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Ａ凶F回ｕＣ▲HMAnBEMATuCALsoC1ETv
vblmtelZO・NumbET2.Ｆ■ｂｍ□Jy1994
ＮＯＮ－ＣＯＨＥＮ－ＭＡＣＡＵＬＡＹＳＹＭＢＯＬＩＣＢＬＯＷ－ＵＰＳ
ＦＯＲＳＰＡＣＥＭＯＮＯＭＩＡＬＣＵＩＷＥＳ
ＡＮＤＣＯUNTERExAMPIESTOCOwSIKSQuESrlON
ＳＨＩＲＯＧＯＴＯ,ＫＯＪＩＮＩＳＨＩＤＡ,ＡＮＤＫＥＩ－ＩＣＨＩＷＡＴＡＮＡＢＥ
(CommunicaledbyEricFnedlandeT）
AnsTRAcT・ＬｅｔＡ＝AIIJr,γ､Ｚ]］ａｎｄｋⅡ、］befbnnalpowersenesnngs
ovemneldと，ａｎｄｌｅｔｎ＞４beanime8ermchlhat〃差Ｏｍｏｄ３･陸ｔ
ｏ:八一Ａ【[nidenotethehomomomhismo「昨algebmsde6nedbyp(Ｘ)＝
77n-ｺ，｡(ｙ）＝ＴＷ－ｚ)河，ａｎｄ｡(Ｚ）＝781,-3．ＷｅｐｕＩｐ＝ＫＣＴ｡、Ｔｈｅｎ
Rn(p)＝①ｌ三op(i）isaNoetheriannngi｢andonlyircM＞0．Henceon
COw3ikosquestionthcに画℃counIeにxamplesamongthepnmeidcalsdenning
spacemonomialcurves,too．
LINTRoDucTIoN
I心ｔ』＝ｋ[[Ｘ，Ｙ,Ｚ]］ａｎｄＳ＝ｋ[[刀］befbrmalpowersenesrmgsovera
he１．ｋ,andlet〃,，〃2,and〃3bepositiveintegc応withGCD("I，〃1,〃〕)＝１．
Wedenotebyp(",，〃２，〃3）thekemelofthehomomolphismp:八一sｏｆ応
algebmsdefinedbyO(Ｘ）＝Ｔ"１，。(Ｙ）＝Ｔ"２，ａｎｄ｡(Ｚ）＝か，、Ｈｅｎｃｅ
ｐ＝ｐ("肋〃1,〃3）istheextendedidealintheringAofthedehningidealfbr
themonomialcurvex＝!"１，J＝ln2,andｚ＝rm，inAii・
AUttlemorcgeneraUy,letpbeaprimeidealofheight2ina3-dimensional
r巳gularlocalringA・ＷｅｐｕｔＲ`(p)＝Ｅ">op(")T〃（herETdenotesanin‐
determinateoverd）ａｎｄcaUitthesymb6iicReesalgebraofp・Then,as
isweUlmown，Ｒ`(p）isaKrullringwiththedivisorclassgroupZ，andif
RS(p）isaNoetherianringitscanonicalmoduleisgivenby［R，(p)](－１）（Ｃｆ，
e,8.,[12,CoToUary3､4])．ConsequenUyR，(p）isaGorensteinring,onceitis
COhen-Macaulay､TheTeadeTsmayconsult[２，３]fbracriterionofRs(p）being
COhen-Macaulay,wheにsevemlexamplesofprimeidealspwithGorenstein
symboUcReesalgebmsaJEexplomed,too・
Nevertheless,aswashrstshownbyMorimotoandthefirstauthor[8],Ｒ`(p）
arenotnecessalilyCohen-Macaulayevenfbrthespacemonomialcuwesp＝
p(",，〃2,〃])．ThisrBseaJchisasuccessionto[8]andtheaimistoprovidethe
fbllowingnewexamples．
Receivedbytheedjto届January22，l9Z2and,hMUvisedfbrm,Ｍａｙ２８，１９１２．
１９９１Ａ血KA2mpUicsSbubjepCⅢｴｆｓｉ〃mIio"・PrimaIyl3HO5,l3Hl3,l3H15;Secondmyl4H20，
14J17．
Theauthorsa定paniallysupporledbyCmnt-in-AjdlbrCO-opcmliveRescar℃h、
、I994AmmcmMmhemmcnlSo句eoy
OOOZ･”W94SLOO＋３．２９perp■GＥ
３８３
牛ⅡローＩｒｔＪ１■■・・・］④エヨ０口－・早足凸い七勺・・□ｒｒ０■砂１㎡殆・・Ｔチー引汁Ⅲ岬’‐（▼が西
s,ＧＯＴ０，Ｋ.NISHIDA,ＡＮＤｌＬｗＡＴＡＮＡＢＥ
３８４
Theorem1.1．Ｃｈｏ“ｅα〃i"(…「〃ｚ４ｓｏｌｈａｒ〃姜Ｏｍｏｄ３ｑ"。”ｌＰ＝
p(7"－３，（5"－２)皿８"－３)．Ｌｅｌｃ＝Ｙ３－Ｘ"Ｚｎα"｡(zsFⅢｍｅｌＡａｌｃｈＡ＝Ｐ＞0.
711Ｉ２ｎｌｈｅ花2xjslsq〃eleme"【ＡＥｐ(3p)ｓａｌｊｑｂ'i"glAee9Ⅸα"ly
lengnhU/(x,ｃ,A))＝3ｐ･lengthW(x)＋p）
１Ｆ・・八・千・・・．叩小咄排朴ｒ・ＬＪⅡかく、命ｆ】１Ｊか‐Ｐ設いＪＩＬ、嵌喉凹肝血幸一㈹肛別仏器篭眠Ⅲ、皿睡攝払、Ⅲ‐隅脚ｆｕ咄蝿即懸山畑砥Ⅲ吋聰鰯幡ⅡⅢ鋤
ﾉｂｒα'､,ｘＥ(Ｘ，Ｙ,Ｚ)Ａｌｐ．】"ｐα"たｗｌｍＲ`(p）ｉｓαﾉＶｂ２ｌｈｅｍｚ〃rmg6m〃α
のｈｅ汁Mzczzl（lbzy・
Ｈｅにletｕｓｎｏｔｅｔｈａｔｔｈｅｉｄｅａｌｐｉｎｔｈｅａｂｏｖｅｌheorcmisgenemtedbythe
maximalmino応ofthematrix
（’V郷:二|)；
hencepisaselfL1inkedidealinthesenseofHerzogandUlrich[6,CoroUaly
1.10】,thatis,ｐ＝(ノ,ｇ):pfbrsomeelemems/,ｇＥｐ・Roughlyspeakins
seMLlinkedidealsenjoymoreexceUentnaturesthanthoseoftheidealswhich
aICnotselfLlinked、FOrmstance,thesubringsA[p7,ｐ(2)Ｔ２】ｏｆＲ`(p）are
alwaysGorenstein,ifthecoITespondingidealspa泥selfLlinked(see[6,Proof
ofTheorCm2.1]).Ｆｒｏｍthispointofviewitseemsmthernaturaltoexpectthat
thewholeringsR`(p）areGorensteinatleastfbrselfLlinkedidealsp;however，
ｔｈｅａｎｓｗｅｒｉｓｎｅｇａｔｉｖｅａｓｗｅｃｌａｉｍｉｎTheo応、１．１．
。■ロロⅡ■■１１｜ケト■■■ＰＩＢＬ２■ＵⅡ日日ｐ■・ＦＵＩ■■ＩⅡ．Ｉｉｒ
１，[8]Morimotoandthehrstauthorconstmcted,fbreachprimenumberP，
spacemonomialcurvespwhosesymbolicReesalgebrasR`(p）areNoetherian
butnotCohen-Macaulay，ifthecharacteristicchAofthegrｏｕｎｄｈｅｌｄｌｃｉｓ
ｅｑｕａｌｔｏｐ、Neverthelesstheirexamplesare〃olselfLlinkedand,fmrthelmore,it
isnotclearfbrtheirexampleswhetherRs(p)areNoetherianornotinthecase
whe妃thecharacteristicisdiHbremfiomthegivenprimenumberp，whileour
examplesareNoetheriana､dnon-Cohen-Macaulaywheneverchk＞０．
ThisadvantagenatumUyenablesus,passingtothereductionmodulopriｍｅ
ｎｕｍｂｅ応，toexplo定thecaseofchamcteristicO，too，MoreoveT，somewhat
surprisingly,asanimmediateconsequenceofTheorem１．１wegetthefbllowing
counterexamplestoCowsik，squestionI1],thataskedwhetherthesymbolicReeｓ
ａｌｇｅｂｍＲ`(p)beaNoetherianringfbranyprimeidealpina定gularloca1ring
A：
ＣＯroUmy1.2.Ｌ２１ｐ６２ａｐｒｉｍ２ｉＵｌｂｚｚﾉｓｌａｌｅｕ！』〃、ＣＯ花、1.1.7111ｍＲ`(p）ｊＳ
ｎｏｌｑﾉＶｂａＡｍα〃mUgbヴchk＝O
WhenCowsikraisedthequestion，heaimedaIsoapossiblenewapproach
towardtheproblemposedbyKronecker,whoaskedwhetheranyirrCducible
allinealgebmiccurveinAecouldbedehnedbyn-1equations、Ｉ、fact，
Ⅶ波．‐研玲・如叩・Ｉ．ｎ㎡．
‐ＤＬ００００■Ⅱ□らＪ■仏叫Ｅｕ昨Ｐｕ・ｂ９ＩⅢＩＤＩⅢ
：
：
Cowsikpointedoutin［1］thatpisaset-theo正ticcompleteinte唾ction,ｉｆ
Ｒ`(p）isaNoetherianringandifdimA/ｐ＝１；however,asisweUknown，
whileKroneckeT､sproblemremamedopenonCowsick，squestiontherewas
al肥adygivenacounterexamp1ebyRoberts[，]．Ｂecausehislirstexampledid
notrCｍａｉｎpnmewhenthelingwascomp1eted，herecentlyconstmctedthe
secondcounterexamples[10]・Theyareactuallyheight3primeidealsinafbrmal
powerseriesringwithsevenvariablesoveraIield；ｎｏｗ，providingnewand
simplercounterｅｘａｍｐｌｅｓａｍｏｎｇｔｈｅｐｌｉｍｅｉｄｅａｌｓｉｎｔhefbnnalpowerseries
nngQ[Pr，ｙ，Ｚ]]，ourCoroUaryLZseltlesCowsik，squestion,thoughitsays
nothingaboutKronecker，sproblemitself
￣
ＮＯＮ厄ＯＨＥＮ･ＭＡＣＡＵｌＡＹＳＹＭＢＯＵＣＢＬＯＷ･ＵＰＳ
｡8５
TbesimplestexamplegivenbyTheorｅｍＬ１ｉｓｔｈｅｐｒｉｍｅｉｄｅａｌｐ＝
p(25,72,29)＝(ＸＩｌ－ＹＺ７，ｙ３－Ｘ４Ｚ４，ＺＩ１－Ｘ７Ｙ２）ｉｎ』＝ＩＣ[[Ｘ，Ｙ,Ｚ]]、
ThesymbolicReesalgebraRs(p）isaNoetherianTingbutnotCohen-Macaulay，
ｉｆｃＭ＞０，ａｎｄＲｓ(p）isnotaNoetherianring,ｉｆｃＭ＝0．Therefbre,the
symbolicblow-upR,(P)＝Ｚ">oＰい)T〃isnotahnitelygeneratedQ-algebm
fbrtheprimeidealP＝(ＸｌＩ－ＹＺ７，ｙ３－Ｘ４Ｚ４，ＺｌＩ－Ｘ７】'2）inthepoly‐
nomialnngO[Ｘ，ｙ,Ｚ],too
NowletusbneOyexplainhowthispapeTisorganized・TheproorofTheorem
1.1andCoroUaryl,Zwillbegivenin§4．Section3isdevotedtoareduction
techniquemoduloprimenumber５．１，§ZweshaUsummanzesomepldiminary
stepslhatweneedtoproveTheorem1.1andCorollary1.2．
Otherwisespecified,inwhatfbllowsletA＝ｋ[[Ｘ，ｙ,Ｚ]］denoteafbrmal
powersenesrmgoverafixedfieldk・止ｔｍ＝(Ｘ,Ｙ,ZMbethemaximal
idealofA・Wedenoteby2AW),fbranA-moduleﾉＶ,thelengthofﾉＶ、ＦＯｒ
ａｇｉｖｅｎｐｒｉｍｅｉｄｅａｌｐｉｎＡｗｅｐｕｔ
Ｒ`(p)＝ZP1jD7mcA[刀，
、＞Ｏ
Ｒｊ(,)＝Ｚｐ⑩１７m(=R`(p)[丁－１])こ｣[7,Ｔ－ＩＬ
ｊＨＥＺ
Ｇｓ(p)＝Ｒ;(p)/丁~IRj(p）
whereTisanindeterminateoverA．
２．PRELIMINARIES
FirstofallletusrecallHuneke，scriterion[7]ｆｂｒＲ`(p）tobeaNoetherian
rin8FbrawhUeletい，、）denotearegularlocalringofdim』＝３ａｎｄｌｅｔ
ｐｂｅａｐｒｉｍｅｉｄｅａｌｏｆＡｗｉｔｈｄｉｍ』/p＝１．
Theo肥、２．１［71.〃ｒｈａで2xjs『雄me"砥/Ｅｐ(Ｉ）α"ｄｇＥｐ(､）ｗｉｌｈｐ“irjve
j"legF7Tノα"ｄｍｓ８４垣ｈｌﾉ、【
２A(』/(ｘ,/,ｇ))＝ﾉ、．cＡｕ/(x)＋p）
ﾉｂｒｍｍｅＸＥｍｌｐ，肋ｅｎＲ`(p）jSdzﾉVDah2rm〃〃"ｇﾘｹﾞﾘiemAeﾉieItfA/ｍｊＳ
ｍ/i"ilelrhecD"ve応ejSdMsolme、
ＴｈｅｎextpropositionallowsustoarbitrarilychoosetheelementJrinTheo‐
ｒｅｍ２・LTheresultisimpliciUyfbundin[7,ProofofTheo1℃、3.1】andisdue
toHuneke
Pmposition2.2.Ｌα/Ｅｐ(ﾉ)α"‘ｇｅｐ(､）（ﾉ,ｍ＞Ｏ）α"‘αmswmerAm
2A(』/(ｘ,／,ｇ))＝ﾉ、･2｣Ｗ(x)＋p）
ﾉｂｒｓｏｍｅｘＥｍｌｐ．〃ｅｎｌｈｅ”ovee984zz"〃holtisﾉb｢α"ｙｘＥｍｌｐ・
ＰｒＵｑｆＷｅｍａｙａｓｓｕｍｅ／＝、、Ｔｈｅｎ（／,ｇ)Aisareductionofp(､)；actuaUy，
[p(､)]２＝(ノ,ｇ).p(､）（Ｃｆ[Z,PToofofProposition(3.1)])．ｌａＲ＝Ａｐａｎｄ
ｎ＝pAp・Thenastheideal（／,ｇ)Ｒｉｓａｒｅｄｕｃｔｉｏｎｏｆｍｍａｎｄａｓ（Ｒ,、）ｉｓａ
２－dimensionalregularlocalring,ｗｅｇｅｔ
２尺(R/(八g)R)＝ど鵬)R(R)＝elh(R)＝、２
ﾎﾟﾆ
。■Ｃ宮⑤⑭四四コ、一戸』ロ蒄二
唾這』畠⑭⑨』堂⑭』⑨吾。二塁Ｕ一周山⑭』口印』◎曼弓⑫ニト・肘』昌凹』ｕ畠】宜己○扇哩囲冨圏彦」。。』。』⑭ＥＣ二爵三．田一
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;11艸可=｣七二=話邑~里由
ＮＯＮ･ＣＯＨＥＮ･ＭＡＣＡＵＬＡＹＳＹＭＢＯＵＣＢＬＯＷ･ＵＰＳ
387
NowrCcaUthatc＝γβ+β,mod(Ｘ）ａｎｄｗｅｇｅｔ
２Ａｕ/(Ｘ,ｃ,９１９２))＝(β＋β')．g`(』/(ｘ,ｙ,9,82)）
＝(β＋β')･２AＵ/(Ｘ，ｙ,ｇｏ))＝ｍ0.2Ａｕ/(Ｘ)＋p)．
Hencetheelememscandglg2satisfyHuneke,scondition（2.1）sothatc
andglsatisfythisconditionasweUlnfact,because２Ａ(A/(Ｘ,ｃ,９，唖))＝
2A(A/(Ｘ,ｃ,８，))＋２Ａ(』/(Ｘ,ｃ,９２))＝ｍ0.2列(A/(Ｘ)＋p)，wegelbythein‐
equalities2A(｣/(Ｘ,ｃ,９１)）三ｉ・［’1Ｗ(Ｘ)＋ｐ）ａｎｄ２Ａ(A/(Ｘ,ｃ,唖)）三
(耐o-i)･gAU/(Ｘ)＋p）ｔｈａｔノ`,(A/(Ｘ,ｃ,９，))＝ｊ･山(A/(Ｘ)＋p)．Ｔｈｕｓｗｅ
ｈａｖｅｊＥ夕，whichcontmdictstheminimalityofmo・
ThenextresultisthekeyinouTproofofCorollary1.2
ComUmy2､6.ＡＳＳＷｍｅｌｈｑｌβ＋β'＝３α"ｄｌＡａｊ３１〃,、Ｌｄｐｂｅａｐ"ｍ２ｍＪ碗be「
ﾉbrMijdiwe(zsswmelhα〃ｈ罪axmanekme"ｌｇＥｐｐｐ)ｓα【幼mglAe珂凹α"〃
2A(｣/(Ｘ,ｃ,ｇ))＝3p･（`,(』/(Ｘ)＋P)．、“ヴｇＴ３ｐｅｄ[{p(")T''},≦,,<卯],ｗｅ
ｈｑｖｅｍｏ＝３口"。Ｒ`(p)＝Ａ[p7,ｐ(Z)72,ｐ(3)Ｔ３ｌ
ＰｍｑｆＡｓ２ＡＵ/(Ｘ)＋p)＝〃，ａｎｄａｓｃ＝Y3mod(Ｘ),wehavethat
2A(A/(Ｘ,ｃ,go))＝、ｏｎ,＝３．２Ａ“/(Ｘ,Ｙ,ｇｏ))．
Hence31mo，because３１〃Ibyourassumption・Ｉａｍｏ＝３ｍ,、Ｔｈｅｎａｓ
ｍｏｌ３ｐｂｙ止ｍｍａ２４,weseemI＝１ｏＴｐ・Ifmlwercequaltop,thcnmo＝
3PandsowemusthavebyProposition２．５ｔｈａｔｇＴ３ＰＥＡＩ{p(")T"},≦"<3pL
whichcontmdictsourstandardassumption，Ｔｈｕｓｍｏ＝３ａｎｄｗｅｇｅｔｂｙ[２，
Proposition(3.7)(4)]thatRs(p)＝Ａ[p丁,ｐ(2)Ｔ２,ｇＯＴ３]．
３．ＲＥＤｕｃＴＩｏＮＴｏｍＥｃＡｓＥｗＨＥＲＥｃｈＡ＞Ｏ
Ｉａ〃山〃２，ａｎｄ〃３bepositiveintege応ｗｉｔｈＧＣＤ(",，〃２，〃3）＝１．Ｌｅｔ
Ｃ＝Ｚ[Ｘ，Ｙ,Ｚ］ａｎｄＺ[nbepolynomialringsoverZ・Wedenoteby9D:Ｃ一
Z[刀thehomomorphismofrmgssuchthatp(Ｘ)＝Ｔ"'，。(Ｙ)＝Ｔ"ﾕ，ａｎｄ
｡(Ｚ)＝Ｔ"３．匹tＩ＝KerpandconsidertheringCtobeagradedringwhose
graduationisdehneｄｂｙＣｂ＝Ｚ，ｑＩヨＸ，Ｑ２ヨァ，ａｎｄＣｈｊヨＺ・Then
thehomomonphismppreservesthegraduationsothatJisnatumllyagmded
pIimeidealinC、
FOragivenheldk，４k＝ｋ[[Ｘ，Ｙ,Ｚ]］denotesafbrmalpowersenesrmg・
WeputBk＝kIjr，Ｙ,Ｚ],thatweshaUidentifywithA②ｚＣ、止tpjtdenote
theidealp(",，〃1，〃〕）conside冗dinAA；hence,pJt＝“〃WeputBt＝Jα・
ThusPhisthedenningpnmeidealofthemonomialcuwex＝r"'，ｙ＝『"２，
andｚ＝["jinAi，
Thepurposeofthissectjonistoprovethefbllowing
Theorem3.1.Ａ"ⅨmeZharR`(pq）jSqﾉV02lherWz"rj"８，１１“ﾋﾟﾙe庵exな【p“"jv2
j"【率晒ノV,、α"。e佗me"ｕｓ/,ｇｑ／Ｊ(､）ｓｗｃＭｉａｔ〃ｐ曲ａｐｒｊｍｅ”ｍ６２ｒ
α"。〃ｐｚｊｖ,ｗＥａﾉMuys他泥
（１）ｐ『)＝J(噸)｡&｡"‘
（２）２Ak(`４k/(ｘ,/,ｇ)Aと)＝、2.2A‘(八k/(ｘ)＋pk）
ﾉbr1heﾉｉ２ＩＷＥＺ/ｐＺ、
ＰｍｑｆＷｅｐｕｔＢ＝８０，Ｐ＝化，ａｎｄ〃＝（Ｘ,γ,Ｚ)β、ＴｈｅｎａｓＡ＝
AQisafmhfUUynatextensionofBM，thesymbolicReesalgebraR`(PＢＭ
li1
388
！
S・ＧＯＴ０.ＫＮＩｇＨＵＤＡ･ＡＮＤＸＷＡＴＡＮＡＢＥ
isaNoethelianlingtoo；therCfbre，ｂｙＴｈｅｏｒｅｍ２・landPToposition2・Zwe
maychoosetwoelements/,ｇＥ(PB」v)(､）withmapositiveintegersothat
2B"(BM/(Ｘ,／,ｇ)B")＝伽2.2恥(β"/ｘＢＭ＋ＰＢＭ)．ＬｅｔⅣ＝(Ｘ，γ,Ｚ)Ｃ、
ThenasCAr＝Ｂ〃ａｎｄＪｑｖ＝PBAf,wecantake/ａｎｄｇｔｏｂｅｉｎノ(､)．
陸ｔｓＥＣＶｂｅａｎｅｌｅｍｅｎｔｓｕｃｈｔｈａｔｓＩ(､）ＣＪｍ・ＷｅｅｘｐａｎｄＯ(s）＝
ELsjTjinZ[T]withsb≠OandchooseapnmenumbcrpsothatPl３．．
Thentheelement了＝１②ｓｏｆＢｋ＝ｋ②zCisnotcontainedinPi＝ノＢＡａｎｄ
５沖)BjEJ碗必＝JW0・HenccJ(凧)ＢｋＥＢ(碗)．
WenowTecaUthat［(PB")(､)]ｚ＝（/,ｇ)BJM・(PB刑)(､）（Ｃｆ[2,Proofof
Proposition(3.1)])andchooseanelement5＝ご(Ｘ，Ｙ,Ｚ）ＥＣＶＶｓｏｔｈａｔ
§･[I(､)]2Ｅ(ノ,ｇ)Ｃ･Ｉ(､)．ThenifpisaprimenumbeTandifpl5(0,0,0)，
theelement§＝１②：ofBkisnotcontainedinMルー(Ｘ，ｙ,Ｚ)ＢＡａｎｄ
Ｆ･[”)Ｂｋ]ﾕﾆ(/,ｇ)Ｂｋ･I(､)Bk・Thuswehave[I(､)｡k]2＝(/,ｇ)｣ルノ(''Ｍｋ
ｉｎｄｋ，
Ｌｅｔｕｓｃｈｏｏｓｅａｐｎｍｅｎｕｍｂｅｒｐｓｏｔｈａｔｐisanon-zero-divisoronD＝
Ｃ/ＸＣ＋J(､）（thischoiceispossible,becauseAsszDisanniteset).Thenas
Xisanon-zem-divisoronC/I(､),wegetacommutativediagmm
O
I
O-C〃(､）当Ｃ/J(､）－Ｄ－Ｏ
ｐＩＰ↓ｐ１
Ｏ－Ｃ〃(､）当Ｃ〃(､）一D-O
withexactrows・ConsequentlyXisanon-ze｢o-divisoronBk/I(､)Bk，while
Pl:＝､/7T祠万万because〃二J(､)BにＢｋ;theにfbrc,thegradedidcalJ(､)Ｂｋ
ｉｓ且-primary(noticethatdimBk/J(､)Ｂｋ＝１，ａｓｄｉｍＢｋ/PX＝Ｉ）andsowe
havCthatPW")ＥＪ(碗)Bk、
SummarizingtheaboveobservationsandchoosingapnmenumbeTpsothat
PISb,Ｐｌ《(0,0,0),andpEUqE…q,wegetthatpF1＝”]Albandthat
[pFml]2＝(/,g)ＡＩ`W1fbrthehcldk＝Z〃Z・Th…condasscrtion(2)now
fbUowsfiommeequality[pＦ１]ユー(/,g)｣ルpF1simUmlyasinthcpm･fof
Pmpositionユ2．ThiscompletestheproofofTheo肥、３．１．
４．PRooFoFTHEoREML1ANDCoRoLLARY1･Ｚ
Ｌｅｔｎ≧４beanintegersuchthat〃差Ｏｍｏｄ３ａｎｄｌｅｔｋｂｅａｌｉｅｌｄ・In
thissectionweexploretheprimeidealp＝ｐ(7〃－３，（5〃－２)恥８〃－３）ｉｎ
』＝ｋ[[Ｘ，Ｙ,Ｚ】]・ThepurposeistoproveTheoreml・landCoroUary1.2．
Firstofallr己callthatpisgeneratedbythemaximalminorsofthematrix
（顎二葱二|）
(Ｃｆ［5])．ＷｅｐｕｔＱ＝Ｚ３"-1-Ｘｚ"-1ＹＺ，６＝Ｘ３ｎ－ｌ－ＹＺｚ''-1,ａｎｄｃ＝
ｙ３－Ｘ風Ｚ〃（hencep＝(α,６，c))．NoticethatanypairofzJ,６，andcfbrms
aregularsystemofparametersinAp，sincethereistheobviousrclation
(4.1）Ｘ"α＋ＹＺｂ＋Ｚ２"-ＩＣ＝０．
ＮＯＮＣＯＨＥＮ･ＭＡＣＡＵしＷＳＹＭＢＯＵＣＢＬＯＷ･ＵＰＳ
３８９
Ｌｅｔ
ｄ２＝Ｘ"-ｌｙ５Ｚ"-ｌ－３Ｘﾕ"-1ＹｚＺ２'1-1＋Ｘ，"-2ｙ＋Ｚ５"－２，
．３＝＿Ｘ]"-ZY7＋２Ｘ"-ｌＹ５Ｚ３"-1＋Ｘ４"-ＺＹ４Ｚ〃
－５jr2Jv-lY2Z4"-1＋３Ｘ５"-ＺＹＺｚ'ｌ－Ｘ８"-3Ｚ＋Ｚ７"－２，
．』＝Y8Z2"-2-4X"Y5Z3"－２＋Ｘ４"-1Y4Z"~!＋6Xz"y2Z4"~ｚ
－４Ｘ５"－ｌＹＺｚ"－１＋Ｘ８ｌｕ－２－ＸＺ７"－３．
ThenadirCctcomputationeasilychecks
X"d2-y62＋Ｚ'w-lqc＝０，
(4.2）Xn-Ib2c＋ｑｄｚ－Ｚ"-ld3＝０，
Xd3＋Ybc2＋ZdB＝0．
HcncediEp(2)ａｎｄｄ３,ｄｊＥｐ(3)．
ThenextlemmaisaspecialcaseofmuchmoTegeneralresults(Ｃｆ[3,CoTol‐
1ary(2.6)]ａｎｄ[4,Theo妃ｍ(5.4)]),however,webrieflygiveadiJEctplDoffbr
thesakeofcomplcteness・
陸mma43.ｐ(2)＝p2＋(｡z)α"。p(3)＝pp(2)＋(｡〕,｡』)．
ＰｍｑｆＡｓ(Ｘ)＋p2＋(｡h)＝(Ｘ)＋(】'６，y4Z2"-1,y3Z3n-l，Ｙ２Ｚ４"-2,Ｚ５"-2)，
ｗｅｇｅｔ
２八(｣/(Ｘ)＋pZ＋(の))＝３(7〃－３)．
Ontheotherhandbytheadditivefbrmulaofmultiplicityweget
［ん“/(Ｘ)＋p(2))＝24,(ＡＰ/p2Ap)･2A(A/(Ｘ)＋p)＝３.(7〃－３)．
Ｈｅｎｃｅ２Ａ“/(Ｘ)＋p(z))＝2Ａｕ/(Ｘ)＋pz＋(d2)）sothatwehave（Ｘ)＋p(2)＝
(Ｘ)＋p2＋(ぬ),whichimpliesp(2)＝ｐ２＋(｡Z)＋Ｘｐ(2)．Ｔｈｕｓｐ(2)＝p2＋(｡z）
byNakayama，slemmaConsequently
（X)＋pp(2)＋(d3,.,＝(X)＋(Y，,Y8Zm-2,ｙ７Ｚｚ"-1,Y6Z3"-1,
y5Z4"-Z，Ｙ３Ｚ５"-2,ｙＺ７"-3,Ｚ７"-2)，
whence
２Ａ似/(X)＋pp(2)＋(ぬ,｡』))＝6(7"－３)．
BeCaUSe2A仏/(Ｘ)＋P(3))＝均,(ＡＰ/P3Ap)･（』仏/(Ｘ)＋P),ＷｅｈａＶｅ
２`(｣/(x)＋p(3))＝2`Ｗ(x)＋pp(2)＋(｡〕,｡』)）
and,thus,ｐ(3)＝pp(2)＋(｡],`』)asrEquirCd
Pmposi回０，４．４．，２㎡"９Ａ/(c)＋p(3)心〃αＣＤＡe"-MzcmUm'．
ｎ℃q八匹ｔｅｈ“/(c)＋p(3)）dcnotethcmultiplicityofXAinA/(c)＋p(3)．
ＴｈｅｎａｓＡｐ/ｃＡｐｉｓａＤＶＲ,wegetbytheadditivefbrmulaofmultiplicitythat
e3`Ｗ(c)＋p(3))＝3(7"－３)．
Ｏｎｔｈｅｏｔｈｅｒｈａｎｄｗｃｇｅｔｂｙｌだ、ｍａ４．３anditsproofthat
（Ｘ)＋に)＋p(3)＝(Ｘ)＋(Ｙ３，ＹＺ７"－３，Ｚ７"-2)．
．Ｎ己田Ⅱ些胃⑪管．（、－踵ト）・刷冨Ⅱ（寺（関・負・淫）「膏）ぞ『宮君：：ご普昼豊爵
（鼻員」◎曲ご目』巴ロ①Ｅ⑩一⑭◎２一画詞⑭豊の召《篭加亘』一己目』⑭色目ヨロ⑭冒疽旦回摺
里」宮昌一。、這已目竜曾⑭駅冒』：冨由ｏｐＵｍｏＣ二○局。⑭」寡『、冒曾ｃ⑭ニト語Ｐ。。』
。Ⅱ芸・菌邑⑪【目⑫恩已冨里自口◎憲一◎官⑪邑②冒冒ご日．、．『昼昌。』。Ｕ忘宕。（『
胃疸目曰恒の二』⑭。ｚ回呂（○二）均曙８巨窃５二．伊函巨恒目恒⑪二⑫ｃｚ呵呂（畠）崎“一旬二口息
。『。『【■⑪凶。⑭二・田』。』。。』ロ
④己切⑩司己日ＣＯ偶ニト．寸．寸目◎冨圀色。占宣訂冒甸烏冨‐色③二．Ｑ一．兵呂（、）曰＋（。）｝『
台＄シ◎』。⑫ヨ僧．（ロ十（※）一再）『》・身Ⅱ（、－竃ト）烏Ⅱ（（ミ’（淫）言）『》菌昌。⑫
場口甸。⑭二㎡・御巨昌。⑭二国円託二ｍ目酒託呵百回。“］ニー員⑪二。。■旨臣旨二『Ｎ目曾。⑪ニト諾ムロロ官
‐①冒⑭。ｚ智（ロ）宮曽眉昌Ｐ．『『【息』Ｃｕ二・閃』ＣｐＣ旨⑭溺甸』曾垣⑭昌刷・内呂◎富的。：占
（Ｕ愈涜）己。目旦（〔Ｉ属｝ＮⅢ鐘
］①函⑭」夢（⑭○員⑭二ワ⑭由出国ロ函昌‐『
：日【巳□Ｎ冨呵ｊ鏡浅沼已冨（。〈※）□。Ｅ》、Ⅲ害国是．（烏）白山亀⑫Ｅｏ２巳
⑭○口⑭二」房
』十島。。「亀一‐．ごニー‐高）凶・‐｛了．一角（》）。ｖ脈高一・（一‐）＋ざⅡ言
．（．Ｎ）で・日。Ⅷ》蔦。．‐込一‐・・二一‐萬亘一一‐・一員（》）．ｖ脈員一・｛『‐）＋智員
⑭息二②」参の８日圃二僧昌已目今（》川罠冒瑁
①邑勝二）》Ⅵ』Ⅵ。二烏○出巳邑川』（－－局）』◎亘川（》－》）堂一旦』８』】。ｐ⑭』⑫田
．（已凶百．日。Ⅲ
…。．‐亀丁．。『一一‐局）日（Ｔ・）皇（》）州．（’１）＋軍※Ⅱ
⑤（Ｑ－１高Ｎ＋口唇淫）□も⑤‐旦寅』（『－）＋等勇Ⅱ島Ｑ１良十》、臭
］甸二］（［・》）計二］⑩凹瞥客
一身。》１塁済・』（凸軋）Ⅱ骨口亀艮８口扇已目（□駒）己ＣＥＣⅢこ、亀臭十》雪涜８眉②
臣。■Ｐ・』畠⑭冒召⑭尹筥⑫○口面》畠］Ｅ夢『＋写刷Ⅱ己⑭一宮夢ごロ甸嗣川凰蔚昌⑭日■陽ぐ
ご①揖二。①』の⑱（角－廷ト）垣Ⅱ
（（【Ｉ塵ト）ＮＮ歸昌（環）、『）更》Ⅱ｛（ペ．。崇涜）「『）『》咀昌・牌．更昌８５号陽』昌畠。』甸負」臣＆
ＮＮｐ目’（鶉湯目○昌争（Ｑ皐浸）己◎日（【Ｉ富（）ＮＮⅢ量芯四②夢５二』．（。←鶉）ロ◎日智
Ⅲ寄凹一塁』旨回（Ｋ）己。日刷Ｉ噌卜ＮⅢざ冨昌②。君。Ｚ・（。）ａ叫べ⑪日。”』＆寄倒
Ⅱ寸Ｕ倉【‐臭十暫８５四・（ＮＮ）ご◎【目。Ⅲ、倉員十暫黄①弄圏：《（（『寸）』・）
（副Ｎ）己ＣＢＣⅢ鈩門十口冨蔑已目ＣⅢ可・亀藍十暫副璃湯目８。息昌。剣Ⅱ＆』目
．。Ⅱ曹旦Ｎ＋身。亀罠＋等罠、
】呵呂》（Ｎ・寸）
回巨◎富コワ①冒二・己倉蔑：ご目。Ａ皇Ⅱ差・可司・コミ曙息言きき。と
．『『ＥＵ８①二牌⑪毎．』・旦語ご綴』⑭』輔⑭」夢湧◎Ｚ
・凹昌疽歸星。⑤○口】二，臼⑪二○Ｕ句②。』Ｃ巨自面○
□ひ、
ざ后出
一【｝巳＋（。）一『⑬。白．．（（、｝且十（・）一雨）更守八（（［）二十（』）＋（浅）一『）更》』罰・彦』昌一。⑰
（一十（、－踵ト）、Ⅱ（（、）己十（。）十（鶉）一再）ご
四餌ペア員昆三型□冨迂窯定（】■湯屋望。』〔己．⑫
鑿鑿鑿
…霧懸鷺騨璽顧議襲鱒ﾛ鰯轤
＝－－■
鍔』刷鐵fi5蝋iUi鵠露3F 萬鐸時
T･UDB￣亡Ｂ
ＮＯＮ－ＣＯＨＥＮ･ＭＡＣＡＵＬﾍＹＳＹＭＢＯＬＩＣＢＬＯＷ－ＵＰＳ
391
ｗｅchooseapnmenumberpsothatpZmax{Ⅳ’２ｍ/3｝andchoosean
clcmcmhofpF，)sothatf劇催いk/(X,c,力))＝3p･(7"－３),whercc＝
Ｙ３－Ｘ"Ｚ〃ｉｍｄｋ（thesecondchoiceispossiblebyTheorem1.1).Thenboth
thepairsc,ｈａｎｄ/,gsatisfyHuneke,scondition(2.1),Whencewegetby[Z，
Proposition(3.1)(2)]that
Ｇ＋＝
(ｃＴ,ＡＴ］P)Ｇ
(/Tm,gTm)Ｇ
whereＧ＝ｑ(p)ａｎｄＧ+＝Ｚｊ>oGi,Consequently/Ｔｍ,gTmisaG-regular
sequencebecausesoisthesequencec7,ｈＴ３ｐｂｙｌ２,Proposition(3.7)(3)]、
Ｈｅｎｃｅｗｅｈａｖｅ
（/,g)np11＝(/,g)･p1-'''１
fbralliEZ,while(/,g)。pｒ碗－１)by[Z,Proposition(3.4)];thus,p1)＝
(/,g)･p1-“)fbrallj〉Zm-LThisparticularlyimpliesRs(p)＝
A[{p11Tj},≦…_ﾙﾉT碗,9丁碗]sothatwChaveAT3,Ｅ』[{p11Tj}!≦…]．
ThusAT"EAI{p1)Ti}!≦i＜3小because2ｍ≦3pbyourchoiceofp・Sincc
317〃－３byourstandardassumption,wegetbyCoroUary2､６ｔｈａｔＲ３(p)＝
』[p7,ｐ(z)Ｔ２,ｐ(3)Ｔ３]．Consequentlyoneoftheconditionsstatedin[4,Theo‐
rem(6.1)(2)]mustbesatishedfbrthedataα＝恥β＝１，γ＝恥｡'＝２"－１，
β'＝２，a､。γ'＝２"－１，whichisobviouslyimpossible・ＨｅｎｃｅＲ`(pk）cannot
beaNoetherianring,ｉｆｃｈｋ＝０．ThiscompletestheproofofCoroUary(1.2)．
Ｒｅｍａｒｋ４､５．丁hesameproofworksfbrthcfbUowingexamples,too・Ｌｅｔ〃Ｚ
Sbeanintegersuchthat317〃－１０ａｎｄ〃菫－７ｍｏｄ５９・唾ｔｐ＝
p(7〃－１０，５"z－７〃＋１，８〃－３)．ＴｈｅｎＲＳ(p）isnotaNoetherianring,ｉｆ
ｃＭ＝0.Thesimplestexampleofthiscascisthepmmeidealp＝ｐ(25,91,37）
＝(Ｘｌ４－ＹＺ７，Ｙ３－Ｘ５Ｚ４,Z11-X9Y2）ｉｎＱ[[Ｘ，ｙ,Ｚ]]・ThecorTcsponding
matrixisgivenｂｙ
（李郷）
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DEPARTMEluToFMATHEMATucs,MEuJHUIwuvERsuTY，HIoAsHIMmh､TAMA-Ku,KJLwAsAnu-sHl，
JAPAN
GRADuATEScHooLoFScIEUwcEApIDTEcHNoLooY,CHuBAUNIvERsITY､YAYoI-cwo,CHIBA･sH1、
JAPAN
DEPARTMENToFMATnuEMATIcALScIENcEs,TOxJuUNuvERsITY,HlRATsuKA-sHu,KANAcAwA，
JAPAN
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