Luận văn Tích tensor của các đại số đơn

Ch~((Jng1 : T6ngquanv~vimhkh6nggiao hodn Trang3 CHUaNG 1 ,,? ;:;,~" , TONG QUAN VE VANH KHONG GIAO HOAN ~ ~" ?",,:> lI: 1.1.CAC KHAI NIEM CeJBAN VA CAC BO DE : 1.1.1.Dinhnghiamodun: ChoRIa mQtvanhtuyy, nhomcQngAben NtduQcgQilaR-modunphaine"ucomQtanhX?: f: Ntx R ~ 'f bO'" - ( --\ th? \-I '!.,f \-I b R l' 1V1 lcn C?P m,l) ~ mr loa: v m E tv.!,v a, E till (i) m(a+b)=ma+mb (ii) (mj+m2)a =fila +m2a (iii) (ma)b=m(ab) y Ne"uR cochaaph~ntltdonvi vam.l=m,V m E M thitagQiM la modununitary 1.1.2.Dinhnghiamoduntrungthanh: MQt R-modunM duQcg9i la trungthanhne"uMr =0 thi keG rhea r =0 1.1.3.Dinhnghia: ChoM laR- moduntadinhnghlaA(M) = {x E R / Mx =(0) } Nhanxet: * A(M)laideanhaiphiacuaR vaM laR/A(M)lamoduntrung thanh * M laR-moduntrungthanhA(M) =(0) 1.1.4.B6 d~: Ta kyhi~u:E(M)={e:M ~ M /elat1;td6ngca"ucaenhomcQngM} Thi E(M) la nh6mcQngcaet1!d6ngca"uti1'M ~ 'N1 Chuang1 ..T6ngquanv~vimhkhonggiaohodn Trang4 Nhiin xet.. (i) Voi V a E R anhX~Ta: M ~ M m~m Ta=ma thi tacoanhX~Tala tv d6ngcfu cuanhomcQngM lienTaE E(M) (ii) TrenE(M) tadinhnghTa2 pheptoan: V cp,'¥ E E(M) , V m EM: m(cp+'¥)=mcp+m'¥ m(cp\fl)=(mcp)\fI Luc doE(M) clingvoi 2pheptmintrenl~pthanhm(>tvanh (iii) RI A(M) d~ngcfu voi vanhconcuavanhcactv d6ngcfu E(M) 1.1.5.DinhnghiaHim: ChoM laR-modun,tagQitamcuaM trenR lat~p C(M) ={cpE E(M) / cpoTr=Tr OCP, V r E R },voi Tr : M ~ M ill ~ mTr= mr 1.1.6.Dinhnghlamodunba'tkhaqui: M du<jcgQilaR-modunbftkhaquine'uMR of.(0)vaM khangco moduliconth~tSvnaG 11 7 M" K b,:!d~. .. otso 0 e: 1.1.7.1B6 d~Schur's Ne'uM Ia R-modunbft khaquithiC(M) la vanhchia Ta matabanchftcuaR modulibft khaqui 1.1.7.2.B6d~ Ne'uM la R-modunbft khaquithiM ding cfu (nhula 1moduli) voi modulithuangRIptrongdop Ia ideanphaitoi d~inaGdocila R.Hanm1'a,t6nt~i1phfintti'aER:x-aXEp,VXER Ch~(ang1: T6ngquanv~vimhkh6nggiaohodn Trang5 +Ng~((jclai :n6um6iideanph,htoi d~ip cuaR thoamantinhchtlt trenthlRIp la mQtR-modunbtltkhaqui 1.1.8.Dinhnghla: MQtideanphaipthoatinhchtltcuab6d~1.1.7.2tilcla: :1aER : VXER thlx-aXEp duqcgQila ideanphaichinhqui ? , 1.2.RADICAL CUA VANH 1.2.1.Dinhnghla: Ta gQiradicalcuavanhR la t~phqpnhungph~nttrcuaR linhhoa duc;5CttltcacacmodulibtltkhaquitrenR Ki hi~u:J(R) ho~cfadR J(R) ={aE R /Ma={O},Mch~yquata'tca cacmoduliba'tkha quy} -NeuR khongcomodulibtltkhaqui,tad~tJ( R )=R ..T1" ~, LVnail xel: *J(R)= n A(M) , V MIa R-moc1unbit ilia qui *J(R) la idean2 phiacuaR * VI M duc;5chi~utheoR-modunphai DenJ(R) con duqcgQila radicalJacobsonphaituynhien2khaini~mnaytrungnhauDenta khongconnha'nm~nhtinhphaivatraicuaradical 1.2.2Dinhnghla: chopIa ideanphaicuaR.Tadinhnghia:(p:R)={XER/ Rxc p } 1.2.3.Mot soHnhcha't: . N6u pIa ideanphaitoid~i, chinhquy.B~tM =RIp thl A(1-1)=(p : R) . J(R) =(\ (p :R) trongdopch~yquamQiideanteid~ichinhqui,(p : R) la idean2phiaWnnhtltcuaR namtrongp Chuang1: T6ngquanv.~vimhkh6nggiaohoan Trang6 . J (R) =() p,pchc;tyguakhapcacideanphait6idc;tichinhguycuaR 1.2.4.Dinhnghla: . Phgntll' a E R duqcgQila tv'achinhguyphain€u :3a' E R : a+a'+aa'=0 tacongQia' la tl!anghichdaDphaicuaa . MQtideanphaitrongR duqcgQila tl!achinhguyphain€u illQi phgntll'cuanod6utl!achinhguyphai Nhan xet: N€u R codonvi thlphgnill'aER 1aWachinhguyQ 1+a langhich daDphaitrongR 1.2.5.Dinh Iv : J( R) la ideanphaiWachinhguyphaicuaR vachilaillQiideanphai tl!achinhguyphai 1.2.6.Dinhnghlaph~ntU'hiyHuh- Ideanlilylinh- Nil idean: .:. a E R duqcgQilaphgntll'lGylinhn€u :3n EN: an=0 .:. Ideantnli ( phai,2phia) cuaR duqcgQila nil ideantrai(phai,2 phia) n€u illQiphgntll'cuanod6ulily linh .:. Ideantnli ( phai,2 phia) p duqcgQila IllY lint n€u :3ill >0 : a[.az am=0, '\!al,aZ ,amE p. Tilc la :3ill> 0 : pm=(0) Nhanxet: (1) N€u pIa ideanlGylinhthlnola nil idean.Nguqclc;tikhongdung (2) MQiphgntll'lGylint d6utl!achinhguy 1.3.7.B43d~: J( R) chilaillQinil ideanillQtphia Noi tomlq.itacothi xacdtnhradicalJacobsonblingnhducachtheos(/ dr5sau: Chuang1 : T6ngquanv~vcmhkhongglaDhoan Trang7 J(R) {aE R /Ma={O},V M - ffiodunb.1tkhaguy}I "A(M), A(M) ={XE RfMx=(O}, 'i M-modun biltkha quy} ~R) ~ n p,peh~yquakhipcaeideanphait6id~iehinhquyeuaR ~J(R) =II (p :R) trongdopch<;lYquaffiQiideantoi d<;lichinhqui, \ Vdi(p:R)={XER/Rxc p } Ideantl;1'achinhguyldnnh.1tcuaR 1.3.MOT s6 VANH DAc BIET : 1.3.1.VanhnU'adon: 1.3.1.1.Dinh nghia: VanhR du<;5cgQila vanhmladonne'uJ ( R) =(0) 1.3.1.2.DinhIf :R/ J(R) lavanhmladon D~caveLO tinhchCi'tcua vemhR/ J( R) fa vimhnT£ad(frztaco thi motel? radicalJacobsonnta meltidean2phiacuavanhA 1.3.1.3.DinhILNe'u A la idean2phiaeuaR thlJ(A) =J(R) II A He qua: Ne'uR la vanhmladonthlt.1teacaeidean2phia d~um1a don 1.3.2..VanhArtin : 1.3.2.1.Dinhngrna: MQtvanhdU<;5egQiIa VflllhArtinphaine'uffiQit~pkhongr6ng caeideanphaieuanod~ucophgntt toiti~u Di ngdngQntagQivanhArtinphdifavanhArtin * Tacothi dtnhnghzavanhArtin bdngeachkhac: Chuang1 ..T6ngquanv~vimhkh6nggiaohodn Trang8 VanhA duqcg9i la vanhArtinphaiffi9idaygiafficuacac ideanphaithl se dungsauhUllh';lnbudcnghlala d€n ffiQtdi€ffi naad6cacPid~ub~ngnhau Vi du: Vanhs6nguyenZ khongla vEwhArtin . Tz£djllh llghiacuavimhArtill tacomi)tsf)'nhijnxetsau : >-Truong,Th€ ( vanhchia) la vanhArtin >-T6ngtn!cti€p ffiQts6hUllh';lnvanhArtinla Artin >-Anhd6ngca'ucuavanhArtinla vanhArtin VcmhArtincoradicalrtJtdqicbi?tdola : 1.3.2.2.DinhIy: N€u RIa vanhArtinthlJ(R) la ffiQtideanlUy linh He qua: Ne'uRIa vanhArtinthlba'tkynil idean(phai,tnii,2phia) cila R d~ulUy linh Lr..lhanxet: Gia Sl(RIa illvt vanhLilyy,n€u R c6ideanphailUylinh khac0thlR sec6ideanhaiphialilylinhkh.ac0 1.3.2.3.Dinhnghlaph~ntti'myd~ng: MQtph~nta e =I:0 trangR la ffiQtph~nt11lUy d~ngn€u e2=e 1.3.2.4.B6 d~: Gias11RIa vanhkhongc6ideanlily Iinh=I:O.GiasaP =1=0la idean phait6iti€u cilavanhR thlPc6d';lngP=eRvdie Iaph~ntalily d~ngcuae trongR 1.3.2.5.DinhIv: N€u RIal vanhArtinva P=I:( 0)laideanphaikhongIllYIinh cuaR thl p chila 1ph~nt11IllY d~ng=I:0 1.3.2.6.DinhIi GiasaR lavanhtuyy, giasae1a1phc1ntti'liiy d~ngtrongR.Khi do J (eRe) =e J(R) e 1.3.2.7.DinhIf: Gia saR 1avanhkhongco ideanlily 1inh*- 0 va gia sae*-0 ,e liiy d~ngtrongR. Khi do eR ( ideanphaichinhsinhbdi e) 1aidean t6i ti~ucuaR 6>vanheRe 1avanhchia Hequa.. Ne'uR khongco idean *- 0 va e2=e trongR thleR 1aideanphai t6i ti~ucua R ~ Re 1aideantnii t6i ti~ucua R 1.3.2.8.Dinh If : Gia saR 1avanhArtinnaadon,giasa p*- ( 0) 1aideanphai bat ky cuaR thi p =eR vdi e 1aphc1nta liiy d~ng 1.3.3.Vanh figureDthuy : 1.3.3.1.Dinhnghla : VanhR du'<;5cgQi1avanh nguyenthiiyne'uno co modulibatkha quy trungthanh Nhan xet.. (1) Ne'u R 1avanh nguyen thiiy thi Ker <p=A(M)= (0) vdi anh Xc;l<p: R ~ E(M) bie'nr thanh Tr vdi Tr : M ~ M m~mTr=mr (2) Ne'uR 1avanhnguyenthiiy ~ J(R) =(0) Nen taco th~noi mQivanhnguyenthuyd~u1avanh naadon (1)Ne'uR 1avanhbit ky , giasaM la R- modulibatkhaquy ~ A(M) 1aidean2 phia ciia R va R/A(M) nguyenthiiy Chucrng1 : T6ngquanv~vimhkhonggiaohoan Trang10 (2) Ne'uM la R-modunba"tkhaquy,pia ideanphaitoi d£;lichinh quycuaR vane'uM =RIp thi A(M) =(p :R) ideanhai phia IOn nha"tn~mtrongp. Theonh~nxet(3)nentaco:RI(p:R)lavanhnguyenthuy 1.3.3.2.DinhIi : Ne'uR la vanhnguyenthuyt6nt£;lip la ideanphai toi d£;lichinh quy trongR saGcho (p :R) =(0) . Ne'uvanhnguyenthuyR giao hmlnthiR la tru'ong 1.3.4.Vanhdon-lVloilienhegiii'avanhmiadOn.artin vavanhdon: 1.3.4.1.Dinhnghia: VanhR dU<;5cgQila vanhdonne'uR2;/: (0)vaR khongcoideannaG khac(0)vaR Nhanxet:M6ivanhchialavanhdon 1.3.4.2.M6i lienhegiii'avanhmi'adon-VanhArtin-Vanhmi'adOn (1) Ne'uR Lavanh drJn c6 drJnvi thiR Lavanhnl,t:adrJn Th~tv~y:Gia sdR lavanhdon =>J(R) =(0) ho~cJ(R) =R GQipia ideanphait5id£;licuaR =>pia ideanphait5id?ichinhqui(viRia vanhcodonvi) =>pc J( R) maJ(R) =n (p :R) =>J(R) =( 0)V~yR la vanhndadon (2)Ne'uR vaaLavanhdrJn, vaa La vcmhArtinthiR LavanhntladrJn Th~tv~y: R la vanhdon R2;/:0 maR21aideancuavanhR =>R2 =R ( vi R la vanhdon) Chuang1: T6ngquanvi vimhkh6nggiaohodn Trang11 Ta giasa: J (R) *-0 maJ( R) la ideancuaR =>J(R) =R =>{J(R)}2=R2=R .Tl1ongtl!: [J(R)t =Rn=R*-0 maRIa vanhArtinkhongcophftntU' lily linh*-0 =>J (R) =(O).V~yR la vanhnaadon (3) R favanhnguyenthttythiR favanhmladr.m ThrJtw;Zy: Ne'uR la vanhnguyenthuythlt6nt£;lip la ideanphai t6i d£;lichinhquisaocho(p : R ) =( 0) maI( R)=n (p :R)=>J(R) =( 0).V~yRIa vanhmYa don (4)Ne'uR vt'ta favanhdr.m,vitafavanhmladr.mthiR favanhnguyen thtty Th(ztv(zy: R2*-0 vakhongcoideannaokhacngoaiR va0 maR 1avanhnaadon=>J(R) =(0) =>(\ (p :R)=(0) voi 1aideant6i d£;li Ta co : (p :R)1aideancuaR =>(p :R)=(0)ho~c(p :R)=R Ne'u(p :R)=R thl(\ (p :R)=R ( vo 11') =>(p :R) =(0) V~yR 1avanhnguyenthuy 1.4.cAe BINDLY : GiasaR 1avanhnguyenthuyvaM laR-modunbit khaquytrung thanh.Theob6d~Schur's:11={q>E E(M) / q>oTr=~oTr, "ifrE R } voiTr :M ~ M thll1 lavanhchia(haycongQila th~) . m.~ mTr=mr Voi vanhchia11,tacothSxet M lakhonggianvectotren11. Th~tv~y: taxetpheptoaD:M/1~ M t-)H.J<~~;'" THlf VIEN 00767 q> Chuang1 : T6ngquanv~vcmhkh6nggiaohoan Trang12 Voi pheptmlnnayM cocffutrUckhonggianvectOtren~,do~la vanhchianenMIa khonggianvectOtrenL1.VItht taco th~dinhnghlas1;1' dQcl~ptuyentinhva s1;1'phl;!thuQctuytntinhnhusau: >- ffir,ffi2, ,ffikdQcl~ptuyentinh( neuffil<Pl+illz<pz+ +illk<Pk=0 thl «>1 =«>2==«>k=0) >- ffi[,ffi2, ,ffikphl;!thuQCtuytn tinh 3«>r*-0 : illl«>l+ill2«>Z+ +illk«>k=0 ( l~r ~k ) 1.4.1.Dinhnghlataedongdayaile: VanhR gQila tacdQngdayd~ctrongM (ho~cR dayd~ctrongM ) ntu voi ffi6ih~vectc5VI,VZ, ,VnE M dQcl~ptuyentinhtren~vaba'tky n phftn tU' WI,WZ, , Wn trong M thl t6n t:;ti rE R sao cho Wi =vir, i=1,2, ,n Nhanxet: (1) 0 daysl!dayd~cduQchi~urheanghla: Iffytliy Y ffiQth~huu h:;tndQcl~ptuytntinh,voi illQth~hii'llh:;tnbfftky , baagid cling16nt:;tiphepbiend6ituyentinhbitn h~dQcl~ptuyen tinhnaythanh ~kia.VIv~ykhitanoiR dayd~ctrongM tuc la R dayd~ccacphepbiend6ituyentinhtrongkhonggian vectdM trenvanhchiaL1 (2) Ntudimt1M =n( huuh:;tn) thethiHOillt1(M,M) =R Th~tv~y: 'iff E Homt1(M,M),giasaEl,E2,"",EnlacdSacuaM D6ng cffuf hoanroanduQcxacdinhntu bitt cacanhElf,Ezf, , Enf Theotinhdayd~ctaco: 3r E R saocho"i/ Wr,W2"""Wn EM: Chuang1 : T6ngquanv~vcmhkhonggiaoholm Trang13 E i r =Wi ( l~ i ~n) ~ r ==f V~yHomtl(M,~i) c R (**) Tli (*) va (**) ~ Homtl (M,M) =R 1.4.2.DinhIt daydac: GiasaRIa Valinnguyenthuy, M IaR-modunb§tkhaguytrung thanh, n€u ~=C(M) tillR la Valincondayd~ccacphepbi€n d6i tuy€ntinhtrongM tren~(n6itat:R dayd~ctrongM) chLtngminh : D€ chungminhdinh ly trentac~nchungrninh: Vc M la khonggianvectOhUllh~nchi~u, mEM,m~Vthl t6nt~ir E R :Vr =0vamr*0 Th~tv~y:N€u tac6di~utrentill :mrR* 0 d~dangchungrninh:mrRla moduliconcuaM trenR DoM-b§tkhaguytrungthanh~ mrR=M Ta tlmdu'<;1Cs E R saocho : mrstuy ytrongM vaVrs=0 Gia sa Vl,VZ,...,Vnla h~dQcl~ptuy€n tinhtren ~ WI,WZ,...,WnE M GQiVi la khonggiancuaM tren~sinhbdicacVj( i * j ) ~ Vi = ~ Vi~Vi Do h~Vl,VZ, ,VndQcl~ptuy€n tinhlien '1/i t6nt~itiE R: Wi =Vitiva Viti =(0 ) D~tt=tl+tz+ ,.+tn. Khi d6 : Wi=Vit, i = l,...,n Chuang 1 ..T6ngquanv~vimhkh6nggiao hodn Trang 14 Ta chungminhtinh tu'angduang,tUGla n€u c6 di~utrenthl tu'ang du'angvdiV la khonggianvectcJconhUllh;~lllchi~ucila M trenLl : mE M, m flY th13r E R : Vr =0, mr::/=0 Ta chungminhdi~unayguyn(;lptheosf)chi~ucilaV . n€u V la0chi~uV ={O}dung . GiasadungvdiV c6sf)chi~u~n- 1chi~u . ChungminhdungvdiV lanchi~u f)?tV =V0 +w6. ~ dimV0=dimV -I, Wfl V0 Ap d\lllggiathi€t guyn(;lpvdi A (Vo)={XE V / Vox=(O)} vdi y fl V0 ~3 r E A(V 0): yr::/=0 ~ yA(V 0) ::/=0 N6i cachkhac: mA(vo)=0~ mEVo D~dangkiSmtraA(Vo)Ia ideanphaicuaR laywA...(vo)"*0 ( do \11~Vo),va wA...(vo)la moduncon1\1 ~ wA(vo)=M (doM batkhaquy) Dungphanchung: GiasamEM, mflV vavdimliirmaVr=0thlmr=0 (*) Ta chungminh(*) khongthSxliyra : £)?t :T:M~M, x ~ xT =waT( vdix=wa) vdia E A(V0): xT =IDa ~ waT =IDa giasa:x=x' ~ wa=wa'~ w(a- a') =0 ~ a-a'linhh6atoaDbQm9iphffntlrA(w) MaVo(a-a')=0~ yea-a')=0(doVoeVe W) ~ m(a-a') =0 ~ ma=ma'=>x T =x'T ~ T III anhX(;l Chuang1 ..T6ngquanv~vcmhkhonggiao hoan Trang 15 D~d~mgki~mtra:T E E(M) (Xl+X2)T=(wal +wa2)T=w (al+a2)T =mal +ma2 =Xl T +X2 T =>T la1tvd6ngca"u * chungminh:T E ,1 '\!x EM, x =wa, a E A(V 0) '\!r E R =>arEA (Vo)( doA(Vo)la ideanphaicuaR) xet :xr=(wa)r=wear) ma(xr)T =((wa)r)T =wear)T =m(ar)=(ma)r=(xT)r * mE V : '\!a E A(Vo) =>ma=(wa)T =(wT)a =>(m- wT ) a=0,aE A(V0)=>m- wT E V0 =>m E Vo+wT c Vo+w,1=V (vo 19).V~y mr=I:-0 1.4.1.3.DinhIv: GiasuR lavanhnguyenthuy.Khid6voivanhchia ,1naGd6thiho~cR ==,1n( v?mhmatr?llca'pn x n tren~)ho~cydi InQisf) nguyenduongill t6nt<;1ivanhconSmcuaR ma d6ngca"ulen ,1m,We la L1mla anhd6ngca"ucuaSm 1.4.1.4.DinhIv Wedderburn- Artin : ChoR la vanhArtindonthlR d~ngca"uvdi Dn,Dnla t~pta"t cacaeIDatr~n x n trenvanhehiaD.Honnua, n duynha"t, D sai khaemQtphepd~ngea"u.Ngu'<;1el<;1i: ba"tky th€ D naGthl Dnla vanh Artindon ChLtngminh..GiasuR lavanhArtindon=>J(Rt =( 0) Ta c6 :R don=>R =I:-0 =>R2 =R => R khongIllY linh =>J(R) =I:-R =>J(R) =0=>R - nuadon =>R la vanhnguyenthuy( doR VITadon, VITanuadon) Chuang 1 : T5ngquanvi vimhkh6nggiaohoan Trang16 =>R coM -modunbfttkhi!quytrungthanh GQiMIa khonggianvectdtrenth~D =C(M) =>R dayd~ctrongHomD(M,M) * Ne'uM huuh~nchi~utrenD :R =HomD(M,M) ==Dn * Ne'uM voh~n Lfty t~p{VI,VZ,"",vn}dQcl~ptuye'ntinhtrongMtren D Pm={X E R / ViX =(0),"1i =1,2,...,m}.Tagia sli' PI~ Pz~ ~ Pm Chungminh:Pmla ideanpilCh v X,X'E Pm: ViX- ViX'=Vi(X- X' ) =0 =>x- X' E Pm v X E Pm, "IrE R :ViX=0 , V i =1,2, , m =>(vix)r=0 , V i = 1,2,...,m =>vixr=0=>xr E Pm=>Pmideanphi!i cua R * Pk+I C Pk : Xet VI,VZ,".,Vk,Vk+1dQcl~ptuye'ntinhtrongM WI,Wz, ,wbwk+llaphftntli'cuaM saocho: WI=W2 = =Wk =0 va Wk+1=Vk+1 "* 0 Dotinhdayd~c uaR , 3 r E R : Wk+1=vk+lr => vir =0, , vkr=O,vk+lr=Wk+1 '* 0 =>r E Pknhangr ~Pk+1 V~y Pk+lC Pkmall thu~nR la vanh Artin =>M huuh~nchi~uvadimDM=n =>R =HomD(M,M) ==Dn