CHARACTERIZATIONS OF UNRAMIFIED SIMPLE
SPOTS AND ABSOLUTELY SIMPLE SPOTS
OVER REGULAR DOMAINS
BYMAMoRu FURUYA
IntrOduction． The main purpose of this paper is to prov6 that the fbnowi皿g state− ment holds：Let I be a regular domain and let A＝1［￥1，…， Xn］be the polynomial 血gin〃variables Xl，…， X。． with coe伍cients in I． AS’S’i me that D⊆磐are prime ideals of．4， a＝（F【，…，Fm）・A is an ideal having n as a’ prime diVisor and D∩1＝（o）． Set R＝（．4／α）・C（撃1a）−1． Then the fb110wing statements are equivalent：（1）Ris an absolutely simple spot over 1．（2）Ris smooth over L （3）The differentia1 module M（」Rμ）is a free R−module and rankハイ（R／1）十height D＝n． （4）rank（J（且，…，Fヵ） modulo率）＝height D． （S㏄Theorem 4） In§1， we introduCe the notions of spots and unramified simple spots over a regular domain and prove some result which is generalization of Theorem 4．2 in［3］． The results in§2 are generalizations of some results i皿［1］． In§3， we introduce the notion of absolutely simple spot and・prove the Theorem 4 which is the main result、 At last， we shall investigate the relations between the absolute simpHcity in our sense and the absolute simplicity in the sense of［4］． Acknowledge〃tents． I wish to express my sincere gratitude to Professor Y． Kawa− hara for his pr㏄ious advices and co皿stant encouragement． Terminologア． All rings in this paper are commutative Neotherian血gs with iden− tity eiements． The terminology is in general the same as that in［2エ A血ghomomorphism will always mean a血g homomorphism which sends identity ele−
ment to identity element． Let I be a ring． A ring R will be called an、1−algebra if lis an operator domain of R and there exists a血g homomorphism f from∬into R』唐浮モ?@that the operation on R of an element a in I is given by the rule ar＝∫てのr fbr r in R． f is called the structural homomorphism． Let l be a ring， R be an I−algebra with the structural homomorphism f and let abe an ideal of R． Then we denote by R∩athe idea1ノ㌔1（a）of l and denote by M（R／1）the dif正brential module of R over l in the sense of［5］． Let p be a prime ideal of a ring、4． Then we shall denote by A・C（p）−1 the quo− tient ring of A with respect to p． We say that an integral domain is a regular domain if it is a regular ring．［5］
6
CHARACTERIZATIONS OF UNRAMIFIED SIMPLE SPOTS
§1・Unramified simple spots Let I be a regular domain． We say a local ri皿g R is a spot over I if there eXists afinitely generated ring A oVer l which contai皿s∬and there exists a p血e ideal P of A such that R−A・C（p）−1． If a spot R over I is a regulaf local ring， then we say that R is a simple spot over 1． − Let R be a spot over l and let f be the canonical homomorphism fro皿1 into R． Then we shall always refer to R as the 1−algebra with the structural homomorphism f． Let R be a simple spot over 1 with the maxima1 ideal IM． When ln班キ（o）， we denote by｛a1，…， at｝aregular system of parameters of I・C（1∩班）−1． Then we say that」R is an unramified simple spot Over I if either ∫∩班＝（o）， or ｛9（a1），… ， g（at）｝can be se1㏄ted to be members of a regular system of paramet6rs pf R， where gis the canonical homomorphism from I・C（1∩班）−！into R． Clearly this definition is independent of the choice of the regular system of parameters｛la1，…， qt｝of 1・C（1∩號）−1． THEOREM 1． Let R ろe an unra〃2ifiedぷ」〃rple sp ot oγer aハegular do〃tain I and let ‖3●θαpr加te idea1（ゾR． ・び1μ∩ll3 is a regular ring， then R・C（率）−1 is an unra〃zi− fied simple spot over L PRooF． Let班be the maximal ideal of．R．． Wg set甲∩1・一 p，．班∩1・・q and J＝． 1・C（q）−1．Then．Jlp・J＝（nyp）・αq／p）−i is a regular loCal血g． Therefore p・Jis gen−． erated by a subset｛d，，…， at｝（’＝height p）of a regular system of pafdmeters of J・Since P・11・C（P）−1＝（al，…， at）・1・C（P）−1，｛4，，…， ai｝is a regular system of para．m− eters of I・C（p）−i． By our assumption on R， we see that｛g（a1），…， g（at）｝is a sub・・ set of a regular system of parameters of R， where g．：」→、R is the canonical homo− morphism． Thus R／p・R＝R／（g（ai），…， g（at））・R is a regular local血g．．Therefore R・C（撃）−1／（9（a1）， … ， 9（at））・R・C（甲）−1＝（RIP・R）・C（割3／P・・R）−1 is a regUlar local ring． Let f be the canonical homomorphism from 1・C（p）−1 into R・C（撃）−1． Then， since （g（al）， … ， g（at））R・C（率）『1＝（ノてal）， ．…， f（at））R・C（撃）−1 and he逗ht （g（al）， … ， 9（・・））R・C（8）−1−t・we have th・t｛，f（a・），…， f（d・）｝i・a・ub・et・f a regul・・sy・te血 of parameters of R・C（率）−1． Thus we complete the proof． COROLLARY・ Let R胞a〃1’nramii17edぷ伽ρたぷρo’oγθ’a1）ε凌え’ぬl do〃lain．工 Then R・C（8）−1∫ぷ・n unramifadぷ卿le sp・t・ve〃f・r eve・y prime ideal as ・f R． We shall use later the next two Lemmas． LEMMA 1・Lε’1お． a・eg〃la・・d・main a〃d・let A−1［X・，…， Xヨbe the P・lyn・mial ・ing・in・n va・iableぷX・i『・…，工W乃c・e177cients in L Then A・C（P）−1∫ぷan un・amified ぷ’〃rple spot・ve’r 1 f・・ eyery prime ideal P〈ザメ． ・ PRooF． We can obtain this proof in a㎞ost the same way as Theorem 40 in［2］． ’ 啄M．FURUYA
7 Therefore we shall omit the proof． Let l be a regular domain and let A・−1［Xl，…， X．］be the polynomia1血g祖π variables Xi，…， Xn With coeMcients in L Let Fl，…， Fm be elbments ofん Th㎝ the Jacobian matrix（∂F 1∂Xi）wm be denote by J（Fl，…， F．）． Assume thatαis a皿ideal of．4 which isα∩∬＝（o）and率is a prime ideal of A containillg a． Set R＝（．4／a）・C（131a）−1． Then LEMMA 2． （1）∬〃陀晒陀吻1・物dule M（Rμ）’∫afree R−〃2・dule and・’rank・砥R11）一ち then there exist n−r elements F1，…， Fn−．げαぷuch that rank（」但，…， Fn−．） modulo宰）一π一r． （2） びthere exis’η一r ele〃昭η’ぷ且，… ，」Fn＿7げαぷuch that rank（」（F，，・「・・， Fs；r） m・dul・ 8）−n−r，血πrank M（R1∫）≦r a〃∂｛4・，…， at，凡…，Fn−．｝匡ぷas〃bぷet ・f αアθ9〃吻卵τε〃！ ・f pa・a〃ieters ・f A・C（8）−1， where｛a・，．…，α’｝匡ぷαγθ9〃』鋼絃m of para〃tetersげ1・C（∫∩撃）−1． PRooF． The proof of（1）and the五rst ass《rtion of（2）is obtained in almost the saine way as the proof of Theorem l in［1］． Therefbre we shall prove the second assertion of（2）． Let撃∩∬＝p，」＝∫・C（p）−1 and let｛41，・・．・，侮｝be a regular system of parameters of 」． Furthermore， let P＝A・C（宰）−1 and let貌be the maximal ideal of P． Then it su伍ces to pτove that the residue classes of the elementsα1，…， ate t ＃−r F，，…，、Fn−． modulo M2 are linearly independent over P／Se． LetΣγia‘＋Σ．zjFi t se2 ゴ＝1 iコ1 w地y‘，Zi∈P． Th㎝， sin㏄∂1∂X．（Σyiai＋Σz∫乃）∈況，Σz∫（∂ち1∂石）∈班（1≦m≦π）・ Therefbre， by the assumption rank（」（Fl，…， Fn−r）modulo＄）＝π一r， we have that る∈班（ノー1，…，〃一’）．ThusΣγ‘α‘∈貌2． Smce｛a・，…，at｝is a subset of a reg− Ular system of parameters of P by Lemma 1，’we have yi∈貌（i＝1，…，’）． §2．Criteria for皿ramified simple spot Let 1 be a regUlar domain and let A＝1［X，，…， L］be the polynomial ring in n variables X，，…， Xn wi血coeMcients in L Assume that D⊆撃are p6me ideals of A，a is an『ideal haVi皿g D as a prime divisor and that￡∩1＝（o）・ Let R＝（Ala） ・C（4｝／a）−1and let班be the maximal ideal of R． Then we have that rank M（RII） 十11eight D≧n． THEOREM 2． こJnder’んsa〃te〃otationSαぷabove， the／b〃bwing sta’θ〃re〃’ぷ乃o〃ご （1）if M（RII）匡ぷa∫たθR−〃iodule and rankルt（」R／1）十height ￡』＝〃，’舵π 、R iぷ an un・amified si．〃rple spot・ver∬αη40㈹（1）匡ぷ卿arably generated over e（1）・ （2） るrRis an〃hra〃iifiedぷimple spo’over l and R1批∫ぷぷeθarably gene「ated ove「 ρ（∫μ∩班），’乃θπルf（Rμ）iぷaノアθθR−〃todule and rankハ4（R11）十height D＝η・ ’（1）Let R．be an血tegral domain． Then we shall denote by e（R）the quotient field of R．8
CHARACTERIZATK）NS OF UNRAMIFIED SIMPLE SPOTS
parameters of 1・α1∩撃）−1． Fn一プof a such that rarik（」（F，， Fl， melements G、， F，、．，，G、，…， ￡・P一α・P一伍 m』…M／a・P−（a、， ramified simple spot over五 Next we consider the second assertion of（1）． Since（∼（R）＝e（．4／D）and tr． deg． ρω（1（AID）十height D＝n， we see that tr． deg． Qω（∼（R）十height 9・＝n． Thus rank M（」R／1）＝tr． deg． Qσ｝ρ（R）． Therefbre ρ（R）is separably generated overρ（1）． （2）By Theorem 2 h1［1］， M（RII）is a froe R−module andρ（R）is separably g㎝一 e「ated・v・・C（1）・Th・F・f・・e・ank・M（Rμ）−t・卿・・…e㈹一・−h卿t O．PROOF．
（1）Let rankルf（Rμ）＝r， P・・A・C（率）−1 and let｛d1，…，αま｝be a regular system of Then， by Lemma 2， there exist〃一アele血㎝ts F，，’…， …，Fn−r）血odulo磐）＝n−rand therefbre｛d1，… ， ai， …，F．一．｝is a subset bf a regular system of parameters of P． Thus there exist …，Gm of the maximal ideal況of P such that｛d1，…， ai， F，，…， G伽｝is a regular system of parameters of P． Since height D・P＝〃−r， ， …，Fn＿r）」P． Therefbre R＝P／（」肩， …， Fn＿r）P is regular． Since …，at， G1，…， G”）R alld dim R＝t十m， we have that R is an皿一 COROLLARY． 〃「i’h t乃eぷa〃昭〃otationぷaぷ加Theore〃12， we a∬卿昭吻τRis a〃 i・t・g・al・d・m・in・〃M（R／∬）i・afre・．R−％肋鋤ρ（R）z∬epa・aゐly generated。，。． ρω，τんπR∫ぷan〃n・amii17ed si，〃rple sp・τ・昭’エ PRooF・ Si lce（］（R）＝（2（AID）， tr・deg． Qω（∼（R）十height D＝・＝n． On the other hand， we have that rank M（R／1）＝tr． deg． Q（1）0（R）． Thus rankハ4（、Rμ）十he㎏ht D＝・n． Therefbre， by Theorem 2， the assertion is proved． §3． Absolutely simple spOts Let R be a spot over a regular domai11∬and let J be a regular domain which has a structure as anみalgebra．‘ket乃be the canonical homomorphism from R into R⑧iJ and let 13＊be a p血e ideal of 1∼（〔×））∬ノsuch that h−1（率＊）＝班， where班is the maximal ideal of R． Then．the 10cal血g（R⑧1」）・C（pt＊）−i is cailed an extention of ．R・・er 」・A・p・t R・v・・．a・eg・1・・d・m・i・1i・caH・d・b・bl…ly卿1・・if・f・・any regular domain J which has a structure as an I−algebra， any extension．of R over J『 iS’ a simple spot over J： ・ THEOREM 3・ ”「ith’舵ぷa〃le notations aぷ加Theore〃12，ゲRis an absolutelyぷ加2− ple sp・’・ver 1， then R∫ぷa〃unramillfedぷ卿le sp・t・γθ・エ PRooF． We may assume that l is a regular loca1血g・with the maximal ideal m＝1∩競・・and dim．1＝’＞0． Let｛al，…，α，｝be a regular system of parameters 6f五 First off we shall prove that・．α1．庄班2． Assume that ai∈班2． Set J＝∫［b］晒th●2＝al． Then， shlce J is integtal overちJis a regular local r㎞g． The extension R＊of」R over J is R⑧1」＝R［勾・By assumption，」R＊is a simple spot over J．・Let｛y1，．…， ル｝be a regular system of parameters of R． Then， smce the maximal ideal i疏＊of、
M．FURUYA
9 ア ’ ア R＊is generated byゐ，γ、，…，ル， there is a non−trivial relation●∂o＋Σγ《c‘−b・Σア遁 ア ’ i＝1 ’ まコ ＋Σア￡乃εξ」with ci， di， eii∈RFand ci−O if cゴ∈班． Since 1， b are 1inearly independ一 輌，ゴ＝1 ent oveLR， we have co rΣγ‘φ，Σyici∈搬2， hence all ci must be zero， which． is a contradiction． Thus we prove ai庄￡m2． From this， al can be se1㏄ted to be a melhber of a regular system of． parameters of R． Let hl be the canonical homoinorphism from l onto nyail， li＝1／all and let lfti be the maximal ideal of．五． Then．the regular domah1五can be regarded as the I−aigebra with the structural homomorphism．乃1． The extension RI of R over五is R1α1R． Thus we have that R1＝」R／a1R， d㎞ 」R＝dim R1十1， M1＝（a2，… ，αオ）1， and RI is an al）solutely simple spot over五．． In the same way， h，（α2）can be sel㏄ted to be a member of a regular system of parameters of RI and the extension R20f RI over Ii∫a21i is R，／h1（α2）R1． Let h2 be the canonical homomorphism臼om l onto．l／（α1， a2）1，ム＝五／α必and let m2 be the maximal ideal of、1，． Then we have that R2，−R1（α、， a2）R， dim R＝dim R2＋2， 12＝11（α1，α2）1，1飢2＝（α3，… ，α∂1，and」R2 is an absolutely s㎞ple spot over I，． By this method we eventua皿y obtain that R童＝R／（al，…， a∂R， dim R＝dim R汁’， It＝1／m and・R， is an absolutely simple spot over the丘eld口． 1・et g be the canon− ical homomorphism加m R onto R， and let｛γ1，”直，Ym｝be a set of elements of R such that｛g（ア1），…， g（γ餌）｝is a regular system of papameters of R，． Th㎝｛a1， …，at，ア1，…，γ伽｝is a regular system of parameters of R． THEOREM 4． 〃Tithτ舵sa〃絃notationぷas in Theore〃12， the fo〃ρ｝vin9／bur condi− tionぷare equivalent to eaeh other： （1） （2） （3） （4） R∫ぷan absolutely 8吻ρ1βspot over正 R 匡s 3〃iooth over l in ’乃θ ぶθηぷe ｛ゾ［2］． M〔Rμ）∫ぷ〆｝θθR−〃2α加た励rankハ4（R11）＋height D一力． rank（」（且，…， Fm）〃todulo 13）＝height￡』， whereα＝（昂，… ， Fm）・∠4． PRooF．（3）⇒（2）． By Lemma 2， there eXist Fi，．…，Fn一夕（r−・rank M（R／1））i⑪ ・such that rank（」（F，，…，F．−r）modulo撃）−n−’． By the same way of the proof of Theorem 2， we have a・．4・C（甲）−1＝（呪，…， Fn．．）A・C（率）−1． Therefbre（2）fbllows加mTheorem 640f［2］．
（2）⇒（3）．By Theorem 640f［2］， there exist F，，…， Fm inαand D1，…，1）勿in Der 1（．4，．41α）such that．α・ン4・C（撃）一』（F，，…， Fm）A・C（墾）−1 and det（1）iFj）庄申／α． Therefore we ha鴨immediately that rank（」（F，，…， Fm）modulo雫）＝m．’By． Lemma 2，we have rankルt（RII）≦n−m． On the other hand， sin◎e the unmixedness theorem holds in A・C（撃）−1， m ・＝ heightα・．P＝height． D， where Pr．4・C（率）−1． Therefbre rank ハ41〔R／1）十height stA」』…≦〃， and hence rankルt（Rμ）十height￡』＝n． By．正￡rrirna 2，｛Fl， …，FSi｝is a subset． of a regular’system of parameters of P． Thus（Fl，…， Fh）P ＝a・P＝D・P． Therefore． R＝（A／D）・C（墾／D）−1， and e（R）＝＠（Atn）and therefbre tr． deg． Qω0（R）十height D＝n． Hence tr． deg． Qωρ（R）＝rank」腰（R／1）． Therefbre M（R／1）10
