Asanuma Teruo

University of Toyama Repository

Hath. I. Toya m a Univ .Vol. 1 4く1 9 91J, 1 6 7- 175N O N- INVAR工AN T T W OD工M EN S工ONA L A F F 工NE DOM A工NSTe r u oA S ANUM Aintr odu ct io nA c o m m utat iv e rin g A w i t h i de nt i ty is s ai d tobe in v a ria nt i fw he n e v e rち is a ring s uch that tw opolyn o mia l rings AtX. . . . . , X n コ a nd B亡Y . , . . . , Y n コin n v a ria ble s a r eis o m o rph ic , t he n A a nd B a r eis o m o rph ic . 工n t he ir wo rk 亡1コ. A b hya nka r, Eak in a ndHe in z e rp ro v e d t hat a n in te gr al do m ain of tr a ns c en d -e n c ede gr ee on e o v e r af iel d is in v a ria nt , a nd a ske dw het he r tw o d im e n sio n al a f f in e do m ain s o v e r af ie l da r e in マ a ria nt . T he pu rpo se of the pr e s e nt pa pe r lsto stu dy t he in v a ria n c e o n s o m etw o dime n sio n alaf f in e do rEtain s o v e r af ield .工n s e ct io n 1 w e stu dy the in v a ria n c e o n,RtZコ,w he r e R is a o n ed im e n sio n al a f f in e do m ain o v e r af iel d k a nd Z is a n in dete r min ate . In pa rt ic ula r,w he n k is a pe rfe ct f iel d, w e gi v e a C O mPlete de -s c ript io n o n a c o m mutat iv e rin g a w h ic h s at is f ie sR亡Z , X. , . . . , Xn コ 筈 BEY. , . . . , Yn l .1 6 8 A S A NtJhIn s eCt io n 2 w e c o n str u ct a no n - in va ria nt tw od im e n sio n al af f in e do m ain o y e r af iel d o f c ha r a cte r-ist ic z e r o.1. エn v a ria n c e o nR亡ZコWe f ir st c o n si de r t he ihv a ria n c e o ntw o d im e n-sio n al a f f in e do m ain s of t he fo r m RtZコ whe r e Z isa n in dete r min ate .L ユ. We r e c al l t he de f int io n of F- rin gs fr o m t21 .Let R c S be a r edu c ed rin g e xte n sio n . We s ay t hatR is F - clo s ed in S i f a ny e le m e nt t E S s u ch thatt2 ,t3 , nt E R fo r s o m epo si t iv e inte ge r n くn mayde pe n d o n tl is alw ays c o ntain ed in 釈 . 工f a r edu c edrin g R is F - clo s ed in a ny r edu c ed rin g e xte n sio n+R c S , t he n R is c al le d a n F - ring . I t s ho ul d ben ot ic ed t ha t, in t he c a s e whe r e a re du c ed rin g Rc ontain s only a f ini te n u mbe r of m in im al prim ei de als , R is a n F - ring i f a nd o nly i f R is F - clo s edin its to tal qu ot ie nt rin g QくRl . 工 n t h is c a s ethe r eis t he s m al le st F- ring in Q くRI c o ntain in g R , w h ic his de n ote d by FくRト1 . 2 . If a r edu c ed rin g R c o ntain s a f iel d ko f cha r-a cte ris t ic p 皇 0 , t he n t he fo l lo win g tw o c o nditio n sa r e equ iv ale nt to e a ch ot he r 二til 氏 is a n F - rin g .くi il R is s em in o r m al o r p ニ 0 .A SA NtJM AFo r the pr o of s e e亡9ト1 6 91 . 3 . A n inte gr al do m ain R is s ai d to be str o ngly土n v a rla n t if t he equ al i tyRtXi , . . . , Xn コ ニ B亡Yl , . . . , Yn コof polyn o mial rin gs alw ay s im pl ie s t he equ al ity R ニB o f c o ef f ic ie nt ring.s R a nd B .1 . 4 . The o r e m.do m ain .Le t R be a str o ngly in v a ria nt integ.r aユ聖主星聖 皇 POIyn o mia 1 互阜畢g REZコ in o n e v a ria b leis in v a r土a nt i fa nd o nly i f R 皇室 旦些 F 一 旦阜些g .PTOPf. Su pp o s e RtZ, X. , . . . , Xn コ ニ BEY. , . . . , Yn l. T he nB c o ntain s R a s a s ubring by 亡5l . F ir st w e cla imt hat B is is o m o rph ic to R亡Zコ i f a nd o nly i f B isR -ris o m o rph ic to RtZl . 工t is s uf f icie nt to sh. w t heTTo.nly i fIIpa rt o f t he cla im . Let a s s u m e B isis om o rph ic to RtZj. The n B is a polyn o mial rin gS亡Tコ in o n e v a ria b le T o v e r a c o efficie nt ringS wh ic h is is o mo rph ic to R I Re pla cin g B by StTコ,w e ge tR亡Z , Xl 一 . . . , Xn コ こ SET , Yl , . . . , Y n コ,a nd he n c e w eha ve R こ S , wh ic h m e a ns t hat B is R -is o m o rph ic to RtZl. W it h t h is cla im , w e s e et hato u r a s s e rt io n is a n im m ed iate c o n s equ e n c e of E23.1 . 5 . Re m a rk . We c a n not e xte nd T he o re m 1 . 4 to t heC a s e Of a po lyn o mial rin g in tw o v a ria b le s a s fo1 -1 70 A S A m A1o w s ニLet R be a str o ngly in v a ria nt inte gr al do m ain .Sup po s e R do e s n ot c o ntain a f ie l d of c ha r acte risticz e r o. The n a po lyn o mial rin g REZl , Z2 コ in tw o v a r-ia b le s is n ot in y a ria nt by 亡4ト1 . 6 . Co r ol la ry . Letdo m ain o v e r af iel d kR be a o n e9fd im e n sio n al a f f in echa r a cte ris t ic p i 0 . Su p -po s e k is alge br aic al ly clo s ed in R . 工f R is n ot旦 pO Iyn o mial 互主些望 旦竺阜旦 k, t he n t he fol lo wing c o n-ditio n s are equ iv ale nt二くり RtZ , Xl , . . . , Xn コ 筈 BtY . , . . . ,Yn コ f52f 堅竺堅n ン1,くi il RtZ , Xlコ てぎ BtY lコ,くi i iJ B 富 RtZq , Z+ a. Zp + a2 Z2 p + . . .+ as Zs p コ.w he r e q ニ pe聖坦 a i E FくRl くi ニ 1, . . . , sI 室竺 旦望竺空望聖- n e gativ e intege r s e a nd s . In pa rt ic ula r B %RtZコ 主圭 型坦 出 il a i E 氏 .Pr o of . Ac c o rd ing to 亡11 R is str o ngly in v a ria nt .Thu s t he c oro l la ry is a n i m ed iate c o n s equ e n c e ofThe o re m 1 . 4 a nd 亡3ト1 . 7 . Re m a rk . 工f k is a pe rfe ct f iel d, t he n a po lyn o -mial rin g ktX , Yl in tw o v a ria b le s is in va ria ntt6 , 8l . The r efo r e, in t h is c a s e, a tw o d im e n sio n al-k -af f in-e do m ain of the fo r m REZコ is in v a ria nt i f a ndo nly i f 氏 is a n F - ring..2 . No n - in va ria nt a f f in e do m ain sA S A m A 1 71Th is s ec tio n is de v ote d to giv e a n e x a mple ofn o n- in va ria nt tw o d im e n sio n al a f f ine do m ain s o v e r af ie l d ko f char a cte rist ic z e r o.2 . 1 . The o r e m. Let k be a f iel d of c ha r a cte rist icz e r o. We s etA ニ kEX . Y+ Y3 ,くX - りくX - 2JY , くX 1 11 くX 1 21Y 2コ,B ニ ktX , X Y+X3 Y3 ,くX - 11 くX - 2JY , くX - 1JくX - 2IY2コ,w he re X, Y a r e in dete r min aLte s . The n w eha v eくiI AtZコ 筈 BEZコ.くi il A 声 B .Pr o of . We be gin wi t hs o m e r e m a rks o n A a nd B . Pu tF 芦 くX - 11くX - 2l. S in c e F a nd X a r e c oprim e to e a chDt.he r in k亡Xコ, w e c a n cho o s e f, g E kEXコ s u ch t hat芦3 f+ F3 g ニ 1 . T hu sX Yf+Y3 こ X Yf+くX3 f+ F3 glY3くX Y+X3 Y3 1f+くF YI3 g E B ,wh ic h s ho w s t hat Y is in te gr al o v e r B . Als o, Y isinte gr al o v e r A be c a u s eY +Y 3 E A . T he r efo r e i f w ede n ote by AIa nd B l the inte gr al clo s u r e sof A a nd Br e spe ct iv ely in the qu otie nt f ie ld Q くAl ニ QくBl ニkくx , yl, t he n w e ha v e A T こ B - こ ktX , Yl . No t ic e t hatt he prin c ipal i de a1 くFl ニ FkEX , Yコ o f ktX . Yコ is als oa n ide al o f bot h A a nd B. Putktx , y , zコ ニ kEX, Y , ZコIFktX , Y , Zコ,1 72 A S A mAwhe r e x, y a nd z a r e t he r e si du e cla ss e s m oduloF ktX , Y, Zコ of A, Y a nd Z, r e spe ctiv ely . T he n ktx , y , 2Ilis a po lyn o mial ring in tw o v a ria b le s y an d a o v e ra r edu c ed A rt in rin g ktxコ def in ed by くx - り くx- 21 ニ 0 .By t he.c a n o nic al map w e r ega rd AltFl a nd 卑t I くFl a ss ubrin gs of ktx , y , zl. So i f w e w ri te cくRtIRl fo rt he c o ndu cto r of a n inte gr al rin g e xte n sio n R cR l , t he ncくくA,I くFl lJ くAI くFl llこ 0 ,a nd he n c e w eha v e cくATIAlこ くFl . A s imi la r argu m e ntc a n be appl ie d to B to get cくBTIBl ニ くFl . Fu rt he r m o r eit is e a sy to s e eth at A a nd B ha v e the fo r m sA ニ ktX , Y+Y3コ+くFl a nd B ニ ktX , XY + X3 Y3 1+くFl.No w w es hal l pr o v eくil . Put G ニ く十X+3I 12 , t he nGX 三 1 くm od くFll.Let u s de n ote by H. t he im age of a n y e le m e nt H Ek亡X.Y , Zコ u nde r t he n atu r al ho m o m o rph is m ktX , Y , Z トk亡x , y , 21 . Thu s G.こ X- I. If w e def in e a 2 x 2ma trix くGi j l bytGG三1.GG三22ト1三三げG三1一三三1r三1,the n くGi j l is in v e rt i b le in ktXコ a ndiGG三1...GG三22+.1x 00 A- ISo t he k亡Xコー a utO mO rpb is mA SA NtJ %申 ニt三卜t三二173of kEX , Y , Zコ is w el l de f in ed. . Notic e t ha t 申くYl.ニx y a nd 申くZI■ ニ X1 1z . T.he im age of AEZコ by くp is申くAEZコ1 ニ 申くkEX , Y +Y 3 , Zコ1+ 申くFk亡X , Y , ZコI. こ kEX, 申くYl+ 申くYl3 , 申くZJコ+ Fk亡X , Y , Zl.T he r efo r e申くAtZコ1, ニ k亡x , xy+ x3 y3 , x- 1 zコ ニ ktx , x y+ x3 y3 , zl .On t he ot he r ha nd w e ha v eB亡Zコ.ニ kEx , x y+ x3 y 3 , zコ .Thu s w eha v e 申くAtZコI. ニ BtZコ. , a nd s o AtZコ 冨 BEZl .T h is c om plete s the pro of o f くil . Ne xt w e sha l l pr o v eくitil. A s s u m e t hat t he r e e xi.sts a n is o m o rph is mV ニ A 筈 B . A s is e a sily s e e n, V c a n be exte nde dto an is omo rph is m AT 富 B T o f inte gr al c lo s ure s ofA a nd B , i . e . , V m ay be c o n si de r ed a s a k - a uto -mo rph is m of kEX , Yコ s at isfyin g t he c o nd i t io n 叫 Alニ B .No w くFl is a c o ndu cto r of bot h inte gr al e xte n sio n sA -1A a nd BtノB . S in c e くFl is stab le u nde r t he a uto -m o rphis m V , i t fo l lo w s that VtFJ ニ a F fo r s omea E k - く0ト Th is sho w s t hat VtXIE ktXコ, a nd he n c ek亡X , Yl ニ k亡VくXJ , 判 Yl コ ーー kEX , VtYll.So tlltYl is w ri t te n in the fo r m vくYl ニ bY+lt w he r eb E k- くOl a nd H E ktXl . Re c al l t hat VくAJ.こ B.is w el l def in ed a s a s ubrin g of kEx , yコ くc KEx , y , zコトThu s w eha v e1 74 A S A N V比良kt 叫XI. ,tVtYJ. +叫 Yl. 3コ ニ ktx , xy+ x3 y3 トThe ele m e nt xy+ x3y3 is .algebr aic al ly in depe nde nto v e r a r edu c ed rin g ktxコ こ kt 叫Xl暮コ . s o.t hattPtYl. +叫 Yl+ 3 is of the fo r mtPtYl.+tVtYJ+ 3こ くby+H暮1+くby+H,13 ニ Pくx y+ x3 y3 1+Q,whe r e,P is a u ni t o f k亡xコ a nd Q is a n ele m e nt o fktxl . 工t fo l lo w s fr o m e a sy 占alc ula tio n t hatby+ b3y3ニ Pxy +Px3y3.a nd s o b ニ Px , b3 こ Px 3 , w h ic h im ply t hat X 三 b2くm o d くFll in ktXコ, a c o ntr ad ic t io n.2 . 2 . Re m a rk . 工n t7コ, Ho chste r n ot ic ed t hat the po ly -n o mial ring in tw o v a riab le s o v e rt he c o o rd in ate上ip g o f a r e al tw o sphe r e is n ot in v aria nt . T he rin gof h is 占x a mple is a fo u r d im e n sio n al r egula r af f in ed わm ain o寸e土 the r e al f iel d. On t he ot he r ha nd, byRe m a rk 1. 5 i f k is i f ield o f po si t iv e cha r a cte r-ist ic , the n kEX , X- 1, Y , Zコ is n ot in v a ria nt . Th ise x a mple is a r egu la r t hre e d im e n sio n al af f in edo m aif1こ We do n ot kn o w whet he r the re e xist n o n-in v a ria nt r egula r af f in e k - do ma in s of d im e nsio n tw o .Re fe r e n c e stll S . A b hya nka r, P . Ea k in a n d W. He in z e r, On theu niqu e n e s s of t he c o ef f ic ie nt r ing. in a polyn o -mial rin g, J . A lge br a, 2 3く1 9 7 2l, 3 1 0- 34 2.A S A m A 17 5t21 T . A s a n u m a, D - alge br a s wh ich a r e stab ly equ iv a -1e nt to DtZコ, Pr o c e ed in gs of t he 工nte r natio n alSy mpo siu m o nAlge br aic Ge o m et ry, Kyo to く1 9 7 7I,K 工NOK U N 工Y A BO OK - STOR E, To kyo . 1 9 8 0, 4 4 7- 4 7 6.亡31 T . A s a n u m a, On stab ly polyn o mial alge bra s, J .Pu re Ap p1 . Alge br a, 2 6t1 9 8 21, 2 3 5- 2 4 7.t4コ T . A s a n u m a, Poly no mial f ibre rin gs of alge br a so v e r n o et he ria n rin gs , エn v e nt . H at h. , 8 7り 9871.101 - 1 2 7.E5コ S . Bhatw ade ka r, A n ote on str o n gly in v a ria ntr in gs J . Algebr a, 5 0く1 9 7 8l, 2 9 7- 2 9 8.t61 T . Fuji ta , On Za ris k i pro ble m , Pro c - Japa nAc ad . , 5 5く1 9 7 91, l o b- 1 1 0.t71 M . Ho chste r. No n- u niqu e n e s s of c o ef f ic ie ntrin gs in poly n o mia l rin gs , Pr o c. Am e r- Math .So c . , 3 4く1 9 7 2l, 8 1- 8 2.1E8コ M . M iya nis h i a nd T . Su gie , A f f in e s u rfa c e s c o n-ta in in g cyl in de rl i ke ope n s ets , I . Ha th . Kyo toUn iv り 2 0く19 8 0I. ll - 4 2.t9コ R . Sw a n. On se mino r mali ty, J . Alge br a, 6 7く1 9 8 0l, 2 1 0- 2 2 9.Fa c ul ty o f Edu c at io nToya m a Un iv e r sity3 1 9 0 Go fu ku , To ya m a 9 3 0JAPA NくRe c eiv ed Ju n e 8, 1 9 9 11

Non-invariant two dimensional affine domains

https://toyama.repo.nii.ac.jp/?action=repository_action_common_download&item_id=3025&item_no=1&attribute_id=18&file_no=1

Non-invariant two dimensional affine domains

Abstract

Similar works

Full text

Available Versions

University of Toyama Repository