; TeX output 2002.04.02:1700y?JDtGGcmr17Notes7tonAzumaqya7talgebras%rۍJXQ cmr12YVorcrkSommerh auserDҤVVersion13e>&Nff cmbx12Contentsrۍ>'"V cmbx101MCen9tralTseparablealgebras1rڍMK`y cmr101.1dTheUUde nitionofseparability....................1M1.2dTheUUseparabilityidempGotent[.....................2M1.3dCentersUUofseparablealgebrasٍ....................2M1.4dTheUUVillamayor-Zelinskytheoremč..................3M1.5dIdealsUUincentralseparablealgebrasi܍.................4M1.6dT*ensorUUproGductsofcentralseparablealgebras㍍...........5M1.7dGeneratorsUUandfaithfulness .....................8M1.8dCharacterizationUUofcentralseparablealgebraso...........9M1.9dBimoGdulesUUovercentralseparablealgebras[.............11>2MAzuma9yaTalgebras814M2.1dLiftingUUidempGotentsj!..........................14M2.2dCentralUUidempGotentsU.........................14M2.3dCriteriaUUforAzumayaUUalgebrasG....................16M2.4dLoGcalUUrings ԍ..............................20M2.5dMatrixUUringsƶ..............................20>Bibliograph9y T201*y?>1VLCentralffseparablealgebras r>1.1 Inthefollowing, b> cmmi10R*cdenotesacommutativering.T*ensorproGductswithout >anUUindexaretakenoverRDz.>De nitionx|SuppGoseUUthatAisanRDz-algebra.o>J81.WAUUiscalledcentralifthemap úR !", cmsy10!Z(A);r57!rG1 0ercmmi7AWisUUaringisomorphism.tJ82.WAisseparableifitispro8jectiveasanAf A^ٓRcmr7op-moGdule,)whereAisconsid-WeredUUasanA8 A^op-moGduleUUviacA8 Aop A!A;(a8 a O!cmsy709;a00r)7!aa00a0ZJ83.WAhItoisthepurpGoseofthissectiontoproveothatAisanAzumayaoalgebraifand>onlyĝifitiscentralseparable.zThisgoalwillbGeachievedinParagraph1.8.zInthe>de nition ofcentrality*,*note that,incontrasttothecaseof elds,theinjectivity>ofUUthemapr57!rG1A isnotautomatic.&W>1.2 W*eUUneedanequivqalentcharacterizationofseparability:>PropQosition,F*orUUanRDz-algebraA,thefollowingassertionsareequivqalent:o>J81.WAUUisseparable.tJ82.WTheexistsanelemente,2AaE AsuchthatA(e),=1AAand(aaE 1A ֲ)e,=We(1A q 8a)UUforalla2A.>ProQof.eJThemultiplicationmapA :nsAH A!AisAH A^op-linear.fIfA>isn&pro8jectiveoverA A^op,nmtheren&mustbGeamapf:A!A An&satisfying>Ab;ҵf=uid #̟A\.YTheYelementeu=f(1A )Ysatis estherequirements.YTherefore,Ythe> rstUUassertionimpliesthesecond.&W>F*orUUtheconverse,UUde ne +fڧ:A!A8 A;a7!(a8 1A q)e>ThisomapisAJQ A^op-linearoandsatis esAJQfB=id H A˚.oTherefore,oAisadirect>summandUUofthefreeA8 A^op-moGduleUUA A,andthuspro8jective.0Tq lasy1022y?>Note,thatanelementewiththesepropGertiesisanidempotentinAo A^op:ݔIf >e=u cmex10P USiTGai, 8biTL,UUwehaveײ(e2C=(X tοiqai, 8biTL)e=(X tοiaiTLbi, 81A q)e=e捑>ItUUisthereforeoftencalledaseparabilityidempGotent.彍>The AabGovecharacterizationimpliesdirectlythatseparabilityispreservedunder>passageUUtoquotients:f>Corollaryv(ҲSuppGoseM.thatI- d*Aisatwo-sidedM.ideal.NIfAisaseparableRDz->algebra,UUthenA=I7isaseparableRDz-algebra.>ProQof.eJIfe 2A} AisaseparabilityidempGotentforAand : A!A=Iis>the\quotientmap,\then( =[ٲ)(e)isaseparabilityidempGotentforA=I%x(cf.[2],>Chap.UUIGI,Prop.1.11,p.46).2彍>1.3 ItVisclearthatanA A^op-moGduleVisthesameasanA-bimodule.WF*or>everyUUA-bimoGduleM,wecanconsiderthecentralizerM^A ofM:WȀMA aò:=fm2M3jam=maforUUallja2Ag>IfUUweconsiderAasanA8 A^op-moGduleUUasabove,UUwehaveamapETHomA A⬱op(A;M)!MA;fڧ7!f(1A )>whichUUisbijectivesincetheinversemapis(MA aø!HomqA A⬱op3J(A;M);m7!(a7!am)=h>ByVde nition,|Aisseparableifitispro8jectiveasanA6 A^op-moGdule.;ThisVis >equivqalentLtotherequirementthatHom"VA A⬱op4Ж(A;)isanexactfunctor.NzW*e>thereforeUUcansaythefollowing:>PropQosition,SuppGose thatAisaseparableRDz-algebra."7Thenforeachexact>sequence{M3!M0l!M00g>theUUsequence#MA aø!M0TA /!M003A>isUUalsoexact.fp>Moreover,QwePhaveseenthatthispropGertycharacterizesseparablealgebras>(cf.[2],Chap.IGI,Lem.1.2,p.42,Cor.1.5,p.43,[6],xIGII.5,Lem.(5.1.2),>p.UU135).彍>W*ewanttoapplythispropGositiontodeterminethecenterofaquotientalgebra:3y?>Corollaryv(ҲSuppGoseethatAisaseparableRDz-algebraandthatI⡵Aisatwo- >sidedUUideal.If":A!A=I>isUUthecanonicalpro8jection,thenZ(A=I)=[ٲ(Z(A)).>ProQof.eJConsidersIA=I<+asanA-bimoGdulebypullbackvia[ٲ.tSince"issurjective,>itUUfollowsfromtheprecedingpropGositionthattheinducedmap˺AA J!(A=I)A>isKalsosurjective.cButthisispreciselytheassertion.(cf.[2],zChap.IGI,Prop.1.11, >p.UU46[6],xIGII.5,UULem.(5.1.4),p.136).2 p>1.4 Next,UUwegivetheVillamayor-Zelinskytheorem:>PropQosition,Aa?separableaBRDz-algebraAthatispro8jectiveasanR-moGduleisa> nitelyUUgeneratedRDz-moGdule.>ProQof.eJSinceAisapro8jectiveRDz-moGdule,wehaveforeveryfamily(aiTL)i2IpRof>generatorsUUcorrespGondinglinearfunctionsfid:A!RisuchUUthatqˍ.ҵa=X 7Ŵi2I㉵fiTL(a)ail>whereGinthissumonly nitelymanycoGecientsarenonzero.G-ChoGoseasepara->bilityUUidempGotentuXe=>n X tjg=1㉵xjo 8yj>W*eUUthenhaveUUthatthesetεJQ:=fi2Ij(idUWA fiTL)(e)6=0g>isUU nite.F*ora2A,UUwehaveuXlOa=ajδn X t5jg=1xj6yjIJ=ajδn X t5jg=1X 7Y8i2J!㊵xj6fiTL(yj)aid=>n X tjg=1㉟X 7ui2J"xjfi(ayj)ai >and#thereforetheelementsxj6aiwͲforj@=1;:::;nandi2JgenerateAasan >RDz-moGduleUU(cf.[2],Chap.II,Prop.2.1,p.47,[14 ],Lem.9.2.12,p.311).2 p>1.5 F*oracentralseparableRDz-algebraA,thereisabijectivecorrespGondence>bGetweenkidealsofRRandtwo-sidedkidealsofA,kaswewillseeinParagraph1.9.>Here,UUwebGeginwithsomepreparation:4(wy?>Lemmak SuppGosexthatAisaseparableRDz-algebra. ThenZ(A)isanR-direct >summandUUofA.>ProQof.eJChoGoseUUaseparabilityidempotente2A8 A^op.UUSincethemapi@fڧ:A8 Aop 7!EndƟRA((A);a a0Q7!(a00㊸7!aa00ra09)>isanalgebrahomomorphism,f(e)isapro8jection.Itfollowsfromthede ning>propGerties(ofethatf(e)mapsAtoitscenterandthatf(e)istheidentity(on>theUUcenter.Therefore,wehavethedecompGosition:A=Z(A)8kerS(f(e))>(cf.UU[2],Chap.IGI,Lem.3.1,p.51,[6],xIGII.5,UULem.(5.1.6),p.136).2p>TheonextpropGositionisstillpreparatory*,althoughitalreadyprovidesthees->sentialUUpartofthecorrespGondence:>PropQosition,SuppGoseUUthatAisacentralseparableRDz-algebra.J81.WF*orUUanidealIRDz,wehaveI=RL\8IA.J82.WF*oramaximalidealI/dfA,J\:=Rk\xI}޲isamaximalidealofRDz,andweWhaveUUI=J9A.>ProQof.eJIn·the rstassertion,itisclearthatI;RH\4VIA.F*orthereverse>inclusion,UchoGose?anRDz-submoduleLofAsuchthatAW=RrѵL;kthis?existsby>theUUprecedinglemma.W*ethenhave1bIA=IRL8ILI¸8L>andUUthereforeRL\8IAI7(cf.UU[2],Chap.IGI,Cor.3.2,p.51).p>TheproGofofthesecondstatementismorecomplicated.·ThealgebraA=Iissim->plebyassumption.\F*romCorollary1.3,weknowthatthecanonicalpro8jection>induces asurjectivemapfromthecenterofAtothecenterofthequotient.This>showsUUthatZ(A=I)T͍+3= UNR=J9.>T*oshowthatJ ismaximal,wehavetoshowthatR=J isa eld.3Ifrx20Rvnc3J9,>itsAequivqalenceclassr ¸2R=J7ϲisnonzero.ASinceA=I xissimple,Athenonzeroideal>sD>rB(A=I)mustcoincidewiththewholequotientalgebra,whichshowsthatr is>invertible$inA=I.rChoGoseanelementaƸ2A$thatsatis eshrvвR8a.=1:A=I .F*romthe>lemma]abGove,]weknowthatR=JT#isadirectsummandofA=I,]i.e.,wecan nd>anUUR=J9-submoGduleLsuchthatA=I=R=J/8L53-y?>W*e(writeMa intheformMa 94=a1lɲ+a2 ,(whereMa12R=J%andMa22L.)0W*ethenhaveJ 1:A=Iz!=:\rIJ,a1+$8r'a22R=J/8L>Thisimplies1:A=Iz=Dqr  a1a(,andthereforer 璲isinvertibleinR=J9,whichproves >thatUUR=JKisa eld.ɗ>TheidealJ9AisobviouslycontainedinI.ConsiderthefactoralgebraA=J9A.F*or>thesamereasonsasabGove,thisisaseparablealgebra.F*romthe rststatement,>we?AhaveR \ J9A=J,?Gand?AthereforewegetasabGovethatthecenterofA=J9Ais>isomorphicXtoR=J9.2ButaseparablealgebraoverXa eldissemisimple,}andifthe>center.Xisone-dimensionalthealgebramustbGesimpleitself..Therefore,.bA=J9Ais>simple,UUandthesurjectivealgebrahomomorphismJ̠A=J9A!A=I>isanisomorphism,whichprovesthatI=J9A(cf.[2],Chap.IGI,Lem.3.5,p.53, >[6],UUxIGII.5,UULem.(5.1.7),p.137).2ɗ>1.6 Inthisparagraph,weshowthatthetensorproGductoftwocentralsepa->rablehalgebrasiscentralseparable.Again,weneedsomepreparation:SuppGose>thatAandBaretwoRDz-algebrasandthatMŲandM^0haretwoA-moGdulesand>thatUUNlpandN^0:aretwoUUBq-moGdules.Asusual,wehaveacanonicalmapJZQݵ ":HomqA (M;M0T)8 Hom9BE(N;N0T)!HomqA B*(MO 8N;M04 N0T),>Lemmak IfMʨisa nitelygeneratedpro8jectiveA-moGduleandNisa nitely>generatedUUpro8jectiveBq-moGdule,then .isanisomorphism.>ProQof.eJChoGoseگgeneratorsm1|s;:::;mk?ofMʲandgeneratorsn1;:::;nlwofN,as>wellascorrespGondinghomomorphismsfid2HomqA (M;A)andgjĸ2HomqB~}(N;Bq)>suchUUthat\tm=5k X tմi=1㉵fiTL(m)miTNn=l X tjg=1gj6(n)nj>for(?allm2M?Zand(?alln2N.(If(?':MѸ ޶N3!M^0 ޶N^0 is(?A Bq-linear,(Kwecan,>forUUalli=1;:::;kandUUjY=1;:::;l2`,UUwriteG'(mi, 8nj6)=j,qO \cmmi5ij YX Rp=1㉵m0፴ijgp c* n0፴ijgp>NowUUde neJX/ijgp b:M3!M0T;m7!fiTL(m)m0፴ijgp*Lijgp:N3!N0T;n7!gj6(n)n0፴ijgp>W*eUUclaimthatOϵ [ٲ(k X ti=1 rl qX tijg=1j"hqij Y 8X Rp=10USijgp c* 8ijgp *J)='6?y?>ThisUUholdssince鍍^ŵ [ٲ(k X ti=1~/l x.X tyEjg=1jDqij YX -ٴp=11ijgp c* 8ijgp *J)(m n)=5k X tմi=1ӊl ㉟X t1jg=1j%0qij Y"X #4p=13kijgp(m)8 ijgp(n)$Yz=5k X tմi=1ӊl ㉟X t1jg=1j%0qij Y"X #4p=11qò(fiTL(m)8 gj6(n))(m0፴ijgp c* n0፴ijgp *J)z=5k X tմi=1ӊl ㉟X t1jg=1!UR(fiTL(m)8 gj6(n))'(mi, nj)z='(k X ti=1 rl qX tijg=1 8fiTL(m)mi, 8gj6(n)nj)='(m8 n)Iu>Thisjprovesthatthemapissurjective.`T*oproveinjectivity*,weconsiderthemapSqTA J:HomqA (M;A)8 M0l!HomqA(M;M0T);fLo m0Q7!(m7!f(m)m09)>andUUtheanalogousmap\SѵTB $:HomqB~}(N;Bq)8 N0l!HomqB(N;N0T);g n0Q7!(n7!g[ٲ(n)n09)>NowUUsuppGosethat< [ٲ(Cq 獍X :p=1qp2 8pR)=0j>ThisimpliesPލ?q%?p=1pR(miTL)vV p(nj6)`=0.W*ehavepR(m)`=Pލ k% i=1 fiTL(m)p(mi)@>andUUpR(n)=Pލ USl% USjg=1Vgj6(n)p(nj),UUwhichimplies!L]l[8q 獍gX g0شp=1w4p2 8pfj= q 獍X Rp=1㉵TA(k X ti=1qfi, pR(miTL)) TB (l X tMjg=1gjo pR(nj6)) >ThisUUshowsthatPލ 㐴q% p=1Mp2 8pistheimageofthetensorTq 獍YX rp=1yHk v+X t4i=1l X t)jg=1ʯ fi, 8gjo pR(miTL) p(nj6)!>withUUrespGecttothemapuFHom[zAa(M;A)8 Hom9BE(N;Bq) 8M04 N0l!HomqA (M;M0T) Hom9BE(N;N0T)fLo 8g m0 n0Nj$7!TA(fLo 8m09) TB (g n0)>ButGwthisiszero,GzsincealreadythetensorPލ ղq% ղp=1?pR(miTL)$ p(nj6)Gwiszero.G(cf.[2], >Chap.UUI,Sec.2.4,p.14,[7],Lem.1.4.1,p.15).27O@y?>ThisUUlemmaisusedinthetreatmentofthecenterinthefollowingcontext:S>PropQosition,If2UAandBƲarecentralseparableRDz-algebras,2thenA5 Bis2Ua >centralUUseparableRDz-algebra.>ProQof.eJLetus rstobservethatthecanonicalmapR߸!Z(AaB Bq)isinjective.>This#followsfromthefactthatR7isadirectsummandofAbyLemma1.5,#and>similarly&adirectsummandofBq,EwhichimpliesthatRT͍߸+3߲= iR`K LRisadirectsum->mandofA Bq.T*oseethatA B;Pisseparable,choGoseaseparabilityidempGotent>eA J=Pލ USn% USi=1tJai /ǵa^0;Zi%forȵAandaseparabilityidempGotenteB $=Pލ USm% USjg=1Vbjfs /ǵb^0;ZjtforBq.>RecallUUfromParagraph1.2thatthismeansthatwehave7Ώn iX t&i=1*ڵaai, 8a0፴id=>n X tմi=1㉵ai a0፴iTLajдn X tigi=1%aia0፴id=1Ac>forUUalla2A,UUandsimilarlyforBq.ThisimpliesthatleA B쵲:=>n X tմi=1 m ㉟X t1jg=1"ai, 8bjo a0፴i b0፴j>is\aseparabilityidempGotentforA= Bq.\T*o\seethatA= B^is\central,\notethat>weUUhavefromtheabGovelemmathatthecanonicalmapߍDTHomY}A A⬱opu+(A;A)8 Hom9BW= B⬱op3yq(Bq;B)!HomqA A⬱op BW= B⬱opUu(A8 Bq;A Bq)>isabijection.ButaswehavediscussedinParagraph1.3,theabGovehomo->morphisms(}aregivenbymultiplicationwithcentralelements.(Therefore,(wesee>thatZ(A Bq)=Z(A) Z(Bq),,whichimpliestheassertion(cf.[2],Chap.IGI,>Prop.UU3.3,p.52,[6],xIGII.5,UU(5.1),p.134).2Yݍ>It9maybGenotedinthiscontextthatforacentralseparablealgebraAthealge->bra"A^op isalsocentralseparable.#9Namely*,"thecenterofA^op obviouslycoincides>with\thecenterofA,ȀandaseparabilityidempGotentforA^op {isgivenbyexchang->ingUUthetensorfactorsinaseparabilityidempGotentforA. u4>1.7 Before#weproGceed,#weneedanothercriterionwhenamoGduleoveracom->mutativeUUringisagenerator.S>De nitionx|SuppGoseUUthatMlpisamoduleoverUUthecommutativeUUringRDz.,LJ81.WTheUUannihilatorofMlpistheidealAM t:=fr52R߸jam=0forUUalljm2MgیJ82.WTheUUtraceidealistheidealTM t:=hff(m)jm2M;fڧ2HomqR+Ӳ(M;RDz)gi,M>W*eUUneedthefollowingvqariantofNakqayama'slemma:8 ^y?>Lemmak SuppGoseZ*thatMqEisa nitelygeneratedRDz-module.Z2IftheidealI&R >satis esUUIM3=M,wehaveI²+8AM t=RDz.F>ProQof.eJSuppGoseCthatM^isgeneratedbym1|s;:::;mnq~.Byassumption,[wecan> ndUUelementsaij 2I7suchthat*$ڵmid=>n X tjg=1㉵aij mjs>NowtheCayley-Hamiltontheorem(cf.[12 ],Thm.2.44,p.27)impliesthat>detK(EnA)2AM\,whereoEnuistheunitmatrixandA=(aij )i;jgnw.Expanding>detK(En6A)SviatheLeibnizformula,weseethatdet7(En6A)21R~+I,Swhich>provespTtheassertion.p}(F*oradi erentproGof,pZsee[2],Chap.I,Lem.1.7,p.6,[6],>Chap.UUIGII,Lem.(5.1.11),p.139).2>F*orUUpro8jectivemoGdule,thisimpliesthefollowing:>Corollaryv(ҲSuppGosepthatPԅisa nitelygeneratedpro8jectiveRDz-module.q Then>weUUhaveTM <+8AM t=RDz.>ProQof.eJSuppGoseMthatPaܲisgeneratedbyp1|s;:::;pnq~.PSincePispro8jective,xwe>canUU ndelementsandlinearformsf1|s;:::;fn82Pc^Ë=HomqR+Ӳ(PG;RDz)UUsuchthatʹʐp=>n X tմi=1㉵fiTL(p)pi^>forallpPq2Pc.=ThisshowsthatTM\Mg=PqM,andthereforetheassertionfollows>fromUUtheprecedinglemma(cf.[2],Chap.I,Prop.1.9,p.7).2>ThisUUprovidesuswiththefollowingcriterionforgenerators:>PropQosition,F*or?a nitelygeneratedpro8jectiveRDz-moGdulePc,thefollowing>assertionsUUareequivqalent:"J81.WPisUUagenerator.J82.WPisUUfaithful.>ProQof.eJIfPisagenerator,weknowfrom[1],Lem.1,p.6thatRisadirect>summand?XinadirectsumofcopiesofPc.?yTherefore,?]anelementvqanishingonP>vqanishesonRDz,andisthereforezero.Thisshowsthatthe rststatementimplies>theUUsecond.>Conversely*,9ifM5isfaithful,thenwehaveAM t=f0g,9andthereforeTM=RDz.By>thereferenceabGove,thistellsthatM/isagenerator.D(cf.[2],Chap.I,Cor.1.10,>p.UU8).29 m5y?>1.8 F*orranRDz-algebraA,wecanloGokatthespaceA~H ArasanA-bimodule. >TheUUcentralizerthenisȼa(A8 A)A J=fx2A8 Aj(a8 1A q)x=x(1A 8a)forUUallja2Ag>Ininthisspace.Ofcourse,wecanalsoviewthisspaceasasubsetofAy A^op,>whereitisobviouslyarightideal.pIfAisacentralseparablealgebra,wecan>sayUUsomethingmore:׍>Lemmak SuppGosethatthealgebraAiscentralseparable.FThentheleftideal>ofUUA8 A^op ProQof.eJSince(A A)^A visalreadyarightideal,thegeneratedleftidealisinfact>agtwo-sidedideal.IfitdoGesnotcoincidewiththewholealgebra,itmustbGecon->tainedinamaximaltwo-sidedidealIA^ A^op. ItfollowsfromPropGosition1.6>thatAf A^op isacentralseparablealgebra.OIfwede neJ/:=9]I/w\fRDz,itfollows>fromEPropGosition1.5thatI=J9(A( A^op).F*orEaseparabilityidempotenteofA,>thisSimpliese2I=J9(A5 A^op).SButSifA misthemultiplicationmapofA,this>wouldUUimply1A=A(e)2J9,UUwhichisacontradiction.2t>ThisUUfactcanbGerestatedasfollows:>Corollaryv(ҲSuppGosethatthealgebraAiscentralseparable.ThenAisan>A8 A^op-generator.>ProQof.eJW*eChavetoshowthatthetraceidealcoincideswithA A^op.ELThedualof>theA A^op-moGduleAisHomZA A⬱op3S(A;A A^op).AsexplainedinParagraph1.3,>this]spaceisisomorphicto(A A)^A.GThe]traceideal,i.e.,the]idealgeneratedby>the (vqaluesofelementsfromHomA A⬱op3(A;A A^op) (onelementsofA, ;therefore>coincideswithA(A A)^A.͝Usingthede nitionofthecentralizer,weseethat>thisisexactlytheleftidealgeneratedby(A_C A)^A lin膵A A^op,whichisA_C A^op>byUUtheabGovelemma.2t>W*eaxcannowgivethepromisedcharacterizationofcentralseparablealgebras.bRe->call thatamoGduleiscalledaprogeneratorifitisa nitelygeneratedpro8jective>generator.>PropQosition,F*orUUanRDz-algebraA,thefollowingassertionsareequivqalent:XmJ81.WAUUiscentralseparable.䍍J82.WAUUiscentralandanA8 A^op-progenerator.J83.WA isanRDz-progeneratorandthecanonicalmapAi A^op 7!EndƟRA((A) isanWisomorphism.10 yy?J84.WAUUisanAzumayaUUalgebra.>ProQof.jJ2:*)1:{DzThisisobvious,{sincebyde nitionanalgebraisseparableif >itUUispro8jectiveasanA8 A^op-moGdule.C1:)2:LByassumption,LwehavethatAispro8jectiveoverA'y A^op,LandLobviously> nitelygeneratedbythegeneratingelement1A*T.@Thatitisageneratorfollows>fromUUtheprecedingcorollary*.C2:F))3:;-Thisfollowsfromthe rstMoritatheorem(cf.[1],;hp.9),whichas->serts(winparticularthatprogeneratorisalsoaprogeneratorasamoGduleover(wits>endomorphismring,3andthattheendomorphismringwithrespGecttothenew>moGduleJzstructureisisomorphictotheoriginalring.KHere,Jtheendomorphism>ringoftheA A^op-progeneratorAisEndA A⬱op1h߲(A)T͍p+3p=Z(A)T͍p+3p=RDz,soAisan>RDz-progenerator,4and4~theendomorphismringwithrespGecttothenewmodule>structureUUisEnd@Re(A),whichshowsthatA8 A^opT͍ 7+3 7=isUUanRDz-progenerator,thistheoremassertsthatEnd:End6ϧRx:(A) (A)T͍+3= UNR>and,thatAisanEndڟR}<(A)-progenerator.UndertheassumptionthatA A^opT͍ 7+3 7= >EndOꮟRVz(A),)this)impliesEndzA A⬱op0º(A)T͍+3= UNRDz.*ButsinceEndzA A⬱op(A)T͍+3= UNZ(A),)we>seeUUthatRiiscentral,andalsothatAisanA8 A^op-progenerator.C3:,4:RThisfollowsdirectlyfromPropGosition1.7(cf.[2],RChap.II,Thm.3.4,>p.UU52).2 p>1.9 If AisanA¸ A^op-progenerator andthespaceHomA A⬱op4B(A;A)isiso->morphictoRDz,weknowfromthe rstMoritatheoremthatthecategoryof>A-bimoGdules\isisomorphictothecategoryofRDz-modules.] T*omakethisequivqa->lenceUUexplicit,weneedthefollowinglemma:>Lemmak SuppGoseH|thatAandBarerings.JSupposethatP isa nitelygenerated>pro8jectiveleftA-moGdule,thatM#isanA-Bq-bimodule,andthatN#isaleftBq->moGdule.UUThenthemapM95:qHomGAʩ(PG;M)8 B N3!HomqA (P;MO B N);fLo 8n7!(p7!f(p) n)>isUUanisomorphism.>ProQof.eJThespacePc^ Jڲ:=NgHom#AP(PG;A)isarightA-moGdule,andwecan,simi- >larlyUUtotheproGofofLemma1.6,considerthemappTM t:Pc5S ApM3!HomqA (PG;M);fLo 8m7!(p7!f(p)m)11 y?>which4isrightBq-linear.IfwechoGosegeneratorsp1|s;:::;pk~IJofPc,Daswellascor- >respGondinghomomorphismsfi2DiPc^ suchthatp=Pލ Ҥk% Ҥi=1fiTL(p)piҲforallp2Pc,>weUUseethattheinverseofTM 5isgivenasu;y5T䍑c17sM:HomqA (PG;M)!Pc5S ApM;g"7!5k X tմi=1㉵fi, 8g[ٲ(piTL);>InsertingthemoGduleM5" BNinsteadofM, wehavethefollowingcommutative>diagram:h7 ~HomAV(PG;M)8 B N ZHom A'C(PG;MO B N)532fd0O line10-އpڅ*(Pc^5S ApM)8 B NڅzPc^5S Ap(MO B N)Ӄ<8҄fd8+Lά-莎oXfe?sTM < B idpCN5[؟Xfe5 ?Bύ9zTM, N"#Y>Thisshowsthat isacompGositionofisomorphisms,Eandthereforeitselfan>isomorphismUU(cf.[2],Chap.I,2.7,p.15).2荑>One mayviewthefactthatTM hisanisomorphismasthespGecialcaseofthe>abGoveklemmainwhichMV=;A=Bq._ThekcorrespondencebetweenkRDz-modules>andUUA-bimoGdulesnowtakesthefollowingform:}͍>PropQosition,SuppGoseUUthatAisacentralseparableRDz-algebra.iJ81.WIfUUMlpisanA-bimoGdule,thenthemap'(A8 MA aø!M;a m7!amWisUUabimoGduleisomorphism.4J82.WIf`qNwisanRDz-moGdule,`thenthecentralizeroftheA-bimoduleA Nwis WisomorphicUUtoNlpunderthemapN3!(A8 N)A;n7!1A n>ProQof.eJBy=assumption,>AisanAո A^op-progenerator.?*Recall=fromthe rst >MoritaRtheoremthatthecategoryequivqalenceisgivenbytensoringwiththe>progenerator,resp.bytensoringwiththedualoftheprogenerator.ZTheproGof>of ?thepropGositionnowconsistsinthereinterpretationofthetensoringwiththe>dual}oftheprogeneratorHom\֟A A⬱op4 (A;AZP A^op).F*rom}thespGecialcaseofthe>lemmaqdiscussedabGoveqandthediscussioninParagraph1.3,wehaveforan>A-bimoGduleUUMlpthat'_xHomtN9A A⬱opy(A;A8 Aop) A A⬱op MT͍3+33=liHom#AŸA A⬱op>(A;M)T͍+3= UNMA8(*)12 "y?>whichUUisgivenonelementsbyfLo 8m7!f(1A )m.n>LetussetupindetailtheMoritacontextthatarisesfromthissituation.W*e >letSP :=ozA,T8consideredasaleftA⡸ A^op-moGduleSandarightRDz-module.U~Let>Q:=HomqA A⬱op3J(A;A A^op),~mconsidered~6asaleftRDz-moGduleandarightA A^op->moGdule.UUW*ethenhaveUUthemappings Iq](;):Po RBQ=A8 RHomA A⬱op9K۲(A;A Aop)!A8 Aop;a fڧ7!f(a)>asUUwellas>[;]:QHո A A⬱opP*=HomqA A⬱op3J(A;AHո Aop) A A⬱opA!R;f\d Hյa7!f(1A )(a)>TheЅsecondmapneedsalittleexplanation.BAepriori,ФthesquarebracketsЅmap >fLo 8aUUtotheA A^op-endomorphismA!A;a0Q7!f(a09):a>Now,anιA͸ A^op-endomorphismisgivenbymultiplicationwithacentralele->ment,%whichinthiscaseisf(1A )(a).~SinceAiscentral,thiselementisinfact>inUURDz.n>F*or6anA-bimoGduleM,6wenowgetfromthe rstMoritatheoremthatthemap>AS5 Hom(A A⬱op1β(A;A Aop) A A⬱opuM3!M;'a ffĸ m7!f(a):m=af(1A ):m>isanisomorphism.Inviewoftheisomorphism(*),-thisisthecompGositionof>theUUmap>AZ HomﳟA A⬱op2(A;A Aop) A A⬱opȚM3!A MA;a f- m7!a f(1A )m>consideredUUabGoveandthemap?'A8 MA aø!M;a m7!am>giveng:inthe rststatement.hAsthe rstmapisanisomorphism,gthesecond>mustUUbGeanisomorphism,too.ThisprovesUUthe rststatement.n>IfŲMx=]Aȸ NͲforanRDz-moGduleN,the rstMoritatheoremalsoassertsthat>theUUmapyaHom7(A A⬱oph(A;A8 Aop) A A⬱op A N3!RL NqAfy 8a n7![fV;a]8 n=(f(1A ):a)8 n=f(1A ):(a8 n)>isXanisomorphism.XOntheotherhand,Xwegetfromtheisomorphism(*)abGove>thattheimageoftheabGovemapis(AQ N)^A.v(Notethatthecanonicalmap>from7RC z|NNܲtoA NNܲisinjective,8=sinceRKisadirectsummandofAby>Lemma1.5.)ThesecondassertionnowfollowsbycompGositionwiththeiso->morphismUUN3!RL 8N;n7!1A nUU(cf.[2],Chap.IGI,Cor.3.6,p.54).213y?>F*romthiscorrespGondence,5wecaninfertherelationbetweentheidealsofAand >theidealsofRDzthatwealludedtoinParagraph1.5.Denotethesetoftwo-sided>idealsUUofAbyIA andthesetof(two-sided)idealsofRibyIRb.>Corollaryv(ҲIfUUAisacentralseparableRDz-algebra,themappingsIA J!IRb;I7!RL\8I>and IR Vz!IA;JQ7!J9A>areUUmutuallyinversebijections.>ProQof.eJNoteythatatwo-sidedyidealinAisnothingelsethanasubbimoGdule>ofA,0andsimilarlyanidealinthecommutativeringR̲isnothingbutasub->moGdule.MSinceM Aisapro8jectiveRDz-module,M itisinparticular at,andtherefore>weUUhaveforanidealJQRithattheinducedhomomorphism^J/ 8A!RL A>is5}injective.5ThisimpliesthatthecanonicalisomorphismbGetweenR 1AandA >inducesUUanisomorphismbGetweenUUJ/ 8AandJ9A.p>F*romUUtheprecedingpropGosition,wenowgettheisomorphismصJQ!(J/ 8A)A J!(J9A)A=Z(A)8\J9A=RL\8JA>whichUUprovesthatRL\8J9A=J.>On theotherhand,ifI%TCAisatwo-sided ideal,we knowfromthepreceding >propGositionUUthatthemap A8 IA !I;a r57!ar>is0qanisomorphism.1SinceI^A =4DZ(A)\I&=R޹\I,0this0qproves(R޹\I)A4D=I>(cf.UU[2],Chap.IGI,Cor.3.7,p.54).2p>InAtheprecedingproGof,QwehaveusetheRDz-pro8jectivityofAtogettheisomor->phismbGetweenJX bAandJ9A.oItmaybGenotedthatonecanalternativelyuse>the8Aw A^op-pro8jectivityofAtoinsurethatthesurjectivityofJ wA!J9A>impliesUUthesurjectivityofZ(J/ 8A)A J=HomqA A⬱op3J(A;J A)!(J9A)A J=HomqA A⬱op3J(A;JA)>asUUpGointedoutinProposition1.3.14ay?>2VLAzumayaffalgebras>2.1 Thegoalofthissectionisto ndcriteriawhichallowsustoverifywhether >agivenalgebraisanAzumayaalgebra.yInthesequel, Rdenotesacommutative>ring.UUW*egivecompleteproGofsexceptforthefollowingfact:>PropQosition,SuppGoseqxthatR?isacompletenoetherianlocalringwithmaximal>idealbI,andthatAisa nitelygeneratedalgebraoverbRDz.Ifp2A=IAbisan>idempGotent,thenthereisanidempotentep2A芲whoseresidueclassJe ysatis es>b:>eEo=p.>ProQof.eJSee([8],XChap.7,Thm.(21.31),p.330,andProp.(21.34),p.332.JA>moreUUgeneralresultisstatedin[2],Chap.IGI,x7,p.73.2 p>2.2 OfIqcourse,IPropGosition2.1appliesinparticulartocentralidempotents.>However,cwebcansaymoreabGoutcentralidempGotents:cTheirliftingsarealso>central,?and,moreimpGortantly*,theirliftingsareunique.JThe rstassertion>followsUUfromthefollowingfact,whichisknownasDade'slemma:>Lemmak SuppGosethatIsisanidealofRXwhichiscontainedintheJacobson>radical.eSuppGoseŸthatAisa nitelygeneratedalgebraoverŸRandthateg2A>isfanidempGotent.fTheneiscentralinAifandonlyifitsequivqalenceclasse uis>centralUUinA=IA.>ProQof.eJItthate YiscentralinA=IA.6ThenwehaveeW(A=IA)(1uSe)]=f0g,andtherefore>eA(1cFe)12IA.PMultiplyingthiswitheontheleftand1cFeontheright,we>seeUUthateA(18e)2I(eA(18e))>Since51eA(1e)isobviously nitelygenerated,59wegetfromNakqayama'slemma>thatgeA(1Ee)=f0g.gNow,gfora2A,gtheequationea(1Ee)=0isequivqalent>toea=eae. Bythesamereasoning,wegetthat(1qe)Ae=f0g,whichimplies>(1Se)aen=0Sresp.aen=eae.U Therefore,SõeSiscentralinA.(cf.[8],SChap.7,>(22.10),UUp.342).2p>DenoteWbyJc(A)resp.Jcp23(A)thesetofcentralresp.centrallyprimitiveidem->pGotentsUUofA.W*ethencansaythefollowing:>PropQosition,SuppGoseqxthatR?isacompletenoetherianlocalringwithmaximal>idealUUI7andthatAisa nitelygeneratedalgebraoverUURDz.ThenthemappingsT.~Jc(A)!Jc(A=IA);e7!)Reoand?Jcp23(A)!Jcp(A=IA);e7!)Re>areUUbijective.15y?>ProQof.eJAstheJacobsonradicalistheintersectionofallmaximalideals,۵Izis >the`JacobsonradicalofRtinthissituation.`ItfollowsfromPropGosition2.1and>Dade'stlemmaabGovetthatthe rstmapissurjective.KLetusseethatthemap>isinjective.-Ifeande^0ɲaretwocentralidempGotentsofAwhichhavethesame>residueclassmoGduloIA,thene^00㊲=e(1e^09)isacentralidempotentwithresidue>classdvzero.dThisimpliesthate^002OIA,dzandthereforee^00rAIA.dAsintheproGof>ofKDade'slemma,Lmultiplicationbye^00 hIgivese^00rAa=I(e^00A),LandKNakqayama's>lemmatgivese^00rAײ=f0g,uwhichtinturnimpliese^00I=0.u+Byde nitionofe^00r,uthis>impliese޲=ee^09. Interchangingtherolesof"eande^09,0weseethate޲=e^09,0and>thereforeUUthe rstmapisinjective.xM>F*orothesecondmap,the rstquestioniswhetheritiswell-de ned.\Byde nition>(cf.ə[8],ɽChap.7,x22,p.336),anonzeroidempGotentiscalledcentrallyprimitive>ifT:itcannotbGewrittenasthesumoftwoT:nonzeroorthogonalcentralidempotents.>Now\suppGosethate2A\isacentrallyprimitiveidempGotent.^7Ife awcouldbGewritten>asvthesumoftwovnonzeroorthogonalcentralidempGotents,wecouldliftthese>idempGotents,UUsothatwewouldhave)Ǔe6=)Re0 v2+8e00>forNtwocentralidempGotentse^09;e^00 Ӹ2faA.PSincee^0>ande^00awareorthogonal,Ntheir >proGductiszero,OandthesameargumentthatweusedabGovethenshowsthat>e^09e^00#isualsozero,uwhichshowsthate^0Cande^00#areorthogonal.wButthene^0GԲ+ye^00isan>idempGotent,XandXtherefore(1?e)(e^0 +e^00r)Xande(1?e^0e^00r)XareidempGotentswhose>residueclassesarezero,whichimpliesasabGovethattheyarezerothemselves.>InUUdi erentterms,thesetwoequationsassertthat,e0+8e00㊲=e(e0+e00r)and8ye=e(e0+e00r)>whichsimpliese`=e^0D+vIe^00inscontradictiontotheassumption.Thisestablishes >thatUUthemapiswell-de ned.>The,injectivityofthesecondmapfollowsfromtheinjectivityofthe rstmap,>and:itisalsoeasytoseethatthemapissurjective::$Given:acentrallyprimitive>idempGotentcinA=IA,cwecanliftittoacentralidempGotentofA,candthequestion>isKwhetherthisidempGotentiscentrallyprimitive.ButadecompGositionofthis>idempGotentBintoasumoftwononzeroorthogonalcentralidempGotentswould>giveasimilardecompGositioninA=IA,sincewehaveseenacoupleoftimesthat>nonzeroidempGotentshavenonzeroresidueclasses.1SincesuchadecompGosition>doGesqnotexistbyassumption,theliftedidempotentmustbGecentrallyprimitive>itselfUU(cf.[8],Chap.7,Prop.(21.22),p.327,andThm.(22.11),p.342).2_>This[resultimpliesthattheW*edderburndecompGositionforthe nitedimen->sional[algebraA=IAover[the eldF\:=|͵R=I$زleadstoacorrespGondingblock>decompGositionofA.Notethat,+ifAispro8jectiveoverRDz,+andthereforefree,and>ifߵe2AisacentrallyprimitiveidempGotent,thenthecorrespGondingblockAeis>pro8jective,UUandthereforealsofree,andinparticularfaithful.16]y?>2.3 W*e]aregettingclosertothemainresultofthissection,]acriterionfor >Azumayaalgebras.First,weneedanauxiliaryresult,whichisanalogoustothe>factUUfromlinearalgebrathatI"ker&(fLo 8g[ٲ)=ker#(f) Wo+Vqĸ kerS(g[ٲ)>forUUtwolinearmapsfڧ:V!V8^0\randg":W*!Wc^0bGetweenvectorspaces.ō>Lemmak SuppGoseHthatI*isanidealinthecommutativeHringRDz,landthatM >andUUNlparetwoUURDz-moGdules.ThenthesequenceI(MO 8N)!M 8N3!(M=IM) (NA=IN)>isUUexact.>ProQof.eJSincethetensorproGductisrightexact(cf.[5], Prop.III.7.3, p.110), >weUUthenhavetheexactsequence(IM)8 N3!MO N!(M=IM) N!0>Itr_iseasytoseethattheimageofthemap(IM) NA!&M Nzcoincides>withdsI(MZ BN).dOntheotherhand,dwwehaveagainbyrightexactnessthatthe>sequencefs(M=IM)8 IN3!(M=IM) N3!(M=IM) (NA=IN)!0>isexact.Sincetheleftmapisidenticallyzero,$weseethatthecanonicalmap>(M=IM)ø N3!(M=IM) (NA=IN):isanisomorphism.:Thisproves:theasser->tion.UU2,>TheUUmainpieceofworknowisthefollowingpropGosition:ō>PropQosition,SuppGoseqxthatR?isacompletenoetherianlocalringwithmaximal>idealI˲andthatAisa nitelygeneratedalgebraoverRDz.JThenAisseparableif>andUUonlyiftheR=I-algebraA=IAisseparable.>ProQof.eJIf\Aisseparable,]"thenA=IAisseparablebyCorollary1.2.^^SuppGosenow>thatLJA=IAisseparable.LXW*eproGceedinsteps,LLfollowing[2],Chap.IGI,Thm.7.1,>p.UU72:卍>(1)OLetus rstseethatIAistheJacobsonradicalJ9(A)ofA.Byde nition,>theJacobsonradicalofAistheintersectionoftheannihilatorsofthesimple>A-moGdules.yXNow,y*ify!Morsince3{Aisa nitelygeneratedRDz-moGdule,3weseethatMJis nitelygeneratedas>anRDz-moGdule.0Therefore,:thecaseM3=IM#isimpossiblebyNakqayama'slemma,>and#weseethatIannihilatesM./Therefore,OIAnJ9(A).On#theotherhand,>A=IACisseparablebyassumption;Ctherefore,Citissemisimpleandhasvqanishing>Jacobsonradical.SinceJ9(A=IA)=J(A)=IA(cf.[8],Chap.2,Prop.(4.6),p.55,>[3],UUChap.2,Exerc.8,p.68),wegetJ9(A)=IA.17iy?>(2)OChoGoseUUaseparabilityidempotent˚8xeҧ=>n X tմi=1!㉵ai \q58bixD>F*romUUtheabGoveUUlemma,weknowthatthecanonicalmapfrom 5W(A8 Aop)=I(A Aop)!(A=IA)8 (A=IA)op>is}anisomorphism,andthereforewecanusePropGosition2.1toliftetoan >idempGotentUUofA8 A^op,UUi.e.,wecanassumethattheelement/ae:=>n X tմi=1㉵ai, 8bi->isgitselfanidempGotent.gHowever,gwegdonotyethavexݲ:=Pލ tn% ti=1aiTLbi:)=1Ain,gand>thereforeUUwewanttomoGdi ythisidempotent.}>(3)OAsUUwehavePލ 㐴n% i=1'ai\q%EX%bi0=1:A=IJA,itfollowsthatԍF1Aʷ8x=1An X ti=1UQaiTLbid2IA=J9(A)}>Thereforex=1AC(1A -Ըx)isinvertible(cf.[8],Chap.2,Lem.(4.3),p.54,[3],>Prop.UU2.8,p.63),andwecanloGokattheelementXwe0Q:=>n X tմi=1㉵x1 taiTLx8 bixD>Obviously*,80this8)isagainanidempGotentinA A^op,80but8)furthermore,usingthat>e2A8 A^op n X tմi=1㉵x1 taibi=x1 tx=1Ag~>InUUaddition,since1AƸ8x2IA,UUwehavee^0 =)Reo.}>(4)OW*e5arestilllackingthesecondpropGertyoftheseparabilityidempGotent,>namelyf{thate^0\2U(A A)^A.hF*orf{this,fweneedalittlepreparation.Consider>the&multiplicationmapA Q:#AW A!A,&and&denoteitskernelbyL.'Itisa>subbimoGdule7ofAv A7whichisadirectsummandasaleftA-module,7sincethe>map":A8 A!L;a a0Q7!a a0aa0 1Ais6apro8jectionontoLthatisleftA-linear.7IfMN isthekernelof[ٲ,6wetherefore>haveUUA8 A=L8M.ύ>SubbimoGdulesofAW A۲canbeconsideredalternativelyasleftidealsinAW A^op,>anditisthisviewpGointthatwewilladoptnow.F*romthisviewpGoint,right18?y?>multiplicationwithe^05isleftA A^op-linear.Moreover,$sinceA(e^09)ʃ=1AN,we >havehy[ٲ((a8 a09)e0)=(a8 a09)e0aa0 1A>andUUtherefore[ٲ(l2`e^09)=le^0#forUUlx2L.Thisimpliesthatthemap*Lx(A8 Aop)e0Q!Le09;b7![ٲ(b)e0>isUUapro8jectionontoLe^09.IfM^0:isitskernel,weget(A8 Aop)e0Q=Le0M0卍>(5)OStill}takingtheabGove}viewpoint,~ we}notethate^0L isaseparabilityidempotent >ifaandonlyifLe^0Q=f0g.T*oseethis,notethataR 1A)1A a2L,soaLe^0Q=f0g>implies(ag@ 1AR1A a)e^0 A=;0,whichispreciselythemissingconditionfora>separabilityUUidempGotent.Conversely*,if*{=(1A q 8a0a0 1A q)e0Q=(a0 1AQ1A a09)e0Q=0>weUUalsohavem(a8 1A q)(1A 8a0a0 1A q)e0Q=(a a0aa0 1A q)e0Q=0>AsUUa8 a^0aa^0 1AƲisUUthegeneralelementofL,thisshowsLe^0Q=f0g.>(6)OT*oseethatthisisreallythecase,denotethekernelofthemultiplication >mapvA=IA A=IA!A=IAvbyx䍑L .YSinceKe^0= 3e OpisaseparabilityidempGotent,>wehavex䍑L 2 `e^0L=f0g.F*romtheabGovelemma,wehavethatthekernelofthe>canonicalCmappingfromA ACtoA=IA A=IACisI(A A),CandCthereforewe>haveUULe^0QI(A8 A),andevenLe^0QI(A A^op)e^09.BytheabGove,weget*^Le0QI(A8 Aop)e0=ILe0IM0>andthereforeLe^0*=IILe^09.PAsListheimageofAm7 Aزunderthepro8jection[ٲ, >itisa nitelygeneratedRDz-moGdule,6andthisimpliesthatalsoLe^0Lisa nitely>generatedHRDz-moGdule.JEButthenNakqayama'sHlemmaimpliesthatLe^0Q=f0g,Hwhich> nallyUUshowsthate^0#isaseparabilityidempGotentandAisseparable.2Ѝ>A8more8generalversionofthispropGositioncanbefoundin[2],9,Chap.II,Thm.7.1,>p.72,[13 ],Chap.5,x5,Thm.7,p.122,[6],Chap.IGII,x5,Thm.(5.1.1),p.135>and.[7],MChap.IGII,.Prop.2.6,p.80,althoughtheabGove.argumentcontainsan>essentialUUpartoftheproGof.@>W*eUU nallyhaveUUthefollowingcriterionforAzumayaalgebras:򍍍>Corollaryv(ҲSuppGoseuthatRideal˩IandthatAisa nitelygeneratedalgebraover˩RDz.|ThenAisanAzumaya>algebraifandonlyifitisfaithfuland nitelygeneratedasanRDz-moGduleand>theUUR=I-algebraA=IAiscentralsimple.19y?>ProQof.eJRecallm%thatweknowfromPropGosition1.8thatAzumayaalgebrasare >thesameascentralseparablealgebras.IfAisanAzumayaalgebra,1itisbydef->initionfaithfuland nitelygeneratedoverRDz.-Moreover,weknowfromPropGo->sitionX:1.2thatthequotientA=IAisseparable,andthereforesemisimple.X>F*rom>PropGosition1.3,BweknowthatthecenterofA=IAistheimageofthecenterofA>under-thecanonicalepimorphism.Thekerneloftherestrictionofthisepimor->phismRtoRisR6\"ܵIA,vwhichisequaltoI4byPropGosition1.5.'Therefore,vwehave>Z(A=IA)T͍+3= UNR=I.Butasemisimplealgebrawithaone-dimensionalcentermust>bGeUUsimpleitself,whichprovestheeasierdirectionintheassertedequivqalence.p>F*ortheconverse,itfollowsfromtheabGovepropGositionthatAisseparable,and>itremainstobGeshownthatAiscentral.SincewehaveassumedthatAis>faithful,thecanonicalmapR!"Z(A)isinjective.SinceZ(A)isanRDz-direct>summandofAbyLemma1.5,Z(A)isa nitelygeneratedRDz-moGdule.Consider>theUUdiagram-JR=I Z(A)=IZ(A)32fda%ά-څ5fRڅZ(A)8҄fdvά-莎Xfe .?&LXfe&~?p>It>isaconsequenceofNakqayama's>lemmathattheuppGermapissurjectiveifthe>lowerjmapissurjective(cf.[3],|Cor.2.12,p.66;notethatthestatementthere>is$incorrect;$itmustbGeassumedthatM^0 6is nitelygenerated).%,Thisshowsthat>theUUcanonicalmapR߸!Z(A)isbijective,i.e.,Aiscentral.2 p>2.4 SuppGoseUUthatRiisalocalringwithmaximalidealI7andresidue eldFc.>PropQosition,SuppGosefl:oݵPl!QisanRDz-linearmapbGetween nitelygener->atedUUpro8jectiveRDz-moGdules.IftheinducedFc-linearmap\qõfʳ:PV=IP*!Q=IQ>isUUanisomorphism,thenfhisanisomorphism.>ProQof.eJNotethat,pinthissituation,thepro8jectivemoGdulesareinfactfree >(cf.x[10 ],xRx2,Thm.2.5,p.9).ykChoGoseabasisv1|s;:::;vn闲ofPۨandw1;:::;wm ofQ.>ThentheresidueclassesUSv1 I/;:::; #vnformabasisofPV=IPWgandtheresidueclasses@ (>w1I2;:::;вwmXformVabasisofQ=IQ.!Iftheinducedmap\qwfisanisomorphism,>wemusthavenIJ=m.SuppGosethatd2R"Qisthedeterminantof(thematrix>representationnSof)f.pThenthereisanelementuh2RthatnSsatis es/u\q 츲 'ѵd=h1.>Therefore,wedhave1ud/2I.JBy[3],Chap.2,Prop.2.8,p.63,thisimplies>that{udisinvertible,|Cand{thereforedisinvertible,which{showsthatfisan>isomorphismUU(cf.[3],Chap.2,Exerc.37,p.75,[11 ],x1,p.6).220Ay?>2.5 SuppGoseNothatRb6isacompletenoetherianlocalringwithmaximalidealI >andUUresidue eldFc.>PropQosition,IfUUAisa nitelygeneratedpro8jectiveRDz-algebrasuchthatRA=IAT͍+3= UNM(n8n;Fc);>thenUUAT͍+3= UNM(n8n;RDz).>ProQof.eJSuppGosethate11 ߸2A=IAcorrespondstothe rstmatrixunitinthe >abGove}isomorphism. ByProposition2.1,wecanlifte11 pctoanidempotente2A.>Ae䖲isaleftidealinA,andthereforealeftA-moGdule,whichmeansthatwehave>anUURDz-algebrahomomorphism6fڧ:A!EndƟRA((Ae)>As=%statedabGove,=+since=%Aeispro8jectiveandRPislocal,=+Aeisfree.=JBypassingto >the quotient,.weseethatitsrankmustbGen.Therefore,.EndܟRK>(Ae)isalsofreeof>rankun^2|s.uTheinducedmap\qf isanisomorphismbyconstruction,uandtherefore>PropGositionUU2.4showsthatfhisalsoanisomorphism.2p>ThisproGofistakenfrom[6],Chap.III,x5,Lem.(5.1.16),p.142.&W*egetasa>corollaryUUtheresultstatedthere:>Corollaryv(ҲIfUUAisanAzumayaUUalgebraoverUURisuchthatRA=IAT͍+3= UNM(n8n;Fc);>thenUUAT͍+3= UNM(n8n;RDz).p>NotethatitwasnotnecessaryinthepropGositiontoassumethatAisfaithful; >thisUUfollowsfromthefactthatitispro8jective,andthereforefree.)4>ReferencesC[1]RNY.Lam:?A> rstcourseinnoncommutative>Nrings,>Grad.T*extsMath.,R[10]R[11]R[12]R[13]R[14]R cmmi10 0ercmmi7O \cmmi5K`y cmr10ٓRcmr7O line10u cmex10('