REMARKS ON THE CHARACTERIZATION OF SIMPLE
SPOTS OVER REGULAR RINGS
BYM▲MoBu FURUYA and ATsusHI ARAKI
Introduction． An illtegral domain S is said to be an afnne rillg over a subdomain l when l is a Noetherian domain and S is finitely generated over ∫・ If s is an af五皿e ring over l and if q is a prhne ideal in s， then the quotient ring R＝S・C（q）−1 D of S with respect to q is said to be a spot over 1．If a spot R is a regular local ring， then we say that R is a 8励pZθspot． We say that a l Noetherian domain 1 is aγθσ％Zαゲ吻σif the quotient ring 1・σ（P）−1is a regular 1・ca1・ring・f・r each prime ideal p・f 1． If盈is a spot over a regular r血191and if班1s the maximal ideal of R， then 1己is a spot over 1・0（∫n⑰）−1． Let R be a simple spot over a regular ring l and let m be the maximal idea1 of R． We say that R is unraMified with respect to 1 if either R con・ taills the quotient丘eld． of∫， or a regular system of parameters of 1・0（1∩i肌）−1 can be selected to be the members of regular system of parameters bf R． 111the contrary case R is said to beγα彿鏡θd with respect to∫． A simple sPot R over l is called an eqnrαmifie（1 8i？nple spot （rαmified simple 8poの over∬if it is unrami丘ed（rami丘ed）With respect to∫． Let R be a rami丘ed simple spot over a regular rillg∫and let貌be the maximal ideal of R． We say that R is of order t（1≧t＞0）if Z一τelements of the members of a regular system of parameters of 1・0（1∩班）−1 can be selected to be the members of a regular system of parameters of−R and call be selected no more， where l＝rank 1・0（1∩班）−1． Let R be a rqmified simple spot of order t over a regular rillg 1，班the maxi血al ideal gf R and let｛π1，…，π‘｝be a regular system of parameters ・fτ・C（∫・SW）−1・Then畢＊＝（π・，…，π・）R＝（1∩皿）R． Theref・re， rank」・C（∫。班）一・ −dim響｝＊十ヨ［）尾2／班22）こ＝＝t． Let A be a commutative rillg alld B a subring of．4， then we shall denote by 1匠（A／B）the d遊erential mqdule oC A over B． In this paper，「we generaUze the Theorem l in［4］． ’ §1・Le．t S be a commutative． rihg with 1， and let 1 be a subring of S c°ntaining 1・Let Ep be a p・ime idea1・f 5 such that∫∩率二〇， and・et・S＝S／撃． ［23］ “24
MFURUYA AND A． ARAKI
W6． denote by le， K the quotient且elds of 1， S respectivly and set dim兎K＝ア． We assume that（近丘erentiahhOdule 1∬（s／1）is a freeぷ・module of rank n ・With the basis dX1，…，．aXπwhere 4 is the differential operator of 1∬（s／J）． Then we 6an「??垂窒?唐刀@M（ぷ／∬）as the direct sum， that is， M（s／1）ニs砥i㊤…（DS‘urn In this paper， we．de丘pe《一∂P／∂，X乞．（4＝．1六〔．・， n）by the、 formula ∂Fニ（∂R／∂x、）ay、＋…＋（∂F／∂x。）班。 for any element F Of 8． NT且EOBEM 1． Let◎bθαPγ伽θ鋤砿o∫Sc曜吻ing…β．1・θ彦q＝◎ββ，
R＝S・σ（q）−1 a⑫d let m bθ τ九ρ mα励？π戚i《leql O∫R・ Then the−followingconditioπts（1）αnd（2）αreθαμ迦Zθ砿． ・
（1）M（R／τ）isαfree R・？nodule and K is sepαTαbly generated over ic． ﹁ （．2）．欠ひアβexist n−r elem．en．es． P・…r，F・一・of．8 sueh鋤t ・theゲα励0∫ the・’idc・biαn matTix（∂Ri／∂xゴ）（iニ1プ…．㍉，n−r；2”＝1，一，％）is n−r励dul・◎・ PRooF． We’ can obtain this proof in almost the same way as Theorem l in〔4］． Therefore we shall show an out1血e of the proof． AssUm・the c・n（近ti・n（1）． L・t・u・・Set P ＝S・C（⑥）一・and皐＝8P． Then R＝廓， and we．have the exact sequence 一柳・」→（り へP／sp）⑧迦（η∫）−MR／1）一・o by Proposition 90f［3］． Since P＝ぶ・0（⑤）−i， we have M（？！1）二戸⑧さ1ば（Sll）＝P」孜1㊥…㊥PdXrn by P・・P・・iti・n．10・f［3］・nd h・n・e「・nk・M（η∫）rn・On ，th・・ther・h・nd・by our assumption on M（R／∫）， we have R⑧三M（P／1）＝1冊（ρ）㊥A where A is the submodule of R⑭P M（P／1）such that ∠1㌫M（R／∫）， hence rank Im（ρ）＝カーT．Then we have
． （R乃醗）⑧（R⑧PM（P／1））＝（．R／E！Yl）⑧1仇（ρ）㊥（R／m）⑧五 Since（R仰）⑧lm（ρ）is−a’vector space ”bf dimension％−r over．R／皿， there e】dstπ一r elements R1，…， Fh一アin ll｝such that 1⑧dF’1，…，1⑧dF’nrr are basis of（R／m）⑧tm（ρ）over R／皿where d ．is the ’differentia1 operatbr of M（Pr！1）， F，z’奄刀@the’image of 1アi by the・natural』 ?盾高盾高盾窒垂?奄唐香@fromぷmto P． Hence We have ；ank｛（∂Pi／∂X」）m・d◎｝＝n−・・ThuS we Py・ve the cOnClusi・n（2）・ 1Now a詰u血e the co血dition’ i2）． We may assume that det（∂珊∂X」）eD（i＝1ド・t−；％−r；」コ＋i，…，．n） we shall den・te・by・ftジx・・and”∂fl／∂x｝the class、 bf Pi；x・ and∂F・1∂xゴm・dU1・ 畢．We set・1）＝det（∂fi／∂窃）（iニ1，…， h： r；ゴ＝r十1，…， n）， then D is．not eohta血ed in q．． SinCe瓶＝（∂的∂X、ぱ、＋…＋（∂F・／∂X。ぱ。（iニ1；一㍉n一め， we have df，＝（∂ft／∂ω1）dω1十…十（∂∫‘／∂xn）dxn＝0（乞二1，．…，n一り．・Therefore・We can express（i￠」（ゴ・＝r十1，… ，7り as a linear combination of （lxl，… ，dxrREMARKS ON THE C且ARACTERIZATIQN．．OF SIMPL、E SPOTS
25 ．．with the coeMcients in．R， where d is the differentialσperator of「M（S／i）． From this 「’ ’ 』 ・r＝’rank・M（R／∫）＝rE血k M（K／ic）＝djm形K Thus M（R／∫）is a・free R−module and K is separably generated Over k．”・ ／COROLLARYL W批the sα？ne no彦ationsαs勿T九θorem 1，ωθdenote吻m
either仇θP・deσree O∫Kover ic O”九θtranscendence d．egree O∫Kover−・ic αccordingα8 the charactePti⑤励pO∫肋s pO8ε物θ0γZθγ0． Then・t九e following conditions（1）and、（2）’α7θθq％勿αZθπ‡． ． （1）M（Rμ）isαfree R・？nodule． （2） TんeTe exist％r．m elements 1ρ1， … ， Pn＿抗 o∫率8％c九 that t九θ Tαnk o∫ 仇θJacobian？πα‡吻（∂Pi／aX，）（⇔1，…，n−m；ゴ＝1，一プn）is・n−m伽d％lo◎． §2． In this section We use the following notations ●■ 1 ◆●R
●● ・．狐f
P： aregular r並ig with the quotient field k．aspot over∫with the quotient且eldπand dimicK＝r．
the maximal ideal of R alld R餌＝θ． the q｛10tient丘dld of 1／1∩班． 「” the characteristic of岳．THEOREM 2．（1）1∫疏8απ脇ア励ified simple spot andθ．εS separdbly
generated over f， then M（R／1）iS a free R・Module and K is seParαbly gene・ rated OVθr 元． （2） ’Assμ？ne thαt R ‘8 αra？ni∫ied simple spot o∫order t an〔15？飴SθPαOfabl2／ generated over f．’ Then M（R／1）is a free R・伽励θZf㈱d o晦乞∫pis not 2eroαηd p−degree of K over． k is equα1 to r＋t．’ln tゐis￠α8θ， K tsπρ‡ 8θPα㌘αbl！ノ9θnθγatθd over盈｝． PRooF． when∫∩EMti『潤C we may assume that I”is「a丘eld． Then our aSsertiOn follows fro血Theorem l of［4］． Theirefore we’ assume that 1∩班≠0． Then we can consider R as a spot over 1・σΦ）−i where p＝1∩班． Henc6 we have 、y・m、］：・L。、 U、。e、遮R；こ㌫霊∫IC⑩一1㌔・…（’）
rank M（R／∫）＝rank、班／撃＋m2十・a4k M（恥 by［1］， since庇is separably generated over f． Moreover We ha†e、 rank皿／皿2＝rank…mP十EIJ｝2／迎2十rahk班！撃十『班2’ Therefore by（i），（ii），（iii）alld our assumption onθ， we have rank M（R／1）＝r十rank 1・C（p）−1−rank準十皿2／’M2 1f．R is皿ramified， then we have rank 4｝十9皿2／Sn2＝rank 1・CΦ）−1．． ra耳k、M（R！イ｝L：ア： Frgm this we get M（R／∫）is a free．R・module and X is separably lgenerated・overん． （ii） （iii） （iv） Therefore rank M（R／1）＝rank M（K／lc） ＝’r． Thus The first26
M．FURUYA AND A． ARAKI
assertion is obtained． Next let us suppose that、R is a rami丘ed simple spot with order t． If ルt（R／∫）is a free R・module， then we have t＝rank∫・C（p）一一rank耳｝十班2／双2． Therefore we have rank M（R／1）ニr十τfrom（iv）． By our assumption on 1匪（R／1）， we have rank M（K／iC）＝rank 1∬（R／∫）＝デ十t． Ifκis separably generated over h，we get t＝O which contradicts to t≠0． Thus K is not separably generated overんand so p≠0． Therefore we have P・deg鳶K＝rankルτ（幻紛二r十t Conversely assume that p≠O and p−degk頁二丁十t， then we have rank」匪（R／∫）ニT十t＝P−degicK from（iv＞． Therefore rank M（．R／∫）＝rank 1匠（K／ん）． Hence M（Rμ）is a free R−module． Thus second assertion is proved． THEOREM 3． 1∫M（R／1）isα∫ree R・module αnd K i8 8epαγably generαted over k， then R i8α8imple spot． PRooF． Let S＝1［x1，…， xn］and let q be the prime ideal of S such that R＝S・0（⑥）−1．．Let 5』1［Xl，…，Xn］be the polynomial ring such that Ses＝S for some prime idea1畢of 5， where｛X1，…， Xn｝is a set of algebraica11y indepelldent elements over 1． ’ Let文）be the prime ideal ofぷsuch that◎⊃8 and D／lP＝q． We set P＊＝S・C（③一1， 8＊＝…pP＊，戸＝S・0（⑥）一・and畢二8P． Th・n R＝廓． Sinee 1 is a regula・ ring， P alldlP＊are regular local rings（cf．［2］）． L、et血be the maximal ideal of戸． Then from Theorem 1， there exist n−r elements∫1，…，∫n−r in撃such that the rank、 of the Jacobian rPatriX（∂∫‘／∂xゴ）（iニ1，一・， n−r；ゴ＝1，… ，n）is n−r ny へ modulo◎． Therefore， sillce rank P＊＝．n−r， we have iβ＊＝i笈＊and P戊is the regular local ring， where班is the ideal of S which is generated by∫1，…， へ N s モ へ fn−r，班＊＝衷P＊and衷＝鞭． Furthermore製is prime ideal and 4｝＊二衷＊⊃零⊃瓢． ．v へ ら ヤ Thus畢＝衷． 且ence R＝P犀＝P／班is the regular local ring which completes oUr aSSertiOn． ユ） Let pbe a prime ideal ofaring R， thell we shall denote by R・σ（p）−1 the quotiellt ring of R with respect to p． 2） dim p＊十IM2／SIJt2 is the dimension of撃＊十双2／il）t20ver続＝R／i肌． 3）dim∫芒is the dimensiqn of㊧over f．REFERENCES
［1］ R．Berger und E． Kunz：Uber die Struktur der Differentialmoduln von diskrenten Bewertungsringen． Math． Z．77，314−338（1961）．［2］ ［3］ ［4］ ［5］
