Asanuma Teruo

University of Toyama Repository

富山大学教育学部紀要 No.56:43-51(平成14年)on Al_ F O RM ST E R IJO A S A NtJM Ae - m ailas a n u m a@edu .toya m a- u .a c.jpAB S T R A C T, We s u r v ey a stru ctu re of algebraic c u rv es who s eba sefieldextension to analgebraic clo s u reis an aLRn elin e.Letk be afielda nd letAbe a n algebr aic clo s u re of k･_A c_om m utativ ek- algebr aA(ora.n algebr aic s che m eSpe cA)is c alled a nAl-for mif A@kk isk-is o m o rphic to a polynomialrlng k[x]in o n e v ari ble･ T he p.u rpo se ofthis a rticleis to su r v ey a k- algebr aic str u ctu reof Al-for m s･ In partic ular we glV ea necess ary a nd s u氏 cie nt c o nditio nfo r aglV e npla n ec urv e(i.e. Spe cA ofa k- algebr aA ge n er ated by tw o ele m e nts o v erk)to be a nAl- for m･Fo rthe detailsee[1]･An Al-for mA is saidto be s epa rable(resp. pu r elyins epa rable), ifthere exists afiniteseparable(r esp. pu relyin s epar able)algebraic exten sio nk'/k s u ch thatA ⑳k k' 望 k'[x].Itis w ell-know nthat any s eparable Al-fo r mis trivial,i. e.,A ;茎k[x].Let A be a nAl_fo r m､ T he n A is an a氏 n ek_do m ain a nd he n c ethere exists afiniteGalois extension kloverk su ch thatA ⑳k k'望 k'[x]･So w e can find a ninter m ediate field kl betw ee nk'and ksuch that k'/klis s epa rable a ndk1/k is pu r elyin separ able. Note that(A ⑳k kl)⑳kl k'望 A ⑳k k' 竺 k'[x]･So A @k kl is a sepa r able kl-for m a nd w ehav eA ⑳k kl 空 kl[x].T his sho w sthat e寸ery Al-for mis pu r elyin separ able･FrorTm O W O nk is alw ays as su m ed to be aheldbfcha ra cte risticp > 0.Give n a nA l-for mA o verk,a s m e ntio n ed abo v ethere exists a ninteger e≧0s u chthatA ⑳k kP-e 望 kP~ e[x],where kP- ぞde n ote sthe s ub field of有 co nsits of allele m ents a ∈ k with ape∈ k. T hes m allest s a m o ng su ch integers eis c alled the height of Aa nd is de n oted heightA - s(c･f･[2】). Itsho uld be n otic edthat a ny Al-fo r mA is a n o r m alan n ek-do m ain su chthat thediffere ntial m odule nk(A)is a pr oje ctiv eA- m odule ofr a nk o n e･ So Ok(A)isis o m o rphicto s om eide alof A.1. Al - FORMS O FBu - T Y PELetk be afieldofchar acteristic p > 0 and let(e,i,,A)be a set ofpositiv eintegers su chthatl/^ = 1(m odpe). No wlet u s setBy - k[xPe,u(I)"v(I), u(I)v(x)入】- 43-a s ak- s ubalgebr a ofkP- e[x]wher e u(I), v(I)are el m e nts ofkP- e[x]s atisfying the differ-e ntialequ atio n〝u(I)'v(x)+ u(I)v(I)I- 1.Le m m al･1･ (1)Give n a ninte9er O≦ n ≦ e w eha vekP- ntBv]-′kP- n[xPm, u(I)iv(I), u(I)v(I)''](⊂ kP- e[x])fo r a ny n o n- n egativeintegers m,i,is u ch thatm ≧ e - n, i ≡ L, ･(m odpe-n), j ≡ Å (m odpe-n)･In pa rtic ula r w eha vekP- e[Bv] - kP- e[x].(2)Bv is afreek[xPeトm odule ofra nkpe with a basis(u(I)(vi)v(I)(i)Ii - 0,1, … ,pe- 1)whe re(i)m ea n stheintege r S u ch that 0≦(i)< pea nd(i)≡j(m od pe)･(3)Ifw e s etI - (u(I)v(I), u(x)v(I)A)theide alofBv ge n erated by u(I)yu(I) and u(I)v(I)A, then w ehavenk(Bv)空 Ias B.,- m odule s.T he o re m1.2. Bv is-a nAl -fo r m ofheightBy ≦ e.(1.1)In T he ore m1.2the iso m orphis mBv ⑳k kPle 望 kP-e[x]is glVe nby the c a n o nic al m apBv ⑳k kP-e → kP-e[Bv]- kP-e[x].D efinitio n l･3･ An Al-fo r mA which is k-is o mqrphic to so m eBv is cal led a nAl-for mof B〝 -type.Co r olla ry l A. Thefollo wingtw o c o nditio n s a re equiv ale nt:(1)A is a nAljo r m su ch thatthe qu otie ntSeld KofA is ratio n al, i,e. , K 望k k(I).(2)A is a nAl-for m ofBv -typefo r u- pe- 1 s u chthatA 望k k[xPe,xV(a x+ 1), I(a x+ 1)"】fo r s o m e aE kP-ek-iso m orphic to Sfor s o m eI>pOlyn o mialI(I)A In particular we have thefollo wingCo rolla ry 2.1. (1)Fo r a nyintege ri≧0 w eha vekP- e[S] - kP-e[x].(2)S is afree k[xPe]- m odule o/ra nkpe with afreebasis(1,I(I), … ,I(I)pe~1)T bisis a v e ry spe cialc as e of Le m m a1.1 a nd the pro ofofthis specialc as eis e asy.C o r olla ry 2.2. (1)S is a nAljo r m ofBl-ty pe OfheightS ≦ e.(2)heights - e ifa nd o nly if- 44 -On Al_F O R M Sf(I +b)卓kPllx]fo r a nyb E kP-eln Co rollary 2.2(1)theiso m orphis mS@kkP- e空 kP~ e[x]is glV enby the c a n o nic al m aps⑳k kP-e→ kP-e[B]- kP- e[x].Definitio n 2･3･ An Al-for m which is k-iso m orphic to S is c alled a n Al-for m ofp-polyn o mialtype.Itsho uld be n oted thatby Co rollary 2.2(2)the Al-for mSis trivial ifa nd o nlyiftheree xists b ∈ kPes u chthatI(I +b)∈ k[x].Le m m a2･4(S. M . B hatw adekar). Let K[x,y]be apolyr w mialringin tw o v a riables. I,yo ve rafield Kofa ny cha ra cteristic a nd let B be aK - algebra which is K -is o m o rphic tothe residu eK[x,y]/(II,y))fo r s om .eI(I, y)∈ K[x,y]･ Sup po seI(I,y)卓 K a nd nK(B)is aprojectiveB - m odule, the nOK(B)is afreeB - m odule･Proof. Co n siderthe c a n o nicalex a ct sequ e n c e(I(I,y))/(I(I,y))2-も ox(K[x,y])⑳ 柵 y]B A oK(B) → 0ofprojectiveB- m odules whe re(I(I,y))/(I(I,y))2 a nd OK(K[x,y])⑳K[x ,y]B ar efr eeB-m odules ofr a nk o n e a nd ra nk tw o respectiv ely. Note that the r a nk rk O^,(B)of theB - m odulesOK(B)islesstha n o r equ alto tw o･ Ifrk OK(B) - 2, thenめis the z ero m apa nd he n c e4,is anis o m o rphis m, which im pliesthatrkOK(B)is alsofree. Ifrk r2K(B) - 1,the nめis injective a nd he n cetheis o m o rphis mOK(K[x,y])⑳K[x ,y]B 空(I(I,y))/(I(I,y))2 o oK(B)of B- m odulesfollo w sfro mthe ex acts equ e n ceju st abo v e. Takingthe e xterio rpr odu ct o nboth sides ofthisiso m orphis m w e e asily v erifythat OK(B)望 B, asdesired. □Lem m a2.5. IfA is a nAlrfo r m with OK(A) - A, the nthe re exists ak-de riv atio nd ofA s u chthatd(I) - 1fo r s o m ef与 ･4･proof. We m ay a su m e that A is ak-s ubalgebr a of有[x]a nd the ca n o nic al m apA ⑳k有 ー 有【A]is a nis o m orphis m. Ifwe w rite Derk(A)the k-derivatio nA- m odule of A, thenby ass u mp-tio n D_e rk(A)is als o afre eA- m odule ofr a nk o n e･ On the otherha nd the is o m o rphis mA ⑳k k 窒 k[x]yields c a n o nic ally a niso m orphis mDerk(A)㊨k有望 Derk(有[x]). (2･1)Sin c eDerk(A)m ay be c o n sidered as a s ubm odule of Derk(A)⑳k有thr o ughthe c a n o nic alinjectio nDerk(A) → Derk(A)⑳k有,So ide ntifying Derk(A)_c*k有 with Derk(k[x])u nderthe is o m o rphis m 2･1 w e m ay r egardd as a_k-deriv atio n of k[x]･ Fu rther m or eitis e_a sy to se ethat d is also afr eebasis ofDer元(k[x]), thatis a(I) - c for s o m e0≠ c ∈ k. Replacing I by c-1x w e m ay ass u m ec = 1 fr o mthe 丘rst. Note thatdP~ 1: A ー Ais w ell-defin ed as ak[AP]- m odule ho m o m orphis m. No wby the c a n o nic alringho m o m o r-- 45-phis m sk[AP]空 kP- 1【A]竺 A ⑳kkP- 1w e s e ethat k[AP]is a nAl-for m o v erthe field kP,s othat k[AP]is n or m al. Rec allthatthe c a n o nic al m apk[AP]⑳k有 ー A ⑳k有isinjectiv e a nd有[xp]竺司AP], A ⑳k有望 有[x]･T his sho w sthat thedegre e ofthe qu otie ntfieldqt(A)of Aov e rthe quotie ntもeldqt(k[AP])is equ alto the char acteristic p, So w ehavedP- 1(A)⊂ k[AP]- A n有[xp].In other w ords dP~ 1(Al m ayセe c o n sider ed a s aprin cipal ide al of k[APrsu ch that itse xtension 有dP~ 1(A)to k[A](- k[x])is_a u nit ide alof k[A]･ T hisimpliesthatdP-1(A)is au nitide al bec a u sethe ring e xte nsio nk[A]/k[AP]isintegr al･ In particular w e c a n cho os ea n ele m entJ ∈ ♂~ 2(A)⊂ As othat♂(∫)- 1, a s r equired.T he ore m 2･6 ･ Let k be aβeldofcharmte risticp > 0 a nd let A be ak- a19ebra･ The nthefollo w m9three co nditio ns a re equiv alent:(1)A is a nAljo r m9e n e raled bytw o ele m e nts o ve rk, i. e. , SpecA is aplan e c u r ve.(2)A is a nAljo r m s u ch that Ok(A)is aJreeA m odule .(3)A isbfpIPOlyn o mialtype.Pro of･ T hefa ct(1)i (2)follow sfro m Lem m a2 A. No w s up pos eA s atisfiesthe co nditio n(2)I W e?ay a ssu m ethat A is ak-s ubalgebr a of kP-e[x]s u ch that kP-e[A]- kPT e[x]a ndthe c ano n lC als u rje ctio nA @k kP- e) kP- e[A]is a nis o m orphis m･ T he nby Le m m a2.5 there e xists ap-polyn o mialI ∈ A. W e s etk[xPe,f] - D fo r sho rt. NoticethatD is ofI>pOlyn o mialty pe a ndthereforeBl -type, Whichim pliesthat D is a nAl-for mby T heo re ml･2･ So w eha v ethe c a n o nic al is o m orphis mD ⑳kkP l 竺 kPIx]空 A ⑳kkP- eT his sho w sthat D - A bec a u seD c A, which c o mpletesthe pr o of of both im plic atio ns(2)⇒ (3)a nd(3)⇒ (1)･ □Co r olla ry 2. 7. IfA is a nAljo r m s u ch that the gro up Autk A of allk- a uto m orphis m sofA is inPnite, the nA is ofpIPOlyn o mialtype.T her e e xists a n o n-trivial k-fo r m s ofthe ad dit ve gro up Ga o v e rk. Su ch k-fo r m占 areco mpletely des cribed by P. Russ ell. Using T he or em 2･6 we obtain a n otherpro ofof hisr esult.Cor ollary 2･8(P･ Ru ss ell)･ Thefollo wingtw o c o nditio n s a re equiv alent:(1)A is ak-fo r m ofGa .- 46-On Al-F O R M S(2)A is a nAljo r m ofpIPOlyn o miaZtype s u ch thatA 望 k[xPe,I + alXP+ a2XP2+ I - + an xpn](ai ∈ kP- c).Co rolla ry 2.9. Letk be aPeldofch k - p > 0 and let A be a nAljo r m. Thenthefollo wingtw o co nditio n s a re equivale nt.･(1)A 窒 k[x].(2)A @k A is U F D.3･ Al- FORM S O F C HAR A CT E RIS TI Cp - 2T he str u cture of Al-fo r m s ofch k- p - 2 is quite differe ntfro mtho seofch k- p > 2a nd in this s ectio n w etre at o nlythe c a se ofp - 2. First w e sho wthe c as e ofheight o ne･T he o r e m3.1. Let k be aSeldof ch w a cte ristic p - 2 a ndlet A be a nAl-fo r m. IfheightA - 1, thenA is ofp-polyn o mialtype.In ge n er althere exists a clos e c o n n ectio nbetw e enA1-for m sA ofch k- p - 2 a nd theAl-for m s of B3-tyPe･Rec allthat B3is defin ed a s a specialca se of Bu . T hatisB3 - k[xPe,u(I)3v(I),u(I)v(I)入】(⊂ kP~e[x]),wherea nd u(x), v(I)a re el m e nts of kPIx]with(e ≡1 (m od 2))1)/3 (e ≡ 0 (m od 2))u(I)tv(I)+ u(I)v(I)I- 1. (3.1)Re m a rk 3.2 .,In c a s e ofp - 2, thefollo w lngthre e co nditio n s ar e equivale nt:(1)(u(I), v(I))is a s olutio n ofthe differ entialequ atio n(3.1).(2)T he pr odu ct u(I)v(I)is ap-polyn o mial.(3)u(I)a nd v(I)a re w ritten asu(I) - uo(xp)+ u1(xP)I, v(I) - vo(xP)+ vl(xP)I,whereuo(xp), ul(xP), v.(xP), vl(xP)∈kP~e[xP】withuo(xp)v1(xP)+ ul(xP)vo(xP) - 1.Thefollo w ingthe or e mgivesthe stru ctu r e of A@kkP-I fo r a nAl-fo r mA in the c a se-ofch k- p - 2.T he o r e m3.3. Let k be aβeldofcha ra cte risticp - 2･ Then thefollo win9three co nditio n sa re equiv alent:(1)A is an Aljo r m oJheightA ≦ e.(2)A is ak- algebra s u ch thatA ⑳た kP- 1空 B3 ⑳k kP~1(3)A @k kP- Iis ofB3-type a S a nAl-fo r m o v erkP- IEx a mple 3 A. Let ko be aperfect fieldofch ko - p - 2 a nd let e be a ninteger with- 4 7 -e > 1. Ifw e setk - ko(tPe)as a s ub fieldofthe r atio nalfu n ction field ko(i)in o ne variablei, then w eha v ekP- e- ko(i). We c o n siderthe Al-fo r mA - k[xPe,x3(1 +tx), x(1+tx)入】ofB3-type. In this c ase u(x)a nd v(I)ar e s et as u(I) - x a nd v(I) - 1 +tx･ Note thatu(x)v(I) (- x +txp)is a升pOlyn o mial･ T he nheightA - e andtheidealI - (u(I)3u(I), u(x)v(I)A)of Ais n ot prin cipal a nd he n c eby Le m m a1.1(3)a nd T he or e m2.6 we se ethat A is notof㌢polyn o mial type･ In particularSpecA is n ot a pla n e cu rv e･No win orderto des cribe the k- algebr aic stru ctu re of an Al-for m ofch k- p - 2 itselfw e n e ed to extend the n otio n of B3-ty pe aSfollo w s:Let k be afieldofcharacteristic p - 2 andlet e be apo sitiv einteger･ Let usdefin e ak-subalgebraC3 0fa polyn o mialring kPIx]byc3 - k[xPe,u(I)2(u(x)v(I)+9(I)2), v(x)入~ 1(u(I)v(I)+9(I)2)】(⊂ kP- e[x]),whereA is apositiv eintege r withA -〈Yp:Il'i;′3 (e ≡ 1 (m od 2))(e ≡0 (m od 2))a nd u(I), v(I),9(I) ar ethe ele m ents of kP- e[x]such that the produ ct u(I)v(I)is ap-polyn o miala nd u(I)g(I)is containedinkP- 2[xpe~2, u(I)3v(I), u(I)v(I)入].Re m a rk 3.5. In ca se ore > 1 ifw e s etR - kP~ 2[xpe - 2,u(I)3v(x), u(I)v(I)^],the nby Le m m a1.1(1)w eha v eR - kP- 2[xpe - 2,u(I)3v(I),u(I)v(I)r]for a ny integer r≧ 0 with 3r ≡ 1(m odpe- 2), a nd 9(I)ju st above is obtain ed as a nele m e nt ofthe(r atio n al)ide al(u(x)pe-2 - 1,u(I)2v(I), v(I)')Rof ILT he o r e m3.6.(1)C3is a nAljo r m.(2)heightC3 - e ifa nd o nlyi/J(I +b)4kPl - e[x]for a nyb ∈ kP- eT beis o m o rpbismC3 ⑳k kP~ e空 kP~ e[x]isglVen e xplicitly a sfollo w s:W e ha v ekP1 1[c3]- kP- 1[B3]a nd the c a n o nical m apc3 ⑳kkP~ 1ぅ kP- 1【c3]- kP- 1[B3]-4 8-On Al-F O R M Sis a niso m orphism . T his sho w sthatc3 ⑳kkP- 1望 B3 ⑳k kP-1Definitio n 3.7. If A is a nAl-fo r mA ofheightA - e which is k-is o m o rphic to C3, thenA is c alled a n Al-for m ofC31tyPe･Re m a rk 3.8. Withr espect tothek- algebraC3ifw e set9(x) - 0, then w eha v eC3 - B3 .T hu s a nAl-for m of B3-typeis alw ays ofC3-ty Pe･T he o r e m3.9j Letk be aβeldofcha ra cte risticp - 2. The n ev e ry Al-for mis ofC3-type.4. Al- FO RMS O F C H A R A CT ERIST IC p > 2Let k be afieldof cha ra cteristic p > 2･ Rec allthat a n Al-fo r m of B2-type is k-iso m orpbic toBl - k[xPe, u(I)2v(I), u(I)v(I)入](A - (pe+ 1)/2)where u(I), v(I)are el m ents of kP-e[x]satisfyingthe differ entialequ atio n2u(I)'v(I)+ u(I)v(x)I- 1. (4.1)Re m a rk 4. 1. Ev ery solutio n(u(I), v(x))of(4.1)isgiv en a sfollo w s:Cho o seh(I)∈kP- e[x]s u chthatgcd(h(I)I,h(I)′′) - 1･ T henh(I)'is separ able a nd he n cew ehav ekP~ c[x]≡ kP- c[xP] (m od h(I)lkP- e[x]).So h(x)is ofthe for mh(I) - 9(xP)+ h(x)'w(x)for s o m e9(xP)∈ kPIxp]a nd w(∬)∈ kP- e[x]. T hu sh(x)I- h(I)"w(3;)+ h(I)'w(x)'･Fro mthe a ssu mptio n w eha v ew(∬)∈h(I)'kP- e[x]･Ifw e set u(I) - h(I)la nd w(I) - u(I)v(x), the n w e ca n e a sily s eethat(u(I), v(x))is arequired s olutio n. Co n v ersely v ery s olutio nis glV e nthr o ugh the pr o cess･T he o r e m4.2. Let k be aJleldofcharyICte risticp > 2. The n every Al-for mis ofB2-tyPe･Letk[X, Y, Z]be a polyn o mialringin thr ee v a riable sX ･Y, Z a nd let¢: k[X , Y, Z] 1 Blbe a s urje ctive k-ho m o m orphis mdefin ed by¢‥(x , y, z) 1 (xPe, u2v, u v入).Notic ethat u(I)Pe,v(I)Pe∈ k[xPe]. T her efo re w e c a ntake U(X),V(X) ∈ k[X]sothat¢(U(X))- u(I)Pea nd ¢(V(X))- v(x)Pe. Note that U(X), V(X)also satisfy theco ndition2U(X)'v(X)+ U(X)V(X)I- 1･Ifw e setP -(Y^- U(X)Z, Z2 I V(X)Y, Y^ - 1z I U(X)V(X))(A - (pe+ 1)/2)(4･2)a占a nidealofk[X, Y, Z], then w ehav eke r¢ - P and hen ce-49-B2 望 k[X, Y, Z]/P･T her efor e as a c o rollay of T heo re m4･2 w ehav ethe follo w ingCo rolla ry 4･3･ Le.ik be aβeldofchara cte ristic p > 2 andletA be a k- algebra. The nthefollo wingtw o co nditio n s a re equivale ntI(1)A is a nAl-fo r m ofheightA ≦ e.(2) The re ecists U(X), V(X)∈ k[X]s uch that2U(X)lv(x)+ u(x)v(x)I- 1a ndA 空 k[X, Y, Z]/Pwhe reP is the prim eide al dePn ed a sin4.2.Itsho uld be n otic ed thatP - r ad(YPe- U(X)2v(x), zpe- u(x)v(x)A).In particularP is a s et the or etic co mpleteinters ectio n.T he o r e m4･4 ･ Let P be asin Co rolla ry4.3. Thenthefollo wing tw o c o nditio n s a reequ w ale ntI(I)P is a nideal theo retic c o mplete inte rs ectio n.(2)A isgen e rated by tw o ele m e nts o ve rk.Re c allthat the c o nditio n(2)is equivalentto sayingthatA is a nAl-fo r m of2>pOlyn o mialty pe byT he o re m2.6.Ex a mple 4･5･ For a ny αi(xP)∈kP- e[xP] (i - 1, . . ., n), w e s etu(I) - I,2v(I) - 1 + α1(xP)xP12 + - + α n(xp)xnp ~ 2T hen(u(I), v(I))is a solutio n ofthe differe ntialequ atio n(4.1)a nd hen cethe Al-for mB2is w ell-defin ed･ Further mor eifai(xP)卓k[xP]for s o m ei - 1, … , n, the nA is n otge n er atedby tw o ele m ents o v erk･ T he prim eide al Pc orrespo nding to this B2 is not a c o mpleteinterse ctio n.5･ A1- FO R M S O FC H A R A CT ERIS TI Cp - 3In this s ectio n w e c o n sidertheis o m orphis m class of all Al-fo r m s of height o n e whench k - p - 3.Ifk is afieldofcha ra cte ristic p - 3, then(u(I),v(I))(u I), v(I)∈ kP-e[x])is a solutio nofthe differe ntialequ atio n(4･1)ifand o nlyifu(I)a nd v(I)are w ritte n asu(I) - α(xp)+β(xp)I, v(I) - 7(xP)+ 6(xP)Ifo r(;('xxpp,'g('xxpp,')∈sL2(kP- e[xP"･In this.w ay a m atrix ofSL2(kP- e【xP])deter min es a s olutio n(u(I), v(x)), in other w o rdsthere e xists a m ap fr o mSL2(kPI e[xp])o nto the is o m o rphis m class esβ2 0f al l k- algebr aβ2, Wbicb willbe de noted◎‥sL2(kP~e【xP])1 β2.Note thatbyT heo re m4･2β2 m aybe c o n side red a5is o m o rphis m clas ses ofall Al-fo r ms-50-On Al-F O R M SA of heigbtA < ど.T he equiv ale-n c e r elatio nin S L2(k3- e[xp])indu ced fro mthe m ap ◎ willbe den oted byi 五W he n e- 1,this equiv ale n ce r elation - is giv e nexplicitly asfollo w s:Let q be akP- 1- a uto m o rphism of kP- 1【x]. T he n we ha v eq(I) - a x+ b(a,b ∈ kP~l,a≠∧ o).Let us s et the 2 × 2 m atrix【q]byEH■ー■n一〇αl†0れ川ⅦⅦ一川uニ桝^4=(∈ G L2(kP-1)⊂ G L2(kP~ 1[xp]).α(xp) β(xp)7(xP) 6(xP)∈ sL2(kP- e[xP]),Giv en a m atrixthe a ctio n q M by q o nM is defined asq M =(q(α(xp))q(β(xp))q(7(xP))q(6(xP))∈ GL2(kP- 1【xp]).T he o r e m5･1. Let k be aheldofcha racte ristic p - 3 a nd let M , N ∈ GL2(kP-1[xp]).The nthe/ollo win9tw o c o nditio n s a re equiv ale nt:(1)〟 - 〟 .(2)M - α Lq N[q]fo r s o m eL ∈ GL2(k[xP])a nd s o me a,b∈ kP- 1, a ≠o, where qis akP-I- a uto m o rphやm ofkP- l【x】dePn ed by q(I) - a x+ b.As a nim m ediate c o n sequ en ce ofthis the ore m, forexam ple, w e c a n s e eth c ardinaln u mber of the set of B2 is equ alto orla rgertha nthat of k in c asek is n ot perfect.In partic ula rthe c ardinalnu mber ofthe set of k-isom orphis m class es ofall Al-fo r m s ofheight on eis equ alto that ofk.RE F ER E NC E S【1】T. As 弧 u m a, A str u ctu r e of Al-br m s. pr eprint.【2】T･ Ka mbaya Bhiand M, M iyanishi, On form s ofthe a艮n elin e o v e r afield, Le ctu r esin M athe m atic sKyoto Univ e rsity lg 77- 51 -

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