Tích Tensor của các đại số đơn

la vanhgiaohoanvoidonvtA, B la K- d(;lis6 : ( ' ) M ~ d ',.( ? A ]' ~ ' h ? A. ' - l'), Qt (;11so con cua - _a mQtvan4 con cua, va cung la K-modunconcuaA (ii)MQtideand(;lis6cuaA vilaIa ideancuavanhA vaclinglaK - moduliconcuaA. (iii) MQtd6ngca'"ucuaK - d(;lis6f : A ~ B la mQtd6ngca'"uvanh vavilala d6ngca'"uK - moduli. Tu'ongtV: MQt d~ngdiu cuaK - d(;1is6 f : A~ B la mQtd~ng ca'"uvanhvala mQtd~ngca'"uK- modUli. 2.1.3.wIenhd~ Ch~(angII : Tichtensorcuacaedr;zisf)' Trang18 Ne'uA la.mQtd(;tis6 trenvanhgiaohoanK thl t~pcac idean chinhquycuad(;tis6 A phiii trungvoi t~pcacideanchinh quytren vanhA. 2.1.4- lVIenhd~ ChoA la d(;tis6trenvanhgiaohoanK, khi d6 miA-modun ba'tkhii quyrheanghiad(;tis6 la A-modunba'tkhii quyrheanghla vanh .Ngu'Qcl(;tim6i modUliba'tkhii quyrheanghlavanhco th~ du'QcxemxetmQtcachduynha'tnhu'mQtmoduliba'tkhii quyrhea nghiac1~is6. 2.2.DAI s6 TREN TRU<JNG: Caekhaini~md(;tis6c1on,d(;tis6 moduli,ideand~is6,d6ng ca'uc1£;lis6trentru'dngK c1u'Qcc1inhnghlatu'ongtt!nhu'dadinhnghla trenvanh.Trongph§nnaytagiiisuK la tru'dng. Ne'uA la d(;tis6 khac0 co c1onvi trentru'dngK thl anhX(;t 0::K --"'A c1inhnghlaboi k -c)k.l~la 1d6ngdfu K-d(;tis6. Nhan xel ~Ne'uAla cl(;tiso "*0 codonvi trentru'dngK, thl a: K -c)A voi a du'<;1cdinhnghlanhu'tren,taco Ima la d(;tis6 con n~mtrongHimcuaA. 2.2.1Dinhnghia MQtph§ntu a cua d(;tis6 A trentru'dngK dUila s6 d(;tis6 trenK ne'ua la nghit%mcuamQtdathuctrongK[x]. A c1uQcgi'la mQtc1~is6c1~is6trenK ne'u'\faeA c1~ula d~is6trenK. Nhdn xet: Ne'uA huu h(;tnchi~uthl A Ia d(;lis6 c1(;lis6 tren K. Thijt vijy : Giii su dimKA =n '>-I L 1 3 n+l } 1, h" ( 1) h~ h?. h th " "" ,,00vaeA=»!-l,a-,a, ,a a ~ n+ p an~u,p\l uctuyenti. k k ) k n+1 0 '>-Ik K t "" ? " k kh" dl:: h' . 0=> la+ ?a-+ + n+,a =, Vie at ca cac i ong ong t (11= . ChuangII : Tfchtensorcuacaed(liso' Trang19 =>f(x) =k,x+kzxz+ +kn+lXn+'=O,Vk;E K[x] la da thuenh~na lam nghi~m. ~ a lasod£;lisotrenK ~ Ala d£;lisod£;lisotrenK. 22 " B~ d~. 0 e Gia saK la tntongd6ngd£;liso.Ne'uD la mQtd£;liso ehiad£;liso tren K thiD =K Ch~(ngminh: Tac6:K ehuatmngtameuaD =>Kc D Ne'uaED thi f(a) =0, "If E K[x] Ma KIa tnlC1ngd6ngd£;liso =>f(x)=k (x-kl)(x-k2) (x-kn) ( vdi k, ki E K, k ;f:0) Khi d6: f(a)=(a-kl)(a-k2) (a-kn)=0 DoD lad£;lisochia=>3i : a-ki =0=> a=ki E K ~ Dc K. V~yD=K 2.3.DAI SONHAN - CENTROID " 31 D . ,," h""--- al son an Gia saR la d£;lisotrenvanhgiaohmlnK c6donvi. GQiE(R) la caetVd6ngcftucuacaenh6mcQngeuaR ,'<:faER. Ta dinhnghia Ta: R~ R X H XTa=xa Va La:R ~ R X H XLa=ax '<:fa,bERtae6Ta,Lac E(R) ,. GQiL(R) la vanhcaneuaE(R) sinhbdiTaLa L(R)= {Ta+Lb+~Ta.Lb.,a,b,ai,bi E R} I 1 1 ChucmgII : Tichtensorcuacaedt;lisrI Trang20 =>' L(R) du<lcgQila d?i s6nhan. 2.3.2.DinhnghiaCentroid: Centroidcila R la t~pnhungphftntli'trongE(R) magiaohoan vdinhungphftntli'cilaL(R). Kyhi~u: ~={<;oE E(R)/ qX:J=(}(P,O"E L(R)} 2.3.3.B6 d~: Ne'uR2=R thiCentroidcilaRIa giaohoan 2.3.4DinhnghiatamcuaR . Tam cua R ky hi~u:C(R)={xE R/xy=lX,'IIyE R} 2.4.DAI SO D<JN 2.4.1Dinhnghla R duQcgQilad?is6dontrentruongK ne'uRIa d?is6trenK , R2:t{O}vaR khongcoideanth1!Cs1,1'haiphianao Nhiill xel.. ChoR la motdaisotrentru'dngK. ne'uR la dais6don- . . . thiR la L(R) -modunba'tkhaguy. 2.4.2.DinhIV : ChoR la d?i s6donthi : (i) CentroidJ cilaR 1ftmQt'1l'ong. (ii) TamcuaR :f.{O}q R codonvi (iii) B?i s6nhanL(R) dayd~ccaephepbie'nd6ituye'ntinhtrong R trencentroid:5 Chungminh (i) Ta co:R la don' =>R-L(R)lamocunba'tkhaguy=>C(R)Iavanhchia =>:3lavanhchia Ma R don=>R2 =R giaohoan =>:3cotinhcha'tgiaohoan.V~y:5cilaR la mQtru'dng. Chuang II ..Tichtensorcuacaed{liso Trang21 (ii) (~ ) hi€n nhien (~) Oia saC(R) *-{OJ,v c EC(R) , c*-o ,taco : - a(xc)=x(ac)vfta(cx)=(ac)x ~ x(ac) =(ac)x( vi c EC(R) ) ~ ac EC (R) OQi 'R ={Tc / c E C (R) },vdiTc du'Qcdinh nghla nhu'(j ml.;lc 2.3.1.Taki€m tradu'Qc'R 1ftmQtideancua~ Mft J 1ftmQttru'ong~ 'R= {O}hay 'R= J Neu "1={OJ=?C (R)={OJ(kh6ngthoaVIC(R) *- {OJ) ~ 'R=~~ 'R codonvi.V?yR cochuadonvi (iii) Suyratuh~quacuadinh19dftyd~c. 2.4.3.lYlenhd~: NeuR 1ftd<;lis6doncodonvi thico th€ d6ngnhatcentroidJ vdi tamcuaR. Changminh Ta co : C(R) c ~( hi€n nhien) \:-I C'"j '1"' "'h'~~ . b . e-. r'O/ R' V IT E '-. ~ a. 1 Ullg illln 1 . :j C '-, ) Va E ~,V r E R, r *- 0 taxet: (lr) a=(lTr)a =(la) Tr=(la)r D~ta=La ~ (Ir) a =ar~ r a =ar Tu'ongtl!: r cr=(d) a =(1.Lr) a =(lcr)Lr=r(1.cr)=fa ~ ar=ra~ a E C(R) dor a =ra~ r (a - a) =0 Do ~1ftru'ongliena - a =0~ a=a E C(R) ~ :5c C(R) V?y:5=C(R) 2.5. Tich tensorcuacaemodun Oia saR 1ftvftnhcodonvi,M 1ftR-modunphai,N 1ftR-moduntnii. Chuang1/: Tichtensorcuacaedfli s6' Trang22 2.5.1- Dinhnghia Gia sli'( P, +) 1anhomAben,anhx~f: M XN ~ P bie'n(m,n)thanh . f(m,n)dU<;5cg9i1aanhx~songtuye'ntinhne'uthoa: . f ( m[ +mz, n) =f ( mt,n) +f(mz,n) , Villi, mz E M . f (m, nl +nz ) =f (m, nl ) +f(m,nz,) ,Vnl,nz E M . f ( m,rn) =f ( mr,n ) ,v r E R 2.5.2-Dinhnghia: Gia slr f: M x N ~ P (voiP, T 1anhomcQngAben) cp: M x N ~ T.,va f ,cp1aanhx~songtuye'ntinh. Ta noi : f coth€ phantichquaT neu:3d6ngCelUh saocho:f =h.cp: f(m,n) =h«p(m,n», V (m,n)EM x N. f Tacosod6giaohoan: M x N~ P <p~/ h T 2.5.3- DinhIi ( slft6nt~icuatichtensor) Voi M, N nhutrenset6nt~iT 1anhomAbenvaanhx~songtuye'ntinh t :M x N ~ T saocho: (i) t (m,n)sinhra(T,+)tl1c1aVUE T tacod~ng: u=I t(ffij,'J)' mi E M, nlE N hh (ii) m6ianhx~canbang<p:M x N ~ P (P nhomAbellmyy) co th€ phantichdU<;5CquaT. ChuangII : Tichtensorcuacaedr;ziso' Trang23 2.5.4- DinhIil..Nh6m T dl1<;5cxaydl,I'ngnhutrongdinh1:92.5.3dU<;5c gQi1atfchtensorcuaM vaN. Ky hit%uM Q9RN. 2.5.5- He qua(Tinhduynha'tcuatfchtensor) Giasa(M Q9'RN, t') voi M Q9'RN 1atfchtensorcuaM vaN d6va anhX?: t': M xN ~ M Q9'RN 1aanhX?songtuye'nHnhthoa 2.6.3khid6: t6nt?i d~ngca'uA :M Q9RN ~ M Q9'RN saocho: v(m,n)E M x N tac6: A(t(m,n)=t' (m,n) Nhanxet: Anh cua(m,n)ky hit%u1a:m @n,dot 1aanhX?songtuye'nHnhDen: (mj + mz) <2)n =mj <2)n + mz <2)n m[<2)(nl+nz)=ml <2)nl +m[<2)nz \. mr@n=m<2)rn 2.5.6.Hequa: M6i anhX? songtuye'ntinh<p: M X N ~ P, P la nh6m Aben. Anh X? (.pco th€ duQcphan tkh duy nha'tqua T=MQ9RN Nhanxet : Tichtensordu<;5cxacdinhduynha'tsaikhac1d~ngca'u. 2.5.7.DinhIvtichtensoreuahaid6ngea'u: Chof :M ~ M', g :N ~ N' lacacR-d6ngca'u.C5d6M,M' 1a cacR-modunphai,N,N' 1acacR-moduntrai.Khid6:t6nt?iduynha't d6ngca'u . f <2)g: M0R N ~ M' 0RN' thoa : (fQ?>g)(mQ?>n)=f(m)Q?>g(n). 2.5.8.Caetinheha'teuatichtensor Chl£cJng II : Tichtensorcuacaedt;lis6' Trang24 1)Gia saN Ia R-moduntreEvaRIa vanhgiaohminthiR@RN :::N 2)Tinhgiaohoan:ChoM IaR-modun,N IaR-moduntrai,gQiR' la vanhphilod~ngca'uvdiR tac6:M@RN ==N@R'M ,. ?" A 2.6.TICH TENSORCUA CAC KHONGGIAN VECTOR: CaedfnhnghTavaxaydlfngrichtensorcuacaekhonggianvectatren traiJnghoanloan tl((jrzgtT!nha eachxay drpzgcua rich tensorcua cae moauntrenvanh. 2.6.1.DinhIi : Gia sa E, F la hai khonggianvectOtrentHrongT, t6n tC;!imQt khonggianvectOM trenT va anhxC;!songtuyentinhcp: E x F ~ M saocho: neuf 18.anhxC;!songtuyentinhba'tky cuaE x F ~ G trenT thi :3!AnhxC;!tuyentinhg :M ~ Gthoadi~uki~nf =gocp. Nhanxet: NeuM vacpt5ntC;!ithiM chilacp(ExF) Ma M lamQtkhonggianvectOnenM luauchilat6hqptuyentinh cuacaephtintU'cuacp(ExF).N6ieachkhacM chilakhonggiansinh bdi cp(ExF). KhonggianM dlfngttrkhonggianE vaF thoacaedi~uki~ntren gQiIatichtensorcuakhonggianE vaF.Ky hi~u:E @ F Gia tri cp(x,y)cuaanhx~cplingvdi x E E, yE F gQiIa tichtensor cuavectoxvay.Kyhi~ux @y. 2.6.2.Tinhcha't (1)E @ F :::F 0 E.(tinhgiaohoancuatichtensor) ChuangII ..Tfchtensorcilacaedr,ziso Trang25 (2)T 0 E ==E 2.6.3.Menh d~: Chotru'ongD vahaikh6nggianvectdM,N trenD. TrongM @DNtaco : (i) Ne"u f1,f2,. . ,fnla t~phii'uh~ncacvectddQcl~ptuye"ntinh n trongM trenD thl It; 0xi=0=>Xi =0(i=l, . . .,n),V XiEN i=1 (ii) ne"uel ,e2,. . .enla t~phii'uh~ncacvectddQcl~ptuye"ntinh trongN trenD thl LYj 0ei =0=>yj=0 0=1,. '.' n),V YjE M , ?, ",'", 2.7.TICH TENSOR CUA CAC DAI SO TREN VANH. MQidinhnghfavatinhchatcilatichtensorcilamoduntrenvanhta cothi v(mdf!,ngd6'ivaitichtensorciladq.iS(Jtrenvanh Nhu'cachKaydlfngtichtensorcuamoduntrenvanhtrongtru'ong h<;5pM,N la 2 d~isotrenvanhR thltaclingcotichM@RN. 2.7.1.Dinhnghia ChoM,N la 2d~isotrenvanhgiaohoanR. Tadinhnghiaphep nhancuaM vaN nhu'sau: "ifai,Cj E M, Ifbi, dj E N ta co: (Iai 0R bJ(ICj 0R d) =(Iaicj 0R bid)(1) i j i,j 2.7.2Dinhnghia Gia saM,N la 2 d~isotrenyanhgiaohoanR (coddnvi) thltich tensorM@RNclingvdi pheptoan dU<;5Cdinhnghianhu(1) l~p thanhmQtd~isodU<;5cgQila tichtensord~isocuaM vaN. 2.7.3Menhd~: * Gia saM, N la d~isoconcuaA, giasa ab=ba,V aE M, V bEN. B~t~=MN={Laibi /ai E M,bi EN} thl ann X£;l: i ChU:cJngII : Tfchtensorcuacaed(lis6' Trang26 <p:M 0 N ~ ~ lamQtd6ngca'u ""a Q9b ~"" a.b.L.. IlL.. I 1 ~ ? ~ "~,, , 2.8.TICH TENSOR CUA CAC DAI SO TREN TRUdNG. TrangphftnnaytagiasaK la mQttntong.Taco : K - d,~lisosela K- modUlitrungthanh.Gia saA,B la haid(;liso trenK * Ne'uB codonvi la IB d~tA ={a01B/ aEA} Xet a'X(;l: <p:A ~ A a ~ a 0 IB Ta co : <pla mQtd£ngca'u Tu'ongtl!:Ne'uA codonvi la lA,d~t B ={IA 0b / b E B}thl \jf:B ~ B: la d£ngca'u b~IA0b V~ 1 - 1, ()() L l a ' ph6n h?" It,;n " I' ,..,,',n A ""t)(>B- ~- 'J:j'>' LAU CU UVll v, ~ua r1.~ Nhfmxet..TacoA,Bd~uchuaI 'II ; E A, bE B :; =a01B, b=IA0b, ;.b =(a0IB)(1A0b) =alA0hb =a0b =>A0B =A.B= {L;Ji;la;E A,~E Jj } I A0B =A.B duqcxetd trengQila tkh tensorcuamoduli A,B trenK. VI v~ytacoth~coid(;lisoA 0 B la tichtensorcua 2d(;lisoconAvaB. ._.