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〔Fonnerly TR∫Mathematics) Vblume 25, NUUil)eT 1 (1989〕, 33−38ANOTE ON p−BASIS OF A POLYNOMlAL RING
IN TWO VARIABLESTetsuzo I(IMURA and Hiroshi NIITSUMA
〔Received April 7, 1989;・Revised June 30,1989〕 ABSTRACT. Let k be a perfect field of characteristic p. Let Aニ k[x1,x21 and B=k【Y1,Y2】be the polynomial ri皿gs over k in two variables such that A⊇B2/lp=k【v?i,弓】・The皿we prove that A has a p−basis over B a皿d B ha3 a p−basis over/IP.1980Mathematics Subject Classi五cations. Pri血ary 13F20,13B10;
Secondary 14E35.
Key words and phrases. p.basi『, polynomial ring,】Frobe皿ius sand− wich. §o.Introduction Let P b e always a prime皿umber, S a commutative ri皿9 with identi.ty of charaCteristic p a皿d SP={xP l. x∈5}. Let 5’be a su1)ring of 5. A subset r of S is said to 1)eρ一independent over 5’if, fbr a皿y subset{ろ1,....,bn}of F, the set of monomials b;1_b;n(0≦e‘<p)is linearly・independent over 51’【SPI. r is ca皿ed a p−1)asis of S over S’, if it is p_i皿depe皿dent over 3’and S=SP[5’, r】. In this note, we prove the f()lowi皿g theorem. THEOREM.五εは6eαρeげect field.o∫ characteristi¢p一五ε‘、4=k【エ1梁21and B=砲、,Y2】5e抗り・∼卿m輌α1 r拘3・uerねn‘ω・varia‘rε8 SU¢ゐ毒ゐα‡
A.スB⊃AP=綱,姥】, A5sum・tゐ・‘1Φ(A)・Φ(B)1=P,ωゐ・r・Φ(ノ1)・ndΦ(B) αrθ‡ゐ¢guotiept∬ε∼ds〔リノ1 and B respectively一τゐeπ‡ゐe∫01わt〃ing statements ゐold. (1)ノ1ゐα5αρ一ba5i・・㊨εrB. (2)Bゐ・5・P−basis・y・・AP. R.Ganong[3]has prOved this theorem and obta1皿ed mo士e pfecise result(s㏄ Theorem[3D under the assumption tha洗is algebraica皿y closed. But, his proof 433
depe皿ds deePly on his ow五the6ry df ph皿e ci’ rVes and on』acohcePt, Hamburger− Noether expansion. The authors Wahtどo have a mOre aJgebralc proof of above theorem as an application of local eXiste皿ce of p−basis(i.e. Kll丑z’s conjecture). §1.Pre五min泣ies O丑eof the importa皿t part()f our proof dep ends o皿Y皿an,si皿separable Galois Theory[6】. So, We rewrite its defi皿itio皿a皿d a theorem. DEFINITION([6D. Let 1)be a ri皿g of prime char㏄teristicρ. A D−algebra Ois called a Gaユo呈s exte皿sio玖of刀provided 、 , ,. (i)Ois finitely generated projective as 1)−module, (ii)tP∈Df・・all t t C, (iii)give皿any prime idea1 P i皿0, the皿Cp adm玉ts aρ一basis over 1)Q,.whele
QニP∩1). ・ ・.・
Theorem ll of S.Y旺a皿[61. Letσ⊇1)⊇E5eαteωer o∫ rings sucゐthat O is a Gα1。is exten3i・n・ver・E and ever 1), Then the fo∼1・wing statement5 ho∼d. (1)1)is a Gα∼・i5 extensi・n・ver E’ (2).Let H={d∈1)erE(0)ld1)⊆D}’Tゐen there ・isα1)−m.oぬ∼εゐom・omor一 輌mD・rE(ヱ))一・H.whi・h f・ll・ω・輌・‡ゐ・∬・・加輌鋤η卿丑一→D・・E(D)givenあy d→41が・the identity map・n刀erE(1))’ ・
(3)五・‡砺(D)b・オゐ・imag・・∫D・・E(1))in・H・Th・n . ・
C・碗(1))㊥D・・D(C)=D・rE(C)・ The fblowing theorem dne to Matsulm皿ra is aユso iMportamt for our pro6f ofTheote皿. 』
Theo士em 2730f H.Matsu’mura【5】.五e‡C5εαア畑《)f characteristi¢p, and SUPP…tゐ・t x∈0, D∈1)・・(0)・ati・fy Dx=1・nd・1)Pニ0;・・‡σ・={・∈OID・ニ0},τλ・・0ぷカ・・m・d・1・・…(る嚇↓・・ξ・1,x,_,x,−1’
L醐MA.1, P9老A・n品6・ω・老・オr」硫・.Th・一・丁九・n・
(1)D・τB(A)i・ψee 4一励d・1・.0加斑・ (2)1)eアAρ(B)輌sα∫アεε1ヲーmodule o∫rank 1◆T.KIMURA AND H. NI ITSUMA
35
PROOF. Let P be any prime ideal of.A. Set(?=P.∩B. Then,、4p a皿d BQ .are reg皿lar local rings. Since Ap is a finitely generate(1」BQ−modUle,ノlp ha3 a『o’basi・・ver BQ by ,Th…em。f【21: ΩB④⑧A・Ap, w・hwr th・姐・wi・9・. On th・・the・h・pd・・i亘ceΩβ。(4・)= D・rB④⑧・A・r∬g頑Ωβ(4), A)⑧・A・ = H・mAi(ΩB(A)⑧A・Ap,ノ1P) =1)eアBQ(ノ1P)・Th・敵e, D…④⑧μ・i・a f・e・ Ap−m・a・1・・f・rank…th・t D・・日(A)
is a pr()jective A−module of rank 1, by Theorem 20f§5, Chap.H,【1エ. Thus, ・i・ce・A i・ap・1y・・㎡al・ri・g, D…(A)i・a・f・ee・A−m・d・1・・f…k1 by Si・rel・conjecture(T.Y.Lam[4D.
LEMMA 2.五e‘F6εαpo lynomiα1 in A=克【σ1,司sucゐ‡ゐαf F¢.4p=
嘱,弓】・・ば∼・オG‘・…∼・m・・‘み.lf bF/∂・、ニ・、σ・・nd∂F/∂x、’= α2G,(α・,α2∈A),tゐ・nω・ゐ・・eF=αGP+午ρ(α,7EA). PROOF. Since F〆.4P, we may assume∂F/∂σ1=α1( P≠0. Hence, wehave
F=α3Gp+β, (α3∈ノ1,β∈AP【x2]・), by i・t・9r・ti・・with ・e・bect t・x、,beca・・e・th・・c・・伍・i・・t・・f・㌘・一’i・∂F/∂。、ニ α1(Pmust be equal to O. Th釘eforel we have ∂F/∂X2=G?(∂α3/∂X2)+∂β!∂X2.Combining this with∂F/∂x2=α2Gp, we obtain
βニ『θ’δザ・(7,δ∈A): Puttihgα∫=α3十δ,尋ve get the ’reqnited identity FニαGρ’十午P, (α,午・∈A).§2.Proof of Theo;em First, w6・h・ll・P・・∀・th・t・B h・s砕b品i・・ve・Ap・L・t.g b6・p・il・・’ id・al
・fB㎝d g=Q∩Ap・Th・・βg ha・ap−basi・・ver(A’), by Th…em・f l21・
Obviously, B is a finitely generated.4P.mod皿le. By virtue of Serre,s conl㏄ture (c£Lam[4D, B is a free AP−module. That is, B is a Gaユois extensio皿of、AP in the sense of S.Yua皿[6】. Similairly,ノ1 is a GaJois extensio皿of B and A is a Galois extension of/IP. So we have ’ D・rA。(A)=A・Gf.A・(B)㊥D・・B(A) by Theore皿110f[61. Further,もy Lemma 1, D eアB(A)is a free A−module of rank 1 and DerA・(B)i・als・a・free・B−m・dule・frank 1・ Hence we haveand
PerAp(B)=Bdl,
DeアB(A)ニノld2, dl∈1)eア.4ρ(B) d2∈1)erB(A). Identifyi皿g dl with the derivation d;i皿1)erAp(A)such that the restrictio皿 ・fdl is di, we・have DerA・(A)ニノld・㊥Ad2・ On the other ha皿d, si皿ce∂/∂x1,∂/∂x2∈1)erAp(A), we have the following ide皿titieS: ’ ∂!∂Xl=αidi+fi,d,, (α1,β1∈A), ∂/∂エ2ニcr 2dl+β2 d2, (α2,β2∈A). Let F be an element of B−AP such that the degree of the polymmi泣Fi皿 工1,ω2is the smaillest o皿e in B一ノ4P. − ApPlyi・g th・d・・ivati…∂/∂x・・エd∂/∂x・t・th・p・1y・・㎡a F, w・g・t ∂F/∂Xlニαi.d1F, ∂Fソ∂X2=α2diF,whereα輌∈A(i=1,2)and dlF∈B.
T.KIMURA AND H. NIITSUMA
37
Then, by the choice of F, we conclllde that the element diF of B must l)e an element of 4P, b㏄a皿se the degree of 41F is sma皿er thah that of F, anddiF∈B.
.Si皿ce dl F∈.4P, by Lemma 2, there eXist elements a’.a皿d 7 of.A such thatF=α41F+7P.
O皿the other ha皿d, since the poly皿olnial ring」B is integrally closed i皿itsq・・ti・・t五・ldΦ(B)and・∈Φ(B),・i・・n・1・me・t・f B・W…laim・th・t.
ば1F∈た*, where k*ニk−0・It is clear that《ll F≠0. SupPose that di F¢ん.Put
・=.Σ・e、。、x:’・・s2. 0≦ε1,ε2≦P−1 Th・・, th・d・g・ee・f・一・∼。。)i・・sm・ll・・th・n・th・d・g・ee・f F・H・・ceαb・1・ng・ to/IP and so F also belongS to AP. This is a, contradiction. . S・,・i・ce・d・F≠0, F 4Φ(Ap), h・・ceΦ(B)=Φ㈹(F)=Φ(A・)【珂. Therefore, we get d?1=0. Thus, by Theorem 27.3 of [5】,・褒ソdiF is a p−basis of Bover Ker dl:ニ{x l x∈B,dlx=0}. Fllrther, since B is a Galois extension over 4P, by Theorem 90f【6】, Ker dl=AP. It fbllows thatアソdi F is a p−basis of B over/IP. Obviously, F is a p−1)asis of B over A,. Next, we prove that.4 has a p−basis over B. Applying the same argument to the situatio皿B⊇AP⊇」BP, there exists a p−basis GP of AP over BP. Then,G is a p−basis of A over B. ・
REMARK Let F be an dement of B−AP such that the degr㏄of the
polynomi al F i皿xl,σ2 is the sma皿est one in B−A,. The proof of Theorem shows us that a p−basis of B over/4P is given by such a polynomial F.・Furtherrhore, sinceΩAρ(B)=BdF is a free B−module of rank 1, a皿y other p.basis of B over AP is of the・f()rm cF十7P(C∈先*,7∈A).REFERENCES
lii N.Bgurbaki, A. lgさbre commutative, Chap.1,2, Hermann, Paris,1961. {21T.Kimura and H.Niitsuma, On Kunz,s conjecture, J.Math. Soc. Japan, 34(1982),371−378. [3]R.Ga皿o皿g, Plane Frobe皿ius sandwiches, Proc. Amer. Math. Soc.,84(1982),474−478. {41T.Y.La皿,’Serre,s COnjecture,i Lectnre −Notes ’in Mathemat}cs VoL635 Spri丑ger−Verlag,1978. 【51H.Mats血mura, Co㎜ntative ring theory, Cambridge’U皿iversity Press,1986. 161S.Yua皿, hseparable Galois仙eory of expo丑ent one,. Trans. AmeL Math. Soc.,149(1970),163−170.