       ●       ■       ●       ●       ・       ，       ・ ●       o       ●       ●       ●       ●       ●       ・       ●       ●       ○       ・       ．

4 Quasi−socle ideals in regular local rings

  4．1  1ntroduction ．．．．。．．．，．．．，．．．．．．．。．．．．   4．2 Proof of Theorem 4．1．1．．．．，．．．．．．．．．．．．．．   4．3 Proof of Theorem 4．1．2−Gorensteinness in G（1）and R（1）

  4．4When 1＝Q？．．，．．．．．．．．・・・・・・・・・・・…

●−

V

−占−占9召4占

  1⊥−←り乙

5 Quasi−socle id

5．1 5．2 5。3

        eals in local rings with Gorenstein tangent cones

Introduction ．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．． Proof of Theorems 5．1．1 and 5．1．2．．．．．。．．．．．．．．．．．．．．．

Examples and apPlications．・・・・・・…  ，・・・・・・・・・・・…

5．3．1 The case where A＝k［［ta，tb】］ ．．．．．＿．．．．．．．．＿．．

5．3．2 The case where A＝RM ．，．．．．．．．．．．．．．．．．．．．．

References

67

PREFACE

This thesis deals with the Cohen−Macaulayness and／or Gorensteinness of the graded

rings associated to some special kind of ideals in Noetherian local rings． First of all， let me state necessary notations． Let A be a commutative local ring with the maximal ideal m and I be an ideal in A． We put π（1） R’（1） G（1） F（1） and call them the Rees algebra， ノ1［ltl⊆A［t］， R（1）［ガ1］⊆A［ちt−1］， π’（1）／オー1π’（1），αnd R（1）／mR（1）望G（1）／mG（1） ring， and the丘ber cone of I， respectively． structure of these graded rings associated to ideals over I if there exists an equation

the extended Rees algebra， the associated graded

       In my researches I study the ring−theoretic       ．Let x∈A． We say that x is integral

♂＋c、xn−1＋…＋Cn＝0

in A with n＞Oand c乞∈Iz． We put 1＝＝｛X∈AIXiS integral OVer l｝ and call it the integral closure of the ideal I． We naturally have I⊆Iand we say that Iis integrally closed if the equality 1ニIholds true．    My researches are originated at the researches［G2］and［GSh2］of S． Goto and Y． Shimoda． The paper［G2］gave a characterization of a regular local ring in terms of the integral closedness of a parameter ideal． The paper［GSh2］showed that the structure of the Rees algebras R（Q）of parameter ideals Q re且ects the structure of the base ring

・A・These researches［G2］and［GSh2］suggest that the analysis of parameter ideals Q

V

1eads that the structure of the base ri孕9 A・．．The purpose of my reserches is to answer

the problem what hapPens when”parameter ideals”is replaced with”quasi−parameter

ideals”，here”quasi−parameter ideals”mean ideals constructed by a parameter ideal Q （e．g． the powers Qn of（2， Q：m， Q：m2， etc＿）． I think that quasi．parameter ideals have more infbrmations of the structure of the base ring than parameter ideals． So， the main purpose is to construct a structure theory of quasi−parameter ideals（especially， ring−theoretic structures of the graded rings associated to quasi−parameter ideals）・    In Chapter 1， I consider the powers of a parameter ideal Q in a Noetherian Iocal ring、4． The purpose of Chapter l is， by using the method of m−fulhdeals， to generalize the result of［G2］as Corollary 1．1．2． The method is a theorem of the following fbrm． Theorem 1．1．1． Let Aわεαノ＞oetherian locα1 ring with the maximal ideα1 mαnd let（？ 6eαpαγ「αmeter ideαl in／1．ノlssume thαt depth／1＞0αn（i Qn is m−full for some integer n≧1．Then・the l・cα‘渤g A乞8 Teg漁丁απd m／Q is cyclic．   Corollary 1．1．2 has been known if、A is excellent and n is su伍ciently large（［MTV， Th60r益me 3］）or if depth、A＞0（［HUV， Corollary 2．11］）， but the method of my proof is different from theirs and provides a new sight of m−full powers of parameter ideals．    Chapters 2，3，4， and 5 aim at studies of quasi−socle ideals which are ideals of the fbrm Q：mq where q is a positive integer． Now， let me here explain the reason why I am interested in quasi−socle ideals． Let A be a Noetherian local ring with the maximal

ideal m and d＝dim、A＞0． Let Q be a parameter ideal in A and q be a positive

integer． Then we put I＝Q：mq and refer to those ideals as quasi−socle ideals of Q． When q＝1， especially， we call 1 ・ Q：m the socle ideal of Q． The purpose of Chapter 2，3，4，and 5 is to solve the following problems．

Problem A．

（1）Find the conditions under which I⊆Q，

Q． where Q stands fbr the integral closure of

（2）When 1⊆Q， estimate or describe the reduction number

・Qの一mi・｛0≦n∈ZIIn＋’＝Qln｝

of 1 with respect to Q in terms of some invariants of Q or A・

（3）Clarify what kind of ring−theoretic properties of the graded rings associated to the ideal I

         即）一㊥・n，G（・）一㊥・n／・n＋1， and F（・）−e・n／m・n

      n＞O      n＞O      n＞O enJoy・    Befbre going ahead， let me note results on the socle ideals Q：m． The study of the socle ideals Q：m． dates back to the research of L． Burch［B］， where she explored socle ideals of finite projective dimension and gave a very nice characterization of regular local rings（cf．［GH1， Theorem 1．1］）． More recently， A． Corso and C． Polini［CP1， CP2］ showed， with the interaction to linkage theory of ideals， the fbllowing beautiful theorem． Theorem B（［CHV， CP1， CP2， CPV， G2］）． Let（A，厩）δeαOoんeη．ハ4acaulay locα1 r吻

ω伽d＝dim・4＞0．五et QδeαPαrameter ideαl in・4αnd let 1＝Q：m， Then the

μ0ω吻three conditions are equivαlent to eαCんothεr．

（1）12≠QI．

（2） （2＝（2，thαt is the parameter ideαl Q i8 integrαlly closed in／4。 （3） 0・n8equθntly， if（A， rrし）is a O・んen− QI for every pαrαmeter ideα1 Q rings， d＝ dim A＞2． A is a regular local r吻ωんich c・ntains a regulαr system Xi，ω2，…，Xd・f param− eter8・sucんthαt Q＝＠・，…，xd．1，娼）！・r s・me integer・q＞0，       ルfαCα駕吻ZOCαZ禰9ωんich is not regular， then 12＝       伽、4，30tんαt G（∫）and F（1）αrθboth Oohen−Mαcαulαy

ωhere 1＝Q：m． The Rees吻ebra R（∫）isα18・αOoんen−Mαcαulay r吻，ザ

  The main theorem of Chapter 2（Theorem 2．1．1）is a natural generalization of The− orem B． On the other hand， there might be other directions of generalization． In fact，

the equality I2＝QI in Theorem B remains true in certain cases， even though the

base local rings、4 are not Cohen−Macaulay． For example， S． Goto and H． Sakurai investigated the case where、4 is a Buchsbaum local ring and gave the fbllowing． See ［GSa1， GSa3］fbr further developme耳ts of this direction．

Theorem C（［GSa1， GSa3］， c￡ ［GN］）．五θオ（A，m）beαBuchsbαum locαl r吻αnd

αs8ume thαtθ励θr dim、4≧20r dim A＝1but eX（A）≧2． Then tんere existsαη傭eger

n＞O such・tんαt for every parameter ideα1 Q of A which is contained in m”， one・hαs・tんe eqtiαlity I2＝QI，30 thαt the grαded rings G（1）αnd F（1）αre Bucんsbαum r乞η95，ωんere

∫＝Q：m．

   However， a more important thing is the fbllowing． If J is an equimultiple

Cohen−Macaulay ideal of reduction number one， the associated graded ring G（」）＝ ㊥。〉。」n／Jn＋1 i・aC・h・n−M・・aul・y・i・g and，・・i・th・Ree・alg・b・a即）一㊥。〉。」・， provided htA J≧2． One also knows the number and degrees of the de丘ning equations ofフ宅（」）， so that one can understand fairly explicitly the process of desingularization of Spec、4 along the subscheme V（」）． This observation motivated the ingenious research of C． Polini and B． Ulrich［PU］， where they posed， among many important results， the fbllowing conjecture．

Conjecture D（［PUD． Let（A，m）beαOoんθη一ハ4acαulαy local ringω伽dim、4≧2．

As8ume thαt dimA≧3ωんen．A i8 regulαr． Let q≧2わθan integer and let Q beα

Pαrαmeter ideαl in・4 sucん彦んαt Q⊆mq． Then

Q：mq⊆mq．

  This conjecture was recently settled by H．−J． Wang［Wan］，whose beautiful theorem says： Theorem E（［Wan］）． Let（A， m）δeαOoんen−Mαcαulay locα1 r吻ωi彦んd＝dim A≧2． 五et q≧16e an integerαnd QαPαrαme彦er i（ieαl in A． A8sume tんαt Q⊆mqαnd put

I＝Q：m9． Then

      ∫⊆mσ，mql＝mqQ， and I2＝Q∫， provi〔ied thαt／4 is not regulαr｛ゲd≧2 αn（i tんαt 9≧2 if『（i≧3。    Wang，s Theorem E is certainly closely related to results in Chapter 2，3，4， and 5． In the introduction of each chapter， the relation between Wang，s Theorem E and my results is described．    In Chapter 2， we will study about quasi−socle ideals in the case where the base local ring、4 is Gorenstein and g＝2． The main purpose of Chapter 2 is to prove the fbllowing

theorem：

Theorem 2．1．1．ゐεオ（A，m）δθα（］orenstein locα1 r吻ωith dim／A＞0αnd assume thαt ・焦（A）≧3，ωh・r・ eX（A）d…te8・th・認鋤吻・ノ肋ith・r・・pect・t・tん・m・xim・1・id・αl m・Then for every parαmeter ideα1 Q in A，・ne・hαS・tんe f・ll・ω吻，ωんere∬＝Q・m2．

（1）m21＝m2Q and 13＝α2，

 （2）7「んθαssociαted grαded ring G（1）（ザ∫and tんe flber coηe F（1）（ゾ1αre both Ooんen−     Mαcαulay r吻5． Hencθ，抗e Reesαlgebra R（1）o／1￠5αlsoαOoんen−Mαcαulαy ring，げdim．4≧3．    In Chapter 3， we will compute some special kind of quasi−socle ideals in numerical

semigroup rings which are typical examples of one−dimensional Cohen−Macaulay local

rings． Let O＜α1＜α2＜…  ＜αe be positive integers and set H＝〈α1，α2，…  ，αe＞the numerical semigroup generated byα1，α2，…，αe． We put       k［［H］］＝k［［tal，舌α2ヂ・・，tαt］］⊆k［［オ］］ where k［剛is the fbrmal powers series ring over a field k． The main result of Chapter 3is the fbllowing．

Theorem 3．2．1．3卯po8ε伽オ．4＝k［［珂］is a Goren8tein ring． Let q＞0δeαη翻θgeT

αnd・a8sume thα彦孟んe fo〃oω吻伽coη伽・ns（01）αnd（02）αre sαtisfied for q，ωんere c＝c（H）’ （01）tn∈mq／br all integers n≧cノ （02）五et・n∈H． Then・n＜α1（q−1）， if t” ￠ mq−1． Leオ0〈5∈H。1ンet Q＝（が）αn（i l＝Q：mq． Then the folloωin9α58ertion8んold true・

（1）mσ1＝mqQ and Q∩12＝QI。

      ■  （2） 12＝QI，霊ブ3≧c．

 （3）13＝Q∫2and theαssociαted grαded r伽g G（1）＝e．＞o∬n／ln＋1 is Oohen−

     Mαcaulay， if s≧α1（q−1）．

   In Section 30f Chapter 3， we will consider special numerical semigroups∬＝

〈α，α十1＞generated byα，α十1fbrα＞Oand give some remarkable results on quasi−

socle ideals in k［［H］］．    Chapter 4 is a starting point of．the research of Chapter 5． In Chapter 4， we will study what happen on quasi−socle ideals in a regular local ring． Let、4 be a regular local ring with the maximal ideal m and d＝dim、4＞0． Let xi，x2，… ，xd be a regular system of parameters in A Letα1，α2，…，αd， and q be positive integers． We put

Qと＠li，錫2，…，場d）and∫＝Q：m9． We then have the following theorems which

are the main results in Chapter 4．

Theorem 4．1．1． The folloω吻伽eεconditionsαre equivαlent toεαcんother．

（1）1⊆Q・

 （2） mq∫＝：mgQ・ （3）e≧αi！bγα”1≦乞≦d・ 肺・・伽・漁・Cα8e・th・f・ll・ω吻α88e・伽・ん・ld t・u・・

（i）・・（・）−mi・｛n∈Zil≦n｝・H・・ce1≦・・（・）≦d−・・

 （ii）The graded rings G（∫）， R（1）， and F（1）areαll Ooんen−Mαcαulαy rings． Theorem 4．1．2． Suppose孟んαt e≧αi for a〃1≦i≦d． Thenωe hαve tんeメo〃oω吻．   （i）G（1）isα（］oren8tein ring if and onlyげ引q・  （ii）フ≧（1）isα（］orenstein ring ifαnd onlyげ（1ニ（d−2）乏・    The main purpose of Chapter 4 is to prove this theorems． In Section 40f Chapter 4，we consider about a question of when I＝Q：nτq are integrally closed and give an answer to this question．    Chapter 5 is a continuation result of Chapter 3 and 4． The main results of Chapter 5is the following． Theorem 5．1．1． Tんε∫o〃oω吻伽ee cond伽nsαre equivαlent toεαcんotんer・

（1）1⊆Q．

（2）m9∫＝mqQ．

（3）e≧αi fc）r・all・1≦乞≦d．

陥eη齢乞3孟んθCα8θ，the fo〃0ω吻α8sertionsんoZd蜘ε．

  （i）rQ（∫）＝「多1：＝min｛n∈Zl多≦n｝・   （ii）Tんe grαded rings G（1）αnd F（1）αre Oohen一ハ4acαulαy． Theorem 5．1．2． SupPose that e≧αi／bTα〃1≦i≦d． Then weんαve the folloω吻・   （i）G（∫）isαGorenstein ring if and onty if乏iq，

（ii）フ己（1）isαGorenstθin ringげand only勾F 9＝（d−2）乏・   In Chapter 5， we will prove these theorems and explore a few examples in order to see how Theorems 5．1．1 and 5．1．2 work fbr the analysis of concrete examples．   The results in Chapter l are already published in the paper［Ma］． The researches in Chapter 2 are joint works［GMT］with S． Goto and R． Takahashi， which has been accepted for publication in Journal of Pure and Applied Algebra． The researches in

Chapter 3（resp． Chapter 5）are joint works［GKM］（resp．［GKMP］）with S． Goto and

S．Kimura（resp． S． Goto， S． Kimura， and T． T． Phuong）， which are already submitted fbr publication． The results in Chapter 4， which are joint works with S． Goto， can be seen as a typical example of the results in Chapter 5． But this is a starting point of the research of Chapter 5， so I shall state these results independently・

November，2007

NAoYuKI MATsuoKA

CHAPTER 1

ON m−FuLL PowERs oF PARAMETER IDEALs

1．1

Introduction

Let、4 be a Noetherian local ring with the maximal ideal m and d＝dim／L Let I be an ideal in．4． Then we say that l is m−full， if鳳1：x＝Ifbr some x∈rn． The notion of m−full ideal was introduced by D。 Rees and played， since integrally closed ideals are m−full under a certain mild condition（［G2， Theorem（2．4）D， an important role in the analysis of integrally closed ideals（cf［G2， GH1， GH2， GHK， HUV， MTV］）．   The present purpose is to prove the fbllowing． Theorem 1．1．1．五et・A be a N・ether伽Z・cα」航9ω乞出んe maximα1 ideα1 m and let Q bθαpαrameter i（1θαl in／1．、48sume tんαt depth／4＞0αηd Qn is m−full foγ゜30mε乞ηオegεγ゜ η≧1．Then・the l・cα1・r吻A is regular and rn／Q is cyclic・   This theorem provides a new sight of m−full powers of parameter ideals and gives rise to a suHiciently simple proof of the fbllowing result， which has been known if A is excellent andηis suHiciently large（［MTV， Th60r6me 3］）or if depth．4＞0（［HUV， Corollary 2．11］）． Corollary 1．12．五e乱4わeαNoetheriαn locαl ringω伽tんe mαximαl ideα1 mαnd let Q わεαP・rαmeter ideα1・in・A． Then・th・μ・ω吻tん・εθC・顧伽・αr・e卿α1・醐・・α・ん otんer． （1）A is a regular l・cα1 ring and m／Q is cyclic・ （2）（［2 i8 integrally closed in A・ （3）Q” is integrally cl・sed・in・A！bT 8・mθn≧1・

When・this i・tんθcα3θ， th・id・als Qe are i吻rall〃・」・8θ4乞一抄α〃鶴・9・rs・e≧1．   In Corollary 1．1．20ur contribution is the implication（3）⇒（2）；the equivalence of conditions（1）and（2）is due to［G2， Theorem（3．1）］as well as the last assertion． Thus， as fbr the parameter ideals Q in a Noetherian local ring・4， the integral closedness of αny power of Q implies that ofαJJ the powers of Q and the regularity of、4 as well．   Aglobal version of Corollary 1．1．2 is as follows． We suspect that the assumption AssA・4／1＝MinA・A／1 in condition（2）of Proposition 1．1．3 is super旦uous． Proposition 1．1．3． Let／1δθαハ「oetheriαn ring． Let Iδe an ideat in、4αndαssume tんαtμA（∬）＝htA I，ωんereμ蓋（1）αη（オhtA∫denotθtんe numbeγ・6ゾgenerαtorsαnd tんe んeight of l， respectively， Then tんe fo〃oω吻conditions are equivαlent。 （1）1 is integrally closed in A． （2）AssA．4／1＝MinA．4／I and ln is integrαlly closed in．4 for some integer n≧1． 陥eη伽5ゴ3診んecα3e， le is integrally cl・sed in A for every integer e≧1．   The proof of Theorem 1．1．1 and Corollary 1．1．2 shall be given in Section 3． Section 2is devoted to some preliminaries． In our proof of Theorem 1．1．1， some results on Ratliff−Rush closures， FLC rings（that is， generalized Cohen−Macaulay local rings），and m−full ideals will play key roles， which we will briefly summarize in Section 2．    In what fbllows， unless otherwise specified， let．4 be a Noetherian local ring with the maximal ideal m and d＝dim A． LetμA（＊）and eA（＊）denote the number of generators and the length， respectively． For each ideal I in A let htA（1）be the height of 1． We denote by e（A）＝ek（A）the multiplicity oL4 with respect to the maximal ideal m． Let Hk（＊）（i∈Z）stand fbr the i Lh local cohomology functor of、4 with respect to m．

1．2

Preliminaries

Let、A be a commutative Noetherian ring and let ．7’A denote the set of ideals in．A which contain at least one nonzerodivisor in A． For each I∈JE’` let       アーU（ln＋1・ln）        n≧0        At  え be the Ratliff−Rush closure of 1． Then 1⊆1⊆1 and 1＝1（cf．［Mc， Lemma 8．2（vi）D， where I denotes the integral closure of J，

proposition 12．1． （1）（Y． Shimoda）Lθt I⊆」6εideαls in．4 and assume thαt

    In＝Jn！bγ’80mε integθrη≧ 1． Then Ie＝Je！bγ・all integer8 e≧η．

（2）Let・1∈」rcA・nd・ssum・・th・t・ln＝万力・5・me・n≧1． Th，n・7＝ア。nd le ＝ 7e

   プbr all integers e≧n．

1）ro（’f，（1）Since In⊆In−1」⊆Jn−1」＝Jn， we get In＝In−IJ＝Jn． Therefbre

In＋1＝Iln＝1（ln−1」）＝In J＝Jn JニJn＋1． Thus Ie＝Je fbr all乏＞n．

  （2）We have 1”＝ア， sinceア⊆万＝In． Hence 7⊆アby［Mc， Lemma 8．2（iv）］so

       that 1＝∫ by［Mc， Lemma 8・2（vi）］・The latter equality fbllows from assertion（1）．［コ   Now let A be a Noetherian local ring with the maximal ideal m and d ＝ dim A． Let H鉛（＊）（i∈Z）be the local cohomology functors of．A with respect to m． Then we say that A has FLC（or equivalently， A is a generalized Cohen−Macaulay local ring）， if all the local cohomology modules H振（A）（i≠のare finitely generated．   Foreach ideal I in A we put π（1）−A［∬オ］一㊥ln，        n≧0 where t denotes an indeterminate． Let       G（1）一π（1）／∫即）一㊥ln／ln＋1        n≧O be the associated graded ring of I．

  Let G＝G（m）and M＝（］＋the unique graded maximal ideal in G， Let e（A）＝

eX（A）denote the multiplicity of A． We then have the following． Proposition 1．2．2． （1）Suppose thαt、4んα8 F1｝0． Then．4 isαregulαr locαl ring，ザ     e（A）＝1 αn（オdepth／1＞0． （2） 7「んelocα1γ・ing／4 んα3 Fl乙（り㌧ げtゐθlocal cohomology modules        H聾（G）一無E・tむ（σ／Mn，G） ・f G with respect t・M are finitely generated f・r・all・i≠d． Proof．（1）The local ring A is unmixed， since A has FLC and depth．A＞0（cf． ApPendix， Proposition 16］）。 Hence A is regular by［N， Theorem 40．6］．   （2）See［G1， Proposition（3．1）］． ［SV， 口

  The notion of m．−full ideal was introduced by D． Rees， who showed that every in． tegrally closed ideal I is m−full， provided I is not nilpotent and the residue class field ・tl．／m of A is infinite（［G2， Theorem（2・4）D・The readers may consult［G2］about basic results on m−full ideals． Here let us note two of them， which we later need to prove Theorem 1．1．1． proposition 1．2．3． Let．τbeαn ideαl in、4αnd assume thαt I i8 m一ノ勧”． Then the

μoω吻αssertions・hold・true・

 （1）Let J be an i（ieαl in A・Assume 1⊆Jan（オ4A（」／1）〈Oo． ThenμA（1）≧μA（」）．  （2） ノlssume that／4／I isαn／lrtinian （］orenstein locαl ring． Then m／I is cyclic。 Pro〔）f，（1）See［G2， Lemma（2．2）（2）］．

   （2）Let x∈msuch that m∫：x＝1． Then l：m＝（ml：x）：m＝（ml：m）：x⊇1：

xso that I：m＝1：x． Thus we get the exact sequence

      O→［1：m］／1→．4／1−9＞A／1→・4／［1十＠）］→0． Hence eAし4／［∬十（x）］）＝eA（［1：m］／1）＝1，because、4／I is Gorenstein． Thus m＝1十（勾， whence m／I is cyclic．       口

1．3

_{Proofs of Theorem 1．1．1 and Corollary 1．1．2}

Let A be a Noetherian local ring with the maximal ideal m and d＝dim．4． Let Q be aparameter ideal in．4． Pro（ザo／Theorern 1ユ．1。 Passing to the local ring．4［X］mA［x］where X is an indetermi− nate over．4， we may assume that the residue class且eld k＝、4／m of、4 is infinite． Let qbe a minimal reduction of m． Hence q is a parameter ideal iL4 and mT＋1＝qm「for some r≧0（such an ideal q must exist， because the field k＝．4／m is infinite）， and then R（m）is a module−finite extension of R（q）， Let gフ：17ヒ（q）／m71（q）→フ1（m）／m7己（m） be the homomorphism of graded k−algebras induced from the inclusion 7it（q）⊆R（m）， Hence the homomorphism 9 is also finite． We put

P＝＝R（q）／m7Z（q）and G＝R（m）／mlR（m）．

For each integer i≧Olet pi＝qi／mqi and Gi＝mi／mi＋1 denote the homogeneous

components of P and G of degree i． Then， because q is a parameter ideal in A， the ring P is the polynomial ring with d variables over the field k・Therefbre g is a

monomorphism， sinceψis finite and dim．P＝dim G＝d．

  We now look at the m− full ideal Qn， Then        μ・（m・）≦μ・（Q・）一（噛1）一μ・（q・） by Proposition 12．3（1）， so that the monomorphism g induces an isomorphism

1『h＝qn／mqn→（］n＝mn／mn＋1

of vector spaces over k． Hence mη＝qn十mn＋1 and so mn＝qn by Nakayama，s lemma． Thus mゑ＝qe for all integers e≧n by Proposition 1．2．1（1）． Hence the homomorphism

g：P→G induces an isomorphism between the vector spaces Pe and Ge over k and so

      e・（m・／me＋・）一 e・（q・／mqe）一幡1）

for every 4≧n． Thus e（A）＝ek（A）＝1by definition． Let O ＝Coker（ρ． Then

dimle C＜Oo， since Ce＝（0）if e≧n． Therefore， because the ring P is the polynomial ring over k， thanks to the exact sequence       O→P珍σ→0→0 of finitely generated graded P−modules， we get HMθ）＝（0）fbr all i≠0，d， where M＝σ＋，Hence by Proposition 1．2．2（2）the local ring、4 has FLC， so that the local ring A is regular by Proposition 1．2．2（1）， because e（A）＝1and depth／1＞0．    Since A／Q is an Artinian Gorenstein local ring， to see that m／Q is cyclic， by

Proposition 1．2．3（2）it is enough to show that Q is m−full． Let x∈msuch that

mQn：x＝Qn and letα∈mQ：x． Then x（αQn−1）＝（xα）Qn−1⊆mQn whence

αQn−i⊆mQn：x＝Qn． Therefbreα∈Qn：Qn−1＝Q， because Q is generated by an

A−regular sequence． Thus mQ：x＝’Q and so Q is m．−full．       口   We are in a position to prove Corollary 1．12． The last assertion and the equivalence of conditions（1）and（2）in Corollary 1．1．2 are due to［G2， Theorem（3．1）］． Let us include brief proofs of the last assertion and the implication（2）⇒（1）for the sake of

completeness．

Proof（）f Corollαry 1．1．2． We may assume that d＝dim A＞0． Passing to the local ring 川X］mA［x］where X is an indeterminate over A we may also assume that the residue class field k＝ノ1／m． of／1 is infinite・    （3）⇒（2）We will show that A is a regular local ring and （？ is integrally closed． Let VI！＝HX（A），β＝．4／W， and n＝m／W． Then QnB is integrally closed in B， because

W⊆〉便可and Qn is integrally closed in A Hence by Theorem（1．1）the local ring

Bis regular， because depth B＞Oand Q”B is n−full． We must show that W＝（0）．

Since W⊆＞E（可， we have W⊆び＝Qn． Let乏＞Obe an integer and assume that

W⊆Qe． L・t Q−（α1，・，，…，・、）and・h…ew∈W． W・w・it・ω一Σ・。aαwith

       國＝e        d ・。∈A，・wh・・e a・一・7・α9・…・：d and・1・・1一Σα・ f・・ea・hα一（α・，α・，…，αのwith       iニユ

0≦α￠∈Z．Let莱denote the image in B． Then

Σ可犀1げ2…曜d−th−0・

1α1 ＝e

Since the system可，砺，…，砺of parameters in B forms a regular sequence， we get

iiiEg∈QB for everyα＝（α1，α2，… ，αd）with lαi＝e． Thus cα∈Q十Wso that

ω∈（Qe＋1＋llVQe）∩W＝（Qe＋1∩W）＋WQe． Consequently， W＝Qe＋1∩Vゾ⊆Qe＋1

by N・k・y・m・’・1・mm・． H・nce・W⊆∩Qe−（0）and・・th・1・・al・i・g A・i・ ・eg・1…        e＞0           Because Qn is integrally closed， we haveσ＝Qby Proposition 1．2．1（2）． The ideal Qis generated by a regular sequence， whence

Q−U（Qn＋1・α）−Q，

    n≧0        AV

so that we have Q＝Q＝Q．

   （2）⇒（1）Thanks to the above proof，〆1 is a regular local ring・Hence rn／Q is cyclic by Proposition 1．2．3（2）， because Q is m−full and／1／Q is an Artinian Gorenstein local ring．    （1）⇒the last assertion． We may assume that d≧2。 Since m／Q is cyclic by our assumption， we may choose a regular systemα1，α2，…，αd of parameters oL4 so that Q＝（α1，…，αd＿1，α匿）f（）rsome q≧1・Let 8＝π（Q）［孟一1］−A［α1ち…，αd．、ち・匿ちオー1］

be the extended Rees algebra of Q（here t denotes an indeterminate over・4）． Let

u＝t−1．Then G（Q）＝8／u8． We must show that 8 is an integrally closed integral

domain． We且rstly recall that the sequenceα1，…，αd−1，αl is regular・Hence the

associated graded ring G（Q）is the polynomial ring with d indeterminates over A／Q， that is        G（Q）＝（A／（［2）［∼冠ヂ・・，ad＿lt，αZt］ and the elements｛αit｝1〈iくd＿1 andαZt are algebraically independent over A／Q， where 9 denotes the image in G（Q）， In particular， the ring G（Q）is Cohen−Macaulay． Hence the ring 5 is also Cohen−Macaulay， because u is a nonzerodivisor in 5．    Let P be a prime ideal in 5 with ht5 P＝1． We will show that the localization

8p of 8 is a discrete valuation ring． We may assume that u∈P（because the ring

8［u−1］＝、4［t，t−1］is regular）． Then P＝（m， u）8＝（α1，α2，… ，αd，u）5， since P／2L8 is a unique minimal prime ideal in the polynomial ring G（Q）． Hence P＝（αd，u）5， because

αi＝αit・u fbr all 1≦i≦d−1． Therefbre P5p＝ad5p， because a匿t≠P＝（m， u）5

and・・一 ﾋ・H・nce 8・i・adi・c・et・valuati・n・i・g with th・・eg・1・・p・・am・terα・・Th・・ the Cohen−Macaulay ring 5 satis丘es Serre，s condition（R1）， so that 5 is an integrally closed integral domain． Hence Qn is integrally closed in A fbr every integer n≧1，

which completes the proof of Corollary 1．1．2。       口

  Before closing this chapter let us note a brief proof of Proposition 1．1．3． We suspect the assumption that AssA．A／1＝MinA 4／I in condition（2）is superfluous． Proof of Proposition 1．1．3，（1）⇒（2）and the last assertion． This is due to［G2， Theo− rem（1．1）］．

   （2）⇒（1）Assume that I≠Iand choose P∈AssA 1／1． Then P∈AssA．A／1＝

MinA A／1． Hence the ideal I 4p is a parameter ideal in the local ring、4p， because htA I＝μA（1）． Since

（IAp）n＝1”Ap＝1”Ap ＝＝ lnAp＝（IAp）n，

by Corollary 1．1．2 the local ring Ap is regular and L4p ：14p＝IAp． This is

impossible．      口

CHAPTER 2

   QuAsI−socLE IDEALs IN

AGoRENsTEIN LocAL RING

2．1

Introduction

The purpose of this chapter is to prove the fbllowing theorem． Theorem 2．1．1． Let（A，m）わεα（］orenstein locα1 r吻ω痂dim！1＞0αnd assume tんαt eX（ノ4）≧3，ωんeγ・e eg（ノ4） denotes tんεmultipticity qノノ4ωith respect to孟んe mαximαl ideαl m・Then f・r every Pαrαmeter ideα1 Q in A，・ne hαS the f・〃・ω吻，ωんθrε1＝Q：m2，

（1）m21＝m2Q and 13＝QI2．

 （2）Theαssociαted grαded ring G（1）oノ∫and the fiber cone F（∫）oノ∫are both Oohen−

   Mαcaulay r吻8，

Hence， the Reesαlgebrα R（∫）o／Iis alsoαOoんθη一ハ4acαulαy ring，ガdim．4≧3． Here we define R（1） π’

ﾌ

G（1） F（1） A［IT］ ⊆A［T］， A［IT， T−1］ ⊆A［T， T−1】， R’（1）／T−1フ1’（1），and R（．τ）／mフこ（1）（窪G（1）／mG（1）） with T an indeterminate over A．   Our Theorem 2．1．1 is a natural generalization of Theorem B， which is due to A． Corso， C． Polini， C． Huneke， W． V． Vasconcelos， and S． Goto．

   Here let me explain the conhections between Theorem 2．1．1 and Wang’s Theorem E．Wang，s result is certainly closely related to Theorem 2．1．1， although Theorem E， apparently， does not cover Theorem 2．1．1． The two researches were perfbrmed indepen− dently and our proof of method is， heavily depending on the facts that the base ring A is Gorenstein， q＝2， and eX（A）≧3， totally different from Wang，s method， and despite the restrictions， Theorem 2．1．1 holds true fbr every parameter ideal Q in．4， even in the case where dim A＝1． For this reason， Theorem 2．1．1 may have its own significance， suggesting a possible modification of the Polini−Ulrich conjecture．   We now explain how this chapter is organized． Section 2 is devoted to some pre− liminary steps， which we will need later to prove Theorem 2．1．1． Theorem 2．1．1 will be proven in Section 3． Our method of proof is， unfortunately， applicable only to the case where the local ring．4 is Gorenstein and the situation seems totally different， unless A is Gorenstein． In order to show that the non−Gorenstein case of dimension l is rather wild， we shall explore three examples ill the last Section 4． One of them will show the quasi−socle ideals I＝Q：m2 are never integral over parameter ideals Q in certain Cohen−Macaulay local rings A of dimension 1， even though e9（A）≧2． The other two will show that unless A is a Gorenstein ring， one can not expect that rQ（1）≦2， even if I is integral over Q．    Unless otherwise specified， in what follows， let（A， m）be a Gorenstein local ring with dim A＝d． We denote by eX（A）the multiplicity of A with respect to the maximal ideal m． Let Q＝（α1，α2，…，αd）be a parameter ideal in．4 generated by the system

α1，α2，…，αdof parameters iL4． For each丘nitely generated A−module M we denote

byμA（M）and 4A（M）， respectively， the number of elements in a minimal system of generators fbr M and the length of M． Let v（A）＝乏A（m／m2）stand f（）r the embedding dimension of．4．

2．2

Preliminaries

Let A be a Gorenstein local ring witれthe maximal ideal m． The purpose of this section is to summarize some preliminaries， which we need in Section 3 to prove Theorem 2ユ．1． Let us begin with the case where dimA＝0．

   Suppose that dimA＝0． Let n＝v（A）＞Oand let xl，x2，…，xn be a system of

generators for m． We choose a socle element z in A． Hence O≠z∈m． and mz＝（0）． Let 1＝（0）：m2， We then have the following．

Lemma 2・2・1・Th・re ・xist・」・m・nt・ y・，y・，…，Yn∈A・u・ゐtん・ε鋤＝ intege7・51≦乞，ゴ≦n。 レVeノを〃tんεγ゜morθhave tんeプblloωin9． δiゴzforα〃 （1）1＝（Yl，Y2，…，Y。），μ（1）＝n，αnd eA（1）＝n＋1。 （2） 1アn＞1，then I⊆ノ4． proof． The existence of elements Yl，Y2，…，Yn is exactly the dual basis lemma． Let us note a brief proof fbr the sake of completeness。 Let 1≦」≦n be an integer． We look at the fbllowing diagram

m

b

蛎

−げv孟

  

   ︶

   ︵

  

  

@脚

  

   2

レ蜘

of A−modules， whereεis the canonical epimorphism， p is the projection map such

that p（勾＝δiゴfbr all 1≦i≦nwhere婿＝1i mod m2 denotes the image of xi in

m／m2 andδiゴis Kronecker’s delta，んis the isomorphism of vector spaces oveL4／m

defined byん（1）＝z， and b，s denote the embedding maps． Then， since the ring A is

self−injective， we have a homothety map！＝祷：A→、4 with防∈、4 such that the

above diagram is commutative． Hence∬助＝δ歪〆fbr all integers 1≦i，ゴ≦n． We put

J＝（Yl，Y2，…，yn）． Then J⊆1＝（0）：m2， because m．z＝（0）and靱ゴ＝δ￠〆． We

have eA（1）＝n十1， since

11

HomA（A／m2，A） and 乏A（A／m2）＝n十1．

Therefbre， to see that 1＝」， we have only to show 4A（」）＝n十1，0r equivalently eA（」／（・））−n・L・t｛6ゴ｝・≦ゴ≦・be el・m・nt・i−and assum・th・tΣ量．、 bゴ〃ゴ∈（・）・Th・n        n       わ・・一δ（鰯一x・・2b、yゴーo・       ゴ＝1 Hence bi∈m。 Thus the images of｛防｝1≦ゴ≦n in J／（z）form a basis of the vector space J／（z）over 4／m， so thatμA（」／（z））＝6A（」／（の）＝n． Hence eA（」）＝n十1and assertion （1）follows。 Assertion（2）is now obvious．      □

   For the rest of this section we throughout assume that d＝dim．A＞0． Let

Q＝（α1，α2，…，αd）be a parameter ideal iL4 generated by a systemα1，α2，…，αd

of parameters for A and let I＝Q：m2． We assume n＝v（A／Q）＞Oand write

m＝Q＋（xl，x2，… ，xn）with ci∈AThen ml⊆Q：mand ml IZ Q（recall that

（［2≠m．，since n＞0）． Let us choose z∈ml so that z￠Q， whence        Q：m＝Q＋m1＝Q＋（z）． Then， apPlying Lemma 2．2．1 to the Artinian local ring・4／Q， we get the elements Y1，Y2，…，Yn G4 such that銑〃ゴ≡δ乞〆mod Q fbr all integers 1≦i，ゴ≦n． Hence        I＝Q十（Yi，Y2，… ，Yn），  μA（1／Q）＝n，  and  乏A（1／Q）＝η十1， so that we haveμ／1（1）≦n十（t．   We now look at the fbllowing inclusions n十1

1

＼戦≦n＋d

ml

Q

1 d

    ml∩Q

mQ

and notice that［Q十ml］／Q望ml／［m∫∩Q］． Then eA（ml／［ml∩Q］）＝1since Q：m＝

Q十ml， so that we have

      μAの＝n＋d⇔m∫∩Q＝mQ・

  We furthermore have the following。 Proposition 2．2．2． Suppose thαt n＝v（．4／Q）＞1。 Then tんe folloωing！b解condition8 αre equivαlent to eαCんother．

（1）1⊆Q．

 （2） ml∩Q＝mQ。

 （3）μ、4（．τ）＝n十（オ．

 （4）m21＝m2Q．

Here Q den・tes the integrα1 closure・f Q． proof． The implication（1）⇒（2）is clear， since Q is a minimal reduction of I． The

equivalence（2）⇔（3）follows from the above observation．

  （4）⇒（1）This is wel1−known（cf．［NR］）． Use the determinantal trick．

  （2）⇒（4）Because z∈ml⊆Q：m＝Q＋（z）， we get

       m∫＝（ml∩Q）＋（z）        ＝mQ＋（z）． Therefore， in order to see the equality m21＝m2（2， we have only to show that        mz⊆m2 Q．

Since z∈搬∫⊆m2（recall that∫≠且；cf． Lemma 2．2．1（2））， we get Qz⊆m2Q．

Hence， because m＝Q十＠1，ω2，… ，xη）， it suf丑ces to show thatコcez∈m2Q for every

1≦e≦n．Choose an integer 1≦乞≦nso that i≠乏and write z＝niyi十g乞with

qi∈Q． Then xez＝xi（xeYi）十xeqi． Because qi＝z−ciyi∈ml∩Q＝mQ and

xeYi∈r匝∫∩Q＝mQ， we certainly have xez∈m2Q． Thus m21＝m2Q．      口

  As a consequence of Proposition 2．2．2 we have the following． Corollary 2．2．3． Suppose thαt n＝v（ノ1／Q）＞1αnd that I is integrαl over Q． Then

 （1）Q歪∩Ii＋1＝QilプbTαll伽孟eger8 i≧1。 Hence I2＝Q∫げ1⊆m2．

 （2） （α1）∩∬2＝α11．

 （3）12＝Q∫ヴQ⊆m2．

Proof．（1）The second assertion follows from the且rst， since I2⊆m21⊆Q． To see the first assertion， notice that m21i＋1＝m2Q乞＋1， since m21＝m2Q by Proposition 2．2．2．

Let！∈Q乞∩Ii＋1 and write

      ！一Σ・ti ’ ah2…・纏、i、．id       il＋i2＋…＋id＝i with fili2．．．id∈A． Letα∈m2． We then have α！一 Σ・11α峯2…・7（・・f・、i、．i、）∈m21’＋1⊆Q’＋1．     Zl十z2十…十td＝z

13

Henceαfiii2…id∈Qbecauseα1，α2，… ，αd is an／1−regular sequence， so that fiii2＿id∈∫．

Thusノ∈Qzl， whence Qz∩Iz＋1＝Qzl．

   （2）Let！∈（α1）∩12 and write！＝・α1g with g∈、4． Then fbr allα∈m2， we

haveα！＝α1（α9）∈m212⊆Q2． Henceα9∈Qso that g∈1， and so f∈αil． Thus

（α1）∩∫2＝α、五    （3）Let us prove the assertion by induction on d． Assume that d＝1． Let b∈m2 be a non−zerodivisor in A． Then， thanks to the isomorphisms        ［（b）：m2｝／（b）yHomA（A／m2，A／（わ）・）i≧Ext弧（、4／m2，A） of 4−modules， we see the length乏A（［（b）：m2］／（b））＝eA（Ext気（A／m2，A））is independent of the choice of the element b∈m2． We putα＝α1． Let Q’＝（α2）and I’＝（2’：m2，

Let

       9・A／（α）→A／（α2） be the monomorphism defined by g（房）＝齋， where R denote the images of the corre． sponding elements x andαx． Thenψ（1／（α））＝1’／（α2）， since（ρ（1／（α））⊆1’／（α2）and ぞA（1／（α））＝eA（1’／（α2））（recall thatα∈m2）． Therefbre       （＃） 1’＝α1＋（α2）＝α1， whenceμA（∫’）＝μA（1）＝n十1， where the last equality follows from Proposition 2。2．2。 Hence I’is also integral over Q’by Proposition 2．2．2， because v（．4／（［2’）＝v（A）＝

v（A／Q）＝n＞1．Therefore（1’）2＝α21’by assertion（1）， since∫’⊆m2． Hence by

equality（＃）we getα212＝（1’）2＝a21’＝α3∫， so that I2＝α1．    Assume now that d≧2and that our assertion holds true for d−1． Letノ隻＝A／（α1）， 而＝m／（α1），ll）r＝Q／（α1）， andア＝1／（α1）． Then Q「：而2＝7， v（IZii／Q「）＝v（A／Q）＝ n＞1，and J is integral over Q． Hence the hypothesis of induction on d yields that

72＝Q7， since Q⊆iii2． Thus I2⊆QI＋（α1）． Therefbre

       I2＝［QI＋（α、）］∩12＝QI＋［（α、）∩12］＝・ QI＋α、1 ・ QI by assertion（2）．      □

Corollary 2．2．4．3鴛ppo3e v（、4／Q）』＞1αnd I￠3 integrαl over Q， Then I⊆m2ザ

Q⊆m2．

Proof Suppose Q⊆m2． Then I2⊆Qsince I2＝Q∫by Corollary 2．2．3（3）． On the

other hand we have Q：（Q：m2）＝m2， because Q is a parameter ideal in the Gorenstein

local ring・4． Hence∫⊆Q：∫＝Q：（Q：m2）＝凱2 as is claimed．      □

  Unless Q⊆m2， the equality I2＝QI does not necessarily hold true．1．et us note one example． Example 2．2．5． Let H＝〈6，7，15＞be the numerical semi−group generated by 6，7，15 and let、4＝k［［オ6，孟7，オ15］］⊆k［［孟］］， where k［［診］］denotes the formal power series ring with one indeterminateオover a field k．Then．4 is a Gorenstein local ring with dim五＝1．

Let O＜5∈H＝〈6，7，15＞， Q＝（が）in、4， and∫＝Q：m，2． Then∫is integral over Q

and rQ（1）≦2． However，12＝QI if and only if 8≠7．

Proof Letη∈H． Then it is direct to check thatオπ∈∫if and only if n＝3，5十

6，5十7，8十8，0r 5十乏fbr some 12≦4∈Z． Thanks to this observation， we get

∫＝（オ8，が＋8，が＋16，が＋17）if 5≧12 but 5≠15． We also have 1＝（舌6，孟14，オ22）if 8＝6， 1＝（オ7，オ15，オ24）if 8＝7， and I＝（オ15，オ31，オ32）if 5＝15． Hence I⊆t8k［［司】∩．4， so that Iis integral over Q＝（の， in any case． It is routine to check that 12＝QI when 8≠7．

If 8＝7， then I3＝Q∫2 but I2＝Q∫十（孟30）and砺（12／QI）＝1， whence∫2≠QI． 口

  Here let us note one example to clarify our arguments．

Example 22．6． Let（A， m）be a regular local ring with d＝dim A≧2and let

xl，x2，…，ωd be a regular system of parameters of A． Let ci≧2（1≦i≦のbe

integers and put Q＝（xl1，錫2，…，xSd）． Let I＝Q：m2 ． We then have the following． （1）The fbllowing conditions are equivalent． （i）1⊆Q．

（ii）d＝2and min｛c1，c2｝＝2．

 （2）12＝Qlifl⊆Q．

Here Q denotes the integral closure of Q．

P…fL・tz−n乳1・夢一1，・、一婿・，and・y、一三f・・ea・h1≦i≦d． Th・nQ・m−Q＋

      Xi （のand xiyj≡δ乞〆modulo Q fbr all integers 1≦i，ゴ≦d． Hence∫＝Q十（Yl，Y2，…，Yd） andμA（1／Q）＝d by Lemma 2．2。1． We put J＝（Yl，Y2，…，Yd）．

  Suppose now that∫⊆Q． Then；since v（A／Q）＝d＞1， by Proposition 2．2．2 we

haveμA（1）＜2d． Henceαi∈（αゴ11≦ブ≦d，ゴ≠の十Jfbr some 1≦i≦d， because

μA（1／Q）＝d．We may assume that乞＝1． Let us write       d       d        α1＝Σαゴξゴ＋Σ防ηゴ        ゴ＝2    ゴ＝1

15

withξ）andηゴ∈A． Thenηゴ∈mfbr all 1

μ。（1／Q）−d．L・t・一Σ匙、 Ci． Th・n

≦ゴ≦d，・inceΣ1。I Yゴηゴ∈

 d

 卜

 ㎞

 ∈Q

 協

・Σ飼

 ＝

 噛

・Σ卸

 一  〇1    d 一Σ・ゴmC−（d＋c・）．   ゴ＝1

Qand

Hence

       d       d       ・・一Σ・ゴξゴーΣ・ゴρゴ        ゴ＝2     ゴ＝1 fbr someρゴ∈mc−（d＋cゴ）， so thatα1（1一ρ1）∈（αゴ12≦ブ≦の． Thereforeρ1 is a unit of 、4，sinceα1≠（αゴ12≦ブ≦の． Thus（オ＝2and c2＝2， becauseρ1∈m（c2＋c3＋”9＋cd）一d and cゴ≧2for all 2≦ブ≦（オ．

  Conversely， assume that d＝2and c2＝2． We then have

1＝Q＋」＝（酋1−1，X：1“2X、，場）． HenceμA（1）＜4＝2（i and so J望Qby Proposition 2．2．2． Thus assertion（1）is proven． Sillce Q⊆m2， the second assertion readily fbllows from Corollary 22．3（3）．    口   The following result is the heart of this paper． Theorem 2．2．7． Lθt n＝v（A／Q）＞1αnd assume that 1 is not integrα1 over Q． Then eX（ノ1） ≦2 αnd n＝2．

・Proof． Suppose that d＝＝1and letα＝α1． Then I＝ （α）十（Y1，Y2，… ，Yn）and

m＝（α）十（x1，x2，…，ωn）． We haveμA（1）≦n by Proposition 2．2．2， because 1 is not integral over Q， whileμA（1／Q）＝nby Lemma 2．2，1（1）． Hence I＝（Yl，Y2，…，Yn） andα∈m・（Y1，Y2，… ，Yn）⊆m2． There」rore m．＝（ω1，x2，… ，ωn）． We put

J：＝Q：m＝Q＋ml＝Q＋（z）．

Then mJ＝mQ（cf ［CP1， Proof of Theorem 2．2］；recall that、4 is not a discrete

valuation ring， because n＞1）． HenceμA（」）＝2， because eA（」／mJ）＝4．4（」／Q）十

eA（Q／rnQ）＝2． We・have J ・ml＝（ω乞防11≦i，ブ≦η）， because Q⊆ml⊆」．

  We divide the proof into two cases．

  Case 1．＠助≠mQ fbr some 1≦i，ブ≦ηsuch that i≠ゴ．）

Without loss of generality we may assume that i＝1andブ＝2． Then， because

xlY2∈Qbutω1Y2≠mQ， we have Q＝＠1Y2）． Hence J＝＠1Yl）十Q＝（X1！ノ1，Xl！ノ2）＝

xl．（Yl，Y2）⊆（x1）because z≡ωIYI mod Q， whence xl is a non−zerodivisor in、4．

We have xlye∈mI＝」＝xl・（Y1，Y2）， so that Ye∈（Yl，Y2）for all 1≦6≦n． Thus

J＝（Yl， Y2）． H・nce・n＝2． Becau・e ml−・・l andμ・（1）−2， w・hav・m2−・1M，

just thanks to the determinantal trick（cf．［DGH， Proposition 5．1］）． Hence e焦（A）＝2， because、4 is a Gorenstein local ring of maximal embedding dimension・    Case 2．（銑防∈mQ fbr all 1≦i，ゴ≦nsuch that i≠ゴ．）

In this case， we have J＝（xiyi l 1≦i≦n）， because J＝ml＝＠乞防11≦i，ゴ≦η）

and mJ＝mQ． SinceμA（」）＝2， without loss of generality， we may assume that

」＝（X1！ノ1，X2Y2）．Becau・e・・〃r・・〃・≠m」＝mQ and・・〃・ヨ・〃・≡・m・d Q， w・

have xlyl＝ω2Y2＋αεwith a unitεin・4， while x1Y2＝ααand x2Yl＝・αβwithα，β∈m・

Hence

       （Xl＋X、）（ッ、・一〃、）＝・（ε一α＋β） withε一α十βa unit of A． We put

Xi−

o農＋究；B

and

Yi−

o1；一究；劣．

Then m＝（X1，X2，… ，Xn），1＝（Yl，Y2，…，Yn）， and Xi Y2≠ mQ clearly． Thus thanks t・Case 1， we・have・n＝eX（A）＝2．

  Now assume that d≧2， Then， by Proposition 2．2．2， we haveμA（1）＜n十d．

SinceμA（1／Q）＝n， we may assume that∫＝（α2，α3，… ，αの十（Yl，Y2，…，Yn）・Let L−（・、，・、，…，・、），A−A／L，・iii−m／L， Q−Q／L， andアー∫／五・Th・n 7−Q・iff2

and万is a Gorenstein local ring of dimension l with v（、4／Q）＝v（、4／Q）＝n＞1．

W・hav・μπの≦n， wh・nce by P・・P・・iti・n 2・2・2，7i・n・t i・t・g・al・v・・Q・Th・・ef・・e・ thanks to the result of the case where d＝1， we have n＝e害（万）＝2． We see ek（A）≦2

becau・e ・k（211）≧・k（A）， whi・h・・mpl・t・・th・p…f・f Th…em 22・7・   □

   The following assertion readily fbllows from Theorem 22．7． C。。。ll。。y 2．2．8．3叩P・・e診ん・オ・k（A）≧3． Th・・1￠・乞・オ・grα1…rQ，伽一・（A／Q）＞ 1．

2．3

Proof of Theorem 2．1．1

Th・・ugh・ut this secti・n l・t（A，m）b・aG・・en・t・i・1・・al・i・9・with・d−dim A＞Oand Q−（・、，・，，…，・d）・p・・am・t・・id・ali・A・W・p・t l−Q・m2・

17

  The purpose of this section is to prove Theorem 2．1．1． Let us begin with the fol− lowing． Theorem 2．3．1．θ即po8e thαt n＝v（A／Q）＞1αnd I i8 integrα1 over Q． Then （1）13 ・ QI2． （2）G（1）αnd F（1）αre Oohen−Mαcαulαy rings． Hencθ・IR（1）i8 als・α0・hen−Mαcαulay ring， if d≧3。 Proof． The last assertion directly fbllows from assertions（1）and（2）， because theα一

invariant a（G（1））of G（1）is at most 2−d（cf．［GSh1， THEOREM（1．1）， REMARK

（3．10）］）．   We may assume that I2望Q， thanks to Corollary 2．2．3（1）． Choose the element

z∈ml so that z∈12． Hence Q：m＝Q十∬2＝Q十（z）and so I2＝QI十（z）， because

Q∩12＝QI by Corollary 2．2．3（1）． Thus I3＝QI2十zl and we get the required

equality I3＝QI2 modulo the following claim， because

（Q2＋zQ）∩13＝（Q2∩J3）＋zQ＝Q21＋zQ⊆QI2

by Corollary 2．2．3（1）．

Clαim 1． zl⊆Q2＋zQ，

Proofofαafmヱ． Since 1＝Q十（Yl，Y2，…，Yn），it su伍ces to show that zye∈Q2十zQ for all integers 1≦e≦n． Let 1≦i≦n be an integer such that i≠乏 and write z＝ciyi十qi

with qi∈mQ． Then zyeニ（viye）yi＋yeqi∈（ml）Q． Since ml⊆Q：m＝Q＋（z）， we

have zye∈［Q＋（z）】・Q＝Q2＋zQ． Thus zJ⊆Q2＋zQ．       □

   As I3＝QI2 and Q∩12＝Q∫by Corollary 2．2．3（1）， we have Q∩Ii＋1＝Qli fbr

every i∈Z， whence G（1）is a Cohen−Macaulay ring． To show that F（1）is a Cohen− Macaulay ring， we need the following。 The equality rnl2＝mQI in Claim 2 yields， since

I3＝QI2， that the elementsα1T，α2T，…，αdT∈R（1）constitute a regular sequence in

F（∫）． Clαim 2． m∫2＝mQI．

proof ofαajm 2． Let J＝（Yl，Y2，…，Yn）． Hence I2＝α十」2 because I＝Q十」．

It suHices to show that r薩」2⊆mQ1． Since m＝Q＋＠1，x2，…，xn）and QJ2⊆mQ∫，

we have only to show物跳防∈mQ∫fbr all integers 1≦乏，乞，ゴ≦η． Let us write

XeYi＝δeiZ十％with％∈mQ． Then

       娩跳防 ＝（δeiZ十9ゼ∂防＝δ乏乞〃〆十qeiyゴ∈13十mQI＝mQI， because I3＝QI2． Hence ml2＝mQI．      口   We are now in a position to prove Theorem 2．1．1． Proof of Theorem 2。1．1． By Proposition 2．2．2， Corollary 2．2．8， and Theorem 2．3．1 we

may assume that n＝v（A／Q）＝1． Hence v（A）＝d十1． Let m＝Q十＠）with x∈m；

henceα1，α2，…，αd，x is a minimal basis of m． We put

A＝A／Q，而＝m／Q＝＝（Zli），1＝1／Q， and e＝乏A（A），

where房＝xmod Q be the image of x in 4． Then， since頭＝（房）， we have       4−1＝max｛t∈Zl而孟≠（0）｝and xe∈Q．

H・nce 7−（0）・iii2一話2 s・th・t∫−Q＋me−2−Q＋（xe”2）・N・tice th・t e−・5（A）≧

eX（A）≧3， where e＆（A）denotes the multiplicity of五with respect to Q． We then have       m21＝［Qm．＋（X2）］・［Q＋（xe−2）］⊆m2Q＋（xe），

because m2＝Qm十（x2）and乏≧3． Consequently， in order to see that m21＝m2Q， it

sufHces to show the following．

Claim．〆∈m2Q．

P…f・fα・im． L・t・・w・it・xe ・．Σ窪1α幽withω、∈A． L・tλb・th・m−adi・

       A

completion of A and take an epimorphism‘，t’：B→A， where（B，n）is a regular

local ring of dimension d十1． Then Kerψis a principal ideal in B generated by

asingle elementξ∈ne such thatξ≠ne＋1 where e＝eX（A）；hence Ker 9⊆ne．

Choose elements｛Ai｝1＜i＜d， X， and｛VVIi｝1くiくd of B such that they are the preimages of｛αi｝1＜i＜d，x， and｛wi｝1〈iくd， respectively． Then we have n＝（Al，．42，… ，Ad，X）and

XLΣ匙1ん琳∈Ker g⊆ne． HenceΣ窪1！魅隅∈ne， because e≧e． Consequently，

・ince（A、，A、，…，Ad）∩・・＝（Ai，A、，…，Ad）…’1，w・・ee th・tΣ窪、 A、Wi一Σ匙、 A、Vi f…s・me el・m・nt・Vi∈・・−1， wh・nce ・e ・・ Z）1・。1 aivi wh・・e・vi−9（Vi）． Th・・〆∈

Qme−1⊆Qm2 as is wanted， because e≧3，      □

19

  Since m21ニm2Q， we have Q∩12＝＝QI similarly as in the proof of Corollary

2．2．3（1）．Therefore， to finish the proof of Theorem 2．1．1， we may assume I2望Q．

Since xe∈Qand J2＝QI十（x2e−4）， we have 24−4＜ewhence乏＝e＝3， so that

J＝Q十（x）＝m．Thus m3＝m21＝Qm2 and so G（m）＝F（m）is a Cohen−Macaulay

ring． As a（G（m））≦2−d， R（m）is a Cohen−Macaulay ring if d≧3． This completes

the proof of Theorem 2．1．1．      ［コ

2．4

Examples

In this section we explore three examples to show that the non−−Gorenstein case is rather wild． Example 2．4．1． Let n≧2be an integer and let A＝κ［［X1，X2，…，X。】］／（X， Xゴ 11≦i＜ゴ≦n） where k［［X1，X2，…，Xn］］denotes the formal power series ring over a field k． Then A is a one−dimensional reduced local ring with ek（A）＝n． F（）r every parameter ideal Q in A， we have        Q：m21（？， where（［？denotes the integral closure of Q． Proof． Let 1＝： （？：m2 and assume that I⊆σ． We write Q＝（α）． Thenα＝Σ葉＝1婿輩ε乞 fbr some units Ei in、4 and some integers ci≧1． Let 1≦乞≦nbe an integer． If c信≧2， we then have婿‘−1∈Ibut婿ぜ一1 is not integral over Q． Hence ci＝1for all 1≦i≦n and soα＝Σ農＝1 xiεi． Therefore m2＝Qm so that we have 1＝A， which is absurd．口   Letting n ＝2， this Example 2．4．1 shows the assumption that e鑑（A）≧3in Theorem 2．1．1is not superfluous．   It seems natural and quite interesting to ask what happens in the case where A is anumerical semi−group ring． Let us explore one example・ Example 2．4．2． Let H＝〈4，7，9＞be the numerical semi−group generated by 4，7， and 9 and let、4＝k［［オ4，オ7，孟9］］⊆k［［オ］］， whereγ＝k［岡］denotes the fbrmal power series ring

with one indeterminateオover a field k． TheL4 is a one−dimensional non−Gorenstein

Cohen．Macaulay local ring． Let O＜8∈H． We put Q＝（tS）and I＝Q：m2． Then

I⊆Ii？． We have∫⊆m2 if 5≧11， whence I2⊆Q． However

 1 1

 ＞一

 8

 r O

 8

Qり47・

＝＝＝

888

fflfl

olo−°1 1←n∠り0 ＝

 

∼

く

Q

r

︶

1

︵

（2）G（1）is a Cohen−Macaulay ring if and only if 8＝4，8，9． （3）F（1）is a Cohen−Macaulay ring if and only if s＝4，9． （4）F（1）is always a Buchsbaum ring． （5）G（1）is a Buchsbaum ring if and only if s≠7．

（6）m21≠m2Qif8＝8，11．

Proof We haveη∈Hfbr all integersη≧11 but 10￠H． Hence the conductor of H

is 11． Notice thatオη∈m．2 fbr all n∈Zsuch thatη≧11， where m＝（孟4，オ7，オ9）denotes

the maximal ideal iL4． Hence 1⊆Q． In fact， let n∈Hand assume that孟η∈∫but

η＜8．Thenが一n＋10∈m2 because 3一π十10≧11， so thatが＋10＝オη孟8｝π＋10∈Q＝（tS）

whenceオ10∈、4， which is impossible． Thus， fbr everyη∈∬with tn∈1， we have

オπ∈がV∩．4＝Q，whence I⊆Q（recall that I is a monomial ideal generated by the

elements｛孟円n∈．H such thatオπ∈1｝）． In particular we have I⊆m2 if 8≧11，

whence 12⊆Q．

  W6 note the following．

α蜘1・Let s2≧5・≧11 be integersαnd let q＝82−5、． We郷Q乞＝（がりαnd

Ii：＝Q￠：m2！∂γ・i＝1，2． Thenωeんαve the f（）”oωing， （1）12河9∫1． （2） 7ヒ（Il）⊆とR（J2） αs grαded A一αlgebγ・as， （3）F（11）i≧iF（12）αs graded A／m一αlgebrαs．  （4） rQ1（ll）＝r（∼2（12）．        ム Proof of Claim 1． Let （p＝tq：V→V be the Vlinear map de丘ned by g（x）＝tqx for all

x∈V．Then， since g（Q1）＝Q， andρ（ll）⊆12， the map g induces a monomorphism

ξ：11／Q，→12／（？2，

_{xmodQ，｝→tqxmodQ2}